EVOLUTION OF MICROSTRUCTURES
This chapter surveys some issues involved in phase-separation kinetics and the dynamics of interfaces. This is a natural subject for collaboration between the mathematical and materials sciences: on the one hand, although the physics is mostly understood, the processes are often quite complicated and of importance for the practicing materials scientist; on the other hand, the new models of the processes are of mathematical interest, and modern machinery in partial differential equations is helpful in studying them.
Energy reduction (involving chemical, elastic, or magnetic energy, or for surface energy) is the driving force. Transport issues (heat and chemical diffusion, surface diffusion, interface kinetics) help determine the resulting dynamics. For shape-memory alloys and magnetic materials, the important questions concern domain structure. For these materials, it is therefore crucial to understand how microstructures arise and change with changes of temperature, stress, and fields.
SPINODAL DECOMPOSITION AND NUCLEATION
Phase-separation kinetics at the coarse-grained microscopic level involves a continuum description down to a scale slightly greater than atomic dimensions. One is not attempting to study details at the atomic scale but is searching for universal behavior in the fluctuations on larger scales.
Several lines of attack have been used. (1) Forward integration in time from a high-temperature initial condition to a temperature substantially below the phase-separation critical point leads to phase ordering, spinodal decomposition, or nucleation depending on the particular transition. Spinodal decomposition happens without an energy barrier for nucleation (second-order phase transition often modeled by the Cahn-Hilliard equation; Cahn and Hilliard, 1958); otherwise, there is a barrier to nucleation of regions of the new phases determined by competition between an increase in surface energy and a reduction in bulk energy. Nucleation is often studied using statistical mechanics on lattice models. The aim, frequently, is to study the systematic behavior of the growth of domains. Domain growth presents interesting nonequilibrium problems without nearby steady states, and is relevant to metallurgy, polymer processing, and other areas in materials research. Significant opportunities exist for mathematical sciences research; for example, there are still unsolved aspects of the Cahn-Hilliard equation in one dimension! (2) Beginning from the coarse-grained Ginzburg-Landau description (see Chapter 8), the related macroscopic equations and boundary conditions have been derived using matched asymptotics thanks to the mathematical contributions of a number of researchers; see, for instance, Cahn and Hilliard (1958) and Pego (1989). Generally the mathematical and physics challenges involve connecting descriptions involving different scales, and gaining analytical understanding of the
evolution of growth patterns into the nonlinear regime. (3) If a substrate is present, nucleation will often occur on it since the energy harrier height for heterogeneous nucleation is usually much lower than that for homogeneous nucleation. From the coarse-grained viewpoint and taking into consideration surface energy and elastic energy, the energy barrier height is a function of orientations of the interface, substrate, and nucleating crystal. The barrier height is one factor involved in determining the orientation of crystals deposited on substrates in the preparation of thin films. By reducing the context to a competition between volume energies and surface energies, the orientations with lowest barrier height can he determined geometrically (Cahn and Taylor, 1988), although elastic considerations usually dominate when the substrate is itself a crystal; see the review and many references cited in Grinfeld (1993); also, see especially Srolovitz (1989).
GRAIN GROWTH AND OTHER INTERFACE MOTION CONTROLLED BY INTERFACE KINETICS
The progression toward equilibrium of a system of spatially distinct domains, produced, for instance, by phase nucleation, proceeds by the motion of the domain interfaces. Examples include the growth of a new phase into an old one, domain growth in spinodal decomposition, or in an ordering system, grain growth, and solid-state or liquid-phase sintering. Driving forces for the motion are bulk or surface energy reduction, and the response is governed by diffusion or interface control. Either the surface energy or the mobility can be anisotropic (that is, can vary depending on the normal directions of the interface). Diffusion may appear to occur extremely rapidly, or it may appear that nothing is present to diffuse, so that interface kinetics control the rate of interface motion. Such diffusionless evolutions are discussed in this section; evolutions involving diffusion are discussed later in this chapter.
In grain growth, there is no bulk energy reduction since (it is assumed that) crystals of a variety of orientations grow until they fill space. Further growth occurs only to reduce surface energy. When the surface energy is assumed to be isotropic, the result is motion by mean curvature, with many junctions where multiple grains meet. "Soliton" self-similar solutions and a growth-shrinkage theorem for network cells were investigated quite early (Mullins, 1956). The first major mathematical consideration of motion by mean curvature treated this multiple-grain case in a highly abstract setting (Brakke, 1978). A flurry of mathematical effort was set off by a study of curves in the plane moving by curvature (Gage and Hamilton, 1986), and many of the techniques developed there apply more generally to interface-kinetics-controlled motion.
Collaboration between materials scientists and mathematical scientists have led to understanding of the general classes of equations to which such interface problems belong as well as to understanding of the properties of such equations, the different methods for finding solutions, and the conditions under which each method becomes the preferred one to use. Recent mathematical advances have been surveyed for the situations in which diffusion does not limit the rate of surface motion (Taylor et al., 1992). The methods include mapping of a fixed manifold (such as a circle or line in 2-space, or a sphere or plane
in 3-space), Brakke's varifold formulation, Taylor's crystalline method, least-time method, method of characteristics, regularization by addition of a smooth (nonphysical) term to the evolution equation, viscosity solutions of a Hamilton-Jacobi partial differential equation, phase-field methods for an order parameter, and order-disorder transformation via Monte Carlo simulations with a Q-state Potts model on a lattice. There are both theoretical and computational versions of most of these methods, although many computational methods are fully developed only for one-dimensional interfaces in 2-space. An extensive list of references and open problems, including that of the convergence of solutions for these diffusionless formulations to those for formulations with diffusion, has been recently compiled (Taylor et al., 1992). Still more recent is the development of a variational method (Almgren et al., 1993) and two applications (Almgren and Taylor, 1992; Roosen, 1993).
Many issues remain open in the case of surfaces with facets. In fact, the relative importance of facet formation to the motion of interfaces is currently a hotly debated topic in materials science. Here growth occurs most easily by the spreading of existing layers of atoms (from places where the surface is locally not the boundary of a convex region of crystal or from ledges emanating from where screw dislocations emerge to the surface) and otherwise by the nucleation of new layers. If surface energy is not directly included in the equations, this situation can be handled by anisotropic mobility (Frank, 1972; Cahn et al., 1991). Surface energy is invoked to determine the barrier height to nucleating new layers; it is possible to incorporate surface energy into the equations by defining an analog of weighted mean curvature (Taylor, 1992) for faceted surfaces. However, the relationship between surface energy and dynamics is not as clear. For example, a recent Ising model computation indicates that the most probable path to nucleating a new layer does not yield a nucleus of the minimum energy barrier height.
Overall, a major mathematical challenge is to produce a comprehensive and practical theory of surfaces and their evolution. A considerable part of the work done during the past 30 years on questions in the geometric calculus of variations arose in modeling physical phenomena in interfaces. In this work, the configuration of interfaces is not specified in advance but emerges as part of the calculated solution. Some earlier work studied the structure of energy-minimizing surfaces and interfaces. More recent work has been on evolution issues and on developing algorithms for computing both static and dynamic solutions (Roosen and Taylor, 1992; also, Computer Algorithms subsection, below).
Interfaces between immiscible liquids of different densities inside a jar, grain boundaries in polycrystalline materials, and a soap bubble froth are examples of physical surfaces that arise when a combination of interface and bulk energies must be minimized subject to constraints. The regularity of solutions of elliptic variational problems is well understood; regularity almost everywhere of solutions to elliptic variational problems with constraints has been studied in Almgren (1976), and a theory of quasi-minimizing functions is being developed. Modeling their time-evolution under changing conditions is a challenge on which some progress is beginning to be made (see Computer Algorithms, below). Models of crystal growth from a melt, with differing heat capacities or heat conductivities, present challenging research opportunities in the evolution of geometric configurations and dendritic features (Almgren and Wang, 1992).
Given a question concerning a surface, one would like to be able to put the boundary conditions, forces, and constraints for a situation into an algorithm and have the answer pop out, much as one can do now in linear programming contexts. But surfaces are nonlinear, and such a ''black box'' solver is still quite elusive. There already are many existence and regularity theorems, but these theorems do not deal with all circumstances and do not always help in solving particular problems. The general algorithms available are computer models, all of which have severe limitations. Even when analytic representations of surfaces are known to exist, such as the Weierstrass representation for minimal surfaces, finding that actual representation is usually impossible as a practical matter. Being able to analyze an arbitrary surface will take truly outstanding mathematical breakthroughs.
Some easily stated questions exemplify how far off a good practical theory of surfaces and their evolution lies:
Given a wire, what is the shape of the soap film it bounds? Analytically, there may exist a Weierstrass representation, but in practice it may not be findable. For something as simple as the soap film on a cubical wire frame, the equations of the surfaces are not known. However, a good computational tool is now available (Brakke, 1992a, b), and a proof exists that the surface produced by this tool is near a true area-minimizing surface (Underwood, 1993). Soap films are often used as models for grain boundaries, assuming that the grain boundaries have isotropic surface energy; in the cubical frame case, the soap film would separate six appropriately chosen grains. The lack of known solutions in such simple contexts illustrates the difficulty of seeking analytical solutions in general.
How many different soap films can a particular wire bound? This is connected to the question of how many different grain-boundary configurations there can be under various circumstances. Even for the case of an octahedral frame, one does not know how many different soap films can form (five are known), or which has the least total area.
What is the best way to partition space into regions of volume 1 with the least interface area? Lord Kelvin conjectured a solution a century ago, and nobody since has been able to prove it or improve upon it. In grain boundaries, Kelvin's solution is not observed, but that is probably because there are many local minima with sizable energy barriers between them for the context. Nevertheless, it would be interesting to know whether or not Kelvin's solution is indeed the "ground state."
A more general version of this question is, When seeking a minimal surface can one prove that a given local minimum is (or is not) a global minimum? Techniques such as calibrations are useful when they work; see, for example, Morgan (1988). One can perhaps prove that the minimum has a calibration, but how can it be obtained in a particular case?
Finally, in an annealing metal (modeled by a surface with 3-fold and 4-fold grain junctions, moving by mean curvature), do the grains that shrink and disappear reach a limiting stable shape before they disappear, and what can such shapes be? This question has been studied in the materials science literature for some time, but the confusion between motion by mean curvature and the type of motion that soap films undergo (involving diffusion of gas and instantaneous rearrangement to a locally area-minimizing configuration) seems to have clouded the issue. The complete mathematical solution is not known.
There are also fundamental questions about why there are surfaces and regular arrays of atoms in crystals. Even how regular arrays form is an open issue, and the existence of stable quasicrystals (Shechtman et al., 1984; Cahn and Gratias, 1987; Senechal and Taylor, 1990) indicates that it is of more than abstract interest. Quasicrystals are alloys whose electron diffraction patterns exhibit a clear and sharp symmetry (typically icosahedral) that is impossible for a periodic translation lattice to have. A quasicrystal "tiles" space non-periodically, with the simplest case possessing "prohibited" (by classical crystallography theory) 5-fold symmetry with respect to some axis (that is, possessing invariance with respect to two-fifths π of a rotation about some axis). A clear and unmistakable diffraction pattern of any sort is evidence of long-range order, since the diffraction pattern is a picture of a Fourier transform. In spite of the surprise of crystallographers at the existence of these alloys, there is in fact no contradiction with any physical or mathematical theory, since it has never been proved that the stable energy state of matter must be a periodic array of atoms, and mathematicians have long known that almost-periodic functions have Fourier transforms that are atomic measures. Number theorists have pointed out the relevance of Pisot numbers and other number theoretic concepts to understanding quasicrystals, and mathematicians involved in tilings and Fourier analysis have made contributions to quasicrystal theory and likewise have had their own mathematical fields enriched by the discovery of this class of alloys.
In developing algorithms and computer programs for solving geometric optimization and evolution problems, the creation of new mathematical ideas and approaches has often been indispensable. Some programs serve primarily mathematical purposes. Some aim to simulate crystal growth and are based on new mathematical approaches such as those mentioned above. Other programs, such as Surface Evolver (Brakke, 1992a, b), are also being used by others for purposes beyond those for which they were initially designed. For example, Surface Evolver now has scores of users:
A Martin-Marietta engineer is using it to simulate liquid fuel shape in weightlessness for a space shuttle small fuel tank (Tegart, 1991).
Scientists at MIT have used it in analysis of liquid-solid geometries that occur in soldered connections (Racz and Szekely, 1993a, b, c; Racz et al., 1993; Abbott et al., 1993).
Scientists at the National Institute of Standards and Technology are using it in studies of sintering processes. A NIST metallurgist has said of it, "I found new materials science problems to solve after seeing the ideal tool to tackle them with."
SHAPE EVOLUTION CONTROLLED BY SURFACE DIFFUSION
Surface shape evolution due to surface diffusion leads to an equation in which the normal velocity is proportional to the surface Laplacian of the mean curvature (in the
isotropic case). This is a fourth-order (instead of a second-order) equation. In general, features of such equations are not well established. Solutions are known (Mullins, 1963) for the linearized equation corresponding to nearly flat surfaces; some of these are shape preserving. Recent work on thin films and small-scale sintering (for example, of ceramic powders) has revived interest in the solutions and properties of the surface evolution equation.
Simple cases described by the above surface evolution equation are the smoothing of an initially undulating surface (Mullins, 1959), the evolution of a general closed body toward a sphere or the blunting of a cone (Nichols, 1965) (for example, field emitter tip). A further example is the development of a groove where a grain boundary meets an external surface (Mullins, 1957) or an internal void. A case described by related equations for materials whose surface energy is sufficiently anisotropic is the faceting of an initially planar surface.
Examples of important practical situations modeled in part by the isotropic surface evolution equation described above are sintering, grain growth in thin films, the breakup of thin films into an island structure, and texture development in thin-film growth from vapor. These situations arise in such materials science processes as molecular beam or liquid-phase epitaxy, and deposition by ion beam or sputtering or chemical vapor. Such issues are important in, for instance, the production of artificially structured materials for such applications as semiconductors, insulators, and new combinations of metals. In sintering, the total surface energy is reduced as the initial set of voids among an aggregate of particles is ultimately eliminated by diffusion of matter into them from the grain boundaries and the concomitant reduction of interparticle distances. In thin-film grain growth, migrating grain boundaries are inhibited by drag exerted by the surface grooves, which can completely trap them. If the deepening grooves reach the substrate, the film breaks up into an island structure. Finally, in polycrystalline thin-film growth from vapor, the migration of grain-boundary groove systems can determine which grains encroach on their neighbors and hence which orientations survive.
In cases involving grain boundaries, as in grooving or sintering, the only known solutions to the surface evolution equation are those for a static planar grain boundary perpendicular to the initial surface plane and those for a few simple translating groove-boundary systems (Mullins, 1958). The general solution for such systems, central to some of the processes described above, has not been studied. Given a simple rule of grain-boundary motion (for example, the mean curvature rule), the situation can be described as two coupled free-boundary problems; see Chapter 8.
The interplay between mathematical sciences concepts and the materials science issue of microstructure formation during solidification deserves emphasis. During solidification of a typical alloy, the rate-limiting step of the crystal formation process is the diffusion of impurities and/or heat. Thirty years ago, it was demonstrated that diffusively controlled growth would be unstable to the "complexification" of the advancing crystal-liquid phase
boundary (Mullins and Sekerka, 1963, 1964); this can, depending on geometry, materials, and so on, lead to cellular solidification, lamellar eutectics, or dendritic growth of fractal structures. Since the resultant microstructure is directly traceable to the growth process, one must come to grips with these inherently nonlinear, nonlocal, and nonequilibrium dynamics.
Mathematics in the broadest sense is crucial for making real progress in this field of materials science. Simulation techniques are essential in the study of any nonequilibrium system. All successful methods for simulating free-boundary problems require sophisticated mathematics. In this regard, work on boundary-integral methods (for steady growth), variational formulations (for nonsteady growth) and phase-field considerations is proceeding apace (Taylor et al., 1992; Taylor, 1993). Mathematicians are not just pursuing existence theorems; they are actually computing real growth patterns.
In certain moving-boundary situations, ranging from viscous fingering to solidification, self-similar growth patterns develop following a morphological instability. These have been studied using methods based on boundary integrals that range from large-scale numerical simulation to qualitative approaches based on renormalization group ideas developed in studies of critical phenomena (see Chapter 8). The mathematical challenges are to overcome computational limitations, extract asymptotic behavior from the relevant integral equations, and take into account that one envisions a statistical process (such as an ensemble of initial conditions).
Phase Transformations and Pattern Formation
An area where there has been considerable mathematical effort, with and without collaboration with materials scientists, is in solidification. One example involving strong interaction between materials scientists, mathematicians, and numerical analysts concerns a new theory and method of calculation for pattern formation during alloy solidification (Wheeler et al., 1992; see also Chapter 8). A method has been developed that predicts the growth patterns of solids (crystals) forming from a melt during alloy solidification. It has come to be known as the "phase-field model." The phase field model is a set of phase field equations. These are time-dependent partial differential equations (usually parabolic) describing the phase transition in which the interface is determined by level sets of a function (called the phase or order parameter); the transition from one phase to another is gradual. In extreme limits, the interface thickness shrinks to zero, and the sharp interface is referred to as the free boundary separating the two phases. This method treats the system as a whole using a continuous field variable to describe the state (liquid or solid) of the various regions of the system. The interfacial region is modeled as a diffuse layer. This phase-field model approach involves solving a system of nonlinear coupled parabolic partial differential equations, and incorporates a wide variety of physical phenomena such as nonequilibrium kinetics and capillarity. Previously developed for pure materials (Wheeler et al., 1992), this method has been extended to the more useful case of binary alloys. Growth-pattern and associated Oswald-ripening calculations that yield the composition segregation pattern in final solid alloy products can now be done. These segregation patterns are responsible for defects that limit the mechanical and corrosion properties of
many alloys. Understanding such defects is one of the basic issues of metallurgy and materials science to which mathematical sciences research can contribute. See Caginalp (1989) and Kobayashi (1992) for further discussion and sources.
While these equations present an elegant formulation of alloy solidification, many mathematical challenges arise in their solution. The equations are "stiff"; that is, they involve small parameters multiplying the highest-order derivatives. Simple finite-difference methods with implicit time stepping currently in use are not sufficiently accurate. New solution procedures are necessary to obtain results that will provide realistic microstructure predictions of use to materials scientists. Significant questions remain about the stability of the solution with respect to mesh size, orientation, numerical noise, and time step.
Computer modeling of solidification of alloys from a melt is an active area of materials research; see, for example, Hurle and Jakeman (1981) and Brown (1988). It is also important for materials processing (see Chapter 7). In some cases it is desirable to prevent inhomogeneous regions in a crystal (as is the case for semiconductors), while in other cases the presence of such inhomogeneities is desirable because it improves the mechanical strength of the solid. Trial-and-error procedures are often used to try to determine optimal processing conditions. For large-scale processing operations, trial and error can be quite costly. It would be advantageous to have a computer model of a crystal furnace that can be used in a fashion analogous to the "numerical wind tunnels" provided by computational aerodynamics, which allow significant savings over design validation using conventional wind tunnels.
While a crystal furnace computer model is a laudable goal, practical difficulties make computational simulation of crystal furnaces unattainable with current resources. Convection occurs in the melt and the description of crystal growth requires the solution of the Navier-Stokes equations with both heat and solute transport. The observed flow is often unsteady and three dimensional; turbulent flow conditions are not uncommon. In addition, the solidification process is a free-boundary problem, of which the classical Stefan problem is a simple example. On the latter, see, for instance, Rubinstein (1971) and Meirmanov (1992); also see Crank (1975) and Caginalp (1989). The geometry can be quite complicated: dendritic growth with secondary and tertiary sidearm formation, inclusions in the crystal, and so on (see Chapter 8). In attempts to control the process better, the systems are often subjected to various types of external forcing, including bulk rotation of the sample, vibration of the sample, and imposition of magnetic or electric fields. Realistic computational simulation of actual processes under these conditions is currently infeasible.
Piecemeal progress is possible under various simplifications, which, in light of the above discussion, may omit important physical processes. A natural approach is to consider simplified geometries, for example, near-planar growth, in which the complications associated with the free-boundary nature of the process are limited. One then has recourse to classical approaches such as linear stability, perturbation theory, and conventional numerical computations. Even in this setting, the difficulties are formidable: the length scales of transport phenomena, flow fields, and crystal-melt interface shapes in real materials can differ by orders of magnitude, ranging from tens of microns for microstructure associated with interfacial instabilities to centimeter-sized flow patterns on the scale of the container size. This variety of length scales makes a brute-force approach by large-scale computing
unrealistic even in idealized geometries.
One approach to this difficulty might be to use analytical means to reformulate parts of the problem. For example, one could conceivably make use of the different length scales in a systematic asymptotic expansion, as is done in homogenization approaches for multiphase flow. To date, little work has been performed along these lines.
Another possibility is to make use of numerical algorithms more sophisticated than those used to date. By and large, work thus far has used conventional finite-difference, finite-element, and boundary integral methods to model solidification. The amount of research addressing the unsteady and three-dimensional situations has been small, and the questions of mesh generation for complicated time-dependent geometries that may produce changes in topology, are formidable. Recent algorithmic developments that offer advances in these directions include fast multipole methods (Greengard and Rokhlin, 1987), efficient time-dependent boundary integral methods (Sethian and Strain, 1991; Brattkus and Meiron, 1992), Hamilton-Jacobi formulations of interface motion (Osher and Sethian, 1988), and implementations of phase field models (Fife, 1988).
Crystal growth from the melt illustrates that successful mathematical modeling in materials science requires an interdisciplinary effort in which applied mathematicians play an important role. Under extreme processing conditions, how to derive the appropriate governing equations and boundary conditions is often an open question, one that requires mathematical as well as physical insight. Guidance is required from both ends of the research spectrum.
Consider the example of the single dendritic crystal. Much work has been devoted to demonstrating that the shape and growth rate of a single dendrite can be calculated via a novel "solvability" mechanism arising due to non-perturbative effects in the surface energy. See, for example, Segur et al. (1992), Brener and Mel'nikov (1991), Kessler et al. (1988), Pelce (1988), and Langer (1987). This mechanism consists in adding (anisotropic) surface energy to the consideration in a way that fundamentally alters the mathematics of the diffusive growth equation and leads to the formation of distinctive patterns. The theory (especially in the real three-dimensional case) remains somewhat controversial within the materials community. It has helped create the entirely new applied mathematics concept of asymptotics beyond all orders (Segur et al., 1992). Citing this example does not imply advocating the theory (or the "asymptotics beyond all orders" concept, which does not contribute to the real case); it is offered only to point out that this issue is likely to be resolved by the combined efforts of materials scientists and mathematical scientists. The mathematical sciences will be enrichened in the process.
Some years ago, the Institute for Theoretical Physics (ITP) in Santa Barbara hosted a program concerned with understanding dendritic crystal growth in a quantitative way. The focus was on predicting the properties of dendrites, including growth speed, dimensions, and side-branching behavior. Dendritic growth is the generic growth mode under diffusion-controlled conditions, and progress in understanding dendrites will lead to an ability
to predict and control in a scientific manner the microstructures of metals, including alloys, and other materials. Although this context had previously attracted the attention of materials scientists and mathematicians, the ITP program approached the questions afresh, using physical principles to construct simplified (two-dimensional) models of the process that were amenable to numerical methods and some analytic work. It was discovered that the existing formulation was physically inadequate—it ignored the role of crystal anisotropy, which was shown by the research of the program to be crucial. Once the correct physics had been identified, it was necessary to solve the resulting mathematical models. This turned out to be a subtle issue in singular perturbation theory, which involved asymptotics beyond all orders. M. Kruskal and H. Segur, who were visitors to ITP that year, worked out the general mathematical framework for this difficult situation (Segur et al., 1992), and their contribution formed the basis for subsequent calculations by physicists in the field.
So-called mushy zones present an interesting challenge to mathematical scientists. These layers between the liquid and solid phases, which consist of tiny parallel needles separated by fluid regions, have been the subject of intensive research in recent years (Huppert, 1990). Currently, it is not fully understood why they form, but there has been success in describing the evolution of the microstructured layer, viewed as a continuum: there is structure that emerges at several levels. The needles themselves have stable side branches but, on a larger scale, organize themselves into uniform regions populated by chimneys. Here, a challenge is to provide a better understanding of how macroscopic equations are related to transport processes near the needles, while accounting for the complex forest of needles. The types of mushy regions best understood by mathematical scientists are those arising in pure material that is being melted under volumetric heating (Atthey, 1974). In this setting the mushy zone has no structure, but what is possible in the distribution of crystal size has been considered (Lacey and Taylor, 1983).
PRECIPITATION AND COARSENING
Many of the properties of multiphase materials depend strongly on the morphology and spatial distribution of minority precipitate phases within a majority matrix phase. The morphologies and spatial distribution of these precipitates are in many cases governed by the dynamics of a phase transformation process. For example, cooling a single-phase alloy to a lower temperature can result in the formation of small nuclei within the high-temperature phase that then proceed to grow. It is this growth and the subsequent coarsening process that frequently determine the morphology and spatial distribution of the particles. In many systems, ranging from transformation-toughened ceramics to high-temperature superalloys, the minority-phase particles possesses a density that differs from the density of the matrix phase. This difference in density engenders elastic stress fields in both the matrix and the particles that can strongly influence the nature of the transformation
process and the morphology of the particles.
Describing the dynamics of diffusional growth and of the coarsening of second-phase particles in the presence of elastic stress has all of the complications of the classical Stefan free-boundary problem augmented by the fact that the boundary conditions at the particle-matrix interface are a function of the local curvature (through the interfacial energy) and a function of the entire shape of the particle and the distribution of the other particles in the system (Voorhees, 1992). The presence of a long-range elastic stress field gives rise to strong spatial correlations between particles during coarsening and to diffusional migration of particles in solids. Although some solutions in this context have been found, much of the mathematical structure of these elastochemical free-boundary problems remains unexplored.
An example of the effect of stresses on the evolution of a two-component system is the later stages of a phase transformation. In this phase, particles usually coarsen (decrease in number and increase in size while maintaining a nearly constant total volume). In several technologically interesting cases, however, this coarsening ceases and the particles begin to split. The stresses that are presumed to give rise to these phenomena are caused by compositional inhomogeneity, external loads, or particle misfit strains.
The elastic stresses manifest themselves in the diffusional and mechanical field equations and in the boundary conditions for diffusion. The long-range elastic fields render the diffusion equation a function of the composition field, the size and spatial distribution of the "second-phase" particles, and the system geometry. There have been many attempts to generalize the Lifshitz-Slyozov theory of grain growth and coarsening (Lifshitz and Slyozov, 1961) during the last thirty years. By modeling a collection of spherical particles under a number of simplifying assumptions, including interface isotropy and equal molar volumes as a homogeneous (single-phase) system in which the particles are replaced by multipole sources and sinks of strengths determined by solution of Poisson's equation, particle growth and migration rates can be determined by computer simulations (Abinandanan and Johnson, 1992). The approach is limited by the inability to solve for strongly coupled elastic and diffusion fields. It also suffers from the inability to treat particle shape changes; compare with Boettinger et al. (1993) and Roosen (1993), which do allow for shape changes.
The simplest models of coarsening omit any influence of stress or texture. These models apply to solid-fluid systems and to solid-solid systems in which there is a negligible misfit strain. Theoretical treatments of these models by the Lifshitz-Slyozov method (Lifshitz and Slyozov, 1961) and its generalizations show that the distribution of particle sizes divided by the (increasing) average particle size becomes asymptotically time invariant. When the Lifshitz-Slyozov model is assumed, the system becomes self-similar in a statistical sense (as defined, for example, in Mullins and Vinals, 1989). The same property is observed in grain-growth models that treat all boundaries as being the same. There is no general mathematical theory that details the conditions under which an evolving system will develop a statistically self-similar structure.
Evolution of Microstructures; Stress and Current Effects
Stress affects both the (final) equilibrium state of crystalline solids and the evolution of a solid not originally in equilibrium. The local state of the material is typically characterized by a displacement gradient and several scalar parameters such as composition and temperature. In the equilibrium setting, one seeks the state that minimizes a particular integral (total energy) over the volume of the material, given the temperature, overall composition, other parameters, and boundary conditions (displacements, applied tractions, and so on). Surfaces of discontinuities can exist within this volume, and their existence and shape are usually part of the question. Ever since Gibbs's famous chapter on solids in contact with fluids (Gibbs, 1876, 1878), the first step in obtaining solutions has been derivation of coupled partial differential and algebraic equations that must be solved at an extremum (Larché and Cahn, 1978). The evolution situation is often formulated as a set of nonlinear partial differential equations with boundary conditions on moving boundaries; see Larché and Cahn (1992) for a simple example. The theory combines classical elasticity, thermodynamic effects on solutions (as a notoriously nonlinear ingredient in the energy of the solid), and the interfaces (that usually involve stronger conditions than simple contact) between parts of solids with different orientations or different properties.
Some open questions in this field are the following: (1) How can a line of discontinuity, such as a dislocation, he incorporated into the energy minimization approach of the example mentioned above (Larché and Cahn, 1978)? (2) What are the necessary conditions for a minimum? (3) What are the general properties of the solutions? (4) When elastic effects are present, a modification of the shape of the regions containing different phases changes the energy of the system (Cahn and Larché, 1984). One can then ask: What geometric arrangement in space produces a minimum in the appropriate energy (including surface energy)? How sensitive is the evolution of such a system to initial conditions? Corollary: What can really he predicted? (5) The presence of a displacement gradient as a local variable to describe the state of a solid often introduces nonlocal effects in the partial differential equations describing the evolution of the system (Larché and Cahn, 1992). What are the general properties of the solutions of these equations?
Stress-induced diffusion often causes voids to grow. Similarly, current-induced diffusion (electromigration) leads to void growth, extrusions, and other microstructural changes. These problems are particularly severe in the microelectronics industry, where they occur in metal interconnects on chips. The theoretical and mathematical models of induced diffusion are currently quite elementary. More comprehensive formulations are needed.
MARTENSITE AND SHAPE-MEMORY MATERIALS
Martensite transformations are phase transformations that produce a change of shape and a change of crystal symmetry. Shape-memory materials are materials that are extremely malleable in the martensite phase below a transformation temperature, but that return to a "remembered" original shape when heated above the transformation temperature. In this orderly change of phase, nearest neighbors of atoms are maintained. One can picture a
cubic lattice, for example, that deforms by elongating along one of the cube edges. Since the three cube edges are crystallographically equivalent, there are three different ways the cubic crystal can deform; these are the variants of martensite. After transformation, the variants arrange themselves in complex patterns characteristic of the material. The goal of mathematical theories of martensitic transformations is to predict these patterns of microstructure, to understand why certain materials exhibit certain patterns, and to explain the macroscopic properties of these materials. One of the most intriguing, yet-to-be-understood aspects of shape-memory materials is that they occur for special compositions (Liu, 1992; Otsuka and Shimizu, 1989).
Coherent phase transitions of crystalline solids lead to mixtures of distinct phases or phase variants with characteristic fine-scale structures. Such transitions can be modeled using continuum elasticity (see Chapter 8). Microstructures arise due to elastic energy minimization. This topic is closely linked to the study of structural optimization and to the identification of extremal composites (see Chapter 6).
The materials science literature in this area is extensive. For martensite, research is dominated by the phenomenological "crystallographic theory" (Christian, 1975). There is also the work of Khachaturyan (1983), Roitburd (1978), and others based on geometrically linear elasticity. Most work oriented toward understanding the shape of coherent precipitates has dealt with isolated inclusions and has taken as a starting point the work of Eshelby (1975).
Recent mathematical work has introduced new ideas and opened new doors. One important development is the modeling of microstructure based on geometrically nonlinear elasticity (Ball and James, 1987). In geometrically linear theory, free energy depends only on linear elastic strain and is therefore invariant only under infinitesimal rotations. In the geometrically nonlinear theory, the free energy depends on the nonlinear Cauchy-Green elastic strain, since it is invariant under all rigid body rotations. This amounts to a marriage of the "crystallographic theory of martensite," which is nonlinear but strictly phenomenological, and the earlier work of Khachaturyan and Roitburd, which uses energy minimization but is geometrically linear. This new approach has suggested specific quantitative predictions of energy minimization (Ball and James, 1992) and new criteria for the design of shape-memory materials (Bhattacharya, 1991, 1993); see also Chapter 8.
Another development is the impact of surface energy as a selection mechanism. It is widely appreciated in the materials literature that the length scale of microstructure is determined by the interaction of bulk-energy and surface-energy effects. A recent attempt to make the theory in this area rigorous led to an entirely unexpected prediction of twin branching in martensite (Kohn and Muller, 1992).
Much remains to be done. In practice there are at least two small parameters: surface energy and geometrical nonlinearity. How they interact and how they together serve to select between otherwise equivalent microstructures are not understood. Combining the strengths of geometrically linear and geometrically nonlinear approaches would he a worthwhile development. The recent progress on extremal microstructures in linear elasticity has yet to be exploited in the setting of coherent precipitates. All important is the modeling of local minima and dynamical effects, a topic that has received current attention (Ball et al., 1991a; Semenovskaya and Khachaturyan, 1992) but that is far from well understood.
On the dynamic side, there are outstanding challenges for mathematical scientists.
One involves a marriage between microstructure and kinetics. It is known that the microstructure of domains is history dependent. Minimizing the energy does give the right kind of microstructure qualitatively, but many key questions involve the dynamics of microstructure. Among the family of minimizing sequences of domain patterns for a given boundary condition, which one will be chosen (see Chapter 8)? Given the one that is chosen for time zero and given a deformation as a function of time, what will the deformation be later in time? We need a theory of parallel transport, of dynamics, on the space of minimizing sequences. Perhaps this will prove impossible: some domain patterns may not be able to change smoothly after sufficiently large deformations.
These dynamic and kinetic considerations may be crucial to the understanding of hysteresis in martensitic materials. Among different martensitic materials, there is a huge variation in the size of hysteresis loops—the transition temperatures on heating and cooling can vary by as little as 1°C and as much as 100°C. The dynamics and kinetics of martensitic materials can also be understood in terms of metastability. In this regard, the current understanding of relative minimizers of energy is inadequate. Recently, this has become an active area of research, with many ideas and models put forth about the origins of hysteresis. Mathematical work that improves the models, suggests new approaches, and discovers underlying connections between the current models is needed.
The computation of the microstructure of materials is important in the development of many emerging technologies such as shape-memory materials (Collins and Luskin, 1989; Collins et al., 1993) and magnetostrictive materials (Luskin and Ma, 1992). The role of computation is both to facilitate the development of theory and to aid in the design of materials that are important for technological applications. The challenge of computing complex material microstructures has brought these concerns to the forefront of modern numerical analysis.
Three-dimensional equilibrium computations for crystalline microstructure constitute a relatively recent phenomenon. This setting is not unlike that of the protein folding problem (see Chapter 3) in there being extreme nonlinearity and huge numbers of relative minima. Here, there is a twist: some (but clearly not all) of these relative minima may be important for the understanding of metastability. The first such computations proved that the deformation of macroscopic crystals could be modeled with effective continuum theory and advanced computational methods (Collins and Luskin, 1989). New techniques have been introduced to give a rigorous error analysis for the numerical approximation of microstructure (Collins and Luskin, 1991; Collins et al., 1991). Future challenges for the mathematical sciences are to create more efficient algorithms that will make possible the computation of more complex microstructures, and to extend error analysis to multidimensional contexts where there is not a unique solution.
Even more challenging is the computation of the dynamics of crystalline microstructure. It remains a fact that, in many branches of the study of phase transformations, the kinetics of transformation is often fit by an essentially empirical relation. A major challenge is to develop kinetic models that relate directly to material parameters. This will not be possible without the ability to account accurately for the full complexity of evolving microstructures. Progress in this area is in its infancy but is encouraging. Recently, dynamical solutions have been computed for the development of microstructure in
three-dimensional martensitic crystals and for the movement of the interface separating martensitic and austenitic phases (Kloucek and Luskin, 1993).
Magnetic materials offer promising opportunities for interactions among materials scientists, applied physicists, and mathematical scientists. Important issues include the prediction of complex domain structures, the sizes and shapes of hysteresis loops, the quantitative effects of defects, and understanding the fascinating properties of thin-film and nanocomposite magnetic materials.
Micromagnetics deals with the prediction of magnetic domains at the micron level. This theory (Brown, 1963) has been influenced strongly by Landau and Lifshitz (1969) and has experienced both remarkable success and spectacular failure. On the successful side, it gave a quantitative prediction of the behavior of small magnetic particles and provided the basis for the design of the strongest magnets, which are obtained by sintering these particles in a way that avoids exchange interactions. It was less successful in dealing with large magnets. Its failure (by a factor of 5000) to predict the size of the hysteresis loop of single crystals of iron came to be known as the coercivity paradox. This was resolved by a series of careful experiments that revealed the huge influence of defects, both scratches on the surface and internal defects. The validity of the theory of micromagnetics was confirmed by growing defect-free iron whiskers, the hysteresis loops of which closely agreed with the micromagnetic predictions. However, except for studies of a single domain or a single defect, a workable form of micromagnetics for defective crystals remains a challenge. Some methods have emerged for coping with complex microstructure (James and Kinderlehrer, 1990), and there has been recent progress on establishing a relation between micromagnetics and phase theory (De Simone, 1993), but these methods fall short of comprehensively treating defective crystals.
Magnetostriction is the phenomenon whereby magnetization produces deformation and, conversely, deformation produces magnetization. Of particular interest because of its comparatively large magnetostriction is the iron/rare-earth alloy Terfenol-D, the magnetostriction of which under the influence of even an extremely small field can be as much as 1 part in 1000. It has a complicated microstructure whose role in magnetostrictive properties is yet to be clarified. In conjunction with experimental work, a theory has been developed that is being applied to this material (James and Kinderlehrer, 1993). There is also a related computer model. Many open issues remain, especially issues related to the mechanism of magnetostriction and possible hindering mechanisms.
Terfenol-D is one of a family of materials that in recent years have been called ''smart materials''; see, for example, Gandhi and Thompson (1992), Rogers (1989), Travis (1993), and Nanavati and Fernandez (1993). These materials can be used in sensors and actuators. They can be used in sensors to understand the behavior of small volumes of material, which can differ quite markedly from the behavior of large volumes. One trend is the application of smart materials to micromachines, that is, tiny mechanical machines.
Although there has been interesting progress in the ability to compute configurations,
computational issues continue to present great difficulties. These are not mere technical obstacles but fundamental obstacles connected with the nature of nonconvex optimization (see Chapter 8). They are most readily resolved in collaborations between materials scientists and analysts. For example, anisotropy of the computational grid and anisotropy of the material can compete to give poor results, especially when many potential wells are separated by low barriers. Such mismatches can be avoided by increased cross-disciplinary understanding.
The basic model in present use for dynamic situations is the Landau-Lifshitz-Gilbert equation. This time-dependent partial differential equation for the magnetization vector is a gradient flow for the micromagnetic energy. Much remains to be learned about the properties and applicability of this equation. The equation exhibits nonuniqueness, and it is likely that with certain applied fields and geometry a smooth solution develops increasingly fine oscillations (see Blue and Scheinfein, 1991; Hoffend, 1993; James and Kinderlehrer, 1990; also see Visintin, 1985, and the references given therein).
Discovery of high-Tc superconducting materials has revitalized the study of all superconducting materials and has focused attention on applications of superconductivity at a level not seen since the 1960s. The modeling of both the macroscopic and microscopic properties of superconducting materials and of the processes for preparing them provides important scientific opportunities for the mathematics community, opportunities that affect both fundamental and applied aspects of the subject. These opportunities involve both conventional metal-based materials and the new high-Tc materials that have been found within the last 6 years.
The macroscopic properties of the new superconducting materials are more complicated than those of metallic superconductors due to anisotropy, layered character, short coherence lengths, and high values of the upper critical magnetic field. As a consequence, the macroscopic equations for the superconducting order parameter, namely, the Ginzburg-Landau equations, are different from and more complex than those used to describe metallic materials. These equations are in need of serious study by mathematical scientists (see Chapter 8). Understanding the solutions of these equations is essential for quantitatively modeling the response of superconductors to magnetic fields and currents; see, for example, Bishop et al. (1993). These are fundamental issues that have an impact on the development of applications.
As a consequence of this complexity, the phenomena observed when high-Tc superconductors are subjected to magnetic fields are far richer than those found in metallic type II superconductors. Both kinds of materials exhibit the Meissner effect; that is, they have a flux-excluding state at low fields, and they both have a mixed state at higher magnetic fields characterized by the penetration of field through the presence of Abrikosov vortices. Because of the anisotropy and layered nature of high-Tc materials, vortices in these materials are more complicated than those in conventional metallicsuperconductors. High-Tc a materials can exhibit, with increasing temperature or magnetic field, either vortex lattice
melting (in ordered materials) or vortex glass melting (in disordered systems and thin films). Although the precise physics of the transitions from a lattice or a glass to a fluid of vortices is currently a subject of physical investigation and is not resolved, any of the proposed models that may ultimately describe these transitions represent significant opportunities for mathematical scientists.
Modeling the dynamical properties of superconducting vortices in the presence of transport currents and magnetic fields is a key to producing materials and configurations in which transport is dissipationless and critical currents are high, features that are required in many applications. The key to building stable systems for high-power applications such as magnets will be determining how vortex dynamics and energy dissipation are related to heat. This is complicated by the fact that ductile conductors may be composite materials, as is the case for metallic superconductors.
On a microscopic level, the details of vortex dynamics depend phenomenologically on the particular nature of pinning sites. The study of the interaction of Abrikosov vortices with structural defects of various types such as point defects, dislocations, grain boundaries (see Chapter 5), and twin planes, all of which can serve as pinning sites, may also be an opportunity for mathematical scientists. The fundamental understanding of this interaction phenomenon has benefited over recent years from the development of various microscopic theories that now permit the quantitative characterization of defects.
A variety of devices incorporating superconductors have been proposed over the years. Development of these devices provides a plethora of research opportunities for mathematical scientists. The most ubiquitous devices are those involving Josephson effects, which have been proposed as computer-processor and computer-memory elements as well as sensors of electromagnetic radiation and magnetic fields. Individual junctions can be modeled as simple dynamical systems or sine-Gordon systems. In physical applications, junctions are perturbed by high-frequency fields or are coupled to other junctions. The detailed modeling of the behavior of Josephson-effect devices, and of diverse geometric arrays of devices subjected to a assortment of purturbing fields, presents additional opportunities for mathematical scientists.
Conventional superconductors can he described microscopically by the Bardeen-Cooper-Schrieffer (BCS) theory (see Chapter 8). The new high-temperature superconductors are likely to be understood using either a modified version of the BCS theory, such as one in which there is nonzero angular momentum paring, or an entirely new theory. The issue at this point has not been resolved by the physics community. Whatever theory comes out of the physical investigations, it will involve new challenges for applied mathematicians.
Another potential application of the mathematical sciences to the study of superconductors is in modeling the materials' synthesis and processing, both in bulk and in thin-film form. Such processing details determine the morphology and chemistry of the materials along with their ultimate physical properties and usefulness. For example, in producing thin films and thin-film devices, it would be invaluable to be able to model film growth and predict the growth mode and such properties as structure, surface smoothness, and chemical composition. This is a research opportunity in surfaces and interfaces that has experimental, analytical, and computational aspects. There are similar issues in controlling
the processing of bulk materials. The solution of synthesis and processing problems for superconductors is key to the development of a competitive superconducting technology.
There has been a recent surge of activity in macroscopic modeling for superconductivity. The mathematical methods involved include the use of asymptotic analysis, free-boundary formulations, variational principles, and finite-element approximations. For comprehensive reviews, see Chapman et al. (1992) and Du et al. (1992).