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).

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