modynamics predicts stable, homogeneous solid solutions. Segregation occurs because, during the solidification process, the liquid and solid phases fall out of equilibrium with each other, and the chemical constituents are rearranged by being driven across the moving solidification front. Thus the last bits of liquid to solidify may be compositionally very different from those that solidified first. The result is often an intricate pattern of cellular or dendritic (treelike) structures, on the scale of tens or hundreds of microns, that are easily observed through an optical microscope.

Control of these microstructural patterns has long been understood to be essential in materials technology. The processes by which the microstructures form are important in determining the grain structure of the solidified material. The microstructures themselves, within each grain, affect the mechanical properties of the material; for example, they may pin or impede the motion of dislocations. They also determine the way the material will behave under further processing such as heating, deformation, aging, or exposure to corrosive environments. The strength of a weld depends on the microstructure in the resolidified material and on microstructural changes in the heat-affected zone. The suitability of a semiconductor crystal for use in electronic devices depends on careful suppression of microstructural solute segregation. Other examples of the technological importance of microstructural properties appear frequently, either explicitly or implicitly, throughout this report.

To the aspiring modeler, the basic ingredients of the microstructural problem may seem pedestrian. Generally, one is being asked simply to solve well-understood diffusion equations subject to apparently simple boundary conditions. The trouble is that the boundaries are moving; in fact, the essence of the problem is to compute their motion. To make matters yet more interesting, microstructural patterns are caused by morphological instabilities of these boundaries. Initially smooth shapes naturally develop grooves and fingers, the fingers split or develop side branches, the side branches split or develop tertiary side branches, and so on. What has been discovered only very recently is that the patterns generated by this process are controlled by an extremely delicate interplay between a basic diffusional instability and a number of ostensibly much weaker effects, most notably surface tension, but also crystalline anisotropy, interfacial attachment kinetics, and even ambient noise. In more technical terms, the instabilities produce intrinsically “nonlinear” behavior, and the weak, controlling effects are “singular perturbations.”

The recent developments in this area have been brought about by an interactive combination of mathematical analysis, numerical computation, and careful experimentation. At the time that this report is being written, there is a sense among workers in the field that they may be narrowing in on a new understanding of fundamental principles—for example, that it may finally be possible to compute the growth rates and geometries of dendritic



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