present in a polycrystal from a technique such as texture analysis. Further advances in statistical approaches to these questions are required in order to advance the recent concept of "grain boundary engineering." that is, tailoring the boundary structure in order to improve the overall properties (Raj and Sass, 1988).

Statistical Mechanics Models

Relatively little is known about the fundamentals of how grain boundary structure depends upon temperature. Some primitive models have been developed (Kikuchi and Cahn, 1980). More realistic statistical mechanical models involving new mathematical approaches would be useful. See also two Chapter 8 sections, Microscopic Scale and Potentially Applicable Mathematical Sciences Developments.

Computer Simulation

Computer simulation has made an enormous contribution to the complex study of grain boundaries; see, for example, Alber et al. (1992). There is a great need for increasing our capacity to work with atomistic simulation models of increasing size using more realistic (that is, more complicated) interatomic force laws (Vitek and Srolovitz, 1989). Mathematicians can be of assistance in developing more efficient ways of minimizing N-dimensional functions and in creating faster and more robust algorithms for the calculations. Effective computer simulation of three-dimensional grain growth in the continuum remains a challenge.



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