The shapes of interfaces between phases at the microscopic level are of critical importance in predicting macroscopic material characteristics such as strength and electronic properties. Thus, understanding the evolution of microstructures in solids is a fundamental problem in materials science. Simulations in the presence of numerous evolving interfaces pose mathematical and computational challenges, especially in three dimensions. This interdisciplinary research and training program focuses on developing mathematical models and computational techniques to overcome the challenges of computationally studying surface defects in solids and coarsening in solid-solid and solid-liquid systems. The mathematical impacts of the study will include accurate and fast numerical methods for microstructural evolution in three dimensions and for evolution of defects on solid surfaces, using boundary integral methods and incorporating both diffusional and elastic effects. These techniques will be applied to (1) investigate coarsening in solid-solid and solid-liquid systems by performing two- and three-dimensional simulations, and (2) examine step flows (motion of atomic-height steps) on solid surfaces in three dimensions. These investigations will provide insights into materials issues encountered in high-temperature alloys and semiconductor thin film growth. Close collaborations with experimentalists will allow comparisons of the simulation and experimental results. The equations governing the proposed material systems are closely related to a host of mathematical descriptions for pattern formation in moving boundary problems. Therefore, the proposed modeling and computational techniques will have broad applications and will provide new tools for other research areas. Importantly, the proposed program will provide excellent training opportunities for graduate students interested in research at the interface of mathematics, computation, and engineering.
