Most of my research is interdisciplinary and is concerned with theoretical problems in materials science that lead to challenging problems in physics and mathematics. Examples are the thermodynamics of stressed solids, transport phenomena, surfaces and interfaces, phase transformations, the precise definition of chemical potentials in stressed solids, the fundamental basis of the Onsager reciprocal relations in multi-component diffusion and heat flow, and the influence of anisotropic surface tension on crystal shape. Problems dealing with phase transformations lead to difficult free boundary problems that are generalizations of the classical Stefan problem because of boundary conditions that depend on the curvature of the free boundary. One seeks to calculate and understand the factors that determine the shapes of the interfaces that separate the growing phase from the nutrient phase.

Linear stability theory is used to analyze the conditions under which bodies of simple shape evolve spontaneously into more complex patterns. Non-linear analyses, frequently requiring numerical techniques, are used to track freely growing shapes and to ascertain fundamental aspects of the cellular and dendritic patterns that often result. Recent work involves the phase field model (diffuse interface) in which an additional PDE is solved in lieu of boundary tracking. The phase field model has been used to calculate the operating state (tip speed and radius of curvature) of dendrites grown at large supercoolings. Finally, there is interest in modeling the effects of g-jitter on interdiffusion in the microgravity environment of space as a stochastic process as well as the influence of fluid convection on dendritic growth on Earth.