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 (including definition of chemical potentials), transport phenomena, surfaces and interfaces, phase transformations, the functional application of the Onsager reciprocal relations in multi-component diffusion and heat flow, the influence of anisotropic surface tension on crystal shape and faceting, and the theory of morphological stability 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. We employ the phase field model (diffuse interface) in which an additional PDE is solved in lieu of boundary tracking, to calculate the operating state (tip speed and radius of curvature) of dendrites grown at large supercoolings, as well as cell shapes and solute segregation during directional solidification of alloys. We have also used Lattice-Boltzmann models to treat buoyancy driven convection, interdiffusion and the patterns that form when one immiscible fluid displaces another. More recent work includes calculation of surface morphologies near moving grain boundaries, irreversible thermodynamics of creep in crystalline solids, and the theory of multicomponent diffusion.