Dr. Yip plans to investigate the motion of interfaces in solid materials from a mathematical point of view. He will concentrate on two topics. The first one concerns the modeling of epitaxial thin film growths. These films can have important semi- or super-conducting properties. The goal of this work is to accurately describe the thin film evolutions at large length scales. The investigator plans to capture the surface diffusion and nucleation phenomena on the film surface at the continuum level. Both of these mechanisms can have profound effects on the film growth and the resultant film properties. The starting point of the research is the Burton-Cabrera-Frank model for the behavior of atomic layers and its generalizations. The mathematical content of this work involves nonlinear partial differential equations and numerical simulations. The second topic is to investigate the effects of noise on motion by mean curvature in the case of non-uniqueness. Such type of motion is a simplified model for surface evolutions in solidification processes. It is well known that the equations can possess more than one solution. The question arises which is the physically relevant one. This non-uniqueness is a major obstacle to a better mathematical understanding of the interface motion. In particular, if the solutions are not unique, it is not clear which limiting solution will be picked by approximation schemes. Dr. Yip plans to investigate the selection effects brought out by the presence of noise, that is small random perturbations. Such an approach has potential applications in many other types of nonlinear evolution problems.
The mathematical modeling of materials growth phenomena has long been an active research area. New materials are being continuously searched for and manufactured. A correct mathematical model can predict and control in positive ways the outcome of the production processes. This research has two goals. The first is to bridge in an accurate manner the microscopic and macroscopic descriptions in epitaxial thin film growths. These films have great applications in electronic materials. The second is to investigate the effects of "noise" (that is, random perturbative effects) in the modeling of the motion in interfaces in materials, such as the interface between different components in alloys. Even though such noise is known to be present in materials growths, its consequences still have not been understood completely. Both topics will involve mathematical analysis and physical intuition.
