This proposal strives to understand the energy landscape experienced by materials phase boundaries. This is a tremendously interesting and important problem in the regimes of applied analysis and modeling in materials science. It is due to the fact that phase boundaries, during their formations, will inevitably encounter complicated microstructures coming from the defects and impurities embedded in the environment. The investigator will study the stable and unstable patterns of the phase boundaries. These correspond to the local minimizers and saddle points on the materials energy functionals. The main emphasis of this proposal is to study the time evolution of the phase boundaries on the energy landscape, in particular, the transition from one stable configuration to another and from the pinning and de-pinning phenomena in response to some external forcing. The outcome of the scientific activity will provide a solid mathematical framework for the understanding of the dynamical response of the materials with respect to the underlying spatial environment and external parameters. The mathematical techniques involved in this proposal include variational methods, asymptotic analysis, and bifurcation theory.
The understanding of material properties is becoming more and more instrumental in modern materials science, especially in the current pursuit of small scale (nano-)materials. A crucial factor is to relate the materials properties to their microstructures. The outcome of the proposed activity can help the description of the stable and unstable pattern formations of the materials phase boundaries and their dynamical behavior when external conditions changes. This gives a better understanding of the materials response to the external environment, which in turn gives better design and manufacture of the materials. The topic of this investigation is highly interdisciplinary, which can lead to close collaboration between mathematicians and materials scientists. It also provides good opportunities for human resources and educational development.
