Solidification and melting processes can generate complicated and unstable material microstructures which evolve with time. This project is concerned with the mathematical study of complex microstructures that can be observed in fundamental phase-field models for solidification and melting processes. In addition to classical and thermodynamically consistent models, the study considers recent stochastic and nonlocal extensions. The latter incorporate the effects of thermal noise and long-range interactions, thereby addressing two key shortcomings of the classical phase-field models. The PI obtains a detailed mathematical description of the microstructure dynamics, which explains both the formation of the microstructures and key characteristics of their complex geometry. He studies how noise and long-range interactions influence the pattern geometry. Furthermore, in collaboration with a graduate student, the PI develops numerical simulation tools to treat stochastic and nonlocal effects.
In recent years, phase-field modeling has proved to be a valuable tool for simulating and predicting a variety of processes in materials science. The proposed research provides tools that can be used for precisely predicting microstructure changes and therefore is of interest for materials design and the understanding of small-scale biological and physical processes. Since the microstructure geometry of modern materials has direct implications for macroscopic material properties, the results of this project can offer help in the development of new high performance materials, which is vital to economic and other national interests. Especially due to the inclusion of thermal noise and long-range interactions, the study allows for the treatment of more realistic models. While the proposed research is focused on a specific situation, its impact reaches well beyond the materials science application. Immediate applications of the project can be expected in the field of mathematical biology. Both the formation of complicated protein patterns inside bacteria and aspects of neural pattern formation are described by phase-field-type models. In addition, the proposed activities offer an excellent framework for integrating research and education at several levels, including undergraduate and graduate research.
