Interfacial Dynamics in Multi-phase Transitions

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The investigator studies mathematical models for interfacial phenomena. The objective is to provide a systematic study for multi-phase transitions based on both existing and newly developed models. These are macroscopic free boundary models, based on observable quantities and needed to deal with singularities due to topological changes of free boundaries, and microscopic continuum models, derived at the molecular level and (always) well-posed. One goal of the project is to derive critical information from continuum models for free boundary models when the latter become indeterminate. Another is to develop new microscopic continuum models for phase transitions among phases such as solid/liquid and different grains in polycrystals. The project is aimed at the fusion of mathematics and material science and new developments for systems of differential equations. The main tools used here are theories of partial differential equations, singular perturbations, asymptotic expansions, complex analysis, bifurcation, center manifolds, global analysis, dynamical systems, and geometric measure. Computational simulations are used to stimulate, validate, and further extend theoretical conclusions. The interfacial phenomena studied in this project are commonplace in nature. They occur whenever there is a continuum that can exist in at least two different phases and there is some mechanism that generates a spatial separation of these phases. The spatial boundaries that separate these phases, referred to as interfaces or free boundaries, evolve with time due to some mechanisms of transitions between phases. A typical two-phase transition is a solidification process of a liquid. A classical multi-phase transition arises from evolutions of grains in a polycrystal where alignments of atoms in different grains do not match at grain boundaries. The growth of a solid polycrystal from a liquid involves a two-phase transition between liquid and solid and a multi-phase transition among grains. The use of microscopic continuum models to study phase transitions has undergone a strong development in recent years, but major advances have been made mostly in the case of two-phase transitions. This project focuses on multi-phase transitions. These problems motivate new theoretical development in free boundary problems, systems of parabolic/elliptic partial/ordinary differential equations, and dynamical systems. Part of the project, involving the Riemann mapping theorem, is integrated into a graduate/undergraduate course on complex analysis to provide students with a real-life application of mathematical theory in material science. In addition, graduate students are involved in the project. New models developed here provide scientists with tools in understanding and designing new materials.