Interfacial Dynamics in Multi-phase Transitions

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The investigator studies mathematical models for interfacialphenomena. The objective is to provide a systematic study formulti-phase transitions based on both existing and newly developedmodels. These are macroscopic free boundary models, based onobservable quantities and needed to deal with singularities due totopological changes of free boundaries, and microscopic continuummodels, derived at the molecular level and (always) well-posed. One goal of the project is to derive critical information fromcontinuum models for free boundary models when the latter becomeindeterminate. Another is to develop new microscopic continuummodels for phase transitions among phases such as solid/liquid anddifferent grains in polycrystals. The project is aimed at thefusion of mathematics and material science and new developmentsfor systems of differential equations. The main tools used hereare theories of partial differential equations, singularperturbations, asymptotic expansions, complex analysis,bifurcation, center manifolds, global analysis, dynamical systems,and geometric measure. Computational simulations are used tostimulate, validate, and further extend theoretical conclusions. The interfacial phenomena studied in this project arecommonplace in nature. They occur whenever there is a continuumthat can exist in at least two different phases and there is somemechanism that generates a spatial separation of these phases. The spatial boundaries that separate these phases, referred to asinterfaces or free boundaries, evolve with time due to somemechanisms of transitions between phases. A typical two-phasetransition is a solidification process of a liquid. A classicalmulti-phase transition arises from evolutions of grains in apolycrystal where alignments of atoms in different grains do notmatch at grain boundaries. The growth of a solid polycrystal froma liquid involves a two-phase transition between liquid and solidand a multi-phase transition among grains. The use of microscopiccontinuum models to study phase transitions has undergone a strongdevelopment in recent years, but major advances have been mademostly in the case of two-phase transitions. This project focuseson multi-phase transitions. These problems motivate newtheoretical development in free boundary problems, systems ofparabolic/elliptic partial/ordinary differential equations, anddynamical systems. Part of the project, involving the Riemannmapping theorem, is integrated into a graduate/undergraduatecourse on complex analysis to provide students with a real-lifeapplication of mathematical theory in material science. Inaddition, graduate students are involved in the project. Newmodels developed here provide scientists with tools inunderstanding and designing new materials.