Interfacial Dynamics in Multi-phase Transitions

研究者研究界面现象的数学模型。 目的是提供一个系统的研究公式，相变的基础上现有的和新开发的模型。 这些是宏观自由边界模型，基于可观测的量，需要处理由于自由边界的拓扑变化而引起的奇异性，以及微观连续模型，在分子水平上导出，并且（总是）适定性。该项目的一个目标是当自由边界模型变得不确定时，从连续体模型中获得关键信息。 另一个是发展新的微观连续模型，用于多晶体中的相之间的相变，如固/液和不同晶粒。 该项目旨在融合数学和材料科学以及微分方程系统的新发展。 这里使用的主要工具是偏微分方程，奇异摄动，渐近展开，复分析，分歧，中心流形，全球分析，动力系统和几何措施的理论。 计算模拟被用来刺激，验证，并进一步扩展理论结论。 本课题所研究的界面现象在自然界中是很常见的。 只要有一个连续体存在于至少两个不同的阶段，并且有某种机制产生这些阶段的空间分离，它们就会发生。分隔这些阶段的空间边界，称为界面或自由边界，由于阶段之间的过渡机制而随时间演变。 典型的两相转变是液体的凝固过程。 一个经典的多相转变是由多晶体中晶粒的演化引起的，其中不同晶粒中的原子排列在晶界处不匹配。 从液体到固体多晶体的生长涉及到液体和固体之间的两相转变和颗粒之间的多相转变。 近年来，用微观连续介质模型研究相变有了很大的发展，但主要的进展主要是在双相变的情况下。 该项目的重点是多阶段过渡。 这些问题激发了自由边界问题、抛物/椭圆型偏微分/常微分方程组和动力系统的理论发展。 该项目的一部分，涉及黎曼映射定理，被整合到研究生/本科课程的复杂分析，为学生提供一个现实生活中的应用数学理论在材料科学。 此外，研究生也参与了该项目。 这里开发的新模型为科学家提供了理解和设计新材料的工具。

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.