Collaborative Research: Morphological Evolution in Materials

相间界面在微观层面上的形状对于预测材料的宏观特性，如强度和电子性能至关重要。因此，了解固体微观结构的演化是材料科学中的一个基本问题。在存在大量不断变化的界面的情况下进行模拟提出了数学和计算方面的挑战，特别是在三维方面。这一跨学科的研究和培训计划专注于开发数学模型和计算技术，以克服通过计算研究固体中的表面缺陷和固体-固体和固体-液体系统中的粗化的挑战。这项研究的数学影响将包括准确和快速的三维微结构演化和固体表面缺陷演化的数值方法，使用边界积分方法，并考虑扩散和弹性效应。这些技术将被应用于(1)通过执行二维和三维模拟来研究固体-固体和固体-液体系统中的粗化，以及(2)研究三维固体表面上的台阶流动(原子高度台阶的运动)。这些研究将为高温合金和半导体薄膜生长中遇到的材料问题提供深入的见解。与实验者的密切合作将允许对模拟和实验结果进行比较。所提出的材料系统的方程与移动边界问题中图案形成的一系列数学描述密切相关。因此，所提出的建模和计算技术将具有广泛的应用前景，并将为其他研究领域提供新的工具。重要的是，拟议的计划将为对数学、计算和工程相结合的研究感兴趣的研究生提供极好的培训机会。

The shapes of interfaces between phases at the microscopic level are of critical importance in predicting macroscopic material characteristics such as strength and electronic properties. Thus, understanding the evolution of microstructures in solids is a fundamental problem in materials science. Simulations in the presence of numerous evolving interfaces pose mathematical and computational challenges, especially in three dimensions. This interdisciplinary research and training program focuses on developing mathematical models and computational techniques to overcome the challenges of computationally studying surface defects in solids and coarsening in solid-solid and solid-liquid systems. The mathematical impacts of the study will include accurate and fast numerical methods for microstructural evolution in three dimensions and for evolution of defects on solid surfaces, using boundary integral methods and incorporating both diffusional and elastic effects. These techniques will be applied to (1) investigate coarsening in solid-solid and solid-liquid systems by performing two- and three-dimensional simulations, and (2) examine step flows (motion of atomic-height steps) on solid surfaces in three dimensions. These investigations will provide insights into materials issues encountered in high-temperature alloys and semiconductor thin film growth. Close collaborations with experimentalists will allow comparisons of the simulation and experimental results. The equations governing the proposed material systems are closely related to a host of mathematical descriptions for pattern formation in moving boundary problems. Therefore, the proposed modeling and computational techniques will have broad applications and will provide new tools for other research areas. Importantly, the proposed program will provide excellent training opportunities for graduate students interested in research at the interface of mathematics, computation, and engineering.