Collaborative Research: Morphological Evolution in Materials

在微观水平上，相间界面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，了解固体中微观结构的演变是材料科学的一个基本问题。在众多不断变化的界面中进行模拟，对数学和计算提出了挑战，特别是在三维空间中。这个跨学科的研究和培训计划侧重于发展数学模型和计算技术，以克服计算研究固体表面缺陷和固体-固体和固体-液体系统粗化的挑战。该研究的数学影响将包括精确和快速的三维微观结构演变和固体表面缺陷演变的数值方法，使用边界积分方法并结合扩散和弹性效应。这些技术将应用于(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)在三维空间中检查固体表面上的阶梯流动（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见解。与实验人员的密切合作将使模拟和实验结果得以比较。控制所提出的材料系统的方程与移动边界问题中模式形成的大量数学描述密切相关。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，该计划将为对数学，计算和工程界面研究感兴趣的研究生提供良好的培训机会。

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.