Collaborative Research: Morphological Evolution in Materials

在微观水平上，相之间的界面形状在预测宏观材料特性（如强度和电子特性）方面至关重要。 因此，理解固体中微观结构的演化是材料科学中的一个基本问题。 在众多不断发展的接口的存在下的模拟提出了数学和计算的挑战，特别是在三维。 这个跨学科的研究和培训计划的重点是开发数学模型和计算技术，以克服计算研究固体表面缺陷和固-固和固-液系统粗化的挑战。 该研究的数学影响将包括三维微观结构演变和固体表面缺陷演变的准确和快速数值方法，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）通过进行二维和三维模拟来研究固-固和固-液系统中的粗化，以及（2）在三维中检查固体表面上的台阶流（原子高度台阶的运动）。 这些调查将提供洞察高温合金和半导体薄膜生长中遇到的材料问题。 与实验人员的密切合作将允许模拟和实验结果的比较。 所提出的材料系统的方程是密切相关的移动边界问题的图案形成的数学描述主机。 因此，提出的建模和计算技术将有广泛的应用，并将为其他研究领域提供新的工具。 重要的是，拟议的计划将为有兴趣在数学，计算和工程的接口研究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.