Atoms, Defects and the Kinetics of Phase Transformations

对相边界动力学的基本理解，以及更广泛的滞后，仍然是活性材料研究中的一个开放挑战。缺乏动力学信息是包括位错力学和薄膜生长在内的各种问题的一贯困难。当前奖项的研究虽然侧重于活性材料，但将对这些领域产生影响。现在有一个公认的基于动力学关系概念的模型框架，但它是现象学的，因为它没有解决为什么一些相边界比其他相边界更具流动性以及如何改变流动性。该项目通过计算和严格的数学来检验是否可以从位移相变背景下更基本或更小尺度的材料描述中推导出动力学关系。有两个尺度是特别有趣的，原子和缺陷。首先，该项目试图理解原子-连续体的联系，从能够相变的原子（离散）模型开始，以捕捉相变过程的基本动力学的方式传递到连续体。其次，该项目旨在了解空位、杂质和第二相沉淀等缺陷在确定相边界整体扩展中的作用。这就导致了非局部自由边界问题的均匀化问题。形状记忆材料和铁电材料等活性材料具有不同寻常但有用的特性，这使得它们在诸如牙套、心脏支架、空间天线、超声波设备、压力传感器和微致动器等各种应用中至关重要。这些不寻常的特性源于这些材料在微观尺度下可以形成的非常独特和复杂的图案。在过去的十年中，人们对微观结构的本质、与基本晶体学的关系及其对材料性能的影响的理解取得了很大的进展。特别是，一个复杂的静态微观结构的数学理论已经出现，这一理论通过预测和解释新的现象和应用具有重要的实际影响。然而，当我们对这些材料施加应力时，微观结构如何变化仍然未知。这很重要，因为它决定了这些材料的一个重要特性，即迟滞或形状记忆。该奖项支持的工作将建立微观结构演化的数学理论。它还将培养研究生和本科生在新兴和重要的多尺度建模领域的研究。

A fundamental understanding of the kinetics of phase boundaries, and more broadly hysteresis, remains an open challenge in the study of active materials. The lack of kinetic information is a consistent difficulty in various problems including dislocation mechanics and growth of thin films. The research in the current award, though focussed on active materials, will impact these areas. There is now an accepted framework to model this based on the notion of kinetic relations, but it is phenomenological in that it does not address why some phase boundaries are more mobile than others and how one can change the mobility. This project examines through computation and rigorous mathematics whether one can derive a kinetic relation starting from a more fundamental or smaller scale description of materials in the context of displacive phase transformations. Two scales are of particular interest, atoms and defects. First, the project seeks to understand the atomic-continuum linkage by starting from an atomistic (discrete) model capable of phase transformations and passing to the continuum in such a manner that captures the essential dynamics of the phase transformation process. Second, the project seeks to understand the role of defects such as vacancies, impurities and second phase precipitates in determining the overall propagation of a phase boundary. This leads to the mathematical problem of homogenization of a non-local free boundary problem.Active materials like shape-memory materials and ferroelectric materials possess unusual but useful properties that make them vital for a variety of applications like dental braces, cardiac stents, space antennas, ultrasonic devices, pressure sensors and microactuators. The unusual properties arise from very characteristic and intricate patterns that these materials can form at a microscopic scale. Over the last decade, much advance has been made in understanding the nature of the microstructure, its relation to basic crystallography and its consequences for material properties. In particular, a sophisticated mathematical theory for static microstructures has emerged, and this theory has had important practical impact by predicting and explaining new phenomena and applications. However, much remains unknown regarding how the microstructure changes as we apply stress to these materials. This is important since it determines an important property of these materials known as hysteresis or shape memory. Work supported by this award will build a mathematical theory of the evolution of microstructure. It will also train graduate and undergraduate students in research in the emerging and important area of multiscale modeling.