Mathematical Modeling of Materials Interfacial Motions

叶博士计划从数学角度研究固体材料中界面的运动。他将专注于两个主题。第一个主题涉及外延薄膜生长的建模。这些薄膜可以具有重要的半导性或超导性。这项工作的目标是准确地描述大长度尺度下的薄膜演化。研究人员计划在连续介质水平上捕捉薄膜表面的表面扩散和成核现象。这两种机制都可以对薄膜的生长和所得到的薄膜性能产生深远的影响。研究的起点是原子层行为的Burton-Cabrera-Frank模型及其推广。这项工作的数学内容涉及到非线性偏微分方程和数值模拟。第二个主题是在非唯一性的情况下，通过平均曲率来研究噪声对运动的影响。这种运动是凝固过程中表面演化的简化模型。众所周知，方程可以有多个解。问题出现了，哪一个是与物理相关的问题。这种非唯一性是更好地从数学上理解界面运动的主要障碍。特别是，如果解不是唯一的，则不清楚近似方案将选择哪个极限解。叶博士计划研究噪音带来的选择效应，也就是小的随机扰动。这种方法在许多其他类型的非线性演化问题中都有潜在的应用。材料生长现象的数学建模长期以来一直是一个活跃的研究领域。人们正在不断地寻找和制造新材料。一个正确的数学模型能够以积极的方式预测和控制生产过程的结果。这项研究有两个目标。首先是准确地在外延薄膜生长的微观和宏观描述之间架起桥梁。这些薄膜在电子材料方面有很大的应用。二是研究了“噪声”(即随机扰动效应)对材料界面运动模拟的影响，如合金中不同组元之间的界面运动。尽管众所周知，这种噪音存在于材料生长中，但它的后果仍然没有完全被理解。这两个主题都将涉及数学分析和物理直觉。

Dr. Yip plans to investigate the motion of interfaces in solid materials from a mathematical point of view. He will concentrate on two topics.The first one concerns the modeling of epitaxial thin film growths. These films can have important semi- or super-conducting properties. The goal of this work is to accurately describe the thin film evolutions at large length scales. The investigator plans to capture the surface diffusion and nucleation phenomena on the film surface at the continuum level. Both of these mechanisms can have profound effects onthe film growth and the resultant film properties. The starting point of the research is the Burton-Cabrera-Frank model for the behavior of atomic layers and its generalizations. The mathematical content of this work involves nonlinear partial differential equations and numerical simulations.The second topic is to investigate the effects of noise on motion bymean curvature in the case of non-uniqueness. Such type of motion isa simplified model for surface evolutions in solidification processes. It is well known that the equations can possess more than one solution. The question arises which is the physically relevant one. This non-uniquenessis a major obstacle to a better mathematical understanding of the interface motion. In particular, if the solutions are not unique, it is not clear which limiting solution will be picked by approximation schemes. Dr. Yip plans to investigate the selection effects brought out by the presence of noise, that is small random perturbations. Such an approach has potential applications in many other types of nonlinear evolution problems.The mathematical modeling of materials growth phenomena has long beenan active research area. New materials are being continuously searchedfor and manufactured. A correct mathematical model can predict and control in positive ways the outcome of the production processes. This research has two goals. The first is to bridge in an accurate manner the microscopic and macroscopic descriptions in epitaxial thin film growths. These films have great applications in electronic materials. The second is to investigate the effects of "noise" (that is, random perturbative effects) in the modeling of the motion in interfaces in materials, such as the interface between different components in alloys. Even though such noise is known to be present in materials growths, its consequences still have not been understood completely. Both topics will involve mathematical analysis and physical intuition.