Differential-Difference Equations and Their Application to Crystalline Growth

PI建议研究空间离散的反应扩散方程(SDRDEs)及其在晶体材料中的生长和界面运动中的应用。PI打算通过寻找和分析比行波解更一般的解以及通过分析高维平面波解的稳定性来实现这一点。这些目标包括证明SDRDE是一个比许多现有的具有晶格结构的材料中的界面运动和生长模型更好的模型。具体包括研究空间离散的一维Allen-Cahn方程的稳定斑图形成并将结果应用于多界面问题，研究二维和三维SDRDE的平衡形状，表明在二维或三维SDRDE的平衡平面界面解中的微小局部扰动可以并且确实导致解演化到原始平衡界面的空间平移，以及精确地模拟氦-4晶体的生长。PI将使用的技术包括使用积分变换、线性弗雷德霍尔姆理论和隐函数定理的构造，展示解的存在和连续，使用自由能泛函寻找和分类平衡形状，以及分析两个平面界面(小平面)相交处的边(3D)或角(2D)的稳定性。虽然用SDRDEs模拟晶体材料中的相变是很自然的，并且可以从物理系统的基本性质出发，但到目前为止，还缺乏有效研究所得到的方程的数学工具；因此，SDRDEs现在已经准备好用作建模工具。本项目旨在开发、求解和应用模拟相变(凝固或熔化、晶界移动等)的数学方程。在晶体材料中，原子以有序的排列排列。例如水、金属和盐。现有的模型要么在原子结构(微观)尺度上，要么在整个系统的(宏观)尺度上。(一些混合模型将这两种比例结合在一起)。相变的微观模型必须通过计算来求解，并且计算资源非常密集。它们还依赖于原子相互作用，如果建模得当，往往会使问题变得太“大”，无法计算。宏观尺度模型可以用一对演化方程来描述相变，但在原子水平上无法解释有序化的影响。已经做出了一些成功的尝试来回收订购信息，但以现象学的方式。这个项目中研究的建模工具是一对简单的演化方程(可以通过分析和计算进行研究)，其中包含从材料的基本物理派生的宏观尺度和微观尺度属性。虽然使用这种方程的想法已经存在了40多年，但研究和求解这些方程所需的数学才刚刚成熟。结果是对晶体材料进行了更准确的数学描述和理解。应用范围从预测机械部件的故障(例如，在喷气式涡轮机中)到控制晶体的生长(例如，在锻钢或生长宝石中)。

The PI proposes to study spatially discrete reaction-diffusion equations (SDRDEs) and their application to growth and interface motion in crystalline materials. The PI intends to do this by finding and analyzing solutions that are more general than traveling plane wave solutions as well as by analyzing the stability of plane wave solutions in higher dimensions. The goals include demonstrating that the SDRDE is a better model than many existing models for interface motion and growth in materials with a crystalline lattice structure. Specifics include studying stable pattern formation for the spatially discrete one-dimensional Allen-Cahn equation and applying the results to multiple interface problems, studying equilibrium shapes for the SDRDE in two and three dimensions, showing that a small local perturbation in an equilibrium planar interface solution for a two- or three-dimensional SDRDE can and does cause the solution to evolve to a spatial translate of the original equilibrium interface, and accurately modeling the growth of helium-4 crystals. The techniques the PI will be using include construction using integral transforms, linear Fredholm theory, and the implicit function theorem, showing existence and continuation of solutions, finding and classifying equilibrium shapes using the free-energy functional, and analyzing the stability of an edge (3D) or corner (2D) where two planer interfaces (facets) meet. Although modeling phase changes in crystalline materials with SDRDEs is natural and can be done from the fundamental properties of the physical systems, the mathematical tools to effectively study the resulting equations have been lacking until now; hence, SDRDEs are now ready to be used as a modeling tool.This project intends to develop, solve, and apply mathematical equations which model phase transitions (solidification or melting, movement of a grain boundary, etc.) in crystalline materials, where the atoms line up in an ordered arrangement. Examples include water, metals, and salts. Existing models are either at the atomic structure (micro-) scale or the (macro-) scale of the entire system. (Some hybrid models combine the two scales). Micro-scale models of phase transitions have to be solved computationally and are extremely computational-resource intensive. They also depend on atomic interactions that, if properly modeled, often make the problem too "large" to compute. Macro-scale models can represent phase transitions with a pair of evolution equations but lose the ability to account for the influences of the ordering at the atomic level. Somewhat successful attempts have been made to reclaim the ordering information, but in a phenomenological manner. The modeling tools that are being studied in this project are a simple pair of evolution equations (which can be studied both analytically and computationally) that contain both the macro-scale and micro-scale properties derived from the basic physics of the materials. Although the idea for using such equations has been around for over 40 years, the mathematics necessary to study and solve these equations is just now reaching maturity. The result is a more accurate mathematical description and understanding of crystalline materials. The applications range from predicting the failure of mechanical parts (for example, in jet turbines) to controlling the growth of crystals (for example, in forging steel or growing gem stones).