Differential-Difference Equations and Their Application to Crystalline Growth

PI建议研究空间离散反应扩散方程（SDRDES）及其在晶体材料生长和界面运动中的应用。 PI打算通过寻找和分析比行进平面波解更一般的解以及通过分析更高维平面波解的稳定性来做到这一点。 目标包括证明SDRDE是一个更好的模型比许多现有的模型界面运动和生长的材料与晶格结构。 具体内容包括研究空间离散一维Allen-Cahn方程的稳定模式形成，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，一个小的局部扰动的平衡平面界面解的两个或三个-三维SDRDE可以并且确实导致解演化到原始平衡界面的空间平移，并且精确地模拟氦-4晶体的生长。 PI将使用的技术包括使用积分变换，线性Fredholm理论和隐函数定理的构造，显示解的存在性和连续性，使用自由能泛函找到和分类平衡形状，以及分析两个平面界面（小平面）相遇的边缘（3D）或拐角（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来完成，但直到现在还缺乏有效研究所得方程的数学工具;因此，SDRDE现在已经可以作为一种建模工具来使用了。本项目旨在开发、求解和应用模拟相变的数学方程（凝固或熔化、晶界移动等）在晶体材料中，原子以有序的方式排列。 实例包括水、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。 (Some混合模型联合收割机结合了这两种尺度）。 相变的微尺度模型必须通过计算来解决，并且是非常计算资源密集型的。 它们还依赖于原子相互作用，如果正确建模，通常会使问题太“大”而无法计算。 宏观尺度模型可以用一对演化方程来描述相变，但不能考虑原子级有序性的影响。 有些成功的尝试已被用于回收有序信息，但以现象学的方式。 该项目正在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学中获得的宏观尺度和微观尺度特性。 虽然使用这些方程的想法已经存在了40多年，但研究和解决这些方程所需的数学才刚刚成熟。 其结果是对晶体材料进行了更准确的数学描述和理解。 其应用范围从预测机械部件的故障（例如，在喷气涡轮机中）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).