Homogenization of liquid phase epitaxy

该项目侧重于新的液相外延双尺度模型，该模型能够同时描述外延微观结构的演变以及溶质、热量和流体的宏观输运。该模型采用均匀化的方法推导出微观模型，该微观模型由Burton-Cabrera-Frank模型和溶质、热量和物质输运的对流扩散方程推导而来。工作计划包括对双尺度模型的数学分析，解决该模型的可解性及其对小尺度给定情况的近似，模型的数值近似和广泛的参数研究，包括与实验的比较，以验证模型并评估其适用范围。这些任务将在外延生长模型的相场版本中进行。计划分几个步骤进行，从一个简单的模型开始，该模型只描述固体中附着原子的扩散和外延固体层的生长。然后将该模型逐步扩展，最终描述包括流体传热和流体流动在内的整个物理过程。

The project focuses on new two-scale models for liquid phase epitaxy that are capable to simultaneously describe the evolution of epitaxial microstructures and the macroscopic transport of solute, heat and fluid. The model is derived by homogenization from a microscopic model that is given by a Burton-Cabrera-Frank model for the growth of the epitaxial solid layer on a part of the boundary of a given threedimensional domain, and by convection-diffusion equations for the transport of solute, heat and matter. The working plan contains a mathematical analysis of the two-scale model that addresses the solvability of the model and its approximation of a given situation with small scale, the numerical approximation of the model and extensive parameter studies including comparisons with experiments in order to verify the model and evaluate its range of applicability. These tasks shall be carried out for a phase field version of the epitaxial growth model. It is planned to proceed in several steps, starting from a simple model that only describes diffusion of adatoms in the solid and the growth of the epitaxial solid layer. Then this model shall be gradually extended to finally describe the full physics including heat transfer in the fluid and fluid flow.