圆盘状晶体生长问题的摄动分析

圆盘状晶体生长是自由晶体生长中典型的生长模式,在其生长过程中顶部和底部的生长速度缓慢,而边缘部分由于热扩散机制的作用而生长速度较快,容易形成花样.由于边缘界面上存在小扰动的作用,从而表现为数学模型的边界条件中出现一个小参数.通过用摄动方法对边界条件中含有小参数的问题进行分析,将其转化为较容易求解的问题,然后用数学物理的方法求出这些问题的解,最后对解的形式进行分析,得到了与实验相符的结果,即固相中的温度场呈现出指数型振荡衰减.

The Asymptotic Analysis of Disc-like Crystal Growth

Disc-like crystal growth is a typical prototype of free crystal growth. The growth of the top and bottom interface is very slow while the growth of its side-interface is much faster owing to heat diffusion mechanism, thus producing a pattern formation. Because of the perturbation on the side-interface, the boundary condition of mathematic model appears with a small parameter. We anayze the problem whose boundary condition produces a small parameter by asymptotic method and convert several problems to easily solved ones. Then mathematic and physical methods are used to work out the problems. Finally by analyzing the form of the mathematic result, we find that the temperature filed of crystal growth presents an oscillatory and exponentially descending pattern, a result corresponding to the experiment.