Elastic Effects in the Phase Separation of the System with Finite Size

(a)晶体相场模型有一个本质的缺点，即缺乏取向信息，不能描述多晶的形成。我们提出了一个矢量化的相场模型，使我们能够模拟同时结晶的许多颗粒具有不同的取向和晶界的形成。(b)再结晶通过两个基本过程发生，即（I）晶界迁移和（ii）晶粒旋转。几乎所有的再结晶模型都只考虑了（I）过程。我们提出了一个全新的相场模型，它可以同时处理过程（I）和（II）。在上述再结晶模型中，通过引入一个新的数学概念“奇异扩散率”，角变量的演化方程表达了一种非局部相互作用。(d)众所周知，颗粒材料的混合物可以在旋转的圆筒中分离。本文通过实验和模拟研究发现，这是由于二元颗粒混合物表面流动的强烈分凝特性造成的。(e)本文提出了生长晶体端面台阶动力学的数学模型。该模型能够描述任意数量螺位错引起的台阶运动。(f)用实验数学的方法明确了产生自复制图形的数学结构。我们证明了有序鞍结分岔分支和连接它们的路径是形成自复制模式的必要条件。

(a) Usual phase field models have an essential short coming that they cannot describe the formation of poly-crystals since they are lacking in the information of orientation. We proposed a vectorized phase field model which enables us to simulate a simultaneous crystallization of many particles with various orientations and a formation of grain boundaries.(b) Recrystallization takes place through the two basic processes, say, (I) grain boundary migration and (ii) grain rotation. Almost all the model of recrystallization handle the process (I) only. We proposed a totally new phase field model which can treat both of the processes (I) and (il) simultaneously.c In the above recrystallization model the equation for the evolution of angle variable expresses a non-local interaction by introducing a new mathematical concept "singular diffusivity" We justified this new equation and analyzed it mathematically.(d) It is well known that the mixture of granular materials can segregate in the rotating cylinder. We investigated this phenomena through experiments and modeling and found that it is caused by the property of strong segregation in the suface flow of binary mixture of granular materials.(e) A mathematical model of step dynamics on the facet of growing crystals are presented. This model is able to describe the motion of steps caused by the arbitrary number of screw dislocations.(f) Mathematical structure which produces self-replicating patterns were made clear by the approach of experimental mathematics. We demonstrated that the ordered saddle node bifurcation branches and the paths connecting them are essential for the formation of self-replicating patterns.