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

(a)通常的相场模型由于缺乏取向信息而不能描述多晶的形成，这是一个重要的缺点。我们提出了一个矢量相场模型，使我们能够模拟具有不同取向的许多颗粒同时结晶和晶界的形成。(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处理(1)过程。在上述再结晶模型中，角变量的演化方程通过引入新的数学概念“奇异扩散率”来表达非局部相互作用，并对新方程进行了证明和数学分析。(d)众所周知，颗粒状物料的混合物在旋转圆筒内会发生离析。我们通过实验和模型研究了这一现象，发现这是由于颗粒材料二元混合物在表面流动中具有强烈的偏析特性造成的。(e)提出了生长晶体面阶跃动力学的数学模型。该模型能够描述任意数量的螺杆位错引起的台阶运动。(f)产生自我复制模式的数学结构是通过实验数学的方法弄清楚的。我们证明了有序鞍节点分叉分支和连接它们的路径对于形成自我复制模式是必不可少的。

(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.