The study of the phase field model designed for recrystallization process

(1)多晶相场模型我们首先提出了新的相场模型，η-θ模型和φ-η-θ模型，通过在常规相场模型中引入角度变量来处理多晶结构。然后，我们将这两个模型统一起来，构造φ-θ模型，使我们能够描述以下过程的顺序：2 .晶界的形成；晶粒的晶界迁移和旋转。该模型的特点是晶界的迁移和晶粒的旋转同时呈现，是唯一能够做到这一点的模型。并编制了该模型的程序代码，进行了各种情况下的仿真。(2)奇异扩散率理论上述模型中最具特色的角变量方程，用Giga和Kobayashi的奇异扩散率理论在数学上进行了证明。将奇异微分方程推广到在圆内取值的函数。并给出了该方程的数值格式。(3)振荡结晶抗坏血酸甲醇溶液在蒸发结晶过程中，根据湿度等条件，形成不同类型的结晶模式。首先，对振荡结晶引起的靶型进行了深入的研究。我们澄清了针状晶体内部的空隙与流体之间的相互作用是振荡结晶的原因。

(1)Phase field model of polycrystalsWe, at first, presented new phase field models, η-θ model and φ-η-θ model, which can handle the polycrystalline structure by introducing an angle variable to the conventional phase field model. Then we constructed φ-θ model by unifying these two models, which enables us to describe the following sequaence of processes : 1.Crystallization after multi nucreation, 2 Formation of grain boundaries, 3.Grain boundary migration and rotation of grains.The feature of our model is that both of the migration of grain boundaries and the rotation of grains are simultaneously presented, and our model is the only model which can do it. Also, we developed the numenal code of this model and performed the simulations which are corres-ponding various situations.(2)Singular diffusivity theoryThe equation of an angle variable in the above models, which is the most characteristic part of our models is justified mathematically by the theory of singular diffusivity developed by Giga and Kobayashi. We extended the equation of singular difftisivity to the functions which takes the value in the circle. In addition, the numerical scheme for this equation was presented.(3)Oscillatory crystallizationVarious types of patterns are formed when the methanol solution of ascorbic acid crystallize by evaporation according to the conditions such as humidity. Above all, target patterns was intensively studied, which is caused by oscillatory crystallization. We clarified that the interaction between the void whithin the needle crystals and the fluid is responsible for the oscillatory crystallization.