The study of the phase field model designed for recrystallization process

（1）多晶体的相场模型首先提出了新的相场模型η-θ模型和φ-η-θ模型，它们通过在传统的相场模型中引入一个角度变量来处理多晶体结构。然后，我们将这两个模型统一起来，构造了φ-θ模型，使我们能够描述以下过程序列：1.多次成核后的晶化，2.晶界的形成，3.晶界的迁移和晶粒的旋转，本模型的特点是晶界的迁移和晶粒的旋转同时出现，我们的模型是目前唯一能做到这一点的模型，我们还开发了该模型的数值代码，并进行了相应于各种情况的模拟。（2）奇异扩散率理论上述模型中的角变量方程是我们模型中最有特色的部分，它是由Giga和小林发展的奇异扩散率理论在数学上证明的。将奇异扩散率方程推广到取值于圆上的函数。此外，还给出了该方程的数值格式.（3）振荡结晶当抗坏血酸的甲醇溶液通过蒸发结晶时，根据诸如湿度的条件，形成各种类型的图案。首先，深入研究了由振荡结晶引起的目标图案。阐明了针状晶体内部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.