Many technologically useful materials are polycrystals composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. The energetics and connectivities of the grain boundaries play an essential role in defining the effective properties of materials across multiple scales. In this paper we derive a Fokker-Planck model for the evolution of the planar grain boundary network. The proposed model considers anisotropic grain boundary energy which depends on lattice misorientation and takes into account mobility of the triple junctions, as well as independent dynamics of the misorientations. We establish long time asymptotics of the Fokker-Planck solution, namely the joint probability density function of misorientations and triple junctions, and closely related the marginal probability density of misorientations. Moreover, for an equilibrium configuration of a boundary network, we derive explicit local algebraic relations, a generalized Herring Condition formula, as well as formula that connects grain boundary energy density with the geometry of the grain boundaries that share a triple junction. Although the stochastic model neglects the explicit interactions and correlations among triple junctions, the considered specific form of the noise, under the fluctuation-dissipation assumption, provides partial information about evolution of a grain boundary network, and is consistent with presented results of extensive grain growth simulations.
翻译:许多技术上有用的材料都是由小单晶状谷物组成的多晶状材料,这些谷物由晶晶体的颗粒边界以不同的花纹取向而分离。谷物边界的活力和连接性在确定各种规模材料的有效特性方面起着关键作用。在本文中,我们为平面谷物边界网络的演进而得出了一个Fokker-Planck模型。拟议模型考虑的是依赖拉特方向的细微微单晶状谷物边界能量,考虑到三重交叉点的移动性以及方向错误的独立动态。我们建立了Fokker-Planck解决方案的长期停滞性,即错误方向和三重交叉点的热度联合概率密度功能,与方向错误方向的边缘概率密度密切相关。此外,对于边界网络的平衡配置,我们提出了明确的当地测深线关系,一种通用的热调调公式,以及将谷物边界的能量密度与三重连接的谷物边界的几何测量性动态联系起来的公式。尽管分解型模型忽视了方向方向和三重连接的谷物网络的深度变化的精确度,但是,在所考虑的深度变化模型中断面的精确度的模型提供了一种精确的模型的模型,并且提供了一种精确的精确的模型,在考虑的模型下提供了一种精确变化变化的模型的模型的模型的模型的精确的模型,提供了一种明确的联系。