In polycrystals, a new phase nucleation usually occurs on the grain boundaries. Cahn (J.W. Cahn: Acta Metall. 1956, 4, 449–459) proposed analytical expressions for transformations nucleated on the grain faces, edges, and vertices. Notwithstanding, there are no models for transformations that nucleate simultaneously on, for example, grain faces and vertices. Modeling simultaneous transformations fundamentally depends on whether the transformations are independent. For example, transformations located on random points and random faces are independent. Independent transformations can be modeled using the methodology developed by Rios and Villa (2011). The situation becomes much more difficult when the transformations are not independent but dependent. In this work, we study transformations nucleated simultaneously at the vertices and faces of Poisson-Voronoi tessellations employing analytical methods and computer simulation. We develop an analytical mathematical treatment for the situation in which simultaneous transformations are not independent but are dependent in the probabilistic sense. Our mathematical approach specifically applies to the simultaneous nucleation on the faces and vertices of a Poisson Voronoi tessellation.

中文翻译：

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在 3D Poisson-Voronoi 镶嵌的顶点和面处成核的同步相关变换


在多晶中，新相成核通常发生在晶界上。Cahn （J.W. Cahn： Acta Metall. 1956， 4， 449–459） 提出了在颗粒面、边和顶点上成核的变换的解析表达式。尽管如此，没有同时在粒度面和顶点上成核的变换模型。对同步变换进行建模从根本上取决于变换是否独立。例如，位于随机点和随机面上的变换是独立的。可以使用 Rios 和 Villa （2011） 开发的方法对独立转换进行建模。当转换不是独立的而是相互依赖的时，情况会变得更加困难。在这项工作中，我们采用分析方法和计算机模拟研究了在 Poisson-Voronoi 镶嵌的顶点和面上同时成核的变换。我们针对同时变换不是独立的，而是在概率意义上依赖于的情况开发了一种分析数学处理方法。我们的数学方法特别适用于 Poisson Voronoi 镶嵌的面和顶点上的同步成核。