Simultaneous dependent transformations nucleated at vertices and faces of 3D Poisson-Voronoi tessellations

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 PoissonVoronoi 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.