The stochastic geometry of the crystallization process of polymers

A Johnson-Mehl tessellation arises as a random division of a given bounded region in a d-dimensional Euclidean space, generated by a stochastic birth-and-growth process, also known as a germ-grain process in stochastic geometry. A typical example is crystallization of a polymer from an amorphous liquid phase, by cooling; in this case a grain (crystal) is formed by growth of a germ (nucleus) born at a random time and space location. The kinetic parameters , i.e. the nucleation rate and the growth rate, may both depend upon space and time. By assuming that at contact points of the growth fronts grains stop growing (impingement), the spatial region will be divided into cells, and interfaces (n-facets) at different Hausdorff dimensions (cells,faces, edges, vertices) appear. Here we extend previous results regarding the evolution of the morphology of the resulting tessellation to the general case of space and time heterogeneous birth-and-growth processes. In particular evolution equations for the n-facet densities are provided and a technique to estimate these densities from digitized images is proposed.