Stochastic geometry of spatially structured birth and growth processes : applications to crystallization processes.

Birth-and-growth processes provide a large class of mathematical models for many phase change phenomena. Their peculiarity with respect to other spatially structured stochastic processes lies in their property of space invasion, so that the kinetic parameters of the process may depend upon the history of the geometric structure of the process itself. As a working example we shall refer to the crystallization process of polymers. If an experiment is started with a liquid (the polymer melt) and the temperature is subsequently decreased below a certain point (the melting point of the material), crystals appear {\em randomly in space and time} and start to grow. Growth processes may be very complicated, but usually, with polymers we have growth of either spherical crystallites ({\em spherulites}) or of cylindrical crystallites (so called {\em shish-kebabs}). In the following the restriction to the case of spherulitic growth is made, which is also a good assumption for the crystallization of relaxed polymer melts. The nucleation process, being random both in time and space, is modelled as a \index{marked point process}{\it marked point process}, that is a marked counting process, the mark being the (random) spatial location of the germ. Immediately after birth, nuclei start to grow in all directions according to some surface growth process, thus filling the space of {\em spherulites}. Each spherulite will stop its grow at points of contact with other crystals . This phenomenon, called {\em impingement}, causes the morphology of the crystalline phase; the resulting division of space into cells is called {\em Johnson-Mehl tessellation} . Because of the randomness in time and location of the birth of crystals, the final morphology is random. A mathematical theory is needed to describe the time evolution of the relevant geometric measures related to the tessellation. If we ignore impingement the space filling due to the growth process may be described as a {\em dynamic Boolean model} (a brief introduction of the basic concepts of stochastic geometry is given). For the sake of generality we must take into account that the kinetic parameters of nucleation and growth may depend upon space and time via some underlying field (such as temperature in the polymer crystallization process). The dynamic Boolean model, coupled with the concept of causal cone in presence of spatial heterogeneities, provides a generalization of the well known Kolmogorov-Avrami-Evans formula \cite{K37} that provides the evolution of the {\em degree of crystallinity}, that is the mean volume fraction of the space occupied by crystals. In this set up a crucial role is played by the {\em hazard function} corresponding to the rate of {\em survival} of a point to capture by the crystalline phase. A quantitative description of the spatial structure of the random spatial tessellation can be given in terms of the mean densities of interfaces (n-facets: cells, faces, edges, vertices), at all different Hausdorff dimensions, with respect to the usual d-dimensional Lebesgue measure. Evolution equations for the above densities are given here in terms of the hazard function. A relationship between the hazard function and the local spherical contact distribution function is also given, so that estimates of $d-1$-facet densities can be obtained in terms of the spherical contact distribution.