Birth-and-growth stochastic processes modelling polymer crystallization

Birth-and-growth stochastic processes play a relevant role in many applications. Here we emphasize the modelling of polymer crystallization processes, which is of major industrial interest. Under rather general assumptions, a birth-and-growth process eventually leads to a random division of the space R^d into cells, known as Johnson-Mehl tesselation. In this paper we provide an overview of basic results obtained in recent literature by various authors, with the scope of revisiting birth-and-growth processes in the framework of the theory of marked point processes and of stochastic geometry of Dynamic Boolean models. Relevant quantities for the tesselation are related to the parameters of the stochastic intensity-kernel of the process and the growth rate of the crystals. In particular we provide an expression of the spatial densities of the interfaces (faces, edges or vertices) of d-dimensional crystals, at any time t of the birth-and-growth process (called also incomplete tesselation) and give a rather general geometric-probabilistic interpretation of the achieved results. Prof. V. Capasso, University of Milano, Italy, was guest professor at the Industrial Mathematics Institute, Johannes Kepler University, Linz, Austria, from October 1997 until January 1998.