Mathematical modelling of the crystallization process polymers

Crystallization of polymers is composed of two processes, nucleation (birth) and subsequent growth of crystallites, which are in general stochastic both in time and space. If we assume that at points of contact between two growing crystallites they stop growing, a random division of the relevant region in a d-dimensional space is obtained, known as a random Johnson-Mehl tessellation, which has been studied in previous literature with homogeneous parameters. A complete characterization of the final spatial structure of the crystallization (tessellation) can be given in terms of the mean densities of interfaces (n-facets: cells, faces, edges, vertices) of the random tessellation, at all different Hausdorff dimensions, with respect to the usual d-dimensional Lebesgue measure. Here an analysis of the above quantities in terms of the kinetic parameters of the process is presented coupled with the evolution equations of the underlying temperature field. In order to make the analysis more treatable an hybrid model is given that makes the temperature field and consequently the kinetic parameters of nucleation and growth deterministic. In this set up the classical theory of Kolmogorov-Avrami and Evans has been extended to space and time inhomogeneous kinetic parameters via the concept of hazard function. Simulation of both deterministic and stochastic models are provided, together with estimates of geometric quantities related to the random tessellation.