Moving bands and moving boundaries with decreasing speed in polymer crystallization

A deterministic model of polymer crystallization, derived from a previous stochastic one, is considered. The model describes the crystallization process of a rectangular sample of a material cooled at one of its sides. It is a reaction–diffusion system, composed of a PDE for the temperature and an ODE for the phase change of a polymer melt from liquid to crystal. The two equations are strongly coupled since the evolution of temperature depends on a source term, due to the latent heat developed during the phase change, the nucleation and growth rates are functions of the local (in time and space) temperature. The main difference with respect to the previous model is the introduction of a critical temperature of freezing in these functions. The paper does not contain detailed analytical aspects, that are left to subsequent investigations. A qualitative analysis of the proposed model is carried out, based on numerical simulations. An interesting feature shown by the simulations is that the solution exhibits an advancing moving band of crystallization in the mass distribution, as well as a moving boundary in the temperature field, both advancing with the same decreasing velocity. For some values of the parameters, which are typical of the physical problem, the advance takes place by jumps due to regular stops of the most advanced point of crystallization. The duration of these halts increases as the applied temperature decreases. This may indicate that the crystallization time is not a monotone function of the applied temperature. A simplified mathematical model is eventually proposed which reproduces the same patterns