Crystal Growth and Impingement in Polymer Melts

Crystallization of polymeric materials is a solidification process in strong interaction with heat conduction. Both of the basic mechanisms involved in the solidification from a melt, namely the nucleation and growth of crystals, are strongly influenced by the temperature and its variation. Vice versa, the latent heat is rather large for polymeric materials, so that it causes a considerable change in the heat transfer process. The interaction of solidification and heat conduction is a well-known phenomenon in models of phase-change, and involves important problems in the theory of free and moving boundary problems. In typical examples of moving boundary problems for the heat equation such as the well-studied Stefan problem (cf. [18]), the unknown boundary is assumed to be an isotherm, which leads in a mathematical model to a homogeneous Dirichlet condition for the (appropriately scaled) temperature. For polymers, the situation is different, since the phase change does not take place at a fixed temperature (or with kinetic undercooling close to this temperature), but in a rather large temperature range between the thermal melting point Tm and the glass transition temperature Tg. The nucleation of new crystals can be modeled as a heterogeneous Poisson process in space and time, with rate α = α(x, t). For many important materials such as isotactic Polypropylene (i-pp), a model for the nucleation rate in dependence of the temperature u is given by α(x, t) = ∂ ∂t Ñ(u(x, t)) (cf. [6]), with some material function Ñ , which varies only in the interval (Tg, Tm), i.e., no crystals nucleate if the temperature is below the glass transition temperature or above the melting point. Usually it is assumed that new crystals nucleate as small spherical objects with initial radius R0 determined from classical nucleation theory. The nucleation process has been well investigated with respect to its stochastic properties