Mathematical Modelling for Polymer Processing

Polymer science is an extraordinary source of very interesting mathematical problems, as it is largely testified by the recent literature. Research is extremely active in various directions: understanding the mechanical and thermodynamical behaviour of known polymers, improving production processes, designing new molecules and new materials with specific properties. A strong and continuous stimulus comes from technological applications in various fields, in which the demand of better performing materials is more and more intense.

The purpose of this chapter is to explain the classical kinetic theory of nucleation in a context simpler than polymer crystallization. Many theories start by assuming that polymer crystallization is an activate d process involving crossing of a free energy barrier [1]. The latter separates two accessible stable states of the system such as monomer solution and crystal. This general setting for activated processes can be used to describe the formation of a crystal from a liquid cooled below its freezing point [2], precipitation and coarsening of binary alloys [3], colloidal crystallization [4] chemical reactions [5], polymer crystallization [1, 6, 7], etc. In all these cases, the theory of homogeneous isothermal nucleation provides a framework to study the processes of formation of nucleii from density fluctuations, and their growth until different nucleii impinge upon each other. In the early stages of these processes, nucleii of solid phase are formed and grow by incorporating particles from the surrounding liquid phase. There is a critical value for the radius of a nucleus that depends on a chemical drive potential, which is proportional to the supersaturation for small values thereof. In this limit, the critical radius is inversely proportional to the supersaturation. At the beginning of the nucleation process, nucleii have small critical radius and new clusters are being created at a non-negligible rate.

The concept of nucleation as a first step in phase transitions, was originally developed in nineteen twenties and thirties by Volmer and Weber [1], Kaischew and Stransky [2], Becker and Doring [3], Zeldowich [4], Frenkel [5], Turnbull and Fisher [6], and others. With some modifications, the original theory has been applied to crystallization of polymers by Lauritzen and Hoffman [7], Mandelkern [8], Frank and Tosi [9]. The original theory concerned one-dimensional growth of molecular (atomic) clusters and could not explain more complex processes, like crystallization in oriented systems, effects of potential fields, anisotropic growth, etc.

Crystallization of polymers is a fascinating branch of polymer physics which has a significant relevance in industrial applications. Its importance arises from the fact that mechanical properties of any crystal polymer are determined by its morphology and internal structure, which in turn is dictated by the crystallization kinetics. That is the reason why, mathematical modeling aiming to describe and control the kinetics of polymer crystallization has achieved great interest.

In this chapter we introduce a new approach that thanks to the multiple-scale structure, allows us to use mathematical techniques of averaging at the lower scale.

Polymer researchis divided over such diverse fields as there are: