Kinetics of Crystallization
The rate of crystallization in polymeric materials is of paramount importance. For some polymers, like atactic polystyrene or some rubbers, rapid cooling can lead to the glassy state without any formation of crystallites. The amount of crystallinity actually depends very much upon the thermal history of the material. The amount of crystallinity, in turn, influences the mechanical properties of the material. Microscopic observation of the growth of the spherulites as a function of time will yield information of the crystallization rate. The rate is a function of the temperature. As the temperature is lowered, the rate increases. This growth is usually observed as being linear with time. Presence of impurities will slow down the growth rate. When the growth rate is plotted against crystallization temperature, a maximum is observed. This is due to the fact that as the temperature is lowered the mobilities of the molecules decrease and the process eventually becomes diffusion-controlled. According to the Av rami equation, the fraction that crystallizes during the time t, and defined as 1-λ (t), can be written as [42]:
where, N(t) is the nucleation frequency per untransformed volume, V(t,t) is the corresponding volume of the growing center, and pc and p1 represent the densities of crystalline and liquid phases. Based on that, the rate constant for crystallization kinetics can be described [42]:
where, V∞, Vt, and V0 are specific volumes at the times shown by the subscripts, and wc is the weight fraction of the polymer crystallized. k is the rate constant for crystallization It was found, however, that crystallization continues in polymeric materials for much longer periods of time than the Avrami equation predicts. For all homopolymers the rate of crystal growth increases linearly with time, or G = dr/dt. Mandelkern defines the steady-state nucleation rate, N as follows [42]:
where ED is the energy of activation for transporting the chain segments across the crystal–liquid interface. If the crystallization takes place over an extended temperature range, most if not all homopolymers display a maxima in rates of spherulitic growth and in the overall crystallization. The equation for spherulitic growth is written as follows [42]:
where T∞ is the temperature at which all molecular and segmental motion stops.
