Further theoretical developments overall crystallization

Closely following the Avrami expression is an empirical relation introduced by Austin and Rickett, based on the experimental results for the decomposition of austenite steel.(51) The relation can be expressed as 

Equation (9.47) is compared with the derived Avrami, Eq. (9.31a), in Fig. 9.21. Here the extent of the transformation is plotted against the log time for integral values of the exponent n. There are only small differences between the two relations, particularly in the usual range of polymer crystalhzation. Analysis of typical kinetic data indicates that deviations from either theory occur at about the same crystallinity level. 

In an alternative approach, Lee and Kim (52) proposed that impingment can be described by a modification of Eq. (9.24). They suggest that 

Tobin (53) has treated the impingement problem in a somewhat different manner from that of Avrami. The premise that impingement is the cause of the cessation of growth and the termination of the crystallization is still the underlying principle involved. This analysis was initially developed for a constant homogeneous nucleation rate accompanied by either two- or three-dimensional growth. It was subsequently extended to heterogeneous nucleation. For the two-dimensional problem, the transformed area at time t, A(t), can be represented as 

it is assumed that at time / each growing center has its effective area reduced by the same factor, irrespective of when it was formed. Substituting Eq. (9.50) into Eq. (9.49) and integrating over r, the nonlinear Volterra integral equation 

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