Extension of the Kolmogorov–Johnson–Mehl–Avrami theory to growth laws of diffusion type

A solution of the problem of calculating the volume fraction of a phase growing by a diffusion-type law is given. This is meant that the growth velocity R˙ of a nucleus is a decreasing function of its radius R. The growth law R˙~1/Rn−1, n > 1, is employed for demonstrative calculations. The solution is obtained in the framework of the classical Johnson–Mehl–Avrami approach which uses the concept of non-physical phantom nuclei. Probabilistic treatment of this approach is offered and the necessity of phantom nuclei is confirmed. The Johnson–Mehl–Avrami approach is compared with Kolmogorov's method and its extension — the differential critical-region method; the latter yields the same equations for the volume fraction. In the case of the growth law considered, phantom nuclei contribute to the incrementing of the transformed volume fraction. The obtained equations for the volume fraction are shown to cancel this contribution; hence they yield the true value of this quantity. Two successive approximations for the volume fraction are considered analytically and the numerical evaluation of the effect of phantom nuclei is given for different values of n.