Fokker-Planck equation for the crystal-size probability density in progressive nucleation and growth with application to lognormal, Gaussian and gamma distributions

The Fokker Planck (FP) equation for the probability density function (PDF) of crystal size in phase transformations ruled by progressive nucleation and growth, has been derived. Crystals are grouped in sub-sets, we refer to as $\tau$-crystals, where $\tau$ is the birth time of the set. It is shown that the size PDF is the superposition of the PDF of the crystal sub-sets ($\tau$-PDFs), with weight given by the nucleation rate. The growth and diffusion coefficients entering the FP equations are estimated as a function of both $\tau$-PDFs and nucleation rate. The functional form of these coefficients is studied for solutions of the FP equation for $\tau$-crystals given by the lognormal, Gaussian and gamma distributions. For the first two distributions, the effect of fluctuations, nucleation rate and growth rate, on the shape of the distribution has been investigated. It is shown that for an exponential decay of the fluctuation term, the shape of the PDF is mainly governed by both the time constant for nucleation and the strength of the fluctuation. It is found that $\tau$-PDFs given by the one-parameter gamma distributions are suitable to deal with KJMA (Kolmogorov Johnson Mehl Avrami) compliant phase transformations, where the fluctuation term is proportional to crystal size. The connection between the FP equation for the size PDF and the evolution equation for the density of crystal populations is also discussed.