Fokker-Planck equation for the particle size distribution function in KJMA transformations

TL;DR

This paper derives a Fokker-Planck equation to model the evolution of particle size distribution during KJMA transformations, showing it aligns with Gamma distributions and confirming previous conjectures through simulations.

Contribution

It introduces a Fokker-Planck framework for particle size distribution in KJMA transformations, linking it to Gamma distributions and validating with simulations.

Findings

PDF is a superposition of Gamma distributions with time-dependent mean sizes

Asymptotic behavior confirms nuclei formed at the same time are Gamma-distributed

Simulation results agree with Johnson-Mehl PDF for constant nucleation and growth rates

Abstract

The Fokker-Planck (FP) equation has been derived for describing the temporal evolution of the particle size probability density function (PDF) for KJMA (Kolmogorov-Johnson-Mehl-Avrami) transformations. The classical case of transformations with constant rates of both nucleation and growth, in 3D space, has been considered. Integration of the equation shows that the PDF is given by the superposition of one-parameter Gamma distributions with time dependent mean size given by the KJMA theory. The asymptotic behavior of the FP solution offers a demonstration of the conjecture, previously proposed by Pineda et al [E. Pineda, P. Bruna, D. Crespo, Phys. Rev. E 70 (2004) 066119], according to which the set of nuclei formed at the same time are Gamma-distributed, with parameter depending on nucleus birth time. Computer simulations of the transformation with constant nucleation and growth rates, do show that the temporal evolution of the PDF, in the volume domain, is in good agreement with the Johnson-Mehl PDF. The approach based on the FP equations provides a particle size PDF that exhibits such behavior.