On the equations describing the grain size distribution change for KJMA kinetics

A partial differential equation is proposed governing the particle size probability density function during grain growth. The equation is derived by using a mean field hypothesis under the same assumptions as the classical Kolmogorov–Johnson, Mehl–Avrami (KJMA) model for nucleation and growth kinetics. The resulting set of equations should give the final grain size distribution after an arbitrary nucleation and growth procedure, provided that the time dependence of the nucleation and growth rates are known, as for example in nanocrystalline materials. Results for the simplest case of initial nucleation and constant growth rates (pre-existing nuclei) are shown.