Time-evolution of grain size distributions in random nucleation and growth crystallization processes

TL;DR

This paper provides an analytical study of how grain size distributions evolve over time during crystallization processes, highlighting the influence of model parameters and dimensionality, and demonstrating the emergence of a lognormal distribution.

Contribution

It introduces an analytical solution for the time-dependent grain size distribution in RNG crystallization, accounting for nucleation and growth rates in any dimension.

Findings

Distribution evolves into a lognormal form under certain conditions

Analytical expressions for grain size distribution and moments are derived

The model describes both thin films and bulk materials

Abstract

We study the time dependence of the grain size distribution N(r,t) during crystallization of a d-dimensional solid. A partial differential equation including a source term for nuclei and a growth law for grains is solved analytically for any dimension d. We discuss solutions obtained for processes described by the Kolmogorov-Avrami-Mehl-Johnson model for random nucleation and growth (RNG). Nucleation and growth are set on the same footing, which leads to a time-dependent decay of both effective rates. We analyze in detail how model parameters, the dimensionality of the crystallization process, and time influence the shape of the distribution. The calculations show that the dynamics of the effective nucleation and effective growth rates play an essential role in determining the final form of the distribution obtained at full crystallization. We demonstrate that for one class of nucleation and growth rates the distribution evolves in time into the logarithmic-normal (lognormal) form discussed earlier by Bergmann and Bill [J. Cryst. Growth 310, 3135 (2008)]. We also obtain an analytical expression for the finite maximal grain size at all times. The theory allows for the description of a variety of RNG crystallization processes in thin films and bulk materials. Expressions useful for experimental data analysis are presented for the grain size distribution and the moments in terms of fundamental and measurable parameters of the model.