A Mathematical Treatment to Determine Transient Growth Kinetics from a Given Size Distribution

TL;DR

This paper develops a mathematical approach to determine the transient growth kinetics of particles or grains from any given size distribution, extending classical theories to non-steady-state conditions.

Contribution

It introduces a method to calculate instantaneous growth rates from arbitrary size distributions, bridging the gap between steady-state theory and real transient growth scenarios.

Findings

Analytical solutions for growth kinetics from arbitrary distributions.

Numerical integration methods for exact transient growth calculation.

Validation of the approach with known growth laws.

Abstract

Particle coarsening and grain growth take place to minimize the total interfacial energy. The classical mean-field treatments by Lifshitz, Slyozov, [1] Wagner [2] and Hillert [3] predicted cubic growth law under bulk-diffusion controlled precipitation coarsening and parabolic growth law under interface controlled grain growth, as well as their steady-state size distribution. When the size distribution is the steady-state one, the average grain size satisfies the following dependence: under bulk-diffusion control and under interface control, with correct slopes definitely given by the theory. However, when the size distribution does not satisfy the steady-state one, the growth kinetics would deviate from theoretical prediction. As is shown by numerical simulations in Ref. 4 and 5, the deviation is less obvious in the growth law, but more evident in the slope of the growth curve. Specifically, starting from a non-steady-state size distribution, the parabolic/cubic growth law would be recovered quickly, yet the convergence of the slope in the or curve is extremely slow. Therefore, it would be interesting to know how to calculate the instantaneous slope of the growth kinetics from an arbitrarily given size distribution. As will be shown below, assuming a known growth law, the analytical solution can be easily obtained and the exact solution can be obtained by numerical integration.