Revisiting time-dependent growth and nucleation rates in the Johnson-Mehl-Avrami-Kolmogorov equation

The Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation is widely used to model phase transformation kinetics-a topic central to materials in which solidification, precipitation or recrystallization is occurring. The classic derivation of this reaction equation primarily assumes that nucleation is either instantaneous or sporadic at a constant rate, and that growth rate is constant; consideration of possible time dependence of these rates is limited to the physically uncommon cases of the time dependence being linear. In contrast, many common phase transformations involve kinetics in which the growth rate varies in proportion to time raised to a power that can range from -0.5 (diffusion control; Fickian diffusion; case I diffusion), through values that represent anomalous diffusion, to zero (interface control; case II diffusion) and beyond (super case II diffusion). We have extended the classic derivation by generalizing the formulation of the growth and nucleation rates to include these contexts, and we introduce and justify some additional refinements while retaining the overall mathematical accessibility of the classic derivation. The reaction equations derived in this process explicitly demonstrate how the time dependence of nucleation rate and growth rate affects the constants in the JMAK equation, allowing the possibility of values beyond their classical range.