On the Effect of Nucleation Undercooling on Phase Transformation Kinetics

We carry out an extensive comparison between Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory of first-order phase transformation kinetics and phase-field (PF) results of a benchmark problem on nucleation. To address the stochasticity of the problem, several hundreds of simulations are performed to establish a comprehensive, statistically-significant analysis of the coincidences and discrepancies between PF and JMAK transformation kinetics. We find that PF predictions are in excellent agreement with both classical nucleation theory and JMAK theory, as long as the original assumptions of the latter are appropriately reproduced - in particular, the constant nucleation and growth rates in an infinite domain. When deviating from these assumptions, PF results are at odds with JMAK theory. In particular, we observe that the size of the initial particle radius $r_0$ relative to the critical nucleation radius $r^*$ has a significant effect on the rate of transformation. While PF and JMAK agree when $r_0$ is sufficiently higher than $r^*$, the duration of initial transient growth stage of a particle, before it reaches a steady growth velocity, increases as $r_0/r^*\to 1$. This incubation time has a significant effect on the overall kinetics, e.g. on the Avrami exponent of the multi-particle simulations. In contrast, for the considered conditions and parameters, the effect of interface curvature upon transformation kinetics - in particular negative curvature regions appearing during particle impingement, present in PF but absent in JMAK theory - appears to be minor compared to that of $r_0/r^*$. We argue that rigorous benchmarking of phase-field models of stochastic processes (e.g. nucleation) need sufficient statistical data in order to make rigorous comparisons against ground truth theories. In these problems, analysis of probability distributions is clearly preferable to a deterministic approach.