Dynamic scaling and stochastic fractal in nucleation and growth processes

TL;DR

This paper investigates nucleation and growth models with various velocities, revealing power-law decay and fractal self-similarity under certain conditions, contrasting with exponential decay in classical models.

Contribution

It introduces a comprehensive analysis of nucleation and growth with dynamic scaling and fractal properties, extending understanding beyond classical constant velocity models.

Findings

Power-law decay of M-phase for specific velocity scalings

Dynamic scaling and fractal self-similarity observed in snapshots

Exponential decay without scaling in classical and inverse velocity models

Abstract

A class of nucleation and growth models of a stable phase (S-phase) is investigated for various different growth velocities. It is shown that for growth velocities $v\sim s(t)/t$ and $v\sim x/\tau(x)$, where $s(t)$ and $\tau$ are the mean domain size of the metastable phase (M-phase) and the mean nucleation time respectively, the M-phase decays following a power law. Furthermore, snapshots at different time $t$ are taken to collect data for the distribution function $c(x,t)$ of the domain size $x$ of M-phase are found to obey dynamic scaling. Using the idea of data-collapse we show that each snapshot is a self-similar fractal. However, for $v={\rm const.}$ like in the classical Kolmogorov-Johnson-Mehl-Avrami (KJMA) model and for $v\sim 1/t$ the decay of the M-phase are exponential and they are not accompanied by dynamic scaling. We find a perfect agreement between numerical simulation and analytical results.