Dynamic scaling and stochastic fractal in nucleation and growth processes

A class of nucleation and growth models of a stable phase is investigated for various different growth velocities. It is shown that for growth velocities v ≈ s ( t ) / t and v ≈ x / τ ( x ), where s ( t ) and τ 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 that are taken to collect data for the distribution function c ( x , t ) of the domain size x of the 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 = const ., such as in the classical Kolmogorov-Johnson-Mehl-Avrami model, and for v ≈ 1 / t, the decays 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.