Scaling relations for grain autocorrelation functions during nucleation and growth

The kinetics of grain growth associated with a first-order phase transformation is studied with use of the Kolmogorov model (constant nucleation rate within the metastable phase; constant growth rate of the nucleated grains) in one and two dimensions (d=1 and 2). The time-dependent grain-size distribution function and grain autocorrelation function, $G_s$(r,t), are calculated exactly for d=1. For d=2, $G_s$(r,t) is estimated by Monte Carlo simulation. Scaling relations, $G_s$(r,t)=F(r/R(t)), using forms for F(x) which are exact for t→0 are investigated. It is found that while such scaling relations are no longer exact in late stages of growth, they continue to provide accurate simple approximations.