From early nucleation past the percolation threshold: Status of the Kolmogorov-Avrami theory on a cold Ising lattice

A high-precision test of the Kolmogorov-Avrami (KA) model of “geometric” interactions between growing nuclei is performed for an Ising ferromagnet driven by Metropolis dynamics. As long as the interface boundaries of interacting nuclei are rough, the KA model is adequate for arbitrary large densities of the new phase, well beyond the percolation transition, as in the case of homogeneous nucleation. For a prenucleated system the KA description breaks down at sufficiently low temperature, when the length of a kinetically smooth interface exceeds the distance between seeds.

Figure 1

Coalescence of growing squares with rough (upper insets, $T=0.8$) and smooth (lower insets, $T=0.4$) interfaces. In each case the arrows indicate the instant the corresponding snapshot was taken; $n$ is the total number of up spins and the scaled time is $t/exp\left[(2-h)/T\right]$. The higher $T$ case is consistent with the KA model, while the lower $T$ case is not (see text).

Figure 2

Homogeneous nucleation at $T=0.8, h=0.88$ (left, nonscaled time $t$) and $T=0.25, h=2.5$ (right, time scaled as in Fig. 1). In each case the insets show snapshots of a fraction ($500\times 500$) of the lattice when time and scaled time are close to 150. In both cases the KA model (lines) is in good agreement with MC data (symbols), including magnetizations beyond the percolation threshold—see text.

Figure 3

Same as in Fig. 2 but for prenucleated systems (all times are scaled as in Fig. 1). The KA model (lines) is in good agreement with MC data for both stable (left) and unstable (right) interfaces of growing nuclei.

Figure 4

Breakdown of the KA model at small $T$ and $h$ and high seed density. Symbols: MC data for initial seed size $m_0=5$ (circles) and $m_0=11$ (squares, time shifted to match the $m_0=5$ data—see text). Inset: Snapshot of the full lattice ($1000\times 1000$) at $t=1000$, shortly before percolation. The KA prediction (line) based on Eq. (1) underestimates the rate of transformation due to the neglect of direct interactions between nuclei which lead to an avalanchelike completion of the phase transformation.