Kinetics of ordering in two dimensions. II. Quenched systems

TLDR

The study relates Monte Carlo findings to experimental observations of surface atom ordering and polycrystalline grain growth. An efficient Monte Carlo simulation of quenched low‑ and high‑Q Potts models tracks domain size, shape, and grain‑size distributions over time. Domain growth follows a power law R∼t^n, with n decreasing from ½ at Q=2 to ≈0.41 for large Q (triangular lattice at all temperatures and square lattice above 0.5Tc); for large Q the grain‑size distribution is time‑independent, while on the square lattice at low temperatures the growth exponent is essentially zero due to nucleation of pinned domains.

Abstract

An efficient Monte Carlo procedure is applied to the study of the kinetics of low- and high-$Q$ Potts models on surfaces quenched from a high temperature ($T\ensuremath{\gg}{T}_{c}$) to low temperatures ($T<{T}_{c}$). The kinetics of these models are analyzed in simulations which yield the time dependence of the size and shape of the domains. After an initial transient period, the mean domain size $R$ increases with time as $R\ensuremath{\sim}{t}^{n}$. For the triangular lattice at all temperatures and for the square lattice at temperatures $T>0.5{T}_{c}$, the exponent $n$ decreases from $\frac{1}{2}$ for $Q=2$ (Ising model) to approximately 0.41 for large $Q$. The distribution of grain sizes and shapes is also calculated from the results of the simulations. For large $Q$, these distributions are time independent. On the square lattice at low temperatures the growth exponent is approximately equal to 0, due to the nucleation of pinned domains. The Monte Carlo results are discussed in terms of experimental studies of the ordering of adsorbed atoms on surfaces and of the growth of grains in polycrystalline materials.