Ostwald ripening in two dimensions: Time dependence of size distributions for thin-film islands

Thin film phase transitions can involve monomer deposition and dissociation on clusters (islands) and island coalescence. We propose a distribution kinetics model of island growth and ripening to quantify the evolution of island size distributions. Incorporating reversible monomer addition at island edges, cluster coalescence, and denucleation of unstable clusters, the governing population dynamics equation can be transformed into difference-differential or partial differential forms with an integral expression for coalescence. The coalescence kernel is assumed proportional to $x^v x'^\mu $, where x and x' are masses of coalescing clusters. The deposition and dissociation rate coefficients are proportional to cluster mass raised to a power, $x^{\lambda}$. The equation was solved numerically over a broad time $(\theta)$ range for various initial conditions. The moment form of the equation algebraically yields asymptotic long-time expressions for the power law dependence of decreasing number concentration $(\theta^{-b})$ and increasing average cluster size $(\theta^b)$. When coalescence is negligible the power is $b = 1/(1- \lambda + d^{-1})$, where d = 2 or 3 is the dimensionality. If coalescence dominates the asymptotic time dependence, the power is $b = 1/(1-v- \mu)$