Upper Critical Dimension for Irreversible Cluster Nucleation and Growth

Abstract We compare the results of kinetic Monte Carlo (KMC) simulations of a point-island model of irreversible nucleation and growth in four-dimensions with the corresponding mean-field (MF) rate equation predictions for the monomer density, island density, island-size distribution (ISD), and capture number distribution (CND) in order to determine the critical dimension dc for meanfield behavior. The asymptotic behavior is studied as a function of the fraction of occupied sites (coverage) and the ratio D/F of the monomer hopping rate D to the (per site) monomer creation rate F . Excellent agreement is found between our KMC simulation results and the MF rate equation results for the average island and monomer densities. For large D/F , the scaled CND does not depend on island-size in good agreement with the MF prediction, while the scaled ISD also agrees well with the MF prediction except for a slight difference at the peak values. Coupled with previous results obtained in d = 3, these results indicate that the upper critical dimension for irreversible cluster nucleation and growth is equal to 4.