Crystal Growth Inhibition by Mobile Randomly Distributed Stoppers

Crystal growth is inhibited by the presence of impurities. Cabrera and Vermilyea introduced a model in 1958, in which the impurities are modeled as immobile stoppers. The quantitative consequences of this model have mainly been explored for the special case where the stoppers are immobile and arranged in a periodic array. Here we use numerical simulation to explore what happens when the stopper locations are randomly distributed and the stoppers have finite lifetimes. As this problem has just two nondimensional parameters, namely, nondimensionalized versions of the mean stopper distance and the mean stopper lifetime, we are able to explore a large region of the parameter space using simulation. The stopper density is measured by the percolation parameter, a nondimensionalized inverse distance between stoppers, ζ. Our results show that when the stopper density is relatively small (ζ below about 0.8), the macroscopic velocity of the step is roughly the same for randomly located stoppers as for a periodic array of stoppers. Moreover, in this regime the average velocity is almost independent of the stopper lifetime. For large stopper densities (more precisely, when the percolation parameter ζ is above about 0.8), the situation is entirely different. For periodically placed immobile stoppers, the average velocity drops sharply to 0 at ζ = 1. For randomly located immobile stoppers, by contrast, the average velocity remains positive for ζ well above 1, and it approaches 0 gradually rather than abruptly. For randomly located stoppers with finite lifetimes, the average velocity has a nonzero asymptote for large ζ; thus, for large stopper densities, the average velocity depends mainly on the mean stopper lifetime. In this regime, the inhibition kinetics predicted by our model resemble those of the Bliznakov kink blocking mechanism.