First passage time Markov chain analysis of rare events for kinetic Monte Carlo: double kink nucleation during dislocation glide

Abstract

We describe a first passage time Markov chain analysis of rare events in kinetic Monte Carlo (kMC) simulations and demonstrate how this analysis may be used to enhance kMC simulations of dislocation glide. Dislocation glide is described by the kink mechanism, which involves double kink nucleation, kink migration and kink–kink annihilation. Double kinks that nucleate on straight dislocations are unstable at small kink separations and tend to recombine immediately following nucleation. A very small fraction (<0.001) of nucleating double kinks survive to grow to a stable kink separation. The present approach replaces all of the events that lead up to the formation of a stable kink with a simple numerical calculation of the time required for stable kink formation. In this paper, we treat the double kink nucleation process as a temporally homogeneous birth–death Markov process and present a first passage time analysis of the Markov process in order to calculate the nucleation rate of a double kink with a stable kink separation. We discuss two methods to calculate the first passage time; one computes the distribution and the average of the first passage time, while the other uses a recursive relation to calculate the average first passage time. The average first passage times calculated by both approaches are shown to be in excellent agreement with direct Monte Carlo simulations for four idealized cases of double kink nucleation. Finally, we apply this approach to double kink nucleation on a screw dislocation in molybdenum and obtain the rates for formation of stable double kinks as a function of applied stress and temperature. Equivalent kMC simulations are too inefficient to be performed using commonly available computational resources.