Statistical simulations of diffusional coarsening in finite clusters

Abstract

The problem of diffusional interactions in a finite-sized cluster of spherical particles is studied. Simulations of diffusional interactions with size distribution and volume fraction ${V}_{v},$ as input parameters, referred to as snapshot (static) simulations, are compared with dynamic (time-dependent) simulation results. The precise size distribution information in the snapshot simulations is obtained on the basis of a perturbation technique proposed recently by Fradkov et al. [Phys. Rev. E 53, 3925 (1996)]. Robust iterative solution schemes for the quasistatic diffusion equation facilitate investigations of coarsening systems comprised of one million particles at ultralow ${(10}^{\ensuremath{-}13})$ to moderate (0.25) volume fractions. The objective of carrying out simulations at such low volume fractions is to analyze the first-order volume-fraction-dependent correction to the effective coarsening rate predicted by the Todes, Lifshitz, and Slyozov (TLS) theory at infinite dilution. When volume fraction is considered as an input parameter, the deviation of coarsening rates from that of the infinite dilution limit of TLS varies as $\sqrt[\mathrm{A}\mathrm{P}\mathrm{S}::\mathrm{X}\mathrm{M}\mathrm{L}::\mathrm{G}\mathrm{e}\mathrm{n}\mathrm{P}\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{e}=\mathrm{H}\mathrm{A}\mathrm{S}\mathrm{H}(0\mathrm{x}512990)]{{V}_{v}}$ for a finite cluster and $\sqrt{{V}_{v}}$ (Debye screening) for an infinite system. Accurate numerical investigations of the rollover volume fraction ${(V}_{v}^{*})$ above which the $\sqrt[\mathrm{A}\mathrm{P}\mathrm{S}::\mathrm{X}\mathrm{M}\mathrm{L}::\mathrm{G}\mathrm{e}\mathrm{n}\mathrm{P}\mathrm{h}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{e}=\mathrm{H}\mathrm{A}\mathrm{S}\mathrm{H}(0\mathrm{x}514\mathrm{c}44)]{{V}_{v}}$ behavior changes to $\sqrt{{V}_{v}}$ showed that ${V}_{v}^{*}$ varies as ${n}^{\ensuremath{-}2},$ where $n$ is the number of particles in the spherical cluster. The deviation of coarsening rates from TLS observed from dynamic simulations agrees with that predicted by snapshot simulations for ${V}_{v}<0.01.$ The dynamic results are higher than the snapshot results for ${V}_{v}>0.01.$ The coarsening rate of the average particle can be calculated directly from dynamic simulations and indirectly from snapshot simulations by a perturbation relation. A different type of snapshot-ensemble averaging is suggested on the basis of the functional nature of the individual particle monopole source or sink strengths. Dynamic and snapshot simulations with dipoles were carried out up to a volume fraction of 0.25, and departure from Debye screening behavior was observed. The inclusion of dipole terms affects the deviations noticeably only above a volume fraction of 0.1.