Unusual features of coarsening with detachment rates decreasing with cluster mass

TL;DR

This paper investigates one-dimensional particle models with size-dependent detachment rates, revealing unusual coarsening dynamics, including logarithmic corrections and dominance of small clusters for certain parameters, supported by analytical and simulation results.

Contribution

It introduces a new class of coarsening models with decreasing detachment rates and derives their scaling laws, cluster distributions, and dominant behaviors through heuristic, analytical, and simulation methods.

Findings

Typical cluster size scales as (t/ln(t))^{1/(k+2)}

Small clusters dominate for k<1, affecting average size scaling

Cluster size distribution exhibits power-law decay and skewed peaks

Abstract

We study conserved one-dimensional models of particle diffusion, attachment and detachment from clusters, where the detachment rates decrease with increasing cluster size as gamma(m) ~ m^{-k}, k>0. Heuristic scaling arguments based on random walk properties show that the typical cluster size scales as (t/ln(t))^z, with z=1/(k+2). The initial symmetric flux of particles between neighboring clusters is followed by an effectively assymmetric flux due to the unbalanced detachement rates, which leads to the above logarithmic correction. Small clusters have densities of order t^{-mz(1)}, with z(1) = k/(k+2). Thus, for k<1, the small clusters (mass of order unity) are statistically dominant and the average cluster size does not scale as the size of typically large clusters does. We also solve the Master equation of the model under an independent interval approximation, which yields cluster distributions and exponent relations and gives the correct dominant coarsening exponent after accounting for the effects of correlations. The coarsening of large clusters is described by the distribution P_t(m) ~ 1/t^y f(m/t^z), with y=2z. All results are confirmed by simulation, which also illustrates the unusual features of cluster size distributions, with a power law decay for small masses and a negatively skewed peak in the scaling region. The detachment rates considered here can apply in the presence of strong attractive interactions, and recent applications suggest that even more rapid rate decays are also physically realistic.