Grain growth from homogeneous initial conditions: Anomalous grain growth and special scaling states

Abstract

We used the large-g Potts model on a two-dimensional lattice to study the evolution of the disordered cluster developed from a perfect hexagonal lattice with a single defect. The distribution functions were not stable, while the average area and the number of grains in the cluster grew linearly in time. However, the grains at the boundary of the cluster formed a well defined region which reached a special scaling state with time invariant distributions but no scale change, contrary to the result of Levitan da" = tr(n -6) dt where~is a system-dependent diffusion constant, i.e. , the rate of change of area of a grain a"depends only on its number of sides n. Therefore grains with more than six sides expand, while grains with fewer than six sides shrink and eventually disappear. When a grain disappears, its neighboring grains may gain or lose sides. In such processes, side redistributions occur that determine the topological evolution of the pattern. In particular, experiments have revealed that the pattern evolves into a scaling state in which the side distribution functions p(n), and the rescaled size distribution functions p(a/(a)), where (a) is the mean area, remain constant, while the length scale increases with time [1,2) In a recent study of time evolution of two-dimensional soap froth with a single defect [6], Levitan challenges the common wisdom that the scaling state dynamics does not depend on the initial condition. Using a mean-field treatment, Levitan claims that the long-time distribution function p(n) is different for generic initial conditions or for an initial hexagonal lattice with only one defect. He also found the scaling law n, (t) -t [7], where n, is the number of grains in the cluster. In a comment, Weaire restates the evidence for scaling in experiment and draws attention to the importance of p, 2, the second moment of sides distribution, which is not mentioned in Levitan's work [8]. Sire concentrates on the discrepancy between the topological model and the numerical result from simulations of the area model [9].In order to account for the discrepancy, Sire introduces a model that gives a scaling law n, (t)-t ln(t ). Unfortunately in this model, Sire assumes that grains in the disordered cluster are in the usual scaling state. This assumption is neither trivial nor obvious. Both comments address the convergence of the convolution. However, we think this question cannot be answered by any mean-field theory, since the final state consists of distinct classes with fixed spatial relations.We evolve a special initial condition: a single big grain (area greater than the mean area of the hexagons) in a perfect regular hexagonal lattice. Without a defect, the lattice would be stable for all time. The defect functions as a seed for the evolution. Therefore the froth consists of two parts: the evolving neighborhood of the defect, and the rest of the lattice which does not evolve. As the froth evolves in time, the boundary of the disordered region propagates outwards. We would like to find out whether...

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