Exact results for two-dimensional coarsening

Abstract

We consider the statistics of the areas enclosed by domain boundaries (‘hulls’) during the curvature-driven coarsening dynamics of a two-dimensional nonconserved scalar field from a disordered initial state. We show that the number of hulls per unit area, n _h (A, t)dA, with enclosed area in the range (A,A + dA), is described, for large time t, by the scaling form n _h (A, t) = 2c _h /(A + λ _h t)², demonstrating the validity of dynamical scaling in this system. Here $c_h = {1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-\nulldelimiterspace} 8}\pi \sqrt 3$ is a universal constant associated with the enclosed area distribution of percolation hulls at the percolation threshold, and λ _h is a material parameter. The distribution of domain areas, n _d (A, t), is apparently very similar to that of hull areas up to very large values of A/λ _h t. Identical forms are obtained for coarsening from a critical initial state, but with c _h replaced by c _h /2. The similarity of the two distributions (of areas enclosed by hulls, and of domain areas) is accounted for by the smallness of c _h . By applying a ‘mean-field’ type of approximation we obtain the form n _d (A, t) ≃ 2c _d [λ _d (t+t ₀)]^τ−2/[A+λ _d (t+t ₀)]^τ, where t ₀ is a microscopic timescale and τ = 187/91 ≃ 2.055, for a disordered initial state, and a similar result for a critical initial state but with c _d → c _d /2 and τ → τ _c = 379/187 ≃ 2.027. We also find that c _d = c _h + O(c ² _h ) and λ _d = λ _h (1 + O(c _h )). These predictions are checked by extensive numerical simulations and found to be in good agreement with the data.