Domain growth morphology in curvature-driven two-dimensional coarsening

We study the distribution of domain areas, areas enclosed by domain boundaries ("hulls”), and perimeters for curvature-driven two-dimensional coarsening, employing a combination of exact analysis and numerical studies, for various initial conditions. We show that the number of hulls per unit area, nh (A,t) dA, with enclosed area in the interval (A,A+dA), is described, for a disordered initial condition, by the scaling function nh (A,t) =2ch/(A+ λht) 2, where ch=1/8 π3 ≈ 0.023 is a universal constant and λh is a material parameter. For a critical initial condition, the same form is obtained, with the same h but with ch replaced by ch /2. For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form nd( A,t) =2cd (λdt)t'−2/(A+ λdt)t', where cd and λd are numerically very close to ch and λh, respectively, and t'=187/91 ≈ 2.055. For critical initial conditions, one replaces cd by cd/ and the exponent is t=379/187 ≈ 2.027. These results are extended to describe the number density of the length of hulls and domain walls surrounding connected clusters of aligned spins. These predictions are supported by extensive numerical simulations. We also study numerically the geometric properties of the boundaries and areas