Domain growth morphology in curvature driven two dimensional coarsening
Résumé
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, $n_h(A,t) dA$, with enclosed area in the interval $(A,A+dA)$, is described, for a disordered initial condition, by the scaling function $n_h(A,t) = 2c_h/(A + \lambda_h t)^2$, where $c_h=1/8\pi\sqrt{3} \approx 0.023$ is a universal constant and $\lambda_h$ is a material parameter. For a critical initial condition, the same form is obtained, with the same $\lambda_h$ but with $c_h$ replaced by $c_h/2$. For the distribution of domain areas, we argue that the corresponding scaling function has, for random initial conditions, the form $n_d(A,t) = 2c_d (\lambda_d t)^{\tau'-2}/(A + \lambda_d t)^{\tau'}$, where $c_d=c_h + {\cal O}(c_h^2)$, $\lambda_d=\lambda_h + {\cal O}(c_h)$, and $\tau' = 187/91 \approx 2.055$. For critical initial conditions, one replaces $c_d$ by $c_d/2$ (possibly with corrections of ${\cal O}(c_h^2)$) and the exponent is $\tau = 379/187 \approx 2.027$. 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.