Geometrical models of interface evolution. II. Numerical simulation

Abstract

We continue our study of local interface models for dendritic growth by presenting detailed numerical simulations of the evolution of snowflake patterns. The local model we employ is quantitatively valid for early stages of the dendritic growth process and is qualitatively similar to the true diffusion-controlled dynamics as far as single-tip behavior is concerned. We show that a critical value of the crystal anisotropy must be exceeded before the system settles into stable tip and repeated side-branching behavior. Varying the anisotropy and measuring the dendrite velocity enables us to get a quantitative picture of the tip dynamics. In addition, we study global features of the evolving patterns, and find exponential growth for the curve complexity, $\xi \equiv \left[\frac{\left(\mathrm{arclength}\right)}{{\left(\mathrm{area}\right)}^{\frac{1}{2}}}\right]$. Our results are not consistent with a simple version of the marginal stability hypothesis of Langer and Müller-Krumbhaar.