On the Modeling of Snowflake Growth Using Hexagonal Automata

Snowflake growth is an example of crystallization, a basic phase transition in physics. Studying snowflake growth helps gain fundamental understanding of this basic process and may help produce better crystalline materials and benefit several major industries. The basic theoretical physical mechanisms governing the growth of snowflake are not well understood: whilst current computer modeling methods can generate snowflake images that successfully capture some basic features of actual snowflakes, so far there has been no analysis of these computer models in the literature, and more importantly, certain fundamental features of snowflakes are not well understood. A key challenge of analysis is that the snowflake growth models consist of a large set of partial difference equations, and as in many chaos theory problems, rigorous study is difficult. In this paper we analyze a popular model (Reiter’s model) using a combined approach of mathematical analysis and numerical simulation. We divide a snowflake image into main branches and side branches and define two new variables (growth latency and growth direction) to characterize the growth patterns. We derive a closed form solution of the main branch growth latency using a one dimensional linear model, and compare it with the simulation results using the hexagonal automata. We discover a few interesting patterns of the growth latency and direction of side branches. On the basis of the analysis and the principle of surface free energy minimization, we propose a new geometric rule to incorporate