Towards a Theory of Interfacial Pattern formation

An understanding of the origin of snowflake shapes, and other such non-equilibrium growth patterns1, has long eluded physicists. Although the governing differential equations and boundary conditions have been known for some time2,3, at least to a working approximation, the nonlinearities and instabilities present therein have made it difficult to understand even qualitatively how these shapes arise. In a variety of physical systems including solidification, multiphase fluid flow, electrochemical deposition etc. one finds interfacial patterns that in one or another aspect vary between compact and branched, between symmetric and irregular, and between stable and unstable. Recent experimental and theoretical developments have revealed unexpected similarities among these ostensibly different physical systems and viable calculational approaches to the elucidation of these issues. This article aims to review these developments, in their current incomplete state, placing particular emphasis on dendritic crystal growth