Models of crystal growth
There are two common types of models for crystal growth: ones based on the physics of individual atoms, and ones based on continuum descriptions of large collections of atoms. In the former category, it was recognized in the 1940s that a single atom is very unlikely to stick to a completely flat surface, so growth will always tend to occur at steps on a crystal surface, often associated with screw dislocations in the crystal structure. In practice, however, as scanning tunnelling microscopes have revealed, most crystal surfaces that are not grown at an extremely slow rate tend to be quite rough at an atomic scale—and so it seems that for example the aggregation model from page 331 may be more appropriate. In snowflakes and other crystals features such as the branches of tree-like structures are much larger than atomic dimensions, so a continuum description can potentially be used. It is possible to write down a nonlinear partial differential equation for the motion of the solidification front, taking into account basic thermodynamic effects. The first result (discovered by William Mullins and Robert Sekerka in 1963) is that if every part of the front is at the same temperature, then any deviations from planarity in the front will tend to grow. The shape of the front is presumably stabilized by the Gibbs-Thomson effect, which implies that the freezing temperature is lower when the front is more curved. The characteristic length for deformations of the front turns out to be the geometric mean of a microscopic length associated with surface energy and a macroscopic length associated with diffusion. It is this characteristic length that presumably determines the size of an individual cell in the cellular automaton model.
Dendritic crystals are commonly seen in ice formations on windows, and in pieces of aluminum of the kind found at typical hardware stores.