  Snowflake growth provides us with a fascinating example of spontaneous
pattern formation in nature. Attempts to understand this phenomenon have led to
important insights in non-equilibrium dynamics observed in various active
scientific fields, ranging from pattern formation in physical and chemical
systems, to self-assembly problems in biology. Yet, very few models currently
succeed in reproducing the diversity of snowflake forms in three dimensions,
and the link between model parameters and thermodynamic quantities is not
established. Here, we report a modified phase field model that describes the
subtlety of the ice vapour phase transition, through anisotropic water
molecules attachment and condensation, surface diffusion, and strong
anisotropic surface tension, that guarantee the anisotropy, faceting and
dendritic growth of snowflakes. We demonstrate that this model reproduces the
growth dynamics of the most challenging morphologies of snowflakes from the
Nakaya diagram. We find that the growth dynamics of snow crystals matches the
selection theory, consistently with previous experimental observations.
