I've taken the scientific background primarily from Ken Libbrecht, who studies the science of snow crystals. He developed a scientific model of snow crystal formation.

What I'm working on here is a computational model for the formation and growth of snow crystals. My aim is to emulate the natural physics as much as I can. The model is much-simplified, as you would expect, since crystallization at the molecular level is complex and not yet well understood, not to mention the computational complexity of emulating the process faithfully.

Snow crystals form as plates, stellar plates, dendrites, prisms, needles, and columns. Scientists have worked out how the variation in temperature and humidity will form different types of snow crystals in the wild. The varying conditions affect the growth in 3D, including the growth of facets and branches.

My aim with the computational model is to explore how different conditions reliably grow different types of crystal, and thus learn more about snow crystal growth. At this stage, I have a working model that can generate different shapes of snow crystal, very reminiscent of real snow crystal, however I haven't been able to model real-world conditions like temperature and humidity. That is for future work.

This section is summarized from Libbrecht's page on Snowflake Science.

A note on terminology: 'Snowflake' is the everyday name for just about anything that falls from the sky in the winter, whereas 'snow crystal' is the more specific technical name for an individual crystal of ice, what we normally call a snowflake.

A snow crystal forms as the water vapour around a seed ice crystal freezes onto the seed, as it falls through the clouds.

Snow crystals have a hexagonal, or six-fold, symmetry because of the angular 3D geometry of water molecules (H₂O). This means new water molecules can only attach in certain patterns.

As a snow crystal falls through the clouds, the temperature and humidity change unpredictably, and these changes make the arms grow differently. But at any one instant the conditions around all six arms are the same, so each arm grows the same. Since every path through the clouds is different, every snow crystal is unique.

Two properties determine how water molecules freeze and attach to the growing crystal:

• The surface physics of the crystal itself, including the surface structure, temperature, and humidity
• The vapour saturation near the surface of the crystal.

My model is a huge simplification, but it is nonetheless based on physics principles:

Setup

• I simplify to 2D (flat) crystals, rather than 3D
• A hexagonal lattice is hard-coded (like hex graph paper), which simulates the hex structure of real ice crystals
• The 'molecules' are cells on the lattice, much much larger than real molecules would be
• You can think of each cell in the lattice as a large volume of water molecules that has a saturation, which condenses altogether as a single 'modecule' onto the growing crystal
• There are two hex arrays:
1. The vapour saturation at each cell; this models the 'world'
2. The crystal itself (1 = ice, 0 = vapour), surrounded by vapour
• The crystal starts as a small hexagon seed crystal
• The algorithm them iterates over two phases: crystal growth and vapour diffusion

Growth phase

• During the growth phase, the algorithm checks every vapour cell on the crystal surface (ie next to already crystallized cells). Each potential attachment point will have a growth velocity, dependent on the surface physics and the vapour saturation. If the velocity is high enough, then the vapour cell freezes onto the crystal
• The algorithm calculates the velocity using a simplified version of Libbrecht's key formula:

  velocity = alpha x sigma

  where alpha is the Condensation Coefficient and sigma is the Vapour Saturation, at that potential attachment point
• The Condensation Coefficient (alpha) is based on the number of surrounding already condensed cells (more ice means higher alpha and higher chance to condense nearby cell)
• Initially, the Vapour Saturation (sigma) of every cell in the vapour is set to a uniform initial value. When a cell condenses, it captures the vapour from the cell and adjoining cells, ie taking water out of the air and into the ice, thus reducing the saturation. If the saturation is too low, because it's all been captured in the snow crystal, then a cell cannot condense (because the velocity will be low)

Diffusion phase

• In the diffusion phase, the algorithm moves vapour saturation from highly saturated cells to lower saturated cells over the whole vapour array, simulating how the vapour pressure drives to equalize saturation.

Changing conditions

• To model changing conditions, a single variable representing the conditions is factored into the sigma calculation of the velocity formula. After each iteration, the conditions variable is modified by a random walk, up or down, simulating unpredicable temperature and humidity changes. The snowflake keeps its symmetry because conditions change only after a complete iteration of growth and diffusion condensation. If conditions were to change continuously, then the snowflake would become degenerate.
• There are several user-specified parameters:
  • Attachment_Power: A constant factor that affects the contribution of the Condensation Coefficient (alpha) to the velocity
  • Velocity_Limit: A minimum threshold on velocity (expressed as a percentage of the maximum velocity in the iteration) for a cell to condense into ice
  • Vapour_Reduction_Factor: The vapour saturation left in the surroundings cells, after a cell condenses into ice, capturing some of the vapour.