Crystal growth
A seed in a supersaturated solution. Molecules attach one by one, guided by symmetry and thermodynamics. Watch facets emerge, dendrites branch, and a crystal take shape in real time.
ΔG = −nkT ln(S) + γA → nucleation barrier → anisotropic growth → crystal facets
Nucleation: how crystals begin
A crystal cannot simply appear in a solution. First, a cluster of molecules must randomly assemble into a tiny nucleus that exceeds a critical size. Below this critical radius, the surface energy penalty outweighs the bulk energy gain, and the cluster dissolves. Above it, growth becomes thermodynamically favorable and the crystal expands spontaneously. This energy barrier explains why solutions can remain supersaturated for long periods before crystallization suddenly begins — they are waiting for a fluctuation large enough to push a nucleus past the critical size. In this simulation, we bypass the stochastic nucleation step and plant a seed directly.
Crystal symmetry
The symmetry of a crystal reflects the symmetry of its underlying lattice. In a cubic lattice (4-fold symmetry), atoms sit at the corners of squares, and the crystal tends to develop flat faces perpendicular to the four main axes. In a hexagonal lattice (6-fold symmetry), atoms sit at the vertices of hexagons, producing the classic six-pointed symmetry of snowflakes. The anisotropy of the lattice — the fact that bond energies differ along different crystallographic directions — is what creates facets. Without anisotropy, a crystal would grow into a featureless sphere.
Faceted vs dendritic growth
At low supersaturation (the driving force for crystallization), growth is slow and orderly. Molecules have time to find the energetically optimal positions on flat crystal faces, producing smooth facets. At high supersaturation, growth outpaces surface diffusion. Any small bump on a face grows faster than its neighbors because it protrudes into fresher solution. This Mullins-Sekerka instability amplifies perturbations into branches, and branches spawn sub-branches, creating the elaborate tree-like structures called dendrites. Snowflakes are the most familiar example: their intricate arms are dendrites grown from a hexagonal seed under high supersaturation.
The role of temperature
Temperature affects crystal growth in two competing ways. Higher temperature increases the thermal energy available for molecules to detach from the crystal surface, making growth more reversible and producing smoother, more faceted crystals. But temperature also affects the supersaturation (since solubility increases with temperature), which can reduce the driving force for growth. In this simulation, the temperature parameter controls thermal noise: higher temperature means more random attachment and detachment, producing rougher but more compact crystals. Lower temperature means more deterministic growth, favoring the anisotropy and producing sharper facets and more elaborate dendrites.
From simulation to reality
This simulation uses a lattice-based growth model inspired by diffusion-limited aggregation (DLA) with anisotropic attachment probabilities. Real crystal growth is far more complex: it involves diffusion of solute through the solution, heat release at the growing interface, surface tension effects, and the detailed atomic structure of crystal faces. Nevertheless, the basic competition between bulk thermodynamic driving force and surface energy, modulated by lattice anisotropy, captures the essential physics that determines whether a crystal grows into a compact faceted shape or an intricate dendritic one.