Crystal Growth in a Dish (3D)

What you see

Dissolved particles drift at random through a thin liquid film. When one touches the crystal it may stick, and the crystal grows. Tips stick out into the supply of particles and catch more of them than the sheltered gaps between branches, so a small bump grows faster than a dent. This screening instability turns a round seed into a branching dendrite. The same process shapes copper deposited at an electrode, zinc dendrites in batteries, frost on glass and soot clusters.

What the 3D scene shows

A thin-film electrodeposition cell, the experiment in which DLA-like copper dendrites were first photographed. The lattice crystal from the 2D page is drawn as one copper block per cell, taller near the seed only to give the dendrite depth; the blue dots are dissolved ions. The dish and bench are drawn enlarged, and the power supply and counter electrode ring are decoration around the same simulation code as the 2D page.

The model: diffusion-limited aggregation (DLA)

Each particle is a random walker on a square lattice. Its density obeys the diffusion equation dc/dt = D lap c and, because walkers are slow compared with the growth of the surface, the density near the crystal is close to the Laplace field lap c = 0 with c = 0 on the crystal surface. The growth speed at a point is proportional to the local gradient of c, which is largest at tips. One lattice step per tick gives D = a^2 / (4 dt) for lattice spacing a; the page converts ticks to model seconds with a = 0.05 mm (a 20 mm thin-film cell) and D = 7e-10 m^2/s (a typical ion in water).

Fractal dimension

The mass inside radius r grows as M(r) ~ r^D. A solid disc has D = 2; classic DLA has D = 1.71. The fit in the second graph measures the slope of log M against log r for your crystal. Smaller sticking probability, more required contacts, or corner contacts push D towards 2 because particles have time to settle into gaps.

Temperature, sticking and supersaturation

Attachment is a thermally activated step, so the sticking probability follows an Arrhenius law, p(T) = p25 exp(-(E_a / k_B)(1/T - 1/298.15 K)) capped at 1, where k_B = 8.617e-5 eV/K. Warmer solutions stick more often per contact (in this simple model), which gives thinner branches. The number of walkers stands for the supersaturation: more dissolved particles means a faster growth rate.

Anisotropy

Counting only the four side neighbours as contacts builds the square lattice into the crystal, so branches follow the lattice axes and the angle histogram shows four peaks. Counting corners as well makes growth nearly isotropic. Real crystals have anisotropy from their lattice too (snowflakes six-fold, salt four-fold).

Honest simplifications

• The model is two-dimensional, has no convection, no depletion of the bath, and no electric field, and it uses one fixed particle size. It illustrates the mechanism; it does not predict a particular salt.
• Real crystal shape also depends on surface tension and facet kinetics, which are only represented here by the contact rule.
• Model time is a conversion of lattice ticks using a typical diffusion coefficient; the concentration is not computed explicitly.

Numerical checks on this page

• A free random walk has mean squared displacement equal to the number of steps, which fixes the diffusion constant used for model time.
• A separate classic DLA run measures the fractal dimension and compares it with the published value 1.71.
• An exact bookkeeping check: contacts, site counters, crystal cells and attached particles must agree.