Cool a saturated salt solution: ions must first beat a nucleation barrier, then crystals grow face by face while they use up the dissolved salt.
Solid: equilibrium solubility. Dashed: the nucleation limit (where about one nucleus per minute appears). Between them the solution is supersaturated but stable (metastable zone).
Each row is a fresh run with the current start/end temperature, nucleation settings and no seeds, followed by 3 h at the end temperature. Faster cooling gives a higher peak supersaturation, so more nuclei and smaller crystals; the yield must equal the lever-rule prediction.
The science
Solubility. van 't Hoff: Cs(T) = Cs0 exp[−(ΔH/R)(1/T − 1/T0)] with Cs0 = 316 g/L at 293.15 K and ΔH = 25.5 kJ/mol, a two-point fit to typical textbook values of a nitrate-type salt (about 316 and 1100 g/L at 20 and 60 °C). Supersaturation ratio S = C / Cs(T), relative supersaturation σ = S − 1.
Nucleation (classical theory). A spherical cluster of radius r costs ΔG(r) = 4πr²γ − (4π/3) r³ kT ln S / v, with surface energy γ and molecular volume v = M/(ρNA). The maximum is at r* = 2γv/(kT ln S) with barrier ΔG* = 16πγ³v² / [3 (kT ln S)²]. The rate is J = J0 exp(−f(θ) ΔG*/kT); contact with a wall or dust (wetting angle θ) lowers the barrier by f(θ) = (2 + cos θ)(1 − cos θ)²/4. Nuclei appear as Poisson events. J0 = 10⁶ L⁻¹s⁻¹ and γ are illustrative; real values depend strongly on the system.
Size effect (Gibbs–Thomson). A crystal of size a sees the effective ratio Seff = S exp(−2γv / (kT a)), so crystals smaller than r* dissolve; new nuclei start at 1.3 r*.
Faceted growth. Each crystal is a hexagonal prism (circumradius a, length L) with two pyramidal caps of height 0.6a. Side faces advance as da/dt = ka σeff (rough face, linear). End faces follow a Burton–Cabrera–Frank spiral law vc = kc σ²/σ1 tanh(σ1/σ) (parabolic at low σ, linear at high σ) and dL/dt = 2vc. An adsorbing impurity multiplies kc by 1 − 0.92 × (impurity level), turning needles into plates (the habit changes with the environment, the lattice does not). Rate constants ka = 2.5, kc = 5 µm/s are illustrative.
Mass balance. The vessel holds 0.2 L. Every volume gained by a crystal, V = (3√3/2) a² (L + 0.4a), is removed from the dissolved mass at density 2110 kg/m³, so supersaturation falls as crystals grow. At the end of a long hold the lever rule must hold: crystal mass = total − CsV.
Simplifications. Perfectly stirred solution (no concentration gradients), all crystals see the same bulk S, no agglomeration, breakage or secondary nucleation, no heat of crystallisation, identical orientation of every crystal, one constant γ. Nuclei are tracked as cohorts: each time step creates one cohort whose weight is the number of identical crystals, and beyond 300 cohorts new nuclei merge into the youngest one (the counter in the HUD is the weighted total; the picture shows one crystal per cohort). Crystals are drawn magnified and standing on the floor, and the lattice inset is schematic. Models built with Blender, scene by Three.js.