Rock candy - 2D jar and solubility diagram

What is modelled

Rock candy is a crystal of sucrose grown from a supersaturated syrup: boiling-hot water dissolves far more sugar than a cool jar can hold at equilibrium, so when the syrup cools the excess is ready to deposit on any seed. The model follows one jar of 150 g of water with a chosen amount of dissolved sugar, a number of strings each carrying six crystals, evaporation of water, and the sugar balance between the syrup, the crystals and any fine crystals that crash out.

Equations

• Supersaturation: S = c / csat(T) with c = 100 msugar / mwater in g per 100 g water. The solubility csat(T) is the Charles/Vavrinecz polynomial for sucrose in water, valid from 10 to 60 C.
• Growth is two resistances in series. Transport through the syrup: Gd = kd ρs (w − wsat) / ρc with kd = Sh D / L and Sh = 2 + 0.6 Re1/2 Sc1/3; the diffusion coefficient falls steeply with syrup viscosity (D ∝ T / η0.8). Surface integration: Gr = Gref exp[−Ea/R (1/T − 1/313 K)] (S − 1)1.5. The face velocity is G = Gd Gr / (Gd + Gr).
• Sucrose is monoclinic (space group P2₁, β = 102.9°). The crystal is a parallelepiped with plane separations H₁, H₂, H₃, volume V = H1 H2 H3 / sinβ; each pair of faces moves outward at G times a habit ratio (1 : 1.7 : 0.8), so dHi/dt = 2 ri G. The sugar taken from the syrup is exactly the crystal density times the volume gained, so mass is conserved by construction.
• When S rises above the metastable limit (1.30, lowered by stirring) fine crystals nucleate spontaneously: dmfines/dt = k (S − Smet)2 msugar. They sink and form the sediment layer, and they drain the syrup so S falls again.

Checks run in the page

• A 200-day run with growth, evaporation and a nucleation crash is checked for exact mass conservation (dissolved sugar + crystals + fines + evaporated water).
• The solubility polynomial is compared with independent handbook values from 0 to 60 C.
• The crystal density derived from the sucrose unit cell (10.86 x 8.70 x 7.76 A, β = 102.9°, two molecules) is compared with the density used in the model.

Honest simplifications

• The surface-kinetics constant, activation energy, reaction order, habit ratios, metastable limit and the nucleation rate constant are illustrative values of the right order of magnitude, not fitted to a data set. The model shows how syrup strength, temperature and stirring trade against each other; it does not predict the exact days a real jar needs.
• The viscosity and diffusion correlations are approximate fits (roughly ±10 to 30 %) and are extrapolated to the very strong syrups of the supersaturated range.
• All crystals are identical copies of one mean crystal; strings do not shade each other, crystals do not grow together, and there is no secondary nucleation from crystal collisions.
• Temperature is held constant (no cooling jar), impurities and colour are ignored, and the stirring velocity is only a simple parameter in the mass-transfer law.
• The jar is drawn to scale as a side cross-section with the crystals at their computed size; the 3D page shows the same model in a modelled kitchen jar.

What to try

• Choose Too concentrated: crash and watch S pass the dashed metastable line, the cloud of fine crystals appear and the sediment build up.
• Compare the Classic preset with Stirred jar: stirring thins the transport layer but also lowers the metastable limit.
• Choose Under-saturated: dissolves and see S below 1 shrink the crystals.