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Water and ink
Heat the water or shrink the molecules to speed diffusion. Tap the water to drop in more ink.
Layers
Model checks
The science
Fick's law
Molecular diffusion obeys ∂c/∂t = D ∇²c. A drop that starts as a 2D Gaussian of width s0 stays Gaussian: s²(t) = s0² + 2Dt per axis and peak concentration M/(2πs²). The simulation integrates the equation on an 80 × 48 grid (125 µm cells) with the explicit scheme and reflecting (no-flux) walls, so total ink is conserved.
Stokes-Einstein
For a sphere of radius r in a fluid of viscosity η: D = kBT / (6πηr). The viscosity of water is taken from the Vogel form η = 2.414×10-5 · 10247.8/(T-140) Pa s (T in kelvin), so warm water has a much smaller η and a larger D. A small molecule (0.5 nm) at 25 °C has D ≈ 5×10-10 m²/s: it needs hours to move a few millimetres, which is why the time warp is in hours per second.
Honest simplifications: real ink in a glass of water mixes mainly by convection, surface tension and density differences within seconds; this page shows only the molecular (diffusive) part, which is the slow limit. Dye molecules are treated as spheres, the species do not interact, and D does not depend on concentration.
What to try
- "Cold water" versus "Hot water": watch how much faster the blue ink spreads at 90 C.
- "Tiny vs huge molecules": the orange ink with the larger radius spreads far more slowly.
- Compare the measured variance in the first graph with the dashed s0² + 2Dt lines.
- Tap the water to add more drops and watch them merge.