Runs the current red particles through 1, 2, 4 and 9 pores (membrane present), fits the decay of the left–right imbalance exp(−t/τ) and compares τ with the Hille pore formula. Three independent ensembles of 400 particles are averaged; takes about 30 s.
The science
Brownian motion. A small sphere in water is kicked ~1020 times a second by solvent molecules. At times far longer than the momentum relaxation time (nanoseconds) the motion is the overdamped Langevin equation, i.e. a Gaussian random walk. Each time step Δt every coordinate changes by a normal random number with variance 2DΔt.
D = kBT / (6 π η r) (Stokes–Einstein)
⟨|r(t) − r(0)|²⟩ = 6 D t (Einstein, 3-D, free space)
η(T) = 2.414×10−5 · 10247.8/(T−140) Pa s (water, ±3 %)
Hotter water is thinner and the thermal energy is higher, so D grows quickly with T; a larger sphere or a thicker liquid slows diffusion. Example: r = 50 nm at 20 °C gives D ≈ 4.3 µm²/s.
Fick's laws. Flux J = −D ∇c, and ∂c/∂t = D ∇²c. With the membrane removed, the right-hand fraction after a step start follows the exact cosine series fR(t) = ½ − (4/π²) Σn odd e−D(nπ/2L)²t/n² (L = 1.5 µm): the dashed curve in the first graph.
Pores. A solute must find a pore. For one cylindrical pore of radius a and length h the diffusive conductance is (Hille)
1/G = 1/(4 D a) + h/(π a² D) [µm³/s]
Δc decays as exp(−2 n G t / V) (n pores, V = 6 µm³ per side)
The first term is the “access resistance” of the two mouths, the second the channel itself. A sphere of radius r only fits if r < a: the model uses the accessible radius a − r, so the blue spheres (160 nm) are completely blocked by 150 nm pores — a molecular sieve.
Honest simplifications. Particles are point-centre spheres that do not collide with each other (dilute limit), the membrane is a rigid slab with straight cylindrical pores and reflecting faces, and walls are specular. Real pores also slow the sphere hydrodynamically (Renkin hindrance, here ignored; for r/a = 1/3 the real diffusivity inside the pore is roughly 40 % of free), have charges, and membranes are never perfectly flat. Hille's formula assumes an isolated pore in an infinite medium; here pores interfere with each other and with the chamber walls, so the measured τ is 1.2–1.7× slower than the formula (see the experiment). The chamber is drawn at a magnified scale, particle sizes are in the correct ratio. Illustrative viscosity multipliers. Independent educational model.
Independent educational model. Made with Blender and Three.js.