2D companion to the 3D tortuous-path model: the same Nielsen tortuosity physics, read off a top-down diffusion map instead of a rendered 3D box. A gas molecule diffusing upward through the film must go around every impermeable platelet it meets instead of straight through it. For thin, well-aligned platelets (aspect ratio α = L/t, volume fraction φ), Nielsen's model predicts:
τ = 1 + (α/2)·φ
P/P₀ = D_eff/D₀ = 1/τ
This isn't just the formula plotted — it's a real 2D random walk. Each molecule takes independent Gaussian steps of variance 2D₀·dt on both axes (Fickian diffusion). When a step would land inside a platelet's rectangle, the step is rejected and the molecule gets a small nudge along the platelet's long axis instead, exactly like a real obstacle deflection. The vertical mean-squared displacement of the whole population gives a measured effective diffusivity:
D_eff = ⟨Δy²⟩ / (2Δt) (sampled every second, excluding molecules
that just respawned mid-window)
τ_measured = D₀ / D_eff
- φ — how much of the film area is nanoclay platelets. Nielsen's formula is linear in φ; the curve panel plots it across the full slider range.
- Aspect ratio L/t — thinner, wider platelets (higher α) force a longer detour per obstacle, steepening the curve.
- Alignment — Nielsen's formula assumes platelets lie flat, perpendicular to the diffusion direction (100%). Drop alignment toward 0% (randomly tumbled, poorly exfoliated filler) and the measured barrier falls well short of the Nielsen line.
- D₀ — the molecule's diffusivity in the pure, unfilled matrix; it sets the animation's step size and is the denominator of every ratio above.
Real-world relevance: this "brick-and-mortar" tortuosity mechanism is why a few weight-percent of exfoliated montmorillonite or graphene in a polymer film can cut oxygen or water-vapor permeability several-fold — the basis of nanocomposite food-packaging and barrier-coating design.