j.mino.mobi/model — Fickian diffusion fit

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Synthesises an analytic radial curve so each fit can be seen to win its own physics. Image-derived curves are blurrier, where Weibull is robustly best.

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Model comparison (lower AIC = better)

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ΔAIC<2 ≈ tie · 4–7 clearly better · >10 decisive. AIC trades fit quality against parameter count.

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Physical scale → D

Convert the dimensionless fit to a diffusion coefficient.

sphere R (µm)

elapsed t (s)

D = τ·R²/t—

What power law should this follow?

Fickian diffusion into a sphere held at fixed surface concentration:

φ(ρ,τ) = 1 + (2/πρ)·Σ (−1)ⁿ/n · sin(nπρ)·e^(−n²π²τ)

Total uptake:

Mₜ/M∞ = 1 − (6/π²)·Σ (1/n²)·e^(−n²π²τ) (τ = Dt/R²)

Early-time power law: Mₜ/M∞ ≈ (6/√π)·√(Dt/R²) − 3Dt/R² → ∝ t^½.
Korsmeyer–Peppas Mₜ/M∞ = k·tⁿ: a sphere's Fickian exponent is n ≈ 0.43 (slab 0.50, cylinder 0.45).
Penetration depth grows as λ ≈ √(Dt) — the semi-empirical exp model captures this directly.