Chronoamperometry: Fick's-Law Diffusion Profile Solver

Potential Step

Cell Parameters

Bulk concentration C₀ 2.0 mM

Diffusion coefficient D 1.0 ×10⁻⁵ cm²/s

Electrode area A 0.10 cm²

Live Readouts

I(t): FD solver (solid) vs. Cottrell (dashed)

Current I(t) — FD solver0.0 µA

Current I(t) — Cottrell0.0 µA

Diffusion front (0.9·C₀)0 µm

Charge passed Q(t)0.0 µC

Elapsed time0.00 s

How it works

This is a genuine numerical solve of Fick's second law, not a scripted curve. The electrolyte between the electrode (left, x=0) and the bulk reservoir (right, x=L) is discretized into a 1D grid of concentration values C(x). Every animation frame the solver advances the grid forward in time with an explicit finite-difference (FTCS) update:

∂C/∂t = D · ∂²C/∂x²
C[i] ← C[i] + r·(C[i+1] − 2C[i] + C[i−1]),  r = D·Δt/Δx²

Boundary conditions while stepped:
  C(0,t) = 0            (instant reaction consumes every ion on contact)
  C(L,t) = C₀           (bulk reservoir stays full)

Electrode current from the LOCAL gradient (Fick's first law):
  J(t) = D · (C[1] − C[0]) / Δx
  I(t) = n F A · J(t)

The concentration field above is drawn straight from the grid — the dark band spreading from the electrode is the numerically depleted region, not an animation.
The current readout is computed independently from the gradient at every step, then plotted against the closed-form Cottrell equation I(t) = nFAC₀√(D/πt) as a dashed reference curve — the two should track closely once the initial step transient settles (see Diffusion front below for how each defines the layer edge).
Diffusion front marks the grid point where C(x) first climbs back to 90% of C₀ — a direct read of the profile, distinct from the Nernst linear-extrapolation thickness δ = √(πDt) used inside the Cottrell formula itself; the two are related by a fixed factor (≈1.31), not equal.
Real-world relevance: this finite-difference approach is exactly how simulation software predicts sensor response before hardware exists — the same PDE and boundary conditions govern amperometric biosensors such as the glucose sensor in a glucometer.

C(x) profile Electrode (x=0)