Every electrode reaction consumes reactant right at the surface, carving out a depleted shell that grows with time as ions diffuse in to replace it — the Cottrell diffusion layer, thickness δ ≈ √(π·D·t). A macro electrode is flat compared to that shell, so ions arrive in straight lines (linear diffusion) and the current spikes then decays as δ grows — the classic peaked cyclic voltammogram.
Once the electrode radius r₀ shrinks below δ, its edges curve away faster than the depletion shell can grow flat against it. Diffusion lines converge from every direction onto the tiny electrode (radial / hemispherical diffusion), constantly resupplying it. The current stops decaying and settles into a steady state — the voltammogram flattens into a sigmoid, and the current density per unit area rises sharply because a nanoelectrode's surface-to-volume ratio (∝ 1/r₀) is enormous.
δ(t) ≈ sqrt(π·D·t)
i_lin(t) = n·F·A·D·C / δ(t) (r₀ >> δ, decaying)
i_ss = n·F·D·C·r₀ · f(shape) (r₀ << δ, steady)
i(r₀) = i_lin + i_ss · (δ / (δ+r₀)) — smooth crossover used here
S/V ∝ 1/r₀ — this is the "size effect" nanoelectrochemistry exploits
- Electrode radius — drag toward nano: watch the depletion shell (translucent dome) shrink relative to the electrode's own curvature and the voltammogram trace flatten.
- Applied overpotential — sets how far the electrode is driven from equilibrium; larger |η| pushes more of the theoretical limiting current, following Butler–Volmer kinetics.
- Bulk concentration — more reactant in solution raises both the linear and steady-state branches proportionally.
- Auto-sweep — cycles the overpotential automatically so you can watch the trace redraw live as you drag the radius slider.
Real-world relevance: this crossover is exactly why nanostructured electrodes dominate modern batteries, fuel-cell catalysts and electrochemical sensors — shrinking the active particle below the diffusion-layer scale multiplies the current (and hence power, or sensitivity) obtainable per gram of material.