Theory for staircase voltammetry and linear scan voltammetry on fractal electrodes: emergence of anomalous Randles–Sevčik behavior

Abstract

A theory for staircase voltammetry (SCV) on randomly rough electrode is developed under reversible charge transfer conditions. An elegant general expression for the statistically averaged current that relates single potential step current transient to the multipotential step (staircase) current response is obtained using properties of Heaviside unit step function. Theory emphasizes the use of power spectrum to characterize the surface geometric disorder. Explicit expression for the roughness averaged current is obtained for finite fractal electrode. The result for linear scan voltammetry (LSV) is obtained by collecting current value at a specific time on each potential step. The characteristic peak current and peak potential in LSV, are dependent on fractal dimension (D_H), topothesy length (ℓ_τ) and finest length scale of fractality (ℓ). Roughness can induce significant (∼15 mV) peak potential shift. Emergent anomalous Randles–Sevčik behavior is observed for intermediate scan rates that depends not only on DH but also on ℓ and ℓτ. The inner (ν_i) and outer (ν_o) crossover scan rates proportional to diffusion coefficient (D) and potential step size (ΔE) are, πΔED/ℓ_τ² and πΔED/ℓ², respectively. Classical Randles Sevčik behavior is observed beyond two crossover scan rates.