Theory of Generalized Cottrellian Current at Rough Electrode with Solution Resistance Effects

A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface.