General theory of arbitrary potential sweep methods on an arbitrary topography electrode and its application to random surface roughness

Abstract

A general theory for an arbitrary potential sweep voltammetry on an arbitrary topography of an electrode operating under diffusion-limited or reversible charge-transfer conditions is developed. The generalized concentration and current expressions are obtained for a deterministic surface, namely, the exact mathematical function of topography is known, as well as for a stochastic surface, namely, the statistical properties of rough topography are known. Random surface roughness is expressed in term of a general statistical property - the power spectrum of roughness. The elegant mathematical structure of the equation obtained here is the time derivative of a convolution integral of Nernstian boundary constraint with arbitrary potential function and the potentiostatic current transient for a rough electrode. The general results obtained here are a solution to several theoretical problems of potential controlled techniques in nanostructured as well as rough electrodes. Explicit results are reported for linear scan voltammetry and cyclic voltammetry for realistic fractal as well as nonfractal roughness. Finally, this theory provides a quantitative description of the role of roughness on chronoamperometric and voltammetric measurements, therefore opening up the way to explore and understand various anomalies in their response.