Generalization of the Anson equation for fractal and nonfractal rough electrodes

Abstract

We have generalized the Anson equation for chronocoulometry at deterministic and randomly rough electrode/electrolyte interface morphologies. The randomly rough electrode surface is characterized by the power spectrum of roughness. An elegant mathematical formula between the surface roughness power spectrum and the charge transient is obtained. Detailed analysis of this formula is performed for the chronocoulometric response of fractal and nonfractal electrodes using an approximate power-law and Gaussian power spectra, respectively. Realistic finite fractal roughness consists of the following four morphological characteristics: fractal dimension (D_H); the strength of fractality (μ), which is the magnitude of the power spectrum at the unit roughness wavenumber; and the lower (ℓ) and upper (L) cut-off lengths, which are related to cut-off wavenumbers in the power spectrum. Nonfractal Gaussian roughness consists of two morphological characteristics, the mean square width of roughness (h²) and the transversal correlation length (a). The chronocoulometric response for such rough electrodes shows three time regimes in double logarithmic plots. An increase in roughness increases the magnitude of charge transients, which introduce non-linear behavior in traditional Anson plots. Finally, this generalization provides insight into the effect of fractal and nonfractal morphological disorder on the chronocoulometric measurements of Nernstian charge transfer under the potential step method.