Diffusion-controlled potentiostatic current transients on realistic fractal electrodes

Abstract

We analyze the problem of diffusion to irregular electrode whose irregularity is characterized as statistically isotropic self-affine fractals on limited length scales. The power spectrum of a surface fractal is approximated in terms of a white noise for low wave-numbers and a power law function for the intermediate wave-numbers. This power spectrum has four fractal morphological parameters. They are fractal dimension (D_H), lower (ℓ) and upper (L) cutoff length scales of fractality, and the proportionality factor (μ) related to topothesy or strength of roughness. Our explicit results for the potentiostatic current transient and its limiting laws are presented. These limiting laws are: (i) short time expansion, (ii) long time expansion and (iii) intermediate time expansion. The intermediate time limiting law for the current transient captures the classical anomalous power law behavior which is usually observed in experimental data. Our results show that the scaling exponent of anomalous region is dependent on D_H as well as on ℓ and μ. These results also unravel the connection between the crossover times and the roughness characteristics of realistic fractal surfaces. We demonstrate an excellent comparison between the theoretical results and the experimental potentiostatic current transient. Finally, we also show the localization of current density on a rough corrugated electrode where surface corrugation is taken as a band-limited Weierstrass–Mandelbrot random function.