Anomalous Localization of Electrochemical Activity in Reversible Charge Transfer at a Weierstrass Fractal Electrode: Local Electrochemical Impedance Spectroscopy

The knowledge of local electrochemical activity over an electrode surface is of utmost importance in understanding several processes like electrocatalysis, dendrite growth, and corrosion. For a redox system with unequal diffusivities, we have developed a second order perturbation theory in the surface profile for the local electrochemical impedance spectroscopy (LEIS) for a reversible charge transfer system at an electrode with arbitrary surface profile corrugations. Detailed analysis of LEIS is performed for the simple roughness model of the sinusoidal surface and the realistic roughness model of the finite fractal Weierstrass function as surface corrugation. At a low roughness surface, local electrochemical activity is localized at peaks and depleted at valleys. However, at moderate and high roughness surfaces, there is anomalous bifurcation beside peak positions, making local electrochemical activity localized at cliff (high slope) locations. Maximum depletion is observed at valleys and remains more depleted at valleys as compared to peak positions of the surface even at a high aspect ratio of roughness and frequency of the external signal. Complex multifurcations behavior of local activity and inactivity occur in Weierstrass corrugation for moderate and high roughness. The localization phenomenon of impedance on a Weierstrass fractal surface depends upon the fractal dimension size of finest feature and mean square width of roughness. The influence of various features of a Weierstrass surface is also studied through local impedance distribution functions.