Experimental validation of roughness power spectrum-based theory of anomalous cottrell response

Abstract

We experimentally validate theoretical relation between the roughness power spectrum (PS) and electrochemical current transient for a reversible charge transfer system under a single potential step. Roughness features at the electrochemically roughened electrode are characterized using standard measurements such as scanning electron microscopy (SEM), atomic force microscopy (AFM) and cyclic voltammetry (CV). The PS obtained from AFM shows composite finite fractal and nonfractal nature in roughness, whereas the PS from SEM shows only a finite fractal nature. AFM or SEM measurements provide knowledge of fractal dimension (D_H) and two fractal cutoff lengths (ℓ and L). Topothesy length (ℓ_τ) or the related proportionality factor (μ≣ ℓ_τ^2D_H-3) from PS data of AFM requires extrapolation of data for unit wavenumber, but this method usually provides unphysical values of μ. We provide a novel method to determine the topothesy of electrodes from CV measurements of electroactive area in conjunction with SEM or AFM measurements. Chronoamperometric measurements were made on morphologically characterized Pt-electrodes for a solution of K₄[Fe(CN)₆] and K₃[Fe(CN)₆] in 3 M NaNO₃. The transient response observed experimentally is validated using the measured PS in the theoretical equation for the current. The transient response does not show contributions from Gaussian PS in the low wavenumber region; this is due to the fact that the effective lower cutoff wavenumber is usually limited up to the inverse of the width of roughness (or topothesy length). Fractal dimensions obtained through chronoamperometric measurement on electrodes using Pajkossy’s approach do not correspond to the one obtained from AFM and SEM measurements. Finally, the anomalous response in the Cottrell measurements can be understood through PS-based theory.