We present the theoretical results elucidating the influence of uncompensated solution resistance on anomalous Warburg’s impedance. Here, we obtain the mathematical expression which incorporates the diffusion at the rough electrode/electrolyte interface and bulk solution resistance via phenomenological length scalesdiffusion length ((D/ω)1/2) and pseudoreaction penetration length (L Ω). The roughness at the interface is contained in the expression through the surface structure factor, which can be used to describe statistically any random surface morphology. Detailed analysis for realistic fractal electrodes, characterized as a finite self-affine scaling property with two lateral cutoff lengths, is presented. Limiting behavior in higher frequency is attributed to resistive effects of the electrolyte, and surface roughness is seen through the roughness factor. In the intermediate frequency regime, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness (marking impedance loss) along with the interplay of L Ω and at lower frequencies the response crossover to classical Warburg’s behavior. This crossover frequency is dependent on the smallest of root-mean-square width (h) or lateral correlation length (L), while high-frequency crossover is dependent largely on L Ω. Phase response also shows the maximum phase gain for intermediate frequencies owing to the roughness features and is a signature to the fractal or nonfractal roughness at the interface. Our results show that owing to solution resistance the impedance response can mimic pseudo-quasireversibilty inducing a delay in the onset of diffusion-controlled regime and can hack the anomalous response due to roughness partially or completely.

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A theory for effect of uncompensated solution resistance on Nernstian (reversible) charge transfer at an arbitrary rough electrode is developed. The significant deviation from the Cottrellian behavior is explained, as arising from the resistivity of the solution and geometric irregularity of the interface. Results are obtained for various electrode roughness models, viz., (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, and (iii) random functions with limited self-affine fractal properties. Expressions for concentration, current density, and total current transients have a systematic operator structure in Fourier transformed (deterministic) surface profile function. For a randomly rough electrode, the statistically averaged response is obtained by ensemble averaging over all possible surface configurations. An elegant mathematical formula between the average electrochemical current transient and surface structure factor of roughness is obtained. Realistic fractal roughness is characterized as self-affine scaling property over limited length scales. Limiting behavior in short time, is attributed to resistive effects of the electrolyte, roughness factor and surface curvatures. In the intermediate time, the anomalous power law behavior is attributed to the diffusion length weighted spatial frequency features of roughness. Our results help with the quantitative understanding of generalized Cottrellian response of moderately supported electrolytic solution at rough electrode/electrolyte interface.

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 ( scriptl .25em and .25em L ). Topothesy length ( scriptl τ ) or the related proportionality factor ( μ ≡ scriptl τ 2 D normalH − 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.

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