Anomalous Warburg Impedance: Influence of Uncompensated Solution Resistance

Abstract

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 (LΩ). 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 LΩ 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 LΩ. 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.