Theory of Generalized Gerischer Admittance of Realistic Fractal Electrode

Abstract

We developed a theory for the generalized Gerischer admittance for an irregular interface, operating under diffusion and homogeneous kinetics coupled with a fast heterogeneous charge transfer reaction. The generalized admittance expressions are obtained for a deterministic surface (viz, the exact mathematical function of roughness profile is known) as well as for a stochastic surface (viz, the statistical properties of roughness profiles are known). A roughness power spectrum or structure factor is sufficient to characterize the statistical properties of stochastic geometrical irregularities. An elegant expression for the generalized Gerischer admittance is obtained as a functional of the roughness power spectrum. This equation is applicable for fractal as well as nonfractal stochastic roughness. Realistic fractal electrodes have a fractal nature over limited length scales and possess an approximate self-affine property, which is characterized in terms of a band-limited power law function for the power spectrum. The generalized Gerischer impedance shows three frequency regimes, viz, (i) low frequency region, which has a kinetic-controlled frequency (ω) independent impedance and phase angle that follows constraint, 0 ≤ ϕ(ω) < 45°, (ii) anomalous power law behavior for intermediate frequency and their phase angle ϕ(ω) > 45° and shows an approximately constant phase angle region, and (iii) high frequency limiting Warburg impedance behavior.