Theory of generalized Gerischer impedance for quasi-reversible charge transfer at rough and finite fractal electrodes

Abstract

A theoretical model for the generalized Gerischer impedance at an irregular interface under quasi-reversible heterogeneous charge transfer reaction has been developed. The dynamic impedance response is predicted to depend on three phenomenological components, viz. diffusion, charge transfer reaction and bulk homogeneous kinetics as well as roughness. The theory has been developed for both deterministic and stochastic electrode roughness. Statistical properties of the stochastic electrode roughness are characterized through their power spectral density. The impedance response is influenced by the finite self-affine fractal roughness which is analyzed in detail. Three governing phenomenological length scales are diffusion length, charge transfer length and reaction layer thickness. The anomalous impedance response of the finite fractal electrode is due to the synergistic effect of the phenomenological lengths and finite-fractal morphological parameters, namely, fractal dimension (DH), lower cut-off length (ℓ) and topothesy length (ℓτ). The impedance response has three distinct characteristic frequency regimes: the low frequency bulk kinetics controlled regime, roughness dependent intermediate frequency region and heterogeneous charge transfer controlled high frequency regime. A contrasting diffusive behavior is predicted, wherein the system switches from sub-diffusive to super-diffusive response with increase in electrode roughness. This theory unravels the influence of both bulk and interfacial kinetics on the impedance response of a rough electrode.