Generalization of Randles-Ershler admittance for an arbitrary topography electrode: application to random finite fractal roughness

Randles-Ershler admittance equation is generalized for an arbitrary topography electrode. This generalization incorporates various phenomenological components involved in complete dynamical response of an electrode, viz. the diffusion, the charge transfer reaction, the uncompensated solution resistance effect and the capacitance of the electric double layer. Generality of results allow their application to all frequency impedance response to both, stochastic and deterministic electrode roughness. The stochastic electrode roughness is characterized through its statistical property of structure factor or power spectral density. A detailed analysis of roughness effect is carried out for a finite self-affine fractal electrode. The dynamics of the system is found to depend on phenomenological lengths, viz. diffusion length, charge transfer kinetics-diffusion length and ohmic-diffusion length; and topographic lengths, viz. tiniest length scale of fractality, topothesy length and the self-similarity index (the fractal dimension). Various anomalous responses emerge through the coupling of phenomenological length scales with topographical scales of roughness. These responses of the rough electrode are described by three characteristic frequencies: charge transfer frequency, anomalous Warburg frequency and the electric double layer charging frequency. Delay and curtailment in anomalous superdiffusion regime is seen due to the influence of quasireversibility in charge transfer process and pseudo-quasireversibility (arising out of uncompensated solution resistance). Finally, one of the most widely employed model in electrochemical impedance analysis is generalized with inclusion of ubiquitous electrode disorder