Admittance of diffusion limited adsorption coupled to reversible charge transfer on rough and finite fractal electrodes

Abstract

A theoretical model is developed to quantify the effect of diffusion controlled adsorption of electroactive species on (reversible) charge transfer admittance at a rough electrode. Results are obtained for the linearized isotherm and applied on various electrode roughness models, viz. (i) deterministic surface profile, (ii) roughness as random surface profile with known statistical properties, viz. the power spectrum of roughness and (iii) random profile function with finite self-affine fractal properties. The general expressions for admittance density and total admittance have a systematic operator structure in Fourier transformed surface profile function. Realistic model of roughness requires inclusion of randomness in surface morphology, hence the quantity of interest, the mean admittance is obtained (by taking ensemble average over all possible random configurations). Mathematical expression for the mean admittance of random morphology is the functional of the power spectrum of roughness. A special case of finite fractal roughness is characterized as self-affine scaling property over limited length scales and hence possess two lateral cutoff lengths along with the fractal dimension and a topothesy length (ℓτ). Our analysis explicitly relates the problem of rough electrode admittance with diffusion length (D/ω), isotherm (length) parameter (K) and fractal morphological characteristics. The three characteristic frequencies in this theory are: (a) lower adsorption crossover frequency –ωad≈D/K2, (b) adsorption–roughness topothesy dependent weak pseudo-quasireversibility characteristic frequency –ωq≈D/(ℓτK) and (c) high frequency small cutoff length (ℓ) dependent Warburg – anomalous Warburg crossover frequency –ωw≈D/ℓ2. Finally, theory shows emergence of pseudo-quasireversibility in intermediate frequency region due to electrode roughness in adsorption–diffusion system and curtailment of high frequency anomalous diffusion regime.