Theory of quasi-reversible charge transfer admittance on finite self-affine fractal electrode

Abstract

Response of an electrode under finite charge transfer rate is strongly influenced by morphological and electrochemical heterogeneity of the surface. Here, we developed a theoretical model for quasi-reversible charge transfer admittance of a random finite fractal electrode. These random heterogeneities are characterized by their roughness power spectrum. The power spectrum of the finite fractal roughness is approximated in terms of a power-law function for the intermediate wave numbers (spatial frequency components in roughness). The mathematical expression for the admittance response of a finite fractal electrode depends on diffusion length ( D / ω ) , charge transfer resistance ( R CT ) and its fractal morphological characteristics. Fractals are characterized by the fractal dimension ( D H ) and the surface roughness amplitude ( μ), but finite fractals includes two lateral cutoff lengths. These lengths are the smallest length of roughness ( ℓ ) and the lateral correlation length ( L). The impedance of the electrode shows three frequency regimes, viz., (i) low frequency region, which has a classical Warburg impedance, (ii) anomalous Warburg behavior for intermediate frequency and their phase angle, ϕ( ω) > 45°, and (iii) high frequency impedance is controlled by charge transfer kinetics, and its phase angle follows the constraint: 0 ≤ ϕ( ω) < 45°. Finally, we can say that finite fractal model accounts for fractal as well as nonfractal roughnesses of the electrode, and it strongly influences the admittance response.