Theory of anomalous diffusion impedance of realistic fractal electrode

Abstract

We developed a theory for the anomalous admittance and the constant phase angle behavior of an irregular interface operating under diffusion controlled charge-transfer condition. Interfacial irregularity is modeled as a realistic random fractal, which is characterized as statistically isotropic self-affine fractals on limited length scales. The power spectrum of such a surface fractal is approximated in terms of a power law function for the intermediate wave numbers. This power spectrum has four fractal morphological parameters. They are fractal dimension (D_H), lower (l) and upper (L) cutoff length scales of fractality, and the proportionality factor (μ) related to topothesy or strength of roughness. Theoretical result obtained for the admittance on such realistic fractal electrode is an indispensable step in the quantitative description of role of roughness and is applicable for all frequency regimes. This result can also be simplified to three limiting laws for the admittance or impedance. The anomalous admittance/impedance and an approximately constant phase angle behavior is explained through a limiting law for the intermediate frequencies. The intermediate frequency limiting law is dependent on the fractal dimension as well as on the lower cutoff length and strength of roughness. Finally, these results also offer an explanation for difficulties in classical power-law form for the diffusive impedance with incomplete characterization of exponent with fractal morphological parameters.