Generalized Warburg impedance on realistic self-affine fractals: comparative study of statistically corrugated and isotropic roughness

Abstract

We analyse the problem of impedance for a diffusion controlled charge transfer process across an irregular interface. These interfacial irregularities are characterized as two class of random fractals: (i) a statistically isotropic self-affine fractals and (ii) a statistically corrugated self-affine fractals. The information about the realistic fractal surface roughness has been introduced through the band-limited power-law power spectrum over limited wave numbers. The details of power spectrum of such roughness can be characterized in term of four fractal morphological parameters, viz. fractal dimension (D_H), lower (ℓ), and upper (L) cut-off length scales of fractality, and the proportionality factor (μ) of power spectrum. Theoretical results are analysed for the impedance of such rough electrode as well as the effect of statistical symmetries of roughness. Impedance response for irregular interface is simplified through expansion over intermediate frequencies. This intermediate frequency expansion with sufficient number of terms offers a good approximation over all frequency regimes. The Nyquist plots of impedance show the strong dependency mainly on three surface morphological parameters i.e. D_H, ℓ and μ. We can say that our theoretical results also provide an alternative explanation for the exponent in intermediate frequency power-law form.