Fractal dynamics of polarized bioelectrodes

This study is concerned with mathematical modelling of the fundamental relationship which exists between the current density and the overpotential across the metalsolution interface in the linear range using methods of system theory enhanced by ‘fractal’ concepts. A primer for both 1/f-type scaling and ‘anomalous’ relaxation/dispersion concepts is provided, followed by a brief review of the research history pertinent to the metal electrode polarization dynamics. Next, the ‘fractal relaxation systems’ approach is introduced to characterize, systems which attenuate with a fractional power-low dependence on frequency through a ‘scaling exponent’. The ‘singularity structure’ which is a scaling, rational system function is proposed to expand fractal systems in terms of basic subsystems individually representing elementary exponential relaxations and collectively exhibiting scaling properties. We stress that the ‘singularity structure’ carries scaling information identical to the conventional ‘distribution of relaxation times’ function. ‘Structure scale’ and ‘view scale’ concepts are presented in the due course to streamline the analysis of scaling phenomena in general and the polarization impedance in particular. System theory-wise, the notable result is that the fractional power function attenuation, or equivalently, the logarithmic nature of the distribution function translates into the ‘self-similar’ pattern replication of the system singularities in the s-plane. The singularity arrangement is governed by a recursive rule solely based on the knowledge of the fractional power factor or the scaling exponent.