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Numerical analysis of the impedance of fractal electrodes
Abstract
The constant-phase-angle (CPA) impedance observed in electrochemical cells is often thought to be due to fractal roughness on the electrode surface. This idea was pursued by numerous theoretical and experimental studies in the last decade but there is no consensus on the quantitative relationship between the roughness and the impedance. In this study, we consider the partial differential equations that govern the electrostatic potential and the concentrations of anions and cations between two blocking electrodes which have no chemical reactions. We assume that diffusion and conduction are the only transport mechanisms and the Poisson-Boltzmann equation is obeyed. These equations are linearized and solved analytically in one dimension and numerically in two dimensions. For the latter, we used electrodes shaped like Koch curves and saw-tooth curves. A special grid was generated by conformal mapping to fit these boundaries with singularities and the equations are solved by finite-difference method on this grid. The numerical results are compared to the one-dimensional solution that give the behavior of the flat electrode. We find that the only observable effect of surface roughness is that it increases the interfacial capacitance due to the increased surface area. No evidence of the CPA impedance could be seen in our numerical data. We also studied the problem with the boundary-element method. It confirms that the numerical results are rigorously correct in the high and low frequency limit. Requiring the impedance in the intermediate frequency regime to match smoothly with these limits rule out the possibility of a CPA impedance. We suggest that the CPA impedance observed in many experiments is caused either by the adsorption and desorption of ions on the surface, or by oxidation and corrosion on the surface that changed the boundary conditions in the system.
Subject Area
Electromagnetics
Recommended Citation
Cao, Qi-Zhong, "Numerical analysis of the impedance of fractal electrodes" (1993). Doctoral Dissertations Available from Proquest. AAI9329578.
https://scholarworks.umass.edu/dissertations/AAI9329578