Electrical response of fractal and porous interfaces

Abstract

The electrical response of porous electrodes is calculated in several particular cases, which permit one to approach the response of a realistic model for a porous interface. The case of nonblocking surfaces and the case of the diffusion impedance of a fractal electrode are also considered. The use of Bode diagrams is shown to provide a very simple means for calculating phase angles and algebraic values for the impedance. It is demonstrated that for a blocking deterministic Sierpiński electrode the impedance presents oscillations around a constant phase angle (CPA). Various electrochemical regimes (blocking, nonblocking, and diffusive) are considered, giving rise to a variety of exponents. For blocking electrodes it is shown that at a given frequency, the power is dissipated in certain parts of the electrodes having a characteristic size which is a direct function of frequency. The fact that the response of the system is linear permits one to relate in general the dc response to the phase angle in the blocking regime and to study certain diffusive cases. It also permits one to deal with cases very common practically where the response of a flat surface would itself exhibit a CPA. In the case of a pure diffusion impedance the response is shown to be related directly to the Minkowski-Bouligand exterior dimension of the interface through the exponent (D-1)/2. This approach can be generalized to any type of irregular electrode independently of its fractal character. If both diffusion and Faradaic (electrochemical) impedance play a role, a CPA response exists for a porous electrode with an exponent equal to D-2. We discuss various regimes in which diffusion plays a role together with Faradaic, resistive, and capacitive effects. It is shown that there is in general no relation between fractal dimension and constant phase angle except in the case of diffusion. The response of irregular electrodes is shown to be related to the fractal dimension when the electrochemical regime is local.