Inverse-Cantor-bar model for the ac response of a rough interface

It has been previously shown that models based on the Cantor-bar fractal for a rough interface between a metal and an electrolyte display constant-phase-angle (CPA) behavior. The exponent $\eta$ of the frequency dependence of the CPA element satisfies the relation $\eta =3-{\overline{d}}_s$ where ${\overline{d}}_s$ is the fractal dimension of the interface. In this paper the generality of the $\eta =3-{\overline{d}}_s$ relation is tested by examining the inverse-Cantor-bar structure defined by interchanging the metal and electrolyte. Despite the fact that the electrical problem and the corresponding mathematics are vastly different for the inverse structure, $\eta$ remains unchanged. A new model for rough interfaces, called the porous electrode, consisting of a fractal distribution of transmission lines is also described.