On a possible physical origin of the constant phase element

Despite the numerous use of the constant phase element (CPE) in the modeling of the impedance characteristics of the electrochemical systems, the physical reasoning of this non-intuitive element is not clear. In this paper, the CPE impedance is analytically calculated using the anomalous diffusion theory. The fractional calculus and the anomalous diffusion are first reviewed. It is shown that the chance inequality in the random walk in a porous media can result in an anomalous diffusion. Then, the Boltzmann distribution of the particles used in the Gouy-Chapman theory of the double layer is modified to determine the double layer capacitance. Finally, the impedance of the double layer is calculated which is equivalent to the CPE impedance reported in literature. It is shown that this novel theory covers the interpretations previously presented for the CPE and its relation to the fractal dimension and the pore size distribution of the porous media.