Theory of the electrochemical impedance of anomalous diffusion

This paper addresses the electrochemical impedance of diffusion in a spatially restricted layer. A physically grounded framework is provided for the behavior Z(i omega) proportional to (i omega)(-beta /2) (0 < <beta> < 2), thus generalising the Warburg impedance (<beta> = 1). The analysis starts from the notion of anomalous diffusion, which is characterized by a mean squared displacement of the diffusing particles that has a power law dependence on time <r(2)> <proportional to> t(beta). Using a theoretical approach to anomalous diffusion that employs fractional calculus, several models are presented. In the first model, the continuity equation is generalised to a situation in which the number of diffusing particles is not conserved. In the second model the constitutive equation is derived from the stochastic scheme of a continuous time random walk. And in the third, the generalised constitutive equation can be interpreted within a non-local transport theory as establishing a relationship of the flux to the previous history of the concentration through a power-law behaving memory kernel. This third model is also related to diffusion in a fractal geometry. The electrochemical impedance is studied for each of these models, and the representation in terms of transmission lines is established. The main finding is that, while models with quite different non-trivial diffusion mechanisms behave similarly in a semi-infinite situation, the consideration of the effect of the boundaries gives rise to neatly different impedance spectra. (C) 2001 Elsevier Science B.V. All rights reserved.