Theoretical models for ac impedance of finite diffusion layers exhibiting low frequency dispersion

This paper is concerned with frequency dispersion in the low frequency range of electrochemical impedance measurements in thin layer cells such as electrochromic devices, conducting polymer-coated electrodes, ion exchange membranes, and in general any type of diffusion layer which exerts some hindrance to mass transport at the boundaries. New theoretical models are developed for diagnostic applications and treatment of cases in which systematic deviations from the standard models for spatially restricted diffusion impedances are found. This is done by using a generalized boundary condition in the solution of Fick’s law for a small ac perturbation. The resulting model has several satisfactory properties: (a) it generalizes in effect classical boundary conditions related to absorbing and reflecting boundaries, (b) it provides exact analytical solutions which can be tested experimentally, and (c) it provides a very simple physical picture of the origin of low frequency dispersion in film electrodes in terms of interfacial transfer functions. The properties of the generalized diffusion impedance imply that boundary effects cannot influence the impedance for frequencies in excess of the characteristic diffusion frequency ωd=D/L2. On the other hand, at low frequencies the response is a mixture of ‘volume’ and ‘boundary’ properties of the layer. Several particularized examples of blocking and non-blocking dispersive boundary conditions are studied in detail. An extended discussion is focused on a blocking interface that presents a capacitive dispersion describable as a constant phase element (CPE). Approximating expressions are derived which allow separation of boundary and volume contributions in the extreme low frequency range. This is expected to provide a powerful analytical tool for analysis in those instances where a sloped line is found in the low frequency region of the measured impedance.