Transfer matrix method for the electrochemical impedance of inhomogeneous porous electrodes and membranes

The method presented here is based on the two-phase model of a porous syste m with two continuous subsystems, electrons in the porous material and ions in the pore electrolyte. Both are continuously interconnected via the pore surfaces e.g. by the double layer capacity and/or the charge transfer resi stance. The equivalent circuit for this system is the transmission line mod el. The method applies to systems with parameters which are not constant ac ross the layer. The layer is divided into a number of slabs and in each sla b all parameters are replaced by their mean values. The potentials and the currents of two adjacent slabs are connected by a matrix, in the general ca se a 4 x 4 matrix. The potential propagation in the whole layer is determin ed by the product matrix. The impedance for both a layer coating a metallic current collector and a porous membrane embedded in the electrolyte (or th e porous layer with electrolyte-filled pores in between two metallic curren t collectors) can be expressed by the elements of the product matrix. The m atrix is reduced to a 2 x 2-form if one of the resistivities is negligible. In this case for a system of two homogeneous sublayers an analytical formu lation is given. The method is applied to a system with an interconnection consisting of double layer capacity, charge transfer resistance and its hin drance by finite diffusion (applicable to polymers). Here the inhomogeneity gradients of the resistivities are considered. It is demonstrated that the y can result in significant qualitative modifications of the impedance. Thi s concerns especially the low frequency pseudo-capacitive behaviour which i s transformed into a dependence resembling the well known empirical descrip tion by constant phase elements often used to interpolate experimental data . (C) 1999 Elsevier Science S.A. All rights reserved.