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

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The method presented here is based on the two-phase model of a porous system 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 resistance. The equivalent circuit for this system is the transmission line model. The method applies to systems with parameters which are not constant across the layer. The layer is divided into a number of slabs and in each slab 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 case a 4×4 matrix. The potential propagation in the whole layer is determined by the product matrix. The impedance for both a layer coating a metallic current collector and a porous membrane embedded in the electrolyte (or the porous layer with electrolyte-filled pores in between two metallic current collectors) can be expressed by the elements of the product matrix. The matrix is reduced to a 2×2-form if one of the resistivities is negligible. In this case for a system of two homogeneous sublayers an analytical formulation is given. The method is applied to a system with an interconnection consisting of double layer capacity, charge transfer resistance and its hindrance by finite diffusion (applicable to polymers). Here the inhomogeneity gradients of the resistivities are considered. It is demonstrated that they can result in significant qualitative modifications of the impedance. This concerns especially the low frequency pseudo-capacitive behaviour which is transformed into a dependence resembling the well known empirical description by constant phase elements often used to interpolate experimental data.