Derivation of the diffusion impedance of multi-layer cylinders. Application to the electrochemical permeation of hydrogen through Pd and PdAg hollow cylinders

The purpose of this research paper is to report on the diffusion impedance of multi-layer cylinders. Analytical expressions of the diffusion impedance in multi-layer solid cylinders (sorption/desorption experiments) and through multi-layer hollow cylinders (permeation experiments) have been obtained by integration of Fick's diffusion equation in cylindrical coordinates. The resulting impedances are expressed as a function of the characteristics of individual active layers (layer thickness, diffusion coefficient and solubility of diffusing species in each layer). These impedance expressions can be used to characterize a large variety of experimental systems. In this work, model impedances have been used to fit experimental impedances of hydrogen permeation through Pd and Pd77Ag23/Pd mono- and bi-layer hollow cylinders, yielding the values of microscopic rate parameters (surface charge transfer resistances, diffusion coefficient). Analytical expressions of low frequency limits (stationary conditions of flow) have been derived. Practical conditions required for the separation of the contributions of the different layers are analyzed. Some limitations and perspectives offered by this work are discussed.