An analytical model for the conductivity of polymeric sulfonated membranes

In this work, we present a new approach to model the conductivity of polymeric sulfonated membranes. In the first part, we develop a model for the proton conductivity in a bulk acid solution. The model describes the vehicular and structural diffusion mechanisms at different proton concentrations as a function of four parameters with well defined physical meanings. These parameters are adjusted by fitting the experimental proton conductivity in a hydrochloric acid solution. In the second part, the effects of the presence of the porous membrane on the proton conductivity are added to the model. As a result, the model predicts the membrane conductivity as a function of its hydration level and porous structure. In the absence of experimental data on the membrane porous structure, the model can be used to predict the membrane porous structure once membrane conductivities are experimentally known. In this scope, in the third part, we provide a simple analytical description of the porous membrane based on the percolation theory depending on two structure parameters. A side result of the conductivity model is to provide analytical expressions for the drag coefficient, pore diameters and water permeability. All obtained results compare very well with experimental data.