Diffuse Charge Effects in Fuel Cell Membranes

It is commonly assumed that electrolyte membranes in fuel cells are electrically neutral, except in unsteady situations, when the double-layer capacitance is heuristically included in equivalent circuit calculations. Indeed, the standard model for electron transfer kinetics at the membrane/electrode interface is the Butler-Volmer equation, where the interfacial overpotential is based on the total potential difference between the electrode and bulk electrolyte. Here, we develop an analytical theory for a solid-state proton-conducting membrane that accounts for diffuse charge in the electrostatic polarization layers and illustrate its use for a steady-state hydrogen concentration cell. The theory predicts that the total membrane charge is nonzero, except at a certain hydrogen pressure, which is a thermodynamic constant of the fuel cell membrane. Diffuse layer polarization introduces the Frumkin correction for reaction rates, where the overpotential is based on the potential difference across only the compact (Stern) part of the polarization layer. In the Helmholtz limit of a relatively thin diffuse layer, we recover well-known results for a neutral membrane; otherwise, we predict significant effects of diffuse charge on the electron-transfer rate. Our analysis also takes into account the excluded volume of solvated protons, moving in a uniform charge density of fixed anions.