Exact solution of an electro-osmotic flow problem in a cylindrical channel of polymer electrolyte membranes

The electric potential of counter-ions (protons) in an infinite cylindrical channel is presented as a solution of the Poisson-Boltzmann equation, involving a constant ion charge density along the wall. The distribution of protons is derived and used subsequently to compute the velocity profile and mass flow rate of the corresponding electro-osmotic flow, driven by an electric field. Analytical expressions are derived for all quantities, including the conductivity and water drag coefficient. This analysis relates especially to cylindrical nano-channels of polymer electrolyte membranes such as Nafion and addresses the validity of continuum models for these materials.