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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.
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Berg et al. (2009) studied this question.
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