A first principles model is developed to describe and predict the protonic conductivity of fully hydrated Nafion membranes and its peculiar non-linear dependence on membrane thickness, potential and P-H2. The model focuses on the surface migration of protons between adjacent sulfonate groups and utilizes the Poisson-Boltzmann charge distribution around each proton combined with the basic Gamow equation of quantum mechanics for proton tunneling. It is shown that the proton tunneling distance equals the proton wavelength and that each proton surrounded by its Debye-Huckel cloud behaves as a leaking nanobattery. The model, which contains no adjustable parameters, is solved analytically and its predictions are in semiquantitative agreement with experiment, including the magnitude of the conductivity, its linear increase with membrane thickness, its exponential increase with potential and its strong dependence on P-H2. (c) 2006 Elsevier Ltd. All rights reserved.