Energy Barrier for an Ion Crossing an Intra-Membrane Channel
  We present a simple approximate analytical estimate for self-energy of a charge in the middle of cylindrical channel of a high permittivity epsilon_1 in a media of a low permittivity epsilon_2 (for the cases of infinitely long and comparatively short channels) and show that this estimate is in a good quantitative agreement with exact solution of Poisson equation. Further, using these estimates, we explain the observed a lower conductivity, caused by an increased the self-free-energy for ions, whose diameter is by ~1 angstrom less than that of the channel (as compared to ions, whose diameter is equal to that of the channel). 
Alexei V. Finkelstein (1), Dmitry N. Ivankov (1), Alexander M. Dykhne   (2) ((1) Institute of Protein Research, Russian Acad...