Ions and nematic surface energy: beyond the exponential approximation for the electric field of ionic origin

We present a general model to describe the influence of the ionic adsorption on the anisotropic part of the surface energy of a nematic liquid crystal in contact with a substrate. We show that in the limit of small adsorption energy, the exponential approximation for the electric field of ionic origin works well. In this limit, the dielectric and flexoelectric contributions to the surface energy are quadratic and linear on the density of adsorbed ions, respectively. In the opposite limit of large adsorption energy, the exponential approximation for the electric field does not work, and the two contributions to the surface energy are both found to depend linearly on the surface density of adsorbed charges. Approximated formulas reported in literature are derived from our general equations as particular cases, and their limits discussed. An expression for the surface polarization in nematic liquid crystal due to the ionic adsorption is also deduced. Our analysis is performed in the framework of the Poisson-Boltzmann theory, where dimensionless ions are treated within a mean field approach. Possible extensions of our model are indicated.