Theoretical analysis of the magnetic Freedericksz transition in the presence of flexoelectricity and ionic contamination

We have derived an approximate analytical expression for the static director distortion of a planar nematic layer subject to a magnetic field H immediately above the critical Freedericksz transition H=H-c. The layer contains a voltage-independent density of positively and negatively singly charged ionic species that interact with the flexoelectric and dielectric polarizations which appear when the director is distorted. The analytical solution is shown to correspond closely to a full numerical calculation when H/H-c=1.01. The analytical approach allows a quantitative insight into how the mobile charge shields the polarization for different values of the elastic constants, the ionic density, the flexoelectric coefficients, and the layer thickness.