Statistical theory of the flexoelectric polarization for the uniaxial nematic phase

A statistical theory for the dipole flexoelectric (FE) polarization in liquid crystals is derived in the thermodynamic limit at small distortions and a small density. General microscopic expressions for the FE coefficients are rederived in the case of the uniaxial nematic phase. The expressions involve the one-particle distribution function and the potential energy of two-body short-range interactions. It is shown that, for wedge-shaped polar molecules, the FE coefficients are proportional to the longitudinal component of the molecule dipole moment. The FE coefficients include a term linear in < P-2 > and a quadratic form in < P-j > (j-even). The theory is used to calculate the temperature dependence of the elastic constants and the FE coefficients for a system of rigid molecules similar to cones.