The formula of anchoring energy for a nematic liquid crystal

Abstract

The interface energy for a nematic liquid crystal (NLC) is considered as the sum of potential energy between LC molecules and molecules of the substrate surface, and a formula for anchoring energy is derived by elementary principles. The anchoring energy for a NLC should have two terms, the first term is the same as the Rapini–Papoular expression, the second is related to the normal of interface and resultes from the biaxial property of a NLC induced by interface. Hence there are two anchoring coefficients, W ₁ and W ₂. We demonstrate that W ₁ is equal to the tilt angle strength A_θ , and W ₂ corresponds to the difference between A_θ and the azimuthal strength A_ϕ . Thus A_θ –A_ϕ is due to the biaxial property of the NLC near the interface. Applying this formula to the twisted NLC cell, we discuss the threshold and saturation field, as well as the maximal tilt angel θ _m with respect to A_θ /A_ϕ . Previously proposed formulae are discussed from our point view.

Notes

^†If P is not small enough, then. P is function of tilt angle θ of n, it can be expressed by P=δ ₁ cos² θ+δ ₂ cos⁴ θ+…≈δ ₁ cos² θ by considering invariance of n=−n and for, P=0. Then for one­dimensional planar anchoring