Nonlinear curvature elasticity of nematic liquid crystals

The nonlinear elastic properties of nematic liquid crystals have acquired new interest with the recent experimental observation of bulk modulated nematic phases which are composed by achiral molecules. We extend the Oseen-Zocher-Frank's elastic theory for nematic liquid crystals by including gradients of the nematic strain tensor in the elastic deformation energy. The invariants of the elastic tensor fields, up to the fourth order in the nematic director spatial derivatives, are calculated. An alternative approach that consists in the extension of the linear elastic energy to higher powers of the nematic strain tensor, as in classical elasticity of solids, is also developed. The twist-bend nematic modulated phase is investigated by both approaches and the results are critically compared. The conical angle of the twist-bend phase is calculated as function of the elastic constants. Surface-like effects are considered. Finally, we demonstrate that a splay-bend nematic phase with small oscillations of the nematic director around an axis is prohibited.