Fréedericksz Transitions In Nematic Liquid Crystal Cells Based Upon Spatially Anisotropic Potential

The essential task of liquid crystal statistical physics is studying macroscopic properties of liquid crystal based on molecular interaction. The molecule of liquid crystal is organic macromolecule, so we usually describe the interaction with pair potential model, such as Lebwohl-Lasher model (L-L model). However, this model only describes spatially isotropic interaction.Recently, Gruhn and Hess proposed a spatially anisotropic pair potential, which approximately reproduced the elastic free energy density. Then Romano and Luckhurst obtained two pair potential models, i.e. model I and model II. In this paper, Fréedericksz transitions of nematic liquid crystal cells are studied based upon the molecular pair potential model which is spatially anisotropic and dependent on elastic constants of liquid crystals.The perfect nematic order is assumed in the theoretical treatment, which means the orientation of the molecular long axis coincides with the director of liquid crystal and the total free energy equals to the total interaction energy. Three kinds of the basic Fréedericksz transition are investigated by means of the Fourier series expansion of variables describing the orientation of the director, upon which threshold magnetic field of TN liquid crystal cells is mainly studied. Tilt and azimuth angels of the director are expanded respectively through Fourier series and then threshold magnetic field of TN cells can be ascertained by way of the second derivative matrix. The magnetic field depends on the total twist angle and elastic constants of liquid crystals,which is compared with that of continuum theory. The equilibrium equation, which is satisfied by the director of liquid crystal, can be obtained by the way that the systematic free energy should be minimized with the state of equilibrium. Based upon the equilibrium equation, numerical calculations are made for the director profiles in a TN liquid crystal cell under the action of magnetic field. Numerical results verified the equilibrium equation satisfied by the director of liquid crystal. It is further proved that one of two mapping schemes from the elastic energy to the pair potential proposed recently in the literature, i.e. model I, can give the result consistent with the continuum theory.