Biaxial Nematic Liquid Crystals, Periodic Deformation

In an external field, a uniaxial nematic liquid crystal can have bend Frederick's transition. For the nematic liquid crystal confined between two parallel plates, with strong homeotropic anchoring conditions on both upper and lower plates, if an external field is applied along the vertical direction of the surface plane, and its strength increases and exceeds a critical value, the director of nematics will have a uniform distortion (where the director vary along the vertical axis and remain unchanged in the surface plane). However, in experiments, people discovered that in some systems, instead of having uniform distortions, the director undergoes a periodic distortion where the polar and azimuthal angles vary along an axis in the surface plane periodically. The theoretical derivation indicates that this periodic distortion is due to small ratio of twist and bend elastic constants. When a bend distortion occurs, the relatively large magnitude of bend energy will make the director escaping into a twist distortion to lower the total distortion energy.Experiment found that in a lyotropic biaxial nematics, there exists a dynamic director periodic distortion in an external field. Considering the fact that if the time is long enough, the director will change from dynamic distortion to static distortion, therefore, a static periodic distortion should be able to occur in a biaxial nematic system. In order to find the conditions, discuss the relation between the periodic distortion threshold field and elastic constants of biaxial nematics, and derive the relation among the elastic constants for which the periodic distortion can occur, we start from Saupe's elastic energy form of biaxial nematics and use variation principle, differential equations of directors are obtained. By assuming trial solutions, and substituting these solutions into boundary conditions, we obtained a relation among external field, periodic distortion wave number and elastic constants. The numerical calculations give the threshold fields of periodic distortion, and from these we discuss the elastic constants conditions on which the periodic distortion can occur. An analytic method is also used to derive the elastic constants condition mentioned above. We gave the area of elastic constants where the periodic distortion can occur, the area where the uniform distortion will occur, and the boundary between these two.