Deformation of nematic liquid crystals in an electric field

The behaviour of liquid crystal materials used in display devices is discussed. The underlying continuum theory developed by Frank, Ericksen and Leslie for describing this behaviour is reviewed. Particular attention is paid to the approximations and extensions relevant to existing device technology areas where mathematical analysis would aid device development. To illustrate some of the special behaviour of liquid crystals and in order to demonstrate the techniques employed, the specific case of a nematic liquid crystal held between two parallel electrical conductors is considered. It has long been known that there is a critical voltage below which the internal elastic strength of the liquid crystal exceeds the electric forces and hence the system remains undeformed from its base state. This bifurcation behaviour is called the Freedericksz transition. Conventional analytic analysis of this problem normally considers a magnetic, rather than electric, field or a near-transition voltage since in these cases the electromagnetic field structure decouples from the rest of the problem. Here we consider more practical situations where the electromagnetic field interacts with the liquid crystal deformation. Assuming strong anchoring at surfaces and a one dimensional deformation, three nondimensional parameters are identified. These relate to the applied voltage, the anisotropy of the electrical permittivity of the liquid crystal, and to the anisotropy of the elastic stiffness of the liquid crystal. The analysis uses asymptotic methods to determine the solution in a numerous of different regimes defined by physically relevant limiting cases of the parameters. In particular, results are presented showing the delicate balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.