Stationary states of hybrid aligned flexoelectric nematic layers

Nematic layers, with planar anchoring conditions on one boundary plate and homeotropic anchoring conditions on the other, subjected to an external electric field may adopt planar, distorted or homeotropic structure depending on the anchoring strength and dielectric anisotropy. The influence of flexoelectric properties on stability of the planar and homeotropic structures with respect to small distortions is studied. The case of a positive sum of the flexoelectric coefficients e>0 is considered under the assumption that the anchoring strength on the planar wall, W 1, is greater than that on the homeotropic boundary, W 2. In the presence of flexoelectricity, one sign of the bias voltage U is distinguished. In the case considered, the planar and homeotropic structures are favoured if the higher electric potential is applied to the homeotropically aligning electrode. If the applied voltage has the reverse sign, the distorted structure prevails. In a flexoelectric nematic possessing positive dielectric anisotropy, Δε>0, the planar structure may exist in thicker layers and in a wider range of voltages than in the non-flexoelectric nematic. The homeotropic state is favoured by weak flexoelectricity, i.e. the range of thickness in which this state is stable at given voltage is wider than in the non-flexoelectric case. For moderate values of e, the homeotropic state appears for a range of thickness that increases with U but narrows with increasing e. When the flexoelectric properties are too strong, i.e. when e exceeds a critical value dependent on W 1/W 2, the homeotropic structure is excluded. If Δε≤0, the planar structure is realized if the thickness belongs to some range the width of which increases with U. The homeotropic state does not occur. These results, obtained for the insulating nematics, are valid quantitatively also for the case of conducting nematics only if the ion concentration is smaller than ca. 5×1018m-3. A method of measurement of e based on the above results is proposed. The examples of representative transitions between the planar, distorted and homeotropic structures obtained numerically are presented. Possible advantages of hybrid aligned flexoelectric nematic layers for display applications are discussed.