Flexoelectricity and pattern formation in nematic liquid crystals.

We present in this paper a detailed analysis of the flexoelectric instability of a planar nematic layer in the presence of an alternating electric field (frequency ω), which leads to stripe patterns (flexodomains) in the plane of the layer. This equilibrium transition is governed by the free energy of the nematic, which describes the elasticity with respect to the orientational degrees of freedom supplemented by an electric part. Surprisingly the limit ω→0 is highly singular. In distinct contrast to the dc case, where the patterns are stationary and time independent, they appear at finite, small ω periodically in time as sudden bursts. Flexodomains are in competition with the intensively studied electrohydrodynamic instability in nematics, which presents a nonequilibrium dissipative transition. It will be demonstrated that ω is a very convenient control parameter to tune between flexodomains and convection patterns, which are clearly distinguished by the orientation of their stripes.