Flexoelectricity and pattern formation in nematic liquid crystals

Abstract

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 respects 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 electro-hydrodynamic instability in nematics, which presents a non-equilibrium 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.

Suppression of spatially periodic patterns by dc voltage

“…In contrast to that, in the experiments a substantial increase of the ac threshold upon dc bias was detected (for both conductive and dielectric EC), similarly to the recent findings on the nematic Phase 5 [22]. The recent theoretical analysis of flexodomains [6] has pointed out that in case of pure ac driving the equations have two solutions with different time symmetries: the conductive mode implies stationary modulation of the out-of-plane director component, while for the dielectric mode it oscillates with the ac frequency. The threshold voltages and critical wave numbers of the two solutions are different; which of them has a lower threshold depends on the flexoelectric coefficients, on the dielectric anisotropy, on the elastic constants and on the frequency of driving.…”

“…As already mentioned in Section I, by now the theoretical description of electroconvection and of flexodomains (the extended SM) has been worked out for pure ac or dc drivings [6,10]. Since the patterns emerge continuously from the initial state at onset (corresponding to a forward bifurcation), the onset characteristics (U c and q c ) can be obtained from a linear stability analysis of the underlying nemato-hydrodynamic equations.…”

“…The pattern has been interpreted as a flexoelectricity induced periodic director modulation [5]: it appears when the free energy reduction due to the flexoelectric polarization overcomes the elastic and dielectric increase of the free energy. This may only occur if certain relations between the material parameters of the nematic fulfils [6].…”

“…By now theoretical description has been developed to describe the patterns discussed above, which is known as the standard model of electroconvection including flexoelectricity [6,10] (we will refer to it as the extended SM). The linear stability analysis of the extended SM provides the threshold U c (f ), the critical wave vector q c (f ) and the spatio-temporal dependence of the material flow v(r, t), the electric potential, and the director n(r, t) at the onset, in a good agreement with experiments [11].…”

“…For f > f c (in the dielectric mode), in contrast, n and v oscillate with the ac frequency in leading order, while ̺ e is stationary. It should be noted that due to these differences in time symmetries there is no smooth transition from the ac f → 0 limit to the dc case [6,12].…”