Continuum modelling of hybrid-aligned nematic liquid crystal cells: optical response and flexoelectricity-induced voltage shift

A continuum model is employed to study systematically the optical response of hybrid-aligned nematic (HAN) liquid crystal cells under the application of an external electric field. The influence of the flexoelectric effect is discussed for a large range of anchoring strengths at the homeotropic alignment layer. It is shown that the optical response of HAN cells is governed by a complicated interplay between the flexoelectric coefficient and homeotropic anchoring strength. In particular, the calculations reveal that, for weak homeotropic anchoring, the flexoelectric effect leads to a non-linear voltage shift of the optical transmittance as a function of flexoelectric coefficient, and gives rise to an asymmetry in the transmittance-voltage curve. Finally, a comparison of the continuum-model simulations with recent experimental observations indicates that both the flexoelectric coefficient and the anchoring strength of the nematic liquid crystal MBBA on a homeotropic polyimide alignment layer are significantly lower than previously reported.     

Continuum modelling of hybrid-aligned nematic liquid crystal cells: optical response and flexoelectricity-induced voltage shift

A continuum model is employed to study systematically the optical response of hybrid-aligned nematic (HAN) liquid crystal cells under the application of an external electric field. The influence of the flexoelectric effect is discussed for a large range of anchoring strengths at the homeotropic alignment layer. It is shown that the optical response of HAN cells is governed by a complicated interplay between the flexoelectric coefficient and homeotropic anchoring strength. In particular, the calculations reveal that, for weak homeotropic anchoring, the flexoelectric effect leads to a non-linear voltage shift of the optical transmittance as a function of flexoelectric coefficient, and gives rise to an asymmetry in the transmittance–voltage curve. Finally, a comparison of the continuum-model simulations with recent experimental observations indicates that both the flexoelectric coefficient and the anchoring strength of the nematic liquid crystal MBBA on a homeotropic polyimide alignment layer are significantly lower than previously reported.     

Flexoelectro-optic effect in a hybrid nematic liquid crystal cell

W e analyse the influence of charged impurities and flexoelectric polarization on the optical transmission of a hybrid aligned nematic liquid crystal cell. The theoretical results obtained within the framework of the Poisson-Boltzmann equation and Frank elastic theory are compared with the observed optical response [N. V. Madhusudana and G. Durand,J. Phys.Lett. 46, L-195 (1985)]. We show that impurities can be very important for the behaviour of the system in the low field regime where the flexoelectric effect is relevant, and we determine the flexoelectric coefficient, the anchoring strength, and the concentration of impurities in the sample previously studied by Madhusudana and Durand.     

Detection of flexoelectric effect from 4-heptyloxy-4'-cyanobiphenyl monolayers at an air-water interface by Maxwell displacement current and optical second harmonic generation

The flexoelectric effect of 4-heptyloxy-4'-cyanobiphenyl (7OCB) monolayers at the air-water interface is studied by Maxwell displacement current (MDC) and optical second harmonic generation measurements. Though MDC was expected to increase during the compression of 7OCB monolayers in L2L2' phase from the MDC theory developed previously, decrease of MDC was detected in these phases. This abnormalous phenomenon is found to be due to the quench of flexoelectric effect by the flow orientation of monolayers.     

Optical measurement of flexoelectric polarisation change in liquid crystals doped with bent-core molecules using hybrid-aligned structure

We proposed an optical measurement method for determination of flexoelectric polarisation change in liquid crystals (LCs), which can be induced in highly distorted LC geometries. A hybrid-aligned nematic LC (NLC) mode was introduced to evaluate the direction and magnitude of the flexoelectric polarisation. We measured the DC offset amounts for equivalent brightness levels between forward and reverse bias vertical electric fields to estimate the sign and magnitude of e_s?e_b of flexoelectric coefficients. Additionally, the optical incident angle (α_max) for the maximum effective birefringence was investigated to predict the depth distribution of the LC director affecting the magnitude of the flexoelectric polarisation. The relationship between the variations of the DC offset and α_max by the flexoelectric polarisation changes was examined using the NLC mixtures doped with three selected bent-core LCs.     

Optical response of a nematic liquid crystal cell at the splay – bend transition: a model and dynamic simulation

We study the dynamical optical response of a nematic liquid crystal cell that undergoes the splay–bend transition after applying a voltage across the cell. We formulate a simplified model that takes into account both the flexoelectric coupling and the surface rotational viscosity. The dynamic equations of the model are solved numerically to calculate the temporal evolution of the director profile and of the transmittance. We evaluate the response time as a function of a number of parameters, such as dielectric and elastic anisotropies, asymmetry of the surface pretilt angles, anchoring energy, surface rotational viscosity and flexoelectricity.     

Optical response of a nematic liquid crystal cell at the splay - bend transition: a model and dynamic simulation

We study the dynamical optical response of a nematic liquid crystal cell that undergoes the splay-bend transition after applying a voltage across the cell. We formulate a simplified model that takes into account both the flexoelectric coupling and the surface rotational viscosity. The dynamic equations of the model are solved numerically to calculate the temporal evolution of the director profile and of the transmittance. We evaluate the response time as a function of a number of parameters, such as dielectric and elastic anisotropies, asymmetry of the surface pretilt angles, anchoring energy, surface rotational viscosity and flexoelectricity.     

Gradient flexoelectric switching response in a nematic phenyl benzoate

We report on the inverse flexoelectric effects observed in a nematic liquid crystal with a small positive dielectric anisotropy subject to static and very low frequency (<1 Hz) a.c. fields. The Bobylev–Pikin flexobands appear at a temperature‐dependent d.c. threshold. Under square wave excitation, a new type of transient optical response occurs soon after each polarity reversal, and we ascribe it to the gradient flexoelectric distortion explicable on the basis of the presence of intrinsic double layers. This instability is characterized by a threshold voltage that decreases with temperature. Its maximum amplitude increases linearly with voltage close to threshold, and occurs after polarity reversal at a time τ_m that scales inversely as the voltage; τ_m decreases exponentially with frequency and temperature. After each polarity change, the ionic current following the charging current decreases almost exponentially to a non‐zero value; the residual current increases monotonically with the applied bias.     

The flexoelectro-optic effect in cholesterics

The flexoelectro-optic effect in short-pitch cholesterics [1] is analysed in terms of applied electric field strengths and material parameters such as the two flexoelectric coefficients es and eb and the three elastic constants. Starting from the free energy density of the uniformly lying-helix (ULH) configuration, including the flexoelectric polarization term, the equation describing the field-induced tilt angle of the bulk optic axis is derived. It is convenient to introduce the flexoelectric 'anisotropy' Deltae as the difference between the splay and bend flexoelectric coefficients, hence defined by Deltae es eb. Our results then show that Deltae is the essential material parameter controlling the sign and magnitude of the electrically induced tilt. In the region of linear approximation, the tilt is proportional to Deltae and to the electric field E, and inversely proportional to the helical wave vector k, as well as the effective elastic constant (K11 K33). The individual values of the elastic constants K11, K22 and K33 do have an influence on the magnitude of the effect, but not on its linearity. The Deltae dependence conforms in the simplest way to the physical requirement that the flexoelectro-optic effect must be particularly pronounced in the case that es and eb are of opposite sign.     

The influence of the flexoelectric effect on the electrohydrodynamic instability in nematics

A rigorous three-dimensional linear analysis of the electrohydrodynamic instability in nematic liquid crystals including the flexoelectric effect is presented for the case of an applied d.c. voltage. The flexoelectric effect leads to an appreciable reduction of the threshold and to the appearance of oblique rolls at threshold for the standard material MBBA. We discuss the influence of a magnetic field and test several approximations against the rigorous results     

Molecular shape and flexoelectricity

Monte Carlo simulations were performed of systems of wedge-shaped objects formed from Gay-Berne ellipsoids joined to Lennard-Jones spheres. We studied two different wedge shapes, one more asymmetric than the other. The bend and splay flexoelectric coefficients were measured in the isotropic and smectic phases using linear response theory, and found to be negligibly small in the isotropic phase. We found a close connection between the properties of the intermolecular potential and the flexoelectric coefficients measured in the smectic phase. In particular, negligible bend coefficients were found for both shapes and a larger magnitude of the splay coefficient for the more prominent wedge, in accord with Meyer's original mechanism for flexoelectricity. The less prominent wedge produced a splay flexoelectric coefficient with the opposite sign due to the attractive tail of the intermolecular potential and the relative narrowness of the molecular head.     

Molecular shape and flexoelectricity

Monte Carlo simulations were performed of systems of wedge-shaped objects formed from Gay-Berne ellipsoids joined to Lennard-Jones spheres. We studied two different wedge shapes, one more asymmetric than the other. The bend and splay flexoelectric coefficients were measured in the isotropic and smectic phases using linear response theory, and found to be negligibly small in the isotropic phase. We found a close connection between the properties of the intermolecular potential and the flexoelectric coefficients measured in the smectic phase. In particular, negligible bend coefficients were found for both shapes and a larger magnitude of the splay coefficient for the more prominent wedge, in accord with Meyer's original mechanism for flexoelectricity. The less prominent wedge produced a splay flexoelectric coefficient with the opposite sign due to the attractive tail of the intermolecular potential and the relative narrowness of the molecular head.     

Threshold flexoelectric deformations in homeotropic nematic layers

Elastic deformations of nematic liquid crystal layers subjected to a d.c. electric field were studied numerically. The flexoelectric properties of the nematic material and the presence of ionic space charge were taken into account. Homeotropic alignment with finite surface anchoring strength was assumed. The director orientation and the electric potential distribution were calculated; the space charge density was also determined. It was found that the threshold voltage strongly depended on the parameters of the system. In particular, a threshold as low as a few tenths of a volt occurred under suitable circumstances. In the case of a negative dielectric anisotropy, Δ ε, such low values of the threshold voltage existed when the ion concentration was sufficiently high, and given sufficiently large magnitudes of the flexoelectric coefficients and a sufficiently small anchoring energy. If the ion concentration was low or if the flexoelectric coefficients were small or if the surface anchoring was strong, the threshold was equal to several volts. In the case of positive dielectric anisotropy, the threshold amounted to several tenths of a volt for a weakly anisotropic and highly conductive material. If the dielectric anisotropy was sufficiently high or if the ion concentration was sufficiently low, the threshold voltage increased with Δ ε and reached tens of volts. These results can be explained as the effect of the inhomogeneous electric field arising in the vicinity of the surfaces, due to the ionic space charge redistributed by the external voltage. They are qualitatively consistent with earlier experiments which show the effect of the ion concentration on the elastic deformations in flexoelectric nematics. They correspond also with theoretical results concerning the effect of the electric field produced by the surface polarization or by the adsorption of ions.     

Polarized states of hybrid aligned nematic layers

Electric polarization arising in hybrid aligned nematic liquid crystal layers with rigid boundary conditions is studied numerically by solving the torques equation and Poisson equation. Three phenomena that give rise to the polarization are taken into account: flexoelectricity, surface polarization and adsorption of ions. The director orientation within the layer, as well as the distribution of electric potential and space charge density are calculated for layers deformed by an external magnetic field. The role of the ionic space charge is investigated. For a particular set of parameters of a model substance, the voltage arising between the layer surfaces varies from 10^-1 V (in an extremely pure nematic) to 10^-3 V (in material with a typical ion concentration). The surface polarization yields an additional voltage (of the order 10^-2 V) nearly independent of the ion concentration. The effect of simultaneous flexoelectric polarization and ion adsorption is evidently different from a linear superposition of their separate contributions. The flexoelectric polarization leads to partial separation of ions of opposite signs. In the case of positive flexoelectric coefficients, a thin sublayer of positive charge arises at the planar-orienting boundary plate. The negative charge is displaced towards the homeotropically aligning plate. The magnitude of this effect increases with the magnetic field. The surface phenomena introduce additional subsurface charges.     

On the flexoelectric effect in nematics

Flexoelectricity is a general and fundamental phenomenon in liquid crystals. It describes the linear coupling between an applied electric field and gradients in the director field. Whereas flexoelectricity has for decades been regarded as only of academic interest, we think it is time to point out its considerable application potential, for instance in the case of the flexoelectrooptic effect, and to urge a revival of interest in the subject. As a result of long-time neglect, published data on flexoelectric coefficients are scarce and inconsistent, even with regard to the sign of the reported effect. In this paper we critically review the possible definitions of flexocoefficients in order to propose an international standard. We point out that the absence of such a standard obstructs the understanding of the physical basis, microscopically as well as macroscopically, of the effect, and leads to the introduction of nonsensical concepts like 'flexoelectric anisotropy'. Based on the only natural convention, we finally propose a simple method for measuring sign and magnitude of the effective flexoelectric coefficient which is the control parameter in electro-optic effects.     

Influence of anchoring at a nematic cell surface on threshold spatially periodic reorientation of a director

We have analysed the influence of surface director anchoring in a planar flexoelectric nematic cell on the threshold spatially periodic reorientation of the director in an external dc electric field. By minimizing the free energy of the nematic cell we obtained the equations for a director and numerically solved them in the one elastic constant approximation. The dependences of the threshold electric field and the spatial period of director structure on the azimuthal and polar anchoring energy, as well as the flexoelectric parameters, are determined. It is shown that the domain of the flexoelectric parameter values, at which the spatially periodic reorientation of a director takes place, increases with decreasing azimuthal anchoring energy and increasing polar anchoring energy.     

Determination of the difference of flexoelectric coefficients in a nematic liquid crystal using a conoscopic technique

The conoscopic images of twisted nematic liquid crystal devices filled with E7 are analysed under the application of in-plane electric fields. The differences observed between the images for positive and negative applied fields are attributed to the flexoelectric effect. By comparison of the conoscopic images with theoretical predictions made using an extended Jones technique, the sign and magnitude of the difference between the splay and bend flexoelectric coefficients is determined for E7.     

Second-harmonic generation and the influence of flexoelectricity in the nematic phases of bent-core oxadiazoles

Second-harmonic generation (SHG) in the nematic phase of bent-core oxadiazole-based liquid crystals (LCs) was studied and compared to that for the rod-like compound 4-cyano-4?-n-octylbiphenyl (8CB). Weak, isotropically scattered second-harmonic (SH) light was observed for all materials, consistent with SHG by nematic director fluctuations. The SH intensity produced by the bent-core materials was found to be up to ~ 3.4 times that of 8CB. We discuss this result in terms of the dependence of SH intensity on temperature, elastic constants and flexoelectric coefficients. We have calculated the latter by using a molecular field approach with atomistic modelling, thus demonstrating how molecular parameters contribute to the flexoelectric coefficients and illustrating the potential of this method for predicting the flexoelectric behaviour of bent-core LCs. We show that the increased SH signal in the bent-core compounds is partly due to their nematic phases being at a much higher temperature, and also potentially due to them having greater flexoelectric coefficients, up to ~1.5 times those of 8CB. These estimates are consistent with reports of increased flexoelectric coefficients in bent-core compounds in comparison to rod-like compounds.     

Rod-shaped dopants for flexoelectric nematic mixtures

We report the synthesis and liquid crystalline behaviour of two series of para-substituted terphenyls as dopants with a rigid rod-like shape, rather than a wedge-, pear- or banana-shape, for guest–host nematic mixtures with flexoelectric properties. One series of liquid crystalline dopants is of low-to-strongly negative dielectric anisotropy and the other is of low-to-strongly positive dielectric anisotropy. The usefulness of apolar and polar rod-like dopants as components of flexoelectric nematic mixtures of positive dielectric anisotropy for use in LCDs is investigated in general and the dependence of the flexoelectric properties of the doped nematic mixtures on the polarity of the dopants is studied in particular. The correlation between the concentration of the dopant and the magnitude of the flexoelastic ratio of several guest–host nematic mixtures is investigated.     

The influence of chirality on the difference in flexoelectric coefficients investigated in uniform lying helix,Grandjean and twisted nematic structures

Measurements of the difference in flexoelectric coefficients (e ₁ – e ₃), using the sign convention as originally defined by Meyer, are reported from three experiments employing the flexoelectro-optic effect in different geometries. The uniform lying helix (ULH) structure is used to measure the tilt angle of the liquid crystal director with respect to the helix axis for an applied electric field, in order to infer a value for (e ₁ – e ₃). Alternatively, measurements of the flexoelectric difference can be made by considering the transmission through a device with an in-plane electric field aligned in either the Grandjean structure for highly chiral materials, or a twisted nematic (TN) structure for largely achiral materials. The results from the Grandjean and ULH structures show the equivalence of the measurement techniques with helix axis either perpendicular or parallel to the substrates. Further comparison of these results with the measurement from the achiral TN device shows that the difference in flexoelectric coefficients displays no dependence on chirality, demonstrating that flexoelectricity is purely associated with splay and bend director deformations, as expected from symmetry considerations.