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.     

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.     

Investigation of flexoelectric properties of a palladium-containing nematic liquid crystal, Azpac, and its mixtures with MBBA

Flexoelectricity of pure Azpac (an HOAB-palladium complex) was studied using planar nematic layers under an in-plane electric field. Longitudinal domains were observed with a period inversely proportional to the applied d.c. electric field. These domains were considered as a first experiment manifestation of the theoretical prediction of R. B. Meyer (1969) and their study has permitted the evaluation of the difference in flexo-coefficientse_1z - e_3x. Their appearance can also be followed for the non-complexed HOAB, using instant video-microscopy frames.

Mixtures of Azpac, up to 10wt%, and MBBA were oriented homeotropically and band flexoelectric deformations were observed, both in d.c. and in a.c. (1 to 1000 Hz) electric fields normal to the director. The dynamics of director reorientation were studied by the method of flexoelectric light modulation. A system comprising a He-Ne laser and a lock-in amplifier interfaced by a PC was developed. By operating this in a frequency sweep regime, viscoelastic spectra of director dynamics were recorded. These spectra were excited using a linear flexoelectric coupling mechanism. Breaks in the spectra were observed in the range 200 to 600 Hz, indicating a cross-over from bulk to surface dissipation of energy. For the first time, a surface viscosity of 2.6 × 10^-8Jsm^-2 was determined for MBBA homeotropically anchored on a DMOAP-coated glass surface.

In concentrations at low as 2.5 wt%, Azpac was found to cancel the bend flexo-coefficient of MBBA and at higher concentrations, a steep rise in the flexo-coefficient of the mixture was observed, but with an opposite sign. Thus, the application of Azpac as an effective additive for adjustment of the value and sign of the flexo-coefficient in flexoelectro-optic displays or light modulators could be suggested.     

Converse flexoelectric effect in homeotropic nematic layers with asymmetric surface anchoring

The dc electric field-induced deformations of conducting flexoelectric nematic layers were studied numerically. Asymmetric boundary conditions expressed by different anchoring strengths on the limiting surfaces were assumed. Nematic material was characterised by negative dielectric anisotropy. Both signs of the sum of flexoelectric coefficients were taken into account. The electric properties of the layer were described in terms of a weak electrolyte model. Mobility of cations was assumed to be one order of magnitude lower than that of anions. Quasi-blocking electrode contacts were assumed. The threshold voltages for the deformations and the director distributions in the deformed layers were calculated for low, moderate and high ion concentrations. The director distributions were also determined. The results show that asymmetry caused by difference between the anchoring strengths and by difference between mobilities of anions and cations lead to two threshold values for a given layer corresponding to two polarities of the bias voltage. Additionally, the values of both thresholds depend on the sign of the flexoelectric parameter. In every case under consideration, the threshold is significantly lowered when the ion concentration is high.     

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.     

Threshold voltage for purely flexoelectric deformations of conducting homeotropic nematic layers

Elastic deformations induced by an electric field in homeotropic nematic layers with finite anchoring energy were studied numerically. A nematic material possessing flexoelectric properties and characterized by a positive dielectric anisotropy was considered. The ionic space charge and the ion transport across the layer were taken into account. The director orientation, the electric field strength and the ion concentrations were calculated as functions of the coordinate normal to the layer. The calculations show that the electric field distribution, which determines the form of the deformations, is influenced by the ionic current and therefore depends on the ionic content and on the properties of the electrodes. Several types of deformations were distinguished. When the electrode contacts are well conducting or when the ionic content is low, the threshold voltage is very close to the value U _f valid for an insulating nematic. When the electrodes are poorly conducting or blocking at high ion concentration, the threshold voltage decreases much below U _f. At moderate ion concentrations, i.e. between 10¹⁹ and 10²⁰ m^?3, two different behaviours were found depending on the sign of the sum of flexoelectric coefficients e ₁₁+e ₃₃. In the case of e ₁₁+e ₃₃<0, the threshold voltage decreases with the ionic content; in the case of e ₁₁+e ₃₃>0, the deformations occur in two separate voltage regimes. They arise above a certain threshold voltage, disappear at some higher voltage and reappear at an even higher threshold.     

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.     

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.     

Dielectric enhancement of chiral flexoelectro-optic switching

The chiral flexoelectro-optic effect is a candidate for realising sub-millisecond electro-optic technologies. Here, we discuss how flexoelectro-optic switching in cholesteric structures can be enhanced by using the dielectric coupling of the field to materials with negative dielectric anisotropy. An experiment is described that can measure the dielectric effect on flexoelectric switching, and the technique is applied to a chiral nematic mixture E7+3.5%R5011. We explore numerically the optimal liquid crystal parameters to best exploit the effect. An enhancement of up to 1° is demonstrated numerically.     

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.     

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.     

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.     

Elastic properties of bent-core nematic liquid crystals: the role of the bend angle

ABSTRACT

The nematic phase formed from bent-core liquid crystals has been the focus of intensive research for more than a decade. With the potential of biaxial nematic phase and other interesting features, such as high Kerr constants, large flexoelectric coefficients and anomalous elastic constants, these bent-core materials have been in the limelight of research. This paper presents a mini-review of the interesting elastic behaviour reported in various bent-core compounds. The review further focusses on two different types of bent-core compounds as exemplars: one derived from an oxadiazole and other a thiadiazole, highlighting the importance of bend angle in bent-core compounds. The universality of the unique behaviour of bend elastic constants via molecular field theory and atomistic modelling has also been summarised.     

Flexoelectric deformations of homeotropic nematic layers in the presence of ionic conductivity

Elastic deformations of homeotropic nematic liquid crystal layers subjected to a d.c. electric field were studied numerically in order to find the dependence of threshold voltage on the properties of such a system. A nematic material characterized by a negative sum of flexoelectric coefficients and by a small negative dielectric anisotropy was considered. The flow of ionic current was taken into account. The electric properties are described in terms of a weak electrolyte model. Finite surface anchoring strength was assumed. The director orientation, the electric potential and the ion concentrations were calculated as functions of the coordinate normal to the layer. It was found that the threshold for the deformation depends on the distributions of the ions, governed by the generation constant and by the properties of the electrodes. The effects observed may be interpreted as a consequence of the separation of the ions. When the electrodes have pronounced blocking character, a high and non-uniform electric field, created by the subelectrode ion space charges, causes drastic decrease of the threshold voltage, much below the value U _f valid for the insulating nematic. On the other hand, the electric field gradient arising in the bulk at moderate concentrations has a stabilizing effect and remarkably enhances the threshold above U _f. When the electrodes are conducting there are no significant space charges and the threshold voltage remains close to U _f. These results indicate that phenomena related to the charge transport should be taken into account in the analysis of the elastic deformations of ion-containing flexoelectric nematics.     

Characterization of large tilt-angle flexoelectro-optic switching in chiral nematic liquid crystal devices

The ‘flexoelastic ratio’ is commonly used to characterise the electro-optic behaviour of chiral nematic liquid crystal (LC) devices that exhibit flexoelectro-optic switching. There has been renewed interest in this electro-optic effect of late as new LC materials and mixtures have been developed that exhibit large tilt angles, Ø, of the optic axis (Ø ≥ 45°) whilst maintaining a fast response time (< 1 ms). In this study, we compare the different flexoelastic ratios that are obtained for fixed and variable-pitch chiral nematic LCs for materials with a tilt of the optic axis as large as Ø = ± 45°. We show that for large tilt angles of the optic axis the values for the two different flexoelastic ratios measurably diverge. Of the two ratios, we propose that for large tilt-angle mixtures it is more appropriate to use the flexoelastic ratio that is derived for the case when the pitch of the helix is constrained. In addition, a simple way of determining the ‘pitch-constrained’ flexoelastic ratio is presented that involves identifying the electric field amplitude at the point for which the transmission levels are the same for both positive and negative electric field polarities.     

Non-linear solutions for smectic C liquid crystals in wedge and cylinder geometries

This paper discusses some non-linear problems for smectic C liquid crystals based on the continuum theory proposed by Leslie et al. New restrictions on the nine elastic constants are also derived. Attention is restricted to samples involving concentric cylindrical layers in which both the layer thickness and the tilt angle are assumed to be constant. Non-linear solutions are presented for a sample contained in a wedge with an electric field applied across the bounding plates, extending earlier work by Carlsson et al., and for a sample between two coaxial concentric circular cylinders to which an azimuthal magnetic field is applied. Freedericksz thresholds, which may lead to the experimental determination of some of the elastic constants, are deduced. In the absence of an applied field it is found that, under suitable restrictions on the elastic constants, there is a critical wedge angle (or critical radius ratio in the concentric cylinder case) above which a variable non-linear symmetric solution satisfying the zero boundary conditions is energetically more favourable than the zero solution.     

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.     

Nematic twist-bend phase under external constraints

ABSTRACT

The recently discovered twist-bend nematic phase, N_TB, has short-pitched heliconical structure with doubly degenerate handedness. In contrast to the classic nematic, in the N_TB phase the director is spontaneously distorted, resulting in unusual elastic properties. The response of the N_TB phase to external stimuli, like chiral doping or applied fields might provide further information about its structure and can find utilisation in practical applications. Here, the N_TB behaviour is theoretically investigated under chiral doping and strong electric fields. We show that the chiral doping removes the N_TB degeneration and modifies the conical tilt angle, leaving the pitch unchanged. Thus, the N_TB helical twisting power is very high and strongly non-linear. Under electric field, we consider separately the ferroelectric, flexoelectric and dielectric couplings. We show that the experiments reported so far disagree with the ferroelectric behaviour, indicating that the N_TB phase is not spontaneously polarised. On the contrary, the observed polar effects fit well with the flexoelectric coupling, confirming the degenerated heliconical structure of the phase. Under very strong fields, we predict a second-order twist-bend nematic – nematic phase transition due to the dielectric torque on the director.     

A molecular theoretical derivation of the Landau free energy of chiral smectic C liquid crystals

A molecular theoretical derivation of the Landau free energy of chiral smectic C liquid crystals is presented on the basis of a molecular theoretical approach. The excess free energy concerned with the ferroelectric helicoidal structure is derived within a mean field approximation. Several material constants such as the Curie temperature, the elastic constant, the helical pitch, the piezo- and flexoelectric constants and the dielectric susceptibility are expressed in terms of some effective intermolecular interaction parameters between molecules. A molecular theoretical consideration of the chiral smectic C-smectic A transitions is given in comparison with a few previous works. By comparing with some available experimental data for DOBAMBC, effective interaction parameters are numerically estimated.