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.     

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.     

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.     

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.     

Dynamics of electric field induced deformations in flexoelectric nematic layers

The dynamics of deformations induced by DC electrical fields in homeotropically aligned layers containing a flexoelectric nematic material with negative dielectric anisotropy has been studied numerically. The rise time constants, characterising the development of deformations after switching on the external voltage, and the decay time constants, describing the decay of deformation after switching off the voltage, were calculated as a function of the parameters essential for the behaviour of the layer. In particular, the influence of flexoelectricity was studied. It was found that the stronger the flexoelectric properties of the nematic, the lower is its viscosity, the higher is the bias voltage, the weaker is the surface anchoring, the thinner is the layer and the higher is the ion concentration, the more rapid was the onset of deformation. Similarly, the lower the viscosity, the thinner is the layer, the stronger the anchoring and the larger the ion content, the more rapid was the decay of deformation. Neither the voltage previously applied nor the flexoelectric properties were found to affect the decay time.     

Strong deformations induced by a DC electric field in homeotropic flexo-electric nematic layers

Elastic deformations of homeotropic nematic liquid crystal layers subjected to a DC electric field were studied numerically in order to determine their development under increasing voltage. Both signs of the dielectric anisotropy, Δ?, and of the sum of flexo-electric coefficients, e ₁₁ + e ₃₃, were considered, as were also low, moderate and high ion concentrations. The electrical properties of the layer are described in terms of a weak electrolyte model. Quasi-blocking electrode contacts and a finite surface anchoring strength were adopted. Director orientation, the electric potential and the ion concentrations were each calculated as a function of the coordinate normal to the layer. The director distributions turned out to depend not only on the sign of Δ? but also on the sign of e ₁₁ + e ₃₃, due to the difference in the mobility of anions and cations. The importance of ion content was also confirmed.     

Ion adsorption and its influence on direct current electric field induced deformations of flexoelectric nematic layers

The influence of ion adsorption on the behavior of the nematic liquid crystal layers is studied numerically. The homeotropic flexoelectric layer subjected to the dc electric field is considered. Selective adsorption of positive ions is assumed. The analysis is based on the free energy formalism for ion adsorption. The distributions of director orientation angle, electric potential, and ion concentrations are calculated by numerical resolving of suitable torques equations and Poisson equation. The threshold voltages for the deformations are also determined. It was shown that adsorption affects the distributions of both cations and anions. Sufficiently large number of adsorbed ions leads to spontaneous deformation arising without any threshold if the total number of ions creates sufficiently strong electric field with significant field gradients in the neighborhood of electrodes. The spontaneous deformations are favored by strong flexoelectricity, large thickness, large ion concentrations, weak anchoring, and large adsorption energy.     

Inversion walls in homeotropic nematic and cholesteric layers

We describe the experimental properties of metastable domains associated with the presence of an Inversion Wall (IW ) and observed with homeotropically anchored nematic and cholesteric liquid crystals sandwiched between parallel glass plates. Such a distorted situation, stabilized by the application of an electric field parallel to the plates as described in reference [1], can also be obtained transiently either when filling a sample cell by capillarity or in some studies of directional solidification at the N-I interface [2]. The application of an electric field perpendicular to the plates with Delta epsilon 0 materials allows control of the reversal region thickness in the bulk of the sample and its associated birefringence. This IW can be stabilized in the particular case of low lateral extension globules in which the line tension of the looped disclination separating the pi wall regions from the homeotropic regions counterbalances the unfavourable bulk free energy. Particular attention is devoted to the defects of these walls, whose appearance using polarizing microscopy is similar to the umbilics of the Freedericksz transition. The structure of these 'four brush' defects is nevertheless here singular, corresponding to point defects of a 3D uniaxial nematic medium. In the case of a chiral nematic, these IW undergo a particular undulation instability which is also observed in 2D simulations.     

Influence of the surface pretilt angle on spatially periodic deformations in nematic layers

The deformations induced by electric field in twisted or untwisted flexoelectric nematic layers can be homogeneous (i.e. one-dimensional) or spatially periodic (i.e. two-dimensional). The periodic deformations are undesirable from an applicative point of view since they destroy the homogeneous appearance of the area of an excited pixel of a display. They are particularly favoured when the nematic material possesses flexoelectric properties. In order to check whether the unwanted periodic deformations can be eliminated by means of suitable surface pretilt angle, the small deformations arising just above the corresponding threshold voltage were investigated numerically. The nematic materials exhibiting both weak and strong flexoelectricity were taken into account. The surface pretilt angles ranging from 0° to 30° were adopted. It was shown that the periodic patterns, arising in the case of planar surface alignment, disappear if sufficiently large surface pretilt angle is applied.     

Experimental determination of the flexoelectric coefficients of some nematic liquid crystals

Abstract

We report measurements of the temperature variation of the flexoelectric coefficient (e ₁–e ₃) of a number of nematic liquid crystals like phenylcyclohexanes, cyanobiphenyls, etc. We have also measured (e ₁+e ₃) of a few systems using appropriate methods of applying an electric field gradient to the sample. In most of the systems, (e ₁–e ₃)/k, where k is a curvature elastic constant, is found to be positive and independent of temperature, as expected. However, in 4-heptyl-1-(4-cyanocyclohexyl)cyclohexane and a few other compounds with relatively flexible parts, |(e ₁–e ₃)/k| increases with temperature. We discuss the possible molecular origin of the sign and temperature dependence of the flexoelectric coefficients of the systems studied.     

Calculation of flexoelectric coefficients for a nematic liquid crystal by atomistic simulation

Equilibrium molecular dynamics calculations have been performed for the liquid crystal molecule n-4-(trans-4-n-pentylcyclohexyl)benzonitrile (PCH5) using a fully atomistic model. Simulation data have been obtained for a series of temperatures in the nematic phase. The simulation data have been used to calculate the flexoelectric coefficients e(s) and e(b) using the linear response formalism of Osipov and Nemtsov [M. A. Osipov and V. B. Nemtsov, Sov. Phys. Crstallogr. 31, 125 (1986)]. The temperature and order parameter dependence of e(s) and e(b) are examined, as are separate contributions from different intermolecular interactions. Values of e(s) and e(b) calculated from simulation are consistent with those found from experiment.     

Stability of the periodic deformations in planar nematic layers

The periodic deformations induced by external fields have been analysed by means of the Taylor expansion method based on the theorems of catastrophe theory. The analysis is restricted to the planar nematic slab influenced by a magnetic field. Two different configurations of the field which lead to periodic deformations with prevailing splay or twist we considered. The ranges of material constants at which the periodic state is stable and the threshold magnetic field strength have been found. The problem of transitions between the undeformed, uniformly deformed and periodic states is discussed.     

The magnetic field-induced deformations in nematic layers with non-zero splay-bend elastic constant

The magnetic field-induced deformations of weakly anchored nematic layers in the presence of the non-zero splay-bend elastic constant K₁₃ are analysed by use of the Pergamenshchik approach. Most of the anomalous phenomena reported by other authors, that do not occur if k₁₃ = 0, have been confirmed. New effects suitable for experimental verification are predicted: the discontinuous transition with hysteresis between the uniform undistorted state and the state uniformly aligned along the field, and the rotation of the director in the tilted layer during continuous increase and decrease of the perpendicular field.     

Effect of weak anchoring on the electric field induced deformations of nematic layers

The stationary deformations of nematic layers with a twisted structure are analysed by means of the Taylor expansion method based on catastrophe theory. The role of weak anchoring is investigated. Variations of the polar and azimuthal angles describing the surface director orientation are allowed. The stability of two equilibrium states, the twisted and the homeotropic, is studied. Several types of continuous and discontinuous transitions between them are revealed. The threshold voltages are calculated.     

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.     

Periodic deformations induced by a magnetic field in planar twisted nematic layers

Magnetic field-induced spatially periodic deformations of planar nematic layers twisted by an angle Φ were investigated numerically. Chiral nematics with pitches compatible with the twist angle and non-chiral nematics twisted by Φ ? π/2 were considered. Two different modes of deformation, taking the form of stripes, were found: the so called Mode X, with periodicity parallel to the mid-plane director in the undisturbed structure, and Mode Y with periodicity perpendicular to the mid-plane director. The static director distributions were calculated for various magnetic field strengths, twist angles and elastic parameters. The influence of surface tilt was also investigated. Mode X appeared for sufficiently large Φ and was possible in nematics with typical elastic properties. Mode Y appeared provided that the k ₂₂/k ₁₁ elastic constant ratio and the twist angle Φ were sufficiently small. Both modes arose from the undistorted state when the magnetic field exceeded a threshold value. The spatial period of the patterns increased with field strength. At high field, regions with almost homogeneous deformation arose in the two halves of each stripe. Their width and, simultaneously, the spatial period diverged to infinity at some critical field. This divergence corresponds to the transition to a homogeneously deformed state. Diagrams were constructed showing the ranges of parameters favouring the periodic distortions.     

Periodic deformations induced by a magnetic field in planar twisted nematic layers

Magnetic field-induced spatially periodic deformations of planar nematic layers twisted by an angle Φ were investigated numerically. Chiral nematics with pitches compatible with the twist angle and non-chiral nematics twisted by Φ ≤π/2 were considered. Two different modes of deformation, taking the form of stripes, were found: the so called Mode X, with periodicity parallel to the mid-plane director in the undisturbed structure, and Mode Y with periodicity perpendicular to the mid-plane director. The static director distributions were calculated for various magnetic field strengths, twist angles and elastic parameters. The influence of surface tilt was also investigated. Mode X appeared for sufficiently large Φ and was possible in nematics with typical elastic properties. Mode Y appeared provided that the k₂₂/k₁₁ elastic constant ratio and the twist angle Φ were sufficiently small. Both modes arose from the undistorted state when the magnetic field exceeded a threshold value. The spatial period of the patterns increased with field strength. At high field, regions with almost homogeneous deformation arose in the two halves of each stripe. Their width and, simultaneously, the spatial period diverged to infinity at some critical field. This divergence corresponds to the transition to a homogeneously deformed state. Diagrams were constructed showing the ranges of parameters favouring the periodic distortions.     

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.     

Critical radius of loop defects in homeotropic nematic liquid crystals

We describe an experimental situation with a looped line defect in nematic liquid crystals observed by polarizing optical microscopy. We measured the critical size of the loop below which it spontaneously shrinks and transforms into a point defect. The experiment was done with 5CB which gives rise to twist disclinations as do most of the usual nematics. For this kind of disclination an in-plane force due to the boundary conditions acts on the line and influences the critical radius. W e have constructed a model which is in good agreement with experimental measurements and deduced the line tension of the disclination.     

Synthesis and mesomorphic behaviour of wedge-shaped nematic liquid crystals with flexoelectric properties

The synthesis and mesomorphic behaviour of a series of wedge-shaped liquid crystals and some reference compounds are reported. These unusual liquid crystals possess smectic C, smectic A and nematic phases. These new wedge-shaped materials with a high degree of shape anisotropy and a large dipole moment can be used to induce an increase in the flexoelectric effects of nematic guest-host mixtures as dopants at low concentrations.