Fluctuations in liquid crystals in the presence of a flexoelectric effect

We examine director fluctuations in nematic liquid crystals in an electric field in the presence of a flexoelectric effect. For the planar and homeotropic orientations we calculate the correlation functions and analyze these functions near the Freedericksz transition and the flexoelectric instability threshold. For both geometries we calculate the angular dependence of the intensity of light scattered by director fluctuations and examine its behavior at electric field strengths close to critical. Zh. éksp. Teor. Fiz. 112, 1675–1693 (November 1997)     

Inhomogeneous threshold director reorientation in a planar nematic flexoelectric cell with finite anchoring energy

The dependence of the threshold parameters and the period of the electric-field-induced spatially periodic reorientation of the director in a flexoelectric nematic liquid crystal (NLC) on the anchoring conditions at the surface of a planar NLC cell has been studied. The threshold electric field and the corresponding wave-number of the periodic structure of the director field have been numerically calculated for arbitrary values of the anchoring energy. In the case of strong anchoring, the corresponding analytical expressions are obtained in a single-constant approximation. A decrease in the azimuthal anchoring energy leads to an increase in the intervals of possible values of the flexoelectric parameter ν and the ratio K₂/K₁ of the Frank elastic constants. A decrease in the polar anchoring energy leads to narrowing of these intervals as compared to the case of infinitely strong anchoring at the NLC cell surface.     

Threshold voltages and optical retardation of deformed flexoelectric nematic layers with asymmetric surface anchoring

Deformations of homeotropically aligned flexoelectric nematic layers induced by dc electric fields were simulated numerically. Two 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 deformations were determined by means of calculations of the phase difference Φ between ordinary and extraordinary light rays passing through a layer placed between crossed polarisers. The threshold values depended on the polarity of the bias voltage U. When the threshold value was exceeded, the phase difference increased with the voltage. Two different Φ(U/U_threshold) dependencies for the two polarities of the voltage were found for each layer if the nematic possessed the flexoelectric properties. The possibility of using this effect to detect the flexoelectricity in the nematic was explored by simulated experiments. The effectiveness of the proposed method is discussed.     

Flexoelectric polarization in nematic liquid crystals measured by a field on-off pyroelectric technique

A novel technique based on the pyroelectric effect is developed for making direct measurements of the flexoelectric polarization P _f in hybrid aligned nematic cells. The pyroelectric response is measured first in the field-off regime and then with the bias field applied. The latter allows a direct comparison of P _f with the field-induced polarization over the whole range of the nematic phase of the standard compound 5CB. The sum of the flexoelectric coefficients (e ₁+e ₃) is shown to be negative, and its dependence on the nematic order parameter does not follow a simple law (linear or quadratic), as has been predicted theoretically. The dynamics of P _f is discussed in terms of the order parameter fluctuations. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 3, 220–225 (10 February 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Electrical Freedericksz transitions in nematic liquid crystals containing ferroelectric nanoparticles

A new theoretical approach was elaborated to explain the contradictions reported in many papers about the electric threshold for Freedericksz transition in nematic liquid crystal (NLC) with ferroparticles additives. The free energy density of the mixture was estimated and the contributions of the interaction energy of NLC molecules with ferroparticles surface were calculated. Experimental results for 5CB-BaTiO₃ mixture are given.     

Nematic ordering in a cell with modulated surface anchoring: effects of flexoelectricity

We have analyzed molecular ordering in a nematic sample sandwiched between two parallel substrates, characterized by a periodically varying anchoring easy axis. If the periodicity lambda is smaller than the Debye screening length l(D) and the nematic material possesses flexoelectric properties, it is necessary to take into account also the electrostatic and flexoelectric contributions in the thermodynamical potential when the actual director field is determined. In this framework, for small deviations from the homeotropic alignment we have derived analytical expressions for the tilt angle (theta) and the electrical potential. To establish a connection with experimentally observable quantities, we have related the theta profile to the average and investigated its behavior for different values of lambda, the flexoelectric coefficient, and the anchoring strength w. Our results indicate that in a nematic with pronounced flexoelectric properties for small enough lambda, a kind of subsurface deformation appears, which substantially decreases . Therefore, effects of flexoelectricity cannot be neglected in treating nematic cells with modulated anchoring which allows bistable ordering.     

Numerical investigation of influence of ionic space charge and flexoelectric polarization on measurements of elastic constants in nematic liquid crystals

Deformations of nematic layers caused by magnetic field allow determination of the elastic constants of liquid crystal. In this paper, we simulated numerically the deformations of planar and homeotropic nematic layers. The flexoelectric properties of the nematic and presence of ions were taken into account. Our aim was to show the influence of flexoelectricity on the results of the real measurement of the elastic constants k₃₃ and k₁₁. In these simulations, we calculated the optical phase difference ΔΦ between the ordinary and extraordinary rays of light passing through the layer placed between crossed polarizers as a function of the magnetic field induction B. One of the elastic constants can be calculated from the magnetic field threshold for deformation. The ratio k₃₃/k₁₁ can be found by means of fitting theoretical ΔΦ(B) dependence to the experimental results. The calculations reveal that the flexoelectric properties influence the deformations induced by the external magnetic field. In the case of highly pure samples, this may lead to false results of measurement of the elastic constants ratio k₃₃/k₁₁. This influence can be reduced if the nematic material contains ions of sufficiently high concentration. These results show that the flexoelectric properties may play an important role, especially in well purified samples.     

Colloidal interactions and transport in nematic liquid crystals

We describe a new nematic liquid-crystal colloid system which is characterized by both charge stabilization of the particles and an interaction force. We estimate the effective charge of the particles by electrophoretic measurements and find that in such systems the director anchoring energy W is very low and the particles have little director distortion around them. The interaction force is created by producing a radial distribution of the nematic order parameter around a locally isotropic region created by ir laser heating. We theoretically describe this as being due to the induced flexoelectric polarization, the quadrupolar symmetry of which provides the required long-range force acting on charged particles.     

Theoretical analysis of the influence of flexoelectric effect on the defect site in nematic inversion walls

Based on the experimental phenomena of flexoelectric response at defect sites in nematic inversion walls conducted by Kumar et al., we gave the theoretical analysis using the Frank elastic theory. When a direct-current electric field normal to the plane of the substrate is applied to the parallel aligned nematic liquid crystal cell with weak anchoring, the rotation of ± 1 defects in the narrow inversion walls can be exhibited. The free energy of liquid crystal molecules around the +1 and-1 defect sites in the nematic inversion walls under the electric field was formulated and the electric-field-driven structural changes at the defect site characterized by polar and azimuthal angles of the local director were simulated. The results reveal that the deviation of azimuthal angle induced by flexoelectric effect are consistent with the switching of extinction brushes at the +1 and-1 defects obtained in the experiment conducted by Kumar et al.     

Giant flexoelectricity of bent-core nematic liquid crystals

Flexoelectricity is a coupling between orientational deformation and electric polarization. We present a direct method for measuring the flexoelectric coefficients of nematic liquid crystals (NLCs) via the electric current produced by periodic mechanical flexing of the NLC's bounding surfaces. This method is suitable for measuring the response of bent-core liquid crystals, which are expected to demonstrate a much larger flexoelectric effect than traditional, calamitic liquid crystals. Our results reveal that not only is the bend flexoelectric coefficient of bent-core NLCs gigantic (more than 3 orders of magnitude larger than in calamitics) but also it is much larger than would be expected from microscopic models based on molecular geometry. Thus, bent-core nematic materials can form the basis of a technological breakthrough for conversion between mechanical and electrical energy.     

Surface driven transition in a nematic liquid crystal cell

Surface driven reorientation effects in a nematic liquid crystal cell caused by light-induced changes of the anchoring parameters were studied. Theoretical consideration of one-dimensional flat distributions of the director has shown that the director can undergo threshold reorientation between hybrid, homeotropic, and planar alignments as the anchoring energy varies continuously. The threshold reorientation takes place when the reference and light-induced easy axes are perpendicular. In the one-elastic-constant approximation the light-induced transition was found to be of second order as shown by a critical increase of the director thermal fluctuations in the vicinity of the transition point. These effects were experimentally studied in the cells containing 5CB liquid crystal aligned by the photosensitive azo-containing polymer layer. Zh. éksp. Teor. Fiz. 112, 2045–2055 (December 1997) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Enhancement of Nematic Order and Global Phase Diagram of a Lattice Model for Coupled Nematic Systems

We use an infinite-range Maier–Saupe model, with two sets of local quadrupolar variables and restricted orientations, to investigate the global phase diagram of a coupled system of two nematic subsystems. The free energy and the equations of state are exactly calculated by standard techniques of statistical mechanics. The nematic–isotropic transition temperature of system A increases with both the interaction energy among mesogens of system B, and the two-subsystem coupling J. This enhancement of the nematic phase is manifested in a global phase diagram in terms of the interaction parameters and the temperature T. We make some comments on the connections of these results with experimental findings for a system of diluted ferroelectric nanoparticles embedded in a nematic liquid-crystalline environment.     

Flexoelectric-Induced Voltage Shift in Hybrid Aligned Nematic Liquid Crystal Cell

Flexoelectric-induced voltage shift in a weak anchoring hybrid aligned nematic liquid crystal cell is investigated theoretically. Based on the elastic theory of liquid crystal and the variation method, the equations for the bulk and the boundary of the cell are derived. By computer simulation, the dependence of the
shift voltage on the sum of the flexoelectric coefficients and the anchoring energy strength is obtained. As a result, a novel method to determine the sum of the flexoelectric coefficients by measuring the shift voltage is put forward.     

Flexoelectric polarization and second order elasticity for nematic liquid crystals

The flexoelectric properties of nematic liquid crystals are analysed. It is shown that in the frame of the usual elasticity two coefficients characterize completely the flexoelectric properties of the liquid crystal. These coefficients, in the limit of small scalar order parameter, i.e. near the clearing point, are approximately equal. More precisely they have the same linear term in the scalar order parameter, and differ for terms quadratic in this parameter. Their difference behaves, hence, as the usual nematic liquid crystal elastic constants, whereas their sum depends on the temperature, like the mixed splay-bend elastic constant. It is shown furthermore that in the frame of a second order elastic theory in the flexoelectric polarization there are no terms from the second order spatial derivatives of the nematic director, or the nematic tensor order parameter. Consequently also in the frame of the second order elasticity the flexoelectric polarization is given by the usual expression. This conclusion is important in connection with the surface polarization recently discovered in pretilted nematic liquid crystal samples.     

Influences of surface and flexoelectric polarization on the effective anchoring energy in nematic liquid crystal

The physical effects on surface and flexoelectric polarization in a weak anchoring nematic liquid crystal cell are investigated systematically. We derive the analytic expressions of two effective anchoring energies for lower and upper substrates respectively as well as their effective anchoring strengths and corresponding tilt angles of effective easy direction.All of these quantities are relevant to the magnitudes of both two polarizations and the applied voltage U. Based on these expressions, the variations of effective anchoring strength and the tilt angle with the applied voltage are calculated for the fixed values of two polarizations. For an original weak anchoring hybrid aligned nematic cell, it may be equivalent to a planar cell for a small value of U and has a threshold voltage. The variation of reduced threshold voltage with reduced surface polarization strength is also calculated. The role of surface polarization is important without the adsorptive ions considered.     

Theory of sonoluminescence in single bubbles: Flexoelectric energy contribution

An equation of motion for a cavitating gas bubble immersed in a liquid is introduced which includes a flexoelectric energy term. This energy is deduced from the electric field produced by the bubble wall acceleration (pressure gradient) in the fluid (the flexoelectric effect). We show that under conditions of sonoluminescence, this electric field reaches values typical of the electric breakdown field in water. Our theoretical results are consistent with the duration of light emission, minimum bubble radius, and energy release as measured in sonoluminescence experiments in water. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 6, 442–448 (25 September 1998)     

Analysis of deformations of flexoelectric homeotropic liquid crystal layers with various anchoring strengths

Deformations of homeotropic layers of nematic liquid crystal possessing flexoelectric properties and subjected to external dc voltage were studied numerically. The influence of the anchoring strengthWwas investigated. The calculations were performed for two types of layers, one with positive dielectric anisotropy and positive sum of flexoelectric coefficients and the other with opposite signs of these parameters. Strongly blocking electrodes were assumed. The threshold voltages for deformations were computed for several anchoring strength values and for low, moderate, and high ion contents. The director distributions were also determined. They were influenced by joined effects of anchoring strength and space charge of ions. For increasing ion contents and increasing anchoring strength, remarkable change of deformation character was noticed. In the case of negative dielectric and flexoelectric parameters, the threshold was nearly proportional to W for low anchoring strength and saturated at some high value of W. In the case of positive dielectric and flexoelectric parameters, the threshold voltage increased strongly with W. In the range of enhanced ion concentrations, the deformation arose in two voltage ranges. The increasing anchoring strength damped the deformations in the whole layer, however, this effect was minor at high ion content.     

Surface electric-field-induced instabilities in a homeotropically oriented nematic liquid crystal sample

The effect of a surface electric field produced by ionic adsorption on the molecular orientation of a nematic liquid crystal sample is analyzed. The eigenvalue problem for a semi-infinite medium is analytically solved both for strong and weak anchoring situations. The threshold instabilities are numerically determined and it is shown that the homeotropic pattern can be destabilized also in the situation of strong anchoring. The dependence of the threshold field on the anchoring strength and on the surface polarization is determined by taking into account also the coupling of the quadrupolar component of the flexoelectric coefficient with the field gradient. Received 4 November 1999 and Received in final form 4 April 2000     

Detection of the contribution of the inverse flexoelectric effect to the photorefractive response in a bismuth titanium oxide single crystal

Relations determining the contribution of the inverse flexoelectric effect to the photorefractive response have been determined for the case of the interaction of a steady pump wave conserving left-hand circular polarization with a phase-modulated signal wave that has circular polarization of the opposite sign on the input face of a crystal that has the (100) orientation and symmetry group 23. The experimental studies of such an interaction in a Bi₁₂TiO₂0:Fe,Cu crystal make it possible to estimate its flexoelectric coefficient as f ₁₁ = 5.3 nC/m.     

On the role of flexoeffect in synchronization of electroconvective roll oscillations in nematics

We describe the dynamics of zigzag oscillations in a system of convective rolls in a nematic liquid crystal above the electroconvection threshold under the action of an ac voltage with a biased position of the mean value. It is found that an increase in the contribution from the constant component leads to a substantial increase in the spatiotemporal ordering of zigzag rolls and their synchronization with the homogeneous twist mode. The results confirm the flexoelectric mechanism of locking.