Scaling effect of flexoelectric (Ba,Sr)TiO3 microcantilevers

The flexoelectric microcantilever offers an alternative approach for the development of micro/nano‐sensors. The transverse flexoelectric coefficients µ₁₂ of barium strontium titanate microcantilevers were measured at room temperature, and found to keep the same value of 8.5 µC/m for microcantilevers with thickness ranging from 30 µm to 1.4 mm. The calculated effective piezoelectric coefficient and electrical energy density of flexoelectric cantilevers are superior to those of their piezoelectric counterparts, suggesting that the flexoelectricity‐induced polarization can be significantly increased as structures are scaled down due to the scaling effect of strain gradient, holding promise for flexoelectric micro/nano cantilever sensing applications. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Flexoelectricity induced increase of critical thickness in epitaxial ferroelectric thin films

Flexoelectricity describes the coupling between polarization and strain/stress gradients in insulating crystals. In this paper, using the Landau–Ginsburg–Devonshire phenomenological approach, we found that flexoelectricity could increase the theoretical critical thickness in epitaxial BaTiO₃ thin films, below which the switchable spontaneous polarization vanishes. This increase is remarkable in tensile films while trivial in compressive films due to the electrostriction caused decrease of potential barrier, which can be easily destroyed by the flexoelectricity, between the ferroelectric state and the paraelectric state in tensile films. In addition, the films are still in a uni-polar state even below the critical thickness due to the flexoelectric effect.     

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.     

Influence of Surface Energy on the Direct Flexoelectric Effect in a Plate

The influence the surface energy introduced earlier by Kretschmer and Binder has on flexoelectric polarization induced by bending a dielectric plate is studied in the context of phenomenological theory. In contrast to the familiar effect this energy has on permittivity, its influence on the flexoelectric response does not vanish within a thick plate.     

Threshold effects in homeotropically oriented nematic liquid crystals in an external electric field

The electric field-induced orientational transition in a homeotropically oriented nematic liquid crystal cell is investigated. The interaction with the field as a result of anisotropy of the permittivity and flexoelectric polarization is taken into account. For an arbitrary energy of interaction of the nematic with the substrate simple relations are derived for determining the threshold characteristics of the phase transition. It is shown that, in contrast with planar orientation, in fields above a critical value a periodic structure can occur only by virtue of the flexoelectric effect. The resulting dependences for the threshold parameters in the given geometry are exceptionally useful for determining experimentally the surface energy and the difference in the flexoelectric coefficients. Zh. éksp. Teor. Fiz. 116, 543–550 (August 1999)     

Surface and flexoelectric polarization in a nematic liquid crystal 5CB

The temperature dependence of the surface polarization has been measured for both the planar and homeotropic orientation of a nematic liquid crystal at a solid substrate. A conventional liquid crystal 5CB, pure and doped with an azo-dye, was used in cells with controlled asymmetry for light absorption. The measurements have been made by a pyroelectric technique using short pulses of a YAG laser to create a temperature increment. The latter, in turn, was measured independently by a novel time-resolved “optical thermometer” technique monitoring temperature-dependent birefringence by a He- Ne laser beam. In accordance with the symmetry of the order parameter, the surface polarization has different sign for the two orientations, its magnitude ranges from -4 to +2pC/m. The same technique has been used for the measurement of the flexoelectric polarization in hybrid cells. The sum of the flexoelectric coefficients is e ₁ + e ₃ = - 13pC/m at 25°C. Received 28 February 2000 and Received in final form 5 September 2000     

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.     

Strain-gradient-induced polarization in SrTiO3 single crystals

Piezoelectricity is inherent only in noncentrosymmetric materials, but a piezoelectric response can also be obtained in centrosymmetric crystals if subjected to inhomogeneous deformation. This phenomenon, known as flexoelectricity, can significantly affect the functional properties of insulators, particularly thin films of high permittivity materials. We have measured strain-gradient-induced polarization in single crystals of paraelectric SrTiO3 as a function of temperature and orientation down to and below the 105 K phase transition. Estimates were obtained for all the components of the flexoelectric tensor, and calculations based on these indicate that local polarization around defects in SrTiO3 may exceed the largest ferroelectric polarizations. A sign reversal of the flexoelectric response detected below the phase transition suggests that the ferroelastic domain walls of SrTiO3 may be polar.     

One- and two-dimensional deformations in nematic liquid crystals: Influence of the flexoelectric polarization

Summary The elastic deformations in nematic liquid crystals are analysed in one and two dimensions. The influence of the flexoelectric polarization on the elastic properties of the medium is studied. It is shown that for one-dimensional deformations, the presence of the flexoelectric polarization introduces a renormalization of the elastic constants. On the contrary in the bidimensional case, the effect of the flexoelectricity is not a simple renormalization of the elastic constants. The profiles of the electric potential and of the tilt angle are evaluated for the cases in which the substrate is an insulating or conducting material. The validity of the calculations is discussed by considering the value of Debye's screening length of the nematic liquid crystal.     

表面序电极化、挠曲电极化与向列液晶盒饱和点的双稳态

理论分析和数值模拟相结合,研究了表面序电极化、挠曲电极化对非对称弱锚定向列液晶盒在饱和点双稳态的影响,得到了两种极化共存时液晶指向矢满足的方程和边界条件.通过引入新的状态参量,从能量的角度对两种极化的物理效应进行了分析.结果表明,在一定条件下,两种极化耦合,在饱和点通过改变锚定能影响边界条件,从而诱导双稳态.文中给出了判断是否存在双稳态的一般条件,此条件与数值计算结果符合. 关键词： 表面序电极化 挠曲电极化 饱和点 双稳态     

Elastic boundary conditions in the presence of the flexoelectric effect

Boundary conditions for the equations of the theory of elasticity modified by the presence of the flexoelectric effect have been obtained. It has been shown that these conditions do not generally coincide with the boundary conditions of the standard theory of elasticity. In particular, stresses that are proportional to the product of the polarization and surface curvature appear on the surface. Due to the boundary conditions, the uniform polarization deforms the crystal, although this polarization does not enter into the three-dimensional equations of equilibrium.     

Variational principles and governing equations in nano-dielectrics with the flexoelectric effect

The flexoelectric effect is very strong and coupled with large strain gradients for nanoscale dielectrics. At the nanoscale, the electrostatic force cannot be ignored. In this paper, we have established the electric enthalpy variational principle for nanosized dielectrics with the strain gradient and the polarization gradient effect, as well as the effect of the electrostatic force. The complete governing equations, which include the effect of the electrostatic force, are derived from this variational principle, and based on the principle the generalized electrostatic stress is obtained, the generalized electrostatic stress contains the Maxwell stress corresponding to the polarization and strain, and stress related to the polarization gradient and strain gradient. This work provides the basis for the analysis and computations for the electromechanical problems in nanosized dielectric materials.     

Converse flexoelectric effect in the SrTiO3 single crystal

The converse flexoelectric effect in the SrTiO₃ single crystal as a response of inhomogeneous strain (bending strain) to an applied electric field has been studied. The temperature dependence of the effect in the temperature range of 77–450 K has been obtained.     

Prediction of flexoelectricity in BaTiO3 using molecular dynamics simulations

Flexoelectric effect, referring to the strain gradient induced polarization, widely exists in dielectric materials, but its molecular dynamics has not been studied so much so far. In this work, the radial distribution function of BaTiO₃ and the phase transition temperatures have been investigated, and the results show that the core-shell potential model is effective and the structure of BaTiO₃ is stable in a temperature range of 10 K-150 K. Molecular dynamics simulated hysteresis loops of BaTiO₃ show that anisotropy can play an important role in the coercive field. Based on the rational simulation process, the effects of cantilever beam bent angle and fixed length on the polarization are analyzed. It is found that the small bent angle of the curved cantilever beam can give a proportional relationship with a fixed end length and a non-linear relationship is presented when the bent angle is much larger. The prediction of flexoelectric coefficient in BaTiO₃ is 18.5 nC/m. This work provides a computational framework for the study of flexoelectric effect by using molecular dynamics.     

Converse flexoelectric effect in comb electrode piezoelectric microbeam

We demonstrate the converse flexoelectric effect in a lead zirconate titanate microbeam. The fringe electric field of a comb electrode induces converse flexoelectric responses in uniformly poled and depoled beams. The simulated electric field distribution shows that bending of the beam is induced by piezoelectric and _μ11${\mu }_{11}$, _μ12${\mu }_{12}$ flexoelectric coefficients. Simulations indicate that piezoelectric displacement occurs in different directions in the two opposite poled samples while flexoelectric displacement remains the same. This finding is verified by the displacement measurement results.     

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 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.     

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.     

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