Flexoelectric Response Research Articles

Flexoelectric effect in PVDF-based polymers

Flexoelectricity is a gradient electromechanical coupling effect that exists in all dielectrics and is important for the understanding of a variety of gradient-induced physical phenomena and the design of new electromechanical devices. At present, the flexoelectric effect in polymer materials has not been well studied. In this work, thick rectangular poly(vinylidene fluoride) (PVDF)-based polymer samples were fabricated and the flexoelectric coefficient was measured. Our results show that the flexoelectric coefficient of the PVDF, which is on the order of several nC/m, is more than twice higher than that of P(VDF-CTFE) and P(VDF-HFP) polymers. All these materials exhibited a non-polar α phase, but the copolymers showed much smaller crystallinity values than the PVDF homopolymer. The difference in the flexoelectric response in these polymers is believed to be related to the crystallinity of the polymers.

Measurement of flexoelectric response in polyvinylidene fluoride films for piezoelectric vibration energy harvesters

This study presents an investigation on the measurement of flexoelectric response in β-phase polyvinylidene fluoride (PVDF) films attached on cantilever beam-based flexible piezoelectric vibration energy harvesters (PVEHs). The flexoelectric response associated with negative strain gradients was simulated through harmonic response analysis by using the finite element method (FEM). The polarization frequency response functions (FRFs) modified by direct flexoelectric effect of PVDF films was experimentally validated by multi-mode FRFs. From quantitative comparisons between experimental observations and simulated estimation of FRFs, it is demonstrated that the direct flexoelectric response can be observed in PVDF films attached on PVEHs.

Analysis of flexoelectric response in nanobeams using nonlocal theory of elasticity

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nonlocal flexoelectric nanobeams. Considering both the direct and the converse flexoelectric effects, and employing the nonlocal theory of elasticity, the governing equations and the associated boundary conditions of the beams are derived to obtain the exact solutions for the displacements, nonlocal stresses and the electric potential in the beams. Both the direct and the converse flexoelectric effects are influenced by the nonlocal parameter. Active beams significantly counteract the applied mechanical load by virtue of the converse flexoelectric effect. The normal and the transverse shear deformations in the beams are affected by the converse flexoelectric effect resulting in the coupling of bending and stretching deformations in the beams even if the beams are homogeneous. Because of the consideration of the nonlocal theory of elasticity, the nonlocal stiffness of the beam appears to be less than the classical stiffness of the beam. The nonlocal elasticity does not influence the stresses of the passive beam while the nonlocal stresses in the active beam due to converse flexoeletric effect are less than the local or classical stresses in the active beam. The benchmark results presented here may be useful for verifying the numerical model and experimental results for nonlocal flexoelectric nanobeams. The present study suggests that the flexoelectric nanobeams may be effectively exploited for developing advanced smart nanosensors and nanoactuators. The research work carried out here also conveys that the nonlocal theory of elasticity must be employed for accurate analysis of flexoelectric solids.

Experimental method research on transverse flexoelectric response of poly(vinylidene fluoride)

Flexoelectricity describes the strain-gradient-induced electric polarization existing in dielectric materials. The coefficient that exists between the strain-gradient and the induced electric polarization defines the flexoelectric coefficient tensor. It is necessary to analyze different experimental methods to evaluate the procedure of measuring the transverse flexoelectric coefficient tensor component. In this work, the transverse flexoelectric coefficient tensor component of poly(vinylidene fluoride) (PVDF) is studied using three different experimental methods and the effects of the mentioned methods are evaluated. The results presented in this work are helpful for the design of experiments of different dielectric materials, including ceramics and polymers on flexoelectricity.

Improved approach to measure the direct flexoelectric coefficient of bulk polyvinylidene fluoride

We experimentally studied the built-in polarization induced effective piezoelectric constant and direct flexoelectric coefficient in α-phase bulk polyvinylidene fluoride (PVDF). This phenomenon was detected and discussed based on the compression of a truncated cone. An improved mechanical formulation of flexoelectricity was presented and discussed in this study, and the experiment was carried out based on the charge measurement. From the experiment study, a flexoelectric coupling coefficient 202.3 V was calculated from the flexoelectric coefficient μ11=1.6×10−8C/m for bulk polyvinylidene fluoride. We measured the flexoelectric response of bulk PVDF with consideration of the residual piezoelectric contributions and geometry-dependent calibration, which affect the flexoelectric measurement.

Theoretical analysis of the influence of flexoelectric effect on the defect site in nematic inversion walls**Project supported by the National Natural Science Foundation of China (Grant Nos. 11374087, 11274088, and 11304074), the Natural Science Foundation of Hebei Province, China (Grant Nos. A2014202123 and A2016202282), the Research Project of Hebei Education Department, China (Grant Nos. QN2014130 and QN2015260), and the Key

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.

Strong surface effect on direct bulk flexoelectric response in solids

In the framework of a continuum theory, it is shown that the direct bulk flexoelectric response of a finite sample essentially depends on the surface polarization energy, even in the thermodynamic limit where the body size tends to infinity. It is found that a modification of the surface energy can lead to a change in the polarization response by a factor of two. The origin of the effect is an electric field produced by surface dipoles induced by the strain gradient. The unexpected sensitivity of the polarization response to the surface energy in the thermodynamic limit is conditioned by the fact that the moments of the surface dipoles may scale as the body size.

Atomistic modeling of flexoelectricity in periclase

Flexoelectricity is evolution of a macroscopic polarization due to a strain gradient. We present a molecular dynamics study of flexoelectricity in the cubic ionic MgO periclase phase. Using an effective interaction force field with polarizable oxygen atoms and applying it to an inhomogeneously strained periclase sample, we detect a collective flexoelectric response of the oxygen dipole moments in visualizations. This induced polarization depends linearly on the strength of the strain gradient as well as the primary polarization, which is caused by the displacement of the ion charges. By three different inhomogeneous deformation modes all three flexoelectric coefficients of periclase are determined.

Bimesogen-enhanced flexoelectro-optic behavior of polymer stabilized cholesteric liquid crystal

We explore flexoelectro-optic effect in polymer stabilized cholesteric liquid crystals (CLC) with uniform lying helix texture. A method to enlarge the flexoelectric response of the CLC mixture is demonstrated by mixing two nematic liquid crystal host materials with large and small effective flexoelastic coefficients. By a choice of material composition of CLC mixture, a large angle of flexoelectrically driven in-plane rotation of helical axis with a submillisecond response time is obtained. Our findings will be useful for material design of future CLC mixtures for applications in flexoelectro-optic displays.

Finite-size scaling of flexoelectricity in Langmuir-Blodgett polymer thin films

The flexoelectric effect, which is a linear coupling between a strain gradient and electrical polarization, is a fundamental electromechanical property of all materials with potential for use in nanoscale devices, where strain gradients can be quite large. We report a study of the dependence of the flexoelectric response on thickness in ultrathin films of polar and non-polar polymers. The measurements of the flexoelectric response in non-polar polyethylene and the polar relaxor polymer polyvinylidene-co-trifluoroethylene-co-chlorofluoroethylene were made using a bent cantilever method and corrected for the contribution from the electrode oxide. The results show that the value of the flexoelectric coefficient increases with decreasing thickness, by up to a factor of 70 compared to the bulk value, reaching such enhanced values in films of only 10 nm thickness. These results are consistent with a model accounting for interfacial contributions, and underline how large electromechanical coupling can be produced at the nanoscale. The results also distinguish the surface flexoelectric response from that coming from the volume.

对向列相液晶反转壁中+1缺陷处挠曲电效应的理论分析

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.

A flexoelectricity effect-based sensor for direct torque measurement

In this study, a direct torque sensor based on the flexoelectricity generated by un-polarized polyvinylidene fluoride (PVDF) via electromechanical coupling is developed as a novel torque measurement mechanism that does not require external electric power excitation. The sensing method is developed based on the shear strain gradient and the shear flexoelectric response of PVDF. A theoretical analysis is primarily presented for the design of the sensing structure. Then the structure of the PVDF sensing module is discussed and designed. The radius ratio of the sensing module is defined and then discussed according to the load, the strain gradient, the electrode area and the general electric charge output. The finite element method is used to analyze the mechanical properties of the designed PVDF sensing module. Then the theoretical sensitivity of the sensor is predicated as 0.9441 pC Nm−1. The experiment system setup is developed, and the sensing properties of the measurement mechanism are tested at frequencies of 0.5 Hz, 1 Hz, 1.5 Hz and 2 Hz using identical modules. The measurement range of the designed sensor is 0–1.68 Nm and the average sensitivity is measured as 0.8950 pC Nm−1. The experimental results agree well with the theoretically predicted results. These results prove that the torque sensing method based on un-polarized PVDF is suitable for measurement of dynamic torque loads with a flexoelectricity-based mechanism. When using this method, external electric power excitation of the sensing module is no longer required.

Shear flexoelectric response along 3121 direction in polyvinylidene fluoride

Flexoelectricity describes the strain gradient-induced electric polarization. Due to the restrictions of experiment technologies, some of the components of flexoelectric coefficient have not been experimentally obtained. In this letter, an experimental method for the measurement of the shear flexoelectric response along 3121 direction of polyvinylidene fluoride (PVDF) is presented. An experiment is conducted on various unpolarized specimens, where shear strain gradient is generated along the radial direction by applying torque to 3 specially designed specimens. The generated shear strain gradient is calculated via finite element analysis and the corresponding induced electrical response is measured. Dynamic torque is exerted on the specimens with a static bias value and at different frequencies. The shear flexoelectric coefficient μ3121 is found to have an average value of 1.037 × 10−8 C/m. With this method, the shear flexoelectric response along 3121 direction of PVDF is experimentally obtained. The experimental results show good agreement with the current research results and indicate the potential value of this material property for electromechanical device fabrication.

Distributed flexoelectric structural sensing: Theory and experiment

“Distributed” sensors are sensitive to spatially distributed structural behaviors, as compared with conventional “discrete” sensors measuring only discrete and local behaviors. The phenomenon of electric polarization induced by strain gradients in solid dielectrics is known as the direct flexoelectric effect. In this study, distributed dynamic sensing using flexoelectric materials is proposed and the flexoelectric signal of distributed flexoelectric sensors laminated on an arbitrary shell structure is defined. Both the open-circuit and the close-circuit models of distributed flexoelectric sensing are developed. An equivalent flexoelectric constant is measured using the proposed open-circuit voltage model. Flexoelectric responses of various locations of a BST beam exhibit consistency and linearity in the case of small deflection. The inferred flexoelectric constant is comparable with the reported maximal coefficient and it also proves that the proposed model is effective to the flexoelectric constant measurement. Distributed sensing behaviors are also experimentally validated on the flexoelectric cantilever beam. Results show that theoretical predictions match well with the experimental results for the fundamental mode. Experiments of distributed sensing signals of beams suggest that distributed flexoelectric sensors are competent to provide the mode shape of strain gradients. Thus, distributed sensing based on the flexoelectric effect is effective to monitoring structural dynamic behaviors.

Enhanced flexoelectricity through residual ferroelectricity in barium strontium titanate

Residual ferroelectricity is observed in barium strontium titanate ceramics over 30 °C above the global phase transition temperature, in the same temperature range in which anomalously large flexoelectric coefficients are reported. The application of a strain gradient leads to strain gradient-induced poling or flexoelectric poling. This was observed by the development of a remanent polarization in flexoelectric measurements, an induced d33 piezoelectric response even after the strain gradient was removed, and the production of an internal bias of 9 kV m−1. It is concluded that residual ferroelectric response considerably enhances the observed flexoelectric response.

Interaction between a +1 defect in a nematic thin film and the surface wall and flexoelectric response at the defect site

Within the Landau-de Gennes theory, we have investigated the interaction between a +1 defect whose nuclei is ‘circular’ nuclei (i.e. the director flux lines around the defect nuclei are circular) and the surface wall, using the one-dimensional finite-difference iterative method. +1 point defects will evolve into line defects perpendicular to the sample plane due to the existence of the surface wall. Therefore, we consider cylindrically symmetric solutions containing the line defect. The free energy of liquid crystal molecules around the defect nuclei and the surface energy produced by the surface wall are competing with each other. This competition makes polar angle of the director around the defect nuclei change considerably in a certain region, and the scale of this region will decrease with the surface anchoring strength increasing. On application of a DC electric field normal to the sample plane, the defect exhibits a flexoelectric response. We demonstrate that the field-driven structural changes at the defect site involve changes in both azimuthal and polar angles defining the local director.

Fabrication and measurement of a flexoelectric micro-pyramid composite

A fabrication method by combining precision mechanical dicing and wet etching was developed to prepare micro-pyramid structures based on (Ba0.67Sr0.33)TiO3 ceramics. The effective piezoelectric properties of flexoelectric pyramid structures in ten micrometers scale were investigated and measured through converse flexoelectric effect. The scaling effect of the flexoelectric response was demonstrated as the structure size shrinks down. The results do suggest the great potential of flexoelectric micro pyramids as an alternative to lead-free piezoelectric material.

Breaking of macroscopic centric symmetry in paraelectric phases of ferroelectric materials andimplications for flexoelectricity.

A centrosymmetric stress cannot induce a polar response in centric materials; piezoelectricity is, for example, possible only in non-centrosymmetric structures. An exception is metamaterials with shape asymmetry, which may be polarized by stress even when the material is centric. In this case the mechanism is flexoelectricity, which relates polarization to a strain gradient. The flexoelectric response scales inversely with size, thus a large effect is expected in nanoscale materials. Recent experiments in polycrystalline, centrosymmetric perovskites (for example, (Ba,Sr)TiO3) have indicated values of flexoelectric coefficients that are orders of magnitude higher than theoretically predicted, promising practical applications based on bulk materials. We show that materials with unexpectedly large flexoelectric response exhibit breaking of the macroscopic centric symmetry through inhomogeneity induced by the high-temperature processing. The emerging electro-mechanical coupling is significant and may help to resolve the controversy surrounding the large apparent flexoelectric coefficients in this class of materials.

Giant flexoelectricity in Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite

Enhanced flexoelectricity in perovskite ceramics and single crystals has been reported before. In this letter, 3-3 ceramic-ceramic Ba0.6Sr0.4TiO3/Ni0.8Zn0.2Fe2O4 composite with a colossal permittivity was employed in the conventional pure bending experiment in order to examine the transverse flexoelectric response. The measured flexoelectric coefficient at 30 Hz is 128 μC/m and varies to 16 μC/m with the frequency increasing from 30 Hz to 120 Hz, mainly due to the inverse correlation between the permittivity and the frequency. This result reveals the permittivity dependence of flexoelectric coefficient in the frequency dispersion materials, suggesting that the giant permittivity composites can be good flexoelectric materials.

Temperature dependence of flexoelectric response in ferroelectric and relaxor polymer thin films

We report the temperature dependence of the flexoelectric response in thin films of both ferroelectric and relaxor forms of vinylidene fluoride polymers. The ferroelectric samples were depoled to minimize piezoelectric response by heating them beyond their Curie temperature and then cooling in zero applied electric field. In both the relaxor ferroelectric polymer and the paraelectric state of the ferroelectric copolymer, the flexoelectric coefficient was proportional to the dielectric constant over a limited range of temperatures, in agreement with general theoretical principles. The enhancements in flexoelectric response were also observed near the Curie transition temperature for the ferroelectric polymer and near the dielectric relaxation temperature for the relaxors. The broad dielectric anomaly in these systems provides greater temperature stability for these enhancements.