Converse Flexoelectric Effect Research Articles

Modeling mechanical waves propagation in flexoelectric solids

In this paper, the propagation of mechanical waves in flexoelectric solids with the consideration of both the direct and converse flexoelectric effects is studied via a collocation mixed finite element method (MFEM). The dynamic effects associated with mechanical waves propagation are accounted by introducing the kinetic energy in the Hamilton’s principle. In the proposed collocation MFEM, a quadratic polynomial is independently assumed for each component of the mechanical strain and electric field. The independently assumed mechanical strain and electric field are collocated with their counterparts computed from the displacement and electric potential at 9 Gaussian quadrature points. Thus, except for the fundamental field variables, no additional degrees of freedom (DOFs) are introduced. By performing the numerical experiments using the collocation MFEM, it is found that due to the direct flexoelectric effect, the propagation of mechanical waves can result in electric polarization in materials. Besides, the converse flexoelectric effect can induce mechanical waves when there are non-uniform transient electric field applied to the material. Numerical results indicate that by increasing the loading speed of the time varying mechanical displacement load, the direct flexoelectric effect associated with the mechanical strain gradient could be significantly enhanced.

Modeling the flexoelectric effect in semiconductors via a second-order collocation MFEM

In this paper, the flexoelectric effect in flexoelectric semiconductors (FSs) is studied via a second-order collocation mixed finite element method (MFEM) with the consideration of the strain gradient, electric field gradient, and drift-diffusion motion of carriers. In the present second-order collocation MFEM, the mechanical strain is assumed to be an independent quadratic polynomial function of local coordinates, and the kinematic constraint between the mechanical strain and displacement is approximately satisfied through a collocation method at 9 Gaussian quadrature points in each element. The setup for the electric field and electric potential also follows similar collocation strategy. Thus, except for the fundamental field variables, no additional DOFs are introduced. The accuracy of the proposed second-order collocation MFEM is verified by comparing numerical results with analytical solutions for a cantilever beam example. By performing the numerical experiments using the second-order collocation MFEM, the flexoelectric effect in FSs is simulated. Besides, we propose a novel mechanism for the converse flexoelectric effect in FSs which is induced by the diffusion motion of electrons. Numerical results indicate that by intensifying the diffusion motion of electrons, the converse flexoelectric effect, associated with the electric field gradient, could be enhanced.

Dynamic Analysis of Flexoelectric Actuated Truncated Conical Shells

Conical shells are widely used in aerospace engineering, such as for rocket fairings, satellite-rocket joints, load-bearing structures of solid rocket motor cases, and so on. In this study, the truncated conical shell with a flexoelectric actuator was discussed, and the precision flexoelectric actuation and control effectiveness were evaluated. A flexoelectric patch covered with an electrode layer was attached to the external surface of an elastic truncated conical shell, and a line electrode was longitudinally attached to the external surface of the patch. When a control voltage was applied to the line electrode and the electrode layer, an inhomogeneous electric field was generated; consequently, the stresses were induced in the flexoelectric patch due to the converse flexoelectric effect, thus regulating the vibration of the conical shell. The dynamic model of the truncated conical shell with a flexoelectric actuator was established. Then, the actuation effects of the flexoelectric actuator on the conical shell were investigated in terms of the microscopic behaviors, that is, flexoelectric-induced stresses, membrane control forces, bending control moments, and modal forces. Furthermore, the influences of several design parameters (that is, the line-electrode radius, conical shell thickness, flexoelectric patch thickness, and conical shell half-apex angle) on the flexoelectric actuation behaviors were also investigated.

Flexoelectricity and piezoelectric effects in poly (vinyl alcohol)-SrTiO3 nanocomposites

This work investigates the structural, dielectric, mechanical, and apparent piezoelectric properties of PVA-SrTiO3 nanocomposite films by varying the concentration of SrTiO3 nanoparticles. It was shown that the piezoelectric coefficient d33, dielectric constant, elastic modulus, and hardness strongly depend upon nanoparticle concentration; the strong molecular interaction between constituents leads to the formation of an interface layer between PVA matrix and SrTiO3. We propose that the apparent piezoelectric coefficient relates to the converse flexoelectric effect and Maxwell stress induced by an electric field. The proper interface layer formation strongly depends upon nanoparticle concentration; when the concentration increases, their agglomeration drives the destruction of the interface such that the properties of the nanocomposites are diminished. By fine-tuning the interface layer of PVA nanocomposite with nanoparticles functionalizing by polyvinylpyrrolidone, the films exhibit an unexpectedly high apparent piezoelectric coefficient (d33=35 pC/N); this value is higher than neat SrTiO3 crystals and polymer composites with flexoelectric properties.

Dynamic Analysis of Cylindrical Shells with Flexoelectric Actuation

The converse flexoelectric effect can be utilized to actuate and control flexible structures. Assessing the precision flexoelectric actuation and control effectiveness of cylindrical shells is the main objective of this study. The application of voltage to the line electrode on the external surface of a flexoelectric patch, coupled with the bottom surface electrode layer, causes an inhomogeneous electric field to be generated, which consequently induces internal actuation stresses. The stress-induced membrane control force and bending control moment generate modal control effects, which regulate the cylindrical shell vibration. For an elastic cylindrical shell with flexoelectric actuation, the modal forces and microscopic behaviors are studied to evaluate the flexoelectric actuation effects. For the transverse direction, the electric field gradient drops sharply with increasing distance from the line electrode; thus, the induced internal actuation stresses concentrate on the upper surface. Due to the inhomogeneous electric field gradient distributions, the membrane control force and bending control moment concentrate like spikes. To evaluate the actuation effects, modal force distributions of the elastic cylindrical shell with flexoelectric actuation are analyzed with respect to design parameters, such as the flexoelectric patch thickness, line-electrode radius, cylindrical shell thickness, and shell radius.

Continuum model for converse flexoelectricity in a thin plate

Flexoelectricity – the effect where inhomogeneous polarization causes deformation – substitutes symmetry-forbidden piezoelectricity in nano-scale centrosymmetric materials. Its mathematical treatment in terms of continuum mechanics represents a challenge: correct models are subjects of debates during the past decades. In the present article we formulate a model with reasonable approximations for converse flexoelectric effect in a plate. Except for very specific boundary conditions the theory inevitably include nonclassical higher order terms. These are treated using variational calculus and are essentially reduced to a modification of the boundary conditions for the elastic variables. Analytical solutions are obtained for a circular plate with round electrodes, where deflections are proportional to the inverse square of plate thickness. For mechanically free boundary conditions at plate’s edges deflection grows monotonically with the electrode radius. In contrast, a plate with clamped edge does not deform at all when fully electroded, an optimal electrode size is found rendering maximal deflection.

A gradient electromechanical theory for thin dielectric curved beams considering direct and converse flexoelectric effects

This work presents the development of a unified gradient electromechanical theory for thin flexoelectric beams considering both direct and converse flexoelectric effects. The two-way coupled electromechanical theory is developed starting from 3D variationals formulation by considering an electric field-strain-based free energy function. The formulation incorporates mechanical as well as electrical size effects. The coupled 3D theory is specialized to isotropic materials, and a 1D beam theory for composite flexoelectric curved beams is derived using the classical Kirchhoff assumptions. The beam theory is solved using a novel \(C^2\) continuous finite element framework for different loading and boundary conditions. Our finite element results are verified with analytical solutions for a simply supported flexoelectric beam operating in both actuator and sensor modes. The results are also compared with existing literature for the special case of a passive micro-beam. Our computational framework is subsequently used to perform various parametric studies to analyze the effect of electrical and mechanical length scale parameters, geometric parameters like the radius of curvature, flexoelectric layer thickness etc., on the response of the beam. Also, contribution of converse flexoelectricity in the overall response of the flexoelectric beam is compared with that of the direct effect. Our simulation results predict that the converse effect is significant (\(\approx \) 10–25% of the direct effect) for a wide range of thickness and length scale parameter values. It is also observed that the effective electromechanical coupling coefficient, calculated in terms of the voltage developed across the flexoelectric layer thickness, is higher in flexoelectric materials compared to piezoelectric materials at smaller length scales (thickness of the order of a few microns). Our simulation results also agree well with the trends observed in recent experiments [1].

Flexoelectric properties of multilayer two-dimensional material MoS2

Two-dimensional (2D) materials exhibit a high strength and flexibility along with unique electrical–mechanical multiphysics properties. In this study, we experimentally demonstrated the electromechanical response of a multilayer 2D material, 2H-phase MoS2, by using a piezoresponse force microscope. In particular, the dominant physical quantity of the deformation response was determined by independently controlling the electric field and electric field gradient by changing the probe shape and material thickness (number of layers). The multilayer MoS2 exhibited an out-of-plane electrical–mechanical deformation response that followed and was inverted with respect to positive and negative voltages, respectively. Moreover, the relationships between the electric field gradient and strain were similar for all shapes of the probe tip and film thickness values. This result indicated that the electrical–mechanical response of this material was dominated by the electric field gradient, and the strain could be attributed to the converse flexoelectric effect. The findings can provide guidance for the realization of ultrathin electromechanical devices.

Direct and Converse Flexoelectricity in Two-Dimensional Materials.

Building on recent developments in electronic-structure methods, we define and calculate the flexoelectric response of two-dimensional (2D) materials fully from first principles. In particular, we show that the open-circuit voltage response to a flexural deformation is a fundamental linear-response property of the crystal that can be calculated within the primitive unit cell of the flat configuration. Applications to graphene, silicene, phosphorene, boron nitride, and transition-metal dichalcogenide monolayers reveal that two distinct contributions exist, respectively of purely electronic and lattice-mediated nature. Within the former, we identify a key metric term, consisting in the quadrupolar moment of the unperturbed charge density. We propose a simple continuum model to connect our findings with the available experimental measurements of the converse flexoelectric effect.

Strongly enhanced electromechanical coupling in atomically thin transition metal dichalcogenides

Flexoelectricity in thin films has emerged as an effective electromechanical response owing to appealing scaling law and universal existence. However, current studies show limited out-of-plane converse flexoelectric effect (CFE) of ultra-thin transition metal dichalcogenides (TMDs) when compared to their conventional in-plane piezoresponse. Here, we report converse flexoresponse of atomically thin TMDs such as molybdenum disulfide (MoS2) and tungsten diselenide (WSe2) which exceeds their intrinsic in-plane piezoresponses. Our piezoresponse force microscopy (PFM) measurements revealed strongly enhanced CFE of the atomically thin MoS2 and WSe2 than their bulk counterpart (∼700% enhancement in MoS2, ∼400% enhancement in WSe2). We observed an anomalous reduction in converse flexoresponse in the monolayer structure attributed to a puckering deformation. By inducing a built-in in-plane tension to reduce puckering, we estimated the CFE of monolayer WSe2 to be 8.14 pm/V, the highest among the atomically thin TMDs.

Observation of a stable fractionalized polar skyrmionlike texture with giant piezoelectric response enhancement

We report the creation of stable fractionalized polar skyrmionlike structures via flexoelectric effect in a prototypical multiferroic ${\mathrm{BiFeO}}_{3}$ thin film, which are directly imaged by angle-resolved piezoresponse force microscopy technique. We also demonstrate an approach to enhance piezoresponse at the nanoscale. Phase-field simulations reveal that a tip-driven flexoelectric effect allows the formation of an ordered polar skyrmionlike configuration that is topologically protected by the converse flexoelectric effect. Our findings open up avenues for designing and controlling spatially localized topological textures in rhombohedral ferroelectrics, and also could contribute to future developments for ferroelectric-based nanoelectronic devices.

Strategies to instigate superior electromechanical response in dielectric materials via converse flexoelectricity

Various strategies to achieve superior converse flexoelectric response, qualitatively and quantitatively, in dielectric materials are presented. For the qualitative aspect, a novel approach for achieving converse flexoelectric effect independent of geometry or material composition of dielectric materials, by having modified electrical boundary conditions is presented. The theoretical investigation of the proposed methodology is presented and compared with the existing methods (functional grading and trapezoid-shaped sample). Later, the hybrid configurations of existing methods with the proposed strategy are analyzed and are found to enhance the actuation in some cases while decreasing in others, suggesting the possibility of a quantitatively superior effect. While the proposed method of varying electric field direction is established as an alternative to existing methods to induce electric field gradient, the FGM is found to exhibit up to 380% actuation of the closest non-hybrid strategy, i.e. trapezoid geometry.

Electromechanical analysis of direct and converse flexoelectric effects under a scanning probe tip

Flexoelectricity refers to phenomena that polarization is coupled with strain gradient and stress is coupled with electric gradient. It occurs in all dielectrics, and becomes significant under a sharp conductive tip in piezoresponse force microscopy (PFM), under which large gradients in electromechanical field naturally arise. Here we analyze direct and converse flexoelectric effects under such a scanning probe, using mixed finite element method (FEM) developed under the phenomenological continuum framework. The FEM was first validated by analytic solutions for simple axially-symmetric tubes, and then applied to analyze two different modes of PFM experiments. It reveals that the flexoelectric effect accounts for less than 20% of the measured piezoresponse in a typical piezoelectric material, while mechanical switching via flexoelectricity is only possible for ferroelectric materials having upward polarization, with the switching zone confined to a small region near surface. These analyses explain a number of experimental observations well, and shed insight into complex electromechanical phenomenon under a sharp PFM tip. The theoretical and computational framework developed can also be applied to study flexoelectric effect in other structural configurations.

Geometry Independent Direct and Converse Flexoelectric Effects in Functionally Graded Dielectrics: An Isogeometric Analysis

In the present study, a functionally graded material (FGM) based dielectric composite is investigated for direct and converse flexoelectric effects induced due to material inhomogeneity. An Isogeometric analysis based formulation is employed to effectively capture the gradient terms appearing in the electro-elastic coupling constitutive laws of flexoelectricity. Non-uniform stresses and strains generated due to non-uniform elastic properties in the domain induce flexoelectricity inherently without any special geometric and/or boundary conditions. A two-dimensional (2D) FGM structure made by the combination of polyvinylidene fluoride (PVDF) and barium titanate (BaTiO3) is studied under mechanical and thermal loading conditions for its flexoelectric response. The induced polarization and voltage along with its dependence on grading index (n) is analyzed. Further, to extend the study to converse flexoelectricity, the mechanical response of the FG flexoelectric material under an applied electric potential is studied. Finally, the actuation of the structure by an applied electric field is studied. As high as 10% actuation of original displacement is observed for higher values of n. Effectiveness of the grading concept to induce flexoelectric response is established through numerical studies. Longitudinal and transverse modes of flexoelectricity are scrutinized and the concept of grading is proved to inherently induce the flexoelectricity in dielectrics without special geometric modifications.

Carbon nanotube effect on the reverse flexoelectricity of a planar liquid crystal cell

ABSTRACT Unique properties of carbon nanotubes (CNTs) such as high anisotropy could be utilised in liquid crystal (LC) medium and it may improve the physical properties of the host medium. In this regard, enhancement of permittivity and flexoelectric coefficients of the medium is calculated through doping an LC planer cell with single-walled carbon nanotubes (SWCNTs), here. These calculations consider the effect of the applied electric field. A splay type curvature orientation is induced in the medium as a result of the converse flexoelectric effect. It is reported experimentally by Petrov et al. that the presence of SWCNTs enhances the curvature orientation and its polarisation. We discussed the case theoretically and calculated the variations of the mentioned physical properties through doping with SWCNTs. A generalised Maxwell-Garnett mixing rule is considered to calculate the effective permittivity of the medium and extend it to estimate the modified flexoelectric coefficients of the medium. In low concentrations of SWCNTs, the model calculates the increase of the dielectric anisotropy and flexoelectric coefficient up to twofold and fivefold, respectively, which is in good agreement with reported data by other researches.

Flexoelectric control of beams with atomic force microscope probe excitation

Based on the converse flexoelectric effect, flexoelectric actuator is designed and used to control the dynamic displacement of cantilever beams. First, shell-type stress expression based on double-curvature shell induced by the converse flexoelectric effect is developed, which can be simplified to a flexoelectric-laminated cantilever beam by applying two Lamé parameters and beam radius of curvature. Then, the flexoelectric actuator is designed with a conductive atomic force microscope probe and a flexoelectric layer. An inhomogeneous electric field is generated when the external voltage is applied on the atomic force microscope probe and the flexoelectric layer, which leads to stress in the longitudinal direction of beam and control moment. With the flexoelectric-induced bending moment, displacement induced by the external force and flexoelectric actuator is derived. The displacement is related to many parameters, such as actuation voltage, atomic force microscope probe radius and flexoelectric layer thickness. Cases are studied to optimize the control effect with different parameters. Results show that vibration control effect is enhanced with a higher actuation voltage and a smaller atomic force microscope probe radius for each mode. Besides, the thicker flexoelectric layer enhances the control effect with a larger bending moment arm for each mode. Dynamic vibration is controlled effectively by converse flexoelectric effect.

Investigation of the converse flexoelectric effect in KTaO3 using the interferometric microscopy method

The inhomogeneous strain induced by an external electric field (the converse flexoelectric effect) has been studied in thin KTaO3 single crystal plates by a scanning interferometric microscopy method. It has been shown that the type of inhomogeneous strain (spherical and cylindrical bending) is determined by the direction of the applied field along [001] and [011] crystallographic directions.

A Scanning Interferometric Method for Studying the Converse Flexoelectric Effect in Thin Plates of Ferroelectrics and Related Materials

A scanning interferometric method for studying inhomogeneous deformations of thin plates of dielectric crystals under the action of a homogeneous electric field (converse flexoelectric effect) is proposed. The results of using this method to determine the type and magnitude of inhomogeneous deformations (spherical and cylindrical bending deformations) with an accuracy of up to 10 nm in model perovskite ferroelectric single crystals and related materials are demonstrated.

An actuation method by a biconcave beam structure with converse flexoelectric effect

Converse flexoelectricity describes the linear relationship between an electric field gradient and the mechanical stress in dielectric materials. It is not restricted by the Curie temperature limit, does not require advanced electrical poling and a wide range of candidate materials exist; hence, there is growing interest in converse flexoelectricity for application in actuators. In this work, a biconcave beam structure and actuation method based on the converse flexoelectric effect is presented. Theoretical and finite element analyzes were developed for the relationship between the electric field gradient and the geometric parameters. A non-piezoelectric dielectric material, polyvinylidene fluoride, was experimentally applied to validate the designed effect. The experimental results show that the designed structure outputs markedly larger displacements magnitudes than the control specimens. This work provides a design method for converse flexoelectric actuators and further enhances the application prospects of converse flexoelectricity in all solid dielectric materials.

Multiflexoelectric Actuation and Control of Beams

Flexoelectric materials subject to inhomogeneous electric field can actuate and control flexible structures based on the converse flexoelectric effect. An atomic force microscope (AFM) probe placed on top of a flexoelectric patch can generate a nonuniform electric field with an excitation control voltage, and consequently, an internal actuation stress is generated in the flexoelectric material. The induced flexoelectric membrane force and bending moment, in turn, influence the vibration of the structure. Because of high stress concentration of the flexoelectric actuation, multiple AFM probes/patches are necessary to alleviate potential fracture. In this study, the vibration control effectiveness of an elastic cantilever beam with multiple AFM probes/patches placed at optimal locations is evaluated. The beam is excited initially with an external mechanical force. In case studies, both open-loop and closed-loop control methods are imposed, and optimal combinations of three actuator positions are chosen to minimize the tip displacement. For the open-loop responses, amplified tip displacements are also observed due to overactuation. Displacement feedback control is proposed. Free vibration, and open-loop and closed loop vibration control are compared for the first three beam modes. The results prove that multiple flexoelectric actuators can effectively accelerate the beam damp process with closed-loop control.