Prediction of flexoelectricity in BaTiO_(3) 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 BaTiO_(3) and the phase transition temperatures have been investigated, and the results show that the core-shell potential model is effective and the structure of BaTiO_(3) is stable in a temperature range of 10 K–150 K. Molecular dynamics simulated hysteresis loops of BaTiO_(3) 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 BaTiO_(3) is 18.5 nC/m. This work provides a computational framework for the study of flexoelectric effect by using molecular dynamics.

Computational Machine Learning Representation for the Flexoelectricity Effect in Truncated Pyramid Structures

In this study,machine learning representation is introduced to evaluate the flexoelectricity effect in truncated pyramid nanostructure under compression.A Non-Uniform Rational B-spline(NURBS)based IGA formulation is employed to model the flexoelectricity.We investigate 2D system with an isotropic linear elastic material under plane strain conditions discretized by 45×30 grid of B-spline elements.Six input parameters are selected to construct a deep neural network(DNN)model.They are the Young's modulus,two dielectric permittivity constants,the longitudinal and transversal flexoelectric coefficients and the order of the shape function.The outputs of interest are the strain in the stress direction and the electric potential due flexoelectricity.The dataset are generated from the forward analysis of the flexoelectric model.80%of the dataset is used for training purpose while the remaining is used for validation by checking the mean squared error.In addition to the input and output layers,the developed DNN model is composed of four hidden layers.The results showed high predictions capabilities of the proposed method with much lower computational time in comparison to the numerical model.

Three-Dimensional Isogeometric Analysis of Flexoelectricity with MATLAB Implementation

Flexoelectricity is a general electromechanical phenomenon where the electric polarization exhibits a linear dependency to the gradient of mechanical strain and vice versa.The truncated pyramid compression test is among the most common setups to estimate the flexoelectric effect.We present a three-dimensional isogeometric formulation of flexoelectricity with its MATLAB implementation for a truncated pyramid setup.Besides educational purposes,this paper presents a precise computational model to illustrate how the localization of strain gradients around pyramidal boundary shapes contributes in generation of electrical energy.The MATLAB code is supposed to help learners in the Isogeometric Analysis and Finite Elements Methods community to learn how to solve a fully coupled problem,which requires higher order approximations,numerically.The complete MATLAB code which is available as source code distributed under a BSD-style license,is provided in the part of Supplementary Materials of the paper.

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Nonlinear thickness-shear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales

This paper presents a nonlinear thickness-shear vibration model for onedimensional infinite piezoelectric plate with flexoelectricity and geometric nonlinearity.The constitutive equations with flexoelectricity and governing equations are derived from the Gibbs energy density function and variational principle.The displacement adopted here is assumed to be antisymmetric through the thickness due to the thickness-shear vibration mode.Only the shear strain gradient through the thickness is considered in the present model.With geometric nonlinearity,the governing equations are converted into differential equations as the function of time by the Galerkin method.The method of multiple scales is employed to obtain the solution to the nonlinear governing equation with first order approximation.Numerical results show that the nonlinear thickness-shear vibration of piezoelectric plate is size dependent,and the flexoelectric effect has significant influence on the nonlinear thickness-shear vibration frequencies of micro-size thin plates.The geometric nonlinearity also affects the thickness-shear vibration frequencies greatly.The results show that flexoelectricity and geometric nonlinearity cannot be ignored in design of accurate high-frequency piezoelectric devices.

Flexoelectricity in oxide thin films

Flexoelectric effect describes the electromechanical coupling between the strain gradient and its internal polarization in all dielectrics.Despite this universality,the resulting flexoelectric field remains small at the macroscopic level.However,in nanosystems,the size-dependent effect of flexoelectricity becomes increasingly significant,leading to a notable flexoelectric field that can strongly influence the material’s physical properties.This review aims to explore the flexoelectric effect specifically at the nanoscale.We achieve this by examining strain gradients generated through two distinct methods:internal inhomogeneous strain and external stimulation.In addition,advanced synthesis techniques are utilized to enhance the properties and functionalities associated with flexoelectricity.Furthermore,we delve into other coupled phenomena observed in thin films,including the coupling and utilization of flexomagnetic and flexophotovoltaic effects.This review presents the latest advancements in these areas and highlights their role in driving further breakthroughs in the field of flexoelectricity.

Flexoelectricity-enhanced photovoltaic effect in trapezoid-shaped NaNbO_(3)nanotube array composites

In this work,we successfully prepared vertically aligned NaNbO_(3)nanotube(NN-NT)with trapezoidal shapes,in which the orthorhombic and monoclinic phases coexisted.According to the structure analysis,the NN-NT/epoxy composite film had excellent flexoelectric properties due to the lattice distortion caused by defects and irregular shape.The flexoelectric effect is the greatest in the vertical direction in the flexible NN-NT/epoxy composite film,and the flexoelectric coefficient()is 2.77×10^(−8)C·m^(−1),which is approximately 5-fold higher than that of the pure epoxy film.The photovoltaic current of the NN-NT/epoxy composite film increased from 39.9 to 71.8 nA·cm^(−2)in the direction of spontaneous polarization when the sample was bent upward due to the flexoelectricity-enhanced photovoltaic(FPV)effect.The flexoelectric effect of the NN-NT/epoxy composite film could modulate the photovoltaic response by increasing it by 80%or reducing it to 65%of the original value.This work provides a new idea for further exploration in efficient and lossless ferroelectric memory devices.

Static and Dynamic Analysis of a Piezoelectric Semiconductor Cantilever Under Consideration of Flexoelectricity and Strain Gradient Elasticity

In this work,the static and dynamic response of a piezoelectric semiconductor cantilever under the transverse end force with consideration of flexoelectricity and strain gradient elasticity is systematically investigated.The one-dimensional governing equations and the corresponding boundary conditions are derived based on Hamilton’s principle.After that,combining with the linearized equations for the conservation of charge,the effects of characteristic length and flexoelectric coefficient on the working performance of a ZnO nanowire are demonstrated as a numerical case,including the static mechanical and electric fields,natural frequencies,and the frequency–response characteristics at resonances.The results indicate that the flexoelectric effect has a great influence on the electric properties of the nanowire,while the strain gradient effect directly contributes to its mechanical properties.To some extent,the increase in characteristic length is equivalent to the stiffness strengthening.The qualitative results and quantitative data are beneficial for revealing the underlying physical mechanism and provide guidance for the design of piezoelectric semiconductor devices.

Flexoelectricity in dielectrics:Materials,structures and characterizations

Flexoelectricity in dielectrics suggests promising smart structures for sensors,actuators and transducers.In this review,dielectric materials,structures and the associated flexoelectric characterization methods are presented.First of all,we review structures and methods to measure different flexoelectric coefficients,including
μ1122;μ1111;μ1211;μ3121;μ2312;μ1123,etc.,via direct or converse flexoelectric effect.The flexoelectric materials in the form of bulk,thin films and 2D materials and the reported flexoelectric properties of these dielectrics will then be discussed.Semiconductor materials and the associated flexoelectric studies will also be reviewed.The progress of flexoelectric device study will next be presented,followed by the flexoelectricity research challenges and future trend.

Universal converse flexoelectricity in dielectric materials via varying electric field direction

Flexoelectricity is a symmetry independent electromechanical cou-pling phenomenon that outperforms piezoelectricity at micro and nanoscales due to its size-dependent behavior arising from gradi-ent terms in its constitutive relations.However,due to this gradient term flexoelectricity,to exhibit itself,requires specially designed geometry or material composition of the dielectric material.First of its kind,the present study put forward a novel strategy of achieving electric field gradient and thereby converse flexoelectri-city,independent of geometry and material composition of the material.The spatial variation of the electric field is established inside the dielectric material,Ba_(0.67)Sr_(0.33)TiO_(3)(BST),by manipulating electrical boundary conditions.Three unique patterns of electrode placement are suggested to achieve this spatial variation.This varying direction of electric field gives rise to electric field gradient,the prerequisite of converse flexoelectricity.A multi-physics cou-pling based theoretical framework is established to solve the flexo-electric actuation by employing isogeometric analysis(IGA).Electromechanically coupled equations of flexoelectricity are solved to obtain the electric field distribution and the resulting displace-ments thereby.The maximum displacements of 0.2 nm and 2.36 nm are obtained with patterns I and II,respectively,while pattern III can yield up to 85 nm of maximum displacement.

A Deep Learning Approach to Shape Optimization Problems for Flexoelectric Materials Using the Isogeometric Finite Element Method

A new approach for flexoelectricmaterial shape optimization is proposed in this study.In this work,a proxymodel based on artificial neural network(ANN)is used to solve the parameter optimization and shape optimization problems.To improve the fitting ability of the neural network,we use the idea of pre-training to determine the structure of the neural network and combine different optimizers for training.The isogeometric analysis-finite element method(IGA-FEM)is used to discretize the flexural theoretical formulas and obtain samples,which helps ANN to build a proxy model from the model shape to the target value.The effectiveness of the proposed method is verified through two numerical examples of parameter optimization and one numerical example of shape optimization.

Flexoelectricity enhanced water splitting and hydrogen evolution reaction on grain boundaries of monolayer transition metal dichalcogenides

Our extensive first-principles calculations reveal that the chemical activities of monolayer transition metal dichalcogenides(TMDs)MX_(2)(M=Mo or W,and X=Te,Se,or S)for water splitting and hydrogen evolution are modified and promoted on their grain boundaries(GBs)when in-plane tensile loadings are applied.Compared with monolayer TMDs without GBs,the flexoelectricity induced by nonuniform deformation and strain gradient significantly enhances the charge polarizations of X and M atoms at the GB sites of monolayer TMDs,which facilitates the dissociation of water molecules on the GB sites and reduces the reaction barrier of hydrogen evolution reaction.The energy barriers of splitting water molecules and hydrogen adsorption free energies on the GB sites decrease with increasing the flexoelectric effect.These results highlight an attractive way of utilizing the flexoelectric effect of GB-containing TMDs to improve their surface catalytic capability for hydrogen generation.

Bandgap Analysis of Periodic Composite Microplates with Curvature-Based Flexoelectricity:A Finite Element Approach

A finite element model is proposed permitting prediction of elastic wave bandgaps of periodic composite microplates incorporating flexoelectric effect.In this model,we applied curvature-based flexoelectricity and Mindlin plate theories and derived a finite element formulation that has been implemented for bandgap analysis.The finite element model utilizes a three-node triangle element with 30 degrees of freedom satisfying Mindlin kinematics assumptions.It is based on a non-conforming interpolation scheme which provides nodal C^(1) continuity and ensures compatibility with curvature-based flexoelectricity.The approach accounts for microstructure effects and,owing to the triangular element topology,can be used to assist the design of microplates with complex microstructures.Validation of the approach is performed through comparison with both analytical and numerical models,in which the effect of flexoelectricity on the bandgap is studied based on cases demonstrating size dependence.

Determining the sum of flexoelectric coefficients in nematic liquid crystals by the capacitance method

A detailed theoretical analysis of determining the sum of flexoelectric coefficients in nematic liquid crystals using the capacitance method is given. In the strong anchoring parallel aligned nematic （PAN） and hybrid aligned nematic （HAN） cells, the dependences of the capacitance on the sum of flexoelectric coefficients and the applied voltage are obtained by numerical simulations, and the distributions of the director and the electric potential for different applied voltages and flexoelectric coefficients are also given. Based on this theoretical analysis, we propose an experimental design for measuring the capacitance of a liquid crystal cell using the improved precision LCR meter E4980A （Agilent）. Through comparing the experimental data with the simulated results, the sum of flexoeletric coefficients can be determined.

Size-dependent electromechanical properties in piezoelectric superlattices due to flexoelectric effect

Piezoelectric superlattice is a potential component for nanoelectromechanical systems. Due to the strong nonlocal effect such as flexoelectric effect at interfaces, classical piezoelectric theory is unable to accurately describe the electromechanical response of piezoelectric superlattice at nanoscale scale. Based on the previous nonlocal thermodynamics theory with flexoelectric effect Liu et al. （2016）, the size- dependent electromechanical properties of piezoelectric superlattices made of BaTiO3 （BTO） and PbTiO3 （PRO） layers are investigated systematically in the present work. Giant strain gradient is found near the interface between BTO and PTO layers, which leads to the significant enhancement of polarization in the superlattice due to the flexoelectric effect. For the piezoelectric BTO-PTO superlattices with different unit- cell sizes, the thickness of interface with nontrivial strain gradient is almost constant. The influence of strain gradient at the interface becomes significant when the size of superlattice decreases, As a result, a strong size dependence of electromechanical properties is predicted for the piezoelectric BTO-PTO superlattices, In particular, for the superlattices with a specific thickness ratio of BTO and PTO layers, the piezoelectric response can be several times larger than that of bulk structure. The present work demonstrates a practical wast to design the piezoelectric superlattices with high piezoelectric coefficient by using the nonlocal effect at nanoscale.

Coupling Enhancement of a Flexible BiFeO_(3) Film-Based Nanogenerator for Simultaneously Scavenging Light and Vibration Energies

Coupled nanogenerators have been a research hotspot due to their ability to harvest a variety of forms of energy such as light,mechanical and thermal energy and achieve a stable direct current output.Ferroelectric films are frequently investigated for photovoltaic applications due to their unique photovoltaic properties and bandgap-independent photovoltage,while the flexoelectric effect is an electromechanical property commonly found in solid dielectrics.Here,we effectively construct a new form of coupled nanogenerator based on a flexible BiFeO_(3) ferroelectric film that combines both flexoelectric and photovoltaic effects to successfully harvest both light and vibration energies.This device converts an alternating current into a direct current and achieves a 6.2% charge enhancement and a 19.3%energy enhancement to achieve a multi-dimensional"1+1>2"coupling enhancement in terms of current,charge and energy.This work proposes a new approach to the coupling of multiple energy harvesting mechanisms in ferroelectric nanogenerators and provides a new strategy to enhance the transduction efficiency of flexible functional devices.

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 4-1 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.

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.

Flexo-photocatalysis in centrosymmetric semiconductors

The separation of photogenerated electron–hole pairs is vitally important for photocatalysis,which can be effectively promoted by polarization field.However,it only manifests in piezoelectric/pyroelectric/ferroelectric materials that have a non-centrosymmetric structure.Here,we demonstrate that the polarization enhanced photocatalysis(with wide spectra from ultraviolet(UV)light to visible light)can be achieved in centrosymmetric semiconductors,such asδ-MnO_(2) and TiO_(2) nanosheets integrated nanoflowers,by using the strain-gradient-induced flexoelectric polarization that is always overlooked in polarization-enhanced catalysis.Under ultrasonic and illumination excitation,the organic pollutants(methylene blue(MB),etc.)can be effectively degraded within 30 min with excellent stability and repeatability.Compared with photocatalysis,the flexo-photocatalytic performance of above centrosymmetric semiconductors is substantially increased by 85%.Moreover,the factors related to flexo-photocatalysis such as material morphology,mechanical stimuli source,and adsorption are explored to deeply understand the mechanism of flexo-photocatalysis.This work opens up a way for high-performance photocatalysis in centrosymmetric semiconductors.

Flexoelectric materials and their related applications: A focused review

Flexoelectricity refers to the mechanical-electro coupling between strain gradient and electric polarization, and conversely, the electro-mechanical coupling between electric field gradient and mechanical stress. This unique effect shows a promising size effect which is usually large as the material dimension is shrunk down. Moreover, it could break the limitation of centrosymmetry, and has been found in numerous kinds of materials which cover insulators, liquid crystals, biological materials, and semiconductors. In this review, we will give a brief report about the recent discoveries in flexoelectricity, focusing on the flexoelectric materials and their applications. The theoretical developments in this field are also addressed. In the end, the perspective of flexoelectricity and some open questions which still remain unsolved are commented upon.