Flexoelectric Energy Harvester Research Articles

Reduced-order modeling and usefulness of non-uniform beams for flexoelectric energy harvesting applications

Energy harvesting at micro- and nanoscales has recently seen a renewed interest that the flexoelectric effect can counter the inability of piezoelectric energy harvesters to generate enough energy at small scales. Almost all small-scale energy harvesters use uniform rectangular geometries, whereas at the macroscale energy harvesters use a wide array of geometries including tapered rectangular geometries. The incorporation of non-uniform effects into a piezoelectric system considering the flexoelectric effect should give insight into how these systems can benefit from different geometries. A non-uniform flexoelectric Euler–Bernoulli cantilever energy harvester is modeled using classical continuum theories and is examined at the microscale. The non-uniformity of the energy harvester is governed by linear and nonlinear tapering effects, with the nonlinearities represented by high-order polynomials. The system is assumed to be linear, only undergoing harmonic base excitation. The varied tapering ratios and powers of the geometric tapering, considering that only the thickness and the width of the beam are tapered, are compared with uniform systems. The results show that non-uniform beams exhibit more harvested power than their uniform counterparts and also increase the range of resonant frequencies where significant power can be generated. Nonlinear tapering increases the amount of power that could be harvested compared to linear tapering; however, the nonlinearity of the tapering effects is limited to cubic and quadratic forms. It is demonstrated that higher-order tapering effects reduce the amount of harvested power compared to the linear taper counterpart. Non-uniform beams prove to be more effective than their rectangular counterparts within a linear system, whereas optimal resistive loads decrease as the tapering effects increase.

Theoretical study of micro/nano-scale bistable plate for flexoelectric energy harvesting

Flexoelectricity offers a number of advantages in micro/nano-scale devices when compared with piezoelectricity as confirmed through numerous experiments and theoretical analyses. Snap-through behavior in bistable plate would convey a higher conversion efficiency for energy harvesting. However, there is no theoretical research on micro/nano-scale energy harvester with flexoelectricity based on bistable plate. This paper describes an electro-thermo-mechanical coupling system of micro/nano-scale bistable plates for piezoelectric energy-harvesting applications. The nonlinear geometric theory of Von Karman, strain gradient theory, the flexoelectric effect, and Hamilton’s principle were used to derive nonlinear dynamic formulas. The ambient vibration frequency and snap-through behavior both have an effect on the output voltages calculated by Matlab® software. An expression for the energy conversion efficiency of the bistable plate was derived and the influence of resistance on this efficiency was investigated. The result showed that the snap-through behavior delivered a large-amplitude vibration and a higher output voltage at the range of one-third super-harmonic resonance (2.8–3.5 MHz) for the micro/nano-scale bistable plate. This work will be helpful in the design of micro/nano-scale bistable plates with higher output voltages and improved conversion efficiencies for energy harvesting.

Nonlinear size dependent modeling and performance analysis of flexoelectric energy harvesters

In this study, the effect of flexoelectricity on rectangular prismatic beams is analytically investigated when considering various boundary conditions including fixed–fixed, fixed–hinged, and hinged–hinged. The dynamical system undergoes harmonic loading and has the von Karman strain nonlinearity due to the mid-plane stretching. The beams are assumed to be long and slender, allowing for Euler–Bernoulli beam analysis to be conducted. To account for the size dependent effects of the material, Eringen’s nonlocal elasticity theory is employed. After determining the exact mode shapes, the Galerkin discretization is utilized to derive the nonlinear reduced-order model. The fixed–hinged model is the only model that can harvest power due to its lack of boundary condition symmetry and is the model that is heavily examined. The harvested power from the macro to the micro scales is small, as is expected; however, at the nanoscale the softening effect due to the nonlocality increases the nonlinear hardening behavior and decreases the system’s natural frequency. This allows for lower excitation frequencies to harvest energy, with a broader range of frequencies being able to harvest energy. Micro and macro scales do not exhibit any variation based on nonlocal effects, nor can they harvest significant amounts of power through flexoelectricity alone. These nano-devices can be developed and can produce significantly more energy than previously thought; but they require more advanced modeling techniques that necessitate a more thorough understanding of non-classical continuum mechanics and its coupling to nonlinear dynamics.

Nonlinear analysis of flexoelectric energy harvesters under force excitations

This work focuses on theoretical modeling and dynamic analysis of a flexoelectric beam energy harvester with the consideration of geometric nonlinearity. We assume that the beam energy harvester has a thickness at nanoscale and a width and length at microscale. Under a harmonic concentrated force, the proposed energy harvester generates electric power due to the phenomenon of flexoelectricity. Based on the theory of flexoelectricity and Euler–Bernoulli beam assumption, the nonlinear electromechanical coupling equations are derived. Galerkin method is employed to obtain the discrete nonlinear equations, which are then solved numerically. Frequency response curves are plotted to reveal the nonlinear characteristics of the energy harvester and it is found that the frequency response curves of the flexoelectric energy harvester exhibit hardening behaviors. Case studies are provided and we emphasize on the influences of load resistance, tip mass, concentrated force and damping ratio on the output performance of the energy harvester. In addition, it is suggested that the flexoelectric energy harvester has a better output performance under a square waveform force among different forms of the concentrated force. The results obtained are significant for designing optimal flexoelectric energy harvesters.

Energy harvesting properties of the functionally graded flexoelectric microbeam energy harvesters

Flexoelectricity may be an excellent candidate for piezoelectricity in micro and nano scaled energy harvesters due to its strong size dependency. In this paper, a size dependent analytical model of the cantilever-based functionally graded flexoelectric energy harvesters is developed. The Hamilton's principle is used to derive the governing equations. By means of the Galerkin's method, the approximated closed-form solutions of electrical output and energy conversion efficiency are obtained. For the functionally graded flexoelectric energy harvester, the effective flexoelectric response is controlled by not only the flexocoupling coefficient but also its first derivative. Numerical results of a 3 μm-thick Polyvinylidene Fluoride/Strontium Titanate composed functionally graded flexoelectric energy harvester demonstrate that when the volume fraction exponent varies from zero to infinity, the optimal working frequency gradually reduces from 41407 Hz to 7761 Hz and the optimal load resistance gradually increases from 0.99 MΩ to 83.91 MΩ. Meanwhile, shrinking the thickness from 3 μm to 0.3 μm will highly increase the normalized power density and the energy conversion efficiency about one and two orders, respectively. Moreover, when the strain gradient elastic coefficient becomes larger, the natural frequency will increase while the corresponding maximum electrical output will decrease.

Comparison of flexoelectric and piezoelectric ring energy harvester

Flexoelectric/piezoelectric effect is an electromechanical coupling effect occurring in dielectrics. In this study, a flexoelectric/piezoelectric ring energy harvester is proposed based on the direct flexoelectric/piezoelectric effect. The flexoelectric/piezoelectric ring energy harvester is made of an elastic ring and a flexoelectric/piezoelectric patch laminated on its surface. The electromechanical coupling mechanism of the flexoelectric/piezoelectric ring energy harvester is explored. Then the voltage and power output across the load resistance are derived in the closed-circuit condition for the energy harvester. The distinctive characteristics between the flexoelectric and the piezoelectric energy harvesters are discussed and compared in detail. The output power/voltage is related to various parameters, such as flexoelectric/piezoelectric patch size, load resistance, and flexoelectric/piezoelectric patch thickness, which are discussed to improve the power output across the load resistance. The flexoelectric ring energy harvester is more effective than the piezoelectric ring energy harvester in the transverse oscillation-bending dominant vibration, since the flexoelectric effect is sensitive to the strain gradient (bending strain). This study, including theoretical derivations and simulation plots, provide design guidelines in engineering applications for flexoelectric/piezoelectric effect.

The flexodynamic effect on nanoscale flexoelectric energy harvesting: a computational approach

Nanoscale flexoelectric energy harvesting is one of the most practical and tantalizing applications of flexoelectricity. Since flexoelectricity scales with sample's size, the energy conversion efficiency of a flexoelectric energy harvester is believed to be high at the nanometer scale. However, at such small length scale, only considering the static bulk flexoelectric effect is not sufficient. Previous works usually neglect the flexodynamic effect, a coupling between the polarization and the accelerated motion of the material, whose strength increases with the vibration frequency of the system. In this work, we examine the impact of the flexodynamic effect on the performance of a nanoscale flexoelectric energy harvester based on a general flexoelectric formulation. Through numerical simulations, we find that, when the cantilever beam's thickness reduces to several nanometers, the flexodynamic effect could be comparable to the static flexoelectric effect due to the sharp increase of its natural frequencies. This effect is negative to the flexoelectric energy harvesting. The simulation results show that the flexodyanmic effect can lead to 50% drop of a 10 nm thickness flexoelectric beam's conversion efficiency. In this work, we also give a possible solution to the problem. Properly adjusting the nanobeam's aspect ratio can effectively reduce the natural frequency and consequently diminish the negative impact of the dynamic flexoelectric effect without sacrificing the conversion efficiency.

A numerical study on flexoelectric bistable energy harvester

A flexoelectric energy harvesting can be a viable solution of energy source for low power devices and sensors due to its higher performance at nano/micro domain size. Numerical study has been performed on energy harvester based on flexoelectric phenomenon of dielectric materials. Cantilever type structure was opted here as it induces the polarization due to the breaking of lattice symmetry upon bending. Host layer of cantilever is made of barium strontium titanate (BST) as it has high flexoelectric coefficient, and electrodes are attached with the host layer to collect the charges. In this study, nonlinearity has been introduced using pair of magnets at the free end of the cantilever. Characteristics of the harvester performance (linear–non-linear) changes by varying the distance between magnets. Results revealed that the bistable energy harvester gives more operating frequency range when excitation is random as compared to the linear energy harvester. For the given dimension of the harvester, when magnets distance d = 6 mm, effective harvesting frequency ranges are 5–17.3 and 17.6–26 Hz as compared to linear harvester. Further, role of load resistance was investigated to understand the impact on the performance. Hysteresis loop between voltage and displacement significantly varies with the resistance. This hysteresis loop confirmed the backward coupling of flexoelectric layer, in which voltage affects the displacement due to actuation. Area under the hysteresis loop is maximum for optimum resistance value (20.4 kΩ) which confirms the maximum extraction of power during vibration.

Energy gathering performance of micro/nanoscale circular energy harvesters based on flexoelectric effect

At micro/nanoscale, flexoelectric energy harvesters (FEHs) are more effective than piezoelectric energy harvesters (PEHs). Calculating the voltage and efficiency of such small-scale FEHs is a challenge for their applications. Because of huge number of atoms, molecular dynamics simulation and first-principle theory are difficult to be used. It is essential to develop an analytical model that can provide a direct calculation of the voltage and the efficiency of FEHs. This paper develops an analytical model for micro/nanoscale circular FEHs which are a composite structure having a flexoelectric layer attached to an elastic substrate. Mechanical and electrical governing equations are derived based on the Hamiltonian principle and with consideration of the flexoelectric effect. Approximate closed-form solutions for voltage output, power output and optimal load resistance are obtained. The optimal values of the inner and outer radii of the flexoelectric layer for maximum voltage and power outputs are identified. The power output and the energy conversion efficiency of the present model are much larger than those of classical model which only includes piezoelectric effect. This research is helpful for materials scientists and mechanical engineers for designing high-performance micro/nanoscale energy harvesters.

Energy harvesting by exploiting the enhanced flexoelectric-like response of reduced (Na0.5Bi0.5)0.92Ba0.08TiO3 ceramics

By exploiting the flexoelectric effect, flexoelectric piezoelectric composites (or piezoelectric metamaterials) with a high-temperature-stable piezoelectric response can be designed, but the practical applications of these materials have rarely been explored. We show that after the reduction, the flexoelectric (or flexoelectric-like) response of Na0.5Bi0.5TiO3-based ferroelectric ceramics can be enhanced by 100 times, and the materials exhibit a high apparent piezoelectric response even at a temperature above the depolarization temperature (Td) of the materials. The performance of the reduced ceramics as an energy harvester was evaluated, especially at a temperature above Td. We found that the chemical reduction enhanced the output power of the ceramics above Td by more than 100 times. Although the performance of the flexoelectric energy harvester is inferior to that of conventional piezoelectric harvester below Td, flexoelectric harvester based on the reduced NBBT8 ceramics provides an approach to harvest the mechanical energy above Td.

Analysis of an array of flexoelectric layered nanobeams for vibration energy harvesting

Comparing with a single energy harvester, an array of energy harvesters can provide higher electrical power output and wider frequency bandwidth. In this article, with consideration of surface effect, an array of layer flexoelectric nanobeams with proof mass connected in parallel and/or series are investigated. Governing equations are derived based on the Hamiltonian principle. Reduced order models for both parallel and series systems are obtained. Closed-form expressions for voltages are presented. Results show that the power, the frequency and the optimal resistances of the system are dependent of surface effect. The flexoelectricity has more significant effect on the normalized power however its effect on the frequency and the optimal resistances can be ignored. The optimal resistance of the system depends on surface effect, the number of array elements and its circuits. Moreover, increasing proof mass delivers a wider bandwidth at the cost of peak normalized power. The peak normalized voltage and power of the parallel system is small than those of the series system, but the bandwidth of the two systems is the same. This is important information for the design of an array of flexoelectric energy harvesters.

Electroelastodynamics of flexoelectric energy conversion and harvesting in elastic dielectrics

Flexoelectricity is the generation of electric polarization by the application of a non-uniform mechanical strain field, i.e., a strain gradient. This phenomenon is exhibited by all elastic dielectrics, but is expected to be significant only at very small scales. Energy harvesting is a potential future application area of flexoelectricity to enable next-generation ultra-low-power MEMS/NEMS devices by converting ambient vibrations into electricity. In this paper, an electroelastodynamic framework is presented and analyzed for flexoelectric energy harvesting from strain gradient fluctuations in centrosymmetric dielectrics, by accounting for the presence of a finite electrical load across the surface electrodes as well as two-way electromechanical coupling, and capturing the size effect. The flexoelectric energy harvester model is based on the Euler-Bernoulli beam theory and it assumes the main source of polarization to be static bulk flexoelectricity. Following recent efforts on the converse flexoelectric effect in finite samples, the proposed model properly accounts for thermodynamically consistent, symmetric direct and converse coupling terms. The transverse mode flexoelectric coupling coefficient (k) is obtained analytically as a direct measure of energy conversion; its dependence on the cantilever thickness and a material Figure of Merit (FoM) is shown. Size effects are further demonstrated by simulations of the electromechanical frequency response for a Strontium Titanate (STO) energy harvester at different geometric scales. It is obtained that the flexoelectric coupling coefficient of an STO cantilever for the fundamental bending mode increases from k≈3.5×10−7 to k≈0.33 as the thickness is reduced from mm- to nm-level. A critique of the experimentally identified large flexoelectric coefficient for Barium Strontium Titanate (BST) from the literature is also given with a coupling coefficient perspective.

Flexoelectric energy harvesters based on Timoshenko laminated beam theory

Different from piezoelectricity which is restricted to certain materials, flexoelectricity is a universal electromechanical coupling in all dielectrics. In this work, mechanical energy harvester models were developed based on Timoshenko laminated beam theory, in which the flexoelectric and piezoelectric mechanisms were discussed. For a three-layered energy harvester in parallel configuration, the mechanical vibration energy can be converted into electrical energy due to flexoelectricity, and for the three-layered energy harvester in series configuration, the energy conversion is enhanced by the flexoelectricity. Resonance frequency shifts were observed in the calculations due to flexoelectricity and external circuit resistance. It is found that the electromechanical coupling displayed from the electrical responses versus resonance frequency and resistance. The energy conversion for the three-layered energy harvester system was found to be increased with the decrease in the laminated beam thickness. The energy conversion calculated for different numbers of layers also indicates that laminated energy harvester systems excel single-layered energy harvesters. This work therefore might help in designing flexoelectricity-based energy harvesters.

Modeling of a nanoscale flexoelectric energy harvester with surface effects

This work presents the modeling of a beam energy harvester scavenging energy from ambient vibration based on the phenomenon of flexoelectricity. By considering surface elasticity, residual surface stress, surface piezoelectricity and bulk flexoelectricity, a modified Euler-Bernoulli beam model for the energy harvester is developed. After deriving the requisite energy expressions, the extended Hamilton's principle and the assumed-modes method are employed to obtain the discrete electromechanical Euler-Lagrange's equations. Then, the expressions of the steady-state electromechanical responses are given for harmonic base excitation. Numerical simulations are conducted to show the output voltage and the output power of the flexoelectric energy harvesters with different materials and sizes. Particular emphasis is given to the surface effects on the performance of the energy harvesters. It is found that the surface effects are sensitive to the beam geometries and the surface material constants, and the effect of residual surface stress is more significant than that of the surface elasticity and the surface piezoelectricity. The axial deformation of the beam is also considered in the model to account for the electromechanical coupling due to piezoelectricity, and results indicate that piezoelectricity will diminish the output electrical quantities for the case investigated. This work could lead to the development of flexoelectric energy harvesters that can make the micro- and nanoscale sensor systems autonomous.

A Flexoelectric Double-Curvature Nonlinear Shell Energy Harvester

Flexoelectricity possesses two gradient-dependent electromechanical coupling effects: the direct flexoelectric effect and the converse flexoelectric effect. The former can be used for sensing and energy generation; the latter can be used for ultraprecision actuation and control applications. Due to the direct flexoelectricity and large deformations, theoretical fundamentals of a generic nonlinear distributed flexoelectric double-curvature shell energy harvester are proposed and evaluated in this study. The generic flexoelectric shell energy harvester is made of an elastic double-curvature shell laminated with flexoelectric patches and the shell experiences large oscillations, such that the von Karman geometric nonlinearity occurs. Flexoelectric output voltages and energies across a resistive load are evaluated using the current model in the closed-circuit condition when the shell is subjected to harmonic excitations and its steady-state voltage and power outputs are also calculated. The generic flexoelectric shell energy harvesting theory can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and nonshell (beam, plate, ring, arch, etc.) distributed harvesters and the simplification procedures are demonstrated in three cases, i.e., a cylindrical shell, a circular ring and a beam harvester. Other shell and nonshell flexoelectric energy harvesters with standard geometries can also be defined using their distinct two Lamé parameters and two curvature radii.

Nanoscale flexoelectric energy harvesting

One of the most tantalizing applications of piezoelectricity is to harvest energy from ambient mechanical vibrations for powering micro and nano devices. However, piezoelectricity is restricted only to certain materials and is severely compromised at high temperatures. In this article, we examine in detail, the possibility of using the phenomenon of flexoelectricity for energy harvesting. The flexoelectric effect is universally present in all dielectrics and exhibits a strong scaling with size. Using a simple beam-based paradigmatical design, we theoretically and computationally examine flexoelectric energy harvesting under harmonic mechanical excitation. We find that the output power density and conversion efficiency increase significantly when the beam thickness reduces from micro to nanoscale and flexoelectricity-based energy harvesting can be a viable alternative to piezoelectrics. Specifically, the conversion efficiency in flexoelectric transduction at sub-micron thickness levels is observed to increase by two orders of magnitude as the thickness is reduced by an order of magnitude. The flexoelectric energy harvester works even for a single layer beam with a symmetric cross section which is not possible in piezoelectric energy harvesting. Our results also pave the way for exploration of high temperature energy harvesting since unlike piezoelectricity, flexoelectricity persists well beyond the Curie temperatures of the high electromechanical coupling ferroelectrics that are often used.