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Universal converse flexoelectricity in dielectric materials via varying electric field direction
Universal converse flexoelectricity in dielectric materials via varying electric field direction
Sharma, Saurav; Kumar, Rajeev; Vaish, Rahul
INTERNATIONAL JOURNAL OF SMART AND NANO MATERIALS 2021, VOL. 12, NO. 1, 107–128 https://doi.org/10.1080/19475411.2021.1880491 Universal converse flexoelectricity in dielectric materials via varying electric field direction Saurav Sharma, Rajeev Kumar and Rahul Vaish School of Engineering, Indian Institute of Technology Mandi, Mandi, India ABSTRACT ARTICLE HISTORY Received 11 December 2020 Flexoelectricity is a symmetry independent electromechanical cou- Accepted 19 January 2021 pling phenomenon that outperforms piezoelectricity at micro and nanoscales due to its size-dependent behavior arising from gradi- KEYWORDS ent terms in its constitutive relations. However, due to this gradient Converse flexoelectricity; term flexoelectricity, to exhibit itself, requires specially designed electrical boundary geometry or material composition of the dielectric material. First conditions; electric field of its kind, the present study put forward a novel strategy of direction; isogeometric analysis 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 Sr TiO (BST), by manipulating 0.67 0.33 3 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. CONTACT Saurav Sharma email@example.com School of Engineering, Indian Institute of Technology Mandi, Mandi, India © 2021 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 108 S. SHARMA ET AL. 1. Introduction Flexoelectricity has gained increasing interest from researchers in recent times as a symmetry independent physical phenomenon exhibiting electromechanical coupling in dielectric materials. A linear relationship between mechanical strain and induced electrical polarization is defined as direct flexoelectricity, whereas the phenomenon of generation of mechanical strain in linear proportionality to the gradient of the electric field is termed as converse flexoelectricity. Due to gradient terms in its constitutive relationships, flexoelec - tricity shows a strong size-dependent behavior. Researchers have used flexoelectric response in micro and nano electromechanical systems (MEMS/NEMS) where the piezo- electric response is of little significance. Kogan et al.  presented the first phenomenolo- gical report on a deformation gradient-based electromechanical coupling. A rough estimate of flexoelectric coefficients was provided, but it was termed as a type of piezo- electricity only. Mindlin  extended the linear theory of piezoelectricity by accounting for the contribution of polarization gradient in the stored energy function. Mindlin showcased the existence of electromechanical coupling even in centrosymmetric materials. Since then, an increasing amount of effort and time has been spent in investigating the physical behavior of flexoelectric materials, both theoretically and experimentally. Tagantsev  was the first to separately represent piezoelectric and flexoelectric effects in crystalline dielec- trics. However, for several decades of its first evidence in solids, flexoelectricity was given a little attention, primarily because of its weak response. It was only after systematic experimental works conducted by the group of Cross [4–8] in the early 2000’s that the flexoelectricity was revived as an alternative of piezoelectricity at micro and nano scales. Theoretical evaluation of gradient elasticity-based electrotechnical response in dielectric materials has been a subject of interest for the research community even before these experimental works. Flexoelectric effect is a gradient-induced phenomenon, i.e. it requires inhomogeneous mechanical strain (for direct effect) or inhomogeneous electric field (for converse effect) to generate a net output. This inhomogeneity can be induced in different ways. One of the most common ways employed to get the required gradient is by having the specially designed geometry of the sample. Truncated pyramid shape was employed for the computational evaluation of flexoelectric characteristics of dielectric solids by Abdollahi et al. . Smooth meshfree basis functions were employed to deal with the higher-order electromechanical coupling of flexoelectricity. Later, Abdollahi et al.  reinvestigated the evaluation of flexoelectric coefficients by compression of the truncated pyramid to address the issue of overestimation of properties by this popular method. The order of overestimation of the properties was systematically established as a function of pyramid configuration. Qi et al.  presented a unique design by generating a wavy piezo ribbons embedded silicon rubber-based micro structure for improvement in energy har- vesting. The curved structure of piezoelectric ribbons was used as the source of inhomo- geneous deformations resulting in a flexoelectric response. Qi et al.  numerically investigated a curved flexoelectric microbeam for direct and converse flexoelectric effect. Several other studies [13–21] have been reported using special geometry as a source of nonuniform deformation field to induce flexoelectricity. Another way extensively exploited for attaining the prerequisite of flexoelectricity, i.e. the gradient of strain, is by having appropriate boundary and loading conditions. For this purpose, cantilever-shaped energy harvesters and sensors, due to the presence of nonuniform strain in bending, have been INTERNATIONAL JOURNAL OF SMART AND NANO MATERIALS 109 vastly investigated in the past [22–33] Variable mechanical characteristics can result in varying strain distribution within the domain. A few studies have suggested the use of special composite materials to achieve variation in mechanical properties and thereby a strain gradient [34–39]. This method eliminates the need for special designs and dimen- sions of the devices to be employed, however, it requires an additional effort at manu- facturing stage for compositional variation. The variety of computational techniques and experimental efforts observed from the literature focus mainly on investigating, optimizing, and improving the direct flexoelectric effect while studies on the converse flexoelectric effect are comparatively less. Fu et al.  experimentally evaluated the converse flexo - electric coefficients in a highly electrically susceptible Ba Sr TiO (BST) trapezoidal 0.67 0.33 3 block. The resulting µ was found to be in good agreement with that measured by direct effect in their previous studies. Abdollahi et al.  computationally evaluated the direct and converse flexoelectricity in a trapezoidal block in a two-dimensional (2D) domain. Experimental analysis of converse flexoelectric effect in a SrTiO crystal was performed by Zalesskii and Rumyantseva , under an inhomogeneous strain produced in bending and varying temperature. In a comparative study of piezoelectricity and flexoelectricity Abdollahi and Arias  presented the potentials of piezoelectric and flexoelectric electro- mechanical coupling phenomena, in both direct and converse effects. The size depen- dence of net output from both phenomena was compared and flexoelectricity was found to be more effective at smaller scales. A mixed finite elements-based FEA formulation was developed by Mao et al. , for direct and converse flexoelectric effects while considering the electromechanical coupling of polarization with both symmetrical and rotational strain gradients. Additional nodal degrees of freedom were introduced to reduce the C contin- uous problem to a C continuous problem. An actuation method employing a biconcave curved beam of polyvinylidene fluoride (PVDF), using the finite element method was presented by Wu et al. . Converse flexoelectric effect has also been utilized in vibration control application by Fan and Tzou  as an actuator mechanism. A cantilever beam laminated with a flexoelectric layer was used to demonstrate the influence of actuating effect of the flexoelectric layer. Very recently, Liu et al.  presented a theoretical and experimental study on direct and converse flexoelectric effects. A mixed finite element method for numerical simulations was used and piezoresponse force microscopy (PFM) was used in experimental studies. Upon reviewing the existing literature, it is revealed that although a fair amount of attention is being given to the computational evaluation of direct flexoelectricity, there exists a lack of focus on converse flexoelectricity as compared to direct effect. Moreover, the previous attempts on studying the converse flexoelectric effect use engineered structures or composition as a source of inhomogeneity to induce the effect. This method, although used widely, puts a constraint on the choice of geometry and/or material composition for the use in relevant applications. In this study, we report an approach for generating converse flexoelectricity in a regular shaped homogeneous dielectric material by spatial variation of electric field direction. The present method incorporates a spatial variation of the electric field vector in the material by manipulating electrical boundary conditions. This is done by employing different configurations of electrode placement. A multi-physics coupled field model is developed and solved using isogeo- metric analysis (IGA) on a 2D material sample of BST. The switching of a polarization vector field in a varying fashion occurs due to the nonconventional electroding, which is 110 S. SHARMA ET AL. captured by solving electrostatic equilibrium. The resulting actuation is computed by employing coupled flexoelectric equations. Three different patterns of varying electric field vector are attained by changing the location of application of electric potential and grounding on the sample. The proposed method provides a relatively easy to implement a way of inducing converse flexoelectric effect for micro- and nano-scale applications. 2. Methodology A variety of methods can be utilized to generate a varying electric fields in order to achieve converse flexoelectricity in dielectric materials. Figure 1(a,b) shows the methods that have been utilized in the past for this purpose, namely, geometry variation and material composition variation. In Figure 1(c), the variation of electric field direction is demonstrated schematically to achieve electric field gradient. To demonstrate the con- verse flexoelectric effect generated due to spatial variation of electric field direction, a 2D rectangular geometry is selected for computation. Assumptions of plane strain elasticity and electrostatics in 2D are employed while formulating the equations used in IGA. The multi-physics computational framework required for the analysis of the converse flexo - electric effect will be developed using flexoelectric material constitutive law. The IGA formulation is developed by keeping these equations as the basis. The detailed formula- tion is discussed in subsequent sections. 2.1. Flexoelectric material law Flexoelectricity is described by a symmetry independent and size-dependent correlation between electrical and mechanical field variables. Linear proportionality between polar- ization and strain gradient is termed as direct flexoelectricity whereas the linear propor- tionality of mechanical stress with electric field gradient is termed as converse flexoelectricity. In this section, we will derive this correlation and later based on the constitutive equations of flexoelectricity, an isogeometric numerical model will be devel- oped. The electrical Gibbs free energy density of a dielectric material having piezoelectric and flexoelectric effects can be written as , 1 1 H ¼ C S S e E S þ f E S þ d E S 2 E E (1) ijkl ij kl ikl i kl ijkl i jk;l ijkl i;j kl ij i j 2 2 Figure 1. Schematic of arrangements to induce converse flexoelectricity, (a) geometry-dependent electric field gradient, (b) varying material composition for electric field gradient, and (c) varying direction of electric field for generating electric field gradient. INTERNATIONAL JOURNAL OF SMART AND NANO MATERIALS 111 where E and S are the electric field vector and strain tensor, respectively. C; f; d; and 2 denote fourth-order material elastic stiffness tensor, fourth-order direct flexoelectric tensor, converse flexoelectric tensor, and second-order dielectric tensor, respectively. Subscripts i; j; k, and l are the representatives of orthogonal cartesian coordinate direc- tions. The first term denotes the contribution of mechanical strain energy, while the second term is the contribution of piezoelectricity. Third and fourth terms are the direct and converse flexoelectricity contributions to the energy density function. The last term is the electrostatic contribution arising from applied potential. It has been shown that the terms pertaining to direct and converse flexoelectric contributions can be represented by a single material tensor in the following form , 1 1 H ¼ C S S e E S μ E S 2 E E (2) ijkl ij kl ikl i kl i jk;l ij i j ijkl 2 2 Here μ ¼ d f , is the fourth-order flexoelectric coefficients’ tensor. We write stress iljk ijkl ijkl and electric displacement as, @ H T ¼ ¼ C S e E (3) ij ijkl ij ikl k @ S kl and @ H D ¼ ¼ e S þ2 E þ μ S (4) i ikl kl ij j jk;l ijkl @ E Next, the higher-order stress (moment stress or hyper stress) and higher-order electric displacement (electric quadrupole) are defined as, @ H T ¼ ¼ μ E (5) ijk i ijkl @ S jk;l and @ H D ¼ ¼ 0 (6) ij @ E i;j The physical stress can be written as, ^ ~ T ¼ T T ¼ C S e E þ μ E (7) ij ij ijk;k ijkl ij kij k ijkl i;k Similarly, the physical electric displacement can be written as, ^ ~ D ¼ D D ¼ e S þ P E þ μ S (8) i i ij;j ikl kl ij j ijkl jk;l For non-piezoelectric material, the piezoelectric terms in Equations (7) and (8) can be omitted. Rewriting Equations (7) and (8), the constitutive equations of flexoelectric effect can be written as, D ¼ P E þ μ S (9) i ij j jk;l ijkl and T ¼ C S þ μ E : (10) ij ijkl ij i;j ijkl 112 S. SHARMA ET AL. Equations (9) and (10) are the constitutive equations for direct and converse flexoelectric effects, respectively. The electrical Gibbs free energy, can be written as, ^ ~ ^ H ¼ ∫ T S þ T S D E dΩ