Journal of Nanomaterials

Volume 2017, Article ID 1924651, 14 pages

https://doi.org/10.1155/2017/1924651

## Evaluation of Effective Elastic, Piezoelectric, and Dielectric Properties of SU8/ZnO Nanocomposite for Vertically Integrated Nanogenerators Using Finite Element Method

School of Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology Bhubaneswar, Toshali Bhawan, Bhubaneswar, Odisha 751007, India

Correspondence should be addressed to Kaushik Das; ni.ca.sbbtii@kihsuak

Received 24 January 2017; Accepted 13 April 2017; Published 15 May 2017

Academic Editor: R. Torrecillas

Copyright © 2017 Neelam Mishra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A nanogenerator is a nanodevice which converts ambient mechanical energy into electrical energy. A piezoelectric nanocomposite, composed of vertical arrays of piezoelectric zinc oxide (ZnO) nanowires, encapsulated in a compliant polymeric matrix, is one of most common configurations of a nanogenerator. Knowledge of the effective elastic, piezoelectric, and dielectric material properties of the piezoelectric nanocomposite is critical in the design of a nanogenerator. In this work, the effective material properties of a unidirectional, unimodal, continuous piezoelectric composite, consisting of SU8 photoresist as matrix and vertical array of ZnO nanowires as reinforcement, are systematically evaluated using finite element method (FEM). The FEM simulations were carried out on cubic representative volume elements (RVEs). Four different types of arrangements of ZnO nanowires and three sizes of RVEs have been considered. The volume fraction of ZnO nanowires is varied from 0 to a maximum of 0.7. Homogeneous displacement and electric potential are prescribed as boundary conditions. The material properties are evaluated as functions of reinforcement volume fraction. The values obtained through FEM simulations are compared with the results obtained via the Eshelby-Mori-Tanaka micromechanics. The results demonstrate the significant effects of ZnO arrangement, ZnO volume fraction, and size of RVE on the material properties.

#### 1. Introduction

Mechanical energy is one of the most ubiquitous and abundant among sources of energy. Mechanical energy can be harvested from wind flows [1], from ambient vibrations [2], and from the human movements [3]. Advanced piezoelectric materials, especially piezoelectric composites, are central in harvesting mechanical energy and converting it into electrical energy. The mechanical strain in these material leads to the development of electric potential which can be used to charge a battery in portable electronic devices or be used for stand-alone, self-powered nanodevices [4]. A mechanical energy-harvesting nanodevice based on piezoelectric zinc oxide (ZnO) nanowires was first demonstrated by Wang and Song in 2006 [5]. Since then, a new group of energy-harvesting nanosystem, known as nanogenerator, has been developed [6–9]. The most common configuration of a nanogenerator involves piezoelectric ZnO nanowire grown vertically on a silicon substrate [10]. The vertically grown array of ZnO nanowires is encapsulated with a polymer, which may be active or passive [11]. Combination of photopatternable polymer SU8 with ZnO nanowires is expected to create a nanocomposite that can be easily integrated with the microfabrication process flow established for fabrication of microsystems and flexible electronics. Design and optimization of a nanogenerator with nanocomposites as the active material require knowledge of the effective elastic, piezoelectric, and dielectric properties of the nanocomposite.

Prediction of effective properties of a piezoelectric composite can be accomplished by numerous approaches. Simple Voigt and Reuss approaches [12], developed for uncoupled elastic composites, have been extended to predict upper and lower bounds of effective dielectric and piezoelectric constants of composites with specific reinforcement arrangements [13, 14]. Bisegna and Luciano have extended Hashin-Shtrikman type of variational approaches [15] to electroelastic problems and have estimated bounds of effective material properties [14, 16]. Several authors have extended Eshelby’s classical solution [17] of an ellipsoidal inclusion in a homogeneous matrix to the field of linear piezoelectricity and have calculated the effective electroelastic constants [18–20]. Odegard has proposed a new model by generalizing the Mori-Tanaka [21, 22] and self-consistent [23, 24] approaches, for predicting the coupled electromechanical properties of piezoelectric composites [25]. Using asymptotic homogenization method Guinovart-Díaz et al. have derived exact expressions for binary piezoelectric composites where both components have 6 mm symmetry for cases where the reinforcement is a piezoelectric ceramic and the polymeric matrix may or may not be piezoelectrically active [26].

In addition to these analytical and semianalytical approaches, computational methods based on finite element modeling have also been employed to evaluate effective properties of piezoelectric composites. The approaches based on FEM are extensions of the representative volume element (RVE) approach commonly employed to solve problems of linear elasticity in case of composites and foams [27, 28]. Gaudenzi employed a RVE-based FEM to evaluate the electromechanical response of a 0–3 piezoelectric composite [29]. Poizat and Sester have evaluated effective longitudinal and transverse piezoelectric constants of 0–3 and 1–3 composites with embedded piezoelectric fibres using FEM [30]. Pettermann and Suresh have developed a RVE-based model and predicted the complete set of moduli for piezoelectric composites with periodic hexagonal and square array of continuous fibres [31]. The effect of fibre distribution, effect of fibre shape and geometric connectivity, and effect of poling on the effective electromechanical properties of 1–3 piezoelectric composites have been reported by Kar-Gupta and Venkatesh [32–34]. Unit cell models of unidirectional periodic 1–3 piezoelectric composites using periodic boundary conditions have also been reported by Berger et al. [35, 36] and by Medeiros et al. [37]. Recent years have seen an interest in nonlinear finite element modeling of laminated piezoelectric composites [38, 39].

Literature review revealed a gap in the experimental as well computational study of process-structure-property relationships of nanocomposites for applications in nanogenerators. Previous work on nanogenerators with piezoelectric ZnO nanowires embedded in a PMMA matrix used simplistic Reuss and Voigt analyses to predict effective properties of the composite, assuming both the matrix and reinforcement to be isotropic [8]. This work presents a rigorous and systematic analysis based on FEM which can be applied for piezoelectric nanocomposites with anisotropic constituents, as well as with different geometries of the reinforcement phase, as long as the RVE has three mutually perpendicular planes of mirror symmetry passing through the centroid of the RVE. To the best of our knowledge, this work represents the first systematic study of the effect of ZnO nanowire volume fraction and ZnO nanowire arrangement on the effective elastic, piezoelectric, and dielectric properties of a unidirectional, unimodal SU8/ZnO nanocomposites. The effective properties of the SU8/ZnO nanocomposite were obtained by FEM and by an approach based on Eshelby-Mori-Tanaka (EMT) micromechanics.

The article is arranged in the following fashion. First, the linear theory of piezoelectricity is introduced in Section 2. Next, two approaches of prediction of effective properties of a piezoelectric composite are discussed. Section 2.1 discusses the approach based on EMT micromechanics, while Section 2.2 presents the FEM study on idealised RVE. Section 3 presents the different types of RVEs considered in this study. The material properties of the constituent phases are provided in Section 4. The boundary conditions applied to the RVEs to calculate the effective material properties are presented in Section 5. Finally, the results of the finite element analyses and EMT micromechanics-based approach are presented and compared in Section 6. The results are discussed in Section 7 and final conclusions are drawn in Section 8.

#### 2. Theory

The theory of linear piezoelectricity couples the interaction between the mechanical (elastic) and electric variables by the following constitutive equations [40]:where the indices , , , and range from 1 to 3. Here, are the components of the elastic strain, are the components of the electric field, are the components of the stress tensor, and are the components of the electric displacement. In this stress-charge form of piezoelectric constitutive equations, the elastic strain tensor and the electric field vector are the independent variables. are components of the fourth-order elastic stiffness tensor measured in the absence of an applied electric field. are components of the piezoelectric modulus tensor measured in the absence of an applied strain. are components of the dielectric modulus measured in the absence of an applied strain. In addition, solution of any problem in the domain of linear piezoelectricity requires that the equations of elastic equilibrium and Gauss’ law of electrostatics be satisfied. In the absence of body forces, the equation for elastic equilibrium is given byIn (2) and in the following, the Einstein summation convention is used. The comma signifies partial differentiation with respect to the coordinate following it. The Gauss equation of electrostatics in the absence of free charges can be expressed asThe components of the strain tensor, , and the components of the electric field, , are related to displacement and electric potential () by the following equations:In modeling of piezoelectric behaviour, it is convenient to treat the elastic and the electric variables in a similar fashion. This is accomplished by employing a notation introduced by Barnett and Lothe [41], which involves uppercase indices that range from 1 to 4, as well as lowercase indices that range from 1 to 3. According to this notation, the stress and electric displacement are represented byAnd, the elastic strain and the electric field are represented byThe electroelastic moduli relating and can be represented asIn short-hand notation,It becomes convenient to represent the constitutive equations in a generalized Voigt two-index notation (with two indices, each ranging from 1 to 9) as [19, 42]. The generalized Voigt notation involves first representing and as matrices and transforming the index of the electroelastic moduli to a two-index form. The following transformation of subscripts is as follows:Thus, would transform to with a value of , while would transform to with a value of . Therefore, the constitutive equations can now be represented asThe electroelastic modulus matrix () can now be defined asFor an isotropic nonpiezoelectric solid, (i) there are two nonzero, independent constants for the stiffness matrix, that is, and , (ii) all components of the piezoelectric modulus matrix of this type of solid are zero, and (iii) there is only one independent constant for the dielectric modulus matrix. The electroelastic modulus for an isotropic passive matrix phase is given byFor a transversely isotropic piezoelectric solid, where 1-2 plane is the plane of isotropy, there are (i) five independent constants for the stiffness matrix, that is, , , , , and , (ii) three independent constants for the piezoelectric modulus, that is, , , and , and (iii) two independent constants for the dielectric modulus, that is, and . The electroelastic modulus for the reinforcement phase is given byFor a 1–3 piezoelectric composite, composed of an isotropic nonpiezoelectric matrix and with transversely isotropic piezoelectric reinforcements, the effective values of the components of the electroelastic matrix can be evaluated using (i) a micromechanics-based semianalytical approach and (ii) finite element method, as discussed in detail in the following subsections.

##### 2.1. Micromechanics-Based Approach

The effective electroelastic modulus of a two-phase piezoelectric composite can be expressed in terms of the electroelastic moduli and the volume fractions of the constituent phases. Under homogeneous elastic displacement and electric potential boundary conditions, generalizing the average strain theorem of elasticity [43] for piezoelectricity, the effective electroelastic modulus of a piezoelectric composite is obtained as follows [19]:Here, the subscripts and refer to the matrix and the reinforcement, respectively. is the volume fraction of the reinforcement, and is the strain-potential gradient concentration matrix. relates the average strain and potential gradient in the reinforcement phase to the applied homogeneous displacement and electric potential in the composite. Estimation of the concentration tensor can be achieved through several micromechanical schemes, for example, by using the dilute approximation scheme [17], the self-consistent scheme [23], the differential scheme [24], and the EMT mean field approach [21, 22]. In this work, we have considered EMT mean field approach, which will be discussed in detail.

Consider the case of an ellipsoidal inclusion in an infinite matrix wherein the electroelastic fields of the reinforcement phases do not interact. The electromechanical response of such “dilute” composites can be obtained using the Eshelby method, which is also known as the dilute approximation method. For dilute approximation, the concentration tensor, , is given bywhere is the identity matrix. is analogous to Eshelby tensor [17] and is represented by a matrix. Similar to Eshelby tensor, components of , also known as the constraint tensor, are functions of the geometry of the reinforcement phase and the electroelastic moduli of the matrix phase. Explicit expressions of are available in the literature [18, 19, 44]. Now, for the nondilute approximation, when there are multiple inclusions present in the matrix, the electroelastic fields of the inclusions interact. Mori-Tanaka approach takes these interactions into account by modeling the interacting electroelastic field as a background electroelastic field. The modified concentration factor can be expressed as follows [22]:Combining (14) and (16), the effective electroelastic modulus of a piezoelectric composite can be estimated as follows:The inverse of the effective electroelastic modulus matrix is represented by , that is, The components of the piezoelectric strain coefficients matrix are defined as follows:

##### 2.2. FEM Approach

The first step towards prediction of effective properties of a composite using FEM is to select an RVE or a unit cell. The RVE must be small enough compared to the macroscopic body, but it should be large enough to capture the major features of the microstructure [37]. An RVE may be of several geometries, for example, cubic, hexagonal, and spherical. A cubic RVE, in spite of several deficiencies like accommodation of low volume fraction of reinforcement, is most commonly used due to the relative ease of applying boundary conditions. A cubic RVE can be further reduced in size by taking into account the symmetry in the arrangement of the reinforcement phase(s). The micromechanics-based method, discussed in the previous section, assumes the stress and strain fields inside the inclusions to be constant, whereas finite element approach predicts a more-realistic picture of nonuniform stress and strain fields at each nodal point. The evaluation of the effective properties of a composite is then accomplished by employing the homogenization method. This method relates the volume average stress, strain, electric displacement, and electric field to the effective properties of the composite. The composite is thus modeled as a homogenized medium. Using FEM, the volume averages can be calculated as follows:In (20), is the volume of the RVE. , , , and are the volume-averaged values of stress, strain, electric displacement, and electric field, respectively. is total number of finite elements in the RVE, is the volume of the th element, is th component of the stress tensor calculated in the th element, and is th component of the strain tensor calculated in the th element. Similarly, and are the th component of the electric displacement field and of the electric field, respectively, each calculated in the th element. Consider a unidirectional composite, where the matrix is passive and isotropic and the piezoelectric reinforcement is transversely isotropic. The local 3-axis of the piezoelectric reinforcement is aligned with the global -axis. In terms of these average values, discussed earlier, the constitutive equations of linear piezoelectricity for this material can be expressed in the matrix form as follows:where the superscript refers to the effective properties of the composite. Equation (21) represents the basis on which the boundary conditions are applied on the RVE. For homogeneous applied strain and homogeneous applied electric field , the boundary conditions applied on the surfaces of the cubic RVE are of the following form:where refers to the coordinates of the surfaces of the RVE.

#### 3. Types of RVE

The RVEs considered are cubes of fixed dimensions with centroid at the origin and with the boundaries given by the following::, , , :, , , :, , , :, , , :, , , :, , , – are the six bounding surfaces of the representative cubic volume element. The nanowires are oriented along the -direction. The RVEs are chosen such that reflectional symmetries are present about -, -, and -axes, and hence only (1/8)th of the cubes have been considered for finite element analysis. The boundaries – of the reduced structures are as follows:::::::Here, represents the outward unit normal vector to the bounding surface. Six types of cubic RVEs are considered in this study. The RVEs differ from one another in terms of size of the unit cube and the arrangement of the nanowires on the - plane. The sizes of the RVEs are chosen such that the maximum diameter of the ZnO nanowires, corresponding to the maximum volume fraction for each RVE, is in the range of 80–110 nm. Three different sizes and four different arrangements are considered, with an aim to study the effect of RVE-size as well as the effect of the distribution of the ZnO nanowires on the effective properties of the nanocomposite. Details of these RVEs are provided in Table 1. RVE1 and RVE2 both have the identical square array in terms of distribution of the nanowires on the - plane. Images of RVE1 and RVE2 are provided in Figures 1(a) and 1(b), respectively. In RVE3, the nanowires are distributed in a face-centred rectangular array, as shown in Figure 1(c). In RVE4 and in RVE5, the nanowires are distributed in a face-centred square array, as shown in Figures 1(d) and 1(e), respectively. In RVE1–RVE5, the maximum reinforcement volume fraction considered is 0.7 since the maximum theoretical volume fraction of cylindrical reinforcements in a cubic volume is 0.785. In RVE6, as shown in Figure 1(f), the nanowires are distributed in a quasi-random fashion, while maintaining the reflectional symmetries. The maximum volume fraction of the reinforcement in RVE6 is limited to ~0.52 to ensure that the fibre geometries do not intersect each other.