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Theoretical Study of the BaTiO_{3} Powder’s Volume Ratio’s Influence on the Output of Composite Piezoelectric Nanogenerator

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## Abstract

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_{3}/polydimethylsiloxane (PDMS) composite PENG was demonstrated as having an optimal volume ratio (46%) at which the PENG can output its highest voltage, and this phenomenon can be ascribed to the trade-off between the composite PENG’s top electrode charge and its capacitance. These results are of practical importance for the composite PENG’s performance optimization.

## 1. Introduction

_{3}/PDMS composite PENG with different volume ratios of evenly distributed BaTiO

_{3}cubes, and found the existence of an optimal volume ratio (46%) of the BaTiO

_{3}cubes at which the PENG can output the highest voltage, resulting from the trade-off between the surface charge and capacitance of the composite PENG.

## 2. Methods and Structure

_{p}= c

_{pq}ε

_{q}− e

_{kp}E

_{k},

_{i}= e

_{iq}ε

_{q}+ κ

_{i}

_{k}E

_{k},

_{p}is the stress tensor, and ε

_{q}is the strain tensor. To keep the tensor equation compact, the Voigt notion was used to reduce the 3 × 3 symmetric stress tensor σ

_{mn}and strain tensor ε

_{mn}where m, n ∈ (x, y, z), to 6-dimensional vectors σ

_{q}and ε

_{p}where q, p ∈ (xx, yy, zz, yz, zx, xy), c

_{pq}is the linear elastic constant, e

_{kp}is the linear piezoelectric coefficient, κ

_{i}

_{k}is the dielectric constant, E

_{k}is the electric field, and D

_{i}is the electric displacement. In this study, these equations are solved by the COMSOL software package (5.1, COMSOL Co., Ltd., Shanghai, China).

_{3}/PDMS composite PENG is schematically shown in Figure 1, where the BaTiO

_{3}/PDMS composite is sandwiched between the top and bottom electrodes. The size of the PDMS matrix is 460 × 460 × 40 μm, and the size of the individual BaTiO

_{3}cube poling along the z axis is 30 × 30 × 30 μm, and the cubes are uniformly distributed in the matrix. In practice, it is inevitable for the BaTiO

_{3}particles/polymer based PENG to have polymer between the electrode and BaTiO

_{3}. As shown in Figure 1, PDMS layer between the electrode and the BaTiO

_{3}cubes in 5 μm thickness was used to mimic this situation. In practical calculations, the electrodes are not added, the bottom of the composite was ground and fixed, the charge on the side walls and top is set to be zero, a stress of 0.5 MPa is exerted on the top surface. BaTiO

_{3}/PDMS composite containing, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225 BaTiO

_{3}cubes are respectively calculated. The corresponding volume ratios are 2.9%, 5.1%, 8.0%, 11.5%, 15.6%, 20.4%, 25.8%, 31.9%, 38.6%, 45.9%, 53.9%, 62.5%, and 71.8%, respectively. After the calculation, the average electric potential of the top surface is the PENG’s open circuit voltage.

## 3. Results and Discussion

_{3}/PDMS composites with different volume ratios of BaTiO

_{3}cubes are calculated. At the volume ratio of 45.9%, the composite PENG reached its max voltage of 1.275 V. To interpret this phenomenon, the composite PENG is approximated as a capacitor whose charge is generated by piezoelectric effects. Below this approximation, the voltage of the PENG can be calculated.

_{3}cubes’ ratio is larger than 31.9%. Below this volume ratio, the voltage obtained by Q/C is higher than that obtained by solving the coupled piezoelectric governing equation. This discrepancy is due to the fact that the external forces on the deformation of PENG are not considered. When the volume ratio of BaTiO

_{3}cubes is 31.9%, the composite is stiff and the PENG’s deformation under external force is small, as shown in Figure 2c. In contrast, when the volume ratio of the BaTiO

_{3}cubes in the composite is low, under the external force, the deformation of the PDMS filling between the BaTiO

_{3}cubes is large as shown in Figure 2d. The capacitance of the plate capacitor is

_{0}κS/d

_{0}is the vacuum permittivity, and κ is the composite’s permittivity. The calculated capacitance of the PENG with low BaTiO

_{3}cubes is smaller than the actual value without consideration of the deformation of the PENG under external force. Therefore, the voltage calculated on the basis of this capacitance is larger than the actual value. With the increase of BaTiO

_{3}cubes’ volume ratio, the composite becomes stiffer and stiffer, so the PENG’s deformation can be neglected and the voltages calculated by the above two methods is consistent, as shown in Figure 2b. As the optimal volume ratio locates at 45.9%, it is sufficient to interpret this phenomenon by comparing the volume ratio ranging from 31.9% to 71.8%. In this volume ratio range, the Equation (3) has a rather good accuracy in describing the PENG’s voltage.

_{3}cube’s ratio, the value of PENG’s capacitance increases gradually and toward a saturated value, because the permittivity of BaTiO

_{3}is much higher than that of the PDMS. In contrast, the surface charge first increases with the BaTiO

_{3}’s ratio and then decreases when the BaTiO

_{3}’s ratio is further increased, because the surface charge of the composite PENG is induced by the charge generated by the BaTiO

_{3}cubes under external force and the electric charge in the ferroelectric materials composed of two parts [23]. One is the surface charge caused by the discontinuous polarization across the ferroelectric material’s surface and this value is proportional to the surface stress exerted on the ferroelectric material. The other one is the body charge caused by the gradient of the polarization, and this value is proportional to the gradient of the stress field in the ferroelectric material. The calculated average pressure and total force on the BaTiO

_{3}cube’s top surface are shown in Figure 3a. When the BaTiO

_{3}’s ratio is increased, the average stress applied on the BaTiO

_{3}cubes decreases, because the BaTiO

_{3}cube in the polymer matrix has the property to concentrate the stress on it. As the cubes increase, the stress is more dispersed. On the one hand, the total force applied on the BaTiO

_{3}cubes reaches its max at the volume ratio of 62.5%. On the other hand, as the BaTiO

_{3}cubes’ ratio increases, the stress gradient in BaTiO

_{3}cubes decreases. So the charge of the top has a maximum within this volume ratio range. With the trade-off of the capacitance and top electrode charge, the composite PENG reached its maximum output, when the BaTiO

_{3}ratio is 45.9%.

_{3}cubes volume ratio around the optimal value is studied. The results are shown in Figure 4a,b. When the polymer’s permittivity was increased, the voltage decreased notably, which is ascribed to the PENG’s increased capacitance. Although the voltage of the PENG is altered when the polymer’s Young’s modulus or permittivity are changed, the PENG with BaTiO

_{3}volume ratio of 45.9% still has the max voltage. According to the experimental results [21] published recently, a paper-based PENG composed of BaTiO

_{3}nanoparticles and bacterial cellulose got its maximum voltage when the mass ratio of uniformly distributed BaTiO

_{3}nanoparticles reached 80%. The Young’s modulus, relative permittivity, and density of the bacterial cellulose are 10 GPa, 7.8, 1.2 g/cm

^{3}respectively [24,25]. The PDMS’s Young’s modulus and relative permittivity are tuned to 10 GPa and 7.8, the optimal volume ratio of BaTiO

_{3}is still 45.9% which can be seen in Figure 4a,b. In consideration of the density of BaTiO

_{3}and bacterial cellulose at 6.02 g/cm

^{3}and 1.2 g/cm

^{3}, the optimal mass ratio is 81%. Therefore, our simulation results are in consistent with those experimental results. The distance between the electrode and the BaTiO

_{3}cubes was studied. As shown in Figure 4c, the PENG’s voltage increases with the increase of this distance, which may be ascribed to more stress delivered from the PDMS to the BaTiO

_{3}cubes.

_{33}f(t)/C

_{33}is the piezoelectric constant of the PENG, and f(t) is the dynamic force exerted on the PENG. When the PENG is driven by a harmonic force f(t)e

^{iωt}, where ω is the angular frequency of the force, the voltage drop across the external load has an analytic expression,

## 4. Conclusions

_{3}/PDMS composite PENG with different volume ratios of evenly distributed BaTiO

_{3}cubes, and find that an optimal volume ratio (46%) of the BaTiO

_{3}cubes exists at which the PENG can output the highest voltage. The optimal ratio is a result of the trade-off between the surface charge and capacitance of the composite PENG. This optimal volume ratio is stable even if the Young’s modulus and permittivity of the polymer matrix are changed. Finally, after a 0.5 MPa, 1000 Hz square wave force was exerted, the PENG at the optimal volume ratio can output a current of 23.2 nA, voltage of 1.16 V on the external load with resistance of 50 MΩ.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic illustration of the BaTiO

_{3}/polydimethylsiloxane (PDMS) composite piezoelectric nanogenerator (PENG).

**Figure 2.**(

**a**) The open circuit voltage of the composite PENG containing different volume ratios of BaTiO

_{3}cubes; (

**b**) The comparison of the voltage obtained Q/C and that directly calculated from the coupled piezoelectric governing equation. The distribution of stress on the BaTiO

_{3}cubes’ top surface with volume ratio of 31.9% (

**c**) and 2.9% (

**d**).

**Figure 3.**(

**a**) The PENG’s top surface charge and capacitance with different BaTiO

_{3}ratios. (

**b**) The averaged stress and total force on the top surface of the embedded BaTiO

_{3}cubes. The distribution of stress in the BaTiO

_{3}at 45.9% (

**c**) and 62.5% (

**d**).

**Figure 4.**The open circuit voltage of the PENG at different polymer Young’s modulus (

**a**), permittivity (

**b**), and the distance between electrode and BaTiO

_{3}cubes (

**c**).

**Figure 5.**(

**a**) The equivalent circuit of the composite PENG. (

**b**) The spectral analysis of the square wave force. The output voltage (

**c**) and current (

**d**) at an external load with a resistance of 50 MΩ.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, X.; Xu, Q.; Bai, S.; Qin, Y.; Liu, W. Theoretical Study of the BaTiO_{3} Powder’s Volume Ratio’s Influence on the Output of Composite Piezoelectric Nanogenerator. *Nanomaterials* **2017**, *7*, 143.
https://doi.org/10.3390/nano7060143

**AMA Style**

Zhou X, Xu Q, Bai S, Qin Y, Liu W. Theoretical Study of the BaTiO_{3} Powder’s Volume Ratio’s Influence on the Output of Composite Piezoelectric Nanogenerator. *Nanomaterials*. 2017; 7(6):143.
https://doi.org/10.3390/nano7060143

**Chicago/Turabian Style**

Zhou, Xi, Qi Xu, Suo Bai, Yong Qin, and Weisheng Liu. 2017. "Theoretical Study of the BaTiO_{3} Powder’s Volume Ratio’s Influence on the Output of Composite Piezoelectric Nanogenerator" *Nanomaterials* 7, no. 6: 143.
https://doi.org/10.3390/nano7060143