Finite element calculation of electromechanical coupling coefficient of piezoelectric materials
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https://www.eduzhai.net International Journal of M aterials and Chemistry 2013, 3(3): 59-63 DOI: 10.5923/j.ijmc.20130303.03 Evaluation of Electromechanical Coupling Factor for Piezoelectric Materials Using Finite Element Modeling Vo Thanh Tung1,*, Nguye n Trong Tinh2, Nguye n Hoang Ye n1, Dang Anh Tuan1 1Hue University of Sciences, Vietnam, 77 N guyen Hue Str., Hue city, Vietnam 2Institute of Applied Physics and Scientific Instrument of Vietnamese Academy of Science and Technology, Vietnam Abstract This work uses a finite element modeling method to study the electromechanical coupling factor of piezoelectric materials. Two types of piezoelectric materials were chosen, piezoelectric materials (PZT) and Pb-free piezoelectric materials Ba(Zr0.2Ti0.8)O3-50(Ba0.7Ca0.3)TiO3 (BZT-50BCT) that measure the resonance and anti-resonance frequencies for the calculation of the planar and thickness coupling factor values. For PZT5A, in rad ial v ibration mode, these frequencies were 178 and 211 kHz, and in thickness mode, 2.182 M Hz, respectively. For the BZT-50BCT, the frequencies were 224 and 250 kHz, and in thickness mode, 2.539M Hz, respectively. The coupling factor values were 0.57 (kp) and 0.33 (kt) for PZT5A and 0.50 (kp) and 0.28 (kt) for BZT-50BCT, respectively. Keywords PZT, Electro mechanical Coupling Factor, Pb-free Piezoelectric Materials 1. Introduction Piezoelectric materials are widely used for various applications such as actuators, sensors, sonar transducers, accelero meters,[1-3] etc. Materials like PZT, BaTiO3 and PVDF are popular candidates for use as piezoelectric materials. For half a century, the PZT (lead zirconate titanate) family has been the icon of a large class of technologically important materials piezoelectrics, wh ich dominated almost all p iezoelectric applications ranging fro m cell phone to high-tech scanning-tunneling microscope[4, 5]. Because of Pb toxicity to the environment and to human body, recently many Pb-free ferroelectric systems, such as Ba(Zr0.2Ti0.8)O3-x(Ba0.7Ca0.3)TiO3 (hereafter abbreviated as BZT-xBCT), (K0,436Na0,5Li0,064)Nb0,92Sb0,08O3... have been reported. Especially, the BZT-xBCT material is particularly interested because it exhibits the equally excellent piezoelectricity as in Pb-based systems. In the present study, we used a finite element modeling method to measure the electromechanica l coupling factor, or k-value of p iezoelectric materials (PZT) and Pb-free piezoe lectric materia ls Ba(Zr0.2Ti0.8)O3-50(Ba0.7Ca0.3)TiO3 (BZT-50BCT). The k-value of piezoelectric samp les both two types were calculated and demonstrated for applications. In addition, fro m the received results, we analyze and compare the physic properties of piezoceramics in PZT family and Pb -free piezoceramic (BZT-50BCT) and show the advantage properties of new material BZT-50BCT. * Corresponding author: email@example.com (Vo Thanh Tung) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved 2. Finite Element Modeling Theory Fro m the theory of piezoelectric material[7-9], the mechanica l properties (the relat ion between stress and strain) are defined in the stress-charge form, in which the user has to specify the elasticity matrix, the coupling mat rix, the relative permittiv ity matrix, the piezoelectric coefficients matrix and the density. These matrices of these parameters are defined as in Table 1. The co mp lete set of constants of the PZT5A material shown in Table 1 is taken out fro m the data of the lib raries of the simulation program. A full set of elastic, piezoelectric, and dielectric parameters of BZT-50BCT are measured and calculated by Dezhen Xue and et.al. when using a resonance method. Investigations using computational fin ite element methods (FEM method) have successfully analysed the behaviour of piezoceramic discs[11-13]. In our study, we used a simp lified 3-D axis-symmet ric model in the FEM of simu lation program for design the model geo metry. The modelling dimensions of piezoceramic d iscs with radius 5.5mm, thickness 1mm is illustrated in Fig.1 (a). Modelling conducted is meshed of by using the standard meshing tool (the free tetrahedral mode) at 13380 ele ments, 16 numbers of vertex elements, 208 nu mbers of edge elements and 4352 numbers of boundary elements in Fig 1.b. Since the analysis involves piezoelectric material, there are two different set of boundary conditions the user has to set. First is the mechanical boundary conditions, second is the electrical boundary conditions. In terms of the mechanical boundary conditions, all surfaces are free. For the electrical boundary condition, the bottom surface of the 60 Vo Thanh Tung et al.: Evaluation of Electromechanical Coupling Factor for Piezoelectric M aterials using Finite Element M odeling cantilever beam is set as ground, the top surface set as conditions are illustrated in Fig 1.a. Terminal/ Electrical with the voltage 0.5V. These boundary Table 1. Measured and derived piezoelectric, dielectric constants and elastic constants of poled BZT-50BCT ceramic (density 5200 kg/m3) comparedto the the PZT-5Aceramic (density 7750 kg/m3) (Ref. 7) Sam ple Elastic Stiffness Constants, cij (1010 N/m2) C11E C12E C13E C33E C44E C66E Elastic C ompliance Constants, sij (10-12 m2/N) S11E S1 E 2 S1 E 2 S33E S4 E 4 S66E BZT -50BCT 13.6 8.9 8.5 11.3 2.66 2.44 15.5 -5.5 -7.4 19.7 37.6 42.0 PZT5A 12.1 7.7 7.7 11.1 2.1 2.3 16.4 -5.74 -7.22 18.8 47.5 44.3 Sam ple BZT -50BCT PZT5A Piezoelectric Coefficients dij (10-12 N/C) d33 d31 d15 546 -231 453 374 -171 584 Piezoelectric Coefficients e ij (C /m2) e33 e31 e15 22.4 -5.7 12.1 15.8 -5.4 12.3 Dielectric constants εij (ε0) ε33T 4050 1700 ε11T 2732 1730 ε33S 2930 830 ε11S 1652 916 (a) (b) Figure 1. (a) The fully expanded model of the disk ceramic samples (b) the meshed model of the designed disk ceramic samples 3. Electromechanical Coupling Factors The electro mechanical coupling factor, k-value, were calculated fro m equations of the IEEE Standard on Piezoelectricity 1961 and 176-1987[14, 15]. The k-value and its high mechanical quality factor (Qm) defined in terms of by: Coupling factor ( ) k 2 p ( ) 1 − k 2 p = 1−σ E j1 η1 1+ ∆f fs −η1 1+ ∆f fs j0 η1 1+ ∆f fs 1+σ E j1 η1 1+ ∆f fs (1) International Journal of M aterials and Chemistry 2013, 3(3): 59-63 61 kt2 = π 2 fs fp π tg 2 ∆f fp (2) Mechanical quality factor = Qm 2π f 2 p fsZmC f ( f 2 p − f 2 s ) ≈ 1 4π∆fZmC (3) Where: fp is parallel resonance frequency, fs is serial (motional) resonance frequency, Zm is minimu m impedance of sample at fs , Cf is capacitance of sample at frequency of 1 kHz and Δf = fp – fs , j0 : Bessel function of first kind and zero order, j1: Bessel function of first kind and first order, η1 : lowest positive root of (1 − σ E ) j1 (η ) = η. j0 (η ) ; ε 0 =8.85.10-12 F/m. For IREE Standards on Piezoelectric Crystals 1961: σ E = 0.31 and η = 2.05. For IEEE 1 Standard on Piezoelectricity 176-1987: σ E , η1 are defined by using a polynomial to represent the data described by Meitzler et al.: η =a0 + a1rS + a2rS2 + a3rS3 (4) σ E =b0 + b1rS + b2rS2 + b3rS3 + b4rS4 (5) where rs =fs1/fs2 is the series resonance ratio, fs1, fs2 is measured for the first and second series resonance frequency. The coefficients of polynomials are shown in the Table: Table 2. Coefficients for the polynomials shown in equation (4) and (5) n 0 1 2 3 4 an 11.2924 -7.63859 2.13559 -.215782 bn 97.527023 -126.91730 63.400384 -14.340444 1.2312109 The formulas (1-5) are mathemat ically accurate for determining k; however, in many cases, appearing the sideband resonance around the fundamental resonance and the impedance curve is disturbed. The disturbance gives the error in determination of fs and fp and hence of k. In this case, the method of Onoe is suggested. 4. Simulation Results and Discussions Figure 2 (a-f) shows the results of the dependence of the impedance on frequency in rad ial and thickness vibration modes for piezoceramics the PZT family and BZT-50BCT materials respectively. Fro m these results, the resonant and anti-resonant frequencies, the min imu m impedance Zmin and other parameters are defined. Using the IEEE Standard on Piezoelectricity 1961 and 176-1987 and the equations (1) to (3) and the method Onoe, the calculated k-values are listed in Table 3. (a) (b) (c) (d) 62 Vo Thanh Tung et al.: Evaluation of Electromechanical Coupling Factor for Piezoelectric M aterials using Finite Element M odeling (e) (f) Figure 2. Variations of impedance and its log as a function of frequency for radial vibration mode (a, b) PZT5A (c, d) BZT-50BCT and thickness vibration mode with 1.00 mm thickness (e) PZT5A (f) BZT-50BCT Sample PZT5A BZT -50BCT Table 3. Simulation data of the electromechanical coupling factor value for PZT5A and BZT-50BCT fp (kHz) 211 250 fs (kHz) 178 224 Radial Vibration Mode Zmin Cs (pF) Kp 0.6029 6.81 4764 (IREE61) 0.6118 (IREE87) 0.4951 1.78 5677 (IREE61) 0.5046 (IREE87) Qm Np (kHz.cm) F1 (MHz) 95 195 2.182 356 246 2.539 Thickness Vibration Mode F2 (MHz) t (mm) Kt 7.65 1 0.33 8.634 1 0.28 Nt (kHz.cm) 218 253 Figure 3. Comparison between experimental and theoretical normalized frequency spectra of impedance of BZT-50BCT in radial vibration mode The k-value of the radial vib ration mode for PZT family (PZT5A) are about 0.61 (kp), Np = 195 (kHz.cm) (Radial frequency constant) and 0.33 (kt), Nt = 218(kHz.cm) (Thickness Mode Frequency Constant). These results are similar to the previous research works. They d iffer fro m the reported value about 3 percent error fo r radial v ibration mode and about 30 percent error for thickness of 1.0 mm. Therefore, the fin ite element method is an accurate and easy method for evaluating the electro mechanical coupling factor values. For BZT-50BCT materials, the values are about 0.50 (kp) and 0.28 (kt), respectively. Although the electro mechanical coupling factors of BZT-50BCT materials is smaller than for PZT5A, however these k-values are rather good. Especially, the mechanical quality factor of BZT-50BCT materials is high (Qm = 356). Such these results reveal that BZT-50BCT ceramic has good piezoelectric p ro p erties . The fundamental resonance frequency of PZT5A (rad ial and thickness vibration modes) are at about 178 kHz and 2.18 M Hz, respectively, whereas the resonance frequency for the BZT-50BCT are at 224 kHz and 2.2 MHz. Further, the results show the appearance of the resonant peaks in the simu lation curve and the minimu m-to-maximu m impedance ratio of BZT-50BCT specimen are similar to the specimens of the materials in PZT family. However, it notes that in two vibration modes, the maximu m amp litude at the resonant frequencies of BZT-50BCT material is lo wer than PZT5A. Especially, in thickness modes, the maximu m impedance for BZT-50BCT samp le is about 1200 Ω, but for PZT5A, is about 60000 Ω. Co mparing the normalized frequency spectra of impedance obtained by the experimental and simulated methods for BZT-50BCT material is shown in Fig. 3. It can be observe two similar peaks of the resonance curves. The fundamental resonance frequency of experimental data is International Journal of M aterials and Chemistry 2013, 3(3): 59-63 63 about 282 kHz and of simu lation data is about 249 kHz. It is ceramics”, J. Appl. Phys. 25, 809 1954. possible to see that the simulated resonant frequency match  N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevorgian, S. quite well with the experimental ones (up to 10% error). Hong, A.Kingon, H. Kohlstedt, N. Y. Park, G. B. Stephenson, Furthermore, the amplitude of resonant peaks is not similar. I. Stolitchnov, A. K. Taganstev, D. V. Taylor, T. Yamada, and Possible reasons for the difference and error could be the S. Streiffer, “Ferroelectric thin films: Review of materials, lack in manufacture o f material and sample, the d ifference between theoretical and real boundary conditions. As a result, properties, and applications”, J. Appl. Phys.100, 051606, 2006. the numerically evaluated FEM do not match exact ly the  W. Liu and X. Ren, “Large Piezoelectric Effect in Pb-Free e xperimental ones. Ceramics”, Phys. Rev. Lett.103, 257602 2009. 5. Conclusions Two types of piezoelectric materials have been studied, PZT5A and BZT-50BCT. The simulat ion results for PZT5A material are shown to agree well with experimental results. Furthermore, it shows and evaluates the good factor values of BZT-50BCT materials. Consequently, these properties make BZT-50BCT to be used as a piezooelectric transducer. In addition, it is found that the FEM model connected the simu lation program has excellent agreement to the experimental data in studying the properties of the new materia l groups. ACKNOWLEDGEMENTS The work was carried out in the frame of the “Basic Project of Hue University 2013-2015”.  Jaffe B., W. R. Cook Jr., Jaffe H., Piezoelectric Ceramics, (Academic Press, New York) 1971.  R. E. Newnham,Properties of M aterials: Anisotropy, Symmetry, Structure (Oxford University Press, New York) 2004.  C. Z. Rosen, B. V. Hiremath, and R. Newnham, Piezoelectricity, (American Institute of Physics, New York) 1992.  Dezhen Xue, Yumei Zhou, Huixin Bao, Chao Zhou, Jinghui Gao and Xiaobing Ren, “Lar ge piezoelectric effect in Pb-free Ba(Ti,Sn)O3-x(Ba,Ca)TiO3 ceramics”, J. Appl. Phys. 109, 054110, 2011.  N. Guo, P. Cawley, and D. Hitchings, “The finite element analysis of the vibration characteristics of piezoelectric discs”, Journal of Sound and Vibration, 159, 115–138, 1992.  C.H.Huang,Y. C. 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