A two-dimensional (2D) plane-strain finite element method (FEM) model of polycrystalline film wherein the film thickness is much greater than the microstructure is presented. A finite element based solution for the rigorous homogenization problem to quantify the microstructural features, such as orientation and spatial distribution of crystallographic grains on the global piezoelectric response of unclamped polycrystalline ferroelectric films is implemented. The homogenized material parameters of the piezoelectric films are calculated using the FEM by the solution of the coupled equilibrium electrical and mechanical fields in the linear domain. The dependence of macroscopic electromechanical properties on domain orientation and texture of single-crystalline as well as polycrystalline ferroelectric BaTiO3 and PZN—PT films are demonstrated based on this model. The orientation dependent anisotropy of homogenized electromechanical moduli of polycrystalline and single-crystalline films are examined. The polycrystalline films can be improved in their out-of-plane global piezoelectric behavior by engineering microstructures that favor {111} texture in the case of BaTiO3 and {001} texture in the case of PZN—PT.
Arlt, G. and Sasako, P.1980. ``Domain Configurations and Equilibrium Size of Domains in BaTiO3 Ceramics,''J. Appl. Phys., 51(9):4956-4960.
2.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.1978. Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam .
3.
Benveniste, Y. and Dvorak, G.J.1992. ``Uniform-Fields and Universal Relations in Piezoelectric Composites,''J. Mech. Phys. Solids, 40(6):1295-1312.
4.
Benveniste, Y.1994. ``On the Micromechanics of Fibrous Piezoelectric Composites,'' Mech. Mater., 18(3):183-193.
5.
Berger, H., Gabbert, U., Koppe, H., Rodriguez-Ramos, R., Bravo-Castillero, J., Guinovart-Díaz, R., Otero, J.A. and Maugin, G.A.2003. ``Finite Element and Asymptotic Homogenization Methods Applied to Smart Composite Materials,'' Comput. Mech., 33(1):61-67.
6.
Bhattacharya, K. and Suquet, P.M.2005. ``A Model Problem Concerning Recoverable Strains of Shape-Memory Polycrystals,'' Proc. R. Soc. A, 461(2061):2797-2816.
7.
Buehler, M.J., Bettig, B.P. and Parker, G.G.2003. ``Numerical Homogenization of Active Material Finite-Element Cells,'' Commun. Numer. Meth. Eng., 19(12):977-989.
8.
Cao, H., Schmidt, V.H., Zhang, R., Cao, W. and Luo, H.2004. ``Elastic, Piezoelectric, and Dielectric Properties of 0.58Pb(Mg1/3Nb2/3)O3-0.42PbTiO3 Single Crystal,''J. Appl. Phys., 96(1):549-554.
9.
Cheng, Z.-Q., Lim, C.W. and Kitipornchai, S.2000. ``Three Dimensional Asymptotic Approach to Inhomogeneous aqnd Laminated Piezoelectric Plates,''Int. J. Solids Struct., 37(23):3153-3175.
10.
Damjanovic, D., Budimir, M., Davis, M. and Setter, N.2003. ``Monodomain Versus Polydomain Piezoelectric Response of 0.67 Pb(Mg1/3Nb2/3)O3-0.33PbTiO3 Single Crystals along Nonpolar Directions,''Appl. Phys. Lett., 83(3):527-529.
11.
Damjanovic, D., Budimir, M., Davis, M. and Setter, N.2006. ``Piezoelectric Anisotropy: Enhanced Piezoelectric Response Along Nonpolar Directions in Perovskite Crystals,''J. Mater. Sci., 41(1):65-76.
12.
Du, X.-H., Wang, Q.-M., Belegundu, U., Bhalla, A. and Uchino, K.1999. ``Crystal Orientation Dependence of Piezoelectric Properties of Single Crystal Barium Titanate,'' Mater. Lett., 40(3):109-113.
13.
Dunn, M.L. and Taya, M.1993. ``Micromechanics Predictions of the Effective Electroelastic Moduli of Piezoelectric Composites,''Int. J. Solids Struct. , 30(2):161-175.
14.
Dunn, M.L.1995. ``Effects of Grain Shape Anisotropy, Porosity, and Microcracks on the Elastic and Dielectric Constants of Polycrystalline Piezoelectric Ceramics,''J. Appl. Phys., 78(3):1533-1541.
15.
Fuentes, L., Rodriguez, A., de los Rios, G.A. and Munoz-Romero, A.2005. ``Modeling the Influence of Texture on the Properties of Electroceramics,''Int. Ferroelectrics , 71:289-301.
16.
Galka, A., Telega, J.J. and Wojner, R.1992``Homogenization and Thermo Piezoelectricity,'' Mech. Res. Comm., 19(4):315-324.
17.
García, R.E., Huey, B.D. and Blendell, J.E.2006. ``Virtual Piezoforce Microscopy of Polycrystalline Ferroelectric Films,''J. Appl. Phys., 100(6):064105-1- 064105-10.
18.
Gibiansky, L.V. and Torquato, S.1997. ``On the Use of Homogenization Theory to Design Optimal Piezocomposites for Hydrophone Applications,''J. Mech. Phys. Solids, 45(5): 689-708.
Guedes, J.M. and Kikuchi, N.1990. ``Preprocessing and Postprocessing for Materials Based on the Homogenization Method with Adaptive Finite Element Methods,'' Comput . Methods Appl. Mech. Eng., 83(2):143-198.
21.
Guinovart-Díaz, R., Bravo-Castillero, J., Rodríguez-Ramos, R., Sabina, F.J. and Martínez-Rosado, R.2001. ``Overall Properties of Piezocomposite Materials 1-3,'' Mater. Lett., 48(2):93-98.
22.
Harris, N.R., Hill, M., Torah, R., Townsend, R., Beeby, S., White, N.M. and Ding, J.2006. ``A Multilayer Thick-film PZT Actuator for MEMS Applications,'' Sens. Actuators A, 132(1):311-316.
23.
Hwang, S.C., Huber, J.E., McMeeking, R.M. and Fleck, N.A.1998. ``The Simulation of Switching in Polycrystalline Ferroelectric Ceramics,''J. Appl. Phys., 84(3):1530-1540.
24.
Hwang, S.C. and McMeeking, R.M.1999. ``A Finite Element Model of Ferroelastic Polycrystals,''Int. J. Solids Struct., 36(10): 1541-1556.
25.
Jiang, B. and Batra, R.C.2001. Effective Electroelastic Properties of a Piezocomposite with Viscoelastic and Dielectric Relaxing Matrix,''J. Intel. Mater. Syst. Str., 12(12):847-866.
26.
Kalamkarov, A.L. and Kolpakov, A.G.2001. ``A New Asymptotic Model for a Composite Piezoelastic Plate, Int. J. Solids Struct., 38(34-35): 6027-6044.
27.
Karaki, T., Yan, K. and Adachi, M.2007. ``Barium Titanate Ceramics Manufactured by Two-step Sintering,''Jpn. J. Appl. Phys. Part 1, 46(10B):7035-7038.
28.
Kim, H.-S., Hyun, T.-S., Kim, H.-G., Kim, I.-D., Yun, T.-S. and Lee, J.-C.2006. ``Orientation Effect on Microwave Dielectric Properties of Si-integrated Ba0.6Sr0.4TiO3 Thin Films for Frequency Agile Devices,''Appl. Phys. Lett., 89(5):052902-1-052902-3.
Li, J.Y., Dunn, M.L. and Ledbetter, H.1999. ``Thermoelectroelastic Moduli of Textured Piezoelectric Polycrystals: Exact Solutions and Bounds for Film Textures,''J. Appl. Phys., 86(8):4626-4634.
Li, J.Y., Rogan, R.C., Ustundag, E. and Bhattacharya, K.2005. ``Domain Switching in Polycrystalline Ferroelectric Ceramics,''Nature Mater., 4(10):776-781.
33.
Li, Z., Chan, S.-K., Grimsditch, M.H. and Zouboulisa, E.S.1991. ``The Elastic and Electromechanical Properties of Tetragonal BaTiO3 Single Crystals,''J. Appl. Phys., 70(12): 7327-7332.
34.
Lines, M.E. and Glass, A.M.1977. Principles and Applications of Ferroelectrics and Related Materials, Clarendon Press, Oxford .
35.
Liu, J.J., Zhou, Y.C., Soh, A.K. and Li, J.Y.2006. ``Engineering Domain Configurations for Enhanced Piezoelectricity in Barium Titanate Single Crystals,''Appl. Phys. Lett., 88(3):032904-1- 032904-3.
36.
Maugin, G.A. and Attou, D.1990. ``An Asymptotic Theory of Thin Piezoelectric Plates,''Ouart. J. Mech. Appl. Math. Part 3, 43:347-362.
37.
Nye, J.F.1985. Physical Properties of Crystals: Their Representation by Tensors and Matrices, Clarendon, Oxford.
38.
Otero, J.A., Castillero, J.B. and Ramos, R.R.1997. ``Homogenization of Heterogeneous Piezoelectric Medium,'' Mech. Res. Commun., 24(1):75-84.
39.
Otero, J.A., Rodriguez-Ramos, R., Monsivais, G. and Perez-Alvarez, R.2005. ``Dynamical Behavior of a Layered Piezocomposite Using the Asymptotic Homogenization Method,'' Mech . Mater., 37(1):33-44.
40.
Park, B.H., Kang, B.S., Bu, S.D., Noh, T.W., Lee, J. and Jo, W.1999. ``Lanthanum-Substituted Bismuth Titanate for use in NonVolatile Memories,'' Nature (London), 401(6754):682-684.
41.
Park, S.-E. and Shrout, T.R., 1997. ``Ultrahigh Strain and Piezoelectric Behavior in Relaxor Based Ferroelectric Single Crystals'', J. Appl. Phys., 82(4):1804-1811.
42.
Ruglovsky, J.L., Li, J., Bhattacharya, K. and Atwater, H.A.2006. ``The Effect of Biaxial Texture on The Effective Electromechanical Constants of Polycrystalline Barium Titanate and Lead Titanate Thin Films,''Acta Mater., 54(14):3657-3663.
43.
Sanchez-Palencia, E.1980. Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, Berlin.
44.
Scott, J.F. and de Araujo, C.A.P.1989. ``Ferroelectric Memories ,'' Science, 246(4936):1400-1405.
45.
Setter, N., Damjanovic, D., Eng, L., Fox. G., Gevorgian, S., Hong, S., Kingon, A., Kohlstedt, H., Park, N.Y., Stephenson, G.B., Stolitchnov. I., Taganstev, A.K., Taylor, D.V., Yamada, T. and Streiffer, S.J.2006. ``Ferroelectric Thin Films: Review of Materials, Properties, and Applications,''J. Appl. Phys., 100(5):051606-1-051606-46.
46.
Shimakawa, Y., Kubo, Y., Tauchi, Y., Asano, H., Kamiyama, T., Izumi, F. and Hiroi, Z.2001. ``Crystal and Electronic Structures of Bi4-xLaxTi3O12 Ferroelectric Materials,'' Appl. Phys. Lett.,79(17):2791-2793.
47.
Shu, Y.C.2000. ``Heterogeneous Thin Films of Martensitic Materials,''Arch. Rational Mech. Anal., 153(1):39-90.
48.
Shu, Y.C. and Bhattacharya, K.2001. ``Domain Patterns and Macroscopic Behaviour of Ferroelectric Materials,''Phil. Mag. B, 81(12):2021-2054.
49.
Silva, E.C.N., Fonseca, J.S.O., de Espinosa, F.M., Crumm, A.T., Brady, G.A., Halloran, J.W. and Kikuchi, N.1999a. ``Design of Piezocomposite Materials and Piezoelectric Transducers Using Toplogy Optimization-Part I,''Arch. Comput. Methods Eng., 6(2):117-182.
50.
Silva, E.C.N., Nishiwaki, S., Fonseca, J.S.O. and Kikuchi, N.1999b. ``Optimization Methods Applied to Material and Flextensional Actuator Design Using the Homogenization Method,'' Comput. Methods Appl. Mech. Eng., 172(1-4):241-271.
51.
Takahashi, H., Numamoto, Y., Tani, J. and Tsurekawa, S.2006. ``Piezoelectric Properties of BaTiO3 Ceramics with High Performance Fabricated by Microwave Sintering,''Jpn. J. Appl. Phys. Part 1, 45(9B):7405-7408.
52.
Telega, J.J.1990. ``Piezoelectricity and Homogenization: Application to Biomechanics,'' In: Maugin, G.A. (ed.), Continuum Models and Discrete Systems, No. 2, Longman, London, pp. 220-230.
53.
Trolier-McKinstry, S. and Muralt, P.2004. ``Thin Film Piezoelectrics for MEMS,''J. Electroceramics, 12(1-2):7-17.
54.
Uetsuji, Y., Nakamura, Y., Ueda, S. and Nakamachi, E.2004. ``Numerical Investigation on Ferroelectric Properties of Piezoelectric Materials Using a Crystallographic Homogenization Method,''Modelling Simul. Mater. Sci. Eng., 12(4):S303-S317.
55.
Wada, S., Suzuki, S., Noma, T., Suzuki, T., Osada, M., Kakihana, M., Park, S.-E., Cross, L.E. and Shrout, T.R.1999. ``Enhanced Piezoelectric Property of Barium Titanate Single Crystals with Engineered Domain Configurations,''Jpn. J. Appl. Phys. Part 1, 38(9B):5505-5511.
56.
Wada, S., Takeda, K., Muraishi, T., Kakemoto, H., Tsurumi, T. and Kimura, T.2007. ``Preparation of [110] Grain Oriented Barium Titatanate Ceramics by Templated Grain Growth Method and Their Piezoelectric Properties,'' Jpn. J. Appl. Phys., 46(10B):7039-7043.
57.
Wakiya, N., Shinozaki, K. and Mizutani, N.2001. ``Preparation of Heteroepitaxial Pb(Mg1/3Nb2/3)O3 (PMN) Thin Film by Pulsed Laser Deposition on Si(001) Substrate Using La0.5Sr0.5CoO3 (LSCO)/CeO2/YSZ Triple Buffer,''Thin Solid Films, 384(2): 189-194.
58.
Wang, B.1992. ``Three-Dimensional Analysis of an Ellipsoidal Inclusion in a Piezoelectric Material,''Int. J. Solids. Struct., 29(3):293-308.
59.
Wang, J., Li, Y., Chen, L.-Q. and Zhang, T.-Y.2005. ``The Effect of Mechanical Strains on the Ferroelectric and Dielectric Properties of a Model Single Crystal - Phase Field Simulation,''Acta Mater., 53(8):2495-2507.
60.
Wang, Y.L., Tagantsev, A.K., Damjanovic, D., Setter, N., Yarmarkin, V.K. and Sokolov, A.I.2006. ``Anharmonicity of BaTiO3 Single Crystals,''Phys. Rev. B, 73(13):132103-1-132103-4.
61.
Wu, T.-L.2000. ``Micromechanics Determination of Electroelastic Properties of Piezoelectric Materials Containing Voids,'' Mater. Sci. Eng. A, 280(2):320-327.
62.
Xu, F., Trolier-McKinstry, S., Ren, W., Xu, B., Xie, Z.-L. and Hemker, K.J.2001. ``Domain Wall Motion and its Contribution to the Dielectric and Piezoelectric Properties of Lead Zirconate Titanate Films,''J. Appl. Phys., 89(2):1336-1348.
63.
Zgonik, M., Bernasconi, P., Duelli, M., Schlesser, R., Gunter, P., Garrett, M.H., Rytz, D., Zhu, Y. and Wu, X.1994. ``Dielectric, Elastic, Piezoelectric, Electro-optic, and Elasto-optic Tensors of BaTiO3 Crystals,''Phys. Rev. B, 50(9):5941-5949.
64.
Zhang, R., Jiang, B., Jiang, W. and Cao, W.2003. ``Complete Set of Properties of 0.92Pb(Zn1/3Nb2/3)O3-0.08PbTiO3 Single Crystal with Engineered Domains,'' Mater. Lett., 57(7):1305-1308.
