Optimization of crystal microstructure in piezoelectric ceramics by multiscale finite element analysis

doi:10.1016/j.actamat.2007.12.040

Acta Materialia

Volume 56, Issue 9, May 2008, Pages 1991-2002

https://doi.org/10.1016/j.actamat.2007.12.040 Get rights and content

Abstract

A multiscale finite element method based on homogenization theory is employed to optimize the microstructure of polycrystalline piezoelectric ceramics so as to maximize the homogenized macrostructural piezoelectric response. This analysis reveals that two specific microstructures, layered or alternating [1 1 1]-oriented structures, result in a piezoelectric response that exceeds that of the single-crystal. The layered structure consists of ordered 120°-rotated layers maximizing d₃₃₃, while the alternating structure consists of adjacent grains rotated by 180° in three dimensions optimizing d₃₁₁. These heterogeneous structures maximize the internal strain in polycrystalline aggregates by optimizing electrical and mechanical effects.

Introduction

Advanced piezoelectric ceramics, such as perovskite compounds with a tetragonal crystal structure, are used as actuators in various electronic and mechanical devices. As new micro- and nanotechnologies begin to incorporate such actuators, the performance requirements of piezoelectric ceramics will become more demanding. Piezoelectric ceramics are polycrystalline materials consisting of randomly orientated crystals. The non-symmetric crystal structure induces spontaneous polarization in the [0 0 1] direction and produces strongly anisotropic mechanical and electrical behavior on the crystal scale. There is thus substantial scope for improving the macroscopic piezoelectric response through the microscopic design of the crystal orientation and morphology. Highly oriented thin piezoelectric films have been successfully fabricated by epitaxial growth techniques. For example, [0 0 1]-oriented thin film on a (1 0 0)SrRuO₃/(1 0 0)SrTiO₃ substrate has been fabricated by metalorganic chemical vapor deposition [1], and [1 1 1]-oriented thin film on Si(1 0 0) has been fabricated by radiofrequency magnetron sputtering [2]. These films exhibit superior piezoelectric performance due to control of crystal morphology. However, most of the highly oriented piezoelectric thin films fabricated to date are designed on the basis of single-crystals with uniform orientation. An alternative method for improving the macroscopic piezoelectric response is to optimize the microstructure by considering the heterogeneous orientation of crystals in a polycrystalline aggregate.

In the present study, a method for optimizing the heterogeneous orientation of crystals in polycrystalline piezoelectric ceramics so as to maximize the macroscopic piezoelectric response is proposed based on a multiscale finite element analysis [3], [4], [5], [6]. The microstructural crystal orientations are employed as design parameters, and the homogenized piezoelectric strain constants of the macrostructure – the dominant factors governing piezoelectric actuation – are adopted as the objective function. Using this analysis scheme, computations are performed for barium titanate (BaTiO₃) and lead titanate (PbTiO₃) as representative examples of piezoelectric ceramics. The mechanism by which the optimized microstructure maximizes the macroscopic piezoelectric response is discussed in terms of the orientational dependence of piezoelectricity.

Section snippets

Multiscale finite element method

Consider a polycrystalline piezoelectric material composed of many non-uniformly oriented crystal grains. The macroscopic polycrystalline electromechanical characteristics are governed by the microscopic crystal morphology, and it can usually be assumed that the overall macroscopic structure is formed locally by the spatial repetition of very small microstructures. In other words, it can be assumed that the material properties are periodic functions of microscopic variables. The coupled field

Orientation dependence of BaTiO₃ crystal grains

The characteristics of crystal grains in the microstructure are examined here using the material properties of a BaTiO₃ single crystal [9]. The dependence of the piezoelectric strain constants ^microd₃₃₃ and ^microd₃₁₁ on the orientation of crystals for single-crystal BaTiO₃ is shown in Fig. 4. These values were computed by tensor transformation for Euler angles ϕ of 0–180° with θ = 0° and ψ = 45°. The piezoelectric strain constant is defined in the microscopic coordinate system, and the Euler angle ϕ

Orientation dependence of PbTiO₃ crystals

Fig. 14 shows the orientation dependence of the piezoelectric strain constants ^microd₃₃₃ and ^microd₃₁₁ for single-crystal PbTiO₃ [11]. The maximum values of these constants, ^microd₃₃₃ = 156.73 pC N⁻¹ and ^microd₃₁₁ = −25.40 pC N⁻¹, are obtained at a Euler angle ϕ of 0°, corresponding to the [0 0 1] direction under spontaneous polarization. As single-crystal PbTiO₃ has smaller ^crystald₁₃₁ and ^crystald₂₂₃ values than single-crystal BaTiO₃, the shearing strain due to ^crystald₁₃₁ and ^crystald₂₂₃ under an

Conclusion

A scheme for maximizing the macroscopic piezoelectric strain constants ^macrod₃₃₃ and ^macrod₃₁₁ of polycrystalline piezoelectric ceramics based on a multiscale finite element method was presented. Analysis using this scheme revealed that a piezoelectric strain constant higher than that for single-crystal can be realized in a polycrystalline structure consisting of [1 1 1]-oriented crystals. Two structures achieve this enhanced behavior: an ordered layer structure consisting of 120°-rotated layers,

Acknowledgements

The authors thank Professor Eiji Nakamachi of Doshisha University for helpful discussions and comments. Y.U. was financially supported by a Grant-in-Aid for Young Scientists (B) (No. 17760095) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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Optimization of crystal microstructure in piezoelectric ceramics by multiscale finite element analysis

Abstract

Introduction

Section snippets

Multiscale finite element method

Orientation dependence of BaTiO3 crystal grains

Orientation dependence of PbTiO3 crystals

Conclusion

Acknowledgements

Appl Phys Lett

J Appl Phys

Comp Meth Appl Mech Eng

Archiv Comput Meth Eng

Trans Jpn Soc Mech Eng

Orientation dependence of BaTiO₃ crystal grains

Orientation dependence of PbTiO₃ crystals