0
Review Article

Advances in Developing Electromechanically Coupled Computational Methods for Piezoelectrics/Ferroelectrics at Multiscale

[+] Author and Article Information
Daining Fang

State Key Laboratory for
Turbulence and Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China;
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: fangdn@pku.edu.cn

Faxin Li

State Key Laboratory for
Turbulence and Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China
e-mail: lifaxin@pku.edu.cn

Jiawang Hong

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Xianghua Guo

State Key Laboratory Explosion and
Safety Science,
Beijing Institute of Technology,
Beijing 100081, China

1Corresponding author.

Manuscript received November 19, 2012; final manuscript received September 23, 2013; published online October 28, 2013. Assoc. Editor: Bart Prorok.

Appl. Mech. Rev 65(6), 060802 (Oct 28, 2013) (52 pages) Paper No: AMR-12-1062; doi: 10.1115/1.4025633 History: Received November 19, 2012; Revised September 23, 2013

Piezoelectrics and ferroelectrics have been widely used in modern industries because of their peculiar electromechanical coupling properties, quick response, and compact size. In this work, we give a comprehensive review of our works and others' works in the past decade on the multiscale computational mechanics methods for electromechanical coupling behavior of piezoelectrics and ferroelectrics. The methods are classified into three types based on their applicable scale (i.e., macroscopic methods, mesoscopic methods, and atomic-level methods). In macroscopic methods, we first introduce the basic linear finite element method and employ it to analyze the crack problems in piezoelectrics. Then, the nonlinear finite element methods are presented for electromechanically coupled deformation and the domain switching processes were simulated. Based on our developed nonlinear electromechanically coupled finite element method, the domain switching instability problem was specially discussed and a constrained domain-switching model was proposed to overcome it. To specially address the crack problem in piezoelectrics, we further proposed a meshless electromechanical coupling method for piezoelectrics. In mesoscopic methods, the phase field methods (PFM) were firstly presented and the simulation results on the defects effect and size effect of deformation in ferroelectrics were given. Then, to solve the computational complexity problem of PFM in polycrystals, we proposed an optimization-based computational method taking the interactions between grains in an Eshelby inclusion manner. The domain texture evolution process can be calculated, and the Taylor's rule of plasticity has been reproduced well by this optimization-based model. Alternatively, the domain switching in polycrystalline ferroelectrics can be simulated by a proposed Monte Carlo method, which treated domain switching as a stochastic process. In atomic-level methods, we firstly introduce the first-principles method to calculate polarization and studied the topological polarization and strain gradient effect in ferroelectrics. Then, we present a modified electromechanically coupled molecular dynamic (MD) method for ferroelectrics based on the shell model and investigated the size effect of electromechanical deformation in ferroelectric thin films and nanowires. Finally, we introduced our recently proposed novel atomic finite element method (AFEM), which has higher computational efficiency than the MD. The deformation as well as domain evolution processes in ferroelectrics calculated by AFEM were also presented. The development of electromechanically coupled computational mechanics methods at multiscale is greatly beneficial, not only to the deformation and fracture of piezoelectrics/ferroelectrics, but also to structural design and reliability analysis of smart devices in engineering.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

Schematic depictions of typical domain structures observed in ferroelectric (a) single crystal and (b) polycrystal ceramic

Grahic Jump Location
Fig. 2

Variations of (a) σθ and (b) Dθ on the rim of the hole subjected to tension at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 3

Variations of (a) Er and (b) Eθ on the rim of the hole subjected to tension at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 4

Variations of (a) Dθ and (b) Eθ on the rim of the hole subjected to electric displacement loading at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 5

Stable electric field versus electric displacement hysteresis loops (the thin line is the simulation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 6

Stable longitudinal strain versus electric field curves (the thin line is the calculation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 7

The simulated D–E hysteresis loop (a) and butterfly loop (b) for an unpoled specimen with marked loading points. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 8

The whole process of domain switching under uniaxial electric loading. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 9

Domain-switching process when the applied electric field is close to the coercive field. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 10

Longitudinal strain versus uniaxial stress curves (the thin line is the calculation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 11

Simulated depolarization versus uniaxial stress curve. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 12

Illustrations of Newton–Raphson iterative instability in nonlinear FE models: (a) taking the electric potential as nodal variables and (b) taking the electric displacement potential as nodal variables. Reprinted from Ref. [75] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 13

High depolarization electric field and mechanical field induced by (a) 180 deg switching or ferroelectric switching and (b) 90 deg switching or ferroelastic switching. Reprinted from Ref. [75] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 14

An infinite PZT-5 H plate with circular hole

Grahic Jump Location
Fig. 15

Comparison of EFG results with analytical results: the variation of hoop (a) stress σθ and (b) dielectric displacement Dθ with angle θ. Reprinted from Ref. [78] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 16

Schematic diagram of interface crack

Grahic Jump Location
Fig. 17

The distribution of stress σyy near the crack tip [76]

Grahic Jump Location
Fig. 18

The distributions of electric field Ey around the crack tip [76]

Grahic Jump Location
Fig. 19

The electric field vector distribution near the crack tip [76]

Grahic Jump Location
Fig. 20

The variation of hoop stress around the crack tip when different electromechanical loadings are applied [76]

Grahic Jump Location
Fig. 21

The engineering domain structures with both 90 deg and 180 deg domain walls: (a) calculated by the phase field simulation and (b) observed by experiments in BaTiO3 single crystal. The arrow represents the polarization direction. Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 22

The distributions of electrostatic potential (a) and oxygen vacancy density (b) in the rank-2 domain structure. The initial oxygen vacancy density is uniformly distributed with a magnitude of Nd = 1 × 1024 m–3. Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 23

The electric hysteresis loops (a) and butterfly loops (b) of rank-2 domain structures with different oxygen vacancy densities, and the domain evolutions for Nd = 1.2 × 1024 m–3 under a positive alternating field (c). Domains indexed by (I) and (II) in (c) correspond to the two states (I) and (II) in (a) and (b). Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 24

Two extreme cases simulated by the phase field method: (a) The domain pattern of a ultrathin film calculated by the phase field method with 2.4 nm < h < 4.8 nm [46]; (b) the domain pattern of the ultrathin film calculated by molecular dynamics with h = 4.4 nm [152]; (c) the domain pattern of a thin film calculated by the phase field method with h > 20 nm without considering the out-of-plane depolarization [46]; (d) the domain pattern of the thin film calculated by the 2D phase field simulation without concerning the size effect [141]. The symbol ⊙ (⊗) denotes the direction of polarization flowing out of (into) the x1x2 surface. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 25

The size-dependent distribution of the out-of-plane polarization at different thicknesses. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 26

The size-dependent out-of-plane polarization and domain pattern. The domain exhibits a zigzag pattern with eight variants coexisting when h > 7.6 nm, a zigzag pattern with four variants coexisting when 6.8 nm < h < 8.0 nm, a vortex pattern with four variants coexisting when 4.4 nm < h < 7.2 nm, and a stripe pattern with two variants coexisting when 2.4 nm < h < 4.8 nm; when the thickness is smaller than 2.8 nm, the domain disappears, indicating that the ferroelectricity is suppressed. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 27

2D Illustration of material model for morphotropic ferroelectric ceramics

Grahic Jump Location
Fig. 28

(a) Six types of tetragonal domains and (b) eight types of rhombohedral domains in ferroelectric crystals

Grahic Jump Location
Fig. 29

Simulated D-E hysteresis loops (a) and butterfly curves (b) of tetragonal PZT ceramics during the first three cycles of electric loading. The overlapped curves after 1.5 cycles loading indicates the good convergence of the proposed model. Reprinted from Ref. [163] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 30

Simulated (a) polarization and (b) strain curves of tetragonal and rhombohedral PZT ceramics. Reprinted from Ref. [53] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 31

(a) (001) pole figure of tetragonal PZT and (b) (111) pole figures of rhombohedral PZT ceramics during cyclic electric loading. Reprinted from Ref. [53] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 32

Switching strain curves of tetragonal and rhombohedral PZT ceramics under uniaxial tension/compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 33

(a) (001) pole figures of tetragonal PZT and (b) (111) pole figures of rhombohedral PZT ceramics during uniaxial tension/compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 34

(a) Switching strain-stress curves and (b) the cumulative pole figures of (001) and (111) axes in morphotropic PZT ceramics under uniaxial tension and compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 35

Comparison of switching strain versus stress curves of tetragonal, rhombohedral, and MPB PZT ceramics (the nominal coercive stress σC = 70 MPa). Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 36

Simulated switching strain versus stress curves of tetragonal, rhombohedral, and MPB PZT ceramics using a small nominal coercive stress of σC = 5 MPa. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 37

(a) D-E hysteresis loops, (b) butterfly curves, and (c) reversed butterfly curves of a morphotropic PZT ceramic under electric loading with different levels of precompression. Reprinted from Ref. [54] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 38

The cumulative pole figures of [001] and [111] axes of a morphotropic PZT ceramic with different levels of precompression: (a) before electric loading; (b) under maximum electric field; (c) after removing the electric field. Reprinted from Ref. [54] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 39

The simulated hysteresis (left) and butterfly (right) loops of PZT ceramics. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 40

Evolution of March coefficient of PZT ceramics in tetragonal (left) and rhombohedral (right) phases during electric poling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 41

Comparison between simulation and experiment to hysteresis (left) and butterfly (right) loops of soft PLZT during electric poling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 42

Depolarization (left) and stress–strain (right) curves of PZT ceramics during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 43

Evolution of March coefficient of PZT ceramics in tetragonal (left) and rhombohedral (right) phases during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 44

Comparison between simulation and experiment of depolarization (left) and stress-strain (right) curves of soft PLZT during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 45

Calculation of the polarization for different unit cells in one-dimensional chain of alternating anions and cations. In the top panel, anions and cations are spaced a distance a/2 apart (a is the lattice constant). In the bottom panel, the cations are displaced by a distance Δ. The dashed rectangles indicate unit cells that are used for calculation of the polarization (see text).

Grahic Jump Location
Fig. 46

The largest balls are for Ba ions, the intermediate ones represent Ti ions, and the smallest ones are for O. (a) is TiO2 terminated; (b) is BaO terminated, except the edges, where the corner Ba atoms have been removed; (c) is stoichiometric, with two BaO surfaces and two TiO2 ones. (d) is BaO terminated. Reprinted from Ref. [177] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 47

Left column (a)–(d): Ti off-center displacements for the four respective nanowires in Fig. 46. (e) is similar to (d) but for a thinner wire. The right column sketches the corresponding 2D field lines around topological point defects of winding numbers n = 1, 1, −1, and −3, respectively, (j) showing the n = −3 decomposition into a central n = +1 and four n = −1 defects. The respective largest arrows correspond to (a) 21 pm, (b) 35 pm, (c) 23 pm, (d) 6.8 pm, and (e) 6.6 pm off-centering displacement. Reprinted from Ref. [177] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 48

Supercell and strain profile used in the calculations. For BaTiO3, the unit cell contains only one formula unit (a), while for SrTiO3, due to its antiferrodistortive transition at low temperature, the in-plane size of the unit cell is doubled (b). (c) and (d) supercell in xz plan, (e) cosine gradient strain in the supercell, (f) atom displacement in the supercell. Reprinted from Ref. [200] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 49

Longitudinal FEC μ3333 of rhombohedral BaTiO3 for different supercell sizes (N = 6, 10, 14) under various strain gradients (εmax = 0.5%, 1.0%, 1.5%, 2.0%). In each panel, the line with square symbols (up) is for the applied strain, the line with dot symbols (down) is for the “relaxed strain” (see text). The two values converge as the supercell size is increased. Reprinted from Ref. [200] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 50

Size effect of elastic stiffness C11 for BaTiO3 films. Reprinted from Ref. [180] with the kind permission of Taylor & Francis.

Grahic Jump Location
Fig. 51

The piezoelectric properties of BaTiO3 bulk and thin films, e33 for bulk and e11 for films of n = 2 and 4. Reprinted from Ref. [180] with the kind permission of Taylor & Francis.

Grahic Jump Location
Fig. 52

Two different representative unit cells of BaTiO3

Grahic Jump Location
Fig. 53

(a) The cell-by-cell out-of-plane polarization profile of a z-direction chain in film with different thicknesses. (b) Schematic side views of the out-of-plane polarization pattern for film in which h ≥ 3.2 nm. (c) Schematic side views of the out-of-plane polarization pattern for film in which h ≤ 2.8 nm. Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 54

The evolution of domains in the 4.4-nm-thick film under the external electric field. (a) and (b): The average polarization of the film as a function of the external electric field. (c) ∼ (k): The sequent top views of the out-of-plane polarization patterns under different electric fields marked in (a) and (b). The value in each grid represents the average z-direction polarization over a chain along z-direction (unit: μC/cm2). Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 55

The hysteresis loops (a) of BaTiO3 films with thicknesses of 4.4 nm, 3.6 nm, and 2.8 nm and the variations of remnant polarization (b) and coercive field (c) with the reciprocal of film thickness. Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 56

(a) Out-of-plane average polarization as a function of temperature for stress-free BaTiO3 nanofilms with different thicknesses, and (b) the Curie temperature as a function of the reciprocal of film thickness. Reprinted from Ref. [64] with the kind permission of American Scientific Publishers.

Grahic Jump Location
Fig. 57

The size-dependent polarization components of BaTiO3 nanowire. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 58

The size-dependent ferroelectric hysteresis loops of BaTiO3 nanowires. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 59

Hysteresis loop of nanowire with diameter d = 2.0 nm and the domain structures at different electrical loading levels. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 60

A comparison of computational efficiency between the developed AFEM and MD method. The inset table demonstrates the ratio of time cost in the MD to that in the AFEM. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 61

Three types of stable nanoscale domain structure: (a) single domain; (b) 90° domain; (c) vortex domain. The three figures above illustrate the polarization distributions with each arrow representing the polarization of a unit cell, while the ones below demonstrate the schematic domain patterns. Notice that only 1/4 area of each figure are calculated by the present AFEM, and based on the periodical boundary conditions, four identical patterns are packed together to obtain a better image of the polarization topology. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 62

Evolution of single domain structure under electric fields with different loading directions. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In