Role of grain orientation distribution in the ferroelectric and ferroelastic domain switching of ferroelectric polycrystals

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Acta Materialia 61 (2013) 6037–6049 www.elsevier.com/locate/actamat

Role of grain orientation distribution in the ferroelectric and ferroelastic domain switching of ferroelectric polycrystals J. Wang a,b,⇑, W. Shu a, T. Shimada b, T. Kitamura b, T.-Y. Zhang c a

Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China b Department of Mechanical Engineering and Science, Kyoto University, Nishikyo-ku, Kyoto 615-8540, Japan c Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China Received 6 February 2013; received in revised form 25 March 2013; accepted 22 June 2013 Available online 20 July 2013

Abstract The non-linear electromechanical behavior of ferroelectric polycrystals stems from polarization/domain switching, which are affected by the grain boundaries and grain orientations. The effects of grain orientation distribution on the domain switching and non-linear behavior of a two-dimensional ferroelectric polycrystal subjected to an electric or/and mechanical load are investigated by computer simulations with a real-space phase-field model. Phase-field simulations indicate that the macroscopic coercive field, remanent polarization and remanent strain in the polycrystal with a random distribution of grain orientation are correspondingly smaller than those in the polycrystal with a uniform distribution of grain orientation. However, the polycrystal with randomly distributed grain has a larger strain variation with the electric field than the polycrystal with uniformly distributed grains, which suggests that the random polycrystal has a better piezoelectric property than the uniform one. The different macroscopic non-linear behaviors of the ferroelectric polycrystals are attributed to different microscopic domain switching processes. For the polycrystal with randomly distributed grains, the domain switching takes place from the regions near the large angle grain boundaries, while new domains nucleate from the cross sections between the grain boundaries and the material surface in the polycrystal with uniform grain orientation. Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Domain switching; Phase-field model; Ferroelectric polycrystals; Crystallographic orientation

1. Introduction Because of their distinguished electromechanical properties, ferroelectric ceramics have been used widely in electromechanical devices such as sensors, actuators and transducers [1]. The electromechanical properties of ferroelectric ceramics are highly dependent on the domain configuration. Domain switching takes place when an external electrical and/or mechanical load exceeds a critical value [2]. The domain switching is the origin of non-linear electromechanical response of ferroelectric materials, which involves ⇑ Corresponding author at: Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China. Tel./fax: +86 571 87953110. E-mail addresses: [email protected], [email protected] (J. Wang).

the nucleation of new domain and domain wall motion [3]. Most ferroelectric ceramics are polycrystals and composed of different grains. The grains are arranged in different orientations. In each grain, there are different domains with specific polarization orientations. Because of the anisotropy of a ferroelectric crystal, the grain orientation influences the polarization orientation in the domains [4]. However, the domain switching of ferroelectric polycrystals is a collective behavior of all grains, in which the long-range elastic and electrostatic interactions between grains play a critical role. The long-range elastic and electrostatic interactions between the grains are dependent on the grain orientations. Therefore, the domain configuration and domain switching of ferroelectric polycrystals are strongly affected by the grain orientations and much more complicated than their single-crystal counterparts [5]. In

1359-6454/$36.00 Ó 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2013.06.044

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order to effectively utilize the electromechanical properties of ferroelectric ceramics, it is of importance to understand the role of grain orientation distribution in the domain switching and non-linear behavior of ferroelectric polycrystals subjected to electrical and/or mechanical loadings. The domain switching and non-linear behavior of ferroelectric materials was first described by the phenomenological Landau theory [6,7]. In the Landau theory, the free energy is expressed as a high-order polynomial of polarization. In one-dimensional simplification, the free energy has a double-well type profile in the absence of an external field, as shown in Fig. 1a. The two wells in the energy profile correspond to the spontaneous polarization states. The polarization state switches from one state to the other when the anti-parallel external field exceeds a critical value. The critical value, i.e. the coercive field, is determined by the non-linear relation between the electric field and polarization in Fig. 1b, which is derived from the free energy polynomial. The Landau theory predicts the intrinsic coercive field for the ideal single crystal and single domain ferroelectrics. However, the intrinsic coercive field predicted from the Landau theory is usually much larger than that observed in experiments [8]. The significant difference between the predicted and observed coercive fields is the so-called Landauer’s paradox [9]. One of the reasons for Landauer’s paradox is that the effect of crystal imperfections and thermal fluctuation on the domain nucleation is not considered in the Landau theory. There are always different crystal imperfections such as the surface, grain boundaries and vacancies in ferroelectric polycrystal materials. These imperfections can be the nucleation sites for new domains in the presence of an external field, which can dramatically reduce the coercive field and thus affect the non-linear behavior of ferroelectric materials [10]. In order to predict the experimentally observed domain switching and non-linear behavior of ferroelectric materials, many micro-electromechanical models have been proposed [11–16]. The models successfully predicted the electric displacement vs. electric field hysteresis loops and

strain vs. stress loops of ferroelectric materials, including the ferroelectric single crystal [17,18] and polycrystal materials [19–21]. The micro-electromechanical models usually need to assume a switching criterion in advance. The domain switching and non-linear behaviors predicted from the micro-electromechanical models are determined by the presupposed switching criterion, which should be obtained from the experiments. Furthermore, the domain in the model is often assumed as a simple inclusion with specific geometry [22] and the Ruess approximation is employed [11,23,24] for ferroelectric polycrystal materials. Domain configuration is described by the volume fraction of different domains, in which domain walls are fixed and treated as sharp interfaces. The detailed polarization distribution and domain wall motion are absent in the micro-electromechanical models. To predict the detailed polarization distribution and domain wall motion, phase-field models with diffuse interfaces have been developed for ferroelectric materials in the framework of the time-dependent Ginzburg–Landau equations (TDGL) [25–28]. The phase-field models do not make any prior assumptions on the domain wall location and switching criteria. Domain switching is a direct consequence of the minimization process of the total free energy of an entire simulated system. Phase-field models make a significant contribution to the understanding of the surface effects [29], nucleation of the new domain [28], size dependence of domain structure [30] and the role of defects in switching [31] in ferroelectric materials. Phase-field models also successfully predict the polarization vs. electric field hysteresis loop and strain vs. stress loop of ferroelectric single crystal [28,32]. For ferroelectric polycrystals, the domain switching and non-linear behavior under an electric field are investigated by a phase-field model with the periodic boundary condition [33,34]. However, the periodic boundary condition is valid only when the simulated size is much smaller than the material size. It is also difficult to directly include the surface effect on the domain evolution with the periodic boundary condition. To overcome the

Fig. 1. (a) Dependence of Landau energy on the polarization in PbTiO3 ferroelectric crystal subjected to different electric fields. The two wells in the energy profile without electric field correspond to two spontaneous polarization states. The polarization state will switch from one state to the other when the anti-parallel external field exceeds a critical value. (b) The P–E hysteresis loop of PbTiO3 ferroelectric crystal predicted from the Landau single domain model. The coercive field is determined by the non-linear relation between the electric field and polarization.

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limitation of the periodic boundary condition, Shu et al. [4] developed a real space phase-field model for finite-size ferroelectric polycrystals and employed it to investigate the effect of grain boundary on the electromechanical response of PbTiO3 ferroelectric polycrystals. In this paper, the real space phase-field model is employed to investigate the effect of grain orientation on the domain switching and non-linear behavior of PbTiO3 ferroelectric polycrystals subjected to an external electric field or stress. The remainder of the paper is organized as follows. First, the real space phase-field model of ferroelectric polycrystals is briefly described in Section 2, and the simulation model is introduced in Section 3. Then, the simulation results on the multi-rank domain structures, ferroelectric and ferroelastic domain switching of ferroelectric polycrystals with different grain orientation distributions are given and discussed in Section 4. Finally, Section 5 concludes. 2. Real-space phase-field model of ferroelectric polycrystals A ferroelectric material changes from the paraelectric phase into the ferroelectric phase when the temperature is lower than its Curie point. To describe the ferroelectric phase transition, the polarization is used as the order parameter in the Landau theory. For ferroelectric polycrystals, different grains have different crystallographic orientations. The Landau free energy at each grain is described by the local polarization vector, which has three components ðP L1 ; P L2 ; P L3 Þ, and the superscript L denotes the local coordinates within the individual grain. Based on the Landau theory, the bulk Landau free energy density for a ferroelectric material is commonly expressed by a six-order polynomial of spontaneous polarization as        2 2 2 fLandau ¼ a1 P L1 þ P L2 þ P L3        4 4 4 þ a11 P L1 þ P L2 þ P L3              2 2 2 2 2 2 þ a12 P L1 P L2 þ P L2 P L3 þ P L1 P L3        6 6 6 þ a111 P L1 þ P L2 þ P L3 h              4 2 2 4 2 2 þ a112 P L1 P L3 þ P L2 P L1 þ P L3 þ P L2   4   L 2  L  2  2  2  2 þ P L3 P1 þ P2 þ a123 P L1 P L2 P L3 ; ð1Þ where a1 ¼ ðT  T 0 Þ=2j0 C 0 is the dielectric stiffness, a11 ; a12 ; a111 ; a112 and a123 are higher-order stiffness coefficient, T and T 0 denote the temperature and the Curie– Weiss temperature, respectively, C 0 denotes the Curie constant, and j0 is the dielectric constant of the vacuum. Spontaneous strain usually accompanies spontaneous polarization when the material transforms from the paraelectric phase to the ferroelectric phase. Thus, the strain couples with the polarization, and the free energy density related to the strain includes the pure elastic energy density

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and the coupling energy density between the strain and polarization: 1  2  2  2  fstrain ¼ c11 eL11 þ eL22 þ eL33 2   þ c12 eL11 eL22 þ eL22 eL33 þ eL11 eL33    2  2  2 þ 2c44 eL12 þ eL23 þ eL13     2  2  2  q11 eL11 P L1 þ eL22 P L2 þ eL33 P L3 h      2   2  2 2  q12 eL11 P L3 þ P L2 þ eL22 P L1 þ P L3    2 i 2 þ eL33 P L1 þ P L2    2q44 eL12 P L1 P L2 þ eL13 P L1 P L3 þ eL23 P L2 P L3 ð2Þ where c11 ; c12 ; c44 are the elastic constants, q11 ; q12 ; q44 are electrostrictive coefficients, and eLij is the total strain tensor in the local coordinates of individual grain. In the phase-field model, a domain wall is modeled by the polarization gradient, and the wall across which the spontaneous polarization is continuous is called a diffuse interface with a finite thickness. Therefore, the contribution of the domain walls to the total free energy is named the gradient energy, given by  2  2  2  1 fgrad ¼ G11 P L1;1 þ P L2;2 þ P L3;3 2   þG12 P L1;1 P L2;2 þP L2;2 P L3;3 þP L1;1 P L3;3  2  2  2  1 þ G44 P L1;2 þP L2;1 þ P L1;3 þP L3;1 þ P L2;3 þP L3;2 2  2  2  2  1 þ G044 P L1;2 P L2;1 þ P L2;3 P L3;2 þ P L1;3 P L3;1 2 ð3Þ G11 ; G12 ; G44 ; G044

where are the gradient coefficients, and P Li;j denotes the derivative of polarization component P Li with respect to coordinate xLj . The gradient energy represents the penalty in energy for spatially inhomogeneous polarization in the ferroelectric materials. A polarization field, in general, produces an electric field. The electrostatic energy for a ferroelectric material comes from the long-range electric interaction between different polarizations and external electric fields, if any. In the present phase-field model, the electrostatic energy density is expressed as 1  2  2  2  felec ¼  jc EL1 þ EL2 þ EL3  EL1 P L1  EL2 P L2  EL3 P L3 2 ð4Þ In which jc is the dielectric constant of the background materials, and ELi are the total electric field in the local coordinates of individual grain. Following previous work [4], the grain boundaries are assumed as dielectrics in the present study. For the dielectric grain boundaries, the spontaneous polarizations are zero. Without spontaneous polarization, the Landau energy of Eq. (1) and the gradient

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energy of Eq. (3) vanish. The pure elastic energy and electrostatic energy remain in Eqs. (2) and (4) without the polarization. Then, the free energy at grain boundaries can be expressed as 1   2  2  2  fB ¼ c11 eL11 þ eL22 þ eL33 2   þ c12 eL11 eL22 þ eL22 eL33 þ eL11 eL33    2  2  2 þ 2c44 eL12 þ eL23 þ eL13 1  2  2  2   jc EL1 þ EL2 þ EL3 ð5Þ 2 In the free energy formula of grain boundaries, the local strain and electric field are assumed the same as the global ones, which are independent of the Euler angles of grains. For a polycrystalline ferroelectric model, it is necessary to establish a common global coordinate system to describe the total energy for all grains. Therefore, the local polarization P Li , strain eLij and electric field ELi of each grain in Eqs. (1)–(5) are transferred into the global polarization P i , strain eij and electric field Ei , respectively, in a common global coordinate system. In the polycrystalline ferroelectric model, the grain orientation is described by three Euler angles ða; b; hÞ with respect to the global coordinates ðx1 ; x2 ; x3 Þ. The transformation matrix from the global to local coordinate system is given by 0

cos½a cos½h  cos½b sin½a sin½h

F ¼

Z

in which, f ¼ fLandau þ fstrain þ fgrad þ felec and f ¼ fB for the grains and grain boundaries, respectively. With the total free energy as a function of the global polarization, strain and electric field, the temporal evolution of polarization is obtained through the TDGL: @P i ðx; tÞ dF ¼ L @t dP i ðx; tÞ

and Maxwell’s (or Gauss’s) equation   @ @f  ¼ 0; @xi @Ei

ð7Þ

in which Rij denotes the ij-component of the transformation matrix R. Similarly, the electrical field in the local coordinate system ELi is expressed by Ei in the global system as ELi

¼ Rij Ej

ð8Þ

The second-rank tensor of strain in the local coordinate system can be expressed by the strain in the global coordinate system as eLij ¼ Rki Rlj ekl

ð9Þ

Up to now, all variables in Eqs. (1)–(5) are transferred from the local coordinate system to the global coordinate system. Hereafter, all variables denote the values in the global coordinate system. The total free energy of a ferroelectric polycrystal is obtained by integrating the total free energy density of f over the whole volume as follows:

ð11Þ

ð13Þ

must be simultaneously satisfied for body force-free and charge-free ferroelectric materials. With the free energy density, Eq. (11) can also be expressed as

B R ¼ @  cos½b cos½h sin½a  cos½a sin½h cos½b cos½a cos½h  sin½a sin½h sin½a sin½b  cos½a sin½b

P Li ¼ Rij P j

ði ¼ 1; 2; 3Þ;

where t represents time, L is the kinetic coefficient related to the domain mobility, dF =dP i ðx; tÞ denotes the thermodynamic driving force, and x is the spatial vector x ¼ ðx1 ; x2 ; x3 Þ. In addition to the governing Eq. (11) for polarization, the following mechanical governing equation   @ @f ¼ 0; ð12Þ @xj @eij

sin½a cos½h þ cos½b cos½a sin½h

With the transformation matrix, the polarization in the local coordinate system P Li is expressed by Pi in the global system as

ð10Þ

f dv V

sin½b sin½h

1

C sin½b cos½h A cos½b

  @P i ðx; tÞ @f @ @f  ¼ L @t @P i ðx; tÞ @xj @P i;j

ð6Þ

ð14Þ

In order to solve the governing Eqs. (12)–(14), the present authors employ a non-linear multi-field coupling finite element method. Using the variation or principle of virtual work, the integral form (or weak form) of governing equations have the form Z

 @f @f 1 @P i @f @f de þ dEi þ dP i þ dP i þ dn dv @eij @Ei L @t @P i @nij ij V Z ¼ fti dui  wd/ þ pi dP i gdA: S

ð15Þ where nij ¼ P i;j , ti denotes the surface traction, w denotes surface charge, and @P@fi;j nj ¼ pi is the surface gradient flux, where nj denotes the components of normal unit vector of surfaces. To solve Eq. (15) by the finite element method, an eight-node brick element with seven degrees of freedom at each node is employed for the space discretization, and the backward Euler iteration method is adopted for the time integration. The detailed derivation of the three-dimensional

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Fig. 2. (a) Schematic drawing of a ferroelectric polycrystal composed of seven grains with random orientations. The local coordinate axes are shown in each grain. The grains with labels r, s, t, u, v, w and x correspond to the orientations of 73.3°, 81.5°, 11.4°, 82.2°, 56.9°, 8.8° and 25.1°, respectively. (b) Mesh partition of the ferroelectric polycrystal. The electric field and stress are applied to the ferroelectric polycrystal to investigate the ferroelectric and ferroelastic domain switching, respectively.

finite element formulation for Eq. (15) can be found in previous work [4]. 3. Simulation model The equation of phase-field model described in Section 2 is solved in real space by the finite element method. The real space phase-field model is applicable for the ferroelectric polycrystals with arbitrary geometries and boundary conditions. In the present work, the domain switching and the non-linear behavior of three-dimensional ferroelectric polycrystal with two grain orientation distributions are simulated using the phase-field model. The first ferroelectric polycrystal consists of seven grains with random distribution of grain orientations, as shown in Fig. 2a, which is called the random polycrystal. For simplicity, the crystallographic axis of grains rotates only in the x1  x2 plane, in which the Euler angle of a is non-zero and other Euler angles are zero. The a values of the seven grains are signed by computer with random numbers. Note that the random polycrystal is the general ferroelectric polycrystal, which is taken as an example of numerous polycrystals with different random grain orientations. Another ferroelectric polycrystal studied has the same grain orientation in the seven grains, which is called the uniform polycrystal. It should be note that the uniform polycrystal is different from the single crystal because of the presence of dielectric grain boundaries between different grains in the former. The accurate description of random orientation distribution needs a large number grains. Because of the limitation of computer capability, the ferroelectric polycrystals with a small number of grains are simulated in the present study. The simulated ferroelectric polycrystal model can be regarded as part of the ferroelectric polycrystal with a large

number of random grains. The simple model provides an example for studying the effect of grain orientation distribution on the domain structures and domain switching of ferroelectric polycrystals. The sizes of the simulated ferroelectric polycrystals in the x1 ; x2 and x3 directions are 302, 345 and 3 nm, respectively. The detailed mesh partition of finite element is shown in Fig. 2b, in which 7760 brick elements are employed. The various electric fields are applied in the x2 direction by giving different electrical potentials to the top surface and zero electrical potential to the bottom surface. The electric boundary conditions on other surfaces are open-circuited, i.e. D  n ¼ 0. For the mechanical boundary conditions, the center node is fixed, and the other nodes can move in the x1  x3 plane. The upper and lower surfaces are applied by a uniform tensile stress and other surfaces of the ferroelectric polycrystal are traction free. The initial values of polarizations are given randomly, and the dimensionless time step is set at Dt ¼ 0:2 in the simulation. In this paper, the polarization components are presented mainly in the x1  x2 plane, and the out-of-plane component in the x3 direction is neglected, because it was found that the in-plane components are independent of x3, and the out-of-plane component is close to zero. The following normalized variables with the material properties for PbTiO3 crystals are employed in the present study. Table 1 Values of the normalized coefficients used in the simulation. Landau coefficient

Elastic constants

Coupling constants

Gradient coefficient

a11 ¼ 0:24 a12 ¼ 2:5 a111 ¼ 0:49 a112 ¼ 1:2 a123 ¼ 7:0

c11 ¼ 1766:0 c12 ¼ 802:0 c44 ¼ 1124:0

q11 ¼ 63:3 q12 ¼ 2:6 q44 ¼ 41:6

G11 G12 G44 G0 44

¼ 1:6 ¼0 ¼ 0:8 ¼ 0:8

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x ¼

J. Wang et al. / Acta Materialia 61 (2013) 6037–6049

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ja1 j=G110 x; t ¼ ja1 jLt; P  ¼ P =P 0 ;

a1 ¼ a1 =ja1 j; a11 ¼ a11 P 20 =ja1 j; a12 ¼ a12 P 20 =ja1 j; e ¼ 1=ð2ja1 jÞ; a111 ¼ a11 P 40 =ja1 j; a112 ¼ a12 P 40 =ja1 j; a123 ¼ a123 P 40 =ja1 j; Q11 ¼ Q11 P 20 ; Q12 ¼ Q12 P 20 ; Q44 ¼ Q44 P 20 ;

ð16Þ

e 2 E ¼ E =ðja1 jP 0 Þ; re; kl ¼ rkl =ðja1 jP 0 Þ; c11 ¼ c11 =ðja1 jP 20 Þ; c12 ¼ c12 =ðja1 jP 20 Þ; c44 ¼ c44 =ðja1 jP 20 Þ; G11 ¼ G11 =G110 ; G12 ¼ G12 =G110 ; 0 G44 ¼ G44 =G110 ; G0 44 ¼ G44 =G110 ; e;

e

in which P 0 ¼ jP s j ¼ 0:757 C=m2 is the magnitude of the spontaneous polarization at room temperature,

a1 ¼ ðT  T 0 Þ=ð2e0 C 0 Þ ¼ ð25  479Þ  3:8  105 m2 N =C 2 and G110 ¼ 1:73  1010 m4 N =C 2 . The dielectric constant of the background materials is set as jc ¼ 66j0 , which is taken from Ref. [30]. The asterisk  represents the dimensionless value of the corresponding variable. Table 1 shows the values of the dimensionless or normalized material coefficients employed in the simulations. 4. Results and discussion 4.1. Multiple-rank domain structure in ferroelectric polycrystals The microscopic polarization distribution or domain structure has a significant influence on the macroscopic

x2

x1

(a)

(b)

(c) Fig. 3. The simulated domain structures from a random initial state in the ferroelectric polycrystal with the random (a) and uniform (b) grain orientations in the absence of external loading. (c) The enlarged plot of the rectangle region in (b). The {1 2 3 4} type rank-2 domain structure is obtained.

J. Wang et al. / Acta Materialia 61 (2013) 6037–6049 0.9 E

0.6

A

*

0.3

P2

material properties, such as the piezoelectric coefficients, elastic constants and dielectric permittivity [35,36]. The ferroelectric polycrystal may have different domain patterns, depending on the boundary conditions and grain orientations. Fig. 3a shows the simulated domain structures from a random initial state in the random polycrystal of Fig. 3 in the absence of external loads. The magnitude and orientation of polarization are presented by the length and direction of arrows in the plots. An abundance of domain patterns exhibit in different grains in the ferroelectric polycrystal. Because of the mechanical constraint by the neighboring grains, 90° domain walls dominate inside the grains. To reduce the depolarization field inside the grains, there are also 180° domain walls. Most polarization orientations in Fig. 3a coincide with the crystallographic axis of grains. The coincidence between the polarization orientation and grain orientation shows the effect of Landau energy on the domain pattern. To investigate the effect of grain orientation distribution on the domain pattern, Fig. 3b shows the domain structures when the distribution of grain orientations is uniform, i.e. a = 0 for all grains. Because the crystallographic axes of all grains coincide with the global coordinate axes, all polarizations are along either the x1 or x2 axis. The polarizations form four domain variants: þP L1 ; P L1 ; þP L2 and  P L2 . To show the details of the domain structure, Fig. 3c gives the enlarged domain pattern in the center grain surrounded by the dotted rectangular box in Fig. 3b. The phase-field simulation shows that four domain variants form a rank-2 laminate structure. It should be noted that the rank-2 domain structure is predicted by the phase-field model from an initially random polarization distribution without any assumption on the location and profile of domain walls. The rank-2 domain structures are classified into eight types by Tsou et al. [37]. According to the classification, the predicted rank-2 domain pattern in Fig. 3c is the {1 2 3 4} type. The domain pattern of {1 2 3 4} has four distinct in-plane domain variants in a herringbone arrangement. Two 180° domain walls are truncated by two 90° domain walls in the middle of the grain. This domain structure has been observed by lateral piezoresponse force microscopy in a ferroelectric single crystal [38].

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F

B

0.0 G

-0.3

-0.9

random uniform

C

-0.6 D H

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

*

E2 Fig. 4. The simulated hysteresis loops between the average polarization and applied electric field in the x2 direction for the ferroelectric polycrystals with random and uniform grain orientations. The normalized electric field is defined as E2 ¼ E2 =E0 . The characteristic electric field is defined as E0 = |a1|P0 = 1.3  108V/m, where P0 = 0.757 C/m2.

of loading is taken as the initial state for the next level of loading in the simulations. After 40 time steps of calculation at each level of the applied electric field, the domain structure has reached a stable state. Fig. 4 shows the average polarization in the x2 direction vs. the applied electric field for the random and uniform grain orientation distributions. The most important difference between the P–E hysteresis loops is that both the remanent polarization and coercive field in the random polycrystal are correspondingly smaller than those in the uniform polycrystal. This difference is attributed to the effect of grain orientation on the domain switching. The smaller coercive field in the random polycrystal is due to the fact that domain switching is easier when the grain crystallographic axis deviates from the direction of external electric field. It should be noted that the shape of the hysteresis loop of the random polycrystal is closer to experimental observation than that of the uniform polycrystal. If one takes another random distribution of grain orientation, the hysteresis loop is expected to be similar. However, if the grain

0.040 0.035

4.2. Ferroelectric domain switching

0.030 0.025

The domain structures of ferroelectrics will change or switch when an external electric field is high enough. The ferroelectric domain switching induces the non-linear electrical behavior of ferroelectric materials. In this section, the effect of grain orientation distribution on the domain switching and the non-linear behavior is investigated under an external electric field. The domain states in Fig. 3a and b are taken as the initial states, and a cyclic electric field is applied in the x2 direction. Because of the non-linear properties, the response of a ferroelectric polycrystal to an applied load is related to the loading history. Therefore, the final state of the domain structure under a given level

0.020 0.015 0.010 0.005

random uniform

0.000 -0.005 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

*

E2 Fig. 5. The simulated butterfly loops between the average strain and applied electric field in the x2 direction for the ferroelectric polycrystals with random and uniform grain orientations.

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(a)

(b)

x2

x1

(c)

(d)

Fig. 6. The process of domain switching under external electric fields for the ferroelectric polycrystal with random grain orientation. The external electric fields are opposite to the initial polarization orientation. The domain structures in (a), (b), (c) and (d) correspond to the dimensionless electric fields at points A (E2 ¼ 0), B (E2 ¼ 0:29), C (E2 ¼ 0:87) and D (E2 ¼ 1:45), respectively, in Fig. 4. The domain switching takes place from the top-right and middle grains with large crystallographic angles.

orientations are uniform, the shape of the hysteresis loop becomes a bilinear curve, as shown in Fig. 4. The reason for the difference in hysteresis loops in the random and uniform polycrystals is determined by different domain switching processes, which will be discussed in the following. The piezoelectric property of ferroelectric materials is characterized by the slope of the butterfly curve between the electric field and strain. The average strain in the x2 direction is calculated and taken as the macroscopic response to the applied electric field, which gives the butterfly curves in Fig. 5. The linear piezoelectric constant is defined as d 33 ¼ De22 =DE2 and calculated proximately from the slopes of butterfly curves at the zero point of the applied electric field. The piezoelectric constant d33 of the random polycrystal is 30.1% larger than that of the

uniform polycrystal. The result implies that the ferroelectric polycrystal with a non-uniform grain orientation has a better piezoelectric property than that with a uniform grain orientation. Fig. 6 shows the detailed domain structures at different electric fields for the random polycrystal. The domain structures in Fig. 6a–d correspond to the electric fields at points A, B, C and D in Fig. 4. Because of the random distribution of grain orientations, the polarizations in some grains do not form a single domain when the electric field decreases to zero. Fig. 6a shows a multidomain structure with 90° domain walls in grain v at the top-right corner, in which crystallographic axis and polarizations deviate from the global coordinate axis maximally among the seven grains in the ferroelectric polycrystal because grains

J. Wang et al. / Acta Materialia 61 (2013) 6037–6049

(a)

(b)

(c)

(d)

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x2

x1

Fig. 7. The process of domain switching under external electric fields for the ferroelectric polycrystal with uniform grain orientation. The external electric fields are opposite to the initial polarization orientation. The domain structures in (a), (b), (c) and (d) correspond to the dimensionless electric fields at points E (E2 ¼ 0:29), F (E2 ¼ 0:58), G (E2 ¼ 0:87) and H (E2 ¼ 1:45), respectively, in Fig. 4. New domains nucleate from the cross sections between the grain boundaries and the surface.

with Euler angles of a = 90° and a = 0° have the same crystallographic axis, owing to the crystal symmetry. The polarizations in the top-right grain v affect the polarization distribution in its neighboring top and middle grains, w and r. The grains on the left and at the bottom-right corner are the single domain state, because their crystallographic axes are very close to the global coordinate axis. When the electric field is reversed, the reversed domains grow, and more domains appear, as shown in Fig. 6b. Because of the multidomain state in the top-right grain, the domain switching becomes easier than that in a single domain state. It is the reason that the coercive field with the random grain orientation distribution is much smaller than that with the uniform grain orientation, as illustrated

in Fig. 4. The grains that have a crystallographic axis close to the global coordinate axis more easily become the single domain state under a reversed field, as shown by the bottom-left grain in Fig. 6c. For the top-right grain, the polarizations do not form a single domain, even when the reversed electric field increases to 1.45, as shown in Fig. 6d, which is also due to the large deviation in crystallographic axis from the global coordinate axis. Fig. 7 gives the detailed domain structures at different electric fields for the uniform polycrystal. The domain structures in Fig. 7a–d correspond to the electric fields at points E, F, G and H in Fig. 4. The domain structures of grains change from the multidomain state of Fig. 3b to the single domain state when the electric field increases

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J. Wang et al. / Acta Materialia 61 (2013) 6037–6049 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 G -22 C -24 -0.03 -0.02

H

D

A

E

F B

-0.01

uniform random

0.00

0.01

0.02

0.03

Fig. 8. The stress vs. strain curves during the mechanical depolarization in the x2 direction for the ferroelectric polycrystals with random and uniform grain orientations. The variation in strain with uniform grain orientation is larger than that with random grain orientation, which results in a larger residual strain. The normalized stress is defined as r ¼ r=r0 . The characteristic stress is defined as r0 = |a1|P 20 = 9.89  107Pa, where P0 = 0.757 C/m2.

0 -2 -4 -6 -8 -10 -12 -14 -16

uniform random

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0.2

0.3

*

0.4

0.5

0.6

0.7

P

Fig. 9. Depolarization due to the compressive stresses in the x2 direction. The average polarization remains zero during the unloading process for both cases. The normalized polarization is defined as P  ¼ P =P 0 , where P0 = 0.757 C/m2.

from 0 to 1.45. During the unloading process from 1.45 to 0, the domain structure remains a single domain. When the reversed electric field increases to 0.29 at point E in Fig. 4, the single domain begins to switch, as shown in Fig. 7a. The domain switching takes place from the intersection between the grain boundaries and top surface of the polycrystal. As the reversed electric field increases, the reversed domains grow and expand along the grain boundaries, as shown in Fig. 7b. The 90° domain switching occurs near the boundary between the top and middle grains to reduce the elastic energy in the grains. The 90° domain switching near the grain boundary has been also predicted in the ferroelectric polycrystal model with the periodic boundary condition [33]. When the electric field reaches 0.87 at G in Fig. 4, the region with unreversed polarization get smaller, while the region with reversed polarization expands as shown in Fig. 7c. When the electric field further increases to 1.45 at point H in Fig. 4, all

polarizations are reversed, and it becomes a single domain again along the direction of electric field in Fig. 7d. Figs. 6 and 7 show that the grain orientation distribution has a significant influence on the switching process of ferroelectric polycrystals subjected to an external electric field. The detailed domain switching processes provide the microscopic explanation to the macroscopic non-linear behaviors of ferroelectric polycrystals. In the present phase-field study, the domain patterns and domain switching are simulated in the two-dimensional ferroelectric polycrystal, in which only one Euler angle of each grain is allowed to change and the out-ofplane polarization is absent. For a three-dimensional ferroelectric polycrystal, the out-of-plane polarization will appear, and the misorientation between grains will be described by three Euler angles. Therefore, the domain patterns and their switching behavior in the three-dimensional ferroelectric polycrystal are expected to be more complicated than that in the two-dimensional ferroelectric polycrystal. The variation in three Euler angles and the appearance of out-of-plane polarization can result in more types of domain patterns, which make the domain switching easier. Therefore, the two-dimensional simulations might over-estimate the coercive field in the three-dimensional real polycrystal. Nevertheless, the effect of the grain orientation on the domain switching behavior illustrated by the present simulations should be generally true qualitatively. Although the simulation results are not directly related to the experimental observations, the present phase-field study provides the fundamental mechanism on the influence of grain orientation distribution on the domain patterns and domain switching of ferroelectric polycrystal. The domain switching induces an internal stress field. When the switching-induced stress is high enough, it will cause electric fatigue and cracking, which is an important topic for future study. 4.3. Ferroelastic domain switching The 90° domain switching occurs when the external stress exceeds a critical value, which is responsible for the ferroelastic hysteresis loop between the stress and strain. To obtain the strain–stress hysteresis loop, different levels of compressive stress are applied in the x2 direction to the ferroelectric polycrystal in Fig. 2. The remanent polarization states in Fig. 5 are employed as the initial states. In the simulations, the dimensionless stress increases from zero to 21.5 with an increment of 2.39 and then decreases from 21.5 to zero with the same increment. Although the inhomogeneous strains are obtained from the simulations, the average strains in the x2 direction are calculated to plot the strain–stress curves. At each level of stress, the calculations with 40 time steps are conducted to ensure the simulated ferroelectric polycrystals reach the stable domain states. Fig. 8 shows the stress–strain hysteresis loop of the simulated ferroelectric polycrystals with the random and uniform grain orientation distributions. In

J. Wang et al. / Acta Materialia 61 (2013) 6037–6049

(a)

(b)

(c)

(d)

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x2

x1

Fig. 10. Domain switching under different compressive stresses for the ferroelectric polycrystal with the uniform grain orientation. The compressive stresses are applied in the x2 direction, and the initial domain state is the remanent state at point A in Fig. 4. The domain structures in (a), (b), (c) and (d) correspond to the dimensionless stress at points A (r* = 2.4), B (r* = 14.3), C (r* = 21.5) and D (r* = 0), respectively, in Fig. 8. The 90° domain switching takes place in the middle grain through the domain wall motion.

both cases, the strains change rapidly at the beginning of the loading process. The rapid change of the strains is due to the dramatic change in domain structure. The strain changes almost linearly with the stress when the stress increases from zero to 16.7 for the random polycrystals. During the stress loading process from zero to 21.5, the change of strain is 0.057 in the uniform polycrystal, which is larger than 0.046 in the random polycrystal. The result suggests that the ferroelectric polycrystal with a non-uniform grain orientation distribution is harder than that with a uniform grain orientation in the loading process. However, the slopes of the two curves are very close during the unloading process, as shown in Fig. 8, implying that the unloading elastic properties of the two ferroelectric

polycrystals are independent of the grain orientation distribution. To investigate the effect of grain orientation distribution on the mechanical depolarization, Fig. 9 shows the dependence of average polarization in the x2 direction on the compressive stress. The polarization decreases almost linearly with the stress for the random polycrystal. For the uniform polycrystal, the polarization decreases very fast at the beginning, and then decreases slowly when the compressive stress increases from 7.2 to 11.9. As the compressive stress increases to 9.5, the polarization is close to zero in the random polycrystal, while there is still a value of 0.2 for the uniform polycrystal. The result shows that the ferroelectric polycrystal with the non-uniform grain

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(a)

(b)

(c)

(d)

x2

x1

Fig. 11. Domain switching under different compressive stresses for the ferroelectric polycrystal with uniform grain orientation. The compressive stresses are applied in the x2 direction, and the initial single domain state is the remanent state with zero electric field in Fig. 4. The domain structures in (a), (b), (c) and (d) correspond to the dimensionless stresses at points E (r* = 4.8), F (r* = 11.9), G (r* = 21.5) and H (r* = 0), respectively, in Fig. 8. The 90° domain switching occurs from the cross sections between the grain boundaries and the surface.

orientation distribution can be depolarized by a smaller mechanical load. Fig. 10 gives the domain structures of the random polycrystal during the loading and unloading processes illustrated in Fig. 8. The domain patterns in Fig. 10a–d correspond to stress states at points A, B, C and D in Fig. 8. As the applied compressive stress increases, the domain switching takes place through the domain growth and 90° domain wall motion in the middle grain, as shown in Fig. 10. The motion of domain walls promotes the domains parallel to the stress shrink, and the domains perpendicular to it grow. The 90° domain wall dominates in the middle grain, which reduces the elastic energy greatly.

The appearance of the 90° domain in the bottom-left grain shows a strong long-range elastic interaction between the middle and bottom-left grains across the grain boundary. More polarizations change their orientations and become perpendicular to the stress when the external stress further increases to 21.5, as shown in Fig. 10c. Polarizations in the grains form 180° domain walls to reduce the depolarization energy. The Landau energy is the minimum when the polarizations are along the crystallographic axis of grain. Therefore, most polarizations are along the local coordinate axis, which makes the 180° domain walls parallel to the crystallographic axis of each grain. When the compressive stress decreases to zero, the domain structure

J. Wang et al. / Acta Materialia 61 (2013) 6037–6049

shows less change, as shown in Fig. 10d, which makes the average polarization remain zero during the unloading process illustrated in Fig. 9. Fig. 11 shows the domain structures at different compressive stresses in the uniform polycrystal. When the compressive stress increases, the new domains nucleate from the cross sections between the top surface and the grain boundaries, which is similar to the case when the electric field is applied in Fig. 7. But the polarization orientations in the new domains are different in the two cases. Most polarizations in the new domains are perpendicular to the stress in Fig. 11, while they are nearly parallel to the electric field in Fig. 7. When the compressive stress increases, the domains parallel to the stress shrink or vanish, the domains perpendicular to it grow and expand. This process is the same as that in the random polycrystal. Fig. 11d shows the domain structures after unloading, which are totally different from the initial single domain structure before loading. As a result, a large remanent strain exists after the loading and unloading processes, as shown in Fig. 8. Because of the effect of grain orientation, the remanent strain in the uniform polycrystal is much larger than that in the random polycrystal. 5. Conclusions A real-space phase-field model is employed to simulate ferroelectric and ferroelastic domain switching and the related non-linear electromechanical behaviors of ferroelectric polycrystals with different grain orientation distributions. The phase-field simulations give detailed information on the evolution of polarization during the domain switching, which provides the microscopic explanation for macroscopic material properties. It is found that the grain orientation distribution plays an important role in the domain switching of ferroelectric polycrystals. For the ferroelectric polycrystals with random grain orientation distribution, the domain switching takes place from the grain with the maximum deviation from the applied field. When the grain orientations are uniform, new domains nucleate from the cross sections between the grain boundaries and the material surface. Because of the effect of grain orientation distribution on domain switching, the coercive field, remanent polarization and remanent strain in the uniform polycrystal are correspondingly larger than those in the random polycrystal. However, the random polycrystal has a larger strain variation than the uniform polycrystal under the same electric field, which implies that the polycrystal with a non-uniform grain distribution is more suitable for the large strain actuators than the uniform one.

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Acknowledgements The work was financially supported by Grant-in-Aid for Scientific Research (No. 21226005) and Research Fellowship (No. P12058) from the Japan Society of the Promotion of Science, the Nature Science Foundation of China under Grants (11002123, 11090333) and Zhejiang Provincial Natural Science Foundation under Grant R6110115. References [1] Haertling GH. J Am Ceram Soc 1999;82:797. [2] Fridkin VM. Ferroelectrics 2012;426:139. [3] Nelson CT, Gao P, Jokisaari JR, Heikes C, Adamo C, Melville A, et al. Science 2011;334:968. [4] Shu WL, Wang J, Zhang TY. J Appl Phys 2012;112:064108. [5] Li JY, Rogan RC, Ustundag E, Bhattacharya K. Nat Mater 2005;4:776. [6] Landau L. Nature 1936;138:840. [7] Chandra P, Littlewood PB. Top Appl Phys 2007;105:69. [8] Ducharme S, Fridkin VM, Bune AV, Palto SP, Blinov LM, Petukhova NN, et al. Phys Rev Lett 2000;84:175. [9] Landauer R. J Appl Phys 1957;28:227. [10] Jesse S, Rodriguez BJ, Choudhury S, Baddorf AP, Vrejoiu I, Hesse D, et al. Nat Mater 2008;7:209. [11] Huber JE, Fleck NA, Landis CM, McMeeking Rm. J Mech Phys Solids 1999;47:1663. [12] Haug A, Huber JE, Onck PR, Van der Giessen E. J Mech Phys Solids 2007;55:648. [13] Kim SJ, Jiang Q. Int J Solids Struct 2002;39:1015. [14] Hwang SC, Lynch CS, McMeeking RM. Acta Metall Mater 1995;43:2073. [15] Chen X, Fang DN, Hwang KC. Acta Mater 1997;45:3181. [16] Kamlah M. Continuum Mech Therm 2011;13:219. [17] Yen JH, Shu YC, Shieh J, Yeh JH. J Mech Phys Solids 2008;56:2117. [18] Li WF, Weng GJ. J Appl Phys 2011;90:2484. [19] Su Y, Weng GJ. Proc Roy Soc A – Math Phys 2006;462:1573. [20] Belov AY, Kreher WS. Acta Mater 2006;54:3463. [21] Hwang SC, Huber JE, McMeeking RM, Fleck NA. J Appl Phys 1998;84:1530. [22] Rodel J, Kreher WS. Comput Mater Sci 2000;19:123. [23] Landis CM, McMeeking RM. Ferroelectrics 2011;255:13. [24] Li J, Weng GJ. J Intel Mater Syst Str 2001;12:79. [25] Hu HL, Chen LQ. J Am Ceram Soc 1998;81:492. [26] Ahluwalia R, Cao WW. Phys Rev B 2011;63:012103. [27] Li YL, Hu SY, Liu ZK, Chen LQ. Acta Mater 2002;50:395. [28] Wang J, Shi SQ, Chen LQ, Li YL, Zhang TY. Acta Mater 2004;52:749. [29] Hong L, Soh AK, Song YC, Lim LC. Acta Mater 2008;56:2966. [30] Wang J, Zhang TY. Phys Rev B 2006;73:144107. [31] Hong L, Soh AK, Du QG, Li JY. Phys Rev B 2008;77:094104. [32] Zhang W, Bhattacharya K. Acta Mater 2005;53:185. [33] Choudhury S, Li YL, Krill CE, Chen LQ. Acta Mater 2005;53:5313. [34] Choudhury S, Li YL, Krill CE, Chen LQ. Acta Mater 2007;55:1415. [35] Li JY, Liu D. J Mech Phys Solids 2004;52:1719. [36] Rodel J. Mech Mater 2007;39:302. [37] Tsou NT, Potnis PR, Huber JE. Phys Rev B 2011;83:184120. [38] Kalinin SV, Rodriguez BJ, Jesse S, Karapetian E, Mirman B, Eliseev EA, et al. Ann Rev 2007;37:189.