Domain switching criterion for ferroelectric single crystals under uni-axial electromechanical loading

Mechanics of Materials 93 (2016) 246–256

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Domain switching criterion for ferroelectric single crystals under uni-axial electromechanical loading Y.W. Li a,b, F.X. Li a,c,∗ a

State Key Lab for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China Department of Engineering mechanics, School of Civil Engineering, Wuhan University, Wuhan 430072, China c Center for Applied Physics and Technologies, Peking University, Beijing 100871, China b

a r t i c l e

i n f o

Article history: Received 2 July 2015 Revised 25 October 2015 Available online 21 November 2015 Keywords: Ferroelectric single crystal Domain switching Electromechanical loading Hardening effect

a b s t r a c t Domain switching is a long concerned topic in ferroelectrics while currently an accurate domain switching criterion (DSC) is still lacking. In this work, both 90° and 180° domain switching were carefully investigated in perfectly poled tetragonal BaTiO3 crystals. Pure 180° domain switching was realized during bipolar electric loading, and pure 90° switching was achieved by mechanical depoling and subsequent electric repoling. Results showed that the energy barrier for 180° domain switching is considerably larger than that for 90° switching. In addition, during 180° domain switching, only slight hardening effect was observed while the 90° switching process showed significant hardening behavior. Moreover, for 90° domain switching, more energy was dissipated during mechanical depoling than that during electric repoling. Based on these experimental results, an incremental DSC was proposed for 90° domain switching and the sharp DSC proposed by Hwang et al. (1995) was adopted for 180° switching. An electric field–stress–domain structure diagram of BaTiO3 crystal was then plotted to address the switching process. It is predicted that pseudoelasticity and large actuation strain can be realized in BaTiO3 crystals via reversible 90° domain switching under uni-axial electromechanical loading. These predictions were further verified by subsequent testing, demonstrating the validity of the proposed DSC. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the past decades, ferroelectric devices such as piezoelectric actuators, vibration dampers, high power acoustic transducers have been developed (Uchino, 2000), and been widely used in modern industries due to their peculiar electromechanical coupling properties, ultrafast response, and compact size. During operation, ferroelectrics were usually subjected to medium-to-high electric fields and/or mechanical stresses to obtain large actuations or output

∗ Corresponding author at: State Key Lab for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China. Tel.: +86-13366341755. E-mail addresses: [email protected] (Y.W. Li), [email protected] (F.X. Li).

http://dx.doi.org/10.1016/j.mechmat.2015.11.005 0167-6636/© 2015 Elsevier Ltd. All rights reserved.

power (Uchino, 2000). The applied electric fields and/or mechanical stresses can drive ferroelectric/ferroelastic domain switching when they are larger than the corresponding critical values. Due to domain switching, the polarization and strain responses of ferroelectrics will show significant nonlinearities (Jaffe et al., 1971). Such nonlinearities are normally harmful as they not only complicate the structure design (Uchino, 2000) but also may lead to property degradation of ferroelectric devices (Schneider, 2007; Yang and Zhu, 1998; Zhu and Yang, 1999). However, some recent works concerning making the nonlinear strains useful in actuators were also conducted (Burcsu et al., 2000; Yen et al., 2008; Ren, 2004; Li et al., 2013; Li and Li, 2013). Since it has been widely accepted that the nonlinearities of ferroelectrics are mainly caused by domain switching, it is necessary to establish a suitable domain switching criterion (DSC) to

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predict the complicated responses of ferroelectrics under electric and/or mechanical loading. Up to now, intensive efforts have been conducted on this topic and quite a few DSCs have been proposed using different parameters as the driving force for domain switching (Hwang et al., 1995; Lu et al., 1999; Shaikh et al., 2006; Sun and Jiang, 1998; Huang and McMeeking, 2000; Sun and Achuthan, 2004; Fotinich and Carman, 2000; Li et al., 2004; Fang et al., 2006). Among all these DSCs, the DSC proposed by Hwang et al. (1995) got the widest applications. They treated domain switching as an energy dissipation process, i.e., when the total work done by external loading reached a critical value, domain switching occurred and finished immediately. In their DSC, the coercive electric fields for non-180° and 180° domain switching were thought to be equal and only the spontaneous parts of the strain/polarization was considered. However, Hwang’s DSC cannot explain the decrease of the coercive electric field during cyclic bipolar electric loading with the increase of compressive stresses. Moreover, the evolution of polarization and strain in ferroelectric crystal predicted by Hwang’s DSC is sharp at the coercive field, which is inconsistent with the hardening effect observed in BaTiO3 crystal (Li et al., 2013). Some other DSCs were also developed after Hwang et al. (1995). Sun et.al. (1998) suggested using the total polarizations and strains rather than the spontaneous part to calculate the work done during domain switching. Lu et al. (1999) and Hwang and McMeeking (2000) suggested using the Gibbs free energy as the reference parameters, respectively. According to this type of criterion, domain switching occurs when the decrease of the Gibbs free energy reaches a critical value between the initial and the final state of the specific domain. In addition, Sun et al. (2004) also suggested using the internal energy density as the driving force for domain switching. That is, domain switching occurs when the associated internal energy reaches a critical value. From a different point of view, Fotinich and Carman (2000) suggested using the change of the polarization as a criterion for 180° domain switching. Other different DSCs were also developed by Shaikh et al. (2006), Li et al. (2004), and Fang et al. (2006). Although great success has been achieved in modeling the nonlinear hysteretic behavior of ferroelectrics by using the DSCs mentioned above, similar to Hwang’s DSC, none of them can reproduce the hardening effect observed in ferroelectric single crystals. The limitation of the DSCs mentioned above was thought to be caused by their developing process. They were all developed by analyzing the experimental results of ferroelectric ceramics, whose macro response is a complex collective process of a very large number of variously oriented grains (Jaffe et al., 1971). However, what DSC describes is the reorientation behavior of a ferroelectric single domain. Meanwhile, in ferroelectric ceramics, both 180° and non-180° domain switching coexist during loading which cannot be clearly distinguished from material’s macro responses. Thus based on the responses of ferroelectric ceramics, it is difficult to determine the coercive electric field of purely non-180° and/or 180° domain switching. Furthermore, significant grain-tograin interactions exist during domain switching in ceramics (Li and Rajapakse, 2007), which couples with the effect of domain-to-domain interactions (Pramanick et al., 2012),

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making the understanding of the later effect on domain switching behavior becomes impossible. Moreover, in ferroelectrics ceramics, the phase structure is normally impurity and phase transition may also exist during loading, which further complicates the understanding of domain switching. Thus to establish an accurate DSC for ferroelectrics, the first work should be realizing pure non-180° and pure 180° domain switching in ferroelectric single crystal in which the grain boundary effects can be removed. It should be noted that some experimental results concerning ferroelectric crystals have been reported (Burcsu et al., 2000; Yen et al., 2008; Webber et al., 2008; Liu and Lynch, 2006a, 2006b). However, as bi-polar electric fields were used in these works, both 180° and non-180° domain switching (Burcsu et al., 2000; Yen et al., 2008; Zhang and Bhattacharya, 2005; Weng and Wong, 2008) coexist and even phase transition may occur (Webber et al., 2008; Liu and Lynch, 2006a; Liu and Lynch, 2006b). Thus further experimental and theoretical works are still required in order to establish an accurate DSC for ferroelectrics. The objective of this paper is to establish a more accurate and practical DSC for ferroelectrics. By specific design and using pre-poled BaTiO3 crystals as model material, pure 90° and 180° domain switching was firstly realized. Significant hardening effects were observed during 90° domain switching both under mechanical depoling and electric repoling. While for 180° domain switching, only slightly hardening effect was observed. An incremental DSC was then proposed for 90° domain switching and the sharp DSC proposed by Hwang et al. (1995) was adopted for 180° switching. Based on the incremental 90° DSC, it is predicted that under suitable uniaxial electromechanical loading, pseudoelasticity and large actuation strain can be realized in prepoled BaTiO3 crystals. These predictions were further verified by subsequent testing. 2. Experiment 2.1. Specimen BaTiO3 crystals were chosen as model material in this study as it is in tetragonal phase at room temperature and can be poled into a single domain state (Garrett et al., 1991). The BaTiO3 blocks were provided by the Institute of Ceramics, Chinese Academy of Sciences, cut along the [001] direction with the dimensions of 5 × 5 × 5 mm3 . At room temperature (24 °C), the spontaneous strain (S0 ) is 1.04%, which was determined by X-Ray diffraction method. All the faces of the samples were polished first in order to prevent crack damage during electric poling and mechanical depoling. Two opposite 5 × 5 mm2 faces were spread with silver paste as electrodes for electric loading. Electric poling was conducted at 110 °C, slightly below the Curie temperature (120 °C) of BaTiO3 crystal. A DC electric field of 500 V/mm was first applied to the specimen along the [001] direction for about 2 h. Then the specimen was gradually cooled to room temperature with the electric field holding. After poling, the longitudinal piezoelectric constant d33 was measured by a Berlincourt d33 meter, with the value of 115 ± 5 μC/N, which is in accordance with the values reported by others (Wada et al., 1999; Zgonik et al., 1994).

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Fig. 1. Testing setup of mechanical depoling, electric repoling, and coupled electromechanical loading.

Fig. 2. The principle of realizing pure 90° domain switching in BaTiO3 crystal. (a) The domain structure of a poled BaTiO3 crystal; (b) The domain structure of a partially depoled or partially repoled BaTiO3 crystal (c) The domain structure of a BaTiO3 crystal after completely mechanical depoling. The red arrows represent the polarization directions of domains. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.2. Testing setup

2.3. Experimental procedure

The testing setup for mechanical depoling, electric repoling and electromechanical loading is illustrated in Fig. 1. Electric fields were provided by a high-voltage amplifier (Trek) controlled by a functional generator. In order to prevent arcing, the specimen was immersed into silicon oil during testing. Compressive stresses were applied by a Shimadzu testing machine and a spherical hinge was used to avoid any bias compression. Alumina blocks were used to insulate the specimen from the loading equipment. Two brass plates of 0.3 mm-thick were pasted on the alumina blocks as electrode for electric loading. Two strain gauges, with the dimension of 3 × 3 mm2 , were glued on the opposite 5 × 5 mm2 faces to measure the longitudinal strain during compression. A large capacitor was connected in series with the testing specimen and an electrometer with high-input resistance was connected in parallel with the capacitor to detect the polarization change. During mechanical depolarization, the electrode beneath the specimen was connected with the ground (the switch “B” is on); whereas during electromechanical loading, it was connected with the high voltage amplifier (the switch “A” is on). In addition, a thick rubber block is put beneath the silicon oil spacer to reduce the mechanical stiffness of the whole loading system, which can reduce the prestress fluctuations during electromechanical loading.

Our recently experimental results (Li and Li, 2014) demonstrated that for a perfectly poled BaTiO3 crystal, 180° domain switching was accomplished by 180° domain nucleation and forward domain wall motion during antiparallel electric field loading. Thus to characterize pure 180° domain switching, the polarization and strain response of a pre-poled BaTiO3 crystal were measured under bipolar electric field loading with triangular waveform. The loading period and amplitude are 5 s and 800 V/mm, respectively. Fig. 2 shows the principle of realizing pure 90° domain switching in a pre-poled BaTiO3 crystal by mechanical depoling and subsequent electric repoling. As shown in Fig. 2, during mechanical depoling, the domains parallel to the prepoling direction (hereafter referred as “c+ ” domains) are driven to lie down (hereafter referred as “a” domains) by 90° domain switching gradually from Fig. 2(a) to Fig. 2(b) and finally to Fig. 2(c). While during electric repoling (Electric field with triangular waveform was used during repoling), a vice versa process occurs. In this way, pure 90° domain switching induced by stresses or electric fields can be realized in BaTiO3 crystals. During mechanical depoling, the loading and unloading rate was 1 MPa/s. After mechanical depolarization, the depoled specimen was repoled by a uni-polar electric field with triangular waveform. The loading amplitude and frequency

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Fig. 3. (a) Electric field–polarization loop and (b) electric field–strain curve of a poled BaTiO3 crystal during cyclic electric field loading. The black line represents the measured results. The red line denotes the calculated results by Hwang’s DSC. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

were 0.8 kV/mm and 0.2 Hz, respectively. Both the polarizations and the strains were recorded to characterize 90° domain switching. 3. Results and discussions 3.1. 180° domain switching in BaTiO3 crystal Fig. 3(a) shows the rectangle-like electric field– polarization loop of a pre-poled BaTiO3 crystal during bipolar electric field loading. Such electric field–polarization loop has also been reported by Merz (1953), Burcsu et al. (2000). The polarization shows a sharp reversal at the coercive field, indicating the effect of domain-to-domain interaction on 180° domain switching is relatively small. The measured maximum and remnant polarizations are 26.5 μC/cm2 and 26 μC/cm2 , respectively, which is in accordance with the results reported previously (Burcsu et al., 2000; Merz, 1953). By calculating the electric field–polarization loop, the energy dissipation density (here after represented by W180◦ ) during one cycle of electric field loading can be calculated, with a value of 74 kJ/m3 . Then the energy barrier U180◦ for 180° domain switching can be calculated by W180◦ /2, which is 37 J/m3 . The strain response of the BaTiO3 crystal during bipolar electric field loading is shown in Fig. 3(b). The total strain variations are only about 0.01%. According to equation d33 = S/E, the inverse longitudinal piezoelectric constant can be evaluated, with a value of 120 μC/N, which is in accordance with the value measured by the Berlincourt meter. This implies that the tested BaTiO3 crystal has been poled to a single domain state and hardly 90° domain switching was involved during electric field loading. As polarization showed sharp change near the coercive electric field and only 180° domain switching occurred during loading, thus the macroscopic constitutive behavior of a perfectly poled BaTiO3 crystal can be modeled by Hwang’s 180° DSC (Hwang et al., 1995). During pure 180° domain switching in BaTiO3 crystal, the evolution of polarizations and strains can be calculated by Eq. (1) and Eq. (2),

respectively.

P3 = k33 · E3 + P3domain

(1)

S33 = E3 · d33

(2)

where P3 and S33 represent the total polarizations and longitudinal strains, respectively; P3domain denotes the polarization induced by domain switching; k33 is the relative dielectric constant, with a value of 68 here; E3 is the applied electric field; d33 is the longitudinal piezoelectric constant, and can 0 × P /P (d 0 means the longitudinal be expressed by d33 3 0 33 piezoelectric constant of BaTiO3 crystal in a single domain state, with a value of 115 μC/N according to the afore measured results; P0 means the spontaneous polarization of BaTiO3 crystal, which is 26 μC/cm2 ). The red lines in Fig. 3 show the calculated electric field–polarization and electric field–strain curves based on Hwang’s 180° DSC, their good fit with the experimental results demonstrates the validation of Hwang’s DSC in modeling 180° domain switching in BaTiO3 crystal. 3.2. 90° domain switching in BaTiO3 crystals 3.2.1. Mechanical depoling and electric repoling Fig. 4 shows the polarization and strain responses of the pre-poled BaTiO3 crystals during mechanical depoling and subsequent electric repoling. Induced by 90° domain switching, both polarization and strain show significant nonlinear behavior in these two processes. During mechanical depoling, nonlinearity starts at about 1.8 MPa, indicating the critical stress needed to drive domains switching from “c+ ” domain to “a” domains is at 1.8 MPa. During electric repoling, nonlinearity starts at about 70 V/mm, demonstrating the critical electric field needed to drive domains switching from “a” domains to “c” domains is about 70 V/mm. Along with the increase of polarizations and strains, the amplitude of stress and electric field needed to drive further domain switching increases. That is, BaTiO3 crystal shows significant hardening effect during 90° domain switching. For instance, during mechanical loading, the stresses needed to drive “c+ ”

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Fig. 4. The polarization and strain response of BaTiO3 crystal during (a) mechanical depoling, and (b) subsequent electric repoling.

domain to “a” domains increases from 1.8 to 5.8 MPa with the increasing applied stress. When the applied stresses or electric fields reach a critical value, 5.8 MPa for mechanical depoling and 1.84 kV/mm for electric repoling, both the polarizations and strains become nearly unchanged with increasing applied stress/field, indicating saturation of the 90° domain switching. Here it should be noted that significant increase of polarizations were observed between 6 and 13 MPa during mechanical depoling. A possible explanation is that some charges were trapped by domain walls and were difficult to be released to the electrodes. During mechanical loading, the maximum strain and polarization at 15 MPa are 1.01% and 24.5 μC/cm2 , respectively, among which about 0.99% is the switchable strain and the switchable polarization. This demonstrates that about 95% “c+ ” domains switched to “a” domains. During stress unloading, the polarization keeps nearly constant at 24.5 μC/cm2 . However, for the strains, a slightly nonlinearity was observed below 5 MPa due to back domain switching. During electric repoling, the maximum strain and polarization induced by electric field are 0.91% and 27.5 μC/cm2 , respectively. The remnant polarization upon removing the electric field is about 27 μC/cm2 , slightly larger than the remnant polarization of 24.5 μC/cm2 measured during depolarization, which can be attributed to the charge leakage effect. The remnant strain measured during electric repoling is smaller than that observed during mechanical depoling, which can be attributed to the effect of back domain switching after stress unloading. By calculating the area of the stress–strain curve with the horizontal axis in Fig. 4(a) and the electric field-polarization curve with the vertical axis in Fig. 4(b), the energy dissipation destiny of BaTiO3 crystal during mechanical depoling and electric repoling can be obtained, which is 36 kJ/m3 and 28 kJ/m3 , respectively. That is, less energy was dissipated during electric repoling than that during mechanical depoling. (It should be noted that during electric repoling, only about 26 kJ/m3 is associated with 90° domain switching. The other 2 kJ/m3 is induced by the leakage effect.) For a perfectly poled BaTiO3 crystal, it is in the singledomain state (Fig. 4(a)) and has the lowest energy. During mechanical depolarization, the “c+ ” domains gradually switch to the four equivalent types of “a” domains. Simulta-

neously, some elastic energy is stored in the 90° a–a domain walls. That is also why the polarizations and strains show hardening effect during mechanical depoling. During electric repoling, these elastic energies will be released to assist the “a” domains switch back to “c+” domains. Thus less energy is dissipated during electric repoling than that during mechanical depoling. 3.2.2. Incremental 90° domain switching criterion In this section, an incremental energy-based criterion for 90° domain switching is proposed based on our experimental results. Compared with Hwang’s DSC (1995), two new physical phenomena were included in this proposed DSC, i.e., (i) Different energy dissipation between stress induced switching and electric field induced switching; (ii) The hardening effect during domain switching. By taking these two effects into account, during mechanical depolarization and electric repoling, the domain switching criterion can be described by Eq. (3) and Eq. (4), respectively. d σ33 S0 ·  f1 − Wsto = U90 o ( f1 ) ·  f1

(3)

e E3 P0 ·  f1 + Wsto = U90 o ( f1 ) ·  f1

(4)

where σ 33 is the applied compressive stress; S0 denotes the spontaneous strain; f1 means the volume fraction of “a” d ( f ) and U e ( f ) are the energy barrier (per domains;U90 o 1 90o 1 unit volume) for 90° domain switching during mechanical depoling and electric repoling, respectively; Wsto is the stored strain energy (per unit volume) in 90° domain walls. When f1 = 0, there is no 90° domain wall at all and Wsto = 0. When f1 reaches to the maximum value, the domain structure turns to be close to that in Fig. 2(b), i.e., most “c+ ” domains switch to “a” domains and the elastic energy density stored in BaTiO3 crystal reaches the maximum value. By integrating Eqs. (3) and (4) and assuming that no domain switching occurs during compression unloading and electric field unloading, we get

 Wσ − Wsto =

0

 WE + Wsto =

f1

f1 0

d U90 o ( f 1 )d f 1

(5)

e U90 o ( f 1 )d f 1

(6)

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where Wσ and WE are the loop areas of the stress–strain curve and electric field–polarization curve, respectively. If we further assume that the energy dissipated during mechanf d ical depoling and electric repoling, i.e., 0 1 U90 o ( f 1 )d f 1 and  f1 e U90o ( f1 )d f1 are same with each other. Then, from Eqs. (5) 0 and (6), we get the maximum stored elastic energy density and the energy density dissipated by domain switching

Wsto = (Wσ − WE )/2 = 5 kJ/m

3

(7)

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3.2.3. Electric field–stress domain structure phase diagram and its evolution process under electromechanical loading In this section, the domain structure evolution process under electromechanical loading will be discussed. When mechanical stress dominates the domain switching process, the domain switching in BaTiO3 crystal can be described by d (σ33 S0 − E3 P0 ) ·  f1 − Wsto = U90 o ( f1 ) ·  f1

(17)

“c+ ”

In addition, if we further assume and increase linearly with the volume fraction of the switched domains. Then the volume fraction dependent energy barrier for 90° domain switching during compressive loading and during electric repoling can be calculated by

That is, to drive domains to “a” domains, the work done by the mechanical stress is not only dissipated by the d ( f )) and transformed domain switching energy barrier (U90 o 1 ° to elastic energy stored by 90 domain walls (Wsto ), but also has to overcome the negative work done by the bias electric field (E3 P0 ). Based on Eq. (17), the “c+ ” domain evolution process in BaTiO3 crystal under electromechanical loading can be plotted out, as shown in Fig. 6(a). If the tested BaTiO3 crystal is in pure “c+ ” domain state, the critical stress needed to drive “c+ ” domain to “a” domain increase with the increase of the pre-load electric field. If the initial domain structure of the tested BaTiO3 crystal is composed by “c+ ” domains and “a” domains, the critical compressive stress needed to drive “c+ ” domains to “a” domains increases with the increase of f1 . When the compressive stress is larger than (E3 P0 + Wsto + d (1 ))/S , all the “c+ ” domains can be driven to “a” doU90 o 0 mains and BaTiO3 crystal changes to “a” domain state. When electric field dominates the domain switching process, the domain switching in BaTiO3 crystal can be described by

d 3 U90 o ( f 1 ) = (1.8 + 4.04 f 1 )MPa × 1.04% − 5.26 kJ/m

e (E3 P0 − σ33 S0 ) ·  f1 + Wsto = U90 o ( f1 ) ·  f1



f1 0

d U90 o ( f 1 )d f 1 =



f1 0

e 3 U90 o ( f 1 )d f 1 = 31 kJ/m

(8)

If Wsto is assumed increases linearly with the volume fraction of the “a” domains, i.e.,f1 , then

Wsto ( f1 ) = 5.26 f1 kJ/m3 f1 ∈ [0, 0.95 )

(9) 90°

Obviously the energy barrier for initial domain switching can be determined from the coercive stress measured in Fig. 3(a), i.e., d 3 3 U90 o (0 ) = 1.8 MPa × 1.04% − 5.26 kJ/m = 13.46 kJ/m

(10) d U90 ◦

= (13.46 + 42 f1 )kJ/m

3

e U90 o

(11)

e 2 3 U90 o ( f 1 ) = (184−161 f 1 ) V/mm × 26μC/cm + 5.26 kJ/m

= (53.3 − 42 f1 ) kJ/m3

(12)

Here it should be noted that the domain-to-domain interaction increases during mechanical depoling, but decreases d ( f )increases but U e ( f ) during electric repoling, thus U90 o 1 90o 1 decreases with the increase of f1 . For BaTiO3 crystal, its constitutive behavior along the loading direction can be calculated by

P3 = d33 · σ33 + k33 · E3 + P3domain

(13)

domain S33 =s33 · σ33 +d33 · E3 +S33

(14)

where s33 is the compliant coefficient, which can be determined by calculating the target of the unloading stress–strain domain denotes the curve, with a value of 8.3 × 10−11 m2 /N; S33 domain can strains induced by domain switching. P3domain and S33 be calculated by

P3domain = P0 ∗ (1 − f1 ) + P0 ∗ f1

(15)

domain S33 = S0 ∗ (1 − f1 ) + S0 ∗ f1

(16)

Using Eqs. (3), (4), (11), (11) and Eqs. (13)–(16), the polarization and strain response of BaTiO3 crystal during mechanical depoling and repoling can be calculated, as shown in the red-line curves in Fig. 5. It can be seen that the predictions can fit well with the measured results.

“c+ ”

(18)

That is, to drive “a” domains back to domains, the work done by the electric field has to overcome the domain switching energy barrier and the negative work done by the preload compression. However, the stored strain energy will be released to help “a” domains switch back to “c+” domains. Based on Eq. (18), the “a” domains evolution process in BaTiO3 crystal can also be predicted, as shown in Fig. 6(b). If the tested BaTiO3 crystal is in one “a” domain state, the critical electric field needed to drive “a” domains to “c+ ” domains increase with the increase of the preload compression. If the initial domain structure of the tested BaTiO3 crystal composed by “c+ ” domains and “a” domains, the critical electric field needed to drive “c+ ” domains to “a” domains increase with the decrease of f1 . When the electric e (0 ))/P , all the “a” field is larger than (σ33 S0 − Wsto + U90 o 0 domains can be poled to “c+” domains. Based on Fig. 6(a) and (b), the domain evolution process in BaTiO3 crystal under arbitrary loading condition can be predicted if the domain structure and the loading path are known, for instance, during mechanical depolarization with preload positive bias electric field or unipolar electric field loading with constant preload compressive stress. Here it should be noted that similar electric field–stress domain structure phase diagram of BaTiO3 crystal has been plotted out by Burcsu et al. (2004). In their work, the domain structure phase diagram was established based on the energy state of domains with different polarization direction. Therefore their phase diagram is suitable to predict the existing possibility of domains in BaTiO3 crystal under static loading, and cannot exactly predict the hysteretic domain switching process during successive loading.

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Fig. 5. Comparison of the predicted and the measured nonlinear behavior of BaTiO3 crystal during mechanical depoling and electric repoling. (a) Stress– polarization curves; (b) Stress–strain curve; (c) Electric field–polarization curve; (d) Electric field–strain curves (the black lines and the red lines represent the experimental curves and the predicted curves, respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. The electric field–stress domain structure phase diagram of BaTiO3 crystal under electromechanical loading. (a) The evolution of “c+ ” domains and (b) the evolution of “a” domains. The black line, the brown line, and the dot lines between them represent the volume fraction of “a” domains. The red line and blue line represent the loading path of mechanical and electric field loading (solid line) and unloading (dot line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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253

Fig. 7. The predicted (left) and measured (right) stress–polarization curves (up) and stress–strain curves (bottom) of BaTiO3 crystals under mechanical loading with a series of bias electric field.

In the following, the domain switching process in BaTiO3 crystal under uniaxial electromechanical loading will be discussed by using Fig. 6(a) and (b). During mechanical depolarization with preload positive bias electric field, the stress dominates the domain evolution process first. The “c+ ” domain evolution path during mechanical loading can be predicted using the red line shown in Fig. 6(a). During mechanical unloading, the electric field dominates the domain switching process. The “a” domain evolution path can be predicted using the dot red line shown in Fig. 6(b). During unipolar electric field loading with a constant preload compressive stress (the domain structure of BaTiO3 crystal depended on the amplitude of the applied stress before loading and can be predicted using Fig. 6(a)), the electric fields dominate the domain switching process first. The “a” domain evolution path can be predicted by the blue line shown in Fig. 6(b). During electric field unloading, the stress dominates the domain evolution process, to drive all the “c+ ” domain back to “a” domains, the preload stress has to be larger than a critical value. Based on Fig. 6(a) and (b), it can be predicted that during mechanical loading/unloading with suitable bias electric fields, pseudoelasticity behavior can be realized in BaTiO3 crystal by reversible 90° domain switching from “c+ ” domains to “a” domains and then to “c+ ” domains; while during unipolar electric field loading with constant preload compressive stress, large actuation strain can be achieved in BaTiO3 crystal by reversible 90° domain switching from “a” domains to “c+ ” domains and then to “a” domains.

3.2.4. Validation of the incremental 90° domain switching criterion In this section, the hysteretic behavior of BaTiO3 crystal during mechanical loading/unloading with bias electric fields and that during cyclic unipolar electric field loading with preload compressive stresses was tested to verify the 90° DSC proposed above. During mechanical depoling, the stress amplitude and loading/unloading rate were 15 MPa and 1 MPa/s, respectively. Bias electric fields of 100, 200, 400 and 600 V/mm were used. During cyclic electric field loading, the maximum applied electric field was 800 V/mm with a loading frequency of 0.2 Hz. The preload compressive stresses were 4, 6, 8, and 10 MPa. Fig. 7(b) and (d) show the measured stress–polarization curves and stress–strain curves of BaTiO3 crystals with a series of bias electric field. For comparison, the depolarization curves without preload bias electric field are also plotted in Fig. 7. Fig. 7(a) and (b) show the predicted stress–polarization curves and stress–strain curves using Fig. 6. Experimental results show that the coercive stress increases steadily with the increase of bias electric fields, which is in accordance with the predicted results. When the applied bias electric field is 100 V/mm, both the predicted and the experimental results show that partial switching strain and polarization can be recovered upon removal of the stress. When the bias electric fields reach 200 V/mm, as predicted, all the domains switch back and BaTiO3 shows pseudoelastic behavior. When the bias electric field is at 400 V/mm, the pseudoelastic strains

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Fig. 8. The predicted (left) and measured (right) electric field–polarization loops (up) and electric field–strain curves of poled BaTiO3 crystals during electric loading with a series of pre-load compressive stresses.

decrease as only part “c+ ” domain can be driven to “a” domains during mechanical loading. When bias electric field is at 600 V/mm, predictions show BaTiO3 should reveal linear stress–polarization curve and stress–strain curve. However, recoverable strain about 0.2% and polarization of 5 C/cm2 were still observed in experiment, indicating that a small volume fraction of domains switched. This phenomenon was thought to be caused possibly by the stress concentration near the corner of the specimen. Fig. 8 shows the predicted and the measured electric field–polarization and electric field–strain curves of BaTiO3 crystal with preload compressive stresses. As predicted, the measured results show that the polarizations and actuation strains first increase and then decrease with the increase of the pre-load stresses. When the pre-load stresses are 6 MPa, the maximum measured strains are 0.93%, slightly smaller than the calculated result. However, differ from the predicted results, the achievable actuation strains drops to 0.82% and 0.68% when the pre-load stresses reach 8 MPa and 10 MPa, respectively. This suggests only part of domains can be reoriented during electric repoling. However, predictions show that the decrease of polarization and strain should start at about of 15 MPa. In addition, the measured coercive electric field with preload stresses at 6, 8, and 10 MPa are significantly larger than the predicted results.

3.2.5. Discussions As shown above, the DSC proposed above can well predict the depolarization behavior of BaTiO3 crystal with bias electric field, but cannot fully reproduce the behavior under electric loading with a constant stresses. This difference is thought to be caused by the time dependent domain switching process in BaTiO3 crystals. The single-domain state in Fig. 2(a) remains stable when a positive dc is applied. With the electric field loading, no creep phenomena will occur. However, the multiple-domain state in Fig. 2(b) is metastable. Creep will occur if not all the domains have switched into the plane perpendicular to the loading direction. This can be seen clearly from the depolarization curves when the bias electric field applied is 400 V/mm (Fig. 7(d)). In comparison, the domain evolution during electric repoling with pre-load compression is realized from “a” domains to “c+ ” domains and then to “a” domains. When a moderate pre-compression (say 4 MPa) is applied, the specimen is at a metastable state, deformation can also happen by the creep phenomena. When a large electric field is applied, some domains can be reoriented by the electric field. However, it cannot be repoled to its original state when the pre-load stresses reach to a critical value, for instance at 8 MPa. Then the domain structure evolution in BaTiO3 crystal is between two different metastable states during electric repoling with

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pre-stresses, which makes it less repeatable if the applied pre-stresses are kept for a period. This is because the aging process during compression holding will gradually increase the electric coercive field (Li et al., 2013). Therefore, the proposed DSC cannot predict the actuation curves well. In addition, the above investigations also reveal two different phenomena between 90° and 180° domain switching. One is that the energy barrier for 90° and 180° domain switching is different with each other; the other is that only slight hardening effect was observed during 180° domain switching but 90° domain switching revealed significant hardening process. These phenomena can be attributed to the differences between 90° and 180° domain switching processes. Our recent in-situ observations showed that 180° domain switching is accomplished by successive 180° domain nucleation and forward domain wall motion during antiparallel electric field loading. As the reorientation of the spontaneous strain was not involved, the domain-todomain interaction is relatively small. Thus only slight hardening effect was observed during 180° domain switching and can usually be neglected. While during 90° domain switching, not only the polarizations but also the spontaneous strain change their direction, which will induced significant domain-to-domain interaction. Therefore, obvious hardening effect was observed during 90° domain switching. 4. Conclusions In summary, 180° and 90° domain switching were investigated in perfectly prepoled BaTiO3 crystals to build the domain switching criterion for ferroelectrics. Pure 180° domain switching was realized during bipolar electric field loading and pure 90° domain switching was realized during mechanical depoling and subsequent electric repoling. Results show that critical energy barrier for 180° and 90° domain switching in BaTiO3 is different. In addition, during 180° domain switching, only slight hardening effect was observed but significant hardening effect appeared during 90° domain switching. Moreover, for 90° domain switching, more energy was dissipated during mechanical depoling than that during electric repoling. By taking these factors into account, an incremental DSC was proposed for pure 90° domain switching. Based on this criterion, the electric field–stress domain structure phase diagram of BaTiO3 crystals was built, and the domain structure evolution process in BaTiO3 crystal under two specific electromechanical loading conditions was discussed. It is predicted that during mechanical depoling with bias electric field, BaTiO3 crystal can show pseudoelastic behavior; while during unipolar electric field loading with preload compress, it can output large actuation strain. These predictions were further verified by subsequent testing, demonstrating the validity of the proposed DSC. Acknowledgment Financial support from the National Natural Science Foundation of China under grant no. 11422216 and 11402177 is acknowledged.

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