The coordinate of particle in R phase BaTiO3 ferroelectric single crystal cell [34]
1. Introduction
Compared to polycrystalline ferroelectric ceramics such as Pb(Zr1/2Ti1/2)-O3 (PZT), domain engineered relaxor ferroelectric single crystals Pb(Zn1/3Nb2/3)O3-
Note that PZN-PT and PMN-PT single crystals usually experience electric and/or mechanical loading during their in-service life. It has been shown that externally applied loading has significant effect on the properties of these crystals [1], [5]-[28]. A number of studies have focused on the loading induced behavior of <001> and <110> oriented anisotropic PZN-
Rhombohedral phase of PZN-
A number of studies have focused on experimental, but the constitutive model of ferroelectric single is absent. Huber [29] and Bhattacharya[30], [31] et.al established a model based on the micromechanical method, but the simulation is not well compare to experimental result.
At present, a mature model to explain the stress-strain behavior of ferroelectric single is absent all along. In this study, electric field induced “butterfly” curves and polarization loops for a set of compressive bias stress of {001}, {011} and {111} poled PMN-0.32PT single crystals will be explored by systematical experiment study. The effects of the compressive bias stress on the material properties along these three crystallographic directions of PMN-0.32PT single crystals will be quantified. The underlying mechanisms for the observed feature will be explained in terms of phase transformation or domain switching,depending on the crystallographic direction.The stress-strain curves along <001> crystallographic direction of ferroelectric single crystals BaTiO3 will be calculated in the first principle method to validate polarization rotation model. Finally, Based on the experimental phase transformation mechanism of ferroelectric single crystal, a constitutive model of ferroelectric single crystal is proposed based on micromechanical method. This constitutive model is facility and high computational efficiency.
2. Experimental methodology
At the test room temperature, PMN-0.32PT single crystals used in this study are of morphotropic composition, and in the rhombohedral phase, very close to MPB. The pseudo-cubic {001}, {011} and {111} directions of these crystals are determined by x-ray diffraction (XRD). Pellet-like specimens of dimensions
Since the focus of this study is to explore the effect of bias stress on the electromechanical properties of PMN-0.32PT single crystals along different crystallographic directions, experimental setup is adapted from Ref. [31] (Fig.2) to allow simultaneously imposing uniaxial stress and electric field to the specimen along the thickness direction. Mechanical load is applied by a servo-hydraulic materials test system (MTS) and electric field is applied to the specimen using a high voltage power amplifier. Once the specimen is placed in the fixture, a compressive bias stress with magnitude of at least 0.4MPa is maintained throughout the test to ensure electrical contact. Stress controlled loading instead of displacement controlled loading is adopted during the test, so that the specimen is not clamped but is free to move longitudinally when electric field
The first set of tests is performed for electric field loading of triangular wave form of magnitude 0.5kV/mm and frequency 0.02Hz, free of stress loading. The low frequency is chosen to mimic quasistatic electric loading, which is of particular interest in this study [32]. Unless stated otherwise, this loading frequency for the electric field is used throughout the following test. The second set of tests consists of mechanical loading upon short circuited samples. The samples are compressed to –40MPa and unloaded to –0.4MPa at loading and unloading rate of 5MPa/min, followed by an electric field which is sufficiently large to remove the residual stress and strain to re-polarize the samples. In the third set of tests, triangular wave form electric field is applied to the samples which are simultaneously subject to co-axial constant compressive stress preload. The magnitude of the preload is varied from test to test and is in the range between 0 and -40MPa. Note that there is a time-dependent effect of the depolarization and strain responses under constant compressive stress. To minimize this effect, each electric field loading starts after a holding time of 150 seconds for a new stress preloading. It is found that three cycles of electric field loading and unloading are sufficient to produce stabilized response for each constant prestress, and the results for the last cycle are reported in the following.
3. Experimental results and discussion
3.1. Crystallographic dependence of electric behavior and piezoelectric properties
The measured electric field induced polarization hysteresis loops and butterfly curves for {001}, {011} and {111} oriented poled PMN-0.32PT single crystals without stress loading are shown in Figs. 3a and 3b. The remnant polarizations
It is also seen from Figs. 3a and 3b that the coercive field is lowest for the {111} oriented crystals, and becomes successively higher for {001} and {110} orientations. This is same to Ref.25 except for {110} orientation. This trend of coercive field is due to two reasons: One is due to reorientation driving force being proportional to the component of electric field aligned with the rhombohedral direction; The other one is due to the domain switching process. In {111}-oriented PMN-0.32PT single crystals, there are two types of domains (shown as type 1 and type 2 in Fig.1c). When the electric field is decreased from 0.5kV/mm to –0.05kV/mm, the strain first decreases linearly (see Fig.3b). When the electric field is decreased further, the type 1 domain switches to type 2 domains, leading to abrupt displacement change. When the electric field exceeds the coercive field –0.216kV/mm, the type 2 domains switch back to type 1 domain, recovering the deformation. In type 2 domain state, three equivalent polar vectors with an angle of 71º from the {111} direction can coexist and are separated by domain walls across which the normal components of electric displacement and displacement jump are zero. Ideally, this type of domain walls has no associated local stress or electric field. So the existing of type 2 domain state and the largest component of electric field along polarization direction induces the lowest coercive field in {111} orientation poled crystal. In the {011} orientation crystals, there are also two types of domains (i.e., type 1 and type 2 domain in Fig. 1b), the domain switching process is similar to {111}-poled crystals. In the type 2 domain state, however, the four possible polar vectors are perpendicular to the applied electric field (Fig. 1b). It is thus difficult to switch type 2 domain to type 1 domain only by applying electric field. Both the type 2 domain state and the smallest component of
Some researchers attribute the high piezoelectric coefficients along {001} and {011} oriented ferroelectric single crystals to the engineered domain state [4, 11]. It has also reported that, however, the piezoelectric coefficient along the {001} direction of single crystals with mono-domain structure is comparable to that of crystals with multi-domain structure [17], implying the origins of the high piezoelectric constants of PMN-0.32PT single crystals may not be due to the engineered domain state. Instead, it could be due to the effect of crystal lattice properties. To further explore this issue, we follow Ref. [17] to calculate the piezoelectric coefficients
With
3.2. Crystallographic dependence of stress induced strain and polarization responses
Figure 4 shows the measured
The stress cycles of {001} poled crystals can be explained by the polarization vector rotation mechanism sketched in Fig. 5 as follows. In general two most possible mechanisms responsible for the observed features of {001} poled crystals in Figs.4 are domain switching and polarization rotation associated phase transformation. Recall that the PMN-0.32PT single crystals considered here are in the R phase close to MPB (although in reality there could also be M or T phase in these crystals near MPB [8], [9] the R phase is nevertheless the dominant phase). It is noted that that, upon application of a field along the <001> poling axis of R phase domain engineered PMN-0.32PT single crystals, only four of the eight polarization orientations are possible, i.e., <111>, <
Fig. 5 illustrates the polarization rotation mechanism of PMN-0.32PT single crystals under stress loading and unloading cycle. The polarization vector states shown in Figs. 5(a)-5(e) correspond to the stress levels marked on the curves in Figs. 4 for the -40MPa stress cycle. In the initial poled state, PMN-0.32PT single crystals possess four equivalent <111> polarizations, with only one shown in Fig. 5a for the sake of clarity. Upon loading, polarization vector starts to rotate from R to O and T phases through intermediate phases MA and MC, when the compressive stress exceeds about 15MPa in magnitude (Fig. 5(b)). This gives a mixture of R and T phases and remarkable augment in the polarization and strain change around
3.3. Crystallographic dependence of electric field induced behavior at constant bias compressive stress
The
The influence of the preloaded compressive stress on the aggregate strain
As suggested by Fu et al. [7], for {001} oriented crystal origins of large aggregate strain and high piezoelectric coefficient at a moderate compressive bias stress (i.e., around -20MPa for the single crystals considered here) may be attributed to the stress induced intermediate states between rhombohedral and tetragonal phases. Under a bias stress of about -20MPa, PMN-0.32PT single crystals, after a phase transformation, are in a state of monoclinic phase which has a larger
As is noted, the polarization rotation introduced by compression gives rise to
Under zero stress, the initial state of {011}-oriented crystals is of multi-domain with two equivalent polarization directions (Fig.1b). For {111}-oriented crystals, the initial state is of mono-domain with polarization direction in the {111} direction (Fig.1c). When the applied electric field decreases from 0.5kV/mm to –
Note that compressive stress can induce domain switching in {011} and {111}-orientated crystals. When compressive stress is superimposed on the samples, the shapes of
4. The first principle calculation of stress-strain
4.1. Calculation methodology
PMN-PT and BaTiO3 have the similar ABO3 structure, so they have the similar ferroelectric properties. There is only Ti4+ particle in B site of BaTiO3, however, there is not only Ti4+ but also minim Mn4+ and Ni2+ particle in B site of PMN-PT. So the single cell of BaTiO3 is convenient in calculation and it keeps the similar ferroelectric to PMN-PT.
In this paper, we calculate a single cell of BaTiO3, the single cell should be the smallest periodic reduplicate cell, it includes one Ti4+ particle, three O2- particles and one Ba2+ particle (Fig.8). It is suggested that the initialized state of BaTiO3 ferroelectric single crystal is R phase after {001} oriented polarization. Refer to literature [34], the crystal lattice constant of R phase BaTiO3 ferroelectric single crystal is 4.001
4.2. Discussion
In order to validate that the method a mentioned in this paper is correct, we calculate the elasticity constant
Stress-strain curve of {001} orientated R phase BaTiO3 calculated in first principle method is shown in Fig.10(a). Fig.10(a) shows that the result has simlar nonlinear behavior to the experimental result of PMN-0.32PT during the loading that sketched in Fig.4(a), namely, there is obvious “a,b,c” steps during loading. We know that the nonlinear behavior of PMN-0.32PT shown in Fig.4(a)should be polarization rotation (R→M→O and R→M→T). The R→M→O is corresponding to the processing of polarization vector
5. The viscoplastic model of ferroelection single
5.1. The viscoplastic model
From the experimental analysis, it is known that the <001>-oriented PMN-0.32PT single crystal undergo the R→M→T and R→M→O phase transformation under the compression. In order to establish a compact model and keep the essence of experiment, R→O phase transformation of PMN-0.32PT is considered in this study. Polarization rotation cause bulk deformation and shear deformation associated with the slip planes. It is assumed that the corresponding deformation associated with the slip planes is shear dominated, a feature similar to that of the multi-slip system of the crystal plasticity. This similarity renders possible to use crystal plasticity models to describe the transformation deformation of PMN-0.32PT single crystal.
The poled ferroelectric single is four domain state, the polarization vector is along <111>, <
In general, a criterion (i.e., the phase transformation criterion) exists for the phase transformation of ferroelectric single crystal, and the material is assumed to undergo phase transformation when at least one of the 8 variants satisfies the phase transformation criterion. This is detailed as follows. Upon loading, the condition to produce R→O polarization rotation on a specified habit plane is that the driving force G of that plane reaches the critical value G0O. The driving force is composed of the chemical driving force Gchem and mechanical driving force Gmech[36]
A similar condition holds for reverse transformation from O→R polarization rotation with a critical value G0A
In Eqs. (2) and (3), the mechanical driving force can be expressed by
Where
where the Schmid factor is defined as follow
with
The chemical driving force in Eqs. (2) and (3) is assumed to be a linear function of the temperature
where
When applying the rate independent crystal theory based model, Eqs. (2) and (3), to simulate the behavior of ferroelectric single crystals, one of the most computationally consuming tasks is to determine the set of instantaneously active transformation systems among the 8 possible variants at crystal level. This determination is usually achieved by an iterative procedure and must be carried out at each loading step, requiring extensive computation. Note that in the crystal theory of plasticity, a similar problem exists whilst in a rate dependent viscoplastic version of crystal theory of plasticity, determination of the set of active transformation systems is not necessary. As a result, computation effort can be reduced significantly. Following this idea, a viscoplastic version of Eqs. (21) and (32) are proposed and employed in this study. In the viscoplastic crystal model for PMN-0.32PT ferroelectric single crystals all transformation systems are assumed to be instantaneously active of varying extent, which is governed by a rate dependent viscoplastic law. In this paper, the phase transformation of variant “
where
The phase fraction of variant “
Summation over all possible variants provides the phase fraction for the whole ferroelectric single crystal, that is
The incremental form of Eq. (12) can be written as
The transformation strain tensor
where
Elastic strain and electrical field induced strain during loading can expressed as
Increment of the total strain
5.2. Numerical results
To validate the model, the corresponding calculation result is compared to experimental stress-strain curve. The material parameters used in this study are as follow [37, 38]. Material parameters of R phase are
|
|
|
|
Ba2+
O1 2- O2 2- O3 2- Ti4+ |
0 0.5133 0.5133 0.0192 0.489 |
0 0.5133 0.0192 0.5133 0.489 |
0 0.0192 0.5133 0.5133 0.489 |
|
|
|
|
Ba2+
O1 2- O2 2- O3 2- Ti4+ |
0 0.514 0.514 0.016 0.4846 |
0 0.514 0.016 0.521 0.4846 |
0 0.01 0.521 0.499 0.499 |
|
|
|
|
Ba2+
O1 2- O2 2- O3 2- Ti4+ |
0 0.5144 0.5 0.0162 0.4857 |
0 0.5144 0.0162 0.523 0.4857 |
0 0 0.523 0.5 0.5 |
6. Conclusion and discussion
In this paper, Stress induced strain and polarization, and electric field induced “butterfly” curves and polarization loops for a set of compressive bias stress for {001}, {011} and {111} poled PMN-0.32PT single crystals are experimentally explored. Obtained results indicate that high piezoelectric responses of PMN-0.32PT single crystals are controlled by the anisotropy of the crystals and the multi-domain structure (i.e., engineered domain structure) has a relatively minor effect. Analysis shows that in all three directions the electric field induced aggregate strain
Acknowledgments
The authors are grateful for the financial supported by the Natural Science Foundation of China (No. 10425210,10802081) and the Ministry of Education of China.
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