Crystal plasticity behavior of single-crystal pure magnesium under plane-strain compression

Bai-Li Xi Gang Fang

Department of Mechanical Engineering,Tsinghua University

State Key Laboratory of Tribology

收稿日期：10 September 2016

基金：supported by the National Natural Science Foundation of China(No.51375256);

Crystal plasticity behavior of single-crystal pure magnesium under plane-strain compression

Bai-Li Xi Gang Fang

Department of Mechanical Engineering,Tsinghua University

State Key Laboratory of Tribology

Abstract：

A phenomenological crystal plasticity constitutive model for magnesium single crystal was presented.Four deformation mechanisms(including basal〈a〉,prismatic〈a〉,pyramidal〈c+a〉 slip and tension twin)and their interactions were considered.Twin-induced lattice reorientation was also incorporated in the model.The proposed model was then applied to the simulation of plane-strain compression deformation for different orientations.Related material parameters were calibrated at first according to the classical channel-die tests.The predicted macro-and microscopic responses,along with the experimental results,show strong orientation-dependent properties.It is also found in the simulation that basal slip in the twinned region is active even before the saturation of twin activity in a twin-favored case.Furthermore,the effect of an initial deviation angle on the mechanical responses was evaluated,which is proved to be also orientation-dependent.Basal slip is found to be easily activated due to a slight deviation,while a slight deviation in the twin-favored case could result in a significant difference in the mechanical behavior after the reorientation.The effort on the study of magnesium single crystal in the present work contributes to further polycrystalline analysis.

Keyword：

Magnesium; Crystal plasticity; Slip and twin; Orientation deviation;

Author： Gang Fang e-mail:fangg@tsinghua.edu.cn;

Received： 10 September 2016

1 Introduction

Magnesium alloy continues to draw attention as a lightweight material in automotive industry due to its relatively high specific strength [ 1] and in biomedical devices due to its attractive biocompatibility [ 2] .Despite these promising aspects,its poor formability as well as strong deformation anisotropy at room temperature (RT) once hindered its wide application.Such deficiencies can be overcome by alloying with proper elements,such as rare-earth (RE)elements,which influences the intrinsic deformation mechanisms [ 3] .Thus,a sound understanding of deformation mechanisms of magnesium is required as a prerequisite.

In magnesium,the deformation perpendicular to the caxis direction is realized by the slip of<a>type dislocation,while the deformation parallel to the c-axis direction is provided by the slip of<c+a>type dislocation and the deformation twin.The critical resolved shear stresses(CRSSs) among various slip modes are significantly different.There is also a pronounced polarity exhibited by twin.These two orientation-dependent factors result in the aforementioned deformation anisotropy in the macroscopic response for monocrystalline and textured polycrystalline magnesium and its alloys.

Efforts were made to identify the micro-scale deformation modes in magnesium and to evaluate the CRSSs for them.In general,such tasks were accomplished by experimental studies on single crystal.In the early stage,Kelley and Hosford [ 4] (referred to as K-H in the paper) as well as Wonsiewicz [ 5] performed comprehensive investigations on pure magnesium single crystal at RT through planestrain compression tests.They also conducted metallographic examination,providing fundamental insights in orientation-dependent macroscopic mechanical behavior and microscopic deformation mechanisms.In their experiments,the prismatic and pyramidal slip traces were not observed due to the limitations of experimental techniques.Recently,Sulkowski [ 6] studied the activation of basal and non-basal slips in magnesium single crystals through tension tests at RT,with the ratio of activation volumes of the soft to the hard slip systems.Drozdenko et al. [ 7] conducted uniaxial and plane-strain compression tests of single-crystal pure magnesium for limited types of orientations at RT,applying acoustic emission (AE) technique in studies of dislocation and twin processes during deformation.Pronounced AE response was observed in the twin-favored orientations,while lower AE signals were related with the basal and pyramidal slips.Spherical nanoindentation,as another experimental research,was also applied in the investigation of the plasticity of magnesium single crystal [ 8, 9, 10] .Indentation morphology in the case of (0001) indentation was reported to result from basal and (c+a) pyramidal slips,while the tension twins along with basal slip were active in other cases.

Crystal plasticity (CP) modeling has been proved to be an appropriate computational method for the deformations of crystalline materials at the continuum scale.It connects macroscopic mechanical behavior with microscopic slip/twin activities.Once constitutive parameters for each slip/twin mode are identified by experimental tests,the model can be used for predicting several mechanical properties,such as stress-strain response,strain hardening and texture evolution in single as well as poly crystals.

CP theory originated from Taylor's work [ 11] .Hill and Rice [ 12] proposed a constitutive model based on the kinematics description of CP deformation.Peirce et al. [ 13] raised a rigorous rate-dependent framework,investigating the effects of material's rate sensitivity and hardening properties.Kalidindi [ 14] presented a fully implicit time integration scheme to predict the large and non-uniform deformation in single crystal as well as polycrystalline fcc metals.This model was further improved by incorporating deformation twin in the framework (based on pseudo slip assumption [ 15] ),which was applied in the prediction of texture evolution in low stacking fault energy fcc metals and in hcp metals [ 16] .Thereafter,more extensive researches were conducted in aspects of macroscopic shape changes and texture evolution [ 17, 18, 19] ,yield surface prediction [ 20, 21] ,and damage and fracture related [ 22, 23, 24] .

A variety of CP-based investigations were performed to describe the complex mechanical behavior for magnesium and its alloys [ 25, 26, 27, 28] .However,relatively limited efforts were made to numerically capture the orientation-dependent mechanical characteristics of pure magnesium single crystals and corroborate macroscopic behaviors as well as slip/twin activities with experiments.Graff et al. [ 29] presented a CP model for magnesium single crystal,considering four deformation mechanisms:basal<a>slip,prismatic<a>slip,pyramidal<c+a>slip and tension twin.Stress-strain responses for different crystal orientations tested in the K-H experiments [ 4] were predicted,and relative slip/twin activity in each case was analyzed quantitatively.Nevertheless,the twinning-induced lattice reorientation was absent in their numerical model.Zhang and Joshi [ 30] proposed an improved CP model based on the framework of Kalidindi [ 16] .One of the highlights in their work was the description for the evolutions of slip,twin and their interactions.In addition,pyramidal<a>slip and contraction twin were also included apart from the four basic mechanisms mentioned above.Trends and features of the stress-strain curves were captured by the simulated results,and the relative slip/twin activities were discussed as well.Gan et al. [ 31] also investigated the plane-strain compression of magnesium single crystal based on an elastic-viscoplastic CP model.Their model was similar to that of Zhang and Joshi [ 30] ,while a more sophisticated description of hardening moduli for dislocation slip was employed.Despite acceptable predicted results of the macro-and microscopic responses achieved in these works,the effect of initial orientation deviation on these responses was rarely investigated.In consideration of the strong orientation dependence in hcp magnesium,this effect deserves an in-depth evaluation.A thorough understanding of the single-crystal behavior contributes to the subsequent poly crystalline analysis.

In the present study,a phenomenological CP model was used to simulate the plane-strain compression of singlecrystal pure magnesium for different crystal orientations.First,CP-related parameters were carefully calibrated according to the K-H experiments.Then,the predicted results were compared with the experimental results,and the activity of various deformation mechanisms for each crystal orientation was discussed together with corresponding macroscopic mechanical behavior.Finally,the effect of initial deviation angle on the macro-and microscopic responses was evaluated and discussed.

2 Modeling framework

2.1 Deformation mechanisms

Four deformation mechanisms of magnesium were considered in the present work,including:basal<a>slip,prismatic<a>slip,pyramidal<c+a>slip and tension twin.Figure 1 shows schematic illustrations of these deformation mechanisms of magnesium,while Table 1 lists details of such slip and twin systems.

Fig.1 Schematic illustrations of main deformation mechanisms of magnesium,with one of slip/twin planes (shaded face) and directions(dark red arrow) for each mode depicted:a basal<a>slip,b prismatic<a>slip,c pyramidal<c+a>slip and d tension twin

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Table 1 Details of deformation mechanisms in magnesium consid-ered in present work

Although some researches [ 30, 31] took pyramidal<a>slip and contraction twin into account,the present calculation did not consider them.It was based on the following two considerations:(1) pyramidal<a>slip can be treated as a combination of basal and prismatic<a>cross-slip [ 32] ;(2) contraction twin was reported to have more influence on fracture than ductility [ 33] .

2.2 Constitutive formulations

Considering a crystalline material undergoes elastic stretching,lattice rotation and plastic deformation,the total deformation gradient (F) can be multiplicatively decomposed as:

where F^*is the deformation gradient due to both elastic stretching and lattice rotation and F^p is the plastic deformation gradient generated by dislocation slip and deformation twin.Correspondingly,the total velocity gradient L can be decomposed into elastic part (L^*) and plastic part (L^p):

where and F^-1 refer to the material derivative and inverse of the deformation gradient (F),respectively.Based on the assumption of pseudo slip for deformation twin [ 15] ,the plastic velocity gradient is expressed as [ 16] :

where the subscripts s,tw and s-tw denote slip in the parent region (i.e.,untwined region),twin in the parent region and slip in the twinned region,respectively;the superscriptsαandβrefer to the certain slip/twin systems,respectively;N represents the total number of slip/twin systems;m and n denote the slip/twin direction vector and the corresponding slip/twin plane normal vector,respectively; is the plastic shear rate;f refers to volume fraction of twin; denotes the tensor product.Here,m_s-tw and n_s-tw are calculated by:

where R_tw denotes the transformation matrix of orientation between twin and matrix;I is the unit tensor.

Elastic constitutive relation of the crystalline material is expressed by a hyper elastic formulation [ 16] :

where the superscripts tw and p denote the twinned and parent regions,respectively;C is the fourth-order elasticity tensor;T is the second Piola-Kirchhoff stress;E^*is the Green strain (conjugate to the second Piola-Kirchhoff stress).The elastic tensors in twinned and parent regions can be related to each other by transformation matrixes R:

Then,the stress in the whole crystal is calculated by a volume average of the stresses in parent and twinned regions:

A rate-dependent plastic constitutive rule is employed to relate the plastic shear strain rate ( ) with the resolved shear stress (RSS)(τ) [ 13] :

where mn and n denote the rate sensitivity parameter for slip and twin,respectively; and are the reference strain rate for slip and twin;g represents the resistance of slip/twin systems (initially equivalent to CRSS).It may be noted that only the positive RSS is allowed to activate the tension twin due to the polarity.In addition,the activity of twin is forced to terminate when the total volume fraction of twin reached an upper bound (f_cr),followed by the lattice reorientation.The evolution of the volume fraction of twin( ) can be related to the shear rate ( ) via a constant twin shear (y^tw):

whereγ^tw=0.129 for tension twin in magnesium.Furthermore,the RSS in the flow rule is calculated by the classical Schmid law:

whereσis the Cauchy stress.

To complete the CP constitutive formulation,the evolution of the slip/twin resistance ( ) is described as:

where q^αβdenotes the self/latent hardening coefficient matrix,including not only the slip-slip and twin-twin interaction but also the interaction between slip and twin;h^βdenotes the self-hardening rate of the slip/twin systemβand was defined as:

with

where is the initial hardening rate; is the initial slip/twin resistance (i.e.,CRSS); is the saturation slip/twin resistance; is the total plastic shear strain accumulated due to slip or twin at time t.

2.3 K-H experiments and finite element model

Since various deformation modes of pure magnesium can only be well identified using oriented single crystals and plane-strain compression,the K-H experimental data were adopted here as the basis of our simulation.A brief introduction of their experimental procedure is presented as follows.The single crystals were firstly grown using oriented seed crystals and then fabricated to cuboid specimens for the channel-die tests.A Teflon film was used in the tests,serving as a lubricant between the specimens and test fixture as well as protecting the specimen surfaces.After the tests,active deformation modes were identified by a three-surface trace analysis.More details can be found in the original paper [ 4] .

The identical schedule was simulated in the present work,with five different orientations being considered,as shown in Table 2.The above-mentioned CP constitutive model was implemented in ABAQUS/Standard as a userdefined material subroutine (UMAT),in which a time integration scheme proposed by Kalidindi [ 14] was employed.A symmetric cube meshed with one C3D8R element was used to model the specimen,and the channel die along with the indenter were processed for several rigid walls meshed with R3D4 elements,as shown in Fig.2.The specimen was compressed in the z direction and constrained in the x direction,while the y direction was free.The friction coefficient between the specimen and the rigid walls was set as 0.05 for all the tests.The load-displacement response obtained from the calculation was transferred to stress-strain response,following the definitions from Graff et al. [ 29] .

2.4 Constitutive parameters calibration

The constitutive parameters were calibrated by carefully comparing the macro-and microscopic responses with the K-H experiments.Since the elastic deformation in the entire process was so small that they may be negligible,the isotropic elasticity was assumed with Young's modulus (E)of 45 GPa and Poisson ratio (v) of 0.29.It was a strategic work to determine the parameters for various slip/twin systems,namely the hardening rate,CRSS and saturation resistance.Basal<a>slip,which was the only activated deformation mode in Case V,was the easiest one to calibrate.CaseⅠfavored pyramidal<c+a>slip,thus corresponding parameters were calibrated in such case.The same procedure worked for prismatic<a>slip corresponding with CaseⅡ.For CasesⅢandⅣ,tension twin dominated the deformation process until the lattice reorientation.Therefore,parameters for tension twin were calibrated with CaseⅢand verified by CaseⅣ.Since other deformation modes were activated in the reoriented material,CasesⅢandⅣwere also selected as an authentication for slip-related parameters.Furthermore,it should be noted that the effect of tension twin on slip hardening due to transmutation mechanism [ 34] or texture hardening mechanism was considered,while a slip-independent latent hardening for twinning was suggested [ 35] and applied in the present framework (through q^αβ=0 forα∈[1,N_tw]andβ∈[1,N_s]).The constitutive parameters calibrated by K-H experiments along with corresponding constitutive equations are listed in Table 3.

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Table 2 Single-crystal orientations used in plane-strain compression simulation

Fig.2 Finite element model of plain-strain compression test by a channel die

3 Results and discussion

3.1 Stress-strain responses and deformation mechanism analysis

The comparison of stress-strain responses between the K-H experiments and the simulation with the present model is shown in Fig.3.The simulated macroscopic responses are in good agreement with the experimental results.CP-related simulations also allow for an analysis of microscopic mechanisms.Figure 4 shows the evolution of the relative activities of various slip/twin systems with strain for each case,while details are discussed below.The relative activity of each deformation mode was defined as the ratio of the respective shear strain increment to the total shear strain increment.

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Table 3 Slip/twin constitutive equations and corresponding cali-brated parameters

Fig.3 Comparison of stress-strain responses in different cases from plane-strain compression simulations (Sim.,lines) with K-H exper-iments [4] (Exp.,symbols)

Pyramidal<c+a>slip is the dominant deformation mode for CaseⅠ,as shown in Fig.4a,and such a hard mechanism results in high yield stress and strong hardening in the stress-strain curves as shown in Fig.3.Since the Schmid factor of either basal or prismatic slip is zero in this case,neither of them is activated.Tension twin is also suppressed due to the compressive strain parallel to the caxis direction.Notably,Kelley and Hosford [ 4] observed basal slip traces during the initial 0.04 strain under this case and ascribed that to a slight misalignment of the lattice orientation.Such a misalignment is not introduced in the present calculation,and therefore,no basal slip is activated.However,the effect of misalignment (i.e.,initial deviation)on macro-and microscopic response will be discussed in the subsequent section.Although contraction twin was also observed in the K-H experiments at stresses above～280 MPa [ 4] ,it is not considered in the present calculation for its limited contribution to the ductility [ 33] .

Fig.4 Relative activities of slip and twin modes in plane-strain compression simulation:a CaseⅠ,b CaseⅡ,c CaseⅢ,d CaseⅣand e CaseⅤ

As shown in Fig.4b,prismatic<a>slip dominates the whole deformation.However,no prismatic slip traces were reported in the K-H experiments [ 4] .Wonsiewicz [ 5] made an identical observation and attributed that to limitations of experimental techniques rather than the absence of prismatic slip.Since prismatic slip is relatively softer than pyramidal slip,lower yield stress and modest hardening for CaseⅡ(compared to CaseⅠ) are observed in the stressstrain curves,as shown in Fig.3.Besides,tension twin is active during the initial stage of deformation and then decreases with the strain increasing,leaving the dominant role to the prismatic slip as shown in Fig.4b.This phenomenon is also in agreement with the K-H experiments [ 4] and other simulations [ 29, 30] .Kelley and Hosford [ 4] believed that the activation of tension twin was resulted by the imperfect constraint in the channel-die tests.However,this experimental imperfection is excluded in the idealized simulation.The twin occurrence could be explained in perspective of the RSS.Though tension twin is favored by the RSS in the loading direction and opposed by the RSS in the constraint direction,the net RSS proves to be a decisive one,resulting in the twin occurrence [ 30] .

For CasesⅢandⅣ,tension twin is the dominant deformation mode in the beginning.Subsequently,other deformation modes are activated,due to the twinning-induced reorientation,to accommodate the further deformation,as shown in Fig.4c,d.Relatively low yield stresses and almost no hardening at first are observed for both cases in the stress-strain curves,followed by a rapid increase in stress level,as shown in Fig.3.Despite the similar orientations for these two cases,their macro-and microscopic responses are quantitatively different in following aspects.(1) The higher level stress for the initial strain in CaseⅣ

can be explained by the lower Schmid factor for the twin.The Schmid factor of the four equally favored twin systems in CaseⅣis 0.347,while CaseⅢfavors two of the possible twin systems with a Schmid factor 0.499.(2)CasesⅢandⅣpossess various lattice reorientation configurations.Based on the Cartesian coordinate system shown in Fig.2,the initial c-axis directions in both cases are (0,1,0).After the reorientation,the c-axis direction changes to (0,0.064,-0.998) in CaseⅢwhile (-0.483,0.066,0.873) in CaseⅣ.The c-axis in CaseⅢforms a final angle of 3.7°with the loading direction,while the included angle is 29.2°in CaseⅣ.The almost identical angles are observed in the K-H experiment with～3.7°for the former case and 31°for the latter case [ 4] .(3) The macro-and microscopic responses after the lattice reorientation also show marked differences.Since the reoriented c-axis for CaseⅢis only～4°away from the loading directions,the updated orientation is analogous to that in CaseⅠ.Accordingly,pyramidal<c+a>slip is activated after the reorientation as shown in Fig.4c,and its subsequent dominance results in a rapid increase in stress and a strong hardening as shown in Fig.3.The situation for CaseⅣis a bit more sophisticated,for all the slip modes are successively activated,as shown in Fig.4d.Basal slip dominates at first,and then other non-basal slip modes are involved,along with a modest hardening behavior,as shown in Fig.3.

A further Schmid factor analysis for CaseⅣreveals that the basal slip in the twinned region (with the highest Schmid factor of 0.432) is active.With such a high Schmid factor and an extremely low CRSS (～0.5 MPa),the basal slip in the twinned region contributes to the deformation from the very beginning,as shown in Fig.4d.However,slip in the twinned region before the saturation of twin activity (the nominal reorientation) was not reported in other simulations [ 30, 31] ,which might be because of some simplification or special treatments in their numerical simulation.

For Case V,the crystal is so aligned that all of the plastic deformation can be accommodated by a single basal(a) slip with no other deformation modes activated as shown in Fig.4e.The Schmid factor of the active basal slip system reaches the maximum.Since basal slip is the softest mechanism available to hep magnesium (i.e.,possesses the lowest CRSS),the overall stress level for Case V is the lowest as shown in Fig.3.

3.2 Effect of initial deviation angle

A slight misalignment of the lattice orientation (i.e.,an initial deviation) may result in huge variations in the macro-and microscopic responses,such as basal slip traces observed under the compressive loading nearly along the caxis [ 4] .Thus,its effect deserves to be analyzed in details.However,considering all the possible misalignments in a single analysis is a formidable task.In order to keep the problem tractable,only a certain kind was considered in the present work.Deviation was supposed to exist in the loading direction,while the constraint direction was accurate as pre-expected.To evaluate the effect of the misalignment,the initial deviation angle (θ) was defined as shown in Fig.5,with three different orientations (CasesⅠ-Ⅲ) being selected as representatives.

Figure 6 shows the effect of initial deviation on the stress-strain responses for different cases,with three deviation angles (θof 0°,2°and 5°) being considered in each case.The initial deviation is found to have limited effect on the macroscopic mechanical behaviors for CasesⅠandⅡ.However,the stress-strain curve for CaseⅢdiffers significantly with the initial orientation.The stress level after the reorientation is much lower forθ=5°,along with a modest hardening behavior,compared to that forθ=0°or 2°.For the certain misalignment (i.e.,deviation only in the loading direction) considered,it can be concluded that the larger initial deviation angles are related with the lower stress level (within the range of 0°-5°for the deviation angle).Notably,it is the underlying evolution of microscopic deformation mechanisms that results in all the features and variations in stress-strain curves.Therefore,corresponding microscopic analysis is carried out subsequently.

Despite the similar stress-strain responses for CaseⅠunder different deviation angles,the deviation has evident influences on the microscopic scale.Basal slip is activated due to the slight deviation,which was also reported in the K-H experiment [ 4] .Figure 7 shows the evolution of the relative activities of various deformation modes with strain for CaseⅠunder an initial deviation angle of 2°.In comparison with the non-deviation case shown in Fig.4a,the sharp distinction is that basal slip dominates the deformation in the beginning and then leaves the dominant role to the pyramidal slip.Even though such a slight deviation angle leads to quite a small Schmid factor of basal slip (～0.035 for the maximum),this deformation mode is still active,mainly due to its extremely low CRSS(～0.5 MPa).Further investigations appear to have the conclusion that relative proportion of basal slip rises with the deviation angle increasing (within the range of 0°-45°).For instance,Case V can be viewed as an extreme case with an initial deviation angle of 45°,and basal slip in that case dominates the whole deformation process with no other modes activated,as shown in Fig.4e.

Fig.5 Schematic diagrams of initial deviation:a CaseⅠ,b CaseⅡand c CaseⅢ

Fig.6 Variations in predicted stress-strain responses with initial deviation angle (θ) of 0°(solid lines),2°(dash lines) and 5°(short dash lines) for CasesⅠ(blue),Ⅱ(green) andⅢ(red)

Fig.7 Relative activities of slip and twin modes in plane-strain compression simulation for CaseⅠwith an initial deviation angle of 2°

From Fig.6,the macroscopic response for CaseⅡseems also unaffected by the deviation,just like CaseⅠ.However,basal slip stays inactive,for the Schmid factors of all the three slip systems possess a zero value.The slight difference in the stress-strain curves for this case is probably due to the various activities of the two favored prismatic slip systems with various deviation angles.For nondeviation case,two of the prismatic slip systems are equally favored (with a Schmid factor of 0.433);for the case with an initial deviation angle of 5°,one of them is more favored (with a Schmid factor of 0.470) than another(with a Schmid factor of 0.383).

Fig.8 Relative activities of slip and twin modes in plane-strain compression simulation for CaseⅢwith an initial deviation angle of5°

Different from CasesⅠandⅡ,the stress-strain curve for CaseⅢvaries with various initial deviations.The evolution of the relative activities of various deformation modes with strain for CaseⅢunder an initial deviation angle of5°is shown in Fig.8.Firstly,basal slip (in the parent region) is activated before the reorientation due to the deviation.However,the involvement of limited basal slip has little effect on the first half of the stress-strain curve,as shown in Fig.6.When the saturation of twin activity is reached,the twinning-induced reorientation makes an included angle of 5.2°with the loading direction (compared with 3.7°in the non-deviation case).In the deviation case,pyramidal slip still dominates the deformation after the reorientation,while limited prismatic slip is also found to be active,as shown in Fig.8.In comparison with the non-deviation case,the larger included angle after the reorientation is probably the primary cause for the great reduction of the subsequent stress level,as shown in Fig.6.

4 Conclusion

In the present study,a phenomenological CP model was presented to simulate the plane-strain compression of single-crystal pure magnesium.According to the K-H experimental results,the parameters related were carefully calibrated,and the accuracy of the model was validated.Besides with the stress-strain responses,the relative activities of deformation modes were also investigated critically.Since the twin-induced reorientation was considered,the crystal orientation in the twinned region as well as the macro-and microscopic responses after the reorientation was well predicted.Furthermore,the effect of initial orientation deviation on the mechanical responses was also discussed.Basal slip is found to be easily activated due to deviation,in spite of its relatively low Schmid factor.A slight deviation in the twin-favored case could result in a significant difference in the mechanical behavior after the reorientation.The effort on the study of magnesium single crystal in the present work contributes to further polycrystalline analysis.

Acknowledgements This study was financially supported by the National Natural Science Foundation of China (No.51375256).

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