Prediction of Pressure-Induced Structural Transition and Mechanical Properties of MgY from First-Principles Calculations

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1 Commun. Theor. Phys. 65 (206) Vol. 65, No., January, 206 Prediction of Pressure-Induced Structural Transition and Mechanical Properties of MgY from First-Principles Calculations Chun-Ying Pu (îë ), Xian-Chao Xun (Û ), 2 Hai-Zhen Song (Ý ), Fei-Wu Zhang (ì ), 3,4 Zhi-Wen Lu ( ã ), and Da-Wei Zhou ( å), College of Physics and Electronic Engineering, Nanyang Normal University, Nanyang 47306, China 2 Basic Department, Aviation University of Air Force, Changchun 30022, China 3 State Key Laboratory of Ore Deposit Geochemistry, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang , China 4 Nanochemistry Research Institute, Curtin University, Perth, WA-6845, Australia (Received May 5, 205; revised manuscript received September 2, 205) Abstract Using the particle swarm optimization algorithm on crystal structure prediction, we first predict that MgY alloy undergoes a first-order phase transition from CsCl phase to P4/NMM phase at about 55 GPa with a small volume collapse of 2.63%. The dynamical stability of P4/NMM phase at 55 GPa is evaluated by the phonon spectrum calculation and the electronic structure is discussed. The elastic constants are calculated, after which the bulk moduli, shear moduli, Young s modui, and Debye temperature are derived. The brittleness/ductile behavior, and anisotropy of two phases under pressure are discussed in details. Our results show that external pressure can change the brittle behavior to ductile at 0 GPa for CsCl phase and improve the ductility of MgY alloy. As pressure increases, the elastic anisotropy in shear of CsCl phase decreases, while that of P4/NMM phase remains nearly constant. The elastic anisotropic constructions of the directional dependences of reciprocals of bulk modulus and Young s modulus are also calculated and discussed. PACS numbers: 7.20.Eh, 7.5.Mb, dj Key words: high pressure, phase transition, first-principles calculations, mechanical properties Introduction Magnesium and magnesium alloys have drawn considerable attention for their unique physical properties such as the low density, high specific strength, Young s modulus and good stiffness. However, the application of magnesium alloys is still limited due to their poor creep resistance, low yield strength and low ductility. Much effort has been devoted to improve the mechanical and thermodynamics properties of Magnesium alloys. It has been reported that addition rare earth (RE) elements such as Gd, Dy, and Y in the materials could significantly improve its mechanical properties at both room and high temperatures. [ 3] In the case of the precipitated CsCltype structured B2-MgRE intermetallic compounds, RE elements are stably distributed on the grain boundaries, which can effectively block the movement of dislocations, and increase the strength at high temperature. [4 5] Therefore, various studies have been carried out for these B2- MgRE intermetallics, including the stability, thermodynamically and elastic properties. For example, Chen et al., [6] calculated the thermal stabilities, elastic properties and electronic structures of B2-MgRE (RE = Sc, Y, La) intermetallic compounds with first-principles calculations. Wu and Hu [7] investigated the brittle and elastic properties of the B2-MgRE (RE = Sc, Y, Ce, Pr, Nd, Gd, Tb, Dy, Ho, Er) intermetallic compounds. Villars et al., [8] have given intermetallic phases in their handbook of crystallographic data. Tao et al., [9] calculated the elastic properties of B2-MgRE (RE = Sc, Y, La Lu) intermetallic compounds by using first-principles calculations, and the characteristics of elastic constants of B2- MgRE intermetallic compounds are explained by the insight of electronic structures under deformation. Wang et al., [0] further studied the third-order elastic constants of the B2-MgRE (RE = Y, Tb, Dy, Nd) intermetallic compounds. Wu et al. [] investigated the generalizedstacking-fault energy (GSFE) surfaces for B2-MgRE (RE = Y, Tb, Dy, Nd) intermetallic compounds using firstprinciples calculations. Using the density functional perturbation theory in combination with the quasi harmonic approximation, Wang et al. [2] investigated the thermodynamic properties of B2-MgRE (RE = Y, Dy, Pr, Tb) intermetallic compounds. Using the same method, The temperature and volume dependent elastic modulus of B2- MgRE (RE = Y, Dy, Pr, Sc, Tb) intermetallic compounds were calculated. [3] Supported by the Henan Joint Funds of the National Natural Science Foundation of China under Grant Nos. U30462, U404608, the National Natural Science Foundation of China under Grant Nos , , and the Special Fund of the Theoretical Physics of China under Grant No , Postdoctoral Science Foundation of China under Grant No. 205M58767, and Young Core Instructor Foundation of Henan Province under Grant No. 205GGJS-22 Corresponding author, zhoudawei@nynu.edu.cn c 206 Chinese Physical Society and IOP Publishing Ltd

2 No. Communications in Theoretical Physics 93 As far as we know, almost all of the previous research focused on the properties of B2-MgRE intermetallic compounds at ambient pressure, but these compounds under high pressure have been paid relatively little attention. It is well known that pressure could effectively modify the physical properties or induce the formation of new structure. Moreover, high-pressure synthesis is one of the most effective techniques to prepare new compounds or to obtain unique properties. Therefore, it is desirable to investigate the high-pressure behaviors of MgRE intermetallic compounds, and the mechanical properties might be fundamentally revised. Among the rare earth elements, Y element is believed to play a very important role in modifying the properties of Mg alloys. [4 6] The addition of Y not only improves the mechanical properties but also increases the corrosion resistance. [7] Up to now, the properties of MgY under ambient pressure are basically clear, but those under high pressures have never been fully investigated. Therefore, it is necessary to design and understand the high pressure behavior of MgY alloy, which can provide theoretical guidance for experiment. In this paper, using a structure search method based on CALYPSO methodology and density functional total energy, we first explore the high-pressure phases of MgY alloy, then study the pressure influences on the properties of MgY alloy. The structural stability, electronic structure, and mechanical properties of MgY alloy under pressure are investigated systematically in this paper. 2 Computational Methods Structural prediction for MgY is based on a global minimization of enthalpy surfaces merging first-principles total energy calculations as implemented in the Crystal structure AnaLYsis by Particle Swarm Optimization (CA- LYPSO) code in the pressure range 0 50 GPa, [8 9] which has been successfully applied to predict high pressure structures. [20 22] The structural relaxations were performed using density functional theory of Perdew, Burke, and Ernzerhof (PBE) of the generalized gradient approximation (GGA) [23] as implemented in the Vienna firstprinciples simulation package (VASP) code. [24] The allelectron projector-augmented wave (PAW) method [25] was adopted with 2p3s and 4s4p4d5s as valence electrons for Mg and Y atoms, respectively. The energy cutoff 500 ev and appropriate Monkhorst Pack k meshes were chosen to ensure that enthalpy calculations are well converged to better than mev/atom. The phonon calculations are carried out by using a supercell approach, as implemented in the phonopy code. [26] To evaluate the elastic constants of MgY, stress-strain method was used in this work. [27] 3 Results and Discussion To explore the high-pressure phases of MgY alloy, structure predictions were performed at 0 50 GPa using variable-cell simulations with 8 formula units (f.u.) in the unit cell. At 0 GPa, the experimental observed CsCl structure was successfully reproduced (Fig. ), validating our method here. At higher pressures, our structural prediction revealed an energetically favorable tetragonal structure with space group P4/NMM (No. 29) containing 4f.u./unit cell, the conventional cell of which is shown in the inset of Fig.. To determine the structural phasetransition pressure of MgY alloy, the enthalpy curve of the P4/NMM structure with respect to the CsCl structure as a function of pressure is shown in Fig.. The structural transition pressure from CsCl to P4/NMM phase is predicted at 55 GPa. The P4/NMM phase at 55 GPa has the optimized lattice parameters a = Å and c = Å, with Mg atoms occupying 8j (0.5, 0, ) and 4d (0.5, 0.5, 0.0) and Y atoms occupying 8j (0.0, 0.5, ) and 4e (0.0, 0.0, 0.5) positions. The P V relation of CsCl and P4/NMM for MgY is also shown in the inset of Fig.. It can be seen that the phase transition has a first order nature accompanied by a volume collapse equal to 2.63%. Fig. Enthalpy as a function of pressure for MgY in P4/NMM and CsCl structures at T = 0 K, where the CsCl structure is taken as the reference. The inset is predicted P4/NMM and CsCl structure and the pressure dependence of volume for MgY. Fig. 2 (Color online) Calculated phonon dispersion curves and vibrational DOS (black solid line) for predicted P4/NMM structure at 55 GPa, along with partial DOS from Mg (red dashed line) and Y (green dotted line).

3 94 Communications in Theoretical Physics Vol. 65 To further check the dynamical stability of P4/NMM phase, the calculated phonon dispersion curves and partial phonon density of states (DOS) at 55 GPa are shown in Fig. 2. As seen from the phonon dispersion curves, the new P4/NMM phase is dynamically stable in view of the absence of any imaginary phonon frequency in the whole Brillouin zone. In Fig. 2, we also plot the partial phonon DOS with i = (Mg, Y), respectively. Due to their different atomic mass, the low-energy phonon modes are mainly associated with Y atoms, while for the high-energy phonon modes frequencies, the phonon DOSs mainly come from the Mg atoms. To study the electrical characteristics of the MgY under pressure, the total and partial density of states for CsCl and P4/NMM structures at 55 GPa are shown in Fig. 3. Both phases exhibit metallic behavior, and the Y-d, Y-p states are mixed with Mg-p states at the Fermi energy, indicating large Y-d, Y-p, and Mg-p hybridization. However, the contribution to the total DOS at Fermi level comes from Mg-p states in the P4/NMM is larger than that in CsCl phase. The value of number of DOS at Fermi level N(E F ) for P4/NMM phase is slightly higher (.386 states/ev) than the value for CsCl phase (.289 states/ev), implying the metallic character of MgY in P4/NMM phase may be greater than CsCl phase. Fig. 3 (Color online) The total and partial density of states for MgY at 55 GPa in (a) CsCl phase and (b) P4/NMM phase. The elastic constants of solids are significant parameters, which can provide important information concerning the nature of the forces operation in solids, such as binding characteristic between adjacent atomic planes, anisotropic character of binding and structural stability. Using the pressure dependence of elastic constants, we can judge the mechanical stability of a crystal under high pressure. The Born mechanical stability criteria for cubic and tetragonal crystal are given below: [28] Cubic phase (C, C 2, C 44 ) C > 0, C 44 > 0, C > C 2, (C + 2C 2 ) > 0. () Tetragonal phase (C, C 2, C 3, C 33, C 44, C 66 ) C > 0, C 33 > 0, C 44 > 0, C 66 > 0, C C 2 > 0, (C + C 33 2C 3 ) > 0, 2(C + 2C 2 ) + C C 3 > 0. (2) Figure 4 presents the hydrostatic pressure dependence of elastic constants of MgY. We concentrate our investigation on the pressure ranging 0 GPa 55 GPa for CsCl phase and GPa for P4/NMM phase. We have found that both phases are mechanical stable in the pressure range studied. It is notable that C is the largest among elastic constants for CsCl and P4/NMM phases, which indicate that the two phases are very incompressible under uniaxial stress along x-axis. All of the elastic constants increase monotonically with pressure, while that of C 44 and C 66 of P4/NMM phase and C 44 of CsCl phase vary slowly. In the case of C 44 and C 66, one can interpret that it indicates the resistances of the crystal with respect to the shear strain, and which are the important parameters relating to the shear modulus. So the C 44 and C 66 in P4/NMM phase and C 44 in CsCl phase are related to vary of the shear modulus and Debye temperature with pressure as we will discuss later. Once the single-crystal elastic constants C ij are obtained, the isotropic aggregate bulk modulus B and shear modulus G can be obtained using the Voigt Reuss Hill (VRH) approximations according to the following

4 No. Communications in Theoretical Physics 95 equations [29 32] B = (B V + B R )/2, G = (G V + G R )/2. (3) For cubic structure B V = B R = (C + 2C 2 )/3, (4) G V = (C C 2 + 3C 44 )/5, (5) G R = 5(C C 2 )C 44 /4[C (C C 2 )]. (6) For tetragonal structure B R = (C + C 2 )C 33 2C 2 3 C + C 2 + 2C 33 4C 3, (7) B V = 9 (2C + C 33 ) (C 2 + 2C 3 ), (8) G R = 5 8B V /[(C + C 2 )C 33 2C 2 3 ] + 6/(C C 2 ) + 6/C /C 66, (9) G V = 5 (2C + C 33 C 2 2C 3 ) + 5 (2C 44 + C 66 ). (0) Fig. 4 (Color online) Calculated elastic constants of MgY with pressure in CsCl and P4/NMM structures. Then Young s modulus E is connected to B and G by the relation E = 9BG/(3B + G). () Having calculated the Young s modulus E, bulk modulus B, and shear modulus G, one can estimate the Debye temperature θ D, which is an important fundamental parameter closely related to many physical properties such as elastic constants, specific heat and melting temperature. One of the standard methods to calculate the Debye temperature is from the data of elastic constant, since the Debye temperature may be estimated from the averaged sound velocity v m by the following equation [32] θ D = h 3 3nNA ρ k B 4πM v m, (2) where h is Planck s constant, k B is Boltzmann s constant, n is the number of atoms per formula unit, M is the molecular weight, N A is Avogadro s number, ρ is the density and ν m is the average sound velocity. In fact, ν m can be obtained from the longitudinal wave velocities ν l and transverse wave velocities ν s [ ( 2 v m = 3 vs 3 + )] /3, vl 3 v l = [(B + 4G/3)/ρ] /2, v s = (G/ρ) /2. (3) The plots of bulk modulus, shear modulus, Young s modulus and Debye temperature for the CsCl and P4/NMM structures as a function of pressure are depicted in Fig. 5. It can be seen that the pressure has an important influence on the bulk modulus, shear modulus, Young s modulus as well as Debye temperature. The bulk modulus of MgY in both phases increase linearly in the similar slope when the pressure is enhanced, indicating it is difficult to be compressed with pressure. It is interesting to notice that the variation tendency of B and θ D are approximately the same for CsCl phase, indicating the bulk modulus is major factor for change of Debye temperature with pressure. However, the variation of Debye temperature in P4/NMM phase has nearly the same tendency with shear modulus G, revealing that the shear modulus plays a more important role on the changes of Debye temperature with pressure. Unfortunately, as far as we know, there are no experimental data and theoretical results available in the literature on these properties for MgY alloy under pressure. We hope future experimental work will test our calculated results. Fig. 5 (Color online) Pressure dependence of bulk modulus, shear modulus, Young s modulus, and Debye temperature. The ratio of shear modulus to bulk modulus (G/B) has been proposed by Pugh [33] to roughly predict brit-

5 96 Communications in Theoretical Physics Vol. 65 tle or ductile behavior of materials. A lower G/B ratio denotes better ductility, and the critical value which separates ductility from brittleness is about For MgY alloy at 0 GPa, we found that the Voigt approximation (the maximum value for Voigt result) is better to describe the brittle behavior, which is in agreement with experimental result. [34] Using the Voigt approximation, the pressure dependence of the G/B ratio of MgY is shown in Fig. 6. The G/B ratio of CsCl phase begins to be less than 0.57 at 0 GPa, implying the transition from the brittle to ductile state occurs at about 0 GPa. However, the G/B ratio of CsCl phase decreases rapidly with pressure increasing up to 30 GPa, indicating the degree of ductility of CsCl phase increases under pressure. Interestingly, the G/B ratio of P4/NMM phase remains the ductile behavior at pressure range studied. Therefore pressure can enhance the ductility of MgY alloy. than zero, indicating that it cannot be regarded as being elastic and isotropic. We note that the value of A G has a small decrease with the increasing pressure, which indicating the isotropy in shear increases with the increasing pressure, while that of P4/NMM phase remains nearly unchanged above 55 GPa. It can be easily seen that A U is larger than 2.2 for both phases, indicating a high anisotropy in the two phases. Furthermore, as pressure increases, A U of CsCl phase drops rapidly, but that of P4/NMM phase remains nearly unchanged except for pressure 90 GPa. Fig. 7 (Color online) Pressure dependence of percent anisotropy A B and A G and the universal anisotropic index A U (inset). Fig. 6 (Color online) Pressure dependence of shear modulus and bulk modulus ratio. For a more comprehensive estimation of MgY elastic anisotropy, we adopt the anisotropy indexes of bulk modulus and shear modulus (A B and A G ) proposed by Chung and Buessem, [35] and the universal elastic anisotropy index A U developed by Ostoja Starzewski [36] for crystal with any symmetry. To estimate the anisotropic characteristic of the system, A B, A G, and A U can be written as: A B = B V B R B V + B R, A G = G V G R G V + G R, (4) A U = 5 G V + B V 6 0, (5) G R B R where A B = A G =0 represents the elastic isotropic, and A B = A G = represents the maximum elastic anisotropy. Equation (5) indicates that the larger deviations from zero imply the higher anisotropic mechanical properties. A U is identically zero for locally isotropic single crystal. Figure 7 shows the pressure dependence of results. For both structures of MgY alloy, we can see that the anisotropy factor A B is zero on the whole pressure range, indicating a complete elastic isotropy in compressibility. However, the anisotropy factor A G of both phases is larger As we know, the elastic anisotropy of a crystal can be characterized by many different ways. As another valid method to describe the elastic anisotropic constructions of the directional dependences of reciprocals of bulk modulus and Young s modulus are practically useful. We have built a curved surface of a three-dimensional (3D) to state the elastic anisotropy of crystal structure on crystallographic directions. [37] For cubic crystal B = S + 2S 2, (6) ( E = S 2 S S 2 S ) 44 (l 2 2 l2 2 + l2 2 l2 3 + l2 l2 3 ). (7) For tetragonal crystal B = S + S 2 + S 3 (S + S 2 + S 3 S 33 )l3 2, (8) E = (l4 + l2)s 4 + l3s ll 2 2(2S S 66 ) + l 2 3 ( l2 3 )(2S 3 + S 44 ), (9) where S ij are the elastic compliance constants and l, l 2, and l 3 are the directional cosines to the x-, y- and z- axes, respectively. From the above relation the estimated directional dependent bulk modulus and Young s modulus for CsCl phase at 0 GPa and P4/NMM phase at 55 GPa, using the elastic compliance constants, are shown in Fig. 8. For an isotropic system, the 3D directional dependence would show a spherical shape, while the deviation degree from the spherical shape reflects the content

6 No. Communications in Theoretical Physics 97 of anisotropy. As can be seen from Fig. 8, we find that the 3D figures of the bulk modulus for CsCl phase at 0 GPa and P4/NMM phase at 55 GPa have almost no deviation in shape from the sphere, which indicates that the bulk modulus show a smaller anisotropy. However, it is obviously found that MgY alloys exhibit a strong anisotropic character in Young s modulus, especially for P4/NMM structure. The projections on the (00), (00), and (00) planes show more details about the anisotropic properties of Young s modulus. It can be seen that Young s modulus of P4/NMM phase has a stronger directional dependence on these planes. The planar contours in these three planes are similar, which implies the analogous anisotropy for Young s modulus. Fig. 8 (Color online) The surface constructions and planar projections of bulk modulus (a) and Young s modulus (b) of CsCl phase at 0 GPa, bulk modulus (c) and Young s modulus (d) of P4/NMM phase at 55 GPa. 4 Conclusions In conclusion, a pressure-induced first-order phase transition of MgY has been predicted by using a particle swarm optimization algorithm in crystal structural prediction. The structural phase transition was predicted at 55 GPa, and the new high-pressure phase is identified to be P4/NMM structure. The structural stability, density of states, and mechanical properties of MgY under pressure are systematically investigated using first-principles calculations in combination with the quasi-harmonic Debye model. Both phases are mechanical stable in the pressure range studied. The G/B ratio of both phases indicate that transition from the brittle to ductile state occurs at about 0 GPa for CsCl phase. With the pressure increase, the isotropy in shear increases while that of P4/NMM phase remains nearly unchanged. Universal elastic anisotropy index A U imply that both phases exhibit relatively higher anisotropy. Young s modulus and shear modulus as a function of crystal orientations have been systematically investigated in order to understand the elastic anisotropy of MgY. The density of states and elastic modulus as well as Debye temperature of two phases under pressure were also discussed in detail. We hope that our results will stimulate further experimental and theoretical works on this technologically material in the future. References [] G.W. Lorimer, P.J. Apps, H. Karimzadeh, and J.F. King, Mater. Sci. Forum (2003) 279. [2] M. Suzuki, H. Sato, K. Maruyama, and H. Oikawa, Mater. Sci. Eng. A 252 (998) 248. [3] M. Suzuki, H. Sato, K. Maruyama, and H. Oikawa, Mater. Sci. Eng. A (200) 75. [4] X. Tao, Y. Ouyang, H. Liu, Y. Feng, Y. Du, Y. He, and

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