Main reinforcement effects of precipitation phase Mg2Cu3Si, Mg2Si and MgCu2 on Mg-Cu-Si alloys by ab initio investigation

Physica B 521 (2017) 339–346

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Main reinforcement effects of precipitation phase Mg2Cu3Si, Mg2Si and MgCu2 on Mg-Cu-Si alloys by ab initio investigation

MARK

⁎

Xue-Feng Shia, Hai-Chen Wanga, Ping-Ying Tangb, Bi-Yu Tanga, a

School of Chemistry and Chemical Engineering, Guangxi University, Nanning 530004, China Key Laboratory of New Electric Functional Materials of Guangxi Colleges and Universities, Guangxi Teachers Education university, Nanning 530023, China b

A R T I C L E I N F O

A BS T RAC T

Keywords: Mechanical properties Electronic structure Precipitation phase Ab initio calculation Reinforcement effect

To predict and compare the main reinforcement effects of the key precipitation phases Mg2Cu3Si, Mg2Si and MgCu2 in Mg-Cu-Si alloy, the structural, mechanical and electronic properties of these phases have been studied by ab initio calculations. The lowest formation enthalpy and cohesive energy indicate that Mg2Cu3Si has the strongest alloying ability and structural stability. The mechanical modulus indicates that Mg2Cu3Si has the strongest resistance to reversible shear/volume distortion and has maximum hardness. The characterization of brittle (ductile) behavior manifests that MgCu2 has favorable ductility. Meanwhile the evaluation of elastic anisotropy indicates that Mg2Si possesses elastic isotropy. Debye temperature prediction shows that Mg2Si and Mg2Cu3Si have better thermal stability. To achieve an unbiased interpretation on the phase stability and mechanical behavior of these precipitation phases, the density of states and differential charge densities are also analyzed. The current study deepens the comprehensive understanding of main reinforcement effects of these precipitation phases on Mg-Cu-Si alloys, and also benefits to optimize the overall performances of Mg-Cu-Si alloy from the hardness, ductility and thermal stability by controlling these second precipitation phases during the heat treatment process.

1. Introduction

improved with addition of these alloying elements owing to the mechanism of precipitation strengthening. Hence, comprehensive understanding of the strengthening mechanisms of how alloying elements affect the mechanical properties and phase stability of MgCu-Si alloy at the atomic level would provide crucial guidance for optimal design and adoption of this promising Mg alloys. It is noted that the application of Mg-Cu-Si alloys in the modern industry is evidently limited. Hence, to further improve the comprehensive performance of Mg-Cu-Si alloys and expand their application, it is essential to optimize microstructure and properties by control the aging conditions of alloys [12]. Experimentally, Zhao et al. [13] have determined the phase equilibrium of Mg-Cu-Si system from x-ray diffraction analysis. Despite many types of binary and ternary phases in Mg-Cu-Si alloys depending on the component (Mg, Cu, Si) and temperature, the key precipitation phases Mg2Si, MgCu2 and Mg2Cu3Si play an important role in strengthening the mechanical performance of Mg-Cu-Si alloys because of their respective superior performance [13–16]. It has been shown that formation of Mg2Si can obviously improve the heat resistance of Mg-Al alloys [17], and formation of MgCu2 can overcome the disadvantages of brittleness of

To improve fuel economy and reduce environmental pollution, Mg alloys have been attracted extensive attention in automotive and aeronautical industries due to their excellent physical properties, e.g. low density, high stiffness, excellent creep resistance and good vibration performance, etc. [1–3]. Currently the limited formability and tensile strength as well as the low ignition resistance are major hinder for the application of Mg alloys, especially the restrained mechanical properties at elevated temperature [4,5]. Therefore, to bypass the limitations of mechanical properties of Mg alloys as engineering materials, and tremendous effort has been conducted toward developing new Mg alloys with higher strength [6,7]. Mg alloys with age hardening response and lightweight have been wide studied [8,9]. The main strengthening mechanisms of Mg alloys include precipitation strengthening, solid solution strengthening and dispersion strengthening [5,10], and addition of alloying elements is an effective approach to optimize the microstructure and improve the mechanical properties of Mg alloys [8,9,11]. Copper and silicon are the frequently-used alloying elements, and the strength of Mg-Cu-Si alloys can be considerably

⁎

Corresponding author. E-mail address: [email protected] (B.-Y. Tang).

http://dx.doi.org/10.1016/j.physb.2017.07.023 Received 12 June 2017; Received in revised form 10 July 2017; Accepted 11 July 2017 Available online 11 July 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.

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code [29] based on density functional theory [30,31]. The exchange– correlation potential was evaluated by the PBE version of the generalized gradient approximation [32,33], and the ion-electron interactions for Mg, Cu and Si atoms were treated by PAW potential [34]. The valence electron configurations in pseudo-potentials are respectively 2p63s2 for Mg, 3d104s1 for Cu, 3s23p2 for Si, and the plane-wave cut-off energy was chosen as 370 eV for all calculations. For k-points sampling in the irreducible Brillouin zone [35], the Gamma centered MonkhorstPack grids of 11×11×7 for Mg2Cu3Si, 10×10×10 and 8×8×8 for Mg2Si and MgCu2 were employed for optimization of geometry. During calculation of the density of states, the k-points for Mg2Cu3Si, Mg2Si and MgCu2 are respectively 13×13×9, 12×12×12 and 12×12×12. The geometrical relaxation was considered optimized when the difference in total energy is below 10−6 eV/atom and the Hellmann–Feynman force is within 10−3 eV/Å. The total energy was calculated by integration in k-space using the linear tetrahedron method with the Blöchl correction [36]. Test results show that the above parameters can provide a sufficient accuracy in the present work. Structural stability and alloying ability of these precipitation phases in Mg-Cu-Si alloys were evaluated by cohesive energy ΔEcoh and formation enthalpy ΔHform, which can be calculated respectively according to equations ΔEcoh=(ET−∑NiEi-iso)/N and ΔHform=(ET∑NiEi-soid)/N (i=Mg, Cu, Si) [5,24]. The single crystal elastic constants Cij for hexagonal Mg2Cu3Si, cubic MgCu2 and Mg2Si were obtained using energy–strain method [18,24]. To assess the crucial engineering properties, bulk modulus B, shear modulus G, Young's modulus E, Poisson's ratio ν and anisotropy index AU etc. are derived by executing the VRH (Vioght-Reuss-Hill) approximation [21,24].

Mg alloys [18], the ternary C14 Laves phase Mg2Cu3Si (τ phase) can act as precipitation hardener [13]. This gives hint that the overall performance of Mg-Cu-Si alloys would be improved by optimal combination of the respective superiority of the three precipitation phases. Hence, it is very meritorious to study the main reinforcement effects of these crucial precipitation phases on Mg-Cu-Si alloys. Up to now, many investigations of Mg2Si (cubic anti-fluorite structure) and MgCu2 (C15 Laves phase) have been performed extensively. Theoretical studies from ab initio calculations [19] have shown that Mg2Si possesses elastic isotropy. The brittle behavior of Mg2Si was indicated by Huang et al. [17]. Chen et al. [20] have shown that MgCu2 Laves phase is ductile and elastic anisotropy by first principle calculation. The ductile characteristics of MgCu2 Laves phase also have indicated by Liu et al. [21] from first-principles method. Furthermore, many closely related binary C14 and C15 Laves phase have also been studied within density functional theory. Especially, the C15 Laves phases Al2X (X=Mg, Ca) [22] and MAl2 (M=Mg, Ca, Sr and Ba) [23] are investigated from first-principles calculations, as well as the C14 Laves phases Al2M (M=Mg, Ca, Sr, Ba) [24] and Mg2Sr [8] are also studied. These extremely important studies are beneficial to extend our knowledge of materials performance and will inevitably advance our understanding of the closely related Mg alloys. To our knowledge, the ternary C14 Laves phase in Mg-based alloys has been almost unexplored. Meanwhile, the main reinforcement effects of Mg2Cu3Si, Mg2Si and MgCu2 in Mg-Cu-Si alloys also have not been studied comprehensively. Experimental investigation of the reinforcement effects is generally time-consuming and resource-consuming. Ab initio calculation has become a reliable and powerful tool to predict phase stability, mechanical and other physicochemical properties, so deepens the understanding of the reinforcement effects for these precipitation phases in Mg-Cu-Si alloys. Therefore, the current work concentrates on the mechanisms of alloying elements on the improvement of mechanical properties of MgCu-Si alloys. Our goal is to comprehensively understand the main reinforcement effects of crucial precipitation phase Mg2Cu3Si, MgCu2 and Mg2Si on Mg-Cu-Si alloys by predicting and comparing the phase stabilities, mechanical and electronic properties of these precipitation phases from ab initio calculations. Current study would provide theoretical guidance to obtain excellent performances of good-quality Mg-Cu-Si alloys in engineering by controlling the second precipitation phases Mg2Cu3Si, Mg2Si and MgCu2 during the heat treatment process, and accelerate the development and application of magnesium alloys.

3. Results and discussion 3.1. Structural stability and alloying ability Structural optimization of Mg2Cu3Si, MgCu2 and Mg2Si are performed initially by GGA with the PBE version, based on known experimental data [13,27,37]. The results of the current work are tabulated in Table 1, together with the available known data for comparison [13,15,27]. The excellent consistencies of the results for MgCu2 and Mg2Si with the known data validate the current parameter settings. It's should be noted that there is no theoretical report on the novel ternary Laves phase Mg2Cu3Si (τ phase) up to now, so we also hope that the current results will accelerate its further theoretical and experimental research. To evaluate the structural stability and alloying ability of these precipitation phases in Mg-Cu-Si alloys, cohesive energy ΔEcoh and formation enthalpy ΔHform are calculated respectively by the aforementioned equations in section of theoretical models and methodology. The ΔEcoh is usually defined as the energy requirement when the compound is decomposed into an isolated constituent atom [14,23]. Thus, structural stability can be reflected via ΔEcoh. The larger absolute value corresponds to a higher structural stability. The obtained ΔEcoh of Mg2Cu3Si, MgCu2 and Mg2Si are tabulated in Table 1, one can see that the excellent consistencies of the results for MgCu2 and Mg2Si with the known data [14,38], confirming again that the current results are accurate and credible. The negative ΔEcoh indicates that these phases are structurally stable from thermodynamics point of view. Meanwhile, current results also imply the strongest stability of Mg2Cu3Si among these precipitation phases owing to the largest absolute value of ΔEcoh. Alloying ability can be evaluated by ΔHform, similarly the structural stability also can be reflected via ΔHform. The negative ΔHform generally represents an exothermic process [5], the lower ΔHform corresponds to a stronger alloying ability and structural stability. The calculated ΔHform of Mg2Cu3Si, MgCu2 and Mg2Si are illustrated visually in Fig. 2 along with the known data [15,17,39]. As shown in Fig. 2, the alloying ability and structural stability of these precipitation phases are gradually weakened from Mg2Cu3Si to Mg2Si and MgCu2 because the

2. Theoretical models and methodology Experimental studies have shown that ternary C14 Mg2Cu3Si (τ phase) has two kinds of crystalline configurations [13,25]. In this article, our study is focused on the τ phase crystallizes in the ordered C14-type Laves phase MgZn2 with the space group P63/mmc (No.194), the conventional cell contains 4 Mg at Wyckoff position 4f sites, 6 Cu at 6 h and 2 Si at 2a sites [13]. The coordination number of Mg and Cu (Si) respectively is 16 and 12, and the Frank-Kasper polyhedron around Mg and icosahedra around Cu (Si) are respectively formed [26]. Mg2Si crystallizes in cubic anti-fluorite structure with space group Fm-3m (No.225), in which the Si atoms are at 4a Wyckoff sites and Mg at 8c sites [27]. For binary cubic C15 Laves phase MgCu2 with space group Fd-3m (No.227), the Mg atoms occupy Wyckoff position 8b sites and Cu atoms occupy 16c sites [21]. The lattice structure are rendered using VESTA software [28] for visually describe and illustrated in Fig. 1(a–c). Considering the fact that the study of lattice structure for new ternary Laves τ phase is scarce in experimentally and theoretically, to provide detailed guidance to further understand the structure characteristics of τ phase, the coordination polyhedrons around Mg, Cu and Si in τ phase are also drawn in Fig. 1. Present investigations of precipitation phases Mg2Cu3Si, MgCu2 and Mg2Si were performed by Vienna Ab initio Simulation Package 340

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Fig. 1. Structure models for precipitation phases: (a) Mg2Cu3Si, (b) Mg2Si, (c) MgCu2.

absolute value of ΔHform is gradually reduced. The current analysis show that Mg2Cu3Si and Mg2Si play an important role in improving the structural stability of Mg-Cu-Si alloy, which is consistent with the previous experimental description [13,17]. 3.2. Mechanical properties 3.2.1. Elastic constants of single-crystal Elastic constants (Cij) of single-crystal can provide pertinent mechanical information of solid material, especially the stiffness and mechanical stability [5,24,40]. Thus, in order to further understand the fundamental strengthening mechanism of these precipitation phases in Mg-Cu-Si alloy, it's very significant to study the elastic constants of these phases. Herein, the elastic constants for hexagonal Mg2Cu3Si, cubic Mg2Si and MgCu2 are acquired using energy–strain method, the detailed calculations are described in Ref. [18,24]. The acquired results are tabulated in Table 2, together with the known data [17,20,27,41]. Notably the agreement between current results and known data is excellent. Meanwhile, these precipitation phases are mechanical stability because their elastic constants satisfy the Born-Huang criteria [42,43]. Since the available theoretical and experimental studies on the mechanical properties of ternary Laves phase Mg2Cu3Si are scarce, so current work will furnishes meritorious references for its further research. From Table 2, C33 of τ phase is slightly larger than C11, implying that incompressibility along [0001] axis between neighboring layers is stronger than one within layer because the elastic constants C11 and

Fig. 2. Formation enthalpy (ΔHform) of Mg2Cu3Si, Mg2Si and MgCu2 phases.

C33 represent the ability of a crystal to resist the compressive strain [5,24], also the bonding strength along the [0001] direction is stronger than that along the [1010] and [0110] directions within layer. The C66 is higher than C44, indicating that the [1120 ](1010) shear is harder than the [0001](1010) shear [24]. Evidently, C11 (C33) of τ phase is larger than ones of Mg2Si and MgCu2, connoting that τ phase has the strongest incompressibility along x- (z-) axis among these precipitation phases. Meanwhile, C44 and C66 of τ phase are also higher, thus the resistance to shear of τ phase is larger than Mg2Si and MgCu2 phases.

Table 1 Equilibrium lattice parameters (L.P.) and cohesive energy ΔEcoh (−kJ/mol atom), together with known data for these precipitation phases. Mg2Cu3Si (τ phase)

Mg2Si

MgCu2

L. P. /ΔEcoh

a=b

c

ΔEcoh

a=b=c

ΔEcoh

a=b=c

ΔEcoh

Current Cal. Exp.

5.02 / 5.00 [13]

7.89 / 7.87 [13]

337.23 / /

6.36 6.36 [27] 6.35 [27]

286.95 281.90 [14] /

7.04 7.05 [15] 7.04 [37]

302.15 279.90 [38] 284.70 [38]

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ductile behavior is further analyzed from Pugh's index G/B [44], the results of G/B in Table 3 also show that MgCu2 is ductile, while τ phase and Mg2Si are brittle because the G/B > 0.57 ( < 0.57) corresponds to brittle (ductile) behavior [40]. These results are supported by the calculated Cauchy pressure. The results of Cauchy pressure (C13-C44) for hexagonal τ phase, (C12–C44) for cubic Mg2Si and MgCu2 are tabulated in Table 3. Because positive Cauchy pressure connotes ductility of crystal and the negative value insinuates brittleness [4]. The results of Cauchy pressure in Table 3 also show MgCu2 is ductile, while τ phase and Mg2Si are brittle and the bonding between atoms is covalent [44]. Moreover, the ratio of B to C44 is also calculated to quantify plasticity of these precipitation phases [46]. The larger the B/ C44 is, the better the plasticity is. The acquired results in Table 3 also show that MgCu2 has the best plasticity owing to the largest B/C44, then τ phase and finally Mg2Si.

Table 2 Elastic constants along with and known data for these precipitation phases. Material

τ phase Mg2Si

MgCu2

Source

Elastic constants Cij (GPa)

Current Current Cal. [17] Exp. [41] Current Cal. [27] Exp. [20]

C11

C12

C13

C33

C44

C66

192.01 117.18 121.45 126.00 117.17 107.90 125.00

55.92 23.35 26.11 26.00 81.57 79.00 71.70

47.72 23.35 26.11 26.00 81.57 79.00 71.70

205.94 117.18 121.45 126.00 117.17 107.90 125.00

55.67 46.39 49.58 48.50 37.78 34.60 42.30

68.04 46.39 49.58 48.50 37.78 34.60 42.30

3.2.2. Mechanical modulus of polycrystalline Based on the potential application of polycrystalline mechanical modulus in practical engineering, the Vioght-Reuss-Hill approach [21,24] is executed in current work to determine the polycrystalline elastic modulus of these precipitation phases, including bulk modulus B and shear modulus G. Then the Young's modulus E, Poisson's ratio ν and micro-hardness H are acquired respectively through formulas E=9BG/(3B+G), ν=(3B−2G)/(2 G+6B) and H=(E−2Eν)/(6+6ν) [44]. The calculated mechanical modulus of τ phase, Mg2Si and MgCu2 are tabulated in Table 3, together with the known data [17,20,27,41]. It can be seen that τ phase has highest B, followed by MgCu2 and Mg2Si. Because B is assumed to be a measure of resistance to volume deformation by applied pressure [4], so τ phase possesses the strongest resistance to volume deformation among these precipitation phases. Shear modulus G is also a measure of resistance to reversible deformations by shear stress [5]. The large G corresponds to larger resistance to reversible shear deformations, also implies the stronger directional bonding [5]. Because the G increases from MgCu2 to Mg2Si and τ phase, the resistance to reversible shear deformations in τ phase would be the strongest. Further, Young's modulus E can be applied to appraise the stiffness of solid material [45]. The larger E implies the stronger stiffness. From the acquired E in Table 3, the stiffness of τ phase is the largest, then Mg2Si and finally MgCu2. Accordingly, the micro-hardness H is further calculated, as shown in Table 3. Overall, the τ phase has the strongest resistance to reversible shear/volume distortion and its stiffness is greatest among these precipitation phases. Thus, the ternary Laves τ phase plays a significant role in improving the mechanical properties of Mg-Cu-Si alloy.

3.2.4. Anisotropy of elasticity The anisotropy of elasticity has significant effect on mechanical durability of alloys and nanometer precursor textures [4]. Consequently, in order to seek effective routes to improve performance, proper assessment of elastic anisotropy is very meaningful for engineering science. In this paper, several criterions are performed to appraise the elastic anisotropy of these precipitation phases. The compression percentage AB and shear percentage AG, universal elastic anisotropy index (AU), compressibility anisotropy (ABa, ABc) are acquired according to the detailed formula in Ref. [21,24,40], the results of (AB, AG), (AU) and (ABa, ABc) are tabulated in Table 4. The values of AB (AG) in Table 4 demonstrate that cubic Mg2Si is isotropic in compression and shear owing to AB=AG=0 [24], τ phase is nearly isotropic in compression (AB=0.01%) and very slightly anisotropic in shear (AG=0.86%), MgCu2 phase is isotropic in compression (AB=0) and larger anisotropy in shear (AG=6.60%). As shown in Table 4, AU shows that Mg2Si is isotropic materials (AU=0) and τ phase exhibits slightly anisotropy (AU=0.1), while MgCu2 has the conspicuous anisotropy owing to the larger deviation from zero, which is similar to the previous calculation [20]. The compressibility anisotropy ABa and ABc further indicating that cubic Mg2Si and MgCu2 exhibit elastic isotropy for linear bulk modulus due to the value of ABa=ABc=1 [40], whereas τ phase exhibits the small anisotropy. To quantify the degree of shear anisotropy on planes in more details, the shear anisotropy factors Ac=(C12+2C44)/C11 for cubic Mg2Si and MgCu2 [21], A1=A2=4C44/ (C11+C33−2C13) and A3=4C66/(C11+C22−2C12) for hexagonal τ phase [24] are calculated. The material is isotropic when the value of factors Ai is unity, otherwise it's anisotropic [47]. As can be seen clearly in Table 4, the shear anisotropic factors Ac demonstrates Mg2Si is almost isotropic (Ac=0.99), while anisotropy of MgCu2 is relatively obvious (Ac=1.34). For hexagonal τ phase, the factors A1, A2 and A3 respectively correspond to the shear anisotropy between the [0111] and [0110] directions on plane (1010), the [1010] and [0001] directions on plane (0110), the [1120 ] and [0110] directions on plane (0001). From the results in Table 4, the (0001) plane of τ phase exhibit isotropy (A3=1.00), whereas the (1010) and (0110) plane exhibit small anisotropy (A1=A2=0.74). Compared to the above elastic anisotropy arguments, the three-

3.2.3. Brittle/ductile behavior Now our attention is shifted to predict the brittle/ductile behavior of these precipitation phases via Poisson's ratio ν, Pugh's index G/B, Cauchy pressure and B/C44. The Poisson's ratio ν of polycrystalline aggregates is an important quantity in the design of solid materials, which ranges from −1 to 0.5. The value of ν can reflect the ductile (brittle) behavior [24], the boundary of ν differentiating ductility and brittleness of materials is 0.26 [4], and larger ν corresponds to better ductility. The acquired results in Table 3 demonstrate that MgCu2 has the best ductility owing to the largest ν and also has definite central inter-atomic force due to the ν is within the ranges 0.25–0.5 [21], τ phase is slightly brittle whereas Mg2Si is evidently brittle. The brittle/ Table 3 The mechanical modulus and known data for these precipitation phases. Material

Source

B

G

E

H

ν

G/B

Cauchy

B/C44

τ phase Mg2Si

Current Current Cal. [17] Exp. [41] Current Cal. [27] Exp. [20]

99.18 54.63 57.88 59.00 93.44 88.60 89.40

64.56 46.60 48.82 / 27.95 26.50 36.00

159.15 108.84 114.32 / 76.25 72.30 95.30

11.65 10.23 / / 2.62 / /

0.23 0.17 0.17 / 0.36 / 0.32

0.65 0.85 0.84 / 0.30 0.30 0.40

−7.95 −23.04 −23.47 / 43.69 / /

1.78 1.18 / / 2.47 / /

MgCu2

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Table 4 The anisotropic factors (AB, AG), the compressibility (ABa, ABc), the shear anisotropy factors (Ai) and the universal elastic anisotropy index (AU) for these precipitation phases.

Table 5 The density (ρ), transverse (νt), longitudinal (νl), average sound velocity (νm) and Debye temperature (ΘD) for these precipitation phases.

Material

Source

AB (%)

AG (%)

Au

ABa

ABc

Ac

A1,2

A3

Materials

Source

ρ (g/cm3)

νt (m/s)

νl (m/s)

νm (m/s)

ΘD (K)

τ phase Mg2Si MgCu2

Current Current Current

0.01 0.00 0.00

0.86 0.00 6.60

0.10 0.00 0.70

1.00 1.00 1.00

1.03 1.00 1.00

0.99 1.34

0.74 / /

1.00 / /

τ phase Mg2Si

Current Current Cal. [17] Current Cal. [20]

5.14 1.94 1.94 5.72 5.72

3544.05 4901.08 4865.77 2212.00 2343.61

6003.57 7758.05 7722.50 4781.00 4691.13

3926.67 5392.64 5355.36 2492.00 2629.23

480.07 573.37 581.67 303.00 320.45

MgCu2

dimension (3-D) graphic dependence of the elastic modulus is more intuitive and comprehensive for the description of elastic anisotropy. Because isotropy of bulk modulus B for cubic Mg2Si and MgCu2 phases has been sufficiently described based on the above elastic anisotropy arguments, and very slightly anisotropic of bulk modulus B for hexagonal τ phase are also described adequately. Hence the focus here is on the 3-D graphic dependence of G and E for these precipitation phases. For cubic Mg2Si and MgCu2 phases, the 3-D directional dependence of shear modulus G and Young's modulus E can be determined using the following equations [4,48]:

1/G = S44 + (4S11 − 4S12 − 2S44 )(l12 l22 + l12 l32 + l22 l32 ) 1/E = S11 − [(2S11 − 2S12 ) −

S44](l12 l22

+

l12 l32

+

l22 l32 )

direction cosine. If the 3-D graph of elastic modulus manifests a spherical shape, the crystal materials possess elastic isotropy. The degree of deviation from sphere connotes the magnitude of elastic anisotropy. The obtained 3-D graphic dependence of shear modulus G and Young's modulus E for these precipitation phases are shown in Fig. 3(a–f). Conspicuously, the 3-D graphic dependence of G and E for cubic Mg2Si is spherical shape in Fig. 3(a, b), suggesting that G and E for cubic Mg2Si are isotropic. For the cubic MgCu2 phase, the deviation degree of 3-D graphic dependence of G and E are very conspicuous in Fig. 3(c, d), so the elastic anisotropy of G and E for cubic MgCu2 is large. As shown in Fig. 3(e, f) for τ phase, the shear modulus G and Young's modulus E are anisotropic due to deviation of 3-D figuration from sphere. To illustrate the elastic anisotropy more explicitly, the projections of the 3-D graph of shear modulus G and Young's modulus E on xy-/yz-/xz-plane for these precipitation phases are shown in Fig. 3(g-l). It is clear that in Fig. 3(g, h) the projections of the 3-D direction dependence of G and E for Mg2Si phase is identical on xy-/ yz-/xz-plane, and on each atomic plane the G (E) for cubic Mg2Si phase demonstrate isotropy of elasticity [40]. As can be seen from Fig. 3(i, j), the projections of the 3-D direction dependence of G and E for MgCu2 phase are overlap on xy-/yz-/xz-plane, the significant and similar

(1) (2)

For hexagonal τ phase, the calculation formulae are as following [24]:

1/G = S44 + (1 − l32 )(S11 − S12 − 0.5S44 ) + (1 − l32 )(2S11 + 2S33 − 4S13 − 2S44 ) l32 2

(3) 2

1/E = S11 (1 − l32 ) + S33 l34 + (2S13 + S44 )(1 − l32 ) l3

(4)

where Sij (i,j=1,2,3,4) is the compliance matrices and li (i=1, 2, 3) is the

Fig. 3. The 3-D graphic dependence of G and E for Mg2Si (a, b), MgCu2 (c, d), τ phase (e, f), along with the projections of the 3-D directional dependence of G and E for Mg2Si (g, h), MgCu2 (i, j), τ phase (k, l) on xy-/yz-/xz-planes, the units are GPa.

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Fig. 4. The total and partial density of states for τ phase (a), Mg2Si (b) and MgCu2 (c).

Fig. 5. The differential charge densities in (11 2 0) plane for τ phase (a), (1 1 0) plane for Mg2Si (b) and MgCu2 (c).

tities such as specific heat (Cv), elastic constants (Cij) and thermal expansion (ɑ) are closely related with the Debye temperature [24,49]. The ΘD of these precipitation phases can be obtained by Eqs. (5)–(8) [20,24].

deviation from the circle is observed, so elastic anisotropy is analogous in three planes. As shown in Fig. 3(k, l) for τ phase, the isotropic for G and E occurs on xy-plane (red color), and the anisotropic on yz-/xzplanes (blue color) are conspicuous and analogous. From the above brittle/ductile and anisotropic analysis, Mg2Si possesses elastic isotropy although it is brittle, while MgCu2 has favorable ductility despite conspicuous anisotropy of elasticity.

Θ D = [3nNA ρ /4πM ]1/3 hνm / k −1/3

3.3. Debye temperature The Debye temperature (ΘD) is a momentous parameter of solid materials. It can characterize the bonding feature between atoms and reflect the thermodynamic properties of solids. Many physical quan-

(5)

νm = [(2/ νt3 + 1/ νl3)/3]

(6)

ν l = [(B + 4G/3)/ρ]1/2

(7)

νt = (G /ρ)1/2

(8)

where h, k, NA, M stand respectively for Planck's constant, Boltzmann's constant, Avogadro number and molecular mass, n denotes the number 344

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of atoms per conventional cell and ρ(=M/V) is the density, νm /νl /νt are respectively represent average/ longitudinal/ transverse sound velocity in polycrystalline material. The calculated results of νm, νt, νl and ΘD are listed in Table 5, together with the known data [17,20]. The obtained Debye temperature ΘD of τ phase is higher than one of MgCu2 but lower than one of Mg2Si, indicating the covalent bonding in τ phase is stronger than MgCu2 and weaker than Mg2Si because the ΘD can reflect the strength of covalent bonding in solid materials [48]. Furthermore, the higher ΘD for Mg2Si phase and τ phase showing better thermal stability of both. Due to the available data on the Debye temperature of τ phase is lacking in literature, current results require support of further research.

4. Conclusions The main reinforcement effects of precipitation phases Mg2Cu3Si, Mg2Si and MgCu2 on Mg-Cu-Si alloys are studied by predicting and comparing the phase stabilities and mechanical properties of these compounds from ab initio calculations. The excellent consistencies of the current results with the known data validate the reliability of the current calculation. The key research results are as following: (I). Mg2Cu3Si has the strongest alloying ability and structural stability among these precipitation phases. (II). Polycrystalline elastic modulus indicates that Mg2Cu3Si has the strongest resistance to reversible shear/volume distortion and also has the maximum hardness. (III). Brittle/ductile and anisotropy analysis manifest that MgCu2 has favorable ductility and Mg2Si has conspicuous elastic isotropy. (IV). Debye temperature reflects that Mg2Si and τ phase have better thermal stability. (V). Electronic structure reveals the strong Cu-Si and Si-Si covalent character in Mg2Cu3Si, the strong Mg-Mg metallic character and Cu-Cu covalent character in MgCu2, and the strong Mg-Si covalent bonding in Mg2Si. In summary, current work studies hardness, ductility and thermal stability of the key precipitation phases Mg2Cu3Si, Mg2Si and MgCu2, and gains the comprehensive insights into main reinforcement effects of these precipitation phases on MgCu-Si alloys, so it is beneficial for obtaining good-quality Mg-Cu-Si alloy by controlling the second precipitation phase Mg2Cu3Si, Mg2Si and MgCu2 during the heat treatment process.

3.4. Electronic structure The electronic structures are further studied to achieve an unbiased interpretation for micro-mechanisms of structural stability and mechanical properties of these precipitation phases. The acquired total and partial density of states (TDOS and PDOS) for τ phase, Mg2Si and MgCu2 are shown respectively in Fig. 4(a–c), in which the Fermi level (EF) is represented by zero energy, as marked by the vertical dashed lines. Conspicuously, the TDOS of τ phase and MgCu2 exhibits nonzero value at Fermi level, revealing the metallic feature of the two phases, while Mg2Si exhibits semiconducting behavior due to the existence of band-gap. The derived band-gap for Mg2Si is about 0.28 eV, deviating slightly from theoretical calculation of 0.16 eV using PBE method [50], the discrepancy of band-gap for Mg2Si may be owing to the different k-points [5]. In comparison with the experimental result of 0.77 eV [51], the underestimation of the theoretical value is attributed to the restriction of PAW potential and/or GGA approximation [5]. A pseudo-gap is observed near the Fermi level in the TDOS for τ phase and MgCu2 phase, which unambiguously distinguishes the bonding states and anti-bonding states, and also reveals the existence of covalent bonding in both [40]. The pseudo-gap of τ phase is more apparent than MgCu2. Furthermore, the Fermi level in τ phase is situated at the bottom of pseudo-gap, manifesting its higher stability [15]. It can be further observed from Fig. 4(a) and Fig. 4(c) that for τ phase and MgCu2, the main bonding peaks at low energy are mainly from Cu(3d) and Mg(2p) states. And the hybridization from Si(3p)Cu(4s)/Mg(2p) and Cu(4s)-Mg(2p) is very conspicuous at low energy for τ phase. For MgCu2, Mg(3s)-Cu(4s) hybridization is also observed. For Mg2Si in Fig. 4(b), hybridization from Mg(3s)-Si(3p) and Mg(2p)Si(3s) is obvious. Overall, the strong covalent bonding in τ phase and Mg2Si phase may reasonably reveal the brittleness [14]. To obtain more insight into the electronic characteristics of these precipitation phases, the differential charge densities in (11 2 0) plane for τ phase and in (1 1 0) plane for Mg2Si and MgCu2 phases are analyzed respectively in Fig. 5(a–c). The positive value in Fig. 5 corresponds to higher density and connotes charge accumulation, while negative value connotes charge depletion. From the differential charge densities of τ phase in Fig. 5(a), the Cu-Si and Si-Si bonding show covalent character due to the obvious accumulation of charge. Meanwhile, the obvious positive value of charge density around Cu (Si) and negative value around Mg indicate that the Mg-(Cu, Si) covalent bonding possesses somewhat ionic component. Conspicuously, in Fig. 5(b), the Mg-Si covalent bonding also exhibits certain ionic component. For MgCu2 phase in Fig. 5(c), the Cu-Cu bonding shows covalent character and Mg-Mg bonding exhibits obvious metallic character due to the relatively uniform charge distribution. Further, one can see that electron cloud distribution of Mg2Si on the whole plane is more uniform than other two phases, which may be the main reason of the elastic isotropy of Mg2Si. In short, the electronic structure can well reveal the underlying mechanism for stability and mechanical properties of these precipitation phases.

Acknowledgement This work was supported by the Natural Sciences Foundation of China under Grant no. 51461002 and No. 51561005. The financial supports from Open Project of Key Laboratory of New Electric Functional Materials of Guangxi Colleges and Universities are gratefully appreciated. References [1] E. Aghion, B. Bronfin, D. Eliezer, The role of the magnesium industry in protecting the environment, J. Mater. Process. Technol. 117 (2001) 381–385. [2] M. Dhahri, J.E. Masse, J.F. Mathieu, G. Barreau, M. Autric, Laser welding of AZ91 and WE43 magnesium alloys for automotive and aerospace industries, Adv. Eng. Mater. 3 (2001) 504–507. [3] A. Suzuki, N.D. Saddock, J.W. Jones, T.M. Pollock, Solidification paths and eutectic intermetallic phases in Mg-Al-Ca ternary alloys, Acta Mater. 53 (2005) 2823–2834. [4] X.D. Zhang, W. Jiang, Elastic, lattice dynamical, thermal stabilities and thermodynamic properties of BiF3-type Mg3RE compounds from first-principles calculations, J. Alloy. Compd. 663 (2016) 565–573. [5] Y.W. Cheng, D. Jiang, X.Q. Zeng, K.G. Wang, J. Lu, First-principles calculations of strengthening compounds in magnesium alloy: a general review, J. Mater. Sci. Technol. 32 (2016) 1222–1231. [6] S. Groh, Mechanical, thermal, and physical properties of Mg-Ca compounds in the framework of the modified embedded atom method, J. Mech. Behav. Biomed. 42 (2015) 88–99. [7] K.Q. Qiu, N.N. Hu, H.B. Zhang, W.H. Jiang, Y.L. Ren, P.K. Liaw, Mechanical properties and fracture mechanism of as-cast Mg77TM12Zn5Y6 (TM=Cu, Ni) bulk amorphous matrix composites, J. Alloy. Compd. 478 (2009) 419–422. [8] D.W. Zhou, J.S. Liu, S.H. Xu, P. Peng, First-principles investigation of the binary intermetallics in Mg-Al-Sr alloy: stability, elastic properties and electronic structure, Comput. Mater. Sci. 86 (2014) 24–29. [9] S. Ganeshan, S.L. Shang, Y. Wang, Z.K. Liu, Effect of alloying elements on the elastic properties of Mg from first-principles calculations, Acta Mater. 57 (2009) 3876–3884. [10] L.J. Zhou, K.H. Su, Y.L. Wang, Q.F. Zeng, Y.L. Li, First-principles study of the properties of Li, Al and Cd doped Mg alloys, J. Alloy. Compd. 596 (2014) 63–68. [11] M. Yuasa, N. Miyazawa, M. Hayashi, M. Mabuchi, Y. Chino, Effects of group II elements on the cold stretch formability of Mg-Zn alloys, Acta Mater. 83 (2015) 294–303. [12] A. Biswas, D.J. Siegel, D.N. Seidman, Compositional evolution of Q-phase precipitates in an aluminum alloy, Acta Mater. 75 (2014) 322–336. [13] J. Zhao, J. Zhou, S. Liu, Y. Du, S. Tang, Y. Yang, Phase diagram determination and thermodynamic modeling of the Cu-Mg-Si system, J. Min. Metall. B 52 (2016) 99–112. [14] D.W. Zhou, J.S. Liu, S.H. Xu, P. Peng, Thermal stability and elastic properties of Mg2X (X=Si, Ge, Sn, Pb) phases from first-principle calculations, Comput. Mater. Sci. 51 (2012) 409–414. [15] J. Zheng, X. Tian, L. Shao, X.Z. Pan, P.Y. Tang, B.Y. Tang, Point defects and Zn-

345

Physica B 521 (2017) 339–346

X.-F. Shi et al.

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24] [25] [26]

[27] [28] [29] [30]

[31]

[32] [33]

doping in defective Laves phase C15 MgCu2: a first-principles study, Comput. Mater. Sci. 122 (2016) 159–166. M.F. Ibrahim, E. Samuel, A.M. Samuel, A.M.A. Al-Ahmari, F.H. Samuel, Metallurgical parameters controlling the microstructure and hardness of Al-Si-CuMg base alloys, Mater. Des. 32 (2011) 2130–2142. Z.W. Huang, Y.H. Zhao, H. Hou, P.D. Han, Electronic structural, elastic properties and thermodynamics of Mg17Al12, Mg2Si and Al2Y phases from first-principles calculations, Physica B 407 (2012) 1075–1081. M.M. Wu, Y. Jiang, J.W. Wang, J. Wu, B.Y. Tang, L.M. Peng, W.J. Ding, Structural, elastic and electronic properties of Mg(Cu1−xZnx)2 alloys calculated by firstprinciples, J. Alloy. Compd. 509 (2011) 2885–2890. G. Murtaza, A. Sajid, M. Rizwan, Y. Takagiwa, H. Khachai, M. Jibran, R. Khenata, S.B. Omran, First principles study of Mg2X (X=Si, Ge, Sn, Pb): elastic, optoelectronic and thermoelectric properties, Mater. Sci. Semicond. Process 40 (2015) 429–435. Q. Chen, Z.W. Huang, Z.D. Zhao, D.Y. Shu, First principles study on elastic properties, thermodynamics and electronic structural of AB2 type phases in magnesium alloy, Solid State Commun. 162 (2013) 1–7. Y. Liu, W.C. Hu, D.J. Li, X.Q. Zeng, C.S. Xu, X.J. Yang, First-principles investigation of structural and electronic properties of MgCu2 Laves phase under pressure, Intermetallics 31 (2012) 257–263. E. Deligoz, K. Colakoglu, H. Ozisik, Y.O. Cifti, The first principles investigation of lattice dynamical and thermodynamical properties of Al2Ca and Al2Mg compounds in the cubic Laves structure, Comput. Mater. Sci. 68 (2013) 27–31. Y.Y. Kong, Y.H. Duan, L.S. Ma, R.Y. Li, Phase stability, elastic anisotropy and electronic structure of cubic MAl2 (M=Mg, Ca, Sr and Ba) Laves phases from firstprinciples calculations, Mater. Res. Express 3 (2016) 106505. L.S. Ma, Y.H. Duan, R.Y. Li, Structural, elastic and electronic properties of C14-type Al2M (M = Mg, Ca, Sr and Ba) Laves phases, Physica B 507 (2017) 147–155. J. Miettinen, G. Vassilev, Thermodynamic assessment of the Cu-Mg-Si system in its copper-rich region, Cryst. Res. Technol. 46 (2011) 1122–1130. F. Stein, M. Palm, G. Sauthoff, Structure and stability of Laves phases. Part I. Critical assessment of factors controlling Laves phase stability, Intermetallics 12 (2004) 713–720. S. Ganeshan, S.L. Shang, H. Zhang, Y. Wang, Elastic constants of binary Mg compounds from first-principles calculations, Intermetallics 17 (2009) 313–318. K. Momma, F. Izumi, Vesta3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Cryst. 44 (2011) 1272–1276. T. Bučko, Tkatchenko scheffler van der waals correction method with and without self-consistent screening applied to solids, Phys. Rev. B 87 (2013) 1664–1667. P. Shao, L.P. Ding, D.B. Luo, J.T. Cai, C. Lu, X.F. Huang, Structural, electronic and elastic properties of the shape memory alloy NbRu: first-principle investigations, J. Alloy. Compd. 695 (2017) 3024–3029. R.M. Shabara, S.H. Aly, A first-principles study of elastic, magnetic, and structural properties of PrX2 (X=Fe, Mn, Co) compounds, J. Magn. Magn. Mater. 423 (2017) 447–452. J.P. Perdew, K. Burke, M. Ernzerhof, Errata: generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868. H.M. Le, T.T. Pham, V.D. Dat, Y. Kawazoe, First-principles modeling of metal ferrocyanide: electronic property, magnetism, bulk moduli, and the role of C≡N-

defect, J. Phys. D. Appl. Phys. 50 (2017) 035004. [34] A.H. Larsen, M. Vanin, J.J.R. Mortensen, K.S. Thygesen, K.W. Jacobsen, Localized atomic basis set in the projector augmented wave method, Phys. Rev. B 80 (2013) 2665–2668. [35] R.A. Evarestov, V.P. Smirnov, Modification of the monkhorst-pack special points meshes in the brillouin zone for density functional theory and Hartree-Fock calculations, Phys. Rev. B 70 (2004) 155–163. [36] M. Kawamura, Y. Gohda, S. Tsuneyuki, Improved tetrahedron method for the brillouin-zone integration applicable to response functions, Phys. Rev. B 89 (2014) 82–84. [37] Y. Kubota, M. Takata, M. Sakata, T. Ohba, K. Kifune, T. Tadaki, A charge density study of the intermetallic compound MgCu2 by the maximum entropy method, J. Phys. Condens. Matter 12 (2000) 1253–1259. [38] C. Li, J.L. Hoe, P. Wu, Empirical correlation between melting temperature and cohesive energy of binary Laves phases, J. Phys. Chem. Solids 64 (2003) 201–212. [39] M.H. Braga, J.J.A. Fereira, J. Siewenie, T. Proffen, S.C. Vogel, L.L. Daemen, Neutron powder diffraction and first-principles computational studies of CuLix Mg2−x(x≅0.08), CuMg2, and Cu2Mg, J. Solid State Chem. 183 (2010) 10–19. [40] X. Hao, Y. Xu, Q. Gao, S. Liu, J. Wang, F. Gao, Elastic anisotropy of novel YAlC from an ab initio study, J. Alloy. Compd. 694 (2017) 982–988. [41] J.I. Tani, H. Kido, Lattice dynamics of Mg2Si and Mg2Ge compounds from firstprinciples calculations, Comput. Mater. Sci. 42 (2008) 531–536. [42] B. Zhang, L.L. Wu, B. Wan, J.W. Zhang, Z.H. Li, H.Y. Gou, Structural evolution, mechanical properties, and electronic structure of Al-Mg-Si compounds from first principles, J. Mater. Sci. 50 (2015) 6498–6509. [43] M. Shakil, M. Zafar, S. Ahmed, M.R.U.R. Hashmi, M.A. Choudhary, T. Iqbal, Theoretical calculations of structural, electronic, and elastic properties of CdSe1−xTex: a first principles study, Chin. Phys. B 25 (2016) 324–330. [44] A. Sari, G. Merad, H.S. Abdelkader, Ab initio calculations of structural, elastic and thermal properties of TiCr2 and (Ti,Mg)(Mg,Cr)2 Laves phases, Comput. Mater. Sci. 96 (2015) 348–353. [45] Z.Y. Zhang, H.Z. Zhang, H. Zhao, Z.S. Yu, L. He, J. Li, Prediction of the electronic structures, thermodynamic and mechanical properties in manganese doped magnesium-based alloys and their saturated hydrides based on density functional theory, J. Power Sources 280 (2015) 147–154. [46] L. Vitos, P.A. Korzhavyi, B. Johansson, Stainless steel optimization from quantum mechanical calculations, Nat. Mater. 2 (2003) 25–28. [47] Y. Liu, W.C. Hu, D.J. Li, K. Li, H.L. Jin, Y.X. Xu, C.S. Xu, X.Q. Zeng, Mechanical, electronic and thermodynamic properties of C14-type AMg2 (A=Ca, Sr and Ba) compounds from first principles calculations, Comput. Mater. Sci. 97 (2015) 75–85. [48] Y.H. Duan, Z.Y. Wu, B. Huang, S. Chen, Phase stability and anisotropic elastic properties of the Hf–Al intermetallics: a DFT calculation, Comput. Mater. Sci. 110 (2015) 10–19. [49] B.M. Ilyas, B.H. Elias, A theoretical study of perovskite CsXCl3 (X=Pb, Cd) within first principles calculations, Phys. B 510 (2017) 60–73. [50] T.D. Huan, V.N. Tuoc, N.B. Le, N.V. Minh, L.M. Woods, High-pressure phases of Mg2Si from first principles, Phys. Rev. B 93 (2016) 094109. [51] T. Kato, Y. Sago, H. Fujiwara, Optoelectronic properties of Mg2Si semiconducting layers with high absorption coefficients, J. Appl. Phys. 110 (2011) 063723.

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