Ab initio calculations on the third-order elastic constants for selected B2–MgRE (RE = Y, Tb, Dy, Nd) intermetallics

Intermetallics 18 (2010) 2472e2476

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Ab initio calculations on the third-order elastic constants for selected B2eMgRE (RE ¼ Y, Tb, Dy, Nd) intermetallicsq Rui Wang*, Shaofeng Wang, Xiaozhi Wu, Yin Yao, Anping Liu Institute for Structure and Function and department of physics, Chongqing University, Chongqing 400044, People’s Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 June 2010 Received in revised form 19 August 2010 Accepted 20 August 2010 Available online 25 September 2010

We present the third-order elastic constants for the magnesium-rare-earth alloys MgY, MgTb, MgDy, and MgNd with CsCl-type B2 structure. Density functional theory (DFT) within generalized-gradientapproximation (GGA) combining with the method of homogeneous deformation is employed. The predictions for the elastic constants are obtained from the nonlinear least-squares polynomial fit to the calculated straineenergy relation from first-principles total-energy calculations. To judge that our computational accuracy is reasonable, we compare the ab initio calculated results for the third elastic constants with experimental data and previous theoretical results for Si. Our calculated second-order elastic constants for the selected magnesium-rare-earth alloys agree well the previous calculations and the experiments. The third-order effects really matter when the finite strains are larger than approximately 2.5%. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: A. Rare-earth intermetallics B. Elastic properties E. Ab-initio calculations

1. Introduction Magnesium alloys are being actively developed for structural applications due to their light weight and high strength [1], and are especially attractive for aeronautical and automotive industry applications. However, magnesium alloys often poor strength at high temperature, which is a very serious problem in application of magnesium alloys. The development and design of new magnesium alloys need more fundamental physical data of elements and alloys. Recently, it has been reported that the MgRE (where RE indicates a rare-earth element) have good strength at high temperature [2e4]. The intermetallic compounds MgRE with CsCltype B2 structure are extremely attractive structural materials for applications in automobile parts, electric appliances and aerospace industries. So various studies have been undertaken of the magnetic properties, elastic and thermodynamic properties for some rare-earth-magnesium B2-MgRE intermetallics [5e15]. In comparison with the ductile rare-earth intermetallic compounds YAg and YCu, the previous first-principles calculations concluded that the B2eMgRE alloys have brittle behavior [7]. q The work is supported by the National Natural Science Foundation of China (10774196) and Fundamental Research Funds for the Central Universities (CDJRC10100003). * Corresponding author. Tel.: þ8613527528737. E-mail address: [email protected] (R. Wang). 0966-9795/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.08.039

For practical applications, applying finite deformation to materials is also very important due to the intermetallics MgRE with superior mechanical properties. Nonlinear elasticity theory is the theory of elasticity under finite deformations [16e19]. In the linear theory of elasticity, the infinitesimal deformations are assumed, and the second-order elastic constants (SOECs) are sufficient to describe the elastic stress-strain response and wave propagation in solids [20]. In nonlinear elastic theory, high-order elastic constants, such as third-order elastic constants (TOECs), play an important role as well as SOECs [21]. TOECs are not only used in describing mechanical phenomena when applying large stress or strain, but also can be used to describe the anharmonic properties such as thermal expansion, the interaction of acoustic and thermal phonon, changes in acoustic velocities due to elastic strain, etc [16,22,23]. In theoretical calculations, it is difficult to obtain TOECs and there are quite a few methods available, including molecular-dynamics simulation [24,25], empirical interatomic force-constant model [26,27], and the first-principles quantum mechanics calculations [28,29]. Recently, the methods of first-principles calculations have been employed to determined TOECs through applying series of finite deformations [30e32], and their results show good agreement with experiments. To understand physical properties of these compounds and providing significant information with respect to application and design of Mg-rare-earth alloys, it is necessary to study further the nonlinear elastic constants of these compounds.

R. Wang et al. / Intermetallics 18 (2010) 2472e2476

2. Methods of homogeneous deformation and ab initio calculations

2473

−6.2565

−6.2566

The relation between the elastic constants and the straineenergy can be written as

ijkl

(1)

ijklmn

Here the nth-order (n  2) elastic constants was defined by Brugger [19] as

  v3 U  Cijklmn. ¼ CIJK. ¼ r0 h ¼ 0 i;j;k;/ ¼ 1;2;3 vhij vhkl vhmn /

−6.2566

Internal energy (eV)

1X 1 X r0 DUðhÞ ¼ Cijkl hij hkl þ Cijklmn hij hkl hmn þ .; 2! 3!

! 1 X Jik Jjk  dij ; 2

−6.2567

−6.2568

(2) −6.2568

where I, J, and K are Voigt subscripts; r0 is the density of the unstrained crystal; U is the energy of the crystal per unit undeformed; hij is Lagrangian strain tensor, which is defined as [21]

hij ¼

−6.2567

(3)

−6.2569

3.782

3.784

3.786

3.788

3.79

3.792

3.794

3.796

Lattice constant (A) Fig. 1. The equilibrium lattice parameter for MgTb is found to be 3.789 Å, which is determined from the corresponding minimum value of the internal energy.

k

and the deformation tensor Jij ¼ vxi =vaj relates the initial and final coordinate xi and ai of infinitesimal element in the deformed solid. The Lagrangian strain tensor h links the Voigt notation by

0

h1

h h¼ B @ 26 h5 2

h6 2

h2 h4 2

h5 2

1

h4 C 2

A

(4)

h3

Because of B2-MgRE (RE ¼ Y, Tb, Dy, Nd) intermetallics with cubic symmetry, there are three independent SOECs (C11, C12, C44) and six TOECs (C111, C112, C123, C144, C155, C456). As shown below, we will obtain the elastic constants in first-principles calculations by applying to the bulk-crystal six simple deformation strains. To obtain a solvable system for the TOECs, the number of applied strain tensors must be as large as the number of independent TOECs. Hence, we consider six sets of deformations:

1

0

1

0

1

0

x 00 x 00 x00 hA ¼ @0 0 0 A; hB ¼ @ 0 x 0 A; hC ¼ @ 0 x 0 A; 000 000 00 x 1 0 0 x x1 0 1 x x00 0 x02 2 2 xC B C hD ¼ B @ 0 0 2 A; hE ¼ @ 0 0 0 A; hF ¼ @ 2x 0 2x A: x

0 2x 0

2 00

x x

2 2

(5)

0

The corresponding elastic energy on deformation parameter x for considered each type of strain modes, ha , a ¼ A, B, ., F, can be expressed as

1 2

1 6

r0 DUðhA Þ ¼ C11 x2 þ C111 x3 ¼ fA ðxÞ; 



In every case forha, x is varied between 0.08 and 0.08 with step 0.008 to obtain accurate TOECs. The elastic constants will be obtained from the least-square polynomial fit to the straineenergy relation from first-principles total-energy calculations. This method was developed by LePage and Saxe [33]. In order to obtain the unit cell of strained crystal, the deformation tensor Jij is applied to the undeformed lattice vectors ai, where i is the lattice index. The P deformed crystal is obtained then from a0i ¼ Jij aj . To implement j the different deformation modes in our calculation, we need to have the deformation tensor Jij, which is determined from the Lagrangian strain by inverting Eq. (3),

Jij ¼ dij þ hij 

1X 1X hik hkj þ hik hkl hlj þ / 2 2 k

(7)

kl

For a given h, in general, J is not unique but this is not a problem since the Lagrange strain brings rotational invariance. We carry out first-principles total-energy calculations based on the density functional theory (DFT) level, using the Vienna ab initio simulation package (VASP 4.6) developed at the Institut für Materialphysik of Universität Wien [34e36]. The Perdew-Burke-Ernzerhof (PBE) [37,38] exchange-correlation functional for the generalized-gradient-approximation (GGA) is used. A plane-wave basis set is employed within the framework of the projector augmented wave (PAW) method [39,40]. For the Billouin zone integrals, reciprocal space is represented by Monkhorst-Pack special k-point scheme [41]; in Si, we have used 15  15  15 grid meshes, while for MgRE (RE ¼ Y, Tb, Dy, Nd), we have applied 21  21  21 sampling. We took the cutoff energy set at 600 eV for MgRE (RE ¼ Y, Tb, Dy, Nd) and Si. The equilibrium theoretical crystal structures are determined by minimizing the Hellmann-Feynman

r0 DUðhB Þ ¼ ðC11 þC12 Þx2 þ 13C111 þC112 x3 ¼ fB ðxÞ; r0 DUðhC Þ ¼



3C þ3C 12 2 11



x2 þ



1C 2 111 þ3C112 þC123

!

!

!

!



x3 ¼ fC ðxÞ;

1 1 1 1 x2 þ C111 þ C144 x3 ¼ fD ðxÞ; C þ C 2 11 2 44 6 2

r0 DUðhD Þ ¼

MgY MgTb MgDy MgNd

1 1 1 1 r0 DUðhE Þ ¼ C11 þ C44 x2 þ C111 þ C155 x3 ¼ fE ðxÞ; 2 2 6 2 3 2

r0 DUðhF Þ ¼ C44 x2 þC456 x3 ¼ fF ðxÞ;

Table 1 The equilibrium lattice constants for B2eMgRE (RE ¼ Y, Tb, Dy, Nd) in our calculation comparison with the previous study and the experiment. The unit of all data is Å.

(6)

a b

Reference [7]. Reference [42].

This work

Previous study

Experiment

3.795 3.789 3.778 3.882

3.796a 3.781a 3.765a 3.872a

3.796b 3.781b 3.759b 3.860b

2474

R. Wang et al. / Intermetallics 18 (2010) 2472e2476

a −100

b

C

−100

C112

112

C144

C144

−110

TOECs (GPa)

TOECs (GPa)

−110

−120

−130

−120

−130

−140

−140 −150

−150 −160

−160

300

400

500

600

700

10

800

12

14

16

18

20

22

24

26

28

K−point grid size

Energy cutoff (eV)

Fig. 2. Sample convergence tests for the TOECs in MgY. (a) The dependence of the TOECs C112 and C144 on the cutoff energy (Monkhorst-Pack sampling 21  21  21 is applied for all points.) (b) The dependence of the TOECs C112 and C144 on the Monkhost-Pack k-points grid size with EMgY cutoff ¼ 600 eV. The relative fluctuation between two successive values of examined constants in our test with EMgY cutoff ¼ 600 eV and 21  21  21 k-point mesh is lower than 0.5 GPa.

force on the atoms and the stress on the unit cell. The convergence of energy and force are set to 1.0  106 eV and 1.0  103 eV/Å, respectively. For example, The calculated lattice of the MgTb at the equilibrium state is shown in Fig. 1. In Table 1, we give the equilibrium lattice constants for MgRE (RE ¼ Y, Tb, Dy, Nd) in our calculation, and it is shown that the results agree well with the previous calculation [7] and experiment [42]. The convergence test for the calculated elastic constants C112 and C144 of MgY is shown in Fig. 2. For the chosen parameters (EMgY cutoff ¼ 600 eV and 21  21  21 k-point mesh) in our calculations, the fluctuation between successive values of examined constants is lower than 0.5 GPa. 3. Results and discussion We present our calculated results in Tables 2 and 3. Table 2 contains our finding for benchmark material Si, which is a wellstudied cubic crystal having the diamond structure, accompanied by available experimental data and previous theoretical findings within DFT theory. In the previous calculations, the all third-order elastic constants of Si were obtained. Overall, our ab initio results agree well with the previous calculations and the experimental data for both SOECs and TOECs. Table 3 gives our prediction for the unknown values of TOECs for B2eMgRE (RE ¼ Y, Tb, Dy, Nd). For completeness, we also provide there our prediction for secondorder elastic moduli and compare them with previous calculations

Table 2 Comparison of the calculated SOECs and TOECs for Si with the previous theoretical values and experimental results. The unit of all data is GPa.

C11 C12 C44 C111 C112 C144 C155 C123 C456 a b c d e

Reference Reference Reference Reference Reference

This work

Previous study

Experiment

153.94 62.21 75.20 734 443 18 305 68 72

153a, 162.07b, 159c 65a, 63.51b, 61c 73a, 77.26b, 85c 698a, 810b, 750c 451a, 422b, 480c 74a, 31b 253a, 293b 112a, 61b, 480c 61a, 57b, 80c

165.77d, 165.64e 63.92d, 63.92e 79.62d, 79.62e 825d, 795e 451d, 445e 12d, 15e 310d, 310e 64d, 75e 64d, 86e

[30]. [31]. [28]. [43]. [44].

and experiment. It is found that our values of SOECs provide extremely good agreement with the results obtained from the previous calculations and experiments. The strain-energies for MgY, including the results of the first-principles calculations and the fitted polynomials, are shown Fig. 3. It shows that the strainenergies with negative strains are always larger than those with positive strains, so the values of TOECs are typically negative. It is worth noting that for the studied intermetallics with Lagrangian strains up to 8.0%, including the terms up to third-order in energy expansion sufficed to obtain good agreement with our ab initio results. Next we focus on examining for which range of strains the third-order effects dominant the properties of solids. In Fig. 4, we also show the curves of the linear elasticity comparison with the nonlinear elasticity and the DFT results particular deformation hA in MgY. It is clear to see that the third-order effects must be considered and it is not sufficient for linear elasticity when applied to finite deformations larger than approximately 2.5%. Table 3 Predictions for the third-order elastic constants (TOECs) of B2eMgRE (RE ¼ Y, Tb, Dy, Nd). We compare the second-order elastic constants (SOECs) with the previous theoretical values. The unit of all data is GPa. This work C11 C12 C44 C111 C112 C144 C155 C123 C456

MgY 52.17 36.66 38.87 450.42 115.76 146.76 172.21 80.62 124.32

C11 C12 C44 C111 C112 C144 C155 C123 C456

MgDY 50.64 37.27 38.07 511.26 122.33 126.80 179.28 77.68 146.26

a b

Reference [7]. Reference [8].

Previous study 53.37a, 53.07b 36.39a, 36.10b 39.05a, 39.26b

51.86a, 51.85b 37.57a, 36.85b 38.35a, 39.74b

This work MgTb 53.13 36.01 39.45 514.00 121.45 115.03 186.15 68.65 143.82 MgNd 52.09 31.21 38.49 420.71 121.04 92.96 167.44 64.63 132.00

Previous study 53.32a, 52.46b 36.18a, 36.20b 39.82a, 39.95b

51.73a, 50.83b 32.15a, 31.99b 38.69a, 37.65b

R. Wang et al. / Intermetallics 18 (2010) 2472e2476

a

b

8

8

x 10

fA(ξ)

7

2475

8

18

x 10

16

f (ξ)

14

fE(ξ)

12

f (ξ)

C

f (ξ) B

6

Energy (Jm−3)

−3

Energy (Jm )

fD(ξ) 5

4

3

F

10 8 6

2 4

1

2

0 −0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0 −0.08

−0.06

Lagrangian Strain ξ

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Lagrangian Strain ξ

Fig. 3. The straineenergy relations for MgY. The discrete points denote the DFT results; the solid lines represent the curves fitted by third-order polynomial function in Eq. (6) for definitions of fA(x), fB(x), ., fF(x).

deformations applied are larger than approximately 2.5%. It is also worth noting that for the studied intermetallics and examined range of deformations, including the terms up to third-order in energy expansion sufficed to obtain good agreement with our ab initio results. We believe that our ab initio results for the predictions of TOECs can be a very useful tool in applying these compounds to practical engineering, in which third-order effects and nonlinear elasticity often really matter. In experiments, the unknown TOECs of B2eMgRE can be determined from wave velocity with small amplitude in statically stressed media.

8

2.5

x 10

DFT results nonlinear elasticity

2

Energy (Jm−3)

linear elasticity 1.5

1

References

0.5

0 −0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Lagrangian strain ξ Fig. 4. Energy as a function of Lagrangian strain parameter x for particular deformation Eq. (15) for MgY. Full points denote results of DFT computations; solid and dashed lines indicate the curves obtained from nonlinear and linear elasticity theory, respectively.

4. Conclusions In this work, we have computed the second- and third-order elastic constants for intermetallics MgRE (RE ¼ Y, Tb, Dy, Nd) with B2-type structure, using the density functional theory (DFT) within generalized-gradient-approximation (GGA) and homogeneous deformation method. From the nonlinear fitting, we obtained the predictions for SOECs and TOECs from the coefficients of the fitted polynomials of the internal energy-strain functions. To benchmark the reliability results of the presented method, we have compared our theoretical results for Si with available experimental and calculated data. Our theoretical results for SOECs in B2eMgRE are in excellent agreement with experimental and previous ab initio results. Comparison with linear elasticity, our work shows that the third-order effects must be considered when the finite

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