Ab initio study of the elastic properties of sodium chloride at high pressure

ARTICLE IN PRESS Physica B 405 (2010) 2175–2180

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Ab initio study of the elastic properties of sodium chloride at high pressure Lei Liu a,, Yan Bi a,, Jian Xu a, Xiangrong Chen b a b

National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, P.O. Box 919-102, Mianyang, Sichuan 621900, PR China School of Physical Science and Technology, Sichuan University, Chengdu 610064, PR China

a r t i c l e in fo

abstract

Article history: Received 11 January 2010 Accepted 1 February 2010

The equation of state and elastic properties for B1- and B2-NaCl up to 160 GPa have been studied by using the density functional simulation within the generalized gradient approximation (GGA). The calculated lattice constants of NaCl agree well with experimental values in a precision of 0.1% over the pressure range studied. It is found that the cell volume decreases 5.5% at the phase transition point. All three independent elastic stiffness coefficients, c11, c12 and c44 for B1- and B2-NaCl are evaluated by a calculated stress tensor which was generated by forcing small strain to the optimized unit cell. The calculated zero-pressure elastic moduli, wave velocities, and their initial pressure dependences of B1-NaCl are in excellent agreement with experiments. Systematic investigation on the elasticity of NaCl has been done through four parameters, the Zener anisotropy ratio (AZ), the acoustic anisotropy factor (Aa), the Cauchy deviation (d), and the normalized elastic constants (cij0 ). With the pressure, the Zener anisotropy ratio AZ decreases in the B1-phase, but increases in the B2-phase and reaches 1 at about 174 GPa, it suggests that NaCl would become elastic isotropic at this pressure range. The acoustic anisotropy factor Aa shows the similar pressure behavior as AZ. The Cauchy deviation (d)) increases with pressures, it demonstrates that in the interatomic interaction, the many-body contribution becomes more important at higher pressures. A discussion on the normalized elastic constants is also presented. & 2010 Elsevier B.V. All rights reserved.

Keywords: Ab initio NaCl Elasticity High pressure

1. Introduction Sodium chloride is one of the most important materials in high-pressure research since it is one of the most widely used internal pressure standards [1–3] in in situ high pressure experiments due to its high compressibility and simple structure. However, since NaCl transforms from B1-pahse (NaCl-type structure) to B2-phase (CsCl-type structure) at about 30 GPa [4–6], the accurately determined equation of state (EOS) of B1NaCl is valid as a pressure gauge only below 30 GPa, which leads to a serial studies on the EOS of B2-phase very recently [7–11]. However, the discrepancies of the EOS of B2-NaCl are still found among these works. The 300 K isothermal curve of the B2-NaCl was measured by Sata et al. [7] (based on the Pt pressure scale by Holmes et al. [12] and the MgO pressure scale by Speziale et al. [13]), Ono et al. [8] (based on the Au pressure scale by Jamieson et al. [14] and Anderson et al. [15]) and Fei et al. [9] (based on the Pt pressure scale by Fei et al. [9]) recently. The laser-annealing technique had been used in all of these works to reduce any possible nonhydrostatic stress in the sample chamber. However, the discrepancies between the Au pressure scale proposed by Jamieson et al. [14] and Anderson et al. [15] increase with pressure and reach about 6 GPa at 135 GPa, which dramatically effecting

 Corresponding authors.

E-mail address: [email protected] (Y. Bi). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.02.001

the determination of the EOS of B2-NaCl. Further more, Fei et al. [9] considered that Pt scale of Holmes et al. [12] overestimated pressure by 4 GPa at 100 GPa at room temperature, which caused the discrepancies of B2-NaCl EOS between Sata and Fei’s results. It is known that the elastic constants and their pressure dependence provide insight into the nature of forces in the crystal. Cauchy deviation and the elastic anisotropy of cubic crystals have been of great interest from many investigators [16–18]. For cubic crystals, if all atoms occupy centrosymmetric positions and all interatomic interactions are central and pairwise, the Cauchy relation c11 c44 ¼ 2P holds. NaCl was deemed as a crystal satisfied the Cauchy relation well at ambient condition [19]. However, the Cauchy deviation has been observed by Kinoshita and Voronov from experimental determination [20,21]. Elastic anisotropy of crystal has been receiving a great deal of attention because the elastic anisotropy of Earth-forming minerals is one of the origins of the anisotropy of seismic wave in the Earth. Although experimental studies about the elasticity of NaCl has been performed [20–22], the pressure ranges of these experiments has been limited to a narrow area near-ambient pressure condition (o10 GPa, in the pressure range of B1-phase). Up to now, in our knowledge, no experimental results of the elasticity of B2-NaCl have been reported yet. We aim at two topics: to present the EOS of NaCl, both for B1- and B2-phase, and to investigate their elastic properties. For the latter purpose, the Zener anisotropy ratio, the acoustic anisotropy factor, the Cauchy deviation, and the normalized elastic constants are discussed.

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2. Computational method We have performed calculations of the electronic structure in the framework of the density functional theory (DFT) using a plane wave basis set with cutoff energy of 1000 eV. The electronion interactions are described by the Troullier–Martins norm-conserving pseudopotentials [23]. The effects of exchange-correlation interaction are treated within the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) with Wu-Cohen (WC) exchange [24]. Pseudo atomic calculations are performed for Na 2s22p63s1 and Cl 3s23p5. The Brillouin zones are sampled with 13  13  13 Monkhorst–Pack mesh, where the self-consistent convergence of the total energy is 5  10  7 eV/atom. We determine the equilibrium structure of the unit cell by minimizing the Hellman–Feynman forces and stress acting on nuclei and the lattice parameters respectively at each given pressure. During the geometrical optimization, the total stress tensor is reduced to the order of 0.01 GPa. All the total energy electronic structure calculations are implemented with the CASTEP code [25,26]. According to Hooke’s law, the elastic stiffness tensor cijkl, the stress tensor sij, and the strain tensor ekl for small deformations to a crystal are linearly related by:

sij ¼ cijkl ekl ði; j; k; l ¼ x; y; zÞ

ð1Þ

The cubic crystal has three elastic components: c11  cxxxx ; c12  cxxyy ; c44  cyzyz ðin Vogit notationÞ

ð2Þ

These elastic constants can be determined by calculating the stress response which was generated by forcing a small strain to ! the optimized unit cell [16,27]. The strained lattice vector a0i and !0 ! ! the optimized lattice vector ai are related by ai ¼ ðI þ eÞ ai , where I is identity matrix and e is the strain tensor as following: 0

e

e¼B @ e=2 0

e=2 0 0

0

1

C 0A

ð3Þ

0

So c11 , c12 and c44 are determined by Eq. (1) as following:

sxx ¼ c11 e; syy ¼ szz ¼ c12 e; syz ¼ c44 e

3.2. Elastic moduli

each pressure and fitted to a parabolic function of e to remove non-linear contribution. The bulk modulus of cubic crystal with an isotropic polycrystalline aggregate is related to the elastic constants by c11 þ 2c12 3

scale in order to avoid the affect of Au pressure gauge and illustrated them in Fig. 1. Our P–V curve of B2-phase is fairly consistent with the experimental results mentioned above. A series of theoretical calculations on the EOS of NaCl by different methods yields the bulk modulus of B1-phase from 21.44 to 28.50 [1,2,5,10,11,31,35], and from 24.4 to 38.96 [7–11,33–35] of B2-phase. These results are summarized in Table 1. A 5.5% volume reduction has been obtained in our calculations at the phase transition boundary (31 GPa). Comparing to Hofmeister’s EOS data, Sata et al. found the volume reduction is 4.6% and Ono’s value is 2.2%. When comparing to Brown’s B1-phase EOS data, Sata et al.’s value is 3.5% and the re-estimated Ono et al.’s value is 2.9%, all slightly smaller than that of Bassett et al. [36] (5.7%) which is very consistent with ours.

ð4Þ

sij are calculated with e=  0.02,  0.00667, 0.00667 and 0.02 at

K¼

Fig. 1. The equation of state for NaCl. The solid curve represents the 3rd Birch–Murnaghan fit to the calculated pressure–volume data in this work.

ð5Þ

3. Results and discussion 3.1. Equation of state Fig. 1 shows our calculated volume–pressure curve compared with previous experimental and theoretical results [3,7–9]. Brown [3] recalibrated the NaCl pressure scale after the work of Decker [1,28,29]. Our calculated P–V curve of B1-phase agrees very well with Brown’s 300 K results as shown in Fig. 1. Very recently, Takemura and Dewaele[30] checked the stress state in the sample chamber very carefully and proposed a new Au pressure scale. Thus, we re-estimate Ono et al.’s results based on this new Au

Fig. 2a shows the pressure dependence of the calculated elastic stiffness coefficients, c11, c12 and c44 of NaCl. The c11 decreases when the phase transformed from B1 to B2 at the transition pressure while c12 and c44 increase. All the elastic stiffness coefficients increase with pressures except c44 of B1-phase, which has a slightly negative pressure dependence. The downward trend in c44 has been theoretically predicted by Cowley et al. [31] and Srinivasa et al. [37] respectively, but has not been observed by experiments. All the experiment results indicated that c44 increased with pressure. c12 and c44 increase more rapidly at the phase transition while c11 decreases obviously. The experiment results of Kinoshita et al. [20], Voronov et al. [21] and Bassett et al.[22] are illustrated in Fig. 2b. Our results agree well with the results of Kinoshita et al. and Voronov et al. However, Bassett et al. gave larger c12 and c44 but smaller c11. Bulk modulus K change at the B1 B2-phase transition has confused researchers for many years. In 1997, Hofmeister [5] suggested that the bulk modulus of NaCl decreases by 1673% based on observed principal infrared frequencies. However, previous results, in theoretical [33,34,38] and in experimental studies [7,8], indicate a small increase in K at the phase transition. Sim et al. [38] discussed the difference between Hofmeister and others, they attributed the discrepancy into the second-neighbor effect which was not included in the model used in IR experiment.

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Table 1 Zero-pressure lattice constant, bulk modulus (K0) and its pressure derivative ðK00 Þ of B1- and B2-phase of NaCl. This study

B1-Phase (Fm3-m)

This study* a

Cowley et al. LD * Cowley et al. MDa* Cohen et al.b Ueda et al.c Ueda et al.c* Bukowinski et al.d Apra et al.e Penda´s et al.f Ono et al. LDAg Ono et al. GGAg Ono et al. AIMDg Brownh Hofmeisteri Cowley et al.j* Decker et al.k Birch et al.l Heinz et al.m Fei et al.n* Sata et al.o** Ono et al.p**

K00

a

K0

5.636

24.94

5.640

23.74

5.27

Theory 5.646 21.44 5.650 21.65 5.640 28.50

5.91 5.86

5.429 5.670 5.645

34.1 24.2 24.0

Experiment 5.634 25.87 5.640 23.8 5.618 27.05 5.640 24.2 23.88

4.71 4.72 4.95

B2-Phase (Pm3-m) a

K0

3.458

28.11

3.413

33.22

K00

5.11

3.452 3.452 3.458 3.410 3.435 3.345 3.483

28.45 30.25(99) 26.6 24.4–35.6 25.6 38.96 29.12

4.45 4.42

3.469 3.458 3.442 3.444

36.2(42) 30.69(29) 35.40 30.62

4(fixed) 4.33(26) 4(fixed) 5.08

2177

where r is the density of NaCl. Fig. 4 is a plot of calculated VP and VS vs. pressure relationship in comparison with experiment data. The discrepancy between present study and the experiment data [39] is 1.1% for VP and 3.5% for VS at about 27 GPa, respectively. At the phase transition point, VP increases by 4.6% (VP  B1 = 7.52 km/s, VP  B2 =7.87 km/s), while VS increases by 18.5% (VS  B2 = 3.32 km/s, VS  B2 = 3.94 km/s). 3.4. Elastic anisotropy The elastic anisotropy of a cubic crystal can be characterized by the Zener anisotropy ratio AZ, which is the ratio of shear moduli in the (1 0 0) and (1 1 0) planes at the [1 0 0] direction:

5.16 4.83(7) 5.2

AZ ¼

c44 ðc11 c12 Þ=2

ð8Þ

The ratio AZ becomes unity for isotropic crystal. Fig. 5a shows the pressure dependence of Zener anisotropy ratio AZ. The calculated value of AZ of B1-NaCl has very large negative value, whose value is  0.55 at zero pressure and reaches  0.06 at 31 GPa. It indicates that NaCl becomes more anisotropic with pressures for B1-phase. However, the anisotropy of NaCl drastically decreases at the transition pressure. The ratio AZ has positive pressure dependence for B2-NaCl. The value of AZ is 0.45 at 31 GPa and increases to 0.95 at 160 GPa, demonstrating that B2-NaCl becomes more and more isotropic. Extrapolating AZ to higher pressure, we found the value of AZ would be unity at about 174 GPa for B2-NaCl, suggesting that B2-NaCl is an absolute isotropic solid at 174 GPa. For comparison, experimental results are also illustrated in Fig. 5. The experimental results also had negative pressure dependence for B1-NaCl. Our calculations show larger anisotropy of B1-NaCl than the experimental data from Kinoshita and Voronov, because of the smaller c12 and c44 we calculated. Combining Hooke’s law with Newton’s second law of motion, the well-known Christoffel equation which describes the wave velocities of single crystal in different directions is derived [40]:

4.54 5.10 4.76 4.71 5.2

*Least-squaes fitted to Vinet EOS [42] by EosFit V5.2 [43]. **Weighted Least-squaes fitted [44] to Vinet EOS by EosFit V5.2. a

Ref. [31]. Ref. [32]. c Ref. [10]. d Ref. [33]. e Ref. [34]. f Ref. [35]. g Ref. [11]. h Ref. [3]. i Ref. [5]. j Ref. [31]. k Ref. [1]. l Ref. [2]. m Ref. [45]. n Ref. [9]. o Ref. [7]. p Ref. [8]. b

!! detðcijkl nj nl rV 2 dik Þ ¼ 0

The bulk moduli of NaCl obtained in our present work are shown in Fig. 3a. In the present work, the bulk modulus are determined of 141.99 and 145.06 GPa of B1- and B2-phases, respectively at 30 GPa, and the latter one has larger about 2%. It is in good consistent with the results from Sim et al. and Sata et al.’s [7]. Bulk moduli of B1-phase in this study is also consistent with ultrasonic results [5] very well although Sata et al. suggested steeper pressure dependence of the B2-phase than this study (Fig. 3).

ð9Þ

! Where cijkl is the single crystal elastic constant tensor, n is the unit vector of the propagation direction, r is the density, V is the velocity, and dik is the Kroenecker symbol. The elastic wave anisotropy is characterized by the ratio of two longitudinal elastic wave velocities (VP) propagating along [1 1 1] and [1 0 0], 2 V½1 1 1 2 V½1 0 0

¼

2Aa þ 1 3

ð10Þ

Where Aa is acoustic anisotropy factor. Aa becomes unified in the isotropic material. For cubic crystal 2c44 þ c12 c11

3.3. Isotropic wave velocities

Aa ¼

The longitudinal and shear-wave velocities, VP and VS, of NaCl are calculated respectively as following:

Acoustic anisotropy factor Aa as a function of the pressure is plotted in Fig. 5b for both B1- and B2-type NaCl. The value of Aa has the similar pressure behavior with AZ: decreasing from 0.64 at zero pressure to 0.20 at 31 GPa, and then increasing from 0.63 at 31 GPa to 0.98 at 160 GPa. However, the value of Aa is lager than AZ at any pressure. Both parameters demonstrate that B1-NaCl is elastic anisotropy at ambient pressure and the elastic anisotropy increases with pressure, however, B2-NaCl would be isotropic at high pressures.

VP ¼

VS ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3K þ 4G 3r sffiffiffiffi G

r

ð6Þ

ð7Þ

ð11Þ

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Fig. 2. Pressure dependence of the elastic stiffness coefficients, c11, c12 and c44 of NaCl.

Fig. 3. Pressure dependence of bulk modulus K of NaCl.

Fig. 4. Pressure dependence of isotropic wave velocities, VP and VS of NaCl.

3.5. Cauchy relation Cauchy relation c12 c44 ¼ 2P

ð12Þ

is valid only when a crystal is constructed by the central force, in which all atoms are in the center of symmetry. Present work indicates c12 c44 ¼ 0:55 GPa at zero pressure, in comparison with the experimental values of  1.4 GPa by Kinoshita et al. [20] and 1.4 GPa by Voronov et al. [21]. These results are small and lead an assumption that all interatomic forces would be almost central for NaCl at zero pressure. However, the deviation from Cauchy relation has been found at high pressure, the deviation d ¼ c12 c44 2P increases with pressure and reaches about  100 at 160 GPa as shown in Fig. 6, which indicates that the non-central component from the many-body force becomes more and more important at higher pressures. For a typical ionic crystal, the

origin of the many-body force should be attributed the charge transfer energy [17]. It is interesting that the deviation decreases dramatically across the transition point. This decrease would be caused by the coordination changes from 6 of B1-phase to 8 of B2phase, which may lead to the decrease of charge transfer potential barrier from B1-phase to B2-phase of NaCl. Present theoretical prediction of deviation from Cauchy relation is in good consistent with experimental results by Kinoshita et al. and Voronov et al. below 10 GPa.

3.6. Normalized elastic constants In order to investigate the elasticity of transition metal in more detail, Fast et al. [41] introduced normalized elastic constants cij0 and Tsuchiya et al. [17] extended this concept to high pressure. cij0 are defined by dividing a specific elastic constant with the bulk

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Fig. 5. Pressure dependence of the elastic anisotropy of NaCl.

Fig. 7. Pressure dependence of normalized elastic constants Cij0 of NaCl.

Fig. 6. Pressure dependence of the Cauchy violation in NaCl.

4. Conclusion

modulus as cij0 ¼

cij 3cij ¼ K c11 þ2c12

ð13Þ

Then the interatomic forces are normalized with an average restoring force of the system. In an elastically ideal cubic crystal, 0 0 ¼ 1:8, c12 ¼ 0:6 viz., the isotropic Cauchy solid, conditions that c11 0 0 and c44 ¼ 0:6 are satisfied at zero pressure. The values of cij of NaCl as a function of pressure are plotted in Fig. 7. The values of 0 0 0 ¼ 2:14, c12 ¼ 0:43, and c44 ¼ 0:46 are found at zero pressure for c11 0 B1-NaCl, which are significantly deviate from the ideal values. c12 0 and c44 have negative pressure dependence while the pressure 0 is positive, which is because that c11 has the dependence of c11 0 is most sensitive to pressure largest pressure derivative. c44 because that c44 decreases with pressure while K increases with pressure in the B1-phase pressure range. For B2-NaCl, the pressure dependences of cij0 are opposite to those of B1-phase, respectively.

In present study, ab initio plane-wave pseudopotential density functional theory method was used to investigate the EOS and elasticity of B1- and B2-NaCl under high pressure. A 5.5% volume reduction was found at the phase boundary. The calculated EOS of B1- and B2-NaCl were fairly consistent with previous results. It is interesting that the Zener anisotropy ratio AZ of B2-NaCl approached unity at high pressure and reached 1 at 174 GPa, indicating that B2-NaCl was elastic isotropic at this pressure. The deviation of the Cauchy relation increased with pressure. This was because many-body effect of NaCl played more and more important role at high pressure.

Acknowledgments This work was supported by the Foundation of Laboratory for Shock Wave and Detonation Physics Research under Grant nos.

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