Characterization of the High-Pressure Structural Transition and

13898

J. Phys. Chem. B 2008, 112, 13898–13905

Characterization of the High-Pressure Structural Transition and Thermodynamic Properties in Sodium Chloride: A Computational Investigation on the Basis of the Density Functional Theory Cheng Lu,† Xiao-Yu Kuang,*,†,‡ and Qin-Sheng Zhu§ Institute of Atomic and Molecular Physics, Sichuan UniVersity, Chengdu 610065, China; International Centre for Materials Physics, Academia Sinice, Shenyang 110016, China; and Department of Applied Physics, UniVersity of Electronic Science and Technology of China, Chengdu 610054, China ReceiVed: July 6, 2008; ReVised Manuscript ReceiVed: August 24, 2008

Using first-principles calculations, the elastic constants, the thermodynamic properties, and the structural phase transition between the B1 (rocksalt) and the B2 (cesium chloride) phases of NaCl are investigated by means of the pseudopotential plane-waves method. The calculations are performed within the generalized gradient approximation to density functional theory with the Perdew-Burke-Ernzerhof exchange-correlation functional. On the basis of the third-order Birch-Murnaghan equation of states, the transition pressure Pt between the B1 phase and the B2 phase of NaCl is determined. The calculated values are generally speaking in good agreement with experiments and with similar theoretical calculations. From the theoretical calculations, the shear modulus, Young’s modulus, rigidity modulus, and Poisson’s ratio of NaCl are derived. According to the quasi-harmonic Debye model, we estimated the Debye temperature of NaCl from the average sound velocity. Moreover, the pressure derivatives of elastic constants, ∂C11/∂P, ∂C12/∂P, ∂C44/∂P, ∂S11/∂P, ∂S12/∂P, and ∂S44/∂P, for NaCl crystal are investigated for the first time. This is a quantitative theoretical prediction of the elastic and thermodynamic properties of NaCl, and it still awaits experimental confirmation. 1. Introduction With the rapid development of high-pressure techniques, there has been considerable interest in recent years in the investigations of the structural phase transition and high-pressure behavior in material because the knowledge of physical and chemical properties of material at high pressure is important for a variety of scientific and technological applications.1-7 For instance, the optical properties, the thermodynamic properties, and the equation of state (EOS) of mineral material at high pressure is essential in the investigations of the interior of the earth in geophysics and for the investigations of the evolutions of stars in astrophysics. NaCl is one of the most widely used nonmetallic mineral materials. It plays an important role in biology, chemistry, and several other scientific disciplines as well as being a material of obvious importance to many aspects of daily life.8 As a simple ionic crystal, investigating the physical properties of NaCl, such as its phase transformation and elastic properties, can provide valuable information for understanding the physical and chemical properties of more complex ionic crystals. The mechanism of transition from rocksalt structure (the B1 phase) to cesium chloride structure (the B2 phase) of NaCl has been extensively investigated both experimentally and theoretically in the pase decades.9-19 On the basis of X-ray diffraction investigations under high pressure, Bassett et al. observed that the phase transformation of NaCl from B1 to B2 phases at a pressure of ∼30 GPa at room temperature.9 In order to elucidate the phase boundary between B1 and B2 phases of NaCl, Nishiyama et al. have developed experimental techniques and * Corresponding author. E-mail [email protected]. † Sichuan University. ‡ Academia Sinice. § University of Electronic Science and Technology of China.

performed a series of X-ray diffraction experiments at temperatures between 1150 and 2000 K using a Kawai-type apparatus.8 Theoretically, several groups have made to interpret the highpressure behavior and calculated the EOS of NaCl by using the ab initio model. Bukowinski et al. have carried out a firstprinciples calculation of the equation of state and charge distributions for the B1 and the B2 phases of NaCl.12 On the basis of the lattice theory of ionic solids and analytical functions for the volume dependence of the short-range force constant, Shanker et al. have discussed the equation of state and pressure derivatives of bulk modulus for NaCl crystal.13 However, comparing with the availability of precise experimental date, an adequate analysis of the complete physical origin of NaCl under pressure is still lacking, especially for the B2-type NaCl. Furthermore, the investigations of the elastic and thermodynamic properties of ionic crystals at high pressure has been described as an important area of research in solid state physics to investigate the essential features of interatomic binding forces at the phase transformation. In this paper, we describe a systematical investigation of the high-pressure behavior and structural phase transition of NaCl by the first-principles quantum-mechanical calculations, which are very successful in predicting the phase stability and highpressure equations of state for a wide class of crystals. The aim of the present work is twofold. On the one hand, it is to provide a reasonable interpretation about the structural phase transition between the B1 phase and the B2 phase of NaCl under highpressure. On the other hand, it is to provide powerful guidelines for future experimental investigations and hope that such a investigation might contribute some further understanding of the thermodynamic properties of NaCl and its isomorphs. This paper is organized as follows: In section 2, a brief description of the technical details and model in the theoretical

10.1021/jp805945v CCC: $40.75  2008 American Chemical Society Published on Web 10/15/2008

Thermodynamic Properties in Sodium Chloride

J. Phys. Chem. B, Vol. 112, No. 44, 2008 13899

determination of the structural phase transition and thermodynamic properties is reported. The calculated pressure-dependent structural, elastic, and thermodynamic properties as well as the experimental and theoretical results for the high-pressure behavior of NaCl are discussed in section 3. Conclusions are summarized in section 4. 2. Method The equation of state and elastic properties calculations are performed using the pseudopotential plane-wave method within the framework of the density functional theory and implemented through the Cambridge Serial Total Energy Package (CASTEP) program.20-24 This technique has become widely recognized as the method of choice for computational solid structural properties investigations.25-27 The thermodynamic properties for the B1 and B2 phases of NaCl are calculated by the quasi-harmonic Debye model.28 The exchange correlation energy is described in the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional.29 The Na (2s2 2p6 3s1) and Cl (3s2 3p5) states are treated as valence electrons. Interactions of electrons with ion cores are presented by the norm conserving pseudopotential for all atoms. In all the highprecision calculations, the cutoff energy of the plane-wave basis set is 500 eV for both rocksalt and cesium chloride structures of NaCl. The special points sampling integration over the Brillouin zone are carried out using the Monkhorst-Pack method with a 20 × 20 × 20 special k-point mesh. The kinetic energy cutoff and mesh of k-points are optimized by performing self-consistent calculations. The self-consistent is considered to be converged when the total energy is 10-6 eV/atom. These parameters are sufficient in leading to well-converged total energy and elastic stiffness coefficients calculations. 3. Results and Discussion 3.1. Pressure-Induced Structural Phase Transition. In order to obtain a more general insight into the phase stability of NaCl, we have carried out calculations of the Gibbs free energy G for the two involved phases. In general, the Gibbs free energy G can be expressed as30

G ) E + PV - TS

(1)

where E is the internal energy, S is the vibrational entropy, and P and V are the pressure and volume. Since the theoretical calculations are performed at T ) 0 K, the Gibbs free energy G becomes equal to the enthalpy H

H ) E + PV

(2)

Under compression, the calculation shows that NaCl will undergo a structural phase transition from the B1 phase to the B2 phase by increasing the pressure (see Figure 1). According to the third-order Birch-Murnaghan equation of states30

[ ( ) ( ) ]{

V0 3 P ) B0 2 V

7/3

-

V0 V

5/3

3 1 + (B0 - 4) × 4 V0 2/3 -1 V

[( )

]}

(3)

we calculated the total energy of the two-ion primitive cell of NaCl in both the B1 structure and the B2 structure as a function of pressure from 0 to 70 GPa. The transition pressure Pt for the

Figure 1. The two structures of NaCl.

Figure 2. Enthalpy of NaCl in both the B1 and B2 structures. The phase transition appears at 29.73 GPa.

TABLE 1: Pressure of the B1-B2 Phase Transformation of NaCl experiments

calculations

Pt (GPa)

references

28.20 29.00 29.20 30.00 30.60 29.73 25.00 31.00 30.00 27.20 28.90-39.20

Bassett et al. (1968) (ref 9) Sato-Sorensen (1983) (ref 14) Piermarini et al. (1975) (ref 15) Li et al. (1987) (ref 16) Nishiyama et al. (2003) (ref 8) present work Ono et al. (2008) (ref 11) Galamba et al. (2007) (ref 10) Zhang et al. (2003) (ref 17) Sims et al. (1999) (ref 18) Apra` et al. (1993) (ref 19)

structural phase transitions is determined by the usual condition of equal enthalpies in the both phases, i.e., HB1(Pt) ) HB2(Pt). The variations of the enthalpy versus the pressure of the primitive cell of NaCl are presented in Figure 2. As can be seen from Figure 2, the intersection of the two enthalpy curves implies that the B1-B2 phase transition in NaCl occurs at 29.73 GPa. This result is in good agreement with experimental data and theoretical work.8,10,11,17 The comparisons of the transition pressure Pt between the our theoretical values and previous investigations are shown in Table 1. 3.2. Structural Properties. The structural properties are very important for understanding the solid properties from a microscopic point of view. The reason is that the structural properties can provide highly detailed information about the essential features of interatomic binding forces in solids. Using the firstprinciples calculations, we have calculated the structural properties for both the B1-type and the B2-type NaCl. The calculated lattice constant a, the primitive cell volume V0, the bulk modulus B0, the first pressure derivative of bulk modulus B′0, and the

13900 J. Phys. Chem. B, Vol. 112, No. 44, 2008

Lu et al.

TABLE 2: Calculated and Experimental Lattice Constant (Å), Primitive Cell Volume (Å3), Bulk Modulus (GPa), and Its First and Second Pressure Derivative of Bulk Modulus of NaCl at Zero Pressurea a

V0

B0

B′0

B0B′′0

B′′0

5.647 5.640b 5.610e 3.501 3.469g 3.407e

45.985 44.83c 45.65f 42.913 41.35b 39.55,e 42.24f

24.684 23.40-26.40d 28.64,e 24.20f 26.723 26.60b 33.47,e 29.12f

4.134 3.90-4.92d 4.72f 4.031 5.20b 4.42f

-7.677

-0.311

-7.156

-0.268

structure B1

present work experiment theory present work experiment theory

B2

c

a The values of B0, B′0, and B′′0 are evaluated from the fitting of the pressure-volume data to the analytical equation of state. b Reference 12. Reference 9. d Reference 31. e Reference 32. f Reference 11. g Reference 33.

second pressure derivative of bulk modulus B′′0 for B1 and B2 phases are listed in Table 2, together with the available experimental, and theoretical work for comparison.31-33 From Table 2, we can see that the calculated lattice constant a and primitive cell volume V0 agree well with the experimental data. The overestimation of lattice constant a parameter is 0.12% for B1 phase and 0.92% for B2 phase, and the primitive cell volume V0 parameter is 3%. However, the calculated first pressure derivative of bulk modulus 4.031 of B2-type NaCl is smaller than the experimental value 5.20 obtained by Bukowinski et al.,12 but it is in good agreement with the theoretical value 4.42 obtained by Ono et al.11 The good agreement between them shows the accuracy of the present electronic structure calculations. 3.3. Elastic Properties. For a cubic NaCl crystal, there are three independent elastic constants (C11, C12, and C44) which are believed to be related to the second-order change in the internal energy of a crystal under deformation. The mechanical stability criteria for a cubic crystal are given by34

C11 > 0,

C44 > 0,

C11 > |C12 |,

( ) ∂σij(x) ∂ekl

(5) X

where σij and ekl are the applied stress and Eulerian strain tensors and X and x are the coordinates before and after the deformation. Under the hydrostatic pressure P, we have35,36

cijkl )

(

1 ∂2E(x) V(x) ∂eij ∂ekl

)

+ X

P (2δijδkl - δikδjl - δilδjk) (6) 2

where δ is the finite strain variable. For the case of isotropic stress, the three nonindependent elastic constants (c11,c12, and c44) of the isotropic aggregate can be transform into the three independent elastic constants (C11, C12, and C44) as follows:

c11 ) C11,

c12 ) C12 + P,

BS )

C11 + 2C12 , 3

GR )

5C44(C11 - C12) , 4C44 + 3(C11 - C12)

c44 ) C44 -

P 2

(7)

GV )

C11 - C12 + 3C44 5 G)

GV + GR 2

(8)

where GV is the Voigt shear modulus and GR is the Reuss shear modulus. The elastic anisotropy of a crystal is the orientation dependence of the elastic modulus or sound velocities. Essentially, all known crystals are elastically anisotropic. A convenient method of describing the degree of elastic anisotropy for a cubic crystal has been defined as38

(C11 + 2C12) > 0 (4)

Hence, to investigate the stability of NaCl crystal, we have calculated the elastic constants under hydrostatic pressure by a direct method, i.e., ab initio stress-strain relations.31 In this method, the stress is calculated as a function of the strain with the internal coordinates optimized under each strain condition, and the elastic constants are the derivates of the stress with respect to the strain. On the basis of the Hooke’s law, the elastic stiffness tensor cijkl can be expressed as34,35

cijkl )

The adiabatic bulk modulus BS and the shear modulus G of the NaCl can be expressed as a function of the three independent elastic constants, which are in the following forms:37

A* )

3(A - 1)2 [3(A - 1)2 + 25A]

(9)

where A ) 2C44/(C11 - C12) is the usual anisotropy factor. Equation 9 is an important result of the present analysis because it represents the degree of elastic anisotropy for NaCl crystal. Generally, A* is zero for elastically isotropic crystal (i.e., A ) 1). For an anisotropic crystal, A* is always positive and a single valued measure of the elastic anisotropy regardless of whether A smaller or larger than 1. Further, it is shown that A* gives the relative magnitude of the actual elastic anisotropy possessed by the crystal.38 The anisotropic factors as calculated from the elastic constants of NaCl are given in Table 3. It can be seen from Table 3 that NaCl crystal is characterized by a profound anisotropy since the anisotropy factor A < 1 for both B1 phase and B2 phase, and the anisotropy factor A ) 0.583 for B1 phase is larger than A ) 0.035 for B2 phase. This means that the degree of elastic anisotropy for B1 phase is smaller that the B2 phase of the NaCl. On the basis of the ab initio stress-strain relations, the Young’s modulus E, rigidity modulus Gr, and Poisson’s ratio V may be estimated by the following relations34,35

E)

1 , s11

V)-

s12 , s11

Gr )

1 2(s11 - s12)

(10)

where sij is the elastic compliance constants. The calculated elastic constants, shear modulus, Young’s modulus, rigidity modulus, and Poisson’s ratio of NaCl are listed in Table 3. The results indicate the elastic constants (C11, C12, and C44) are positive for both B1 phase and B2 phase. From the theoretically



TABLE 3: Calculated Elastic Constants Cij (GPa), Anisotropic Factor A, Adiabatic Modulus BS (GPa), Voigt Shear Modulus GV (GPa), Reuss Shear Modulus GR (GPa), Shear Modulus G (GPa), Young’s Modulus E (GPa), Rigidity Modulus Gr (1/GPa), and Poisson’s Ratio W for NaCl at Zero Pressure B1 present work 57.546 11.903 13.315 0.583 0.034 27.117 17.118 15.977 16.547 53.466 0.171 22.822

C11 C12 C44 A A* BS GV GR G E V Gr a

B2

experiment

theory

75.996 3.903 1.252 0.035 0.763 27.934 15.170 2.039 8.605 75.615 0.049 36.047

Reference 14. b Reference 39. c Reference 32. d Reference 40.

obtained elastic constants, we can seen that C11 - C12 > 0 and C11 + 2C12 > 0. This implies that the both structures are elastically stable. 3.4. Thermodynamic Properties. To investigate the thermodynamic properties of NaCl, we apply the quasi-harmonic Debye model, in which the nonequilibrium Gibbs function G/(V;P,T) of NaCl can be written in the form of28,41

G*(V;P, T) ) E(V) + PV + AVib(Θ(V);T)

(11)

where E(V) is the total energy per unit cell for NaCl, PV corresponds to the constant hydrostatic pressure condition, and AVib(Θ(V);T) is the vibrational term, which can be written as28

+ 3 ln(1 - e ( 9Θ 8T

-Θ/T

( ΘT ))

)-D

(12) where n is the number of atoms in the molecule, and the Debye integral D(Θ/T) is defined as28

D(y) )

3 y3

3

∫0y ex x- 1 dx

(13)

One of the standard methods of calculating the Debye temperature, Θ in equation 12, is from elastic constant. It is related to an average sound velocity, since the vibrations of the solid are considered as elastic waves in Debye’s theory. For NaCl crystal, the Debye temperature can be estimated from the average sound velocity Vm, using the following equation42

Θ)

(

h 3nNAF kB 4πM

)

1/3

Vm

(

1 2 + 3 3 VP VS

)

-1/3

(15)

where VP and VS are the longitudinal and transverse elastic wave velocities, respectively, which can be obtained from Navier’s equation42

present work

57.300a,b 58.430,c 41.700d a b 11.200, 11.230 13.730,c 11.200d 13.300,a 13.310b 13.730,c 13.300d

AVib(Θ(V);T) ) nKT

3

Vm ) √3

(14)

where h is Planck’s constant, kB is Boltzmann’s constant, NA is Avogadro’s number, M is the molecule mass, F is the density, and the average sound velocity Vm is approximately given by30

VP )

3BS + 4G , 3F

VS )

GF

(16)

where G is the shear modulus and BS is the adiabatic bulk modulus. 3.5. Pressure Dependence of Structural, Elastic, and Thermodynamic Properties. In order to investigate the pressure-dependence behaviors of the structural, elastic, and thermodynamic properties of NaCl, we concentrate our investigation to the pressure range below 70 GPa. In Figures 3-9, we present the dependence of elastic constants, shear modulus, Young’s modulus, and rigidity modulus on hydrostatic pressure. From Figures 3-5, we can note that C11 and C12 vary largely under the effect of pressure as compared with the variations in C44. Both C11 and C12 increase monotonically with the pressure, whereas the C44 decrease in the range of P < Pt and increase rapidly in the range of P > Pt. It can be seen from Figure 6 that, in both B1 and B2 phases of NaCl, the adiabatic bulk modulus BS increase rapidly with increasing pressure. Furthermore, it is noted from Figure 6 that the adiabatic bulk modulus BS are found to show a linear variation with the pressure. From Figures 7-9, we can also observe that the pressure has an important influence on the shear modulus, Young’s modulus, and rigidity modulus. In Figures 10-12, we have plotted the variation of the anisotropic factor and the Poisson’s ratio of NaCl crystal at different pressures. As shown in Figures 10 and 12, we find that both anisotropic factor A and Poisson’s ratio V in B1-type NaCl decrease with increasing pressure, while for B2-type NaCl, A increases strongly with increasing pressure. From Figure 11, we can also find that the degree of elastic anisotropy A* in B1type NaCl increases as the pressure increases and gradually tends to a linear increase at P < Pt. Meanwhile, in Figures 13 and 14, we have indicated the results of sound velocity, wave velocity, and Debye temperature for NaCl crystal. It is worth pointing out that the sound velocity, wave velocity, and Debye temperature increase as the pressure increase. From Figures 13 and 14, it is interesting to note that the increased tendency of average sound velocity Vm and transverse sound velocity VS, longitudinal sound velocity VP, and Debye temperature Θ are, to some extent, similar. Some of the quantitative calculation results are listed in Tables 4 and 5. Finally, we have also, for completeness, calculated the pressure derivatives of elastic constants, ∂C11/∂P, ∂C12/∂P, ∂C44/ ∂P, ∂S11/∂P, ∂S12/∂P, and ∂S44/∂P, for NaCl crystal. The theoretical results are presented in Table 6. Unfortunately, as far as we know, there are no experimental and theoretical data available related to the pressure derivative of elastic modulus of NaCl in the literature for our comparison. Consequently, our work is a first attempt in this direction and our results can serve as a prediction for future investigations. 3.6. Structural Properties of B2-Type NaCl at High Pressure. The investigation of NaCl crystal at high pressure is of considerable current interest, particularly in the case of elements. The knowledge of structural properties of NaCl at high pressure is important for high-pressure behavior inves-


Lu et al.

Figure 3. Pressure dependence of elastic constants C11 in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.

Figure 6. Pressure dependence of adiabatic bulk modulus BS in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.


Figure 7. Pressure dependence of shear modulus G in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.


Figure 8. Pressure dependence of Young’s modulus E in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.

tigations of solids.43-46 For a long time, experiments to measure the structural properties of NaCl at high pressure remain a formidable challenge with existing measurement facilities. Furthermore, most of the theoretical work about the structural properties of NaCl is only performed at

pressures below 70 GPa.14,33,47,48 We therefore think that it is worthwhile to perform a theoretical calculation for the structural properties of NaCl under high pressure in order to provide reference data for the experimentalists. Recently, it has become possible to compute with great accuracy an


Figure 9. Pressure dependence of rigidity modulus Gr in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.


Figure 12. Pressure dependence of Poisson’s ratio V in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.

Figure 13. Pressure dependence of average sound velocity Vm and transverse wave velocity VS of NaCl. Figure 10. Pressure dependence of anisotropic factor A in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.

Figure 14. Pressure dependence of longitudinal wave velocity VP and Debye temperature Θ of NaCl. Figure 11. Pressure dependence of the degree of elastic anisotropy A* in both the B1 and B2 structures of NaCl: (a) omitting the phase transition; (b) considering the phase transition.

important number of electronic and structural parameters of solids from first-principles calculations.49-53 This kind of development in computer simulations has opened many interesting and exciting possibilities in condensed matter investigations. For example, it is now possible to explain

and predict properties of solids which were previously inaccessible to experiments. In order to investigate the structural properties of B2-type NaCl under high pressure, we have carried out a theoretical investigation for the lattice constant a and primitive cell volume V0 up to 500 GPa. The first-principles calculations are performed within densityfunctional theory, using the pseudopotential method. The exchange correlation effects are treated with the generalized


Lu et al.

TABLE 4: Dependence of Elastic Constants, Shear Modulus, and Adiabatic Bulk Modulus on Hydrostatic Pressure of NaCl, where P, Cij, BS, GV, GR, and G Are in Units of GPa, a Is in Units of Å, and Sij Is in Units of 10-11 m2/N P

a

V/V0

C11

C12

C44

S11

S12

S44

Bs

GV

GR

G

0 5 10 15 20 22 24 26 28 29.73 30 32 34 40 45 50 60 70

5.649 5.412 5.250 5.133 5.041 5.009 4.979 4.952 4.925 4.904 3.034 3.020 3.007 2.970 2.943 2.918 2.875 2.838

1.000 0.881 0.804 0.751 0.712 0.698 0.686 0.675 0.664 0.656 0.608 0.599 0.591 0.570 0.554 0.541 0.517 0.497

57.546 97.442 143.004 186.558 228.865 245.802 262.006 277.803 294.404 308.193 256.624 267.339 276.475 299.827 332.595 355.898 403.220 449.031

11.903 18.797 25.174 31.674 37.662 40.036 42.701 44.446 47.029 49.104 80.855 86.189 90.048 104.599 118.789 131.492 156.030 181.071

13.315 13.005 12.852 12.358 11.764 11.663 11.271 11.147 10.921 10.691 34.076 36.231 38.387 45.739 51.425 57.381 68.839 81.189

1.870 1.095 0.738 0.564 0.458 0.426 0.340 0.376 0.355 0.339 0.459 0.443 0.431 0.407 0.370 0.351 0.316 0.290

-0.321 -0.177 -0.110 -0.082 -0.065 -0.059 -0.056 -0.052 -0.049 -0.047 -0.109 -0.108 -0.106 -0.105 -0.097 -0.095 -0.088 -0.083

7.510 7.689 7.781 8.092 8.500 8.574 8.872 8.971 9.157 9.354 2.935 2.760 2.605 2.186 1.945 1.743 1.453 1.232

27.117 45.012 64.450 83.302 101.396 108.624 115.802 122.232 129.487 135.467 139.445 146.572 152.190 169.675 190.057 206.294 238.426 270.391

17.118 23.532 31.277 38.392 45.299 48.151 50.624 53.360 56.028 58.232 55.599 57.969 60.318 66.489 73.616 79.310 90.741 102.305

15.977 17.759 18.700 18.616 18.120 18.073 17.580 17.466 17.190 16.889 45.128 47.672 50.197 58.087 64.897 71.320 83.666 96.379

16.547 20.646 24.989 28.504 31.710 33.112 34.102 35.413 36.609 37.561 50.363 52.821 55.258 62.288 69.256 75.315 87.204 99.342

TABLE 5: Dependence of Young’s Modulus (in GPa), Poisson’s Ratio, Rigidity Modulus (in 1/GPa), Anisotropic Factor, Longitudinal Wave Velocity (in km/s), Transverse Wave Velocity (in km/s), Average Sound Velocity (in km/s), and Debye Temperature (in K) on Hydrostatic Pressure of NaCl P

E

V

Gr

A

A*

VP

VS

Vm

ΘD

0 5 10 15 20 22 24 26 28 29.73 30 32 34 40 45 50 60 70

53.466 91.363 135.468 177.364 218.221 234.587 250.038 265.543 281.448 294.696 217.880 225.314 232.229 245.721 270.073 284.948 316.156 344.963

0.1714 0.1617 0.1497 0.1451 0.1413 0.1401 0.1400 0.1379 0.1377 0.1374 0.2396 0.2438 0.2457 0.2586 0.2632 0.2698 0.2790 0.2874

22.822 39.323 58.915 77.442 95.602 102.883 109.653 116.678 123.688 129.545 87.885 90.575 93.214 97.614 106.903 112.203 123.595 133.980

0.583 0.331 0.218 0.160 0.123 0.113 0.103 0.096 0.088 0.083 0.388 0.400 0.412 0.469 0.481 0.511 0.557 0.606

0.034 0.140 0.252 0.347 0.429 0.454 0.484 0.507 0.530 0.550 0.104 0.097 0.092 0.067 0.063 0.053 0.041 0.030

10.307 12.518 14.532 16.187 17.616 18.166 18.664 19.131 19.625 20.020 24.675 25.288 25.799 27.290 28.848 30.065 32.331 34.456

2.768 3.092 3.401 3.633 3.831 3.915 3.973 4.049 4.117 4.170 4.134 4.234 4.330 4.597 4.848 5.055 5.440 5.806

3.158 3.530 3.885 4.151 4.378 4.474 4.541 4.628 4.705 4.766 4.729 4.842 4.953 5.259 5.545 5.782 6.222 6.641

333.239 372.489 409.953 437.948 461.996 472.122 479.158 488.294 496.491 502.922 553.372 566.708 579.630 615.395 648.910 676.697 728.150 777.176

TABLE 6: Predicted Pressure Derivatives of the Elastic Modulus Cij and Sij for NaCl, Where Pressure P Is in Units of GPa P

∂C11/∂P

∂C12/∂P

∂C44/∂P

∂S11/∂P

∂S12/∂P

∂S44/∂P

0-5 5-10 10-15 15-20 20-22 22-24 24-26 26-28 28-30 30-32 32-34 34-40 40-45 45-50 50-60 60-70

7.979 9.112 8.711 8.461 8.469 8.102 7.899 8.301 18.890 5.358 4.568 3.892 6.554 4.661 4.732 4.581

1.379 1.275 1.300 1.198 1.187 1.132 0.873 1.292 16.913 2.667 1.930 2.425 2.838 2.541 2.454 2.504

0.062 0.031 0.099 0.119 0.051 0.196 0.062 0.113 11.578 1.077 1.078 1.225 1.137 1.191 1.146 1.235

0.155 0.071 0.035 0.021 0.016 0.043 0.018 0.011 0.052 0.008 0.006 0.004 0.007 0.004 0.004 0.003

0.029 0.013 0.006 0.003 0.003 0.002 0.002 0.002 0.030 0.001 0.001 0.000 0.002 0.000 0.001 0.001

0.036 0.018 0.062 0.082 0.037 0.149 0.050 0.093 3.111 0.088 0.078 0.070 0.048 0.040 0.029 0.022

gradient-corrected exchange-correlation functionals given by Perdew et al.29 The calculated results are shown in Figure 15. It worth noting that the pressure dependence of the lattice constant a and primitive cell volume V0 for B2-type NaCl

are in good agreement with recent experimental and theoretical results published in the literature.11,43 Using the pseudopotential plane-waves method based on the density functional theory, with the generalized gradient approximation, we have systematically investigated the structural, elastic, and thermodynamic properties of NaCl under high pressure. The obtained results might be interesting from the viewpoint of technological applications because, in addition to the traditional use of the elastic properties, the thermodynamic properties may have potential applications in material science. Furthermore, the accurate structural parameters of NaCl, which can be considered as a simple ionic crystal, are an important first step that helps to provide a complete interpretation of the essential features of interatomic binding forces in more complex ionic crystals. Significantly, the theoretical calculations clearly confirm that the structural phase transition between the B1 phase and the B2 phase of NaCl is 29.73 GPa. Another important conclusion of the present work is that the elastic properties sensitively depend on the pressure. The bulk modulus is found to shown a linear variation with the pressure. These findings are of significant importance for a better understanding of the high-pressure behavior of NaCl crystal.


Figure 15. Pressure dependence of the primitive cell volume V0 and lattice constant a for B2-type NaCl at 300 K.

4. Conclusions From the above investigations, we have the following conclusions: (i) The B1-B2 structural phase transition of NaCl has been investigated by using the first-principles calculation. The calculated results show that the transition pressure Pt for the structural phase transition from the B1 phase to the B2 phase is 29.73 GPa. (ii) The structural parameters and thermodynamic properties for both the B1-type and the B2-type NaCl have been derived. The theoretical values are found to be in good agreement with the experimental finds. (iii) The pressure dependence of the structural parameters and elastic properties are investigated. It is found that the pressure has an important influence on the structural parameters and a linear dependence of the bulk modulus versus applied pressure. (iv) We have calculated the shear modulus, Young’s modulus, rigidity modulus, Poisson’s ratio, sound velocity, and Debye temperature of NaCl at high-pressure and derived the pressure derivatives of elastic constants, ∂C11/∂P, ∂C12/∂P, ∂C44/∂P, ∂S11/ ∂P, ∂S12/∂P, and ∂S44/∂P, for NaCl crystal. However, there are no experimental data available related to the pressure derivative of elastic modulus of NaCl. Hence, careful experimental investigations are required in order to clarify the pressure derivatives of elastic constants of NaCl in detail. Acknowledgment. The authors express their gratitude to Dr. Yang Ze-Jin for many helpful discussions. This work was supported by the National Natural Science Foundation (No. 10774103 and No. 10374068) and the Doctoral Education Fund of Education Ministry (No. 20050610011) of China. References and Notes (1) Lee, Y.; Hriljac, J. A.; Vogt, T.; Parise, J. B.; Edmondson, M. J.; Anderson, P. A.; Corbin, D. R.; Nagai, T. J. Am. Chem. Soc. 2001, 123, 8418. (2) Wenger, O. S.; Salley, G. M.; Gu1del, H. U. J. Phys. Chem. B 2002, 106, 10082. (3) Lapenã, A. M.; Wu, J. J.; Gross, A. F.; Tolbert, S. H. J. Phys. Chem. B 2002, 106, 11720. (4) Imai, M.; Kikegawa, T. Chem. Mater. 2003, 15, 2543. (5) Dreger, Z. A.; Gupta, Y. M.; Yoo, C. S.; Cynn, H. J. Phys. Chem. B 2005, 109, 22581. (6) Davydov, V. A.; Rakhmanina, A. V.; Rols, S.; Agafonov, V.; Pulikkathara, M. X.; Vander Wal, R. L.; Khabashesku, V. N. J. Phys. Chem. C 2007, 111, 12918.

J. Phys. Chem. B, Vol. 112, No. 44, 2008 13905 (7) Zhu, J. L.; Jin, C. Q.; Cao, W. W.; Wang, X. H. Appl. Phys. Lett. 2008, 92, 242901. (8) Nishiyama, N.; Katsura, T.; Funakoshi, K.; Kubo, A.; Kubo, T.; Tange, Y.; Sueda, Y.; Yokoshi, S. Phys. ReV. B 2003, 68, 134109. (9) Bassett, W. A.; Takahashi, T.; Mao, H. K.; Weaver, J. S. J. Appl. Phys. 1968, 39, 319. (10) Galamba, N.; Costa Cabral, B. J. J. Chem. Phys. 2007, 126, 124502. (11) Ono, S.; Brodholt, J. P.; Alfe`, D.; Alfedsson, M.; Price, G. D. J. Appl. Phys. 2008, 103, 023510. (12) Bukowinski, M. S. T.; Aidum, J. J. Geophys. Res. 1985, 90, 1794. (13) Shanker, J.; Kushwah, S. S.; Kumar, P. Physica B 1997, 239, 337. (14) Sato-Sorensen, Y. J. Geophys. Res. 1983, 88, 3543. (15) Piermarini, G. J.; Block, S. ReV. Sci. Instrum. 1975, 46, 973. (16) Li, X.; Jeanloz, R. Phys. ReV. B 1987, 36, 474. (17) Zhang, S.; Chen, N. X. Modell. Simul. Mater. Sci. Eng. 2003, 11, 331. (18) Sims, C. S.; Allan, N. L.; Barron, T. H. K. Phys. ReV. B 1999, 60, 2968. (19) Apra`, E.; Causa`, M.; Prencipe, M.; Dovesi, R.; Saunders, V. R. J. Phys.: Condens. Matter 1993, 5, 2969. (20) Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip, P. J.; Probert, M. J.; Refson, K.; Payne, M. C. Z. Kristallogr. 2005, 220, 567. (21) Segall, M. D.; Lindan, P. J. D.; Probert, M. J.; Pickard, C. J.; Hasnip, P. J.; Clark, S. J.; Payne, M. C. J. Phys.: Condens. Matter2002, 14, 2717. (22) Sun, J.; Wang, H. T.; He, J. L.; Tian, Y. J. Phys. ReV. B 2005, 71, 125132. (23) Hu, C. H.; Wang, Y. M.; Chen, D. M.; Xu, D. S.; Yang, K. Phys. ReV. B 2007, 76, 144104. (24) Stojic´, N.; Corso, A. D.; Zhou, B.; Baroni, S. Phys. ReV. B 2008, 77, 195116. (25) Lippens, P. E.; Chadwick, A. V.; Weibel, A.; Bouchet, R.; Knauth, P. J. Phys. Chem. C 2008, 112, 43. (26) Luo, X. G.; Zhou, X. F.; Liu, Z. Y.; He, J. L.; Xu, B.; Yu, D. L.; Wang, H. T.; Tian, Y. J. J. Phys. Chem. C 2008, 112, 9516. (27) Fan, C. Z.; Wang, Q.; Li, L. X.; Zhang, S. H.; Zhu, Y.; Zhang, X. Y.; Ma, M. Z.; Liu, R. P.; Wang, W. K. Appl. Phys. Lett. 2008, 92, 101917. (28) Blanco, M. A.; Francisco, E.; Luanã, V. Comput. Phys. Commun. 2004, 158, 57. (29) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865. (30) Poirier, J. P. Introduction to the Physics of the Earth’s Interior; Cambridge University Press: Cambridge, UK, 2000. (31) Birch, F. J. Geophys. Res. 1978, 83, 1257. (32) Zhang, S.; Chen, N. X. Phys. ReV. B 2002, 66, 064106. (33) Heinz, D. L.; Jeanloz, R. Phys. ReV. B 1984, 30, 6045. (34) Nye, J. F. Physical Properties of Crystals; Clarendon Press: Oxford, 1985. (35) Karki, B. B.; Ackland, G. J.; Crain, J. J. Phys.: Condens. Matter 1997, 9, 8579. (36) Wang, J. H.; Li, J.; Yip, S.; Phillpot, S.; Wolf, D. Phys. ReV. B 1995, 52, 12627. (37) Wu, J. Z.; Zhao, E. J.; Xiang, H. P.; Hao, X. F.; Liu, X. J.; Meng, J. Phys. ReV. B 2007, 76, 054115. (38) Chung, D. H.; Buessem, W. R. J. Appl. Phys. 1967, 38, 2010. (39) Hardy, J. R.; Karo, A. M. The Lattice Dynamics and Statics of AlKali Halides Crystals; Plenum Press: New York, 1975. (40) Rao, B. S.; Sanyal, S. P. Phys. ReV. B 1990, 42, 1810. (41) Francisco, E.; Recio, J. M.; Blanco, M. A.; Martıń Penda´s, A.; Costales, A. J. Phys. Chem. A 1998, 102, 1595. (42) Anderson, O. L. J. Phys. Chem. Solids 1963, 24, 909. (43) Ono, S.; Kikegawa, T.; Ohishi, Y. Solid State Commun. 2006, 137, 517. (44) Archer, D. G.; Carter, R. W. J. Phys. Chem. B 2000, 104, 8563. (45) Wenger, O. S.; Salley, G. M.; Gudel, H. U. J. Phys. Chem. B 2002, 106, 10082. (46) Bondarenko, G. V.; Gorbaty, Y. E.; Okhulkov, A. V.; Kalinichev, A. G. J. Phys. Chem. A 2006, 110, 4042. (47) Sata, N.; Shen, G.; Rivers, M. L.; Sutton, S. R. Phys. ReV. B 2002, 65, 104114. (48) Hofmeister, A. M. Phys. ReV. B 1997, 56, 5835. (49) Zhan, C. G.; Dixon, D. A. J. Phys. Chem. A 2004, 108, 2020. (50) Jose, R.; Zhanpeisov, N. U.; Fukumura, H.; Baba, Y.; Ishikawa, M. J. Am. Chem. Soc. 2006, 128, 629. (51) Zhao, M.; Xia, Y.; Liu, X.; Tan, Z.; Huang, B.; Song, C.; Mei, L. J. Phys. Chem. B 2006, 110, 8764. (52) Xu, X. G.; Li, C.; Li, J. X.; Kolb, U.; Wu, F.; Chen, G. J. Phys. Chem. B 2003, 107, 11648. (53) Yang, Y.; Geng, H.; Yin, S.; Shuai, Z.; Peng, J. J. Phys. Chem. B 2006, 110, 3180.

JP805945V

Characterization of the High-Pressure Structural Transition and

Recommend Documents