Calculation of the thermodynamic properties of B2 AlRE (RE=Sc, Y, La, Ce–Lu)

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Physica B 399 (2007) 27–32 www.elsevier.com/locate/physb

Calculation of the thermodynamic properties of B2 AlRE (RE ¼ Sc, Y, La, Ce–Lu) Xiaoma Taoa, Yifang Ouyangb,, Huashan Liua, Fanjing Zengb, Yuanping Fengc, Zhanpeng Jina a

School of Materials Science and Engineering, Central South University, Changsha, Hunan 410083, PR China b Department of Physics, Guangxi University, Nanning, Guangxi 530004, PR China c Department of Physics, National University of Singapore, Singapore 119260, Singapore Received 12 March 2007; received in revised form 13 May 2007; accepted 14 May 2007

Abstract First-principles calculations for the total energy and elastic constants of the B2-type AlRE (RE ¼ Sc, Y, lanthanide) have been performed at T ¼ 0 K by using the projector augmented wave (PAW) method within the generalized gradient approximation (GGA). The Debye temperatures, Gru¨neisen constants, the temperature dependences of the Gibbs free energy, coefficients of thermal expansion, heat capacities are obtained for the B2-AlRE within the Debye–Gru¨neisen model. The activation energy of self-diffusion, Poisson’s ratio, Debye sound velocities are also evaluated for the B2-AlRE in the present work. r 2007 Published by Elsevier B.V. Keywords: First principles; B2-AlRE; Debye model; Gibbs free energy

1. Introduction The simplest way to investigate the thermal properties on the basis of binding curves obtained by first principles is to use the Debye–Gru¨neisen model, which is a simple and very effective method to describe the vibrational contribution to the Gibbs free energy of the elements and of simple compounds and alloys [1,2]. The Gibbs free energy can be used in CALculation of PHAse Diagrams (CALPHAD) applications by simple polynomials representation [3]. Chen et al. [4] has pointed out, the parameters of the simple polynomials in the CALPHAD applications are lacking any physical significance. Such parameters often make the extrapolation of the data outside their temperature range and the estimation of parameters for metastable phases is very difficult. Generally, Gibbs energy of a crystalline phase [5] should include the ground state energy, the lattice vibrational contribution, the electronic excitation, etc., which can be Corresponding author. Tel.: +86 771 3232666; fax: +86 771 3237386.

E-mail address: [email protected] (Y. Ouyang). 0921-4526/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physb.2007.05.037

done for stable phases by experiment, but for the metastable phase, their physical properties are not experimentally accessible. Fortunately, the theoretical calculation can play a major role in this field. During the last decades, the first principles methodology has become increasingly sophisticated, with which the total energy and other physical properties of a crystalline phase can be accurately calculated and only the atomic number and atoms position are input variants [6]. And the total energy and elastic constants calculation from first principles can provide the ground state properties and often used together Debye– Gru¨neisen model to derive thermophysical properties and phase stabilities at finite temperatures [7,8]. In fact, the Debye temperature can be yielded from phonon density of states, which can be obtained by using linear response theory [9], the frozen phonon method [10], or first principles force constant approach [11]. However, the complicated computation is required in those schemes. In the Debye–Gru¨neisen model [12], the Debye temperature can be calculated from elastic constants, and which can be obtained from first principles calculation easily [13–19]. Moruzzi et al. [1] have investigated the thermal

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X. Tao et al. / Physica B 399 (2007) 27–32

properties of the 14 nonmagnetic cubic metals by using the first principles electronic structure calculation and Debye– Gru¨neisen model [12]. This method has been gained wide use for its simplicity. In this method, the Debye temperature can be obtained from the calculated bulk modulus, and the Gru¨neisen constant can be obtained from anharmonicity of the binding curve. With these two quantities, the vibrational energy can been described easily, and the Gibbs free energy can be yielded at finite temperature. Recently, Chen et al. [4] extended this method to hexagonal structure, and calculated the Debye temperatures of the Ti, Zr and Hf. The calculated values agree very well with the values derived from the high-temperature experimental entropy data. In the present work, the thermal properties for the B2AlRE are studied by combined the first principles calculation with the Debye–Gru¨neisen model. The total energy and elastic properties of the B2-AlRE have been calculated from first principles calculations. And the vibrational energy has been obtained within the Debye–Gru¨neisen model. The Debye temperature, Gru¨neisen constant, coefficients of thermal expansion, heat capacity and activation energy of self-diffusion are evaluated in the present work. 2. Method of calculation 2.1. First principles method First principles calculations were performed by using the scalar relativistic all-electron Blo¨chl’s projector augmented wave (PAW) approach [20,21] within the generalized gradient approximation (GGA), as implemented in the highly efficient Vienna ab initio simulation package (VASP) [22,23]. For the GGA exchange-correlation function, the Perdew–Wang parameterization (PW91) [24,25] was employed. A plane-wave energy cutoff of 420 eV was held constant for all the AlRE compounds. The k-point meshes for Brillouin zone sampling were constructed using the Monkhorst–Pack scheme [26] and the 15  15  15 kpoints mesh are used in all AlRE systems, and the total energy is converged to better than 1 meV/atom. Spin polarization was used in all calculations, and all structures were fully relaxed. In the present calculation, the potpaw_GGA pseudopotentials of Al, Sc, Y_sv, La, RE_3(RE ¼ Ce, Pr, Nd, Pm, Sm, Gd, Tb, Dy, Go, Er, Tm, Lu), Eu_2, Yb_2 are used.

where the h and kB are the Plank’s and Boltzmann’s constants, and YD is the Debye temperature. And the Debye cutoff frequency oD can be obtained by the Debye sound velocity uD oD ¼

 2 1=3 6p uD , V

(2)

where the V is the volume of the solid. In the real solid, there are three different types of the sound velocities and they are generally anisotropic. For an elastically isotropic medium, the sound velocities associated with the transverse and longitudinal modes are related to the elastic constants of the system in complicated manner [27,28]. The sound velocities are independent of crystallographic directions, but different for the longitudinal and the two degenerate transverse branches. Some authors [29,30] express the lowtemperature average for the sound velocity for an isotropic crystal as   1 1 1 2 ¼ þ , (3) u3D 3 u3L u3S where uL and uS are longitudinal and transverse pffiffiffiffiffiffiffiffisound ffi velocities, and they can be expressed as u ¼ L=r and L pffiffiffiffiffiffiffiffiffi uS ¼ S=r, respectively. With the above equation, a much simpler method has been suggested for the calculation of the Debye sound velocity and Debye temperature. The elastic modulus can be obtained from the first principles calculation [31] and Poisson’s ratio can be obtain as n¼

3B  2G 2ð3B þ GÞ

(4)

and then they are used to calculate longitudinal and transverse moduli by following formula: L¼

3ð1  nÞ B; 1þn

S¼

3ð1  2nÞ B. 2ð1 þ nÞ

And then, one can obtain sffiffiffiffi B uD ¼ kðnÞ , r

(5)

(6)

where ( "    #)1=3 1 1 þ n 3=2 2ð1 þ nÞ 3=2 kðnÞ ¼ þ2 . 3 3ð1  nÞ 3ð1  2nÞ

(7)

The lattice vibrations is assumed can be obtained from Debye theory, the Debye temperature defined as a measure of the Debye cutoff frequency oD, and as the following equation:

Anderson [29] has shown that this method can obtain reasonable Debye sound velocity. So, the Debye temperature can be calculated from the bulk modulus and density using the equation rffiffiffiffiffiffiffi h 5 1=6 r0 B YD ¼ kðnÞ ð48 p Þ , (8) 2 pkB M

h oD ¼ kB YD , 2p

where the M is the atomic weight, the r0 is the equilibrium Wigner–Seitz radius.

2.2. Debye temperature

(1)

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2.3. Gru¨neisen constant

terms of the Debye function D(YD/T)

Anharmonic effects in the vibrating lattice are usually described in terms of a Gru¨neisen constant, g, which can be defined as q ln YD (9) q ln V and which gives the volume dependence of Debye temperature (YD). Using the theoretical expression for YD, the Gru¨neisen constant can be expressed as

E Vib D ðV ; TÞ ¼ E 0 þ 3 kB TDðYD =TÞ,

SVib D ðV ; TÞ

¼ 3 kB

g¼

1 1 q ln B . g¼  6 2 q ln V As we know, the bulk modulus can be defined as

(10)

qP . (11) B ¼ V qV And then, the Gru¨neisen constant can be obtained as  2 V q2 P qV 2 g¼  . (12) 3 2 qP=qV The above expression derived by Slater [32] firstly, which contains the implicit assumption that Poisson’s ratio is a constant and that all vibrational modes are excited. Moruzzi et al. [1] point out that the Gru¨neisen constant is dominated by lower-frequency transverse modes at low temperature. The Slater’s expression for g assumes equal excitation of all modes and is effectively a high-temperature average and, in general, yields g values that are larger than values derived from low-temperature specific heat data by an additive factor of about 13. Barron [33] has shown that 1 (13) 3 for cubic close-packed lattices with central forces between nearest neighbors. So the low-temperature g can be expressed as  V q2 P qV 2 g ¼ gL ¼ 1  , (14) 2 qP=qV gH  gL ffi

2.4. Gibbs free energy The Gibbs free energy of the vibrating system can be expressed as the sum of the total energy of the rigid lattice and the free energy of the vibrating lattice. Since the electronic entropy is expected to be negligible, the Gibbs free energy can be expressed as Vib F ðV ; TÞ ¼ EðV Þ þ E Vib D ðV ; TÞ  TS D ðV ; TÞ,

29

(15)

where E(V) is the total energy at 0 K, and can be obtained directly from first principles calculations as a ground-state Vib energy, E Vib D ðV ; TÞ and S D ðV ; TÞ are the thermal vibrational energy and the vibrational entropy, respectively, at volume V and temperature T. The thermal vibrational energy and the vibrational entropy can be expressed in the

 4 DðYD =TÞ  ln 1  expðYD =TÞ 3

(16)  (17)

and the zero-point energy is 9 E 0 ¼ kB YD . 8 The Debye function is given by Z 3 y x3 DðyÞ ¼ 3 dx. y 0 ex  1

(18)

(19)

The final expression for the Gibbs free energy is F ðV ; TÞ ¼ EðV Þ

  kB T DðYD =TÞ  3 ln 1  expðYD =TÞ ð20Þ þ 98kB YD .

2.5. Other thermodynamic properties After the Debye temperature and the Gru¨neison constant are obtained, the heat capacity, thermal expansion within the Debye–Gru¨neison model are given by   3YD =T C V ¼ 3 kB 4DðYD =TÞ  , (21) expðYD =TÞ  1

a¼

gC V . B0 V

(22)

3. Results and discussion Using the first principles calculations, the total energy of the B2-AlRE is calculated. To obtain the equilibrium volume and bulk modulus, we fitted the first principles calculated the total energies at seven different volumes to a Vinet’s [34] equation of state. The tetragonal shear modulus C0 ¼ C11–C12 and trigonal shear modulus C44 are calculated and the elastic properties of the B2-AlRE are detailed described in previous work [31]. The Poisson’s ratios of the B2-AlRE are obtained by Eq. (4) and listed in the Table 1. As mentioned above, the Debye sound velocity and the Debye temperature can calculated approximately from the elastic constants, which can be evaluated with first principles method. Moruzzi et al. [1] found an empirical relations between the shear and longitudinal modulus for nonmagnetic cubic elements: LE1.42 B and SE0.30 B, which correspond to Poisson’s ratio is approximately constant and approximately equal to 13. Because of its

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Table 1 The calculated K(v), Debye sound velocity, Debye temperature, Gru¨neisen constants and activation energy for self-diffusion of the B2-AlRE K(v)

uD (m/s)

(YD)0 (K)

(YD)0M (K)

(YD)0C (K)

(g)0

ESD (eV)

0.295 0.286 0.278 0.306 0.304 0.302 0.299 0.298 0.251 0.292 0.290 0.288 0.288 0.288 0.289 0.242 0.286

0.761 0.774 0.784 0.745 0.748 0.752 0.756 0.758 0.821 0.765 0.768 0.770 0.771 0.771 0.770 0.831 0.774

3857 3049 2501 2411 2425 2425 2440 2414 2102 2395 2395 2381 2368 2355 2335 2027 2291

402 297 231 223 227 229 232 230 194 232 233 233 233 232 231 192 229

326 237 182 185 187 188 189 187 146 187 187 187 186 186 185 143 183

354 261 207 199 202 203 206 204 189 204 205 205 205 204 203 186 201

1.192 1.162 0.901 1.026 1.127 1.143 1.122 1.065 1.161 1.024 1.125 1.161 1.209 1.132 1.120 1.319 1.039

1.544 1.551 1.558 1.446 1.467 1.489 1.508 1.516 1.409 1.537 1.544 1.551 1.551 1.551 1.543 1.419 1.534

simplicity, this approach has been used in many investigations to derive thermal properties at finite temperature for pure elements and intermetallic compounds [1,7,8]. But for the other systems, the Poisson’s ratio could deviate from 13, and then this approach can not be used to predict the thermal properties for those systems accurately. Chen et al. [4] examined the approach detailed and modified scaling factor for the expression of the average sound velocity in terms of the bulk modulus, and consequently the Debye temperature of cubic and hexagonal elements can be calculated from the knowledge of all the elastic constants of the systems. Siethoff et al. [35–37] pointed out that sophisticated methods have been put forward in the literatures to calculate Debye temperatures from the elastic constants. As a simple alternative method for cubic systems, a relation is often used that correlates Debye temperature and bulk modulus (c11+2c12)/3 by a square-root relation. It is shown that, with exception of the alkali metals, such a relation is only poorly fulfilled for other cubic elements and compounds. And they obtained a semi-empirical relation between the Debye temperature and the elastic constants, which is fulfilled with high precision and which allows one to easily calculate unknown Debye temperatures if the elastic constants are available. And the relation was found to be relevant for a great variety of materials with different cubic, hexagonal and tetragonal crystal structures such as metals, alloys and others. For the cubic crystal system, the Debye temperature can be obtained by Y0 ¼ C c n1=6 ðaG c =MÞ1=2 ,

(23)

where Cc is a further constant and n is the number of atoms in the crystallographic unit cell. And a is lattice parameter, M the average atomic weight, which is the weighted arithmetical average of the masses of the species for compounds. The elastic modulus Gc can be

AlYb

0K

-2.75

100K

Free energy (eV)

AlSc AlY AlLa AlCe AlPr AlNd AlPm AlSm AlEu AlGd AlTb AlDy AlHo AlEr AlTm AlYb AlLu

n

Rigid 200K

-2.80

300K

-2.85

400K

-2.90 1.56

1.58

1.60 1.62 RWS (Angstrom)

1.64

Fig. 1. The free energy of the B2-AlYb in finite temperatures.

written as n  o1=3 1=2 ðc11  c12 þ c44 Þ=3 . Gc ¼ c44 c44 ðc11  c12 Þ=2

(24)

And fitting to the data sets of all analyzed cubic materials yielded Cc ¼ (26.0570.81) K(m Kg N1). Siethoff [35,36] pointed out that in this case the square-root relation of Eq. (23) is only met within an error limit of approximately 15%. In the present work, the K(v), Debye sound velocity and Debye temperature are calculated using Eqs. (6)–(8) and Eq. (23) and the results are listed in Table 1. The scaling factor K(v) is changed from 0.745 to 0.831, and which are some what larger than the value of 0.617. This shows that Debye temperature could possibly be underestimated by using the Moruzzi et al.’s scaling factor. But the Debye

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where x ¼ (V/V0)1/3, B0 is the bulk modulus, B00 is the pressure derivative of the bulk modulus and V0 is the equilibrium volume. The calculated Gru¨neisen constants are listed in the Table 1 for the B2-AlRE. The high-temperature Gru¨neisen constant, that is Slater expression for g would yield the present values plus an additive constants of 13. Mukherjee [38] proposed an empirical relationship between vacancy formation energy E1f and Debye temperature for cubic metals firstly. Glyde [39] additionally proposed a similar relationship for the vacancy migration

temperatures of Siethoff are somewhat 12% smaller than the present calculated Debye temperature obtained from bulk modulus, as for the AlEu and AlYb, only about 3% smaller than the present values. The Gru¨neisen constant g0 can been obtained from Eq. (14) at equilibrium volume, and the q2 P=qV 2 and qP=qV can be calculated from Vinet’s equation of state   3B0 ð1  xÞ 3ðB00  1Þ ð1  xÞ , (25) exp P¼ x2 2 20 Sc

3

Y

2

Pr

15 La Gd Ce

5

Yb

0

Pm

Tb

1

Eu

0 3

3 Ho

Dy Sm

2 Nd

2

Tm

Er

Lu

1

1

0 0

100

200

300

0 400 Temperature (K)

100

200

300

Fig. 2. The coefficients of thermal expansion for the B2-AlRE.

3.0

2.5 Yb(Eu) CV (10-4eV/atom∗K)

α (10-5/K)

10

2.0 Y

1.750 La Ce Pr Nd Pm Sm Gd Tb Dy Ho Er Tm Lu

1.5 1.725

Sc 1.700

1.0

1.675

0.5 1.650 80

81

82

83

84

85

0.0 0

100

200 Temperature (K)

300

Fig. 3. The heat capacities of the B2-AlRE.

400

400

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energy Em. Since E1f+Em ¼ ESD, the activation energy for self-diffusion can be eventually expressed as E SD ¼ KMa2 Y20 ,

(26)

where K is a model constant. As usual, the Debye temperature Y0 was obtained from the long-wavelength approximation at low temperature. On the other hand, ESD is measured at higher temperatures. But Siethoff et al. point out that the Y0 and Debye temperatures obtained at room temperature only differ by a constant amount for the FCC metals [35]. And Eqs. (23) and (26) can be combined to yield E SD ¼ Ca3 GC ,

(27)

where C is the future parameters, and after fitting some experimental data, C is obtained as 0.26 for the B2 compounds. Though this relationship cannot describe the BCC metals, the intermetallics with the B2 structure’s data also exhibit a slope closed to the unity. In the present work, the activation energies for self-diffusion of the B2-AlRE are calculated by employing the Siethoff et al.’s relationship, and listed in Table 1. Gibbs free energies are obtained at finite temperatures in this work. The calculated Gibbs free energy curves for B2AlYb (as a example) at several temperatures are shown in Fig. 1. And the total energy for the rigid lattice is also shown in the Fig. 1. From Fig. 1, the Gibbs free energy curves for 0 and 100 K are higher than the rigid lattice energy due to the zero point energy. As the temperature increases, Gibbs free energy becomes more negative because of the vibrational entropy’s contribution. The calculated coefficients of the thermal expansion are also shown in Fig. 2. From Fig. 2, the coefficients of the thermal expansion of the AlSc and AlY are larger than the others, and AlEu and AlYb’s coefficients of the thermal expansion are also larger than the other Al-lanthanides, whose coefficients of the thermal expansion are similar. The heat capacities for the B2-AlRE are also obtained and shown in the Fig. 3. The heat capacities for the AlSc and AlY are smaller than the others, and the heat capacity of AlYb is larger than the other Al-lanthanides. There are not available experimental thermal expansion coefficients and heat capacities or other theoretical calculations for the B2AlRE, and so some experimental works are needed for the comparison in the future. 4. Conclusion Debye temperatures, Gru¨neisen constants, Gibbs free energies, coefficients of thermal expansion and heat capacities of the B2-AlRE are obtained by using the Debye–Gru¨neisen model and first principles calculations. The activation energies for self-diffusion, Poisson’s ratio of the B2-AlRE are also calculated in the present work. There

are not experimental values or other theoretical calculations for the thermal properties of the B2-AlRE, and the detailed comparison cannot be performed in the present and some further experiments should be needed for the verification. Combining the first principles elastic constants calculation and Debye–Gru¨neisen model, the thermal properties of the metastable phase or intermetallic compounds can be obtained and it is useful in the materials research. References [1] V.L. Moruzzi, J.F. Janak, K. Schwarz, Phys. Rev. B 37 (1988) 790. [2] M.A. Blanco, E. Francisco, V. Luan˜a, Comput. Phys. Commun. 158 (2004) 57. [3] A. Dinsdale, CALPHAD 15 (1991) 317. [4] Q. Chen, B. Sundman, Acta Mater. 49 (2001) 947. [5] M.W. Chase, I. Ansara, A. Dinsdale, G. Eriksson, G. Grimvall, L. Ho¨glund, H. Yokokawa, CALPHAD 19 (1995) 437. [6] W. Kohn, L. Sham, Phys. Rev. 140 (1965) 1133. [7] S.A. Ostanin, V.Y. Trubitsin, Phys. Rev. B 57 (1998) 13485. [8] J.M. Sanchez, J.P. Stark, V.L. Moruzzi, Phys. Rev. B 44 (1991) 5411. [9] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [10] K. Kunc, R.M. Martin, Phys. Rev. Lett. 48 (1982) 406. [11] W. Frank, C. Elsa¨sser, M. Fa¨hnel, Phys. Rev. Lett. 74 (1995) 1791. [12] P. Debye, Ann. d. Physik 39 (1912) 789. [13] L. Fast, J.M. Wills, B. Johansson, O. Eriksson, Phys. Rev. B 51 (1995) 17431. [14] C.L. Fu, J. Zou, M.H. Yoo, Scripta Metal. Mater. 33 (1995) 885. [15] S. Hong, C.L. Fu, Intermetallics 7 (1999) 5. [16] G.S. Neumann, L. Stixrude, R.E. Cohen, Phys. Rev. B 60 (1999) 791. [17] C.L. Fu, X.D. Wang, Y.Y. Ye, K.M. Ho, Intermetallics 7 (1999) 179. [18] Y.B. Lee, B.N. Harmon, J. Alloys Compds. 338 (2002) 242. [19] C. Jiang, B. Gleeson, Scripta Mater. 55 (2006) 759. [20] P.E. Blo¨chl, Phys. Rev. B 50 (1994) 17953. [21] G. Kresse, J. Joubert, Phys. Rev. B 59 (1999) 1758. [22] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169. [23] G. Kresse, J. Furthmuller, Comput. Mater. Sci. 6 (1996) 15. [24] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [25] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, et al., Phys. Rev. B 46 (1992) 6671. [26] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1972) 5188. [27] G. Leibfried, W. Ludwig, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 12, Academic, New York, 1961, p. 275. [28] G.A. Alers, in: W.P. Mason (Ed.), Physical Acoustics, vol. III-B, Academic, New York, 1965, p. 1. [29] O.L. Anderson, in: W.P. Mason (Ed.), Physical Acoustics, vol. III-B, Academic, New York, 1965, p. 43. [30] G. Grimvall, Thermophysical Properties of Materials, Amsterdam, North-Holland, 1986. [31] X.M. Tao, Y.F. Ouyang, H.S. Liu, F.J. Zeng, Y.P. Feng, Z.P. Jin, Comp. Mater. Sci. (2007), doi:10.1016/j.commatsci.2006.12.001. [32] J.C. Slater, Introduction to Chemical Physics, McGraw-Hill, New York, 1939. [33] T.H.K. Barron, Philos. Mag. 46 (1955) 720. [34] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, J. Phys.: Condens. Matter 1 (1989) 1941. [35] H. Siethoff, K. Ahlborn, Phys. Stat. Sol. (b) 190 (1995) 179. [36] H. Siethoff, K. Ahlborn, J. Appl. Phys. 79 (1996) 2968. [37] H. Siethoff, Intermetallics 5 (1997) 625. [38] K. Mukherjee, Philos. Mag. 12 (1965) 915. [39] H.R. Glyde, J. Phys. Chem. Solids 28 (1967) 2061.