first principles calculations of thermodynamic quantities and phase diagrams of high temperature bcc taw and mota alloys

Fig.2 The calculated equilibrium phase diagram and melting temperatures of Ta-W alloys (a) and Mo-Ta alloys (b): cal-2 (or cal-1) represents the phase boundaries between B2 [r]

First Principles Calculations of Thermodynamic Quantities and Phase Diagrams of High Temperature BCC Ta-W and Mo-Ta Alloys

K Masuda-Jindo1,a*, Vu Van Hung2,b and P.E.A Turchi3,c

1

Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta 4259, Midori-ku, Yokohama 226-8503, Japan

2

Department of Physics, Hanoi National Pedagogic University, km8 Hanoi-Sontay Highway, Hanoi, Vietnam

3

Lawrence Livermore National laboratory, PO Box 808, L-353 LLNL, Livermore CA 94551 U.S.A

a

kmjindo@issp.u-tokyo.ac.jp, b bangvu57@yahoo.com, c turchi1@llnl.gov

Keywords: statistical moment method, cluster variation method, TB-LMTO method, coherent potential approximation, Ta-W and Mo-Ta alloys

Abstract The thermodynamic properties of high temperature metals and alloys are studied using the statistical moment method, going beyond the quasi-harmonic approximations Including the power moments of the atomic displacements up to the fourth order, the Helmholtz free energies and the related thermodynamic quantities are derived explicitly in closed analytic forms The configurational entropy term is taken into account by using the tetrahedron cluster approximation of the cluster variation method (CVM) The energetics of the binary (Ta-W and Mo-Ta) alloys are treated within the framework of the first-principles TB-LMTO (tight-binding linear muffin tin orbital) method coupled to CPA (coherent potential approximation) and GPM (generalized perturbation method) The equilibrium phase diagrams are calculated for the refractory Ta-W and Mo-Ta bcc alloys

1 Introduction

The calculations of the thermodynamic quantities and the equilibrium phase diagrams are of great importance for the purpose of materials designs and developments of high temperature technological alloys It is the purpose of the present article to study the thermodynamic quantities of metals and alloys using the moment method in the quantum statistical mechanics, hereafter referred to as the statistical moment method (SMM) [1-4] We firstly derive the Helmholtz free energy Ψ(V,T), of metals and alloys using the fourth order moment approximation, and then calculate the thermodynamic quantities, such as the thermal lattice expansions, root mean square atomic displacements, specific heats, Grüneisen constants and elastic moduli The application calculations using the SMM will be done for the high temperature bcc alloys, like Ta-W and Mo-Ta systems Recently, much attention has been paid to alloy systems made of refractory metals of columns VB and VIB of the Periodic Table [5,6], and in particular, Nb, Mo, Ta, and W that display high melting temperature suitable for space and nuclear applications In view of this, we calculate the equilibrium phase diagram of Ta-W and Mo-Ta alloys including the effects of thermal lattice vibrations

2 Theory

We derive the thermodynamic quantities of metals and alloys, taking into account the higher (fourth) order anharmonic contributions of the thermal lattice vibrations which goes beyond the quasi-harmonic (QH) approximation [7] Within the fourth order moments approximation of the SMM the free energy of the system is given by

Advanced Materials Research Vols 26-28 (2007) pp 205-208 online at http://www.scientific.net

© (2007) Trans Tech Publications, Switzerland Online available since 2007/10/02

( )

( ) ( )

2

2 2

0 2

3

2 1

4

2 coth

3 coth ,

3

2 coth coth

coth 2 1 coth

3 2

X X X

U N X n e N X X

k

X X X X

X X X X

k θ θ γ γ θ γ γ γ γ −       Ψ = +  + − +   −  +              +   + − +  +  +        l 　 (1)

where X = hω θ/ , θ being kBT k and γi are second and fourth order vibrational constants [1-10],

respectively With the aid of the free energy formula Ψ=E-TS, one can find the thermodynamic quantities of metals and alloy systems The specific heats and elastic moduli at temperature T are directly derived from the free energy Ψ of the system

The configurational entropies of bcc alloys are calculated using the tetrahedron cluster approximation of the cluster variation method (CVM) [8,9] The nearest-neighbour and next-nearest-neighbour pair probabilities are taken into account in accordance with the effective pair interaction energies derived from the TB-LMTO-CPA formalism The entropy expression S (N) for bcc lattice is given by

( ) ({ } ) { } { } ( ) {( } ) ({ } ) 12 Point exp

Ttrh Pr Pr

N

N N

st nd

N N N

S

k n n

∆   =       (2)

In the present study, we will use the so-called generalized perturbation method (GPM) for the energetics of the bcc alloys composed of Ta, Mo and W elements [5,6] Within the GPM, only the configuration-dependent contribution to the total energy is expressed by an expansion in terms of effective pairwise and multisite interactions Then, the ordering contribution to the total energy of an A-B alloy is given by

{ }

( )i ( ) ( ){ }i

n dis ord n

E p =E c + ∆E p , (3)

{ }

( ) (2) (3)

, , , , ,

, ,

1

2

ij i j ijk i j k

ord n nm n m nml n m l

ij ij

n m n m n l m

n m m l n l

E p V c c V c c c

N δ δ N δ δ δ

≠

≠ ≠ ≠

∆ = ∑ + ∑ +L, (4)

( ) ( )

1

Im EF ,

ij

s s s ij ij

V dE G Gλµ µλ tλ tµ

λµ

π

= − ∫ ∑ ∆ ∆ K (5)

where

i

n

c

δ refers to the fluctuation of concentration on site ni, δcni = pni −c , (c is the

concentration in B species), and pi is an occupation number associated with site ni, equal to or

depending on whether or not site ni is occupied by a B species The ( )1, k k

n n

V L corresponds to a

kth-order effective cluster interaction (ECI) involving a cluster of k sites Results and Discussions

In Fig.1(a) we show the thermal expansion coefficients, αv, of bcc Mo, Ta and W crystals at zero

pressure as a function of temperature T, together with the experimental results [10] The thermal expansion coefficients αv of bcc Mo, Ta and W crystals are shown by dot-dashed, solid and dashed

lines, respectively, and they are in good agreement with the corresponding experimental results In particular, the calculated thermal expansion coefficients of the bcc Ta crystals are in fairly good agreement with the experimental results expect for higher temperature region than ~2000K For this higher temperature region, experimental results, by symbols ×, show the anomalous increase of the thermal expansion coefficients as the temperature increases (which might be attributed to the extrinsic causes such as the oxidiation of the specimen) Instead, the present SMM calculations of the thermal lattice expansion coefficients of bcc Ta crystal (solid curve) are in good agreement with the ab initio theoretical calculations of Ref.[11], symbols ○, using the anharmonic PIC (particle in a cell) model

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Fig.1 a) Thermal lattice expansion coefficients calculated for bcc Mo, Ta (×) and W (▲) crystals b) Temperature dependence of root mean square displacements u2j (solid lines) and root mean square relative displacements σ (dashed lines) for bcc Mo, Ta and W crystals 2j

In Fig.1 (b), we present the calculated root mean square atomic displacements u2j , by solid lines, and root mean square relative displacements

j

σ , by dashed lines, for bcc Mo, Ta and W crystals, as a function of the temperature The relative magnitudes of

j

u and j

σ among the bcc elements are such that u2j Ta > u2j Mo > u2j W and

2 2

j Ta j Mo j W

σ > σ > σ For the lower temperature region, i.e., T <1500K

% , both

2 j

u and σ2j have the linear temperature dependence, but for higher temperature region, they increase nonlinearly as the temperature increases The nonlinear increases of u2j and σ2j indicate the importance of the anharmonicity of thermal lattice vibrations in the higher temperature region

To calculate the thermodynamic quantities and the equilibrium phase diagram of bcc Mo-Ta and Ta-W alloys, we use the cluster variation method (CVM) [8,9] and the first-principles TB-LMTO method coupled to the coherent potential approximation (CPA) and the generalized perturbation method (GPM) [5,6] We calculate the change in the free energy ∆Ψ(eV/atom) due to the inclusion of the thermal vibration effects of bcc Mo-Ta and Ta-W bcc alloys as a function of the temperature T; the concentrations of tantalum are chosen to be 0.0, 0.1, 0.2, 0.25, 0.3, 0.33, 0.4 0.5, 0.6, 0.67, 0.7, 0.8, 0.9 and 1.0 Here, the change in the free energy ∆Ψ corresponds to the ordering (free) energy defined by "∆Ψord = Ψ + ΨAA BB- 2Ψ " as in the conventional treatments without thermal AB lattice vibration effects

The resulting equilibrium phase diagrams of bcc Ta-W alloys are presented in Fig.2 The dark circles connected by solid lines in Fig.2(a) represent the phase boundaries between B2 and A2 phases of bcc Ta-W alloys, including the thermal lattice vibration effects (cal-2) while the white circles connected by dashed lines are the phase boundaries without including the thermal lattice vibration effects (cal-1) [5,6] It can be seen in Fig.2(a) that the B2 phases of Ta-W alloys are stabilized more strongly by including the anharmonicity effects of thermal lattice vibrations at the higher Ta concentration The similar tendencies are more prominent for Mo-Ta alloys, as can be seen in Fig.2(b) This type of theoretical findings is of great interest since the inclusion of the thermal lattice vibration effects is believed in most cases to destabilize the ordered phases as in the CuAu alloys [12] In addition, we have calculated the melting temperatures Tm (critical temperature

of the crystalline stability) of the disordered Mo-Ta and Ta-W alloys using the SMM [1-4], and the results of melting temperatures Tm of Ta-W and Mo-Ta alloys are presented in Fig.2, in comparison

with the experimental liquidus (dashed line) and solidus (solid line) curves Although the direct comparison between the theoretical Tm and experimental liquidus and solidus curves is not possible,

one sees that there are good correlations between the calculation and experimental results

Fig.2 The calculated equilibrium phase diagram and melting temperatures of Ta-W alloys (a) and Mo-Ta alloys (b): cal-2 (or cal-1) represents the phase boundaries between B2 and A2 phases, calculated by including (or not including) the effects of thermal lattice vibrations

4 Conclusions

We have presented the SMM formalism combined with the CVM and investigated the thermodynamic properties of high temperature bcc metals and alloys composed of Mo, Ta and W elements The linear thermal expansion coefficients, bulk modulus and root-mean-square atomic displacements are calculated as a function of the temperature as well as a function of the alloy compositions The calculated results of the thermodynamic quantities are in good agreement with the corresponding experimental results The equilibrium phase diagrams are calculated for bcc Ta-Mo and Ta-W alloys, including the anharmonicity of thermal lattice vibrations It has been shown that the B2 phases of Ta-W alloys are stabilized more strongly by including the anharmonicity of thermal lattice vibrations for higher Ta concentration region The similar tendency has also seen found for Ta-Mo alloys

References

[1] K Masuda-Jindo, V V Hung and P D Tam, Phys Rev B67, 094301 (2003) [2] K Masuda-Jindo, S R Nishitani and V V Hung, Phys Rev B70, 184122 (2004)

[3] V V Hung, K Masuda-Jindo and P H M Hanh, J Phys.: Condens Matter 18, 283 (2006) [4] V V Hung, J Lee and K Masuda-Jindo, J Phys Chem Solids, 67, 682 (2006)

[5] P E A Turchi, A Gonis, V Drchal and J Kudrnocsky, Phys Rev B64, 085112 (2001)

[6] P E A Turchi, V Drchal, J Kudmocský, C Colinet, Larry Kaufman and Zi-Kui Liu, Phys Rev B71, 094206 (2005)

[7] V L Moruzzi, J F Janak and K Schwarz, Phys Rev B37, 790 (1988) [8] R Kikuchi, Phys, Rev 81, 988 (1951)

[9] R Kikuchi and K Masuda-Jindo, Comp Mat Sci 8, (1997)

[10] Y S Touloukian, R K Kirby, R E Taylor, and P D Desai, Thermophysical properties of Matter ( Thermal Expansion-Metallic Elements and Alloys, Vol 12) (Plenum Press, New York, 1975)

[11] R E Cohen and O Gülseren, Phys Rev B63, 224101 (2001)

[12] T Horiuchi, S Takizawa, T Suzuki and T Mohri, Metall Mater Trans 26, 11 (1995)

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