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Acta Materialia 57 (2009) 2657–2664

Influence of vibrational entropy on structural stabilityof Nb–Si and Mo–Si systems at elevated temperatures

Yue Chen a,b, T. Hammerschmidt b, D.G. Pettifor b, Jia-Xiang Shang a,*, Yue Zhang a

a School of Materials Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinab Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK

Received 23 January 2009; received in revised form 10 February 2009; accepted 10 February 2009Available online 1 April 2009

Abstract

The heats of formation of stable and metastable phases of the Nb–Si and Mo–Si systems were studied using density functional theory(DFT). The high-temperature behavior of the competing phases was studied by performing additional phonon calculations. Our theo-retical results rationalize the major differences observed in the behavior of the Nb–Si and Mo–Si systems: Nb3Si is only stable at tem-peratures above 2043 K, whereas Mo3Si is always stable; Nb5Si3 and MoSi2 undergo phase changes at elevated temperatures, in contrastto Mo5Si3 and NbSi2. These differences are qualitatively explained by including the vibrational entropy to the free energies within theharmonic approximation. In particular, the softer shear moduli of the Nb5Si3 and MoSi2 b phases cause their stabilities over the a phasesat elevated temperature.� 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Transition metal silicides; Ab initio electron theory; Phase transformations; Elastic behavior

1. Introduction

A central goal in the development of high-temperaturematerials for the next generation of turbine blades is toovercome the current barriers in operating temperatureassociated with Ni-based superalloys. Recently, Nb–Siand Mo–Si-based superalloys have been extensively studiedas candidate materials due to their high melting points andlow densities [1–9]. Although the atomic numbers of Nband Mo differ by only 1, the Nb–Si and Mo–Si binary sys-tems show very different microstructural and mechanicalproperties. In order to understand this difference, it is nec-essary to investigate the origin of the thermodynamic dif-ferences that are observed in their phase diagrams.

The binary phase diagrams of Nb–Si and Mo–Si inFig. 1 show three different intermetallic compounds in bothsystems, namely TM3Si; TM5Si3 and TMSi2, where TMrefers to the transition metal, Nb or Mo. The predominant

1359-6454/$36.00 � 2009 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2009.02.014

* Corresponding author.E-mail address: shangjx@buaa.edu.cn (J.-X. Shang).

differences in the phase diagrams are: Nb3Si becomes stablefor temperatures above 2043 K, whereas Mo3Si is alwaysstable; Nb5Si3 and MoSi2 change their crystal structuresat elevated temperatures, but Mo5Si3 and NbSi2 do not.Furthermore, the corresponding low-temperature phasesin the two systems have different crystal structures. A sys-tematic first-principles study of the thermodynamics ofthe Nb–Si and Mo–Si binary systems is, therefore, essentialfor understanding the origin of the above differences.

In this paper, we first discuss the heats of formationcorresponding to the eight different observed structure-typeswith 3:1, 5:3 and 1:2 stoichiometries that were found in thedatabase of the Pauling File [11]: cP4-Cu3AuðL12Þ,cP8-Cr3SiðA15Þ, tP32-Ti3P, hP16-Mn5Si3ðD88Þ, tI32-Cr5B3

ðD81Þ, tI32-W5Si3ðD8mÞ, hP9-CrSi2ðC40Þ and tI6-MoSi2(C11b). The structures are denoted by Pearson Symbols to-gether with crystal structure prototypes and availableStrukturbericht Symbols in brackets. In addition, thebody-centered cubic (bcc)-based structures cF16-BiF3

ðD03Þ and cP2-CsCl(B2), as well as the diamond-basedstructure cF8-ZnS(B3), have been investigated. Finally,

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Fig. 1. Equilibrium phases diagrams of Nb–Si and Mo–Si systems [10].

2658 Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664

we have used the phenomenological Pettifor Map [12] toidentify three other possibly metastable structure types,namely cP8-FeSi(B20), oP8-FeB(B27) and tI12-CuAl2

ðC16Þ. We have used density functional theory (DFT) tocalculate the heats of formation at absolute zero of Nb–Si and Mo–Si binary systems with these stoichiometriesand structure types. We have then calculated the phonondensity of states of competing phases and evaluated theirfree energies as a function of temperature. We find thatthese free energies are qualitatively reproduced by the sim-ple Debye model [13]. Our DFT and phonon calculationsrationalize the observed differences between the Nb–Siand Mo–Si phase diagrams.

2. Methodology

The DFT calculations are carried out with the ViennaAb initio Simulation Package (VASP) [14] within the gen-eralized gradient approximation (GGA) [15] using projec-tor augmented wave (PAW) pseudopotentials. Weverified our results of the heats of formation of competingphases by additional calculations with the local densityapproximation (LDA) [16]. We found in convergence teststhat an energy cutoff of 360 eV is sufficient to reduce theerror in the total energy to less than 1 meV/atom. Weemployed 12� 12� 12k-point meshes generated by theMonkhorst–Pack scheme [17] for numerical integrationsover the Brillouin zone of the bcc unit cell, and used thesame k-point density for all other structures. The cell shapeand the internal atomic coordinates were fully relaxed in all

our total energy computations until the change in totalenergy from the electronic relaxation was less than10�4 eV/atom.

The phonon calculations employed the small displace-ment method using the PHON package [18] in conjunctionwith VASP. We performed more accurate relaxations priorto the phonon calculations to ensure that the change intotal energy from the electronic and ionic relaxation is lessthan 10�8 and 10�7 eV, respectively. Due to the differentsize of different primitive cells, phonon dispersions werecomputed using a 4� 4� 4 supercell for the bcc structure;a 3� 3� 3 supercell for the tI6-MoSi2 and diamond struc-tures; a 2� 2� 2 supercell for the tI12-CuAl2, hP9-CrSi2,tI32-Cr5B3, tI32-W5Si3, cP8-Cr3Si and hP16-Mn5Si3 struc-tures; and a 1� 1� 2 supercell for the tP32-Ti3P structure.

3. Heats of formation

The Nb–Si phase diagram in Fig. 1 displays four stableintermetallic phases: Nb3SiðtP32-Ti3PÞ, a-Nb5Si3 ðtI32-Cr5B3Þ; b-Nb5Si3ðtI32-W5Si3Þ and NbSi2ðhP9- CrSi2Þ. Onthe other hand, the Mo–Si phase diagram displays differentstable intermetallic phases: Mo3SiðcP8-Cr3SiÞ;Mo5Si3ðtI32-W5Si3Þ; a-MoSi2 ðtI6-MoSi2Þ and b-MoSi2ðhP9-CrSi2Þ. In addition to these six distinctstructure types, we considered hP16-Mn5Si3 and cP4-Cu3Au (observed in Nb–Si systems [2,20]), the bcc-basedstructures cF16-BiF3 and cP2-CsCl, as well as the dia-mond-based structure cF8-ZnS(B3). We completed ourset of investigated structures by cP8-FeSi(B20), oP8-FeB(B27) and tI12-CuAl2ðC16Þ, which are observed innearby stable domains within the AB and AB2 structuremaps of Pettifor [12].

We have calculated the heats of formation of Nb–Si andMo–Si phases with respect to these 14 structure types forstoichiometries 3:1, 2:1, 5:3, 1:1, 3:5, 1:2 and 1:3, respec-tively. The heat of formation per atom at 0 K of a transi-tion metal silicide TMmSin is obtained from

DH elf ¼ ETMmSin � mETM � nESi

� �=ðmþ nÞ; ð1Þ

where m and n are the number of TM and Si atoms, respec-tively, in the unit cell. The three values of the energy E onthe right-hand side of Eq. (1) correspond to the total en-ergy of the (m + n) atoms in the unit cell of TMmSin andthe total energies per atom of the ground state structuresof elemental TM (Nb or Mo) and Si, respectively. Thesuperscript el attached to the heat of formation is to remindus that it is the value resulting from solving the total elec-tronic energy at 0 K of the system. The calculated heats offormation are shown in Fig. 2. We see that the Nb–Si andMo–Si systems have very different convex hulls. (Only thephases with heats of formation on or below the convex hullcan be stable; all others will decompose.) In order to verifyour GGA results for the stability of phases with heats offormation close to the convex hull, we performed addi-tional LDA calculations for these phases. The LDA results

Fig. 2. Heats of formation of various structures with different compositions in Nb–Si and Mo–Si systems. Solid lines denote convex hulls, whereas dottedand dot-dashed lines join bcc-based and diamond-based phases, respectively. Energy of above structure types also calculated at end points relative to bccand diamond ground states by replacing all atoms in unit cell by single element, denoting each value by symbol of particular structure type.

Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664 2659

show a general shift in the heats of formation towards low-er values compared to GGA, but the relative positions ofthe heats of formation with respect to the convex hull showthe same phase equilibria. This is shown in Table 1, wherewe see that the values calculated by GGA are in betteragreement with the experimental results than LDA. Themore negative values of LDA are the result of its well-known over-binding [19].

The low-temperature stable phases observed in the phasediagrams of Fig. 1 are reproduced by our DFT calculationsat T = 0 K in Fig. 2 for all Si concentrations. For the Nb–Sisystem, the stable phases change from bcc Nb throughNb5Si3ðtI32-Cr5B3Þ and NbSi2ðhP9-CrSi2Þ to diamond Si.For the Mo–Si system, the stable phases change frombcc Mo through Mo3SiðcP8-Cr3SiÞ, Mo5Si3 ðtI32-W5

Si3Þ and MoSi2ðtI6-MoSi2Þ to diamond Si.Our DFT calculations also rationalize one of the main

differences between the Nb–Si and Mo–Si systems: Nb3Siis not stable at low temperature but Mo3Si is stable. Wefind that Nb3Si will decompose to Nb and Nb5Si3 at lowtemperature since its heat of formation lies above the con-vex hull, whereas Mo3Si lies below the tie line between Moand Mo5Si3, so that it becomes the stable ground statephase. The instability of Nb3Si is mainly caused by thelower relative position of the heat of formation of

Table 1Heats of formation (eV/atom) of stable intermetallic phases in Nb–Si andMo–Si systems as obtained from our DFT calculations and fromexperiment (the numbers with star symbols are average values).

Stable intermetallics GGA LDA Experiment

Nb–Si Nb5Si3ðD81Þ �0.6521 �0.7119 �0.6657* [21,23]NbSi2ðC40Þ �0.5078 �0.6367 �0.5568 [23]

Mo–Si Mo3SiðA15Þ �0.2911 �0.3570 �0.2675 [22]Mo5Si3ðD8mÞ �0.3737 �0.4527 �0.3899* [22–25]MoSi2ðC11bÞ �0.4520 �0.5961 �0.4713* [22–28]

Nb5Si3ðtI32-Cr5B3Þ as compared to Mo5Si3ðtI32-W5Si3Þ.This effect can be understood in terms of the electronic den-sities of states which are shown in Fig. 3: the Fermi energyof Nb5Si3ðtI32-Cr5B3Þ is located closer to a deep pseudo-gap arising from the d electrons on the Nb atoms comparedto Mo5Si3ðtI32-W5Si3Þ, which gives it a higher stability,and leads to the instability of Nb3Si at low temperature.

The heats of formation of the three possible metastablestructures cP8-FeSi, oP8-FeB and tI12-CuAl2, which havebeen located in the Pettifor Maps, lie above the convexhulls. However, the heats of formation of Nb2Si and

Fig. 3. Total and partial density of states of Nb5Si3 and Mo5Si3 withtI32-Cr5B3 and tI32-W5Si3 crystal structures, respectively.

Table 2Zero-point energies (eV/atom) of different phases.

Phase Nb Si Nb3Si Nb2Si a-Nb5Si3 b-Nb5Si3

Structure cI2-W cF8-C tP32-Ti3P tI12-CuAl2 tI32-Cr5B3 tI32-W5Si3ZPE 0.027 0.060 0.039 0.043 0.045 0.042

Phase Mo Mo3Si Mo5Si3 Mo5Si3 a-MoSi2 b-MoSi2Structure cI2-W cP8-Cr3Si hP16-Mn5Si3 tI32-W5Si3 tI6-MoSi2 hP9-CrSi2ZPE 0.036 0.041 0.037 0.043 0.052 0.051

Fig. 4. Phonon contributions to the heat of formation of different phasesin Nb–Si and Mo–Si systems.

2660 Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664

Mo2Si with the tI12-CuAl2 structure are very close to thetwo convex hulls and could possibly, therefore, be detectedexperimentally. We also found that the experimentallyobserved Nb3Si with the cP4-Cu3Au structure [20] is unli-kely since its theoretical heat of formation lies far abovethe convex hull.

The good agreement between our DFT results andexperiment demonstrates the power of first-principles cal-culation for investigating phase stability at low tempera-ture. However, the investigation of the high-temperaturephases in the Nb–Si and Mo–Si systems, e.g. the transitionfrom a-Nb5Si3 to b-Nb5Si3 at 2213 K, and from a-MoSi2 tob-MoSi2 at 2173 K, as well as the appearance of the Nb3Siphase above 2043 K, requires the inclusion of the phononcontribution to the entropy, as discussed in the nextsection.

4. Free energy

The observed stability of Nb3Si at finite temperature andthe phase transformations between a-Nb5Si3=b-Nb5Si3 and a-MoSi2=b-MoSi2 suggest that the entropiccontributions play an important role in their structural sta-bility. For strongly ordered intermetallic compounds, themain contribution is the vibrational entropy, since the con-figurational and electronic contributions are generally rela-tively small [29]. The free energy per unit cell of atransition metal silicide TMmSin can then be written as:

F TMmSinðT Þ ¼ ETMmSin þ F phonðT Þ; ð2Þwhere F phonðT Þ is the phonon contribution to the free en-ergy. It is given within the harmonic approximation by[13,30,31]:

F phonðT Þ ¼ 1

2

Z xmax

0

gðxÞ�hxdxþ kBTZ xmax

0

gðxÞ

� ln 1� exp � �hxkBT

� �� �dx; ð3Þ

where gðxÞ is the phonon density of states (DOS) per unitcell, �h is the reduced Planck constant and kB is the Boltz-mann constant. The first term on the right-hand side ofEq. (3) is the zero point energy.

The DFT calculations of the phonon DOS have beencarried out for the competing phases with the tI32-Cr5B3,tI32-W5Si3, tP32-Ti3P and tI12-CuAl2 structures in theNb–Si systems, and the phases with the hP16-Mn5Si3,tI32-W5Si3, cP8-Cr3Si, hP9-CrSi2 and tI6-MoSi2 structures

in the Mo–Si systems, as well as for bcc-Nb, bcc-Mo anddiamond-Si. The vibrational entropic contributions to thefree energy were then obtained by inserting the DFT resultsof the phonon DOS in Eq. (3). Following Eq. (1) the finitetemperature heat of formation per atom is defined asfollows:

DH f ðT Þ ¼ DH elf þ DH phon

f

¼ F TMmSinðT Þ � mF TMðT Þ � nF SiðT Þ�

ðmþ nÞ:ð4Þ

The zero-point energies of ground state or competingphases in the Nb–Si and Mo–Si phase diagrams are sum-marized in Table 2, where we observe that they contributeapproximately 40 meV/atom to the binding energy of eachsystem. This usually results in only small corrections to theheats of formation at T = 0 K as can be seen in Fig. 4.

It can also be seen from Fig. 4 that the temperaturedependence of the phonon contribution to the heats of for-mation favors the b phase over the a phase for both the

Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664 2661

Nb5Si3 and the MoSi2 stoichiometries, as expected experi-mentally. This behavior can be understood within the sim-ple Debye model [32,33] for the phonon DOS. It assumesthat the dispersion relation is that of a classical continuum,namely:

x ¼ tk; ð5Þwhere t is the velocity of sound, x is the angular phononfrequency and k is the wave vector. The correspondingphonon DOS is given by:

gðxÞ ¼ 3V x2

2p2Nt3; ð6Þ

where V is the volume of the specimen. This phonon DOSis cut off at the Debye frequency xD when the total numberof phonon states in a solid with N atoms have been ac-counted for, namely:

xD ¼ ð6p2N=V Þ1=3t: ð7ÞThe Voigt–Reuss–Hill (VRH) approximation [34] has

been shown [35,36] to give errors of less than 2% in theDebye temperature HD (where kBHD ¼ �hxD) by writingthe velocity of sound as:

3

t3¼ 1

t3L

þ 2

t3T

; ð8Þ

with a longitudinal velocity

qt2L ¼ Bþ ð4=3ÞG; ð9Þ

Table 3Calculated elastic constants (in GPa) of different phases in the Nb–Si and Mo

C11 C12

Nb Theoretical 244.5 128.0Experimental [38] 253 133

Mo Theoretical 449.5 163.3Experimental [38] 450 173

Si Theoretical 162.4 59.1Experimental [39] 166.0 64.0

a-Nb5Si3 Theoretical 372.9 96.0b-Nb5Si3 Theoretical 378.3 119.3Nb3Si Theoretical 290.2 150.5a-MoSi2 Theoretical 400.9 108.4

Experimental [40] 404.8 116.7b-MoSi2 Theoretical 427.0 78.6

Table 4Calculated bulk moduli (in GPa), shear moduli (in GPa) and Debye temperatuapproximation and their reported Debye temperatures obtained from heat cconstants ða-MoSi2Þ.

Nb Nb3Si a-Nb5Si3 b-Nb5

B 166.8 184.1 193.1 193.1G 36.6 96.4 127.9 109.8HD 266 462 555 515

275 [32] 430 [41]

and a transverse velocity

qt2T ¼ G; ð10Þ

where q is the density. The bulk modulus B and shear mod-ulus G are written explicitly in terms of the elastic constantsCij in Ref. [34].

Table 3 gives the elastic constants of the elemental constit-uents Nb, Mo and Si, the a and b phases of Nb5Si3

and MoSi2, and the high-temperature Nb3Si phase that werecalculated by DFT using CASTEP [37]. We see that our pre-dicted values agree well with experiments for the a phase ofMoSi2 and the elemental constituents. Table 4 gives the cor-responding values of the bulk modulus B, shear modulus G

and the Debye temperature HD using the VRH approx-mation.

The resultant Debye phonon DOS are shown in Fig. 5,where we find good agreement with the DFT acoustic spec-trum in the frequency range of up to 2–3 THz. The phononcontributions to the free energies within the Debye approx-imation are plotted together with those of the DFT resultsin Fig. 6. We observe that the Debye model reproducesthe phonon contribution to the free energies better for theelemental phases than for the compounds. Although thephonon contribution to the free energies calculated withinthe Debye model for the a and b phases of Nb5Si3

and MoSi2 are noticeably smaller than those found in theDFT, the offsets are consistent for the whole range of tem-peratures considered. In both approaches the b phases ofNb5Si3 and MoSi2 are stabilized at elevated temperatures.Within the Debye model the stabilization can be explained

–Si systems using DFT and their available experimental values.

C13 C16 C33 C44 C66

31.63197.9

12575.079.6

115.5 338.1 133.5 122.1107.9 318.6 89.2 132.7117.6 9.8 306.0 101.5 119.486.9 508.3 179.5 195.293.0 508.0 201.8 193.7

128.2 423.4 156.4 174.2

res (in K) of different phases in the Nb–Si and Mo–Si systems using VRHapacity measurements (Nb, Si, Mo and Nb3Si) and experimental elastic

Si3 Si Mo a-MoSi2 b-MoSi2

93.5 258.7 207.6 216.958.2 102.8 178.2 159.7608 423 734 695645 [32] 450 [32] 759 [42]

Fig. 5. Phonon density of states per degree of freedom (solid line, reference to left axes) and its integral (dotted line, reference to right axes) for differentphases in Nb–Si (left-hand panel)and Mo–Si (right-hand panel)systems. Debye phonon DOS are given by dashed lines.

Fig. 6. Phonon contribution to free energies calculated by smalldisplacement method (curves) and Debye model using predicted Debyetemperatures (discrete symbols) for Nb–Si and Mo–Si systems.

2662 Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664

as due to the approximately 10% softer sheer moduli of theb phases (Table 4) which leads to lower Debye frequencies(Eqs. (7)–(10)); more of the transverse acoustic modes areexcited at the same temperature thereby driving down theb phase relative to the a phase. While the Debye modelreproduces the stabilization trend for the two sets of isosto-ichiometric compounds, it should, in general, be used withcaution as the Debye temperature is determined only bythe elastic properties and carries little or no informationabout the optical phonon modes.

5. Phase transitions

Phase equilibria at finite temperatures can now be inves-tigated by including the phonon contributions in the heatsof formation DHf ðT Þ. For the Nb–Si system, using theelectronic heats of formation in Fig. 2, we find that for3:1 stoichiometry Nb3Si decomposes to Nb and Nb5Si3 atlow temperature since its heat of formation lies above theconvex hull. When the temperature increases, the heats offormation of both Nb5Si3ðtI32-Cr5B3Þ and Nb3Si ðtP32-Ti3PÞ phases increase due to the phonon contribution ascan be seen in Fig. 4. However, the former phase increasesmuch faster, and, at approximately 2100 K, the Nb3Siphase lies below the tie line between Nb and Nb5Si3 ascan be seen in Fig. 7, so that it becomes stable. This resultis in good agreement with the phase transition temperature

Fig. 7. Temperature dependence of heat of formation differences betweencompeting phases of a and b Nb5Si3 as well as a and b MoSi2. For TM3Si,we demonstrate their stability with respect to the tie line between TM andTM5Si3 by plotting DH TM3Si

f ðT Þ � 2=3DH TM5Si3f ðT Þ.

Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664 2663

of Nb3Si of 2043 K given in the phase diagram (Fig. 1). Onthe other hand, for Mo3Si we notice from Fig. 4 that achange in the temperature leaves the heat of formation ofMo3SiðcP8-Cr3SiÞ nearly unchanged. Even though the heatof formation of Mo5Si3ðtI32-W5Si3Þ decreases slightlywhen the temperature increases, it is not enough to driveMo3Si above the tie line between Mo and Mo5Si3

(Fig. 7). Thus from our calculations we find that Mo3Siis stable at low and elevated temperatures, in line withthe Mo–Si phase diagram in Fig. 1.

For 5:3 stoichiometry, Nb5Si3, we find a phase transitionfrom aðtI32-Cr5B3Þ to bðtI32-W5Si3Þ at elevated tempera-ture (Fig. 7). The vibrational entropy contributions ofNb5Si3ðtI32-Cr5B3Þ increase faster with temperature thanthe competing Nb5Si3ðtI32-W5Si3Þ phase (Fig. 4), and thetotal heats of formation cross at 3000 K; this compares withthe experimental upper and lower transition temperatures of2213 and 1923 K [10], respectively. We attribute this discrep-ancy to the harmonic approximation which we employed inthe DFT calculation of the phonon DOS. On the other hand,for Mo5Si3 there is no phase change: the same crystal struc-ture tI32-W5Si3 is observed up to the melting temperature.From our results in Fig. 4, we come to the same conclusionthat there is no transition between the two competing phasesMo5Si3ðtI32-W5Si3Þ and Mo5Si3ðhP16-Mn5Si3Þ, since withincreasing temperature the heat of formation ofMo5Si3ðtI32-W5Si3Þ goes more negative, whereasMo5Si3ðhP16-Mn5Si3Þ goes more positive, so that they nevercross each other.

For 1:2 stoichiometry, the Nb–Si phase diagram showsthat NbSi2 has the same crystal structure at low and ele-vated temperatures. The vibrational entropic contributionat elevated temperature is too small to affect the relative sta-bility of the two competing structures hP9-CrSi2

and tI6-MoSi2. In the case of Mo–Si, however, the phasediagram shows a transition of MoSi2 from tI6-MoSi2 tohP9-CrSi2 at 2173 K. This value is overestimated in our cal-culations, which yield a phase transition temperature of3460 K.

6. Conclusions

The elements Nb and Mo differ by only one electron intheir valence shell, but the technologically relevant binarysystems Nb–Si and Mo–Si display very different phaseequilibrium behavior and hence microstructure. Accordingto their binary phase diagrams, Nb3Si becomes stable fortemperatures above 2043 K, whereas Mo3Si is always sta-ble; Nb5Si3 and MoSi2 undergo phase transitions at ele-vated temperature, but Mo5Si3 and NbSi2 do not. OurDFT calculations of the heats of formation reproduce thestable intermetallic phases in Nb–Si and Mo–Si systemsthat are observed in the experimental phase diagrams andalso rationalize the instability of Nb3Si at low temperature.By taking into account the vibrational entropic contribu-tions to the heats of formation within the harmonicapproximation, we have successfully explained the differentbehavior of Nb–Si and Mo–Si systems at elevated temper-ature. In particular, the softer shear moduli of theNb5Si3 and MoSi2 b phases cause their stabilities over thea phases at elevated temperature. However, the calculatedphase transition temperatures require careful considerationdue to the harmonic approximation in the phononcalculations.

Acknowledgments

We would like to thank Aleksey Kolmogorov and RalfDrautz for helpful discussions, the National Nature Sci-ence Foundation of China (50771004) and the Foundationfor the Author of National Excellent Doctoral Dissertationof China (200334) for financial support. Y.C. is grateful tothe China Scholarship Council for financial support andthe Department of Materials, University of Oxford fortheir hospitality and use of computing facilities in theMaterials Modelling Laboratory.

References

[1] Rodrigues G, Nunes CA, Suzuki PA, Coelho GC. Intermetallics2004;12:181.

[2] Kim JH, Tabaru T, Hirai H, Kitahara A, Hanada S. Scripta Mater2003;48:1439.

[3] Kim WY, Yeo ID, Ra TY, Cho GS, Kim MS. J Alloy Compd2004;364:186.

[4] Li W, Yang HB, Shan AD, Zhang LT, Wu JS. Intermetallics2006;14:392.

2664 Y. Chen et al. / Acta Materialia 57 (2009) 2657–2664

[5] Zhao JC, Peluso LA, Jackson MR, Tan LZ. J Alloy Compd2003;360:183.

[6] Fu CL, Schneibel JH. Acta Mater 2003;51:5083.[7] Chu F, Thoma DJ, Mcclellan K, Peralta P, He Y. Intermetallics

1999;7:611.[8] Chen Y, Shang JX, Zhang Y. J Phys: Condens Matter

2007;19:016215.[9] Chen Y, Shang JX, Zhang Y. Phys Rev B 2007;76:184204.

[10] Massalski TB. Binary alloy phase diagrams. Materials Park(OH): ASM International; 1990.

[11] http://crystdb.nims.go.jp/.[12] Hafner J, Hulliger F, Jensen WB, Majewski JA, Mathis K, Villars P,

et al. The structures of binary compounds. Amsterdam: North-Holland; 1989.

[13] Grimvall G. Thermophysical properties of materials. Amster-dam: North-Holland; 1986.

[14] Kresse G, Hafner J. Phys Rev B 1993;47:558;Kresse G, Hafner J. Phys Rev B 1994;49:14251.

[15] Perdew JP, Wang Y. Phys Rev B 1992;45:13244.[16] Ceperley DM, Alder BJ. Phys Rev Lett 1980;45:566.[17] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188.[18] Alfe D. Program available at http://chianti.geol.ucl.ac.uk/~dario;

1998.[19] Kohanoff J. Electronic structure calculation for solids and mole-

cules. Cambridge: Cambridge University Press; 2006.[20] Tarte P. Acta Crystallogr 1963;16:228.[21] Gorelkin CS, Dubrovin AS, Kolesnikova OD, Chirkov NA. Zh Fiz

Khim 1972;46:754.[22] Kubaschewski O. Thermodynamics of nuclear materials. In: Pro-

ceedings of a symposium, Vienna; 1968. p. 685.

[23] Meschel SV, Kleppa OJ. J Alloy Compd 1998;274:193.[24] Chart TG. Met Sci 1974;8:344.[25] Ohmori S, Hashimoto Y, Kohyama K. Koon Gakkaishi 1982;8:113.[26] Maslov VM, Neganov AS, Borobinskaya IP, Merzhanov A. G Fiz

Goreniya Vzryna 1978;14:73.[27] O’Hare PAG. J Chem Thermodyn 1993;25:1333.[28] Robins DA, Jenkins I. Acta Metall 1955;3:598.[29] Wang Y, Woodward C, Zhou SH, Liu ZK, Chen LQ. Scripta Mater

2005;52:17.[30] Maradudin AA, Montroll EW, Weiss GH. Theory of lattice dynamics

in the harmonic approximation, vol. 3. New York: Academic Press;1963. p. 45.

[31] Mounet N, Marzari N. Phys Rev B 2005;71:205214.[32] Kittel C. Introduction to solid state physics. 8th ed. New York: Wiley;

2005. p. 116.[33] Garanin DA. Phys Rev B 2008;78:020405(R).[34] Hill R. Proc Phys Soc A 1952;65:349.[35] Anderson OL. J Phys Chem Solids 1963;24:909.[36] Anderson OL. In: Mason WP, editor. Physical acoustics, vol.

3B. New York: Academic Press; 1965. p. 43.[37] Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MJ, Refson K,

et al. Z Kristallogr 2005;220:567.[38] Soderlind P. Phys Rev B 1993;48:5844.[39] Dargys A, Kundrotas J. Handbook on physical properties of Ge, Si,

GaAs and InP. Vilnius: Science and Encyclopedia Publishers; 1994.[40] Pechenik A, Kalia RK, Vashishta P. Computer-aided design of

high-temperature materials. New York: Oxford University Press;1999. p. 125.

[41] Stewart GR, Olinger B, Newkirk LR. Solid State Commun 1981;39:5.[42] Nakamura M, Matsumoto S, Hirano T. J Mater Sci 1990;25:3309.