A modeling investigation for pseudo-ternary (Ti,Mo,W)(CN) solid-solution: Thermodynamic and elastic properties

Computational Materials Science 83 (2014) 51–56

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

A modeling investigation for pseudo-ternary (Ti,Mo,W)(CN) solid-solution: Thermodynamic and elastic properties Zhe Gao, Shinhoo Kang ⇑ Department of Materials Science and Engineering, Seoul National University, Seoul 151-742, Republic of Korea

a r t i c l e

i n f o

Article history: Received 4 July 2013 Received in revised form 15 October 2013 Accepted 18 October 2013 Available online 22 November 2013 Keywords: Pseudo-ternary First-principles Thermodynamic Elastic

a b s t r a c t We employ ordered-VCA (virtual crystal approximation) models to characterize [Ti0.75(Mo1xWx)0.25] (C0.75N0.25) pseudo-ternary system, using density functional theory, and compare the results with experiment. The ordered structure is found suitable to represent TiC-based solid solution systems. The chemical similarity of W and Mo is also noted in the calculations. The ordered-VCA models provide a good agreement with experimentally measured elastic moduli, and show the effect of nuclear charge number on the increase of elastic quantities. The ternary system becomes the most stable when x = 0.6. The value of x for the most stable phases increases with temperature. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Owing to outstanding hardness, thermal stability and economic efficiency, TiC-based cermet is considered as a replacement for WC-based cermet in cutting tools area [1,2]. However, when TiC–Co/Ni is prepared, the carbide particles undergo coalescence caused by relatively poor wettability between TiC and metallic binder phase. This results in a low density after liquid-phase sintering, and reduces the mechanical properties and reliability of the products. By adding nitrogen to form Ti(C1xNx) solid solution, the particle growth is effectively controlled due to the much higher thermal stability of carbon-nitride phase than that of TiC [3]. This has been proven by experiments and theoretical calculations. In this system, however, toughness was not improved significantly, as it lacks a continuous metal-network in the system [4]. In order to overcome this problem, (Ti,W)(C,N) solid solutions are produced by adding tungsten to Ti(CN) [5]. Thus far, Mo and Ta have been used frequently as solutes elements to enhance the hardness and deformation resistance by forming ternary carbide or ternary carbonitride. [6,7]. Along with large quantities of experimental works, the effect of nitrogen on TiC properties has been clearly revealed by first-principles [8,9]. The connection between mechanical and elastic properties was developed and the mechanism was explained by changes in electronic structure due to the extra electron introduced by nitrogen [10,11]. A similar behavior has been also found in (Ti0.75W0.25)(C1xNx) system at finite temperatures [12]. How⇑ Corresponding author. Tel.: +82 2 880 7167. E-mail address: [email protected] (S. Kang). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.10.026

ever, there has been little study on [Ti,(Mo,W)](C,N) due to the complex structure of pseudo-ternary systems. Within the framework of density functional theory (DFT) [13], there are several methods to simulate substitutional solid solutions. The detailed comparison for these methods has been shown in elsewhere [14]. In present work, we employ virtual crystal approximation (VCA) in the framework of ordered supercell to approach pseudo-ternary carbonitride system, where the total mole fractions of (Mo + W) and N are set at 0.25, respectively. The validity of ordered supercell method for this calculation will be examined in the terms of free energy of formation and electronic structure. Thermodynamic properties of VCA-supercell are calculated via linear response method, and elastic constants by strain-energy curve fitting. Calculated shear and Young’s moduli of pseudo-ternary were compared with experimental data from ceramic bulk sintered by spark plasma sintering (SPS).

2. Methodology First-principles calculations are mainly done by QuantumESPRESSO code [15], which is based on PW-PP (plane-wave pseudo-potential) DFT scheme. The universal ground-state for pseudo-binary solid-solution is organized by ATAT code [16]. The k-points density is settled at 1500 per reciprocal atom, and cut-off energy for kinetics is 95Ry, since the Troullier Martins type norm-conserving pseudo-potentials with semi-states, e.g., (n1)s2(n1)p6 states for Ti, W and Mo, are used, where n is the maximum principal quantum number of corresponding element. GGA-PBE exchange correlation functional [17] describes the

52

Z. Gao, S. Kang / Computational Materials Science 83 (2014) 51–56

x=0.0 x=0.2 x=0.4 x=0.6 x=0.8 x=1.0 20

30

40

50

60

70

80

2θ Fig. 1. XRD patterns of ternary [Ti0.75(Mo1xWx)0.25](C0.75N0.25) sintered by SPS.

interaction between core and valence states. Because transition metal carbides and carbonitrides are conductive, the Marzari–Vanderbilt method [18], so-called cold smearing, is introduced for integration at Fermi level with the broadening parameter 0.02Ry. Electronic structure was computed by ELK code [19], based on the full potential linear augmented plane wave (FP-LAPW). In order to correct the error rising from the potential mixing, linear response is introduced for VCA thermodynamic calculation [20,21]. There are three basic steps in this method, i.e., elastic energy, chemical energy and relaxation energy calculations. When temperature effect is considered, free energy will consist of three terms, i.e., configurational, vibrational and electronic free energies, where configurational energy is provided by the zero-temperature total energy calculation [22]. The vibrational energy is calculated from lattice dynamic by quasi-harmonic approximation (QHA) [23]. Since the electronic DOS (density of states) near Fermi level varies slowly relative to Fermi distribution, the electronic contribution to free energy can be simplified by Sommerfeld model. The strain-energy curve fitting for elastic constants is reported by Mehl [24], where bulk modulus, B0, is obtained from the 4th order Birch– Murnaghan equation of states (EOS). Other elastic properties, including the shear modulus, G, and Young’s modulus, Y, were derived according to 8 < GV ¼ 1 ðC 11  C 12 þ 3C 44 Þ; GR ¼ 5ðC 11 C 12 ÞC 44 ; GH ¼ 1 ðGV þ GR Þ 5 2 4C 44 þ3ðC 11 C 12 Þ : Y i ¼ 9B0 Gi 3B0 þG

i

where GV, GR and GH stand for Voigt, Reuss, and Hill approximations, respectively. In present paper, we used Hill approximation for both of shear and Young’s moduli. The pseudo-ternary carbonitrides in this study were prepared by means of a conventional powder metallurgy technique. After weighing the raw materials, which were TiO2 (anatase, >99%, Sigma–Aldrich, USA), WO3 (99+%, Sigma–Aldrich, USA), MoO3 (P99.5%, Sigma–Aldrich, USA) and graphite, the powders were

mixed with WC–Co balls and ground using a planetary mill (Fritsch Pulverisette 7, Germany). The ball-powder ratio and rotation speed were 30:1 and 250 rpm, respectively. The mixed powders were carbothermally reduced at 1500 °C for 2 h under 10 torr nitrogen atmosphere, where the heating rate was 10 °C/min. The reduced powders were dispersed by ball milling in ethanol for 20 h and the slurry was dried at 90 °C for 6 h. The samples were heated at a rate of 100 °C/min and then sintered instantly at 1850 °C by SPS. The specimens were cross-sectioned and polished from 6 to 1 lm with diamond slurry for microstructure analysis. RINT2000, made in Japan, was used for the X-ray diffraction (XRD) measurement of the sintered bulk from 20° to 80° with a scanning speed of 5°/min. The average shear and Young’s moduli were measured via ultrasonic pulse-echo method (TDS 220, Tektronix, US and 5800, Panametrics, Korea). The microstructures of polished specimens were observed by scanning electron microscopy (SEM, Model JSM-6360, JEOL, Japan). 3. Results and discussion 3.1. Phase formation and microstructure Fig. 1 shows the XRD results of six different compositions of pseudo-ternary samples. Three main peaks, are observed at 35.5°, 41.7° and 60.3°, and these are shifted to higher angles than those of Ti(C,N), implying a decrease in the lattice parameters. Also, Fig. 1 shows that there is little variation in lattice parameters as a function of the W-to-Mo ratio. The lattice parameter was calculated as 4.325 Å according to Bragg’s law. The actual peak shift between (Ti0.75W0.25)(C0.75N0.25) and (Ti0.75Mo0.25)(C0.75N0.25) is 0.03 Å. Since it was shown that the lattice parameters of the solid-solution carbide and nitrides from the same group cations do not deviate greatly from Vegard’s law [25,26], a significant change in the lattice parameters of ternary carbonitrides is not expected as well. Further, peak splitting was not observed, indicating that the pseudo-ternary solid solution is a single phase in this system. Fig. 2(a) illustrates an SEM–BSE (back scattering electron) image of ternary carbonitrides ceramic [Ti0.75(Mo0.4W0.6)0.25](C0.75N0.25), solid solution sintered without metallic binder phase by SPS. The black line in Fig. 2(a) is the crack introduced by an indent for the property measurements such as hardness and toughness. The ternary phase has a homogeneous composition, thus the microstructure is almost featureless. In contrast, Ti(CN)-based cermets are normally known to exhibit a core–rim structure due to the difference in thermodynamic stability of various carbides or carbonitrides in the system. For example, Fig. 2(b) shows the microstructure of Ti(C0.7N0.3)–10WC–5Mo2C–15(Co,Ni) cermet sintered at 1510 °C for 1 h of Ref. [31]. In Fig. 2(b) a typical core–rim structure is shown, where the black core is undissolved Ti(C,N), and white inner rim and grey outer rim are W-rich and ternary solid solution, respectively. This complex microstructure often

Fig. 2. SEM–BSE microstructures of: (a) [Ti0.75(Mo0.4W0.6)0.25](C0.75N0.25) solid solution ceramic sintered by SPS and (b) Ti(C0.7N0.3)–10WC–5Mo2C–15(Co,Ni) cermet sintered at 1510 °C for 1 h [31].

Z. Gao, S. Kang / Computational Materials Science 83 (2014) 51–56

53

causes low toughness values of the cermets. The cermets with complete solid-solution carbide are far advantageous over ordinary cermets with various carbide or carbonitride phases in terms of providing desirable mechanical properties with enhanced toughness [32]. 3.2. Ordered models framework As a powerful method to determine the ground state of the [Ti0.75(Mo1xWx)0.25](C0.75N0.25) system for both ordered/disordered structures, the pseudo-binary results from cluster expansion method (CEM) are used first as reference to evaluate the reliability of super-cells: (Ti0.75Me0.25)C (Me = W, Mo, Ta, Zr etc.) and Ti(C0.75N0.25). They are often investigated as ordered models with the space group of Pm-3m. Further, the data are reasonably matching with experimental works [8,9,27]. All CEM calculations are performed with 60–70 clusters for the convergence of cluster expansion interaction (ECI), which is significantly higher than in typical alloy systems [28]. As shown in Fig. 3, the formation energies of (Ti1xWx)C, (Ti1xMox)C and Ti(C1xNx) were calculated by CEM with ordered models. The WC and MoC phases were treated as in Fm-3m symmetry (B1), and this led to positive formation energies of WC and MoC as shown in Fig. 3(b and c). This differs from the formation energies of stable HCP-WC and MoC [29]. Given the solubility of WC and MoC in TiC are higher than 0.25 [30] [Ti0.75(Mo1xWx)0.25](C0.75N0.25), system is located well within a single phase region. The overall tendency of ordered and CEM for the ground state is identical in all of three pseudo-binary solutions, especially for (Ti1xWx)C and (Ti1xMox)C. The deviation between two methods with ordered and disordered models is less than 0.01 eV/f.u. (0.97 kJ/mol). In the case of Ti(C0.75N0.25) the difference is slightly larger, but it is still within 0.015 eV/f.u. (1.45 kJ/mol). Therefore, the ordered models can reliably represent the properties of TiCbased pseudo-binary systems. In order to determine the relative position of W and N, two typical structures were built within the framework of ordered models, Pm-3m and P4/mmm, as illustrated in Fig. 4(a and b), representing the non-neighboring and neighboring situation between W and N, respectively. We selected (Ti0.75W0.25)(C0.75N0.25) as an example to compare the difference between two structures in terms of the finite temperature free energies, reflecting the vibrational and electronic contributions. At 0 K, the P4/mmm structure (W–N neighboring) has a much higher free energy, by 15 kJ/mol, than Pm-3m (W–N non-neighboring). Obviously, the relative thermal stability of the P4/mmm structure decreases more rapidly with an increase in the temperature as compared to Pm-3m. Thus, the latter structure is chosen for the following calculation. 3.3. Chemical similarity of W and Mo in Ti(C0.75N0.25) Band structures, so-called fat bands with atomic characteristics [33], are calculated within FP-LAPW scheme for the ordered (Ti0.75W0.25)(C0.75N0.25) and (Ti0.75Mo0.25)(C0.75N0.25). It is along the C–X– M–R–C–M high symmetry line in the First Brillouin Zone (FBZ) of a simple cubic as shown in Fig. 5. Around band lines, the error bars represent the contributions from Mo or W to a certain eigenvalue at a specified wave vector. There are some small mismatches between these two fat bands that are caused by the difference in the type of element. Such as three states with T1u symmetry rely on Fermi level at C point in (Ti0.75W0.25)(C0.75N0.25) [12], but they are not occupied by electrons in (Ti0.75Mo0.25)(C0.75N0.25). The general valence states in (Ti0.75Mo0.25)(C0.75N0.25) locate at higher energy levels than those in (Ti0.75W0.25)(C0.75N0.25). However, the general tendency and even the curvatures of bands are shown to be analogous. Furthermore, the angular momentum characteristics

Fig. 3. Differences in the formation energy calculated using CEM and orderedsupercell methods: (a) Ti(C1xNx), (b) (Ti1xMox)C and (c) (Ti1xWx)C systems.

for Mo and W are also similar, implying the electronic similarity of W and Mo in Ti(C0.75N0.25)-based solid-solution. As the direct solution of Kohn–Sham equations, band structure only contains information about the eigenvalues. In contrast to the energy-based calculations, lattice dynamic is more sensitive to changes in electronic structure because it is originated from the second order differential of energy [34]. In particular, information

54

Z. Gao, S. Kang / Computational Materials Science 83 (2014) 51–56

Fig. 4. Two different structures exhibiting the neighboring positions between W–N: (a) non-neighboring W–N in Pm-3m, and (b) neighboring W–N in P4/mmm, where blue, white, yellow and orange balls stand for Ti, W, C and N, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Fat band structures of (a) Mo in (TiMo0.25)(CN0.25) and (b) W in (TiW0.25)(CN0.25).

Fig. 6. Phonon dispersion and corresponding PDOS for ordered (a) (Ti0.75Mo0.25)(C0.75N0.25) and (b) (Ti0.75W0.25)(C0.75N0.25).

about atomic mass is also involved in phonon frequency [35,36]. Fig. 6 shows the phonon dispersion and corresponding project DOS (PDOS) for (Ti0.75Mo0.25)(C0.75N0.25) and (Ti0.75W0.25)(C0.75N0.25). The three acoustic branches are mainly contributed by Mo or W according to PDOS. It is notable that the frequencies of acoustics branch in (Ti0.75Mo0.25)(C0.75N0.25) are significantly higher than in (Ti0.75W0.25)(C0.75N0.25). This can be interpreted to the mass effect as in the case of Ga1xAlxAs, where the approximated phonon

dispersion can be calculated with reasonable accuracy by the same interatomic force constants (IFCs) with different atomic masses [37]. It is clear that the curvatures of the acoustic branches in both compositions are very close. The contributions of W and Mo are also analogous according to phonon PDOS, where W and Mo have limited contributions in optical branches associated with the vibrations of Ti. Since present calculations considered only harmonic vibration, the oscillator-like atoms express a simple relationship

55

Z. Gao, S. Kang / Computational Materials Science 83 (2014) 51–56 0

(a)

0K 273K 1773K

-1 id, 273K

ΔG

-2 -3 -4 -5

id, 1000K

-6

ΔG

-7 -8 -9 -10

(b)

-11 0.0

id, 1773K

ΔG 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1. 0

Fig. 8. Free energy of mixing for [Ti0.75 (Mo1xWx)0.25](C0.75N0.25) system at various temperatures along with those of ideal systems.

Random distribution of W and Mo at this site is represented by Ramer–Rappe type VCA. 3.4. Elastic properties of ternary system

Fig. 7. Elastic properties as a function of W concentration in [Ti0.75(Mo1xWx)0.25](C0.75N0.25): (a) elastic constants and (b) elastic moduli.

Table 1 The fitted b values for lattice [Ti0.75(Mo1xWx)0.25](C0.75N0.25).

parameter

and

elastic

properties

Quantities

a0 (Å)

C11 (GPa)

C12 (GPa)

C44 (GPa)

B0 (GPa)

G (GPa)

Y (GPa)

b

0.00502

41.06

1.62

5.80

5.18

12.02

25.24

of

between bonding strength and mass. Only the mass-dependent frequencies and the same distributions indicate the similarity of W and Mo in Ti(C0.75N0.25) in bonding behavior. Generally, the formation energy of defect (alloying) can be expressed as:

   tot  tot tot Efdef ¼ Etot ¼ EfA1x Bx C  EfAC A1x Bx C  EAC  x EB  EA where the superscript tot and f denote total energy and formation energy, respectively. The formation energies of defect (alloying) are 39.97 kJ/mol, 31.52 kJ/mol and 8.45 kJ/mol, respectively, for the alloying processes of Ti(C0.75N0.25)-to-(Ti0.75W0.25)(C0.75N0.25), Ti(C0.75N0.25)-to-(Ti0.75Mo0.25)(C0.75N0.25), and (Ti0.75W0.25)(C0.75N0.25)-to-(Ti0.75Mo0.25)(C0.75N0.25). This result indicates that the effects of Mo and W are similar to each other on free energy of a solid solution. It is fairly obvious from their degree of chemical similarity, as they are in the same group in the periodic table. Therefore, we assume that W and Mo share W’s site in the Pm-3m structure.

Fig. 7 shows the results of elastic constants and moduli from VCA models with the experimental results. Due to the limitation of ultrasonic measurements, only shear and Young’s moduli were obtained directly. According to the relationship between bulk modulus and shear-Young’s moduli as mentioned in Section 2, the bulk modulus can be obtained from the other two measured moduli. The experimental results are shown in Fig. 7(b). This method can circumvent the necessity of complicated and expensive bulk modulus measurements. Although there is no report about elastic constants for the ternary carbon-nitride system, the reliability of our calculated results can be proven by the comparison of elastic moduli from first-principles calculations with experimental results. Not only does the trend observed from our ordered-VCA models agree with experiments, but also the absolute values match well within an acceptable error range. This indicates the feasible rational of our method of employing ternary models. Further discussions about the predicted results originated from the ordered-VCA models are expected to be reliable. In general, all elastic properties increase monotonously with an increase in the tungsten concentration in nearly the entire range except C12. From 0.9 to 1.0, the elastic properties are reduced somewhat, but this decrease is limited to C11 within 5.0 GPa. It is notable that in all concentration range, the elastic quantities show the positive deviation from Vegard’s law, which indicates the enhancement effect of solid solution as noted in other alloys [38,39]. Quadratic fitting is usually utilized in the mixing or solid solution forming cases [40]:

Q ternary ðxÞ ¼ ð1  xÞQ ðTi0:75 Mo0:25 ÞðC 0:75 N0:25 Þ þ xQ ðTi0:75 W 0:25 ÞðC0:75 N0:25 Þ þ bð1  xÞx where Q stands for the elastic quantity for specified composition and b is the undetermined coefficient. The fitted b values for the lattice parameter and elastic properties are shown in Table 1. The small negative b value for the lattice parameter agrees with the

56

Z. Gao, S. Kang / Computational Materials Science 83 (2014) 51–56

XRD results. This indicates that there is no strong interaction between two binary solutions and that the ternary solid solutions remain stable. Since various compositions of [Ti0.75(Mo1xWx)0.25] (C0.75N0.25) have the same valence electron concentration, these ternary solid solutions do not behave as Ti(C1xNx). The change of elastic properties may be affected by nuclear charge number. On average, W has eighty-eight more proton and neutrons compared to Mo does, and the ionic radius of W is larger than Mo at three valence states, i.e., +4, +5 and +6. Therefore, it is much easier for valence electron in W to delocalize and form the ionic bond with C or N. As a result, the bond strength between W and C is expected to be slightly stronger than that between Mo and C. This phenomenon has also been observed among IVB and VB group mono-carbides according to their electronic structure and thermodynamic properties [41].

tion of the most stable phase remains almost the same when temperature increases from 0 K to 1773 K, the latter being a normal sintering temperature.

3.5. Thermodynamic properties of ternary system

[1] L.E. Toth, Transition Metal Carbides and Nitrides, Academic Press, New York and London, 1971. [2] H. Kwon, S. Kang, Mater. Sci. Eng. A 520 (2009) 59–79. [3] J. Jung, S. Kang, J. Am. Ceram. Soc. 90 (2007) 2178–2183. [4] J.M. Humenik, N.M. Parikh, In: The American Ceramic Society 57 Annual Meeting, Cincinnati, Ohio, April 27, 1955. [5] S. Ahn, S. Kang, Int. J. Refract. Metals Hard Mater. 26 (2008) 340–345. [6] J. Zackrisson, H.-O. Andrén, Int. J. Refract. Metals Hard Mater. 17 (1999) 265– 273. [7] P. Lindahl, P. Gustafson, U. Rolander, L. Stals, H.-O. Andrén, Int. J. Refract. Metals Hard Mater. 17 (1999) 411–421. [8] I. Jung, S. Kang, S. Jhi, J. Ihm, Acta Mater. 47 (1999) 3241–3245. [9] S. Jhi, J. Ihm, Phy. Rev. B 56 (1997) 13826. [10] Q.M. Hu, K. Kadas, S. Hogmark, R. Yang, B. Johansson, L. Vitos, Appl. Phys. Lett. 91 (2007) 121918. [11] S. Jhi, J. Ihm, S.G. Louie, M.L. Cohen, Nature 399 (1999) 132–134. [12] Z. Gao, S. Kang, Solid State Commun. 156 (2013) 25–30. [13] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1131–A1138. [14] L. Vitos, Computational Quantum Mechanics for Materials Engineering: The EMTO Method and Applications, Springer Press, 2008. [15] P. Giannozzi, S. Barroni, N. Bonini, et al., J. Phys.: Condens. Matter 21 (2009) 395502. [16] A. van de Walle, CALPHAD 33 (2009) 266. [17] J.P. Perdew, K. Burke, M. Emzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [18] D. Shin, A. van de Walle, Y. Wang, Z. Liu, Phys. Rev. B 76 (2007) 144204. [19] ELK Code. . [20] S. de Gironcoli, Ab-inito Numerical Studies in Semiconductor Alloys, Ph.D Dissertation, École Polytechnique Fédérale de Lausanne (EPFL), 1992. [21] S. de Gironcoli, P. Giannozzi, S. Baroni, Phys. Rev. Lett. 16 (1991) 2116. [22] S. Yip, Handbook of Materials Modeling, Springer Press, 2005. [23] S. Baroni, P. Giannozzi, E. Isaev, Rev. Min. Geol. 71 (2010) 39–57. [24] M.J. Mehl, J.E. Osburn, Phys. Rev. B 41 (1990) 10311. [25] B. Alling, C. Hoglund, R.H. Wilton, L. Hultman, Appl. Phys. Lett. 98 (2011) 241911. [26] K. Bouamama, P. Djemia, D. Faurie, G. Abadias, J. Alloys Compd. (2011), http:// dx.doi.org/10.1016/j.jallcom.2011.12.034. [27] J. Kim, S. Kang, J. Alloys Compd. 528 (2012) 20–27. [28] A. van de Walle, M. Asta, G. Ceder, CALPHAD 26 (2002) 539–553. [29] J. Weidow, S. Johansson, H.-O. Andrén, G. Wahnstrom, J. Am. Ceram. Soc. 94 (2011) 605–610. [30] J. Weidow, H.-O. Andrén, Acta Mater. 58 (2010) 3888–3894. [31] J. Wang, Y. Liu, P. Zhang, J. Peng, J. Ye, M. Tu, Int. J. Refract. Metals Hard Mater. 27 (2009) 9–13. [32] S. Park, S. Kang, Scripta Mater. 52 (2) (2005) 129–133. [33] O. Jepsen, O.K. Andersen, Z. Phys. B 97 (1995) 35–47. [34] S. Baroni, P. Giannozzi, A. Testa, Phys. Rev. Lett. 58 (1987) 1861. [35] P. Pavone, Lattice Dynamics of Semiconductors from Density-Functional Perturbation Theory, Ph.D Dissertation, Scuola Internazionale Superiore di Studi Advanzati (SISSA), 1991. [36] S. de Gironcoli, Phys. Rev. B 46 (1992) 2412. [37] S. Baroni, S. de Gironcoli, P. Giannozzi, Phys. Rev. Lett. 65 (1990) 84. [38] Q.M. Hu, R. Yang, J.M. Lu, L. Wang, B. Johansson, L. Vitos, Phys. Rev. B 76 (2007) 224201. [39] C.M. Li, Q.M. Hu, R. Yang, B. Johansson, L. Vitos, Phys. Rev. B 82 (2010) 094201. [40] L.E. Koutsokeras, G. Abadias, Ch E. Lekka, G.M. Matenoglou, D. Anagnostopoulos, G.A. Evangelakis, P. Patsalas, Appl. Phys. Lett. 93 (2008) 011904. [41] Z. Gao, S. Kang, Density-Functional Investigations for solid-solutions between TMC (TM=Ti, Zr, Hf, V, Nb and Ta) and WC. (unpublished). [42] A. Predith, G. Ceder, C. Wolverton, K. Persson, T. Mueller, Phys. Rev. B 77 (2008) 144104.

Thermodynamic properties have been calculated for ZrO2-based multi-component materials by CEM using both ordered and disordered models [42]. Our models are related to VCA approach, but the basic structures are built on the ordered framework. Thus, it is possible for our cases to calculate free energy at a finite temperature with the same approach as in the ZrO2 case above along with summing up the configurational, vibrational and electronic free energies. To simplify the calculations while also avoiding the error associated with the temperature effect of graphite and N2 phase, we simply compare the formation energy of mixing in the present work at 0 K, 273 K and 1773 K as shown in Fig. 8 along with those of ideal mixing. The result shows a negative deviation from ideal solution behavior at all compositions and temperatures below 1000 K. This is consistent with the positive deviation in the elastic properties, showing the tendency to form solid solutions. Above this temperature it show a positive deviation from ideality, that is, the energy values at 1773 K are less than 10.2 kJ/mol of ideal mixing case. This implies that Mo and W can be mixed moderately without a strong tendency to form solid solution in the presence of nitrogen when total mole fraction of W and Mo remains constant, for example, at 0.25 in the ternary [Ti0.75(Mo1xWx)0.25](C0.75N0.25). However, even at 1773 K, the phase separation is not expected based on the shape of parabolic curves we obtained. In the case of phase separation, so-called spinodal decomposition phenomenon, the energy curves start exhibiting two minima at a temperature the phase separation starts. Furthermore, the most stable composition, corresponding to the lowest energy point, does not change much, i.e., x = 0.55–0.60, when temperature increases from 0 K to 1773 K. 4. Conclusions In present work, the ordered-based VCA models are developed to understand the physical and thermodynamic properties of [Ti0.75(Mo1xWx)0.25](C0.75N0.25) ternary system. The XRD result showed the existence of homogeneous and single-phase ternary solid solution. We justify the use of ordered binary models and demonstrated the chemical similarity of W and Mo in Ti(C0.75N0.25) through electronic structure, lattice dynamic and thermodynamic properties. By comparing the elastic properties from VCA models with experimental results, excellent agreements are obtained, implying that our modeling method is acceptable for ternary cases. Elastic properties generally increase monotonously with W concentration due to the nuclear mass effect, a maximum point at x = 0.9. Further, thermodynamic calculations show the composi-

Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MEST) (No. 2011-0017168) and partially by grants-in-aid for the National Core Research Center Program from MOST/KOSEF (No. R15-2006022-03001-0). We also acknowledge the use of the facilities of the Research Institute of Advanced Materials, Seoul National University. References