First-principles study of NiAl alloyed with Co

First-principles study of NiAl alloyed with Co

Computational Materials Science 111 (2016) 34–40 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.els...

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Computational Materials Science 111 (2016) 34–40

Contents lists available at ScienceDirect

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

First-principles study of NiAl alloyed with Co Yong Cao a,⇑, Peixian Zhu a, Jingchuan Zhu b, Yong Liu b a b

School of Materials Science and Engineering, Kunming University of Science and Technology, Kunming, Yunnan 650001, PR China School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150001, PR China

a r t i c l e

i n f o

Article history: Received 14 April 2015 Received in revised form 16 August 2015 Accepted 27 August 2015

Keywords: Co-doped NiAl First principles Elastic properties Thermal properties

a b s t r a c t The site preference of Co in NiAl and its effects on structural, elastic, electronic and thermal properties were investigated by performing first principles calculations using density functional theory (DFT). The site preference was investigated by calculating the transfer energy of NiAl alloys with Co. The result shows that Co tends to occupy Ni site. By analyzing changes in electronic density of states, Mulliken population, overlap population and valence charge density, the electronic property and bond characters were discussed. The elastic properties calculation shows that Co increases alloy hardness. Moreover, the pressure and temperature dependences of the thermal expansion coefficient, bulk modulus, Debye temperature and heat capacity in a wide temperature (0–1600 K) and pressure (0–30 GPa) ranges are presented in this study. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction The intermetallic compound B2-NiAl is one of the most promising engineering materials with attractive chemical and mechanical properties, and it is a contender of next generation of high temperature structure material [1]. The properties include high melting temperature (1638 °C), low density (5.86 g/cm3), high modulus, high thermal conductivity and excellent environmental resistance. However, room temperature brittleness and high temperature strength shortage seriously limit its practical application, but it can be improved by alloying additions [2–4]. It is known that solid solubility varies with different dopant atoms, and Co has high solubility in NiAl. The understanding of the alloying effects requires the knowledge of site distributions of ternary elements. The site preference of Co has been investigated extensively by various experimental techniques, such as X-ray diffraction (XRD) [5], Extended X-ray absorption edge fine structure (EXAFS) [6]. Thermal conductivity measurement [7]. The results showed that Co substitute for a Ni site. Theoretical evaluations of the site preference of Co have been conducted by many researchers [8–13]. They get the same conclusion that Co has a consistent preference for the Ni sublattice. There are some investigations about the effect of alloying on properties of NiAl. Kovalev et al. [14] pointed out that alloying with Co has a beneficial effect on the micro-mechanism of fracture and ductile– brittle transition temperature, and this improvement is tied with the change of density of electron states N(EF). The microhardness ⇑ Corresponding author. Tel.: +86 0871 5126513; fax: +86 0871 5160625. E-mail address: [email protected] (Y. Cao). http://dx.doi.org/10.1016/j.commatsci.2015.08.053 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

of NiAl alloys was increased due to solid solution hardening of Co [15]. Through the theoretical and experimental research above, it is worth noting that the theoretical calculations only gave material properties at 0 K and zero pressure, without any thermal effects included. As a potential high-temperature material, the thermal properties cannot be over-emphasized. Practical use of NiAl alloy often occurs at high temperature. Therefore, it is necessary to investigate the influence of Co on the thermal properties. To our knowledge, there is no correlative report about this aspect. To address this interest, we investigate the structural and thermodynamic properties at high pressures and temperatures, by using first-principle calculations combined with the quasi-harmonic Debye model. In this paper, the site preference of Co in NiAl and its effects on structural, elastic, electronic and thermal properties are investigated by performing first principles calculations using density functional theory (DFT), which would be useful for the understanding and design of the relevant alloys. The influence of Co on the thermal expansion coefficient, bulk modulus, Debye temperature and heat capacity at different pressure and temperature were studied for the first time. 2. Theoretical methods 2.1. Calculation parameters NiAl has an ordered B2 structure that consists of two interpenetrating simple cubic sublattices, with Ni and Al atoms occupying the corners and the center of the body, respectively. The lattice

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Y. Cao et al. / Computational Materials Science 111 (2016) 34–40

Fig. 1. Original model used in the calculation.

constants are a0 = 0.2887 nm, a = b = c = 90. A supercell consisting of 16 atoms is devised for the present study, a 2  2  2 cubic representation of the ordered structure, with one single point defect such as an antisite (NiAl or AlNi) or the ternary element Co on the sublattice. One alloying atom in each supercell corresponds to a fraction of 6.25 at.%. Fig. 1 shows the Al-site and Ni-site defect model respectively. All the calculations in this study were performed with the CASTEP code [16], based on the first principles density functional theory, within the generalized gradient approximation (GGA). For the GGA exchange–correlation function, the Perdew–Wang parameterization (PW91) [17,18] was employed. The ultra-soft pseudopotential and plane wave basis sets [19] were implemented. After a series of tests, the cutoff energy was set at 400 eV. The k-points is 8  8  8 in all calculations and the self-consistent convergence of the total energy is at 5  107 eV/atom. 2.2. Debye model In order to investigate the thermodynamic properties of NiAl alloys, the quasi-harmonic Debye model [20] was applied, in which the non-equilibrium Gibbs function G⁄(V;P,T) can be written in the form of

G ðV; P; TÞ ¼ EðVÞ þ PV þ AVib ðHðVÞ; TÞ

ð1Þ

where EðVÞ is the total energy per unit cell, PV corresponds to the constant hydrostatic pressure condition, and the vibration term AVib is the Helmholtz free energy for lattice vibrations and can be written as

AVib ðHðVÞ; TÞ ¼ nkT

  9H þ 3 lnð1  eH=T Þ  DðH=TÞ 8 Y

ð2Þ

where D(y) is the Debye integral defined as

3 DðyÞ ¼ 3 y

Z 0

y

x3 dx x e 1

ð3Þ

ð4Þ

where M is the molecular mass per formula unit, Bs the adiabatic bulk modulus, and f ðrÞ is given by following forms:



3B  2G 6B þ 2G

8 " 9  3=2  3=2 #1 =1=3 < 2 1þr 1 1þr þ f ðrÞ ¼ 3 2 : ; 3 1  2r 3 1r

2

Bs ffi BðVÞ ¼ V

d EðVÞ

ð7Þ

dV 2

Therefore, the non-equilibrium Gibbs function G  ðV; P; TÞ as a function of (V; P,T) can be minimized with respect to V

   @G ðV; P; TÞ ¼0 @V P;T

ð8Þ

By solving Eq. (8), one can get the thermal equation of state (EOS). The isothermal bulk modulus BT, the heat capacity C V and the thermal expansion coefficient a are given by

BT ðpP; TÞ ¼ V

! @ 2 G ðV; P; TÞ @V 2

ð9Þ P;T

  3H=T C V ¼ 3nk 4DðH=TÞ  H=T e 1



cC V BT V

ð10Þ ð11Þ

where c is the Grüneisen parameter, which is defined as:

c¼

d ln HðVÞ d ln V

ð12Þ

The thermodynamic properties of NiAl and doping system were calculated through the quasi-harmonic Debye model. 3. Results and discussion 3.1. Site preferences of ternary elements in NiAl at ground state

H is the Debye temperature, and n is the number of atoms per formula unit. For an isotropic solid, with Poisson ratio r, H can be expressed as rffiffiffiffiffi h h 2 1=2 i1=3 Bs H ¼ 6p V n f ð rÞ k M

Bs is the adiabatic bulk modulus, which is approximated given by the static compressibility:

ð5Þ

Formation enthalpy is the entropy of the compound or solution in relation to the composition-weighted pure elements. The bulk stability of the compound is determined by formation enthalpy. The negative formation enthalpy value implies the structure is thermodynamically stable and the positive value means the structure is instable. In this work, the formation enthalpy of (Ni7Co)Al8 was calculated by using the following expression:

DHðNi7 CoÞAl8 ¼

ð13Þ

The formation enthalpy of Ni8(Al7Co) is calculated by using the following expression:

DHNi8 ðAl7 CoÞ ¼ ð6Þ

 1  Al Co Etot  7ENi solid  8Esolid  Esolid 16

 1  Al Co Etot  8ENi solid  7Esolid  Esolid 16

ð14Þ

Al Co where Etot refers to the total energy of unit cell. ENi solid , Esolid and Esolid are the energy per atom of bulk Ni, Al and Co, respectively. The

Y. Cao et al. / Computational Materials Science 111 (2016) 34–40

Table 1 Calculated equilibrium lattice constants, ground state energies (eV/atom) and formation enthalpy (eV/atom). Alloys

Designation

a (Å)

E (eV/atom)

DHf (eV/atom)

Ni8Al8 Ni8(Al7Ni) (Ni7Al)Al8 (Ni7Co)Al8 Ni8(Al7Co)

Ni8Al8 NiAl AlNi CoNi CoAl

5.793 5.765 5.860 5.781 5.767

707.318 788.516 625.926 687.828 769.005

0.671 0.600 0.548 0.664 0.571

lattice parameters and formation enthalpy are summarized in Table 1. The results of DH are negative and clearly verify the stability of the studied compounds. In order to investigate site preference of ternary alloying elements in NiAl at 0 K, the Ruban and Skriver method [21] was applied. At T = 0 K, the site preference of ternary alloying elements in NiAl is solely governed by enthalpy. A reaction was supposed to react: XNi + AlAl ? XAl + AlNi, which means moving an X atom from

total

40

Density of state /electrons eV-1

36

20 0

Ni-s Ni-p Ni-d

40 20 0 6

Al-s Al-p

3 0 -10

-5

0

5

Energy /eV Fig. 2. The density of states (DOS) of NiAl.

) in this reaction Ni site to Al site in NiAl. The energy change (ENi!Al X can be calculated:

ð15Þ

EXNi!Al ,

There are three situations of and every situation corresponds to a site preference of ternary alloying elements.

EXNi!Al

< 0; X atoms occupy Al site;

EXNi!Al > HAlNi þ HNiAl ¼ 3:10 eV; X atoms occupy Ni site; 0 < ENi!Al < HAlNi þ HNiAl ¼ 3:10 eV; X atoms may occupy both sites: X HAlNi

Hf ðAlNi Þ  Hf ðNiAlÞ Hf ðNiAl Þ  Hf ðNiAlÞ ; HNiAl ¼ ¼ 1=16 1=16

ð16Þ

40

Density of state /electrons eV-1

EXNi!Al ¼ EðNi8 Al7 XÞ þ EðNi7 AlAl8 Þ  EðNi7 XAl8 Þ  EðNi8 Al8 Þ

The values of ENi!Al is 3.45, and Co substitute for a Ni site. This is Co agrees well with others [5–13].

total

(a)

20 0 40

Ni-s Ni-p Ni-d

20 0 6

Al-s Al-p

3 0 6

Co-s Co-p Co-d

3 0

-10

-5

0

5

10

Energy /eV 3.2. Electronic properties

1 For interpretation of color in Figs. 4 and 5, the reader is referred to the web version of this article.

40

Density of state /electrons eV-1

To clearly analyze the main electronic compositions of band structures, the total and partial density of state (DOS) of NiAl and with the addition of Co at zero pressure were calculated as presented in Figs. 2 and 3. It is found that there were few changes of TDOS and PDOS of Ni and Al after alloying. As shown in Figs. 2 and 3, the density of states of electrons at Fermi energy is nonzero, implying that these phases exhibit some metallic behavior. The PDOS of NiAl shows that the main characteristic of the electronic structure is dominated by the hybridization between the Ni 3d and Al 3p orbital. The p–d bond is one of the reasons why NiAl is brittleness. Fig. 3 shows the DOSs of the substitution of CoNi and CoAl. For the case of Ni site substitution, the DOSs of Co has two bonding peaks below the Fermi level near at about 1.5 eV and 0.25 eV. For the other case, there are three bonding peaks below the Fermi level at about 2.75 eV, 1 eV, 0.25 eV. The crest value of density of states in the latter case is much lower than that of the former case. In addition, the density of states of electron at Fermi energy in the former case is just one half of the latter case. The results clearly imply that Co shows strong preference to the Ni sites in NiAl. The bonding charge density has been calculated to study the nature of the bonds. The charge density distributions in the (1 1 0) plane of NiAl and doping system are shown in Figs. 4 and 5. The red1 color means a high charge density in these places and blue means the charge density is low. It is clear that charge density

total

(b)

20 0 40

Ni-s Ni-p Ni-d

20 0 6

Al-s Al-p

3 0 6

Co-s Co-p Co-d

3 0

-10

-5

0

5

10

Energy /eV Fig. 3. The density of states (DOS) of (Ni7Co)Al8 and Ni8(Al7Co). (a) (Ni7Co)Al8 and (b) Ni8(Al7Co).

around Ni atom and Co atom is high. It can be seen from the bonding charge density of NiAl on the (1 1 0) plane that the electrons enrich on h1 1 1i direction, and relatively deplete on h1 0 0i direction. The main reason is that the formation of p–d polarization bond causes the electronic depletion on h1 0 0i and results in intrinsic brittleness of NiAl crystal. From Figs. 4 and 5, we can see that enrichment phenomenon of bonding charge density on h1 1 1i are more serious for the case of CoNi, CoAl, indicating that these site substitutions increase the brittleness of alloy.

37

Y. Cao et al. / Computational Materials Science 111 (2016) 34–40

Fig. 4. The charge density distributions on (1 1 0) plane for NiAl.

Population analysis results can provide more insightful information on chemical bonding, many ground state properties such as magnetic state, stability and elastic parameters are all determined by chemical bonds. It is seen from Table 2 that the total electrons of Ni decrease, when the Al is replaced by Co. However, the total population of Al is not change, when the Ni is replaced by Co. It can be seen from Table 2 that Mulliken charge is changed after alloying. The substitution in which the Co atom loses electrons is CoAl and the Co atom gets electrons is CoNi. The bonding (anti-bonding) states are related to positive (negative) values of bonds population, while the low (high) values imply that the chemical bond exhibits strong ionic (covalent) bonding. The bonds population between Co and its nearest atom of (Ni7Co)Al8 and Ni8(Al7Co) are 0.12, 0.04 and bonds population between Ni and Al of Ni8Al8 is 0.14. The values of bonds population become smaller after alloying. For the case of Al site substitution, the value of bonds population is negative, the reason may be that the rejection between 3 d of Co and nearest Ni is strong, and the stability is reduced. This conclusion is in good agreement with the analyzing results of formation enthalpy. 3.3. Elastic properties The elastic constants provide valuable information about the bonding character between adjacent atomic planes and the anisotropic character of the bonding and structural stability. For a cubic crystal, the generally accepted requirements of mechanical stability on the elastic constants [22] are: C11  C12 > 0, C11 > 0, C44 > 0, C11 + 2C12 > 0. In this paper, three elastic constants (C11, C12 and C44) for NiAl and the doping system were calculated and the results were listed in Table 3. It can be seen from Table 3 that, the elastic constants satisfy the above criteria, indicating that these phase compound is mechanically stable. It has been found that material hardness and elastic constant C44 exist monotone corresponding [23]. The larger the C44, the higher hardness is. In our calculation, the C44 is changed with alloying of Co, indicating that the hardness is changed. The sequence of hardness from high to low is: (Ni7Co)Al8 > Ni8Al8 > Ni8(Al7Co). Co

Fig. 5. The charge density distributions on (1 1 0) plane for (Ni7Co)Al8 and Ni8(Al7Co). Table 2 Population and electronic charges of atoms in nondoped and impurity-doped. Designation

atom

s

p

d

Total population

Mulliken charge (e)

Ni8Al8

Ni Al Co Al Co Ni

0.43 0.80 0.34 0.81 0.38 0.44

0.95 1.87 0.74 1.88 0.56 0.94

8.95 0.00 8.07 0.00 7.72 8.92

10.32 2.68 9.15 2.68 8.66 10.30

0.32 0.32 0.15 0.32 0.34 0.30

CoNi CoAl

has a consistent preference for the Ni sublattice and the hardness of (Ni7Co)Al8 is bigger than NiAl. It can be concluded that the addition of Co increases hardness of NiAl, which is proved by experimental result [14,15].

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Y. Cao et al. / Computational Materials Science 111 (2016) 34–40

Table 3 Calculated elastic constants (GPa) as well as the elastic parameters (B, G, E, B/G, m). Designation

C11

C12

C44

B

G

E

B/G

m

Ni8Al8 CoNi CoAl

177.39 242.97 209.73

149.96 127.27 138.50

115.39 123.08 113.08

159.10 165.84 162.24

51.91 90.90 71.28

140.46 230.58 186.53

3.065 1.824 2.276

0.353 0.268 0.308

The bulk modulus (B) is the represent of the resistance of a material to volume change [24]. It can be directly calculated with these elastic constants as follows:

C 11 þ 2C 12 3

C 11  C 12 þ 3C 44 5

5ðC 11  C 12 ÞC 44 GR ¼ 3ðC 11  C 12 Þ þ 4C 44 G¼

GV þ GR 2

1400

ð18Þ

CoNi 10GPa

NiAl 20GPa

CoNi 20GPa

NiAl 30GPa

CoNi 30GPa

1300

1200

ð19Þ

1150

ð20Þ

0

300

600

900

1200

1500

Tempreture/(K) Fig. 6. The variations of volume as a function of temperature and pressure.

Poisson’s ratio (m) is used to quantify the stability of the crystal against shear, and determined by the following relation [25]:

30

ð22Þ

The calculated results are shown in Table 3. The material hardness is closely related to the value of elastic modulus E and shear modulus G [26]. The larger the value of elastic modulus E and shear modulus G, the higher hardness is. It can be seen from Table 3 that the value of elastic modulus E and shear modulus G of (Ni7Co)Al8 is larger than the value of Ni8Al8. The result obtained by analyzing the value of E and G has the same the conclusion as according to the C44. The ratio of the bulk modulus to shear modulus of crystalline phases introduced by Pugh [24] can predict the brittle and ductile behavior of materials. A high (low) B/G value is associated with a good ductility (brittleness) and the critical value is about 1.75. In this calculations, the B/G of all the system are bigger than 1.75, indicating that NiAl and doping system are ductile material. The values of B/G shown that alloying reduce the ductility of alloy. The sequence of plasticity from high to low is: Ni8Al8 > Ni8(Al7Co) > (Ni7Co)Al8. It is well known that, the larger the Poisson’s ratio, the better the plasticity. The same conclusion can be drawn according to the analysis of m and B/G. 3.4. Thermal properties Co has a consistent preference for the Ni sublattice, so we investigate the thermodynamic properties of Ni8Al8 and (Ni7Co)Al8 under high temperature and high pressure, the quasi-harmonic Debye was applied. The temperature range is from 0 to 1600 K and pressure range is within 0–30 GPa. Fig. 6 demonstrates the relationships between the volume with temperature and pressure. The volume decreases with increasing pressure and increases with increasing temperature. The volume of Ni8Al8 is bigger than

Thermal expansion (10-6K-1)

35

3B  2G m¼ 2ð3B þ GÞ

CoNi 0 GPa

NiAl 10GPa

1350

ð21Þ



NiAl 0GPa

1250

Young’s modulus (E) reflects the resistance of materials against uniaxial tensions, the larger value of E, the stiffer is the material. It can be calculated as follows:

9BG 3B þ G

1450

ð17Þ

The shear modulus (G) is the measure of resistance to reversible deformations upon shear stress, is given as:

GV ¼

1500

V (bohr3)



1550

25

NiAl 0GPa

Co Ni 0 GPa

NiAl 10GPa

Co Ni 10GPa

NiAl 20GPa

Co Ni 20GPa

NiAl 30GPa

Co Ni 30GPa

Touloukian et al. 20 15 10 5 0

0

200

400

600

800

1000

1200

1400

1600

Tempreture (K) Fig. 7. The variations of thermal expansion coefficient as a function of temperature and pressure.

(Ni7Co)Al8 at the same temperature and pressure. Fig. 7 shows the effects of temperature and pressure on the thermal expansion coefficient (a). The thermal expansion coefficient of NiAl in this work is higher than experimental values [27] and agrees well with other calculation [28]. It can be seen that a decrease with pressure and increase with temperature. Obviously, the a is more sensitive to the temperature at low temperature than high temperature. The a of Ni8Al8 is bigger than (Ni7Co)Al8 at the same temperature and pressure, and the difference reduces with increasing pressure. The dependences of the bulk modulus B on temperature T (0–1600 K) and pressure (P = 0 GPa, 10 GPa, 20 GPa, 30 GPa) are shown in Fig. 8. It well known that, the temperature dependence of bulk modulus partially reflects the inharmonic interactions because the bulk modulus would be independent of temperature for a purely harmonic crystal. At low temperature, the bulk modulus is nearly constant, indicating that the volume nearly

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Y. Cao et al. / Computational Materials Science 111 (2016) 34–40

750

280

700

Debye temperatures (K)

240

B (GPa)

200 160 120 80 40

0

NiAl 0GPa

Co Ni0 GPa

NiAl 10GPa

Co Ni10GPa

NiAl 20GPa

Co Ni 20GPa

NiAl 30GPa

Co Ni30GPa

300

600

900

650 600 550 500 450 400 350

1200

1500

NiAl 0GPa

Co Ni 0 GPa

NiAl 10GPa

Co Ni 10GPa

NiAl 20GPa

Co Ni 20GPa

NiAl 30GPa

Co Ni 30GPa

0

Fig. 8. The variations of bulk modulus as a function of temperature and pressure.

keeps constant. When T > 100 K, the bulk modulus decreases linearly with increasing temperature, which indicates that the volume varies significantly as the temperature increases. These conclusions are in agreement with what is shown in Fig. 6. From Fig. 8, we can see that bulk modulus is sensitive to both temperature T and pressure P. The bulk modulus decreases and increases with increasing temperature and pressures, respectively. These results are due to the fact that the effect of increasing pressure on the material is the same as decreasing temperature of the material. We also found that bulk modulus of Ni8Al8 is smaller than (Ni7Co)Al8 at the same temperature and pressure, and the difference is increase with the temperatures and decrease with pressures. The dependence of heat capacity of on temperature at different pressures is illustrated in Fig. 9. It can be seen that C V changes little after doping. At low temperature, the temperature dependence follows a T3 law. At high temperature (T > 900 K), C V approaches the Dulong–Petit limit (390 J mol1 K1), which is common to all solids at high temperature [29]. The pressure has little influence on heat capacity. It is well known that the Debye temperature hD of solid is an important physical quantity for a solid. Fig. 10 displays the temperature-dependence of the Debye of Ni8Al8 and (Ni7Co)Al8 under different temperatures. The Debye temperature increases and decreases with increasing pressure and temperature, respectively. The reason is that hD is related to bulk modulus and volume.

450 400 350

CV (J/mol*K)

300

600

900

1200

1500

Tempreture (K)

Tempreture/(K)

300

Fig. 10. The variations of Debye temperature as a function of temperature and pressure.

The bulk modulus decreases and increases with increasing temperature and pressure respectively, and the volume is on the contrary. It can be seen from Fig. 10 that hD is nearly constant at low temperature and decreases linearly with increasing temperature when the temperature is above 100 K. hD of Ni8Al8 is smaller than (Ni7Co)Al8 at the same temperature and pressure, and the difference is increase with increase of temperatures and decrease of pressures. 4. Conclusions In this paper, the site preference of Co in NiAl and its effects on electronic and elastic properties were investigated by first principles. The formation enthalpy is negative, implying that the compounds can exist from the energetic point of view. The formation enthalpy of doping system is bigger than NiAl means that stability is reduced after Co doping. The site preference were investigated through calculating the transfer energy of NiAl alloyed by Co element. The results show that Co shows the tendency to occupy Ni site. By analyzing changes in electronic density of states, valence charge density, Mulliken population and overlap population, the electronic property and bond characters were discussed. The enrichment phenomenon of bonding charge density on h1 1 1i become more serious for the substitution indicating that Co increases the brittleness of alloy. The present elastic constants satisfy the stability criteria, so single crystals are mechanical stable. The values of B/G and m show that Co reduces alloy plasticity. The variations of the volume, expansion coefficient, heat capacities and Debye temperature of Ni8Al8 and (Ni7Co)Al8 as functions of pressure and temperature have been predicted by quasi-harmonic Debye model. Acknowledgment

250 200 150

NiAl 0GPa

CoNi 0 GPa

NiAl 10GPa

CoNi 10GPa

NiAl 20GPa

CoNi 20GPa

NiAl 30GPa

CoNi 30GPa

This work is supported by program of excellent team at Kunming University of Science and Technology. References

100 50 0

0

300

600

900

1200

1500

Tempreture (K) Fig. 9. The variations of heat capacity as a function of temperature and pressure.

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