Effects of Transition Elements on the Structural, Elastic Properties and Relative Phase Stability of L1 2 γ (cid:48) -Co 3 Nb from First-Principles Calculations

: In order to explore novel light-weight Co-Nb-based superalloys with excellent performance, we studied the effects of alloying elements including Sc, Ti, V, Cr, Mn, Fe, Ni, Y, Zr, Mo, Tc, Ru, Rh, Pd, Hf, Ta, W, Re, Os, Ir and Pt on the structural stability, elastic and thermodynamic properties of γ (cid:48) -Co 3 Nb through ﬁrst-principles calculations. The results of transfer energy indicate that Y, Zr, Hf and Ta have a strong preference for Nb sites, while Ni, Rh, Pd, Ir and Pt have a strong tendency to occupy the Co sites. In the ground state, the addition of alloying elements plays a positive role in improving the stability of γ (cid:48) -Co 3 Nb compound. The order of stabilizing effect is as follows: Ti > Ta > Hf > Pt > Ir > Zr > Rh > V > Ni > W > Sc > Mo > Pd > Re > Ru. Combining the calculation results of elastic properties and electronic structure, we found that the addition of alloying elements can strengthen the mechanical properties of γ (cid:48) -Co 3 Nb, and the higher spatial symmetry of electrons accounts for improving the shear modulus of γ (cid:48) -Co 3 Nb compound. At ﬁnite temperatures, Ti, Ta, Hf, Pt, Ir, Zr and V signiﬁcantly expand the stabilization temperature range of the γ (cid:48) phase and are potential alloying elements to improve the high-temperature stability of the γ (cid:48) -Co 3 Nb compounds. in the matrix. The results suggest several potential novel γ / γ (cid:48) Co-Nb-based superalloys, such as Co-Nb-Ti, Co-Nb-Ta and Co-Nb-Hf.


Introduction
With the continuous development of modern aviation industry, Ni-based superalloys cannot meet the increasing performance needs because their operating temperature is close to the melting point. Therefore, the development of the superalloys that have a higher melting point becomes an urgent task [1,2]. Co-based superalloys are expected to be the next generation of commercial superalloys owing to the higher melting point of Co compared to Ni (1495 • C vs. 1455 • C). In 2006, Sato et al. [3] discovered the Co-Al-W ternary superalloy system, where coherent γ -Co 3 (Al, W) precipitates from the γ-Co matrix. The microstructure of γ/γ two-phase equilibrium, which is extremely similar to that in commonly used Ni-based superalloys, allows Co-Al-W superalloys to exhibit excellent high-temperature mechanical properties while having a high melting point [4][5][6]. However, due to the high content of heavy element W, the density of Co-Al-W superalloys is much higher (9.5 g/cm 3 for Co-9Al-9W) than that of Ni-based superalloys (~8.5 g/cm 3 ), which limits the practical application of Co-Al-W superalloys [7,8]. In order to explore W-free type Co-based superalloys, numerous experiments have been carried out with great progress in recent years, and γ/γ two-phase microstructure was found in Co-Al-Mo-Nb/Ta [9,10], Co-Ti-Mo [11], Co-Ti-Cr [12], Co-Ti-V [13], Co-V-Ta [14][15][16], and Co-Al-V [17]. after short aging time. The addition of alloying elements can significa properties of the alloy, which is an effective way to improve the perfo the new Co-based superalloy. Moreover, transition metal elements are elements in the research of new Co based superalloys. For instance, the Ni, Mo, Ru, Ta and Ir can improve the thermal stability and volume fr [20]; the addition of Zr and Hf can improve the grain boundary bond enhance the ductility of the alloy [21,22]. Cr plays an important role in i idation resistance, reducing the alloy density and γ/γ′ phase mismatch [ ing basic system, the effect of alloying transition elements on the prop phase is fundamentally not clear, and there is little relevant report on system. Therefore, further research on Co-Nb-based superalloys is of for the development of light-weight superalloys.
In this work, the alloying effect of transition metal elements (X) in Cr, Mn, Fe, Ni, Y, Zr, Mo, Tc, Ru, Rh, Pd, Hf, Ta, W, Re, Os, Ir and Pt investigated. Specifically, the atomic configurations of X-substituted γ termined according to the site preference of transition metal element X on this basis, the structural stability and mechanical properties of tern γ′-Co3Nb were evaluated. By considering the contribution of entropy, th ture range of γ′-Co3Nb phase against D019 phase is determined, which p ical basis for the development of the ternary light-weight Co-based sup system.

Site Preference Criteria
As showed in Figure 1, we adopted a 2 × 2 × 2 supercell of 32 atom Co3Nb in this work and its chemical formula is Co24Nb8. The labeled sph Co site and Nb site, which can be substituted by one transition metal ele in compositions of (Co23X1)Nb8 and Co24(Nb7X1), respectively.  We introduce transfer energy E Co→Nb X and exchange antisite energy E ant [24,25] to explore the site preference of ternary alloying elements in γ -Co 3 Nb. E Co→Nb respectively, where E(Co 3 Nb) is the total energy of hP24-Co 3 Nb, E(Co) is the total energy of hcp-Co and E(X) is the total energy of X in its most stable state. This investigation does not consider whether a solution of X in the Co-matrix may be more stable. Taking Ta occupying Nb site as an example, if the stable formation energy is negative, E Ta Nb stab < 0, it indicates that Ta tends to stabilize γ -Co 3 Nb rather than the mixture of three phases (i.e., hP24-Co 3 Nb + hcp-Co + bcc-Ta). A larger negative E stab means that the formed phase is more stable; whereas if E Ta Nb stab > 0, it indicates that Ta is unfavorable to the stability of γ -Co 3 Nb, and it is difficult to form Ta-substituted Co 3 Nb phase compared with the mixture of three phases.

Elastic Properties
The elastic properties of alloys can be evaluated on the basis of elastic stiffness constants C ij s. The values of C ij s at equilibrium were calculated by employing stress-strain method proposed by Shang et al. [27]. A given set of strains ε = (ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 )(where ε 1 -ε 3 refer to normal strains and ε 4 -ε 6 refer to shear strains) was imposed on the crystal to generate the slight deformations. Then, one corresponding set of stress σ = (σ 1 , σ 2 , σ 3 , σ 4 , σ 5 , σ 6 ) of the deformed crystal can be obtained by first-principles calculations. In the present work, the linearly independent sets of strain were used as: with x = ±0.01. Through the n sets of strain and their corresponding stresses, the elastic stiffness can be calculated based on Hooke's law. According to the symmetry of L1 2type crystal structure, the number of independent components of elastic stiffness tensor decreases to three (C 11 , C 12 , C 44 ). However, it is worth noting that the crystal structure of γ -Co 3 Nb with transition element X occupying Nb site still maintains the original cubic symmetry. When Co is substituted by transition elements X, the cubic supercell of γ -Co 3 Nb will be slightly distorted, which changes from cubic symmetry to tetragonal symmetry, resulting in the increase of the number of independent C ij s. Therefore, Hooke s law can be simplified to the following formula by: The average C ij were used, which ensures the comparability of our calculated results: where S ij s represents elastic compliance constants, which can be obtained from the inverse matrix of C ij s. Furthermore, the polycrystalline properties, including the bulk modulus (B), shear modulus (G), Young's moduli (E) and Poisson's ratio (ν), can be calculated by Voigt-Reuss-Hill (VRH) method [28]:

Details of First-Principles Calculations
Herein, first-principles calculations were performed by Vienna Ab initial Simulation Package (VASP) [29,30] based on the projector augmented wave (PAW) method [31] and density functional theory (DFT) [32]. The exchange and correlation function were described by using the Perdue-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) [33]. The plane wave basis was truncated at a kinetic energy cutoff of 450 eV to ensure that the total energy of all structures converges within 1 meV/atom. For the k-point sampling, Gamma centered grid of 7 × 7 × 7 for L1 2 and 3 × 3 × 7 for D0 19 was used. The reciprocal space integration was executed with the Methfessel-Paxton technique [33] with a smearing width of 0.20 eV. Throughout the calculations, the convergence thresholds of total energy and the maximum force acting on ions was set to less than 10 −5 eV/atom and 10 −2 eV/Å, respectively. The spin polarized calculations were performed due to the ferromagnetic nature of Co.
The conjugate gradient algorithm (IBRION = 2) was employed to optimize the structures, allowing the change of ionic positions, cell shape and cell volume (ISIF = 3). In order to obtain an accurate and stable atomic structure, ISIF = 3/IBRION =2 was repeatedly used for ion relaxation until the structural optimization converged in only one loop. For evaluating the accuracy of this method, fitting a state equation were performed. By changing the scale factors, 10 structures were selected near the structure optimized with ISIF = 3. Then, the optimization (ISIF = 4) and static calculation were carried out. The P-V data of these structures were fitted with the third-order Birch-Murnaghan equation of state to find total energy at the lowest point. The difference in total energy calculated by two methods is about 0.2 meV/atom, which ensures the accuracy of the structure.
When calculating the energy of pure elements, the most stable structures at 293 K and 1 atm were employed as the initial structure. For Ni, Rh, Pd, Ir and Pt with fcc structure containing 4 atoms, a gamma centered 14 × 14 × 14 k-point mesh was adopted; for V, Cr, Fe, Nb, Mo, Ta and W with bcc structure containing 2 atoms, a gamma centered 18 × 18 × 18 k-point mesh was adopted; for Sc, Ti, Co, Y, Zr, Tc, Ru, Hf, Re and Os with hcp structure containing 2 atoms, a gamma centered 23 × 23 × 12 k-point mesh was adopted. Mn has a complicated structure of 58 atoms, and a gamma centered 6 × 6 × 6 k-point mesh was used due to its large unit cell. In particular, Cr was initialized with an antiferromagnetic state. Figure 2 plots the normalized transfer energy of X-substituted γ -Co 3 Nb compounds, which reveals the site preference of transition metal X in γ -Co 3 Nb compounds, and their corresponding values of E Co→Nb X are listed in Table 1. It can be seen from Figure 2 that Y, Zr, Hf and Ta have a strong Nb-site preference, while Sc, Ti, V, Cr, Mo, W and Re have a weak Nb-site preference; Mn, Fe, Tc, Ru and Os have a weak Co-site preference, while Ni, Rh, Pd, Ir and Pt have a strong Co-site preference. It is apparent that the site preference shows regular variation according to the periodic table of the elements, i.e., the tendency of elements to occupy Co site becomes progressively stronger with increasing atomic number in each period, which is similar to the findings in the Ni 3 Al phase [24].

Site Preference and Structural Stability
The lattice parameter is an important physical quantity for the γ -strengthened superalloys, which determines the degree of lattice mismatch between the γ -Co 3 Nb precipitate and the γ-Co matrix, and thus controls the morphology and strength of the γ -Co 3 Nb precipitate. The addition of transition element X will cause different degrees of changes in the lattice parameters of γ -Co 3 Nb. Figure 3 illustrates the relationship between the lattice parameters of X-substituted γ -Co 3 Nb and the atomic radius of X. The atomic radius of X is metallic radius, taking from Reference [34]. It can be seen that the lattice parameter of γ -Co 3 Nb is roughly proportional to the atomic radius of element X, which leads to a lattice mismatch of 2.60%~3.72% between the X-substituted γ -Co 3 Nb and the γ-matrix (lattice parameter is 3.52Å). Although the degree of mismatch in the present Co-Nb-X system is much larger than that of a typical Ni-based superalloy (about 0.5%) [35], the γ /γ two-phase microstructure can still be formed in the Co-Nb-based system [14]. Table 1. It can be seen from Figure 2 that Y, Zr, Hf and Ta have a strong Nb-site preference, while Sc, Ti, V, Cr, Mo, W and Re have a weak Nb-site preference; Mn, Fe, Tc, Ru and Os have a weak Co-site preference, while Ni, Rh, Pd, Ir and Pt have a strong Co-site preference. It is apparent that the site preference shows regular variation according to the periodic table of the elements, i.e., the tendency of elements to occupy Co site becomes progressively stronger with increasing atomic number in each period, which is similar to the findings in the Ni3Al phase [24].  (4) and (5)   of γ′-Co3Nb is roughly proportional to the atomic radius of element X, which leads to lattice mismatch of 2.60% ~ 3.72% between the X-substituted γ′-Co3Nb and the γ-matr (lattice parameter is 3.52Å). Although the degree of mismatch in the present Co-Nb-X sy tem is much larger than that of a typical Ni-based superalloy (about 0.5%) [35], the γ′ two-phase microstructure can still be formed in the Co-Nb-based system [14]. Based on the site preference of the alloying elements, the stable formation energ stab E was further calculated in order to understand the relative structural stability of substituted γ′-Co3Nb. Figure 4 illustrates the calculated stab E for X-substituted Co3Nb.
can be seen that the elements of Sc, Ti, V, Ni, Zr, Mo, Ru, Rh, Pd, Hf, Ta, W, Re, Ir and play a positive role in enhancing the structural stability of γ′-Co3Nb, in the order of T Ta > Hf > Pt > Ir > Zr > Rh > V > Ni > W > Sc > Mo > Pd > Re > Ru. In agreement with t reported experimental and theoretical studies of Co-based superalloys [6,[36][37][38], the a dition of Ta and Ti can also greatly improve the structural stability of γ′-Co3Nb precip tates in the matrix. The results suggest several potential novel γ/γ′ Co-Nb-based supera loys, such as Co-Nb-Ti, Co-Nb-Ta and Co-Nb-Hf. Based on the site preference of the alloying elements, the stable formation energy E stab was further calculated in order to understand the relative structural stability of Xsubstituted γ -Co 3 Nb. Figure 4 illustrates the calculated E stab for X-substituted Co 3 Nb. It can be seen that the elements of Sc, Ti, V, Ni, Zr, Mo, Ru, Rh, Pd, Hf, Ta, W, Re, Ir and Pt play a positive role in enhancing the structural stability of γ -Co 3 Nb, in the order of Ti > Ta > Hf > Pt > Ir > Zr > Rh > V > Ni > W > Sc > Mo > Pd > Re > Ru. In agreement with the reported experimental and theoretical studies of Co-based superalloys [6,[36][37][38], the addition of Ta and Ti can also greatly improve the structural stability of γ -Co 3 Nb precipitates in the matrix. The results suggest several potential novel γ/γ Co-Nb-based superalloys, such as Co-Nb-Ti, Co-Nb-Ta and Co-Nb-Hf.

Alloying Effects on Mechanical Properties
The ij C ′s of the X-substituted γ′-Co3Nb compounds were calculated according effective stress-strain method to investigate the effect of alloying element on the mechanical properties of γ′-Co3Nb (see Table 2). According to Born′s theory [39], t chanical stability of cubic crystal systems can be determined by the following form

Alloying Effects on Mechanical Properties
The C ij s of the X-substituted γ -Co 3 Nb compounds were calculated according to the effective stress-strain method to investigate the effect of alloying element on the elastic  Table 2). According to Born s theory [39], the mechanical stability of cubic crystal systems can be determined by the following formula: From Table 2, it can be seen that all X-substituted γ -Co 3 Nb satisfies the above mechanical stability criterion, indicating that they are mechanically stable in the ground state. The polycrystalline elastic mechanical parameters such as bulk modulus B, shear modulus G, Young's modulus E and Poisson's ratio ν calculated for each compound using Voigt-Reuss-Hill scheme [28] are also given in Table 2. In order to investigate the change pattern of elastic properties after the transition metal atom X occupies different sites in γ -Co 3 Nb, the volume change of X-substituted Co 3 Nb compounds was plotted in relation to the change of bulk modulus and shear modulus (Figure 5a,b, respectively). As can be seen, the addition of transition elements that preferentially occupy the Co site increases the supercell volume of γ -Co 3 Nb while the opposite is true for elements that tend to occupy the Nb site. This phenomenon can be explained by the atomic radius of transition elements X with respect to the atomic radius of the Co/Nb atom [34]: in general, the radii of alloying atoms (Rh, Ir, Ni, Pd, Pt, etc.) are larger than that of Co atom, but smaller than that of Nb atom. Therefore, the volume of the compound expands when the alloying atoms occupies the Co site and shrinks when the alloying element occupies Nb site. As can be seen from Figure 5, the B and G decrease linearly with the increasing volume when occupying Nb site or Co site, in agreement with previous studies [40]. The addition of transition element V results in the largest volume deformation of the Co 3 Nb compound, which led to a large increase in both the elastic modulus and bulk modulus of the Co 3 Nb compound. The addition of Ni is effective in increasing the bulk modulus of the Co 3 Nb compound even though it hardly changes the volume of the Co 3 Nb compound. Figure 5, the B and G decrease linearly with the increasing volume when occupying Nb site or Co site, in agreement with previous studies [40]. The addition of transition element V results in the largest volume deformation of the Co3Nb compound, which led to a large increase in both the elastic modulus and bulk modulus of the Co3Nb compound. The addition of Ni is effective in increasing the bulk modulus of the Co3Nb compound even though it hardly changes the volume of the Co3Nb compound. Then, we predicted the ductile/brittle nature of X-substituted γ′-Co3Nb by both Pugh's classical criterion (B/G) [41] and Cauchy pressure ( C C ) [42]. The solid materials with B/G greater than 1.75 indicate ductility otherwise brittleness. According to Table  2, the B/G ratio of all X-substituted γ′-Co3Nb compounds except the element Sc is higher than the critical value of 1.75, indicating that most of the X-substituted γ′-Co3Nb compounds have good ductility, among which the alloying element such as Tc, Os and Hf significantly improve the ductility of the γ′-Co3Nb compounds. On the other hand, the Cauchy pressure ( C C ) can be used as an indicator of binding properties, which can be associated with the Cauchy pressure. A positive value of Cauchy pressure indicates that the system is dominated by metallic bonds, otherwise it is dominated by covalent bonds. For metallic systems, a higher Cauchy pressure represents a more ductile feature. Then, we predicted the ductile/brittle nature of X-substituted γ -Co 3 Nb by both Pugh's classical criterion (B/G) [41] and Cauchy pressure (C 12 − C 44 ) [42]. The solid materials with B/G greater than 1.75 indicate ductility otherwise brittleness. According to Table 2, the B/G ratio of all X-substituted γ -Co 3 Nb compounds except the element Sc is higher than the critical value of 1.75, indicating that most of the X-substituted γ -Co 3 Nb compounds have good ductility, among which the alloying element such as Tc, Os and Hf significantly improve the ductility of the γ -Co 3 Nb compounds. On the other hand, the Cauchy pressure (C 12 − C 44 ) can be used as an indicator of binding properties, which can be associated with the Cauchy pressure. A positive value of Cauchy pressure indicates that the system is dominated by metallic bonds, otherwise it is dominated by covalent bonds. For metallic systems, a higher Cauchy pressure represents a more ductile feature. To elucidate the intrinsic relationship between the bonding nature and the ductility/brittleness of materials, the B/G ratio is plotted versus Cauchy pressure (C 12 − C 44 ) in Figure 6. It is clear that there is a linear relationship between the B/G ratio and C 12 − C 44 . The Cauchy pressure is positive for B/G > 1.75, indicating that the ductile characteristics of X-substituted γ -Co 3 Nb are essentially contributed from the metallic bonding nature. To elucidate the intrinsic relationship between the bonding nature and the ductility/brittleness of materials, the B/G ratio is plotted versus Cauchy pressure (    These shear moduli can be calculated by the following formula [43]: (20) According to the results listed in Table 2, for all X-substituted γ -Co 3 Nb compounds, the shear modulus on each crystal plane satisfies the following relationship: which demonstrates that, the X-substituted γ -Co 3 Nb compounds are less resistant to shear sliding in the {110} and {111} planes than that in the {100} plane. By comparing with the shear modulus of the pure γ -Co 3 Nb, we classified these alloying elements into three categories (see

Electronic Structure
According to previous studies [44,45], the valence electrons of compound play a significant role in the shear modulus. Here, we use the charge density difference (CDD) to reveal the relationship between the electronic structure and mechanical properties of the X-substituted γ′-Co3Nb compounds. The CDD in the (110) plane is intercepted for analysis because of its lowest shear modulus (see Table 2). Figure 8

Electronic Structure
According to previous studies [44,45], the valence electrons of compound play a significant role in the shear modulus. Here, we use the charge density difference (CDD) to reveal the relationship between the electronic structure and mechanical properties of the X-substituted γ -Co 3 Nb compounds. The CDD in the (110) plane is intercepted for analysis because of its lowest shear modulus (see Table 2). Figure 8 plots the CDD contours in the (110) plane for Re, Ti, Pt and Tc substitutions, by considering the relatively large and low G {110} when Re and Ti occupying the Co site or Pt and Tc occupying the Nb site, respectively. X-substituted γ′-Co3Nb compounds. The CDD in the (110) plane is intercepted for analysis because of its lowest shear modulus (see Table 2). Figure 8 plots the CDD contours in the (110) plane for Re, Ti, Pt and Tc substitutions, by considering the relatively large and low {110} G when Re and Ti occupying the Co site or Pt and Tc occupying the Nb site, respectively. From Figure 8a, it can be seen that the transferred electrons of pure γ′-Co3Nb are mainly distributed between the Co atom and Nb atoms in the (110) plane, which indicates that they form a covalent-like bond. In the case of Re substitution (Figure 8b), transferred electrons homogeneously assembles around the Re atoms, forming a ring-shaped valence bond. This weakens the anisotropy of the (110) plane, and results in an increase in shear From Figure 8a, it can be seen that the transferred electrons of pure γ -Co 3 Nb are mainly distributed between the Co atom and Nb atoms in the (110) plane, which indicates that they form a covalent-like bond. In the case of Re substitution (Figure 8b), transferred electrons homogeneously assembles around the Re atoms, forming a ring-shaped valence bond. This weakens the anisotropy of the (110) plane, and results in an increase in shear modulus on this plane compared with that of pure γ -Co 3 Nb. In Figure 8c, the transferred electrons increase along [110] direction by the substitution of Ti, which leads to a weakened symmetry and thus makes G {110} lower. When the Pt atom occupies the Co site as seen in Figure 8d, it has no obvious d-orbital interaction with the surrounding Nb atoms, and the spatial distribution of the transferred electrons is close to γ -Co 3 Nb, so they have comparable values of G {110} . Furthermore, H-shaped distributed charge can be observed between Tc atoms and two nearest-neighbor Nb atoms as shown in Figure 8e, while the CDD along [110] direction is much more intensive than that along [001] direction. This H-shaped charge distribution increases the anisotropy of bonding, resulting in a significant decrease in G {110} . It turns out from the above analysis that the ring-shaped electron distribution has a positive effect on the increase of shear modulus.

Thermodynamic Properties
To consider the influence of temperature on the phase stability and thermodynamic properties of X-substituted γ -Co 3 Nb, the quasi-harmonic Debye model [46] was adopted to evaluate the contribution of entropy to free energy, which can be well implemented by Gibbs2 code [47]. According to Debye model and neglecting the contribution of hot electrons to free energy, the non-equilibrium Gibbs free energy can be expressed as: where E(V) is the static energy and F vib (Θ; T) is vibrational Helmholtz free energy. At a given temperature T and pressure P, the Gibbs free energy of the equilibrium state can be obtained from the derivative of the Gibbs free energy of the non-equilibrium state to the volume. See [46,47] for more details.
According to a recent experimental study, the γ -Co 3 Nb phase in Co-Nb-V based superalloy will decomposed into D0 19 phase with increasing annealing time [14]. Therefore, in order to evaluate the phase stability of X-substituted γ -Co 3 Nb at finite temperature, the difference in Gibbs free energy between the D0 19 phase ( Figure S1 provides the supercell of X-substituted D0 19 -Co 3 Nb and Table S1 provides relevant data of equilibrium state in the Supplementary) and L1 2 phase should be considered by the following equation: (22) where G D0 19 and G L1 2 are the Gibbs free energy of D0 19 and L1 2 structures, respectively. If ∆G < 0, it means that D0 19 structure is more stable than L1 2 structure, and vice versa. Based on the results in Section 3.1, the γ -Co 3 Nb substituted by Ti, Ta, Hf, Pt, Ir, Zr and V elements were, respectively, selected in the calculations, since these compounds have lower stable formation energy E stab than other compounds. Figure 9 depicts the calculated ∆G as a function of temperature for these X-substituted Co 3 Nb compounds. It can be seen that Co 3 Nb has a negative ∆G over the entire temperature range, which means that γ -Co 3 Nb is a metastable phase relative to the D0 19 structure. It is noted that the ∆G of X-substituted Co 3 Nb is significantly higher than that of Co 3 Nb, suggesting that these alloying elements can improve the phase stability of γ -Co 3 Nb at high temperature to a certain extent. Recent experimental studies have verified that the Co 3 (Nb, V) is a metastable phase [14], which is consistent with our prediction. According to the results, the addition of Pt, Hf, Ir, Ta and Ti should be more effective than V for improving the stability of γ -Co 3 Nb and can be considered as the potential basic ternary system of X-substituted γ -Co 3 Nb.  Figure 10a,b illustrates the calculated specific heats of these X-substituted γ′-Co3Nb compounds at constant pressure ( p C ) and constant volume ( v C ), respectively. In Figure   10a, the curves show typical feature of specific heat, that is, v C of these compounds is   Figure 10a,b illustrates the calculated specific heats of these X-substituted γ -Co 3 Nb compounds at constant pressure (C p ) and constant volume (C v ), respectively. In Figure 10a, the curves show typical feature of specific heat, that is, C v of these compounds is proportional to T 3 (C v ∝ T 3 ) at low temperatures (<300 K), and tends to Dulong-Petit limit of C v = 3nR = 800 J/K · mol at high temperatures. C p depicted in Figure 10b, is larger than C v over the entire temperature range, which can be explained by the relation: C p − C v = αBVT (α = volume thermal expansion coefficient). At low temperatures, C p exhibits Debye T 3 power-law behavior same as C v , then C p , monotonously increases and deviates from C v with increasing temperature. The difference between specific heats is due to the thermal expansion caused by anharmonicity effects. limit of = = ⋅ 3 nR 800 J/ K mol v C at high temperatures. p C depicted in Figure 10b, is larger than v C over the entire temperature range, which can be explained by the relation:

− =α BVT
p v C C (α = volume thermal expansion coefficient). At low temperatures, p C exhibits Debye 3 T power-law behavior same as v C , then p C , monotonously increases and deviates from v C with increasing temperature. The difference between specific heats is due to the thermal expansion caused by anharmonicity effects. Finally, to estimate the effect of alloying elements on the volume change arise from temperature, the volume thermal expansion coefficient α of both the γ′-Co3Nb and X-substituted γ′-Co3Nb compounds at finite temperature was calculated, as shown in Figure  10c. For all X-substituted γ′-Co3Nb compounds, the value of α increases sharply from 0K Finally, to estimate the effect of alloying elements on the volume change arise from temperature, the volume thermal expansion coefficient α of both the γ -Co 3 Nb and Xsubstituted γ -Co 3 Nb compounds at finite temperature was calculated, as shown in Figure 10c. For all X-substituted γ -Co 3 Nb compounds, the value of α increases sharply from 0K to about 300K, gradually approaching a linear increase at high temperatures, and the incremental trend becomes moderate. It is noteworthy that all γ -Co 3 Nb compounds show a similar trend with increasing temperature, except for the Hf-substituted γ -Co 3 Nb. The α of γ -Co 3 Nb substituted by Hf is much lower than that of γ -Co 3 Nb substituted by other elements, which means that addition of Hf element can effectively resist the volume effect of γ -Co 3 Nb caused by temperature.

Conclusions
In this study, we clarified the alloying effect on the atomic structure, elastic mechanical properties and relative phase stability of γ -Co 3 Nb compound by using DFT calculations. From the calculated transfer energy of various alloying elements in different sublattices of γ -Co 3 Nb, Y, Zr, Hf and Ta have a strong preference in Nb sites, while Ni, Rh, Pd, Ir and Pt are more inclined to occupy Co sites. For elements of the same period, the tendency to occupy Co site enhances with the increase of atomic number. In the ground state, the stabilizing effect of transition metal elements on γ -Co 3 Nb are listed in order as follows: Ti > Ta > Hf > Pt > Ir > Zr > Rh > V > Ni > W > Sc > Mo > Pd > Re > Ru. After comparing the shear modulus of different crystal planes, it is found that these X-substituted γ -Co 3 Nb are more likely to shear on the {110} crystal plane along the [110] direction. The analysis of the electronic structure show that when the alloying element, such as Tc, occupies the Nb site, the ring-shaped electron distribution formed is beneficial to the improvement of the shear modulus. At finite temperatures, the addition of Ti, Ta, Hf, Pt, Ir, Zr and V can effectively expand the stabilization temperature range of γ -Co 3 Nb. In particular, the addition of Hf element can further reduce the sensitivity of the volume of γ -Co 3 Nb to temperature.

Conflicts of Interest:
The authors declare there is no conflicts of interest regarding the publication of this paper.