Theoretical Investigation of Effects of Transition Elements on Phase Stabilities and Elastic Properties of the γ′ Phase in Co–V–Ta Superalloys

In order to improve the thermal stability and mechanical properties of W-free light-weight Co–V–Ta-based superalloys, the effects of alloying elements including Sc, Ti, Cr, Mn, Fe, Ni, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Hf, W, Re, Os, Ir, and Pt on γ′-Co₃(V, Ta) stability and elastic properties were studied. The results from the reaction energy indicate that Sc, Ti, Y, Zr, Nb, Mo, Hf, and W tend to occupy the V site, whereas Cr, Mn, Fe, Ni, Tc, Ru, Rh, Pd, Re, Os, Ir, and Pt prefer to occupy the Co site. It was found that Sc, Ti, Y, Zr, Fe, and Mn stabilize the γ′-Co₃(V, Ta) phase by raising the phase-transfer energy. The addition of Mo and W increases the bulk modulus, shear modulus, and Young’s modulus. According to Pugh’s classical criterion, the γ′-Co₃(V, Ta) phase is an intrinsically brittle material, and the addition of elements such as Tc and Re significantly enhances the ductility. At finite temperature, the doping of Sc, Ti, Mn, Fe, and Hf enhances the relative stability of the γ′-Co₃(V, Ta) phase in the whole temperature range from 0 to 1200 K and are candidates for improving the stability of the γ′-Co₃(V, Ta) phase.

1 Introduction

Since the discovery of the γ′ phase in the Co–Al–W ternary system (Sato et al., 2006), extensive attention has been attracted to the γ′-strengthened Co-based superalloys. The coherent γ/γ′ structure makes this new class of Co-based superalloy possess good high temperature strength and creep resistance, and its solidus and liquidus temperatures are 100°C–150°C higher than those of advanced Ni-based alloys (Pollock et al., 2010). Therefore, the development of γʹ-strengthened Co-based superalloys has become a significant subject for research. It has been reported that W can effectively stabilize the γ′ phase and elevate the γ′ solvus temperature in Co-based superalloys (Pyczak et al., 2015). However, the heavy element W significantly increases the density of Co–Al–W-based superalloys (Shi et al., 2015), which is unfavorable for industrial applications, such as turbine blades. Recently, a considerable amount of literature work has been focused on the development of W-free Co-based high-temperature superalloys, aiming at reducing the mass density while maintaining the microstructure stability and high-temperature properties (Bantounas et al., 2019; Reyes Tirado et al., 2019; Migas et al., 2020; Reyes Tirado et al., 2020; Ruan et al., 2020; Davydov et al., 2021). Recently, Ruan et al. (2020) used machine learning algorithms and CALPHAD methods to successfully design a Co–V–Ta alloy with a wide γ/γ′ two-phase region, lower density than that of the Co–9Al–9.8W alloy (8.86 vs. 9.82 g/cm³), and higher yield strength. These advantages make the Co–V–Ta system a promising base alloy for further development.

However, the stability of the γ′ phase in the Co–V–Ta system is relatively low, and the volume fraction of the γ′ phase is not high enough. To improve the γ′ phase stability and volume fraction, an understanding of effects of alloying elements on the γ′ phase stability is required. Ruan et al. (2020) found that Ti effectively increases the γ′ solvus temperature by 70°C. Reyes Tirado et al. (2019) studied the alloying effects of Al, Ti, and Cr additions on the γ/γ′ microstructure and discovered that Al can stabilize the γ′ phase, and the addition of Ti and Ni has no significant effects on γ′ stability. Cr can slightly increase the γ′ volume. However, compared to the commercial Ni-based alloys, which contain upward of ten additional alloying elements, the research studies on alloying effects in the Co–V–Ta system are quite insufficient. The traditional experimental methods are quite expensive and time-consuming. Alternatively, the first-principle method provides an efficient way to explore the alloying effects and can give a deeper insight into the electronical interactions, which is hard to obtain by experiments.

To deeply understand the alloying effects of TM elements (including Sc, Ti, Cr, Mn, Fe, Ni, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Hf, W, Re, Os, Ir, and Pt) on the γ′-Co₃(V, Ta) phase, the first-principle calculations were performed to systematically investigate the phase stabilities and mechanical and thermodynamic properties of alloying effects of TM elements on the γ′-Co₃(V, Ta) phase. First, the site preference of TM elements in the γ′-Co₃(V, Ta) phase was calculated. On this basis, the stability and mechanical properties of γ′-Co₃(V, Ta) were evaluated. Further studies on the electronic structure explain the alloying effects of TM elements. Our research could be highly valuable for the interpretation of the experimental results in the Co-based superalloy, which provides very useful guidelines for the design and preparation of W-free Co–V–Ta-based superalloys.

2 Computational Methods

2.1 Structural Model

The γ′-Co₃(V, Ta) phase is built as a 2 × 2 × 2 supercell with Co at the cubic corner sites and V/Ta at the face center sites, as shown in Figure 1A. The supercell has 24 Co atoms, six V atoms, and two Ta atoms, which is consistent with experimental results (Ruan et al., 2020). There are six non-equivalent substitutional sites, namely, Co₁, Co₂, Co₃, V₁, V₂, and Ta in Figure 1A. Since the D0₁₉ phase is preferred in both Co–V and Co–Ta binary systems (Drapier and Coutsouradis, 1968; Wang et al., 2018), it is rational to consider the D0₁₉ phase as the main competing phase in the Co–V–Ta ternary system. The D0₁₉-Co₃(V, Ta) phase is built as a 2 × 2 × 1 supercell, and 10 non-equivalent substitutional sites, marked as Co₁, Co₂, Co₃, Co₄, Co₅, Co₆, Co₇, V₁, and V₂, Ta are shown in Figure 1B.

FIGURE 1

FIGURE 1. (A) Crystal structure of L1₂-ordered Co₃(V, Ta) with six nonequivalent sites (Co₁, Co₂, Co₃, V₁, V₂, and Ta) for the substitution by TM elements. (B) Crystal structure of D0₁₉-ordered Co₃(V, Ta) with 10 nonequivalent sites (Co₁, Co₂, Co₃, Co₄, Co₅, Co₆, Co₇, V₁, V₂, and Ta) for the substitution by TM elements.

2.2 Reaction Energy

To reveal the preferred substitutional sites of TM elements, the following reaction process is considered:

$\mathit{C}{\mathit{o}}_{\mathbf{24}}{\mathit{V}}_{\mathbf{6}}\mathit{T}{\mathit{a}}_{\mathbf{2}}+\mathit{X}\to \{\begin{array}{c}\mathit{C}{\mathit{o}}_{\mathbf{23}}{\mathit{V}}_{\mathbf{6}}\mathit{T}{\mathit{a}}_{\mathbf{2}}\mathit{X}+\mathit{Co}\\ \mathit{C}{\mathit{o}}_{\mathbf{24}}{\mathit{V}}_{\mathbf{5}}\mathit{T}{\mathit{a}}_{\mathbf{2}}\mathit{X}+\mathit{V}\\ \mathit{C}{\mathit{o}}_{\mathbf{24}}{\mathit{V}}_{\mathbf{6}}\mathit{T}{\mathit{a}}_{\mathbf{1}}\mathit{X}+\mathit{Ta}\mathit{.}\end{array}(1)$

The reaction energy is defined as follows:

${\mathit{E}}_{\mathit{sub}}^{\mathit{X}\to \mathit{i}}=\left({\mathit{E}}_{\mathit{sub}}+{\mathit{E}}_{\mathit{i}}\right)-\left({\mathit{E}}_{\mathit{pure}}+{\mathit{E}}_{\mathit{X}}\right),(2)$

where ${\mathit{E}}_{\mathit{sub}}$ and ${\mathit{E}}_{\mathit{pure}}$ refer to the total energy of the substituted system and pure system, respectively. ${\mathit{E}}_{\mathit{i}}/{\mathit{E}}_{\mathit{X}}$ is the energy per atom of Co, V, Ta, and X (X = Sc, Ti, Cr, Mn, Fe, Ni, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Hf, W, Re, Os, Ir, and Pt) in their most stable bulk structure. The preferred substitutional sites are determined by those with the lowest reaction energy ${\mathit{E}}_{\mathit{sub}}^{\mathit{X}\to \mathit{i}}$.

2.3 Phase Stability

Once the preferred sites are determined, the phase stability can be decided by its formation enthalpy, $\mathrm{\Delta }\mathit{H}\text{.}$

$\mathrm{\Delta }\mathit{H}=\left({\mathit{E}}_{\mathit{sub}}-{\displaystyle \sum }_{\mathit{i}}{\mathit{n}}_{\mathit{i}}{\mathit{E}}_{\mathit{i}}\right)/\mathbf{32},(3)$

where ${\mathit{n}}_{\mathit{i}}$ is the number of i atoms in the phase. A negative formation enthalpy implies the thermal stability at 0 K. Other than $\text{Δ}\mathit{H}$, the stability of the L1₂ structure is also related to the formation enthalpy of the competitive phase in the Co-based superalloys. Here, we considered the D0₁₉ phase as the competing phase and introduced the phase transfer energy to evaluate the relative stability between L1₂ and D0₁₉ structures, which is defined as follows:

$\mathrm{\Delta }\mathit{E}\left(\mathit{L}{\mathbf{1}}_{\mathbf{2}}\to \mathit{D}{\mathbf{0}}_{\mathbf{19}}\right)=\mathrm{\Delta }{\mathit{H}}_{\mathit{D}{\mathbf{0}}_{\mathbf{19}}}-\mathrm{\Delta }{\mathit{H}}_{\mathit{L}{\mathbf{1}}_{\mathbf{2}}}.$

2.4 Mechanical Properties

To understand the effects of TM elements on mechanical properties, the elastic constants are calculated using the stress–strain method (Shang et al., 2007). Based on Hooke’s law, the relationship between strain and stress is defined as follows:

${\mathit{\sigma }}_{\mathit{i}}={\displaystyle \sum }_{\mathit{j}}{\mathit{C}}_{\mathit{ij}}{\mathit{\epsilon }}_{\mathit{j}},(4)$

where ${\mathit{C}}_{\mathit{ij}}$ is the elastic constant.

According to the symmetry of the L1₂ structure, only ${\mathit{C}}_{\mathbf{11}\mathit{ }}$, ${\mathit{C}}_{\mathbf{12}}$,, and ${\mathit{C}}_{\mathbf{44}}$ are independent entries. Due to the relatively small size of the supercell, the TM elements may induce local lattice distortion and break the symmetry. As a result, the average ${\mathit{C}}_{\mathit{ij}}$ was used in this study to ensure the comparability of the results. For the L1₂ structure, the results are averaged as follows:

$\begin{array}{c}{\overline{\mathit{C}}}_{\mathbf{11}}=\left({\mathit{C}}_{\mathbf{11}}+{\mathit{C}}_{\mathbf{22}}+{\mathit{C}}_{\mathbf{33}}\right)/\mathbf{3},\mathit{ }{\overline{\mathit{S}}}_{\mathbf{11}}=\left({\mathit{S}}_{\mathbf{11}}+{\mathit{S}}_{\mathbf{22}}+{\mathit{S}}_{\mathbf{33}}\right)/\mathbf{3}\end{array},(5)$

${\overline{\mathit{C}}}_{\mathbf{12}}=\left({\mathit{C}}_{\mathbf{12}}+{\mathit{C}}_{\mathbf{13}}+{\mathit{C}}_{\mathbf{23}}\right)/\mathbf{3},\mathit{ }{\overline{\mathit{S}}}_{\mathbf{12}}=\left({\mathit{S}}_{\mathbf{12}}+{\mathit{S}}_{\mathbf{13}}+{\mathit{S}}_{\mathbf{23}}\right)/\mathbf{3},(6)$

${\overline{\mathit{C}}}_{\mathbf{44}}=\left({\mathit{C}}_{\mathbf{44}}+{\mathit{C}}_{\mathbf{55}}+{\mathit{C}}_{\mathbf{66}}\right)/\mathbf{3},\mathit{ }{\overline{\mathit{S}}}_{\mathbf{44}}=\left({\mathit{S}}_{\mathbf{44}}+{\mathit{S}}_{\mathbf{55}}+{\mathit{S}}_{\mathbf{66}}\right)/\mathbf{3},(7)$

where ${\mathit{S}}_{\mathit{ij}}$ are elastic compliance constants.

$\left[{\mathit{S}}_{\mathit{ij}}\right]={\left[{\mathit{C}}_{\mathit{ij}}\right]}^{-\mathbf{1}}.(8)$

Based on the calculated results, the aggregate properties can be calculated by

${\mathit{B}}_{\mathit{V}}=\left({\overline{\mathit{C}}}_{\mathbf{11}}+\mathbf{2}{\overline{\mathit{C}}}_{\mathbf{12}}\right)/\mathbf{3},\mathit{ }{\mathit{B}}_{\mathit{R}}=\mathbf{1}/\left(\mathbf{3}{\overline{\mathit{S}}}_{\mathbf{11}}+\mathbf{6}{\overline{\mathit{S}}}_{\mathbf{12}}\right),(9)$

${\mathit{G}}_{\mathit{V}}=\left({\overline{\mathit{C}}}_{\mathbf{11}}-{\overline{\mathit{C}}}_{\mathbf{12}}+\mathbf{3}{\overline{\mathit{C}}}_{\mathbf{44}}\right)/\mathbf{5},\mathit{ }{\mathit{G}}_{\mathit{R}}=\mathbf{5}/\left(\mathbf{4}{\overline{\mathit{S}}}_{\mathbf{11}}-\mathbf{4}{\overline{\mathit{S}}}_{\mathbf{12}}+\mathbf{3}\right),(10)$

${\mathit{B}}_{\mathit{H}}=\left({\mathit{B}}_{\mathit{V}}+{\mathit{B}}_{\mathit{R}}\right)/\mathbf{2},(11)$

${\mathit{G}}_{\mathit{H}}=\left({\mathit{G}}_{\mathit{V}}+{\mathit{G}}_{\mathit{R}}\right)/\mathbf{2},(12)$

${\mathit{E}}_{\mathit{H}}=\left(\mathbf{9}{\mathit{B}}_{\mathit{H}}{\mathit{G}}_{\mathit{H}}\right)/\left(\mathbf{3}{\mathit{B}}_{\mathit{H}}+{\mathit{G}}_{\mathit{H}}\right),(13)$

$\mathit{\nu }=\mathbf{3}{\mathit{B}}_{\mathit{H}}-\mathbf{2}{\mathit{G}}_{\mathit{H}}/\mathbf{2}\left(\mathbf{3}{\mathit{B}}_{\mathit{H}}+{\mathit{G}}_{\mathit{H}}\right),(14)$

where $\mathit{B}$, $\mathit{G}$, and $\mathit{\nu }$ denote the bulk, shear, Young’s moduli, and Poisson’s ratio, respectively.

2.5 Calculation Details

All calculations were performed using the Vienna ab initio simulation package (VASP) (Kresse and Furthmüller, 1996) based on the density functional theory using the projector augmented wave (PAW) pseudopotential method (Kresse and Joubert, 1999). The generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) was used to calculate the exchange–correlation functional. The cut-off energy of the plane-wave basis was set to 450 eV. After careful testing, the Brillouin zone was sampled by 7 × 7 × 7 and 3 × 3 × 7 k-meshes for L1₂ and D0₁₉ structures using the Monkhort–Pack scheme (Monkhorst and Pack, 1976). In the ionic relaxation, the Methfessel–Paxton method (Methfessel and Paxton, 1989) was used for reciprocal integration, and the smearing width was set to 0.12 eV. The energy and force convergence criteria were 10⁻⁵ eV/atom and 10⁻² eV/Å, respectively. Spin polarization was considered.

3 Results and Discussion

3.1 Site Preference and Phase Stability

The reaction energy ${\mathit{E}}_{\mathit{sub}}^{\mathit{X}\to \mathit{i}}$ of TM elements is plotted in Figure 2. The more negative the value of ${\mathit{E}}_{\mathit{sub}}^{\mathit{X}\to \mathit{i}}$, the stronger is the tendency for X to occupy this site. As seen, TM elements on the left side of the periodic table (Sc, Ti, Y, Zr, Nb, Mo, Hf, and W) tend to occupy V sites, whereas elements on the right side (Cr, Mn, Fe, Ni, Tc, Ru, Rh, Pd, Re, Os, Ir, and Pt) prefer the Co site. The results indicate that substitutional sites are closely related to the atomic radius. For TM elements with a large radius, V sites are preferred because the V atom is larger than the Co atom. For TM elements with a smaller radius, Co sites are preferred. The high reaction energy of TM elements at Ta sites indicates that no TM elements occupy Ta sites.

FIGURE 2

FIGURE 2. Reaction energy for the TM-substituted γ′-Co₃(V, Ta). The square, round, and triangle markers denote TM elements occupying the Co, V, and Ta sites, respectively.

The formation enthalpies $\Delta \mathit{H}$ are tabulated in Table 1. Here, the TM elements at their most preferred substitutional sites were considered. Without considering the competing phase like D0₁₉, a negative value of $\Delta \mathit{H}$ indicates that this phase is thermodynamically allowed. It can be seen that $\Delta \mathit{H}$ for all TM-substituted γ′-Co₃(V, Ta) phases are negative, suggesting they are stable or metastable at 0 K.

TABLE 1

TABLE 1. Site preference and formation enthalpies ΔH (eV/atom) of X-substituted γ′-Co₃(V, Ta).

Other than $\Delta \mathit{H}$, the stability of the L1₂ structure is also related to the formation enthalpy of the competitive phase in the Co-based superalloys. The phase transfer energy $\Delta \mathit{E}(\mathit{L}{\mathbf{1}}_{\mathbf{2}}\to \mathit{D}{\mathbf{0}}_{\mathbf{19}})$ is illustrated in Figure 3. $\Delta \mathit{E}(\mathit{L}{\mathbf{1}}_{\mathbf{2}}\to \mathit{D}{\mathbf{0}}_{\mathbf{19}})$ is an indicator of the relative stability between the L1₂ and D0₁₉ structure. A negative value indicates that the D0₁₉ phase is more stable than the L1₂ phase at 0K and vice versa.

FIGURE 3

FIGURE 3. Calculated structure transition energy of X-substituted γ′-Co₃(V, Ta), where 3d-, 4d-, and 5d-elements are denoted by square, round, and triangle, respectively.

3.2 Effect of TM Elements on Mechanical Properties

To investigate the alloying effects of TM elements on the mechanical properties, the elastic constants ${\mathit{C}}_{\mathit{ij}}$, bulk modulus $\mathit{B}$, shear modulus $\mathit{G}$, Young’s modulus $\mathit{E}$, and Poisson’s ratio $\mathit{\nu }$ were calculated, as shown in Table 2. The predicted $\mathit{B}$, $\mathit{G}$, and $\mathit{E}$ of γ′-Co₃(V, Ta) of previous theoretical results (Ruan et al., 2020) are also listed to validate our results.

TABLE 2

TABLE 2. Elastic properties of γ′-Co₃(V, Ta), including elastic constants ${\mathit{C}}_{\mathit{ij}}$, bulk modulus $\mathit{B}$, shear modulus $\mathit{G}$, Young’s modulus $\mathit{E}$, and Poisson’s ratio $\mathit{\nu }$. (${\mathit{C}}_{\mathit{ij}}$, $\mathit{B}$, $\mathit{G}$, and $\mathit{E}$ are given by GPa).

As seen, the elastic constants are remarkably changed by TM elements. We evaluated the mechanical stability of the TM-substituted γ′-Co₃(V, Ta) phase using the Born criterion (Born, 1939): ${\mathit{C}}_{\mathbf{11}}>\mathbf{0}$, ${\mathbf{C}}_{\mathbf{44}}>\mathbf{0}$, ${\mathit{C}}_{\mathbf{11}}>\left|{\mathit{C}}_{\mathbf{12}}\right|$, ${\mathit{C}}_{\mathbf{11}}+\mathbf{2}{\mathit{C}}_{\mathbf{12}}>\mathbf{0}$. It was confirmed that all TM-substituted γ′-Co₃(V, Ta) phases are mechanically stable.

To reveal the changing pattern of elastic properties induced by TM elements, the volume change concerning elastic properties is plotted in Figure 4. The doped TM elements cause the volume to expand despite the different substitutional sites (Co and Ta), which can be explained by a larger atomic radius of TM elements than that of their substituted atom. For 3d and 4d elements, the bulk/shear modulus change with respect to the volume change exhibits negative correlation, namely, expansion in volume and reduction in the bulk/shear modulus. However, for 5d elements, this relationship no longer holds. It is believed that mechanical properties have a strong correlation with the electron density in pure metals and binary alloys (Miedema et al., 1973). Figure 5 plots the relationship between the electron density ${\left(\mathit{B}/{\mathit{V}}_{\mathit{m}}\right)}^{\mathbf{1}/\mathbf{2}}$, where the electron density $\text{n}$ is defined by

$\mathit{n}={\mathit{Z}}_{\mathit{B}}/{\mathit{V}}_{\mathit{m}},(16)$

where ${\mathit{Z}}_{\mathit{B}}$ is the total valence electron number of X-doped Co₃(V, Ta), and ${\mathit{V}}_{\mathit{m}}$ is the mole volume.

FIGURE 4

FIGURE 4. Relationship between the elastic modulus changes and volume changes of X-substituted γ′-Co₃(V, Ta): (A) bulk modulus $\mathit{B}$, (B) shear modulus $\mathit{G}$, and (C) Young’s modulus $\mathit{E}$.

FIGURE 5

FIGURE 5. Relationship between the electron density and ${\left(\mathit{B}/{\mathit{V}}_{\mathit{m}}\right)}^{1/2}$ of TM-substituted γ′-Co₃(V, Ta). The units of $\mathit{B}$ and ${\mathit{V}}_{\mathit{m}}$ are GPa and 10⁻⁶ m³/mol, respectively.

It can be seen that ${\left(\mathit{B}/{\mathit{V}}_{\mathit{m}}\right)}^{\mathbf{1}/\mathbf{2}}$ is proportional to the electron density for 4d- and 5d-elements. However, for 3d elements, this relationship breaks down. Hence, we concluded that both the volume change and electron density affect the elastic modulus, whereas for 3d elements, the volume change is the key factor. For 5d elements, the electron density becomes the key factor. For 4d elements, these two factors have comparable influences.

To further investigate the mechanism that how TM elements affect the elastic modulus of the γ′-Co₃(V, Ta) phase, their charge density difference (CDD) on the (100) plane is plotted in Figure 6. Here, we selected two elements W and Y that significantly changed the shear modulus $\mathit{G}$ (the $\mathit{G}$ of the γ′-Co₃(V, Ta) phase is raised from 144 to 147 when W is doped, and it is lowered to 126 when Y is doped). The CDD of pure γ′-Co₃(V, Ta) was also plotted for comparison. The distribution of interatomic valence electrons on the (100) plane can be judged by the shades of colors, and the darker the color, the more concentrated the valence electrons are, and the stronger the corresponding bonds are. As shown in Figure 6A, the valence electrons of the pure γ′-Co₃(V, Ta) phase on the (100) plane are mainly concentrated between the Co and V atoms, forming a very obvious covalent interaction. The doping of W does not significantly change the shape of the charge distribution but slightly increases the charge density between the W and the nearest neighboring Co atom, which means that the strength of the newly formed Co–W bond is greater than that of the original Co–V, resulting in a small increase of the shear modulus. Y, however, drastically changes the charge distribution on the (100) plane. Since Y has a much larger atomic radius than V, its doping causes significant lattice expansion in the γ′-Co₃(V, Ta) phase, resulting in Y crowding out the Co atoms and concentrating the charge between Co and V on the Y and Co lines, as shown in Figure 6C. The strength of the Co–V bond along the Y–Co direction is much stronger than that without doping, and the charge density around the Y atom is significantly reduced. This asymmetric charge distribution makes the crystal structure less resistant to shear deformation.

FIGURE 6

FIGURE 6. Charge density differences (CDD) of the γ′-Co₃(V, Ta) phase on the (100) plane of (A) pure, (B) W occupying the V site, and (C) Y occupying the V site.

The ductility of the γ′-Co₃(V, Ta) phase after doping with TM elements was further evaluated using the Pugh’s classical criterion and Cauchy pressure. According to Pugh’s theory (Pugh, 1954), the value of B/G represents the ductility of materials. The critical value to distinguish the material ductility from brittleness is 1.75, and the higher the ratio, the more tough the material is, and conversely, the lower the ratio, the more brittle the material tends to be. On the other hand, Pettifor (1992) suggested that the Cauchy pressure (C₁₂–C₄₄) is a sign of atomic bonding within crystals, and systems with positive values of Cauchy pressure are dominated by metallic bonds, while systems with negative values are dominated by covalent bonds. For metallic systems, higher Cauchy pressure means that the system is more ductile. To clarify the intrinsic relationship between atomic bonding and ductility, Figure 7 plots the B/G value versus the Cauchy pressure. It is clear that there is a linear relationship between the B/G value and the Cauchy pressure, which indicates that the ductility of the γ′-Co₃(V, Ta) phase doped with X is mainly due to the influence of the metallic bonding within the structure. As shown in Figure 7, the undoped γ′-Co₃(V, Ta) phase exhibits intrinsic brittleness, and doping of TM elements changes the ductility of the γ′-Co₃(V, Ta) phase, for example, the addition of Tc, Re, and Os increases the Corsi pressure of the γ′-Co₃(V, Ta) phase, which enhances its internal metallicity and thus significantly improves the ductility of the γ′-Co₃(V, Ta) phase, which means that these elements can improve the mechanical properties of the γ′-Co₃(V, Ta) phase. In contrast, doping of Ir, Pt, W, Rh, and Pd causes the brittleness of the γ′-Co₃(V, Ta) phase, which reduces the mechanical properties.

FIGURE 7

FIGURE 7. Relationship between Cauchy pressure ${\mathit{C}}_{12}–{\mathit{C}}_{44}$ and $\mathit{B}/\mathit{G}$ ratio of the X-substituted γ′-Co₃(V, Ta).

3.3 Effect of TM Elements on the Thermodynamic Properties of the γ′-Co₃(V, Ta) Phase

Based on the Debye model of quasi-harmonic approximation (Francisco et al., 2001; Blanco et al., 2004), we investigated the thermodynamic properties of the γ′-Co₃(V, Ta) phase at elevated temperature using the Gibbs2 program (Otero-de-la- Roza et al., 2011). First, we discussed the relative stability of the γ′-Co₃(V, Ta) phase by analyzing the Gibbs free energy difference between the L1₂ structure and the D0₁₉ structure. The definition of the Gibbs free energy difference is as follows:

$\mathrm{\Delta }\mathit{G}={\mathit{G}}_{\mathit{D}{\mathbf{0}}_{\mathbf{19}}}-{\mathit{G}}_{\mathit{L}{\mathbf{1}}_{\mathbf{2}}},(18)$

where ${\mathit{G}}_{\mathit{D}{\mathbf{0}}_{\mathbf{19}}}$ and ${\mathit{G}}_{\mathit{L}{\mathbf{1}}_{\mathbf{2}}}$ are the Gibbs free energy of D0₁₉ and L1₂ structures, respectively. When $\Delta \mathit{G}>\mathbf{0}$, it means that the L1₂ structure is more stable than the D0₁₉ structure and vice versa. Figure 8 plots $\Delta \text{G}$ at 0–1200 K as a function of temperature. It can be seen that phonon contribution helps stabilize the D0₁₉ structure at elevated temperature. When no TM elements are doped, the Gibbs free energy of D0₁₉-Co₃(V, Ta) is lower than that of L1₂-Co₃(V, Ta) from about 600 to 1200 K, indicating that L1₂-Co₃(V, Ta) is unstable at high temperature. By comparing the doped and undoped $\Delta \text{G}$, the effects of TM elements on the relative stability of the γ′-Co₃(V, Ta) phase at finite temperature can be divided into the following three categories:

1) Sc, Ti, Mn, Fe, and Hf can improve the relative stability of the γ′-Co₃(V, Ta) phase in the entire temperature range of 0–1200 K. The calculated results are compared with experimental results. According to Ruan et al. (2020), the γ′ phase decomposition was observed in the Co–12V–2Ta alloy that annealed at 800°C for 720 hours. When doped with Ti, the γ′ phase is maintained under the same anneal condition, which is in good agreement with the prediction in this study.

2) Elements that increase the relative stability of the γ′-Co₃(V, Ta) phase in some temperature ranges, namely, Cr, Y, Zr, and Tc. Although the doping of Cr and Tc will reduce the relative stability of the γ′-Co₃(V, Ta) phase at low temperatures, the relative stability of the γ′-Co₃(V, Ta) phase will increase as the temperature rises from about 700 K.

3) Ni, Nb, Mo, Ru, Rh, Pd, W, Re, Os, Ir, and Pt reduced the relative stability of the γ′-Co₃(V, Ta) phase in the entire temperature range of 0–1200 K. Among them, the doping of the TM Mo will make it negative in the entire temperature range, which means that the Mo-doped L1₂-Co₃(V, Ta) is always in a metastable state. Ruan et al. (2020) showed that the decomposition of the γ′ phase is observed after annealing at 800°C for 48 hours in the Co–12V–2Ta alloy doped with Nb, Ni, Mo, and W, which agrees well with our calculation results.

FIGURE 8

FIGURE 8. Gibbs free energy difference $\mathit{\Delta G}$ of L1₂-Co₃(V, Ta) and D0₁₉-Co₃(V, Ta) doped with: (A) 3d elements, (B) 4d elements, and (C) 5d elements.

Then, the change of isochoric heat capacity ${\mathit{C}}_{\mathit{v}}$ with temperature is discussed as it is an important thermodynamic quantity. As shown in Figure 9, the curve shows the typical characteristics of the isochoric heat capacity, that is, it follows the Debye model (${\mathit{C}}_{\mathit{v}}\left(\mathit{T}\right)\propto {\mathit{T}}^{\mathbf{3}}$) at low temperatures and approaches 800 J/K·mol at high temperatures, which is the Dulong–Petit limit (${\mathit{C}}_{\mathit{v}}=\mathbf{3}\mathit{nR}$) [39]. The ${\mathit{C}}_{\mathit{v}}$ of γ′-Co₃(V, Ta) phases after doping with different TM elements is not much different except for W. When W is doped, the ${\mathit{C}}_{\mathit{v}}$ of the γ′-Co₃(V, Ta) phase is reduced, while other doping elements increase the ${\mathit{C}}_{\mathit{v}}$. Figure 10 shows the change of γ′-Co₃(V, Ta) isobaric heat capacity ${\mathit{C}}_{\mathit{p}}$ with temperature. The ${\mathit{C}}_{\mathit{p}}$ is greater than ${\mathit{C}}_{\mathit{v}}$ in the entire temperature range and satisfies the relationship ${\mathit{C}}_{\mathit{p}}-{\mathit{C}}_{\mathit{v}}=\mathit{\alpha BVT}$ ($\mathit{\alpha }$ is the bulk thermal expansion coefficient, and $\mathit{B}$ is the bulk modulus). At low temperatures, ${\mathit{C}}_{\mathit{p}}$ exhibits the same Debye model law as ${\mathit{C}}_{\mathit{v}}$, and then, ${\mathit{C}}_{\mathit{p}}$ increases monotonously with increasing temperature which deviates from ${\mathit{C}}_{\mathit{v}}$, exceeding the Dulong–Petit limit (Kittel and Hellwarth, 1957). This difference in heat capacity is mainly caused by the non-simple harmonic effect caused by thermal expansion. When doped with Cr, Mn, Y, Tc, Rh, and Hf, the ${\mathit{C}}_{\mathit{p}}$ curve at high temperature will be much higher than that of pure γ′-Co₃(V, Ta), which means that doping these elements will increase the sensitivity ${\mathit{C}}_{\mathit{p}}$ to temperature. When Ti and Fe are doped, ${\mathit{C}}_{\mathit{p}}$ basically remains unchanged. When elements such as Sc, Zr, and W are doped, the isobaric heat capacity of the γ′ phase will be slightly reduced.

FIGURE 9

FIGURE 9. Isochoric heat capacity ${\mathit{C}}_{\mathit{v}}$ of γ′-Co₃(V, Ta) doped with (A) 3d elements, (B) 4d elements, and (C) 5d elements.

FIGURE 10

FIGURE 10. Isobaric heat capacity ${\mathit{C}}_{\mathit{p}}$ of γ′-Co₃(V, Ta) doped with (A) 3d elements, (B) 4d elements, and (C) 5d elements.

Finally, the effect of TM elements on the volume thermal expansion coefficient α of the γ′-Co₃(V, Ta) phase at a finite temperature is calculated, and the results are shown in Figure 11. The $\mathit{\alpha }$ of the γ′-Co₃(V, Ta) phase after doping exhibits the same pattern at low temperature, that is, it increases sharply within 200 K, and then, the growth rate gradually slows down. The difference is that the doping of Sc, Ti, Fe, Zr, Ru, W, and other elements makes the volume thermal expansion coefficient of the γ′-Co₃(V, Ta) phase tends to be a constant at high temperature, indicating that its dependency on temperature is low. Other TM elements will significantly increase the volume thermal expansion coefficient at high temperature, especially Cr, Mn, Y, Tc, Rh, and Hf. When the temperature is greater than 1000 K, the volume thermal expansion coefficient rises exponentially with the temperature, indicating that the doping of these elements will cause the γ′-Co₃(V, Ta) phase to undergo a large volume change with the increase in temperature at high temperatures, thereby reducing its mechanical properties.

FIGURE 11

FIGURE 11. Volume thermal expansion coefficient $\mathit{\alpha }$ of γ′-Co₃(V, Ta) doped with (A) 3d elements, (B) 4d elements, and (C) 5d elements.

4 Conclusion

In this study, the first-principle–based density function theory (DFT) is used to investigate the effect of TM elements X (X = Sc, Ti, Cr, Mn, Fe, Ni, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Hf, W, Re, Os, Ir, and Pt) on the structural stability, mechanical properties, and thermodynamic properties of the γ′-Co₃(V, Ta) phase. The main results are as follows:

(1) The occupancy tendency of TM elements in the γ′-Co₃(V, Ta) phase was obtained by comparing the substitutional reaction energies of each non-equivalent site in the ground state: Sc, Ti, Y, Zr, Nb, Mo, Hf, and W atoms occupied the V atomic site; Cr, Mn, Fe, Ni, Tc, Ru, Rh, Pd, Re, Os, Ir, and Pt atoms occupied the Co atomic site. The structural transition energy showed that X-doped Co₃(V, Ta) phases are all more stable in the L1₂ structure than that the in D0₁₀ structure in the ground state, and some of the TM elements can enhance the relative stability of the γ′ phase, such as Sc, Ti, Y, Zr, Fe, and Mn.

(2) In the ground state, the γ′-Co₃(V, Ta) phases are all mechanically stable after the doping of TM elements. The addition of Mo and W increases the bulk modulus, shear modulus, and Young’s modulus. When doped with 3d elements, the volume change is an important factor affecting its elastic modulus. When doped with 4d elements, the volume change and electron density affect its elastic modulus simultaneously. When doped with 5d elements, the effect of electron density on its elastic modulus is more important than the volume change. According to Pugh’s classical criterion, the γ′-Co₃(V, Ta) phase is an intrinsically brittle material, and the addition of elements such as Tc and Re can significantly enhance the ductility.

(3) The L1₂ structure of Co₃(V, Ta) is more stable than the D0₁₉ structure in the temperature range from 0 to 600 K, and the D0₁₉ structure is more stable in the temperature range from 600 to 1200 K. The doping of Sc, Ti, Mn, Fe, and Hf enhances the relative stability of the γ′-Co₃(V, Ta) phase in the whole temperature range from 0 to 1200 K. The doping of TM elements does not significantly change the isovolumetric heat capacity ${\mathit{C}}_{\mathit{v}}$. The isobaric heat capacity ${\mathit{C}}_{\mathit{p}}$ and the bulk thermal expansion coefficient $\mathit{\alpha }$ of the γ′-Co₃(V, Ta) phase doped with Cr, Mn, Y, Tc, Rh, and Hf elements increased significantly with the temperature at elevated temperature, while the doping of Sc, Ti, Fe, Zr, Ru, and W elements does not significantly change the isobaric heat capacity ${\mathit{C}}_{\mathit{p}}$ and bulk thermal expansion coefficient $\mathit{\alpha }$.

Data Availability Statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

Author Contributions

Conceptualization: XL and CW. Formal analysis: YC and JH. Investigation: YC, CZ, and CY. Original draft preparation: YC and JH. Writing—review and editing: YC and JH. Visualization: YC and JH. Supervision: XL and CW.

Funding

This work was supported by the Key-Area Research and Development Program of Guangdong Province (No. 2019B010943001) and the National Natural Science Foundation of China (Grant No. 51831007).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

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References

Bantounas, I., Gwalani, B., Alam, T., Banerjee, R., and Dye, D. (2019). Elemental Partitioning, Mechanical and Oxidation Behaviour of Two High-γ′ W-free γ/γ′ Polycrystalline Co/Ni Superalloys. Scr. Mater. 163, 44–50. doi:10.1016/j.scriptamat.2018.12.025

CrossRef Full Text | Google Scholar

Blanco, M. A., Francisco, E., and Luaña, V. (2004). GIBBS: Isothermal-Isobaric Thermodynamics of Solids from Energy Curves Using a Quasi-Harmonic Debye Model. Comput. Phys. Commun. 158 (1), 57–72. doi:10.1016/j.comphy.2003.12.001

CrossRef Full Text | Google Scholar

Born, M. (1939). Thermodynamics of Crystals and Melting. J. Chem. Phys. 7 (8), 591–603. doi:10.1063/1.1750497

CrossRef Full Text | Google Scholar

Davydov, D., Kazantseva, N., Ezhov, I., and Popova, E. (2021). Effect of Alloying on the γ-γ′ Microstructure of W-free Co-based Superalloys. Mater. Today Proc. 38, 1971–1973. doi:10.1016/j.matpr.2020.09.131

CrossRef Full Text | Google Scholar

Drapier, J. M., and Coutsouradis, D. (1968). Precipitation Hardening of Co‐Cr‐Ta Alloys. Cobalt 39, 63–74.

Google Scholar

Francisco, E., Blanco, M. A., and Sanjurjo, G. (2001). Atomistic Simulation of SrF₂ Polymorphs. Phys. Rev. B 63 (9), 094107. doi:10.1103/physrevb.63.094107

CrossRef Full Text | Google Scholar

Kittel, C., and Hellwarth, R. W. (1957). Introduction to Solid State Physics. Phys. Today 10 (6), 43–44. doi:10.1063/1.3060399

CrossRef Full Text | Google Scholar

Kresse, G., and Furthmüller, J. (1996). Efficient Iterative Schemes Forab Initiototal-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 54 (16), 11169–11186. doi:10.1103/physrevb.54.11169

CrossRef Full Text | Google Scholar

Kresse, G., and Joubert, D. (1999). From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 59 (3), 1758–1775. doi:10.1103/physrevb.59.1758

CrossRef Full Text | Google Scholar

Methfessel, M., and Paxton, A. T. (1989). High-precision Sampling for Brillouin-Zone Integration in Metals. Phys. Rev. B 40 (6), 3616–3621. doi:10.1103/physrevb.40.3616

PubMed Abstract | CrossRef Full Text | Google Scholar

Miedema, A. R., Boer, F. R. d., and Chatel, P. F. d. (1973). Empirical Description of the Role of Electronegativity in Alloy Formation. J. Phys. F. Metall. Phys. 3 (8), 1558–1576. doi:10.1088/0305-4608/3/8/012

CrossRef Full Text | Google Scholar

Migas, D., Moskal, G., and Maciąg, T. (2020). Thermal Analysis of W-free Co-(Ni)-al-mo-nb Superalloys. J. Therm. Anal. Calorim. 142 (1), 149–156. doi:10.1007/s10973-020-09375-7

CrossRef Full Text | Google Scholar

Monkhorst, H. J., and Pack, J. D. (1976). Special Points for Brillouin-Zone Integrations. Phys. Rev. B 13 (12), 5188–5192. doi:10.1103/physrevb.13.5188

CrossRef Full Text | Google Scholar

Otero-de-la-Roza, A., Abbasi-Pérez, D., and Luaña, V. (2011). Gibbs2: A New Version of the Quasiharmonic Model Code. II. Models for Solid-State Thermodynamics, Features and Implementation. Comput. Phys. Commun. 182 (10), 2232–2248. doi:10.1016/j.cpc.2011.05.009

CrossRef Full Text | Google Scholar

Pettifor, D. G. (1992). Theoretical Predictions of Structure and Related Properties of Intermetallics. Mater. Sci. Technol. 8 (4), 345–349. doi:10.1179/mst.1992.8.4.345

CrossRef Full Text | Google Scholar

Pollock, T. M., Dibbern, J., Tsunekane, M., Zhu, J., and Suzuki, A. (2010). New Co-based γ-γ′ High-Temperature Alloys. JOM 62 (1), 58–63. doi:10.1007/s11837-010-0013-y

CrossRef Full Text | Google Scholar

Pugh, S. F. (1954). XCII. Relations between the Elastic Moduli and the Plastic Properties of Polycrystalline Pure Metals. Lond. Edinb. Dublin Philosophical Mag. J. Sci. 45 (367), 823–843. doi:10.1080/14786440808520496

CrossRef Full Text | Google Scholar

Pyczak, F., Bauer, A., Göken, M., Lorenz, U., Neumeier, S., Oehring, M., et al. (2015). The Effect of Tungsten Content on the Properties of L12-Hardened Co–al–W. J. Alloys Compd. 632, 110–115. doi:10.1016/j.jallcom.2015.01.031

CrossRef Full Text | Google Scholar

Reyes Tirado, F. L., Taylor, S., and Dunand, D. C. (2019). Effect of Al, Ti and Cr Additions on the γ-γ' Microstructure of W-free Co-ta-V-based Superalloys. Acta Mater. 172, 44–54. doi:10.1016/j.actamat.2019.04.031

CrossRef Full Text | Google Scholar

Reyes Tirado, F. L., Taylor, S. V., and Dunand, D. C. (2020). Low-density, W-free Co-nb-V-al-based Superalloys with γ/γ' Microstructure. Mater. Sci. Eng. A 796, 139977. doi:10.1016/j.msea.2020.139977

CrossRef Full Text | Google Scholar

Ruan, J., Xu, W., Yang, T., Yu, J., Yang, S., Luan, J., et al. (2020). Accelerated Design of Novel W-free High-Strength Co-base Superalloys with Extremely Wide γ/γʹ Region by Machine Learning and CALPHAD Methods. Acta Mater. 186, 425–433. doi:10.1016/j.actamat.2020.01.004

CrossRef Full Text | Google Scholar

Sato, J., Omori, T., Oikawa, K., Ohnuma, I., Kainuma, R., and Ishida, K. (2006). Cobalt-Base High-Temperature Alloys. Science 312 (5770), 90–91. doi:10.1126/science.1121738

PubMed Abstract | CrossRef Full Text | Google Scholar

Shang, S., Wang, Y., and Liu, Z.-K. (2007). First-principles Elastic Constants of α- and θ-Al2O3. Appl. Phys. Lett. 90 (10), 101909. doi:10.1063/1.2711762

CrossRef Full Text | Google Scholar

Shi, L., Yu, J. J., Cui, C. Y., and Sun, X. F. (2015). Microstructural Stability and Tensile Properties of a Ti-Containing Single-Crystal Co-ni-al-W-base Alloy. Mater. Sci. Eng. A 646, 45–51. doi:10.1016/j.msea.2015.08.044

CrossRef Full Text | Google Scholar

Wang, C. P., Deng., B., Xu., W. W., Yan., L. H., Han., J. J., and Liu., X. J. (2018). Effects of Alloying Elements on Relative Phase Stability and Elastic Properties of L12 Co3V from First-Principles Calculations. J. Mater Sci. 53 (2), 1204–1216. doi:10.1007/s10853-017-1549-9

CrossRef Full Text | Google Scholar