First-principles study on the preferential sites of Cr in Co7W6

The preferential sites of Cr in the μ-Co7W6 phase and its influence on electronic properties were studied by first-principles calculations based on density functional theory. The calculation results of the formation energy and defect formation energy show that the stability of the system is enhanced when Cr occupies the Co site, which indicates that Cr tends to occupy the Co site of the system. By calculating the density of states, the Hamilton population of crystal orbital, the electron location function and the Bader charge distribution, the reason why Cr preferentially occupied the Co sites is further explained. This is primarily owing to the hybridization of the d-d orbitals of the Cr atom to its adjacent atoms.


Introduction
Owing to the addition of V, Cr, Co, Mo, W and other refractory elements, the nickel-based single crystal superalloy has prominent high-temperature properties and excellent creep properties, so it has been extensively used in contemporary gas turbine engines [1][2][3]. However, excessive refractory elements will cause the generation of Topologically Close-Packed (TCP) phases, such as μ, ρ, σ phases and so on. The TCP phase is a brittle phase that promotes crack initiation and propagation. In addition, the TCP phase would consume solidsolution strengthening elements in γ matrix [4][5][6][7][8]. Because of these adverse factors, it is extremely important to predict the formation law of TCP phase. In general, when the content of W and Co is high, the μ phase of B 7 A 6 structure will replace Laves and σ phases as the primary TCP phase [9,10]. Based experiments and calculations, there are numerous reports on the μ phase. Carvalho [11] studied the stair fault of Co 7 W 6 on a pyramid plane and found that the stair fault was offset by 9°across twin domains. R. Ravi et al studied the interdiffusion of the Co-W alloy and discovered that the activation energy was positively correlated with the content of W in the Co-W alloy [12]. Zhang et al [13] studied the phase equilibrium of the Co-Cr-W ternary system and found that the maximum solubility of Cr in Co 7 W 6 at 1200°C was 46%. Barrows and Newkirk [14] determined the phase equilibrium of the Co-Cr-W system. It is found that the addition of chromium to the Co-W alloy leads to the projection of μ field into the ternary diagram in a direction parallel to the binary edges of Co-Cr. That is, chromium atoms appear to be substituting primarily for the similar size cobalt atoms in the μ structure [14]. The thermodynamic calculation results of the Co-Cr-W system by Peisheng Wang et al [15] showed that increasing the dosage of Cr could significantly increase the range of the μ phase. Bartek Kaplan [16] determined the ground state intermetallic phase in the W-Co system by DFT calculation, and a method for constructing a sublattice model for an intermetallic phase, based on energetics, is proposed. In addition, among these refractory elements, Cr can enhance the oxidation resistance and corrosion resistance of the nickel-based single crystal superalloy and also accounts for a certain proportion in the μ phase [17]. With the solution of Cr element into μ phase, the content of Cr element dissolved in nickel-based superalloy decreases, which leads to the reduction of oxidation resistance and corrosion resistance of the alloy at high temperature. In addition, the proportion of other elements in the nickel-based superalloy will also change. It is well known that Cr has some impacts on the formation of μ phase, but the specific alloying mechanism is still unclear [18][19][20]. On the other hand, for the Cr Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. doped μ phase, it is difficult to experimentally prove the quantitative relationship between the formation energy, bond strength, and doping position at the atomic scale. Therefore, this work take the representative μ phase Co 7 W 6 as the research target, and study the occupancy of Cr element in μ-Co 7 W 6 phase by first-principles calculations. The results have important guiding significance for further understanding the effect of Cr element on the stability of μ phase and optimizing the design of nickel-based superalloy.

Calculation detail and model
All the calculations were performed in the Vienna Ab initio Simulation Package (VASP) based on density functional theory [21], and the PBE exchange-correlation functional in the generalized gradient approximation (GGA) approach was used to characterize the exchange-correlation interaction [22]. Projector augmented wave (PAW) method was used to describe the core electrons [23]. The spin-polarized calculations were conducted under the setting of ferromagnetic parameters [24]. The energy cut-off is 600 eV, and Brillouin k point is set at 7 × 7 × 1 by Monkhorst-pack method [25][26][27][28]. During structural optimization, the energy converges to 1 × 10 -5 eV/atom, and the maximum force converges to 0.003 eV Å. The stress was less than 0.01 GPa, and the displacement was less than 0.001Å. Furthermore, Hubbard corrections, U = 3.5, 3.5 and 6 eV, were added to treat the strong correlation effect of Co-d, Cr-d and W-d orbitals, respectively [29][30][31][32]. According to the characteristics of the μ phase, its equivalent structure Co 21 W 18 [33], a hexagonal protocell inclusive of 39 atoms, was employed. The lattice parameters of μ-Co 7 W 6 are a = 4.723 Å, b = 4.723 Å, c = 25.480 Å, α = 90°, β = 90°, γ = 120°, and its five non-equivalent sites (Co1, Co2, W1, W2, W3) were marked in figure 1(a). In order to better explore the doping effect of Cr element on μ phase, the Cr atom was used to replace five non-equal sites successively, and the doped crystal structures were shown in figures 1(b) ∼ (f).

Lattice constants
The crystal structure of Co 21 W 18 was optimized. The optimized equilibrium lattice constants of this work are listed in table 1 along with the experimental data reported in the research paper. Table 1 shows that the relative error between our results and the experiment [33] is less than 1%, which indicates that our calculation results are reliable. The change of lattice constants a, b, c, and c/a of the Co 21 W 18 doped by Cr at different positions is no more than 0.024 Å, 0.024 Å, 0.1 Å, and 0.05 Å compared with the optimized Co 21 W 18 system. The lattice constant of the Co 21 W 18 doped by Cr at W site decreases, and increases at Co site. This is because the atomic radius (1.28 Å) of Cr is smaller than that of W (1.37 Å) and larger than that of Co (1.25 Å).

Energy analyses
The formation energy (E f ) of the system was calculated to examine the stability of the doping system. The formation energy is negatively correlated with the stability of the system. The formation energy is equivalent to the standard Gibbs free energy change of the reactions. The formula below can be used to determine formation energy [34]: where E t refers to the total energy of the system, and x, y and z represent the atomic number of Co, W and Cr atoms in the crystal structure, respectively. E , s Co E s W and E s Cr represent the average energy per atom of solid fcc ɑ-Co(Fm3mm), bcc-W(Im3m) and bcc-Cr(Im3m), respectively [13].
The defect formation energy (E def ) is commonly used to characterize the difficulty of formation of a doped system, and the lower defect formation energy means a greater chance of stable existence of the doped system [31,35]. Cr preferentially occupies the position with the smallest E def , which can be obtained by the following formula [36]: where E t(doped) expresses the total energy of the Cr-doped system. E t(pure) represents the total energy of the pure Co 21 W 18 system. μ i and μ Cr are the chemical potentials of the substituted atom and Cr atom, severally. μ Cr , μ Co and μ W were obtained by calculating the total energies per Cr, Co and W in their bulk phases. The corresponding crystal structures are fcc ɑ-Co(Fm3m), bcc-W(Im3m) and bcc-Cr(Im3m) [13]. The atomic ground state energy was calculated by placing it in a cubic box with a lattice length of 15 Å.
The calculation results for the E f and E def of the Co 21 W 18 system and the corresponding five doping systems are listed in table 2. As displayed in table 2, when Cr occupies W position, in comparison to the pure system, E f and E def are both high; When Cr occupies Co position, both E f and E def are lower than that of the pure system. It is shown that Cr in Co position makes the μ-phase more stable. According to the analysis of the E f of Cr doping W position, the energy is a bit different when Cr occupancies W1, W2, and W3 sites. However, when Cr replaces W3 position, the E f of the system is the lowest, and E def is also small, so the system of Cr doped W3 was selected to represent Co 21 W 17 Cr. When Cr replaces the Co1 position, the formation energy and defect formation energy of the system are much lower than those of other systems, and its defect energy is the only one below zero Hence, Co 20 CrW 18 was used to represent the system which the Cr replaced Co1 site. Moreover, it can be learned from table 2 that E f and E def of Co1 are lower than that of W3. Therefore Cr will occupy the Co1 site as a priority.

Density of states (DOS)
To better understand the interaction between Cr, W and Co in the system and the influence of Cr on the structural stability of Co 21 W 18 , their densities of states were calculated. The DOS of the pure Co 21 W 18 system, Cr stead of Co1 site and Cr occupying W3 site are drawn, as shown in figures 2 and 3. The Fermi level (E F ) is at 0 eV. As shown in figures 2 and 3, the total state density of an atom is mainly provided by the d orbital of each atom, followed by the p orbital, and finally, the s orbital. The most significant feature of these densities of states is the strong overlap between the 3d orbital of Co and the 5d orbital of W. The DOS at Fermi level is not zero, indicating that both systems exhibit metallic properties. Comparing the occupied energy levels below the Fermi level, in general, the system with more overlapping peaks and lower main peak energy levels is more stable. Figures 2(b) and 3(b) represent the density of states of Cr occupying Co1 and W3 sites, respectively. In figure 2(a), the highest main peak of Co1 atom does not overlap with the highest peak of Co2 atom, which is −2.84 eV and −2.80 eV, respectively. In figure 2(b), it is observed that although the highest peak of the Cr atom shifts to higher energy levels, it overlaps with the highest peak of Co2 atom. The overlap peak is located at −2.80 eV, indicating the strong hybridization between Cr 3d, Co2 3d, and W2 5d states in the Co1 position system.  In figure 3(a), the highest primary peak of W3 atom does not overlap with the highest peak of W2 atom, which is −3.24 eV and −2.94 eV, respectively. In figure 3(b), the highest main peak of Cr atom at −2.63 eV and the highest prominent peak of W2 atom at −2.87 eV. It can be seen that the highest main peak of Cr atom shifts to the high energy level when Cr atom occupies W3 position. However, after the shift, the primary peak of Cr atom does not overlap with the primary peak of W2 atom, so its binding ability becomes weak. In general, the pseudogap directly reflects the bond covalency of the system, the larger its value reflects the stronger covalency of the system [37]. Obviously, compared with Pseudogap of the Cr doped W3 site, Pseudogap of the Cr doped Co site was wider. It can be seen from comparison between figures 2(a) and (b) that the value of DOS at the Fermi level of the Cr doped Co1 system decreased significantly. However, when Cr replaces W3, the value of the DOS of the system at the Fermi level only decreases from 1.17 to 0.56. Compared the DOS of Cr replacing Co1 and W3 positions at the Fermi level, it is obvious that the former is much lower than the latter. At the same time, the substitution of Cr for W3 position increases the number of states at lower energy levels. In conclusion, it is proved that the system is more stable when Cr occupies the position of Co1, while the system stability decreases when Cr occupies the position of W3. The analyses results of density of states are consistent with that of the energy analyses.

The crystal orbital Hamilton population (COHP)
By studying the bond strength with surrounding atoms, the difficulty of bond rupture and its impact on the total energy and structural stability of the system can be estimated. The COHP was further calculated for the sake of displaying the bonding characteristics of atoms more clearly, and the results are presented in figure 4.
It can be noted from figure 4(a) that the COHP of Co-Co, Co-W, Cr-Co, and Cr-W bonds below Fermi level is basically positive, indicating that there is a bonding between them. When Cr substitutes Co1 position, it can be observed that the integral area under the COHP line of the Cr-Co2 bond and Cr-W2 bond and the COHP at the Fermi energy level are larger than the Co1-Co2 bond and Co1-W2 bond. In other words, the bonding with surrounding atoms is more stronger after Cr doping, and the COHP analyses results are consistent with that of state density analysis. However, as shown in figure 4(b), when the W3 position is replaced by Cr, although Cr-W bond and Cr-Co bond are still bonding, their peak height is reduced significantly, indicating that the bonding strength is weakened significantly. Meanwhile, the COHP integral area under the COHP line of Cr-W2 bond and Cr-W4 bond and the COHP at the Fermi level are both smaller than that of W3-W2 bond and W3-W4 bond.

Charge density
By calculating the charge density of the system, the bond strength between atoms can be revealed. The charge densities of the (110) plane of pure system Co 21 W 18 and each doped system were calculated, as shown in figure 5. For the pure Co 21 W 18 system, it can be observed from figure 5(a) that the charge density of W2-W3 atoms is high, so these two atoms show a strong bonding ability. It can be discovered from figure 5(a) and figure 5(b) that the electron density between these three atoms Cr-Co2-W2 is higher than that of Co1-Co2-W2, which indicates that the bonding ability is enhanced. The stability of the system becomes better when Cr atoms occupy Co1 site. In figure 5(c), electrons are localized around Cr atoms, and the charge density between Cr and W2 decreases obviously in comparison with figure 5(a), indicating that the covalent bond between the two atoms are weaker than that of the W3-W2 bond. This proves that when Cr atoms occupy W3 site, the bonding ability between atoms becomes weaker, and the stability of the system becomes worse. This is consistent with the energy and DOS analyses above. It can be known from the above analysis that the system becomes more stable when the Cr atom occupies the Co position. Therefore, Cr preferentially occupies the Co position.

Bader charge
Bader charge analyses can reveal the relative charge transfer between different atoms during phase formation and quantitatively estimate the binding strength of Cr doped Co1 and W3 sites. It is discovered from figure 6 that  the charge is always transferred from W site to Co site. Compared with pure Co 21 W 18 , when Cr occupied the place of Co1, Co2 atom gains more electrons, while W2 atom loses more electrons. In this way, the electron interaction between Cr and Co2 and W2 is enhanced, and the number of electrons involved in bonding increases, which greatly improves the bonding interaction between Cr and surrounding atoms. On the contrary, when Cr replaces W3, the loss of charge of W2 and W3 atoms increases from −0.49 and −0.67 to −2.32 and −2.26, respectively. The increase of delocalized electrons leads to the weakening of the covalent bond, and the bond length with the surrounding atom (W3) is changed from 2.632 Å to 2.644 Å.

Conclusions
The influence of the Cr atom on the μ-Co 7 W 6 system was studied through the first principles method based on density functional theory, according to the results of the calculation, the conclusions can be reached as follows: The formation energy and defect formation energy of the Cr doped unit cell are lower than those of the pure Co 7 W 6 unit cell, and the Cr doped Co1 system is the most stable.
When Cr atom occupies Co1 position, the hybridization ability of d-d orbitals between Cr atom and its neighbors becomes stronger, and the density of states at Fermi level decreases more than that at W3 position and the DOS curve also shifts to the low energy region.
By analyzing the charge density and Bader charge, when Cr atom occupies the position of Co1, the binding strength between Cr atom, W2 atom and Co1 atom becomes stronger. In other words, the results of energy and electronic properties analysis show that chromium readily forms the μβ-Co 7 W 6 phase.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.