First-Principles Study of Mechanical and Thermodynamic Properties of Binary and Ternary CoX (X = W and Mo) Intermetallic Compounds

In this study, the structural, elastic, and thermodynamic properties of DO19 and L12 structured Co3X (X = W, Mo or both W and Mo) and μ structured Co7X6 were investigated using the density functional theory implemented in the pseudo-potential plane wave. The obtained lattice constants were observed to be in good agreement with the available experimental data. With respect to the calculated mechanical properties and Poisson’s ratio, the DO19-Co3X, L12-Co3X, and μ-Co7X6 compounds were noted to be mechanically stable and possessed an optimal ductile behavior; however, L12-Co3X exhibited higher strength and brittleness than DO19-Co3X. Moreover, the quasi-harmonic Debye–Grüneisen approach was confirmed to be valid in describing the temperature-dependent thermodynamic properties of the Co3X and Co7X6 compounds, including heat capacity, vibrational entropy, and Gibbs free energy. Based on the calculated Gibbs free energy of DO19-Co3X and L12-Co7X6, the phase transformation temperatures for DO19-Co3X to L12-Co7X6 were determined and obtained values were noted to match well with the experiment results.


Introduction
Cobalt (Co)-based superalloys are suitable candidates for modern aircraft engines, turbo superchargers, and chemical industrial materials requiring optimal performance at elevated temperatures. This is owing to their prominent creep resistance during longterm high-temperature exposure, which is guaranteed by the addition of the various refractory elements, such as Cr, Mo, W and Re [1][2][3][4][5][6][7]. Hence, the incorporation of W and Mo into Co has attracted widespread research attention [8]. In addition, Co-W and Co-Mo coatings are considered to be an appropriate alternative to replace Cr coatings, due to the superior mechanical, tribological, and corrosion resistance properties of these alloys [9][10][11]. Moreover, cobalt still retains the magnetic properties in cases where the tungsten and molybdenum concentrations in the alloy are less than 20%. Therefore, the Co-W and Co-Mo alloys can be utilized in disk and magnetic film memory devices [12].
Due to the similar characteristics of W and Mo, the Co-W and Co-Mo binary alloys have close physical and chemical properties, which means that they can be thought of as a substitute for each other in many cases. Due to the differences of W and Mo in melting points, density etc., the performance gap between binary alloys of Co-W and Co-Mo also cannot be negligible in specific applications. However, the insufficient studies on Co-W-Mo ternary alloys make it inconclusive as to whether ternary alloys can conform to the performance of binary Co-W/Mo alloys. Therefore, it is meaningful to study the difference on properties of binary and ternary alloys of Co-W/Mo for achieving continuous adjustment of alloy performances.

Computational Method
In the current study, the first-principles calculations have been performed by using the plane wave basis projector augmented wave (PAW) method [22,23] in the DFT framework, as implemented in the Vienna Ab initio Simulation Package (VASP) code [24][25][26]. The local density is described through the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [27]. A plane wave basis set energy cut-off of 500 eV was applied during the calculations to ensure accuracy. Brillouin zone sampling was performed by using the gamma point centered k-meshes 6 × 6 × 8, 9 × 9 × 9, and 7 × 7 × 1 for the DO 19 , L1 2 and µ structures, respectively. Throughout the calculations, the convergence of the total energy and maximum force of the ionic relaxation were set to be less than 10 −6 eV and 10 −2 eV/Å, respectively. The second order elastic constants were calculated using the efficient stress-strain energy method [28,29]. Accordingly, the stresses and strains satisfied Hooke's law for small deformations, as per Equation (1): where σ i , C ij and e i are the stress vector, second order elastic constants and strain vector, respectively. The Taylor expansion of the internal energy of the deformed crystal under the micro-strain component yields can be expressed as: where V 0 and E(V 0 ,0) are the equilibrium volume and energy of the undistorted structure, respectively. In the strain set (e = e 1 , e 2 , e 3 , e 4 , e 5 and e 6 ), e 1 , e 2 and e 3 refer to the normal strains, and e 4 , e 5 and e 6 indicate the shear strains. These are used to generate the small deformations of the unit cell. The crystalline lattice vectors before (R) and after (R') the deformation are related as follows: The elastic constants can be obtained by fitting the energy versus strain curve with the quadratic polynomial function.

Geometry and Structural Properties
In this study, nine intermetallic compounds in the Co-W, Co-Mo, and Co-W-Mo alloys have been studied, i.e., Co 3 W_DO 19 (P6 3 /mmc), Co 3 Mo_DO 19 (P6 3 /mmc), Co 3 W(Mo) _DO 19 (P6 3 /mmc), Co 3 W_L1 2 (Fm-3m), Co 3 Mo_L1 2 (Fm-3m), Co 3 W(Mo)_L 12 (Fm-3m), Co 7 Mo 6 _µ(R-3m), Co 7 W 6 _µ (R-3m) and Co 7 W(Mo) 6 _µ (R-3m). The structural models of the nine alloys are presented in Figure 1. The calculated ground-state lattice parameters a and c of the DO 19 , L1 2 and µ phases of the binary and ternary intermetallic compounds are listed in Table 1 (a = b; thus, b is omitted here) and the site occupation in structures are attached in the Supplementary Materials. In addition, the parameters have been compared with the available experimental data and other theoretical results. As shown in Table 1, the calculated lattice parameters are consistent with the experimental values, with the average deviation less than 1.0%. Subsequently, the cohesive energy (∆E) and formation energy (∆H) have been calculated to investigate the chemical stability of the compounds, as shown in Table 1. The ∆E and ∆H values have been defined by Equations (4) and (5), respectively.
where x, y, E total , E atom and E solid represent the content of element A in the unit cell, content of element B in the unit cell, total energy of the unit cell, energy of the isolated atom and ground state energy of the pure metal, respectively. Generally, ∆H can be used to characterize the phase stability of the intermetallic compounds. The more negative the ∆H value, the higher the phase stability. Due to the calculations carried out at 0 K in first-principles calculations, the results should be effective in low temperature. It can be observed from Table 1 that the formation energy of DO 19 -ordered Co 3 X is more negative than L1 2 -ordered Co 3 X, thus implying that DO 19 -Co 3 X is more stable at low temperature. This observation is consistent with the experimental data and other theoretical results. In fact, the L1 2 -ordered Co 3 X is detected in Co-W(Al) systems, but it is difficult to be prepared and found in the pure binary Co-W system. Compared with DO 19 -Co 3 X, µ-ordered Co 7 X 6 displays a lower stability. According to the experimental results, the alloying processes of Co-W/Mo obey the orders from Co + X → Co 3 X → Co 7 X 6 (X = W or Mo) with the increasing temperature, as depicted in the reactions of Equations (6) and (7). The inference, therefore, is that Co 7 X 6 should be more stable at high temperature.

Elastic Properties
Elastic stiffness (C ij ) and compliance (S ij ) tensors were used to describe the response of the crystal to the external stress applied in different directions. These are useful for understanding the mechanical and physical properties of the alloys, such as machinability, bonding characteristics, and ductility. There were nine independent elastic stiffness constants for the anisotropic crystal: C 11 , C 12 , C 13 , C 22 , C 23 , C 33 , C 44 , C 55 and C 66 . In addition, the S ij quantities represent the inverse matrix of C ij , as defined in Equation (8): Due to the crystal symmetry, three independent elastic constants (i.e., C 11 , C 12 and C 44 ) exist for the L1 2 -ordered cubic structure, and Equation (8) can be simplified as: For the DO 19 -ordered hexagonal structure and µ phase, two more independent elastic constants (i.e., C 13 and C 33 ) can be added, and Equation (8) can be expressed as: The mechanical stability of the Co-W-Mo alloy can be determined from the calculated elastic constants. For the cubic L1 2 -Co 3 X crystals, the mechanical stability criteria can be written as [33]: The mechanical stability criteria for the DO 19 -Co 3 X and µ-Co 7 X 6 structures can be expressed as follows [34]: The C ij and S ij values of the Co 3 X and Co 7 X 6 compounds have been calculated at 0 K and are tabulated in Table 2. The elastic constants obtained in this study were observed to be in good agreement with the previously reported results [15]. It should be noted that L1 2and DO 19 -Co 3 X and Co 7 X 6 are mechanically stable at 0 K, as per Equations (11)-(15). The bulk and shear moduli are used to evaluate the resistance to the volume change and reversible deformation at a given stress, and the Young's modulus is applied to estimate the stiffness of materials. The Poisson's ratio is associated with the resistance of the material against shear. Based on the Voigt and Reuss theories [35], the bulk and shear moduli can be calculated by using Equations (19)-(24): 15G where V and R denote the values calculated by using the Voigt and Reuss theories, respectively. On the basis of the elastic constants, the bulk, shear and Young's moduli as well as Poisson's ratio at the ground state can be calculated by using the Voigte-Reuss-Hill method: The bulk modulus measures the resistance offered by the material against the changes in its volume. From Figure 2a, the µ-Co 7 X 6 phase is observed to exhibit the highest bulk modulus, whereas L1 2 -Co 3 X possesses the lowest bulk modulus in the system with the same X element (X = Mo or W). The results also indicate that Co-W has a higher bulk modulus than Co-M and Co-W-Mo possessing the same structures, thus demonstrating the superior strength of the Co-W alloys as compared to Co-Mo. The Young's modulus is an effective indicator of the stiffness of the material. The higher the Young's modulus, the stiffer the material. As observed in Figure 2c, the Young's modulus decreases in the following sequence: L1 2 > DO 19 > µ, thus indicating that L1 2 -Co 3 X is stiffer than the other structures, which corresponds with the experimental findings that the L1 2 phase exists as a precipitation strengthening compound in the Co-based alloys.
The bulk modulus measures the resistance offered by the material against the changes in its volume. From Figure 2a, the μ-Co7X6 phase is observed to exhibit the highest bulk modulus, whereas L12-Co3X possesses the lowest bulk modulus in the system with the same X element (X = Mo or W). The results also indicate that Co-W has a higher bulk modulus than Co-M and Co-W-Mo possessing the same structures, thus demonstrating the superior strength of the Co-W alloys as compared to Co-Mo. The Young's modulus is an effective indicator of the stiffness of the material. The higher the Young's modulus, the stiffer the material. As observed in Figure 2c, the Young's modulus decreases in the following sequence: L12 > DO19 > μ, thus indicating that L12-Co3X is stiffer than the other structures, which corresponds with the experimental findings that the L12 phase exists as a precipitation strengthening compound in the Co-based alloys. The brittleness and ductility of Co-W/Mo have been studied by taking the ratio of the bulk modulus to the shear modulus (B/G) [36]. A high B/G value indicates an optimal ductility, whereas a low B/G value suggests the material brittleness. The critical borderline The brittleness and ductility of Co-W/Mo have been studied by taking the ratio of the bulk modulus to the shear modulus (B/G) [36]. A high B/G value indicates an optimal ductility, whereas a low B/G value suggests the material brittleness. The critical borderline value between the ductile and brittle materials has been evaluated to be 1.75. The Poisson's ratio is employed to estimate the stability of a material against the shear deformation and usually ranges from −1 to 0.5. A large Poisson's ratio represents an effective plasticity. Basically, the material is considered to be ductile in cases where σ is higher than 0.26. Hence, based on the obtained B/G ratio for Co-W/Mo in Table 2, DO 19 -Co 3 X and µ-Co 7 X 6 possess a ductile character, and L1 2 -Co 3 X represents more brittle materials. Moreover, the value of the Poisson's ratio stands for the degree of directionality of the covalent bonds. For the ionic crystals, the σ value is usually close to 0.25, whereas the σ value is around 0.1 for the covalent materials [37]. As can be observed from Table 2, the computed values of the Poisson's ratio are greater than 0.25. Therefore, the ionic bond interactions significantly contribute to the interatomic bonding in the Co-W-Mo binary and ternary compounds.
In comparison with the previous study results of the Ni-based intermetallic phase, the bulk, shear, and Young's moduli of Co-W/Mo in this work and the previous works were all higher than the Ni-based alloys such as Ni 2 Mo 2 , Ni 2 Mo, Ni 3 Mo, Ni 4 W, Ni 3 W, etc. However, the B/G of Ni-based alloys are usually higher than Co-W/Mo alloys [38][39][40]. This means that the Co-W/Mo would perform better on strength but worse on ductility tests than Ni-based alloy. This may be caused by the stronger Co3d-W5p/Mo4p orbital hybridization interaction between Co and W/Mo atoms than Ni3d-W5p/Mo4p, because the difference of electronegativity between Ni-W/Mo is smaller than Co-W/Mo.
Comparing the isomorphous compounds with different elements, the Co-W alloys exhibit higher strength than the Co-Mo alloys, whereas the strength of the Co-W-Mo alloys is located between them. The similar structures and properties of the Co-W and Co-Mo alloys enable them to exhibit the continuous solid solution phase-like behavior of the tungsten and molybdenum alloys. Therefore, the method involving the continuous adjustment of the mechanical properties leads to an effective design of the Co-W/Mo alloy. However, it is worth noting that the mechanical properties of the Co 6 WMo phases are closer to the Co-W alloy instead of the Co-Mo alloy. The W atoms have a larger radius than the Mo atoms; thus, the W atoms in the ternary alloys would play a vital role during the deformation process.

Thermodynamic Properties
In this section, the quasi-harmonic Debye model has been applied to investigate the thermodynamic properties of the Co-W/Mo compounds in the L1 2 , DO 19 and µ structures at a finite temperature [41]. The non-equilibrium Gibbs energy of the crystal phase at fixed temperature and hydrostatic pressure can be expressed as: where E(V) is the total energy of the crystal in a given volume, A vib is the Helmholtz vibrational free energy, and Θ is the Debye temperature. Adopting the Debye model of the phonon density of states, the vibrational contribution of A vib can be written as [42,43]: where n is the number of atoms per formula unit, and k is the Boltzmann constant. D(y) is the Debye integral, which is defined as: The Debye temperature Θ, related to the average sound velocity, can be calculated as: where M refers to the molecular mass per unit cell, and k is the reduced Planck constant. B s is the adiabatic bulk modulus, approximated by using the static compressibility, and can be written as [44,45]: Hence, the (p, T) thermal equilibrium can be obtained by solving the non-equilibrium Gibbs function, given as: For the equilibrium state at a given (p, T), the thermodynamic properties such as isochoric heat capacity (C v ), isobaric heat capacity (C p ), vibrational entropy (S vib ) and volume thermal expansion coefficient (α) can be calculated by using the following equations: where B T is the isothermal bulk modulus, and the Grüneisen parameter γ is defined as: The calculated C v and C p values of the Co-W/Mo compounds in the L1 2 , DO 19 and µ structures at 0 GPa are illustrated in Figure 3. Herein, the thermodynamic properties have been calculated by ignoring the thermal electronic contribution and effect of the zero-point energy.
Hence, the (p, T) thermal equilibrium can be obtained by solving the non-equilibrium Gibbs function, given as: * ( ; , ) , = 0 For the equilibrium state at a given (p, T), the thermodynamic properties such as isochoric heat capacity (Cv), isobaric heat capacity (Cp), vibrational entropy (Svib) and volume thermal expansion coefficient (α) can be calculated by using the following equations: where BT is the isothermal bulk modulus, and the Grüneisen parameter γ is defined as: The calculated Cv and Cp values of the Co-W/Mo compounds in the L12, DO19 and μ structures at 0 GPa are illustrated in Figure 3. Herein, the thermodynamic properties have been calculated by ignoring the thermal electronic contribution and effect of the zeropoint energy.  As can be observed from Figure 3a, the C v values of the phases increase rapidly at sufficiently low temperatures, which coincides with the Debye model theory. As the temperature reaches 900 K, the C v values become almost constant (25 J/K mol), which is known as the Dulonge Petit limit [46]. In Figure 3b, the C p values of the compounds are noted to be proportional to T 3 at temperatures near 0 K. Furthermore, the C p values of the Co-W phases are higher than Co-W-Mo and Co-Mo, which is consistent with the C p value of W being larger than Mo at 300 K. Co 7 X 6 and DO 19 -Co 3 X exhibited the lowest and highest C p values at high temperatures, respectively (shown in Figure 3b). The observed phenomenon can be associated with the stability and melting points of the different phases at high temperatures, and Co 7 W 6 generally has the highest melting points in these phases due to the high melting point of W.
As is well known, the enthalpy, as a function of temperature, is a crucial parameter in thermodynamic modeling. The calculated enthalpies of the Co 3 X compounds in the DO 19 , L1 2 and µ structures at zero pressure are shown in Figure 4. Similar to the heat capacity, the findings are almost identical with the order of enthalpy of Co, W and Mo reported earlier [47] (the enthalpy of W is higher than that of Mo in the range 300-1500 K). be proportional to T 3 at temperatures near 0 K. Furthermore, the Cp values of the Co-W phases are higher than Co-W-Mo and Co-Mo, which is consistent with the Cp value of W being larger than Mo at 300 K. Co7X6 and DO19-Co3X exhibited the lowest and highest Cp values at high temperatures, respectively (shown in Figure 3b). The observed phenomenon can be associated with the stability and melting points of the different phases at high temperatures, and Co7W6 generally has the highest melting points in these phases due to the high melting point of W.
As is well known, the enthalpy, as a function of temperature, is a crucial parameter in thermodynamic modeling. The calculated enthalpies of the Co3X compounds in the DO19, L12 and μ structures at zero pressure are shown in Figure 4. Similar to the heat capacity, the findings are almost identical with the order of enthalpy of Co, W and Mo reported earlier [47] (the enthalpy of W is higher than that of Mo in the range 300-1500 K). The calculated Gibbs free energy of DO19-, L12-and μ-ordered Co-W/Mo are shown in Figure 5. In order to verify the reliability of the obtained values, the calculated and experimental Gibbs free energy [47] values of pure solid Co, Mo and W from 300 K to 1500 K are displayed in Figure 5a. Herein, the Gibbs free energy of the pure elements at 300 K was set as the benchmark. The calculated ΔG values were noted to be smaller than the experimental values, which can be associated with the contribution of the thermal electrons to entropy (S) at high temperatures. However, the atoms are conserved before and after the alloying process, and the S of the thermal electrons is primarily determined by the atoms. The relative errors generated from the thermal electrons to ΔS of the alloying reaction at a specific temperature should be smaller than the errors originating from enhancing the temperature. Figure 5b presents the calculated Gibbs free energy of the DO19-, L12-, and μ-ordered Co-W/Mo compounds as a function of temperature. The Co-W phases are noted to have a lower Gibbs free energy in comparison with the Co-Mo and Co-W-Mo alloys, in accordance with the findings for the pure elements. The calculated Gibbs free energy of DO 19 -, L1 2 -and µ-ordered Co-W/Mo are shown in Figure 5. In order to verify the reliability of the obtained values, the calculated and experimental Gibbs free energy [47] values of pure solid Co, Mo and W from 300 K to 1500 K are displayed in Figure 5a. Herein, the Gibbs free energy of the pure elements at 300 K was set as the benchmark. The calculated ∆G values were noted to be smaller than the experimental values, which can be associated with the contribution of the thermal electrons to entropy (S) at high temperatures. However, the atoms are conserved before and after the alloying process, and the S of the thermal electrons is primarily determined by the atoms. The relative errors generated from the thermal electrons to ∆S of the alloying reaction at a specific temperature should be smaller than the errors originating from enhancing the temperature. Figure 5b presents the calculated Gibbs free energy of the DO 19 -, L1 2 -, and µ-ordered Co-W/Mo compounds as a function of temperature. The Co-W phases are noted to have a lower Gibbs free energy in comparison with the Co-Mo and Co-W-Mo alloys, in accordance with the findings for the pure elements.
In order to effectively analyze the stability of the phases at different temperatures, the formation Gibbs free energy ∆G f has been calculated using Equation (37) and is shown in Figure 5c: where G total (A x B y ), G solid (A) and G solid (B) are the Gibbs free energy values of the unit cell of A x B y , A and B, respectively. As shown in Figure 5c, the ∆G f values are significantly related to the crystal structure, and the DO 19 -Co 3 X phases have the lowest ∆G f as compared with the other two phases possessing the same elements. However, it can be noticed that the ∆G f difference between the DO 19 and µ structures is inversely proportional to the increasing temperature. According to the phase transformation from DO 19 -Co 3 X to Co 7 X 6 in the experiments. The ∆G f of the follow reactions (38)(39)(40) are calculated and shown in Figure 5d: Co 3 Mo + Mo → Co 7 Mo 6 (39) In order to effectively analyze the stability of the phases at different temperatures, the formation Gibbs free energy ΔGf has been calculated using Equation (37) and is shown in Figure 5c: where Gtotal (Ax By), Gsolid (A) and Gsolid (B) are the Gibbs free energy values of the unit cell of Ax By, A and B, respectively. As shown in Figure 5(c), the ΔGf values are significantly related to the crystal structure, and the DO19-Co3X phases have the lowest ΔGf as compared with the other two phases possessing the same elements. However, it can be noticed that the ΔGf difference between the DO19 and μ structures is inversely proportional to the increasing temperature. According to the phase transformation from DO19-Co3X to Co7X6 in the experiments. The ΔGf of the follow reactions (38)(39)(40) are calculated and shown in Figure 5d: As can be observed from Figure 5d, Co3X and X react to form Co7X6 in the temperature range 1000-1200 K, and the transformation temperature of Co-Mo from Co3Mo to Co7Mo6 is noted to be lower than that of Co-W by about 100 K. Comparing the Co-X binary diagram, DO19-Co3Mo and Co3W disappear after heating at about 1050 °C and 1100 °C, respectively. Hence, the phase transformation temperature of Co3Mo was lower than Co6WMo and Co3W in both experimental and theoretical analyses. In addition, the ΔG of the Co-W system for the phase transformation from Co3W to Co7W6 was higher than the As can be observed from Figure 5d, Co 3 X and X react to form Co 7 X 6 in the temperature range 1000-1200 K, and the transformation temperature of Co-Mo from Co 3 Mo to Co 7 Mo 6 is noted to be lower than that of Co-W by about 100 K. Comparing the Co-X binary diagram, DO 19 -Co 3 Mo and Co 3 W disappear after heating at about 1050 • C and 1100 • C, respectively. Hence, the phase transformation temperature of Co 3 Mo was lower than Co 6 WMo and Co 3 W in both experimental and theoretical analyses. In addition, the ∆G of the Co-W system for the phase transformation from Co 3 W to Co 7 W 6 was higher than the Co-Mo system. It indicates the high stability of Co 7 W 6 at high temperatures, which is consistent with the previously observed improvement in the stability of the µ-phase of Co 7 Mo 6 on incorporating tungsten.

Summary
The DO 19, L1 2 and µ structural phases in the Co-W, Co-Mo and Co-W-Mo compounds have been considered in this study for calculating the mechanical and thermodynamic properties. Possessing the same elements, the DO 19 -and µ-phases exhibited a lower shear modulus and higher ductility than the L1 2 -phases. On the other hand, with the same structure and different elements, Co-W displayed higher strength than Co-W-Mo and Co-Mo. Moreover, the mechanical properties of Co-W-Mo with the same amount of W and Mo were noted to be closer to Co-W than Co-Mo. The assessment presented in this study reveals the lack of the experimental data with respect to the thermodynamic properties of Co-W/Mo. The C v and C p values of the nine phases are noted to match well with the regular pattern of the elements obtained experimentally. Coinciding with the phase transformation, DO 19 -Co 3 X transforms to Co 7 X 6 at high temperatures. Besides, the properties of the Co-W-Mo ternary alloys are noted to hover between the Co-W and Co-Mo binary systems. The findings obtained in this study are expected to provide valuable clues for the design of novel Co-based materials and can be helpful in stimulating future experimental and theoretical research in the field of Co-W/Mo alloys.