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First-principles calculations of thermal expansion path of the Co7Mo6 Mu-phase

The Mu-phase, which is an intermetallic compound, can form in steels and superalloys, and significantly degrade their mechanical properties. The thermal expansion of Co7Mo6 Mu-phase have been investigated by first-principles based quasi-harmonic Debye-Gr\"uneisen approximation and by comparing the free energies calculated along different paths of thermal expansion. The calculations were carried out for temperatures from 0 K to 1500 K which below the decomposition temperature of Co7Mo6. The electronic, vibrational and magnetic energy contributions to the free energy were accounted. The thermal expansion of Co7Mo6 is not isotropic. The influence of physical factors on the stability of Co7Mo6 was analyzed. The isobaric heat capacity, thermal expansion, elastic constants, bulk modulus, sound wave velocities, Debye and Curie temperatures were reported. The calculated results analyzed and are in an agreement with the available experimental and theoretical data.


Introduction
The Co7Mo6 compound is the topologically close-packed (TCP) intermetallic phase existing in the Co-Mo systems. In the Co-Mo phase diagram the µ-phase has a homogeneity range from approximately 45 to 50 at. % Mo as presented by Davydov et al. [1,2]. Thus, the Co7Mo6 with its stoichiometry composition of 46.2 at. % of Mo is a stable compound.
The Co7Mo6 compound is the model µ-phase which used to study the effect of precipitation of TCP phases on mechanical properties of the cubic matrices of cobalt (or nickel) based superalloys, because it is prone to precipitation during thermal exposure at high temperature. A large number of refractory elements have been adding to superalloys to improve their creep and stability at elevated temperatures, but the drawback of it is a precipitation of TCP phases [3].
The effect of a possible precipitation of TCP phases brings significant obstacles in design of superalloys which are widely used in production of turbine blades for aircraft engines. It's because, these materials have excellent high-temperature mechanical properties, creep and corrosion resistance [4].
The µ-phase is a hard-brittle compound and is detrimental because its precipitation induces a local depletion of a matrix of superalloys and its needle morphology can easily initiate a rupture and influence in particular the creep resistance, yield strength and ductility. Therefore, the range of stability of µphase should be accurately investigated. Especially, Gibbs potentials, thermodynamic properties and a path of thermal expansion of µ-phase must be calculated to model phase boundaries and simulate properties of superalloys.
The alloy development depends on existing phase diagrams and reliable thermodynamic description of Gibbs energies of constituent phases. Due to slow diffusion and the presence of other phases, it can be difficult to conduct an experiment at low temperatures, lower than 1073 K. Therefore, quantum mechanical calculations together with models can significantly help in obtaining such a thermodynamic description of the concerning phases.
In this work, the calculations are restricted to Co7Mo6 compound and does not take into account the disorder associated with possible deviation from the stoichiometry.
The µ-phase due to its technological importance attracts a lot of attention. A number of works studding the phase equilibria, structural properties and lattice parameters of µ-phases were conducted over the last years. The phase boundaries of the µ-phase in the Co-Mo system were studied in the work [1,2]. The experimental calorimetric study of the Co7Mo6 was reported by Spencer et al. [5]. The phase equilibria in the Co7Mo6 µ-phase were studied by Oikawa et al. [6]. Zhao et al. [7] studied a precipitation of µ-phase in nickel base alloys by scanning, transmission electron microscopy and X-ray diffraction. Qin et al. [8] investigated the µ-phase in superalloys to applied stress at T = 800 C. The site occupation in the Ni-Nb µ-phase and configurational thermodynamic properties have been studied by Sluiter et al. [9]. The formation enthalpies of elements of Fe and Mo in metastable µ-phases were reported by Sluiter in [10], using the DFT calculations. Joubert et al. [11] studied the atom distribution on the five different sites of the crystal structure of µ-phases. The structure and lattice parameters of Co7Mo6 µ-phase have been studied by Forsyth et al. [12].
As one could see from above, that these studies are mainly focusing on structure properties of µ-phases. But the thermodynamic and physical properties of Co7Mo6 compound are not reported yet. The factors influencing on stability of µ-phase have not been investigating so far. But, Grabowski et al. [13] have shown that the magnetic entropy can significantly increase, up to 30 %, the free energy of alloys and it should be accounted in calculations. Moreover, the answer to the question of whether the thermal expansion of these compounds is isotropic or not is also unknown.
Insofar as answering these questions and obtaining these data are important for calculating the Gibbs potentials and predicting the properties of compounds at various temperatures, this was the motivation for this study.
In this work, the thermal expansion of the Co7Mo6 µ-phase was studied by first-principles calculations based on density functional theory (DFT) and combined with the quasi-harmonic Debye-Grüneisen approximation (QHA), as used in [13,14]. Since, the crystal lattice of Co7Mo6 has two parameters a and c, and in order to find out how the lattice parameters of Co7Mo6 change during heating up, the approach used to calculate thermodynamic properties of Laves phase Fe2Mo in the work [15], was utilized.
The electronic and the vibration energies, the magnetic entropy of local magnetic moments of atoms were accounted to study the thermodynamic properties and evaluate their effect on the stability of the Co7Mo6. The thermal expansion path, isobaric heat capacity, structural properties, elastic constants of the strain tensor, bulk, shear, Young's modulus, Debye and Curie temperatures are calculated. The anisotropic thermal expansion in the Co7Mo6 compound is predicted. The results of this work can be useful for further modelling the Gibbs energies of the µ-phase as a function of composition, for calculating the phase diagrams and further theoretical and experimental research. In addition, it may provide valuable data which difficult to obtain from experiments.

Free energy subsystems
The electronic energy Fel(V,T) was described as in [17] ( , ) = ( , ) − ( , ), With the energy of electrons due to their excitations Eel(V,T) is formulated by [18] ( , ) = ∫ ( , ) ( , ) where n(ε,V) is the electronic density of states DOS, f(ε,T) is the Fermi-Dirac distribution. The electronic entropy Sel(V,T) takes the form where kB is the Boltzmann constant.
The magnetic energy Fmag(V,T) was described by the assumption: Where F'mag(T) is the magnetic energy term of Hillert and Jarl model as given in [19] ′ ( ) = ( + 1) ( ) where τ = T/Tc; the function f(τ) is defined in [19]; β is the average magnetic moment of atoms. The expression in square brackets of (10) is to compensate the constant contribution of magnetic internal energy persistently existing in the total energy.
The theoretical estimation of Curie temperature was calculated using the mean-field approximation borrowing in [13] = 2( ( 0 )− ( 0 )) 3 (12) where E PM tot(V0) and E FM tot(V0) are the total energies of the compound calculated at T = 0 K for paramagnetic (PM) and ferromagnetic (FM) states at the corresponding equilibrium volumes V0.
The magnetic entropy Smag(V) and configurational entropy Sconf were calculated as follow: where µi is the local magnetic moment of atom i, ci is the atomic concentration.

First-principles calculations
The calculations were carried out using the Full Potential-Linear Augmented Plane Wave (FP-LAPW) Method [20] within the (DFT) [21,22], as implemented in the WIEN2k package [20]. The exchangecorrelation effects were treated as a function within the Generalized Gradient Approximation (GGA) [23] and GGA along with Perdew-Bruke-Eruzerhof (PBE) [24]. The muffin-tin radius (RMT) were set as RMT = 2.05 for Co, RMT = 2.15 for Mo. The parameter RKmax was set as 8. To model the Co7Mo6 compound the two kinds of cells, in hexagonal [11] and rhombohedral [12] notation of 39 and 13 atoms, were used with the 15 x 15 x 3 and 12 x 12 x 12 k-point mesh in the first irreducible Brillouin zone using Monkhorst-Pack scheme [25]. The energy between successive iterations is converged to 1x10 -6 eV/atom and forces are minimized to 5 meV/Å.

Structural properties
The crystal lattice of Co7Mo6 µ-phase belongs to the R3m space group and typified by Fe7W6 compound. The unit cell of Co7Mo6 can be represented in rhombohedral and hexagonal notations, with 13 and 39 atoms per unit cell, according [12,11]. The hexagonal notation of Co7Mo6 is shown in Figure 1.

Figure 1.
Unit cell of Co7Mo6 µ-phase, in the hexagonal notation. Co atoms occupy 3a and 18h Wyckoff sites are shown in red and brown respectively while Mo atoms occupy 6c, 6c' and 6c" sites are shown in blue, green and pink respectively.
The computations in this work were carried out with spin-polarized calculations. The geometry optimizations and the full structural relaxation procedures were used to obtain lattice parameters and atoms positions of Co7Mo6 compound. The optimized lattice parameters are listed in Table 1 and Figure  3, together with the available experimental data [12,26]. The lattice parameters of Co7Mo6 calculated in this work at T = 1423 K are close to the experimental data [26] obtained at T = 1423 K after 912 hours of annealing treatments, as listed in Table 1 and shown in Figure 3. The enthalpy formation ∆H of Co7Mo6 was calculated by the following expression where E Co7Mo6 total is the total energy of Co7Mo6, E Co hcp and E Mo bcc are the energies of Co and Mo atoms in the hexagonal close-packed (hcp) and the body centred cubic (bcc) lattice, respectively; x is the concentration of Mo atoms (x = 0.462). The ∆H of Co7Mo6 calculated in this work have the negative value, this suggests that the compound is stable at T = 0K, the obtained value is listed in Table 1 with the other theoretical value [27] for comparison. -2.51 1) calculated in this work at T = 1423 K. 2) experimental data obtained at T = 1423 K after 912 hours of annealing treatments.

Elastic and thermal properties
The elastic constants Cij of the strain tensor were determined by successively imposing distortion matrices Di [33] on the lattice of Co7Mo6, according to the expressions where, R is the lattice of µ-phase in a matrix form described by Bravais lattice vectors of a hexagonal crystal with parameters a and c, Rʹ is the deformed matrix containing distorted lattice vectors.
The variations in total energies ΔE(δ) as functions of applied strains for the different types of distortions Di are plotted in Figure 2.
The calculated elastic constants Cij are listed in Table 2, together with the available theoretical data [27] for comparison. As can be seen from Table 2, Cij satisfy the mechanical stability criterion for hexagonal crystals at zero pressure [28] 11 > 0; ( 11 • 33 − 2 13 2 + 12 • 33 ) > 0; 11 − | 12 | > 0; 44 > 0 (18) and, as a consequence, the Co7Mo6 is mechanically stable compound at zero pressure.  From the Cij obtained for a single crystal, using the Voigt-Reuss-Hill approximation [29], the elasticity parameters for polycrystals were calculated: bulk B, shear G moduli, Young's modulus E, and Poisson's ratio ν according [33]. The obtained values presented in Table 3 together with data [27] for comparison. The values of the B/G > 1.75 and ν ≥ 0.26 ratios, according [30,31], indicate that the Co7Mo6 polycrystalline aggregate is a ductile material at T = 0K. The average (Vm), shear (vs), and longitudinal (vl) elastic wave velocities calculated by the formulas [32], predicted Debye θD and Curie temperatures calculated by (9,12), elastic wave velocities calculated by [33] in the [001] and [100] directions for the Co7Mo6 µ-phase, are given in Table 4. The Debye temperature is a fundamental parameter of thermodynamics and is found in the equations describing physical processes such as heat capacity, thermal expansion, elastic properties, vibrational entropy, and melting temperature that result from theories involving atomic vibrations. θD is necessary to use the quasi-harmonic theory of Debye -Gruneisen to calculate the thermodynamic properties of Co7Mo6.
Anisotropic sound waves velocities calculated for Co7Mo6 are listed in Table 4. The fastest propagation of sound waves was obtained along the [001] direction by longitudinal waves.
For the best knowledge, there are no experimental or theoretical studies on the Curie, Debye temperature and elastic wave velocities for Co7Mo6. So, the obtained predictions may be useful in further studies of the µ-phase.

Scheme of the calculations
In order to investigate the stability of Co7Mo6 with different lattice parameters and find out whether the thermal expansion paths of this compound is isotropic or not, the approach of searching a thermal expansion path (STEP) [15] was applied.
The layout used in these calculations is shown in Figure 3. The intersection point of the d0 ÷ d13 paths (routes) corresponds to the coordinates (a0, c0), these are the optimized parameters of Co7Mo6 lattice obtained in this work by DFT at T = 0K, which are listed in Table 1. A number of di paths are presented here as an example, in Figure 3. The d0 corresponds to the direction of isotropic expansions of Co7Mo6, where the c/a = 5.359 ratio remains constant. The d1 route is passing through coordinates of the experimental point obtained at T = 1423 K after 912 hours of annealing treatments [26], as shown in Figure 3 by a blue triangle. Along the d10 the a-parameter of Co7Mo6 lattice remains constant, as shown in Figure 3.
Since, the volume of the µ-phase, V(a,c) = (√3/2)•a 2 •c, is a function of lattice parameters a and c, and in order to avoid dealing with the energy surface E(a,c), the total energies E(V) were calculated along the d0 ÷ d13 routes, as shown in Figure 4. This approach simplifies the calculations and allows us to reduce the problem to a one-dimensional case and consider the free energy as depending on one variable, the volume. On the one hand, it allows to avoid the differentiation of the total energy E(a,c) with respect to variables a and c, in order to obtain the pressure P(a,c) and the other functions, like the bulk modulus B(a,c). And, on the other hand, it allows to use the equation of state (EOS), like the Vinet EOS [34], and apply the quasi-harmonic Debye-Grüneisen model [14] for each paths di to simulate the effect of temperature on thermodynamic properties of compounds by calculating the free energies F(V,T), depending only on the volume and temperature, calculated by (1) along each routes. Then, by comparing these free energies between themselves one can find the most energetically favourable thermal expansion route with the lowest energy.  [12], and obtained at T = 1423 K [26] are presented as well.  If adding more routes, one can improve an accuracy of the calculations by comparing the energies calculated along these extra routes. In effect, the thermal expansion route may turn out to be curved. To study the vibrational entropy contribution to the free energy of the Co7Mo6 compound the Debye temperatures θD(V) were calculated for the d0 ÷ d13 routes by (9) and shown in Figure 5. As follows from the Figure expansion and contraction relatively the other routes as follows from Figure 4, therefore, according the Debye-Grüneisen model [14] the contribution to the free energy along the d4 will be smallest. Along the d10 route the Debye temperature has the smallest value, but the total energy E(V) calculated along the d10 don't have the smallest value at expansion. And it is difficult to see in advance which route wins this peculiar competition. Thus, all these routes will compete with each other for the most energetically stable thermal expansion path of Co7Mo6.

Accounting for magnetic excitations
If the magnetic energy is not taken into account, then the thermal expansion, as was shown in [15] using the Laves phase Fe2Mo as an example, will be isotropic, which contradicts the experimental data. So, in order to investigate the influence of magnetic entropy on the stability of Co7Mo6 compound the dependences of local magnetic moments, in Bohr magnetons (µB), on volumes were obtained along d0 ÷ d13 routes and shown in Figure 6.   Analyzing the curves of the distribution of magnetic moments over the sublattices obtained along different routes, it was found that at equilibrium volume V0, which is related to T = 0 K, the largest contribution to the magnetic energy of Co7Mo6 is made by Co atoms located on the 18h sub-lattice, about 35 %, then follow Co atoms located on the 3a sub-lattice and the interstitial space, they contribute 28 %, each. The rest comes from Mo atoms accommodated on 6c, 6c' and 6c", these are 1%, 6% and 2% respectively.

Helmholtz free energy and its constituents
The calculations of free energies F(V,T) of Co7Mo6 were carried out by (1) at finite temperatures along the d0 ÷ d13 routers shown in Figure 3. The example of free energies calculated along the d12 route, for T = 100 ÷ 1500K, is shown in Figure 7. Total energy Etot(V) calculated by DFT at T = 0 K along the d12 route is shown in Figure 7 by the red dashed line together with equilibrium volumes V0(T) obtained at each temperature as minimum values of the thermodynamic functions and shown by blue dotted line.  All the energy contributions to the Helmholtz free energy F(V,T): electronic, vibrational and magnetic were calculated by (2 -14) at the corresponding equilibrium volumes V0.
It's essential to note that the configurational entropy Sconf (14) in the stoichiometry case, such as Co7Mo6, gives the same energy contribution for all the routes, and does not affect the comparative analysis. Therefore, by analyzing the free energies F(V0(T)) of Co7Mo6 shown in Figure 8 (d), one can find out that the d12 route is the most energetically favorable one. Therefore, at heating the lattice parameters of Co7Mo6 will be increased from (a0, c0), calculated at T = 0K, towards values lying on the d12 route shown in Figure 3. Thus, this is the pathway of thermal expansion of Co7Mo6, in accordance with Figure  7.   Figure 9. The graphs of free energies F(V0(T)) for Co7Mo6 with correspondent terms: total, magnetic, electronic, magnetic entropy and vibrational energies calculated along the d12 path.
Analysis shows that at heating the spread of the magnetic energies of Co7Mo6 in different directions is about 5%, as shown in Figure 8 (c), while the electronic energy is about 2%, as follows from Figure 8 (a). Both the electronic and magnetic energies equally affect the stability of Co7Mo6. But, the main contributor in the free energy of Co7Mo6 compound is the vibrational energy. And, the calculation shows that all the energy contributions should be accurately accounted for correct prediction of a thermal expansion path.
It is follow from the obtained free energies of Co7Mo6 shown in Figure 8 (d) that the other closest stable routes to the d12 are the d10, d2 and d0, thus these routes lose this peculiar competition.

Thermal expansion and heat capacity
The thermodynamic properties of Co7Mo6 were obtained along the most energetically favourable thermal route d12. The volume expansion V(T) of Co7Mo6 is shown in Figure 10 (a) together with the experimental result [26] obtained at 1423 K after 912 hours of annealing treatments for the sake of comparison. The heat capacity Cp(T) of Co7Mo6 calculated along the route of thermal expansion d12 is shown in Figure 10 (b) together with the experimental data taken from [5]. The figure also shows the upper and lower error of the experiment, which is given in the work [5]. The heat capacity jump at TC, shown in the graph, is the ferromagnetic/paramagnetic phase transition occurring in Co7Mo6 upon heating, as the used model [19] predicts. But, since the experimental data on Curie temperatures for Co7Mo6 is unknown, the search for the exact temperature value is necessary and it is a subject for further study.  Figure 10. a) The volume V(T) of Co7Mo6 calculated along the thermal expansion path is shown in comparison with the experiment data obtained in [26] at T = 1423 K after annealing of 912 hours; b) the heat capacity Cp(T) of Co7Mo6 calculated along the route of thermal expansion is presented together with the experimental data taken from [5] and the scatter of experimental data, which provided in the work [5], is shown as "+ error" and "-error".

Discussion
Thus, DFT calculations with quasi-harmonic Debye -Grüneisen approximation and by employing the approach of searching a thermal expansion path (STEP) of a compound [15] which compares free energies calculated along different routes of thermal expansion, can correctly predict the thermal expansion of Co7Mo6 compound. The lattice parameters of Co7Mo6 calculated in this work at T = 1423 K are in a good agreement with the experimental data reported in [12,26] as shown in Figure 3, Figure  10 (a) and listed in Table 1. The heat capacity calculated in this work and the experimentaly estimated data reported in work [5], as shown in Figure 10 (b), are in satisfactory agreement at T = 800 K, at the higher temperatures the calculated heat capacity increases slowly then the estimated values [5].
The results of this work show that if we neglect the electronic and magnetic energies, then the d10 route, along which the lattice parameter a remains constant, will be the most energetically favourable, as shown in Figure 8 (b) and presented in Figure 5 by the most lower Debye temperature along the d10. But the experimental data on the lattice parameters obtained in [26] after annealing at T = 1423 K, and in [12], show that this is not the case. The calculated d12 path of thermal expansion is passing close to the experimental data [12,26] as shown in Figure 3. Therefore, the electronic and magnetic entropies have an essential effect on the stability of this compound, and in no case should they be neglected.
The phonon calculations of the vibrational energy and their comparison with energies calculated in this work using the Debye-Grüneisen model, which theoretically should match each other, may be a valuable argument for checking the results obtained in this work.
Similar calculations were carried out in work [35], where the direction of thermal expansion of the Fe2Mo Laves phase calculated using the phonon spectrum was compared with the direction calculated using the Debye-Gruneisen approximation. Comparative analysis showed that these two methods completely coincide with each other and can be used in similar STEP calculations.

Conclusions
The finite-temperature DFT calculations have been carried out to calculate electronic, vibrational and magnetic free energy contributions for the Co7Mo6 µ-phase. By employing the approach of searching a thermal expansion path of compounds the thermal expansion of Co7Mo6 is correctly calculated in accordance with the experimental data. The nature of the thermal expansion is not isotropic. The elastic constants, bulk modulus, elastic sound velocities, Curie and Debye temperatures were calculated at ground state. The heat capacity calculated along the thermal expansion path of Co7Mo6 is in satisfactory agreement with the experimental data. Thus, the approach of searching a thermal expansion path of compounds may be used to study thermodynamic properties of µ-phase. The work shows that the vibrational energy is the main factor influencing on the stability of Co7Mo6, but the electronic and magnetic entropies should be accurately accounted for correct thermodynamic description of the µphase.