Investigation of the thermodynamic, structural, electronic, mechanical and phonon properties of D0c Ru-based intermetallic alloys: an ab-initio study

We present the structural, elastic, electronic, magnetic, and phonon properties of D0c X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) alloys in their respective ground-states at zero pressure using first-principles density functional theory (DFT). The calculated heat of formation for Sc3Ru, Ti3Ru, V3Ru, Mn3Ru, and Zn3Ru are negative, signifying their thermodynamic stability. Meanwhile, we find that Sc3Ru, V3Ru, Mn3Ru, Co3Ru, Ni3Ru, Cu3Ru and Zn3Ru alloys are mechanically stable. The electronic properties indicate a metallic nature in all the X3Ru alloys due to valence-conduction band overlap at the Fermi energy. Additionally, the phonon dispersion curves suggest that Cr3Ru, Fe3Ru, Ni3Ru, Cu3Ru, and Zn3Ru are dynamically stable. These results provide a comprehensive overview of the stability, electronic, and mechanical properties of D0c Zn3Ru structures, suggesting their suitability for engineering novel alloys in high-temperature structural applications.


Introduction
High-temperature structural materials have garnered significant attention due to their role in advancing industrial applications [1,2].Currently, Nickel-based super-alloys (NBSAs) are the predominant materials employed in the hot segments of turbine engines, primarily because of their exceptional high-temperature, physical and mechanical properties [3,4].However, the application of NBSAs is constrained in application in the next generation of turbine engines, primarily due to the limitation on the low melting point temperature of Ni (1543 °C).In this context, various metallic systems, including Refractory metals (RMs) such as Ti, Mg, Pt, Ru, and Cr-based alloys and the platinum group of metals (PGMs) have been proposed as viable alternatives to overcome the melting point temperature limitations associated with NBSAs.
Refractory metals (RM) based on Mo, Nb, Ta, and W suffer substantial oxidation challenges in air at elevated temperatures (500 °C), commonly referred to as pesting [5].In addition, the specific strength of titanium alloys tends to deteriorate with increasing temperature, while the high-temperature application of magnesium alloys is limited by their constrained mechanical properties, including deficient strength, internal fatigue, and creep resistance [6].Recent investigations have delved into Mg-X (X = La, Nd, and Sm) intermetallic alloys across various crystallographic phases such as D0c, A15, and L1 2 , utilizing an ab initio density functional theory calculation [7,8].Notably, the heats of formation data indicate the thermodynamic stability in all phases.Additionally, these alloys demonstrate mechanical stability, except for the L1 2 and A15 phases.
The Platinum group metals (PGMs), such as Pt, Ru, Os, Rh, Pd, and Ir, for high-temperature structural applications has been investigated [9][10][11][12].Despite their chemical properties resembling those of Nickel-based super-alloys (NBSAs) [13], some PGMs are inherently brittle and face challenges related to weight and cost.This limits their potential use in high-temperature applications.In a prior study by Chauke et al, [14] the phase stability of Pt 3 Al across various crystallographic phases, including L1 2 , D0c' (tI16-Ir 3 Si), and tP16 (Pt 3 Ga) indicated that that Pt 3 Al in the tP16 phase exhibits the highest thermodynamic stability.Meanwhile,
The ground state electronic and magnetic properties of the D0c X 3 Ru alloys were modeled using their respective unit cells, as illustrated in figure 1. Plane wave basis sets were utilized to represent electronic wave functions, with a converged cut-off energy of 800 eV., while optimized Monkhorst-Pack k-grid sampling of 15 × 15 × 10 was used for spin and geometry optimization [39].To ensure the accuracy of the results, a convergence criteria for energy, force displacement, and stress were set at 5.0 × 10 −6 eV per atom, 0.01 eV Å, and 0.02 GPa, while a temperature smearing width of 0.001 eV was set to ensure accuracy of the calculated magnetic moments.Similar convergence criteria were applied in the calculation of electronic and mechanical properties.For phonon calculations, a finite displacement method was employed [40][41][42][43] using a larger supercells of 4 × 4 × 4, a cut-off radius of 5.0 Å, and a k-grid sampling of 5 × 5 × 3.

Heats of formation and structural stability
The crystal structures of X 3 Ru alloys (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) belong to the body centered tetragonal D0c phase with the I4/mcm space group.Within the unit cell, the Ru atoms are positioned at the Wyckoff site 4a (0, 0, 0.25), while X atoms are located at 4b (0, 0.5, 0.25) and 8 h (0.25, 0.75, 0) positions, as illustrated in figure 1.
To determine the equilibrium ground state properties of X 3 Ru alloys, structural optimization was executed under zero temperature-pressure conditions.The results for the equilibrium lattice constants, volume, heats of formation, magnetic moments, and density are presented in table 1. Notably, an observed trend in X 3 Ru (X = Sc-Cu) reveals a consistent decrease in calculated lattice parameters (hence volume) across the 3d series, aligning with the trends in atomic radii of the 3d elements across the series.Table 2 shows the calculated and experimental lattice parameters for all 3d-transition metal elements such as Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn.We note that the calculated and experimental data agree.
The heat of formation (ΔH f ) for an alloy is defined as the energy required to either form or break chemical bonds and is calculated as: where, E(A x B y ), E(A), and E(B) are the calculated equilibrium total energies of the alloy system AB and that of the individual elemental species A and B, with atomic concentrations x and y respectively.A negative value for ΔH f indicates chemical stability, while a positive value of ΔH f indicates instability.Table 1 displays the calculated heat of formation for X 3 Ru alloys.The calculated heat of formation for Cr 3 Ru is observed to be comparable to the previous theoretical value.The slight discrepancy can be primarily attributed to the use of different exchange correlation functionals.In the previous study, the GGA functional was employed, whereas in this investigation, the GGA + U (U = 2.5 eV) was used.Notably, Sc 3 Ru, Ti 3 Ru, V 3 Ru, Mn 3 Ru, and Zn 3 Ru exhibit negative heat of formation, signifying thermodynamic stability.Therefore, these alloy systems can readily be synthesized experimentally under equilibrium conditions.Considering that the stability of intermetallic alloys is intertwined with their magnetic moments through orbital hybridization, we also computed the magnetic moments of X 3 Ru alloys.It was observed that all X 3 Ru alloys are magnetic except Table 1.Equilibrium lattice constants a and c, equilibrium volume V 0 , heats of formation, magnetic moments, density and X-Ru bondlengths of X 3 Ru, calculated at zero pressure.
Magnetic moments (μ B /atom) Density (g/cm Cu 3 Ru and Zn 3 Ru.Additionally, it is noteworthy that most of the stable systems display significant magnetic moments, with Mn 3 Ru having the highest magnetic moment.
The density of materials is a vital tool used to characterize its use in lightweight applications such as in aerospace, and is calculated as: where Vol represents the volume of a unit cell, M W is the average molecular weight of the elements in the unit cell, N is the total number of atoms and A 0 is the Avogadro's number (6.022 × 10 23 ).Table 1 gives the calculated densities of D0c X 3 Ru alloys.Importantly, it is observed that the densities of the most stable alloys, namely Sc 3 Ru (4.54 g/cm 3 ), Ti 3 Ru (5.86 g/cm 3 ), V 3 Ru (6.31 g cm −3 ), and Mn 3 Ru (6.57g cm −3 ), are comparable to that of L1 2 -Ni 3 Al (6.14 g cm −3 ) [12] commonly employed in the aerospace industry.Therefore, these structures are promising candidates for high-temperature lightweight structural applications.
Research on D0c X 3 Ru alloys is relatively new and as such there is limited data of these materials systems in literature.Our calculated lattice constant of Cr 3 Ru-which is the prototypical alloy in this class of materials agrees with literature.To further validate our theoretical model, our calculated lattice parameters for the individual elements (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu) are agreement with experimental values (table 2).

Electronic and magnetic properties
Understanding the density of states (DOS) is crucial for clarifying the link between the stability of a material and its magnetic properties [46].The partial density of states (PDOS) for X 3 Ru is shown in figure 2, showing the contribution arising from the s, p and d orbitals in the unit cell.Significantly, there is an electronic overlap between the valence and conduction bands near the Fermi energy level, suggesting metallic conductivity in these alloys.In Cu 3 Ru and Zn 3 Ru, the spin up and spin down bands are symmetric, hence zero spin polarization.This symmetry leads to the elimination of the magnetic moment linked to electronic spin, explaining the absence or minimal calculated magnetic moments (table 1).Conversely, the density of states in Sc 3 Ru, Ti 3 Ru, V 3 Ru, Cr 3 Ru, Mn 3 Ru, Fe 3 Ru, and Co 3 Ru indicates spin polarization, confirming the existence of magnetic moments in these systems.The existence of magnetic moments in these alloys makes them potentially viable for applications in spintronics and spin injection.These findings align with our earlier research in similar material systems [32,[47][48][49].
Notably, around the Fermi energy, a prominent feature is the presence of a valley referred to as the pseudogap in both the spin-up and spin-down bands, which indicates covalent bonding [50,51].The existence of the pseudo-gap arises from strong hybridization in X-3d and Ru-3d states, effectively separating the bonding states from the anti-bonding states.

Mechanical stability
Mechanical stability serves as a metric to gauge a material's strength and is employed for characterizing the structural stability and deformation of a system under external load [52].This stability is defined in terms of elastic constants C ij , Bulk Modulus (B), Shear Modulus (G), and Young's modulus (E), providing insights into a material's hardness and ductility.For tetragonal crystals, mechanical stability criteria at zero pressure are defined by the following set of equations [53];  The melting temperature (Tm) of a material depends on its mechanical properties, and it follows a linear relationship with its elastic constants [54][55][56][57].For tetragonal structures, the melting point is given by In table 3, we noticed a rising pattern in melting temperatures from Sc to Zn alloys, with Fe 3 Ru exhibiting the highest melting temperature.Moreover, the melting temperatures of Fe 3 Ru, Co 3 Ru, and Ni 3 Ru exceed those of the presently employed L1 2 Ni 3 Al (1691 K [12] and 1668 K [58]).As a result, these particular alloys present themselves as promising candidates for high-temperature structural conditions.
The Bulk Modulus (B) is a measure for a material's resistance to compression, and its magnitude is influenced by the crystal structure of the material.Materials with high compressibility exhibit large values of bulk modulus, while low bulk modulus values indicate materials with low compressibility.In the context of tetragonal structures, the expression for B is: and the Reuss bounds are: where, with B R , B = B H and B V being the Bulk modulus for Reuss, Voigt and Hill approximations [59].The shear modulus (G) of a material describes its response to shear stress and is a measure of a material's stiffness.For tetragonal structures, G is expressed as where, where the G R and G V are the Reuss and Voigt bounds [60], Young's modulus and Poison's ratio ʋ are independent of the type of a material's crystal structure and are given by: where, X = Voigt, Reuss and Hill approximations.
From the calculated elastic constants provided in table 3, the bulk modulus (B), shear modulus (G), and Young's modulus (E) are computed and presented in table 4. Across all the X 3 Ru alloys, it is noteworthy that B H,R consistently surpasses G H,R , indicating that the principal parameter governing the stability of base-centered tetragonal X 3 Ru alloys is the shear modulus [61].On the other hand, Young's modulus characterizes a material's strain response to uniaxial stress, reflecting its stiffness.Higher values of Young's modulus are associated with We find that the elastic moduli across the 3d-series (X) is generally lower in the early 3d-elements and higher in the middle 3d-element, with Fe 3 Ru, Co 3 Ru and Ni 3 Ru having the largest B and G values.Interestingly, we therefore conclude that, high elastic moduli in X 3 Ru alloys correspond to the high melting temperature (figure 3).Further, the bulk, shear, young's moduli have a monotonic relationship with the melting temperature.Therefore, increased stiffness, hardness, and compression resistance in these structures explains the higher melting temperatures correspond.These findings align with the trends previously observed by Popoola et al [12].
Applying Pugh's criterion [62], the ratio of bulk to shear modulus (B H /G H ) is employed to predict the ductility/brittleness of a material, with the critical value set at 1.75.Ductile behavior is predicted if B H /G H is greater than 1.75; otherwise, the material is deemed brittle.Notably, Sc 3 Ru Mn 3 Ru, Fe 3 Ru, Co 3 Ru, Ni 3 Ru, Cu 3 Ru, and Zn 3 Ru exhibit ductile behavior, while Ti 3 Ru, V 3 Ru, and Cr 3 Ru display brittleness.Additionally, the ductility/brittleness of materials can be assessed using Poisson's ratio, with the threshold value set at 0.33 [63].Employing the Poisson's ratio criteria, it is observed that all X 3 Ru alloys are ductile except V 3 Ru and Cr 3 Ru, aligning with the results from Pugh's criterion, as shown in the figure 4.
In evaluating the crystal's stability against shear deformation, Poisson's ratio has been considered [64,65].A higher Poisson's ratio indicates better plasticity in crystals, and it also serves as a characterization of the bonding   4. Calculated bulk modulus B (GPa), shear modulus (GPa), Young's modulus E (GPa), a ratio of bulk to shear modulus (B H /G H ), Poisson's ratio (υ) and Vickers hardness (H V ) of X 3 Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) alloys at zero pressure.V, R and H are the Voigt, Reuss and Hill approximations.UND = undefined.forces within crystals [66].The central forces in a solid typically fall within the range of 0.25 υ 0.5.The computed Poisson's ratios (υ) for X 3 Ru range from 0.14 to 0.54.All the structures exhibit a Poisson's ratio within the range of 0.25 υ 0.5, confirming that the interatomic forces in these structures are central, except for Ti 3 Ru, V 3 Ru and Cr 3 Ru.
Hardness is a critical attribute in high-temperature environments, particularly in aerospace applications.It defines a material's resistance to localized plastic deformation caused by either mechanical indentation or abrasion.Hardness, while not a fundamental property, relies on tensile strength, yield strength, and the modulus of elasticity.It is determined through the Vickers hardness equation (12) [67] where K represents the ratio of shear modulus to bulk modulus (G H /B H ), and H denotes the Hill approximation.
The calculated hardness values for tetragonal X 3 Ru structures are presented in table 3. It is evident that V 3 Ru exhibits the highest hardness, while Sc 3 Ru has the lowest.Notably, the hardness for Ti 3 Ru is undefined due to the negative square root in G H /B H , indicating instability associated with a phase change in this structure.This type of instability is also observed in ferro-elastic phase transformations [68], as explained by Landau theory [69], where two local minima form in a strain energy function.

Elastic anisotropy
Understanding the anisotropic behavior of a material is crucial in engineering science and crystal physics, as it is closely related to micro-cracks in materials.Calculating elastic anisotropy provides valuable information about a material, including details about micro-cracks, phase transformation, precipitation, and dislocation dynamics [70].For tetragonal alloys, elastic anisotropy can be expressed through three elastic factors: shear anisotropy factors (such as A 1 , A 2 and A 3 ) for different crystallographic planes, the universal anisotropic index (A U ), and the percentage in compression (A B ) anisotropy as given below: where A 1 , A 2 and A 3 represent the shear anisotropic factors for the (001), ( 010) and (100) shear planes.In the case of locally isotropic structural materials, the shear anisotropy factors A 1 , A 2 and A 3 are expected to be equal to one.Additionally, the universal anisotropic index A U should be zero in such materials.Any deviations from one or zero, indicates the degree of elastic anisotropy [71,72].The B V , B R , G V and G R represents the Voigt and Reuss approximation for bulk modulus (B) and shear modulus (G).The maximum value of (A B ) is 100%, which corresponds to maximum anisotropy [73,74] while the minimum is zero which corresponds to an isotropic material.Table 5 summarizes the calculated values of A 1, A 2 , A U andA B in the X 3 Ru alloys.It is evident that A 1 values in the X 3 Ru alloys are smaller than one, indicating elastically anisotropic structures.On the other hand, the A 2 values are larger than one (except in Ti 3 Ru and Fe 3 Ru) suggesting that they are slightly isotropic.Regarding A U , it is observed that V 3 Ru and Ni 3 Ru and Zn 3 Ru exhibit a small degree of universal elastic isotropic behavior as they are close to unity.The (A B ) results clearly indicate that X 3 Ru (X = V, Mn, Fe, Co, Ni, Cu, and Zn) structures are close to zero, suggesting they are slightly isotropic, while Sc 3 Ru (0.0) is isotropic, indicating that these structures are predicted to have nearly identical values of A B in all x, y and z directions, attributing to external forces.

Phonon dispersion
Phonons play a vital role in the dynamical behavior and thermal conductivities, which are critical aspects in the research advancements of novel materials.The phonon dispersion curves provide insights into dynamic stability or instability of a material by showcasing positive (real branches) and negative (imaginary branches) phonon frequencies.Positive phonon frequency modes signify dynamic stability in a material, while negative frequency modes indicate dynamic instability in a compound [75,76].Figure 5 shows the dispersion curves of D0c X 3 Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) at plotted along the highest symmetry k-points of the Brillouin zone.The phonon dispersion curves of Cr 3 Ru, Fe 3 Ru, Ni 3 Ru, Cu 3 Ru, and Zn 3 Ru are dynamically stable due to the absence of imaginary frequencies.Conversely, the phonon dispersion plots of Ti 3 Ru, V 3 Ru, Co 3 Ru, and Mn 3 Ru exhibit negative phonon modes, indicating dynamic instability, which can be drawback in the application of these materials where dynamic stability is critical.
In summary, by considering the thermodynamic, mechanical and dynamic stability of the X 3 Ru alloys presented in this study, our findings suggest that: (1) D0c X 3 Ru (X = Sc, Ti, V, Cr, Mn, and Zn) alloys exhibit negative heat of formation, hence are thermodynamical stable (2) X 3 Ru (X = V, Mn, Co, Ni, Cu, and Zn) alloys are mechanically stable, with Co 3 Ru and Ni 3 Ru exhibiting elevated melting temperatures compared to the currently utilized alloy L1 2 Ni 3 Al (1691 °C) [12] (3) X 3 Ru (X = Cr, Fe, Ni, Cu, and Zn) are dynamically stable.Therefore, Zn 3 Ru meets all three stability criteria, which could give a slight advantage for structural applications over the other systems examined in this work.

Conclusion
In conclusion, we employed Density Functional Theory to calculate the electronic, magnetic, elastic, and thermodynamic properties of binary D0c X 3 Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) tetragonal alloys.Our investigation reveals that D0c X 3 Ru (X = Sc, Ti, V, Cr, Mn, and Zn) alloys exhibit negative heat of formation, indicating structural stability.Notably, Ti 3 Ru, V 3 Ru, Cr 3 Ru, Mn 3 Ru, Fe 3 Ru, Co 3 Ru, and Ni 3 Ru compounds display high magnetism attributed to strong spin polarization.Mechanical property analysis demonstrate that X 3 Ru (X = V, Mn, Co, Ni, Cu, and Zn) alloys are mechanically stable.Additionally, the mechanically stable structures of Co 3 Ru and Ni 3 Ru exhibit elevated melting temperatures compared to the currently utilized alloy L1 2 Ni 3 Al (1691 °C) [12], positioning them as potential candidates for high-temperature structural applications.Phonon calculations indicated that Cr 3 Ru, Fe 3 Ru, Ni 3 Ru, Cu 3 Ru, and Zn 3 Ru are dynamically stable.These results provide a comprehensive overview of the stability, electronic, and mechanical properties of D0c Zn 3 Ru structures, suggesting their suitability for engineering novel alloys in high-temperature structural applications.

Figure 2 .
Figure 2. Calculated Partial density of states of X 3 Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) alloys.The zero indicates the Fermi energy (0 eV).The vertical scale for all X 3 Ru alloys is the same for comparison's sake, except in Zn 3 Ru.

Figure 4 .
Figure 4. Calculated trends in the ratio of the bulk to shear modulus and Poisson's ratio across the 3d series in X 3 Ru X 3 Ru alloys (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn).

Figure 5 .
Figure 5. Dispersion curves for D0c X 3 Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) compounds at high symmetric points in the Brillouin zone.

Table 2 .
The calculated and experimental lattice parameters for Sc, Ti,V,Cr, Mn, Fe, Co, Ni, Cu and Zn elements.
In tetragonal X 3 Ru (X = Sc-Zn) alloys, the six independent elastic constants C 11 , C 33 , C 12 , C 44 , C 13 , and C 66 are outlined in table 3. It is noteworthy that the elastic constants of V 3 Ru, Ni 3 Ru, Co 3 Ru, Cu 3 Ru, Mn 3 Ru, Cr 3 Ru, and Zn 3 Ru meet the Bohr mechanical stability criteria (C ij > 0), signifying their mechanical stability.In contrast, Sc 3 Ru, Ti 3 Ru, and Fe 3 Ru structures have C 44 and C 66 values below zero, indicating mechanical instability.

Table 3 .
Calculated independent elastic constants (C ij ) in GPa and melting temperature (T m ) in Kelvin for X 3 Ru (X = Sc, Ti, V, Fe, Co, Ni, Cu and Zn) alloys.whereaslowervalues indicate less stiffness.Table3reveals that Ni 3 Ru stands out as the stiffest, while Ti 3 Ru is the least stiff.

Table 5 .
The shear anisotropic factors (A 1 , A 2 ), universal elastic anisotropy index (A U ) and the anisotropy percentages (A B ) of X 3 Ru 3 (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) alloys under zero pressure.