First-Principles Study on Thermodynamic, Structural, Mechanical, Electronic, and Phonon Properties of tP16 Ru-Based Alloys

Abstract

Transition metal-ruthenium alloys are promising candidates for ultra-high-temperature structural applications. However, the mechanical and electronic characteristics of these alloys are not well understood in the literature. This study uses first-principles density functional theory calculations to explore the structural, electronic, mechanical, and phonon properties of X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) binary alloys in the tP16 crystallographic phase. We find that Mn₃Ru, Sc₃Ru, Ti₃Ru, V₃Ru, and Zn₃Ru have negative heats of formation and hence are thermodynamically stable. Mechanical analysis (C_ij) indicates that all tP16-X₃Ru alloys are mechanically stable except, Fe₃Ru and Cr₃Ru. Moreover, these compounds exhibit ductility and possess high melting temperatures. Furthermore, phonon dispersion curves indicate that Cr₃Ru, Co₃Ru, Ni₃Ru, and Cu₃Ru are dynamically stable, while the electronic density of states reveals all the X₃Ru alloys are metallic, with a significant overlap between the valence and conduction bands at the Fermi energy. These findings offer insights into the novel properties of the tP16 X₃Ru intermetallic alloys for the exploration of high-temperature structural applications.

1. Introduction

Intermetallic alloys offer potential as structural materials in extreme environments. Nickel-based superalloys (NBSAs) are the current material of choice in high-temperature structural applications such as in aerospace, marine, nuclear reactor, and chemical industries. This is attributed to the distinctive surface and mechanical properties of NBSAs [1], which arise from the $\gamma /{\gamma }^{\prime }$ microstructures in the L1₂ cubic phase ($\gamma$) and ordered tetragonal phase (${\gamma }^{\prime }$), where the trans-granular and grain boundary weakness are found to play a critical role [2]. For this reason, research on high-temperature structural materials has revolved around NBSAs for a number of years [3,4,5]. Despite the accomplishment of NBSAs, they are currently limited by their high-temperature capability due to nickel’s low melting point [6,7,8], thus hindering the development of next-generation, high-temperature structural engineering systems.

In this context, several intermetallic structures, including the refractory metals (RMs), platinum group metals (PGMs), as well as their binaries, have been studied in an effort to raise the melting point of NBSAs. However, RMs based on Mo, Nb, Ta, and W suffer intense oxidation in the air above 500 °C (normally referred to as pesting [9]), while the strength of titanium alloys deteriorates with increasing temperature. Although the PGMs (consisting of Pt, Ru, Os, Rh, Pd, and Ir) have comparable chemical properties and similar mineral deposits to NBSAs [10], their use is hampered by weight and cost.

The phase diagrams of the binaries of these metals have also been studied in an effort to find suitable stable alternatives to NBSAs. McAlister and Kahan [11] studied the phase diagram of Pt-Ga, and their findings [8] suggest that L1₂-Pt₃Ga transforms to tetragonal tP16-Pt₃Ga at excessive temperatures of around 1290 °C [12]. On the other hand, Mishima et al. [13] found a phase transformation in Pt₃Al from L1₂ to D0c and L1₂ to D0c’, at around 420 °C and 130 °C, respectively [14,15]. Furthermore, Tibane et al. [16] studied the phase stability of Pt-Cr and Cr-Ru in various phases including A15, B2, tP16, D0c, D0c’, and L1₂ and found tP16 Pt₃Cr to be the most energetically favorable. In a similar study by Chauke et al. [17], the tP16 phase of Pt₃Al was found to be the most stable. Other studies have similarly found the tP16 phase to be stable in a number of alloys [12,18].

Recently, we have studied a number of [19,20,21] 3d-transition metal Ru-based alloys due to Ru potential as a high-temperature structural material [16]. Specifically, in A15 Cr₃Ru and Ru₃Cr alloys doped with 3d and 4d transition metals (Mn, Mo, Pt, Pd, Fe, Co, Re, and Zr), it was established that doping leads to thermodynamic stability. Furthermore, we investigated A15 X-Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) [20], and it was found that Mn₃Ru has a negative heat of formation, which may lead to its experimental realization, among other attractive mechanical properties. In D0c X₃Ru, we found [19] that the calculated heat of formation for Sc₃Ru, Ti₃Ru, V₃Ru, Mn₃Ru, and Zn₃Ru is negative, signifying their thermodynamic stability. The utilization of 3d transition metals is motivated by their capability to form alloys with relatively higher melting points and low densities [22]. Furthermore, the incorporation of transition metal elements is known to improve thermodynamic stability [21] and increase room temperature ductility in various materials [23]. It is therefore of interest to study the properties of Ruthenium-based transition metal alloys in other crystallographic phases in order to understand their full potential for high-temperature structural applications.

In this paper, we investigate the structural, electronic, and mechanical properties of tetragonal tP16 X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) alloys using the density functional theory (DFT) technique [24,25]. Specifically, we compute their thermodynamic stability, mechanical stability, and electronic properties to determine their potential for high-temperature structural applications.

2. Methods

In this work, the properties of tP16 X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) are studied by first-principles density functional theory method using the CASTEP code [25]. The interaction between the core and valence electrons is described by the Vanderbilt ultrasoft pseudopotential [26], with a converged plane wave cut-off of 800 eV, while the electronic exchange correlation is treated by the Perdew–Burke–Ernzerhof Generalized Gradient Approximation (PBE-GGA) functional incorporating the Hubbard parameter (U = 2.5 eV) [27]. The value of U in this study has been chosen to reproduce the experimental lattice constants and magnetic moments of the individual elements X (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) and Ru. The properties of the tP16 X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) alloys were modeled from their respective unit cells (Figure 1), using a well-converged Monkhorst-Pack [28] k-point sampling of 15 × 15 × 11.

For each system, geometry optimization was performed using the BFGS algorithm, with the self-consistent convergence of the total energy and the forces set to 5.0 × 10⁻⁶ eV/atom and 0.01 eV/Å. To ensure an accurate representation of the ground state magnetic properties, the smearing width was set to 0.001 eV, allowing an initial random magnetic moment corresponding to a ferromagnetic state. It should be noted that, by symmetry and periodicity of the supercell approach, any calculated magnetic moment is ferromagnetic, since the spin alignment is replicated in all neighboring image supercells. Phonon calculations were conducted using finite displacement method [29,30,31,32], utilizing larger supercells size of 4 × 4 × 4, with a cut-off radius of 5.0 Å, and a k-grid sampling of 5 × 5 × 3.

3. Results and Discussion

3.1. Crystal Structure

The X₃Ru alloy crystallizes in the space group P4/mbm (Pearson symbol tP16) in a primitive tetragonal phase. The crystal structure consists of 16 atoms per unit cell (4Ru and 12X) with Ru on 4f (0, 0.5, 0.258), X1 on 4e (0, 0, 0.251), X2 on 4 g (0.231, 0.731, 0), and X3 on 4 h (0.29, 0.29, 0.5) sites.

The calculated equilibrium lattice constants, volume, heat of formation, magnetic moments, density, and tetragonality (c/a ratio) for these alloys are given in

Table 1. Optimized lattice constants (a, c), volume (V₀), ratio of c/a, heat of formation (ΔH_f), magnetic moment (µ_B), and density of tP16 X₃Ru (X = Sc − Zn) structures.

Alloy	a (Å)	c (Å)	V0 (Å3)	ΔHf (eV/Atom)	Magnetic Moments (µB/Atom)	Density (g/cm3)	c/a
Sc3Ru	6.23	8.97	348.07	−0.18	0.02	4.50	1.44
Ti3Ru	5.40	9.28	270.39	−0.41	0.89	6.01	1.72
V3Ru	5.73	8.11	266.05	−0.27	2.19	6.34	1.42
Cr3Ru Cr3Ru [16]	6.04 5.20	7.28	265.89	0.11 0.24	0.68	6.42	1.21
Mn3Ru	5.75	8.14	269.02	−1.61	3.25	6.56	1.42
Fe3Ru	5.07	8.02	206.15	0.07	0.95	8.65	1.58
Co3Ru	5.20	7.34	198.30	0.10	1.82	9.31	1.41
Ni3Ru	5.15	7.28	192.89	0.92	1.08	9.55	1.41
Cu3Ru	5.21	7.37	200.43	0.31	0.00	9.67	1.41
Zn3Ru	5.36	7.57	217.32	−0.23	0.00	9.08	1.41
tI16-Pt3Al [17]	5.4			−0.79			1.45
tP16-Pt3Ga [12]	5.5	7.80				6.14
L12-Pt3Al [37]	3.9

. We note that the proposed tP16 X₃Ru systems are relatively new; hence, their fundamental data are limited in the literature. In order to validate our model, we find that the calculated lattice constant of Cr₃Ru is in accord with the experiment [33] and literature [21]. Similarly, our calculated lattice constants of the individual metals are in agreement with experimental data [34] (see Supplementary Materials (references [16,35,36] are cited in the Supplementary Materials)).

For a given material system, the heat of formation ΔH_f describes the strength of atomic interactions, which indicate the thermodynamic stability of the material, and it is given by Equation (1):

$$\mathsf{\Delta }{H}_f=\left[\frac{E_{structure}-{\displaystyle \sum _i{x}_i{E}_i}}{x_T}\right]$$

(1)

where E_(structure) and E_(i) are the calculated equilibrium total energies of the alloy system and that of the individual elemental species i, with total atomic concentrations x_T. The calculated heat of formation for each alloy system is presented in Table 1. Negative heat of formation suggests thermodynamic stability, whereas a positive value indicates thermodynamic instability under equilibrium conditions. The calculated heat of formation is comparable to previous theoretical data in the case of Cr₃Ru, with a slight discrepancy that can be attributed to the use of different exchange correlation functionals. In this study, the on-site Hubbard U was incorporated into the generalized gradient approximation (GGA) in order to accurately describe the ground-state magnetic moments, whereas the previous studies employed plain GGA [16].

We find that the heat of formation for Sc₃Ru, Ti₃Ru, V₃Ru, Mn₃Ru, and Zn₃Ru is negative. This result indicates that these alloys are thermodynamically stable and hence can be synthesized under equilibrium experimental conditions. Conversely, Ni₃Ru, Co₃Ru, Cu₃Ru, and Fe₃Ru exhibit positive heat of formation, rendering them unstable, and can only be achieved under non-equilibrium conditions. It is noteworthy that the stable systems of Ti, V, Cr, Mn, Co, Ni, and Cu possess relatively high magnetic moments, which can potentially expand the use of these alloys in spintronic applications.

The density of a material describes its lightness/heaviness and holds an essential responsibility in aerospace applications specifically for spinning components in aircraft wings and turbine blades. It is calculated as follows:

$${\rho }^{cal}=\left[\frac{M_W\ast N}{Vol\ast A_0}\right]$$

(2)

where $M_W$ is the molecular weight of the alloy, $N$ is the number of atoms, $Vol$ is the volume of the unit cell, and $A_0$ is Avogadro’s number (6.022 × 10²³). Table 1 indicates that the densities for X₃Ru alloys range from 4.5 g/cm³ to 9.1 g/cm³. It is noted that the thermodynamically stable Sc₃Ru and Ti₃Ru have the lowest densities compared to the currently used L1₂-Ni₃Al density.

We have further determined the tetragonality (c/a) ratio of the X₃Ru alloys in order to evaluate possible phase changes that might exist in them. The tetragonality ratio is found to range from 1.21 to 1.72, with X = Co, Ni, Cu, and Zn possessing a constant value of 1.41, which indicates the absence of phase transition, while in Ti₃Ru, c/a = 1.7, indicating a possible phase change.

3.2. Mechanical Properties

3.2.1. Elastic Constants

Mechanical properties describe the resistance of crystal structures to external stress or strain applied to them. It was evaluated by applying a minimum strain to the unit cell to enable the total energy difference to be determined [38,39]. The elastic strain is calculated using Equation (3).

$$U=\frac{\mathsf{\Delta }E}{V_0}\frac{1}{2}{\displaystyle {\sum }_i^6}{\displaystyle {\sum }_j^6}{C}_{ij}{e}_i{e}_j$$

(3)

where $\mathsf{\Delta }E$ is the total energy change of the deformed unit cell relative to the initial cell, V₀ is the unstrained volume, $C_{ij}$ (i,j = 1 to 6) are the elastic constants, and $e_i$ or $e_j$ corresponds to the strain. In this study, the stress–strain method based on Hook’s law was used to calculate the elasticity and the elastic stiffness tensors, expressed as follows [40]:

$$\left\{{\delta }_{ij}\right\}=[C_{ij}]\left\{{\epsilon }_{ij}\right\}$$

(4)

where $\left\{{\delta }_{ij}\right\}$ and $\left\{{\epsilon }_{ij}\right\}$ are the stress and strain tensors, respectively. For primitive tetragonal X₃Ru (X = Sc − Zn) structures, there are six independent components (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, C₆₆) calculated at zero pressure. Based on the calculated elastic constants, the Bohr mechanical stability criteria were examined for all the structures as follows:

$$\begin{array}{l}C11>0,C33>0,C44>0,C66>0,C11-C12>0,\\ (C33+C11-2C13)>0,(2C11+2C12+C33+4C13)>0\end{array}$$

(5)

Table 2. Elastic constants (C_ij) and melting temperatures (MT) of X₃Ru (X = Sc − Zn) structures. The data for the currently used alloy L1₂ Ni₃Al structure are included for comparison.

Phase	C11	C12	C13	C33	C44	C66	MT ± 300 K
Sc3Ru	120.46	64.81	62.42	95.48	38.42	8.42	858.60
Ti3Ru	172.26	77.43	91.89	162.47	154.62	3.50	1114.49
V3Ru	192.82	24.35	73.52	154.00	58.86	47.65	1163.46
Cr3Ru	51.53	36.48	53.03	5.39	−3.65	15.73	516.68
Mn3Ru	190.79	29.40	83.55	130.89	77.64	25.21	1122.71
Fe3Ru	361.31	77.92	141.14	237.96	39.08	−42.41	1794.87
Co3Ru	330.98	68.68	170.26	270.15	130.85	36.12	1602.33
Ni3Ru	337.15	115.37	166.10	286.49	117.29	63.19	1795.19
Cu3Ru	262.26	87.26	153.90	192.07	94.72	28.84	1428.89
Zn3Ru	247.59	68.25	122.39	177.61	89.16	53.27	1363.19
L12-Ni3Al [45] L12-Pt3Al [37]	235 317	157 177			128 111		1691.00

presents the elastic constants (C_ij) and melting temperatures for X₃Ru compounds. Notably, all structures exhibit C_ij > 0, indicating compliance with the mechanical stability criteria. However, the Fe₃Ru and Cr₃Ru structures are mechanically unstable due to C₆₆ < 0 and C₄₄ < 0. A discernible correlation emerges between the elastic constants and the melting temperature of materials [41,42,43,44]. Noteworthy is the observation that the melting temperature of the mechanically stable Ni₃Ru surpasses that of the currently used L1₂ Ni₃Al (Table 2). Additionally, Ni₃Ru has the highest melting temperature, outperforming Cr₃Ru. Consequently, Ni₃Ru proves to be a viable candidate for high-temperature structural applications.

To assess the suitability of the X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) alloys for high-temperature structural application, the melting temperature (T_m) was calculated using the C_ij elastic parameters. For tetragonal structures, the melting temperature (T_m) was calculated from Equation (6):

$$T_m=354 \mathrm{K}+\left(\frac{450 \mathrm{K}}{Mbar}\right)\ast \left[\frac{1}{3}\left(2C_{11}+C_{33}\right)\right]\pm 300 \mathrm{K}$$

(6)

where Mbar is a mega bar. There exists a direct proportionality between the elastic constants and the melting temperature of materials, as evidenced in references [41,42,43,44]. The melting temperature data of the X₃Ru alloys range from 516.68 K to 1795.19 K, with Ni₃Ru having the highest melting temperature. Notably, the thermodynamically and mechanically stable Sc₃Ru, Ti₃Ru, V₃Ru, Mn₃Ru, and Zn₃Ru alloys exhibit melting temperatures of 858.60 K, 1114.49 K, 1163.46 K, 1122.71 K, and 1363.19 K, respectively, which are in the range of the currently utilized L1₂ Ni₃Al.

3.2.2. Bulk, Shear, and Young’s Moduli

Based on the C_ij, other elastic parameters can be calculated such as bulk modulus (B), shear modulus (G), and Young’s modulus (E), which are obtained from the Voigt, Reuss, and Hill approximations [46,47]. Bulk modulus (B) highlights the resistance to compression depending on the crystal structure of a material. High compressibility in materials attributes to large bulk modulus, while low bulk modulus constitutes low compressibility. For tetragonal structures, B is given as follows:

$$B_V=\frac{1}{9}\left((2C_{11}+C_{12})+C_{33}+4C_{13}\right)$$

(7)

and the Reuss bounds are given as follows:

$$B_R=\frac{C^2}{M}$$

(8)

where

$$M=C_{11}+C_{12}+2C_{33}-4C_{13}$$

(9)

with B_R, B = B_H, and B_V being the Bulk modulus for Reuss (R), Voigt (V), and Hill (H) approximations [47]. 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 follows:

$$G_V=\frac{1}{30}(M+3C_{11}-3C_{12}+12C_{44}+6C_{66})$$

(10)

where, $M=C_{11}+C_{12}+2C_{33}-4C_{13}$

$$G_R=15{\left[\left(\frac{18B_V}{C^2}\right)+\left(\frac{6}{C_{11}-C_{12}}\right)+\left(\frac{6}{C_{44}}\right)+\left(\frac{3}{C_{66}}\right)\right]}^{-1}$$

(11)

where $C^2=(C_{11}+C_{12})C_{33}-2C_{13}^2$.

G_R and G_V are the Reuss and Voigt bounds [46]. Young’s modulus and Poison’s ratio ʋ are independent of the type of crystal structure of a material and are given as follows:

$$\begin{gathered} E=\left(\frac{9B_X{G}_X}{3B_X+G_X}\right) \\ \upsilon =\left(\frac{3B_X-2G_X}{2(B_X+G_X)}\right) \end{gathered}$$

(12)

where X = V, R, and H approximations.

The Voigt, Reuss, and Hill approximations [46,47] utilize elastic constants to calculate macroscopic quantities, such as bulk (B) modulus, shear (G) modulus, and Young’s (E) modulus. For tetragonal structures, the bulk, shear, and Young’s modulus were derived from the elastic constants [48,49], as summarized in

Table 3. Bulk (GPa), shear (GPa), Young modulus (GPa), B_H/G_H, Poisson’s ratio ($\upsilon$), and Vickers hardness (H_V) of X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) structures, where V, R, and H stands for Voigt, Reuss, and Hill approximations. The UND = undefined. Data for L1₂-Ni₃Al and L1₂-Pt₃Al are included for comparison.

tP16 Phase	BV	GV	BR	GR	BH	GH	EH	BH/GH	$\mathit{\upsilon }$	HV
Sc3Ru	79.52	26.84	78.21	19.61	78.86	23.22	63.43	3.40	0.366	0.01
Ti3Ru	114.38	78.93	114.35	14.34	114.37	46.64	83.28	2.45	0.321	3.63
V3Ru	98.05	57.62	97.94	53.86	97.99	55.74	140.57	1.76	0.261	8.35
Cr3Ru	43.73	−0.58	45.44	12.43	44.58	−6.51	−20.53	−6.85	0.577	UND
Mn3Ru	100.61	57.16	100.56	42.22	100.58	49.69	127.99	2.02	0.288	5.61
Fe3Ru	186.78	47.18	184.48	96.20	185.63	71.69	190.54	2.59	0.329	5.00
Co3Ru	194.50	94.42	193.07	67.62	193.79	81.02	213.33	2.39	0.317	6.43
Ni3Ru	206.21	93.77	206.21	85.46	206.21	89.61	234.82	2.30	0.310	7.46
Cu3Ru	167.41	65.09	167.39	42.36	167.40	53.72	145.59	3.12	0.355	2.44
Zn3Ru	144.32	70.30	144.01	58.49	144.17	64.39	168.14	2.24	0.306	5.90
Li2-Ni3Al [45]					183.00	92.00	237.00	1.98	0.280
Li2-Pt3Al [37]								2.43

. The bulk and shear modulus represent volume and shape changes, respectively, while Young’s modulus indicates material stiffness. Table 3 illustrates that the Ni₃Ru structure demonstrates the highest bulk, shear, and Young’s modulus, whereas Mn₃Ru exhibits the lowest values. In addition, Cr₃Ru possesses a negative shear modulus and Young’s modulus. The negative shear and Young moduli are due to phase change leading to instability. This instability is also observed in ferro-elastic phase transformation [50]. The negative elastic modulus is due to Landau theory [51] when two local minima form in a strain energy function. Furthermore, solids with negative elastic modulus can be stabilized with sufficient constraint.

Consequently, Ni₃Ru displays superior resistance to volume and shape changes, emerging as the most compression-resistant material under zero-pressure conditions. At the same time, the G values in X₃Ru (X = Mn, Fe, Ni, Co, and Zn) alloys are lower than the B values, consistent with observations in reference [52]. This difference suggests that G limits the stability of these alloys, aligning with trends documented in the aforementioned reference.

The assessment of material ductility/brittleness can be effectively performed using the bulk-to-shear modulus ratio (B/G), as introduced by Pugh [53]. A high B/G ratio is associated with ductility, while a low ratio corresponds to brittleness, with the critical value between them being 1.75. The calculated B/G ratios are detailed in Table 3, revealing that all X₃Ru structures, except Cr₃Ru, exhibit ductility. Notably, the B/G ratios for all structures surpass those of L1₂-Ni₃Al, except for Cr₃Ru and V₃Ru. Furthermore, Poisson’s ratio (υ) serves as an additional indicator for ductility or brittleness, as outlined by Frantsevich [54]. Materials with υ > 0.3 are considered ductile, while those with υ < 0.3 are deemed brittle. The values for Poisson’s ratio for all structures exceed 0.3, aligning with the ductile nature predicted by the B/G ratios above, except for the Cr₃Ru structure.

Additionally, we conducted calculations for Vickers hardness, a metric that gauges a material’s resistance to localized plastic deformation caused by either mechanical indentation or abrasion. The hardness of the X₃Ru alloys is determined using the Vickers hardness equation [55]:

$$H_V=2{(K^{2}GH)}^{0.585}-3$$

(13)

where K represents the G_H/B_H ratio of the shear modulus to the bulk modulus. In mechanical properties, Vickers hardness assumes significant importance, particularly in high-temperature applications, where micro-cracks are prevalent. The calculated Vickers hardness values for the primitive tetragonal X₃Ru structures are outlined in Table 3. Notably, V₃Ru exhibits the highest hardness, whereas Cu₃Ru has the lowest value. Consequently, V₃Ru showcases superior resistance to localized plastic deformation induced by abrasion or mechanical indentation.

3.2.3. Elastic Anisotropy

Anisotropic behavior refers to a material’s directional dependence on a physical property, a phenomenon linked to micro-cracks, phase transformation, precipitation, and dislocation dynamics [40]. This directional dependence is represented by three elastic factors A₁, A₂, and A₃ corresponding to different crystallographic planes, as well as the universal anisotropic index (A^U) and the compression percentage (A_B) anisotropy, respectively. This holds true particularly for tetragonal structures as given in Equation (14).

$$\begin{array}{l}A1=\frac{2C66}{C11-C12}, A2=A3=\frac{4C44}{C11+C33-2C13}\\ A^U=5\frac{GV}{GR}+\frac{BV}{BR}-6 and AB=\frac{BV-BR}{BV+BR}\ast 100\%\end{array}$$

(14)

In the context of shear anisotropic factors, A₁, A₂, and A₃ correspond to the (001), (010), and (100) shear planes, respectively. In isotropic structural materials, these factors (A₁, A₂, and A₃) should ideally be equal to one. Similarly, the value of the universal anisotropic index (A^U) should be zero for isotropy. Deviations from these values, either one or zero, indicate the degree of elastic anisotropy in a crystal. The terms B_V, B_R, G_V, and G_R represent the Voigt and Reuss approximations in bulk modulus (B) and shear modulus (G). The maximum value for the compression percentage (A_B) is 100%, and the minimum is zero. A value of zero signifies isotropy, while 100% indicates maximum anisotropy. The calculated Zener anisotropy factor (A₁) values indicate that they are both less than and greater than one for A₂ and A^U, signifying that all the structures exhibit elastic anisotropy under zero-pressure conditions, as shown in

Table 4. Shear anisotropy factors (A₁, A₂), universal elastic anisotropy index (A^U), and the percentage of anisotropy (A_B) of X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) structures.

Phase	A1	A2	AU	AB
Sc3Ru	0.30	1.69	1.86	0.83
Ti3Ru	0.07	4.10	22.53	0.01
V3Ru	0.57	1.18	0.35	0.06
Cr3Ru	2.09	0.30	−4.80	−1.92
Mn3Ru	0.31	2.01	1.77	1.77
Fe3Ru	−0.30	0.49	−2.53	3.45
Co3Ru	0.28	2.01	1.99	0.37
Ni3Ru	0.57	1.61	0.49	0.00
Cu3Ru	0.33	2.59	2.68	0.01
Zn3Ru	0.59	1.98	1.01	0.11

. Additionally, the A_B value for the bulk modulus percentage is zero for Ni₃Ru, Ti₃Ru, and Cu₃Ru, suggesting isotropy in their structures, while the rest are characterized by elastic anisotropy.

3.3. Electronic and Magnetic Properties

To gain insights into the bonding characteristics of tP16 X₃Ru alloys (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn), their electronic and magnetic properties were investigated. The calculated magnetic moments of these alloys are detailed in Table 1. Notably, the thermodynamically and mechanically stable V₃Ru and Mn₃Ru exhibit the highest magnetic moments of 2.19 μ_B and 3.25 μ_B, respectively. Additionally, Co₃Ru and Ni₃Ru, which are mechanically stable, demonstrate magnetic moments of 1.82 μ_B and 1.08 μ_B.

The relationship between a material’s structural stability and magnetic moments relies on its electronic properties [56]. Figure 2 and Figure 3 show the total density of states (TDOS) and partial density of states (PDOS) for X₃Ru alloys in the tP16 phase. An electronic overlap is observed between the valence and conduction bands around the Fermi energy level of X₃Ru alloys, signifying a metallic nature. These findings are consistent with previous theoretical studies [16,18,24]. To evaluate the individual atom contributions to the TDOS, we calculated the PDOS for X₃Ru alloys, as depicted in Figure 2 and Figure 3. It is evident that the transition metal and Ru 3d orbitals are the major contributors to the total density of states around the fermi level, with the s, p orbitals making minimal contribution.

Moreover, the symmetry of the spin-up and spin-down bands is evident in Sc₃Ru, Cu₃Ru, and Zn₃Ru alloys. This symmetrical pattern signifies non-spin polarization, confirming the absence of magnetic moments (zero magnetic moments) in these alloys. In contrast, V₃Ru, Cr₃Ru, Mn₃Ru, Fe₃Ru, Co₃Ru, Ni₃Ru, and Zn₃Ru alloys exhibit asymmetric spin-up and spin-down channels. This asymmetry indicates electron spin polarization at the Fermi energy, leading to a spin polarization effect. This observation aligns with the presence of non-zero magnetic moments, as indicated in Table 1.

3.4. Phonon Dispersion Curves

Phonons are crucial for understanding the dynamic behavior and thermal conductivities, which are key areas in the advancement of novel materials research. Phonon dispersion curves offer valuable information about a material’s dynamic stability or instability by revealing both positive and negative phonon frequencies. Positive frequencies indicate dynamic stability, whereas negative frequencies signify dynamic instability within a compound [57,58]. Figure 4 shows the dispersion curves of tP16 X₃Ru (where X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) alloys, plotted across the highest symmetry k-points of the Brillouin zone. Among these alloys, the phonon dispersion curves of Cr₃Ru, Co₃Ru, Ni₃Ru, and Cu₃Ru demonstrate dynamic stability, as there are no imaginary frequencies present. In contrast, the phonon dispersion plots of Sc₃Ru, Ti₃Ru, Fe₃Ru, V₃Ru, Zn₃Ru, and Mn₃Ru display negative phonon modes, indicating dynamic instability and, therefore, which may be a limiting factor in applications where dynamic stability is vital.

4. Conclusions

In summary, we have utilized first-principles calculations to investigate the structural, electronic, and mechanical properties of tetragonal X₃Ru (X = Sc − Zn) binary alloys in the tP16 phase. Analysis of the heat of formation results indicates that Mn₃Ru, Sc₃Ru, Ti₃Ru, V₃Ru, and Zn₃Ru alloys are thermodynamically stable. Furthermore, the electronic density of states reveals a strong overlap between the valence and conduction bands in Sc₃Ru, Ti₃Ru, V₃Ru, Mn₃Ru, Fe₃Ru, Co₃Ru, Ni₃Ru, Cu₃Ru, and Zn₃Ru alloys, which exhibit a metallic character. All alloys except Cr₃Ru demonstrate ductility, as evidenced by high B/G ratios exceeding 1.75, along with high melting temperatures. We observe a direct correlation between these elastic moduli (B, G, and E) and the melting temperatures of the proposed structures, consistent with previous theoretical data. Phonon calculations indicate that Cr₃Ru, Co₃Ru, Ni₃Ru, and Cu₃Ru are dynamically stable. Our findings suggest that tetragonal Ru-based alloys are promising candidates for ultra-high-temperature structural applications.