Insight into Physical and Thermodynamic Properties of X₃Ir (X = Ti, V, Cr, Nb and Mo) Compounds Influenced by Refractory Elements: A First-Principles Calculation

Abstract

The effects of refractory metals on physical and thermodynamic properties of X₃Ir (X = Ti, V, Cr, Nb and Mo) compounds were investigated using local density approximation (LDA) and generalized gradient approximation (GGA) methods within the first-principles calculations based on density functional theory. The optimized lattice parameters were both in good compliance with the experimental parameters. The GGA method could achieve an improved structural optimization compared to the LDA method, and thus was utilized to predict the elastic, thermodynamic and electronic properties of X₃Ir (X = Ti, V, Cr, Nb and Mo) compounds. The calculated mechanical properties (i.e., elastic constants, elastic moduli and elastic anisotropic behaviors) were rationalized and discussed in these intermetallics. For instance, the derived bulk moduli exhibited the sequence of Ti₃Ir < Nb₃Ir < V₃Ir < Cr₃Ir < Mo₃Ir. This behavior was discussed in terms of the volume of unit cell and electron density. Furthermore, Debye temperatures were derived and were found to show good consistency with the experimental values, indicating the precision of our calculations. Finally, the electronic structures were analyzed to explain the ductile essences in the iridium compounds.

1. Introduction

Ir-based superalloys have received intensive interest in the last decades due to their high melting temperature as well as their improved strength, oxidation resistance and corrosion resistance at higher temperatures [1,2,3,4,5]. Consequently, these intermetallics can be deemed a suitable choice for high-temperature applications. For example, the cubic L1₂ intermetallic compounds Ir₃X (X = Ti, Zr, Hf, Nb and Ta) containing refractory elements could be proposed as “refractory superalloys” based on their higher melting points and superior mechanical properties at higher temperatures [3,4,5]. Terada et al. [6] conducted measurements on thermal properties (i.e., thermal conductivity and thermal expansion) from 300 to 1100 K, and found that the L1₂ Ir₃X (X = Ti, Zr, Hf, Nb and Ta) compounds were characterized by a larger thermal conductivity and a smaller thermal expansion. Chen et al. [7] exhibited the elastic constants and moduli of binary L1₂ Ir-based compounds at ground states by first-principles calculations, and reported the higher elastic moduli of these compounds together with their brittle characteristics in nature. Liu et al. [8] studied the elastic and thermodynamic properties of Ir₃Nb and Ir₃V under varying pressure (0–50 GPa) and temperature (0–1200 K), and found that both compounds were stable without phase transformations.

Meanwhile, the typical refractory intermetallics should also include A15 cubic structure compounds with refractory metal elements. For example, Pan et al. [9] reported the mechanical and electronic properties of Nb₃Si using the first-principles method. By combining the first-principles method with quasi harmonic approximation, Papadimitriou et al. [10,11,12] critically investigated the mechanical and thermodynamic properties of Nb₃X (X = Al, Ge, Si and Sn). Chihi et al. [13] theoretically evaluated the elastic and thermodynamic properties of V₃X (X = Si, Ge and Sn) intermetallics utilizing the first-principles calculations. Jalborg et al. [14] determined the electronic structure of V₃Ir, V₃Pt and V₃Au using the self-consistent semi relativistic linear muffin-tin orbital (LMTO) band calculations. Paduani et al. [15] investigated the chemical bonding behavior and estimated the electron-phonon coupling constants of V₃X (X = Ni, Pd, Pt) by the full-potential linearized augmented-plane-wave (FP-LAPW) method.

Therefore, the Ir-based intermetallics with A15 crystal structure should also have the potential to be applied as structural materials. These compounds have been studied for their structural and electronic properties. For instance, Standanmann et al. [16] determined the lattice parameters of A15 X₃Ir (X = Ti, V, Cr, Nb and Mo) compounds. Meschel et al. [17] reported the experimental standard enthalpy of formation for V₃Ir. Paduani and Kuhnen [18] studied the band structure and Fermi surface of V₃Ir using the FP-LAPW method, and discussed Knight shift behavior in the compound. Paduani et al. [19] reported the electronic properties of Nb₃Ir via FP-LAPW calculations. Nevertheless, to our knowledge, the elastic and thermodynamic properties of X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics have rarely been discussed.

This research is divided into the following parts. In the second section, the computational methods of X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics are offered in detail. In the third section, the results and discussions are presented and discussed based on the effects of refractory metals on the physical and thermodynamic properties of X₃Ir compounds, including structural properties, elastic propertie, anisotropic behaviors, anisotropic sound velocities, Debye temperatures, and electronic structures. In the fourth section, the conclusions are drawn and presented in detail.

2. Materials and Methods

The first-principles calculations were performed using the CASTEP code, which is based on the pseudopotential plane-wave within density functional theory [20,21]. Using the ultrasoft pseudopotential [22] to model the ion-electron exchange-correlation, both the generalized gradient approximation (GGA) with the function proposed by Perdew, Burke and Ernzer (PBE) [23,24] and the local density approximation (LDA) with Ceperley–Alder form [25] were used. Additionally, the basis atom states were set as: Ti3s³3p⁶3d³5s², V3s²3p⁶3d³5s², Cr3s²3p⁶3d⁵4s¹, Nb4s²4p⁶4d⁴5s¹, Mo4s²4p⁶4d⁵5s¹ and Ir5d⁷6s². Through a series of tests, the cutoff energy of 400 eV was determined. In addition, a 10 × 10 × 10 k-point mesh in the Brillouin zone was set for the special points sampling integration for the intermetallics. Both lattice constants and atom coordinates should be optimized via minimizing the total energy. Furthermore, the Brodyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme was used for the geometric optimization [26,27]. Overall, the maximum stress has to be within 0.02 GPa, the maximum ionic force has to be within 0.01 eV/Å, the maximum ionic displacement has to be within 5.0 × 10⁻⁴ Å and the difference of the total energy has to be within 5.0 × 10⁻⁶ eV/atom for the geometrical optimization. Finally, the total energy and electronic structure were calculated, followed by cell optimization with a self-consistent field tolerance (5.0 × 10⁻⁷ eV/atom). Using the corrected tetrahedron Blöchl method, the total energies at equilibrium structures were derived [28].

3. Results

3.1. Structural Properties

The X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics have a A15 cubic structure with the cP8 (No. 223) space group. In a unit cell, six X atoms and two Ir atoms are dominated at the sites of 6c (0.25, 0, 0.5) and 2a (0, 0, 0), respectively. For the sake of performing structural property optimizations on IrX₃ compounds, the GGA method as well as the LDA method were utilized. The results are exhibited in

Table 1. The optimized and experimental lattice parameters, the calculated deviations and densities for X₃Ir (X = Ti, V, Cr, Nb and Mo) compounds.

Compounds	a0 (Å)	aexp (Å)	Calculated Deviation (%)	Density (g/cm3)
Ti3Ir	5.010 a	5.012 c	−0.041 a	8.872 a
	4.901 b		−2.223 b	9.479 b
V3Ir	4.7842 a	4.7876 d	−0.072 a	10.463 a
	4.6913 b		−2.012 b	11.099 b
Cr3Ir	4.652 a	4.685 e	−0.712 a	11.489 a
	4.651 b		−0.732 b	11.496 b
Nb3Ir	5.1585 a	5.135 f	0.457 a	11.394 a
	5.0777 b		−1.116 b	11.946 b
Mo3Ir	4.9874 a	4.9703 g	0.344 a	12.986 a
	4.9199 b		−1.014 b	13.387 b

^a: From the GGA method in this work: Theoretical values. ^b: From the LDA method in this work: Theoretical values. ^c: From Reference [29]: Experimental values. ^d: From Reference [30]: Experimental values. ^e: From Reference [31]: Experimental values. ^f: From Reference [32]: Experimental values. ^g: From Reference [33]: Experimental values.

, showing that the derived lattice constants using both methods are close to the experimental values [29,30,31,32,33].

In most compounds, the lattice constants generated by the GGA method offer much smaller calculated deviations than the LDA method (Table 1). For instance, the a₀(GGA) has the calculated deviation of −0.041%, and a₀(LDA) has the calculated deviation of −2.223% in comparison with a_exp in Ti₃Ir. Clearly, the GGA method exhibited better reliability for structural optimization, and thus giving a superior quality of calculation over the LDA method. As a result, the following calculation work was accomplished only by GGA method.

3.2. Elastic Constants

In the crystalline materials, the elastic constant represented the capability of resisting the exterior imposed stress. In such manners, a whole package of elastic constants was achieved to characterize mechanical properties of crystals. By imposing small strains to the equilibrium unit cell, elastic constants can be computed by determining the corresponding variations in the total energy. Theoretically, the elastic strain energy was formulated by Equation (1):

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

(1)

where V₀ represents the cell volume at equilibrium state; ΔE represents the energy difference; e_i and e_j represent the strains; C_ij (ij = 1, 2, 3, 4, 5 and 6) represent the elastic constants.

In cubic structures, C₁₁, C₁₂ and C₄₄ are nonzero elastic constants without mutual dependence. In

Table 2. The elastic constant (C_ij), Cauthy pressure (C₁₂-C₄₄), bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (v) and B/G ratio for X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics.

Compounds	Cij			C12–C44 (GPa)	B (GPa)	G (GPa)	E (GPa)	v	B/G
	C11 (GPa)	C44 (GPa)	C12 (GPa)
Ti3Ir	183.8	52.8	166.0	114.2	171.9	26.5	75.7	0.427	6.483
	207.2 a	48.8 a	153.1 a	104.3 a	171.1 a	38.5 a	107.4 a	0.395 a	4.446 a
V3Ir	471.5	109.4	136.8	27.4	248.4	129.8	331.7	0.277	1.913
					279.89 b
Cr3Ir	478.6	89.6	190.2	100.4	286.3	108.5	289.0	0.332	2.639
Nb3Ir	433.7	84.5	123.7	39.2	227.0	108.0	279.7	0.295	2.102
					216.4 c
Mo3Ir	512.7	87.6	175.8	88.2	288.1	114.2	302.6	0.325	2.523
					297.5 d

^a: From Reference [34]: Theoretical values. ^b: From Reference [18]: Theoretical values. ^c: From Reference [35]: Theoretical values. ^d: From Reference [36]: Theoretical values.

, the calculated elastic constants (C_ij) for X₃Ir intermetallics are shown, accompanied by the available theoretical values [18,34,35,36] for comparison.

In the elastic constant, a larger C₄₄ corresponds to a stronger resistance to monoclinic shear in the (100) plane, and therefore symbolizes a larger shear modulus. For instance, V₃Ir has the largest C₄₄ and shear modulus, exhibiting a superior capability to resist the shear stress. Furthermore, the compressive resistance along the x axis is reflected by C₁₁. For each compound, the derived C₁₁ exhibited the biggest value among elastic constants, suggesting that is has the greatest incompressibility under x uniaxial stress. Among the compounds, Mo₃Ir was the least compressible along the x axis because it had the biggest C₁₁ (512.7 GPa), and Ti₃Ir was the most compressible owing to its small C₁₁ (183.8 GPa)

Utilizing Born’s criteria [37,38], the essence of mechanical stability should be evaluated for cubic crystals:

C₁₁ > 0; C₄₄ > 0; C₁₁ − C₁₂ > 0; C₁₁ + 2 C₁₂ > 0

(2)

Using the values in Table 2, all X₃Ir compounds were found to have mechanical stability by satisfying the Born’s criteria at the ground state.

The Cauchy pressure, illustrated as (C₁₂–C₄₄) [39], should be an effective indicator to evaluate the ductile/brittle nature of cubic crystals. In Pettifor’s work [40], a more positive Cauchy pressure symbolized better ductility in the compound [41]. In Table 2, the Cauchy pressures for X₃Ir compounds were all positive in the order of V₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < Ti₃Ir, which means that X₃Ir compounds are naturally ductile. Such a result is in good compliance with the Cauchy pressure of Ti₃Ir provided by Rajagopalan [34]. Similarly, other A15 cubic crystals (i.e., V₃X (X = Si and Ge) [13], Nb₃X (X = Al, Ge, Si and Sn) [10] and Nb₃X (X = Al, Ga, In, Sn and Sb) [42]) have ductile characters owing to their positive Cauchy pressures.

3.3. Elastic Properties

Once the elastic constants were achieved, the elastic moduli (i.e., bulk modulus (B) and shear modulus (G)) could be computed by means of the Voigt–Reuss–Hill (VRH) method [43]. In cubic structures, the equations are exhibited as [44,45,46]:

$$B_V=B_R=\frac{1}{3}(C_{11}+2C_{12})$$

(3a)

$$G_V=\frac{1}{5}(C_{11}-C_{12}+3C_{44})$$

(3b)

$$G_R=\frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})}$$

(3c)

$$B=\frac{B_V+B_G}{2}$$

(3d)

$$G=\frac{G_V+G_G}{2}$$

(3e)

When the elastic moduli are achieved, the Young’s modulus (E) and Poisson’s ratio (ν) should be calculated in the second step [47]:

$$E=\frac{9BG}{3B+G}$$

(3f)

$$\nu =\frac{3B-2G}{2(3B+G)}$$

(3g)

Lastly, the computed elastic moduli for X₃Ir compounds using the VRH method are tabulated in Table 2 in combination with the available theoretical results for comparison [18,34,35,36]. Comparably, our calculated bulk moduli showed satisfactory agreement with the theoretical values for Ti₃Ir [34], Nb₃Ir [35] and Mo₃Ir [36], and a value slightly smaller smaller than the theoretical one for V₃Ir [18].

Analytically, the resisting capability against volume fluctuation under pressure is determined by the bulk modulus. For X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics, the bulk moduli showed the sequence of Ti₃Ir < Nb₃Ir < V₃Ir < Cr₃Ir < Mo₃Ir. In References [48,49], a larger equilibrium cell volume was reported to correspond to a lower bulk modulus in the cubic crystal. Observably, such a conclusion is effective when the alloying elements are in the same cycle of the periodic table of elements (Figure 1a). For example, the bulk moduli are improved in the order of Ti₃Ir < V₃Ir < Cr₃Ir depending on the reduced equilibrium cell volume. Similarly, the bulk modulus of Nb3Ir is smaller than that of Mo₃Ir with the larger equilibrium cell volume of Nb₃Ir (Figure 1a). Nevertheless, when the alloying elements are in the same group of the periodic table, the conclusion is valid for Nb₃Ir < V₃Ir, but ineffective for Cr₃Ir < Mo₃Ir, where Mo₃Ir actually has a larger equilibrium cell volume.

In order to further illustrate the relationship between the equilibrium cell volume and the bulk modulus of X₃Ir intermetallics, the linear dependence of the electron density on the bulk modulus is exhibited in Figure 1b. Clearly, dividing the bonding valence (ZB) by the volume per atom (VM) can deduce the electron density (n) in metallic compounds [50]. For X₃Ir compounds, the electron density (n) can be formulated as:

$$n(X_{3}Ir)=Z_B(X_{3}Ir)/V_M(X_{3}Ir)$$

(4a)

where VM(X₃Ir) represents the volume (cm³/mol) of X₃Ir.

Rationalized by Vegard’s law [51], ZB(X₃Ir) showed a bonding valence in (el/atom), and the Reference [52] tabulated the bonding valence of the pure element:

$$Z_B(X_{3}Ir)=(3Z_B(X)+Z_B(Ir))/4$$

(4b)

Using this method, the linear dependence of the electron density on the bulk modulus was identified through the calculated values. Conclusively, it is more precise to rationalize the bulk modulus from the electron density, rather than the equilibrium cell volume.

Shear modulus (G) symbolizes the capability to resist shape fluctuation [44], and Young’s modulus (E) is a measurement of resistance to tension and compression in the elastic regime [53]. Notably, there is a linear dependence of the Young’s modulus on the shear modulus following the order of Ti₃Ir < Nb₃Ir < Cr₃Ir < Mo₃Ir < V₃Ir (Figure 2).

Overall, the bigger bulk modulus over shear modulus for each X₃Ir compound should reflect that the X₃Ir compound has an improved capability to resist volume fluctuation over shape fluctuation (Table 2). This conclusion complies well with the available data regarding the dependence of the bulk modulus on the shear modulus in other A15 intermetallics, i.e., Ti₃Ir (X = Ir, Pt and Au) [34], V₃X (X = Si and Ge) [13], Nb₃X (X = Al, Ga, In, Sn and Sb) [42] and Mo₃X (X = Si and Ge) [54].

The Poisson’s ratio (−1 ≤ v ≤ 0.5) is used to quantify the stability of crystals against the shear deformation [55]. Materials with improved plasticity should possess a larger Poisson’s ratio. The X₃Ir compounds have Poisson’s ratios in the order of V₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < Ti₃Ir. This means that Ti₃Ir should be the most ductile, while V₃Ir is most brittle. Additionally, the Poisson’s ratio provides information on the bonding forces in solids [56]. The lower and higher limits are 0.25 and 0.5 for the central force in a solid, respectively. For XIr₃ intermetallics, the interatomic forces of intermetallics should be central forces, since all the obtained values are located on this scale (Table 2).

The B/G ratio formulated by Pugh [57] is commonly adopted to quantitatively estimate the brittle or ductile essence of metallic compounds. The critical B/G ratio to distinguish the brittle from ductile material is 1.75. A smaller value is connected with the brittle nature, whereas a larger B/G ratio is related to ductility. Furthermore, the revised Cauchy pressure ((C₁₂–C₄₄)/E) [58] was plotted against the B/G ratio to clarify the extent of ductility intuitively (Figure 3). As a result, the ductility was found to be enhanced in the order of V₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < Ti₃Ir. This conclusion agrees well with the analysis of the Poisson’s ratio. Clearly, Ti₃Ir should be much more ductile than the other X₃Ir compounds (Figure 3).

3.4. Elastic Anisotropy

The universal anisotropic index (A^U) can be used to evaluate the elastic anisotropy, which is also referred to as the probability to introduce materials’ micro-cracks [59]. The index can be calculated via Equation (5) [60]:

$$A^U=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6$$

(5)

where B_V (B_R) and G_V (G_R) represent the symbols of the bulk modulus and the shear modulus at Voigt (Reuss) bounds, respectively.

In the calculated elastic anisotropies, B_V/B_R, should be equal to 1 for cubic crystals (

Table 3. The computed bulk and shear moduli at Voigt (Reuss) bounds, and the universal anisotropic index (A^U) for X₃Ir compounds.

Compounds	BV	BR	GV	GR	BV/BR	GV/GR	AU
Ti3Ir	171.9	171.9	35.3	17.8	1	1.984	4.922
V3Ir	248.4	248.4	132.6	127.0	1	1.044	0.220
Cr3Ir	286.3	286.3	111.4	105.6	1	1.055	0.277
Nb3Ir	227.0	227.0	112.7	103.3	1	1.091	0.455
Mo3Ir	288.1	288.1	119.9	108.4	1	1.106	0.532

).

Indeed, G_V/G_R has a decisive effect on the universal anisotropic index (A^U). Figure 4 shows that the universal anisotropic index increases linearly with the increment of the G_V/G_R value. A compound with a smaller A^U represents a weaker extent of anisotropy. Therefore, the universal anisotropy was found to be reduced in the sequence of V₃Ir < Cr₃Ir < Nb₃Ir < Mo₃Ir < Ti₃Ir. Generally, Ti₃Ir has the largest universal anisotropy, and V₃Ir has the smallest. Because the experimental value is lacking for comparison in these compounds, this calculation has to be evaluated in later research.

In addition, to further describe the anisotropy of X₃Ir compounds, the directional dependence of the reciprocal of the Young’s modulus was constructed for a three-dimensional (3D) surface according to Equation (6) [48]:

$$\frac{1}{E}=S_{11}-2(S_{11}-S_{12}-\frac{S_{44}}{2})(l_1^2{l}_2^2+l_2^2{l}_2^3+l_1^2{l}_3^2)$$

(6)

where S_ij represents the usual elastic compliance constant obtained from the inverse of the matrix of the elastic constant; l₁, l₂ and l₃ represent the direction cosines in the sphere coordination.

If a crystal has ideal isotropic performance, the 3D directional dependence of the Young’s modulus would show a spherical shape. In fact, the extent of deviation from the spherical shape symbolizes the anisotropic extent. In Figure 5, X₃Ir compounds showed the distinctive 3D figures of Young’s moduli with various deviations from a sphere. This confirmed that X₃Ir compounds have anisotropic behaviors. Obviously, Ti₃Ir shows the largest deviation from the sphere shape along the <111> direction. On the contrary, other X₃Ir compounds exhibited different forms of deviation, and the most visible deviations were observed along the zone axes. Finally, the extent of the elastic anisotropy for X₃Ir obeyed the sequence of V₃Ir < Cr₃Ir < Nb₃Ir < Mo₃Ir < Ti₃Ir. This conclusion complies well with the result obtained from the universal anisotropic index.

3.5. Anisotropic Sound Velocity and Debye Temperature

In the crystalline material, the sound velocities should depend on the crystalline symmetry in combination with the propagating direction. In the cubic structure, [111], [110] and [001] directions exhibited the pure transverse and longitudinal modes, accordingly. Regarding other directions, both quasi-transverse and quasi-longitudinal waves can work as the main sound propagating modes. Therefore, the sound velocities formulated in the principal directions are listed as follows [61]:

$$\begin{gathered} [100]v_l=\sqrt{C_{11}/\rho };[010]v_{t1}^{}=[001]v_{t2}^{}=\sqrt{C_{44}/\rho } \\ [110]v_l=\sqrt{(C_{11}+C_{12}+C_{44})/(2\rho )} \\ [1\stackrel{-}{1}0]v_{t1}=\sqrt{(C_{11}-C_{12})/\rho };[001]v_{t2}^{}=\sqrt{C_{44}/\rho } \\ [111]v_l=\sqrt{(C_{11}+2C_{12}+4C_{44})/(3\rho )} \\ [11\stackrel{-}{2}]v_{t1}=[11\stackrel{-}{2}]v_{t2}=\sqrt{(C_{11}-C_{12}+C_{44})/(3\rho )} \end{gathered}$$

(7)

where v_l (v_t) represents the longitudinal (transverse) sound velocity; ρ represents the density (see Table 1).

Overall, the longitudinal sound velocity along the [100] direction was only decided by C₁₁. The transverse modes along [010] and [001] directions were dependent on C₄₄. The longitudinal sound velocities along both the [110] and [111] directions were influenced by C₁₁, C₁₂ and C₄₄.

Along the [100], [110] and [111] directions, the longitudinal sound velocities and the transverse sound velocities are exhibited in

Table 4. The anisotropic sound velocities (m/s), average sound velocities (m/s) and Debye temperatures (K) for X₃Ir intermetallics.

Crystalline Orientation		Ti3Ir	V3Ir	Cr3Ir	Nb3Ir	Mo3Ir
[111]	[111]vl	5226.6	6138.7	5942.8	5460.3	5583.6
	[11$\stackrel{-}{2}$]vt1,2	1629.2	3762.0	3311.6	3397.4	3301.0
[110]	[110]vl	4763.3	5856.6	5744.8	5307.3	5466.2
	[1$\stackrel{-}{1}$0]vt1	1416.9	5656.7	5010.3	5216.1	5093.6
	[001]vt2	2440.3	3234.2	2792.4	2723.7	2597.2
[100]	[100]vl	4551.4	6713.4	6454.1	6169.4	6283.3
	[010]vt1	2440.3	3234.2	2792.4	2723.7	2597.2
	[001]vt2	2440.3	3234.2	2792.4	2723.7	2597.2
	vL	4833.4	6346.8	6124.7	5706.4	5822.9
	vT	1728.8	3522.6	3073.2	3079.0	2965.2
	vD	1962.7	3920.4	3444.0	3434.1	3320.0
	Θ	233.3	487.9	441.0	396.4	397.8
		238 a, 262.6 b	460 ± 10 c, 445 d	449 e	409 ± 8 c, 377 d	452 f, 325 g, 497.06 h

^a: From Reference [64]: Experimental values. ^b: From Reference [34]: Theoretical values. ^c: From Reference [65]: Experimental values. ^d: From Reference [66]: Experimental values. ^e: From Reference [67]: Experimental values. ^f: From Reference [16]: Experimental values. ^g: From Reference [68]: Experimental values. ^h: From Reference [36]: Theoretical values.

for each X₃Ir compound. For each compound, the longitudinal sound velocity followed the rising sequence of [100] < [110] < [111]. The sound velocities showed anisotropic properties, further confirming the elastic anisotropic behaviors of the compounds.

These theoretically computed physical properties (i.e., elastic moduli and Poisson’s ratio) and structural properties (i.e., density) should be adopted to calculate the Debye temperature (Θ) using the following formula [54,62,63]:

$$\mathrm{\Theta }=\frac{h}{k}{[\frac{3n}{4\pi }(\frac{N_A\rho }{M})]}^{\frac{1}{3}}{V}_D$$

(8a)

where ρ represents the density (see Table 1); h represents the Planck’s constant (h = 6.626 × 10⁻³⁴ J/s); k represents the Boltzmann’s constant (k = 1.381 × 10⁻²³ J/K); n represents the number of atoms per formula unit; N_A represents the Avogadro’s number (N_A = 6.023 × 10⁻²³/mol); M represents the molecular weight (M(Ti₃Ir) = 335.8 g/mol, M(V₃Ir) = 345 g/mol, M(Cr₃Ir) = 347.8 g/mol, M(Nb₃Ir) = 470.9 g/mol, M(Mo₃Ir) = 480 g/mol); v_D represents the average sound velocity in polycrystalline materials. The latter is formulated as:

$$v_D={[\frac{1}{3}(\frac{1}{V_L^3}+\frac{2}{V_T^3})]}^{-\frac{1}{3}}$$

(8b)

where v_T and v_L represent the transverse and longitudinal sound velocities, respectively, as formulated by the equations below:

$$v_T=\sqrt{\frac{G}{\rho }}$$

(8c)

$$v_L=\sqrt{\frac{B+\frac{4}{3}G}{\rho }}$$

(8d)

For each X₃Ir compound, the derived Debye temperature (Θ) is tabulated in Table 4. Also, the published experimental [16,64,65,66,67,68] and theoretical [34,36] values are included for comparison.

Generally, V₃Ir had the largest Debye temperature, and Ti₃Ir had the smallest. The calculated Debye temperatures were in the order of Ti₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < V₃Ir. Clearly, our results revealed the reduced tendency of Debye temperatures with the M atom in the same group, i.e., Cr (Lighter element) and Mo (Heavier element) are from Group-VIB, and V (Lighter element) and Nb (Heavier element) are from Group-VB.

Comparably, the obtained Debye temperatures for Ti₃Ir, V₃Ir, Cr₃Ir and Nb₃Ir were all in excellent agreement with the available experimental results [16,64,65,66,67,68]. For instance, the calculated Θ was 233 K for Ti₃Ir. This agrees well with the experimental values reported by Junod et al. [64] with the calculated deviation of 2.01%. Notably, the Debye temperature reported by Rajagopalan et al. [34] had the calculated deviation of 10.3%, indicating the poor quality of the prediction in this work. However, for Mo₃Ir, the published experimental [16,68] and theoretical [36] Debye temperatures were quite scattered, although our calculated values were closer to the experimental values from Staudenmann’s report [16]. Nevertheless, more works are required on this compound.

Because both structural parameters and elastic moduli are incorporated to calculate the Debye temperature, the superior quality of our calculation on these structural and elastic parameters using the GGA method was evidenced by the smaller differences between the estimated and experimental values of Debye temperatures.

3.6. Electronic Structures

Figure 6a–e exhibit the density of states (DOS) spectra representing the calculated electronic structures for X₃Ir compounds. The DOS spectra for these A15 cubic phases were similar to each other. In a typical DOS spectrum, there are normally three regions, including the lower electron band, the upper electron band, and the conduction unoccupied states around the Fermi level (E_F). For example (Figure 6a), the lower the electron band was mainly contributed by 4s electrons of Ti ranging from −55 to −57.5 eV. The upper electron band was occupied by 3p electrons of Ti ranging from −32 to −35 eV. Around the Fermi level, the conduction unoccupied states were created through the hybridization of mainly Ti3d electrons with Ti3p, Ir5d and Ir4p electrons. X₃Ir compounds were plotted around the Fermi level at zero in all the total DOS (TDOS) and partial DOS (PDOS) spectra. Clearly, no any energy gap can be found near the Fermi level. Therefore, their nature of metallicity was confirmed.

Furthermore, the electron density values can provide quantitative evidence of the metallic nature in the bonding characteristics. Even at the Fermi surface, the electron density values were much larger than zero. According to the electronic Fermi liquid theory [69], the metallicity of the compound has to be estimated using Equation (9) [70]:

$$f_m=\frac{n_m}{n_e}=\frac{k_{B}T{D}_f}{n_e}=\frac{0.026D_f}{n_e}$$

(9)

where

• k_B represents the Boltzmann constant (k = 1.381 × 10⁻²³ J/K);
• T represents the absolute temperature;
• D_f represents the DOS value at the Fermi level;
• n_m and n_e represent the thermally excited electrons and valence electron density of the cell, respectively;
• n_e is calculated by n_e = N/V_cell (N represents the total number of valence electrons; V_cell represents the cell volume).

Using the calculated metallicity values (f_m), the correlation between metallicity and Poisson’s ratios can be constructed for X₃Ir compounds (Figure 6f). It was found that the Poisson’s ratios were diminished with the reduction in metallicity in compounds with the order of V₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < Ti₃Ir. This indicated that a compound with higher metallicity in its bonds should possess better ductility.

4. Conclusions

The effects of refractory metals on physical and thermodynamic properties of X₃Ir (X = Ti, V, Cr, Nb and Mo) intermetallics were investigated utilizing first-principles calculations. The conclusions are listed as follows:

(1)

Using the GGA method to structurally optimized the unit cell, smaller calculation deviations for lattice constants were achieved as compared to those achieved using the LDA method.

(2)

The calculated bulk moduli exhibited the increasing sequence of Ti₃Ir < Nb₃Ir < V₃Ir < Cr₃Ir < Mo₃Ir. Furthermore, the bulk moduli showed a linear relationship with electron densities. The Young’s modulus showed a linear dependence on shear modulus following the order of Ti₃Ir < Nb₃Ir < Cr₃Ir < Mo₃Ir < V₃Ir.

(3)

Based on the discussions on the Cauchy pressure, Poisson’s ratio and B/G ratio, the ductile essence was found to be enhanced in the order of V₃Ir < Nb₃Ir < Mo₃Ir < Cr₃Ir < Ti₃Ir.

(4)

For X₃Ir compounds, the extent of the elastic anisotropy for X₃Ir obeyed the increasing sequence of V₃Ir < Cr₃Ir < Nb₃Ir < Mo₃Ir < Ti₃Ir via the analyses of the universal anisotropic indexes and 3D surface constructions.

(5)

The Debye temperatures obtained for Ti₃Ir, V₃Ir, Cr₃Ir and Nb₃Ir were all in good agreement with the results from experiments. Such good compliance proved the superior quality of our calculations of the structural and elastic properties, since the computation of Debye temperature is concerned with both structural and elastic parameters.

(6)

The calculated electronic structures for X₃Ir compounds showed similar features in the DOS spectra. Furthermore, the metallicity of the compounds was calculated, and was correlated with the Poisson’s ratios. This indicated that a compound with higher metallicity in its bonds should possess better ductility.