Theoretical Approach of Stability and Mechanical Properties in (TiZrHf)_1−x(AB)_x (AB = NbTa, NbMo, MoTa) Refractory High-Entropy Alloys

Abstract

The stability and mechanical properties of (TiZrHf)_1−x(AB)_x (AB = NbTa, NbMo, MoTa) refractory high-entropy alloys have been investigated by combining the first-principles with special quasi-random structure (SQS) method. It is found that with the increase in solute concentration x, the ΔH_mix of (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) linearly decreases, whereas both ΔH_mix and ΔS_mix of (TiZrHf)_1−x(NbTa)_x increase initially and subsequently decrease, with the crossover occurring at x = 0.56. The ΔH_mix of (TiZrHf)_1−x(NbTa)_x and (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) alloys are larger and lower than that of TiZrHf, respectively, while the ΔS_mix of all (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf. The formation possibility parameter Ω of all (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) first decreases sharply, followed by a gradual decrease. And the local lattice distortion (LLD) parameter δ remains relatively stable around x = 0.56 for all cases, after which it decreases sharply until x = 0.89. The δ value of (TiZrHf)_1−x(AB)_x is higher than that of TiZrHf for x < 0.56 but becomes lower beyond this composition. The valence electron concentration (VEC), a possible indicator for a single-phase solution, of (TiZrHf)_1−x(AB)_x increases nearly linearly, while the formation energy ΔH_f of (TiZrHf)_1−x(AB)_x shows the opposite tendency, except for (TiZrHf)_0.67(NbTa)_0.33. Furthermore, the VEC of all (TiZrHf)_1−x(AB)_x alloys increases, whereas their ΔH_f decreases compared to that of TiZrHf. The ideal strength σ_p of (TiZrHf)_1−x(AB)_x increases linearly, reaching approximately 2.12 GPa. The bulk modulus (B), elastic modulus (E), and shear modulus (G) also exhibit linear increases, and their values in all (TiZrHf)_1−x(AB)_x alloys are higher than those of TiZrHf, with some exceptions. The Cauchy pressure (C₁₂–C₄₄) and Pugh’s ratio G/B of all (TiZrHf)_1−x(AB)_x alloys increase, whereas the Poisson’s ratio ν exhibits the opposite trend. Moreover, the C₁₂–C₄₄ and G/B ratio of TiZrHf are lower and higher, respectively, than those of (TiZrHf)_1−x(AB)_x, and the ν of TiZrHf is lower than that of (TiZrHf)_1−x(AB)_x. This study provides valuable insights for the design of high-performance TiZrHf-based refractory high-entropy alloys.

1. Introduction

Refractory high-entropy alloys (RHEAs) are a novel class of materials that have been widely studied due to their high yield strength and excellent high-temperature softening resistance, which significantly broadens the application range of traditional alloys [1,2], and possibly replace nickel-based superalloys as a candidate material [1,3,4]. Additionally, coating technology integrates the material advantages of RHEAs with practical application requirements. Through advanced surface engineering techniques such as laser cladding, plasma spraying, and vapor deposition, high-performance protective coatings can be fabricated on substrate surfaces, including superalloys, titanium alloys, and carbon-based materials. This approach effectively enhances the durability and service life of components operating in extreme environments, such as high-temperature, corrosive, or abrasive conditions, without significantly increasing production costs or compromising the intrinsic properties of the base materials [5,6]. Popular RHEAs with high mixing entropy ΔS_mix > 1.5R (R = 8.314 J/(mol·K) [7,8] are usually composed of five refractory elements containing Nb, Mo, W, Ta, etc. to form single-phase solid solutions with simple face-centered cube (FCC) or body-centered cube (BCC) structures [1,9]. Although majority of RHEAs exhibit good strength and resistance to high temperatures, they are rather brittle at room temperature. For example, the yield stress of a high-entropy Nb₂₀Mo₂₀Ta₂₀W₂₀V₂₀ alloy at room temperature can achieve 1246 MPa, while the ductility along the almost parallel compression direction is only 1.5% [10].

Fortunately, the design and development of TiZrHf-based RHEAs bring new hope to solve the brittleness of RHEAS as Hf effectively improves the ductility in the BCC system [11,12]. Dirras et al. [13] adopted vacuum arc melting technology to prepare a Ti₂₀Zr₂₀Hf₂₀Nb₂₀Ta₂₀ high-entropy alloy in experiments, and its elastic and plastic behaviors are investigated by the pulse-echo technique and tensile experiments. Their results show that the bulk modulus (B), shear modulus (G), and Young’s modulus (E) of the Ti₂₀Zr₂₀Hf₂₀Nb₂₀Ta₂₀ alloy are 134.6, 28.0, and 78.5 GPa, respectively. The results further showed that the Ti₂₀Zr₂₀Hf₂₀Nb₂₀Ta₂₀ alloy has a low Pugh ratio (B/G) and positive Cauchy pressure (C₁₂–C₄₄) of 0.208 and 80 GPa, respectively, suggesting that the Ti₂₀Zr₂₀Hf₂₀Nb₂₀Ta₂₀ alloy is a ductile material. The tensile test results show that the yield strength of the Ti₂₀Zr₂₀Hf₂₀Nb₂₀Ta₂₀ alloy can reach 800~840 MPa. Sheikh et al. [12] designed a high-entropy Hf_0.5Nb_0.5Ta_0.5Ti_1.5Zr alloy with excellent strength-ductility by combining theory and experiment. They compared the valence electron concentration (VEC) of a range of medium- and high-entropy alloys, showing that reducing VEC can lead to a transition from brittle to plastic in multi-component alloys. Thus, they prepared Hf_0.5Nb_0.5Ta_0.5Ti_1.5Zr RHEAs with low electron concentration (4.25) with arc melting technology. Tensile experiments showed that the yield stress, fracture stress, and elongation of Hf_0.5Nb_0.5Ta_0.5Ti_1.5Zr RHEA are 903 MPa, 990 MPa, and 18.8%, respectively. Furthermore, achieving the correct chemical composition is an effective way to maintain the BCC structure of TiZrHf-based RHEAs, and avoid the formation of brittle ω phase with the hexagon close-packed (HCP) phase [14,15]. In addition, the stability of TiZrHf-based RHEAs is strongly affected by local lattice distortions (LLDs), especially for BCC structure [16]. Therefore, it is urgent to study novel TiZrHf-based alloys in a variety of chemical compositions for achieving better RHEAs and to establish the relevant database.

Generally speaking, investigating the mechanical properties of materials can be divided into two types: (1) small strain response in Hooke’s law; (2) nonlinear response stress at larger strains [17]. The small strain response mainly studies the elastic modulus of materials containing bulk, shear, and Young’s moduli, and the nonlinear response stress at larger strains called is ideal strength. However, because of the multi-variable nature and complexity of the experiment, accurately investigating the mechanical properties of RHEAs is quite difficult [18,19,20]. In the past decade, with the emergence of theoretical calculations, e.g., first-principles (FP) calculation, they have played a crucial role in multiple material fields [21,22,23,24,25,26,27,28,29].

Very recently, Li et al. [30] investigated the phase stability and the micromechanical properties of (TiZrHf)_1−xTM_x (TM = (V, Nb, Cr, Mo, W)) based on first-principles method combing coherent-potential approximation (CPA). Their results suggested that Mo and W have a stronger stability for BCC structure, and the elastic hardening effect of Mo is more pronounced than that of Nb. In the framework of density functional theory (DFT), Chen et al. [31] employed thermodynamic models to investigate the stability of the Hf-Nb-Ta-Ti-Zr HEA alloy with BCC structure. The results showed that the equiatomic HfNbTaTiZr HEA suffers from phase decomposition below critical temperature of 1298 K, and the HEA decomposes most favorably to BCC NbTa-rich and HfZr-rich phases. Meng et al. [32] employed the DFT calculation based on special quasi-random structure (SQSs) to study the influences of LLDs on elastic properties for both hexagonal TiZrHf(Sc) alloys. They discovered that the LLDs significantly reduce the shear elastic properties, and the influence of LLDs on elastic properties for TiZrHf is more significant than that for TiZrHfSc.

However, until now, a comprehensive study of the stability and mechanical properties in relation to the chemical composition of TiZrHf-based RHEAs has not been reported. Here, we selected three substituted elements (Nb, Mo, and Ta) to construct (TiZrHf)_1−x(AB)_x (x = NbMo, NbTa, MoTa) five-element RHEAs based the SQS method and investigated their stability and mechanical property by using the first-principles calculation. Meanwhile, we chose eight chemical components, containing x = 0.11, 0.22, 0.33, 0.44, 0.56, 0.67, 0.78, 0.89, for (TiZrHf)_1−x(AB)_x alloys, and calculated the formation energy H_f, entropy of mixing ΔS_mix, enthalpy of mixing ΔH_mix, parameter Ω, standard deviation δ-parameter, bulk modulus B, shear modulus G, Young’s modulus E, Cauchy pressure C₁₂–C₄₄ and Poisson’s ratio ν, and ideal tensile strength σ_p as a function of concentration x to evaluate their stability and mechanical properties.

2. Calculation Details

In this work, the first-principles calculation was implemented in the Vienna Ab initio Simulation Package (VASP) [33] based on the density functional theory (DFT). Meanwhile, the plane-wave basis projector augmented-wave (PAW) method [34] and the Perdew–Burke–Ernzerhof (PBE) of generalized-gradient approximation (GGA) [35] were, respectively, applied to account for the core-valence electron interaction and exchange-correlation functional. During the whole relaxation of initial models, the cut-off energy of plane-wave was 350 eV, and the PAW_GGA pseudopotentials of (Ti-Zr)_sv and (Nb-Ta)_pv were adopted. Furthermore, we tested the total energy of (TiZrHf)_0.44(AB)_0.56 (AB = NbMo, NbTa) as a function of cutoff energy, and it is plotted in Figure 1a,b. The results indicate that when the cutoff energy exceeds 250 eV, the total energy exhibits minimal variation. Therefore, a cutoff energy of 350 eV was deemed sufficient and was employed in this study. For the accuracy of all calculations, the total energy and Hellmann–Feynman forces [36] on each atom were, respectively, converged to 10⁻⁶ eV/atom and 0.01 eV/Å by the conjugate gradient (CG) minimization method.

Here, we established the models of (TiZrHf)_1−x(AB)_x RHEAs with the BCC structure by fixing the proportional concentrations of Ti, Zr, and Hf to adding three combinations for A and B elements with proportional concentrations (AB = NbMo, NbTa, MoTa and x = 0.11, 0.22, 0.33, 0.44, 0.56, 0.67, 0.78, 0.89). Note: The proportional concentrations of the elements in the parentheses are equimolar. To solve the random arrangement and chemical disordering of (TiZrHf)_1−x(AB)_x alloys, we adopted the special quasi-random structure (SQS) method carried out in the alloy theoretic automated toolkit (ATAT) [37]. And the corresponding Brillouin zone integrations employed the Gamma centered mesh [38] of 4 × 3 × 3 k-point grid. Figure 2 shows all models of (TiZrHf)_1−x(NbMo)_x as examples. The strain matrix with a maximum strain of 9% was employed to calculate the elastic properties using the energy-strain method, while the ideal strength was determined under the condition of applying a maximum strain of no more than 35% along the [001] direction. All calculations were performed under ambient pressure conditions of 0 GPa.

3. Results and Discussion

3.1. Formation and Stability

To evaluate the possibility of formation of (TiZrHf)_1−x(AB)_x (AB = NbTa, NbMo, MoTa) RHEAs, the enthalpy of mixing ΔH_mix firstly is calculated. Generally, when the ΔH_mix is closer to 0 kJ/mol, single-phase solid solutions will be easier to form. The range of optical critical values of ΔH_mix for forming (TiZrHf)_1−x(AB)_x is generally −15 kJ/mol < ΔH_mix < 5 kJ/mol. The entropy of mixing ΔS_mix is also a main factor, and the whole energy of system can be reduced by higher ΔS_mix to form a simple solid solution. The ΔH_mix and ΔS_mix can be obtained by the following [20,39]:

$${\Delta H}_{\mathrm{m}\mathrm{i}\mathrm{x}}={\sum }_{i=1}^n{4{\mathrm{H}}_{i,j}^{\mathrm{m}\mathrm{i}\mathrm{x}}\mathrm{C}}_i{\mathrm{C}}_j$$

(1)

$${\Delta S}_{\mathrm{m}\mathrm{i}\mathrm{x}}=-\mathrm{R}{\sum }_{i=1}^n{C}_i\mathrm{l}\mathrm{n}{C}_j$$

(2)

where $\Delta H_{i,j}^{mix}$ are, respectively, the ith and jth mixing entropies from Miedema’s mode [40]. C_i and C_j are the atomic concentrations from ith and jth elements. R is the ideal gas constant. Note: All primary data for calculated $\Delta H_{i,j}^{mix}$ can be obtained from ref [40].

To describe the possibility of (TiZrHf)_1−x(AB)_x to form a single solid solution structure (SSSS), we introduce an important formation parameter Ω that was proposed by Yang et al. [41], where a larger value of Ω indicates that a greater ΔS_mix is to reduce more free energy, which makes the system more stable according to Gibbs free energy theory. It can be calculated as follows:

$$\Omega ={\displaystyle \frac{T_{\mathrm{m}}{\Delta S}_{\mathrm{m}\mathrm{i}\mathrm{x}}}{\left|{\Delta H}_{\mathrm{m}\mathrm{i}\mathrm{x}}\right|}}$$

(3)

$$T_{\mathrm{m}}={\sum }_{i=1}^n{C}_{\mathrm{i}}{\left(T_{\mathrm{e}}\right)}_{\mathrm{i}}$$

(4)

where T_m and T_e are the melting temperatures of (TiZrHf)_1−x(AB)_x and the simple substance of the ith element from refs. [42,43].

In addition, for simple multi-component solid solutions, the effects of LLDs should be considered. A standard deviation δ-parameter of first-nearest neighbor distortions can characterize the LLDs. When ranges of Ω and δ are, respectively, greater than 1.1 and less than 6.6%. The simple solid solution phase is easier to form in multi-component alloys with the BCC system, and it can be calculated as follows [44]:

$$\delta =\sqrt{{\displaystyle \frac{1}{n}}}\sum _n{\left(x_n-\mu \right)}^2$$

(5)

where x and µ correspond to the nth data and the average value, respectively.

The formation enthalpy ΔH_f can be calculated to describe the relative stability for different phases, whereas the average valence electron concentration (VEC) can also be used to evaluate the ability to form single-phase solutions. In the BCC system, it was found that BCC phases are stable at lower VEC (<6.87). They can be obtained and defined, as follows [45,46,47,48]:

$${\mathsf{\Delta }H}_{\mathrm{f}}={\displaystyle \frac{\left(E_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-\sum n{E}_i\right)}{N}}$$

(6)

$$VEC={\sum }_{i=1}^n{C}_i{VEC}_i$$

(7)

where the E_i and n correspond to the energy of the ith element and the number of atoms of the ith element, respectively. E_total and N are the total energy of systems and total number of atoms. The VEC_i is the VEC of each element i.

The formation energy of atomic vacancies $E_{vac}$ plays an important role in computational materials science. It can directly or indirectly reflect the thermodynamic stability of materials and their resistance to phase transformation. In this work, the vacancy formation energy of (TiZrHf)_1−x(AB)_x is obtained by averaging the vacancy formation energies of all elements in the system, as follows:

$$E_{vac}={\displaystyle \frac{\left(\sum _1^i{E}_i-E_{per.}+{\mu }_i\right)}{N}}$$

(8)

where $E_i$, $E_{per.}$, N, and ${\mu }_i$, respectively, represent the total energy for the defective cell of the i-th element and perfect cell, the number of times of vacancies, and the chemical potential of the i-th element.

We have calculated the enthalpy of mixing ΔH_mix, entropy of mixing ΔS_mix, melting temperature T_m, SSSS formation possibility parameter Ω, LLD parameter δ, average valence electron concentration VEC, formation enthalpy ΔH_f, and the formation energy of atomic vacancies E_vac of (TiZrHf)_1−x(AB)_x RHEAs; the values of these parameters were summarized in

Table 1. The calculated enthalpy of mixing ΔH_mix (kJ/mol), entropy of mixing ΔS_mix (J/(mol·K)), melting temperature T_m (K), Ω, δ-parameter (%), average valence electron concentration (VEC), and formation enthalpy ΔH_f (kJ/mol).

Alloys	ΔHmix	ΔSmix	Tm	Ω	δ	VEC	ΔHf
TiZrHf	0	9.13	2160.48	−	11.67	4.00	5.40
(TiZrHf)0.89(NbMo)0.11	−0.34	11.66	2233.61	77.18	13.26	4.17	5.05
(TiZrHf)0.78(NbMo)0.22	−0.76	12.79	2306.74	38.96	13.92	4.33	3.89
(TiZrHf)0.67(NbMo)0.33	−1.26	13.30	2379.87	25.14	14.75	4.50	1.96
(TiZrHf)0.56(NbMo)0.44	−1.84	13.35	2453.00	17.76	14.02	4.67	1.51
(TiZrHf)0.44(NbMo)0.56	−2.51	12.97	2526.13	13.05	13.71	4.83	0.77
(TiZrHf)0.33(NbMo)0.67	−3.26	12.18	2599.26	9.71	10.42	5.00	−0.87
(TiZrHf)0.22(NbMo)0.78	−4.09	10.92	2672.39	7.13	8.29	5.17	−2.54
(TiZrHf)0.11(NbMo)0.89	−5.00	9.04	2745.52	4.96	4.99	5.33	−4.92
(TiZrHf)0.89(NbTa)0.11	1.12	11.66	2254.34	23.48	13.10	4.11	6.97
(TiZrHf)0.78(NbTa)0.22	1.96	12.79	2348.19	15.33	13.01	4.22	7.42
(TiZrHf)0.67(NbTa)0.33	2.52	13.30	2442.04	12.90	15.86	4.33	7.71
(TiZrHf)0.56(NbTa)0.44	2.80	13.35	2535.89	12.10	14.62	4.44	5.95
(TiZrHf)0.44(NbTa)0.56	2.80	12.97	2629.74	12.19	15.43	4.56	5.28
(TiZrHf)0.33(NbTa)0.67	2.52	12.18	2723.59	13.17	11.48	4.67	3.39
(TiZrHf)0.22(NbTa)0.78	1.96	10.92	2817.45	15.70	10.22	4.78	2.15
(TiZrHf)0.11(NbTa)0.89	1.12	9.04	2911.30	23.51	5.45	4.89	0.52
(TiZrHf)0.89(MoTa)0.11	−0.52	11.66	2262.95	50.49	12.34	4.17	5.07
(TiZrHf)0.78(MoTa)0.22	−1.05	12.79	2365.41	28.72	13.28	4.33	4.60
(TiZrHf)0.67(MoTa)0.33	−1.59	13.30	2467.87	20.61	13.81	4.50	4.47
(TiZrHf)0.56(MoTa)0.44	−2.14	13.35	2570.34	16.03	14.60	4.67	1.93
(TiZrHf)0.44(MoTa)0.56	−2.70	12.97	2672.80	12.86	12.26	4.83	1.30
(TiZrHf)0.33(MoTa)0.67	−3.26	12.18	2775.26	10.37	11.34	5.00	−0.70
(TiZrHf)0.22(MoTa)0.78	−3.83	10.92	2877.72	8.20	9.15	5.17	−4.43
(TiZrHf)0.11(MoTa)0.89	−4.41	9.04	2980.19	6.11	7.14	5.33	−5.98

. To clearly clarify the correlation between concentration x and the above parameters, Figure 3 shows the ΔH_mix, ΔS_mix, Ω, δ, VEC, and ΔH_f as a function of concentration x. From Figure 3a, we can see that, with the increase in concentration x, the ΔH_mix of (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) linearly decreases from the concentration x = 0.11 to 0.8 whereas the ΔH_mix of (TiZrHf)_1−x(NbTa)_x firstly increases from the concentration x = 0.11 to 0.56, and then decreases to x = 0.89. It can be found that the ΔH_mix of all (TiZrHf)_1−x(NbTa)_x alloys is larger than that of TiZrHf of 0 kJ/mol while the ΔH_mix of all (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) is lower than that of TiZrHf of 0 kJ/mol. Figure 3b shows only a curve about ΔS_mix as a function of concentration because the ΔS_mix does not depend on the type of element. We can see that the calculated ΔS_mix of (TiZrHf)_1−x(AB)_x, with the concentration increasing first, increases from the concentration of x = 0.11 to 0.56 and then decreases to 0.89. And the ΔS_mix of all (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf, except for (TiZrHf)_0.11(NbMo)_0.89 of 9.04 J/(mol·K).

According to Equations (1)–(3), we have further obtained the key parameter Ω based on the above results and drawn the relationship between the concentration and Ω in Figure 3c. From the results, the Ω of (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) first decreases sharply from x = 0.11 to 0.44 and then slowly decreases to 0.89, whereas the Ω of (TiZrHf)_1−x(NbTa)_x first decreases slowly from x = 0.11 to 0.56 and then increases to 0.89. However, the Ω of TiZrHf cannot be calculated as a ΔH_mix of 0 kJ/mol forces when Ω approaches $\infty$. The δ is also an important parameter based on Zhang et al.’s extensive research on multi-components with simple BCC, FCC, and HCP crystals.

In Figure 3d, we draw the curves of δ for (TiZrHf)_1−x(AB)_x alloys with the substituted concentration increasing. From the results, it can be seen that the δ of (TiZrHf)_1−x(AB)_x floats within the 0.56 concentration. After 0.56 concentration, the δ decreases sharply to 0.89. The δ of (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf of 11.67% before x = 0.56. After that, the δ of cases is lower than that of TiZrHf of 11.67%, indicating that a higher substituted concentration may maintain the BCC structure [16]. It is known that when the average valence electron concentration (VEC) is less than 6.87 in BCC system, the structure is easier to form. But the relationship between stability and VEC for (TiZrHf)_1−x(AB)_x alloys has not been proposed. The calculated VEC and formation enthalpy H_f with the concentration increasing are plotted in Figure 3e,f. It can be found that the VEC increases linearly from x = 0.11 to 0.89 while the H_f shows the opposite, except for (TiZrHf)_0.67(NbTa)_0.33 of 7.71 kJ/mol. The VEC of TiZrHf is lower than that of all (TiZrHf)_1−x(AB)_x alloys whereas the ΔH_f of TiZrHf is higher than that of all cases, except for (TiZrHf)_1−x (NbTa)x (x = 0.11, 0.22, 0.33, 0.44). These results show that the lower VEC has a higher stability for (TiZrHf)_1−x(AB)_x RHEAs in the range of VEC. Further, the E_vac of (TiZrHf)_1−x(AB)_x alloys is at the concentration of x = 0.56, drawn in Figure 3g. Obviously, the E_vac of ~2.38 for (TiZrHf)_1−x(AB)_x is much larger than that of −0.24 for TiZrHf, and the (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) shows a larger E_vac.

3.2. Ideal Strength

The ideal strength σ_p presents the upper strength limit along a fixed crystal orientation and provides insights into how the crystal is dissociated. We have calculated the σ_p by plotting stress–strain curves, given by the following:

$$\sigma ={\displaystyle \frac{1}{V\left(\epsilon \right)}}{\displaystyle \frac{\mathrm{d}E}{\mathrm{d}\epsilon }}$$

(9)

where ε, σ and V are the engineering strain, corresponding tensile stress and equilibrium volume of the supercell, respectively.

In Figure 4a–c, the stress–strain curves of (TiZrHf)_1−x(AB)_x are given by using Equation (9). We can find that, with the respectively increasing strains to 0.20 and 0.25 along the [001] direction, the σ of (TiZrHf)₋(NbMo)_x (x = 0.11, 0.67, 0.78, 0.89), (TiZrHf)_1−x(NbTa)_x (x = 0.11, 0.33), (TiZrHf)_1−x(MoTa)_x (x = 0.11, 0.22, 0.33, 0.78) and (TiZrHf)_1−x(NbMo)_x (x = 0.22, 0.33, 0.44, 0.56), (TiZrHf)_1−x(NbTa)_x (x = 0.44, 0.56, 0.67, 0.78, 0.89), and (TiZrHf)_1−x(MoTa)_x (x = 0.56, 0.67, 0.89) gradually increases to the maximum value σ_p, and then dramatically decreases at the 0.25 and 0.30 strains, whereas the maximum strain for (TiZrHf)_0.78 (NbTa)_0.22 and (TiZrHf)_0.56 (MoTa)_0.44 corresponds to the σ_p when it is at 0.30. To describe the ideal strength σ_p of (TiZrHf)_1−x(AB)_x as a function of concentration x, Figure 4d shows the calculated σ_p with the concentration x increasing. We can clearly see that the σ_p of (TiZrHf)_1−x(AB)_x increases linearly in a range of ~ 2.12 GPa, suggesting that the higher the concentration x is, the more advantageous it is for (TiZrHf)_1−x(AB)_x deformation. Moreover, at all concentrations, the additions of NbMo, NbTa, and MoTa element combinations would increase the σ_p of TiZrHf.

To study the effect of stress on the structure and thermodynamics of materials, we use the Debye model implemented in Gibbs2 codes [49] to calculate key thermodynamic parameters, including the heat capacity C_V, entropy S, thermal expansion coefficient α_T and Gibbs free energy E_g, as a function of strains at the NbMo, NbTa and MoTa concentration x = 0.56 under room temperature. In Figure 5a–d, the α_T of (TiZrHf)_1−x(AB)_x containing pure TiZrHf presents slight changes in varying degrees with the increase in strains while the C_V, S, and E_g of all cases increases. Moreover, adding NbMo, NbTa, and MoTa would significantly decrease all thermodynamic parameters of TiZrHf. The (TiZrHf)_1−x(MoTa)_x and (TiZrHf)_1−x(NbMo)_x show, respectively, the lowest α_T, E_g and C_V, S.

3.3. Elastic Property

To study the elastic property of HEA_s, we first calculate their elastic constants using the energy–strain method, and other mechanical parameters in this work are dependent on calculated elastic constants. Elastic constants are a measure of a material’s resistance to small deformations under a constant stress, and they can be calculated as follows:

$$\mathsf{\Delta }E={\displaystyle \frac{V_0}{2}}\sum _{\mathrm{i},\mathrm{j}=1}^6{C}_{\mathrm{i}\mathrm{j}}{\epsilon }_{\mathrm{i}}{\epsilon }_{\mathrm{j}}$$

(10)

where V₀, C_ij, and $\mathsf{\Delta }E$ are the volume of the equilibrium cell, elastic constants, and total energy difference between the deformed cell and the perfect cell, respectively.

It is well known that the number of independent elastic constant components depends on the symmetry of the crystal. In a cubic crystal system, C_ij contains three independents, C₁₁, C₁₂, and C₄₄. And this C_ij can be interpreted as an important judgment of material stability for (TiZrHf)_1−x(AB)_x alloys in the Born criterion, as follows:

$$C_{11}>0,C_{44}>0,C_{11}-C_{12}>0,C_{11}+{2C}_{12}>0$$

(11)

All values of C_ij can be found in

Table 2. The calculated elastic constants C_ij (GPa) and Cauchy pressure C₁₂–C₄₄ (GPa).

Alloys	C11	C12	C44	C12–C44
TiZrHf	155.60	84.20	40.86	43.30
(TiZrHf)0.89(NbMo)0.11	136.86	91.65	38.04	53.61
(TiZrHf)0.78(NbMo)0.22	152.01	94.06	37.99	56.07
(TiZrHf)0.67(NbMo)0.33	172.86	100.04	39.60	60.44
(TiZrHf)0.56(NbMo)0.44	190.43	107.40	39.25	68.16
(TiZrHf)0.44(NbMo)0.56	218.76	110.81	38.63	72.19
(TiZrHf)0.33(NbMo)0.67	251.02	121.27	42.47	78.80
(TiZrHf)0.22(NbMo)0.78	275.92	136.40	51.10	85.30
(TiZrHf)0.11(NbMo)0.89	321.77	143.82	59.66	84.16
(TiZrHf)0.89(NbTa)0.11	124.48	96.16	30.66	65.49
(TiZrHf)0.78(NbTa)0.22	138.71	97.73	33.24	64.49
(TiZrHf)0.67(NbTa)0.33	148.50	102.31	31.06	71.25
(TiZrHf)0.56(NbTa)0.44	165.37	110.51	38.36	72.16
(TiZrHf)0.44(NbTa)0.56	183.87	113.11	38.23	74.88
(TiZrHf)0.33(NbTa)0.67	204.55	120.73	41.49	79.23
(TiZrHf)0.22(NbTa)0.78	223.90	130.25	43.96	86.28
(TiZrHf)0.11(NbTa)0.89	247.21	136.94	44.18	92.76
(TiZrHf)0.89(MoTa)0.11	126.70	97.22	38.12	59.10
(TiZrHf)0.78(MoTa)0.22	152.38	97.07	41.77	55.30
(TiZrHf)0.67(MoTa)0.33	177.56	98.84	42.57	56.28
(TiZrHf)0.56(MoTa)0.44	202.47	108.42	48.03	60.39
(TiZrHf)0.44(MoTa)0.56	218.00	118.67	46.15	72.52
(TiZrHf)0.33(MoTa)0.67	254.04	130.62	51.65	78.97
(TiZrHf)0.22(MoTa)0.78	289.34	142.80	59.41	83.39
(TiZrHf)0.11(MoTa)0.89	324.49	155.10	65.70	89.40

, and the calculated results show that all the (TiZrHf)_1−x(AB)_x phases containing pure TiZrHf can meet the mechanical stability. The C₁₁ of TiZrHf is larger than that of (TiZrHf)_1−x(NbTa, MoTa)_x (AB = NbTa, MoTa; x = 0.11, 0.22) and (TiZrHf)_1−x(NbTa)_x (x = 0.11, 0.22, 0.33), and lower than that of the rest of (TiZrHf)_1−x(AB)_x. The C₁₂ of all (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf, and the C₄₄ of TiZrHf is larger than that of (TiZrHf)_1−x(NbTa, MoTa)_x (AB = NbTa, MoTa; x = 0.11, 0.22, 0.33, 0.44, 0.56) and (TiZrHf)_0.89(NbTa)_0.11, and lower than that of the rest of (TiZrHf)_1−x(AB)_x. Furthermore, the bulk modulus B and shear modulus G have been calculated by the above C_ij, given as follows:

$$B=B_{\mathrm{R}}=B_{\mathrm{V}}{\displaystyle \frac{C_{11}+{2C}_{12}}{3}}$$

(12)

$$G_{\mathrm{V}}={\displaystyle \frac{C_{11}-C_{12}+{3C}_{44}}{5}}$$

(13)

$$G_{\mathrm{R}}={\displaystyle \frac{5\times (C_{11}-C_{12})\times C_{44}}{4\times C_{44}+3\times (C_{11}-C_{12})}};$$

(14)

$$G={\displaystyle \frac{G_{\mathrm{V}}+G_{\mathrm{R}}}{2}}$$

(15)

where ${ B}_{\mathrm{R}}$,$G_{\mathrm{R}}$ and $B_{\mathrm{V}}$, $G_{\mathrm{V}}$ are the lower and upper limits of polycrystalline elastic modulus and come from the Reuss and Voigt methods, respectively.

As previously mentioned, B and G are used to describe the measurement of resistance to volume and shear deformations under loading strains [50]. In addition, the Young’s modulus E and Poisson’s ratio ν are also important, and the E and ν can, respectively, reflect the forward deformation and heterogeneity of materials. They can be calculated by the following:

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

(16)

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

(17)

Additionally, the Debye temperature ${\theta }_{\mathrm{D}}$ is also an important parameter that can be directly calculated from elastic constants. It enables the evaluation of the bonding strength between atoms in a material and provides data support for analyzing the material’s hardness, melting point, and thermal conductivity. It can be calculated as follows [51,52,53]:

$${\theta }_{\mathrm{D}}={\displaystyle \frac{\mathrm{h}}{{\mathsf{\kappa }}_{\mathrm{B}}}}{\left({\displaystyle \frac{3}{4\mathsf{\pi }}}\right)}^{{\displaystyle \frac{1}{3}}}{v}_{\mathrm{m}}$$

(18)

$$v_{\mathrm{m}}={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{v_{\mathrm{L}}^3}}+{\displaystyle \frac{2}{v_{\mathrm{S}}^3}}\right)\right]}^{-{\displaystyle \frac{1}{3}}}$$

(19)

$$v_{\mathrm{L}}={\left({\displaystyle \frac{3B+4G}{3\rho }}\right)}^{{\displaystyle \frac{1}{2}}}$$

(20)

$$v_{\mathrm{S}}={\left({\displaystyle \frac{G}{\rho }}\right)}^{{\displaystyle \frac{1}{2}}}$$

(21)

where $h$, ${\kappa }_B$, $v_{\mathrm{L}}$, $v_{\mathrm{S}}$, and $\rho$ are the Plank’s values, Boltzmann’s constants, longitudinal values, traverse sound velocities, and corresponding density of the crystal, respectively.

We have obtained the Cauchy pressure C₁₂–C₄₄, bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Pugh’s G/B, Poisson’s ratio ν, and Debye temperature ${\theta }_{\mathrm{D}}$, collected in Table 2 and

Table 3. The calculated bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Pugh’s G/B, and Poisson’s ratio ν.

Alloys	B	G	E	G/B	υ	ϴD
TiZrHf	108.00	38.71	103.74	0.358	0.340	260.01
(TiZrHf)0.89(NbMo)0.11	106.72	30.87	84.47	0.289	0.368	233.74
(TiZrHf)0.78(NbMo)0.22	113.38	34.09	92.95	0.301	0.363	245.99
(TiZrHf)0.67(NbMo)0.33	124.31	38.29	104.18	0.308	0.360	261.00
(TiZrHf)0.56(NbMo)0.44	135.08	40.14	109.56	0.297	0.365	267.96
(TiZrHf)0.44(NbMo)0.56	146.79	44.18	120.44	0.301	0.363	281.68
(TiZrHf)0.33(NbMo)0.67	164.52	50.36	137.08	0.306	0.361	301.40
(TiZrHf)0.22(NbMo)0.78	182.91	57.90	157.11	0.317	0.357	323.88
(TiZrHf)0.11(NbMo)0.89	203.14	70.05	188.49	0.345	0.345	356.87
(TiZrHf)0.89(NbTa)0.11	105.60	22.49	62.99	0.213	0.401	196.34
(TiZrHf)0.78(NbTa)0.22	111.39	27.38	75.91	0.246	0.386	212.33
(TiZrHf)0.67(NbTa)0.33	117.71	27.58	76.75	0.234	0.391	209.42
(TiZrHf)0.56(NbTa)0.44	128.80	33.53	92.57	0.260	0.380	226.71
(TiZrHf)0.44(NbTa)0.56	136.70	37.07	101.98	0.271	0.376	234.25
(TiZrHf)0.33(NbTa)0.67	148.67	41.66	114.31	0.280	0.372	244.26
(TiZrHf)0.22(NbTa)0.78	161.46	45.09	123.74	0.279	0.372	250.10
(TiZrHf)0.11(NbTa)0.89	173.70	48.28	132.55	0.278	0.373	254.90
(TiZrHf)0.89(MoTa)0.11	107.05	26.05	72.27	0.243	0.387	210.56
(TiZrHf)0.78(MoTa)0.22	115.50	35.40	96.37	0.307	0.361	239.44
(TiZrHf)0.67(MoTa)0.33	125.08	41.25	111.50	0.330	0.351	252.94
(TiZrHf)0.56(MoTa)0.44	139.77	47.63	128.30	0.341	0.347	266.39
(TiZrHf)0.44(MoTa)0.56	151.78	47.52	129.10	0.313	0.358	261.59
(TiZrHf)0.33(MoTa)0.67	171.76	55.47	150.23	0.323	0.354	277.22
(TiZrHf)0.22(MoTa)0.78	191.65	64.61	174.26	0.337	0.348	293.83
(TiZrHf)0.11(MoTa)0.89	211.56	72.74	195.77	0.344	0.346	306.37

according to Equations (12)–(17) and using the results of C_ij. To clarify the relation between mechanical behavior of (TiZrHf)_1−x(AB)_x and substituted concentration x, we have to first plot the bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa) as a function of x, respectively, as seen in Figure 6a. It can be seen that the B, E, and G of (TiZrHf)_1−x(AB)_x increase linearly with the increase in concentration x, and the B, E, and G of all (TiZrHf)_1−x(AB)_x are larger than that of TiZrHf, except for (TiZrHf)_1−x(AB)_x (x = 0.11) of B, (TiZrHf)_1−x(NbTa)_x (x = 0.11, 0.22, 0.33, 0.44, 0.56) of G and E, (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa; x = 0.11, 0.22) of G and E, and (TiZrHf)_0.67(NbMo)_0.33 of G. The results further show that when the element containing Mo is added, the elastic modulus is significantly improved.

To evaluate the atomic bonding characteristics and ductile behavior of (TiZrHf)_1−x(AB)_x, we have drawn the curves of the Cauchy pressure C₁₂–C₄₄ and Pugh’s G/B as a function of concentration x in Figure 6b,c. Research showed that when C₁₂–C₄₄ is greater than 0 GPa, the atomic bonding of the material exhibits a metallic characteristic; otherwise, it presents as a directional covalent characteristic [54,55]. And the larger absolute value indicates the stronger metallic or directional covalent characteristic of materials, whereas the larger G/B shows the more brittle. From the results, the C₁₂–C₄₄ of all (TiZrHf)_1−x(AB)_x increases with the substituted concentration x, except for (TiZrHf)_0.11(NbMo)_0.89 of 84.16 GPa and (TiZrHf)_0.78(NbTa)_0.22 of 64.49 GPa and are greater than 0 GPa. Similarly, the G/B of all (TiZrHf)_1−x(AB)_x increases with the increasing concentration x, except for (TiZrHf)_0.67(NbMo)_0.33 of 0.308, (TiZrHf)_0.56(NbMo)_0.44 of 0.297, and (TiZrHf)_0.67(NbTa)_0.33 of 0.234. This phenomenon indicates that atomic bonds of (TiZrHf)_1−x(AB)_x present stronger metallic characteristics with the increase in concentration x, and the ductility improves. Further, the C₁₂–C₄₄ and G/B of TiZrHf are lower and larger than that of (TiZrHf)_1−x(AB)_x, indicating that the atomic bonds of TiZrHf have a weaker metallic characteristic and it presents worse ductility. Poisson’s ratio ν is widely used to show the heterogeneity of materials [56], and when the ν is greater/smaller than 0.26, it shows uniform/nonuniform behaviors in response to applied stress. Figure 6d shows the calculated ν as a function of concentration x, and the ν of (TiZrHf)_1−x(AB)_x shows the opposite trend. Clearly, all (TiZrHf)_1−x(AB)_x and TiZrHf present nonuniform behaviors, and the ν of TiZrHf is lower than that of (TiZrHf)_1−x(AB)_x. Further, Figure 6e shows the ${\theta }_{\mathrm{D}}$ of (TiZrHf)_1−x(AB)_x as a function of concentration x, and the results show that the ${\theta }_{\mathrm{D}}$ linearly increases with the increasing concentration x, except for a few scattered points. Among these alloys, the ${\theta }_{\mathrm{D}}$ of (TiZrHf)_1−x(NbMo)_x (x = 0.33–0.89) and (TiZrHf)_1−x(MoTa)_x (x = 0.44–0.89) is larger than that of TiZrHf of 260.01 K. In Figure 6f, we show the T_m as a function of ${\theta }_{\mathrm{D}}$ for (TiZrHf)_1−x(AB)_x RHEAs, and it is seen that the T_m increases significantly with the increase of ${\theta }_{\mathrm{D}}$, indicating that there is a strong positive correlation between T_m and ${\theta }_{\mathrm{D}}$.

To further show the underlying effect of NbMo, NbTa, MoTa to TiZrHf on mechanical stability, the orbital Hamiltonian populations (COHP) [57,58] of (TiZrHf)_0.44(AB)_0.56, including pure TiZrHf, are calculated using the Local Orbital Basis Suite Towards Electronic-Structure Reconstruction (LOBSTER) program [59], and the COHP curves are plotted in Figure 7a–c. The red, blue, navy, and violet curves are, respectively, the COHP curves of TiZrHf, (TiZrHf)_0.44(NbMo)_0.5, (TiZrHf)_0.44(NbTa)_0.5, and (TiZrHf)_0.44(MoTa)_0.5. We focus on the chemical bonds of the first nearest neighbor with lengths less than 3.5 Å because the strengths of longer bonds are very weak. It can be found that, in pure TiZrHf, the chemical bonds of Ti-Zr, Ti-Hf, and Zr-Hf exhibit obvious antibonding states under the fermi level, indicating that the chemical bonds between them are very weak, and pure TiZrHf exhibits poor mechanical stability. When NbMo, NbTa, and MoTa are added to TiZHf, the chemical bonds of Nb-Mo, Nb-Ta, and Mo-Ta exhibit a strong bonding state, indicating that the addition of NbMo, NbTa, and MoTa significantly improves the mechanical stability.

4. Conclusions

The stability and mechanical properties of (TiZrHf)_1−x(AB)_x (AB = NbTa, NbMo, MoTa) refractory high-entropy alloys have been investigated by combining the first-principles with special quasi-random structure (SQS) method. Here are some main results as a function of concentration x, as follows:

(1)

The ΔH_mix of (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) linearly decreases, whereas both ΔH_mix and ΔS_mix of (TiZrHf)_1−x(NbTa)_x increase initially and subsequently decrease, with the crossover occurring at x = 0.56. The ΔH_mix of (TiZrHf)_1−x(NbTa)_x and (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) alloys are larger and lower than that of TiZrHf, respectively, while the ΔS_mix of all (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf.

(2)

The Ω of all (TiZrHf)_1−x(AB)_x (AB = NbMo, MoTa) first decreases sharply, followed by a gradual decrease. And the δ remains relatively stable around x = 0.56 for all cases, after which it decreases sharply until x = 0.89. The δ value of (TiZrHf)_1−x(AB)_x is higher than that of TiZrHf for x < 0.56 but becomes lower beyond this composition.

(3)

The VEC increases linearly from x = 0.11 to 0.89 while the ΔH_f shows the opposite, except for (TiZrHf)_0.67(NbTa)_0.33. The VEC of all (TiZrHf)_1−x(AB)_x alloys increases, whereas their ΔH_f decreases compared to that of TiZrHf.

(4)

The σ_p of (TiZrHf)_1−x(AB)_x increases linearly, reaching approximately 2.12 GPa, and the σ_p of all (TiZrHf)_1−x(AB)_x is larger than that of TiZrHf. B, E, and G also exhibit linear increases, and their values in all (TiZrHf)_1−x(AB)_x alloys are higher than those of TiZrHf, with some exceptions.

(5)

The C₁₂–C₄₄ and G/B of all (TiZrHf)_1−x(AB)_x alloys increase, whereas the ν exhibits the opposite trend. Moreover, The C₁₂–C₄₄ and G/B ratio of TiZrHf are lower and higher, respectively, than those of (TiZrHf)_1−x(AB)_x, and the ν of TiZrHf is lower than that of (TiZrHf)_1−x(AB)_x.