First-Principles Calculations of the Effect of Ta Content on the Properties of UNbMoHfTa High-Entropy Alloys

Abstract

Uranium-containing high-entropy alloys (HEAs) exhibit great potential as a novel energetic structural material, attributed to their excellent performance in impact energy release, superior mechanical properties, and high density. This study investigates the effects of Ta content on the phase stability, lattice constant, density, elastic constants, polycrystalline moduli, and electronic structure of (UNbMoHf)_54−xTa_x high-entropy alloys (where x = 2, 6, 10, 14, 18), utilizing a combination of density functional theory (DFT) calculations and the special quasi-random structure (SQS) approach. Our findings confirm that these alloys maintain stable body-centered cubic structures, as evidenced by atomic radius difference and valence electron concentration evaluations. Analysis of elastic modulus, Cauchy pressure, and Vickers hardness indicates that Ta incorporation enhances mechanical properties and increases the anisotropy of these alloys. Furthermore, investigations into the electronic structure reveal that adding Ta reduces metallic character while increasing covalent characteristics, enhancing the contribution of Ta’s d-orbitals to the total density of states and intensifying covalent bonding interactions between Ta and other elements such as Nb, Mo, and U. These findings provide theoretical guidance for the design of high-performance UNbMoHfTa HEAs with tailored properties.

1. Introduction

Energetic structural materials represent an innovative category of materials that merge the explosive properties with the mechanical strength and stability characteristic of metals [1,2,3]. These materials often encounter inadequate strength, plasticity, and toughness despite their promise. High-entropy alloys (HEAs) have broken through the traditional alloy design concept and creatively distribute multiple main elements evenly in the middle area of the phase diagram. This unique design endows HEAs with excellent comprehensive properties, such as mechanical properties [4], corrosion resistance [5,6], resistance to irradiation swelling properties [7], superconducting properties [8,9], and catalytic performance [10]. Particularly, body-centered cubic (BCC) HEAs, enriched with refractory alloying elements like Ti, Zr, V, Hf, Nb, Ta, Cr, and Mo, exhibit notable combustion enthalpy [11,12,13,14,15] and can release substantial thermal energy upon fracture triggered by impact. The characteristics of refractory high-entropy alloys (RHEAs) offer fresh perspectives on advancing energetic structural materials with superior performance.

The density of existing RHEA systems remains inferior to that of conventional tungsten alloys, resulting in greater velocity decay at low speeds and consequently limiting impact energy release [13]. While incorporating tungsten—a high-density element—can elevate alloy density, its extremely high melting point and resistance to oxidation and heat release pose significant challenges [14]. In contrast, uranium offers a compelling alternative as a high-density element. At room temperature, uranium forms a stable γ-body-centered cubic solid solution with alloying elements such as Al, Hf, Nb, Mo, Si, Ta, V, and Zr, thereby enhancing the alloy’s strength, toughness, and workability [10,16,17,18]. Additionally, uranium exhibits unique self-sharpening properties and a high exothermic capacity [17,19,20,21], further amplifying its utility. Consequently, uranium-containing HEAs effectively address the issue of insufficient density while simultaneously optimizing the impact energy release efficiency of the alloy components. This dual advantage positions uranium-based HEAs as a promising solution for advancing the performance of energetic structural materials.

Research on uranium-containing HEAs as energetic structural materials remains limited, with existing studies primarily focusing on their mechanical properties [17,18,19,20,21]. Shi Jie et al. synthesized equimolar UNbMoTaHf and UNbMoTaTi HEAs via arc melting with a single-phase BCC structure. Room-temperature compression tests revealed that titanium additions enhance the alloys’ plasticity [22]. Further, Zhang et al. investigated two single-phase BCC HEAs, UNb_0.5Zr_0.5Mo_0.5 and UNb_0.5Zr_0.5Ti_0.2Mo_0.2, demonstrating high hardness and strength at room temperature [23]. However, challenges persist, such as the brittleness induced by the Laves phase in UMoNbZr HEAs, as reported by Michael Aizenshtein, which limits ductility despite its hardness exceeding that of α-uranium [24]. Notably, uranium-based HEAs exhibit exceptional resistance to irradiation damage and swelling [25], and some systems have been explored for advanced nuclear fuel applications [26].

Despite these advancements, the traditional trial-and-error approach to identifying HEAs with optimal mechanical properties remains resource-intensive and time-consuming. Additionally, depleted uranium materials pose risks due to their α-radioactivity and biotoxicity. The advent of modern computational tools has introduced a more efficient alternative: first-principles calculations enable rapid, accurate predictions of alloy properties with minimal experimental overhead. By leveraging the Exact Muffin-tin Orbitals (EMTO) software, high-throughput screening, and machine learning, researchers can efficiently identify promising uranium-containing HEAs [27,28]. Among the proposed systems, the quinary HEA UNbMoHfTa stands out with a 14.7 g/cm³ of theoretical density, excellent isotropy, and ductility, positioning it as a leading candidate for next-generation high-performance energetic structural materials. The complex composition–property relationships in multicomponent systems necessitate precise theoretical guidance to optimize performance. In contrast, previous studies have demonstrated that equimolar UNbMoTaHf HEAs form single-phase BCC structures, and the fixed compositional designs may limit further performance enhancement. Theoretical studies suggest that non-equimolar composition adjustments can improve the strength–plasticity balance in such alloys [29,30,31,32,33,34,35,36]. However, EMTO method has limitations in describing lattice distortion and atomic interactions caused by atomic positional deviations in alloys, unlike the special quasi-random structure (SQS) method [37].

To address these challenges, this study investigates the influence of (Ta content on the properties of (UNbMoHf)_54−xTa_x HEAs by combining first-principles calculations based on density functional theory (DFT) with the SQS method. The Ta element was chosen for several reasons: (1) the Ta element possesses a higher density, and its compositional alterations exert a diminished influence on the density of uranium-based alloys; (2) the Ta element has a significant solid-solution strengthening ability, which enhances the strength and hardness of the alloys; and (3) Ta may be able to enhance the oxidation resistance of the alloys. Initially, the effect of Ta content on the relevant mechanical properties was calculated, and the hypothesis was tested to determine whether different Ta content compositions of (UNbMoHf)_54−xTa_x HEAs could form stable BCC structures based on the empirical criterion. Subsequently, the effect of Ta content on the lattice constant, density, elastic constant, elastic modulus, Pugh ratio, Poisson’s ratio, anisotropy, and Vickers hardness of (UNbMoHf)_54−xTa_x HEAs are examined. Finally, the underlying mechanisms governing Ta’s influence on mechanical properties were analyzed through electronic structure calculations.

2. Methods

Density functional theory (DFT) calculations were carried out using the Vienna Ab initio Simulation Package (VASP) [38], combined with the SQS method. Firstly, the Alloy Theoretic Automated Toolkit (ATAT) algorithm [39] was employed to generate a BCC structure of 3 × 3 × 3 cells supercell model ABCD_54−xE_x (x = 2, 6, 10, 14, 18) with a fixed unit cell of 54 atoms. SQS represents the best periodic supercell approximation of the real disordered state for a given number of atoms per supercell. The minimum objective function of the SQS algorithm ensures that the generated structure has the best degree of disorder. However, due to the absence of differences in elemental properties at the structural sites of the initial supercell, for each SQS model of (UNbMoHf)_54–xTa_x HEAs, there are a total of ${\mathrm{A}}_4^4$ = 24 ways of elemental arrangement. The configuration with a lattice constant closest to the experimental value and relatively low energy was selected for further calculations of mechanical properties. This selection criterion ensured structural stability and realistic lattice parameters for subsequent analysis.

The SQS cell model before structure optimization is shown in Figure 1. The Generalized Gradient Approximation (GGA) of the Perdew–Burke–Ernzerhof (PBE) function [40] was employed to characterize the exchange–correlation interactions between electrons. The Projected Augmented Wave (PAW) pseudopotential [41] method was used to describe the interactions between ions and electrons. The plane-wave cutoff energy was set to 400 eV, and the k-points in the Brillouin zone were sampled on a 3 × 3 × 3 grid using the Monkhorst–Pack-centered grid [42]. The maximum tolerance for total energy and Hellmann–Feynman force convergence criteria were set to 1 × 10⁻⁶ eV/atom and −1 × 10⁻² eV/Å, respectively. For the calculation of U, Nb, Mo, Ta, and Hf, the k-points were set as 15 × 15 × 15, and all the calculations of the elastic constant were carried out by the stress–strain method. To deeply explore the nature of chemical bonding in alloy systems, the LOBSTER 5.1.0 [43] software package was used to calculate and analyze the crystal orbital Hamilton populations (COHP) [44].

The theoretical density of the alloy was calculated by Vegard’s mixing rule [45]:

$${\rho }_{\mathrm{mix}}={\displaystyle \frac{{\displaystyle \sum _i^n{c}_i{A}_i}}{{\displaystyle \sum _{i=1}^n(c_i{A}_{\mathrm{i}}/{\rho }_i)}}}$$

(1)

Here, c_i, A_i, and ρ_i are the molar ratio, atomic weight, and pure metal density of the i-th element, respectively, and n is the number of elements in the solid solution.

The elastic constants reflect the relationship between stress and strain when an applied external force deforms a material. The original high symmetry of the BCC crystal structure is destroyed because of the severe lattice distortion in HEAs modeled by the SQS supercell. In this case, the C₁₁, C₁₂, and C₄₄ obtained from the SQS model are generally calculated by averaging them according to Equation (2) [46]:

$$C_{11}={\displaystyle \frac{C_{11}^{\prime }+C_{22}^{\prime }+C_{33}^{\prime }}{3}}{C}_{12}={\displaystyle \frac{C_{11}^{\prime }+C_{23}^{\prime }+C_{13}^{\prime }}{3}}{C}_{44}={\displaystyle \frac{C_{44}^{\prime }+C_{55}^{\prime }+C_{66}^{\prime }}{3}}$$

(2)

The ability of a material to resist volumetric, elastic, and shear deformation can be reflected by the bulk modulus B, shear modulus G, and Young’s modulus E, respectively. According to the Voigt–Reuss–Hill method, the polycrystalline elastic properties are calculated as follows [47,48,49]:

$$B={\displaystyle \frac{C_{11}+C_{12}}{3}}$$

(3)

$$G={\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{C_{11}-C_{12}+3C_{44}}{5}}+{\displaystyle \frac{5C_{44}(C_{11}-C_{12})}{3(C_{11}-C_{12})+4C_{44}}}\right]$$

(4)

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

(5)

Based on the single-crystal elastic constants C₁₁, C₁₂, and C₄₄, the corresponding flexibility constants S₁₁, S₁₂, and S₄₄ can be calculated to obtain the single-crystal Young’s modulus, E_[h,k,l], on the specified crystallographic direction [h, k, l] [50,51]:

$$1/E_{[hkl]}=S_{11}-2(S_{11}-S_{12}-S_{44}/2)(n_1^2{n}_2^2+n_2^2{n}_3^2+n_1^2{n}_3^2)$$

(6)

where S₁₁ = (C₁₁ + C₁₂)/[(C₁₁ – C₁₂)(C₁₁ + C₁₂)], S₁₂ = −C₁₂/[(C₁₁ – C₁₂)(C₁₁ + 2C₁₂)], and S₄₄ = 1/C₄₄. n₁ = h/d, n₂ = k/d, and n₃ = l/d are the cosines of the directions concerning the crystallographic directions [h, k, l], with d = $\sqrt{h^2+l^2+k^2}$.

The shear modulus of a single crystal in a specified crystallographic direction is related to the crystallographic direction n and the direction m on the crystallographic plane perpendicular to n. For cubic crystals, the single crystal shear modulus G_{(n, m)} can be expressed as follows:

$$\begin{array}{c}1/G_{(n,m)}=4S_{11}(n_1^2{m}_1^2+n_2^2{m}_2^2+n_3^2{m}_3^2)+8S_{12}(n_1{n}_2{m}_1{m}_2+n_1{n}_3{m}_1{m}_3+n_2{n}_3{m}_2{m}_3)\\ +S_{44}[{(n_1{m}_2+n_2{m}_1)}^2+{(n_1{m}_3+n_3{m}_1)}^2+{(n_2{m}_3+m_3{m}_2)}^2]\end{array}$$

(7)

where n₁, n₂, and n₃ are the direction cosines of the crystal orientation n perpendicular to the shear plane, and m₁, m₂, and m₃ are the direction cosines of the crystal orientation m, respectively. For a fixed crystal direction n, infinitely many arbitrary orientations m are on the crystal plane perpendicular to n. The shear modulus is typically characterized by its extremum value in each direction, which requires us to calculate the extreme value of the shear modulus in a specific direction. The maximum (G_max) and minimum (G_min) values of the shear modulus of a single crystal in each direction n can be calculated based on Equation (8) alone.

The Vickers hardness of the BCC high alloy can be calculated using Tian’s hardness model [52]:

$$H_V=0.92(G/B{)}^{1.137}{G}^{0.708}$$

(8)

For BCC crystals, the energy factor K for an edge, screw, and mixed dislocations can be calculated by Equations (9)–(11), where θ is the orientation angle (0 ≤ θ ≤ π) [53,54,55,56]:

$$K_{screw}={\left[{\displaystyle \frac{1}{2}}{C}_{44}(C_{11}-C_{12})\right]}^{{\displaystyle \frac{1}{2}}}$$

(9)

$$K_{edge}=(C_{11}+C_{12})\left[{\displaystyle \frac{C_{44}(C_{11}-C_{12})}{C_{11}(C_{11}+C_{12}+2C_{44})}}\right]$$

(10)

$$K_{mixed}=K_{edge}\cdot {\sin }^2\theta +K_{screw}\cdot {\cos }^2\theta$$

(11)

3. Results and Discussion

3.1. Phase Structure, Lattice Constants, and Density

The phase stability of HEAs follows the Hume–Rothery rules [57], and the phase formation rules of (UNbMoHf)_54–xTa_x HEAs can be predicted based on empirical Formulas (12)–(16):

$$\delta =\sqrt{{\displaystyle \sum _{i=1}^n{c}_i}{(1-{\displaystyle \frac{r_i}{\overline{r}}})}^2}\hspace{1em}\overline{r}={\displaystyle \sum _{i=1}^n{c}_i}{r}_i$$

(12)

$$\Delta H_{mix}={\displaystyle \sum _{i=1,i\ne \mathrm{j}}^{n}4H_{AB}^{\mathrm{mix}}}{c}_i{c}_j$$

(13)

$$VEC={\displaystyle \sum _{i=1}^n{c}_i}{\left(VEC\right)}_i$$

(14)

$$\Delta S_{mix}=-R{\displaystyle \sum _{i=1}^n\left(c_i\ln c_i\right)}$$

(15)

$$\Omega ={\displaystyle \frac{\left|T_m\right.\left.\Delta S_{\mathrm{mix}}\right|}{\left|\left.\Delta H_{mix}\right|\right.}}\hspace{1em}{T}_{\mathrm{m}}={{\displaystyle \sum _{i=1}^n{c}_i\left(T_m\right)}}_i$$

(16)

The parameter δ in Equation (12) denotes the atomic radii difference [58], where n, c_i, r_i, and $\overline{r}$ are the ordinal number, mole fraction (at.%), atomic radius (Å), and average atomic radius (Å) of the alloying elements, respectively. In Equation (13), ΔH_mix denotes the mixing enthalpy (kJ/mol), and Δ$H_{AB}^{mix}$ is the mixing enthalpy in the binary liquid phase of the i-th and j-th element at equal molar concentration (kJ/mol) [59]. In Equation (14), VEC is the valence electron concentration parameter [60]. In Equation (15), ΔS_mix is the mixing entropy (J/(mol·K)), R is the ideal gas constant (8.314 J/(mol·K)), and only the configurational mixing entropy is considered. In Equation (16), the Ω parameter is a comprehensive effect parameter that considers both ΔH_mix and ΔS_mix, and T_m is the effective melting temperature. According to previous studies on a large number of HEAs, the conditions for forming stable BCC-HEAs are as follows [61,62]: (i) δ < 6.6%; (ii) –15 kJ/mol < ΔH_mix < 5 kJ/mol; (iii) 12 J/(mol·K) < ΔS_mix < 17.5 J/(mol·K); (iv) VEC < 6.87; and (v) Ω > 1.1. The calculated values of the empirical parameters of the (UNbMoHf)_54−xTa_x HEAs with different Ta contents, such as the atomic radii difference δ, mixing enthalpy Δ$H_{AB}^{mix}$, valence electron concentration VEC, mixing entropy ΔS_mix, and Ω parameter, are shown in

Table 1. Calculated values of empirical parameters δ, ΔH_mix, VEC, ΔS_mix, Ω.

Alloy	δ × 100	ΔHmix(kJ/mol)	VEC	ΔSmix J/(mol·K)	Ω
U13Nb13Mo13Hf13Ta2	6.18	−0.75	4.52	12.42	27.30
U12Nb12Mo12Hf12Ta6	6.15	−1.19	4.56	13.15	19.20
U11Nb11Mo11Hf11Ta10	5.75	−1.54	4.59	13.38	15.78
U10Nb10Mo10Hf10Ta14	5.67	−1.81	4.63	13.30	13.95
U9Nb9Mo9Hf9Ta18	5.24	−2.00	4.67	12.98	12.87

. All the above conditions can be met within the studied composition variation range, indicating that all (UNbMoHf)_54−xTa_x HEAs can form a stable single-phase BCC solid solution.

The enthalpy of formation (H_form) and the cohesive energy (E_coh) can represent alloys’ formation ability and structural stability. The calculation formulae for the H_form and the E_coh of (UNbMoHf)_54–xTa_x HEAs are as follows:

$$H_{f\mathrm{orm}}={\displaystyle \frac{E_{tot}-\left[(E_{solid}^U+E_{solid}^{Nb}+E_{solid}^{Mo}+E_{solid}^{Hf})\times (54-x)+x{E}_{solid}^{Ta}\right]}{54}}$$

(17)

$$E_{coh}={\displaystyle \frac{E_{tot}-\left[(E_{atom}^U+E_{atom}^{Nb}+E_{atom}^{Mo}+E_{atom}^{Hf})\times (54-x)+x{E}_{atom}^{Ta}\right]}{54}}$$

(18)

Here, E_tot represents the total free energy of the optimized (UNbMoHf)_54−xTa_x HEAs. $E_{solid}^i$ refers to the total energy of the average i atoms in the single-crystal cell of each component within (UNbMoHf)_54–xTa_x HEAs. $E_{atom}^i$ represents the total energy of isolated atoms in the unit cell.

Table 2. The formation enthalpy (H_form), cohesive energy (E_coh), and the total energy (E_tot) of the (UNbMoHf)_54−xTa_x RHEA.

Alloy	Hform (eV/atom)	Ecoh (eV/atom)	Etot (eV)
U13Nb13Mo13Hf13Ta2	−0.0061	−9.1864	−566.96
U12Nb12Mo12Hf12Ta6	−0.0046	−9.2079	−572.35
U11Nb11Mo11Hf11Ta10	−0.0096	−9.2360	−578.10
U10Nb10Mo10Hf10Ta14	−0.0021	−9.2516	−583.17
U9Nb9Mo9Hf9Ta18	−0.0088	−9.2815	−589.01

summarizes the variation in formation enthalpy, cohesive energy, and total energy of (UNbMoHf)_54−xTa_x HEAs with a BCC structure as the Ta content changes.

Table 3. The total energy ($E_{solid}^i$) of the average i atoms in the single-crystal cell. The $E_{atom}^i$ represents the total energy of isolated atoms in the unit cell (a = 15 Å).

Alloy	U	Nb	Mo	Hf	Ta
$E_{solid}^i$ (eV/atom)	−10.8737	−10.9432	−10.2138	−11.8116	−9.739
$E_{atom}^i$ (eV)	−1.0726	−0.4372	−0.8203	−2.3309	−2.7649

shows the total energy ($E_{solid}^i$) of the average i atoms in the single-crystal cell and the total energy of isolated atoms($E_{atom}^i$) in the unit cell (a = 15 Å) for every component in (UNbMoHf)_54−xTa_x HEA. The calculated results reveal that both the formation enthalpy and cohesive energy of the BCC-structured alloys are negative, which unambiguously confirms the thermodynamic stability of the alloy system and its ability to form stable BCC solid solutions. Furthermore, the magnitude of the negative cohesive energy and total energy increases with rising Ta content [36], and this trend indicates that Ta addition enhances the structural stability of the alloy by strengthening interatomic bonding, thereby lowering the system’s energy to a more favorable state.

The lattice constants of the SQS model for the (UNbMoHf)_54−xTa_x HEAs after structural optimization are shown in Figure 2. The U₁₁Nb₁₁Mo₁₁Hf₁₁Ta₁₀ alloy exhibits a lattice constant of 3.354 Å, which aligns closely with the experimentally reported value of 3.321 Å for the equimolar UNbMoTaHf alloy [22], differing by less than 1%. The lattice constant decreases with increasing Ta content, reflecting stronger atomic bonding and enhanced structural stability. This behavior is attributed to Ta’s smaller atomic radius than the other constituent elements (U, Nb, Mo, Hf). As Ta’s proportion increases, the average atomic radius of the alloy decreases, resulting in a more compact lattice structure. A reduced lattice constant correlates with stronger interatomic interactions, thereby improving mechanical and thermal stability. The compact lattice structure (higher density) also enhances the alloy’s kinetic energy density at equivalent velocities. This reduces velocity decay during high-velocity impacts, thereby improving the alloy’s target-penetration capability—a critical factor for applications in ballistic and aerospace engineering. As shown in Figure 2, the theoretical density of the alloy increases with Ta addition. This is due to Ta’s significantly higher pure-metal density compared to Nb, Mo, and Hf, as well as the base alloy’s average density. The incorporation of Ta shifts the alloy’s overall density toward that of the denser element, further reinforcing its structural integrity and mechanical performance.

3.2. Elastic Properties and Mechanical Properties

Figure 3a–d demonstrates the evolution of elastic constants C₁₁, C₁₂, C₄₄, and C₁₂ − C₄₄ with Ta content in the (UNbMoHf)_54−xTa_x HEAs. The elastic constants C₁₁, C₁₂, and C₄₄ gradually rise with increasing Ta content, indicating enhanced stiffness and resistance to deformation. The Cauchy pressure (C₁₂ − C₄₄) decreases as Ta content increases, reflecting changes in interatomic bonding properties. A positive Cauchy pressure is characteristic of metallic bonding, while a negative value indicates covalent bonding [63]. All studied HEAs exhibit positive Cauchy pressures, confirming the dominance of metallic bonding. This metallic bonding is critical for achieving optimal alloy strength and plasticity. For BCC crystals, their mechanical stability follows the Born’s rules [64]: C₁₁ > 0, C₄₄ > 0, C’ = (C₁₁ – C₁₂)/2 > 0, C₁₁ + 2C₁₂ > 0. As shown in Figure 3, the elastic constants of (UNbMoHf)_54−xTa_x HEAs satisfy the stability criterion, indicating that this series of alloys exhibits favorable mechanical stability. Table 4 lists the calculated lattice constants and elastic moduli of pure metals (U, Nb, Mo, Hf, Ta) used in this study. These values align well with experimental measurements and prior computational studies. The results indicate that, since the C₁₁, C₁₂, and C₄₄ values of Ta are significantly higher than those of (UNbMoHf)_54−xTa_x HEAs, the addition of more Ta elements leads to an increase in the C₁₁, C₁₂, and C₄₄ of alloys.

As shown in Figure 4a–c, the bulk modulus B, shear modulus G, and Young’s modulus E of (UNbMoHf)_54−xTa_x HEAs demonstrate a clear trend. As the content of Ta increases, the resistance of the alloy to volumetric deformation and elastic deformation increases. This is due to Ta’s relatively small atomic radius, and the elements with smaller atomic radii form stronger chemical bonding. At the same time, the addition of smaller atoms makes the atomic arrangement more compact; the lattice constant decreases and forms a denser lattice structure so that the alloy’s modulus of elasticity and hardness increases [65]. Moreover, the B, G, and E values of pure Ta metal are higher than those of (UNbMoHf)_54−xTa_x HEAs, and the overall B, G, and E values of the alloys increase with the incorporation of more Ta. Since bcc-Hf is unstable, the Hf elastic constants of the hcp structure are compared. The hardness of a material is an essential physical property index that expresses the ability of a material to resist plastic deformation under external forces. The hardness of the material is reflected in Figure 4d as the variation in the hardness of the alloy with Ta content. The addition of Ta increases the hardness of the alloy tremendously. However, this comes at the expense of ductility, a common trade-off in metallic alloys. Figure 4e,f show the Pugh B/G [66] and Poisson’s ratio v [67] for (UNbMoHf)_54−xTa_x with different Ta contents. The alloys generally exhibit toughness characteristics when v > 0.26 or B/G > 1.75 [68,69]. As Ta content increases, B/G and v decrease, indicating reduced toughness. This aligns with the observed hardness–ductility trade-off: higher hardness reduces the material’s ability to deform plastically before failure. The values of E and H_V of the U₁₁Nb₁₁Mo₁₁Hf₁₁Ta₁₀ alloy are 137 GPa and 4.3 GPa, respectively, which are relatively close to the experimental values of 171.14 GPa and 5.29 GPa.

Table 4. The calculated values of lattice constants a (Å), elastic constants C₁₁, C₁₂, C₄₄, bulk modulus B (GPa), shear modulus G (GPa), and Young’s modulus E (GPa).

		a	C11	C12	C44	B	G	E
U	This work	3.432	86.93	155.06	38.47	132.35	134.49	249.27
	Other	3.427 a	86 a	155 a	37 a	132	113	265
Nb	This work	3.304	250.53	135.57	20.08	173.89	31.09	87.96
	Other	3.310 b	243.7 b	135.4 b	19 b	173.00	40.51	112.72
Mo	This work	3.162	473.50	162.24	105.80	265.99	123.54	320.91
	Other	3.169 b	454.2 b	169.2 b	96 b	265.33	130.24	335.77
Hf	This work	3.540	142.50	84.30	66.20	108.760	56.820	145.170
	Other	3.539 c	-	-	-	103 c	67 c	-
Ta	This work	3.319	265.13	161.57	76.38	196.09	65.36	176.48
	Other	3.306 b	266 b	158 b	87 b	194	71.86	191.86
UNbMoTaHf	This work	3.354	224.0	114.69	48.50	150.97	50.78	136.99
	Other	-	275.58	131.58	62.76	179.58	66.31	175.87 d
		3.359 e	256.83	102.33	67.20	153.83 e	70.96 e	184.5 e
	Experiment	3.321 f	-	-	-	-	-	171.14 f

^a Reference [70]; ^b reference [71]; ^c reference [72]; ^d reference [28]; ^e reference [38]; ^f reference [73].

Table 4. The calculated values of lattice constants a (Å), elastic constants C₁₁, C₁₂, C₄₄, bulk modulus B (GPa), shear modulus G (GPa), and Young’s modulus E (GPa).

		a	C11	C12	C44	B	G	E
U	This work	3.432	86.93	155.06	38.47	132.35	134.49	249.27
	Other	3.427 a	86 a	155 a	37 a	132	113	265
Nb	This work	3.304	250.53	135.57	20.08	173.89	31.09	87.96
	Other	3.310 b	243.7 b	135.4 b	19 b	173.00	40.51	112.72
Mo	This work	3.162	473.50	162.24	105.80	265.99	123.54	320.91
	Other	3.169 b	454.2 b	169.2 b	96 b	265.33	130.24	335.77
Hf	This work	3.540	142.50	84.30	66.20	108.760	56.820	145.170
	Other	3.539 c	-	-	-	103 c	67 c	-
Ta	This work	3.319	265.13	161.57	76.38	196.09	65.36	176.48
	Other	3.306 b	266 b	158 b	87 b	194	71.86	191.86
UNbMoTaHf	This work	3.354	224.0	114.69	48.50	150.97	50.78	136.99
	Other	-	275.58	131.58	62.76	179.58	66.31	175.87 d
		3.359 e	256.83	102.33	67.20	153.83 e	70.96 e	184.5 e
	Experiment	3.321 f	-	-	-	-	-	171.14 f

^a Reference [70]; ^b reference [71]; ^c reference [72]; ^d reference [28]; ^e reference [38]; ^f reference [73].

The Zener anisotropy index A_Z = C₄₄/C′ quantifies directional mechanical property differences in materials. A_Z ≈ 1 indicates isotropic behavior, while deviations from 1 reflect anisotropy [74]. Young’s modulus E_[h,k,l] anisotropy describes the material’s directional resistance to elastic deformation under tension or compression [50,51]. Figure 5 shows the variations in E_[h,k,l] and A_Z for (UNbMoHf)_54−xTa_x HEAs across different crystallographic directions as Ta content increases. The results show that the shape of E_[h,k,l] is closer to that of a sphere when A tends to 1. E_[h,k,l] is sensitive to both elemental contents and generally behaves in two categories according to the direction of maximum E_[h,k,l]. When A_Z < 1, the largest (smallest) of E_[h,k,l] is along the <100> (<111>) direction; when A_Z > 1, the largest (smallest) of E_[h,k,l] is along the <111> (<100>) direction [28]. As the Ta content increases, the anisotropy A_Z value of the alloy gradually increases from 0.814 to 1.008, which indicates that the alloy tends to be isotropic. The anisotropy of the alloys depends on the type and concentration of alloying elements, leading to complex interplay effects. Ta addition enhances isotropy by balancing elastic constants, demonstrating that optimal composition design can mitigate anisotropy. The findings suggest that Xu’s screening process for uranium-based HEAs may be suboptimal [28]; it overlooks the significant potential for composition optimization to tailor anisotropy. By strategically adjusting Ta content, the alloy’s mechanical behavior can be fine-tuned toward isotropy, a critical factor for uniform performance in structural applications.

The shear modulus quantifies a material’s resistance to shear deformation. Crystallographic directions with high shear modulus exhibit tightly packed atomic arrangements, requiring dislocations to overcome significant interatomic forces to move—thus resisting deformation. Conversely, directions with lower shear modulus allow easier dislocation motion, making the material more susceptible to shear failure. For high-strength HEAs, achieving both isotropy and high shear modulus is critical. Isotropic shear modulus ensures uniform resistance to deformation across all directions, while high modulus enhances overall strength. Three-dimensional plots of the maximum value G_max and minimum value G_min of the single-crystal shear modulus G_{(n, m)} are shown in Figure 6 and Figure 7. Similarly to the anisotropic 3D plots of E_[h,k,l], the shape of G_{(n, m)} approaches spherical shape when the A_Z approaches 1. In Figure 6, when A_Z < 1, the G_max peaks along the <111>(<100>) direction; when A_Z > 1, the largest (smallest) of G_max shifts to the <100>(<111>) direction. Similarly, the G_min (Figure 7) follows the same directional trend: highest along <111> and lowest along <100> when A_Z < 1; the G_min peaks along <100> and is lowest along <111> when A_Z > 1. Xu et al. highlighted that both E_[h,k,l] and G_(n,m) can exhibit complex alloying effects in HEAs [28]. This study further validates this by demonstrating that Ta content significantly alters the anisotropy of both moduli. By controlling Ta content, the anisotropy of (UNbMoHf)_54−xTa_x can be systematically tuned toward isotropy.

Dislocation nucleation and movement are primary mechanisms of plastic deformation in alloys. The energy factor K quantifies the energy required for dislocation nucleation, with lower K indicating easier nucleation [53]. Figure 8a shows the energy factors of edge dislocations and screw dislocations in (UNbMoHf)_54−xTa_x HEAs as a function of Ta content. The results show that edge dislocations dominate deformation in (UNbMoHf)_54−xTa_x HEAs, as their energy factor K is significantly higher than that of screw dislocations. This suggests edge dislocations require more nucleation energy and are the primary deformation mechanism. Both edge and screw dislocation energy factors increase with rising Ta content, implying that higher Mo content enhances resistance to dislocation nucleation. However, adding Ta further inhibits the nucleation of both dislocation types, as indicated by the upward trend in K with increasing Ta content. To further study the dislocation nucleation characteristics, we calculated the mixed dislocation factor K_mixed for different Ta element contents (Figure 8b). The results show that the energy factor increases monotonically with Ta content, indicating that higher Ta concentrations demand more tremendous energy for dislocation nucleation. This aligns with the observed trend in pure edge and screw dislocation energy factors. The enhanced resistance to dislocation nucleation explains why the alloy U₉Nb₉Ta₉Hf₉Mo₁₈ exhibits the highest hardness.

3.3. Electronic Properties

Figure 9a presents the total density of states (TDOSs) and the integral density of states (ITDOSs) plots of (UNbMoHf)_54−xTa_x high-entropy alloys (HEAs). The TDOSs exhibit no significant variation with increasing Ta content, indicating that the electronic structure stability of the alloy is insensitive to Ta concentration. Non-zero TDOS and projected density of states (PDOSs) near the Fermi level confirm the alloy’s metallic bonding character [31,56,75,76]. The pseudo-gap is the energy difference between the two dominant peaks near E_F. The wider the pseudo-gap, the stronger the covalency of the material. The TDOSs plot in Figure 9a does not exhibit a distinct pseudo-gap, which may be caused by the significant lattice distortion that occurs after the relaxation of the SQS model. With increasing Ta content, both the TDOSs (Figure 9a) and ITDOSs (Figure 9b) significantly decrease. This indicates a reduction in electronic states near E_F. Figure 9c,d are, respectively, the locally magnified plots of the TDOSs at the Fermi level and in the high-energy level region. By combining the Fermi level value and the density of states value at the Fermi level given in

Table 5. The calculated values of Femi energy (eV) and the total dos at E_F = 0 (states/eV).

Alloy	EF/eV	Total Dos (States/eV)
U13Nb13Mo13Hf13Ta2	10.29	76.44
U12Nb12Mo12Hf12Ta6	9.94	73.98
U11Nb11Mo11Hf11Ta10	9.57	71.27
U10Nb10Mo10Hf10Ta14	9.21	70.24
U9Nb9Mo9Hf9Ta18	8.88	68.46

, it can be seen that the addition of Ta lowers the Fermi level, decreases the density of states value at the Fermi level, and shifts the TDOSs towards the high-energy levels.

A higher density of states value at the Fermi level means that there are more electronic states available for electrons to occupy at this energy position. In an alloy system, this is usually related to stronger metallicity. When there are strong covalent bonds between atoms in the alloy, electrons will be more localized between atoms to form covalent bonds, resulting in a decrease in the density of state value at the Fermi level. This is because the formation of covalent bonds makes electrons bound between specific pairs of atoms, reducing the number of electronic states that can move freely near the Fermi level. If the Fermi level enters the high-energy level region where the original density of states is relatively low, it may lead to a decrease in the number of electrons participating in conduction, because electrons at higher energy levels need to overcome greater energy barriers to become free electrons, which will reduce the electrical conductivity of the alloy and weaken its metallicity. The shift in the density of states plotted towards the high-energy levels may imply that the types and strengths of the chemical bonds between atoms in the alloy have changed, which is often related to the enhancement of covalency. When stronger covalent bonds are formed between atoms, electrons will be more strongly bound between atoms, and the energy of electrons will increase, resulting in the shift in the density of states plotted towards the high-energy levels.

Figure 10 presents the projected density of states (PDOSs) of (UNbMoHf)_54−xTa_x HEAs, revealing distinct orbital contributions to its electronic structure. The f-orbital of U is quite prominent near the Fermi level. The d-orbitals of Nb and Mo show strong peaks near E_F, consistent with their status as transition metals. Their d-electrons are critical for bonding stability and material property determination. While the Hf-d orbital contributes to the density of states near E_F, its contribution is relatively minor. This reflects Hf’s tendency to form Hf-Hf metallic bonds within the alloy, reducing its electronic interaction with other elements. As the Ta content increases, the PDOS of Ta’s d, p, and s orbitals gradually become more prominent. The change in the density of states of the Ta-d orbital near the Fermi level is particularly worthy of attention. It reflects the change in the electronic structure within the alloy matrix and orbital overlap dynamics between Ta and other elements (such as in U-Mo-Nb-Ta interactions, etc.). A greater overlap between Ta-d-orbitals and those of neighboring elements implies strengthened chemical bonding between Ta and its alloying partners.

Figure 11a shows the relationship between the -COHP values and energy of all atomic pairs of the first and second nearest neighbors in the unit cell of the (UNbMoHf)_54−xTa_x alloy. COHP is weighted by the Hamiltonian function for the density of states (DOSs). The larger the negative value of COHP, the higher the degree of overlap of the electron densities of the two atoms [77,78]. This means that electrons have a higher probability of hopping between the two atoms and being shared by them, which is also a description of the bonding state. Moreover, the formation of a covalent bond involves the overlapping of atomic orbitals between atoms and the sharing of electron pairs, resulting in a lower-energy bonding state. In this study, we use the average -COHP value (-pCOHP) of all atomic pairs to describe the bonding characteristics of the entire system and calculate the integrated value (-IpCOHP) from −10 eV to the Fermi level (Figure 11b). The calculation results show that as the Ta content increases, the -pCOHP curve shifts towards the bonding region relative to the Fermi level. In addition, the -IpCOHP value also increases with the increase in Ta content, indicating that the addition of Ta is beneficial for enhancing the bonding effect in the (UNbMoHf)_54−xTa_x alloy system.

Figure 12 illustrates the charge density distribution diagram of (UNbMoHf)_54−xTa_x HEAs along the (110) crystal plane. Hf exhibits limited bonding with other metal atoms in the alloy, forming strong Hf-Hf metallic bonds. This self-bonding behavior enhances ductility by reducing interatomic stress concentrations. This aligns with prior studies on Hf-containing alloys, such as those by H on UNbTiMoHf systems [79], where Hf addition reduced strength but increased ductility. Similar observations in refractory high-entropy alloys (RHEAs) with Hf and V [56] further validate Hf’s role in modulating mechanical properties via reduced electron cloud overlap between dissimilar elements. Pronounced electron cloud overlap is observed in specific crystallographic directions for U-Mo, U-Nb, Mo-Nb, and U-Mo-Nb atomic pairs. This indicates strong directional covalent bonding among these elements, reinforcing alloy cohesion. With increasing Ta content, Ta atoms form significant electron cloud overlaps through interactions such as Ta-Mo, Ta-U, Ta-Nb, and U-Mo-Nb-Ta. This enhances covalent bonding strength. Notably, weaker electron interactions between other elemental pairs (e.g., U-Hf or Nb-Hf) are observed, suggesting Ta’s role in reorganizing the alloy’s electronic structure. The increased covalent bonding strength (via Ta’s orbital overlaps) directly correlates with the observed rise in alloy hardness with Ta content. This is consistent with covalent bonding’s role in strengthening materials through localized electron sharing.

The electron-localization function (ELF) shows the degree of electron localization in space. When a covalent bond is formed, the atomic orbitals of the bonding atoms overlap, and the electron cloud is densely distributed in the region between the two atomic nuclei, thus forming a region of high electron density. Figure 13 shows the 3D ELF diagram of HEAs. The shape and coverage of the yellow regions reflect the degree of electron localization. If the yellow part in a certain region is compact and concentrated between atoms, it indicates a high degree of electron localization at these positions, suggesting the possible existence of strong chemical bonding; if the yellow region is more diffuse, the degree of electron localization is low and the electron distribution is more dispersed. It can be seen that U atoms have a relatively large number of outer-shell electrons and easily form bonds with Mo, Nb, and Ta. As the content of Ta increases, the degree of electron localization in the alloy significantly increases, indicating that Ta can enhance the inter-atomic bonding within the system, resulting in an increase in the covalent nature and a decrease in the metallic nature of the system. This is consistent with the results of the total density of states diagram.

This study has successfully revealed the electron interaction mechanism at the atomic scale of uranium-containing high-entropy alloys and clarified the correlation between it and the macroscopic mechanical properties. However, the first-principles calculation method ignores the influences of complex factors such as compositional segregation, precipitated phases, phase transformations, and processing history. Due to the restrictions of nuclear material regulatory policies, some of the calculation results of this study cannot be experimentally verified for the time being. Nevertheless, the theoretical results obtained from this study provide crucial theoretical evidence for a deep understanding of uranium-containing high-entropy alloys, marking a breakthrough that previous studies have not achieved.

4. Conclusions

To improve the mechanical properties of the UNbMoHfTa HEAs, the effects of Ta alloying on the phase structure, lattice constant, density, elastic properties, and electronic structure were calculated based on first principles. The results show that the (UNbMoHf)_54−xTa_x high-entropy alloy can form a stable body-centered cubic (BCC) high-entropy alloy. With the Ta element content increase, the lattice constant of the (UNbMoHf)_54−xTa_x high-entropy alloy decreases and the density increases. As the Ta content increases, the ability of the (UNbMoHf)_54−xTa_x alloy to resist volumetric deformation, elastic deformation, and shear deformation is enhanced, and the hardness increases accordingly. According to the Cauchy pressure, Pugh ratio B/G, and Poisson’s ratio, adding Ta will reduce the toughness of the alloy and increase its anisotropy. The dislocation energy factor results indicate that adding Ta can inhibit dislocation nucleation and produce a remarkable strengthening effect. The incorporation of Ta increases the covalency of the alloy (as demonstrated by the lower electron density of states near the Fermi level) while reducing its metallic character. The d-orbitals of Ta make a significant contribution to TDOS. As the content of Ta in the alloy increases, the bonding characteristics of the alloy system are enhanced. Ta forms a large number of electron cloud overlaps with Nb, Mo, and U, which can be evidenced by the enhanced orbital interactions (for example, the interactions among Ta-Mo, Ta-Nb, and U-Mo-Nb-Ta). Moreover, as the content of Ta in the alloy increases, the localization degree of electrons in the alloy is significantly improved, ultimately leading to an enhancement of the covalency and a weakening of the metallic character of the system.