Numerous alloying elements can improve the mechanical properties of NbMoTaW refractory high-entropy alloys (RHEAs), but the relationship between the alloying of different elements and the changes in the microstructure and mechanical properties of RHEAs is ambiguous. The first principles based on density functional theory are used to calculate the lattice parameters, electronic properties, and elastic properties of NbMoTaW-based RHEAs to reveal the microstructure and mechanical properties of NbMoTaW RHEAs with alloying elements of the same period or subgroup. The melting point, lattice constant, and mass density of NbMoTaW-based RHEAs are controlled by the alloying elements such as Cr, V, Ti, Zr, and Hf. Valence electron concentration (VEC) is a key factor affecting the electronic structure and mechanical properties of NbMoTaW-based RHEAs. High VEC can improve the mechanical properties of RHEAs but reduce the ductility. Cr-alloying has a significant effect on mechanical properties of NbMoTaW RHEAs, and Ti has a significant effect on ductility. The orbital electronic interactions between the alloying elements and Nb, Mo, Ta, and W atoms shown by the density of states and electron density difference may explain the relationship between VEC and the mechanical properties of RHEAs. Our results provide the underlying insights needed to guide the design of NbMoTaW RHEAs with excellent mechanical properties.
High-entropy alloys (HEAs) are composed of at least four or five elements with equal or almost equal molar ratios.1 HEAs are prone to form single-phase body centered cubic (BCC), face centered cubic (FCC), or hexagonal close-packed (HCP) structures and exhibit excellent material properties, such as high strength, hardness and wear resistance, corrosion resistance, and high temperature oxidation resistance.2–4 Among the numerous HEA systems, refractory high-entropy alloys (RHEAs) have attracted extensive attention due to their excellent thermal stability,5 high strength,6 and improved friction properties7 at high temperatures.
Senkov8 first proposed a new type of RHEA in 2010: Mo25Nb25Ta25W25; it is completely composed of refractory metal elements. In particular, NbMoTaW-based RHEAs have higher melting points and mechanical strength, excellent thermal stability, and high softening resistance at high temperature than nickel-based alloys and other superalloys.9,10 Therefore, they are considered potential high-temperature load-bearing structure and thermal protection system material candidates in aerospace, nuclear energy, defense, metallurgy, and other fields of extreme environments at high temperatures.11–13 However, the difficulty in the application of NbMoTaW RHEAs as a structural material is that they are very brittle at room temperature. Alloying is an effective method to improve the mechanical properties of NbMoTaW RHEAs, such as vacuum arc melting, mechanical alloying, and the sputtering technique.14 V-alloying increases the compressive yield strength and fracture strength of NbMoTaW alloys,6,15 enhances the bonding strength of NbMoTaW RHEAs, and improves their mechanical properties,16 and the hardness of nanocrystalline VNbMoTaW RHEAs reaches 11.4 GPa at 1150 °C compared with NbMoTaW alloys.17 Ti-alloying can increase the ductility of NbMoTaW RHEAs by 9.6% and the yield strength by 459 MPa at room temperature,13 and the yield strength of NbMoTaWTi RHEAs is as high as 586 MPa at 1200 °C.18 Re-alloying increases the hardness, strength, and ductility of NbMoTaW alloys first and then decreases them. Re0.5NbMoTaW alloys exhibit the best mechanical properties, and their yield strength and room temperature ductility are increased by 89 MPa and 4.4%, respectively, compared with NbMoTaW.11
The good agreement between simulated results and the experimental data provides a feasible method for designing and predicting the performance of RHEAs by first-principles calculation.16 Many alloying elements can improve the mechanical properties of NbMoTaW RHEAs, but how to improve them and the rules between the alloying elements and properties are ambiguous. Therefore, we adopted the first-principles method to systematically study the effect of alloying elements of the same period or subgroup on the microstructure and mechanical properties of NbMoTaW RHEAs, and the alloying elements include Cr, V, and Ti of the same period and Ti, Zr, and Hf of the same subgroup.
A. Calculation models and theoretical methods
In this work, five elements, namely, Ti, V, Cr, Zr, and Hf are alloyed to NbMoTaW RHEAs. Ti, V, and Cr belong to the fourth period, Ti, Zr, and Hf belong to the fourth subgroup, and their atomic numbers gradually increase. As shown in Fig. 1, supercell solid solution alloy models of 1 × 1 × 2 and 1 × 1 × 5 dimensions containing four and ten atoms were established on the basis of a single BCC cell using special quasi-random supercell (SQS) methods.19 All properties of the solid solution are calculated using the CASTEP package based on the plane wave pseudopotential method and density functional theory (DFT).20 The electronic exchange-correlation functional was described using the Perdew–Burke–Ernzerhof (PBE) function21 in Generalized Gradient Approximation (GGA). The electron–ion interaction of all the elements is described using norm conserving pseudopotential in reciprocal space. The valence electrons of the alloying elements are Cr 3d54s1, V 3d34s2, Ti 3d24s2, Zr 4d25s2, and Hf 5d26s2. The energy cutoff of the plane-wave basis set was 600 eV, the tolerance of the self-consistent field using the Pulay density mixing method was 5 × 10−7 eV/atom, and the special k-point sampling in the Brillouin zone was integrated using the Monkhorst–Pack method. The k-point meshes in the reciprocal space of different crystal models were set to be 12 × 12 × 6 for BCC NbMoTaW and 12 × 12 × 2 for BCC NbMoTaWCr, NbMoTaWV, NbMoTaWTi, NbMoTaWZr, and NbMoTaWHf. The geometry optimization of all crystal models adopted the Broyden–Flecher–Goldfarb–Shanno scheme. The criteria of self-consistent convergence are as follows: the total energy was 5 × 10−6 eV/atom, the maximum force was 0.01 eV/Å, the maximum stress was 0.02 GPa, and the maximum displacement was 0.0005 Å.
B. Benchmark calculations
To assess the performances of the DFT total-energy approach in our study, test calculations of the lattice constants and elastic constants of NbMoTaW RHEAs were performed. The calculation results of the lattice constant are between the experimental results,8,11,13,14,16,18,22 3.193 Å–3.226 Å, and the error range is 0.09%–0.93%. The differences in lengths of a (b) and c axes between the calculated results and the corresponding calculated results are 0.09% and 0.33% compared with results by Hu et al.16 (Table I), and the tiny differences can be ignored. Since no experimental results of elastic constants are available, the calculated values of elastic constants16 are adopted for comparison. The differences in C11, C22, and C44 between this work and the previous study are 7.91%, 0.63%, and 11.39%, respectively (Table I). Notably, even if the same package and pseudopotential are used, we have a larger solid solution model and more refined calculation parameters, which may lead to discrepancies in elastic constants. The difference between our calculated elastic modulus and the results by Hu et al.16 is 0.61%–1.69% (Table I), which shows that although the crystal structure and calculated parameters have a greater influence on the elastic constants, the elastic modulus can be in good agreement. Such close agreement between the experimental and calculated values of lattice constants and elastic constants demonstrates the validity of this computational project.
|.||Lattice constants (Å) .||Elastic constants (except B/G and ν, the unit is GPa) .|
|a = b .||c .||C11 .||C12 .||C44 .||B .||G .||E .||B/G .||ν .||Reference .|
|.||Lattice constants (Å) .||Elastic constants (except B/G and ν, the unit is GPa) .|
|a = b .||c .||C11 .||C12 .||C44 .||B .||G .||E .||B/G .||ν .||Reference .|
III. RESULTS AND DISCUSSION
A. Crystal structures
Solid solution characteristic parameters such as valence electron concentration (VEC), atomic size difference (δ), and ratio of thermodynamic entropy to enthalpy (Ω) are usually used to describe the phase structure of RHEAs. VEC theory23 proposes that the alloy forms a BCC structure when 4.33 < VEC < 7.55. The VEC of the six calculated alloys are all between 4.33 and 7.55 [Fig. 2(a)], indicating that they are BCC structures. The VEC of the alloy increases with the alloying of Ti, V, and Cr elements and does not change with the alloying of Ti, Zr, and Hf elements. The VEC change in the alloys should be controlled by the VEC of the alloying elements. Atomic size difference and the ratio of thermodynamic entropy to enthalpy are used to predict the formation of random solid solutions for various multi-component alloys. The atomic size difference of the alloy increases with the alloying of Ti, V, and Cr elements and increases first and then decreases with the alloying of Ti, Zr, and Hf elements [Fig. 2(b)]. The change in the difference value between the radius of the alloying element and the average atomic radius of NbMoTaW is the same as the atomic size difference of the alloy, that is, the atomic size difference increases as the difference value increases. Yang and Zhang24 proposed that the alloy can form a stable random solid solution when δ ≤ 6.6% and Ω ≥ 1.1. The δ and Ω of six refractory high entropy alloys meet the conditions of δ ≤ 6.6% and Ω ≥ 1.1, indicating that all the alloys are random solid solutions [Fig. 2(c)]. The melting point of the alloy increases first and then stabilizes with the alloying of Ti, V, and Cr elements and increases with the alloying of Ti, Zr, and Hf elements [Fig. 2(d)]. However, the melting point of the other five types of alloys is lower than that of NbMoTaW alloys. This is because the lower melting point of the alloying element leads to a decrease in the melting point of the corresponding refractory high-entropy alloy.
Structure stability is the basis for the first-principles calculation of microstructure and properties of alloys. As shown in Fig. 3, our phonon calculations of NbMoTaW-based RHEAs show stable phonons without an imaginary frequency, indicating that all the alloys are stable. It is worth noting that the phonon dispersion of NbMoTaW RHEAs has 12 branches, while that of NbMoTaWX (X = Cr, V, Ti, Zr, and Hf) RHEAs has only ten branches due to the degeneracy of the acoustic modulus. For NbMoTaWX (Cr, V, Ti, Zr and Hf) RHEAs, the phonon dispersion should have 15 branches in each wavevector, but the five groups of branches have the same frequency and degeneracy and finally only ten branches. Degeneracy exists in all wavevectors in the Brillouin zone. Therefore, the phonon dispersion of the five-element RHEAs has only ten branches. The lattice constants and mass density of the alloy are calculated and shown in Fig. 4. The arithmetic mean cell length aavg25 of the supercell is used as the lattice constant of the alloy. aavg is calculated as follows:
where n represents the number of unit cells in the supercell and ai, bi, and ci are the lattice constants of the supercell. The lattice constant of the alloy decreases with the alloying of Ti, V, and Cr elements and increases first and then decreases with the alloying of Ti, Zr, and Hf elements. The density of the alloy increases with an increasing atomic number. The changes in the lattice constant and density of the alloys are the same as the changes in the atomic radius and mass density of the alloyed Cr, V, Ti, Zr, and Hf elements, indicating that the radius and density of the alloying elements control the lattice constant and density of the corresponding alloys. Our results show that the parameters of the alloying elements play a vital role in the physical properties of the alloy.
B. Electronic structure
The band structure of NbMoTaW-based RHEAs and the high symmetrical direction of the Brillouin zone are shown in Fig. 5. The horizontal red dotted line in the figure indicates the position of the Fermi level (level = 0 eV). The bandgap is zero due to the overlap between the bottom of the conduction band and the top of the valence band, which indicates that all RHEAs exhibit metallic characteristics. After alloying Cr, V, Ti, Zr, and Hf to NbMoTaW RHEAs, the band overlap increased significantly, indicating that the alloying elements enhance the bonding strength of the alloy system. The other five types of RHEAs have a more compact energy band than NbMoTaW RHEAs, which may be caused by the strong interaction of orbital electrons between Cr, V, Ti, Zr, and Hf atoms and the other four atoms.
The total and partial densities of states of NbMoTaW-based RHEAs are shown in Fig. 6. The Fermi level is set at 0 eV. The value of the total density of states at the Fermi level is not zero, which means that all RHEAs exhibit metallic characteristics, which is consistent with the band structure results. All the six calculated RHEAs have a similar pattern of the density of state, but the total densities of states of the other five types of RHEAs are higher than that of NbMoTaW RHEAs, indicating that NbMoTaW RHEAs have no phase transition and stronger metal properties after alloyed elements. The alloying elements make the peak of the total density of states of NbMoTaW-based RHEAs at low energy levels shift to the direction of high energy levels, and the peak is more obvious near the Fermi level, which indicates that the stability of the crystal structure of NbMoTaW-based RHEAs decreases. The peak positions of the total density of states of NbMoTaWTi, NbMoTaWZr, and NbMoTaWHf are basically the same, indicating that their crystal structure stability is similar. At the same time, the alloying elements make the strong peaks sharper, and the conduction band of the high-energy region becomes smoother from sawtooth, indicating that the alloying elements make the bonding more sufficient and the interaction of orbital electrons is stronger.
The pseudogap is the energy difference between the two strong peaks on both sides of the Fermi level, and the covalency of the alloy system becomes stronger with a wider pseudogap.26 The pseudogap of NbMoTaW RHEAs and RHEAs alloyed with Cr, V, Ti, Zr, and Hf is 5.6875 eV, 3.3049 eV, 3.5298 eV, 3.9954 eV, 3.7911 eV, and 3.9699 eV, respectively. The pseudogap decreases with the alloying of Ti, V, and Cr elements and is basically similar with the alloying of Ti, Zr, and Hf elements. That is to say, after alloying Cr, V, Ti, Zr, and Hf to NbMoTaW RHEAs, the covalency of RHEAs is weakened, which may increase the number of metal bonds between atoms. This may mean that the bonding strength is enhanced and the mechanical properties of NbMoTaW RHEAs are improved.
The partial densities of states are calculated to further understand the contribution of the electrons of the alloying element to the total density of states of RHEAs (Fig. 6). The results show that the valence band at the Fermi level is formed by the hybridization of the d orbital electrons and the contribution of the s orbital electron is slight. The electron hybridization of the alloyed Cr, V, Ti, Zr, and Hf elements and other elements of the p orbital and d orbital forms the conduction band of the five-component RHEAs, which may be the reason for the flattening of the total density of states, which means that there is a strong interaction between atoms.
Electron density and electron density difference are used to describe the bond characteristics and electron transfer between atoms. The electron density and the electron density difference of NbMoTaW-based RHEAs along the (110) crystal plane are shown in Fig. 7. The electron cloud is evenly distributed around the atom’s surface and has no obvious directionality, indicating that a metal bond is formed [Figs. 7(a)–7(f)].27 After alloying Cr, V, Ti, Zr, and Hf, the overlap of electron clouds between Nb and Mo, Mo and Ta, and Ta and W becomes stronger, indicating that their electron interaction is enhanced. As shown in Figs. 7(g)–7(l), red and blue represent gain and loss of electrons, respectively. The color change indicates that the ability of atoms to lose electrons decreases with Cr, V, and Ti, while the ability of Ti, Zr, and Hf atoms to lose electrons is close, which is related to the valence electrons of the atoms. That is to say, the strong orbital electron interaction between the alloying elements and other atoms may cause the increase in the metal bonds formed between Nb, Mo, Ta, and W.
C. Mechanical properties
Mechanical properties are important performance indicators of materials and are affected by crystal structure and electronic structure. We calculated the elastic properties of NbMoTaW-based RHEAs to evaluate their mechanical properties and analyzed the relationship between mechanical properties and internal structures. The mechanical stability criteria of a refractory high entropy alloy of the tetragonal phase are given by28
The elastic constants Cij of NbMoTaW-based RHEAs are shown in Table II. All alloys are mechanically stabile at ground state because their elastic constants meet the stability criterion. The tetragonal shear constant (c′) can also be used to describe the structural stability of the alloys. As shown in Table II, c′ increases with the alloying of Ti, V, and Cr elements and slightly changes with the alloying of Ti, Zr, and Hf elements but is basically similar. Among these alloys, NbMoTaWCr maintains the most stable structure corresponding to the largest c′. This change is consistent with the result of the total density of states. c′ is determined by BCC–FCC energy difference of a given system, which in turn is determined by band filling.29 Our result indicates that the BCC structural stability increases with the alloying of Ti, V, and Cr elements and the stability increases with the alloying of Ti, Zr, and Hf elements. In the same period, the number of filled electrons in the d orbital band increases with the increase in the atomic number of the alloying elements, and the corresponding structural stability increases. However, the number of filled electrons in the d orbital band of the same subgroup remains unchanged, and the structural stability is similar.
|Alloys .||C11 .||C33 .||C44 .||C66 .||C12 .||C13 .||c′ .|
|Alloys .||C11 .||C33 .||C44 .||C66 .||C12 .||C13 .||c′ .|
The elastic properties of the alloy including bulk modulus (B), shear modulus (G), and Young’s modulus (E) are important parameters of the mechanical properties of the material, which are evaluated by the Voigt–Reuss–Hill approximation method based on the elastic constants.30,31 As shown in Fig. 8(a), B, G, and E increase with the alloying of Ti, V, and Cr elements and are basically close with the alloying of Ti, Zr, and Hf elements. This change is consistent with VEC, pseudogap, electron density difference, and c′. The elastic modulus is mainly used to describe the ability of the alloy to resist elastic deformation. The ability of the alloy to resist deformation increases as the elastic modulus increases, as does stiffness. Our results indicate that the stiffness and the ability to resist volume or shear deformation of the alloy increase with the alloying of Ti, V, and Cr elements, but there is basically no change with the alloying of Ti, Zr, and Hf elements. From a microperspective, the elastic modulus is determined by the strength of chemical bonds between constituent elements, which in turn is determined by the valence electron of the alloying element. Enhanced bonding strength and strong interaction between atoms can improve the mechanical properties of the alloys. Therefore, the elastic properties have the same changes as the VEC, pseudogap, and electron density difference of the alloy. The elastic modulus of NbMoTaWCr is the largest of all alloys and corresponds to its most stable structure and highest VEC. This means that Cr-alloying can improve the stiffness of the NbMoTaW RHEA.
The Debye temperature correlates elastic properties with thermodynamic properties and can be obtained from the average sound velocity based on the elastic constants.31–33 Hardness is a reflection of the comprehensive mechanical properties of the material, which can be predicted through bulk modulus and shear modulus.16 As shown in Fig. 8(b), the Debye temperature of the alloy has the same change as hardness, which is consistent with previous work.34 Debye temperature and hardness increase with the alloying of Ti, V, and Cr elements, but the difference is smaller with the alloying of Ti, Zr, and Hf elements. Among all alloys, NbMoTaWCr has the highest Debye temperature and hardness, which is a 17.91% increase in hardness compared to NbMoTaW. The hardness and Debye temperature of NbMoTaW-based RHEAs have the same changes in the elastic modulus and VEC. It can be seen from the density of states and electron density difference that the enhancement in the metal bond and the strong electronic interaction related to the d orbital control the change in the mechanical properties of the alloy. Therefore, the hardness and Debye temperature of NbMoTaW-based RHEAs are also consistent with the changes in the density of states and electron density differences.
Parameters on toughness and brittleness of the material reflect its ability of plastic deformation under external forces, including Cauchy pressure (C12–C44), Pugh ratio (B/G), and Poisson’s ratio (ν). RHEAs show a toughness characteristic when B/G > 1.75, ν > 0.26, and (C12–C44) > 0; otherwise, it is brittle.35 All the RHEAs calculated here are tough [Fig. 9(a)]. The changing trends of B/G, ν, and C12–C44 of the alloy indicate that the toughness or plastic deformation ability of the alloys decrease with the alloying of Ti, V, and Cr elements and the change is more stable with the alloying of Ti, Zr, and Hf elements. Our results provide supporting evidence for the higher ductility of alloys containing molybdenum and tungsten as the VEC decreases.36–39 The reduction in VEC leads to the reduction in the Fermi level of the system, the earlier occurrence of shear instability, and the increase in the ductility of the alloy,36 which means that the lower tetragonal shear modulus of the alloy has higher ductility. Therefore, the toughness changes of NbMoTaW-based RHEAs after alloyed elements are opposite to the tetragonal shear modulus and VEC. It is worth noting that NbMoTaWCr has the smallest B/G, ν, and C12–C44 among the six calculated RHEAs, that is, Cr-alloying will make the NbMoTaW RHEA more brittle, while other elements will improve the toughness of the alloy, and Ti has the most significant effect.
We evaluated the elastic anisotropy of all alloys.40 The smaller difference between the elastic anisotropy coefficient (AU) and 0 indicates smaller elastic anisotropy. The results show that the anisotropy of the alloy increases with the alloying of Ti, V, and Cr elements [Fig. 9(b)]. AU is between 1.58 and 1.84 with the alloying of Ti, Zr, and Hf elements. The elastic anisotropy of the alloy has the same trend as the toughness but the opposite of the VEC change. That is to say, the elastic anisotropy of NbMoTaWCr is the smallest among all alloys, and the corresponding VEC is the largest, and the toughness is the worst. For TiZrNbMoVx RHEAs, Tian et al.41 showed that RHEAs become elastically isotropic for a VEC of 4.72. This is different from the research results in this article, which may be caused by different alloys. More about the relationship between the alloy’s VEC and anisotropy requires further theoretical and experimental exploration.
First principles were used to calculate the phase structure, electronic structure, and mechanical properties of NbMoTaW-based RHEAs. The solid solution characteristic parameters indicate that NbMoTaW-based RHEAs are single-phase BCC solid solutions after alloying Cr, V, Ti, Zr, and Hf elements. The VEC, atomic size difference, melting point, lattice constant, and mass density of RHEAs are controlled by alloying elements. The mechanical properties of NbMoTaW RHEAs increase with the alloying of Ti, V, and Cr elements and are basically similar to the alloying of Ti, Zr, and Hf elements, but the toughness changes are opposite. That is, a higher VEC improves the mechanical properties of NbMoTaW-based RHEAs but reduces the ductility of the material. The electronic structure shows that the improvement in mechanical properties is attributed to the orbital electron hybridization of the alloying elements with Nb, Mo, Ta, and W atoms, which enhances the interaction between atoms and the enhancement of metal bonds. Among all the alloying elements, Cr has a significant effect on stiffness and hardness, and Ti has a significant effect on toughness. Our results show that first principles are an effective method for designing and predicting RHEAs’ performance.
This work was supported by the Foundation of the Key Laboratory of Earthquake Forecasting, the Institute of Earthquake Forecasting, CEA (Grant No. 2019IEF0101-1), and the National Natural Science Foundation of China (Grant Nos. 41174071 and 41573121).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The data that support the findings of this study are available from the corresponding author upon reasonable request.