Ab initio-predicted micro-mechanical performance of refractory high-entropy alloys

Recently developed high-entropy alloys (HEAs) consisting of multiple principal elements represent a new field of metallurgy and have demonstrated appealing properties for a wide range of applications. Using ab initio alloy theory, we reveal the alloying effect on the elastic properties and the ideal tensile strength (ITS) in the [001] direction of four body-centered cubic (bcc) refractory HEAs based on Zr, V, Ti, Nb, and Hf. We find that these HEAs show high elastic anisotropy and large positive Cauchy pressure, suggesting good extrinsic ductility. Starting from ZrNbHf, it is found that the ITS decreases with equimolar Ti addition. On the other hand, if both Ti and V are added to ZrNbHf, the ITS is enhanced by about 42%. An even more captivating effect is the ITS increase by about 170%, if Ti and V are substituted for Hf. The alloying effect on the ITS is explained by the d-band filling. An intrinsic brittle-to-ductile transition is found in terms of the failure mode under uniaxial tension. These investigations suggest that intrinsically ductile HEAs with high ideal strength can be achieved by controlling the proportion of group four elements to group five elements.

material 14 . The ideal strength is involved in fracture theory and the nucleation of defects 13,[15][16][17] and has been accepted as a mechanical parameter for the design of high performance materials, especially for nano-or micro-size applications 13,[18][19][20][21][22] . Today, no information on the intrinsic tensile properties of HEAs is available.
In this work, we reveal the compositional effect on the micro-mechanical properties of four HEAs in bcc phase composed of the refractory elements Ti, Zr, Hf, V, and Nb. The studied elastic constants and polycrystalline elastic moduli describe the mechanical properties in the small deformation region, and allow conclusions on extrinsic ductility/brittleness. The ideal tensile strength (ITS) displays the mechanical behavior for large elongations of ideal defect-free crystals where stress-strain relations are nonlinear. The ITS of the constituent refractory elemental solids in bcc structure (V, Nb, and Ta) is well understood 15 , the description of the ITS in multicomponent solid solutions is, however, rather limited. Recently, Li et al. determined the compositional dependence of bcc ternary V-based and binary Fe-based solid solutions from first-principles employing the coherent-potential approximation (CPA) to model chemical disorder 23,24 . Here we follow a similar computational procedure to establish the ITS of the present HEAs.

Results and Discussion
The equilibrium volume (expressed here via the Wigner-Seitz radius) is the first physical parameter which can be used to assess the accuracy of the ab initio calculations. Table 1 compares the Wigner-Seitz radius from our calculation (w t ) to those from experiments (w exp ) 7,9,10,12 . For all HEAs considered here, the theoretical value is close to the experimental data. Furthermore, for all HEAs, the bcc structure is predicted to be energetically more stable than the fcc one (cf. also discussion below), in line with the observations.
The single-crystal elastic constants and polycrystalline elstic moduli are listed in Table 1. The calculated elastic constants demonstrate that all present refractory HEAs are elastically stable (c 44 > 0, c c 11 12 > , and c 11 + 2c 12 > 0). The structure stability of a cubic solid under tetragonal deformation is described by the tetragonal shear constants c′ (c′ = (c 11 − c 12 )/2). Compared to ZrNbHf, titanium addition decreases c′ , meaning that the ability of ZrTiNbHf to resist tetragonal shear decreases. However, if we substitute Hf in ZrNbHf by Ti and V, c′ becomes two times larger than that of ZrNbHf, which indicates that V strongly stiffens the bcc structure under tetragonal deformation. The Zener anisotropy ratio measures the anisotropy of elastic behavior, A Z ≡ c 44 /c′ (for an isotropic cubic crystal, A Z = 1). According to our results, c 44 is almost constant for the investigated HEAs. The trend of the Zener anisotropy ratio is, hence, mostly dictated by the trend of c′ . The A Z data listed in Table 1 imply that all HEAs possess high elastic anisotropy. The anisotropy of ZrNbHf increases with adding Ti, however, decreases with introducing an equilmolar concentration of V simultaneously to Ti. The ZrVTiNb HEA possesses the largest Young's modulus Y, and ZrTiNbHf exhibits the lowest one. The same behavior was found for the bulk modulus B and shear modulus G, however, their changes are smaller than Y.
The extrinsic ductility/brittlness of materials is connected with the magnitude of the B/G ratio and the sign of the Cauchy pressure (c 12 − c 44 ) by Pugh 25 and Pettifor 26 , respectively. A simple physical picture behind the Pugh criterion is that B reflects the crystal cohesion and thus the resistance against cleavage, whereas G gives the opposition against dislocation movement. Polycrystalline materials are usually ductile when their B/G ratio is greater than 1.75 and when (c 12 − c 44 ) is positive; otherwise they are in the brittle regime. From Table 1, one can see that the B/G ratios are larger than 3.5, and (c 12 − c 44 ) are positive and large, suggesting that all investigated HEAs possess good extrinsic ductility.
Turning in the following to the ITS of HEAs, we note that the [001] direction was already identified as the weakest direction of bcc systems 23,24,[27][28][29] . Hence, here we consider the ITS of the four HEAs upon uniaxial loading along the [001] direction. The calculations followed the same quasi-static procedure as previously described 15,24 . Accordingly, the tensile stress σ(ε) was obtained from where E is the total energy per atom and Ω (ε) is the volume per atom at a given tensile strain, ε. At each value of the strain, the two unit cell lattice vectors perpendicular to the [001] direction were relaxed allowing for a possible symmetry lowering deformation relative to the initial body-centered tetragonal (bct) symmetry (see Fig. 1(b)), e.g., a branching to an orthorhombic (ort) symmetry (see Fig. 1(c)). The latter bifurcation was previously reported to occur for the refractory elements V, Nb, and Ta 15 . The first maximum on the stress-strain curve defines the ITS (σ m ) with corresponding maximum strain (ε m ). Figure 2 displays the tensile stress-strain relationships for bcc ZrNbHf, ZrVTiNb, ZrTiNbHf, and ZrVTiNbHf random alloys. The calculated ITSs and the corresponding maximum strains are listed in Table 1. We found that the ITS of the ternary ZrNbHf alloy amounts to 2.64 GPa, and slightly decreases to 2.28 GPa by adding the fourth element Ti. However, if we add Ti and V to ZrNbHf, the ITS is enhanced by nearly 42% with respect to the value of ZrNbHf. An even more pronounced impact of alloying on the ITS occurs if Hf in ZrNbHf is substituted with equiatomic Ti and V, which increases the ITS by about 170% (Table 1).
We found that three alloys, namely ZrNbHf, ZrTiNbHf, and ZrVTiNbHf, maintain bct geometry during tension along the [001] direction for strains up to slightly larger than ε m . Hence, these alloys fail by cleavage of the (001) planes and are representatives of bcc materials termed intrinsically brittle owing  For ZrVTiNb, a branching from the tetragonal (bct) to the orthorhombic (ort) deformation path occurs at a strain of 12.7% < ε m as shown more clearly in the inset; for larger strains, both the fully-relaxed orthorhombic and the constrained to bct stress-strain curves are plotted. Branching was not observed for the other alloys in the considering strain interval.
to their particular deformation and failure behaviors during tension 14 . In the case of the ZrVTiNb alloy, a symmetry lowering branching from bct to orthorhombic symmetry, caused by an elastic shear instability, occurs immediately before the maximum stress on the bct deformation path could be reached. This branching leads to a slight decrease of the ITS of ZrVTiNb with respect to the maximum value on the stress-strain curve assuming that branching is prohibited (the geometry was constrained to bct for these additional calculations). The constrained stress-strain curve and the stress-strain curve accounting for branching of ZrVTiNb are compared in Fig. 2. The aforementioned elastic instability can also be understood as an activated shear instability of the 111 {112} slip system (the resolved shear stress reaches the ideal shear stress under superimposed tension in [001] direction) 30 . The resolved shear stress for the 111 {112} slip system at the instability is 3.5 GPa for ZrVTiNb, which is a good estimate of its ideal shear strength (τ m ) for the same slip system without superimposed tension 29  = ′/( ′ + ), amounts to 0.1 in agreement with other bcc systems 15,31 . Owing to this particular failure behavior, the ZrVTiNb alloy is referred to as being intrinsically ductile, which represents the preferred failure mode over the intrinsically brittle one 14 . Thus, ZrVTiNbHf and ZrVTiNb display an intrinsic brittle-to-ductile transition in terms of their intrinsic failure mode which is altered upon changing the composition of the alloy, i.e., removal of Hf.
In order to understand the trend of the ITS for the present HEAs, we first realize that the alloy with the highest ITS (ZrVTiNb) contains two elements from the fourth group and two from the fifth group, which can be denoted by the compositional ratio 2:2. In contrast, ZrTiNbHf, possessing the lowest ITS, has a composition dominated by elements from group four (ratio 3:1). The alloy with moderate ITS (ZrVTiNbHf) consists of totally five elements with ratio 3:2. This simple analysis indicates that the ITSs of refractory HEAs are determined by the ratio of constituent transition metal elements belonging to the fourth group and fifth group. Obviously, the smaller is the ratio the higher is the ITS. It further suggests that the ITS of refractory HEAs could be tuned by varying the total number of equimolar transition metals elements. Both observations are explicated in the following.
It is well known that the cohesive properties of the three transition metal series are mainly determined by the d-electrons, and the observed sequence of stable crystal structures as well as trends of cohesive properties originate from the gradual filling of the d-band [32][33][34][35] . For a more elaborate analysis of the ITSs, we show in the following that the ITS of the present HEAs can be understood by band-filling arguments. To this end, we employ the structural energy difference (SED) model of the ITS recently suggested for nonmagnetic V-Ti-Cr solid solutions 23 Accordingly, the ITS of an intrinsically brittle, bcc stable material is correlated with the fcc-bcc SED, σ m ∝ Δ E SED ≡ E fcc − E bcc . In this model, the energy difference is computed at constant volume equal to the bcc equilibrium volume. Although ZrVTiNb was found to be intrinsically ductile, we also apply the SED model to ZrVTiNb, since its ITS is only slightly lowered by branching as discussed above. Figure 3(a) shows a very strong correlation between the ITS and the SED for the presently investigated HEAs, i.e., higher fcc-bcc SED corresponds to higher ITS. Encouraged by the strong correlation between the ITS and the SED, we plotted the determined Δ E SED s for the present HEAs as a function of the d-band filling in Fig. 3(b). The d-band filling of alloys was obtained by averaging the number of valence d-electrons of the constituent in the alloy. The data for the HEAs in Fig. 3(b) clearly reveals that the SED is a monotonic, nearly linear function of the average d-occupation.
Electronic structure theory showed that the fcc-bcc SED itself follows a regular pattern for all three transition metal series, as a function of the number of valence d-electrons 32,36 . For a d-band filling corresponding to bulk Ti (Zr, Hf), Δ E SED < 0, but for a d-band filling corresponding to bulk V (Nb, Ta), Δ E SED > 0. This implies that Δ E SED increases as the number of d 3 (4d, 5d) electrons increases and changes sign. Both properties of Δ E SED are confirmed by the present theory and are shown for the 3d (Ti, V) and 4d (Zr, Nb) elements and their equimolar random solid solutions (Ti-V, Zr-Nb) in Fig. 3(b). Surprisingly, the SED of the multicomponent HEAs follows very closely the trend of the SED inherent to the 3d or 4d transition metal series (for the 5d series, see Refs 32,36). This close correlation confirms the fact that the SED of the present refractory HEAs is mainly determined by their average d-occupation number. Combining the above discussion and the data in Fig. 3, our results show that the ITS of the present refractory HEA is correlated with their average d-occupation, i.e., the higher the d-occupation, the larger the ITS. We notice that c′ increases monotonically with the ITS, and consequently with the average d-occupation number. No similar correlation was found in the case of magnetic bcc alloys 24 .
It is important in this context to realize that, although Ti, Zr, and Hf are not stable in the bcc structure (at ambient conditions), the HEAs composed to a major part of elements from the fourth group are, in fact, bcc stable. This in turn suggests, that there is great flexibility to tune the ITS of refractory HEA by varying composition and number of constituent elements as illustrated by the present work.
The highest values of the ITS are obtained for HEA containing a large amount of group five elements due to their high intrinsic ITS (approximately 12-14 GPa 15 ). The benefit of forming HEA by alloying group four elements to group five elements is a wide stability range of the bcc phase (in terms of the average d-occupation number, see Fig. 3(b)) at the cost of lowering the ITS. The ZrVTiNb alloy with d-occupation approximately equal to 3.15 is intrinsically ductile as are the elements V, Nb, and Ta with larger d-occupation approximately equal to 3.6. Given the present finding that the d-occupation is the controlling variable for the ITS, one might expect, that HEAs with d-band fillings larger than ~3.2 are also intrinsically ductile and reach ITS values that approach those of the group five elements, as the d-occupation approaches ~3.6.

Conclusion
We have employed ab intio alloy theory to investigate the equilibrium volume, elastic properties, and the ideal tensile strength for the single bcc-phase HEAs: ZrNbHf, ZrVTiNb, ZrTiNbHf, and ZrVTiNbHf. The predicted equilibrium volumes are consistent with the available experimental data. All investigated HEAs exhibit good extrinsic ductility shown by the large B/G ratio and the positive Cauchy pressure (c 12 − c 44 ). For the HEAs, we also found that the ideal tensile strengths increase with decreasing the ratio of the fourth group elements to fifth group elements. The alloying effect on the ideal tensile strength of HEAs is explained by the d-band filling. In light of the recent attempts connecting the intrinsic materials properties to the extrinsic ductility 37 , the present results offer a guideline to design ductile refractory HEAs with tailored strength level.

Methods
The present calculations were performed with the exact muffin-tin orbitals (EMTO) method in combination with the CPA [38][39][40] , which were previously employed to study the structure, mechanical properties, and phase transformation of other HEAs 41,42 . In the present application, the EMTO-CPA calculations were solved within the frozen-core scheme using the local-density-approximation (LDA) exchange-correlation density functional 43 . The Green's function was calculated for 24 complex energy points around the valence states. The basis set included s, p, d, and f states. Brillouin zone integrations were performed on a 27 × 27 × 27 k-points mesh. The electrostatic correction to the single-site CPA was described using the screened impurity model. The 5d, 6s and 4f electrons were treated as valence states for Hf. The technical details of the elastic constants/modulus calculations are identical to those described in Ref. 41