Ab Initio Predicted Alloying Effects on the Elastic Properties of Al X Hf 1−x Nbtatizr High Entropy Alloys

Using ab initio alloy theory, we investigate the equilibrium bulk properties and elastic mechanics of the single bcc solid-solution AlxHf1−xNbTaTiZr (x = 0–0.7, 1.0) high entropy alloys. Ab initio predicted equilibrium volume is consistent with the available experiment. We make a detailed investigation of the alloying effect of Al and Hf on the equilibrium volume, elastic constants and polycrystalline elastic moduli. Results imply that the partial replacement Hf with Al increases the stability of the bcc phase and decreases the ductility of the AlxHf1−xNbTaTiZr HEAs. The inner ductility of Al0.4Hf0.6NbTaTiZr is predicted by the calculations of ideal tensile strength.


Introduction
In the past decade, a simple solid-solution phase was found in a new alloy [1,2].Yeh et al. named the new class of materials "high entropy alloys" (HEAs) [1].HEAs are composed of at least five multiple principal elements in equimolar or near-equimolar ratios [1].In spite of a considerable number of principal elements, some HEAs tend to form single-phase solid solution due to the small atomic size difference, high mixing entropy, and relatively low mixing enthalpy [3].Single-phase HEAs adopt the OPEN ACCESS face-centered cubic (fcc) [1], body-centered cubic (bcc) [1] or Hexagonal Close-Packed (hcp) crystallographic structures [4].Due to the unique microstructure, the excellently mechanical and corrosion resistant performances, HEAs, as the potential engineering materials, have attracted rapidly increasing attention in research community of materials and condensed physics.
The HEAs consisting of refractory elements satisfy the superior mechanical and functional properties in the high-temperature applications.Since the single bcc-phase WTaMoNb and WTaMoNbV HEAs were reported by Senkov et al., many other experiments focus on the HEAs based on the refractory elements (Ti, Zr, Hf, V, Nb, Ta, Cr, Mo and W).For instance, the microstructures and mechanical properties were investigated for HfNbTaTiZr, HfTiZr, HfNbTiVZr and HfNbTiZr, etc. [5][6][7][8].Due to the low density and similar atomic radius with the refractory elements (except for Cr), an Al element was used to dope refractory HEAs.Some researchers have investigated the effect of alloying with Al on the microstructure, composition and mechanical properties of these new refractory HEAs.For example, the addition of aluminum was found to increase hardness from 440.1 to 556.4 Hv as going from MoNbTiV to Al1.5MoNbTiV [9].The light-weight AlNbTiV has the specific yield strength, which can be comparable with strength of multi-phase refractory HEAs [10].The Al additions have excellent beneficial effects on the microstructure and properties of parent CrMo0.5NbTa0.5TiZr and HfNbTaTiZr [11].For instance the hardness (yield strength) increases 3.8 to 4.9 GPa (929 to1841MPa) for HfNbTaTiZr and Al0.4Hf0.6NbTaTiZr.
Today, ab initio electronic structure methods have become powerful tools for predicting the atomic-level physical properties of multi-component metallic materials.However, HEAs present a great challenge for conventional ab initio calculations due to the complicated chemical and magnetic disorder.As an effective method to treat substitutional solid solutions, the exact muffin-tin orbital method in combination with the coherent potential approximation (EMTO-CPA) has been used to successfully predict the equilibrium bulk properties of some HEAs [12][13][14].In the present work, we employ the EMTO-CPA ab initio density functional method to study the effect of Al addition on the equilibrium bulk properties and elastic parameters of AlxHf1−xNbTaTiZr refractory HEAs.
The ideal strength provides the upper limit of the stress attainable during a specific loading condition, and is one of the importantly mechanical properties of metallic structure materials.For the calculations of ideal strength, there are often two special loading directions, such as ideal strength in tension, e.g., for cubic crystals under uniaxial load, and ideal strength in shear, e.g., for common slip systems (<111>{112} and <111>{110}) in the bcc crystal [15,16].Due to the large strain behavior predicted from ab initio, researchers have investigated some elementary substances (Ti, Mo, Nb, V, W, Cu, Pt and Au etc.) and ordered alloys (TiAl and Ni3Al) from ab initio calculations [16][17][18][19].So far, the calculations of ideal strength of HEAs have not been found.In the present work, we investigate the ideal tensile strength of HEAs along bcc<001> direction.
The present theoretical predictions contribute to the further theoretical as well as experimental studies.The structure of this paper is as follows.In Section 2, we describe the computational tool and important numerical details.The equilibrium bulk properties, elastic mechanics and the ideal tensile strength are presented and discussed in Section 3. The paper ends with conclusions.

Computational Methods
The EMTO-CPA method is an efficient and accurate ab initio method for solving the Kohn-Sham (KS) equation in the density functional theory [20].The exact one-electron potential is accurately discussed by using optimized overlapping muffin-tin potential.For total energy, the EMTO-CPA method uses the full charge density (FCD) technology [21][22][23], which improves the calculation efficiency and makes sure the accuracy is similar to the full-potential methods.Nowadays, CPA is the most effective alloy theory for calculating electronic structure in multi-component random solid solutions [24][25][26][27][28], despite of the single-site nature of the CPA limiting its applicability to system with insignificant short-range order and local lattice relaxation effects.It has been proven that using the EMTO-CPA method can obtain the structural energy differences related to the lattice distortion in complex alloys of arbitrary number of components and compositions [29][30][31][32][33][34].
In the present calculations, we used Local Density Approximation (LDA) density function to solve the KS equation within the scalar-relativistic approximation [35].The Green's function was calculated for 16 complex energy points including the valence states.The EMTO basis set included s, p, d, and f states.The irreducible wedge of the bcc Brillouin zone was sampled by 285 inequivalent k points.The electrostatic correction to the single-site CPA was described using the screened impurity model [36].According to the standard methodology and the Viogt-Reuss-Hill averaging method [37], the three independent single-crystal cubic elastic constants (c11, c12, c44) and the polycrystalline elastic moduli (B, G, E, v) were obtained.For all alloy components, the potential sphere radii were chosen to be equal to the corresponding average atomic sphere radius.All calculations were performed for a static lattice (that is, neglecting all thermal contributions).
In the calculations of ideal tensile strength, the stress σ is given by [38]     where E is the total energy per atom and V is volume per atom at a given tensile strain.ε denotes the train of the simulation cell in the direction of the applied uniaxial force.It is defined as where l  and 0 l is the length of the cell along the direction of uniaxial stress in the final state and in the initial state, respectively.The uniaxial tensile stress is loaded along bcc<001> direction to obtain energy difference and relevant strain.In this work, we only consider the uniaxial tensile stress of Al0.4Hf0.6NbTaTiZralong the Bain path.

Equilibrium Volume
Experiments indicated that Al0.4Hf0.6NbTaTiZrand HfNbTaTiZr adopt single-phase bcc structure [11].It is noted that Ti, Zr and Hf are stable in the hexagonal close packed (hcp) phase at ambient conditions, whereas these refractory elements become bcc phase at elevated temperature.In this part, we firstly study the atomic Wigner-Seitz (WS) radius and bulk modulus of each alloying element, and then turn to the AlxHf1−xNbTaTiZr HEAs.In Table 1, we list theoretically calculated WS radii and the bulk moduli along with the available experiment data [28].Here wt and we represent the theoretical (EMTO) WS radius and experimental WS radius of alloy elements and HEAs respectively.Bt and Be are the theoretical and experimental bulk moduli of the alloy elements.We can see from Table 1 that the theoretical results and experimental data are in good agreement with each other.The relative deviations of all elements between calculations and experiments are within the range of the errors obtained within the present density function approximation [35].
For HEAs, Table 1 shows the theoretical equilibrium WS radius.Unfortunately, the experimental equilibrium WS radii are only available to AlxHf1−xNbTaTiZr (x = 0, 0.4) [11,39].With the increase of Al content, the theoretical WS radius of HEAs decreases from 3.216 to 3.127 Bohr, which is in line with the trend of experimental WS radius measured by x-ray diffraction.The symbol Δ stands for the relative deviation (in %) between the theoretical and experimental WS radius.The Δ is about 1.5% for AlxHf1-xNbTaTiZr (x = 0, 0.4).These small errors might be due to the effect of temperature neglected in the present calculations.Although the similar measure method was employed, the experimental Wigner-Seitz radius w = 3.168 Bohr from Reference [11] is smaller than w = 3.178 Bohr from Ref [39].To further assess the theoretical equilibrium WS radii, we make use of Vegard's law to estimate the equilibrium WS radius by using the WS radius of the alloy components.In Table 1, t w stands for the estimated radius based on the present theoretical values and e w is the one obtained from the experimental data.As expected, t w and e w agree with each other.Both t w and e w are slightly smaller than wt.For all HEAs considered here, the results show a systematic negative deviation relative to Vegard's rule.The atomic size difference is one of important parameters that are used to determine the formation of the solid solution phases [40,41].The atomic size difference parameter δ is defined as where , with ri and ci being the atomic radius and atomic percentage of the individual alloy components, respectively, and n the number of components.For the solid solutions with bcc crystals, δ should be smaller than 8.5 from Ref [40] and 6.6 from Ref [41].Here, we estimate δ from the WS radii of alloying elements.In Table 1, for all HEAs, δt derived from the ab initio calculations is very close to δe obtained from experiments.Both δt and δe are below 6.6%, which are in line with the rule of single-phase solid solutions with bcc phase [3,40,41].As an indicator to determine the crystals of single-phase solid solutions, the valence electron concentration is often used to predict the solid solution phase of HEAs.According to statistical law [3], HEAs with valence electron concentration (VEC) < 7.5 adopt the bcc phase.VEC of the present HEAs is about 4.2-4.4,which remains below the phenomenological threshold value.

Cubic Elastic Constants
The elastic constants c11, c12 and c44 as well as the Zener ratio Az, the Cauchy pressure (c12-c44) and the tetragonal shear modulus c' of AlxHf1-xNbTaTiZr are listed in Table 2.The elastic constants (c11, c12, c44, (c12-c44) and Az) of AlxHf1−xNbTaTiZr are plotted in Figure 1 as a function of Al content.Table 2. Theoretical elastic constants c11, c12, c44, cʹ (units of GPa), polycrystalline elastic moduli B, G, E (unit of GPa) as well as the B/G ratio, the Zener ratio Az, the elastic anisotropy ratio AVR, the Poisson's ratio v and Cauchy pressure (c12-c44) (unit of GPa) for the body-centered cubic (bcc) phase AlxHf1−xNbTaTiZr HEAs (x = 0-0.7,1).The dynamical stability condition is satisfied by the theoretical single-crystal elastic constants for the present HEAs, i.e., c44 > 0, c11 > |c12| and c11 + 2c12 > 0 [37].From Table 2, we can see that all HEAs considered here are mechanically stable.As shown in Figure 1, the partial replacement of Hf with Al in AlxHf1-xNbTaTiZr (x = 0-0.7, 1) HEAs slightly increases the tetragonal elastic constant c' with the increase of Al content.The trend implies that Al enhances the mechanical stability of the bcc phase against tetragonal deformation for the present HEAs.The positive Cauchy pressure (c12-c44) is associated with the covalent nature of the metallic bond and the character of ductile alloy [42].Due to the large positive Cauchy pressure (c12-c44) shown in Figure 1, the present HEAs are predicted to have a strong metallic character and good ductility.We observe that the Cauchy pressure (c12-c44) slightly increases with the addition of Al content.

Polycrystalline Elastic Moduli
Table 2 shows the polycrystalline elastic moduli, including the bulk modulus B, the shear modulus G, Young's modulus E and the Poisson ratio v as well as the Pugh ratio B/G and elastic anisotropy ratio AVR of AlxHf1−xNbTaTiZr.The theoretical results for AlxHf1−xNbTaTiZr are plotted in Figure 2. To our knowledge, there are is available experimental information on the polycrystalline elastic moduli of the AlxHf1−xNbTaTiZr HEAs, expect for Al 0.4 Hf 0.6 NbTaTiZr [11] and HfNbTaTiZr [39].For the present HEAs, the bulk modulus B changes from 134.0 to 149.7 GPa, G from 38.0 to 44.9 GPa, E from 104.2 to 122.6 GPa.The partial replacement Hf with Al increases these polycrystalline elastic moduli.For the ductile-brittle behavior of alloys, B/G is one of indexes [43].Generally the materials with B/G > 1.75 are ductile, whereas those with B/G < 1.75 are considered to be brittle.According to Table 2 and Figure 2, the Pugh ratio B/G is larger than the critical value 1.75 for the present HEAs.The Poisson ratio v is considered as an indicator to predicate the ductile-brittle behavior of metallic materials.In some literature, bulk metal glasses are reported to have good ductility with the v > 0.31 [44].The Poisson ratio of the present HEAs is v = 0.36-0.37.Combining B/G and v, we find that the AlxHf1−xNbTaTiZr HEAs are predicted to be ductile, and the ductility slightly decreases with the increase of Al content.It should be noted that these phenomenological correlations between elastic moduli and the ductile/brittle behavior of materials have not been confirmed in the case of HEAs.
The Zener ratio AZ (c44/cʹ) and the elastic anisotropy ratio AVR are used to predict the elastic anisotropy of materials.For instance, AZ = 1 and AVR = 0 correspond to the elastically isotropic materials.The Zener ratio AZ > 1 shown in Table 2 implies that the present HEAs are anisotropic.From Figure 2 we can see that partial substitution of Hf with Al has a small effect on the elastic anisotropy of AlxHf1−xNbTaTiZr HEAs.With increasing of Al content, AZ decreases from 3.431 to 3.035 and AVR is from 0.1713 to 0.1407.Due to the anisotropy, the Young's modulus changes with the crystalline direction.For the cubic crystal, the Zener ratio AZ > 1 implies that the largest (smallest) Young's modulus is along the <111>(<001>) crystalline direction.For instance for Al0.4Hf0.6NbTaTiZr the Young's modulus E are 167 GPa, 59.5 GPa and 115GPa along <111>, <001> and <110> crystalline directions, respectively.Our calculated average Young's modulus E = 111.9GPa is slightly large, comparing with the experimental E = 78.1 GPa at room temperature [11].The large deviation is between our calculated E = 104.2GPa and the experimental E ≈ 55 GPa [39] for HfNbTaTiZr.It should be emphasized that all present calculations were carried out at static conditions (0 K) and the present HEAs are assumed an ideal solid solution phase with bcc underlying lattice.

The Ideal Tensile Strength of Al0.4Hf0.6NbTaTiZr
As an inherent property of a material, the ideal tensile strength (ITS) does not change with the grain boundaries, cracks, dislocations, and microstructural defects, ITS offers insight into the correlation between the intrinsic chemical bonding and the crystal symmetry [45].Figure 3 shows the energy difference ΔE and stress σ as a function of relaxed tensile strain and unrelaxed tensile strain along bcc<001> direction, respectively.We can see from Figure 3 that the relaxed (unrelaxed) maximum stress is 3.86 (4.20) GPa at the critical strain of ε = 10.6% (10.4%), which is bigger than the experimental value of the 2.27 GPa at the strain ε = 10%.The reason for the phenomenon may be that the experiment-measured value is the strength of real crystal, considering the crystal dislocation and temperature in the process of measurement.Owing to the Poisson contraction, the relaxed strain ε = 27.0%,leading to formation of an fcc structure (the second zero-stress point), is smaller than the unrelaxed one of 27.5%.The energy curve shown in Figure 3 implies that the deformation of the cubic structure is unstable at ε = 27.0%[46].With further increase of the strain, a stress-free tetragonal structure is reached at the strain of 32.7%, which is in line with the third zero-stress point in the stress-strain curve.For the unrelaxed strain, the third zero-stress point is ε = 32.3%.

Conclusions
We employed the ab initio EMTO-CPA method to study the equilibrium bulk properties, elastic parameters and ideal tensile strength of AlxHf1−xNbTaTiZr (x = 0-1) HEAs.The calculated equilibrium volumes are consistent with available experimental data and the estimated Wigner-Seitz radii.According to the atomic size difference and valence electron concentration, the present HEAs adopt a single-phase bcc crystalline structure.The trends of the calculated elastic parameters imply that the addition of Al increases the thermodynamic stability of the bcc phase of the present HEAs.
According to the ab initio B/G ratio and Poisson ratio v, theoretical calculations suggested that the ductility of the present HEAs slightly decreases with the increasing Al content, whereas all HEAs remain ductile.The anisotropic AlxHf1−xNbTaTiZr HEAs are predicted from the criteria of Zener ratio and polycrystalline elastic ratio.Furthermore, the change of Young's moduli E is from 59.5 GPa to 160 GPa along the different crystalline directions.Considering the effect of temperature, our ab initio calculated E = 111.9GPa is acceptable compared to the experimental E = 78.1 GPa at room temperature.For the ideal tensile strength of Al0.4Hf0.6NbTaTiZr, the maximum stress is 3.85 GPa at the critical strain of ε = 10.6%.

Figure 1 .
Figure 1.Theoretical elastic constants of AlxHf1−xNbTaTiZr as a function of Al content.

Figure 2 .
Figure 2. Polycrystalline elastic moduli for AlxHf1−xNbTaTiZr as a function of Al content.

Figure 3 .
Figure 3.The energy difference ΔE and tensile stress σ as a function of tensile strain ε under <001> tensile loading for Al0.4Hf0.6NbTaTiZr.The rectangles stand for ΔE and σ of the relaxed structure, the cycles represent ΔE and σ of the unrelaxed structure.

Table 1 .
WS radii (w, units of Bohr) for the pure metals (M) in their equilibrium crystal structure and for AlxHf1−xNbTaTiZr in the bcc structure shown as a function of Al content.wt and we represent the theoretical (EMTO) and the experimental WS radius, respectively, Bt is calculated bulk modulus, Be is the experimental one.VE stands for the number of valence electrons.twand e w are the estimated WS radii of the HEAs from wt and we , where ei is the number of valence electrons and ci the atomic fraction of alloy component i.