Alloying effect on the elastic properties of refractory high-entropy alloys

    Alloying effect on the elastic properties of refractory high-entropy alloys Li-Yun Tian, Guisheng Wang, Joshua S. Harris, Douglas L. Irving, Jijun Zhao, Levente Vitos PII: DOI: Reference:

S0264-1275(16)31479-4 doi: 10.1016/j.matdes.2016.11.079 JMADE 2515

To appear in: Received date: Revised date: Accepted date:

2 October 2016 3 November 2016 21 November 2016

Please cite this article as: Li-Yun Tian, Guisheng Wang, Joshua S. Harris, Douglas L. Irving, Jijun Zhao, Levente Vitos, Alloying effect on the elastic properties of refractory high-entropy alloys, (2016), doi: 10.1016/j.matdes.2016.11.079

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Alloying effect on the elastic properties of refractory high-entropy alloys

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Li-Yun Tiana,b , Guisheng Wangc , Joshua S. Harrisd , Douglas L. Irvingd , Jijun Zhaob , Levente Vitosa,e,f a

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Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden b Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams of Ministry of Education, Dalian University of Technology, Dalian 116024, China c School of Applied Mathematics and Physics, Beijing University of Technology, Beijing 100124, China d Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695-7907 e Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75121, Uppsala, Sweden f Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, Budapest H-1525, P.O. Box 49, Hungary

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Ab initio total energy calculations are used to determine the elastic properties of TiZrVNb, TiZrNbMo and TiZrVNbMo high-entropy alloys in the body centered cubic (bcc) crystallographic phase. Calculations are performed using the Vienna Ab initio Simulation Package and the Exact Muffin-Tin Orbitals methods, and the compositional disorder is treated within the frameworks of the special quasi-random structures technique and the coherent potential approximation, respectively. Special emphasis is given to the effect of local lattice distortion and trends against composition. Significant distortion can be observed in the relaxed cells, which result in an overlap of the first and second nearest neighbor (NN) shells represented in the histograms. When going from the four-component alloys TiZrVNb and TiZrNbMo to the five-component TiZrVNbMo, the changes in the elastic parameters follow the expected trends, except that of C44 which decreases upon adding equiatomic Mo to TiZrVNb despite of the large shear elastic constant of elemental Mo. Although the rule of mixtures turns out to be a useful tool to estimate the elastic properties of the present HEAs, to capture the more delicate alloying effects one needs to resort to ab initio results. Preprint submitted to Materials & Design

November 22, 2016

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Keywords: High-entropy alloys, Lattice parameter, Elastic constant, Local lattice distortion, Alloying effect.

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1. Introduction

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High-entropy alloys (HEAs) are equiatomic or nearly-equiatomic multicomponent solid solutions, which attracted substantial attentions in recent years [1–6]. They crystalize in body centered cubic (bcc) [7–13], face centered cubic [14–16] or hexagonal close-packed (hcp) lattice [17], and often show a good combination of high ductility and high strength. Due to the rich physical, chemical and mechanical properties [9, 14, 18], HEAs are considered as the primary choice in many future high-technology applications. The large number of possible combinations of five or more metallic elements makes the number of potential HEAs enormous [19, 20]. Despite of the substantial experimental [21, 22] and most recently also theoretical [19, 23–30] efforts to explore the HEAs, today the available information on their properties is rather limited. It is often assumed that the bulk parameters of random solid solutions may be estimated from the rule of mixtures [31–33] representing a weighted mean of the parameters of pure constituents. This is especially the case for non-magnetic alloys consisting of elements which in their pure form have the same crystal lattice. A well-known example is Vegard’s law [34], which is used to evaluate the equilibrium lattice parameter of alloys using data available for the end members. Such simple rules are very useful for an initial screening of the alloying effects for unknown systems such as new HEAs and may be used for categorizing the alloy components for design purposes. In theoretical condensed matter physics, subtle electronic structure mechanisms are traced which are responsible for the small deviations relative to these rules. Unfortunately, today the available experimental data on HEAs has no sufficient volume and accuracy to provide a solid ground for assessing the rule of mixtures for these multi-component alloys. Some initial attempts of using first principles approaches to evaluate simple mixing rules have been done for composition variations NiFeCrCo [35] and MoNbTaVW [36] but little has been done for refractory systems. The TiHfZrTaNb refractory high entropy alloy was found to show superior mechanical properties [37]. The corresponding elastic properties were also investigated by ultrasound measurements yielding C44 = 28 GPa and C11 2

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= 172 GPa, Young modulus E = 78.5 GPa, the bulk modulus B = 134.6 GPa and Poisson ratio µ = 0.402 [38]. Unfortunately, experimental data on single-crystal or polycrystalline elastic moduli for other refractory HEAs are very limited, making all experimental and theoretical attempts to derive and understand such data highly timely and welcome. First-principles methods built on density functional theory (DFT) [39– 41] have been widely used to access the properties of solids including multicomponent disordered alloys. Special quasi-random structure (SQS) [42, 43] and coherent potential approximation (CPA) [44–46] are two frequently used techniques in combination with DFT methods for modeling the random distribution of alloy components on a given crystal lattice. SQS involves large supercells constructed in terms of atomic correlation functions mimicking a random distribution within the first few coordination shells. The atomic positions within the supercells may be relaxed by minimizing the interatomic forces during the self-consistent calculations and by that properly accounting for the local lattice distortion (LLD) effect. Hence, SQS combined with a suitable underlying DFT method may provide highly accurate results for solid solutions. The main limiting factor of this technique is the computational load associated with large supercells needed to model (nearly) arbitrary compositions. In addition, these supercells usually have lower symmetry than the underlying lattice which further increases the computational efforts. The CPA is a very powerful mean-field technique to treat disordered systems. The real alloy is substituted by an effective medium obtained by solving the impurity problem for each alloy constituent within the single-site approximation. This approach is the ideal choice for complex multi-component alloys such as HEAs since the size of the fundamental DFT equations scales linearly with the number of alloy components. In addition, the symmetry of the alloy matrix is retained which facilitates the calculations of symmetry dependent properties such as elastic constants. The main drawback of CPA is that the LLD can not be taken into account within the single-site approximation. Because of that, most theoretical research on HEAs using CPA focused on the alloying effect but completely omitted the LLD. On the other hand, severe lattice distortion decreases the XRD peak intensity [47] and contributes significantly to properties of HEAs. Since nowadays CPA is commonly used in the description of HEAs, it is highly desired to establish the impact of LLD on the properties in question and check whether the associated errors can overwrite the chemical effects. High entropy alloys containing refractory elements (Ti, Zr, Hf, Nb, Mo, 3

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W, V and Ta) display a single-phase bcc structure [7, 8]. These alloys exhibit superior mechanical properties in compression at high temperature compared to the conventional high-temperature alloys. Experimentally, Senkov et al. showed that the NbTiVZr alloy adopts a single bcc phase [48]. The XRD patterns of TiZrNbMoVx (x = 0, 0.25, 0.5, 0.75, 1) show only a single bcc phase and those of TiZrNbMoVx (x = 1.5, 2, 3) exhibit both bcc and Zrrich (β-Zr) phases with increasing V molar ratio [22, 49]. These alloys have been in the focus of many experimental and theoretical studies [4, 9, 23, 48, 50]. Dirras et al. investigated the elastic properties of TiHfZrTaNb using ultrasound measurements [38]. Tian et al. [23] studied the elastic anisotropy and ductility of TiZrVNb and TiZrNbMoVx (x = 0-1.5) adopting the bcc structures using ab initio method. In this work, we carry out ab initio SQS and CPA calculations to establish the elastic properties of three refractory HEAs: TiZrVNb, TiZrNbMo and TiZrVNbMo all crystalizing in the bcc structure. We place special attention on the impact of LLD on the bulk properties and parallel establish the accuracy of single-site CPA for these HEAs. The theoretical data is used to assess the rule of mixtures for the lattice parameter and elastic constants. Alloying effects are discussed starting from the two base alloys (TiZrVNb and TiZrNbMo) and focusing on the effect of Mo and V (TiZrVNbMo). The rest of the paper is arranged as follows. The calculational methods and related numerical details are given in Section 2 and the two sets of ab initio data are compared in Section 3. The results for the lattice parameter, mixing energy and elastic constants are given in Section 4. Section 5 presents the discussions on the alloying effect and rule of mixtures. The paper ends with a brief summary in Section 6. 2. Computational method 2.1. Total-energy calculations All calculations were performed within the framework of density functional theory formulated within the generalized gradient approximation (GGA) by Perdew, Burke, Ernzerhof (PBE) [51] for the exchange-correlation functional. The Kohn-Sham equations were solved using the Vienna Ab initio Simulation Package (VASP) [52–54] and the Exact Muffin-Tin Orbital (EMTO) methods [55–58]. The substitutional disorder was treated using the SQS technique in VASP calculations and the CPA in EMTO calculations.

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In EMTO calculations, the single-electron equations were solved with the scalar-relativistic approximations and soft-core scheme. The EMTO total energy was obtained by the full charge density technique (FCD) [59, 60], and the Kohn-Sham equations were solved for the optimized overlapping muffin-tin (OOMT) potential. The SQS structures for VASP calculations were constructed using the Alloy Theoretic Automated Toolkit (ATAT) [61, 62]. The disordered SQS supercells were either used as rigid bcc supercells (denoted by SQSu ) or taking into account the local chemical environment induced LLD (SQSr ). Comparing the so obtained sets of data can give direct evidence for the impact of LLD on the computed physical properties. For the VASP calculations, we adopted the projector augmented wave (PAW) method [63].

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2.2. Elastic constants For bcc structures, there are three independent elastic constant: C11 , C12 and C44 and the associated parameters of the tetragonal shear modulus C ′ =(C11 -C12 )/2 and the bulk modulus B=(C11 +2C12 )/3. In this work, all elastic constants were calculated according to the energy-strain relation. The theoretical equilibrium volume and bulk modulus were derived from an exponential Morse-type function [64] fitted to the total energies calculated for nine different atomic volumes. In order to obtain the two cubic shear moduli C ′ and C44 , we used volume-conserving orthorhombic and monoclinic deformations as described   0 0 1 + εo  0 , 1 − εo 0 (1) 2 0 0 1/(1 − εo ) and

 1 εm 0 .  εm 1 0 2 0 0 1/(1 − εm ) 

(2)

The strains εo and εm vary between 0 and 0.025 (0.05) for six energy points for VASP (EMTO) calculations. In studies based on SQS, the crystal symmetry is lowered due to the quasi-random distribution of the atomic species. Therefore, the elastic constants tensor has a lower symmetry than cubic. Here we dropped the non-cubic components and made an arithmetical average between C11 , C12 and C44 calculated for the three main crystallographic directions following the scheme described in Refs. [65–67]. 5

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The polycrystalline shear modulus (G) was obtained according to the Hill averaging method G = (GV + GR )/2, where the Reuss and Voigt bounds are GR = 5(C11 − C12 )C44 (4C44 + 3C11 − 3C12 )−1 and GV = (C11 − C12 + 3C44 )/5. The Young modulus (E) and the Poisson ratio (ν) are connected to B and G by the relations E = 9BG/(3B + G) and ν = (3B − 2G)/(6B + 2G). The Zener anisotropy ratio is defined as AZ = 2C44 /(C11 − C12 ).

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2.3. Numerical details The EMTO Green’s function was calculated for 16 complex energy points distributed exponentially on a semicircular contour including the valence states. In the basis set s, p, d and f orbitals (lmax = 4) were included. The h number of orbitals for the full-charge density (FCD) was chose as lmax =8. The electrostatic correction to the single-site CPA was described using the screened impurity model [68] with a screening parameter of 0.6. To obtain the accuracy needed for the calculation of elastic constants we used about 1968326973 inequivalent k-points in the irreducible wedge of the orthorhombic and monoclinic Brillouin zones. The disordered SQS structures were generated with 128 atoms (4 × 4 × 4) for TiZrVNb and TiZrNbMo and 250 atoms (5 × 5 × 5) for TiZrVNbMo. The atomic positions of the SQS supercells can be found in Ref. [69]. The VASP calculations were performed using an energy cutoff of 400 eV for the plane wave basis set. Integrations in the Brilouin zone were performed using k points generated with 5 × 5 × 5 mesh grids for the 128-SQS structures and 3×3×3 for the 250-SQS structure. The changes of the total energies with the A number of k points and the cutoff energy were tested to ensure 10−2 eV/˚ convergence for the Hellmann-Feynman forces used for the optimization of the atomic positions. The energy tolerance for the electronic relaxation was set as 1 × 10−6 eV. For both rigid and relaxed SQS structures, the volumes were optimized while keeping the shape fixed. 3. Assessing the accuracy In order to assess the accuracy of our theoretical methods, first we test the lattice constants and elastic parameters for the five elemental metals. The low-temperature crystal structure is hcp for Ti and Zr and bcc for V, Nb, Mo and the three HEAs considered here [3, 19]. With increasing temperature, both Ti and Zr undergo an hcp-bcc structural transition. The experimental lattice parameters (for all present metals) and bulk moduli (for V, Nb and 6



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Figure 1: (Color online) Comparison between the theoretical (VASP and EMTO) and the experimental (extrapolated to static conditions [70]) lattice parameters (in ˚ A) (a) and bulk moduli (in GPa) (b) of bcc Ti, Zr, V, Nb and Mo.

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Mo) quoted below are those measured for the bcc phase, extrapolated to 0 K and corrected for the zero point phonon effect. For details about the extrapolation see Ref. [70] and references therein. In Figure 1, the VASP and EMTO lattice parameters and bulk moduli of Ti, Zr, V, Nb and Mo are compared with the experimental values. The data are also listed in Table 1. The two sets of theoretical lattice constants are in excellent agreement with each other and also with the experimental values. The EMTO bulk moduli are slightly smaller than the VASP (Ti, Zr, V, Nb and Mo) and experimental values (Nb and Mo), the largest relative deviations being around 11%. The lattice constants and bulk moduli follow the trends aV < aMo < aTi < aNb < aZr and BMo > BV > BNb > BTi > BZr , respectively. That is, the bulk moduli of the 3d and 4d metals, when considered separately, correlate well with the size of the corresponding lattice constants. Figure 2 compares the present theoretical elastic constants of elemental metals to the room-temperature experimental data. No experimental values are given for bcc Ti and Zr (being unstable at room temperature). The three sets of C11 values are in good agreement with each other, except for V where both methods slightly underestimate the elastic constant. The agreement is less perfect for the other two elastic constants. In particular, EMTO gives lower C12 for Nb and Mo relative to the VASP and experimental values. The underestimated C12 by EMTO is reflected in positive C ′ = (C11 − C12 )/2 for 7

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C11 111.2 96.8 92.3 87.0 279.6 266.2 232 252.1 243.7 253 476.0 454.2 450

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B 104.5 107.9 82.3 91.0 174.9 184.0 161 155.5 171.4 174 247.2 264.4 278

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Method EMTO VASP Expt. EMTO VASP Expt. EMTO VASP Expt. EMTO VASP Expt. EMTO VASP Expt.

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˚) and elastic Table 1: Theoretical (EMTO and VASP) equilibrium lattice parameters a (A constants (GPa) for Ti, Zr, V, Nb and Mo in bcc structure. Shown are also the hightemperature (Ti and Zr) and room-temperature (V, Nb and Mo) experimental lattice parameters and bulk moduli extrapolated to 0 K and corrected for zero point phonon effect (see Ref. [70] and references therein). The experimental single-crystal elastic constants are from Ref. [71].

C12 101.2 113.5 77.3 92.9 122.5 142.9 119 107.1 135.3 133 132.8 169.2 173

C44 64.9 43.6 49.2 28.2 36.1 24.9 46 34.6 19.0 31 117.3 96.0 125

Ti (5 GPa) and Zr (7 GPa), which is in contrast to the small but negative C ′ values predicted by VASP for these two metals (-8 and -3 GPa, respectively). We recall that mechanical stability requires C11 & C12 , meaning that EMTO predicts elastically barely stable Ti and Zr. Nevertheless, the above absolute deviations between VASP and EMTO C ′ values are below those found for V, Nb and Mo. The EMTO C44 values are higher than those by VASP and eventually closer to the experimental values. For V, Nb and Mo, VASP underestimates C44 relative to the experimental values by 46%, 39% and 23%, respectively, compared to 22%, 12% and 6% errors found for EMTO. Based on the above findings, we conclude that in general the two ab initio methods provide consistent results for elemental metals. Somewhat larger deviations between VASP and EMTO predictions are observed in the 8

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Figure 2: (Color online) Comparison between theoretical (VASP and EMTO) and experimental (room temperature) C11 (a), C12 (b) and C44 (c) of bcc Ti, Zr, V, Nb and Mo.

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case of C12 and C44 elastic constants. Since none of the methods performs clearly better for all parameters when compared to the experimental data, we continue to present results obtained by both approaches and discuss the alloying effects and rule of mixtures separately for the two sets of theoretical data. 4. Results

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4.1. Lattice constant Figure 3 displays the lattice constants and bulk moduli of bcc TiZrVNb, TiZrNbMo and TiZrVNbMo alloys calculated using the SQS and CPA approximations. Numerical data are also listed in Table 2. It is found that the CPA lattice constants are in perfect agreement with the SQSu values obtained without relaxing the atomic positions. The differences between these two sets of theoretical predictions are 0.12%, 0.15% and 0.12%, respectively. Similar deviations are also observed for the pure elements (see Table 1) and thus they may be considered as the characteristic differences between EMTO and VASP results for the lattice parameter. Actually, the same conclusion holds also for the bulk modulus. CPA systematically underestimates the SQSu bulk moduli of bcc TiZrVNb, TiZrNbMo and TiZrVNbMo (by about 9-11 GPa), which is consistent with (even smaller than) the deviations seen for pure metals (Figure 1 (b)). Comparing the results from the two sets of SQS calculations, we find that the lattice constants are increased by LLD for all three HEAs. The differences between CPA and SQSr lattice parameters are 0.40%, 0.48% and 0.43% for TiZrVNb, TiZrNbMo and TiZrVNbMo, respectively. These deviations should still be considered very small indicating that the present theoretical equilibrium lattice parameters are robust and they represent the correct DFT values obtained at PBE level. LLD slightly increases the bulk moduli of all three HEAs. Nevertheless, the changes are very small (below 3 GPa) so that the good agreement between the three sets of theoretical bulk moduli remains. We conclude that the two ab initio approaches, CPA and SQS (with and without LLD), predict very similar equation of states for the present HEAs. Furthermore, LLD is found to have insignificant influence on the equilibrium volume and bulk modulus of these HEAs, despite of the nearly 20% difference between the largest and the smallest lattice parameters of the alloy constituents. 10

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˚, Figure 3: (Color online) Theoretical lattice constants of the underlying bcc unit cell (A upper panel) and bulk moduli (GPa, lower panel) obtained for TiZrVNb, TiZrNbMo and TiZrVNbMo alloys using the EMTO-CPA and VASP-SQS methods. For the latter, both the unrelaxed (SQSu ) and relaxed (SQSr ) results are shown. The experimental data for TiZrVNb and TiZrVNbMo are taken from Ref. [48] and Ref. [72], respectively.

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4.2. Mixing energy The enthalpy of formation of HEAs is one of the key parameters controlling the solid solution phase formation [26]. For the present alloys, the estimated enthalpy of formation are -0.03 (TiZrVNb), -2.50 (TiZrNbMo) and -2.72 (TiZrVNbMo) kJ/mol [73]. These values are all located inside the empirical limits from -15 to +5 kJ/mol [74] for solid solution formation. Hence, it is expected that these systems form a solid solution. The enthalpy of formation is defined with respect to the ground state structure of the alloy constituents, i.e. hcp for Ti and Zr, and bcc for V, Nb and Mo. Here we focus on the mixing energy defined with respect to the total energies of the alloy components in the bcc structure. In this way, when comparing different theoretical results we can exclude effects coming from the hcp-bcc structural energy differences between various methods. We define the mixing energy as X ∆E = Ealloy − ci Eibcc (3) i

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where Ealloy is the total energy per atom of the HEA and Eibcc is the equilibrium energy of the ith alloy component calculated for the bcc structure. ci stands for the concentration. The calculated mixing energies for TiZrVNb, TiZrNbMo and TiZrVNbMo alloys are shown in Figure 4. One can see that ∆E calculated from CPA and SQSu are in reasonable agreement with each other. The largest deviation of ∼ 4 kJ/mol (corresponding approximatively to 25% difference) is obtained for TiZrVNb. The differences between SQSu and CPA data are due to the singlesite approximation in CPA. Here we use the screened impurity model [68] with a fixed screening parameter to describe the electrostatic potential within CPA. Adjusting the screening parameter, one could in principle reproduce with very high accuracy the SQSu results by CPA. On the other hand, the ∆E values obtained by the rigid-lattice SQSu approximation turn out to be far too large when compared to the relaxed SQSr results, which makes such tuning of the screening parameter rather useless. As seen in Figure 4, the theoretical ∆E values are significantly decreased when we take into account the LLD. Actually, the LLD effect on the mixing energies is substantially larger than the CPA-SQSu differences, which makes the CPA results for the mixing energy less reliable. We observe that the fully relaxed mixing energies follow the trend ∆E 12

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Figure 4: (Color online) Theoretical mixing energies of bcc TiZrVNb, TiZrNbMo and TiZrVNbMo alloys obtained by unrelaxed SQS (SQSu ), relaxed SQS (SQSr ) and CPA calculations. In all cases the reference states are the pure elements in bcc structure.

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(TiZrNbMo) < ∆E(TiZrVNbMo) < ∆E(TiZrVNb), which is different from the above quoted trend of the formation enthalpies [73]. The present VASP (EMTO) total energy differences between bcc and hcp Ti and Zr are 10.4 (8.1) and 7.7 (5.5) kJ/mol, respectively. These figures are in line with the Local Density Approximation results reported previously [75]. Using the above VASP bcc-hcp structural energy differences in combination with the SQSu mixing energies, for the formation enthalpies of TiZrVNb, TiZrNbMo and TiZrVNbMo we get 11.1, 0.15 and 6.3 kJ/mol, respectively. The rather high formation enthalpy of TiZrVNb may indicate that the solid solution formation is at least questionable for this HEA in the light of the empirical upper limit of 5 kJ/mol by Yang et al. [74]. We notice that the CPA and SQSu mixing energies yield formation enthalpies far above the Yang’s upper limit. Hence, it is clear that without proper account for the LLD effects no meaningful theoretical prediction can be made on the thermodynamics of HEAs. Finally, we would like to note that the above-quoted empirical solid solution formation limits have to be treated with precautions in the mirror of recent developments by Laurent-Brocq et al. [76]. Probing the present systems against phase separation, segregation or intermetallic formation is, however, beyond the scope of this work.

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C44

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92.8 50.5 94.7 53.8 111.7 21.6 114.3 18.5

CPA CPAa SQSu SQSr

3.306 3.304 3.311 3.322

209.6 209.9 198.9 203.4

98.8 101.0 122.1 121.5

49.9 52.6 26.9 29.6

CPA CPAa SQSu SQSr

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216.3 213.7 203.7 209.3

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C′ AZ C12 -C44 B G E TiZrVNb 36.5 1.385 42.2 117.1 44.4 118.2 35.9 1.500 41.0 118.6 45.7 121.1 22.6 0.956 90.0 126.7 22.0 62.4 22.8 0.810 95.8 129.5 20.1 57.3 TiZrNbMo 55.4 0.901 48.9 135.8 52.0 138.4 54.4 0.966 48.4 137.3 53.3 141.7 38.4 0.702 95.1 146.9 31.1 87.1 41.0 0.723 91.8 148.2 33.8 94.1 TiZrVNbMo 58.1 0.824 52.4 138.9 51.7 137.9 56.5 0.900 49.8 138.5 53.2 141.1 41.3 0.583 97.1 148.7 29.9 84.1 43.1 0.623 96.2 151.8 32.5 91.0

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Table 2: The theoretical lattice constant for the bcc underlying lattice (a, ˚ A), elastic constants C11 , C12 , C44 and C ′ (GPa); the Zener anisotropy (AZ = C44 /C ′ ) and Cauchy pressure (C12 -C44 ) (GPa), polycrystalline elastic moduli (B, G, E, GPa), Poisson’s ratio (ν) and the Pugh ratio (B/G) of the TiZrVNb, TiZrNbMo and TiZrVNbMo alloys calculated using the EMTO-CPA and VASP-SQS methods. For SQS both unrelaxed (SQSu ) and relaxed (SQSr ) results are shown. For reference, we also list the CPA results (CPAa ) obtained by Tian et al. [23].

ν

B/G

0.332 0.330 0.418 0.426

2.639 2.604 5.756 6.447

0.330 2.608 0.328 2.575 0.401 4.729 0.394 4.408 0.335 2.688 0.330 2.608 0.406 4.969 0.400 4.669

4.3. Elastic parameters Figure 5 displays the single-crystal elastic constants (Cij ) from CPA and both sets of SQS calculations. The CPA and the average SQSu and SQSr values are listed in Table 2 as well. The individual SQS elastic constants due to the low symmetry induced by the chemical quasi-randomness in SQS calculations are provided in Table 3. By comparing the individual values one can learn about the effect of reduced-symmetry on the elastic parameters. For instance, the deviation is about 12.8 GPa for C44 of TiZrNbMo, representing about 65% of the average value. Sizable scatter is seen for C11 as well, especially for TiZrVNb. One expects that increasing the size of the SQS cell should reduce the differences between the individual elastic constants and eventually restore the cubic symmetry for very large cells. In the following we discuss only the average SQS elastic constants. According to the numerical values in Table 2, we find that the mechanical stability criteria (C ′ > 0, C44 > 0, B > 0) are fulfilled by all three sets of data 14

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TiZrNbMo

C33 153.4 (155.9) 202.1 (197.7) 201.3 (202.4)

C12 109.5 (110.1) 123.3 (121.6) 121.0 (121.3)

C12 C13 117.5 (112.2) 122.1 (121.6) 127.0 (121.4)

C23 117.2 (112.7) 118.9 (123.0) 121.1 (120.7)

C44 18.5 (21.0) 32.8 (32.6) 20.2 (21.2)

C44 C55 18.8 (20.8) 34.3 (28.5) 32.0 (25.2)

C66 18.1 (23.0) 21.8 (19.8) 26.9 (25.7)

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C11 C22 153.9 (154.9) 208.5 (201.4) 213.2 (203.7)

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C11 169.3 (160.0) 199.6 (197.6) 213.4 (205.0)

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Table 3: The relevant elastic parameters of TiZrNbMo, TiZrVNb and TiZrVNbMo alloys calculated using VASP-SQS method. The unrelaxed data are listed in the parentheses.

Figure 5: (Color online) Single-crystal elastic constants of bcc TiZrVNb, TiZrNbMo and TiZrVNbMo alloys obtained by unrelaxed SQS (SQSu ), relaxed SQS (SQSr ) and CPA calculations. For SQS, shown are the average values.

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and thus the TiZrVNb, TiZrNbMo and TiZrVNbMo alloys are predicted to be dynamically stable. However, when comparing the CPA and SQS results, we find that some elastic constants are sensitive to the employed method. For example, C ′ and C44 obtained in CPA calculations are by ∼ 15 and ∼ 30 GPa, respectively, larger than those calculated using the SQS method. We should recall that similar differences between VASP and EMTO single-crystal elastic constants are found for the pure metals as well (Table 1). This observation suggests that the main deviations between the CPA and SQS values in Table 2 in fact originate from the errors associated with the underlying DFT methods rather than from the way how the chemical randomness is treated by CPA or SQS. This is in line with previously reported conclusions. For instance, Zaddach et al. studied the elastic constants of CoCrFeMnNi 3d-high entropy alloy based on the EMTO-CPA and VASP-SQS methods [77]. They found the elastic coefficients in EMTO-CPA calculations in close agreement to those obtained in the VASP calculations for ferromagnetic configuration. Niu et al. compared the elastic constants of non-equiatomic NiFeCrCo high entropy alloy (i.e., 10-40 at.%) by the EMTO and VASP [35]. Good agreement was found between the two methods. The lattice distortion induced by the size of different elements can affect the strength of bcc HEAs [78]. Therefore, it is essential to consider the influence of LLD on the mechanical properties of HEAs. Our results can reveal this effect in the case of elastic parameters. Comparing the SQSu and SQSr data in Table 2, it is concluded that the elastic constants of the present HEAs remain almost unchanged upon taking into account the LLD. For instance, the average C11 increases by 2.9 GPa (1.8%), 4.5 GPa (2.3%) and 5.6 GPa (2.7%) for TiZrVNb, TiZrNbMo and TiZrVNbMo, respectively, with local lattice relaxation. The relative LLD induced changes in C44 are somewhat larger due the small absolute value. In Table 2, the Zener anisotropy ratio AZ = C44 /C ′ , the Young modulus E, the shear modulus G, the Cauchy pressure (C12 − C44 ), the Pugh ratio B/G and the Poisson ratio ν are also listed. Unfortunately, no experimental information on these elastic moduli is available. In general the SQS and CPA values for elastic moduli are consistent with each other. The large differences obtained for the shear and Young moduli are primarily due to the strong underestimation of the C44 elastic constant of pure metals (also HEAs) by VASP as compared to the EMTO and experimental data (Table 1). Notice that all HEAs considered here are relatively isotropic (AZ is close to 1), which places G close to both single-crystal shear elastic constants (C ′ 16

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and C44 ). The B/G ratio is often used as an indicator of the brittle-ductile behavior. High values of B/G (> 1.75) indicate that the material is ductile, while low values are associated with brittleness. The Poisson ratio ν may also be used to predict the brittle-ductile behavior. It has been reported that bulk metallic glasses with ν > 0.31 are ductile. Similarly, positive Cauchy pressure indicates ductility as well. According to Table 2, for the present systems B/G is always larger than the critical value of 1.75, ν is above 0.31 and the Cauchy pressure is positive. Therefore, the TiZrVNb, TiZrNbMo and TiZrVNbMo HEAs are predicted to be ductile by both CPA and SQS calculations. The above phenomenological ductility indicators are very weakly affected by LLD.

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4.4. Local lattice distortions The magnitudes of the LLDs are directly evaluated from the atomic coordinates obtained through the VASP-SQS geometry optimization. For each relaxed HEA system, a histogram of the radial distributions for every atom in the cell is presented in Figure 6. Significant distortion may be observed in the relaxed cells, which results in an overlap of the first and second nearest neighbor (NN) shells represented in the histograms. This overlap is more likely in the present alloys √ due to the close proximity of the first two NN shells in the bcc lattice ( 3ao /2 and ao ). Before the LLDs are analyzed, we will briefly describe the procedure by which the results were obtained. The histograms in Figure 6 were generated by calculating the distance to each atom around every site and retaining only those within a radial cut off of approximately 4 ˚ A. The histograms count atoms in bins of width 0.03 ˚ A between 2.49 and 3.69 ˚ A. Because this prescription double counts every pair, the number of atoms in each bin was then reduced by half. Due to the overlap of NN shells in the relaxed systems, the distances determined from the relaxed supercells are by themselves insufficient to determine whether a given pair belongs in the first or second NN shell. Here, we use the ideal unrelaxed SQS to determine the indices of first and second NNs, i.e. we make use of the unrelaxed indexing to associate the atoms to initial coordination shells after lattice relaxation. These indices are then used to identify the pair distances plotted in Figure 6. Pairs of atoms that were first NNs in the unrelaxed system are binned in one histogram, while those that were second NNs in the unrelaxed system are binned in a separate histogram, both of which are represented on the same plot. It should be noted that the standard deviations determined for the radial distribution of all sites for the 17

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Figure 6: (Color online) Histogram plot of LLDs for TiZrVNb (top), TiZrNbMo (middle), and TiZrVNbMo (bottom). The 1st and 2nd nearest neighbors (NNs) are colored red and gray, respectively. The ideal (unrelaxed) positions of the first and second NNs based on the lattice parameter are indicated with dashed lines.

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first and second NNs in each system are identical to 4 decimal places, which should be the case for a periodic bulk supercell and indicates the degree of relaxation from the ideal lattice. From inspection of Figure 6, it is clear that the most significant LLDs are found in the TiZrVNb HEA. Smaller, yet still significant, LLDs are found in TiZrNbMo and TiZrVNbMo alloys. It is evident in Figure 6 that the radial distribution histograms for first and second NN pairs in TiZrVNb are more spread out than the corresponding histograms for the other HEAs. Moreover, the histograms for TiZrVNb seem to be superpositions of small subpeaks, whereas the histograms for the other HEAs are more centered around a single peak. In order to understand the more significant distortion found in TiZrVNb as compared to TiZrNbMo and TiZrVNbMo, we have calculated the average pair distance and standard deviation for first NN pairs at every site and also at each site of a given element in each of these alloys. These results are presented in Table 4. A similar approach based on the matrix description of the LLDs effect in bcc MoNbTaVW high entropy alloy was recently used in Ref. [36]. Each four-component alloy has 32 unique chemical sites, whereas there are 50 unique chemical sites in the five-component alloy. With 8 first NNs in the bcc unit cell, this leads to sample sizes of 256 and 400 for the fourand five-component systems, respectively. Looking at the results in Table 4, it is apparent that within each HEA there is little variation of the standard deviations for different chemical sites. The average radial distance around each site, however, varies more significantly from site to site. Generally, the average radial distance is largest around the largest atom, Zr, and smaller around the smaller atoms Mo and V. Ti and Nb are intermediate in average radial distance as compared to the other elements, and are generally closer in average radial distance to the overall average of the bulk. The more significant standard deviations in TiZrVNb originates from the variation in average radial distances around each atomic site. This is a product of both atomic size and the resultant electronic structure of the alloy. TiZrVNb and TiZrNbMo are very close in lattice spacing, which is clear from inspection of the ideal first and second NN distances in Figure 6 or the average radial distance values of all sites in Table 4. The average radial distance for first NN pairs around V sites is smaller as compared to Mo sites. Comparison of the four-component alloys shows that Ti and Nb have smaller average radial distances in the alloy containing V, while Zr has a slightly larger average radial distance in the alloy containing V. This is 19

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Table 4: Average distances (ravg ) and the standard deviations (σ) for the first nearest neighbor (NN) pairs around each chemical site in TiZrVNb, TiZrNbMo, and TiZrVNbMo HEAs.

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TiZrVNbMo

Site All Ti Zr V Nb All Ti Zr Nb Mo All Ti Zr V Nb Mo

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1st NN ravg ± σ 2.866 ± 0.123 2.875 ± 0.111 2.946 ± 0.111 2.785 ± 0.107 2.856 ± 0.106 2.880 ± 0.090 2.884 ± 0.072 2.944 ± 0.080 2.872 ± 0.082 2.819 ± 0.072 2.832 ± 0.098 2.850 ± 0.085 2.914 ± 0.088 2.782 ± 0.087 2.833 ± 0.077 2.780 ± 0.087

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consistent with Ref. [36] and it is due to the smaller V atom relative to the rest of the alloy constituents. If we envision the histograms in Figure 6 as the superposition of a set of element-specific histograms for radial distributions, we can readily explain their shapes. In such a picture, an average value in Table 4 would represent the location of the peak of the distribution around a given element, while the standard deviation would represent the width of that peak. We see from Table 4 that the average values, and therefore the peak locations, are more spread out for TiZrVNb than for the other two HEAs; and the standard deviations, that is, the width of each peak, are higher. Therefore, we expect the overall distribution of radial pair distances in TiZrVNb to feature multiple broad peaks, whereas the overall distributions for the other two HEAs should feature relatively narrow peaks at roughly the same location. This is exactly what we observe in Figure 6: multiple flat peaks for the first NN distribution in TiZrVNb, and a single narrow peak for the first NN distributions in TiZrNbMo and TiZrVNbMo. A similar analysis applies to the shape observed for the histograms of the second NN distributions. In summary, significant LLDs are found in TiZrVNb, TiZrNbMo, and TiZrVNbMo. These have been illustrated by the presentation of radial distribution histograms as well as quantitative analysis. How the LLDs influence other physical properties of the alloys will be analyzed through the use of both the EMTO-CPA and the VASP-SQS approaches. 5. Discussion

5.1. Alloying effect Table 5 lists the calculated changes in the single-crystal elastic constants for TiZrVNbMo relative to TiZrVNb and TiZrNbMo. These changes reveal the “HEA-type of alloying” effect when going from a four component to a five component equiatomic alloy. We should emphasize that this process means that the atomic fraction of all constituents in the base alloy decreases from 0.25 to 0.20 when 20% V or Mo is introduced. In Table 5 alloying effects are shown for CPA, SQSu and SQSr . It is obvious that adding equiconcentration V (Mo) to TiZrVNb (TiZrNbMo) strongly affects some of the elastic constants. In all three sets of theoretical data, the largest changes are obtained for C11 when Mo is added to TiZrVNb. Actually, the theoretical C11 of Mo is much larger than that of TiZrVNb, which could partly explain the large 21

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∆C11

∆C12

TiZrVNbMo TiZrNbMoV

50.5 6.7

7.4 1.4

TiZrVNbMo TiZrNbMoV

46.8 4.8

9.4 -1.0

TiZrVNbMo TiZrNbMoV

49.5 5.9

∆C44

∆C ′

∆(C12 − C44 )

-2.7 -2.1

21.5 2.7

10.2 3.5

2.5 -2.8

18.7 2.9

7.1 2.0

8.4 -2.7

20.3 2.1

0.4 4.4

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Table 5: Changes in the single-crystal elastic constants (GPa) as going from TiZrVNb or TiZrNbMo to TiZrVNbMo. TiZrVNbMo (TiZrNbMoV ) stands for the effect of Mo (V) on the elastic constants of TiZrVNb (TiZrNbMo). Results are shown for CPA, SQSu and SQSr (all data taken from Table 2).

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positive slope. The nearly 50 GPa increase in C11 upon Mo addition to TiZrVNb is reflected by ∆C ′ being about half of ∆C11 . The second largest change is seen for C12 again upon Mo addition, which is also in line with the individual elastic constants of Mo and TiZrVNb. The changes are much smaller for C44 and for all elastic constants when V is added to TiZrNbMo. The alloying effects by V may be understood on the same footing as above, i.e. by comparing the elastic parameters of pure V to those of the TiZrNbMo host. The trend of C44 is much less obvious when Mo is introduced in TiZrVNb. Pure Mo has the largest C44 among all metals considered here (96.0 GPa in VASP and 117.3 GPa in EMTO, Table 1) whereas C44 of TiZrVNb (18.5-21.6 GPa in SQS and 50.5 GPa in CPA, Table 2) is similar to those of the other two HEAs. Simple linear mixing for C44 of TiZrVNbMo predicts 35.0-36.5 GPa in VASP-SQS and 63.7 GPa in EMTO-CPA. If one uses C44 of Mo computed at the equilibrium volume of TiZrVNb (72.4 GPa), linear mixing still yields large positive change for C44 when going from TiZrVNb to TiZrVNbMo. On this ground, one might expect that adding equiatomic Mo to the base alloy should strongly enhance C44 . However, ab initio calculations show a completely different trend. Actually, both CPA and SQSu give small changes (-2.7 and 2.5 GPa, respectively) and the somewhat larger ∆C44 predicted by SQSr (8.4 GPa) is also only ∼ 50% of what the linear mixing suggests. The reason behind this “inconsistency” should be associated with the complex 22

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chemical interactions within the HEAs which are to some extend captured by both alloy theories (CPA and SQS) but fall outside of the linear rule of mixtures. In the next section, we make a more robust assessment of the rule of mixtures for the present HEAs and investigate if at least the general trends could be reproduced by such simple estimate.

p

est

N 1 X = pi , N i=1

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5.2. Rule of mixtures for refractory HEAs The rule of mixtures have often been employed to make an initial screening of the properties of alloys and compounds. It gives a simple estimate of the selected and often unknown property of the alloy based on the accessible properties of the alloy constituents. A widely used example is Vegard’s rule where the lattice constant of a binary alloy is estimated form a linear interpolation between the lattice parameters of the constituents (assuming same crystal lattice for both). Here we possess sufficient amount of reliable ab initio data for the alloys and their components to be able to test such rules in the case of refractory HEAs. Based on data obtained for the elemental metals, we estimated the lattice constants and elastic parameters of HEAs according to (4)

where pi stands for elastic parameters or lattice parameter of pure metal i in bcc structure and N is the number of components in the alloy. We compare the so estimated physical parameter to the one computed in fully self-consistent ab initio calculation. The results are shown in Figure 7. Upper panel contains data obtained using the EMTO method for pure metals and the EMTO-CPA method for HEAs. Lower panel shows results obtained by VASP and VASP-SQS both without (SQSu ) and with (SQSr ) LLD. Since bcc Zr and bcc Ti are dynamically unstable (at static conditions) one cannot define a physically meaningful shear modulus for these two hypothetical metals. Hence, G (same applies to E) cannot be obtained directly from the rule of mixture. But these polycrystalline elastic parameters can be computed from the averaged single-crystal data. The so estimated and calculated values are also shown in Figure 7. Note that the Poisson ratio can easily be derived from G and E. We observe that in general the trends of the lattice parameters and elastic constants are well reproduced by the rule of mixtures. The largest errors (de23

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Figure 7: (Color online) Comparison between calculated and estimated single-crystal and polycrystalline elastic constants and lattice parameters of TiZrVNb, TiZrNbMo and TiZrVNbMo HEAs. Upper panel shows data obtained with EMTO-CPA method and lower panel with VASP-SQS both with and without LLD.

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viations between calculated and estimated data) are obtained for the lattice parameters. The differences are not systematic: Eq. (4) overestimates the CPA lattice parameters of TiZrNbMo and TiZrVNbMo and slightly underestimates that of TiZrVNb. Very similar deviations are found for the SQS lattice parameters as well. On the other hand, the estimated elastic constants are surprisingly close to the theoretical values in both CPA and SQS calculations. One can conclude that the linear rule of mixtures provides a rather useful estimate for the elastic parameters of unknown HEAs assuming that the elastic constants of the alloy constituents are known for the same crystal structure. Nevertheless, one should notice that this nice “agreement” strictly holds only on the scale of the present elastic constants (0-250 GPa), but fails when one looks at the individual elastic parameters on their own scales.

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The elastic properties of bcc TiZrVNb, TiZrNbMo and TiZrVNbMo random alloys were studied using two first-principles methods in combination with two techniques for random alloys. First, the DFT tools (VASP and EMTO) were assessed by comparing the two sets of ab initio bulk parameters with the available experimental data for pure elements. The two methods provide consistent equation of states for bcc Ti, Zr, V, Nb, and Mo, but systematic deviations are observed for the single-crystal elastic constants, especially for C12 and C44 . None of these methods in combination with the present DFT approximation (PBE) is able to provide highly accurate results for all elastic parameters. This means that alloying effects should always be discussed with respect to the data for end members obtained using the same underlying DFT tool. In general, the ab initio results obtained with CPA and SQS for the present HEAs are consistent with each other. Local lattice distortions were investigated by use of SQS cells and the VASP method. LLD has significant impact on the mixing energy. Neglecting this effect can lead to strongly overestimated enthalpy of formation and thus such rough data is completely useless to draw any conclusions concerning the solid solution formation. On the other hand, LLD has very small effect on the equation of state and elastic constants. For these properties, CPA is clearly superior to SQS as it gives more flexibility for the number of components and deviations from equiatomic compositions. The two DFT tools in combination with the two 25

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alloy theories efficiently complement each other in ab initio description of complex alloys. The magnitude of the LLDs were measured for the three HEA systems here using histograms for showing the data. Here we ultimately were interested in quantifying the degree of the distortion. An overlap of the first and second nearest neighbor shell was observed in TiZrVNb, TiZrNbMo and TiZrVNbMo. That means that significant distortions exist in the relaxed cells. We also studied the alloying effects and found large deviations from the trends expected from simple estimates. In particular, when equiatomic Mo is added to TiZrVNb, the C44 elastic parameter decreases or slightly increases (depending on the employed DFT solver and alloy theory) despite the fact that Mo has the largest C44 among all metals and alloys considered here. Having computed all necessary data for the pure end members and alloys, we had the possibility to assess the rule of mixtures for HEAs. The estimated elastic parameters are found to be rather close to the values obtained from the CPA and SQS calculations especially on the scale of the present values (0-250 GPa). But this simple estimate turned out to be too rough to resolve delicate alloying effects like the one seen for C44 of TiZrNbMo and TiZrVNbMo. Nevertheless, when based on data obtained for the same crystal lattice, the rule of mixtures can provide very useful first-level estimates for the values of the lattice parameters and elastic constants including general trends. Acknowledgments The authors acknowledge the Swedish Research Council (VR), the Swedish Foundation for Strategic Research (SSF), the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Carl Tryggers Foundations, the Swedish Innovation Agency (VINNOVA), the Hungarian Scientific Research Fund (OTKA 109570), and the China Scholarship Council for financial supports. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at Link¨oping. References [1] Y. Y. Chen, T. Duval, U. D. Hung, J. W. Yeh, H. C. Shih, Microstructure and electrochemical properties of high entropy alloys-A comparison with type-304 stainless steel, Corros. Sci. 47 (9) (2005) 2257–2279. 26

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[2] H. P. Chou, Y. S. Chang, S. K. Chen, J. W. Yeh, Microstructure, thermophysical and electrical properties in AlxCoCrFeNi high-entropy alloys, Mater. Sci. Eng. B 163 (3) (2009) 184–189.

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[3] V. Dolique, A. L. Thomann, P. Brault, Y. Tessier, P. Gillon, Thermal stability of AlCoCrCuFeNi high entropy alloy thin films studied by insitu XRD analysis, Surf. Coatings Technol. 204 (12-13) (2010) 1989– 1992.

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[7] J. P. Couzini´e, G. Dirras, L. Perri`ere, T. Chauveau, E. Leroy, Y. Champion, I. Guillot, Microstructure of a near-equimolar refractory highentropy alloy, Mater. Lett. 126 (2014) 285–287. [8] O. N. Senkov, G. B. Wilks, D. B. Miracle, C. P. Chuang, P. K. Liaw, Refractory high-entropy alloys, Intermetallics 18 (9) (2010) 1758–1765. [9] O. N. Senkov, J. M. Scott, S. V. Senkova, D. B. Miracle, C. F. Woodward, Microstructure and room temperature properties of a high-entropy TaNbHfZrTi alloy, J. Alloys Compd. 509 (20) (2011) 6043–6048. [10] Y. J. Zhou, Y. Zhang, Y. L. Wang, G. L. Chen, Microstructure and compressive properties of multicomponent Alx(TiVCrMnFeCoNiCu)100-x high-entropy alloys, Mater. Sci. Eng. A 454 (2007) 260–265. [11] Y. J. Zhou, Y. Zhang, Y. L. Wang, G. L. Chen, Solid solution alloys of AlCoCrFeNiTix with excellent room-temperature mechanical properties, Appl. Phys. Lett. 90 (18) (2007) 181904.

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[12] Y. Zhou, Y. Zhang, F. Wang, Y. Wang, G. Chen, Effect of Cu addition on the microstructure and mechanical properties of AlCoCrFeNiTi0.5 solid-solution alloy, J. Alloys Compd. 466 (1) (2008) 201–204.

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[14] B. Cantor, I. T. H. Chang, P. Knight, A. J. B. Vincent, Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A 375-377 (1-2 SPEC. ISS.) (2004) 213–218.

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Highlights

• Local lattice distortion has significant impact on the mixing energy but its effect is negligible for the elastic properties of HEAs.

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• Local lattice distortion may lead to overlap of the first and second nearest neighbor shell in bcc HEAs.

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• Alloying shows unexpected trends highlighting the importance of complex chemical interactions between constituents in HEAs.

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• The rule of mixtures can provide very useful first-level estimates for the values of the lattice parameters and elastic constants of HEAs.

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