Abstract
Nickelbased superalloys and nearequiatomic highentropy alloys containing molybdenum are known for higher temperature strength and corrosion resistance. Yet, complex solidsolution alloys offer a huge design space to tune for optimal properties at slightly reduced entropy. For refractory MoWTaTiZr, we showcase KKR electronic structure methods via the coherentpotential approximation to identify alloys over fivedimensional design space with improved mechanical properties and necessary global (formation enthalpy) and local (shortrange order) stability. Deformation is modeled with classical molecular dynamic simulations, validated from our firstprinciple data. We predict complex solidsolution alloys of improved stability with greatly enhanced modulus of elasticity (3× at 300 K) over nearequiatomic cases, as validated experimentally, and with higher moduli above 500 K over commercial alloys (2.3× at 2000 K). We also show that optimal complex solidsolution alloys are not described well by classical potentials due to critical electronic effects.
Introduction
Nickelbased superalloys exhibit hightemperature strength, toughness, and oxidation resistance in harsh environments.^{1} Improving existing singlecrystal alloys is unlikely as melting is near 1350 °C, and, heat treatment lowers this to ~ 1270 °C. In highspeed turbines, melting reduces below 1250 °C at the zone between the bond coat (e.g., NiAl) and the singlecrystal blade.^{2} As such, the engine efficiency and thrusttoweight ratio can be improved by a guided search for new materials. Highentropy alloys based on refractory elements may achieve higher temperature operation with superior creep strength.^{3} Typical refractory highentropy alloys exhibit a yield strength of 500–700 MPa at 1200 °C, surpassing Nibased superalloys.^{4} Indeed, at elevated temperatures Mobased alloys show good thermal (higher conductivities with lower strains^{5}) and mechanical (machinability)^{6} properties, making them promising candidates.
Almost all highentropy alloys for which mechanical properties have been reported are based on CrFeCoNi with other elements added, e.g., Al,^{7} Mn,^{8} Mo,^{9} and Ti.^{10} CoCrFeNi exhibits very high compression strength at 300 K, often exceeding 1500 MPa. Strains in ascast condition do not often exceed 5–7%, albeit a few exhibit 25–33%.^{10,11} Annealing does improve ductility of ascast alloys.^{11} As with conventional alloys, a rapid decrease in strength (i.e., Young’s modulus, E) occurs above 0.6 of the melting temperature T_{m}, and the strength of alloys approaches 100 MPa at 1273 K.^{7}
Highentropy alloys consist of N (≥5) elements in nearequiatomic compositions (c_{α} ~ 1/N), giving maximal point (mixing) entropy (S_{pt} = −\(\mathop {\sum}\nolimits_{\alpha = 1}^N {\kern 1pt} c_\alpha {\kern 1pt} {\mathrm{ln}}{\kern 1pt} c_\alpha \mathop{\longrightarrow}\limits^{{{\mathrm{max}}}}{\mathrm{ln}}{\kern 1pt} N\)), that may better form solid solutions due to a compromise between the large S_{pt} and a formation (mixing) energy ΔE_{form} that is not too positive (strongly clustering) nor too negative (strongly ordering).^{7} As for binary solid solutions, HumeRothery’s rules^{12} for atomic size difference (δ), crystal structure, valence electron concentration (VEC), and electronegativity difference (Δχ) play a similar role in highentropy alloy formation. The production of several single or multiphase alloys with facecentered cubic (A1), bodycentered cubic (BCC or A2), hexagonal closepacked (A3), or cubic diamond (A4) structures exhibiting enhanced hightemperature strength, ductility, fracture and creep resistance to corrosion,^{7,8,9,10,13,14} and thermal stability^{15} validates the concept of HEAs.^{7} MoWVNbTa, e.g., with a density of 12.2 g/cm^{3}, has a reported usable strength up to 1873 K.^{4}
Nonetheless, from an alloy design perspective, complex solidsolution alloys (CSAs) offer a huge design space to tune properties, especially considering the strong effects alloying has on electronic properties (“band” filling, hybridization, Fermisurface nesting, …), phase stability, and structure. The CSAs comprised of whole composition (Gibbs) space, however, highentropy alloys are a subset of it. Optimized CSAs offer a slightly reduced entropy with a singlephase region, or twophase region for enhanced mechanical properties, existing in a desired operational temperature range.^{16,17,18}
Here we narrow the design of highstrength, refractory (MoW)Ta(TiZr) alloys via KKR electronic structure methods within density functional theory (DFT) using the coherentpotential approximation (CPA) to handle chemical disorder and thermodynamic averaging.^{16,17,18,19} The wellestablished KKRCPA predicts structural properties [e.g., Young’s (E) or bulk modulus (B)], and phase stability (ΔE_{form} vs. {c_{α}}), as well as shortrange order (SRO) via thermodynamic linear response,^{18,20,21,22,23} a method, in particular, that revealed the origin for HumeRothery’s sizeeffect rule.^{21,24} Notably, global stability (ΔE_{form}) and local instability (SRO) should be jointly assessed: while segregation is expected for ΔE_{form} > 0, SRO can be segregating from local compositional instabilities even if ΔE_{form} < 0. To predict mechanical behavior (e.g., E) vs. temperature (T) rapidly, we performed extensive molecular dynamics (MD) simulations based on semiempirical potentials, validated in part by firstprinciples results (and also highlighting limitations of such methods). The tuned and proposed refractory quinary alloys and their properties are placed in context to HumeRotherytype design targets and compared to experiments.
Results and discussion
HumeRothery design targets
Highentropy alloys contain elements with c_{α} ~ 35–12 at.% (N = 3–8). Trial and error has led to alloys with simple crystal structures, and a few with extraordinary properties,^{25} e.g., formability using size disparate elements for confusion by design.^{26} For CSA design of phase stability and of electronic and mechanical behavior, targets for DFTbased KKRCPA are limited by extending HumeRothery^{12} criteria:

1.
Size: solute and host atomic radii (in elemental solid) must differ by <15%.^{12,21} For CSAs, with \(\bar r\) = \(\mathop {\sum}\nolimits_{i = 1}^N {\kern 1pt} c_ir_i\), size limit in terms of standard deviation is sensible: 0 ≤ δ ≤ 6%, with δ = 100% × \(\left[ {\mathop {\sum}\nolimits_{i = 1}^n {\kern 1pt} c_i\left( {r_i^2  \bar r^2} \right){\mathrm{/}}\bar r^2} \right]^{1/2}\).

2.
Lattice: similar crystal structures for solute and host.

3.
VEC: large solubility when solute and host have the same VEC. A metal dissolves one of higher (lower) valency to a greater (lesser) extent.

4.
χ’s: if Δχ is too great, metals tend to form intermetallic compounds, not solid solutions.

5.
ΔE_{form}: for −11 ≲ ΔE_{form} ≲ 5 mRy CSAs stabilized in usable T’s.
A few comments are warranted. In #1, the 6–6.8% achieves 15% rule for CSAs with ~50% confidence level, an inequality also found empirically.^{27,28} Extending #3 via electronic density of states concepts, A2 forms for 4 < VEC < 6 as stability increases when bonding dstates fill, and is maximal when halffilled (VEC ≈ 6); antibonding states fill with VEC > 6 (above a pseudogap, see Results) and stability decreases. Indeed, A2 CSAs are observed when VEC is 5 ± 1.^{29} For 6.8 ≤ VEC ≤ 8 other phases compete, e.g., FeCr has VEC = 7 (like Mn) and constituent’s structure are both A2, yet the CSA is unstable to the σphase, as often appears.^{27} Again, from band filling, A1 becomes more stable for VEC > 8.^{30} CSAs are indeed observed to form within these rules.^{31} ΔE_{form} lower limit in #5 is set by −T_{a}S_{pt}, where annealing temperature (needed for kinetics) is T_{a} ~ 0.55 T_{m} (~1000–1650 K for refractories); upper limit is set such that miscibility gap \({T}_{\mathrm{c}}^{{\mathrm{MG}}}\) < T_{a} (where 158 K ~ 1 mRy). For δ > 5%, CSAs with ΔE_{form} > 5 mRy form complex phases, but tend to form metastable metallic glasses for ΔE_{form} < −11 mRy.^{32} Considering binaries and supercells, these limits for CSA formation are supported.^{33}
As we have shown, transition temperatures from α (CSA) to β (ordered or segregated) phases are well estimated from calculated ΔE_{form}’s:^{34,35} For segregating CSAs (ΔE_{form} > 0), \({T}_{\mathrm{c}}^{{\mathrm{MG}}}\) ≈ ΔE_{form}/S_{pt};^{34} and, for ordering CSAs with \({\Delta E}_{{\mathrm{form}}}^{\alpha \to \beta }\) = \({\Delta E}_{{\mathrm{form}}}^\alpha  {\Delta E}_{{\mathrm{form}}}^\beta\) > 0, the orderdisorder transition is \({T}_{\mathrm{c}}^{{\mathrm{od}}}\) ≈ \({\Delta E}_{{\mathrm{form}}}^{\alpha \to \beta }\).^{35} Notably, #4 also reveals if vibrations are important, as vibrational entropy in binaries correlates as ΔS_{vib} = −Δχ/3 (±0.06Δχ).^{36} Thus, we may quickly estimate T_{c} for dband CSAs as \({T}_{\mathrm{c}}^{\alpha \to \beta } \approx {T}_{{\mathrm{c}},{\mathrm{pt}}}^{\alpha \to \beta }\) \(\left[ {1 + {\mathrm{\Delta }}S_{{\mathrm{vib}}}^{\alpha \to \beta }{\mathrm{/\Delta }}S_{{\mathrm{c}},{\mathrm{pt}}}^{\alpha \to \beta }} \right]^{  1}\), which reproduces measured trends without phonon calculations.^{34} These estimates are within 5–10%.^{34,35}
As a predictive guide, we use KKRCPA results to tune (ΔE_{form}, δ, VEC, and Δχ) vs. {c_{α}} to find (MoW)Ta(TiZr) alloys in fivedimensional (5D) space with better stability and mechanical properties. Results identify the stability of competing phases, possible multiphase regions, electronic properties, and practical design limits. We use the above criteria to restrict search space for mechanical simulations.
Here, via the KKRCPA, we search all CSAs without restrictions on {c_{α}}, or the need for large supercells, as A1, A2 (A3) have only 1 (2) atoms per cell. In this quinary, atomic size of Zr (1.60 Å) is largest, followed by Ti, Ta, and W, Mo (1.46, 1.43, and 1.37, 1.36 Å), where bandwidths (inversely related to atomic size) and alloy hybridization determine the effect of size.^{21} For χ (or Δχ’s), reflecting solubility and vibrational entropy,^{34,36} (W, Mo) have largest χ (2.36, 2.16), followed by (Ti, Ta, Zr) with (1.54, 1.50, 1.33). From Δχ (Mo, W) would have the largest solubility (mixing) range, while %Zr is smaller based on δ. Larger %W increases “E” for engineering needs, but increases weight, and %Zr reduces Ti content while positively impacting flow stress.^{37}
Design and assessment
First, we exemplify in Fig. 1, top panel, our accuracy for ΔE_{form} vs. x in Ta_{1−x}W_{x}, which agrees well with measured values (within 5%), and ordering enthalpies are low (<310 K) compared to melting. Also, we show results for specific ordered cells, which are compared to and agree well with other reliable bandstructure methods [e.g., VASP pseudopotential^{38} and fullpotential linearaugmented plane wave^{39}]. For Mo_{x}(WTaTiZr)_{1−x} in Fig. 1 (bottom panel), we find that A2 is favored over A1 or A3, and that increasing %Mo (larger x) helps stabilize A2. So, in this work, we focus on A2 phase of (MoW)Ta(TiZr) alloy. For x ≳ 0.4 A4 phase competes with A2, and FrankKasper phases, like C15Mo_{2}(TiZr), may be anticipated. Usually, elements from group IIA/IVA of the periodic table, e.g., Al/Si, are added to stabilize or change, e.g., oxidation resistance. However, we find that adding Al stabilizes A2 phase up to 20% Mo, and similar behavior of Al addition has been seen in other alloys too.^{19} On the other hand, Si addition comes out to be energetically less favorable than Al addition.
Highthroughput assessments
For (MoW)Ta(TiZr) results are most easily presented in a cut through 5D {c_{α}} space to visualize with only two parameters (x, y) along lines or planes (Fig. 2), changing {c_{α}} in obvious ways. For fast screening of (MoW)Ta(TiZr) 5D composition design space, we used an estimate for lattice constant to perform “highthroughput”^{40,41,42,43,44,45,46} calculations to discover the best alloys in terms of phase stability and/or mechanical behavior. Specifically, we estimated alloy lattice constants via Vegard’s rule, which is the concentrationweighted sum of volumeoptimized elemental lattice constants in the parent alloys (A2) phase, or, simply, a_{alloy} = \(\mathop {\sum}\nolimits_i \left[ {c_i^Xa_i^X} \right]\), where i = 1, 5 and X = Mo, W, Ta, Ti, Zr. The estimated lattice constants are within 1–3% with respect to the optimized lattice constants for all considered compositions. To downselect regions of interest, we perform the calculation over the entire design space and chose increments in {c_{α}} every 5% to sweep whole 5D space (Fig. 2). For selected alloys, we perform full lattice optimization to determine ΔE_{form} and B, and detail the electronic structure (dispersion and density of states), and the thermodynamic SRO (incipient ordering) for alloy design.
Assessment and validation
We now assess CSAs that best satisfy design criteria, and local stability. Along with other targets, KKRCPA ΔE_{form} vs. {c_{α}} for (MoW)_{x}Ta_{y}(TiZr)_{1−x−y} are shown in Fig. 2. Clearly, ΔE_{form} for equiatomic case is too positive (+12.7 mRy), and decomposition is expected (with \({T}_{\mathrm{c}}^{{\mathrm{MG}}}\) = 1244 K from estimates in HumeRothery section). Our calculated SRO also indicates phase decomposition at spinodal T_{sp} = 1240 K, agreeing with \(T_{\mathrm{c}}^{{\mathrm{MG}}}\), in the nearequiatomic and TiZrrich alloys (Fig. 3). This predicted segregation is corroborated by our Xray diffraction experiments (Fig. 4a) that indicate presence of two (major/minor) phases. The phases were indexed as a disordered A2 phase with \(Im\bar 3m\) space group, and a minor phase of \(Fd\bar 3m\) space group. The A2 lattice parameter was measured as 3.1713 Å (standard deviation: 0.0002 Å). Figure 4b shows the scanning electron microscope (SEM) micrograph of the alloy with a twophase alloy evident. The major phase (A2) is Mo, W, and Tarich, with small amounts of Ti and Zr incorporated in it. Given the higher melting temperatures of Mo, W, and Ta, the major phase is likely to be the primary solidifying phase during the final step of casting. As this phase forms during casting, Ti and Zr are rejected into the surrounding liquid, which subsequently freezes. Hence, the minor phase is Ti and Zr rich and incorporates small amounts of the refractory metals.
To visualize key alloying effects for these CSAs, we plot in Fig. 5 the electronic dispersion and projected total density of states (TDOS), referenced to each alloy’s Fermi energy, E_{F}. With disorder, dispersion exhibits broadening in E and k, showing that k is a “good” (on the scale of the Brillouin zone (BZ)) but not an exact quantum number (as for zerowidth, ordered bands); the width \({\mathrm{d}}{\bf{k}}\sim l_{\mathrm{e}}^{  1}\) (l_{e} is the electronscattering length) and gives rise to increased residual resistivity, as may be calculated.^{47}
Guided by such details, we can improve CSA properties. The equiatomic alloy has a TDOS with E_{F} not yet in the pseudogap between bonding and antibonding states (top, Fig. 5), so this alloy does not satisfy the design criteria. The VEC (the average electrons per atom outside the closed shells of the component atoms) is a dominant factor in controlling the phase stability of the alloys. The electronic states present on/near the E_{F} are chemically most active, which affect the chemical property of the alloy, i.e., more states at E_{F} destabilizes the alloy. This means, for such cases, adding or removing electron we can manipulate the electronic properties very quickly. By integrating states from E_{F} to the pseudogap for equiatomi case, 0.2 electrons are needed to fill bonding states and improved stability. More at.%MoW (VEC = 6) adds electrons, moving E_{F} up (Fig. 5), and ΔE_{form} reduces to stabilize CSAs (Fig. 2). Adding small %Ta helps in altering states near E_{F}: the flat bands near Γ in Fig. 5 are moves from E_{F} for C7, lowering ΔE_{form}, shown in Supplementary Figs. 2 and 3. We show in Fig. 3 that the SRO changes from clustering in C6 to ordering in C7, while ΔE_{form} ~ 0. Here ΔE_{form} reduces quickly for MoW with a small %Ta and %TiZr, while bulk modulus (B) increases quickly (inset Fig. 2). Notably, the dispersion of A2 metals is canonical when scaled by bandwidth (inverse atomic size), and so the behavior of the alloy dispersion is fairly generic and predominantly determined by relative composition and size, hybridization, and band filling.
To promote oxidescale formation for protection, and lightweighting, Al is often added. In Fig. 5, 5% Al added at the expense of Ta to C6 (whose ΔE_{form} = +0.70 mRy) increases disorder broadening (from Al spd hybridization) and causes dstate around Γ (predominantly TiZr) to again straddle at E_{F}. This Fermisurface feature energetically destabilizes the alloy making ΔE_{form} much more positive (+6.8 mRy, with \({T}_{\mathrm{c}}^{{\mathrm{MG}}}\) = 828 K), which is also visible in SRO showing strong clustering behavior at T_{sp} = 780 K (see Supplementary Fig. 4). The most significant AlMo pair suggest that Al will segregate to surfaces due its faster kinetics, as needed for oxide formation, i.e., adding Al at the expense of Ta or TiZr decreases VEC and drops E_{F} into localized dstates, reduces stability; so a balance must be struck by keeping some Ta and TiZr and making VEC high enough to be near ΔE_{form} ~ 0 but with a large B (Fig. 2). Unlike in other systems, Al is not generically a good A2 stabilizer, as it leads to larger electron scattering for reduced stability, increased resistivity, and decreased thermal transport, see ref. ^{18} and references therein.
Chemical SRO
From KKRCPA linear response (see Methods), we predict (Fig. 3) WarrenCowley SRO (or atomic pair correlations) α_{μν}(k;T), whose largest peak at wavevector k_{0} reveals the unstable (Fourier) modes to ordering, or clustering at Γ = (000). As an alloying guide, SRO identifies pairs driving the instability, and predicts the spinodal T_{sp}, where \(\alpha _{\mu \nu }^{  1}\left( {{\bf{k}}_0;T_{{\mathrm{sp}}}} \right)\) = 0 signifying the absolute instability to this chemical fluctuation.^{18,19} In real space, pair probabilities are \(P_{ij}^{\mu \nu } = c_\mu ^ic_\nu ^j\left( {1  \alpha _{\mu \nu }^{ij}} \right)\), with \(\alpha _{\mu \nu }^{i \ne j}\) = 0 for no SRO, and α < 0 (α > 0) indicates ordering (clustering) with bounds of −\(\left[ {{\mathrm{min}}\left( {c_\mu ,c_\nu } \right)} \right]^2\left( {c_\mu c_\nu } \right)^{  1} \le \alpha _{\mu \nu }^{i \ne j} \le 1\).
Nearequiatomic alloys in Fig. 3 have maximal SRO peaks in α_{μν}(k_{0} = Γ;T > T_{sp}) signaling spinodal (infinite wavelength) decomposition in specified pairs at T_{sp} of 1240 K for C1, and at 500 K for C6. This 60% drop in T_{sp} is unsurprising given that ΔE_{form} reduces with MoW and Ta addition (Fig. 2). For C7, where ΔE_{form} has become slightly negative due to movement of bands present at Γ away from E_{F}, a weak incommensurate (longperiod) ordering is found with SRO peak (Fig. 3) at 70% along N–H at k_{0} = (0.85, 0.15, 0). This SRO arises from Fermisurface nesting,^{48} with contributions at a radius of \(\left {{\bf{k}}_0} \right\) ~ 0.86, as confirmed along ΓH (Fig. 3). [SRO is B2 type if it peaks at k_{0} = H = {100}, commensurate with A2 lattice.] For theory and detailed examples, see refs. ^{18,19} We also show, in Supplementary Fig. 4, that 5% Al addition to the C6 alloy instigates a clustering instability. The AlMo pair drives spinodal decomposition at T_{sp} of 780 K, which shows the tendency of Al to phase separate from Mo, an indication that Al’s clustering tendency might be helpful in promoting stable oxide layer at high temperatures. These results indicate that alloying may improve oxidation behavior, just as for FeCr with a narrow window for chromia formation. Clearly, KKRCPA methods address profound electronic and alloying effects not possible from effective potentials, or methods that approximate disorder by ordered configurations.
Deformation analysis
Mechanical properties in CSAs have been studied at macro and microscopic levels,^{49,50} but deformation analysis is key to establish highT structural candidates. We perform quasistatic uniaxial loading via MD simulations (see Methods) by deforming an ideal singlecrystal alloy in small but finite steps and equilibrating after each step. For equiatomic case, <100> compression (Fig. 6a) reveals a smooth stress–strain curve signaling simple plastic flow. In contrast, C3 (0.425 at.%Mo) ideal crystal has stress drops and strainhardening triggered by <111> dislocations; a stress drop at 0.065 strain marks the initiation of dislocation with A2 Burgers vector, b = \(\frac{1}{2}\)<111>, from 77 to 2000 K.
Snapshots of the evolution show that dislocations (edge and screw type) triggered these instabilities (Fig. 6b). The defect mobility is affected by local distortions caused by the different sizes and modulus of the solutes. The rise and drop in stress with increasing strain in the ideal crystal corresponds to the defect evolution where new dislocations occur after every major stress drop followed by strainhardening due to dislocation interactions and drag. An investigation of the local structural environment (Fig. 6b) reveals deviation from perfect A2, as yielding occurs for 300 K. Shear bands (black) are promoted, denoting deformed regions with higher compression. At very high strain the interplay between edge and screw dislocations can be visualized via the band dynamics (Fig. 6b4).
For engineering, Young’s modulus E = 3(1–2ν)B is pertinent, so Poisson’s ratio ν is also key. Smalldeformation MD simulations determined E and ν at 300 K, and E vs. T was found from the elastic stress–strain curve (Fig. 6a). The KKRCPA energy vs. a (A2 lattice constants) at {c_{α}} determines the equilibrium a_{0}, ΔE_{form}, and B (Fig. 6d), all used in Fig. 2. We compare temperaturedependence of E, in Fig. 6c, calculated from MD, KKRCPA (using Grüneisen approximation at low T), and experiments for commercial Morich TZM alloy,^{3} which again validate theory results. As MD is performed on ideal crystals, an “ideal” yield strength is obtained, with a qualitative relative change vs. temperature.
To confirm our predicted E in equiatomic C1, we performed indentation on samples prepared by arcmelting (see Methods), in Fig. 6c, and show that value from measurement 104 ± 12 GPa at 300 K compared very well to our predictions 115 GPa from KKRCPA and 120 GPa from MD. As B changes slowly for binary MoW (Fig. 6d), the Poisson effect (variation of ν) controls strength, which requires W or Morich alloys for larger E values (Fig. 6d). Similarly, for quinary (Mo_{1−z}W_{z})_{0.85}Ta_{0.10}(TiZr)_{0.05}, we find that C10 (z = 0.50) has strength similar to C4 (Fig. 6c). Whereas we predict a region around C\(\bar 10_{}^{}\) (z = 0.05, highlighted in Fig. 6d, which is perpendicular to plane in Fig. 2 at C10) that shows both enhanced stability and E (Fig. 6c). For these CSAs, we find 3× larger E than highentropy alloys at 300 K, and alloys like C\(\bar 10\) have a much larger, less temperaturedependent modulus (Fig. 6c) above 500 K (2.3× at 2000 K) over existing commercial TZM alloys, and lie midway between pure Mo and W, unlike TZM alloys.
Finally, one notable point, albeit not surprising, the classical MD simulations fail to represent properly the alloys that crossover from ΔE_{form} positive to negative (e.g., C4 to C7 or C3 to C10) in which electronic dispersion (not addressed by semiempirical potentials) is controlling the materials physics. Hence, we plot only C\(\bar 10\) KKRCPA values in Fig. 6c, as MD values of E vs. T for C10 and C\(\bar 10\) are similar to C4, whereas C\(\bar 10\) values from first principles increases over C10, as expected from Fig. 6d.
From a design perspective, in general, alloys in CSAs have superior properties over nearequiatomic alloys (socalled highentropy alloys), although the design space becomes enormous. Using a firstprinciples KKRCPA, we predicted the relative phase stability, dispersion, SRO (i.e., incipient longrange order, including T_{c}) and its electronic origin, and mechanical properties over all compositions as a design guide. Using electronic alloy design concepts and criteria, we identified higherstrength refractory (MoW)Ta(TiZr) alloys from materials physics and engineering perspectives. Temperaturedependent deformation (most relevant the elastic behavior) in selected set of alloys was modeled using classical MD simulations, validated from firstprinciples data; we also identified failures in classical potentials that arose from dispersion effects.
Based on our calculation, we designed a Morich region of improved stability with enhanced Young’s moduli over highentropy alloys, as we confirmed experimentally, and an improved temperaturedependence above 500 K (2.3× at 2000 K) over existing commercial alloys. Our electronic structure approaches and analysis of alloying and stability (formation energies, dispersion, and shortrange ordering) highlight how instructive these details are in guiding design. The techniques are quite general for assessing any arbitrary CSAs, where alloying and nontrivial electronic effects play a key role.
Methods
DFT methods
KKR electronic structure is used with the CPA to handle chemical disorder;^{16,17} screened CPA addresses Friedel screening from charge correlations.^{17} Scalarrelativistic effects are included (no spin orbit). Generalized gradient approximation to exchange correlation was included through use of
libXC libraries.^{51} CSAs require only oneatom (twoatom) cells for A1, A2 (A3). BZ integrations were performed with MonkhorstPack kpoint method,^{52} with 12 × 12 × 12 (6) for A1, A2 (A3) meshes. We used 300 kpoints in the irreducible BZ to visualize dispersion along symmetry lines. Each scatterer’s radii were defined by neutral “atoms in cell”, with interstitial divided proportionally to each scatterer, to improve radial density representation near saddle points in the electronic density.^{53,54} We chose L_{max} = 3 sphericalharmonic basis to include s, p, d, and forbital symmetries. Shallow core states were included in the valence in all calculations. A variational potential zero v_{0} was used to yield kinetic energies nearing those of fullpotential methods.^{55} For selfconsistent densities, complexenergy contour integration^{56} used 20point GaussLegendre semicircular contour.Chemical SRO
From KKRCPA linear response, we calculate SRO parameters, α_{μν}(k;T), for μν pairs,^{18,19} as detailed elsewhere.^{20,21,22,23,48} Dominant pairs driving SRO are identified from pairinterchange energies, \(S_{\mu \nu }^{(i,j)}(T)\), or curvature (concentration second variation) of the KKRCPA grand potential, yielding energy cost for concomitant fluctuations of \(c_\mu ^i\), \(c_\nu ^j\) at atomic sites i, j. \(S_{\mu \nu }^{(2)}({\bf{k}};T)\) reveals the unstable (Fourier) modes with ordering wavevector k_{0} (or clustering at (000)), identifies the origin for phase transitions, and dictates the SRO: \(\alpha _{\mu \nu }^{  1}({\bf{k}};T)\) = [c_{μ}(δ_{μν} − c_{ν})]^{−1}\(\left[ {\left( {\delta _{\mu \nu }c_\mu ^{  1} + c_N^{  1}} \right)} \right.\) − \(\left. {\left( {k_{\mathrm{B}}T} \right)^{  1}S_{\mu \nu }^{(2)}({\bf{k}};T)} \right]\). The spinodal temperature, where \(\alpha _{\mu \nu }^{  1}\left( {{\bf{k}}_0;T_{{\mathrm{sp}}}} \right)\) = 0, signifies an absolute instability to this fluctuation and provides an estimate for \({T}_{\mathrm{c}}^{{\mathrm{MG}}}\) or \({T}_{\mathrm{c}}^{{\mathrm{od}}}\).^{18,19,57} For N > 2, pairs driving ordering (clustering) will not necessarily be the same pairs that peak in the SRO due to the matrix inversion that relates them (Fig. 3).
MD simulations
Deformation is evaluated using Largescale Atomistic/Molecular Massively Parallel Simulator package.^{58} The KKRCPA structural parameters are used to validate potentials for finiteT modeling. The forcefield parameters for the quinary are established from available ternary EAM potentials.^{59} We verified similar hybrid potential parameter combinatorial technique for highentropy alloy, like Al_{10}CrCoFeNi.^{19} In A2 lattice^{60}, with dimensions (30 × 30 × 30) a (54 000 atoms), we distributed five elements via composition to form (MoW)Ta(TiZr) solid solutions. Initially, the lattice was melted at 4000 K for 90 ps, followed by a quench to 300 K within 10 ns. Uniaxial deformation was performed after equilibrating and relaxing the structure at high strain, as detailed in Supplementary Video 1.^{19}
Synthesis and characterization
The equiatomic MoWTaTiZr was synthesized by arcmelting pellets of elemental powder blends (Alfa Aesar, purity ≥99.9%) in an ultrahighpurity argon atmosphere on watercooled copper hearth. Powders were used to reduce the large macrosegregation that occurs during casting when using elemental chips. With the significant difference in melting temperatures (3695 K for W vs. 1941 K for Ti) a threestep melting process was adopted. Step 1: W and Ta powders were mixed thoroughly in a SPEX 8000 mill, and pressed using a Carver hydraulic press; and the pellet was then arcmelted. Step 2: elemental blends of Mo, Ti, and Zr were similarly mixed, pressed, and arcmelted. Step 3: both arcmelted buttons were remelted together for a total of four times to ensure better homogeneity.
Phase analyses were carried out using a Philips PANalytical XRay Diffractometer, in a BraggBrentano geometry using CuKα radiation. Microstructure and phase compositions were analyzed using a FEI Helios NanoLab G3UC SEM, equipped with Oxford Energy Dispersive Spectroscopy system. Accelerating voltages of 10–15 kV were employed for imaging and compositional analyses. Compositions were measured at seven different locations for each phase, with the average composition and Xray diffraction shown in Fig. 4. The diffraction pattern indicated the presence of two phases, indexed as a BCC (A2) phase with lattice parameter 3.1713(2) Å and a minor phase with \(Fd\bar 3m\) space group, like for B32 (NaTl prototype) or C15 (MgCu_{2} prototype) structures, with lattice constant 7.6148(9) Å.
Nanoindentation
Nanoindentation utilized a triboindenter HYSITRON TI900 with a Berkovich (3 μm) tip. The indenter control module applies a trapezoidal load on the sample for 10 s, followed by 5 s rest, and unloads in 10 s. To calibrate the sample measurements, which also determines the best applied load and optimum contact depth, the alloy was scanned (15 measurements) on an arbitrarily chosen sample location to optimize for force vs. displacement. For a minimum of 200 nm of contact depth, a 6000 μN load was found to suffice. With these set, we indented the alloy at 30 manually chosen locations to measure the sample’s elastic response; for the equiatomic case, the mean values were as follows: Young’s modulus of 103.73 ± 11.49 GPa; hardness of 4.6 ± 0.34 GPa; and contact depth of 231.18 ± 8.74 nm.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and supplement. Also, the data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Change history
09 July 2018
The caption of Fig. 6 and the main text contained an error in the chemical formula of the “(Mo_{z}W_{1−z})_{0.85}Ta_{0.10}(TiZr)_{0.05}” alloy; it has now been corrected to “(Mo_{1−z}W_{z})_{0.85}Ta_{0.10}(TiZr)_{0.05}”. Figure 5 also contained an erroneous text box, which has now been removed. This has now been corrected in both the PDF and HTML version of this article.
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Acknowledgements
Work supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science & Engineering Division for theory/code development, and by the Office of Fossil Energy, Crosscutting Research for application and validation for specific HEAs. Research was performed at Iowa State University and Ames Laboratory, which is operated by ISU for the U.S. DOE under contract DEAC0207CH11358. Work by A.S., M.S.D., and G.B. supported by the Office of Naval Research (grant N000141612548), with computing resources from the Department of Defense HighPerformance Computing Modernization Program.
Author information
Affiliations
Ames Laboratory, U.S. Department of Energy, Iowa State University, Ames, IA, 50011, USA
 Prashant Singh
 , A. V. Smirnov
 , Pratik K. Ray
 & Duane D. Johnson
Mechanical Engineering, Iowa State University, Ames, IA, 50011, USA
 Aayush Sharma
 & Mouhamad S. Diallo
Materials Science & Engineering, Iowa State University, Ames, IA, 50011, USA
 Pratik K. Ray
 & Duane D. Johnson
Mechanical Engineering & Mechanics, Lehigh University, Bethlehem, PA, 18015, USA
 Ganesh Balasubramanian
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Contributions
P.S., A.S., G.B., and D.D.J. designed project. P.S. performed firstprinciples calculations. A.S. performed MD simulations. M.S.D. and P.K.R. performed experiments. P.S., A.S., and D.D.J. analyzed data and drafted manuscript. All authors contributed to the manuscript.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Prashant Singh or Duane D. Johnson.
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