Strength can be controlled by edge dislocations in refractory high-entropy alloys

Energy efficiency is motivating the search for new high-temperature (high-T) metals. Some new body-centered-cubic (BCC) random multicomponent “high-entropy alloys (HEAs)” based on refractory elements (Cr-Mo-Nb-Ta-V-W-Hf-Ti-Zr) possess exceptional strengths at high temperatures but the physical origins of this outstanding behavior are not known. Here we show, using integrated in-situ neutron-diffraction (ND), high-resolution transmission electron microscopy (HRTEM), and recent theory, that the high strength and strength retention of a NbTaTiV alloy and a high-strength/low-density CrMoNbV alloy are attributable to edge dislocations. This finding is surprising because plastic flows in BCC elemental metals and dilute alloys are generally controlled by screw dislocations. We use the insight and theory to perform a computationally-guided search over 107 BCC HEAs and identify over 106 possible ultra-strong high-T alloy compositions for future exploration.

, and d, [11 ̅ 2] g vectors. The g vectors were marked on the images with arrows.   The CrMoNbV alloy shows a typical dendritic and interdendritic microstructure with a bodycentered-cubic (BCC) structure, resulting from the different melting points of constituent elements.
The detailed microstructure information of the CrMoNbV alloy can be found in the previous work 1 .

Supplementary Note 2. In-situ neutron diffraction and mechanical testing
The lattice strains ( ℎ ) were calculated from the variation of the diffraction-peak positions, during loading by the following equation: where ℎ is the hkl lattice spacing as a function of the applied stress, and ℎ 0 is the reference hkl

Supplementary Note 3. TEM Analysis of Burgers vector and character in NbTaTiV
The TEM contrast of a dislocation is related to g⋅b, and when |g⋅b|/gb > 1/3, the contrast is visible in the TEM. Supplementary Table 1

Supplementary Note 4. Analysis of elastic response
Supplementary Figure 5 shows the reciprocal diffraction elastic constants (1/ ℎ and ℎ / ℎ ) , calculated by the Kröner model 2 , plotted as a function of the elasticanisotropy factor, ℎ = { ℎ 2 2 + 2 2 + 2 ℎ 2 (ℎ 2 + 2 + 2 ) 2 }, at both room and elevated temperatures [2][3][4] . It is found that the theoretical Kröner model fits the experimental data very well at all temperatures. Based on this good agreement with the in-situ neutron experimental data, the single-crystal elastic constants ( ), macroscopic bulk modulus ( ), and shear modulus ( ) were calculated, using the Kröner's self-consistent model 3, 4 : where is the diffraction shear modulus, and , , and are constants given by 3 : where is the bulk modulus, which is given by (   Supplementary Figures 6a and 6c shows the atomic-resolution HAADF images of the 11.8 % -strained NbTaTiV and 4.2 % -strained CrMoNbV. Both samples presented the extra (11 ̅ 1) plane as indicated in Supplementary Figures 6a and 6c with "T", and the dislocation line was along the viewing direction [110]. It implied that the dislocations had a (11 ̅ 1) Burgers vector and (11 ̅ 2 ̅ ) slip plane. By the definition of the dislocation-core width, which was the distance between two atomic columns site with ±1/4 b Burgers vectors displacements on the extra plane, the corresponding values were 1.2 nm and 1.6 nm for NbTaTiV and CrMoNbV, respectively. Supplementary Figures 7a and 7b  The ratio of edge dislocation increased in both NbTaTiV and CrMoNbV at a high temperature, which can further support our claim that edge dislocations played the key role in high temperature strengthening.  Supplementary Fig. 8a is obtained. The curve is naturally "smooth" because the alloys have been numbered from the lowest to the highest strength (or strength/weight, Supplementary Figure 8b).

Supplementary Note 7. Finding high-temperature strengths in the whole Cr-Mo-Nb-Ta-V-W-Ti-Zr-Hf-Al composition space.
By following the alloy ranking from the lowest to the highest strength (or strength/ratio in Supplementary Figure 8b), it is known that on top of the yield strength, what is the alloy composition. By plotting directly, the Nb, Mo, etc. contents as a function of the alloy label (the weakest to strongest), the composition vs. alloy number is not a smooth function, but is rather noisy, as presented in the zoom-in of Supplementary Figure 8a. This is because one can find alloys that have similar yield strengths, but very different compositions. Therefore, for the sake of the visualization, the elemental content vs. compositions plot has been "smoothened", by following this operation: take the first 1,000 lowest strength compositions. Compute the average Nb, Mo, ... concentrations among these first 1,000 compositions, and store the result. Take the next 1,000 compositions, compute the average, and store the result. Proceed until one covers all the alloy compositions. Thus, as a final result, one obtains the average contents of Nb, Mo, etc... per "bins" of 1,000 compositions. Hence, the bin-averaged composition can be plotted as a function of the bin number, which goes from 1 to > 10,000. During this averaging operation, the "noise" is lost.
In order to assess how much an element can be varied within a bin, to obtain a similar yield strength, For the case of Al, the atomic volume is 14,075 Å 3 , based on the work by Chen et al. 5 . The atomic volumes of all other elements are the same, as reported in the Ref. 6 . For the Ti and Zr, the values are obtained by extrapolating high-temperature (high-T) measurements to room temperature (RT), while Hf is obtained, employing the Vegard's law on Hf-HEAs. The atomic volumes adopted for Ti, Zr, and Hf are similar to those estimated in Ref. 7 , which were instead obtained by extrapolating the elemental values from binary alloys in the literature.
The cubic elasticity constants of the BCC Al are assumed to be equal to the FCC values.

Description
Energy efficiency is motivating the search for new high-temperature metals. Some new bodycentered-cubic random multicomponent "high-entropy alloys (HEAs)" based on refractory elements (Cr-Mo-Nb-Ta-V-W-Hf-Ti-Zr) possess exceptional strengths at high temperatures, but the physical origins of this outstanding behavior are not known.
Here, using a recent mechanistic theory, we have computed the high-temperature (T = 1,300 K) yield strengths based on solute strengthening of over 10 million alloys within the whole Al-Cr-Mo-Nb-Ta-V-W-Hf-Ti-Zr alloy family. In addition, the yield strength/density has been computed.
This database enables the efficient search of new alloys with exceptional high-temperature strengths.

Materials Cloud sections using this data
No Explore or Discover sections associated with this archive record.  Table 3. Single-element properties used for theoretical predictions.

Supplementary Note 8. Energetic competition within the Cr-Mo-W-Zr alloy system
In order to achieve a BCC solid solution, we require the free energy to be lower than competing phases. Two factors enter consideration, enthalpy and entropy. We will compute enthalpies within the Cr-Mo-W-Zr quaternary alloy system utilizing density functional theory Such configurations exist from which we take Nsamples = 20 representative configurations in order to obtain a distribution of instability energies, ΔEk. Our energies, ΔEk, range from 232 up to 314 meV/atom, with an average value of 266 and standard deviation of 15 meV/atom. The energies define a partition function 14 = ( 1 , 2 , 3 , 4 ) ∑ − ⁄ =1 (8) and its associated free energy, = − ( ), in which varying degrees of short-range chemical order are appropriately weighted. The BCC solid-solution gains stability, relative to competing phases above the temperature, T0, at which F vanishes. Vibrational and electronic entropies are neglected, as these are found to be small effects, relative to the entropy of chemical substitution when comparing phases of similar structures (i.e., BCC phase) 15 .
For the equiatomic CrMoWZr, we predict phase separation at low temperatures into a mixture of BCC-Cr, Cr2Zr.cF24, W2Zr.cF24, and BCC-Mo. The transition to a single-phase BCC structure occurs at T0 = 2,300 K. The melting temperature is not precisely known. We estimate it as 2,300 K by averaging the melting temperatures of the six equiatomic binaries. Thus, we predict the equiatomic BCC phase to be unstable at all temperatures below melting. Because Laves phases are major competitors to the HEA, we estimated the free energy of the cF24 Laves phase assuming a concentration, Mo2Zr6, on site 8a and Cr6Mo4W6, on site 16d. This structure lies 82 meV/atom above the convex hull, suggesting that it could be stabilized by the entropy of mixing on the sublattices above T0 = 1,300 K.
Cr and Zr are especially prone to the Laves-phase formation. Hence, we investigated the effect of moving off-stoichiometry on CrMo2W2Zr within a 24-atom supercell. Because the composition moves away from the Cr2Zr and W2Zr Laves phases, the distribution of ΔEk values shifts downward by approximately 100 meV/atom, and we predict the formation of a single-phase BCC structure at T0 = 1,600 K. At the same time, the melting temperature should rise because the composition is enriched in elements, Mo and W, whose melting temperatures are high. Thus, we obtain a thermally-stable non-stoichiometric HEA over a wide temperature range. However, this analysis considers only a subset of potentially-competing phases that omit binary and ternary solid solutions. Hence, further investigation will be required to validate this prediction.