Short-range order and compositional phase stability in refractory high-entropy alloys via first-principles theory and atomistic modeling: NbMoTa, NbMoTaW, and VNbMoTaW

Using an all-electron, first-principles, Landau-type theory, we study the nature of short-range order and compositional phase stability in equiatomic refractory high-entropy alloys, NbMoTa, NbMoTaW, and VNbMoTaW. We also investigate selected binary subsystems to provide insight into the physical mechanisms driving order. Our approach examines the short-range order of the solid solutions directly, infers disorder/order transitions, and also extracts parameters suitable for atomistic modeling of diffusional phase transformations. We find a hierarchy of relationships between the chemical species in these materials which promote ordering tendencies. The most dominant is a relative atomic size difference between the $3d$ element, V, and the other $4d$ and $5d$ elements which drives a $B32$-like order. For systems where V is not present, ordering is dominated by the difference in filling of valence states; pairs of elements that are isoelectronic remain weakly correlated to low temperatures, while pairs with a valence difference present $B2$-like order. Our estimated order-disorder transition temperature in VNbMoTaW is sufficiently high for us to suggest that SRO in this material may be experimentally observable.

Figure 1

Examples of ordered structures based on the bcc lattice which can be described by concentration waves. $A2$ represents a disordered alloy. $B2$ can be described by modes $\mathbf{k}=\{0,0,1\}$, while $B32$ is described by modes $\mathbf{k}=\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\}$.

Figure 2

Plots of the total and species-resolved density of states for NbTa, NbMo, VTa, NbMoTa, NbMoTaW, and VNbMoTaW around the Fermi energy. (The Fermi energy for each system is denoted by a grey, dashed, vertical line.) The total DoS curve is given by the average of the components from the separate species. NbTa is selected as an example of a $4d/5d$ binary where the two components are isolectronic. NbMo is selected as an example of a $4d/4d$ system where there is a valence difference. VTa is selected as an example of a $3d/5d$ system where there is an atomic size and bandwidth difference. It can be seen that, as for the binaries, in the multicomponent systems, isoelectronic species have DoS curves lying almost on top of one another, while species where there is a valence difference are separated. Being a $3d$ transition metal, the curves for V have a different profile those of the $4d$ and $5d$ elements and show the $3d-4/5d$ hybridized bonding states at the lower energies.

Figure 3

Plots of the eigenvalues of the chemical stability matrix around the IBZ for NbTa, NbMo, VTa, NbMoTa, NbMoTaW, and VNbMoTaW at $T=1200\mathrm{K}$. The isoelectronic binary, NbTa, has one very flat mode, associated with weak ordering tendencies. The NbMo system, in which there is a valence difference, has a mode which dips at $H$ and rises at $\mathrm{\Gamma }$, indicative of $B2$-like ordering. The VTa system is dominated by atomic size and bandwidth differences and its mode dips at $P$ and peaks at $H$, indicative of $B32$-like ordering. The modes present in the multicomponent systems can be interpreted as combinations of these behaviors. Flat modes are associated with correlations between isoelectronic species, modes dipping at $H$ are associated with valence differences and associated with $B2$-like ordering, while modes dipping at $P$ are V-dominated and associated with $B32$-like ordering.

Figure 4

Visualized configurations from Monte Carlo simulations for the three considered multicomponent systems at temperatures of 1200, 300, and 10K. V, Nb, Mo, Ta, and W are coloured blue, green, pink, yellow, and grey respectively. A cut has been made through the simulation cell to make ordered structures more clearly visible. In the case of NbMoTaW, the emergence of a layered, $B2$-like structure can be seen in the $T=300\mathrm{K}$ configuration. For the ternary NbMoTa, some layering can be seen relative to the 1200K configuration, but it is less clear than for the quarternary. For the five component system, the system demonstrates multiphase behavior, with patches of $B32$-like order between V and W, and other patches of order akin to that observed in the ternary NbMoTa. Images generated using ovito [78].

Figure 5

Plots of the Warren-Cowley SRO parameters and configurational contribution to the SHC as a function of temperature for the three multicomponent systems considered from lattice-based Monte Carlo simulations using our extracted pairwise parameters. We show ${\alpha }_n^{pq}$ for $n=1,2$. NbMoTa shows little SRO, with the only notable feature being towards pairs with a valence difference on the first shell, and away from those pairs on the second shell, indicative of $B2$-like ordering. The same trend can be seen more strongly in NbMoTaW. The most notable trend on the five component plots is that V-W is largely indifferent on the first shell, but highly favored on the second, a precursor to a $B32$-like structure.