Atomic short-range order and incipient long-range order in high-entropy alloys

Within density-functional theory, we apply an electronic-structure-based thermodynamic theory to calculate short-ranged order (SRO) in homogeneously disordered substitutional $N$-component alloys, and its electronic origin. Using the geometric properties of an $\left(N-1\right)$ simplex that describes the Gibbs (compositional) space, we derive the analytic transform of the SRO eigenvectors that provides a unique description of high-temperature SRO in $N$-component alloys and the incipient low-temperature long-range order. We apply the electronic-based thermodynamic theory and the new general analysis to ternaries ($A1$ Cu-Ni-Zn and $A2$ Nb-Al-Ti) for validation, and then to quinary Al-Co-Cr-Fe-Ni high-entropy alloys for predictive assessment.

Figure 1

Gibbs space for (a) ternary shown by an equilateral triangle (two-simplex) and (b) quaternary shown by a regular tetrahedron (three-simplex), where $\left\{\delta c_{\text{i}}\right\}$ are the directions of concentrations fluctuations. Barycentric points “$o_2$” and “$o_3$” are centroids of two- and three-simplex, respectively and $\left\{x_i\right\}$ are Cartesian labels.

Figure 2

For $A2$ Nb-Al-Ti, concentration-wave polarizations (OTLs) are plotted for lines of short black (old theory), short blue (new theory), and green (ALCHEMI [43, 44, 45, 46] extended to maximal permitted values). Neutron Rietveld-refinement results [41, 42] are plotted by dotted lines. CVM $B2$-sublattice concentrations at $T/T_c=0.9$, 0.8, 0.7 given by circles, squares, and triangles, respectively.

Figure 3

For ${\text{Al}}_{\mathrm{\Delta }/5}{\left[\text{CoCrFeNi}\right]}_{1-\mathrm{\Delta }/5}$; top: stability of $A2$ phase relative to $A1$ phase, for KKR-CPA VP and ASA calculations, see text. Bottom: Common tangent (solid-red) line to free-energy curves shows %Al composition region where mixed $A1+A2$ phase occurs.

Figure 4

For $\mathrm{\Delta }=1.0$ (20% Al) in $A1$ phase, ${\alpha }_{\mu \nu }\left(\mathbf{\text{k}}\right)$ (upper) and $S_{\mu \nu }^{\left(2\right)}\left(\mathbf{\text{k}},T\right)$ (lower) plotted along $\mathrm{\Gamma }\text{−}X\text{−}W\text{−}K\text{−}L\text{−}\mathrm{\Gamma }$ of the fcc Brillouin zone.

Figure 5

For $\mathrm{\Delta }=1.0$ (20% Al) in $A2$ phase, ${\alpha }_{\mu \nu }\left(\mathbf{\text{k}}\right)$ (upper) and $S_{\mu \nu }^{\left(2\right)}(\mathbf{\text{k}},T$) (lower) plotted along $\mathrm{\Gamma }\text{−}P\text{−}N\text{−}\mathrm{\Gamma }\text{−}H$ of the bcc Brillouin zone.

Figure 6

For $\mathrm{\Delta }=0.395$ (8% Al) in $A1$ phase, ${\alpha }_{\mu \nu }\left(\mathbf{k}\right)$ (upper) and $S_{\mu \nu }^{\left(2\right)}(\mathbf{k},T$) (lower) plotted along $\mathrm{\Gamma }\text{−}X\text{−}W\text{−}K\text{−}L\text{−}\mathrm{\Gamma }$ of the fcc Brillouin zone.

Figure 7

For $\mathrm{\Delta }=1.6$ (32% Al) in $A2$ phase, ${\alpha }_{\mu \nu }\left(\mathbf{\text{k}}\right)$ (upper) and $S_{\mu \nu }^{\left(2\right)}(\mathbf{\text{k}},T)$ (lower) plotted along $\mathrm{\Gamma }\text{−}P\text{−}N\text{−}\mathrm{\Gamma }\text{−}H$ of the bcc Brillouin zone.

Figure 8

For ${\text{Al}}_{\mathrm{\Delta }/5}{\left[\text{CoCrFeNi}\right]}_{1-\mathrm{\Delta }/5}$, partial density of states vs energy (relative to $E_f$) at $\mathrm{\Delta }=0.395$, 1.0, and 1.6. For comparison, lattice constant (volume) is kept constant in $A1$ (6.80 a.u.) and $A2$ (5.47 a.u.) phases.

Figure 9

Height $h$ of triangle $\mathrm{\Delta }{A}_1{A}_2{A}_3$ can be generalized to evaluate the height of regular polygon, here, $\left(N-1\right)$ simplex. $P\text{−}{O}_2$ is shift from composition to centroid of the simplex.