First-principles concentration-wave approach to predict incipient order in high-entropy alloys: case of Ti$_{0.25}$CrFeNiAl$_{x}$

High-temperature disordered multi-component alloys, including high-entropy alloys, experience either segregation or partially-ordered phases to reach low-temperature phases. For Ti$_{0.25}$CrFeNiAl$_{x}$ ($0 \le x \le 1$), experiments suggest a partially-ordered B2 phase, whereas CALculation of PHAse Diagrams (CALPHAD) predicts a region of L2$_{1}$+B2 coexistence. We employ first-principles KKR-CPA to assess stability of phases with arbitrary order and a KKR-CPA linear-response theory to predict atomic short-range order (SRO) in the disorder phase that reveals the competing long-range ordered (LRO) phases in a given Bravais lattice. The favorable SRO provides a specific concentration-waves (site occupation probabilities and partially-ordered unit cells) and estimated energy gains that can then be assess directly by KKR-CPA formation enthalpies. Our results are in good agreement with experiments and CALPHAD in Al-poor regions ($x \le 0.75$) and with CALPHAD in Al-rich region (1$\ge{x}> $0.75). Our first-principles KKR-CPA and SRO-based concentration-wave analysis is shown to be a powerful and fast method to assess competing LRO phases in complex solid-solution alloys, and our results suggests more careful experiments in Al-rich region are needed.