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Multi-principal-element alloys, including high-entropy alloys, experiencesegregation or partially-ordering as they are cooled to lower temperatures. ForTi$_{0.25}$CrFeNiAl$_{x}$, experiments suggest a partially-ordered B2 phase,whereas CALculation of PHAse Diagrams (CALPHAD) predicts a region ofL2$_{1}$+B2 coexistence. We employ first-principles density-functional theory(DFT) based electronic-structure approach to assess stability of phases ofalloys with arbitrary compositions and Bravais lattices (A1/A2/A3). Inaddition, DFT-based linear-response theory has been utilized to predictWarren-Cowley short-range order (SRO) in these alloys, which revealspotentially competing long-range ordered phases. The resulting SRO is uniquelyanalyzed using concentration-waves analysis for occupation probabilities inpartially-ordered states, which is then be assessed for phase stability bydirect DFT calculations. Our results are in good agreement with experiments andCALPHAD in Al-poor regions ($x \le 0.75$) and with CALPHAD in Al-rich region($0.75 \le {x} \le 1$), and they suggest more careful experiments in Al-richregion are needed. Our DFT-based electronic-structure and SRO predictionssupported by concentration-wave analysis are shown to be a powerful method forfast assessment of competing phases and their stability inmulti-principal-element alloys.

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