χ and σ phases in binary rhenium-transition metal systems: a systematic first-principles investigation

The Frank-Kasper phases, known as topologically close-packed (tcp) phases, are interesting examples of intermetallic compounds able to accommodate large homogeneity ranges by atom mixing on different sites. Among them, the χ and σ phases present two competing complex crystallographic structures, the stability of which is driven by both geometric and electronic factors. Rhenium (Re) is the element forming the largest number of binary χ and σ phases. Its central position among the transition metals in the periodic table plays an important role in the element ordering in tcp phases. Indeed, it has been shown that Re shows an opposite site preference depending on which elements it is alloyed with. In the present work, χ- and σ-phase stability in binary Re-X systems is systematically studied by a first-principles investigation. The heats of formation of the complete set of ordered configurations (16 for χ and 32 for σ) have been calculated in 16 well-chosen systems to identify stability criteria. They include not only the systems in which χ-Re-X (X = Ti, Mn, Zr, Nb, Mo, Hf, Ta, W) or σ-Re-X (X = V, Cr, Mn, Fe, Nb, Mo, Ta, W) exist but also the systems in which both phases are not stable, including systems in which X is a 3d element from Ti to Ni, a 4d element from Zr to Ru, and a 5d element from Hf to Os. Careful analysis is done of the energetic tendencies as a function of recomposition, size effect, and electron concentration. Moreover, the site preference and other crystallographic properties are discussed. Conclusions are drawn concerning the relative stability of the two phases in comparison with the available experimental knowledge on the systems.