Phase stability and short-range order of refractory high entropy alloys
For this purpose, we employ two complementary first-principles strategies. The first one is based on the generalized perturbation method (GPM) in combination with the coherent potential approximation (CPA) . Based on this effective pair interactions are computed on the ideal lattice i.e. without taking local lattice distortions into account. Based on subsequent Monte Carlo calculations the phase decomposition as well as short-range order parameters are calculated. Application to bcc NbMoTaW revealed a long-ranged nature of these interactions. The ground state of bcc NbMoTaW is predicted to consist of B2(Mo;Ta) and B32(Nb;W). At ambient temperatures a B2(Mo,W;Ta,Nb) ordering is found.
More recently novel machine-learning methods  have been utilized which allow in combination with explicit supercell calculations to account also for explicit relaxations in a similar spirit as cluster expansion techniques, being, however, computationally more efficient. Application to the same alloy revealed that while the impact on local relaxations on short-range order parameters is not crucial, it is significant in revealing the decomposition and phase stability at moderate temperatures, where also a new intermediate relaxation-stabilized layered-structure has been found .
 Körmann, F., Ruban, A. V. & Sluiter, M. H. Long-ranged interactions in bcc NbMoTaW high-entropy alloys. Materials Research Letters 5, 35–40 (2017).
 Shapeev, A. Accurate representation of formation energies of crystalline alloys with many components. Computational Materials Science 139, 26–30 (2017).
 T. Kostiuchenko, F. Körmann, J. Neugebauer, and A. V. Shapeev, Impact of local lattice relaxations on phase stability and chemical ordering in bcc NbMoTaW high-entropy alloys explored by ab initio-based machine-learning potentials, arXiv: 1810.10820.