Competition between long- and short-range order in size-mismatched medium-entropy alloys

Chemical short-range order (SRO) is known to alter a wide array of alloy properties. Here, we investigate the SRO of binary and ternary alloys in the Cr-Mo-W system using density functional theory (DFT) calculations, coupled with Monte Carlo simulations and two distinct approaches: the real-space cluster expansion (CE) and a machine learned interatomic potential (MLIP) based on the moment tensor potential (MTP) form. The size-mismatched binary, Cr-W, exhibits phase-separating long-range order (LRO) in the phase diagram, but surprisingly, the SRO is predicted to be ordering-type from both CE and MTP. We rationalize this apparent discrepancy by accounting for the large coherency strain present in this system, which significantly suppresses the coherent phase-separating tendency relative to the incoherent phase diagram. This competition between LRO and SRO persists in the ternary Cr-Mo-W system, where we find that SRO tendencies can qualitatively differ from those in the corresponding binary alloy. The real-space CE can efficiently capture the correct nearest neighbor SRO tendencies in this system, despite failing to account for long-range strain effects, provided it is trained on the energies of sufficiently large structures to capture short-range strain contributions over the nearest neighbor ranges; however, we suggest MTP, or more generally MLIP, may provide a more general approach for comprehensive studies in disordered alloy systems, especially in medium- and high-entropy alloys exhibiting large lattice mismatch.