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.