Competition between Long- and Short-Range Order via Cluster Expansion and Machine Learned Interatomic Potentials*

We calculate the phase stability in the Cr-Mo-W system with Density Functional Theory, Cluster Expansion (CE), and Moment Tensor Potentials (MTP) combined with Monte Carlo simulation. We find a surprising competition between the long-range order (LRO) present in the phase diagram and the short-range order (SRO) in the solid solutions.  In the size-mismatched binary subsystems, Cr-W and Cr-Mo,  the calculations reproduce the experimentally observed miscibility gap in both systems; however, SRO in the high-temperature disordered solid solutions are predicted to be ordering type. This apparent contradiction is explained by the coherency strain energies that are even larger than the mixing energies of the solid solutions.  The phase-separating LRO is determined by the incoherent competition between phases, and is not subject to the coherency strain constraints. Hence, the energy of the random alloy is above the incoherent phase separation between the elements, but is below the coherent phase separation energy, explaining the predicted competition between LRO and SRO.  The significant coherency strain energy provides an energetic penalty for clustering in the solid solution for systems with a large size mismatch, leading to the phenomenon of mechanically/incoherent clustering but chemically/coherent ordering.