Hybrid Monte Carlo–metadynamics approach to solid–liquid equilibrium in substitutional alloys

Molecular dynamics (MD) simulations are inherently limited in their accessible time scales, making it difficult to reproduce equilibrium solute partitioning in substitutional alloys. Here, hybrid Monte Carlo (MC)–metadynamics approaches are developed to obtain free-energy surfaces (FESs) of solid–liquid coexistence under equilibrium solute partitioning conditions, using Cu Ag as a model system. An adaptive semi-grand canonical Monte Carlo (SGCMC) method is also proposed to efficiently determine the chemical-potential difference corresponding to solid–liquid equilibrium. Atom-swap MC–metadynamics simulations yield convex FESs with minima at solid fractions satisfying the equilibrium solid and liquid compositions. In contrast, SGCMC–metadynamics simulations yield nearly constant FESs throughout the solid–liquid coexistence region while maintaining the equilibrium compositions independently of the solid fraction. These results demonstrate that combining MC with metadynamics enables FES calculations under equilibrium solute partitioning conditions for slow solute diffusion, extending the applicability of metadynamics-based FES calculations to substitutional alloys.