Alloy Phase Diagrams
An alloy phase diagram maps which phases (ordered compounds, solid solutions, or mixtures of them) are stable as a function of composition and temperature. First-principles approaches compute it by fitting a lattice model such as a cluster expansion to DFT energies and running Monte Carlo simulations to obtain free energies and phase boundaries, sometimes adding vibrational or other non-configurational contributions.
The Cu-Pd fcc/B2 phase diagram (2017)
Geng, Boes and Kitchin computed the Cu-rich Cu-Pd phase diagram, relevant to Pd-Cu hydrogen separation membranes, using DFT-fitted fcc and bcc cluster expansions and ATAT Monte Carlo boundary tracing. Configurational free energy alone gave the right topology, with L12 and B2 as the stable ordered phases, but an order-disorder temperature near 720 K versus 823 K in experiment, a critical composition near x(Pd) = 0.48 versus 0.4, and too narrow a B2 field. Adding vibrational free energy widened the B2 region and shifted the critical point toward the Cu-rich side, while lowering Tc slightly to about 700 K (Geng et al. 2017). Summary: geng2017.