  Building on our previously introduced Multi-cell Monte Carlo (MC)^2 method
for modeling phase coexistence, this paper provides important improvements for
efficient determination of phase equilibria in solids. The (MC)^2 method uses
multiple cells, representing possible phases. Mass transfer between cells is
modeled virtually by solving the mass balance equation after the composition of
each cell is changed arbitrarily. However, searching for the minimum free
energy during this process poses a practical problem. The solution to the mass
balance equation is not unique away from equilibrium and consequently the
algorithm is in risk of getting trapped in nonequilibrium solutions. Therefore,
a proper stopping condition for (MC)^2 is currently lacking. In this work, we
introduce a consistency check via a predictor-corrector algorithm to penalize
solutions that do not satisfy a necessary condition for equivalence of chemical
potentials and steer the system towards finding equilibrium. The most general
acceptance criteria for (MC)^2 is derived starting from the isothermic-isobaric
Gibbs Ensemble for mixtures. Using this ensemble, translational MC moves are
added to include vibrational excitations as well as volume MC moves to ensure
the condition of constant pressure and temperature entirely with a MC approach,
without relying on any other method for relaxation of these degrees of freedom.
As a proof of concept the method is applied to two binary alloys with
miscibility gaps and a model quaternary alloy, using classical interatomic
potentials.
