  The thermodynamics of phase transitions between phases that are
size-mismatched but coherent differs from conventional stress-free
thermodynamics. Most notably, in open systems such phase transitions are always
associated with hysteresis. In spite of experimental evidence for the relevance
of these effects in technologically important materials such as Pd hydride, a
recipe for first-principles-based atomic-scale modeling of coherent, open
systems has been lacking. Here, we develop a methodology for quantifying phase
boundaries, hysteresis, and coherent interface free energies using
density-functional theory, alloy cluster expansions, and Monte Carlo
simulations in a constrained ensemble. We apply this approach to Pd--H and show
that the phase transition changes character above approximately 400 K,
occurring with an at all times spatially homogeneous hydrogen concentration,
i.e., without coexistence between the two phases. Our results are consistent
with experimental observations but reveal aspects of hydride formation in Pd
nanoparticles that have not yet been accessible in experiment.
