First-principles modeling of temperature and concentration dependent solubility in the phase separating Fe$_x$Cu$_{1-x}$ alloy system

We present a novel cluster-expansion (CE) approach for the first-principles modeling of temperature and concentration dependent alloy properties. While the standard CE method includes temperature effects only via the configurational entropy in Monte Carlo simulations, our strategy also covers the first-principles free energies of lattice vibrations. To this end, the effective cluster interactions of the CE have been rendered genuinely temperature dependent, so that they can include the vibrational free energies of the input structures. As a model system we use the phase-separating alloy Fe-Cu with our focus on the Fe-rich side. There, the solubility is derived from Monte Carlo simulations, whose precision had to be increased by averaging multiple CEs. We show that including the vibrational free energy is absolutely vital for the correct first-principles prediction of Cu solubility in the bcc Fe matrix: The solubility tremendously increases and is now in quantitative agreement with experimental findings.