Vibrational entropy changes the solid solubility of a random alloy at high temperatures

We have developed a method to accurately and efficiently determine vibrational entropy as a function of temperature and volume for substitutional alloys from first principles. Using Ti$_{1-x}$Al$_x$N metal alloy as a model system we calculate the isostructural phase diagram by minimization of the free energy, solving the original Gibbs problem of finding its global minimum corresponding to the true equilibrium state of the system. We demonstrate that the vibrational contribution to the free energy has a decisive impact on the calculated phase diagram of Ti$_{1-x}$Al$_x$N alloy, lowering the maximum temperature for the miscibility gap from 9000 K to 2400 K. The solubility limit of the predicted phase diagram is experimentally verified by local chemical composition measurements of thermally aged Ti$_{50}$Al$_{50}$N alloys.