The effect of lattice vibrations on substitutional alloy thermodynamics

A long-standing limitation of first-principles calculations of substitutional alloy phase diagrams is the difficulty in accounting for lattice vibrations. A survey of the theoretical and experimental literature seeking to quantify the effect of lattice vibrations on phase stability indicates that they can be significant. Typical vibrational entropy differences between phases are of the order of 0.1 to ${0.2k}_{B}/\mathrm{atom},$ which is comparable to the typical values of configurational entropy differences in binary alloys (at most ${0.693k}_{B}/\mathrm{atom}).$ This article presents the basic formalism underlying ab initio phase diagram calculations, along with the generalization required to account for lattice vibrations. The authors review the various techniques allowing the theoretical calculation and the experimental determination of phonon dispersion curves and related thermodynamic quantities, such as vibrational entropy or free energy. A clear picture of the origin of vibrational entropy differences between phases in an alloy system is presented that goes beyond the traditional bond counting and volume change arguments. Vibrational entropy change can be attributed to the changes in chemical bond stiffness associated with the changes in bond length that take place during a phase transformation. This so-called ``bond stiffness vs bond length'' interpretation both summarizes the key phenomenon driving vibrational entropy changes and provides a practical tool to model them.