Theoretical Investigation of Lattice Thermal Vibration Effects on Phase Equilibria Within Cluster Variation Method

First-principles phase equilibria calculations often overestimate an order-disorder transition temperature due to the neglect of local lattice distortion effects originated from the mixture of elements of different atomic sizes. The lattice vibration effects introduced through the Debye-Grüneisen theory within the quasi-harmonic approximation has proven to be quite effective in circumventing the inconveniences. With the preferential enhancement of the stability of a disordered phase by introducing the lattice vibration effects, the transition temperature was reduced considerably. In order to gain further insight into the lattice vibration effects, a systematic investigation of the vibrational free energy of the Debye-Grüneisen theory is attempted on the two-dimensional square lattice which constitutes a prototype study prior to the first-principles calculations on realistic alloy systems. A particular focus of the present study is placed on the effects of Debye temperatures of constituent phases on the transition temperature. It is shown that lattice softening by lattice vibration stabilizes the disordered phase by reducing the energy expended to accommodate atoms of different sizes, which is manifested by the reduction of the curvature of the atomic potentials. It is, however, predicted that an opposite case can also take place. When the Debye temperature of an ordered phase is lower than that of the pure metals, the ordered phase is more stabilized and the inclusion of the lattice vibration effects in the free energy raises the resultant transition temperature.