Vacancies in ordered and disordered binary alloys treated with the cluster expansion

First-principles investigations of the thermodynamics of binary alloys using a cluster expansion have so far neglected the presence of vacancies. Here, we invoke a local cluster expansion as a perturbation to the standard binary cluster expansion to model the equilibrium vacancy concentration in a binary alloy as a function of temperature and alloy composition. We apply this approach to a first-principles investigation of the fcc ${\mathrm{Al}}_{1\ensuremath{-}x}{\mathrm{Li}}_{x}$ alloy (for $x$ less than 0.3) which at $x=0.25$ exhibits $\mathrm{L}{1}_{2}$ superstructure ordering. The equilibrium vacancy concentration is predicted to be sensitive to the bulk alloy composition $x$ in the ordered $\mathrm{L}{1}_{2}$ phase, varying by more than an order of magnitude in a narrow interval of $x$ at intermediate temperatures. Both in the solid solution and in the ordered $\mathrm{L}{1}_{2}$ phase, the vacancy prefers a nearest neighbor shell rich in aluminum. In the $\mathrm{L}{1}_{2}$ ordered phase, the vacancy predominantly occupies the lithium sublattice. The type of short-range order around a vacancy should affect the mobility of the constituents of the alloy and hence its interdiffusion coefficient.