Temperature and particle-size dependence of the equilibrium order parameter of FePt alloys

First, second, and third nearest-neighbor pair mixing potentials for equiatomic FePt alloys were calculated from first principles by the Connolly-Williams method within the canonical cluster expansion formalism. It was demonstrated that these Connolly-Williams potentials (based on completely ordered states) and the corresponding Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA) potentials (based on completely disordered states) can be brought into very close correspondence to each other simply by increasing the magnitude of the strain-induced interactions added to the KKR-CPA potential. Using the mixing potentials obtained in this manner, the dependence of equilibrium L1₀ ordering on temperature was studied for bulk and for (approximately) spherical nanoparticles ranging in size from 2.5 to 6 nm. The order parameter was calculated using Monte Carlo simulations and the analytical ring approximation. The calculated order-disorder temperature for bulk (1495--1514 K) was in relatively good agreement (4% error) with the experimental value (1572 K). For nanoparticles of finite size, the (long-range) order parameter changed continuously from unity to zero with increasing temperature. Rather than a discontinuity indicative of a phase transition, we obtained an inflection point in the order as a function of temperature. This inflection point occurred at a temperature below the bulk phase transition temperature and decreased as the particle size decreased. Our calculations predict that 3.5-nm-diam particles in configurational equilibrium at 600 ^∘{}C (a typical annealing temperature for promoting L1₀ ordering) have an L1₀ order parameter of approximately 0.84 (compared to a maximum possible value equal to unity). According to our investigations, the experimental absence of (relatively) high L1₀ order in 3.5-nm-diam nanoparticles annealed at 600 ^∘{}C or below is primarily a problem of kinetics rather than equilibrium.