First-principles model study of the phase stabilities of dilute Fe-Cu alloys: Role of vibrational free energy

Abstract

The aim of our first-principles study is to model dilute Fe-Cu alloys by including vibrational properties for the determination of phase boundaries. Single-atom and pairwise defects of Cu in bcc Fe and Fe in fcc Cu are considered in terms of structurally relaxed supercells, for which the corresponding total energies are calculated by a density-functional theory (DFT) approach. Based on DFT total energy differences strongly nonbonding substitutional formation energies for all defects are derived. For determining the entropy of mixing, a mechanical statistical model is designed, which is also able to include pairwise and larger defects. By the same DFT approach and within the harmonic model vibrational properties and, in particular, the temperature-dependent vibrational free energies are calculated. Based on these results a vibrational free energy for each defect species is obtained, which in combination with the corresponding DFT formation energies and the entropy of mixing is added up to a grand canonical potential. Minimizing this potential as a function of defect concentrations yields the desired free energies as a function of concentration and temperature, from which the phase boundaries are derived. Analyzing our results we find that the vibrational entropies are determining the temperature dependency of the vibrational free energies. We find, that including the vibrational free energy via the vibrational entropy is very important for explaining the rather wide miscibility range of mixing Cu to bcc Fe as observed experimentally. On the Cu-rich side of the Fe-Cu system including the vibrational entropy narrows the miscibility range.