First-Principles Calculation of the Solubility of Er in Al for Ordered Al3Er

The solubility curves of Er in Al-Er alloys were obtained by first principles calculations based on the density functional theory. The solution energies of Er atoms in these Al-Er alloys were calculated using the frozen core approximation and standard potential approximation for the 4f electrons. The calculated results show that the solution energies of hR20, cP4 and hP8-Al3Er are -1.003 and -0.767 eV/Er atom, -0.987 and -0.757 eV/Er atom, -0.967 and -0.713 eV/Er atom, respectively, obtained from the two approximations. The lattice dynamics calculation shows that the excess enthalpies of hR20, cP4 and hP8-Al3Er are 3.301, 3.229 and 3.309 k(B)/Er atom. The simulated solubility curves were obtained by combining the lattice dynamics values and the weighted average of the solution energy values. The calculated solubility curves of cP4-Al3Er are consistent with the experimental values, which indicates that the 4f electrons play a very important role. In addition, the solubility curve of cP4-Al3Er is very close to that of the hR20-Al3Er, but lower than that of hP8-Al3Er at the same temperature. The chemical driving forces corresponding to the solubility curves of hR20 and cP4-Al3Er are also close to each other, but larger than that of the hP8-Al3Er. Due to the smaller interfacial energy in Al matrix of cP4-Al3Er than that of hR20-Al3Er, it could be deduced that the cP4-Al3Er precipitation is the first in priority order, which is consistent with the experimental observations.