Angular forces and melting in bcc transition metals: A case study of molybdenum

Using multi-ion interatomic potentials derived from first-principles generalized pseudopotential theory (GPT) together with molecular-dynamics (MD) simulation, a detailed study of melting and related high-temperature solid and liquid properties in molybdenum has been performed. The energetics in such bcc transition metals are dominated by d-state interactions that give rise to both many-body angular forces and enhanced electron-thermal contributions. The angular forces are accounted for in the GPT through explicit three- and four-ion potentials, ${\mathit{v}}_3$ and ${\mathit{v}}_4$, which in the present work are applied in analytic model-GPT (MGPT) form. With the MGPT potentials, ion-thermal melting in Mo has been investigated both mechanically, by cycling up and down through the observed MD melting point at constant volume, and thermodynamically, by calculating solid and liquid free energies. In the former approach, parallel MD simulations have also been done with a corresponding effective-pair potential ${\mathit{v}}_2^{\mathrm{eff}}$ in which the angular dependence of ${\mathit{v}}_3$ and ${\mathit{v}}_4$ has been averaged. The multi-ion angular forces, which are essential to an accurate description of the bcc solid, are found to lower the dynamically observed melting point by about 1000 K. Above the melting transition, however, ${\mathit{v}}_2^{\mathrm{eff}}$ gives a reasonbly good account of the structure and thermal energy of the liquid and the accuracy of this description improves with increasing temperature.

Both the multi-ion and effective pair potentials also permit a large amount of supercooling of the liquid before the onset of freezing. With ${\mathit{v}}_2^{\mathrm{eff}}$ a bcc structure is nucleated at freezing, while with the multi-ion potentials an amorphous glasslike structure is obtained, which appears to be related to the energetically competitive A15 structure. In our second approach to melting, the multi-ion potentials have been used to obtain accurate solid and liquid free energies from quasiharmonic lattice dynamics and MD calculations of thermal energies and pressures. The resulting ion-thermal melting curve exactly overlaps the dynamically observed melting point, indicating that no superheating of the solid occurred in our MD simulations. To obtain a full melting curve, electron-thermal contributions to the solid and liquid free energies are added in terms of the density of electronic states at the Fermi level, ρ(${\mathit{E}}_{\mathit{F}}$). Here the density of states for the solid has been calculated with the linear-muffin-tin-orbital method, while for the liquid tight-binding calculations have been used to justify a simple model. In the liquid ρ(${\mathit{E}}_{\mathit{F}}$) is increased dramatically over the bcc solid, and the net effect of the electron-thermal contributions is to lower the calculated melting temperatures by about a factor of 2. A full melting curve to 2 Mbar has thereby been obtained and the calculated melting properties near zero pressure are in generally good agreement with experiment.