First-principles thermodynamics of La${}_2$O${}_3$-P${}_2$O${}_5$ pseudobinary system

Phase stabilities in the La${}_2$O${}_3$-P${}_2$O${}_5$ pseudobinary system have been theoretically analyzed. Phonon modes of five crystals, i.e., La${}_2$O${}_3$, La${}_3$PO${}_7$, LaPO${}_4$, LaP${}_3$O${}_9$, and LaP${}_5$O${}_{14}$, and vibrational modes of gaseous P${}_2$O${}_5$(g) are computed from first principles in order to obtain the contribution of vibrations to the free energy. Additional dynamical contributions, i.e., rotations and translations are also taken into account for the gaseous P${}_2$O${}_5$(g). Vibrational states strongly reflect the crystal structures and bonding states. In this system, the strong P-O covalent bonds in PO${}_4$ units and the relatively weak La-O bonds are found to be the key factors determining the vibrational spectra. In the oxyphosphate, La${}_3$PO${}_7$, the two bonding states are coexisting, and the vibrational spectrum is approximately an average of La${}_2$O${}_3$ and LaPO${}_4$. On the other hand, the P${}_2$O${}_5$-rich compounds, i.e., LaP${}_3$O${}_9$ and LaP${}_5$O${}_{14}$, cannot be treated in the same manner. Their PO${}_4$ units form corner-sharing networks whose vibrations are strongly correlated. The networks raise the vibrational frequencies, leading to high-frequency modes up to 40 THz. The Gibbs energies using the calculated vibrational spectra are in reasonable agreement with the available data of La${}_2$O${}_3$, LaPO${}_4$, and P${}_2$O${}_5$(g), e.g., the differences between the calculated and reported values are less than 2 kJ/mol-atom (20 meV/atom) at 1500 K. The Gibbs energies and the phase stabilities of the other three compounds, La${}_3$PO${}_7$, LaP${}_3$O${}_9$, and LaP${}_5$O${}_{14}$, are evaluated, whose data are yet unknown so far.

Figure 1

Crystal structures of La${}_2$O${}_3$, La${}_3$PO${}_7$, LaPO${}_4$, LaP${}_3$O${}_9$, and LaP${}_5$O${}_{14}$. The red (small) and green (large) balls denote La and O, and the purple tetrahedra denote PO${}_4$ units. La${}_3$PO${}_7$ has a stacked structure of La${}_2$O${}_3$-like and LaPO${}_4$-like structures.

Figure 2

Calculated static energy of each phase in the La${}_2$O${}_3$-P${}_2$O${}_5$ system. The energy reference is the static energy of the mixture of La${}_2$O${}_3$ and P${}_2$O${}_5$(g) in each ratio of $x$.

Figure 3

Total and partial density of states for lattice vibrations of each phase in the La${}_2$O${}_3$-P${}_2$O${}_5$(g) system. The black lines denote the total density of states, and the green (light gray), blue (gray), orange (medium gray), and red (dark gray) lines are contributions of lanthanum, phosphorous, oxygen with La-O bonds, and oxygen with P-O bonds, respectively. The O contribution with P-O bonds is subdivided into two, i.e., corner shared and unshared by PO${}_4$ tetrahedra, shown by the solid and dotted lines, respectively. Oxygen atoms with La-O bonds are also distinguished by the coordination number, 4 and 6, by the solid and dotted lines. The regions below 0 THz correspond to imaginary frequencies.

Figure 4

Vibrational free energies of La${}_2$O${}_3$, La${}_3$PO${}_7$, LaPO${}_4$, LaP${}_3$O${}_9$, LaP${}_5$O${}_{14}$, and P${}_2$O${}_5$(g) as a function of temperature. The broken line denotes the average vibrational free energy of La${}_2$O${}_3$ and LaPO${}_4$.

Figure 5

The vibrational, rotational, and translational free energies of P${}_2$O${}_5$(g) under 1 atm (broken lines). The summation of the three contributions is also shown by the solid line.

Figure 6

The standard Gibbs energies of La${}_2$O${}_3$ (black [top] lines), LaPO${}_4$ (blue [middle] lines), and P${}_2$O${}_5$(g) (red [low] lines) with reference to those at 298 K. The solid and broken lines denote the calculated and experimental values, respectively.

Figure 7

Mixing Gibbs energies under 1 atm of La${}_2$O${}_3$, La${}_3$PO${}_7$, LaPO${}_4$, LaP${}_3$O${}_9$, LaP${}_5$O${}_{14}$, and P${}_2$O${}_5$(g) at 0, 500, 1000, 1500, and 2000 K, with reference to the two end members.