First-principles calculations of mechanical and thermodynamic properties of tetragonal Be12Ti

Abstract

The elastic and thermodynamic properties of tetragonal Be₁₂Ti under high temperature and pressure are investigated by first-principles calculations based on pseudopotential plane-wave density functional theory (DFT) within the generalized gradient approximation (GGA) and quasi-harmonic approximation (QHA). The calculated lattice parameters and bulk modulus are in good agreement with the available experimental data. The calculated elastic constants of Be₁₂Ti increase monotonously with increasing pressure, and the elastic stability criterion and the phonon dispersion calculation show that the Be₁₂Ti crystal satisfies the mechanical and dynamic stability under applied pressure (0-100 GPa). The related mechanical properties such as bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) are also studied for polycrystalline of Be₁₂Ti; the calculated B/G value shows that Be₁₂Ti behaves in a brittle manner, and higher pressure can significantly improve the brittleness of Be₁₂Ti. The elastic anisotropy is demonstrated by the elastic anisotropy factors. The direction-dependent Young's modulus and bulk modulus of Be₁₂Ti are dealt with in detail under pressure from 0 GPa to 100 GPa. The pressure and temperature dependencies of the relative volume, the bulk modulus, the elastic constants, the heat capacity and the thermal expansion coefficient, as well as the entropy are obtained and discussed using the quasi-harmonic approximation in the ranges of temperature 0-1600 K and pressure 0-100 GPa.

Fig. 1. Crystal structure of the tetragonal Be₁₂Ti.

Fig. 2. The calculated total energy E varies unit cell volume V for tetragonal Be₁₂Ti.

Fig. 3. The dependence of normalized lattice constants a/a₀, c/c₀, and cell volume V/V₀ on pressure for tetragonal Be₁₂Ti.

Fig. 4. The dependencies of elastic constants (a) and increasing rate (b) for each elastic constant for Be₁₂Ti.

Fig. 5. Phonon dispersion curves for tetragonal Be₁₂Ti at 0 GPa (a), 20 GPa (b), 40 GPa (c), 60 GPa (d) and 100 GPa (e).

Fig. 6. The dependencies of B/G ratio on pressure for Be₁₂Ti.

Fig. 7. Directional dependence of the Young's modulus for Be₁₂Ti under 0 GPa (a) and 40 GPa (b). The corresponding projections in different planes (XY plane, and XZ plane, and YZ plane) of Young's modulus are also shown in (c) and (d).

Fig. 8. Directional dependence of the Bulk modulus for Be₁₂Ti under 0 GPa (a) and 40 GPa (b). The corresponding projections in different crystal planes (XY plane, and XZ plane, and YZ plane) of Bulk modulus are also shown in (c) and (d).

Fig. 9. Dependence of the Helmholtz free energy F(V, T) on crystal volume at various temperatures and the locus of the minimum of the free energy for Be₁₂Ti.

Fig. 10. Temperature dependence of the bulk modulus for Be₁₂Ti.

Fig. 11. Heat capacity of constant volume C_v and constant pressure C_P as a function of temperature at 0 GPa.

Fig. 12. Dependencies of the constant volume heat capacity C_v on temperatures at high pressure.

Fig. 13. Dependence of the entropy S on temperature at high pressure.

Fig. 14. The thermal expansion α of tetragonal Be₁₂Ti as a function of temperature.