First-principles thermoelasticity of transition metals at high pressure: Tantalum prototype in the quasiharmonic limit

The thermoelastic properties of tantalum have been investigated over its theoretical high-pressure bcc solid phase (up to $26000\mathrm{K}$ at $10\mathrm{Mbar}$) using an advanced first-principles approach that accurately accounts for cold, electron-thermal, and ion-thermal contributions in materials where anharmonic effects are small. Specifically, we have combined ab initio full-potential linear-muffin-tin-orbital electronic-structure calculations for the cold and electron-thermal contributions to the elastic moduli with phonon contributions for the ion-thermal part calculated using model generalized pseudopotential theory. For the latter, a summation of terms over the Brillouin zone is performed within the quasiharmonic approximation, where each term is composed of a strain derivative of the phonon frequency at a particular $\mathbf{k}$ point. At ambient pressure, the resulting temperature dependence of the Ta elastic moduli is in excellent agreement with ultrasonic measurements. The experimentally observed anomalous behavior of $C_{44}$ at low temperatures is shown to originate from the electron-thermal contribution. At higher temperatures, the main contribution to the temperature dependence of the elastic moduli comes from thermal expansion, but inclusion of the electron- and ion-thermal contributions is essential to obtain quantitative agreement with experiment. In addition, the pressure dependence of the moduli at ambient temperature compares well with recent diamond-anvil-cell measurements to $1.05\mathrm{Mbar}$. Moreover, the calculated longitudinal and bulk sound velocities in polycrystalline Ta at higher pressure and temperature in the vicinity of shock melting $(\sim 3\mathrm{Mbar})$ agree well with data obtained from shock experiments. However, at high temperatures along the melt curve above $1\mathrm{Mbar}$, the $B^{\prime }$ shear modulus becomes negative, indicating the onset of unexpectedly strong anharmonic effects. Finally, the assumed temperature dependence of the Steinberg-Guinan strength model obtained from scaling with the bulk shear modulus is examined at ambient pressure.

Figure 1

Zero-temperature shear elastic moduli in bcc Ta over the 10-Mbar pressure range of interest in this paper, as calculated from the FP-LMTO method and from the present MGPT multi-ion interatomic potentials.

Figure 2

For single-crystal Ta at ambient pressure, the present calculated adiabatic elastic moduli (solid lines) and corresponding ultrasonic data (Refs. 4, 30) (solid circles and triangles) are compared as a function of temperature up to melt. Previous calculations via the PIC method (Ref. 10) (dashed lines) are also shown. Note that $C_{44}^S=C_{44}^T=C_{44}$ and that $C_{11}^S>C_{12}^S>C_{44}^S$ at all temperatures, which is consistent with bcc lattice stability up to melt.

Figure 3

The present calculated Ta shear modulus $C^{\prime }$ and its components as a function of temperature at ambient pressure. Top panel: the full calculated $C^{\prime }$ (solid line) and its cold component (dotted line) together with experimental data (Ref. 4) (solid circles). Bottom panel: the relative electron-thermal (dot-dashed line) and ion-thermal (dashed line) components to $C^{\prime }$, as defined in Eq. (19).

Figure 4

The present calculated Ta shear modulus $C_{44}$ and its components as a function of temperature at ambient pressure. Top panel: the full calculated $C_{44}$ (solid line) and its cold component (dotted line) together with experimental data (Ref. 4) (solid circles). Bottom panel: the relative electron-thermal (dot-dashed line) and ion-thermal (dashed line) components to $C_{44}$, as defined in Eq. (19). Note the rapid negative increase in the electron-thermal contribution below $500\mathrm{K}$ and the corresponding rapid decrease above $3000\mathrm{K}$, which explain the observed anomalous thermal behavior of $C_{44}$ discussed in the text.

Figure 5

The present calculated high-pressure elastic moduli $B_{ijkl}^S$ compared to DAC-SAX data (Ref. 3) as a function of pressure at $T=300\mathrm{K}$.

Figure 6

Longitudinal, bulk, and shear sound velocity in Ta along its principal Hugoniot. Present results: open squares and solid line fits to the calculated points for the bcc polycrystal based on the bulk modulus $B_S$ and the average shear modulus $G$ obtained from Hashin-Shrikman bounds. Experimental data from Ref. 22: solid points. The sharp drop in measured sound velocity near $3\mathrm{Mbar}$ is indicative of shock melting, where the shear modulus $G$ has dropped to zero in the liquid above that point.

Figure 7

The pressure dependence of the single-crystal elastic anisotropy is shown for the $T=300\mathrm{K}$ isotherm (top) and along the melt curve $T_m$ (bottom). The minimum in $A$ at $T=300\mathrm{K}$ and $1\mathrm{Mbar}$ (top) is removed by the electron and ion thermal components at high temperature along the melt.

Figure 8

High-pressure and -temperature shear elastic moduli and their components in bcc Ta, as calculated along the melt curve to $10\mathrm{Mbar}$. The temperature range along the melt line is from $3400\mathrm{K}$ at ambient to $26000\mathrm{K}$ at $10\mathrm{Mbar}$ (Ref. 13). The top panel displays $B^{\prime }$ (solid line) and its cold component $B_0^{\prime }$ (dotted line). The second panel from the top shows the additional electron- and ion-thermal relative components ${\alpha }_{\mathrm{el}}^{\prime }$ (dot-dashed line) and ${\alpha }_{\mathrm{ion}}^{\prime }$ (dashed line). The third and fourth panels from the top display the corresponding results for $B_{44}$ and its components. Note that $B^{\prime }$ becomes negative beyond about $1\mathrm{Mbar}$ in pressure.

Figure 9

Temperature dependence of the average shear modulus in Ta at ambient pressure, as applicable to constitutive models of strength. Included for comparison are the Steinberg-Guinan model (Ref. 2) (dotted line) and the present Voigt average of single-crystal bcc moduli (solid line) together with experimental data (open triangles and solid circles). The open triangles represent polycrystalline data from Ref. 5 while the solid circles are Voigt-averaged single-crystal data (Ref. 4).