Ramp compression of tantalum to multiterapascal pressures: Constraints of the thermal equation of state to 2.3 TPa and 5000 K

We report measurements of the compressibility of ramp compressed tantalum to a final stress of 2.3 TPa corresponding to threefold volumetric compression. Using these data, we extended the experimental constraint on the Ta cold compression curve by an order of magnitude in pressure. By combining the resulting data with previous measurements of shock compression and ambient pressure heating, we construct an experimentally bounded and thermodynamically consistent equation of state model for Ta which has 2% uncertainty in pressure at 1 TPa. We therefore propose Ta as an in situ pressure scale for laser-heated static compression experiments which were recently able to reach terapascal pressures and thousands of degrees Kelvin. Our new equation of state of Ta is experimentally constrained at extreme pressures and temperatures relevant to a wide range of planetary interiors and will allow for more accurate comparison between experimental measurements and theory at extreme conditions.

Figure 1

The temperature-pressure phase diagram of Ta. Recent advances in static compression have enabled the study of matter beyond 1 TPa at elevated temperatures (white dashed box). Laser-driven ramp compression allows materials to be studied in their solid form to extreme pressures close to an isentrope (blue line). Shock compression achieves states along a thermodynamic path called the Hugoniot (purple line). The ramp compression measurements presented here, in addition to previous experimental data sets, allow for the construction of an accurate high-temperature Ta EOS far beyond the conditions constrained previously [10, 11, 12, 13] (orange box). Temperature-pressure conditions similar to those found in planetary interiors are now within the operating limits of our EOS (gray dashed lines) [14, 15]. The melt curve, density color map, and isochores are outputs of our high-temperature EOS.

Figure 2

(a) A temporally shaped laser pulse irradiates an Au hohlraum which generates an x-ray bath and launches a gradual compression wave into the stepped Ta sample. A spatially resolving VISAR records the free surface velocity history from each of the four steps (colored traces). (b) Ta Lagrangian sound speed versus particle velocity for seven NIF experiments on Ta is shown with $1\sigma$ uncertainties. The averaged stress-density response is shown in black. Data from previous laser experiments are shown for comparison [41].

Figure 3

The percent correction applied to reduce the measured stress-density response to a hydrostatic pressure-density isentrope. The total correction is $\sim 4$% at 2.3 TPa.

Figure 4

The pressure-density response of Ta to pressures up to 2.3 TPa and 53 g/${\mathrm{cm}}^3$. The measured stress (black dashed line) and 298 K isotherm (black line) are shown. The error bars represent $1\sigma$ standard deviation of multiple measurements collected and averaged into a single result, which also includes measurement uncertainties. Previous ramp compression data from pulsed power [47, 48] (blue and purple) and laser [41] (red) experiments and shock wave reduced isotherms [49, 50] (pink and green) are shown as colored lines. Shock wave [20, 21, 22, 23, 24, 25] (gray symbols) and static [51, 52] (red and orange symbols) data are shown for comparison. The computed cold curve and Hugoniot of the new EOS are shown by cyan and gray curves, respectively. Inset: Comparison of room temperature isotherms from previous studies [49, 50, 51, 52, 53, 54, 55] and the 298 K isotherm determined in this work. Solid lines denote the regions where the isotherms are constrained by data, and dashed lines indicate extrapolations. The asterisk indicates that the data of Cynn have been reinterpreted using a different EOS of Au [17] (see main text.)

Figure 5

Comparison of high-temperature experimental and theory data for Ta with our EOS model. (a) Ambient pressure heating data from Touloukian et al. [62]. (b) A 30.5927 g/${\mathrm{cm}}^3$ isochore determined from DFT-based molecular dynamics simulations. (c) Shock Hugoniot data obtained with two-stage gas gun and convergent explosive drive experiments. [20, 21, 22, 23, 24, 25] (d) Shock release data from two-stage gas gun experiments [63]. The parameters of our high-temperature EOS model are varied until excellent agreement is found with these data (see text).

Figure 6

Temperature-pressure phase diagram of Ta which shows the experimental constraints used in the construction of the high-$T$ EOS. Ramp compression (blue line) and 300 K DAC [52] (cyan line) data provide constraints on $P_{\mathrm{cold}}\left(\rho \right)$. Existing ambient pressure heating [62] (green line), laser-heated DAC [65] (cyan squares), Hugoniot (purple line), and shock melting [19] (red bars) data are used to constrain $P_{\mathrm{thermal}}(\rho ,T)$. Comparisons with isochores calculated at high pressure using DFT-based molecular dynamics (purple dotted line) and release isentropes measured from plate impact experiments [63] (black dashed line) provide scrutiny of the performance of the EOS.