Thermodynamically complete equation of state of MgO from true radiative shock temperature measurements on samples preheated to 1850 K

Plate impact experiments in the 100–250 GPa pressure range were done on a $\langle 100\rangle$ single-crystal MgO preheated before compression to 1850 K. Hot Mo(driver)-MgO targets were impacted with Mo or Ta flyers launched by the Caltech two-stage light-gas gun up to 7.5 km/s. Radiative temperatures and shock velocities were measured with 3%–4% and 1%–2% uncertainty, respectively, by a six-channel pyrometer with 3-ns time resolution, over a 500–900-nm spectral range. MgO shock front reflectivity was determined in additional experiments at 220 and 248 GPa using $\approx 50/50$ high-temperature sapphire beam splitters. Our measurements yield accurate experimental data on the mechanical, optical, and thermodynamic properties of B1 phase MgO from 102 GPa and 3900 K to 248 GPa and 9100 K, a region not sampled by previous studies. Reported Hugoniot data for MgO initially at ambient temperature, $T=298$ K, and the results of our current Hugoniot measurements on samples preheated to 1850 K were analyzed using the most general methods of least-squares fitting to constrain the Grüneisen model. This equation of state (EOS) was then used to construct maximum likelihood linear Hugoniots of MgO with initial temperatures from 298 to 2400 K. A parametrization of all EOS values and best-fit coefficients was done over the entire range of relevant particle velocities. Total uncertainties of all the EOS parameters and correlation coefficients for these uncertainties are also given. The predictive capabilities of our updated Mie-Grüneisen EOS were confirmed by (1) the good agreement between our Grüneisen data and five semiempirical $\gamma \left(V\right)$ models derived from porous shock data only or from combined static and shock data sets, (2) the very good agreement between our 1-bar Grüneisen values and $\gamma \left(T\right)$ at ambient pressure recalculated from reported experimental data on the adiabatic bulk modulus $K_s\left(T\right)$, and (3) the good agreement of the brightness temperatures, corrected for shock reflectivity, with the corresponding values calculated using the current EOS or predicted by other groups via first-principles molecular dynamics simulations. Our experiments showed no evidence of MgO melting up to 250 GPa and 9100 K. The highest shock temperatures exceed the extrapolated melting curve of Zerr and Boehler by $>3300$ K and the upper limit for the melting boundary predictions of Aguado and Madden by $>2600$ K and those of Strachan et al. by $>2100$ K. We show that the potential for superheating in our shock experiments is negligible and therefore out data put a lower limit on the melting curve of B1 phase MgO in $P\text{−}T$ space close to the set of consistent independent predictions by Sun et al., Liu et al., and de Koker and Stixrude.

Figure 1

Pressure-temperature diagram of MgO to 250 GPa showing the melting data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and conventional shock temperature data [25, 26]. Typical uncertainties of the predicted melt lines are 100–200 K. Thicker lines indicate several melting curves that are close enough to be grouped and shown together. Dashed lines are the results of our numerical extrapolation of the reported data. All crosses are the actual error bars.

Figure 2

Schematics of hot MgO targets (not to scale; horizontal dimensions are stretched relative to vertical in order to show the details of high-$T$ sapphire beam splitter): (a) “unwindowed” configuration for conventional shock temperature measurements (used in shots 389–405 and 407–410 and with no Ti foil in shots 383–387) and (b) “windowed” configuration for shock front reflectivity measurements (shots 406 and 411 only).

Figure 3

Maximum likelihood fits to the reported 298-K [29, 44, 45, 46] and our 1850-K $D$ vs $U$ data for MgO (bottom) and analytically computed values of ${\gamma }_0$ from the double-constrained fit to hot Hugoniot (top). The $D\text{−}U$ fit from the last line of Table 4 was used to compute ${\gamma }_0$ for $U_1$ from 2.2 to 4.8 km/s. Note the particle velocity (horizontal) offset between the matching ${\gamma }_0\left(U_1\right)$ and ${\gamma }_0\left(U_2\right)$ curves and its monotonic increase with $U$. All crosses indicate the actual error bars. Although the 1850-K Hugoniot of Molodets [56] shown here evidently diverges from the data, his 300-K Hugoniot (not shown) nearly coincides with our best fit line to 298-K data.

Figure 4

Comparison of our $\gamma \left(V\right)$ model with other reported empirical [76], semiempirical [77, 78], and analytical models [56] for shock-compressed MgO. The shaded areas indicate uncertainties estimated by us.

Figure 5

Comparison of our macroscopic Grüneisen parameters at 1 bar with the reported experimental $\gamma \left(T\right)$ values [70, 71], results of our calculations from the reported measurements of $K_s\left(T\right)$ [69, 70, 71], and the values predicted by other reported EOS [60, 72, 73, 74, 79, 80, 84, 85]. Two shaded areas indicate the uncertainties of our data. Some curves are not plotted because they overlie our model fits precisely and the curves would obscure one another: $\gamma \left(T\right)$ computed from the third-order Birch-Murnaghan EOS of Tange et al. [74] matches the solid red line below 1000 K; that obtained from the Vinet EOS of Tange et al. [74] matches the solid red line above 1000 K; and Kennett and Jackson's second model [73] coincides with the upper thin solid red line representing ${\gamma }_1\left(T\right)+{\sigma }_{{\gamma }_1}$.

Figure 6

Brightness temperature profiles (analyzed assuming $r=0.02)$ from all our experiments. Records are labeled with the corresponding shot numbers; see Sec. I A of Ref. [33] for description of salient features of each shot. Triangles indicate arrival times of shock waves at the free surfaces of MgO samples.

Figure 7

Brightness and true temperature profiles from shots 405 and 406 done with the same pyrometer settings and at nearly the same impact velocity. The noise on records indicates the level of data precision. The brightness temperature from the windowed shot 406 is about 200 K higher than that from its unwindowed counterpart 405. Both temperature profiles match over the overlapped useful portions of both records at a shock front reflectivity value of $r=0.22\pm 0.04$.

Figure 8

Brightness and true temperature profiles from shots 410 and 411 done with the same pyrometer settings and at nearly the same impact velocity. The noise on records indicates the level of data precision. The brightness temperature from the windowed shot 411 is about 200 K higher than that from its unwindowed counterpart 410. Both temperature profiles match over the overlapped useful portions of both records at shock front reflectivity value of $r=0.21\pm 0.04$.

Figure 9

Summary of our shock temperature data from MgO preheated to 1850 K, the reported melting data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and conventional shock temperature data [25, 26]. All unlabeled data are the same as those shown in Fig. 1. Typical uncertainties of the predicted melt lines are 100–200 K. Thicker lines indicate several melting curves that are close enough to be grouped and shown together. Dashed lines are the results of our numerical extrapolation of the reported data. All crosses are the actual error bars.