Equation of state of CH${}_{1.36}$: First-principles molecular dynamics simulations and shock-and-release wave speed measurements

We report the computation and measurement of the equation of state of a plastic with composition CH${}_{1.36}$. The computational scheme employed is density functional theory based molecular dynamics, at the conditions: 1.8 g/cm${}^3$ $<\rho <10$ g/cm${}^3$, and 4000 K $<T<$ 100 000 K. Experimental measurements are of the shock speeds in a geometry in which the plastic is directly abutting a different material, liquid deuterium, from which release wave behavior in the plastic can be deduced. After fitting our computed pressure and internal energy with a Mie-Grüneisen free energy model, we predict the principal shock Hugoniot and various shock-and-release paths and show that they agree with both recently published laser-shock data and our new data regarding the shock speeds on release. We also establish that, at least in the particular $(\rho ,T)$ range considered, the equation of state of this complex two-component material is well described by an equal pressure and temperature mixture of pure C and H equations of state with a composition-weighted additive-volume assumption. This observation, together with our fit to the limited-range simulation data, can form the basis for the construction of an accurate wide-range equation of state model for this plastic. Implications for its use as an ablator in inertial confinement fusion capsules are discussed.

Figure 1

(a) Polymer model constructed with Material Studio (472 atoms). (b) High-temperature configuration cooled down to ambient conditions (236 atoms).

Figure 2

At a temperature of 4000 K and a density of 1.837 g/cm${}^3$, the carbon and hydrogen atoms are atomically mixed. The pressure is 36 GPa. Left panel is the atomic geometry (C: green balls, H: gray balls). Right panel shows the radial distribution functions.

Figure 3

At a temperature of 4000 K and a density of 4 g/cm${}^3$, the carbon component of the plastic exhibits a large degree of $s{p}^3$ bonding. The pressure is 423 GPa. Left panel is the atomic geometry (C: green balls, H: gray balls). Right panel shows the radial distribution functions.

Figure 4

At a temperature of 11000 K and a density of 1.837 g/cm${}^3$, the carbon and hydrogen atoms are atomically mixed. The pressure is 65 GPa. Left panel is the atomic geometry (C: green balls, H: gray balls). Right panel shows the radial distribution functions.

Figure 5

At a temperature of 50000 K and a density of 10 g/cm${}^3$. The pressure is 5612 GPa. Left panel is the atomic geometry (C: green balls, H: gray balls). Right panel shows the radial distribution functions.

Figure 6

At a temperature of 100 000 K and a density of 10 g/cm${}^3$. The pressure is 7178 GPa. Left panel is the atomic geometry (C: green balls, H: gray balls). Right panel shows the radial distribution functions.

Figure 7

Pressure vs temperature isochores of the CH${}_{1.36}$ plastic over (a) the full range of our consideration, and (b) a closeup showing the lowest-$\rho$ isochores in detail. Corresponding energy vs temperature isochores, (c) and (d). Lines indicate the Mie-Grüneisen EOS, points are the FPMD data. Each isochore is indicated with a different color to guide the eye.

Figure 8

(a) Cold curve for the liquid phase of the CH${}_{1.36}$ system. (b) Zoom of the lower-density part of the cold curve. Line is the third-order Birch-Murnaghan fit, points are the 0 K extrapolation of the finite temperature FPMD data.

Figure 9

Pressure vs temperature isochores of the CH${}_{1.36}$ plastic over (a) the full range of our consideration, and (b) a closeup showing the lowest-$\rho$ isochores in detail. Corresponding energy vs temperature isochores, (c) and (d). Thin dashed lines indicate the Mie-Grüneisen EOS, fit to the FPMD data shown as the dark points. Red points indicate the same isochores for the mixture EOS.

Figure 10

Pressure vs temperature isochores of the CH${}_{1.36}$ plastic over (a) the full range of our consideration, and (b) a closeup showing the lowest-$\rho$ isochores in detail. Corresponding energy vs temperature isochores, (c) and (d). Thin dashed lines indicate the Mie-Grüneisen EOS, fit to the FPMD data shown as the dark points. Green points indicate the same isochores for the QEOS-based LEOS-5350.

Figure 11

Cutaway drawing of the NIF target used in the shock-impedance-matching measurements. The target provides line-of-sight access for a VISAR probe beam to measure the Doppler shift of light reflected from the shock front.

Figure 12

(a) shows a streaked line-imaging VISAR data record for one of the shock-impedance-matching experiments. The shock front is launched near $t=0$, and is observed within the range of positions [$0,-210$] $\mu$m (vertical scale). It becomes visible (reflecting fringes) in the GDP starting around $t=3$ ns. The abrupt fringe shift near 13.8 ns marks the passage of the shock front from the GDP into the D${}_2$. (b) Quantitative determination of the shock velocity at the shock transit from the GDP into the D${}_2$. The blue curves show the continuous velocity extracted from the fringe phase; solid black lines and symbols show the velocity jump inferred from the data.

Figure 13

Principal Hugoniot comparison with the experimental data from M. Barrios et al..[11] Red data points were analyzed using the older interpretation of the EOS of the quartz standard.[36] Cyan points are the re-analyzed data using the updated information regarding the quartz standard.[37] Orange line indicates the Hugoniot of the FPMD-derived Mie-Grüneisen EOS; black line indicates the Hugoniot from LEOS-5350. The initial density for the predicted Hugoniots is taken to be ${\rho }_0=1.05$ g/cc. The experimental samples had initial densities clustered around 1.05 g/cc (from 1.044 to 1.06 g/cc). Note that we account for these differences by plotting pressure as a function of compression ratio, $\rho /{\rho }_0$.

Figure 14

Graphical description of shock-impedance matching in the $P-U_P$ plane. See text for details.

Figure 15

NIF shock-impedance-matching data (red symbols), together with the theoretical results from three EOS models for the GDP: blue: LEOS-5310 (not fit to these data); black: LEOS-5350 (fit to these data); orange: Mie-Grüneisen EOS (fit to ab initio FPMD EOS calculations, but not to these data). For all three theoretical results, the EOS of Ref. 8 was used for deuterium.