One-dimensional radiation-hydrodynamic simulations of imploding spherical plasma liners with detailed equation-of-state modeling

The initial conditions used in this work are shown schematically in Fig. 1 and summarized in Table 1 (columns 2–5). The rationale for these choices are described in detail in Awe et al.Awe et al. (2011) In short, they are intended to span a range of initial plasma liner velocities $v_{0}$ and kinetic energies $KE_{0}$ from PLX-relevant values (50 km/s, hundreds of kJ) at the lower end to what might be needed for an MIF standoff compression driverThio et al. (1999, 2001); Hsu et al. (2012) ($>50$ km/s, $>$ tens of MJ) at the higher end. The initial liner ion density values $n_{0}$ are what can be expected based on plasma jet densities that have already been achieved by the plasma guns designed and built by HyperV TechnologiesWitherspoon et al. (2011) for use on PLX. In this work, we assume that discrete plasma jets have merged at $R_{m}=24.1$ cm and formed a spherically symmetric imploding plasma liner with uniform mass density $\rho_{0}$, thickness 25.5 cm, temperature $T_{0}=T_{e}=T_{i}=1.0$ eV, and uniform inward radial velocity $v_{0}$. The effects of non-uniform initial liner profiles are not studied here but are a potential way of optimizing imploding plasma liner performance.Kagan et al. (2011) The void region (Fig. 1) is modeled using very low density plasma with mass densities of $10^{-8}$ kg/m³ (tabular EOS runs) and $10^{-9}$ kg/m³ (polytropic EOS runs), temperature of 1.0 eV, and the same species as the liner. The different initial void mass densities led to slightly different computational zone setups but did not affect the overall liner evolution.

Although this work does not focus on the physical evolution of a spherical plasma liner imploding on vacuum, which was described in detail in Awe et al.,Awe et al. (2011) we provide here a brief summary of its 1D evolution: (1) the imploding liner converges toward the origin, and the liner density rises as $R^{-2}$, (2) upon the liner reaching the origin, an outward going shock propagates radially outward, shock-heating the incoming liner material, and (3) the outgoing shock reaches the outer edge of the incoming liner, at which time the high pressure of the post-shocked liner material can no longer be sustained, and the entire system disassembles.

This work used the HELIOS-CRMacFarlane, Golovkin, and Woodruff (2006) 1D radiation-hydrodynamic code with LTE and non-LTE EOS tables generated using the PROPACEOSMacFarlane, Golovkin, and Woodruff (2006) code. HELIOS-CR is a Lagrangian code that can use both a polytropic EOS (with adiabatic constant $\gamma$) or a tabular EOS. The HELIOS-CR code has been applied previously to the modeling of inertial confinement fusion (ICF) implosions.MacFarlane et al. (2005); Welser-Sherrill et al. (2007) PROPACEOS EOS and frequency-dependent opacities are computed using a combination of an isolated atom model and, at high densities at which cohesive effects are important, a quotidian-EOS-like model.More et al. (1988) At the densities of interest in this paper, energies and pressures are computed for an isolated atom model in which atomic level populations are computed using Boltzmann statistics and the Saha equation (LTE case) or a collisional-radiative model (non-LTE case). The energies and pressures of the EOS tables are based on the free energy, thus providing for thermodynamically consistent energy and pressure values. For the non-LTE case, the effects of photoionization and photoexcitation are ignored when computing the temperature- and density- dependent tabular EOS data, as the radiation field is a non-local quantity, i.e., the radiation can originate in other parts of the plasma, and therefore is unknown when generating the tables. Detailed atomic models are utilized in which all ionization stages are included, e.g., for Ar, a total of $>10^{4}$ energy levels are included. More than $10^{6}$ transitions were used for the opacity calculations. Frequency-dependent radiation transport is treated using a radiation diffusion model; fifty frequency groups were used in this work. When modeling a plasma that is not optically thick (our case) with radiation diffusion, it is possible that radiation losses could be overestimated. Finally, we checked energy conservation at $t=5$ $\mu$s (near peak compression) for the polytropic, LTE tabular, and non-LTE tabular cases, using run 6 of Table 1 as an example. Energy is conserved to within 0.1%, 0.1%, and 1.1% for the polytropic, LTE, and non-LTE runs, respectively. These are very small effects compared to the differences we report in this paper, and thus we conclude that energy conservation does not impact our results.

We previously verifiedAwe et al. (2011) a HELIOS-CR calculation against the convergent shock Noh problemNoh (1987) and got better than 5% agreement for the highest resolution case we ran (50 $\mu$m zone resolution over a sphere with 25 cm initial radius). We also did a grid resolution convergence study showing that an average grid resolution of 250 $\mu$m gave $\sim 10$% accuracy.Awe et al. (2011) In this work, we used 400 computational zones over the initially 255 mm thick liner (average of 637.5 $\mu$m/zone), which was chosen based on achieving a balance between getting reasonable agreement in the benchmarking studies described in Sec. II.3 and the run time of each simulation (typically a few hours to a day per run). We used the automatic zoning feature of HELIOS-CR, and thus the zone size was much smaller than the average near boundaries and much larger elsewhere.

When using a polytropic EOS, HELIOS-CR only allows use of a one-temperature (1T) model. The tabular EOS simulations allow for separate ion $T_{i}$ and electron $T_{e}$ temperatures. All reported HELIOS-CR results with polytropic EOS are 1T, and all reported results with tabular EOS are 2T with radiation diffusion. To verify that the use of 1T versus 2T does not have a great impact on our results, we compared the time evolution of $T_{e}$ and $T_{i}$ for the zone initially at $R=25.1$ cm and saw no difference. Furthermore, the $T_{i}$ evolution from a 1T run was virtually the same as that from the 2T run.

Because Awe et al.Awe et al. (2011) used the 1D Lagrangian radiation-hydrodynamics code RAVEN,Oliphant (1981) we first benchmarked HELIOS-CR against RAVEN by performing runs 1–8 of Table 1 using a polytropic EOS and comparing with the RAVEN results. In this paper, all polytropic EOS runs used $\gamma=1.6667$. We also included thermal tranport and radiation diffusion to be as consistent as possible with Awe et al.,Awe et al. (2011) which found that in these spherically convergent simulations, non-physical temperature and pressure spikes occur near the origin due to compression of the leading edge material of the liner. Including thermal and radiation transport in the simulations helps prevent these non-physical spikes, which prevent the imploding plasma liner from reaching as small a radius as it otherwise would, leading to a much-reduced peak pressure.Awe et al. (2011)

We compared the quantities $P_{stag}$ and $\tau_{stag}$, where $P_{stag}$ is defined as the thermal pressure in the simulation zone that is initially 1.0 mm from the inner edge of the liner ($R=24.2$ cm) averaged over a time duration $\tau_{stag}$, which is defined (and also shown graphically in Fig. 2A) as the time duration between maximum pressure to the time when pressure sustainment ends due to the outward going shock reaching the outer edge of the incoming liner. We found that RAVEN $P_{stag}$ results (Table 2) are 1.2–3.5 times higher than the HELIOS-CR results (Table 3). The $\tau_{stag}$ results agree to within 20%. These differences were considered to be small enough to proceed with this study.

We also compared HELIOS-CR and RAVEN simulation results for the time evolution (Fig. 2A) of the pressure $P$, mass density $\rho$, and ion temperature $T_{i}$ of a single computational zone (initially 1.0 cm from the inner edge of the liner, to be consistent with Fig. 4 of Awe et al.Awe et al. (2011)), and the radial profiles (Fig. 2B) of $P$, $\rho$, and electron temperature $T_{e}$ at a single time ($t=6.5$ $\mu$s), for run 6 of Table 1. Both comparisons indicate good qualitative agreement with some discrepancies. One difference is the increase in both $P$ and $T_{i}$ near the end of $\tau_{stag}$ for the HELIOS-CR but not the RAVEN results. The magnitude of the rise in $P$ increases with increasing $v_{0}$ and is present throughout the entire stagnated liner, and is thus not a localized numerical effect in a single zone. The reason for this rise is not yet understood, but because this does not affect the conclusions of this paper, we have set the issue aside. There are also differences in the radial profiles at $t=6.5$ $\mu$s, largely due to the slight difference in time evolution between the two simulations. At this time, there is an outward going shock for both the HELIOS-CR (at $\sim 0.6$ cm) and RAVEN (at $\sim 0.5$ cm) simulations, but the HELIOS-CR shock has propagated farther outward. The discrepancy in $T_{e}$ inside the shock also appears to be due to the difference in time evolution, as slightly later radial profiles from the RAVEN results also fall toward the origin (see Fig. 4 of Awe et al.Awe et al. (2011)). These differences do not affect the conclusions of this paper. The rest of the paper focuses on comparing HELIOS-CR simulations using a polytropic EOS versus tabular EOS data.

The initial conditions we used (see Table 1) are in a parameter space in which LTE is not obviously correct. Non-LTE can have an effect on EOS (i.e., ionization/excitation) and radiation losses (excited states tend to be less populated). Below an argon ion density of $\sim 10^{17}$ cm^-3 at $\sim 1$ eV, the fractions of Ar I, Ar II, and Ar III as a function of temperature are noticeably different between LTE and non-LTE calculations. Above $\sim 10^{17}$ cm^-3, corresponding to later stages of liner implosion and stagnation, the EOS data are similar. We performed run 6 of Table 1 using both LTE and non-LTE tabular EOS data to explicitly assess the difference in results. Figure 3 shows a comparison of $P$, $\rho$, and $T_{i}$ versus time for the computational zone that is initially 1.0 cm from the inside edge of the plasma liner. Note that for the tabular EOS runs, the pressure is calculated as $n_{i}T_{i}+n_{e}T_{e}$, where $n_{i}$ and $n_{e}$ are the ion and electron densities, respectively, $n_{e}=Zn_{i}$, and $Z$ is the mean charge state. The LTE and non-LTE results are virtually indistinguishable over the entire duration of the simulation, and there are also no appreciable differences in $P$, $\rho$, and $T$ over the entire liner at several time steps (not shown). This indicates that either LTE or non-LTE EOS models could be used without altering the relevant simulation results for the purposes of this study, and thus we used LTE for all subsequent results presented in the paper.

To assess the effects of ionization and other effects included in the LTE tabular EOS data, we compared HELIOS-CR simulation results using LTE tabular EOS with those using a polytropic ($\gamma=1.6667$) EOS for runs 1–8 of Table 1. For the computational zone that is initially 1.0 mm from the inside edge of the liner, the maximum pressure $P_{max}$ is 3.6–6.1 times larger and $P_{stag}$ 3.9–8.6 times larger for the polytropic EOS results. Complete results for both the tabular LTE and polytropic EOS’s are shown in Tables 1 and 3, respectively. The $T_{i}$ and $\rho$ at peak compression were also higher in the polytropic EOS results as well (not shown). Figure 4 shows $P$, $\rho$, $T_{i}$, and radial position versus time of the computational zone initially at 25.1 cm (1.0 cm from the inside edge of the liner) for the initial conditions of run 6 of Table 1. The lower $P_{max}$ ($\sim$ zone thermal energy divided by zone volume) for the tabular EOS result is consistent with the combined effects of lower zone thermal energy ($\sim\rho T$) and higher zone volume at peak compression, as compared to the polytropic result. Next, we investigate in more detail the reasons for the lower zone thermal energy and higher zone volume at peak compression for the tabular EOS run.

Figure 5 shows various components of the total liner energy versus time for run 6 of Table 1. The liner kinetic energy $KE$ falls quickly around 5 $\mu$s when the leading edge of the liner reaches peak compression and stagnates. In the polytropic EOS case, the difference between the initial $KE$ and the stagnated liner thermal energy $TE$ is due mostly to radiative losses. In the tabular EOS case, around half of the initial liner $KE$ goes to ionization/excitation energy, which is stored in free and excited electrons in the stagnated liner. The mean charge peaks around 7 (see Sec. III.4). The result of including the ionization/excitation physics is a drastic reduction in the liner compression at stagnation, leading to lower $P_{stag}$ (see Tables 1 and 3).

Awe et al.Awe et al. (2011) found the approximate scalings $P_{stag}\sim v_{0}^{15/4}$ and $P_{stag}\sim n_{0}^{1/2}$ for the cases in Table 1 using a polytropic EOS model, and gave an heuristic argument for why $P_{stag}\sim v_{0}^{5}$ ought to be an upper bound on the scaling dependence of $P_{stag}$ on $v_{0}$. The heuristic argument invoked the following assumptions: (1) perfect conversion of liner $KE$ to liner $TE$ at stagnation, (2) no radiation and thermal losses, and (3) no entropy production due to the outward going post-stagnation shock. Because the RAVEN runs of Awe et al.Awe et al. (2011) included radiation and thermal losses, the observed weaker $P_{stag}\sim v_{0}^{15/4}$ scaling is consistent with the heuristic upper bound of $P_{stag}\sim v_{0}^{5}$. Our HELIOS-CR runs using a LTE tabular EOS violate assumption (1) above due to the major ionization/excitation energy channel discussed in Sec. III.2. Thus, it is expected we should see an even weaker scaling than $P_{stag}\sim v_{0}^{15/4}$.

Figure 6 shows $P_{stag}$ versus $n_{0}$ and $v_{0}$ and fits to power law functions for HELIOS-CR simulations of the cases in Table I using a LTE tabular EOS model. While the fitted exponents are very similar across density and velocity groups in Awe et al.,Awe et al. (2011) i.e., $P_{stag}\sim n_{0}^{0.54\pm 0.02}$ and $P_{stag}\sim v_{0}^{3.71\pm 0.08}$, our fitted exponents have a much wider spread, i.e., $P_{stag}\sim n_{0}^{0.64\pm 0.14}$ and $P_{stag}\sim v_{0}^{2.91\pm 0.30}$. This larger spread in exponents is likely due to the fact that the LTE tabular EOS is sensitive to density and temperature in the imploding plasma liner, thus introducing a degree of variability that depends on the liner initial conditions not expected in the polytropic EOS simulations of Awe et al.Awe et al. (2011) Our $P_{stag}\sim v_{0}^{2.91\pm 0.30}$ scaling (depending on $n_{0}$) is indeed weaker than the approximate $P_{stag}\sim v_{0}^{15/4}$ scaling of Awe et al.,Awe et al. (2011) as expected. Our $P_{stag}\sim n_{0}^{0.64\pm 0.14}$ scaling agrees with the $P_{stag}\sim n_{0}^{1/2}$ scaling of Awe et al.Awe et al. (2011) within our observed variation across velocity groups.

Due to the significant effect of ionization/excitation on achievable $P_{stag}$, we explored the use of different plasma species for the same initial liner $\rho_{0}$ and $v_{0}$ (and hence $KE$) of run 6 of Table 1. The only quantity that was varied was the initial number density $n_{0}$ to keep $\rho_{0}$ constant. We performed HELIOS-CR simulations with LTE tabular EOS data to investigate the effect of species on $P_{stag}$. The species chosen for study are those of interest as fusion fuel, blanket materials, or MIF implosion drivers for fusion energy applications: H, D-T (50%-50% by atom), ⁴He, ⁶Li, ¹¹B, Ne, Ar, Kr, and Xe.

Figure 7A shows HELIOS-CR simulation results using a LTE tabular EOS model for $P$, volume, $TE$, and ionization/excitation energy at peak compression versus atomic mass number for the zone initially 1.0 cm from the inner edge of the liner. There is a clear trend of increasing peak $P$ with atomic mass number, quantitatively consistent with the combined effects of decreasing volume and thermal pressure with atomic mass number at peak compression. The thermal and ionization/excitation energies are not monotonic with atomic mass number, and this is related to the different atomic and radiative properties of the different species.

We also examined the mean charge and electron count versus time (Fig. 7B) for the same computational zone as above. Although the heavier species have more electrons to ionize, there are a larger number of the lighter species present by the ratio of their atomic mass numbers. It turns out that the peak mean charge for the heavier species reaches only about 5–10 (Fig. 7B), much less than the mass ratios between the heavier and lighter species, and thus the electron count is higher for the lighter species. Thus, the ionization/excitation energies are in general higher for the lighter species, in part explaining why they reach lower peak pressures for the initial conditions imposed.