The dynamics, mixing, and thermonuclear burn of compressed foams with varied gas fills

Inertial confinement fusion¹ (ICF) capsule performance relies on the symmetric implosion of deuterium–tritium (DT) fuel to efficiently couple energy to a “hot spot,” where alpha particles produced from fusion reactions deposit their energy to sustain and propagate thermonuclear burn into a surrounding high density fuel layer. While laboratory ignition has been demonstrated,² performance at available laser energies is highly sensitive to capsule non-uniformities and engineering features, even at sub-micron scales that are difficult to characterize,³ which in turn makes it challenging to predict capsule performance.⁴ Such non-uniformities can seed hydrodynamic instabilities⁵ or jetting^6,7 that can disrupt hot spot formation⁸ and introduce contaminant into the fuel.^9,10 The impact of asymmetries is amplified for high performing designs that typically require high-convergence, high velocity, and low fuel entropy,^11,12 all of which amplify hydrodynamic instabilities and increase susceptibility to jetting.^13–15 

Accurate numerical simulation of ICF capsule implosions, particularly high performance targets, often requires the use of detailed high-resolution three-dimensional simulations using accurate target and drive characterization.^16–29 Nevertheless, the computational expense of such calculations can necessitate the use of surrogate perturbations instead of the accurate representation of geometries, and it is known that implosions are sensitive to features that are too small to feasibly include in a full-physics three-dimensional simulation.^8,30,31 Furthermore, the detailed arrangement of materials and temperatures below the grid scales used for design simulations can have a strong impact on thermonuclear reactivities.^{17,27,32–38} As a result, there is limited evidence to credibly evaluate whether radiation-hydrodynamics simulations are capable of accurately capturing ICF capsule performance when the geometry is faithfully represented in the initial conditions, sufficient resolution is employed, and all physics models that are known to be important are included. Nevertheless, it is necessary to undertake such simulations in order to disentangle the effects of inadequate characterization of the initial conditions, uncertainties in the physical data used in simulations (see, e.g., Refs. 39–44) and unmodeled physics (see, e.g., Refs. 45–48). Furthermore, while current capsule designs⁴⁹ aim to minimize all sources of asymmetries, practical concepts for inertial fusion energy, such as wetted foams,^50,51 may require heterogeneous materials, so that it is critical to understand any potential limitations in our ability to model such materials during implosions.

We report on detailed highly resolved three-dimensional radiation-hydrodynamics simulations of foam-filled capsule implosions performed on the National Ignition Facility⁵² as part of Los Alamos National Laboratory's MARBLE campaign^{27–29,53,54} (some results from these simulations were previously reported in Ref. 29). These are separated reactant capsules fielded in an indirect drive configuration, in which lasers illuminate the inside of a cylindrical Au hohlraum to generate X-rays that drive the capsule implosion. The experiments and capsules are diagramed along with diagnostic lines of sight in Figs. 1(a)–1(c). They contain deuterated plastic (CD) foams^55,56 whose pores are filled with a mixture of hydrogen and tritium (HT) or argon and tritium (ArT) gas. The pores are fabricated to be in a close-packed arrangement but exhibit inherent variations. This complex foam geometry is intended to serve as a strong seed for hydrodynamic instabilities. Because the deuterium and tritium ions reside in different materials, during the course of the implosions, thermonuclear DT fusion reactions occur primarily when the gas mixes with the foam material. The foams are engineered to have macropores of a desired size⁵⁷ in order to control the dominant characteristic length scale of perturbations influencing hydrodynamic instability growth. Smaller pores are expected to seed flows at small scales that rapidly produce atomic mixing, whereas larger pores will seed larger scale flows that take longer to cascade to the smaller scales where material mixing occurs.⁵⁸ We have performed two 3D simulations of capsules with 90 μm pore foams—one with an HT gas fill and one with an ArT gas fill—in order to evaluate our understanding of the physics of the development of mixing in capsule implosions with varied gas fills. We chose to simulate the 90 μm pore case since this is the largest pore size fielded in the experiment, which produces mixing conditions that are hardest to reproduce in reduced-dimension simulations.²⁸ Previous work has demonstrated the need to accurately account for the geometry when modeling shocked heterogeneous media in high energy density conditions,^{27–29,59–71} though in many cases, simulations do not employ full physics, adequate resolution, or three-dimensionality due to the computational expense involved. Our simulations, which employ highly accurate capsule and foam characterization data^72,73 in the initial conditions, exhibit remarkable agreement with experimental data—the level of agreement between the simulations with experimental observables is better than the level of reproducibility of the experiments for nearly all observables. Our results provide evidence that when radiation-hydrodynamics simulations are performed in 3D and include well characterized and well resolved initial conditions, they can accurately predict the outcome of the most complex implosions. Nevertheless, consistent with previously reported results in Refs. 26 and 74–76, the comparison between the simulation and experiment suggests that some disagreement may arise due to radiation-hydrodynamic modeling of shocks, which is known to not reproduce experimentally observed behavior^77–79 even when viscosity is modeled. During shock flash, the temperature in the center of the implosion reaches $≈ 10$ keV, and we can estimate the ion mean free path $l mfp = ν π m 2 k B T i ≈ 0.2 – 2.0 μ$m depending on the species, where ν is the kinematic viscosity, m is the ion mass, k_B is the Boltzmann constant, and T_i is the ion temperature. This is comparable to or larger than the resolved shock width and grid resolution of $0.25 μ$m. Furthermore, the comparison between simulations and experiment suggests that the magnitude of non-radial flows is too large in simulation and that viscosity, which was neglected in simulations, could explain discrepancies in ion temperatures, which are generally too low in simulations when flows are neglected and too high when their impact on the apparent ion temperature is accounted for using methodology developed in Refs. 80 and 81. Indeed, our estimates of the level of viscosity in the burn region indicate that it is six orders of magnitude larger than the effective numerical viscosity of the simulation calculated from the dissipation of non-radial flows. Another possibility is that simulations underestimate the impact of radiative preheat from the shock front, which causes foam expansion ahead of the shock front. This could lead to a more symmetric shock at bang time in the HT-filled implosion, which would improve agreement with X-ray imaging and hence produce higher temperatures at bang time, which would improve agreement with the level of X-ray emission. In addition, this would reduce the density gradients that seed the strong non-radial flows at bang time. These uncertainties have recently been quantified in dedicated experiments.^82,83

Our simulations uncover many interesting effects related to mixing in ICF in the presence of different gas fills and the intricate coupling between physics—particularly, hydrodynamics, radiation, thermal conduction, equation of state, and thermonuclear burn. Temperature separation between ion species, previously observed in,^{27,28,38,84,85} is strongly influenced by the gas species. The species heat to different temperatures during the implosion and do not achieve thermal equilibrium except in regions where they mix atomically. In addition, a radiative precursor from the shock front accelerates mixing between CD and HT due to pore collapse, whereas the ArT fill prevents pore collapse through the combined impact of a higher ionized gas pressure and higher gas opacity (which leads to more gas heating). Dedicated planar experiments have recently quantified our ability to accurately model preheat from the shock precursor and pore collapse.^82,83 The density variation between the gas fills also impacts the shock speed and shock front uniformity, consistent with dedicated experiments reported in Ref. 86. In addition, the ArT gas fill suppresses shock flash yield (yield obtained during the intense heating that occurs as the shock converges to a point at the center of the implosion). Finally, the comparisons between the simulations and experiments suggest that viscosity, which was not explicitly modeled in the present simulations, may play an important role in limiting non-radial velocities in the burn region and spreading out the burn region during shock flash.

We will also use our 3D simulation results to explain in detail results that were reported in Ref. 29 related to the differences in MARBLE implosions when using HT and ArT gas fills. In the experiment, the HT gas-filled implosion exhibited signatures of complete atomic mixing between the foam and gas material (inferred from the neutron yield ratio) as well as somewhat paradoxically, persistent temperature separation between the foam and gas (inferred from the burn-weighted ion temperatures). The ArT gas-filled implosion exhibited signatures of incomplete mixing between the foam and gas but exhibited thermal equilibration between materials. These results are somewhat surprising since one would expect the development of mixing to rapidly bring the materials into thermal equilibrium. Nevertheless, simulations indicate that the combination of four competing factors is able to explain these different behaviors. First, the use of Ar inhibits shock flash yield in the ArT-filled implosion. Second, radiation from the shock front is preferentially absorbed by the foam in the HT-filled experiments, causing the macropores to collapse before the burn phase. This foam expansion is prevented by a combination of the higher opacity of the gas and higher pressure in the gas in the ArT-filled implosion. Third, the HT gas-filled implosion develops temperature separation between materials due to their disparate molar specific heats. The gas and foam specific heats are much closer in the ArT-filled implosion, which prevents the development of temperature separation. Finally, the ArT gas-filled implosions exhibit a lower density gradient at foam/gas interfaces, which reduces the distortion of the shock by interaction with pores. This has a complex but important impact on the development of mixing between foam and gas in these implosions.

This paper proceeds as follows. In Sec. II, we will describe the simulation strategies employed and experimental details. Next, we present comparisons to available experimental data in Sec. III. We then discuss details of the simulation results in Sec. IV. Finally, we conclude in Sec. V.

Simulations were performed with Los Alamos National Laboratory's (LANL) xRAGE Eulerian radiation-hydrodynamics code.^88–91 xRAGE has recently undergone significant improvements for ICF and high energy density physics modeling capabilities, including modifications to adaptive mesh refinement (AMR) and three-temperature (3T) plasma physics to increase the accuracy and reduce numerical seeding of implosion asymmetries,⁸⁹ the implementation of laser ray-tracing physics,^90,92,93 as well as advancements to heat conduction, non-local thermodynamic equilibrium (NLTE) physics, and plasma equation of state.⁹¹ xRAGE uses a custom approximate Godunov-type solver for the Euler equations reminiscent of the method of Harten–Lax–van Leer.⁹⁴ A multi-group multi-frequency gray diffusion (MGMFG) approximation is available for the radiation transport equations.⁹⁵ In the present simulations, we employ a Gray approximation and adjust energy distributions in materials as a function of time and space in order to achieve an agreement between 60 group MGMFG and Gray simulations in two dimensions (this methodology is described in more detail in Ref. 21). Simulations use the LANL OPLIB opacity data⁹⁶ through the TOPS code⁹⁷ and the LANL SESAME tabular equation of state data.⁹⁸ Simulations use a three-temperature plasma model with electronic and ionic thermal conductions with coefficients based on the formulas of Lee and More⁹⁹ with modifications as in Ref. 100. Simulations are performed with adaptive mesh refinement using a $0.25 μ$m maximum resolution, so that there can be as many as 360 cells across a pore. Two-dimensional simulations were used to demonstrate that this resolution was sufficient to converge mixing and integrated performance metrics. Differences between 2D and 3D hydrodynamics^27,105,106 suggest that it would be better to perform the resolution study in 3D. However, this was not possible due to the computational expense. One-dimensional simulations of these and other indirectly driven simulations suggest that the implosion hydrodynamics and burn physics are converged at a $0.25 μ$m resolution but not at lower resolutions. To estimate whether there are flows below the grid scale that we are not capturing, we calculate the Kolmogorov lengthscale at which viscous effects are dominant: $η : = ( ν 3 ϵ ) 1 / 4 = ( ( 0.28 cm 2 / s ) 3 3.33 × 10 15 cm 2 / s 3 ) 1 / 4 = 0.5 μ$m, where ν is the viscosity and ϵ is the average rate of dissipation of non-radial kinetic energy. Here, we have used the values calculated inside the region with $T i > 500$ eV at peak implosion velocity and we have used xRAGE's plasma viscosity model^102–104 for one time step to calculate the viscosity. This suggests that the resolution is adequate to capture the 1D and 3D flow physics, but that lower resolution 3D simulations would not adequately capture the implosion and burn physics. It is also critical to ensure that meshing strategies do not impact simulation results for all hydrodynamics simulation tools. Our meshing criteria were evaluated for relevant test problems in Ref. 88, refined and evaluated for ICF implosions in Ref. 89, and evaluated through code comparison for an implosion with a fill tube in Ref. 101. To evaluate the impact of meshing criteria on our MARBLE implosions, we have compared 1D simulations with AMR to simulations with a fixed 0.25 μm resolution grid, and these simulations exhibit negligible differences in their trajectories, yield production rates over time, and integrated burn metrics. Two three-dimensional (3D) simulations were performed to evaluate the impacts of different gas fills. Each 3D simulation used approximately 400 million CPU hours on Lawrence Livermore National Laboratory's (LLNL) Sierra supercomputer and employed as many as 12 × 10⁹ cells (the cell counts varied throughout the simulations due to the use of AMR). Simulations did not use an explicit model for sub-grid mixing, diffusion, or viscosity but instead employed an implicit large eddy simulation (ILES) approach.¹⁰⁷ Simulations were driven using a frequency dependent boundary X-ray flux source (FDS) derived from integrated capsule and hohlraum simulations performed using LLNL's HYDRA radiation-hydrodynamics code.¹⁰⁸ Details of the HYDRA simulations, which were constrained to experimental X-ray self-emission data from similar experiments, were previously reported in Ref. 52. Two-dimensional FDS-driven xRAGE simulations were also reported in Ref. 52, where it was shown that these also reproduce the experimentally observed burn region size and shape as a function of time.

xRAGE is capable of modeling plasma transport phenomena using a diffusion approximation as well as viscosity,^102–104 but these capabilities were not employed in the present simulations except to evaluate the magnitude of the viscosity in the burn region. Transport coefficients scale roughly as $T 5 / 2 / Z 2 ¯ 2$⁠,¹⁰⁹ where T is the temperature, $Z ¯$ is the average number of free electrons per atom, and $Z 2 ¯$ is the average number of electrons squared per atom, so that their impact is expected to be limited in the mixing regions containing substantial amounts of carbon (see, e.g., Ref. 110). Previous work⁴⁷ has demonstrated that such physics could be important in small regions of MARBLE implosions but the integrated impact on a full implosion has yet to be quantified. The plasma transport and viscosity models were neglected in these simulations due to their computational expense, which would have made the simulations intractable. As we will discuss below in Sec. III, comparison to the experiment suggests that the absence of viscosity could explain discrepancies between simulated and experimentally measured burn-weighted ion temperatures as well as the size of the emitting region during shock flash.

In order to elucidate capsule dynamics and shock timing, we will present some results below from one-dimensional (1D) simulations. Since it is not possible to accurately render pore geometry in 1D, these simulations employ a uniform homogeneous fill similar to that used in some simulations reported in Refs. 52 and 86. The 1D fill assumes a uniform mixture of gas and foam material throughout the capsule interior with densities and atomic compositions set to equal their bulk values. Opacity and equation of state data for the mixture are derived using the TOPS code.⁹⁷ The validity of this method for predicting bulk dynamics (such as shock speeds) in heterogeneous media in conditions similar to our experiment was demonstrated in Ref. 86. The same method is used to set the material parameters for the foam matrix in the 3D simulation. Indeed, since the foam matrix is a foam itself with nanometer-scale pores,⁵⁷ the fill gas occupies these pores. We have used 2D simulations to confirm that these pores will collapse due to the radiative precursor from the strong shock that traverses the foam. We calculate the volume of the micropores from the bulk density of the foam and the direct measurement of the macropores from 3D X-ray tomography.

Three experiments were fielded on the National Ignition Facility relevant to our two 3D simulations. Each experiment was fielded at 150 K in an indirect-drive configuration, in which lasers illuminate the inside of a cylindrical Au hohlraum, producing a uniform X-ray bath to drive the capsule implosion. The capsules, diagramed in Fig. 1, consisted of 190 μm thick 1800 μm outer diameter hemispherical shells that were glued together surrounding machined 40 mg $/$cm³ CD foams.^55–57 The shells were made of plastic that was doped with silicon at 1% by atom fraction in order to prevent high-energy X-ray emission from the hohlraum from preheating the foam. Shot N180729 used a 7600 Torr HT gas fill (95% hydrogen and 5% tritium by atom fraction; 1.8 ± 1 mg $/$cm³) and shots N181028 and N200305 used a 7600 Torr ArT gas fill (91% argon and 9% tritium by atom fraction; 34 ± 1 mg $/$cm³). Shot N200305 was intended as an exact repeat of N180128 in order to estimate shot-to-shot variability. The tritium content in the gas fill was set at these levels in order to ensure that down-scattered DT neutrons, produced at 14 MeV, would not prevent accurate measurement of the neutrons from the DD reaction, which have a birth energy of 2.5 MeV, due to the significantly higher cross section of the DT reaction. The 1.2 MJ, 400 TW, 8 ns laser pulse was shaped to produce two separate shocks in the ablator that would coalesce in the foam shortly before bang time (the time of peak neutron production). This pulse shape was chosen to reduce radiative emission from the shock front, which may preheat the foam, and to more efficiently compress the foam and gas, thereby increasing neutron yields obtained from compression relative to shock flash yields.⁵² 

Simulations employed detailed target metrology obtained prior to the experiment. Measured foam densities, determined gravimetrically and from transmission measurements using a narrow-band X-ray source,⁸⁷ were 37.9 ± 1.9, 35.3 ± 1.8, and 39.2 ± 2.0 mg $/$cm³ for shots N180729, N181028, and N200305, respectively. The foam deuteration was measured to be 88.2  $± .2$% for all shots. The as-shot gas fill densities were measured to be 1.86, 34.5, and 35.3 mg $/$cm³, respectively. The pore geometry, joint, and glue geometry were obtained from 3D X-ray tomography of the capsule used in shot N180729.^72,73 We show the tomographic data and simulation initial conditions in Fig. 2. We used identical pore geometry in both 3D simulations in order to isolate the impact of the gas fill in simulations.

In order to understand the progression of the experiments, it is necessary to understand the capsule dynamics and burn, which we present from 1D simulations. We show Lagrangian shock plots as well as trajectories of the ablation front and foam/shell interface and neutron production rates in Fig. 3. The drive produces two shocks that break out of the shell at $t ≈ 6$ and $t ≈ 7$ ns, respectively. These shocks coalesce inside the foam/gas fill at $t ≈ 7.5$ ns to form a stronger shock. This combined shock reflects off the origin at t = 8.2 and t = 8.3 ns for the HT and ArT gas fills, respectively (the higher density of the ArT fill slows the shock somewhat). A significant fraction of the yield from the HT-filled implosion is produced at shock flash (when the shock reflects off the origin and is at its strongest), whereas this is suppressed with the ArT fill. This suppression occurs primarily due to reduced heating from the shock since the higher Z gas has a higher molar heat capacity and density. Both implosions then achieve peak compression at t = 8.5 ns. Due to the higher density fill, the ArT-filled implosion achieves a convergence ratio (CR) of 7.5, whereas the HT-filled implosion achieves a CR of 10. As a result, the HT gas-filled implosion exhibits somewhat more compression yield. Radiative cooling from the Ar does not significantly suppress the yield since, for both gas fills, there is significantly more carbon in the burn region and this dominates radiative losses. Furthermore, in the 1D simulations, the temperatures are lower in the burn region with the ArT gas fill, so that the total radiative losses are actually reduced compared to the simulation of the implosion with the HT gas.

We compare integrated metrics from the 3D simulations to experimental data in Table I. In this table, we include yield uncertainties for the simulations that are estimated from characterization uncertainties and how these affect the total number of deuterons and tritons that are present in the experiment. Most of the simulated data are in good agreement with the experimental data. It is notable that while the burn widths are longer in simulations than in the corresponding experiments, they are still within the uncertainty of the measurement.

The most notable discrepancies are that the DD yield is 37% high for the HT simulation and the apparent DT burn-weighted ion temperature is high for the ArT simulation. The high DD yield could be caused by over-heating of deuterium ions due to the radiation-hydrodynamics approximation during shock flash and is consistent with previous results;^26,74–76 this effect is expected to be significantly reduced for the ArT gas fill due to the suppression of shock flash yield in the ArT-filled implosion.

The high apparent DT temperature for simulations with both gas fills is likely due to the absence of viscosity in the calculations, so that the magnitude of non-radial flows is overestimated in simulations. In the 3D simulations, the total kinetic energy (KE) in the foam and gas materials is 1.2 and 3.6 kJ at bang time for the HT and ArT gas fills, respectively (the higher KE for the ArT gas fill is due to the smoothing of density gradients ahead of the shock by radiation for the HT gas fill, discussed in more detail below). This corresponds to approximately 26% and 48% of the internal energy, respectively, so that the non-radial flows have a larger impact on the apparent ion temperatures for the ArT gas fill. Since viscosity scales as $≈ T 5 / 2 / Z 2 ¯ 2$⁠,¹⁰⁹ one would expect the effect to be much larger for the HT gas fill, but the dominance of carbon in the burn region again limits the deviation so that, based on simulated temperatures (which are somewhat higher in the HT burn region) and the difference in Z in fully mixed regions, the viscosity in the burn region for the HT gas-filled implosion should only be $≈ 1.8$ times the corresponding value for the ArT gas-filled implosion.

We compare simulated and experimental down-scattered ratios (DSR, the ratio of down-scattered to primary DT neutrons) in Tables IV and V for HT- and ArT-filled implosions, respectively. The simulated DSR is calculated by assuming that the DT neutrons are generated as a point source of 14 MeV neutrons at the origin. We then evaluate the elastic scattering and (n,2n) cross sections separately for the shell, the foam, and the gas based on their integrated conditions along the nTOF line of sight and sum the fraction of neutrons scattered and undergoing (n,2n) reactions with the different materials. Measured DSRs for both exhibit an increase in the line of sight that includes the glue spot. This is to be expected, since the glue density is higher than the plastic density. However, the simulations underpredict the amount by which the DSR is increased in this line of sight for both experiments. In simulations, the joint and glue are aligned exactly with the axis, whereas in the experiment, this alignment is only approximate. Therefore, the comparison is sensitive to the exact alignment of the joint in the experiment, which was not measured, and this may be the reason for the discrepancy. We note that the quality of the alignment depends on the distribution of material during the burn phase rather than its initial orientation. In simulation, the material jetted by the glue spot is spread out over a much larger line of sight than in the initial conditions. As a result, in simulation, the highest ρR is not along the line of sight that goes through the initial location of the glue spot but is slightly offset from it. We also note that our calculation methodology, which only accounts for single elastic scattering events and (n,2n) reactions, is likely unsuitable to modeling a situation where the DSR is 10%, as is the case where the discrepancy arises for the HT-filled experiment.

We note that the assumption of a 14 MeV point source does not account for the thermal broadening of the neutron spectrum. The down-scatters occur primarily due to elastic scattering and (n,2n) reactions, and the probability of these events is proportional to the ρR and the reaction cross sections. The thermal broadening of the neutron spectrum is $≈ 177 T i = 321.5$ KeV, using the hottest thermal burn-weighted T_i from simulations. The DT elastic scattering cross section varies by no more than 2% over all of the target isotopes in the problem in this energy range, and the variation of the (n,2n) cross sections is much smaller. This suggests that the error due to the approximation that the neutron source is monoenergetic is $≲ 2$%. The point source assumption is likely to be a larger source of error, though properly accounting for the source distribution would tend to smooth out the DSR vs angle.

The X-ray timing data from simulation is in generally good agreement with experimental data obtained from the SPIDER diagnostic.¹¹⁵ We compare the simulated and experimental X-ray emission vs time in Fig. 4. Since the SPIDER data are not absolutely calibrated in flux, we normalize each curve so that its maximum is 1.0. For the HT gas fill comparison in Fig. 4(a), the relative amount of X-ray emission during shock flash (⁠ $≈ 8.2$ ns) to that during compression burn (⁠ $≈ 8.5$ ns) is about one third less than that obtained in the experiment. This could result from the simulation not accurately capturing the heating of the material in the center of the implosion, which would be consistent with the results from an earlier study comparing detailed 3D simulation data to the experiment.²⁶ Additionally, the time of peak X-ray emission for the ArT simulation is earlier than either of the corresponding experiments by $120 – 180$ ps. Nevertheless, considering the measurement error (⁠ $40 – 80$ ps) and the shot to shot variability (60 ps), this is a small discrepancy and could be accounted for by the fact that this simulation did not use the as-shot geometry for either of these shots. Despite the fact that we cannot compare absolute X-ray yields, we can compare the ratio of the peak emissions from the ArT- to HT-filled experiments. This is done in Fig. 4(c). The value obtained from the two 3D simulations is in good agreement with the ratio from the N200305 ArT- and N180729 HT-filled experiments, though somewhat lower than the ratio from the N181028 ArT- and N180729 HT-filled experiments. The underestimation of emission at shock flash in simulation is also consistent with a low burn-weighted T_i in simulation. Indeed, shock flash occurs at a higher temperature, so that a higher level of shock flash yield relative to compression yield would result in a higher HT SPIDER signal and a higher burn-weighted T_i.

We compare synthetic and experimental time-resolved X-ray self-emission images from the GXD (gated X-ray detector) diagnostic¹¹⁶ in Fig. 5 for the HT-filled experiment and Fig. 6 for the ArT-filled experiment. These images are from the equatorial line of sight; the exact orientation is diagramed relative to the capsule in Figs. 1(a) and 1(b). The images also indicate the size and Legendre mode 2 shape of the 17% contour, which is the contour where emission is reduced to 17% of its peak value. The experimental images in Fig. 6 are from shot N200305 and are shifted in time by $≈ 100$ ps to account for the bang time offset shown in Table I and the total emission shown in Fig. 4. In this comparison, each image is independently normalized to its maximum value. The comparison is generally good, indicating that the simulations capture the burn region dynamics and shape. The biggest difference occurs at shock flash (8.2 ns for the HT-filled experiment and 8.3 ns for the ArT-filled experiment), where the simulations underpredict the size of the emitting region. This is qualitatively consistent with the impact of viscosity in reduced dimension simulations—viscosity is predicted to smooth out the shock and hence the temperature profile at shock flash.

The GXD comparison between the simulation and experiment in Fig. 5 is suggestive of another explanation for differences between the simulation and experiment. The reported size and shapes are very consistent from 8.3 ns through later times, yet the emission at the time of shock flash (8.2 ns) as well as the shape of the highest levels of emission from 8.2–8.3 ns are much more round in the experimental images. This could indicate that the simulations underpredict the level to which radiative preheat from the shock front homogenizes the foam and pores before shock flash. The density gradients at the interfaces of the pores are the leading cause of the loss of shock front symmetry in simulations. If the shock front were more symmetric at shock flash, this would correspond to a stronger shock, which would simultaneously improve the agreement of the shape of the experimental GXD with the experiment in Fig. 5, increase the radiative emission at shock flash, compared in Fig. 4, increase the thermal burn-weighted T_i compared in Table I, and would correspond to a decrease in seeded flows that lead to the high apparent T_i also reported in Table I. A similar effect for the ArT implosion could also simultaneously explain why the simulated GXD images show less uniformity than thee experimental images in Fig. 6 and would tend to enhance the DT $/$DD yield ratio to bring it more in line with the experiment in Table I by introducing more early preheat-driven mixing between the foam and gas.

In this section, we compare details of the 3D simulations with the HT gas fill and ArT gas fill in order to understand how differences in gas fill impacted the results of these experiments. In the experiment, the HT gas-filled implosion exhibited signatures of complete atomic mixing between the foam and gas material (inferred from the neutron yield ratio) as well as persistent temperature separation between the foam and gas (inferred from the burn-weighted ion temperatures). The ArT gas-filled implosion exhibited signatures of incomplete mixing between the foam and gas but exhibited thermal equilibration between materials.

We compare the development of the pores at the time of shock merger (t = 7.3 ns) in Fig. 7. At this time, the pores in the HT gas-filled simulation are significantly smaller than those in the ArT gas-filled simulation (recall that the initial pore distribution is the same in both simulations, specifically to improve the quality of this comparison). This arises due to foam expansion. The foam matrix is heated before shock arrival due to a radiative precursor from the shock front—the source of this heating has been verified in 1D simulations by artificially suppressing the M-band component of the FDS, which is highly shielded by 190 μm of Si-doped plastic. The HT gas is also heated by this precursor, but the relatively low pressure of the gas cannot hold back pore collapse. The ArT gas, on the other hand, achieves significantly higher pressures than the HT as it ionizes, which substantially reduces the pore collapse.

Here, $c CD$ and $c gas$ are the mass fractions of the gas fill and CD matrix materials, respectively, and $c CD , tot$ and $c gas , tot$ are the ratios of CD and gas mass, respectively, to the sum of the CD and gas masses. This metric gives θ = 0 if the materials are completely separated and θ = 1 if the materials are uniformly atomically mixed. This metric was chosen in an attempt to isolate the mixing between the gas and foam materials from mixing with ablator material. With this metric, mixedness appears to drop when the foam or gas material mixes with a third material (primarily the shell). This is observed for the HT gas-filled implosion, where the Richtmyer–Meshkov (RMI) instability^117,118 induces mixing between the foam and gas material with the shell. This is also observed during shell deceleration due to Rayleigh–Taylor mixing at the same interface. Both instabilities are suppressed substantially by the lower Atwood number at the interface in the ArT-filled implosion. It should be noted, however, that mixing with shell material, as well as jetting due to the joint feature and the fill tube, does not impact the burn directly since the burn region is spatially isolated from the regions impacted by these effects. For the HT implosion, the burn region has a radius of $71 μ$m at bang time and the inner shell radius is $≈ 80 μ$m. For the ArT implosion, the burn region has a radius of $54 μ$m at bang time and the inner shell radius is $≈ 200 μ$m.

For the HT-filled implosion, there is significant mixing seeded by preheat. Indeed, all of the mixing before the shock breakout at 5.7 ns is seeded by preheat and much of the subsequent mixing occurs ahead of the shock front. This mixing is diffusive and occurs at the interface between the gas and the foam and pores as this interface expands. In the ArT-filled implosion, the pressure gradient at this interface is reduced substantially and thus preheat-induced mixing is nearly eliminated. In both cases, the foam–gas mixing rate is accelerated as the shocks break out into the foam as well as during shock merger. The apparent decrease in mixing at the second shock breakout in the HT-filled implosion is caused by RMI at the inner interface of the ablator, which induces mixing between the ablator material, the foam, and the gas. As noted above, this RMI is much more pronounced than for the ArT-filled implosion since the ArT density is much higher, which reduces the Atwood number at the interface.

The resulting neutron production rates for the DT and DD reactions are plotted in Fig. 9(a), and the DT/DD yield ratio is plotted in Fig. 9(b), indicating the impact that the development of mixing has on the yields. Shock flash yield is observed for the HT-filled implosion around $8.1 – 8.2$ ns and is significantly suppressed for the ArT-filled implosion due to the reduced heating of the gas, which has higher density and more degrees of freedom. The high DT $/$DD neutron production ratio for the HT-filled implosion during this phase reflects both the high temperature (this reaction ratio increases sharply with temperature), the high level of mixedness that has been achieved (see Fig. 8), and the temperature separation between reactants (the DT $/$DD reactivity ratio is enhanced when the tritium is hotter than the deuterium). Only a small spike in the DT yield production rate and in the DT $/$DD neutron production ratio indicates the shock flash yield for the ArT-filled implosion at 8.24 ns. The second neutron production peak for the HT-filled implosion, and the primary peak for the ArT-filled implosion, indicates compression yield, which occurs during 8.25–8.6 ns, with peak production rates at 8.46 ns and 8.40 ns for the HT-filled and ArT-filled implosions, respectively. The DT $/$DD neutron production rate is nearly monotonically decreasing after shock flash. This behavior is dominated by the gas and foam materials coming into thermal equilibrium as clean pockets of materials mix (we show material temperature plots in Fig. 10). On the other hand, the DT $/$DD neutron production rate for the ArT-filled implosion shows a slow increase during compression and for 100 ps after bang time when it is cooling due to expansion, a signature that the mixedness is increasing in the burn region.

The mass-weighted temperatures for the gas fill and CD foam matrix are shown as a function of time in Fig. 10(a). We show the same plot restricted to clean regions of the materials in Fig. 10(b). Despite the high level of atomic mixing by t = 8.1 ns for the simulation with an HT gas fill, we see in Fig. 10(a) that the mass-weighted HT temperature is 10% higher than the mass-weighted CD temperature at t = 8.1 ns and 15% higher in the burn region (defined as $T i > 500$ eV, not shown) at t = 8.3 ns. As observed previously,^27,28 temperature separation in the HT gas-filled simulation arises during shock and compression heating of the materials due to their disparate molar specific heats $C V ≈ ( Z + 1 ) R$ (the number of atoms of H in the HT gas fill was roughly consistent with the number of atoms of Ar in the ArT gas fill), where Z is the average number of free electrons per atom and R is the ideal gas constant. However, in the present simulations, this effect competes with high levels of mixing between the HT gas and the CD foam matrix (shown in Fig. 8) that is accelerated due to preheat in the present simulations, so that the temperature separation arising from shock heating is quickly equilibrated behind the shock front and temperature separation is localized near the shock front until shock reflection. Compression heating also contributes to temperature separation from t = 7.5 to 8.4 ns, and this persists until the CD foam matrix and HT gas fill atomically mix completely at approximately t = 8.5 ns, at which point rapid local ion collision rates quickly bring the ions into thermal equilibrium. As a result, even though temperature separation is locally much higher in the HT simulation, the mass-weighted average over the entire domain shows more temperature separation between ion species for the ArT gas fill due to a significant reduction of atomic mixing, and it is only possible to observe the enhanced temperature separation throughout the simulation in the clean regions shown in Fig. 10(b).

The mass-weighted gas temperature in the HT-filled implosion is always higher than or comparable to the mass-weighted foam temperature [Fig. 10(a)], but this same behavior is not reflected in the comparison for clean regions in Fig. 10(b). The brief time period where the clean foam regions are hotter than the clean gas regions occurs as the shock is rebounding before peak compression. During this phase, RMI from the outgoing shock induces rapid mixing of the remaining clean regions of either material, and due to the location of the shock, most of these remaining regions are located in the colder regions closer to the ablator that have not yet been reshocked. However, owing to the overwhelming dominance of foam mass, which up to this point has also been expanding due to preheat, a few clean regions of foam material persist for a short time (⁠ $≲ 100$ ps) after shock flash near the center of the implosion, so that their temperature is hotter in clean regions for this period of time.

Finally, to give a better idea of the combined dynamics of the implosion, we show visualizations of the capsule implosions at comparable times in Fig. 11, where we show the locations of pores, the shock front, and the burn region. At the earliest time, 100 ps before shock flash, the unshocked pores are much smaller in the HT simulation and many of the shocked pores have been enveloped by the surrounding foam matrix through mixing (here, pores are only visible when the gas is the dominant material in at least one cell). At the same time, the shock interaction has caused substantial evolution of the pores in the ArT simulation, though many clean pockets of gas remain. This behavior continues to evolve in a similar fashion through shock flash, where we note that the interactions with the pores have highly distorted the shock front, creating the asymmetry observed in the simulated GXD image during 8.0–8.3 ns in Fig. 5. By bang time (peak compression), the pores are nearly entirely mixed with the foam matrix in the HT-filled simulation, whereas significant materials of clean ArT persist in the ArT-filled simulation. This is consistent with the mixing behavior observed quantitatively in Fig. 8.

We have performed detailed ultra high-resolution simulations of MARBLE separated reactant experiments on the National Ignition Facility with varied gas fills. The agreement between the simulations and experiments is generally better than experimental reproducibility. The biggest discrepancies between the simulations and experiments include the DD yield for the HT gas-filled experiment, which is 37% high in simulation, the apparent DT burn-weighted ion temperature (including broadening due to fluid motion) that is too high for the ArT-filled experiments, and the X-ray self-emission size during shock flash, which is consistently small in simulations. These results suggest that viscosity could be the most important unmodeled physics in the simulations. Another potential explanation for the differences between the simulations and experiments is that the simulations may underpredict the amount to which the fill is homogenized due to radiative preheat from the shock front in the HT gas-filled implosion. This would tend to increase the symmetry of the shock at shock flash, thereby increasing the level of emission and its symmetry at shock flash, and reduce the magnitude of flows that develop in simulation and hence the level to which they enhance the simulated apparent T_i.

We used our simulation results to understand the results of HT and ArT gas-filled experiments. In the experiment, the HT gas-filled implosion exhibited signatures of complete mixing between the foam and gas material (inferred from the neutron yield ratio) as well as persistent temperature separation between the foam and gas (inferred from the burn-weighted ion temperatures). The ArT gas-filled implosion exhibited signatures of incomplete mixing between the foam and gas but exhibited thermal equilibration between materials. These results are somewhat paradoxical since mixing tends to rapidly bring materials into thermal equilibrium. Our simulations elucidate a delicate competition between four factors that are able to explain these different behaviors. First, the use of Ar inhibits shock flash yield in the ArT-filled implosion, primarily due to the higher molar heat capacity and the density of the ArT fill relative to the HT fill. Second, radiation from the shock front is preferentially absorbed by the foam in the HT-filled experiments, causing the macropores to collapse before the burn phase. This foam expansion is prevented by a combination of the higher opacity of the gas, so that the radiation energy absorbed into the foam and gas is much closer, and higher pressure in the gas in the ArT-filled implosion, due to its larger density and additional degrees of freedom as it ionizes. Third, the HT gas-filled implosion develops temperature separation between materials due to their disparate molar specific heats $∝ ( Z + 1 ) R$⁠. The gas and foam specific heats are much closer in the ArT-filled implosion, which prevents the development of temperature separation. Finally, the ArT gas-filled implosions exhibit a lower density gradient at foam/gas interfaces, which reduces the distortion of the shock by interaction with pores. This reduces both the density gradient and the misalignment between the pressure and density gradients that drive the Richtmyer–Meshkov instability.

The authors would like to thank L. Kuettner and R. B. Randolph for their contributions to target metrology and fabrication, Lawrence Livermore National Laboratory's computing support for supercomputer operations and support, and LANL's Eulerian Applications Project for xRAGE support. The MARBLE campaign was supported by the National Nuclear Security Administration (NNSA) Office of Experimental Sciences. Computing time on LLNL's Sierra supercomputer was awarded through the NNSA Advanced Simulation and Computing Program's Advanced Technology Computing Campaign. Los Alamos National Laboratory is managed by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218CNA000001.

The authors have no conflicts to disclose.

Brian Michael Haines: conceptualization (equal); data curation (equal); formal analysis (lead); methodology (lead); software (lead); validation (lead); visualization (lead); and writing—original draft (lead). Tana Morrow: methodology (equal); project administration (equal); and supervision (equal). Brian Patterson: formal analysis (equal); methodology (equal); and visualization (equal). Thomas J. Murphy: conceptualization (lead); data curation (equal); formal analysis (lead); investigation (equal); and methodology (equal). Richard E. Olson: conceptualization (equal); formal analysis (equal); investigation (equal); methodology (equal); and writing—review and editing (equal). Yongho Kim: conceptualization (equal); data curation (equal); formal analysis (equal); investigation (equal); methodology (equal); and writing—review and editing (equal). Brian J. Albright: conceptualization (equal); formal analysis (equal); funding acquisition (equal); investigation (equal); project administration (lead); supervision (lead); and writing—review & editing (equal). Brian D. Appelbe: data curation (equal); formal analysis (equal); methodology (equal); and software (equal). Thomas H. Day: methodology (equal); project administration (equal); and supervision (equal). Mark Gunderson: data curation (equal); formal analysis (equal); investigation (equal); methodology (equal); validation (equal); and writing—review and editing (equal). Chris Hamilton: methodology (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.