The effects of convergence ratio on the implosion behavior of DT layered inertial confinement fusion capsules Free

Achieving ignition via inertial confinement fusion (ICF) is a major goal of the National Ignition Facility (NIF).¹ The standard ignition capsule design¹ is a spherical shell that contains a layer of solid cryogenic deuterium-tritium (DT) fuel and a central volume filled with low-density DT vapor. For a DT ice layer, the initial temperature of the capsule must remain below the DT triple point temperature of 19 °K. This limits the density of the DT vapor to <0.4 mg/cm³ so that the hot-spot convergence ratio (CR), defined as the ratio of the initial radius of the DT layer to the radius of the hot-spot, is typically >30. Although high CR is desirable in an idealized one-dimensional (1D) sense, capsule implosions are inherently hydrodynamically unstable, and they become more unstable as the convergence ratio is increased. Indeed, this has contributed to the current inability to achieve ignition on the NIF.^2,3

The wetted foam Inertial Confinement Fusion (ICF) capsule design is an alternative approach to achieving a high yield on the National Ignition Facility (NIF). This design includes a foam layer wetted with DT liquid in lieu of an ice layer⁴ and has recently been successfully fielded in several subscale experiments on the NIF.⁵ These capsules have an outer diameter of 1.81 mm with a $50 μ$m wetted foam layer and use a laser pulse that employs 869 kJ of laser energy. The use of a liquid DT layer enables the capsules to be fielded at temperatures higher than 19 °K, which in turn enables higher DT vapor densities in the central volume of the capsule. It is therefore possible to achieve a wide range of convergence ratios (12 <CR <25 for the subscale variant) using the wetted foam design. In 1D simulations, it is predicted that the capsule yield will increase as the convergence ratio is increased due to higher hot-spot temperatures and increased fuel areal density. However, the implosions also become increasingly hydrodynamically unstable as the convergence ratio is increased. Therefore, it is plausible that there should be an ideal convergence ratio for the capsule implosion given fixed asymmetry seeds that balances the benefits of increased convergence ratio with the deleterious effects of hydrodynamic instabilities.

In this paper, we present a full-scale wetted foam capsule design for the NIF using a high density carbon (HDC) ablator. These capsules have an outer diameter of 2.22 mm with a 55 μm wetted foam layer and use a laser pulse that employs 1.7 MJ of laser energy. By varying the initial DT vapor density from 4.0 mg/cm³ to 0.3 mg/cm³, the convergence ratio varies from 14 to 31. In idealized 1D simulations, the yield of these capsules ranges from 115 kJ (CR $≈$ 14) to 14.3 MJ (CR $≈$ 31). We also present high-resolution two-dimensional (2D) simulations of these capsules using detailed and adequately resolved models for the capsule support tent, fill tube, drive asymmetry, and surface roughness. 2D simulations predict that the deleterious impacts of all asymmetry seeds increase with the convergence ratio. However, at the highest convergence ratios, robust alpha heating adequately compensates for the impacts of the fill tube, drive asymmetry, and surface roughness in 2D (though the impact of some or all of these asymmetries would likely be enhanced in 3D simulations at high convergence ratios). Nevertheless, when simulations also include the capsule support tent, the yield is nearly constant for CR < 20 and decreases as the convergence ratio is increased for CR > 20.

At the lowest convergence ratio (CR $≈$ 14), simulations including all asymmetries predict that the full scale wetted foam design will produce 47 kJ of yield. At this CR, similar simulations of the fielded sub-scale design produce a yield within 10% of that observed in experiment.⁵ This suggests that 3D flows do not significantly impact performance at this CR given expected asymmetries, and so it may be feasible to attain this yield in experiment. However, this relies on the ability to achieve a sufficiently symmetric radiation drive to achieve a round implosion, which is not ensured due to the lack of predictability in hohlraum calculations.⁶ As the level of capsule and drive asymmetries is reduced, the wetted foam platform will enable layered implosions to be fielded at convergence ratios that optimize the trade-off between enhanced 1D performance and increased implosion instability as the convergence ratio is increased. Furthermore, enhanced stability at a low convergence ratio will enable layered capsule implosions that are more robust to asymmetries than ice layer designs.

In the simulations of the full scale design presented in this paper, the only parameters that are varied are the initial DT vapor density and initial temperature (the temperature is adjusted in experiment to achieve the desired DT vapor density). The drive, all other capsule parameters, and all asymmetry seeds are the same for all simulations. This reflects an idealized experimental configuration that is not currently realizable due to shot-to-shot variations in capsule manufacturing and laser performance.

By only varying the initial temperature and the initial DT vapor density, it is possible to isolate the effects of the convergence ratio on the present design. With all other parameters fixed, four factors contribute to the increased impact of asymmetries with the increased convergence ratio. First, for a fixed implosion time (set by the density of the gas fill of the hohlraum, which determines the amount of time before blow-off from the hohlraum interaction with the capsule implosion), a higher convergence ratio requires a higher peak shell velocity. As a consequence, the magnitudes of shell acceleration and deceleration are both increased, which enhances both ablative and stagnation Rayleigh-Taylor (RT) instabilities. Second, the higher convergence ratio enhances Bell-Plesset effects, which modify the growth rate of perturbations subject to hydrodynamic instabilities.⁷ Third, the higher density shell at peak compression results in a higher Atwood number at unstable interfaces. Finally, with a fixed drive, the higher convergence ratio implosion has a lower fuel adiabat, which, combined with the higher shell density, results in a decrease in the density gradient scale length of the ablator.

It is important to note that the choice of altering the DT vapor density and fixing all other parameters is not unique, and this choice affects the mechanisms that make the implosion more unstable at high CR. For example, the peak implosion velocity could be fixed instead of the implosion length, in which case ablative instabilities would have more time to grow, which would produce larger seeds for late-time instabilities. From a practical standpoint, this would also risk allowing the hohlraum blow-off to interact with the imploding capsule. The adiabat is also known to have an impact on capsule stability,⁸ so the drive could be varied with CR in order to fix the adiabat in an attempt to further isolate the impacts of CR. However, this is typically done by altering the strength of the first shock, which would result in different amounts of ablative stabilization⁹ and different magnitudes of the Richtmyer-Meshkov (RM) instability at different CR, making it difficult to disentangle these effects from the impacts of CR. In general, since the adiabat determines the compressibility of the fuel, it is not possible to adjust the CR and the adiabat independently.

Beyond decreasing the CR, several techniques have been demonstrated to stabilize implosions. For example, shell thickness and adiabat both play an important role in determining capsule stability. For a fixed capsule design, the adiabat can be adjusted by changing the strength of the first shock. Through a higher initial radiation temperature (which increases ablative stabilization of the Rayleigh-Taylor (RT) instability⁹) and increasing the density gradient scale length, the stability of the capsule implosion is increased.⁸ This enabled the higher performance of the “high foot” campaign on NIF.¹⁰ It should be noted, however, that the adiabat cannot be adjusted independently of the convergence ratio, since increasing the fuel adiabat makes the fuel less compressible. Indeed, some of the stability benefits of the “high-foot” design derive from a modest reduction in the CR compared to the previous “low foot” design.¹¹ Increasing the shell thickness will also stabilize capsules by decreasing the in-flight aspect ratio (IFAR) of the capsule,¹² by decreasing the peak implosion velocity and hence the magnitude of deceleration seeding stagnation RT, due to added shell mass, and by increasing the amount that perturbations must grow in order to seed late-time instabilities. As with the convergence ratio, there is a trade-off for each of these adjustments between increased stability and increased performance in one-dimensional simulations.

It should be noted that many of the stabilization strategies have a different impact on different wavelength ranges of the asymmetries. For example, in simulations, it is observed that the effects of surface roughness and the tent, which are effectively short wavelength perturbations, are stabilized by increasing the adiabat of the implosion and increasing the shell thickness. These, however, have a minimal impact in simulations on the growth of long-wavelength perturbations, such as x-ray flux asymmetries and the fill tube, whereas decreasing the convergence ratio stabilizes all asymmetries. Therefore, adjusting the convergence ratio can be expected to have the largest impact on implosion stability.

The physics of the impact of the fill tube is different from that of other asymmetries, as pointed out previously in Refs. 13 and 14. Indeed, in addition to shadowing the drive,¹⁵ the fill tube collapses early in the implosion and allows shocks to propagate through the low-density material at a higher speed than through the ablator material. Shear due to an ensuing jet traveling through the low density material can entrain fill tube, glue, and ablator material, and the jet can then bring this material into the center of the capsule. In the present simulations, the effect of the fill tube is nearly eliminated at all convergence ratios by a closure of the fill tube hole ahead of the jet. However, HDC experiments performed on NIF typically exhibit an x-ray emission feature that is believed to be associated with the fill tube jet, and this feature is especially prominent in experiments where the HDC ablator is doped with tungsten,^16,17 as in the full-scale wetted foam design. Nevertheless, several differences between the full-scale wetted foam design and the fielded tungsten-doped HDC capsules could account for this. For example, several fielded HDC capsules use a depleted uranium hohlraum, which decreases the amount of preheat in the tungsten doped layer. Additionally, the thinner ablator used in the fielded HDC capsules likely results in a faster interaction between the fill tube jet and the tungsten-doped layer.

The present full-scale design is particularly sensitive to the impact of the capsule support tent. As the tent ablates, an impulse at the tent lift-off point, which circles the capsule at two locations, imprints a distortion on the shock shape. The magnitude of this imprint is enhanced by ablative RT and can be limited by ablative stabilization. However, due to the small magnitude of the first shock, ablative stabilization, which increases with ablation velocity,⁹ is minimal for this capsule as the first shock is formed, making it particularly susceptible to the tent perturbation compared to other capsule designs with HDC ablators (e.g., Refs. 18 and 19). It may be possible to mitigate this sensitivity by altering the pulse shape⁸ or using an alternative capsule support.²⁰ 

It should be noted that the x-ray flux asymmetries used in these simulations were obtained from hohlraum simulations. Typically, simulations must be tuned in order to match the in-flight shape of capsules.²¹ However, this procedure could not be done for the present simulations since no relevant experimental data are available. Instead, tuning was performed to the most similar capsule implosions that have been performed to date. Nevertheless, since the design is for near-vacuum hohlraums, it is expected that the effects of unmodeled laser-plasma interactions that are believed to contribute to discrepancies in the prediction of these asymmetries are negligible.⁶ 

The pulse shape (the laser power as a function of time) was designed to produce a round implosion in simulations. It is expected that even if the experiment results in an implosion with an asymmetric shape, the drive could be adjusted to correct this. This does not eliminate the detrimental effects of drive asymmetry, however, since it is known that symmetry swings can degrade yield even when the resulting implosion is round.²² The effects of these symmetry swings increase with the convergence ratio. However, at the highest convergence ratio, these effects on their own are completely offset by alpha heating in 2D simulations. It is possible that symmetry swings in real experiments may exceed those predicted in simulations. It is also plausible that 3D effects may enhance the impacts of these symmetry swings. For example, in 3D simulations of direct-drive capsule implosions, 3D shock interactions enhance asymmetries in the shock shape²³ and 3D flows increase the separation between fuel mass and internal energy during the burn phase of the implosion.²⁴ 

Simulations are performed using the xRAGE radiation-hydrodynamics code developed at Los Alamos National Laboratory (LANL).²⁵ Significant improvements have recently been made to xRAGE to enable improved modeling of indirect-drive ICF experiments, and many of these improvements are outlined in Ref.14. xRAGE performs all necessary physics in an Eulerian reference frame and employs adaptive mesh refinement (AMR) using square cells. In the present simulations, we use xRAGE's multigroup radiation diffusion package, three-temperature (3T) plasma model, electron and ion thermal heat conduction models, and thermonuclear burn model. The xRAGE code has been used to perform detailed high-resolution simulations of sub-scale HDC wetted foam capsule implosions and demonstrated good comparison to experiment.⁵ 

In Fig. 1(a), we show a diagram of the full scale (1.7 MJ laser energy) wetted foam capsule design, which is based on an ice layer design presented in Ref. 26 and is similar to HDC capsules with ice layers fielded in experiments in Refs. 18 and 19. The shell is made of high-density carbon (HDC) with a layer that has a tungsten dopant. The use of HDC enables shorter laser pulses compared to more common CH ablators due to its higher density (3.5 g/cm³ for HDC versus 1 g/cm³ for CH), which allows the use of thinner shells. The tungsten dopant reduces fuel preheat by absorbing M-band emission from the gold wall. Excessive preheat would cause an increase in the adiabat of the implosion and hence reduce fuel compression during the implosion.

The design employs a 10 ns three-shock pulse shape, shown in Fig. 1(b). Longer laser pulses often require a gas fill in the hohlraum to prevent the interaction of the capsule blow-off with the hohlraum wall, whereas the 10 ns pulse allows for the use of near vacuum hohlraums, which are less susceptible to laser-plasma interactions that are not modeled in radiation-hydrodynamics codes, such as backscatter and hot electron generation. This enables more predictable implosions than gas-filled hohlraums. The pulse is designed so that the three shocks coalesce in the DT vapor, allowing the fuel to remain on a low adiabat. The peak laser power is 420 TW, and the peak hohlraum radiation temperature is 300 eV.

The capsule is designed to be filled with DT vapor with a density ranging from 0.3 to 4.0 mg/cm³. By varying the cryogenic fielding temperature (and, hence, DT vapor density), from approximately $19° to 26°$ K, and leaving all other aspects of the design fixed, the CR of the implosion can be varied from 14 to 31. The peak implosion velocity ranges from 363 μm/ns to 395 μm/ns. For each vapor density, the density of the DT liquid is the saturated vapor density, which, along with the pressure, varies with temperature. These variations impact the interactions with the three shocks that pass through the fuel layer, causing the fuel adiabat to vary from $α=3.3$ to $α=2.1$ for the lowest and highest CR implosions, respectively. Here, α is defined as the ratio of the fuel pressure to the minimum DT pressure at a density of ρ  =  1000 g/cm³. Note that this definition is used due to the fact that the degenerate pressure is not readily available from the tabular equations of state (EOS) used in simulation, and for DT the cold pressure at this density corresponds approximately to the ideal Fermi degenerate pressure.¹¹ To put these adiabats in perspective, the original NIF point design had an adiabat of $α≈1.45$ and the newer “high-foot” design has an adiabat of $α≈2.5$⁠.¹⁰ 

It should be noted that, unlike changes in adiabat due to changes in the pulse shape, the difference in adiabat between the lowest and highest CR implosions here does not result in a change in ablative stabilization, since this difference is due to differences in the fielding temperature required to achieve the different vapor densities. In particular, when increasing the adiabat by increasing the radiation temperature of the “foot” of the laser pulse (the part of the pulse used to generate the first shock), the amount of ablative stabilization is increased. However, in the present design, the pulse shape is the same for all convergence ratios, so the impact of the different adiabats on stability in these capsules is limited to the impact of an increase in the density gradient scale length. One might attempt to isolate the impact of CR further by adjusting the drive so that the implosions have the same adiabat at each CR. However, this would impact the amount of ablative stabilization and RM instability, making it difficult to disentangle these effects from the impacts of CR.

The present simulations employ the xRAGE radiation-hydrodynamics code.^14,25 The standardized post-shot capsule simulation model outlined in Ref. 27 was followed as closely as possible for implosion simulations in xRAGE. The simulations are driven by an x-ray flux source derived from hohlraum calculations using Lawrence Livermore National Laboratory's HYDRA code.²⁸ Typically, hohlraum calculations must be tuned to experimental data in order to match implosion characteristics, including shock-timing,²¹ shell trajectory,²⁹ and in-flight shape.³⁰ However, since no full-scale wetted foam experiments have been performed, hohlraum calculations were tuned based on simulations of the most similar ice-layer experiments using HDC ablators. Radiation transport is performed using 60 group radiation diffusion with LANL OPLIB opacities.³¹ A transverse flux limiter is applied between the ablation front and the simulation boundary in order to avoid artificial symmetrization of the radiation field. This is necessary because the diffusion approximation is not valid in this region. A diffusion approximation is also used for electronic and ionic thermal conduction, with coefficients determined by the analytic formulae of Lee and More.³² LANL SESAME tabular equations of state (EOS)³³ are used for all materials. For the tungsten-doped HDC layer and wetted foam material, mixed material EOS and opacities are generated using the TOPS code.³⁴ 1D xRAGE simulations using Lawrence Livermore National Laboratory's LEOS tables^35,36 for DT and HDC show very little difference with 1D xRAGE simulations using SESAME EOS tables for these materials. It is not currently possible to directly perform this comparison for mixed material EOS. However, experiments have recently been performed to evaluate the mixed EOS used for the wetted foam material, though results are not yet available.

Simulations are performed using a maximum resolution of $0.25μ$m in all materials except the tent material. For the tent material, a maximum resolution of $0.0625μ$m is used until 400 ps into the simulation, at which point these cells are dezoned to $0.25μ$m. Higher resolution simulations were performed for selected variants and did not exhibit appreciable differences in any of the presented results.

The foams used in fielded capsules^37,38 are made of CH and have pore sizes that are approximately 1 μm in diameter. The foams can be manufactured with no apparent defects or other features that can impact capsule performance. Fielded capsules have had foam layer thickness variations as large as $1.4 μ$m, but simulations suggest that even the largest of these nonuniformities have a smaller impact on performance than engineering features. Therefore, in the present simulations of the full-scale design, we assume that the foams have uniform thickness.

The model for the fill tube, shown in Fig. 2(a), was based on data from Ref. 39. This consists of a 10 μm diameter glass (SiO₂) tube with an inner diameter of 5 μm inserted into a 12 μm diameter hole drilled 40 μm into the capsule (note that the figure shows the simulated configuration for the larger 30 μm diameter fill tube used in fielded sub-scale wetted foam experiments⁵). A fillet of glue 10 μm wide and $6 μ$m high surrounds the fill tube at the entrance to the capsule and fills the extra space in the drilled hole. Since the present simulations are performed in 2D, they cannot account for the azimuthal shadowing pattern observed in recent experiments producing face-on radiography of the growth of the fill tube perturbation due to the distribution of laser spots on the hohlraum wall.¹⁵ 

To evaluate the accuracy of our fill tube modeling, we compare simulated gated x-ray images to those obtained using the NIF's HGXD diagnostic⁴⁰ for the sub-scale wetted foam shot N160421 in Fig. 3. In the experimental images shown here, the resolution is approximately $10 μ$m and the images are convolved with a Gaussian with a width of 8–12 μm in order to smooth out noise due to photon statistics. When all asymmetries are accurately modeled, these simulations compare favorably with the experimental measurements of neutron yield, nuclear bang time, nuclear burn width, DT burn-weighted ion temperature, DD burn-weighted ion temperature, and hot spot radius from both X-ray and neutron imaging systems. These comparisons were presented in Ref. 5 and are repeated in Table I. For the sub-scale capsules, a 30 μm fill tube is used, whereas for the full-scale capsules, a 10 μm fill tube is assumed. Simulated HGXD images of N160421 show good agreement with experiment (the calculated quantities are within the experimental error bars) in terms of size (quantified by the radius of the 17% emission contour), shape (quantified by the second Legendre moment of the 17% emission contour), and the location and size of a high-emission spot. In simulations, this high-emission spot is caused by the jetting of the SiO₂ (fill tube) material into the hot spot. This suggests that xRAGE correctly calculates the trajectory of the jet caused by the fill tube and the resulting distribution of materials qualitatively, though a quantitative comparison would require more careful analysis. It is notable that the dominant presence of the SiO₂ material in the hot spot contrasts with the results of simulations of plastic shell capsules in xRAGE (e.g., Ref. 14) in which plastic from the outermost shell layer is the primary material transported into the hot spot due to the fill tube jet. Additional nonuniformities in the experimental HGXD images at early time are caused by noise due to the fact that the signal is very weak. Closer to bang time, other nonuniformities appear in experimental images that have similar wavelengths to features that appear in the simulated images that are seeded by the growth of surface defects. Nevertheless, these features show up as lines in the simulated images due to the axisymmetric nature of the calculations.

The shape of the Formvar (C₃₁H₅₆O₁₃) support tent, shown in Fig. 2(b), is based on the description in Ref. 41: it sits on the outside of the capsule between 45° and 135° from the pole, where it lifts off at an angle that varies from 10° to 15°. The shape of the tent between the lift off point and the location of the hohlraum wall is a segment of an ellipse. It is important to note that, in order to model both the fill tube and the tent simultaneously in 2D simulations that assume axisymmetry, the orientation of these features had to be altered in simulation since their combined geometry is not axisymmetric in experiment. In experiment, the fill tube enters the capsule at a point along its equator (with respect to the hohlraum's axis of symmetry), whereas the support tent is symmetric about the pole. Therefore, in 2D simulations, the fill tube enters the capsule at the pole and the support tent is oriented such that it lifts off the capsule towards the poles. In order to evaluate this procedure, 2D simulations including only the support tent have been performed in both configurations and show very little difference in terms of integrated quantities (<2% change in yield, ion temperature, hot spot size, integrated aerial density, etc.), suggesting that this technique does not have a significant negative impact on the results.

In order to evaluate the fidelity of our simulations including the support tent, simulations were performed of NIF shot N120321 (see, e.g., Ref. 42 for details), a low-foot plastic-shell DT layered capsule implosion, that include this feature. These are compared to experimental data from Ref. 43, in which areal density modulations are inferred from x-ray images of the implosion. As shown in simulations in Ref. 44, the induced areal density perturbation due to the tent is sensitive to the lift-off angle and tent thickness. When using a lift-off angle of 14° and the tent thickness appropriate to shot N120321, the xRAGE simulations show a favorable comparison to the experimentally observed areal density perturbations relevant to this shot. Indeed, the actual areal density perturbation of 34.1% in simulation at peak implosion velocity falls inside the $≈$ 30%–40% range inferred from radiography data relevant to this shot.⁴⁴ This corresponds to an inferred value of $≈15$% from a 20 μm experimental resolution.⁴⁴ 

The fill tube is a uniquely important feature for capsules with HDC ablators due to the observation of a high-emission feature in x-ray self-emission images of these capsule implosions that emanates from the location of the fill tube.^17,45 Indeed, this feature is observed in both experimental and simulated images of wetted foam shot N160421, shown in Fig. 3. The impact of the fill tube has been studied in the literature for plastic^14,46 and beryllium¹³ ablators. In both cases, the fill tube quickly ablates and the low density material inside provides a faster medium through which the shock can propagate. This produces a jet that shears the wall of the fill tube hole, seeding the Kelvin-Helmholtz (KH) instability and entraining glue, fill tube, and ablator material, bringing it into the fuel region and contaminating the hot spot. Due to the increased density of HDC compared to CH and Be, the density contrast between the ablator and material in the fill tube hole is larger in this study than in previous studies. This causes a larger difference in velocity between the shock propagating in the ablator and through the inside of the fill tube. In CH capsules, the dominant material that contaminates the hot spot due to the fill tube jet is CH (along with some glue material) from the outer layer of the capsule. In HDC capsules, however, the fill tube material is the main component of the fill tube jet, as shown in Figs. 4(a) and 4(b), where we show material distributions at two different times from simulations of a capsule with an un-doped HDC ablator. Indeed, the high Z of the fill tube material (SiO₂) could explain why the fill tube is more prominent in X-ray self-emission images of HDC capsule implosions than CH capsule implosions.⁴⁵ 

In Fig. 4(c), we show the impact of adding a tungsten-doped HDC layer to the capsule, as in the presently considered full-scale wetted foam design. This image visualizes the density at t = 2.0 ns, before the shock has traversed the shell and before the fill tube jet has reached the tungsten-doped ablator layer. In this simulation, preheat in the tungsten-doped layer causes it to expand and close off the fill tube hole before the jet arrives, preventing it from propagating further. Indeed, in Fig. 4(c), one can see the fill tube hole beginning to close around the location of the tungsten-doped layer. While this does not eliminate the fill tube effect entirely, it does significantly reduce the distance that the jet propagates into the fuel region.

This result seems to contradict the results of tungsten-doped layered HDC capsule implosion experiments in Ref. 17, which exhibit a high-emission feature consistent with that caused by the fill tube in Fig. 3. However, it is likely that doping the HDC with tungsten alone is not sufficient to suppress the fill tube jet; the speed at which the doped layer expands due to preheat and the time at which the shock arrives at the doped layer, which are sensitive to many design parameters, are likely to determine whether the expansion of the doped layer is sufficient to inhibit the propagation of the fill tube jet.

In Fig. 5, we show density profiles for simulations with surface roughness only as well as simulations with surface roughness, the support tent, and the fill tube. HDC capsules are fabricated with smoother surfaces than CH, which results in implosions that exhibit significantly lower levels of short-wavelength perturbations in simulations including only surface roughness. Jetting due to isolated defects is minimal for all initial DT vapor densities. Jetting of the material seeded by the support tent brings an increasing amount of the high-density material into the center of the implosion as the initial DT vapor density is decreased. The support tent perturbation is the biggest source of shell distortion in these simulations and, as noted above, the shell distortion due to the fill tube is largely mitigated due to expansion of the tungsten-doped layer due to preheat. At low convergence ratios, the tent causes a distortion in the foam layer and a small jet into the gas. Unlike CH capsules studied in Ref. 14, the jet due to the tent contains the HDC material. As the convergence ratio is increased, the distance the jet travels in the gas and the amount of the entrained material are increased.

In Fig. 6, we show the hot spot asymmetry, quantified by a ratio of Legendre moments (⁠$P2/P0$⁠), during the burn phase for the simulations including surface roughness and drive asymmetry. The drive asymmetry imposed in these simulations leads to a fuel cavity shape that is slightly prolate, and the magnitude of this asymmetry is approximately 60% larger for the highest CR implosion compared to the lowest CR implosion as the hot spot forms. For all but the lowest CR implosion, the hot spot shape swings sharply oblate as the capsule disassembles, and this procedure is accelerated as the CR increases.

In Fig. 7(a), we show the mass of different materials in the hot spot at bang time. In these simulations, the HDC material is the dominant contaminant of the hot spot, and the amount is large due to the low adiabat of the implosion. Nearly all of the HDC in the hot spot was entrained by the jetting due to the tent. Simulations indicate that the magnitude of the perturbation resulting from the tent is highly sensitive to the adiabat of the implosion. Therefore, it is likely possible to mitigate some of the negative impacts of the tent by increasing the adiabat of the implosion—a similar approach has succeeded for CH ablators.¹⁰ Simulations of sub-scale high-adiabat wetted foam implosions including the support tent exhibit significantly smaller areal density modulations at peak compression for the same convergence ratio.⁵ The relatively low power of the foot (the part of the pulse driving the first shock) in this full scale design reduces the amount of ablative stabilization compared to HDC capsules imploded previously in experiment, making this design particularly unstable to instabilities seeded by the tent.

The ratio of hot spot mass as calculated by 2D simulations to hot spot mass as calculated by 1D simulations is also shown in Fig. 7(a) (here, the hot spot is defined as the region inside the isocontour at 17% of peak emission in simulated x-ray images). This indicates that the reduced initial fuel mass is not sufficient to explain the reduction of fuel mass in the hot spot with CR. Good agreement between 1D and 2D simulated hot spot sizes suggests that the reduced fuel mass in the hot spot is also not explained by a reduction in the size of the hot spot due to emission by the material mixed into the hot spot. In the present simulations, fuel mass escapes primarily through the hole in the fuel caused by the support tent. Drive asymmetry also contributes to the movement of the fuel material into colder regions of the implosion (this mechanism is discussed in detail in Ref. 47).

In Fig. 7(b), we show DT neutron yields for all simulations. More detailed burn quantities are shown in Table II. 1D simulations show a trend of increased yield with convergence ratio due to higher hot-spot temperatures and increased fuel areal densities, and this trend is maintained for simulations including surface roughness as well as the fill tube and drive asymmetry. Note that in these simulations, as discussed above, the effect of the fill tube is largely mitigated by expansion of the tungsten-doped shell layer due to preheat. The impact of the fill tube on the yield would therefore be much larger if the tungsten dopant were removed or if the amount of preheat were not sufficient to close the fill tube hole before the shock arrives. Additionally, the amount of drive asymmetry is uncertain, and the simulations may be underestimating its impact as a result. The increase in yield with CR is lost in simulations including the support tent, which exhibit nearly constant yield for CR < 20 and decreasing yield with convergence ratio for CR > 20. This is a result of jetting of the HDC material into the hot spot as well as the formation of a hole in the shell due to the support tent. The impact of each of these effects grows larger as the convergence ratio increases. Indeed, in Fig. 7(a), we see that the amount of the non-fuel material in the hot spot doubles as the CR varies from 14 to 31 and the ratio of fuel mass in the hot spot to the amount in 1D simulations decreases by a factor of 3 over the same range. Based on the model outlined in Ref. 14, increased losses due to conduction and radiation account for only 25% of the decrease in the yield between 2D simulations of the lowest and highest CR capsules, whereas decreased hot spot mass accounts for 30% and accelerated disassembly accounts for 40%.

In Fig. 8(a), we show yield rates for 2D simulations including surface roughness, fill tube, and support tent with different initial DT vapor pressures. For the high fill pressures, corresponding to initial vapor densities of 2.0 mg/cm³ to 4.0 mg/cm³, these show a competition between increased reactivity and accelerated disassembly that results in a nearly constant total yield throughout this range of initial vapor densities. For lower fill pressures, the capsule disassembles faster and, at the lowest fill pressure, the reactivity is decreased significantly.

In Fig. 8(b), we show yield rates for select 2D simulations with drive asymmetry compared to 1D. We show 1D and 2D results for the initial DT vapor density of 1.0 mg/cm³, since this case results in the largest impact on yield. Compared to the 1D simulation, the 2D simulation exhibits reduced reactivity. In this case, 80% of the yield reduction in 2D compared to 1D is a result of a reduced conversion of shell kinetic energy to fuel internal energy. The remaining 20% of the yield reduction is due to the displacement of fuel mass from the hot spot.

The wetted foam capsule design for implosions on the National Ignition Facility (NIF), which includes a foam layer wetted with DT liquid, enables layered capsule implosions with a wide range of hot-spot convergence ratios. We presented a full-scale wetted foam capsule design that demonstrates high gain in one-dimensional simulations. In these one-dimensional simulations, increasing the convergence ratio leads to an improved capsule yield due to higher hot-spot temperatures and increased fuel areal density. However, in detailed high-resolution two-dimensional simulations, despite exhibiting robustness to the capsule fill tube, surface roughness, and asymmetries in the x-ray drive, the effects of the support tent negate all of the benefits of increasing the convergence ratio. Indeed, when the support tent is included in simulations, the yield is nearly constant for CR < 20 and decreases as the convergence ratio is increased for CR > 20.

These results suggest that, given the currently achievable level of asymmetry seeds, a full-scale DT wetted foam capsule at a low convergence ratio has the potential to outperform current high convergence ratio DT ice layer designs on the NIF. As the level of these asymmetries is reduced, the wetted foam platform will enable layered implosions to be fielded at convergence ratios that optimize the trade-off between enhanced 1D performance and increased implosion instability as the convergence ratio is increased.

The authors would like to thank S. Batha for advice and encouragement and B. Hammel for useful discussions. The authors would also like to thank the NIF shape team for providing diagnostic support on NIF shot N160421. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC for the U. S. Department of Energy NNSA under Contract No. DE-AC52-06NA25396.