Improving ICF implosion performance with alternative capsule supports

To obtain the conditions necessary for high fusion yield using inertial confinement on the National Ignition Facility (NIF),¹ the capsule needs to be compressed by factors of 30–45×. This process strongly amplifies any perturbation present on the capsule through the ablative Richtmyer-Meshkov and Rayleigh-Taylor instabilities.^2,3 Past inertial confinement fusion (ICF) experiments suffered from excess instability growth that contaminated the hot-spot with high-Z ablator material and reduced the implosion performance.^4,5 More recent experiments demonstrated that reduced instability growth with the same compression can improve implosion performance.^6,7

The largest perturbation on the capsule is expected to be that left by the thin plastic membrane that holds the capsule inside the hohlraum, shown in Fig. 1. This feature, known as the “tent,” leaves a large perturbation at the location where the tent departs from the capsule. The inset of Fig. 1 shows a back-lit radiograph of the shell using the 2DconA platform,⁸ which first identified the tent to be a significant problem. To confirm that these features were caused by the tent, an experiment was performed where the capsule was held by a 30 μm diameter stalk instead of the tent.⁹ Radiographs of this experiment showed that the scar was removed, but the large stalk degraded the fusion to yield a similar amount as the tent.

Fig. 2 shows a simulation of the tent and the perturbation that arises.¹⁰ This simulation used the code HYDRA¹¹ and modeled a low-foot¹² experiment that used a 100 nm thick tent, but experiments on NIF have used a thickness of 15–300 nm. The resolution required to resolve this feature is demonstrated by the upper-left image, which is zoomed into the central image by 100×, enabling the tent to be seen. As the tent absorbs X-rays, it expands outwards and interacts with the capsule. This is shown in the pressure field in the right two images. This interaction generates vorticity at the capsule surface through baroclinic vorticity deposition. This occurs as $d ω / d t = ( ∇ ρ × ∇ p ) / ρ 2$ or $d ω / d t = | ∇ ρ | | ∇ p | sin θ / ρ 2$⁠, where θ is the local angle between the tent and the capsule. Here the density gradient, $∇ ρ$⁠, at the ablation front and the pressure gradient, $∇ p$⁠, from the tent explosion leads to the generation of vorticity which will drive the growth of a perturbation. Originally it was assumed that the tent departed the capsule tangentially, which would make θ zero (but larger further away). Measurements of this angle, however, have shown it to be 10°–20° for CH (and maybe larger for other capsules). This change, included in this calculation (14° used here), increases the simulated perturbation size.¹⁰ The perturbation that develops on the ablation front is amplified by the ablative Rayleigh Taylor instability and becomes significant late in time. The perturbation near peak velocity (22 ns in Fig. 2) has been measured with radiography and shows good agreement with simulations.^10,13,14 At bang-time, the latest time shown in Fig. 2, two damaging effects are occurring. The thin spot in the shell allows the hot spot to vent and release some of its pressure. Also, a finger of cold dense material is poking into the hot spot. This finger gives the hot spot significantly more surface area, allowing it to conductively cool. This effect is observed in experiments as a mode 4 shape on the X-ray self-emission contour and is removed in the experiment mounted by a large stalk.⁹ In the image shown here, this finger consists of DT fuel, but some simulations show significant amounts of CH ablator coming with it, which could further cool the hot spot through radiative losses.¹⁵ This is believed to be the source of the large amounts of mix observed in low-foot experiments.⁵ 

To replace the tent as a capsule support, several alternatives are being considered and are shown in Fig. 3. On the left of Fig. 3 shows the standard setup, with a capsule inside the hohlraum, filled by a fill-tube, and held by the tent. One would hope to eliminate the tent entirely and support the capsule using only the fill-tube. The fill-tube is not rigid enough to do this alone, so it either needs to be thicker or given its own support. This second option, termed the supported fill-tube, uses a rod running perpendicular to the fill-tube and attached to it some distance away from the capsule. A multi-part fill-tube is also an option, which would have a standard diameter near the capsule and transition to a larger diameter further away. A concept called the tetra-cage uses four wires, two placed below and parallel to each other and two above, perpendicular to those below. To leave a minimal perturbation, the wires needs to be $≲$1 μm diameter and spider silk or carbon nano-tubes spun into yarn may accomplish this. The next option is to use a layer of low-density foam on the outside of the capsule and hold the capsule/foam target with the standard tent. If the foam is transparent to the X-ray radiation, ablation at the capsule surface will start before the tent blow-up can reach the capsule. Finally, moving the location where the tent departs the capsule closer to the poles will reduce the area over which the capsule is perturbed and reduce the lift-off angle of the tent.

Three of these concepts have been tested to observe their imprint on the capsule and will be discussed in detail below. Of these, the supported fill-tube appears to be the most promising, and detailed analysis has gone into understanding the effect of the resultant perturbation. This will be discussed in Section III. Modeling and experiments of the fill-tube and foam-shell supports will be discussed in Sections IV and V. A comparison of the performance improvement from these options and how the degradation varies with pulse-shape is provided in Section VI, and concluding remarks are given in Section VII. But first, Section II discusses the methodology that is used to evaluate a capsule support option.

A successful replacement capsule mount not only needs to impart a smaller perturbation on the capsule than the existing tent, but also needs to survive all of the engineering steps necessary for a successful experiment. The final assembly needs to be strong enough to survive handling between the fabrication stage and its final home at the center of the NIF target chamber. It also needs to behave predictably during cooling to cryogenic temperatures. Different materials contract differently during cooling, and this can lead to a misaligned capsule or stress buildup that causes the assembly to fail. Finally, the target must damp vibrations present in the system, such as those coming from the cryo pumps. Even small vibrations could inhibit imaging of the target while the DT ice layer is forming or de-center the capsule in a short time. These requirements ruled out some of the options not listed in Fig. 3.

To evaluate whether the replacement is an improvement over the tent, a combination of modeling and experimental validation is used. Ideally one would like to test the replacement and observe the predicted improvement in yield, but reproducibility issues and other degradation sources may make this as an ambiguous test. For this reason, confirming that the shell integrity is improved is an important aspect of this project.

Some of these support options have been tested using the hydro-growth radiography (HGR) platform.^14,16–20 This platform, shown in Fig. 4 uses a standard hohlraum and capsule with a gold cone cutting through a portion of the hohlraum and capsule. Two NIF quads are used to irradiate a backlighter foil, sending X-rays up the axis of the cone and providing an image of the optical depth of one side of the imploding shell. The 2D images are recorded using an array of 20 μm pinholes and a gated X-ray camera.²¹ Multiple images are combined to reduce the signal-to-noise level.²² An example target is shown in Fig. 4. This target is mounted on a cone, seen at the bottom of the image. Fixed to the top of the plastic (CH) capsule are two fill-tubes with support rods attached at two different distances from the capsule surface. As these features blow up inside the hohlraum, their impact on the capsule will be measured. The pair of fill-tubes and supports allows a direct comparison between stand-off distance of the supports. This will be discussed in Sec. III.

These HGR experiments are based on the “low-foot” laser pulse and capsule²³ used during the National Ignition Campaign (but scaled down by 0.8× to operate at reduced laser energy). This laser pulse had more instability growth than other designs,^18–20 but this additional growth is useful here in diagnosing the perturbations. We focus here on plastic ablators, given their continued use, the dataset of past experiments, and their ability to obtain high fusion performance,²⁴ but other materials, such diamond²⁵ or beryllium,²⁶ may have better stability properties and be more resilient to the tent.

An integrated implosion that includes a DT ice layer and attempts to achieve high fusion performance is the ultimate test of the alternative support. This is complicated by the presence of other degradation sources, such as low-mode radiation flux asymmetries in the hohlraum,²⁷ oxygen non-uniformities in the ablator,^28,29 or hot-electron pre-heat.^7,30 While the tent may be a 15× performance degradation to a symmetric implosion,¹² its degradation in the presence of other perturbation is much smaller and not well understood. However, removing or reducing the impact of the tent perturbation will help clarify these other perturbation sources.

The supported fill-tube is one of the most promising, albeit more challenging, alternatives to replace the tent. It allows the fill-tube to be the only feature touching the capsule and, if the support rod is far-enough removed from the capsule, it may provide no additional degradation source. Conversely, to keep the capsule from undue drooping and vibration, the rod needs to be as close as possible. Preliminary tests suggest that a 12 μm diameter SiC rod offset by $≤$300 μm is needed to achieve the necessary control over the capsule's position (although the coefficient of thermal expansion may not be compatible and other materials are being explored). Modeling and HGR experiments are used here to determine if there is a stand-off distance that can meet the two competing requirements.

Modeling this type of feature is a challenge because it is inherently three dimensional. Fill-tubes and tents can be modeled in two dimensions with axial symmetry, but the supported fill-tube cannot. Additionally, the small size of the rod makes a full three dimensional simulation prohibitively expensive. To construct an approximate model, two dimensional simulations can be used. Consider the geometry of Fig. 5(a), with a rod running next to the capsule but displaced from its surface by some distance. Cross sections of this geometry would look like that in Fig. 5(b) with the intersection of the rod, a further distance from the capsule at each slice. These cross sections can be simulated in 2D with axial symmetry, but they represent a ring running around the equator at a fixed distance, rather than a straight rod. This model makes the assumption that this is an adequate approximation for the hydrodynamics occurring in this plane. This methodology also neglects the impact of the fill-tube, so it considers only the additional perturbation caused by the rod. These capsule-only simulations use a spectrally resolved radiation source located far from the capsule and have a resolution of 0.06° on a domain of $θ = 60 ° – 90 °$⁠. Simulated density images in Fig. 5(c) show the result of this 2D approximation as the rod varies between 100, 200, and 300 μm away from the capsule. These images are at a similar time as in the experimental radiograph that will be discussed in Fig. 7, 16.7 ns or when the capsule has converged by 2.8×. At 100 μm offset, a significant ρR perturbation exists, but the perturbation reduces quickly as the rod is displaced further from the capsule.

The development of this perturbation is shown in Fig. 6 from the simulation of the support rod offset from the capsule by 200 μm. The left half of the frame shows the different material regions and the right shows $log ( ρ )$⁠. In this simulation, the rod heats up from the incoming radiation and expands in the form of a blast wave. The blast wave can be seen at 1.8 ns having reached the wave blowing off from the ablation front and at 3 ns having reached halfway between the ablation front and the original rod location. In this simulation, the blast wave never actually hits the capsule and therefore does not hydrodynamically couple to the capsule. A perturbation does arise from the shadow that the rod casts on the ablation front. The 12 μm size of the initial rod is too small to cast a meaningful shadow, but the simulation shows that the rod grows up to ∼50× its original size in the first few ns. Thus the radiation is partially blocked by this expanded feature and less ablation pressure is felt in its shadow.

This model is tested against an HGR experiment, and the results are shown in Fig. 7. In the experiment, two (glue-plugged) fill-tubes are attached to the capsule ±20° from each other and support rods are attached to each at different distances from the capsule. This first experiment tested rods of 100 μm and 200 μm from the capsule. A follow-up experiment repeated the 200 μm case but with an extended length rod and another rod with a 300 μm stand-off distance.³¹ 

This experiment showed that the support rod 100 μm from the capsule leaves horizontal features of large ρR. As the rod is displaced further from the capsule, the magnitude of the perturbation not only becomes smaller, but the horizontal feature also splits into two.

To compare with the experiment, a simulated radiograph is generated by combining the set of 2D simulations in the geometry described by Fig. 5(a) and ray-tracing through the simulations at the energy of the Fe backlighter, 6.7 keV. Comparing between the experiment and simulation shows that the 100 and 200 μm offset rods perturb a similar horizontal extent as predicted by the models. A lineout through the radiographs is shown in Fig. 7(c). Simulations were performed with multi-group radiation diffusion and with implicit Monte-Carlo (IMC) transport. The diffusion approximation neglects the directional dependence of radiation transport that is retained with IMC transport. This approximation is commonly used in capsule modeling for computational efficiency but has inaccuracies, as this experiment illustrates. The case with the 100 μm offset rod is modeled well by both diffusion and IMC but at 200 μm offset, IMC produces a ∼50% larger perturbation and a good match to the data. The reason for these differences seems to be that the blast wave from the 100 μm offset rod actually interacts with the ablation front and leaves a perturbation. This hydrodynamic interaction is well modeled with either of the radiation models. For the 200 μm offset case, there is not a hydrodynamic interaction, but there is a shadowing of radiation, and the diffusion approximation is inadequate to model this correctly.

A clear difference between the model and the experiment is the existence of a secondary feature in the experiment of the 200 μm offset rod which does not show up in the model (seen in Fig. 7(a) as a second high ρR feature and at y = 180 μm in Fig. 7(c)). Fig. 8 shows a 2D simulation of a support rod offset by 200 μm, but this simulation also models the lasers and the hohlraum.³² Doing this adds significantly more computational expense, but allows the source locations of radiation energy to be accurately modeled. Fig. 8 shows the material region at 1.8 ns and a laser intensity super-imposed on top. This shows the split of the lasers into “inners” that hit near the equator and “outers” that hit near the laser entrance hole. This split results in bright emission sources where the lasers hit the hohlraum wall. With the support rod located 20° from the equator (as it was in the HGR experiment) two bumps appear on the ablation front (seen in Fig. 8(b)). These two bumps seem to come from the two shadows cast by the inners and outers (the shadow from the outers on the left side appear to be over the horizon of the capsule). Work is ongoing to compare this model with the experiment.³² 

Three dimensional simulations are required to evaluate the impact of these features on the performance of the implosion. As mentioned earlier, a direct simulation of a supported fill-tube in three-dimensions is prohibitively expensive, but this 2D approximation can be used to initialize the 3D simulations. Here, 2D simulations are interpolated onto a 3D domain at the time when the shock wave reaches the origin. The capsule-only simulations are used, since they can be run more rapidly than the hohlraum simulation and capture the dominant perturbation. The 2D simulations' domain occupies slices of constant $ϕ$ on a spherical/polar mesh, and intermediate values of $ϕ$ are filled by interpolating between the 2D simulations. This methodology fills the density, material region, temperature, and velocity arrays and is simulated on a quarter of a sphere (⁠ $θ = 0 ° – 90 ° , ϕ = 0 ° – 180 °$⁠) using symmetry planes.

Fig. 9 shows the results of these simulations when modeling a support rod offset by 300 and 100 μm. These simulations are using a capsule with a DT ice layer and a high foot drive (as opposed to the previous HGR simulations where the capsule was a 0.8× scale gas-filled capsule). The first row shows the initial setup, constructed from the 2D simulations. The second row shows the condition of the shell at bang time. The 300 μm offset rod leaves a very small perturbation, whereas the 100 μm offset rod starts as a large ρR perturbation that, upon deceleration, sends a jet of material into the hot spot. The simulation of the 100 μm offset rod produces a yield-over-clean value of 0.43 or 0.83 without including α heating. (Note that α heading is very sensitive to modeling choices in this yield regime of 10¹⁷ neutrons, so the no-α heating number is more meaningful). The case with a 300 μm offset gives full yield, both with and without α heating. These results are compared to other options in Section VI.

Since capsules on NIF need to be filled with DT fuel using a fill-tube, the ideal support solution would be to use the existing fill-tube to also hold the capsule. The fill-tube is made of borosilicate glass and is 10 μm in outer diameter and 6 μm in inner diameter.³³ It is strong enough to hold the capsule but cannot hold it rigidly. A fill-tube of 30 μm in outer diameter and 22 μm in inner diameter, however, is large enough to support the capsule.

Simulations of a larger fill-tube show it imparting a similar yield degradation as the tent. Fig. 10(a) shows the bang-time images from simulations of a standard 10 μm fill-tube and one with a 40 μm diameter (based on an earlier design). In the simulation, the standard fill-tube sends a narrow jet through the center of the capsule and leaves a small ρR perturbation on the shell. In this simulation of high-foot experiment N140520, the yield is degraded by 0.5× compared to the symmetric yield. The larger fill-tube develops a significant ρR perturbation, and the jet entering the hot-spot is more substantial. This degrades the yield to 0.12× the symmetric yield, close to the 0.11× reduction caused by the tent alone in this series of simulations. Despite these predictions, testing a larger fill-tube is worthwhile given its simplicity and the modeling uncertainty.

HGR experiments have begun testing the impact of fill-tubes^31,34 and the results are shown in Fig. 10(b). The left image shows an experiment with a fill-tube along the axis of the HGR cone. The right image shows an experiment that compared 10 μm and 30 μm diameter fill-tubes located ±20° from the center and included two tents. These experiments found that the fill-tube perturbation is more complex than previously understood. Past experiments³⁵ and modeling^36,37 showed the in-flight fill-tube perturbation to be a localized bump, while these perturbations extend wider and have more low ρR content or “bubbles” than expected. These differences could be from the non-uniform radiation environment of the hohlraum (similar to what produced the dual shadow in the supported fill-tube experiment) or from UV uptake during assembly when the fill-tube bonding epoxy was cured with a UV light source.²⁹ Modeling work is underway to try to model these patterns and understand their impact on implosion performance.

Another option for reducing the imprint of the tent is to surround the capsule with a layer of low density foam so that the tent is only in contact with the foam. The radiation front in the foam can travel faster than the shock wave from the foam ablation (and tent blow-off) if the density is less than a critical value, which scales as³⁸  $ρ 2 / 5$⁠. Fig. 11(a) shows low foot simulations that include 200 μm thick layers of foam and a 100 nm thick tent compared with a simulation without foam. In these calculations, the foam significantly reduces the ρR perturbation caused by the tent if the density is $≤ 30$ mg/cm³. Fig. 11(b) shows that at 30 mg/cm³, radiation is able to get through the foam at early time and reach the capsule. This enables the ablation of the capsule and decreases the density gradient by the time the pressure wave from the tent reaches the capsule. Additionally, the spatial offset of the tent from the capsule allows the perturbed pressure wave to weaken by the time it reaches the capsule.

An HGR experiment with a foam layer and a tent was performed to test this concept. This experiment used a 200 μm thick layer of SiO₂ foam with a density of 30 mg/cm³. While lower density foams have been made, this density is the lowest that can be machined into hemisphere shells and attached to the capsule. The experiment, shown in Fig. 11(c), shows no sign of the tent perturbation (as observed in the right image of Fig. 10(b)), but does show other features that come from a hair-line crack, handling perturbations, and from internal modulations in the foam. Analysis is underway to understand the effect of these features on the capsule and implosion performance.³⁹ 

To compare the expected performance improvements among these options, here the yield-over-clean is reported from simulations that do not include α-particle energy deposition, i.e., the additional energy deposited by the slowing down of α particles from DT reactions. The boot-strapping effect that can occur with α-deposition can produce significantly different yields based on subtle difference in simulation parameters, like resolution or how the α-particle slowing is modeled.⁴⁰ Therefore, no α-deposition calculations are a more faithful representation of the integrity of implosion.

Table I compares the yield-over-clean values for several capsule support options. These calculations use the high-foot pulse shape, since experiments have demonstrated good performance^24,41 and a large experimental dataset exists to compare with. In particular, three experiments with 1.75 MJ of laser energy and using a 175 μm thick ablator have produced neutron yields of 6.9 ×  $10 15 ±$10%, and the simulation model used here is tuned to one of these experiments, N140520. While Table I shows that the 40 μm diameter fill-tube worsens the performance slightly compared to the tent, performance of the tent simulation would reduce if it also included a standard fill-tube (as would be required in the experiment). It is not feasible, however, to directly simulate multiple supports and fill features in one calculation. The foam shell along with a 45 nm thick tent attached to the foam is calculated to have almost no yield degradation. However, this number will likely reduce once a realistic model of foam perturbations is developed. Finally, the support rod appears as the most promising support alternative. At 100 μm offset, the model suggests a similar yield degradation as the tent. As the support rod is moved to 200 and 300 μm, there is almost no reduction in performance.

The first two experiments to test these alternatives with a layered DT implosion used a similar high-foot pulse shape and thickness ablator. While simulations suggest that the yield is degraded by 5–6× in these implosions from a symmetric 1D implosion, the largest degradation source is predicted to be low-mode asymmetries, which degrade the yield by ∼15×.⁴⁰ Thus removing the tent perturbation alone without improving other degradation sources should result in more modest performance improvements. These first two layered DT experiments used a 30 μm diameter fill-tube and a polar tent (as discussed in Fig. 3), but did not show a yield improvement. This could be because these supports are similarly perturbative as the original tent, because some other degradation source was present and counteracted the improvement, or because this experiment was less sensitive to the tent than others. Detailed analysis of these experiments is ongoing. But these experiments demonstrate that fielding challenges, such as capsule centering and vibration, can be overcome.

Other experiments are modeled to be much more sensitive to the tent and may show a larger improvement by reducing its impact. Table II shows the effect of a 45 nm thick tent for various pulse shapes and ablator thicknesses. The high-foot campaign found that by using thinner ablators, higher velocity implosions could be obtained for a given laser energy.^41,42 Table II shows that as the ablator is thinned, the effect of the tent increases in simulations.⁴⁰ Reducing the foot level to achieve a lower adiabat^43,44 (the so-called 3-shock or high foot adiabat shaping) is more sensitive to the tent than the high-foot at the same thickness. At a higher convergence, the low foot and adiabat shaped low foot⁷ pulse shapes are the most sensitive to the tent. Testing these implosions with an improved capsule support should give the clearest evidence of an improvement in performance.

New methods of holding the capsule inside the hohlraum are being developed that will leave less of a perturbation than the tent. Testing the imprint of some of these options has shown that promising support alternatives exist, but has also transformed the understanding of how perturbations from engineering features develop on the capsule. In the test of the support rods and the fill-tubes, the perturbations that are attributed to shadowing are more complex than previously understood. Simulating the capsule in isolation from the hohlraum with a uniform radiation source, as traditionally done to model the instability growth, may not be adequate for these features. Additionally, the presence of these shadows as significant perturbation sources may require the development of new numerical methods to simulate radiation transport. Radiation diffusion will not correctly model these effects accurately, and the IMC calculations can be more costly and introduce noise. Work is underway to give HYDRA the capability to simulate radiation transport on an arbitrary mesh using discrete ordinates formulated in a polar geometry.⁴⁵ 

The HGR experiment with a foam layer showed that foam is effective not only in reducing the tent perturbation, but also showed the other perturbation sources that can arise with the foam. While the foam may not be the best alternative for supporting the capsule, it is being used or planned for in other ICF platforms, such as wetted foams⁴⁶ and double shell capsules.⁴⁷ The results here may influence the use of foams in those platforms.

The three dimensionality of some of these features required the development of new simulation methods to understand and quantify their effect. Without being able to perform a direct 3D simulation, a model made up of 2D slices may suffice. Comparisons of this model to experimental data showed good agreement. This methodology is also used to assess the impact on implosion performance. This last stage needs to be simulated in 3D, but can be initialized using this 2D approximation.

The experiments and modeling of the supported fill-tube suggest that it can be an effective replacement for the tent support. When the support rod is placed 200–300 μm from the capsule, little to no performance degradation is expected. To observe this improvement in a layered DT implosions, other degradation sources need to be reduced, either by improving the hohlraum⁴⁸ or by using a capsule designed to mitigate low-mode asymmetries.⁴⁹ The high convergence of low foot and adiabat shaped low foot pulses are more sensitive to the tent and may be candidates to demonstrate a performance improvement.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 and by General Atomics under Contract No. DE-NA0001808.