The engineered macro-pore foam provides a new way to study thermonuclear burn physics by utilizing capsules containing deuterated (D) foam and filling tritium (T) gas in the engineered macro-pores. The implosion of a thermonuclear capsule filled with an engineered macro-pore foam will be complex due to the interaction of a shock wave with the engineered macro-pores. It is our goal to quantify how substantially complex foam structures affect the shape of shock and bulk shock speed. A cylinder-shape shock tube experiment has been designed and performed at the Omega Laser Facility. In order to examine how a foam structure will affect shock propagation, we performed several tests varying (1) engineered macro-pore size, (2) average foam density, and (3) with/without neopentane (C5H12) gas. X-ray radiographic data indicate that shock speed through engineered macro-pore foams depends strongly on average foam density and less on pore size. Experimental shock propagation data helped guide two numerical simulation approaches: (1) a 2D simulation with homogenizing foams rather than explicitly simulating engineered macro-pores and (2) a 2D toroidal-pore approximation adopting a toroidal-tube geometry to model engineered macro-pores.

Low-density porous materials or foams are useful for Inertial Confinement Fusion (ICF) and High Energy Density Plasma (HEDP) research because foams offer a well-defined geometry and density.1,2 Low-density foams are unique materials having intrinsic pores, the sizes of which are varied from nano-meter to micro-meter scales, depending on the synthesis conditions.3 There have been considerable interest in studying how a laser or an ionization wave interacts with intrinsic pores or microstructure of foam4–6 and effects of microstructure on shock propagation7,8 and equation-of-state.9 Foams with intrinsic pores offer additional applications, for example, as a matrix for liquid deuterium–tritium (DT) fuel for thermonuclear fusion study.10,11

The MARBLE campaign12 of Los Alamos National Laboratory (LANL) was able to control the macro-pore sizes of a foam in the range of 30–100 μm in diameter (shown in Fig. 1). A divinylbenzene (DVB) foam was used, and its intrinsic pore sizes are measured to be hundreds of nano-meters.13 Foams with a fixed, engineered macro-pore size are made by placing a fixed size of glass shells (i.e., 30, 50, or 100 μm in diameter) in a DVB aerogel precursor, and then chemically leaching out once the foam matrix is polymerized.13,14 The engineered macro-pore foam provides a new way to study thermonuclear burn physics by utilizing capsules containing deuterated DVB and filling tritium gas in the engineered macro-pore.15–17 The initially separated D and T will create thermonuclear burn as a mix between foam material and gas progress. One can change the mix scale length by simply adjusting the size of engineered macro-pore foam.

FIG. 1.

MARBLE foams with engineered, macro-pores. Images are obtained from a scanning electron microscope. All scale bars shown in white color are 100 μm in length. This figure is modified with permission from Hamilton et al., Fusion Sci. Technol. 70, 226 (2016). Copyright 2016 Taylor & Francis Ltd. and from Haines et al., Nat. Commun. 11, 544 (2020). Copyright 2020 Springer Nature Ltd.

FIG. 1.

MARBLE foams with engineered, macro-pores. Images are obtained from a scanning electron microscope. All scale bars shown in white color are 100 μm in length. This figure is modified with permission from Hamilton et al., Fusion Sci. Technol. 70, 226 (2016). Copyright 2016 Taylor & Francis Ltd. and from Haines et al., Nat. Commun. 11, 544 (2020). Copyright 2020 Springer Nature Ltd.

Close modal

The implosion of a thermonuclear capsule filled with an engineered macro-pore foam will be complex due to the interaction of a shock wave with the engineered macro-pores. The engineered macro-pores can alter the shape or propagation pattern of the shock wave. Haas and Sturtevant showed that when a pore or bubble has a lower density than that of its surroundings (i.e., an interstitial foam matrix), the pore acts like a diverging lens.18 As an initial uniform shock front passes by a pore, the shape of the shock front may become non-uniform because the shock moves quickly through a low-density pore and slowly through high-density interstitial foam matrix. Since the MARBLE foam consists of multiple engineered macro-pores, individual diverging shock fronts will superpose each other and produce an averaged or bulk shock front. Kotelnikov and Montgomery suggested that instabilities on foam/pore interface and transverse flows in pores may dissipate the shock propagation energy, thereby reducing shock speed.19 It is our goal to quantify how substantially complex foam structures affect the shape of shock and bulk shock speed.

The ICF and HEDP research community have established a depth of capability on radiation-hydrodynamics simulations. However, there is significant uncertainty in computational modeling of experiments involving engineered macro-pore foams. Macro-pore simulation will be sensitive to how physical properties such as equation of state20 or conductivities21 are mixed for disparate materials at sub-grid scales. Improperly modeled, geometric effects such as shock lensing and the development of hydrodynamic instabilities at foam/pore interfaces may not accurately simulate the shape of the shock front and propagation speed. Furthermore, radiative emission from the shock can cause expansion of the interstitial foam matrix material ahead of the shock front and inhibit hydrodynamic evolution that would otherwise occur at a sharp foam/pore interface.22 These effects combine to influence the shock speed in ways that may not be accounted for in simulations. For MARBLE implosion experiments at National Ignition Facility (NIF1), these effects can impact experimental timing (e.g., time of peak compression) as well as the amount of energy deposited into the fuel. It is therefore important to validate the integrated modeling capabilities for shock propagation through engineered macro-pore foams.

We aim to address this question by experimentally measuring the shock speed passing through an engineered macro-pore foam and providing data to determine the accuracy of numerical simulations. In the MARBLE implosion at NIF, however, it is not straightforward to study shock propagation due to its spherical geometry. As a result, a cylinder-shape shock tube experiment has been designed and performed at the Omega Laser Facility.23 Foam-filled shock tubes were driven directly by lasers on one end, while x-ray radiographs were generated at various times, which enabled the direct measurement of shock front location and shape. In order to examine how a foam structure will affect shock propagation, we performed several tests varying (1) engineered macro-pore size, (2) average foam density (bulk density of interstitial foam and voids), and (3) with/without gas. For gas-filled experiment, a neopentane (C5H12) was chosen for its high density (3.26 mg/cc) at 1 atm. X-ray radiographic data indicate that shock speed through engineered macro-pore foams depends strongly on average foam density and less on macro-pore size.

For the shock propagation study through engineered, macro-pore foams, xRAGE (LANL's Eulerian radiation-hydrodynamics) simulation code was chosen because it was used to design the MARBLE implosion at the NIF. Enabled physics include hydrodynamics, radiation diffusion, electron thermal conduction, ion conduction, and three-temperature (ion, electron, radiation).24–26 Due to the importance of laser ray-tracing for simulating the complex, time-dependent energy deposition pattern, the Mazinisin laser ray-tracing package was incorporated into the xRAGE.27 However, simulated laser drives were adjusted to account for scattered laser light, such as cross-beam energy transfer, that were not directly modeled in simulations. The resultant laser drives on target were determined by matching a simulated shock speed with the measured shock speed for a foam without engineered macro-pores. The same amount of laser drives was then consistently applied to simulation for engineered macro-pore foam targets.

In our previous work, xRAGE demonstrated the capability to simulate engineered macro-pores foam implosions in three-dimensions.16,17 However, 3D simulation remains expensive, and it is practical to develop alternative ways to simulate engineered macro-pores contained in a cylinder shock tube. In this work, all simulations were performed in 2D axisymmetric geometry, where two approaches were tried to approximate the engineered macro-pores. The first approach aimed to homogenize a foam with a mixture of carbon and hydrogen with an average density rather than explicitly simulating engineered macro-pores. The homogenizing foam approach does not simulate pore effect and only simulates the average foam density effect on shock speed. In the second approach, 2D simulation adopts a toroidal-tube geometry to model engineered macro-pores. The macro pores were simulated by generating circular voids that were placed throughout the two-dimensional foam. The number and position of the voids were determined based on the experimental void fraction of the foam, which was 49% ± 2%.13,14 Due to the cylindrical symmetry of the problem, the circular voids placed in the foam became toroidal once rotated around the central axis. The toroidal-tube geometry cannot determine the pore volume fraction, pore diameter, and interstitial foam matrix density simultaneously. Therefore, in 2D simulations, while average foam density (or bulk density of interstitial foam and voids) is maintained, the interstitial foam density is artificially increased and the pore radius is reduced. The toroidal-pore approximation is expected to over-estimate the pore impact on shock lensing or pore collapse since the surface area to volume ratio is larger than for the actual pores, which are approximately spherical. Both approaches have their own limitations; therefore, the results of these simulations should be considered an approximation case rather than an accurate representation of the engineered macro-pore geometry.

Figure 2 compares the predicted shock speed for both numerical approximations: (1) homogeneous foam approach shown in the red curve, where no engineered macro-pores were simulated explicitly and (2) a toroidal-pore approach shown by the blue curve. Both approaches show that shock speed decreases as average foam density increases. For a fixed average foam density, there is a discrepancy between two methods on the order of 5%. Experimental data are needed to validate if both 2D practical approaches are acceptable in predicting a shock speed through engineered macro-pore foams. However, given the limited precision of an experimental setup, it will be difficult to determine which approximate is more accurate.

FIG. 2.

Comparison of 2D numerical models between homogeneous approximation and toroidal-pore approximation.

FIG. 2.

Comparison of 2D numerical models between homogeneous approximation and toroidal-pore approximation.

Close modal

For a shock-tube experiment, 50 and 90 μm DVB macro-pore foams are machined into a cylinder shape. For comparative purposes, a fine-pore foam without engineered macro-pores was additionally prepared. Figure 3(a) shows a cut-away view of a foam target that was machined into a cylinder shape (1650 μm in length and 500 μm in diameter). The foam cylinder was contained by a beryllium tube, serving as a shock tube, with a wall thickness of 100 μm. The ends of the tube were capped with a Rexolite ablator disk (75 μm thick). The disk was doped with 0.5% iodine to reduce potential laser-driven preheat (density of 0.5% iodine-doped ablator = 1.15 g/cc). The ablator was directly irradiated by a total of eight Omega laser beams (∼4 kJ) for 1-ns. The laser spot size is approximately 600 μm in diameter and intensity is 1.4 PW/cm2. Using a point-projection radiographic technique (magnification ∼22–24), up to two x-ray radiographs [Fig. 3(b)] were obtained for each target to measure the shock position at a given time. As an x-ray backlighter material, a titanium was chosen which has a property with H-α and L-α emission lines at 4.727 and 4.977 keV, respectively.

FIG. 3.

(a) A cut-away view of foam target and (b) experimental layout including two orthogonal x-ray backlights.

FIG. 3.

(a) A cut-away view of foam target and (b) experimental layout including two orthogonal x-ray backlights.

Close modal

For the majority of experiments, the engineered macro-pore foam target was placed in a vacuum condition, and no gas-fill was present. During the later experiment series, an additional 1-atm neopentane (C5H12) gas-fill was achieved. Adding the gas into the engineered macro-pores foams can increase the pressure inside pore and may prevent the engineered macro-pores from collapsing, maintaining a heterogeneous foam structure ahead of the shock. We aim to study if the addition of gas affects the propagation of shock through engineered macro-pores foams.

Machining low-density foams, particularly with engineered macro-pores, was challenging and created density variation across the radius of the foam cylinder. Figure 4 shows the radial density profiles of an engineered macro-pore foam (with a nominal uniform density of 100 mg/cc in this case) obtained by using a Cr (5.4 keV) x-ray transmission method.28 The foam densities increased radially outward. It is hypothesized that the high density seen at the edges of the cylinder is caused by debris from dry machining collecting on the macro-pores. The intensity of this localized densification (or edge effect) has been varied with the amount of air flow during the machining operation. The air flow must be optimized to reduce debris collection and not blow the part off the machined mandrel prematurely. A study has not been done at this time with the purpose of removing this layer. The shock speed is inversely proportional to the mass density, causing the shock speed to vary with the radius of the foam. This may notably affect the shape of the shock front.

FIG. 4.

(a) Density variation across the radius of foam cylinder and (b) radial foam density profile measured by x-ray transmittance method.

FIG. 4.

(a) Density variation across the radius of foam cylinder and (b) radial foam density profile measured by x-ray transmittance method.

Close modal

In simulations, two foam density profiles were tested: (a) uniform and (b) radial density variation, shown in Fig. 4(b). In Fig. 5, shock waves propagate from top to bottom. Figure 5(a) shows that a simulated shock front is uniform along the radius of foam cylinder (uniform density case). When actual radial density distribution was used, the higher density on the periphery of the foam decreased shock speed, while the lower density near the centerline resulted in a relatively faster shock speed. A curved shape appears on the shock front near the axial centerline. Figure 5(c) shows the experimentally measured x-ray image for the 50 μm macro-pore foam. Modeling actual density profile of foam improves to match the shape of shock front.

FIG. 5.

Modeling a measured radial distribution of foam density improves shock front simulation: (a) numerical simulation with uniform density profile, (b) numerical simulation with non-uniform density profile, and (c) experimental x-ray radiograph with 50 μm-marble foam (shot number 90877).

FIG. 5.

Modeling a measured radial distribution of foam density improves shock front simulation: (a) numerical simulation with uniform density profile, (b) numerical simulation with non-uniform density profile, and (c) experimental x-ray radiograph with 50 μm-marble foam (shot number 90877).

Close modal

Shock propagation through engineered macro-pore foams was studied to examine if the pore size affects shock speed. Figure 6 shows the location of shock front measured by x-ray radiography as a function of time in nano-seconds for two different type of foams: (a) a 50 μm foam with 47.4 mg/cc ± 4.8% and (b) a fine-pore foam with 49.4 mg/cc ± 4% (November 2017, shot series from 87748 through 87762). Time-zero is defined as the moment when the incident lasers irradiate the Rexolite ablator. In both foam types, shock front positions increase linearly with a time. The position of the shock front was determined by averaging across the cross section of the cylinder excluding the edge effects. A total uncertainty in determining the position of shock front is approximately 35 μm, mainly caused by the three following limitations. A spatial resolution of radiographic images was limited by 20 μm due to a pinhole size used (20 μm). Each image was spatially calibrated by measuring the inner diameter of the radiographic image of shock tube. Spatial calibration was uncertain on the order of 25 μm. The variation in shock front position was also caused by asymmetries in the shock front (∼15 μm). An average shock speed was obtained from four snapshot images over 7 ns to 11.5 ns, for each foam type. An average shock speed of 98.5 ± 9.1 μm/ns (or km/s) was obtained for the 50 μm pore-size [Fig. 6(a)] and 99.5 ± 9.2 μm/ns for the fine-pore [Fig. 6(b)], demonstrating that engineered macro-pore size does not affect shock propagation within experimental error bars. In Figs. 6(a) and 6(b), solid lines are the best linear fit to the data, and the dashed lines indicate a one-standard deviation (or 1-σ) bounding of the position based on uncertainty analysis. In this work, the authors adopted a linear fit tool developed by Ref. 29. In Fig. 6(b), the first three data points describe a line with a shallower slope than the linear fit when including the fourth point. The four data points shown in Fig. 6(b) are obtained from separate shots on nominally identical conditions. However, there were small amounts of shot-to-shot variations especially in average foam density and laser energy on target. Shock front locations from the second and the third data points are slightly lower than the linear fit shown in Fig. 6(b). Although it is not entirely responsible, a slightly higher foam density (51.2 mg/cc vs 49.4 mg/cc average) may contribute to a slower shock speed. In Fig. 6(c), data from Fig. 6(b) were compared with simulation results using a homogeneous approach. In the simulation, the shock speed does not decrease significantly over time even after the laser turns off at 1-ns. Figure 6(c) shows that the decrease in shock speed in simulations is lower than the error bars from experiment.

FIG. 6.

Measured shock speed through (a) a 50 μm macro-pore foam and (b) a fine-pore foam, showing that pore-size does not affect shock speed, when they have identical density. In (c), data from (b) were compared with simulation results using a homogeneous approach.

FIG. 6.

Measured shock speed through (a) a 50 μm macro-pore foam and (b) a fine-pore foam, showing that pore-size does not affect shock speed, when they have identical density. In (c), data from (b) were compared with simulation results using a homogeneous approach.

Close modal

To extend the investigation on the effect of engineered macro-pores and average foam density on the shock speed, three additional foam types were tested including a 90 μm foam with 35.1 mg/cc, a 50 μm foam with 114.0 mg/cc, and a fine-pore foam with 52.8 mg/cc. Table I shows the list of the five total foam types used in Sec. IV B. In this work, foam density means average foam density or bulk density of interstitial foam matrix and voids. The five foam types shown in Table I come from five separate batches. Because all engineered macro-pores foam has a void fraction of 49% ± 2%, the macro-pore foam is a mix of void and interstitial foam matrix at roughly twice the average density. For example, for the fine foams, we aimed to produce 50 mg/cc density, therefore the desired monomer (divinylbenzene, or DVB) mass was added to the appropriate volume of solvent. For example, 500 mg DVB is added to 10 ml of dibutyl phthalate solvent. To aim 50 mg/cc foam with the engineered macro-pore, we doubled the initial monomer mass (i.e., 1000 mg DVB) in 10 ml solvent. The last foam type (50 μm pore-size with the average density of 114 mg/cc) used a 2280 mg DVB monomer in 10 ml solvent.

TABLE I.

Foam types used in Sec. IV B.

Average density (mg/cc)Pore-size (μm)Shock speed measured (km/s)
35.1 ± 11 % 90 110.0 ± 9.3 
47.4 ± 4.8 % 50 98.5 ± 9.1 
49.4 ± 4 % <1 99.5 ± 9.2 
52.8 ± 3.9 % <1 92.1 ± 10.4 
114 ± 2.5 % 50 70.2 ± 8.8 
Average density (mg/cc)Pore-size (μm)Shock speed measured (km/s)
35.1 ± 11 % 90 110.0 ± 9.3 
47.4 ± 4.8 % 50 98.5 ± 9.1 
49.4 ± 4 % <1 99.5 ± 9.2 
52.8 ± 3.9 % <1 92.1 ± 10.4 
114 ± 2.5 % 50 70.2 ± 8.8 

Figure 7 shows the shock speed measured for above five foams types as a function of average foam density. Overall, shock speed decreased as average foam density increased. For comparative purposes, numerical simulation results shown in Fig. 2 are scaled to overlay with experimental data. In the current simulation work, the scattered laser light, such as cross-beam energy transfer, is not directly modeled. The resultant laser drives on target were determined by matching the simulated shock speed with the measured shock speed for the standard fine pore foam. We kept the same amount of laser driver for the case of engineered macro-pore simulation. Experimental data agree with simulation within error bars, confirming that shock speed depends strongly on average foam density regardless of engineered macro-pore size. We did not observe an experimentally measurable effect of engineered macro-pore size on shock speed.

FIG. 7.

Shock speed through engineered, macro-pore foams depends strongly on average foam density, less on macro-pore size.

FIG. 7.

Shock speed through engineered, macro-pore foams depends strongly on average foam density, less on macro-pore size.

Close modal

Shock-tube experiments shown in Secs. IV A and IV B are all performed under vacuum conditions, where no gas was added to the shock tube. Since the MARBLE implosion experiment at NIF uses a mixture of tritium and hydrogen gases, it is useful to develop a gas-filled shock-tube experiment at Omega and benchmark numerical simulations accordingly. A typical NIF MARBLE experiment uses up to 10 atm tritium/hydrogen gas at 150 K temperature, as a result producing a gas density of ∼2 mg/cc. However, the gas-filled shock tube design that we developed for the Omega experiment could not hold more than 1 atm gas. Instead of using 10 atm hydrogen gas, neopentane (C5H12) gas was chosen simply because it has a high enough gas density (3.26 mg/cc) at 1 atm, which provides a similar gas density condition at NIF (∼2 mg/cc). In addition to the experimental simplicity that neopentane offers, uncertainty is reduced in the numerical simulation when calculating shock speed through a gas-filled foams because the elemental composition of neopentane is the same as the MARBLE foam (DVB). Before this gaseous experiment series, the shock-tube design was upgraded to seal gaseous C5H12. To field a gas-filled shock tube, we have to modify the target holder design, while keeping the same shock-tube length and ablator thickness. By an error in our target design, the new target holder came out 50 μm longer than the original holder, which led to the ablator position being offset relative to the beam pointing. Although we cannot directly compare the new data set using the gas-filled shock-tube with the previous dataset shown in Secs. IV A and IV B, Sec. IV C uses only the upgraded shock-tubes for both (1) with neopentane gas and (2) without gas experiment. Figure 8 shows the measured shock speeds for two types of foam targets: (1) fine-pore foams with 50 mg/cc (shown in blue, closed square) and (2) engineered macro-pore foam with 50 μm in diameter with 75 mg/cc (shown in red, closed circle). Adding an 1 atm C5H12 did not affect shock speed for both fine- (blue) and engineered macro-pore foam (red) within error bars. Dashed lines in Fig. 8 show the numerical simulation with homogeneous foam approximation. The numerical simulation agrees with experiment within error bars, showing that the homogeneous foam approximation is still valid to predict shock speed through engineered macro-pore foams filled with 1 atm C5H12 gas.

FIG. 8.

2D numerical simulation with homogeneous foam approximation is still valid to predict shock speed for both 50 mg/cc and 75 mg/cc foam filled with 1 atm neopentane gas.

FIG. 8.

2D numerical simulation with homogeneous foam approximation is still valid to predict shock speed for both 50 mg/cc and 75 mg/cc foam filled with 1 atm neopentane gas.

Close modal

In order to examine how foam structure consisting of engineered macro-pores will affect shock propagation, we performed several tests varying (1) engineered macro-pore size, (2) average foam density, and (3) with/without C5H12 neopentane gas. We found that radial density variation of the foam target play a role in setting the shape of shock front. X-ray radiographic data indicate that shock speed through engineered macro-pore foams depend strongly on average foam density (bulk density of interstitial foam matrix and voids) and less on engineered macro-pore size. After accounting for average foam density, the engineered macro-pore size and gas content has no measurable impact on experimental shock speeds. Experimental shock propagation data helped guide two numerical simulation approaches: (1) a 2D axisymmetric simulation with homogenizing foams rather than explicitly simulating engineered macro-pores and (2) a 2D toroidal-pore approximation adopting a toroidal-tube geometry to model engineered macro-pores. While simulated shock speeds from both approximations agree with experiment within error bars, we were unable to verify which approximation represents experimental data best due to limited experimental resolution.

The authors acknowledge the support of the OMEGA Laser Facility operations team at Laboratory for Laser Energetics. This work was performed by the Los Alamos National Laboratory, operated by Triad National Security, LLC for the National Nuclear Security Administration (NNSA) of U.S. Department of Energy (DOE) under Contract No. 89233218CNA000001. MARBLE is supported by the DOE NNSA Office of Experimental Sciences (No. NA-113) SAT and PAT Programs.

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

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