Experimental quantification of the impact of heterogeneous mix on thermonuclear burn

In inertial conﬁnement fusion, deuterium–tritium (DT) fuel is brought to densities and temperatures where fusion ignition occurs. However, mixing of the ablator material into the fuel may prevent ignition by diluting and cooling the fuel. MARBLE experiments at the National Ignition Facility provide new insight into how mixing affects thermonuclear burn. These experiments use laser-driven capsules containing deuterated plastic foam and tritium gas. Embedded within the foam are voids of known sizes and locations, which control the degree of heterogeneity of the fuel. Initially, the reactants are separated, with tritium concentrated in the voids and deuterium in the foam. During the implosion, mixing occurs between the foam and gas materials, leading to DT fusion reactions in the mixed region. Here, it is shown that by measuring the ratios of DT and deuterium–deuterium neutron yields for different macropore sizes and gas compositions, the effects of mix heterogeneity on thermonuclear burn may be quantiﬁed, supporting an improved understanding of these effects.


I. INTRODUCTION
Inertial confinement fusion (ICF) ignition involves compressing thermonuclear fuels to densities and temperatures where fusion heating exceeds losses from radiation, thermal conduction, and hydrodynamic work. 1,2 Though recent experiments on the National Ignition Facility (NIF) indicate that we may be on the cusp of ignition, 3 achieving this has proven difficult, in part because of the presence of contaminant material in the fuel (mix). In ICF implosions, ablator material may mix into the fuel, diluting and cooling it, and reducing the rate of thermonuclear reactions. [4][5][6][7][8][9][10] This mix is not uniform and may be incomplete, as the time scale 11 for achieving a mixing transition, 12 where flow drives the rapid development of atomic mixing, is comparable to the implosion time scales. 13 At present, the ability to predict how much mix occurs in ICF experiments, how this mix is distributed, and what the effects are on yield is limited. The MARBLE campaign addresses this by providing quantitative data for validating models and improving our understanding of the physics of mix and burn.
The degree to which mix affects yield depends on the morphology, i.e., how heterogeneous the medium is at "bang time," the time of maximal yield production. Morphology is very difficult to diagnose directly in ICF settings, however. MARBLE was designed to study mix morphology and its effects on thermonuclear burn in ICF in a controlled way. In MARBLE, morphology is controlled in the initial conditions through the use of capsules containing deuterated divinylbenzene (DVB) foam fabricated with "macropores," voids of known sizes and locations (see Fig. 1), in which tritium gas resides. As the implosion progresses, a hierarchy of scales develops in the medium. At the finest scales, atomistic mixing at the interfaces between tritium gas in the pores and deuterium in the foam leads to deuterium-tritium (DT) reactions in the mixed regions. The smaller the macropore diameter, the more rapid and complete the mix at bang time and the higher the DT yield relative to the deuterium-deuterium (DD) yield. (DD reactions in the interstitial foam plasma proceed in the absence of atomistic mix.) In a volume V of homogeneously mixed fuel, the thermonuclear reaction rate is where for reactant species a, and b, N a , and N b are the respective numbers of particles in the burn volume V and hrvi ab is the product of the fusion cross section r and the relative velocity v averaged over the distribution functions of ion velocities of species a and b. In a separated reactants experiment such as MARBLE, the deuterium and tritium reactants are spatially separated initially. Heterogeneity leads to partial mixing of the deuterium and tritium, reducing the DT reaction rate relative to _ R hom DT while reducing somewhat the volume occupied by the deuterium and thus increasing the DD reaction rate relative to _ R hom DD . The ratio of DT to DD yield would therefore be expected to decrease with heterogeneity.
Previous separated reactants experiments 14,15 studied mix in layered capsules containing deuterium with a tritium gas fill. These experiments suffer from a degree of ambiguity, however, as the mixing occurs far from the hot spot and difficult to diagnose hydrodynamic features such as fill tube jetting and perturbations from surface defects may complicate interpretations of the mixing dynamics. 16

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scitation.org/journal/php In MARBLE experiments, in contrast, the bulk of the thermonuclear burn occurs in plasma generated from macroporous foam in the central volume of the capsule far from the capsule walls, thus avoiding many of these complications. This article describes the MARBLE experimental campaign and the key physics results obtained from this study. In Sec. II, we discuss the development of the MARBLE experimental campaign. Section III presents the key experimental results from this campaign and Sec. IV describes numerical modeling of the campaign, both large-scale three-dimensional (3D) simulations and reduced-scale simulations employing a subgrid mix and burn model. We conclude with a brief discussion of the results.

II. EXPERIMENTAL DESIGN
Our MARBLE experiments employ a two-shock implosion 17 on the National Ignition Facility (NIF) in a gas-filled Hohlraum, a design shown to exhibit superior reproducibility. 18 In these experiments, a 1.2 MJ, 400 TW, 8 ns, two-shock laser pulse ( Fig. 2) was directed into a gold Hohlraum to produce x rays that were absorbed by the capsule ablator, leading to implosion of the capsule. The design desiderata includes (1) high x-ray conversion efficiency (85%-90% of incident laser energy converted to x rays); (2) low laser backscatter (<2%); (3) minimal hot electron production, which causes undesirable heating of the fuel (preheat) early in the implosion; and (4) implosion symmetry tuned through choice of laser cone fraction (i.e., without relying on cross-beam energy transfer, a nonlinear laser-plasma interaction process that is imperfectly understood [19][20][21] ). This design led to Hohlraum drive conditions that could be reliably calculated using integrated 2D rad-hydro simulations (laser þ Hohlraum þ capsule) with the HYDRA radiation hydrodynamics code, 22 as evidenced by our ability to predict simultaneously the time-resolved Hohlraum x-ray drive, the capsule bang time, and the time-resolved capsule implosion symmetry. Further confidence in the design resulted from a series of shock-tube experiments on the Omega laser facility at the Laboratory for Laser Energetics that validated our understanding of shock propagation through fine-pore and engineered foams [23][24][25] as well as directly driven MARBLE capsule implosions on the Omega laser that were described well by simulations. 13 In our experiments, the target capsules consisted of 40 mg/cm 3 (the density volume-averaged over the macropores and interstitial foam matrix), engineered-pore-size, deuterated plastic foam inside machined silicone-doped plastic hemispheres that, when joined together, served as the ablator. The capsules were inserted into a 5.75 mm diameter, 9.44 mm long gold Hohlraum with a 0.3 mg/cm 3 helium fill gas and were filled by a fill tube with hydrogen-tritium (1.8 6 1 mg/cm 3 HT, 5% T by atom) or argon-tritium (34 6 1 mg/cm 3 ArT, 9% T by atom) gas mixtures at 7600 Torr. The capsules were fielded at a temperature of 150 K to allow higher gas density at the set pressure. (Lower temperatures were not used in order to limit adsorption of hydrogenic species in the foam. 26 ) The MARBLE experiments required the synthesis of engineered, fully deuterated DVB foams with a tunable pore diameter from 30 to 90 lm. The engineered foams were made by dispersing hollow SiO 2 beads into an aerogel precursor, creaming the SiO 2 under gravity, polymerizing the dilute gel network in the interstitial regions, and etching the SiO 2 beads using hydrofluoric acid. 27 The process for synthesizing the foam is described in Appendix A, along with a discussion of foam machining, fabrication of the capsules, and characterization of the targets, all of which proved critical for the reproducibility and analysis of our experiments.

III. EXPERIMENTAL RESULTS
The DD and DT neutron yields and inferred ion temperatures were measured for both the HT and ArT gas-filled implosions. As shown in Fig. 3, when a HT fill was used, ion temperatures for the DT and DD neutrons, obtained from the widths of the neutron time-offlight (nToF) detector signals, were found to be quite different, with the inferred ion temperatures from DT neutrons of around 3 keV, roughly twice the temperature of the inferred DD neutrons. (Note that these implosions are unlike more typical shock-dominated implosions due to their high mass density and modified temperature equilibration times.) In contrast, when ArT gas was used, the inferred DT ion

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scitation.org/journal/php temperatures dropped to around 1.6 keV and were in much better agreement with the DD ion temperatures (around 1.4 keV).
Simulations indicate that this discrepancy is likely primarily a result of contributions of the initial shock yield, which occurs at higher temperature than the subsequent compression yield. The DT reactivity increases with temperature more quickly than does the DD reactivity, with the result that the burn-averaged ion temperature is weighted by shock yield more for the DT neutrons than for the DD neutrons, leading to a large observed temperature discrepancy. In contrast, for ArT implosions, radiative cooling of the shock flash nearly eliminates the shock yield, so both DD and DT inferred ion temperatures are representative of the ion temperature during the compression yield. This behavior is observed both in simulations and in time-resolved x-ray emission data from the Streaked Polar Instrumentation for Diagnosing Energetic Radiation (SPIDER) diagnostic 28 on the NIF shown in Fig. 3. Data are shown for NIF shots N180729-001 and N181028-002, which are representative of the observed behavior of HT and ArT gas fills, respectively. X-ray emission from the capsule with an HT gas fill shows two peaks, associated with shock flash and compression, whereas emission from the capsule with an ArT fill has negligible shock flash and only a compression yield peak. These SPIDER data are consistent with bang-time analysis using the particle time-of-flight (PToF) diagnostic. 29 For HT fills, the PToF recorded a DTn bang time that preceded the DDn by 0.2 6 0.09 ns averaged over three shots, and for the experiments with ArT fills, the bang times are simultaneous within the statistical uncertainty.
To analyze the effects of mix morphology for different macropore sizes and gas fills, we compared the ratio of DT to DD neutron yield normalized by twice the ratio of tritium density to deuterium density, a quantity referred to as the normalized yield ratio (NYR). 30,31 The use of NYR allowed us to control for variations in foam density and gas composition and density and to decrease systematic variability from shot-to-shot variations. For a uniform atomic mix at a single ion temperature, the NYR should be equal to the ratio of DT to DD reaction rates at the hot spot temperature at bang time. Heterogeneity (partial mixing) would be expected to produce a lower NYR in the absence of species separation. Note that in our experiments, the qR for neutron scattering is negligible and does not change the NYR appreciably. This is in contrast with high-qR shots, where DT/DD ratios can show significant variability from down-scattered neutrons since the scattering length for 14 MeV neutrons is larger than for 2.5 MeV neutrons.
In experiments with an ArT gas fill, the NYR decreased from the atomically mixed value for fine pore foams to a lower value, consistent with our expectations for a heterogeneous mix (Fig. 4). For capsules with an HT gas fill, however, the expected decrease in NYR with increasing macropore diameter was not observed. Instead, the data may show a slight increase in NYR. This behavior may be a result of thermal fluctuations brought on by complicated material distributions with unequilibrated contaminants in the pore plasma, a feature observed in high-resolution calculations of similar experiments fielded on the Omega laser system at the Laboratory for Laser Energetics. 13 In contrast, for ArT fills, the specific heat is much closer for ArT and carbon-deuterium (CD) than for HT and CD, so the materials would heat to less disparate temperatures. Such temperature differences can survive in the presence of incomplete atomic mixing over the short time scales of MARBLE implosions relative to the thermal conduction time scales. The HT fill experiments were also much more susceptible to preheat from non-thermal M-band radiation from the Hohlraum walls and radiation from the shocks in the foam, leading to violent expansion of the foam material, compression of the macropores prior to compression burn, and the possibility of substantial hydrodynamic stirring prior to bang time, all of which would increase the NYR. A comparison of highresolution three-dimensional radiation hydrodynamics simulations (described in the next section) confirms these differences for ArT and HT fills, the former showing, e.g., 82% less pore compression from Mband preheat. Moreover, in the HT fill experiments, lower mean ion charge states and smaller collapsed macropore diameters lead to mean free paths of the reactant ions being comparable to the diameters of the pores after preheat, introducing the possibility of Knudsen-layer [32][33][34] and other plasma kinetic effects 21,35 on thermonuclear burn rates.

IV. NUMERICAL MODELING
Numerical simulations have proven invaluable in interpreting our experiments. We obtained from integrated two-dimensional HYDRA simulations a time-and symmetry-resolved frequency

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scitation.org/journal/php dependent radiation source (FDS source) that was used as a drive input for capsule implosion simulations using the LANL xRAGE code. 10,36 This enabled comparison with both high-resolution calculations of the implosion, which resolve the as-shot geometry of the macropores and other features of the assembly, as well as calculations at lower resolutions and dimensions that employ statistical treatments of the mix and its effects on thermonuclear burn. Two high-resolution, three-dimensional, xRAGE simulations, one with an HT fill and another, ArT fill, were performed of MARBLE implosion N180729, which had 90-lm pores and an HT gas fill. The simulations employed full physics with initial conditions that included accurate representations of the fill tube geometry as well as the gluefilled gap between the carbon-hydrogen (CH) hemispheres used to surround the foam. Additionally, the initial conditions included an accurate representation of the engineered foam: the location, geometry, and size of every macropore as well as residual pieces of glass in the asshot capsule were characterized by x-ray tomography and faithfully reproduced in the initial conditions. The large-scale xRAGE simulations were run with a maximum resolution of 0.25 lm. This level of fidelity is enabled by xRAGE's adaptive mesh refinement (AMR). The resulting simulations used between 5.6 and 11.4 Â 10 9 cells and more than 400 Â 10 6 central processing unit (CPU) hours on Lawrence Livermore National Laboratory's Sequoia and Sierra supercomputers.
The distributions of materials in the HT capsule implosion are shown in Fig. 5 at t ¼ 0, 7.9, 8.1 (shock flash), and 8.35 ns (compression burn). The shell (shown in the left panel) is red, the HT-filled pores are green, the glue is yellow, the glass (from both the fill tube and residual material from foam fabrication) is blue, the position of the shock front is magenta, and the burn region with ion temperature T i > 1 keV is cyan. The shock front and burn region have a P4 asymmetry resulting from the glue joint between the hemispheres (the glue has higher density than the ablator and slows the shock somewhat near the equator). The inner shell radius is 705 lm at t ¼ 0 and 270 lm at t ¼ 7.9 ns. The burn volume has radius 80 lm. By 7.9 ns, the pores have collapsed from heating of the foam, caused primarily by radiative emission from the shock front, and the right panel shows the effect of shock interaction with the pores behind the shock front. In contrast, high-resolution simulations of the same capsule with an ArT gas fill instead of HT show that the presence of Ar increases the electron pressure substantially and prevents pore collapse. In both, the thick shell and highdensity capsule fill prevent detrimental effects from the fill tube and the capsule seam, though some mild jetting of the glass material from the fill tube is evident at 7.9 ns. In our analysis, the simulated burnweighted ion temperatures (BWTIs) and NYR were found to be in good agreement with the data. The difference in BWTI is dominated by the different temperatures from shock and compression yields. The DT and DD BWTIs remain the same during shock flash and though temperature separation as in Refs. 13 and 37 is observed during compression burn (757 eV in the gas and 735 eV in the foam matrix), the effect of this separation is minor compared with temporal variations in the BWTI. This contrasts with previous results observed for MARBLE experiments on the Omega laser facility 13 because the Omega experiments did not exhibit preheat-induced pore collapse. In simulations of our NIF HT-filled experiments, pore collapse leads to near complete atomic mixing before thermonuclear (TN) burn occurs and this reduces temperature separation substantially.
High-resolution, three-dimensional simulations are impractical for ICF design, which requires rapid turn-around. One must instead rely on models that treat the mix statistically. In the xRAGE code, this is done by employing the Besnard-Harlow-Rauenzahn (BHR) 3.1 Reynolds-averaged Navier-Stokes (RANS) model for turbulent mixing 38,39 with a probability distribution function (PDF) burn model that accounts for the effects of heterogeneous mix using moments of the probability distribution functions of the concentrations of reacting species. 40

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scitation.org/journal/php reactants (the formula for non-separated reactants is different) 31,41 is modified according to wherec 1 andc 2 are the mass-weighted Favre average concentrations of the mixing materials, q is the Reynolds-average density of the mixed fluid, q 02 is the variance of this density, r ¼ ðq 2 =q 1 Þ À 1, and a and b indicate the reacting species (deuterium and tritium) which are components of the mixing fluids. To account for changes in the mix morphology, the BHR 3.1 mix model evolves the density-specific volume covariance where brackets denote an ensemble average. This quantity encodes the heterogeneity of the medium, 39 making it possible to compute q 02 = q 2 in (2) and modify the thermonuclear reaction rates accordingly.
In a hydrodynamic description of the plasma, assuming the reactants are in thermal equilibrium, dividing the NYR by the expected NYR for homogeneous mix gives a quantity from 0 to 1 known as the mixedness of the medium over the burn interval (see the Appendix B). The mixedness for the ArT and HT experiments is shown in Fig. 6 as a function of the macropore diameter. The data are compared with the mixedness obtained from one-dimensional xRage simulations using the BHR model for turbulent mixing with a PDF burn fusion reactivity. The BHR model was initialized with a turbulent scale length S 0 , a model parameter that we set equal to the macropore diameter d pore times a scaling factor. As shown in Fig. 6, the MARBLE ArT results are in good quantitative agreement with the use of a scaling factor of 0.07, corresponding to an S 0 of size roughly that of the radial size of the macropores after compression by the main shock as it passes through the foam. In contrast, the sub-grid mix model fails to explain the HT inferred mixedness data, even qualitatively, evidencing no reduction of mixedness with the pore size. (If anything, the data show a slight increase.) This is an indication of missing physics from the FIG. 5. Distributions of materials from xRAGE simulations of MARBLE implosion N180729, which had 90 lm pores and an HT gas fill. The shell is red, the HT-filled pores are green, the glue is yellow, the glass is blue, the shock front (upper right panel) is magenta, and the burn volume (lower left, at shock flash time; lower right, at compression burn) is cyan. For scale, the upper right and lower images reside within the white and yellow dashed regions, respectively. modeling or invalid assumptions in computing the mixedness. Specifically, the data are consistent with the hypothesis that preheat in the HT experiments from non-thermal M-band radiation from the Hohlraum and radiation emanating from the shock may have caused vigorous mixing prior to the bang time, obviating the dependence of yield on initial inhomogeneity. By virtue of their higher opacity and higher electron pressure in the macropores, capsules with ArT fills would be much less susceptible to such preheat-induced mixing. The time evolution of the BHR b parameter (the covariance of density and specific volume) from one-dimensional xRage simulations is shown in the top panel of Fig. 7, along with the product of b and density, shown in the bottom panel. These simulation data were obtained at a radius of 40 lm from the center of the capsule for the ArT two-shock design. (This radius was chosen to lie within the burn volume; the results are similar at radii throughout the burn volume.) Here, the BHR model has been initialized with three different turbulent length scales: S 0 ¼ 0:03 (red), 1 (blue), and 7 lm (green). The larger the S 0 , the less complete the mixing at bang time (indicated by the larger values of b), consistent with the inferred and simulated mixedness results shown in Fig. 6.

V. DISCUSSION
MARBLE experiments allow for a detailed study of the effects of mix morphology on thermonuclear burn, a problem crucial to the success of ICF ignition and important in other high energy density physics settings. By varying the macropore size (the degree of initial heterogeneity) and gas fill (the ion mean free paths), MARBLE experiments enable experimental control of the mix morphology as well as of the introduction of effects such as ion temperature nonequilibration and plasma kinetic behavior.
This platform provides valuable data for validating computational models of mix and burn. Indeed, comparisons with mix and burn models in the LANL xRAGE code show quantitative agreement with the ensemble of ArT fill data for a common, fixed ratio of turbulent scale length to macropore diameter. However, the inability of this FIG. 6. Inferred mixedness data (blue squares) for ArT and HT experiments as a function of macropore diameter d pore . Also shown are simulated results with three relations of the turbulent scale length S 0 in the BHR 3.1 mix model to d pore : S 0 ¼ 0:07d pore (solid curve), S 0 ¼ 0:03d pore (dashed), and S 0 ¼ 0:2d pore (dot-dashed). For a uniform atomic mix, the mixedness is unity. The initial mix assumes that the gas within the foam mixes with the foam, but that the gas in the macropores remains separated.

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scitation.org/journal/php model to match the HT fill data raises questions about how well we understand how hydrogenic plasmas mix with carbon (e.g., ablator mix into DT ice in ICF settings). Such a mix affects the compressibility and fuel qR, both long-standing issues in ICF, and an improved understanding of the interplay of mix and burn will therefore have important implications for future ICF fusion experiments on the NIF.

ACKNOWLEDGMENTS
The

Conflict of Interest
The authors have no conflicts of interest to disclose.

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

Synthesis of fully deuterated foam
Deuterium labeled styrene was copolymerized with fully deuterated 1,4-divinylbenzene, synthesized by the following method: First, 1,4-dibromobenzene D4 (deuterated para-dibromobenzene) was subjected to Sonogashira coupling with two equivalents of trimethylsilyl acetylene. The trimethylsilyl groups were then removed with potassium fluoride in D labeled methanol. Finally, a Lindlar type catalyst (Pd nanoparticles on carbon) with bubbled deuterium gas was used to reduce the acetylene groups to vinyl groups. The final product, 1,4 divinylbenzene D10 was obtained in about 70% yield. The level of deuteration was measured using nuclear magnetic resonance (NMR).

Target machining
For experiments at Omega and the NIF, 20-and 120-lmthickness capsules were machined and the engineered joint was sealed with Epon 815C adhesive, chosen for its high adhesive strength. After the sleeves and mandrels were glued together, the assembly was cured for several days at room temperature to prevent adhesive shrinkage. 42 Machining the foam sphere required a new dry machining method developed using turn milling 43 to avoid difficulties with wax fill (e.g., non-uniform shrinkage and incomplete removal of the wax 44 ).

Foam characterization
The control of systematic errors in the MARBLE experiments required accurate foam characterization, 45 including precise measurements of the tritium and deuterium content. The former was obtained from measuring capsule pressure and temperature just before shot time. The latter required pre-shot characterization of the foam density, measured gravimetrically and with x-ray adsorption, and the deuterium fraction, measured by nuclear magnetic resonance. The heterogeneity arises from the macropore structure, so the size distribution and volume fraction of the macropores must be known precisely. This information was obtained using scanning electron microscopy and computed tomography (CT). Impurities, such as silicon left in the foam as a result of incomplete etching of the glass microballoons used to form the macropores, were characterized using confocal x-ray fluorescence. Finally, the pressure and temperature ranges for fielding experiments are affected by the foam's ability to hold excess amounts of hydrogenic species through surface adsorption. Therefore, the specific surface area of the foam must be measured along with its ability to hold hydrogen, characterized through nitrogen gas sorption porosimetry measurements. Previous experiments utilizing a single-shock pulse with fine-pore foam at different temperatures and pressures support the calculations indicating that the effect of surface adsorption was minimal at the conditions used in the experiments reported here. In these experiments, an initial implosion (N170625-002) was fielded at room temperature, followed by an implosion (N170713-004) with the same fill gas density, and then (N180313-001) with twice the fill gas density. The normalized yield ratios for the three were all within 11% of the average, and the room temperature implosion and high gas density shots had normalized yield ratios with 3% of each other.

APPENDIX B: MEASURES OF MIX
Assuming a mixture of D and T, the neutron yield is dependent on the average of the products of the densities of D and T. In general, hn D n T i hn D ihn T i and we can define the mixedness h 2 ½0; 1 such that hn D n T i ¼ hhn D ihn T i. We can measure both the DT and DD neutron yields, given by where V DT and V DD are the burn volumes for DT and DD burn, respectively, and s DT and s DD are the burn times for the same. If we assume that the burn volume and burn time for the two reactions are approximately equal, the ratio of yields is given by For the target parameters used in these experiments, hn 2 D i % hn D i 2 , allowing us to write

Physics of Plasmas
We refer to the term on the left as the normalized yield ratio (NYR). For a uniformly atomic mix, NYR is equal to the reaction rate ratio at a given temperature, and is less by a factor of h for a heterogeneous mix. Dividing the NYR by the reaction rate ratio then provides a measure of the mixedness. The uncertainty in the mixedness is dependent on uncertainties in the measurement of ion temperature (from neutron time-of-flight), the yield for each reaction, the deuterium and tritium densities, and the degree to which the burn volumes and burn times for the two reactions are indeed equal.