Numerical Study of Carbon Nanofoam Targets for Laser-Driven Inertial Fusion Experiments

. Porous materials have peculiar characteristics that are relevant for inertial confnement fusion (ICF). Among them, chemically produced foams are proved to be able to smooth the laser inhomogeneities and to increase the coupling of the laser with the target. Foams realized with other elements and techniques may prove useful as well for ICF applications. In this work, we explore the potential of a novel class of porous materials for ICF, namely, carbon nanofoams produced with the pulsed laser deposition (PLD) technique, by means of hydrodynamic numerical simulations. By comparison with a simulation of solid-density carbon, PLD nanofoams show a higher pressure at the shock front, which could make them potential good candidates as ablators for a capsule for direct-drive fusion.


Introduction
Porous materials, or foams, are the subject of an intense research activity in the context of inertial confnement fusion (ICF) and, more generally, of laser-matter interaction. Foams of low-Z elements have been proposed for use in ICF capsules, to generate bright X-ray sources [1], for the study of equations of state [2], and more recently for efcient electron acceleration [3].
In the context of ICF, foams are considered as constituents of the outer layer of the capsule, because of their ability to smooth the inhomogeneities in the laser energy [4][5][6][7], to enhance the laser absorption efciency [8,9], and to increase the ablation loading on a substrate [10]. Tese features make a foam ablator a potentially important choice to increase the laser-target coupling, thus transferring the laser energy into compression of the inner layers of the capsule.
Most of the experimental data available regard plastic foams, whose internal structure is constituted by features in the micrometric scale, such as flaments and membranes.
Typically, these foams are produced by chemical methods. However, the investigation of the behavior of porous materials constituted by diferent elements and materials is needed to explore other potential applications. Recent studies [11,12] indicate the advantages of using mid-Z ablators, which can reduce the impact of laser plasma instabilities (LPIs), therefore increasing the performance of the target. Among them, high-density carbon (HDC), typically used in indirect-drive experiments, can be benefcial for enhancing the performances of capsules, not only for the indirect-drive scheme but also for the direct-drive approach. In this context, low-density carbon nanofoams could potentially combine the benefts coming from the use of an ablator made of mid-Z elements and from the features of plasma behavior deriving from its random internal structure.
In this work, we present the frst numerical investigation of the behavior of nanostructured carbon foams produced with the pulsed laser deposition (PLD) technique [13][14][15] under the action of a high-power nanosecond-long laser beam, at conditions relevant to ICF. PLD nanofoams have already been studied in the framework of high-intensity laser-matter interaction as a near-critical layer in doublelayer targets [16] for ion acceleration with ultrafast lasers (pulse duration from ∼ 30 fs to 100s fs) [17,18].
In this context, a solid theoretical understanding of the physical processes granting an enhanced acceleration has been developed thanks to particle-in-cell kinetic simulations [19][20][21], and the optimal nanofoam parameters (namely, thickness and density) have been determined as a function of laser intensity [22]. However, there are signifcant diferences in the laser parameters between typical ultrafast laserdriven acceleration and ICF experiments, especially in terms of pulse duration (typically from 30 fs to 100s fs for the former, few ns for the latter) and irradiance (typically 10 18 -10 22 W/cm 2 and 10 14 -10 15 W/cm 2 , respectively), and dedicated studies are thus required. Particle-in-cell codes are not suited to simulate dynamical processes evolving on the longer timescales relevant for ICF (up to hundreds of nanoseconds).
Here, we use the hydrodynamic MULTI-FM code [23], validated with experiments on microstructured plastic foams [24], to simulate the behavior of the PLD carbon nanofoams. Te need for a dedicated code has been confrmed in the last years since modeling the foam as an equivalent medium of the same density in the simulations proved to be unsatisfactory [6,23]. Te MULTI-FM code features a model for reproducing the laser absorption in the foam and the behavior of the laser-generated plasma, by taking into account the randomly arranged internal structure of the material.
We consider a set of diferent nanofoam parameters, which depend on the manufacturing process used and in particular on the time duration of the laser employed. From the simulations, we fnd that the shockwave in all the foams is signifcantly slower than in a homogeneous medium with the same density as the nanofoam, as expected from the studies with the plastic foams [24,25]. Moreover, we fnd the indication that the pressure at the front of the shockwave is signifcantly larger than the one obtained in the equivalent homogeneous medium.

Materials.
Te carbon nanofoam presented in this work is produced by pulsed laser deposition (PLD), a physical vapour deposition technique that employs a pulsed laser to remove material from a target and deposit it on a substrate in a controlled gas atmosphere. If the background gas pressure is sufciently high with respect to the energy of the ablated species, they can be slowed to a difusive regime and aggregate while in fight, giving rise to porous, nanostructured, low-density flms. In the case of carbon, what can be obtained are carbon nanofoams, nanoparticle-assembled flms with a fractal-like morphology, whose basic constituents are nanoparticles with dimensions in the order of 10 nm and densities as low as a few mg/cm 3 .
Two diferent deposition systems are employed, one exploiting a nanosecond duration laser (ns-PLD, conventional technique) and the other an ultrashort femtosecond laser (fs-PLD). Te frst is a Q-switched Nd : YAG laser (second harmonic λ � 532 nm), with 5-7 ns pulse duration, 1 J maximum pulse energy, 10 Hz repetition rate; the second is a Ti : sapphire-based CPA laser (Coherent Astrella, λ � 800 nm), ∼ 100 fs pulse duration, 5 mJ maximum pulse energy, 1 kHz repetition rate. As shown in previous work [15], the diferent laser pulse duration leads to very diferent ablation regimes and enables better control of the foam properties. Notably, carbon nanofoams with the same density and diferent spatial uniformity (i.e., diferent microscale homogeneity) can be obtained.
Te PLD target is pyrolytic graphite, the laser incidence angle is 45°, and the distance between target and substrate is fxed at 7 cm. We set the fuence to 360 mJ/cm 2 for both techniques, obtained with 8 mm spot size (top hat) and 260 mJ for ns-PLD and 0.8 mm spot size (FWHM) and 2.6 mJ for fs-PLD. By suitably tuning the vacuum chamber pressure in the range from 50 to 200 Pa (argon is used as an inert gas), carbon nanofoams of diferent densities are obtained. Combining the efect of the pressure with the efect of the diferent techniques (ns-PLD and fs-PLD), four kinds of representative carbon nanofoam are produced, as shown in Table 1.
Te nanofoam morphology is characterized with a feld emission scanning electron microscope (SEM, Zeiss Supra 40) with 3-10 kV accelerating voltage. Te microscope electron is also exploited to perform EDXS (energydispersive X-rays spectroscopy) on the foam samples and to obtain information about the elemental composition and average density. Tis is done through the EDDIE method [26]: a theoretical model of the electron transport in the flm and substrate allows to retrieve the flm mass thickness starting from the ratio between the respective X-rays line intensities. Te nanofoam thickness is measured from the SEM cross-sectional images, and the density is obtained as their ratio.
Te determination of the nanofoam pore size δ 0 , one of the parameters needed for the foam model implemented in the MULTI-FM code, is not straightforward: contrary to most of the chemically produced foams, pulsed laser-deposited nanofoams are cluster-assembled and fractal, with disordered voids without a clear shape. One possibility is to estimate the average pore size with the characteristic uniformity scale length of the nanofoam, that is, the average distance between two adjacent high-density regions of the foam (or equivalently the distance between two less-dense regions, i.e., the void dimension). Te EDDIE method can be used point-bypoint on the samples to build mass thickness maps of the foams, and by performing a Fourier transform analysis on the maps, the nanofoam uniformity scale length is obtained. In principle, it would be possible to perform the Fourier transform analysis directly on the SEM images, but they would be afected by electronic efects and arbitrary postprocessing (i.e., contrast and brightness), while the mass thickness maps cleanly convey the useful physical information. Te structure factor S(q) is the radius-averaged squared Fourier transform of the mass thickness image, and since the spatial frequency q max -corresponding to the maximum value of S(q)-is the predominant spatial frequency in the image, its inverse (1/q max ) is the uniformity scale length [15], and that value is used as an estimate for δ 0 .
Four distinct categories of carbon nanofoams are produced and investigated in this work, as reported in Table 1 Figure 2 shows the respective EDXS mass thickness maps, highlighting the same nanofoam microstructure. From the Fourier transform image in the insets, the ring at q max can be appreciated (note the wider scale for the Figure 1(c) inset).
Since carbon nanofoams grow through the random stacking of aggregates which are fractal in nature, it is possible to relate the density of the nanofoams to the characteristics of the fractal aggregates [15]: where ρ p and ρ s are, respectively, the foam and the solid element (nanoparticle) density and k is a proportionality factor related to the packing of the aggregates in the tridimensional foam structure. d np , R g , and D f are the fractal aggregates properties: nanoparticle diameter, gyration radius, and fractal dimension. Since all the carbon nanofoams considered in this work have a fractal dimension D f ≈ 2 [15], Te foam model implemented in the MULTI-FM code is still based on a fractal description but expresses the ρ p over ρ s ratio as a function of diferent parameters: the thickness of the solid elements b 0 , the pore size δ 0 , and the fractal parameter α.
By comparing the two equations, one could seek a way to express each term of (3) as a function of the parameters of the other, namely, d np , 2R g , and D f , since the left-hand term is the same in both equations. Given the number of parameters and their possible combinations, a complete term-by-term correspondence is not univocal. A very straightforward relationship between α and D f can be derived, as they are both a measure of the fractality of the system, and it is reasonable to assume they should be independent of the other parameters. Under this hypothesis, equating the exponents of the two equations leads to α � (3 − D f ) − 1 ≈ 1. Ten, since the pore dimension δ 0 is obtained from the Fourier analysis of the mass thickness maps, the only remaining parameter b 0 can be calculated by inverting (3) 3 is taken as solid element density ρ s , coherent with the disordered sp 2 structure of the carbon nanoparticles composing the nanofoam [13,14,27]. Te main parameters for the diferent carbon nanofoams considered in this study (Series A, B, C, and D) are shown in Table 1: their density, pore dimension δ 0 , and solid element dimension b 0 , along with the respective production technique (ns-or fs-PLD). It can be noted that the value of the solid element thickness b 0 , in the order of some tens of nanometers, is coherent with the foam microstructure as seen in high magnifcation SEM images. Since the nanoparticle diameter is around 10 nm, the carbon nanofoam solid element can be identifed with the strands of connected nanoparticles that make up the short-range structure of the nanofoams, which delimit the voids-of dimension δ 0 -from one another. Terefore, the foam model implemented in MULTI-FM is representative of the nanofoam microstructure, while it is not sensitive to the short-range nanostructure (namely, the nanoparticles composing the foam solid element).

MULTI-FM Simulations.
We simulated the behavior of the laser-generated plasma with the 1D MULTI-FM code [23]. Te MULTI-FM code is a modifcation of the MULTI code [28] specifcally developed for reproducing the interaction of powerful laser pulses with porous materials. It is based on an efective model for laser absorption accounting for the efect of solid elements and empty spaces on the plasma behavior. In the model, the pore size δ 0 , the thickness of the solid elements b 0 , the average density of the foam ρ p , and the density of the solid parts ρ s are related as in equation (3). In the code, the homogenization status of the plasma is determined through a parameter called ISFOAM, defned for each numerical cell at each time step as In this equation, Laser and Particle Beams 3   Table 1 for the (a), (b), (c), and (d) samples. Te images can be directly compared since the magnifcation is the same in all cases.

Laser and Particle Beams
where τ H is the characteristic homogenization time given as follows: where Z is the efective ionization, A is the mass number, and T is the temperature of the plasma. In the last step, equation (3) was substituted in place of b 0 to better illustrate the dependence of the homogenization time on pore dimension and density. Due to the low density of the nanofoams, (ρ p /ρ s ) ≪ 1, therefore, τ H ∝ δ 2 0 ρ p . When the homogenization is reached, As the plasma is heated and homogenized, the parameter ISFOAM changes from 1 for the cold, not yet irradiated foam, to 0, when the plasma in the cell is completely homogenized. In the model, the thermal conduction is limited by the same parameter ISFOAM, to account for the inhomogeneous distribution of free electrons in the pores during homogenization as follows: where q(x, t) is the heat fux, χ(x, t) is the Spitzer conductivity, and T e (x, t) is the electron temperature. In the same fashion, the force acting on the mesh nodes is limited by the homogenization as where v(x, t) is the node velocity, P(x, t) is the total pressure, and m is the Lagrangian coordinates used in the calculations in the code. Te MULTI-FM code has been validated with an experimental campaign carried out at the ABC laser facility [24].
Here, the laser parameters are taken as the ones of the ABC laser facility, in view of an experimental campaign to be conducted there in the near future. Te laser pulse has a sin 2 temporal profle with a full-width half maximum (FWHM) τ L � 3 ns, at the fundamental wavelength λ L � 1054 nm, with an intensity on target of 10 14 W/cm 2 . Te parameters for the carbon nanofoam samples used in the simulations are the ones indicated in Table 1. In all the simulations, a tabulated equation of state from the SESAME library has been used. Te carbon ions are considered to be fully ionized, Z eff � 6. For all the simulations, the thickness of the sample has been taken as 300 μm, the largest possible to obtain by PLD in a reasonable production time.

Results
We report a series of simulations performed with the MULTI-FM code. Te foam parameters have been chosen as the ones reported in Table 1. We will refer to each simulated nanofoam sample with the corresponding series name reported in that table. As explained in the Methods section, the laser wavelength in the simulations is taken as λ L � 1054 nm. Te corresponding critical mass density is, therefore, 3.3 mg/ cm 3 , which means that all the foams we are considering are overcritical. Figure 3 shows the results of the simulations for Series A and B, having both the density of 6 mg/cm 3 and the pore size of 10 and 20 μm, respectively. Te shockwave in the nanofoam is slower than in the homogeneous medium. Tis behavior is well known and theoretically and experimentally documented in the literature for plastic foams [5,6,[23][24][25]29]. It is due to the time required for the lasergenerated plasma to homogenize and fll the empty spaces in the foam. Te lack of free electrons and the inhomogeneity of the plasma lead to a reduced heat conduction and hydrodynamic motion, until the plasma is homogenized. At that point, the plasma behaves as an ordinary homogeneous plasma.
Te characteristic time of homogenization is afected by the pore size δ 0 as in (6). Because of this dependence, the homogenization time is increased in the case of the nanofoams of Series B, compared to the ones of Series A, and the shockwave is slower in the former than in the latter.
Te plots on the right column show that the pressure of the plasma attained in the foam is noticeably higher than that one obtained in the homogeneous medium of the same density. As analytically demonstrated in Ref. [7], an overcritical foam is more efcient than a solid homogeneous material in converting the laser energy into ablation loading. Te lower thermal conduction and the inhibited hydrodynamic motion of the cold part of the foam lower the fraction of the laser energy absorbed into the plasma converted into compression or heating of the material. Terefore, a large fraction of this energy is left in the plasma as thermal energy, increasing the pressure in the foam. Tis property of the foam plasma can be relevant for ICF because it will increase the efciency of compression on the frst inner layer in the fusion target. Tis feature of foams will be benefcial for ICF, especially considering also the other properties of foam materials, the most important being the high absorption efciency of the laser energy, which can be as high as 90%. Figure 4 shows the results of a simulation of a soliddensity carbon sample obtained with the MULTI code and with the same equation of state used for the foams, and the density is equal to 2.0 g/cm 3 Table 1. Te homogeneous material has the same density as the average density of the A and B. Te laser comes from the right. in the nanofoam Series A exceeds the pressure obtained in the diamond sample. Tis efect is even more evident in the case of the nanofoam Series B, where on average it exceeds 20 Mbar. Figure 5 shows the results of the simulations performed with the nanofoams Series C and D. Te oscillations which can be seen in the plots of the density and pressure, in the plasma behind the shock front, are due to the interplay between the homogenization of the plasma and the propagation of the shockwave during the simulation. Tey refect the inhomogeneous nature of the plasma realized inside the foam, where some inhomogeneities on large scales are known to remain long after the homogenization time [30].
In nanofoams Series C and D, the shockwave in the foam is slower than in the case of Figure 3. Te pore size is smaller, leading to a reduction in the homogenization time, but, at the same time, the density is increased, with the opposite efect (see (6)). Te homogenization time is thus mostly unchanged, and the slower shockwave propagation can be attributed to the higher density of the material. Te pressure at the shock front in the nanofoam Series C exceeds 20 Mbar on average, an efect even more noticeable in Series D, where it exceeds 30 Mbar.
Tese results show how a foam can improve the ablation loading, by giving a higher pressure at the front of the shock wave compared with a homogeneous medium. Te foams show in all cases a frst narrow peak of the pressure at the shock front, followed by a thicker zone of pressure which has generally the same value attained in the homogeneous material. Tis behavior is related to the limiters in the model, which act on the thermal conductivity and on the compression of the material, reducing both efects (see (8) and (9) in Ref. [23]). Te fraction of the plasma energy transformed into the motion of the fuid, and the one transported by heat conduction is reduced in comparison with a homogeneous medium, leading to an increase in the pressure at the front of the shock, which travels at a lower speed.
Te simulations can also give an estimate of the most efective choice for the thickness of the sample, to maximize the pressure at the shock front when it exits the ablator. While for Series A, the highest pressure is attained at the end of the laser pulse, in Series B, the peak pressure is reached at 1.7 ns, before the peak power of the laser. By looking at Series C and D, we see that the peak pressure is always reached at about half of the laser time, i.e., when the laser is at its maximum power. Terefore, by properly choosing the  Table 1. Te homogeneous material has the same density as the average density of the C and D nanofoams. Te laser comes from the right.
thickness of a carbon nanofoam ablator, depending on the specifc laser temporal profle, one can increase the pressure at the shock front and maximize the ablation loading, analogously to what has been observed in Ref. [7] for plastic foams. When considering carbon nanofoams of 18 mg/cm 3 , for both Series C and D, a thickness of the sample of about 150 μm would be the optimal to improve the compression efciency from the ablator. Samples with this thickness can easily be obtained with the PLD technique. We will perform dedicated experiments on the ABC facility in the near future to characterize the laser-generated plasma with these nanofoams and to test the results obtained with the MULTI-FM code described in this work.

Conclusion
In this work, we reported a frst study on the use of carbon nanofoams obtained with the pulsed laser deposition technique as ablators for ICF experiments. We considered four kinds of samples obtained through the use of nanosecond and femtosecond pulsed lasers, reported in Table 1.
Taking advantage of the fexibility provided by the two diferent PLD techniques, two pairs of carbon nanofoams samples with the same average density but diferent microscale structures (and thus pore size) have been produced: samples A and B with a density of 6 mg/cm 3 and C and D of 18 mg/cm 3 . Samples A and C, produced with ns-PLD, have smaller pore size, while larger pore size is obtained for samples B and D, produced with fs-PLD. Te nanofoams are characterized in terms of morphology, and a connection between the fractal nanofoam model [15] and the foam description implemented in the MULTI-FM code is proposed. We investigated the behavior of these samples under irradiation of a high-power laser by numerical simulations performed with the MULTI-FM code. In particular, we considered the laser parameters as one of the ABC laser facilities at ENEA Centro Ricerche Frascati. Te simulations showed that the shockwave into the nanofoam propagates sensibly slower than in a homogeneous medium of the same average density because of the internal structure of the foam. Tis efect is enhanced in the case of the nanofoams of Series C and D with an average density of 18 mg/cm 3 due to the larger average density compared to Series A and B, having an average density of 6 mg/cm 3 . In all the nanofoams, the pressure at the front of the shockwave is larger than the one obtained in the equivalent homogeneous medium and also higher than the one obtained in a compact carbon sample, whose behavior has been simulated with the same code and equation of state. Tese frst simulations on compact carbon, while limited in scope, are preparatory for future development toward the study of HDC ablators, and the implementation of the proper equation of state into the MULTI-FM code is foreseen. Tis efect can be of use for realizing an ablator layer for a fusion capsule, increasing the ablation loading on a subsequent layer, thus increasing the compression efciency. Tis can be important for directdrive inertial fusion with carbon-based ablators, such as HDC ablators. Teoretical works showed that mid-Z ablators have a better overall performance than plastic ablators, also reducing the LPI detrimental efects [12]. Te use of carbon nanofoams could improve the performances of fusion targets, combining the benefts that can be obtained from a mid-Z ablator with the ones that come from the use of a porous material, such as the reduction of laser imprint and the smoothing of the laser energy on the capsule. A potential issue related to this fnding could reside in the ability of the model implemented in the MULTI-FM code, which presently does not include the possible efects deriving from the interaction of the laser with the peculiar nanoscale structure of the material. Dedicated experiments are foreseen in the near future to confrm the results of this work, through an experimental campaign to be held at the ABC laser facility.

Data Availability
Te data that support the fndings of this study are available from the corresponding authors upon reasonable request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.