High energy density physics (HEDP) and inertial confinement fusion (ICF) research typically relies on computational modeling using radiation-hydrodynamics codes in order to design experiments and understand their results. These tools, in turn, rely on numerous charged particle transport and relaxation coefficients to account for laser energy absorption, viscous dissipation, mass transport, thermal conduction, electrical conduction, non-local ion (including charged fusion product) transport, non-local electron transport, magnetohydrodynamics, multi-ion-species thermalization, and electron-ion equilibration. In many situations, these coefficients couple to other physics, such as imposed or self-generated magnetic fields. Furthermore, how these coefficients combine are sensitive to plasma conditions as well as how materials are distributed within a computational cell. Uncertainties in these coefficients and how they couple to other physics could explain many of the discrepancies between simulation predictions and experimental results that persist in even the most detailed calculations. This paper reviews the challenges faced by radiation-hydrodynamics in predicting the results of HEDP and ICF experiments with regard to these and other physics models typically included in simulation codes.

High energy density physics (HEDP) experiments pursue an understanding of the physics underlying inertial confinement fusion1 as well as fundamental plasma physics and astrophysics.2 Inertial confinement fusion (ICF) involves the laser-, x-ray-, or magnetically driven implosion of deuterium and tritium (DT) fuel to conditions where the temperatures and densities are high enough to produce DT fusion in which the alpha particle (a He nucleus) products deposit their energy and amplify the thermonuclear reaction rates. If such conditions are maintained for long enough, it is possible to produce gain3,4—i.e., the energy of fusion products escaping the system exceeds the driver energy. These experiments involve a multitude of coupled physics and the results typically require computational modeling to interpret.

Laser indirect-drive (LID) experiments, in which lasers illuminate the inside of a high-Z cylindrical hohlraum to produce a uniform x-ray drive for capsule implosions, are representative of the complexity in these experiments. Many of the modeling challenges are shared by all approaches, so we will focus on these experiments, though we will also discuss challenges specific to magnetic direct drive (MDD) and laser direct drive (LDD). MDD involves the use of high currents, typically produced by a pulsed power driver, through cylindrical targets, generating high magnetic fields that compress the target due to J× B forces (where J is the current density and B is the magnetic field strength). LDD involves the direct illumination of targets by lasers, causing them to rapidly heat and ablate, generating an inward rocket-like implosion.

The LID approach to ICF is illustrated in Fig. 1. Laser beams illuminate the inside of a high-Z (typically gold or depleted uranium) cylinder called a hohlraum and rapidly heat it to high temperatures, causing it to emit x-rays. The hohlraum wall is an efficient reflector of x-rays, so that the x-ray flux is fairly uniform throughout the cavity. X-rays incident on the capsule are absorbed, bringing the shell material to high pressures that drive mass ablation and a rocket-like inward acceleration of the capsule. Shocks traversing the capsule and subsequent compression heat the DT inside the capsule to several KeV, initiating thermonuclear fusion reactions. Each DT fusion reaction produces a 14.1 MeV neutron as well as a 3.5 MeV charged alpha particle, which may be stopped in the surrounding plasma, heating it further. As the pressure in the DT increases, the shell disassembles and the system cools, quenching the DT burn.

FIG. 1.

Diagram of the laser indirect drive approach to inertial confinement fusion.

FIG. 1.

Diagram of the laser indirect drive approach to inertial confinement fusion.

Close modal

The fidelity at which such experiments can be simulated is determined by the amount of physics modeled in the code and its accuracy, the quality of the initial conditions and physical data used in simulation, and the adherence to validated modeling practices. State-of-the-art modeling of the hohlraum and implosion physics currently employs highly complex radiation-hydrodynamics codes such as HYDRA,5 Lasnex,6 xRAGE,7–10 and TROLL.11 Similar tools have been developed for LDD such as DRACO,12,13 LILAC,14 ASTER,15,16 FAST2D,17 SARA,18 DUED,19 Chimera,20 and CHIC,21 as well as for MDD,22 such as ALEGRA,23 Ares,24–27 GORGON/Kraken,28,29 and PERSEUS.30 The publicly available FLASH code31–33 also offers similar capabilities that enable the simulation of high energy density physics experiments. There is significant overlap between the physics modeling capabilities of these codes, such that some of these codes can be used for multiple approaches. At a minimum, these codes model hydrodynamics or magnetohydrodynamics (MHD) coupled to thermal radiation transport, equation of state (with separate ion and electron temperatures), and electronic and ionic thermal conduction. When modeling ICF implosions, these additionally require thermonuclear burn coupled to charged fusion product transport. For laser-driven problems, these codes also model laser propagation and energy deposition. For pulsed-power driven implosions, the ability to model circuits coupled to the MHD is required. Each of these computational models relies on physical data in the form of coefficients that must be tabulated or evaluated analytically. Significant effort is expended in pursuit of improved coefficients that might enable more predictive modeling of experiments. The goal of this paper is to catalog the extant evidence for why improvements to specific transport coefficients or to the methodologies used for mixing transport coefficients could improve this predictive capability.

The ability of simulations to reproduce experimental results is in some cases remarkable,34–37 yet radiation-hydrodynamics simulations still fail to predict the integrated behavior of capsules and hohlraums,38,39 struggle to model target energy coupling in MDD implosions,40–42 struggle to model energy coupling and laser imprint and suprathermal electron generation, particularly at high laser intensity, in LDD,43–47 and are sensitive to inherently three-dimensional features that are too small to model routinely.48–50 Despite a wide body of evidence demonstrating the necessity of performing simulation in three dimensions,34–36,51–63 such simulations remain a rarity due to the computational expense required. Furthermore, state-of-the-art simulations frequently use analytic formulas for coefficients like thermal conduction and electron-ion coupling when significant theoretical advances have been made to improve our understanding of them. This is typically done because the most accurate models require tabulating coefficients for conditions that could be encountered in simulations and such tables have not been produced, because using them would require prohibitive memory consumption, or because it has not been demonstrated that the improvement would have a meaningful impact for the problems the code is used to simulate. When simulations are able to capture capsule performance, accurate modeling based on high fidelity initial conditions is critical. Remaining discrepancies could be caused by unmodeled physics, physical data, or inadequate target characterization. It is therefore critical to ensure that high quality physical data are available with quantifiable uncertainties. At conditions of interest to HEDP and ICF experiments, comparisons of state-of-the-art first-principals models for calculating these coefficients have demonstrated differences of roughly a factor of five.64 

We focus on uncertainties in charged particle transport coefficients—i.e., coefficients related to the transport of electrons or ions through plasmas. These coefficients are used in radiation-hydrodynamic codes to calculate laser absorption, viscous dissipation and mass transport, thermal conduction, charged fusion product transport, equilibration between species, and electron–ion equilibration. We have opted to be inclusive in this list, including relaxation properties, since these utilize similar methodologies for computation and the relevant challenges are common. In many cases, we discuss coefficients for which improved models have already been developed because they are still widely used in-state-of-the-art simulations. While it is well known that there are important uncertainties in other physical data used in radiation-hydrodynamics modeling, including ionization,65–67 opacities,68 and equation of state,69–71 we focus on uncertainties in charged particle transport coefficients in this paper. There are three reasons to anticipate that improvements to the fidelity of these coefficients would benefit the quality of our computational modeling.

First, in the process of ICF implosions and HEDP experiments, the materials typically transit the warm dense matter (WDM) regime for a majority of the implosion/experiment (see, e.g., Ref. 72) as demonstrated in Fig. 2 for a typical ICF implosion. The WDM regime is roughly described in terms of the plasma ionic coupling parameter Γ and the degeneracy parameter Θ,
Γ = q e 2 4 π ϵ 0 k B T e ( 4 π n e 3 ) 1 / 3 1 , Θ = k B T e E F 1
(1)
(see, e.g., Refs. 73 and 74), where ne is the electron density, ϵ0 is the permittivity of free space, kB is the Boltzmann constant, qe is the electron charge, Te is the electron temperature, and EF is the Fermi energy. It is a regime where temperatures are hot but quantum effects are still important. According to the 2009 Fusion Energy Science Advisory Committee report,75 it is, “an intermediate state between condensed matter (solids and liquids), gases, and ideal plasmas. It exists in the lower-temperature portion of the HED regime, under conditions where the assumptions of both condensed-matter theory and ideal-plasma theory break down, and where quantum mechanics, particle correlations, and electric forces are all important.” In terms of physical observables, it spans temperatures of roughly 1–100 eV and densities of roughly 0.01–100 g/cm3.76 This regime is difficult to probe experimentally and is in some cases beyond the range of validity of analytic coefficients—and the few experiments that have been done that measure transport coefficients (e.g., Ref. 77) suggest the need for more sophisticated models to calculate transport coefficients. In the WDM regime, condensed matter physics, superdense matter physics, ideal plasma physics, and dense chemistry all compete to determine the behavior of the system.
FIG. 2.

Phase space trajectory of materials in the course of an ICF implosion. This figure is reproduced from Hu et al., Phys. Plasmas 25, 056306 (2018),72 with the permission of AIP publishing.

FIG. 2.

Phase space trajectory of materials in the course of an ICF implosion. This figure is reproduced from Hu et al., Phys. Plasmas 25, 056306 (2018),72 with the permission of AIP publishing.

Close modal
Second, computational modeling involves resolving physics at a grid scale that is often set by computational expediency rather than physics considerations or numerical convergence. In many cases, the length-scale where the continuum approximation breaks down varies greatly across the computational domain and complex hydrodynamics results in material compositions that also vary greatly across the domain. Physical data are typically only tabulated for single materials or the most common mixtures and the coefficients may not combine using simple averaging rules. The way that coefficients combine is impacted by geometric considerations, the way that collision rates change in the presence of multiple species, and the local plasma conditions. Furthermore, the distribution of materials may also cause an isotropic coefficient to behave anisotropically (consider thermal conduction through parallel bands of low and high conductivity materials). The geometric impacts are illustrated for heat conduction in Fig. 3 for two materials with the same density and volume and uniform conductivities κ1 and κ2. Analogous to combining resistors in simple circuits, when the materials are in series, the conductivities combine using harmonic averaging. When the materials are in parallel, the conductivities are averaged. In a random checkerboard arrangement, the expected value of the conductivity over all realizations is given by the intermediate result κ 1 κ 2.78 On the other hand, if the materials are mixed atomically, taking into account plasma collision rates between species gives more complicated mixing rule (see, e.g., Ref. 79),
{ κ n i , 1 Z 1 + n i , 2 Z 2 n i , 1 Z 1 2 + n i , 2 Z 2 2 T e 5 / 2 ln Λ e i , κ l T e 5 / 2 Z l ln Λ e i , l = 1 , 2 ,
(2)
where n i , l is the ion density, Zl is the charge state of species l, ln Λ e i is the Coulomb logarithm, which quantifies the relative impact of small-angle collisions to large-angle collisions as a particle traverses a plasma. Integrated radiation-hydrodynamics systems typically show at least as much sensitivity to the mixing methodology employed to combine coefficients as to the source of the single-material coefficients employed. Methods for accounting for geometric variations in coefficients have been developed for radiation transport80,81 and reactive flows,82–91 but such methodologies have not been adopted for charged particle transport coefficients. Such approaches, however, require modifications to computational algorithms in addition to a detailed understanding of how coefficients combine in different geometric configurations.
FIG. 3.

The geometric arrangement of materials impacts how transport coefficients should be combined. Here, we illustrate the impact on heat conduction for two materials with the same density and volume and uniform conductivities κ1 and κ2. Here, we assume the heat flux is from left to right. Three limiting cases arise: (a) when the materials are in series, the conductivities combine using harmonic averaging; (b) when the materials are in parallel, the conductivities are averaged; (c) in a random checkerboard arrangement, the expected value of the conductivity over all realizations is given by κ 1 κ 2.78 

FIG. 3.

The geometric arrangement of materials impacts how transport coefficients should be combined. Here, we illustrate the impact on heat conduction for two materials with the same density and volume and uniform conductivities κ1 and κ2. Here, we assume the heat flux is from left to right. Three limiting cases arise: (a) when the materials are in series, the conductivities combine using harmonic averaging; (b) when the materials are in parallel, the conductivities are averaged; (c) in a random checkerboard arrangement, the expected value of the conductivity over all realizations is given by κ 1 κ 2.78 

Close modal
Finally, coefficients exhibit coupling to additional physics. It is well known, for example, that thermal conduction perpendicular to a magnetic field is inhibited according to79,92
κ κ 1 1 + χ 2 ,
(3)
where κ is the conductivity perpendicular to the magnetic field, κ is the uninhibited conductivity parallel to the magnetic field, the Hall parameter χ : = Ω e τ e i, the electron cyclotron frequency Ω e : = e B / ( m e c ), and τei is the electron–ion collision time. This makes it difficult to disentangle the effects of unmodeled or inaccurately coupled physics from inaccuracies in transport coefficients. We will focus on coupling of transport physics to magnetic fields, but other coupling could be important—for example, in-flight charged particles will modify the opacity and equation of state of the background material in non-equilibrium ways that have not been studied in detail.
The challenges among charged particle transport coefficients are related because they rely on similar basic parameters. Particle interactions are mediated by a pair potential, which describes a screened Coulomb interaction. Commonly, the Yukawa potential is used, which is parametrized here by λ DH, the Debye–Hückel screening length,
{ Φ j k ( r ) = Z j Z k q e 2 r e r / λ DH , λ DH = k B 4 π q e 2 ( j n i , j Z j 2 / T i , j + j n i , j Z j / T e , j ) .
(4)
Here, T i , j is the ion temperature for species j and n i , j is the ion density for species j. Transport coefficients are sensitive to the choices made in how these parameters are calculated and, particularly for the warm dense matter regime, how quantum effects are incorporated.93 The calculations of transport coefficients rely on collision integrals Ω j k n , m, cross sections σ j k n, and scattering angles θjk (see, e.g., Ref. 94) calculated as
{ Ω j k n , m ( T ) = k B T 2 π m j k 0 V 2 m + 3 exp ( V 2 ) σ j k n ( V ) d V , σ j k n = 2 π 0 b { 1 cos n [ σ j k ( b , v ) ] } d b , θ j k = π 2 b r 0 d r r 2 1 ( b / r ) 2 2 Φ j k / ( m j k v 2 ) ,
(5)
where mjk is the reduced mass. From the Yukawa pair potential, one can calculate the binary mass diffusivity Djk, dynamic viscosity μj, and thermal conductivity κj,
D j k = 3 k B T 16 n i m j k Ω j k ( 1 , 1 ) , μ j = 5 k B T 8 Ω j j ( 2 , 2 ) , κ j = 25 c v , j k B T 16 Ω j j ( 2 , 2 ) .
(6)
These formulas can be generalized to binary mixtures (see Ref. 94) though extension to generalized multi-species plasmas is an open challenge.95 The integrals in Eq. (5) cannot be evaluated analytically for most cases, so differences in coefficients ultimately derive from the choice of screening length λ, the ionization model (which introduces uncertainties inherited from atomic physics), what methods are used to integrate quantum effects, and what approximations are used to evaluate Ω j k ( n , m ). Since the methodology used to calculate these coefficients is similar, the uncertainties in their evaluation and the challenges in applying them to the modeling of physical systems are related. Analytic models are the simplest to implement and can be used for any material, though their range of validity is the most restrictive. Molecular dynamics (MD) calculations provide a wider range of validity, but using them in radiation-hydrodynamics simulations requires a separate table for every species or the development of a fit to an analytic model. Due to the wide number of materials used and wide range of states accessed in ICF simulations, this requires a large amount of computation to build the tables and using tables for many coefficients can be prohibitive in terms of memory usage. Ideally, tables would be constrained by experimental data, but there are very few measurements of coefficient values available in the WDM regime. The quality of a table will depend on how well the MD simulations used to produce it cover the region of phase space reached in a given simulation, how well the MD simulations recover the hot, low density plasma and low temperature limits, and how well the WDM physics is captured in between. Both approaches struggle to account for how coefficients change in the presence of multiple species or due to variations in the distribution of species.

A caveat is warranted for the reader. Most of the sensitivities observed in radiation-hydrodynamics modeling derive from repeating simulations and varying the available coefficient data and models used or varying the coefficient mixing rule among implemented options. Therefore, if there is a systematic error across the coefficients or mixing paradigms that are available, an important sensitivity could be missed. Furthermore, it is difficult to disentangle uncertainties in transport coefficients from uncertainties in physics implementation details and coupling to other transport coefficients. For example, errors in heat fluxes could arise from conduction coefficients, inadequate accounting for geometric effects relating to material distributions and mixing, unmodeled self-generated magnetic fields, inaccuracies in the magneto-hydrodynamic transport coefficients used to determine self-generated magnetic field strengths, inaccurate coupling between magnetic fields and heat fluxes, inadequacies in the modeling of non-local transport effects, etc. For this reason, throughout the manuscript, we discuss physics sensitivities and discrepancies that could potentially be resolved by improved transport coefficients rather than focusing on known uncertainties in the coefficients themselves.

In the following sections, we will focus on different physics that are modeled in radiation-hydrodynamics codes for ICF and HEDP that rely on charged particle transport coefficients. In each section, we will discuss the impact of the coefficients, sensitivities to coefficient values, and specific challenges. We note that some liberty must be taken to determine what counts as a charged particle transport coefficient, since nearly any coefficient applied to a plasma involves the transport of a charged particle. While we tried to be as inclusive as possible both in the coefficients considered and the challenges discussed, neither is exhaustive. Furthermore, many of the challenges, such as mixing methodologies, apply to all or many of the coefficients, but are only discussed where we are aware of detailed studies of the impacts. In Sec. II, we discuss laser propagation and absorption. Next, in Sec. III, we discuss viscosity and mass transport. We then discuss thermal conductivity in Sec. IV, non-local ion (including charged fusion product) transport in Sec. V, non-local conductivity in Sec. VI, and electrical conduction in Sec. VII. In Sec. VIII, we focus on modifications to transport coefficients due to magnetic fields. Finally, we discuss multi-species thermalization in Sec. IX and electron-ion coupling in Sec. X. We present our conclusions in Sec. XI.

Radiation-hydrodynamics codes rely on laser ray-tracing algorithms to accurately account for the deposition of laser energy into targets.9,43,96–100 Lasers propagate according to geometric optics with the refractive index n refr : = 1 n e / n e , crit, where ne is the electron density and n e , crit : = π m e c 2 e 2 λ 2 is the critical electron density (where the plasma frequency equals the laser frequency), where me is the electron mass, c is the speed of light, and λ is the laser wavelength. At laser wavelengths and intensities relevant to modern laser facilities, lasers deposit energy primarily via inverse bremsstrahlung (IBS), a process in which an electron absorbs energy from an electromagnetic wave and then collides with an ion. The IBS absorption E(s) along a ray path is calculated by numerically integrating96,97,101
E ( s ) = E 0 exp ( d τ IBS ) , d τ IBS d s = 4 3 2 π m e e 4 c k B 3 / 2 n e , crit Z 2 ¯ Z ¯ ln Λ n e 2 T e 3 / 2 n refr ,
(7)
where ln Λ is the Coulomb logarithm, Te is the electron temperature, kB is the Boltzmann constant, and Z is the number of free electrons per atom. The Coulomb logarithm quantifies the relative impact of small-angle collisions to large-angle collisions as a particle traverses a plasma. It has been known for a long time that IBS deposition is sensitive to the form chosen for the Coulomb logarithm,102 and this remains an active area of research.103,104 A popular choice is to use the formula from Lee and More,105 but recently it has been shown that the Coulomb logarithm for IBS needs to account for the laser frequency (it is common, as when using the Lee and More variant, to use the plasma frequency instead).104 In addition, non-Maxwellian electron distributions have been observed to decrease IBS deposition. For example, the Langdon effect106,107 involves the modification of IBS absorption rates due to non-Maxwellian electron distributions that are formed when laser heating time scales are shorter than electron–electron collision time-scales. This effect needs to be accounted for in order to explain experimentally measured absorption. This is shown in Fig. 4. The Langdon effect can also impact cross-beam energy transfer.108,109 This has the effect of reducing the laser absorption and also serves as a source of non-thermal electrons. Nevertheless, current models for the Langdon effect only account for IBS as a driver of non-Maxwellian distributions and neglect the non-thermal electron production. In general, large temperature gradients, other heating sources, and transport phenomena can induce non-Maxwellian distributions, and their impact on laser absorption rates is not accounted for. It is not currently clear how these might affect simulation results.
FIG. 4.

Ratios of simulated to measured absorption for OMEGA experiments to measure IBS absorption: (a) without Langdon factor and without screening; (b) with Langdon and without screening; and (c) accounting for both the Langdon effect and ion screening and showing experimental uncertainty. This figure is reproduced with permission from Turnbull et al., Phys. Rev. Lett. 130, 145103 (2023).104 Copyright 2023, the American Physical Society.

FIG. 4.

Ratios of simulated to measured absorption for OMEGA experiments to measure IBS absorption: (a) without Langdon factor and without screening; (b) with Langdon and without screening; and (c) accounting for both the Langdon effect and ion screening and showing experimental uncertainty. This figure is reproduced with permission from Turnbull et al., Phys. Rev. Lett. 130, 145103 (2023).104 Copyright 2023, the American Physical Society.

Close modal

It is typical to neglect the physics associated with the solid to plasma transition induced by high intensity laser illumination of a target. This is justified by the observation that capsule dynamics can be captured accurately without carefully accounting for these physics, though very high resolution of the laser deposition region is required at early times.110 Some reduced models have been developed that attempt to account for the complex physics that occur, typically well below the grid resolution, during this phase.111–118 However, it is more common to use simplified models that deposit a fraction of the laser energy when rays encounter overdense cells.119 It is during this phase when these models dominate energy deposition that laser nonuniformities are imprinted onto targets, and it has been shown that simulations that do not account for the microphysics of these interactions do not correctly capture the level of imprint onto the target.44,45,117 We demonstrate this in Fig. 5, where experimental velocity imprint spectra are compared between simulations and experiments performed on Nike45 [Fig. 5(a)] and OMEGA44 [Fig. 5(b)]. In the case of the OMEGA experiments, the inclusion of a model for multi-photon ionization (MPI) in simulations is able to improve agreement with experiment. This will have important consequences both for directly driven capsule implosions, where laser imprint is a primary asymmetry seed,120–124 and in indirectly driven implosions, where laser imprint could seed mixing between hohlraum and gas material125 that can impact transport10 and thermodynamic properties126 of the radiating gold plasma where the x-ray drive for the capsule is produced. Furthermore, many materials are initially transparent to the laser light, which could introduce a mechanism for target preheat.127 While there has been rapid development in this area recently, the models that have been developed have not been widely adopted and still need to be benchmarked against experiment. It is likely that their use and potential future improvements to the coefficients they require will improve predictive capability for a wide class of laser-driven problems.

FIG. 5.

Comparison of velocity imprint spectra from laser nonuniformity between simulations and experiments: (a) experiments performed on the Nike laser and simulations performed with FAST3D;45 (b) experiments performed on the OMEGA laser and simulations performed with DRACO with and without a model for multi-photon ionization (MPI).44 Figure in (a) was reproduced from Oh et al., Phys. Plasmas 28, 032704 (2021)45 with the permission of AIP Publishing. Figure in (b) was reproduced with permission from Peebles et al., Phys. Rev. E 99, 063208 (2019).44 Copyright 2019, the American Physical Society.

FIG. 5.

Comparison of velocity imprint spectra from laser nonuniformity between simulations and experiments: (a) experiments performed on the Nike laser and simulations performed with FAST3D;45 (b) experiments performed on the OMEGA laser and simulations performed with DRACO with and without a model for multi-photon ionization (MPI).44 Figure in (a) was reproduced from Oh et al., Phys. Plasmas 28, 032704 (2021)45 with the permission of AIP Publishing. Figure in (b) was reproduced with permission from Peebles et al., Phys. Rev. E 99, 063208 (2019).44 Copyright 2019, the American Physical Society.

Close modal

Inertial confinement fusion implosions are sensitive to sub-micron features and hot, low density regions can exhibit ion mean free paths comparable to or exceeding the size of these features. Ion viscosity, which scales as T i 5 / 2 / Z 2 ¯ 2 (compared to T 0.7 for gases128) is important in hot, low-Z regions and can substantially suppress non-radial flows in ICF hot spots.129 Viscosity is also important for determining the energy partitioning of shocks,130,131 and for this it is critical to accurately capture both the electronic and ionic viscosities. In the hot, low density region between the hohlraum and capsule blowoff inside a hohlraum, the flow can be non-hydrodynamic132 and exhibit interpenetration.133,134 Here, by non-hydrodynamic, we mean that the Knudsen number Kn = λ L 1, where λ is the ion mean free path and L is a representative physical lengthscale. When the Knudsen number is large, kinetic effects like interpenetration become important. In addition, strong shocks in low density gas can produce non-hydrodynamic conditions that separate ion species behind the shock.135 The discrepancies between detailed high resolution 3D radiation-hydrodynamics simulations of ICF implosions and experiments tend to occur during shock flash, when the shock reflects off the center of the implosion,36,60 which is when simulations exhibit the largest temperature gradients. These conditions are not captured by the single-fluid inviscid assumptions built into radiation-hydrodynamics codes and it is known that hydrodynamic approximations do not reproduce experimentally observed shock structures136–139 even when viscosity is explicitly modeled.

Significant success has been achieved in understanding implosions and optimizing performance despite uncertainties in the details of the hydrodynamics and the impacts of small features (density variations, surface defects, high-Z inclusions, etc.). In current state-of-the-art implosion modeling, it is computationally prohibitive to model the small scale features that implosions are sensitive to48 and these are typically either neglected or treated using surrogate perturbations. Therefore, there is ambiguity as to the true impact of these features and the impact of viscosity. Comparisons of detailed, high resolution three-dimensional modeling to experiment has provided evidence that flows may be over-predicted without modeling viscosity,36 suggesting that predictive modeling likely requires the inclusion of viscosity and that the associated coefficients are accurate.

Plasma transport models attempt to account for viscosity and non-hydrodynamic mass transport physics. Several models have been proposed with varying levels of complexity, and it is still an open question to determine exactly when such models are needed and how sophisticated they need to be in order to predict experimental observables. The most complex, multi-fluid models, allow different ion species to have their own velocities,140,141 and thus enable interpenetration in appropriate regions. These require the development of closure models to account for inter-species interactions as well as corresponding transport coefficients. The simplest models use diffusion approximations to account for non-hydrodynamic effects as a perturbation to the single-fluid solution142–151—i.e., thermodiffusion, frictional heating, concentration diffusion, barodiffusion, electrodiffusion, and the corresponding enthalpy flux. Such models differ primarily in the forms chosen for the coefficients used in these terms. A uniquely challenging aspect of developing and utilizing these models is that the strong temperature and Z ¯ dependence of the viscosity and transport coefficients means that the coefficients can change by orders of magnitude within a mixing layer.74 Since the coefficients change rapidly as the material composition evolves within a region and the equations are highly nonlinear, these models typically require high resolution and subcycling to achieve numerically converged results, so that they can be very computationally expensive. Because of this, analytic coefficients that are quick to evaluate are advantageous.

Uncertainties in mass transport coefficients is also important for the understanding of white dwarf stars, which consist of a degenerate core and outer convected plasma separated by a layer of warm dense matter.152 Gravity-driven diffusion competes with accretion to determine the composition of the atmosphere visible to telescopes. Understanding the spectrum of the emission therefore requires accurate diffusion coefficients,153 subject to the same challenges.

Plasma transport and viscosity can strongly impact the hydrodynamic development of inertial confinement fusion implosions. Hydrodynamic instabilities154,155 play a key role in determining performance by impacting the efficiency of energy transfer to the hotspot48,156 as well as introducing contaminant that can radiatively cool the hotspot after it has formed.157,158 These instabilities grow due to misaligned density and pressure gradients, which can cause the growth of any target imperfections and play a critical role in the development of the impacts of engineering features.159–163 Misaligned density and pressure gradients drive shear flows between heavy and light portions of fluid, whereas viscosity dissipates shear flows, so that viscosity tends to inhibit hydrodynamic instability growth even for room temperature fluids.164–168 In the plasma regime, the ion viscosity is proportional to T i 5 / 2 / Z 2 ¯ 2 (electron viscosities are smaller by a factor of m e / m i but can become important in situations when T e T i), so that even highly turbulent flows could be suddenly dissipated as they heat.169 In addition to impacting the implosion dynamics,170 it has been shown that viscosity has a significant impact on dissipating non-radial hotspot flows,129 reducing the growth of high-wavelength modes171 and even some long-wavelength modes172 in ICF implosions, and that Rayleigh-Taylor growth rates in ICF implosions are sensitive to the coefficients used for both viscosity and plasma transport173 and relevant viscosity coefficients can vary by orders of magnitude depending on the model used.174,175 Mass flux can also inhibit the growth of hydrodynamic instabilities through two mechanisms.176–178 First, mass flux increases the density gradient scale length over time. Second, the mass flux alters the amplitude of the perturbations. Radiation-hydrodynamic simulations are typically performed at resolutions above the diffusive scale length, in which case species diffusion arises from implicit179 or explicit sub-grid models.180 When diffusive scales are resolved, the plasma transport models provide physical mechanisms for kinetic mass flux due to gradients in the species concentrations, electron and ion pressures, and electron and ion temperatures.148,150 151 

We show the combined impact that plasma transport and viscosity have on Rayleigh-Taylor growth at different scales in Fig. 6 from Ref. 171. Here, we plot the helium concentration in various box sizes L for a problem that would be invariant to the Euler equations—i.e., the spatial distribution of materials would be identical in these plots if viscosity and mass transport were not enabled. The problem is initialized with Xe above He at T = 1 keV accelerated downward at a = 3000 L 1 cm 2 / s2 with an interface initialized with perturbations with wavelengths λ = 2 L , L / 5, and L / 20, each with an amplitude of L / 40. When L = 1 cm, the solution with plasma transport exhibits only very small deviation from the pure Euler solution. As the box size is decreased, the impacts of viscosity and plasma transport become increasingly important, and the instability is nearly entirely stabilized for L = 0.001 cm.

FIG. 6.

Helium mass concentration plot in a He/Xe mixture at an initial temperature of 1 keV demonstrating the impact of plasma transport and viscosity on Rayleigh–Taylor instability growth at different scales. Each simulation uses fixed initial conditions that are scaled to the box size δx and the end time δt is scaled with the size of the box so that δ x / δ t is constant. This figure is reproduced from Haines et al., Phys. Plasmas 21, 092306 (2014),171 with the permission of AIP Publishing.

FIG. 6.

Helium mass concentration plot in a He/Xe mixture at an initial temperature of 1 keV demonstrating the impact of plasma transport and viscosity on Rayleigh–Taylor instability growth at different scales. Each simulation uses fixed initial conditions that are scaled to the box size δx and the end time δt is scaled with the size of the box so that δ x / δ t is constant. This figure is reproduced from Haines et al., Phys. Plasmas 21, 092306 (2014),171 with the permission of AIP Publishing.

Close modal
For the purpose of modeling instability growth, it is particularly important to accurately capture viscosities for individual ion species as well as how viscosities evolve when different species mix. Several methods for calculating coefficients for mixed species are available: Grad–Kagan–Baalrud (GKB)181 viscosity coefficients, which use the effective potential theory framework182 motivated by the work in Ref. 145 to self-consistently calculate the ion viscosity for a mixture of multiple ion species. The GKB method uses a Grad method, in which a closure is postulated for a system of moment equations (the GKB method uses 21 equations per species). The simpler Vold model148,150,151 combines viscosities for each individual species using the following formula:
η vold : = α n α T α β ν α β * ,
(8)
where the sums are over plasma species and ν α β * is an effective collision rate between plasma species. The theory of Braginskii–SimakovMolvig (BSM)183–185 generalize the Braginskii results92 and involve expanding the system of equations from the Braginskii closure scheme to multiple ions and solving it asymptotically. The multi-material viscosity is defined as
η : = α k η ( n l T i ν l l ) ,
(9)
where α k η is a coefficient specific to the particular mixture and density of the plasma, l corresponds to the lightest ion species in the mixture, and νll is the self-collision rate of the lightest ion. The BSM methodology is valid for small Knudsen number in non-magnetized weakly-coupled plasmas. Finally, the Molvig–Simakov–Vold scheme makes additional approximations in order to obtain a general analytic form for α k η.149 

We demonstrate the sensitivity of Rayleigh–Taylor growth rates to viscosity mixing methodologies in Fig. 7, taken from Ref. 173, where Rayleigh–Taylor growth rates are compared as a function of mode number in a deuterium-carbon mixture at T = 5 keV using an analytic model (fluid model), xRAGE simulations,7,8,148,151,186 and VPIC simulations.187–189 Deviations between VPIC and xRAGE are substantial at low wavenumber, which could be representative of a failure of the mixing rule, details of the kinetic mixing physics, or differences in the effective single species viscosity.173 

FIG. 7.

Comparison of Rayleigh–Taylor instability growth rate vs mode number for a deuterium-carbon mixture at T=5 keV using different analytic viscosity formulas and different coefficient mixing methodologies. This figure is reproduced from Keenan et al., Phys. Plasmas 30, 072106 (2023),173 with the permission of AIP publishing.

FIG. 7.

Comparison of Rayleigh–Taylor instability growth rate vs mode number for a deuterium-carbon mixture at T=5 keV using different analytic viscosity formulas and different coefficient mixing methodologies. This figure is reproduced from Keenan et al., Phys. Plasmas 30, 072106 (2023),173 with the permission of AIP publishing.

Close modal

Based on Ref. 186, viscosity pre-factors α k η are compared directly for the different mixing methods in Fig. 8, which demonstrate that calculated viscosities for mixed materials can vary substantially depending on the closure model that is assumed, even when the mixture is assumed to be uniform. The MSV model assumes m l m h, where ml is the mass of the light fluid and mh is the mass of the heavy fluid, which is why it fails to capture the limiting behavior when the light species mass fraction is 0. Even ignoring this curve, differences in the mixed material viscosities can exceed 50%.

FIG. 8.

Viscosity prefactor α k η calculated vs mass fraction using several viscosity mixing techniques. This figure was provided by Keenan based on work performed in Ref. 186.

FIG. 8.

Viscosity prefactor α k η calculated vs mass fraction using several viscosity mixing techniques. This figure was provided by Keenan based on work performed in Ref. 186.

Close modal

In addition to the viscosities derived in the literature cited above, many additional models are available for single species plasmas. Viscosities are derived using using a Yukawa model in Ref. 190 and using an effective Boltzmann approach.94 It is not currently possible to measure plasma or even warm dense matter viscosities experimentally (though there are platforms under development to do so),191 though some work has been done to benchmark models using quantum molecular dynamics (QMD) simulations.192–195 These generally show poor agreement, though these comparisons are only possible at relatively low temperatures where the viscosity is very small and potentially beyond the range of validity of the analytic models.

Diffusion-based models only account for near-hydrodynamic regimes, where ion mean free paths are comparable to the relevant gradient scale lengths. In hohlraum interiors, however, ion mean free paths are estimated to be much longer and multi-fluid models may be necessary. Recent experiments196 were performed to evaluate the extent of interpenetration between plasma species in conditions similar to those in laser-driven hohlraums. xRAGE plasma transport10 and Lasnex multi-fluid197 simulations suggest that the primary impact of such physics is diffusion of helium into the gold holhraum blowoff as well as into the blowoff from the capsule ablator. This, in turn, will impact transport coefficients calculated in the mixed material cells, including the viscosity. Recent work has shown that the coupling between multi-fluid hydrodynamics and cross-beam energy transfer (a laser plasma interaction in which energy is resonantly exchanged between beams via the background plasma) is critical to determining the shape of indirectly-driven capsule implosions.197 Dedicated experiments have been performed on the OMEGA laser facility to study Au-C and Au-He-C interpenetration physics in conditions relevant to laser-driven hohlraums.196 Nevertheless, xRAGE simulations suggest that much of the interpenetration observed in these simulations may have been hydrodynamic, driven by the oblique incidence of the lasers, but that the plasma transport model underpredicts the interpenetration for the Au-He-C experiments.198 It is unclear if the discrepancy is caused by a deficiency in the plasma transport model, the coefficients, or in the mixing methodology employed for the coefficients.

Implosions involving low density gas fills and strong shocks have been observed to exhibit behavior that radiation-hydrodynamics codes struggle to replicate, and various kinetic effects, many of which are captured by multi-fluid and plasma transport models, likely explain these discrepancies since these conditions enhance ion mean free paths.199 For example, stratification of deuterium and tritium ions has been proposed as an explanation for anomalous DT to DD reaction yield ratios in DT gas-field implosions.200 Simulations have shown that ion species can cause stratification,135,201–203 though it has also been observed that species de-stratify when the shock rebounds.204 Shock-acceleration of ablator ions into the fuel region has also been hypothesized as a mechanism for introducing contaminant into ICF hot spots.205 Hydrodynamically equivalent implosions with varied gas fills have also exhibited yields that are inconsistent with hydrodynamic predictions.206,207 Barodiffusion (pressure-driven diffusion)142,208,209 and uncertainty in species equation of state mixing207 (discussed in more detail in Sec. IX) have been proposed as explanations. Diffusion of fuel into the ablator has also been observed in simulations.210 Finally, “superdiffusive” behavior has been observed in simulations relevant to implosions at material interfaces.211 

Nevertheless, it is difficult to evaluate these hypotheses numerically. Dedicated kinetic codes such as iFP,212–215 FPion,204,216–219 and VPIC187,189 can accurately account for the kinetic plasma physics, but cannot couple to accurate cold and atomic physics such as equation of state and opacity, respectively, and would be computationally prohibitive to apply to the entire spatial and temporal extent of an implosion. On the other hand, radiation-hydrodynamics codes can only apply approximate reduced models for the physics. One particular challenge for radiation-hydrodynamics codes is how to consistently handle changing material properties of molecules such as plastic (CH) and DT both when they are cold, during the early phase of a problem when accurate physical properties such as equations of state are needed to capture their behavior, and when they are hot and subject to disassociation. For example, an accurate CH equation of state is needed to capture the implosion dynamics, but is inappropriate to describe the behavior of C and H ions after they have dissociated. None of the available multi-fluid or plasma transport models addresses this transition and there is always a trade-off between the accuracy at which cold physics is captured and the accuracy at which plasma physics is captured. Nevertheless, it is clear that improved coefficients, mixing methodologies, and, critically, transitioning models between cold and hot conditions are warranted.

Thermal conduction is a critical process in all HEDP and ICF experiments. Thermal conduction transports deposited laser energy, in the ablator or in the hohlraum wall blowoff, to the ablation front. In addition, thermal conduction is critical to hotspot formation in ICF.220 As the low density DT in the center of the implosion gets compressed to high temperatures, heat transport into the high density fuel layer causes mass ablation of DT into the hotspot.221 Due to the low initial vapor density of DT ice layer designs, a majority of the hotspot mass must derive from the fuel layer. Heat conduction, along with radiation, is one of the most important energy loss mechanisms from the hotspot. Finally, thermal conduction impacts hydrodynamic instability growth.222–229 Indeed, thermal conduction has a strong impact on the density gradient at the ablation front, where ablated mass from the capsule mixes with vacuum material. During shell acceleration and deceleration, the low density blow-off and low-density DT, respectively, are hotter than the higher density shell. Thermal conduction into the higher density material leads to ablation that smooths high mode perturbations and increases the density gradient scale length, thus reducing instability growth.

The use of Lee–More conductivities,105 which is based on the theory of Spitzer and Härm230 for high temperature low Z plasmas, sometimes with corrections to further extend regimes of applicability,231,232 is still common. Nevertheless, the theory behind these conductivities breaks down in the warm dense matter regime. For strongly coupled plasmas, One Component Plasma models233 can be used to produce both analytically182,234,235 and numerically236–240 evaluated conductivities. To obtain conductivities valid over a wider range of conditions, relativistic average atom codes, such as the Purgatorio code241,242 based on the Inferno algorithm,243 can be used to generate tables.244–246 Classical- and Quantum-Molecular Dynamics-based conductivities can be used at low temperatures down to the limit for liquid metals,247 and more recently these have been used to generate tables that are more accurate in the WMD regimes.248–259 Nevertheless, QMD models struggle to reproduce the high temperature plasma limit, since these methods tend to converge more slowly at higher temperatures. Furthermore, neither Lee-More nor existing QMD-based conductivities account for Electron–electron collisions, which can reduce conductivities by 30%–70%.77 A correction for this can be estimating using a model Ref. 260. Intermediate approaches such as density functional theory average-atom (DFT-AA)261 and average-atom two-component plasma (AA-TCP)262–265 attempt to bridge the gap. The differences in conductivities calculated by different models in the WDM regime is substantial.

Only limited experimental data are available to directly constrain thermal conductivities in the warm dense matter regime,77,266,267 and these data highlight the need for the more sophisticated models. The differences in models and comparison to experimental data are illustrated in Fig. 9 reproduced from Ref. 77, where conductivities from various models are compared to experimental data for CH and Be. This includes Quantum Molecular Dynamics (QMD),258,259 Purgatorio,241,242 Spitzer,230 Lee–More–Desjarlais (LMD),105,232 and DFT-AA,261 with some calculations including corrections for electron–electron scattering (ee) based on a model from.260 Several models agree with the experimental data point for CH when electron–electron collisions are accounted for, though many of the same methods underpredict the Be conductivity, so that further work on this correction is warranted.252,265,268,269

FIG. 9.

Comparison of experimentally measured conductivities77 for (a) CH at T = 7.8 ± 1.8 eV and (b) Be at 4.4 ± 0.7 eV compared to various analytic and computationally derived values: Quantum Molecular Dynamics (QMD),258,259 Purgatorio,241,242 Spitzer,230 Lee–More–Desjarlais (LMD),105,232 and Density Functional-Average Atom (DFT-AA),261 with some calculations including corrections for electron–electron scattering (ee) based on a model from Ref. 260. This figure has been reproduced with permission from Jiang et al., Nat. Comm. Phys. 6, 98 (2023).77 Copyright 2023 Authors, licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

FIG. 9.

Comparison of experimentally measured conductivities77 for (a) CH at T = 7.8 ± 1.8 eV and (b) Be at 4.4 ± 0.7 eV compared to various analytic and computationally derived values: Quantum Molecular Dynamics (QMD),258,259 Purgatorio,241,242 Spitzer,230 Lee–More–Desjarlais (LMD),105,232 and Density Functional-Average Atom (DFT-AA),261 with some calculations including corrections for electron–electron scattering (ee) based on a model from Ref. 260. This figure has been reproduced with permission from Jiang et al., Nat. Comm. Phys. 6, 98 (2023).77 Copyright 2023 Authors, licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

Close modal

The use of recently developed QMD-based tables in radiation-hydrodynamics simulations has resulted in notably improved agreement between simulations and integrated experiments. For example, the use of improved thermal conduction coefficients for DT from Ref. 258 in detailed 3D modeling of layered NIF implosions has improved comparisons between simulated and measured burn-weighted T ion by 200 eV.34 In addition, the use of QMD-derived thermal conduction coefficients for CH and DT has had a significant impact on predictions of capsule compression ( 20%) for directly driven layered implosions on OMEGA,259 as shown in Fig. 10. The overprediction of capsule compression by radiation-hydrodynamics simulations is a long-standing problem48,270–274 that is clearly impacted by the quality of the thermal conductivity coefficients.

FIG. 10.

Pressure and density profiles from one-dimensional simulations of layered directly-driven capsule implosions performed on the OMEGA laser, at (a) the end of the laser pulse and (b) peak compression, comparing the use of state-of-the-art analytic and QMD-based conductivity coefficients with the default coefficients used by LILAC. This figure has been reproduced from Hu et al., Phys. Plasmas 23, 042704 (2016),259 with the permission of AIP Publishing.

FIG. 10.

Pressure and density profiles from one-dimensional simulations of layered directly-driven capsule implosions performed on the OMEGA laser, at (a) the end of the laser pulse and (b) peak compression, comparing the use of state-of-the-art analytic and QMD-based conductivity coefficients with the default coefficients used by LILAC. This figure has been reproduced from Hu et al., Phys. Plasmas 23, 042704 (2016),259 with the permission of AIP Publishing.

Close modal

In several cases, simulations have been observed to show a higher level of sensitivity to the methodology used to determine multi-material conductivities than to the coefficients themselves. This has been shown to be true for wetted foam implosions,275,276 in which DT liquid wets a foam shell on the inside of the ablator, in Ref. 277. This has also been shown to be true for mixtures of helium and gold in hohlraum interiors in Ref. 10. We demonstrate this in Fig. 11 (reproduced from Ref. 10) where we compare the x-ray flux exiting the laser entrance hole from simulations of two different experiments conducted on the National Ignition Facility, N090728278–280 and N181209,281 using different conductivity mixing methodologies–mass-weighted average and mass-weighted harmonic average—as well as to experimental measurements. The simulations using mass-weighted average is labeled “default mix” in Fig. 11(a) and “alt. cond.” in Fig. 11(b). The simulations using mass-weighted harmonic averaging are labeled as “harmonic mix” in Fig. 11(a) and “baseline” in Fig. 11(b). The different mixing methodologies primarily impact the x-ray flux during the initial rise [this difference is more pronounced in Fig. 11(b)] as well as when the flux is reduced after the laser is turned off. It is notable that the mass-weighted average produces better agreement with experimental measurements during early times, whereas the mass-weighted harmonic average produces better agreement with experimental measurements during late time. This could reflect changes in plasma conditions (the mixing region is denser and cooler at early times) or changes in the three-dimensional geometric configuration that is not accounted for in these two-dimensional simulations (the gold bubble growth is three-dimensional282 and regions of discrete bubble growth will appear as mixed regions in axisymmetric two-dimensional simulations). We note that the inclusion of plasma transport in hohlraum simulations [labeled “diffm” in Fig. 11(b)] enhances the sensitivity to the mixing methodology by diffusing more helium into the gold bubble. Predicting the shape of capsule implosions inside hohlraums is an ongoing challenge38,132,283 and simulations in Ref. 10 show that the shape, as measured by the ratio of Legendre mode magnitudes P 2 / P 0 measured from the 17% contour of the x-ray self-emission image at bang time, can vary by more than 40% due to the choice of conductivity mixing methodology. These results indicate that it is important to properly account for how conductivities combine in mixed material computational cells and how they are impacted by geometries, even in very high resolution simulations, and it is not clear that current methodologies are appropriate for all conditions encountered throughout the course of a simulation.

FIG. 11.

(a) Comparison of x-ray flux escaping the laser entrance hole vs time from simulations of vacuum hohlraum experiment N090728278–280 compared to experimental measurements. Simulations are presented using two different conductivity mixing methodologies: default (mass-weighted average) and harmonic (mass-weighted harmonic average). (b) Comparison of x-ray flux escaping the laser entrance hole vs time from simulations of layed capsule implosion experiment N181209281 compared to experimental measurements. Simulations are presented using a number of variations, including comparing the same conductivity methodologies: default (labeled as baseline) and harmonic (labeled as alt. cond.). Figures are reproduced from Haines et al., Phys. Plasmas 29, 083901 (2022),10 with the permission of AIP publishing.

FIG. 11.

(a) Comparison of x-ray flux escaping the laser entrance hole vs time from simulations of vacuum hohlraum experiment N090728278–280 compared to experimental measurements. Simulations are presented using two different conductivity mixing methodologies: default (mass-weighted average) and harmonic (mass-weighted harmonic average). (b) Comparison of x-ray flux escaping the laser entrance hole vs time from simulations of layed capsule implosion experiment N181209281 compared to experimental measurements. Simulations are presented using a number of variations, including comparing the same conductivity methodologies: default (labeled as baseline) and harmonic (labeled as alt. cond.). Figures are reproduced from Haines et al., Phys. Plasmas 29, 083901 (2022),10 with the permission of AIP publishing.

Close modal

There is evidence that suggests that electron heat transport is inhibited by some unknown physics in the gold blowoff (the “gold bubble”) in hohlraums.39 Artificially reducing heat conduction in this region using a flux limiter improves agreement with many experimental measurements.284 For example, temperature measurements using spectroscopic tracers suggest that hohlraum simulations underpredict the temperature in the gold bubble unless electron heat conduction is restricted.285,286 In addition, artificially restricting the heat conduction in the bubble improves agreement between simulated and experimental images of emission through the laser entrance hole.287 Nevertheless, simply using a restrictive heat flux limiter tends to improve agreement with experimental observables by increasing the amount of “glint,” light that escapes the hohlraum without getting absorbed, to levels in excess of that observed in experiment.288 The use of such flux limiters also precludes agreement with experiments where solid spheres are illuminated with lasers to obtain data to constrain modeling of heat transport.289,290 Resolving these discrepancies could require improvements to thermal conductivities, non-local effects (discussed below in Sec. VI), or potentially coupling these physics to self-generated magnetic fields (discussed below in Sec. VIII).

Ion thermal conduction is typically negligible compared to electron thermal conduction since the ratio of the associated conductivities in the former to the latter is m e m i. Nevertheless, ion thermal conduction can be important in situations where energy deposition strongly favors the ions so that T i T e, such as near strong shocks. For this reason, ion conduction sets the temperature profiles in ICF hot spots in the period between shock flash (when the shock reflects off the origin) and when the sound speed has grown large enough that it becomes isobaric. During this period, the temperature profile is very sensitive to the value of the ion conduction flux limiter, if it is used.172 Variations of Spitzer230 and Lee-More105 are commonly used for ion conductivities, but Boltzmann approaches have been applied to derive coefficients that have a wider range of validity.94 Because ion conduction is particularly important near strong shocks, uncertainties could contribute to anomalous DT/DD yield ratios observed in some experiments.206,207 Because it tends to be dominated by electron conduction, ion conduction is much more difficult to measure and less effort has been expended to improve coefficients. Furthermore, available models are very similar, so sensitivity studies do not indicate how much impact improved ion conductivities would have.

The success of inertial confinement fusion in the laboratory3 relies on the production of alpha particles by DT fusion, which are then re-absorbed into the burning plasma. This process generates a thermal instability that is quenched by capsule disassembly. Modeling this process generally involves the use of Monte Carlo charged particle transport of the reaction products. The rate at which charged particles lose energy to the background plasma, dE/dx, is referred to as the stopping power, and these are typically pre-tabulated using one of several analytic models that have been developed. The available experimental data are too sparse and the error bars too large to enable the creation of an experimentally derived table, so that the extant experimental data are used only to distinguish between the accuracy of the different analytic models in various regimes. Widely used stopping power models include Maynard–Deutsch292–294 (MD), LiûPetrasso295,296 (LP), and Brown–Preston–Singleton297,298 (BPS), and these models exhibit variations in the alpha particle range on the order of 20%–30% in simulated hotspot conditions.298 Other available models include Refs. 299–301. The MD model is based on the Lenard–Balescu kinetic equations, which is expected to be accurate in the high-velocity, weak-scattering regime. BPS combines both Lenard–Balescu and Boltzmann physics, extending its validity into the strong-scattering regime, though it does not include quantum degeneracy effects. LP attempts to accurately cover both the strong- and weak-scattering regimes by applying a generalized Fokker–Planck equation for moderately coupled plasmas.302 The parameterization of the Li–Petrasso model has recently been updated in order to improve agreement with experimental data.303,304 An alternate Parameterization of MD is also frequently used.305 When applying these models to the simulation of igniting layered inertial confinement fusion implosions on NIF, the variation in yields across models typically exceeds 20%, and in some situations can be orders of magnitude for a marginally igniting capsule due to the yield amplification potential of alpha heating.298 Data from the different models is plotted in Fig. 12 from Ref. 303, along with the results of molecular dynamics simulations,306 showing the broad range of predictions from different models across particle energies and plasma conditions.

FIG. 12.

Comparison of proton stopping powers vs particle energy in various ICF-relevant plasmas. This figure has been reproduced from Zylstra et al., Phys. Plasmas 26, 122703 (2019),303 with the permission of AIP publishing.

FIG. 12.

Comparison of proton stopping powers vs particle energy in various ICF-relevant plasmas. This figure has been reproduced from Zylstra et al., Phys. Plasmas 26, 122703 (2019),303 with the permission of AIP publishing.

Close modal

Numerous experiments have been done to constrain stopping power models, mostly for highly ionized plasmas.307–311 When particles are fast ( v p v th e, where vp is the particle velocity and v th is the thermal velocity), ion stopping powers are primarily influenced by weak long-range interactions with the electrons, and all models agree reasonably well with the experimental data. Very little data are available for low-velocity particles—i.e., the Bragg peak, v p v th i—where models exhibit large variations,300,312 but this is where charged particles deposit the bulk of their energy into the background plasma. Some experiments performed in this regime did not have sufficient measurements of the plasma conditions to discriminate between models.313–315 More recent experiments have been able to discriminate between models and demonstrate the need for models to include detailed accounting for close binary collisions in the beam-plasma interaction description.304,316 Therefore, BPS tends to produce the best agreement with data, but still exhibits large discrepancies when v p 0.3 v th i. In the regime v p v th i, TD-DFT calculations are required to account for experimental data.317 The amount of alpha heating can also be inferred from surrogate THD implosions318 to high yield DT implosions, where the amount of tritium is intentionally suppressed to achieve hydrodynamically equivalent implosions without alpha heating, which could help to constrain stopping power models. Improved stopping powers have the potential to improve the accuracy of ICF simulations with strong alpha heating, particularly in the regime where ignition is marginal—i.e., the capsule ignites very close to peak compression so that performance is very sensitive to asymmetries.

In many situations, it is important to account for non-Maxwellian electron distributions when calculating thermal conduction. Large temperature gradients291 and laser-plasma instabilities46,47 can seed non-negligible flows of suprathermal electrons. Capsule implosions rely on maintaining low entropy in the fuel layer in order to achieve efficient energy coupling to the hotspot, so excess heat conduction into the fuel layer can inhibit implosion performance.319 In the coronal plasma, temperature gradients can be large enough that calculated heat fluxes exceed the free streaming limit.320 In some cases, it is sufficient to use an electron conduction flux limiter to account for this, but this approach effectively throws away the non-local electrons and some laser-driven experiments require more detailed treatment321,322 in order to accurately capture energy coupling and the resulting distribution of energy within the target. In order to account for these effects, non-local thermal conduction models have been developed321–328 that are suitable for coupling to radiation-hydrodynamics. The most commonly used model is the Schurtz-Nicolaï-Busquet (SNB) model and variants thereof,321–324 which uses a diffusion approximation of the kinetic equations, discretize electron energy groups, and calculates first order corrections to the electron fluxes for each group. Another approach325,326 uses a linearized steady-state Vlasov-Fokker-Planck equation (VFP) and calculates the electron distribution as a Maxwellian and first order perturbation. The resulting equations are solved using an orthogonal eigenfunction basis. In practice, this method requires a large number of eigenfunctions for convergence.326 A third method327,328 again separates the heat flux into a Maxwellian and non-local first order correction and calculates the non-local heat flux as a sum of Lorentzians, whose coefficients are determined by solving the system in Fourier space. These models are compared in detail and benchmarked against VFP simulations in Ref. 329, where the SNB model exhibited the best agreement with VFP in a set of test cases. Other models have been proposed,330–337 but more work is needed to evaluate the utility of these models for modern ICF applications.

The data required for non-local models are group mean free paths λ g , e (i.e., the mean free paths of electrons within some energy band). Though many of the methodologies used to calculate fusion product stopping powers discussed in Sec. V, including Li–Petrasso (LP)303 and Brown–Preston–Singleton (BPS),297,298 can be applied to non-local electrons, the most popular method for calculating these is to use the Lee–More analytic formula,105,
λ e g = E g 2 4 π Z + 1 n e e 4 ln Λ ,
(10)
though some effort has been made to improve this formula.329 Nevertheless, it is known that these coefficients disagree with available low temperature data, and recent time-dependent density functional theory (TD-DFT) suggests this formula overestimates the nonlocal electron stopping power and underestimates their stopping range.338 The comparison to low temperature data, reproduced from Ref. 324, is shown in Fig. 13, where λ e g is plotted based on Eq. (10) along with available experimental data. Electron stopping powers in a CH plasma calculated with various models are plotted vs electron energy in Fig. 14 for the LP, BPS, dielectric function (DF) formalism,339 and various flavors of TD-DFT calculations.338 The analytic models (LP, BPS, DF, and Lee–More) fail to account for partial ionization effects, the Barkas effect,340,341 in which collision rates depend on the signs of the colliding particles, nor observed Z1 oscillations of the stopping power342 (i.e., a periodic dependence of the electronic contribution to the stopping power on the charge of the incident ion). In recent work, electron stopping powers in DT plasmas have been calculated using TD-DFT.268,343–345 When coupled to non-local conduction models, these significantly change capsule compression in directly-driven layered capsule implosions.338 The choice of λ e g has also been shown to impact hohlraum ablation346 and temperature profiles in laser-irradiated spheres.347 Similar to local thermal conduction, a leading uncertainty in these calculations is how to account for electron–electron scattering. These uncertainties could contribute to unexplained fuel decompression in some direct-drive layered implosions.348 
FIG. 13.

Comparison of group mean free paths as calculated by Eq. (10) to available low temperature data. This figure is reproduced from Cao et al., Phys. Plasmas 22, 082308 (2015),324 with the permission of AIP publishing.

FIG. 13.

Comparison of group mean free paths as calculated by Eq. (10) to available low temperature data. This figure is reproduced from Cao et al., Phys. Plasmas 22, 082308 (2015),324 with the permission of AIP publishing.

Close modal
FIG. 14.

Comparison of electron stopping powers vs energy in a CH plasma calculated using various models, including Li–Petrasso (LP),303 dielectric function (DF),339 Brown–Preston–Singleton (BPS),297,298 and various flavors fo time-dependent density-functional-theory (TD-DFT).338 This figure is reproduced with the permission from Nichols et al., Phys. Rev. E 108, 035206 (2023) [338], Copyright 2023, the American Physical Society.

FIG. 14.

Comparison of electron stopping powers vs energy in a CH plasma calculated using various models, including Li–Petrasso (LP),303 dielectric function (DF),339 Brown–Preston–Singleton (BPS),297,298 and various flavors fo time-dependent density-functional-theory (TD-DFT).338 This figure is reproduced with the permission from Nichols et al., Phys. Rev. E 108, 035206 (2023) [338], Copyright 2023, the American Physical Society.

Close modal

Finally, many non-thermal electrons arise from laser-plasma interactions such as cross-beam energy transfer, two-plasmon decay, etc.46,47 These can preheat targets and hence cause decompression of the fuel in inertial confinement fusion implosions, but reduced models have not been developed that can couple to non-local thermal conduction algorithms. Therefore, an important source term could be missing.

Electrical conductivity is critical for pulsed-power-driven high energy density physics experiments and inertial confinement fusion implosions. Accurate calculation of the coupling of the driver to the target requires accurate resolution of the materials along the circuit path and their conductivities; uncertainties in our knowledge of electrical conductivities remains a major challenge for the development of inertial fusion energy approaches utilizing Z-pinches.349 Indeed, pulsed power drivers act like voltage sources so that the current (and hence energy) delivered to the target depends on the conductivity of the target. Furthermore, the development of the current-driven electrothermal instability (ETI)350 depends on how the electrical conductivity varies with temperature, amplifying the sensitivity of such implosions to electrical conductivities. The methodologies used to calculate electrical conductivities are generally similar to those used to calculate electron thermal conductivities: analytic formulas are frequently produced in tandem with thermal conductivities,105,230,251 though more recently some have been calculated independently.351 Methods such as DFT have been applied to produce more accurate calculations.252,256,352–360 Indeed, free electrons are responsible for both transport mechanisms. As a result, the empirically derived Wiedemann–Franz law states that
κ σ = L T ,
(11)
where κ is the thermal conductivity, σ is the electrical conductivity, and L = 2.44 × 10 8 V2 K−2. This value can vary for different materials and breaks down in the WDM regime. Nevertheless, the close relationship means that similar theoretical and simulation methodologies can be used to evaluate both quantities. Therefore, the challenges and ranges of validity for calculations of electrical conductivity are largely the same as those discussed for electron thermal conductivity in Sec. IV.

One unique challenge to electrical conductivity is how to treat vacuum or near-vacuum conditions. Large gradients in material properties are problematic for numerical simulation because they generally increase the stiffness of numerical systems and thereby make them more expensive to calculate; this problem is acute for pulsed power since vacuum regions have zero conductivity while plasmas have nearly infinite conductivity.42 Furthermore, since sound speeds are 1 / ρ, low density regions reduce the time step required by the Courant condition to maintain stability in the hydrodynamics, which has the side effect of making the hydrodynamics more diffusive in regions where the sound speed is lower. Therefore, it is common to place bounds on the density and conductivity in order to prevent numerical instabilities and avoid runaway computational expense for what can be a marginal improvement to simulation accuracy361,362 (in a rigorous analysis, the floors are varied to ensure the simulation results are not sensitive to the value used, as in Ref. 42). We note that similar approaches are frequently employed for opacities when the electron density exceeds the critical density for a given electromagnetic wave frequency n e , c = ϵ 0 m e q e 2 ω 2. In this situation, the plasma becomes perfectly reflective (i.e., the opacity becomes infinite). However, to avoid making the radiation system too stiff, the opacity is limited to some large number instead.

For electrical conductivity, despite having no charge carriers to conduct, there is no repulsive force to hinder the propagation of charged particles through the vacuum. Therefore, in the presence of an electric field, charged particles that escape from materials adjacent to vacuum will experience no resistance. This is a situation that continuum equations are not suited to model, but can have important consequences for physical systems of interest. In practice, any material that ablates into the vacuum increases its conductivity and thereby affects the target-driver coupling. Therefore, simulations of pulsed power-driven experiments exhibit high levels of sensitivity to vacuum cutoffs and how these are implemented in the code.40–42 In addition to the density floor, one approach that is used is to artificially reset momentum and energy to “floor” values in a buffer region (defined as the region where the density exceeds the floor density by no more than a small factor ϵ) to prevent unphysical ablation. Again, such techniques are not unique to modeling pulsed-power driven systems; this is reminiscent of the quiet start technique used in ICF to avoid unphysical interface smearing due to the use of hydrodynamics when the target is still solid (though simulations tend to be very insensitive to quiet start trigger parameters). Simulation results are highly sensitive to the value of the floor density as well as ϵ. Indeed, any current in a nominally vacuum region will generate J × B forces that induce flows that reduce the vacuum density in their wake, requiring mass to be artificially added to the problem in order to maintain the density floor (if this is enforced). Mass accumulates near vacuum interfaces and eventually shrinks the buffer region, allowing unphysical ablation. It has been hypothesized that this sensitivity could be reduced by performing simulations routinely in three-dimensions (currently computationally prohibitive) or by including more extended magneto-hydrodynamic (MHD) terms in simulations. One of the biggest challenges, however, to the implementation and use of more extended terms is uncertainty in the corresponding transport coefficients that are introduced.

Relative to the other charged particle transport coefficients discussed in this paper, experimental electrical conductivity measurements are more readily available at conditions approaching relevancy to HEDP and ICF.363–372 Unfortunately, electrical conductivities also exhibit extreme variability across excitation conditions, suggesting that more data and highly sophisticated models are required to constrain the conductivities needed for simulations. Furthermore, there is a wide body of literature where discrepancies between experimentally measured conductivities and theoretical models are documented.373–385 

Magnetic fields can strongly alter transport coefficients and do so anisotropically. Magnetic fields are generated by large currents in Z-pinch experiments to drive cylindrical implosions, but more generally, they can be imposed externally or generated via the Biermann battery mechanism, which occurs due to misaligned gradients in ne and Te,
B t = T e × n e e n e .
(12)
There is indirect evidence of such fields in hohlraum laser entrance holes (though it is not possible to distinguish if these are electric or magnetic fields).386 Such conditions are exhibited by hydrodynamic instabilities387–393 and jetting.394 Self-generated magnetic fields of order 100 T have been inferred from proton radiography in various laser-produced plasmas395–400 and are predicted in some situations to grow exponentially through the thermomagnetic instability.401 In this way, they can also alter density gradients by inhibiting heat flows and enhance the growth of hydrodynamic instabilities as well as inhibit heat losses into the ablator and into contaminant in the hotspot of ICF implosions. Nevertheless, mechanisms such as the Nernst effect can also strongly suppress or dissipate magnetic fields.402 We show the predicted impact on heat fluxes from self-generated magnetic fields in an ICF hotspot from Ref. 394 in Fig. 15, where in some regions the heat flux is reduced by more than 70%. Self-generated magnetic fields have been shown to reduce electron heat conduction in the gold blowoff plasma in hohlraums and improve agreement with experimental data.403 As noted previously, there is significant evidence of inhibited heat transport in the gold bubble in hohlraums relative to that predicted in radiation-hydrodynamics codes,39 and self-generated magnetic fields could potentially resolve this discrepancy, though discrepancies between code predictions and uncertainties in transport coefficients have made it difficult to resolve this question. We demonstrate the simulated impact of self-generated magnetic fields in Fig. 16, reproduced from Ref. 403, where the temperature profile is compared for simulations with and without MHD. Self-generated magnetic fields limit the heat flux from the gold bubble, causing the temperature to remain hotter there. Imposed magnetic fields have been proposed as a way of reducing heat losses and suppressing hydrodynamic instabilities.404,405 The inhibition of transport perpendicular to the magnetic field is critical to MDD concepts such as MagLIF406 and the presence of large magnetic fields in MDD implosions has a strong impact on instability growth, such as the magnetic Rayleigh-Taylor instability.407–413 
FIG. 15.

Modification to heat flux due to self-generated magnetic fields near the fill tube jet from a simulation of a layered implosion on the National Ignition Facility. This figure is reproduced with the permission from Sadler et al., Phys. Plasmas 27, 072707 (2020),394 with the permission of AIP Publishing.

FIG. 15.

Modification to heat flux due to self-generated magnetic fields near the fill tube jet from a simulation of a layered implosion on the National Ignition Facility. This figure is reproduced with the permission from Sadler et al., Phys. Plasmas 27, 072707 (2020),394 with the permission of AIP Publishing.

Close modal
FIG. 16.

Temperature profiles from hohlraum simulations with and without MHD demonstrating the simulated impact of self-generated magnetic fields on heat flow. This figure is reproduced from Farmer et al., Phys. Plasmas 24, 052703 (2017),403 with the permission of AIP publishing.

FIG. 16.

Temperature profiles from hohlraum simulations with and without MHD demonstrating the simulated impact of self-generated magnetic fields on heat flow. This figure is reproduced from Farmer et al., Phys. Plasmas 24, 052703 (2017),403 with the permission of AIP publishing.

Close modal

Extended hydrodynamics models used to simulate magnetic fields in ICF and HEDP experiments require transport coefficients for heat flux, resistive, and thermoelectric processes that depend on the magnetic field. The most common approach is to use the forms derived in Ref. 79, which uses fits to kinetic theory for resistive ( α , α , and α ) and thermoelectric ( β , β , and β ) transport coefficients. Recent work has shown that the coefficients from Ref. 79 can overestimate Nernst advection and magnetic dissipation414 and proposed corrections. Different implementations of the Nernst effect in different codes result in differences in the impact of magnetic fields on heat fluxes in hohlraum simulations such that simulated temperatures in the gold bubble vary by about 1 keV.286 

Despite its importance, there remains significant uncertainty in how to couple magnetic fields to heat conduction. The relationship in Eq. (3) typically used to capture the interaction between magnetic fields and the heat conduction is only valid in the limits χ 0 and χ , whereas the formula underpredicts the reduction in the intermediate regime where real systems exist. This is demonstrated in Fig. 17, reproduced from Ref. 403, where the commonly used heat reduction factor from Eq. (3) is compared to results from Ref. 79. This shows that the reduction in heat flow can be significantly underpredicted at intermediate values of χ.403 

FIG. 17.

Impact of a magnetic field on the heat conduction perpendicular to the field line relative to that parallel to the field line vs the Hall parameter. The dashed line corresponds to Eq. (3) and the solid black, blue, green, and magenta lines correspond to Z = 1 , 2 , 8 , , respectively. This figure is reproduced from Farmer et al., Phys. Plasmas 24, 052703 (2017),403 with the permission of AIP publishing.

FIG. 17.

Impact of a magnetic field on the heat conduction perpendicular to the field line relative to that parallel to the field line vs the Hall parameter. The dashed line corresponds to Eq. (3) and the solid black, blue, green, and magenta lines correspond to Z = 1 , 2 , 8 , , respectively. This figure is reproduced from Farmer et al., Phys. Plasmas 24, 052703 (2017),403 with the permission of AIP publishing.

Close modal

Challenges related to coupling heat transport to magnetic fields are only exacerbated in the presence of nonlocal transport.415 It has been observed that magnetic fields quench the flow of non-local electrons416 and the current state-of-the-art is to use an additional flux limiter on the Nernst term in order to account for the interaction of these physics.417 Some work has been done to develop models that could be implemented in a radiation-hydrodynamics code323,418 but these have not been widely adopted or tested.

Magnetic fields also impact the behavior of charged fusion products. The magnetic fields in MDD help to trap the energy of these particles in order to amplify fusion yields.419,420 The impact of magnetic fields, both imposed and self-generated, on alpha particle transport could be important for inertial confinement fusion, yet this has only recently been considered.421 Similar to electron thermal conduction, the presence of magnetic fields will suppress alpha particle fluxes, so that the distribution of alpha deposition in ICF hot spots and corresponding hotspot evolution will be sensitive to this. As a result, uncertainties in alpha particle deposition could relate to uncertainties in MHD transport coefficients and coupling details.

Multi-species thermalization—i.e., the rate by which dissimilar ions that are out of equilibrium come into equilibrium—is typically treated only by thermal conduction in radiation-hydrodynamics codes. When the resolution is adequate to capture material distributions (set by the diffusive length scale, typically 0.25 μm for ICF) and thermalization rates are rapid relative to the time step, this may be reasonable. Nevertheless, the latter condition is never satisfied, since the latter condition is equivalent to saying that a sound wave crosses many cells within a time step, yet the Courant–Friedrichs–Lewy condition422 used to ensure stability of the hydrodynamics solve restricts the time step so that a sound wave cannot cross the smallest cell during a simulation time step.

Instead of using ion kinetics to determine the rate of thermalization between ion species, radiation-hydrodynamics codes assume all species within a computational cell achieve some form of equilibrium. The state of each species is governed using an independent equations of state (EOS) that is typically read from a table (often SESAME423 or LEOS424,425). The independent EOS are coupled by using equilibrium closure models that ensure consistency of the thermodynamic state of multiple materials occupying a single computational cell.7,426–429 Each material is consistent with the EOS, but this system is underdetermined without a closure model. These closure models use assumptions by necessity to ensure unique solutions, yet these closures do not account for sub-grid material distributions nor thermalization rates. The closure models that have been developed for plasmas assume thermal equilibrium between ion species coupled with either pressure or number density equilibrium. The different approaches have different ranges of validity. Only pressure-temperature equilibration produces exact state relations for neutral (non-ionized) ideal gases428 and only the electron number density equilibration approach429 directly considers the impact of ionization and electron and ion species that are out of equilibrium. MD simulations430–433 and simulations employing free-energy minimization434 show reasonable broad agreement with the electron number density approach,429,434 less broad agreement with pressure-temperature equilibration, and the worst agreement with the use of Dalton's law. Dedicated experiments435 have been unable to distinguish between mixing rules.

These mixing rules and the simulations used to validate them all assume that mixed materials have sufficient time to achieve equilibrium. As noted above, this is impossible within a single time step in any computational cell, since the time step must be set lower than the sound crossing time of any cell to ensure numerical stability. Recent experiments performed on the National Ignition Facility have provided evidence of persistent ion species temperature separation in dynamic ICF mixtures.35,36,61,62,436 This effect has also been predicted independently in simulation.90,91 Transient species temperature separation is also observed in the wake of strong shocks.437–439 Without allowing for the possibility of persistent nonequilibrium temperatures in thermodynamic closure models, it is only possible to account for this in simulation using computationally prohibitive extremely high-resolution three-dimensional simulations and allowing thermal conduction to mediate the thermalization. It has been shown that neglecting incomplete thermalization on estimates of contaminant in ICF hot spots increases the uncertainty by a factor of 2 or more.35 

We demonstrate persistent ion species temperature separation in Fig. 18, reproduced from.62 Here, the burn-weighted ion temperature is plotted for DT and DD reactions for two series of MARBLE implosions.35,61,62,440 These implosions contain engineered deuterated plastic foams whose pores are filled with mixtures of hydrogen and tritium (HT) or argon and tritium (ArT). For the HT gas fill, the DT and DD burn-weighted ion temperatures are separated by 2 keV, whereas for the ArT gas fill, temperatures are separated by <200 eV. This indicates that for the implosions with HT fill, the HT and CD foam do not achieve thermal equilibrium during the course of the implosion. This interpretation is backed up by simulations performed in Ref. 36.

FIG. 18.

Burn-weighted ion temperature measured for DT (red squares) and DD (blue circles) reactions in MARBLE implosions on the National Ignition Facility using capsules containing deuterated foams and ArT (solid) or HT (open) gas fills.62 This figure has been reproduced from Albright et al., Phys. Plasmas 22, 022702 (2022).62 Copyright 2022 the Authors, licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

FIG. 18.

Burn-weighted ion temperature measured for DT (red squares) and DD (blue circles) reactions in MARBLE implosions on the National Ignition Facility using capsules containing deuterated foams and ArT (solid) or HT (open) gas fills.62 This figure has been reproduced from Albright et al., Phys. Plasmas 22, 022702 (2022).62 Copyright 2022 the Authors, licensed under a Creative Commons Attribution 4.0 International License, http://creativecommons.org/licenses/by/4.0/.

Close modal

Most work on temperature relaxation in plasmas has focused on electron-ion equilibration, discussed below in Sec. XI, and this work relies heavily on their disparate mass. Some models have recently been developed to model the relaxation of temperatures in binary mixtures.441–443 However, these have not yet been extended to arbitrary mixtures nor adapted to replace existing closure models in radiation-hydrodynamics codes. The Bosque experimental campaign on the National Ignition Facility aims to provide data to help in the development and evaluation of such models.444 The development and implementation of such models will undoubtedly enable improved understanding of material mixing and its consequences in ICF.

It is standard practice in radiation-hydrodynamics models to treat ions and electrons as separate species with independent temperatures and pressures, even when different ion species are assumed to be in equilibrium.445–447 Supplemented with the radiation temperature, this is commonly referred to as a three temperature model. These models allow separate energy flows to and from the electrons and ions, which is critical to the accurate modeling of plasma shocks, laser energy deposition, thermonuclear burn, and non-local ion and electron transport, among other physics. These models require the evaluation of equilibration rates between species, which come in the form of opacities for electron-radiation coupling and electron-ion equilibration rates for electron-ion coupling.

Electron-ion equilibration rates are important in regions where energy is rapidly deposited preferentially into one species over another. In particular, shocks deposit energy preferentially into ions, lasers deposit energy preferentially into electrons, and alpha particles preferentially heat electrons. It is critical to simulate this physics correctly since it determines the split between Te and Ti and, for example, ICF capsule performance can be very sensitive to this split since Ti determines reaction rates and Te determines radiative loss rates and the ability for alpha heating to effectively increase reaction rates relies on rapid electron-ion energy coupling. There is evidence that radiation-hydrodynamics simulations do not accurately capture the ratio of emissivity weighted Te to burn-weighted Ti measured in experiment448 for some classes of experiments, and inaccurate modeling of electron-ion equilibration rates will contribute to this.

Radiation-hydrodynamics codes typically employ simple classical formulas for electron-ion coupling from Refs. 230, 449, and 450, which are valid in the limit of high temperatures and low densities, due to the ease of implementation for a wide range of materials and mixtures. Nevertheless, there is evidence that these formulas over-estimate equilibration rates.451–459 Several efforts have produced formulas with a broader range of applicability,297,460–467 though these produce widely varying predictions of equilibration rates that exhibit differences as large as three orders of magnitude.468 Molecular dynamics simulations have been used to discriminate between models469–473 and these tend to show the best agreement with the Gericke–Murillo–Schlanges462 and Brown-Preston–Singleton297 models. More recent MD-based models provide a more complete coverage of materials and phase space, including Daligault–Simoni465,468 and Medvedev–Milov.466 Nevertheless, comparisons between models and experiments have only been done for a fairly limited set of elements and discrepancies with experiments remain even for the best models.

A systematic exploration of sensitivities of simulations of ICF or HEDP experiments to electron-ion equilibration models has not been carried out, likely because the more sophisticated models are rarely implemented into codes, so it is not clear how sensitive integrated results will be to these results. Nevertheless, since electron-ion equilibration powers are comparable to those of hydrodynamic compression and radiation in ICF hot spots (see, e.g., Ref. 474) it is unlikely that improvements to the accuracy of electron-ion equilibration coefficients will not improve simulation quality.

We have surveyed uncertainties in charged particle transport coefficients important to the modeling of high energy density physics (HEDP) and inertial confinement fusion (ICF) experiments, with a focus on those that could explain important remaining discrepancies between radiation-hydrodynamics simulation and experiment. We have discussed challenges related to coefficients used for laser propagation and absorption, viscosity and mass transport, thermal conductivity, non-local ion (including fusion product) transport, non-local electron thermal conduction, electrical conduction, magnetohydrodynamics, multi-ion-species thermalization, and electron-ion equilibration.

Most of the challenges in providing accurate charged particle transport coefficients for radiation-hydrodynamics simulations fall into three categories. First, materials in HEDP and ICF experiments transit the warm dense matter (WDM) regime, in which both kinetic and quantum effects are important. As a result, it is particularly difficult to derive analytic formulas for coefficients and to develop accurate simulation methodologies for calculating coefficients in this regime. Experimental measurements of coefficients that can constrain models are also exceedingly rare in the WDM regime. Second, transport coefficients are highly sensitive to material distributions and material mixing, which may not be accurately captured in simulation. Simulations utilize analytic formulas or tables to evaluate transport coefficients given local state data, and when multiple materials reside within a computational cell, these coefficients must be combined using mixing rules. Unless simulations are performed in three-dimensions using extremely high-resolution, the coefficient for the combined material will depend on how the materials are distributed. Furthermore, the distribution of materials may also cause an isotropic coefficient to behave anisotropically. Finally, coefficients exhibit coupling to disparate physics. For example, nearly all charged particle transport coefficients will be impacted by imposed or self-generated magnetic fields. Models for this coupling are typically inadequate and those that are available are frequently neglected in radiation-hydrodynamics codes. Inadequate or inaccurate coupling between physics could easily be misinterpreted as errors in transport coefficients.

Despite the challenges, great strides have been made in charged particle transport coefficients, and the full impact of many of the improvements will only be known as they become more widely adopted over the coming years. Improvements to laser absorption modeling, for both inverse bremsstrahlung and during the solid to plasma transition, have improved predictions of energy coupling and laser imprint. Implementation of plasma mass transport physics, which has relied on the recent development of analytic coefficients suitable for implementation into highly non-linear physics algorithms, have offered potential explanations for many observed phenomena such as anomalous yield ratios. Improved thermal conductivities have brought simulated predictions of ICF capsule compression and temperatures closer to experimental measurements. It would not have been possible to design the capsules that achieved ignition and gain on the National Ignition Facility without high quality ion stopping power models.

Our survey suggests several low-risk paths forward for improving the predictive capability of radiation-hydrodynamics simulations for HEDP and ICF. For example, improvements to magneto-hydrodynamics (MHD) modeling and coefficients and coupling to other physics, which will rely on improving coupling coefficients, will help to improve predictions of energy flows in hohlraums and pulsed power targets. Improved understanding of material distributions and how these impact conductivities will also improve the modeling of energy flows in hohlraums. The wider adoption of improved inverse bremsstrahlung coefficients and models for the solid-to-plasma transition in laser-illuminated targets will improve predictions of energy coupling and laser imprint. In addition, the development of laser-plasma instability source terms and improved coefficients (including coupling to MHD) for non-local electron conduction, could help to improve predictions of laser-target coupling and compression. The development of non-equilibrium thermalization models for multiple ion species will be critical to understanding the development and impact of material mixing. The development, implementation, and routine usage of improved models for plasma transport and viscosity will improve predictions of hydrodynamic instabilities and associated flows. Improved electrical conductivities and models for the target-vacuum interface will enable better predictions of target-driver coupling for pulsed power. Constraining charged particle, particularly alpha particle, stopping powers would improve the ability of simulations to predict the yields of igniting capsules. Finally, the implementation of improved electron-ion coupling coefficients will improve predictions of strong shock and hotspot energetics in ICF.

The author would like to thank the attendees of the Second Charged Particle Transport Coefficient Code Comparison Workshop held in Livermore, California on July 24–27, 2023 for useful discussions and input. The author would also like to thank B. Keenan, P. Knapp, O. Schilling, and E. Vold for useful discussions. Los Alamos National Laboratory is managed by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218CNA000001.

The authors have no conflicts to disclose.

Brian Michael Haines: Conceptualization (lead); Data curation (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

1.
J.
Nuckolls
,
L.
Wood
,
A.
Thiessen
, and
G.
Zimmerman
, “
Laser compression of matter to super-high densities: CTR applications
,”
Nature
239
,
139
(
1972
).
2.
H.
Takabe
and
Y.
Kuramitsu
, “
Recent progress of laboratory astrophysics with intense lasers
,”
High Power Laser Sci. Eng.
9
,
e49
(
2021
).
3.
H.
Abu-Shawareb
,
R.
Acree
,
P.
Adams
,
J.
Adams
,
B.
Addis
,
R.
Aden
,
P.
Adrian
,
B. B.
Afeyan
,
M.
Aggleton
,
L.
Aghaian
et al, “
Lawson criterion for ignition exceeded in an inertial fusion experiment
,”
Phys. Rev. Lett.
129
(7)
,
075001
(
2022
).
4.
H.
Abu-Shawareb
,
R.
Acree
,
P.
Adams
,
J.
Adams
,
B.
Addis
,
R.
Aden
,
P.
Adrian
,
B. B.
Afeyan
,
M.
Aggleton
,
L.
Aghaian
et al, “
Achievement of target gain, larger than unity in an inertial fusion experiment
,”
Phys. Rev. Lett.
132
,
065102
(
2024
).
5.
M. M.
Marinak
,
G. D.
Kerbel
,
N. A.
Gentile
,
O.
Jones
,
D.
Munro
,
S.
Pollaine
,
T. R.
Dittrich
, and
S. W.
Haan
, “
Three-dimensional HYDRA simulations of National Ignition Facility targets
,”
Phys. Plasmas
8
,
2275
(
2001
).
6.
G. B.
Zimmerman
and
W. L.
Kruer
, “
Numerical simulation of laser-initiated fusion
,”
Comments Plasma Phys. Controlled Fusion
2
,
51
60
(
1975
).
7.
M.
Gittings
,
R.
Weaver
,
M.
Clover
,
T.
Betlach
,
N.
Byrne
,
R.
Coker
,
E.
Dendy
,
R.
Hueckstaedt
,
K.
New
,
W. R.
Oakes
,
D.
Ranta
, and
R.
Stefan
, “
The RAGE radiation-hydrodynamic code
,”
Comput. Sci. Discovery
1
,
015005
(
2008
).
8.
B. M.
Haines
,
C. H.
Aldrich
,
J. M.
Campbell
,
R. M.
Rauenzahn
, and
C. A.
Wingate
, “
High-resolution modeling of indirectly driven high-convergence layered inertial confinement fusion capsule implosions
,”
Phys. Plasmas
24
,
052701
(
2017
).
9.
B. M.
Haines
,
D. E.
Keller
,
J. A.
Marozas
,
P. W.
McKenty
,
K. S.
Anderson
,
T. J. B.
Collins
,
W. W.
Dai
,
M. L.
Hall
,
S.
Jones
,
M. D.
McKay
, Jr.
,
R. M.
Rauenzahn
, and
D. N.
Woods
, “
Coupling laser physics to radiation-hydrodynamics
,”
Comput. Fluids
201
,
104478
(
2020
).
10.
B. M.
Haines
,
D. E.
Keller
,
K. P.
Long
,
M. D.
Mckay
, Jr.
,
Z. J.
Medin
,
H.
Park
,
R. M.
Rauenzahn
,
H. A.
Scott
,
K. S.
Anderson
,
T. J. B.
Collins
,
L. M.
Green
,
J. A.
Marozas
,
P. W.
Mckenty
,
J. H.
Peterson
,
E. L.
Vold
,
C.
Di Stefano
,
R. S.
Lester
,
J. P.
Sauppe
,
D. J.
Stark
, and
J.
Velechovsky
, “
The development of a high-resolution Eulerian radiation-hydrodynamics simulation capability for laser-driven hohlraums
,”
Phys. Plasmas
29
,
083901
(
2022
).
11.
E.
Lefebvre
,
S.
Bernard
,
C.
Esnault
,
P.
Gauthier
,
A.
Grisollet
,
P.
Hoch
,
L.
Jacquet
,
G.
Kluth
,
S.
Laffite
,
S.
Liberatore
,
I.
Marmajou
,
P.-E.
Masson-Laborde
,
O.
Morice
, and
J.-L.
Willien
, “
Development and validation of the TROLL radiation-hydrodynamics code for 3D hohlraum calculations
,”
Nucl. Fusion
59
(
3
),
032010
(
2019
).
12.
D.
Keller
,
T. J. B.
Collins
,
J. A.
Delettrez
,
P. W.
Mckenty
,
P. B.
Radha
,
B.
Whitney
, and
G. A.
Moses
, “
DRACO—A new multidimensional hydrocode
,”
Bull. Am. Phys. Soc.
DPP 41 BP1.39 (
1999
).
13.
P. B.
Radha
,
V. N.
Goncharov
,
T. J. B.
Collins
,
J. A.
Delettrez
,
Y.
Elbaz
,
V. Y.
Glebov
,
R. L.
Keck
,
D. E.
Keller
,
J. P.
Knauer
,
J. A.
Marozas
,
F. J.
Marshall
,
P. W.
Mckenty
,
D. D.
Meyerhofer
,
S. P.
Regan
,
T. C.
Sangster
,
D.
Shvarts
,
S.
Skupsky
,
Y.
Srebro
,
R. P. J.
Town
, and
C.
Stoeckl
, “
Two-dimensional simulations of plastic-shell, direct-drive implosions on OMEGA
,”
Phys. Plasmas
12
,
032702
(
2005
).
14.
J.
Delettrez
,
R.
Epstein
,
M. C.
Richardson
,
P. A.
Jaanimagi
, and
B. L.
Henke
, “
Effect of laser illumination nonuniformity on the analysis of time-resolved x-ray measurements in UV spherical transport experiments
,”
Phys. Rev. A
36
,
3926
(
1987
).
15.
I. V.
Igumenshchev
,
V. N.
Goncharov
,
F. J.
Marshall
,
J. P.
Knauer
,
E. M.
Campbell
,
C. J.
Forrest
,
D. H.
Froula
,
V. Y.
Glebov
,
R. L.
McCrory
,
S. P.
Regan
,
T. C.
Sangster
,
S.
Skupsky
, and
C.
Stoeckl
, “
Three-dimensional modeling of direct-drive cryogenic implosions on OMEGA
,”
Phys. Plasmas
23
,
052702
(
2016
).
16.
I. V.
Igumenshchev
,
D. T.
Michel
,
R. C.
Shah
,
E. M.
Campbell
,
R.
Epstein
,
C. J.
Forrest
,
V. Y.
Glebov
,
V. N.
Goncharov
,
J. P.
Knauer
,
F. J.
Marshall
,
R. L.
Mccrory
,
S. P.
Regan
,
T. C.
Sangster
,
C.
Stoeckl
,
A. J.
Schmitt
, and
S.
Obenschain
, “
Three-dimensional hydrodynamic simulations of OMEGA implosions
,”
Phys. Plasmas
24
,
056307
(
2017
).
17.
A. J.
Schmitt
,
D. G.
Colombant
,
A. L.
Velikovich
,
S. T.
Zalesak
,
J. H.
Gardner
,
D. E.
Fyfe
, and
N.
Metzler
, “
Large-scale high-resolution simulations of high gain direct-drive inertial confinement fusion targets
,”
Phys. Plasmas
11
,
2716
2722
(
2004
).
18.
J. J.
Honrubia
, “
A synthetically accelerated scheme for radiative transfer calculations
,”
J. Quant. Spectrosc. Radiat. Transfer
49
,
491
(
1993
).
19.
S.
Atzeni
, “
2-D Lagrangian studies of symmetry and stability of laser fusion targets
,”
Comput. Phys. Commun.
43
,
107
(
1986
).
20.
J. P.
Chittenden
,
B. D.
Appelbe
,
F.
Manke
,
K.
Mcglinchey
, and
N. P. L.
Niasse
, “
Signatures of asymmetry in neutron spectra and images predicted by three-dimensional radiation hydrodynamics simulations of indirect drive implosions
,”
Phys. Plasmas
23
,
052708
(
2016
).
21.
J.
Breil
and
P.-H.
Maire
, “
A cell-centered diffusion scheme on two-dimensional unstructured meshes
,”
J. Comp. Physiol.
224
,
785
823
(
2007
).
22.
S. A.
Slutz
and
R. A.
Vesey
, “
High-gain magnetized inertial fusion
,”
Phys. Rev. Lett.
108
(
2
),
025003
(
2012
).
23.
A. C.
Robinson
,
T. A.
Brunner
, and
S.
Carroll
, “
ALEGRA: An arbitrary lagrange-eulerian multimaterial, multiphysics code
,” AIAA Paper No. 2008-1235,
2008
.
24.
R. M.
Darlington
,
T. L.
Mcabee
, and
G.
Rodrigue
, “
A study of ALEGRA simulations of rayleigh-taylor instability
,”
Comput. Phys. Commun.
135
,
58
73
(
2001
).
25.
B. E.
Morgan
and
J. A.
Greenough
, “
Large-eddy and unsteady RANS simulations of a shock-accelerated heavy gas cylinder
,”
Shock Waves
26
,
355
383
(
2016
).
26.
C.
Leland Ellison
,
H. D.
Whitley
,
C. R. D.
Brown
,
S. R.
Copeland
,
W. J.
Garbett
,
H. P.
Le
,
M. B.
Schneider
,
Z. B.
Walters
,
H.
Chen
,
J. I.
Castor
,
R. S.
Craxton
,
M.
Gatu Johnson
,
E. M.
Garcia
,
F. R.
Graziani
,
G. E.
Kemp
,
C. M.
Krauland
,
P. W.
McKenty
,
B.
Lahmann
,
J. E.
Pino
,
M. S.
Rubery
,
H. A.
Scott
,
R.
Shepherd
, and
H.
Sio
, “
Development and modeling of a polar-direct-drive exploding pusher platform at the national ignition facility
,”
Phys. Plasmas
25
,
072710
(
2018
).
27.
J. D.
Bender
,
O.
Schilling
,
K. S.
Raman
,
R. A.
Managan
,
B. J.
Olson
,
S. R.
Copeland
,
C. L.
Ellison
,
D. J.
Erskine
,
C. M.
Huntington
,
B. E.
Morgan
,
S. R.
Nagel
,
S. T.
Prisbrey
,
B. S.
Pudliner
,
P. A.
Sterne
,
C. E.
Wehrenberg
, and
Y.
Zhou
, “
Simulation and flow physics of a shocked and reshocked high-energy-density mixing layer
,”
J. Fluid Mech.
915
,
A84
(
2021
).
28.
J. P.
Chittenden
,
S. V.
Lebedev
,
S. N.
Bland
,
F. N.
Beg
, and
M. G.
Haines
, “
One-, two-, and three-dimensional modeling of the different phases of wire array Z-pinch evolution
,”
Phys. Plasmas
8
,
2305
2314
(
2001
).
29.
J. P.
Chittenden
,
S. V.
Lebedev
,
C. A.
Jennings
,
S. N.
Bland
, and
A.
Ciardi
, “
X-ray generation mechanisms in three-dimensional simulations of wire array Z-pinches
,”
Plasma Phys. Controlled Fusion
46
,
B457
(
2004
).
30.
C. E.
Seyler
and
M. R.
Martin
, “
Relaxation model for extended magnetohydrodynamics: Comparison to magnetohydrodynamics for dense Z-pinches
,”
Phys. Plasmas
18
,
012703
(
2011
).
31.
B.
Fryxell
,
K.
Olson
,
P.
Ricker
,
F. X.
Timmes
,
M.
Zingale
,
D. Q.
Lamb
,
P.
MacNeice
,
R.
Rosner
,
J. W.
Truran
, and
H.
Tufo
, “
FLASH: An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes
,”
Astrophys. J., Suppl. Ser.
131
,
273
(
2000
).
32.
P.
Tzeferacos
,
M.
Fatenejad
,
N.
Flocke
,
C.
Graziani
,
G.
Gregori
,
D. Q.
Lamb
,
D.
Lee
,
J.
Meinecke
,
A.
Scopatz
, and
K.
Weide
, “
FLASH MHD simulations of experiments that study shock-generated magnetic fields
,”
High Energy Density Phys.
17
,
24
31
(
2015
).
33.
A.
Reyes
,
D.
Lee
,
C.
Graziani
, and
P.
Tzeferacos
, “
A variable high-order shock-capturing finite difference method with GP-WENO
,”
J. Comput. Phys.
381
,
189
217
(
2019
)
34.
D. S.
Clark
,
C. R.
Weber
,
J. L.
Milovich
,
A. E.
Pak
,
D. T.
Casey
,
B. A.
Hammel
,
D. D.
Ho
,
O. S.
Jones
,
J. M.
Koning
,
A. L.
Kritcher
,
M. M.
Marinak
,
L. P.
Masse
,
D. H.
Munro
,
M. V.
Patel
,
P. K.
Patel
,
H. F.
Robey
,
C. R.
Schroeder
,
S. M.
Sepke
, and
M. J.
Edwards
, “
Three-dimensional modeling and hydrodynamic scaling of National Ignition Facility implosions
,”
Phys. Plasmas
26
,
050601
(
2019
).
35.
B. M.
Haines
,
R. C.
Shah
,
J. M.
Smidt
,
B. J.
Albright
,
T.
Cardenas
,
M. R.
Douglas
,
C.
Forrest
,
V. Yu.
Glebov
,
M. A.
Gunderson
,
C. E.
Hamilton
,
K. C.
Henderson
,
Y.
Kim
,
M. N.
Lee
,
T. J.
Murphy
,
J. A.
Oertel
,
R. E.
Olson
,
B. M.
Patterson
,
R. B.
Randolph
, and
D. W.
Schmidt
, “
Observation of persistent species temperature separation in inertial confinement fusion mixtures
,”
Nat. Commun.
11
,
544
(
2020
).
36.
B. M.
Haines
,
T. J.
Murphy
,
R. E.
Olson
,
Y.
Kim
,
B. J.
Albright
,
B.
Appelbe
,
T. H.
Day
,
M. A.
Gunderson
,
C. E.
Hamilton
,
T.
Morrow
, and
B. M.
Patterson
, “
The dynamics, mixing, and thermonuclear burn of compressed foams with varied gas fills
,”
Phys. Plasmas
30
,
072705
(
2023
).
37.
G. A.
Shipley
and
T. J.
Awe
, “
Three-dimensional magnetohydrodynamic modeling of auto-magnetizing liner implosions on the Z accelerator
,”
Phys. Plasmas
30
,
102707
(
2023
).
38.
D. A.
Callahan
,
O. A.
Hurricane
,
J. E.
Ralph
,
C. A.
Thomas
,
K. L.
Baker
,
L. R.
Benedetti
,
L. F.
Berzak Hopkins
,
D. T.
Casey
,
T.
Chapman
,
C. E.
Czajka
,
E. L.
Dewald
,
L.
Divol
,
T.
Döppner
,
D. E.
Hinkel
,
M.
Hohenberger
,
L. C.
Jarrot
,
S. F.
Khan
,
A. L.
Kritcher
,
O. L.
Landen
,
S.
Lepape
,
S. A.
Maclaren
,
L. P.
Masse
,
N. B.
Meezan
,
A. E.
Pak
,
J. D.
Salmonson
,
D. T.
Woods
,
N.
Izumi
,
T.
Ma
,
D. A.
Mariscal
,
S. R.
Nagel
,
J. L.
Kline
,
G. A.
Kyrala
,
E. N.
Loomis
,
S. A.
Yi
,
A. B.
Zylstra
, and
S. H.
Batha
, “
Exploring the limits of case-to-capsule ratio, pulse length, and picket energy for symmetric hohlraum drive on the National Ignition Facility laser
,”
Phys. Plasmas
25
,
056305
(
2018
).
39.
N. B.
Meezan
,
D. T.
Woods
,
N.
Izumi
,
H.
Chen
,
H. A.
Scott
,
M. B.
Schneider
,
D. A.
Liedahl
,
O. S.
Jones
,
G. B.
Zimmerman
,
J. D.
Moody
,
O. L.
Landen
, and
W. W.
Hsing
, “
Evidence of restricted heat transport in National Ignition Facility hohlraums
,”
Phys. Plasmas
27
,
102704
(
2020
).
40.
C. E.
Seyler
,
M. R.
Martin
, and
N. D.
Hamlin
, “
Helical instability in MagLIF due to axial flux compression by low-density plasma
,”
Phys. Plasmas
25
,
062711
(
2018
).
41.
N. D.
Hamlin
and
C. E.
Seyler
, “
Power flow in pulsed-power systems: The influence of hall physics and modeling of the plasma–vacuum interface
,”
IEEE Trans. Plasma Sci.
47
(
5
),
2064
2073
(
2019
).
42.
R. L.
Masti
,
C. L.
Ellison
,
J. R.
King
,
P. H.
Stoltz
, and
B.
Srinivasan
, “
Cross-code verification and sensitivity analysis to effectively model the electrothermal instability
,”
High Energy Density Phys.
38
,
100925
(
2021
).
43.
J. A.
Marozas
,
M.
Hohenberger
,
M. J.
Rosenberg
,
D.
Turnbull
,
T. J. B.
Collins
,
P. B.
Radha
,
P. W.
Mckenty
,
J. D.
Zuegel
,
F. J.
Marshall
,
S. P.
Regan
,
T. C.
Sangster
,
W.
Seka
,
E. M.
Campbell
,
V. N.
Goncharov
,
M. W.
Bowers
,
J.-M. G.
Di Nicola
,
G.
Erbert
,
B. J.
MacGowan
,
L. J.
Pelz
, and
S. T.
Yang
, “
First observation of cross-beam energy transfer mitigation for direct-drive inertial confinement fusion implosions using wavelength detuning at the National Ignition Facility
,”
Phys. Rev. Lett.
120
,
085001
(
2018
).
44.
J. L.
Peebles
,
S. X.
Hu
,
W.
Theobald
,
V. N.
Goncharov
,
N.
Whiting
,
P. M.
Celliers
,
S. J.
Ali
,
G.
Duchateau
,
E. M.
Campbell
,
T. R.
Boehly
, and
S. P.
Regan
, “
Direct-drive measurements of laser-imprint-induced shock velocity nonuniformities
,”
Phys. Rev. E
99
(
6
),
063208
(
2019
).
45.
J.
Oh
,
A. J.
Schmitt
,
M.
Karasik
, and
S. P.
Obenschain
, “
Measurements of laser-imprint-induced shock velocity nonuniformities in plastic targets with the nike KrF laser
,”
Phys. Plasmas
28
,
032704
(
2021
).
46.
M. J.
Rosenberg
,
A. A.
Solodov
,
J. F.
Myatt
,
W.
Seka
,
P.
Michel
,
M.
Hohenberger
,
R. W.
Short
,
R.
Epstein
,
S. P.
Regan
,
E. M.
Campbell
,
T.
Chapman
,
C.
Goyon
,
J. E.
Ralph
,
M. A.
Barrios
,
J. D.
Moody
, and
J. W.
Bates
, “
Origins and scaling of hot-electron preheat in ignition-scale direct-drive inertial confinement fusion experiments
,”
Phys. Rev. Lett.
120
,
055001
(
2018
).
47.
A. A.
Solodov
,
M. J.
Rosenberg
,
W.
Seka
,
J. F.
Myatt
,
M.
Hohenberger
,
R.
Epstein
,
C.
Stoeckl
,
R. W.
Short
,
S. P.
Regan
,
P.
Michel
,
T.
Chapman
,
R. K.
Follett
,
J. P.
Palastro
,
D. H.
Froula
,
P. B.
Radha
,
J. D.
Moody
, and
V. N.
Goncharov
, “
Hot-electron generation at direct-drive ignition-relevant plasma conditions at the National Iignition Facility
,”
Phys. Plasmas
27
,
052706
(
2020
).
48.
B. M.
Haines
,
J. P.
Sauppe
,
B. J.
Albright
,
W. S.
Daughton
,
S. M.
Finnegan
,
J. L.
Kline
, and
J. M.
Smidt
, “
A mechanism for reduced compression in indirectly driven layered capsule implosions
,”
Phys. Plasmas
29
,
042704
(
2022
).
49.
S. C.
Miller
and
V. N.
Goncharov
, “
Instability seeding mechanisms due to internal defects in inertial confinement fusion targets
,”
Phys. Plasmas
29
,
082701
(
2022
).
50.
S.
Davidovits
,
C. R.
Weber
, and
D. S.
Clark
, “
Modeling ablator grain structure impacts on ICF implosions
,”
Phys. Plasmas
29
,
112708
(
2022
).
51.
D. S.
Clark
,
D. E.
Hinkel
,
D. C.
Eder
,
O. S.
Jones
,
S. W.
Haan
,
B. A.
Hammel
,
M. M.
Marinak
,
J. L.
Milovich
,
H. F.
Robey
,
L. J.
Suter
, and
R. P. J.
Town
, “
Detailed implosion modeling of deuterium-tritium layered experiments on the National Ignition Facility
,”
Phys. Plasmas
20
,
056318
(
2013
).
52.
B. M.
Haines
,
F. F.
Grinstein
, and
J. R.
Fincke
, “
Three-dimensional simulation strategy to determine the effects of turbulent mixing on inertial-confinement-fusion capsule performance
,”
Phys. Rev. E
89
,
053302
(
2014
).
53.
D. S.
Clark
,
M. M.
Marinak
,
C. R.
Weber
,
D. C.
Eder
,
S. W.
Haan
,
B. A.
Hammel
,
D. E.
Hinkel
,
O. S.
Jones
,
J. L.
Milovich
,
P. K.
Patel
,
H. F.
Robey
,
J. D.
Salmonson
,
S. M.
Sepke
, and
C. A.
Thomas
, “
Radiation hydrodynamics modeling of the highest compression inertial confinement fusion ignition experiment from the national ignition campaign
,”
Phys. Plasmas
22
,
022703
(
2015
).
54.
C. R.
Weber
,
D. S.
Clark
,
A. W.
Cook
,
D. C.
Eder
,
S. W.
Haan
,
B. A.
Hammel
,
D. E.
Hinkel
,
O. S.
Jones
,
M. M.
Marinak
,
J. L.
Milovich
,
P. K.
Patel
,
H. F.
Robey
,
J. D.
Salmonson
,
S. M.
Sepke
, and
C. A.
Thomas
, “
Three-dimensional hydrodynamics of the deceleration stage in inertial confinement fusion
,”
Phys. Plasmas
22
,
032702
(
2015
).
55.
D. S.
Clark
,
C. R.
Weber
,
J. L.
Milovich
,
J. D.
Salmonson
,
A. L.
Kritcher
,
S. W.
Haan
,
B. A.
Hammel
,
D. E.
Hinkel
,
O. A.
Hurricane
,
O. S.
Jones
,
M. M.
Marinak
,
P. K.
Patel
,
H. F.
Robey
,
S. M.
Sepke
, and
M. J.
Edwards
, “
Three-dimensional simulations of low foot and high foot implosion experiments on the national ignition facility
,”
Phys. Plasmas
23
,
056302
(
2016
).
56.
B. M.
Haines
,
G. P.
Grim
,
J. R.
Fincke
,
R. C.
Shah
,
C. J.
Forrest
,
K.
Silverstein
,
F. J.
Marshall
,
M.
Boswell
,
M. M.
Fowler
,
R. A.
Gore
,
G.
Jungman
,
A.
Klein
,
R. S.
Rundberg
,
M. J.
Steinkamp
, and
J. B.
Wilhelmy
, “
Detailed high-resolution three-dimensional simulations of OMEGA separated reactants inertial confinement fusion experiments
,”
Phys. Plasmas
23
,
072709
(
2016
).
57.
D. S.
Clark
,
A. L.
Kritcher
,
J. L.
Milovich
,
J. D.
Salmonson
,
C. R.
Weber
,
S. W.
Haan
,
B. A.
Hammel
,
D. E.
Hinkel
,
M. M.
Marinak
,
M. V.
Patel
, and
S. M.
Sepke
, “
Capsule modeling of high foot implosion experiments on the National Ignition Facility
,”
Plasma Phys. Controlled Fusion
59
,
055006
(
2017
).
58.
D. S.
Clark
,
C. R.
Weber
,
A. L.
Kritcher
,
J. L.
Milovich
,
P. K.
Patel
,
S. W.
Haan
,
B. A.
Hammel
,
J. M.
Koning
,
M. M.
Marinak
,
M. V.
Patel
,
C. R.
Schroeder
,
S. M.
Sepke
, and
M. J.
Edwards
, “
Modeling and projecting implosion performance for the National Ignition Facility
,”
Nucl. Fusion
59
,
032008
(
2019
).
59.
B. M.
Haines
,
R. E.
Olson
,
W.
Sweet
,
S. A.
Yi
,
A. B.
Zylstra
,
P. A.
Bradley
,
F.
Elsner
,
H.
Huang
,
R.
Jimenez
,
J. L.
Kline
,
C.
Kong
,
G. A.
Kyrala
,
R. J.
Leeper
,
R.
Paguio
,
S.
Pajoom
,
R. R.
Peterson
,
M.
Ratledge
, and
N.
Rice
, “
Robustness to hydrodynamic instabilities in indirectly-driven layered capsule implosions
,”
Phys. Plasmas
26
,
012707
(
2019
).
60.
M.
Gatu Johnson
,
B. M.
Haines
,
P. J.
Adrian
,
C.
Forrest
,
J. A.
Frenje
,
V. Y.
Glebov
,
W.
Grimble
,
R.
Janezic
,
J. P.
Knauer
,
B.
Lahmann
,
F. J.
Marshall
,
T.
Michel
,
F. H.
Séguin
,
C.
Stoeckl
, and
R. D.
Petrasso
, “
3D xRAGE simulation of inertial confinement fusion implosion with imposed mode 2 laser drive asymmetry
,”
High Energy Density Phys.
36
,
100825
(
2020
).
61.
B. M.
Haines
,
R. C.
Shah
,
J. M.
Smidt
,
B. J.
Albright
,
T.
Cardenas
,
M. R.
Douglas
,
C.
Forrest
,
V. Y.
Glebov
,
M. A.
Gunderson
,
C. E.
Hamilton
,
K. C.
Henderson
,
Y.
Kim
,
M. N.
Lee
,
T. J.
Murphy
,
J. A.
Oertel
,
R. E.
Olson
,
B. M.
Patterson
,
R. B.
Randolph
, and
D. W.
Schmidt
, “
The rate of development of atomic mixing and temperature equilibration in inertial confinement fusion implosions
,”
Phys. Plasmas
27
,
102701
(
2020
).
62.
B. J.
Albright
,
T. J.
Murphy
,
B. M.
Haines
,
M. R.
Douglas
,
J. H.
Cooley
,
T. H.
Day
,
N. A.
Denissen
,
C.
Di Stefano
,
P.
Donovan
,
S. L.
Edwards
,
J.
Fincke
,
L. M.
Green
,
L.
Goodwin
,
R. A.
Gore
,
M. A.
Gunderson
,
J. R.
Haack
,
C. E.
Hamilton
,
E. P.
Hartouni
,
N. V.
Kabadi
,
S.
Khan
,
P. M.
Kozlowski
,
Y.
Kim
,
M. N.
Lee
,
R.
Lester
,
T.
Morrow
,
J. A.
Oertel
,
R. E.
Olson
,
B. M.
Patterson
,
T.
Quintana
,
R. B.
Randolph
,
D. W.
Schmidt
,
R. C.
Shah
,
J. M.
Smidt
,
A.
Strickland
,
C.
Wilson
, and
L.
Yin
, “
Experimental quantification of the impact of heterogeneous mix on thermonuclear burn
,”
Phys. Plasmas
29
,
022702
(
2022
).
63.
J. P.
Sauppe
,
Y.
Lu
,
P.
Tzeferacos
,
A. C.
Reyes
,
S.
Palaniyappan
,
K. A.
Flippo
,
S.
Li
, and
J. L.
Kline
, “
On the importance of three-dimensional modeling for high-energy-density physics experiments
,”
Phys. Plasmas
30
,
062707
(
2023
).
64.
L. J.
Stanek
,
A.
Kononov
,
S. B.
Hansen
,
B. M.
Haines
,
S. X.
Hu
,
P. F.
Knapp
,
M. S.
Murillo
,
L. G.
Stanton
,
H. D.
Whitley
,
S. D.
Baalrud
,
L.
Babati
,
A. D.
Baczewski
,
M.
Bethkenhagen
,
A.
Blanchet
,
R. C.
Clay
,
I. I. I. K. R.
Cochrane
,
L. A.
Collins
,
A.
Dumi
,
G.
Faussurier
,
M.
French
,
V. V.
Karasiev
,
S.
Kumar
,
M. K.
Lentz
,
C. A.
Melton
,
K. A.
Nichols
,
G. M.
Petrov
,
V.
Recoules
,
R.
Redmer
,
G.
Röpke
,
M.
Schörner
,
N. R.
Shaffer
,
V.
Sharma
,
L. G.
Silvestri
,
F.
Soubiran
,
P.
Suryanarayana
,
M.
Tacu
,
J. P.
Townsend
, and
A. J.
White
, “
Review of the second charged-particle transport coefficient code comparison orkshop
,”
Phys. Plasmas
31
,
052104
(
2024
).
65.
K. L.
Wong
,
M. J.
May
,
P.
Beiersdorfer
,
K. B.
Fournier
,
B.
Wilson
,
G. V.
Brown
,
P.
Springer
,
P. A.
Neill
, and
C. L.
Harris
, “
Determination of the charge state distribution of a highly ionized coronal au plasma
,”
Phys. Rev. Lett.
90
,
235001
(
2003
).
66.
R. F.
Heeter
,
S. B.
Hansen
,
K. B.
Fournier
,
M. E.
Foord
,
D. H.
Froula
,
A. J.
Mackinnon
,
M. J.
May
,
M. B.
Schneider
, and
B. K. F.
Young
, “
Benchmark measurements of the ionization balance of non-local-thermodynamic-equilibrium gold plasmas
,”
Phys. Rev. Lett.
99
,
195001
(
2007
).
67.
M. J.
May
,
S. B.
Hansen
,
J.
Scofield
,
M.
Schneider
,
K.
Wong
, and
P.
Beiersdorfer
, “
Gold charge state distributions in highly ionized, low-density beam plasmas
,”
Phys. Rev. E
84
,
046402
(
2011
).
68.
J. E.
Bailey
,
T.
Nagayama
,
G. P.
Loisel
,
G. A.
Rochau
,
C.
Blancard
,
J.
Colgan
,
P.
Cosse
,
G.
Faussurier
,
C. J.
Fontes
,
F.
Gilleron
,
I.
Golovkin
,
S. B.
Hansen
,
C. A.
Iglesias
,
D. P.
Kilcrease
,
J. J.
MacFarlane
,
R. C.
Mancini
,
S. N.
Nahar
,
C.
Orban
,
J.-C.
Pain
,
A. K.
Pradhan
,
M.
Sherrill
, and
B. G.
Wilson
, “
A higher-than-predicted measurement of iron opacity at solar interior temperatures
,”
Nature
517
,
56
59
(
2015
).
69.
S. X.
Hu
,
B.
Militzer
,
V. N.
Goncharov
, and
S.
Skupsky
, “
First-principles equation-of-state table of deuterium for inertial confinement fusion applications
,”
Phys. Rev. B
84
,
224109
(
2011
)
70.
S. X.
Hu
,
V. N.
Goncharov
,
T. R.
Boehly
,
R. L.
Mccrory
,
S.
Skupsky
,
L. A.
Collins
,
J. D.
Kress
, and
B.
Militzer
, “
Impact of first-principles properties of deuterium-tritium on inertial confinement fusion target designs
,”
Phys. Plasmas
22
,
056304
(
2015
).
71.
D. I.
Mihaylov
,
V. V.
Karasiev
,
S. X.
Hu
,
J. R.
Rygg
,
V. N.
Goncharov
, and
G. W.
Collins
, “
Improved first-principles equation-of-state table of deuterium for high-energy-density applications
,”
Phys. Rev. B
144
,
144104
(
2021
).
72.
S. X.
Hu
,
L. A.
Collins
,
T. R.
Boehly
,
Y. H.
Ding
,
P. B.
Radha
,
V. N.
Goncharov
,
V. V.
Karasiev
,
G. W.
Collins
,
S. P.
Regan
, and
E. M.
Campbell
, “
A review on ab initio studies of static, transport, and optical properties of polystyrene under extreme conditions for inertial confinement fusion
,”
Phys. Plasmas
25
,
056306
(
2018
).
73.
J.
Clérouin
,
P.
Noiret
,
P.
Blottiau
,
V.
Recoules
,
B.
Siberchicot
,
P.
Renaudin
,
C.
Blancard
,
G.
Faussurier
,
B.
Holst
, and
C. E.
Starrett
, “
A database for equations of state and resistivities measurements in the warm dense matter regime
,”
Phys. Plasmas
19
,
082702
(
2012
).
74.
O.
Schilling
,
Evaluation of Transport Coefficient Models for Plasma and Warm Dense Matter Applications
(
Lawrence Livermore National Laboratory
,
2023
).
75.
See https://science.osti.gov/-/media/fes/fesac/pdf/2009/Fesac_hed_lp_report.pdf for “
Fusion Energy Science Advisory Committee
.
76.
See https://www.osti.gov/servlets/purl/15009836 for “
Warm Dense Matter: An Overview
.”
77.
S.
Jiang
,
O. L.
Landen
,
H. D.
Whitley
,
S.
Hamel
,
R.
London
,
D. S.
Clark
,
P.
Sterne
,
S. B.
Hansen
,
S. X.
Hu
,
G. W.
Collins
, and
Y.
Ping
, “
Thermal transport in warm dense matter revealed by refraction-enhanced x-ray radiography with a deep-neural-network analysis
,”
Nat. Comm. Phys.
6
,
98
(
2023
).
78.
S. M.
Kozlov
, “
Averaging of random operators
,”
Mat. Sb.
109
,
188
202
(
1979
).
79.
E. M.
Epperlein
and
M. G.
Haines
, “
Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker-Planck equation
,”
Phys. Fluids
29
,
1029
1041
(
1986
).
80.
G. C.
Pomraning
, “
Radiative transfer and transport phenomena in stochastic media
,”
Int. J. Eng. Sci.
36
,
1595
1621
(
1998
).
81.
E. D.
Fichtl
and
A. K.
Prinja
, “
The stochastic collocation method for radiation transport in random media
,”
J. Quant. Spectrosc. Radiat. Transfer
112
,
646
659
(
2011
).
82.
A. W.
Cook
and
J. J.
Riley
, “
A subgrid model for equilibrium chemistry in turbulent flows
,”
Phys. Fluids
6
,
2868
(
1994
).
83.
See https://books.google.com/books?id=cqFDkeVABYoC for “
Theoretical and Numerical Combustion
.”
84.
See https://books.google.com/books?id=JIdxQgAACAAJ for “
Turbulent Reacting Flows
.”
85.
P. E.
Dimotakis
, “
Turbulent mixing
,”
Annu. Rev. Fluid Mech.
37
,
329
356
(
2005
).
86.
J. R.
Ristorcelli
, “
Passive scalar mixing: Analytic study of time scale ratio, variance and mix rate
,”
Phys. Fluids
18
,
075101
(
2006
).
87.
J.
Bakosi
and
J. R.
Ristorcelli
, “
Exploring the beta distribution in variable-density turbulent mixing
,”
J. Turbul.
11
,
N37
(
2010
).
88.
J.
Bakosi
and
J. R.
Ristorcelli
, “
Extending the langevin model to variable-density pressure-gradient-driven turbulence
,”
J. Turbul.
12
,
N19
(
2011
).
89.
J. R.
Ristorcelli
, “
Exact statistical results for binary mixing and reaction in variable density turbulence
,”
Phys. Fluids
29
,
020705
(
2017
).
90.
B. E.
Morgan
,
B. J.
Olson
,
W. J.
Black
, and
J. A.
Mcfarland
, “
Large-eddy simulation and Reynolds-averaged Navier–Stokes modeling of a reacting Rayleigh-Taylor mixing layer in a spherical geometry
,”
Phys. Rev. E
98
,
033111
(
2018
).
91.
B. E.
Morgan
, “
Simulation and Reynolds-averaged Navier–Stokes modeling of a three-component Rayleigh-Taylor mixing problem with thermonuclear burn
,”
Phys. Rev. E
105
,
045104
(
2022
).
92.
S. I.
Braginskii
, “
Transport processes in a plasma
,” in
Reviews of Plasma Physics
, edited by
M.
Leontovich
(
Consultants Bureau
,
New York
,
1965
).
93.
C. S.
Jones
and
M. S.
Murillo
, “
Analysis of semi-classical potentials for molecular dynamics and Monte Carlo simulations of warm dense matter
,”
High Energy Density Phys.
3
,
379
394
(
2007
).
94.
L. G.
Stanton
and
M. S.
Murillo
, “
Ionic transport in high-energy-density matter
,”
Phys. Rev. E
93
,
043203
(
2016
).
95.
L. G.
Stanton
,
S. D.
Bergeson
, and
M. S.
Murillo
, “
Transport in non-ideal, multi-species plasmas
,”
Phys. Plasmas
28
,
050401
(
2021
).
96.
See https://books.google.com/books?id=ideuDwAAQBAJ for “
The Physics of Laser Plasma Interactions
.”
97.
See https://books.google.com/books?id=CnW6EAAAQBAJ for “
Introduction to Laser-Plasma Interactions
.”
98.
T. B.
Kaiser
, “
Laser ray tracing and power deposition on an unstructured three-dimensional grid
,”
Phys. Rev. E
61
,
895
(
2000
).
99.
J. A.
Marozas
,
M.
Hohenberger
,
M. J.
Rosenberg
,
D.
Turnbull
,
T. J. B.
Collins
,
P. B.
Radha
,
P. W.
Mckenty
,
J. D.
Zuegel
,
F. J.
Marshall
,
S. P.
Regan
,
T. C.
Sangster
,
W.
Seka
,
E. M.
Campbell
,
V. N.
Goncharov
,
M. W.
Bowers
,
J.-M. G.
Di Nicola
,
G.
Erbert
,
B. J.
MacGowan
,
L. J.
Pelz
,
J.
Moody
, and
S. T.
Yang
, “
Wavelength-detuning cross-beam energy transfer mitigation scheme for direct drive: Modeling and evidence from national ignition facility implosions
,”
Phys. Plasmas
25
,
056314
(
2018
).
100.
A.
Colaïtis
,
I.
Igumenshchev
,
J.
Mathiaud
, and
V.
Goncharov
, “
Inverse ray tracing on icosahedral tetrahedron grids for non-linear laser-plasma interaction coupled to 3D radiation hydrodynamics
,”
J. Comp. Physiol.
443
,
110537
(
2021
).
101.
T. W.
Johnston
and
J. M.
Dawson
, “
Correct values for high–frequency power absorption by inverse bremsstrahlung in plasmas
,”
Phys. Fluids
16
,
722
(
1973
).
102.
S.
Skupsky
, “
‘Coulomb logarithm’ for inverse-Bremsstrahlung laser absorption
,”
Phys. Rev. A
36
,
5701
(
1987
).
103.
R.
Devriendt
and
O.
Poujade
, “
Classical molecular dynamic simulations and modeling of inverse Bremsstrahlung heating in low Z weakly coupled plasmas
,”
Phys. Plasmas
29
,
073301
(
2022
).
104.
D.
Turnbull
,
J.
Katz
,
M.
Sherlock
,
L.
Divol
,
N. R.
Shaffer
,
D. J.
Strozzi
,
A.
Colaïtis
,
D. H.
Edgell
,
R. K.
Follett
,
K. R.
McMillen
,
P.
Michel
,
A. L.
Milder
, and
D. H.
Froula
, “
Inverse Bremsstrahlung absorption
,”
Phys. Rev. Lett.
130
,
145103
(
2023
).
105.
Y. T.
Lee
and
R. M.
More
, “
An electron conductivity model for dense plasmas
,”
Phys. Fluids
27
,
1273
(
1984
).
106.
A. B.
Langdon
, “
Nonlinear inverse Bremsstrahlung and heated-electron distributions
,”
Phys. Rev. Lett.
44
,
575
(
1980
).
107.
J. P.
Matte
,
M.
Lamoreaux
,
C.
Möller
,
R. Y.
Yin
,
J.
Delettrez
,
J.
Virmont
, and
T. W.
Johnston
, “
Non-Maxwellian electron distributions and continuum x-ray emission in inverse Bremsstrahlung heated plasmas
,”
Plasma Phys. Controlled Fusion
30
,
1665
1689
(
1998
).
108.
C. J.
Randall
,
J. J.
Thomson
, and
K. G.
Estabrook
, “
Enhancement of stimulated Brillouin scattering due to reflection of light from plasma critical surface
,”
Phys. Rev. Lett.
43
,
924
(
1979
).
109.
D.
Turnbull
,
A.
Colaïtis
,
A. M.
Hansen
,
A. L.
Milder
,
J. P.
Palastro
,
J.
Katz
,
C.
Dorrer
,
B. E.
Kruschwitz
,
D. J.
Strozzi
, and
D. H.
Froula
, “
Impact of the Langdon effect on crossed-beam energy transfer
,”
Nat. Phys.
16
,
181
185
(
2020
).
110.
B.
Scheiner
and
M.
Schmitt
, “
Considerations for the modeling of the laser ablation region of ICF targets with Lagrangian simulations
,”
AIP Adv.
11
,
105208
(
2021
).
111.
A.
Bourgeade
and
G.
Duchateau
, “
Time-dependent ionization models designed for intense and short laser pulse propagation in dielectric materials
,”
Phys. Rev. E
85
,
056403
(
2012
).
112.
S.
Weeratunga
,
Energy Deposition into Overdense Plasma with Laser Ray Tracing
(
Lawrence Livermore National Laboratory
,
2015
).
113.
M. M.
Basko
and
I. P.
Tsygvintsev
, “
A hybrid model of laser energy deposition for multi-dimensional simulations of plasmas and metals
,”
Comput. Phys. Commun.
214
,
59
70
(
2017
).
114.
G.
Duchateau
,
S. X.
Hu
,
A.
Pineau
,
A.
Kar
,
B.
Chimier
,
A.
Casner
,
V.
Tikhonchuk
,
V. N.
Goncharov
,
P. B.
Radha
, and
E. M.
Campbell
, “
Modeling the solid-to-plasma transition for laser imprinting in direct-drive inertial confinement fusion
,”
Phys. Rev. E
100
,
033201
(
2019
).
115.
A.
Kar
,
S. X.
Hu
,
G.
Duchateau
,
J.
Carroll-Nellenback
, and
P. B.
Radha
, “
Implementing a microphysics model in hydrodynamic simulations to study the initial plasma formation in dielectric ablator materials for direct-drive implosions
,”
Phys. Rev. E
101
,
063202
(
2020
).
116.
A.
Pineau
,
B.
Chimier
,
S. X.
Hu
, and
G.
Duchateau
, “
Modeling the electron collision frequency during solid-to-plasma transition of polystyrene ablator for direct-drive inertial confinement fusion applications
,”
Phys. Plasmas
27
,
092703
(
2020
).
117.
E.
Kaselouris
,
I.
Fitilis
,
A.
Skoulakis
,
Y.
Orphanos
,
G.
Koundourakis
,
E. L.
Clark
,
J.
Chatzakis
,
M.
Bakarezos
,
N. A.
Papadogiannis
,
V.
Dimitriou
, and
M.
Tatarakis
, “
The importance of the laser pulse-ablator interaction dynamics prior to the ablation plasma phase in inertial confinement fusion studies
,”
Philos. Trans. R. Soc., A
378
,
20200030
(
2020
).
118.
A.
Pineau
,
B.
Chimier
,
S. X.
Hu
, and
G.
Duchateau
, “
Improved modeling of the solid-to-plasma transition of polystyrene ablator for laser direct-drive inertial confinement fusion hydrocodes
,”
Phys. Rev. E
104
(
1
),
015210
(
2021
).
119.
W. C.
Mead
,
R. A.
Haas
,
W. L.
Kruer
,
D. W.
Phillion
,
H. N.
Kornblum
,
J. D.
Lindl
,
D. R.
MacQuigg
, and
V. C.
Rupert
, “
Observation and simulation of effects on parylene disks irradiated at high intensities with a 1.06 μm laser
,”
Phys. Rev. Lett.
37
,
489
(
1976
).
120.
S. X.
Hu
,
P. B.
Radha
,
J. A.
Marozas
,
R.
Betti
,
T. J. B.
Collins
,
R. S.
Craxton
,
J. A.
Delettrez
,
D. H.
Edgell
,
R.
Epstein
,
V. N.
Goncharov
,
I. V.
Igumenshchev
,
F. J.
Marshall
,
R. L.
McCrory
,
D. D.
Meyerhofer
,
S. P.
Regan
,
T. C.
Sangster
,
S.
Skupsky
,
V. A.
Smalyuk
,
Y.
Elbaz
, and
D.
Shvarts
, “
Neutron yield study of direct-drive, low-adiabat cryogenic D2 implosions on OMEGA laser system
,”
Phys. Plasmas
16
,
112706
(
2009
).
121.
S. X.
Hu
,
V. N.
Goncharov
,
P. B.
Radha
,
J. A.
Marozas
,
S.
Skupsky
,
T. R.
Boehly
,
T. C.
Sangster
,
D. D.
Meyerhofer
, and
R. L.
McCrory
, “
Two-dimensional simulations of the neutron yield in cryogenic deuterium-tritium implosions on OMEGA
,”
Phys. Plasmas
17
,
102706
(
2010
).
122.
P. B.
Radha
,
C.
Stoeckl
,
V. N.
Goncharov
,
J. A.
Delettrez
,
D. H.
Edgell
,
J. A.
Frenje
,
I. V.
Igumenshchev
,
J. P.
Knauer
,
J. A.
Marozas
,
R. L.
Mccrory
,
D. D.
Meyerhofer
,
R. D.
Petrasso
,
S. P.
Regan
,
T. C.
Sangster
,
W.
Seka
, and
S.
Skupsky
, “
Triple-picket warm plastic-shell implosions on OMEGA
,”
Phys. Plasmas
18
,
012705
(
2011
). 0
123.
S. X.
Hu
,
D. T.
Michel
,
A. K.
Davis
,
R.
Betti
,
P. B.
Radha
,
E. M.
Campbell
,
D. H.
Froula
, and
C.
Stoeckl
, “
Understanding the effects of laser imprint on plastic-target implosions on OMEGA
,”
Phys. Plasmas
23
,
102701
(
2016
).
124.
D. T.
Michel
,
S. X.
Hu
,
A. K.
Davis
,
V. Y.
Glebov
,
V. N.
Goncharov
,
I. V.
Igumenshchev
,
P. B.
Radha
,
C.
Stoeckl
, and
D. H.
Froula
, “
Measurement of the shell decompression in direct-drive inertial-confinement-fusion implosions
,”
Phys. Rev. E
95
,
051202
(
2017
).
125.
C.-K.
Li
,
F. H.
Séguin
,
J. A.
Frenje
,
M.
Rosenberg
,
R. D.
Petrasso
,
P. A.
Amendt
,
J. A.
Koch
,
O. L.
Landen
,
H. S.
Park
,
H. F.
Robey
,
R. P. J.
Town
,
A.
Casner
,
F.
Philippe
,
R.
Betti
,
J. P.
Knauer
,
D. D.
Meyerhofer
,
C. A.
Back
,
J. D.
Kilkenny
, and
A.
Nikroo
, “
Charged-particle probing of x-ray-driven inertial-fusion implosions
,”
Science
327
(
5970
),
1231
1235
(
2010
).
126.
P.
Amendt
, “
Entropy generation from hydrodynamic mixing in inertial confinement fusion indirect-drive targets
,”
Phys. Plasmas
28
,
072701
(
2021
).
127.
D. H.
Edgell
,
W.
Seka
,
R. E.
Bahr
,
T. R.
Boehly
, and
M. J.
Bonino
, “
Effectiveness of silicon as a laser shinethrough barrier for 351-nm light
,”
Phys. Plasmas
15
,
092704
(
2008
).
128.
W.
Sutherland
, “
The viscosity of gases and molecular force
,”
London, Edinburgh Dublin Philos. Mag. J. Sci.
36
(
223
),
507
531
(
1893
).
129.
C. R.
Weber
,
D. S.
Clark
,
A. W.
Cook
,
L. E.
Busby
, and
H. R.
Robey
, “
Inhibition of turbulence in inertial-confinement-fusion hot spots by viscous dissipation
,”
Phys. Rev. E
89
,
053106
(
2014
).
130.
A. L.
Velikovich
,
K. G.
Whitney
, and
J. W.
Thornhill
, “
A role for electron viscosity in plasma shock heating
,”
Phys. Plasmas
8
,
4524
4533
(
2001
).
131.
D. S.
Miller
, “
Splitting shock heating between ions and electrons in an ionized gas
,”
Comput. Fluids
210
,
104672
(
2020
).
132.
L. F.
Berzak Hopkins
,
S.
Le Pape
,
L.
Divol
,
N. B.
Meezan
,
A. J.
Mackinnon
,
D. D.
Ho
,
O. S.
Jones
,
S.
Khan
,
J. L.
Milovich
,
J. S.
Ross
,
P.
Amendt
,
D.
Casey
,
P. M.
Celliers
,
A.
Pak
,
J. L.
Peterson
,
J.
Ralph
, and
J. R.
Rygg
, “
Near-vacuum Hohlraums for driving fusion implosions with high density carbon ablators
,”
Phys. Plasmas
22
,
056318
(
2015
).
133.
C.
Chenais-Popovics
,
P.
Renaudin
,
O.
Rancu
,
F.
Gilleron
,
J.-C.
Gauthier
,
O.
Larroche
,
O.
Peyrusse
,
M.
Dirksmöller
,
P.
Sondhauss
,
T.
Missalla
,
I.
Uschmann
,
E.
Förster
,
O.
Renner
, and
E.
Krousky
, “
Kinetic to thermal energy transfer and interpenetration in the collision of laser-produced plasmas
,”
Phys. Plasmas
4
,
190
208
(
1997
).
134.
J.
Denavit
, “
Collisionless plasma expansion into a vacuum
,”
Phys. Fluids
22
,
1384
1392
(
1979
).
135.
C.
Bellei
,
P. A.
Amendt
,
S. C.
Wilks
,
M. G.
Haines
,
D. T.
Casey
,
C.-K.
Li
,
R.
Petrasso
, and
D. R.
Welch
, “
Species separation in inertial confinement fusion fuels
,”
Phys. Plasmas
20
,
012701
(
2013
).
136.
H. M.
Mott-Smith
, “
The solution of the Boltzmann equation for a shock wave
,”
Phys. Rev.
82
,
885
892
(
1951
).
137.
B.
Schmidt
, “
Electron beam density measurements in shock waves in argon
,”
J. Fluid Mech.
39
,
361
373
(
1969
).
138.
L. G.
Margolin
, “
Nonequilibrium entropy in a shock
,”
Entropy
19
(
7
),
368
(
2017
).
139.
B. D.
Keenan
,
A. N.
Simakov
,
L.
Chacón
, and
W. T.
Taitano
, “
Deciphering the kinetic structure of multi-ion plasma shocks
,”
Phys. Rev. E
96
,
053203
(
2017
).
140.
R. W.
Schunk
, “
Mathematical structure of transport equations for multispecies flows
,”
Rev. Geophys.
15
(
4
),
429
445
, https://doi.org/10.1029/RG015i004p00429 (
1977
).
141.
T.
Vazquez-Gonzalez
,
A.
Llor
, and
C.
Fochesato
, “
A mimetic numerical scheme for multi-fluid flows with thermodynamic and geometric compatibility on an arbitrarily moving grid
,”
Int. J. Multiphase Flow
132
,
103324
(
2020
).
142.
P.
Amendt
,
O. L.
Landen
,
H. F.
Robey
,
C.-K.
Li
, and
R. D.
Petrasso
, “
Plasma barodiffusion in inertial-confinement-fusion implosions: Application to observed yield anomalies in thermonuclear fuel mixtures
,”
Phys. Rev. Lett.
105
,
115005
(
2010
).
143.
N. M.
Hoffman
,
G. B.
Zimmerman
,
K.
Molvig
,
H. G.
Rinderknecht
,
M. J.
Rosenberg
,
B. J.
Albright
,
A. N.
Simakov
,
H.
Sio
,
A. B.
Zylstra
,
M.
Gatu Johnson
,
F. H.
Séguin
,
J. A.
Frenje
,
C.-K.
Li
,
R. D.
Petrasso
,
D. M.
Higdon
,
V. Y.
Glebov
,
C.
Stoeckl
,
W.
Seka
, and
T. C.
Sangster
, “
Approximate models for the ion-kinetic regime in inertial-confinement-fusion capsule implosions
,”
Phys. Plasmas
22
,
052707
(
2015
).
144.
C.
Paquette
,
C.
Pelletier
,
G.
Fontaine
, and
G.
Michaud
, “
Diffusion coefficients for stellar plasmas
,”
Astrophys. J., Suppl. Ser.
61
,
177
(
1986
).
145.
V. M.
Zhdanov
, “
Transport processes in multicomponent plasma
,”
Plasma Phys. Controlled Fusion
44
,
2283
(
2002
).
146.
G.
Kagan
and
X.-Z.
Tang
, “
Electrodiffusion in a plasma with two ion species
,”
Phys. Plasmas
19
,
082709
(
2012
).
147.
G.
Kagan
and
X.-Z.
Tang
, “
Thermo-diffusion in inertially confined plasmas
,”
Phys. Lett. A
378
(
21
),
1531
1535
(
2014
).
148.
E. L.
Vold
,
R. M.
Rauenzahn
,
C. H.
Aldrich
,
K.
Molvig
,
A. N.
Simakov
, and
B. M.
Haines
, “
Plasma transport in an Eulerian AMR code
,”
Phys. Plasmas
24
,
042702
(
2017
).
149.
K.
Molvig
,
A. N.
Simakov
, and
E. L.
Vold
, “
Classical transport equations for burning gas-metal plasmas
,”
Phys. Plasmas
21
,
092709
(
2014
).
150.
E. L.
Vold
,
R. M.
Rauenzahn
, and
A. N.
Simakov
, “
Multi-species plasma transport in 1D direct-drive ICF simulations
,”
Phys. Plasmas
26
,
032706
(
2019
).
151.
E. L.
Vold
,
L.
Yin
, and
B. J.
Albright
, “
Plasma transport simulations of Rayleigh-Taylor instability in near-ICF deceleration regimes
,”
Phys. Plasmas
28
,
092709
(
2021
).
152.
S.
Blouin
,
N. R.
Shaffer
,
D.
Saumon
, and
C. E.
Starrett
, “
New conductive opacities for white dwarf envelopes
,”
Astrophys. J.
899
,
46
(
2020
).
153.
D.
Koester
, “
Accretion and diffusion in white dwarfs. New diffusion timescales and applications to GD 362 and G 29
,”
Astron. Astrophys.
498
(
38
),
517
525
(
2009
).
154.
Y.
Zhou
, “
Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing
,”
I. Phys. Rep.
720
,
1
136
(
2017
).
155.
Y.
Zhou
, “
Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing
,”
II. Phys. Rep.
723
,
1
160
(
2017
).
156.
R. H. H.
Scott
,
D. S.
Clark
,
D. K.
Bradley
,
D. A.
Callahan
,
M. J.
Edwards
,
S. W.
Haan
,
O. S.
Jones
,
B. K.
Spears
,
M. M.
Marinak
,
R. P. J.
Town
,
P. A.
Norreys
, and
L. J.
Suter
, “
Numerical modeling of the sensitivity of x-ray driven implosions to low-mode flux asymmetries
,”
Phys. Rev. Lett.
110
,
075001
(
2013
).
157.
B. A.
Hammel
,
H. A.
Scott
,
S. P.
Regan
,
C.
Cerjan
,
D. S.
Clark
,
M. J.
Edwards
,
R.
Epstein
,
S. H.
Glenzer
,
S. W.
Haan
,
N.
Izumi
,
J. A.
Koch
,
G. A.
Kyrala
,
O. L.
Landen
,
S. H.
Langer
,
K.
Peterson
,
V. A.
Smalyuk
,
L. J.
Suter
, and
D. C.
Wilson
, “
Diagnosing and controlling mix in National Ignition Facility implosion experiments
,”
Phys. Plasmas
18
,
056310
(
2011
).
158.
A.
Pak
,
L.
Divol
,
C. R.
Weber
,
L. F.
Berzak Hopkins
,
D. S.
Clark
,
E. L.
Dewald
,
D. N.
Fittinghoff
,
V.
Geppert-Kleinrath
,
M.
Hohenberger
,
S.
Le Pape
,
T.
Ma
,
A. G.
MacPhee
,
D. A.
Mariscal
,
E.
Marley
,
A. S.
Moore
,
L. A.
Pickworth
,
P. L.
Volegov
,
C.
Wilde
,
O. A.
Hurricane
, and
P. K.
Patel
, “
The impact of localized radiative loss on inertial confinement fusion implosions
,”
Phys. Rev. Lett.
124
,
145001
(
2020
).
159.
J.
Edwards
,
M.
Marinak
,
T.
Dittrich
,
S.
Haan
,
J.
Sanchez
,
J.
Klingmann
, and
J.
Moody
, “
The effects of fill tubes on the hydrodynamics of ignition targets and prospects for ignition
,”
Phys. Plasmas
12
,
056318
(
2005
).
160.
T. R.
Dittrich
,
O. A.
Hurricane
,
L. F.
Berzak-Hopkins
,
D. A.
Callahan
,
D. T.
Casey
,
D.
Clark
,
E. L.
Dewald
,
T.
Döppner
,
S. W.
Haan
,
B. A.
Hammel
,
J. A.
Harte
,
D. E.
Hinkel
,
B. J.
Kozioziemski
,
A. L.
Kritcher
,
T.
Ma
,
A.
Nikroo
,
A. E.
Pak
,
T. G.
Parham
,
H.-S.
Park
,
P. K.
Patel
,
B. A.
Remington
,
J. D.
Salmonson
,
P. T.
Springer
,
C. R.
Weber
,
G. B.
Zimmerman
, and
J. L.
Kline
, “
Simulations of fill tube effects on the implosion of high-foot NIF ignition capsules
,”
J. Phys. Conf. Ser.
717
,
012013
(
2016
).
161.
A. G.
MacPhee
,
D. T.
Casey
,
D. S.
Clark
,
S.
Felker
,
J. E.
Field
,
S. W.
Haan
,
B. A.
Hammel
,
J.
Kroll
,
O. L.
Landen
,
D. A.
Martinez
,
P.
Michel
,
J.
Milovich
,
A.
Moore
,
A.
Nikroo
,
N.
Rice
,
H. F.
Robey
,
V. A.
Smalyuk
,
M.
Stadermann
, and
C. R.
Weber
, “
X-ray shadow imprint of hydrodynamic instabilities on the surface of inertial confinement fusion capsules by the fuel fill tube
,”
Phys. Rev. E
95
,
031204
(
2017
).
162.
C. R.
Weber
,
D. S.
Clark
,
A.
Pak
,
N.
Alfonso
,
B.
Bachmann
,
L. F.
Berzak Hopkins
,
T.
Bunn
,
J.
Crippen
,
L.
Divol
,
T.
Dittrich
,
A. L.
Kritcher
,
O. L.
Landen
,
S.
Le Pape
,
A. G.
MacPhee
,
E.
Marley
,
L. P.
Masse
,
J. L.
Milovich
,
A.
Nikroo
,
P. K.
Patel
,
L. A.
Pickworth
,
N.
Rice
,
V. A.
Smalyuk
, and
M.
Stadermann
, “
Mixing in ICF implosions on the National Ignition Facility caused by the fill-tube
,”
Phys. Plasmas
27
,
032703
(
2020
).
163.
B. M.
Haines
,
D. S.
Clark
,
C. R.
Weber
,
M. J.
Edwards
,
S. H.
Batha
, and
J. L.
Kline
, “
Cross-code comparison of the impact of the fill tube on high yield implosions on the National Ignition Facility
,”
Phys. Plasmas
27
,
082703
(
2020
).
164.
S.
Chandrasekhar
, “
The character of the equilibrium of an incompressible heavy viscous fluid of variable density
,”
Proc. Cambridge Philos. Soc.
51
,
162
(
1955
).
165.
K. O.
Mikaelian
, “
Effect of viscosity on Rayleigh-Taylor and Richtmyer-Meshkov instabilities
,”
Phys. Rev. E
47
,
375
383
(
1993
).
166.
A. R.
Piriz
,
O. D.
Cortázar
, and
J. J.
López Cela
, “
The Rayleigh-Taylor instability
,”
Am. J. Phys.
74
,
1095
(
2006
).
167.
S.
Gerashchenko
and
D.
Livescu
, “
Viscous effects on the Rayleigh-Taylor instability with background temperature gradient
,”
Phys. Plasmas
23
,
072121
(
2016
).
168.
C.
Xie
,
J.
Tao
, and
J.
Li
, “
Viscous Rayleigh-Taylor instability with and without diffusion effect
,”
Appl. Math. Mech.
38
(
2
),
263
270
(
2017
).
169.
S.
Davidovits
and
N. J.
Fisch
, “
Sudden viscous dissipation of compressing turbulence
,”
Phys. Rev. Lett.
116
,
105004
(
2016
).
170.
T.
Yabe
and
K. A.
Tanaka
, “
Long ion mean-free path and nonequilibrium radiation effects on high-aspect-ratio laser-driven implosions
,”
Laser Part. Beams
7
,
259
265
(
1989
).
171.
B. M.
Haines
,
E. L.
Vold
,
K.
Molvig
,
C.
Aldrich
, and
R.
Rauenzahn
, “
The effects of plasma diffusion and viscosity on turbulent instability growth
,”
Phys. Plasmas
21
,
092306
(
2014
).