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.

## I. INTRODUCTION

High energy density physics (HEDP) experiments pursue an understanding of the physics underlying inertial confinement fusion^{1} 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 gain^{3,4}—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.

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 code^{31–33} 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.

*n*is the electron density,

_{e}*ϵ*

_{0}is the permittivity of free space,

*k*is the Boltzmann constant,

_{B}*q*is the electron charge,

_{e}*T*is the electron temperature, and

_{e}*E*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,

_{F}^{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/cm

^{3}.

^{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.

*κ*

_{1}and

*κ*

_{2}. 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 $ \kappa 1 \kappa 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),

*Z*is the charge state of species

_{l}*l*, $ ln \u2009 \Lambda 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 transport

^{80,81}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.

^{79,92}

*τ*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.

_{ei}*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 $ \Omega j k n , m$, cross sections $ \sigma j k n$, and scattering angles

*θ*(see, e.g., Ref. 94) calculated as

_{jk}*m*is the reduced mass. From the Yukawa pair potential, one can calculate the binary mass diffusivity

_{jk}*D*, dynamic viscosity

_{jk}*μ*, and thermal conductivity

_{j}*κ*,

_{j}^{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 $ \Omega 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.

## II. LASER PROPAGATION AND ABSORPTION

^{9,43,96–100}Lasers propagate according to geometric optics with the refractive index $ n refr : = 1 \u2212 n e / n e , crit$, where

*n*is the electron density and $ n e , crit : = \pi m e c 2 e 2 \lambda 2$ is the critical electron density (where the plasma frequency equals the laser frequency), where

_{e}*m*is the electron mass,

_{e}*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 integrating

^{96,97,101}

*T*is the electron temperature,

_{e}*k*is the Boltzmann constant, and

_{B}*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 effect

^{106,107}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.

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 Nike^{45} [Fig. 5(a)] and OMEGA^{44} [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 material^{125} that can impact transport^{10} and thermodynamic properties^{126} 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.

## III. VISCOSITY AND MASS TRANSPORT

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 \xaf 2$ (compared to $ T 0.7$ for gases^{128}) 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-hydrodynamic^{132} and exhibit interpenetration.^{133,134} Here, by non-hydrodynamic, we mean that the Knudsen number $ Kn = \lambda L \u2273 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 structures^{136–139} 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 to^{48} 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 solution^{142–151}—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 \xaf$ 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 instabilities^{154,155} play a key role in determining performance by impacting the efficiency of energy transfer to the hotspot^{48,156} 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 \xaf 2$ (electron viscosities are smaller by a factor of $ m e / m i$ but can become important in situations when $ T e \u226b 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 modes^{171} and even some long-wavelength modes^{172} in ICF implosions, and that Rayleigh-Taylor growth rates in ICF implosions are sensitive to the coefficients used for both viscosity and plasma transport^{173} 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 implicit^{179} 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 \u2212 1$ $ cm 2 /$ s^{2} with an interface initialized with perturbations with wavelengths $ \lambda = 2 L , \u2009 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.

^{181}viscosity coefficients, which use the effective potential theory framework

^{182}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 model

^{148,150,151}combines viscosities for each individual species using the following formula:

^{183–185}generalize the Braginskii results

^{92}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

*l*corresponds to the lightest ion species in the mixture, and

*ν*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 $ \alpha k \eta $.

_{ll}^{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}

Based on Ref. 186, viscosity pre-factors $ \alpha k \eta $ 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 \u226a m h$, where *m _{l}* is the mass of the light fluid and

*m*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%.

_{h}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 experiments^{196} were performed to evaluate the extent of interpenetration between plasma species in conditions similar to those in laser-driven hohlraums. xRAGE plasma transport^{10} and Lasnex multi-fluid^{197} 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 mixing^{207} (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 VPIC^{187,189} 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.

## IV. THERMAL CONDUCTIVITY

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 Härm^{230} 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 models^{233} can be used to produce both analytically^{182,234,235} and numerically^{236–240} evaluated conductivities. To obtain conductivities valid over a wider range of conditions, relativistic average atom codes, such as the Purgatorio code^{241,242} 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}

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 $ \u2248 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 ( $ \u2248 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 problem^{48,270–274} that is clearly impacted by the quality of the thermal conductivity coefficients.

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, N090728^{278–280} 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-dimensional^{282} 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 challenge^{38,132,283} 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.

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 $ \u2248 m e m i$. Nevertheless, ion thermal conduction can be important in situations where energy deposition strongly favors the ions so that $ T i \u226b 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 Spitzer^{230} and Lee-More^{105} 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.

## V. NON-LOCAL ION TRANSPORT

The success of inertial confinement fusion in the laboratory^{3} 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–Deutsch^{292–294} (MD), LiûPetrasso^{295,296} (LP), and Brown–Preston–Singleton^{297,298} (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.

Numerous experiments have been done to constrain stopping power models, mostly for highly ionized plasmas.^{307–311} When particles are fast ( $ v p \u226b v th e$, where *v _{p}* 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 \u2248 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 \u2248 0.3 v th i$. In the regime $ v p \u226a 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 implosions

^{318}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.

## VI. NON-LOCAL ELECTRON THERMAL CONDUCTION

In many situations, it is important to account for non-Maxwellian electron distributions when calculating thermal conduction. Large temperature gradients^{291} and laser-plasma instabilities^{46,47} 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 treatment^{321,322} 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 developed^{321–328} 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 approach^{325,326} 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 method^{327,328} 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.

^{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}

^{,}

^{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 $ \lambda 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

*Z*

_{1}oscillations of the stopping power

^{342}(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 $ \lambda e g$ has also been shown to impact hohlraum ablation

^{346}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}

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.

## VII. ELECTRICAL CONDUCTION

^{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*is the thermal conductivity,

*σ*is the electrical conductivity, and $ L = 2.44 \xd7 10 \u2212 8$ V

^{2}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 $ \u221d 1 / \rho $, 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 accuracy^{361,362} (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 = \u03f5 0 m e q e 2 \omega 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}

## VIII. TRANSPORT COEFFICIENT COUPLING TO MAGNETIC FIELDS

*n*and

_{e}*T*,

_{e}^{386}Such conditions are exhibited by hydrodynamic instabilities

^{387–393}and jetting.

^{394}Self-generated magnetic fields of order 100 T have been inferred from proton radiography in various laser-produced plasmas

^{395–400}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 MagLIF

^{406}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}

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 ( $ \alpha \u2225 , \u2009 \alpha \u22a5$, and $ \alpha \u2227$) and thermoelectric ( $ \beta \u2225 , \u2009 \beta \u22a5$, and $ \beta \u2227$) transport coefficients. Recent work has shown that the coefficients from Ref. 79 can overestimate Nernst advection and magnetic dissipation^{414} 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 $ \chi \u2192 0$ and $ \chi \u2192 \u221e$, 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}

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 electrons^{416} 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 code^{323,418} 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.

## IX. MULTI-ION-SPECIES THERMALIZATION

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 $ \u2248 0.25$ *μ*m 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 condition^{422} 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 SESAME^{423} or LEOS^{424,425}). 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 gases^{428} and only the electron number density equilibration approach^{429} directly considers the impact of ionization and electron and ion species that are out of equilibrium. MD simulations^{430–433} and simulations employing free-energy minimization^{434} 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 experiments^{435} 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 $ \u2248 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.

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.

## X. ELECTRON–ION EQUILIBRATION

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 *T _{e}* and

*T*and, for example, ICF capsule performance can be very sensitive to this split since

_{i}*T*determines reaction rates and

_{i}*T*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

_{e}*T*to burn-weighted

_{e}*T*measured in experiment

_{i}^{448}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 models^{469–473} and these tend to show the best agreement with the Gericke–Murillo–Schlanges^{462} and Brown-Preston–Singleton^{297} models. More recent MD-based models provide a more complete coverage of materials and phase space, including Daligault–Simoni^{465,468} 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.

## XI. CONCLUSIONS

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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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

## DATA AVAILABILITY

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

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_{2}implosions on OMEGA laser system