The increase in entropy from the physical mixing of two adjacent materials in inertial confinement fusion (ICF) implosions and gas-filled hohlraums is analytically assessed. An idealized model of entropy generation from the mixing of identical ideal-gas particles across a material interface in the presence of pressure and temperature gradients is applied. Physically, mix-driven entropy generation refers to the work done by the gases in expanding into a larger common volume from atomic mixing under the condition of no internal energy change, or work needed to restore the initial unmixed state. The effect of a mix-generated entropy increase is analytically shown to lead to less compression of the composite ICF fluid under adiabatic conditions. The amount of entropy generation is estimated to be ∼10 J for a Rayleigh–Taylor-induced micrometer-scale annular mixing layer between the solid deuterium–tritium fuel and (undoped) high-density carbon pusher of an imploding capsule at the National Ignition Facility (NIF). This level of entropy generation is consistent with lower-than-expected fuel compressions measured on the NIF [Hurricane *et al.*, Phys. Plasmas **26**, 052704 (2019)]. The degree of entropy increase from mixing of high-*Z* hohlraum wall material and low-*Z*, moderate- to high-density gas fills is estimated to lead to ∼100 kJ of heat generation for NIF-scale experiments [Moody *et al.*, Phys. Plasmas **21**, 056317 (2014)]. This value represents a significant fraction of the inferred missing x-ray drive energy based on observed delays in capsule implosion times compared with mainline simulations [Jones *et al.*, Phys. Plasmas **19**, 056315 (2012)].

## I. INTRODUCTION

The demonstration of ignition at the National Ignition Facility (NIF) remains a grand challenge with the collective experimental evidence to date suggesting that the peak stagnation pressure *P*_{stag} of the deuterium tritium (DT) fuel falls modestly short of the ignition threshold of ∼450 Gbars.^{1} Recent progress toward approaching the burning plasma state with high-density carbon (HDC) ablators has achieved a *P*_{stag} of ∼360 Gbars,^{2} while plastic (CH) pushers have reached 280 ± 40 Gbars.^{3} Moreover, a persistent anomaly has been the low fuel areal densities experimentally inferred relative to standard modeling, with a deficit of ∼20%;^{2,4} in contrast, this discrepancy has been found to not carry over to shell compressibility.^{5} Several hypotheses have been advanced to understand these shortfalls in fuel pressure and compression—all based on the notion that the implosions proceed on higher fuel adiabats than calculated due to possible missing or incomplete physics in our mainline design tools. These candidate physical effects include—but are not limited to—hot electron preheat of the main fuel layer,^{6} shock front resistive heating,^{7} generation of late-time non-radial mass flows from drive flux asymmetry,^{8} and large perturbation growth from capsule supports and fill-tube shadowing,^{9} but none of these has been universally accepted by the inertial confinement fusion (ICF) community. This paper expands on one further candidate mechanism for preheat generation and reduced implosion performance: atomic mixing of materials across a material interface and the associated internal generation of entropy.^{10,11} During the capsule implosion phase, perturbations at the various material interfaces are subject to accelerations—whether impulsive or continual—leading to Richtmyer–Meshkov or Rayleigh–Taylor growth, respectively. If the amplitude of a perturbed interface grows to a level on the order of the wavelength, the memory of the initial conditions becomes lost, and material mixing across the classical interface may take place with an associated increase in entropy. Since the mixing process is inherently a multi-fluid (and possibly kinetic) phenomenon, mainline simulation models that are based on an average-atom single-fluid description often invoke an in-line mixing model with adjustable (but phenomenologically constrained) parameters to approximate the multi-component flows occurring within a presumptive mix layer. These in-line mix models generally do not include a mixing-induced entropy increase, which comes at the expense of compressional energy change −*P*d*V* in the mix region. Such an increase in entropy should ideally be contained in the equation-of-state (EOS) for the mixed material, but this “lookup” is constructed from a concentration-weighted mean of the two (pure) constituent EOSs and arguably neglects any entropy generation that occurs during the mixing transition. Our purpose here is to estimate the amount of entropy increase from mixing generated at several candidate interfaces of interest in ICF and to assess its potential impact on target performance.

It is well known that when two initially segregated materials are mixed across a permeable interface, an entropy increase must occur consistent with the 2^{nd} law of thermodynamics. An empirical necessary condition for two fluids to turbulently mix is that the Reynolds number should exceed 10^{4} in a statistically stationary flow^{12}—with an even greater threshold value under transient conditions.^{13} During an ICF implosion, the episode most susceptible to exceeding the mixing threshold is near deceleration onset of the DT fuel ablator interface when instability growth is greatest. At this stage, the fuel is in a “warm dense matter” state with temperature ∼10 eV and density ∼10 g/cc. The plasma dynamic viscosity *ν* under these conditions is approximated by ∼0.001 cm^{2}/s,^{14} while the scale length Δ*x* for the fluid shear flow to vary by a terminal bubble velocity $\Delta \upsilon \u2248g/3k$ is taken as a bubble vortex size ∼3/*k*,^{15} where *g* is the interface peak acceleration ∼1000 *μ*m/ns^{2} and *k* is the perturbation wavenumber. Upon taking a 20 *μ*m wavelength perturbation (approximately equal to the thickness of the fuel and remaining ablator for the dominant perturbation according to linear growth-factor simulations) at deceleration onset of a shell at ∼300 *μ*m radius gives a resulting Reynolds number $Re\u2261\Delta x\xb7\Delta \upsilon \nu \u22451.6\xd7106$—far exceeding the (fluid) threshold for mixing. This analytic estimate is largely consistent with recently reported three-dimensional simulation results for the ignition DT hot spot^{16} that used a viscous dissipation model based on a Yukawa system.^{17}

The interface between the fuel and ablator is susceptible to the Rayleigh–Taylor instability following deceleration onset ($g\u2261d\upsilon \u2192/dt$ ≤ 0), provided the product of Atwood number [dimensionless density mismatch parameter $\rho H\u2212\rho L/\rho H+\rho L]$ and acceleration *g* as defined herein is negative. However, the effects of x-ray preheat in hohlraum-driven central hot-spot ignition capsule designs can reverse the Atwood number well before deceleration onset, leading to instability growth during acceleration (*g* > 0).^{18} If the initial conditions ensure robust linear growth and rapid nonlinear saturation, then a transition to an atomically mixed state may result. For the purpose of this work, we assume that this atomically mixed state is achieved—even if its existence is by no means assured. This assumption serves to establish an upper bound on the degree of entropy generation from mixing in the vicinity of a perturbed interface and to address whether mixing significantly impacts predicted implosion performance compared with simulation tools that ignore this effect. It should be mentioned that for the case of an unperturbed interface, binary mass diffusion in principle always leads to a mixing layer on some scale, *independent* of a Reynolds number threshold. Moreover, a recently suggested kinetic mechanism for generating nearly instantaneous mix across an unperturbed interface with a step-down density discontinuity by a transiting high Mach number shock is also independent of a Reynolds number criterion.^{19}

This work studies the degree of potential entropy generation from mixing across an ideal fluid interface for three cases of ICF interest with distinct Reynolds number: (1) the interface between the main DT fuel and the ablator—which can be any material, though we focus here on CH and high-density carbon (HDC), (2) the (ablating) solid–gas DT fuel interface, and (3) the interface between the high-*Z* hohlraum wall and the low-*Z* gas fill that is intended to retard the motion of the ablating wall. The fuel/ablator interface is very susceptible to mixing from hydrodynamic instability growth, and the extent of the mixing layer is expected to reach 10–25 *μ*m based on multi-mode radiation-hydrodynamic simulations^{20} and analytic arguments applied to (undoped) HDC implosions. Analytic estimates of the degree of entropy generation in the solid DT fuel layer suggest that a 10-*μ*m-thick mix layer may generate on the order of 100 J of entropy increase *T*Δ*S* (in energy units). If this entropy generation comes at the expense of compressional energy from work done in mixing −*P*d*V* (or, equivalently, the work required to de-mix or restore the original configuration), the resulting reduced fuel compression may appreciably erode the ∼200 J of ignition margin in low-adiabat ignition capsule designs.^{6} The same formalism is applied to the intrinsically unstable interface between the high-*Z* hohlraum wall and the low-*Z* gas fill. Using analytic representations of the acceleration-induced pressure gradient and a self-consistent temperature gradient, over 100 kJ of energy-equivalent entropy *T*Δ*S* may be generated at high gas-fill density (1.6 mg/cc). This effect is driven by the large difference in ionization states between the hohlraum wall and gas fill, potentially leading to large entropy changes from the interchange of electrons across the unstable interface from mixing. This estimate represents a significant fraction of the inferred level of unaccounted (or missing) drive energy from routinely measured delayed capsule implosion times in (high-density) gas-filled hohlraums relative to state-of-the-art simulation predictions.^{21}

This article is organized as follows. Section II develops the foundation for estimating the degree of entropy generation from mixing in the presence of temperature and pressure differences in hohlraum-driven ICF targets. The effect of entropy generation from mixing on reducing the adiabatic compression of an ICF implosion is analytically demonstrated. The formalism is then applied to the DT ice—ablator interface in ICF implosions in Sec. III for a variety of ablator types and mixing scenarios. An estimate of the degree of entropy generation for doped HDC ablators is shown to coincide with nearly half the available ignition margin to preheat sources and capsule imperfections, thereby putting at higher risk the demonstration of ignition at the 1–2 MJ driver scale. An Euler scaling of driver energy to nearly 10 MJ is argued to significantly reduce the relative effects of entropy generation from Rayleigh–Taylor-induced mixing. Section IV considers the potential for entropy generation at the unstable DT ice—gas interface and concludes that the risk is likely small due to incomplete mixing—if mix even occurs. A formalism to describe the conditions leading to full atomic mixing is developed and shown to scale with material partition number (or inversely with partition width). As the partition width (e.g., a bubble or spike width) decreases, the rate of thermal equilibration increases and leads to less (temperature gradient-driven) entropy generation from subsequent atomic mixing. In the case where the transition to atomic mixing is delayed and the partition width reached is relatively small, the dominant entropy production is now attributed to thermal conduction—a phenomenon arguably well-captured by mainline simulation tools—instead of mixing. Thus, a longer duration of hydro-instability growth before the mixing transition is attained helps mitigate the exposure to entropy increases from mixing in ICF implosions. Finally, Sec. V applies the formalism of mixing-induced entropy generation to the hohlraum high-Z—gas fill boundary to assess the effects of electron interchange from either instability growth or binary diffusion. Nearly 100 kJ of entropy generation *T*Δ*S* is found to roughly coincide with a significant fraction of the inferred energy unaccounted for in implosion experiments with moderate- to high-density gas-filled hohlraums. We conclude in Sec. VI.

## II. ENTROPY OF MIXING FORMULATION

### A. Analysis

We assume that two ideal gases occupying two adjacent vessels of volume *V*_{j} (*j* = 1, 2) are characterized by total (ion and electron) pressure *P*_{j}, equal ion and electron temperature *T*_{j}, and total number of particles *N*_{j} in each vessel. An impermeable (non-conducting) partition between the two vessels is removed or ruptured and the gases are allowed to fully mix, providing a potential increase in combined entropy that we now calculate.

We denote the final mixed state quantities by a primed notation. The mixed temperature $T\u2032$ follows from (internal) energy conservation $\Delta E=0$ (neglecting kinetic energy from transient flows), assuming a number-weighted specific heat at constant volume for the mixed gases $cV\u2032=N1cV,1+N2cV,2N1+N2$ and instantaneous atomic mixing:

while the mixed total pressure $P\u2032P\u2032$ is readily obtained

where *c* is the concentration fraction of species “1”:$\u2009N1/N1+N2$ For an ideal gas, the specific heat per particle is 3*k _{B}*/2, but here we allow for the more general case of partial ionization, for example, carbon (ablator) or gold (hohlraum wall), where the specific heat per ion can be much larger [see Figs. 2(a) and 2(b)].

The total entropy of the mixed state can be written as

where *e* is the base of the natural logarithm, $cV\u2032$ is in units of the Boltzmann constant *k _{B}*, $V=V1+V2$ is the total volume, and $\zeta \u2032=c\zeta 1+(1\u2212c)\zeta 2$ is the number weighted chemical constant of the mixed gas.

^{22}The initial entropy before mixing is

giving for the entropy change from mixing

where $N=N1+N2$ and $cP\u2032=cV\u2032+kB$ is the mean specific heat of the mixture at constant pressure. The specific heats in Eq. (4) are all normalized to the number of free particles (parent ion plus stripped electrons)—and not just the number of ions—to maintain consistency with the total number of free particles *N* appearing in Eq. (4). The last factor in Eq. (4) denotes the change in entropy alone from the mixing of two *different* gases at equal temperature and pressure; otherwise, this factor is unity.^{22} Equation (4) reduces to zero when *c* = 0 or *c* = 1. When $P1=P2$ and *c* = 1/2, Eq. (4) becomes for equal specific heats $cV,j$

while for $T1=T2$ and *c* = 1/2 we have

in agreement with previous work for the case of identical ideal gases.^{22} For $P1=P2$ and $T1=T2$, Eq. (4) reduces to $\Delta S=\u2212NkB\u2009ln\u2009[cc(1\u2212c)1\u2212c]\u22650$ for the “entropy of mixing” between two different gases.^{10,22} This latter source of entropy change—depending on whether identical or sufficiently “distinct” gas species are mixed or not—leads to the Gibbs' paradox.^{23} The “entropy of mixing” usage is normally reserved for the mixing of different gases, while mixing of identical gases at different pressures or temperatures is the focus of this paper and serves as an additional source of entropy generation from mixing. The entropy-of-mixing formalism was recently applied to the mixing of ablator and fuel material in an ICF implosion.^{10} Figure 1 compares these two sources of entropy generation obtained from Eq. (4). For distinct gases, the maximum entropy change per particle is *k*_{B}ln2; for identical gases, the entropy change depends on the temperature difference and is unbounded. In practice the dominant (identical) particle type by sheer number is the electron and may contribute the majority of entropy generation from mixing. For example, in mixing of carbon ablator ions at an ionization level of ∼4 with hydrogen fuel, the number of electrons available for mixing can be 2.5 $\xd7$ higher than from the different ions alone (see Sec. III). The contribution from electrons in the mixing process becomes far greater for the case of the high-*Z* hohlraum wall mixing with the low-*Z* fill gas (See Sec. V).

In this paper, we shall treat all particles—electrons and ions—as identical in the definition of *N*, thereby providing a lower bound on the change in entropy from mixing of all species, identical or not. The inclusion of “entropy of mixing” from distinct particles adds an additional *k _{B}*ln2 per ion as an upper bound on this particular source of entropy generation.

The *enthalpy* of mixing Δ*H* is given by $\Delta \Phi +T\u2032\Delta S+S\u2032\Delta T,$ where ΔΦ is the change in Gibbs free energy between the mixed state and the initially segregated states. For a thermally insulated pair of ideal gas vessels considered here, the enthalpy change from mixing is zero since Δ*H* = Δ*Q*, and no heat energy is delivered by definition.

In many ICF applications, the hohlraum wall material and capsule ablator (both labeled *j* = 2 in the following) are not fully ionized, leading to elevated values of specific heat compared with their respective fully stripped states of matter. For example, Figs. 2(a) and 2(b) show the calculated specific heat at constant volume (per free particle) and the adiabatic ratio $\gamma \u2261d\u2009ln\u2009P/d\u2009ln\u2009\rho |s$ vs temperature for pure carbon in Saha equilibrium at several values of density. The range of temperatures and densities expected at the fuel-ablator interface near the time of deceleration onset is from 10 to 100 eV and greater than 10 g/cc, respectively. Figures 2(a) and 2(b) show that the specific heat per free particle over this range of temperatures and extrapolated to densities exceeding 1 g/cc is only mildly increased over 3*k _{B}*/2, and the adiabatic ratio is slightly reduced from 5/3. For laser-ablated gold as in a high-

*Z*hohlraum, a similar trend is found when the electron temperature is several keV and the density is above 0.01 g/cc. Thus, the power law factor

*c*(1-

*c*)(

*c*

_{V}_{,1}-

*c*

_{V}_{,2})/

*k*in Eq. (4) is expected to be small for the conditions of interest in this paper: carbon mixing with fully ionized DT fuel (Sec. III), and gold mixing with fully ionized hohlraum gas fill

_{B}^{4}He at several keV (Sec. V). By contrast, in the limit of

*c*

_{V}_{,2}≫

*c*

_{V}_{,1}occurring at lower density (≪0.01 g/cc) and temperatures below 100 eV, Eq. (4) reduces to

where $cP\u2032=cV\u2032+kB$ (per free particle) is used. Under these conditions, the change in entropy is significantly reduced, for example, by ∼66% at *T*_{2}/*T*_{1} = 10 for *c*_{V,1} = 3*k*_{B}/2 and *c* = 0.5 under isobaric conditions.

Equation (4) states that temperature and pressure gradients are an important source for generating additional entropy from mixing. Both of these gradients have familiar roles in entropy generation, respectively: (1) thermal conduction and thermodiffusion, and (2) barodiffusion. In the former case, the rates of specific entropy production are proportional to $\kappa \u2207lnT2$ and $\kappa T\u2207lnT2$, where *κ* is the electron or ion thermal conductivity and $\kappa T$ is the electron or ion thermodiffusion coefficient; in the latter case, the rate $\u221d\kappa P\u2207lnPi2$, where *κ _{P}* is the barodiffusion coefficient and $Pi$ the ion pressure.

^{24}Of these transport processes only thermal conduction is a single-species effect; barodiffusion and thermodiffusion are intrinsically multi-species in nature. Because this present work treats exclusively the entropy generation from mixing of

*identical*particles in the presence of temperature and pressure gradients, only the role of thermal conduction enters here. The potential mitigating effect of thermal conduction on the mix-generated entropy considered herein is described in Sec. IV B.

### B. Evaluation of −*P*d*V* work from mixing

We earlier asserted in Sec. I that the generation of entropy from mixing is not a new source of internal energy $\Delta E$, but instead comes at the expense of compressional energy or −*P*d*V* work when the internal energy is conserved: $dE=0=T\Delta S\u2212P\Delta V$. To understand this further, we proceed to evaluate the amount of work $Wmix\u2261\u2212P\Delta V$ required to instantaneously mix two identical ideal gases at different temperatures under isobaric conditions, that is, at the same physical conditions underlying Eq. (1). Recall that the temperatures in vessels “1” and “2” before mixing are *T*_{1} and *T*_{2}, respectively, and that the mixed state temperature $T\u2032$ is $T1+T22$ for equal (and contant) numbers of particles *N*_{1} = *N*_{2} ≡ *N/*2. The (initial) volumes of the two gases before mixing are *V*_{1,0} and *V*_{2,0}, and the respective volumes after mixing are both *V* = *V*_{1,0} + *V*_{2,0}. To proceed, we make the following ansatz for the “temperatures” during mixing:

where $\varpi $ is a parameter that will be constrained shortly. The mixing process is intrinsically non-equilibrium and as such the identification of *T*_{j} in Eqs. (6a) and (6b) with temperature lacks a thermodynamic basis; rather, *T*_{j} should be construed as a transient mixing parameter that coincides with temperature only in the initial (unmixed) and final (mixed) equilibrium states. Since both gases are ideal and nominally non-interacting, the rate of expansion for each gas will differ. However, Eqs. (6a) and (6b) describe a particular mixing scenario where the two gases are decoupled to lowest order. The partial pressures during mixing *P*_{j}(*V*_{j}) follow from Eqs. (6a) and (6b) as $NkBTj(Vj)/2Vj$ The work done by both gases then readily follows:

where the ratio of initial volumes $V1,0/V2,0=T2/T1$ follows from the assumed isobaric conditions for the two ideal gases prior to mixing. The first two terms on the right-hand side of Eq. (7) represent the maximum work done by each expanding isothermal gas against a confining piston (as if the other gas were not present): both terms are nonzero in the limit as *T*_{1} goes to *T*_{2}. By contrast, the third (or last) term is second order in $T2T1\u22121$ and describes the work done in mixing: *W _{mi}*

_{x}. In comparison, the amount of heat generated upon mixing $Tmix\Delta S$ is easily shown from Eq. (5a) to vary as $\u2009NcPT2/T1\u221212Tmix/8$—also second order in $T2T1\u22121$. In the limit of $T2T1\u21921$, the mix temperature

*T*

_{mix}must also approach

*T*

_{2}(and

*T*

_{1}) as well. We use this asymptotic condition on

*T*and internal energy conservation $0=Tmix\Delta S+Wmix$ to constrain the value of $\varpi \u22450.14.$ For arbitrary $T2/T1$,

_{mix}*T*can now be evaluated from $\u2212Wmix/\Delta S$. Figure 3 shows the variation of

_{mix}*T*over a wide range of $1\u2264T2/T1\u226410$. Overall,

_{mix}*T*(solid line) is below but remains fairly close to the mean temperature $T\u2032$ [dashed line; see Eq. (1)] of the two vessels before mixing for $T2/T1$ < 5. This exercise establishes that the entropy generated from mixing of the two vessels is consistent with an estimate of the internal “work” done in mixing. This value of work can also be notionally identified as the energy required to “demix” the composite mixture and restore the initial binary configuration.

_{mix}^{22}Other realizations for the mixing history (and

*T*) are innumerably possible, given that the integral of $\u2212P\Delta V$ (and $\Delta S$) depends on the choice of quasi-static path taken to reach the fully mixed state.

_{mix}^{25}Still, the ansatz represented by Eqs. (6a) and (6b) is reasonably consistent with ideal agreement between

*T*and $T\u2032\u2009\u2261$ (

_{mix}*T*

_{1}+

*T*

_{2})/2 over a significant range of $T2/T1$.

### C. Effect of entropy generation from mixing on compression

A key question is whether the generation of entropy from mixing has macroscopic consequences for an ICF implosion: for example, a reduced compressibility of the imploding fuel. Preheat of the fuel from hohlraum-generated hot electrons and x-ray preheat absorption are well-known examples of *external* sources of heat energy Δ*Q* that can deleteriously raise the fuel adiabat and limit the fuel compression. However, the entropy of mixing represents a potential *internal* source of degraded compressibility without the addition of heat energy as we now show.

Consider again the idealized case from above (Sec. II B) of two vessels with equal number of ideal gas particles at different temperatures but at the same pressure. The two unmixed vessels are then adiabatically compressed to a final volume, and the work done in this process is evaluated. A second calculation allows the two vessels to first atomically mix, followed by an adiabatic compression to the same final (total) volume. Comparison of the work done under both scenarios isolates the role of entropy generation from mixing in potentially affecting the compressed state of the composite ideal gas.

In the first scenario of adiabatically compressing the two unmixed gases, the work done is

where $PjVj\gamma =bj$ is a function of the entropy only, *b*_{1}/*b*_{2} = (*V*_{1}/*V*_{2})^{γ} follows from the isobaric condition *P*_{1} = *P*_{2}, and *C _{V}* =

*V*

_{2}/

*V*

_{2,f}=

*V*

_{1}/

*V*

_{1,f}is the final volume convergence of each compressed gas. We now evaluate the work done in compressing the gases

*after*mixing to the same volume convergence

where $b\u2032$/*b*_{2} = (*V*_{1} + *V*_{2})^{γ}/*V*_{2}γ follows from the condition $P\u2032$ = *P*_{1} = *P*_{2}. A crucial distinction must be made in how the final state $I\u2032$ is reached, whether by atomically mixing the two vessels or by separately bringing the two initial gases in thermal contact with a temperature reservoir at $T\u2032=(T1+T2)/2.$ In the latter case, the change in entropy Δ*S* can be shown to vanish, while for the mixing scenario, the change in entropy is given by Eq. (5a). To demonstrate that an adiabatic path to $T\u2032$ for both gas vessels exists, we increase $V1$ to $V1T\u2032/T1$ and decrease $V2$ to $V2T\u2032/T2$ to maintain constant pressure *P*. The corresponding work done by the first vessel is $\u2212P\Delta V1=P(V1\u2212V2)/2,$ while for the second vessel, $\u2212P\Delta V2=P(V2\u2212V1)/2$; thus, the total work done $\Delta W$ is zero. The change in internal energy for the first vessel $\Delta E1$ as the temperature is raised to $T\u2032$ is $NcV,1T\u2032\u2212T1=NcV,1T2\u2212T1/2$; similarly $\Delta E2=NcV,2T\u2032\u2212T2=\u2212\Delta E1$ for *c*_{V,1} = *c*_{V,2}, and the total change in internal energy $\Delta E$ is also zero. From $\Delta E=T\Delta S+\Delta W$, we find $\Delta S=0$ in the process of bringing both vessels to *T'* at constant pressure. Note that with the exception of entropy the same values of *N*, *P*, $T\u2032$, *V _{f}* for the two samples are maintained across the two different processes. (Applying the adiabatic law $PVf\gamma =const.$ would at first glance lead to a contradiction, but this is resolved by realizing that this law applies to a

*process*and not the description of a unique state—which is the defining role of an equation of state.)

The extra work required to compress the mixed gas to the same final volume as the two separate vessels (or unmixed composite vessel at *T'*) follows from Eq. (9) as $I12e\Delta S/cV\u2032\u22121$. Using Eq. (5a) we may rewrite this required extra work as

Figure 4 displays the relative increase in required work to overcome the entropy increase resulting from mixing vs temperature ratio of the two gases and ratio of specific heats. For a factor-of-2 ratio of initial temperatures of the gases before mixing, nearly 20% more work is required to achieve the same volume compression as the unmixed gases. The extra work required to compress a mixed gas Δ*W* compared with its unmixed constituent gases is tantamount to adiabatically introducing an equivalent amount of internal energy Δ*E* to the mixed state, for example, preheat.

## III. EVALUATION OF Δ*S* FOR FUEL-ABLATOR MIX

### A. Rayleigh–Taylor mix under isothermal conditions (*T*_{1} = *T*_{2}, *P*_{1} ≠ *P*_{2})

We now apply Eq. (4) to a previously reported central hot-spot ignition design (“Rev. 5.0”).^{18} In this target design, an innermost undoped (pure) CH layer of the ablator is adjacent to the solid DT fuel layer. As discussed in the Introduction, this interface may be a candidate for instability-induced mixing due to the presumed high Reynolds number that occurs near and after deceleration onset. In the following, we presume that strong mixing occurs across this interface with establishment of a well-defined mix layer that follows a Youngs-type of self-similar evolution^{26} in time *τ*:

where $\alpha b\u22450.06$ is based on experiments and direct numerical simulation studies with sufficiently long wavelength,^{27,28}$At\u2261\rho 2\u2212\rho 1/\rho 2+\rho 1$ is the Atwood number at the CH/DT interface, *ρ*_{1} (*ρ*_{2}) is the DT (CH) mass density, Δ_{b} refers to the “bubble” side of the mix layer with buoyant DT material rising into the CH material for positive Atwood number, Δ_{s} refers to the “spike” side of the mix layer where CH material penetrates the DT fuel (*At* > 0), and *g >* 0 is the acceleration of the interface. The total mix width is denoted by $\Delta mix=\Delta b+\Delta s.$ The Atwood number of the unperturbed interface can be estimated by asserting (total) pressure and temperature continuity at the interface separating two ideal gases to ensure finite interface acceleration and heat transport $\u221d\u2207T$ across the interface, giving

where *Z*_{1} is the mean ionization state of an averaged D and T ion—assumed to be unity, *Z*_{2} is the ionization state of an average CH ion, and *A*_{1}(*A*_{2}) is the mean atomic weight of an average DT (CH) ion. For the case of $\u224515$ eV CH plasma, we have $Z2\u22452$ and an Atwood number ∼0.27, while for a high-density carbon (HDC) ablator for the same conditions *At* = 0.53. Equation (12) is appropriate for infinitesimally small excursions away from the interface or for bubble and spike amplitudes much less than both the pressure- and temperature-gradient scale lengths.

The assumption of temperature continuity at the interface underlying Eq. (12) can be justified as follows. The total pressure must be continuous across the interface for a finite acceleration, but the temperature and density need not be—at least initially. In the case of temperature, the difference in specific heats of the two adjacent materials will lead to an initial jump in temperature that diffusively relaxes in time due to thermal conduction. The scale length Δ over which the temperature profile can relax in a time *τ* is $4\tau \chi $, where $\chi =\kappa /\rho $ is the thermometric conductivity.^{22} Similarly, the time scales for the temperature *gradients* to relax from an infinite value at *τ* = 0 to the values of temperature-gradient scale length on either side of the interface $LT,j$ are $LT,j2/2\chi j$ with *j* = 1, 2.^{24} For typical conditions expected on the DT/CH interface (*T* ∼ 15 eV, *ρ* ∼ 1 g/cc) early in the implosion, the time scales for relaxation of the temperature gradients on either side of the interface are on the order of 1 ns when $LT,j$ is on the order of 1 *μ*m. This means that there is sufficient time for the temperature profile across the material interface to become quasi-continuous while maintaining a difference in temperature-gradient scale lengths. By contrast, the timescale for diffusive mass flow across the interface is 10–100 times slower, as given by the Lewis number (or ratio of thermometric conductivity to the diffusion coefficient *D*). Thus, the original interface remains essentially intact over the timescale for the temperature profiles to collisionally equilibrate as argued above. Radiation–hydrodynamics simulations (with flux-limited electron transport) are also consistent with this physical picture of the temperature profile across a classical interface becoming continuous by the time the interface has begun accelerating.

We write for the average pressure in the DT fuel penetrated by a spike distance Δ_{s}: $P\xaf1\u2245P0\u2212\u2207P1\xb7\Delta s/2$ and for the average pressure in the surrounding CH material over a bubble distance Δ_{b}: $P\xaf2\u2245P0+\u2207P2\xb7\Delta b/2$, where $P0$ is the pressure at the classical interface. The factor-of-2 arises since we are taking a spatial average of the (linearly varying) pressure changeover the spike or bubble penetration length on either side of the interface. From $\u2207P=\u2212\rho g$ and using that both sides of the interface must accelerate at *g* (by definition), we obtain $\u2207P1/\u2207P2=1\u2212At/1+At$. Thus, $P\xaf2\u2245P\xaf1+\u2207P2\xb7\Delta mix+At\Delta b\u2212\Delta s/21+At$, which simplifies to $P\xaf2\u2245P\xaf1+\u2207P2\xb7\Delta mix/2$ with use of Eqs. (11a) and (11b) in the limit of small Atwood number. In addition to the mix scale length $\Delta mix=\Delta b+\Delta s$, another natural scale length for the quasi-isothermal conditions at hand is the usual gravitational (or acceleration) scale length $\Delta g,j=C\u2009s,j2/g,$ where the isothermal sound speed $Cs,j$ is defined as $Zj+1kBT/mpAj$ with *m _{p}* the proton mass. Using that the acceleration

*g*must be continuous across the interface, the gravitational scale lengths on each side of the interface must satisfy: $\Delta g,2\Delta g,1=1\u2212At1+At,$ after using Eq. (12). According to 1D radiation hydrodynamics simulations of the Rev. 5.0 (CH) design, near the time of deceleration onset the ion temperature is about 15 eV and the mass density is nearly 15 g/cc. Under these conditions, Fig. 2 establishes that the specific heats (per free particle) are the same to a good approximation. Thus, the change in entropy from Eq. (4) under isothermal conditions and taking identical specific heats can now be written as

where

follows from Eq. (12), and Eqs. (11a) and (11b) are used to evaluate $\Delta b/\Delta s$. Here, *N*_{1i} (*N*_{2i}) is the total number of ions in region 1(2), and $n1i(n2i)$ is the ion number density in region 1(2) with volume $V1(V2)$.

We now apply Eqs. (13) and (14) to the case of the Rev. 5.0 implosion design.^{18} After deceleration onset, the average value of *g* over the ∼500 ps remaining implosion time is simulated to be nearly $7\xd71016$ cm/s^{2}. As stated earlier, the temperature of the material near the interface early in the deceleration phase is ∼15 eV, giving $\Delta g,2\u22452\u2009$*μ*m. Similarly, $\Delta mix\u22456\u2009$*μ*m under the same deceleration conditions after using $Z2\u22452$ in Eq. (12) to evaluate the Atwood number. However, we cannot know with confidence how soon before or after deceleration onset a presumptive mix layer describable by Eqs. (11a) and (11b) is formed. For this reason, we have parameterized the entropy change in terms of the dimensionless mix parameter $\Delta b\Delta g,2<2$. If a mix layer is established, say, halfway between deceleration onset and hot spot ignition, then the total mix layer thickness is also close to 2 *μ*m. Figure 5 displays the amount of entropy (in units of energy) generated by mix (at the expense of compressional energy −*P*d*V*) across the DT/pusher interface for two choices of $Z2$ for CH and $Z2$ = 2 for HDC. The number density of *interchanged* ions $nmix,i$ in the Rayleigh–Taylor-induced mix layer is estimated as $nmix,i=N1i+N2i/V1+V2\u2245\u2009n2iV1+n1iV2V1+V2=n2i\Delta s+n1i\Delta b/\Delta mix$. The number density of mixed ions and electrons follows as $nmix\u2245n2i\Delta s1+Z2+n1i\Delta b1+Z1/\Delta mix$, which simplifies with use of Eq. (12) to $nmix\u2245\rho 11+Z1A1.$ Thus, the total number of ions and electrons in the mixed layer follows as:

where *R _{fuel}* is the radius of the DT–pusher interface near deceleration onset. The mass density on the fuel side of the unstable interface is estimated to be ∼10 g/cc since an expected peak fuel density of ∼1000 g/cc in the main fuel layer is achieved at a convergence of ∼35, compared with ∼3.5 near deceleration onset. This density value agrees with 1D radiation hydrodynamics simulations near the time of deceleration onset. Figure 5 shows that the level of “heating” from entropy generation remains safely below several Joules if $\Delta b/\Delta g,2$ for both ablator types and two values of CH ablator ionization state $Z2$.

Finally, we should state the limits of applicability of applying non-isobaric conditions across the evolving mix layer. The process of mixing tends to homogenize the hydrodynamic quantities, for example, *P _{j}*, but a sufficiently short sound-speed transit time allows the information from the surrounding unmixed material to “maintain” or update the gradient profiles within the mix layer from outside the mix region. The applicability of the present analysis with maintained pressure gradients across a mix layer can be justified to roughly hold when the bubble transit time $\Delta b/Cs,j\u227c\Delta g,j/Cs,j$.

### B. Rayleigh–Taylor mix under isobaric conditions (*P*_{1} = *P*_{2}, *T*_{1} ≠ *T*_{2})

For the Rev. 5.0 capsule design,^{17} x-ray preheat of the CH layer adjacent to the DT ice layer is predicted to give a negative Atwood number *before* deceleration onset (when averaged over several micrometers or more away from the interface), thereby satisfying the necessary conditions for Rayleigh–Taylor growth of surface perturbations. The preheated CH layer also gives rise to a temperature gradient across the interface that may be important near deceleration *onset* $g\u22610$ when the gravitational scale length is large compared to a bubble or spike amplitude. To account for nonzero temperature gradients near deceleration onset, we derive in place of Eq. (12) for the Atwood number

where the local approximation reads

Δ_{j} is the bubble or spike penetration into region *j* depending on whether the Atwood number is positive or negative, respectively, and Δ_{T}_{,j} > 0 is the temperature-gradient scale length on the *j*th side of the interface. The amount of entropy generated from mixing follows from Eq. (4) when: $\Delta g,2\u226b\Delta j:$

where the dependence of *c* and *N* on Atwood number is obtained as follows:

We require that the thermal energy flux $\u2212\kappa \u2207T$ be continuous across the interface,^{29} where *κ* is the Spitzer–Harm thermal conductivity. Using that *κ* is proportional to *s*(*Z*)/*Z*, where *s*(*Z*) is a relatively weak function of *Z*,^{30} we have that $\Delta T,2/\Delta T,1\u22450.85$ for *Z*_{CH} = 1.5 and $\Delta T,2/\Delta T,1\u22450.75$ for *Z*_{CH} = 2.5. Figure 6 shows the amount of predicted entropy generation from mixing of DT and CH in the presence of a temperature gradient from hohlraum-generated x-ray preheat near deceleration onset $\Delta i\Delta g,2\u226a1$ vs the ratio of spike (*At* > 0) or bubble (*At* < 0) size to a temperature-gradient scale length on the CH side of the interface. The Atwood number is also shown, which is found by iteratively solving Eqs. (11a), (11b), (16), and (17). A strong temperature gradient drives a negative Atwood number, which significantly enhances the entropy of mixing to levels approaching ∼20 J under the conditions shown. In the case of undoped ablators such as high-density carbon (HDC), the higher levels of x-ray preheat absorbed by the carbon near the interface will lead to a smaller temperature-gradient scale length, a larger (negative) Atwood number and potentially larger values of entropy generation, for example, when $\Delta b/2\Delta T,2$ is near unity or greater. In assessing Fig. 6, it is important to realize that at early time when the mix width is still small ($\u22721\u2009\mu m),\u2009T1\u2245T2$ and the Atwood number achieves its maximum (positive) value. As the mix layer grows further, the ratio $T2/T1$ as defined by Eq. (17) increases, and the Atwood number decreases toward zero or even negative values. By this time, the entropy generation from mixing has reached significant levels even with a reduced Atwood number, as depicted in Fig. 6.

### C. Rayleigh–Taylor mix under non-isothermal (*T*_{1} ≠ *T*_{2}) and non-isobaric conditions (*P*_{1} ≠ *P*_{2})

Given the significant level of entropy of mixing from a temperature gradient alone, cf. Fig. 6, it is useful to assess the amount of entropy heating in the presence of *both* pressure and temperature gradients. From $T2T1=A2A1\xb71\u2212At1+At\xb71+Z11+Z2\xb7P2P1$ for two contiguous ideal gases, approximating

and recalling the ratio of gravitational scale lengths

we obtain the solution for *T*_{2}/*T*_{1}

Figure 7 shows the (reciprocal) solution of Eq. (22) for *T*_{1}/*T*_{2} and the associated change in entropy from mixing for several choices of Atwood number before deceleration onset (*g* ≤ 0). A real solution exists when $\Delta 2\Delta g,2\u22641+1+At0.661\u2212At0.66\u22121$, leading to a restricted range of admissible values for $\Delta 2\Delta g,2$. A clear trend of increasing entropy change with larger (negative) Atwood number and pressure gradient is indicated. Thus, the change in entropy from preheat-induced temperature gradients as shown in Fig. 6 is further enhanced by the additional effects of finite (nonzero) pressure gradients.

### D. Application to HDC implosions under isobaric conditions (*P*_{1} = *P*_{2}, *T*_{1} ≠ *T*_{2})

Simulation studies of Rev. 5.0 low-adiabat CH implosions routinely show that Atwood numbers of ∼ −0.1 (when averaged over several micrometers away from the interface) or less up to the time of deceleration onset are expected. For undoped (ablator) HDC implosions,^{31} the susceptibility to preheat is greater and the interface (DT/C) temperature is closer to 35 eV or more, according to 1D radiation–hydrodynamics simulations. The analysis of the entropy generated from mix [based on Eqs. (16)–(19)] for (undoped) HDC as shown in Fig. 8 suggests that for an average Atwood number equal to −0.26 prior to deceleration onset, close to 950 J of entropy change (in energy units) is predicted for a mix width of 25 *μ*m at an average electron temperature of 35 eV. Here, the temporal average is taken over times up to deceleration onset that satisfy *At* < 0. High-resolution 2D multi-mode simulations of undoped HDC implosions using the radiation-hydrodynamics code HYDRA^{32} show the effective combined range of bubble and spike penetration approaches the range of 20–30 *μ*m by the instant of deceleration onset.^{20,33} Such a potentially high level of DT ice heating from the entropy of mixing would compromise ignition, for example, see Fig. 9(a), but its effects could also be manifested as a strongly decompressed fuel or anomalously reduced areal density. To gauge the level of decompression we consider an adiabatic (γ = 5/3) implosion analysis that relates the final fuel stagnation pressure *P*_{stag} and radius *r*_{stag} to the fuel pressure *P*_{d} and radius *r*_{d} at deceleration onset

where *M _{p}* is the pusher mass and

*υ*is the pusher peak velocity. The density of the DT ice layer $\rho shell\u2248rdrstag2\u2261C2,$ where

_{p}*C*is the fuel radius convergence relative to the time of deceleration onset. From Eq. (23), we readily obtain $\u22481/Pd1/2$, so that $\Delta \rho shell\rho shell=\u2212\Delta Pd/Pd$ at constant

*r*. At a simulated average temperature of ∼20 eV throughout the DT ice layer near deceleration onset, the internal energy $\epsilon =3PV/2$ is ∼400 J, including the electron internal energy. From Fig. 8 with

_{d}*At*= −0.26, the change in entropy is ∼950 J for HDC, compared with only ∼80 J for a CH (Rev. 5.0) ablator. The assumed amount of mix (25

*μ*m) for undoped HDC is based on the following argument. According to 1D radiation hydrodynamics simulations, the duration of acceleration for a (3-shock) undoped HDC implosion after final shock launch into the fuel is about 4.5 ns and the peak implosion speed attains ∼320

*μ*m/ns, giving an average acceleration of nearly 63

*μ*m/ns

^{2}. From Eqs. (11a) and (11b), the mix width is about 16

*μ*m if the instability growth is due only to Rayleigh–Taylor. Since nearly half of this entropy increase resides in the fuel (with the other half in the ablator), we infer from above that the fuel density fractionally decreases by a factor on the order of two or more. A key role for Richtmyer–Meshkov instability early in the evolution of the presumptive mix layer will contribute further to the mix width.

A number of uncertainties in the above estimate for *T*Δ*S* in undoped HDC ablators should be expressly noted, including values of the mix width in 3D compared with 2D, the average temperature taken at the interface, and the Atwood number. If our understanding of thermal conductivity is significantly in error, this uncertainty would directly impact the temperature and Atwood number, for example.^{18} Given that Fig. 8 at an Atwood number of −0.26 for HDC lies close to a resonance of the temperature profile near $\Delta 2/\Delta T,2\u22452$ in the local approximation, cf. Eq. (17) and its near vanishing denominator, the profiles used in evaluating Δ*S* should be reassessed. To this end, we choose a more general profile for the temperature ratio free of resonances at all finite values of Δ_{j}/Δ_{T,j}

Figure 10 shows an adaptation of Fig. 8 with this assumed temperature ratio parameterization for undoped and doped^{33} HDC ablators. Clearly, the amount of entropy change ∼350 J has significantly dropped and leads to a reduced density in the DT ice layer of ∼44% for the undoped case. If the converging fuel layer has a nearly constant thickness, that is, $\rho \u223c1R2,$ this leads to a ∼22% drop in areal density, compared with the observed deficit of 21% relative to 2D multi-mode simulations for shot N160120.^{33} For the W-doped HDC ablator an average Atwood number of −0.08 up to deceleration onset is predicted from simulations, giving nearly 100 J of entropy generation from Eq. (7), and nearly half of that value localized to the DT fuel layer. The corresponding drop in fuel density is ∼12%, or ∼6% in areal density if $\rho \u2009\u223c\u20091/R2$ is again taken. This compares favorably to within the error bars of shot N160313, where a 2% deficit in areal density was inferred relative to postshot 2D multi-mode simulations.^{33} Although this amount of entropy change is small, it may still significantly degrade the available ignition margin from inspection of Fig. 9(a). Whether the ∼50 J of entropy is deposited uniformly throughout the DT ice layer or only the outer half, the yield is still reduced by several MJ. In both cases, the energy is deposited over 3 ns at a rate that is linearly ramped in time to emulate the Rayleigh–Taylor mix layer growth rate (∼*τ*^{2}). Figure 9(b) shows a strong sensitivity of the yield the onset of energy deposition. The baseline case of 3 ns for Fig. 9(a) was chosen due to the comparably long duration of negative Atwood number seen in the 1D simulations.

### E. Scaling of entropy of mixing with capsule size and mitigation strategy

The data for undoped HDC implosions and the estimated amount of entropy generated in doped HDC ablators collectively argue that the entropy of mixing may be an important source of performance margin degradation and a potential challenge to achieving ignition at the 1–2 MJ driver scale. It is natural to ask at what scale could the entropy of mixing be rendered relatively unimportant. Here, we derive some relations that may suggest such a scale if the origin of the mix is primarily generated by Rayleigh–Taylor instability. The chosen strategy is to reduce fuel convergence with increasing scale size and driver energy by sacrificing target energy gain.

If we introduce the Euler scaling parameter *σ* such that all hydrodynamic lengths and times scale with driver energy as $Edriver1/3$ = *σ*, then the required density $\rho gas$ in the gaseous (hot spot) fuel of radius $Rgas$ follows from the compressed central DT fuel mass $Mgas=4\pi 3\rho gasRgas3=4\pi \rho gasRgas33\rho gas2\u2009\u223c\u2009\sigma 3$ and the minimum areal density constraint $\rho gasRgas$ of ∼0.3 g/cm^{2} for spark or hot-spot ignition,^{30} giving $\rho gas\u2009\u223c\u2009\sigma \u22123/2$ and $Rgas\u2009\u223c\u2009\sigma 3/2.$ In a similar fashion, the solid fuel radius *R*_{shell} follows from writing the mass of the solid fuel layer *M*_{shell} in terms of the minimum areal density requirement for efficient alpha particle heating and propagating thermonuclear burn in the thin-shell approximation: $Mshell\u22454\pi Rshell2(\rho shell\Delta shell)\u2009\u223c\u2009\sigma 3.$ Using that the minimum value of $\rho shell\Delta shell\u2245$ 2 g/cm^{2} should be met across all driver scalings of interest,^{30} we have that $Rshell\u2009\u223c\u2009\sigma 3/2$ and $\rho shell\u2009\u223c\u20091/\Delta shell$ for the peak shell density. The fuel convergence *C*_{f}, defined as the initial ablator radius $\u223c\sigma $ over the final fuel radius, thus scales favorably as $\sigma \u22121/2$ instead of $\sigma 0.$ This relation implies that $\rho shell\u2009\u223c\u2009\rho 0Cf2\u2009\u223c\u2009\sigma \u22121.$ Next, the requirement of a robust ignition boundary forces the peak hot-spot temperature *T*_{gas} to remain sufficiently high: $\rho gasRgasTgas\rho shell/\rho gas>6$ keV, over the *T*_{gas} range 5–15 keV.^{35} From the above scalings for $\rho shell$ and $\rho gas$, we find that $Tgas\u2009\u223c\u2009\sigma \u22121/4$, which is a fairly weak function of driver energy $Tgas\u2009\u223c\u2009Edriver\u22121/12.$ The peak implosion speed $\upsilon imp$ follows from applying energy conservation between the time of peak shell kinetic energy and final fuel stagnation: $\upsilon imp\u223cNi,fTgas/\rho shellVgas\u2009\u223c\u2009\sigma \u22123/8$, where *N _{i,f}* is the number of hot-spot fuel ions and $Vgas\u2009\u223c\u2009Rgas3$ is the hot spot volume. The acceleration

*g*of the solid fuel/ablator interface (or $Rfuel)$ then follows from $g\u2009\u223c\u2009\upsilon imp\tau imp\u2009\u223c\u2009\upsilon imp2\Delta R\u2009\u223c\u2009\upsilon imp2Rshell(Cf\u22121)\u2009\u223c\u2009\sigma \u221294,$ where $\tau imp$ is the implosion time and Δ

*R*is the deceleration distance of the Rayleigh–Taylor unstable fuel–pusher interface. This scaling is in contrast to $g\u2009\u223c\u20091/\sigma $ for a standard Euler scaling where the convergence is

*not*a function of Euler scale

*σ*. From Eqs. (11a) and (11b), the mix layer thickness Δ

_{mix}scales in turn as $g\tau imp2\u2009\u223c\u2009\sigma 1/2$. We now can write for the scaling of number of mixed ions

*N*on a presumptive Rayleigh–Taylor unstable interface between the fuel and ablator: $Nmix\u223c\u2009\Delta mix\rho shellRshell2\u2009\u223c\u2009\sigma 2$, compared with $\sigma 3$ from a usual Euler scaling. Thus, $\Delta S\u2009\u223c\u2009Nmix$ varies with scale in a relative sense as 1/σ according to this prescription for reduced convergence with scale. For example, a factor-of-2 decrease in the role of mix on degraded ignition margin requires a 2 $\xd7$ larger scale or 8 $\xd7$ in driver energy.

_{mix}The strategy set forth here for curbing the effects of entropy generation by mix with reduced target convergence comes at the price of reduced energy gain and larger energy drivers. This trade-off is unavoidable unless measures are taken to directly reduce Rayleigh–Taylor-induced mix by pulse shaping, hohlraum preheat control or improved target fabrication protocols. On the other hand, if the underlying mix has an inherent transport origin, that is, finite ion mean free paths, then a strategy of reduced convergence can aggravate the effects of mix. For example, binary mass diffusion leads to an effective mix width $\Delta x\u223cD\tau \u2245Cs\lambda ii\tau $, where the ion–ion mean free path $\lambda ii$ scales as $T2/\rho shell$. Thus, a goal of reducing the shell density can increase the ion mean free path and lead to an unfavorable scaling of mix with size of the target: $\Delta x\u2009\u223c\u2009\sigma 3/2.$ However, if the required peak shell density and shell convergence remain unchanged, then a favorable (Euler) scaling with size is found: $\Delta x\u2009\u223c\u2009\sigma 1/2.$ Clearly, the most effective means of controlling the effects of mix will depend on the origin of the mixing, that is, whether it is hydrodynamic-like (Rayleigh–Taylor or Richtmyer–Meshkov) or generated by finite *λ _{ii}* as in shock-generated mixing.

^{19}

Another approach to help mitigate the potential role of entropy of mixing in ICF is to relax the above (fuel) shell areal density requirement ($\rho shell\Delta shell\u2245$ 2) and operate at a higher adiabat by design.^{36} This option provides reduced hydro-instability growth from greater ablative stabilization and may altogether avoid the Rayleigh–Taylor mix transition. The trade-off of this strategy is reduced energy gain and 1D performance margin *M*, but the advantage is less exposure to all sources of preheat, whether known or anomalous. This strategy can be most readily realized by using a shortened laser pulse with raised laser “picket”—which sets the initial shock strength or pedestal drive temperature—to markedly raise the pusher adiabat and fuel adiabat $\alpha $. The margin is related to the ignition threshold factor, ITF = *M* + 1 $\u221dEcap/\alpha 1.9.$^{35} Thus, the 1D margin is appreciably eroded by operating at a high adiabat, but this can be offset by substantially increasing the capsule absorbed energy $Ecap$. Increasing the capsule size is the most direct and efficient way of increasing $Ecap$, but the challenge is ensuring adequate hohlraum drive symmetry near the end of the laser pulse. Cylinder-shaped hohlraums are limited in accommodating a significantly larger capsule due to (1) the restricted volume above the capsule for ample late-time laser propagation, and (2) the need to maintain sufficient drive temperature. Other hohlraum shapes such as the rugby^{36,37} and Frustraum^{38} are intended to allow nearly $3\xd7$ higher $Ecap$ by designing a large volume above the capsule while maintaining sufficient drive and symmetry. In this way, a high adiabat can be well-tolerated as a direct means of withstanding all sources of hohlraum- and capsule-generated preheat.

## IV. EVALUATION OF Δ*S* FOR ISOBARIC ICF CONDITIONS

### A. Evaluation of entropy of Rayleigh–Taylor mixing at gas–solid fuel interface

Another interface of potential interest for mix in an ICF implosion is the DT gas/solid interface. Compared to the DT/CH and DT/HDC cases considered in Sec. III, this interface that defines the boundary of the “hot spot” is isobaric to a good approximation and is characterized by a stronger temperature gradient. Equation (4) with $P1\u2245P2$ is the appropriate limit to use in evaluating the generation of entropy from mixing across this boundary.

To obtain a representative value of temperature-gradient scale length at this interface, we recall the hot-spot dynamics analysis of Betti *et al.* for obtaining the gradient density scale length $L\rho $^{39}

Because of the isobaric assumption and assumed ideal gas equation of state for DT, the temperature-gradient scale length at the ablation front $LT\u22611/\u2207lnT$ achieves its maximum (absolute) value and is equal and opposite to $L\rho $. Near the time of deceleration onset $\rho gas$ is on the order of 0.1 g/cm^{3}, and $\rho shell$ is nearly 2 g/cm^{3} for a shell convergence of ∼5, giving a temperature-gradient scale length on the order of 1 *μ*m. At the time of fuel stagnation (or shell fuel convergence of ∼35), $\rho gas$ is on the order of 100 g/cm^{3} and $\rho shell$ is nearly 1000 g/cm^{3}, after noting that the ablated DT shell mass provides a nearly $10\xd7$ increase in igniting hot spot mass. The corresponding temperature-gradient scale length is still on the order of 1 *μ*m.

Near the interface or fuel ablation front we approximate the mass density and temperature two-step distributions as

where $\rho 0$ is the density at the fuel ablation front (defined where $\u22022\rho \u2202R2=0$) and $T0$ is the corresponding temperature. Using Eqs. (4), (11a), (11b), and (14), we obtain for the entropy generated by mixing under isobaric conditions

where the factor $\rho 0\xb7Rgas2$ remains of order unity throughout the implosion from mass conservation. The Atwood number in Eq. (28a) is found by iteratively solving the equation

with use of Eqs. (11a), (11b), and (26). Figure 11 shows the entropy change generated near the ablation-front “interface” where the conditions for achieving a full mixing transition are more likely to occur on the bubble (high density) side because of the higher density and lower temperature compared with the spike (low density) side. At a hot-spot radius of ∼170 *μ*m, the energy in the hot-spot fuel is nearly 350 J according to radiation-hydrodynamic simulations. Thus, the case of a bubble amplitude of nearly 0.5 *μ*m in Fig. 11 generates less than 1% of the total energy of the hot-spot fuel in $T\Delta S$ from complete mixing.

### B. Mitigation of mix-induced entropy generation by thermal conduction

The degree of mixing at the fuel ablation-front interface—or at any unstable interface—warrants further discussion. According to multi-mode simulations of the perturbation growth near the DT gas/ice interface, the dominant Legendre mode numbers in the highly nonlinear regime often lie in the range of 10–15. The Reynolds number on the bubble side of the interface can be evaluated from $Re=\Delta x\Delta \upsilon \nu ,$ where $\Delta x$ is the radius of one of the pair of counterrotating vortices $\pi Rgas/\u2113$ within a bubble of dominant perturbation mode number $\u2113$, $\Delta \upsilon \u2245gRgas3\u2113$ is a classical 2D bubble velocity, and $\nu =1.91T052eV/(2.5ln\Lambda \xb7ni01019cc).$^{14} In convenient (normalized) units, we have

Under steady-state conditions, the mixing transition for a fluid occurs near a Reynolds number of ∼10^{4}, so the “bubble” or heavy side of the interface is marginally subject to large-scale mixing. For conditions of incomplete or marginal mixing, the available entropy change from mixing can be estimated as follows. The process of mixing can be visualized by successively adding more contiguous and alternating regions of pure materials in analogy with delineated bubble and spike regions in hydrodynamic growth experiments and simulations (e.g., see Ref. 33). For each manifestation as depicted in Fig. 12, the regions remain distinct and unmixed, but are further susceptible to transport effects as the number of partitions increases (or region width decreases). In particular, thermal transport can lead to a smaller temperature difference between adjacent regions and limit the available amount of entropy of mixing once a transition to full atomic mixing occurs.

We define a temperature equilibration mix parameter $0<\epsilon m<1$, where $\epsilon m\u2261\tau H/(\tau H+\tau eq),$ *τ*_{H} is a hydrodynamic timescale of interest and *τ*_{eq} the timescale for effective temperature equilibration of two contiguous regions. The equilibration of temperature is inherently a diffusive process, which we write as the square of the width of a region over the thermometric conductivity $\chi $: $\tau eq\u223c\Delta x2\chi .$ The width Δ*x* in turn scales as the reciprocal of the number of partitions *m*. The hydrodynamic timescale is taken as the duration of Rayleigh–Taylor growth, beginning with the linear stage and continuing through the fully nonlinear stage of bubble and spike growth. Often the bubbles and spikes become stretched with time, particularly in a spherical converging geometry where Bell–Plesset effects can contribute.^{40,41} As the size of each delineated region is reduced, the potential for temperature equilibration increases. For a sufficiently long hydrodynamic timescale *τ*_{H}, a reduced temperature difference translates into less available entropy increase from mixing—once it arguably takes place.

For a given value of *ε*_{m}, the intermediate temperatures of the two species types can be derived using energy and number conservation: $T1=T1,0+\u03f5mT\u2032\u2212T1,0$ and

The corresponding available change in entropy from subsequent mixing then follows from Eq. (18):

Figure 13 shows that the available entropy of mixing as a function of *ε _{m}* drops off more steeply than linear. Depending on the physical circumstances,

*ε*may approach unity and appreciably curtail the available entropy change from mixing. If the instability growth occurs following deceleration onset,

_{m}*τ*

_{H}is on the order of 1 ns compared with an equilibration time on the order of 10 ns for a several micrometer scale length Δ

*x*. For instability growth before deceleration onset as in the case of an HDC ablator (see Sec. III), the hydrodynamic timescale of interest is several ns and

*ε*can be significantly greater than zero. The scale at which mix sets in largely determines the available entropy change from mixing. Prior to a mixing transition for Rayleigh–Taylor growth, there is entropy generated from thermal conduction $\u221d\kappa \u2207lnT2$. However, mainline simulations do capture this source of entropy generation. Thus, from the standpoint of entropy generation the process of mixing can be viewed as loosely mimicking a highly enhanced thermal conduction when the equilibration timescale is long compared to a hydrodynamic growth time.

_{m}Finally, we mention that although the fuel ablation-front “interface” is unlikely to be subject to full atomic mixing induced by hydrodynamic instability growth, for example, Rayleigh–Taylor, a recently reported shock-induced phenomenon may promote mixing in 1D, particularly for high Mach number shock traversal of this interface.^{19} In this case, the Reynolds number threshold for mixing [Eq. (29)] does not apply, and the potential for higher values of entropy generation than shown in Fig. 10 exists.

## V. EVALUATION OF Δ*S* FOR HOHLRAUM WALL-GAS MIXING

### A. Entropy of RT mixing at wall–gas interface

Similar arguments for assessing the entropy of mixing near the hohlraum wall and gas-fill interface can also be applied. For long drive pulses (∼15–20 ns) and moderately high gas-fill densities (>1 mg/cc), the necessary conditions for driving high Rayleigh–Taylor growth rates are in place: high Atwood numbers (>0.5), strong acceleration ($\u2272$ 5 *μ*m/ns^{2}), and long duration (∼10 ns).^{42} In addition to the potential for large hydrodynamic instability growth, diffusive mixing of high-*Z* wall material and the gas fill across this interface can also be important. A numerical study of gas-filled hohlraum instability growth initiated by the Kelvin–Helmholtz instability has been reported.^{43} The seed for this instability can be direct laser non-uniformities such as speckle and misaligned density and pressure gradients at the bubble-gas interface from successive shock transits. The resulting baroclinicity can generate vorticity and the Kelvin–Helmholtz instability. Later, the Kelvin–Helmholtz and Rayleigh–Taylor instabilities combine and enter a nonlinear stage where the amplitudes are on the order of the wavelength of the perturbation wavelength or greater. Presumably, the nonlinear evolution of the perturbed interface has sufficiently progressed to erase any memory of the numerical seeds of instability growth and establish a self-similar mix layer.

Since the plasma conditions in the vicinity of the wall–gas interface are neither isobaric nor isothermal in general, Eq. (4) with unequal temperatures and pressures must be applied. As mentioned in Sec. II, the Au/He interface is sufficiently dense and hot to justify ignoring the differences in specific heats between ^{4}He and Au in Eq. (4) that arise from the partially ionized high-*Z* hohlraum wall. From a Taylor expansion of the total pressure on either side of the interface, we have

where the subscript “2” refers to the hohlraum side of the interface and “1” to the gas side, and the ratio of gravitational scale lengths is used

Next, we use that $T2T1\u22451+Z11+Z2\xb7A2A1\xb71\u2212At1+At\xb7P2P1$ to obtain

Equations (31) and (33) define the pressure and temperature ratios for use in Eq. (4) once the Atwood number, charge states and bubble size normalized to the gravitational scale length, that is, $\Delta b\Delta g,2$, are specified. The number fraction of “light” ions and electrons in the mix layer *c* and the total number of particles in the mix layer *N* are obtained from Eq. (19), assuming a Rayleigh–Taylor origin to the presumptive mix layer. Here, the number of gas fill ions *N*_{1i} is estimated from $2\pi RHzH\Delta mix\rho 1/A1mp$, where *R _{H}* is the radius of the gas-wall interface at the time of the mixing transition, and

*z*is the hohlraum length. The majority of hohlraum experiments on the NIF to date have used cylindrical gold cans of initial radius of 0.2875 cm and length of 1 cm. The amount of wall (radial) motion depends on the initial density of the low-

_{H}*Z*gas fill and laser pulse length, but we take as a starting point ∼500

*μ*m of wall motion based on simulations to estimate the degree of entropy of mixing. The efficient ablation of the capsule acts to further compress the hohlraum

^{4}He gas fill, which we approximate by a near doubling of the density. Figure 14 shows the estimated degree of entropy generated by the mixing process for a range of pressure-gradient scale lengths, Atwood numbers and ionization state of the laser heated gold, assuming an average hohlraum (gas) temperature of ∼2 keV, initial gas-fill density of 1 mg/cc

^{4}He, and a total mix width of 500

*μ*m. For the conditions near the end of the laser power trough, the charge state of the gold is close to 20, the temperature is on the order of 1 keV, the acceleration of the interface is ∼10

*μ*m/ns

^{2}, and the bubble scale is less than 100

*μ*m. The resulting entropy generated by mix

*T*Δ

*S*is only on the order of 10 kJ at this early stage, as indicated by the lower red circle in Fig. 14 at $\Delta b\Delta g,2\u22450.1$. Shortly after the instant when peak power is reached, the motion of the “interface” is now in a deceleration phase ∼ −50

*μ*m/ns

^{2}, the temperature is now approaching 2 keV, the charge state of the gold is near 50 and the bubble size is ∼200

*μ*m. The amount of energy released by mixing as

*T*Δ

*S*is approaching 100 kJ or more, as shown by the upper red circle at $\Delta b\Delta g,2\u2245\u22120.2$. Note that at both times, the gravitational scale length $\u2009\Delta g,2$ is on the order of 1000

*μ*m, which considerably exceeds a temperature-gradient scale length ∼100

*μ*m as determined from radiation-hydrodynamics simulations. Physically, much of the entropy change is attributed to the large exchange of electrons across the interface, owing to the high degree of ionization of the gold blowoff.

### B. Entropy generation from diffusive mixing at wall–gas interface

A long-standing feature of long-pulse, gas-filled hohlraum experiments is the anomalous reduction of drive on the order of 100–200 kJ, based on delayed capsule implosion times.^{44} Dedicated experiments have shown that the delayed capsule response is primarily due to reduced drive conditions in the hohlraum.^{45} A previous analysis has suggested that mixing of Au and He may lead to a temperature inversion and reduced x-ray drive.^{42} This scenario required the establishing of a mix layer and did not account for an entropy increase from mixing. Figure 14 suggests that the entropy generated by mixing may have a role in the inferred drive deficit in gas-filled hohlraum experiments to date if the mix layer is initiated by Rayleigh–Taylor instability growth. Although the conditions for robust Rayleigh–Taylor growth, that is, high Atwood number and interface acceleration, are present early in the hohlraum drive,^{42} the subsequent heating and compression of the gas fill leads to a significantly higher plasma viscosity ($\nu \u221dT5/2/n$) and a lower Reynolds number—not to mention reduced instability growth. Thus, the conditions for reaching and exceeding a fluid mixing transition threshold from hydrodynamic instability growth may not be met in a hohlraum around the time of peak power. However, the multi-keV temperatures near the wall/gas fill interface may be responsible for high levels of mixing from binary mass diffusion. The diffusion scale length of a gold ion into the helium gas fill can be estimated^{46} from

where *τ* is a time interval, $DAu/He$ is the diffusion coefficient, and $\nu Au/Hes$ is the collision frequency in the slow test particle limit, that is, $mHe\upsilon Au22kBTHe\u226a1.$^{14} The ratio of diffusion scale lengths for gold and helium ions is similarly found

For *Z*_{Au} = 40 and *Z*_{He} = 2, the ratio of scale lengths is 0.38 for *At* = 0.9, and 0.21 for *At* = 0.7. The diffusion scale lengths for Au and He ions include an enhancement by the high electron mobility in Eq. (34), that is, “ambipolar” diffusion: $DAu/He\u2192DAu/He\xb71+ZAuTe/Ti$ and $DHe/Au\u2192DHe/Au\xb71+ZHeTe/Ti$ because of the use of an ion sound speed instead of an ion thermal speed in Eq. (34). According to radiation–hydrodynamic simulations, the channel electron temperature *T _{e}* is often nearly twice the ion temperature

*T*, but very close to the ion temperature near the channel–wall interface where mix can occur.

_{i}Evaluating Eqs. (34) and (35) gives $\Delta xAu\u224585\u2009\mu m$ and $\Delta xHe\u2245226\u2009\mu m$ for *τ* = 3 ns, *T* = 2 keV, *n _{e}* = 10

^{20}/cc, lnΛ

_{Au/He}= 5,

*Z*= 40, and

_{Au}*At*= 0.9. We next adapt Eq. (19) to mixing from ion diffusion (instead of Rayleigh–Taylor instability), using the prescription:$\u2009\Delta b\u2192\Delta xHe$ and $\Delta s\u2192\Delta xAu$, to obtain

Figure 15 shows the change in entropy from mixing due to ion diffusion based on Eqs. (7), (31), and (35)–(37). Although the change in entropy curves are steeper near small values of bubble widths, cf. Fig. 14, the overall change is similar to what was found for mixing induced by Rayleigh–Taylor instability.

In Sec. II, the entropy of mixing was shown to lead to reduced compression due to the extra work required to adiabatically compress two mixed vessels compared with the unmixed case [see Eq. (10)]. Moreover, the two vessels were thermally isolated and no association of *T*Δ*S* with Δ*Q* was made, or even allowed, according to the 2^{nd} law of thermodynamics. The case at hand with a hohlraum heated by a laser is neither adiabatic nor thermally isolated, so *T*Δ*S* generated from mixing may now lead to Δ*Q*. This possibility is important in that the inferred drive deficit may correlate with extra (or parasitic) generation of coronal (*T*_{e} > *T _{R}*) plasma heating from mixing, where

*T*

_{e}is the electron temperature and

*T*

_{R}the radiation temperature. However, such a case with the hohlraum going from one equilibrium state (just before the rise to peak laser power) to another equilibrium state (well past peak power) and being far from equilibrium during this heating process leads to the inequality Δ

*Q*<

*T*Δ

*S.*

^{22}Thus, the effect of entropy generation from mixing on the inferred hohlraum drive deficit and any additional coronal plasma heating may range from nil to having a significant impact. In this respect, the entropy of mixing calculated in Figs. 14 and 15 can only be strictly interpreted as an upper bound on the amount of coronal plasma heating from mixing at the potential expense of x-ray drive energy. In support of this entropy-of-mixing scenario in gas-filled hohlraums, recent

*T*

_{e}measurements in gas-filled hohlraums near the ablating capsule plasma show a systematically higher temperature by 500–600 eV compared with no-mix simulations.

^{47}More recently, evidence for restricted heat transport in the high-

*Z*coronal plasma regions of laser heated hohlraums could be consistent with higher electron temperatures

^{48}than expected from the “high flux model.”

^{49}However, the physical mechanism for how the inferred drive deficit can potentially translate into auxiliary electron heating and more coronal plasma is an open question.

## VI. SUMMARY AND CONCLUSIONS

Ignition experiments on the NIF to date have shown a trend of unexplained deficits of stagnation pressure in the fuel and unaccounted but inferred drive energy losses in gas-filled hohlraums. The theme of this paper is that material mixing, whether induced by hydrodynamic instability growth or binary species diffusion, may be a common denominator for both seemingly disparate phenomena. In the case of capsule implosions, the fuel–ablator interface appears to be a potential candidate for high instability growth and a subsequent transition to the fully mixed state. In the area of hohlraum dynamics, the high-*Z* wall and low-*Z* gas–fill interface are strongly Rayleigh–Taylor unstable—at least early in the drive history. For both phenomena, we develop and apply thermodynamic arguments on the generation of entropy from mixing when gradients in pressure and temperature are present. Application of the model to CH and undoped HDC ablators in contact with DT fuel reveals the potential for significant levels of entropy generation if mixing across the interface is present. Multi-mode simulations in 2D suggest appreciable levels of bubble and spike growth, which are necessary—but not sufficient—conditions for achieving a mixed state across an initially perturbed interface. Estimates of the amount of entropy generated for undoped HDC ablators suggest on the order of 100 J are deposited in the DT fuel layer, which largely coincides with the available ignition margin from other sources of preheat and entropy generation in a capsule, for example, hot electrons, shock mistiming. Such a large level of deposited energy in the fuel, besides potentially thwarting ignition, may be manifested as a decompressed fuel or lower areal density at the time of peak stagnation pressure. The presence of a mid- to high-*Z* dopant in a HDC ablator may help alleviate the levels of entropy from mixing by reducing the impact of x-ray preheat outside the interface and achieving lower Atwood numbers for reduced instability growth.^{33}

The methodology from generating entropy from mixing is applied to the gas–wall interface in hohlraum experiments. This interface has previously been reported to be fertile ground for hydrodynamic instability growth and binary mass diffusion in potentially leading to the establishment of a mix layer.^{42} Estimates of an upper bound for the amount of entropy generated at a mix layer of 500 *μ*m extent give a level consistent with inferred levels of “missing energy” on the order of 100 kJ from comparisons of mainline single-fluid radiation–hydrodynamic simulations with the data.

The treatment of entropy generation from mixing described here is analytical by intent in order to provide a conceptual basis for wide application to ICF problems. The natural next step is to define a suite of test problems to both check the approximations used herein and to understand the limits of mainline simulation tools in incorporating the physics of entropy generation from mixing. Standard average-atom-based radiation hydrodynamics simulations include an assortment of phenomenological mix models to choose from Ref. 32, but the extent to which thermodynamic consistency is maintained for an underlying multi-species effect is natural to address. Models that solve the multi-fluid equations often involve some approximations for numerical tractability and efficiency; however, the impact of these approximations on preserving key properties of the mixing process warrants an assessment. For example, the Navier–Stokes multi-component code MIRANDA uses an approximation scheme to limit species velocities to small deviations from a mean flow speed (or altogether ignoring kinetic energy differences among species).^{50} As mixing is an intrinsically multi-species effect, it is not clear to what degree—if any—such approximations have on thermodynamic consistency. A truer test would be to exercise particle-in-cell, for example, LSP,^{51} and molecular dynamics techniques to track the interpenetration of species and the resulting entropy generation from a microphysics perspective. The challenge becomes separating entropy generation from mixing and like-particle transport, for example, thermal conduction. A goal of a candidate test problem would seek to suppress the available venues for transport and to isolate the entropy of mixing contribution for comparison with analysis.

## ACKNOWLEDGMENTS

This work benefitted from numerous discussions with J. Milovich. The author acknowledges probing questions and valuable suggestions from the referees.

This work was performed under the auspices of Lawrence Livermore National Security, LLC (LLNS) under Contract No. DE-AC52–07NA27344. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring the United States government of Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

## DATA AVAILABILITY

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