Entropy generation from hydrodynamic mixing in inertial confinement fusion indirect-drive targets

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 −PdV 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 10⁴ in a statistically stationary flow¹²—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 cm²/s,¹⁴ while the scale length Δx for the fluid shear flow to vary by a terminal bubble velocity $Δυ≈g/3k$ is taken as a bubble vortex size ∼3/k,¹⁵ where g is the interface peak acceleration ∼1000 μm/ns² and k is the perturbation wavenumber. Upon taking a 20 μm 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 μm radius gives a resulting Reynolds number $Re≡Δx·Δυν≅1.6×106$—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 spot¹⁶ 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 (⁠$g≡dυ→/dt$ ≤ 0), provided the product of Atwood number [dimensionless density mismatch parameter $ρH−ρL/ρH+ρL]$ 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 μm based on multi-mode radiation-hydrodynamic simulations²⁰ 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-μm-thick mix layer may generate on the order of 100 J of entropy increase TΔS (in energy units). If this entropy generation comes at the expense of compressional energy from work done in mixing −PdV (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 TΔS 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 TΔS 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.

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 $T′$ follows from (internal) energy conservation $ΔE=0$ (neglecting kinetic energy from transient flows), assuming a number-weighted specific heat at constant volume for the mixed gases $cV′=N1cV,1+N2cV,2N1+N2$ and instantaneous atomic mixing:

while the mixed total pressure $P′P′$ is readily obtained

where c is the concentration fraction of species “1”:$N1/N1+N2$ For an ideal gas, the specific heat per particle is 3k_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, $cV′$ is in units of the Boltzmann constant k_B, $V=V1+V2$ is the total volume, and $ζ′=cζ1+(1−c)ζ2$ 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 $cP′=cV′+kB$ 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 $ΔS=−NkB ln [cc(1−c)1−c]≥0$ 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_Bln2; 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 $×$ 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_Bln2 per ion as an upper bound on this particular source of entropy generation.

The enthalpy of mixing ΔH is given by $ΔΦ+T′ΔS+S′ΔT,$ 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 ΔH = ΔQ, 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 $γ≡d ln P/d ln ρ|s$ 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 3k_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_B 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 ⁴He 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 $cP′=cV′+kB$ (per free particle) is used. Under these conditions, the change in entropy is significantly reduced, for example, by ∼66% at T₂/T₁ = 10 for c_V,1 = 3k_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 $κ∇lnT2$ and $κT∇lnT2$⁠, where κ is the electron or ion thermal conductivity and $κT$ is the electron or ion thermodiffusion coefficient; in the latter case, the rate $∝κP∇lnPi2$⁠, 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.

We earlier asserted in Sec. I that the generation of entropy from mixing is not a new source of internal energy $ΔE$⁠, but instead comes at the expense of compressional energy or −PdV work when the internal energy is conserved: $dE=0=TΔS−PΔV$⁠. To understand this further, we proceed to evaluate the amount of work $Wmix≡−PΔV$ 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 T₁ and T₂, respectively, and that the mixed state temperature $T′$ is $T1+T22$ for equal (and contant) numbers of particles N₁ = N₂ ≡ 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 $ϖ$ 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 T₁ goes to T₂. By contrast, the third (or last) term is second order in $T2T1−1$ and describes the work done in mixing: W_mix. In comparison, the amount of heat generated upon mixing $TmixΔS$ is easily shown from Eq. (5a) to vary as $NcPT2/T1−12Tmix/8$—also second order in $T2T1−1$⁠. In the limit of $T2T1→1$⁠, the mix temperature T_mix must also approach T₂ (and T₁) as well. We use this asymptotic condition on T_mix and internal energy conservation $0=TmixΔS+Wmix$ to constrain the value of $ϖ≅0.14.$ For arbitrary $T2/T1$⁠, T_mix can now be evaluated from $−Wmix/ΔS$⁠. Figure 3 shows the variation of T_mix over a wide range of $1≤T2/T1≤10$⁠. Overall, T_mix (solid line) is below but remains fairly close to the mean temperature $T′$ [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.²² Other realizations for the mixing history (and T_mix) are innumerably possible, given that the integral of $−PΔV$ (and $ΔS$⁠) depends on the choice of quasi-static path taken to reach the fully mixed state.²⁵ Still, the ansatz represented by Eqs. (6a) and (6b) is reasonably consistent with ideal agreement between T_mix and $T′ ≡$ (T₁ + T₂)/2 over a significant range of $T2/T1$⁠.

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 ΔQ 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 $PjVjγ=bj$ is a function of the entropy only, b₁/b₂ = (V₁/V₂)^γ follows from the isobaric condition P₁ = P₂, and C_V = V₂/V_2,f = V₁/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 $b′$/b₂ = (V₁ + V₂)^γ/V₂γ follows from the condition $P′$ = P₁ = P₂. A crucial distinction must be made in how the final state $I′$ 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 $T′=(T1+T2)/2.$ In the latter case, the change in entropy ΔS 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 $T′$ for both gas vessels exists, we increase $V1$ to $V1T′/T1$ and decrease $V2$ to $V2T′/T2$ to maintain constant pressure P. The corresponding work done by the first vessel is $−PΔV1=P(V1−V2)/2,$ while for the second vessel, $−PΔV2=P(V2−V1)/2$⁠; thus, the total work done $ΔW$ is zero. The change in internal energy for the first vessel $ΔE1$ as the temperature is raised to $T′$ is $NcV,1T′−T1=NcV,1T2−T1/2$⁠; similarly $ΔE2=NcV,2T′−T2=−ΔE1$ for c_V,1 = c_V,2, and the total change in internal energy $ΔE$ is also zero. From $ΔE=TΔS+ΔW$⁠, we find $ΔS=0$ 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, $T′$⁠, V_f for the two samples are maintained across the two different processes. (Applying the adiabatic law $PVfγ=const.$ 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 $I12eΔS/cV′−1$⁠. 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 ΔW compared with its unmixed constituent gases is tantamount to adiabatically introducing an equivalent amount of internal energy ΔE to the mixed state, for example, preheat.

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 evolution²⁶ in time τ:

where $αb≅0.06$ is based on experiments and direct numerical simulation studies with sufficiently long wavelength,^27,28$At≡ρ2−ρ1/ρ2+ρ1$ is the Atwood number at the CH/DT interface, ρ₁ (ρ₂) 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 $Δmix=Δb+Δs.$ 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 $∝∇T$ across the interface, giving

where Z₁ is the mean ionization state of an averaged D and T ion—assumed to be unity, Z₂ is the ionization state of an average CH ion, and A₁(A₂) is the mean atomic weight of an average DT (CH) ion. For the case of $≅15$ eV CH plasma, we have $Z2≅2$ 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 $4τχ$⁠, where $χ=κ/ρ$ 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 $LT,j2/2χj$ 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 μm. 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: $P¯1≅P0−∇P1·Δs/2$ and for the average pressure in the surrounding CH material over a bubble distance Δ_b: $P¯2≅P0+∇P2·Δb/2$⁠, 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 $∇P=−ρg$ and using that both sides of the interface must accelerate at g (by definition), we obtain $∇P1/∇P2=1−At/1+At$⁠. Thus, $P¯2≅P¯1+∇P2·Δmix+AtΔb−Δs/21+At$⁠, which simplifies to $P¯2≅P¯1+∇P2·Δmix/2$ with use of Eqs. (11a) and (11b) in the limit of small Atwood number. In addition to the mix scale length $Δmix=Δb+Δs$⁠, another natural scale length for the quasi-isothermal conditions at hand is the usual gravitational (or acceleration) scale length $Δg,j=C s,j2/g,$ 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: $Δg,2Δg,1=1−At1+At,$ 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 $Δb/Δs$⁠. 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 $7×1016$ cm/s². As stated earlier, the temperature of the material near the interface early in the deceleration phase is ∼15 eV, giving $Δg,2≅2$μm. Similarly, $Δmix≅6$μm under the same deceleration conditions after using $Z2≅2$ 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 $ΔbΔg,2<2$⁠. 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 μm. Figure 5 displays the amount of entropy (in units of energy) generated by mix (at the expense of compressional energy −PdV) 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 $nmix,i=N1i+N2i/V1+V2≅ n2iV1+n1iV2V1+V2=n2iΔs+n1iΔb/Δmix$⁠. The number density of mixed ions and electrons follows as $nmix≅n2iΔs1+Z2+n1iΔb1+Z1/Δmix$⁠, which simplifies with use of Eq. (12) to $nmix≅ρ11+Z1A1.$ 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 $Δb/Δg,2$ 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 $Δb/Cs,j≼Δg,j/Cs,j$⁠.

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 $g≡0$ 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 jth side of the interface. The amount of entropy generated from mixing follows from Eq. (4) when: $Δg,2≫Δj:$

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

We require that the thermal energy flux $−κ∇T$ 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 $ΔT,2/ΔT,1≅0.85$ for Z_CH = 1.5 and $ΔT,2/ΔT,1≅0.75$ 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 $ΔiΔg,2≪1$ 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 $Δb/2ΔT,2$ 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 (⁠$≲1 μm), T1≅T2$ 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.

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 $T2T1=A2A1·1−At1+At·1+Z11+Z2·P2P1$ for two contiguous ideal gases, approximating

and recalling the ratio of gravitational scale lengths

we obtain the solution for T₂/T₁

Figure 7 shows the (reciprocal) solution of Eq. (22) for T₁/T₂ and the associated change in entropy from mixing for several choices of Atwood number before deceleration onset (g ≤ 0). A real solution exists when $Δ2Δg,2≤1+1+At0.661−At0.66−1$⁠, leading to a restricted range of admissible values for $Δ2Δg,2$⁠. 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.

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 μm 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 HYDRA³² show the effective combined range of bubble and spike penetration approaches the range of 20–30 μm 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_p is the pusher peak velocity. The density of the DT ice layer $ρshell≈rdrstag2≡C2,$ where C is the fuel radius convergence relative to the time of deceleration onset. From Eq. (23), we readily obtain $≈1/Pd1/2$⁠, so that $Δρshellρshell=−ΔPd/Pd$ at constant r_d. At a simulated average temperature of ∼20 eV throughout the DT ice layer near deceleration onset, the internal energy $ε=3PV/2$ is ∼400 J, including the electron internal energy. From Fig. 8 with 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 μm) 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 μm/ns, giving an average acceleration of nearly 63 μm/ns². From Eqs. (11a) and (11b), the mix width is about 16 μm 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 TΔS 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 $Δ2/ΔT,2≅2$ in the local approximation, cf. Eq. (17) and its near vanishing denominator, the profiles used in evaluating ΔS 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 doped³³ 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, $ρ∼1R2,$ 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 $ρ ∼ 1/R2$ 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 (∼τ²). 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.

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 $ρgas$ in the gaseous (hot spot) fuel of radius $Rgas$ follows from the compressed central DT fuel mass $Mgas=4π3ρgasRgas3=4πρgasRgas33ρgas2 ∼ σ3$ and the minimum areal density constraint $ρgasRgas$ of ∼0.3 g/cm² for spark or hot-spot ignition,³⁰ giving $ρgas ∼ σ−3/2$ and $Rgas ∼ σ3/2.$ 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: $Mshell≅4πRshell2(ρshellΔshell) ∼ σ3.$ Using that the minimum value of $ρshellΔshell≅$ 2 g/cm² should be met across all driver scalings of interest,³⁰ we have that $Rshell ∼ σ3/2$ and $ρshell ∼ 1/Δshell$ for the peak shell density. The fuel convergence C_f, defined as the initial ablator radius $∼σ$ over the final fuel radius, thus scales favorably as $σ−1/2$ instead of $σ0.$ This relation implies that $ρshell ∼ ρ0Cf2 ∼ σ−1.$ Next, the requirement of a robust ignition boundary forces the peak hot-spot temperature T_gas to remain sufficiently high: $ρgasRgasTgasρshell/ρgas>6$ keV, over the T_gas range 5–15 keV.³⁵ From the above scalings for $ρshell$ and $ρgas$⁠, we find that $Tgas ∼ σ−1/4$⁠, which is a fairly weak function of driver energy $Tgas ∼ Edriver−1/12.$ The peak implosion speed $υimp$ follows from applying energy conservation between the time of peak shell kinetic energy and final fuel stagnation: $υimp∼Ni,fTgas/ρshellVgas ∼ σ−3/8$⁠, where N_i,f is the number of hot-spot fuel ions and $Vgas ∼ Rgas3$ is the hot spot volume. The acceleration g of the solid fuel/ablator interface (or $Rfuel)$ then follows from $g ∼ υimpτimp ∼ υimp2ΔR ∼ υimp2Rshell(Cf−1) ∼ σ−94,$ where $τimp$ is the implosion time and ΔR is the deceleration distance of the Rayleigh–Taylor unstable fuel–pusher interface. This scaling is in contrast to $g ∼ 1/σ$ 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 $gτimp2 ∼ σ1/2$⁠. We now can write for the scaling of number of mixed ions N_mix on a presumptive Rayleigh–Taylor unstable interface between the fuel and ablator: $Nmix∼ ΔmixρshellRshell2 ∼ σ2$⁠, compared with $σ3$ from a usual Euler scaling. Thus, $ΔS ∼ Nmix$ 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 $×$ larger scale or 8 $×$ in driver energy.

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 $Δx∼Dτ≅Csλiiτ$⁠, where the ion–ion mean free path $λii$ scales as $T2/ρshell$⁠. 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: $Δx ∼ σ3/2.$ However, if the required peak shell density and shell convergence remain unchanged, then a favorable (Euler) scaling with size is found: $Δx ∼ σ1/2.$ 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 (⁠$ρshellΔshell≅$ 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 $α$⁠. The margin is related to the ignition threshold factor, ITF = M + 1 $∝Ecap/α1.9.$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 Frustraum³⁸ are intended to allow nearly $3×$ 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.

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 $P1≅P2$ 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 $Lρ$³⁹ 

Because of the isobaric assumption and assumed ideal gas equation of state for DT, the temperature-gradient scale length at the ablation front $LT≡1/∇lnT$ achieves its maximum (absolute) value and is equal and opposite to $Lρ$⁠. Near the time of deceleration onset $ρgas$ is on the order of 0.1 g/cm³, and $ρshell$ is nearly 2 g/cm³ for a shell convergence of ∼5, giving a temperature-gradient scale length on the order of 1 μm. At the time of fuel stagnation (or shell fuel convergence of ∼35), $ρgas$ is on the order of 100 g/cm³ and $ρshell$ is nearly 1000 g/cm³, after noting that the ablated DT shell mass provides a nearly $10×$ increase in igniting hot spot mass. The corresponding temperature-gradient scale length is still on the order of 1 μm.

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