Impact of different electron thermal conductivity models on the performance of cryogenic implosions [29]

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Impact of different electron thermal conductivity models on the performance of cryogenic implosions Cite as: Phys. Plasmas 29, 042702 (2022); https://doi.org/10.1063/5.0066708 Submitted: 12 August 2021 • Accepted: 15 March 2022 • Published Online: 01 April 2022 Chuanying Li,

Jianfa Gu, Fengjun Ge, et al.

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Impact of different electron thermal conductivity models on the performance of cryogenic implosions Cite as: Phys. Plasmas 29, 042702 (2022); doi: 10.1063/5.0066708 Submitted: 12 August 2021 . Accepted: 15 March 2022 . Published Online: 1 April 2022 Chuanying Li,a)

Jianfa Gu,

Fengjun Ge, Zhensheng Dai,a) and Shiyang Zou

AFFILIATIONS Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, People’s Republic of China a)

Authors to whom correspondence should be addressed: [email protected] and [email protected]

ABSTRACT The electron thermal conduction strongly affects the hot-spot formation and the hydrodynamic instability growth in inertial confinement fusion implosions. A harmonic-mean flux-limited conductivity model has been widely used in implosion simulations. In this paper, using the high foot implosion N140520 as an example, we have performed a series of one-dimensional (1D) no-alpha simulations to quantify the impact of different conductivity models including the Spitzer–Harm model, the Lee–More model, and the recently proposed coupled Gericke-MurilloSchlanges model [Ma et al., Phys. Rev. Lett. 122, 015001 (2019)] with the flux limiter fe ranging from 0.03 to 0.15 on the performance of cryogenic implosions. It is shown that varying fe has a bigger impact on the performance than changing conductivity models. Therefore, we have only performed two-dimensional (2D) no-alpha simulations using the Lee–More model with different flux limiters feLM to quantify the effect of the electron thermal conduction on the performance, with single-mode velocity perturbations with different mode numbers L seeded on the inner shell surface near the peak implosion velocity. We find that in both the 1D implosions and the 2D implosions with the same L, increasing fe leads to more hot-spot mass and lower hot-spot-averaged ion temperature, resulting in approximately constant hot-spot internal energy. In addition, the no-alpha yield Yna is dominated by the neutron-averaged ion temperature Tn in these two cases. Increasing feLM from 0.0368 to 0.184 reduces Tn by 15% in 1D and by 20% for the 2D implosions with the same L, both leading to a 20% reduction in Yna . Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0066708

I. INTRODUCTION In inertial confinement fusion (ICF) implosions, a spherical shell of cryogenic deuterium–tritium (DT) ice filled with DT gas is imploded at a high velocity to compress the central fuel to high ion temperature and areal density (qR) for ignition.1,2 The electron thermal conduction plays an essential role in the hot-spot formation and affects the hydrodynamic instability growth in ICF implosions. For the National Ignition Facility (NIF) capsules, most of the hot-spot mass comes from ablation of the cold shell material during compression. Meanwhile, the interface between the hot spot and the cold shell (i.e., the inner shell surface) is unstable to the Rayleigh–Taylor instability (RTI) during the deceleration phase (dp).3,4 The dp-RTI spikes penetrate into the hot spot, which can cool the hot spot, reduce the hotspot volume, and cause degradation in the compression efficiency.5,6 The growth rate of the dp-RTI is approximately given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kg c¼a (1)  bkVa ; 1 þ kLq

Phys. Plasmas 29, 042702 (2022); doi: 10.1063/5.0066708 Published under an exclusive license by AIP Publishing

where k, g, Lq , and Va are the perturbation wave number, average acceleration, minimum density-gradient scale length, and average ablation velocity, respectively. Typically, the values of a and b satisfy a ¼ 0:9–0:95 and b ¼ 1:2–1:5 in ignition implosions.7 The dp-RTI can be reduced by increasing the electron thermal conduction, which provides ablative stabilization as the inner shell surface moves into the cold fuel during compression and also establishes a density gradient on the inner shell surface. Different theoretical investigations on the electron thermal conductivity9–16 have been conducted since the early work of Spitzer.8 The Spitzer–Harm (SH) conductivity model was intended for fully ionized nondegenerate plasmas, which is suitable for dilute, weakly coupled plasmas. Based on the first-order approximation to the Boltzmann equation, Lee and More (LM) presented an electron conductivity model9 for dense plasmas by including the electron degeneracy effects. This model gives enhanced electron thermal conductivities at temperatures below the Fermi temperature compared to the SH model. Recently, first-principle (FP) simulations using quantum molecular dynamics (QMD)11,15 have shown that the electron thermal

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conductivity calculated from the traditional analytic models deviates from the FP-based conductivity significantly in the warm dense matter (WDM) regime. In 2019, Ma et al. proposed the coupled GerickeMurillo-Schlanges model (referred to as the CGMS model), in which the Coulomb logarithms are in good agreement with their nonequilibrium molecular dynamics results using the constrained electron force field (CEFF) method. A flux-limited analytic model for the electron thermal conductivity has been widely used in our implosion simulations. In this paper, using the high foot implosion N140520 (Refs. 17–22) as an example, we have quantitatively assessed the impact of different analytic models including the SH model, the LM model, and the CGMS model with the flux limiter fe ranging from 0.03 to 0.15 on the performance of cryogenic implosions. Since the motivation of this paper is to investigate the effect of the electron thermal conduction on the performance, which is strongly dependent on the temperature, and the alphaparticle deposition affects the temperature significantly, we do not consider heating by alpha-particle deposition throughout this paper, unless specified otherwise. First, we have performed a series of onedimensional (1D) simulations for N140520 using the SH, the LM, and the CGMS models with fe ranging from 0.03 to 0.15. It is shown that the difference in fe has a bigger impact on the performance than the change in the analytic model. Therefore, we have only performed twodimensional (2D) simulations using the LM model with different fe to quantify the effect of the electron thermal conduction on the performance, with single-mode velocity perturbations with different mode numbers L ¼ 8, 16, 24 seeded on the inner shell surface near peak implosion velocity. This paper is organized as follows. In Sec. II, we give a brief description of the three flux-limited conductivity models. In Sec. III, we introduce our numerical simulation codes and methods. In Sec. IV, we analyze the impact of different conductivity models on the implosion performance of N140520 in 1D and 2D. In Sec. V, we summarize our findings. II. FLUX-LIMITED ELECTRON THERMAL CONDUCTIVITY MODELS In our implosion simulations, the electron heat flux Fe is given by Fe ¼

F^e ; jF^e j 1þ fe Q f

(2)

jSH

(3)

where k is the Boltzmann constant, me is the electron mass, e is the electron charge, Z is the average ionization state, and ðln KÞSH   bmax ¼ ln bmin is the Coulomb logarithm in the SH model. The maximum impact parameter bmax is set to be the electron Debye length

Phys. Plasmas 29, 042702 (2022); doi: 10.1063/5.0066708 Published under an exclusive license by AIP Publishing

qffiffiffiffiffiffiffiffiffiffi kTe 4pne e2 , with ne the electron number density. The minimum

impact parameter bmin is set by the larger of the classical distance of 2 the closest approach mZee v2 and the quantum uncertainty-principle limqffiffiffiffiffiffiffi e ited length 2mhe v, where v ¼ 3kT me is the electron thermal velocity. The limit of the free streaming heat flux in the SH model QSH f is given by rffiffiffiffiffiffiffiffi 3 2 2n QSH : (4) ¼ ðkT Þ e e f pme The electron thermal conductivity in the LM model jLM is given by jLM ¼

ne kðkTe Þs b T A d ; me

(5)

where the coefficient Ab is given by Ab ¼ ð13:512 þ 1:33f þ0:5692f2 Þ=ð1 þ 0:5097f þ 0:1715f2 Þ; f ¼ ð0:7531x3 þ 0:1679x4 þ0:3108x5 Þ=ð1 þ 0:2676x þ 0:228x2 þ 0:3099x3 Þ, x is defined as qffiffiffiffiffi 2 EF h2 x ¼ kT ð3p2 ne Þ3 is the Fermi energy, the non-Lorentz ; EF ¼ 2m e e correction factor dT used to include the electron–electron collision contribution is a function of ne, Te, and Z. The electron–ion relaxation time s is expressed as 3 pffiffiffiffiffiffi 3 me ðkTe Þ2 expðfÞ F1 ðfÞ; s ¼ pffiffiffi 2 2pne Ze4 ðln KÞLM expðfÞ  1 2

where ðlnKÞLM

(6)

F12 ðfÞ ¼ ð0:8844f þ 0:1768f2 þ 0:0008f3 Þ=ð1 þ 0:0296fÞ;   b2 ¼ 12 ln 1 þ bmax is the Coulomb logarithm in the LM model. 2 min

The maximum impact parameter bmax is set by the larger of the sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi

Debye–H€ uckel screening length kDH ¼

k 4pe2

ne pffiffiffiffiffiffiffiffiffiffi þ nTi Z 2 2 Te þTF

2

1

i

1

3 3 Þ , where ni and Ti are the ion and the ion-sphere radius R0 ¼ ð4pn i F number density and temperature and TF ¼ 2E 3k is the Fermi temperature. The minimum impact parameter bmin is also set by the larger 2 of mZee v2 and 2mhe v, yet the electron thermal velocity is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3k Te2 þ259 TF2 v¼ . The limit of the free streaming heat flux in the LM me

is given by model QLM f

where F^e ¼ jrTe , j is the electron thermal conductivity, Te is the electron temperature, fe is the flux limiter, and Qf is the limit of the free streaming heat flux. Different analytic models differ in j and Qf. The electron thermal conductivity in the SH model jSH is given by  32 7 5 2 k2 Te2 ¼ 9:44 ; p m12e e4 ðZ þ 4Þðln KÞSH

kD ¼

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QLM f

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 u  u pffiffiffi uk3 Te2 þ 9 TF2 3 3 t 25 : ne ¼ 8 me 3

At the high-temperature limit, ðTe2 þ 259 TF2 Þ2 ! Te3 ; feLM

(7) QSH f QLM f

 1:2284;

feSH

to is fixed at 1.2284 to ensure therefore, the ratio of LM SH feLM QLM ¼ feSH QSH f f throughout this paper, with fe ; fe representing the flux limiters for the LM model and the SH model, respectively. In this work, we have implemented the CGMS conductivity model which includes dynamical electron–ion coupling in our radiation-hydrodynamics codes. The electron thermal conductivity in the CGMS model jCGMS has a format similar to jSH, which is given by

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FIG. 1. (a) The Coulomb logarithm vs the electron number density at Ti ¼ 1 eV and Te ¼ 10 eV for H obtained from different theoretical models. (b) The Coulomb logarithm vs the electron number density at Ti ¼ 10 keV and Te ¼ 5 keV for DT obtained from different theoretical models.

 32 7 5 2 k2 Te2 jCGMS ¼ 9:44 : p m12e e4 ðZ þ 4Þðln KÞCGMS

(8)

The Coulomb logarithm in the CGMS model ðln KÞCGMS is defined as12,13 2 3 1 k2D þ R20 CGMS ðln KÞ ¼ ln61 þ  (9) 2 7; 2 4 5 h þ b2c 2me v where kD and R0 are the electron Debye length and the ion-sphere 2 radius, respectively, bc ¼ 1ðkTZe , and the electron thermal velocity is e þkTi Þ 2 qffiffiffiffiffi kTe given by v ¼ me . The limit of the free streaming heat flux in the has the same format as QSH CGMS model QCGMS f f ; thus, in the following analysis, the flux limiter for the CGMS model feCGMS equals feSH . Figure 1 compares the Coulomb logarithm ln K vs the electron number density ne calculated from different theoretical models, in

which the black, green, and red solid lines correspond to the SH, the LM, and the CGMS models, respectively. Figures 1(a) and 1(b) show ln K at Ti ¼ 1 eV and Te ¼ 10 eV for H, and ln K at Ti ¼ 10 keV and Te ¼ 5 keV for DT, respectively. It is found that ln K decreases with increasing ne. At low temperatures below 100 eV typical in the imploding shell, results from different models differ significantly, and ðln KÞCGMS is smaller than ðln KÞSH and ðln KÞLM by orders of magnitude at most. The overestimation of ðln KÞSH and ðln KÞLM leads to the underestimation of jSH and jLM in the WDM regime.10,15,16 At high temperatures of keV typical in the hot spot, ðln KÞLM and ðln KÞCGMS approach ðln KÞSH . Figures 2(a) and 2(b) show the ratio of jLM to jSH and the ratio CGMS of j to jSH in ( log10 q; log10 Te ) space, respectively, in which the density q and Te are in units of g/cm3 and MK. The density/temperature paths of representative meshes in the DT gas and ice regions from a 1D simulation of N140520 are shown in red and blue, with the acceleration and deceleration phases marked by dashed and solid lines, respectively. It is found in the acceleration phase, the DT shell

SH FIG. 2. (a) The ratio of jLM to jSH in ( log10 q; log10 Te ) space. (b) The ratio of jCGMS to jSH in ( log10 q; log10 Te ) space. (c) The ratio of 1.2284 QLM f to Qf vs the electron temperature Te at q ¼ 104 g/cm3 for DT.

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conditions pass through the WDM regime, in which jLM and jCGMS are higher than jSH by two orders of magnitude at most. Overall, the difference between jCGMS and jSH is larger than that between jLM and jSH. In the WDM regime, the DT plasma is coupled and degenerated 2 2 e (C > 1 and h < 1; C ¼ RZ0 kT ; h ¼ TTFe ); thus, the SH model, which e was originally developed for ideal plasmas, fails to give accurate thermal conductivity. In the deceleration phase, the difference between jCGMS (or jLM) and jSH is reduced. Under most of the DT shell conditions, jCGMS (or jLM) is larger than jSH by 50% at most. Overall, the difference between jCGMS and jSH is smaller than that between jLM and jSH. The ratio of 1.2284 QLM to QSH f f vs the electron temperature Te at q ¼ 104 g/cm3 for DT is shown in Fig. 2(c), which approaches 1 when Te ⲏ 0:1 eV. III. SIMULATIONS OF THE IMPLOSION DECELERATION PHASE 5

Since the electron thermal conductivity exhibits a Te2 power law in analytic models, and Te is relatively low in the acceleration phase, the impact of the electron thermal conduction on the implosion performance in the acceleration phase is much smaller than that in the deceleration phase. Several implosion characteristics for N140520 from 1D multigroup transport simulations using the code RDMG23 with different conductivity models are compared with available measured and simulated results in Refs. 21, 24, and 25 in Table I. The flux limiters were set as feSH ¼ feCGMS ¼ 0:05; feLM ¼ 0:0614 in these 1D simulations. The x-ray drive profile for N140520 used in RDMG has been adjusted to match experimental shock-timing and bang time. It is shown that the change in the analytic model has little impact on the implosion performance in the acceleration phase. Therefore, in the following analysis, the 1D hydrodynamic profiles near peak implosion velocity obtained from RDMG with the LM model and feLM ¼ 0:0614 were used as initial conditions for the deceleration-phase 1D and 2D simulations. The deceleration phase was simulated using the code LARED-S,26,27 which has been widely used in the studies of hydrodynamic instabilities5,6 and low-mode asymmetries.28,29 It is a multidimensional massively parallel Eulerian-mesh-based code including multigroup diffusion radiation transport, flux-limited Spitzer–Harm thermal conduction for electrons and ions, and multigroup diffusion for alpha particles. In this work, we have implemented the LM and the CGMS electron conductivity models in LARED-S. The opacities used in the radiation transport calculation (with 32 radiation groups) were derived from the code OPINCH30 based on the relativistic Hartree–Fock–Slater (HFS) self-consistent average atom model. Tabular equations of state (EOSs) for DT, CH, and Si-doped CH were obtained by patching several physical models to produce the required database.31

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The 2D simulations used a r  h geometry mesh including 2400 meshes in the r direction and 40 meshes in the h direction. To keep high spatial resolution and reduce the computational cost, the radial meshes near the ablation front, the ablator–fuel interface, and the inner shell surface were locally refined with a minimum mesh size of 0.05 lm, and the mesh convergence was verified through further refinement. A single-mode velocity perturbation with a given mode number L (L ¼ 8, 16, 24) and an initial amplitude n was introduced on the inner shell surface near the peak implosion velocity,32–34 which is given by V2D ðr; hÞ ¼ V1D ðrÞ½1 þ n cosðLhÞ expðkjr  r0 jÞ;

(10)

where r0 is the position of the inner shell surface and k is the perturbation wave number. The inner shell surface, corresponding to the hotspot boundary in the deceleration phase, is defined as the boundary of qmax ðt;h Þ

the imploded core region satisfying qðt; h0 Þ < DT 2 0 , with qmax DT ðt; h0 Þ the maximum DT density in the h ¼ h0 direction. IV. IMPACT OF DIFFERENT ELECTRON THERMAL CONDUCTIVITY MODELS ON THE IMPLOSION PERFORMANCES A. The effects of different electron thermal conductivity models on the implosion performances in 1D We have performed a series of 1D no-alpha simulations for N140520 using the SH, the LM, and the CGMS models with the flux limiters feSH ¼ feCGMS ¼ 0:03; 0:05; 0:10; 0:15, and feLM ¼ 0:0368; 0:0614; 0:123; 0:184. Figure 3 compares the hot-spot mass Mhs, the hot-spot-averaged ion temperature Tihs , and the hot-spot internal energy Ehs at bang time tb. Tihs was calculated by taking the mass average of the Ti profile over the entire hot spot. It is found that increasing fe can enhance the electron heat flux, leading to more Mhs and lower Tihs . Since the heat flux leaving the hot spot is converted into the internal energy and kinetic energy of the material ablated from the inner shell surface, Ehs is nearly independent of the conductivity model and fe for symmetric implosions. As described in Ref. 4, the yield without alpha heating Yna scales with the neutron-averaged quantities and the burnwidth s as ! 2 hrvi Yna / Pn (11) Vn s; Tn2 where the neutron-averaged pressure Pn, ion temperature Tn, and volume Vn are defined as

TABLE I. Comparison of implosion characteristics for N140520 from 1D simulations using different conductivity models with available measured and simulated results. The bang time tb is shown in a format of tbna =tba , with the superscripts “na” and “a” representing the results from 1D simulations without and with alpha heating, respectively. The other implosion characteristics were all obtained from simulations without alpha heating.

Peak implosion vel. (km/s) Adiabat Bang time (ns) Neutron yield (1015 )

SH model

LM model

CGMS model

NIF simulation

392 2.2 15.86/15.92 17.2

391 2.2 15.86/15.93 16.5

392 2.3 15.86/15.92 16.6

391 2.3

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Experiment

15.96 6 0.03 8.98 6 0.17

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FIG. 3. Comparison of (a) the hot-spot mass, (b) the hot-spot-averaged ion temperature, and (c) the hot-spot internal energy at bang time tb for N140520 obtained from 1D noalpha simulations using the LM, the SH, and the CGMS models with the flux limiters feSH ¼ feCGMS ¼ 0:03; 0:05; 0:10; 0:15 and feLM ¼ 0:0368; 0:0614; 0:123; 0:184.

ð

ð dt dVnD nT hrviP Pn ¼ ð ð ; dt dVnD nT hrvi ð dt dVnD nT hrviTi ; Tn ¼ ð ð dt dVnD nT hrvi

ð

ð dt (12)

Vn ¼ ð

2 dVnD nT hrvi

ð ; dt dV ½nD nT hrvi2

(14)

ð

(13)

with nD, nT the deuterium and tritium particle number densities and hrvi the fusion reactivity.35 hrvi is a function of Tn, which can be approximated in the temperature range 3  Tn  6 keV by a power

FIG. 4. Comparison of (a) the neutron-averaged pressure, (b) the neutron-averaged volume, (c) the neutron-averaged ion temperature, (d) the burnwidth, and (e) the yield for N140520 obtained from 1D no-alpha simulations using the LM, the SH, and the CGMS models with the flux limiters feSH ¼ feCGMS ¼ 0:03; 0:05; 0:10; 0:15 and feLM ¼ 0:0368; 0:0614; 0:123; 0:184.

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FIG. 5. Contour plots at bang time tb for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0614 for perturbations with different mode numbers L on the inner shell surface and the initial amplitude n ¼ 0:05. (a) The ion temperature Ti (left half) and the density q (right half) distributions for L ¼ 8, (b) L ¼ 16, (c) L ¼ 24, ej (d) the electron temperature Te (left half) and the jjrT (right half) distributions for L ¼ 8, (e) L ¼ 16, and (f) L ¼ 24. The contours of Ti ¼ 1; 2; 3 keV are shown in magenta in f LM Q e

f

ej distribution in (d)–(f). each Ti distribution in (a)–(c). The contours of Te ¼ 1; 2; 3 keV are shown in white in each jjrT f LM Q e

law of Tn as hrvi / Tn3:57 . Combining this power law with Eq. (11) leads to the following scaling of the yield: Yna / Pn2 Tn1:57 Vn s:

(15)

The burnwidth can be derived from the full width at half maximum (FWHM) of the neutron-rate history.36 Figure 4 compares the neutron-averaged pressure, volume, ion temperature, burnwidth, and yield for N140520 obtained from 1D no-alpha simulations using the LM, the SH, and the CGMS models with the flux limiters feSH ¼ feCGMS ¼ 0:03; 0:05; 0:10; 0:15 and feLM ¼ 0:0368; 0:0614; 0:123; 0:184. We find that Pn, Vn, and s are nearly independent of the conductivity model and fe, and the noalpha yield Yna is dominated by Tn. Our simulations show that increasing feSH from 0.03 to 0.15 can reduce Tn by 15%, leading to a 20% reduction in Yna , which is in good agreement with the scaling of Yna in Eq. (15). In addition, Figs. 3 and 4 show that the difference in fe has a bigger impact on the performance than the change in the analytic model. Therefore, we have only performed 2D simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 to explore the effect of the electron thermal conduction on the performance.

Phys. Plasmas 29, 042702 (2022); doi: 10.1063/5.0066708 Published under an exclusive license by AIP Publishing

f

B. The effects of different electron thermal conductivity models on the implosion performances in 2D Figure 5 shows the contour plots at bang time tb for N140520 obtained from 2D no-alpha simulations using the LM model with

FIG. 6. The growth factor on the inner shell surface vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05.

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FIG. 7. (a) The hot-spot mass, (b) the hot-spot-averaged ion temperature, and (c) the hot-spot internal energy at bang time tb vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05.

feLM ¼ 0:0614 for perturbations with different mode numbers L on the inner shell surface and the initial amplitude n ¼ 0:05. Figures 5(a)–5(c) show the Ti (left half) and q (right half) distributions for L ¼ 8, L ¼ 16, and L ¼ 24, with the contours of Ti ¼ 1; 2; 3 keV shown in magenta in each Ti distribution. Figures 5(d)–5(f) show the Te (left ej half) and the jjrT feLM Qf (right half) distributions for L ¼ 8, L ¼ 16, and L ¼ 24, with the contours of Te ¼ 1; 2; 3 keV shown in white in each jjrTe j feLM Qf distribution. Since the steep temperature gradient near the tips of the spikes induces enhanced electron heat flux there, most of the mass ablation occurs near the spike tips, resulting in the moving of the ej ablated materials from the spikes into the bubbles. Also, jjrT feLM Qf peaks in the hot spot and near the tips of the spikes, due to both the steep rTe and the high Te there. As described in Ref. 32, the surface-to-volume

ratio of the bubbles is increased with increasing L, leading to the enhanced conduction cooling on the bubble surface for the higher mode. Therefore, the bubble temperature declines with increasing L.4 In addition, the spike density declines and the bubble density increases with increasing L, which also indicates the enhanced mass ablation due to the enhanced thermal conduction for the higher mode. Figure 6 shows the growth factor (GF) on the inner shell surface vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05, with the GF defined as the ratio of the perturbation amplitude at tb to that at the peak implosion velocity. It is found that for the same fe, the GF for the L ¼ 24 perturbation is smaller than those for the L ¼ 8 and L ¼ 16 perturbations, indicating the enhanced ablative stabilization for

FIG. 8. Comparison of (a) the neutron-averaged pressure, (b) the neutron-averaged volume, (c) the neutron-averaged ion temperature, (d) the burnwidth, and (e) the yield vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05.

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the higher mode. Meanwhile, for the same L, the GF is reduced with increasing fe, due to the increased ablation velocity resulting from the enhanced heat flux. Figures 7(a)–7(c) compare Mhs, Tihs , and Ehs at tb vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05. It is similar to the 1D case that Mhs increases and Tihs declines with increasing fe due to the enhanced electron heat flux. Consequently, Ehs stays nearly constant with increasing fe for the asymmetric implosions with the same L. Figure 8 compares the neutron-averaged pressure, volume, ion temperature, burnwidth, and yield vs the perturbation mode number L for N140520 obtained from 2D no-alpha simulations using the LM model with feLM ¼ 0:0368; 0:0614; 0:184 with the initial perturbation amplitude n ¼ 0:05. We find that for the asymmetric implosions with the same L, Pn and s stay nearly constant with increasing fe, and the yield Yna is still dominated by Tn. Increasing feLM from 0.0368 to 0.184 can increase Vn by 8% and reduce Tn by 20%, leading to a 20% reduction in Yna . In addition, for the asymmetric implosions with the same fe, the yield Yna increases with increasing L. When the perturbation mode number is increased from L ¼ 8 to L ¼ 16, Pn increases by 10%, and s declines by 10%, with Tn and Vn varying slightly, resulting in a 10% increase in Yna . When the perturbation mode number is increased from L ¼ 16 to L ¼ 24, the 10% increase in Yna is dominated by the 15% increase in Vn, with Pn and s decreasing slightly and Tn staying nearly constant. Overall, the change in Yna can be explained by the scaling in Eq. (15) rather well. V. CONCLUSIONS To summarize, we have performed a series of 1D and 2D noalpha simulations for N140520 to quantify the impact of different harmonic-mean flux-limited conductivity models on the performance of cryogenic implosions. The 1D simulation results show that the difference in fe has a bigger impact on the performance than the change in the analytic model. We find that for the 2D implosions with the same L, the GF is reduced with increasing fe, due to the increased ablation velocity resulting from the enhanced heat flux. In both the 1D implosions and the 2D implosions with the same L, the increase in fe leads to more Mhs and lower Tihs , resulting in approximately constant Ehs. In addition, the no-alpha yield Yna is dominated by Tn in these two cases. Increasing feLM from 0.0368 to 0.184 reduces Tn by 15% in 1D and by 20% for the 2D implosions with the same L, both leading to a 20% reduction in Yna . For the 2D implosions with the same fe, Yna increases with increasing L. Overall, the change in Yna for both the 1D and 2D implosions can be explained by the scaling in Eq. (15) rather well. ACKNOWLEDGMENTS This work was supported by the National Key R&D Program of China under Grant No. 2017YFA0403204. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose.

Phys. Plasmas 29, 042702 (2022); doi: 10.1063/5.0066708 Published under an exclusive license by AIP Publishing

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DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1

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