First-principles thermal conductivity of warm-dense deuterium plasmas for inertial confinement fusion applications

Thermal conductivity (κ) of both the ablator materials and deuterium-tritium (DT) fuel plays an important role in understanding and designing inertial confinement fusion (ICF) implosions. The extensively used Spitzer model for thermal conduction in ideal plasmas breaks down for high-density, low-temperature shells that are compressed by shocks and spherical convergence in imploding targets. A variety of thermal-conductivity models have been proposed for ICF hydrodynamic simulations of such coupled and degenerate plasmas. The accuracy of these κ models for DT plasmas has recently been tested against first-principles calculations using the quantum molecular-dynamics (QMD) method; although mainly for high densities (ρ > 100 g/cm${}^3$), large discrepancies in κ have been identified for the peak-compression conditions in ICF. To cover the wide range of density-temperature conditions undergone by ICF imploding fuel shells, we have performed QMD calculations of κ for a variety of deuterium densities of ρ = 1.0 to 673.518 g/cm${}^3$, at temperatures varying from $T$ = 5 × 10${}^3$ K to $T$ = 8 × 10${}^6$ K. The resulting ${\kappa }_{\mathrm{QMD}}$ of deuterium is fitted with a polynomial function of the coupling and degeneracy parameters Γ and θ, which can then be used in hydrodynamic simulation codes. Compared with the “hybrid” Spitzer-Lee-More model currently adopted in our hydrocode lilac, the hydrosimulations using the fitted ${\kappa }_{\mathrm{QMD}}$ have shown up to ∼20% variations in predicting target performance for different ICF implosions on OMEGA and direct-drive–ignition designs for the National Ignition Facility (NIF). The lower the adiabat of an imploding shell, the more variations in predicting target performance using ${\kappa }_{\mathrm{QMD}}$. Moreover, the use of ${\kappa }_{\mathrm{QMD}}$ also modifies the shock conditions and the density-temperature profiles of the imploding shell at early implosion stage, which predominantly affects the final target performance. This is in contrast to the previous speculation that ${\kappa }_{\mathrm{QMD}}$ changes mainly the inside ablation process during the hot-spot formation of an ICF implosion.

Figure 1

The equation-of-state comparison between QMD and PIMC calculations for deuterium density at ρ = 5.388 g/cm${}^3$: (a) Pressure versus temperature and (b) internal energy versus temperature.

Figure 2

The QMD-calculated reflectivity of deuterium shock as a function of shock speed along the principal Hugoniot, which is compared to both the previous Nova measurement [55] and the recent OMEGA experiment for different VISAR wavelengths: (a) λ = 404 nm, (b) λ = 532 nm, (c) λ = 808 nm, and (d) λ = 1064 nm.

Figure 3

The generalized Coulomb logarithm, derived from QMD-calculated thermal conductivities for different densities and temperatures, is fitted with the polynomial function [Eq. (10)] of the coupling parameter (Γ) and the degeneracy parameter (θ). The values of lnΛ at high temperatures [i.e., ln(lnΛ) > 0] are converged to the standard lilac ones.

Figure 4

The thermal conductivities from first-principles QMD calculations, the QMD-fitting formula [Eq. (10)], and the hybrid model used in lilac are plotted as a function of temperature for different deuterium densities of (a) ρ = 2.453 g/cm${}^3$ and (b) ρ = 10.0 g/cm${}^3$.

Figure 5

The thermal conductivities from first-principles QMD calculations, the QMD-fitting formula [Eq. (10)], and the hybrid model used in lilac are plotted as a function of temperature for different deuterium densities of (a) ρ = 43.105 g/cm${}^3$ and (b) ρ = 199.561 g/cm${}^3$. The Hubbard + Spitzer interpolation results are also depicted by the black dash-dotted line in (b).

Figure 6

(a) The laser pulse shape used for a high-adiabat (α = 4) cryogenic-DT implosion on OMEGA (the $\varphi$ = 868.8-μm capsule consists of 47 μm of DT ice with an 8.4-μm-thick plastic ablator); (b) comparisons of the density and ion-temperature profiles at peak compression for the two hydrodynamic runs using ${\kappa }_{\mathrm{lilac}}$ (blue dashed lines) and ${\kappa }_{\mathrm{QMD}}$ (red solid lines), respectively. Very little difference is seen in target performance for the two thermal-conductivity models used for such a high-adiabat implosion.

Figure 7

(a) The laser pulse shape used for a low-adiabat (α = 2.2) cryogenic-DT implosion on OMEGA (the $\varphi$ = 860.6-μm capsule consists of 49 μm of DT ice with an 8.3-μm-thick plastic ablator). (b) and (c) Comparisons of the density and temperature profiles at the beginning of deceleration phase and at the peak compression, respectively, for the two hydrodynamic simulations using ${\kappa }_{\mathrm{lilac}}$ (blue dashed lines) and ${\kappa }_{\mathrm{QMD}}$ (red solid lines). (d) The neutron yields as a function of time are plotted for the two cases. A modest variation (∼6%) in target performance is seen in such low-adiabat OMEGA implosions, when ${\kappa }_{\mathrm{QMD}}$ is compared to the hybrid lilac model.

Figure 8

Similar to Fig. 6 but for a NIF-scale implosion: (a) The laser pulse shape for a mid-adiabat (α = 3.2), 1.5-MJ direct-drive NIF design (the $\varphi$ = 3460-μm capsule consists of 220 μm of DT ice with a 30-μm-thick plastic ablator); (b) comparisons of the density and ion-temperature profiles at the peak compression for the two hydrodynamic runs using ${\kappa }_{\mathrm{lilac}}$ (blue dashed lines) and ${\kappa }_{\mathrm{QMD}}$ (red solid lines), respectively. The effects of using different κ are small for such mid-adiabat designs.

Figure 9

Tests on a high-implosion-velocity NIF design: (a) The laser pulse shape (α = 2.5) has a total 1.6-MJ energy (the $\varphi$ = 3294-μm capsule consists of 125 μm of DT ice with a 22-μm-thick plastic ablator). (b) and (c) Comparison of the density and temperature profiles at the beginning of the deceleration phase and at peak compression, respectively, for the two hydrodynamic simulations using ${\kappa }_{\mathrm{lilac}}$ (blue dashed lines) and ${\kappa }_{\mathrm{QMD}}$ (red solid lines). The neutron yields as a function of time are plotted in (d) for the two cases. The use of ${\kappa }_{\mathrm{QMD}}$ changes the one-dimensional (1D) prediction of implosion performance modestly (∼6%).

Figure 10

The predicted shock conditions during the shock transit stage in the DT ice, for the NIF design plotted in Fig. 9. The density and electron temperature are plotted as a function of the Lagrangian cell numbers for times at (a) $t$ = 4.0 ns and (b) $t$ = 4.8 ns. Again, the two cases of using ${\kappa }_{\mathrm{lilac}}$ (blue dashed lines) and ${\kappa }_{\mathrm{QMD}}$ (red solid lines) are compared.

Figure 11

Similar to Fig. 9 but for a relatively lower adiabat (α = 1.7) and lower-implosion-velocity ($v_{\mathrm{imp}}$ = 3.3 × 107 cm/s) NIF design: (a) The laser pulse shape has a total energy of 1.2 MJ and the $\varphi$ = 3420-μm capsule consists of 180 μm of DT ice with a 30-μm plastic ablator. (b) and (c) Comparison of the density and temperature profiles at the beginning of the deceleration phase and at the peak compression, respectively. (d) Comparison of the neutron yields for the three cases, which shows about ∼20% variation in the 1D predictions of target performance using ${\kappa }_{\mathrm{lilac}}$ and ${\kappa }_{\mathrm{Brysk}}$ (green dash-dotted lines) in comparison to the ${\kappa }_{\mathrm{QMD}}$ modeling.