Interfacial mixing in high-energy-density matter with a multiphysics kinetic model

We have extended a recently developed multispecies, multitemperature Bhatnagar-Gross-Krook model [Haack et al., J. Stat. Phys. 168, 822 (2017)], to include multiphysics capabilities that enable modeling of a wider range of physical conditions. In terms of geometry, we have extended from the spatially homogeneous setting to one spatial dimension. In terms of the physics, we have included an atomic ionization model, accurate collision physics across coupling regimes, self-consistent electric fields, and degeneracy in the electronic screening. We apply the model to a warm dense matter scenario in which the ablator-fuel interface of an inertial confinement fusion target is heated, but for larger length and time scales and for much higher temperatures than can be simulated using molecular dynamics. Relative to molecular dynamics, the kinetic model greatly extends the temperature regime and the spatiotemporal scales over which we are able to model. In our numerical results we observe hydrogen from the ablator material jetting into the fuel during the early stages of the implosion and compare the relative size of various diffusion components (Fickean diffusion, electrodiffusion, and barodiffusion) that drive this process. We also examine kinetic effects, such as anisotropic distributions and velocity separation, in order to determine when this problem can be described with a hydrodynamic model.

Figure 1

Interdiffusion coefficient. Shown are the predictions of the nondimensionalized interdiffusion coefficients $D_{ij}^{*}$ for a mixture of hydrogen and helium over a range of $\mathrm{\Gamma }$. The BGK-ET model (small circles) coincides with the SM model (solid line), and the MD results (large circles), which is expected since the formulas are based on the same collision integral ${\mathrm{\Omega }}_{ij}^{\left(11\right)}$. The BGK-EM model (plus symbols) predicts less diffusion than the SM model. However, it tracks the trend of the MD results into the moderately coupled regime, unlike the classical Coulomb logarithm theory (dashed line).

Figure 2

Viscosity coefficient. Shown above are predictions of the nondimensionalized viscosity $\eta$ for a hydrogen plasma over a range of $\mathrm{\Gamma }$. The BGK model (lower, blue line) again predicts a smaller transport coefficient but follows the trend of the SM model (upper, black line). (In the single-species case, the BGK-EM and BGK-ET versions are the same.) Both the BGK and SM models fail to capture the turning point of the viscosity for stronger coupling that is seen in the MD results (large circles).

Figure 3

Electric field models. A comparison of the electric field models is shown at 0 fs, 500 fs, 1 ps, and 5 ps for an instantly heated 10-eV interface. The spike in electric field due to the discontinuity at the initial time rapidly decreases but the field strength remains significant. The PTF model (green upper line) notably produces a much stronger electric field than the PB model (brown lower line). The DH and LTF solutions are excluded from the figure as they track very closely to the PB profile.

Figure 4

Density evolution for 10- to 50-eV interface test problem. Number density plots are shown at 500 fs (first column) and 10 ps (second column). The instant heating case (first row) shows a separation between the hydrogen (brown, lightest line) and carbon (green, next darkest line) species, while this effect is somewhat less evident for the 10- to 50-eV temperature ramp case (second row). In the 500 fs plots, there is a compression of the deuterium species (blue, darkest line) as the interface moves into the gas portion of the domain. In the instant heating case at 10 ps, the simulation has run long enough that the data from the right side of the domain have reached the shell-gas interface due to the periodic boundary condition. The tritium and oxygen plots are excluded as there is very little tritium separation from the deuterium and the oxygen is a trace species.

Figure 5

Diffusion velocity evolution for 10- to 50-eV interface test problem. The normalized diffusion velocities $({\mathbf{v}}_i-{\mathbf{v}}_0)/v_{th,i}$ are shown for hydrogen (brown, lightest line), carbon (green, next darkest line), and deuterium (blue, darkest line) at 500 fs (first column) and 10 ps (second column). For reference, the number densities are shown in the background as dashed lines. In both the instant heating (top row) and 10- to 50-eV ramp cases (second row), there are large deviations from the mixture velocity inside the mixing layer, which shows that the usual Chapman-Enskog transport theory does not apply.

Figure 6

Density and diffusion velocity evolution for the 100- to 300-eV interface test problem. Number density (first row) and diffusion velocity ${\mathbf{v}}_i-{\mathbf{v}}_0$ (second row) plots are shown at 5 ps (first column) and 50 ps (second column). This simulation reaches time, length, and temperature scales that would be infeasible for a molecular dynamics calculation.

Figure 7

Electric field for the 100- to 300-eV interface test problem. The self-consistent electric field is shown at four times: 0 fs, 5 ps, 25 ps, and 50 ps. The electric field persists at a similar order of magnitude for the entire simulation.

Figure 8

Ionization states for the 100- to 300-eV interface test problem. The ionization states of carbon (lower green line) and oxygen (upper brown line) are shown at four times: 0 fs, 5 ps, 25 ps, and 50 ps. The multispecies kinetic model dynamically updates the ionization state, which affects both the electric fields and the collision rates. The hydrogenic species ionization states are all near unity in this density and temperature range, so they are excluded from this plot.

Figure 9

Temperature component ratio for the 100- to 300-eV and 20-keV case. Ratios of the normal and transverse temperature components for the 100- to 300-eV ramp test case and 20-keV test case is shown for H (brown, lightest line), C (green, next darkest line), and D (blue, darkest line) at 5 and 0.5 ps, respectively. For reference, the number densities are plotted as dashed lines in the background. This suggests that in the low-temperature case, the distributions remain close to Maxwellian in the mixing layer.

Figure 10

Comparison of kinetic and hydrodynamic number densities for the 100- to 300-eV case. The relative difference of the number densities between the kinetic solution and a multivelocity hydrodynamic solution is shown for the 100- to 300-eV ramp test case. The relative difference in number density for H (brown, lightest line), C (green, next darkest line), and D (blue, darkest line) is shown at 500 fs. At these low temperatures, a multivelocity hydrodynamics code may be enough to accurately capture the dynamics.

Figure 11

Comparison of kinetic and hydrodynamic number densities for the 1- and 20-keV cases. In the left plot, we show the relative error in the number density between the kinetic and hydrodynamic models at 1 keV and 5 ps. This appears to be a transition point where a multicomponent hydrodynamic model would begin to break down for this problem. In the right plot, we show the number densities from the kinetic (solid lines) and hydrodynamic (dashed lines) models. At this high temperature, the kinetic effects are strong and the hydrodynamic description no longer applies.

Figure 12

Diffusion components for the 100- to 300-eV test problem. The Fickean (brown, lightest line), barodiffusion (green, next darkest line), and rlectrodiffusion (blue, darkest line) contributions to the carbon (left), hydrogen (center), and deuterium (right) diffusive fluxes are shown at 5 ps for the 100- to 300-eV ramp test case. All three types of diffusion appear to have a significant effect on the mass flux.