Shear viscosity for dense plasmas by equilibrium molecular dynamics in asymmetric Yukawa ionic mixtures

We present molecular dynamics (MD) calculations of shear viscosity for asymmetric mixed plasma for thermodynamic conditions relevant to astrophysical and inertial confinement fusion plasmas. Specifically, we consider mixtures of deuterium and argon at temperatures of 100–500 eV and a number density of ${10}^{25}$ ions/cc. The motion of 30 000–120 000 ions is simulated in which the ions interact via the Yukawa (screened Coulomb) potential. The electric field of the electrons is included in this effective interaction; the electrons are not simulated explicitly. Shear viscosity is calculated using the Green-Kubo approach with an integral of the shear stress autocorrelation function, a quantity calculated in the equilibrium MD simulations. We systematically study different mixtures through a series of simulations with increasing fraction of the minority high-$Z$ element (Ar) in the D-Ar plasma mixture. In the more weakly coupled plasmas, at 500 eV and low Ar fractions, results from MD compare very well with Chapman-Enskog kinetic results. In the more strongly coupled plasmas, the kinetic theory does not agree well with the MD results. We develop a simple model that interpolates between classical kinetic theories at weak coupling and the Murillo Yukawa viscosity model at higher coupling. This hybrid kinetics-MD viscosity model agrees well with the MD results over the conditions simulated, ranging from moderately weakly coupled to moderately strongly coupled asymmetric plasma mixtures.

Figure 1

Effective coupling for D-Ar mixture at 100 eV and ion density ${10}^{25} \mathrm{cm}{}^{-3}$ as a function of Ar mole fraction. The ionized charges of the two species are kept constant at $Z_{\mathrm{Ar}}^{*}=13$ and $Z_{\mathrm{D}}^{*}=1$. With black solid line we plot the value of the Coulomb effective coupling ${\mathrm{\Gamma }}_{\mathrm{eff}}$ from Eq. (7). The red dashed line represents values of the Yukawa coupling $e^{-\kappa }{\mathrm{\Gamma }}_{\mathrm{eff}}$, and the blue dashed-dot line corresponds to values computed for ${\mathrm{\Gamma }}_{12}$ in Eq. (8). In the same figure we show the system coupling as measured directly from MD.

Figure 2

The stress autocorrelation function SAF as a function of time that enters in the shear viscosity plotted for different compositions of binary ionic mixture at 200 eV and ${10}^{25}$ ions/cc. In the inset same results are plotted in log scale. SAF were calculated out to a time 40 fs (not shown).

Figure 3

Partial integrals in time of the correlation functions shown in Fig. 2. These are the average over different initial conditions of the cumulative sums of the SAF, as highlighted in Fig. 4, for each respective case. Each SAF was calculated out to a time of 40 fs (not shown).

Figure 4

The partial integrals in time of SAF for five different initial configurations represent with dotted lines. The solid line is the average. This particular case corresponds to a mixture with $10\%$ Ar at $T=200$ eV. The viscosity is found to be $0.3706 \pm$ 0.033 Pa s.

Figure 5

Viscosity versus composition for $T=100,200$, and 500 eV. The MD results are shown with symbols compared to the viscosity calculated from Chapman-Enskog collision integrals shown without symbols.

Figure 6

Shear viscosity versus composition for $T=100$ eV at ${10}^{25}$ ion/cc, as computed from the Chapman-Enskog collision integrals using the tabular values in Paquette et al. [3]. With black circular symbols results from the MD simulations for the full shear viscosity are presented. The kinetic component of the shear viscosity is presented by red square symbols.

Figure 7

Shear viscosity of a mixed D-Ar system from MD at $T=100$ eV and ion density ${10}^{25}$ ion/cc. The charge of D and Ar are respectively $Z_{\mathrm{D}}=1$ and $Z_{\mathrm{Ar}}=13$. The results from MD are shown with symbols. Estimates from Bastea mixing model are shown with the green dot-dashed line. Results from the extension to mixtures of the Murillo's YVM prescription are shown with the solid red line.

Figure 8

Viscosity for a deuterium-argon mixture at temperatures 100, 200 and 500 eV and ion density $n={10}^{25}$ ion/cc. The MD values of the shear viscosity $\eta$ are plotted with symbols. The solid lines represents values of viscosity ${\eta }_{\mathrm{KMD}}^{}$ computed by the KMD viscosity model in Eq. (43). This model represents in an approximate fashion contribution of the kinetic, cross and potential terms of viscosity. Results from the KMD model and from the equilibrium MD are in good agreement (within 16%) over the aforementioned conditions.

Figure 9

A plot of the shear viscosity of deuterium as determined from the eYVM over a range of temperatures at the densities indicated in the legend.