Diffusivity in asymmetric Yukawa ionic mixtures in dense plasmas

In this paper we present molecular dynamics (MD) calculations of the interdiffusion coefficient 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 $\sim {10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$. 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. The species diffusivity is then calculated using the Green-Kubo approach using an integral of the interdiffusion current autocorrelation function, a quantity calculated in the equilibrium MD simulations. Our MD simulation results show that a widely used expression relating the interdiffusion coefficient with the concentration-weighted sum of self-diffusion coefficients overestimates the interdiffusion coefficient. We argue that this effect due to cross-correlation terms in velocities is characteristic of asymmetric mixed plasmas. Comparison of the MD results with predictions of kinetic theories also shows a discrepancy with MD giving effectively a larger Coulomb logarithm.

Figure 1

Effective coupling ${\Gamma }_{\mathrm{eff}}$ using relation (4) as a function of Ar mole fraction in a D-Ar ionic mixture at $T$ = 100, 200, and 500 eV density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$ and ionization $Z_{\mathrm{D}}^{*}=1$ and $Z_{\mathrm{Ar}}^{*}=13$. As a reference the values of ${\Gamma }_{12}$ are $6.5,3.25$, and $1.3$ for $T$ = 100, 200, and 500 eV, respectively. These are represented with the short horizontal lines around the value of the mole fraction where ${\Gamma }_{12}={\Gamma }_{\mathrm{eff}}$.

Figure 2

Electron degeneracy $\Theta$ and warm dense matter parameter $\mathcal{W}$ [43] as a function of Ar mole fraction in a D-Ar ionic mixture at $T$ = 100, 200, and 500 eV density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$ and ionization $Z_{\mathrm{D}}^{*}=1$ and $Z_{\mathrm{Ar}}^{*}=13$.

Figure 3

Electron degeneracy $\Theta$ and warm dense matter parameter $\mathcal{W}$ [43] as a function of Ar ionization in a D-Ar ionic mixture at $T$ = 100 eV, density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$, and Ar mole composition $X_{\mathrm{Ar}}=0.05,0.1$, and $0.2$. The ionization of D was kept fixed at $Z_{\mathrm{D}}^{*}=1$.

Figure 4

Electron screening length ${\lambda }_{D,e}$ computed using Eq. (31) as a function of Ar mole fraction in a D-Ar mixture at $T$ = 100, 200, and 500 eV, density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$, and ionization $Z_{\mathrm{D}}^{*}=1$ and $Z_{\mathrm{Ar}}^{*}=13$.

Figure 5

The interdiffusion current correlation function $C_{\mathrm{MS}}$ as a function of time that enters in the MS diffusion plotted for different compositions of a binary ionic mixture at 100 eV and ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$.

Figure 6

Time-dependent MS diffusivities from the partial integrals in time of the correlation functions shown in Fig. 5.

Figure 7

MS diffusion for different temperatures and Ar composition in the mixtures. Ionization of both D and Ar was entered as a free parameter and kept constant at values 1 and 13, respectively.

Figure 8

Values of Ar and D ionization as a function of composition of a D-Ar mixture at $T=100\mathrm{eV}$ and ion number density ${10}^{25}{\mathrm{ion}/\mathrm{cm}}^3$. The values were computed using an average atom Thomas-Fermi approach.

Figure 9

Values of self-diffusivities and mutual diffusivity as a function of the ionization level of Ar. The case presented here corresponds to a $10\%$ Ar binary mixture with deuterium, kept at $T=100\mathrm{eV}$ and number density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$. The ionization of deuterium was fixed at 1, while that of Ar was varied.

Figure 10

Velocity cross-correlation function for $1\%,5\%$, and $10\%$ Ar. The black solid line is related to the MD calculated MS diffusivity, the red dashed line is the Darken relation, and the blue dot-dashed lined is the common force model.

Figure 11

MS diffusivities from MD in black, compared with the Darken combination of self-diffusivities in red and CFM in blue for different composition of the D-Ar mixture at 100 eV and ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$. The ionizations of D and Ar were 1 and 13, respectively.

Figure 12

MS diffusivities from MD in black, compared with the Darken combination of self-diffusivities in red and CFM in blue for a D-Ar mixture with 10% Ar, as a function of Ar ionization $Z_{\mathrm{Ar}}^{*}$. The mixture was kept at 100 eV and number density ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$. The ionizations of D was kept at 1.

Figure 13

Comparison of MS diffusion values from MD at 100 eV, ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$ for the D-Ar mixture with kinetic theories based on binary collision approach. The ionizations of D and Ar are 1 and 13, respectively. MD results are represented by symbols connected with the black dashed line. Results from the CC kinetic model assuming a pure Coulombic interaction are on the solid red curve. The dotted blue curve and the green dashed-dot curve correspond to results from assumed screened Coulomb with electron screening only (as in MD) and all-particle screening, respectively. For the screened Coulomb cases we used the collision integrals tabulated in Ref. [14].

Figure 14

Comparison of MS diffusion values from MD at 100, 200, and 500 eV, ${10}^{25} \mathrm{ions}/{\mathrm{cm}}^3$ for D-Ar mixtures with kinetic theories based on the binary collision approximation. The ionizations of D and Ar are taken to be 1 and 13, respectively. MD results are represented by symbols connected with lines as shown in the legend. Results from kinetic theories based on a screened Coulomb effective ionic interaction are shown with solid curves. For the screened Coulomb cases we used the collision integrals tabulated in Ref. [14]. The effective screening length $\lambda$ is calculated from Eq. (59).

Figure 15

Generalized Coulomb logarithm $ln\overline{\Lambda }$ from MD as a function of Ar mole fraction for three temperatures $T=100,200,$ and 500 eV. For comparison we also show the value $ln\overline{\Lambda }$ from the kinetic theories that use SC potential models with solid lines. The values of $ln\overline{\Lambda }$ from the kinetics are lower than the ones derived from MD.

Figure 16

Generalized Coulomb logarithm $ln\overline{\Lambda }$ as a function of ionization for a 10% Ar mixture at 100 eV. Values extracted from MD (symbols connected with dashed line) are compared with the ones derived from kinetic theory that use SC potential models (red solid line).