Ionic transport in high-energy-density matter

Ionic transport coefficients for dense plasmas have been numerically computed using an effective Boltzmann approach. We have developed a simplified effective potential approach that yields accurate fits for all of the relevant cross sections and collision integrals. Our results have been validated with molecular-dynamics simulations for self-diffusion, interdiffusion, viscosity, and thermal conductivity. Molecular dynamics has also been used to examine the underlying assumptions of the Boltzmann approach through a categorization of behaviors of the velocity autocorrelation function in the Yukawa phase diagram. Using a velocity-dependent screening model, we examine the role of dynamical screening in transport. Implications of these results for Coulomb logarithm approaches are discussed.

Figure 1

Relevant structure of kinetic theory. From a many-body Hamiltonian, one solves the equations of motion either directly, as in molecular dynamics, or through an approximation of the BBGKY hierarchy [20, 21]. Most approximations made in the latter case are of two types, either binary-collision approximations, which handle strong collisions well using a cross section, or correlation expansions, which include many-body physics only in the weak-scattering limit. Here we aim to exploit both avenues via the use of an effective potential, motivated by the right (Lenard-Balescu) branch, in a numerical cross section from the left (Boltzmann) branch. Finally, this approach is compared with the Landau (weak, binary-scattering) approach, which uses a so-called Coulomb logarithm.

Figure 2

Collision trajectories for various impact parameters in the reference frame of one of the particles placed at the origin. All lengths are in units of the screening length and all trajectories are for the same fixed initial energy of $E=4Z^2{e}^2/\lambda$. In both panels, the dashed line indicates the trajectory for a full Yukawa potential. In the top panel, the solid lines are trajectories computed by assuming a pure Coulomb potential everywhere for particles that enter with impact parameters below unity (in these dimensionless units), as indicted by the green region, while particles not entering in this range of impact parameters experience no force anywhere. In the bottom panel, the impact parameter cutoff is replaced by a distance cutoff (below the screening length), as indicated by a circular green region, and a potential of zero elsewhere; this is the TC model described in the text. Each of the three trajectory types yields different scattering angles, with some improvement offered by the TC model over the more common impact parameter cutoff model.

Figure 3

Reduced cross section ${\sigma }^{\left(1\right)}/2\pi {\lambda }^2$ as a function of the dimensionless velocity $w={(\mu \lambda /2Z_i{Z}_j{e}^2)}^{1/2}v$ for truncated Coulomb interactions (green lower curve) and Coulomb interactions using the truncated impact parameter $b_{\text{max}}=\lambda$ (blue upper curve). While the two interactions result in qualitatively similar cross sections, these cross sections differ for moderate values of $w$.

Figure 4

Ratio of the collision integral ${\mathrm{\Omega }}_C^{\left(11\right)}$ (velocity-dependent CL), calculated using (9), to the collision integral ${\mathrm{\Omega }}_{\text{th}}^{\left(11\right)}$ (velocity-independent CL), calculated using (14), as a function of the plasma parameter $g=Z_i{Z}_j{e}^2/\lambda T$. The thermal-velocity approximation within the CL is a singular perturbation and thus approaches the Coulomb form logarithmically as $g\to 0$. This illustrates the magnitude of the error that results from neglecting the full velocity dependence of the cross section.

Figure 5

Radial distribution functions $g\left(r\right)$ (in the hypernetted-chain approximation) for a one-component plasma ($\kappa =0$) with $\mathrm{\Gamma }=\{0.01,0.1,1,10,100\}$. Note that the hole in the ion density around the origin (where there is implicitly an ion) changes very little for $\mathrm{\Gamma }>10$ and that the peak does not shift, revealing that the screening-scale length is approximately $a_i$ over most of the range of strong coupling.

Figure 6

Reduced cross sections ${\varphi }_n={\sigma }_{ij}^{\left(n\right)}/2\pi {\lambda }^2$ versus dimensionless velocity $w={(\mu \lambda /2Z_i{Z}_j{e}^2)}^{1/2}v$ shown for $n=1$ (top) and $n=2$ (bottom). The numerical results (black circles) are compared to fits generated with a least-squares method (blue lines); these fits are presented in Eqs. (C15, C16, C17, C18).

Figure 7

Comparison of dimensionless cross sections. The screened Coulomb (black solid curve) cross section from Eq. (C15) and the Coulomb cross section using a truncated impact parameter (blue dashed curve), as in Eq. (9), have the same limiting behavior at high velocities but deviate significantly from each other at low velocities. The Coulomb cross section using a thermal velocity in the logarithm, as in Eq. (14), is shown for several plasma parameters (green dash-dotted curve), for a range of values of $g=\{0.1,1,10\}$.

Figure 8

Reduced collision integral (43) as a function of the plasma parameter $g=Z_i{Z}_j{e}^2/\lambda T$ for the case $(n,m)=(1,1)$. The numerical results (black circles) are compared to fits generated with a least-squares method (blue lines); these fits are shown in Eqs. (C22, C23, C24). The cases of $(1,2), (1,3)$, and $(2,2)$ yield visually similar results and have been omitted for brevity.

Figure 9

Comparison of collision integrals as ratios of these integrals to the EB collision integral calculated from (42). Here ${\mathrm{\Omega }}_{\text{tip}}$ (black solid curve) is calculated using the cross section with a truncated impact parameter in (9), ${\mathrm{\Omega }}_{\text{th}}$ (blue dashed curve) is calculated using the cross section with a thermal velocity in (14), and ${\mathrm{\Omega }}_{\text{WC}}$ (red dotted curve) is calculated using the cross section in the weakly coupled limit from (16). Each approximation results in significant deviations from unity for even smaller values of the plasma parameter $g$.

Figure 10

Self-diffusion coefficient $D^{*}=D/{\omega }_p{a}^2$ versus $\mathrm{\Gamma }$ for $\kappa =\{0.5,1.0,1.5,2.0\}$. Shown are three variants of the EB approach based on an electron-only screening length (solid yellow curve), the total DH screening (dashed red curve), and the effective screening length of (28) (solid blue curve). Black points are our MD results. Also shown are the results obtained using the CL of (15) (dash-dotted green curve).

Figure 11

Same quantities as in Fig. 10, but shown here using an abbreviated linear scale for $\mathrm{\Gamma }$ to emphasize the weak-coupling limit.

Figure 12

Comparison of our EB solution of the interdiffusion coefficient to MD data from [64]. Two models for the effective screening length are used: total (red dashed curve) and effective (blue solid curve).

Figure 13

Dimensionless viscosity ${\eta }^{*}=\eta /mn{\omega }_p{a}^2$ shown for $\kappa =\{1,2,3\}$ using the MD data of Donko and Hartmann (black triangles) [70] and the EB approach with both the total screening length (red upper curve) and the effective screening length of (28) (blue lower curve). The effective screening length result agrees well with the MD data for $\mathrm{\Gamma }<10$ at $\kappa =1$ and for even larger values of $\mathrm{\Gamma }$ as $\kappa$ increases, to nearly $\mathrm{\Gamma }\sim 100$ with $\kappa =3$.

Figure 14

Dimensionless thermal conductivity $K^{*}$ versus $\mathrm{\Gamma }$ for four values of $\kappa =\{0.1,1,2,3\}$. The EB result using (28) is shown as a solid blue line. A fit to MD results [78] is shown in black; we use a solid line for the fit over the $\mathrm{\Gamma }$ range where MD results contributed to the fit and a dashed line in the range where the fit is extrapolated, revealing the large error associated with extrapolating MD results to weak coupling. For this transport coefficient, the MD fit provides much less of a validation than for the others.

Figure 15

Variation in functional forms of the VACF. In the left panel, the Yukawa phase diagram for three different behaviors is shown: the bottom (green) region is exponential decay, the middle (blue) region is the modulated decay regime in which oscillations in the VACF begin to arise, and the top (purple) region is the caged regime. In the right panel, we show several characteristic VACFs across the phase diagram corresponding to the $(\kappa ,\mathrm{\Gamma })$ values of $(0.5,70), (1,20)$, and $(1.5,5)$, respectively.

Figure 16

Transition from exponential decay to algebraic $t^{-3/2}$ decay versus coupling. Several VACFs on a log-log scale are shown for fixed $\kappa =1$ and various values of $\mathrm{\Gamma }$. Note that the top two curves show exponential decay and this decay is faster than the dashed $t^{-3/2}$ line. However, near $\mathrm{\Gamma }\sim 10$, the decay remains mostly exponential but with important modulations at later times. At $\mathrm{\Gamma }=20$, the VACF has well-defined minima and decays very slowly at late time, a behavior that increases for large $\mathrm{\Gamma }$.

Figure 17

Impact of a velocity-dependent screening length $\lambda \left(v\right)$ on the first-order cross section shown by taking the ratio of the dynamically screened cross section using ${\lambda }_{\text{eff}}\left(v\right)$ (denoted by ${\sigma }_{\text{DS}}^{\left(1\right)}$) to that using only the static ${\lambda }_{\text{eff}}$ as a function of $v/v_i$. The ratios are calculated at $\kappa =0.5$ for the values $\mathrm{\Gamma }=\{0.5,1,2,5,10\}$. Deviations are greatest for $v\sim v_i$ and increase for larger coupling values.

Figure 18

Ratio of the dynamically screened self-diffusion (top) and viscosity (bottom) coefficients to their static counterparts as a function of coupling $\mathrm{\Gamma }$ for $\kappa =\{0.5,1,1.5,2\}$. Note that for weaker screening (smaller $\kappa$), the dynamical correction can be significant.

Figure 19

Diagram outlining the algorithm to calculate transport properties using the fits of this work. The calculation begins with a determination of the atomic structure of the possibly partially ionized mixture, followed by a determination of the relevant plasma properties (electron screening length and ion screening length) to obtain the effective screening length. This can then be used to compute transport coefficients either near equilibrium (right branch) or using cross sections and a non-Maxwellian distribution (left branch).