Multicomponent mutual diffusion in the warm, dense matter regime

We present the formulation, simulations, and results for multicomponent mutual diffusion coefficients in the warm, dense matter regime. While binary mixtures have received considerable attention for mass transport, far fewer studies have addressed ternary and more complex systems. We therefore explicitly examine ternary systems utilizing the Maxwell-Stefan formulation that relates diffusion to gradients in the chemical potential. Onsager coefficients then connect the macroscopic diffusion to microscopic particle motions, evinced in trajectories characterized by positions and velocities, through various autocorrelation functions (ACFs). These trajectories are generated by molecular dynamics (MD) simulations either through the Born-Oppenheimer approximation, which treats the ions classically and the electrons quantum-mechanically by an orbital-free density-functional theory, or through a classical MD approach with Yukawa pair-potentials, whose effective ionizations and electron screening length derive from quantal considerations. We employ the reference-mean form of the ACFs and determine the center-of-mass coefficients through a simple reference-frame-dependent similarity transformation. The Onsager terms in turn determine the mutual diffusion coefficients. We examine a representative sample of ternary mixtures as a function of density and temperature from those with only light elements (D-Li-C, D-Li-Al) to those with highly asymmetric mass components (D-Li-Cu, D-Li-Ag, H-C-Ag). We also follow trends in the diffusion as a function of number concentration and evaluated the efficacy of various approximations such as the Darken approximation.

Figure 1

Upper panel: Normalized velocity autocorrelation functions (VAC) as a function of time associated with the RM Onsager coefficients ${\mathrm{\Lambda }}_{ij}^{\mathrm{RM}}$ for the Yukawa potential for a ternary 1:1:1 mixture of C, Li, and D at $10\mathrm{g}/{\mathrm{cm}}^3$ and 100 eV. Curves: DLi (dashed blue line), DC (solid red line), and LiC (within the DC line width). Lower panel: Unnormalized center-of-mass velocity autocorrelation (c.m.-VAC) as a function of time for same case as in upper panel. Curves: CC (solid blue line), LiLi (red long-dashed line), DD (green dash-dotted line), DLi (purple dash-double-dot line), and DC and CLi within the DLi width.

Figure 2

Onsager coefficient ${\mathrm{\Lambda }}_{ij}\left(t\right)$ as a function of time for Yukawa potential for a ternary 1:1:1 mixture of C, Li, and D at $10\mathrm{g}/{\mathrm{cm}}^3$ and 10 eV. Curves: LiC (green solid line); DC (red dashed line); DLi (blue dash-dotted line). The fits lie within the line widths.

Figure 3

Self-diffusion coefficients as a function of temperature for Yukawa model for a ternary 1:1:1 mixture DLiC mixture at $10\mathrm{g}/{\mathrm{cm}}^3$ for 399 atoms with C (green triangle), Li (blue circle), and D (red square).

Figure 4

Ternary mutual diffusion coefficients $D_{ij}$ as a function of temperature for a Yukawa model at $10\mathrm{g}/{\mathrm{cm}}^3$ for a 1:1:1 mixture with 399 atoms of C, Li, and D: Comparison of full Maxwell-Stefan (symbols) and Darken (lines) formulations. $D_{\mathrm{CLi}}$ (green), $D_{\mathrm{CD}}$ (blue), $D_{\mathrm{DLi}}$ (red).

Figure 5

Mutual diffusion coefficients as a function of temperature for D-Li-Ag for a 1:1:2 sample of 800 atoms at $20\mathrm{g}/{\mathrm{cm}}^3$ result for coefficients DLi (blue upper group), DAg (green middle group), and LiAg (red lower group). Within each group: Yukawa model for full MD simulations (solid line) and Darken approximation (dash line); OFMD Darken results shown as symbols.

Figure 6

Mutual diffusion coefficients as a function of temperature for H-C-Ag for 1:1:2 sample of 800 atoms at $20\mathrm{g}/{\mathrm{cm}}^3$ result for coefficients HC (upper group, blue), HAg (middle group, green), and CAg (lower group, red); OFMD Darken results shown as symbols.

Figure 7

Yukawa center-of-mass-frame Onsager ${\mathrm{\Lambda }}_{\mathrm{DLi}}\left(t\right)$ [Eq. (9)] and mutual diffusion $D_{\mathrm{DLi}}$ (inset) coefficients for a DLiX mixture as a function of the heavier species $X$ = carbon ($Z=6$), aluminum (13), copper (29), and silver (47) for 20 g/cc and a number concentration of 1:1:2 for 800 atoms at 100 eV. Error bars at 1 standard deviation.

Figure 8

OFMD self-diffusion and Darken mutual diffusion coefficients (${\mathrm{cm}}^2/\mathrm{s}$) for the ternary mixture D-Li-Ag at 100 eV (upper panel) and 200 eV (lower panel) for a fixed total pressure at each temperature as a function of concentration of the heavy species Ag. Labels: $D_D$ (blue dashed line), $D_{\mathrm{Li}}$ (green dashed line and triangles), $D_{\mathrm{Ag}}$ (red dashed line and circles); $D_{\mathrm{DLi}}$ (red solid line), $D_{\mathrm{DAg}}$ (green solid line and squares), and $D_{\mathrm{LiAg}}$ (blue solid line and diamonds).

Figure 9

OFMD self-diffusion and Darken mutual diffusion coefficients (${\mathrm{cm}}^2/\mathrm{s}$) for the ternary mixture H-C-Ag at 100 eV for a fixed total pressure at temperature of 100 eV as a function of concentration of the heavy species Ag. Labels: $D_H$ (blue dashed line), $D_C$ (green dashed line and triangles), $D_{\mathrm{Ag}}$ (red dashed line and circles); $D_{\mathrm{HC}}$ (red solid line), $D_{\mathrm{HAg}}$ (green solid line and squares), and $D_{\mathrm{CAg}}$ (blue solid line and diamonds).