Soft mean spherical approximation for dusty plasma liquids: One-component Yukawa systems with plasma shielding

The structure and thermodynamics of strongly coupled dusty plasmas are investigated with the soft mean spherical approximation. This integral theory approach is analytically solvable for Yukawa pair interactions yielding a closed-form solution for the direct correlation function. The pair correlation function, the structure factor, and basic thermodynamic quantities are calculated for a wide range of parameters. Exact consistency between the “energy”-“virial” thermodynamic routes and approximate consistency between the “energy”-“compressibility” paths is demonstrated. Comparison with extensive molecular dynamics results is carried out and a remarkable agreement from the Coulomb limit to the strongly screened limit is revealed. The soft mean spherical approximation is concluded to be particularly well suited for the study of dusty plasma liquids, uniquely combining simplicity and accuracy.

Figure 1

Plot of the least-square fit for ${\Gamma }_{\mathrm{crit}}\left(\kappa \right)$ together with the numerical data of Table 1.

Figure 2

The packing fraction $\eta$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$, calculated with the SMSA. Results for different values of $\kappa$. The lines switch from continuous to dashed at ${\eta }_{\mathrm{c}}$, i.e., when the sign in front of the discriminant of $v_{\lambda }$ switches. (The top curve corresponds to $\kappa =5.0$ and the bottom curve corresponds to $\kappa =1.0$.)

Figure 3

The pair correlation function $g\left(r\right)$ for $\kappa =2.0$ and varying values of $\Gamma$ as a function of the distance normalized by the hard-core diameter $(r/R_{\mathrm{h}})$. See also Ref. [42]. (In the left column, the top curve corresponds to $\Gamma =50$ and the bottom curve corresponds to $\Gamma =0.6$. In the right column, the top curve corresponds to $\Gamma =440$ and the bottom curve corresponds to $\Gamma =70$.)

Figure 4

The pair correlation function $g\left(r\right)$ for $\kappa =2.0$ and varying values of $\Gamma$ as a function of the distance normalized by the cubic mean interparticle distance $(r/\Delta )$. See also Ref. [42]. (In the left column, the top curve corresponds to $\Gamma =50$ and the bottom curve corresponds to $\Gamma =0.6$. In the right column, the top curve corresponds to $\Gamma =440$ and the bottom curve corresponds to $\Gamma =70$.)

Figure 5

The structure factor $S\left(q\right)$ for $\kappa =2.0$ and varying values of $\Gamma$ as a function of the normalized wave number $q=R_{\mathrm{h}}k$. (The top curve corresponds to $\Gamma =400$ and the bottom curve corresponds to $\Gamma =20$.)

Figure 6

The reduced excess internal energy due to dust-dust interactions $u_{\mathrm{ex}}^{\mathrm{dd}}$ normalized by the coupling parameter $\Gamma$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$, calculated with the SMSA. The data points correspond to the MD results [14]. The vertical lines correspond to $\Gamma ={\Gamma }_{\mathrm{crit}}/{\Gamma }_{\mathrm{melt}}$, in order to identify the sign switching of the closed-form solution. Results for low $\kappa$.

Figure 7

The reduced excess internal energy due to dust-dust interactions $u_{\mathrm{ex}}^{\mathrm{dd}}$ normalized by the coupling parameter $\Gamma$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$, calculated with the SMSA. The data points correspond to the MD results [14, 15, 16]. The vertical lines correspond to $\Gamma ={\Gamma }_{\mathrm{crit}}/{\Gamma }_{\mathrm{melt}}$, in order to identify the sign switching of the closed-form solution. Results for intermediate $\kappa$. (The top curve corresponds to $\kappa =1.0$ and the bottom curve corresponds to $\kappa =2.6$.)

Figure 8

The reduced excess internal energy due to dust-dust interactions $u_{\mathrm{ex}}^{\mathrm{dd}}$ normalized by the coupling parameter $\Gamma$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$, calculated with the SMSA. The data points correspond to the MD results [16]. The vertical lines correspond to $\Gamma ={\Gamma }_{\mathrm{crit}}/{\Gamma }_{\mathrm{melt}}$, in order to identify the sign switching of the closed-form solution. Results for high $\kappa$. (The top curve corresponds to $\kappa =3.0$ and the bottom curve corresponds to $\kappa =5.0$.)

Figure 9

The reduced excess free energy due to dust-dust interactions $f_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ as calculated with the SMSA (solid line) and the Hamaguchi et al. fitting expression (dashed line). The SMSA results for low $\kappa$ are totally indistinguishable from the MD results. (The top curve corresponds to $\kappa =0.2$ and the bottom curve corresponds to $\kappa =0.8$.)

Figure 10

The reduced excess free energy due to dust-dust interactions $f_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ as calculated with the SMSA (solid line) and the Hamaguchi et al. fitting expression (dashed line). The SMSA results for intermediate $\kappa$ are totally indistinguishable from the MD results. (The top curve corresponds to $\kappa =1.0$ and the bottom curve corresponds to $\kappa =2.6$.)

Figure 11

The reduced excess free energy due to dust-dust interactions $f_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ as calculated with the SMSA (solid line) and the Hamaguchi et al. fitting expression (dashed line). The SMSA results for high $\kappa$ are practically indistinguishable from the MD results. (The top curve corresponds to $\kappa =3.0$ and the bottom curve corresponds to $\kappa =5.0$.)

Figure 12

The reduced excess pressure due to dust-dust interactions $p_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the virial route (solid line) and the energy route (dashed line), with the SMSA. Results for low $\kappa$. The curves are indistinguishable. (The top curve corresponds to $\kappa =0.2$ and the bottom curve corresponds to $\kappa =1.0$.)

Figure 13

The reduced excess pressure due to dust-dust interactions $p_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the virial route, with the SMSA. Results for intermediate $\kappa$. (The top curve corresponds to $\kappa =1.0$ and the bottom curve corresponds to $\kappa =2.6$.)

Figure 14

The reduced excess pressure due to dust-dust interactions $p_{\mathrm{ex}}^{\mathrm{dd}}$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the virial route, with the SMSA. Results for high $\kappa$. (The top curve corresponds to $\kappa =3.0$ and the bottom curve corresponds to $\kappa =5.0$.)

Figure 15

The reduced excess inverse isothermal compressibility ${\mu }_{\mathrm{p}}^{\mathrm{dd}}(\Gamma ,\kappa )$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the statistical route (solid line) and the energy route (dashed line), with the SMSA. Results for low $\kappa$. The curves are practically indistinguishable. (The top curve corresponds to $\kappa =0.2$ and the bottom curve corresponds to $\kappa =1.0$.)

Figure 16

The reduced excess inverse isothermal compressibility ${\mu }_{\mathrm{p}}^{\mathrm{dd}}(\Gamma ,\kappa )$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the statistical route with the SMSA. Results for intermediate $\kappa$. (The top curve corresponds to $\kappa =1.2$ and the bottom curve corresponds to $\kappa =2.6$.)

Figure 17

The reduced excess inverse isothermal compressibility ${\mu }_{\mathrm{p}}^{\mathrm{dd}}(\Gamma ,\kappa )$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$ calculated from the statistical route with the SMSA. Results for high $\kappa$. (The top curve corresponds to $\kappa =3.0$ and the bottom curve corresponds to $\kappa =5.0$.)

Figure 18

The pair correlation function $g\left(r\right)$ for Coulomb interactions and varying values of $\Gamma$ as a function of the distance normalized by the hard-core diameter $(r/R_{\mathrm{h}})$. See also Ref. [42]. (In the left column, the top curve corresponds to $\Gamma =10$ and the bottom curve corresponds to $\Gamma =0.2$. In the right column, the top curve corresponds to $\Gamma =171.8$ and the bottom curve corresponds to $\Gamma =14$.)

Figure 19

The pair correlation function $g\left(r\right)$ for Coulomb interactions and varying values of $\Gamma$ as a function of the distance normalized by the cubic mean interparticle distance $(r/\Delta )$. See also Ref. [42]. (In the left column, the top curve corresponds to $\Gamma =10$ and the bottom curve corresponds to $\Gamma =0.6$. In the right column, the top curve corresponds to $\Gamma =171.8$ and the bottom curve corresponds to $\Gamma =14$.)

Figure 20

The reduced excess internal energy due to the presence of dust $u_{\mathrm{ex}}^{\mathrm{d}}$ normalized by the coupling parameter $\Gamma$ as a function of $\Gamma /{\Gamma }_{\mathrm{melt}}$, as calculated with the SMSA for a Coulomb plasma. The data points correspond to the MD results [14].