Screening in Ionic Systems: Simulations for the Lebowitz Length

Simulations of the Lebowitz length, ${\xi }_{\mathrm{L}}(T,\rho )$, are reported for the restricted primitive model hard-core (diameter $a$) $1:1$ electrolyte for densities $\rho \lesssim 4{\rho }_c$ and $T_c\lesssim T\lesssim 40T_c$. Finite-size effects are elucidated for the charge fluctuations in various subdomains that serve to evaluate ${\xi }_{\mathrm{L}}$. On extrapolation to the bulk limit for $T\gtrsim 10T_c$ the exact low-density expansions are seen to fail badly when $\rho >\frac{1}{10}{\rho }_c$ (with ${\rho }_c{a}^3\simeq 0.08$). At higher densities ${\xi }_{\mathrm{L}}$ rises above the Debye length, ${\xi }_{\mathrm{D}}\propto \sqrt{T/\rho }$, by 10%–30% (up to $\rho \simeq 1.3{\rho }_c$); the variation is portrayed fairly well by the generalized Debye-Hückel theory. On approaching criticality at fixed $\rho$ or fixed $T$, ${\xi }_{\mathrm{L}}(T,\rho )$ remains finite with ${\xi }_{\mathrm{L}}^c\simeq 0.30a\simeq 1.3{\xi }_{\mathrm{D}}^c$ but displays a weak entropylike singularity.

Figure 1

Reduced charge fluctuations, $⟨Q_{\Lambda }^2⟩/q^2$, for given $(T,\rho )$ in cubes $\Lambda$ of edges $\lambda L$ vs reduced area, $A_{\Lambda }/L^2=6{\lambda }^2$.

Figure 2

Quadratic fits to finite-size Lebowitz length data for sizes up to $L^{*}=24$ at $T^{*}=0.5$ and various densities.

Figure 3

Density variation of the bulk Lebowitz length extrapolated from various subdomains at $T^{*}=0.5$. The dashed, solid, and dotted curves represent GDH theory [10], and approximants that become exact at low density; see text.

Figure 4

Lebowitz length for $L^{*}=8$ and 12 on the critical isochore compared with the Debye length.

Figure 5

(a) Temperature derivative of reduced Lebowitz lengths and (b) specific heats on the critical isochore. The dashed curve approximates the bulk specific heat [13].

Figure 6

Derivatives on the critical isotherm of (a) the reduced Lebowitz lengths and (b) the energy densities with respect to the chemical potential ${\mu }^{*}$ where ${\mu }_c^{*}\simeq -1.36218$; see text.