Charge fluctuations and correlations in finite electrolytes

Charge fluctuations, $\langle Q^{2}_{\Lambda}\rangle$, for the 1:1 equisize hard-sphere electrolyte with the diameter $a$ are computed via grand canonical Monte Carlo simulations, where $Q_{\Lambda}$ is the total charge inside a subvolume $\Lambda$ contained in a simulation box of dimensions $L\times L\times L$ with periodic boundary conditions. The charge fluctuations increase like the surface area $|\partial\Lambda|$ as $\Lambda$ increases, even for small system sizes $L\leq 12a$. For slabs of dimensions $L\times L\times \lambda L$ with $0 < \lambda < 1$, the scaled charge fluctuations, $\langle Q^{2}_{\Lambda}\rangle/|\partial\Lambda|$, approach the thermodynamic limits exponentially fast. The extrapolations to $L\rightarrow\infty$ then yield the Lebowitz length, $\xi_{\mbox{\scriptsize L}}(T,\rho)$, where densities $\rho\alt 3\rho_c$ and temperatures $T\agt T_c$ have been studied. An exact asymptotic expression is obtained for $\langle Q^2_\Lambda \rangle$. This enables one to compute the charge correlation length $\xi_{Z}(T,\rho)$ precisely. The results for $\xi_Z(T,\rho)$ agree with Debye-H\"{u}ckel-type theories at low densities, but show deviations as the density increases. Charge oscillations at higher densities are also observed, as anticipated theoretically. \newline \noindent [1] Y. C. Kim, E. Luijten, and M. E. Fisher, Phys. Rev. Lett. {\bf 95}, 145701 (2005).