Diverging correlation lengths in electrolytes: Exact results at low densities

Abstract

The restricted primitive model of an electrolyte (equisized hard spheres carrying charges $\ifmmode\pm\else\textpm\fi{}{q}_{0})$ is studied using Meeron's expressions [J. Chem. Phys. 28, 630 (1958)] for the multicomponent radial distribution functions ${g}_{\ensuremath{\sigma}\ensuremath{\tau}}(\mathbf{r};T,\ensuremath{\rho}),$ that are correct through terms of relative order \ensuremath{\rho}, the overall density. The exact second and fourth moment density-density correlation lengths ${\ensuremath{\xi}}_{N,1}(T,\ensuremath{\rho})$ and ${\ensuremath{\xi}}_{N,2}(T,\ensuremath{\rho}),$ respectively, are thereby derived for low densities: in contrast to the Debye length ${\ensuremath{\xi}}_{D}{=(k}_{B}T/4\ensuremath{\pi}{q}_{0}^{2}\ensuremath{\rho}{)}^{1/2},$ these diverge when $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}0$ as $(T\ensuremath{\rho}{)}^{\ensuremath{-}1/4}$ and $(T/{\ensuremath{\rho}}^{3}{)}^{1/8},$ respectively, with universal amplitudes. The asymptotic expressions agree precisely with those obtained by Lee and Fisher [Phys. Rev. Lett. 76, 2906 (1996)] from a generalization of Debye-H\uckel (GDH) theory to nonuniform ion densities. Other aspects of this GDH theory are checked and found to be exact at low densities. Specifically, with the further aid of the hypernetted-chain resummation, the corresponding charge-charge correlation lengths ${\ensuremath{\xi}}_{Z,1}$ and ${\ensuremath{\xi}}_{Z,2}$ and the Lebowitz length, ${\ensuremath{\xi}}_{L}$ (which restricts charge fluctuations in large domains), are calculated up to nonuniversal terms of orders $\ensuremath{\rho}\mathrm{ln}\ensuremath{\rho}$ and \ensuremath{\rho}. In accord with the Stillinger-Lovett condition, one finds ${\ensuremath{\xi}}_{Z,1}={\ensuremath{\xi}}_{D}$ although the ratios ${\ensuremath{\xi}}_{Z,2}/{\ensuremath{\xi}}_{D}$ and ${\ensuremath{\xi}}_{L}/{\ensuremath{\xi}}_{D}$ deviate from unity at nonzero \ensuremath{\rho}.