Strong-Coupling Theory for Counter-Ion Distributions

– The Poisson-Boltzmann approach gives asymptotically exact counter-ion density profiles around charged objects in the weak-coupling limit of low valency and high temperature. In this paper we derive, using field-theoretic methods, a theory which becomes exact in the opposite limit of strong coupling. Formally, it corresponds to a standard virial expansion. Long-range divergences, which render the virial expansion intractable for homogeneous bulk systems, are shown to be renormalizable for the case of inhomogeneous distribution functions by a systematic expansion in inverse powers of the coupling parameter. For a planar charged wall, our analytical results compare quantitatively with extensive Monte-Carlo simulations. Recent years have witnessed a revival of the interest in classical charged systems[1, 2]. Specific attention has been paid to the failure of the Poisson-Boltzmann (PB) approach[3, 4, 5, 6, 7], which is known to give reliable results only in the limit of low-valency ions or high temperatures. Corrections to PB have been attributed to correlations between ions, or, more precisely, correlated ion-density fluctuations, and, if present, additional non-electrostatic interactions between ions. These corrections are particularly important for the interaction between macroscopic similarly charged objects, where they can lead to attractions[3, 4, 5, 8, 9]. In as much as the PB approach is accurate for weakly charged systems, no systematic theory is available for the distribution of counter-ions around charged objects in the opposite limit of high-valency ions; moreover, it was not clear whether such a limit exists and whether it is physically meaningful. In this paper we show, using field-theoretic methods, that while PB corresponds to the asymptotically exact theory in the weak-coupling limit, (corresponding to low-valency ions or high temperatures), our novel strong-coupling theory becomes asymptotically exact in the opposite limit of high-valency ions or low temperatures and constitutes a physically sound limit. For the case of a planar charged wall, we give explicit results for the asymptotic density profile in the strong-coupling limit. We also have performed extensive Monte-Carlo (MC) simulations of this system. The resulting density profiles agree for weak and for strong coupling with predictions from PB theory and our strong-coupling theory, respectively. The strong-coupling limit is experimentally easily reached at room temperatures with highly charged walls and/or multivalent counter ions and thus relevant from the application point of view. Typeset using EURO-TEX 2 EUROPHYSICS LETTERS To proceed, consider the Hamiltonian of a system of N ions of valency q at an impenetrable and oppositely charged wall of number density of surface charges σs, H kBT = ∑ j<k lBq 2 |rj − rk| + 2πqlBσs N