Dressed-ion theory for electrolyte solutions: A Debye–Hückel-like reformulation of the exact theory for the primitive model

Abstract

A detailed derivation of the dressed-ion theory—a formally exact theory for primitive model Coulomb fluids—is presented for the case of bulk electrolyte solutions. It is shown that the exact average electrostatic potential, ψ av(r), in the ion atmosphere around each ion satisfies a linear Poisson–Boltzmann (PB) equation for ‘‘dressed ions,’’ each of which consists of a central ion together with a specific part of the surrounding ion cloud. The dressed-ion charge distribution—a renormalized charge for each ion—takes the role that the bare ionic charge has in the usual PB equation. Apart from this, virtually the only difference between the exact dressed-ion and the approximate Debye–Hückel (DH) theories for the pair distribution function is that the former theory is nonlocal; the spread-out nature of the dressed-ion charge distribution gives rise to a nonlocal polarization response to the average potential. The linear response function relating the polarization and the average potential is investigated in the general case and is found to be intimately related to the dressed-ion charge distributions. A close relationship is also demonstrated between these charge distributions and the electrostatic susceptibility of the electrolyte solution. The theory gives a rigorous definition of the concept of effective point charges for ions. Except at high coupling, the long-range asymptotic behavior of the pair distribution functions is of the Debye–Hückel form, but with effective values of the ionic charges (qi*) and a decay length (κ−1) different from the Debye length (κ−1D). A simple formula relating qi* and κ is derived. Explicit formulae for qi* and κ in terms of κD are given in the limit of low electrolyte concentrations. The long range asymptotic behaviors of the bridge function and the short range parts of the direct and total correlation functions are analyzed in some detail.