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.

Modelling of morphology and proton transport in PFSA membranes

“…Clearly, at these lower water contents, the effect of these 'traps' becomes more significant and hence PFSA ionomers with very high sulfonic acid densities are unlikely to give the high proton conductivity needed in fuel cells operated with little water in the electrolyte. Finally, in what constitutes a culmination of their work, 138 these authors incorporated their extensive modeling of the dielectric saturation of the water in PEMs [139][140][141] and extended their model to include ionic screening with a quasiparticle or 'dressed' ion formulation 142 with two distinct treatments for the screening parameter (i.e. Debye-Hu¨ckel and Attard 143 ).…”

A modified Poisson-Boltzmann approach to homogeneous ionic solutions

“…Investigations of the hypernetted chain (HNC) equation [8,[26][27][28] indicate that y C is dependent on the electrolyte valence. Also, from diagrammatic techniques [29] and dressed ion theory [30], the asymptotic behaviour of the screening length for small values of the coupling parameter Γ = βκe 2 /4πε 0 ε r , e elementary unit of charge, has a purely electrostatic contribution for a → 0. Unequal ion sizes further complicate the screening length behaviour [31].…”

“…The neglect of (i) leads to incorrect weighting for the short range interaction of two ions, and hence shortfalls when concepts such as ion pairing are important. An adequate treatment of (i) should give the required variation of y C with different valences, and further corrections could be given by including in the fluctuation potential problem the exclusion volume term (30) or the non-linear term for unequal valences [12]. The point ion contribution in the screening length for small Γ should come from a solution of the fluctuation potential problem using the more accurate equation (37) rather than equation (38).…”

Mesoscopic Simulation Methods for Studying Flow and Transport in Electric Fields in Micro- and Nanochannels

“…For highly charged macroions, the Debye-Hückel theory can still be applied far from the macroions, but the macroion charge must be replaced by an effective "renormalized charge" (Alexander et al (1984); Trizac et al (2002)). For high ion concentrations, detailed theoretical studies have lead to the conclusion that the results of the Debye-Hückel can still be used in a wide parameter range if λ D is replaced by a modified effective screening length (Kjellander & Mitchell (1994); McBride et al (1998);Mitchell & Ninham (1978)). …”