Comments on the linear modified Poisson-Boltzmann equation in electrolyte solution theory

Abstract

Three analytic results are proposed for a linear form of the modified Poisson-Boltzmann equation in the theory of bulk electrolytes. Comparison is also made with the mean spherical approximation results. The linear theories predict a transition of the mean electrostatic potential from a Debye-Hückel type damped exponential to a damped oscillatory behaviour as the electrolyte concentration increases beyond a critical value. The screening length decreases with increasing concentration when the mean electrostatic potential is damped oscillatory. A comparison is made with one set of recent experimental screening results for aqueous NaCl electrolytes.

A multiple decay-length extension of the Debye–Hückel theory: to achieve high accuracy also for concentrated solutions and explain under-screening in dilute symmetric electrolytes

“…24 The leading modes give rise to the oscillatory behavior at high ion densities in accordance with the scenery above. Many other approximate theories for electrolytes also predict such decay modes and the occurrence of the Kirkwood cross-over, for example the Mean Spherical Approximation (MSA), [25][26][27][28][29] the Linearized Modified Poisson-Boltzmann (LMPB) approximation by Outhwaite, 16,30,31 the Modified Debye-Hückel (MDH) approximation by Kjellander, 32 the closely related ''Local Thermodynamics'' (LT) approximation by Hall, 33 the Generalized Debye-Hückel (GDH) approximation by Lee and Fisher, 34,35 the Modified MSA by Varela and coworkers, [36][37][38][39] the charge renormalization theory by Ding et al 40 and the ionic cluster model approach by Avni and coworkers. 41 These theories, from GMSA onwards, are linear approximations, meaning that c i (r) and r i (r) are proportional to the ionic charge q i .…”

“…Most of these approximations have been used to obtain thermodynamical quantities, distribution functions and/or the mean electrostatic potential. 15,16,27,28,[31][32][33]37,39,[42][43][44][45][46][47] Xiao and Song [48][49][50] have developed a linear theory that exploits all decay modes obtained in many of these approximations to calculate thermodynamical quantities for electrolytes. We will return to some of these linear, approximate theories later.…”

“…Another approach is that of Outhwaite and Bhuiyan, 31 who have used the LMPB approximation to calculate c i (r) in manner that also shares features with the simplest version of the MDE-DH approximation in the present work. They obtain k and k 0 as the two smallest solutions of the LMPB equation for the decay parameters, whereby c i (r) is approximated for r Z 2a like in eqn (3) and obtained for r o 2a by other means.…”

“…This equation for k has also been obtained in a different manner by Hall 33 in his LT approximation and recently by Ding et al in their the charge renormalization theory. 40 It can be solved numerically to give k as function of k D and when this solution inserted into eqn (31), one obtains q eff as a function of k D . For low ion densities, where k E k D , eqn (30) agrees with the Debye-Hückel expression (11) for r i , but it differs otherwise.…”

Size and charge correlations in spherical electric double layers: a case study with fully asymmetric mixed electrolytes within the solvent primitive model

“…These include liquid state analytical theories, viz. modied PB theory, 32 integral equation theory, 33 and density functional theory, 34 and also simulation methods. [35][36][37] In recent times, a number of delicate experiments also point to the detailed structural behavior of ionic clouds of double layers around colloidal macroions.…”