Thermodynamics of primitive model electrolytes in the symmetric and modified Poisson-Boltzmann theories. A comparative study with Monte Carlo simulations

Abstract

Osmotic coefficients, individual and mean activity coefficients of primitive model electrolyte solutions are computed at different molar concentrations using the symmetric Poisson-Boltzmann and modified Poisson-Boltzmann theories. The theoretical results are compared with an extensive series of Monte Carlo simulation data obtained by Abbas et al. [Fluid Phase Equilib., 2007, 260, 233; J. Phys. Chem. B, 2009, 113, 5905]. The agreement between modified Poisson-Boltzmann predictions with the "exact " simulation results is almost quantitative for monovalent salts, while being semi-quantitative or better for higher and multivalent salts. The symmetric Poisson-Boltzmann results, on the other hand, are very good for monovalent systems but tend to deviate at higher concentrations and/or for multi-valent systems. Some recent experimental values for activity coefficients of HCl solution (individual and mean activities) and NaCl solution (mean activity only) have also been compared with the symmetric and modified Poisson-Boltzmann theories, and with the Monte Carlo simulations.

Activity Coefficients of Concentrated Salt Solutions: A Monte Carlo Investigation

“…The striking features, which are apparent from these figures, are that the proton activity coefficient from MC is the lowest whereas the chloride ion activity coefficient is the highest compared with the experimental values [25]. In a recent study Quinones et al [56] applied a modified Poisson-Boltzmann theory and obtained good agreement with our simulation data for single and mean activity coefficients of 1:1, 2:1 and 3:1 electrolytes. The upward bending behavior of experimentally determined activity or osmotic coefficients in sufficiently concentrated solutions is generally considered to be due to ion size effects.…”

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

“…They include the HNC approximation [51][52][53][54][55] and other integral equation theories. 54,[56][57][58][59][60][61] There are also improved versions of the PB approximation that correct for the neglect of correlations in the ion cloud around each ion, like the Modified PB approximation by Outhwaite et al 16,[62][63][64] and the correlation-enhanced PB theory by Su and coworkers. 65 Furthermore, various field theories 4,[66][67][68][69][70] have been developed for the study of bulk electrolytes.…”

“…7, where the ions are large, are calculated in this approximation. Furthermore, in the Simple MDE-DH approximation, there are simple analytical expressions for E eff r and E 0 eff r [eqn (46) and (47)] and for the thermodynamic properties: the activity coefficient g AE is given by ln g AE = ln g el AE + ln g core AE with ln g el AE from eqn (64) and ln g core AE from eqn (60) and the osmotic coefficient f = 1 + f el + f cont with f el from eqn (69) and f cont from eqn (68) [see also eqn (124)]. These expressions are valid also for complex-valued k and k 0 , but they are reformulated in Appendix C into equivalent formulas that are more easy to use for that case.…”

“…where we have made use of eqn (19) and its primed counterpart eqn (22). Using eqn (33), we can alternatively write this expression as eqn (64). When the charge density is oscillatory and k and k 0 are complex-valued, these expressions for ln g el AE can be used as they stand, but they can alternatively be written in a more convenient manner as…”

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

“…Little attention has been paid to the linear modified Poisson-Boltzmann (MPB) equation in the electrolyte solution theory, mainly due to the ready availability of the mean spherical approximation (MSA) analytical results [1]. By contrast, the non-linear MPB approach along with the hypernetted chain (HNC) [2] theory have proved two of the more successful theories in predicting the structure and thermodynamic properties of the primitive model (PM) (arbitrary sized charged hard spheres moving in a continuum dielectric) electrolyte solution [3][4][5]. It can be shown that the MSA is the linearized version of the HNC theory (see for example, the references [6,7]).…”