A modified Poisson–Boltzmann study of the singlet ion distribution at contact with the electrode for a planar electric double layer

Abstract

The properties of the singlet ion distributions at and around contact in a restricted primitive model double layer are characterized in the modified Poisson-Boltzmann theory. Comparisons are made with the corresponding exact Monte Carlo simulation data, the results from the Gouy-ChapmanStern theory coupled to an exclusion volume term, and the mean spherical approximation. Particular emphasis is given to the behaviour of the theoretical predictions in relation to the contact value theorem involving the charge profile. The simultaneous behaviour of the coion and counterion contact values is also examined. The performance of the modified Poisson-Boltzmann theory in regard to the contact value theorems is very reasonable with the contact characteristics showing semi-quantitative or better agreement overall with the simulation results. The exclusionvolume-treated Gouy-Chapman-Stern theory reveals a fortuitous cancellation of errors, while the mean spherical approximation is poor. KeywordsElectric double layer; Contact value theorems; Density profile; Charge profile; Monte Carlo Alongside exact numerical simulations, exact analytical conditions are invaluable in statistical mechanics of Coulomb fluids in comparative assessment of different theories and/ or further theoretical development. These exact conditions can also help provide additional, useful checks on the consistency of simulations. One such condition in the electric double layer phenomenon is our principal concern in this paper. An electric double layer is formed when a charged electrode is brought in contact with a charged fluid and an ionic atmosphere develops in the vicinity of the electrode. The phenomenon has practical significance for a spectrum of systems in biology and industrial chemical processes. Here ρ s , d s and g s are, respectively, the average number density, the diameter and the electrode-ion distribution function of ionic species s, p is the bulk osmotic pressure and σ is the uniform surface charge density on the electrode with ε 0 and ε r , respectively, being the vacuum permittivity and relative permittivity of the solvent. Also, k B is the Boltzmann constant and T the absolute temperature. The relation was derived from force balance considerations at the electrode-electrolyte interface. It is local, easy to implement and, as a consequence, has been useful in checking theoretical descriptions of the electric double layer over the past three decades 1 .The other, relatively recent condition concerns the contact value of the charge profile in the planar double layer. This was derived by Holovko, Badiali and di Caprio4,5 (HBC), and Holovko and di Caprio6 starting from the Bogoliubov-Born-Green-Yvon (BBGY) hierarchy of equations7. For a double layer containing a restricted primitive model (RPM) electrolyte (the ions of the PM are now restricted to having a common diameter), and for symmetric valency salts -our interest in this paper, their relation is particularly simple, viz.,where g sum (x) = (1/2)(g ctr (x) + g co (x)), g diff (x...

Charge reversal and surface charge amplification in asymmetric valence restricted primitive model planar electric double layers in the modified Poisson-Boltzmann theory

“…The MPB theory does not satisfy exactly the contact condition [60][61][62] which holds for no imaging. Previous work [63][64][65] has shown that the MPB5 theory fairly accurately satisfies the contact condition for 1:1 electrolytes and gives a good agreement for 1:2/2:1 and 2:2 electrolytes. A similar conclusion holds in general when the new F is used, but the MPB4 theory can be poor.…”

Evidence for a second, local contact condition for a planar electric double layer containing size and valency asymmetric ions

“…The MC data is again well described by Equation (17). Figure 9 shows an overall view of the changing pattern of asymmetry in the double layer through the plots of MC g diff ðct, bÞ, the GCS (or the PB) g diff ðct, bÞ, and the HB relation [Equation (17)]. Although some of the individual results have been observed in previous publications [12,21,22], it is revealing to have a global view.…”

“…The essential behaviour of the results follow the earlier patterns in Figures 1-4, although perhaps slightly less pronounced because of the lower densities. The MC data is again well described by Equation (17). Figure 9 shows an overall view of the changing pattern of asymmetry in the double layer through the plots of MC g diff ðct, bÞ, the GCS (or the PB) g diff ðct, bÞ, and the HB relation [Equation (17)].…”

“…However, the precise form of this dependence is unclear. At any rate, the fact that the form of g diff ðct, bÞ given by Equation (17) suffices to describe the electrode charge dependence of this quantity at low electrode charge, is the more important result. Another useful improvement to the HB expression would be an extension to higher values of b apropos of the usual range of b used in electrochemical experiments.…”

“…On the other hand Equation (10) is easy to apply and is also consistent within the numerical uncertainties of the simulations, although admittedly, the equation only holds at low charge densities (small b) on the electrode. The modified Poisson-Boltzmann (MPB) theory [16] has also been applied vis-a`-vis to the two relations [12,17]. Henderson and Bhuiyan [18], and Silvestre-Alcantara et al [19] have further considered a semi-empirical extension of Equation (10) to large charge densities (large b), namely,…”