Ion and potential distributions in charged and non-charged primitive spherical pores in equilibrium with primitive electrolyte solution calculated by grand canonical ensemble Monte Carlo simulation. Comparison with generalized Debye–Hückel and Donnan theory

Abstract

Grand canonical ensemble Monte Carlo simulations (GCEMC) have been performed for dilute to moderately concentrated restricted primitive model electrolytes (1 : 1) in equilibrium with spherical micropores with radii from 1.5 to 10 times the ionic diameter. The pores are primitive : hard walls, with the same relative permittivity as the pore fluid and a smeared out fixed charge on the walls. The fixed charge is set to zero, one or five elementary charges. The constraining chemical potentials in the bulk solution are found by canonical Monte Carlo simulations by HNC calculations.

The following topics are emphasized : 1, The need to move ions independently without regard to electroneutrality. 2, The deviation from electroneutrality in isolated small pores. 3, Electroneutrality may be artificially induced by the application of an overall ‘Donnan potential’. 4, Electroneutrality is a collective phenomenon of a ensemble of many pores. 5, Average mean activity coefficients in the pores depend slightly on the total applied electric potential for the dilute solutions corresponding to electrosorption. The dependence is even more pronounced for the average single ion activity coefficients. 6, Except in dilute solutions, the mean ionic singlet distribution functions G_±(r) are unaffected by the presence of the wall charge. 7, G_± near the wall is a compromise between the hard sphere ‘piling up’ at high concentrations (an ionic diameter effect) and the tendency of the ions at lower concentrations to avoid the zone, where symmetric ionic clouds cannot be formed (a Debye length effect). The latter effect points to the need for a generalisation of the simple Onsager–Samaras theory of surface adsorption and surface tension due to image charges. 8, The electric potential may be effectively found and smoothed by a Poisson equation integration of the GCEMC data for G₊ to G_–. 9, At low concentrations, a simple analytic generalisation of the Debye–Hückel theory explains well the observed potential distribution. 10, At higher concentrations, the potential distribution may still be well fitted by one or two eigenfunctions of the Laplace operator. 11, There is evidence of a quasi-crystalline structure induced by the wall charge in the middle of large pores at higher ionic concentrations. 12, The present simulation method may be used also for simulation of bulk properties of electrolytes without introduction of periodic boundary conditions, since bulk densities are obtained in a great volume fraction in a pore with radius only ca. 10 times the ionic diameter.

We discuss some implications of the present model calculations and future generalisations for the theory of real desalination membranes with an alveolar structure of the skin layer and for the theory of ion-exchange membranes.