Expansion and shrinkage of the electrical double layer in charge-asymmetric electrolytes: A non-linear Poisson-Boltzmann description

Abstract

The spatial extension or thickness of the electrical double layer determines many thermodynamic and electrokinetic properties of charged colloids in solution. In the classical Debye-Hückel formalism of point ions, the thickness of the electrical double layer around a spherical macroion or next to an infinite planar electrode can be characterized by the bulk Debye length of the supporting ionic fluid. As a result, that approach neglects, at least, the influence of the colloidal charge on the spatial extension of the ionic cloud. Given that the Debye-Hückel formalism of point ions is valid only in the limit of very weak colloidal charges, in this work we use the non-linear Poisson-Boltzmann equation to study the thickness of the electrical double layer near a positively charged electrode in spherical and planar geometries, in the presence of several binary charge-asymmetric −1:z+ point-ions electrolytes with monovalent counterions and multivalent coions. The properties of counterions are maintained fixed, whereas the properties of coions (such as their valence and concentration) are varied fulfilling the bulk electroneutrality condition. The thickness of the ionic cloud is quantified here via the recently introduced capacitive compactness idea (Phys. Chem. Chem. Phys. 20 (2018) 262). Physically, this length represents essentially the separation distance between two electrodes associated to the corresponding effective electrical double layer capacitor, in both planar and spherical geometries. Our numerical calculations show that the capacitive compactness obtained via the non-linear Poisson-Boltzmann equation reduces to the Debye length of the supporting bulk electrolyte at the point of zero charge. In the presence of a charge symmetric −1:+1 electrolyte, the capacitive compactness always decreases as a function of the colloidal charge. Contrastingly, the electrical double layer may expand or shrink as a function of the surface charge density, in the presence of multivalent coions, which is confirmed here by primitive model Monte Carlo simulations. This last non-monotonic behaviour of the capacitive compactness for multivalent coions, depending on the colloidal charge, is related to the microscopic behaviour of the local electric field, the mean electrostatic potential, and the net ionic charge per unit volume close to the colloidal surface. At very large colloidal charges, the capacitive compactness of all −1:z+ electrolytes collapses onto a single curve illustrating the dominance of counterions in the non-linear Poisson-Boltzmann theory.