An Electrical Double Layer between Spherical Particles. The Effect of Ion Sizes

Abstract

Using the modified Poisson–Boltzmann theory, which includes restrictions on the maximum attainable concentration of ionic species in the solution Cmax, determined by their effective dimensions, the spatial distributions of the electric potential and ions between spherical nanoparticles immersed in a 1 : 1 electrolyte solution have been investigated. It has been found that the pattern of the aforementioned distributions is governed by surface electric potential ψ_s of the particles, interparticle distance, and maximum concentration C_max. Between weakly charged particles, the electric potential profiles are almost independent of the value of C_max. As absolute value \(\left| {{{\psi }_{{\text{s}}}}} \right|\) increases, an effect associated with the ionic sizes manifests itself; i.e., for high \(\left| {{{\psi }_{{\text{s}}}}} \right|\) values, the electrical double layer expands into the solution, and, the lower C_max (the larger the effective size of counterions), the higher the degree of expansion. This behavior is related to the fact that, near a particle surface, the profile of counterion concentration is developed with the following distinctly pronounced regions: the region of a plateau with width l_c, which corresponds to the condensed part of the profile with C(x) = C_max for 0 < x < l_c, and the diffusion region with C(x) < C_max for x > l_c. When highly charged particles approach to each other up to such distances that their electrical double layers appear to be strongly overlapped, coions are completely displaced from the interparticle electrolyte layer, and only counterions are present in the layer.