Finite thickness and charge relaxation in double-layer interactions

We extend the classical Gouy-Chapman model of two planar parallel interacting double layers, which is used as a first approximation to describe the force between colloidal particles, by considering the finite thickness of the colloids. The formation of two additional double layers due to this finite thickness modifies the interaction force compared to the Gouy-Chapman case, in which the colloids are semi-infinite objects. In this paper we calculate this interaction force and some other size-dependent properties using a mean-field level of description, based on the Poisson-Boltzmann (PB) equation. We show that in the case of finite-size colloids, this equation can be set in a closed form depending on the geometrical parameters and on their surface charge. The corresponding linear (Debye-Hückel) theory and the well-known results for semi-infinite colloids are recovered from this formal solution after appropriate limits are taken. We use a density functional corresponding to the PB level of description to show how in the case where the total colloidal charge is fixed, it redistributes itself on their surfaces to minimize the energy of the system depending on the aforementioned parameters. We study how this charge relaxation affects the colloidal interactions.