Numerical Studies on Electrical Interaction Forces and Free Energy between Colloidal Plates of Finite Size

By solving the nonlinear Poisson-Boltzmann (PB) equation with a finite element method (FEM), three-dimensional (3D) spatial distributions of the electric potential (ψ, scaled) in electrolyte solutions having two charged parallel finite plates (including cubes and prismatic rods) are determined for various separations (d, scaled by the Debye length, κ^-1), surface potentials (ψ_s), and plate dimensions (length × width × thickness, each scaled by κ^-1). The total interaction force between two plates, F, is the sum of the electrostatic double-layer (EDL) repulsion (the osmotic pressure, F_osm) and the Maxwell electrostatic stress (F_es). The EDL repulsion is estimated using the distribution of ψ not only between the facing surfaces of two parallel plates but also around the other extremities of the plates. The Maxwell stress (F_es) is localized near the extremities to act as a repulsive force on the midplane between the two plates. The ratio F_es/F is 0.07-0.5, depending on d, ψ_s, and dimensions. It is found that, with increasing dimensions, the total F values per unit area calculated for finite plates, F̃, decreasingly approach the exact ones for parallel infinite plates, F̃^inf; for example, at d = 1 and ψ_s = 5, the ratio F̃/F̃^inf is 2.83 for plates with dimensions of 1 × 1 × 1 and 1.18 for plates of 10 × 10 × 1. The repulsions arising from the extremities cannot be neglected for plates with dimensions <10 × 10 × 1. Furthermore, the total interaction forces (F) are calculated at a series of discrete d values, respectively, for parallel plates. We introduce a force fitting function, F_f(d), with parameters that can be determined so that F_f(d) fits well to the calculated serial F values. By integrating the F_f(d), we obtain the interaction free energy, G(d), for finite parallel plates that consists of two Γ functions.