Finite difference solution of the 2-dimensional Poisson–Boltzmann equation for spheres in confined geometries

Published experimental data has shown a long-range attractive interaction between identical colloidal particles close to a charged surface. However, previous numerical calculations for a capillary geometry, which indicated that such attractions could arise from solution of the non-linear Poisson–Boltzmann equation (PBE) have been shown to be both qualitatively and quantitatively in error. The present paper uses a finite difference method for solving the PBE in 2-dimensions. Test case calculations are performed on two spheres interacting within a charged capillary. These calculations verify that, as long as the confining walls are parallel to the line joining the centres of the particles, the interaction between identical particles is always repulsive irrespective of whether or not the particles are isolated or confined in a capillary. The finite difference solution method also enables the calculation of the interaction between two spheres when there is a very small slope or protrusion on the confining walls. An apparent attractive interaction is predicted under such conditions. These calculations indicate that surface roughness (protrusion) is unlikely to be the origin of the experimentally determined attraction, but that great care is needed in the alignment of surfaces in such experiments.