Analytical and numerical computations of the van der Waals force in complex geometries: Application to the filtration of colloidal particles

Particle capture during the filtration of colloidal dispersions depends on a complex balance between repulsive forces, such as hydrodynamic or electrostatic effects, and attractive forces, amongst them the van der Waals interaction forces. Satisfactory expressions for the latter are thus required in complex geometries. Exact expressions for the geometrical factor involved in the van der Waals interaction energy based on Hamakerʹs additivity hypothesis are derived for a sphere in interaction with a square wedge, a semi-infinite or finite slit, a semi-infinite slab, a 2D pillar, a rectangular rod, a corner and a rectangular channel. A numerical tool based on an adaptive mesh refinement strategy is presented and used to validate the analytical results. The analytical result for a sphere/wedge system is used to assess the domain of applicability of the sphere/plane model in the vicinity of the edge. The interaction between a sphere and a cylindrical pore in a plate of finite thickness is then simulated and the range of validity of the sphere/wedge system as a model of the sphere/pore system is deduced from the numerical results.