The van der Waals Interaction between a Spherical Particle and a Cylinder

Abstract

Based on the Hamaker approach, this paper presents a general method to compute the retarded van der Waals (vdW) interaction potential and force between a spherical particle and a cylinder. The effects of the relative dimensions of the cylinder to the sphere were examined by this general method. First, the unretarded vdW interaction potential between these two bodies is obtained by pairwise summation of all the relevant intermolecular interactions and evaluated by accurate multiple numerical integrations. The interaction potential is then modified to account for the retardation effect by incorporating a correction factor which depends on the separation distance and the characteristic wavelength of the interactions. The numerical predictions indicate that the vdW interaction between a sphere and a finitely long cylinder can be approximated as the interaction between a sphere and an infinitely long cylinder only if the ratio of the cylinder length to its radius, B = L/R, is greater than a certain lower limit, say, B > 10. At smaller dimensionless separation distances, H = D/a ≤ 1, the vdW interaction between a sphere and a cylinder can be approximated by that between a sphere and a flat plate. However, such a commonly used flat plate approximation is found to be acceptable only if the ratio of the cylinder radius to the sphere radius, A = R/a, is larger than 10, regardless of the B value. Otherwise, it will seriously overestimate the vdW interaction for the sphere–cylinder system.