Approximate Scheme for Calculating van der Waals Interactions between Finite Cylindrical Volume Elements

A successful approach to calculating van der Waals (vdW) forces between irregular bodies is to divide the bodies into small cylindrical volume elements(1-3) and integrate the vdW interactions between opposing elements. In this context it has been common to use Hamaker's expression for parallel plates to approximate the vdW interactions between the opposing elements.(1,3) This present study shows that Hamaker's vdW expression for parallel plates does not accurately describe the vdW interactions for co-axial cylinders having a ratio of cylinder radius to separation distance (R/D) of 10 or less. This restricts the systems that can be simulated using this technique and explicitly excludes consideration of topographical or compositional variations at the nanoscale for surfaces that are in contact or within a few nm of contact. To address this limitation, approximate analytical expressions for nonretarded vdW forces between finite cylinders in different orientations are derived and are shown to produce a high level of agreement with forces calculated using full numerical solutions of the corresponding Hamaker's equations. The expressions developed here allow accurate calculation of vdW forces in systems where particles are in contact or within a few nm of contact with surfaces and the particles and/or surfaces have heterogeneous nanoscale morphology or composition. These calculations can be performed at comparatively low computational cost compared to the full numerical solution of Hamaker's equations.