Van der Waals attraction between cylinders, rods or fibers

Abstract

In a first step, we calculate the van der Waals energy between two cylinders by pairwise integration of unscreenedr−6 interactions between any two molecules. It turns out proportional tod−3/2 at small separations and proportional tod−5 at large separationsd of the cylinders. In a second step, we use an integration method for multiplet interactions, which relates the latter to screening and represents the van der Waals energy by macroscopic reaction fields. We expand these reaction fields in terms of modified Bessel functions. The van der Waals energy evolves from a sum over the numberl of field reflections, from a frequency integral over the dielectric constants involved, and from a wave number integral over the radii and the separation of the cylinders. The lowest order term verifies the results found by integration of pair interactions, yet replaces the unscreened polarizabilities of the atoms by the screened dielectric constants of the media. The higher order reflection termsl≧2 likewise turn out to be proportional tod−3/2 at small separationsd, but decrease in weight more rapidly than 1/l3. Their contribution at large separation is proportional tod−(4l+1). From a comparison of our results with those obtained for spheres and half-spaces, we conclude that retardation entails ad−5/2 and ad−6 law at small and large separations, respectively. This suggestion is confirmed by preliminary calculations based on the Helmholtz equation.