Effects of surface roughness on the van der Waals force between macroscopic bodies. II. Two rough surfaces

Abstract

We extend the method used in an earlier paper by the present authors to calculate here the effect of surface roughness on the van der Waals force between two different, semi-infinite dielectric media, separated by a region of vacuum of nominal thickness $l$, when both surfaces of the two media are rough. The result obtained has the form $f(l)=\ensuremath{-}\frac{{c}_{3}}{{(\frac{l}{a})}^{3}}\ensuremath{-}(\frac{{\ensuremath{\delta}}^{2}}{{a}^{2}})[\frac{{c}_{4}}{{(\frac{l}{a})}^{4}}+\frac{{c}_{5}}{{(\frac{l}{a})}^{5}}+\ensuremath{\cdots}]+O(\frac{{\ensuremath{\delta}}^{4}}{{a}^{4}})$ in the limit $\frac{l}{a}$ is large. Here $a$ is the transverse correlation length, the mean distance between consecutive peaks and valleys on the rough surface, while $\ensuremath{\delta}$ is the root-mean-square departure of the surface from flatness. For simplicity, $a$ and $\ensuremath{\delta}$ are assumed to be the same for both surfaces. The terms in the series multiplying the factor ($\frac{{\ensuremath{\delta}}^{2}}{{a}^{2}}$) have been calculated through terms of $O({(\frac{l}{a})}^{\ensuremath{-}7})$ for three different assumptions about the correlation between the roughness profile functions on the two surfaces. In addition, in the Appendix we show that a simple effective medium model of surface roughness, that has previously been shown to reproduce accurately the effects of surface roughness on the image potential and on the surface-plasmon dispersion curve, also yields the leading term in the roughness-induced contribution to the van der Waals force for large ($\frac{l}{a}$) with semiquantitative accuracy.