Effects of surface roughness on the van der Waals force between macroscopic bodies

Abstract

We have calculated the van der Waals force between two different, semi-infinite dielectric media, separated by a region of vacuum of nominal thickness $l$, when the surface of one of the two media is rough. The calculation is carried out by the method of van Kampen, Nijboer, and Schram [Phys. Lett. 26A, 307 (1968)] to yield the force at the absolute zero of temperature in the regime $l\ensuremath{\ll}\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the smaller of the principal absorption wavelengths of the two dielectric media. The result obtained has the form $f(l)=\ensuremath{-}\frac{{C}_{3}}{{(\frac{l}{a})}^{3}}\ensuremath{-}\frac{{\ensuremath{\delta}}^{2}}{{a}^{2}}\left(\frac{{C}_{4}}{{(\frac{l}{a})}^{4}}+\frac{{C}_{5}}{{(\frac{l}{a})}^{5}}+\ensuremath{\cdots}\right)+O\left(\frac{{\ensuremath{\delta}}^{4}}{{a}^{4}}\right)$ 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. Explicit expressions have been obtained for the coefficients ${C}_{4}$ and ${C}_{5}$, and numerical estimates of the magnitude of the roughness-induced contribution to the van der Waals force are obtained in the case that the two media are the same, and their common dielectric constant is given by $\ensuremath{\epsilon}(\ensuremath{\omega})=1\ensuremath{-}(\frac{{{\ensuremath{\omega}}_{p}}^{2}}{{\ensuremath{\omega}}^{2}})$, where ${\ensuremath{\omega}}_{p}$ is a plasma frequency. It is found that surface roughness increases the magnitude of the van der Waals force over its value when the surfaces of both media are flat.