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 $l\ll \lambda$, where $\lambda$ is the smaller of the principal absorption wavelengths of the two dielectric media. The result obtained has the form $f\left(l\right)=-\frac{C_3}{{\left(\frac{l}{a}\right)}^3}-\frac{{\delta }^2}{a^2}\left(\frac{C_4}{{\left(\frac{l}{a}\right)}^4}+\frac{C_5}{{\left(\frac{l}{a}\right)}^5}+\cdots \right)+O\left(\frac{{\delta }^4}{a^4}\right)$ 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 $\delta$ 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 $\epsilon \left(\omega \right)=1-\left(\frac{{{\omega }_p}^2}{{\omega }^2}\right)$, where ${\omega }_p$ 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.