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

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≪λ, where λ is the smaller of the principal absorption wavelengths of the two dielectric media. The result obtained has the form f(l)=-C₃/(l/a)³-δ²/a²(C₄/(l/a)⁴+C₅/(l/a)⁵+⋯)+O(δ⁴/a⁴) in the limit l/a is large. Here a is the transverse correlation length, the mean distance between consecutive peaks and valleys on the rough surface, while δ is the root-mean-square departure of the surface from flatness. Explicit expressions have been obtained for the coefficients C₄ and C₅, 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 ε(ω)=1-(ω_p²/ω²), where ω_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.