The distance upon contact: Determination from roughness profile

The point at which two random rough surfaces make contact takes place at the contact of the highest asperities. The distance upon contact d_0 in the limit of zero load has crucial importance for determination of dispersive forces. Using gold films as an example we demonstrate that for two parallel plates d_0 is a function of the nominal size of the contact area L and give a simple expression for d_0(L) via the surface roughness characteristics. In the case of a sphere of fixed radius R and a plate the scale dependence manifests itself as an additional uncertainty \delta d(L) in the separation, where the scale L is related with the separation d via the effective area of interaction L^2\sim\pi Rd. This uncertainty depends on the roughness of interacting bodies and disappears in the limit L\to \infty.