  We use molecular simulations to study the nonadhesive and adhesive
atomic-scale contact of rough spheres with radii ranging from nanometers to
micrometers over more than ten orders of magnitude in applied normal load. At
the lowest loads, the interfacial mechanics is governed by the contact
mechanics of the first asperity that touches. The dependence of contact area on
normal force becomes linear at intermediate loads and crosses over to Hertzian
at the largest loads. By combining theories for the limiting cases of nominally
flat rough surfaces and smooth spheres, we provide parameter-free analytical
expressions for contact area over the whole range of loads. Our results
establish a range of validity for common approximations that neglect curvature
or roughness in modeling objects on scales from atomic force microscope tips to
ball bearings.
