The 2/3 power law dependence of capillary force on normal load in nanoscopic friction

During the sliding of an atomic force microscope (AFM) tip on a rough hydrophilic surface, water capillary bridges form between the tip and the asperities of the sample surface. These water bridges give rise to capillary and friction forces. We show that the capillary force increases with the normal load following a 2/3 power law. We trace back this behavior to the load induced change of the tip-surface contact area which determines the number of asperities where the bridges can form. An analytical relationship is derived which fully explains the observed interplay between humidity, velocity, and normal load in nanoscopic friction.