Size dependence of static friction between solid clusters and substrates

The existence of the static friction force Fs implies that two surfaces in contact are locked into a free energy minimum. The microscopic origin of this minimum can be easily understood if considering commensurate interfaces where all the slider particles occupy the minima of the substrate potential. On the contrary, in incommensurate interfaces the particle arrangement on the substrate gives rise to an interaction potentialbetweenthesurfaceswhichisflatexceptforboundary effects. 1,2 One proposed explanation for the origin of static friction in macroscopic interfaces is that small molecules, the so-called “third bodies,” adsorbed on surfaces exposed to air can arrange to lock together the contacting surfaces. 3 Yet some observed frictional phenomena remain puzzling, even at the nanoscale. An example is provided by the stick-slip motion of tips sliding on crystalline substrates: The recorded friction force maps usually present the substrate periodicity independently from the tip structure (often amorphous) and environmental conditions, including vacuum. In this Rapid Communication we describe the atomistic mechanisms that determine the effective commensurability of two surfaces in contact, which we show can differ highly from the geometrical commensurability of the separated surfaces. We consider monolayer (ML) islands of krypton on copper. This idealized system has been studied experimentally by means of a quartz crystal microbalance (QCM). 4 We use molecular dynamics simulations based on a first-principles derived potential for the film-substrate interaction. A direct visualization of the particle positions relative to the minima of the substrate potential allowed us to resolve domain structures. Domains are formed in incommensurate interfaces becauseofthecompetitioninminimizingtheinterfacialenergy and elastic strain energy. When the size of the contact is reduced below a critical radius Rc, domains coalesce and the interface becomes commensurate. This structural transition is accompanied by a sharp increase of static friction. It is interesting to notice that the complementary process, i.e., the depinning of a commensurate interface, occurs by the nucleation and growth of domains. 5 We verify that the sizeinducedfrictionalappearanceoccursalsointhree-dimensional systemswithcovalentbondsbyconsideringclustersofsilicon. We describe these results within the nucleation theory and we derive an analytical expression for the critical dimension. It