Scale- and load-dependent friction in commensurate sphere-on-flat contacts

Contact of a spherical tip with a flat elastic substrate is simulated with a Green's-function method that includes atomic structure at the interface while capturing elastic deformation in a semi-infinite substrate. The tip and substrate have identical crystal structures with nearest-neighbor spacing d and are aligned in registry. Purely repulsive interactions between surface atoms lead to a local shear strength that is the local pressure times a constant local friction coefficient α. The total friction between tip and substrate is calculated as a function of contact radius a and sphere radius R, with a up to 103d and R up to 4×104d. Three regimes are identified depending on the ratio of a to the core width of edge dislocations in the center of the contact. This ratio is proportional to αa2/Rd. In small contacts, all atoms move coherently and the total friction coefficient μ=α. When the contact radius exceeds the core width, a dislocation nucleates at the edge of the contact and rapidly advances to the center where it annihilates. The friction coefficient falls as μ∼α(αa2/Rd)−2/3. An array of dislocations forms in very large contacts and the friction is determined by the Peierls stress for dislocation motion. The Peierls stress rises with pressure, and μ rises with increasing load.