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

Abstract

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 $\ensuremath{\alpha}$. The total friction between tip and substrate is calculated as a function of contact radius $a$ and sphere radius $R$, with $a$ up to ${10}^{3}d$ and $R$ up to $4\ifmmode\times\else\texttimes\fi{}{10}^{4}d$. 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 $\ensuremath{\alpha}{a}^{2}/Rd$. In small contacts, all atoms move coherently and the total friction coefficient $\ensuremath{\mu}=\ensuremath{\alpha}$. 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 $\ensuremath{\mu}\ensuremath{\sim}\ensuremath{\alpha}{(\ensuremath{\alpha}{a}^{2}/Rd)}^{\ensuremath{-}2/3}$. 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 $\ensuremath{\mu}$ rises with increasing load.