Analytical Expression for the Atomic Friction of a Single Asperity Based on the Prandtl–Tomlinson Model

The Prandtl–Tomlinson (PT) model has been widely applied to interpret the atomic friction mechanism of a single asperity. In this study, we present an approximate explicit expression for the friction force in the one-dimensional PT model under quasi-static conditions. The ‘stick–slip’ friction curves are first approximated properly by sawtooth-like lines, where the critical points before and after the ‘slip’ motion are described analytically in terms of a dimensionless parameter η. Following this, the average friction force is expressed in a closed form that remains continuous and valid for η > 1. Finally, an analytical expression for the load dependence of atomic friction of a single asperity is derived by connecting the parameter η with the normal load. With the parameters reported in experiments, our prediction shows good agreement with relevant experimental results.