Analytical expressions for the kinetic friction in the Prandtl-Tomlinson model

Analytical expressions are presented for the motion of a point mass driven on a periodic potential by a lateral spring, approximating an atomically sharp tip rubbing over a crystal surface. The tip position before and after jumping between two consecutive minima, the kinetic friction force ${F}_{\mathrm{kin}}$, and the energy barrier $\ensuremath{\Delta}E$ preventing the tip jumps are expressed in terms of a parameter $\ensuremath{\eta}$, which is the ratio between the maximum curvature of the substrate potential and the elastic spring constant. In the two limiting cases of $\ensuremath{\eta}\ensuremath{\rightarrow}1$ (superlubric transition) and $\ensuremath{\eta}\ensuremath{\gg}1$, we demonstrate that ${F}_{\mathrm{kin}}\ensuremath{\propto}{(\ensuremath{\eta}\ensuremath{-}1)}^{2}$ and ${F}_{\mathrm{kin}}\ensuremath{\propto}\ensuremath{\eta}$, respectively. We also show that the relation $\ensuremath{\Delta}E\ensuremath{\propto}{({F}_{c}\ensuremath{-}F)}^{3/2}$, which is valid in the case of a strong interaction, is replaced by $\ensuremath{\Delta}E\ensuremath{\propto}{(F\ensuremath{-}{F}_{c})}^{3}$ close to the superlubric transition. The proportionality coefficients are determined in all cases. The case of multiple jumps is also studied, and numerical results reproducing the influence of the finite temperature and the viscous damping accompanying the tip slippage are shown.