Velocity dependence of kinetic friction in the Prandtl-Tomlinson model

The Prandtl-Tomlinson model for friction has been used extensively for the interpretation of atomic force microscopy data during the past decade. Up to this point, the kinetic friction Fₖ has nevertheless not been studied in a range of velocities v that would be sufficiently broad to cover the crossover from the high-velocity logarithmic to the low-velocity linear Fₖ(v) dependence. This gap will be closed here through a combination of an asymptotic analysis and direct simulations of the relevant Langevin equation. The simulations span three decades in temperature T and up to six decades in v. All numerical data can be fit quite accurately with a Fₖ=a(T)arsinh[v/vc(T)] law, where the prefactor $a(T)$ scales with T2/3. Correction terms proportional to odd powers of arsinh(v/vc), only need to be included at vvc. Reasons are given as to why it is difficult to confirm meticulously the (lnv)2/3 dependence of kinetic friction predicted by recent rate theories, although they can be easily modified to produce the correct prefactor to the a(T)∝T2/3 law.