Statistics of Sliding on Periodic and Atomically Flat Surfaces - Srbulovic, Maja; Gkagkas, Konstantinos; Gachot, Carsten; Vernes, András (Frontiers in Mechanical Engineering, 2021)
Among the so-called analytical models of friction, the most popular and widely used one, the Prandtl-Tomlinson (PT) model in one and two dimensions is considered here to numerically describe the sliding of the tip within an atomic force microscope (AFM) over a periodic and atomically flat surface. Because in these PT-models, the Newtonian equations of motion for the AFM-tip are Langevin-type coupled stochastic differential equations (SDEs), the resulting friction and reaction forces must be statistically correctly determined and interpreted. For this, it is firstly shown that the friction and reaction forces as averages of the time-resolved ones over the sliding part, are normally (Gaussian) distributed. Then based on this, an efficient numerical scheme is developed and implemented to accurately estimate the means and standard deviations of friction and reaction forces without performing too many repetitions for the same sliding experiments. The used corrugation potential is the simplest one obtained from the Fourier series expansion of the two-dimensional (2D) periodic potential, e.g., for an fcc(111) surface, which permits sliding on both commensurate and incommensurate paths. In this manner, it is proven that the PT-models predict both frictional regimes, namely the structural superlubricity and stick-slip along (in)commensurate sliding paths, if the ratio of mean corrugation and elastic energies is properly set.