Three-dimensional Prandtl-Tomlinson model of nanoscale friction

Friction is a complex phenomenon that involves interaction between microsized asperities on the surfaces of two bodies in contact. Friction force microscopy allows investigating friction forces that arise at the level of a single asperity, which is the tip of an atomic force microscope (AFM). In this thesis, the Prandtl-Tomlinson model of single-asperity friction is developed that fully incorporates the three-dimensional character of the problem. An algorithm is derived that allows integrating the resulting equations of motion. Special care is taken to select the model parameters close to the values that can be deduced from the experimental results published in the literature. The effect of periodic actuation of the AFM cantilever on the resulting friction forces is studied within this model. Three actuation modes are considered: transverse, normal, and lateral. Transverse actuation has no effect whatsoever on the friction force, whereas the effect of normal actuation is somewhat weaker than the effect of lateral actuation. Due to the finite mass of the cantilever, its motion may proceed in many different regimes which depend on the actuation amplitude and frequency. Hence, the dependence of friction force on actuation frequency at fixed amplitude and on the actuation amplitude at fixed frequency is obtained numerically. Both dependences are not simple and consist of a series of irregular peaks. Those peaks are more pronounced at zero temperature than at room temperature. Finally, the limitations of the model are discussed. It is suggested that two effects must be incorporated into the model in order to provide a more realistic picture of the nanoscale friction: the finite elasticity of the apex of the cantilever tip and the aging of the tip-sample contact