Frictional properties of quasi-two-dimensional materials from the Prandtl-Tomlinson model
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Tribology, the study of friction, is both an old theoretical problem in physics and an area of great practical importance. The invention of experimental instruments such as Atomic Force Microscope (AFM) has lead to the emergence of the eld of nanotribology, the exploration of friction phenomenon at the nanoscale. While more complete descriptions of friction make use of density functional theory (DFT) and molecular dynamics (MD) simulations, many essential features of frictional phenomena are accurately modeled by so called "reduced order models" such as the Prandtl-Tomlinson (PT) Model. We illustrate the PT model in both one-dimensional and two-dimensional forms via application to various crystal lattice surfaces (cubic, planar hexagonal) and reproduce important results from the literature by solving the resulting Langevin equation within the PT model. We also discuss the parameter dependence in this model via relevant simulations. We then generalize the PT model to a three-dimensional case and analyse quasi-two-dimensional systems. These systems thus exhibit a small amount of "buckling" - i.e. with out-of-plane basis atoms. The equations of motion of the Prandtl-Tomlinson model are solved numerically and the resulting friction force curves, tip path and lattice are analysed comparatively. The results agree with underlying theory and make testable predictions. We conclude that our generalized, three-dimensional PT model is a good approximation to the frictional dynamics at this scale for these systems and has the advantage of being computationally less intensive than full scale MD or DFT calculations.
Prandt- Tomlinson Model