Friction Laws for Elastic Nano-Scale Contacts

– The effect of surface curvature on the law relating frictional forces F with normal load L is investigated by molecular dynamics simulations as a function of surface symmetry, adhesion, and contamination. Curved, non-adhering, dry, commensurate surfaces show a linear dependency, F ∝ L, similar to dry flat commensurate or amorphous surfaces and macroscopic surfaces. In contrast, curved, non-adhering, dry, amorphous surfaces show F ∝ L similar to friction force microscopes. In our model, adhesive effects are most adequately described by the Hertz plus offset model, as the simulations are confined to small contact radii. Curved lubricated or contaminated surfaces show again different behavior; details depend on how much of the contaminant gets squeezed out of the contact. Also, it is seen that the friction force in the lubricated case is mainly due to atoms at the entrance of the tip. Introduction. – Amontons law, which connects the frictional force F between two solids in relative motion linearly with the normal load L, was suggested more than 300 years ago. This law has proven applicable since for many sliding interfaces [1, 2]. Nevertheless, it is still discussed controversially whether or not Amontons’ law is valid on the micrometer or the nanometer scale as well, e.g., whether it holds for individual asperity contacts. If a contact deforms plastically, there is a simple popular, yet, phenomenological argument why this should be the case [3]: The local normal pressure p⊥ in the contact is everywhere close to the yield pressure py and the shear stress of the junction is limited through the yield stress σc. Omitting adhesive effects, this results in a static friction coefficient of μs = σs/py. μs is commonly defined as the ratio of the force Fs needed to initiate sliding and L. This argument is of rather limited predictive power for various reasons, e.g., py depends strongly on the size of the asperities in contact [2]. Furthermore, it can not be applied to elastic, wearless friction, which is the subject of many friction force microscope (FFM) [4–6] and surface force apparatus (SFA) [7, 8] experiments. In the following we will focus on wearless, elastic friction. Even in the elastic regime, SFA and FFM experiments are often consistent with the interpretation of a yield stress σc that is (relatively) independent of the normal pressure p⊥, e.g., very careful SFA experiments observe that F is mainly proportional to the real area of contact Ac [9]. Similarly, strong deviations from Amontons’ laws are observed in FFM. If the FFM tip has a reasonably well-defined shape and adhesive forces are included into the load L in terms of a so-called Hertz-plus-offset model, F ∝ L is observed [5]. This is again consistent with the assumption of a normal-pressure independent value of σc. However, simulations of