The contact area and sliding friction in nanotribological systems

Although friction, wear, and lubrication have been investigated for hundreds of years, many important questions in the field, now dubbed tribology, remain open. Two of these are the determination of the real area of contact and the direction dependence of friction forces on the nanoscale, which are investigated in this thesis. The invention of the atomic force microscope (AFM) in 1986 by Binnig , Quate, and Gerber was invaluable for the investigation of frictional forces on the atomic scale, giving a boost to the emerging field of nanotribology. While experimental AFM set-ups where quickly modeled with classical molecular dynamic (MD) simulations, the use of 'ab-initio' methods on tribological problems remained scarce until recently. In this thesis density functional theory (DFT) is used to develop parameter free methods in the field of nanotribology. Since the discovery that the apparent area of contact can be orders of magnitude larger than the true area of contact, the definition and determination of the latter has been an important topic of research. While classical contact mechanics provides satisfactory theories and results for various macroscopic systems, the application of these methods to atomistic systems is dubious, as the contacting bodies are not continuous at this length scale. We developed a parameter free approach to define and calculate the real area of contact between two bodies depending on distance. Strong relaxations at distinct distances, like the ^jump to contact^, which is often observed in AFM experiments, are used to define the onset of contact and Bader-s Quantum Theory of Atoms in Molecules is used to calculate the real area of contact at a given distance. Bader's method partitions the charge density ρ unambiguously into atoms, which theoretically fill all space and thus give non zero contact areas for all distances. It is therefore necessary to use a density cutoff ρ_cut which assigns all regions in space where ρ < ρ_cut to the vacuum, resulting in surface atoms of finite size. In our proposed method the parameter ρ_cut is calculated by allowing contact only after the 'jump to contact', or a similar discontinuity, has taken place, which can be clearly observed in our DFT simulations. We demonstrate the method by lowering a ten atom tungsten pyramid, which serves as a model of an AFM tip, onto a smooth surface. Two systems are examined, the first being moiré graphene on iridium (111), the second a clean copper (111) surface. Although the surfaces are very different, a similiar cutoff parameter ρ_cut of about 5 x 10^-2 electrons per ų is computed in both cases. The calculated area of contact is found to increase linearly with lowering of the tip support while increasing exponentially with the true relaxed distance between the tip apex atom and the surface atom below it. Although there exist a number of models that tackle the problem of calculating friction forces on the atomic level, providing a completely parameter-free approach remains a challenge. To examine the direction dependence of dry sliding friction we developed a quasi-static grid method with a mechanism to allow dissipative sliding, which relies on atomic relaxations. We define two different ways of calculating the mean nanofriction force, both leading to an exponential friction-versus-load behavior for all sliding directions. Since our approach does not impose any limits on lengths and directions of the sliding paths, we investigate arbitrary sliding directions for several metal interfaces and detect two periodic paths which form the upper and lower bound of nanofriction in all cases. For long aperiodic paths the friction force convergences to a value in between these limits. For low loads we retrieve the Derjaguin generalization of the Amontons-Coulomb kinetic friction law which appears to be valid all the way down to the nanoscale. We observe a non-vanishing Derjaguin-offset even for atomically flat surfaces in dry contact. Incorporating our approach to determine the true area of contact into the evaluation of the loading force leads to higher loads for each selected pressure. This influences the friction versus load curves by stretching them laterally and thus reducing the coefficients of friction by about 25% - 35%. (author)