We review the historical and modern understanding of the most basic equation of friction, Amontons' law, which describes phenomena that were already understood and studied by Leonardo da Vinci 500 years ago. This law states that for any two materials the (lateral) friction force is directly proportional to the (normal) applied load, with a constant of proportionality, the friction coefficient, that is constant and independent of the contact area, the surface roughness, and the sliding velocity. No theory has yet satisfactorily explained this surprisingly general law; all attempts have been model or system dependent. We review the experimental evidence and find, for example, that the same friction coefficient is often measured for the same system of materials with junctions whose areas differ by more than 6 orders of magnitude. The trends obtained through molecular dynamics (MD) simulations agree with recent and past experiments and with Amontons' law, and they suggest that the local energy-dissipating mechanisms are not merely "mechanical", as assumed in most models, but "thermodynamic" in nature, like miniature irreversible compression-decompression cycles of the trapped molecules between the surface asperities as they pass over each other. The MD analysis reveals that, for such dynamic, nonequilibrium, energy-dissipating processes, a proper statistical description can be formulated through the use of the Weibull distribution of the local friction forces, which may be regarded to serve in this context a similar purpose as the Boltzmann distribution for classical systems at equilibrium. Another important conclusion is that the concept of the "real" area of contact is a nonfundamental quantity, whether at the nano-, micro-, or macroscale. However, it may serve as a convenient scaling parameter for describing the really fundamental parameters, which are the number density of atoms, molecules, or bonds involved in an adhesive or frictional interaction. Brief History of the Concept of the "Coefficient of Friction"

Direct and simultaneous measurements of the normal and lateral forces encountered by a nanosize spherical silicon tip approaching a solid surface in purified water are reported. For tip-surface distances, 0 ± 0.03 nm Ͻ d Ͻ 2 nm, experiments and grand canonical molecular-dynamics simulations find oscillatory solvation forces for hydrophilic surfaces, mica and glass, and less pronounced oscillations for a hydrophobic surface, graphite. The simulations reveal layering of the confined water density and the development of hexagonal order in layers proximal to a quartz surface. For subnanometer hydrophilic confinement, the lateral force measurements show orders of magnitude increase of the viscosity with respect to bulk water, agreeing with a simulated sharp decrease in the diffusion constant. No viscosity increase is observed for hydrophobic surfaces.

Using grand-canonical molecular dynamics simulations and free energy calculations of spherical molecular confined films, we investigate the internal energy and entropic origins of density layering and solvation force oscillations. We show that these properties, as well as diffusion and rheological characteristics of such films, depend on interfacial commensurability, with commensurate films exhibiting abrupt solidification for relatively thick, 5 or 6 layer, films.