Frictional forces and Amontons' law: From the molecular to the macroscopic scale

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.