Understanding the friction laws of Amontons and Coulomb by evaluating the real contact area

Friction is a complex phenomenon that depends on many parameters. Despite this, we still rely on and describe friction in the vast majority cases with a single value, namely the coefficient of friction (µ), as first proposed by Amontons in 1699. Later, Coulomb introduced a two-parameter description by separating the adhesive and load-dependent terms. However, experimental evidence that determines under what conditions either of the two historic models is more appropriate has not been investigated in detail. In particular, to take full advantage and achieve better accuracy with the two-parameter equation, the real contact area must be well characterized to determine the constant adhesive component sufficiently accurately. In this study we performed sliding experiments and measured the friction, but at the same time we also measured the real contact area with sub-micron lateral resolution, which allowed us to design a two-parameter (Coulomb-type) friction equation. We compared these results with the historical friction models of Amontons and Coulomb to better understand the actual differences between them, and how this corresponds to the linearity between the friction and the normal force, which is a key assumption in the more common and simpler Amontons relation. A strong scaling effect of roughness was observed, as well as the related non-linearity between the normal load, the friction and the real contact area. Under high loads and roughnesses, the one- or two-parameter friction descriptive models differed from experiments by only a few percent and the variation among them was small in the same range. However, for very low roughnesses and loads (close to the nano-scale region), the two-parameter Coulomb model was required for any relevant friction prediction due to the strong adhesive contribution, while the one-parameter description was not appropriate.