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In many textbooks static and kinetic friction is explained by surface roughness and interlocking asperities. However this website: http://amasci.com/miscon/miscon4.html#fric which is cited in Wikipedia:

Kinetic friction is now understood, in many cases, to be primarily caused by chemical bonding between the surfaces, rather than interlocking asperities;[23] however, in many other cases roughness effects are dominant, for example in rubber to road friction.

So Wikipedia mentions one example where the roughness theory should be correct. Nevertheless don't get the big picture, in which cases the chemical bonding theory applies and in which cases the roughness theory gives the correct explanation.

In particular I am looking for references of peer-reviewed papers and books which give a correct overview of the theory of friction with respect to my question above.

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The linked website from 1995 or 2005 is correct in that it says that "interlocking asperities" is not the universal explanation that it was once naively hoped to be (as in Coulomb's model of friction). Micro-scale asperity interlocking seems to be rare for typical surfaces that are sliding because, for example, sliding often tends to smooth the highest solid surface peaks.

But the answer to your question is that there is no known simple characterization of when roughness is more important than chemical bonding. In fact, it is known the picture is much more complex. The transmission of force between the two solids is due to many complex mechanisms, as your website alludes to. Here is an overview of some of the different mechanisms and systems. (Urbakh's review article http://dx.doi.org/10.1103/RevModPhys.85.529)

The website seems to be aware of a popular complicated model that takes into account only roughness (Persson's model doi:10.1016/j.surfrep.2006.04.001) and was first proposed for rubber and asphalt, where the roughness is large and the contact area nonetheless also large. If the surfaces are well-lubricated and the sliding speed moderate then I agree with your website that surface roughness will clearly dominate over chemical bonding.

Given that we teach friction in high school it's amazing how complex the origins are.

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Both chemical bonding at surfaces and roughness play an interconnected role in governing static friction. This is examined in some detail here. Overall, the value of mean roughness, commonly denoted as Ra, shows little correlation to frictional behaviour.

One way of looking at friction is that the junctions of true contact area are being sheared. The area of true contact depends on surface structure, and tends to vary linearly with load, across a broad range of surface fractality.

Overall the frictional behaviour is goverend by surface structure, which tends to be fractal as well as the molecular scale friction that arises from chemical bonding. As the area of contact and asperity interlocking behaviours are governed by multi-scale surface structure, it is hard to disentangle the two phenomena.

Refs

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