# Can the coefficient of friction be derived from fundamentals?

It is common to want to derive macroscopic laws from what we know microscopically - after all, given a (correct) microscopic description, everything larger should follow.

Has it ever been done to calculate a coefficient of static/kinetic friction simply from fundamental properties of materials? (say, the crystal structure or intermolecular forces). I searched for some time on the internet and found nothing (although my results were saturated by classical physics problems).

If it has been done, what was the general strategy?

• No. Friction is a thermodynamic force, so it depends on the thermodynamics of the process that causes it, and that's not universal. One can make friction forces so large by simple patterning of the surface that the materials will break or abrade before the surfaces will start moving against each other. That's how tools like a file work. May 1 '15 at 19:25
• The coefficient of friction $\mu$ is not for a specific material. It is a combined property for the two surfaces touching - a property of the current system. If you wish to look at roughness of one material alone, have a look at the $R_a$ value e.g., which is most commen, or any of the other mathematical means of describing that. May 1 '15 at 19:52
• @CuriousOne I thought there were only four fundamental forces. 'Thermodynamic Force' is a new one to me. I would have guessed friction is fundamentally an electromagnetic force. And there is another option over breaking or shearing - you can have elastic yielding. May 1 '15 at 20:15
• I'll bet that if you thought long and hard enough about it there might be a an approach to derive a model that would lead to a statistical result. The 'coefficient of friction' comes from a highly idealized, linear model. There are nonlinear models that provide much better predictions of frictional forces, such as the Dahl Model addressed among others in the paper here: mate.tue.nl/mate/pdfs/11194.pdf . But these models were not derived from the fundamentals. May 1 '15 at 20:21
• Yes, it has been done. See First-principles theory of atomic-scale friction Phys. Rev. Lett. 64, pg. 3054 – Published 18 June 1990 May 1 '15 at 20:31

## 2 Answers

Fundamentally, this is no different from computing the friction in a fluid (shear viscosity). The theory of viscosity goes back to Maxwell and Boltzmann, and microscopic calculations are possible for many fluids. Solid friction is more complicated, because the exact preparation of the surface obviously matters. First principles theories therefore concentrate on idealized crystalline surfaces, see for example these two papers. These days, systems like this can be studied, and theories tested, using nano-scale devices. This is field is known as nano-tribology.

In general, yes. But it has been done only for static friction coefficients.

The manner of which two surfaces in contact interact is highly investigated by the Tribology community.In particular, the field exploring the mechanics of the interaction is called contact mechanics.

Tackling problems of contact mechanics analytically/numerically is often done by solving the elasticity equations. By predicting quantitatively the forces and deformation of the bodies in contact, the coefficient of friction can be calculated. Much research has been carried on the subject: starting from Hertz's groundbreaking work, to more contemporary papers such as A Static Friction Model for Elastic-Plastic Contacting Rough Surfaces, and Static Friction Coefficient Model for Metallic Rough Surfaces.