# Is gecko-like friction Coulombic? What is the highest known Coulombic $\mu_s$ for any combination of surfaces?

Materials with large coefficients of static friction would be cool and useful. Rubber on rough surfaces typically has $\mu_s\sim1-2$. When people talk about examples with very high friction, often they're actually talking about surfaces that are sticky (so that a force is needed in order to separate them) or wet (like glue or the tacky heated rubber used on dragster tires and tracks). In this type of example, the usual textbook model of Coulombic friction with $\mu_s$ and $\mu_k$ (named after Mr. Coulomb, the same guy the unit was named for) doesn't apply.

I was looking around to try to find the largest $\mu_s$ for any known combination of surfaces, limiting myself to Coulombic friction. This group at Berkeley has made some amazing high-friction surfaces inspired by the feet of geckos. The paper describes their surface as having $\mu\sim5$. What's confusing to me is to what extent these surfaces exhibit Coulombic friction. The WP Gecko article has pictures of Geckos walking on vertical glass aquarium walls, and it also appears to imply a force proportional to the macroscopic surface area. These two things are both incompatible with the Coulomb model. But the Berkeley group's web page shows a coin lying on a piece of glass that is is nearly vertical, but not quite, and they do quote a $\mu$ value. This paper says gecko-foot friction involves van der Waals adhesion, but I think that refers to microscopic adhesion, not macroscopic adhesion; macroscopic adhesion would rule out the Coulomb model. (The WP Gecko article has more references.)

So my question is: what is the highest coefficient of Coulombic static friction ever observed, and does the Berkeley group's substance qualify?

-
Could You explain what "coulombic friction" means? Googeling did not help really. Is that related to Coulombs law of friction? – Georg Jul 29 '11 at 19:34
It means the usual textbook model of friction with $\mu_s$ and $\mu_k$, as described in the first paragraph. If that part needs to be rewritten to clarify it, let me know -- or did you just miss that on a quick read-through? This paper naun.org/journals/mechanics/19-239.pdf has a treatment of various models of friction. – Ben Crowell Jul 29 '11 at 20:32
I idn't know that it is called "Coulombic" friction. For me that was just friction. – Georg Jul 29 '11 at 21:40
I didn't know either, but your explanation isn't too bad, Ben. I added one link that I think may clarify it for others, though. – David Z Jul 29 '11 at 22:17
@David Zaslavsky: Thanks for the link and the general cleanup! – Ben Crowell Jul 29 '11 at 22:24

Short answer: Yes, I'd buy the Berkeley group's work; their value of $\mu$ is the highest I've seen...

Crudely, adhesion is a force that resists the separation of in-contact surfaces in the normal direction. Friction is a force that opposes relative tangential motion between two in-contact surfaces. One need not imply the other, their causative mechanisms are distinct and in fact most models exclusively address one or the other.

Adhesion is driven by Van der Waals kinda forces.
"Coulomb" friction (solid-solid) is caused by the presence of asperities (think small bumps or protrusions) on surfaces. Due to the presence of these asperities, the "real" area of contact between two surfaces is much smaller than the "apparent" area of contact. The Coulomb model is a phenomenological fit to experiments that was shown to be deducible assuming this type of contact. In some sense this sets an upper bound on the resistance to the tangential relative motion between surfaces.

If you somehow ensure these areas are very nearly the same, would $\mu$ then increase?

There are, indeed, "intimate" contacts where the apparent and real areas of contact are very nearly the same and the resistance to sliding large. In such cases, the frictional behavior is intimately linked to the mechanisms of deformation at the small scale (e.g. plasticity in metals). However, even this is not enough to get the largest possible $\mu$.

For instance, the maximum shear stress resisting relative motion in metals is capped to a maximum value, beyond which it cannot increase. (i.e.) Instead of

Shear stress = $\mu\times$ Normal stress ...(I)

You would've

Shear stress = min ($\tau_{max}$ , $\mu\times$ Normal stress) ...(Ia)

Equation (I) is simply Coulomb's law applied locally.

Q = $\mu N$ ...(II)

Now, if someone devised a contact / material system that produces very high $Q$ for a given $N$ in equation (II) in an experiment, they could claim that they had devised a material with high friction coefficient.

This is essentially what the Berkeley group seems to have done. As I said, this kind of thing is hard to do with metals - even if intimate contact is achieved (say, under a state of severe deformation), something like Eqn. (Ia) kicks in and prevents the shear resistance from rising further. Previously, therefore, people achieved high $\mu$ using compliant, soft materials, but these guys use a microfiber array to engineer a surface with high $\mu$.

Their main advances are (1) High $\mu$ than reported in soft materials (2) Allowing control of $\mu$ by controlling the fiber layout etc (3) Achieving high $\mu$ in combination with low adhesion, which was not the case in softer materials. This is the kind of property combination you'd need for automobile tyre.

Coming to the Gecko paper, it has far more to do with adhesion that friction. The Gecko paper and the Berkeley friction paper have little to do with each other. Also, contrary to popular myth, the Gecko mechanism has nothing to do with "suction". See these papers in Nature - it is largely adhesion driven.

Adhesive force of a single gecko foot-hair K Autumn, YA Liang, ST Hsieh, W Zesch, WP Chan… - Nature, 2000

Micro-fabricated adhesive mimicking gecko foot-hair AK Geim, SV Dubonos, IV Grigorieva… - Nature materials, 2003

For physicists interested in these areas - I understand that tribology and solid mechanics are not taught in US physics departments - it might help to refer to standard texts on Tribology by Bowden and Tabor, Kendall, Israelachvili, Persson, Maugis etc.

Or better still, talk to your colleagues who work in tribology (they're usually to be found in mechanical engineering, materials science and chemistry). They will be eager and willing to help, if only for the opportunity to brag at faculty meetings that a physicist asked them for advice :-)

-