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?
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