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The scaling of friction with the contact size $A$ and (in)commensurabilty ofnanoscopic and mesoscopic crystals on a regular substrate are investigatedanalytically for triangular nanocrystals on hexagonal substrates. The crystalsare assumed to be stiff, but not completely rigid. Commensurate andincommensurate configurations are identified systematically. It is shown thatthree distinct friction branches coexist, an incommensurate one that does notscale with the contact size ($A^0$) and two commensurate ones which scaledifferently (with $A^{1/2}$ and $A$) and are associated with variouscombinations of commensurate and incommensurate lattice parameters andorientations. This coexistence is a direct consequence of the two-dimensionalnature of the contact layer, and such multiplicity exists in all geometriesconsisting of regular lattices. To demonstrate this, the procedure is repeatedfor rectangular geometry. The scaling of irregularly shaped crystals is alsoconsidered, and again three branches are found ($A^{1/4}, A^{3/4}, A$). Basedon the scaling properties, a quantity is defined which can be used to classifycommensurability in infinite as well as finite contacts. Finally, theconsequences for friction experiments on gold nanocrystals on graphite arediscussed.

(In)commensurability, scaling and multiplicity of friction in nanocrystals and application to gold nanocrystals on graphite