  The scaling of friction with the contact size $A$ and (in)commensurabilty of
nanoscopic and mesoscopic crystals on a regular substrate are investigated
analytically for triangular nanocrystals on hexagonal substrates. The crystals
are assumed to be stiff, but not completely rigid. Commensurate and
incommensurate configurations are identified systematically. It is shown that
three distinct friction branches coexist, an incommensurate one that does not
scale with the contact size ($A^0$) and two commensurate ones which scale
differently (with $A^{1/2}$ and $A$) and are associated with various
combinations of commensurate and incommensurate lattice parameters and
orientations. This coexistence is a direct consequence of the two-dimensional
nature of the contact layer, and such multiplicity exists in all geometries
consisting of regular lattices. To demonstrate this, the procedure is repeated
for rectangular geometry. The scaling of irregularly shaped crystals is also
considered, and again three branches are found ($A^{1/4}, A^{3/4}, A$). Based
on the scaling properties, a quantity is defined which can be used to classify
commensurability in infinite as well as finite contacts. Finally, the
consequences for friction experiments on gold nanocrystals on graphite are
discussed.
