The water-graphite interaction potential proposed recently (Gonzalez et al.emph{J. Phys. Chem. C} textbf{2007}, emph{111}, 14862), the three TIP$N$P ($N=3,:4,:5$) water-water interaction models, and basin-hopping global optimization are used to find the likely candidates for the global potential energy minima of (H$_{2}$O)$_{n}$ clusters with $nleq21$ on the (0001)-surface of graphite and to perform a comparative study of these minima. We show that, except for the smaller clusters ($n<6$), for which ab-initio results are available, the three water-water potential models provide mostly inequivalent conformations. While TIP3P seems to favor monolayer water structures for $n<18$, TIP4P and TIP5P favor bilayer or volume structures for $n>6$. These $n$ values determine the threshold of dominance of the hydrophobic nature of the water-graphite interaction at the nanoscopic scale for these potential models.
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