the substrate. Since the pinned clusters are smaller than the TEM resolution limit their diameters must be smaller 1 nm or $N<50$ atoms.[@Urba98]

Surface adsorbate interaction: DFT calculations {#SubstrateDFT}
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A quantitative description of Au/a-C substrate interaction is needed for the MC simulations described in Sec. \[sectionMCfilm\]. To this end we studied the interaction of individual Au atoms with graphite and a model a-C surface using ab-initio DFT calculations.

For the calculation of the Au/graphite-surface interaction we used a supercell consisting of 2x2 graphite unit cells in the $x$-$y$ plane and four carbon layers in the $z$-direction. A large vaccuum distance of 19 Å was used.[@DFTquote] Calculations were performed for Au in the top, bridge, and hollow sites. Using as reference a Au atom at the center of the vacuum we calculated binding energies of 0.065 eV, 0.062 eV and 0.041 eV, respectively. The Au-graphite potential for Au in the top-site as a function of Au atom surface distance is shown by line “g” in Fig. \[Au-substrate-potentials\]. The result of the DFT calculations was fitted with a modified Lennard-Jones potential $$\label{LJfitgraphite}
V_{\rm LJ,graph} = V_{{\rm LJ},0}\left( \frac{0.31}{(z/r_0 - 0.1)^6} - 
  \frac{1.11}{(z/r_0 - 0.1)^3} \right),$$ where $V_{{\rm LJ},0}$ is the binding 