atom in the center of the depression can bind to several C atoms and the binding energy is smaller but of the same order as the Au-Au binding energy (case c in Fig. \[Au-substrate-potentials\]).

To model the binding energy of Au atoms in clusters with a contact area larger than the surface intrusions the resulting Au/a-C substrate potentials were averaged using approximate relative surface areas as weights and fitted to a modified Lennard-Jones potential $$\label{LJfitmean}
V_{\rm LJ,mean} = V_{{\rm LJ},0} \left( \frac{297}{(z/r_0-1.2)^{12}} - 
  \frac{34.5}{(z/r_0-1.2)^6} \right).$$ where $V_{{\rm LJ},0}$ is the binding energy and $r_0 = 2.885$ [Å]{}. Using weights of c:d:b:a=1:4:9:16 results in an average binding energy of $V_{{\rm LJ},0}=0.34$eV. The MC simulations in Sec.\[sectionMCfilm\] do not depend qualitatively and only little quantitatively on the specific value in a range of $0.2 <
V_{{\rm LJ},0}/{\rm eV} < 0.4$ and hence an average value of $V_{{\rm LJ},0} = 0.3$eV was adopted.

The value of $V_{{\rm LJ},0} \sim 0.3$ eV is only appropriate for sufficiently large clusters. Small clusters with contact areas smaller than the intrusion size have larger weight of the strongly binding intrusion center c and are easily pinned consistent with the observations discussed in Sec. \[SubstrateHolography\].

Long time-scale dynamics 