Extended Data Fig. 5: Comparison between the dispersion forces on a flake suspended on an untreated silica substrate and on a gold-coated substrate.

a, Casimir–Lifshitz force acting on a gold flake suspended at a height h above a gold coated substrate (orange line) or an untreated silica substrate (grey line). The magnitude of the force is represented in a log-log plot and is calculated for a reference hexagonal flake with side length equal to 700 nm. The force is attractive for both substrates, but in the case of an untreated silica substrate the force is one order of magnitude smaller than in the gold-coated case. b, Experimental values of ‹h› (see Eq. (9) for its definition) of the flake for the various cases, that is, gold flake suspended on a gold-coated substrate with hydrophilic SAM (red symbols), hydrophobic SAM (blue symbols), and uncoated silica substrate (green symbols). The solid lines represent the theoretical model. The resulting fit is in good agreement with the experimental measurements for all substrates. It is worth noting that the confidence interval for the values of \(\left\langle h\right\rangle\) for the uncoated silica substrate are much larger than those for the gold-coated substrates. This happens as the minimum of the total potential is very shallow for an uncoated silica substrate, because the Casimir–Lifshitz forces are much weaker in this case. At T ≪ T_c, that is, in the absence of attractive critical Casimir forces, the gravity plus buoyancy is the only force that is effective in pushing the flake towards the substrate. In the case of a gold-coated substrate, the Casimir–Lifshitz forces are very relevant for T ≪ T_c, and the resulting total potential has a very pronounced minimum, which is reflected in the very narrow error intervals. In the figure, the darker shade of each colour represents the 68% confidence interval. The error on the experimental data is the standard deviation from the mean of three measurements. The data are presented as mean values.