$d_0$, which is half of the distance between lowest and highest points of the rough profile (full width of the histogram) for the sphere plus the same for the plate. In this way we found $d_0=7.5\pm 1\; nm$ from multiple AFM scans at different locations.

It has to be stressed that this direct way (from the definition) to determine $d_0$ is in good correspondence with what we would expect from the electrostatic calibration. In Refs. [@Zwo08a; @Zwo08b] the relation $d_0\approx (3.7\pm 0.3)\times (w+w_{sph})$ was found from the electrostatic calibration. In our case the total rms roughness (sphere and plate) is $w+w_{sph}=1.3+1=2.3\; nm$. Using this relation we can find $d_0$ within one standard deviation from that found above from the first principle definition.

![Experimental data for the force vs distance (circles) and theoretical prediction without water layer (red curve). Errors in the absolute separation are shown for some points by the bars. The continuous black curve is the prediction for $1.5\;
nm$ continuous water layer. The dashed black curve corresponds to the same water layer with 50% of voids.[]{data-label="fig3"}](fig3.eps){width="45.00000%"}

The calibration of the deflection sensitivity, cantilever stiffness $k$, and contact potential $V_0$ was done in the same way as in previous 