$k$. For self-affine roughness $\sigma(k)$ scales as [@Kri95] $\sigma (k)\propto k^{-2-2H}$ for $k\xi\gg 1$ and $\sigma (k)\propto const$ for $k\xi\ll 1$. The parameters $\xi$ and $H$ are the correlation length and roughness exponent, respectively. For the roughness calculations we used the roughness model in Fourier space [@Pal93]: $$\label{r_model}
    \sigma(k)=\frac{AHw^2\xi^2}{(1+k^2\xi^2)^2},
    \ \ \ A=\frac{2}{\left[1-(1+k_c^2\xi^2)^{-H}\right]},$$ where $w$ is the rms roughness, and $k_c\sim 1\;nm^{-1}$ is a lower roughness cutoff.

For actual calculations we took for the roughness parameters in $\sigma (k)$ the values: $w_{sph}=1.0\; nm$, $w=1.3\; nm$, lateral correlation lengths $\xi=20\; nm$, and roughness exponent $H=0.9$ [@Zwo08b]. At separations $d>10\; nm$ roughness can still play significant role by increasing the force. It has to be noted that relation (\[corr\]) derived within the scattering theory [@Mia05] is applicable for $d\gg w+w_{sph}$ and small local surface slopes. For smaller $d$ a much stronger roughness effect is expected increasing the force up to five times or more with respect to that of flat surfaces as former studies indicated [@Zwo08a].

We performed calculations of the force between a sphere of radius $R=50\; \mu m$, and a plate without any water layers using Eqs. (\[E\_conv\])-(\[r\_model\]). As the dielectric function of $Au$ film we used the data for 