Reading about cloud formation, I learned that to a droplet to form, the water vapour needs a Cloud Condensation Nuclei, which is an aerosol with a size in the order of 0.0001mm. And if no CCN is found temperatures as low as -13°C are needed to form the droplets. Why this is happening? What does the contact of the solid with water do to make it condense?
The correct answer is that Surface tension prevents the formation of small water droplets.
You see, as the surface energy is proportional to the droplet area ($r^2$) and the bulk energy - to its volume ($r^3$), their ratio ($1/r$) will rise to infinity as the droplet radius $r$ goes to zero. Therefore, the droplet cannot grow steadily from $r=0$. Even below 0°C there is a certain critical droplet radius $r_c(T)$, below which the Gibbs free energy of droplet is higher than that of vapor so the droplet evaporates.
Condensation nuclei are a shortcut to the droplet formation: if they are hygroscopic and have a radius above $r_c$, water surface energy will be too small to prevent condensation and the droplet will grow quickly (assuming that vapor is supersaturated).