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I have found hundreds of papers describing the contact angle of water droplet sitting on hydrophobic surface and the change between Wenzel Regime and the Cassie-Baxter regime.

Drop of liquid on surface and contact angle

Now, as I understand when the water droplets are sitting on super-hydrophobic surfaces, the roll off and pickup dust along the way. That's because the energy required by the water to pick them up is much less than the energy required to stick them to the hydrophobic and so they are washed off.

Sliding droplets take away dust

Does anyone know what equations govern the adhesion between the dust particles and the water droplets? And how is the adhesion between the droplets and the dust is more than the adhesion between the dust and the surface?

The adhesion between dust and surface is probably Van der Waals force, but I am not sure about the interaction of the dust particles and the water droplets.

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As you already mention yourself, the energy to pick up the dust particles by the droplets is less than the energy required for the dust particles to remain at the (repelling) surface. The reason for this, just as why the droplet doesn't 'like' to be there, is surface tension.

This means that you can remove dust only if it favours water over the surface (i.e. if the surface tension is lower between water and dust than water and surface), but I guess this will be the case for (somewhat) polar dust particles

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