# What equations govern the formation of droplets on a surface?

When some smooth surface (like that of a steel or glass plate) is brought in contact with steam (over e.g. boiling milk) then water is usually seen to condense on that surface not uniformly but as droplets. What are the equations which govern the formation and growth of these droplets ? In particular what role does the geometry of the surface plays in it? Also it is possible to prepare experimental conditions where no droplets are formed but water condenses uniformly ?

## 2 Answers

It is more about properties of the material and surface roughness/texture, than geometry of the surface on the large scale. There is a nice Wikipedia article on wetting, which you may find useful. In brief, that is the difference in surface tension at liquid-solid ($\gamma_{LS}$), liquid-air ($\gamma_{LA}$), and solid-air ($\gamma_{SA}$) interfaces which define the value of contact angle $\theta=\arccos\left(\left(\gamma_{SA}-\gamma_{LS}\right)/\gamma_{LG}\right)\neq0$, and lead to droplet formation. If the angle is zero, the liquid will tend to cover the whole surface uniformly.

• Is the contact angle the only factor, or is there any other factor? – Bernhard Jul 14 '12 at 18:04
• It is not completely clear what do you mean by "factor" in this context. This expresion for the contact angle describe wetting of smooth, homogeneous surfaces. For textured surfaces it should be modified, in some cases wetting properties are determined by surface structure rather than properties of the material, see lotus effect. Even for the homogeneous surface calculation of surface energy from first principles seems to be quite hard, so such phenomenological "factors" are used. – straups Jul 15 '12 at 5:17
• Well, I can imagine that droplet size is also dictated by gravity (orientation of the plate) and the surface tension or the liquid-air interface. The maximum size of the droplets do not follow directly from your equation. – Bernhard Jul 15 '12 at 6:44

I would suggest you the following well written review articles, by Jens Eggers, who is one of the most renowed and acknowledged researchers on this field:

Drop formation – an overview

Nonlinear dynamics and breakup of free-surface ﬂows