## Bubbles in Coating

### Quick Start

Bubbles are a big problem in coatings. Here we do two calculations.

- How long it will take for bubbles or radius r to rise a height h in a liquid of viscosity η and density ρ; i.e. how long do you need to leave a pan of liquid to de-bubble?
- How many bubbles of your chosen radius will appear if you raise the temperature of your coating solution from T
_{store}to T_{coat}?

The second question sounds obscure. But many coaters (myself included) have suffered from "unexplainable" bubble problems caused simply by going from a colder storage to a hotter coating machine.

### Bubbles

When you have a bubble of radius r in a liquid of viscosity μ and density ρ and a gravitational force g then the particles will rise at a velocity v and cover a distance h in time t. The formula is the Stokes equation:

`v=(2ρgr^2)/(9η)`

`t=h/v`

The typical use for the calculation is when you have a coating solution with bubbles in it and you want to know the typical time required for them all to come to the surface.

If you have a volume fraction of bubbles φ then Richardson and Zaki tell us that the velocity is reduced by a factor of `(1-φ)^4.65`. If you have a "foam" rather than just a few bubbles, this can be a large effect.

Bubbles can appear for many reasons. One reason that is very common but surprisingly little-known is that if the temperature of the coating solution rises then the solubility of the air decreases and bubbles appear. The calculation here assumes that you are using water because its solubility characteristics (Henry's law constants) are well-known. Simply choose a starting and ending temperature and the calculation gives the number of bubbles of the chosen radius r that would appear in each ml of solution. Clearly this is an upper limit as the bubbles don't all come out at the same time or in the same place, but it's a good reminder of why there is a golden rule of coating:

If the storage temperature of your solution is higher than the coating temperature life is easy, if the storage temperature is lower then bubble defects are very likely.

The strongest influence is your choice of bubble radius. If it is large then the number of bubbles is relatively small because for a given volume of dissolved gas, the number of bubbles depends on volume per bubble which goes as r³. The calculations, therefore, are not to be taken too literally. But the point remains that dissolved gas can, and often does, give coating defects because of bubble formation.