## Levelling

### Quick Start

It is common for coatings to start out with regularly-spaced defects from the coating process or paintbrush. Fortunately, many of these defects level out before drying. The app shows that the levelling time depends weakly on surface tension σ and viscosity η, strongly on thickness, h, and very strongly on the defect wavelength λ. So if you want fast-levelling, focus on h and λ before worrying about σ and η!

### Levelling

h μm
λ μm
η cP
σ mN/m
t

A coating will often contain some sort of line defect visible at the coating head but invisible once dry. As the wet coatweight is reduced the same wet defect now shows up in the dry coating. Why? Because the defect has not had time to level out. The reasons for this are not obvious until Orchard levelling theory is invoked.

The levelling time for a defect of wavelength λ in a coating of thickness h with viscosity η and surface tension σ to level by 1/e is given by:

t=(3η(λ/(2π))^4)/(σh^3)

It immediately becomes clear that relatively small decreases in thickness can give a large increase in levelling time because of the h3 term. On the positive side, increasing the sharpness (decreasing λ) of the defect greatly decreases levelling time. That's why those who buy cheap paint brushes with coarse bristles (large λ) have to put on more paint (larger h) than those who buy better brushes with finer bristles.

As seen in the Pinholes app, thin coatings also tend to make pinholes worse, so life is tougher for nanocoaters.

Earlier versions of the app omitted the factor of (2π)4 due to a confusion between a notional "length" l and "wavelength" λ.