# Disjoining Pressure

Entry by Kelly Miller

The disjoining pressure is due to an attractive interaction between two surfaces. It is a pressure due to the attractive force between two surfaces, divided by the area of the surfaces. For two flat, parallel surfaces the disjoining pressure can be calculated as the derivative of the Gibbs energy per unit area.

In the case of a film on a substrate, deGennes defined the energy function P(e) as the excess energy of a film on a substrate as a function of film thickness (e). Derjaguin defined the disjoining pressure as the derivative of this energy P(e) as a function of e.

The disjoining pressure is related to the stability of films. When P(e) is a decreasing function of e (favoring a thick film), $\pi (e)$ is positive and the film is considered to be stable.

d(e)= increase in thickness e by adding a number of molecules dn=de/Vo where Vo= volume/molecules

If $\mu$ is the chemical potential of the liquid, the energy varies by $\mu$ dn

The chemical potential is related to the disjoining pressure by:

Sheluelko devised a method to measure the disjoining pressure using a system under equilibrium. The system is a porous ring filled with the same liquid as in the bulk, in equilibrium with the thin film:

The ring is subjected to a hydrostatic pressure $\rho g H$ where

p=density of the liquid g= acceleration due to gravity H= height of the column

The chemical potential of the column of water is equal to the chemical potential of the thin film: