  We calculate density profiles of a simple model fluid in contact with a
planar surface using density functional theory (DFT), in particular for the
case where there is a vapour layer intruding between the wall and the bulk
liquid. We apply the method of Hughes et al. [J. Chem. Phys. 142, 074702
(2015)] to calculate the density profiles for varying (specified) amounts of
the vapour adsorbed at the wall. This is equivalent to varying the thickness
$h$ of the vapour at the surface. From the resulting sequence of density
profiles we calculate the thermodynamic grand potential as $h$ is varied and
thereby determine the binding potential as a function of $h$. The binding
potential obtained via this coarse-graining approach allows us to determine the
disjoining pressure in the film and also to predict the shape of vapour
nano-bubbles on the surface. Our microscopic DFT based approach captures
information from length scales much smaller than some commonly used models in
continuum mechanics.
