  Classical density functional theory is applied to investigate the validity of
a phenomenological force-balance description of the stability of the Cassie
state of liquids on substrates with nanoscale corrugation. A bulk free-energy
functional of third order in local density is combined with a square-gradient
term, describing the liquid-vapor interface. The bulk free energy is
parameterized to reproduce the liquid density and the compressibility of water.
The square-gradient term is adjusted to model the width of the water-vapor
interface. The substrate is modeled by an external potential, based upon
Lennard-Jones interactions. The three-dimensional calculation focuses on
substrates patterned with nanostripes and square-shaped nanopillars. Using both
the force-balance relation and density-functional theory, we locate the
Cassie-to-Wenzel transition as a function of the corrugation parameters. We
demonstrate that the force-balance relation gives a qualitatively reasonable
description of the transition even on the nanoscale. The force balance utilizes
an effective contact angle between the fluid and the vertical wall of the
corrugation to parameterize the impalement pressure. This effective angle is
found to have values smaller than the Young contact angle. This observation
corresponds to an impalement pressure that is smaller than the value predicted
by macroscopic theory. Therefore, this effective angle embodies effects
specific to nanoscopically corrugated surfaces, including the finite range of
the liquid-solid potential (which has both repulsive and attractive parts),
line tension, and the finite interface thickness. Consistently with this
picture, both patterns (stripes and pillars) yield the same effective contact
angles for large periods of corrugation.
