  We present a theoretical study of the intrusion of an ambient liquid into
pores of a nano-corrugated wall w. The pores are prefilled with a liquid
lubricant which adheres to the walls of the pores more strongly than the
ambient liquid. The two liquids are modeled as a binary mixture of two types of
particles, A and B. The mixture can decompose into an A-rich(ambient) liquid,
and a B-rich(lubricant) liquid. The wall attracts B particles more strongly
than A particles. The ratio of w-A to w-B interaction strengths is changed to
tune the contact angle ${\theta}_{AB}$ formed by the A-rich/B-rich liquid
interface between the two fluids and the corresponding planar wall. We use
classical density functional theory, in order to capture the effects of
microscopic details on the intrusion transition as a function of composition
and pressure of the ambient liquid, for various values of ${\theta}_{AB}$ and
different pit sizes.
  We also studied the reverse process in which a pore initially filled with the
ambient liquid is refilled with the lubricant. For small ${\theta}_{AB}$ ,there
is no hysteresis. However, beyond a certain ${\theta}_{AB}$, we find the
presence of hysteresis, which increases with increasing ${\theta}_{AB}$. The
dependence of location of the intrusion on ${\theta}_{AB}$ and on the pit size,
qualitatively follows the corresponding shift of the capillary-coexistence line
away from the bulk liquid-liquid coexistence line, as predicted by a
macroscopic capillarity model. The quantitative discrepancies become larger for
narrower cavities. For the considered geometry, macroscopic capillarity theory
predicts an open hysteresis loop above a threshold value of ${\theta}_{AB} =
54.7^\circ$. This threshold, however, shifts to much higher values if widths of
the pores become smaller than roughly ten times the diameter of the fluid
particles.
