  We consider a simple fluid confined between two parallel walls (substrates),
separated by a distance L. The walls exert competing surface fields so that one
wall is attractive and may be completely wet by liquid (it is solvophilic)
while the other is solvophobic. Such asymmetric confinement is sometimes termed
a `Janus Interface'. The second wall is: (i) purely repulsive and therefore
completely dry (contact angle 180 degrees) or (ii) weakly attractive and
partially dry (the contact angle is typically in the range 160-170 degrees). At
low temperatures, but above the bulk triple point, we find using classical
density functional theory (DFT) that the fluid is highly structured in the
liquid part of the density profile. In case (i) a sequence of layering
transitions occurs: as L is increased at fixed chemical potential (mu) close to
bulk gas--liquid coexistence, new layers of liquid-like density develop
discontinuously. In contrast to confinement between identical walls, the
solvation force is repulsive for all wall separations and jumps discontinuously
at each layering transition and the excess grand potential exhibits many
metastable minima as a function of the adsorption. For a fixed temperature
T=0.56Tc, where Tc is the bulk critical temperature, we determine the
transition lines in the L, mu plane. In case (ii) we do not find layering
transitions and the solvation force oscillates about zero. We discuss how our
mean-field DFT results might be altered by including effects of fluctuations
and comment on how the phenomenology we have revealed might be relevant for
experimental and simulation studies of water confined between hydrophilic and
hydrophobic substrates, emphasizing it is important to distinguish between
cases (i) and (ii).
