  We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard
equations to describe phase separation in thin films. The equations we derive
underscore the coupled behaviour of free-surface variations and phase
separation. We introduce a repulsive substrate-film interaction potential and
analyse the resulting fourth-order equations by constructing a Lyapunov
functional, which, combined with the regularizing repulsive potential, gives
rise to a positive lower bound for the free-surface height. The value of this
lower bound depends on the parameters of the problem, a result which we compare
with numerical simulations. While the theoretical lower bound is an obstacle to
the rupture of a film that initially is everywhere of finite height, it is not
sufficiently sharp to represent accurately the parametric dependence of the
observed dips or `valleys' in free-surface height. We observe these valleys
across zones where the concentration of the binary mixture changes sharply,
indicating the formation of bubbles. Finally, we carry out numerical
simulations without the repulsive interaction, and find that the film ruptures
in finite time, while the gradient of the Cahn--Hilliard concentration develops
a singularity.
