We study two-dimensional steady concentration and film thickness profiles for isothermal free surface films of a binary liquid mixture on a solid substrate employing model-H that couples the diffusive transport of the components of the mixture (convective Cahn-Hilliard equation) and the transport of momentum (Navier-Stokes-Korteweg equations). The analysis is based on minimising the underlying free energy equivalent to solving the static limit of model-H. Additionally, the linear stability (in time) of relevant layered films is analyzed. This allows for a comparison of the position of certain branching points in the bifurcation diagrams of steady solutions with the value predicted as onset of a linear instability. Results are presented for the cases of (i) a flat film without energetic bias at the free surface, (ii) a flat film with energetic bias, (iii) a height-modulated film without energetic bias, and (iv) a height-modulated film with energetic bias. In all cases we discuss symmetries of the various steady solutions allowing us to order them and to infer properties of solution branches and relations between them.

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We present steady non-linear solutions of films of confined polymer blends deposited on a solid substrate at off-critical concentrations with a free deformable surface. The solutions are obtained numerically using a variational form of the Cahn-Hilliard equation in the static limit, which allows for internal diffuse interfaces between the two components of the mixture. Existence of most of the branches of non-linear solutions at off-critical concentrations can be predicted from the knowledge of the branching points obtained with a linear stability analysis plus the non-linear solutions at critical concentrations. However, some families of solutions are found not to have correspondence at critical compositions. We take a value for surface tension that allows strong deformations at the sharp free upper surface. Varying the average composition and the length and thickness of the films we find a rich morphology of static films in the form of laterally structure films, layered films, droplets on the substrate, droplets at the free surface, and checkerboard structures. We show that laterally structured solutions are energetically favorable over homogeneous and other structured solutions within the whole spinodal region and even close to the absolute stability binodal boundary.

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