A phase-field model for a modulated-disordered interface with varying density

Soft modulated phases have been shown to undergo complex morphological transitions at high temperatures, in which the orientation of their layers and the Gaussian curvature play a major role. This is the case of smectic films under thermal treatment, where focal conics can be reshaped into conical pyramids and concentric ring structures. While evaporation-condensation mechanisms have been theoretically and numerically studied for a smectic-isotropic interface, hydrodynamic stresses at the interface and their resulting flows are yet to be analyzed. This is particularly challenging in the case of a smectic-air boundary, due to the large density ratio between the phases. We derive a phase-field model that accounts for a varying density field and represents the smectic layering by an order parameter. The resulting equations govern the evolution of an interface between a modulated and a disordered phase with distinct densities, being able to account for compressibility effects at the interface and accommodate topological transitions. By integrating the equations in time, we investigate the interfacial flow on a disturbed smectic, and verify the implementation based on the derived dispersion relation.