Modulated hexagons in Marangoni convection with deformable surface covered by surfactant

Hexagons are typical patterns generated by the Marangoni instability in a liquid layer. In the vicinity of the instability threshold, the pattern evolution can be described by a system of Ginzburg-Landau-type amplitude equations. Due to the asymmetry of the boundary conditions at the top and the bottom of the layer, those equations contain additional non-gradient quadratic terms with spatial derivatives. These terms are especially significant for non-equilateral hexagons based on the resonance of wave vectors with slightly different lengths.

We consider the onset of Marangoni convection created by thermocapillary and solutocapillary stresses in a layer with deformable free interface covered by surfactant. In that case, the description has to include two additional variables, a large-scale deformation of the boundary and disturbance of the surfactant, which evolve slowly because of the conservation of the

liquid volume and of the surfactant amount. The interaction of longwave disturbances near the bifurcation point can create a modulational instability of periodic patterns.

We study the joint action of the deformability of the liquid free surface and the effect of insoluble surfactant spread over the surface on the regular (equilateral) and deformed (non-equilateral) hexagons and their modulational instability. The governing parameters of the problem are Biot number characterizing the heat transfer resistance of the surface and Galileo number indicating the role of gravity. Stability maps are plotted for different concentration of insoluble surfactant.