Ginzburg-Landau theory of oil-water-surfactant mixtures

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Abstract
A simple Ginzburg-Landau free-energy functional for oil-water-surfactant mixtures, with a single, scalar order parameter, is studied in detail. We show that the free energy of swollen spherical and cylindrical micelles obtained from this model can be described in terms of the Helfrich Hamiltonian of surfactant monolayers. The surface tension σ, the spontaneous curvature modulus λ, the saddle-splay modulus κ¯, and the bending rigidity κ of the surfactant sheet at the oil-water interface can all be calculated from the order-parameter profile of the planar oil-water interface. It is demonstrated that for stable droplets and cylinders, these expressions of κ and κ¯ give reliable predictions for the free energy only if the system is at oil-water coexistence. Off coexistence, the distortion of the profile due to the finite curvature of the interface has to be taken into account. The results are used to discuss the phase diagram of the Landau model. In addition to the bending elasticity, the interaction between monolayers plays an important role. This interaction is found to be always attractive in our model. We show that the simple Ginzburg-Landau theory describes various spatially modulated phases: the lamellar phase, the hexagonal phase of cylinders, a cubic crystal of spherical micelles, and bicontinuous cubic phases. Finally, we discuss the behavior of oil-water-surfactant mixtures near a wall. We find that under suitable conditions the lamellar phase wets the wall-oil (or wall-water) interface.

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