Statistical mechanics of microemulsions: Droplet phases and macroscopic interfaces

We discuss the statistical mechanics of a droplet microemulsion within the context of the Canham-Helfrich model of interfacial elasticity. We focus on the behavior of Winsor I and Winsor II states in which a dilute droplet phase coexists with an excess phase of the dispersed fluid. By taking into account fluctuations of the size, position, and shape of the microemulsion droplets on an equal footing, our treatment resolves existing ambiguities regarding the correct definition of the conformational entropy within such a droplet phase, and thus allows quantitative predictions to be made for, e.g., the dependence of average droplet size and interfacial tension upon the elastic parameters of the interfaces. Our results for the droplet phase may be summarized in terms of a phenomenological model in which, if the renormalized bending energy of a spherical droplet is used as the statistical weight for droplets of constrained size and center-of-mass position, then a length scale of order of the thermal roughness of a droplet's surface due to shape fluctuations appears as the natural measure of surface displacements used to calculate the excess entropy arising from variations of droplet radius (i.e., polydispersity) and position. The renormalized bending energy of a droplet is found to contain contributions arising from the renormalization of both the mean and Gaussian bending rigidities, and from a separate renormalization of the bending torque, or the spontaneous curvature, of the monolayer. We also reconsider the interpretation of experiments probing the fluctuations of a macroscopic interface such as that separating the microemulsion and excess phases in such systems, and discuss the effect of fluctuations upon the measured interfacial tension of systems with very low tensions. We show that the fluctuations of this interface can be described in terms of a renormalized bending rigidity κ_R and surface tension γ that both depend upon a crossover length ξ ∼ √κ_R/γ and upon a microscopic cutoff length, but that, in contrast to the wave number dependencies predicted in some earlier studies, are both essentially independent of the wavelength of the fluctuations.