Smoothening Transition of Rough Surfactant Surfaces

We propose a model for surfaces in mixtures of oil, water, and surfactants with strong electric dipoles. The dipole interactions give rise to a non-local interaction with a negative stiffness between surface elements. We show that, for large space dimension D, this model has a phase transition from rough to smooth surfaces. Contrary to models with a simple quadratic curvature energy of a positive sign, which always have a finite persistence length typical for rough surfaces, the smooth surfaces in our model exhibit long-range correlations of a generalized antiferromagnetic type. These correlations might be related to the recently observed “egg carton” superstructure of membranes.