2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/28332We 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 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 positive sign, which always have a finite persistence length typical of 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.11 pages, 1 figure, harvmac, minor correctionsSoft Condensed MatterStatistical Mechanicstext