Physica A 290 (2001) 360–378 www.elsevier.com/locate/physa Fluctuations and instabilities of model amphiphile interfaces C. Varea ∗ , A. Robledo Instituto de F sica, Universidad Nacional Aut onoma de M exico, Apartado Postal 20-364, M exico D.F. 01000, Mexico Received 21 June 2000 Abstract We study the stability of planar, cylindrical and spherical interfaces with respect to shape and width uctuations for a model amphiphile solution described by a free energy density func- tional with square-gradient and square-Laplacian terms. That is, we determine the stability matrix when the stationary state consists of an interface with given geometry that separates two im- miscible solvent phases. From the spectrum and the related eigenfunctions of this matrix we establish where lamellar and micellar domain-structured phases occur, and contrast our results with those for a simple square-gradient uid model for which these phases are always unstable. We also characterize some instability properties such as the buckling of lamella, the undulation of cylindrical structures and the nucleation of micelles. c  2001 Elsevier Science B.V. All rights reserved. PACS: 05.40.+j; 68.10.−m; 82.65.Dp; 64.60.Qb Keywords: Interfaces; Amphiphiles; Fluctuations; Instabilities 1. Introduction The properties of curved interfaces have proved to be crucial for the understanding of equilibrium phases that have mesoscopic structures, like those that occur in solu- tions of amphiphilic molecules with otherwise immiscible solvents, as in water and oil microemulsions. Microemulsions [1,2] form interfaces with very small tensions, and in this case the pertinent term in the phenomenological free energy is that which measures the elastic properties of curved interfaces. The density functional approach is a power- ful tool that can give base to this term and provide statistical–mechanical expressions for the quantities involved, such as the spontaneous curvature and the bending rigidities * Corresponding author. 0378-4371/01/$ - see front matter c  2001 Elsevier Science B.V. All rights reserved. PII: S0378-4371(00)00461-1