Four-point correlation functions and average Gaussian curvature in microemulsions

Abstract

The structure of microemulsions is studied in the Landau-Ginzburg model [A. Ciach, J. Chem. Phys. 104, 2376 (1996)], in which all the coupling constants are expressed in terms of temperature $T,$ surfactant volume fraction ${\ensuremath{\rho}}_{s},$ and amphiphilicity. The extension of surface-averaged Gaussian curvature, $\overline{K}$ [A. Ciach and A. Poniewierski, Phys. Rev. E 52, 596 (1995)], is calculated in the Landau-Ginzburg approximation. In the neighborhood of the liquid-crystal phases $\overline{K}l0.$ Thermal fluctuations destroy the bicontinuous structure and cause a transition to $\overline{K}g0.$ A dimensionless average radius of curvature $\overline{R}=|\overline{K}{|}^{\ensuremath{-}1/2}/\ensuremath{\lambda},$ where \ensuremath{\lambda} is a period of damped oscillations of the two-point correlation function, is calculated. Comparison with a corresponding quantity for periodic minimal surfaces shows that microemulsions have (for $\overline{K}l0$) a structure resembling different periodic minimal surfaces with low genus for different $T$ and ${\ensuremath{\rho}}_{s}.$