Crumpling and second sound in lyotropic lamellar phases

Abstract

We study relative concentration fluctuations in two component lamellar smectic liquid crystals consisting of surfactant layers of width w separated by a background fluid and show that these fluctuations are dominated by crumpling fluctuations of the surfactant layers when (w/l)2 <<; 1 where f is the average layer spacing. Our results are based on generalizations of the Helfrich theory in which the crumpling ratio k-2, densities πs and πb of the surfactant and background fluid, and w as well as l are allowed to vary. In the incompressible limit with πs, πb and w fixed, concentration fluctuations at constant f are determined entirely by fluctuations in k-2 . In this limit, the ratio of the high-frequency, constant concentration compressibility modulus B to the low-frequency, constant chemical potential modulus B is of order (κ /T)2 (l-ω) 2/l2 where k is the layer bending rigidity and T is the temperature. We show that corrections to the incompressible limit arising from fluctuations in w are of relative order (ω/l)2 (κ/U) where U is a molecular energy.