Statistical mechanics of closed fluid membranes

We discuss the statistical mechanics of systems, such as vesicle phases, that consist of many closed, disconnected fluid membranes. We show that the internal undulation free energy of a single such membrane always contains a logarithmically scale-dependent ''finite-size'' contribution, with a universal coefficient, that arises essentially from the absence of undulation modes of wavelength longer than the size of the surface. We show that the existence of this contribution is closely related to the existence of additional collective degrees of freedom, such as translations and polydispersity, that are present in a system of disconnected surfaces but absent in a system of infinite surfaces such as a lamellar or bicontinuous phase. The combination of undulation and collective mode free energy contributions found here is shown to yield thermodynamic properties (e.g., size distributions) for a vesicle phase whose parameter dependence is consistent with the scale invariance of the Helfrich bending energy. We briefly discuss the conceptually related problem of a polydisperse ensemble of fluctuating one-dimensional aggregates, such as rodlike micelles, for which we obtain the experimentally observed scaling of micelle size with concentration. We conclude by discussing the application of our results to the interpretation of recent experiments on equilibrium vesicle phases.