Gaussian curvature and the equilibrium among bilayer cylinders, spheres, and discs.

In mixtures of cetyltrimethylammonium bromide (CTAB) and sodium perfluorooctanoate (FC(7)) in aqueous solution, novel bilayer cylinders with hemispherical end caps and open, flat discs coexist with spherical unilamellar vesicles, apparently at equilibrium. Such equilibrium among bilayer cylinders, spheres, and discs is only possible for systems with a spontaneous curvature, R(o), and a positive Gaussian curvature modulus, kappa. We have measured the size distributions of the spherical vesicles, cylinders, and discs by using cryo-electron microscopy; a simple analysis of this length distribution allows us to independently determine that the mean curvature modulus, kappa approximately 5 +/- 1 k(B)T and kappa approximately 2 +/- 1 k(B)T. This is one of the few situations in which R(o), kappa, and kappa can be determined from the same experiment. From a similar analysis of the disk size distribution, we find that the edges of the discs are likely stabilized by excess CTAB. The fraction of discs, spherical vesicles, and cylinders depends on the CTABFC(7) mole ratio: increasing CTAB favors discs, while decreasing CTAB favors cylinders. This control over aggregate shape with surfactant concentration may be useful for the design of templates for polymerization, mesoporous silicates, etc.