In this paper we give an overview of cubic liquid-crystalline mesophases formed by amphiphiles. In section 1 we present brief descriptions of the principal types of translationally ordered lyotropic phases, and describe the locations in the phase diagrams where the different types of cubic phase occur. In section 2 we discuss the various forces that act between bilayers. These transverse interactions are relatively straightforward to quantify in the case of lamellar phases, but are more complex for cubic phases, because of the non-planar geometry. In section 3 we show how an intrinsic desire for interfacial curvature can lead to a state of physical frustration. We then introduce the curvature elastic energy, and describe how this may be related to the stress profile across the bilayer. In the following sections we focus attention on the inverse (water-in-oil) versions of the non-lamellar phases, although analogous effects also operate in the normal topology (oil-in-water) structures. In section 4 we briefly describe the inverse hexagonal phase, which is the simplest inverse phase with curved interfaces. This allows us to illustrate the role of hydrocarbon chain packing frustration in a rather clear way before coming on to the more subtle interplay between packing and curvature frustration, characteristic of the bicontinuous cubic phases, which is discussed in section 5. In section 6 we describe an entirely different class of cubic phases, with positive interfacial gaussian curvature. These cubic phases are composed of complex packings of discrete micellar or inverse micellar aggregates, which may be quasi-spherical and/or anisotropic in shape. Finally, in section 7 we discuss geometric aspects of transitions between lamellar, hexagonal and cubic phases, and show how determination of the epitaxial relations between phases can shed light on the precise mechanisms of the phase transitions.
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