Amphiphiles are molecules such as surfactants or lipids that have a polar head group (hydrophilic) attached to nonpolar hydrophobic alkyl chains. Because of this characteristic they self-assemble in water and give rise to a wide range of phases with different structures and properties. Aqueous dispersions of amphiphiles are present in every aspect of day-to-day life-e.g., forming biological cell membranes, stabilizing emulsified food, or being used as soap. Time-resolved x-ray diffraction has been used to study the hexadecylhexa(oxyethylene glycol) ether (C 16 EO 6 )͞water system, which shows an intermediate phase whose structure depends on the thermal path between lamellar and hexagonal structures. Heating the hexagonal phase from room temperature leads to a lamellar phase via an Ia3d cubic structure. Cooling from the lamellar phase initially leads epitaxially to an intermediate R3 m before the hexagonal phase is reached. Both cubic and R3 m phases are formed by very similar rod units, but the overall structures differ because of their spatial distribution and they both bridge morphologically the hexagonal and lamellar phases. The Ia3d does so on heating, whereas the R3 m does on cooling. The structural path during the phase transitions is determined by topological similarities between the forming phase and the one from which it originates. Although the estimated curvature energies for these two phases are similar, on cooling, kinetics and topology are initial factors determining the path for the phase transitions, whereas on heating energy is the dominant factor.Aqueous suspensions of nonionic surfactants of the type poly(oxyethylene glycol) alkyl ether, C n EO m , are widely used in, e.g., extraction of membrane proteins, oil recovery, and cleaning processes. They show a rich polymorphism governed by the hydrophobic effect (1), forming among others lamellar, hexagonal, and cubic phases. In some cases the cubic phase is replaced by the so-called intermediate (or fluid) phases whose structures are still controversial. Characterization of these structures and their building units is important for the understanding of lyotropism. Moreover, these phases are also believed to occur as intermediate structures during biological processes-e.g., cell fusion (2).The mesogenic units can have different sizes and shapes, and from their organization different phases originate. They can range from disordered (liquid) to ordered structures in one (lamellar), two (hexagonal), and three dimensions (cubic). The most common cubic phases are formed by bicontinuous networks of rods (3). Gel and lamellar phases are stacked bilayers of amphiphiles and water, as shown in Fig. 1. The transition between these phases can be induced by, e.g., change in temperature or concentration. Most important, during such process there are significant changes in the curvature and morphology of the units (4).From the different amphiphile͞water systems studied it has been found that C n EO m molecules having long alkyl chains form stable intermed...