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The phase diagram of the ternary surfactant system tetradecyldimethylamine oxide/heptanol/water is reported for small surfactant concentrations. With increasing cosurfactant/ surfactant ratio X, the generally observed sequence of the phases Li, La, and L3 is found. These single phases are separated by narrow two-phase regions. Detailed investigations show however that the wide L"-phase region consists of at least two subphases, a L«i and a L"h phase which have different macroscopic properties. Freeze fracture electron micrographs which were produced from the Lai phase show clearly multilamellar vesicles which have large holes in their bilayers and are therefore called "perforated vesicles". For small surfactant concentrations around 50 mM, two more isotropic phases are observed besides the Li and L3 phases which are called Li* and L3*. Electron micrographs of the L3 phase show that it consists of continuous bilayers which form a three-dimensional multiply connected network with saddle-shaped structures. Typical for the micrographs are fractures across the tubular bilayers, resulting in oval-shaped patterns. The images furthermore show that the preferred fracture of the L3 phase follows the midplane of the bilayer. The electron micrographs of the L3 phase are reported for concentrations between 50 and 400 mM. The photos show that the typical length scale of the L3 phase, that is, the average diameter across a tubular structure, is of the same size as the interlamellar spacing in the neighboring La phase and varies inversely with the volume fraction of the surfactant (d *=~1 2345). The bilayers in the L3 phase and the La phase which in some cases were produced in the preparation process for freeze fracturing were generally very flat on a local scale and showed no undulations on a microscopic level. The phase boundary between the La and the L3 phases was very sharp and smaller than the length scale of the L3 phase. Electric birefringence data which were obtained on the L3 phase show that both the amplitude and the time constant follow simple scaling laws. The relaxation time is inversely proportional to the third power of the volume fraction while the amplitude is inversely proportional to the second power.

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