Films of amphiphiles: packing constraints and phase diagrams

The polymorphism of systems of amphiphilic molecules in liquid crystalline and micellar regions of their phase diagrams is characterized by similar sequences of phases, whatever the details of the chemical structures of the molecules. For instance, varying the dominant parameter of the phase diagram and starting from a lamellar phase, the most general sequence continues with cubic and/or cylindrical phases and ends with micellar phases, each step of the sequence being characterized by a dramatic change of the topology of the organization of the molecules. As this behavior is independent of the chemical details of the amphiphilic molecules, it must be related to their common property of building interfaces organized in symmetric films. We have therefore applied our recent analysis of systems of amphiphiles as periodic systems of frustrated fluid films to this problem. We recall that, in our view, a frustration arises from the fact that symmetric interfacial curvatures different from zero and constant interfacial distances are not compatible in Euclidean space. In our preceding papers we demonstrated that the observed structures can be understood as the geometrical configurations adopted by the systems to optimize the frustration, i.e., to conciliate the two constraints, by spontaneous creations of disclinations. We now examine the properties of these geometrical configurations in detail. We show that, for each of them, any interfacial curvature cannot be fitted with any interfacial distance. They are indeed related by specific functions in each case, and the set of these relations is such that the configurations must appear along the sequence described above when the frustration varies with the parameters of the phase diagrams.