Quantifying packing frustration energy in inverse lyotropic mesophases

In this paper we investigate a simple model of inverse lyotropic mesophase energetics. The total energy is constructed by first minimizing the curvature elastic energy for the interface and then calculating the energy tied up in the chain extension variations that result for the various interfacial shapes and crystallographic space groups. We have calculated the chain packing energy in the harmonic approximation and find that we can separate this into two distinct terms. The first of these we call the packing factor, which is a constant for each interfacial shape and its associated crystallographic space group. The second term describes the variation in the packing frustration energy with mean curvature and monolayer thickness for the different interfacial shapes, i.e. spherical, cylindrical, and hyperbolic. Using this formalism and optimizing the mean interfacial curvature, we are able to build a global phase diagram in terms of the spontaneous mean curvature and the molecular length. The phase diagram we construct from the model places the phase boundaries between the inverse bicontinuous cubic, inverse hexagonal, and inverse micellar cubic phases in the expected regions of the diagram. This gives some encouragement to the widely held notion that the competition between interfacial curvature and hydrocarbon packing constraints can be used to explain lyotropic mesomorphism. However, the model is overly simplistic and breaks down in the regimes where the average interfacial curvature is at its greatest. Specifically it predicts that a body-centered cubic arrangement of inverse micelles is of lower energy than an Fd3m packing, but the latter are the only arrangements which have been found to date.