Scattering form factors for self-assembled network junctions

The equilibrium microstructures in microemulsions and other self-assembled systems show complex, connected shapes such as symmetric bicontinuous spongelike structures and asymmetric bicontinuous networks formed by cylinders interconnected at junctions. In microemulsions, these cylinder network microstructures may mediate the structural transition from a spherical or globular phase to the bicontinuous microstructure. To understand the structural and statistical properties of such cylinder network microstructures as measured by scattering experiments, models are needed to extract the real-space structure from the scattering data. In this paper, we calculate the scattering functions appropriate for cylinder network microstructures. We focus on such networks that contain a high density of network junctions that connect the cylindrical elements. In this limit, the network microstructure can be regarded as an assembly of randomly oriented, closed packed network junctions (i.e., the cylinder scattering contributions are neglected). Accordingly, the scattering spectrum of the network microstructure can be calculated as the product of the junction number density, the junction form factor, which describes the scattering from the surface of a single junction, and a structure factor, which describes the local correlations of different junctions due to junction interactions (including their excluded volume). This approach is applied to analyze the scattering data from a bicontinuous microemulsion with equal volumes of water and oil. In a second approach, we included the cylinder scattering contribution in the junction form factor by calculating the scattering intensity of Y junctions to which three rods with spherical cross section are attached. The respective theoretical predictions are compared with results of neutron scattering measurements on a water-in-oil microemulsion with a connected microstructure.