Bicontinuous cubic surfactant liquid crystals are thought to have structures consisting of bilayers of surfactant molecules centred on periodic minimal surfaces. The surfactant structures have been widely studied, and despite the growing wealth of demonstrated similarities with minimal surfaces there has been no conclusive evidence to support this theory. The most promising avenue for more convincing evidence is the comparison of theoretical and experimental single-crystal X-ray diffraction integrated intensities, which requires advanced theoretical models and difficult to obtain experimental samples.
Recent work by McGrath and Tate has produced single-crystal experimental diffraction data suitable for comparison with theoretical results. Our research focuses on creating mathematical models based on minimal surfaces for the three bicontinuous surfactant structures observed by McGrath and Tate. Corresponding numerical methods and theoretical advances for the simulation of X-ray diffraction of liquid crystals have also been developed.
We model the liquid crystals for X-ray diffraction purposes in two ways; as a continuous distribution of charge, and as a discrete set of atoms. The continuous model is a standard approach widely used in electron density reconstruction. It requires the diffracting region of the liquid crystal to be large. The atomic model provides upper and lower bounds on the relative intensities corresponding to small and large diffracting regions of the liquid crystal. These bounds confirm that the diffracting portion of the experimental crystals is comprised of a large number of unit cells, and that the assumptions of the continuous model are valid.
We test the suitability of each model together with the associated numerical methods by comparing theoretical and experimental single-crystal diffraction data. The results in these comparisons correlate very well, providing strong evidence that the bicontinuous cubic surfactant liquid crystal structures are based on mathematical minimal surfaces. These models provide the basis for an ongoing investigation into surfactant liquid crystal structures, crystal imperfections and diffuse scattering.
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A "unit cell" (repeating unit) of the double diamond, gyroid and primitive minimal surfaces. Minimal surfaces have minimum surface area for a given perimeter.
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