Bicontinuous Cubic Morphologies in Block Copolymers and Amphiphile/Water Systems: Mathematical Description through the Minimal Surfaces

Block copolymers and amphiphile/water systems both exhibit very rich polymorphism. The bicontinuous cubic morphologies mediate the transformation from a lamellar phase to a hexagonally-packed cylinder phase. However, certain bicontinuous cubic morphologies can theoretically transform smoothly (without disruption or tearing) to other bicontinuous cubic morphologies in response to variation in temperature and concentration. These bicontinuous phases are best understood in terms of their associated minimal surfaces. The minimal surfaces D (i.e, ordered bicontinuous double diamond OBDD for block copolymer; cubic phase Q 224 for amphiphile/water system), G (i.e., gyroid G* for block copolymer; cubic phase Q 230 for amphiphile/water system), and P (cubic phase Q 229 for amphiphile/water system; not yet reported for block copolymers) were computed and their two-dimensional projections on the plane reveals various 4-fold and 3-fold symmetries that are at times indistinguishable from that of the hexagonal phase. Moreover, because the surfaces are homotopic, certain 2-D projections of the three bicontinuous cubic phases are remarkably similar. However, the identification of bicontinuous cubic morphologies from each other by various microscopy techniques could still be achieved provided that the number of domains present in an experimental sample is large enough. Experimentally-obtained electron tomographs of sections of suspected bicontinuous phases may be compared with relative ease to the computed slices. These methods extend the range of concentrations in which bicontinuous cubic phases may be classified without the use of X-ray or neutron diffraction since diffractograms are generally difficult to obtain for the dilute samples commonly employed in amphiphile/water systems.