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 p olymorphism. The bicontinuous cubic morphologies mediate the transform ation from a lamellar phase to a hexagonally-packed cylinder phase. Ho wever, certain bicontinuous cubic morphologies can theoretically trans form smoothly (without disruption or tearing) to other bicontinuous cu bic morphologies in response to variation in temperature and concentra tion. These bicontinuous phases are best understood in terms of their associated minimal surfaces. The minimal surfaces D (i.e, ordered bico ntinuous double diamond OBDD for block copolymer; cubic phase Q(224) f or 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 copolymer s) were computed and their two-dimensional projections on the plane re veals various 4-fold and 3-fold symmetries that are at times indisting uishable from that of the hexagonal phase. Moreover, because the surfa ces are homotopic, certain 2-D projections of the three bicontinuous c ubic phases are remarkably similar. However, the identification of bic ontinuous cubic morphologies from each other by various microscopy tec hniques could still be achieved provided that the number of domains pr esent in an experimental sample is large enough. Experimentally-obtain ed electron tomographs of sections of suspected bicontinuous phases ma y 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 sin ce diffractograms are generally difficult to obtain for the dilute sam ples commonly employed in amphiphile/water systems.