Gyroid Materials: triply periodic minimal surfaces, skeletal graphs, mesoatoms and crystallographic defects*

Triply periodic minimal surfaces (TPMS) have zero mean curvature and their associated constant mean curvature families are all quite good structural models for many self-assembled, bicontinuous amphiphilic soft materials, ranging from water and cation-soap systems to block copolymers. The G or gyroid TPMS discovered by Schoen in 1970 subdivides space into two intertwined labyrinths. These labyrinths can be reduced to a pair of enantiomorphic 3-connected skeletal graphs. Structural aspects of crystalline materials have been depicted with respect to the networks. Laves first used network graphs to describe silicate structures in 1932. In his 1954 paper on 3 connected nets, Wells described this same periodic graph, denoted the 10-3a, indicating the smallest closed loop contained 10 segments, each of which was 3 connected. Luzzati and Spegt used small angle xray scattering to solve the structure of a self-assembled strontium soap in 1967. The polar groups formed the two interpenetrating networks of 3-connected rods embedded in the hydrocarbon matrix having the cubic Ia3d space group. Schoen's 1970 NASA report debuted the gyroid TPMS and referred to the Laves' periodic graph. Only in 1994 did researchers find that block copolymers could also form a gyroid structure. Here the G TPMS bisects the majority A block domain while the two minority B domains surround the pair of skeletal graphs. In soft matter, the structural units correspond not to atoms as in hard matter but to large groups of molecules aggregating into mesoatoms. In the case of block copolymers, a mesoatom can contain millions of atoms. Mesoatoms exhibit point group symmetries, shapes and sizes that satisfy space group symmetry and pack together to fill the unit cell. As with all crystalline materials, symmetry breaking crystallographic defects occur during growth and from mechanical forces. Point, line and surface defects induce alteration of the malleable mesoatoms near the defects. Because the bicontinuous gyroid presents interesting topology as well as geometry, the structure and its various defects afford multiple opportunities for technological applications ranging from photonics to batteries.