Structure of Twin Boundaries in Tubular Block Copolymers

The appearance of twin boundary (TB) mirror defects in tubular double gyroid (DG) and double diamond (DD) networks suggests the networks can readily adapt into new low energy structures.   In the DG, the networks are trihedral with the smallest loop of 10 nodes; in the DD, the networks are tetrahedral with the smallest loop of 6 nodes. There are no mirror symmetries in DG, while there are many types of mirror planes present in the DD.  A twin plane in the DG acts as a topological mirror, causing the right-handed and left-handed DG networks to reverse chirality across the TB. The TB occurs on the (422) plane and intersects and alters the nodes in both DG networks via formation of 3 new types of achiral, mirror-symmetric nodes (2 trihedral and 1 tetrahedral) and the appearance of 2 new classes of achiral loops. In the DD, the 2 networks are identical and achiral, just translationally shifted. One of the networks has its nodes offset from the (222) TB plane while the other has nodes on the mirror.   The offset network has precisely the same structure as a (111) twin in carbon diamond.  For the 2^nd network of the DD, the twin results in formation of 2 new types of mirror symmetric nodes (alternating pentahedral and trihedral nodes) which form planar hexagonal loops.