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We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval
such that, for each
, all Birkhoff minimizers with rotation number
do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.
Complex phase transformation between the hexagonal cylinder (Hex) and double gyroid (G) phases in a polystyrene-block-poly(ethylene oxide) (PS-b-PEO) diblock copolymer was investigated using two-dimensional (2D) synchrotron small-angle X-ray scattering (SAXS), and transmission electron microscope (TEM). The PS-b-PEO sample contained a small population of another bicontinuous cubic phase having an Im3m symmetry. These two bicontinuous cubic phases (G and Im3m) had the same unit cell dimensions. Under a large-amplitude reciprocating shear, the bicontinuous cubic phases transformed into a “single-crystal”-like Hex phase. When annealed at 150 °C for 40 min, the Hex phase partially transformed into well-oriented twinned structures of the G and Im3m phases without significant loss of orientation in 2D SAXS measurements. Epitaxial phase transformation relationships between the Hex/G and Hex/ Im3m phases were identified. The mechanism of the Hex → G transformation was examined by TEM.
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