Hexagonally ordered cylinders

Block copolymers with well-defined segments often show microphase-separated morphologies (such as lamellar layers, hexagonal ordered cylinders, and micelle formation). If we use SCLCP blocks together with non-liquid crystalline segments, the mesophases are formed within one of the separated microdomains. If the non-SCLCP block has a higher Tg than the phase transition temperature of the mesophase, the amorphous block should physically support the SCLCP microdomains, forming a self-supported SCLCP system. 

Behavior that is intermediate between that of a solid and that of a liquid is perhaps not surprising for a block copolymer with hexagonally ordered cylinders, since such a material has solid-like order in the two directions perpendicular to the cylinders and liquid-like order parallel to the cylinders. Similar behavior is observed in lamellar block copolymers, which has solid-like order in the direction normal to the lamellae and has liquid-like order in the other two directions. For lamellar block copolymers, solid-like behavior at low frequencies typically arises from the disrupting effect of defects, such as those present in smectic liquid crystals (see Section 10.4.8). 

Also in bulk block copolymers microphase-separate into ordered liquid crystalline phases. A variety of phase morphologies such as lamellae (LAM), hexagonally ordered cylinders (HEX), arrays of spherical microdomains (BCC, FCC), modulated (MLAM) and perforated layers (FLAM), ordered bicontinuous structures such as the gyroid, as well as the related inverse structures have been documented. The morphology mainly depends on the relative block length. If, for instance, both blocks are of identical length, lamellar structures are preferred. 

The development of patterns is not necessarily a manifestation of a nonequilibrium process. A spatially non-uniform state can correspond to the minimum of the free-energy functional of a system in thermodynamic equihbrium, as Abrikosov vortex lattices, stripe ferromagnetic phases and periodic diblock-copolymer phases mentioned above. In the latter, a hnear chain molecule of a diblock-copolymer consists of two blocks, say, A and B. Above the critical temperature Tc, there is a mixture of both types of blocks. Below Tc, the copolymer melt undergoes phase separation that leads to the formation of A-rich and B-rich microdomains. In the bulk, these microdomains typically have the shape of lamellae, hexagonally ordered cylinders or body-centered cubic (bee) ordered spheres. On the surface, one again observes stripes or hexagonally ordered spots. 

Figure 1 Common morphologies of microphase-separated block copolymers body-centered cubic packed spheres (BCC), hexagonally ordered cylinders (HEX), g)Toid (Ia3d), hexagonaUy perforated layers (HPLs), modulated lamellae (MLAM), lamellae (LAM), cylindrical micelles (CYL), and spherical micelles (MIC). (Reproduced from Ref. 2. Wiley-VCH, 1998.)...

Figure 7.16 Experimental phase diagram of a lamellar (LAM) hexagonally ordered cylinders diblock copolymer melt polystyrene-poly- (HEX) bodycentered cubic (BCC) modulated...

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

Fig. 9 Schematic representation of three approaches to generate nanoporous and meso-porous materials with block copolymers, a Block copolymer micelle templating for mesoporous inorganic materials. Block copolymer micelles form a hexagonal array. Silicate species then occupy the spaces between the cylinders. The final removal of micelle template leaves hollow cylinders, b Block copolymer matrix for nanoporous materials. Block copolymers form hexagonal cylinder phase in bulk or thin film state. Subsequent crosslinking fixes the matrix hollow channels are generated by removing the minor phase, c Rod-coil block copolymer for microporous materials. Solution-cast micellar films consisted of multilayers of hexagonally ordered arrays of spherical holes. (Adapted from [33])...

In subsequent work, ordering in solutions of the same matched diblock and triblock spanning a broader range of volume fractions, 0.1 < < 0.4, was explored (Hamley et al. 1997). For liquid-like and SAXS showed that there was no inter-micellar order in the liquid. Above a crossover concentration 0.2, ordering of micelles was shown by the presence of a structure factor peak. The ordered micellar structure, identified as hexagonal-packed cylinders for more concentrated solutions, persisted up to an order-disorder transition located from a discontinuity in the... 

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