Honeycomb symmetry

CL-DNA complexes form spontaneously when solutions of cationic liposomes (typically containing both a cationic lipid and a neutral helper lipid) are combined. We have discovered several distinct nanoscale structures of CL-DNA complexes by synchrotron X-ray diffraction, three of which are schematically shown in Fig. 1. These are the prevalent lamellar phase with DNA sandwiched between cationic membranes (Lo,c) [22], the inverted hexagonal phase with DNA encapsulated within inverse lipid tubes (Hnc) [23], and the more recently discovered Hj0 phase with hexagonally arranged rod-like micelles surrounded by DNA chains forming a continuous substructure with honeycomb symmetry [24]. Both the neutral lipid and the cationic lipid can drive the formation of specific structures of CL-DNA complexes. The inverse cone shape of DOPE favors formation of the... 

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

The single cylindrical pore is of course not the geometry we are interested in for porouS catalysts, which may be spheres, cylinders, slabs, or flakes. Let us consider first a honeycomb catalyst of thickness It with equal-sized pores of diameter cfp, as shown in Figure 7-14. The centers of the pores may be either open or closed because by symmetry there is no net flux across the center of the slab. (If the end of the pore were catalytically active, the rate would of course be sHghtly different, but we will ignore this case.) Thus the porous slab is just a collection of many cylindrical pores so the solution is exactly the same as we have just worked out for a single pore. 

Single crystals of /S-A1203 are essentially two dimensional conductors. The conducting plane has hexagonal symmetry (honeycomb lattice). This characteristic feature made -alumina a useful model substance for testing atomistic transport theory, for example with the aid of computer simulations. Low dimensionality and high symmetry reduce the computing time of the simulations considerably (e.g., for the calculation of correlation factors of solid solutions). 

This also applies to the macroscopic properties of composite materials with underlying symmetry—like honeycomb, wood, and woven materials—for which the crystal structure, if any, may play no direct role. 

Fig. 60 Examples of anchor-shaped (bent-core) bolaamphiphiles and comparison of the hexagonal honeycombs (a) 6-molecule hexagons (p6mm symmetry) as formed by 186b) and (b) 3-molecule hexagons (p iml symmetry) as formed by 186a (dark gray = fluorinated cores, small light gray dots = glycerols) [344]...

PPO repeating units self-assemble into a supramolecular honeycomb-like layered structure, in which perforations are filled by coil segments. When cast from dilute CHCR solution onto a carbon support film, honeycomb-like supramolecular structure was observed, as revealed by transmission electron microscopy (TEM), in which coil perforations are packed on a hexagonal symmetry with distances between perforations of approximately 10 nm (Figure 14a). 

The 2-D hexagonal materials with honeycomb arrays of nonintersecting primary channels, its symmetry (space group) is 2-D hexagonal p6mm. Typical materials include ... 

A strategy to manipulate supramolecular structures assembled from rod segments may be accessible by the alteration of the coil architecture (linear (5) versus branched (6)) in the rod-coil system [36]. On the basis of SAXS and TEM results, rod-coil molecules (5) with a linear PPO coil showed a honeycomb-like lamellar rod assembly with hexagonally arrayed PPO coil perforations, while the rod-coil molecules (6) with a dibranched PPO coil self-organized into rod-bundles with a body-centered tetragonal symmetry surrounded by a PPO coil matrix (Fig. 3). The notable feature is that a sim-... 

Both well-ordered tetrapropylammonium cations in the asymmetric unit occupy special positions of site symmetry 2 in space group C222i atoms N(5) and N(6) are situated at Wyckoff positions 4(b) at (1/2, y, 1/4) and 4(a) at (., 0, 0), respectively. Figure 20 shows a honeycomb-like double layer at z = 1/4 with large octagonal windows and (ti-C3H7)4N(5) cations trapped within it. When the crystal structure of 2.3 is viewed parallel to the a axis, the ( -C3H7)4N(6) cations are seen to be concentrated about the (002) planes and sandwiched between adjacent double layers. 

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