Neovius surfaces

The polyhedron corresponding to the Neovius surface has the same arrangement of points as that for the infinite semi-regular polyhedral surface 6.43 discussed above, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration of 8.4.8.6 and this leads to a surface of genus of 9. This surface has two kinds of flat points and is thus not regular (Mackay Terrones 1991). 12 tubes in the [110] directions connect cavities. 

Figure 1.27 A conventional unit cell of the Neovius surface.

Figure 7b. Solution wth H = 1.05, same space group and topological type as the Neovius surface.

However, H. A. Schwarz found before 1865 that patches of varying negative gaussian curvature and constant H = 0 could be smoothly joined to give an infinite triply periodic surface of zero mean curvature. About five different types were found by Schwarz and Neovius, but now about 50 more have been described (Schoen 1970 Fischer Koch, 1989 a e). 

Figure 7a. The Neovius minimal surface C(P), space group Pm3m and Euler characteristic — 16 per lattice-fundamental region. One unit cell is shown, which is also a lattice-fundamental region. 

Fig. 2 BCP-derived morphologies (note that Lidinoid, I-WP, K surface, and Neovius structures have not been observed in BCP SA yet). The knitting pattern was reprinted with permission from [8] Copyright 2001 American Chemical Society. The woodpile structure was reprinted with permission from [9] Copyright 2008 American Chemical Society...

Although a few 3D isotropic cubic network phases have already been observed, other cubic network phases have stayed elusive in BCP SA, such as the I-WP [25], Neovius [26], K surface [27], and Lidinoid [28] structures (see Fig. 2). Packing frustration of polymer chains in the network nodes of these structures is a primary hurdle for formation of these phases. However, as shown in the cases of the plumber s nightmare and double diamond structures, BCP co-assembly with additive molecules or ordered local packing of monomer units may open new routes to such cubic structures. 

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