Gyroid surfaces

W. Gozdz, R. Holyst. High genus gyroid surfaces of non-positive gaussian curvature. Phys Rev Lett 76 2726-2729, 1996. 

Figure 1.21 The gyroid surface discovered by Schoen in the 1960 s. (Top view down [111] axis of a larger partion of the surface bottom solid model.)...

Another critical factor is the overall molecular shape, or local packing requirement, discussed in Chapters 3 and 4. The volume per unit surface (expressed as v2/3/a) for the P-, D- and gyroid surfaces of the same local... 

However, there is a structure consistent with both the required space group and the optical properties. The gyroid surface, which occurs frequently in lipid-water systems, provides such a possibility. If we assume that cholesterol skeletons form rod-like infinite helices, this structure represents an effective three-dimensional packing of such helices. Thus, the rods form a body-centered arrangement as shown in Fig. 5.5. In this structure, there is a helical twist between the rods, in addition to the cholesteric twist within each rod. The h)rperbolic structure is a consequence of the chirality of the esters, which induces torsion into the packing arrangement. A racemic mixture does not exhibit this phase natural cholesteric esters contain a single enantiomer only. 

Fig. 2.2 Single-gyroid surfaces calculated using Eq. (2.5). a Minimal gyroid surface for 1=0. b Two helical interpenetrating single-gyroid networks separated by the minimal gyroid surface, c Single-gyroid for 1 = 1.3. d Pinch-off surface for t = 1.413...

The minimal gyroid surface divides space into two helical regions or networks, each claiming a volume fraction of 50 %, see Fig. 2.2b. The surface embeds an inversion center which interchanges the two sides of the surface and also the two regions which it partitions. The separated networks are enantiomorphic, which means that one network is left-handed, the other one right-handed [9]. 

In the interval 0 < t < 1.413 the DG level surface effectively divides space into three embedded continuous subvolumes. This phase which is located between the two SG surfaces, centered on the minimal gyroid surface, forms the so-caUed matrix phase through which the two networks run. Each one of the three networks is periodic in all three principle directions. The matrix phase volume increases with increasing values of t, and thereby gradually reducing the strut diameters of the two enclosed networks. 

The values of x, y, z) that solve the following quadruple level surface equation generate the core-shell gyroid surface... 

Unfortunately there are no lUPAC or lUCr recommendations, or even a consensus among the scientists in the field, about the nomenclature of 3D-nets, and several naming systems are currently in use. As noted by O Keeffe et al. some have many names and symbols...other structures have no names at all and they give the example of the srs net that is also know as (10,3)-a , Laves net , Y , 3/10/cl , SrSi2 and labyrinth graph of the gyroid surface [1,2]. It can also be described by various sets of numbers, the most complete being lOs lOs lOs, referred to as the vertex symbol. Unfortunately, there is no present nomenclature that creates a set of numbers for each net that can be proven to be unique. 

Figure 4.4 Left The srs or (10,3)-a net also known as Laves net , Y , 3/IO/ct , SrSii and labyrinth graph of the gyroid surface" [1J. Right A 10-ring contained in this net.

The most intriguing cases of lower symmetry relatives to known mesophases are anisotropic bicon-tinuous mesophases. These are sponges whose homogeneity lies one rank below the cubic genus-three P, D and gyroid surfaces. The most likely candidates are tetragonal and rhombohedral variants. These include the rPD, tP, tD, tG and rG triply periodic minimal surfaces (18). These surface are deformations of their cubic parent structures, and can be modelled as perturbations of the known bicontinuous cubic mesophases. 

In the case of metal complexes, Bmce et al. [27] proved the existence of space group laid on the basis of well developed cubic monodomains. The structural model is discussed on the basis of the gyroid surface G, where the silver atoms are located on the rods (Fig. 8) and the terminal... 

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