Tetrahedral frameworks Three- or two-dimensional structures

Clearly, the hyperbolic two-dimensional picture, which assumes constant surface density (area per Si02, H2O, Si or Ge group) is more realistic than the classical Euclidean three-dimensional model, which supposes fixed bond angles, lengths and torsion. 

In all the cases analysed here, those frameworks realised in the laboratory (or in nature) do lie near to the dotted curves deduced from reasonable bond lengths and angles. Remarkably though, the variation in bulk density follows the locus of the curves assuming constant area. The intersection of the two curves determines the preferred intrinsic curvature, characterised by the ring size. 

If then we abandon the standard three-dimensional Euclidean perspective and adopt this non-Euclidean two-dimensional view, it can be seen that stable polymorphs are characterised by a global geometric constraint surface density 2 1, and a local constraint Gaussian curvature, K . We shall see in Chapter 4 that this description is identical to one that accounts for the mesophase behaviour of lyotropic liquid crystals in amphiphile-water mixtures. 

How do these unconventional ideas link with the standard view of a solid as a close packed array of atoms Evidently most of the frameworks discussed above cannot be so characterised. The two-dimensional hyperbolic picture does break down for very dense structures. Thus the densest four-coordinated silicate, coesite, violates this universality (see Fig. 2.12). (Its ring size is less than that of trid5m[ ite, cristobalite, keatite or quartz, in spite of its higher density.) This polymorph is too dense for a two-dimensional description to be useful and the Aree-dimensional description takes over. The notion of intrinsic curvature is less rigid for silicates than for the other frameworks, because the Si-O-Si angle usually differs from 180 . 

The hyperbolic description implies that to a reasonable approximation, tetrahedral water, silicate, silicon and germanium frameworks are characterised by a preferred area per vertex group and a preferred Gaussian curvature. Thus, identical tessellations of isometric surfaces, with equal areas and curvatures at corresponding points on the surface, should offer alternative possibilities for stable frameworks. Indeed this is the case for the zeolite frameworks, faujasite and analcime, which are related to each other through the Bonnet transformation. Within an intrinsic two-dimensional description, these two frameworks are indistinguishable. We have seen in section 2.6 that the Bonnet transformation describes well a number of characteristics of the fee - bcc martensitic phase transformation in metals and alloys. The success of this model suggests that the hyperbolic picture, intuitive and obvious for zeolites, is also valid for other atomic structures. 

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