Crystal structures, polymers symmetry group

A number of selected crystal structures of polymers are shown in this section in projection along the helix axes. The poly(ethylsilylethylene) of Fig. 5.23 has its silicon atoms marked by solid circles. Neighboring helices are related by a center of symmetry, so that they must be enantiomorphous and also anticlined, i.e., the helix pairs have different handedness (d-RH and f-LH helices) and opposite inclinations of side-groups (up- and down-helices) as discussed in Sect. 5.1.8. The coordination number for the helices is three instead of the expected four because the 31 and 32 screw axes of the 2 3/1 helices match the trigonal lattice symmetry and permit a closer overall packing with CN = 3 rather than 4 (see Fig. 5.21). 

A nnmber of techniques are appropriate to investigate the hierarchy of structnres formed by crystalline polymers. Crystallized polymer chains form crystal structures with lattices built up by translation of unit cells, just like crystals formed by low molar mass compounds. The space group symmetry depends on the polymer under consideration and also the conditions of the sample. For example, polyethylene usually forms a structure belonging to the orthorhombic crystal system, but at high pressures it is possible to obtain a hexagonal structure. Because it can adopt more than one crystal structure, polyethylene is said to be polymorphic. The best way to determine the crystal structure of a polymer is to perform wide-angle x-ray scattering (WAXS) experiments. WAXS on oriented polymers also provides information on the orientation of crystalline stems (chains). 

Polymers may exhibit liquid crystalline, ordered structures similar to those of small molecular liquid crystals. As polymer chains are very long, containing about 10 -10 monomer units, only parts of them or side groups can be ordered in oeir tic. cholesteric, and different kinds of smectic structures. As in small-molecular nematic liquid crystals, in nematic polymer structures groups of cylindrical-like symmetry are arranged statistically with their long axes along a certain direction. 

It is worth noting that the domain containing the center of symmetry is empty for proteins (Figure lOd), as it is the case for synthetic polymer crystals where the presence of an inversion center is rare. The helicoidal conformation of proteins (as well as that of synthetic polymers) implies a high frequency of groups with 2X screw axes [31], It is important to note, however, that the statistics for proteins have been done only on about two hundred structures. 

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