Polymer chains symmetry operators

Smallest structural unit of a polymer chain with a given conformation that is repeated along that chain through symmetry operations [5]. 

The stereocenters in all three stereoregular polymers are achirotopic. The polymers are achiral and do not possess optical activity. The diisotactic polymers contain mirror planes perpendicular to the polymer chain axis. The disyndiotactic polymer has a mirror glide plane of symmetry. The latter refers to superposition of the disyndiotactic structure with its mirror image after one performs a glide operation. A glide operation involves movement of one structure relative to the other by sliding one polymer chain axis parallel to the other chain axis. 

The symmetry operations of rotation, reflection, inversion and glide reflection obey all the tenets of group theory [3,5]. Matrix equations can express the results of these operations. The trace or spur of diagonal matrices (Appendix 3D) is particularly valuable in condensing the information contained in symmetry operations on polymer chain subunits. The terms irreducible representation, symmetry type, and species are also used in denoting the trace [12]. 

The structural characteristics of the polymer chain constructed as described above are that translational or roto-translational symmetry can be found along the chain in addition to the traditional point group symmetry operations (rotation axes, symmetry planes, inversion center, and identity) [18]. 

Metallocene catalysts of type 12.8 have been found to be highly selective fw the formation of isotactic polypropylene. Structure 12.8 is chiral because it lacks a plane of symmetry. Like the asymmetric hydrogenation catalysts discussed in Chapter 9,12.8 also has a Cz symmetry axis, so both binding sites, both occupied by Cl ligands in 12.8 and 12.9 have the same chirality. Each new propylene monomer that is incorporated is therefore expected to mter into the polymer chain with the same chirality, giving isotactic polymer, whichever binding site is operative for any givai step. 

The chain program was developed to be able to take into account simple translation as well as helical symmetry. In the case of a helical polymer we can define a screw operator S(a, 0)88 with a translation along the z-axis (the long axis of the polymer) and with a right-handed rotation about the z-axis. Applying this operator to the position vector r one obtains... 

In most theoretical investigations a stereoregular polymer is described by an infinite, extended and isolated chain constructed from a periodic sequence of monomer units. In addition to translational symmetry, stereoregular polymers possess some other symmetry elements like screw axes, mirror, or glide planes. The related operations combine into groups, the line groups (I6a-c). 

The most straightforward application of the entropic principle is the driving force that induces crystallization of polymers with chains in s M/N) helical conformation in crystalline lattices containing the screw M/N operator, so that the local helical symmetry of the chains becomes a crystallographic symmetry. According to this... 

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