Polymers, chain type symmetry

The parameters K1/ K2/ and K3 are defined by the refractive indices of the crystal and sample and by the incidence angle [32]. If the sample has uniaxial symmetry, only two polarized spectra are necessary to characterize the orientation. If the optical axis is along the plane of the sample, such as for stretched polymer films, only the two s-polarized spectra are needed to determine kz and kx. These are then used to calculate a dichroic ratio or a P2) value with Equation (25) (replacing absorbance with absorption index). In contrast, a uniaxial sample with its optical axis perpendicular to the crystal surface requires the acquisition of spectra with both p- and s-polarizations, but the Z- and X-axes are now equivalent. This approach was used, through dichroic ratio measurements, to monitor the orientation of polymer chains at various depths during the drying of latex [33]. This type of symmetry is often encountered in non-polymeric samples, for instance, in ultrathin films of lipids or self-assembled monolayers. 

A few examples of the moduli of systems with simple symmetry will be discussed. Figure 1A illustrates one type of anisotropic system, known as uniaxial orthotropic. The lines in the figure could represent oriented segments of polymer chains, or they could be fibers in a composite material. This uniaxially oriented system has five independent elastic moduli if the lines (or fibers) ara randomly spaced when viewed from the end. Uniaxial systems have six moduli if the ends of the fibers arc packed in a pattern such as cubic or hexagonal packing. The five engineering moduli are il-... 

This brings us to double stereoselection and reinforcement of the mechanisms. If the site (a)symmetry were to control the orientation of the chain, and if, then, the orientation of the incoming propene is controlled by both the chain and the site, the highest stereoselection is obtained when the two influences reinforce one another. For 1,2-insertion this can be done most effectively for isotactic polymerization, since chain-end control naturally leads to isotactic polymer and this we can reinforce by site control with ligands of the bis(indenyl)ethane type. The chain-end influence of short chains is smaller than that of longer polymer chain and therefore short chain ends lead to lower selectivities. It may also be irrferred that making syndiotactic polymer via a 1,2-insertion mechanism on Ti or Zr complexes is indeed more difficult than making an isotactic polymer, because the two mechanisms now play a counterproductive role. 

As an example for a one-dimensional property transfer, the preparation of optically active co-polymers using chiral template molecules is described. In this case, the optical activity in the polymer arises from the chirality of the configurational arrangement of the main chain (main chain chirality). Symmetry considerations show that there are several possible structures both for stereoregular homo- and for co-polymers in which this type of chirality can be expected [4]. 

In the electrochemical sense the p-type doping reaction consists of anodic oxidation of neutral polymer chains to polycations with simultaneous insertion, to the polymer matrix, of electrolyte originating anions which preserve the electrical neutrality of the system. By symmetry, n-type doping is achieved by cathodic reduction of the polymer chains to polyanions, with... 

The formation of stable crystalline regions (crystallites) in a polymer requires (a) that an efficient close-packed arrangement of the polymer chains can be achieved in three dimensions and (b) that a favorable change in internal energy is obtained during this process. This imposes restrictions on the type of chain that can form crystallites easily and one would expect linear chains with high degree of symmetry such as polyethylene, polytetrafluoroethylene, various polyesters, and polyamides to crystallize most readily. 

Two important types of polymer chain are the planar zigzag and the helix, but whatever form the chains take in crystallites, they must be straight on a crystallite-size scale and the straight chains must pack side by side parallel to each other in the crystal. The ideas of standard bond lengths, angles and orientations around bonds discussed in chapter 3 can be used to predict likely possible low-energy model chain conformations that exhibit translational symmetry (see example 4.3). As discussed in the... 

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]. 

A second, and often more important, type of configurational variation is stereoisomerism (Flory, 1953 Lenz, 1967, pp. 252-260 Schultz, 1974), which arises from differences in symmetry between substituted carbons along a polymer chain. In small molecules, carbon atoms that are attached to four different groups are clearly asymmetric, and mirror image isomers and optical activity are observed. For example, assuming tetrahedral bonding to the carbon atom,... 

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