Translational repeat unit

In this study, we investigated a set of model polysilane chain systems that illustrate the basic physics and chemistry of some optical properties of these materials. In particular, we looked at the band structure for unsubstituted polysilane in an all-trans conformation, as well as in a 4/1 helical conformation with four silicon atoms contained in one translational repeat unit. In addition, we compared results for the dimethyl-substituted polysilane in an dl -trans conformation with the results for the unsubstituted poly silane. 

Stereoregularity means that the chain must be capable (by rotations about single bonds) of taking up a structure with translational symmetry. Translational symmetry means that the chain is straight on a scale large compared with the size of the chemical repeat unit and consists of a regular series of translational repeat units, so that the chain can be superimposed upon itself by translation parallel to the axis of the chain by the length of one translational repeat unit (see fig. 4.1). The translational repeat unit may be identical to the chemical repeat unit, but often there is more than one chemical repeat unit in a translational repeat unit. The most important... 

The important consequence of the polycondensation reaction is that the chain can only consist of DCH2CH20 units joined to -(-CO)— —(CO units in an alternating sequence. The single bonds along the backbone of the molecule are not collinear as shown in the chemical formula above, but it is nevertheless possible for the chain to acquire translational symmetry by suitable rotations aroxmd them and, for this molecule, the chemical and translational repeat units are the same. Another example is a polyamide such as nylon-6,6 (Bri-nylon)... 

More generally, isotactic means that all repeat units have the same handedness and syndiotactic means that the handedness alternates along the chain. Alternate chemical repeat units along the syndiotactic chain are mirror images of each other. In the translationally symmetric conformations shown in fig. 4.2 the translational repeat unit for the syndiotactic chain is two chemical repeat units, whereas it is only one for the isotactic chain. 

CF2 units in one translational repeat unit, as shown in fig. 4.7(b). This repeat unit corresponds to a twist of the backbone through 180°. In polyethylene a rotation of 180° is required to bring one CH2 group into coincidence with the previous one but, because of the twist, the angle required for PTFE to bring one CF2 group into coincidence with the next is 180° less 180°/13, or 12u/13, so that the structure is a 13g helix. 

Figure 3-4 shows the drawing of the actual molecule as well as of the simplified model with the labeling (1) of the sites of the translational repeating unit and the labeling of the internal coordinates of the 1 = 0 unit and the equivalent ones at sites -y/— s. The unit cell is made up by two masses (p = 2) and the in-plane internal coordinates are 2x2 = 4 which is also the size of the secular equation. We expect to calculate four branches in the dispersion relation, two of which are optical and two acoustical. 

The graph connecting the R, vciticcs (nodes) of the smallest translational repeating unit of a net. 

Figure 5. Crystalline substructures obtainable in 1 1 co-crystals of dmvatives of barbituric acid (B) and melamine (M). T = the number of B-M dimers that constitute a translational repeat unit along a tape (boxed) S = the number of tapes that constitute a translational repeat unit in a sheet 6 = the angle between tape axes in adjacent sheets. Tlie structures in this figure are representative examples of possible geometries and not an exhaustive list of all possible orientations of M andB.

Let us consider as example PHTh in the anti-coplanar-centrosym-metric structure with the alkyl side chains in the all transplanar conformation and lying in the plane of the Th rinqs. In such a model the translational repeat unit in the 1-d crystal contains two Th rings as in fig. 13 Similar considerations can be made for POTh. 

Fig.13 Translational repeat unit for polyhexylthiophene with centrosymmetric structure.

One example of an erroneous unit cell is the small square outlined near the center of the figure drawn in part (a). But notice that simply repeatedly translating this unit cell toward the right, so that its left edge sits where its right edge is now, will not generate the lattice. 

Polyelectrolyte complexes are formed by the ionic association of two oppositely charged polyelectrolytes [60,117-119]. Due to the long-chain structure of the polymers, once one pair of repeating units has formed an ionic bond, many other units may associate without a significant loss of translational degree of freedom. Therefore the complexation process is cooperative, enhancing the stability of the polymeric complex. The formation of polyelectrolyte complexes... 

Fig. 10.1. Illustration of a two-dimensional repeating pattern. The repeating unit cell, outlined with heavy lines, contains four symmetry-related flowers. The thin lines outline the lattice of translational symmetry. X marks the intersection of a two-fold axis with the plane of the page.

It is not always possible to choose a unit cell which makes every pattern point translationally equivalent, that is, accessible from O by a translation a . The maximum set of translationally equivalent points constitutes the Bravais lattice of the crystal. For example, the cubic unit cells shown in Figure 16.2 are the repeat units of Bravais lattices. Because nt, n2, and w3 are integers, the inversion operator simply exchanges lattice points, and the Bravais lattice appears the same after inversion as it did before. Hence every Bravais lattice has inversion symmetry. The metric M = [a, a ] is invariant under the congruent transformation... 

X-ray diffraction analysis can be used to characterize the crystallites themselves.15,68 First, it is possible to obtain information about the chain conformation adopted by the polymer chains in the crystallites. Many polymers crystallize in helical conformations. These can be defined by the following helical parameters. The designation pn tells how to rotate and simultaneously translate along an axis to generate the specified helix. First a point of reference is rotated around the proposed axis by nip times a complete rotation of In. Simultaneously, there should be a translation of nip of the crystallographic repeat distance, a quantity that is also obtained from diffraction data. Repetition of this scheme gives a helix having p repeat units in n turns of the helix. 

The most widely known ordered structure in polypeptides is the a-helix. When the repeating units are amino acid residues of l configuration, the a-helix is right-handed i.e., the chain backbone follows the pattern of the thread of a right-hand screw. Each turn of the helix is made up of 3.6 residues, so the helix is nonintegral. An exact repeat of the backbone structure occurs every 18 monomeric units. This repeat corresponds to five turns of the helix, with a linear translation along the axis of 27 A and a pitch of 5.4 A. The helix diameter, neglecting side chains, is about 6 A. 

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