Polyethylene structure: orthorhombic unit cell with

The crystal structures of C2Ht—CO copolymers with QH4/CO ratios of 1, 1.3, 2.2 and 3.5 have been determined49,50). For the 1 1 copolymer an orthorhombic unit cell of dimensions a = 7.97, b = 4.76, c (fiber axis) = 7.57 A was observed 49). The main chain had a planar zigzag form. For copolymers with higher C2Ht contents, the fiber periods were essentially identical with that of polyethylene (c 2.54 A) 50). Also, the higher the C2H content, the shorter the a axis and longer the b axis. 

Commercial PVC is very nearly atactic and therefore crystallises only to a very small extent. Syndiotactic PVC, which can be made only by a rather special polymerisation technique, crystallises with an orthorhombic unit cell. Like the unit cell of polyethylene, there are two chains passing through it, but this time they are oriented at 180° to each other, with the planes of both backbones parallel to the -axis. The cell dimensions are a = 1.026, b = 0.524 and c = 0.507 nm. The structure is shown in fig. 4.18. 

The results of such calculations for semi-crystalline polyethylene have been reviewed elsewhere [37]. A rather wide range of predicted values is obtained, due to the choice of force constants and also to sensitivity to detailed assumptions on the unit cell structure. In spite of these limitations the principal predictions for the elastic anisotropy are clear. These include the anticipated high values for C33 and the very low values for the shear stiffnesses C44, C55 and cee, which reflect the major differences between bond stretching and bond bending forces that control C33 and the intermolecular dispersion forces that determine the shear stiffnesses. It is therefore of value to compare such theoretical results with those obtained experimentally. Table 7.3 shows results for polyethylene where data for the orthorhombic unit cell at 300 K are used to calculate these constants for an equivalent fibre (Voigt averaging procedure see Section 7.5.2 below) compared with ultrasonic data for a solid sheet made by hot compaction. It can be seen that... 

A third, less obvious limitation of sampling methods is that, due to the heavy computational burden involved, simpler interatomic potential models are more prevalent in Monte Carlo and molecular dynamics simulations. For example, polarizability may be an important factor in some polymer crystals. Nevertheless, a model such as the shell model is difficult and time-consuming to implement in Monte Carlo or molecular dynamics simulations and is rarely used. United atom models are quite popular in simulations of amorphous phases due to the reduction in computational requirements for a simulation of a given size. However, united atom models must be used with caution in crystal phase simulations, as the neglect of structural detail in the model may be sufficient to alter completely the symmetry of the crystal phase itself. United atom polyethylene, for example, exhibits a hexagonal unit cell over all temperatures, rather than the experimentally observed orthorhombic unit cell [58,63] such a change of structure could be reflected in the dynamical properties as well. 

It can be seen that a rather wide range of predicted values is obtained that is partly due to choice of different force constants. The results are also sensitive to the details of the assumed crystal unit cell structure, especially the angle made by the plane of the planar zigzag polyethylene chain with the b-axis of the orthorhombic unit cell. The overall pattern of elastic anisotropy is however clear. The stiffness in the chain axis direction C33 is by far the greatest value, and the shear stiffnesses C44, C55 and Cee are the lowest values. This reflects the major differences between the intramolecular bond stretching and valence bond bending forces and the intermolecular dispersion forces, which determine the shear stiffnesses. The lateral stiffnesses also relate primarily to dispersion forces and are correspondingly low. 

The most widely studied crystalline polymer is polyethylene. The lowest-energy chain conformation of this macromolecule can be evaluated from considerations of the conformations of the n-butane molecule (Fig. 3.5). A series of all trans bonds produce the lowest-energy conformation in n-butane because of steric interactions and this is reflected in polyethylene where the most stable molecular conformation is the crystal with all trans bonds, i.e. the planar zig-zag. These molecules then pack into the orthorhombic unit cell given in Table 4.1. The arrangement of the molecules in the polyethylene crystal structure is illustrated in Fig. 4.5 with the polymer molecules lying parallel to the c axis. The molecules are held... 

Polyethylene is orthorhombic, with two chains per unit cell. The cell dimensions are a = 0.742, b = 0.495 and c = 0.255 nm. The angle between the plane of the zigzag backbone and the )-axis is approximately 45° and the planes of the two chains passing through the unit cell are at right angles to each other. The chains are almost close-packed, as discussed in section 4.2.3. The structure is shown in fig. 4.17. 

The unit cell structure of polyethylene was first investigated by Bunn (20). A number of experiments were reviewed by Natta and Corradini (21). The unit cell is orthorhombic, with cell dimensions of a = 7.40, b = 4.93, and c = 2.534 A. The unit cell contains two mers (see Figure 6.5) (22). Not unexpectedly, the unit cell dimensions are substantially the same as those found for the normal paraffins of molecular weights in the range 300 to 600 g/mol.The chains are in the extended zigzag form that is, the carbon-carbon bonds are trans rather than gauche. The zigzag form may also be viewed as a twofold screw axis. 

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). 

The crystalline state may exist in polymeric materials. However, because it involves molecules instead of just atoms or ions, as with metals and ceramics, the atomic arrangements will be more complex for polymers. We think of polymer crystallinity as the packing of molecular chains to produce an ordered atomic array. Crystal structures may be specified in terms of unit cells, which are often quite complex. For example. Figure 14.10 shows the unit cell for polyethylene and its relationship to the molecular chain structure this imit ceU has orthorhombic geometry (Table 3.2). Of course, the chain molecules also extend beyond the unit cell shown in the figme. 

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