Thermal Analysis of Polymeric Materials

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regions of stability. Also, since the poly(L-alanine) is truly optically active (see Appendix 14). The symmetry about the 360 to 360o diagonal in the potential-energy diagram, present for polypropylene, is disturbed. The right-handed helix (RH) has a lower energy than the left-handed helix (LH).

Poly(L-alanine) can stabilize the RH 3*18/5 helix, the well-known -helix, by intramolecular hydrogen bonds, indicated by the dotted lines in Fig. 5.16 A. Another stable conformation, called the pleated sheet, must make the H-bonds inter-

Fig. 5.16

molecularly, as shown by the sketch in Fig. 5.16 B. The different conformations of the amino acid residues have a marked influence on their structural functioning in biological systems, such as in globular and structural proteins.

It is of interest to note that one may change the translation lattice of Fig. 5.3 by

replacing the translation lattice vector c with the molecular helix lattice, keeping the

translation symmetries a and b . This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal structure symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1*1/1 helix has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case.

5.1.7 Packing in Crystals

To produce crystals of lowest free enthalpy, a hierarchy of steps is necessary, involving a decreasing amount of interaction energy [6]. First, all strong bonds of the motif must be given proper length and angle (see Sect. 1.1). Next, the lowest-free- enthalpy helix is produced by minimizing conformational energy and steric

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interaction. Finally, the helices are packed densely, taking care of any possible H- bonds, dipoles and possibly even isolated ionic charges.

The packing fraction, k, of solid rods, described in Fig. 4.24, can be easily derived from Fig. 5.17. As indicated, k is larger than possible for the three-dimensionally closely-packed spheres (CN = 12), or for the lower coordination number of rods in two dimensions (CN = 4). The closer approach of the covalently bound atoms along the chain is the reason for these high packing-fractions of rod-like motifs.

Fig. 5.17

The closest approach of polyethylene molecules in a crystal is probed in Fig. 5.18. The packing has a CN = 6. Four of the chains approach more closely (triple contacts) than the other two (single contacts). The packing fraction, k, of 0.70 is based on the experimental data listed, and the corresponding van der Waals radii defined in Fig. 4.23. Several crystals with zig-zag chains result in an average of 0.72 for k, to which the data on crystals with true helices in Sect. 5.1.8 have to be compared.

The unit cell of polyethylene is drawn in Fig. 5.19 in two projections. The zig-zag chains are represented such that the carbon atoms are located at the intersections of the backbone bonds and the hydrogen atoms are located at the ends of the heavy lines. All symmetry elements are marked by their symbols as also given in Figs. 5.7–9. With help of the symmetry elements all atom-positions can be fixed in both projections. The symbols are standardized and will be used for the description of all the other crystal structures. Since the structure in the chain direction is best known, most projections will be made parallel to the chain axis which is also the helix axis. Before continuing with the study, it is of value to take the time to check the operation of every symmetry element in Fig. 5.19 with respect to its operation, since the later examples will similarly be displayed and a facility of three-dimensional visualization of crystal structures is valuable for the understanding of the crystal structures.

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Fig. 5.18

Fig. 5.19

The orthorhombic space group for polyethylene has been given the symbol Pnam by Bunn in 1939 [7]. The P indicates a primitive space group, n, a, and m indicate the characteristic symmetry elements that determine the nature of the space-group. There are two chains in the unit cell and along the c-axis each chain repeats with two CH2- units, i.e., the unit cell contains four CH2-units.

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Polyethylene has two other crystal structures, one is monoclinic and somewhat less stable, the other is hexagonal and represents a mesophase (condis crystal, see Sect. 2.5). The mesophase is stable only at elevated pressure and temperature, as discussed in Sect. 5.5. The monoclinic crystal structure (formerly erroneously called triclinic) is frequently found in drawn polyethylene, and then identified by its different X-ray reflections. Its zig-zag planes are all parallel and angle = 107.9°.

5.1.8 Packing of Helices

To pack a helix closely is more involved than the packing of spheres. The density depends on the degree of overlap that can be achieved between neighboring helices, i.e., the interpenetration that can be achieved by having the turns of the helix intermesh.

An example of a very close pack is metallic selenium, shown in Fig. 5.20. One can look upon the polymeric chains in the trigonal crystal as being related to a close packing of metallic atoms with a coordination number CN = 12 that overlap along the

Fig. 5.20

path of the helix by formation of the much shorter covalent bonds. The low-energy helix of selenium is of type 1*3/1, as is illustrated in Fig. 5.12. The crystal-structure parameters are listed in Fig. 5.20. Its trigonal symmetry lets the helices intermesh and gives each Se, in addition to the normal six laterally nonbonded neighbors, four nonbonded neighbors with shorter secondary bonds, two each in the higher and lower layers. The remaining two neighbors are the very short-distance covalently bonded atoms of the helix proper. Overall, this gives the CN of 12 as in a close pack of spheres. The unit cell of Fig. 5.20 shows the shortened van der Waals bonds from atom 2 of chain 2 to atoms 1 of chains 1 and 4. These less favorable shorter van der

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Waals bonds are offset by the six lateral neighbors. The only other choice of packing would have been to reduce the neighboring chains from six to four, as shown for the organic polymers with more complicated helix geometries.

The helices of organic macromolecules have their side-groups stick out sufficiently without matching the lattice symmetry that intermeshing as in Se is not possible. In these cases the coordination number must be reduced to four, as shown in Fig. 5.21. The raised part of the schematic screws, the threads, can only intermesh (approxi-

Fig. 5.21

mately) if the contacting screws are of opposite hand, as illustrated in the right-hand sketch of Fig. 5.21. With a coordination number CN = 6, at least two of the neighboring helices must be of the same hand. A further reduction to CN = 3 is observed for some of the vinyl polymers with helix symmetry three, where a CN of three permits more regular and denser packing due to the symmetric intermeshing. The same type of reduction of coordination number occurs on packing of ions. Instead of a CN of 12, that is possible on packing of the atoms of a metal, the CN is reduced to eight seen for the CsCl structure. With a larger difference in the radii of the two ions, CN decreases to six to the NaCl structure shown as Fig. 5.2. The directive covalent bonds as in carbon, naturally, fix the CN to four, the number of permitted covalent bonds. In addition, the covalent bond is directive as pointed out in Fig. 1.5. It directs the positions of the neighbors into the corners of the circumscribed tetrahedron with bond angles to its corners of 109 degrees.

Inspecting the helical molecules of isotactic polypropylene, as discussed with Figs. 5.13 and 5.14 (see also Fig. 1.20), one finds that in crystals the same helix can assume the four different shapes drawn in Fig. 5.22. For closest packing, any given crystal position can accommodate only one of these shapes, not the other three. To produce the four helices of Fig. 5.13, one can orient a polypropylene molecule so that

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Fig. 5.22

all CH3-groups point to the right or to the left. One can place, for example, the molecule of Fig. 5.13 so that all CH3-groups point to the right which yields the d- arrangement when progressing in direction C-1 C-2 or to the left, giving the - arrangement in the opposite direction C-1 C-3. A horizontal rotation normal to the plane of the paper by 180 degrees interchanges the two alignments. Next, each of the two differently aligned chains can be wound into a rightor a left-handed helix of equal energy (RH or LH). The equal energetics of the interchange of the handedness of the helix can be seen from the potential energy diagram of Fig. 5.13.

Once a given shape is included in a crystal, a rewinding of the helix with different handedness is possible without changing the helix direction. Such reorganization of the helix can change a polypropylene condis crystal (see Fig. 2.107) to a crystal, or remove on annealing similar, isolated helix defects in crystals. In both cases a more stable crystal results. The mesophase is described in Sect. 5.5.5 as an aggregate of helix segments of improper handedness. The annealing becomes possible above the glass transition of the mesophase at about 380 K (see Fig. 5.146).

To change the helix direction from d to , however, takes melting and recrystallization with turning of the molecule or finding some suitable position elsewhere within the crystal. One expects, thus, that quickly grown crystals may show both types of defects, annealed crystals may improve the perfection of the handedness (for polypropylene the larger defect), but only slow crystallization or recrystallization can lead to truly perfect crystals with both, proper directiveness and helicity.

Further inspection of Fig. 5.22 reveals that the d-RH/d-LH and -LH/ -RH helix pairs have a plane of symmetry between them, i.e., they are enantiomorphous (see Sect. 5.1.5), and the two d-chains have all CH3-groups pointing up, while the -chains have their CH3-groups pointing down. All four alignments may occur in crystals, and examples are discussed in the next section.

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5.1.9 Selected Unit Cells

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 -LH helices) and opposite inclinations of side-groups (upand 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).

Fig. 5.23

The unit cell of poly(o-fluorostyrene) crystals is represented in Fig. 5.24. The three visible carbon atoms of the vinyl helix are dotted. Note that, as in all 2*3/1 vinyl helices, the not substituted CH2-group is located directly underneath the CHR-group and does not show on projection into the ab-plane (see Fig. 5.14). The fluorinesubstituted phenyl group can be used to easily locate the chains. The neighboring helices are separated by glide planes, i.e., the helix pairs must be enantiomorphous and isoclined. Note that all side-groups in the figure point downwards. A comparison of the packing fraction of the two similar structures of Figs. 5.23 and 5.24 shows a somewhat better packing of the second. This is due to the phenylene groups that can better fill the space between the helices.

Figure 5.25 shows the unit cell of isotactic poly(1-butene). It illustrates a special problem to polymer crystallography. Its space group requires simultaneous presence of centers of inversion and glide planes between the chains. One produces isoclined neighbors, the other anticlined neighbors, an obvious impossibility. One solution to

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Fig. 5.24

Fig. 5.25

this puzzle is to assume that the crystals consist of mosaics of sufficiently small domains of the different inclinations, so that the X-ray probing the lattice averages over both substructures and records the higher symmetry. Another possibility is that the crystal has complete disorder in inclination of the side groups. Only a detailed discussion of the packing density for the two cases may resolve the actual structure of isotactic poly(1-butene) and similar vinyl polymers.

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Fig. 5.26

The crystal structure of polypropylene of Fig. 5.26 is closely related to that of the poly(1-butene) of Fig. 5.25. By shortening the CH2 CH3-group to CH3, the helices can pack more closely and do not need to interpenetrate as much. The pairs of polypropylene helices that are related through the glide planes are then able to touch the next-neighbor pairs because of the smaller side chains, in contrast to the poly(1- butene) in Fig. 5.25. The CN increases, as a result, from three to five, a CN rarely found in crystals. The chains drawn in the figures are all isoclined up-chains. Across the glide planes, the helices are enantiomorphous (LH facing RH and vice versa). The centers of inversion ( ) require, as for the poly(1-butene), that pairs of helices on either side of the center of inversions are anticlined. One assumes, as before, that the actual crystal contains a mosaic of domains of opposite orientation, or a random mixture of the inclination of the chains. Either arrangement is averaged by the X-ray beam and mimics the shown, higher symmetry. The crystal structure of polypropylene is monoclinic, i.e., one of the angles of the unit cell is 90o. Crystallographic convention makes this the angle and the unique axis b (see Fig. 5.4). For polymers, however, one frequently breaks this tradition and keeps the c-axis parallel to the unique helix axis, as shown for polypropylene ( = 99.6o). A final point to recognize is because of this choice polypropylene cannot include the helix symmetry 2*3/1 as part of the crystal symmetry. This provides the reason for the lower overall symmetry which, in turn, permits the higher and unusual CN of five. Two of the five neighboring chains have, in addition, the same handedness.

Figure 5.27 illustrates the crystal structure of -trans-1,4-poly(2-methyl butadiene), also known as trans-polyisoprene or gutta-percha. The -crystal is more stable, but not fully identified. The CH3-group is attached to position 2 in the helix 4*1/1, drawn in the sketch of Fig. 5.27. The methyl group is in this case on the carbon atom that has undergone a right-handed rotation out of the planar zig-zag by about