Thermal Analysis of Polymeric Materials

26 1 Atoms, Small, and Large Molecules

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Fig. 1.22

incorporated in flexible polymers to enhance liquid-crystal-like behavior as is discussed in Sects. 2.5.3 and 5.5). Since liquid crystals align their mesogens spontaneously when being cooled below a certain transition temperature, these linear macromolecules can straighten spontaneously from their random-coil shape. On further cooling into the solid state, these molecules may retain the once produced partial order or crystallize further. In this way strong fibers can be made because of a more perfect alignment of the molecules along the fiber axis on spinning. An example of such a material is Kevlar ® (trade name DuPont), source-based name poly(p-phenylene terephthalamide), and the structure-based name is poly(imino-1,4- phenylene-iminoterephthaloyl).

The ball-and-chain polymer has been proposed to be made by attaching, for example, a fullerene (C60, see Sect. 2.5.3) to a polymer chain. This shows that there are no limits to the structures to which chains can be attached. Adding flexible chains to rather rigid structures can enhance the solubility of the often poorly soluble rigid molecules and change the processing and the physical properties.

The ribbon polymer is related to the less-flexible ladder and sheet polymers discussed before. One might expect very much different viscous behavior of such molecules in the melt. The interpenetrating networks [8] can have interesting elastic properties since each network may respond differently and interact with the other. The two-dimensional flexible polymers have recently been explored. They also belong to the sheet-like polymers.

Very specific properties, thus, can be achieved with these funny polymers. The question of nomenclature, however, is for most of them formidable. New rules or naming must be formulated to satisfy the goal to have not only an empirical name for a polymer, but to be able to systematically link a name to the chemical structure of the macromolecule.

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deviations more heavily. The kurtosis, finally is again considering positive and negative deviations about the mean equally (because of the fourth power). The greater influences of the larger deviations make it an overall measure of the flatness of the distribution.

Fig. 1.25

The next step involves the application of the mathematical moments to the special needs of macromolecular mass distributions. Figure 1.25 shows how to express the distribution of molecules of length x (x-mers) and the total number of molecules. The number-average molecular mass M n is derived in the figure. It is the ordinary average. Figure 1.25 also illustrates the link of the number-average molecular mass to the first moment 1. Its limitation for a full characterization of a broad molecular mass distribution can be seen by calculating Mn for a mixture of an equal number of moles of molar mass 10,000 and 1,000,000 Da. This leads to an Mn of only 505,000, despite the fact that the lower-mass polymer is only about 1% of the mass of the mixture. If the mass (or volume) of the molecules determines a property, knowing Mn is not very useful.

A mass-average molecular mass, Mw, is derived in Fig. 1.26. It is proportional to the second moment. The above mixture of equal mole fractions of largely different molar masses yields a mass-average molar mass, Mw, of 990,198. This is a better representation of the molecules in the mixture than given by M n. Since most applications in polymer science deal with effects that scale with molar mass or volume and not number of molecules, it is more important to know Mw than Mn. The second equation for Mw lists a more detailed expression. You may want to check all four equations for the one most suitable for a given calculation. Finally Fig. 1.26 introduces the term polydispersity, a measure of the disparity between the size of molecules in the mixture.

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1.3.2 Random Flight

The next statistics problem in characterizing a flexible, linear macromolecule is to describe its overall shape (see also Fig. A-13.3). Changing the rotation angle about the backbone bonds lets the molecule take on many shapes. This multiplicity of arrangements leads to high entropy and permits the appearance of polymers as liquids and in solutions (see Sect. 1.1.3). A simple estimate for the shape of a molecule is based on the statistical model of a random flight, a walk of x steps of constant length in three dimensions. One assumes that in this model for a molecule of x flexible bonds (and x + 1 chain units) each bondand rotation-angle is chosen randomly. Figure 1.28 illustrates how such a random coil is generated.

Treating each bond as a vector, an end-to-end-distance vector r can be calculated. To simplify the rather lengthy computation for which all bondand rotation-angles

Fig. 1.28

must be known, one is usually satisfied with the mean square end-to-end distance <r2>. A random choice for every angle has just as many positive as negative advances of the vector r i. The double sum for the square of r, a scalar quantity, retains thus, on averaging, only the terms i2 which have an angle of ab = 0 (i.e.,

cos ab = 1.0) and do not have negative counterparts. For polyethylene (CH2 )20,000, with a number of bonds of x = 19,999, one can compute the root-mean-square end-to-

end distance of:

<r2>1/2 = 21.8 nm

since the C C bond length is 0.154 nm (Mw = 280,000 Da). Fully extended, the molecule would be 141 times as long (contour length, assuming 180° bond angles).

32 1 Atoms, Small, and Large Molecules

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This simple calculation gives an idea of the possible amount of coiling of the molecule. The root-mean-square end-to-end distance increases with the square root of the number of bonds, while the contour length grows linearly. To fully extend such random coil, one has to use a draw-ratio of 141×, much more than is usually possible in drawing of polymeric fibers for textile applications which is 5 10×. But note, that the gel-spun, high-molar-mass polyethylenes, which are discussed in Sect. 6.2.6 have a draw ratio of more than 100×.

1.3.3 Mean Square Distance from the Center of Gravity

The equation that defines a center of gravity is shown in Fig. 1.29. The sum of the vectors from the center of gravity to all elements of the object (assumed to be of equal mass) is zero. The figure shows the correlation of the vectors from the center of gravity of a macromolecule to the end-to-end vectors. The equation for the mean square distance from the center of gravity is illustrated, but not derived. The sums

Fig. 1.29

are similar to the random-flight case, but it is somewhat more involved to reach the simple, final equation. There is no such obvious solution to the double sums of the vector products as in the derivation of the expression for <r2>.

The density of matter within the random coil is a quantity that lets one better understand the nature of macromolecules. The overall conformations is also called the macroconformation (see Fig. 5.42). The density of a given molecule in a random coil macroconformation can be estimated by assuming that the coil locates in a spherical volume V with a radius of its root-mean-square end-to-end distance. Taking the example of the standard polyethylene of the previous section of molar mass 280,000 Da (4.65×10 22 kg molecule 1) and <r2>1/2 = 21.8 nm, one finds that

34 1 Atoms, Small, and Large Molecules

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must be <r2> = x 2, as computed before. The integrations for the proper values of a and b are written at the bottom of Fig. 1.30. They go over all vector orientations (yielding the multiplication with 4 r2) and lengths dr.

The distribution function just derived is an expression for the probability that an end-to-end vector r terminates in the volume element centered about the coordinates x,y,z (at the end of the vector r ). Figure 1.31 is drawn for a random flight of 10,000 steps, each step being 0.25 nm, almost double the length of a chain segment of polyethylene. Because of bond restrictions that will be discussed later in this section,

Fig. 1.31

the curve is close to the dimensions of the 280,000 Da polyethylene molecule described above. The general shape of the curve indicates that it is most probable that the two ends of the molecule meet (x = y = z = 0) and for larger r, W(x,y,z) continuously decreases. The probability to find an end-to-end vector of length r, regardless of the direction, is shown in Fig. 1.32. The number of vectors of length r is represented by the surface of a sphere with radius r, which is 4 r2, and yields a maximum when the exponential decrease of W(x,y,z) overtakes the second-power increase of the number of vectors.

The chain statistics can be brought closer to reality by a number of improvements to be discussed next. In Fig. 1.33 the results from the random flight model for a 280,000 Da polyethylene molecule are listed once more. Its ultimate extension is the 141 times larger contour length of 30.8 m. The distribution function of the previous figure gives, however, small probabilities for even longer separations which are physically impossible. An inverse Langevin function must be used to assess the extension of macromolecules and the retractive forces it produces. The inverse Langevin function reaches infinity for this force when r reaches the contour length of the molecule.