Thermal Analysis of Polymeric Materials

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Fig. 5.116

Before an analysis of the more common, nonequilibrium melting is possible in Sect. 2.4 and Chap. 6, the empirical information that can be extracted from equilibrium melting of small and large molecules in crystals needs to be understood. For such a discussion, equilibrium data have first to be extrapolated from nonequilibrium. For the equilibrium values of the structure-insensitive, thermodynamic functions, such as enthalpy, volume, and heat capacity, it is sufficient to extrapolate to 100% crystallinity, as discussed in Sect. 5.3.2. The equilibrium melting temperature, Tmo, is more difficult to obtain with precision. Equilibrium crystals have been made and measured only for few macromolecules, such as polyethylene in Fig. 4.18, some polyoxides, and polytetrafluoroethylene. Extrapolations of melting temperatures with increasing perfection as linked to fold lengths or crystallization temperatures are often chosen as approximation for Tmo. The final quantity considered is the entropy of

melting ( Sfusion = Hfusion / Tmo). The enthalpy of fusion is structure insensitive and the melting temperatures of polymers are typically in the range of 300 to 600 K, errors

of 5 20 K in Tmo are, thus, acceptable for entropy estimates since most thermodynamic functions are known only with errors of ±3%. The often heard argument that equilibrium information is not available for polymers is not valid in this range of precision, and it will be shown in this section that valuable information can be deduced from a discussion of the equilibrium entropy of fusion.

The discussion of Tmo must involve four independent variables, namely the enthalpies and entropies of both the melt and the crystal. It, however, will be shown

that good estimates of Tmo can be made by using empirical rules for Sfusion, to be derived next. For similar macromolecules Hfusion is usually also similar and derivable

from the chemical structure. For aliphatic polyesters, for example, Hfusion 2.3 kJ (mole of backbone atoms) 1. For a summary of extrapolated equilibrium data the ATHAS Data Bank, summarized in Appendix 1, should be consulted.

538 5 Structure and Properties of Materials

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5.4.2 Crystals of Spherical Motifs

The melting of crystals of spherical motifs is the easiest to describe. It is assumed that before melting, the crystal is in perfect order in a cubic or hexagonal close-pack with long-range order and undergoes only vibrational motion. The melting can then be broken into three steps: (1) An increase to the usually larger volume of the melt. (2) The disordering of the crystalline arrangement to the liquid which shows short-range order only. (3) The introduction of additional defects into the liquid structure which, at least for spherical motifs, is often a quasi-crystalline structure and can have defects of the type described for crystals in Sect. 5.3. Of the three types of disorder of Fig. 2.102 which can be introduced on melting of different molecules, spherical motifs can gain only positional disorder. If the motifs are not spherical, rotation adds the orientational disorder. Finally, after fusion, flexible molecules may carry out internal rotations that lead to conformational disorder.

Figure 5.117 lists as the first four substances the entropy of melting of monatomic noble gases. Quite clearly, all noble gases have close to the same entropy of fusion. A simple corollary of this observation is that the value of the melting temperature is linked mainly to the heat of fusion. This, in turn, suggests that the packing for different noble gases is not only similar in the crystals, but also in the melts, so that the increase in interaction energy for atoms with larger number of electrons via the

Fig. 5.117

London dispersion forces is the cause of the higher Tmo. To explain the similar entropies of fusion, one can assume that the entropy of going from positions fixed in the crystal lattice to full access for the motifs to the liquid volume is R, the gas constant of 8.31 J K 1 mol 1, the communal entropy [38]. An additional 0.7 R was estimated for the two other stages of melting [39]. Assuming that the latter

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contributions to the entropy of fusion are approximately proportional to the introduced disorder, one can include all within the rather wide, empirically derived, limits of the positional entropy of fusion. Similar proportionalities are assumed for the other types of disorder included in Fig. 2.103.

The metals in Fig. 5.117 expand the table of entropies of fusion. Perhaps it is not surprising that the metals have an entropy of melting similar to the noble gases. Their crystal structures are frequently also hexagonal or cubic and closely packed with a coordination number of 12 as mentioned in Sect. 1.1.2. A comparison of chromium and sodium in Fig.5.117 shows that in metals, enormous differences in melting temperatures can be produced with the same entropy of fusion. The maximum in variation of entropies of melting is illustrated between tin and sodium. Some additional metals are shown in Fig. 5.118. In all cases shown, the wide variation of melting temperatures is based mainly on differences of the intermolecular forces and demonstrated by the comparison of the metals tungsten and potassium.

Fig. 5.118

Larger changes in entropies of fusion can be discussed in terms of changes in coordination number between the crystal structure and the quasi-crystalline structure in the melt. The changes in heat of fusion, in turn, are to be linked to the bond energies that will change with coordination number, CN, the changing number of electrons in the conduction band, as well as changes of the character of the bonds from the metallic CN (usually 12) to the covalent CN (often 4, see Fig. 1.5).

The rule of constant entropy of melting was first observed by Richards [40], and should be compared to the rule of constant entropy of evaporation of 90 J K 1 mol 1 by simple liquids as proposed by Trouton (see Sects. 1.1 and 2.5.7). Both rules are helpful in developing an understanding of the differences of transition temperatures for different materials. A general application of Richards’s rule requires, however,

540 5 Structure and Properties of Materials

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modification whenever other than positional disorder is produced on melting. Before exploring the melting of such different molecules, Fig. 5.118 shows also some entropies of melting of larger, but close to spherical motifs (molecules). Surprisingly, they still seem to follow Richards’s rule. A more detailed investigation, however, reveals that most of these molecules have one or more additional transitions at lower temperatures where the molecules take on a mesophase structure in form of a plastic crystals as defined in Sect. 2.5. At the melting temperature, or better, the isotropization temperature, the motifs rotate within the crystal and, thus, average to a sphere, and satisfy Richards’s rule.

Figure 5.119 extends the list of almost spherical motifs to much larger, organic molecules with a molecular structure which is still close to spherical. Again, these molecules are plastic crystals and have gained orientational mobility before ultimate

Fig. 5.119

isotropization. For the discussion of the entropies of fusion these results indicate that the size of the motif does not significantly influence the entropy of fusion. All data treated up to now, thus, can be summarized by stating that under the given conditions the entropy of fusion, or better isotropization, is made up of mainly positional disordering, and that the entropy of disordering varies little among all these almost spherical and truly spherical molecules.

Figure 5.120 presents, next, the entropy of melting of 20 alkali halides. All of these crystals have the Fm3m space group of the NaCl structure which is depicted in Fig. 5.2. The average entropy of fusion of these 20 salts is 24.43 ±1.7 J K 1 mol 1. Since their formula mass refers to two ions, the average positional entropy of fusion is 12.2 J K 1 mol 1, in good accord with Richards’s rule. A total of 76 other salts, with up to four ions per formula, was similarly analyzed [41]. The majority of these salts follow the same rule of constant entropy of fusion per mole of ions, irrespective of the

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Fig. 5.120

charge of the ions. There are, however, a reasonable number of exceptions. Smaller as well as larger entropies of fusion have been reported. Lower entropies of fusion can be observed, for example, if the melt is made up of covalently bonded dimers, or if the crystal is disordered at lower temperatures. Higher entropies of fusion are possible if the aggregates in the melt attain more orientational freedom than in the crystal with its correspondingly higher entropy of fusion, as will be discussed next. Excluding 14 such special salts from consideration permitted the analysis of 82 salts and yielded an average of 10.9 ±2.0 J K 1 mol 1 for the entropy of fusion per mole of ions, in good accord with Richards’s rule.

5.4.3 Crystals of Non-spherical Motifs

Non-spherical motifs that do not rotate in the crystal before melting, in contrast, should gain an orientational contribution to S in addition to the positional disorder. Figure 5.121 illustrates the rather sizable increase on examples of small molecules. This observation of a different, but again largely constant entropy of fusion, was made by Walden [42]. Increasing the size of the motifs, as shown in Fig. 5.122, does not seem to affect the entropy of melting much, as observed also in Fig. 5.119 for the rotating large organic motifs. Assuming now that the entropy of fusion is made of two major contributions, leads to the range for Walden’s rule listed in Fig. 5.122.

Finally, one can derive from Richards’s and Walden’s rules combined that

Sorientational should be about 10 to 50 J K 1 mol 1. This uncertainty of the entropy change is larger than for Spositional, but the rotational degrees of freedom have a larger variation of rotational motion. The rotation may require different amounts of volume,

be coupled between neighboring molecules, and, depending on the symmetry of the molecules, may have a reduced entropy contribution.

542 5 Structure and Properties of Materials

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Fig. 5.121

Fig. 5.122

5.4.4 Crystals of Linear Hydrocarbons

Flexible molecules have an additional, size-dependent entropy contribution that can also be derived from the molar entropy of fusion. Figure 5.123 shows the melting temperatures and entropies of melting of the homologous series of normal paraffins

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Fig. 5.123

CnH2n+2. The first three members of the series with n = 1 3 illustrate Richards’s rule with methane, a close to spherical molecule, and Walden’s rule with ethane and

propane, two non-spherical, rigid molecules. The threefold increase in the entropy of fusion of ethane and propane is caused by the added orientational motion. The linear increase of mass with the number of C-atoms also increases the interaction-energy and, thus, keeps Tm almost constant.

The special change that occurs with the longer molecules (n 4) is connected with their flexibility, i.e., the possible rotation about the interior bonds of the molecule. It becomes first possible in butane with its three conformational isomers, as illustrated in Fig. 1.37. Figure 5.123 shows that for the paraffins with even n from hexane (n = 6) to dodecane (n = 12), there is an increase in S 20 J K 1 mol 1 for each pair of additional bonds around which rotation is possible. Similar observations can be made for the paraffins with odd numbers of carbon atoms. The difference in absolute level of S between the odd and even series arises from differences in crystal packing as was analyzed with thermodynamic parameters in Fig. 4.52 for decane. The few paraffins with exceptionally low S are, at present, not fully understood. Their lowS may be caused by experimental error caused by incomplete crystallization, mobility in the crystal through mesophase formation, or by differences in the packing in the crystal or structure in the melt.

The entropy of melting per carbon atom is given in Fig. 5.123 in parentheses and approaches the entropy of melting of polyethylene which is 9.91 J K 1 mol 1. For polyethylene and other flexible polymers the positional and orientational contributions to the melting process are negligible because of the many bonds about which rotation is possible in contrast to a single translational and rotational contribution for the molecule as a whole. This simplification allows for an easy assessment of the chain flexibility, as is shown in Section 5.4.5.

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From entropy data on polymers discussed in the next section and the just presented

small molecules, one can derive that Spositional 7 14 J K 1 mol 1, Sorientational 20 50 J K 1 mol 1, and Sconformational 7 12 J K 1 mol 1. Only the last contribution is sizedependent, i.e., it is proportional to the number of rotatable bonds in the molecule or,

for polymers, the repeating unit. The proportionality of the conformational entropy of melting to the number of bonds that allow rotation is also the reason for dividing flexible macromolecules into beads of which each contributes 7 12 J K 1 mol 1 to the conformational entropy. These empirical limits of entropy gain on disordering can also be used for the discussion of mesophases, as is shown in Fig. 2.103 (see also Sects. 2.5 and 5.5).

5.4.5 Crystals of Macromolecules

Turning to macromolecules with Figs. 5.124–126, the positional and orientational contributions to the entropy of melting can be neglected for the present discussion since they contribute only little when considering that the total molecule of thousands of chain atoms can have only one positional and one orientational contribution (see Fig. 2.103). To discuss the remaining conformational contributions one can divide the molecules into beads, defined by the bonds causing molecular flexibility. The data in the tables were calculated per mole of beads, with the number of beads indicated in

Fig. 5.124

parentheses in the column for Sf. The tables include also information on packing fractions in the liquid, k (see Fig. 4.24), and the change of packing fractions on melting, , as well as cohesive energy densities, CED. The cohesive energy density is calculated per mole of interacting groups, i.e., per mole of CH2 , NH , O , CO , which maybe different from the number of beads. The CED is derived from the heat

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Fig. 5.125

Fig. 5.126

of evaporation of small model compounds or the heat of solution into solvents of known CED and corrected for volume changes. Out of such tables, omitting the obvious exceptions, the conformational entropy of melting per bead was derived, as shown in Figs. 5.124–126 and mentioned already in the discussion of paraffins in Sect. 5.4.4.

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Combining the results of all presented data, a number of valuable conclusions can be drawn about polymers from their melting characteristics. Furthermore, this experience helps to predict the melting properties from information about the chemical structure. Adding this information about melting to the predictions possible from the heat-capacity analysis (see Sect. 2.3 and Appendix 1), a rather complete profile can be developed for the thermal properties of a given chemical structure.

Some examples of the discussion of melting transitions of flexible macromolecules are given next. For polyethylene and a number of all-carbon-and-hydrogen vinyl polymers in Fig. 5.124 one observes that the short side-groups do not contribute to S. Next, one can see that going from polyethylene to polypropylene adds one more interacting unit, i.e., it increases the heat of fusion. As a result, the melting temperature increases, yielding a material useable to higher temperature. For poly(1- butene) and poly(1-pentene), the added CH2-groups in the side chain do not seem to be able to increase the interaction in the crystal as derivable from Figs. 5.25 and 5.26, so that the melting temperature decreases again. Of special interest is the poly(4- methyl-1-pentene). The second methyl group at the end of the side chain interacts in this case sufficiently to increase the heat of fusion and melting temperature. Also, it can be derived from Fig. 5.124 that all these polymers have the same packing density in the melt, k , and that the helical polymers pack less-well in the crystal, as was already found in Sect. 5.1.8 when analyzing their crystal structure.

The top three polymers in Fig. 5.124 show a large variation in properties. Polytetrafluoroethylene, PTFE, has a much higher Tm due to the low S. A more detailed analysis shows that at lower temperature PTFE becomes a conformationally mobile mesophase, as is discussed in Sects. 2.5 and 5.5 and illustrated in Fig. 2.63. Adding the entropy of disordering of PTFE of 2.85 J K 1 mol 1 at the lower transitions at 292 and 303 K, leads to a similar total S as for polyethylene. The somewhat higher S of Se is linked to the chemical depolymerization equilibrium in the melt, analyzed with Fig. 2.69. The much higher CED of Se, listed in the last column of Fig. 5.124 is largely compensated by the higher entropy, so that the densely packed Se, whose crystal structure is shown in Fig. 5.20, melts at only 494 K.

Figure 5.124 shows also the similarity between polyethylene and cis-1,4- polybutadiene when representing the chains by their bead structure. The C=C-group is a rigid single bead. The reason of the low melting point of the cis-1,4- polybutadiene is, however, not obvious. The solution of this puzzle lies with the large intramolecular contribution on melting of polyethylene from the change of the trans to gauche conformations, not present in cis-1,4-polybutadiene.

The series of five polymers with particularly low entropies of melting in Fig. 5.124 show crystals that are disordered, most likely to condis crystals (see Sect. 2.5), as shown above for PTFE. Of special interest is the difference between cis- and trans- 1,4-polybutadiene. Their entropies are shown in Fig. 2.113.

Figure 5.125 illustrates the changes in the homologous series of polyoxides. As always in such series, the first member can have a more suitable packing for the chemically different groups. In this case, the higher melting temperature is due to the higher CED. A similar effect can be seen in the polyesters in Fig. 5.125. Note how the two added methyl groups in poly(dimethyl propiolactone) increase the melting point because of better packing in the crystal.