Thermal Analysis of Polymeric Materials

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macromolecule must be at its -temperature so that there is no expansion of the random coil due to polymer-solvent interaction as described in Sects. 1.3 and 1.4 (the expansion coefficient = 1.0). At sufficiently low shear-rate, there will be Newtonian behavior, i.e., no deviation from the random coil due to the friction is expected. The energy dissipated due to the frictional force of the macromolecule can then be calculated by summing the contributions arising from all beads P. The shear-gradient in the volume occupied by the molecule causes tumbling. In addition, the tumbling molecule is carried along with the velocity of the solvent at the center of rotation of the macromolecule.

Figure 5.164 lists the frictional coefficient of one bead moving through the medium of viscosity o, approximated by the Stokes expression for a sphere of radius R. The equation for ( W/ t)i represents then the energy dissipated per second by bead i of the molecule on rotation in the shear-gradient indicated. The viscosity contribution due to the tumbling random coil of the molecule was summed by Debye as explained in Sect. 1.4.5, and is given in the boxed equation with lo representing the segment length of the random coil; NA/M, the number of molecules per unit volume,= polymer density, M = molar mass, NA = Avogadro’s number, and Mo, the molar mass of one bead. This result gives only a first-power dependence of viscosity on molar mass. To correct this wrong exponent it was suggested in Sect. 1.4.5 to use the equivalent-sphere approach. To reach an understanding of more concentrated solutions, and ultimately the melt, the entanglement effect or the reptation behavior has to be assessed.

The bottom three equations of Fig. 5.164 return to the melt viscosity and its temperature dependence through the factor k which is introduced in Fig. 5.163. Above the glass transition, a standard Arrhenius expression, as is used for the description of chemical and crystallization kinetics in Chap. 3 can be used to express the temperature dependence, as summarized in Appendix 7. As the glass transition temperature, Tg, is approached on cooling, the motion becomes more cooperative and the viscosity increases much more rapidly than given by a simple exponential expression. At the glass transition, the viscosity reaches often about 1012 Pa s, a value also characteristic of the plastic deformation of solids. The general WLF equation, given as the last equation in Fig. 5.164 and introduced in Sect. 2.5.5 can be used for the description of the viscosity in this temperature region.

5.6.4 Energy Elasticity

The definitions needed for the description of the elastic behavior of a solid are summarized in Fig. 5.165 together with its symbolic spring. The connection to other types of deformation and compressibility is given in Figs. 4.143 and 4.144. For a constant stress-strain ratio, the deformation is reversible and follows Hooke’s law. Hookean behavior is achievable for solids when the deformation is small, usually less than 1%. On a molecular level, the load-bearing network of atoms is assumed to extend proportional to the external strain, i.e., it undergoes an affine deformation by extending the appropriate bonds as sketched in the figure and also depicted in Fig. 4.144. The energy of extension fd , thus, is stored as energy of bond deformation, and the system is said to be energy elastic.

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Fig. 5.165

The thermodynamic description of energy elasticity is easily assessed with the equations at the bottom of Fig. 5.165, using the expressions of Fig. 2.17. The free

energy, dF (= dwreversible), gained on extension has two contributions. A small amount comes from a minimal increase in volume ( pdV, due to a Poisson’s ratio < 0.5, see

Fig. 4.144). The major amount is linked to the work of extension, fd . Since the order changes little, there is little change in entropy. The boxed thermodynamic equation states that the extensive force, f, for energy elasticity is caused mainly by the change of U with .

5.6.5 Entropy Elasticity

The tensile deformation of polymers is more complicated than described in Fig. 5.165. Only below the glass transition temperature and for small strain is the response similar to the energy elasticity of Sect. 5.6.4. At higher strain, the deformation becomes plastic, i.e., irreversible. To describe such plastic deformation, the detailed molecular deformation mechanism must be known. A typical example is treated in the discussion of drawing of crystalline fibers or films which involves necking, as seen in Figs. 5.108–111. Above the glass transition, however, polymers are highly viscous liquids, as described in Sect. 5.6.3. Liquids should respond to a stress with viscous flow and not be elastic. In this section, it is described that liquid macromolecules have special elasticity, the entropy or rubber elasticity [50].

On tensile deformation above the glass transition, the flexible macromolecule reacts differently from the rigid macromolecule. It changes its random coil to a more extended conformation, as shown in Fig. 5.166. This contrasts also the rotation on shear deformation, illustrated in Fig. 5.164. The main effect on extension is a decrease in conformational disorder, i.e., a decrease in entropy. This property was made use

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Fig. 5.166

of by applying extensive flow to grow more extended crystalline fibers from solution. The extension of the molecule also reduces the barrier to molecular nucleation, as discussed with Figs. 5.66 and 67. To change the viscous to an elastic response, it is necessary to arrest the slip of the molecules. Otherwise, the initial, external retracting force will reduce eith time to zero, as seen in Fig. 4.145. Even the ever present entanglements can retard this slip. More permanent are chemical cross-links which produce a network with an elastically responding mesh. One must think of the molecule of Fig. 5.166 as being attached at the entanglement or cross-link points of the network. The following discussion of the entropy elasticity is simplified by using only the single chain depicted in the figure, and furthermore, by dealing with an ideal rubber which shows no energy elasticity, but only the change in entropy.

The thermodynamic expression for the extension of an ideal rubber is given in Fig. 5.166. It can be easily derived from Fig. 2.17 (see also Fig. 5.165). In contrast to the energy elasticity, there is no change in energy U, but a change in entropy. To fulfill the conditions of an ideal rubber, all conformations must have the same energy, only then can the force be solely entropic. In any real case one expects, thus, some energy contribution. The condition of no volume work is approached by most rubbers. Since the rubber is liquid, it does not change its volume on deformation, i.e., Poisson’s ratio is very close to 0.5 (see Fig. 4.144).

A typical entropy-elastic material is cross-linked natural rubber, cis-poly(1-methyl- 1-butenylene) or cis-1,4-polyisoprene, as summarized in Fig. 5.166 (see also Fig. 1.15). Its extensibility is 500 to 1,000%, in contrast to the 1% of typical energyelastic solids. Natural rubber has a molar mass of perhaps 350,000 Da (about 5,000 isoprene monomers or 20,000 carbon backbone bonds) and is then vulcanized to have about 1% cross-links (see Fig. 3.50). A rubber with a Young’s modulus of 106 Pa (depending on cross-link density) must be compared to its bulk modulus (= 1/ ,

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inverse of the compressibility) of about 109 Pa (see Fig. 4.143). This leads by insertion into the equation of Fig. 5.166 to a Poisson’s ratio of 0.4998 for this specific rubber. A typical Poisson’s ratio for a solid ( = 0.33) would, in contrast, yield similar bulk and Young’s moduli.

Kinetically one can compare the rubber elastic extension to the compression of a gas, as illustrated in Fig. 5.167. The thermodynamic equations reveal that reversible rubber contraction can just as well drive a heat pump as reversible gas expansion. Raising temperature, increases the pressure of a gas, analogously it takes a greater force to keep a rubber band extended at higher temperature. The two equations for this fact are known as the ideal gas law p = poTVo/(ToV) with poVo/To = R, the gas

Fig. 5.167

constant (see Figs. 2.8 and 2.9) and the analogous ideal rubber equation which is written = Eo T /(To o). The graph in Fig. 5.166 gives the results of a kinetic model of an ideal rubber molecule with <r2> representing the mean-square end-to-end distance between cross-link points, and r, the extension of the random coil.

The change of the melting temperature of crystals of macromolecules as a function of tensile force f can be treated the same as the effect of pressure which is described by the well-known Clausius-Clapeyron equation. Figure 5.168 lists at the top the change of free enthalpy with changes in pressure and extension, as can be derived from Fig. 2.19. Equation (a) is the Clausius-Clapeyron equation and Eq. (b) the ideal rubber equation. The ratios of the changes of volume and length on fusion over the corresponding entropies of fusion give information of the change in melting temperature. On melting, the volume of most substances increases, while a molecule of a macromolecular, extended-chain crystal shrinks to the random coil on fusion. Considering the opposite signs of the two equations, one finds that both, increasing pressure and extension, will increase the melting temperature. For crystallizable

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crystallization of polymer molecules when the corresponding liquid phase is under strain. In Fig. 5.168 the liquidus line is drawn. Representing the last crystals melting, it should be closer to equilibrium. Three marked paths are distinguishable for further discussion. Path 1,2 indicates melting at constant force, path 3,4, at constant temperature. Both paths lead to a constant melting temperature. At constant length, path 5,6, one finds a melting range, T. The same paths are shown more clearly in Fig. 5.169 a schematic length-temperature phase diagram, which is compared to the analogous V-T plot. More detailed p-V-T plots are shown in Fig. 4.20–22. Such diagrams permit the description of the behavior of a polymer under different conditions of force, pressure, length, and volume. A second set of experimental data is shown in Fig. 5.170. At not too high extensions, the phase diagram looks as

Fig. 5.170

expected, but only the liquidus line is identified, the phase boundary indicating the end of melting. The solidus line would mark complete crystallization, a state that is never reached in polymers.

5.6.6 Viscoelasticity

The analyses of several polymers by dynamic mechanical analysis, DMA, are described in Sect. 4.5 in connection with the brief description of the DMA equipment. It was observed in such experiments that neither the viscosity nor the modulus are constant, as is assumed for the discussion of energy and entropy elasticity, outlined in Sects. 5.6.4 and 5.6.5, respectively. One finds a stress anomaly when the elastic limit of a material has been exceeded and plastic deformation occurs. Other deviations have the stress depend both on strain and rate of strain. Finally, a time anomaly exists whenever the stress/strain ratio depends only on time and not on the stress magnitude.

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A sample which shows such time anomaly only is said to be viscoelastic and can be analyzed as described in this section. At small strains most polymers show such viscoelastic behavior.

Based on the measurement of the stress, , resulting on the application of periodic strain, , with equipment as shown in Fig. 4.155, one can develop a simple formalism of viscoelasticity that permits the extraction of the in-phase modulus, G', the storage modulus, and the out-of-phase modulus, G", the loss modulus. This description is analogous to the treatment of the heat capacity measured by temperature-modulated calorimetry as discussed with Fig. 4.161 of Sect. 4.5. The ratio G"/G' is the loss tangent, tan . The equations for the stress are easily derived using addition theorems for trigonometric functions. A complex form of the shear modulus, G*, can be used, as indicated in Fig. 4.160.

A number of DMA analyses are discussed in Sect. 4.5.5. They deal with amorphous poly(methyl methacrylate) and a polycarbonate and their frequency dependencies as functions of temperature (Fig. 4.162–165 and Fig. 4.166–169, respectively); and poly(vinyl chloride) with a glass transition (Tg = 370 K) and some local motion at about 200 K (see Fig. 4.170).

A typical example of a DMA experiment on a series of semicrystalline polyethylenes is shown in Fig. 5.171 [51]. The peaks in the tan plot are labeled ,, and . From the changes of the amplitudes of G and G with crystallinity in

Fig. 5.171

Fig. 5.171 one can see that the -transition is connected to the crystals and begins at about 260 K. The -transition is best seen in the tan plot. The tan increases with decreasing crystallinity and, thus, has been linked to the glass transition of the amorphous part of the semicrystalline samples. Its maximum can be placed at about 250 K. Calorimetrically, the glass transition of polyethylene is rather difficult to

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establish because of its breadth, seen in the heat-capacity plot of Fig. 2.46. The - transition increases in amplitude of G and tan . It marks the first drop in the storage modulus G from the value of the solid and increases with decreasing crystallinity. Its temperature range in Fig. 5.172 falls into the low-temperature region of the extended range of the glass transition when measured by heat capacity, which stretches in Fig. 2.46 from 120–200 K. The interpretation of the dynamic mechanical analyses of the three relaxation regions in semicrystalline PE [51] can now be combined with the knowledge gained by the calorimetric experimentation, which is assembled in Chaps. 2 and 5 and was first summarized in [52].

Based on evidence as shown in Fig. 5.171, the -relaxation is linked to local mobility in the crystalline lamellae, which mobilizes the chain translationally, and affects the amorphous regions tied to the crystals. Similarly, the increase in heat capacity beyond the vibrational Cp of crystalline PE and paraffins starts at about 250 300 K and can be discerned from Figs. 2.52 and 2.65. It was linked in Sects. 2.1.6 and 5.3.4 (Fig. 5.100) to the increase in gauche conformations in the crystal. The conformational defects are also at the root of the diffusion of the chains within the crystal which were simulated using the molecular dynamics calculations shown in Figs. 5.103–106. This motion of the chain within the crystals is now generally referred to as sliding diffusion [10] and is also at the root of the chain-extension of the crystals on annealing.

The -transition is a broad relaxation in the temperatureor frequency-domain, interpreted as a localized crankshaft-like motion of the backbone of the chain. This interpretation of the DMA result [51] agrees well with the calorimetry. The calorimetric results were interpreted already in 1962 based on an energy estimation as a local relaxation of gauche conformations in the amorphous phase [53,54]. The broad increases in the heat capacity beyond the vibrational Cp of amorphous PE seen in Fig. 2.46 were interpreted as a gradual, local unfreezing of the gauche-trans equilibrium.

On further heating, the -transition is followed by the major part of the cooperative glass transition, the -relaxation. Again, the calorimetry agrees with the DMA interpretation, but allowing several more quantitative statements. The mid-point of the glass transition in Fig. 2.46 is at 237 K [53]. The end of the glass transition is reached at about 250 K, with an overall Cp of 10.5 J K 1 mol 1 (see Appendix 1), in agreement with the expected value of Fig. 2.103. Further broadening of the glass transition up to about 300 K was seen for gel-spun fibers of polyethylene (see Sect. 6.2.6), as is expected for the strained noncrystalline fraction of the fibers. The properties of high-molar-mass, gel-spun polyethylenes are discussed further in Sect. 5.5.5 with Figs. 5.156–158.

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References

General References

Sect. 5.1. The fundamental tabular summary of crystallography is:. Henry NFM, Lonsdale K, eds (1969) International Tables for X-Ray Crystallography, Vol I. The International Union of Crystallography, Publ Kynoch Press, Birmingham, see also later reprints.

Specific books treating the crystal structures of linear macromolecules are: Wunderlich B (1973) Macromolecular Physics, Vol I, Crystal Structure, Morphology, Defects. Academic Press, New York; Alexander LE (1969) X-Ray Diffraction Methods in Polymer Science. Wiley-Interscience. New York; Tadokoro H (1979) Structure of Crystalline Polymers. Wiley, New York; Miller RL (1989) Crystallographic Data for Various Polymers. In Brandrup J, Immergut EH, Polymer Handbook, 3rd edn. Wiley, New York, Sect VI, pp 1–277.

Proteins are special macromolecules. For their crystal structure see, for example,: Blundell TL, Johnson, LN (1976) Protein Crystallography. Academic Press, New York.

Specific books of interest to the history of crystallography are: Burke JG (1966) Origins of the Science of Crystals. University of California Press, Berkeley; Kepler J (1611) Strenua seu, de nive sexangula. Frankfurt. Reprint in English (1966) The Six-cornered Snowflake. Oxford Univ Press (Clarendon), London; Hooke R (1665) Micrographia, or some Physiological Descriptions of Minute Bodies Made by Magnifying Glasses with Observations and Inquiries Thereupon. London.

The first enumerations of the 14 translation lattices, 32 point groups, and 230 space groups can be found in: Bravais A (1850) Mémoire sur les systèmes formès par des points distribués regulièrement sur un plan ou dans l’espace. J Ecole Polytech 19: 1 128, English translation by Shaler A (1949) Cryst Soc Amer Memoir, Vol 1; Hessel JFC (1831) Kristallometrie oder Kristallonomie und Kristallographie. Leipzig; Schoenflies A (1891) Kristallsysteme und Kriststallstruktur. Leipzig. A general text on symmetry groups is: Flurry RL, Jr (1980) Symmetry Groups, Theory and Chemical Applications. Prentice Hall, Englewood Cliffs.

Sect. 5.2. The main reference on which this Sect was developed is: Wunderlich B (1973) Macromolecular Physics, Vol I, Crystal Structure, Morphology, Defects. Academic Press, New York. A more recent source of excellent pictures and brief discussions of polymer morphology is given by Woodward AE (1988) Atlas of Polymer Morphology. Hanser, München.

The first major treatise on polymer crystallization is the three volume series: Stuart HA, ed (1955) Die Physik der Hochpolymeren. See Vol 3 Ordnungserscheinungen und Umwandlungserscheinungen in festen hochpolymeren Stoffen. Springer, Berlin.

General books on optical microscopy, a technique often sufficient to evaluate the morphology of polymers and general crystal morphology are: Hemsley DA (1989) Applied Polymer Light Microscopy. Elsevier, New York; Sawyer L, Grubb DT (1987) Polymer Microscopy. Chapman and Hall (Methuen), New York.

Discussions of the problems of chain folding are published in the following places: (1979) Special issue of the Farad Disc Chem Soc 68; Dosiére M, ed (1993) Crystallization of Polymers. NATO ASI Series C, Volume 405, Kluver, Dordrecht; (2002) Proceedings of the International Symposium on Polymer Crystallization in Mishima, Japan, June 9–12, also published in part in a special issue (2003) J Macromolecular Science B42.