Thermal Analysis of Polymeric Materials

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The eutectic point is located at v2 = v1 = 0.5 as long as the two values of Tmo are taken to be identical. As soon as the melting temperatures of the two components are different, the eutectic moves to the ordinate of the lower-melting component, i.e., the higher melting component is practically independent of the lower melting component, a typical behavior of macromolecules.

Figure 7.14 contrasts the liquidus line of a polyethylene with a molar mass of 25,000 Da with those of its oligomers. A paraffin C9H20, for example, would cause a value of x = 200 in Eq. (13) of Fig. 7.6, and has an equilibrium melting temperature of 219.6 K. Such low melting temperature is still far below any reasonable concentration of paraffin, v1, in the phase diagram, and the eutectic point lies on the ordinate at v1 1.0 or v2 0. The melting curve of the polymer, however, is considerably broadened due to the dissolution of the polymer into the liquid paraffin.

Fig. 7.14

The other curves in Fig. 7.14 complete the phase diagrams for a series of oligomers. The eutectic points are still located at the right ordinate. The Tmo of the lower mass components are for x = 4, 408 K, and for x = 85, 311 K.

High-molar-mass, extended-chain crystals melt sharply, as is discussed with Figs. 3.89, 4.18, and 6.28. Broader molar-mass distributions crystallized under conditions of chain extension can be checked for attainment of equilibrium using the phase diagrams displayed in Fig. 7.14 after adjustment for the change from a twocomponent system to multi-components. Equations (1) and (2) of Fig. 7.15 indicate the needed alterations [8]. The equations are derived from the Flory-Huggins equation of Fig. 7.6 by using suitable averages for x in Eq. (13) to handle the multiple components in the melt. With the larger number of components, the phase diagram can only be calculated using numerical iteration, as shown in the flow sheet of Fig. 7.15. The first step is to approximate the composition of the polymer by division

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Fig. 7.15

into typically 10 to 30 components, rather than using the continuous distribution function. Next, a low concentration limit of v2 is chosen to be 0.0002 to avoid the very large negative values for ln v2 which would occur in Eq. (1) as v2 approaches zero. After setting up this description of the melt, an experiment with increasing temperature is simulated. At low temperature, all polymer fractions are crystallized. As soon as on raising the temperature, an intersection of any of the phase diagrams calculated with Eq. (1) and shown in Fig. 7.16 occurs at v2 > 0.0002, its fraction in the melt is iteratively increased until it matches the phase diagram. If the sample does not contain enough crystals of the given fraction, all is assumed to have melted.

Going through all fractions and temperatures, leads to the melting curves, illustrated in Fig. 7.17. Comparisons with the experiments on samples of close to 100% crystallinity show that the low-molar-mass fraction, FR1, follows the calculation throughout the melting range within the limits of error. One concludes that all components in this distribution follow an equilibrium phase diagram. For the two other samples, the match between experiment and calculation stops after partial melting. The crystals melt sharper than predicted for eutectic crystallization, an indication that solid solutions were formed with shorter molecules. The first deviations from equilibrium can be traced to the melting temperatures of molecules with a molar mass of about 20,000. One concludes that under the given crystallization conditions, eutectic separations can occur below a molar mass of 20,000. Above, nonequilibrium, mixed crystals grow. It is of interest that similar separations into solid solutions of different, high-molar-mass components and eutectic crystallization of components with low-molar-mass components occurs also on melt and solution crystallization to lower crystallinity into the folded-chain macroconformations. The reasons for this behavior are linked to the need of molecular nucleation (see Sect. 3.5.6) and the chain-folding principle, discussed in Sect. 5.22.

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Fig. 7.16

Fig. 7.17

The analysis by TMDSC of the melting of extended-chain crystals of polyethylene of low-molar-mass oligomers, and paraffins is discussed with Figs. 6.24–28. It is concluded in Sect. 6.2.1 that the melting transition of extended-chain crystals is irreversible as long as their molar mass exceeds 1,000 Da. This lower limit of irreversible melting is established with experiments depicted in Fig. 3.75 and linked to the need for molecular nucleation for the longer chain lengths in Sect. 3.5.6. Such

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molecular nucleation is not needed for melting, so that on heating, the melting temperature is usually observed at the zero-entropy-production condition, as discussed in Sect. 2.4.2. Only for large, extended-chain crystals is the melting sufficiently slow that superheating is observed in DSC experiments, as shown in Fig. 6.22. To document all these effects, Fig. 7.18 presents a comparison of standard DSC and quasi-isothermal TMDSC for the extended-chain crystals M of Fig. 7.17 [9]. Their morphology is illustrated in Fig. 5.76 [10]. On first heating, when the extended-chain macroconformation is still intact, the reversing melting by quasi-isothermal TMDSC is minimal, almost all crystals melt irreversibly. A comparison to the lower-molar- mass samples of extended-chain macroconformation is shown in Fig. 6.28. Quite clearly, the reversible melting decreases with molar mass.

Fig. 7.18

The crystal structure and perfection of the extended-chain crystals was measured by X-ray diffraction, the crystal size distribution was assessed by electron microscopy and compared to the molar mass distribution determined by size-exclusion chromatography. The melting, finally was followed by slow dilatometry to avoid superheating which is common for extended-chain crystals of more than one micrometer thickness [8,11,12]. The slow dilatometry, free of superheating, and the long-time corrected reversible melting on quasi-isothermal TMDSC yield a baseline for the reversible melting, as shown in Fig. 7.19 [9]. The reversible heat of fusion is indicated by the shaded area. It is only about 7% of the total heat of fusion. Figure 7.20, finally, illustrates the molar-mass distribution, as obtained from size-exclusion chromatography (see Sect. 1.44), and crystal-size distribution from electron microscopy [12].

From Fig. 6.20, the limits of molecular nucleation of Fig. 3.75, and the phase diagram evaluated from dilatometry, a separation of the various contributions to the reversible melting is possible. The polyethylene of Fig. 7.18 has 3% of its molar-mass

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Fig. 7.19

Fig. 7.20

distribution shorter than the critical length given in Fig. 3.75 ( 1,000 Da). These 3% of the crystals melt fully reversibly at temperatures below 385 K, i.e., the ratio of reversible latent heat absorbed per kelvin of change of temperature to the total is one, DSC and reversing TMDSC give the same apparent heat capacities. This small low- molar-mass fraction accounts for about 40% of the reversible melting seen in

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Fig. 7.19. The next portion of the molecules, up to about 10,000 Da in molar mass, accounts for 22% of the polyethylene, and still segregates on crystallization at elevated pressure into separate crystals of uniform mass (see the marking in Fig. 7.20). This second fraction agrees also with the number of crystals of the same length which have a known melting range up to 408 K. In this temperature range between 385 and 408 K, the amount of reversible melting decreases rapidly as also seen in Fig. 6.28. The percentage of reversible latent heat absorbed per kelvin of change of temperature decreases from 75% K 1 to 7% K 1 of the total latent heat per kelvin in this temperature range. Beyond 408 K, 73% of the total crystallinity remain. This large fraction melts sharply and gives rise to the major irreversible DSC peak in Fig. 7.18. In this fraction, the crystals consist of a solid solution of different molar masses and lamellar thicknesses between 100 and 4,000 nm, as deduced from Fig. 7.20. The TMDSC experiments display very little reversible melting, going to zero at about 409.5 K, while the dilatometry shows the last melting at about 411 K, i.e., the highest melting, most perfect, extended-chain crystals show no reversibility. On cooling after melting of the extended-chain crystals, normal folded-chain crystals grow and the reversing melting reaches the higher levels discussed in Sect. 6.2.1 with Figs. 6.29–34.

7.1.6. Block Copolymers

The effect of a second component in the phase structure is treated in this chapter in several stages. First, it was assumed that both components were of identical size and behaved ideally, i.e., they do not interact. Assuming further that there is no cocrystallization, the typical eutectic phase diagram of Fig. 2.27 results. The discussion of the experimental evaluation of phase diagrams indicated the effect of formation of solid solutions, as seen in Fig. 7.2. The deviation from an ideal solution can be treated by introduction of an activity as in Fig. 7.3. Next, in Sect. 7.1.2 the shape-difference of two components was treated using the free-volume argument of Hildebrandt, and finally, the interaction parameter 0 was introduced to treat real solutions in Fig. 7.4–6. The examples treated in Sects. 7.1.3 5 dealt then with components that were isolated molecules.

In this section the case of two components being part of a single molecule is treated with the example of a block copolymer as described in Fig. 1.19 and Sect. 3.4. The treatment of shape effects and the consideration of parts of the same molecule as different components are typical for such polymeric systems. Multi-component molecules are of interest if the different components phase separate without breaking the molecular integrity and yield phase boundaries that contain the intramolecular junctions between the components. There exists a full spectrum of systems in the treatment of phases of large molecules. Each molecule may either need to be described as the constituent part of a component, as in Sects. 7.1–5, or it may be separated into its smallest identities, the constitutional repeating units (CRU), which are then the constituent parts of the components, as discussed in Sect. 7.2. Finally, it is possible to introduce variable or fixed points of decoupling into the molecule, which can locate at phase boundaries. For semicrystalline polymers, variable points of decoupling were useful for the description of the rigid-amorphous fractions. The points of decoupling are in this case located at the crystal-amorphous interface,

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enabling stress-transfer between the phases (see Sect. 6.1.3). Similarly, it was suggested that reversible melting, seen in folded-chain polymers, may need points of decoupling as illustrated in Figs. 6.69 and 6.70. In the present section, the concept of fixed points of decoupling is explored, caused by a change in chemical structure along the molecule at positions which are determined by the synthesis.

Typical phase morphologies of block copolymers are illustrated in Figs. 5.38–40 and 5.79. In the classification of phases, the phase-separated block copolymers are considered to be amphiphilic liquid crystals despite the fact that inside the phase areas the typical liquid-crystalline order is missing (see Sect. 5.5). In this section the question will be addressed what happens when the usually microor nano-phase separated block copolymers show solubility, i.e., when the amphiphilic liquid crystal becomes thermotropic, i.e., dissolve at a given temperature.

Figure 7.21 illustrates a phase diagram of an AB-diblock copolymer with two amorphous segments, joined at the fixed point of decoupling. The interaction due to lateral contact between the repeating units of A and B are represented by the Flory-

Fig. 7.21

Huggins parameter 0, with positive values signifying unfavorable interaction. Changes in the phase diagram are caused by changes in 0, the length of the molecular segment, N (ordinate), and the fraction of A in the molecule, f (abscissa). Of special interest is the limit of phase separation for small values of 0N. Depending on the strength of interaction 0 and the length of the decoupled segments, the phases may dissolve for small lengths, N, as indicated in the phase diagram. Since 0 is also temperature-dependent, amphiphilics of such short-segment lengths may also act thermotropically, i.e., they may lose their liquid crystalline structure at a well-defined transition temperature. The typical different phase morphologies exhibited by the amphiphilic liquid crystals are shown at the bottom of Fig. 7.21.

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By changing from diblock copolymers of two, micrometer-long segments A and B to alternating block copolymers of many shorter repeating segments of A and B, the short-range order introduced by the interfaces may become a significant fraction of the sample. When the segment lengths reach nanometer size, the orientational and conformational order caused by the interface may persist throughout the phases. A truly liquid crystalline phase structure results in this case from the molecular-level nanophase separation, not from the orientational ordering of mesogens, just as for low molar mass soaps and lipids[13]. Naturally, in all these molecules one must identify the type of phase or mesophase and realize that there are also intermediate cases where both mesogen and interface can stabilize a liquid-crystal-like structure.

7.2 Melting Transitions of Copolymers

7.2.1. Theory of Copolymer Melting and Crystallization

In the copolymers described in Sect. 3.4, the multiple components of the system are joined by chemical bonds and demixing, needed for complete phase separation of the components, is strongly hindered and may lead to partial or complete decoupling from crystallization. The resulting product is then a metastable microor nanophaseseparated system with arrested, local equilibria. In some cases, however, it is possible to change the copolymer composition during the crystallization or melting by chemical reactions, such as trans-esterification or -amidation. In this case, the chemical and physical equilibrium must both be considered and a phase separation of the copolymer into either crystalline homopolymers or block copolymers is possible.

The thermodynamics of macromolecular solutions with small molecules is described in Sect. 7.1. A term frequently used to describe solutions of macromolecules is: blend. The word is obviously derived from the mixing process and should only be used when the resulting system is not fully analyzed, i.e., one does not know if a dissolution occurred or the phases remained partially or fully separated. The term blend should best be used only if a phase-separated system has changed by vigorous mixing to a finer subdivision, containing microor nanophases. The differences between nanophase separation and solution can be rather subtle, as is seen, for example, in the thermodynamic description of block copolymers (see Sect. 7.1). Microand nanophase-separated systems can often be stabilized by compatibilizers that may be block copolymers of the two components. Their properties can be considerably different from macrophase separated systems or solutions and, thus, of considerable technical importance.

The problems of nonequilibrium crystallization and melting discussed in Chap. 6 and Sect. 7.1 for single-component systems, are even more pronounced for multicomponent systems. The additional problem of separating the components by diffusion into phases of different concentration complicates the description of multicomponent systems even further. Usually equilibrium concentrations cannot be achieved at the moment of crystallization or melting, i.e., the processes of component diffusion and crystallization/melting are not molecularly coupled. One must thus always consider two deviations from equilibrium. One is the formation of non-

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equilibrium crystals, the other, the added influence of limited mixing or demixing of the components. As discussed in Chap. 6, the results of incomplete equilibration are local equilibria, critically dependent on sample history.

Figure 7.22 represents a typical DSC trace of copolymer melting. The poly(ethylene-co-vinyl acetate) is expected to follow a eutectic phase diagram. The melting temperature decreases from the value of the homopolymer, but the crystallinity decreases much more than expected from the small amount of noncrystallizable comonomer. Also, the crystallization of a eutectic component, required by an equilibrium phase diagram, seen for example in Fig. 7.1, is rarely observed in random copolymers with short repeating units, as in vinyl polymers.

Fig. 7.22

Three limiting theories are outlined next. The first is an equilibrium theory of the eutectic phase diagram of copolymers as developed by Flory which has been widely used, even for systems not in equilibrium. The second is the corresponding theory for the formation of solid solutions. The third is the application of cold crystallization to copolymers as a limiting, nonequilibrium theory of melting and crystallization.

The derivation of the melting equation for the eutectic of copolymers is similar to low-molar-mass materials as described by Figs. 2.24–27. The molar mass of the molecule itself is of little concern since the crystallization cannot extend beyond the length of the sequences of crystallizable components. Naturally, one expects a result similar to the eutectic of low molar mass. The main modification in the equation describing the phase diagram is a term accounting for the length of the crystallizable repeating units, usually designated as Gk. zeta, .

For an ideal, random copolymer of repeating units A and B which shows no heat of mixing or demixing, one can write the top two equations in Fig. 7.23. The sequences of consecutive, crystallizable A-units have the length . The term