Thermal Analysis of Polymeric Materials

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Figure 6.106 shows the TMDSC of fibers B using a sawtooth modulation and evaluation with the standard DSC technique described in Appendix 13. On first heating with constraint, reversibility exists to 405 K, which includes a substantial decrease in the orthorhombic crystals and compensating increase in the intermediate phase. On second heating, after recrystallization, the DSC and TMDSC traces are similar to the sample of Fig. 6.102 and contain only orthorhombic crystals. In this

Fig. 6.106

case, complete reversibility reaches to 360 K, as can also be derived from Fig. 6.105. The maximum of the reversing excess heat capacity is double as much as for the polyethylene of lower molar mass in Fig. 6.30, and occurs at higher temperature.

As one goes into the double melting peak of Fig. 6.106, which was ascribed to melting of orthorhombic crystals, transition into the hexagonal phase, and melting of the hexagonal phase, the reversible contributions ( ) are only a fraction of the reversing Cp (+). Most of the reversibility of the transition must involve the orthorhombic, metastable morphology and employ local processes which involve the intermediate, oriented phase since the amorphous phase is still negligible at 413 K. It may even be speculated that the decrease of the reversible excess heat capacity beyond 410 K is a direct measure of the disappearance of orthorhombic and intermediate, mobile phase. The hexagonal phase is melting irreversibly. The special points in Fig. 6.106 were taken out of the experimental sequence by direct heating from room temperature. They indicate changes during the annealing through the stepwise increase in temperature. The curved arrow connects a repeat point ( ) after the analysis at 400 K, proving reversibility of the structure over wider temperatures.

Another fiber with rather well-known structure is drawn poly(ethylene terephthalate), PET. Some typical DSC traces are shown in Fig. 4.151. Again, the restrained fibers melt higher, but for PET the collapse of the fibers that leads to the

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analysis rates because of insufficient time on partial melting of the crystals to disorder the oriented, noncrystalline areas. Even more superheating and no reorganization is seen in the annealed fibers. The structure of the PET fibers was illustrated in Figs. 5.69–72 and the mechanical properties in Figs. 5.114 and 5.115. The three-phase structure, proven by X-ray diffraction and solid-state NMR, yields only one melting peak, as in the gel-spun polyethylene described above. The initial shrinkage in Fig. 6.108 is seen at the glass transition temperature. The amorphous fiber (1) loses its integrity at this temperature since it is not crystalline. Additionally, shrinkage can occur in the region where the rigid amorphous part is reduced (see Sect. 6.1.3) and/or small crystals start melting. The final step in the analysis indicates the flow of the fiber after the main body of crystals and mesophase melt (negative shrinkage).

Some more insight into the thermal properties of an oriented PET is given by the data from quasi-isothermal, temperature-modulated differential scanning calorimetry. Figure 6.109 displays a comparison of a drawn film of PET [25] with a standard bulk sample which is analyzed in more detail in Figs. 4.136–139. Compared to the 44%

Fig. 6.109

crystalline bulk material, the 42% crystalline film has a considerably higher glasstransition temperature. The glass transition range is also broadened, and the heat capacity does not rise to the expected heat capacity for 42% crystallinity. This deficit in heat capacity indicates a rigid amorphous fraction. A reversing latent heat contribution to Cp develops towards the end of the glass transition and increases towards the melting region. These observations point to a large level of strain and orientation in the amorphous fraction, as described in detail for the PET fibers.

The final example of the DSC of fibers deals with nylon 6 [64,65]. Comparing on the right in Fig. 6.110 the DSC traces of drawn fibers before cross-linking to a meltcrystallized sample, one may conclude that there is little difference between the two

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Fig. 6.110

samples, despite the enormous rearrangement the fiber must have undergone on drawing (see Fig. 5.111). An analysis at fixed length increases the melting peak temperature of the fibers, indicating the presence of strained, noncrystalline material that does not relax before melting. A more realistic analysis is possible after chemical cross-linking of the amorphous phase by irradiation in the presence of acetylene. After this treatment the amorphous phase prohibits the crystals from annealing and recrystallizing, as also shown in Fig. 6.80. Although the chemical reaction changes the phase-transition conditions somewhat, one can now observe an approximation to zero-entropy-production melting. Both the fiber and the melt-crystallized polymer have a much lower melting temperature after cross-linking.

The detailed analysis of this method of fixing the crystal morphology by crosslinking is documented in Fig. 6.111. The controls in traces (a) and (d) are to be compared to Fig. 6.110 (note the change from the heat-flow rate to heat capacity). The sample analyzed in (a) was next heated to 417 K, then quenched to room-temperature, cross-linked, and analyzed by DSC (curve a). Next, the same process was repeated with new samples by heating to 437, 459, 480, 485, 489, 492, 495, 498, and 501 K for curves b–j, respectively. The melting peak moves continually to higher temperature than reached before quenching and cross-linking. The quantitative analysis of the heats of fusion after the various partial annealings is depicted in Figs. 6.111(b) and (c). Graph (b) shows that the annealing starts with some new crystallization, and graph (c) reveals that by separating the contributions to the two melting peaks, one can separate annealing and recrystallization steps.

This was followed by a new set of experiments carried out at constant length, reproduced in Fig. 6.111(d). The temperatures chosen were 430, 453, 473, 483, 493, 498, 501, 503, and 507 K for curves 1–9, respectively. No high-temperature melting material is produced in this case.

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6.3 Sample History Through Study of the Glass Transition

Glasses are not in equilibrium, as is discussed in more detail in Sect. 2.5.6. Every path from a liquid to the glass is unique. A description of the internal variables which characterize a particular glass can be accomplished by irreversible thermodynamics (see Sect. 2.4). These internal variables can then yield information about the sample history, i.e., determine how the final, metastable state was attained. Two common histories are easily established: (1) The initial path from the mobile to the glassy state by cooling or increasing of the pressure at given rates which produced the metastable state. (2) Annealing of a given glass close to the glass transition temperature for a specific time or plastic deformation. In both cases the sample must remain unchanged after the setting of the structure until the analysis is carried out. The first type of history is often linked to checking of manufacturing conditions, while the second may allow an analysis of storage conditions or changes that may have occurred during use of the material. Such analyses of sample history are not possible on materials in equilibrium, since all history is erased when equilibrium is reached.

The tools for the analysis of the glass transition are developed in Chap. 4 and general descriptions are given in Sect. 6.1.3. In Sect. 7.3, finally, thermal analysis is applied to the study of the glass transition in multi-component systems. In the present section time-temperature effects are reviewed first (Sect. 6.3.1). This is followed by a description of modeling by using temperature-modulated DSC (Sect. 6.3.2). Finally, applications to the study of pressure and strain, crystallization, and network formation are treated in Sects. 6.3.3 5, respectively.

6.3.1 Time and Temperature Effects

Six of the seven parameters describing the glass transition region in Fig. 2.117 are dependent on time. Only the overall increase in heat capacity, Cp, does not change significantly with time of measurement and thermal history. The heat capacity of the glass is based almost entirely on vibrational motion. As such, it reacts very fast to temperature changes (within picoseconds, see Sect. 2.3) and practically all of the lowfrequency vibrational modes which may be different for different metastable glasses are already fully excited before reaching the glass transition temperature, Tg, i.e., they contribute the same amount to Cp(glass), namely R = 8.314 J K 1 mol 1 (see Sect. 2.3). The Cp(liquid) reacts also sufficiently fast to temperature changes, but only as long as the sample for thermal analysis is sufficiently above Tg. In addition, as an equilibrium state, it has lost all thermal history. Since Cp(glass) and Cp(liquid) change differently with temperature, there is, however, a gradual change in Cp if the glass transition changes, for example, due to cross-linking (see Sect. 6.3.5), or time effects.

In order to identify the history of a glass, one heats the sample through the glass transition region and measures the enthalpy relaxation (hysteresis, see Fig. 6.6) and then relates it to the characteristic internal parameters changed on cooling or annealing given in Fig. 2.117. The experimental separation of the enthalpy relaxation and the glass transition by TMDSC is described in Sect. 4.4.6. The schematic enthalpy diagram is Fig. 4.128. The parallel, thinner lines in Fig. 4.128 are characteristic of the metastable glasses reached by different cooling rates, as indicated on fast cooling from

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T1 to T2. Isothermal annealing, as seen at temperatures T2 and T3, reaches different levels of enthalpy, characterized by “fictive temperatures,” Tf. The Tf is related to a Tg which corresponds to an appropriate cooling rate. The kinetics of annealing to lower and higher enthalpies (fictive temperatures) is asymmetric, giving one of several indications that a single internal parameter is insufficient to describe a given glass (see Sect. 2.4.4). An empirical fit of this asymmetry is given by the so-called ToolNarayanaswamy equation:

where 0 x 1 is the nonlinearity parameter. As Tf approaches T, equilibrium is reached and follows an Arrhenius expression (see Fig. A.7.2). Approaching the enthalpy of a given Tf from higher values, after quenching to temperatures lower than Tf, the parameter x increases with time, i.e., the process is self-retarding. Approaching the enthalpy of Tf from lower enthalpy, after quick heating to temperatures higher than Tf, x decreases with time, i.e., the process is autocatalytic.

Figure 6.113 shows the glass transition of polystyrene as a function of cooling rate, measured with a dynamic DTA technique, DDTA. The glass transition temperature decreases exponentially with cooling rate q and is described with an activation energy, Ea. Since Ea is of the order of magnitude of the energy of strong bonds (see Fig. 1.4), but no strong bonds are broken at Tg, this indicates a cooperative process.

Fig. 6.113

A general evaluation of the experimental results on the glass transition is given in Sect. 4.4.6. Figures 4.125–127 illustrate the treatment of the raw data of polystyrene as obtained by TMDSC. Next, Figs. 4.129–133 are used to document the conversion of such experimental data for poly(ethylene terephthalate) to yield activation energies and extrapolated data to a wider frequency range. In Sect. 6.1.3 the glass transition is described based on the hole theory and documented for the example of polystyrene

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ranges, the activation energies change, i.e., a simple Arrhenius description does not hold. The activation energy increases as temperature decreases.

In comparing glass transition temperatures, one must, furthermore, consider the differences in definition for the glass transition temperature for the different techniques. In Fig. 6.20, for example, the glass transition temperatures are compared to data obtained from DMA (using the maximum in tan ) and TMDSC (using the temperature of half-vitrification) when measured at similar frequencies.

6.3.2 Modeling of the Glass Transition

The simplest model to represent the glass transition is based on the hole theory which was developed by Frenkel and Eyring some 60 years ago and is described in more detail in Sect. 6.1.3 (see also Sect. 4.4.6). The equilibrium number of holes at T is N* and each contributes an energy h to the enthalpy. As given on Fig. 6.5, the hole contribution to the vibrational heat capacity Cpo and its kinetics is represented by:

(1)

(2)

In Eq. (2), N represents the instantaneous number of holes and is the relaxation time towards the hole equilibrium. During a TMDSC experiment, as outlined in Sect. 4.4.6, N* and are temperature and, thus, time dependent. First, a solution of Eq. (2)is attempted for quasi-isothermal temperature-modulated DSC, to be followed by a numerical solution for standard TMDSC. The basic solution of Eq. (2) is:

(3)

where to is the beginning of the experiment, and .(t) is the time-averaged relaxation time. The evaluation of Eq. (2) is rather complicated [12]. Because of the temperature modulation, both and N* are to be inserted into Eq. (3) with their proper temperature/time dependence. In addition, the discussion in Sect. 6.3.1 revealed that is also dependent on N which further compounds Eq. (3).

Figure 6.118 shows the integrated Eq. (3) for the steady state of quasi-isothermal experiments ( ). The measurements needed for this analysis are displayed for polystyrene in Figs. 6.14 and 6.15 and for poly(ethylene terephthalate) in Figs. 4.129 and 4.130. The parameters in Fig. 6.118 were arbitrarily chosen to clarify the three different contributions to the approximation shown in the figure. For a solution fitted to the experiments for poly(ethylene terephthalate), see Fig. 4.131. The parameters A and AN represent the amplitude contributions due to the change in and N* with temperature, and and are phase shifts. The plotted (N No)/N* is proportional to the heat flow (and thus to Cp). The curve ( ), however, is not a sinusoidal response.