Thermal Analysis of Polymeric Materials

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Fig. 6.118

The curve is elevated relative to zero by a constant amount (0.125) and has a contribution of 2 , double the modulation frequency (+, second harmonic). Both these contributions are not included in the experimental reversing heat flow which contains only the contribution to the first harmonic ( , see Sect. 4.4.3). Accepting the present analysis, it is possible to determine and from the reversing heat capacity by matching the last term of the equation in Figs. 6.118 and 4.131, and then use the paramters describing the match to compute ( ), the actual response of the TMDSC to the quasi-isothermal temperature modulation.

To assess standard TMDSC experiments with an underlying heating rate <q>, a numerical analysis has to be found for Eq. (2). Intervals of one second, running from i = 1 to n, are integrated with constant values for Ni* appropriately chosen for the instantaneous temperature, and an average value of is inserted in Eq. (2), as is outlined in Eq. (2) on p. 601. The various Ni are then summed over the whole temperature range of interest. For cooling experiments from above Tg, No is equal to No*, for heating from below Tg, No corresponds to N* at the fictive temperature, Tf. The fictive temperature identifies the temperature at which the actual hole concentration of the metastable glass would be in equilibrium, as shown in Fig. 4.128.

The results based on polystyrene for the number of holes for cooling, on vitrification, and heating, on devitrification, at the given underlying rates, <q>, are shown in Figs. 6.8 and 6.9. For poly(ethylene terephthalate) analogous data are reproduced in Figs. 6.119 and 120. The freezing of the hole equilibrium is clearly visible in Fig. 6.119. Also, it can be seen that after the response to the fast modulation has almost stopped, a considerable further decrease in number of holes occurs due to the slow, underlying cooling rate until the glass becomes metastable after about 1,000 s. Heating, in Fig. 6.120, starts with annealing as the glass transition is approached. This leads first to an exotherm with a decrease in N up to about 500 s,

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Fig. 6.119

Fig. 6.120

an effect that has not been analyzed to a great extend with experiments [12]. Next , one can see the well-known, endothermic hysteresis which overlaps the beginning response to the oscillations (see also Fig. 6.6 for a schematic). Finally, the liquid state is reached after about 900 s. The analogous polystyrene data in Figs. 6.8 and 6.9 show similar responses, just that the concentrations and times are different.

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The total, reversing, and nonreversing heat capacities can then be calculated as described in Sect. 4.43, with the results shown for polystyrene in Figs. 6.10 and 11 (assuming that all instrument lags have been eliminated). The abbreviated Fourier analysis used to extract the reversing part of the heat flow assesses only the first harmonic of the oscillations and only at the modulation frequency , i.e., the reversing heat capacity does not correspond exactly to the apparent reversible heat capacity, as was already shown for the quasi-isothermal analysis in Fig. 6.118. The interference of the time scales of the underlying rates <q> and cause a small shift in (Doppler effect) which explains the additional oscillations of the heat capacities in Figs. 6.10 and 11. The higher harmonics also can be extracted and are small, as shown in Fig. 6.121 which summarize data for poly(ethylene terephthalate) from Fig 6.119.

Fig. 6.121

This modeling reveals a glass-transition-like behavior in systems describable with the hole model. It also shows that the reversing heat capacity by TMDSC is only approximately the apparent reversible heat capacity. In addition, it was shown in Sect. 6.1.3 that an asymmetry exists, not treated in the simple hole model, causing a change of the relaxation time with N (see also Fig. 4.126). Additional observations point to deviations from the exponential response of the first-order kinetics of Eq. (2), above. All these complications point to the fact that the glass transition is cooperative and needs a more detailed model for full representation.

6.3.3 Pressure and Strain Effects

The principal effect of hydrostatic pressure on a substance is an increase in density and a preference of better packed structures, in contrast to tensile strain which mainly causes a chain extension. To describe the change of melting temperature with pressure

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and strain, one uses the Clausius-Clapeyron equation which is written as dTm/dp =Vf/ Sf, and dTm/dp = f/ Sf (see Fig. 5.168). The effect of pressure on Tg, however, cannot be described by the corresponding equation for reversible secondorder transitions ( / = dTg/dp, where is the change in compressibility and the change in expansivity, see also Sect. 2.5.6). Still, the change of Tg with pressure goes usually parallel to Tm.

The experimental increase of Tg with pressure of polystyrene is indicated in Fig. 6.122 by line BE. The liquid D cooled at elevated pressure, freezes at the higher glass transition temperature E than liquid C cooled at atmospheric pressure (Tg = B). On releasing the pressure below Tg, the compressed sample F expands to G, not to the level A of a glass cooled at atmospheric pressure. The glass remains “pressure densified” by the amount AG FF , where F is the volume after isothermal compression from A. This densification corresponds to a volume strain.

Fig. 6.122

Slow cooling leads to glasses of lower free enthalpy and enthalpy (see Figs. 2.118 and 4.128, respectively). Due to the same direction of changes in H, S, and V, schematically shown in Fig. 6.6, denser glasses ought to be more stable. Surprisingly the pressure-densified glass of Fig. 6.122 does not have a lower enthalpy, but rather a higher one, as is shown in Fig. 6.123. Such higher enthalpy, however, is expected for strained samples. The relaxation kinetics depicted in the graph on the right in Fig. 6.123 indicates that the pressure-densified glass relaxes to a less dense glass (larger volume) with lower enthalpy, i.e., it is less stable than a glass of the same density obtained by annealing. In fact, on extended relaxation, the volume will decrease again and the enthalpy decrease even further. Figure 6.123 also reveals that enthalpy and volume follow different kinetics. Enthalpy relaxation is coupled with little volume relaxation and volume relaxation sees little change in enthalpy.

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towards lower temperature and despite the fact that the pressure-densified glasses were cooled with the same slow rate, they lose their endothermic enthalpy relaxation at Tg. Instead, they develop a broad exotherm below Tg of the samples cooled at atmospheric pressure. The annealing of a polystyrene cooled at an intermediate pressure of 276 MPa is illustrated by the DSC traces in Fig. 6.125. The annealing temperature at atmospheric pressure is chosen to be 343 K, 30 K below the Tg at atmospheric pressure of about 373 K. The integral kinetics of this annealing is similar to the data of Fig. 6.123. As the initial enthalpy decrease is completed, the exotherm in the DSC trace disappears and the standard enthalpy relaxation (hysteresis) builds up with further annealing, and the strain history, introduced by the cooling under pressure, is erased. This example of superposition of strain history and thermal history illustrates our presently poor understanding of the glass transition. No set of internal variables and no proper relaxation kinetics have been derived for these more complicated processes of pressure, three-dimensional strain, and temperature/time variations.

Fig. 6.125

Two other examples of the influence of strain on the glass transition are discussed in Sect. 6.1. One deals with two-dimensional surface strain, the other with molecular, one-dimensional strain. Surface strain has been detected in small particles as an exotherm that occurs below Tg for small particles and high strain and above Tg for larger particles. An example for polystyrene is given in Fig. 6.13. The molecular strain can result from molecules crossing a solid/liquid phase boundary (see Sect. 6.3.4). Its presence may broaden the glass transition to a higher temperature, or even shift the glass transition for a part of the amorphous fraction to higher temperature. Then, the affected parts of the molecules make up a rigid-amorphous fraction (see Figs. 6.16–18). The effect of macroscopic strain on the glass transition, as on linear extension of amorphous polymers, is mentioned above.

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6.3.4. Crystallinity Effects

Partial crystallization of a linear macromolecule strongly affects the glass transition of the remaining noncrystalline portion. In fact, the strains introduced by the tiemolecules between the amorphous and crystalline fractions, together with the chainfolding principle, are the cause of the limited crystallization (see Chap. 5). The details of the melting of the semicrystalline polymer poly(oxy-1,4-phenyleneoxy-1,4- phenylenecarbonyl-1,4-phenylene), PEEK, are treated in Sect. 6.2.5. Figure 6.126, in turn, is a plot of the changes in the glass transition temperature as a function of the crystallization history which was set by crystallization at different temperatures. The crystallization at high temperature leads to a constant, relatively low glass transition temperature, Tg, indicative of a relaxed interface between crystals and melt. The achieved Tg, however, is not the glass transition of the fully amorphous PEEK, which is 419 K, four kelvins lower. The lower the crystallization temperature, the more strain remains, and the higher is Tg. The higher glass transition is mainly caused by a broadening of the glass-transition region, as is seen, for example, in poly(ethylene terephthalate), PET, and polydioxanone of Figs. 4.136 and 4.124, respectively. In both figures the amorphous sample is indicated by , the semicrystalline one, by .

Fig. 6.126

Figures 6.127 and 6.128 illustrate that for the two macromolecules the strain on the amorphous fraction is so large that a portion of the amorphous phase remains rigid up to the transition region and a lower increase in heat capacity at Tg results than is expected from the fraction of the amorphous sample, (1 wc). This fraction is called the rigid-amorphous fraction, RAF. The method of its evaluation is described in Figs. 6.16–18 for the case of poly(oxymethylene). For PEEK, the RAF is illustrated in Fig. 6.127. It is calculated as the ratio to the total amorphous fraction and increases

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Fig. 6.127

Fig. 6.128

with increasing cooling rate from 8 to 18%, while the crystallinity decreases from 45 to 39%, i.e., the overall rigid fraction does not decrease, as one might expect, but it increases from 53 to 57%. Similarly, on annealing of a macromolecule with a rigidamorphous fraction, the overall rigid fraction may decrease since the interfaces relax on annealing by physical changes or ester interchange. Figure 6.128 illustrates a similar behavior for poly(thio-1,4-phenylene), PPS, or poly(phenylene sulfide).

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Besides Tg, the breadth of the transition region, and Cp, partial crystallization affects also the hysteresis behavior (enthalpy relaxation) of the amorphous fraction. Figure 6.129 in its bottom curve depicts a typical enthalpy relaxation in amorphous PET that can be used to study the thermal history, as outlined for polystyrene in Figs. 4.125–127. The DSC curves for the samples with increasing crystallinity show that the hysteresis disappears faster than the step in Cp at the glass transition. The top sample in Fig. 6.129 of 31% crystallinity shows a smaller change in heat capacity than expected for an amorphous content of 69%, due to some rigid-amorphous fraction, but the hysteresis peak seems to have disappeared completely.

Fig. 6.129

Since development of the temperature-modulated DSC (see Sect. 4.4) it is possible to quantitatively describe the kinetics of the glass transition, as is illustrated in Sects. 6.3.1 and 6.3.2. Figures 6.130 and 6.131 Illustrates the glass transition regions of a number of PETs with different thermal histories which leads to different crystallinities (wc) and rigid-amorphous fractions (wr) [66]. The standard DSC traces reveal that the broadening of the glass transition region and the disappearance of the hysteresis are connected to the crystallinity alone. The rigid amorphous fraction, by not taking part in the glass transition, affects only the Cp.

A drawn film was also included in the analysis [66]. The quasi-isothermal TMDSC of this drawn sample is reproduced in Fig. 6.109. It was produced out of practically amorphous PET by biaxial drawing at 368 K. The sample retains no residual cold crystallization and has a higher rigid-amorphous fraction than the semicrystalline reference PET of Fig. 3.92. Long-time annealing causes an annealing peak of the crystals, as described in Sect. 6.22, but displays no hysteresis peak, as was also observed for the slowly cooled, undrawn PET samples with similar crystallinity used as an example in Fig. 6.129.

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Fig. 6.130

Fig. 6.131

A more detailed analysis of the kinetics of the glass transition was carried out by quasi-isothermal TMDSC [66,67]. Figure 6.132 is a representation of plots of the reversing, apparent heat capacities of some of these samples after extrapolation to zero modulation amplitude. All traces of hysteresis are absent. The same methods as described in Sects. 6.3.1 and 6.3.2 were used for the analyses of the glass transitions. For the amorphous PET, the analysis is shown in Figs. 4.129–133 and 6.119–121.