Thermal Analysis of Polymeric Materials

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The three-phase structures #12, #13, and #14 are of particular interest because of the even more severe violation of the phase rule. At constant pressure, in equilibrium, only one-phase areas should be stable. Drawn fibers of poly(ethylene terephthalate) (PET) are discussed as three-phase structure with Figs. 5.68–72 and 5.113–115. As in polyethylene, the drawing to fibers or films orients the amorphous nanophase and achieves an arrested mesophase order, proven with Fig. 5.72. Since the drawing of PET fibers is much less efficient in extending the molecules than gel-spinning, there remains a sizeable portion of the amorphous phase, as shown in Fig. 5.71. The mobile mesophase of PET has not been found as a stable phase, as in polyethylene. Copolymers of PET with stiffer repeating units, such as oxybenzoate, however, have stable mesophases (see Chap. 7).

The final phase area #15 of Fig. 6.1 is quite common for macromolecular mesophases that do not crystallize well. It is, for example, easily possible to quench poly(oxy-2,2'-dimethylazoxybenzene-4,4'-dioxydodecanedioyl), DDA-12, from the liquid state (#1) to the LC glass (#15). The lower-right DSC trace of Fig. 6.2 reproduces the subsequent DSC heating trace. Above the glass transition temperature, the supercooled mesophase, the liquid crystalline area #2, crystallizes partially to a condis crystal, another mesophase which is more similar to a crystal (see Fig. 2.103). This phase arrangement of two mesophases is not described in the scheme of Fig. 6.1, which was restricted to one mesophase only. In Fig. 6.2 this new area is marked as #2/2, to indicate the simultaneous presence of two mesophases. This condis crystal/liquid crystal state is similar to area #9. Further heating produces the stable liquid crystalline state #2 and, on isotropization, the melt #1. This example indicates the rather complicated paths on heating low-temperature structures when larger numbers of polymorphs are possible.

The large multiplicity of phases, of which many are mesophases, is typical for macromolecules. The old two-phase model for semicrystalline polymers is in need of considerable extension. Most of the phase areas are metastable and have, in addition, only nanometer dimensions [7]. Only when a full characterization has been completed, can an attempt be made to link structure and properties, the central goal of thermal analysis of materials.

As a final point, one must remember that first-order phase transitions are based on equilibrium and require a sharp transition at the intersection of the free enthalpy curves as seen in Figs. 2.84–88. As indicated in Fig. 2.120, the observed broadness comes mainly from a distribution of areas of different size and perfection causing the distributions of subsystems in Fig. 6.3 (for the definition of subsystems see Fig. 2.80). Combining the different types of molecules, phases, and sizes with the range of metastable phase structures yields an enormous number of materials that must be explored to find the perfect match for the application on hand.

6.1.3 Glass Transition

The glass transition is much more subtle than the first-order transition. As is illustrated in Sect. 2.5.6. The classical glass transition occurs in a sample that can undergo large-amplitude motion, i.e., liquids or mesophases (see Fig. 2.103). On cooling, the large-amplitude motion freezes cooperatively, and the sample becomes

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glassy over a relatively narrow temperature range. A glass transition with its seven characteristic parameters as measured by DSC is shown in Fig. 2.117. A typical freeenthalpy diagram for two glasses cooled at different rates is given in Fig. 2.118. The experimental analysis of glass transitions as a function of frequency can also be done with a temperature-modulated DSC, as illustrated in Sect. 4.4.5. Dilatometry can identify glass transitions as shown in Fig. 4.27, and the effects of glass transitions on mechanical measurements are discussed with Figs. 4.170 and 5.171. Details of the DSC analyses are described next. This is followed by a discussion of the changes that occur when glassy phase-areas are restricted in size and if there are constraints on the amorphous phase so that there remains an amorphous, rigid phase above Tg of the unrestrained amorphous material.

The hole model of the glass transition. To get a better feeling for the nature of the glass transition to polymers, the simple, 70-year old hole-model of the liquid [8,9] was used for its description [10,11]. The model is shown schematically on the upper right of Fig. 6.5. The main distinction between liquid and solid is the low viscosity of the liquid (see Sect. 5.6). This low viscosity is linked to the long-range mobility of the holes through their ability to diffuse or, after collapse, to reform elsewhere.

Fig. 6.5

The model is linked easily to the heat capacity of the liquid, as seen in the upper box of Fig. 6.5. The Cp arises mainly from vibrations, Cp(vib), and a contribution from the holes, Cp(h). In Cp(h) one finds the energy needed to change conformational isomers to accommodate the hole, as well as the potential energy required to create the extra volume of the holes that permits the large-amplitude molecular motion. Below Tg, Cp(h) is zero, as shown in the second box, because the hole-equilibrium is arrested. Above Tg, it contributes the hole energy, h, multiplied with the change in equilibrium number of holes with temperature N*/ T at constant pressure. The equations can be

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region. The glass superheats and follows the V, H, and S of the glass beyond the liquid equilibrium along curve 2 and catches up only above Tg with a higher slope in H and a peak in Cp. This endothermic hysteresis peak is heating-rate dependent and must not be mistaken for a first-order transition. It can be made to disappear by matching cooling and heating rates, as is shown in the experiments of Fig. 6.5. The two dotted areas in the lower curve of Fig. 6.6 are equal, as required by the first law of thermodynamics for any cyclic process that starts and ends with the same equilibrium state which, in the present case, is the liquid (see also Sects. 2.1 and 2.2).

The kinetics of the change in number of holes. The glass-transition range with its gradual change in heat capacity from the value of the solid to that of the liquid can be based on the differential equation of Fig. 6.5. The number of holes refers then not to the equilibrium number of holes, N*, which changes only with temperature, but to the actual change of holes, N, with time during heating or cooling with rate q. The solution of this equation is given in Sect. 4.4.6 with Fig. 4.131 under the assumption of first-order kinetics, and applied to the analysis of TMDSC experiments on polystyrene [13]. Figure 6.7 illustrates a collection of TMDSC data with underlying heating and cooling rates and under quasi-isothermal conditions. The discrepancies

Fig. 6.7

between the three analysis methods are to be resolved next. For this purpose quasiisothermal polystyrene and poly(ethylene terephthalate) data were generated [14] and analyzed as indicated in Fig. 4.131 and 6.118 [15].

As indicated in Fig. 6.5, the equilibrium number of holes at a given temperature is N*, and its contribution to Cp is listed in the boxed equation on the right. Creation, motion, and destruction of holes in the glass-transition region lead to deviations if the measurement is carried out faster than the kinetics allows. Applied to the glass transition, one can write the simple first-order kinetics expression given in Sect. 4.4.6:

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(1)

with representing the relaxation time for the change in number of holes. Due to the modulation of temperature, the values of N* and are also time dependent and solutions could only be found for quasi-isothermal analyses and, as usual, for the conditions that steady state could be maintained [15]. Figure 4.113 illustrates that steady state could be kept at the glass transition under typical analysis conditions [14]. The activation energies j of Fig. 4.132 could be evaluated as 345.5 and 328 kJ mol 1 for polystyrene and poly(ethylene terephthalate), respectively. The corresponding values for the pre-exponential coefficients were 1.876×10 46 and 5.59×10 49 s, the hole energies, h 6.00 and 4.70 kJ mol 1, respectively. With these results, the coefficients of the equation in Fig. 4.131 could be evaluated.

To attempt to describe the TMDSC data with an underlying cooling and heating rate, it turns out that the above kinetics equation is best solved numerically with the parameters derived from the quasi-isothermal analyses. Each of a series of short intervals, running from 0 to i, is integrated with a constant N*, appropriate for the instantaneous temperature, and an average value of is set equal to [ i × ( i 1)]½ and lead to:

(2)

For a heating experiment, No* at i = 0 is chosen at the fictive temperature for the given prior thermal history. The fictive temperature identifies the temperature at which the actual hole concentration equals equilibrium. It is also found at the intersection of the heavy line of the enthalpy of the equilibrium liquid with the thin lines for the glass in Fig. 4.128. For a cooling experiment No* is the equilibrium value at To, the starting temperature in the liquid region.

Typical results for the number of holes for cooling through vitrification and on heating, illustrating devitrification are shown in Figs. 6.8 and 6.9. The glass transitions are modeled on data for the polystyrene shown in Fig. 6.7. For the calculation, data were calculated in intervals of one second. Figure 6.8 shows that the fast modulation with a frequency of 0.01 Hz freezes first. The fastest rates of temperature change for modulations are ±3.8 K min 1, to be compared to the slow, underlying rate of cooling or heating of 1.0 K min 1. In the glass-transition region, the amplitude of the changing number of holes due to modulation decreases rapidly. This is followed by the drift to the constant N, characteristic of the glass, caused by the underlying cooling rate <q>. Stopping the cooling within the glass transition region also yields annealing, discussed qualitatively with Fig. 4.128. The simulation of the modulation with an underlying heating rate in Fig. 6.9 is started at the same number of holes reached on cooling in Fig. 6.8. From Fig. 6.5, one expects the small exotherm before Tg and a small endotherm above. Indeed, this is observed. Overall, on heating, the changes on modulation are more asymmetric than on cooling.

The next step after evaluation of the change of hole numbers with time (and temperature) is to calculate the heat capacities obtained from the reversing signal.

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Fig. 6.8

Fig. 6.9

Again, it is assumed that all instrument lags are minimized and steady state is maintained. The number of holes is multiplied with the hole energy for polystyrene of 6 kJ mol 1 and entered in a spread sheet which can calculate instantaneous heatflow rates, HF(t). Figures 6.10 and 6.11 represent the total, reversing, and nonreversing heat capacities due to the glass transition on cooling and heating.

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Fig. 6.10

Fig. 6.11

The initial observation is that the deconvolution of the HF(t) as outlined in Figs. 4.97–99 does not remove all the modulation in the glass transition region. By counting the cycles in a given time, one finds that the linear temperature change and the modulation cause a change in frequency, a Doppler-like effect, similar to the change in frequency of a moving source of sound relative to a stationary observer.

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The irreversible contribution shows a maximum in heat capacity, as is expected from the difference in time scale of the modulation and the underlying change in temperature. While cooling, there are gradual changes in hole concentration and irreversible heat capacity, as displayed in Figs. 6.8 and 6.10. On heating, there are a minimum and an asymmetry in hole concentration in Fig. 6.9, and the heat capacity in Fig. 6.11 mirrors this change with an exotherm at low temperature followed by an endothermic enthalpy relaxation. At least qualitatively, the simulations have thus duplicated the experiments. The remaining differences are pointed out next.

Figure 6.12 displays the calculations of the change in heat capacity of polystyrene in the glass-transition region for comparison with the experiments in Fig. 6.7. The changes in frequency in the glass-transition region, which cause the remaining

Fig. 6.12

modulation of the results in Figs. 6.10 and 6.11 are not obvious in the experiments of Fig. 6.7. One reason is a further smoothing done in the evaluation of the experiment, as discussed in Figs. 4.98–102 (curves I). Another difference in Fig. 6.7 is the crossover between heating and cooling with underlying changes of temperature. Only close to the liquid state is the calculation comparable to the experiment. Clearly the relaxation time, , is shorter on cooling than on heating. This observation was already made early when following the glass transition with dilatometry [16]. Combining all DSC data, which reach from the early efforts to model the glass transition as measured in Fig. 6.5 with the hole theory [12] to the results in Fig. 4.126 [17] and the extensive measuring and modeling efforts of Figs. 6.7–12 [13–15], one must reach the conclusion that the relaxation time is not only temperature-dependent, but also dependent on the hole concentration itself, i.e., it shows the cooperative character pointed-out throughout the discussion of the glass transition. Some discussion of cooperativity based on DMA can also be found, for example in Ref. [18].

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Effect of the size of the phase on the glass transition. Changing the size of the amorphous phase from macrophase to microphase, and finally to nanophase dimensions, has a strong influence on the glass transition. A free surface allows the large-amplitude motion to occur at a lower temperature and broadens the glass transition range because of a gradual change to the bulk sample. If molecules cross the interface, Tg increases in case the adjacent phase has a higher Tg, and it decreases in case of a lower Tg. Examples of such changes of the glass transition are discussed in Sect. 7.3 for immiscible block copolymers. In semicrystalline polymers the strong interaction across the interface causes also a higher glass transition in the interface [19]. In case the interaction is strong, a rigid amorphous phase may occur and remain rigid far above Tg of the unrestrained, amorphous phase. The effect of a free surface is discussed next, followed by an analysis of rigid-amorphous fractions.

Figure 6.13 illustrates the glass transition temperature of small beads of polystyrene, measured by standard DSC [19]. One can observe, besides the glass transition with a small hysteresis peak, that on first heating, shown by the solid curves,

Fig. 6.13

there is a sizeable exotherm of strain release when the beads start coalescing. The dashed lines allow a comparison with the same sample after completed coalescence to a compact sample as it existed on second heating. The glass transition of the small beads can be seen to start about 30 K earlier than after the fusion to a macrophase, and the exotherm has disappeared, as expected. Increasing the bead sizes changes the glass transition on first heating, without affecting the second heating scans. The exotherm due to the coalescence of the beads moves to increasingly higher temperature, as does the beginning of the glass transition. When approaching the limit of the microphase size of one micrometer, the broadening of the glass transition to lower temperature becomes negligible.