Thermal Analysis of Polymeric Materials

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Fig. 4.122

The problem of maintaining modulation and keeping the other basic conditions of a DSC, makes for a difficult interpretation of a large transition over a narrow temperature range (see Sect. 4.4.7). Still, in many cases the nonreversing effects can be eliminated and heat capacities can be measured where no other method could succeed. If, in addition, the irreversible change to the new material produces a change in heat capacity, the kinetics of transformation can be followed by Cp measurement, as is illustrated in Sects. 4.4.7 and 4.4.8 (see Figs. 4.139 and 4.141). Melting of polymers is often more problematic. An attempt to separate melting and heat capacity is illustrated in Fig. 4.123 for a poly(oxyethylene) oligomer of 1,500 Da molar mass. This oligomer crystallizes about 8 K below Tm as seen from Fig. 3.91. The figure shows that melting is largely not followed by crystallization, and TMDSC can establish the heat capacity over much of the melting range. Additional information about TMDSC of the melting and crystallization of poly(oxyethylene) is given in Figs. 3.88 and 3.89, Fig. 4.114, and in Chap. 5.

A final data set of heat capacity measurements is shown in Fig. 4.124 for poly-p- dioxanone, a linear polymer with a repeating unit of CH2 CH2 O CH2 COO . Data by adiabatic calorimetry (open circles), standard DSC (open squares), and quasiisothermal TMDSC (filled circles) are plotted together. The sample for adiabatic calorimetry was initially semicrystalline, that for TMDSC, amorphous. The glass transition is visible at 264 K, and cold crystallization occurs at 345 K and melting at about 400 K. The adiabatic calorimetry seems to fail at high temperature, and the TMDSC discovers a small amount of reversible melting, to be discussed in Sect. 4.4.7. The interpretation of the heat capacities in terms of the molecular motion is indicated using the calculations described in Sect. 2.3. This graph illustrates the large amount of information the measurement of heat capacity can provide, and the good agreement between the various methods of measurement.

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Fig. 4.123

Fig. 4.124

4.4.6 Glass Transition Measurement

An analysis of the glass transition of polystyrene by TMDSC is illustrated in Fig. 4.125. These measurements can be compared to the DSC results in Sect. 4.3.7. The left half of the figure shows besides the modulated heat-flow rate, T(t), the three

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Fig. 4.125

computed output curves of the TMDSC (nonreversing, reversing, and total heat-flow rate, top-to-bottom). The total heat-flow rate, <HF(t)>, is the appropriate average of HF(t) (proportional to T(t), as discussed in Sect. 4.4.3). On the right, an enlarged graph of the left is shown, but without the instantaneous values of T(t). The reversing heat-flow rate <A (t)> is proportional to the heat capacity, the nonreversing heat-flow rate, defined as <HF(t)> <A (t)>, indicates the existence of a hysteresis, an irreversible, slow drift of the sample towards equilibrium, caused by the thermal history and treated in more detail in Sect. 6.1.3. The TMDSC is able to approximately separate these two effects. It permits the simultaneous characterization of the glass transition and the thermal history, not possible by DSC.

Figure 4.126 illustrates that the reversing glass transition measurement is almost independent of the annealing (thermal) history. The still little-explored, small differences are an indication of the changes in relaxation kinetics on annealing of glasses. The corresponding nonreversing components are shown in Fig. 4.127. They permit the quantitative characterization of the thermal history of the glass as mentioned above, the differences, however, between the curves of Fig. 4.126 are not considered in this separation.

A preliminary explanation of the behavior of glasses is now attempted using the characterization of the transition in Figs. 2.117 and 2.118, and the quasi-isothermal TMDSC data. This is followed by extraction of quantitative data. At all temperatures the liquid represents equilibrium, illustrated by the heavy line in Fig. 4.128. On cooling at different rates, the glass transitions occur close to the intersections with the thinner enthalpy curves of the corresponding glasses. The heat capacity is equal to the slope of the enthalpy. Starting from a glass, any drift towards the liquid is irreversible and slows as the temperature decreases. Drifts from the liquid towards the glassy states are forbidden by the second law of thermodynamics (see Fig. 2.118).

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Fig. 4.126

Fig. 4.127

Figure 4.128 illustrates that on cooling a liquid from high temperature, one measures the proper, reversible Cp of the liquid over a wide frequency and cooling rate range. On reaching T1 the large-amplitude molecular motion that characterizes the liquid has slowed so much, that it can only be followed by slow experiments. Fast modulation, on the other hand, elicits a glass-like response. On quickly cooling to T2,

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Fig. 4.128

to begin a quasi-isothermal TMDSC run, a glassy response is observed, characterized by the indicated fictive temperatures (see Sect. 6.3). With time, a slow, irreversible drift towards the liquid enthalpy continues as an exothermic enthalpy relaxation. The fast modulation measures, however, only the low-value of the vibrational Cp of a glass, as indicated by the insert. Even when ultimately the stable liquid is reached, relative to the time constant of the fast modulation, the large-amplitude motion is too slow to respond to the reversing temperature and only the vibrational Cp is measured. After further quenching to T3, initially some annealing is possible toward an enthalpy corresponding to a lower fictive temperature. At T4, the glass is finally in a fully metastable, arrested state and does not change with time. On reheating to T3, annealing gives first some further exothermic relaxation, but heating to T2 produces a superheated glass that relaxes endothermically toward the liquid. Final heating to T1 shows superposition of the heat capacity and the irreversible enthalpy relaxation since now the time scales of measurement and relaxation are similar and one observes a hysteresis. The degree to which relaxation and reversible heat capacity are separable by TMDSC are shown next. Further details about the glass transition region in homopolymers, including the behavior of polystyrene and the quantitative assessment of hysteresis due to aging of the glass, are discusses in Sects. 6.1.3 and 6.3.

The TMDSC results, displayed in Figs. 4.125–127, illustrated for the first time a possible separation of the heat capacity and relaxation processes in the critical temperature range between Tb and Te, defined in Fig. 2.117. We expect the reversing heat-flow rate to register the thermodynamic heat capacity, and the nonreversing heatflow rate, the relaxation. The analysis will show to what degree this separation is possible. Because the time constants of the physical processes change rapidly with temperature, we will attempt first to approximate the kinetic parameters of the glass transition by quasi-isothermal TMDSC which involves only one time scale, expressed

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by the modulation frequency, , or period, p. This is followed by a look at the standard TMDSC with its two time scales defined by p and <q(t)>. The description is more involved than described here and, thus, the superposition of the overlapping processes with different temperature as well as structure-dependent relaxation times are considered to be not well understood. More about the description of the glass transitions can be found in many places of this book, underlining the importance of the transition for the thermal analysis of polymeric materials. For major discussions see, in addition, the following Sects.: 2.5.6, 4.3.7, 5.6, 6.1.3, 6.3, and 7.3.

Figure 4.129 displays on the left some quantitative, quasi-isothermal data at different frequencies for amorphous poly(ethylene terephthalate), PET, but at equal

Fig. 4.129

maximum heating rates q(t) in the glass-transition region. Since for a kinetic analysis it is important to not lose steady state, the sample mass was reduced to 6 mg. To match the higher precision possible when measuring outside the glass transition region with larger sample masses, the raw data were calibrated internally by normalizing to the change of the heat capacity at the glass transition Cpo, as shown on the right side of the figure. Next, the equations of Fig. 4.130 are used to extract a phase angle ' for the description of the frequency-dependent, apparent heat capacity, Cp#.

The simplest model for the representation of the glass transition, perhaps, is the hole theory. With it, the larger expansivity of liquids and the slower response to external forces is said to be due to changes in an equilibrium of holes. These holes are assumed to be all of equal size, and their number depends on temperature. The equilibrium number of holes at a given T is N*, each contributing an energy h to the enthalpy. The hole contribution to Cp is then given under equilibrium conditions by:

(1)

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Fig. 4.130

Creation, motion, and destruction of holes are cooperative kinetic processes and may be slow. This leads to deviations from Eq. (1) if the measurement is carried out faster than the kinetics allows. Applied to the glass transition, one can write and solve a simple, first-order kinetics expression [40] based on the equilibrium expression, above:

(2)

where N is the instantaneous number of holes, t is the time, and , a relaxation time. Details about the solution of Eq. (2) are described in Sect. 6.3.2. Similarly, one can derive an analogous relaxation time ' for the formation of holes from TMDSC experiments [41] as in Fig. 4.129. The process is illuminated by Fig. 130. The experimental relaxation times in the center graph are dependent on modulation amplitude, but can be extrapolated to zero amplitude, as seen in the bottom graph. At larger distance from equilibrium, however, ' does not remain a constant. Some five to ten kelvins above Tg, most liquids show no kinetic effects for typical DSC heating

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Fig. 4.132

Fig. 4.133

heating and cooling data are different and separate from the quasi-isothermal data taken after a steady state had been reached. The reason lies in the incomplete separation of the apparent, reversing heat capacity from the total heat capacity. An attempt at a more detailed separation of the various contributions is given in Sect. 6.1 by numerical solution of the rather complicated kinetic equations.

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The small difference of the data sets at <q> = 0.5 K prove the usefulness of TMDSC in separating reversing and nonreversing contributions to the glass transition (for the latter see Fig. 4.127). Furthermore, a quantitative kinetic analysis is possible that lets us model the behavior of the glass transition. And finally, the remaining small discrepancies in Fig. 4.126 and Fig. 4.133 can be quantitatively assessed.

4.4.7 First-order Transition Analysis

The analyses of the first-order transitions, such as melting and crystallization, bring a major new problems and opportunities for analysis to TMDSC. Due to the usually larger heat of transition in comparison to the heat capacity one must consider greater lags in the heat-flow rate, but can now study the closeness of the transitions to equilibrium. For example, at 400 K the heat of fusion of polyethylene, 4.1 kJ mol 1, is 120 times larger than the heat capacity of the crystal, 34.2 J K 1 mol 1. For the standard DSC this problem was solved by the baseline method, described in Figs. 4.71 and 4.72. The sample temperature was assumed to remain at the transition temperature as long as the sample needed to add or lose heat of transition, and then approach exponentially the steady state that is characteristic of the heat capacity of the new phase. Except for very broad transitions with small heats of transition, loss of steady state is inevitable and must be dealt-with in the data interpretation. As glass transitions, first-order transitions are of major importance in thermal analysis of polymeric materials (see Sects. 2.5.7, 3.6, 4.3.7, 5.4, 5.5, 6.1, 6.2, and 7.2).

A simple analysis of an irreversible first-order transition is the cold crystallization, defined in Sect 3.5.5. For polymers, crystallization on heating from the glassy state may be so far from equilibrium that the temperature modulation will have little effect on its rate, as seen in Fig. 4.122. The modeling of the measurement of heat capacity in the presence of large, irreversible heat flows in Fig. 4.102, and irreversible melting in Figs. 3.89 and 4.123, document this capability of TMDSC to separate irreversible and reversible effects. Little needs to be added to this important application.

A second type of analysis pertains to sharp equilibrium transition, such as seen for In. Figures 4.82 and 4.83 illustrate the change of onset of melting with heating rate and the need of supercooling by about 1.0 K. On TMDSC, melting will occur as soon as the modulation reaches the melting temperature, Tm. The extent of melting will then be determined by the time Ts(t) stays at Tm, or the sensor temperature exceeds Tm, as illustrated in the time-domain recording of the heat-flow rates in Figs. 4.106 and 4.107. As soon as Ts(t) drops below Tm, recrystallization start without supercooling as long as crystal nuclei remain in the sample. The shape of the heat-flow-rate curve is, however, far from sinusoidal, so that the data treatment of Sect. 4.4.3 is not applicable. The curve must be analyzed in the time domain, as in the baseline method of Figs. 4.71 and 4.72 for the standard DSC, but modified for a sinusoidal baseline. If then steady state is approached after melting and crystallization during the modulation cycle, quantitative analysis is possible. Small sample masses and appropriate modulation are needed for such studies. Extraction of a reversing amplitude gives higher values for the apparent Cp, as seen in Figs. A.13.13 and 4.108. Its interpretation is better done by the integration of HF(t) in the time domain, as is illustrated in Fig. A.13.15 with the example of melting of the paraffin C50H102.