Thermal Analysis of Polymeric Materials
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Fig. 4.108
in Figs. 4.82 and 4.83, i.e., no nuclei are left. This sequence of figures on melting of indium has shown that much can be learned about the melting and crystallization process by performing well-controlled TMDSC experiments. At the same time, the instrumental limitations become obvious when analyzing such a sharp-melting substance like indium. Figure 4.109 reveals the close control possible with the quasiisothermal mode of analysis with very small modulation amplitudes. A step-wise
Fig. 4.109
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increase in To of 0.1 K with a temperature modulation of ±0.05 K brackets the melting point of indium. The figure reproduces five steps of the experiment. Only in the center step, melting occurs. Integrating the melting peaks in the time domain yields about 25% of the heat of fusion. The remaining 20 J g 1 can be detected on heating to the next quasi-isotherm (at about 30 min). The small modulation amplitude did not set up a large-enough temperature-gradient between Tb and Ts to conduct enough heat into the sample calorimeter for complete melting. The jump from the endotherm of partial melting to the exotherm goes with maximum speed (see also the enlarged Fig. 4.134, below). The observed melting temperature of 429.34±0.05 K agrees reasonably well with the expectation from Fig. 4.83 at <q> = 0.
The use of the data treatment of Sect. 4.4.3 brings an additional problem into the quantitative analysis. By restricting the analysis to measurement of maximum amplitudes, one cannot distinguish between a positive or negative T or heat-flow rate. This can only be done by analyzing the data in the time domain, as with a standard DSC. As long as the difference in heat capacity between reference and sample is large, the problem is minor, because the sign of the differential heat flow is obvious. The baseline runs, however, with empty calorimeters to evaluate the asymmetry correction have very small amplitudes (see Fig. 2.29, amplitude aR) and may be just as often positive as negative. Figure 4.110 presents the results of an experiment to determine this cell asymmetry. A series of pairs of empty aluminum pans of different weights were run in quasi-isothermal experiments at 322 K, and then with the sample pan filled with 41.54 mg of sapphire, represented by open and filled circles, respectively. Negative and positive imbalances give positive heat capacities for correction. The sapphire run shows, because of its always higher heat capacity on the sample side, the expected increasing heat capacity with increasing weight of the empty sample pan. Only if the left empty run amplitudes are added and the right ones
Fig. 4.110
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subtracted is the correct heat capacity found for the heat capacity of the sapphire given by the open squares. The asymmetry measured extrapolates to an equivalent m of +0.6 mg. To resolve this problem, and to account for the change of the imbalance with temperature, it is suggested when using the method of Fig. 4.92 for heat capacity measurement, to overcorrect by using in the present case a two-milligram lighter reference pan for all runs to guarantee a positive C(empty±) at all temperatures.
Figure 4.111 illustrates the effect of the phase difference by comparing the determination of the corrected amplitudes in standard DSC and TMDSC for both, a positive and negative m, in the left and right schematics, respectively. The resulting phase difference between the Al2O3 and empty run is clearly seen. Figure 4.112 shows the results of an actual TMDSC run in a plot of the observed phase angle for the data of Fig. 4.110. An asymmetry correction, thus, must involve either a correction in the time domain (possible in some commercial software), an over-correction as suggested with Fig. 4.110, or it may be possible to use the Tzero™ method of Appendix 11.
Another topic to establish a good TMDSC practice deals with Lissajous figures (or Bowditch curves), plots of the time-dependent heat-flow rate HF(t) or T(t) versus
Fig. 4.111
Ts(t). For perfect steady-state without heat-flow or temperature drift, the phaseshifted, sinusoidal responses of identical frequency, , should define ellipses. Rather perfect Lissajous figures of a quasi-isothermal TMDSC run in the glass transition region of poly(ethylene terephthalate) are displayed in Fig. 4.113. Only the last 10 minutes of 20-min runs are shown, eliminating the approach to steady state. Figure 4.114 shows long-time changes of the Lissajous figures of poly(oxyethylene) in the melting range. A small amount of melting and crystallization can be seen to build up with time on subsequent Lissajous figures (see also Chap. 6). Two further uses of Lissajous figures are given in the application sections, below, one in standard TMDSC
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Fig. 4.112
Fig. 4.113
for measurement of heat capacity with an underlying heating rate with Fig. 4.121, the other illustrating the effect of melting and crystallization in Fig. 4.135 where the indium melting is illuminated using the data of the Figs. 4.109 and 4.134. These examples point to the large amount of information that can be extracted by the interpretation of Lissajous figures.
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Fig. 4.114
Finally, the amplitude and frequency dependence of the TMDSC response is of interest. In Fig. 4.115, the dependence of the reversing, specific heat capacity of polycaprolactone with the repeating unit (–CH2)5–CO–O is displayed as a function of modulation amplitude close to the melting peak [37]. The experimentation involved quasi-isothermal TMDSC at 334 K. Within the experimental error, no amplitudedependence of the reversing specific heat capacity is seen, as is expected for a linear
Fig. 4.115
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response. Only at the highest amplitudes is the melting peak exceeded and the response decreases to the heat capacity of the liquid, as proven by the dashed line which follows the decreasing crystallinity.
The frequency dependence of the TMDSC response is shown in Fig. 4.116 [38]. The example polymer is nylon 6, polycaprolactam, [(–CH2)5–CO–NH]x. The heat capacity at lower temperature shows no frequency dependence, but the apparent,
Fig. 4.116
reversing heat capacity in the melting range decreases drastically with frequency. The cause of this change with frequency is the time-dependence of the processes seen in the melting region. A detailed analysis was attempted using the measured phase lag between the modulation of the temperature and the heat-flow-rate response. All modulations were chosen such that the sample was exposed only to heating and no reversible crystallization should be seen. By extrapolation from outside the transition range, the phase lag due to the phase transformation of the sample was separated from the observed total phase lag which also includes delays due to thermal lags within the calorimeter and the sample. From the phase lag caused by the transition alone, one can calculate the complex heat capacity, Cp*, as described in Sect. 2.3.5, Eq. (3):
Cp* = Cp iCp
with Cp representing the real or storage heat capacity and Cp , the imaginary or loss heat capacity. The reversing heat capacity is then equal to the modulus or the absolute value of Cp* = |Cp*|. Next, the irreversible melting was modeled in [38] by a broad distribution of crystallites of different melting temperatures with the total melting rate represented by the time derivative of the total crystallinity. The latent heat-flow rate was analyzed as a function of modulation frequency and heating rate, using experiments of Fig. 4.116 for the underlying heating rate 1.6 K min 1.
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The frequency dependence yields information on the characteristic time of melting, and from its dependence on the underlying heating-rate, the superheating can be assessed. The extensive reorganization of the nylons, discussed in Chap. 6, provides an additional complication, adding an exotherm to the heat-flow rate, even during the heating-only modulation condition. Figure 4.117 depicts schematically the summation of the modulation effects on the basic heat capacity. Curve a represents the
Fig. 4.117
contribution from the heat capacity, curve b the reorganization exotherm with a negligible modulation, and curve c, the melting endotherm. By a judicious choice of the parameters, the observed total heat-flow-rate curve of Fig. 4.116 can be matched. Forming now the total reversing heat-flow rate at zero frequency from the curve d in the figure, one obtains the mean endothermic heat-flow rate.
Plotting the normalized, imaginary versus the real (storage) heat capacity yields the Cole-Cole plot depicted in Fig. 4.118. The points are calculated from Fig. 4.116 at the various analyzed heating rates, making use of the phase angles. The half-circle was matched to the data points by adjusting the heat capacity (curve a of Fig. 4.117), the mean endothermic heat-flow rate (curve c of Fig. 4.117), and a single (Debye-type) relaxation time ( 7 s). While for higher frequencies the curve is close to a semicircle, as expected for a single relaxation time, deviations become appreciable for lower frequencies at higher heating rates, an indication that then, the assumed steady state is lost. Similarly, it could be shown that higher harmonics should have been included in the analysis to account for nonlinearity of the response. The relaxation time showed an irregular dependence on temperature and an inverse square-root dependence on the heating rate. Figure 4.119, finally, summarizes the various heat-flow rates that can be extracted from this analysis. The total heat-flow rate is represented by the heavy line and is related to the standard DSC signal. The reversing heat-flow rates are frequency
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Fig. 4.118
Fig. 4.119
dependent, as observed also for a number of other polymers [39]. At low frequency they exceed the total heat-flow rate because of the exotherm which is modulated only little. The endothermic, reversing heat-flow rate is given by the circles, and the reversible heat capacity by the triangles. The latter quantities were obtained by fitting the Cole-Cole circle.
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4.4.5 Heat Capacities
The calculation method for heat capacity using the reversing signal of TMDSC is shown in Fig. 4.92. The total heat-flow rate can, in addition, be treated as for the standard DSC, illustrated in Figs. 2.28–30. For reversible processes both methods should give the same results. Differences may arise from instrumental problems, as discussed in Sect. 4.4.4 and in Sect. 4.3.4. If all precautions are taken, the reversing heat flow of TMDSC should be able to yield better data by rejecting spurious heat losses and irreversible heat flows. In Appendix 13 it is, furthermore, proven, that the ratio of A to ATs can cancel the effect due to not reaching steady state, as long as the heat-flow rate is a linear response to the temperature modulation. Quasi-isothermal experiments are particularly well suited for establishing ideal measuring conditions. They take at least 10 min for each temperature, but note, that typical data tables for Cp need only one entry per 10 K. This time is then equivalent to continuous heating at 1.0 K min 1. In addition, the averaging of the many data points in quasi-isothermal analysis eliminates all statistical errors. If data of high quality are needed for both, the reversing and the total Cp, it may be better to couple the quasi-isothermal TMDSC with the faster standard DSC mode at 10 to 40 K min 1. Multiple runs are, however, needed for a high-quality DSC.
Figure 4.120 illustrates how the run parameters affect the quasi-isothermal TMDSC. As the modulation amplitude increases and causes a larger change in the heating rate q(t), the heat capacity shows less noise (after discarding the initial irregularities). For ATs > 0.5 K the precision is quite impressive. Figure 4.104 suggest that for the same conditions of modulation, periods, and temperature To, the limit of measurement is reached with an ATs of about 2 K.
Fig. 4.120
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Adding an underlying heating rate <q> to the measurement leads to a Lissajous figure as is shown in Fig. 4.121. The computation using the pseudo-isothermal method of Sect. 4.4.3 implies the plotting of HF(t) versus Ts(t) <Ts(t)>. A similar plot is obtained when plotting versus dTs/dt.
Fig. 4.121
The determination of heat capacities by using multiple modulation frequencies is illustrated above in detail using the Figs. 4.95–96. A precision of up to ±0.1% has been reported with these measurements [33]. Such precision allows TMDSC to compete with adiabatic calorimetry described in Sect. 4.2. Although DSC and TMDSC are not able to reach as low temperatures as adiabatic calorimeters, they are able to reach up to 1,000 K, not possible for typical adiabatic calorimetry.
Figure 4.122 illustrates heat-flow data on TMDSC of quenched, amorphous poly(ethylene terephthalate). This figure is copied from the original MDSC™ Patent. The total heat-flow rate is similar as in a standard DSC trace. At least outside the transition regions, the thermodynamic heat capacity can be computed from the reversing heat-flow rate. A later, full evaluation on a similar experiment is shown in Sect. 4.4.7 with Figs. 4.136–139. When conditions of linearity, stationarity, and negligible temperature gradient within the sample for TMDSC are not violated, one can also obtain the nonreversing heat-flow rate from the difference between the total and the reversing HF. The heat capacity which is proportional to the reversing HF shows the typical increase at the glass transition (see Fig. 2.117). The hysteresis, as described in Sect. 6.3, is only seen in the total HF, as also shown in Sect. 4.4.6.
The cold crystallization starts at about 400 K at a supercooling not affected by modulation and registers as nonreversing. For separation of such nonreversing transitions, several modulation periods must occur across the transition, otherwise the pseudo-isothermal analysis would not develop the proper sinusoidal oscillations about <Ts>, as can be seen from the modeling in Figs. 4.100–102 (loss of stationarity).