Thermal Analysis of Polymeric Materials

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modulation amplitude seems to have no influence on the dependence on period. The open symbols represent the reversing specific heat capacity. They were calculated using the amplitudes of the first harmonic of the Fourier fit of heat-flow rate and sample temperature to Eq. (2). In the case of the Fourier fit, the deviation from the true heat capacity starts at a period of 96 s. In the case of standard DSC, this critical value is shifted to 48 s. The influence of the sample mass was analyzed for Al. The data are shown on the right in Fig. 4.94. An increasing sample mass, as well as a shorter modulation period reduces the measured heat capacity. Again, the Fourier fit results in larger deviations from the expected value when compared to the heat capacity calculated by the standard DSC equation, Eq. (1).

Figure 4.95 illustrates the evaluation of the constant as the slope of a plot of the square of the reciprocal of the uncorrected specific heat capacity as a function of the square of the frequency as suggested by Eqs. (2) and (3) of Fig. 4.94. Note that the calibration run of the sapphire and the polystyrene have different values of . In

Fig. 4.94

addition, the asymmetry of the calorimeter must be corrected in a similar process as discussed in Sect. 4.4.4, again with a different value of . Online elimination of the effects of asymmetry and different pan mass, as described in Appendix 11, should improve the ease of measurement. This method of analysis increases the precision of the reversing data gained at high frequency, and also the overall precision, since a larger number of measurements are made during the establishment of as a function of frequency and spurious effects not following the modulation are eliminated.

A simplification of the multi-frequency measurements can be done by using not only the first harmonic of sawtooth modulation for the calculation of the heat capacity, but using several, as shown in Fig. 4.96 [32]. A single run can in this way complete many data points. As in Fig. 4.95, the lower frequencies have a constant , while at

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periods later than the time of measurement to. All data between to and t4 are needed for the computation of this smoothed, reversing heat-flow amplitude <A (t3)>.

The curves for the previous screens were modeled with a simple Lotus 1-2-3™ spreadsheet [36]. The input is a table of the to be analyzed heat-flow rate HF or temperature difference T. Since one can check through the columns of data for A I for the propagation of any changes, this spreadsheet has proven a valuable tool for the prediction, analysis, and understanding of the results of TMDSC. Commercial mathematics packages which carry out Fourier analyses can as well be adapted to these calculations and also will give the higher harmonics without need of further programming. Several examples of data analyses are shown next.

Figure 4.100 is a similar analysis as in Fig. 4.99, except that at 200 s a sharp, 50% spike occurs in the heat-flow rate. The maximum reversing heat-flow-rate amplitude was chosen for all simulations to be 1.0, the phase lag is 45° and p = 100 s. Note that for clarity, the reversing, smoothed heat-flow rate I is moved by +1.0 in amplitude

Fig. 4.100

in Fig. 4.100, and also in all the subsequent figures of this set of computations. The spike causes an instantaneous loss of stationarity which can be traced through the evaluation. The averaging steps smooth this effect greatly. If the spike is to be studied, as in the presence of a small amount of sharply melting material, the just completed data treatment is a poor choice. A detailed analysis of curve C in the time domain must then be made, as seen in Figs. 4.106–109, below.

In Fig. 4.101 an abrupt 50% increase in curve C starts at 300 s. An effect on the reversing heat-flow rate occurs only at the time of increase, broadened and smoothed by triple averaging. The constant change does not affect the measurement at a later time. The baseline shifts due to fluctuations in purge-gas flow or changes in temperature of the calorimeter environment are thus minimized.

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chemical reactions or phase transitions. Despite the major disruption in the heat flow, TMDSC can still evaluate heat capacity quantitatively and follow, for example, the kinetics of the change of state.

These four examples, depicted in Figs. 4.100–103, illustrate the behavior of the heat-flow-rate response on analyzing various changes in the sample. The loss of stationarity, most clearly seen in Fig. 4.100 and 4.101, does not only affect the first harmonic, i.e., the reversing signal, but also the total signal, B in Fig. 4.99, which is not shown in the later figures. Because of the deviation from the sinusoidal response, the sliding average broadens the effect of the sharp peak at 200 s in Fig. 4.100 and the sharp jump in Fig. 4.101 over one whole modulation period. The total signal, thus, does not correspond to the actual effect, possibly broadened by instrument lag, but it is broadened additionally by the deconvolution. A separately run, standard DSC curve and the total heat-flow-rate curve of the TMDSC do not agree in shape, although the enthalpy, obtained by integration over time does ultimately reach the same value because of the conservation of heat. Specially clear examples of this broadening of the total heat-flow rate are illustrated in Figs. 4.108 and A.13.13, below.

The reversing signals H in Figs. 4.100 and 4.101, as the first harmonic of a Fourier series describing the actual response, do not represent the shapes of the response, but also, they do not give the proper magnitudes on integration since the higher harmonics are not considered. Proper analysis is reestablished after stationarity is reached again, or if the effect is gradual, as in Fig. 4.102, and the deviation from stationarity is negligible. The nonreversing signals, being calculated from B and H, contain the same errors due to nonstationarity, but note that Eq. (3) of Appendix 13 allows for a separate determination of the irreversible response.

4.4.4 Instrumental Problems

Considerable calibration and care are needed not to exceed the conditions of steady state and negligible temperature gradients within the sample for the standard DSC, as discussed in Sect. 4.3.4 7. For TMDSC, additional points must be considered, as discussed in Sect. 4.4.3. The conditions of steady state may be relaxed to some degree, as long as linearity and stationarity are kept. Calibrations and checks of compliance with those conditions, thus, are also key to good TMDSC.

To compare the functioning of a TMDSC with that of a DSC, Fig. 4.103 gives data for three DSC runs and a matching TMDSC trace (TA Instruments, MDSC™ 2910). Plotted are the instantaneous (not averaged or smoothed) heating rates q(t) and heatflow rates HF(t) (proportional to T). The DSC was run at heating rates q = 3, 5, and 7 K min 1, the TMDSC at < q > of 5 K min 1. These conditions result in a maximum q(t) of 7.0 and a minimum q(t) of 3.0 K min 1 for <q> ± ATs (2 /p. Note that the time to reach steady state with TMDSC is longer than in a standard DSC. The four curves match well above 340 K. Typically, one should allow 2 3 min to reach steady state for the chosen calorimeter.

The next question to be answered is how large a modulation amplitude can one choose for a given experiment? In order to achieve maximum sensitivity, one wants to use maximum heating rates and large sample masses, but without creating undue temperature gradients within the sample. Figure 4.104 illustrates for two cooling

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Fig. 4.103

Fig. 4.104

conditions the maximum amplitudes that could be used at selected temperatures as a function of p. To use the figures, the amplitude for the lowest temperature of the run is picked and applied to the whole temperature scan, to make sure that the calorimeter can follow the chosen modulation over the chosen temperature range. One should generate these plots for each individual calorimeter and sample-type.

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The effect of increasing sample mass on Cp is shown in Fig. 4.105. As the mass goes beyond the limit of negligible temperature gradient within the sample, part of the sample does not follow the modulation and thus, the linear response is lost. To find the proper conditions, one should run samples with different masses or do a full calibration of , as shown in Figs. 4.93 to 4.96.

Fig. 4.105

An interesting experiment is the temperature calibration of TMDSC with indium. To understand the process, a direct analysis in the time domain is necessary, as shown in Fig. 4.106. The broken curve of the sample-sensor temperature indicates that the calorimeter was still controlled reasonably well. A magnification of the traces is given in Fig. 4.107. In the melting and crystallization ranges a temperature lag of up to 0.2 K can be seen when comparing the dash-dotted line for the programmed temperature with the solid line for the measured temperature. The small heat-flow-rate amplitudes only due to heat capacity can be noticed before and after the melting range (up to 5 min and above 14 min). To create the about 30-fold increase in T during melting and crystallization, the blockand reference-temperature modulations must have increased also by a factor of 30, a situation unique for controlling the modulation at the sample sensor. The first melting occurs at peak 1, close to Ts(t) for zero heating rate, and the last crystallization at peak 18, close to the same temperature (see Figs. 4.82 and 4.83). During melting, sufficient nuclei are seemingly left in these measurements to initiate crystallization with little or no supercooling.

The evaluation of the heat-flow-rate amplitude as modeled in Figs. 4.97–100 gives obviously an apparent heat capacity that yields on integration a too large reversing heat of fusion. An example of such evaluation is shown for a block-temperature- modulated TMDSC in Fig. 4.108. The maximum amplitude <AHF> (H in Figs. 4.99–102) remains high as long as melting and crystallization occurs. Similarly, the

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Fig. 4.106

Fig. 4.107

total heat-flow rate <HF(t)> is broadened over the whole melting and crystallization range, and even the largest maximum at the end of the melting is displaced to a higher temperature and broadened due to the averaging procedures. Only the time-domain recording of the heat-flow rate HF(t) can be interpreted as is seen in Figs. 4.107–108. After the last melting in peak 19 of Fig. 4.106, the melt supercools, as can also be seen