Thermal Analysis of Polymeric Materials

358 4 Thermal Analysis Tools

___________________________________________________________________

a heat-flux DSC as shown in the sketch A of Fig. A.9.2, using different purge gases. The magnitude of change in the onset temperature of melting is similar to Fig. 4.82. The rounding in the vicinity of heating rate zero is a specific instrument effect and does not occur with the operating system of the power-compensated DSC in Fig. 4.82. When controlling the furnace temperature as in the heat-flux DSC of Fig. 4.57, there is a mass dependence of the onset of melting as illustrated in Fig. 4.84.

Fig. 4.84

Note, that some DSCs have a lag correction incorporated in their analysis software. Such corrections are, however, only approximations because of the changes with sample mass, heat conductivity, and environment as is pointed out on pg. 340, above. More applications of the DSC to the analysis of materials are presented in Sect. 4.4 as well as in Chaps. 6 and 7, below.

4.4 Temperature-modulated Calorimetry

A major advance in differential scanning calorimetry is the application of temperature modulation, the topic of this section. The principle of measurement with temperature modulation is not new, the differential scanning technique, TMDSC, however, is. This technique involves the deconvolution of the heat-flow rate into one part that follows modulation, the reversing part, and one, that does not, the nonreversing part. The term reversing is used to distinguish the raw TMDSC data from data proven to be thermodynamically reversible. Reversing may mean the modulation amplitude bridges the temperature region of irreversibility or the modulation causes nonlinear or nonstationary effects, as will be discussed in Sects. 4.4.3 and 4. The first report about TMDSC was given in 1992 at the 10th Meeting of ICTAC [1,22], see also Fig. 2.5.

___________________________________________________________________

4.4.1 Principles of Temperature-modulated DSC

In principle, any DSC can be modulated. As the details of construction of the DSC equipment vary, it may be advantageous to modulate in different fashions. One can modulate the block, reference, or sample temperatures, as well as the temperature difference (proportional to the heat-flow rate). For example, the temperature modulation of the Mettler-Toledo ADSC™ is controlled by the block-temperature thermocouple (see Fig. 4.57), while the modulation of the MDSC™ of TA Instruments in Fig. 4.85 is controlled by the sample thermocouple. Of special interest, perhaps, would be a modulated dual cell as shown in Fig. 4.56.

Fig. 4.85

In the following discussion, most calculations are patterned after the heat flux DSC as developed by TA Instruments. The actual software, however, is proprietary and may use a different route to the same results and also may change as improvements are made. Similarly, for other instruments the derived equations must be adjusted to the calorimeter used. A method to combine standard DSC and sawtooth modulation allows the simultaneous analysis of the standard DSC on linear heating and cooling segments and the averaged total response, and the evaluation of the reversing heatflow rates from the Fourier harmonics of the sawtooth. Details about this versatile modulation are shown in Appendix 13, which also contains a detailed discussion of sawtooth modulation linked to the following description of TMDSC.

The major discussion will be based on the DSC shown in Fig. 4.85 which is a further development of the DSC A of Fig. A.9.2, i.e., TMDSC is an added choice of operation, not a new instrument. It will be advantageous to do some sample characterizations in the DSC mode, others in the TMDSC mode. It was noted in Sect. 4.3 that DSC could be used as a calorimeter or as DTA, similarly TMDSC can

362 4 Thermal Analysis Tools

___________________________________________________________________

modulation can be treated similar to the here discussed sinusoidal modulation, so that a single sawtooth, for example, can generate multiple-frequency experiments. Figure 4.89 illustrates the complex sawtooth of Fig. 4.87 which can be used to perform multi-frequency TMDSC with a standard DSC which is programmable for repeat segments of 14 steps [29]. Each of the five major harmonics possesses a similar modulation amplitude and the sum of these five harmonics involves most of the programmed temperature modulation, so that use is made of almost all of the temperature input to generate the heat-flow-rate response.

Fig. 4.89

4.4.2 Mathematical Treatment

Next, a mathematical description of Ts is given for a quasi-isothermal run. This type of run does not only simplify the mathematics, it also is a valuable mode of measuring Cp as described in Sect. 4.4.5. In addition, standard TMDSC with <q> 0 is linked to the same analysis by a pseudo-isothermal data treatment as described in Sect. 4.4.3.

Figure 4.90 summarizes the separation of the modulated sample temperature into two components, one is in-phase with Tb, the other, 90° out-of-phase, i.e., the in-phase component is described by a sine function, the out-of-phase curve by a cosine function. The figure also shows the description of the time-dependent Ts as the sum of the two components. The sketch of the unit circle links the maximum amplitudes of the two components. Furthermore, the standard addition theorem of trigonometric functions in the box at the bottom expresses that Ts is a phase-shifted sine curve.

The additional boxed equation on the right side of Fig. 4.90 introduces a convenient description of the temperature using a complex notation. The trigonometric functions can be written as sin = ( i/2)(ei e i ) and cos = (1/2)(ei + e i ) where i = 1, so that sin + i cos = ie i . The real part of the shown equation

364 4 Thermal Analysis Tools

___________________________________________________________________

The next step is the analysis of a single, sinusoidal modulation in the DSC environment. In the top line of Fig. 4.91 Eq. (2) of Fig. 4.69 is repeated, the equation for the measurement of heat capacity in a standard DSC. The second line shows the needed insertions for Ts and T for the case of modulation. When referred to Tb, the phase difference of T is equal to and its real part is the cosine, the derivative of the sine, as given by the top equation. Next, the insertion and simplification of the resulting equation are shown. By equating the real and imaginary parts of the equations separately, one finds the equations listed at the bottom of Fig. 4.91. These equations suggest immediately the boxed expression for Cs Cr in Fig. 4.92.

Fig. 4.91

Several types of measurement can be made. First, one can use, as before, an empty pan as reference of a weight identical to the sample pan. The bottom equation of Fig. 4.92 suggests that in this case the sample heat capacity is just the ratio of the modulation amplitudes of T and Ts multiplied with a calibration factor that is dependent on the frequency and the mass of the empty pans. Standardizing on a single pan weight (such as 22 mg) and frequency (such as 60 s) permits measurements of the reversing heat capacity with a method that is not more difficult than for the standard DSC, seen in Figs. 2.28–30. Another simplifying experimental set-up is to use no reference pan at all. Figure 4.92 shows that in this case the pan-weight dependence

___________________________________________________________________

Fig. 4.92

of the calibration disappears. As a disadvantage, the measured quantity is, however, not the sample heat capacity directly, as before, but the measurement is the heat capacity of sample plus sample-pan.

More specific information was derived for the Mettler-Toledo DSC, as described in Figure 4.57 [30]. For the calculations, the DSC was modeled with an analog electrical circuit of the type shown in Appendix 11 for the elimination of asymmetry and pan effects. The Mettler-Toledo DSC is also of the heat-flux type, but the controlthermocouple is located close to the heater. The model considered the thermal resistances between heater and T-sensor, R, sensor and sample, Rs, and a possible cross-flow between sample and reference, R' [30]. Equations (1) and (2) in Fig. 4.93 show the equations which can be derived for the heat capacity, Csm, and the phase angle . A series of quasi-isothermal measurements for different masses, m, and frequencies, are shown in Fig. 4.93 for a sawtooth modulation. Uncorrected data for and are given by the open symbols. They were calculated using the bottomright equation in Fig. 4.92. For the short periods with p less than 100 s, strong deviations occur. The known specific heat capacity of the aluminum sample is marked by the dotted, horizontal line. The constant, negative offset at long periods, p, was corrected by the usual calibration with sapphire (Al2O3) with a constant shift over the whole frequency range. A value of 35.38 mW K 1 was found for K* at 298 K. The deviations at small p were then interpreted with Eq. (1) making the assumption that they originate from the coupling between the various thermal resistance parameters. To determine , Eq. (2) was fitted with the true specific heat capacity of the aluminum. The results of the fitting are indicated in Fig. 4.93 by the dashed lines. The value of Rs + R2/(2R + R') of Eq. (2) has a sample-mass independent value of 1.34/K*. Finally, the corrected values for the specific heat capacities were calculated with Eq. (1) and marked in Fig. 4.93 by the filled symbols. All frequency and mass

366 4 Thermal Analysis Tools

___________________________________________________________________

Fig. 4.93

dependence could be removed by this calculation. The RMS error of the data is 0.77%. Although measurements could now be made down to periods of about 20 s, the data evaluation is still cumbersome and not all thermal resistances one would expect to affect the data could be evaluated, such as the resistances within the sample pan and the heat supplied by the purge gas. If these latter effects cause significant changes or delays in the modulation, the parameters in the just discussed equations in Fig. 4.93 are just fitting parameters.

To find a universal equation which can be used to calibrate the data for different masses, frequencies, sample packing, and purge-gas configuration, the powercompensated DSC was employed [31]. Its operation is described in Figs. 4.58–61. Equation (1) in Fig. 4.94 repeats the approximate heat capacity determination from standard DSC using the total (averaged) heat-flow rate and rate of change of sample temperature (see Fig. 4.76, <HF> = K T, Cs Cr = mcp, <q> = q). Equation (2) progresses to the quasi-isothermal, reversing heat capacity given in Fig. 4.92 at the bottom left. Finally, Eq. (3) shows the substitution of an empirical parameter, , into the square root of the equation in Fig. 4.92 which can be calibrated. For very low frequencies, approaches Cr/K, as expected from Fig. 4.92. This equation was introduced already in Fig. 4.54 as basic equation for the data evaluation.

The experiments on the left of Fig. 4.94 show the results for quasi-isothermal, sawtooth-modulated polystyrene as a function of period and amplitude (12 mg, at 298.15 K) [31]. The filled symbols were calculated using Eq. (1). Data were chosen when steady state was most closely approached, i.e., at the time just before switching the direction of temperature change or over the range of time of obvious steady state. The thin solid line is the expected heat capacity, the dotted lines mark a ±1% error range. The bold line is an arbitrary fit to the open symbols. The different modulation amplitudes were realized by choosing different heating and cooling rates. The