Sawtooth modulation

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 x, while at... 

Pak J, Wunderlich B (2001) Heat Capacity by Sawtooth-modulated, Standard Heat-flux Differential Scanning Calorimeter with Close Control of the Heater Temperature. Thermochim Acta 367/368 229-238. 

Kwon YK, Androsch R, Pyda M, Wunderlich B (2001) Multi-frequency Sawtooth Modulation of a Power-compensation Differential Scanning Calorimeter. Thermochim. Acta 367/368 203-215. 

Figure 6.106 shows the TMDSC of fibers B using a sawtooth modulation and evaluation with the standard DSC technique described in Appendix 13. On first heating with constraint, reversibility exists to 405 K, which includes a substantial decrease in the orthorhombic crystals and compensating increase in the intermediate phase. On second heating, after recrystallization, the DSC and TMDSC traces are similar to the sample of Fig. 6.102 and contain only orthorhombic crystals. In this... 

Calorimeter Perkin Elmer Pyris 1, sawtooth modulation amplitude 1.0 K, period 60 K... 

In this Appendix a number of applications of sawtooth modulations are described with modeling and actual results, starting with the sawtooth modulation by utilizing a standard DSC and analysis without Fourier analysis, followed by the analysis of the sawtooth-modulation data after fitting to a Fourier series. Such temperature modulation can be done with any standard DSC which can be programed for a series of consecutive heating and cooling steps. 

Figure A. 16.6 illustrates the attainment of steady state after the beginning of sawtooth modulation and on changing from a heating to a cooling segment. Based on these equations, the following curves are computed for Figs. A.13.7-10by using the same conditions as in the equations shown in Figs. 4.67 and 4.68.

The sawtooth modulation shown in Fig. A.13.14 allows an easy interpretation of the actual data for a similar pentacontane sample as in Figs. A.13.11 and A.13.12. 

Metaer-Toledo DSC 820 Linear heating 7.24 K/min Linear cooling 5.24 K/min (Sawtooth modulation with an underlying heating rate of 1.0 K/min, and a modulation period p = 90 s)... 

An extensive analysis of the sawtooth modulation brought a number of interesting results. Mathematically, it could be shown that if there were no temperature gradients within the sample and if all other lags and gradients could be assessed with the Fourier heat-flow equation, Eq. (11) does allow the calculation of the precise heat capacities [33]. Temperature gradients are, however, almost impossible to avoid. Especially in the power-compensated calorimeter, the temperature sensor is much closer to the heater than the sample and cannot avoid gradients. The empirical solution to this problem was to modify Eq. (11) as follows [34] ... 

Figure 4.17. Heat flow rate in MTDSC as a function of time for a quasi-isothermal sawtooth modulation, with indicated first, third and fifth harmonic Fourier components.

Figure 4.27 illustrates in its top sketch the simple sawtooth modulation, discussed in Section 3.1, and the response to a sawtooth modulation is shown in Figure 4.17. The amplitudes of the Fourier series of the sawtooth modulation decrease with 1/r, so that the precision of the analysis of higher harmonics decreases rapidly. To overcome this difficulty, the complex sawtooth shown in the center sketch of Figure 4.27, as well as given by the series of Eq. (15), was proposed [45]. Its first four Fourier terms describe practically all the variation shown in Figure 4.17. An overall modulation repeat of 210 s yields almost equal temperature amplitudes with periods of 210, 70, 42 and 23.3 s.

Figure 4.41. Quasi-isothermal melting of indium using a heat-flux calorimeter with control of the modulation at the heater temperature (sawtooth modulation, the indicated sample temperatures are uncorrected) [53].

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