Complex heat capacity

In general, one can represent a complex number, defined in Fig. A.6.3 of Appendix 6 as z = a + ib, with i = V-1. A complex number can also be written as  

The internal degree of freedom contributes at low frequencies the total equilibrium contribution, A Cp, to the specific heat capacity. With increasing o), this contribution decreases, and finally disappears. 

The limiting dissipative parts, Cp (w), without analogs in equilibrium thermodynamics, are  

The dissipative heat capacity Cp (( )) is a measure of A,s, the entropy produced in nonequilibrium per half-period of the oscillation T(t) - T  

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Figure 36. Glass transition temperature 7 (determined by ac calorimetry at 20 Hz) for a film thickness of 93 nm with respect to dependence on the annealing time at 200°C in ambient air. Inset The real and the imaginary part of the complex heat capacity (in arbitrary units) at 20 Hz as a function of temperature.

An alternative nomenclature uses complex heat capacity such that ... 

Gaur U, Pultz G, Wiedemeier H, Wunderlich B (1981) Analysis of the Heat Capacities of Group IV Chalcogenides using Debye Temperatures. J Thermal Anal 21 309-326. Baur H, Wunderlich B (1998) About Complex Heat Capacities and Temperature-modulated Calorimetry. J Thermal Anal and Calorimetry 54 437 65. 

Merzlyakov M, Wurm A, Zorzut M, Schick C (1999) Frequaicy and Temperature Amplitude Dependence of Complex Heat Capacity in the Melting Region of Polymers, J Macromolecular Sci, Phys 38 1045-1054. 

As a result the heat capacity measured by TMDSC can be considered as a complex heat capacity and is denoted complex heat capacity has two components a component that is in-phase with... 

Complex heat capacity (C P) quenched PET sample (courtesy of TA Instruments Inc.)... 

Very often the out-of-phase component C is small, so the reversing heat capacity (modulus of the complex heat capacity) is the same as the in-phase component (phase-corrected reversing or real heat capacity). So, the phase correction can be neglected. This means that the simple deconvolution deflned above can be used. 

This gives rise to a complex heat capacity of... 

The Fourier transform of the rates of change of enthalpy and temperature are related through that of the kernel function, which can be identified with the complex heat capacity. By carrying out a succession of experiments to determine C(a>) and then doing a Fourier inversion it is then possible, in principle, to recover the function x/r, and hence gain information about the physical kinetics. However, to be useful this approach must refer to more specific models of realistic behaviour, which then brings us back to the kinds of results discussed in this chapter. 

To do the calibration for a required frequency in finding the three Ks, it is now necessary to carry out three runs one with an empty sample pan to fix ATi, and then two more with different heat capacities for the sample to determine Kj and K. Once these have been established, the calorimeter can be used to determine the Cp (or rather the complex heat capacity C) for a sample by ... 

The simultaneous measurement of the amplitude (modulus) of the complex heat capacity, the heat flow and the phase angle between heat flow and heating rate (termed heat flow phase) enables a more detailed study of complicated material systems, both in quasi-isothermal and non-isothermal conditions. The extraction of the signals is briefly summarised below. More details on theory and applications are given in dedicated special issues of Thermochimica Acta [6,7] wid Journal of Thermal Analysis and Calorime-try [8] and in other chapters of this book. 

Using the corrected heat flow phase (material contribution) and the modulus of the complex heat capacity, Cp, the components in-phase (Cp and out-of-phase (Cp ) with the modulated heating rate can be calculated using Eq. (4) (for more detailed approaches see Refs. [3,9,10]) ... 

As soon as it was introduced, MTDSC became a controversial technique. This was is large part because, unusually for a new method of characterisation, it was launched as a commercial product with no gestation period in an academic laboratory. Commercial rivalries inevitably led to conflict and a certain amount of misrepresentation. There was also confusion because the first commercial version did not include the ability to use the phase angle to separate the response to the modulation into in- and out-of-phase components. Despite this omission, the fact that this was an option was demonstrated and discussed at the time MTDSC was first described [1-3]. Some workers leapt to the conclusion that this had not been considered and criticised the technique for this reason proposing the alternative method of deriving a complex heat capacity [14] (like Goldbrecht et al. [3]). The debate became polarised into advocacy of one or the other approach when, in reality, this conflict was entirely artificial and the use of the phase angle is completely compatible with the practise and theory of MTDSC [15]. Unfortunately, even today this fallacy persists. 

In the MTDSC technique, Cp (heat capacity) data from the convoluted MDSC is replaced with the term Cp, called a complex heat capacity. Cp can then be... 

Fig. 16.4 Illustration of the use of DSC data for measuring Tg and AH (the activation energy for enthalpy relaxation). Tgon, Tgenj, and ATg indicate the onset, end, and width of the glass transition. Modulated DSC (mDSC) allows the separation of the total heat flow into reversing and nonreversing components. AH can be evaluated from (a) the dependence of Tgon on scanning rate q, (b) ATg, (c) the dependence of the relaxation enthalpy AH (area of the overshoot on annealing time, and (d) the dependence of the complex heat capacity Cp (obtainable by mDSC) on modulation frequency v. (Reproduced with permission from Yu 2001)...

If time-dependent processes are involved, then the process is clearly outside the scope of classical thermodynamics. The time-dependent (apparent) heat capacity, measured with, say, TMDSC would lead to time-dependent potential functions which must be interpreted in terms of irreversible thermodynamics. In such cases, a nonzero imaginary part of the complex heat capacity exists which is linked to the entropy production of the process in question (for details see [32]). Thus, temperature-modulated calorimetry makes it possible to determine time-dependent (irreversible) thermodynamic quantities. 

The measured heat flow rate function consists of two summands. The first one is identical to the heat flow rate one would measure in a common DSC the second one changes periodically with the same frequency as the temperature and an amplitude proportional to the (apparent) heat capacity of the sample. The amplitude and phase of the latter contain additional information from which a frequency-dependent and therefore complex apparent heat capacity can be deduced. Gobrecht, Flamann, and Willers (1971) were the first to use this method to determine the complex heat capacity of polymers in the glass... 

From Eq. (7.24), it follows that the vibrational (static) heat capacity of the sample can be determined in the usual way from the first summand as well as from the amplitude of the modulated component of the measured heat flow rate function. In absence of other processes, the two results must coincide this can be used for calibration purposes. If other processes take place in the sample, the apparent heat capacity from the modulated part, the reversing heat capacity, may differ from the apparent heat capacity, the total heat capacity determined from the first summand of the right-hand side in Eq. (7.24). The difference between these two apparent heat capacities is the so-called nonreversing heat capacity . To obtain the frequency-dependent complex heat capacity from the measured heat flow rate function, different evaluation methods exist. It is not within the scope of this book to explain this ambitious matter in detail interested readers are referred to the special literature (e.g., Hohne, Hemminger, and Flammersheim, 2003 Schick, 2002). 

For such an arrangement, the heat diffusion equation can be solved (see Jeong, 1997), and the (possibly complex) heat capacity of the sample ensemble can be calculated from the amplitude Ta and phase shift

[Pg.206]

In the case of time-dependent processes in the sample, an additional phase shift occurs, and the heat capacity remains complex. The same is true if nonadiabatic conditions exist (Pth < °o)- Minakov (1997) constructed such a calorimeter in which the complex heat capacity of very thin (lOfxm) samples at low temperature at frequencies up to 1 kHz could be measured. 

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