Heat capacity deconvolution

Figure 8.16 (a) IR and (b) Raman spectra for the mineral calcite, CaC03. The estimated density of vibrational states is given in (c) while the deconvolution of the total heat capacity into contributions from the acoustic and internal optic modes as well as from the optic continuum is given in (d). 

Figure 8.23 Heat capacity of Fe2C>3 [18]. The heat capacity is deconvoluted to show the relative magnitude of the main contributions. Qpi = Cp m - Cy m = cc TVIkj.

Deconvolution Analysis of DSC. Data analysis was carried out using the DECONV section of the DA-2 software package. This software, which is based on the deconvolution procedure of Freire and Biltonen (4), allows deconvolution of differential heat-capacity peaks either as the result of simple addition of multiple independent transitions or as the result of more complex mathematical processes representing the combination of transitions that interact in such a way that an obligatory reaction sequence is imposed (sequential transitions). 

Each thermogram was normalized on scan rate, the corresponding (scan-rate-normalized) buffer-buffer baseline was subtracted, and the differential heat capacity values were divided by the number of moles of protein or peptide in the sample, to yield ordinate values in terms of calories moF deg. The resulting files were then analyzed using the deconvolution software. 

Figure 4. Deconvolution of the denaturational endotherm for native CBHI at pH 4.80. Circles represent experimental values for differential heat capacity the solid curves represent the overall best fit model and the two sequential component transitions that contribute to the overall fit (See text).

The upper solid line represents the measured heat capacity, lower lines show the deconvolution with solid lines for the individual melting... 

Dealing with simple deconvolution first, it can be seen from Equation 4.3 that the heat flow signal is composed of two sets of terms. The first, Cp(b + Bto costow)), is dependent on the magnitudes of the terms b, B, and to (all experimental rather than sample parameters) but has no sample dependence other than the value of Cp. Consequently, this first term is a measure of the sample s heat capacity (as defined previously) and, except in the case of the glass transition (see later discussion), can be considered, for the purposes of this discussion, to be an effectively instantaneous sample response. If we focus on the modulation, we see that this is a cosine wave as opposed to the sine wave in the temperature. This means that, when there is no transition, the modulation in heat flow will, in principle, follow this cosine wave (the modulation in heating rate) with zero phase lag when the convention is adopted that endothermic is up. It will be 180° out of phase when the convention is adopted that endothermic is down. 

Proceeding with simple deconvolution, from Equation 4.2, assuming C is negligible, we can calculate heat capacity from the ratio of the amplitude of the modulated heat flow (AMHF) to that of the modulated heating rate (AMHR) via... 

We can define a new quantity, the kinetic heat capacity CpK = C/AMHR. The complete deconvolution then gives rise to a new nomenclature ... 

For MTDSC, it is also essential to calibrate for the reversing heat capacity to allow quantification of the deconvoluted results. There are several approaches by which this may be achieved, with more accurate methods requiring greater sophistication and more time hence, a decision needs to be made with regard to how important accurate heat capacity data are to the objectives of the study. For most pharmaceutical applications, fairly simple calibration procedures such as those about to be outlined are usually sufficient. However, for more accurate work it is essential to use more detailed approaches such as that described in Reference 11. In this summary, we outline only the simple approaches, but readers should be aware of the availability of more complex methods that yield more reliable results. 

More refined models of the heat capacities of polymers can be obtained by deconvoluting the "skeletal vibrations" of chain molecules from their set of discrete "atomistic group vibrations", and by further deconvoluting the "intramolecular" component of the skeletal vibrations from the "intermolecular" (i.e., interchain) component. The major portion of the heat capacity at temperatures of practical interest (i.e., temperatures which are not too low) is accounted for by the atomistic group vibrations. The remaining portion of the heat capacity arises from skeletal modes. Detailed discussion of these issues is beyond the scope of this chapter. The reader is referred to the reviews provided by references [1-3] for further details and lists of the original publications. 

The complete deconvolution then proceeds in the same way as for the simple deconvolution, except that the phase-corrected reversing heat capacity is used instead of the reversing heat capacity. Thus... 

Figure 1.6. Co-plot of reversing and non-reversing heat capacity arising from the simple and full deconvolution procedures apphed to the raw data from Figure 1.2.

The above is intuitively satisfactory when one considers that, in a zero-order reaction, the reaction rate will change only with temperature and will thus follow the B sin cot of the modulation. The contribution from the heat capacity, on the other hand, follows the derivative of temperature and thus follows coB cos cot. The in-phase contribution arises from a signal that depends only on the heat capacity. Thus, this provides a means of separating or deconvoluting these two different contributions to the heat flow. 

Thus, as also demonstrated in Eq. (32), by carrying out this deconvolution procedure it is possible to separate the two fundamentally different contributions to the total heat flow the reversible contribution that derives from the heat capacity (the phase-corrected reversing heat flow) and the contribution that derives from f(t,T) which is, on the time-scale of the modulation, irreversible. In most cases, the phase-corrected reversing heat flow will be the same as the reversing heat flow to an accuracy greater than that of the measurement being made. 

Thus, it is possible to conclude that the non-reversing heat flow contains that part of the underlying signal that comes from the chemical reaction. In most cases, it is also true to a very good approximation that C = CpPCR-Thus, it is not necessary to use the phase correction in order to measure the heat capacity and then calculate the non-reversing signal. So, the simple deconvolution can be used. 

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. 

Figure 4.63 illustrates that the second harmonic is a minor, but not negligible, correction to the total heat capacity. Of additional interest are the remaining small ripples of the various Cp plots. The deconvolution should have removed all periodic contributions of frequency co and also higher harmonics. Inspection of Eq. (24) shows, however, that the underlying heating rate causes a small frequency shift of the type of a Doppler effect, as found in the analysis of sound from moving sources, quantitatively assessable... 

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