Deconvolution of DSC peaks of high-temperature phases in an illite/smectite–CaCO3 mixture

Heat flow curves measured in differential scanning calorimetry (DSC) may contain overlapping peaks corresponding to multiple-step processes. To analyze the individual steps in such a process, a deconvolution of the overlapping peaks into the individual peaks is necessary. Using the Frazer-Suzuki function to describe asymmetric peaks, we apply a nonlinear least squares analysis to perform the decomposition of heat flow curves measured for a mixture of an illite/smectite clay with 19.6 wt.% of CaCO3. The curves contain two overlapping peaks associated with the crystallizations of gehlenite and anorthite in the temperature interval from 900 °C to 1050 °C. Several versions of the decomposition analysis may be used, depending on the number of optimized parameters. These may range from four to all eight parameters (four for either peak). We applied the versions with seven and eight fitting parameters, yielding results that are in very good agreement with the experimental data.


Introduction
Traditional ceramic products are still widely used.Progress in the production technology can be achieved by modifications in the composition of raw ceramic mixtures or by the use of waste materials.The latter are often characterized by a high content of CaO that, when added into clays, is responsible for the formation of anorthite (CaO • Al2O3 • 2SiO2) [1].This leads to an improvement in material properties, such as an increase in the mechanical strength [2,3].This is usually used in the production of tiles, paves, or porcelain [4].It is therefore interesting to investigate the crystallization of new phases in a system of illite/smectite and laboratory CaCO3 mixture for which a non-isothermal kinetic analysis of the corresponding DSC data can provide solid results and good insight.
The DSC measurements of this crystallization process indicate that at high temperatures, between 900 °C and 1050 °C, two steps appear.This is due to two overlapping peaks that are present in the heat flow vs. temperature plots obtained from the DSC measurements.They are associated with the crystallizations of gehlenite and anorthite, respectively [1].To be able to apply the methods of kinetic analysis [5,6] to such a two-step process, it is necessary to decompose the two overlapping peaks into two separate peaks representing the individual crystallization processes by the mathematical deconvolution analysis [7].This is the focus of this paper in which we describe our procedure to carry out this decomposition and apply it to DSC data for an illite/smectite-CaCO3 mixture.
Measurements were carried out using differential scanning calorimetry (DSC) performed by apparatus Netzsch DSC 404F3.First, the heating was from 30 °C up to 830 °C with a heating rate of 20 °C min −1 .Then we applied isothermal regime at 830 °C for 30 min to be sure that all CaCO3 was decomposed into CaO.Finally, heating from 830 °C up to 1100 °C at six heating rates -from 10 °C min −1 to 15 °C min −1 -was applied.Two measurements were performed for each heating rate.All measurements were done in an argon atmosphere with a flow rate of 40 ml min −1 .

Theoretical
We first split each measured DSC curve (Φ) into its baseline (Φbase) and excess curves (Φex), using the procedure from [8].Namely, upon selecting two ranges of temperatures, one to the left and one to the right of the DSC curve cumulative peak, we use the best linear fits (L1 and L2) of the curve Φ in these ranges to get the baseline curve as where 0 < J < 1 is a jump that occurs in the integrated curve, Γ, of the heat flow curve Φ.If l1 and l2 are best linear fits of Γ in the two selected ranges, then J = [2Γ -(l1 + l2)]/(l2 -l1).Calculating Φbase, the excess heat flow is equal to the difference between the heat flow curve and the baseline curve, Φex = Φ -Φbase.The two ranges of temperatures are such that the two DSC curves measured for every sample are as close to each other as possible (obtained by minimizing the sum of squares between the two curves within the range from 895 °C to 1095 °C).
The mathematical deconvolution analysis was applied to decompose the excess heat flow curves Φex into a sum of two individual overlapping peaks.This was done by a nonlinear least squares analysis using the Frazer-Suzuki function [9] where T0, h, ΔT, and a are four fitting parameters.The first two parameters are equal to the peak's maximum position and height, respectively, while the last two parameters are related to the peak's half-width and asymmetry.Note that the logarithm in φ(T) is well defined only when T > T0 -ΔT/2a for a > 0 and when T < T0 -ΔT/2a for a < 0. For the remaining intervals of temperatures, the function fFS is set equal to zero.For a = 0 the exponent φ(T) is set equal to 2(T -T0)/ΔT.We performed two types of fitting calculations.One was with eight fitting parameters, four for either individual peak function fFS.The other fitting was with seven parameters: three parameters (T0, ΔT, and a) for either individual peak function fFS, plus one additional parameter that varied the heights of the two peaks in a pre-fixed ratio.The ratio coincided with that of the input values.
The input values of the fitting parameters were chosen using the experimental DSC data as follows.First, the input maximum positions (T1 and T2) of the two peaks were taken as the positions in the measured DSC curves.Then the two input heights (H1 and H2) were taken as the values of the DSC curves at T1 and T2.The two input peaks' half-widths were estimated as the differences between the center T12 = (T1 + T2)/2 and the temperatures (one below T1 and one above T2) at which the DSC curves are equal to half of H1 and H2.The two input peaks' asymmetries were estimated from the DSC curves as (1) the ratio of the curve areas below T1 and below the center T12 for the left peak, and (2) one minus the ratio of the curve areas above T2 and above the center T12 for the right peak.
The calculation of the apparent activation energy, EA, is based on the Kissinger equation [10] where β is the heating rate, Tm is a temperature of a peak's maximum, R is the universal gas constant, and a is a dimensionless constant.The parameters β0 and T0 are arbitrary heating rate and temperature, respectively, and will be chosen as β0 = 10 °C/min and T0 = 1000 K.

Results and discussion
In figure 1 we show the heat flow vs. temperature plots obtained from DSC measurements of the illite/smectite-CaCO3 mixture at the six heating rates.Between 900 °C and 1050 °C, two overlapping peaks are present in these plots.They are associated with the crystallizations of gehlenite and anorthite, respectively [1].The two peaks into which the excess heat flow curves are decomposed using a nonlinear least squares analysis for eight fitting parameters (see Section 3) are plotted in figure 3. The values of the parameters are listed in table 2. The two peaks into which the excess heat flow curves are decomposed using a nonlinear least squares analysis for seven fitting parameters (see Section 3) are plotted in figure 4. The values of the parameters are listed in table 3.
The coefficients of determination R 2 corresponding to these results ranged from 0.999516 to 0.999981 when eight fitting parameters were used.When seven fitting parameters are used, the fits are slightly less precise, with R 2 ranging from 0.999509 to 0.999943.Nevertheless, in both cases the obtained decompositions of the excess heat flow into two peaks fit the experimental data with very good precision.Table 1.The ranges of temperatures used to calculate the excess heat flow curves.Table 3.The values of the peaks parameters (four for either peak) obtained by a nonlinear least squares analysis in which all eight parameters were varied under the condition that the left and right peaks have pre-fixed ratio (equal to that from the measured data).Using equation (3), we calculated the apparent activation energy EA for the two processes (the crystallizations of gehlenite and anorthite) associated with the two overlapping peaks.To this end, we used that, according to equation (3), EA is equal to minus the slope of the best (least square) linear fit of the plot ln(β T0 2 /β0 Tm 2 ) vs. 1/RTm (see figure 5).The so determined values of EA are listed in table 4. We observe that EA are similar for both fitting procedures (with seven or eight parameters) for a given measurement and a given peak.The average values (over both measurements and both fitting procedures) of EA are almost identical, around 400 kJ/mol.This suggests that both processes have very similar activation energies, nevertheless further thermal analysis will be needed to describe these processes in a more complete picture.

Conclusions
We presented a procedure to decompose excess heat flow curves obtained from DSC measurements within the temperature range from 900 °C to 1050 °C for a mixture of illite/smectite clay and CaCO3 into two overlapping peaks associated with the crystallizations of gehlenite and anorthite.We first described how excess heat flow curves were determined from experimental data.Then a nonlinear least squares analysis was applied to perform the decomposition, using the Frazer-Suzuki function with four fitting parameters as a mathematical model for an asymmetric peak.Two cases were considered: in one all eight parameters (four for either peak) were optimized, while in the other one the ratio of peaks' heights was pre-fixed.Both led to very similar results, the former one being slightly more accurate.The activation energy is very similar for both processes, around 400 kJ/mol.

Figure 1 .
Figure 1.The DSC heat flow curves of the illite/smectite-CaCO3 mixture at the six heating rates.At each rate two measurements were performed. .

Figure 2 .
Figure 2. The excess heat flow curves corresponding to the data in figure 1.

Figure 3 .
Figure 3.The decomposition of excess heat flow curves from figure 2 into two peaks using a sum of two Fraser-Suzuki functions and eight fitting parameters.

Figure 4 .
Figure 4.The decomposition of excess heat flow curves from figure 2 into two peaks using a sum of two Fraser-Suzuki functions and seven fitting parameters.

Figure 5 .
Figure 5. Plots used to calculate the apparent activation energy.

Table 2 .
The values of the peaks parameters (four for either peak) obtained by a nonlinear least squares analysis in which all eight parameters were varied.

Table 4 .
The values of the apparent activation energy EA.