Kinetic analysis of complex solid-state reactions. A new deconvolution procedure
Abstract
The kinetic analysis of complex solid-state reactions that involve simultaneous overlapping processes is challenging. A method that involves the deconvolution of the individual processes from the overall differential kinetic curves obtained under linear heating rate conditions, followed by the kinetic analysis of the discrete processes using combined kinetic analysis, is proposed. Different conventional mathematical fitting functions have been tested for deconvolution, paying special attention to the shape analysis of the kinetic curves. It has been shown that many conventional mathematical curves such as the Gaussian and Lorentzian ones fit kinetic curves inaccurately and the subsequent kinetic analysis yields incorrect kinetic parameters. Alternatively, other fitting functions such as the Fraser-Suzuki one properly fit the kinetic curves independently of the kinetic model followed by the reaction and their kinetic parameters, and moreover, the subsequent kinetic analysis yields the correct kinetic parameters. The method has been tested with the kinetic analysis of complex processes, both simulated and experimental.
Full text
1 Kinetic analysis of complex solid state reactions. A new deconvolution procedure Antonio Perejón, Pedro. E. Sánchez-Jiménez, José M. Criado and Luis A. Pérez-Maqueda* Instituto de Ciencia de Materiales de Sevilla. C. Américo Vespucio 49, Sevilla 41092. Spain EMAIL ADDRESS: [email protected] RUNNING TITLE: Kinetic analysis of complex solid state reactions *Corresponding author . Tel +34954489548 Fax +34954460665
2 Kinetic analysis of complex solid state reactions. A new deconvolution procedure ABSTRACT The kinetic analysis of complex solid state reactions that involve simultaneous overlapping processes is challenging. A method that involves the deconvolution of the individual processes from the overall differential kinetic curves obtained under linear heating rate conditions, followed by the kinetic analysis of the discrete processes using combined kinetic analysis is proposed. Different conventional mathematical fitting functions have been tested for deconvolution, paying special attention to the shape analysis of the kinetic curves. It has been shown than many conventional mathematical curves such as the Gaussian and Lorentzian ones fit inaccurately kinetic curves and the subsequent kinetic analysis yields incorrect kinetic parameters. Alternatively, other fitting functions such as the Fraser-Suzuki one properly fits the kinetic curves independently of the kinetic model followed by the reaction and their kinetic parameters, moreover the subsequent kinetic analysis yields the correct kinetic parameters. The method has been tested with the kinetic analysis of complex processes both simulated and experimental. KEYWORDS: Kinetic Analysis, solid-state reactions, deconvolution, complex processes.
3 1. Introduction Solid state reactions are in many cases complex and involve several overlapping processes. The kinetic analysis of such solid state reactions is challenging, as far as the kinetic parameters, i.e. activation energy, preexponential factor and kinetic model, of each individual process should be determined for a complete kinetic description of the overall reaction. Thus, while a large number of analytical methods are available for determining the kinetic parameters of discrete solid state reactions, the number of procedures for the analysis of complex processes is much more limited. Methods for the analysis of discrete processes include isoconversional or model-free methods,1-3 model fitting procedures,4 master plots,5-7 non parametric analysis,8,9 and combined kinetic analysis.10,11 For complex processes, non linear regression methods are the most commonly used.12-14 In general, a reaction scheme with different processes is assumed, while the kinetic parameters, i.e activation energy and preexponential factor, corresponding to the different individual processes are optimized by an iterative procedure that minimizes an objective function, usually defined as a function of the difference between the experimental curves and the curves reconstructed using the kinetic parameters to be optimized. An interesting alternative for the kinetic analysis of complex processes with overlapping reactions implies separating the individual processes by peak deconvolution, using statistical functions, followed by the kinetic analysis of the separated peaks to calculate the kinetic parameters. Different deconvolution functions have been used in literature, being the Lorentz distribution function one of the most extensively used. Examples of solid state reactions where this latter function is used for the separation of the individual processes include the thermal decomposition of poly[B- (methylamino)borazine] precursor into boron nitride,15 the thermal decomposition of a polymer made from silsesquioxanes that convert into silicon oxycarbide ceramics on pyrolysis,16 the thermal decomposition of hydromagnesite that involves different decomposition steps, i.e. dehydration, dehydroxylation and decarbonation,17 the thermal degradation of cellulose derivatives/starch blends,18
4 and the pyrolysis reactions in industrial waste activated sludge.19 Wagner et al have proposed the use of Gaussian functions for the deconvolution of the overlapping processes in the differential scanning calorimetry traces corresponding to the crystallization of chalcogenide glasses.20 Other statistical functions such as the Weibull and logistic mixture models have been also used for fitting complex solid state reactions, such as thermal degradation of polyurethane, 21,22 wheat straw oxidative pyrolysis,23 and animal bones combustion.24 The objective of the present paper is performing a comparative study of the different deconvolution functions for fitting curves corresponding to the diverse kinetic models proposed in literature for solid state reactions, including a shape analysis of the kinetic curves. Additionally, a new procedure for performing the kinetic analysis of complex solid state reactions will be proposed. The method involves the deconvolution of the differential curves obtained at different linear heating rates followed by the combined kinetic analysis of the resulting individual curves to obtain the kinetic parameters without assumptions about the kinetic model followed by the reaction. This procedure will be tested with both simulated and experimental curves. 2. Theoretical The reaction rate for a single solid state process in conditions far from equilibrium is described by two functions, one of the reaction temperature and another of the extent of conversion, i.e. k(T) and f(α), respectively: (1), being α the reaction fraction, t the time and T the temperature. In general, k(T) is described by an Arrhenius expression: / (2),
5 where A is the Arrhernius preexponential factor, E is the activation energy, and R is the gas constant. For f(α), a number of expressions have been proposed in literature, a selection of the most common ones are included in Table 1. These latter expressions have been proposed considering different physical ideal models that take into consideration certain geometrical and driving forces for solid state processes.25,26 A complete kinetic analysis procedure involves the calculation of both k(T) and f(α) functions that properly describe the solid state process. 2.1 Determination of the activation energy from isoconversional methods The activation energy, E, as a function of the reacted fraction can be determined from isoconversional methods without any previous assumption on the kinetic model fitted by the reaction. One of the most extensively used isoconversional method is that proposed by Friedman1 that provides accurate values of activation energies even if they were function of the reacted fraction.27 Taking into consideration the Arrhenius expression (eq 2), the general kinetic equation (eq 1) can be written in logarithmic form as follows: (3) At a constant value of , f( ) would be also constant and eq 3 would be written in the form: (4) Thus, the activation energy at a constant value, Eα , can be determined from the slope of the plot of the left hand side of eq 4, that is the logarithm of the reaction rate at a constant value of α, versus the inverse of the temperature at the same value of α.
6 2.2 Combined kinetic analysis The combined kinetic analysis allows determining the kinetic triplet (E, A, and f(α)) from the simultaneous analysis of a set of different curves measured under any different temperature programs (not necessarily linear).10,11,28 In this method, the kinetic model is determined in the following general form: 1 (5). This equation can accurately fit every ideal kinetic model included in Table 1 by adjusting the parameters c, n and m. Besides, this equation (eq 5) can also describe deviations of the ideal kinetic models due to inhomogeneities in the shape and size of the solid particles.10 By introducing eq 5 into eq 3 and rearranging terms it follows ln ln1ln (6). This latter equation is the basic equation for the combined kinetic analysis. Thus, the experimental sets of data, i.e. α, dα/dt, and T, corresponding to several temperature programs are substituted into eq 6, while the parameters n and m that provide the best linearity (maximum coefficient of linear correlation, r) to the plot of the left hand side of eq 6 versus the reciprocal temperature, are determined by an optimization procedure. Thus, the entire set of experimental data is used for the determination of the n and m parameters. Once the values of n and m that maximize r are found, the values of E and ln(cA) are estimated respectively from the slope and intercept of the linear plot. Finally, the kinetic model is discriminated by comparing the shape of the f(α) function resulting from the optimization procedure with that of the functions corresponding to the ideal kinetic models included in Table 1.
7 2.3 Deconvolution Different fitting functions have been used for the deconvolution process, namely: Gaussian: (7), where a0, a1, and a2 are amplitude, center, and width of the curve, respectively; Lorentzian: (8), where the parameter a0, a1, and a2 have the same meaning as in the Gaussian function. Weibull:29 (9), where a0, a1, a2 and a3 are amplitude, center, width and shape of the curve, respectively; and Fraser-Suzuki:30,31 2 (10), where a0, a1, a2 and a3 are amplitude, position, halfwidth and asymmetry of the curve, respectively. Two different computer programs were used for nonlinear least squares curve fitting: Peakfit (Systat Software Inc.) and Fityk (distributed under the terms of GNU General Public License). The functions that were not included as standard functions were introduced as user defined functions (Fraser-Suzuki for both programs and Weibull for Fityk). Equivalent results were obtained from both computer programs.
8 3. Results A shape analysis of the kinetic curves is of interest for determining the suitability of the different fitting functions used for deconvolution of complex solid-state processes. The value of the reaction fraction, α, at the maximum reaction rate is indicative of the symmetry of the kinetic curve. Thus, this study is detailed here. Under linear heating rate conditions, the general equation (eq 1) can be written as follows (11), where β is the heating rate. The differentiation of eq 11 yields: ′ (12). At the maximum reaction rate, d2α/dt2 is zero, and therefore: ′ (13), where Tm and αm are the temperature and reacted fraction at the maximum, respectively. Besides, eq 11 can be integrated yielding ∞ (14), being x=E/RT. From eqs 13 and 14, it follows: ′10 (15). Table 2 includes the values of αm obtained from eq 15, using an 8th degree rational approximation to p(x),32,33 as a function of x (E/RT), for the different kinetic models in Table 1. It is clear from Table 2
9 that values of α at the maximum rate are different from 0.5, indicating that kinetic curves are asymmetrical. Therefore, functions such as the Lorentzian or Gaussian ones that yield symmetrical curves seem to be inadequate for fitting kinetic curves, while the Weibull and Fraser-Suzuki functions that allow fitting the asymmetry of the curves might be more adequate. Kinetic curves were simulated by assuming all the different kinetic models in Table 1 and fitted using the Lorentzian, Gaussian, Weibull, and Fraser-Suzuki equations. As a way of example, Figure 1 shows a curve simulated assuming a R2 kinetic model and the resulting fitting curves using Lorentzian, Gaussian, Weibull, and Fraser-Suzuki equations. It is quite clear from Figure 1 that the kinetic curve is asymmetrical and, therefore, the fitting with Lorentzian and Gaussian functions is poor, while Weibull and Fraser-Suzuki equations nicely fit the kinetic curve, being the Fraser-Suzuki equation the one that provides the best fit. Similar results were obtained for the different kinetic models in Table 1. Figure 2 includes a set of four kinetic curves simulated using a fourth order Runge-Kutta numerical integration by assuming three different kinetic models, that is a phase boundary reaction, R3, a nucleation and growth reaction, A2, and a diffusion controlled reaction, D3. These curves were very nicely fitted by the Fraser-Suzuki equation (Figure 2). To further validate the ability of the Fraser-Suzuki equation for fitting kinetic curves that deviate from the ideal kinetic modes due to inhomogeneities in the size of the solid particles, another curve was simulated assuming a D2 kinetic model and a log normal particle size distribution with a standard deviation in logarithmic scale of 0.75.34,35 Figure 2d shows that this latter kinetic curve was properly fitted by the Fraser-Suzuki equation. Similar results were obtained for other particle size distribution functions and other heterogeneities, such as inhomogeneities in particle shape. Therefore, we can conclude that the Fraser-Suzuki equation can accurately fit kinetic curves obtained not only by assuming ideal kinetic models, but also kinetic curves simulated by assuming deviations of the ideal kinetic models due to inhomogeneities in particle size distribution or particle shape. On the contrary, Lorentzian and Gaussian functions fail to fit asymmetrical curves such as kinetic curves. Therefore, to fit kinetic curves obtained for a single process with a Lorentzian or Gaussian fitting function requires of more than a single fitting function, as shown in Figure 3, where a single simulated curve requires at least
16 TABLE 1. f() and f’(), i.e. df()/d, kinetic functions for the most widely used kinetic models, including the newly proposed random scission model. Mechanism Symbol f() f´() g() Phase boundary controlled reaction (contracting area) R2 1/ 1 21 ⁄ 21 )1(12 Phase boundary controlled reaction (contracting volume) R3 1/ 2 31 ⁄ 31 1 13 () Random nucleation followed by an Instantaneous growth of nuclei. (Avrami-Erofeev eqn. n =1) F1 1 -1 )1ln( Random nucleation and growth of nuclei through different nucleation and nucleus growth models. (Avrami-Erofeev eqn ≠1.) An 1ln1 ⁄ ln11 ln1 ⁄ n/1 )1ln( Two-dimensional diffusion D2 1ln1 1 1ln1 ()ln()11 Three-dimensional diffusion (Jander equation) D3 31 ⁄ 211 ⁄ 12 ⁄1 ⁄ 11 ⁄ 2 3/1 11 Three-dimensional diffusion (Ginstling-Brounshtein equation) D4 3 21 1 13 () / 1 ⁄ 21 ⁄1 32 )1(321 Random Scission of polymer chain L=2 L2 2 ⁄ 1 ⁄2 )1ln(2 2/1
17 TABLE 2. Values of αm as obtained from eq. (15) for the different kinetic models included in Table 1 as a function of x (E/RT). x Model 10 20 50 100 R2 0.7058 0.7266 0.7403 0.7451 R3 0.6521 0.6762 0.6922 0.6979 F1 0.5699 0.5985 0.618 0.6249 A2 0.6022 0.6157 0.6251 0.6285 A3 0.6124 0.6212 0.6275 0.6297 D2 0.7528 0.7947 0.8184 0.826 D3 0.5878 0.6452 0.6801 0.6919 D4 0.6801 0.7287 0.757 0.7664 L2 0.5323 0.5511 0.5644 0.5693
18 Figure Captions Figure 1. Overlay of the simulated curve (dots; E = 200 kJ mol-1; A =6 1017min-1, R2 kinetic model, and linear heating rate conditions, 10 K min-1) and the Gaussian (a), Lorentzian (b), Weibull (c) and FraserSuzuki (d) curves (solid lines) used for fitting the simulated curve. Residuals are plotted underneath the plots. Correlation coefficients have been included into the figure. Figure 2. Overlay of the simulated curves (dots) assuming different kinetic parameters (a) R3, E = 150 kJ mol-1, A = 1.2·1015 min-1, and β = 5 K min-1; (b) A2, E = 150 kJ mol-1, A = 6·1013 min-1, and β=2.5 K min-1; (c) D3, E = 120 kJ mol-1, A = 6·107 min-1, and β = 1 K min-1; (d) D2 with a log normal particle size distribution (0.75), E = 100 kJ mol-1, A = 105 min-1, and β = 10 K min-1, and the Fraser-Suzuki function used for fitting the simulated curves (solid lines). Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. Figure 3. Simulated kinetic curve (dots, A2, E = 150 kJ mol-1, A =6 1013 min-1, and β = 2.5 K min-1) fitted with three Gaussian function (solid lines). Figure 4. Overlay of the simulated kinetic curves (dots, F1, E = 180 kJ mol-1 and A =6·1011 min-1) assuming different linear heating rates: (a) 1 K min-1, (b) 2.5 K min-1, (c) 5 K min-1 and (d) 10 K min-1, and the Lorentzian function (solid lines) used for fitting the simulated curves. Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. Figure 5. Overlay of the simulated kinetic curves (dots, F1, E = 180 kJ mol-1 and A =6·1011 min-1) at different linear heating rates: (a) 1 K min-1, (b) 2.5 K min-1, (c) 5 K min-1 and (d) 10 K min-1, and the Fraser-Suzuki function (solid lines) used for fitting the simulated curves. Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure.
19 Figure 6. Activation energy values as a function of α, as obtained from the isoconversional analysis of the Lorentzian and Fraser-Suzuki curves resulting of the fitting of kinetic curves shown in Figures 4 and 5. Figure 7. (a) Combined analysis plot for Lorentzian curves resulting of the fitting of kinetic curves in Figure 4. (b) Comparison of the f(α) functions (lines) normalized at α= 0.5 [f(α)/f(0.5)] corresponding to some of the ideal kinetic models included in Table 1 with the f(α) function with the resulting values of n and m coefficients, i.e., n= 1.684 and m= 0.056 (dots). Figure 8. (a) Combined analysis plot for Fraser-Suzuki curves resulting of the fitting of kinetic curves in Figure 5. (b) Comparison of the f(α) functions (lines) normalized at α= 0.5 [f(α)/f(0.5)] corresponding to some of the ideal kinetic models included in Table 1 with the f(α) function with the resulting values of n and m coefficients, i.e., n= 0.996 and m= 0.00117 (dots). Figure 9. Overal of kinetic curves (dots) simulated by assuming two independent processes (f1(α) = R3; E1 = 125 kJ mol-1; A1 = 108 min-1; Contribution: 30% and f2(α) = A2; E2 = 195 kJ mol-1; A1 = 2·1012 min-1; Contribution: 70% ) at different linear heating rates: (a) 1 K min-1, (b) 2.5 K min-1, (c) 5 K min-1 y (d) 10 K min-1, and the two Fraser-Suzuki functions whose overlapping fit the experimental curves (solid lines). Residuals and correlation coefficients are shown. Figure 10. Combined analysis plot for Fraser-Suzuki curves resulting of the fitting of kinetic curves in Figure 9: (a) first and (b) second process. Figure 11. Comparison of the f(α) functions (lines) normalized at α= 0.5 [f(α)/f(0.5)] corresponding to some of the ideal kinetic models included in Table 1 with the f(α) function resulting of the combined analysis (dots) for the first (a) and second (b) process in Figure 9. Figure 12. Overlay of experimental thermogravimetric curves in its differential form for the thermal dehydrochlorination of PVC (dots) at different linear heating rates: (a) 2.5 K min-1, (b) 5 K min-1, (c) 10
20 K min-1 and (d) 15 K min-1, and the corresponding Fraser-Suzuki function used for fitting the experimental curves (solid lines). Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. Figure 13. Combined analysis plot for Fraser-Suzuki curves resulting of the fitting of experimental curves in Figure 12: (a) first and (b) second process. Figure 14. Comparison of the f(α) functions (lines) normalized at α= 0.5 [f(α)/f(0.5)] corresponding to some of the ideal kinetic models included in Table 1 with the f(α) function resulting of the combined analysis (dots) for the first (a) and second (b) process in Figure 12. Figure 15. Experimental curve in its integral and differential forms corresponding to the dehydrochlorination of PVC obtained at 10 K min-1 (dots). Reconstructed curves using the kinetic parameters resulting of the combined kinetic analysis are plotted as solid lines.
21 Figure 1. Overlay of the simulated curve (dots; E = 200 kJ mol-1; A =6 1017min-1, R2 kinetic model, and linear heating rate conditions, 10 K min-1) and the Gaussian (a), Lorentzian (b), Weibull (c) and FraserSuzuki (d) curves (solid lines) used for fitting the simulated curve. Residuals are plotted underneath the plots. Correlation coefficients have been included into the figure. 480 500 520 540 560 580 600 620 -0.003 0.000 0.003 0.001 0.002 0.003 0.004 0.005 0.006 residuals T (K) r2= 0.898 d/dt (b) 480 500 520 540 560 580 600 620 -0.003 0.000 0.003 0.001 0.002 0.003 0.004 0.005 0.006 residuals T (K) r2= 0.905 d/dt (a) 480 500 520 540 560 580 600 620 -0.003 0.000 0.003 0.001 0.002 0.003 0.004 0.005 0.006 residuals T (K) r2= 0.976 d/dt (c) 480 500 520 540 560 580 600 620 -0.003 0.000 0.003 0.001 0.002 0.003 0.004 0.005 0.006 residuals T (K) r2= 0.998 d/dt (d)
22 Figure 2. Overlay of the simulated curves (dots) assuming different kinetic parameters (a) R3, E = 150 kJ mol-1, A = 1.2·1015 min-1, and β = 5 K min-1; (b) A2, E = 150 kJ mol-1, A = 6·1013 min-1, and β=2.5 K min-1; (c) D3, E = 120 kJ mol-1, A = 6·107 min-1, and β = 1 K min-1; (d) D2 with a log normal particle size distribution (0.75), E = 100 kJ mol-1, A = 105 min-1, and β = 10 K min-1, and the Fraser-Suzuki function used for fitting the simulated curves (solid lines). Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. 420 440 460 480 500 520 540 -0.001 0.000 0.001 0.001 0.002 0.003 residuals T (K) r2= 0.999 d/dt (a) 480 500 520 540 560 -0.0005 0.0000 0.0005 0.0005 0.0010 0.0015 0.0020 residuals T (K) r2= 0.999 d/dt (b) 360 400 440 480 520 560 600 640 680 720 -0.00008 0.00000 0.00008 0.00004 0.00008 0.00012 0.00016 residuals T (K) r2= 0.998 d/dt (c) 300 400 500 600 700 800 900 1000 1100 1200 -0.005 0.000 0.005 0.005 0.010 0.015 0.020 residuals T (K) r2= 0.999 d/dt (d)
23 460 480 500 520 540 560 580 -0.0005 0.0000 0.0005 0.0005 0.0010 0.0015 0.0020 residuals T (K) r2= 0.999 d/dt Figure 3. Simulated kinetic curve (dots, A2, E = 150 kJ mol-1, A =6 1013 min-1, and β = 2.5 K min-1) fitted with three Gaussian function (solid lines).
24 Figure 4. Overlay of the simulated kinetic curves (dots, F1, E = 180 kJ mol-1 and A =6·1011 min-1) assuming different linear heating rates: (a) 1 K min-1, (b) 2.5 K min-1, (c) 5 K min-1 and (d) 10 K min-1, and the Lorentzian function (solid lines) used for fitting the simulated curves. Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. 550 600 650 700 750 800 -0.00005 0.00000 0.00005 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 residuals T (K) r2= 0.951 d/dt (a) 550 600 650 700 750 800 850 -0.0001 0.0000 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 residuals T (K) r2= 0.950 d/dt (b) 600 650 700 750 800 850 -0.0002 0.0000 0.0002 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 residuals T (K) r2= 0.951 d/dt (c) 600 650 700 750 800 850 -0.0005 0.0000 0.0005 0.0005 0.0010 0.0015 0.0020 0.0025 residuals T (K) r2= 0.950 d/dt (d)
25 Figure 5. Overlay of the simulated kinetic curves (dots, F1, E = 180 kJ mol-1 and A =6·1011 min-1) at different linear heating rates: (a) 1 K min-1, (b) 2.5 K min-1, (c) 5 K min-1 and (d) 10 K min-1, and the Fraser-Suzuki function (solid lines) used for fitting the simulated curves. Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. 550 600 650 700 750 800 -0.00005 0.00000 0.00005 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 residuals T (K) r2= 0.999 d/dt (a) 600 650 700 750 800 -0.0001 0.0000 0.0001 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 residuals T (K) r2= 0.999 d/dt (b) 600 650 700 750 800 850 -0.0002 0.0000 0.0002 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 residuals T (K) r2= 0.999 d/dt (c) 600 650 700 750 800 850 -0.0005 0.0000 0.0005 0.0005 0.0010 0.0015 0.0020 0.0025 residuals T (K) r2= 0.999 d/dt (d)
32 Figure 12. Overlay of experimental thermogravimetric curves in its differential form for the thermal dehydrochlorination of PVC (dots) at different linear heating rates: (a) 2.5 K min-1, (b) 5 K min-1, (c) 10 K min-1 and (d) 15 K min-1, and the corresponding Fraser-Suzuki function used for fitting the experimental curves (solid lines). Residuals are plotted underneath the figures. Correlation coefficients have been included into the figure. 450 500 550 600 650 -0.02 0.00 0.02 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 residuals T (K) r2= 0.998 d/dt (a) 500 550 600 650 -0.04 0.00 0.04 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 residuals T (K) r2= 0.999 d/dt (b) 500 550 600 650 -0.1 0.0 0.1 0.05 0.10 0.15 0.20 0.25 0.30 residuals T (K) r2= 0.999 d/dt (c) 500 550 600 650 -0.15 0.00 0.15 0.1 0.2 0.3 0.4 residuals T (K) r2= 0.999 d/dt (d)
33 Figure 13. Combined analysis plot for Fraser-Suzuki curves resulting of the fitting of experimental curves in Figure 12: (a) first and (b) second process. 0.00176 0.00184 0.00192 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 (a) 1 K min-1 2.5 K min-1 5 K min-1 10 K min-1 Fit ln((ddt)/(1-)0.966·0.556) T-1 (K-1) r2 = 0.999 0.0016 0.0017 0.0018 0.0019 0.0020 -10 -8 -6 -4 -2 0 2 ln((ddt)/(1-)1.611·-1.099) T-1 (K-1) (b) 1 K min-1 2.5 K min-1 5 K min-1 10 K min-1 Fit r2 = 0.998
34 Figure 14. Comparison of the f(α) functions (lines) normalized at α= 0.5 [f(α)/f(0.5)] corresponding to some of the ideal kinetic models included in Table 1 with the f(α) function resulting of the combined analysis (dots) for the first (a) and second (b) process in Figure 12. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 2.0 A2 L2 R3 F1 D3 f()/f(0.5) (a) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.4 0.8 1.2 1.6 2.0 A2 L2 R3 F1 D3 f()/f(0.5) (b)
35 Figure 15. Experimental curve in its integral and differential forms corresponding to the dehydrochlorination of PVC obtained at 10 K min-1 (dots). Reconstructed curves using the kinetic parameters resulting of the combined kinetic analysis are plotted as solid lines. 500 550 600 650 700 0.0 0.2 0.4 0.6 0.8 1.0 500 600 700 T (K) T (K)
36 TABLE OF CONTENTS IMAGE 500 550 600 650 -0.1 0.0 0.1 0.05 0.10 0.15 0.20 0.25 0.30 residuals T (K) r2= 0.999 d/dt 1 2