Isoconversional Kinetic Analysis of Complex Solid-State Processes

Nov 26, 2012 - ... a systematic kinetic analysis for discriminating the reactions rather than merely assuming that the activation energy varies with t...
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Isoconversional Kinetic Analysis of Complex Solid-State Processes: Parallel and Successive Reactions Junmeng Cai,*,†,‡ Weixuan Wu,† and Ronghou Liu*,† †

Biomass Energy Engineering Research Centre, School of Agriculture and Biology, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, P. R. China ‡ State Key Laboratory of Heavy Oil Processing, China University of Petroleum, 18 Fuxue Road, Beijing 102249, P. R. China ABSTRACT: A critical study of the use of the iterative linear integral isoconversional method for a parallel reactions process and a successive reactions process has been performed using theoretical simulated nonisothermal data. The activation energies obtained for these complex processes show a strong dependence on the range of heating rates, which is against the assumptions of isoconversional methods. Therefore, a systematic kinetic analysis for discriminating the reactions rather than merely assuming that the activation energy varies with the conversion would be required for understanding the reaction mechanism of these processes. Model fitting methods lead to the uncertainty in estimating the activation energy which primarily originates from the uncertainty of choosing the reaction models.6 This type of uncertainty can be eliminated when isoconversional methods of kinetic analysis are used.7 To use the isoconversional methods, at least three experiments have to be carried out at different heating rates. Expressing eq 4 in logarithmic form and then taking its differential with respect to T−1 yields the equation, below, at fixed α:

1. INTRODUCTION The reaction rate of a thermally stimulated solid state reaction can be kinetically described by the following basic kinetic equation:1 dα = k(T ) f (α) dt

(1)

where α is the extent of reaction ranging from 0 to 1 as the process progresses from initiation to completion, T is the temperature, k(T) is the temperature-dependent rate constant, and f(α) is the differential conversion function. The temperature dependence of the reaction rate is typically parametrized through the Arrhenius law:2

⎛ E ⎞ ⎟ k(T ) = A exp⎜ − ⎝ RT ⎠

⎡ ∂ ln f (α) ⎤ ⎡ ∂ ln(β(dα /dT )) ⎤ ⎡ ∂ ln A ⎤ E +⎢ − α ⎥ ⎢ ⎥ = ⎢⎣ ⎥ 1 1 1 − − − ⎣ ⎦α ⎦α R ∂T ∂ T ⎦α ⎣ ∂ T (5)

The isoconversional methods assume that the reaction model f(α) and the kinetic parameters (A and E) at a certain extent of conversion are independent of the heating rate, and that the reaction rate at constant extent of conversion is only a function of the temperature.8 Therefore, the first and second terms on the right-hand side of eq 5 effectively disappear. From the above assumptions and eq 5, one can derive the relationship, below, from which it is easy to see why isoconversional methods are often called “model free methods”:

(2)

where E is the activation energy, A is the frequency factor, and R is the universal gas constant. f(α) is the term that describes the dependence of the reaction rate with the mechanism of the process. Different function forms have been proposed in the literature for describing the kinetic mechanism of the solid-state reactions.3 These mechanisms were proposed considering different geometrical assumptions for the particles shape and driving forces.4 Some of the most common function forms for f(α) can be found in the literature.5 The kinetic behaviors of thermally stimulated solid state reactions are normally studied under the conditions of isothermal, linear nonisothermal, or nonlinear nonisothermal heating.5 The most common temperature program is the one in which the temperature changes linearly with the time: β=

dT = constant dt

⎡ ∂ ln(β(dα /dT )) ⎤ Eα ⎢ ⎥ =− −1 ⎣ ⎦ R ∂T α

which is the basis of the differential isoconversional method suggested by Friedman.9 The conventional linear integral isoconversional methods, such as the FWO and KAS methods, can be employed to estimate the activation energy from a series of kinetic conversion data conducted at several heating rates.10 They are based on the

(3)

where β is the heating rate. Combining eqs 1, 2 and 3 yields the kinetic equation under the linear heating program: dα A −E /(RT ) e f (α ) = dT β

Received: Revised: Accepted: Published:

(4) © 2012 American Chemical Society

(6)

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The first term on the right-hand side of eq 9 is a constant at any given α. At each given α, the value of Eα is determined from the slope of a plot of the term on the left-hand side of eq 9 against −1/(RTα,i). However, the term on the left-hand side of eq 9 is a function of Eα. Cai and Chen15 developed an iterative procedure to solve the problem. Cai et al.19 gave the method for the estimation of the standard deviations for the activation energy calculated by the iterative linear integral isoconversional method.

integral form of eq 4: g (α ) =

∫0

α

A dα = f (α ) β

∫0

Tα

e−Eα /(RT ) dT

(7)

Many other conventional integral isoconversional methods are also based on the above equation.11 However, in the derivation of the above integral methods, it is assumed that the value of Eα remains constant over the whole interval of the dT integration in eq 7 which smoothes the dependence of Eα on α.12 Therefore, none of these methods is exact when Eα varies with α.13 This source of inaccuracy does not apply to the advanced nonlinear integral isoconversional method proposed by Vyazovkin,14 and the iterative linear integral isoconversional method proposed by Cai and Chen,15 which are derived from the following equation: α

∫α−Δα A·df α(α) = 1β ∫T

Tα

e−Eα−Δα/2 /(RT ) dT

α −−Δα

3. RESULTS AND DISCUSSION 3.1. Parallel Reactions. The process that involves two parallel reactions has been considered. The kinetics of the process can be formulated as follows:20

(8)

In the above equation, the constancy of Eα is assumed only for small intervals of conversion, Δα. The use of integration by segments yields Eα values that are practically identical with those obtained when using the Friedman differential method.16 The analysis by means of isoconversional methods of solidstate processes taking place through competitive reactions is of interest because it would lead to an apparent variation of the activation energy with the conversion. Criado et al.17 performed a critical study of the use of isoconversional methods for the kinetic analysis of nonisothermal data corresponding to competitive reactions and the results showed that the apparent activation energy values obtained for competitive reactions from isoconversional methods depend on the heating rate, which is against the assumptions of isoconversional methods. The systematic analysis of other complex processes (including parallel reactions process and successive reactions process) by means of isoconversional methods is still missing. Thus, the aim of this short communication is to carry out a critical study of these complex processes by isoconversional methods. The iterative linear integral isoconversional method proved to be capable of providing reliable activation energy values when it is applied to analyze the varying activation energy processes.18 So the isoconversional method would be used for all isoconversional kinetic analyses of this work.

(10)

dα 2 A = 2 e−E2 /(RT ) f2 (α2) dT β

(11)

l1 + l 2 = 1

(12)

α = l1α1 + l 2α2

(13)

In the above equations, the subscripts 1 and 2 refer to the first and second processes, respectively, l1 and l2 are the contribution fraction of the first and second processes to the overall reaction, respectively. Equations 10 and 11 can be solved by the numerical integration method of Runge−Kutta in order to simulate a series of α − T and dα/dT − T curves at different heating rates. The thermal dehydrochlorination of PVC has been extensively studied in the literature and the recent results showed that PVC dehydrochlorination is complex and involves two independent parallel reactions.21 The kinetic parameters for the thermal dehydrochlorination of PVC obtained from the literature20 were considered: E1 = 111 kJ mol−1, A1 = 2.2 × 1010 min−1, f1(α1) = (1 − α1)[−ln(1 − α1)]1/2, E2 = 202 kJ mol−1, A2 = 7.2 × 1016 min−1, f 2(α2) = 1.5(1 − α2)2/3/[1 − (1 − α2)1/3] and l1 = l2 = 0.5. The α − T and dα/dT − T curves were calculated for 20 heating rates with values in the range 2−40 K min−1 and plotted. In such a way, it was shown that for 2 K min−1 ≤ β ≤ 10 K min−1 the dα/ dT − T curves exhibit two distinct maxima corresponding to the two reactions (Figure 1) and for 12 K min−1 ≤ β ≤ 40 K min−1 the dα/dT − T curves exhibit a single maximum (Figure 2).

2. ITERATIVE LINEAR INTEGRAL ISOCONVERSIONAL METHOD The iterative linear integral isoconversional method leads to correct values of the activation energy in much less time than the Vyazovkin advanced nonlinear integral isoconversional method.13 It is based on the following equation: ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ βi ⎬ ln⎨ 2 xα , i ⎪ T 2⎡h(x ) − xα ,i e h(xα−Δα ,i) ⎤ ⎪ ⎪ α , i ⎢⎣ α , i (xα −Δα , i 2 e xα −Δα , i) ⎥ ⎦⎪ ⎭ ⎩ ⎡ ⎤ Eα −Δα /2 R ⎢ ⎥ = ln⎢ − α dα ⎥ RTα , i ⎣ Eα −Δα /2 ∫α −Δα A·f (α) ⎦

dα1 A = 1 e−E1/(RT ) f1 (α1) dT β

(9)

which is derived from eq 8. In eq 9, the subscript i represents the ordinal number of a nonisothermal experiment conducted at the heating rate βi, xα = Eα−Δα/2/(RTα), xα−Δα = Eα−Δα/2/ −2 −x (RTα−Δα), and h(x) = x2 ex ∫ ∞ x x e dx.

Figure 1. The α − T and dα/dT − T curves obtained for two parallel reactions (β = 2 K min−1). 16158

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assumption that k1 and k2 are expressed by Arrhenius equation are given below: dα1 A = 1 e−E1/(RT )(1 − α1) dT β

(14)

dα 2 A = 2 e−E2 /(RT )(α1 − α2) dT β

(15)

dα 2 ⎞ dα 1 ⎛ dα ⎟ = ⎜ 1 + ⎝ dT 2 dT dT ⎠

(16)

where the subscripts 1 and 2 refer to the reaction I and reaction II, respectively. Detailed information about the above process can be found in the literature.22 A MATLAB program using the function “ode45” was used to solve the above ordinary differential equations. The following kinetic parameters: A1 = 900 s−1, E1 = 58.5 kJ mol−1, A2 = 5 × 108 s−1, E2 = 125.4 kJ mol−1 which are inputs to the MATLAB program were obtained from the literature.22 In their paper, Budrugeac et al.22 performed a theoretical analysis of the influence of the heating rate on the α − T and dα/dT − T curves. Figures 4 and 5 show the α − T and dα/dT − T curves for the above process involving two successive reactions at different heating rates.

Figure 2. The α − T and dα/dT − T curves obtained for two parallel reactions (β = 20 K min−1).

The plots of Eα determined as a function of α from different ranges of heating rates are compared in Figure 3. The results

Figure 3. Values of Eα determined as a function of α from the iterative linear integral isoconversional method at different ranges of heating rates: (a) 2, 4, 8, and 16 K min−1 and (b) 10, 20, 30, and 40 K min−1 for the process involving two parallel reactions.

demonstrate that the activation energy for two parallel reactions from the iterative linear integral isoconversional method depends on the heating rates, which is consistent with the conclusion of two competitive reactions of Criado et al.17 Therefore, the Eα values obtained from isoconversional methods have no physical meaning if such a parallel reaction process is concerned. 3.2. Successive Reactions. Let us consider the following theoretical process involving two successive decompositions, which was presented in detail in the literature:22 k1

A(s) → B(s) + v1G1(g) k1

B(s) → C(s) + v2G2(g)

Figure 4. The α − T and dα/dT − T curves obtained for two successive reactions (β = 2, 4, 8, and 16 K min−1).

The dependences Eα vs α for simulated α − T curves corresponding to two successive reactions at different ranges of heating rates are shown in Figure 6. The results indicate that the apparent activation energy obtained for successive reactions from the iterative linear integral isoconversional method is dependent on the range of heating rates. That is to say, the conclusion of two competitive reactions of Criado et al.17 is also valid for two successive reactions. In such a case, the Friedman’s activation energies cannot be taken as reference values for checking other isoconversional methods. Therefore, a rigorous kinetic analysis discriminating the successive reactions rather than merely assuming that the activation energy varies with the conversion degree would be required for understanding the reaction mechanism.

(I) (II)

where A, B, and C are solid compounds, and G1 and G2 are gaseous products, k1 and k2 are the reaction rate constant for the reaction I and reaction II, respectively. It is further assumed that both reactions have the same kinetic model f(α) = 1 − α, and v1M1 = v2M2 (where Mi is the molecular mass of the gaseous compound Gi). For nonisothermal linear heating with the heating rate β, the corresponding kinetic equations of the above process with the 16159

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2012-1-02), State Key Lab of Chemical Resource Engineering (Grant No. CRE-2011-C-304), School of Agriculture and Biology, Shanghai Jiao Tong University, China (Grant No. NRC201101) and National Natural Science Foundation of China through contract (Grant No. 51176121).

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(1) Vyazovkin, S.; Wight, C. A. Isothermal and non-isothermal kinetics of thermally stimulated reactions of solids. Int Rev Phys Chem 1998, 17 (3), 407−433. (2) Vyazovkin, S.; Wight, C. A. Kinetics in solids. Annu. Rev. Phys. Chem. 1997, 48, 125−149. (3) Cai, J. M.; Liu, R. H. Kinetic analysis of solid-state reactions: A general empirical kinetic model. Ind. Eng. Chem. Res. 2009, 48 (6), 3249−3253. (4) Perez-Maqueda, L. A.; Sanchez-Jimenez, P. E.; Criado, J. M. Kinetic analysis of solid-state reactions: Precision of the activation energy calculated by integral methods. Int. J. Chem. Kinet. 2005, 37 (11), 658−666. (5) Perez-Maqueda, L. A.; Criado, J. M.; Sanchez-Jimenez, P. E. Combined kinetic analysis of solid-state reactions: A powerful tool for the simultaneous determination of kinetic parameters and the kinetic model without previous assumptions on the reaction mechanism. J. Phys. Chem. A 2006, 110 (45), 12456−12462. (6) Vyazovkin, S.; Wight, C. A. Estimating realistic confidence intervals for the activation energy determined from thermoanalytical measurements. Anal. Chem. 2000, 72 (14), 3171−3175. (7) Vyazovkin, S. Model-free kineticsStaying free of multiplying entities without necessity. J. Therm. Anal. Calorim. 2006, 83 (1), 45− 51. (8) Vyazovkin, S.; Burnham, A. K.; Criado, J. M.; Perez-Maqueda, L. A.; Popescu, C.; Sbirrazzuoli, N. ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data. Thermochim. Acta 2011, 520 (1−2), 1−19. (9) Friedman, H. L. Kinetics of thermal degradation of char-forming plastics from thermogravimetry. Application to phenolic plastic. J. Polym. Sci. Polym. Symp. 1964, 6, 183−195. (10) Farjas, J.; Roura, P. Isoconversional analysis of solid state transformations A critical review. Part I. Single step transformations with constant activation energy. J. Therm. Anal. Calorim. 2011, 105 (3), 757−766. (11) Sbirrazzuoli, N.; Vincent, L.; Mija, A.; Guigo, N. Integral, differential and advanced isoconversional methods. Complex mechanisms and isothermal predicted conversion-time curves. Chemom. Intell. Lab. 2009, 96 (2), 219−226. (12) Burnham, A. K.; Dinh, L. N. A comparison of isoconversional and model-fitting approaches to kinetic parameter estimation and application predictions. J. Therm. Anal. Calorim. 2007, 89 (2), 479− 490. (13) Budrugeac, P. An iterative model-free method to determine the activation energy of non-isothermal heterogeneous processes. Thermochim. Acta 2010, 511 (1−2), 8−16. (14) Vyazovkin, S. Modification of the integral isoconversional method to account for variation in the activation energy. J. Comput. Chem. 2001, 22 (2), 178−183. (15) Cai, J. M.; Chen, S. Y. A new iterative linear integral isoconversional method for the determination of the activation energy varying with the conversion degree. J. Comput. Chem. 2009, 30 (13), 1986−1991. (16) Segal, E.; Budrugeac, P. Is there any convergence between the differential and integral procedures used in chemical kinetics, particularly in nonisothermal kinetics? Int. J. Chem. Kinet. 2006, 38 (5), 339−344. (17) Criado, J. M.; Sanchez-Jimenez, P. E.; Perez-Maqueda, L. A. Critical study of the isoconversional methods of kinetic analysis. J. Therm. Anal. Calorim. 2008, 92 (1), 199−203.

Figure 5. The α − T and dα/dT − T curves obtained for two successive reactions (β = 10, 20, 30, and 40 K min−1).

Figure 6. Values of Eα determined as a function of α from the iterative linear integral isoconversional method at different ranges of heating rates: (a) 2, 4, 8, and 16 K min−1 and (b) 10, 20, 30, and 40 K min−1 for the process involving two successive reactions.

4. CONCLUSIONS The iterative linear integral isoconversional method for the evaluation of the activation energy has been applied for (1) simulated nonisothermal data for two parallel reactions; (2) simulated nonisothermal data for two successive reactions. The results showed that for those complex processes the activation energies obtained by means of the isoconversional method are dependent on the range of heating rates, which is against the assumptions of isoconversional methods.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 21 34206624. E-mail: [email protected]; liurhou@ sjtu.edu.cn. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to acknowledge financial supports from State Key Laboratory of Heavy Oil Processing (Grant No. 16160

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(18) Cai, J. M.; Chen, Y. Iterative linear integral isoconversional method: Theory and application. Bioresour. Technol. 2012, 103 (1), 309−312. (19) Cai, J. M.; Han, D.; Chen, Y.; Chen, S. Y. Evaluation of realistic 95% confidence intervals for the activation energy calculated by the iterative linear integral isoconversional method. Chem. Eng. Sci. 2011, 66 (12), 2879−2882. (20) Perejon, A.; Sanchez-Jimenez, P. E.; Criado, J. M.; PerezMaqueda, L. A. Kinetic analysis of complex solid-state reactions. A new deconvolution procedure. J. Phys. Chem. B 2011, 115 (8), 1780−1791. (21) Sanchez-Jimenez, P. E.; Perez-Maqueda, L. A.; Crespo-Amoros, J. E.; Lopez, J.; Perejon, A.; Criado, J. M. Quantitative characterization of multicomponent polymers by sample-controlled thermal analysis. Anal. Chem. 2010, 82, 8875−8880. (22) Budrugeac, P.; Homentcovschi, D.; Segal, E. Critical considerations on the isoconversional methodsIII. On the evaluation of the activation energy from non-isothermal data. J. Therm. Anal. Calorim. 2001, 66 (2), 557−565.

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