A simple and linear isoconversional method to

A simple and linear isoconversional method to determine the pre-exponential factors and the mathematical reaction mechanism functions Djalal Trache, Amir Abdelaziz & Bachir Siouani

Journal of Thermal Analysis and Calorimetry An International Forum for Thermal Studies ISSN 1388-6150 J Therm Anal Calorim DOI 10.1007/s10973-016-5962-0

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Author's personal copy J Therm Anal Calorim DOI 10.1007/s10973-016-5962-0

A simple and linear isoconversional method to determine the pre-exponential factors and the mathematical reaction mechanism functions Djalal Trache1 • Amir Abdelaziz1 • Bachir Siouani1

Received: 7 June 2016 / Accepted: 9 November 2016 Akade´miai Kiado´, Budapest, Hungary 2016

Abstract A simple and linear integral method which uses multiple heating schedules to evaluate the kinetic parameters has been proposed by Trache–Abdelaziz–Siouani (TAS). This approach is based on the combination of the iterative modified Coats–Redfern equation with the kinetic compensation parameters (ln A = aE ? b). The suggested method was applied to experimental non-isothermal data obtained from the literature for decomposition of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin studied by differential scanning calorimeter at two different pressures (0.1 and 2 MPa). This method leads to consistent pre-exponential factor and kinetic model with those obtained from the accurate approximation of Tang et al. using the activation energy derived from either the integral nonlinear Vyazovkin procedure or the Friedman’s differential method. These kinetic parameters are reliable with those obtained by two integral linear (iterative Kissinger–Akahira–Sunose and iterative Flynn–Wall–Ozawa) methods as well. The superiority of TAS method is due to the possibility of obtaining all the kinetic parameters in an objective manner with a reasonable computation time. Keywords Isoconversional method Pre-exponential factor Reaction model Kinetic parameters Modified Coats–Redfern Compensation effect

& Djalal Trache [email protected] 1

UER Chimie applique´e, Ecole Militaire Polytechnique EMP, BP 17, Bordj El-Bahri, Algiers, Algeria

Introduction In the last few decades, determination of the mathematical reaction mechanism function and computation of the kinetic parameters from non-isothermal data obtained by thermal analysis techniques such as differential scanning calorimetry (DSC) and thermogravimetric analysis (TG) have been the topic of numerous works [1–10]. Nowadays, isoconversional methods are among the most reliable kinetic methods for the treatment of the thermoanalytical data [1, 11–16]. These data for various processes could be collected correctly following the recommendation of the ICTAC kinetics committee [17]. The birth of the isoconversional methods dates back to the 1925 work by Kujirai and Akahira [18], where they employed TG data to investigate the decomposition kinetics of some insulating materials under isothermal conditions. The merit of such kinetic isoconversional procedures is that they provide a way to get kinetic parameters with ignoring completely the reaction mechanism. Consequently, they are often called ‘‘model-free.’’ These methods are based on the isoconversional principle, where the reaction rate at fixed extent of conversion depends only on the temperature [11]. They can broadly be divided in two categories: differential (e.g., Friedman) and integral (e.g., KAS, FWO, Vyazovkin and modified Coats–Redfern). Due to the use of the instantaneous rate value, the differential isoconversional methods are sensitive to experimental noise and cause the instability of the numerical values. This phenomenon, however, can be effectively avoided by using integral methods [1]. Commonly, kinetic analysis is considered incomplete without the determination of pre-exponential factor and reaction mechanism. The estimation of the pre-exponential factor using an isoconversional analysis necessitates the knowledge of the reaction mechanism. To avoid this

123

Author's personal copy D. Trache et al.

deficiency, two main methods based on compensation parameters or z(a) master plots have previously been suggested for evaluation of these kinetic parameters [19, 20]. Recently, Sbirrazzuoli employed several model-fitting methods for the computation of compensation parameters that have been used to compute the pre-exponential factor dependency based on the model-free way [21]. This latter parameter was inserted in model-free methods to compute the values of the mathematical function that describes the reaction mechanism. The author stated that this approach leads to an accurate estimation of the pre-exponential factor and mathematical functions g(a) and f(a). In 1964, Coats and Redfern presented an integral kinetic method [2, 22–24]. Despite its drawbacks [25], it remains one of the methods more extensively used as indicated by more than 1800 Scopus citations received in the last 5 years [26]. This is a model-fitting method that is commonly used for kinetic analysis of data obtained with a single heating rate. However, the International Confederation for Thermal Analysis and Calorimetry (ICTAC) kinetics committee recommended that using multiple heating rate programs will give more reliable kinetic parameters instead of single heating rate programs [1]. To avoid the shortcomings of the Coast–Redfern model-fitting method (CR), Burnham and Braun proposed a modification of this equation to generate an isoconversional method, commonly called the modified Coats–Redfern method (MCR), which requires data from multiple heating rates to determine the activation energy [27]. The present work aims to propose a completely different methodology to determine the mathematical function that describes the reaction mechanism and to compute the preexponential factor. The main approach is based on the combination of the iterative modified Coats–Redfern equation with the kinetic compensation parameters (ln A = aE ? b). To the best of our knowledge, no other report addresses the implementation of the compensation effect on the modified Coats–Redfern method to determine these kinetic parameters. The analysis of non-isothermal data taken from the literature for the non-isothermal decomposition of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin studied by differential scanning calorimeter at two different pressures (0.1 and 2 MPa) using this method is also performed. The so obtained kinetic parameters were compared with those resulting from other approaches.

Theoretical approaches The rate of many thermally stimulated processes can be usually written in terms of T and a as follows [1]:

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da ¼ kðTÞf ðaÞ dt

ð1Þ

where t is the time, T is the temperature, a is the extent of conversion (0 \ a \ 1), k(T) is the rate constant, and f(a) is the mathematical function that represents the reaction mechanism. The value of a is experimentally derived from the thermal analysis technique used as a fraction change of any physical property associated with the reaction progress [20]. When the process progress is monitored as a change in mass by TG, a is computed as a ratio of the current mass change, Dm, to the total mass change, Dmtot, occurred throughout the process: a¼

m0 mt Dm ¼ m0 m1 Dmtot

ð2Þ

where m0, mt and m? are initial sample mass, sample mass at time t and sample mass at the end of reaction, respectively. In its turn, when the progress is determined as a change in heat by DSC, a is calculated as a ratio of the current heat change, DH, to the total heat released or adsorbed DHtot in the process: Rt a¼

t0 Rtf

ðdH=dtÞdt ¼ ðdH=dtÞdt

DH DHtot

ð3Þ

t0

where dH/dt is the heat flow measured by DSC. The temperature dependence of k(T) can be satisfactorily described by the Arrhenius law, which after substitution into Eq. (1) yields da ¼ A expðE=RT Þf ðaÞ dt

ð4Þ

where A is the pre-exponential factor (in s-1), E the activation energy, and R the universal gas constant. Integration of Eq. (4) leads to: gðaÞ ¼

Za 0

da ¼A f ðaÞ

Zt

expðE=RT Þdt

ð5Þ

0

where g(a) is the integral form of the reaction model (Table 1). Integral methods originate from the application of the isoconversional principle to Eq. (5). The integral in this equation does not have an analytical solution for an arbitrary temperature program. A large number of approximate equations have been suggested in the literature for performing the kinetic analysis of solid-state reactions. The most popular are those proposed by Doyle [28], Coats– Redfern [22] and Senum and Yang [29, 30]. Among a variety of isoconversional methods, it-KAS [31–33], it-FWO [31–33], Friedman [34] and Vyazovkin’s

Author's personal copy A simple and linear isoconversional method to determine the pre-exponential factors and the … Table 1 Expressions for f(a) and g(a) functions of the common mechanisms operating in solid-state reactions No.

Model

g(a)

f(a)

Rate-determining mechanism

1. Chemical process or mechanism non-involving equations F1/3

1 ð1 aÞ2=3

F3/4

1 ð1 aÞ

1=4

3

F3/2

1=2

4

F2

1 2

5

F3

ð 1 aÞ

1

1

ð1 aÞ 1 3

F4

ð1 aÞ 1

G1

1 ð1 aÞ

2

8

G2

1 ð1 aÞ3

9

G3

4

1 ð1 aÞ

4ð1 aÞ

Chemical reaction

2ð1 aÞ

3=2

Chemical reaction Chemical reaction

ð 1 aÞ

2

7

Chemical reaction

3=4

2

ð1 aÞ 1

6

ð3=2Þð1 aÞ1=3

ð1=2Þð1 aÞ

3

ð1=3Þð1 aÞ

4

Chemical reaction Chemical reaction

1=½2ð1 aÞ

Chemical reaction

1=½3ð1 aÞ2

Chemical reaction

3

1=½4ð1 aÞ

Chemical reaction

ð2=3Þa1=2

Nucleation (power law)

2. Acceleratory rate equations P3/2

a3=2

11

P1/2

a

1=2

12

P1/3

13

P1/4

a

14

P2

a2

ð1=2Þa1

Nucleation (parabolic law)

15

E1

ln a

a

Nucleation (exponential law)

10

1=2


a1=3

3a2=3


1=4

3=4


2a 4a

a=2 ln a2 3. Sigmoidal rate equations or random nucleation and subsequent growth

16

E2

lnð1 aÞ

Nucleation (exponential law)

1a

17

A1, F1

18

A2/3

½ lnð1 aÞ

ð2=3Þð1 aÞ½ lnð1 aÞ

Random nucleation (Avrami-Erofeev)

19

A3/2

½ lnð1 aÞ2=3

ð3=2ð1 aÞ½ lnð1 aÞ1=3


A3/4

4=3

20 21

A5/2

22

A2

23

A3

24

A4

25

A1/2

26

A1/3

3=2

½ lnð1 aÞ

2=5

½ lnð1 aÞ

1=2

½ lnð1 aÞ

1=3

½ lnð1 aÞ

1=4

½ lnð1 aÞ

2

½ lnð1 aÞ

3

½ lnð1 aÞ

4

Random nucleation/first order (Mampel) 1=2

1=3


3=5


ð3=4Þð1 aÞ½ lnð1 aÞ ð5=2Þð1 aÞ½ lnð1 aÞ 2ð1 aÞ½ lnð1 aÞ

1=2


3ð1 aÞ½ lnð1 aÞ

2=3


4ð1 aÞ½ lnð1 aÞ

3=4


1


2


1=2ð1 aÞ½ lnð1 aÞ 1=3ð1 aÞ½ lnð1 aÞ

3

27

A1/4

½ lnð1 aÞ

1=4ð1 aÞ½ lnð1 aÞ


28

B1

ln½a=ð1 aÞ

a=ð1 aÞ

Branching nuclei (Prout–Tompkins) Contracting disk

4. Deceleratory rate equations (phase boundary reaction) 29

R1, F0, P1

a

1

30

R2, F1/2

1 ð1 aÞ1=2

2ð1 aÞ1=2

R3, F2/3

1=3

2=3

31

1 ð1 aÞ

3ð1 aÞ

Contracting cylinder Contracting sphere

5. Deceleratory rate equations (equations based on the diffusion mechanism) 32

D1

a2

1=ð2aÞ

One-dimensional diffusion 1

½ lnð1 aÞ

D4

a þ ð1 aÞ lnð1 aÞ h i2 1 ð1 aÞ1=3 1 23 a ð1 aÞ2=3

36

D5

h

37

D6

h i1 ð3=2Þð1 aÞ4=3 ð1 aÞ1=3 1 h i1 ð3=2Þð1 þ aÞ2=3 ð1 þ aÞ1=3 1

33

D2

34

D3

35

i2 ð1 aÞ1=3 1 h i2 ð1 þ aÞ1=3 1

Three-dimensional diffusion h

ð3=2Þð1 aÞ2=3 1 ð1 aÞ1=3 h i1 ð3=2Þ ð1 aÞ1=3 1

i1

Three-dimensional diffusion (Jander) Three-dimensional diffusion (Ginstling–Brounshtein) Three-dimensional diffusion (Crank) Three-dimensional diffusion

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Author's personal copy D. Trache et al. Table 1 continued No.

Model

g(a)

38

D7

1þ

39

D8

h

f(a) 2 3

a ð1 þ aÞ2=3

ð1 þ aÞ1=3 1

i2

6. Other kinetic equations with unjustified mechanism h i1=2 40 G7 1 ð1 aÞ1=2 h i1=2 41 G8 1 ð1 aÞ1=3

Rate-determining mechanism

h i1 3=2 ð1 þ aÞ1=3 1 h i1 3=2ð1 þ aÞ4=3 ð1 þ aÞ1=3 1

Evaluation of the pre-exponential factor and reaction model based on the it-KAS and it-FWO methods To increase the accuracy of the kinetic parameters calculated by KAS and FWO, Gao et al. [31] have suggested an iterative procedure. The iterative equations of KAS (itKAS) and FWO (it-FWO), used to compute the activation energy, are given below. The it-KAS equation is expressed as [Eq. (6)]: bi Aa R Ea ¼ ln 2 gðaÞEa RTa;i hðxÞTa;i

bi 0:0048Aa Ea Ea ¼ ln 1:0516 HðxÞ gðaÞR RTa;i

ð7Þ

where x = Ea/RTa,i, h(x) is expressed by the fourth Senum and Yang approximation formula [30], and H(x) is calculated from h(x) and x. x4 þ 18x3 þ 86x2 þ 96x x4 þ 20x3 þ 120x2 þ 240x þ 120 expðxÞhðxÞ x2 HðxÞ ¼ 0:0048 expð1:0516xÞ

hðxÞ ¼

ð8Þ ð9Þ

The detailed iterative procedure used to perform the activation energy computation is given elsewhere [33, 35]. The most correct reaction mechanism function g(a) is often determined with the following equation [32]: Aa Ea ex ln gðaÞ ¼ ln þ ln 2 þ ln hðxÞ ln bi ð10Þ R x The degrees of conversion a corresponding to multiple rates at the same temperature are inserted into the left-hand

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side of Eq. (10), combined with 41 types of mechanism functions given in Table 1. The slope of the straight line obtained from the plot of ln[g(a)] versus ln b, using linear regression of least square method, should be nearly equal to -1.0000 with the better linear correlation coefficient r2. If incidentally a number of g(a) functions satisfy the conditions specified above, the degrees of conversion corresponding to multiple heating rates at several temperatures are applied to compute the most probable mechanism by the same way. Based on the estimated values of activation energy (obtained by it-KAS or it-FWO), the pre-exponential factor (A) value can be evaluated from the intercept of the plot of Eq. (6) or (7) by inserting the g (a) function determined to be the most probable.

ð6Þ

The it-FWO equation is written as [Eq. (7)]: ln

Three-dimensional diffusion

n h i o1=2 4 ð1 aÞ 1 ð1 aÞ1=2 h i1=2 6ð1 aÞ2=3 1 ð1 aÞ1=3

nonlinear integral [1, 20] methods have been reputed to afford considerably more accurate estimation of the kinetic parameters. These methods have been extensively used by several authors in different fields.

ln

Three-dimensional diffusion

Evaluation of the pre-exponential factor and the reaction model based on the Sbirrazzuoli methodology (VYA/CE and FR/CE methods) Vyazovkin has developed an isoconversional nonlinear method to determine the activation energy based on numerical integral to overcome the shortcomings of other integral procedures [1, 20]. This method is free of the approximations used in KAS and FWO equations because it is based on a direct numerical integration of Eq. (5). A further advantage of this isoconversional method is that it is not limited to linear temperature schedules, and it takes into account possible fluctuations in the activation energy. According to this method, for a set of n experiments carried out under non-isothermal conditions at different heating rates, the activation energy can be determined at any particular value of a by finding the value of Ea that minimizes the following function: n X n X IðEa ; Ta;i Þbj ð11Þ UðEa Þ ¼ IðEa ; Ta;j Þbi i¼1 j6¼1 where the temperature integral:

Author's personal copy A simple and linear isoconversional method to determine the pre-exponential factors and the …

IðEa ; Ta Þ ¼

ZTa

ln Ai ¼ aEi þ b expðEa =RT ÞdT

ð12Þ

0

is solved numerically. Minimization is repeated for each value of a to obtain a dependence of Ea on a. b in Eq. (11) denotes the heating rates, the indexes i and j denote set of experiments performed under different heating rates, and n is the total number of experiments performed. The fourth-degree approximation [Eq. (8)] proposed by Senum and Yang [29, 30], which exhibits a very high accuracy (the relative error lower than 0.6% for x C 1), has been used in the present study to evaluate the integral Eq. (12). IðEa ; Ta Þ ¼

Ea f ðxÞ R

ð13Þ

where f ðxÞ ¼

Ea x4 þ 18x3 þ 86x2 þ 96x 4 R x þ 20x3 þ 120x2 þ 240x þ 120

and x = Ea/RTa. MATLAB software was used to perform the activation energy computations. The differential isoconversional method of Friedman is based on the Arrhenius equation. Friedman analysis applies the logarithm of the conversion rate (da/dt) as a function of the reciprocal temperature at different degrees of conversion. From Eq. (4), Eq. (14) can be obtained [34]. Application of this method necessitates the knowledge of the reaction rate and (da/dt)a,i of the temperature Ta,i corresponding to a given extent of conversion, for the i temperature programs employed. The advantages of differential methods like Friedman’s method are that they make no approximations and can be applied to any temperature program [21]. da Ea ln ¼ ln½f ðaÞAa ð14Þ dt a;i RTa;i To estimate the reaction model and the pre-exponential factor, the compensation effect seems to give accurate evaluations [20]. The method was originally proposed for a single-step process [36]. Later, it was proved to work for estimating the pre-exponential factors of multistep processes [37]. The methodology has been perfected by Sbirrazzuoli [21]. The compensation effect was widely discussed, and more detailed information can be found elsewhere [38, 39]. For the present work, it would suffice to mention that the Arrhenius parameters lnAi and Ei evaluated by a single heating rate method are strongly correlated in the form of a linear relationship known as compensation effect:

ð15Þ

where a and b are the compensation parameters and the subscript i refers to a factor generating change in the Arrhenius parameters (temperature program, conversion). Overall, the method of estimating the pre-exponential factor can be summarized as follows: (1) An isoconversional method is applied to compute the activation energy as a function of conversion (in the present work, Vyazovkin (VYA) or Friedman (FR) methods have been used); (2) a single heating rate method (e.g., Tang et al. [40] method) is employed to determine several lnAi and Ei pairs; (3) the Arrhenius parameters values are then fitted to Eq. (15); (4) the activation energy values are substituted into Eq. (15) to yield the respective value of the pre-exponential factor: ln Aa ¼ aEa þ b

ð16Þ

Sbirrazzuoli has demonstrated that highly accurate values of the pre-exponential factor can be obtained when one evaluates the parameters of the compensation effect (CE) from only four pairs of Arrhenius parameters estimated by single heating rate method of Tang et al. [21]. Once both Ea and Aa parameters have been estimated, it becomes possible to reconstruct numerically the reaction model in either integral or differential form. The integral reaction model is obtained using Eq. (17). Aa gðaÞ ¼ b

ZTa

expðEa =RT Þdt

ð17Þ

0

Inserting the values of Ea, Aa and Ta for each value of a in Eq. (17) yields numerical values of g(a) that can be matched to the theoretical g(a) models. Evaluation of pre-exponential factor and kinetic model based on the proposed approach (TAS method) Burnham and Braun proposed a modification of CR equation producing an isoconversional method to avoid the shortcomings of the Coast–Redfern model-fitting method and to increase the accuracy of the kinetic results [27]. The obtained isoconversional equation is commonly called the modified Coats–Redfern method (MCR) [Eq. (18)]. In the present work, based on the MCR method, Trache et al. have suggested a simple approach (TAS method) for the estimation of reaction model g(a) and the pre-exponential factor. The method relies on the application of the compensation effect (ln Aa ¼ aEa þ b).

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3 Ea;jþ1 b Aa R i 4 5

¼ ln ln gðaÞE 2 RTa;i 2RT a;jþ1 a;i Ta;i 1 Ea;j ð18Þ where j is the number of iterations and i is a set of experiments carried out at different heating programs. At a fixed conversion degree (a) for each of the heating rates (bi), the left-hand side is plotted against 1/T, which yields a family of straight lines of slope—Ea,j?1/R. Since the left-hand side is a weak function of Ea,j, the calculation process must be performed iteratively by first assuming a value of Ea,j, then recalculating the left-hand side until convergence (Ea,j?1 - Ea,j \ 0.01 kJ mol-1) occurs at the iteration n. By changing a in step, a series of relating Ea,n may be determined. To increase the accuracy of the method, the use of the approximation (1–2RTa,i/E & 1) is avoided [41]. MCR isoconversional method identifies the dependence of E on a, but does not yield the reaction model and the pre-exponential factor. Therefore, with the proposed approach in the present work, the estimation of the kinetic model and the pre-exponential factor has been carried out. Let us put Ia(n) as the intercept of the MCR linear relationship [Eq. (18)] at the iteration n: Aa R Ia ðnÞ ¼ ln ð19Þ gðaÞEa;n

Using the 41 models presented in Table 1, the left-hand side of Eq. (21) denotes ‘‘Yi (Ea,n)’’ can be computed for each g(a). Therefore, the compensation parameters a and b can be calculated by linear regression. The best fit for an appropriate kinetic model g(a) can be estimated through the correlation coefficient and Fisher’s transformation [42]. The obtained parameters a and b, corresponding to the best model, are the slop and the intercept of the linear fitting of Eq. (22), respectively. The compensation parameters can be then inserted in Eq. (16), in addition to Ea, in order to compute the pre-exponential factor Aa. ð22Þ

The advantages of this approach are as follows: (1) simplicity and accuracy; (2) determination of the E-a

123

The applicability of the suggested approach, based on the intercepts, to determine the mathematical reaction model and the pre-exponential factor was checked for the nonisothermal decomposition of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin

(a)

240

MCR Vyazovkin it-KAS it-FWO Friedman

220

ð20Þ

Taking into account the linear relationship between ln Aa and Ea, Eq. (20) can be transformed to Eq. (21) using the compensation effect (ln Aa ¼ aEa þ b). R Ia ðnÞ ln ð21Þ ¼ aEa;n þ b gðaÞEa;n

Yi ðEa;n Þ ¼ aEa;n þ b

Applications

200

180

160

140 0.0

(b)

0.2

0.4

α

0.6

0.8

1.0

240 MCR Vyazovkin it-KAS it-FWO Friedman

220

Ea/kJ mol–1

Some simple rearrangement gives rise to Eq. (20). R Ia ðnÞ ln ¼ ln Aa gðaÞEa;n

dependency with using the iterative MCR based on one approximation only; (3) estimation of the compensation parameters without using any model-fitting method; (4) estimation of the theoretical reaction model; and (5) accurate evaluation of the pre-exponential factor. All calculations mentioned above were run by a program compiled by means of MATLAB, which is a powerful software for numerical calculation, in a computer that has an Intel CoreTM i7–4710 HQ CPU with frequency of 2.50 GHz and memory capacity of 16 GB.

Ea/kJ mol–1

2

200

180

160 0.0

0.2

0.4

0.6

0.8

1.0

α

Fig. 1 Dependence of E on a for non-isothermal data evaluated by different isoconversional procedures for: a sample 1 and b sample 2

Author's personal copy A simple and linear isoconversional method to determine the pre-exponential factors and the … Table 2 Comparison of the kinetic parameters for the thermal decomposition of samples 1 and 2 obtained by the proposed approach and other isoconversional methods Method

Ea/kJ mol-1

logA/s-1

Sample 1

Sample 2

Sample 1

Sample 2

Original worka

157.90

229.81

15.45

23.38

TAS

167.83 ± 3.53

209.03 ± 9.49

16.40 ± 0.33b

20.56 ± 1.12

208.87 ± 9.51

15.75 ± 0.34c b = 5 C min-1

VYA/CE

FR/CE

167.64 ± 3.54

167.26 ± 7.64

201.50 ± 11.48

16.00 ± 0.40

b = 5 C min-1

20.29 ± 1.06

16.05 ± 0.39

b = 10 C min-1

20.35 ± 1.05

-1

20.33 ± 1.04

b = 10 C min

-1

b = 15 C min

-1

16.02 ± 0.39

b = 15 C min

b = 20 C min-1

16.01 ± 0.39

b = 20 C min-1

20.29 ± 1.03

b = 5 C min-1

15,96 ± 0.86

b = 5 C min-1

19.42 ± 1.27

b = 10 C min-1

16,00 ± 0.85

b = 10 C min-1

19.48 ± 1.26

b = 15 C min-1

15,98 ± 0.84

b = 15 C min-1

19.46 ± 1.25

b = 20 C min-1

15,96 ± 0.83

b = 20 C min-1

19.44 ± 1.24

it-KAS

167.81 ± 3.53

209.01 ± 9.50

16.00 ± 0.37

20.85 ± 1.49

it-FWO

163.02 ± 3.32

202.25 ± 8.96

15.99 ± 0.37

20.84 ± 1.49

a b c

In their works [43], the authors have calculated the values of Ea and logA using Kissinger method Computed using model no. 2 Computed using model no. 17

for the following cases: (1) Sample 1 was studied by DSC at 2 MPa and (2) sample 2 was investigated by DSC at 0.1 MPa [43]. Decomposition of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin studied by DSC at 2 MPa (sample 1) Yi et al. [43] have investigated the non-isothermal decomposition reaction kinetics of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin studied by DSC at 2 MPa (sample 1) using DSC analysis at 2 MPa. Four heating rates (5, 10, 15 and 20 C min-1) have been chosen for the experiments. The authors have used several kinetic methods to compute the kinetic triplet (activation energy (Ea), pre-exponential factor (logA) and the most probable kinetic model function [g(a) and f(a)]. They demonstrated that the reaction mechanism was random nucleation followed by growth and obtained the following kinetic parameters: g(a) = [-ln(1– a)]4/3, f(a) = (3/4)(1 - a)[-ln(1–a)]-1/3, Ea = 157.90 kJ mol-1 and logA = 15.45 s-1.

MCR with those derived from other isoconversional methods reputed to afford more accurate estimation of Ea. The dependence of Ea on a for thermal decomposition of sample 1 is presented in Fig. 1a. It indicates that the activation energy of the decomposition process is almost independent of the conversion degree for the section 0.05–0.98 (a). Table 2 gives the mean values of Ea obtained by MCR method as well as those obtained using Friedman, Vyazovkin, it-KAS and it-FWO methods. From Fig. 1a and Table 2, one notes that, in the limits of inherent experimental errors, a good agreement among the Ea values determined by all considered isoconversional methods (Friedman, Vyazovkin and it-KAS) was obtained. However, the relative lower values of Ea computed by it-FWO method could be explained by worst approximation for temperature integral on which original FWO method is based. Finally, we can conclude that the more accurate methods are MCR, Friedman, Vyazovkin and it-KAS. The value of activation energy reported by Yi et al. and computed by the original Kissinger method is slightly lower with respect to that obtained by the accurate tested isoconversional methods as well.

Determination of the activation energy

Estimation of the pre-exponential factor and the reaction model

In this section, we evaluated the reliability of MCR method against experimental data corresponding to the decomposition of sample 1, to compare Ea values determined by

The main objective of this work is to compute the logarithm of the pre-exponential factor and the reaction model g(a) using TAS method and to evaluate its accuracy.

123


(a)

Based on it-KAS and it-FWO methods, the first step was the evaluation of the best mathematical function using the activation energy derived from these methods, followed by the determination of the pre-exponential factor. The FR/CE and VYA/CE methods firstly require the calculation of the activation energy by Friedman and Vyazovkin methods, respectively, followed by the determination of the compensation parameters using Tang method that employs one dataset obtained with a single heating rate. Equation (17) was then used to obtain the reaction model. The TAS method, however, is based on the computation of the activation energy using MCR, followed by the determination of the reaction model using the compensation parameters obtained with experimental data. The derived compensation parameters using Tang and TAS methods are very close (Fig. 2a; Table 3), which confirms that the heating rate has practically a negligible effect on the values of a and b parameters as previously reported by Vyazovkin [36]. Furthermore, it has been revealed that the obtained reaction models (no. 2 and 17 in Table 1) for the non-isothermal decomposition of sample 1 using TAS method do not significantly affect the values of the compensation parameters and consequently the values of Ln A, as shown in Fig. 3a and Tables 2 and 3. Therefore, the determination of the correct reaction model gives rise to correct values of Ln A. Figures 4a and 5a present the logAa dependency and the reaction model for the decomposition of sample 1 obtained with different methods. It can be deduced that TAS method leads to accurate values of logAa, since the difference with other methods is low and the values obtained are quite close to each other. Broadly, the shape of g(a) models is almost similar to that obtained with other methods. Figure 6 gives the shape of a selection of g(a) and f(a). The values (F3/4 model) of g(a), corresponding to

41 TAS VYA/CE, β = 5 °C min–1

40

VYA/CE, β = 10 °C min–1 VYA/CE, β = 15 °C min–1

Ln A /s–1

39

VYA/CE, β = 20 °C min–1

38

37

36

35 166

164

168

170

172

174

176

Ea/kJ mol–1

(b)

TAS

50

VYA/CE, β = 5 °C min–1 VYA/CE, β = 10 °C min–1 VYA/CE, β = 15 °C min–1

Ln A /s–1

48


46

44

42 190

195

200

205

210

215

220

225

Ea/kJ mol–1

Fig. 2 Correlation between ln A and E computed using VYA/CE at different heating rates and TAS methods for: a sample 1 and b sample 2

Table 3 Compensation parameters obtained with TAS and VYA/CE methods Method

ln A = aE ? b Sample 1

Sample 2

a TAS

2

b

R

0.21653 ± 0.00226a

-1.44139 ± 0.37937a

0.99491a

b

b

b

0.22433 ± 0.00610

-1.38996 ± 1.02339

a

b

R2

0.27296 ± 0.0019

-9.71443 ± 0.39788

0.99772

0.98308

VYA/EC, b = 5 C VYA/EC, b = 10 C

0.25521 ± 0.00137 0.25276 ± 0.00140

-5.83260 ± 0.39167 -5.48276 ± 0.39081

0.99896 0.99889

0.20616 ± 0.00167 0.25405 ± 0.00185

-6.78505 ± 0.40114 -6.19494 ± 0.40083

0.99839 0.99799

VYA/EC, b = 15 C

0.25120 ± 0.00145

-5.25008 ± 0.39013

0.99880

0.25182 ± 0.0018

-5.79426 ± 0.04004

0.99811

VYA/EC, b = 20 C

0.25910 ± 0.00144

-6.58806 ± 0.38992

0.99888

0.25008 ± 0.00176

-5.50442 ± 0.40042

0.99807

a

Computed using model no. 17

b

Computed using model no. 2

123


(a)

19 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

18

Log A /s–1

Fig. 3 Dependence of logA on a for non-isothermal data evaluated by TAS method for: a sample 1 and b sample 2, using different kinetic models

17

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

16

15

14 0.0

(b)

0.2

0.4

α

0.6

0.8

1.0

24 1 2

3

23

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

22

Log A /s–1

21

20

19

18

17

16 0.0

f(a) (model no. 2 in Table 1), obtained by TAS method are in good agreement with that reported by Yi et al. [43]. These results show that, in this case, the reaction model can be identified using TAS method with a high accuracy. Besides, the model no. 17 seems to be appropriate to present the thermal decomposition of sample 1 as well.

0.2

0.4

α

0.6

0.8

1.0

Decomposition of gun propellant containing the mixed ester of triethylene glycol dinitrate and nitroglycerin by DSC at 0.1 MPa (sample 2) The effect of pressure on the decomposition reaction mechanism and kinetics of double-base gun propellant composed of mixed ester of triethylene glycol dinitrate,

123


(a)

(a)

26

24

TAS, model no. 17 TAS, model no. 2 it-KAS it-FWO

4

VYA/CE, β = 5 °C min

–1


22


VYA/CE, β = 15 °C min–1 VYA/CE, β = 20 °C min–1


3


FR/CE, β = 5 °C min

–1

20

g(α )

Log A /s–1

5

TAS it-KAS it-FWO

FR/CE, β = 10 °C min–1 FR/CE, β = 15 °C min

–1

FR/CE, β = 5 °C min–1

2


18

VYA/CE, β = 20 °C min–1 FR/CE, β = 10 °C min–1 FR/CE, β = 15 °C min–1

14 0.0

0.2

0.4

α

0.6

0

1.0

0.0

0.2

0.4

α

0.6

0.8

1.0

0.6

0.8

1.0

32

(b)

TAS it-KAS it-FWO

30

7 TAS it-KAS it-FWO

6


28

Log A /s–1

0.8




26


5




24


22


4


g(α )

(b)


1

16


FR/CE, β = 5 °C min–1 FR/CE, β = 10 °C min–1

3

FR/CE, β = 15 °C min–1 FR/CE, β = 20 °C min–1

20

2

18 1 16 0.0

0.2

0.4

α

0.6

0.8

1.0

Fig. 4 Dependence of logA on a for non-isothermal data evaluated by different isoconversional procedures for: a sample 1 and b sample 2

nitroglycerine and nitrocellulose was studied by Yi e al. using high-pressure differential scanning calorimetry [43]. The present part will be limited to one dataset of their experiments performed at 0.1 MPa. Different heating rates of 5, 10, 15 and 20 K min-1 were employed to obtain their DSC curves. The authors have explored various kinetic methods to calculate the kinetic triplet (activation energy (Ea), pre-exponential factor (logA) and the most probable kinetic model function [g(a) and f(a)]. They revealed that the reaction mechanism was random nucleation followed by growth and obtained the following kinetic parameters: g(a) = (1–a)–1 - 1, f(a) = (1–a)2, Ea = 229.81 kJ mol-1 and logA = 23.38 s-1. Evaluation of the activation energy The values of Ea were computed by MCR, Vyazovkin, Friedman, it-KAS and it-FWO methods with a changing

123

0 0.0

0.2

0.4

α

Fig. 5 Dependence of g(a) on a for non-isothermal data evaluated by different isoconversional procedures for: a sample 1 and b sample 2

from 0.02 to 1.00 for thermal decomposition of sample 2, as shown in Fig. 1b. One can see that the activation energy changes slightly in the section 0.1–0.9 for the different curves. From Table 2, one can find that the mean value of Ea from the non-isothermal DSC curves using MCR is in excellent agreement with the values computed by the accurate Vyazovkin and it-KAS methods. These results confirm the reliability and the consistency of the MCR method to calculate the activation energies. However, the apparent activation energies obtained by Friedman and itFWO are slightly lower than that obtained by other isoconversional methods, which is certainly due to the worst approximation for temperature integral on which the original FWO method is based and to the high sensitivity to experimental noise of Friedman method [44]. In addition, the value of the activation energy of sample 2 computed and reported by Yi et al. using Kissinger method is higher


(a)

(a)

1.50 F4

1.25

F3

F2

F2

F3

A2

1.6 F4

D5

1.00 A3

A3 1.2

F3/2

0.75

g(α )

g(α )

2.0

F1

D5 0.8

0.50

A2

R2

R3 F3/4

0.25

F3/2 R2

D3

0.4

R3 D3

D4

D4

0.00 0.0 0.2

0.0

(b)

0.4

α

0.6

0.8

1.50 1.25

A3

R3

0.0

(b)

F3/4

R2

F3/2

1.0

D3

A2

0.75

1.0

D4

D5 A3

1.2

0.50

0.8

D3

F3/2

D5

F2

0.6

α R3

R2

D4

f (α )

f (α )

F1

0.4

2.0

1.6 1.00

0.2

F2 0.8

A2

F3

R1

0.25 F3

0.4

F4

F4

0.00 0.0

0.2

0.4

α

0.6

0.8

1.0

Fig. 6 Plots of a g(a) and b f(a) versus a for selected models of nonisothermal data evaluated by different isoconversional procedures for sample 1

than that obtained with the accurate isoconversional methods (MCR, Vyazovkin and it-KAS) and is tabulated in Table 2. Determination of logAa and of g(a) dependencies To verify the validity of TAS method, thermal decomposition kinetics of sample 2 were considered, and the obtained results were compared with FR/CE and VYA/CE methods. For the sake of comparison, logAa, g(a) and f(a) have also been computed using it-KAS and it-FWO. Initially, the calculation of compensation parameters was performed by TAS, FR/CE and VYA/CE. From Table 3 and Fig. 2b, it is evident that the parameters a and b are very close and the plot of lnAa against Ea gives linear curves with high correlation coefficients values. Figure 3b demonstrates the effect of reaction mode g(a) on the values of logA. A considerable change of these values can be seen once different models (41 models from

0.0 0.0

0.2

0.4

α

0.6

0.8

1.0

Fig. 7 Plots of a g(a) and b f(a) versus a for selected models of nonisothermal data evaluated by different isoconversional procedures for sample 2

Table 1) are employed. This result reveals that the accurate choice of the kinetic model leads to correct values of logA. Besides, Fig. 4b shows the logAa dependency for the thermal decomposition of sample 2 calculated with different methods. It can be concluded that TAS method gives rise to accurate values of logAa, since the difference with the reliable VYA/CE approach is very low. Figure 5b presents the reaction model g(a) for the decomposition of sample 2 computed with different methods. TAS method gives very similar results, and there is a very good adequacy with the shape of the function g(a) computed with VYA/CE and FR/CE methods, whereas the shape of g(a) obtained by it-KAS and it-FWO methods is completely different. Furthermore, comparing the TAS result with the reported one from Yi et al., it can be deduced that g(a) obtained by TAS belongs to the same model’s category (chemical reaction in Table 1). The results obtained with VYA/CE method are commonly

123


(a)

20.0 17.5


8


15.0

Relative error/%

10

it-KAS it-FWO


Computation time/s

(a)


12.5

FR/CE, β = 5 °C min–1 FR/CE, β = 10 °C min–1 FR/CE, β = 15 °C min

10.0

–1


7.5

6

4

5.0 2

2.5 0

0.0 0.0

(b)

24

0.2

0.4

α

0.6

0.8

TAS

1.0

VYA/CE

it-KAS

it-FWO

FR/CE

Computation method

(b) 10

it-KAS it-FWO VYA/CE, β = 5 °C min–1 VYA/CE, β = 10 °C min–1

20

8

VYA/CE, β = 20 °C min

16

Computation time/s

Relative error/%

VYA/CE, β = 15 °C min–1 –1

FR/CE, β = 5 °C min–1 FR/CE, β = 10 °C min–1 FR/CE, β = 15 °C min–1

12


8

6

4

2

4

0

0

0.0

0.2

0.4

α

0.6

0.8

1.0

Fig. 8 Relative error in the pre-exponential factor obtained from TAS and the different isoconversional methods for: a sample 1 and b sample 2

considered very accurate [20, 21], showing that TAS method is an efficient approach to determine g(a). Figure 7 displays the shape of a selection of g(a) and f(a). The model F3/2 (Table 1) is considered as the best model determined by TAS method which fits the real reaction mechanism. Although this model and that reported by Yi et al. (F2) belong to the same category, they are different. This difference is mainly caused by the absence of the model F3/2 in the list of the 41 models used by Yi et al. [43] to determine the correct mechanism function of the decomposition process of sample 2. These results confirm that, in this case, the reaction model can be determined by TAS method with a high accuracy. Kinetic predictions The precision of TAS method with respect to it-KAS, itFWO and VYA/CE and FR/CE methods has been quantitatively estimated as the relative deviation of the logAa

123

TAS

VYA/CE

it-KAS

it-FWO

FR/CE

Computation method

Fig. 9 Computation time associated with different isoconversional methods for: a sample 1 and b sample 2

values calculated by different approaches. The relative errors of the pre-exponential factor obtained from different methodologies are shown in Fig. 8a, b for samples 1 and 2, respectively. It can be observed that the relative errors of the pre-exponential factor obtained from it-KAS, it-FWO and FR/CE are slightly high and change with the conversion degree. However, lower relative errors of lnAa are found from VYA/CE. This latter method was previously considered by Vyazovkin [36] and recently confirmed by Sbirrazzuoli [21] as a consistent approach to determine the pre-exponential factors and the mathematical reaction model. Consequently, these results demonstrated that TAS approach proposed in the present work is reliable for the determination of these kinetic parameters. In addition, the computation time associated with each methodology form obtaining the kinetic triplet was easily determined with MATLAB. The obtained computation time for different approaches is shown in Fig. 9a, b for samples 1 and 2, respectively. It can be seen that the


computation time for TAS approach is much less than that of it-KAS, it-FWO, FR/CE and VYA/CE methods. According to the aforementioned analyses, it can be deduced that the new TAS method is capable of providing the valid pre-exponential factors and the mathematical reaction mechanism functions.

Conclusions An integral isoconversional procedure has been deduced from the modified Coats–Redfern (MCR) method to evaluate the pre-exponential factor and the reaction mechanism function. TAS method appears to accurately simulate the decomposition reaction kinetics. The proposed approach was experimentally verified by taking two experimental examples of non-isothermal decomposition kinetics of two different energetic materials. The results showed that: 1.

2.

3.

The compensation parameters derived from this method are constant for different heating rates, whereas the accurate methods employed until now (e.g., Tang method) give different compensation parameters. The methodology seems to be more efficient than itKAS and it-FWO methods. This procedure is very simple and precise. The newly proposed method allows the pre-exponential factor and reaction model to be accurately determined in less time than VYA/CE, FR/CE, itKAS and it-FWO methods.

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