Kinetic calculation: Software tool for determining the kinetic parameters of the thermal decomposition process using the Vyazovkin Method

SoftwareX 11 (2020) 100359

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SoftwareX journal homepage: www.elsevier.com/locate/softx

Original software publication

Kinetic calculation: Software tool for determining the kinetic parameters of the thermal decomposition process using the Vyazovkin Method Dmitry Drozin, Sergey Sozykin, Natalia Ivanova, Tatiana Olenchikova, ∗ Tatyana Krupnova , Natalia Krupina, Viacheslav Avdin Institute of Natural Sciences and Mathematics, South Ural State University, 454080 Chelyabinsk, Russia

article

info

Article history: Received 27 May 2019 Received in revised form 18 November 2019 Accepted 19 November 2019 Keywords: Thermal decomposition Thermogravimetric analysis Kinetic parameters Vyazovkin method

a b s t r a c t Thermal analysis is widely applied to determine the composition of materials and predict their thermal stability. This paper introduces the basic features of a new software called Kinetic Calculation. Knowledge of the kinetic parameters of thermal decomposition is used to optimize chemical processes. The software can analyze different types of thermal curves depicting the changes to a given property of a material measured throughout a process. The Kinetic Calculation calculates the substance decomposition parameters using the Vyazovkin method and a simpler Ozawa–Flynn–Wall method for comparison. The substance of copper sulfate pentahydrate decomposition experiment was conducted to demonstrate the operation of the Kinetic Calculation program. This software has several advantages such as an easy interface, the ability to calculate an unlimited number of heating rates and any of their values, and independent selection of temperature intervals at each stage of the decomposition. Unlike analogous programs, Kinetic Calculations can calculate real heating rate according to experimental data. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Code metadata Current code version Permanent link to code/repository used for this code version Code Ocean compute capsule Legal Code License Code versioning system used Software code languages, tools, and services used Compilation requirements, operating environments & dependencies If available Link to developer documentation/manual Support email for questions

v1.0 https://github.com/ElsevierSoftwareX/SOFTX_2019_190 – MIT License git C# Microsoft.Chart.Controls.4.7.2046; OxyPlot.Core.1.0.0; OxyPlot.WindowsForms.1.0.0 – [email protected], [email protected]

Software metadata Current software version Permanent link to executables of this version Legal Software License Computing platforms/Operating Systems Installation requirements & dependencies If available, link to user manual - if formally published include a reference to the publication in the reference list Support email for questions

1 https://ietn.susu.ru/kinetic_calculation MIT Microsoft Windows .NET Framework – [email protected], [email protected]

1. Introduction ∗ Corresponding author. E-mail address: [email protected] (T. Krupnova).

Many substances and things used in everyday life are destroyed over time. An important task is to find out how long this

https://doi.org/10.1016/j.softx.2019.100359 2352-7110/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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D. Drozin, S. Sozykin, N. Ivanova et al. / SoftwareX 11 (2020) 100359

substance will serve at operating temperature. For many materials, this period is tens of years, and, of course, it is impossible to conduct such an experiment. Thermo analytical measurements allow us to calculate theoretically the period of destruction of materials. Mass loss is measured as a function of heating rate. As a result of the experiment, data sets of the mass loss of the substance are obtained as a function of temperature. According to these data, a mathematical model of the thermal decomposition of the substance is parameterized and the thermal decomposition parameters are determined. These parameters allow one to calculate the fracture time of a material at any temperature. Scholars have suggested numerous strategies for calculating kinetic parameters using a differential thermal analysis curve [1– 14] and isoconversional methods [15]. At this time, isoconversional methods are more frequently used. The principle of isoconversion states that with a constant degree of conversion, the reaction rate depends only on temperature. These methods make it possible to find the activation energy E without knowing the exact form of the reaction of a model. It is assumed that kinetic parameters are not constant during the reaction [15]. Isoconversional methods can be further divided into following categories: differential methods, integral methods, and the Vyazovkin method. The most common differential method is the Friedman method [16], but there is a wide variety of integral methods, among which the most commonly used methods are the Ozawa–Flynn–Wall (OFW) method [17,18] and the Kissinger–Akahira–Sunose (KAS) method [19]. Compared to differential methods, integral methods are universal due to their higher tolerance to experimental noise, which propagates the effect of noise, reducing the accuracy of the results, when derivatives are used [20]. There are commercial software for calculating kinetic parameters [21,22]. AKTS (Advanced Kinetic and Technology Solution) program [21] requires input of different heating rates differential thermal analysis (TGA) or differential scanning calorimeter (DSC) curves; heating rates should be below 8 K/min. The software NETZSCH Kinetics Neo [22] allows the analysis of temperaturedependent processes. The output is a kinetics model or method that correctly describes experimental data under different temperature conditions. In this paper, we propose using the free software Kinetic Calculation program [23], since the program 1) allows you to calculate the decomposition kinetics of a substances according to the most modern and popular methods of Vyazovkin and Ozawa–Flynn–Wall; 2) allows you to calculate the actual heating rate during the experiment; 3) has the function of dividing the entire process of multi-stage decomposition into stages, which is important for studying the decomposition of substances, and 4) allows you to calculate accurately the function of the mechanism, not limited to a set of models.

T is a temperature; and t is a time variable [24]. It should be noted that the process may involve several steps, and each of them specifies the overall kinetics. The extent of conversion is experimentally determined, and it increases from 0 to 1 throughout the process, no matter if it is a mass-loss process or a process accompanied by release or heat absorption. The Arrhenius equation E

k(T ) = A · e− RT ,

(2)

defines the temperature dependence of the process rate. Kinetic parameters A and E are the pre-exponential factor and the variable activation energy, respectively, with the universal gas constant R and the absolute temperature T . The temperature program is non-isothermal, and the heating rate β changes linearly:

β=

dT dt

= const .

(3)

The reaction of model f (α ) = α m (1 − α)n − ln (1 − α)p ,

(4)

represents a number of reaction models [25], where m, n, p are some degree parameters. If we combine Eqs. (1), (2), (3), (4), we obtain dα dT

=

A

E

· e− RT · α m (1 − α)n − ln (1 − α)p ,

β

(5)

Thus, the result of the kinetic analysis is the energy of activation E, the pre-exponential factor A. 2.1. Software functionalities Kinetic Calculation v. 1.0 program allows calculating the substance decomposition parameters using the data of the thermal decomposition experiment. Each experiment corresponds to one text file with four columns: temperature (◦ C), time (min), DSC (µV/mg), and the mass (%). The program allows the user to upload any number of initial files with different heating rates. The columns correspond to the heating rates. The head of each column represents the real heating rate, calculated by the method of least squares. Important to note that, devices do not give values of real heating rates, while real values may differ from the set by several percent, especially at values less than 2 K/min. Wherein taking into account the real heating rate increases the accuracy of the calculation, which is calculated by minimizing the following equation. S L ∑ ∑ (

Ti,j − β · ti,j + T0

(

))2

,

(6)

i=1 j=1

2. Software framework In studies, researchers must often know additional parameters, which are hard to measure for example the life of the polymers [15–25]. This section is dedicated to theoretical portion of the Vyazovkin method, which provides procedures for isoconversional computations in thermal gravimetric analysis. The algorithm has been fully realized, programmed and tested. The correctness of the computations is accompanied by accuracy estimates. The equation usually used to describe the kinetics of the reaction of solids in a single-step process is dα

= k(T )f (α ), (1) dt where f (α ) is a reaction of a model dependent on the extent of conversion α ; k(T ) is a rate constant given in Arrhenius form;

where S is the number of files for the corresponding heating rate, L is the number of rows in the i-file, Tij is a value of the temperature in the i-file in the j-row and tij is value of the time in the i-file in the j-row measurements, and β , T0 are unknown parameters of the equation. In the Kinetic Calculation v. 1.0 program, the ‘Calculation’ tab contains three initial data plots averaged in each heating rate: DSC vs T , TG vs T , DTG vs T (Fig. 3). Two movable vertical lines on the plots indicate the temperature range for the maximum heating rate necessary for further calculations of the activation energy E, the pre-exponential factor A, and the most probable mechanism function f (α ) according to the Vyazovkin method. The Kinetic Calculation v. 1.0 program also calculates the substance decomposition parameters using a simpler Ozawa–Flynn–Wall method for comparison.

D. Drozin, S. Sozykin, N. Ivanova et al. / SoftwareX 11 (2020) 100359

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3. Implementation details The first step of calculation is to transfer TG to α for each heating rate with the equation

α=

TGinit − TGcurrent TGinit − TGfinal

,

(7)

where TGinit , TGfinal are initial and final masses of a sample on the selected temperature range for the maximum heating rate, respectively, and TGcurrent is a current mass. In the second step, we calculate the activation energy of the decomposition process from TGinit to TGfinal . According to the Vyazovkin method [20], we need to minimize the following expression

Ω (Eα ) =

) n n ( ∑ ∑ I(Eα , Tα,i ) βj · , I(Eα , Tα,j ) βi i

i̸ =j

where Ω (Eα ) is a functional of the integral isoconversion method [20,24], Eα is the activation energy for each real number α , Tα,i is the thermodynamic temperature for each α and each ith heating rate, n is the number of heating rates, and minimize the integral I(Eα , Tα ) =

Tα

∫ 0

( ) Eα Eα − dT = · p(x). R·T R

(9)

Here, x = E /RT , p(x) is expressed by the fourth Senum and Yang approximation [5]: p(x) =

e−x x

Fig. 1. Values of I(Eα , Tα ) for different heating rates.

(8)

≈

x3 + 18 · x2 + 88 · x + 96 x4

+ 20 · x3 + 120 · x2 + 240 · x + 120

.

(10)

The MatLab code of Eq. (8) is presented below. function S=F(TT,E,b,k)

where β1 = 1.97, β2 = 4.95, β3 = 10.14 K/min. Fig. 1 shows the integrals I(Eα , Tα,1 ), I(Eα , Tα,2 ), I(Eα , Tα,3 ) dependent on the chosen activation energy Eα . By the non-isometric principle, the square between the curves, presented in Fig. 1, leads to the minimum. Then, it is necessary to find a value of Eα , which is held by this condition. The curve Ω (Eα ) has a parabolic form and a unique minimum. Thus, we use one-parameter unconditional optimization method to find the minimum [26]. As an example, for the temperature range of 22–102 ◦ C, α = 0.2, the curve Ω (Eα ) is presented in Fig. 2. On the third step, we find the pre-exponential factor A and the reaction of model f (α ) (see Eq. (4)). The pre-exponential factor A is calculated by equation

% Function of calculating the equation (8) % TT is matrix, where column 1 corresponding the alpha range of

A=−

% 0 1 , column 2,3, ... corresponding the temperature range for % each heating rate, E is the activation energy for a current

β ·E

E

2 R · Tmax · f ′ (αmax )

· e ·Tmax ,

where E = Eα is the average activation energy at α = 0.5 for the maximum heating rate, Tmax and αmax relate to the maximum of the differential thermogravimetric curve, and the derivative f ′ (αmax ) is calculated by

% alpha value, b is the vector of coefficients of the heating % rates, k is the row number of the matrix TT, that related to % a current alpha value S=0; for i=1:1:length(b) for j=1:1:length(b)

f ′ (α ) = −α m−1 · (1 − α)n−1 · ln (1 − α)p−1

if (i~=j) T1=TT(k,i+1);

· (ln (1 − α) · (m (α − 1) + nα) + pα) .

T2=TT(k,j+1); S=S+(I(T1,E).*b(j))./(I(T2,E).*b(1));

(13)

Substituting Eq. (12) to (5) and enumerating degrees of m, n, p of the model reaction, we find A, m, n, p through minimization of the following expression

end end end end function y=I(T,E)

(12)

[ ∑ ( dα )

%Function of calculating the integral value

% T and E is temperature and activation energy for a current

dT

% alpha value, respectively R=8.3144598;

exp

( −

dα dT

]2

)

,

where ddTα exp is an experimental value and from Eqs. (5) and (12).

(

y=E.*p(T,E)./R; end function y=p(T,E)

)

(14)

calc

( dα ) dT

calc

is obtained

R=8.3144598;

4. Example

x=E./(R.*T); y=(exp ( x).*(x^3+18.*x.^2+88.*x+96))./(x.*(x.^4+20.*x.^3+120.*x .^2+240.*x+120)); end

Eq. (8) has the following form for the number of heating rates n = 3 through minimization of the following expression

Ω (Eα ) =

I(Eα , Tα,1 )

·

β2

+

I(Eα , Tα,1 )

·

β3

+

I(Eα , Tα,2 ) β1 I(Eα , Tα,3 ) β1 I(Eα , Tα,2 ) β1 I(Eα , Tα,2 ) β3

+

+ β2 I(Eα , Tα,3 ) β1 I(Eα , Tα,3 ) + · + I(Eα , Tα,1 ) β3 I(Eα , Tα,2 ) I(Eα , Tα,1 ) I(Eα , Tα,3 )

·

·

+ β2 β2 · , β3

(11)

The decomposition reactions of copper quaternary sulfate, CuSO4 ·5H2 O, were used to demonstrate the computations. These reactions are well studied in physical chemistry. There are three stages [27]: (1) CuSO4 ·5H2 O → CuSO4 ·3H2 O + 2H2 O, (2) CuSO4 ·3H2 O → CuSO4 ·1H2 O + 2H2 O, (3) CuSO4 ·1H2 O → CuSO4 + 2H2 O. These data were obtained on a synchronous thermal analyzer (Netzsch STA449F1), which is a combination of a differential scanning calorimeter (DSC) and a thermogravimetric analyzer (TGA). These and similar instruments are available to most research

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D. Drozin, S. Sozykin, N. Ivanova et al. / SoftwareX 11 (2020) 100359

Fig. 2. Parabolic form of Ω (Eα ).

Fig. 3. The first step calculations of kinetic parameters.

groups. However, for special studies, there are high-resolution DSC and TGA devices. The study of this reaction on HR TGA showed that this reaction proceeds in five stages: (1) CuSO4 ·5H2 O → CuSO4 ·4H2 O + 1H2 O, (2) CuSO4 ·4H2 O → CuSO4 ·3H2 O + 1H2 O, (3) CuSO4 ·3H2 O → CuSO4 ·2H2 O + 1H2 O, (4) CuSO4 ·2H2 O → CuSO4 ·1H2 O + 1H2 O, (5) CuSO4 ·1H2 O → CuSO4 + 2H2 O. In fact, each of the first stages contains two stages [27]. Thus, the described program allows identifying the real staged process even when using standard equipment. The presence of several stages that are not detected by DSC and TGA analysis is characteristic of most thermo analytical processes [27]. We used results obtained by Simultaneous TG-DTA/DSC Apparatus STA 449 F1 Jupiter in the Nanotechnology Research and Education Center at South Ural State University. All the tests are carried out for β1 ≈2, β2 ≈5, β3 ≈10 K/min. We should note that ‘‘≈’’ symbolizes an apparatus error. We were guided by the fact that in kinetic calculations of a substance decomposition, the generally accepted heating rates are 2, 5, and 10 K/min. The

experiment was carried out only at these speeds. At the end of the experiment, we obtain three text files for each heating rate. Now, let us consider three thermal decomposition steps presented in the left part of Fig. 3. In the first step, the observed mass loss is 13.59% over the temperature range of 46–88 ◦ C. The DTG peak is at 81.12 ◦ C. The thermal analysis calculations from Kinetic Calculation for this step are presented in Fig. 3. The top right corner of Fig. 3 shows the dependences of the activation energy E(α ) calculated by the Vyazovkin and Ozawa– Flynn–Wall methods. The relative error of the Ozawa method with respect to the Vyazovkin method reaches 15%. In the bottom right corner of Fig. 3, two curves show the correspondence between experimental data and calculated values obtained by the Vyazovkin method. This graph illustrates the relative error of the Vyazovkin method calculated in item 4; for the first stage of the process, this value is 11.6%. This relative error is greater than 10% that indicates a multi-step mechanism [28]. Thus, the kinetic equation of the first step of thermal decomposition can be written as follows dα dT

=

224.49 · 1010 1.97

(

·e

3

·10 − 83.26 R·T

) 1

1

· α 10 · (1 − α) 3 .

(15)

D. Drozin, S. Sozykin, N. Ivanova et al. / SoftwareX 11 (2020) 100359

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Fig. 4. The second step calculations of kinetic parameters.

Fig. 5. The third step calculations of kinetic parameters.

In the second step, the observed mass loss is 14.83% over a temperature range of 88–158 ◦ C. The DTG peak is at 112.74 ◦ C. The thermal analysis calculations for this step are presented in Fig. 4. The relative error of the Ozawa method with respect to the Vyazovkin method reaches 14%. The relative error of the Vyazovkin method is 8.61% that indicates a single-step mechanism. The large discrepancy between the experimental data and the calculated values is explained by the errors of the computational method caused by the selected discretization step of the selected parameters m, n, and p. Thus, the kinetic equation of the second step of thermal decomposition can be written as dα

137.06 · 1010

(

3

·10 − 114.R05 ·T

)

·e · α 0.9 · (1 − α)1.31 . (16) 4.95 In the third step, the observed mass loss is 7.29% over the temperature range of 158–319.87 ◦ C. The DTG peak is at 231.16 ◦ C. The thermal analysis calculations for this step are presented in Fig. 5. The relative error of the Ozawa method with respect to the Vyazovkin method reaches 15%. The relative error of the dT

=

Vyazovkin method is 4.82% that indicates a single-step mechanism. Thus, the kinetic equation of the second step of thermal decomposition can be written as dα dT

=

4.87 · 1014 10.14

(

·e

3

·10 − 103.R38 ·T

)

· α 0.5 · (1 − α)1.15 .

(17)

Let us demonstrate the dependence of the calculation results on the correctness of temperature range. The observed mass loss is 25.09% over the temperature range of 67–186 ◦ C. DTG has several peaks, each of which corresponds to its own thermal decomposition reaction [15–20]. Thus, this multi-step process shows 27% relative error and qualitative discrepancy between theoretical and calculated data (Fig. 6). 5. Impact We offer a free software tool Kinetic Calculation that provides many advantages for a researcher in the field of kinetic analysis.

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D. Drozin, S. Sozykin, N. Ivanova et al. / SoftwareX 11 (2020) 100359

Fig. 6. The dependence of the calculation results on the correctness of temperature range.

The program can determine the kinetic parameters of the thermal decomposition process using the Vyazovkin and the Ozawa– Flynn–Wall methods and accurately calculates the mechanism function, not limited to a set of models. The program is new, and therefore there are no published works with the results obtained with its help. Nevertheless, the program is actively used in a number of scientific groups. Among them are the group of prof. Avdin V.V., Department of Ecology and Chemical Technology, SUSU (Russia, Chelyabinsk), group of prof. Vasin A. Ya., Department of Technosphere Security, RHTU im. DI. Mendeleev (Russia, Moscow). Kinetic Calculation program does not require installation, as it is distributed as an executable file for the Windows operating system. In older versions of Windows, the installation of the .NET Framework may be additionally required. 6. Conclusions The program Kinetic Calculations makes it possible to calculate kinetic substance decomposition parameters for thermogravimetric analysis through the Vyazovkin and the Ozawa–Flynn–Wall methods. The correctness of the calculations was demonstrated using copper sulfate pentahydrate thermal decomposition experiment data. It can be seen that, for a multi-stage process of decomposition of CuSO4 ·5H2 O, the error of the Ozawa method with respect to the Vyazovkin method reaches 15%. Unlike analogous programs, Kinetic Calculations makes it possible to calculate real heating rate according to experimental data. In addition, the program allows researchers to choose a step minimizing a relative error. The program accurately calculates the mechanism function, not limited to a set of models. The free software Kinetic Calculations provides many advantages for a researcher in the field of kinetic analysis. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment The work was supported by Act 211 Government of the Russian Federation, contract No 02.A03.21.0011.

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