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Latvian Journal of Chemistry, No 3, 2012, 209–227 DOI: 10.2478/v10161-012-0013-z

Physical chemistry

EVALUATION OF KINETIC PARAMETER CALCULATION METHODS FOR NON-ISOTHERMAL EXPERIMENTS IN CASE OF VARYING ACTIVATION ENERGY IN SOLID-STATE TRANSFORMATIONS A. Bērziņš, A. Actiņš University of Latvia, Faculty of Chemistry, Krišjāņa Valdemāra 48, Rīga, LV-1013, Latvia e-mail: agris.berzins@lu.lv

Simulations of solid-state transformation kinetics were carried out calculating temperature and conversion degree for non-isothermal experiments with different heating rates. Simulations were divided in two parts: with constant and with variable activation energy. Simulations were analyzed with widely used model–based and model-free activation energy determination methods, frequency factor and kinetic model determination methods. Much of the attention was devoted to the calculation of kinetic models and frequency factors, as a more difficult and less developed step. For simulations where activation energy did not change all activation energy determination methods were found to give correct results. Howe-ver, much attention should be devoted to frequency factor determination, because incorrect results would lead to problems in determination of kinetic models. For simulations where activation energy changes, correct activation energy can be determined only by differential methods or integral methods using numerical integration over small intervals. Iso-kinetic relationship coefficients b and c were more accurately determined with the average linear integral method. Correct kinetic model determi-nation was possible only when coefficients b and c were accurate, and only by analyzing results of all available methods.

Key words: non-isothermal kinetics, activation energy, frequency factor, kinetic model, isoconversional methods, solid-state kinetics.

INTRODUCTION

Solid state reactions are more complicated than reactions in homogenous media, because of various possible rate limiting steps, such as nucleation, nuclei growth and diffusion [1, 2].

The rate of a solid-stage reaction can be described as

( )d kfdtα α= , (1)

where α is the conversion degree; k is the rate constant, time-1; f(α) is the reaction model. Eq. (2) is obtained by integration of above equation

( )g ktα = , (2)

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where g(α) is the integral reaction model. If rate constant dependence on temperature needs to be included, Arrhenius

Eq. (3) is often used [3].

( )exp / ( )ak A E RT= ⋅ − , (3)

where A is a frequency factor, time–1; Ea is the activation energy, J·mol–1; T is the absolute temperature, K; R is the universal gas constant, J·mol–1·K–1.

There are a lot of discussions concerning the meaning of constants A and Ea in this equation, because originally this equation was developed for gaseous reactions [1]. There are many alternative functions known for this relationship [3], although constants of those equations do not provide an understanding of the reaction mechanisms. At the same time, other types of this relationship have been described, which are based on physical models like the Polanyi–Wigner model [4, 5] and the transition state description [4, 6]. As reported, in solid state kinetics A and Ea may have only an empirical interpretation [7] although these constants are necessary to define the reaction rate adequately. One should also note the band theory, which has been developed for band structure characteri-zation across the reaction zone [4, 8], and thus offering a physical meaning to activation energy, together with explaining a Maxwell–Boltzmann-like energy distribution function.

Variation in activation energy

Due to the complicated processes described by the solid state reactions, the activation energy may vary as a function of reaction progress [9, 10]. This statement has caused controversies about the meaning and explanations of variable activation energy over the course of reactions [11, 12]. Most common explanations of variable activation energy are:

a) due to the complex processes [11, 13] (where the apparent activation energy varies, but there are constant activation energy values for indi-vidual reactions);

b) due to multistep processes, such as melting or recrystallization during the reaction [11];

c) due to inconsistent procedural variables or incorrect assumptions in computations [11];

d) due to reversibility of the actual reaction mechanism [14]; e) in reactions complicated by limited diffusion rate [15]. A review is available, where variations in activation energy are explained

with changes in the physical properties of the medium [16]. In this paper, two types of processes are examined: a) processes where activation energy is constant; b) processes where true changes of activation energy happen. Much of the attention is devoted to the calculation of kinetic models and

frequency factors, because the calculation of activation energy has been well described in the literature as can be seen in the further references.

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CALCULATION METHODS OF NON–ISOTHERMAL KINETICS

For determination of kinetic parameters from non–isothermal experiments we introduce the Eq. (4). The experimentally determined non–isothermal reaction rate dα/dT is then described by Eq. (5):

d d dt ddT dt dT dtα α α β= = ⋅ ; (4)

exp ( )aEd A f

dT RTα α

β− =

. (5)

Integration of Eq. (5) gives

0( ) exp

TaEAg dT

RTα

β− = ∫

. (6)

If Ea/RT is replaced with x, Eq. (7) is obtained

2

exp( )( ) ( )a a

x

AE AExg dx p xR Rx

αβ β

∞ −= =∫ , (7)

where p(x) is the exponential integral. For this exponential integral there is no analytical solution [17], but there are

many numerical approximations [17–20].

Model fitting methods

Kinetic parameters of non-isothermal kinetic data can be obtained in two ways: from model fitting methods and from model-free or isoconversional methods. Model fitting methods are based on fitting of obtained data to various known solid state reaction models (most common ones can be found in the literature, like [1, 2, 21]) and so obtaining the activation energy values, frequency factors and reaction model (based on the best fit) from one α – temperature plot. Methods based on this model in non-isothermal studies have been widely criticized [22–24], because only constant values of kinetic triplet (Ea, A, g(α)) are obtained, the three parameters (kinetic triplet) are obtained from only one curve, and just one heating rate is examined for determination of all those parameters. The most popular model fitting methods are the direct differential method [25] and the Coats–Redfern (CR) method [18]. It is self-evident that model fitting methods can be successfully used only for processes with constant activation energies, so for complex processes or processes with varying activation energy isoconversional methods should be used.

Model-free and isoconversional methods

Model-free methods calculate activation energy independently from reaction model, so for calculation of activation energy no assumptions are made. Frequency factor A can be calculated from the intercept of linear equations, but it requires a model-based assumption. Isoconversional methods are model-free methods where activation energy is calculated at progressive conversion values α [26]. These methods require more heating rates, and therefore they are some-times called multicurve methods [27]. From these methods so-called isocon-

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versional curves (Ea versus α) are obtained. These methods usually are divided into two types – differential and integral methods.

Classical widely used methods are differential Friedman (FR) method [28] and integral Ozawa–Flynn–Wall (OFW) method [29] and Kissinger–Akahira–Sunose (KAS) method [30, 31].

More recently, other kinetic parameter determination methods are developed such as Li–Tang (LT) method [32–34], Vyazovkin (VYZ) method (also called the Nonlinear-integral method NL-INT) [35, 36], Advanced Isoconversional (AIC) method (also called Modified nonlinear-integral method MNL-INT) [37, 38], Average linear integral method (ALIM) [39], ‘non–parametric kinetics’ (NPK) method [40, 41] and other calculation methods.

Kinetic compensation effect (KCE). KCE is a linear relationship between lnA and Ea within the set of Arrhenius parameters calculated for each of a series of related or comparable rate processes [42–44], and can be expressed as

ln aA bE c= + , (8)

where b and c are constants. If KCE exists, then an overlay of Arrhenius curves reveals an artificial

isokinetic relationship (IKR) [44]. IKR can be used to determine the frequency factor in isoconversional kinetics by using Eq. (8) constants, calculated from Ea and A values obtained by different kinetic models [45, 46]. This method is successfully used to calculate frequency factor values for activation energies determined through isoconversional methods [47, 48].

Determination of reaction model

Besides model fitting methods [18, 25] there are other methods for reaction model determination using non-isothermal kinetics. One such way relies of usage of activation energies calculated by isoconversional kinetics together with some model fitting method. The correct model is that, for which a good fit is obtained, and the determined activation energy is identical with the one ob-tained by the isoconversional analysis [49]. Another way requires a recon-struction of integral or differential reaction model with independently obtained activation energy (from isoconversional method) and frequency factors (from IKR), using Eq. (1) and Eq. (2), in which Eq. (3) is inserted [45, 47, 48]. With these parameters reconstruction in coordinates α – T is also possible. The third way for reaction model determination entails a use of master plots (MP) by plotting either parameter p(x) [50] or parameter Z(α) [51].

Kinetic model and IKR parameters b and c for reactions where Ea depends on conversion degree can also be determined using average linear integral method [39]. Another linearization calculation method can be found in literature [52], where isothermal plot is used.

EXPERIMENTAL

In this paper, only simulated solid-state transformation kinetics were ana-lyzed, because that was the only way how to test the method suitability for calculating the correct activation energy, frequency factor and reaction model. For these purposes, simulations were divided into two general types:

A) simulations with constant activation energy for all α values;

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B) simulations, where the activation energy varied with conversion degree as in the case of true activation energy change, and the correct activation energy at each point was known.

The data (reaction rate and conversion degree at each temperature) for simulations A and B are calculated using the Runge–Kutta method according to Eq. (5) for each temperature starting from 0.01 K to complete reaction, with a 0.01 K step for heating rates of 0.5, 1, 2, 4 and 6 K⋅min–1. Using these data the temperatures at given conversion degrees were picked out. For activation energies that varied with α, Eq. (9) [53], (10) and (11) were used for calculation of activation energy at each value of α. As shown previously [46, 54], frequency factor can be calculated from Eq. (8).

0 1 ln(1 )E E E α= + − ; (9)

0 1E E Eα= − ; (10)

0 1 2exp( )E E E Eα= + ⋅ . (11)

All simulations with the kinetic parameters used for calculations are given in Table 1.

Table 1. Parameters used for performing simulations

Constant activation energy

Symbol Ea, kJ∙mol–1 ln A Model α step A1 100 27.63 R2 0.001 A2 100 27.63 A2 0.001 A3 100 27.63 D2 0.001 A4 100 27.63 F2 0.001 A5 150 27.63 F2 0.001

Varying activation energy Symbol E0, kJ∙mol–1 E1, kJ∙mol–1 E2 b, kJ–1 c Model α step

B1 150 50 – 0.1842 –1.15 F2 0.001 B2 150 20 – 0.1842 –1.15 F2 0.001 B3 35 300 –6 0.386 –4.00 A2 0.001 B4 35 300 –6 0.386 –4.00 F2 0.001 B5 200 –20 – 0.400 –4.00 F2 0.001 B6 50 –30 – 0.100 –1.00 F1 0.001 B7 35 300 –6 0.386 –4.00 R2 0.001

The data for all simulations were analyzed using the Coats–Redfern model

fitting method, and also with OFW, KAS, FR, LT, AIC, VYZ and ALIM methods. Ea values were calculated using α step of 0.05 for OFW and KAS methods, using α step of 0.02 for FR, LT, AIC and VYZ methods and using α step of 0.01 for the ALIM method. Minima were determined in AIC and VYZ methods by using Microsoft Excel Solver.

Coefficients b and c of IKR were obtained according to the Coats–Redfern method with activation energy and frequency factor values of all kinetic models calculated from the curve with 2 K∙min–1 heating rate from α = 0.1 to α = 0.9. For master plot analysis all heating rates were used. However, only those curves where heating rate is 2 K∙min–1 are shown here.

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RESULTS AND DISCUSSION Constant activation energy (simulations A1–A5)

The simulations where activation energy is constant over all α interval were at first analyzed using the previously described methods. The average activation energies determined from these methods in simulations A1–A5 are shown in Table 2, while the Ea – α plots calculated with all isoconversional methods for simulation A1 are shown in Fig. 1.

Table 2. Average activation energy values for simulations A1–A5

Symbol Ea, kJ∙mol–1

OFW KAS FR LT AIC VYZ CR ALIM

A1 101.0 99.81 100 99.92 99.99 99.994 99.8 100.0 A2 101.2 99.791 101 100.7 100.0 99.996 99.9 100.0 A3 100.8 99.81 100 99.987 99.99 99.992 99.8 99.99 A4 101.3 99.79 100.1 99.990 100.00 99.992 99.8 99.98 A5 151.8 149.68 149.8 149.984 149.99 149.99 149.7 149.97

Fig. 1. Activation energy values at each α value for simulation A1, according to different calculation methods.*

As expected for these simulations, activation energy does not change with α, as illustrated in Fig. 1. The only changes can be observed in the case of using the OFW method, possibly due to the change of temperature integral precision (in this case Doleys approximation), by changes in temperature that affect the parameter x [55]. The number of significant figures in Table 2 allows the determination of result distribution, because all numbers are rounded based on the standard deviation.

All methods can be divided in four groups based on the main calculation approach – those groups are: differential methods (FR); integral methods using numerical integration of differential equations (LT); methods using temperature integral approximation (OFW, KAS, VYZ) and integral methods using numerical integration of integral equation over small intervals (AIC, ALIM). It is clear that the deviations will be higher for differential methods than for integral methods, as it can be proved from the results presented in the Table 2. From the Fig. 1 it can also be estimated that the second highest dispersion is created by the methods which use numerical integration over small intervals.

* Here and further in figures Ea denotes the theoretical values of activation energy.

E a, k

J⋅mol

–1

α

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The most accurate results with the least dispersion are calculated with the VYZ method, which is an integral method and uses the most precise temperature integral.

If activation energy is constant, then there are no problems for calculation of the correct average values using all the above-mentioned isoconversional me-thods, and also using the CR model-based method.

If the correct activation energy is determined, then the frequency factor and kinetic model should also be determined. As stated previously, the frequency factor for model-free analysis should be calculated using IKR [45, 46]. To perform the simulation A1, the calculated Ea and lnA values from all reaction models were plotted in coordinates lnA – Ea and the constants of Eq. (8) were calculated by the least squares method, as shown in the Fig. 2.

Ea, kJ∙mol–1

Fig. 2. Compensation effect for Ea and lnA of simulation A1.

Using the constants b and c calculated from Eq. (8) it is possible to find lnA by using activation energy determined by the AIC method. Together with the frequency factor determined by CR method, these values are shown in the Table 3 along with the determined kinetic models. For constant activation energy it is obvious that the best kinetic model determined according to the CR method is also correct, and the kinetic parameters Ea and lnA determined using this linear equation (see Table 2 and Table 3) are thus correct.

Table 3. Determined frequency factor logarithms and kinetic models for simulations A1–A5

Symbol lnA ALIM Determined model

CR IKR (AIC) Model lnA MP

p(x) MP Z(α) g(α) f(α) CR R2

(CR)

A1 28.27 28.28 R2 28.35 R2 R2 R2 R2 R2 1 A2 28.29 27.81 A2 28.34 A2 A2 (A2)a (A2)a A2b 1 A3 27.58 27.26 D2 27.68 D2 D2 (D2)c (D2)c R2 1 A4 27.58 26.67 F2 27.68 F2 F2 (F2)c (F2)c F2 1 A5 27.58 27.12 F2 27.67 F2 F2 (F2)c (F2)c F2 1

––––––– N o t e s: a Correct model can be determined from the shape of g(α) curve, but appropriate fit is

achieved using theoretical frequency factor. b Also models A1 and A3 are determined, but Ea values are appropriate only for A2. c Model cannot be determined, or can only be guessed with obtained frequency factor

values from IKR, but by inserting values determined by CR method, almost ideal fit is obtained.

ln A

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Widely used methods for reaction model determination if activation energy is calculated using the isoconversional methods, are master plots and line re-construction in some coordinates. In the Table 3, results are shown for model determination in simulations A1–A5 using two master plot methods (p(x) and Z(α)), and from reconstruction of reaction progress in the coordinates g(α) – α and f(α) – α. Of all the used kinetic models the most appropriate theoretical lines and experimental points for simulation A1 are shown in the Fig. 3.

Fig. 3. Kinetic model determination of simulation A1 with: a) master plot p(x); b) master plot Z(α), c) reaction progress reconstruction in the coordinates of g(α) – α; d) reaction progress reconstruction in the coordinates of f(α) – α.

As can be seen from Table 3, only in simulation A1 the reaction model is correctly determined using the all methods, whereas for other simulations there are problems, greatest of which is due to the slightly incorrect value of lnA. For example, the effect of this error can be evaluated in Fig. 4, where lnA value calculated for simulation A4 using IKR method is changed to the value calculated with the CR method, and the correct model can be determined.

Fig. 4. Reconstruction of reaction progress according to the simulation A4, with lnA calculated using IKR (empty circles), or determined using the CR method (triangles).

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For all the initial five simulations, the models can be determined using the master plots, but due to incorrect lnA value it is not possible with reaction progress reconstruction. For simulation A2 the model cannot be determined using neither lnA value from IKR, nor from CR. If the theoretical value of lnA = 27.63 is used, an ideal fit for f(α) and satisfactory fit for g(α) is obtained.

Although no systematic activation energy change is observed, from the fit for the most appropriate models Eq. (8) constants b and c is obtained with ALIM. The lnA values calculated using this method are approximately the same as calculated using the both previously discussed methods.

For all these simulations, the correct model was also calculated using the isothermal plot method with the highest correlation coefficient and slope being equal to unity, an error being less than 0.5%.

As can be seen from all the results of simulations A1–A5, there is no major difference when using either the CR method or isoconversional methods in combination with the IKR method for the determination of frequency factor and kinetic models. However, if only the CR method is used, there can be no proof that the activation energy actually does not change with changes in conversion degree, because the only factor used is the correlation coefficient, and as it will be shown later, it can fit almost perfectly even for reactions with variable activation energy. Also, if correlation coefficients fit equally well in more than one model, as it was with simulation A2, then it is hard to resolve which model and, most importantly, which particular activation energy is correct, so the CR method should be used for the determination of the correct activation energy only in combination with the at least one isoconversional method.

Variable activation energy (simulations B1–B7)

Simulations were also obtained and analyzed for processes where activation energy varies when the conversion degree is changed. For determining the accuracy of used calculation methods plot activation energy – conversion degree should be analyzed and compared with the theoretical values. So, for the simu-lations B1, B2, B3, B4, B5 and B7 Ea – α plots are shown in Fig. 5a–f respecti-vely. Simulation B6 was recalculated using the same kinetic parameters as used by Ortega [39], and the results were the same as shown therein.

Using the comprehensive analysis of all plots it is evident, that the only methods giving the same or similar results are the methods that do not rely on approximations, namely, FR, AIC and ALIM methods. It is clear that none of the methods employing the temperature integral approximations (KAS, OFW and VYZ methods) produce true activation energy values. Also the LT method, where no assumptions are used, fails to give correct Ea values. We also con-clude that if activation energy decreases along with the decrease of α, then the methods producing incorrect results tend to overestimate the activation energy, while if activation energy increases with the increase of the α value, then those methods underestimate the activation energy. The same observations have been made previously [56]. We additionally conclude that the Friedmans method produces high random errors in all cases, while in the simulation B5 there is also relatively high random dispersion also for the AIC and ALIM methods. Therefore the methods FR, AIC or ALIM should be used for determination of activation energy in the processes where the activation energy is changed with

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altering α. The best methods are AIC and ALIM, because the activation energy values calculated according to these methods exhibit smaller random errors.

Fig. 5 Ea – α plots for simulation: a) B1; b) B2; c) B3; d) B4; e) B5 and f) B7, calculated with different methods.

If the correct activation energy values are calculated it is also often neces-

sary to determine correctly the reaction kinetics model and frequency factor. However, when activation energy is not constant, then Eq. (8) connects frequen-cy factor with activation energy thus coefficients b and c should be determined. In the literature where variable activation energy was studied main attention was paid to the correct determination of activation energy, but only a few publiccations discuss the determination of parameters b and c, and kinetic models. For each of the simulations B1–B7, the parameters b and c were deter-mined using the ALIM and IKR methods, while reaction kinetic models were determined using the ALIM, CR and with other methods. Methods used for the determination of reaction model can be divided in three groups:

a) where activation energy and temperature are used in master plots; b) where activation energy and frequency factor or Eq. (8) coefficients are

used by reconstructing reaction progress in coordinates g(α) – α, f(α) – α or α – T);

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c) where correlation coefficient of linearized points in some coordinates is used (CR, ALIM and isothermal plot).

If the methods belonging to the first group are used for determination of reaction kinetic model in the case of varying activation energy, then master plots p(x) are not appropriate, because changes of activation energy affect p(x), and these activation energy perturbations are much higher than the expected effect due to the mechanism. If the activation energy is changed by more than 100%, then point values change by order up to 40 in case of simulations B3, B4 and B7. If master plots of Z(α) are evaluated, then in this case all experimental points are in the same value range as for theoretical lines. However, the shapes of the experimental plots are not the same as for theoretical lines. There are examples where the correct model can be guessed from the curve shape, as in simulation B2 (Fig. 6a), and simulations where the plot shape of the correct model is only slightly similar to experimental points, and it is impossible to predict the correct model from this information as in simulation B7 (Fig. 6b).

Fig. 6. Master plot Z(α) for simulations: a) B2 and b) B7.

Table 4. Model determination for simulations B1–B7 from master plot Z(α)

Simulation Similar form

Similar height Other similar models

Point position compared with theoretical line

B1 F2 F3/2 F1, R2, D3 Higher B2 F2 F3/2 F1, R3, D3 Higher B3 F1 R2 R3, A3/2, A2 Lower B4 – – D5, F2, F3/2, F1 Lower B5 F2, F3/2 F2 D5, F1, R2 Higher B6 D5 – D3, F2, F3/2, F1, R3 Lower B7 R3 A3/2 F2, F3/2, F1, R2, A2 Higher

If these master plots for all simulations B1–B7 are analyzed, the most similar

reaction models are given in Table 4. In some cases it is possible to determine the correct kinetic mode from the

plot shape, but in some cases it is not possible. Also the height of points, if compared with the correct model, is meaningless.

The second group of examined methods consists of methods which calculate coefficients b and c from Eq. (8). Using these methods kinetic models are deduced from the best correlation coefficients. One classical example is the CR method, results of which are shown in the Table 5.

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Table 5. Results from the Coats–Redfern method for simulations B1–B7

Symbol Ea AIC average, kJ∙mol–1

CR with best fit Closest average Ea to AIC method

α Model Ea,

kJ∙mol–1 R2 Model Ea, kJ∙mol–1 R2

B1 125 F2 148 0.998 F3/2 122 0.990 0.1–0.9 B2 131 F2 153 0.99995 F1 125 0.992 0.1–0.8 B3 83 F2 100 0.997 D1 80 0.93 0.2–0.9 B4 83 F2 33 0.995 D5 49 0.98 0.3–0.9 B5 219 F3/2 194 0.99994 F2 229 0.996 0.1–0.8 B6 65 D5 95 0.99990 D6 63 0.990 0.1–0.9 B7 83 F2 119 0.992 F1 81 0.97 0.1–0.9 The average values of activation energy calculated from the AIC method are

also listed in the table. Results fall in two series – those with the highest correlation coefficient, and those with the closest activation energy to the average activation energy calculated using the AIC method. We also provide the α interval for each simulation where points for CR method were used.

From the Table 5 follows, that according to the best fit criterion, the correct model is obtained only for simulations B1, B2 and B4, and it is F2. The activation energy closest to that calculated using the AIC method, in average is not a suitable criterion, because the correct model is found only in the simu-lation B5. Changing the α interval offers no advantage to determination of the correct model.

The factor necessary for validation of the CR and IKR method combination is the regained frequency factor or coefficients b and c in this case. The coefficients calculated for these simulations are shown in the Table 6.

Table 6. Parameters b and c, calculated according to the IKR method Eq. (8)

Symbol Baseline parameters Improved parameters

b, kJ–1 c R2 b, kJ–1 c R2 Criteria

B1 0.182 –2.63 0.994 0.183 –2.16 0.996 R2>0.95 B2 0.182 –2.20 0.998 0.182 –2.00 0.998 R2>0.95 B3 0.348 –2.11 0.995 0.350 –1.64 0.998 R2>0.97 B4 0.329 –3.54 0.92 0.313 –2.04 0.98 R2>0.95 B5 0.378 –0.969 0.9995 0.384 –1.41 0.9996 Ea<250kJ B6 0.116 –3.08 0.94 0.117 –2.92 0.97 R2>0.99 B7 0.357 –1.72 0.998 0.357 –1.40 0.9990 R2>0.90

Parameters have been calculated using the all available kinetic models, and

improvement was attempted by choosing only those models where the cor-relation coefficient was higher than the criteria listed in the Table. In the most cases the coefficient b has been determined with higher accuracy (1–15% relative error) than coefficient c (10–200% relative error). When models with low correlation coefficient are disregarded (in each of the simulations limiting correlation coefficient is set up to enclose approximately the same amount of points except for simulation B5 where enclosure were made by limiting activation energy), approximately the same b and c values are calculated.

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Model determination and parameter b and c calculation with the average linear integral method are reflected in Table 7.

Table 7. Kinetic models with the Eq. (8) parameters b and c, calculated with the average linear integral method

Symbol Model b, kJ–1 c ΔT B1 F2 0.1844 –1.136 220 B2 F2a 0.1838 –1.061 250 B3 A2b 0.384 –3.18 45 B4 F2 0.3863 –4.000 200 B5 F2a 0.4011 –4.177 25 B6 F1 0.0985 –0.873 500 B7 R2b 0.3865 –3.330 40

–––––––– N o t e s : a The correct model cannot be determined using only the correlation coefficient, but can

be determined by comparing the obtained b and c parameters between models with high correlation coefficients and those calculated using the IKR method.

b The correct model can be determined by correlation coefficient, but there are other models with high correlation coefficients and very similar b and c values.

It can be seen that using this method both the coefficient b (0.1–1.5%

relative error) and coefficient c (0–20% relative error) have relative errors reduced tenfold, compared with those obtained using the IKR method. If model determination is done according to the best correlation coefficient method, then the correct models for simulations B1, B3, B4, B6 and B7 are determined, but for simulations B2 and B5 the b and c values calculated using the IKR method must be taken into account. It is also noticed that for simulations with small temperature changes during the whole process there are high correlation coefficients for all models, and therefore there are more models providing linear correlation. Also the highest relative errors for coefficient c are calculated for simulations B3 and B7, where a complete conversion occurs over a small temperature interval.

Table 8. Kinetic models calculated with the isothermal plot method

Symbol Best correlation coefficient Best slope

Model a R2 Model a R2 B1 F2 –1.13 0.9999 D2 –1.01 0.97 B2 F2 –1.09 0.9997 D2 –1.02 0.98 B3 D3 –1.23 0.997 D4 –0.99 0.995 B4 D2 –0.59 0.9998 D5 –0.90 0.996 B5 F2 –1.04 0.99993 P3/2 –1.03 0.990 B6 D5 –1.57 0.99990 F2 –1.02 0.998 B7 D6 –1.10 0.99998 F2 –1.07 0.97

If isothermal plots are used for determination of reaction model, then the two

criteria should be fulfilled: correlation coefficient should be equal to 1, and the slope should be equal to –1 [52]. For the simulations listed in Table 8 the correct model can be determined from the best correlation coefficient for simulations B1, B2 and B5, but when slope values are additionally analyzed, then the correct

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model can be determined only for the simulation B5. Determination of the correct kinetic model is complicated by the fact that correlation coefficients and slopes can be hard to distinguish from those produced by incorrect models.

The third way to determine the reaction model is that using reconstruction of reaction progress using of activation energy and Eq. (8) coefficients. For this purpose, activation energy values calculated using the AIC method were ap-proximated with Eq. (9), Eq. (10) or using a fifth degree polynomial expressed as Eq. (12). The equation constants were obtained by Excel Solver.

Ea = E0α5 + E1α4 + E2α3 + E3α2 + E4α + E5. (12)

It is clear that for these calculations coefficients b and c obtained using the IKR method can not be applied, because of significant errors. When coefficients calculated from the average linear integral method are used, then results are obtained as shown in the Table 9.

Table 9. Calculated reaction models with reaction progress reconstruction

Symbol g(α) – α f(α) – α α – T

B1 F2 F2 F2 B2 F2 F2 F2b B3 –a –a – B4 F2 F2 F2 B5 F2 F2 Fc B6 F1 F1 F1 B7 –a –a (R2)d

–––––– N o t e s : a The correct model cannot be determined using the calculated b and c values, but can

be determined with the original values. b Also the models F1, F3/2 and R1 are appropriate c F1, F2 and F3/2 are almost identical d Only a distant similarity with R2 model is achieved, but this similarity is better than

for other models For simulations B1, B2 and B4–B6 the correct models can be determined (as

in Fig. 7 for the simulation B2), but for simulations B3 and B7 it is not possible (as shown in Fig. 8 for the simulation B3). It should be mentioned that the latter simulations are those where the highest errors of coefficients b and c are ob-tained.

Fig. 7. Reconstructed reaction progress in the coordinates: a) g(α)– α and b) f(α) – α for simulation B2.

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Fig. 8. Reconstructed reaction progress in the coordinates: a) g(α)– α and b) f(α) – α for simulation B3.

For reconstruction in the coordinates of α – T parameters b and c (calculated for each model with the average linear integral method) were used. It is found that the correct model can not be determined for those simulations where erroneus b and c values are obtained, and also for simulations B2 and B5, where more than one model is appropriate in these coordinates. Examples of a successful simulation B4 and an unsuccessful simulation B3 are shown in the Fig. 9.

Fig. 9. Reconstructed reaction progress in the coordinates α – T for simulations: a) B4 and b) B3.

Kinetic model determination and frequency factor calculation can be also carried out using the NPK method. Similar values of Eq. (8) parameters b and c are obtained, as in the case when ALIM method was used, but the mathematical procedure is much more complex. Kinetic model determination from the plot obtained in coordinates u – α proves to be unsuccessful.

From all this information about model determination and calculation of coefficients b and c it can be concluded that the master plot p(x) is not usable, but Z(α) can be used to exclude incorrect models from further analysis. Coefficients b and c and kinetic models should be determined using the average linear integral method, but not using the IKR method, because of significant inaccuracy of the latter one, although the IKR method can be useful for evaluation of approximate limits of these coefficients. If the model can not be determined from the average linear integral method, then the reconstruction in one of the coordinates should be performed, remembering that if the coefficients b and c are inaccurate, then reconstruction will be unsuccessful.

α

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CONCLUSIONS

Simulations of solid-state transformation kinetics where activation energy is constant over the all conversion degrees allow to calculate correct activation energy values using all the available mathematical methods. The largest error is observed for the OFW method, where the least precise temperature integral is used, but the largest random error is observed for the FR method. Model determination attempts using Coats–Redfern method, average linear integral method and both master plot equations are successful, but problems are encountered when determining kinetic models from reaction progress reconstruction in coordinates of g(α) – α and f(α) – α, because imprecise frequency factor values are obtained using some or even all of the used methods. So the most attention during the analysis of results should be paid to frequency factor determination and its accuracy.

From simulations of solid-state transformation kinetics with variable acti-vation energies it is clear, that correct results are obtained using the FR method, AIC method and ALIM, but incorrect values were obtained using the integral methods (OFW, KAS, VYZ and LT), so any data from the integral methods using numerical integration of integral equation should be applicable over short intervals only. In this case there are major problems with determination of kinetic model and coefficients b and c. No clear results are obtained using the Coats–Redfern method and with isothermal plot method, but accurate and careful analysis of data from ALIM allowed the identification of the correct model. Using the latter method the optimal values of coefficients b and c are also calculated. If the calculated parameters b and c are close to the theoretical values (less than 10% deviation), reaction model could be found also by reconstruction in the coordinates of g(α) – α, f(α) – α and α – T. The master plot p(x) was not usable here, but Z(α) could be used in order to choose a limited number of appropriate kinetic models for the examined process.

Acknowledgments

This work has been supported by the European Social Fund within the project “Support for Doctoral Studies at University of Latvia” and by Latvian Academy of Sciences Grant No. 09.1555. We wish to thank Professors Michael E. Brown and Sergey Vyazovkin for kindly supplying the data from the ICTAC kinetic study [22].

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NEIZOTERMISKO EKSPERIMENTU KINĒTISKO PARAMETRU NOTEIKŠANAS METOŽU IZVĒRTĒŠANA MAINĪGAS AKTIVĀCIJAS ENERĢIJAS GADĪJUMĀ CIETFĀŽU PĀRVĒRTĪBĀM A. Bērziņš, A. Actiņš

K O P S A V I L K U M S

Veiktas cietfāžu pārvērtību kinētikas simulācijas dažādiem karsēšanas ātrumiem, aprēķinot temperatūru un pārvēršanās pakāpi neizotermiskiem ekspe-rimentiem. Simulācijas iedalītas divās daļās: ar konstantu un ar mainīgu aktīvā-cijas enerģiju. Simulācijas analizētas ar plašāk lietotajām uz modeli balstītajām un bezmodeļu aktivācijas enerģijas noteikšanas metodēm, pirmseksponenciālā faktora un kinētiskā modeļa noteikšanas metodēm. Pastiprināta uzmanība pie-vērsta sarežģītā un mazāk izstrādātā soļa – kinētiskā modeļa un pirmsekspo-nenciālā faktora aprēķināšanai. Tika noteikts, ka simulācijām, kurās aktivācijas enerģija nemainās, visas aktivācijas enerģijas noteikšanas metodes dod pareizus rezultātus. Daudz uzmanības jāpievērš pirmseksponenciālā faktora noteikšanai, jo nepareizi rezultāti rada problēmas kinētiskā modeļa noteikšanā. Simulācijām, kurās aktivācijas enerģija mainījās, pareiza aktivācijas enerģija noteikta, tikai lietojot diferenciālās metodes vai integrālās metodes, kurās lieto integrēšanu pa maziem apgabaliem. Izokinētiskās sakarības koeficienti b un c precīzāk noteikti ar vidējo lineāro integrālo metodi. Pareizā kinētiskā modeļa noteikšana iespē-jama tikai tajos gadījumos, kuros koeficienti b un c noteikti precīzi un tikai analizējot rezultātus, kas iegūti, lietojot visas metodes.

Received 04.08.2012