New insights on the kinetic analysis of isothermal data: The independence of the activation energy from the assumed kinetic model

1 New insights on the kinetic analysis of isothermal data: the independence of 1 the activation energy from the assumed kinetic model. 2 Pedro E. Sánchez-Jiménez  , Antonio Perejón, Luis A. Pérez-Maqueda, José M. Criado. 3 Instituto de Ciencia de Materiales de Sevilla, C.S.I.C.-Universidad de Sevilla, C. Américo 4 Vespucio nº49, 41092 Sevilla, Spain 5 6 Abstract 7 Isothermal experiments are widely employed to study the kinetics of solid state reactions or 8 processes in order to extract essential kinetic information needed for modeling the processes at 9 an industrial scale. The kinetic analysis of isothermal data requires finding or assuming a 10 kinetic function that can properly fit the evolution of reaction rate with time so that the 11 resulting parameters, i.e. the activation energy and the preexponential factor, can be considered 12 reliable. In the present work, we demonstrate using both simulated and experimental data that 13 the kinetic analysis of a set of isothermal plots obtained at different temperatures, considering a 14 single step solid state reaction, necessarily leads to the real activation energy, regardless the 15 mathematical function selected for performing the kinetic analysis. This makes irrelevant the 16 election of the kinetic function used to fit the experimental data and greatly facilitates the 17 estimation of the activation energy for any single process. 18 Keywords: Kinetic Analysis, Isothermal; Kinetic Model; Activation Energy; Process 19 Modeling 20 21 Corresponding author. Tel +34954489548 Fax +34954460665 e-mail address: pedro.enriq[email protected] 2 1. Introduction 22 Kinetic analysis is widely employed as a tool for obtaining the essential knowledge needed for 23 modeling processes on an industrial scale. This is also true in the field of energy conversion 24 and production, with an important number of papers published every year in which the main 25 objective is determining the kinetics governing processes such as pyrolysis, gasification, 26 combustion or thermal decomposition in order to optimize operating conditions 1-14. The 27 experimental data are usually collected under isothermal or linear heating conditions. While 28 linear heating rate experiments provide quickness and simplicity, in many studies such as those 29 involving long-term aging at operation temperatures 15-17, oxidation processes 18-20, reaction 30 progress followed by spectral or DRX peak intensity measurements 21-23, chemical looping 31 processes 1, 3, 11, 24 or those set-ups that try to replicate industrial operation conditions 9, 12, 13, 32 isothermal experiments are still the most convenient or even feasible option. Moreover, 33 isothermal experiments present the distinct advantage of a higher capability for kinetic 34 mechanism discrimination due to the fact that the shape of the integral α-time curve is directly 35 related to the obeyed model 25, 26. Thus, the α-time traces of phase boundary controlled reaction 36 (so called “n order” reactions) are convex, the diffusion controlled reactions are concave and 37 those whose rate is controlled by the formation and growth of nuclei (Avrami-Erofeev models) 38 have sigmoidal shape. On the other hand, the α-temperature plots recorded under rising 39 temperature are always sigmoidal-shaped, whatever the reaction kinetic model 27, 28 40 A former review 29 on the kinetic dehydroxylation of kaolinite found that similar activation 41 energies had been reported by different authors despite the proposal of different kinetic models. 42 An analysis of those experimental data assuming a set of different kinetic models revealed that 43 the activation energies obtained were independent of the kinetic model previously assumed, 44 although no explanation was given. If such behavior was generalized, it would constitute an 45 extraordinary advantage since it is generally assumed that the activation energy obtained by a 46 3 kinetic analysis is dependent on the kinetic law used to fit the experimental data 27, 30, 31 . Thus, 47 authors must spend a great deal of effort to determine the kinetic function most adequate to the 48 process under study in order to assure a reliable set of kinetic parameters 27. In the present work 49 we explore the influence the kinetic law selected to fit the experimental data has on the 50 activation energy yielded by the kinetic analysis of a set of isothermal curves recorded at 51 different temperatures. It is demonstrated first theoretically and then analyzing sets of both 52 simulated and experimental data, that the activation energy obtained by this kind of analysis 53 would always be the correct one, regardless the mathematical function selected for fitting the 54 data. 55 2. Experimental 56 Thermal degradation experiments were carried out using polytetrafluoroethylene (Aldrich, 57 product number 182478) at temperatures of 480, 490 and 500 ºC in a Q5000IR TA Instruments 58 TGA (TA Instruments, Crawley, UK) connected to a gas flow system to work in inert 59 atmosphere equipment under 100 mL min-1. Samples sizes of ~20 mg were placed in a 60 platinum crucible and heated at 300 ºC min-1 to the final temperature in order to avoid mass 61 loss before the steady state is attained. 62 63 3. Theory 64 It is well known that the kinetics of a solid state reaction, in conditions far from the 65 equilibrium, can be described by the general equation: 66 𝑑𝛼 𝑑𝑡 =𝑘𝑓(𝛼) (1), 67 4 where t represents the time, α is the extent of reaction and k is the rate constant, which depends 68 on the temperature according to the Arrhenius equation, k=Aexp(-Ea/RT). The kinetic model, 69 f(α), is a function describing the relationship between the reaction rate and the reacted fraction. 70 An ample selection of f(α) functions have been published along the last decades, ranging from 71 the widely employed first or nth order laws to more sophisticated diffusion or nucleation models 72 32-34. It is important to remark that Eq. (1) does not consider any particular heating schedule so 73 it should be fulfilled whatever the time-temperature program employed for obtaining the 74 experimental data. Additionally, the reacted fraction or conversion α can be established using 75 data extracted with any technique measuring a property that can be directly related to the 76 reaction rate, most usually the mass loss recorded by thermogravimetry. The standard 77 isothermal method of kinetic analysis follows a model-fitting approach27. Thus, a series of 78 isotherms are recorded at different temperatures and the extracted experimental data are fitted 79 to a set of different kinetic models, according to the following equations, which are obtained by 80 integrating Eq (1): 81 ∫𝑑𝛼 𝑓(𝛼) 𝛼 0= ∫𝑘𝑑𝑡 𝑡 0 (2a) 82 g(α) = kt (2b), 83 where g(α) is the integral form of the kinetic model. The plot of g(α)versus the reaction time 84 (provided that the time for reaching the temperature steady state is negligible with regards to 85 the elapsed time) leads to a straight line whose slope is the rate constant, k. Then, given the 86 Arrhenius dependence of the rate constant with the temperature, the activation energy can be 87 subsequently calculated by plotting the logarithm of the rate constants versus the reverse of 88 their corresponding temperatures: 89 lnk = lnA-Ea/RT (3) 90 5 Let us assume the experimental kinetic data are fitted with a G(α) function different from the 91 one really obeyed by the reaction, g(α), regardless of the quality of the resulting regression 92 coefficient. In such case, an apparent rate constant, ka, would be obtained from the plot of G(α) 93 as a function of t according to the following equation: 94 G(α) = kat (4), 95 The comparison of Eqs (2b) and (4) leads to the conclusion that the acceptance of a reasonable 96 linear correlation between G(α) and t necessarily implies to accept a linear correlation between 97 G(α) and g(α), that would be expressed according to the following equation: 98 G(α) = ag(α) +b (5), 99 a and b being constants. 100 It can be concluded from Eqs. (2b), (4) and (5) that, whatever would be the temperature, the 101 apparent constant reaction rate, ka is related with the actual one, k, through the following 102 relationships: 103 𝑑𝐺(𝛼) 𝑑𝑡 = 𝑘𝑎= 𝑎 𝑑𝑔(𝛼) 𝑑𝑡 =𝑎𝑘 (6). 104 Eq. (6) shows that ka = ak. Moreover, taking into account that k fits the Arrhenius equation, it 105 follows: 106 𝑘𝑎= 𝑎𝑘 =𝑎𝐴𝑒−𝐸/𝑅𝑇 (7), 107 that could be rearranged in the following form: 108 −𝑑𝑙𝑛𝑘𝑎 𝑑(1/𝑇) = −𝑑𝑙𝑛𝑘 𝑑(1/𝑇) = 𝐸/𝑅 (8) 109 6 Thus, Eq. (8) clearly demonstrates that the activation energy determined from a set of 110 isothermal α-time plots obtained at different temperatures is independent of the kinetic model 111 previously assumed for performing the kinetic analysis. 112 113 4. Kinetic analysis of simulated isothermal curves 114 The time at which a given α value is reached at a certain temperature T can be determined, 115 according to Eq. (2b), from the following expression: 116 𝑡 = 𝑔(𝛼) 𝐴𝑒−𝐸 𝑅𝑇 ⁄ (9) 117 Thus, Eq. (9) was used to numerically construct two sets of simulated curves assuming 118 temperatures of 250, 265, 275 and 300 ºC. Figure 1a includes a set of isotherms simulated 119 according to the following kinetic parameters: E=200 kJ/mol, A=1016 min-1 and a first order 120 kinetic model (F1). On the other hand, isotherms in Figure 1b were constructed assuming the 121 same activation energy and pre-exponential factor as the previous set and an Avrami nucleation 122 kinetic model (A2). The figure clearly shows how the first order trace is convex while the 123 nucleation-driven trace is sigmoidal, as aforementioned. Then, the data from all the curves in 124 Figure 1 were linearly fitted to several theoretical kinetic models, as per Eq (2b), producing the 125 rate constant values listed in Tables 1 and 2, together with their corresponding correlation 126 coefficients. In order to better replicate the analysis procedures most commonly employed in 127 the literature, the fit was limited to data comprising the conversion range 0.1≤ α ≤0.9 since the 128 extreme ranges are more sensitive to experimental errors. Figure 2 includes a selection of plots 129 constructed using data from the 265 ºC isothermal curve, providing a clear picture of the results 130 obtained. The linear fits from which slope the rate constants are determined are also marked in 131 the Figure. As expected, only the correct g(α) function produces a flawless linear fit whereas 132 7 the fit to incorrect models yield non-linear plots with different slopes and, therefore, lead to 133 different rate constants. Nevertheless, it is worth mentioning that 5 out of the 8 models tested 134 yielded significant correlation coefficients, over 0.99. Even more striking are the conclusions 135 reached when the activation energy is determined for each set of rate constants (Tables 1 and 2) 136 as per Eq. (3). Table 3 lists the activation energies and pre-exponential factors obtained, 137 depending on the model used to construct the simulated curves and the model used to fit the 138 data. Results are identical for every case, with the analysis yielding the correct activation 139 energy regardless the model assumed. Thus, the erroneous selection of a kinetic law, and 140 consequently, the erroneous estimation of the rate constants have no influence whatsoever on 141 the obtained activation energy. On the other hand, the pre-exponential factors present a slight 142 variability as it is expected from Eq. (7), although it is still well within the accepted error range 143 given the high numerical value of such constant. It should be warned that any prediction 144 attempt requires the knowledge of correct kinetic model driving the proccess. Thus, for a given 145 temperature, each kinetic triplet in Table 3 will produce a different α-time curve, with only the 146 right model been able to accurately predict the experimental curve. Nevertheless, it is still 147 possible to make reliable predictions from the model-independent E values obtained by the 148 isothermal method by employing Vyazovkin’s isoconversional equation.35 149 150 5. Kinetic analysis of experimental isothermal curves 151 Next, for further confirmation, real experimental data are tested. Figure 3 includes three 152 isothermal curves, corresponding to the decomposition of polytetrafluoroethylene (Aldrich, 153 product number 182478) recorded at temperatures of 480, 490 and 500 ºC in a Q5000IR TA 154 Instruments TGA equipment under a 100 mL min-1 flow of N2. The plots of g(α) versus the 155 reaction time, as per Eq (2b), for each isothermal curve were built using eight different kinetic 156 models. A selection of these plots is shown as examples in Figure 4. The rate constants, as 157 8 directly calculated from the slope of the plots, are included in Table 4, as well as the activation 158 energies determined using each set of rate constants, as previously described. Additionally, the 159 plots of the rate constants versus the reverse of the temperature (as per Eq. (3)), from which the 160 activation energies are calculated, are included in Figure 5. These results indicate that, as 161 predicted by the mathematical development presented in a previous section, the same value for 162 the activation energy, 261±1 kJ/mol, is reached whatever the kinetic model used to fit the data. 163 The activation energy here obtained is in agreement with that estimated in a previous study 164 using the same material and employing a combined approach based on bothmodel-fitting and 165 isoconversional methods, thus confirming the validity of the results.36 Additionally, as it 166 happened with the simulated curves, four out of eight models can fit the experimental data with 167 reasonable correlation coefficients. 168 This finding entails significant implications. The fact that the correct activation energy of a 169 single process would always be obtained from a set of isotherms regardless of the kinetic model 170 chosen to fit the data permits the isothermal method of kinetic analysis to provide the activation 171 energy of any single step reaction without needing any previous knowledge regarding the 172 reaction mechanism. In any case, heat and mass transfer limitations typical of isothermal 173 experiments must still be considered in the experiments design since they will inevitably 174 produce interferences with the real proccess if not adequately minimized. The ability of the 175 method to yield the correct activation energy regardless of the model used is especially 176 interesting if we consider that most reactions will rarely follow faithfully any theoretical 177 models, which were built upon several ideal assumptions and constraints which are seldom 178 fulfilled in real reactions. For instance, inhomogenous distribution in size and particle shape 179 have been shown to have an important effect on the shape of the experimental curves.37 Such 180 deviations would have consequences in model-fitting methods of kinetic analysis of linear 181 heating rate experiments because the activation energy and the pre-exponential factors provided 182 9 are heavily dependent on the kinetic model used for the fitting. 31, 38, 39 On the other hand, as 183 shown here, the activation energy and pre-exponential factors provided by the isothermal 184 method are independent of the model used to fit the experimental data, hence preventing the 185 distorting effect of non-ideal models or inhomogenous materials. 186 187 6. Conclusions 188 It has been proven that the actual activation energy of any single step solid state reaction can 189 be determined from a set of isothermal experiments regardless of the kinetic model of function 190 obeyed by the reaction and/or the kinetic equation previously assumed for performing the 191 kinetic analysis. Thus, the isothermal method of kinetic analysis behaves at all effects as a 192 model-free since the activation energy and the pre-exponential factor can be calculated without 193 a previous knowledge of the kinetic model. Moreover, the kinetic parameters thus obtained 194 would be representative of the reaction even when none of the theoretical models could closely 195 represent the studied process. 196 ACKNOWLEDGEMENTS 197 Financial support from projects TEP-7858 and TEP-1900 from Junta de Andalucía, CTQ2011- 198 27626 from the Spanish Ministerio de Economía y Competitividad and FEDER funds is 199 acknowledged. Additionally, one of the authors (PESJ) is supported by a Juan de la Cierva 200 grant. 201 202 References 203 1. Grasa, G.; Martinez, I.; Diego, M. E., et al., Energ Fuel. 2014, 28 (6), 4033-4042. 204 2. Wang, C.; Dou, B. L.; Song, Y. C., et al., Energ Fuel. 2014, 28 (6), 3793-3801. 205 16 302 303 Figure 4. Fit of the PTFE degradation isothermal curves in Figure 3 to four 304 different kinetic models: first order (F1), pase boundary controlled (R2), two- 305 dimensional diffusion (D2) and nucleation (A2). 306 307 17 308 309 310 Figure 5. Plot of lnk versus the reverse of their corresponding temperature, 311 constructed for every set of rate constants in Table 4. Activation energy is 312 extracted from the slope of the plots, being identical in every case. 313 314 315 316 317 318 319 18 Table 1. Rate constants and correlation factors obtained from fitting simulated curves in Figure 320 1a (assuming a F1 model) to different kinetic models, according to Eq. (2a). 321 322 250ºC 265ºC 275ºC 300ºC Model Fitted to k (min-1) r2 k (min-1) r2 k (min-1) r2 k (min-1) r2 F1 0.0062 1.000 0.0224 1.000 0.0507 1.000 0.3441 1.000 R2 0.0038 0.981 0.0136 0.980 0.0309 0.980 0.2097 0.980 R3 0.0044 0.991 0.0160 0.991 0.0362 0.991 0.2456 0.991 A2 0.0034 0.970 0.0122 0.970 0.0275 0.970 0.1868 0.970 A3 0.0024 0.944 0.0085 0.944 0.0192 0.944 0.1305 0.944 D2 0.0020 0.996 0.0073 0.996 0.0165 0.996 0.1124 0.996 D4 0.0005 0.993 0.0020 0.993 0.0044 0.993 0.0300 0.993 L2 0.0145 0.996 0.0524 0.996 0.1186 0.996 0.8060 0.996 323 324 Table 2. Rate constants and correlation factors obtained from fitting the simulated curves in 325 Figure 1b (assuming a A2 model) to different kinetic models, according to Eq. (2a). 326 327 250ºC 265ºC 275ºC 300ºC Model Fitted to k (min-1) r2 k (min-1) r2 k (min-1) r2 k (min-1) r2 F1 0.0111 0.970 0.0400 0.970 0.0905 0.970 0.6147 0.970 R2 0.0069 0.997 0.0250 0.997 0.0565 0.997 0.3836 0.997 R3 0.0080 0.992 0.0290 0.992 0.0656 0.992 0.4458 0.992 A2 0.0062 1.000 0.0224 1.000 0.0507 1.000 0.3441 1.000 A3 0.0044 0.996 0.0158 0.996 0.0358 0.996 0.2430 0.996 D2 0.0036 0.953 0.0130 0.953 0.0293 0.953 0.1994 0.953 D4 0.0010 0.936 0.0034 0.936 0.0078 0.936 0.0529 0.936 L2 0.0263 0.988 0.0947 0.988 0.2143 0.988 1.4563 0.988 328 329 330 331 19 Table 3. Activation energies, pre-exponential factors and correlation coefficients obtained from 332 the plot of the different rate constants in Tables 1 and 2 versus the reverse of their 333 corresponding temperature, as per Eq.(3). 334 Model used to simulate the curves F1 A2 Model fitted to E (kJ mol-1) A (min-1) r2 E (kJ mol-1) A (min-1) r2 F1 200 5x1016 1.000 200 1x1017 1.000 R2 200 5x1016 1.000 200 6x1016 1.000 R3 200 4x1016 1.000 200 8x1016 1.000 A2 200 3x1016 1.000 200 6x1016 1.000 A3 200 2x1016 1.000 200 4x1016 1.000 D2 200 2x1016 1.000 200 4x1016 1.000 D4 200 5x1015 1.000 200 1x1016 1.000 L2 200 1x1017 1.000 200 3x1017 1.000 335 336 Table 4. Rate constants and correlation factors calculated from the fit of the experimental 337 curves corresponding to the degradation of PTFE (Figure 3) to a set of theoretical kinetic 338 models. The activation energy and correlation coefficients obtained from the plot of the rate 339 constants versus the reverse of the temperature (Eq.(3)) are also included. 340 341 480ºC 490ºC 500ºC Result of Isothermal analysis Model Fitted to k (min-1) r2 k (min-1) r2 k (min-1) r2 E (kJ mol-1) r2 F1 0.0047 1.000 0.0092 1.000 0.0164 0.999 261±10 0.997 R2 0.0026 0.978 0.0052 0.979 0.0093 0.982 261±11 0.996 R3 0.0032 0.990 0.0062 0.990 0.0111 0.993 261±11 0.996 A2 0.0023 0.973 0.0046 0.974 0.0082 0.977 260±11 0.996 A3 0.0016 0.951 0.0031 0.952 0.0055 0.955 261±12 0.996 D2 0.0015 0.997 0.0030 0.998 0.0054 0.998 262±11 0.996 D4 0.0004 0.997 0.0008 0.997 0.0015 0.997 260±10 0.997 L2 0.0106 0.996 0.0208 0.997 0.0373 0.997 261±10 0.996 20 342 343 344