Determination of kinetic mechanisms for reactions measured with thermoanalytical instruments
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Abstract
The work considers the methods and techniques, allowing the assignment of the kinetic mechanisms to the chemical reactions evaluated from signals of thermoanalytical measurements. It describes which information about the kinetic mechanisms can be found from either modelfree or modelbased methods. The work considers the applicability of both methods and compares their results. The multiplestep reactions with wellseparated peaks can be equally analyzed by both methods, but for overlapping peaks or for simultaneously running parallel reactions the modelfree methods provide irrelevant results.
Keywords
Modelfree analysis Modelbased analysis Kinetics of multistep reactions NETZSCH ThermokineticsIntroduction
 1.
Find the degree of conversion for given temperature conditions, if the chemical mechanism of reaction is unknown and not really important,
 2.
Determine and describe the kinetic mechanism if the chemical mechanism of reaction is unknown or partially unknown.
The first task is a more technical task and usually could be solved experimentally if the measurement equipment allows to follow the desired temperature conditions. If the measurements cannot be done exactly according to the temperature conditions, then extrapolation is done by kinetic methods without of the detailed description of the chemical mechanisms of the process. This study deals only with the second task, where the kinetic mechanism must be detected.
The thermoanalytical measurements for kinetic analysis must have the measured signal as the function of the time and temperature and must include signal changes caused by the chemical processes in the sample. The common signals are DSC and TG, but other signal types also could be analyzed by kinetic methods.
There are two approaches to kinetic analysis of thermoanalytical data: modelfree analysis and modelbased analysis. Both approaches need several thermoanalytical measurements with different temperature conditions. Usually this is a set of measurements with different heating rates or a set of isothermal measurements with different temperatures. We will consider here the set of different heating rates, because it can be analyzed by all modelfree methods.
Modelfree analysis allows to find activation energy of the reaction process without the assumption of a kinetic model for the process. Usually the knowledge of the reaction type is also not necessary to find the activation energy by modelfree methods.
The second assumption for modelfree analysis: the reaction rate at a constant value of conversion is only a function of temperature [1, 2].
In modelfree analysis the thermoanalytical signal is equal to the reaction rate (1), multiplied by the total effect of reaction: total enthalpy for DSC or total mass loss for TG.
There are several different modelfree methods including Friedman analysis [2], Ozawa–Flynn–Wall analysis. They are wideused for various applications [3, 4, 5, 6, 7, 8], but all of them are based on the above described assumptions.
The approach of the modelbased kinetic analysis is based on other assumptions.
Each step has an own reaction type described by the function \( f_{j} \left( {e_{j} ,p_{j} } \right) \). Some examples of such functions are: secondorder reaction f = e ^{2}, Prout–Thompkins reaction with acceleration f = e ^{ m } p ^{ n }, reaction with onedimensional diffusion 0.5/p.
The second assumption for modelbased analysis: all kinetic parameters like activation energy, preexponential factor, order of reaction, and reaction type are assumed constant during the reaction progress for every individual reaction step.
The third assumption for modelbased analysis: the thermoanalytical signal is the sum of the signals of the single reaction steps. The effect of each step is calculated as the reaction rate, multiplied by the effect of this step like enthalpy change or mass loss.
For singlestep reactions, where the reaction mechanism does not change during the reaction [10], both, modelfree and modelbased approaches result in the same kinetic equation with the same kinetic parameters, which are constant or nearly constant during the reaction progress. Singlestep reactions are well studied in the literature, and therefore are omitted here.
For complex reactions, where the kinetic mechanism changes during the reaction, there is big difference in interpretation of kinetic results, obtained by different approaches. For modelfree approaches, the change of the kinetic mechanism is described by the continuous changing of the activation energy and the preexponential factor with the progress of the reaction. For modelbased approaches, the change of the kinetic mechanism is described by appearing of several reaction steps with own activation energy and with own reaction type.
The highest interest and complexity lies in the analysis of multistep processes, because of the ambiguity of applying different approaches and interpretation of results.
Multiplestep reactions
Usually for reactions with unknown reaction mechanism the number of reaction steps is also unknown. Sometimes several chemical reactions could be proposed from the chemical point of view, but the kinetic parameters of the reaction steps are unknown.
There are first questions which must be answered before a kinetic analysis: how many reaction steps are present in the measured process? How many steps can be analyzed? The answers to the first and second question can be quite different. Processes can chemically have several reaction steps, but corresponding to the thermoanalytical curve, can show only a single peak. In this case only one step, responsible for this peak, can be analyzed, and only for this peak kinetic parameters can be found correctly.
Example: in the chemical process with two consecutive steps (A → B → C) the first reaction step is slow enough to produce the peak on the thermoanalytical curve. If the second process is fast and the intermediate product B reacts immediately to form product C, then the concentration of B is always near to zero, and from a thermoanalytical point of view, the process looks like a singlestep process A → C. The analysis of these data provides the kinetic parameters like activation energy, preexponential factor and reaction order only for the first step, but the area of the DSC peak will have the meaning of the sum of enthalpies of both steps. The kinetic parameters for the first step can be found by both modelfree and modelbased methods. But it is impossible to find parameters for the second step from such measured data by any method, because the experimental data do not contain any kinetic information about the second step.
The kinetic parameters can be found only for those reaction steps, which are visible on the thermoanalytical curve as peaks or shoulders in DSC curves, or as steps in TG curves. Kinetic parameters cannot be found from thermoanalytical data for individual steps taking place during a reaction, without showing the corresponding peaks or part of peaks on the thermoanalytical curve.
Independent reactions
The kinetic model for this process includes severalindependent reaction parameters. The most common example is the process in the mixture of several materials, which react independently of each other. Let us consider the simplest situation, the mixture of two materials, where Peak1 (on DSC or DTG curve) means the reaction in material1, and Peak2—reaction in material2.
 (a)
The temperature of the Peak1 is lower than the temperature of the Peak2;
 (b)
Peak1 and Peak2 overlap at the same temperature range;
 (c)
the temperature of the Peak1 is higher than the temperature of the Peak2.
By the increasing of the heating rate the peaks are shifted to the higher temperatures, and the shift value is higher for lower activation energy. If the activation energies of two processes are not exactly the same, then by changing of heating rate, the distance between peaks is also changed. Therefore in one set of measurements with different heating rates one (a, b, or c), two (a + b or b + c) or all three possible situations could be present. We apply here the modelfree and modelbased analyses to the different data sets for the same process of two independent reactions and compare the results.
The steps are wellseparated for the low and for the high heating rates. Let us analyze separately the set of three curves at low heating rates, the set of three curves at high heating rates, and the total set of curves by the both modelfree and modelbased methods and then compare the results.
In Fig. 4 the data set and results for high heating rates are represented. Figure 4a with the formal concentrations of substances shows that now the Step2 is earlier than the Step1. The steps are still slightly overlapping, but for very high heating rates the overlapping disappears and peaks will be well separated. The modelbased analysis of the two independent steps provides the same results as for the set of low heating rates. The results of modelfree analysis are now not the same as the results for the set of low heating rates. The first part of reaction with α = 0.2 has the higher activation energy than the second part of reaction with α = 0.8. Now the predictions based on modelfree results can be done only for very high heating rates, where the steps are well separated. But for lower heating rates there is the overlapping of steps and modelfree predictions cannot get a result showing the independent character of steps. It is impossible to get the curves shown in Fig. 3b by the predictions based on activation energies from Fig. 4c.
Figure 2b shows the modelfree analysis for complete set of data for independent steps. The steps are separated only for low and high heating rates, but overlapped for the middle heating rates. Modelfree analysis produces very high error bars, now the activation energies for α = 0.2 and for α = 0.8 are the same. But the modelbased analysis based on two independent steps provides here the same results as for the two previous sets of data.
The comparison of the modelfree results for the process with independent steps indicates different dependencies of the activation energy on the degree of conversion for the set of high heating rates, for the set of low heating rates and for the complete set of data. It means that the modelfree result for overlapping independent steps depends on the heating rate and on the number of measured curves. But this fact is in conflict with the above mentioned second assumption of the modelfree analysis, where the reaction rate at a constant conversion must be only a function of temperature. If this assumption of the modelfree analysis cannot be fulfilled, then the modelfree analysis may not be used for the situation with overlapping independent steps.
The reason for the different modelfree results for the range of overlapping peaks can be found by the detailed consideration of applying this analysis to the total data set.
Three sets of data, a set with low heating rates, a set with high heating rates and a complete set of data were analyzed by modelfree and by modelbased analysis. The modelbased results are the same for all three sets. The modelfree results are different for all three sets.
In Fig. 5 the isoconversional lines are drawn separately for each independent step. It is seen that the independent steps go through each other. For low heating rates Step1 appears before Step2 (graphic must be read from right to left because of the reciprocal temperature), and for high heating rate Step1 appears after Step2. Stars mark the points with the same conversion value of 0.95, dashed line is drawn through them. It is clearly seen that the stars with low heating rates belong to the Step2, and are placed on the straight line with slope, corresponding to Step2. The stars with high heating rates belong to Step1 and are placed on the straight line with slope corresponding to Step1. Here the peaks go independently through each other. This fact could be used as the indicator of independent reaction steps.
The dashed line represents the attempt to use the standard modelfree analysis for the complete data set, where isoconversion line must be drawn through all points with the same conversion value (marked with stars). But the points belong to different reactions and therefore are not placed on one line. By using linear regression, a straight dashed line results, which is far from the marked points (stars), especially for the highest and for the lowest heating rates. This fact produces very high error bars in the energy plot. The found activation energy is not the activation energy of the first step, and not the activation energy of the second step, but some value between them. The extrapolation of this data set to the much higher or to the much lower additional heating rates will give more deviation from the drawn straight line and real isoconversional points. In other words, the modelfree predictions for such a situation of independent steps for both very low and very high heating rates will be even more far from the real process.
The order of peaks depends on the heating conditions. The sequential order of peak for dynamic measurements cannot be the same as the order of peaks for isothermal conditions. Therefore an erroneous simulation could happen if the modelfree analysis is done for the set of dynamic data with different heating rates, and then the prediction is done for isothermal conditions. Data for analysis must contain wide range of heating rates, even better including also isothermal measurements to have complete information about independent processes.
Modelfree methods cannot provide the correct values of activation energy for each of simultaneously runningindependent processes. It can be used with reasonable results only for processes, where peaks are well separated, show no overlapping for any heating rate, and where the order of peaks never changes.
 1.
Modelbased kinetic analysis:
(+) can be used for the processes with independent steps. It ensures the stable kinetic results independent on heating rates and overlapping of peaks;
(+) provides correct kinetic results for both wellseparated and overlapping peaks;
 2.
Modelfree kinetic analysis:
(−) provides contradictory kinetic results for the data of the same process with different sets of heating rates;
(+) provides correct kinetic results for separated peaks;
(−) provides incorrect kinetic results for data sets containing overlapping peaks;
(+) provides correct predictions for the temperature conditions, where no overlapping happens;
 3.
If the steps go through each other by changing of heating rate then the steps are independent.
Here the advantages of methods are marked with (+), and disadvantages—with (−).
Consecutive reactions
Let us consider the consecutive reactions, where each reaction step has the corresponding peak on the thermoanalytical curve.
The general kinetic model has the following view: A → B → C → D → …
In Fig. 6a the complete data set and analysis results for the process with consecutive steps are represented. The steps are separated only for low heating rates, but overlapping for low and high heating rates. Modelfree analysis in Fig. 6b produces very high error bars for activation energy values. But for this set of data the modelbased analysis with the kinetic model of two consecutive steps provides again the same results like for the set of data with wellseparated peaks for low heating rates.
For the three sets of data of the same process the modelfree analysis provides three different dependencies E _{a}(α). Let us find the reason for the different modelfree results in the Friedman plot.
If the peaks for consecutive reactions show overlapping, but are still wellvisible, then in the overlapping ranges of α two reaction steps take place simultaneously: previous step is still not finished, and the next step is already started. It corresponds to the state, where simultaneously two reactions take place. But by modelfree analysis only one value of activation energy can be found. For the state where peaks overlap the modelfree analysis provides only one intermediate value, which already does not correspond to the activation energy of the first step, and yet does not correspond to the activation energy of the second step. If really only one reaction runs at any time point in the states where no overlapping occurs, then modelfree analysis can provide a correct result (Fig. 7c).
The modelbased analysis provides the correct kinetic parameters for each reaction step for the data set containing both wellseparated and overlapping steps. If the overlapping is wide, then the parameters of nonlimiting step could not be found or could be found with less accuracy. For the consecutive reactions the order of steps is always the same and independent from heating rate.
The dashed line is drawn through the points with α = 0.95, marked with stars. These points belong to the different chemical reactions and to the different sets of isoconversion lines. And again, like for independent steps, the attempt to draw straight line through stars has no big success. The activation energy, found from the slope of this line, has the meaning of an intermediate value between activation energies of Step1 and Step2, and has very high error bars on the energy plot. The extrapolation of this data set to the higher or to the lower additional heating rates, will give high deviation of real isoconversional points from the drawn straight line. Therefore, the modelfree predictions for such a situation of consecutive steps with overlapping peaks will be very far from the real process.
 1.
Modelbased kinetic analysis:
(+) can be used for the processes with consecutive steps. It ensures the stable kinetic results independent on heating rates and overlapping of peaks;
(+) provides correct kinetic results for both wellseparated and overlapping peaks;
 2.
Modelfree kinetic analysis:
(−) provides contradictory kinetic results for the data of the same process with different sets of heating rates;
(+) provides correct kinetic results for separated peaks;
(−) provides incorrect kinetic results for data sets containing overlapping peaks;
 3.
If one of steps completely disappeared by changing of heating rate then the steps are consecutive.
Competitive reactions

A → B step 1

A → C step 2
If the activation energies of these steps are not the same then the increasing or decreasing of heating rate changes the ratio of the products B and C in the product mixture. If the activation energy of step1 is lower than the activation energy of step 2, then the decreasing of the heating rate moves the step1 to much lower temperatures and step1 becomes to be dominant in the model. Reaction will go mainly by A → B. If the measurements with only low heating rates are analyzed then only activation energy of the first step will be found by both modelfree and modelbased methods. Increasing of the heating rate forces to increase the branch A → C and to increase the amount of product C in the mixture. If the measurements with only high heating rates are analyzed, then only the second reaction takes place, and the activation energy only of the second step can be found. If the measurements with low and high heating rates are analyzed together then the modelfree analysis provides the intermediate value between activation energies of steps with low accuracy. Modelbased analysis provides both values correctly only if the contribution of the each competitive step is known.
The modelfree analysis for all three situations (Fig. 10c) provides nonconstant dependence of activation energy from conversion with high error bars. This is the indicator of multistep process. Moreover, the dependences E _{a}(α) are different for all three data sets. It means that the activation energy for the given conversion value depends not only on temperature, but on the heating rate. This fact is in the contradiction with the second assumption of the modelfree analysis. It means that the modelfree analysis cannot be used for the competitive steps.
The indicator of the presence of the competitive steps is the dependence of the total effect on the heating rate like in Fig. 10b. This dependence cannot be explained by the independent steps like in Fig. 1, because for independent steps the total effect of reaction is independent of heating rates and always the same. The dependence of the total effect on the heating rate also cannot be explained by the consecutive steps like in Fig. 6a, because for consecutive steps the total effect of reaction is always the same for any heating rate. The only explanation for such dependence is the presence of competitive steps with a mixture of final products where the contribution of each component depends on the heating rate. It could be formulated with other words: The dependence of the total effect of reaction on heating rates is the indicator of competitive steps.
 1.
Modelbased kinetic analysis:
(+) can be used for the processes with competitive steps. It can explain, describe and predict the dependence of total reaction effect on the heating rate;
(+) provides correct kinetic results for both wellseparated and overlapping peaks;
 2.
Modelfree kinetic analysis:
(−) provides contradictory kinetic results for the data sets of the same process with different heating rates;
(−) provides incorrect kinetic results for overlapped peaks;
(−) does not take into account the dependence of total reaction effect on the heating rate and therefore cannot provide correct predictions
Selection of the best model
Very often the several different solutions can be found for the same data set. And then the following questions come: What solution is correct? What solution could be used for the predictions?
 1.
Modelfree analysis provides the two plateaus in the plot of activation energy: E = 110 for α < 0.6 and E = 500 for α > 0.6.
 2.
Modelbased analysis provides model of two independent steps with E = 110 and E = 500 kJ/mol.
 3.
Modelbased analysis provides model of two consecutive steps with E = 110 and E = 500 kJ/mol.
Question: What solution could be used for the predictions?
If the predictions must be done from here for the much lower heating rates, then the reaction steps will remain wellseparated and Step1 will be always earlier than the Step2. All three solutions give the same predicted signal for wellseparated peaks, and each of them can be used.
Let us see, what happens if the predictions must be done to the much higher heating rates, where the big overlapping of steps happens. The dependence E _{a}. vs. α for three low heating rates is the same for both set with two independent steps and set with two consecutive steps. Predictions by modelfree analysis do not take into account the interaction of the steps, and produce the same predictions for both cases. This fact must be considered as the disadvantage of the modelfree method. The prediction according to modelfree analysis is very far from the predictions for the model with independent steps. It is clear that modelfree analysis may not be used for the predictions of process with even only two independent processes. It may also not be used for reactions with three or more parallel steps. But some authors [1] still believe that “obtaining E vs. alpha dependence is enough for kinetic predictions … further computations may not be necessary”.
Additionally they say, that “if there is no significant difference between two different mechanisms, then it means that both mechanisms provide the same goodness of fit”. Does it mean that any of suggested models can be taken for predictions? But the predictions for these models are not the same. The modelbased predictions for model with independent steps (Fig. 3b) differ dramatically from the predictions according to model with consecutive steps (Fig. 8b).
If the real mechanism consists of independent reactions, then the model with independent steps can describe the correct behavior of system for high heating rates. And only this model must be used for predictions. If the real mechanism has two consecutive steps, then the model with consecutive steps must be used. But how to recognize which of these two models is correct? If the steps are wellseparated then the data are the same (Figs 3b, 7b). If the data are the same then these data contain no information about the interaction of steps. The difference between data from these two models can be seen in the range of the overlapping of peaks and in the range of high heating rates. Therefore, we can say that these data contain no information about the mechanism of the process. The set of three low heating rates contains only information of the presence of two steps, but not any information about the interaction between them. It has no sense to select the most appropriate model here according to any mathematical criteria like F test or correlation coefficient, because the information about the interaction of steps is not present in the experimental data.

Get the information about mechanism from the chemical knowledge about the system. For example, if decomposition of one substance happens (like Calcium oxalate monohydrate), then it is most probably a reaction with consecutive steps. If originally it was the mixture of noninteracting materials, then probably it is the model with independent steps.

Add the set of experimental data with additional measurements containing information about the interaction of the steps. The good set of data must contain the measurements with wellseparated steps as well as the measurements with the big overlapping of steps. The measurements with wellseparated steps allow estimating accurately the kinetic parameter for each step, but have no information about the step interaction. The measurements with big overlapping of steps contain information about activation energy of each step with very low accuracy, but they allow to find the type of interaction between steps.
The present example of three curves shows not any problem for low heating rate predictions, but has not enough information for predictions for high heating rates. The inverse situation is mostly dangerous, where there is not enough data available for predictions at the low heating rates. Practically, it comes to such situations when only dynamic measurements are done, and not correct isothermal predictions are performed based on them.
Conclusions

If the steps go through each other independently by changing of heating rate then the steps are independent.

If one step completely disappears by changing of heating rates and not coming again by further changing heating rates then this step connected consequently to other steps.

If the value of total effect (mass loss or area) changes by changing of heating rates then the steps are competitive.

For all three situations the modelbased analysis can be used for both searching of kinetic parameters and predictions.

The modelfree analysis can be used for estimation E _{a}(α) only for the data sets with wellseparated steps.

Modelfree results of the previous item can be used only for the conditions where no overlapping happens.

All simulations and both modelfree and model based analysis in this work are performed by the software NETZSCH Thermokinetics 3.1.
Notes
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