Simulation and non-linear optimization of kinetic models for solid-state processes

Numerical simulations and optimizations methods are increasingly used in the field of kinetic analysis of solid-state processes, such as the crystallization of glassy materials. The influence of the simulations accuracy (with the two main factors being the initial value of conversion rate and the density of points) on the kinetic distortions was tested for the major solid-state kinetic models: nucleation-growth Johnson–Mehl–Avrami model, nth order reaction model, autocatalyzed nth order reaction model, diffusion models, contracting cylinder and contracting sphere models. The simulations were performed using a self-developed software based on the LSODA initial-value-problem-solver; the evaluation of the changes in the shape of the kinetic peaks was done using a commercial software that utilizes a standardized multivariate kinetic analysis approach. The accuracy was found to be influenced mainly by initial value of conversion rate. For majority of the tested kinetic models, the simulation accuracy had negligible effect on the consequently determined values activation energy, pre-exponential factor, integrated area of the kinetic peaks, or the asymmetry-determining values of the models kinetic exponents. Significant influence of the simulation accuracy was observed for the models with active autocatalytic features, which were identified to be the main source of the deviations introduced and propagated through the simulation algorithm. Contrary to the previous research, the deviations of the simulated peaks shape cannot be associated solely with the positive asymmetry of the kinetic peaks.


Introduction
Solid-state processes include thousands of chemical reactions and physico-chemical transformations.Kinetics of these processes is of major interest due to the associated possibility to predict the behavior and stability of the related materials under various conditions, including e.g. the long-term storage or exposure to harsh processing conditions.Papers dealing with the kinetics of solid-state processes are thus frequently published not only in thermo-analytical journals [1][2][3][4][5][6][7][8][9][10], but also in journals focused on materials science [11][12][13][14][15], pharmacy [16][17][18][19][20], plastics [21][22][23][24][25], explosives [26][27][28][29][30] etc.In correspondence, the methodology for the kinetic calculations is constantly developed-reviews on the recent advancements in this field can be found e.g. in the papers [31,32] produced by the ICTAC Kinetic Committee.Whereas much attention is paid to the simpler methods, such as the model-free linearization [33][34][35][36] and isoconversional [37][38][39][40][41] methods, the model-based non-linear optimization methods (such as e.g. the multivariate kinetic analysis (MKA) [42]) are considered to be fully developed and very little attention is paid to their performance or further advancement.This is surprising because with the continuous increase of computational power it is these nonlinear optimization methods that represent the future of kinetic analysis.Imagine the software, where with a single click the raw kinetic data are processed (tangential area-proportional baseline [43,44] is subtracted), the model-free kinetic methods are applied to determine the temperature (T) and degree of conversion (α) dependences of the overall process activation energy (E), the modified single-curve MKA [45,46] is applied iteratively to reveal the process complexity, the most suitable reaction mechanism and the corresponding reaction models are determined (based on the best correlation coefficient and consistency) and the data are quantitatively described, and the temperature or heating rate dependences of kinetic data are utilized to construct the parametric matrix for accurate prediction of the material kinetic behavior far outside the measured range of data.Partial accomplishments towards such goal are already available-see e.g.[45][46][47][48][49][50].
Whereas several online projects exist, allowing simpler versions of the solid-state kinetics determination, the commercial softwares properly dealing with this topic are very rare and generally inaccessible for broader audience due to the high price (Thermokinetics Professional by Netzsch is the most relevant software in this regard).This leaves majority of the scientists that want to implement advanced kinetic analysis to program their own softwares and calculations, the intricacies of which are, however, then not generally accessible and may lead to confusion when comparing kinetics results evaluated via multiple self-programmed sources.Note that the programming details and accuracy of the involved simulations are not known even for the commercial and online tools of the kinetic analysis.The importance of the studies of the simulation and non-linear optimization methods was recently introduced in the paper [51] dealing with the influence of the simulation conditions on the shape and parameters of the kinetic peaks modeled according to the autocatalytic Šesták-Berggren (AC) equation [43].It was found that the impact of the simulation conditions largely increases with the asymmetry of the kinetic peaks being shifted towards positive values (corresponding to the peaks being skewed to higher T or α).In the present paper a similar analysis will be performed for all major kinetic models used in the solid-state kinetic analysis in order to generalize the findings and derive the universal conclusions.Main goals of the present paper are as follows: (1) To determine the influence of the accuracy of the kinetic simulations on the results of the consequent kinetic analysis based on these data.(2) To demonstrate the main pitfalls of the non-linear optimization usage in solid-state kinetic analysis, and to identify the particular kinetic models (and/or the responsible qualities/mathematics) most prone to the simulation-induced distortions.(3) To emphasize the importance of stating the accuracy of the solid-state kinetic data simulations (these are used in absolute majority of the non-linear optimization methods, on which the advanced kinetic approaches are based).

Theoretical simulations of the kinetic data-curves
Theoretical simulations of the kinetic data were done using the LSODA function [52], a part of the 'deSolve' package [53].This function provides an interface for the Fortran ordinary differential equation (ODE) solver [54].It also includes fixed and adaptive time-step explicit Runge-Kutta solvers and the Euler method [55], and the implicit Runge-Kutta method RADAU [56].The package deals with differential equations implemented in lower-level languages such as FORTRAN, C, or C++, which are compiled into a dynamically linked library and loaded into R [57].All the details are explained in [53]; the simulations were performed using double precision.The LSODA function switches automatically between stiff and nonstiff methods-the stiffness characteristics of the initial value problem (IVP) does not have to be determined by the user, the solver will automatically choose the appropriate method (always starting with the nonstiff method).The LSODA function utilizes the following linear multistep methods: Adams methods (interpolation of the past dy/dx values) for solving the nonstiff problems and the backward differentiation formulae (interpolation of the past y values) for solving the stiff problems.Our simulation software is described in [58] (links to repositories and free download therein) and extensively tested for the solid-state kinetic simulations e.g. in [51,[59][60][61].Regarding the precision and accuracy of the simulation method, we have not identified any problems since these methods are well-known in the literature, controlled and rechecked, and implemented for several years.The simulated solid-state kinetics was based on the standard equation [43]: where α stands for the degree of conversion, t and T are time and temperature, respectively, I stands for the integrated area of the kinetic peak, A is the pre-exponential factor, E denotes the apparent activation energy of the kinetic process and f (α) corresponds to a kinetic model.In the present paper, the following kinetic models [31,62] will be simulated: the catalyzed nth order reaction (equation (2); denominated C n , C s and C p for the expressions with negative, zero and positive asymmetries, respectively), Johnson-Mehl-Avrami model (equation (3); denominated A1, A2 and A3 for the expressions with the kinetic exponents equal to 1, 2 and 3, respectively), nth order reaction (equation (4); denominated F0.5, F1.0, F1.5 and F2 for the expressions with the kinetic exponents equal to 0.5, 1, 1.5 and 2, respectively), one-dimensional diffusion model (equation (5); denominated D1), two-dimensional diffusion model (equation ( 6); denominated D2), three-dimensional diffusion model (equation (7); denominated D3), Ginstling-Brounstein diffusion (equation ( 8); denominated D4), contracting cylinder model (equation ( 9); denominated R2), contracting sphere model (equation (10); denominated R3), Note that the first three mentioned models contain kinetic exponents N, K, m and n, which further diversify the shape of the kinetic function and provide flexibility to the given model.

Introduction of the kinetic models
Numerical simulations of kinetic curves corresponding to different solid-state models were performed to evaluate the influence of the simulation parameters and conditions (number of points, initial dα•dt −1 value) on the shape of the kinetic data.Throughout the manuscript, an overall term 'simulation accuracy' will be used from now on to denote these conditions.Since the study aims to comprise majority of the solid-state kinetics types for the universal conclusions to be derived, the selected kinetic models were simulated with variable kinetic exponents to represent the commonly encountered cases of kinetic data.On the other hand, the nature of the intended conclusions does not require the model-free kinetic parameters (E, A, I-see equation ( 1)) to be varied, because these parameters do not influence the kinetic peak asymmetry (only its general magnitude and position on the T axis).Only a single set of these three parameters was used for each model; the list of the kinetic parameters used for the different f (α) models is included in table 1.
The theoretical simulations were performed in accordance with a typical setup of the solidstate kinetics measurement, i.e. for a set of heating rates q + = 0.5, 1, 2, 5, 10 and 20 Example sets of the kinetic peaks modeled for the presently tested f (α) functions are shown in figures 1-3.
The nucleation-growth kinetics (models A1-A3; see figure 1) represents one of the most typical and commonly encountered shapes of the kinetic peaks, characteristic for crystallization processes and slower types of physico-chemical reactions and transformations in general.This kinetics is known for its fingerprint, a slightly negative asymmetry that changes only very slightly with different values of the kinetic exponent m JMA [63,64].The catalyzed nth order reaction (models C n , C s and C p ; see figure 1) represents and equal alternative for the probably Table 1.Kinetic parameters used to simulate datasets for the following kinetic models: autocatalyzed nth order reaction C, Johnson-Mehl-Avrami A, nth order reaction F, onedimensional diffusion D1, two-dimensional diffusion D2, three-dimensional diffusion D3, Ginstling-Brounstein diffusion D4, contracting cylinder R2, contracting sphere R3.The four sections of the table correspond to the models with different (or none in the last case) kinetic exponents.best known solid-state kinetic model, the semi-empirical autocatalytic AC model [43].As was shown e.g. in [65], the C x and AC models can, owing to their immense flexibility, equally well describe absolute majority of the solid-state kinetic processes.For the present testing, three types of the C x model asymmetry were chosen (by planned selection of the N and K kinetic exponents) to cover the negative, zero and positive peak asymmetries.The C x model is also the only model in the present study that exhibits autrocatalytic features (K parameter in equation ( 2); this information will be crucial later during the evaluations of the simulations performance).Qualitatively akin kinetic behavior exhibits also the simple nth order model (F0.5-F2)-see figure 2. The F model exhibits a span of negative-to-positive asymmetries without the autocatalytic term being present.However, the span in asymmetries is not that large and the onset peak tail has always a similarly slowly-increasing slope (even for the peak with overall positive asymmetry).The example datasets for the diffusion (D1-D4) and contracting (R2 and R3) models are depicted in figure 3-these models represent the kinetic behavior with highly negative asymmetries, without any exponents further regulating the kinetic response.

Variables in the kinetic simulations and determination of their impact
The accuracy of the kinetic modeling (done via theoretical numerical simulations based on the LSODA method) was tested based on the variable density of points (10 3 -10 5 ) and variable initial dα•dt −1 value for the simulations (10 −2 -10 −12 ); the internal tolerance of the LSODA method was set to 10 −6 .Note that the term 'simulation accuracy' will be used to refer to these conditions.For each kinetic model, 10 datasets of kinetic curves (similar to those depicted in  4)) kinetic model.The horizontal arrows indicate the shift of the simulated peaks with increasing q + .figures 1-3), were simulated for various combinations of the above-mentioned variables.The identification of the particular datasets is listed in table 2.
Example of the consequences of the kinetic modeling accuracy on the shape of the kinetic peaks is shown in figure 4 (where the right-side column depicts left-side graphs zoomed-in on the peaks maxima)-the catalyzed nth order reaction was chosen as a most suitable demonstration as it can adopt all types of asymmetries.The negatively asymmetric C n peaks show a significant difference only for the set no. 1.There is a negligible (below the common level of data noise in thermo-analytical techniques, such as the differential scanning calorimetry, in which the present type of the derivative dα•dt −1 signal is most often encountered) but visually detectable difference also for the sets nos. 2 and 3, but all other peaks (sets nos.4-10) perfectly overlap.For the symmetric C s peaks, the set no. 1 is already largely different, showing not only a decrease in height but also a significant skewing to a more positive asymmetry.Sets nos. 2 and 3 are still significantly lower in their magnitude, rest of the sets perfectly overlaps.Contrary to the two previous cases, the positively asymmetric C p kinetic peaks are massively dependent on the simulation conditions (accuracy) of the simulation algorithm.In particular, at low initial dα•dt −1 , the peaks are shifted to lower T; they are also wider and lower in their absolute magnitude.Only from the set no. 5 onwards, the kinetic behavior unifies and standardizes.
The quantifiable evaluation of the apparent changes in the shape of the kinetic peaks with the numerical simulation variables was done by means of a standardized (done by a benchmark  5)-( 8)) and R (contracting models, equations ( 9) and ( 10)) kinetic models.The horizontal arrows indicate the shift of the simulated peaks with increasing q + .software Thermokinetics 3.1) method of MKA [42], i.e. a simultaneous non-linear optimization of all 6 curves (simulated for different q + within each dataset): where RSS is the residual sum of squares, n is number of simulated curves, j is index of the given simulated curve, First j is the index of the first point of the given curve, Last j is the index of the last point of the given curve, Yexp j,k is the experimental value of the point k of curve j ('experimental' denotes the original data simulated by the LSODA method), Ycal j,k is the calculated (via the benchmark software programmed for MKA) value of the point k of curve j and w j is weighing factor for curve j.The maxima in equation ( 12) then correspond to the lowest and highest Y value for the given data-curve (for the present simulated data, minimum = 0, and maximum = value corresponding to the peak maximum).This expression for the weighing factor allows comparing all kinetic peaks (with potentially changing height as a consequence of the T-dependent kinetics) within the given simulated series with similar impact on the resulting RSS (residual sum of squares).Using this approach (equations ( 11) and ( 12)), the theoretically simulated curves (examples of which are shown in figures 1-4) were for all models evaluated via a standardized (regarding its simulation accuracy and programming algorithm) method, which allows to compare the deviations not only within the framework of each model but also model-to-model.The trends in values of the kinetic parameters E, A, I, and kinetic exponents (n, N, K, m JMA ) determined by the non-linear optimization based on equations ( 11) and ( 12) will be introduced and discussed in the following section.

Evaluation of the simulations accuracy
The standardized MKA method was used to quantify the differences between kinetic parameters (E, A, I, kinetic exponents) of theoretically simulated kinetic peaks as a result of the variable simulation accuracy (number of points per curve, diminutiveness of the initial dα•dt −1 value).For the purpose of discussion, the acceptable errors (considering that the theoretical simulations are evaluated) are arbitrarily set to 1% in the E and I quantities, and 0.1 (absolute value) for logA and the kinetic exponents (n, m JMA , N, logK).These values are based on the ∼20 years' experience of the corresponding author with the measurement, simulation, evaluation and prediction of the thermo-analytical dα•dt −1 signals.Example of the MKA output is shown in figure 5 for the A1 kinetic model.Note that the curve-fitting procedure of the MKA software was not restricted in any way, nor were any parameters values forced towards certain values.Hence the deviations against the 'correct' values (marked by dashed red lines) reflect the deviations caused by the simulation accuracy, rounding up, quality and accuracy of finding the absolute minimum of RSS, and potential differences in the ODE solvers employed for the simulation and evaluation tasks.The latter three are however responsible only for negligible/minor deviations in the <0.1% order of magnitude.In figure 5, these are the reasons why the E, logA and m values do not have to fall onto the red dashed line (indicating the true values input into the simulations-as perceived by the benchmark software, the accuracy of which lies somewhere in-between our sets 4 and 5).As is apparent from figure 5, even the least accurate numerical simulations of the A1 model lead to E, logA and m deviations ⩽0.1% and even small increase in the accuracy further reduces these deviations by an order of magnitude.The only noteworthy deviation is that of ∼1% for the integrated area under the peak evaluated for set no. 1.However, already for set no. 2 this deviation also decreases to ∼0.1%.
Since there are 16 kinetic models (and 61 graphs akin to the 4 depicted in figure 5), only those of interest, where the deviations are higher than ∼0.1% or are of general interest will be commented on in the following text.All 61 graphs can be found in the supplemental online material.Starting with the JMA model, the results for A2 and A3 are similar to those for A1 with regard to E and I. Slightly higher deviation were found for the pre-exponential factor A and then most importantly for the kinetic exponent m-see figure 6.
It is however noteworthy that the trends in logA and corresponding m sort of mirror each other, which indicates an interdependence with the potential for canceling out the deviations consequences caused by each single kinetic parameter alone (this will be proven and further commented on later in this section).This stresses the universal requirement to report on all kinetic parameters in the kinetic studies (which is by far not the case in the present literature), because e.g.reporting the present trends for A3 alone might lead to a belief that the accuracy of the kinetic simulations has significantly greater impact than it truly has.The change of the A3 kinetic peaks shape is indeed non-negligible, however, it has to be born in mind that with increasing kinetic exponent m (A1 → A2 → A3) smaller and smaller deviations are needed to change m by the same amount.
The second model to be commented on is the catalyzed nth order reaction (C n , C s , C p )-see figure 7. The deviations of these models associated with the simulation accuracy are qualitatively similar to those reported for the A2 model: negligible changes of E and I; logK being mirrored/compensated by logA; and, in addition, negligible changes of the exponent N (which represents the apparent reaction order).While the lowest simulation accuracy leads to ∼4% and 9% logK errors for the C n and C s , respectively, the magnitude of this error is ∼60% for the C p model.Interestingly, the reaction order exponent N is barely affected and does not exhibit any trends with the peak asymmetry.These findings can further be confronted with the results for the simple nth order model-see figure 8.
Also here, the reaction order exponent n is practically not affected by the simulation accuracy-similarly to all other kinetic parameters (E, A, I) of this model.Note that at n = 2, the corresponding kinetic peaks are clearly (although not largely) positively asymmetric.These findings are in a striking contradiction to the earlier reported results [51] for the AC model [43], where both its kinetic exponents varied markedly in the case of the positively asymmetric kinetic peaks-the exponent M AC responsible for the autocatalytic behavior and the exponent N AC attributable to the apparent reaction order.This indicates an important difference in the mutual interconnection of the kinetic exponents and the logA parameter.

Impact of the simulations accuracy
The main aim of the kinetic analysis and advanced kinetics description of the solid-state processes is the ability to predict their course under extrapolated conditions.Hence, similar evaluation criterion was chosen in the present case to consider the severity of the consequences of the simulations accuracy.Based on the results summarized in figures 5-8 (and included in the supplemental online material), only the A2, A3, C n , C s and C p models exhibit significant changes of the kinetic parameters directly associated with the quality of the performed theoretical simulations.We have thus calculated for these models the kinetic predictions for a simple long-term isothermal annealing at 50 • C (100 • C for C p ) using the standardized MKA software.The predicted time dependence of the degree of conversion α is for the A models depicted in figure 9.These predictions illustrate the propagation of the errors in the simulations associated with the particular asymmetries and kinetic equations.The cases displayed in these figures were simulated for the worst-case scenarios (simulation-induced distortions), which explains the apparent inaccuracies.The data also demonstrate that relatively minor differences in the results of the kinetic analysis can often transfer into severe consequences for the extrapolated kinetic predictions (which is the main use of the kinetic analysis in practice).In the case of both A models in figure 9, the lowest accuracy of the theoretical simulations (set no. 1) results in non-negligible but very small difference.Such deviation is on par with the commonly obtained experimental errors and would be suitable not only for judging on the long-term stability of the given materials but also for direct control of the process based on these data (e.g.controlled crystal growth in non-crystalline materials during the formation of glass-ceramics).Note the more accurate theoretical simulations of the JMA kinetics have shown 10 times lower deviations compared to the set no. 1.In addition, the graphs also depict the sole effect of incorrect simulation (or determination) of the apparent activation energy E. The ∆E ≈ 0.1%, which is the case of the present data, results in negligible deviations from the 'true' kinetics.Even for the ∆E ≈ 1%, the errors in the kinetic predictions would be acceptable and comparable to the imprecisions associated with the real-life experimental data evaluated by means of the nowadays state-of-the-art methods [45,46].The second series of kinetic predictions was done for the autocatalyzed nth order reaction model-see figure 10.For the C n and C s models, the least accurate simulations translate into negligible errors/differences in the predicted α.On the other hand, the accuracy of the kinetic simulations makes a massive difference for the positively asymmetric C p model-the kinetic description based on the set no. 1 evaluation led to a prediction that is in its initial stage off by practically 80+ % (note that this is solely the effect of the difference in the simulation algorithm).Interestingly, the result for the highest tested accuracy (set no.10) is indeed practically indistinguishable from the prediction based on the 'true' kinetics, although the data in figure 7 might suggest otherwise-this is the evidence of the compensation effect in the theoretical simulations.
Whereas the standard customary approach to the evaluation of the modeling outcomes is for the field of the solid-state analysis represented by the determination of the accuracy of the kinetic predictions, few real-life data examples will be referred to in the following text to demonstrate the relevant thermo-analytical measurement and evaluation precisions/accuracies.One of the highest precisions and reproducibilities of the thermo-analytical measurements are obtained for the polymeric materials.In [66], a complete thermo-kinetic characterization of the amorphous poly(p-dioxanone) was introduced with the following results: errors of the determination of the activation energies of structural relaxation, different types of crystallization, and depolymerization were 1.7%, 0.7/0.9/2.3%, and 2.9%, respectively; crystallization enthalpies were reproducible to ∼0.3%; variability of the kinetic exponents was interpreted with 0.05 precision (in absolute values) in mind.In case of the low-molecular glass indomethacin (amorphous anti-inflammatory drug) [48], the errors in determined activation energies were 2.0% for structural relaxation, 2%-6% for the crystallization of different powders, and 3.5% for thermal decomposition; the errors of crystallization enthalpies varied between 3% and 10%; the kinetic exponents of the corresponding models varied with 0.03-0.07errors (in absolute scale) and their larger changes were already interpreted as temperature-dependent trend in kinetic behavior.For the dehydration of the mixed calcium oxalate hydrates [45], the following errors in the kinetic analysis were reported: the activation energies for the two dehydration steps were determined with errors of 2.6% and 2.2%; the two dehydration enthalpies were determined with 7.3% and 4.7% errors; the kinetic exponents were reproducible with the standard deviations 0.02-0.06(absolute values) and their changes > 0.8 were recognized as trends in the temperature-dependent kinetics.The above-mentioned data demonstrate the relevancy of the simulation-induced changes in the kinetic description of the present data.The variability in accuracy of the present simulations is indeed on par with the precision of the real-life experimental data and the highest observed changes (obtained e.g. for sets no. 1, or for the A3, C s and C p models) would largely influence the corresponding kinetic analyses.

Conclusions
The influence of the accuracy of the solid-state kinetics numerical simulations was tested using the LSODA IVP-solver from the R-package deSolve-the main variables were the density of points and the initial dα•dt −1 value.The tested kinetic models were: JMA, autocatalyzed nth order reaction, simple nth order reaction, diffusion and contracting models.The accuracy was found to be influenced mainly by the initial dα•dt −1 value.Under the least accurate simulation conditions (easily surpassable even using basic programming/editing tools like MS Excel), the activation energy errors were ∆E ≈ 0.1% and errors for the integrated area of the kinetic peaks was ∆I ≈ 1%, which both are still negligible.For the simple nth order reaction, diffusion and contracting models, also the pre-exponential factor A (and n in the case of the F models) changed only negligibly, making these models 'immune' to the negative consequences of low simulation accuracy.For the JMA kinetics, the deviations in A and the kinetic exponent m JMA caused by the inaccurate simulations were occurring only for the cases with higher kinetic exponent m, where even small nuances of the peak shape result in seemingly significant changes of these kinetic parameters.However, the effects of changing A and m compensate each other with regard to the consequent kinetic predictions.This makes these models also 'immune' to the negative effects of the inaccurate kinetic modeling as long as all kinetic parameters are reported and re-simulated simultaneously, in their more-or-less deviated form.The autocatalyzed nth order model in the form of C p (positively asymmetric kinetics) was found to be most influenceable by the accuracy of the numerical simulations.The least accurate simulations led to a large distortion of the predicted kinetics, making such simulations incompensably erroneous as a direct result of the inaccurate kinetic calculations.From the comparison of the distortions observed for the C x and F models, it was concluded that the identification of the kinetic models most prone to the inaccurate theoretical simulations cannot be based on the asymmetry of the kinetic peaks but it has to be based on the influence of the autocatalytic term (represented by a certain exponent within the model equation).It is therefore imperative to perform the kinetic simulations and optimization of the models with active autocatalytic features using similar software (or at least a software of a similar simulation accuracy).

Figure 1 .
Figure 1.Example datasets differential kinetic data simulated for the A (JMA model, equation (3)) and Cx (catalyzed nth order reaction model, equation (2)) kinetic models.The horizontal arrows indicate the shift of the simulated peaks with increasing q + .

Figure 2 .
Figure 2. Example datasets differential kinetic data simulated for the F (nth order reaction, equation (4)) kinetic model.The horizontal arrows indicate the shift of the simulated peaks with increasing q + .

Figure 3 .
Figure 3. Example datasets differential kinetic data simulated for the D (diffusion models, equations (5)-(8)) and R (contracting models, equations (9) and (10)) kinetic models.The horizontal arrows indicate the shift of the simulated peaks with increasing q + .

Figure 4 .
Figure 4. Changes of the shape of the Cx (catalyzed nth order reaction model, equation (2)) kinetic peaks as a direct consequence of the simulation accuracy (datacurves corresponding to the sets nos.1-6 and q + = 20 • C•min −1 ).The right-side column depicts left-side graphs zoomed-in on the peaks maxima.Due to the perfect overlap of sets with higher simulation accuracy (sets.nos.7-10) with the set no. 6, these are not included in the figures.

Figure 7 .
Figure 7. Kinetic parameters evaluated by the standardized MKA method for the Cx (catalyzed nth order reaction model, equation (2)) peaks simulated under different simulation accuracy conditions (sets nos.1-10).

Figure 8 .
Figure 8. Kinetic parameters evaluated by the standardized MKA method for the F (nth order reaction, equation (4)) peaks simulated under different simulation accuracy conditions (sets nos.1-10).

Figure 9 .
Figure 9. Kinetic predictions simulated (using the standardized MKA-based software) according equation (1) for the isothermal annealing at 50 • C and A2/A3 (JMA model with kinetic exponents m JMA = 2 and 3, equation (3)) models.Bases for the predictions were: input kinetic parameters listed in table 1 denoted as 'precise model'; resulting kinetic parameters for the set no. 1; input kinetic parameters with E increased by 0.1%; input kinetic parameters with E increased by 1%.

Figure 10 .
Figure 10.Kinetic predictions simulated (using the standardized MKA-based software) according equation (1) for the isothermal annealing at 50 • C and Cx models (catalyzed nth order reaction model, equation (2)).Bases for the predictions were: input kinetic parameters listed in table 1 denoted as 'precise model'; resulting kinetic parameters for the set no. 1 (and set no. 10 in the case of Cp). -

Table 2 .
Variables used for the theoretical simulations of the kinetic datasets for the present solid-state models.