Application of non-linear regression methods for the estimation of common kinetic parameters from several thermoanalytical curves

Novel methods of unified evaluation of two (or more) thermogravimetric curves have been worked out on the basis of known non-linear parameter estimating procedures (Gauss-Newton-Marquardt-type regression and the direct integral method of Valkó and Vajda were adapted). Their ability to provide estimate for common kinetic parameters of several TG (m?T) or DTG (dm/dt-T) curves were tested for pairs of curves of different heating rates, and for repeated curves of the same heating rate, obtained for the decomposition of CaCO₃ in open crucible. In these cases the Arrhenius terms and then-th order model functions were assumed. The fitting ability of estimations made for single curves and for pairs of curves sharing different number of parameters, was judged on the base of residual deviations (S _res ) and compared to the standard deviation of the measurements. In the case of different heating rates, the two curves could not be described with the assumption of three common parameters, because of the minimum residual deviation was very high. However, sharing of activation energy and preexponential term only, and applying different exponents for the two curves, provided a satisfactory fit by our methods. Whilst in the case of repeated curves, we could find such a common three-parameter set, which has a residual deviation comparable with the standard deviation of the measurements. Because of their flexibility (taking into account the variable number of common parameters and the versatile forms of model equations), these methods seem to be promising means for unified evaluation of several related thermoanalytical curves.     

Ozawa’s kinetic method for analyzing thermoanalytical curves

By reviewing the history of thermal analysis and its application to the kinetic analysis of the solid-state processes, we investigate the theoretical basis and historical perspective of Ozawa’s kinetic method for analyzing thermoanalytical curves. Ozawa’s nonisothermal kinetic method is demonstrated using thermoanalytical data for the thermal decomposition of sodium hydrogencarbonate and the crystallization of anhydrous magnesium acetate glass as examples. Through investigating recent theoretical advancements in nonisothermal kinetic analysis in view of the theoretical fundamentals of Ozawa’s kinetic method, it is indicated that they are in line with Ozawa’s kinetic theory. On the basis of the above investigations, we discuss the role of Ozawa’s kinetic theory in advancing the analysis of complex reaction kinetics.     

Analysis of complex thermoanalytical curves: The thermodynamic and kinetic parameters of isopropylammonium nitrate

A method is presented to determine thermodynamic and kinetic parameters from complex thermoanalytical curves. Such curves are obtained when thermoanalytical events like phase transition and chemical decomposition overlap.Isopropylammonium nitrate was taken as an example to demonstrate how these parameters were determined from non-isothermal TG and DSC curves by constructing DSC-DTG sum curves.

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Zusammenfassung Es wird ein Verfahren zur Bestimmung thermodynamischer und kinetischer Parameter von komplexen thermoanalytischen Kurven beschrieben. Kurven dieser Art entstehen bei der Überlappung thermoanalytischer Ereignisse wie z.B. Phasenumwandlungen und chenliche Zersetzungen.Am Beispiel von Isopropylammoniumnitrat wird demonstriert, wie die einzelnen Parameter der nicht-isothermen TG und DSC Kurven durch Konstruktion von DSC-DTG-Summenkurven ermittelt werden.

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Problems of the characterization of thermoanalytical processes by kinetic parameters

On the basis of many experiments and theoretical reflections, the authors showed earlier that the courses of non-isothermal analytical curves are strongly influenced by the experimental conditions, and therefore the sense of kinetic parameters calculated from these curves is fictitious and their determination is uncertain.In the present work some further problems of this question are discussed. It was found that with combinations of strongly differing parameters nearly identical TG curves can be produced, and this situation cannot be improved even by orthogonal polynomial transformation. Further integral methods, using linearization, the estimation of the parameters is poor.The model transformed according to the conditions of the quasi-isothermal-quasiisobaric technique leads to contradictions, unambiguously showing the correlation existing between the parameters.

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Zusammenfassung Aufgrund zahlreicher Versuche und theoretischer Erwägungen hatten die Autoren früher gezeigt, daß der Verlauf nicht isothermer analytischer Kurven stark von den Versuchsbedingungen beeinflußt wird und deshalb die Bedeutung der aufgrund dieser Kurven berechneten kinetischen Parameter fiktiv und ihre Bestimmung unsicher ist.In der gegenwärtigen Arbeit werden einige weitere Probleme dieser Frage erörtert. Die Autoren fanden, daß mit Hilfe von Kombinationen stark verschiedener Parameter nahezu identische TG-Kurven hergestellt werden können und daß dieser Tatsache selbst durch orthogonale Polynomtransformation nicht abgeholfen werden kann. Weiter wurde gefunden, daß mit Hilfe von Integralmethoden unter Anwendung von Linearisierung erhaltene Schätzungen der Parameter ungenau sind.Das den Bedingungen der quasi-isothermen—quasi-isobaren Technik entsprechend transformierte Modell führt zu Widersprüchen, welche die zwischen den Parametern bestehenden Beziehungen eindeutig zeigen.

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Résumé A partir de nombreuses expériences et réflexions théoriques, les auteurs avaient démontré auparavant que l'allure des courbes TG non isothermes était fortement influencée par les conditions d'expérience. Pour cette raison, la signification des paramètres cinétiques calculés à partir de ces courbes était fictive et leur détermination incertaine.Dans le présent travail les auteurs discutent quelques autres problèmes sur cette question. En se servant de combinaisons de paramètres fortement différents on peut produire des courbes TG presque identiques et cette situation ne peut pas être améliorée même par transformation polynomiale orthogonale. En outre, on a trouvé qu'à l'aide de méthodes par intégration et par linéarisation, l'estimation des paramètres est mauvaise.Le modèle transformé selon les conditions de la technique quasi isotherme — quasi isobare entraîne des contradictions qui indiquent, sans équivoque, les corrélations existant entre les paramètres.

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The authors wish to thank Prof. E. Pungor for valuable remarks in this work.     

On the method of bae for the determination of kinetic parameters from DTA curves

The mathematical error in the method proposed by Bae for the determination of kinetic parameters from DTA curves has been corrected. The proposed equation does not contain thermal constants of the apparatus, and can be applied to DTA curves by an iterative method. The results obtained by the application of this equation to experimental DTA curves for the decomposition of sodium bicarbonate compared well with those from isothermal measurements, even when the DTA sample holder assembly was of the isolated cup-type instead of the block-type assembly recommended by Bae.     

Description of the shape of thermoanalytical curves : Part 3. A Method for estimating kinetic constants from parameters characterizing peak shape

The relationships between empirical parameters characterizing the shape of thermo-analytical curves and the constants of the kinetic equation, dα/dt = A exp (?E/RT)(1 ? α)ⁿ, are studied. A procedure is developed for the estimation of the three constants of this equation from the empirical parameters. The efficiency of this method is compared to that of model fitting. In evaluation of the confidence intervals of estimated constants and in the case of identification, least-squares (or similar) fitting is shown to be inferior because of its low sensitivity to properties other than the position of the peak along the temperature axis. This lack of sensitivity may be a major cause of the apparent kinetic compensation effect often encountered in the field of thermal analysis.     

Description of the shape of thermoanalytical curves

Empirical parameters were chosen for the characterization of peaks (or steps) of thermoanalytical curves. The parameters were applied in studies on the repeatability, the relationship between kinetic constants and peak shape, the effect of sample thermal resistance. Kinetic constants can be estimated on the basis of peak shape parameters. Besides, approximate criteria were formulated for experiments allowing kinetic evaluation with the neglect of the heat transport within the sample. The empirical parameters were also used in checking the suitability of DSC data for purity analysis and in detecting changes of the thermal decomposition of papers caused by ageing.Zur Beschreibung von Peaks bzw. Stufen thermoanalytischer Kurven wurden emperische Parameter ermittelt, die bei der Untersuchung der Reproduziertbarkeit, der Beziehung zwischen kinetischen Konstanten und Peakform so wie des Einflusses des thermischen Wiederstandes der Probe angewendet wurden. Kinetische Konstanten können auf der Basis von Peakformparametern geschÄtzt werden. Weiterhin wurden Kriterien für Experimente ermittelt, die eine kinetische NÄherungslösung unter VernachlÄssigung des WÄrmetransportes innerhalb der Probe ermöglichen. Die empirischen Parameter wurden au\erdem bei der Untersuchung der Eignung von DSC-daten zur Reinheitsanalyse angewendet, weiterhin zum Nachweis von alterungsbedingten Änderungen der thermischen Zerzetzung von Papier.

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The authors express their thanks to Mrs E. Tóth for her valuable technical help. The permission to reproduce Figs 2–6 and 8–10 from Analytica Chimica Acta and Thermochimica Acta (Elsevier, Amsterdam) is also acknowledged.     

Precision of integral methods for the determination of the kinetic parameters

In this paper, a systematic analysis of the errors involved in the determination of the kinetic parameters (including the activation energy and frequency factor) from five integral methods has been carried out. The integral methods analyzed here are Coats-Redfern, Gorbachev, Wanjun-Yuwen-Hen-Zhiyong-Cunxin, Junmeng-Fusheng-Weiming-Fang, Junmeng-Fang and Junmeng-Fang-Weiming-Fusheng method. The results have shown that the precision of the kinetic parameters calculated by the different integral methods is dependent on u (E/RT), that is, on the activation energy and the average temperature of the process.     

Critical investigation of methods for kinetic analysis of thermoanalytical data

Journal of Thermal Analysis and Calorimetry -     

Computer-determined kinetic parameters from TG curves. Part IX

By applying the restriction that α (conversion) = constant to non-isothermal TG (NITG), isothermal TG (ITG) and a combination of both (NITG/ITG), various corresponding expressions have been derived for the evaluation of the activation energy, E. By means of a computer, all the expressions developed were tested against data which was generated from theoretical equations. Also, changes were made in the number of significant figures (s.f.) of data in order to ascertain their effects on values of E. Data were generated and tested for the three theoretical mechanisms, A₂, F₁ and R₂.