Optimizing fitting parameters in thermogravimetry

This study presents an alternative to simple estimation of parametric fitting models used in thermal analysis. The addressed problem consists in performing an alternative optimization method to fit thermal analysis curves, specifically TG curves and their first derivatives. This proposal consists in estimating the optimal parameters corresponding to fitting kinetic models applied to thermogravimetric (TG) curves, using evolutionary algorithms: differential evolution (DE), simulated annealing and covariance matrix adapting evolutionary strategy. This procedure does not need to include a vector with the initial values of the parameters, as is currently required. Despite their potential benefits, the application of these methods is by no means usual in the context of thermal analysis curve’s estimation. Simulated TG curves are obtained and fitted using a generalized logistic mixture model, where each logistic component represents a thermal degradation process. The simulation of TG curves in four different scenarios taking into account the extent of processes overlapping allows us to evaluate the final results and thus to validate the proposed procedure: two degradation processes non-overlapped simulated using two generalized logistics, two processes overlapped, four processes non-overlapped and four processes overlapped two by two. The mean square error function is chosen as objective function and the above algorithms have been applied separately and together, i.e., taking the final solution of the DE algorithm is the initial solution of the remaining. The results show that the evolutionary algorithms provide a good solution for adjusting simulated TG curves, better than that provided by traditional methods.     

Kinetic parameters for solid-phase polycondensation of L-aspartic acid: Comparison of thermal gravimetric analysis and differential scanning calorimetry data

The computer simulation of the kinetic scheme advanced for the thermostimulated polycondensation of aspartic acid in the solid phase is performed. The scheme includes the autocatalytic growth of a polymeric chain followed by polymer dehydration. This process occurs simultaneously in two matrix zones with identical reaction products but different rate constants. Arrhenius parameters for each of the four chemical stages are defined via solution of the inverse kinetic problem. The functions obtained in the simulation almost coincide with kinetic curves obtained for three regimes of two thermochemical research methods: thermogravimetric analysis and differential scanning calorimetry. The parameters thus obtained can be described by the compensation-effect curve with a linear dependence between the logarithm of the pre-exponential factor and the activation energy. It was proposed that the bimolecular chain-growth reaction in one of the zones proceeds by the monomolecular mechanism.     

Thermal characterization of ammonium alum

Thermal decomposition of ammonium alum was studied by simultaneous thermogravimetry (TG)-differential scanning calorimetry (DSC) attached to a Fourier transform infrared (FTIR) spectrometer, so that each mass loss was related with the simultaneous endo- or exothermal behavior and to the FTIR absorption produced by the evolved gases. Apart from some clear dehydration and desulfation processes, other overlapping peaks were observed by DSC, TG, and FTIR. Optimal fitting to logistic mixture models was performed to separate the overlapping processes. Deconvolution of overlapping DTG peaks resulted in single constituent peaks, which were related to plots of some specific FTIR bands along time. Thus, a more accurate insight of the chemical processes taking place was obtained.     

Analysis of S2QA- charge recombination with the Arrhenius, Eyring and Marcus theories

The Q band of photosynthetic thermoluminescence, measured in the presence of a herbicide that blocks electron transfer from PSII, is associated with recombination of the S(2)Q(A)(-) charge pair. The same charge recombination reaction can be monitored with chlorophyll fluorescence. It has been shown that the recombination occurs via three competing routes of which one produces luminescence. In the present study, we measured the thermoluminescence Q band and the decay of chlorophyll fluorescence yield after a single turnover flash at different temperatures from spinach thylakoids. The data were analyzed using the commonly used Arrhenius theory, the Eyring rate theory and the Marcus theory of electron transfer. The fitting error was minimized for both thermoluminescence and fluorescence by adjusting the global, phenomenological constants obtained when the reaction rate theories were applied to the multi-step recombination reaction. For chlorophyll fluorescence, all three theories give decent fits. The peak position of the thermoluminescence Q band is correct by all theories but the form of the Q band is somewhat different in curves predicted by the three theories. The Eyring and Marcus theories give good fits for the decreasing part of the thermoluminescence curve and Marcus theory gives the closest fit for the rising part.     

Comprehensive kinetic studies on isothermal and non-isothermal reactions of some aluminium compounds

Residual differences after model fitting were investigated in both isothermal and non-isothermal kinetics in order to make numerical comparisons between several models and various parameter-estimating methods. Data from two independent experimental series were evaluated. A large data set, collected earlier under isothermal conditions from decompositions and hydrothermal reactions of aluminium hydroxides and oxides, was processed first. It showed that mechanical activation of the starting gibbsite affected reactivity of samples in several subsequent reactions for all model equations tried. The relative residual deviation concept is introduced, and statistics were applied to find a model that fits a certain reaction in most of the cases. In the second study, the sulphate decomposition step of aluminium sulphate octadecahydrate was investigated. TG curves were measured using a constant heating rate. Dynamic models were fitted by three mathematical methods, including a new general purpose one. Fitting ability of the methods with various complexity were compared on the basis of residual deviations obtained after integration of the model equations. As well as evaluating the best fit, this new parameter-estimating method provides a statistical analysis of the reliability of the whole model fitting process.     

Aluminum sulphate hydrates

Three different calculation methods of deriving kinetic parameters (activation energy and preexponential factor) from dynamic TG data have been applied for the sulphate decomposition stage of the aluminum sulphate octadecahydrate. The constant rate experiments were carried out by Derivatograph and DuPont thermobalances. The three parameters estimation methods included a simple differential method, the classical Coats-Redfern and a new direct integral method. The fits of the curves obtained by these procedures were compared both graphically and numerically. It was found that the direct integral method gave the most satisfactory results. With the order type reaction models this method in each case produced the smallest residual deviation values and the best fitting curves compared to those obtained by the other two methods. The activation parameters calculated by the differential method were not acceptable at all, for the estimated curves were very far from the measured ones.     

Weibull mixture model for modeling nonisothermal kinetics of thermally stimulated solid-state reactions: application to simulated and real kinetic conversion data

The possibility of applying Weibull mixture model for the fitting of the nonisothermal kinetic conversion data has been investigated. It has been found that the kinetic conversion data at different heating rates can be successfully described by one or the linear combination of few Weibull distribution functions. Several simulated and real kinetic conversion traces have been analyzed. An optimal fitting of the kinetic conversion data has been obtained by a mixture of Weibull distribution functions. The results obtained have shown that the obtained conversion curves calculated by the model proposed in this paper are in agreement with the raw kinetic conversion data.     

Determination of kinetic parameters from TG curves by non-linear optimization

The results obtained so far by kinetic analysis of non-isothermal experiments indicate that the kinetic parameters found by the conventional methods, in general, do not describe the experimental curve in an optimum manner. This is due to the fact that the initial differential equation is transformed into the logarithmic and, consequently, linear form and that the initial and final weights of the conversion curve cannot be determined exactly, which may falsify the slope of the curve.Investigations have shown that the determination of the kinetic parameters by non-linear optimization (simplex method) results in a better fit of the theoretical conversion curve to the experimental one. But this procedure gives optimum results only when the initial and final weights of the reaction can be determined exactly. If this is impossible, exact parameters can be obtained only by the use of the non-standardized TG curve.Examples are cited to prove that it is possible to evaluate overlapping reactions by the formation of intervals. However, the evaluation of conversion curves merely by the use of mathematical methods can easily result in an erroneous interpretation of the reaction course investigated. Therefore, it is necessary to check the mathematical results as to their physical and chemical meaning.     

Compatibility and decomposition kinetics studies of prednicarbate alone and associated with glyceryl stearate

In this work, TG/DTG and DSC techniques were used to the determination of thermal behavior of prednicarbate alone and associated with glyceryl stearate excipient (1:1 physical mixture). TG/DTG curves obtained for the binary mixture showed a reduction of approximately 37 °C to the thermal stability of drug ( T_{\textdm/\textdt = 0 \textDTG}^\textMax T_{{{\text{d}}m/{\text{d}}t = 0\,{\text{DTG}}}}^{\text{Max}} ). The disappearance of stretching band at 1280 cm⁻¹ (ν_as C–O, carbonate group) and the presence of streching band with less intensity at 1750 cm⁻¹ (ν_s C–O, ester group) in IR spectrum obtained to the binary mixture submitted at 220 °C, when compared with IR spectrum of drug submitted to the same temperature, confirmed the chemical interaction between these substances due to heating. Kinetics parameters of decomposition reaction of prednicarbate were obtained using isothermal (Arrhenius equation) and non-isothermal (Ozawa) methods. The reduction of approximately 45% of activation energy value (E _a) to the first step of thermal decomposition reaction of drug in the 1:1 (mass/mass) physical mixture was observed by both kinetics methods.     

霍林河褐煤热解甲烷生成反应类型及动力学的热重-质谱实验与量子化学计算

应用开放体系的热重-质谱联用技术(TG/MS)对未成熟的霍林河褐煤(镜质组最大反射率ROmax为0.33%)进行了热解模拟, 获得了甲烷的在线析出速率曲线. 曲线拟合结果表明, 甲烷生成速率曲线可以分解为5个峰, 结合化学动力学分析, 发现低温阶段甲烷的峰为吸附甲烷析出峰, 其余四个峰为热解甲烷的生成峰, 代表四类不同的化学反应. 将煤的结构特征及量子化学的理论计算相结合, 认为甲烷的生成包括4类反应, 类型1为与氧等杂原子相连的脂碳断裂; 类型2包含了两种反应, 一种是短链脂肪烃类官能团β位断裂, 另一种是所形成的长链脂肪烃类物质二次裂解; 类型3是与芳核直接相连的甲基热解脱落; 类型4为存在于煤结构中脂肪类物质的芳香化.     

Functional nonparametric classification of wood species from thermal data

In this study, thermogravimetric (TG) and differential scanning calorimetry (DSC) curves, obtained by means of a simultaneous TG/DSC analyzer, and statistical functional nonparametric methods are used to classify different wood species. The temperature ranges, where the highest probability of correct classification is reached, are also computed. As each observation is a curve, a nonparametric functional discriminant technique based on the Bayes rule and the Nadaraya–Watson regression estimator is used. It assigns a future observation to the highest probability predefined class (supervised classification). The smoothing parameter needed in this nonparametric method is selected according to the cross-validation technique. The method proposed is applied to a sample of 49 wood items (7 per wood class) and also to classify between hardwoods and softwoods. In all the cases, the samples have been successfully classified, obtaining better results with the TG curves. The results are compared with those obtained with other nonparametric methods based on boosting algorithm. A discussion about the relation of the obtained results with the referenced wood component degradation temperature ranks is presented.     

Simulation study for generalized logistic function in thermal data modeling

The principal aim of the present study is to describe, analyze, and compare from a statistical standpoint the generalized logistic model with some well-known models used in the solid-state kinetics: power law, Avrami–Erofeev, and reaction order. For this purpose, synthetic conversion curves that simulate the kinetic processes were generated using the power law, Avrami–Erofeev, and reaction order models, where the Arrhenius equation was assumed in all the cases. This comprehensive simulation study allows to describe the relationship between the parameters belonging to the proposed generalized logistic model and the pointed traditional models’ parameters, and also to validate the performance of the generalized logistic model in a wide variety of cases where other methods can be applied. Performing this analysis has been necessary to employ some new statistical techniques in thermal analysis modeling as the generalized additive models, and to perform global optimization evolutionary algorithms as the differential evolution for solving the non-linear regression problem. In order to implement these techniques, R statistical software routines were developed and applied.     

Thermal analysis of sulfurization of polyacrylonitrile with elemental sulfur

Thermal analysis of sulfurization of polyacrylonitrile (PAN) with elemental sulfur was investigated by thermogravimetry and differential thermal analysis of the mixture of polyacrylonitrile and elemental sulfur up to 600°C. Due to the volatilization of sulfur, the different heating rate (10 and 20 K min⁻¹) and different mixture proportion of polyacrylonitrile and elemental sulfur were adopted to run the analysis. The different heating rates make the DSC curves of sulfur different, but make the DSC curves of PAN similar. In the DSC curve of sulfur for the heating rate of 20 K min⁻¹ around 400°C, a small exothermic peak occurs at 400°C in the wide endothermic peak around 380∼420°C, indicative of that there is an exothermic reaction around 400°C. In the DSC curves of the mixture, the peaks around 320°C are exothermic as the content of sulfur is below 3.5:1 and endothermic as the content of sulfur is over 4:1, indicating that one of the reactions between PAN and sulfur takes place around 320°C. In the TG curves of the mixture, the mass losses begin at 220°C, and sharply drop down from 280°C. The curves for the low sulfur content obviously show two steps of mass loss, and curves for the high sulfur content show only one step of mass loss, indicative of more sulfur is benefit for the complete sulfurization of PAN. This study demonstrates that the TG/DSC analysis can give the parameter for the sulfurization, even if the starting mixture contains the volatile sulfur.     

Studies on ignition and afterburning processes of KClO4/Mg pyrotechnics heated in air

Thermal behavior of KClO₄/Mg pyrotechnic mixtures heated in air was investigated by thermal analysis. Effects of oxygen balance and heating rates on the TG?CDSC curves of mixtures were examined. Results showed that DSC curves of the mixtures had two exothermic processes when heated from room temperature to 700?°C, and TG curve exhibited a slight mass gain followed by a two-stage mass fall and then a significant mass increase. The exothermic peak at lower temperature and higher temperature corresponded to the ignition process and afterburning process, respectively. Under the heating rate of 10?°C?min^?1, the peak temperatures for ignition and afterburning process of stoichiometric KClO₄/Mg (58.8/41.2) was 543 and 615?°C, respectively. When Mg content increased to 50%, the peak ignition temperature decreased to 530?°C, but the second exothermic peak changed little. Reaction kinetics of the two exothermic processes for the stoichiometric mixture was calculated using Kissinger method. Apparent activation energies for ignition and afterburning process were 153.6 and 289.5?kJ?mol^?1, respectively. A five-step reaction pathway was proposed for the ignition process in air, and activation energies for each step were also calculated. These results should provide reference for formula design and safety storage of KClO₄/Mg-containing pyrotechnics.     

Kinetic Investigation on the Dehydration of Coprecipitated Mixed Oxide Powders

The authors present an original kinetic model and a computer program in order to determine the mass of various types of water from the oxide powders obtained through co-precipitation, using thermogravimetric data. The model is based on kinetic equations in the framework of the ‘reaction order’ desorption of the water loss. The program minimizes the mean square deviation between the experimental TG curve and the approximation curve; it allows visualizing the experimental and the approximated TG curves for the overall process, as well as the approximated TG and DTG curves for the individual processes. The influence of the aging time and temperature on the mass of various types of water to be found in the co-precipitated manganese ferrite powder was investigated based on this original kinetic model and the suitable computer program. Some correlations between the water release and microstructure changes of the powder during aging in mother solution are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

EPDM accelerated sulfur vulcanization: a kinetic model based on a genetic algorithm

A simple closed form equation for the prediction of crosslinking of EPDM during accelerated sulfur vulcanization is presented. Such a closed form solution is derived from a second order non homogeneous differential equation, deduced from a kinetic model. The kinetic model is based on the assumption that, during vulcanization, a number of partial reactions occurs, both in series and in parallel, which determine the formation of intermediate compounds, including activated and matured polymer. Once written standard first order differential equations for each partial reaction, the differential equation system so obtained is rearranged and, after few considerations, a single second order non homogeneous differential equation with constant coefficients is derived, for which a solution may be found in closed form, provided that the non-homogeneous term is approximated with an exponential function. To estimate numerically the degree of crosslinking, kinetic model constants are evaluated through a simple data fitting, performed on experimental rheometer cure curves. The fitting procedure is a new one, and is achieved using an ad-hoc genetic algorithm, provided that a few points, strictly necessary to estimate model unknown constants with sufficient accuracy, are selected from the whole experimental cure curve. To assess the results obtained with the model proposed, a number of different compounds are analyzed, for which experimental or numerical data are available from the literature. The important cases of moderate and strong reversions are also considered, experiencing a convincing convergence of the analytical model proposed. For the single cases analyzed, partial reaction kinetic constants are also provided.     

Some methodological problems concerning nonisothermal kinetic analysis of heterogeneous solid–gas reactions

Isoconversional methods, those using only one curve α = α(T) (α is the conversion degree and T is the temperature), and invariant kinetic parameter method were applied to estimate the kinetic parameters from the following nonisothermal data: (1) simulated TG curves for a single reaction; (2) TG curves for thermal degradation of PVC; and (3) TG curves for the dehydration of CaC₂O₄·H₂O. The results obtained by applying various methods for the same system are compared and discussed. Finally, a procedure of kinetic analysis is suggested. Its application could lead to kinetic parameter values that can be used to predict either α = α(t) curves for other heating rates or α = α(T) curves for isothermal conditions. © 2001 John Wiley & Sons, Inc. Int J chem Kinet 33: 564–573, 2001     

Incompatible reaction for (3-4-epoxycyclohexane) methyl-3′-4′-epoxycyclohexyl-carboxylate (EEC) by calorimetric technology and theoretical kinetic model

(3-4-Epoxycyclohexane) methyl-3′-4′-epoxycyclohexyl-carboxylate (EEC) is a typical epoxy resin (EP). In Asia, due to the unstable reactive natures of EP, various thermal hazard and runaway reaction incidents have been occasioned by EP in the manufacturing process, such as fire, explosion, and toxic release, resulting in loss of life as well financial catastrophes and social outcries. Certain catalysis substances, H₂SO₄, acetic acid, or NaOH, may accelerate the reaction or curing rate for EP. However, an incompatible reaction with these chemical substances may induce a thermal hazard, causing a runaway excursion during the last stage. We employed thermogravimetry (TG) to obtain thermal stability parameters under non-isothermal conditions to evaluate the runaway reactions for EEC. The experimental data were compared with kinetics-based curve fitting to assess thermally hazardous phenomena by optimizing curve fitting on the kinetic parameters. The aim of this study was to estimate the incompatible hazards for EEC, provide thermal hazard information in order to determine the optimum operation conditions, and diminish the likelihood of fire and explosion accidents incurred by EP.     

A closed form solution for the vulcanization prediction of NR cured with sulphur and different accelerators

The paper presents a novel efficient closed form approach to determine the degree of vulcanization of natural rubber (NR) vulcanized with sulphur in presence of different accelerators. The general reaction scheme proposed by Han and co-workers for vulcanized sulphur NR is re-adapted and suitably modified taking into account the single contributions of the different accelerators, focusing in particular on some experimental data, where NR was vulcanized at different temperatures (from 150 to \(180\, ^{\circ }\hbox {C}\)) and concentrations of sulphur, using TBBS and DPG in the mixture as co-agents at variable concentrations. In the model, chain reactions initiated by the formation of macro-compounds responsible for the formation of the unmatured crosslinked polymer are accounted for. It is assumed that such reactions depend on the reciprocal concentrations of all components and their chemical nature. In presence of two accelerators, reactions are assumed to proceed in parallel, making the assumption that there is no interaction between the two accelerators. Despite there is experimental evidence that a weak process by which each accelerator affects the other, the reaction chemistry is still not well understood and therefore its effect cannot be translated into any mathematical model. In any case, even disregarding such interaction, good approximations of the rheometer curves are obtained. From the simplified kinetic scheme adopted, a closed form solution is found for the crosslink density, with the only limitation that the induction period is excluded from computations. The main capability of the model stands however in the closed form determination of kinetic constants representing the velocities of single reactions in the kinetic scheme adopted, which allows avoiding a numerically demanding least-squares best fitting on rheometer experimental data. Two series of experiments available, relying into rheometer curves at different temperatures and different concentrations of sulphur and accelerators, are utilized to evaluate the fitting capabilities of the mathematical model. Very good agreement between numerical output and experimental data is experienced in all cases analyzed.     

Effect of particle size on pyrolysis characteristics of Elbistan lignite

In this study, the relationship between particle size and pyrolysis characteristics of Elbistan lignite was examined by using the thermogravimetric (TG/DTG) and differential thermal analysis (DTA) techniques. Lignite samples were separated into different size fractions. Experiments were conducted at non-isothermal conditions with a heating rate of 10°C min⁻¹ under nitrogen atmosphere up to 900°C. Pyrolysis regions, maximum pyrolysis rates and characteristic peak temperatures were determined from TG/DTG curves. Thermogravimetric data were analyzed by a reaction rate model assuming first-order kinetics. Apparent activation energy (E) and Arrhenius constant (A _r) of pyrolysis reaction of each particle size fraction were evaluated by applying Arrhenius kinetic model. The apparent activation energies in the essential pyrolysis region were calculated as 27.36 and 28.81 kJ mol⁻¹ for the largest (−2360+2000 μm) and finest (−38 μm) particle sizes, respectively.