Model-free and model-fitting approaches to kinetic analysis of isothermal and nonisothermal data
Introduction
Kinetic analysis of solid state decompositions is usually based on a single step kinetic equation [1]where t is the time, T is the temperature, α is the extent of conversion, and f(α) is the reaction model. The reaction model may take various forms, some of which are shown in Table 1. The explicit temperature dependence of the rate constant is introduced by replacing k(T) with the Arrhenius equation, which giveswhere A (the pre-exponential factor) and E (the activation energy) are the Arrhenius parameters and R is the gas constant. The Arrhenius parameters, together with the reaction model, are sometimes called the kinetic triplet. Under nonisothermal conditions in which a sample is heated at a constant rate, the explicit temporal dependence in Eq. (2) is eliminated through the trivial transformationwhere β = dT/dt is the heating rate.
Compared with isothermal experiments, nonisothermal runs are more convenient to carry out because it is not necessary to perform a sudden temperature jump of the sample at the beginning. However, Arrhenius parameters obtained from nonisothermal data are often reported to disagree with the values derived from isothermal experiments. In our opinion, there are two major reasons for this disagreement. The first is a result of the prevalent use of kinetic methods that involve force fitting of nonisothermal data to hypothetical reaction models. Following this “model-fitting approach”, Arrhenius parameters are determined by the form of f(α) assumed. Because in a nonisothermal experiment both T and α vary simultaneously, the model-fitting approach generally fails to achieve a clean separation between the temperature dependence, k(T), and the reaction model, f(α). As a result, almost any f(α) can satisfactorily fit data at the cost of drastic variations in the Arrhenius parameters, which compensate for the difference between the assumed form of f(α) and the true but unknown reaction model. For this reason, the model-fitting methods tend to produce highly uncertain values of Arrhenius parameters.
The second major reason for this disagreement arises from the fact that isothermal and nonisothermal experiments are necessarily conducted in different temperature regions. If decomposition involves several steps with different activation energies, the contributions of these steps to the overall decomposition rate measured in a thermal analysis experiment will vary with both temperature and extent of conversion. This means that the effective activation energy determined from thermal analysis experiments will also be a function of these two variables. However, the usual implementation of model-fitting methods is aimed at extracting a single value of the activation energy for an overall process. The value obtained in such a way is in fact an average that does not reflect changes in the reaction mechanism and kinetics with the temperature and the extent of conversion.
The aforementioned drawbacks of model-fitting can be avoided with the use of isoconversional methods [2], [3], [4]. Firstly, these methods allow the activation energy to be determined as a function of the extent of conversion and/or temperature. Secondly, this dependence is determined without making any assumptions about the reaction model. Because the model-free isoconversional methods eliminate the causes of the aforementioned disagreement, they are likely to produce consistent kinetic results from isothermal and nonisothermal experiments.
In this paper, we explore an opportunity of employing model-fitting and model-free methods to produce consistent kinetic characteristics from isothermal and nonisothermal experiments. Analysis of isothermal kinetic experiments is traditionally believed to be more reliable because the one variable (T) is held constant during each experiment, thereby reducing the number of kinetic parameters that are determined simultaneously by fitting. Therefore, the results of nonisothermal experiments are expected to agree with the isothermal data. As a definitive test for such agreement we consider the capability of predicting isothermal kinetics from nonisothermal data.
Section snippets
Experimental
As experimental examples we have chosen thermal decompositions of two energetic materials, l,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) and ammonium dinitramide (ADN). The Thiokol Corporation kindly supplied a sample of ADN. The material was used without further purification. An HMX sample was received from the Army Research Laboratory at Aberdeen Proving Grounds, MD. The sample was used after recrystallization from acetone.
The thermogravimetric analysis (TGA) experiments were carried out using
Model-fitting method
Rearrangement and integration of Eq. (1) for isothermal conditions giveswhere is the integrated form of the reaction model (Table 1). The subscript j has been introduced to emphasize that substituting a particular reaction model into Eq. (4) results in evaluating the corresponding rate constant, which is determined from the slope of a plot of gj(α) versus t. For each reaction model selected, the rate constants are evaluated at several temperatures, Ti, and the
Model-fitting method
Examination of Table 2, Table 3 suggests that the Arrhenius parameters determined for the isothermal data using the model-fitting method are rather mildly variable when changing the reaction model. The reduced time plots for isothermal decompositions of HMX (Fig. 1) and ADN (Fig. 2) were subjected to statistical analysis as described above , . The resulting values of F are given in Table 1, Table 2. The F-test allows the Avrami–Erofeev model (number 9) to be identified as the best description
Conclusions
The application of the model-fitting method to nonisothermal data results in highly uncertain kinetic triplets that cannot be meaningfully compared with the triplets evaluated from isothermal measurements. Because of the characteristic uncertainty, the kinetic triplets obtained from nonisothermal data are not capable of reasonably predicting isothermal kinetics. The model-fitting method applied to isothermal data gives rise to unambiguous values of Arrhenius parameters that are likely to
Acknowledgements
The authors thank Pete Lofy for obtaining the HMX data. This research is supported in part by the University of Utah Center for Simulations of Accidental Fires and Explosions, funded by the Department of Energy, Lawrence Livermore Laboratory, under subcontract B341493 and by the Office of Naval Research under contract No. N00014-95-l-1339.
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