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Prediction of Thermal Aging of Materials by Modified Kinetic Analysis based on Limited Amount of Experimental Points

1. Introduction

It is very difficult to determine experimentally the thermal stability of materials at ambient temperatures due to the fact that small changes of physicochemical properties occurring at the beginning of process proceed with very low rates. In addition, any attempt to determine the lifetime of a material is strongly dependent upon the ability to correctly identify the physical properties that are the test criteria. The process of the thermal stability investigation is seeking for an indication of material's ability to retain a particular physical, chemical, mechanical or biological property above a certain level after exposure to elevated temperatures and/or extended periods of time. The knowledge of temperature-dependent behaviour of the properties considered has an unlimited number of applications in the area of e.g. chemicals, energetic materials, biopharmaceutics, vaccines, polymers, etc.

The collection of the data used for the prediction of the thermal stability of the materials proceeds, generally, by two methods:

• (i) by non-isothermal experiments carried out with few heating rates (continuous method of data collection)
• (ii) by isothermal conditions at few temperatures, during relatively long periods of time being sometimes in the range of months, or even years (discontinuous method of data collection).

Independent of the experimental procedure, (i) or (ii), the collection of the data is followed by the kinetic analysis resulting in the evaluation of the kinetic parameters such as the activation energy E, the pre-exponential factor in Arrhenius equation A and the most probable reaction model expressed by the form of the function ƒ(α) where α denotes the reaction extent. Often, see e.g. [1, 2], this last function is assumed to have an arbitrarily chosen form, mainly those characteristic for the first- or n-th order kinetics. In such a case, the prediction of a “life–time” value is based on the simplified kinetic description of the process which, often, may not proceed according to the assumption that the first order kinetics describes exactly the reaction course. In AKTS-Thermokinetics Software we propose the modification of the kinetic analysis which allows the description of the investigated processes by all existing reaction models and including the option that the reaction may proceed not only in one, but also in two or more stages. Additionally we introduce specific model selection tools which are required in the case where available data are in the form of only scarce experimental points (for example those collected in discontinuous mode). We illustrate our method by the results of the elaboration of approximately 30 data points only and present the kinetic and model selection procedures allowing the evaluation of kinetic parameters and rational prediction of the reaction course.
The proposed method delivers a prediction band (e.g. 95%) showing scatter of the data and allows considering the uncertainty of the best-fit curve being particularly valuable for thermal aging predictions. The results of the predictions can be compared with the real experimental results obtained for energetic materials and vaccines.

2. Kinetic elaboration of the experimental data

For the determination of the kinetic parameters we apply in AKTS-Thermokinetics Software the model fitting approach. The I-st ICTAC Kinetic Project [3] proved that model fitting method can be as reliable as iso-conversional, model free methods, as long as the models are fitted simultaneously to multiple data sets obtained under different temperature programs. The sets of kinetic models are widely spread over the kinetic papers, see e.g. [4]. In AKTS-Thermokinetics Software for scarce experimental points, we apply the truncated form of the Sesták Berggren equation [6]

(1)

for exponent p=0.

The simplified equation proposed by Sesták Berggren (called throughout this text as SB-model) matches any of the ƒ(α) functions applied in the literature:

(2)

For example, by taking n and m equal to 1, equation (2) transforms to the known Prout-Tompkins equation [7,8] often used [9-14] for the description of the kinetics of auto-catalytic reactions. However, the correct ƒ(α) function for these types of reactions is more complicated, and the Prout-Tompkins model may be treated as a simplification which may be used under certain boundary conditions only. In the strict sense, the general scheme for autocatalytic reactions includes two steps. For the elaboration of scarce data, AKTS-Thermokinetics Software applies two Sesták Berggren models for reactions built up from two sub-reactions. The general rate expression for the model containing both stages can be depicted as :

(3)

which in general form can be written for I reaction stages as:

(4)

For example, the application of equation (3) by setting n₁=1, m₁=0, n₂=1, m₂=1 allows describing the autocatalytic reactions including two steps:

• i) The primary decomposition reaction of the reactant A with the formation of the product B acting as a catalyst during the further course of the reaction.
• ii) The secondary decomposition reaction having the autocatalytic character.

This case is similar to the approach already proposed by Kamal and Sourour [15,16] and Finke-Watzky [17,18] applied e.g. in [19-24]:

(5)

(6)

(7)

which can also be written as

(8)

with

A0 : initial amount of A(t) at t0=0

(9)

B0 : initial amount of B(t)=B0+A0-A(t) at t0=0

(10)

: reaction progress α of A assuming that A0=1 and B0=0 at t0=0.

(11)

Further illustration of the general application of equation (3) by setting n₁=1, m₁=0, n₂=1, m₂=0 is for example the well-known case of two competitive first order reactions:

(12)

(13)

(14)

By changing the value of the parameters n₁, m₁, n₂, m₂ the number of combination of possible models described becomes infinite, covering not only existing typical kinetic models but also processes that cannot be described by only one of those models. Furthermore, the same method can be applied for higher number of sub-reactions, however, in such a case the number of parameters increases seriously, therefore the possible over-fitting of the data has to be carefully considered. The more complicated expression given by equation (3), which combines two possible models, should be considered only if there exists a physico-empirical reasons to do so. Otherwise, possible simplifications have to be considered to find out the simplest model being still able to describe the recorded data correctly according to an Ockham's razor principle [25]. It states that when several competing theories make exactly the same predictions, the simpler one is the best. This issue is discussed in details in the next section.

3. Model selection: Comparing models using Akaike (AIC) and Bayesian (BIC) Information Criteria

When fitting data with different models, the general objective is often to find the best method for their discrimination [26]. In AKTS-Thermokinetics Software we apply the criteria for comparing models based on information theory developed by Akaike [27] and its Bayesian counterpart [28] which allow finding out the most plausible model fitting the experimental points. The Akaike's and Bayesian's criteria determine not only which model is more likely to fit better the considered data but also quantifies how much more likely. In AKTS-Thermokinetics Software both Information Criteria: AIC and BIC and are considered after following five actions undertaken for each model:

• The number of data points is defined.
• The number of parameters to be fitted is defined. The parameters that are constraint to constant values are not counted.
• The optimization starts with an initial estimated value for each parameter in the equation. The fit of the model parameters takes place using the appropriate optimization algorithm (non-linear regression).
• The sum of residual squares RSS of the distances of the experimental points to the simulated curves is calculated.
• AIC and BIC values and the 'Akaike and Bayesian weights' w, for each model, are calculated. AKTS-Thermokinetics Software gives a relative weight that a given model is the best one among the set of models examined. There is no limit for the number of models under investigation. According to AIC, the best model has the highest (closer to 1) weight. The same approach for the calculation is applied for the BIC weights.

These AIC and BIC approaches are illustrated below in the next figures.

Examples of applied single stages and models

Examples of applied two stages and models

Unlimited number of stages and models combinations

e.g. A+B->C, A+C->D, C->E

N.B.: Possible combinations of all above stages for multi-population systems

Fig. Comparison of model combination based on information theory developed by Akaike and its Bayesian which allows finding out the most plausible model fitting the experimental points. The Akaike's and Bayesian's criteria determine not only which model is more likely to fit better the considered data but also quantifies how much more likely.

In the case of discontinuous data collection, the number of data points is generally limited, moreover, they may contain an experimental error. Taking into account these limitations one has to assume that the more complicated models fit the experimental points better, simply because they have more parameters and will deliver smaller RSS. But the lower RSS may not be optimal as a decisive criterion. Based on principle of parsimony (also known as Ockham's razor principle [25]), the procedure of models discrimination starts to be more complicated. One has to apply such methods that look at the trade-off between criterion of lower RSS and criterion based on the number of parameters used. The illustration of this problem is presented in the previous Figures. The application of the AIC and BIC in the kinetic computations based on the scarce data is illustrated by the simulation of experimental data collected during the investigation of the energetic (propellant), biological (vaccine) and polymer (pipes) materials. The number of applications is however almost unlimited. For the sake of clarity we illustrate the modified kinetic and model selection methodology by considering the case when the reaction proceeds in one or two-steps only and its course may be sufficiently well fitted by one- or combination of two kinetic models (as in eqs. 3, 8 and 14). It is obvious that the presented methodology may be used for the kinetic computations of the reactions proceeding via more stages which require the use of combination of more than two models (see eq. 4). However, such situations in the case of the elaboration of scarce points are rather unrealistic from the model selection point of view because in such a case the number of parameters used for optimization should generally not be too important, limiting our approach to the combination of one or two SB-models (eq. 3).

4. Kinetic and model selection analysis of experimental data

4.1 Investigation of propellants aging

Nitrocellulose-based propellants may decompose slowly which can lead to the decreasing of their chemical stability. To prevent this undesired process, the components reacting with the degradation products (stabilizers) are introduced in the propellants. The determination of the stabilizer depletion by e.g. chromatographic techniques such as High Performance Liquid Chromatography (HPLC) offers, therefore, an efficient tool for monitoring propellant aging process. In this study the single base propellant was aged in temperatures ranging from 40 to 80°C. Not more than 2-6 points characterizing their stabilizer depletion were collected at each temperature. The prediction of the stabilizer depletion was based on the kinetic parameters obtained by fitting experimental data at 50, 60, 70 and 80°C collected over 84 days by best reaction models combination which was chosen by using AIC and BIC. The results of AIC and BIC for a single base propellant containing Akardite-1 (1,1-Diphenylurea) as stabilizer for one and two SB-models are given in Table. Three additional experimental points measured at 40, 50 and 60°C and collected after 252 days were used for the verification of the predictions (see next Fig.). The two-step models having significantly higher AIC and BIC weights with w_A = 96.15% and w_B = 91.12% (see Tab.) for a combination of two nth-order reactions with fixed, constant reaction orders n₁=4 and n₂=0 have been chosen for the simulations. For comparison, AIC and BIC weights amounted to only w_A = 3.54% and w_B = 8.65% for the same two-step model combination with fitted reaction order n₁ and, w_A and w_B < 1% for the one step model approach, respectively. As results from the Akaike and Bayesian criteria, the investigated stabilizer depletion can be best described with a model combination containing two reactions of zero-th and fourth order kinetics. The figure presents the 16 experimental points (open symbols), collected over 84 days, which were used for the determination of the kinetic parameters and, in turn, for the predictions of change of stabilizer concentration in the propellant.

Tab. Kinetic parameters of the depletion of Akardite-1 in a single base propellant calculated by non-linear regression and Akaike and Bayesian criteria. The column containing the recommended model combination with the best scores is depicted in bold. The AIC and BIC weights w_A = 96.15% and w_B = 91.12% suggest the use of a two-steps model combination (zero-th and fourth order kinetics) to describe the process of the stabilizer depletion. (*) means fixed reaction orders and (**) means value of reaction orders fixed to '0' because the optimized parameters amounted to values close to 0.

The prediction bands depicted in the next Figure were determined by the bootstrap method which is based on Monte Carlo approach frequently used in applied statistics. In our case it was used for the estimation of the (asymmetrical) confidence interval of a specific parameter. The bootstrap method allows randomized resampling of the data set S to construct a fictional set of data S*. Each of these data sets is constructed by resampling N points with replacement from the original data set. For each of the fictional set of data S*, we apply the nonlinear regression (see 'step 3' in chapter 3) to obtain the fitted parameters. The random resampling process is repeated (at least) 1000 times to obtain a set of values for each of the estimated parameters. Based on these (bootstrap) sets of estimated parameters, we can estimate the prediction band in the form of e.g. the upper and lower 95 percentiles of each of the fitted parameters. More theoretical insights concerning the bootstrap technique can be found e.g. in [29]. We have computed the bootstrap prediction band for the chosen kinetic model following the methodology of Mishra, Dolan and Yang [29]. The prediction curves at 40, 50 and 60°C were further verified by the experimental points collected after 252 days (marked by the filled circles) which lay inside the prediction bands.

Fig. Prediction of change of stabilizer (Akardite-1) concentration (line) in single base propellant based on the 16 experimental points (symbols) collected during 84 days in the range of 50-80°C. The reaction model used for the predictions has been chosen using AIC and BIC. Prediction curves at 40, 50 and 60°C were verified by the experimental points collected after 252 days marked by the filled circles. The predictions bands (PB), marked by the dashed lines, were determined by the bootstrap method (see text). The temperature values are marked on the curves.

The next Figure presents the results of the determination of one from parameters considered by NATO Allied Ordnance Publication AOP-48 Ed.2 [1] namely, T₁₀ i.e. the temperature for a 10 years storage after which a critical stabilizer depletion of 50% is reached. The plot depicts the fitted curves together with the prediction bands calculated with the bootstrap method.

Fig. Simulation of storage temperature (T₁₀) at which, after 10 years, the stabilizer depletion will reach 50% of the initial concentration value. For the simulated T10 value of 47.4°C the prediction band is spread between 8.15 and 13.24 years. At fixed time of 10 years, the stabilizer depletion varies between 44.4 and 57.2%.

4.2 Investigation of thermal stability of vaccines

The described above procedure of the modified kinetic computations and their validation by AIC and BIC is illustrated below by the results of the investigation of the thermal stability of freeze-dried measles vaccine. Vaccines are made up of proteins, nucleic acids, lipids and carbohydrates, which undergo changes when exposed to higher temperatures. The experimental results presented by Allison [30] were simulated by both, one- and two-steps SB-models. The results of these calculations presented in the next Table indicate that the two-steps model (zero-th and second order kinetics) best describes the process of the deterioration of the investigated vaccine. The combination of two-step models with two nth-order reactions and fixed constant reaction orders n₁=2 and n₂=0 leads to AIC and BIC highest weights with w_A = 86.69% and w_B = 88.49% (see Table 5) compared to w_A = 12.92% and w_B = 10.64% for the fitted reaction order n₁ and, w_A,B < 1% for the one step model approach, respectively. Chosen kinetic model combination with zero-th and second order kinetics can then be applied to predict the shelf-life of the vaccine. The next Figure presents the results of the simulation of the change of the concentration of the viruses in the vaccine (displayed as number of plague forming units per vial) by two-steps model chosen after application of the AIC and BIC criteria.

Tab. Kinetic parameters of the degradation of lyophilized measles vaccine calculated by non-linear regression and verified by Akaike and Bayesian criteria. The column containing the recommended model combination with the best scores is depicted in bold. The AIC and BIC weights w_A = 86.69% and w_B = 88.49% suggest the use of a two-steps model combination (zero-th and second order kinetics) to describe the process of the degradation of lyophilized measles vaccine. (*) means fixed reaction orders and (**) means value of reaction orders fixed to '0' because the optimized parameters amounted to values close to 0.

Fig. Prediction (lines) of change of virus content in the vaccine dose based on the 21 experimental points (symbols) collected during 9.5 months in the temperature range 31-45°C. The reaction model used for the predictions has been chosen using AIC and BIC. With AKTS-Thermokinetics Software [31-32] it can be demonstrated that a two steps model combination with zero-th and second order kinetics is recommended for accurate simulations. The prediction bands, marked by the dashed lines, were determined by the bootstrap method (see text). The temperature values are marked on the curves.

After determination of the best kinetic models it is possible to simulate the reaction progress after arbitrarily chosen aging time and under any temperature profiles such as stepwise variations, oscillatory conditions, temperature shock, or even real atmospheric temperature profiles. This is especially important for precise determining shelf-life of the vaccines which may undergo different temperature variation during their transportation and storage before the final usage. Such simulation is presented in the next Fig. for the case including three temperature profiles:

• (i) constant temperature of 5°C for about 2 years,
• (ii) 3 days of temperature excursion oscillations with a sinusoidal period of 24 h reproducing the day/night temperature variation in Miami (USA) in August,
• (iii) constant temperature of 5°C.

After two years of the storage at 5°C the reaction progress (expressed by the amount of plaque-forming units per vaccine dose) amounts to 2.1% (point I). Three days of temperature oscillations lead to a rapid increase of the reaction progress to 5.4% (point II). Using AKTS-Thermokinetics Software [31-32] it is also possible to simulate the stability of vaccines and extent of their degradation which strongly depends on the exact temperature profile during transport and storage before the final use. The results presented in the next next Figure clearly demonstrate that even a short temperature variation resulting from the temporary storage outside the cold chain can lead to significant decrease of the virus content.

Fig. Variation of the vaccine degradation extent (bottom plot) based on the change of plaque-forming units per vaccine dose as a function of the temperature profiles (upper plot). After storage during ca. 2 years at 5°C when the degradation extent amounted to 2.1% (point I) the sample was exposed for three days to real atmospheric temperature profile of Miami (USA) which results in increase of degradation extent to 5.4% (point II). Finally the sample was once more stored at 5°C.

4.3 Investigation of thermal stability of thermoplastic pipes

Determination of the change of the specific properties of the materials (such as e.g. hydrostatic strength in thermoplastic pipes) in function of time and temperature can be also done by the kinetic analysis of the investigated process. As already mentioned, with AKTS-Thermokinetics it is possible to find by applying one- and two- or more combination of kinetic models, the best fit of experimental data. The method used for models selection takes into account not only the quality of regression fit, but also the number of data points and number of parameters in specific models based on the Akaike's and Bayesian's Information Criteria (AIC and BIC). The next Figure presents the application of the method for the determination of the long-term hydrostatic strength of various thermoplastics materials in pipe form (PE100, β-Nucleated PP-H). The applied procedure was validated by comparing results of the predictions with subsequent measurements performed at various temperatures after longer period (c.a. 1 year) of storage time (depicted in the next Fig.). Due to considering AIC and BIC weights w such procedure allows concluding not only which model is more likely to be correct but even quantifying how much more likely. The proposed method delimits also the borders of the prediction band (e.g. 95% confidence) based on the bootstrap method showing scatter of the data and allows considering uncertainty of the best-fit curve being very important for thermal aging predictions.

Fig. Plastics piping and ducting systems: determination of the long-term hydrostatic strength of thermoplastics materials in pipe form (β-Nucleated PP-H) based on the data (empty circles) collected in experimental domain depicted in the plot. Prediction curves at 60, 80 and 95°C were verified by the experimental points collected after ca. 1 year and marked by filled circles. The prediction band PB is shown for the simulations of the hydrostatic strength at 20 and 80°C. Results obtained by proposed new method are compatible with those determined by actual norm EN DIN ISO 9080 for plastics piping and ducting systems (marked by filled triangles).

5. Conclusion

The prediction of the materials aging based on the discontinuously collected experimental points can be achieved in AKTS-Thermokinetics Software by the modifications of the often applied kinetic and statistical approaches. Contrary to the commonly applied methods which are based on the arbitrarily assumed n-th order kinetics only, we consider all ƒ(α) functions including also the option that the reaction may proceed not only in one, but also in two or more stages. The approach can also consider the specific forms of the kinetic equations describing the peculiarities of the autocatalytic-type reactions (Kamal-Sourour or Finke-Watsky approaches). The difficult issue of discriminating best kinetic reaction model, when having scarce data points only, is solved by the application of Akaike's and Bayesian's Information Criteria (AIC and BIC). These criteria allow successful discrimination between considered kinetic models and their combination even if the number of available data points is as small as 20. Using the bootstrap method AKTS-Thermokinetics Software allows the calculation of the prediction band (95% confidence interval), being particularly beneficial parameter in reliable estimation of long-term properties of the materials.
Proposed kinetic and model selection approaches implemented in AKTS-Thermokinetics Software have been checked with a multitude number of application domains for the determination of the shelf-life of e.g. stabilizers, biophamaceutics, vaccines, pipes, propellants. Despite the fact that the scarce data may contain the scatter, which reflects the range of experimental errors. It was possible, having only 2-6 points collected at four temperatures between 50-90°C during about 90 days to simulate correctly the stabilizer depletion in single base propellant. The best fit was obtained for the two step model composed from combination of two nth-order reactions (zeroth and fourth order). The results of the investigation of the thermal stability of freeze-dried measles vaccine showed that the two-steps model (zero-th and second order kinetics) best describes the process of its deterioration. Determined set of kinetic parameters allowed the exact determination of the biological material degradation extent as a function of temperature profiles together with the respective prediction bands. The calculations performed by means of AKTS-Thermokinetics Software enabled also to predict the change of the reaction extent in any temperature profiles such as isothermal 5°C (e.g. 1 year) followed by few days atmospheric temperature profile and, finally, by the second, one year period at 5°C. The number of applications of proposed method is almost unlimited.
The verification of the method implemented in AKTS-Thermokinetics Software was compared with the investigation of aging by other methods as described in specific norms such as NATO-AOP-48 test procedure for propellants or DIN EN ISO 9080 for Lifetime prediction of Plastics piping and ducting systems. Plastics piping and ducting systems can be analyzed by using the same approach. It is for example possible to predict the long-term hydrostatic strength of thermoplastics materials in pipe form (ß-Nucleated PP-H). Additionally, AKTS-Thermokinetics Software introduces also in the simulations the procedure of determination the prediction band (e.g. 95%) which shows the scatter of the data and allows considering the uncertainty of the best-fit curve being particularly valuable for thermal aging.
For the sake of clarity and because of the scarce number of data points, our modified kinetic and model selection methodology can be best applied by considering the case when the reaction proceeds in one or two-steps only and its course may be sufficiently well fitted by one- or combination of two kinetic models. However, the presented methodology implemented in AKTS-Thermokinetics Software may be used for the kinetic computations of the reactions proceeding via more stages which require the use of combination of more than two models.

6. References

[1] NATO Allied Ordnance Publication (NATO AOP) 48, Ed.2, Military Agency for Standardization, NATO Headquarters, 1110, Brussels, Belgium, 2007.
[2] ASTM Standard E1641-07(2012), Standard test method for Decomposition Kinetics by Thermogravimetry, ASTM International, West Conshohocken, PA, 2003, DOI: 10.1520/E1641-07R12, www.astm.org.
[3] M.E. Brown, M. Maciejewski, S. Vyazovkin, R. Nomen, J. Sempere, A. Burnham, J. Opfermann, R. Strey, H.L. Anderson, A. Kemmler, R. Keuleers, J. Janssens, H.O. Desseyn, Chao-Rui Li, Tong B. Tang, B. Roduit, J. Malek, T. Mitsuhashi, Computational aspects of kinetic analysis Part A: The ICTAC kinetics project-data, methods and results, Thermochim. Acta, 355 (2000) 125-143.
[4] S. Vyazovkin, A. K. Burnham, J. M. Criado, L. A. Pérez-Maqueda, C. Popescu, N. Sbirrazzuoli, ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data, Thermochim. Acta, 520 (2011) 1-19.
[5] L.A. Pérez-Maqueda, J.M. Criado, and P.E. Sànchez-Jiménez, Combined Kinetic Analysis of Solid-State Reactions: a powerful tool for the simultaneous determination of kinetic parameters and the kinetic model without previous assumptions on the reaction mechanism J. Phys. Chem. A, 110 (2006) 12456-12462.
[6] J. Šesták, G. Berggren, Study of the kinetics of the mechanism of solid-state reactions at increasing temperatures, Thermochim. Acta, 3 (1971) 1-12.
[7] E.G. Prout and F.C. Tompkins, The thermal decomposition of potassium permanganate, Trans. Faraday Soc., 40 (1944) 488-498.
[8] E.G. Prout and F.C. Tompkins, The thermal decomposition of silver permanganate, Trans. Faraday Soc., 42 (1946) 468-472.
[9] M.E. Brown, The Prout-Tompkins rate equation in solid-state kinetics, Thermochim. Acta, 300 (1997) 93-106.
[10] M.E. Brown, B.D. Glass, Pharmaceutical applications of the Prout–Tompkins rate equation, Int. J. Pharm., 190 (1999) 129-137.
[11] M.A. Bohn, Prediction of life times of propellants - improved kinetic description of the stabilizer consumption, Propellants, Explos., Pyrotech., 19 (1994) 266-269.
[12] M.A. Bohn, Prediction of in-service time period of three differently stabilized single base propellants, Propellants, Explos., Pyrotech., 34 (2009) 252-266.
[13] M.A. Bohn, The Prout-Tompkins problem and its solution, Proc. 42nd Int. Annual Conf., ICT, Karlsruhe, Germany (2011), paper 52, 1-19.
[14] N. Eisenreich, A. Pfeil, Non-linear least-squares fit of non-isothermal thermoanalytical curves. Reinvestigation of the kinetics of the autocatalytic decomposition of nitrated cellulose, Thermochim.Acta 61, (1983) 13-21.
[15] S. Sourour, M.R. Kamal. Difffrential scanning calorimetry of epoxy cure: isothermal cure kinetics, Thermochim. Acta, 14 (1976) 41-59.
[16] M.R. Kamal, Thermoset characterization for moldability analysis, Polym.Eng. Sci., 14 (1974) 231-239.
[17] M.A. Watzky, A.M. Morris, E.D. Ross, R.G. Finke, Fitting yeast and mammalian prion aggregation kinetic data with the Finke-Watzky two-step model of nucleation and autocatalytic growth, Biochemistry, 47 (2008) 10790-10800.
[18] A.M. Morris, M.A. Watzky, R.G. Finke, Protein aggregation kinetics, mechanism, and curve-fitting: A review of the literature, Biochim. Biophys. Acta, 1794 (2009) 375-397.
[19] V.L. Zvetkov, R.K. Krastev, S. Paz-Abuin, Is the Kamal’s model appropriate in the study of the epoxy-amine addition kinetics?, Thermochim. Acta, 505 (2010) 47-52.
[20] M. Kaiser, U. Ticmanis, Thermal stability of diazodinitrophenol, Thermochim. Acta, 250 (1995) 137-149.
[21] R. Capart, L. Khezani, A.K. Burnham, Assessment of various kinetic models for the pyrolysis of a microgranular cellulose, Thermochim. Acta, 417 (2004) 79-89.
[22] N. Sbirrazzuoli, A. Mititelu-Mija, L. Vincent, C. Alzina, Isoconversional kinetic analysis of stoichiometric and off-stoichiometric epoxy-amine cures, Thermochim. Acta, 447 (2006) 167-177.
[23] P.W.M. Jacobs, Formation and growth of nuclei and the growth of interfaces in the chemical decomposition of solids: new insights, J.Phys.Chem., B, 101 (1997) 10086-10093.
[24] J.D.R. Talbot, The kinetics of the epoxy amine cure reaction from a solvation perspective, J. Polym.Sci, Part A: Polym.Chem., 42 (2004) 3579-3586.
[25] http://www.britannica.com/EBchecked/topic/424706/Ockhams-razor
[26] J.B. Johnson and K. S. Omland., Model selection in ecology and evolution, Trends Ecol. Evol., 19 (2004) 101-109.
[27] H. Akaike, A new look at the statistical model identi?cation, IEEE Trans. Automatic Control, 19 (1974) 716–723.
[28] K.P. Burnham, D.R. Anderson, Model selection and multimodel inference: a practical information-theoretic approach, 2002, 2nd edn. Springer, New York.
[29] D.K. Mishra, K.D. Dolan, L. Yang, Bootstrap confidence intervals for the kinetic parameters of degradation of anthocyanins in grape pomace, J. Food Process Eng. 34 (2011) 1220-1233.
[30] L.M. Allison, G.F.Mann, F.T. Perkins and A.J. Zuckerman, An accelerated stability test procedure for lyophilized measles vaccines, J. Biol. Stand., 9 (1981) 185-194.
[31] B. Roduit, M. Hartmann, P. Folly, A. Sarbach, R. Baltensperger, Prediction of thermal stability of materials by modified kinetic and model selection approaches based on limited amount of experimental points, Thermochim. Acta, 579 (2014) 31-39.
[32] AKTS-Thermokinetics Software, http://www.akts.com.

Possibilities of analysis offered

Abbreviations:
TA: AKTS-Thermal Analysis (Calisto Software)
TK: AKTS-Thermokinetics Software
TS: AKTS-Thermal Safety Software
RC: AKTS-Reaction Calorimetry Software	TA	TK	TS	RC
	Possibilities of analysis offered
Temperature modes allowed
isothermal	yes	yes	yes	yes
non-isothermal linear, non-linear, arbitrary heating or cooling rates	yes	yes	yes	yes
isoperibolic (various constant oven temperatures)	yes	yes	yes	yes
Evaluation of the data collected by the following thermoanalytical techniques at conventional and/or specific conditions:
Differential Scanning Calorimetry (DSC)	yes	yes	yes	yes
Differential Thermal Analysis (DTA)	yes	yes	yes	yes
Simultaneous Thermogravimetry & Differential Scanning Calorimetry / Differential Thermal Analysis	yes	yes	yes	yes
Pressure monitoring / Gas generation: P and dP/dt	yes	yes	yes	yes
TG (m(t)) and DTG (dm/dt)	yes	yes	yes	yes
Hyphenated Techniques: TG-EGA (MS or FTIR)	yes	yes	yes	yes
Dilatometry / Mechanical Analysis: TMA, DMA	yes	yes	yes	yes
Non Destructive Assay: NDA for e.g. Nuclear Waste Characterization (e.g.Setaram LVC-3013)	yes	yes	yes	yes
Gas Humidity Monitoring (e.g. Setaram Wetsys)	yes	yes	yes	yes
Microcalorimetry (e.g.TA Instruments TAM, Setaram C80, MicroSC and many others)	yes	yes	yes	yes
Reaction Calorimetry (e.g. Mettler RC1, Setaram DRC, HEL Simular, ChemiSens CPA 102, 202 and many others)	yes	yes	yes	yes
Thermal Conductivity of liquids and solids (e.g. C-Therm TCI)