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A brief description
The investigation of materials aging at ambient temperatures is experimentally very difficult due to its very low rate, small changes of physicochemical properties and, very often, limited amount of experimental data. Commonly applied methods of thermalaging determination are therefore based on kinetic analysis carried out by measuring material properties at several elevated temperatures. Kinetic analysis can be applied to evaluate not only onestep reactions but also the multistep reactions proceeding by several consequent or parallel steps that can be the combination of chemical or physical substages. With today's computers there are almost no limitations concerning the type of the reaction models applied and number of reaction steps during kinetic computations. The only limitations usually arise from the experimental procedure when the number of experimental points is in the range of ca. 30 or less. If only such scarce experimental points collected in discontinuous mode are available, we propose in the current study to modify both the kinetic analysis and the model selection approach in the way which still allows the correct description of investigated processes despite of experimental limitations. Applying simultaneous combination of two Sesták Berggren models enables to consider all the specific forms of the kinetic equations commonly applied in kinetic computations including also the peculiarities of the models applied to autocatalytictype reactions (KamalSourour (KS) or FinkeWatzky (FW) models). The difficult task of discriminating best kinetic reaction model among all models, when having scarce data points only, is solved by the application of Akaike and Bayesian Information Criteria (AIC/BIC). This approach allows successful discrimination between an unlimited numbers of kinetic models even if the total number of available data points is very limited. Using additionally the bootstrap method it is possible to calculate the prediction band, being particularly useful in reliable estimation of longterm properties of the materials. The method is illustrated by the experimental simulations of the depletion of the stabilizer Akardite1 (1,1Diphenylurea) in a single base propellant, the prediction of the thermal stability of freezedried measles vaccine and the determination of the longterm hydrostatic strength of thermoplastics materials in pipe form (ßNucleated PPH).

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 temperaturedependent 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:
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 preexponential 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 nth 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 AKTSThermokinetics 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 bestfit 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 AKTSThermokinetics Software the model fitting approach. The Ist ICTAC Kinetic Project [3] proved that model fitting method can be as reliable as isoconversional, 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 AKTSThermokinetics 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 SBmodel) 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 ProutTompkins equation [7,8] often used [914] for the description of the kinetics of autocatalytic reactions. However, the correct ƒ(α) function for these types of reactions is more complicated, and the ProutTompkins 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, AKTSThermokinetics Software applies two Sesták Berggren models for reactions built up from two subreactions. 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 n_{1}=1, m_{1}=0, n_{2}=1, m_{2}=1 allows describing the autocatalytic reactions including two steps:
This case is similar to the approach already proposed by Kamal and Sourour [15,16] and FinkeWatzky [17,18] applied e.g. in [1924]:
(5) 
(6) 
(7) 
which can also be written as
(8) 
with
A_{0} : initial amount of A(t) at t_{0}=0 
(9) 
B_{0} : initial amount of B(t)=B_{0}+A_{0}A(t) at t_{0}=0 
(10) 
: reaction progress α of A assuming that A_{0}=1 and B_{0}=0 at t_{0}=0. 
(11) 
Further illustration of the general application of equation (3) by setting n_{1}=1, m_{1}=0, n_{2}=1, m_{2}=0 is for example the wellknown case of two competitive first order reactions:
(12) 
(13) 
(14) 
By changing the value of the parameters n_{1}, m_{1}, n_{2}, m_{2} 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 subreactions, however, in such a case the number of parameters increases seriously, therefore the possible overfitting 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 physicoempirical 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 AKTSThermokinetics 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 AKTSThermokinetics Software both Information Criteria: AIC and BIC and are considered after following five actions undertaken for each model:
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 multipopulation 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 tradeoff 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 twosteps 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 SBmodels (eq. 3).
4. Kinetic and model selection analysis of experimental data
4.1 Investigation of propellants aging
Nitrocellulosebased 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 26 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 Akardite1 (1,1Diphenylurea) as stabilizer for one and two SBmodels 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 twostep 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 nthorder reactions with fixed, constant reaction orders n_{1}=4 and n_{2}=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 twostep model combination with fitted reaction order n_{1} 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 zeroth 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 Akardite1 in a single base propellant calculated by nonlinear 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 twosteps model combination (zeroth 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 (Akardite1) concentration (line) in single base propellant based on the 16 experimental points (symbols) collected during 84 days in the range of 5080°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 AOP48 Ed.2 [1] namely, T_{10} 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 (T_{10}) 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 freezedried 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 twosteps SBmodels. The results of these calculations presented in the next Table indicate that the twosteps model (zeroth and second order kinetics) best describes the process of the deterioration of the investigated vaccine. The combination of twostep models with two nthorder reactions and fixed constant reaction orders n_{1}=2 and n_{2}=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 n_{1} and, w_{A,B} < 1% for the one step model approach, respectively. Chosen kinetic model combination with zeroth and second order kinetics can then be applied to predict the shelflife 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 twosteps model chosen after application of the AIC and BIC criteria.
Tab. Kinetic parameters of the degradation of lyophilized measles vaccine calculated by nonlinear 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 twosteps model combination (zeroth 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 3145°C. The reaction model used for the predictions has been chosen using AIC and BIC. With AKTSThermokinetics Software [3132] it can be demonstrated that a two steps model combination with zeroth 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 shelflife 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:
After two years of the storage at 5°C the reaction progress (expressed by the amount of plaqueforming 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 AKTSThermokinetics Software [3132] 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 plaqueforming 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 AKTSThermokinetics 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 longterm hydrostatic strength of various thermoplastics materials in pipe form (PE100, βNucleated PPH). 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 bestfit curve being very important for thermal aging predictions.
Fig. Plastics piping and ducting systems: determination of the longterm hydrostatic strength of thermoplastics materials in pipe form (βNucleated PPH) 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 AKTSThermokinetics 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 nth 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 autocatalytictype reactions (KamalSourour or FinkeWatsky 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 AKTSThermokinetics Software allows the calculation of the prediction band (95% confidence interval), being particularly beneficial parameter in reliable estimation of longterm properties of the materials.
Proposed kinetic and model selection approaches implemented in AKTSThermokinetics Software have been checked with a multitude number of application domains for the determination of the shelflife 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 26 points collected at four temperatures between 5090°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 nthorder reactions (zeroth and fourth order). The results of the investigation of the thermal stability of freezedried measles vaccine showed that the twosteps model (zeroth 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 AKTSThermokinetics 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 AKTSThermokinetics Software was compared with the investigation of aging by other methods as described in specific norms such as NATOAOP48 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 longterm hydrostatic strength of thermoplastics materials in pipe form (ßNucleated PPH). Additionally, AKTSThermokinetics 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 bestfit 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 twosteps only and its course may be sufficiently well fitted by one or combination of two kinetic models. However, the presented methodology implemented in AKTSThermokinetics 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
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Part A: The ICTAC kinetics projectdata, methods and results, Thermochim. Acta, 355 (2000) 125143.
[4] S. Vyazovkin, A. K. Burnham, J. M. Criado, L. A. PérezMaqueda, C. Popescu, N. Sbirrazzuoli, ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data, Thermochim. Acta, 520 (2011) 119.
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[13] M.A. Bohn, The ProutTompkins problem and its solution, Proc. 42nd Int. Annual Conf., ICT, Karlsruhe, Germany (2011), paper 52, 119.
[14] N. Eisenreich, A. Pfeil, Nonlinear leastsquares fit of nonisothermal thermoanalytical curves. Reinvestigation of the kinetics of the autocatalytic decomposition of nitrated cellulose, Thermochim.Acta 61, (1983) 1321.
[15] S. Sourour, M.R. Kamal. Difffrential scanning calorimetry of epoxy cure: isothermal cure kinetics, Thermochim. Acta, 14 (1976) 4159.
[16] M.R. Kamal, Thermoset characterization for moldability analysis, Polym.Eng. Sci., 14 (1974) 231239.
[17] M.A. Watzky, A.M. Morris, E.D. Ross, R.G. Finke, Fitting yeast and mammalian prion aggregation kinetic data with the FinkeWatzky twostep model of nucleation and autocatalytic growth, Biochemistry, 47 (2008) 1079010800.
[18] A.M. Morris, M.A. Watzky, R.G. Finke, Protein aggregation kinetics, mechanism, and curvefitting: A review of the literature, Biochim. Biophys. Acta, 1794 (2009) 375397.
[19] V.L. Zvetkov, R.K. Krastev, S. PazAbuin, Is the Kamal’s model appropriate in the study of the epoxyamine addition kinetics?, Thermochim. Acta, 505 (2010) 4752.
[20] M. Kaiser, U. Ticmanis, Thermal stability of diazodinitrophenol, Thermochim. Acta, 250 (1995) 137149.
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of a microgranular cellulose, Thermochim. Acta, 417 (2004) 7989.
[22] N. Sbirrazzuoli, A. MititeluMija, L. Vincent, C. Alzina, Isoconversional kinetic analysis of stoichiometric and offstoichiometric epoxyamine cures, Thermochim. Acta, 447 (2006) 167177.
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[25] http://www.britannica.com/EBchecked/topic/424706/Ockhamsrazor
[26] J.B. Johnson and K. S. Omland., Model selection in ecology and evolution, Trends Ecol. Evol., 19 (2004) 101109.
[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 informationtheoretic 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) 12201233.
[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) 185194.
[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) 3139.
[32] AKTSThermokinetics Software, http://www.akts.com.
Possibilities of analysis offered


Abbreviations:  
TA: AKTSThermal Analysis (Calisto Software)  
TK: AKTSThermokinetics Software  
TS: AKTSThermal Safety Software  
RC: AKTSReaction Calorimetry Software  TA  TK  TS  RC  
Possibilities of analysis offered  
Temperature modes allowed  
isothermal  yes  yes  yes  yes  
nonisothermal linear, nonlinear, 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: TGEGA (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 LVC3013)  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. CTherm TCI)  yes  yes  yes  yes  
Adiabatic Data (THT ARC, Fauske VSP, Omnical DARC and many others)  yes  yes  yes  yes  
Additional Thermal hazard data: Radex, Sedex, Sipcon (Grewer, Lütolf, Miniautoclave, Hot storage test), COMonitoring and A16Test, DeflagrationTest  yes  yes  yes  yes  
Data collected discontinuously by e.g. HPLC with only few points for each temperature  yes  yes  yes  yes  
Simultaneously collected data from the same or different instruments and units as e.g.  yes  yes  yes  yes  
Heat flow DSC (W) and reaction calorimetry data of RC1 (W)  yes  yes  yes  yes  
Heat flow DSC (W) and mass loss TG (mg)  yes  yes  yes  yes  
Heat flow DSC (W) and temperature T(°C) and pressure P(bar) in adiabatic conditions (e.g. ARC)  yes  yes  yes  yes  
Features offered  
Subtraction of experimental base line (blank)  yes  yes  yes  yes  
Reconstruction of the "under peak" base line (BL) for reaction rate data e.g DSC, DTA, DTG, etc.  yes  yes  yes  yes  
Baseline types considered: Sigmoid, Tangential Sigmoid, Linear, Horizontal First Point, Horizontal Last Point, Horizontal, Staged, Spline & Polynomial with variable order, Tangential First Point, Tangential Last Point  yes  yes  yes  yes  
Possible adjustments of temperature onset and offset  yes  yes  yes  yes  
Baseline Subtraction with or without normalization (setting the integration value of the signal to one)  yes  yes  yes  yes  
Smoothing data (allows the user to smooth partially or entirely a curve. Methods: Savitzky & Golay or Gaussian)  yes  yes  yes  yes  
Custom Interpolation and Spikes Correction (designed to interpolate a portion of the signal to remove the bad or noisy data points). Interpolation modes: Straight line, Horizontal, tangential first or last points, Spline or Polynomial with variable orders  yes  yes  yes  yes  
Dragging Data Points (for moving a data point in order to manually smooth the noisy part of the signal)  yes  yes  yes  yes  
Removing Vertical Displacement (Signal Step) (allows the user to bring the zone of displacement to the same level as the left limit point)  yes  yes  yes  yes  
Cutting externals, separate points, internal fragments (allows the user to cut a part of a signal which is not required)  yes  yes  yes  yes  
Building complementary responses (integral from derivative and vice versa)  yes  yes  yes  yes  
Derivation with adjustable "Derivative Filter" (the derivative of a curve at a certain point is the slope of the tangent to the curve at that point)  yes  yes  yes  yes  
Integration (generates the integrated curve of a subtracted or normalized subtracted signal)  yes  yes  yes  yes  
Viewing data in form of overall conversion α(t) or dα(t)/dt  yes  yes  yes  yes  
Viewing data in original form (Q(t), dQ/dt, m(t), dm/dt) raw mass or heat data considered instead of reaction extent α  yes  yes  yes  yes  
Deconvolution and/or Temperature Adjustment by Inverse Filtering of DSC, heat flux or any type of thermoanalytical data (allows the user to consider the time constant of the temperature sensor in order to reconstruct the real response of the sample on the temperature change)  yes  yes  yes  yes  
Automatic unit management by changing axis units: from e.g. W to mW, mW, Cal/s, mCal/s, mCal/s (and/or normalization: e.g. W/g, W/mol, etc.)  yes  yes  yes  yes  
Customizable axis unit menu with any signals of user defined units: e.g. count/g, mg/ml, etc.  yes  yes  yes  yes  
Automatic unit management by signal derivative and/or integral: e.g. J, W or K, K/s or K/min, etc.  yes  yes  yes  yes  
Peak separation based on the application of Gaussian and/or FraserSuzuki (asymmetric) types signals (Position; Amplitude; Halfwidth; Asymmetry)  yes  yes  yes  yes  
Thinning out data (reducing number of points without loss of information)  yes  yes  yes  yes  
Statistical analysis of results of parallel runs via Customizing Equation (allows the user to apply a mathematical formula to one or more signals)  yes  yes  yes  yes  
Heat capacity determination via two methods: Continuous Cp or Cp by Step (Both methods with or without reference)  yes  yes  yes  yes  
Phase transition parameters determination  yes  yes  yes  yes  
Glass transition (Tg) determination according to IUPAC procedure  yes  yes  yes  yes  
Thermal conductivity determination of both solids and liquids  yes  yes  yes  yes  
Converting to Natural Logarithm (especially useful when addressing the exponential Heat Flow signals obtained during isothermal studies)  yes  yes  yes  yes  
Crystallinity evaluation of semicrystalline materials  yes  yes  yes  yes  
Oxidation Induction Time calculation based on the ISO 113576 norm  yes  yes  yes  yes  
Purity determination calculated with Van`t Hoff equation  yes  yes  yes  yes  
Setting Signal to Zero (allows the user to set to zero on the Yaxis the value of a selected point of a signal)  yes  yes  yes  yes  
Slope Correction (adjusting the slope of a signal to remove its drift for a better presentation)  yes  yes  yes  yes  
Temperature Correction (allows calibration of the apparatus to adjust the measured and the real temperatures of the sample)  yes  yes  yes  yes  
Temperature Segmentation (generates from an experimental temperature curve a new temperature profile built up from an arbitrarily chosen number (between 1 and 2000) of segments)  yes  yes  yes  yes  
TMATrue and Average Alpha and TMA Correction  yes  yes  yes  yes  
Data Loading (Importing data in the form of ASCII files from files created by any type of apparatus via general interface  yes  yes  yes  yes  
User Rights Management (Controls access to the software's features. The administrator can create the list of the users and decide about their rights)  yes  yes  yes  yes  
Managing the Connection to the Database in which the data are stored in "ressource.adb" file  yes  yes  yes  yes  
Deletion Management (allows to definitively remove zones and experiments (single or series) stored in the database)  yes  yes  yes  yes  
Customizing Menus (to change the visible icons shown on the toolbars)  yes  yes  yes  yes  
Copying Signals, Moving Axes, Merging Axes, Scrolling, Zooming, Magnifying Glass Option, Autoscaling, Cursor Tool, .  yes  yes  yes  yes  
Chart Size Adjustment, Selecting Default Temperature, Merging Multiple Signals  yes  yes  yes  yes  
Draganddrop a signal (from treeview to chart (and vice versa), from chart to chart, from treeview to treeview)  yes  yes  yes  yes  
Saving and Loading Macros (recorded actions performed by the user to be applied again quickly)  yes  yes  yes  yes  
Exporting Chart (available formats: *.png, *.gif, *.bmp, *.jpg, *.emf, automated exportation to MSWord)  yes  yes  yes  yes  
Exporting Points (with or without interpolation, *.txt and *.csv, Excel *.xls)  yes  yes  yes  yes  
Customizing Chart: Background, Border and Margins, Legend, Titles, Themes, Axes, Series, Adding and Customizing Notes and Images, etc.  yes  yes  yes  yes  
Supported languages and translation for all Calisto features: English, French, Chinese  yes  yes  yes  yes  
Types of kinetic analyses supported  
Isoconversional (modelfree) kinetic analysis  yes  yes  yes  
Custom arbitrary chosen formal kinetic models and reaction rates introduced manually  yes  yes  yes  
e.g. da/dt =1e9 * exp(100000/8.314/(T+273.15)) * (1a)^1 + 1e10 * exp(100000/8.314/(T+273.15)) * (1a)^2*a^0.5  yes  yes  yes  
Formal one or multistage modelbased kinetic analysis for discontinuously collected data  yes  yes  yes  
Formal one or multistage and concentration modelbased kinetic analysis  yes  
Data types and their combinations used for kinetic evaluation  
Discontinuous data composed from only few points (sparse data points, e.g. GC, HPLC data collected e.g. at three temperatures only)  yes  yes  yes  
Continuous data:  yes  yes  yes  
Trcontrolled data  yes  yes  yes  
heat flow (e.g. DSC)  yes  yes  yes  
pressure P (dP/dt) data  yes  yes  yes  
mass loss and its rate (TG, DTG)  yes  yes  yes  
all other thermoanalytical data collected continuously such as TGEGA, TMA, etc.  yes  yes  yes  
all microcalorimetric data such as TAM, C80, etc.  yes  yes  yes  
nonisothermal  set of runs at various heating rates  yes  yes  yes  
isothermal  set of runs at various temperatures  yes  yes  yes  
set of runs at various heating rates and temperatures (combination of nonisothermal and isothermal data)  yes  yes  yes  
Adiabatic data (e.g. THT ARC, Fauske VSP, Omnical DARC)  yes  yes  
Tjcontrolled data (isoperibolic) and cascade controlled (PID controller) data of reaction Calorimetry (both batch or semibatch) (e.g.: Mettler RC1, Setaram DRC, HEL Simular, ChemiSens CPA 102, 202 and many others)  yes  
Combination of Trcontrolled data of different types (e.g. DSC and TG data)  yes  yes  yes  
Combination of Adiabatic and Trcontrolled data (e.g. ARC and DSC for calculation of the kinetic parameters) for determination of safety hazard indicators (e.g. TMRad24, SADT)  yes  yes  
Combination of Tr and Tjcontrolled data for thermal safety and process optimization purpose (e.g. DSC and Mettler RC1)  yes  
Methods for estimation of the kinetic parameters  
Arrheniustype dependence of the reaction rate on temperature  yes  yes  yes  
Linear optimization suitable for single stage models  yes  yes  yes  
Nonlinear optimization; applicable to data collected discontinously (sparse data points)  yes  yes  yes  
Model ranking (Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC) and weighted scores (w)) for comparing and discriminating best kinetic models based on information theory  yes  yes  yes  
Nonlinear optimization method; applicable to complex multi stage models  yes  
Simulation of thermal behavior in mg, kg and ton scales  
Temperature profiles applicable for thermal behavior predictions  
Isothermal  yes  yes  yes  
Nonisothermal  yes  yes  yes  
Stepwise  yes  yes  yes  
Modulated temperature or periodic temperature variations  yes  yes  yes  
Rapid temperature increase (temperature shock)  yes  yes  yes  
Real atmospheric temperature profiles for investigating properties (50 climates by default with yearly temperature profiles with daily minimal and maximal fluctuations)  yes  yes  yes  
Customized temperature and humidity profiles: possibility to compare the reaction progress of substances at any temperature and relative humidity (useful in combination with datalogger)  yes  yes  yes  
NATO norm STANAG 2895 temperature profile: Zones A1, A2, A3, B1, B2, B3, C0, C1, C2, C3, C4, M1, M2, M3  yes  yes  yes  
Specific features  
Extended option for High Sensitivity Isothermal Heat Flow Microcalorimetry (e.g. TAM data of propellants, surveillance of ammunitions, quality control) allowing to calculate the kinetic parameters from long term isothermal data for very precise lifetime prediction applying data collected during the first percent of degradation  yes  yes  yes  
Sample Controlled Thermal Analysis: possibility to optimize temperature program in such a way that it allows obtaining the value of the constant reaction rate set by the user (allows creating temperature profiles for achieving e.g. TGAcurves with constant mass loss rates or DSCcurves with rate controlled heat release (or consumption))  yes  yes  yes  
Combination of Trcontrolled data e.g. TG & DSC/DTA & MS data in multiprojects for simultaneous comparison of mass loss, heat flow and volatiles species evolution  yes  yes  yes  
Bootstrap method for evaluation of prediction band (e.g. 95, 97.5 or 99 % confidence intervals), particularly important for longterm predictions (e.g. stabilizers in propellants, vaccines, etc.)  yes  yes  yes  
Heat Accumulation, Thermal Runaway and Explosion  yes  yes  
Simulation of transient heat conduction systems such as thermal explosion in solids (this analysis considers the variation of temperature with time and position in one and multidimensional systems)  yes  yes  
Simulation of lumped systems such as thermal explosion in low viscous liquids (this analysis considers that the temperature of a body varies with time but remains uniform throughout at any time)  yes  yes  
Influence of packaging geometry, material properties and insulations in simulation of the storage of dangerous materials  yes  yes  
Infinite slab  yes  yes  
Infinite axissymmetrical cylinder  yes  yes  
Limited cylinder with given H/D ratio (H:height, D:diameter) and flat lids (e.g. drums, containers,etc.)  yes  yes  
Sphere (application of volume equivalent sphere radius and surfacetovolume ratio S/V, useful for the characterization of any package regardless its specific shape and size)  yes  yes  
Comparative thermal explosion analysis (e.g. cylinder with given H/D ratio vs sphere with equivalent surfacetovolume ratio S/V)  yes  yes  
Others geometries (after exportation of the kinetic parameters into codes like Abaqus, Ansys dedicated for the more complex geometries)  yes  yes  
Inert shell and partitions, multilayer packaging materials (different layers of insulation with different thicknesses)  yes  yes  
Different properties for separate part of an object  yes  yes  
Possibility of considering temperature dependence of physical properties  yes  yes  
Export of material data properties from database (with possible customization of the material property list)  yes  yes  
Heat sources in an object  yes  yes  
Possibility of application of specific kinetic parameters for separate parts of an object  yes  yes  
Heatgenerated by a reaction and or nonreactive heat sources  yes  yes  
Timedependent boundary conditions:  yes  yes  
1st kind  Prescribed temperature at the surface (Dirichlet condition)  yes  yes  
2nd kind  Heat flux at the surface (Neumann condition)  yes  yes  
3rd kind  Heat transfer at the surface (Newton law, convective heat transfer, mixed boundary conditions)  yes  yes  
Determination of hazard indicators  yes  yes  
Time to Maximum Rate under adiabatic conditions (TMRad) for any chosen starting temperature  yes  yes  
Safety diagram: runaway time as a function of process temperature under adiabatic conditions (TMRad = f(T))  yes  yes  
Automatic determination of the starting temperatures corresponding to TMRad of 7 days, 24h, 8h and 4h  yes  yes  
Self heat rate curves dT/dt, dQ/dt and dalpha/dt (dP/dt possible in combination with e.g. ARC data for pressure/gas generation and ventsizing calculations)  yes  yes  
Influence of the different Phi factors (Phi=1 and Phi>1) on the TMRad and on dT/dt, dQ/dt, dalpha/dt and dP/dt  yes  yes  
Total energy release under adiabatic conditions  yes  yes  
Total pressure release under adiabatic conditions (possible in combination with e.g. ARC data)  yes  yes  
Temperature corresponding to ARC detection limit such as 0.02 K/min for any Phi factors  yes  yes  
Automatic determination of the SelfAccelerating Decomposition Temperature (SADT) according to the recommendations of Manual of Tests and Criteria of the United Nations on the transport of dangerous goods  yes  yes  
Automatic determination of the critical hot discharge temperature 'Tin' in e.g. a container or critical surrounding temperature 'Tout'  yes  yes  
Automatic determination of the critical radius 'r' of e.g. a container and the critical thickness 'd' of an insulation layer of such a container  yes  yes  
Determination of the relationship between the input factor Xi (thermal conductivity, density and specific heat) and the output Y (time to thermal explosion) for identifying the physical property of a material (chemical or packaging layer) which will mostly influence the time to thermal explosion  yes  yes  
Setting of time steps, spatial mesh and numerical precision and computation speed  yes  yes  
Variable adaptive time step  yes  yes  
Uniform and Nonuniform spatial mesh  yes  yes  
Second order accuracy in both space and time and numerical stability even for large time steps (to ensure high precision and decreases by orders of magnitudes the calculation time)  yes  yes  
Display of results  yes  yes  
Evolution of the temperature profile T(t) and reaction progress a(t) in the crosssection or in a selected point of an object  yes  yes  
Temperature and conversion distribution on isolines (2D) and/or 3D graphs  yes  yes  
Animated isolines (2D) and/or 3D views of both temperature and reaction progress distribution  yes  yes  
Chemical reactors considered  yes  
Batch  yes  
SemiBatch  yes  
Continuous Stirred Tank Reactor (CSTR)  yes  
PlugFlow (PFR)  yes  
Cascade of reactors including  yes  
Stream (continuous or discontinuous with or without dosing conditions for optimization of feed rate dosing profile)  yes  
Mixing  yes  
Splitting  yes  
Heating  yes  
Temperature modes  yes  
Adiabatic  yes  
Trcontrol  yes  
Tjcontrol (isoperibolic)  yes  
Cascade control (PID controller)  yes  
Customizable temperature profiles (isothermal, nonisothermal, stepwise, own profile, etc.)  yes  
Process Flow Diagram (PFD) modules for an easy saving of various reactor types  yes  
Process optimization (e.g. adjustment of the best feed or temperature profiles for maximum yield and selectivity)  yes  
Specific process control (process parameters (e.g. feed or temperature) can be constraint to remain below or above some critical values at all time during the reaction for achieving inherent safety process)  yes  