Kinetic Analysis of the Thermal Decomposition of Polymer-Bonded Explosive Based on PETN: Model-Fitting Method and Isoconversional Method

This work investigates kinetics and thermal decomposition behaviors of pentaerythritol tetranitrate (PETN) and two polymer-bonded explosive (PBX) samples created from PETN (named as PBX-PN-85 and PBX-PP-85) using the vacuum stability test (VST) and thermogravimetry (TG/DTG) techniques. Both model-free (isoconversional) and model-ﬁtting methods were applied to determine the kinetic parameters of the thermal decomposition. It was found that kinetic parameters obtained by the modiﬁed Kissinger–Akahira–Sunose method (using non-isothermal TG/DTG data) were close to those obtained by the isoconversional and model-ﬁtting methods that use isothermal VST data. The activation energy values of thermal decomposition reactions were 125.6–137.1, 137.3–144.9, and 143.9–152.4kJ · mol − 1 for PBX-PN-85, PETN, and PBX-PP-85, respectively. The results demonstrate the negative eﬀect of the nitrocellulose-based binder in reducing the thermal stability of single PETN, while the polystyrene-based binder seemingly shows no adverse inﬂuence on the thermal decomposition of PETN in our presented PBX compositions.


Introduction
Pentaerythritol tetranitrate (also called PETN), which is a popular nitrate explosive, is extensively used in civilian and military applications due to its high energy characteristics [1,2]. e main drawback of PETN is that it is highly sensitive to impact and friction pulses, and thus there may be explosion risks in the processes of production, storage, and especially in the transportation of PETN. erefore, PETN is often used in the form of a polymer-bonded explosive (also called PBX) [3][4][5] to overcome this shortcoming. e PBX sample based on PETN is a composite explosive in which the PETN particles are covered and bonded by the polymeric binder. e polymeric binder in PBX composition can reduce the sensitivity of explosive to mechanical pulses and make the charge compression easier and safer [6][7][8][9].
Like all other high-energy materials, the PBX composition is also an unstable thermodynamic system [10]. Compared to the single explosive such as PETN, the thermal decomposition of PBX is generally more complex because of the incompatible reactions between explosives and the polymeric binders [11][12][13][14], and the safety and performance of PBX sample may change during aging. erefore, the behaviors and kinetics of thermal decomposition of PBX samples should be evaluated and clarified to assess the safety and the thermal hazard potential during storage. e decomposition kinetics and thermal behaviors of several composite explosives based on PETN have been carefully examined, and it was shown that the presence of the additive materials exerts powerful impacts on the thermal decomposition of these explosives [15][16][17][18][19].
In this study, we investigate the thermal decomposition kinetics of two PETN-based PBX samples named as PBX-PN-85 and PBX-PP-85. ese results were compared with those of a single PETN to evaluate the influence of binders on the thermal decomposition behaviors of explosives. ermal analyses were conducted by thermogravimetry (TG/DTG) and vacuum stability test (VST), where the VST method (developed by STABIL, Czech Republic) is commonly used to examine the chemical stability, compatibility, and shelf-life of energetic materials in recent years [10,14,20]. Based on TG/DTG curves and VST results, we determined kinetic parameters by using isoconversional and model-fitting methods according to the ICTAC kinetics committee recommendations [21]. It is noteworthy that while model-fitting and isoconversional methods are both applicable to isothermal data, the model-fitting method is generally not applicable to non-isothermal techniques since the kinetic parameters obtained for non-isothermal data utilizing model-fitting technique are highly variable [22].

Materials.
We imported Class-1 PETN with a melting temperature of over 139.0°C from South Korea. Good quality nitrocellulose (NC), that has a nitrogen content of 12.20%, was supplied from the Vietnam factory. Polystyrene (PS), which has an average molecular weight of 80,000 u, was produced in our laboratory. e widely used plasticizer, dioctyl phthalate (DOP), was utilized to make the binder from the PS and NC (the ratios of DOP/NC and DOP/PS were 3/1 and 2/1, respectively). Besides, some suitable solvents such as toluene and ethyl acetate are used in this study.

Preparation of PBX Samples.
PS and NC were plasticized by DOP (the ratios of DOP/NC and DOP/PS are 3/1 and 2/1, respectively), then dissolved in the suitable solvents (toluene for PS and ethyl acetate for NC), maintaining the polymer/solvent ratio of 1/15 (w/v), and kept for approximately 5 hours to obtain a homogeneous solution. e sample preparation method was carried out by mixing PETN crystals with the binders based on PS and NC. e mixing process took place on the heating device at 70°C for 30 minutes, and the PETN crystals were covered by the binder layers [6,23]. After that, the mixtures were vacuum-dried at 60 to 70°C for 5 hours to remove the solvents. Table 1 shows the compositions of two proposed PBX samples.

Experimental Techniques.
e thermal decomposition behaviors of PBX samples were determined by employing thermogravimetry (TG/DTG) and vacuum stability test (VST). e experimental conditions are as follows.
TG/DTG analysis was conducted using a NETZSCH STA 409 PC/PG (NETZSCH-Gerätebau GmbH, Selb, Germany). PBX samples (around 5.0 mg) were put in an aluminum oxide crucible and heated at various heating rates of 3, 5, 7, and 10 K·min − 1 . e sample vial was heated from 30 to 350°C. Experimental heating processes were carried on under a dynamic nitrogen atmosphere with a flow rate of 30  e sample mass was 20.0 mg, and we performed the test from 5 to 10 days (depending on the temperature test), until the end of the thermal decomposition reaction.

Kinetics Analysis.
e kinetics of thermal decomposition reactions in solids can be expressed as the following equation: where t is time (s); φ is the extent of conversion; T is the temperature (K); and k(T) is a function of conversion representing the reaction model. For the TG tests, the extent of conversion φ is defined as follows: where m i , m t , and m f are the initial sample mass, sample mass at any time t, and sample mass at the end of the decomposition, respectively. For the VST test, the value of φ is expressed as [13,24] where p max is the pressure at the end of decomposition, p t is the pressure at any time t, and p 0 is initial pressure. Following the Arrhenius equation, k(T) is computed by where R, T, E, and A are the universal gas constant, absolute temperature, activation energy, and pre-exponential factor, respectively.
(1) Model-Fitting Method. Model-fitting methods employed different forms of reaction model for finding the referred decomposition kinetic parameters; some of these reaction models are illustrated in Table 2.
For isothermal conditions, equation (1) can be used in the integral form as follows [22,24]: where i(φ) is the integrated form of the reaction model c(φ). Replacing each reaction model from Table 1 into equation (5), the rate constant can be computed from the slope of the straight line when plotting i(φ) against t. en, we can base on i(φ) and c(φ) formulations in the reaction model to determine k(T). For each selected reaction model, k(T) is determined at different temperatures, and the kinetic parameters can be calculated by the Arrhenius equation: (2) Isoconversional (Model-Free) Method. e reaction model (as in equation (1)) in the isoconversional method is considered as being independent of temperature. Under the isothermal condition of the VST test, combining equations (4) and (5), we obtain [13,22,24] where the activation energy (E φ ) is estimated from the slope of the straight line when plotting (− lnt φ,j ) against (1/T j ). us, the dependencies of activation energy (E φ ) on the conversion of the thermal decomposition (φ) could be obtained.
At a linear heating rate β � dT/dt and under non-isothermal condition, equation (1) can be described as the following: and Kissinger, Akahira, and Sunose introduced the modelfree method using the following linear equation [22,26]: For each extent of conversion φ, the values of E and A can be found from the slope and the intercept when plotting ln(β/T 2 ) against 1/T. e function i(φ) can be determined according to the methodology reported in the literature [27,28].
For comparison, the characteristic parameters of TG/DTG curves of PETN, PBX-PN-85, and PBX-PP-85 are summarized in Table 3.
It has been shown that only a single decomposition step is related to the thermal decomposition of a single PETN (i.e., one-step reaction model) in all samples. e peak temperatures in DTG curves for the thermal decomposition of PETN, PBX-PN-85, and PBX-PP-85 were observed in a range of 186-201, 183-199, and 189-204°C, respectively. Generally, the decomposition residue of all samples increases with the increasing heating rate, and the residue mass of PBX-PN-85 and PBX-PP-85 may be less than that of PETN because of the presence of inert ingredient (i.e., DOP and PS) in their formulation.
As seen, the thermal decomposition of PETN occurred at a higher temperature than PBX-PN-85 but lower than that of PBX-PP-85. DTG curves in Figure 1 also demonstrated the role of the binder on the thermal decomposition behaviors of PETN. ermal stability and decomposition temperature of single PETN are clearly affected by the binders. Specifically, the energetic binder (i.e., NC-based binder) reduces the decomposition temperature of single PETN, while the nonenergetic binder (i.e., PS-based binder) hardly affects the thermal stability of PETN. Interestingly, the PBX-PN-85 sample also shows an early decomposition. is may be caused by the exothermic decomposition effect of NC and the release of the gaseous products in the early decomposition of PETN in the binder matrix [13]. Contrarily, the decomposition temperature of PBX-PP-85 has a slight increase because this PBX sample has less amount of PETN (compared to single PETN) in its composition [8].
According to equation (2), the corresponding curve of the conversion φ versus temperature T was obtained and is Table 2: Different reaction models generally applied to describe the thermal decomposition in solids [21,22,24,25].
Model no.
Reaction model Contracting volume (R3)  T ot -onset temperature of decomposition exotherm peak, T i -the initial temperature of the mass loss, and T p -the peak temperature of the DTG curve. 4 Advances in Materials Science and Engineering presented in Figure 2. It can be observed that all of the φ − T curves have the form of the sigmoid function (i.e., the "S" shape curve), which is often obtained in the thermal analysis of high energetic material [29]. As recommended by the ICTAC kinetics committee [21,30], the modified Kissinger-Akahira-Sunose (KAS) technique was applied for investigating the relationship between the activation energy and the conversion fraction.
e fitting lines at each conversion fraction are presented in Figure 3 by plotting ln(β/T 2 ) against 1/T. e activation energy E φ and the pre-exponential factor A of the thermal decomposition of all samples were evaluated based on the slopes and intercepts of the plots shown in Figure 3 and are expressed in Table 4. e dependence of activation energy on the conversion by the modified KAS method is shown in Figure 4. To verify these results obtained by the KAS method, the kinetic parameters were recalculated using NETZSCH ermokinetic software according to the ASTM E698 [31]. e E φ values of single PETN, PBX-PN-85, and PBX-PP-85 are 138.2, 129.7, and 150.9 kJ·mol − 1 , respectively. e results were close to those obtained from the KAS method with the maximum difference of approximately 5%, which indicates the accuracy and reliability of measurements.
Compared to single PETN, the introduction of NC in PBX-PN-85 formulation leads to the decrease of its Advances in Materials Science and Engineering activation energy because the thermal stability of the nitrate group in NC is less than that in PETN [32,33]. For PBX-PP-85 formulation, besides the reducing mechanical sensitivity of the explosive, an inert binder system (i.e., DOP/ PS) acts as a stabilizer by trapping the active thermal decomposition product [29,34], resulting in the higher activation energy and thus higher stability versus single PETN.

e VST Results.
Results of TG/DTG were verified by VST studies, which were conducted at several isothermal temperatures (e.g., 125, 130, 135, and 140°C). From the relationship of the volume of released gas versus heating time in VST tests, the conversion fraction values (φ) for each time (t) were computed according to equation (3), and the curve of the conversion fraction (φ) versus heating time (t) is plotted for all samples in Figure 5.      It is obvious from Figure 5 that the obtained φ − t curves of PBX-PN-85, PBX-PP-85, and PETN obey the logarithmic trend, and the decomposition rate of all samples increased when the temperature was increased. In the case of PBX-PN-85 and PETN, it is apparent that the conversion grew rapidly during the first 2000 minutes and the conversion rate is low in the next 8000 minutes until the samples were completely decomposed. Similarly, the conversion rate of PBX-PP-85 can also be derived from Figure 5. In the case of PBX-PP-85, the conversion also increased rapidly in the first 2000 minutes at 135 and 140°C, while the conversion took longer time when tested at 125 and 130°C. By comparing the results of all samples, it was found that PBX-PP-85 sample took a longer time for completing the conversion than PBX-PN-85 and PETN samples. For example, at an isothermal temperature 125°C, the time for completing the conversion of PBX-PN-85, PETN, and PBX-PP-85 was 9990, 10485, and 10970 min, respectively. e differences in the thermal stability of three samples were assessed by the activation energy employing the model-free (isoconversional) and model-fitting methods.

Isothermal Isoconversional Method from VST Data.
Next, we utilize VST data to find kinetics parameters of PBX samples. By drawing − ln(t) versus 1/T curve at each temperature according to equation (7) for each conversion φ, the kinetic thermal decomposition parameters of three samples were calculated and are expressed in Table 5. e dependence of activation energy on the conversion is presented in Figure 6.

Isothermal Model-Fitting Method from VST Data.
e model-fitting method includes two fits: the first fit for determination of the model rate constant k(T) that best fits the VST data and the second fit for determination of several kinetic parameters such as pre-exponential factor A from the Arrhenius equation and the activation energy E φ [13,22]. Fourteen reaction models from Table 1 were selected for fitting the conversion of the thermal decomposition. Table 6 shows the results for the pre-exponential factor, the activation energy, and squares of the correlation coefficients obtained by plotting the integrated form i(φ) versus heating time t at each isothermal temperature.
From Table 6, it was observed that the kinetic parameters (specifically, pre-exponential factor and activation energy), which are derived from the VST data utilizing the model-fitting method, are essentially independent of the reaction model used. It means that there is an insignificant difference between the kinetic parameters (i.e., E φ and A) calculated using different model reactions.
For selecting the best model fitting, the correlation coefficient R 2 was used as a parameter. It was found that the best model that describes the thermal decomposition of PETN is contracting volume (R3) and the best model for PBX-PN-85 is 3-D diffusion-Jander (D3) while the best model for PBX-PP-85 is second-order (F2).
On the other hand, we found that the activation energy values calculated by the isothermal isoconversional method are approximate to those derived from the isothermal model-fitting method using the VST technique. To make a  Advances in Materials Science and Engineering      Table 7. e values in Table 7 demonstrate the role of the binder on the initiation activation energy of PETN. e energetic binder based on NC reduces the activation energy (i.e., PBX-PN-85 has lower thermal stability than that of PETN), while the introduction of the nonenergetic material based on PS increases the activation energy of the explosive. e ranges of E φ values obtained by different methods were 125.6-137.1 kJ·mol − 1 for PBX-PN-85, 137.3-144.9 kJ·mol − 1 for PETN, and 143.9-152.4 kJ·mol − 1 for PBX-PP-85. ese ranges are narrow, proving the equivalence of these methods with each other. It is also noteworthy that E φ of PETN obtained in this work is in good agreement with that reported by Pouretedal et al. [15] (135.1-136.9 kJ·mol − 1 ) and by Künzel et al. [16] (137.4 ± 3.4 kJ·mol − 1 ).
According to Brill et al. [35], the E φ and ln A values-determined from isothermal techniques in the same conditions-usually compensate each other. ey found that all of E φ − ln A pairs that lie in the regression line represent legitimate measurement and reliable results. We have plotted all E φ values of PETN, PBX-PN-85, and PBX-PP-85 versus corresponding ln A values (presented in Table 7) in Figure 7.
It can be observed the linear relationship of E φ with ln A values, which confirm the accuracy of the kinetic parameters obtained by the isoconversional method and model-fitting method using the VST test. e results indicate that the NC-based binder accelerates the thermal decomposition of PBX, thus reducing the activation energy of single PETN. One more valuable conclusion of this study is the influence of the PS-based binder. Specifically, the presence of the PS-based binder causes no adverse effect on the thermal stability of the main explosive in PBX composition.

Data Availability
e data used to support the findings of this study are included in the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  10 Advances in Materials Science and Engineering