Introduction

Thermal stability of an organic compound containing energetic groups is an important feature for assessment of its sensitivity and shelf life [1–5]. Thermal analysis methods are frequently applied to estimate thermal stability of different types of energetic compounds [6–8]. They can be used to determine temperature [9], heat [10, 11] and activation energy [12] of decomposition process of energetic materials. Beside heat, the other kinds of stimuli such as impact [13–16], shock [17], friction [18] and electrostatic charges [19, 20] can also cause initiation reaction of energetic compounds. It was shown that activation energy of thermolysis of different classes of energetic compounds can be related to impact [21] electrostatic [22, 23] and friction [24] sensitivities.

Organic peroxides contain hazardous peroxide functional group in their molecular structures that can initiate their decomposition upon external stimuli such as impact [25]. They are unique in their stability characteristics and potential hazards because of the weak oxygen–oxygen bond in the peroxide functional group [26, 27]. However, they can decompose violently and produce free radicals by heat, friction, mechanical shock, low temperature and various contaminants [28–30]. Thus, decomposition of organic peroxides can cause dangerous occurrences. Due to the various incidents caused by organic peroxides in the laboratory or industrial processes during their storage and transportation, the recognition of their thermal stability has the utmost importance from safety point of view [31–36]. Therefore, the theoretical and practical studies of reactivity hazards of organic peroxides can help chemists and chemical engineers for the safe use of these compounds. It is necessary to investigate some related characteristics to thermodynamic and kinetic stability of organic peroxides such as decomposition onset temperature, heat of decomposition and self-accelerating decomposition temperature [37–40].

Organic peroxides are widely used in industrial and military fields such as oxidants, curing agent, polymerization catalyst and bleaching agent [27, 41]. There are several techniques for determining thermal stability hazards of organic peroxides, e.g., differential scanning calorimetry (DSC) and calorimetry [37–44]. The study of thermolysis of various energetic compounds can be used for estimation of their thermal stability and sensitivity parameters [21–23, 45–47]. Since thermal stability of hazardous materials is an important feature in their shelf life and safety aspects [9, 48, 49], several methods were reported recently for prediction of thermal stability of hazardous materials such as nitroaromatic energetic compounds, nitramines, nitroaliphatic energetic materials [46, 50–52], ionic liquids [53, 54] and organic peroxides [37–40].

The quantitative structure–property relationships (QSPR) method was applied to explore the relationship between the heat and temperature of decomposition of organic peroxides and their quantum properties [38, 40]. The QSPR method was also used to derive a correlation between self-accelerating decomposition temperature (SADT) of organic peroxides and their molecular structures [39] or their quantum mechanical properties [37]. Complex descriptors were used in these works, which include dissociation energy, bond length of the oxygen–oxygen bond, the number of peroxide functional groups, molecular hardness, the oxygen balance, the charge on oxygen atoms of peroxide bonds, the average local Fukui function on O atoms of the peroxide bond and the eV energy difference between the LUMO and HOMO orbitals [37–40]. All of these methods require specific computer codes and expert researchers.

The purpose of this work is to introduce two new simple models for evaluation of decomposition onset temperature and heat of decomposition as two important parameters for assessment of thermal stability of organic peroxides [2, 8, 55–68]. These models correlate decomposition onset temperature and heat of decomposition of organic peroxides to their molecular structures using several simple molecular descriptors. The predicted results by this new method are also compared with two of the best available methods, i.e., Lu et al. [38] and Prana et al. [40] models.

Theory

Onset decomposition temperature

The study of organic peroxides has indicated that it is possible to express decomposition onset temperature of these compounds as a function of several structural parameters. The data of decomposition onset temperature of 41 organic peroxides, which were obtained by DSC method [38, 40], were used for constructing and testing the new model. Experimental data of 33 different organic peroxides containing simple molecular structures have been used to derive the new model that is given in Table 1. Multiple linear regression method was used to obtain the relationship between decomposition onset temperature and molecular structure in Eq. (1) [69] as:

$$ T_{\text{Dec}} = 165.81 - 4.92n_{\text{O}} - 29.11 \lambda_{{{\text{C}} = {\text{O}}}} - 14.02 \lambda_{\text{sym}} - 30.81\lambda^{\prime} $$
(1)

where T Dec is decomposition onset temperature in  °C; n O represents the number of oxygen in molecular formula; λ C=O is 1.0 for the presence of carbonyl group; and λ sym is also 1.0 for those peroxides that have the same fragments attached to the –O–O– bond, i.e., R–O–O–R′ where R = R′. The presence of some molecular fragments may also affect the values of T Dec, which are incorporated in \( \lambda^{\prime} \) as correcting factor. Positive and negative contributions of various structural parameters in \( \lambda^{\prime} \) are discussed in the following sections.

Table 1 Comparison of the predicted decomposition onset temperature and heat of decomposition of some organic peroxides by Eqs. (1) and (2), Lu et al. [38] and Prana et al. [40] with experimental data as training test
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Positive values of λ

  1. 1.

    The presence of molecular fragments in any organic peroxide or in acyclic peroxides: The value of \( \lambda^{\prime} \) is 1.0 for these peroxides.

  2. 2.

    The existence of molecular moieties , or as well as : For three initial molecular fragments, \( \lambda^{\prime} \) is 0.7. Meanwhile, the value of \( \lambda^{\prime} \) for the latest molecular moiety is given as: \( \lambda^{\prime} = 0.7 \times \) the number of in molecular structure of peroxide.

Negative values of λ

For those peroxides containing –O–C(R)(R’)–O– molecular fragment structure, where R and R′ can be –CH3 or –CH2–CO, or –C(CO)2 in their molecular structures, the value of \( \lambda^{\prime} \) is −1.0.

Heat of decomposition

The data of heat of decomposition of 41 organic peroxides, which were obtained by DSC method [38, 40], were also used for constructing and testing of the new model. Table 1 also contains 33 experimental data of different organic peroxides with simple molecular structures, which were used as training set. The study of organic peroxides has indicated that it is possible to express the heat of decomposition of these compounds as a function of several structural parameters. Multiple linear regression method was used to obtain the relationship between heat of decomposition and molecular structure in Eq. (2) [69] as:

$$ \Updelta H_{\text{Dec}} = - 1551.45 + 41.13n_{\text{C}} - 1014.05\delta_{{\left( {{\text{HO}} - {\text{O}} \cdots - {\text{O}} - {\text{OH}}} \right)}} - 640.13\alpha - 857.68\beta $$
(2)

where ∆H Dec is the heat of decomposition in J g−1; n C represents the number of carbon in molecular formula; and \( \delta_{{\left( {{\text{HO}}{-}{\text{O}}{-} \cdots {-}{\text{O}}{-}{\text{OH}}} \right)}} \) is 1.0 for the presence of two hydroperoxy functional groups in the molecular structure. The presence of some molecular fragments may also affect the values of ∆H Dec which are incorporated in α and β as non-additive structural parameters.

Definition of α

For the presence of aromatic cycles, non-aromatic cycles and non-aromatic cycles with methyl substitution in them, the values of α are 0.1, 0.2 and 0.6, respectively. If organic peroxide has the fragment in its structure, the value of α is 2.

Definition of β

For the presence of molecular fragment in acyclic peroxides, β is 0.5. If two fragments of (–O–O–R) or exist, the value of β will be 0.4 or 0.6, respectively.

Results and discussion

Roles of different variables

As seen in Eq. (1), the coefficients of \( n_{\text{O}} \), \( \lambda_{{{\text{C}} = {\text{O}}}} \) and \( \lambda_{\text{sym}} \) have negative signs, which confirm that higher values of these parameters can reduce the values of T Dec. For the presence of several molecular fragments given in \( \lambda^{\prime} \), this reduction may be enhanced. However, the predicted results of Eq. (1) have confirmed that the number of oxygen atoms in the formula of organic peroxides has an important contribution in prediction of their decomposition temperature as well as their melting point [70]. The variable n O in Eq. (1) is also an important factor to evaluate hazards reactivity of energetic compounds [21–23]. The presence of λ sym term in Eq. (1) confirms that the decomposition process of peroxides is strongly affected by reactivity of the formed radicals [26, 28]. It was shown that the presence of some polar functional groups such as –O–C(O)–OO–(CO)–O– or –C(O)–OO–(CO)– given in \( \lambda^{\prime} \) can influence the thermal stability of organic peroxides like their melting points [70]. Positive sign of the number of carbon atoms in Eq. (2) shows that heat of decomposition of organic peroxides can increase with increasing the number of carbon atom. Meanwhile, the existence of two hydroperoxy functional groups in the molecular structure of organic peroxides can reduce their heat of decomposition.

Reliability of the predicted results and model validation

The R 2 values or the coefficients of determination of Eqs. (1) and (2) are 0.959 and 0.916, respectively. As seen in Table 1, the predicted decomposition temperatures of organic peroxides have a root-mean-square (rms) and the average absolute deviations (σ) of 6.06 and 5.18 °C, respectively. Moreover, deviation of the estimated onset temperature is more than 10 °C only for one compound. The values of rms and σ of Eq. (2) for the heat of decomposition are 146.41 and 118.19 J g−1, respectively. Tables 2 and 3 indicate statistical parameters of Eqs. (1) and (2), which allow comparing the relative weight of the variables in both models. Standard error shows a measure of the precision of the estimation of a coefficient that can determine precision over repeated measurements. The p value can assess the significance of an observed effect or variation. For p value <0.05, it may confirm that the observed effect is not due to random variations and the effect is significant. In this work, the p values for two models are near to zero. Thus, suitable statistical parameters and relatively good R 2 value validate that the predicted results of these new methods are in good agreement with experimental values.

Table 2 Standardized coefficients and some statistical parameters of Eq. (1)
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Table 3 Standardized coefficients and some statistical parameters of Eq. (2)
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Table 4 contains further eight organic peroxides with complex molecular structures, which have been used for external validation of these new models. As seen in Table 4, the new proposed methods give relatively good results that confirm high reliability of the new models. The comparison of predicted results of Eqs. (1) and (2) is shown in Tables 1 and 4 with Lu et al. [38] and Prana et al. [40] models as two of the best available QSPR methods, which are based on complex molecular descriptors. Table 5 summarizes several important statistical parameters of two new proposed models. As indicated in Table 5, the internal validation of proposed new model has been checked by applying the cross-validation test in which the leave-more-out (25 % out) cross-validation, denoted as Q 2 − CV, was employed.

Table 4 Comparison of the predicted decomposition onset temperature of several organic peroxides containing complex molecular structures by Eq. (1), Lu et al. [38] and Prana et al. [40] with experimental data as test set
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Table 5 Several statistical parameters of the new models
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Table 6 shows the comparison of statistical parameters of training and test sets of Eqs. (1) and (2) with Lu et al. [38] and Prana et al. [40] models, which confirm higher reliability of the new correlation. As indicated in Table 6, the internal validation of proposed new model has been checked by applying the cross-validation test in which the leave-more-out (25 % out) cross-validation was employed. Since the strong models are expected to show a low difference between Q 2 − CV and R 2 coefficients [71, 72], Eqs. (1) and (2) satisfy these conditions.

Table 6 Comparison of the several statistical and validation parameters of the proposed correlations with Lu et al. [14] and Prana et al. [16] models
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Figures 1 and 2 shows the correlation of the predicted results of Eq. (1) and (2) with experimental data of organic peroxides, which is reported in Tables 1 and 4. Thus, as shown in Tables 1, 4 and 6, the new relationships give good results for estimation of decomposition onset temperature and heat of decomposition of organic peroxides with respect to the models of Lu et al. [38] and Prana et al. [40], which are based on complex quantum mechanical descriptors.

Fig. 1
figure 1

Predicted decomposition onset temperature of organic peroxides versus experimental data for both training and test sets, which are given in Tables 1 and 4

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Fig. 2
figure 2

Predicted heat of decomposition of organic peroxides versus experimental data for both training and test sets, which are introduced in Tables 1 and 4

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Conclusions

In this study, two new reliable relationships were introduced for prediction of onset decomposition temperature and heat of decomposition of organic peroxides on the basis of their molecular structures. These novel correlations can predict decomposition onset temperature of organic peroxides and heat of decomposition of organic peroxides using three and two descriptors, respectively. In contrast to the other available QSPR models, which are based on complex molecular descriptors, the descriptors of Eqs. (1) and (2) can be easily found on the basis of the molecular structure of desired organic peroxide. Thus, the new proposed methods can be applied for the prediction of thermal stability parameters of any new organic peroxides only from their molecular structures. It was shown that the number of carbon and oxygen atoms as well as the presence of carbonyl or two hydroperoxy functional groups has significant effect on the thermal stability of organic peroxides. The suitable statistical parameters and relatively good R 2 values of Eqs. (1) and (2) showed that these new methods are reliable. Moreover, they have a good predictive power for the estimation of thermal stability of organic peroxides as compared to the best available methods.