Introduction

Polyhedral oligomeric silsesquioxane (POSS) molecules are inorganic/organic hybrid cage-type nanostructures with the general formula of (RSiO3/2) n , where n is 6, 8, 10, etc., and R can be hydrogen, alkyl, alkylene, aryl or other organofunctional derivatives. Due to the many useful applications, various research works have been done on POSS compounds in recent years [1]. Cubic POSS ((RSiO3/2)8) compounds are especially important because they can be considered as starting materials for the synthesis of a wide range of materials with the desired properties [2, 3]. In octafunctional POSS compounds, the central inorganic core, Si8O12, is surrounded symmetrically by distribution of organic moieties. The composition of silsesquioxane containing inorganic core, Si8O12, and organic groups describes their thermal and mechanical properties. Moreover, the organic periphery on their silicon atom, which can be easily functionalized, allows a facile tuning of the silsesquioxane properties [1, 4]. Figure 1 shows the general structure of an octafunctional POSS compound.

Fig. 1
figure 1

General structure of a cubic POSS compound with the formula of (RSiO3/2)8

Full size image

Cubic POSS materials are widely used in various branches of material science such as catalysis [5,6,7], flame-resistant nanocomposites [8, 9], biomedicals [10, 11], light-emitting diode materials [12,13,14], ionic liquids [15], ionic crystals [16], electrolytes with high thermal stability [17] and functional coatings [18, 19]. Thermal stability is an important characteristic of POSS compounds, which improves the thermal properties of polymer nanocomposites and hybrids. Thermal analysis methods are used to investigate the thermal degradation of POSS materials. Thermal stability and decomposition of cubic POSS have been studied by the thermogravimetric analysis (TGA) [4, 10]. Different thermal analysis methods have been used widely for the evaluation of thermal stability [4, 10, 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. The heat of decomposition shows a relatively large error of about 10%. Meanwhile, the exothermic onset temperature, the temperature at which the first deflection from the baseline is observed, thermal decomposition temperature and the temperature at which maximum of mass loss give better reproducibility than that of the heat of decomposition. For classes of organic compounds containing energetic groups, there are some relationships between activation energy of thermolysis and the other stimuli for initiation of their decomposition [37, 38].

Development of suitable predictive methods on thermal stability parameters such as decomposition onset temperature can help the scientists to increase their knowledge on the synthesis of good candidate. Quantitative structure–property relationship (QSPR) studies are usually used for this purpose [39,40,41,42,43,44,45], but they require some complex molecular descriptors, e.g., constitutional and topological, as well as special computer codes, which need expert users. Some molecular moieties of the desired compounds can also be used to predict thermal stability parameters of classes of organic compounds [46]. Since molecular structure of a desired compound can be used to study its thermal analysis, the purpose of this paper is to introduce a novel simple model for evaluation of the decomposition onset temperature (T decom) as important parameter for the assessment of thermal stability of cubic POSS compounds. It will be shown that there is no need for complex computer code and descriptors as well as expert users. The new model correlates T decom to the molecular structure of POSS materials using several simple molecular descriptors.

Materials and methods

Experimental data of T decom under N2 for 54 cubic POSS compounds were collected from different sources. Among these compounds, 44 cubic POSS compounds containing simple or similar molecular fragments were used to construct the new model, which are given in Table 1. Ten cubic POSS compounds containing complex molecular moieties were also chosen to test the validity of the novel correlation.

Table 1 Predictions of Eq. (1) for the decomposition onset temperature of cubic POSS compounds, compared to the experimental data (training set)
Full size table

Results and discussion

Development of the new model

The study of T decom of different cubic POSS compounds has shown that it is possible to introduce a new correlation on the basis of the number of some specific atoms divided by molecular weight of POSS compound by considering the type and size of substituents without using complex molecular descriptors. It was found that the existence of several molecular fragments can increase or decrease the predicted T decom on the basis of elemental composition by two correcting functions. The following correlation was derived on the basis of the measured values given in Table 1 by using multiple linear regression method (R 2 = 0.9384) [47]:

$$\begin{aligned} T_{\text{decom}} & = 516.0 + 3.034 \times 10^{4} \frac{{n_{\text{C}} }}{\text{Mw}} - 1.021 \times 10^{4} \frac{{n_{\text{H}} }}{\text{Mw}} - 2.560 \times 10^{4} \frac{{n_{\text{O}} }}{\text{Mw}} \\ & \quad + 9.941 \times 10^{4} \frac{{n_{\text{Cl}} }}{\text{Mw}} + 121.3T_{\text{decom}}^{ + } - 108.3T_{\text{decom}}^{ - } \\ \end{aligned}$$
(1)

where n C, n H, n O and n Cl are the number of carbon, hydrogen, oxygen and chlorine atoms in R (Fig. 1), respectively; \(T_{\text{decom}}^{ + }\) and \(T_{\text{decom}}^{ - }\) are two correcting functions, which are used to show increasing and decreasing the contribution of nonadditive structural parameters, respectively; Mw is the molecular weight of the desired POSS compound. To derive Eq. (1) by multiple linear regression method, several steps have been done to find the variables containing statistically significant. At first, various combination of elemental composition, functional groups and structural parameters have been tested. It was found that \(\frac{{n_{\text{C}} }}{\text{Mw}},\frac{{n_{\text{H}} }}{\text{Mw}},\frac{{n_{\text{O}} }}{\text{Mw}}\) and \(\frac{{n_{\text{Cl}} }}{\text{Mw}}\) and the presence of some substituents with specific molecular structures are important. Moreover, four variables \(\frac{{n_{\text{C}} }}{\text{Mw}},\frac{{n_{\text{H}} }}{\text{Mw}},\frac{{n_{\text{O}} }}{\text{Mw}}\) and have significant contributions with respect to the other elements by evaluation of statistical parameters. Finally, the contribution of structural parameters has also been included by two terms and \(T_{\text{decom}}^{ - }\) for increasing and decreasing T decom. The study of T decom for polynitro arenes has confirmed that the same trend exists for coefficients n C, n H and n O [48]. As seen in Table 2, there is a direct dependency of and on the substituent type and size. For example, the values of \(T_{\text{decom}}^{ + }\) and \(T_{\text{decom}}^{ - }\) should be considered for phenyl and alkyl substituents under certain conditions, respectively, because the thermal behavior of phenyl substituents of POSS is completely different from those of alkyl substituted POSS. In fact, some phenyl substituents show considerably a higher high thermal stability. Since the collected data of T decom are under nitrogen atmosphere, cleavage of weak bond in substituents is one of the important factor for the contribution of . Since the coefficients of \(\frac{{n_{\text{C}} }}{\text{Mw}}\) and \(\frac{{n_{\text{Cl}} }}{\text{Mw}}\) in Eq. (1) have positive sign, increment of the number of carbon and chlorine atoms can increase the value of T decom. Meanwhile, decreasing the values of n H and n O can also increase the value of T decom. There is no contribution for the other atoms in Eq. (1) because their contributions do not improve the value of the coefficient of determination (R 2). The values of \(T_{\text{decom}}^{ + }\) and are given in Table 2 for different molecular fragments. They can adjust high deviations of the predicted T decom only on the basis of \(\frac{{n_{\text{C}} }}{\text{Mw}},\frac{{n_{\text{H}} }}{\text{Mw}},\frac{{n_{\text{O}} }}{\text{Mw}}\) and from experimental data. For those POSS compounds where the estimated results of T decom from \(\frac{{n_{\text{C}} }}{\text{Mw}},\frac{{n_{\text{H}} }}{\text{Mw}},\frac{{n_{\text{O}} }}{\text{Mw}}\) and show relatively large underestimated and overestimated predictions with respect to experimental data, respectively, the contributions \(T_{\text{decom}}^{ + }\) and are valid. As seen in Table 2, direct attachment of aromatic rings, hydrogen atoms and saturated or unsaturated hydrocarbon under certain conditions as well as the presence of some specific functional groups to the central inorganic core are important parameters for taking the existence and different values for \(T_{\text{decom}}^{ + }\) and . For those compounds where the conditions of either \(T_{\text{decom}}^{ + }\) or given in Table 2 are not satisfied, their values in Eq. (1) are equal to zero. It should be mentioned that Eq. (1) can be applied only for POSS compounds containing eight similar substituents.

Table 2 Values of \(T_{\text{decom}}^{ + }\) and \(T_{\text{decom}}^{ - }\) for prediction of the onset decomposition temperature of R8(SiO1.5)8 Compounds
Full size table

Statistical evaluation of the new model

Table 3 shows statistical parameters corresponding to variables given in Eq. (1). The values of standard deviation (SD) for variables indicate the significant of individual variables for estimation of the dependent variable. As seen in Table 3, the variables are significant because the values of SD of the variables are small relative to corresponding coefficients. The P value gives the probability that the observed results in a study could have occurred by chance. If the detected effect is not due to random variations for P value < 0.05, the effect will be significant. As indicated in Table 3, the results of P-value of four variables are less than 0.05, which confirm that they have a highly significant impact. Figure 2 shows the predicted T decom using the developed model versus experimental values given in Table 1. Figure 3 also shows the range of the absolute percent errors of new model for these data.

Table 3 Regression coefficients of Eq. (1), as well as their standard deviations (SD), P values and confidence intervals
Full size table
Fig. 2
figure 2

Predicted T decom/K using the developed model versus experimental values for training set

Full size image
Fig. 3
figure 3

Range of the absolute percent errors of new model for training set

Full size image

Table 4 shows reliability of Eq. (1) method for several further POSS compounds containing complex molecular structures. As seen in Table 4, the predicted results are also close to the measured data. Table 5 indicates statistical parameters of the new model in cross-validation. Goodness-of-prediction is measured by Q 2 where Q 2 is smaller than R 2 but if the model is well behaved, there should not be a significant difference between Q 2 and R 2 [49] and in a robust model the Q 2 is independent of partition size and also close to R 2 [50]. For a suitable QSPR model, its R 2 is greater than 0.6 and its Q 2 values are greater than 0.5 [51]. As seen in Table 5, the coefficient of determination for LOO-CV (\(Q_{\text{LOO}}^{2}\)) of Eq. (1) is 0.9189. Meanwhile, the fivefold CV (\(Q_{{ 5 {\text{CV}}}}^{2}\)), which is the average of 100 runs, is 0.8812. Both \(Q_{\text{LOO}}^{2}\) and \(Q_{{ 5 {\text{CV}}}}^{2}\) are smaller than R 2. Since all values of R 2, \(Q_{\text{LOO}}^{2}\), and \(Q_{{ 5 {\text{CV}}}}^{2}\) are significantly greater than the threshold values, i.e., 0.6 for R 2 and 0.5 for Q 2, respectively, Eq. (1) is a reliable predictive QSPR model. Since both the \(Q_{\text{LOO}}^{2}\) and \(Q_{{ 5 {\text{CV}}}}^{2}\) values are close to the R 2, the new model is robust, well behaved and more importantly, not over-fitted model. The mean absolute percent error (MAPE) and root-mean-squared error (RMSE) values and for LOO-CV and fivefold CV of Eq. (1) are close to the MAPE and RMSE of Eq. (1). Table 6 compares some statistical parameters for training and test sets. As seen, RMSE, MAPE and maximum error (Max Error) for test set is lower than training set, which indicates that external validation test for four further POSS compounds containing complex molecular structures is good.

Table 4 Predictions of Eq. (1) for the decomposition onset temperature of cubic POSS compounds, compared to the experimental data (test set)
Full size table
Table 5 Statistical parameters of the new model in cross-validation
Full size table
Table 6 Statistical parameters of the new model in external validation
Full size table

Conclusions

This work introduces a novel method for prediction of T decom of different POSS compounds. As indicated in Eq. (1), the novel model requires the values of \(\frac{{n_{\text{C}} }}{\text{Mw}},\frac{{n_{\text{H}} }}{\text{Mw}},\frac{{n_{\text{O}} }}{\text{Mw}}\) and as well as two correcting functions \(T_{\text{decom}}^{ + }\) and . The new method is a very simple model because there is no need to use computer codes and complex molecular descriptors. Equation (1) can explain important structural parameters to have the desired values of T decom for a new designed POSS compound. In contrast to the other QSPR methods, Eq. (1) gives the simplest and reliable method for calculation of T decom of different POSS compounds with good heat safety.