Introduction

Determination of fire safety and flame retardancy of polymeric materials is important for the development of new polymers with desired flame-retardant properties. Specific heat release rate (HRR), heat release capacity (HRC) and total heat release (THR) are important parameters to reflect the combustion properties of materials. Microscale combustion calorimeter (MCC) is a small-scale flammability testing technique to screen polymer flammability prior to scale-up, which measures the values of HRR, HRC and THR on the basis of principle of oxygen consumption as well as cone calorimeter to determine the rate and amount of heat during combustion [1]. The thermogravimetric analysis (TG) and derivative TG (DTG) data of polymers are also adopted to assist the assessment of data from the MCC [2]. The HRR is the molecular-level fire response of a burning polymer, which can be obtained through analyzing the oxygen consumed by the complete combustion of the pyrolysis gases during a linear heating program. It can be divided by the rate of the temperature rise of a sample during a test to determine the HRC. However, the HRC appears to be a good predictor of the fire response and flammability of polymers. The HRC is a combination of the thermal stability and combustion properties, which can be obtained by the following equation [3–5]:

$$ {\text{HRC}} = \frac{{Q_{\text{c}}^{^\circ } \left( {1 - \mu } \right)E_{\text{a}} }}{{eRT_{\text{p}}^{2} }} $$
(1)

where \( Q_{\text{c}}^{^\circ } \) is the heat of complete combustion of the pyrolysis gases; μ is the weight fraction of the solid residue after pyrolysis or burning; E a is the global activation energy for the single-step mass-loss process or pyrolysis; T p is the temperature at the peak mass-loss rate in a linear heating program at a constant rate; e is the natural number; and R is the gas constant.

Additive molar group contributions and quantitative structure–property relationships (QSPR) methodology are two different approaches, which have been recently developed for prediction of the HRC [6–9]. The molar group contribution method is an easy approach, which uses additive contribution from a variety of functional groups. This method cannot be applied for prediction of the HRC of those polymers containing a particular functional group where it is missed from the used database of functional groups to build the model. Parandekar et al. [9] used QSPR approaches to predict the HRC as well as total heat release and % char using genetic function algorithms. In contrast to available additive molar group contribution methods, two QSPR models of Parandekar et al. [9] for estimation of the HRC are based on complex/unfamiliar descriptors such as AlogP98 and LUMO–HOMO energy, which require specific computer codes and expert users.

Since searching for new heat- and flame-resistant polymers has attracted considerable research activity during recent years, it is important to develop new approach for prediction of the HRC. The purpose of this work is to introduce a simple and reliable model for the prediction of the HRC of different polymers with their repeat units that are comprised of chemical groups/moieties such as methyl, phenyl, carbonyl, ether, amide and ester. The model is based on a molecular basis for polymer flammability that correlates the HRC test results.

Materials and methods

Experimental data of the HRC for 111 polymers with their repeat units containing chemical groups/moieties such as methyl, phenyl, carbonyl, ether, amide and ester are given in Table 1, which were taken from previous works where these data have been used to provide over 40 different empirical molar group contributions as well as (QSPR) methodology for prediction of the heat release capacity [6–9]. These data were taken as training set for building the new model. Since each new model is frequently tested on some chemicals that were not used in the model building, further an external dataset of 11 polymers compiled from several new experimental studies was considered to compare the new model predictions with molar group contribution methods.

Table 1 Comparison of the predicted results of HRC in J g−1 K−1 for the new model as well as two molar group contributions of Walters and Lyon [6] and Lyon et al. [8] with experimental data
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Results and discussion

Development of the new model

Parandekar et al. [9] introduced the following correlations on the basis of complex descriptors for prediction of the HRC:

$$ \begin{aligned} {\text{HRC}} &= [ - 2.03W(19.93 - \% {\text{H}}) + 2.76W(\% {\text{C}} - 87.02)\\ & \quad - 1.05W({\text{AlogP}}98 - 1.615) - 8.55W(0.668 - {\text{AlogP}}98) \\ & \quad- 1.12W(12 - R_{\text{bonds}} ) - 2.48W(H_{{{\text{bond}}\;{\text{donor}}}} ) + 5.46]^{2} \\ \end{aligned} $$
(2)
$$ \begin{aligned} {\text{HRC}} = & [67.55W(E_{{{\text{LUMO}}{-}{\text{HOMO}}}} - 0.104) + 0.743\,R_{\text{bonds}} - 0.647W(R_{\text{bonds}} - 26) + 0.00128{\text{VDE}} \\ & \, - 0.00056{\text{VDM}} - 0.6853Kap + 0.347\% {\text{C}} + 4.38W(\% {\text{C}} - 90.5) - 0.084{\text{MR}} + 0.248{\text{AC}} \\ & \, + 5.473W(0.95 - {\text{DM}}) + 1.07W(5.372 - {\text{S}}\_{\text{sNH}}_{2} ) - 15.73]^{2} \\ \end{aligned} $$
(3)

where W is the ramp function; %H is mass percent of hydrogen atoms; %C is mass percent of carbon atoms; AlogP98 is the log of the octanol–water partition coefficient that is calculated from empirical atomic contributions; R bonds is rotatable bonds; H bond donor is hydrogen bond donor; E LUMO–HOMO is LUMO–HOMO energy (au); VDE is vertex distance/equality; VDM is vertex distance/magnitude; Kap is kappa-1; MR is molecular refractivity; AC is atomic composition (total); and DM is dipole moment (au). Equation (2) was generated by genetic function algorithm containing six variables. Meanwhile, Eq. (3) was obtained by using trimer structures including 12 variables where the geometry was optimized with density functional method GGA/PW91 (L56). Although complexity of Eq. (3) is higher than Eq. (2), it provides more reliable predictions. Beside complexity of descriptors, these QSPR models require special computer codes and expert users.

It was indicated that the flash point and the auto-ignition temperature as two flammability characteristics of organic compounds depend on elemental composition and the contribution of some structural parameters, which are related to intra- and intermolecular interactions [10–13]. A careful examination of the HRC of many polymers revealed that the elemental composition as well as the specific structural parameters can be used to construct the new model. Among different elements, the number of carbon, hydrogen, nitrogen, oxygen, chlorine and silicon atoms is important for prediction of the HRC of a desired polymer. Since the presence of the other elements cannot improve the coefficient of determination (R 2) [14] value, their contributions are zero. Magnitude of R 2 is important for validation of the new correlation because it determines that whether regression accounts for the variation or not. For the value of R 2 equals 1.0, the regression accounts for all of the variations and that the correlation is deterministic. Meanwhile, the value of R 2 equals zero means that the regression accounts for none of the variations [15]. Beside the contribution of the above-mentioned elemental composition, repeat units containing molecular moieties –(CH2)n≥1–C(X)(Y)–Z– or –X–Ar–Y– with specific X, Y and Z groups can increase the value of the HRC on the basis of elemental composition. Moreover, the presence of oxygen and carbonyl groups between two aromatic rings as well as –O–P or >CHCO and –OC(O)– in repeat units can decrease the value of the HRC on the basis of elemental composition. Halogen-containing polymers can also act as heat-resistant polymers with the discovery that some chlorinated organic compounds were highly toxic and/or persistent in the environment. Since the contribution of specific groups in molecular moieties depends on the kind of these groups, different values may be considered, which are given in Table 2. On the basis of the training data set, the correlation of the HRC of a desired polymer can be obtained as follows:

$$ \begin{aligned} {\text{HRC}} \,= \,97.00 + \frac{{5850n_{\text{H}} - 17532n_{\text{N}} - 7495n_{\text{O}} - 19601n_{\text{Cl}} - 83828n_{\text{Si}} }}{{{\text{MW}}_{{{\text{repeat}}\;{\text{unit}}}} }} \\ & +\, 236.5{\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ}},{\text{XArY}}}} - 116.2{\text{HRC}}_{\text{YXArZ,Hal}} \\ \end{aligned} $$
(4)

where HRC is in J g−1 K−1; n H, n N, n O, n Cl and n Si are the number of moles of hydrogen, nitrogen, oxygen, chlorine, and silicon atoms per mole of repeat unit, respectively; MWrepeat unit is the molecular weight of repeat unit in g mol−1; and \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal are two correcting functions in J g−1 K−1 for the presence of molecular fragments \( \text{-}\left( {{\text{CH}}_{2} } \right)_{{{\text{n}} \ge 1}} \text{-}C\left( X \right)\left( Y \right)\text{-}Z\text{-} \) or –X–Ar–Y– and Y–X–Ar–Z or halogens (fluorine) in repeat units. The numerical coefficients for n H, n N, n O, n Cl and n Si have units of J mol−1 K−1. Since aliphatic polymers such as vinyl-based polymers have higher energy gap than aromatic or unsaturated polymers containing nitrogen and oxygen, the contribution of molecular fragment \( \text{-}\left( {{\text{CH}}_{2} } \right)_{{{\text{n}} \ge 1}} \text{-}C\left( X \right)\left( Y \right)\text{-}Z\text{-} \) in these polymers can increase significantly the value of \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \). As seen in Eq. (1), all coefficients of the number of atoms are negative except the coefficient of n H, which indicate that increasing the values of n N, n O, n Cl and n Si in a desired polymer can decrease the value of the HRC. Since the value of the coefficient n H is smaller than the coefficients of n N, n O, n Cl and n Si, its contribution in lowering the HRC is minor. The coefficients of electronegative elements n N and n Cl as well as n Si are much higher than the coefficient of n O, which indicates increasing n N, n Cl and n Si is more effective than n O for reduction in the HRC. Increasing of unsaturation in a new designed heat resistance polymer is one of the appropriate ways for decreasing the value of the HRC because it reduces the value of n H. Since the addition of relatively small amounts of silicon compounds to various polymeric materials can improve their flame retardancy [16–18], the coefficient of n Si in Eq. (4) has the largest negative value with respect to the coefficients of the other atoms. Thus, introducing silicon element and its groups into monomers of suitable polymers such as epoxy can also improve some other properties of the epoxy resins, such as thermal stability, high resistance to thermal oxidation, low surface energy and low toxicity [19, 20]. It was found that phosphorus-containing compounds or resins have been demonstrated as effective flame retardants for epoxy resins. As indicated in Table 2, this situation has been considered in the new correlation because the presence of –O–P can decrease the value of the HRC through the contribution of the HRCYXArZ,Hal. As indicated in Eq. (4), the effects of halogens for decreasing the value of the HRC appear in two terms: (a) n Cl for chlorine as additive term and (b) HRCYXArZ,Hal for fluorine as non-additive contribution. For those polymers such as biphenol phthalonitrile where \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal have no contribution and \( \frac{{5850n_{\text{H}} - 17532n_{\text{N}} - 7495n_{\text{O}} - 19601n_{\text{Cl}} - 83828n_{\text{Si}} }}{{{\text{MW}}_{{{\text{repeat}}\;{\text{unit}}}} }} < 97 \), the value of the HRC should be taken 20 J g−1 K−1.

Table 2 Values of two correcting functions \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal
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The last column of Table 2 indicates the use of two correcting functions in Eq. (4). For example, the value of HRC for poly(2-vinyl naphthalene) is calculated as follows:

Repeating unit:

figure a

Elemental composition and MW repeat unit :

n H = 10, n N = n O = n Cl = n Si = 0 and MWrepeat unit = 154.22 g mol−1.

The various parameters for repeating unit with general molecular fragment \( \text{-}\left( {{\mathbf{CH}}_{{\mathbf{2}}} } \right)_{{{\mathbf{n \ge 1}}}} \text{-}{\mathbf{C}}\left( \varvec{X} \right)\left( \varvec{Y} \right)\text{-}\varvec{Z}\text{-} \)

n = 1, X = –H, Y = carbocyclic aromatic without substituent and Z = –CH2– where as given in Table 2, the value of \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) is 1.5 J g−1 K−1. Since there is no contribution of HRCYXArZ,Hal, the value of HRCYXArZ,Hal = 0. Thus, Eq. (4) gives the value of the HRC as:

$$ {\text{HRC}} = 97.00 + \frac{{5850\left( {10} \right)}}{154.22} + 236.5\left( {1.5} \right) = 831\;{\text{J}}\;{\text{g}}^{ - 1} \;{\text{K}}^{ - 1} $$

Statistical parameters in new model and their significance

Table 3 shows important statistical parameters of the new model including regression coefficients, standard errors, t statistics, P values, as well as the upper and lower bounds of a 95% confidence interval. The statistical significance of the regression coefficients in predicting the HRC values can be evaluated on the basis of the above-mentioned statistical parameters as: (1) standard error—if the standard error is small relative to each coefficient, its variable is significant; (2) t statistic—since it is the ratio of coefficients to their standard errors, higher t statistic values correspond to the more significant coefficients [21, 22]; (3) P value—it shows the probability that a parameter estimated from the measured data should have the value which was determined. However, the effect of variable is significant and the observed effect is not due to random variations for P value of <0.05 [15]. As seen in Table 3, all statistical parameters show that the proposed nine descriptors in Eq. (4) have a highly significant ability to predict the HRC. As shown in Table 1, the predicted results of the new model, Walters and Lyon [6] and Lyon et al. [8], have been compared with experimental data. For several polymers, the group additivity methods of Walters and Lyon [6] and Lyon et al. [8] cannot be used because some particular functional groups in these polymers are absent for these methods. The predictive reliability of the new method has been tested for some new polymers, which are given in Table 4. As seen in Table 4, the results of the model as well as two group additivity methods of Walters and Lyon [6] and Lyon et al. [8] have also been compared with experimental data. Among eleven polymers given in Table 4, two group additivity methods can be applied only for seven polymers because these methods do not contain some particular functional groups.

Table 3 Regression coefficients, standard errors, t statistics, P values and confidence intervals for new model
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Table 4 Comparison of the predicted results of HRC in J g−1 K−1 of the new model as well as two molar group contributions of Walters and Lyon [6] and Lyon et al. [8] for some new polymers with the measured values
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Table 5 shows a comparison between further statistical parameters of Eq. (4) and two group additivity methods for model building and the test dataset. Root-mean-squared (RMS) error provides a reliable indication of the fitness of the model, which is independent of the distribution of data points. RMS values should be low and as similar as possible to ensure both the predictive ability (low values) and generalizability (similar values) [23]. Mean absolute deviation (MAD) is also a linear measure of errors that assesses the average size of errors when negative signs are ignored. Statistical parameters RMS, MAD and maximum of errors of these data for different models are given in Table 5. These parameters for new polymers are close to those obtained for training set. Low values of these statistical parameters confirm high reliability of the new model as compared to two available group additivity methods of Walters and Lyon [6] and Lyon et al. [8].

Table 5 Statistical parameters of the predicted results of HRC in J g−1 K−1 of the new model as well as two molar group contributions of Walters and Lyon [6] and Lyon et al. [8] for model building and testing
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It should also be mentioned that derivation of Eq. (4) was done from an examination of the HRC of different types of polymers given in Table 1 where their repeat units containing chemical groups/moieties by two principal steps as:

  1. 1.

    Elemental composition: It was found that the ratios of the number of moles of some atoms (n H, n N, n O, n Cl and n Si) to MWrepeat unit have important contribution because P values of corresponding coefficients are <0.05 [14].

  2. 2.

    The presence of some specific molecular fragments: It is possible to correct large deviations of the predicted results of step 1 through introducing two correcting functions \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal as well as adjusting their coefficients by minimizing RMS error [14].

Figure 1 shows a graphical comparison between the new model and the group additivity methods for all data. As indicated, the predictions of the new model methods exhibit a lower dispersion with respect to both group additivity methods. This is consistent with the fact that the new method has lower RMS values for model building and testing data as compared to group additivity methods. Figure 2 shows the range of absolute errors of the new model (AE = |HRCexp. − HRCpred.|) for all 122 data points given in Tables 1 and 4, which indicate high reliability of the new method.

Fig. 1
figure 1

Predicted results of HRC in J g−1 K−1 for the new model (122 data) as well as two molar group contributions of Walters and Lyon [6] (117 data) and Lyon et al. [8] (108 data) versus experimental data

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Fig. 2
figure 2

Range of absolute errors of the new model for all data points (122 data)

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Conclusions

A simple and accurate model was developed for prediction of the HRC values of different polymers with their repeat units that are comprised of chemical groups/moieties such as methyl, phenyl, carbonyl, ether, amide and ester. The model is based on the contribution of n H, n N, n O, n Cl and n Si divided by MWrepeat unit as well as two correcting functions of \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal. The predicted results of the new model were compared with the calculated data of two group additivity methods, which confirm higher reliability of the new correlation. The values of \( {\text{HRC}}_{{({\text{CH}}_{2} )_{\text{n}} {\text{CXYZ,XArY}}}} \) and HRCYXArZ,Hal beside elemental composition and MWrepeat unit can be easily obtained from repeat units of polymers.