Toward a Physically Based Quantitative Modeling of Impact

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Toward a Physically Based Quantitative Modeling of Impact Sensitivities Didier Mathieu* CEA, DAM, Le Ripault, F-37260 Monts, France S Supporting Information *

ABSTRACT: Among the subsequent steps leading from impact to explosive decomposition in nitro compounds, the ability of early exothermic reactions to trigger the decomposition of neighboring molecules before the released energy has dissipated away is assumed to be critical. The rate of this process is roughly estimated using as inputs the energy content and the dissociation energy of the weakest X−NO2 bonds. While the sensitivity index thus obtained was previously shown to exhibit striking correlations with gap test pressures, its correlation with drop weight impact test data is poorer. Nevertheless, considering four different subsets of molecules depending on the environment of the most labile nitro groups, straightforward regressions against this sensitivity index yield a practical method to estimate impact sensitivity, whose combination of fair performance and generality is provided by no alternative approach, except purely empirical models based on extensive parametrization. compound and equivalent in practice to OB.8 Over the years, alternative single-variable correlations have been put forward, based on a variety of molecular properties including electronic band gap,17 electronic shakeup promotion energies,18 13C and 15 N NMR chemical shifts,19 molecular electronegativity,20 X− NO2 bond lengths,21 total volumic energy content,22 a parameter derived from the hard and soft acid and bases (HSAB) principle,23 etc. As emphasized by Brill and James for nitroaromatic compounds, such studies based on small data sets are prone to chance correlations and are often misleading.24,25 This problem is even more pregnant for multivariables correlations, especially for those that consider only small data sets. Features related to electron distributions26,27 or molecular electrostatic potentials28−30 are especially popular descriptors. Some authors focus on the electronic structure of covalent X− NO2 bonds (called explosophore bonds).31,32 Some correlations based on constitutive descriptors, in the spirit of group contribution methods, are also available.33,34 Current research increasingly resorts to advanced quantitative structure− property relationships (QSPR) methodology and regression techniques beyond straightforward multilinear regression (MLR), such as partial least-squares (PLS) or artificial neural networks (ANN).35−41 While the use of large data sets should decrease the risk of chance correlations, the fact that input variables are typically selected out of extended pools of descriptors and the flexibility afforded by nonlinear regression techniques does increase this risk. As a result, it is not easy to figure out the true predictive value of such QSPR models.

1. INTRODUCTION The impact sensitivity of energetic materials is most often characterized by the height H50 that a given weight must be dropped onto the sample to trigger an observable decomposition with a 50% probability.1,2 The H50 values measured in this so-called drop weight impact test depend on many factors beyond chemical structure, some of them poorly controlled, such as the detailed microstructure and defects of the sample or the environment of the experiment.3−6 As a result, they exhibit large uncertainties. To some extent, shock sensitivity, as characterized by gap test threshold pressures, provides a more reliable index of safety.7 Both sensitivity tests involve completely different initiation regimes, namely, reaction times in the range of 200−250 μs and pressures between 7−15 kbar for the drop weight impact test, versus corresponding ranges of 0.05−2 μs and 30−200 kbar for the gap test.8 Nevertheless, significant correlations between H50 and gap-test pressures are observed in practice.8 Being more easily amenable to measurements, H50 remains therefore the most popular criterion to characterize mechanical sensitivities of new compounds. A general procedure to evaluate H50 from molecular structure would therefore be of interest not only as a guide to design new explosives9,10 but also to assess reactive hazards associated with organic compounds in the broader context of emerging safety regulations concerning the chemical industry.11,12 For many years, much effort has been devoted to the development of such structure−sensitivity relationships. Early approaches are mainly empirical. Half a century ago, experimental evidence that oxygen rich compounds are especially sensitive stimulated the development of linear correlations between log 10 (H 50 ) and oxygen balance (OB).13−16 In 1990, Storm et al. introduced a sensitivity index defined on the basis of the empirical formula of the © 2013 American Chemical Society

Received: November 27, 2012 Revised: February 14, 2013 Published: February 14, 2013 2253

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models for H50 have already been reported, they usually fail in demonstrating reasonable predictions for an extended test set of explosives. Not surprisingly, the best correlations with experiment are obtained using state-of-the-art QSPR techniques. The most impressive results to date are probably those reported very recently by Xu et al.41 Using an ANN model with input variables automatically picked out of an extended pool of descriptors, these authors obtain R2 = 0.85 for a training set of 127 explosives and R2 = 0.87 for a test set of 29 explosives. Using a very similar approach, Morill and Byrd obtain R2 = 0.69 for a test set of seven molecules.39 This discrepancy illustrates the difficulty in assessing the performance of such purely empirical models. Notwithstanding the recent ANN of Xu et al.,41 there are only very few models with R2 > 0.80.66,69 These correlations are obtained for small data sets with typically about 10 compounds. Notwithstanding the above-mentioned overfitting issues, the quality of such correlations rapidly decreases as the size/diversity of the data set increases. For instance, a carefully designed model based on charge distributions in molecules yields R2 = 0.54 for the 14 compounds in the test set.26 In fact, notwithstanding the Xu et al. results, the highest R2 values obtained are typically close to 0.70 as extended data sets (with N > 100 compounds) are considered. In this work, we focus on a more physically grounded approach that provides similar performance without resorting to an empirical search for optimal input variables and analytical expression. While not as accurate as the Xu et al. method, it yields more valuable insight into relevant aspects of initiation and the role of molecular parameters.

Considering the limited amount of experimental data at hand, a more physically grounded approach is desirable, even if a fully rigorous and quantitative model will be hard to obtain considering the complex phenomena involved. Indeed, an impact wave propagating into a material interacts with defects such as voids, shear bands, or interfaces through various mechanical processes.42 In order to trigger an explosion, these processes must be able to focus the mechanical energy in localized regions of the material and lead to the formation of hot spots. In the simplest picture, hydrodynamic processes around localized defects are sufficient to generate hot spots where temperature is high enough to initiate thermal bond scission reactions. Assuming such processes to be similar for most organic explosives, one might expect H50 to depend on the decomposition temperature Tdec. In fact, correlations between H50 and Tdec are observed only for the most stable explosives and exhibit significant exceptions.43 Therefore, there is a need to better understand the role of hydrodynamic aspects, microstructure and defects in typical granular explosives,44,45 and/or to identify alternative molecular mechanisms. In the seventies, it was proposed that the decomposition might be a result of shock induced electronic excitations.46,47 Theoretical justifications for such processes were subsequently developed.48−50 In fact, calculated probabilities of mechanically induced electronic transitions are significantly overestimated unless they account for the decoherence of the electronic wavefuntion induced by its interaction with the macroscopic sample.51 Quantitative estimates can only be obtained under the assumption that the electron−phonon coupling time scale is shorter than the reaction time scale.52 This condition implies that the approach is best satisfied for inorganic primary explosives, such as metal azides for which some limited correlations have been reported.17 Other attempts to account for the first steps in the decomposition process include the viscoplastic collapse of the porosity,53 jetting or vaporization of material into voids,54,55 pressure-induced proton tunneling,56 mechanical generation of charge transfer pairs that localize at vacancies,57 or intermolecular metastable trigger reactions leading to early chemical energy release.58 However, these assumed mechanisms did not lead to any general quantitative model that would allow to correlate observed H50 data with microscopic features. Alternatively, it has been assumed that the transfer of the mechanical energy from the phonon bath to intramolecular vibrations is the most determinant process influencing observed sensitivities. This energy transfer depends essentially on the excitation of doorway modes, i.e., of the molecular vibrations of lowest frequency. The corresponding rate can be theoretically estimated using standard approximations.59,60 To some extent, H50 data correlates with values thus calculated61 or with related theoretical quantities.62−65 However, such correlations rely only on small sets of compounds. The alternative assumption that sensitivities depend primarily on the energy required to trigger the first bond scissions yields similar results.66−68 The performance of predictive models for H50 is often characterized in terms of the determination coefficient R2 and root-mean-square deviation (rms) between observed and predicted (rather than f itted) log10(H50) values. Considering H50 directly leads to ill-defined statistics as sensitivity data sets typically exhibit a few insensitive compounds with H50 values in the range 300−500 cm, while most explosives exhibit H50 values N−NO2 and −NH−NO2. Finally, ΔG0diss is assumed to exhibit the same value for all O− NO2 bonds. With such transferability assumptions, only 11 ΔG0diss values are needed for the bonds present in our data set. With the help of the ORCA program,76 these 11 values are computed within the harmonic approximation using the model compounds listed in Table 1. HF/3-21G geometries and

as no thermal equilibrium is assumed. It is expressed here as a constant fraction η of the chemical energy content per atom, an approximation that implies that all compounds considered exhibit similar values of their diffusion coefficient. However, the activation energy E†pr for the propagation increases with the strength of the chemical bonds to be broken for new molecules to decompose, hence on the bond dissociation energy (BDE) of the weakest X−NO2 bonds in unreacted molecules, hereafter denoted Dmin. Since E†pr = Dmin for simple gas phase X−NO2 bond cleavages, a scaling coefficient λ is introduced to account for the role of the environment of the nitro group: E†pr = λDmin. For simplicity, λ is first assumed to be constant for all explosives considered. However, it will prove necessary to introduce different values depending on the nature of the weakest X− NO2 bond, as described in section 3. Under these assumptions, the propagation rate kpr is given by ⎛ E† ⎞ ⎞ ⎛ Dmin pr ⎟ = Zpr exp⎜ − λ ⎟ k pr = Zpr exp⎜⎜ − 0 ⎟ ⎝ η 2Δd H /3N ⎠ ⎝ kBT ⎠

Table 1. Definition of Standard X−NO2 Bond Types and Corresponding ΔG0diss Values

(1)

where kB is the Boltzmann constant, ΔdH0 the energy released by the decomposition of one mole of the compound, N the corresponding number of atoms, and Zpr the prefactor of the process. Since the impact energy that eventually leads to decomposition is proportional to H50, it is natural to assume that log10(H50) linearly correlates with log10(kpr), hence with the shock sensitivity index SI defined as follows SI = NDmin /Δd H 0

(2)

and previously introduced to rationalize gap test data.70,71 This linear relationship exhibits a slope equal to 3λ/2η. Assuming λ close to unity, we can estimate the fraction η of the chemical energy that efficiently contributes to propagate the decomposition. In the present work, the decomposition enthalpy ΔdH0 is obtained as the difference between the formation enthalpy ΔfH0 of the explosive compound and corresponding enthalpies for the decomposition products. The latter are estimated with the help of the H2O−CO2 arbitrary.72 In other words, it is assumed that the main decomposition products are, in order of decreasing formation priority, H2O, CO2, and, of course, N2. For the initial molecule, ΔfH0 is approximated by the value derived from the RM1 Hamiltonian73 as implemented in the MOPAC7 program.74 For the products, standard experimental enthalpies compiled in the NIST database are used.75 For simplicity, Zpr is assumed constant. In that case, E†pr should in principle be replaced in eq 1 by the Gibbs energy of activation ΔG†pr for the propagation of the decomposition. There is no way to rigorously evaluate this quantity. However, in an attempt to capture at least partially some entropic contributions, standard Gibbs free energies of dissociation ΔG0diss are used. The systematic calculation of ΔG0diss for all X− NO2 bonds would be tedious, given the need for many calculations on large radicals. Therefore, the present work resorts to a rough approximation, assuming the transferability of a limited number of ΔG0diss values. For aromatic C−NO2 bonds, ΔG0diss is assumed to depend only on the presence of proton donor groups in adjacent positions on the aromatic ring, as in such cases NO2 oxygens are involved in hydrogen bonding. For nitro groups bonded to sp3 carbons, the energy of the C−NO2 bond is assumed to depend only on the number of nitro groups in geminal positions. For nitramines, two distincts

frequencies are used. The frequencies are scaled using the standard 0.9207 coefficient.77 Total electronic energies are calculated at the B3LYP level using the def2-TZVP basis set,78 the chain of spheres acceleration algorithm,79 and van der Waals corrections.80 It must be emphasized that the present approach relies exclusively on the energy content and bond strengths of isolated molecules. As such, it cannot account for the dependence of the impact sensitivity on such parameters as defects, polymorphism, or the crystal orientation with respect to the shock direction.81 However, this dependence may be addressed with approaches complementary to the present one, such as molecular dynamics simulations based on reactive force fields.82,83

3. RESULTS The X−NO2 dissociation energies obtained in this work are listed in Table 1. As expected, they follow the usual stability order, namely, CA−NO2 > N−NO2 > O−NO2 with CA standing for an aromatic carbon. For nitroalkyls, isolated C− NO2 bonds are somewhat stronger than N−NO2 bonds, while geminal C−NO2 bonds are no stronger than O−NO2 bonds. Despite the fact that X−NO2 bonds in similar chemical 2255

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this work leave much room for further improvement, e.g., through the use of more accurate BDE data. The poor correlation obtained for nitramines (set S2) stems from the fact that cyclotrimethylene trinitramine (RDX) and cyclotetramethylene tetranitramine (HMX) prove less sensitive than estimated on the basis of present approximate SI values, while the opposite is observed for N,N′-dinitro-methanediamine O2N−NH−CH2−NH−NO2, which is the smallest nitramine in the data set. The fact that these three outliers do not fit in the present correlation does not appear to stem from the use of standard bond energies from Table 1. Indeed, assuming transferable ΔG0diss values prove reasonable for these compounds. For instance, a value of 26.43 kcal/mol is computed for RDX, instead of the assumed value of 28.61 kcal/mol. Similarly, the value of 34.5 kcal/mol calculated for N,N′-dinitro-methanediamine is reasonably close to the assumed value of 32.98 kcal/mol. For NACs (set S1), the correlation is significantly spoiled by the fact that 2,4,6-trinitrobenzene-1,3,5-triol proves significantly more sensitive than expected on the basis of the corresponding sensitivity index. In contrast, other compounds with strong intramolecular hydrogen bonds, including triaminotrinitrobenzene (TATB), fit nicely into the present scheme. Therefore, there is no obvious explanation for the underestimation of the sensitivity of this compound. Finally, the correlation for set S4 is especially sensitive to uncertainties associated with present SI data since all SI values are very similar (in the range 1−1.3) in this data set, while H50 values range from 1 implies that, as the critical step in the propagation of a decomposition reaction to the surrounding material is attained, a regime where chemical energy is released faster than it diffuses away is already reached. Therefore, values of η larger 2256

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than unity probably arise as a result of present approximations, especially the fact that the prefactor in eq 1 is assumed constant, while ΔG†pr is simply derived from ΔG0diss. Nevertheless, the result that η = 3.4 for NACs is puzzling as long as ΔG†pr ≃ Dmin is assumed. In fact, although this approximation is reasonable for nitramines, nitric esters, and nitroaliphatic compounds, for which the homolysis of X−NO2 bonds is believed to be the dominant process determining the stability of the compound, it might be misleading when applied to NACs for which preliminary rearrangements of ring substituents are often energetically favored.25 In such cases, activation energies significantly lower than Dmin would account for the large value of η presently obtained for NACs. Compared with the Storm et al. sensitivity index defined on the basis of the empirical formula,8 the present one is more physically grounded.70 Figure 2 shows how presently calculated

Table 2. Performance Comparison between Present Linear Regressions and Recent Models Based on MLR, PLS, or ANN Techniques (See Text for Details) training set 2

2

test set 2

model

ref

R (fit)

R (LOO)

rms

R (ext)

rms

MLRa MLRc MLRc PLSa ANNa ANNb ANNc this work

40 41 39 40 40 40 41

0.771 0.768 0.814 0.766 0.816 0.795 0.848 0.710

0.593 0.703 0.795 0.674

0.212 0.194

0.715 0.722 0.691 0.718 0.740 0.756 0.866 0.721

0.251 0.177 0.285 0.250 0.247 0.257 0.130 0.265

0.846 0.658

0.214 0.192 0.203 0.164 0.238

a

Using electrotopological state indices. bUsing the same descriptors as in ref 35. cSelecting variables out of a large pool of descriptors.

combined with automatic variable selection from extended pools of descriptors, as done by Xu et al.41 Unfortunately, such techniques provide good results at the expense of physical insight.

4. CONCLUSIONS AND PERSPECTIVES Although present equations might not be as reliable as neural network models, the present sensitivity index opens new perspectives for the a priori evaluation of drop weight impact test data. A major advantage of the present approach over empirical QSPR methodologies lies in the fact that it can be systematically improved through removing simplifying assumptions. For instance, the present focus on the weakest bond in the compound is clearly a rough approximation. Accounting for all X−NO2 bonds should yield significant improvement. This will be the next step in our present approach. However, in the context of QSPR modeling, the present sensitivity index appears as a promising descriptor in view of its ability to fit a significant body of drop-weight impact test data on the basis of a straightforward and physically motivated partition of the data set. It might be combined with descriptors taking account secondary aspects of impact sensitivity, such as intermolecular interactions, particle size distribution, etc.

Figure 2. Calculated versus measured impact sensitivities for the 156 compounds in the database. Same color code as that in Figure 1. Empty symbols are for compounds in the training set, and filled symbols are for compounds in the test set.

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impact sensitivities compare to experiment. The fact that they are not very accurate is to be expected in view of present approximations and keeping in mind experimental uncertainties. However, it is gratifying to observe that the present model does not lead to any significant outlier, suggesting that SI is indeed the primary factor determining observed sensitivities. Table 2 compares the performances of present correlations with the outcome of recent models taking advantage of state-ofthe-art machine learning techniques. R2(fit), R2(ext), and R2(LOO) are the determination coefficients between measured log 10 (H 50 ) values and calculated counterparts derived, respectively, from the training set, the test set, and the leaveone-out (LOO) cross-validation procedure. Values of the rootmean-square error (rms) are provided for both the training and the test sets. For every model, these subsets contain, respectively, 127 and 29 compounds from the present data set, except for the model from ref 39, which used a somewhat larger data set. Although present regressions rely on relatively poor correlations, their predictive value compares well with the best empirical linear models, according to statistics reported in this table for the external test set. In fact, more accurate predictions can only be obtained using neural networks

ASSOCIATED CONTENT

S Supporting Information *

Table with experimental and calculated log10(H50/cm) values for the 156 explosives considered in this work, referred to by their index as defined in ref 43. For every compound, the corresponding molecular subset (S1 to S4) is indicated, as well as a letter T or V depending on whether this compound belongs to the training or the test (validation) set. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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