Article pubs.acs.org/JPCA
Theoretical Shock Sensitivity Index for Explosives D. Mathieu* CEA, DAM, Le Ripault, F-37260 Monts, France ABSTRACT: On the basis of simple physical arguments, the ratio of the weakest bond dissociation energy of nitro compounds to their decomposition enthalpy per covalent bond is put forward as a practical shock sensitivity index. Without any empirical fitting, it correlates remarkably well (R ≥ 0.95) with shock sensitivity data reported for 16 molecules spanning the most significant families of explosive compounds. This result supports the underlying assumption that this property depends on the ability of decomposition events to propagate into the material. It demonstrates that sensitivity−structure relationships should take the energy content of the material into account. A linear regression against the present sensitivity index yields a predictive method with better performance than previous ones. Its sounder physical bases provide new insight into the molecular determinants of sensitivity and a compelling explanation for the sensitivity values reported for TATB and FOX-7.
1. INTRODUCTION Despite decades of intensive research, understanding and ultimately predicting mechanical sensitivities of explosives remains a particularly challenging issue.1−3 In principle, defects are likely to play a prominent role, because initiation originates from so-called hot spots arising from their interaction with shock waves.4 Nevertheless, because all energetic materials used in practical applications do exhibit defects, the latter are not necessarily the main determinant of sensitivity differences between various explosives. In fact, the search for relationships between sensitivities and molecular structures only makes sense provided the details of how molecules are packed together in the material (i.e., crystal structure, morphology, defects, ...) do not play the primary role. To date, the vast majority of published structure−sensitivity relationships concern impact sensitivities, as measured by the height H50 that a given weight must be dropped onto the sample to trigger an observable decomposition with a 50% probability. This focus on H50 presumably stems from the large body of consistent data available in open literature.5 However, H50 proves especially sensitive to a number of factors beyond the chemical composition of the material, including not only defects but also loading density, porosity, particle and crystal size distribution, crystal morphology, orientation of crystals with respect to the compression wave, surface state, humidity, and actual equipment used (e.g., smoothness of impact hammers).6−9 Although these aspects are likely to influence the responses of energetic materials to any stimulus, it is worth considering alternative sensitivity criteria to gain a better understanding of how decomposition initiates in these systems. In particular, shock sensitivities, as characterized by the threshold pressure needed to initiate an explosion in the small scale gap test,5 are usually considered as more repeatable and more useful sensitivity data than H50. Unfortunately, gap test experiments are more difficult to carry out. As a consequence, relatively few consistent sets of gap test threshold pressures are © 2012 American Chemical Society
available. On the other hand, the empirical approaches most often used to estimate H50 require extensive data sets for their parametrization and thorough validation.10−18 Therefore, they appear difficult to apply to the prediction of gap test threshold pressure owing to the scarcity of data available. In this context, a more physical approach is desirable. In view of the complexity of the ignition process, we must resort to arbitrary approximations to be assessed through a posteriori comparisons with experiment. It may be assumed for instance that a specific step in the decomposition process plays a critical role. Actually, many relationships proposed for H50 assume that the critical step is the energy transfer to intramolecular vibrational modes,19−21 initial X−NO2 bond ruptures,22 or the subsequent propagation of the decomposition process.1 This latter step is also implicitly assumed determinant by models that put emphasis not only on bond dissociation energies but also on oxygen balance, a property closely related to decomposition products.23 Correlations involving the bond dissociation energy (BDE) of the weakest X−NO2 bond and/or the energy content of the compound are especially illuminating.1,22−24 They suggest that H50 depends primarily on the intrinsic stability of isolated molecules rather than on the rate and mechanism by which the impact energy is transferred to intramolecular degrees of freedom.
2. PREVIOUS SHOCK SENSITIVITY MODELS Early studies focused on nitroaromatic compounds (NACs) suggest that their shock sensitivities are primarily determined by the BDEs of the weakest C−NO2 bonds, thus pointing to their cleavage as a possible determining step for decomposition.25 However, this simple scheme does not account for all gap test data at hand. In fact, more general predictive Received: October 10, 2011 Revised: January 18, 2012 Published: January 25, 2012 1794
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Table 1. Comparison of the Present Approach for P90 with Recently Published Models26,27,a name
D
ΔfH0
ΔdH0
NB
Ed
V
ref 26
dev
ref 27
dev
this work
dev
exp
PETN TACOT-Z TNEDV TETRYL HNB DIPAM TNETB HMX RDX HNAB HNS TNT TNB MATB DATB TATB
156 282 208 135 209 293 170 172 170 253 266 261 278 296 321 357
−384 703 −392 162 244 102 −317 294 192 447 269 55 85 29 −24 −58
2158 2367 2172 1848 2529 2599 2573 2048 1508 2898 2765 1349 1333 1300 1270 1258
28 35 32 25 35 39 31 28 21 37 39 21 18 20 22 24
77.1 67.6 67.9 73.9 72.3 66.6 83.0 73.1 71.8 78.3 70.9 64.2 74.1 65.0 57.7 52.4
186 217 213 168 177 251 216 168 126 254 264 139 125 131 137 143
2.30 3.41 2.75 2.63 3.06 3.16 1.70 2.35 2.35 3.06 3.34 2.91 3.00 3.06 3.63 4.29
0.29 −0.13 0.04 0.27 0.16 −0.06 −0.52 −0.03 −0.05 0.51 0.07 0.04 0.29 −0.27 −0.20 0.04
2.09 3.18 2.46 2.33 2.59 3.38 2.39 2.38 2.39 2.91 3.06 3.06 3.12 3.30 3.67 4.25
0.08 −0.36 −0.25 −0.03 −0.31 0.16 0.17 0.00 −0.01 0.36 −0.21 0.19 0.41 −0.03 −0.16 0.00
2.25 3.18 2.70 2.17 2.62 3.27 2.27 2.39 2.40 2.77 3.00 3.13 3.00 3.34 3.77 4.30
0.24 −0.36 −0.01 −0.19 −0.28 0.05 0.05 0.01 0.00 0.22 −0.27 0.26 0.29 0.01 −0.06 0.05
2.01 3.54 2.71 2.36 2.90 3.22 2.22 2.38 2.40 2.55 3.27 2.87 2.71 3.33 3.83 4.25
a
The nomenclature is defined in ref 26. Values in kJ/mol are provided for the following descriptors: weakest bond dissociation energy D, formation enthalpies ΔfH0, and decomposition enthalpies ΔdH0. The number of bonds NB in the molecule, the corresponding volume V in cm3/mol, and the energy density as defined by Ed = ΔdH0/NB are also indicated. The rightmost columns report calculated and observed ln(P90/kbar) values. For calculated values, deviations from experiments are provided.
schemes (not restricted to NACs) have been published only very recently,26,27 taking advantage of gap test threshold pressures P90, P95, and P98 compiled by Storm et al. for samples loaded respectively at 90%, 95%, and 98% of their theoretical maximum crystalline density (TMD).5 This data set includes not only NACs but also other common classes of explosives, such as nitrates and nitramines. This diversity may contributes to the enhanced complexity of models developed on the basis of this data set. Indeed, to obtain reasonable fits of gap test threshold pressures for the 16 compounds studied, empirical expressions involving as many as 4−5 adjustable parameters proved necessary.26,27 As a consequence, their predictive value for new compounds is questionable not only owing to their lack of physical grounds but also in view of the relatively large number of parameters with regard to the size of the data set. On the other hand, it should be noted that diaminotrinitribenzene (DATB) and especially triaminotrinitribenzene (TATB) exhibit threshold pressures well above the other compounds. As a consequence, any model accounting for their enhanced stabilities will lead to spuriously high determination coefficients, as illustrated in ref 26. This problem can be avoided by fitting the model against the logarithms of the threshold pressures, as done in the most recent and reliable procedure presently available.27 Furthermore, this approach is consistent with the exponential dependence of the ignition probability with molecular features determining activation energies and local temperatures.
More specifically, this work assumes that the primary determinant of shock sensitivity is related to the ability of decomposition reactions to propagate and initiate a selfsustained process. This condition is satisfied if the energy ΔdH0 released by a decomposed molecule is sufficient to ensure a high enough temperature over a sufficient period of time to trigger further reactions among neighboring molecules. The sensitivity should therefore correlate with the corresponding propagation rate given by k p = A exp[−E⧧/kBT ]
(1)
where A is a pre-exponential factor, kB the Boltzmann constant, and the rate kp is mainly determined by the ratio of the activation energy E⧧ for decomposition to the local temperature T induced in the vicinity of reacting molecules. This equation points to the ratio E⧧/kBT as the main determinant of shock sensitivity. This ratio was already put forward by Fried et al. as a potentially determinant factor for H50.1 However, although a correlation between E⧧/kBT and H50 could be demonstrated, it did not provide convincing evidence for the primary role of this factor on impact sensitivity. Accordingly, subsequent structure− sensitivity relationships do not consider the possible role of this ratio. The two main reasons for the relatively poor correlation observed between E⧧/kBT and H50 are presumably (1) the large uncertainties associated with H50 data and (2) the assumptions involved in the evaluation of E⧧/kBT. This work reports further investigations along the same lines, considering gap test threshold pressures instead of the less welldefined H50 values, and using a somewhat different procedure to evaluate E⧧/kBT. First, assuming the rate-determining step of the decomposition to be associated with the cleavage of the weakest X−NO 2 bond, E ⧧ should correlate with the corresponding bond dissociation energy D. Indeed, D is clearly a major contribution to E⧧, as the others arise from nonbonded interactions. Noting that E⧧ should get close to zero in the hypothetical case of bonds with very low dissociation energies, it is reasonable to assume E⧧ ∝ D. As for the local temperature T available to propagate the decomposition process, it clearly depends on the chemical
3. PRESENT PHYSICAL ASSUMPTIONS Like previous models,26,27 the present one is parametrized against the Storm et al. database5 reported in Table 1. However, in contrast to earlier empirical methods, it relies on simple but well-defined physical assumptions, in an attempt to keep the number of empirical parameters to a minimum and to obtain a better reliability. Another advantage of this approach is the fact that it provides some insight into microscopic ignition mechanisms, as demonstrated below. 1795
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functional theory27 provide present D values. In our attempts to define Ed as ΔdH0/V, the crystal volumes V are obtained using a recent additivity scheme with an average relative error as small as 2%.34
energy content of the molecules and on how it subsequently diffuses away. The diffusion rate is controlled by the thermal conductivity κ of the material, which undoubtedly affects the shock response of explosives. For instance, the high thermal conductivity of TATB contributes to its insensitivity because it allows energy to diffuse rapidly away from heated regions. However, detailed investigations indicate that this factor is secondary and that activation energies play a more significant role.28 Considering TATB on one hand, and the much more sensitive cyclomethylenetrinitramine molecule (HMX) on the other, this secondary role of κ is reflected by the fact that κ(TATB)/κ(HMX) < 2 whereas the ratio of their gap test threshold pressures is almost 1 order of magnitude higher (≃7).28 Because κ values are lacking for many explosives of interest, we are especially interested in a model that does not explicitly depend on this property. Therefore, the present model neglects thermal conductivity differences between explosives. As a result, T may be assumed to be proportional to a density of chemical energy Ed. At this stage, this model is very similar to the one introduced in 2001 by Fried et al. for impact sensitivities.1 Notwithstanding the fact that the present work is concerned with gap test threshold pressures, its main difference from this earlier investigation lies in the definition adopted for Ed. Indeed, Fried et al. define Ed as the ratio ΔdH0/V of the energy content of a molecule to the volume it occupies in the material.1 Because the released energy tends to get distributed over all degrees of freedom, the energy per atom ΔdH0/NA should more closely reflect the local temperature T arising from early decomposition processes. Alternatively, the energy per covalent bond ΔdH0/NB may exhibit a further advantage as it accounts for the extra stability associated with the formation of additional bonds. In practice, the two latter definitions yield very similar results in view of the close relationship between NB and NA. In contrast, the proportionality relationship between NB and V is more approximate, as reflected by a correlation coefficient R = 0.89 between both quantities. As a result, defining Ed as ΔdH0/NB instead of sticking to the Fried et al. definition may have a more significant impact on the correlation for gap test data. In fact, as further demonstrated in the sequel, this simple modification substantially improves this correlation. This improvement and the above-mentioned assumptions justify the use of the present variant of the Fried et al. ratio D/Ed as a theoretical shock sensitivity index.
5. RESULTS AND DISCUSSION As anticipated from the previous considerations, logarithmic threshold pressures prove to be remarkably correlated with D/ Ed where Ed = ΔdH0/NB, as shown in Figure 1. The
Figure 1. Small scale gap test threshold pressures (in kbar) for samples loaded at 90%, 95%, and 98% of the crystal density (from ref 5) reported as a function of the ratio D/Ed, where Ed is the total energy content per bond. Filled symbols are for explosives mixtures. Inset: zoom on the data with TATB and DATB removed.
corresponding correlation coefficients exhibit values as high as 0.95, 0.97, and 0.96 for 90%, 95%, and 98% loading densities, respectively. This linear relation implies an exponential increase of the threshold pressure P with D/Ed and a concomitant decrease with the rate kp as P ∝ kp−α with α > 0, in line with physical expectations. It may be observed that the correlation appears to be better for higher pressures. As a possible explanation, it may be pointed out that the present neglect of intermolecular features is better justified for insensitive compounds (associated with high pressure values) for which the energies required to activate the decomposition are especially large and thus predominant with respect to energies involved in intermolecular mechanisms. To compare the predictive value of the present correlation with available alternative schemes, P90 values have been calculated using linear regression coefficients derived from the data in Table 1 according to
4. COMPUTATIONAL DETAILS Following Tan et al.,27 shock sensitivities are characterized in this work by the natural logarithms of gap-test threshold pressures reported in Table 1, thus avoiding the pitfalls associated with the specific distribution of threshold pressures in the present data set. Present values of ΔdH0 are obtained as differences between the formation enthalpy ΔfH0 of the compound studied and corresponding enthalpies for the decomposition products derived from the H2O−CO2 arbitrary.29,30 For the initial molecule, ΔfH0 is approximated by the value derived from the RM1 Hamiltonian31 as implemented in the MOPAC7 program. 32 For the products, standard experimental enthalpies compiled in the NIST database are used.33 In view of the small size of these species, reliable theoretical values can readily be obtained from high-level quantum chemical methods. This might prove necessary if more exotic products were to be considered. BDE data computed by Tan et al. at the BLYP/DNP level of density
ln(P90/kbar) = 1.39 + 0.428D/Ed
(2)
where Ed is the energy derived from the H2O−CO2 arbitrary for one molecule divided by the corresponding number NB of covalent bonds. The gap test threshold pressures calculated using this equation for the pure compounds in the data set are reported in Table 1 and compared to recent alternative schemes. For completeness, the number NB of bonds per molecule and calculated values of D, ΔfH, ΔdH, Ed, and V are also reported in this table. 1796
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ill-defined. The present model is not plagued by such linear dependencies. This is because the values of D and Ed, though correlated to some extent, are combined into a single descriptor before carrying out the regression. The remarkable correlation presently obtained might be unexpected in view of the similarity of the model with that applied with only moderate success to impact sensitivities by Fried et al.1 The better correlation presently obtained for gap test data might be due to the relative specificities of the drop weight impact and small scale gap tests. However, it may be observed that such results are very sensitive to the actual definition adopted for Ed. Actually, although the molecular volume V roughly scales linearly with NA or NB, assuming Ed = ΔdH0/V instead of Ed = ΔdH0/NB significantly deteriorates the correlation, with the determination coefficient decreasing from 0.90 to 0.82 for the fit and from 0.88 to 0.77 for the LOO crossvalidation. Finally, the present correlation is complementary to numerical molecular simulations. With state-of-the-art analytic potentials, the latter techniques prove successful to qualitatively explain sensitivity differences between materials with similar chemical compositions, such as crystal polymorphs.35 However, presently available analytic potentials are too approximate to provide good estimates of the bond strengths and energy content involved in the present sensitivity index, and therefore can probably not be used to compare the sensitivities of various compounds. In contrast, ab initio molecular dynamics is more accurate but restricted to very small time scales, a drawback hampering its direct application to the propagation of the decomposition within the material. In short, although molecular simulations are well-suited to study the influence of physical parameters, the present correlation is better for evaluating the role of the chemical composition. As such, it should be useful to screen new synthesis targets that might be considered as promising high energy density materials.
Because DATB and especially TATB are much more insensitive than other compounds presently considered, it is worth mentioning that the present fit is hardly affected when these two molecules are removed from the training set. Indeed, the intercept decreases from 1.39 to 1.37, whereas the slope increases from 0.428 to 0.433. Despite a smaller number of adjustable parameters and the fact that the ratio D/Ed arises from simple considerations rather than being empirically picked out of an extended pool of descriptors, the present procedure yields a better fit of the data than available published methods. Indeed, as summarized in Table 2, it yields a determination coefficient R2 closer to unity Table 2. Comparison of the Performance of the Present Model (Eq 2) to That of the Methods Reported in Refs 26 and 27, According to Statistics Derived Respectively from Fitted (FIT) and Leave-One-out (LOO) Cross-Validated Data eq 2 ref 27 ref 26 eq 2 ref 27
FIT FIT FIT LOO LOO
R2
AAD
rmsd
MIN
MAX
0.90 0.87 0.84 0.88 0.77
0.15 0.17 0.19 0.16 0.23
0.19 0.22 0.24 0.21 0.29
−0.36 −0.36 −0.52 −0.39 −0.41
+0.29 +0.41 +0.51 +0.31 +0.59
and lower values of the average absolute deviation (AAD) and root-mean-square deviation (rmsd) between experimental and estimated values of ln(P90). In addition, the minimum (MIN) and maximal (MAX) deviations are also reduced in the present scheme. The superiority of the present model is even more obvious when the statistics obtained from a leave-one-out (LOO) crossvalidation procedure are considered, which usually provide more reliable estimates of the predictive value of semiempirical models. As indicated in Table 2, the determination coefficient for cross-validated data is only slightly lower (0.88 versus 0.90) than the value derived from the fit. In contrast, for the model described in ref 27, the corresponding value drops from 0.87 to 0.77, thus pointing to a lack of robustness associated with overfitting issues. On the other hand, deviations between predicted and observed values of In(P90) increase by almost 50% on going from the present model to the previous one reported in ref 27, thus reflecting the fact that eq 2 yields a significant improvement over available methods. An even more stringent validation of empirical schemes is provided by their application to an external test set. Therefore, a downgraded version of the present model has been obtained from a fit against a reduced training set made of six compounds, namely PETN, HMX, TNEDV, HNB, MATB, and TATB. Again, the regression coefficients are very close to those obtained from the fit using the whole data set, with values of 1.38 for the intercept and 0.430 for the slope. Predictions thus obtained for the remaining ten compounds in the external test set are consistent with the statistics reported in Table 2, with rmsd = 0.21 and deviations from observed In(P90) values ranging from −0.37 to +0.28. The lower performance of earlier models is no surprise in view of their more empirical character. For instance, the equation put forward in ref 27 includes as linear descriptors two properties that prove highly correlated, namely X−NO2 bond dissociation energies and atomic charges on NO2 groups. As a result, the corresponding regression coefficients are likely to be
6. APPLICATION TO EXPLOSIVE MIXTURES Although the model reported in ref 27 proves the most reliable to date, it exhibits a severe limitation due to the fact that it relies only on local molecular descriptors, namely the BDE of the weakest X−NO2 bond and the most negative charge on a nitro group. As a result, the influence of the other parts of the molecule is completely ignored. For instance, the method cannot be applied to explosive mixtures, because its outcome depends on the compounds present in the material but not on their relative proportions. To some extent, the present model overcomes this limitation. Indeed, any change in the composition of an explosive mixture is reflected by the value of its energy density Ed. As the amount of the most energetic compound increases, Ed increases as well, hence a decrease of the threshold pressure indicating an enhancement of the shock sensitivity. Unfortunately, the influence of the composition is still not satisfactorily accounted for by our model. This stems from its focus on the lowest energy barrier associated with a weak bond. Clearly, given two hypothetical explosives with the same values of D and Ed, the one with the largest number of weak bonds should be more sensitive because the probability to initiate an observable reaction increases with the number of reaction pathways leading to a decomposition of the molecules. Taking advantage of BDEs calculated for all X−NO2 bonds, as done in a prelimiminary attempt to model decomposition temperatures,36 it should be possible to make further improvement. Meanwhile, 1797
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as a result of the strength of its C−NO2 bonds and does not essentially depend on the crystal structure.24,25 However, subsequent investigations clearly indicate that sensitivities are not determined by bond strengths alone, as emphasized by the review of Fried et al.1 It is clear that the environment of an explosive molecule in a condensed phase can have a dramatic influence on its decomposition pathways. A more interesting question is to what extent this influence may affect observable properties, such as thermal stability or shock sensitivity. Though the thermal stability of TATB as a solid is exceptionally good, on par with its mechanical stability, a different picture emerges in solution, where this compound proves less stable than other trinitrobenzene derivatives.38,39 This result has suggested that the thermal stability of TATB might be a function of its crystal lattice energy rather than an intrinsic property of the molecule. However, the superior C−NO2 bond strength in this compound supports a different interpretation, namely that the intrinsic stability of the TATB molecule might be especially sensitive to solvent effects. In addition to the remarkable cohesion of its hydrogenbonded graphite-like crystal structure,40 some properties closely related to the crystal structure and dynamics have been invoked to explain the stability of TATB. In particular, numerical simulations have provided a number of possible explanations, including the formation of a HONO isomer leading to stable benzofurazan and benzofuroxan derivatives that might hinder further decomposition,41 buffering effects against mechanical stimuli,42 or the persistence of extended nitrogen-rich heterocyclic structures.43 However, these explanations remain purely qualitative. Although the mechanisms uncovered are probably realistic and exhibit much fundamental interest, their actual influence on shock sensitivity remains to be quantified. In contrast, the present approach provides not only a straightforward explanation for the stability of TATB but also a quantitative estimate of its shock sensitivity. In fact, the remarkable success of the present sensitivity index D/Ed in correlating a significant body of gap test data suggests that the detailed microscopic mechanisms reported in a number of publications41−43 contribute only marginally to the sensitivity differences between high energy compounds, although they are invaluable to better understand such differences between materials made from the same constitutive molecules but prepared according to different experimental protocols. On the other hand, in contrast to the above-mentioned explanations for the low sensitivity of TATB, the present model is consistent with an approach of the problem from the hot spot theory, which emphasizes the role of the low temperature of explosion products.44 From the viewpoint of this latter model, numerical simulations of the early molecular responses to mechanical stimuli cannot be directly compared with shock sensitivity data for explosives such as TATB, whose sensitivity critically depends on the subsequent step of the propagation of the decomposition from microscopic hot spots. Clearly, present results may account for experimental observations for many explosives beyond TATB. For instance, much work has been devoted in recent years to diaminodinitroethylene (FOX-7). This compound bears much similarity with TATB. Both molecules exhibit the same chemical bonds. In particular, the C−NO2 bond dissociation energies are almost exactly the same.45 Moreover, the crystals of TATB and FOX-7 both lack a melting point and exhibit similar layered structures with extensive hydrogen bonding networks. Assuming the
this model might lead to a systematic overestimation of the sensitivities of explosive mixtures, because only one BDE value (the lowest one) is arbitrarily retained, even if it comes from a compound in very small concentration in the material. This might explain why the present sensitivity index does not satisfactory reflect the variance of the shock sensitivities reported in ref 5 for various mixtures, as illustrated in Figure 1. In fact, calculated shock sensitivities for mixtures exhibit no systematic overestimation. Predicted values are detailed in Table 3 and compared to experiment and to predictions Table 3. Application of the Present Method To Estimate Shock Sensitivities of Explosive Mixtures and Comparison with the Model of Keshavarz et al.26 Comp B-3 Octol 75/25 Octol 65/35 Pentolite
ref 26
dev
this work
dev
exp
2.49 2.43 2.46 2.41
−0.30 −0.11 −0.05 0.31
2.45 2.43 2.44 2.34
−0.33 −0.11 −0.07 0.23
2.78 2.54 2.51 2.11
reported in ref 26. The root-mean-square error of 0.21 presently obtained does not differ significantly from the value of 0.19 reported in Table 2 for pure compounds. In fact, it is even lower than the value of 0.22 obtained using the model of ref 26. This is noteworthy as those mixtures were used in the parametrization of this earlier highly flexible model, whereas they make up an external test set for the present method. However, this success of eq 2 clearly depends on the fact that the concentration of the weakest X−NO2 bonds is not dramatically different in these mixtures and in the pure components. Because the role of this concentration is still neglected, the present model does not account for such effects as the sensitization of nitromethane by small concentrations of amines. Indeed, as compared to the case with neat nitromethane, both Ed and D are left virtually unchanged upon addition of amines, and our model would thus completely fail to predict any change in sensitivity. In fact, none of the presently available models can describe such a phenomenon. Nevertheless, the present one appears as the most promising in view of a generalization to such situations, which would involve (1) considering energy barriers rather than BDEs and (2) taking into account the respective concentrations of the reactive moieties.
7. APPLICATION TO TATB AND FOX-7 In addition to its interest for the design of novel molecules, the present correlation contributes to the current understanding of the sensitivities of common high explosives. In particular, it supports the recent suggestion that BDEs and oxygen balance (considered as a rough measure of the energy content) are the two main factors determining sensitivities.23 Furthermore, the present sensitivity index can be used to rationalize experimental data for specific compounds. For instance, it sheds some light on the exceptional stability of TATB, which has been a matter of controversy over the last decades.37 For many years, this compound has been contrasted to the more powerful but less stable HMX molecule to illustrate the necessary trade-off between performance and safety. As mentioned in section 3, the explanation of the sensitivity difference between TATB and HMX in term of their relative thermal conductivities is not fully satisfactory. In the eighties, early quantum chemical calculations suggested that the shock insensitivity of TATB can be viewed 1798
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stability of such compounds to be determined by such factors as bond strengths, hydrogen bonding networks or lattice stability, they should exhibit similar sensitivities. The fact that FOX-7 proves more sensitive led Kuklja et al. to consider the role of mechanical shear on reactivity.45 Unfortunately, the ab initio techniques employed for simulations make it possible to study only a small number of structures. Therefore, the explanation thus obtained for the higher sensitivity of FOX-7 depends not only on the assumption that it is determined by C−NO2 decomposition barriers but also on a somewhat arbitrary selection of crystal configurations. In contrast, the present procedure naturally accounts for this sensitivity difference. For FOX-7, assuming the same C−NO2 bonds as for TATB, it yields ΔfH0 = −56 kJ/mol, hence ΔdH0 = 822 kJ/mol, D/Ed = 5.65, and ln(P90/kbar) = 3.81. Thus, with a gap test threshold pressure predicted at P90 = 45 kbar, close to that observed for DATB, FOX-7 is among the less sensitive explosives, while being more sensitive than TATB, in line with the experimental results mentioned in ref 45.
8. CONCLUSION AND PERSPECTIVES In summary, present results provide new insight into the sensitivity of explosive mixtures and into the unusual stability of specific materials, such as TATB and FOX-7. Briefly, they support the view that bond strengths and energy content are the two primary factors determining mechanical sensitivities. Accordingly, future structure−sensitivity relationships should take the energy content of the material into account, in contrast to previous ones that rely on descriptors computed on the unreacted compound. The present correlation is valuable in practice to evaluate the sensitivity of target molecules prior synthesis. Though more physically grounded than previous methods for predicting gap test pressures, the present one still involves some arbitrariness, especially with regard to the decomposition scheme assumed and its focus on the lowest energy bond. The original and modified versions of the Kistiakowsky−Wilson rules29 did not yield any improvement with present values of D and ΔfH0. Nevertheless, it should be worth investigating this point using high theoretical levels. In addition, overcoming the limitations of the present approach associated with its focus on the weakest X−NO2 bond should further improve the procedure. On the other hand, it will be of interest to apply a similar approach to impact sensitivity and consider more extended data sets to identify hypothetical outliers that might lead to the identification of specific mechanisms.
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AUTHOR INFORMATION
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REFERENCES
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