Background

A major and continuing objective in the area of energetic materials is reduced sensitivity to accidental detonations due to unintended stimuli. These can include impact, shock, heat [1–3], friction [3], electric spark [4, 5], etc. The challenge is to combine low sensitivity with a high level of detonation performance.

Sensitivity reflects the ease of initiating detonation, which in turn depends upon a complex interplay of a number of factors: molecular and crystal properties, physical conditions, the nature of the stimulus, etc. [1, 2, 6, 7]. One consequence of this is the notorious difficulty of reproducibly quantifying sensitivity. The type most often measured is impact sensitivity; this is usually done by dropping a given mass upon a sample of the compound and determining the height from which 50% of the drops produce evidence of reaction [1–3, 7]. Impact sensitivity is generally reported either as this height in cm, designated h 50, or as the corresponding impact energy in J; for a mass m, this is given by mgh 50, where g is the acceleration due to gravity. With a 2.5 kg mass, an h 50 of 100 cm is equivalent to an impact energy of 24.5 J. The greater the value of h 50 or the impact energy, the less is the sensitivity.

As is discussed in some detail elsewhere [1–3, 7], it is very difficult to obtain reproducible h 50 values. They should accordingly be regarded as, at best, qualitative indicators of relative impact sensitivities. This, together with the variety of factors that can influence detonation initiation, has led some to conclude that meaningful correlations between impact sensitivity and specific molecular or crystal properties are unlikely to be achieved.

In designing and evaluating proposed new energetic compounds, however, it is very important to be able to make at least some estimates of their levels of sensitivity by molecular and/or crystal modeling techniques. This might also help to identify molecular or crystal properties that are involved in detonation initiation. Furthermore, as was discussed by Kamlet and Adolph [8, 9], a relationship of impact sensitivity to some particular molecular or crystal properties might be found if other factors that affect detonation initiation are kept as constant as possible.

There have accordingly been numerous attempts to correlate measured impact sensitivities (usually within a given class of compounds, e.g., nitroaromatics) and various molecular and crystal features; these have included the strengths of certain bonds, NMR chemical shifts, heats of fusion and sublimation, atomic charges, electronic energy levels, the efficiency of lattice-to-molecular vibrational energy transfer, molecular stoichiometry, etc. These efforts, which have often been quite successful, are summarized in more detail elsewhere [6, 7, 10, 11]. It is important to keep in mind, however, that the existence of a correlation does not necessarily indicate a causal relationship; it may simply be symptomatic of something else [6].

A concept that has received a great deal of attention is that the breaking of a certain type of bond, a “trigger linkage,” is a key step in the decomposition process involved in the initiation of detonation [8, 9]. Some possible trigger linkages are C−NO2 in nitroaromatics and nitroaliphatics, N−NO2 in nitramines, O−NO2 in nitrate esters and N−N2 in organic azides. A considerable amount of work, reviewed on several occasions [6, 10–12], has focused upon potential trigger linkages, and there has been interest in factors, e.g., external electric fields, that could affect their properties [13–15].

In the case of nitramines in particular, there is compelling evidence of the significance of N−NO2 rupture in the early stages of detonation initiation [16–23], the weakness of the N−NO2 bond being consistent with this idea [24]. For example, Kohno et al. have pointed out that the crystallographic N−NO2 distances in several nitramines are significantly shorter, by 0.05–0.08 A, than the computed gas phase values [20–22]. This was not observed for C–N or N–O bonds. They suggested that this N−NO2 bond compression introduces strain and thus is destabilizing. A rough correlation was found between the extent of bond shortening and sensitivity.

While the trigger linkage concept may be widely applicable in the case of nitramines, and sometimes for other types of compounds as well, a number of other mechanisms have also been implicated in detonation initiation. Among these are interactions of aromatic NO2 groups with ortho substituents [6], N2 release from 1,2,3-triazoles [25, 26], nitro-nitrite [27], nitro/aci tautomerism [28] and other intramolecular rearrangements [29], etc.

In earlier work [24, 30–33], we have demonstrated that there is a characteristic anomalous imbalance in the molecular surface electrostatic potentials of energetic compounds, and that it can be related quantitatively – albeit perhaps symptomatically – to the impact sensitivities within groups of nitroaromatics, nitramines and nitroheterocycles. In the most recent study [24], we began to explore in greater depth the apparent connection between surface potential imbalance and sensitivity, focusing upon the bond energies of possible trigger linkages. We continue this exploration now, examining the roles of crystal and molecular volumes in conjunction with molecular surface electrostatic potentials.

In doing so, we recognize the limitations that are imposed by the number of factors that can affect impact sensitivity and by the uncertainty in its measurement. Our goal, therefore, is not a precise structure/activity correlation, but rather to continue to try to understand which factors and properties actually do influence sensitivity, as opposed to relationships that are symptomatic (which can also be useful) or even coincidental.

Molecular electrostatic potentials of energetic compounds

The electrostatic potential V(r) that is created at any point r by the nuclei and electrons of a molecule is,

$$ {\text{V}}\left( {\mathbf{r}} \right) = \sum\limits_{\text{A}} {\frac{{{{\text{Z}}_{\text{A}}}}}{{\left| {{{\mathbf{R}}_{\text{A}}} - {\mathbf{r}}} \right|}}} - \int {\frac{{\rho ({\mathbf{r}}\prime ){\text{d}}{\mathbf{r}}\prime }}{{\left| {{\mathbf{r}}\prime - {\mathbf{r}}} \right|}}} $$
(1)

in which ZA is the charge on nucleus A, located at R A, and ρ(r) is the molecule’s electronic density. V(r) is a physical observable, and can be obtained experimentally by diffraction techniques [34, 35] as well as computationally. The sign of V(r) in any given region depends upon whether the positive contribution of the nuclei or the negative one of the electrons dominates there.

While the electrostatic potential has considerable fundamental significance [36], it has also been found to be very effective for analyzing and predicting noncovalent interactions [37–39]. For this purpose, V(r) is commonly computed on the molecular surface defined, as proposed by Bader et al. [40], by the 0.001 au (electrons/bohr3) contour of the electronic density. The resulting surface potential, labeled VS(r), can be characterized by means of several statistically-defined quantities [37–39], including its positive, negative and total variances (\( \sigma_{+}^2 \), \( \sigma_{-}^2 \) and \( \sigma_{\text{tot}}^2 \)) and an electrostatic balance parameter ν. These are given by Eqs. 2–4:

$$ \sigma _ + ^2 = \frac{1}{m}\sum\limits_{i = 1}^m {{{\left[ {{\text{V}}_{\text{S}}^ + ({{\mathbf{r}}_i}) - \overline {\text{V}} _{\text{S}}^ + } \right]}^2}} \quad \sigma _ - ^2 = \frac{1}{n}\sum\limits_{j = 1}^n {{{\left[ {{\text{V}}_{\text{S}}^ - \left( {{{\mathbf{r}}_j}} \right) - \overline {\text{V}} _{\text{S}}^ - } \right]}^2}} $$
(2)
$$ \sigma_{\text{tot}}^2 = \sigma_{+}^2 + \sigma_{-}^2 $$
(3)
$$ \nu = \frac{{\sigma_{+}^2\sigma_{-}^2}}{{{{\left( {\sigma_{+}^2 + \sigma_{-}^2} \right)}^2}}} $$
(4)

In Eq. 2, m and n are the numbers of surface points at which VS(r) is positive, \( {\text{V}}_{\text{S}}^{+} \left( {\mathbf{r}} \right) \), and negative, \( {\text{V}}_{\text{S}}^{-} (r) \), respectively. \( \overline {\text{V}} _{\text{S}}^{\text{ + }} \) and \( \overline {\text{V}} _{\text{S}}^ - \) are the averages of the positive and negative surface potentials.

The quantities \( \sigma_{+}^2 \), \( \sigma_{-}^2 \) and \( \sigma_{\text{tot}}^2 \) are indicators of the strengths and variabilities of the positive, negative and total surface potentials. Since the terms in Eq. 2 are squared, the effects of the positive and negative extrema are emphasized. The degree of balance between the positive and negative surface potentials is measured by ν, which reaches a maximum of 0.250 when \( \sigma_{+}^2 = \sigma_{-}^2 \).

On the surfaces of most organic molecules, the most positive electrostatic potentials are usually associated with hydrogens and sometimes with the σ-holes of Group IV–VII atoms [41], while negative potentials are most often due to lone pairs and π electrons. The positive regions may be more extensive than the negative, but they tend to be weaker, so that in general, \( \sigma_{-}^2 > \sigma_{+}^2 \) [33, 37, 39].

Molecules of energetic compounds are different. They frequently feature several highly electron-attracting components, e.g., NO2, ONO2 or N3 groups and/or aza nitrogens. As a result, the surface potential above the central portion of the molecule is likely to be strongly positive, with negative regions mainly limited to the periphery, reflecting the lone pairs of the NO2 oxygens, any aza nitrogens, etc. For example, whereas the π electrons of benzene produce VS(r) that reach −18 kcal mol−1 above/below the ring, these negative regions are completely gone in 1,3,5-trinitrobenzene, which instead has maximum VS(r) of +29 kcal mol−1 above/below the ring [24].

The electrostatic potentials on the molecular surfaces of energetic compounds are therefore characterized by an anomalous imbalance; the positive regions are typically stronger and more variable than the negative, so that \( \sigma_{+}^2 > \sigma_{-}^2 \) [24, 30–33], in contrast to the majority of organic molecules. This is depicted graphically by Murray et al. [24, 30] and very extensively by Rice and Hare [7].

Molecular electrostatic potentials and sensitivity

We have shown in the past that there is a link – not necessarily a causal relationship – between the anomalous imbalance in the molecular surface electrostatic potentials of energetic compounds and their impact sensitivities [24, 30–33]. Using various measures of this imbalance (some involving \( \sigma_{+}^2,\,\sigma_{-}^2 \) and ν), very good correlations with experimental impact sensitivities have been obtained for nitroaromatics, nitramines and nitroheterocycles, treated separately. The unifying concept, as has been pointed out [7, 24, 33], is that the most sensitive molecules have high levels of the imbalance between the strong positive and weaker negative surface potentials.

Among the most positive portions of energetic molecular surfaces are those related to C-NO2 and N-NO2 bonds [24, 31, 32]. These regions become more positive as the number of NO2 groups, aza nitrogens, etc., increases and further depletes the electronic charge in the central part of the molecule. What is particularly significant is that the C-NO2 and N-NO2 bonds concomitantly become weaker; this was demonstrated computationally at the density functional B3PW91/6-31++G(3d,2p) level [24]. Thus, to the extent that C-NO2 and N-NO2 bonds are trigger linkages in some energetic compounds, the link between the imbalance in their molecular surface potentials and impact sensitivity may be that the former reflects the electronic charge depletion that facilitates the breaking of these bonds.

The N-NO2 bonds in nitramines are quite weak, with bond energies of 37 to 46 kcal mol−1 [24], and nitramines tend to be very sensitive; 80% of those listed by Storm et al. have h50 < 40 cm [2]. These observations, together with the evidence cited earlier [16–23], support the trigger linkage concept in nitramine detonation initiation. Indeed, the impact sensitivities of a group of eight nitramines correlate fairly well (correlation coefficient R = 0.93) with just \( \sigma_{+}^2 \) and ν [24].

In contrast to the N-NO2 bonds in nitramines, the C-NO2 bonds in nitroaromatics and nitroheterocycles are normally considerably stronger, with bond energies greater than 60 kcal mol−1. Thus other mechanisms may be competitive with C-NO2 bond rupture, or sometimes dominant, in the decomposition and detonation initiation process. For example, Kamlet and Adolph concluded, in developing their oxygen balance correlations with impact sensitivity [9], that nitroaromatics with an ortho substituent having a C-H bond must be treated separately from the others. Accordingly, for classes of energetic compounds other than those having particularly weak trigger linkages (e.g., N-NO2, N-N2), some additional factor(s), besides the surface potential imbalance, may need to be taken into account in establishing correlations with sensitivity.

A crystal volume/electrostatic potential relationship with sensitivity

The preceding discussion has focused upon molecular properties in relation to sensitivity. Now we would like to also include an aspect of crystal structure. We have investigated the possibility that one of the factors involved in determining impact sensitivity is the space available to a molecule of the compound in its crystal lattice. How can a measure of this available space be obtained? If the crystal density d is known, then an effective volume per molecule is given by,

$$ {{\text{V}}_{\text{eff}}} = M/d $$
(5)

where M is the molecular mass. Alternatively, Veff can be calculated from the unit cell dimensions and the number of molecules per unit cell. Veff simply apportions the total volume of the unit cell among these molecules; it corresponds to a hypothetical 100% packing efficiency. Thus if Veff could be compared to a more realistic measure of the volume of the molecule, the difference would be indicative of the “free space” available to it.

The volume enclosed by the 0.001 au surface, V(0.001), is not suitable for this purpose, since it has been found to be very similar to Veff for many C,H,N,O energetic compounds [42–44]. In fact, Eq. 5 in conjunction with V(0.001) often yields quite satisfactory estimates of C,H,N,O crystal densities; for a group of 180 compounds, 41% of the predicted densities were within 0.030 g cm−3 of the experimental values [43]. In some instances, however, V(0.001) differs significantly from Veff, and the predicted densities can then be too high or too low by as much as 0.100 g cm−3 or more. It has been demonstrated that the overall results can be improved, and the number of rather bad predictions (error >0.100 g cm−3) considerably decreased by introducing the term \( \nu \sigma_{\text{tot}}^2 \) into Eq. 5 [44]. We will return to this later.

Since V(0.001) is not appropriate for our needs, we have used instead the volume defined by the 0.002 au contour of ρ(r), i.e., V(0.002). All computed results to be presented were obtained with the density functional B3PW91/6-31G(d,p) procedure, and pertain to the 0.002 au surfaces.

In Table 1 are listed 20 energetic compounds of different chemical types: nitroaromatics, nitramines, nitroheterocycles, benzofuroxans, a nitrate ester, a nitroalkane, and a nitroalkene. They are listed in order of decreasing sensitivity, bis(2,2,2-trinitroethyl)nitramine being the most sensitive. The h 50 values are taken primarily from Rice and Hare [7], with a few from Storm et al. [2]. The Veff in Table 1 were calculated from the experimental densities [43], using Eq. 5, and the V(0.002) were obtained computationally, as were \( \sigma_{+}^2 \), \( \sigma_{-}^2 \), and \( \nu \sigma_{\text{tot}}^2 \). Note that in every instance is seen the characteristic anomalous imbalance, \( \sigma_{+}^2 > \sigma_{-}^2 \), which is sometimes considerable.

Table 1 Experimental and computed dataa,b
Full size table

The first point of interest concerns the differences between the Veff and the V(0.002), i.e., ΔV=Veff–V(0.002). Is there evidence of any relationship whatsoever between h 50 and ΔV?

Figure 1 shows that there are in fact two rough experimental relationships, one for the six nitramines (Group A) and the other for the remaining 14 compounds (Group B). The correlation coefficients R are, respectively, 0.69 and 0.80. The general trends are that the compounds become more sensitive (lower h 50) as ΔV increases.

Fig. 1
figure 1

Measured impact sensitivities (h 50) plotted against computed ΔV for the compounds in Table 1. Triangles correspond to nitramines, circles to all others

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For Group A, however, h 50 is seen in Fig. 1 to depend only marginally upon ΔV. This reinforces the idea, already discussed, that the key to nitramine decomposition and detonation initiation is breaking the N-NO2 bond, the ease of which is promoted by the electronic charge depletion that is reflected in the anomalous surface potential imbalance. Thus we found in a previous study that the h 50 of a series of eight nitramines can be expressed in terms of just \( \sigma_{+}^2 \) and ν [24]. Similarly, when multivariable regression analysis [45] is applied to fit the h 50 of the six nitramines in Table 1 to,

$$ {h_{50}} = \alpha \sigma_{+}^2 + \beta \nu + \gamma $$
(6)

R = 0.96, with an average absolute deviation from experiment of 3 cm and a root-mean-square deviation of 6 cm. (The coefficients α, β and γ are given in Table 2.) The database that was used is of course very small, but these results do provide further evidence that the weakness of the N-NO2 trigger linkage, of which the surface potential imbalance is a symptom, is likely to be the critical factor in nitramine sensitivity.

Table 2 Data pertaining to Eqs. 6 and 7
Full size table

Figure 1 suggests that ΔV is significantly more important for group B than for group A, although it certainly does not suffice for the former, as can be seen from the example of FOX-7. Its ΔV of 10.5 Å3 would give an h 50 of about 240 cm, rather than the actual 126 cm. Thus, while Fig. 1 does imply a relationship with ΔV for Group B, some additional correction term is needed.

For this role, we tested the same quantity, \( \nu \sigma_{\text{tot}}^2 \), that was found to produce improved crystal densities [44]. This quantity is a well-established index of electrostatic interactive tendencies, particularly between molecules of the same kind (as in molecular crystals); it is a key term in analytical representations of a variety of liquid and solid phase properties that depend upon noncovalent interactions, e.g., heats of phase transitions, boiling points and critical constants [37–39]. Since \( \nu \sigma_{\text{tot}}^2 \) is effective in correcting crystal densities based upon poor estimates of Veff [44], it seems reasonable to ask if it might in an analogous manner benefit ΔV.

Fitting the experimental h 50 of group B to

$$ {h_{50}} = \alpha {\left( {\Delta V} \right)^{1/3}} + \beta \nu \sigma_{\text{tot}}^2 + \gamma $$
(7)

yielded R = 0.93, an average absolute deviation of 19 cm and a root-mean-square deviation of 27 cm. (We found the fit to be statistically better in terms of (ΔV)1/3 rather than ΔV. The coefficients α, β and γ are in Table 2.) These results are quite satisfactory, given the uncertainty in measuring impact sensitivity. The h 50 predicted by Eq. 7 are compared to the experimental in Fig. 2.

Fig. 2
figure 2

Comparison of impact sensitivities predicted by Eq. 7 with experimental values, for non-nitramines in Table 1

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It might be argued that Eq. 7 can be applied only to existing compounds, since ΔV requires a knowledge of Veff, which is obtained from the crystal density or lattice dimensions. However we have demonstrated that the density can be predicted computationally with an average absolute error of 0.036 g cm−3 [44]. This should allow Eq. 7 to be applied, with reasonable reliability, to proposed compounds and others for which the density is not known.

The coefficients α, β and γ in Eqs. 6 and 7 will change somewhat with the development of larger databases. We are encouraged to believe, however, that these or similar relationships will provide a capacity for qualitatively ranking, with reasonable reliability, the relative impact sensitivities of many energetic compounds.

Discussion and summary

There is need for caution in drawing inferences concerning factors determining impact sensitivity. Different mechanisms may be involved even within a given class of energetic compounds, e.g., nitroaromatics [6, 9]. One consideration must certainly be the ease or difficulty of breaking possible trigger linkages, such as C-NO2 and N-NO2. These become weaker as more NO2 groups and aza nitrogens are introduced into the molecule and withdraw electronic charge [24]. This gives rise to the anomalous imbalance in the molecular surface electrostatic potentials that is characteristic of energetic compounds. Thus, in those cases in which trigger linkage rupture is a key step in detonation initiation, the surface potential imbalance can be symptomatic of sensitivity. Our present and recent [24] analyses indicate that such is often the case for nitramines; this is also consistent with a great deal of earlier work [16–23, 33].

Our current study also draws attention to the possible significance of ΔV with respect to impact sensitivity. Why might this be a factor? Perhaps the availability of more space enhances the molecule’s ability to absorb and localize, vibrationally or translationally, the external energy coming from the impact.

An interesting aspect of ΔV is suggested by Fig. 1, in which it separates the six nitramines from the other compounds. Perhaps a relatively low dependence of h 50 upon ΔV is a characteristic of compounds in which trigger linkage rupture is a dominant factor in determining sensitivity. In any case, the role of the surface potential in Eq. 7 is that of a correction term.

In this context, it is relevant to point out that there are three compounds in Table 1 – Tetryl, TNAZ and 1,4-dinitroimidazole – that have both C-NO2 and N-NO2 bonds. Purely structurally, therefore, it is not obvious whether each of them should or should not be viewed as a nitramine. Figure 1 clearly puts TNAZ and 1,4-dinitroimidazole with the nitramines, and this is confirmed by Eq. 6, which predicts their h 50 to be 25 and 52 cm, respectively, very close to the experimental 30 and 55 cm (Table 1). In contrast, Eq. 7 would assign them h 50 of 225 and 110 cm.

The case of Tetryl is more ambiguous. In Fig. 1, it could be with either group of compounds. We have treated it as a non-nitramine [24], but Eqs. 6 and 7 both give reasonable predictions for it, 35 and 9 cm respectively, compared to the experimental h 50 of 25 cm. The decomposition of Tetryl warrants further investigation.