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Sensitivity refers to explosive phenomena caused by perturbations that are ordinarily insufficient to produce a detonation. As a very active research field, theoretical prediction of sensitivity and thermal stability for high explosives has attracted many scientists' attention. However, the sensitivity of energetic compounds, which is related to the kinetics and thermodynamics of their thermal decomposition, is very complex and difficult to investigate theoretically.1 It is important, nevertheless, to understand the relationship between the molecular structures, including chemical bonds, electronic structure, and energy transfer rate of energetic molecules and their sensitivities to specific stimuli.2 Some of the well-known stimuli include impact, friction, shock, electrostatic charges, and heat. For the complicacy of such sensitivity tests, it is impossible to test the comprehensive performance of all the newly designed energetic materials, it is essential, therefore, to establish some empirical methods to predict various characteristics of an energetic compound, including sensitivity, performance, and physical and thermodynamic properties. Such a study on explosives could provide considerable insight into understanding the factors that affect their behavior.3–9
In the past few decades, many efforts have been made to improve the survivability and safety of munitions, during which a new class of insensitive munitions (IM) explosives has been developed. However, these IM formulations differ somewhat from each other in a variety of aspects especially in sensitivity and stability. The explosive selection process for a given munitions has therefore become more complex, and the models for theoretical sensitivity prediction for such IM explosives need to be further improved. In this article, a large number of theoretical methods that have been used to evaluate the mechanical and spark sensitivities are summarized, in which the advantages and disadvantages of these methods, together with their scope of use are clarified. In addition, the theoretical models for thermal stability of explosives are briefly introduced as a supplement.
Evaluation of Impact Sensitivity
Assessing impact sensitivity of explosives has long been considered a challenging task. The microscopic processes involved in initiation by impact are complicated and not very well understood. In general, it is believed that ignition by impact starts from pockets of hotspots generated from energy localization because of shear band formation.10, 11 Actually, the accurate modeling of impact sensitivity for high explosives requires an understanding of the mechanical, thermal, and chemical responses during impact and the subsequent chemical reactions.
One of the simplest and best-known small-scale evaluations for impact sensitivity is the drop hammer (impact) test, which is routinely applied to many high explosives.12, 13 Results from drop hammer tests cannot be trusted beyond a rough ranking in categories of danger. An alternative to the simple drop hammer test is the ballistic impact chamber (BIC) apparatus (see Fig. 1) developed in recent years.14 A theoretical modeling to understand what can be learned from BIC experiments as well as its limitations has therefore been established,15 and this will be discussed in detail in a later section. As suggested by many researchers, the subthreshold ignition of energetic materials involves the generation of hot spots.16 Though the actual mechanism of hot spot formation has not been experimentally determined, a variety of mechanisms have been proposed for their formation, including adiabatic compression of trapped gas in voids, friction involving sliding or impacting surfaces, shear band formation caused by mechanical failure, sparks, tribioluminescence, and heating at crack tips. Studies of accidental and intentional initiations have shown that impact and friction stimuli, even much less than those required to heat bulk explosive to its deflagration temperature, could still ignite or initiate an explosion. Hence, the highly fruitful idea of the localized hot spot was conceived. Except for hot spot theory, various methods based on bond dissociation energy, electronic structure properties, and energetic transfer rate have been proposed to evaluate and interpret the impact sensitivity of energetic compounds over the years. These are summarized in the following sections.
Bond Dissociation Energy Method
New energetic compounds are usually designed based on their chemical structure, and it is more convenient to predict the performance and sensitivity directly based on their chemical bonds. It has been proposed by Zeman17 that typical empirical approaches to evaluating impact sensitivity could be based on CHNO compositions, namely the ratio of C, H, and O atoms. However, the chemical structure of their functional groups was also important in determining impact sensitivity. For instance, nitro esters containing a CONO2 bond (e.g., NG, nitroglycerine) are more sensitive than nitramines containing CNNO2 (e.g., RDX, 1,3,5-trinitro-1,3,5-triazine), and the compounds with a CNO2 nitro group (e.g., nitromethane) are the least sensitive. In addition, the enthalpy of explosion follows a similar trend.18
In recent years, some high-nitrogen content molecules such as nitroimidazoles, nitropyrazoles, nitrotriazoles, and nitropyrimidines with different impact sensitivity have been designed. It has been found by Keshavarz et al.19 that impact sensitivity of nitrohetrocycles can most suitably be expressed by their elemental composition and the number of CNC and CNNC moieties in the aromatic ring divided by their molecular structure. This approach was based on the elemental composition and two structural parameters of CaHbNcOd nitroheterocyclic energetic compounds. It was found that the root mean squares (rms) of the deviation of different nitroheterocyclic molecules were 58 and 71 cm using the new correlation and neural networks computation methods, respectively. In fact, compared to the neural network method established by Cho et al.,20 the correlation method gave results that are more reliable, but which, however, cannot be applied to insensitive explosives.
To predict impact sensitivities of various nitro energetic compounds, further studies have been carried out by Lai et al.21 They introduced some new and simple correlations based on the atomic numbers of C, H, N, O, and other amending factors, which were determined by the effect of connective positions of function groups. Using this method, a prediction of impact sensitivities for nearly 200 energetic compounds were carried out and compared with the abovementioned empirical computation results presented by Keshavarz et al.19 The root mean squares (rms) of the deviation from these experimental data was 37 cm, which was much lower than that of the calculations from Keshavarz et al.
In fact, the methods mentioned above are commonly used to do some qualitative research on the impact sensitivity of energetic compounds. It is no doubt not sufficient to know only the rank of the impact sensitivity, and it is more important to clarify how much energy is needed to initiate such energetic materials. For such purposes, the bond dissociation energy (BDE) for removing the NO2 group in nitroamine molecules with nitro alkyl, and benzoate with nitro alkyl was calculated at B3LYP/6-31G* and B3P86/6-311G* levels by Song et al.22 It was shown that the strength of the CNO2 bond was weaker than that of the NNO2 bond in nitroamine molecules with nitro alkyl, and CNO2 was also the weakest one. The relationship between the impact sensitivities and the weakest CNO2 bond dissociation energy values was further examined. It was shown that a nearly linear correlation exists between the impact sensitivity and the ratio of the BDE value to the molecular total energy E (see Fig. 2). According to Figure 2, to obtain a better correlation, it is more suitable to calculate the BDE at B3LYP/6-31G* level. Based on these results, BDE/E could be considered as a reasonably practical indicator of explosive sensitivity. According to Odiot,23 however, based on self-consistent field calculations, it was found that the crystalline environment could significantly reinforce the CN bond in nitromethane by about 4.7 kcal mol−1 compared to a gas phase molecule. It was further found by Odiot's et al. that one should be wary of relying on an oversimplified one-to-one correspondence between sensitivity and molecular structure in the context of safety rules, as changes in bond dissociation energies (BDEs) are dependent on the physical state of a substance. In particular, numerous studies now support the idea that the CNO2 and NNO2 bonds are key factors in nitro-containing molecules and it has been suggested that sensitivity increases as the strength of these bonds decreases.24–26 Besides, a good correlation between the impact sensitivity and molecular structures for nitramine explosives was also found using the neural networks method.27
For the purposes of getting some more valuable data on impact sensitivity, a ballistic impact chamber (BIC) apparatus was designed and is, mentioned at the beginning of “Evaluation of Impact Sensitivity” section. Continuum mechanics simulations to examine the behavior of energetic materials in BIC experiments have also been performed by Wu et al.15 In these simulations, an arbitrary Lagrangian-Eulerian program (ALE3D) was used. It was revealed that interface friction played an important role in inducing the formation of shear bands, which resulted in “hot spots” for ignition. It was found that the temperature localization during BIC impact was significant in materials with high yield strength and there were multiple locations inside shear bands, which could achieve temperatures exceeding the threshold temperature for reaction. It was also discovered that the mass of the sample could influence significantly the shape of the pressure profile (see Fig. 3). To avoid generating features in P(t) curves that are not relevant to ignition chemistry, one should use a small sample mass or increase the length of gun chamber (the length of gun was kept at 30 cm here).
However, when the first principle density functional theory method SIESTA was used by Zhang et al.28 to compute the band gap of several polynitroaromatic explosives, such as 1,3,5-triamino-2,4,6-trinitrobenzene (TATB), 2,4,6-trinitro-1,3-benzenediamine (DATB), trinitrotoluene (TNT), and picric acid, the bond dissociation energy (BDE) alone cannot predict accurately the relative sensitivity to impact for these compounds. In these systems, the weakest bond was the one between an NO2 group and the aromatic ring. It was found that their relative impact sensitivity could be explained by considering the BDE and the band gap value of the crystal state together. In other research by Türker et al.,29 certain amino and methyl substituted 1,3,5-trinitrobenzenes have been considered, and structures of their hydrocarbon isoconjugates are depicted (see Fig. 4). The structure-impact sensitivity relation of some substituted 1,3,5-trinitrobenzene type explosives was investigated by using a topological approach. It was found that, except for 2,4,6-trinitrophenylmethylnitramine (TETRYL) which has a nitramine group, the sensitivity increases with a decrease in the upper bound (U) of the deepest lying molecular orbital energy (x1) or the fourth coefficient (a4) of the secular polynomial of the molecular graph of the isoconjugate hydrocarbon system within the Hückel molecular orbital framework. The relationship between these parameters can be expressed as Eq. (1).
where e is the number of edges in the molecular graph (G(N,e)) associated with π-skeleton of the conjugated system having N atoms. Then the sensitivity of the compounds studied could be calculated by Eq. (2).
It is very practical and time saving to use this method to clarify how skeletal changes affect the impact sensitivity of substituted 1,3,5-trinitrobenzenes, some of which are used as secondary explosive materials. Although the approach is simple, it may only be suitable for some secondary explosives without any nitramine group (NNO2).
Molecular properties and micromechanics methods
To find the relationship between impact sensitivity and the electronic structure (molecular properties) of energetic compounds, many studies have been carried out at a molecular level.30–34 A quantitative structure-property relationship (QSPR) study was used by Wang et al.35 for prediction of impact sensitivity of nitramine, and a set of 35 molecular descriptors were calculated to represent its molecular structure. It was found that tunneling occurred from the ground state to excited states during the explosion of benzenoid molecules initiated by impact.36 The impact sensitivity values of these structures are highly dependent on the lowest unoccupied molecular orbital energy and partially dependent on the highest occupied molecular orbital energy. Owens et al. first noted that there is a correlation between sensitivity and the electrostatic potential (ESP) at the midpoint of the CONO2 bond.37–39 To clarify how the energy of a chemical bond linking atoms i and j depends on the electronic charges carried by the bond-forming atoms, an expression has been established as Eq. (3), and the electrostatic potential, V(r), could be defined by Eq. (4).
where ε is for a reference bond with net charges q and q at atoms i and j, respectively, whereas εij corresponds to modified charges, qi = q + Δqi, and qj = q + Δqj. The aij and aij parameters, “measuring” the changes in bond energy accompanying unit charge variations at atoms i and j, respectively, are readily deduced by theory; Zi and Ri denote the charge and position of the nucleus of atom i and j; ρ(r′) represents the electronic density. The electrostatic potential is a property that can be determined through diffraction measurements or evaluated using quantum mechanical theory and has often been used to analyze the electron density distribution in a molecule. ESPs on isosurfaces of electron density have usually been used in identifying sites within molecules that might be conducive to nucleophilic or electrophilic attack.
Some later works have suggested that the dissociation energy of the weakest bond of the explosive molecule might play an important role in initiation events.40 The work by Davis and Brower41 on the initiation chemistry of nitroarenes did not support the assumption that the CONO2 bond was the trigger linkage. It was also pointed out that correlation studies should not be used for interpretation of mechanistic details.42 However, it is still necessary and useful to carry out correlation studies to identify molecular properties that indicate impact sensitivity.43 In fact, Chen and Cheng44 discovered a relationship between the electronic properties (band gap) of cis-1,3,4,6-tetranitrooctahydroimidazo-[4,5-d]imidazole (BCHMX) and its impact sensitivity. Besides, according to Zeman,45 Huang,46 and Wang,47, 48 impact sensitivity is also dependent on the electrotopological state indices. Furthermore, based on bond types and electronic structures, a new method has been introduced by Mohammad49 to predict impact sensitivity of different types of polynitroheteroarenes. In this model, the numbers of carbon and hydrogen atoms as well as specific structural parameters that might affect the impact sensitivity were considered. Calculations have been carried out on some 67 different polynitroheteroarenes, including some nitroheterocyclic explosives and several newly synthesized polynitroheteroarenes, and the results were compared with output from recently calculated results using complex neural network methods. It was shown that the root mean squares (rms) of deviations of these compounds obtained by this novel model were 36 cm, which was closer to the experimental data compared to the 66 cm for the neural network methods.27
However, some Korean scientists50 recently proposed a more precise evaluation method also based on the quantitative structure-property relationship (QSPR). They calculated the impact sensitivity of some energetic compounds according to the electrostatic potential (ESP) values on the van der Waals molecular surface (MSEP). Among various 3D descriptors derived from MSEP, they utilized total and positive variance of MSEP, and devised a new QSPR equation by combining three other parameters. There were 6 models involved in their research which could be expressed as Eqs. (5)–(10) listed below.
where, h50% is the height (in cm) from which 50% of the drops results in detonation of the sample by dropping 2.5 kg weight of a drop hammer. and are the averages of the positive and negative ESPs on molecular surface, respectively. ν is the balance parameter which is described as the degree of balance between positive and negative potentials on the isosurfaces; H, HBD, and PSA are the number of hydrogen atoms, the number of hydrogen bond donor and the polar surface of the molecule, respectively. σtot is the sum of MSEP values and σ, σ are the variance of positive and negative MSEPs defined in Eq. (11). Q is the heat of detonation for CHNO explosives and defined as the heat of the reaction for the reaction in Eq. (14).
As shown in the equations above, the correlations of impact sensitivity with charge distribution in the molecular surface followed exponential equations. In fact, in most QSPR studies, however, linear equations have been widely used for simplicity and ease of interpretation.
Based on the abovementioned models, the Koreans used 37 HEDMs having a benzene scaffold and nitrosubstituents, which have also been calculated by Rice and Hare.30 All the molecular structures were optimized at the 3LYP/6-31G (d) level of theory and confirmed as minima by frequency calculations. Their new QSPR equation provided a good result (see Fig. 5) to predict the impact sensitivities of the molecules in the training set, including zwitterionic molecules. However, according to Figure 5, it seems that it is better to use Model 5 to evaluate energetic materials with lower impact sensitivity, while Model 6 is better for insensitive energetic compounds. Since the sensitivities of the molecules in this study appear to be related to electron deficiency over covalent bonds within the inner structure of the molecule, the global GIPF parameters could not adequately reflect the localized description of charge buildup that seems to be symptomatic of an explosive's sensitivity.
In fact, there is also a good relationship between crystal volume factors and impact sensitivity. It was firstly found by Miroslav etc. and further confirmed by Vavra etc.51, 52 that the available free space of the energetic materials molecules could affect their impact sensitivity, and this space could be defined as ΔV, and
where Veff is the effective molecular volume obtained from the crystal density, and V(int) is the volume that enclosed by the 0.002 au (for gas phase) or 0.003 au contour of the molecule's electronic density which is determined by calculation. To calculate the impact sensitivity, except for the σtot as mentioned above, they defined another parameter, the electrostatic balance parameter v as below.
Based on curve fitting with a yielded R = 0.93, the equation to calculate the impact sensitivity (h50) for the 20 energetic compounds listed (see Ref. 34) was deduced as follows:
However, the dependence upon ΔV may be rather minor for nitramines or even negligible for many organic azides because some members of this group may have a specific mechanistic step in common. Eq. (17) can be applied only to existing compounds, since ΔV requires establishing Veff, which is obtained from the crystal density or lattice dimensions. Besides, the coefficients of Eq. (17) will change when the curve fitting is performed on some other group of energetic compounds. Therefore, this method cannot be widely used, especially for those energetic compounds under design.
As far as we know, during the impact process, the partitioning within a microparticle cluster and its fracture should be considered as the essential factors.53–55 Using the abovementioned comprehensive molecular properties method, Wu and Huang56 recently developed a micromechanics model describing hot-spot formation in the energetic crystal powders subjected to drop-weight impact. In their theory, contact deformation, friction and chemical reactions at the particle level during the impact loading process were considered and three hot-spot sources were included (see Fig. 6).
However, there were some assumptions for this model. These included the microparticle contact deformation between two equal-sized particles with no relative sliding, the contact sites between the impacting surface and the particles, the contact zone between particles with a sliding velocity along the maximum shear-stress direction, and drop-weight impacts on samples composed of equal-sized particle layers. Hot-spot ignition was predicted via thermal explosion, using an Arrhenius thermochemical model and the effects of drop height and particle size on the ignition processes were also considered. It was shown that the hot-spot sources played the most important role during the early ignition stage and the time-to-ignition increased with a decrease in drop height. The total valid calculation time was the time during which the drop weight falls to a maximum distance. Many parameters related to impact sensitivity were calculated, including the motion of the drop-weight during impact [ , calculated by Eq. (18)], the temperature increasing terms contributed by the heat of the chemical reaction [ΔTr, calculated by Eq. (19)], and certain other parameters (see Ref. 34)
where g is gravitational acceleration, and Fc(t) is the force exerted by the whole sample on the weight; qr is the heat of reaction per unit mass for a single step chemical decomposition; Zr is the pre-exponential factor; EA is the activation energy and Rg is the universal gas constant. According to the equations mentioned earlier, but not only limited to that, it was shown from the calculation that the 1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) and pentaerythritol tetranitrate (PETN) crystals demonstrate constantly increasing time-to-ignition with a reduction in particle size, and that samples with a smaller particle size undergo larger localized deformations and lower average pressures. Besides, the calculated time to ignition decreases with the increase in drop height. It is possible to use such a model to evaluate the effects of drop height and particle size on local temperature increase and bulk sample responses. In addition, an approximate method based on such mechanical model concepts of the physicochemical and explosive properties of reactive mixtures was also proposed for calculating impact sensitivity indices, including critical initiation pressure and critical charge thickness for mixtures of an oxidizer with a fuel.57, 58
Molecular modeling method
For some special energetic compounds such as inorganic azides, however, the situation is different. Usually, inorganic azides do not have any C, H, or O atoms, so oxygen balance is irrelevant. Particularly, inorganic azides sometimes display a diversity of sensitivity, such as nonenergetic and insensitive lithium azide and highly sensitive copper(I) azide. Also, a number of other parameters, such as enthalpy of formation, again do not provide a simple correlation with sensitivity. The prediction methods for these molecules should be specific and different from the abovementioned methods. Recently, a molecular modeling program has been developed by Cartwright and Wilkinson59 to meet this requirement, and has been used to determine some of the interatomic distances in the ideal unit cell structure for a number of inorganic azides, with a particular emphasis on the distance between nonbonded nitrogen atoms on neighboring azide ions. Based on this theory, models for 13 azides were successfully constructed and validated, with the exception of lithium and β-sodium azide (see Table 1).
Comparing such a modeling method for the ion positions with the values found in the literature showed the process to be valid. It was further found that there was a very good correlation between impact sensitivity and the minimum nonbonded nitrogen–nitrogen distance across a wide variety of azides, which appears to be consistent with the theory requiring minimum atomic movements to produce the reaction products. Regarding the sodium and copper(I) azides, though both of them have monovalent cations, the former one would generate a controlled gas evolution on decomposition whereas the latter would generate a violent detonation decomposition. To clarify the fundamental difference in the structure, a further study has been attempted by Cartwright60 to examine the possible reasons for this dramatic variation in properties. It has been suggested that the differences arise from both the reduction potential (electron affinities) of the metal cations and the arrangement of the ions in the crystals. To rank the azides by sensitivity, as many of their structures are well characterized a correlation between the structure and sensitivity is possible. This has also been shown by some initiation theories based on the excitonic and divacancy models,61, 62 which essentially concern the presence of crystal defects being crucial to the initiation of azides.
According to these modeling results, there should be a strong correlation in inorganic azides between the nonbond NN distance and impact sensitivity. The closer together the nonbonded nitrogen atoms are, the more sensitive the compounds. This correlation is much stronger than correlations based on enthalpy of formation, lattice energy and ionization potential of the cations involved. Short nonbond distances minimize atomic movements to form products and are vital to sensitivity as they provide for a more rapid decomposition reaction.
Energy transfer rate method
There is some dependence between atomization energy and energy transfer rate, so the impact sensitivity might have some correlation with atomization energy. To clarify this, a theoretical estimation of the number of doorway modes for explosives has been carried out by Tokmakoff et al.63 based on the theory of Dlott and Fayer. It has been suggested that the energy transfer rate is proportional to the number of normal mode vibrations. The frequencies of normal mode vibrations of eight molecules were evaluated by means of density functional theory (DFT) at the B3P86/6-31G(d,p) level. In this way, the number of doorway modes was found to present a linear correlation with the atomization energies of the molecules, and a mechanism for this correlation was further discussed. It is appropriate to note that in those explosives with similar molecular structure and molecular weight, the correlation between the atomization energy and the number of doorway modes is better. Nevertheless, this successful theory does not provide an insight into the molecular level events occurring in the solid state behind the shock front. According to Ge et al.,64 the relatively low frequency, large-amplitude vibrations in large molecules could be excited readily via a low order anharmonic coupling process, typically two-phonon absorption. The energy transfer parameter noted as K, which characterizes the rate of energy transfer from external to doorway mode can be calculated by the following equation [Eq. (20)]:
where j is the number of vibron states in the door-way region, Ω the vibrational frequency of the door-way state, θe the equivalence temperature, at which the rate of up-pumping into the doorway mode is equal to the low temperature rate of relaxation by the two phonon emission. τ1(0) is the lifetime of doorway mode at temperature T = 0 K. θe could be defined by the relation nΩ/2(θe) − nΩ(θe) = 1, in which nΩ(θe) is the occupation number of phonons at frequency Ω and temperature θ. Therefore, based on Eq. (20), the value of k can be evaluated by knowing τ1(0), the lifetime of doorway mode at temperature T = 0 K, and j, the number of doorway modes.
In fact, most secondary explosives are molecular solids consisting of large organic molecules,65 which are stable molecules with large energy barriers to chemical reaction. A sizeable amount of energy therefore must be transferred from the phonons to the molecules' internal vibrations. Vibrational energy is a fundamental aspect of chemical reactivity and a few vibrational energies are energetic barriers that need to overcome for chemical reaction. Shock initiation of explosives is quite a complicated process; wherever a mechanical excitation is created in explosive materials, the excess mechanical energy is eventually dissipated into a bath consisting of the low frequency mode of lattice vibrations. Therefore, the initial energy in the phonons must be transferred to the molecular vibrations that are responsible for the bond breaking.66 Besides, according to Ye et al.,67 the effect of the impact event is to deposit energy into a band of molecules faster than the energy redistribution processes can randomize that energy. Following this rapid and exclusive perturbation of low-energy modes, Fermi's Golden Rule could determine the initial rate of energy transfer from these low-energy modes into higher energy states. Results of calculations suggested that this initial rate of energy transfer, from states within a low-energy “phonon manifold” to that of a higher energy “internal vibrational manifold,” plays an important role in determining impact sensitivities of the explosives.
To investigate the relationship between impact sensitivities and energy transfer rate, the number of doorway modes of explosives has been estimated by a simple theory, in which the rate was proportional to the number of normal mode vibrations. For instance, in the research work of Ge et al.,68, 69 the frequencies of normal mode vibrations of 2,4,6-trinitro-m-cresol (C7H5N3O7), picric acid, TNB, TNAP, TNT, TNA, TBN, DATB, and TATB (their molecular structures can be found in Ref. 69) were evaluated by means of density functional theory (DFT) at the b3p86/6-31g (d,p) level. The number of doorway modes in the region of 200–700 cm−1 was evaluated by the direct counting method, which, together with atomization energy, are summarized in Table 2. It was found that the number of doorway modes showed a linear correlation with the impact sensitivities derived from drop hammer tests. It was also shown that for the secondary explosives with similar molecular structure and weight, the correlation between the impact sensitivity and the number of doorway modes was very high. Besides, some of the calculation results were also obtained by Ge et al. (see Fig. 7). According to their findings, it is very interesting that the j-number is linearly correlated to the impact sensitivities derived from drop hammer tests. This reveals that the energy transfer rates are directly proportional to the number of vibron states in the doorway region and that the energy transfer rates are linearly correlated with the impact sensitivities for secondary explosives that have similar molecular structure and weight.
Table 2. The number of doorway modes and atomization energy.
Most energetic materials, including explosives and propellants, could be considered to contain brittle constituents. A general approach has therefore been proposed by Dienes et al.70 to estimate friction sensitivity, in which they examined the hypothesis that the intense heating by frictional sliding between the faces of a closed crack during unstable growth can form a hot spot, causing localized melting, ignition, and fast burn of the reactive material adjacent to the crack. It was shown that the opening and growth of a closed crack due to the pressure of burned gases inside the crack and interactions of adjacent cracks could lead to a violent reaction, with detonation or combustion as the possible consequence. This approach was also used to model a multiple-shock experiment by Mulford et al.71 involving initiation and subsequent quenching of chemical reactions in a slab of PBX 9501 impacted by a two-material flyer plate. In this research, the effects of crack orientation and the temperature dependence of viscosity of the melt on the result were examined. In this way, the higher shock pressure LX-04 model proposed by Kevin72 was extended to accurately simulate these lower pressure and multiple shock gauge recordings. The shock desensitization effects observed with multiple shock compressions were partially accounted for in such a model by using a critical compression corresponding to a shock pressure of 1.2 GPa. This shock desensitization effect occurs at higher pressures than those of other HMX-based PBXs containing more HMX.
In fact, such a theoretical finding was further confirmed by numerical results73 showing that crack orientation had a significant effect on brittle behavior, especially under compressive loading where interfacial friction plays an important role. With a reasonable choice of crack orientation and a temperature-dependent viscosity obtained from molecular dynamics calculations, the calculated particle velocities compare well with those measured using embedded velocity gauges. The critical temperature (ignition point) for a reactive material under constant energy flux could be obtained via Eq. (21) as follows:
where Ti is the initial temperature, and the other variables and related equations are defined in Ref. 73. Though the energy flux due to frictional heating normally changes with time, the critical temperature given above does provide a useful check on the numerical scheme. Based on this theory, different crack mechanisms would generate different impact or shock sensitivity. In fact, the surface defects of the energetic compounds would affect the crack process, which was confirmed by Bellitto and Melnik's experiments74 using atomic force microscopy (AFM). It was determined that there exists a statistically significant relationship between surface roughness characteristics and the shock sensitivity of the material.
Correlation verification for the predicted results
It is well known that the initiation reactivity of energetic materials is highly dependent on the intermolecular interactions in their crystals.75, 76 In polynitro compounds, for example, the oxygen atoms of the nitro groups, by their dipole–dipole interactions, contact the oxygen and nitrogen atoms of nitro groups in neighboring nitramine molecules in the crystal,77–79 which is the decisive factor governing the crystal structure of nitramines. Nonbinding interatomic distances of oxygen atoms in the nitro compound molecules are shorter than those for the intermolecular contact radii for oxygen in carbonyl or nitro groups (i.e., 1.35–1.63 Å80). This fact corresponds to the general finding in the field of polynitro compounds, which is a topic of new scientific interest at present,81 where quantum chemistry might prove useful.
Conclusions based on conformation of isolated molecules neglect very important crystal-lattice effects that are vital in the determination of explosive properties. As the papers of Vavra et al.51, 52 have shown, an approach on this basis might be acceptable for relatively simple molecules, such as nitramines. Based on these facts, it might be seen that the application of DFT methods to the study of initiation (sensitivity) of energetic materials is limited. It is well known that reactivity in the condensed state is very different from that in the vapor phase and a similar difference exists between reactivity in the liquid and solid states. This basic knowledge is a physical organic chemistry characteristic which can be found in a monograph for energetic materials (EMs).82 Impact sensitivity is specified by means of the hammer weights of 1, 2, 2.5, 5, or 10 kg.83 Presentation of this sensitivity by the drop heights h50% can lead to incorrect selection of the sensitivity data. For prediction and evaluation purposes, sensitivity data of the pure EMs must be taken, which is another source of error.31 Elimination of this incorrect selection is possible based on verifying the correlation of the given sensitivity data with the corresponding molecular structure.75, 84–86
In summary, different methods for impact sensitivity evaluation are suitable for different groups of energetic compounds. As stated above, for instance, the Bond Dissociation Energy method is only applicable for compounds with simple structures based on C, H, N, O compositions, while the Molecular Modeling method is only applicable for inorganic azides. In addition, most of the correlations are made between the molecular properties of similar compounds and their sensitivity. It is impossible to use a single prediction method to get the sensitivity data of an arbitrary group of energetic compounds whose structures are not similar with each other. Therefore, if one wants to make some predictions about new energetic compounds, the most proper method should be selected first and then a comparison made between the theoretical predictions and the experimental results.
Prediction of Spark Sensitivity
Spark sensitivity represents the ease with which an explosion can be initiated by an electrostatic spark. The electrostatic performance of an explosive is an important aspect for estimating its safety in an electrostatic discharge environment that could be helpful in reducing accidents.87 And the theoretical prediction of spark sensitivity is a good way to prejudge the electrostatic performance of explosives. The spark sensitivity of various explosive molecules has been the subject of many articles in the literature. Currently, using Matlab software, some correlations are sought between spark sensitivity and certain molecular orbital characteristics of some nitramine type explosives.88 According to these papers, a new method for estimating the electrostatic sensitivities of nitroaromatic compounds has been developed so that utilizing the current correlation may be used for reliable estimation of the electric spark sensitivity of any proposed new nitroaromatic explosive for which no data currently exists. Therefore, this can be taken as appropriate validation test of the new method for nitroaromatic compounds. It was suggested that the following general Eq. (22) with four variables is suitable for various types of nitroaromatic compounds:
where z1, z2, z3, z4 and z5 are adjustable parameters which can be obtained by best fit to experimental electric spark sensitivities data using a multiple linear regression method, nC, nO, and nH are the number of carbon, oxygen, and hydrogen atoms, respectively, in the explosive's molecular formula, RnH/nO is the ratio of nH to nO, and CR-OR is the presence of certain groups such as alkyl (−R) or alkoxy (−OR) groups attached to an aromatic ring. The optimized correlation is as follows:
The value of CR, OR can be determined as follows: (a) CR-OR = 1.0 for an alkyl group attached to the nitroaromatic ring; (b) CR-OR = −2.0 for an alkoxy group attached to the nitroaromatic ring. The estimated electric spark sensitivity of different nitroaromatic compounds by this new correlation were within ±4.0 J of 26 measured values. Moreover, Eq. (23) can be used easily for nitroaromatic energetic compounds that have complex molecular structures. In addition, since spark sensitivity depends on the shape and size of the crystals that might connect with dislocations in these crystals,89 this method cannot predict electrostatic sensitivity versus the grain size. However, Eq. (23) cannot be applied for nitroaromatic compounds which have N-NO2 groups. It should be mentioned that the results from the correlation may not be reliable if the predictions are outside the energy range of the training set, namely about 4-23 J.
To make the prediction more reliable, and to determine a new correlation between electric spark sensitivity and the molecular structure of nitroaromatic energetic compounds, certain semi-empirical and DFT calculations have been carried out by Keshavarz.90 These research results (see Figs. 8 and 9) revealed that the nitramines considered undergo decomposition in the electric field mainly via their anionic states and a reasonably good correlation was found between electric spark sensitivity and the elemental composition as well as the existence of alkyl or alkoxy groups attached to an aromatic ring. The detailed structure of the compounds in Figures 8 and 9 can be found in Ref. .
When the results of calculation methods used are compared with the results in Refs. [91,92], correlation of the spark energy with the HOMO energy group of anions was better for UHF/PM3 and UB3LYP/6-31G(d,p)//UHF/PM3 treatments than for UB3LYP/6-31G(d,p). However, the UB3LYP/6-31+G(d,p) treatment was the best (see Table 3). In contrast, when the LUMO energy group was considered, the appropriate order would be UHF/PM3 > UB3LYP/6-31G(d,p) > UB3LYP/6-31+G(d,p). Some other computational data obtained from ionic forms of some nitroamines and their spark sensitivity values were theoretically investigated, which could be used as a supplement to help prove the theory mentioned above.85, 93–96
Table 3. The attraction energy values at different level of calculations and the spark energies for the anionic compounds considered.
Corre, correlation with the experiment; Attraction energy is in (unit electron)2/Å.
In fact, Eckhoff proposed an appropriate method that could clarify exactly how much spark energy would be used to ignite gaseous explosives.86 It was found that adequate differentiation of the required maximum permissible currents and/or voltages in intrinsically safe electrical circuits to be used in explosive dust clouds could be achieved by this method. In this way, some conservative capacitive ignition curves for dust clouds were calculated based on the different equations (see Fig. 10). In essence, the concept is to use conservative first-order ignition curves, calculated or estimated from the experimental MIE (minimum ignition energy) value of clouds of the actual dust in the air. Internationally standardized test methods allow MIE for clouds of any dust to be determined, at least down to the range of a few mJ. However, there was a need for a supplementary method covering the range of lower energies, down to 0.01 mJ. It was concluded that conservative experimental determination of MIE of dust clouds requires the use of electric sparks of sufficiently long discharge times to prevent significant disturbance of the dust cloud by the shock wave generated by the spark discharge. This is achieved by introducing a 1-2 mH inductance in the discharge circuit.
In addition, similar to impact sensitivity, Zhang et al.97 proposed a QSPR model to predict the electric spark sensitivity of 39 nitro arenes. To select from the various descriptors the ones that have a significant contribution to electric spark sensitivity, the generic function approximation (GFA) was employed for fitting the relationship between the selected descriptors and electric spark sensitivity. Their predicted electric spark sensitivity values are in good agreement with the experimental data with the correlation coefficients (R2) together with correlation coefficient of the leave-one-out cross validation (Q2CV) of the model are 0.924 and 0.873, respectively. There might be some different mechanisms for the spark sensitivity and impact sensitivity, even though both of them have some correlation with the corresponding molecular structures. In fact, in our preliminary research,98 the spark energy (EES) required for 50 percent initiation probability of 41 polynitro compounds was determined and compared with their impact sensitivity in terms of molecular structure. It was found that, depending on intermolecular interaction factors in crystals of energetic materials, the mechanism of impact energy transition to the reaction center of their molecule can be different from that of transition of energy of the electric spark.
Predictions of Thermal Stability
Understanding the violence of high explosive cookoff events is important for understanding their thermal stability and safety. The severity of these events can range from benign rupture of the confinement to a violent response close to that of a detonation. By predicting the violence of the cookoff one will be able to determine the proper configurations and the level of controls required for safe handling of the explosive systems. However, there are only a few workgroups involved in such projects.99 In the past few decades, some computational tools have been developed to predict the response of Navy ordnance to abnormal thermal (cook-off) events.100 The Naval Air Warfare Center (NAWC) and Naval Surface Warfare Center (NSWC) in the USA are performing cook-off experiments to help validate DOE computer programs and associated thermal, chemical, and mechanical models. Initial work at the NAWC was focused on the cook-off of an aluminized, RDX-based explosive, PBXN-109 that was initially confined in a tube with sealed ends. A modified version of this system was developed at the NSWC. The design of these cook-off systems are relatively simple which facilitates the initial model development.
Recently, the Lawrence Livermore National Laboratories (LLNL) and Sandia National Laboratories (SNL) have started developing computer programs and materials models to simulate cook-off for ordnance safety evaluations. The computer program ALE3D from LLNL is being used to simulate the coupled thermal transport, chemical reactions, and mechanical response during heating and explosion.101 SNL is employing multiple computer programs in a parallel effort.101, 102 It is clear that there must be more work done on the material models associated with the chemically reacting mixture material. The strength of the explosive in the partially decomposed state can have a profound effect on the ensuing reaction. The higher the strength of the explosive, the more it will resist the expansion of the decomposition products. This will increase the rate at which the explosive burns, turning what could have been a benign event into a catastrophic one. In a different direction, the current material model assumes that all of the species are of one uniform temperature and pressure. This is valid during the initial phase of the cookoff, when time scales are long compared to the time it takes to exchange energy between the phases, but is less so as the time scale gets small.
Sensitivity of explosives is a subject of keen interest to all those involved in the handling of these materials. The current ability to predict sensitivity is based on a series of empirical rules, such as the simple oxygen balance and the ratios of C and H to oxygen for different classes of explosive compounds, which are valid only for organic compounds. In addition, some special models have been proposed for some inorganic explosives such as azides. However accurate such results may be, it is safer not to rely solely on this sort of information if one wants to escape accidents, because most of the models are based on the empirical and theoretical calculations using the Kamlet and Adolph method103 or using quantum mechanical methods, and these can only really be used to rank the relative sensitivities of energetic materials.
Though the sensitivities were correlated with direct and indirect molecular properties such as oxygen balance, electronegativity, vibrational states, partial atomic charge, etc., they only show a rough correspondence to external stimuli. In fact, various sensitivities do not correlate well. For instance, impact sensitivity is not a function of molecular structure and decomposition pathways alone. In these cases, further research needs to consider additional details of the stimuli as well as corresponding responses which are used as a measure of sensitivity. An exact standard for sensitivity, therefore, should be experimentally established for all energetic compounds and their mixtures, which was once proposed by Shackelford104 and Zhang105 but not successfully achieved. In fact, the typical sensitivity data could be equivalent to the absorption of thermal energy, including inner energy. To obtain some more reliable characteristic parameters that could reflect all of the sensitivities, relationships between sensitivity and heat capacity, heat conductivity, decomposition point and dielectric properties of explosives and their mixtures should further be experimentally clarified. Therefore, there is plenty of research work that needs to be carried out, especially concerning the composite high explosives such as polymer-bonded explosives, which have seldom been theoretically investigated with regard to their sensitivity and thermal stability.
S. Zeman, born in 1942, head of the Institute of Energetic Materials (since 1994) of the Faculty of Chemical Technology at University of Pardubice, received his M.Sc. in Organic Technology—Technology of Explosives from the University of Pardubice in 1966. Because of political affairs in the then Czechoslovakia in 1968 and later, he had to leave the University in 1969 and went to work in the Slovak enterprise CHEMKO Strážske as a research worker. In 1998, he defended D.Sc. Thesis and in 2000 was appointed full Professor. He has coauthored about 350 papers cited in Chemical Abstracts. His h-factor is 18 (ISI). He is a member of editorial boards of Chinese Journal of Energetic Materials, Journal of Hazardous Materials, vice-chairman of editorial board of the Central European Journal of Energetic Materials. He is a cofounder, and from 1999 chairman, of the International Seminars called “New Trends in Research of Energetic Materials” which are held annually in April. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Q.-L. Yan, born in 1983, as one of the Ph.D. students of Prof. S. Zeman, in the Institute of Energetic Materials (since 2011) of the Faculty of Chemical Technology at University of Pardubice, is a research assistant in the field of thermal safety evaluation of high energy polymer-bonded explosives. Before his Ph.D. studies, he was an assistant professor in Xi'an Modern Chemistry Research Institute in China, during which his research interest was mainly focused on the safety evaluation and thermal analysis of energetic materials; He has co-authored over 30 journal and conference papers. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]