Introduction

In the recent past, a lot of work has been focused on nitrogen-rich energetic compounds for developing new energetic compounds having both low sensitivity and high energetic properties together in the same compound [1,2,3,4,5,6,7,8,9,10]. But all the energetic compounds cannot fulfill this requirement. One among many reasons is that the energetic properties of parent nitrogenous heterocycles are not good enough; hence, they need energetic fragment substitution to enhance their energetic properties, which causes an increase in their sensitivity simultaneously. On the other hand, to decrease the number of energetic fragments will hurt their energetic properties. Therefore, keeping a good balance between sensitivity and energetic properties is still a major challenge in the field of high energy density compounds (HEDCs).

Energetic cage-shaped compounds emerged as one of the important classes of HEDCs due to their high heats of formation (HOFs) and high strain coupled with large crystal densities and compact structure [11,12,13,14,15,16,17,18,19]. For example, hexanitrohexaazaprismane [16] (Fig. 1) has high HOFs (1052.8 kJ/mol) and excellent detonation properties (P = 49.58 GPa, D = 10.13 km s−1, and ρ = 2.10 g cm−3). Because of the superior detonation performance, the synthesis and design of new cage HEDCs have become a hot field. One new energetic cage compound 2, 4, 6, 8, 10, 12, 13, 14, 15-nonanitro-2, 4, 6, 8, 10, 12, 13, 14, 15 nonaazaheptacyclo [5.5.1.13, 11. 15, 9] pentadecane (NNNAHP) [20] was designed recently, as shown in Fig. 1. NNNAHP has high positive HOFs (772.45 kJ/mol) and excellent detonation properties (P = 43.14 GPa, D = 9.55 km s−1, and ρ = 2.01 g cm−3). However, to the best of our knowledge, this compound has not been reported experimentally. Probably the highly nitrated cage structure gives it high sensitivity and makes its synthesis difficult. A large number of nitro (–NO2) groups are also associated with high sensitivity [21, 22]. Therefore, to decrease the sensitivity and make the experimental synthesis possible, it is essential to replace the nitro groups in NNNAHP by other structural fragments.

Fig. 1
figure 1

Molecular structures of HNHAH, CL-20, and NNAHP

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Prior studies [23,24,25] show that the incorporation of oxygen (–O–) into the structure of energetic compounds is a very effective way to get thermally stable and low-sensitive energetic materials. Similarly, combining both nitro (–NO2) and amino (–NH2) groups in the same compound greatly decreases the sensitivity [26,27,28]. This sparks the idea of designing new cage-shaped compounds based on the skeleton of NNNAHP by incorporating –O– and replacing some of the –NO2 groups with –NH2 to combine both low sensitivity and high energy together in the same compounds.

In this work, we studied the HOFs, densities, detonation velocity, detonation pressure, thermal stability, free space, impact sensitivity, and electronic structures of energetic NNNAHP derivatives (Fig. 2). The prime purpose of our study is to investigate the roles of –O– and –NH2 in designing insensitive cage-shaped compounds with superb detonation performance.

Fig. 2
figure 2

Molecular numbering of the designed compounds

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Computational methodology

All the calculations were performed using the Gaussian 09 program package [29]. The DFT-B3LYP method with the 6-311G (d, p) basis set was employed to optimize the structures of the title compounds. Many studies show that the 6-311G (d, p) basis set is suitable to investigate the molecular structure and energy of energetic compounds [2, 30, 31].

Based on the total energies (E0), thermal corrections (ET), and zero-point energies (ZPE), the gas-phase phase HOFs (HOFsf, gas) were calculated via isodesmic reactions. As the number of all sorts of bonds on both reactants and products is the same in isodesmic reaction, therefore, the errors in the HOFs can be effectively decreased. The plotted isodesmic reactions used to evaluate the HOFsf, gas are displayed in Scheme 1.

Scheme 1
scheme 1

Isodesmic reactions for the designed compounds (isodesmic reaction a for the compound P, b for the compound A, c for the compounds B–E, and d for the compounds F–L)

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The heat of reactions ΔH298 K at room temperature can be calculated by the following expression:

$$ {\Delta H}_{298\ \mathrm{K}}=\sum {\Delta H}_{\mathrm{f},\kern0.5em \mathrm{P}}-\sum {\Delta H}_{\mathrm{f},\kern0.5em \mathrm{R}} $$
(1)

where ΔHf, P and ΔHf, R are the HOFs of the product and reactants, respectively at 298 K. The value of ΔH298 K can be calculated using the following expression:

$$ {\Delta H}_{298\ \mathrm{K}}={\Delta E}_{298\ \mathrm{K}}+\Delta \left(\mathrm{PV}\right)={\Delta E}_0+\Delta \mathrm{ZPE}+{\Delta E}_{\mathrm{T}}+\Delta \mathrm{nRT} $$
(2)

where ΔE0 and ΔZPE are the changes in total energies and zero-point energies between the products and reactants at 0 K, respectively, ΔET is the thermal correction from 0 to 298 K, and Δ(PV) is the pressure-volume work term and is equal to ΔnRT for the reactions of ideal gas.

It is well-known that most of the energetic compounds exist are solid-state; hence, the calculation of detonation properties (heat of detonation, detonation velocity, and detonation pressure) needs to calculate the solid-phase HOFs (ΔHf, solid) first. According to Hess’s law of constant heat summation, ΔHf, solid can be calculated from ΔHf, gas and heats of sublimation (ΔHsub).

$$ {\Delta H}_{\mathrm{f},\mathrm{solid}}={\Delta H}_{\mathrm{f},\mathrm{gas}}-{\Delta H}_{\mathrm{sub}} $$
(3)

The ΔHsub can be calculated by the following equation proposed by Politzer et al. [32]:

$$ {\Delta H}_{\mathrm{sub}}=a{A}^2+b{\left(\upsilon {\sigma}_{\mathrm{tot}}^2\right)}^{1/2}+c $$
(4)

where A is the surface area of the 0.001 electrons/bohr3 isosurface of the electronic density of a molecule; \( \upsilon {\sigma}_{\mathrm{tot}}^2 \) is the electrostatic interaction index; and a, b, and c are 2.670× 10−4 kcal mol−1 Å−4, 1.650 kcal mol−1, and c = 2.9660 kcal mol−1 [33].

The crystal densities for the designed compounds were calculated using the method suggested by Politzer et al. [34]

$$ \rho =\alpha \left(\frac{M}{V(0.00)}\right)+\beta \left(\upsilon {\sigma}_{\mathrm{tot}}^2\right)+\varUpsilon $$
(5)

where M is the molecular mass of the molecule (g/mol), V(0.001) is the volume enclosed by 0.001 electrons/bohr3 (cm3/molecule), and \( \upsilon {\sigma}_{\mathrm{tot}}^2 \) is the electrostatic interaction index factor. The constants α, β, and ϒ are equal to 0.9183, 0.0028, and 0.0443, respectively.

The detonation properties (detonation velocities and detonation pressures) were calculated based on the Kamlet-Jacobs equation [35].

$$ D=1.01{\left(N{\overline{M}}^{1/2}{Q}^{1/2}\right)}^{1/2}\left(1+1.30\rho \right) $$
(6)
$$ P=1.558{\rho}^2N{\overline{M}}^{1/2}{Q}^{1/2} $$
(7)

where D and P are detonation velocity (GPa) and detonation pressure (km s−1), respectively, N is the number of detonation gases produced per gram of energetic materials, \( \overline{M} \) is the average of the produced gases, Q is the heat of detonation (cal/g) which can be calculated the HOF differences between products and reactants, and ρ is the density (g/cm3) of the energetic materials.

Bond dissociation energy (BDE) is one of the important parameters to understand the thermal decomposition and bonding strength of energetic materials. The homolytic BDE at 0 K can be calculated by the following expression:

$$ {\mathrm{BDE}}_0\left(A-B\right)\to {E}_0\left({A}^{.}\right)+{E}_0\left({B}^{.}\right)-{E}_0\left(A-B\right) $$
(8)

Finally, the BDE with zero-point energy corrections was evaluated with the help of the following equation:

$$ \mathrm{BDE}{\left(A-B\right)}_{\mathrm{ZPE}}={\mathrm{BDE}}_0\left(A-B\right)+\Delta {E}_{\mathrm{ZPE}} $$
(9)

where ΔZPE denotes the differences between the zero-point energies of products and reactants.

The impact sensitivities for the designed compound can be calculated by using the positive variance (\( {\sigma}_{+}^2 \)) and the balance between the charges (υ) as follows [36]:

$$ {h}_{50}=\alpha {\sigma}_{+}^2+\beta \upsilon +\varUpsilon $$
(10)

where h50 represents the height in centimeters from which 2.5 kg is dropped upon the energetic compound, and the chance of detonation is 50%; \( {\sigma}_{+}^2 \) shows the strength of the positive surface; υ is the balance of the charges on the surface; and α, β, and ϒ are the coefficients of − 0.0064, 241.42, and − 3.43, respectively.

Free space represented as ΔV is another important parameter used to know the impact sensitivity of energetic materials [37]. The higher the ΔV value is, the more sensitive the compound is. The ΔV can be calculated using the following empirical equation:

$$ \Delta V={V}_{\mathrm{eff}}-{V}_{0.003} $$
(11)

where ΔV is the free space, Veff is the effective volume which is needed to exactly fill the crystal, and the V0.003 is the volume enclosed by 0.003 electrons/bohr3. The effective volume can be calculated by dividing molecular mass over crystal density as follows:

$$ {V}_{\mathrm{eff}}=\frac{M}{\rho } $$
(12)

Results and discussion

Molecular geometry and electronic structure

It is important to study the geometrical structures first before studying other properties. Figure 3 displays the optimized structures of the designed compounds. Table 1 enlists the bond lengths of some selective bonds for the title compounds. In every NNNAHP (P) derivative, one of the C–N bonds in the cage was replaced by the C–O bond. As clear in Table 1, in every compound, the average bond length of the C–O bond connecting two rings is smaller than that of the C–N bond connecting two rings. This indicates that connecting two rings in the cage structure via a C–O bond instead of a C–N bond could make it more compact crystal packing. The majority of the designed compounds have smaller average C–N (in the six-member ring) bond length than the parent one. Similarly, the average value of the N–N (linked to the cage) bond length is smaller than that of normal N–N bond (1.45 Å) for every designed compound.

Fig. 3
figure 3

Optimized structures of the title compounds

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Table 1 Selected bond lengths of the title compounds at the B3LYP/6-311G (d, p) level
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Figure 4 shows the 3D plots of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) as well as the HOMO and LUMO energies and their energy gaps for the title compounds. The energy gap is an important parameter which gives valuable knowledge about the reactivity of compounds [30, 38]. All the derivatives have suitable energy gaps ranging from 4.61 to 5.83 eV. Only the compound A has higher energy gap than the parent compound P, which means that all the compounds except for A are chemically more active than P. In the 3D plots of HOMO and LUMO, the negative and positive phases are shown in green and red, respectively. It is clear from Fig. 4 that all of the C–N orbits take part in the HOMO level, while there are only few C–N orbits to participate in the LUMO level. This indicates that the removal of an electron from the HOMO level could make the cage structure more unstable than the acceptance of an electron by the LUMO level. Additionally, in comparison with –NH2, the LUMO is more localized over the –NO2 group.

Fig. 4
figure 4

HOMO and LUMO energy levels and energy gaps of the title compounds

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Molecular electrostatic potentials (MEPs) are critical to know the charge distribution and intermolecular forces in a molecule [39]. Prior research [40] shows that those compounds become more sensitive due to electron deficiency (positive potential) in the cage structure. Figure 5 shows the molecular electrostatic potential of the title compounds. The MEPs are for 0.001 electrons/bohr3 isosurface of electron density. The color varies from − 0.02 to 0.02 hartree. The blue and red indicate the electron-deficient (positive potential) and electron-rich (negative potential) regions, respectively. It can be seen in Fig. 5 that the negative potential is spread mainly around the N–NO2 portion, while the positive potential locates around the center of the cage. The compounds B, C, and D have the same number of –NO2, –NH2, and –O–. However, the positive potential in the compound C is more located at the center of the cage which might give it high sensitivity. Similarly, among E, F, G, H, I, J, K, and L, the positive potential in the case of J focuses more deep at the cage center than that of the other compounds. This might make it more sensitive.

Fig. 5
figure 5

Molecular electrostatic potentials of the title compounds

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The regions with the most positive potential and most negative potential are most likely to be attacked by nucleophiles and electrophiles, respectively. The ratio of positive and negative surface area of the compounds P, A, B, C, D, E, F, G, H, I, J, K, and L are 1.12, 1.16, 1.0, 1.02, 1.08, 0.96, 0.96, 0.99, 0.95, 0.99, 0.99, 1.02, and 1.1, respectively. These values indicate that the introduction of the –O– moiety and increasing the number of –NH2 groups tend to keep the ratio around 1 (a kind of balance between the positive and negative surface areas) and so decrease the chances of nucleophilic and electrophilic attacks. This makes the molecules chemically more stable. The molecular surface areas in each electrostatic potential region were calculated and displayed in Fig. 6. Furthermore, the global molecular electrostatic potential maxima of the compounds A, B, C, D, E, F, G, H, I, J, K, and L are 52.8, 52.8, 55.0, 45.6, 47.6, 52.1, 50.5, 52.7, 50.7, 52.2, 49.9, and 39.4 kcal mol−1, respectively. Similarly, the global molecular electrostatic potential minima of the compounds A, B, C, D, E, F, G, H, I, J, K, and L are − 14.3, − 18.1, − 17.2, − 18.2, − 19.1, − 21.4, − 20.1, − 23.0, − 21.5, − 18.8, − 18.7 and − 19.8 kcal mol−1, respectively.

Fig. 6
figure 6

Percentage area in each electrostatic potential region for the title compounds

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The contour line maps of the title compounds are displayed in Fig. 7. The high peaks are associated with the heavy nucleus’s nuclear charges (such as O and N) which intensify the electron accumulation. Due to the high electronegativity, it is found that the electron density around the oxygen is the highest, while that around hydrogen is the lowest. The decrease in the electronic density is noted in the regions A-1, A-2, B-1, B-2, C-1, C-2, D-1, D-2, E-1, E-2, F-1, F-2, G-1, G-2, H-1, I-1, I-2, I-3, J-1, J-2, K-1, K-2, K-3, L-1, L-2, and L-3 due to the repulsive forces of the heavy nucleuses with O and N.

Fig. 7
figure 7

Contour line map of electronic densities on the title compounds

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Gas- and solid-phase HOFs

Table 2 enlists the total energies (E0), zero-point energies (ZPE), thermal corrections (ET), and HOFs for the reference compounds in the plotted isodesmic reactions. The experimental HOFs of CH4, NH3, CH3CH3, CH3OCH3, H2NNH2, and CH3NHCH3 were available in the literature [42], while the HOFs of C3H9N3 and H2NNO2 were unavailable. Additional calculations were performed at the G2 level [43] for atomization reaction CaHbOcNd → aC + bH + cO + dN to obtain their theoretical HOFs. The G2 theory is able to predict the HOFs of small molecules [44].

Table 2 Calculated total energies (E0), thermal corrections (ET), zero-point energies (ZPE), and HOFs for the reference compoundsa
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Table 3 shows the total energies, ZPEs, ET, heats of sublimation (ΔHsub), ΔHf, gas, and solid-phase HOFs (ΔHf, solid) of NNNAHP (P) and its derivatives. As clear from Table 2, all the title compounds have large positive gas-phase HOFs (478.72–547.27 kJ/mol) and solid-phase HOFs (287.92–369.62 kJ/mol). Among all the title compounds, the compound A has the lowest ΔHf,gas, while G has the highest one. When –O– is incorporated to the parent (P) to construct the compound A, the HOF decreases dramatically. But after introducing one amino (–NH2) (B, C, and D) and two amino groups (E, F, G, H, I, J, K, and L) to P, the HOFs gradually increase. This may be because of the strong interactions between the –NO2 and –NH2 groups. The compounds B, C, and D have the same chemical composition, but their HOFs are different from one another. The same is the case of E, F, G, H, I, J, K, and L. These results show that the substituents’ position plays a very important effect on the HOFs of the title compounds.

Table 3 Calculated total energies (E0), thermal corrections (ET), zero-point energies (ZPE), and HOFs for the title compoundsa
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Figure 8 presents a comparison of solid- and gas-phase HOFs for the title compounds. Qualitatively, the solid-phase HOFs follow the same trend of gas-phase HOFs. This result indicates the variation trend of the gas- and solid-phase HOFs of the NNNAP-based compounds in the presence of –O– and –NH2 groups are consistent with one another.

Fig. 8
figure 8

Comparison of solid- and gas-phase HOFs of the title compounds

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Detonation performance

Table 4 shows the oxygen balance (OB), ρ, the heat of detonation (Q), D, and P of the title compounds. In order to compare our results, the calculated detonation properties of NNNAHP and experimental ones of two famous energetic compounds CL-20 and HMX are also tabulated in Table 4. It is noted that the ρ values from 1.95 to 2.03 g cm−3 are all higher than that of HMX but lower than that of NNNAHP and CL-20. This is because the replacement of –NO2 by –O– and –NH2 brings down the density values in the compounds A–L. The D and P values of the title compounds range from 9.42 to 9.71 km s−1 and 41.2 to 44.48 GPa, respectively. All the title compounds have larger D values than CL-20 and HMX. Similarly, their P values are either greater or very close to that of CL-20 (Fig. 9).

Table 4 Calculated densities (ρ), heat of detonations (Q), oxygen balance (OB), detonation velocities (D), and detonation pressures (P) for the title compounds
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figure 9

Fig. 9

OB is a measure of excess or deficiency of the oxygen content in the HEDC needed to oxidize carbon and hydrogen to carbon mono oxide or carbon dioxide and water, respectively. Generally the high negative and high positive values of OB are not favorable to get high Q values. It is because the high negative value does not let the complete oxidation of carbon and hydrogen. Similarly, the high positive OB values cause the production of additional oxygen gases (O2) which withdraw a lot of energy during the detonation process. Therefore, it is good to keep the OB value around zero while designing HEDCs. All the designed compounds have high Q values than the parent molecule ranging from 1483.98 to 1595.4 cal g−1. It is important to note that the compounds B, C, and D have the OB values very close to zero (−2.94), and their Q values are the highest one among all derivatives. This result depicts that the oxygen balance values play a very important effect on the heat of detonation.

In the energetic compounds CaHbOcNd, CO or CO2, H2O, and N2 are the major gaseous species produced during the detonation. Kistiakowsky and Wilson [46, 47] proposed the rules which can be used to know the number of moles and types of gases produced during the detonation process. Table 5 shows the types of gases, their number of moles, and the total number of gases produced during the detonation for the title compounds.

Table 5 Detonation products of the title compounds
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The volume of gaseous products provides valuable knowledge about work done by the HEDC. Standard conditions (273 K temperature and 1 atm pressure) were applied to calculate the total volume of the gases, because the changing condition causes a change in the volume. The amount of gas produced by 1 g of energetic compound (V) can be calculated by dividing total volume over the molecular volume.

The explosive power of energetic compounds can be obtained by the multiplication of Q and volume of gases produced by 1 g of energetic compound (V) as follow:

$$ \mathrm{Explosive}\ \mathrm{power}=Q\times V $$

Then, the explosive power value of an energetic compound can be compared with a standard explosive picric acid (PA) to calculate the explosive power index of the title compounds.

$$ \%\mathrm{Power}\ \mathrm{index}=\frac{Q\times V}{Q_{\mathrm{PA}}\times {V}_{\mathrm{PA}}} $$

Table 6 presents the explosive power index values of the PA, parent compound (P), and title compounds. In terms of the explosive power index, all the title compounds except for A are better than PA and P. The explosive power index values of the designed compounds range from 97.68 to 120.10.

Table 6 Heats of detonation (Q), volume of the gaseous products per gram of an explosive (V), explosive power (Q × V), and explosive power index for the title compounds
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Thermodynamic properties

Based on the principle of statistical thermodynamics, some important thermodynamic properties such as standard molar entropies\( \left({S}_{\mathrm{m}}^{\theta}\right) \), standard molar heat capacities\( \left({C}_{\mathrm{P},\mathrm{m}}^{\theta}\right) \), and standard molar enthalpies\( \left({H}_{\mathrm{m}}^{\theta}\right) \) were calculated at different temperatures ranging from 100 to 600 K and are tabulated in Table 7. The corresponding equations for these three parameters at various temperatures (100–600 K) can be expressed as follows:

$$ {S}_{\mathrm{m}}^{\theta }=a+ bT+c{T}^2 $$
$$ {C}_{\mathrm{P},\mathrm{m}}^{\theta }=a+ bT+c{T}^2 $$
$$ {H}_{\mathrm{m}}^{\theta }=a+ bT+c{T}^2 $$

where a, b, and c are constants and are displayed in Table 7. It is clear in Table 7 that all the thermodynamic properties obviously increase with the increasing temperature. It is because the vibrational movement becomes intensive at higher temperature. However, their growth rates are different from each other. The growth rate of \( {H}_m^{\theta } \) gradually increases with the increasing temperature, while that of \( {S}_{\mathrm{m}}^{\theta }\ \mathrm{and}\ {C}_{\mathrm{P},\mathrm{m}}^{\theta } \) gradually decreases. It is also observed that at a specific temperature, the \( {S}_{\mathrm{m}}^{\theta }\ \mathrm{and}\ {C}_{\mathrm{P},\mathrm{m}}^{\theta } \) successively decrease after replacing one and two –NH2 groups by one and two –NO2 groups. However, for \( {H}_{\mathrm{m}}^{\theta } \), no such trend is noted.

Table 7 Thermodynamic properties of the title compounds
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Thermal stability

BDC gives important information about the stability of HEDCs. The larger the value of BDE is, the stronger the bond is, and the more difficult the bond breaks. To study the thermal stability of the title compounds, first, the bond with the smallest Mulliken bond order (BO) among all the bonds was determined, and then this bond was broken to calculate its BDE. It is found that all the bonds having the lowest BO values are the N–NO2 bonds. This means that the N–NO2 bonds are the weakest bonds and may rupture in the thermal decomposition process.

Table 8 lists the BDEs for the parent and title compounds. Among the designed compounds, E has the smallest BDE value (127.16 kJ/mol), while G has the largest one (144.37 kJ/mol). All the designed compounds have higher BDE values than the parent one except for the derivative E. When –O– was incorporated into the parent molecule to get the molecule A, its BDE increases, which shows that the incorporation of –O– into the cage helps to thermally stabilize its derivative. Replacing one –NO2 by one –NH2 further increases in its BDE. When the second –NO2 was replaced by –NH2, the situation was not clear as some of the double-substituted –NH2 compounds that have higher BDEs than single NH2-substituted compounds, while the others have lower BDEs.

Table 8 Mulliken bond orders and bond dissociation energies (BDEs) for the title compounds
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It is understood that HEDCs should not only have high detonation properties but should also have suitable thermal stability. The energetic compounds having BDE values greater than 83.0 kJ/mol are considered to be thermally stable [49]. In this regard, all of our designed compounds are thermally stable.

Sensitivity

The impact sensitivity (h50) is the height from which impacting 2.5 kg mass on sample energetic compound leads 50% chance of initiating explosion [50, 51]. The larger the value of h50 is, the more insensitive the compound is. Table 9 lists the h50 values of the title compounds. In order to compare our results, the experimental h50 value of CL-20 is also enlisted in Table 9.

Table 9 Effective volume (Veff), intrinsic volume (Vint), free space (ΔV), and impact sensitivities (h50) of the title compounds
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As shown in Table 9, all the designed compounds have larger h50 values than the parent one. In comparison with CL-20, it is found that only 4 compounds (A, B, C, and D) have smaller h50 values than CL-20, while all others (E, F, G, H, I, J, K, and L) have larger ones. The incorporation of –O– into the parent cage increases the h50 value of its derivatives. When one –NO2 group is replaced by –NH2, a further increase in the h50 value is observed. Similarly, when two –NO2 are replaced by two –NH2 groups, its h50 values continue to increase. These results indicate that incorporation of –O– moiety and –NH2 groups is very helpful to decrease the sensitivity of the energetic derivatives. The h50 values of the designed compounds range from 6.26 to 21.40 cm. It is worthy to note that compounds B, C, and D have the same chemical composition, but their h50 values are different from one another. The same situation is true for E, F, G, H, I, J, K, and L. This shows that the different substitution positions of the substituents can be used to tune the impact sensitivity of energetic materials. The predicted sensitivities of the designed molecules seem to be overall fairly high (low h50) because they still have a lot of the –NO2 groups that are in fact electron-withdrawing groups even after introducing –O– and –NH2 groups into the cage. Due to the electron-withdrawing effect of the –NO2 groups, the cage structure becomes more positive (electron-deficient), leading to the increase in the sensitivity of energetic materials. This is supported by some studies [39, 40]. However, it should be noted that all of our designed compounds are less sensitive as compared with the parent compound.

Free space denoted as (ΔV) is another parameter used to analyze the impact sensitivity of energetic materials [36, 52]. When external force such as shock or impact is applied on energetic compound, it gets compressed [53,54,55,56]. Due to compression, the temperature in the compound increases [56, 57]; the larger is the compression (decrease in volume), the larger is the increase in the temperature. The smaller or larger decrease in the volume is dependent on the free space available in the compound. The smaller the ΔV value is, the more insensitive the compound is. All the designed compounds have smaller ΔV values than that of the parent one but larger than CL-20. This means that all the compounds are less sensitive than the parent molecule but more sensitive than CL-20. However, it is noted that their ΔV values are very close to that of CL-20. The compound A has the highest ΔV value (105.46 Å3), while the compound I has the smallest one (92.07 Å3) among all the title compounds. The incorporation of –O– moiety is found to decrease their ΔV values. Similarly, the introduction of an increasing number of –NH2 groups into the cage can gradually decrease their ΔV values. This shows that our designing strategy is very useful to decrease the impact sensitivity of the designed compounds.

Figure 10 presents the ΔV and h50 profile of the designed compounds. It is clear that the compounds with large h50 values (more insensitive) have small ΔV values and vice versa, which is according to previous studies [26, 37, 52].

Fig. 10
figure 10

ΔV and h50 profile of the title compounds

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Conclusions

In the present study, molecular structures, HOFs, detonation properties, thermodynamic properties, thermal stability, and impact sensitivity of 12 novel cage skeleton compounds based on the skeleton of NNNAHP were examined by employing the DFT-B3LYP method with 6-311G (d, p) basis set. The results show that favorable substitution position of –NH2 is very helpful to increase HOFs, detonation properties, thermal stability, and insensitivity of the designed compounds. All the 12 compounds have densities, detonation velocities, and detonation pressures over than 1.95 g cm−3, 9.42 km s−1, and 41.2 GPa, respectively. The incorporation of the –O– moiety and –NH2 groups is found to increase thermal stability (and to decrease impact sensitivity) for the designed compounds as compared with the parent molecule (NNNAHP). The majority of the compounds (8) have h50 values greater (less sensitive) than CL-20.

In addition, the findings of this work show that our designing strategy of incorporating –O– into the cage and replacing up to two –NO2 groups attached to the cage by the –NH2 groups is very useful to combine large density, excellent detonation property, suitable thermal stability, and low sensitivity all together in the same compound. Our strategy provides new ways to design novel cage skeleton compounds which can hold both low sensitivity and excellent detonation performance in one compound.