Introduction

In recent decades, 1,2,3,4-tetrazine 1,3-dioxide (v-TDO), as a favorable structure unit, has become the choice for designing new high-energy-density compounds (HEDCs) [1,2,3,4,5,6,7,8], and many promising candidates have been obtained [9,10,11,12,13] based on its energetic skeleton. 5,7-Dinitrobenzo-1,2,3,4-tetrazine-1,3-dioxide (DNBTDO) has been studied by Klapötke et al. [11], and they found that the sensitivity of DNBTDO is slightly higher, while the detonation performance is slightly lower than that of RDX. [1,2,5]-oxadiazole-[3,4-e]-[1,2,3,4]-tetrazine-4,6-di-N-oxide (FTDO) [14,15,16] reduced the combustion temperature significantly and the signatures of the outlet gases when it was an additive of propellant compositions.

In our previous work, furoxano-v-tetrazine trioxide-β (FTTO-β) [17] was designed, and it is believed to be a considerable intermediate in the synthesis of monocyclic v-TDO. The structures mentioned above are shown in Fig. 1.

Fig. 1
figure 1

The structures of v-TDO, DNBTDO, FTDO, and FTTO-β

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The v-TDO annulated with a pyrazole ring or 1,2,3-triazole ring, pyrazolo-v-TDOs, and 1,2,3-triazolo-TDOs (TTDOs), have been studied in our previous work [18, 19]. As shown in Table 1, some of them were found to have good detonation performances and acceptable stabilities, which compared with RDX. Their detonation performances and structures are listed in Table 1 and Fig. 2.

Table 1 Detonation performances of the structures in Fig. 2
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Fig. 2
figure 2

Structures of T2, T5, T11, P1, P4, and P11

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In this paper, v-TDO annulated with another five-membered ring was designed to continue our research. Four imidazolo-v-tetrazine 1,3-dioxides (ITDOs) I1–I4 and eight oxazolo-v-tetrazine 1,3-dioxides (OTDOs) O1–O8 were designed and shown in Fig. 3. Their properties were evaluated by the density functional theory (DFT). At the end of this paper, some points of view about design strategy for this kind of molecule are put forward after contrasting all of them, including FTDO, FTTOs, TTDOs, pyrazolo-TDOs, ITDOs, and OTDOs.

Fig. 3
figure 3

Designed ITDOs and OTDOs

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Computational methods

The geometry optimization of the 12 target molecular structures was calculated by the B3LYP/6-311++G(d,p) level in the Gaussian software package [20]. At the same level, the vibrational analysis was performed, and the optimized structures were confirmed to be the local minimum without imaginary frequencies. According to the previous study, the calculated frequencies were scaled uniformly by 0.96 to approximately correct the systematic overestimation.

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)

The homodesmotic reactions [21] designed as Eqs. (1–8) were employed to calculated the enthalpies of formation of I1–I4 and O1–O8 by Eq. (9):

$$ \varDelta H{{}^{\circ}}_{298}=\sum {\varDelta}_{\mathrm{f}}H{{}^{\circ}}_{298,\mathrm{P}}-\sum {\varDelta}_{\mathrm{f}}H{{}^{\circ}}_{298,\mathrm{R}}=\varDelta E{{}^{\circ}}_{298}+\varDelta (pV) $$
(9)

where R or R’ is either NO2 or NH2; ΔH°298 is the change of enthalpy of formation at 298 K; ∑ΔfH°298, P and ∑ΔfH°298, R are the sum of the enthalpies of formation of products and the reactants, respectively; ΔE°298 represents the change of total energy at some temperature; Δ(pV) represents the change of the pressure p and volume V in product. ΔE°298 and Δ(pV) can be obtained by Eqs. (10) and (11), respectively.

$$ \varDelta E{{}^{\circ}}_{298}=\varDelta {E}_0+\varDelta {E}_{\mathrm{ZPV}}+\varDelta H{{}^{\circ}}_{\mathrm{T}} $$
(10)
$$ \varDelta (pV)=\left(\varDelta n\right) RT $$
(11)

where ΔΕ0, ΔΕΖPV, and ΔH°T are the change of total energy at 0 K, the zero-point vibrational energy, and the temperature-dependent enthalpy, respectively.

The solid-phase enthalpy of formation ΔfH°298(s) can be obtained by Eqs. (12) to (15):

$$ {\varDelta}_{\mathrm{f}}H{{}^{\circ}}_{298}\left(\mathrm{s}\right)={\varDelta}_{\mathrm{f}}H{{}^{\circ}}_{298}\left(\mathrm{g}\right)+{\varDelta}_{\mathrm{sub}}H{{}^{\circ}}_{298} $$
(12)
$$ {\varDelta}_{\mathrm{sub}}H{{}^{\circ}}_{298}={\alpha}_1{A_{\mathrm{S}}}^2+{\alpha}_2{\left(\nu {\upsigma_{\mathrm{tot}}}^2\right)}^{0.5}+{\alpha}_3 $$
(13)
$$ \nu ={\sigma_{+}}^2{\sigma_{-}}^2/{\left({\sigma_{\mathrm{tot}}}^2\right)}^2 $$
(14)
$$ {\sigma}_{\mathrm{tot}}={\sigma_{+}}^2+{\sigma_{-}}^2 $$
(15)

where AS is the molecule surface area, σ+2 is the mean variance of the positive molecular electrostatic potential, and σ−2 is the negative one. The empirical parameters α1, α2, and α3 are obtained from [22]. The B3PW91/6-31G(d,p) level of DFT was employed to calculate the surface properties of sublimation enthalpies.

The densities (ρ) of these compounds can be obtained by Eq. (16) suggested by Politzer et al. [23]. The coefficients β1, β2, and β3 are 0.9183, 0.0028, and 0.0043, respectively:

$$ \rho ={\beta}_1\left(M/V\right)+{\beta}_2\left(\nu {\upsigma_{\mathrm{tot}}}^2\right)+{\beta}_3 $$
(16)

where M is the molecular mass and V is the volume of the 0.001 electrons/bohr3 contour of the molecule electronic density.

The Kamlet-Jacobs equation is often employed to estimate the detonation velocity (D) and detonation pressure (P) of energetic materials [24,25,26].

$$ D=1.01\ {\left(N{\overline{M}}^{1/2}{Q}^{1/2}\right)}^{1/2}\left(1+1.30\rho \right) $$
(17)
$$ P=1.558{\rho}^2N{\overline{M}}^{1/2}{Q}^{1/2} $$
(18)

Where N is the moles of gaseous products per gram explosives, \( \overline{M} \) is the products’ average molecule mass, and Q is the detonation heat. They are determined by the most exothermic principle and based on the most exothermic principle. The released energy can reach its maximum when the products are supposed to be only CO2, H2O, and N2.

In order to satisfy the demand of industry and military, insensitive high explosives are imperative. The sensitivity becomes an essential factor in design of energetic materials, and can be evaluated by many methods [27,28,29,30,31,32,33]. According to refs. [34,35,36,37,38,39], the solid compound’s compressibility and the crystal lattice free space (ΔV) are the factors related to sensitivity. According to Politzer’s findings, ΔV can be calculated as follows:

$$ \Delta V={V}_{\mathrm{eff}}\hbox{--} {V}_{\mathrm{int}} $$
(19)
$$ {V}_{\mathrm{eff}}=M/\rho $$
(20)

where Veff is the effective molecule volume when the cell unit was fully filled hypothetically. Vint is the intrinsic space encompassed by the 0.003 a.u. contour of the molecule’s electronic density. M is the molecular mass, and ρ is the crystal density.

The thermal stabilities of the ITDOs and OTDOs were further confirmed by the bond dissociation activation energy (BDAE) obtained by scanning the potential energy surface for pyrolysis.

Results and discussion

Molecular structures

The optimized structures of I1–I4 and O1–O8 are shown in Fig. 4 for comparison.

Fig. 4
figure 4

Molecular geometric parameters of ITDOs and OTDOs

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Under the influence of hydrogen bond interactions, all the ITDOs and OTDOs are nonplanar structures; however, the OTDOs are quite close to fully planar ones. The absolute values of dihedral angles of O(5)-C(6)-N(12)-O(13), N(5)-C(6)-N(12)-O(13), O(5)-C(6)-N(12)-O(14), and N(5)-C(6)-N(13)-O(14) in O1, O2, O5, and O6 are less than 0.1 degree. The dihedral angles of N(5)-C(6)-N(12)-H(13) and N(5)-C(6)-N(13)-H(14) in O4 and O8 are 0.19166° and 0.11849°, respectively. The bond lengths in 12 compounds are all in the reasonable range, and we presume these imidazole and oxazole rings, like the v-TDO ring, are both solid enough.

Enthalpy of formation

In this paper, the enthalpies of formation of the 12 target compounds were calculated by the isodesmic reaction method and the designed homodesmotic reactions (1–8) were proposed properly.

The total energies E0 and enthalpies of formations ΔfH° of the small molecules in reactions (1–8) are listed in Table 2. Of these, ΔfH° values for 2-nitro-1H-imidazol-1-amine, 2-nitrooxazole, and oxazol-2-amine were calculated by G3 theory based on the atomization reaction CaHbOcNd(g) → aC(g) + bH(g) + cO(g) + dN(g).

Table 2 Enthalpies of formation of the compounds in reactions (1–8)
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Table 3 presents the corresponding parameters that are needed to calculate the solid-phase enthalpies of formation of I1–I4 and O1–O8. The results show that I4 has the highest ΔfH°298(s), while O3 has the lowest one.

Table 3 Enthalpies of formation of ITDOs and OTDOs
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Detonation properties

The calculated detonation properties according to Eqs. (16), (17), and (18) are listed in Table 4. The corresponding data of RDX and HMX are also provided. It shows that the detonation performances of O5 and O6 are as good as HMX. I3, I4, O1, and O2 can also achieve the RDX energy level.

Table 4 Detonation performances of ITDOs, OTDOs, RDX, and HMX
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Sensitivity

For most of the v-TDO compounds, according to the value of BDAEs calculations, the N–N bonds on pyrazole rings are above 200 kJ·mol−1, and the N=N bonds of pyrazine rings are above 280 kJ·mol−1. Affected by the substituent N–O, –NO2 or –NH2, the weak bonds of FTTOs [17], TTDOs [18], and pyrazolo-TDOs [19] mainly center on the five-membered rings. In addition, C–NO2 or N–NO2 may also be the trigger bonds for many nitro compounds. The BDAEs of some possible trigger bonds of 12 target compounds are shown in Table 5. The BDAEs of trigger bonds N–NO2 of RDX and HMX are also given in Table 5.

Table 5 BDAE of various bonds and ΔV values of ITDOs and OTDOs
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According to the research of Heming Xiao et al. [25], a BDAE or BDE of 80–120 kJ·mol−1 is usually considered to be an indicator for a stable energetic compound in HEDC molecular design. As shown in Table 5, the most fragile part of the imidazole rings in I1–I4 are bonds between nitro substituent and amino substituent or bonds between nitro substituent and coordinated oxygen atom, the BDAEs of which are 220.54, 194.29, 204.79, and 178.53 kJ·mol−1 respectively. BDAEs of the weakest bonds in nitro-substituented compounds O1, O2, O5, and O6 are near 200 kJ·mol−1, almost 50 kJ·mol−1 higher than the ones of amino-substituented compounds O3, O4, O7, and O8. The BDAEs of C–NO2 bonds in I1–I4 and O1, O2, O5, O6 are above 200 kJ·mol−1, which means they may not be the trigger bonds.

Sometimes using the BDAE or BDE values to measure the sensitivity is not accurate [33], and according to the research [39], the increase of sensitivity tends to follow the increase of the free space in the crystal lattices (ΔV). Table 5 lists the ΔV values of ITDOs and OTDOs, compared with RDX and HMX. It can be found that the OTDOs are more insensitive than the others. Of these, O5 (42.34 Å3) and O6 (42.71Å3) are more sensitive than O1 to O4. However, using the ΔV values to estimate sensitivity is a rough estimating method [41], so the sensitivities of ITDOs and OTDOs are roughly similar to RDX and HMX.

Conclusions

Generally speaking, a molecule of v-TDO annulated with a five-membered nitrogen-rich heterocycle owns good energy characteristics (Fig. 5). Compared to other molecules, ITDOs and OTDOs are no less energetic and are usually more stable. Some of them, such as I3, I4, O1, and O2, are promising materials as good as RDX; O5 and O6 are even as powerful as HMX.

Fig. 5
figure 5

Comparison of the Q (cal·g−1), D (m·s−1), and P (GPa) values of target v-TDOs and RDX

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Compared with other v-TDOs, the v-TDO ring in this kind of compound is rather stable (Fig. 6) because the five-membered ring deformation and the steric hindrance change caused by the different substituents are the main reasons affecting the stabilities of these molecule structures. The more heteroatoms directly connected in the ring, the more obvious this effect performs. For this reason, the FTTOs and some designed TTDOs are quite unstable, compared to which the thermal stability of ITDOs, OTDOs, and the pyrazolo-TDOs are less affected. It is worth mentioning that FTDO and TTDOs without any electron-withdrawing substituents or atoms have good stability as well. N–NO2, which should always be given more attention than C–NO2, is an important potential trigger bond. For example, the average BDAE of the N–NO2 of T1, T2, P5, P6, P7, and P8 is about 110 kJ·mol−1, much lower than 280 kJ·mol−1, which is the average BDAE of the C–NO2 of other pyrazolo-TDOs, ITDOs, and OTDOs. Of course, we believe there are still other important factors to decide the stability of a v-TDO compound, for example, the position of the electron-withdrawing substituents or the position of coordinated oxygen atom, which may be worthwhile to investigate in future work.

Fig. 6
figure 6

Comparison of the BDAE of the weakest bond and the ΔV values of target v-TDOs and RDX

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