Introduction

Energetic materials are used extensively for both military and civil purposes. However, most such materials do not meet the current requirements for insensitive high density energetic materials. The search for new insensitive and high energy explosives has been a primary goal in the field of energetic material chemistry in recent years [1, 2].

Cocrystals are a type of supramolecule, exhibiting intermolecular interactions arising from hydrogen bonds, π-stacking, van der Waals (vdW) forces, and electrostatic interactions, etc. [3–6]. The design of cocrystal explosives is one of the most promising approaches to decreasing sensitivity by favoring intermolecular interactions while maintaining the detonation performance of existing explosives [5, 7, 8]. Recently, much attention has been paid to the investigation of the synthesis, characterization and application of cocrystal explosives [9–14]. In theoretical studies, structural, electronic, and thermodynamic properties as well as intermolecular interactions of 1,3,5,7-tetranitro-1,3,5,7-tetrazacyclooctane (HMX)/1,3-dimethyl-2-imidazolidinone (DMI) cocrystal explosives were investigated using density functional theory (DFT). The calculated results were in reasonable agreement with experimental data [15, 16]. Ab initio and DFT were adopted to investigate the intermolecular H-bonds in HMX/5-nitro-1,2,4-triazol-3-one cocrystals [17]. Plots of the reduced density gradient (RDG) versus sign(λ 2)ρ were used to understand the internal mechanisms of 2,4,6-trinitrotoluene/2,4,6,8,10,12-hexanitrohexaazaisowurtzitane (CL-20) cocrystals [14].

It is well known that the stronger the intermolecular interactions, the more stable the cocrystal complex. Cocrystallization can be realized by combining two or more neutral components in a defined molecular molar ratio. The stability of a cocrystal can be affected by the ratio of the molecular combination because different molecular ratios have different intermolecular interaction energies. The ratio of molecular combination plays an important role in sensitivity and detonation performance of cocrystal explosives. In general, if a cocrystal explosive has too high a content of high energy explosive, the energy is enhanced and the sensitivity increases. However, high sensitivity is an undesirable property for an energetic material. On the contrary, if a cocrystal explosive has too high a content of low energy explosive with low sensitivity, the sensitivity decreases and simultaneously the energy will also be decreased. The latter is also an undesirable property. Thus, the content of high energy and low sensitivity explosive components must be controlled within a reasonable range. Therefore, it is very important for the design of cocrystals to determine the optimal ratio of molecular combination. In most of the published literature, only defined molecular ratios were reported. To our knowledge, few theoretical investigations on the influence of molecular ratios on the stabilities of cocrystal explosives have been presented.

HMX is a high energy and sensitive explosive. In order to reduce the sensitivity, cocrystal explosives involving HMX have been developed and are attracting increasing interest [15–23]. In 2009, Wei et al. [18] reported the theoretical calculation of cocrystal HMX/1,3,5-triamino-2,4,6-trinitro-benzene (TATB) using molecular dynamics (MD) simulation. In 2011, a cocrystal HMX/TATB explosive was prepared with a solvent/nonsolvent process and characterized by several experimental methods. The results indicated that the main properties of the cocrystal originate from the N–O⋯H hydrogen bonding between –NO2 (HMX) and –NH2 (TATB) [19, 20]. In 2012, the structures of seven cocrystals involving HMX were elucidated using single crystal X-ray diffraction, and drop-weight impact sensitivity was determined [21]. In 2013 and 2014, Lin et al. synthesized and characterized some cocrystal explosives involving HMX by using X-ray single crystal diffraction, including HMX/1,3-dimethyl-2-imidazolidinone [15], HMX/2,6-diamino-3,5-dinitropyrazine-1-oxide [22], HMX/5-nitro-1,2,4-triazol-3-one [17], and HMX/N-methyl-2-pyrrolidone [23].

Although its energy is not high in comparison with HMX, nitroguanidine (NQ) is an insensitive explosive [24]. The α-form of NQ comprises –NH2, −NH– and –NO2 groups, and HMX contains –CH2– and –NO2 moieties. If HMX cocrystallizes with NQ, intermolecular H-bonds (N–H⋯O and C–H⋯O) can be formed, which stabilize the cocrystal. Furthermore, the sensitivity of HMX will decrease and the energy defect of NQ will be overcome. Thus, the HMX/NQ cocrystal represents a promising explosive for potentially widespread implementation in both military and civilian fields.

In this work, MD methods were employed to study the binding energies of selected crystal planes of the β-HMX/α-NQ cocrystal at different molecular molar ratios (β-HMX was selected because of its superior density and thermal stability). The mechanical properties of the cocrystal are also discussed. In order to obtain further information about bonding nature of the HMX/NQ cocrystal, the intermolecular interactions and bond dissociation energies (BDE) of the N–NO2 bond in the 1:1 complex of HMX:NQ were calculated by ab initio and DFT methods, and RDG analysis was also carried out. Density and detonation velocity of the cocrystals at different molecular molar ratios were estimated. Molecular surface electrostatic potentials were used to judge the change in sensitivity upon complex formation. These studies will provide some novel insights that will inform the design of HMX cocrystal explosives.

Computational details

Molecular dynamics calculations

The unit cell model of HMX was constructed according to its cell parameters [25, 26]. The initial model was relaxed (COMPASS force-field, Smart algorithm) to produce a stable conformation using the Discover module (see Table S1). An accuracy of 1.0×10−5 kcal mol−1 was required to determine minimization convergence. The crystal morphologies of HMX in vacuum were predicted by the Growth morphology model. The major growth faces of HMX with relative attachment energies [E att (Total), E att (vdW), E att (Elect), %A] are listed in Table S2. There are five major growth faces for HMX: (011), (11–1), (020), (10–2) and (100), in agreement with a previous investigation [27]. These growth faces suggest an important approach towards substituting to build cocrystal models. Therefore, in this paper, these five growth faces and an additional random face were selected to study the binding energies of HMX/NQ cocrystal in different molecular molar ratios in order to evaluate the influence of molecular ratio on stability of the cocrystal.

The HMX supercells were substituted by equal numbers of NQ moieties at 10:1, 9:1, 8:1, 7:1, 6:1, 5:1, 4:1, 3:1, 2:1 and 1:1 molar ratio (HMX:NQ). The selected parameters (molar ratios, mass ratios, supercell patterns and the number of substituted molecules) are listed in Table 1.

Table 1 Molecular molar ratios, mass ratios, super cell patterns and the numbers of substituted molecule in 1,3,5,7-tetranitro-1,3,5,7-tetrazacyclooctane (HMX)/nitroguanidine (NQ) cocrystals
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After building and geometry optimization of the substituted models, NVT ensembles and a temperature of 298 K were employed. Andersen was set as the temperature control method. COMPASS force-field was assigned above all MD simulations, and summation methods for electrostatic and vdW were Ewald and atom-based, respectively. The accuracy for the Ewald method was 1.0×10−4 kcal mol−1. Cutoff distance and buffer width for the atom-based method were 15.5 Å and 2.0 Å, respectively. A 1.0 f. time step was set for MD processes, and the total dynamic time was 100,000 fs. The whole system reached an ideal state when the fluctuation of temperature and energy tended to be smooth with alternation less than 10 %. All MD calculations were carried out using MS 7.0.

An energy correction formula was involved in computing the corrected binding energy (E b*) in order to standardize (make uniform) the differences caused by diverse supercells with different molecular molar ratios.

$$ {E_{\mathrm{b}}}^{*}={E}_{\mathrm{b}}\cdotp {N}_0/{N}_{\mathrm{i}} $$

Where E b denotes the non-corrected binding energy, N i is the number of molecule for different super cell, and N 0 the number of molecules for a standard pattern (molar ratio in 1:1).

Quantum-chemical calculations

According to the literature [14–16], in order to obtain further information about the bonding nature of the HMX/NQ cocrystal, HMX/NQ complexes (1:1) were designed by considering assembly motifs of HMX complexes with NQ. About ten stable complexes were obtained at the HF/6-311G (d,p) level. In these complexes, HMX molecules have the chair–ring conformation. Five structures of complexes were then selected and re-optimized at the B3LYP/6-311++G(d,p) level. Local minima were checked without any imaginary frequencies. Intermolecular interaction energies were calculated using the B3LYP, MP2(full), and M06-2X methods with the 6-311++G(d,p) and 6-311++G(2df,2p) basis sets, with correction of basis set superposition error (BSSE) [28]. Plots of RDG versus sign(λ 2)ρ were carried out at the B3LYP/6-311++G (d,p) level. All the above calculations were performed with Gaussian 09 [29] and Multiwfn 3.3.5 [30].

The BDEs of the N–NO2 bonds involving hydrogen bonds were calculated at the B3LYP/6-311++G(d,p), B3LYP/6-311++G(2df,2p), MP2(full)/6-311++G(2df,2p), and M06-2X/6-311++G(2df,2p) levels. BDE was defined as:

$$ BDE={E}_{\left({\mathrm{R}}^{\cdotp}\right)}+{E}_{\left(\cdotp, {\mathrm{NO}}_2\right)}\hbox{--} {E}_{\left({\mathrm{R}\mathrm{NO}}_2\right)}\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{H}\mathrm{M}\mathrm{X}\ \mathrm{molecule} $$

and

$$ BDE={E}_{\left(\mathrm{R},\cdotp \right)}+{E}_{\left(\cdotp {\mathrm{NO}}_2\dots \kern0.5em \mathrm{N}\mathrm{Q}\right)}\hbox{--} {E}_{\left({\mathrm{RNO}}_2\dots \kern0.5em \mathrm{N}\mathrm{Q}\right)}\kern1em \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{complex} $$

where R· indicates the radical.

Results and discussion

Molecular dynamics analysis

Based on the MD simulation of substituted models, the corrected binding energies, E b*, were calculated and are summarized in Table 2. Values were in the range of −1850.26 to −1064.08 kJ mol−1 in the 1:1 molar ratio (total number of molecules = 32, see Table 1). In the HMX/FOX-7 cocrystal, the binding energies were between −975.79 and −560.09 kcal mol−1 (i.e., −4080.75 and −2342.30 kJ mol−1) in the 1:1 molar ratio (total number of molecules = 72) [27]. Comparing the results of binding energies from the HMX/FOX-7 cocrystal with the data shown in Table 2, it can be seen that our calculations are reliable.

Table 2 Corrected binding energy, E b*, of the substituted models with different molecular molar ratiosa
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The binding energy E b* is a indication of component interaction strength. It influences the structural stability of the cocrystal and thus provides a general evaluation for screening for preferred substituted patterns and molecular ratios. From Table 2, each different substituted pattern at each of the ratios holds a different binding energy. At a ratio of 1:1, the binding energy E b* of the substituted pattern (0 2 0) is stronger than that of (1 0 0). In the ratios of 2:1, 3:1 and 4:1, the binding energies of (0 2 0) are close to those of (1 0 0). At the remaining ratios, the binding energies E b* of (0 2 0) are weaker than those of (1 0 0). For all ratios, the binding energies are in the same order: random > (1 1–1) > (1 0–2) > (0 1 1). The substituted patterns (0 2 0) and (1 0 0) possess higher binding energies than the other patterns and are thus the most stable; cocrystallization is dominated by the (0 2 0) and (1 0 0) facets.

As can also be seen from Table 2, the different ratios in each of the substituted patterns hold different binding energies. For all the substituted patterns, the binding energies E b* are in the same order of 1:1 > 2:1 > 3:1 > 4:1 > 5:1 > 6:1 > 7:1 > 8:1 > 9:1 > 10:1. At the ratio of 1:1, all the substituted patterns possess the strongest binding energies and HMX/NQ cocrystal prefers cocrystalizing in a 1:1 molecular molar ratio, while those at a ratio of 10:1 have the poorest binding energies. Many previous experimental results have also shown that cocrystals at low ratios, such as 1:1 and 2:1, were more stable [31, 32]. For example, Lin et al. [15] synthesized a cocrystal composed of HMX and DMI at a 1:1 molar ratio. A HMX/NMP cocrystal explosive at a 1:1 molar ratio was prepared [23]. The cocrystal of a 2:1 molar ratio of CL-20 and HMX has been reported experimentally [8]. Furthermore, from 1:1 to 5:1, the binding energies decreased sharply, while from 6:1 to 10:1, they decreased gently, and from 8:1 and 10:1, the corresponding values changed only slightly. For instance, for the (0 2 0) substituted pattern, from 1:1 to 5:1, the binding energies ranged from −1850.26 to −938.86 kJ mol−1. However, from 6:1 to 10:1, the binding energy decreased by 312.93 kJ mol−1, and from 8:1 and 10:1, the decrement of the binding energy was only 59.98 kJ mol−1.

Mechanical properties

The mechanical properties are an important performance indicator in the preparation and usage of energetic materials. From the statistical mechanics of elasticity [33], the material stress and strain tensor are denoted by σ and ε, respectively. Hooke’s law is stated as follows:

$$ {\sigma}_{\mathrm{I}}={C}_{\mathrm{I}\mathrm{j}}\;{\varepsilon}_{\mathrm{j}}\cdotp $$

In this equation, C ij is the elastic coefficient, which can be used to describe the stress–strain behavior of an arbitrary material.

Hardness, tensile strength and fracture strength can be related to the elastic modulus [34]. Hardness and tensile strength denoting the resistance to plastic deformation are proportional to the shear modulus G. Fracture strength is proportional to the bulk modulus K. Tensile modulus E and Poisson’s ratio γ can be obtained by K and G with the equation E = 2G(1 + γ) = 3K(1 − 2γ). The K/G and Cauchy pressure (C 12−C 44) can be taken as a criterion with which to evaluate the ductility or brittleness.

On the basis of MD simulations and the elastic static method, the mechanical properties of HMX/NQ co-crystals were estimated using the “constant strain” approach [35]. Table 3 presents the effective isotropic mechanical properties (K, G, C 12−C 44, γ, E and K/G) at different molecular ratios.

Table 3 Mechanical properties of HMX/NQ cocrystal models at different molecular ratiosa
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It can be seen from the values of C 12−C 44 that the ductility of the material differs depending on the different molecular ratios, and the C 12−C 44 values are in the following order: 1:1 > 2:1 > 3:1 > 4:1 > 5:1 > 7:1 > 6:1 > 8:1 > 10:1 > 9:1 (see Table 3). This result indicates that the 1:1 model has the better ductility.

From Table 3, the K, G, E values of the anisotropic cocrystals are in the order 1:1 < 2:1 < 3:1 < 4:1 < 5:1 < 6:1 < 7:1 < 8:1 < 9:1 < 10:1. Except for 7:1 and 10:1, the order of the K, G, E values is in accordance with that of the C 12−C 44 values. The K, G, E values of the model with the 1:1 ratio are smaller than those of the other models, indicating that the rigidity of the model of 1:1 is the smallest. That is, to produce the same strain, less stress is required for the 1:1 model. Furthermore, the K/G value of model 1:1 is larger than that of models with other ratios, suggesting that the 1:1 model can hold the better ductility.

Therefore, the 1:1 cocrystal model can be inferred to have good mechanical properties. This model will deform more easily when subjected to an external force. Combined with the results of binding energy, one can see that the model with a molecular ratio of 1:1 will be more meaningful for experimental research.

Structural analysis

To obtain further information about the bonding nature of the cocrystal, the structures of HMX/NQ (1:1) complexes were investigated (see Fig. 1). The intermolecular O···H and N···H distances were within 1.996 – 2.511 Å at the B3LYP/6-311++G (d, p) level, indicating that hydrogen bonds are formed between HMX and NQ. The C12–H13···N1 and N2–H3···O16 hydrogen bonds were confirmed in complex (A). Hydrogen bonds were also found in the other complexes: C12–H13···O7, N5–H6···O9 and N2–H4···O16 [as seen in (B)], N5–H6···O17 and N2–H4···O9 [as seen in ©)], C12–H13···O7, N5–H6···O17 and N2–H4···O9 [as seen in (D)], and C12–H13···O8 [as seen in (D)], respectively.

Fig. 1
figure 1

Optimized structures and atomic numbers of 1,3,5,7-tetranitro-1,3,5,7-tetrazacyclooctane (HMX), nitroguanidine (NQ) and their complexes at the B3LYP/6-311++G (d, p) level

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Interestingly, as shown in Fig. 1, the bond lengths of the N–NO2 bonds involving hydrogen decrease in comparison with those in the isolated HMX molecule. This result is similar to the change in the C–NO2 trigger-bond length upon formation of the intermolecular H-bonding interaction between the nitro group of CH3NO2 and HF [36]. Shortening of a bond length can lead to strengthening of that bond [37]. Thus, the N–NO2 bond becomes stronger upon formation of the intermolecular H-bonding interaction. Both experimental and theoretical studies have verified that the trigger bond of HMX is N–NO2 [38, 39]. In other words, the trigger bonds of N–NO2 in the cocrystal complexes are more stable than those in the isolated HMX molecule. Thus, the sensitivity of HMX decreases in cocrystal explosives.

H-bonding interaction and BDE of the N–NO2 bond

Table 4 lists the intermolecular H-bonding interaction energies calculated using the MP2(full) method, revealing that these are larger than those obtained with the B3LYP method. Interaction energies at the MP2(full)/6-311++G(2df,2p) level were close to those obtained at the M06-2X/6-311++G(2df,2p) level. At the MP2(full)/6-311++G(2df,2p) level, the intermolecular H-bonding interaction energies were within the range 24.3–39.2 kJ mol−1. In our previous investigation into HF complexes with 1,2,4-triazoles, the H-bonding energies were in the range 20.6–27.7 kJ mol−1 at the same level [40]. In our study of the cocrystal complex CL-20/FOX-7, the intermolecular H-bonding energies were in the range 16.0–29.5 kJ mol−1 at the MP2(full)/6-311++G** level [41].

Table 4 Intermolecular hydrogen-bonding interaction energy [−E int (kJ mol−1)] and bond dissociation energy [BDE (kJ mol−1)] in the N–NO2 bond a
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At four levels of theory, H-bonding interaction energies decrease in the same order of (A) ≈ (B) > ©) > (D) > (E). Hence, HMX/NQ cocrystals packed in complexes (A) and (B) are more stable.

The BDE of the trigger bond provides useful information with which to understand the stability of energetic materials [42, 43]. The BDE values calculated using the B3LYP method were all lower than those obtained by the MP2(full) and M06-2X methods (see Table 4). As mentioned in our previous work [36], the B3LYP method correctly describes the BDE values, but the MP2 method cannot be used to adequately describe BDEs [44, 45]. Therefore, the B3LYP/6-311++G(2df,2p) method was selected to elucidate trends in the calculated BDEs in this work. From Table 4, at the B3LYP/6-311++G(2df,2p) level, the order of the BDEs is (D) > ©) > (A) > (B) > (E).

The BDEs in the N–NO2 bond in the complexes are larger than those in the isolated HMX. This result suggests that the strength of the N–NO2 bond is enhanced upon formation of intermolecular H-bonds between HMX and NQ, which is in agreement with the structural analysis. The stronger the N–NO2 trigger bond, the lower the sensitivity [46]. Thus, the explosive sensitivity decreases in complexes.

Therefore, we conclude that the HMX/NQ cocrystal explosive complex needs more energy not only to destroy the intermolecular H-bonding interactions, but also to counteract the increments of the BDEs of the N–NO2 trigger-bonds in the process of detonation. This change in cocrystal explosive sensitivity originates not only from formation of the intermolecular H-bonding interaction but also from the increment of the N–NO2 BDE. In previous investigations, the influence of the intermolecular H-bonding interaction on the strength of the N–NO2 trigger-bond was not considered in cocrystal explosive complexes, with only the formation of intermolecular H-bonding interaction being regarded as the origin of the sensitivity change in cocrystal explosives [6, 17].

RDG analysis

The RDG [47] can be used to reveal intermolecular and intramolecular interactions in real space based on the electron density. RDG is defined as

$$ RDG={\scriptscriptstyle \frac{1}{2{\left(3{\pi}^2\right)}^{1/3}}}{\scriptscriptstyle \frac{\left|\nabla \rho (r)\right|}{\rho {(r)}^{4/3}}} $$

According to Johnson et al. [47], the sign of λ2 (the second eigenvalue of the electron density Hessian matrix) can be used to distinguish bonded (λ2 < 0) from non-bonded (λ2 > 0) interactions. In order to explore the features associated with small reduced gradients, plots of RDG versus sign [λ2®)ρ(r)] were examined. Figure 2 depicts the RDG results of five complexes. The regions in low density and large reduced gradient correspond to the exponentially decaying tail regions of the density, i.e., far from the nuclei.

Fig. 2
figure 2

Upper panel Plots of reduced density gradient (RDG) versus sign[λ2®)ρ®)], lower panel low-gradient (s=0.5 a.u.) isosurfaces for the HMX/NQ complex. The surfaces are colored on a blue-green-red scale according to values of sign(λ 2)ρ, ranging from –0.04 to 0.04 a.u. Blue Strong attractive H-bond, green van der Waals (vdW) interaction, red strong steric effect

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Most important for our present consideration were the regions of low density and low gradient. For the left-side region in each of complexes, several spikes lying at negative values were found in the low density and low gradient region (see upper panel in Fig. 3)—a signature of noncovalent interactions between HMX and NQ. The strong intramolecular interaction between the nitro-group and amine-group of NQ corresponds to the region of ~ −0.022 a.u. The intermolecular hydrogen-bonding interactions correspond to density values between −0.020 a.u. and −0.015 a.u. The moderate density values indicate that the intermolecular hydrogen-bonding interactions between nitro-groups of HMX and amine-groups of NQ are not strong, in agreement with the results of the molecular interaction energies. Density values between −0.015 a.u. and −0.005 a.u. (low reduced gradient) show vdW interactions.

The interaction regions can also be located by generating RDG isosurface enclosing the corresponding regions in real molecular space (lower panel in Fig. 3). The red isosurface corresponds to the repulsion and steric effect in the HMX ring or among chemical bonds. The blue and green isosurfaces between HMX and NQ correspond to the region in the low density, low-gradient spike lying at negative values, indicative of the attractive intermolecular interactions (i.e., intermolecular hydrogen-bonding interactions and vdW interactions). From Fig. 3, the weak H-bonds and vdW interactions are the main intermolecular interactions in HMX/NQ complexes. Note that, for complex (E), the blue isosurfaces are found in the region between the nitro-groups in HMX and NQ. This result shows that, in the complex (E), O···H and O···O interactions coexist. Indeed, according to natural bond orbital (NBO) analysis, the delocalization interaction E (2) σ(N10–O9)→RY*O7 is up to 6.22 kJ mol−1 at the B3LYP/6-311++G (d, p) level.

Fig. 3
figure 3

Electrostatic potential surface of HMX, NQ and their complexes

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Density and detonation velocity

Density (d mix) and detonation velocity (V D) are two important parameters for the cocrystallized systems of high density energetic materials. We calculated the theoretical mixing densities (d mix) using the following equation [48]:

$$ {d}_{\mathrm{mix}}=\varSigma {m}_I/\left(\varSigma {m}_I/{d}_{298\mathrm{K},I}\right) $$

where m I is the mass of component I.

Since the heat of formation (HOF) of a solid cocrystal is the sum of its lattice energy and the HOF of the gaseous state of molecule, COMPASS field was employed to calculate lattice energy. According to Zhang et al. [48], the HOF of the gaseous state of molecule was derived from PM6 calculation. Then, the heats of detonation reactions were obtained by calculating HOF differences. Finally, V D was evaluated using the Kamlet approximation [49].

All the results are given in Table 5. It was found that crystal density of cocrystal d mix in each of ratios was slightly lower than that of HMX (i.e., the value at a ratio of 1:0), while higher than that of NQ (i.e., the value at a ratio of 0:1). The calculated detonation velocity of isolated HMX (1:0) or NQ (0:1) was larger slightly than the experimental result [49, 50]. The detonation velocity of HMX/NQ cocrystal at each of the ratios was slightly lower than that of HMX due to their low chemical energies, although larger than that of NQ. Although their detonation performance decreases slightly in contrast with HMX, they still exhibit good detonation performance. The d mix (1.864 g cm−3) and V D (9.003 km s−1) of HMX/NQ cocrystal in a 1:1 molecular molar ratio were very close to those of HMX. Therefore, HMX/NQ cocrystal in a 1:1 molar ratio is satisfactory in view of explosive properties.

Table 5 Predicted properties [d mix (g cm−3) and V D(km s−1)] at different molecular molar ratios (HMX:NQ)
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Surface electrostatic potentials

Electrostatic potential (ESP) is a real and fundamentally significant physical property of compounds [52, 53]. Politzer et al. [46] suggested that, in addition to focusing on the trigger bond, the ESP, as a global feature of a molecule, should be taken into account in the analysis of explosive sensitivity. These latter authors found that the strongly positive central regions that characterize molecular surface potentials can be linked to sensitivity: the sensitivity increases as the central ESP becomes more positive. Furthermore, some quantitative relationships between the impact sensitivities and certain features of the molecular surface ESPs, such as the positive variances of VS®) (i.e., σ 2+ ) and electrostatic balance parameter \( {\scriptscriptstyle \frac{\sigma_{+}^2{\sigma}_{-}^2}{{\left({\sigma}_{+}^2+{\sigma}_{-}^2\right)}^2}} \), have been established [54–60]. In 2013, Li et al. [14] used the ESP on the surface to evaluate the sensitivities of TNT and CL-20 both before and after co-crystal formation.

The ESP VS(r) on the 0.001 a.u. molecular surfaces are shown in Figs. S1–S7 and Tables S3–S9. The surface minima (VS,min) were associated mainly with the lone pairs of the O, N atoms in ring of HMX and N atoms in –NH2 and –NH groups of NQ. The surface maxima (VS,max) were located mainly near the H atom or the N atom of the −NO2 group (see Figs. S1–S7 and Tables S3–S9). For all complexes, the largest value of V S,max was larger than that in the corresponding monomer.

Particularly interesting for us was the local maximum above the N−NO2 bond involving intermolecular H-bonds. For the isolated molecules, such local V S,max have been found above the C−NO2 or N−NO2 bonds in nitroaromatics, nitroheterocycles, nitroalkanes or dimethylnitramine by Politzer and colleagues. Furthermore, these latter authors have stated that these V S,max values correlate inversely, to some extent, with C−NO2 and N−NO2 bond energies, and have been linked to impact sensitivities: the bond becomes weaker, i.e., the sensitivity becomes higher, as V s,max (C/N−NO2) becomes more positive [58, 61–63]. From Figs. S1–S7 and Tables S3–S9, in ©), the V S,max above the N−NO2 bond involving the intermolecular H-bonds was submerged by the strongly positive potential of hydrogens and the N atoms of the −NO2 groups. However, the V S,max above the N−NO2 bonds in HMX and NQ were found to be 25.58 and 27.15 kcal mol−1, respectively, and those involving the H-bonds were 24.82, 19.61, 25.70 and 31.30 kcal mol−1 in (A), (B), (D) and (E), respectively. The VS,max above the N−NO2 bonds in (A) and (B) were less positive than that in the isolated HMX, suggesting that the N−NO2 bonds become stronger and the sensitivity becomes lower upon complex formation, in accordance with results from the structure and BDEs of the N−NO2 bonds.

The values of the positive and negative variances of VS®) (i.e., σ 2+ and σ 2‐ ), and the electrostatic balance parameter \( {\scriptscriptstyle \frac{\sigma_{+}^2{\sigma}_{-}^2}{{\left({\sigma}_{+}^2+{\sigma}_{-}^2\right)}^2}} \) (ν) are collected in Table 6. σ 2+  > σ 2‐ is found in HMX, NQ and their complexes, in accordance with the features of energetic explosives with strong NO2 electron-attracting groups [58].

Table 6 Computed molecular surface potentials σ 2+ and σ 2‐ [in (kcal mol−1)2], and electrostatic balance parameter \( {\scriptscriptstyle \frac{\sigma_{+}^2{\sigma}_{-}^2}{{\left({\sigma}_{+}^2+{\sigma}_{-}^2\right)}^2}} \) (ν)
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From the relationship between the impact sensitivities and the features of surface ESPs (σ 2+ and ν) [64], the smaller the value of σ 2+ , the larger the value of ν, and, simultaneously, the lower the impact sensitivity becomes. From Table 6, for complexes (A) and (E), the values of σ 2+ [196.88 and 197.74 (kcal mol−1)2] were smaller than that of isolated HMX [206.20 (kcal mol−1)2], and, simultaneously, the values of ν (0.1430 and 0.1763) were larger than that of isolated HMX (0.1350). Thus, the impact sensitivities of (A) and (E) decrease in comparison with that in isolated HMX. However, for complexes (B), ©) and (D), although the values of ν were larger than that of isolated HMX, the values of σ 2+ were also larger than that of HMX. Thus, for these three complexes, we cannot judge whether the sensitivity decreases or increases in comparison with that in HMX. For complexes (B) and (D), the values of σ 2+ were larger than that of isolated NQ, and, simultaneously, the values of ν were smaller than that of isolated NQ. Thus, the impact sensitivities of (B) and (D) increase in comparison with that in isolated NQ. In previous investigations, although for isolated molecules some relationships between the impact sensitivities and the features of surface ESPs (σ 2+ and ν) were established, unfortunately they were not established for the intermolecular H-bonded complexes. Therefore, the quantitative impact sensitivities h50 were not calculated in this work.

It is worth mentioning that, although for (B), ©) and (D), we cannot judge whether the sensitivity decreases or increases, for the most stable complex (A), which dominates the cocrystallization, it has been confirmed that the sensitivity decreases in comparison with that in HMX. Therefore, it can be concluded that the sensitivity decreases in the cocrystal HMX/NQ in comparison with that in isolated HMX.

Conclusions

In this paper, MD methods were employed to study the binding energies and mechanical properties of selected crystal planes for the HMX/NQ cocrystal at different molecular molar ratios. Density and detonation velocity of the cocrystals at different molecular molar ratios were estimated. The intermolecular interactions and BDE of N–NO2 bond in the 1:1 complex of HMX:NQ were calculated. The results indicate that HMX/NQ cocrystal prefers cocrystalizing in a 1:1 molecular molar ratio, and the cocrystallization is dominated by the (0 2 0) and (1 0 0) facets. The K, G, E values of the model of 1:1 were smaller than those of the other models and the 1:1 cocrystal has the best ductility. The N–NO2 bond becomes stronger upon formation of the intermolecular H-bonding interaction, and the sensitivity of HMX decreases in the cocrystal. The change in cocrystal explosive sensitivity originates not only from the formation of the intermolecular interaction but also from the increment of the N–NO2 BDE. HMX/NQ (1:1) cocrystals exhibit good detonation performance. The surface electrostatic potential analysis further confirms that the sensitivity decreases in complex (or cocrystal) in comparison with that in isolated HMX. These studies provide some novel insights that will aid the design of HMX cocrystal explosives.