Introduction

For most kinds of energetic compounds (ECs), energetic performance (or energy density) and safety are always incompatible with each other, namely, ECs with high power also exhibited high mechanical sensitivity and poor safety, and it was more remarkable and severe for ECs with high energy density (HEDCs) [1, 2]. This problem has troubled researchers a lot and limited the development or application of ECs. At present, it is still an embarrassing difficulty and unavoidable challenge to solve this problem in some degree.

In recent years, co-crystallization [3, 4] has proven to be a superior and promising way of decreasing mechanical sensitivity and ameliorating safety of EMs. At the same time, co-crystallization also had splendid advantages or merits for ECs. For example, co-crystallization could strengthen thermal safety, improve mechanical properties, increase energy density, and reduce mechanical sensitivity and so on. Co-crystallization was a novel branch for ECs and it could be formed among different components through nonbond interaction forces, such as hydrogen bond, electrostatic energy, van der Waals (vdW) force, π-π stacking interaction energy, and halogen bond. Up to now, many different kinds of energetic cocrystals have been synthesized and investigated [5,6,7,8,9,10,11].

2,4,6,8,10,12-hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane (CL-20) was a typical kind of HEDCs and exhibited better energetic performance and power than other explosives. However, CL-20 has not witnessed its potential prospect and wide application in military or civil areas because it was quite sensitive to external stimulus and had high mechanical sensitivity, i.e., CL-20 had poor safety and could not meet the requirement of ammunitions and explosives.

In 2012, Bolton [12] dissolved CL-20 and octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) into acetone and prepared a novel CL-20/HMX cocrystal explosive (molar ratio of 2:1) and explored its properties. The results showed that the detonation velocity of CL-20/HMX cocrystal explosive was about 100 m/s higher than that of pure HMX. However, the impact sensitivity was near to HMX. As we all know, HMX is a high-power explosive with high sensitivity, therefore, the sensitivity of CL-20/HMX cocrystal explosive is also relatively high to some extent. Later, Zongwei Yang [13] synthesized another cocrystal composed of CL-20 and 2,4,6-trinitrotoluene (TNT) with molar ratio of 1:1 and tested its performance. The results showed that the stability of CL-20/TNT cocrystal explosive was greatly enhanced with sensitivity effectively decreased. However, the energy density was not quite desirable. Consequently, if CL-20, TNT, and HMX could be cocrystallized together, the CL-20/TNT/HMX cocrystal explosive might have held the advantages of CL-20/TNT and CL-20/HMX cocrystals, namely, excellent energy density and low mechanical sensitivity. At the same time, the stability could also be improved. In cocrystal explosives, the molar ratio of different components would affect properties such as stability, mechanical properties, and especially the energy density and sensitivity. Therefore, it might be very significant to investigate the properties of cocrystal models with different molar ratios and estimate the influences of molar ratios on the performance of cocrystal explosives.

In this article, we mainly establish CL-20/TNT/HMX cocrystal models with different substituted patterns and molar ratios. Besides, the pure CL-20, TNT, HMX components, and CL-20/TNT, CL-20/HMX cocrystal models were also established. Molecular dynamics (MD) method was selected to optimize the structures and predict the stabilities, sensitivity, energy density, and mechanical properties of different cocrystal models. The effects of molar ratios on the performance of cocrystals were investigated and estimated. The results could help to clarify the co-crystallization mechanism and provide some useful guidance for energetic cocrystal designs.

Calculation models and methods

Molecular models of CL-20, TNT, and HMX

In this work, the crystal polymorph of CL-20 was chosen as ε-CL-20 because this polymorph was more stable and had better energetic performance than other polymorphs (α-, β-, γ-CL-20) [14, 15]. For HMX, we selected β-HMX owing to the fact that among the whole polymorphs of HMX (α-, γ-, δ-, β-), β-HMX had higher density and was the most stable and promising polymorph [16, 17]. The molecular structure and lattice parameters of ε-CL-20, TNT, and β-HMX are presented in Table 1, while the chemical models are illustrated in Figs. 1, 2, and 3, respectively.

Table 1 Molecular structure and lattice parameters of ε-CL-20, TNT, and β-HMX
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Fig. 1
figure 1

Molecular structure of ε-CL-20

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Fig. 2
figure 2

Molecular structure of TNT

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Fig. 3
figure 3

Molecular structure of β-HMX

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Crystal models of CL-20/TNT and CL-20/HMX cocrystals

Based on the experimental results, the crystal models of CL-20/TNT and CL-20/HMX cocrystals were established. The molecular structure and lattice parameters of CL-20/TNT and CL-20/HMX cocrystal explosives are listed in Table 2. The primitive cell and supercell models of CL-20/TNT cocrystal explosive are illustrated in Fig. 4 and CL-20/HMX cocrystal models are presented in Fig. 5.

Table 2 Crystal structure and lattice parameters of CL-20/TNT and CL-20/HMX cocrystal explosives
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Fig. 4
figure 4

Primitive unit cell (a) and (3 × 2 × 1) supercells (b) of CL-20/TNT cocrystal models

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Fig. 5
figure 5

Primitive unit cell (a) and (2 × 3 × 2) supercells (b) of CL-20/HMX cocrystal models

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CL-20/TNT/HMX cocrystal models

In cocrystal models, the molar ratio of different components affects the properties of cocrystal explosives, such as stability, intermolecular interactions, density, mechanical properties, and especially the energetic performance and sensitivity. If the mass percent of high-power component was too much, the cocrystal explosive might have a high crystal density and superior energetic performance. However, the mechanical sensitivity would also be increased. Therefore, too much of a molar ratio of high energy density component would have a positive influence on energetic performance, but a negative effect on safety of cocrystal explosives. On the contrary, if the low power component occupied too much, the cocrystal explosive might present low sensitivity. However, the energy density would be severely weakened at the same time. It was also a negative factor for cocrystal explosives. Therefore, to ensure that the cocrystal explosive has desirable energetic performance and appropriate mechanical sensitivity, the molar ratio of different components should be determined or controlled at a reasonable and proper extent.

Previous studies [13, 20,21,22,23,24,25] have illustrated that cocrystal explosives might be more likely to be formed with relatively low molar ratios. Therefore, the molar ratios of CL-20, TNT, and HMX were determined based on this principle.

At present, the substitution method is very practical and widely applied when establishing energetic cocrystal models. Therefore, in this work, the CL-20/TNT/HMX cocrystal models were established by substitution method. In other words, CL-20 molecules in supercell crystals would be substituted by TNT and HMX molecules based on the molar ratio of different components. The substitution included cleaved surfaces substitution and random substitution, namely, TNT and HMX molecules would replace the CL-20 molecules on the cleaved surfaces or randomly. To investigate the influences of co-crystallization and molar ratios on properties of cocrystal explosives and compare the performances of CL-20/TNT/HMX cocrystals with that of raw materials, we also established the crystal models of CL-20, TNT, HMX, CL-20/TNT, and CL-20/HMX cocrystals.

The molar ratio, supercell pattern, total number of CL-20, TNT, HMX molecules, and atoms in cocrystal models are listed in Table 3.

Table 3 Molar ratio, supercell pattern, total number of molecules and atoms of CL-20,TNT, HMX, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal models
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For example, when the molar ratio of the three components (CL-20:TNT:HMX) was in 2:1:1, the CL-20/TNT/HMX cocrystal model was established as follows: Firstly, the single unit cell of ε-CL-20 was established; then it was expanded to 16 (4 × 2 × 2) supercells and 64 CL-20 molecules were included in the supercell; next, the supercell model was cleaved into (1 0 0), (0 1 0), and (0 0 1) surfaces, and shown in Fig. 6; afterwards, a vacuum layer with the thickness of 0 Å would be added into the three cleaved surfaces along the c crystallographic axis; finally, 16 TNT and 16 HMX molecules would substitute 32 CL-20 molecules on the three cleaved surfaces or CL-20 molecules in the initial supercell crystal model.

Fig. 6
figure 6

Cleaved surfaces (1 0 0), (0 1 0), and (0 0 1) of (4 × 2 × 2) supercell of CL-20

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When the initial CL-20/TNT/HMX cocrystal model was established, the structure would be optimized with total energy of cocrystal model minimized. For example, when the substitution was on (0 0 1) crystal surface, the cocrystal model after optimization is presented in Fig. 7.

Fig. 7
figure 7

CL-20/TNT/HMX cocrystal model on (0 0 1) crystal surface with molar ratio of 2:1:1

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Calculation conditions and details

In this work, the crystal models of pure components (CL-20, TNT, and HMX) and CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives were established. The structures and energies were optimized and properties of different models were predicted through MD method. The influences of molar ratio on stability, sensitivity, energetic performance, and mechanical properties of cocrystal explosives were investigated and evaluated. To accurately predict the properties of cocrystal models, COMPASS force field [26,27,28] was selected because this force field was suitable for numerous materials in condensed phase. At the same time, the parameters in COMPASS force filed were validated with the correlated algorithms modified to improve the applicability and ensure its accuracy. At present, COMPASS force field was regarded as an advanced or splendid force filed and had been applied widely to determine parameters and predict the properties of materials, including energetic materials. The thermostat and barostat was set as Anderson, Parrinello, respectively. To determine the nonbond energies, such as van der Waals (vdW) and electrostatic interactions, we selected atom-based summation method for vdW and Ewald method for electrostatic. In the MD simulation, NPT ensemble was chosen and the temperature and pressure was set as 295 K, 0.0001 GPa, respectively. The temperature, pressure, and total number of atoms in crystal models would be kept constant. The step size was 1 fs and total MD simulation time was 2 ns (2 × 106 fs). At first, the crystal model would be under a MD simulation for 1 ns (1 × 106 fs) to optimize the structures, total energies, nonbond energies, and to make the system reach the equilibrium state. Then, the optimized model would be under another MD simulation for 1 ns (1 × 106 fs) to determine the parameters and energies.

Results and discussion

Choice of force field

The force field would directly affect the parameters of crystal systems and each force field might be only suitable for limited crystal models. To choose the most suitable force field and ensure the precision, we performed MD simulations with PCFF, COMPASS, Universal, and Dreiding force field. Besides, the calculated parameters were compared with experimental results.

The calculated crystal parameters and density of pure components (CL-20, TNT, HMX), CL-20/TNT and CL-20/HMX cocrystal models with NPT ensemble (295 K and 0.0001 GPa) by different force field are presented in Table 4.

Table 4 Calculated crystal parameters and density of CL-20, TNT, HMX, CL-20/TNT, and CL-20/HMX cocrystalsa
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As presented in Table 4, it could be concluded that the calculated results obtained by the COMPASS force field were very accurate and it was in good agreement with experimental results, thus implying that this force field was suitable for CL-20, TNT, HMX, CL-20/TNT, and CL-20/HMX cocrystals. At the same time, the data in Table 4 also indicate that the COMPASS force field was reasonable for MD simulations. Previous studies [29,30,31,32,33] have also pointed out that the COMPASS force field is very practical for predicting parameters and properties of large numbers of energetic materials.

Binding energy

Binding energy (Eb) was mainly defined as the intermolecular interactions between different kinds of molecules. Binding energy was an important criterion to reflect or estimate the stability of explosives. Besides, binding energy could also be applied to evaluate the compatibility or miscibility of different components. Generally speaking, if the value of Eb was higher, it would mean that the interaction energies between different molecules were stronger and the explosive had desirable stability and favorable compatibility [29, 30, 32]. For cocrystal explosives, binding energy could also predict the formation probability. That is to say, cocrystal models with a higher value of Eb might be formed more easily or probably. In this work, the cocrystal explosive consisted of CL-20, TNT, and HMX. We were mainly concerned about the interaction energy and compatibility between CL-20 and other components, i.e., the binding energy between CL-20 and TNT, HMX molecules in cocrystal models.

Binding energy was determined by the total energy of the whole system and individual energy of each component. Binding energy was depicted as follows:

$$ {E}_{\mathrm{b}}=-{E}_{\mathrm{inter}}=-\left[{E}_{\mathrm{total}}-\left({E}_{\mathrm{CL}-20}+{E}_{\mathrm{TNT}/\mathrm{HMX}}\right)\right] $$
(1)
$$ {E}_{\mathrm{b}}^{\ast }=\frac{E_{\mathrm{b}}\times {N}_0}{N_{\mathrm{i}}} $$
(2)

where Einter is defined as the interaction energy, Etotal is defined as the total energy of cocrystal model when it was under equilibrium state, ECL-20 is the total energy of CL-20 molecules with all the TNT and HMX molecules moved away from the cocrystal model, ETNT/HMX is the total energy of TNT and HMX molecules when all the CL-20 molecules were removed, Eb* is called the relative binding energy or corrected binding energy, Ni is the total number of molecules (including CL-20, TNT, and HMX) for ith cocrystal model, N0 was the total number of molecules for a standard cocrystal model. In this work, the CL-20/TNT/HMX cocrystal model with a molar ratio of 1:1:1 was selected as the standard pattern, i.e., N0 = 48.

Binding energy of CL-20/TNT/HMX cocrystal models with different substituted patterns and molar ratios is illustrated in Table 5.

Table 5 Binding energy of different CL-20/TNT/HMX cocrystal explosives (kJ/mol)
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From Table 5, it was concluded that the binding energy for each cocrystal model was different from each other. In other words, the binding energy would be affected or determined by substituted pattern and molar ratio of different components. In most cases, the binding energy varied as (0 1 0) > (0 0 1) > (1 0 0) > random, i.e., (0 1 0) crystal surface was more stable and the interaction energies between CL-20, TNT and HMX molecules on the (0 1 0) surface was stronger. Besides, Table 5 also illustrates that the highest binding energy of (1 0 0) and (0 1 0) crystal surfaces appeared at 3:1:2 (molar ratio), corresponding to 616.17 kJ/mol, 683.31 kJ/mol, respectively and it was 3:1:3 for (0 0 1) crystal surface (640.38 kJ/mol) and random substituted pattern (613.57 kJ/mol). Based on these data, it might be indicated that when the molar ratio was 3:1:2 or 3:1:3, the CL-20, TNT, and HMX molecules in cocrystal model would be combined tighter and the interaction force between these components was stronger. What’s more, the cocrystal explosive with these molar ratios exhibited more desirable stability and might be formed more probably or likely in the same condition.

Sensitivity

Sensitivity was directly related to safety of ECs and it was commonly defined as the relative probability or possibility for ECs to be decomposed or exploded when subjected to different external stimulus. Sensitivity had a great effect on properties of ECs and it might be one of the most important performances for ECs in some sense. Up to now, many theories have been proposed to judge or predict the sensitivity of ECs [34,35,36,37]. In this work, we chose the trigger bond length, interaction energy of trigger bond, and cohesive energy density (CED) to evaluate the sensitivity of CL-20/TNT/HMX cocrystal explosives. This theory was put forward by Jijun Xiao [33, 38,39,40,41,42] and has been verified as an accurate and practical theory. Up to now, this theory has been applied in ECs fields to predict the sensitivity of some explosives consisting of single component or multiple components.

Trigger bond length

The trigger bond of ECs was generally defined as the chemical bond that was the weakest or had the least bond energy. Compared with other chemical bonds, the trigger bond was more active and would be ruptured or broken more easily under external stimulus. For CL-20/TNT/HMX cocrystal explosives, it consisted of three components, namely, CL-20, TNT, and HMX. Among them, CL-20 was the most active component and exhibited the highest mechanical sensitivity. Therefore, CL-20 molecules would be decomposed or exploded more easily than TNT and HMX molecules under external stimulus, which would result in the next chemical reaction of cocrystal explosives. For CL-20, the N-N bond in nitro groups (N-NO2 groups) had the least bond energy and was the weakest chemical bond. In other words, the N-NO2 bond was the trigger bond for CL-20 [43, 44]. Consequently, the N-NO2 bond in CL-20 molecules was chosen as the trigger bond to predict the sensitivity of CL-20/TNT/HMX cocrystal explosives.

The trigger bond length of CL-20/TNT/HMX cocrystal explosive with a molar ratio of 2:3:1 on (0 1 0) crystal surface is shown in Fig. 8. The probable trigger bond length (Lprob), average trigger bond length (Lave), and maximum trigger bond length (Lmax) of pure CL-20, CL-20/TNT, CL-20/HMX, and different CL-20/TNT/HMX cocrystal explosives were presented in Table 6.

Fig. 8
figure 8

Trigger bond length distribution of CL-20/TNT/HMX cocrystal explosive

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Table 6 Trigger bond length distribution of different CL-20/TNT/HMX cocrystal explosives (Å)
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It can be concluded from Fig. 8 that the trigger bond length distribution was nearly the Gauss’s distribution and most of the trigger bond distributed between 1.350~1.450 Å.

Table 6 illustrates that the values of Lprob and Lave were nearly the same for CL-20, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives and both of them did not vary obviously, which might indicate that co-crystallization had no influence or little influence on probable trigger bond length (Lprob) or average trigger bond length (Lave). However, the value of maximum trigger bond length (Lmax) varied obviously for each cocrystal model. For CL-20, CL-20/TNT, and CL-20/HMX cocrystal explosives, the value of Lmax was 1.642, 1.545, and 1.618 Å, respectfully. The maximum trigger bond length of CL-20/TNT cocrystal explosive was decreased by 5.91% than CL-20 and it was 1.46% for CL-20/HMX cocrystal explosive. Although the CL-20 molecules that had larger trigger bond length than Lmax only occupied a small portion, these molecules were particularly active and sensitive to external stimulus, and it might be more likely for the trigger bond to be broken to make the explosive decomposed or exploded. Therefore, CL-20 had higher mechanical sensitivity than CL-20/TNT and CL-20/HMX cocrystal explosives. The Lmax for CL-20/TNT/HMX cocrystal explosives were between that of pure CL-20 and CL-20/TNT cocrystal explosive and the cocrystal model with a molar ratio of 3:1:2 had the least value of Lmax (1.592 Å). Generally speaking, if the value of Lmax was larger, it would mean that the chemical bond was more active and had weaker bond strength. Consequently, the trigger bond in the CL-20/TNT/HMX cocrystal model was stabilized with bond strength enhanced. In other words, the safety of CL-20/TNT/HMX cocrystal explosive was improved or increased.

Interaction energy of trigger bond

Interaction energy of trigger bond could directly reflect the bond strength. If the value of trigger bond energy was higher, it would indicate that the bond strength was stronger and the sensitivity of ECs was lower.

Interaction energy of the trigger bond was illustrated as:

$$ {E}_{\mathrm{N}-\mathrm{N}}=\frac{E_{\mathrm{T}}-{E}_{\mathrm{F}}}{n} $$
(3)

where ET is the total energy of the cocrystal explosive when it is under equilibrium state, EF is the total energy of cocrystal model when all the N atoms in CL-20 molecules is constrained purposely, n is the total number of N-N bond in CL-20 molecules.

The trigger bond energy of CL-20/TNT/HMX cocrystal explosives with different molar ratios and substituted patterns are listed in Table 7.

Table 7 Interaction energy of trigger bond of different CL-20/TNT/HMX cocrystal explosives (kJ/mol)
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As illustrated in Table 7, it can be concluded that CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives had higher values of trigger bond strength than raw CL-20. For example, the trigger bond energy of CL-20, CL-20/TNT, and CL-20/HMX cocrystal explosive was 138.6, 156.7, and 145.6 kJ/mol, respectively. Compared to pure CL-20, the trigger bond energy of CL-20/TNT cocrystal explosive was increased by 13.06%. For CL-20/HMX cocrystal explosive, the trigger bond energy was increased by 5.05% compared to that of CL-20. The increase of trigger bond energy illustrated that the stability and strength of trigger bond was effectively improved or enhanced, which further meant that CL-20/TNT, CL-20/HMX and CL-20/TNT/HMX cocrystal explosives had lower sensitivity and better safety than CL-20.. At the same time, the trigger bond energy also implied that co-crystallization was a superior and effective means to decrease sensitivity of ECs. Table 7 also illustrates that for the different substituted patterns, trigger bond energy ranged as (0 1 0) > (1 0 0) > random > (0 0 1), thus indicates that (0 1 0) crystal surface was more insensitive and had the best safety. What’s more, Table 7 also implies that the molar ratios of CL-20, TNT, and HMX would affect the trigger bond strength. Among the whole CL-20/TNT/HMX cocrystal explosives, the trigger bond energy for cocrystal model with molar ratio of 3:1:2 was the highest, which indicates that this cocrystal model was more insensitive and had the lowest mechanical sensitivity and the best safety.

Cohesive energy density

Cohesive energy density (CED) belonged to the nonbond energy. It was composed of vdW and electrostatic energy. CED could be defined as the total energy that is needed to separate all the explosive molecules from each other. The CED, vdW, and electrostatic energies of different CL-20/TNT/HMX cocrystal explosives are shown in Table 8.

Table 8 CED, vdW, and electrostatic energies of CL-20/TNT/HMX cocrystal explosives (kJ/cm3)a
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As shown in Table 8, it was concluded that CL-20 (molar ratio of 1:0:0) had the least value of CED, vdW, and electrostatic energies, corresponding to 0.638, 0.176, and 0.462 kJ/cm3, respectively. CL-20/TNT cocrystal explosive (molar ratio of 1:1:0) exhibited the largest value of CED (0.855 kJ/cm3), vdW (0.243 kJ/cm3), and electrostatic energy (0.612 kJ/cm3). These energies of CL-20/TNT cocrystal explosive were increased by 34.01, 38.07, and 32.47% when compared to that of pure CL-20, thus illustrating that the sensitivity of CL-20/TNT cocrystal explosive was greatly decreased. Among the different CL-20/TNT/HMX cocrystal models, the value of CED, vdW, and electrostatic energy was higher than pure CL-20 and CL-20/HMX cocrystal explosive, but lower than CL-20/TNT cocrystal explosive. Table 8 also shows that when the molar ratio was 3:1:2, the nonbond energies, including CED, vdW, and electrostatic energies, were the largest, corresponding to 0.797, 0.233, and 0.564 kJ/cm3. The CED was increased by 24.92% compared pure CL-20, and it was 32.39% for vdW energy and 22.08% for electrostatic energy. These data meant that the CL-20/TNT/HMX cocrystal explosive model with molar ratio of 3:1:2 had the lowest mechanical sensitivity, but the best safety.

Detonation performance

The power and energetic performance of ECs could be reflected by correlated detonation parameters, such as detonation velocity (D), detonation pressure (P), and detonation heat (Q). In this work, the nitrogen equivalent coefficient (NEC) method was chosen to calculate the detonation parameters and predict the energy density of cocrystal explosives and raw components. The NEC method was first put forward in 1964 and later revised by Housheng Zhang in 1978 [45]. The NEC method took some factors that might affect or determine the detonation parameters into consideration, such as molecular structure, detonation products, the chemical bond, or chemical groups existing in explosives. Therefore, the NEC method was a very practical and precise method for numerous different kinds of explosives.

For ECs that only contained C-H-O-N element (CaHbOcNd), the oxygen balance (OB) was depicted as follows:

$$ OB=\frac{\left[c-\left(2a+b/2\right)\right]}{M_r}\times 16\times 100\% $$
(4)

where a, b, c, and d are the total number of carbon, hydrogen, oxygen, and nitrogen atoms existing in explosive molecules, and Mr is the molar mass of explosive with the unit of g/mol.

The value of OB for mixed explosives could be obtained as:

$$ OB=\sum {w}_iO{B}_i $$
(5)

where wi is the mass percent of ith component in mixed explosive and OBi is the oxygen balance of ith component.

According to NEC method, detonation parameters (D and P) could be calculated as that:

$$ \Big\{{\displaystyle \begin{array}{l}D=\left(690+1160{\rho}_0\right)\sum {N}_{\mathrm{ch}}\\ {}P=1.106{\left({\rho}_0\sum {N}_{\mathrm{ch}}\right)}^2-0.84\\ {}\sum {N}_{\mathrm{ch}}=\frac{100}{M_r}\left({p}_i{N}_{pi}+\sum {B}_K{N}_{BK}+\sum {G}_j{N}_{Gj}\right)\end{array}} $$
(6)

where ∑Nch is the total nitrogen equivalent coefficient, pi, Npi, BK, NBK, Gj, NGj are the correlated parameters determined by explosive molecules, detonation products, chemical bonds, and chemical groups. All of the parameters in Eq. (6) are interpreted in Refs. [46, 47]. To better understand the NEC method, we could consult Refs. [46, 47] for more information and further help.

The detonation products and nitrogen equivalent parameter of raw components, such as CL-20 (C6H6O12N12), TNT (C7H5O6N3), and HMX (C4H8O8N8) were determined based on the H2O-CO-CO2 principle, i.e., the detonation equations of these three explosives were illustrated as:

$$ \Big\{{\displaystyle \begin{array}{l}{\mathrm{C}}_6{\mathrm{H}}_6{\mathrm{O}}_{12}{\mathrm{N}}_{12}\to 3{\mathrm{H}}_2\mathrm{O}+3\mathrm{CO}+3{\mathrm{C}\mathrm{O}}_2+6{\mathrm{N}}_2\\ {}{\mathrm{C}}_7{\mathrm{H}}_5{\mathrm{O}}_6{\mathrm{N}}_3\to 2.{5\mathrm{H}}_2\mathrm{O}+3.5\mathrm{C}\mathrm{O}+3.5\mathrm{C}+1.{5\mathrm{N}}_2\\ {}{\mathrm{C}}_4{\mathrm{H}}_8{\mathrm{O}}_8{\mathrm{N}}_8\to 4{\mathrm{H}}_2\mathrm{O}+4\mathrm{CO}+{4\mathrm{N}}_2\end{array}} $$
(7)

Detonation heat (Q) is depicted as follows [48, 49]:

$$ Q=\sum {\omega}_i{Q}_i $$
(8)

where Q is the detonation heat of mixed explosive and ωi and Qi are the mass percent and detonation heat of ith component, respectively.

The detonation parameters of raw components (CL-20, TNT, HMX), CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives are presented in Table 9.

Table 9 Detonation parameters of raw components, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives
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From Table 9 it can be concluded that the density (ρ) and detonation parameters (D, P, Q) of raw CL-20 was 2.035 g/cm3, 9.50 km/s, 46.60 GPa, 6230 kJ/kg, respectively. The high value of density and detonation parameters mean that CL-20 had outstanding energetic performance and superior power. Besides, these data also indicate that CL-20 was a kind of splendid and promising HEDMs. For CL-20/TNT, CL-20/HMX, or CL-20/TNT/HMX cocrystal explosives, the density declined and detonation parameters were also decreased, namely, the power and energetic performance was weakened. The CL-20/TNT/HMX cocrystal model with the molar ratio of 1:3:1 had the least value of density and detonation parameters and it was 1.806 g/cm3, 8.27 km/s, 30.98 GPa, and 5423 kJ/kg, respectively. This further implied that the cocrystal model also exhibited the poorest detonation performance and lowest energy density. On the contrary, the cocrystal model with a molar ratio of 3:1:1 had the highest density and maximal detonation velocity and detonation pressure, corresponding to 1.956 g/cm3, 9.34 km/s, 41.29 GPa, and 6018 kJ/kg. Previous studies [50,51,52] have clearly stated that for HEDMs, it was required that ρ > 1.9 g/cm3, D > 9.2 km/s, P > 40 GPa. Among the whole CL-20/TNT/HMX cocrystal models, only the cocrystal model with the molar ratio of 3:1:1, 3:1:2, 3:1:3 could satisfy the requirement.

Mechanical properties

Mechanical properties were generally characterized by five parameters, i.e., tensile modulus (E), shear modulus (G), bulk modulus (K), Poisson’s ratio (v), and Cauchy pressure (C12-C44). Among them, E, K, and G were also called the engineering modulus. E, K, and G was commonly applied as a criterion to evaluate the rigidity, stiffness, or hardness of a material. Besides, K was an important level to reflect the rupture strength and materials with higher value of K would also have greater rupture strength [53]. Cauchy pressure was commonly used to estimate the ductility, plastic property, or brittle property of a material [54]. In other words, a high positive value of (C12-C44) would mean that the material had desirable ductility or superior plastic property. On the contrary, if the value of (C12-C44) was negative, it would indicate that the material exhibited a brittle property.

Mechanical properties were generally depicted by elastic coefficients (Cij), stress (σ), and strain (ε) as follows [55, 56]:

$$ {\sigma}_i={C}_{ij}{\varepsilon}_j $$
(9)

Bulk modulus (K) and shear modulus (G) were illustrated as follows:

$$ {K}_{\mathrm{R}}={\left[{S}_{11}+{S}_{22}+{S}_{33}+2\left({S}_{12}+{S}_{23}+{S}_{31}\right)\right]}^{-1} $$
(10)
$$ {G}_{\mathrm{R}}=15{\left[4\left({S}_{11}+{S}_{22}+{S}_{33}\right)-4\left({S}_{12}+{S}_{23}+{S}_{31}\right)+3\left({S}_{44}+{S}_{55}+{S}_{66}\right)\right]}^{-1} $$
(11)

where the subscript R is the Reuss average and the parameters (Sij) and elastic coefficients (Cij) are illustrated as S=C−1.

Mechanical properties could be related together as follows:

$$ E=2G\left(1+\nu \right)=3K\left(1-2\nu \right) $$
(12)

Based on the above equations, tensile modulus (E) and Poisson’s ratio (v) could be calculated as that:

$$ E=\frac{9 GK}{3K+G} $$
(13)
$$ \nu =\frac{3K-2G}{2\left(3K+G\right)} $$
(14)

The mechanical properties of CL-20, TNT, HMX, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives are listed in Table 10.

Table 10 Mechanical properties of raw components, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal modelsa
Full size table

What can be concluded from Table 10 is that both the raw components (CL-20, TNT, and HMX), CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosives had different mechanical properties. Among the whole crystal models, CL-20 had the highest value of E, K, G, but the lowest Cauchy pressure (C12-C44). In other words, the value of E, K, G for pure CL-20 was very high, while Cauchy pressure (C12-C44) was negative. For example, the value of E, K, G for (0 0 1) crystal surface was 16.165, 9.932, and 6.578 GPa, respectively. The high and positive value of E, K, G might mean that the stiffness, rigidity, hardness, or rupture strength of CL-20 was very good. However, the negative value of Cauchy pressure (− 3.187 GPa) indicates that CL-20 presented poor ductility and plastic property. Consequently, the mechanical properties or raw CL-20 was undesirable. For CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal models, the three engineering moduli (E, K, G) were decreased, while Cauchy pressure was increased, thus implying that the rigidity and hardness of cocrystal models was lower than CL-20, but the plastic property was better than CL-20, namely, the cocrystal model had better mechanical properties than CL-20. Besides, the variation of mechanical properties also illustrated that co-crystallization could effectively improve mechanical properties of ECs. Table 10 also illustrates that the value of E, K, G of different substituted patterns varied as (1 0 0) > (0 0 1) > random > (0 1 0), while Cauchy pressure was on the opposite order. Therefore, (1 0 0) crystal surface exhibited the highest rigidity, but poorest ductility, while (0 1 0) surface held the most desirable mechanical properties. What’s more, for (1 0 0) and (0 0 1) crystal surfaces, when the molar ratio was 3:1:2, the CL-20/TNT/HMX cocrystal model held the lowest E, K, G, but the largest Cauchy pressure, i.e., this cocrystal model had the best mechanical properties. For (0 1 0) crystal surface, the cocrystal model presented the best mechanical properties with a molar ratio of 3:1:3 and it was 3:1:1 for a random substituted pattern.

When taking the stability, sensitivity, energetic performance, and mechanical properties into consideration, it could be concluded that when the molar ratio of different components was 3:1:1, 3:1:2, or 3:1:3, the CL-20/TNT/HMX cocrystal explosive had the most desirable or superior properties and might be formed more probably, especially the cocrystal model with molar ratio of 3:1:2. Consequently, the cocrystal explosives with these molar ratios were very promising and worth more attention.

Conclusions

In this work, the pure CL-20, TNT, HMX, CL-20/TNT, CL-20/HMX, and CL-20/TNT/HMX cocrystal explosive models were established and MD method was applied to predict the stability, sensitivity, energetic performance, and mechanical properties of different crystal models. The influences of co-crystallization and molar ratios on properties of cocrystal explosives were investigated and estimated. The main results and conclusions are listed as follows:

  1. (1)

    The CL-20/TNT/HMX cocrystal explosive with molar ratio of 3:1:2 or 3:1:3 had the largest binding energy and best stability. The cocrystal model might be more likely to be formed with these molar ratios. (0 1 0) crystal surface was more stable than (0 0 1), (1 0 0) and random substituted models.

  2. (2)

    The cocrystal model had less value of trigger bond length, but higher value of trigger bond energy and CED than CL-20, i.e., cocrystal model had lower mechanical sensitivity and better safety. The CL-20/TNT/HMX cocrystal model with a molar ratio of 3:1:2 was the most insensitive cocrystal model and exhibited the best safety.

  3. (3)

    The detonation parameters and energetic performance of CL-20/TNT/HMX cocrystal explosive was lower than pure CL-20, only the cocrystal model with molar ratio of 3:1:1, 3:1:2, or 3:1:3 exhibited desirable detonation performance and could satisfy the requirement of HEDMs.

  4. (4)

    The cocrystal explosive presented better mechanical properties than CL-20 and the CL-20/TNT/HMX cocrystal explosive with molar ratios of 3:1:1, 3:1:2, or 3:1:3 had the most desirable mechanical properties.

In a word, co-crystallization could effectively decrease mechanical sensitivity, strengthen safety, improve mechanical properties, and transform energetic performance. The CL-20/TNT/HMX cocrystal explosive with molar ratio of 3:1:2 exhibited the best stability, lowest mechanical sensitivity, and excellent safety. Besides, this cocrystal model also had superior energetic performance and desirable mechanical properties. Therefore, this CL-20/TNT/HMX cocrystal model had the best comprehensive properties and was very promising. This work could provide some theoretical support and helpful guidance to better clarify the cocrystal mechanism and design new kinds of energetic cocrystals.