Unraveling the impact sensitivity mechanism of energetic materials from vibrational and electronic energy transfer

Summary

Clarifying the sensitivity mechanism of explosives is a key prerequisite for developing energetic materials (EMs) with both low sensitivity and high detonation performance. In this study, we propose a sensitivity mechanism for energetic materials. We find that the acceleration of heat energy transfer from phonons to doorway vibrations results in a reduced transfer time to the target mode, thereby increasing the compound’s impact sensitivity. Delocalization of the initially broken chemical bond’s electrons subsequently facilitates the transfer of energy to neighboring molecules, effectively preventing the formation of hot spots. Remarkably, we establish a robust correlation between impact sensitivities and the parameter ζ, exhibiting a correlation coefficient (R² = 0.985). We expect that this elucidated sensitivity mechanism will serve as a crucial tool in predicting the impact sensitivities of EMs, paving the way for the development of advanced and safer explosive materials.

Graphical abstract

Highlights

• •Rapid phonon to doorway vibration transfer increases compound impact sensitivity
• •Electron delocalization aids energy transfer and prevents hotspot formation
• •Establish correlation between the parameter ζ and impact sensitivity with R² = 0.985
• •Energy reaches the targeting bond-breaking mode, raising the temperature of the system

Applied sciences; Energy systems; Energy Modeling

Introduction

Energetic materials (EMs) containing explosives, pyrotechnic compositions, and propellants possess a great deal of energy and can be instantly released with accidental stimulus, which have widespread applications in military and civilian areas such as space exploration, metal processing, and mine exploitation.^1,2 Safety stands as one of the paramount attributes in the realm of energetic materials, usually measured and represented by sensitivity, with higher safety corresponding to lower sensitivity.³ Finding the elusive balance between outstanding performance and low sensitivity has persistently posed a formidable challenge, primarily due to their inherent contradictory nature.^4,5,6,7,8 To overcome this hurdle and achieve groundbreaking advancements in energetic materials, a profound and all-encompassing comprehension of the sensitivity mechanism is imperative. The design of EMs with precisely tailored properties necessitates a holistic and in-depth understanding of their initiation mechanisms. Against this backdrop, significant endeavors have been dedicated to establishing a correlation between sensitivity and structural or other properties, encompassing facets such as crystal defects,⁹ the formation of hot spots,¹⁰ oxygen balance,¹¹ band gap characteristics,¹² vibrational modes,¹³ rates of phonon energy transfer,¹⁴ and bond dissociation energy.¹⁵ Due to the intricate physical and chemical phenomena that energetic materials undergo upon external energy stimulation, encompassing energy absorption, storage, transfer, lattice vibration, hot spot formation, molecular dissociation, combustion, and detonation, the absence of parameters amenable to experimental validation persists.

It is worth noting that phonon calculation serves an essential role since phonons not only influence molecule vibration and heat energy transfer,^16,17 it also open up the possibility of a more rudimentary mechanism for impact sensitivity. Multi-phonon up-pumping theory was first proposed by Dlott and Fayer¹⁸ to describe the transfer of energy and chemical bond breaking, and was further developed by McNesby and Coffey.¹⁹ In addition, the model, applied by Morrison’s group to a variety of metal azides^20,21 and organic nitroexplosives,^22,23 phenyl diazonium chloride, tetrafluoroborate crystalline salts²⁴ as well as 2,4,6,8,10,12-hexanitrohexaazaisowurtzitane/1-methyl-3,4,5- trinitropyrazole (CL-20/MTNP) cocrystal,²⁵ was also applied by Bidault et al. to polycrystalline phases.²⁶ Specifically speaking, the mechanical impact results in vibrational excitation of the lattice vibrational modes, and equilibration within the phonon modes will be quickly reached. Meanwhile, a series of phonons successively pump energy up the intramolecular vibrational mode through doorway vibration modes (the lowest energy intramolecular vibrational mode). The energy is redistributed among the intramolecular vibration modes. When energy accumulates at a specific vibration mode to the reaction threshold, the chemical bond breaks, and the chemical reaction begins to occur. This sensitivity mechanism is known as the indirect pathway.²⁷ There is an alternative view that a highly excited coherent phonon pumps energy directly into the chemical bond with which it is most strongly coupled, leading to this bond breaking. This mechanism is described as the direct pathway.²⁸ Up to now, the sensitivity mechanism widely accepted by most researchers is the indirect pathway, namely the multi-phonon up-pumping theory, which is favored over the direct pathway. Bondarchuk et al.²⁹ proposed an improved multi-phonon up-pumping model by introducing damping factors a and b, which can better describe the transfer process of mechanical energy to vibrational energy. However, the experimental validation of this model is still lacking, and it fails to account for key aspects such as hot spot formation, molecular dissociation, and electronic energy transfer. Consequently, the multi-phonon up-pumping model necessitates further refinement and enhancement.

In addition, more and more studies suggest that electronic excitations have an important effect on the sensitivity of energetic materials. When a detonation front impinges on energetic materials, they will be compressed and sheared. This transforms the electronic structure of the compound locally. The most important electronic change is the reduction of the LUMO-HOMO energy gap. Therefore, within the tens of femtoseconds, the bonding electrons can delocalize. Dremin et al. investigate that the explosion is caused by the dissociation of molecules within the shock front, which can proceed in three ways: from an accumulation mechanism, thermal ionization, to electronic excitation.³⁰ Kuklja et al. report a mechanism for detonation initiation in energetic explosives based on electronic excitations by using the RDX crystal as an example. They think that the pressure inside the impact wavefront reduces the band gap, creating favorable conditions for the initiation of a chain reaction.³¹ Gilman found that sensitivity is connected with the presence of delocalized electrons, and when the electrons in the chemical bonds become delocalized, the chemical energy will be released.³² Mathieu discussed the relevance of non-adiabatic electronic effects to shock initiation of energetic materials, and concluded that energetic materials are compressed, electron excitation is facilitated by a non-adiabatic process, leading to the decomposition of Ems.³³

In this manuscript, we put forward the notion that the impact sensitivity of energetic materials arises from the synergistic interplay between vibration and electron energy transfer. Our findings reveal a strong correlation between the impact sensitivity and the parameter ζ, as evidenced by a high correlation coefficient of 0.985. In the case of the experimental H₅₀ values for energetic materials, it is preferable to employ consistent equipment (such as ERL Type 12 or BAM) or to rely on measurements from the same laboratory. The accuracy of the results can be substantiated by the fact that the most reliable sources for experimental data of this nature come from the Los Alamos National Laboratory (using ERL Type 12) and the Lawrence Livermore National Laboratory (employing both ERL Type 12 and BAM procedures), as documented in peer-reviewed chapters and individual studies.^34,35 Nonetheless, as indicated in Table S1, for the compounds α-FOX-7, LLM-105, and TATNB, there are no available measurements of impact sensitivity H₅₀ using the ERL Type 12 method. Consequently, this manuscript utilizes the experimental H₅₀ values obtained through the standard BAM procedure.^36,37,38,39 Specifically, the H₅₀ value of LLM-105 compound measured with a 2.5 kg drop weight comes from the Lawrence Livermore National Laboratory, which is also accurate.³⁷ For the compounds α-FOX-7 and α-RDX, the impact sensitivity values H₅₀, measured using the BAM procedure with a 2 kg mass, are recorded as 126 cm and 38 cm, respectively. By applying the formula (E₅₀ = mgH₅₀) and considering a drop weight mass of 2.5 kg, we can deduce corresponding heights for the impact sensitivities of α-FOX-7 and α-RDX, which are 101 cm and 30 cm, respectively. The H₅₀ value for α-RDX at 2.5 kg, as in ref.³⁸ aligns closely with the findings from Los Alamos National Laboratory (H₅₀ = 23–26 cm)³⁰ and Lawrence Livermore National Laboratory (28 cm).³⁵ Using a BAM device with a 2.5 kg hammer, the impact sensitivity of TATNB is measured at 6 cm. The HMX and RDX explosives, when tested under identical conditions, exhibit impact sensitivities of 30.2 cm and 30.6 cm, respectively, as noted in ref.³⁹ These H₅₀ measurements for HMX (30.2 cm) and RDX (30.6 cm) are basically consistent with findings by the Los Alamos National Laboratory and/or Lawrence Livermore National Laboratory, which report values in the range of 26 cm–32 cm for HMX and 23-28 cm for RDX. Consequently, the H₅₀ value reported for TATNB is considered to be reliable. Based on the analysis results, it is logical to combine the H₅₀ sensitivity values of α-FOX-7, LLM-105, and TATNB with those of the other 11 compounds for study. Utilizing the drop energy formula (E₅₀ = mgH₅₀), the impact sensitivity of all 14 energetic materials can be uniformly characterized. This approach effectively reduces variances in impact sensitivity values that could result from using drop weights with different masses. The H₅₀ impact sensitivity, E₅₀ drop energy, and molecular formulas are shown in Table S1.

Computational methods

Simulations of periodic systems

The structural properties based on the density functional theory (DFT) and vibrational properties using density functional perturbation theory (DFPT) with a linear response method have been performed in the CASTEP code.⁴⁰ The generalized gradient approximation (GGA) proposed by Perdew, Burke, and Ernzerhof was employed as the exchange-correlation functional.⁴¹ The van der Waals interaction with the GGA functional was dealt with using the DFT-D2 approach of Grimme.⁴² Norm-conserving pseudopotentials were adopted to describe the core-valence interaction by the electronic configurations of H 1s¹, C 2s²2p², N 2s²2p³, and O 2s²2p⁴ states.⁴³ During the process of geometry optimization, a Quasi-Newton algorithm with the Broyden-Fletcher-Goldfarb-Shanno scheme was used.⁴⁴ Convergence thresholds were as follows: total energy, maximum force, maximum stress, and maximum displacement variation were smaller than 5.0 × 10⁻⁶ eV/atom, 0.01 eV/Å, 0.02 GPa, and 5.0 × 10⁻⁴ Å, respectively. The kinetic energy cutoff of electron wave functions employing plane-wave basis sets was 830 eV. The first Brillouin zone in the reciprocal space was sampled based on the Monkhorst-Pack grid.⁴⁵ The molecular and crystal structures of 14 energetic explosives are shown in Figure 1 and Figure S1, respectively. The experimental and optimized crystal parameters and density are displayed in Table S2. It can be observed that the calculated results agree well with the experimental values, indicating that the chosen calculation methods are valid and reliable.

Figure 1.

Molecular structures of these 14 explosives

Simulations of the gas phase

The optimized molecular structures of all the molecules were obtained based on DFT.^46,47 We have employed the hybrid functional B3LYP^48,49 in conjunction with the 6-311G basis set, which has been supplemented with both polarization (2 days, 2p) and diffuse (++) functions,^50,51 to ensure an accurate representation of the electronic structure in our computation. HOMO-LUMO orbitals and vibrational parameters were displayed at the same level as theory. No imaginary frequency is presented in the result of frequency results of 14 molecules, demonstrating that these are energy minima.

Results and discussion

Vibrational energy transfer and impact sensitivity

Low-frequency and collective vibrations in large molecules can be excited easily through a low-order anharmonic coupling process. Thermal expansion is caused by the anharmonic effect of molecular vibration, and Dlott and Fayer supposed that the cubic term of anharmonic potential plays a dominant role in lattice-dynamic expressions for thermal expansion:^52,53

$${\mathrm{V}}^{\left(3\right)}=\frac{1}{3!}\sum _{\mathrm{\psi }}\frac{{\partial }^3\mathrm{V}\left(\left\{\mathrm{\psi }\right\}\right)}{\partial {\mathrm{\psi }}_1\partial {\mathrm{\psi }}_2\partial {\mathrm{\psi }}_3}$$ (Equation 1)

Here, V({ψ}) expresses the potential energy surface for the molecular crystal with a series of normal coordinates{ψ}, and the partial derivative is computed at the equilibrium position {ψ}₀. However, it is worth noting that the specific relationship between it and thermal expansion still requires further research to clarify.

The most effective path for doorway vibration up-pumping is to absorb a pair of phonons whose frequencies are all Ω/2 (half of the doorway mode frequency Ω).^54,55 The energy transfer process from phonon to doorway mode can be given based on the following formula:^56,57

$${\mathrm{\kappa }}_{2\text{ph}\to \text{vib}}=\frac{36{\mathrm{\Pi }}^2}{ħ}{\langle {\mathrm{V}}^{\left(3\right)}\left(\mathrm{V}\right)\rangle }^2{\mathrm{\rho }}^2\left(\mathrm{\Omega }\right)\left[{\mathrm{n}}_{\text{ph}}\left(\mathrm{T},\mathrm{V}\right)-{\mathrm{n}}_{\text{vib}}\left(\mathrm{T},\mathrm{V}\right)\right]$$ (Equation 2)

Here ⟨V⁽³⁾(V)⟩, ρ²(Ω), n_ph(T,V) and n_vib(T,V)signify the matrix element of the cubic anharmonic Hamiltonian, the two-phonon density of states at the doorway vibration, the occupation number of phonons of frequency ω = Ω/2 and the occupation number of the doorway vibration ω = Ω at definite temperature T and bulk V, respectively.

n_ph(T,V) and n_vib(T,V) can be calculated from the following expression:

$$\mathrm{n}\left(\mathrm{T},\mathrm{V}\right)=\frac{1}{\exp \left(\raisebox{1ex}{$ħ\mathrm{\Omega }$}\!\left/ \!\raisebox{-1ex}{${\mathrm{k}}_{\mathrm{B}}\mathrm{T}$}\right.\right)-1}$$ (Equation 3)

For a doorway vibration with frequency Ω, at ambient temperature and above, ħΩ(V)≤k_BT,

As a consequence, n_ph(T,V)≈ $\raisebox{1ex}{$k_{B}T$}\!\left/ \!\raisebox{-1ex}{$\hslash {\Omega }_{ph}$}\right.$,n_vib(T,V)≈ $\raisebox{1ex}{$k_{B}T$}\!\left/ \!\raisebox{-1ex}{$\hslash {\Omega }_{vib}$}\right.$.

$${\mathrm{n}}_{\raisebox{1ex}{$\mathrm{\Omega }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(T\right)-{\mathrm{n}}_{\mathrm{\Omega }}\left(T\right)\approx \raisebox{1ex}{${\mathrm{k}}_{\mathrm{B}}\mathrm{T}$}\!\left/ \!\raisebox{-1ex}{$ħ\Omega $}\right.$$

For Ω = 200 cm⁻¹, T = 300 K

$${\mathrm{n}}_{\raisebox{1ex}{$\mathrm{\Omega }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(T\right)-{\mathrm{n}}_{\mathrm{\Omega }}\left(T\right)=1$$

The product of ${\langle {\mathrm{V}}^{\left(3\right)}\left(\mathrm{V}\right)\rangle }^2{\text{and}\mathrm{\rho }}^2\left(\mathrm{\Omega }\right)$ in formula 2 is related to the lifetime of doorway vibration τ₁(0) at zero temperature, and it can be written as follows:

$$\frac{1}{{\mathrm{\tau }}_{1\left(0\right)}}=\frac{36{\mathrm{\Pi }}^2}{ħ}{\langle {\mathrm{V}}^{\left(3\right)}\left(\mathrm{V}\right)\rangle }^2{\mathrm{\rho }}^2\left(\mathrm{\Omega }\right)$$ (Equation 4)

The energy transfer rateκ⁵² that represents the transfer of energy from phonon to doorway vibrations is proportional to j and τ₁₍₀₎:

$$\mathrm{\kappa }\propto \frac{\mathrm{j}}{{\mathrm{\tau }}_{1\left(0\right)}}$$ (Equation 5)

Here, j denotes the number of doorway vibrational modes. Previous research has indicated that the lifetime τ₁(0) is approximately constant for EMs with a range of 3–6 ps.⁵² Therefore, in this manuscript, we also assume that the value of τ₁(0) is constant for all explosives investigated. Under this assumption, the energy transfer rate κ is in direct proportion to the number of doorway vibrational modes j. The phonon dispersion curves for TATB, DATB, TNA, o-TNT, LLM-105, α-FOX-7, TNX, PA, TNAZ, α-RDX, PETN, ε-CL-20, HNB, and TATNB compounds are plotted in Figure 2. The results show except for TNX, α-RDX, and HNB compounds, the acoustic phonons of other compounds appear as imaginary frequencies. We have carefully compared our computational method with that used by Michalchuk et al.²³ in their studies on energetic compounds (TATB, α-FOX-7, o-TNT, and ε-CL-20), where imaginary frequencies did not appear. Michalchuk et al. increased the plane wave cutoff kinetic energy to 1800 eV in their research. By contrast, we speculate that the reason for the appearance of imaginary frequencies in our calculations might be due to a lower cutoff kinetic energy setting. Computing the phonon spectrum is a very expensive and time-consuming task, and a key factor influencing the vibrational energy transfer is the value at the top of the phonon bath. We compared the value of the top of the phonon bath in our calculations with the results obtained by Michalchuk et al. We found that for the four compounds TATB, α-FOX-7, o-TNT, and ε-CL-20, their calculated values for the top of the phonon bath are 140, 169, 212, and 217 cm⁻¹, respectively. In contrast, our calculated values for the top of the phonon bath for these four compounds are 140.2, 171.8, 212.3, and 165.8 cm⁻¹, respectively. It can be observed that, except for ε-CL-20, the values for the other three compounds are very close. For the ε-CL-20 compound, although Michalchuk et al. reported the value for the top of the phonon bath as 217 cm⁻¹, based on the phonon density of states graph provided in their article published in the Journal of Chemical Physics, we found that ε-CL-20 exhibits a phonon band gap around the frequency of 170 cm⁻¹. Therefore, we believe that although some of the compounds in this study showed imaginary frequencies in their phonon spectra, the impact on the magnitude of the value for the top of the phonon bath can be considered negligible. Since the imaginary frequencies have almost no effect on the top of the phonon bath, we believe that the number of vibrational modes within the imaginary frequencies actually all fall within the range of the phonon bath. The number of vibrational modes of the center of the Brillouin zone and crystal in these four regions, lattice A (0–0-ω_c), internal B (ω_c-2ω_c), internal C (2ω_c–1000 cm⁻¹), and internal D (1000–2000 cm⁻¹), are listed in Tables 1 and 2 based on the phonon curves. The ω_c is not strictly defined, but it can be described as the top value of the phonon bath. It is commonly recognized by an obvious gap in the phonon dispersion curves. The second frequency range, ω_c-2ω_c, is known as the doorway modes range. In order to verify the relationship between j and impact sensitivity, the plot of the number of doorway vibrational modes versus impact energy of 14 energetic compounds is illustrated in Figure 3. It is very hard to find a correlation between the j in the center of the Brillouin zone and impact energy in Figures 3A–3D. There are only a few compounds whose impact energy decreases as j increases. It can also be observed that a similar phenomenon exists in Figures 3E, 3G, and 3H. The j (the number of vibrational modes of ω_c-2ω_c in the crystal)-E₅₀ dependence displayed in Figure 3F does not show an apparent correlation of decreasing E₅₀ with increasing j, while this tendency becomes clear if the energetic materials are divided into two categories (where TATB, DATB, TNA, o-TNT, LLM-105, TNX, PA, TNAZ, and TATNB belong to a category, α-FOX-7, α-RDX, PETN, and ε-CL-20, HNB are another category). Based on the aforementioned analysis, it is evident that the number within the realm of doorway modes exerts a more pronounced influence on the impact sensitivity of energetic materials when compared to modes in other frequency ranges. Moreover, it signifies that impact sensitivity is not solely determined by vibrational energy transfer, but rather encompasses other underlying processes. (More discussion about vibrational and thermodynamical properties is shown in supporting materials along with Figures S2–S17; Tables S3–S7).

Figure 2.

Phonon dispersion curves of 14 energetic compounds

(A) TATB, (B) DATB, (C) TNA, (D) o-TNT, (E) LLM-105, (F) α-FOX-7, (G) TNX, (H) PA, (I) TNAZ, (J) α-RDX, (K) PETN, (L) ε-CL-20, (M) HNB, and (N) TATNB.

Table 1.

Number of vibrational modes in the center of the Brillouin zone

Explosives	Lattice A(0-ωc)	Internal B(ωc-2ωc)	Internal C(2ωc-1000 cm−1)	Internal D(1000-2000 cm−1)
TATB	24	2	64	42
DATB	22	6	56	38
TNA	48	20	88	68
o-TNT	120	48	144	152
LLM-105	48	28	72	64
α-FOX-7	36	12	56	48
TNX	68	32	64	96
PA	96	48	168	152
TNAZ	104	40	96	136
α-RDX	113	55	144	152
PETN	60	32	124	100
ε-CL-20	78	38	163	129
HNB	91	33	88	76
TATNB	104	48	60	76

Table 2.

Number of vibrational modes in the crystal

Explosives	Impact sensitivity(cm)	Drop Energy(J)	Lattice A 0-ωc	Internal B ωc-2ωc	Internal C 2ωc-1000 cm−1	Internal D 1000-2000 cm−1
TATB	490	120.05	504	42	1344	882
DATB	320	78.40	770	210	1960	1330
TNA	177	43.37	1392	580	2552	1972
o-TNT	160	39.20	2160	864	2592	2736
LLM-105	117	28.67	1296	756	1944	1728
α-FOX-7	126	24.70	1152	384	1792	1536
TNX	100	24.50	2040	960	1920	2880
PA	87	21.32	2208	1104	3864	3496
TNAZ	28	6.86	2912	1120	2688	3808
α-RDX	26	6.37	1356	660	1728	1824
PETN	13	3.19	1620	864	3348	2700
ε-CL-20	13	3.19	1872	912	3912	3096
HNB	12	2.94	2275	825	2200	1900
TATNB	6	1.50	2600	1200	1500	1900

Figure 3.

Plot of impact energy vs. the number of vibrational modes within different frequency ranges

(A) Plot of impact energy vs. the number of vibrational modes of 0-ω_c in the center of Brillouin zone.

(B) Plot of impact energy vs. the number of vibrational modes of ω_c-2ω_c in the center of Brillouin zone.

(C) The plot of impact energy vs. the number of vibrational modes of 2ω_c-1000 cm⁻¹ in the center of Brillouin zone.

(D) The plot of impact energy vs. the number of vibrational modes of 1000-2000 cm⁻¹ in the center of Brillouin zone.

(E) The plot of impact energy vs. the number of vibrational modes of 0-ω_c in the crystal.

(F) The plot of impact energy vs. the number of vibrational modes of ω_c-2ω_c in the crystal.

(G) The plot of impact energy vs. the number of vibrational modes of 2ω_c-1000 cm⁻¹ in the crystal.

(H) The plot of impact energy vs. the number of vibrational modes of 1000-2000 cm⁻¹ in the crystal.

Phonon density of states for the C atom and impact sensitivity

The contribution of various atomic species to the phonon density of states provides insights into their unique vibrational characteristics and their response to thermal energy. Comparing the phonon density of states across different materials can offer valuable information on their thermal transport properties and their sensitivity to external stimuli. The total and partial phonon density of states (PHDOSs) for 14 energetic materials are displayed in Figures S3–S13. The PHDOS are divided into three parts: (i) a low-frequency band from 0 to 600 cm⁻¹, (ii) a middle-frequency band from 600 to 1200 cm⁻¹, and (iii) a high-frequency band from 1200 to 3500 cm⁻¹. In the low-frequency lattice vibration region, the density of phonon states of N and O atoms accounts for the largest proportion for TATB, DATB, TNA, LLM-105, α-FOX-7, TNAZ, α-RDX, PETN, ε-CL-20, HNB, and TATNB crystals, which indicates that for these energetic materials, the stress response of the nitro group to external stimuli will be the most obvious. For o-TNT, TNX, and PA compounds within this region, the density of states is mainly contributed by C and O atoms, demonstrating that C and O atoms are more prominent to external stimuli. The molecular vibrational frequency associated with bond breaking is above the doorway mode (greater than 1000 cm⁻¹). In the middle-frequency band from 600 to 1200 cm⁻¹, it can be observed that the contribution of C atoms to the PHDOS increases, and the contribution of N and O atoms to the decreases within 800-1200 cm⁻¹, which indicates that the heat energy mainly flows to C atoms in this range. In addition, it can be found that in the vibrational frequency of 1200-2000 cm⁻¹, for energetic materials with relatively higher sensitivity (TNAZ, α-RDX, PETN, and ε-CL-20), the contribution of C atoms to the phonon density of states is relatively larger, while for energetic materials with relatively lower sensitivity (TATB, DATB, TNA, o-TNT, LLM-105, α-FOX-7, TNX, and PA), C atoms contribute relatively lower to the phonon density of states. Hence, it is suggested that the impact sensitivity of energetic materials might be partially influenced by the phonon density of states associated with carbon atoms in the molecules.

Electronic energy transfer and impact sensitivity

The frontier molecular orbitals (HOMO and LUMO) are common quantum mechanical parameters, which give insight into molecular chemical stability and reactivity. The HOMO serves as an electron donor, and the LUMO is an electron acceptor. The HOMO-LUMO energy levels for 14 energetic materials are shown in Figure 4, respectively. It is observed from Figure 4 that the band gap values of TNAZ (5.34 eV), α-RDX (6.01 eV), PETN (6.73 eV) and ε-CL-20 (5.77 eV) crystals are greater than those of TATB (4.32 eV), DATB (3.88 eV), TNA (3.83 eV), o-TNT (4.90 eV), LLM-105 (3.51 eV), α-FOX-7 (4.64 eV), TNX (4.85 eV), PA (4.26 eV), HNB (2.79 eV), and TATNB (3.84 eV). The reason is that compounds with smaller band gaps have unsaturated C=C bonds or benzene rings in their molecular structures. We also discovered that for the DATB, TNA, and o-TNT compounds, as the impact sensitivity increases, the energy level of the HOMO orbital monotonically decreases. The same phenomenon has been found in the LLM-105, α-FOX-7, TNX, and PA compounds. For TNAZ, α-RDX, PETN, and ε-CL-20 crystals, their sensitivity values are higher than those of TATB, DATB, TNA, o-TNT, LLM-105, α-FOX-7, TNX, and PA compounds, and their corresponding HOMO levels are lower than those of the eight compounds. In order to find out whether there is a correlation between impact sensitivity and HOMO energy level more intuitively, the plots of impact energy vs. ionization potential (I = -E_HOMO) of 14 energetic materials is displayed in Figure 5. It can be seen that, except for the o-TNT compound, when energetic materials are divided into two groups, the ionization potential tends to decrease as the impact energy increases. Ionization potential is defined as the amount of energy required to remove an electron from a molecule or atom. We can conclude that the easier it is for electrons to delocalize, the easier it is to transfer energy to mitigate the formation of hotspots, ultimately diminishing the compound’s sensitivity.

Figure 4.

HOMO-LUMO plots of 14 energetic materials

Figure 5.

Plot of impact energy E₅₀ vs. ionization potential I of 14 energetic materials

An impact sensitivity mechanism of energetic materials

According to the analysis above, it is not completely accurate to rely on a single parameter to predict or characterize the impact sensitivity of energetic materials. Vibrational energy transfer, the phonon density of states of carbon atoms, and electron energy transfer may affect the impact sensitivity of EMs. In fact, the sensitivity mechanism for EMs is very complicated. Based on the theory of multi-phonon up-pumping, the energy converted from mechanical energy to heat energy in energetic materials under the action of external impact or shock is first transferred to phonon lattice modes, and these excited phonon modes propagate the energy to the higher frequency molecule vibration mode through the lowest frequency molecular vibration mode. In this process, we propose a sensitivity mechanism: if the heat energy from phonon to doorway vibrations transfers faster, the less time it takes for the energy to transfer to the target mode, and the impact sensitivity of the compound will be higher. When the energy transfers to the specific mode that causes the chemical bond in the molecule to break, the temperature of the whole system rises. The electrons in the chemical bond that broke first in the molecule become delocalized, transferring energy to other molecules to avoid hot spot formation. The easier the hot spot is formed, the higher the impact sensitivity of the energetic material is. In order to verify the correctness of this sensitivity mechanism, a parameter ζ is used to represent the impact sensitivity of energetic materials, which includes the contribution of vibrational energy transfer, carbon content, and electron energy transfer. It should be noted that in the previous work, we found that the stronger the delocalization of electrons is, the faster the energy transfer of electrons is. The energy gap E_g can approximately represent the delocalization of electrons. Therefore, this parameter ζ can be expressed as the following formula:

$$\mathrm{\zeta }=\frac{{\mathrm{j}}^{\mathrm{x}}\ast {\mathrm{a}}^{\mathrm{y}}\ast {{\mathrm{E}}_{\mathrm{g}}}^{\mathrm{z}}}{{\mathrm{n}}^{\mathrm{p}}{\mathrm{b}}^{\mathrm{q}}}$$

here, the j, a, b, n, and E_g signify the number of vibrational modes in the crystal, the number of vibration modes on an Acoustic or optical branch, the mass percentage of carbon atoms in the formula, the number of molecules in a unit cell, and the electronic band gap, respectively. A lower parameter ζ shows a higher impact sensitivity. After fitting, as shown in Figure 6, it can be found that the following relationship between the impact energy and the parameter ζ

$${\mathrm{E}}_{50}=134.9-101\ast \left(1-{\mathrm{e}}^{-\mathrm{\zeta }}\right)-31.7\ast \left(1-{\mathrm{e}}^{-3\mathrm{\zeta }}\right)$$ (Equation 6)

$$\mathrm{\zeta }=\frac{{\mathrm{j}}^{0.8}\ast {\mathrm{a}}^{0.7}\ast {{\mathrm{E}}_{\mathrm{g}}}^{1.2}}{{\mathrm{n}}^{1.2}{\mathrm{b}}^{1.7}}$$

Figure 6.

Plots of impact energy vs. the parameter ζ for 14 energetic compounds

There is an obvious correlation between the ζ vs. E₅₀ values, and the correlation coefficient R² is 0.985, which indicates that the sensitivity mechanism is feasible. The values of j, a, b, n, and E_g are listed in Table 3, respectively. Moreover, we also evaluated the impact sensitivity of NTO, DINGU, and HMX using the current model, as detailed in the Supplementary materials in Figures S18 and S19.

Table 3.

Values of the number of vibrational modes in the crystal (j), the number of vibrational modes on an acoustic or optical branch (a), the mass percentage of carbon atoms in the formula (b), the number of molecules in a unit cell (n), and the electronic band gap (E_g)

Explosives	Drop Energy(J)	j	a	b	n	Eg
TATB	120.05	2	21	27.9	2	2.32
DATB	78.40	6	35	29.9	2	2.07
TNA	43.37	20	29	31.6	4	1.90
o-TNT	39.20	48	18	37.0	8	2.52
LLM-105	28.67	28	27	22.2	4	1.24
α-FOX-7	24.70	12	32	16.2	4	2.11
TNX	24.50	32	30	39.8	4	2.46
PA	21.32	44	23	31.5	8	2.08
TNAZ	6.86	40	28	18.8	8	2.55
α-RDX	6.37	44	12	16.2	8	3.35
PETN	3.19	31	27	19.0	4	3.91
ε-CL-20	3.19	38	24	16.4	4	3.42
HNB	2.94	33	25	20.7	4	2.60
TATNB	1.50	48	25	21.4	4	1.95

Conclusions

In this study, the relationship between the vibrational modes in these four regions, Lattice A (0–0-ω_c), internal B (ω_c-2ω_c), internal C (2ω_c–1000 cm⁻¹), and internal D (1000–2000 cm⁻¹), and the impact sensitivities of 14 EMs is investigated. It was shown that the tendency between vibrational modes in internal B and the impact energy becomes clear if the energetic materials are divided into two categories. In addition, it can be found that at the vibrational frequency of 1200-2000 cm⁻¹, with relatively higher sensitivity for EMs, the contribution of C atoms to the phonon density of states is relatively larger, while for energetic materials with relatively lower sensitivity, C atoms contribute relatively lower to the phonon density of states. The ionization potential tends to decrease as the impact energy increases. Based on the definition of the ionization potential, it can be inferred that, to some extent, a compound becomes less sensitive when electrons can more readily escape from its molecular structure. These findings have led to the proposal of a sensitivity mechanism. Although a significant correlation was observed between the parameter ζ and the experimental impact energy of 14 energetic materials, further research is needed to extend the applicability of this sensitivity mechanism to a broader range of energetic materials.

Limitations of the study

Theoretically, the lifetime is not constant and is affected by various factors such as lattice structure, temperature, and phonon-phonon interactions. In this article, in order to greatly simplify the calculation process, we assume that the lifetime is a constant for all energetic materials, which will cause a minor systematic error in the result.

Resource availability

Lead contact

Further information and resource requests should be directed to the lead contact, Wei Zeng (E-mail: zengwei_chemistry@126.com).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •The lead contact will share all data reported in this article upon request.
• •This article does not report the original code.
• •Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request.

Acknowledgments

This work has been supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 2682024GF019, 2682024ZTPY054, and 2682025ZTPY013).

Author contributions

W.-H.L.: writing-original draft, formal analysis, investigation, and visualization; W.Z.: writing-review and editing and supervision; F.-S.L.: validation, methodology, and data curation; Z.-T.L.: software and resource; Q.-J.L.: writing-review and editing, conceptualization, validation, project administration, and funding acquisition.

Declaration of interests

The authors declare no competing interests.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
CASTEP	Clark et al.31	https://www.castep.org/

Method details

Simulations of periodic systems

The calculations were conducted using the plane wave method implemented in the CASTEP code. To simulate the electron-ion interactions, we employed optimized norm-conserving Vanderbilt pseudopotentials (ONCVPSP). The exchange and correlation potential were described using the generalized gradient approximation (GGA-PBE). The van der Waals interaction with the GGA functional was dealt with using the DFT-D2 approach of Grimm e. During the process of geometry optimization, a Quasi-Newton algorithm with the Broyden-Fletcher-Goldfarb-Shanno scheme was used. Convergence thresholds were as follows: total energy, maximum force, maximum stress and maximum displacement variation were smaller than 5.0×10^-6eV/atom, 0.01eV/Å, 0.02GPa and 5.0×10^-4Å, respectively. The kinetic energy cutoff of electron wave functions employing plane-wave basis sets was 830 eV. The first Brillouin zone in the reciprocal space was sampled based on the Monkhorst-Pack grid.

Simulations of gas phase

The optimized molecular structures of all the molecules were obtained based on the DFT . We have employed the hybrid functional B3LYP in conjunction with the 6-311G basis set, which has been supplemented with both polarization (2d, 2p) and diffuse (++) functions, to ensure an accurate representation of the electronic structure. HOMO-LUMO orbitals and vibrational parameters were displayed at the same level of theory.

Quantification and statistical analysis

There are no quantification or statistical analyses to include in this study.

Published: October 24, 2025