An Integrated Experimental and Modeling Approach for Assessing High-Temperature Decomposition Kinetics of Explosives

Abstract

We present a new integrated experimental and modeling effort that assesses the intrinsic sensitivity of energetic materials based on their reaction rates. The High Explosive Initiation Time (HEIT) experiment has been developed to provide a rapid assessment of the high-temperature reaction kinetics for the chemical decomposition of explosive materials. This effort is supported theoretically by quantum molecular dynamics (QMD) simulations that depict how different explosives can have vastly different adiabatic induction times at the same temperature. In this work, the ranking of explosive initiation properties between the HEIT experiment and QMD simulations is identical for six different energetic materials, even though they contain a variety of functional groups. We have also determined that the Arrhenius kinetics obtained by QMD simulations for homogeneous explosions connect remarkably well with those obtained from much longer duration one-dimensional time-to-explosion (ODTX) measurements. Kinetic Monte Carlo simulations have been developed to model the coupled heat transport and chemistry of the HEIT experiment to explicitly connect the experimental results with the Arrhenius rates for homogeneous explosions. These results confirm that ignition in the HEIT experiment is heterogeneous, where reactions start at the needle wall and propagate inward at a rate controlled by the thermal diffusivity and energy release. Overall, this work provides the first cohesive experimental and first-principles modeling effort to assess reaction kinetics of explosive chemical decomposition in the subshock regime and will be useful in predictive models needed for safety assessments.

Introduction

For over 80 years, there has been extensive debate on which properties of explosives control their initiation behavior in safety scenarios.^1,2 These questions are fundamental to developing safe energetic materials in a wide range of fields, including space exploration, mining, and the U.S. stockpile. However, it is complicated by the fact that the behavior of explosives, and the necessary experimental diagnostics, span the fields of fundamental chemistry, physics, materials science, and mechanical and electrical engineering. Under shock initiation, for example, microstructural properties, phase, and density have been shown to influence how heat is generated in the vicinity of the shock front. Hence, mesoscale structure plays a vitally important role in the so-called Pop-plot, which connects the input shock pressure to the run distance needed for a self-sustaining detonation to be established.³ But central to all chemical reactions–specifically the self-propagation of exothermic decomposition and gas formation required for explosive initiation–is the breaking of covalent bonds and release of energy.

Independent of their mesoscale structure, some explosives are intrinsically more sensitive than others, and hence more hazardous to handle and transport. This aspect is critical to understand at the earliest stages of synthesis, though very few methods exist to quantitatively measure safety properties. Specifically, in the subshock initiation regime relevant to most safety scenarios, we are interested in the molecular properties of explosives that dictate their reactivity.

Here we present three significant results that transform how we understand explosive sensitivity: (1) a new microsecond heating experiment allows for the rapid ranking of explosive safety properties by quantitative reaction kinetics information, (2) reaction rates obtained from quantum molecular dynamics (QMD) simulations at very short time scales match the ranking observed in experiment and extrapolate well to long duration (minutes to hours) homogeneous ignition experiments from the literature, and (3) quantitative kinetic Monte Carlo (kMC) simulations capture the transition from homogeneous explosion at low temperatures to the heterogeneous explosions observed in our experiments, based only on the thermal diffusivity and QMD-informed reaction kinetics. Taken together, these findings provide a unified framework for explosive initiation under the high-temperature, low-pressure conditions that are of crucial importance to most accident scenarios. In this manuscript, we define homogeneous reactions and explosions as those which occur after full temperature equilibrium has occurred throughout the sample. Heterogeneous reactions occur before temperature equilibration, due to rapid heating at the edges of a system before the center material can equilibrate.

In the field of energetics development and design, the drop-weight impact test is the only standardized experiment that has been available to determine handling sensitivity properties at the early design stage. This technique, which was developed originally in the early 1900s, derives the height, H₅₀, from which a 2.5 kg mass dropped onto a small 40 mg sample gives rise to a violent reaction in 50% of trials. Although a large amount of research has been devoted to understanding how drop-weight impact results correlate with molecular properties, the test is often based on acoustic threshold diagnostics, and does not provide any physical or chemical understanding of initiation phenomena.⁴ Other tests that have investigated decomposition or time-to-explosion kinetics are either too slow (differential scanning calorimetry), are in unconfined geometries (e.g., Brill’s T-jump experiment)⁵ that may not replicate real safety scenarios, or require long reaction times and larger amounts of pressed material (such as the one-dimensional time-to-explosion test, ODTX), which are not available at the initial design stage.^6,7 In the early 1960s, Wenograd developed a method to measure the time-to-explosion kinetics of explosive samples confined in hypodermic needles that were resistively heated on microsecond time scales.⁸ This particular test created conditions that were relevant in size, confinement, and time scale to the formation of subshock ignition sites, but with an accurately known temperature input. The dependence of the time to explosion on the input temperature, τ(T), was found to closely follow Arrhenius kinetics, 1/τ = Aexp(−E_a/RT), where A is the pre-exponential factor, E_a the effective activation enthalpy, and R the gas constant. More importantly, Wenograd showed that the critical temperatures, T_c, derived from the Arrhenius rates that are needed for a thermal explosion to occur on the 250 μs time scale of the drop weight test were proportional to log(H₅₀). This result showed clearly that handling sensitivity is connected intimately with reaction rates, and that ignition in the subshock regime is primarily thermal in nature.

Wenograd’s results did not get broad attention, and the test was never replicated or updated to incorporate modern energetics or theoretical capabilities. Moreover, in the original test every explosive sample had to be melted to load into the needles, which eliminated many energetics of interest due to safety considerations, and altered particle morphology characteristics. More recently, QMD simulations by our team provided the first theoretical confirmation of Wenograd’s early results, which highlighted the importance of thermally driven reactivity–specifically the weakest bonds, or trigger linkages–under a range of high temperatures and pressures.^9,10 These mechanisms of explosive decomposition have been discussed in numerous previous studies by our group and others.¹¹

In this work, experimental time-to-explosion kinetics data have been collected from the recently developed High Explosive Initiation Time (HEIT) experiment, which is uniquely designed for the solid loading of small explosive samples into hollow stainless steel needles (Figure 1). A high-voltage pulse results in resistive heating of the needle on the order of 20 μs, and the resulting temperature of the needle wall is determined from the calibrated resistance of the needle. The fireset, voltage pulse characteristics, and calibration data are described in a separate report.¹² Due to our development of novel loading techniques, optical probes, access to high-speed imaging, and modeling capabilities, we have explicitly designed this experiment to look at a broad spectrum of energetics loaded in the solid state. Because our ultimate goal is to understand how molecular structure influences reactivity and sensitivity, complementary QMD simulations have been performed on all of the tested explosives, along with coupled heat transport and reaction simulations of the onset and propagation of reactions, which take into account the heterogeneity of the rapid surface heating in this specific experimental geometry.

Figure 1.

Schematic of the High Explosive Initiation Time (HEIT) experiment (top), where a high voltage pulse generates rapid resistive heating in a needle filled with energetic material. Optical probes and rupture of the needle are used to indicate time-to-explosion, and needle resistance calibrations allow for the determination of temperature. QMD calculations (bottom) determine time-to-explosion vs temperature on shorter time scales, resulting in similar trends as found experimentally, as depicted in an Arrhenius plot.

Six explosives, erythritol tetranitrate (ETN), pentaerythritol tetranitrate (PETN), hexanitrohexaazaisowurtzitane (CL-20), trimethylenetrinitramine (RDX), 1,3,5,7-tetranitro-1,3,5,7-tetrazocane (HMX), and 2,4,6-trinitrotoluene (TNT), have been evaluated for the first time with the HEIT experiment. These materials were selected from three distinct chemical families–nitrate esters, nitramines, and nitroaromatics–that span a broad range of performance and sensitivity. They have been tested at temperatures between ∼800 and 1400 K in order to determine time-to-explosion kinetics. With new solid-loading techniques and high-speed imaging, we find that their high-temperature thermal stabilities track closely with their handling sensitivities as measured by the drop-weight impact test. Condensed-phase QMD simulations of the high temperature reaction rates predict the same ordering of thermal stabilities, including the anomalously low activation energy displayed by CL-20, which strongly suggests that the distribution of times-to-explosion in the HEIT experiment is dominated by the incubation time for the onset and propagation of reactions.

The Arrhenius kinetics for homogeneous explosion obtained from the QMD simulations correspond remarkably well with literature values for the slower, homogeneously heated ODTX experiment. These results could have important implications for the longstanding kinetics studies of explosive decomposition in the energetics field, including predictive capabilities that often require extrapolations over many orders of magnitude in time.¹³⁻¹⁷ The rapid heating inherent to the HEIT experiments results in samples that cannot thermally equilibrate on the short experimental time scales. The results have therefore been further evaluated by kMC simulations of heat transport and the onset of deflagration, with excellent agreement. These calculations reveal that at the high temperatures of the HEIT experiment, the time scales for thermal diffusion can have a significant effect on the times-to-explosion, which is manifested in the results by smaller than expected activation enthalpies. By quantifying the intrinsic propensity of energetic molecules to undergo thermal runaway, the results presented here have the potential to inform models and transform predictions on condensed-phase chemical decomposition for newly designed energetic molecules, and direct the future of safety measurements in the field of energetic chemistry.

Results

High Explosive Initiation Time (HEIT) Experiment

High explosive samples were loaded into stainless steel (SS-304), 20 gauge needles purchased from Hamilton (OD 0.9081 mm and ID 0.603 mm), and used as received. A loading fixture (Figure 2a) was utilized to pack the powders consistently and safely, resulting in nominal densities of 1.6 g cm^–3, which varied somewhat between explosives (Table 1). Two optical probes were inserted into each end of the needle, until contacting the explosive solid on each side, and were then bonded to seal each end. The filled needles were loaded into a custom fixture (Figure 2b) which allowed for crimping connections until the resistance (as measured by a milliohmmeter) was minimized. The optical probes were fed through a fixture so light emission from the samples would be visible in the high-speed imaging. A 10 kV, 5 μF capacitor was discharged through the needle over approximately 20 μs, resulting in resistive heating of the needle. The needle temperatures were adjusted by selecting a voltage input in the range 4.5 to 6.5 kV. Voltages above 6.5 kV were avoided because they resulted in temperatures >1300 K, which led to the immediate softening of the needle. Detailed descriptions of the electrical circuit, as well as the heating of the needle and explosive making contact with the walls have been provided in ref (12).

Figure 2.

(a) Powder loading apparatus, (b) firing fixture, (c) representative needles, postfiring, showing the blue optical probes glued inside each end, (d–f) representative high-speed imaging of ETN during ignition.

Table 1. Average Needle Packing Density and Percent Theoretical Maximum Density (TMD) for Each Explosive, along with Needle Burst Times Compared with Literature Drop-Weight Impact Test Values.

explosive	average density (g cm–3)a	TMD (g cm–3)b	% TMD	average needle burst time at 5500 V (ms)c	drop-weight impact value (cm)d
ETN	1.7	1.773	96	2.2	5
CL-20	1.8	2.044	87	1.4	11
PETN	1.6	1.778	91	7.3	12
HMX	1.7	1.902	87	12	32
RDX	1.5	1.806	83	13	19
TNT	1.5	1.653	94	40	220

^aAll errors are ±0.2 g cm^–3, due to the estimated error in the explosive fill length (1 ± 0.1 cm).

^bTMD, theoretical maximum density; taken from refs (18−20).

^cAverage burst time at input of 5500 V, which is ∼1100 ± 50 K.

^dTaken from ref (10), drop-weight values (H₅₀) are lower for more sensitive explosives.

The temperature of the needle is calculated from the measured current, I(t), versus time, t, in the SS-304 needle, combined with the known resistance, R(T), versus temperature, T, calibration of the needle and its thermal mass. We verify the calculated temperature by measuring the voltage across the needle and comparing it with the calculations. From the resistance calibration, the electrical power deposited in the needle, I²R(T), is calculated as a function of time. As the heat capacity of SS-304 is known, the rate of temperature rise of the needle can also be calculated, so that both the temperature and needle resistance can be determined as functions of time. From the measured current and the calculated resistance we can also calculate the voltage across the needle, which can be compared from direct measurement through the voltage probes on the fixture.

After the needle reaches the target temperature, the outermost layer of explosive is heated by conduction, decomposes, and then initiates a subsonic deflagration front that consumes the remaining material. The deflagration results in light emission (detected by optical probes) and enough heat and gas production to burst the needle (Figure 2c,e). Times-to-explosion were evaluated through high-speed imaging with an interframe time of 4 μs (Figure 2d–f). At the onset of current flow, electrical arcing is visible as an intense bright flash, resulting in some amount of spark formation, even though contact resistances have been minimized in our setup. To avoid preliminary ignitions during these arcing events, our loading techniques were developed to insert ∼5 mg into the center of each needle, far from the junctions in the fixture.

Comparison between HEIT and Drop-Weight Impact Test

Reaction times varied for each explosive in the HEIT experiment. At an input voltage of 5500 V the temperature of the needle was generally ∼1100 ± 50 K, resulting in approximate breakout times of 2.2 ms (ETN), 1.4 ms (CL-20), 7.3 ms (PETN), 12 ms (HMX), 13 ms (RDX), and 40 ms (TNT) (Supporting Information). Specifically, the sensitive nitrate esters ETN, PETN, and the high-performance nitramine CL-20 reacted fastest, followed by the nitramine-based conventional high explosives RDX and HMX, with the relatively insensitive trinitrobenzene derivative TNT being the most stable. Table 1 has needle burst times compared with literature drop-weight impact test data, which correlates reasonably well: for example, longer burst times in the HEIT test for TNT correspond to the higher drop heights necessary for ignition in the drop test. In general, the optical probe lights were observed before breakout of the needle. For most of the explosives tested in this series, the probe lights occurred at approximately 4–15 μs before the needle breakout occurred (generally by the next frame; see Supporting Information). However, with TNT, probe lights were observed at an average of 300 μs before breakout, consistent with slower reactions and more time required for effective reaction and light emission.

The needle burst times, τ, were plotted as ln(1/τ) versus the inverse temperature that was derived from the measured needle resistance. From this plot (Figure 3a), activation barriers can be determined from the slopes and intercepts of the fitted lines using the Arrhenius equation ln(1/τ) = ln (A) – E_a/RT, where the prefactor A includes contributions from the activation entropy, collision frequency, and other factors, and the slope E_a/R includes the activation enthalpy (Supporting Information).

Figure 3.

(a) Arrhenius kinetics in HEIT experiment (plotted in s^–1; with reactions generally on millisecond time scales), and (b) Arrhenius kinetics obtained from QMD. The error bars capture the distribution of times to explosion from the 16 independent trajectories run at each temperature for each material.

Quantum Molecular Dynamics Modeling

Quantum molecular dynamics simulations of the dependence of the time-to-explosion on temperature were performed using self-consistent charge transfer density functional tight binding (DFTB) theory²¹ with the lanl31 parametrization²² for systems containing C, H, N, and O. DFTB is a fast, semiempirical electronic structure method that provides an accurate and transferable description of the interatomic forces in organic molecules and materials at a small fraction of the computational cost of ab initio methods like density functional theory (DFT). The interatomic forces are computed on-the-fly from the self-consistent ground state electronic structure, which allows chemical reactions to occur in response to the imposed temperature and pressure.

Small systems undergoing thermally activated reactions exhibit a distribution of times-to-explosion that broadens exponentially as the rates decrease.^23,24 Hence, we obtained an average rate by running 16 independent trajectories at each target temperature by assigning different seeds for the random number generator that sets the initial atomic velocity distributions. The reactive QMD trajectories evolved through 4 stages: (i) a nonreactive incubation period prior to the rupture of the first covalent bonds, (ii) a short period of endothermic or thermal neutral chemistry where reactions continue to propagate slowly, (iii) thermal runaway where exothermic reactions accelerate precipitously, and (iv) rapid, thermal neutral reactions in the high temperature product species. The time-to-explosion, τ, was defined as the simulation time when the kinetic temperature was midway between the initial temperature of the reactants and that of the hot products (Figure 1 and ref (25)).

The rates, κ = 1/τ, derived from the distributions of the times-to-explosion are presented in an Arrhenius plot in Figure 3b. These results are consistent with our earlier theoretical studies, as well as those of Wenograd, since the set of reaction rates reflect clearly the relative sensitivities of the six explosives. In particular, we observe that the sensitive energetic materials ETN, CL-20, and PETN, with H₅₀ values of 5, 11, and 12 cm, respectively, will react on the time scales of the QMD simulations at much lower temperatures than the less sensitive materials HMX and TNT, which have H₅₀ = 32 and 220 cm, respectively. Crucially, the ordering with respect to temperature of a set of reaction rates obtained from the QMD simulations (Figure 3b) matches that obtained from the HEIT experiments (Figure 3a). Despite the disparate length and time scales, the QMD calculations capture the relatively small activation energy of CL-20 with respect to the other explosives and that, based on reaction rates, RDX appears to be less sensitive than HMX, which is at odds with their drop weight impact sensitivities (and could result from the fact that that the drop-weight test convolutes thermal initiation with mechanical properties²⁶). Although there is close agreement between the qualitative trends in the reaction rates obtained from the HEIT test and QMD simulations, (i) the time scales for reaction differ by 7–8 orders of magnitude, and (ii) the activation energies taken from the HEIT experiment appear significantly smaller than those obtained from the QMD simulations. The origins of these differences are explored in the next section.

Effects of Homogeneous versus Heterogeneous Ignition on Time-to-Explosion

The QMD simulations correspond to a homogeneous thermal explosion because the periodic simulation cell is thermalized to a uniform high temperature prior to the onset of reactions. In this regime, the time-to-explosion is described by the Frank-Kamenetskii model for an adiabatic thermal explosion with self-heating,

1

where κ^FKis the inverse of the adiabatic induction time, ΔT = Q/C_V is the total change in temperature during the thermal explosion, Q is the portion of the enthalpy of explosion that increases the thermal energy, and C_V the heat capacity at constant volume.²⁷ Despite the large discrepancies in sample sizes and the times-to-explosion, the ODTX experiment provides a useful comparison for the results of the QMD simulations due to the homogeneous heating environment inherent to this slower test. The ODTX apparatus measures the time-to-explosion of a relatively large, 1.27 cm diameter spherical sample placed within preheated aluminum anvils, where low wall temperatures (400–600 K) lead to incubation times of up to tens of hours. The time scale for a sample of dimension L to reach thermal equilibrium by diffusion is t_eq ≈ L²/α, where α is the thermal diffusivity. For the ODTX experiment, L = 0.635 cm and using a typical value for the low temperature diffusivity of an energetic material of α = 1.66 × 10^–3 cm² s^–1, we obtain t_eq = 243 s. Hence, for induction times, τ, in the ODTX experiment longer than 4–5 min, we can assume that the sample undergoes a homogeneous explosion that is controlled primarily by the chemical kinetics via eq 1 rather than the deflagration front speed.

An Arrhenius plot that combines homogeneous explosion data from high temperature QMD simulations and low temperature ODTX experiments²⁸ (τ > t_eq) for PETN, RDX, HMX, CL-20, and TNT is presented in Figure 4. Remarkably, despite the disparate time scales that span from picoseconds through hours, the rates for all five explosives can each be represented with good accuracy by a single Arrhenius function over the entire temperature range (see Supporting Information). This result implies, albeit with the potential exception of CL-20, that the underlying decomposition chemistry of the molecules during homogeneous thermal explosion does not differ significantly from the low temperature ODTX regime through the high temperatures accessed by the QMD simulations. Instead, the same reactions simply occur more rapidly as the temperature is increased, which can be modeled accurately by eq 1. The pre-exponential factors, A, and activation enthalpies, E_a, obtained from least-squares fits of the Frank-Kamenetskii equation, eq 1, to these data on the Arrhenius plot are presented in Table 2. The ΔT used in the fits were obtained directly from the QMD trajectories of thermal explosion. These are likely to underestimate the experimental temperature changes slightly owing to the classical heat capacity, C_V, inherent to the QMD simulations. Nevertheless, previous studies have demonstrated the accuracy to which the lanl31 DFTB parametrization predicts the heat of formation of the reactants and products that are used to compute the enthalpy of explosion, Q.²⁹

Figure 4.

(a) Arrhenius plot of the reaction rate, κ = 1/τ, for the transition to a thermal explosion from QMD and the ODTX experiments for CL-20, PETN, RDX, HMX, and TNT. The HEIT experimental data are plotted alongside the QMD and ODTX data. (b) The broken lines denote the best-fit of the Frank-Kamenetskii adiabatic induction time, eq 1, to the rates obtained from the QMD simulations and low temperature ODTX experiments that correspond to homogeneous thermal explosions (Supporting Information). (c) HEIT experimental data plotted with ODTX data, revealing a more dramatic flattening of the slope as temperatures increase.

Table 2. Best-Fit Pre-Exponential Factors, A, Activation Enthalpies, E_a, and Temperature Differences, ΔT, for Frank-Kamenetskii Homogenous Thermal Explosion Model.

explosive	A (s–1)	Ea (kJ mol–1)	ΔT (K)
CL-20	9.72 × 1014	180.4	3000
PETN	1.61 × 1014	160.2	3100
HMX	6.10 × 1014	189.1	2600
RDX	1.10 × 1014	170.8	2400
TNT	1.13 × 1014	183.3	1500

Figure 4 shows clearly that the reaction rates, κ, derived from the HEIT experiment are significantly slower than those that would be expected for a homogeneous thermal explosion. The thermal equilibration time for the HEIT experiment, where L = 0.3 mm, is about t_eq = 0.5 s, which is significantly longer (×100) than the times-to-explosion obtained to date (∼5 ms). Hence, ignition in the HEIT experiment is expected to occur primarily at the needle wall, after which the reaction progress is limited by the deflagration velocity through the thermal diffusivity.

Kinetic Monte Carlo (kMC) Simulations of Coupled Heat Transfer and Reaction

We have developed a numerical model to quantify the transition between the homogeneous thermal explosion regime at low temperatures to a heterogeneous explosion regime, where the wall temperatures are sufficiently high that ignition occurs on time scales significantly less than t_eq. The numerical model uses the kinetic Monte Carlo (kMC) method²³ to solve the coupled heat transport and chemistry inherent to the HEIT experiment. First, the sample is discretized on a regular 1D grid. Thermal transport is modeled using the thermal bit transfer (TBT) method of Castonguay and Wang,³⁰ where a rate of heat transfer between neighboring cells i and j is given by,

2

where h is the grid spacing, δT the magnitude of the temperature update, and T_i and T_j are the temperatures of cells i and j, respectively. The temperatures of the two cells are updated such that heat only flows from hot to cold and we implement a temperature-dependent thermal diffusivity, α(T̅), where T̅ = (T_i + T_j)/2. Hanson and Hanson-Parr proposed that the thermal diffusivity of porous, polycrystalline samples, like those used in the HEIT experiment, is related to the thermal diffusivity of a single crystal through

3

where p = 1 – ρ/ρ₀ is the porosity, ρ is the mass density, and the subscripts denote those properties for a single crystal.³¹ Hence, the temperature-dependent thermal diffusivity of a polycrystalline sample can be written

4

where λ₀ is the thermal conductivity of a single crystal. This expression for the thermal diffusivity has the advantage that (i) the factor (1 – p) can be estimated during needle filling (Table 1), and (ii) the remaining terms can be computed routinely from atomistic simulations and complete equations of state, which we illustrate here for PETN.

Perriot computed the temperature dependence of the thermal conductivity of single crystal PETN using reverse nonequilibrium molecular dynamics, which can be described accurately using λ₀(T) = λ₀ + a₁/T, where λ₀ = 0.085 W m^–1 K^–1 and a₁ = 68.442 W m^–1.³² The temperature-dependent heat capacity, C_V(T), of PETN, was parametrized to the heat capacity computed from the set of vibrational normal-mode frequencies presented in ref (33) using the analytic expression developed by Lozano et al.,³⁴

5

where θ = T/T^′, T^′ is a constant that reflects the temperature scale, b is an adjustable parameter, and C^∞_V the high temperature, Dulong-Petit limit of the heat capacity. The best fit parameters for single crystal PETN are C^∞_V = 2.29272 J g^–1 K^–1, T^′ = 253.36 K, and b = 1.34503. Finally, the single crystal density at temperature T was estimated using the volumetric thermal expansion coefficient, α_V, estimated by Perriot by molecular dynamics simulations,

6

where ρ₀ = 1.884 g cm^–3, α_V = 20 × 10^–5 K^–1, with the temperature measured on the Kelvin scale.³² The reaction rate for the thermal explosion of cell i, κ_i, was set equal to the Frank-Kamenetskii rate, eq 1, with the parameters given in Table 2. Upon being selected for reaction by the kMC algorithm, the temperature of the cell, T_i, is increased by the ΔT given in Table 2. Each cell can “react” only once during the simulation, which accounts for the depletion factor as the system transforms from reactants to products.

The kMC simulations were performed in 1D with a distance between constant temperature walls of 0.6 mm, a grid spacing, h = 1 μm, which is on the order of the particle size, and a temperature update for the TBT scheme (eq 2) of δT = 0.1 K. The initial boundary conditions for the temperature distribution were set to mimic the HEIT experiment, that is, the material in all cells between the boundaries was set to a temperature of 300 K and the wall temperatures, T_w, which were held fixed throughout the simulation, were set to an elevated temperature, which for PETN, is 450 ≤ T_w ≤ 1500 K. In kMC, the total simulation time is updated in increments, Δt, that are computed from the total rate,

7

using the standard result,

8

where U₁ is a random number in (0,1]. A second random number, U₂, is drawn to select which event occurs–a thermal bit transfer or reaction–based on the individual rates for all possible events.²³

In Figure 5 we present the reaction rate, 1/τ, computed using the kMC method for PETN as a function of wall temperature in the HEIT experiment. At low temperature, where τ > t_eq, the rate asymptotes to the Frank-Kamenetskii result for a homogeneous thermal explosion. At higher temperature, the thermal runaway is heterogeneous with the reaction starting promptly at the needle wall. In the heterogeneous regime (τ < t_eq), the consumption of the remaining PETN is controlled by thermal diffusivity and energy release due to reaction at the deflagration front, which has a relatively weak dependence on temperature. Not only does our model, which is based only on the rates for homogeneous explosion (Table 2) and thermal transport, capture the general trends at high and low temperature, but it is also able to predict with good accuracy the asymptote at high temperature with respect to experiment.

Figure 5.

(a) Time-to-explosion data from HEIT, alongside the larger-scale, slower ODTX test (an example is shown with PETN and TNT), taken from refs (6,7). (b) Arrhenius plot of the time-to-explosion rates for PETN in the HEIT test calculated using kMC versus experiment and homogeneous thermal explosion.

The kMC algorithm described here is remarkably simple to implement and requires only that we can define rates for the events of interest. Furthermore, the stochasticity inherent to the kMC method accurately captures the distribution of times-to-explosion seen in our QMD simulations. To the best of our knowledge, this is the first example of kMC being applied to explore coupled heat conduction and chemistry in energetic materials.

Discussion

The ability to reliably predict the handling sensitivity of explosives has been a long-term goal in the energetics community. A variety of numerical approaches have been developed for predicting drop weight impact sensitivities, H₅₀, based on the optimization of complex functions of large sets of descriptors of the molecules to experimental data.³⁵ While powerful, these approaches tend to provide little physical insight and therefore cannot guide the design of new materials. The most reliable physically motivated models are based on the reaction kinetics of the molecules, which is consistent with the previous results of Wenograd. For example, models based on the idea of a trigger linkage–the weakest bond in the reactant–capture the observed trends in sensitivity across different classes of explosives surprisingly well.^10,36 Mathieu and Alaime developed a semiempirical rate, based on the bond dissociation energies for the first reactions and the energy release, that accounted for the sensitivity trends of a large set of nitro-based explosives.³⁷ In addition, it is well-known that high performance explosives tend to be relatively sensitive and strong correlations between the enthalpy of explosion, Q, or the oxygen balance of the molecules with H₅₀ have been observed. The performance of an energetic material can influence the reaction rate by accelerating exothermic runaway following the first reactions and/or by reducing the effective activation enthalpy of the reactions via the Bell-Evans–Polanyi principle.^1,10,38 Correlations between handling sensitivity and a variety of electronic, structural, or vibrational properties of energetic materials have been investigated³⁹ but any trends tend to extend only over subsets of molecules and the resulting models lack the transferability of those based on high temperature reaction rates.

Here we have investigated a new method for establishing handling sensitivity data, using more quantitative scientific information than has been available through existing tests implemented by the energetics community. HEIT needle burst times for the six explosives evaluated in this study are on millisecond time scales, ranking as ETN ∼ CL-20 < PETN < HMX ∼ RDX < TNT. This trend in reactivity corresponds closely with drop-weight impact test results, which have been utilized to make preliminary determinations on explosives safety for more than 80 years. These results provide further support for the importance of high temperature reaction rates in understanding and predicting the handling sensitivity of energetic materials. However, the HEIT experiment extends previous studies by providing unambiguous information on reaction temperatures and times-to-explosion, allowing for the critical kinetics measurements needed to inform mesoscopic and continuum models.

The cascade of reactions leading to a thermal explosion in the condensed phase is exceedingly complex and encompasses hundreds of different steps and short-lived intermediate species. Nevertheless, the QMD simulations show that reaction rates can be captured accurately by an effective single step model, that is, with a single pre-exponential factor and activation enthalpy. The QMD simulations reveal similar activation energies among the explosives, with that of CL-20 appearing to be anomalously small. As with the QMD results, the activation energies derived from the HEIT experiment are similar for all of the explosives except for the shallow slope of CL-20. Combined, these results point to CL-20 being more sensitive than PETN, an energetic material used in detonators. These high-temperature results are consistent with the known handling sensitivity of CL-20, and might be attributed to its unusual, cage-like scaffold, which accommodates strained bonds that are not present in the other five molecules. However, at lower temperatures the ODTX experiments indicate that CL-20 is significantly less sensitive than PETN and more closely aligned with the secondary explosive RDX, another nitramine-based molecule. These results suggest that a change in phase or reaction mechanism may occur in CL-20 between the slow “cook-off” experiments probed by the ODTX test and the high temperature HEIT experiment, which imparts stimuli that are more closely aligned with prompt ignitions arising from impact (Table 1). These results emphasize that to avoid misleading information from safety tests (with potentially disastrous results), it is necessary to obtain data under the confined, high temperature conditions that best approach those that are encountered in accident scenarios.

Surprisingly, this study revealed for the first time that the activation barriers extracted from QMD simulations correspond remarkably well with literature ODTX data (Figure 4a,b). The slow reactions for the ODTX test, especially in the lower temperature limits, allow for homogeneous reaction initiation which closely resemble the uniform heating in the QMD simulations. Though the QMD calculations exhibit similar activation barriers as found in the induction time-limited ODTX experiment (see Supporting Information), the kinetics obtained from the HEIT experiment result in significantly more shallow temperature dependences and slopes. These shallow slopes indicate that multiple processes are occurring that involve (1) induction time at the explosive/needle interface, dependent on reaction rates for explosive decomposition, (2) thermal transport into the explosive, dependent on thermal conductivity and microstructure, and (3) deflagration propagation into the unreacted explosive, which depends on pressure and microstructure. In summary, the needle burst time is diffusion limited in the high temperature limit of the HEIT experiment, whereas in the lower temperature limit (seen in experimental conditions such as the lower limits of the ODTX test), the time-to-explosion is limited solely by the induction time for chemical reactions, and should therefore asymptote to the simple isothermal explosion time as described by the Frank-Kamenetskii model (eq 1). This study highlights the importance of knowing the extent of any thermal gradients and temperature heterogeneity within a sample before reaction occurs in a given experimental setup.

Conclusions

We describe here the first integrated experimental and modeling effort that utilizes modern experimental techniques (HEIT) and modeling (QMD and kMC) to quantitatively assess explosive chemical decomposition. The solid explosive sample loading used in the HEIT experiment is safer than previous methods (avoids the need for melting explosives), while retaining particle morphology and density characteristics. Notably, this experiment is useful for energetic synthesis in the early design stage, as sample sizes in this test are at the 5 mg level, and each test requires roughly 30 min to load and fire. This technique has the potential to allow for a more quantitative assessment of explosive sensitivity than provided by the traditional drop-weight impact test, which has been utilized to make crude assessments on molecular properties for many decades. QMD reactions capture only the homogeneous time-to-explosion in pristine material on Å to nm length scales with no contributions from particle morphology or thermal transport. Nevertheless, in the complex HEIT experimental system at millisecond time scales, we observe the same ranking of explosive sensitivity as predicted by the QMD calculations (Figure 3). This important result highlights the fact that thermal runaway in explosives is driven by, and depends sensitively on, the temperature.

In this study, we have determined that QMD results correspond exceedingly well with experimental activation parameters obtained in experiments with slower, equilibrated heating, like the ODTX test. The results from both QMD and the HEIT experiment give us confidence in the importance of Arrhenius kinetics in explosive initiation: that the chemical induction time for simple bond breaking dominates the relative ranking of times-to-explosion in these 6 common explosives. The shallow HEIT experimental slopes suggest that care should be taken in further analysis of the Arrhenius parameters, as thermal transport, deflagration propagation, and gas production are critical to bursting the needles. This combined experimental and modeling effort provides the first example where reaction from homogeneous heating can be clearly contrasted with heterogeneous, deflagration-controlled reactions, with preliminary kMC results allowing for a simple interpretation of this particular system. Additionally, Arrhenius parameters obtained from the QMD simulations provide kinetic information that will be useful in predictive models needed for safety assessments. Initial assessments of grid resolution in the kMC simulations indicate that the results do not qualitatively change with increasing refinement, and future efforts will focus on resolution studies and investigating other explosives beyond PETN. In future experimental work, these slopes will be evaluated at different packing densities, needle diameters, and lower temperature inputs.

Experimental Section

HEIT Experiment

ETN was prepared in our laboratory and purified using literature procedures. CL-20 was purchased from ATK and used as received. Fine, detonator-grade PETN powder was purchased from DuPont and purified through crash precipitation, fine HMX (Class V), and fine RDX (Class V) were purchased from Holston and used as received. TNT was obtained from a LANL historical stock, and purified through crash precipitation.

Explosive powders were loaded using a fixture designed for the small stainless steel needles, and safe loading methods. The fixture uses a small pin (∼0.5 mm diameter) to hold the needle, and position the powder on one side. On the other side, a small explosive sample is held inside a well. A plunger then pushes small amounts of explosive into the center of the needle until making contact with the pin on the other side of the needle. Samples were filled with explosive in the center of the needle, totaling ∼5 mg and 1 ± 0.1 cm in length. The fixture design allows for the operator’s hand to remain separated from the small amount of explosive contained in the well (<20 mg). Small grooved lines at the side of the plunger were utilized to measure the length of loaded solid. The needles were weighed before and after adding the explosive, and density was calculated as the average mass per length of explosive filled. Two operators filled the needles regularly, and all points were included in the Arrhenius plots. An average error of ±0.2 g cm^–3 was applied to all density values, given the estimated error in the length of the explosive fill. Preliminary experiments have indicated that variability in density of 0.2–0.3 g cm^–3 did not result in significant changes in times to explosion or a reordering of the trends observed in the Arrhenius plot.

Phantom PCC software was used to analyze the high-speed imaging videos, with a camera interframe time of 4.29 μs and exposure time of 1.91 μs. The time when first light is observed at the junctions (≤50 μs) is subtracted from the observed time-to-explosion.

Each needle test requires ∼10 min for sample loading, 10 min to connect the fixture, and 5 min to fire the needle, resulting in a fairly quick turnaround time to collect a data set. In 96 tests on 6 explosives over a 4 month period, 7 data points were discarded due to obvious breakout during the sparking event in the initial voltage pulse (3), no visible breakout of needle (3), or creation of a small puncture in the needle resulting in early pressure release (1).

Quantum Molecular Dynamics Modeling

The simulations of ETN, PETN, CL-20, RDX, HMX, and TNT were all performed in the condensed phase under three-dimensional periodic boundary conditions using the same general protocol as ref (25). Supercells containing a few hundred atoms were constructed from their published crystal structures at ambient temperature and density. The supercells were thermalized to their target temperatures, T₀, at constant volume by short, 20,000 time step runs with a velocity rescaling thermostat. Once thermalized, the systems were allowed to evolve in the microcanonical ensemble (constant number of particles, volume, and total energy). The use of microcanonical dynamics is essential for time-to-explosion simulations because it allows the temperature to evolve in response to the endo- or exothermicity of the underlying reactions.⁴⁰ Precise conservation of the total energy throughout the simulations was enabled by our use of the extended Lagrangian Born–Oppenheimer MD formalism that was developed by Niklasson.⁴¹ In addition, our simulations used a finite electronic temperature set equal to the initial ionic temperature to smear occupancies in the vicinity of the electronic chemical potential, an adaptive self-consistent field (SCF) optimization algorithm¹⁰ that promotes stability during the making and breaking of bonds, and a time step for the integration of the equations of motion of 0.25 fs.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.4c08424.

• QMD calculations, raw data from HEIT experiment, slopes and intercepts from Arrhenius plots, and frames of needle bursts for each shot (PDF)

The authors declare no competing financial interest.