Evolution of Carbon Clusters in the Detonation Products of the

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Evolution of Carbon Clusters in the Detonation Products of the Triamino-Trinitro-Benzene (TATB)-Based Explosive PBX 9502 Erik B Watkins, Kirill A Velizhanin, Dana Mcgraw Dattelbaum, Richard L. Gustavsen, Tariq D Aslam, David W Podlesak, Rachel C. Huber, Millicent Anne Firestone, Bryan S Ringstrand, Trevor Michael Willey, Michael Bagge-Hansen, Ralph L Hodgin, Lisa Lauderbach, Anthony van Buuren, Nicholas Sinclair, Paulo A Rigg, Soenke Seifert, and Thomas Gog J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b05637 • Publication Date (Web): 15 Aug 2017 Downloaded from http://pubs.acs.org on August 21, 2017

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The Journal of Physical Chemistry

Evolution of Carbon Clusters in the Detonation Products of the Triamino-trinitro-benzene (TATB)Based Explosive PBX 9502 Erik B. Watkins‡1, Kirill A. Velizhanin‡1, Dana M. Dattelbaum1*, Richard L. Gustavsen1, Tariq D. Aslam1, David W. Podlesak1, Rachel C. Huber1, Millicent A. Firestone1, Bryan S. Ringstrand1, Trevor M. Willey2, Michael Bagge-Hansen2, Ralph Hodgin2, Lisa Lauderbach2, Tony van Buuren2, Nicholas Sinclair3, Paulo A. Rigg3, Soenke Seifert4, Thomas Gog4 1

Los Alamos National Laboratory, Los Alamos, NM 87545

2

Lawrence Livermore National Laboratory, Livermore, CA 94550

3

Washington State University, Pullman, WA 99164

4

Argonne National Laboratory, Lemont, IL 60439

ACS Paragon Plus Environment

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ABSTRACT The detonation of carbon-rich high explosives yields solid carbon as a major constituent of the product mixture and, depending on the thermodynamic conditions behind the shock front, a variety of carbon allotropes and morphologies may form and evolve. We applied time-resolved small angle x-ray scattering (TR-SAXS) to investigate the dynamics of carbon clustering during detonation of PBX 9502, an explosive composed of triaminotrinitrobenzene (TATB) and 5 wt% fluoropolymer binder. Solid carbon formation was probed from 0.1 to 2.0 µs behind the detonation front and revealed rapid carbon cluster growth which reached a maximum after ~200 ns. The late-time carbon clusters had a radius of gyration of 3.3 nm which is consistent with 8.4 nm diameter spherical particles and matched particle sizes of recovered products. Simulations using a clustering kinetics model were found to be in good agreement with the experimental measurements of cluster growth when invoking a freeze-out temperature, and temporal shift associated with the initial precipitation of solid carbon. Product densities from reactive flow models were compared to the electron density contrast obtained from TR-SAXS and used to approximate the carbon cluster composition as a mixture of 20% highly ordered (diamond-like) and 80% disordered carbon forms, which will inform future product equation of state models for solid carbon in PBX 9502 detonation product mixtures.

INTRODUCTION The detonation of high explosives (HE) is a complex process in which an unreacted energetic material undergoes rapid shock-driven decomposition into a chemical product mixture consisting of a dense fluid (H2O, N2, CO, CO2), and solid carbon.1 The region immediately behind the shock front is referred to as


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the “chemical reaction zone” (CRZ), which is defined by the von Neumann spike on its leading edge and Chapman-Jouguet or sonic locus. The energy released within the CRZ serves to drive the detonation shock forward, so that the detonation wave is self-sustained and can steadily propagate with typical velocities of D = 6-9 km/s for solid, high performance explosives.2 The chemical transformations of a typical carbon/hydrogen/nitrogen/oxygen (CHNO)-based HE can be schematically represented as CHNO → H O + H + O + CO + CO + NH + NO + N + C()

where () denotes the solid carbon products and all other products are gaseous.2 In ideal HEs, all the

post-detonation carbon is in the form of CO and CO gases. For TATB specifically, the quantities of O2 and H2 in the products are negligible.

For many conventional high explosives (CHE) and high performance HEs (e.g. pentaerythritol tetranitrate, PETN), chemical transformations are very fast, occurring over tens of nanoseconds3-4 and ~0.1-0.3 mm reaction zones. In modeling CHEs, the reaction zone may be considered infinitely thin, so that the propagation of the shock front and the subsequent chemical reactions take place essentially simultaneously.1 This approximation breaks down for explosives with longer reaction zone length, such as for insensitive carbon-rich solid HEs (e.g., triaminotrinitrobenzene, TATB, or trinitrotoluene, TNT). The reaction zone of carbon-rich HEs often consists of two stages, schematically shown in Fig. 1A.4-6 The first fast stage is referred to as the “fast reaction zone” and, akin to ideal HE detonation, is thought to be associated with formation of gaseous products. The excess carbon, liberated during this fast reaction zone, then undergoes slow “carbon clustering”5-6 (also referred to as carbon condensation7-10 or coagulation1115

), where smaller clusters gradually coalesce into larger ones. The excess carbon has been shown by

recovery studies to forms solid carbon residues of various allotropes and morphologies (e.g., graphite, amorphous carbon, nanodiamonds).16-18 At even longer time scales, carbon clusters may aggregate, without significant fusing, and form fractal networks.15,19-20 While this general framework is largely accepted, quantitative understanding of the dynamics of carbon clustering in detonation of carbon-rich HEs has still been largely inaccessible, and has implications for post-detonation forensics, nanodiamond


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synthesis, and HE performance modeling. Since the allotrope, morphology, and size of post-detonation carbon particles have been observed to vary vastly depending on the chemical composition of the HE and on the environmental conditions of the detonation, forensic analysis of post-detonation nanoparticles can in principle be used to reveal the detonation condition (e.g., type of HE).21 Secondly, detonation nanodiamonds and other explosive synthesized carbon particles are useful in multiple applications16,22 and a predictive approach to synthesis of carbon nanoparticles with desired parameters (e.g., size) requires elucidation of the processes leading to their formation. Lastly, accounting for carbon clustering dynamics is necessary for accurate modeling of detonation. As carbon clusters grow, the surface-to-volume ratio of carbon decreases and the energy, corresponding to the formation of carbon-carbon bonds, is released. This energy release associated with carbon clustering shifts the detonation locus and release isentrope, which leads to uncertainty in the C-J state, and anomalous behavior in the release isentropes.5,23 Theoretical studies of the formation of solid carbon products in detonation by Shaw and Johnson approximated the process as diffusion-limited, non-reversible fusing of carbon clusters, Fig. 1A.5 In particular, carbon cluster fusing was assumed to not have any significant activation barrier due to supposed high surface reactivity of carbon clusters. Under these conditions, the only input parameter for the model was the viscosity of the dense fluid of small gas molecules forming the detonation products, which was evaluated using the Enskog theory.24-25 Recently, experimental observations of carbon precipitation using time-resolved small angle x-ray scattering (TR-SAXS) suggested that the apparent clustering rates were much slower, and in disagreement with those predicted by Shaw and Johnson. Specifically, TR-SAXS based measurements performed by Ten et al., Pruuel et al. and Rubtsov et al. suggest that the formation of carbon clusters several nanometers in diameter occurs over the course of several microseconds after the detonation front,7-8,26-29 whereas the Shaw-Johnson model predicted 1-2 orders of magnitude shorter formation times. For example, Ten et al. reported maximum cluster diameters of ~2.6 nm after >4 µs in a TATB-based explosive in explosive charges larger than those studied here with 0.5 µs sampling rates.8 Implications of


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this disagreement and possible remedies for modeling were recently discussed.14 Feasible mechanisms capable of slowing down carbon clustering to match the experimental observations included: accounting for “evaporation” or fragmentation of carbon clusters,13 a large activation barrier of cluster fusing,30 or an unrealistically large viscosity of the detonation products.11 Less direct estimations of the carbon cluster formation times, based on electrical conductivity measurements,31 interferometry and finite HE stick radius effects,4 as well as gas gun experiments,32 seemed to agree better with theoretical predictions than the TR-SAXS results cited above. Furthermore, recent TR-SAXS results obtained at the Advanced Photon Source (APS) also imply rather fast carbon clustering.10,33-34 Specifically, it was demonstrated that carbon nanoparticles with diameters of ~7 nm are produced in less than 400 ns after the passage of the detonation front in hexanitrostilbene (HNS).10 In this recent study the HNS TR-SAXS patterns were integrated over 250 ns and lacked the time resolution to detect the carbon nanoparticle growth; subsequent single-bunch measurements on HNS detonations demonstrated that the formation or carbon particles of ~7 nm in diameter occurs more rapidly than 300 ns but still did not probe the growth regime. This is significantly faster than previous TR-SAXS measurements,7-8,26-29 and thus much closer to the theoretical predictions of the Shaw-Johnson model14 and indirect experimental measurements.4,31-32,35 Here, we present the first measurements of carbon formation and cluster evolution during detonation of PBX 9502, a plastic bonded explosive containing the insensitive high explosive 1,3,5-triamino-2,4,6trinitrobenzene (TATB) and fluoropolymer binder (Kel-F 800). TR-SAXS were performed at the newlycommissioned Dynamic Compression Sector at the Advanced Photon Source in order to resolve the dynamics of carbon cluster formation during detonation. The SAXS scattering technique allowed the structure of carbon clusters to be probed at the nano to meso length scales, providing details about the average particle size and composition. In particular, TR-SAXS was used to answer whether there is substantial growth of carbon clusters in the ~50-2000 nanosecond time scale for comparison to previous indirect measurements,4,31-32,35 recent TR-SAXS measurements,10 as well as modeling efforts,5,14 or if carbon cluster formation continues into the microsecond time scale.7-8,26-29 The data were interpreted by a modification of the Shaw-Johnson model to understand the interplays between finite charge size, and


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temporal dynamics of thermodynamic fields (e.g., pressure, density) affect the dynamics of carbon clustering. Together with TR-SAXS data, this modeling clarifies the important time scales of carbon cluster formation and provide insights into the chemical and physical transformation associated with detonation of insensitive carbon-rich HEs.

EXPERIMENTAL METHODS Detonation of PBX 9502 samples PBX 9502 is a carbon-rich polymer-bonded explosive composed of 95 wt% 1,3,5-triamino-2,4,6trinitrobenzene (TATB) and 5 wt% poly(chlorotrifluoroethlyene-co-vinylidene fluoride) (Kel-F 800). Cylinders of PBX 9502 with a density of 1.89 g/cm3 and approximate dimensions of 10 mm diameter and 10 mm height were prepared by uniaxial pressing. The cylinders were oriented with the long axis normal to the beam and positioned so that the beam intercepted the center of the cylinder 3 mm below the top surface. Detonation was initiated at the bottom of the cylinder by an electrical detonator. The detonator consisted of three stages: (i) an exploding foil initiator used to initiate (ii) a 100mg LX-16 (96 wt% pentaerythritol tetranitrate, 4 wt% FPC-461 fluoropolymer) which accelerated an aluminum flyer plate into the bottom of (iii) a 10 mm diameter and 6mm tall cylinder of PBX 9501 (95 wt% HMX, 2.5 wt% Estane 5703®, 2.5 wt% bis-dinitroacetal-formal (BDNPA-F) nitroplasticizer). The detonation shock front progressed through the PBX 9501 cylinder and initiated detonation in the PBX 9502 cylinder on top of it.

SAXS experimental setups TR-SAXS experiments were performed at the Dynamic Compression Sector (DCS) special purpose hutch (35-ID-B) at the Advanced Photon Source (APS) (Fig. 1B).36 The x-ray source provided a ~200 µm x 50 µm ‘pink’ beam (~600eV FWHM) consisting of ~80 ps wide x-ray pulses spaced 153.4 ns apart. Measurements were made at either 14.7 keV or 14.3 keV corresponding to lambdas of 0.85 Å and 0.87 Å, respectively. X-rays scattered from the sample were measured as a function of the momentum transfer, Q,


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where Q=4π sin(θ)/λ and 2θ is the scattering angle. Two different sample-to-detector distances were used: ~3m for the low-Q configuration and ~1m for the high-Q configuration yielding momentum transfer ranges of 0.01 < Q < 0.07 Å-1 and 0.02 < Q < 0.3 Å-1, respectively. HE samples were contained within an evacuated (< 200 mTorr) detonation chamber equipped with Kapton windows to pass the x-ray beam and polycarbonate shielding plates to contain explosive debris.10 An evacuated flight tube was installed between the detonation chamber and the detector array to minimize air scattering and a beam stop was located between the downstream window of the flight tube and the scintillator to block the direct beam. Scattered x-rays were converted to visible light by a lutetium oxyorthosilicate:Ce3+ (LSO:Ce) scintillator, amplified by an image intensifier (Photek MCP140), and directed by beam-splitters to four gated CCD cameras (Princeton Instruments PiMax-4). The scintillator’s ~30 ns e-1 decay time allowed scattering from individual x-ray pulses to be discriminated and the cameras were used to collect scattered intensity from four sequential pulses. However, coupling with the decay time of the camera’s anode screen resulted in longer decay times and an 18% after image from the previous x-ray pulse. This after image had significant influence on the SAXS pattern when it originated from the undetonated sample and this may have contributed to the inability to obtain particle sizes at short times after detonation. In general, we found that contamination of the signal from after images did not have a significant effect on the particle sizes obtained from fitting. Static SAXS measurements of recovered products were performed using a Bruker NanoStar operating at 50keV and 0.6mA with a 2-D VÅNTEC-2000 detector. Samples were placed between two pieces of Kapton tape and placed in a 4x10-2 Torr evacuated chamber. Data was collected for 500 seconds covering a Q range from 0.007-0.3 Å-1.

Detonation timing and data collection During the experiment, timing of the measurements relative to the arrival of the detonation front was determined in reference to the current rise from the detonator. The detonator timing was highly reproducible, typically to within 5 ns, allowing precise control over timing of the detonation front arrival


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at the beam position. The four cameras were gated to collect data coincident with individual synchrotron x-ray pulses. After the measurement, the timing was calculated to a higher precision using the signal from a PZT piezo timing pin in contact with the top surface of the sample. Relative to the signal from the timing pin, the time that the detonation front reached the beam position was determined using the steady detonation velocity of 7.47 km/s37 and the distance from the top surface of the sample to the beam position (3 mm). This was correlated with the timing of the x-ray pulses which were coincident with camera gating and data acquisition. Here, we have defined t = 0 to coincide with the arrival of the detonation front at the beam position in the center of the sample. Due to front curvature, there is a ~100 ns delay of the detonation front arrival at the beam position between the center and the edge of the sample. Excluding the front curvature, uncertainty in the timing can be attributed to various factors including uncertainty in the positioning of the x-ray beam, uncertainty in the detonation velocity, and uncertainty in the cross timing between the x-ray pulses and the PZT timing pin circuits.38 For the conditions used here, the timing uncertainty was approximately 15 ns. Following the detonation of each sample, four camera frames were collected generally triggered to collect four sequential x-ray pulses. Timing of subsequent detonations was adjusted so that camera frames were interleaved to achieve higher time resolution than that available from the synchrotron pulse structure. Comparison of scattered intensities from different samples collected at similar times after detonation demonstrated that the sample-to-sample variation was small and did not significantly affect the scattered curve. Three series of measurements were performed: two high Q configuration series and one low Q configuration series. Fourteen measurements were made using the low Q configuration spanning up to t = 571 ns. An additional three measurements were performed using the high Q configuration up to t = 452 ns. The final series of twelve measurements using the high Q configuration extended the measured time range to t = 1876 ns. For t < 500 ns, the average time resolution was 22 ns. For t > 500 ns, the average time resolution was 133 ns. All measurements before arrival of the detonation front were consistent with static measurements even as close as t = -9 ns. Changes in the scattering pattern were observed as early as 31 ns after arrival of the detonation front further demonstrating the timing precision.


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Data processing Data processing procedures were performed using the Nika software package to convert the 2-D scattering images into 1-D profiles.39 A silver behenate standard was used to determine the sample to detector distance and position of the beam center for each camera by fitting the position of the known powder diffraction peaks. Using the beam center and sample to detector distance, data was radially integrated to obtain a 1-D profile and converted to Q space. Masks were applied to remove contributions from pixels corresponding to the beam stop and the camera edges. A glassy carbon standard, calibrated on 15-ID-D at the APS, was used to obtain absolute intensity normalization.40

Background subtraction In order to isolate scattered intensity originating from the sample, contributions from other sources must be eliminated or quantified and subtracted. Dark current contributions were subtracted from the data by recording and subtracting detector images in the absence of x-rays: ([data-dark]). Efforts were made to minimize scattering from air by evacuating the sample chamber and x-ray transit tubes upstream and downstream of the sample chamber. However, other background scattering sources such as the Kapton windows and polycarbonate shielding could not be eliminated. Ideally, by removing the sample such background contributions can be measured independently and directly subtracted from the data: ([datadark]-[empty-dark]). Such an approach requires that the sample does not significantly attenuate the x-rays so that the flux illuminating scattering sources downstream from the sample are equivalent for the two measurements. For the experimental conditions used here, transmission (Tr) through the static PBX 9502 sample was 100ns, this feature was attributed to the primary particles: carbon clusters. At earlier times (t < 100ns), the Guinier region could not be reliably modeled. There are several possible reasons for the inability to determine the particle size at early times including particle length scales too small to be accessed within the measured Q ranges, broad distributions of particle sizes, and low contrast between the carbon clusters and the dense fluid product mixture. Fitting the Guinier region was used to determine the radius of gyration (Rg) of the carbon clusters and values obtained for both the low Q and the high Q configurations were consistent (Fig. 5). The uncertainty in the Rg parameter was calculated by determining the range of Rg values that resulted in a χ2 within 10% of the minimum while simultaneously allowing all other fit parameters to vary within physically reasonable ranges. Larger error bars are associated with the second series of high Q measurements (Fig. 5, blue symbols) due to the lower


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incident flux used during these experiments. Error bars in time were calculated based on cross-timing uncertainty of 2 ns, beam position uncertainty of 50µm, and 2% uncertainty in the detonation velocity (7.47 km/s) but did not include uncertainty associated with the curvature of the detonation front.38 An additional degree of uncertainty can be attributed to the background subtraction method employed. To address this, we performed the same analysis on all data sets for both transmission dependent (Fig. 5, solid symbols) and steady state (Fig. 5, open symbols) background subtraction approaches. The difference in the background subtraction method was most pronounced for data measured in the low Q configuration where the Guinier region was located near the high end of the measurement range. Due to the presence of significant background contributions from downstream windows, we have a higher degree of confidence in the data points corresponding to the transmission dependent background subtraction (solid symbols). At t = 133 ns, an average carbon cluster Rg of 17.3 ±3.9 Å was obtained from the transmission dependent background subtracted data. With the current measurements, it was not possible to unambiguously determine the cluster morphology. As a simplest approximation, we have assumed a spherical particle shape for the carbon clusters in which case the Rg obtained corresponds to a diameter of ! = 2Y5/3#[ =

44.7 ± 10.1 Å. Rapid growth of the clusters was observed between 133 and 200 ns, stabilizing at Rg ~ 32.5 Å which, assuming spherical particles, corresponds to a diameter of ~84 Å. The data clearly demonstrates cluster growth that is outside the parameter errors and errors associated with background subtraction. After the initial growth, the carbon cluster Rg remained stable for all subsequent measurements (up to 1.88 µs) and was consistent with the Rg of 32.8 ±1.0 Å (for spheres, D=84.7 ±2.6 Å) obtained from static SAXS measurements of the recovered products.

Power law of the primary particle Scattering obeying power law dependence associated with the primary carbon particles (carbon clusters) was observed in the momentum transfer range of 0.1 < Q < 0.3 Å-1. This feature could be attributed to carbon clusters for t > 100 ns and was only measured using the high Q instrument


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configuration. For Q values greater than 0.2 Å-1, the scattering curves deviates from the power law dependence due to background contributions. Therefore, the fit to the data in this region was sensitive to both the power law parameters and to a constant background parameter. A -3 power law, corresponding to fractal roughness on the surface of the particle, was consistent with all measurements and the data could not be fit using a -4 power law corresponding to smooth surfaced particles. However, the parameters for the primary particle power law and the constant background were, to some degree, interdependent and it was not possible to rule out power laws between -2 and -3 which would indicate a degree of mass fractal structure. After making these determinations, unified fits were performed by fixing the power law to -3 in order to minimize the number of free parameters. In contrast, a -4 power law was measured for the primary particle in the recovered products, possibly suggesting annealing of the carbon clusters at longer times.

Power law of larger length scale structures Scattering obeying power law dependence was observed at the lowest measured Q values (0.01 < Q < 0.03 Å-1) corresponding to structures with length scales greater than 500 nm. Since an associated Guinier region was not within the measurement ranges, it was not possible to more precisely quantify the length scales associated with these structures. At times close to the detonation front arrival, this scattering was attributed to pores within undetonated material in the beam.48-49 At later times, the scattering was attributed to the formation of larger length scale aggregates composed of the primary carbon particles. Due to the greater uncertainty associated with the high Q data sets, we discuss only the results obtained from the low Q data sets which captured more of this region of the scattering. During fitting, the power law corresponding to larger length scales was allowed to vary between -4 and -2. Up to 200 ns after the arrival of the detonation front, we observed a -4 power law consistent with the initial pore structure. Additionally, there was a rapid decrease in intensity in this region. Between 0 and 100 ns, this intensity decrease can be mostly attributed to the decreasing volume of undetonated sample illuminated by the beam. From 300 to 400 ns, the power law rapidly increased to between -3 and -2, consistent with the


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power law obtained from the recovered products. These higher power laws suggest a transition to a fractal organization of the aggregate structure. As the power law increased, the intensity associated with this region simultaneously increased by approximately a factor of 5. The increasing intensity is consistent with the contrast function between the detonation products and carbon particles having an approximate composition of 80% turbostratic graphite and 20% diamond discussed in the following section.

Estimation of carbon product density In order to estimate the density of the solid carbon products, the scattered intensities as a function of time were compared to contrast functions derived from hydrodynamic simulation and hypothetical carbon compositions. Scattering contrast, ∆ρ, was defined as the difference between the electron density of the solid carbon clusters and the average electron density of the remaining detonation products. To calculate this quantity, product densities were extracted from the hydrodynamic simulation and converted to electron densities using the chemical composition of the starting material adjusted for the quantity of detonation products in solid carbon form. Net reaction of TATB decomposition during detonation was assumed to be CT HT NT OT → 3H O + 1.5CO + 3N + 4.5C(),

(4)

which results in 75% of the starting carbon being “excess” and ultimately converting to solid product form.2 Next, contrast between the detonation products and different solid carbon forms was calculated as a function of time and radial position by taking the average of the contrast values along the x-ray trajectory as a function of time. While a wide variety of carbon allotropes may contribute to the particle’s composition, we limited our analysis to mixtures of turbostratic graphite, a disordered carbon form, and highly ordered diamond for which there are known equations of state. Changes in the solid carbon density as a function of pressure and temperature were accounted for using the equation of state for turbostratic graphite50 E = 7.46(^ T − 1) + 27.12^&_ − 22.67&_ + 3.58^ 5.2 &_

(5)


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where δ=V0/V, 1/a5 = b5 = 1.80 g/cm3 is the density of turbostratic graphite at ambient conditions,

&_ = &/11,600 (p in GPa, T in K); and the equation of state for cubic diamond51 %

f

E = %cd eR fc U

%d

− 1g

(6)

where 1/a5 = b5 = 3.51 g/cm3 is the density of diamond at ambient conditions, B0=442 GPa is the

isothermal bulk modulus, and Mh = 4.438 is its pressure derivative. In the absence of inter-particle spacing contributions (structure factor), the scattered x-ray intensity from a single particle can be described as ? (@) = (∆b) a j (@) ,

(7)

where a is the volume of particle i and Fi(Q) is the form factor of the particle, normalized so that

j (0) = 1. The total scattered intensity will therefore be the sum of Ii(Q) for all particles illuminated by the x-ray beam. For a constant number, size, and shape of the particles, the total scattered intensity is

proportional to the square of the contrast, (∆ρ)2. However, the volume of the particles is also a function of P and T, which change throughout the detonation and need to be accounted for using the carbon equations of state. Here, we assume that these conditions (constant number and size of illuminated carbon particles) are valid for all measurements where the cluster size had stabilized (t > 180 ns). However, there were too many unknown variables to apply this approach during the cluster growth phase. Contributions to the total intensity include scattering from the primary carbon particles as well as from their aggregates which continue to evolve during the measured time range. In order to minimize the scattering contribution from the larger length scale aggregates, intensities were integrated for Q > 0.03 Å-1 which correspond to the Guinier regime and power law of the primary particles. After normalization by the x-ray transmission function, Tr(t), these intensities are proportional to the square of the contrast multiplied by the square of the average particle volume l ∑ mnorO k(G) c.cpÅ

st()

∝ (∆b) av .

(8)


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Intensities were summed from data sets reduced using a steady state background subtraction, so that sample transmission was not accounted for twice, and arbitrarily scaled to compare to the contrast functions (Fig. 6). The shapes of the contrast functions for turbostratic graphite or for diamond do not fit the trend in the summed intensities. The best match to the trend in the summed intensities was obtained from the contrast function for a mixture of 80% turbostratic graphite and 20% diamond.

Simulation of carbon clustering Since the size of fractal aggregates is not directly resolved experimentally in this work, our simulation efforts are focused solely on carbon clustering, i.e., the process where smaller clusters can coalesce to form larger ones. As discussed above, carbon clustering is simulated via kinetic rate modeling within “parcels” of material travelling along streamlines, Fig. 3. Accordingly, reaction rate modeling is performed within the Lagrangian frame,6 whereas the thermodynamic variables are obtained from the reactive flow simulations as functions of Eulerian (laboratory) time, the latter being smaller by ~10-30% in our calculations. Conversion between the two time frames for a specific streamline can be obtained using the axial particle velocity field extracted from the reactive flow simulations. The entire simulation of carbon clustering into primary particles consisted of the following steps: (i)

AWSD reactive flow simulations were performed for a 10 mm diameter stick of PBX 9502, to obtain temperature, pressure and density fields (Fig. 3), as well as radial and axial particle velocities.

(ii)

The particle velocity fields were used to generate 5-10 streamlines with r0 spanning the HE cylinder radius (Fig. 3A). Transformations between Eulerian and Lagrangian time were constructed for all the considered streamlines.

(iii)

Density, pressure, and temperature were evaluated as functions of Lagrangian (reaction) time along the streamlines, and the Enskog theory24-25 was used to determine the viscosity of detonation products. The resulting viscosity was ~3 x/(y ∙ {) immediately after the detonation

front (3 ∼ 10 {), and decreased to ~1 x/(y ∙ {) at 3~200 − 300 ns due to the expansion of ACS Paragon Plus Environment

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the detonation products. These values agree with the previous estimates of viscosity at the Chapman-Jouguet (C-J) point of TATB (HE component of PBX 9502).5,14 (iv)

The time-dependent viscosities, obtained from (iii), were used together with the StokesEinstein relation to evaluate rate constants of clustering. Conversion from the number of carbon atoms in a cluster to the radius of a cluster was performed assuming the average atomic density of b = 0.128 Å , which corresponds to b =2.55 g/cm3 approximated from the experimental data (see Fig. 6). The final results of carbon clustering simulations were only very weakly sensitive to the specific value of this density.

(v)

Eq. (1) was then solved numerically along each streamline, depicted in Fig. 3A, to yield concentrations of carbon clusters of each size as a function of Lagrangian time. Time was subsequently converted to the Eulerian (laboratory) frame. The initial condition for each streamline was set to correspond to excess carbon in the atomic form, i.e., P ≠ 0, ~P = 0 in Eq. (1), at the time when the streamline goes through the detonation front. This time, which is also assumed to correspond to the onset of clustering along each streamline, is zero for the axial streamline and as large as ∼ 100 ns for the outermost streamlines, Fig. 3. Having the excess carbon not in the atomic form, but as small clusters (e.g., C6 or C24) at the onset of clustering was previously shown to affect results only very weakly.14 Before the onset of clustering, all the cluster concentrations are set to zero.

The described simulations of carbon clustering yield time-dependent distributions of the sizes of primary carbon clusters for each of the streamlines shown in Fig. 3A. To directly compare these results to the average size of the clusters obtained from experiment, Fig. 5, one has to average over cluster sizes within a single streamline and then average over all streamlines. Such an averaging procedure should mimic how the experimental SAXS signal, and its subsequent fitting with Eq. (3), average over cluster sizes producing Fig. 5. In particular, one might expect that the averaging could be based on Eq. (7), since this equation determines the magnitude of contribution of a single cluster to the total SAXS signal. To this end, test simulations were performed, where we directly generated SAXS patterns from the reaction


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rate modeling, fitted these patterns with Eq. (3) to obtain average cluster sizes, and then compared these results to several averaging procedures. We found that the following expression mimics well how the SAXS signal “averages” over cluster sizes ()

t av (3) = c() c

∑K K (t,)fKH (>(t,))H

t ∑K K (t,)fK (>(t,))H

,

(9)

where #(3) is radial coordinate, as a function of Eulerian time, of the outermost streamline. As is seen,

the numerator averages over a (b), similar to Eq. (7), whereas the denominator normalizes the entire

expression to obtain the average volume of carbon clusters. The continuous dependence of (4, 3) on 4, required for averaging, was obtained from several representative streamlines via interpolation. If integration is omitted in the both numerator and denominator, Eq. (9) gives the average size of carbon clusters along a specific streamline. The results of such averaging, corresponding to three streamlines in Fig. 3B, are shown in Fig. 7A by solid black, red and blue curves. As is seen, the curves are similar barring the overall time shift corresponding to the arrival of the curved detonation front. The result of carbon clustering simulations with time-independent thermodynamic variables corresponding to the C-J [

point of PBX 9502 (T=3126 K, b=2.52 g/cm3, P=28 GPa, ) = 2.26 *∙)52 are shown by circles in Fig. 7A and are very similar to the axial streamline (solid black curve). The observed weak dependence of carbon clustering dynamics on exact values of thermodynamic variables originates from two factors. First, the growth of diameter of carbon clusters is !(3) ∝ Y3/) in the Shaw-Johnson model,5,30 so that variation of p

viscosity ) by e.g., 20% results in variation of cluster size by only ~7% (shown by dashed lines in Fig. 7A). Viscosity varies with thermodynamic variables approximately as ) ∝ b5.5T 5. & 5./, so it is

almost independent of density and the dependences on pressure and temperature compensate each other to some extent during the expansion of detonation products. Second, the acceleration of carbon clustering with time due to the expansion of detonation products and the resulting decrease in viscosity is partially compensated by the same expansion driving carbon clusters farther away from each other and effectively decreasing the rate constants of clustering.


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DISCUSSION Time resolved small angle x-ray scattering using high brilliance x-rays from the Advanced Photon Source was used to probe the solid carbon products formed in the detonation of TATB-based PBX 9502 high explosives. Two distinct regimes of carbon formation were observed: clustering of carbon into dense primary particles and the subsequent aggregation of these dense particles into a looser network. Here, we focus on the kinetics of the clustering process and the structure and composition of the primary carbon particles because the aggregate structure was not fully captured by the measured momentum transfer ranges. In principle, SAXS is capable of precisely determining nanoscale particle sizes and distinguishing between wide varieties of particle morphologies. While the average cluster size was obtained as a function of time relative to the passage of the detonation’s shock front, the data did not possess sufficiently distinct features, presumably due to either shape or size polydispersity, to determine the particle shape. As a result, we limited our description of cluster morphology to spheres, the simplest possible shape consistent with the scattering results. In addition to morphology and size, consideration of the scattering contrast and intensity of the TR-SAXS measurements enabled the density of the primary carbon particles to be estimated. A wide range of constituent carbon allotropes may be attributed to a given particle density and without x-ray diffraction measurements it was not possible to directly determine the composition. As a result, we invoked a simplifying approximation and limited our consideration of the particle’s makeup to a combination of a highly order carbon allotrope (diamond) and a more disordered allotrope (turbostratic graphite). Using this approach, the allotropic composition of the stable carbon clusters was approximated by a 4:1 turbostratic graphite to diamond mixture. One possible interpretation of this composition is a spherical core-shell structure with turbostratic graphite surrounding a nanodiamond center, although the measurements presented here were not capable of distinguishing the particle morphology. Assuming such a structure, the nanodiamond core of the carbon clusters would have a diameter of 5.3 nm which is consistent with the dimensions of nanodiamond observed in the detonation products of other HEs.16,22


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While morphology was not uniquely determined, the SAXS data precisely measured the evolution of average carbon cluster size as a function of time after detonation The main features of the time dependent carbon evolution shown in Fig. 5 can be summarized as: (i) rapid growth of carbon clusters reaching a diameter of 8.4 nm within the first 200 ns after the arrival of the detonation front, and (ii) the abrupt end of this growth at 3 ≈ 200 ns, with cluster size remaining stable for at least 2 µs and consistent with the sizes of the carbon products in recovered solid products. The saturation of carbon cluster size is in reasonable agreement with the duration of the “slow” evolution of the reaction zone, ~280 ns.32 Recent experimental results indicate rapid clustering kinetics during the detonation of 1,3,5-trinitro-2-[2-(2,4,6trinitrophenyl)ethenyl]benzene (HNS).10 The HNS measurements detected stable carbon clusters of diameter ~7 nm within ~400 ns of the arrival of the detonation front. While this suggests rapid cluster growth, consistent with our results, the actual dynamics of clustering was not observed.10 By probing times closer to the arrival of the detonation front, our measurements were capable of observing the growth of carbon clusters, albeit for a different HE (Fig. 5). These results on PBX 9502 appear to be in stark contrast to earlier and some recent TR-SAXS results which suggested much slower formation of carbon products with growth continuing up to ~1-10 µs after arrival of the detonation front.7-8,26-29 It should be noted that the experiments on HNS10 and the experiments on PBX 9502 presented here were designed to investigate the clustering and formation of the dense primary carbon particles and not their subsequent aggregation into extended networks. While the growth of the primary carbon particles was observed to cease rapidly, carbon aggregation in networks may persist for significantly longer times. Comparison of the experimental and simulated results is shown in Fig. 7B. The solid black line shows the evolution of the average cluster size, obtained via simulations of carbon clustering along the streamlines, Figs. 3 and 7A, and then averaging via Eq. (9). Even though the slopes of the initial diameter growth for this C-J simulation and the experiment can be considered comparable, the calculated cluster growth starts earlier and proceeds to larger sizes. To improve the agreement between simulation and experiment, two empirical but physically motivated modifications to the kinetic rate model are introduced. The first modification addresses the abrupt saturation of the average carbon cluster size at t ≈


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200 ns observed in the experimental data but not captured in simulations (solid black line in Fig. 7B). It has been proposed that the rate of carbon clustering may be strongly temperature-dependent because of the need to overcome a thermal activation barrier for two carbon clusters in contact to fuse.30 In this work, the thermal activation was modeled by introducing a freeze-out temperature of carbon clustering, & , so that rate constants in Eq. (1) are set to once the temperature along a streamline drops below & . The best

fit to experiment, shown by the dashed red line in Fig. 7B, was obtained using & = 2500K which is

consistent with freeze-out temperatures of various chemical reactions within the reaction zone.53 To assess the sensitivity of our calculations with respect to the exact value of the freeze-out temperature, we further plotted results of modeling for freeze-out temperatures of 2450 and 2530 K as dashed black lines in Fig. 7B. As is seen, relatively small changes of the freeze-out temperature can result in large variations of the clustering dynamics. This might be one of the reasons for large disparity between the present and earlier experimental results.7-8,26-28 The second modification to the kinetic rate model addresses the delayed onset of carbon clustering in experiment compared to simulations where clustering was initiated immediately upon arrival of the detonation front. However, carbon clustering cannot start immediately upon front arrival since some finite time is required for the shock-induced decomposition of HE molecules and the liberation of free carbon. Accordingly, carbon clusters are likely to start forming near the end of the fast reaction zone which can take a few tens of nanoseconds.3-4,32 For example, the reactive flow simulations performed here suggest that ~90% of unreacted HE is converted to products by ~50-100 ns (depending on a specific streamline) after the arrival of the shock front. We introduce this delay in the onset of carbon clustering as an ad hoc time shift, 3 , of the computed clustering kinetics with the best fit corresponding to 3 ~75 ns. The result of carbon clustering simulations with these two modifications – freeze-out temperature and the clustering onset delay - is shown by the solid red line in Fig. 7B, producing good agreement with experimental data. It has to be noted here that the introduction of the freeze-out temperature to simulations is not the only possibility to cause the slow down or complete cessation of the growth of carbon clusters. Small charge


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sizes in the experiments result in a more rapid decrease in pressure and temperature behind the detonation front compared with larger charges or supported conditions (such as gas gun experiments), which mayu alter the freeze-out dynamics. Other factors may also influence the late time cluster growth. For example, a reactive diamond-like cluster can heal its surface by forming a graphitic outer shell,54 thus gradually decreasing its surface reactivity resulting in a potential barrier to be overcome to fuse clusters.30 Another possibility is that the finite size of carbon clusters might be determined not by kinetic effects but by the equilibration between clustering and fragmentation processes.9 Further research is required to unequivocally understand the influence of HE type, conditions and charge dimensions on clustering dynamics.

CONCLUSIONS TR-SAXS was used to investigate the structure and dynamics of carbon cluster growth during detonation of the carbon-rich explosive PBX 9502. Analysis of the scattering data revealed rapid initial growth of carbon clusters within 200 ns after passage of the shock front to an average Rg of ~3.3 nm or, assuming spherical particles, a diameter of ~8.4 nm. Cluster growth within the size domain probed in the experiments did not continue after ~200 ns, and the size of carbon particles remained constant up to 2 µs, the longest time measured in the TR-SAXS experiments. The average Rg of the primary carbon particles obtained from analysis of recovered products from PBX 9502 was 3.4 nm, which corresponds to 8.5 nm diameter spheres, confirming that growth of carbon clusters stops at ~200 ns and does not resume at later times. These results agree well with recent TR-SAXS experiments on carbon cluster formation in HNS10 as well as with indirect measurements of the temporal dynamics of the CRZ, typically associated with carbon clustering.4,31-32 In addition to the formation of the primary carbon particles, signatures of aggregation of these particles into extended networks was observed with the structural evolution of the networks persisting for longer times than the cluster growth. While allotropic information about the carbon products was not directly accessible in this experiment, analysis of the scattered intensities allowed an estimation of the carbon density of the clusters which was consistent with a 4:1 mixture of


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turbostratic graphite and diamond. This result is significant, as the detonation conditions are well within the diamond region of the static carbon phase diagram, yet incomplete conversion to diamond is observed. In order to obtain insight into the dynamics of the formation of the primary carbon nanoparticles, we modeled carbon clustering using the Shaw-Johnson approach coupled with reactive flow calculations.5,14,43-44 Results agreed well with the experimental observations once (i) a freeze-out temperature of carbon clustering was set to & = 2500 K, and (ii) the onset of carbon clustering was “delayed” by 75 ns relative to the detonation front to account for the initial chemical transformations leading to the liberation of free carbon. Good agreement between the Shaw-Johnson model and the experimental data suggests that neither a large activation barrier,30 nor significant fragmentation13 occur, so that carbon clustering is an irreversible, diffusion-limited process with reaction radii given by particle sizes at temperatures higher that & ∼ 2500 K. The relatively low C-J temperature of TATB, the explosive component of PBX 9502, is not much higher than the freeze-out temperature obtained here which may explain the rapid saturation of carbon particle sizes. This suggests that performing similar studies for HEs with higher C-J temperatures can delay the saturation and lead to larger time scales for carbon cluster formation.


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Figure 1. (A) Scheme of carbon clustering Figure 2. Static PBX 9502 (+) and empty and aggregation processes. The shock front chamber (o) scattering after subtraction of arrives at t=0 followed by chemical ‘dark’ intensity. Subtraction of [emptydecomposition, clustering of solid carbon dark] scattering from [data-dark] leads to particles, and finally particle aggregation. over subtraction of the background (-). An (B) Schematic of TR-SAXS experiments. empirical coefficient, c(t0), was determined X-ray pulses are timed to intercept the HE so

that

[data-dark]-[c(t0)·empty-dark]

sample relative to passage of the detonation (filled circles) matched a -3.9 power law front

and

scattered

intensity

from (dashed line).

individual pulses is measured by four gated CCD cameras coupled to a scintillator.


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Figure 3. (A) Detonation product density, as a function of radial coordinate and time in the lab (Eulerian) frame, obtained from AWSD simulations. Several streamlines are shown as black lines. (B) Density of detonation

products

along

three

representative streamlines.


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Figure 4. (A) Representative SAXS patterns reduced using transmission dependent background subtraction for low Q (solid symbols) and high Q measurements (open symbols) covering the Guinier region of the primary particles. Low Q measurements also capture a power law corresponding to larger length scales and high Q measures the power law of the primary particles. Times after passage of the detonation front for each measurement are listed in the inset. (B) Fits to low Q data acquired during the growth phase (top) and at steady state (bottom). The unified fits (red solid lines) are composed of a power law corresponding to large length scales (dashed green lines) and a primary particle Guinier (solid blue lines). (C) Fits to high Q data acquired during the growth phase (top) and at steady state (bottom). The total fits (red solid lines) are composed of a constant background (dotted lines), a large length scale power law (dashed green lines), and the Guiner and power law of the primary particle (solid blue lines). Data and fits are shifted vertically for clarity.


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Figure 5. Carbon particle radius of gyration (Rg) or spherical particle diameter (D) as a function of time. Red squares and blue diamonds are values obtained using two series of high Q configuration SAXS measurements. Black circles are values obtained from low Q configuration SAXS measurements. The green triangle is Rg of the recovered products. Data was reduced using either a transmission dependent (solid) or steady state (open) background subtraction. Error bars in Rg correspond to

Figure 6. SAXS intensity compared to SLD contrast between detonation products and carbon particles. Integrated intensities for Q > 0.03 Å-1 were divided by the transmission through the sample and scaled to compare to contrast functions for carbon compositions varying from turbostratic graphite to diamond. The best match (red line) was obtained for a 4:1 graphite to diamond ratio (20% diamond). Diamond fractions of 10% and 30% are shown as dashed lines.

the range of parameter values that satisfy χ2 < 1.1·χ2min.


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Figure 7. (A) Comparison of carbon clustering simulation results for three representative streamlines (solid lines) to the C-J simulations (circles). The onset time of clustering varies for different streamlines due to the finite curvature of the detonation front. Results of C-J simulations with viscosity increased or decreased by 20% are shown by dashed lines. (B) Comparison of particle sizes obtained from low Q TRSAXS using transmission dependent background subtraction (solid black symbols both here an in Fig. 5) to clustering simulations. Averaging of multiple streamlines, using Eq. (9), results in the solid black line. Introducing freeze-out temperatures, Tf, resulted in the red and black dashed lines. The best match to the experimental data (solid red line) was obtained using & = 2500 and a 75 ns time delay.


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AUTHOR INFORMATION Corresponding Author *Dana M. Dattelbaum M-DO: Explosive Science and Shock Physics Los Alamos National Laboratory Los Alamos, NM 87545 [email protected]

Author Contributions ‡E.BW and K.A.V. contributed equally to this work. All authors contributed to the performance of the SAXS measurements. SAXS analysis was done by E.B.W., AWSD simulations were done by T.D.A. and kinetic rate modeling was done by K.A.V. The manuscript was written through contributions of all authors and all authors have given approval to the final version of the manuscript. ACKNOWLEDGMENT The authors acknowledge support from DOE-NNSA and the Dynamic Material Properties program. This publication is based upon work performed at the Dynamic Compression Sector supported by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002442 and operated by Washington State University. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. Prepared by LLNL under Contract DE-AC5207NA27344. We are grateful to Joshua Coe (LANL) and Joshua Hammons (LLNL) for helpful discussions and feedback.


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(35) Gustavsen, R. L.; Aslam, T. D.; Bartram, B. D.; Hollowell, B. C. Plate Impact Rxperiments on the TATB Based Explosive PBX 9502 at Pressures Near the Chapman-Jouguet State. J. Phys. Conf. 2014, 500, 052015. (36) Gupta, Y. M.; Turneaure, S. J.; Perkins, K.; Zimmerman, K.; Arganbright, N.; Shen, G.; Chow, P. Real-Time, High-Resolution X-Ray Diffraction Measurements on Shocked Crystals at a Synchrotron Facility. Rev. Sci. Instrum. 2012, 83, 123905. (37) Campbell, A. W. Diameter Effect and Failure Diameter of a TATB-based Explosive. Prop. Expl. Pyro. 1984, 9, 183-187. (38) Gustavsen, R. L.; Dattelbaum, D. M.; Watkins, E. B.; Firestone, M. A.; Podlesak, D. W.; Jensen, B. J.; Ringstrand, B. S.; Huber, R. C.; Mang, J. T.; Johnson, C. E., et al. Time Resolved Small Angle X-Ray Scattering Experiments Performed on Detonating Explosives at the Advanced Photon Source: Calculation of the Time and Distance Between the Detonation Front and the X-Ray Beam. J. Appl. Phys. 2017, 121, 105902. (39) Ilavsky, J. Nika: Software for Two-Dimensional Data Reduction. J. Appl. Cryst. 2012, 45, 324-328. (40) Zhang, F.; Ilavsky, J.; Long, G. G.; Quintana, J. P. G.; Allen, A. J.; Jemian, P. R. Glassy Carbon as an Absolute Intensity Calibration Standard for Small-Angle Scattering. Metall. Mater. Trans. A 2009, 41, 1151-1158. (41) Schelten, J.; Schmatz, W. Multiple-Scattering Treatment for Small-Angle Scattering Problems. J. Appl. Cryst. 1980, 13, 385-390. (42) Wescott, B. L.; Stewart, D. S.; Davis, W. C. Equation of State and Reaction Rate for Condensed-Phase Explosives. J. Appl. Phys. 2005, 98, 053514. (43) Aslam, T. D. Shock Temperature Dependent Rate Law for Plastic Bonded Explosives. J. Appl. Phys., in preparation. (44) Romick, C. M.; Aslam, T. D. High-Order Shock-Fitted Detonation Propagation in High Explosives. J. Compt. Phys. 2017, 332, 210-235. (45) Romick, C. M.; Aslam, T. D. Two-Dimensional Detonation Propagation Using ShockFitting. In 15th International Detonation Symposium, San Francisco, CA, USA, 2014; pp 380389. (46) Ilavsky, J.; Jemian, P. R. Irena: Tool Suite for Modeling and Analysis of Small-Angle Scattering. J. Appl. Cryst. 2009, 42, 347-353. (47) Beaucage, G. Approximations Leading to a Unified Exponential/Power-Law Approach to Small-Angle Scattering. J. Appl. Cryst. 1995, 28, 717-728. (48) Mang, J. T.; Hjelm, R. P. Fractal Networks of Inter-Granular Voids in Pressed TATB. Propellants, Explos. Pyrotech. 2013, 38, 831-840. (49) Willey, T. M.; van Buuren, T.; Lee, J. R. I.; Overturf, G. E.; Kinney, J. H.; Handly, J.; Weeks, B. L.; Ilavsky, J. Changes in Pore Size Distribution upon Thermal Cycling of TATBbased Explosives Measured by Ultra-Small Angle X-Ray Scattering. Propellants, Explos. Pyrotech. 2006, 31, 466-471. (50) Danilenko, V. V. Nanocarbon Phase Diagram and Conditions for Detonation Nanodiaomanod Formation. In Synthesis, Properties and Applications of Ultrananocrystalline Diamond, Gruen, D. M.; Shenderova, O. A.; Vul, A. Y., Eds. Springer: Netherlands, 2005; pp 181-198. (51) Winey, J. M.; Gupta, Y. M. Shock-Compressed Graphite to Diamond Transformation on Nanosecond Time Scales. Phys. Rev. B 2013, 87, 174104. (52) Menikoff, R. Complete EOS for PBX 9502; 2009; pp LA-UR-09-06529.


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TOC graphic


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Figure1 176x166mm (300 x 300 DPI)



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Figure2 148x142mm (600 x 600 DPI)


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Figure 3

145x140mm (200 x 200 DPI)



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Figure 4 66x20mm (300 x 300 DPI)


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Figure5 165x149mm (300 x 300 DPI)



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Figure6 152x144mm (600 x 600 DPI)


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Figure 7 85x36mm (300 x 300 DPI)



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TOC 44x23mm (300 x 300 DPI)


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Evolution of Carbon Clusters in the Detonation Products of the

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