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

Solid carbon is a major constituent of high explosive (HE) detonation products, particularly when the HE is oxygen-lean (carbon-rich). However, almost no details of the carbon evolution are known. Behind the leading edge of a detonation is the chemical reaction zone (CRZ), in which the energy that drives the detonation front is released on a time-scale of tens to hundreds of ns. Within the CRZ, the explosive molecules: (reactants) evolve into (products) consisting of a dense fluid mixture of small gas molecules like N₂ and CO₂ and solid carbon. It has been posited that excess carbon delays full thermodynamic equilibrium resulting in two rates of energy release: a “fast” rate, of order ns, due to decomposition of explosive molecules, and a “slow” rate, of order μs, believed to be due to carbon coagulation or “clustering.”^1–3 Carbon clustering is also believed to lengthen the CRZ. The environment behind the detonation front is almost experimentally inaccessible, because it is hot, dense, optically opaque, and evolves rapidly. Although some information regarding detonation carbon can be gleaned from recovery experiments,^4–7 significant details are obscured by post-detonation processes. Expansion, cooling, and interaction with the surrounding environment irreversibly alter the carbon cluster morphology and surface chemistry.

In situ time-resolved x-ray scattering measurements are promising for providing direct measurement of carbon precipitation and coagulation timescales, as well as details of the evolution of carbon particle morphology and allotrope composition with time. From Guinier⁸ and Porod^9,10 analysis of x-ray scattering intensity, I, as a function of scattering vector, q,¹¹ estimation of carbon particle size, shape, and fractal dimension may be determined. Previous reports^12–16 of time-resolved small angle x-ray scattering (TR-SAXS) experiments on a variety of explosives, indicated that carbon clustering rates could be quite slow, occurring over several microseconds. These time scales should be compared with the CRZ timescales which are of order 200 ns. On the basis of TR-SAXS experiments on detonating Tri-Nitro-Toluene (TNT), 50/50 wt. % mixtures of 1,3,5-Trinitroperhydro-1,3,5-triazine (RDX) and TNT, and Tri-Amino-Trinitro-Benzene (TATB), the Russian studies^17,18 predict clusters of order 1000 atoms and 2.2 nm in diameter forming on time scales of order 100 ns. After this, the clusters slowly increase in size out to ∼4 μs. The carbon cluster growth rates measured in these experiments are much smaller than estimates derived from theory. The apparent discrepancies have been addressed in a number of ad-hoc ways, including surface “evaporation,”^19,20 surface chemistry,²¹ and an arbitrary increase of the products' fluid viscosity by up to two orders of magnitude.²² 

Recently, significant progress beyond that achieved by the Russian pioneers, has been achieved through use of third generation synchrotron x-ray sources such as the Advanced Photon Source (APS) at Argonne National Laboratory, USA. These sources have beamlines optimized to probe structure at high time resolution, over a full range of q, and with better detection schemes. Experiments by members of this team^23–25 measured TR-SAXS patterns at APS in explosives such as hexanitrosilbene (HNS) and Composition-B3 (60/40 wt. % RDX/TNT). The experiments were carried out at the Dynamic Compression Sector (DCS)²⁶ of the APS. DCS is a new experimental capability that links dedicated dynamic compression drivers, such as gas-guns, to a third-generation synchrotron light source.²⁶ DCS has several experimental hutches, one of which, 35-ID-B, is used for experiments that do not require permanent driver platforms. When the synchrotron is in “standard” 24-bunch mode, x-ray pulses come 153.4 ns apart and diffracted or scattered x-rays can be recorded from up to 4 separate pulses. This apparatus provides higher time resolution scattering patterns with significantly better signal to noise ratios than previously reported. For example, in HNS, carbon clusters were observed to reach their final size within ∼400 ns²³ and cluster size and growth rate was in reasonable agreement with theories such as Shaw–Johnson.¹ 

In this work, we provide details of the TR-SAXS experiments carried out at Sector 35 (DCS) of the Advanced Photon Source using the “standard 24-bunch” operating mode of the synchrotron. A high level experimental schematic is shown in Figure 1. In the lower left is shown the APS synchrotron ring with 24 electron bunches circulating at almost the speed of light. The electron bunches pass through an undulator generating an x-ray pulse which is directed into the DCS/Sector 35 experimental hutch, upper left. The x-ray pulses are directed through the detonation experiment (center) and discrete images from each x-ray pulse (upper right) are recorded. The images are analyzed (lower right) to extract the scattering intensity, I(q) and obtain information about the carbon particles formed behind the detonation front.

Explosive charges, up to 3 g in mass, were detonated in an explosive chamber in the B station of DCS/Sector 35. In the TR-SAXS experiments, the x-ray beam is fixed and is focused into a small, 50 μm vertical by 200 μm horizontal spot near the end of the explosive cylinder. Up to 4 sequential SAXS measurements are collected during detonation, each spaced by the bunch timing of 153.4 ns.

The usual frame of reference for detonation propagation experiments is one in which the detonation front is stationary, and reactants evolve into products with time and distance behind the detonation front. In the TR-SAXS experiments, the x-ray beam is fixed at a spot near the end of the explosive cylinder. A Gallilean translation between the experimental, fixed-frame-coordinates, to the detonation-front-coordinates is used. This allows us to understand the carbon coagulation in the appropriate frame; as a function of time and distance behind the detonation front. We consider the sources of experimental uncertainty and resolution that limit the current experimental approach. As part of the error analysis, we consider uncertainties in detonation velocity, uncertainties in auxiliary diagnostics such as pins, uncertainty in beam to sample positioning and uncertainties due to detonation front curvature. These generate uncertainties in the times and positions of the x-ray diagnostic relative to the detonation front. Consideration of these uncertainties leads to a lower time resolution limit of ∼15 ns, generated largely by an uncertainty in the detonation velocity of ∼0.25 mm/μs.

The overarching goal of our project is to obtain time-resolved SAXS patterns on detonating explosives and use these patterns to understand carbon clustering as a function of time and distance behind the detonation front. Information which can be extracted from the SAXS patterns includes particle size, shape, carbon allotrope, etc. These results will not be presented here but rather in forthcoming publications. The current manuscript presents a subset of this larger project; it introduces a systematic description of the experimental setup for time-resolved SAXS measurements on detonating explosives. Synchronization of the x-ray pulses with recording equipment consisting of four gated CCD cameras, a piezoelectric (PZT) pin, a high speed digitizer, and the detonator result in a SAXS pattern recorded every 153.4 ns within a single experiment. Through a simple transformation, these scattering patterns can be associated with times and distances behind the moving detonation front. Although example scattering patterns from one experiment are shown, these patterns will not be interpreted here, but rather in forthcoming publications.

A simplified schematic of the experiment is shown in Fig. 2. We have left out the details, such as the vessel containing the detonation, in order to make things easier to understand. As described by Luo et al.²⁷ a “slow” upstream shutter opens over about 60 ms allowing the x-ray beam to enter the hutch and interact with the explosive sample. The x-ray beam is composed of pulses 153.4 ns apart in time, and approximately Gaussian in shape. The measure of the pulse width is the standard deviation of the Gaussian: 33.5 ps.²⁸ This equates to a width of 79 ps at FWHM. The time between pulses is controlled by synchrotron operations at the Advanced Photon Source. Dynamic experiments are synchronized to the x-ray pulses by way of a bunch-clock.

An x-ray pulse interacts with the explosive sample and a fraction of the x-rays are scattered; the scattering is determined by the sample microstructure, electron density, and x-ray absorption, all of which are functions of the x-ray energy or wavelength. Scattered x-rays are collected on a scintillator plate with active area ∼75 mm in diameter and 0.5–3 m downstream of the HE sample. On the scintilator plate, scattered x-rays are converted into visible light and amplified by a micro-channel plate image intensifier. The e^–1 decay time for the Lutetium Oxyorthosilicate, Ce³⁺ (LSO:Ce) scintilator is ∼28 ns.²⁷ Visible light from the image intensifier is directed through a series of beam-splitters and then into four electronically gated CCD cameras (Princeton Instruments PiMax-4). Camera gates are synchronized with the x-ray pulses by means of the bunch-clock.

The high explosive (HE) sample is a right circular cylinder typically about 6–10 mm in diameter by 10–20 mm long and oriented vertically as shown. Detonation is initiated at the bottom of the explosive by an electrical detonator. For the experiments described here, the detonator is known as a Livermore Test Detonator or LTD. The LTD uses an exploding foil initiator (EFI),²⁹ often called a slapper, to initiate a 100 mg LX-16 (96 wt. % PETN, 4 wt. % plastic binder) pellet. A high voltage capacitor is discharged through the foil of the exploding foil initiator. The current through the foil is measured using a current viewing resistor. Detonation of the LX-16 pellet accelerates a thin aluminum flyer plate, approximately 5 mm in diameter, into the bottom of the main HE sample. Detonation in the main HE sample is initiated by the impact of this aluminum plate. Function time of the LTD detonator, from current rise to impact of the aluminum plate on the HE sample is ∼1.0 μs.

Once initiated by the aluminum flyer plate, a detonation front (or detonation wave) moves up the HE sample at a characteristic Detonation velocity, D, of 6–9 mm/μs. The detonation velocity, D, depends on the type of explosive. As in the cartoon shown in Fig. 2, the detonation front separates the unreacted explosive from the detonation products. The detonator is fired at a time that is synchronized with the x-ray pulses and the camera gates so that the detonation front reaches the -ray spot at a time predetermined by the experimenter. Typically, one scattering pattern is recorded before the detonation front reaches the x-ray beam position. This pattern will have the characteristics of the pressed unreacted explosive. After the detonation front passes the x-ray beam position, 3 more scattering patterns are collected. These are characteristic of the explosive products. We note that timing of the LTD detonators is very reproducible, typically to within about 5 ns. This allows us a great deal of control over the time when the detonation front crosses the x-ray beam position.

The piezoelectric (PZT) pin located at the top of the HE cylinder is a pressure sensitive electrical transducer. It produces a fast rising electrical pulse when the detonation front arrives at the top of the cylinder.

Figure 3 shows a timing diagram for one of the experiments. X-ray intensity, the four camera gate signals, the detonator current, and the output from the PZT pin are shown for shot# 8, April 2015. The vertical scale in Fig. 3 is arbitrary; the amplitudes of the signals have been scaled. The time origin is the rise in the detonator current. The cameras are gated on sequentially every 153.4 ns with the leading edge of the gate nearly coincident with the arrival of an x-ray pulse. Camera timing is locked to the synchrotron bunch clock. This shot used the standard or 24 bunch mode with 153.4 ns between the x-ray pulses. (APS operates in other modes that provide a different time spacing between x-ray pulses.) After the x-rays from a pulse strike the scintilator, it will glow for at most 84 ns because the e⁻³ decay time for the LSO:Ce scintilator is ∼84 ns.²⁷ This means that light from only one x-ray pulse is recorded on each camera. The last signal on Fig. 3 is from the PZT pin whose pressure sensitive tip is located at the top surface of the HE cylinder in Fig. 2. The rising edge of this signal provides a time“fiducial” of when the detonation wave reaches the end of the charge.

For the remainder of this paper, we will use shot# 8 April 2015, as an example. The HE sample for this shot was a 10.0 mm diameter by 10.0 mm long cylinder of Composition B-3 (60 wt. % RDX, 40 wt. % TNT) with density 1.708 g/cm³.

To recapitulate, the experimental sequence of events is as follows. The ms shutter upstream of the hutch is opened allowing x-rays to reach the sample. At t = 0, current is discharged through the exploding foil initiator in the LTD detonator. At ∼1.00 μs, detonation is initiated in the Comp B and starts traveling upward in the explosive. X-rays scattered by pulses at t = 2.0708, 2.2242, 2.3776, and 2.5310 μs are recorded by cameras 1 through 4, respectively. Note that each “frame” is timed to sequential pulses of x-rays which are 153.4 ns apart in 24 bunch mode. At t = 2.547 μs, the detonation front reaches the top of the sample, and the PZT pin produces a fast rising signal. The x-rays are focused at the HE sample, and the spot size for these experiments is typically 50 μm vertical by 200 μm horizontal.

Figure 4 shows the SAXS scattering patterns for shot #8 April 2015. These measurements were made with an x-ray energy of 14.68 KeV (0.845 Å) and ∼3 m sample to detector distance giving a q range of 0.01 < q < 0.1 Å⁻¹. In all experiments, the absolute q-vector calibration was the silver behenate (AgBeh) scattering standard. The scattering patterns shown employed an absolute minimum of processing. Background subtraction was simple data-dark. Dark was a measurement with the x-ray shutter closed, but the scintillator active and camera exposure times the same as for the data measurement. Scattering patterns presented in forthcoming publications will have undergone more sophisticated processing procedures to subtract parasitic scattering from air, windows, etc. The two-dimensional camera intensity images are reduced to one-dimension by circular integration using the Nika package.³⁰ The 1-D data are then intensity calibrated and analyzed using the Irena package.³¹ 

In Fig. 4, pattern #1 was obtained prior to the detonation front crossing the x-ray beam. The scattering pattern is characteristic of scattering from the voids in the pressed Comp-B. Patterns #2–#4 were obtained after the detonation front crossed the x-ray beam. The scattering patterns are characteristic of nanoscale carbon particles in the products. Pattern #2 may be a superposition of detonated and undetonated explosive. Similar SAXS patterns in detonating Comp-B but with a different relative timing can be seen in Firestone et al.²⁴ 

The explosive charge, Composition B-3 pressed to a density of 1.708 g/cm³, for shot# 8 April 2015, is represented in cross section in Fig. 5. The charge diameter is 10.0 mm and the length 10.0 mm. In 10 mm diameter and at density 1.700 g/cm³, the steady detonation velocity should be 7.786 mm/μs.^32,33 Note that for all explosives, detonation velocity decreases as the charge diameter decreases. This is known as the diameter effect. Furthermore, detonation fails to propagate in charges smaller than a diameter known as the failure diameter. In Comp B-3, the detonation velocity is calculated from

where R is the charge radius in mm, and D(R) is the detonation velocity in mm/μs.³² The failure diameter in Comp-B3 is 4.28 mm. Table I summarizes the definitions used in the text.

In Fig. 5, the explosive is represented as a plot of vertical position z(mm) vs. horizontal position x(mm). The edge of the charge is outlined in cyan. The x-ray beam is directed through the charge 3.00 ± 0.02 mm below the top. The position of the detonation front at four different times (t₁, t₂, t₃, t₄) is represented by the red, green, blue, and black lines. The four discrete times correspond to the four times that the x-ray beam probes the sample, and the SAXS signal is recorded by the cameras. Note that the detonation is traveling from the bottom to the top of the charge. At the time the detonation front reaches the top of the charge, the PZT pin produces an electrical signal with a very fast rise time. This is just after t₄.

Solid lines assume the steady detonation velocity for a 10 mm diameter charge; D₁₀ = 7.786 mm/μs. Dashed lines assume a much slower detonation velocity. The slowest possible steady detonation velocity is just slightly greater than the velocity at which the detonation fails. For Comp B-3, the failure diameter is 4.28 mm and using Eq. (1), D_failure= 6.743 mm/μs. Note that the detonation fronts represented by the solid and dashed lines are at very different positions at early times and low on the charge but converge near the top of the charge. This is because the time at which the detonation front arrives at the end of the HE cylinder is measured by PZT pin signal.

In Fig. 5, we show a fairly flat detonation front. In Sec. IV D we will discuss how the detonation front curvature affects the SAXS data.

Position–time diagrams, often called x–t diagrams, are used as a graphical aid to understand wave interactions at interfaces in shock physics experiments. If time is the horizontal axis and position is the vertical axis, the leading edge of a shock or rarefaction wave will be represented as a straight line with the slope of the line being the wave velocity. Fixed positions can be represented by horizontal lines, and fixed times can be represented by vertical lines. In our case, we have a detonation wave propagating in the vertical or z direction, so this is a z–t diagram, and it is shown in Fig. 6. Horizontal lines at 10 mm and 7 mm represent the position of the PZT pin (end of charge) and x-ray beam position, respectively. Vertical dashed lines are the times, t₁–t₄, an x-ray pulse interacts with the sample. The solid purple line represents the expected steady detonation velocity of D₁₀ = 7.786 mm/μs. In Fig. 6, the positions at which the vertical lines at t₁–t₄ intersect with the purple detonation line correspond to the shock front positions represented in Fig. 5.

In Fig. 6, the purple line is the position z of the detonation front with time t. The equation is

where L_HE is the length of the HE, D is the detonation velocity (the slope of the line), and t_pin is the time at which the detonation wave reaches the end of the HE (z = L_HE), and the PZT pin produces a signal. Note that from Eq. (2), z(t_pin) = L_HE.

The vertical position of the x-ray beam is z_beam. In our example, it is nominally 3.00 mm from the end of the HE or z_beam = 7.00 mm. The time, t_beam at which the leading edge of the detonation front crosses the x-ray beam position z_beam can be determined from Eq. (2).

In our example shown in Fig. 6, t_beam = 2.162 μs for the solid purple line and the nominal detonation velocity of D₁₀ = 7.786 mm/μs. For the dashed purple line and the failure detonation velocity of D_failure = 6.743 mm/μs, t_beam= 2.102 μs.

The x-ray probe times, t_n, relative to the detonation front (df) passing through the x-ray beam position z_beam are

Note that in our example shown in Fig. 6, $t1df$ is negative indicating that the SAXS pattern was taken before the detonation front crossed the x-ray beam position z_beam. Likewise, $t2df$ through $t4df$ are positive indicating that the SAXS patterns were taken after the detonation front crossed z_beam.

Detonation front position at t_n relative to the x-ray beam position, z_beam is obtained by evaluating Eq. (2) at t_n and t_beam;

Again in our example shown in Fig. 6, $z1df$ is negative indicating that the SAXS pattern was taken when the detonation front was below the x-ray beam position z_beam. Likewise, $z2df$ through $z4df$ are positive indicating that the SAXS patterns were taken after the detonation front was above z_beam.

The relative times, $tndf$⁠, and positions, $zndf$⁠, can be expressed as functions of (a) the x-ray to PZT pin times, t_n – t_pin (b) the position of the x-ray beam relative to the end of the charge, L_HE – z_beam and (c) the detonation velocity, D. Using standard uncertainty calculations based on taking the total derivative of Eqs. (5) and (6) with respect to these variables, we find

From these equations, we can see that, even though the 4 different x-ray times are involved, only $Δ(zndf)$ vs. Δ(D) from Eq. (8) will have more than one branch or solution. This will have 4 branches, one corresponding to each value of (t_n–t_pin). Also, if taken variable by variable, we can expect the functional forms of all but one of the uncertainty plots to be linear. The one exception is the uncertainty in $Δ(tndf)$ vs. Δ(D). Because of this nonlinearity, we will complete the uncertainty analysis using Monte-Carlo methods.³⁴ The measure of uncertainty is assumed to be one standard deviation, and the assumed probability distribution function is a normal distribution. We take the mean value of detonation velocity, D, to be 7.786 mm/μs corresponding to the expected velocity for a 10 mm diameter charge. One million Monte-Carlo trials were carried out for each value of uncertainty in each variable.

It is important to understand the uncertainty in relative times and positions as a function of the detonation velocity, D. At present we are limited to studying small explosive charges with total weight less than 3 g. This means the charge dimensions are of order 10 mm diameter × 16 mm long or smaller. In these sizes, the detonation velocity is likely not steady and still changing at the time the detonation front reaches the end of the charge. This uncertainty in D generates uncertainty in the relative time/distance to the detonation front. One of the overall goals of the project is understanding how carbon clusters evolve with time and distance through the chemical reaction zone (CRZ) which begins at the detonation front. For example, the CRZ in Comp-B is ∼50 ns.³⁵ Uncertainties in the calculated relative times and distances will affect how closely we can tie the carbon evolution to the time/distance behind the detonation front.

Figure 7 shows the uncertainty in the relative times, $Δ(tndf)$⁠, as a function of the uncertainty in the detonation velocity, Δ(D). Note that we have overlaid graphs for all four relative times.

Figure 8 shows the uncertainty in the relative detonation front position, $Δ[zndf]$⁠, as a function of the uncertainty in the detonation velocity, Δ(D). Note that we have overlaid plots for all four relative times. The uncertainty in relative position is the least for n = 4, when the detonation front is closest to the end of the HE (and the PZT pin) and increases, the further one moves away from the PZT pin.

We note that Δ(D) = 1 mm/μs is probably rather extreme. For PBX 9502, a TATB based explosive that has been much more thoroughly studied than Composition B, the variation between detonation velocity at failure and detonation velocity in a 10 mm charge is <1%. D_min = 7.408 mm/μs was the observed detonation velocity in an 8 mm diameter stick, just above the failure diameter,³⁶ while D_10mm = 7.469 mm/μs was the observed velocity in a 10 mm diameter stick.³⁶ Note that the difference is only 0.061 mm/μs, or less than 1%. This Δ(D) would result in $Δ[zndf]<0.04 mm$ (Fig. 8) and $Δ[tndf]=5 ns$ (Fig. 7).

In these experiments, it is somewhat difficult to precisely determine the distance from the top of the charge to the x-ray beam. The procedure we used is as follows. An assembly containing the LTD detonator, the HE cylinder, and the PZT pin is mounted on an x–z micro-positioning translation stage. The stage is moved through the x-ray beam and the intersection of the top of the HE cylinder and the PZT pin, z = L_HE, is located by plotting the transmitted x-ray intensity vs. z. With this procedure, the top of the HE is found with an uncertainty of ∼20 μm.²³ After the end (top) of the HE cylinder is found, the assembly is translated vertically so that, in the detonation experiment, x-rays intersect the HE at z = z_beam.

Figure 9 shows the uncertainty in the relative time, $Δ(tndf)$⁠, as a function of the uncertainty in the location of the x-ray beam relative to the end of the HE, Δ(L_HE – z_beam). Note that we have overlaid graphs for all four x-ray times.

Figure 10 shows the uncertainty in the relative detonation front position, $Δ(zndf)$⁠, as a function of the uncertainty in the location of the x-ray beam relative to the end of the HE, Δ(L_HE − z_beam). Note that we have again overlaid graphs for all four x-ray times. The uncertainty in position scales 1:1 with the uncertainty in L_HE – z_beam.

It is important to use the correct timing between the detonation front and when the x-rays strike the sample. In all experiments, an oscilloscope records both the camera gate signals and the PZT pin signal. If t_n is the time, an x-ray pulse strikes the HE sample, then

where $tnscope$ is the camera gate time recorded on the oscilloscope, and C₁ represents all the delays in this signal path. Likewise if t_pin is the time when the detonation front strikes the PZT pin

where $tpinscope$ is the PZT pin signal recorded on the oscilloscope and C₂ represents cable delays, etc., in the signal path between the PZT pin and the oscilloscope. The quantity that appears in Eqs. (5) and (6) is t_n – t_pin. Rearranging Eqs. (9) and (10) results in

This is in the form of a measured time minus a correction factor. The correction factor (C₁–C₂) can be found if we make a measurement of $tnscope−tpinscope$ in a configuration such that t_n − t_pin = 0. In other words, we use the arrival of an x-ray pulse at the PZT pin to generate an electrical signal. In summary, the PZT pin to x-ray pulse cross timing, Δ(t_n − t_pin) was performed by directing x-ray pulses to the end of the PZT pin. Each x-ray pulse presumably generated a small stress pulse in the end of the PZT pin, and small electrical pulses were generated. The x-ray generated electrical pulses from the PZT pin were recorded along with the CCD camera gate signals on an oscilloscope. Cabling, optical geometry, and other instrumentation were identical to that used in the experiments. The time difference between the leading edge of the camera gates and the x-ray pulse induced PZT pin output was recorded and used to determine the time correction factor, (C₁–C₂). We estimate that PZT pin to x-ray pulse cross timing, Δ(t_n − t_pin) is ≈ 2 ns or 0.002 μs.

Figure 11 shows the uncertainty in the relative time, $Δ(tndf)$⁠, as a function of the uncertainty in PZT pin to x-ray pulse cross timing, Δ(t_n − t_pin). Note that we have overlaid graphs for all four x-ray times. The uncertainty in relative time is, perhaps obviously, directly equal to the uncertainty in the cross timing.

Figure 12 shows the uncertainty in the relative detonation front position, $Δ(zndf)$⁠, as a function of the uncertainty in PZT pin to x-ray pulse cross timing, Δ(t_n − t_pin). Note that we have again overlaid graphs for all four x-ray times.

In many cases, the detonation front may have more curvature than is shown in Fig. 5. The curvature can arise from the detonation being initiated in a small volume and spreading outward (roughly) spherically. In this case, the detonation front will be curved in inverse proportion to the distance from the center of initiation. At small distances the front will be highly curved; at large distances the front will be flatter.

Detonation front curvature can also be caused by chemical reaction zone effects, particularly in explosives with a long CRZ.³⁷ The result is that in long cylinders or sticks of explosive, far from the initiation point, the detonation front is still curved. For example, in a 10 mm diameter stick of the TATB based explosive PBX 9502, the detonation front at the center will be 0.78 mm ahead of the edge.³⁸ 

Figure 13 shows calculated detonation fronts for Shot #8, April 2015 at x-ray times t₁ through t₄. The detonation front construction was designed to create a center to edge lag of ∼0.8 mm or ∼100 ns near the end of the HE; black curve, t₄. This is similar to the lag observed in long sticks of PBX 9502³⁸ and in direct numerical simulations of the entire explosive system. (The construction used a virtual point source of initiation 6 mm below the bottom of the HE and a spherical detonation front expanding at 7.786 mm/μs.)

Using the calculated detonation fronts shown in Fig. 13 we see that the x-ray pulse at t₂ (green curve) would sample both reactant and product. This is consistent with the t₂ (green) scattering pattern seen in Fig. 4. If the detonation front had as little curvature as those shown in Fig. 5 the t₂ (green) scattering pattern would not be a superposition of undetonated and detonated HE; it would only be detonated HE. In summary, typically curved detonation fronts result in the x-ray beam passing through HE products which are various distances from the detonation front. Recorded scattering patterns, therefore, are superpositions of many differing states.

PZT (Dynasen CA-1135 × 2) pins are centered by eye on the top of the HE. The pins are 1.63 mm in diameter. We estimate that we can center the pins to ∼ 1/10^th the charge diameter or 1 mm. Assuming curvature as in Fig. 13, this would result in an effective uncertainty in t_pin of ∼ 4 ns.

In this article, we summarized the experimental setup for recording Time-Resolved SAXS measurements on detonating explosives at the Advanced Photon Source, a third generation synchrotron light source. We have presented methods to associate the recorded SAXS patterns with times and positions in the reference frame of the detonation front. Further, we have calculated uncertainties in those times and positions using Monte-Carlo methods.

Table II summarizes the major sources contributing to the relative distance and time to the detonation front. The largest source of uncertainty is the detonation front curvature followed by uncertainties in detonation velocity. Combining all uncertainties, we have uncertainties of 0.8 mm in position and 100 ns in time. Again, this is overwhelmingly generated by the detonation front curvature. By comparison, if the detonation front were planar, the uncertainties in position would be ∼0.1 mm in position and ∼10 ns in time, and the uncertainties would be generated largely by uncertainties in the detonation velocity, Δ(D).

The approach we have outlined for calculating the uncertainties in positions and times, relative to the detonation front, at which the x-ray pulses interrogate the sample can be applied more generally to similar geometries and other explosives. Suggested improvements to the experiment include; (1) higher x-ray fluxes such as those that can be generated by high brilliance fourth generation sources such as x-ray free electron lasers. (2) Larger high explosive charges that would allow the detonation wave to reach a steady velocity. Larger diameter charges would also result in the pressure behind the detonation front being maintained at a higher pressure for a longer time. At present, the size of the explosives charge is limited by APS facility rules. (3) Internal fiducials could be used to better diagnose the detonation flow in-situ. For example, radiographic methods are currently being developed that have the potential to measure the detonation velocity and detonation front curvature.

Los Alamos authors acknowledge the support of the U.S. Department of Energy through the LANL/LDRD Program (LANL No. 20150050DR) and the Dynamic Materials Campaign. Lawrence Livermore authors were primarily funded by LLNL-LDRD (14-ERD-018) and performed their work under the auspices of the U.S. Department of Energy, Lawrence Livermore National Laboratory, Contract No. DE-AC52-07NA27344. The Dynamic Compression Sector at the Advanced Photon Source (DCS@APS) is managed by the Washington State University and funded by the National Nuclear Security Administration of the U.S. Department of Energy under Cooperative Agreement No. DE-NA0002442. 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.