Quantitative x ray phase contrast imaging of oblique shock wave–interface interactions Open Access

Simulation of shock wave–matter interactions with hydrocodes requires careful calibration of physical models, for example, EOS, material strength, and kinetic phase transitions. While these models are commonly tuned from one-dimensional (1D) measures of, for instance, wave speed, particle speed, stress, and strain, they need to be tested and further refined with higher dimensional measurements to accurately simulate complex shock-induced material responses (e.g., energetic materials). Therefore, we develop techniques to measure the mean value of the average mass density (areal density divided by thickness) over region $S$ (⁠ $B S$⁠) and flow deflection angle (⁠ $θ$⁠) using XPCI and test them on a shock wave interacting with an angled interface. These techniques can evaluate the accuracy of numerical methods to predict two-dimensional (2D) behavior of shock waves and phase states around a material interface.

The early work of von Neumann¹ on shock wave reflections from a solid wedge laid the theoretical foundation of oblique shock wave interactions. Many models since then have considered shock wave interaction with other interfacial phases of matter, for example, gas–gas,^2,3 liquid–solid,^4,5 and solid–solid interfaces.^6–8 In some configurations, the analytical approach of shock polar analysis (SPA) dictates a three-shock solution, and in others, there forms a two-shock and a Prandtl–Meyer expansion (PM) fan configuration. In SPA, the Rankine–Hugoniot shock jump relations⁹ are combined with constitutive equations (e.g., Mie–Grüneisen,¹⁰ shock wave-particle velocity,¹¹ and tabulated¹² EOS) to provide a analytical expression between pressure and $θ$ (shock polars) of the shocked states. By imposing uniform pressure and velocity across the interface (impedance matching), and knowing the incident shock speed, the shocked states can be determined. This process can be shown graphically by plotting the shock polars for each shock (and rarefraction) wave and locating where the shock polars intersect. The applicability of SPA lies in assuming that the shock front is steady, that body forces (e.g., surface friction), and diffusion are negligible.

Time-dependent effects such as material jetting,¹³ phase transitions,¹⁴ and material heterogeneity¹⁵ are not captured by SPA, in which case numerical modeling is used.^6,8 Calibrating constitutive models using an oblique shockwave as a test case offers great insight for simulating more complex shockwave–matter interactions, e.g., optical elements subjected to intense synchrotron x rays,^16,17 hydrodynamic instabilities in inertial confinement fusion (ICF),¹⁸ blast-induced damage to vital organs,^19–21 ultrasonic-assisted drug delivery,²² and projectile impact on light weight structural materials.²³ 

Imaging has been an effective tool for visualizing and quantifying the dynamic behavior of oblique shock waves. Early on, high speed optical imaging methods provided first insight into oblique shock waves in shock tubes.^24,25 These images have been further enriched with mass density, pressure, and temperature measurements extracted from the image intensity. However, optical methods are unable to visualize shock waves in optically opaque solids. Consequently, other forms of radiation have been investigated. X ray imaging combined with the emergence of highly brilliant x ray sources (flash x rays,²⁶ betatron x rays,²⁷ synchrotron,^28–34 x ray backlighting^35–39) have been used to study and quantify shock waves.^{31,34,40–42} However, the attenuation contrast of (particularly weak) shock waves is insufficient to overcome the large image noise associated with dynamic imaging. Proton radiography provides better contrast of shock waves^43–46 with a reported measured mass density accuracy of 1%.⁴⁴ However, its spatial resolution is inferior to x ray imaging due to multiple Coulomb scattering.⁴⁷ 

XPCI is ideal for imaging shock waves because of its phase-induced contrast enhancement of shock fronts and material interfaces, and its sensitivity to mass density due to attenuation.^48,49 Specifically, synchrotron XPCI possesses the brilliance and coherence to perform high speed imaging. Antonelli et al.³³ was one of the first to successfully demonstrate the feasibility of XPCI imaging shock waves with a synchrotron source. They demonstrated strong phase contrast of shock waves propagating along a cylindrical nylon wire where their behavior showed good qualitative agreement with hydrodynamic simulations using the 2D Lagrangian fluid code DUED.⁵⁰ Yanuka et al.^30,34 further demonstrated that areal mass densities can be computed from XPCI images. They compared areal mass densities calculated from shock waves generated from exploding wires in water with hydrodynamic simulations. However, they ignored phase contrast effects when computing areal mass density. Conversely, Barbato et al.³¹ performed phase retrieval before quantifying the mass density of a laser-driven shocked plastic material. Outside of synchrotrons, XPCI has also been performed at high energy density physics research facilities such as OMEGA and the National Ignition facility to study nuclear fusion materials and recover electron density.^51,52 In short, although XPCI studies on shock waves in different geometries have been performed and $B S$ measured, investigations have not been done in using XPCI to study oblique shock waves–interface interactions or quantify the accuracy and uncertainty of $B S$ as well as $θ$⁠.

In this work, we dynamically captured propagated-based XPCI images at a synchrotron of impact-driven oblique shock waves impinging on an interface between different combinations of solid polymers polymethylpentene (TPX)^TM, polytetrafluoroethylen (Teflon), and polymethyl methacrylate (PMMA). We developed a Hough transform-based approach⁵³ to calculate $θ$ and segmented the un-shocked, shocked, and rarefracted regions. Within each segmented region, we presented a Monte Carlo-based approach that incorporates the statistical properties of the attenuated image intensity to compute $B S$⁠. The polymers in this work were chosen as they have well-defined EOS models that allowed us to use SPA and hydrodynamic simulations as benchmarks to interpret and evaluate the accuracy of the measured quantities from XPCI images. Based on this work, we anticipate using XPCI in shock polar configurations to accurately determine the EOS of materials with redundant measurements similar to what was done here.

All shock experiments used both the single (hutch D) and two (hutch E) stage light gas guns at the Dynamic Compression Sector (DCS) of the Advanced Photon Source (APS), Argonne National Laboratory (Fig. 1).⁵⁴ A cylindrical polymer impactor was launched at velocities measured by a Photon Doppler Velocimetry (PDV) probe pointed at the impactor of up to 6 km/s toward the sample. The sample was composed of a polymer target that was adhered to a polymer window with a $≤$15  $μ$m thick 1.15 g/cm³ density AngstromBond AB9110 epoxy (Fiber Optic Center, Inc.) to form a material interface. The impactor was a symmetrical impactor, i.e., the same material was used for both the impactor and the target. Also, the window was coated with aluminum (Al) using physical vapor deposition at the material interface end to increase the reflected PDV signal (the results of which are not included in this work). While traveling down the barrel, the impactor broke an optical beam, allowed x rays into the target chamber of the gas gun through the Lexan vacuum window to illuminate the sample. Upon impact, a shock wave was propagated at a nominal angle of $30 °$ with respect to the material interface. Nearly, simultaneously, with the impact of the sample, the impactor struck the lead zirconate titanate (PZT) pin, triggering acquisition on DCS’s in-house built multi-frame phase contrast imaging (MPCI) system, similar to that described previously,^55,56 to record the transmitted x rays exiting the target chamber through another Lexan vacuum window.

APS was operated in the standard 24-bunch top up mode where a 33.5 ps RMS electron pulse bunch arrived every 153.4 ns at a 27 mm period undulator (U27) with a 20 mm gap. Each electron pulse bunch was converted into a x ray pulse with the first harmonic peaking at 14.8 keV [Fig. 2(a)].

MPCI was comprised of four $2048×2048$ pixel [13.5  $μ$m charged-coupled detector (CCD) pixel size] PI-MAX intensified charge-coupled device (ICCD) detectors (Teledyne Princeton Instruments, Inc.) coupled to a $10×$ objective lens to produce an overall magnification of $9×$ and an effective pixel size of 1.499  $μ$m. The objective lens was focused onto a 150  $μ$m thick single crystal lutetium oxyorthosilicate (LSO) scintillator positioned 1 m from the sample. 100s of nanometer thick Al films were deposited on the side of the LSO facing the sample to increase the signal to the detectors. LSO was chosen because its decay rate of 43 ns⁵⁷ was much shorter than the x ray pulse rate of 153.4 ns. This minimized ghosting artifacts. When triggered by the PZT pin, each detector, in turn, recorded a single image of the x ray pulse gated to a 140 ns exposure time, bringing the total number of images recorded per trigger to 4. For each shock experiment of a sample, the following images were recorded from each detector: (1) flat field (sample absent), (2) dark field (detector shutter closed), (3) sample before impact, and (4) sample during impact.

In the following, we describe the steps taken to pre-process and segment XPCI images before calculating $θ$ and $B S$ (Fig. 3):

1. XPCI image of the sample was flat field and dark field corrected to normalize against the incident x ray beam intensity and offset the dark current arising from the detector, respectively.

2. A circular binary mask was created using Hough transform and applied to segment the central part of the x ray beam.

3. Images were denoised using Total-Variation (TV) denoising with a regularization parameter value of $λ=0.3$ and the number of iterations set at 150.⁵⁹ After denoising, the image intensity was rescaled to match the mean intensity that it had before denoising.

4. TV-images were binarized using the Canny edge detection method⁶⁰ with the threshold tuned visually to maximize the number of segmented pixels belonging to the shock fronts and material interface, and minimize pixels that belonged to the background. The threshold needed to be tuned differently for each segmented region due to spatially varying levels of image noise resulting from the non-uniform x ray beam profile.

5. Binarized TV-images were delineated into regions bounded by the shock front and material interface using Hough transform.

6. Delineated regions were morphologically eroded to mask out the phase contrast induced by the wave fronts and material interface.

After completing these steps, the following regions were segmented from the XPCI image (Fig. 3): un-shocked (UST; red), incident shocked (IST; green), and reflected (RST; periwinkle) regions of the target, as well the un-shocked (USW; cyan) and transmitted shocked (TSW; purple) regions of the window. These regions were bounded by the undeflected (blue) and deflected (yellow) material interface, incident shock front (red), reflected wave front (green), and transmitted shock front (pink).

The following is a generalized approach to calculating the $B S$ PDF from a recorded XPCI image. First, we derive an analytical model that accounts for the x ray energy spectrum, attenuation through various materials in the x ray beam path, and non-linear effects of the scintillator. Next, we incorporate the statistical variation of the image intensity arising from the bunch-to-bunch variations in the electron position and population to derive its PDF. Finally, we outline our Monte Carlo method using the analytical model and intensity PDF to determine the $B S$ PDF.

We present an analytical model relating the average mass density (⁠ $B$⁠) to its flat and dark field corrected XPCI image intensity $C ^ i j( r → m;B)$ at the detector plane $r → m=( x m, y m)$⁠, with $( x m, y m)$ representing the $m$th pixel coordinate of the image, recorded over the $i$th x ray pulse, and normalized by the flat field recorded over the $j$th x ray pulse. In this model, we assumed only x ray attenuation and neglected phase contrast and inelastic scattering. Phase contrast effects were neglected since the regions of analysis were chosen far from sharp interfaces (i.e., shock fronts and material interface). Inelastic scattering was also neglected because its contribution to the image intensity was found using the Monte Carlo M-Particle Transport Code (MCNP) to be insignificant, which will be discussed in Sec. IV.

Here, $I i( r → m;B)$ and $I j( r → m;0)$ represent the sample and flat field images recorded over the $i$th and $j$th x ray pulse, respectively. $D( r → m)$ is the dark field image. $μ(E)$ is the mass attenuation. It is calculated using X ray Oriented Programs (XOPs)⁵⁸ under ambient conditions and is assumed unchanged under the high pressures reached in this experiment.⁶²  $B=(1/t) ∫ zρ(z)dz$ is defined as the mass density $ρ(z)$ averaged over the sample thickness $t$ along the $z$ direction. The total conversion gain of the detector $G$ after the scintillator remains unchanged throughout these experiments, thus its effects cancel out in $C ^ i j( r → m;B)$⁠. The unit of measurement assumed for each variable in Eq. (1) and thereafter are listed in Table I.

$w i(E, r → m)$ consists of five terms: (1) x ray flux spectrum of U27 with a 20 mm gap, (2) attenuation by a $t k=125μ$m thick Kapton film (⁠ $ρ k=1.42$ g/cm³)⁶³ that seals the end of the vacuum tube, (3) attenuation by Lexan vacuum windows (⁠ $ρ L=1.2$ g/cm³),⁶³ used to seal the gas gun target chamber, totaling a thickness of $t L=4.77$ mm, (4) attenuation by $t A=126.32$ m of air (⁠ $ρ A=1.205× 10 − 3$ g/cm³)⁶³ between the end of the vacuum tube and detector, and (5) LSO conversion efficiency (defined as the number of visible light photons produced per unit energy of x rays incident on LSO).

Both the LSO conversion efficiency and x ray flux spectrum vary with x ray pulse $i$⁠, the reason for each are elaborated below and in S1 of the supplementary material, respectively.

The LSO conversion efficiency is a product of the number of x rays absorbed by the scintillator [i.e., $exp(− μ ~ L S O(E) t L S O)$ where $ρ L S O=7.4$ g/cm³],⁶⁴ and the number of visible photons emitted per absorbed x ray [light yield; $ζ i(E)$]. Previous studies have shown that $ζ i(E)$ varies between LSO crystals and, for the same LSO crystal, $ζ i(E)$ can change before and after intense exposures to x ray radiation and thermal heating.^65–68 DCS produces extremely bright x ray pulses and, therefore, likely altered $ζ i(E)$ during the course of our experiments on a x ray pulse-to-pulse basis. While $ζ i(E)$ could be measured each time before a sample is tested, it is time consuming. Therefore, we calculated $ζ i(E)$ post hoc by modeling it as: $ζ i(E)= ζ R(E)exp(−μ(E) t i)$⁠. That is, $ζ i(E)$ is equal to that of a reference LSO light yield, $ζ R(E)$⁠, attenuated by a material with attenuation coefficient $μ(E)$ and thickness $t i$ that varies with x ray pulse $i$⁠. Since for most materials $μ(E)$ decreases generally by $E − 4$⁠, any material can be used as long as $μ(E)$ does not include any absorption edges within the energy spectrum of the x ray source.⁶⁹ For this work, we chose Al where its attenuation coefficient and thickness are denoted as $μ A l(E)$ and $t A l , i$⁠, respectively. Al is imaginary in that $t A l , i$ can take on negative or positive values. When $t A l , i>0$ and $t A l , i<0$⁠, LSO light yield is lower and higher than that of the reference LSO, respectively. Figure 4(a) demonstrates how well our model fits to $ζ i(E)$ measured from three different LSO scintillators (LSO 1–3).⁶⁵ We chose LSO 1 as $ζ R(E)$ and fitted our model to LSO 2 and LSO 3 to return $t A l , i$ values of 89 and 40  $μ$m, respectively. The fitted and measured $ζ i(E)$ show good agreement. Therefore, $ζ i(E)$ can be determined by only needing to know $t A l , i$⁠. Section II C 3 will describe how we calculated $t A l , i$⁠.

We incorporate the stochastic variation of the image intensity into Eq. (1) to derive its PDF. Then, the PDF of the mean image intensity sampled over $N$ pixels is derived.

To arrive at Eq. (3), we had made five assumptions: (1) $η i(E, r → m)$ is equal to $η ¯ i(E, r → m)$ scaled by the x ray flux $ψ i( r → m)$ emitted from electron bunch $i$⁠, (2) $ψ i( r → m)$ can be decomposed into a product of $A i$⁠, which is equal to the number of electrons per electron bunch (the scale factor between the $i$th and $j$th x ray pulse is given by $F i j= A i/ A j$⁠), and the spatial x ray flux profile $⟨ P ( r → m ) ⟩ T$ averaged over time $T$⁠, (3) random fluctuations in $C i j( r → m; B S)$ are dominated by $ψ i( r → m)$⁠, (4) $Std [ C i j ( r → m ; B S ) ]$ at an expected value of unity is denoted by $σ 0$ and is constant, and (5) $t A l , i≈ t A l , j$ since both the flat field and sample image were recorded closely between x ray pulses.

The significance of Eq. (6) is that the expected value of $C ¯ i j( r → m ¯; B S)$ is proportional to its standard deviation if the material type and mass density is constant in $S$⁠. As a consequence, if the proportionality factor (i.e., $Std [ C ¯ 0 ( N ) ]$⁠) is known then the standard deviation can be immediately obtained from the expected value, or vice versa. If they were not proportional then the standard deviation would need to be measured independently by recording multiple images of the same material under the same thermodynamic state. This would be particularly difficult to obtain for shocked materials.

Therefore, Eq. (6) provides the following strategy for computing $C ¯ i j( r → m ¯; B S)$ over a given $N$ pixel region: (1) approximate $⟨ C ¯ i j ( r → m ¯ ; B S ) ⟩ T$ as the mean intensity over the $N$ pixel region $S$ and (2) compute $Std [ C ¯ i j ( r → m ¯ ; B S ) ]$ by multiplying $Std [ C ¯ 0 ( N ) ]$ with $⟨ C ¯ i j ( r → m ¯ ; B S ) ⟩ T$ calculated in (1).

$Std [ C ¯ 0 ( N ) ]$ was determined from a $Std [ C ¯ 0 ( N ) ]$ vs $N$ pixel calibration curve. To generate the calibration curve, several recorded flat field images were flat and dark field corrected (note that different flat fields were used to correct the flat fields), then scaled to a mean intensity of unity. Then, the images were partitioned into $N$ pixel regions and in each region the mean intensity was computed. This was repeated for different $N$ size regions. The distribution of mean intensities for each $N$ size region are plotted in Fig. 5(a), where finally calculating their standard deviations generates a $Std [ C ¯ 0 ( N ) ]$ vs $N$ pixel region calibration curve shown in Fig. 5(b).

Returning to Eq. (7), $C ¯ 0(N)$ is normally distributed. In general, the sum of normally distributed correlated random variables is not always normally distributed. For our imaging setup, the pixels are correlated due to effects such as the detector point spread function. It would suggest then that $C ¯ 0(N)$ may not be normally distributed. However, since for our case each correlated pixel draws from the same set of independent normally distributed image intensities in $r → m$⁠, $C ¯ 0(N)$ must be normally distributed. This is supported by a Gaussian function fitted to each histogram in Fig. 5(a) with $R 2>0.99$⁠.

Equipped with the expression for $C ¯ i j$ [Eq. (6)], we can solve for the $B S$ PDF. That is, we can propagate the PDF from $C ¯ i j$⁠, due to the stochastic fluctuation in the x ray flux, to $B S$ so that the expected value and precision of $B S$ can be computed under the framework of our model. Unfortunately, analytically propagating the PDF from $C ¯ i j$ to $B S$ is not trivial. Instead, we develop a Monte Carlo-based method that repeatedly draws from $C ¯ i j( r → m ¯; B S)$ to numerically solve for the $B S$ PDF (Fig. 6).

Hydrodynamic simulations were performed to test the accuracy of $B S$ and $θ$ measured from XPCI images. SPA was also performed to test its accuracy as a first order approximation to the hydrodynamic simulations. Detailed descriptions of SPA and the hydrodynamic models can be found in Romick et al.⁸ Here, only specific details are provided that are pertinent for comparison with the XPCI images.

For SPA, thermodynamic steady states were calculated though the Rankine–Hugoniot equations while utilizing the Mie–Gruneison EOS with a reference isentrope given by a Murnaghan functional form for each material under ambient conditions, and the impactor projectile velocity. At the material interface, the epoxy and Al layers were ignored, while surface tension, diffusive effects, and other body force were assumed negligible. Consequently, continuity in pressure and deflection angle was enforced across the material interface.

For the hydrodynamic simulations, a 2D geometric model of Fig. 1(a) at a depth equal to the center of the detector field-of-view (FOV) was created. That is, from the viewpoint of Fig. 1(b), the model was a horizontal slice into the page crossing the center of the detector FOV. The sample was surrounded in atmospheric air. Detailed time-dependent flow behavior were modeled using a level set based ghost fluid method governed by Euler equations. Temporal discretization of Euler equations was performed with a third order TV diminishing Runge–Kutta method, while spatial discretization was evaluated by a combination of fifth-order mapped weighted essentially non-oscillatory and Lax–Friedrichs finite difference schemes. Initially, simulations with the epoxy layer and a $0.5μ m$ resolution were performed, and it was found that the effects were only noticeable very near the driving shock. However, the thermodynamic states and flow deflection angle, the focus of this work, were negligibly affected. Therefore, hydrodynamic simulations excluded the epoxy (and Al) layer and produced 2  $μ$m resolution output images for achieving computational efficiency.

We investigated three cases of a shock wave interacting with an oblique solid–solid interface, a shock wave traveling from (1) a solid target (TPX; $ρ T P X=0.83$ g/cm³) of lower impedance to a solid window (PMMA; $ρ P M M A=1.186$ g/cm³) of higher impedance, (2) a solid target (Teflon; $ρ T P X=2.152$ g/cm³) of higher impedance to a solid window (PMMA) of lower impedance, and (3) a solid target (PMMA) of equal impedance to another solid window (PMMA). In this section, we first tested and validated our Monte Carlo method on XPCI image recordings of the three cases before impact. Then, we applied our Hough transform and Monte Carlo method on the XPCI images recorded during impact to compare with SPA and hydrodynamic simulations.

Flat and dark field corrected XPCI images of TPX/PMMA, Teflon/PMMA, and PMMA/PMMA material interfaces before impact are presented in Figs. 7(a), 8(a), and 9(a), respectively. The target and window are positioned, respectively, on the left and right side of the material interface.

We segmented the target and window following the steps in Sec. II B. These are shown for the three cases in Figs. 7(b), 8(b), and 9(b). Each shaded region represents the largest area possible that can be segmented without including phase contrast affected regions, particularly, near the material interface, and edge of the x ray beam, where the assumption that $F$ is constant breaks down.

In each shaded region, we calculated $B S$ PDF using our Monte Carlo method, and plotted the expected value of the $B S$ PDF with uncertainty bars corresponding to the 95% confidence interval (CI) range of the $B S$ PDF in Fig. 10. The markers are colored according to the sample ID and shaped to correspond to camera 1 through 4 (C1, C2, C3, C4). Since the ambient $ρ S$ is nominally constant throughout the material, $B S≈ ρ S$⁠. Thus, the known $ρ S$ for each material is plotted as a black horizontal line to directly compare with our Monte Carlo method. Figure 10 shows that the expected $ρ S$ lies within the uncertainty range in $∼$94% of the measured $ρ S$⁠. This is consistent with having assigned the uncertainty to equal the 95% CI range of the $B S$ PDF. Interestingly, there appears to be a random offset in $B S$ between samples. For example, Fig. 10(b) shows that $B S$ for C1–C4 lie in samples 4–18 above and for sample 5–18 below the expected value. One possible reason is variations in the LSO light yield are not perfectly captured by our $ζ i(E)$ model. Regardless, Fig. 10 shows that our statistical model for $C ¯ i j( r → m; B S)$ is able to compute $B S$ with $<0.1%$ accuracy and 1%–2% precision corresponding to a region $S$ of the order of $10 4$ pixels. In Sec. S1 of the supplementary material, we showed that assuming $σ 0$ is constant introduces up to $55.4%$ error in estimating the precision of $C( r → m;B)$ and therefore $B S$⁠. This means the actual precision of $B S$ may be down to 1%–4%. In the dynamic images, the segmented regions are of the order of $10 3$ pixels, further lowering the precision of $B S$ but only to 3%–5%.

Figures 7(c), 8(c), 8(e), and 9(c) show flat and dark field corrected XPCI images of an impact-induced oblique shock wave traversing the material interface for the three cases (their unprocessed XPCI images are shown in Fig. S1 of the supplementary material). We segmented the shocked regions from these figures and shaded them in Figs. 7(d), 8(d), 8(f), and 9(d), respectively. We applied our Monte Carlo method to calculate $B S$ in each of the segmented regions and our Hough transform-based method to compute $θ$⁠. To determine the uncertainty in $θ$⁠, we gradually decreased the angle bin size of the Hough transform until the detected line angle stayed constant. This was found to be $0.1 °$ and since the deflection angle depends on two detected lines, the total uncertainty is $0.2 °$⁠. Here, we added the uncertainties linearly rather than in quadrature because the correlation between the detected angles is not known. Thus, we calculated the maximum possible uncertainty.

All values of $B S$ and $θ$ measured from XPCI, SPA, and hydrodynamic simulations for this work are tabulated in Table S1 of the supplementary material.

Figure 11 compares $B S$ between XPCI and SPA. The black curves represent shock polar solutions for the shocked states behind the incident (solid), reflected (dashed) and transmitted (dotted) waves. The flow deflection angle at $θ= 0 °$ is parallel to the undeflected material interface. There is no way of calculating the deflection angle behind the incident shock front (defined as $θ I$ in Table S1 of supplementary material) from XPCI. Consequently, for the sake of comparing $B S$ behind the incident shock front, we assumed $θ I$ is equal for XPCI, SPA and the hydrodynamic simulations.

In all three cases, the ambient $B S$ and $θ$ calculated from the XPCI images (open circle) and SPA (filled circle) agree within uncertainty as expected.

In the TPX/PMMA case, SPA and XPCI both calculated an increase in $B S$ from the incident to the reflected region. Similarly, for the Teflon/PMMA case, both methods calculated a decrease in $B S$ from the incident to rarefracted region. These results demonstrate that both XPCI and SPA correctly infer that TPX/PMMA and Teflon/PMMA produced a reflected shock and a PM fan, respectively. To directly validate this requires reconstruction of the mass density across the shock front from the XPCI images, which is beyond the scope of this work but is a topic of future work in Sec. V.

In the PMMA/PMMA case, both SPA and XPCI calculated no significant difference in $B S$ between the incident and transmitted shocked region. This was expected since the target and window have matching impedance. However, Fig. 9(c) shows a faint bright line (as pointed out by an orange arrow) approximately where the reflected wave front would be if there was an impedance difference between the target and window material. We investigate possible sources of the bright line in Sec. IV.

Overall, XPCI and SPA showed agreement in the sign change of $B S$ between shocked/released regions, but XPCI predominantly underestimated $B S$ compared to SPA. This underestimation is further explored using time-dependent hydrodynamic simulations in the next section and is discussed in Sec. IV.

We performed detailed time-dependent 2D hydrodynamic simulations of the flow behavior of an oblique shock wave for the three cases. Here, we focused on the simulation result of the TPX/PMMA case in Fig. 12. The Teflon/PMMA and PMMA/PMMA cases show similar results and can be found in Figs. S3 and S4 of the supplementary material, respectively.

Figure 12 (top row) tracks the materials from the impactor, target and window during impact. Corresponding mass density maps are shown in Fig. 12 (middle row), where the TPX impactor makes contact with the TPX target and generates an incident shock wave into the target and back into the impactor. The shock front is curved by dominant edge release waves propagating outwardly from the target boundary. We also see a jet of material from the impactor flowing parallel with the incident shock wave because of the impactor having a larger diameter than the target. When the incident shock wave reaches the TPX/PMMA material interface, it splits into a reflected and transmitted shock wave.

Two red vertical lines drawn 1.9 mm apart in Fig. 12 (middle row) represent the x ray beam width traveling vertically down along the material interface. After applying a 2D Gaussian filter with a standard deviation of two pixels on the mass density map to reduce discrete sampling artifacts along the material/air boundary (see Fig. S2 of the supplementary material for more details about the discrete sampling effects), the mass density map was integrated along the x ray beam and divided by the initial material thickness (i.e., 0.394 in. for the impactor, and 0.135 in. for the target/window) to calculate $B$ (orange curve). In addition, the jetted material was cropped from the right column of images before calculating $B$⁠. The reason was hydrodynamic simulations showed that the jetted material arrived at the material interface at approximately the same time as when the reflected shock wave was created. XPCI images showed otherwise (see Sec. IV for a further discussion). Two additional curves are also plotted in Fig. 12 (bottom row): $B$ computed from SPA (pink curve), and $B$ calculated from simulated XPCI profiles (XPCI profiles were simulated using the angular spectrum method⁴⁸) of the orange curves using Eq. (1) (blue curve). The oscillations seen from the simulated XPCI profiles (blue curve) are due to phase contrast arising from the material interface and shock fronts. As mentioned in Sec. II C 1, our model ignores the effects of phase contrast. Therefore, the shocked regions were chosen away from the phase contrast affected regions as was done in Figs. 7(b), 7(d), 8(b), 8(d), 8(f), 9(b), and 9(d).

Turning our focus onto the shaded regions in Fig. 12 (bottom row), $B$ calculated from both the hydrodynamic simulations (orange curve) and simulated XPCI (blue curve) are in good agreement and are lower than that calculated from SPA (pink curve). It is clear then from these simulations that the curved shock front arising from the release waves is the reason why in Fig. 11 XPCI underestimates $B S$ compared to SPA. SPA intrinsically assumes materials are of infinite size and, therefore, does not include finite sample size effects.

Next, the $B$ values for each shaded region of the hydrodynamic simulations (orange curve) in Fig. 12 (bottom row) are plotted in Fig. 11 as a box. The box for each shaded region is vertically centered at the mean $B$ (i.e., $B S$⁠) with a vertical size equal to the range of $B$ values within the shaded region. It is horizontally centered at the same $θ$ as that calculated from SPA after Romick et al.⁸ had showed that $θ$ calculated from both methods were similar. The horizontal size of the box was optimized for visual clarity and does not represent the uncertainty in $θ$⁠, which is zero. Figure 11 shows XPCI is in close agreement with the hydrodynamic simulations, further validating our Monte Carlo and Hough transform-based methods as a quantitative tool for studying shocked materials.

Several factors were considered but not included in Eq. (1) because they did not contribute significantly to the image. We discuss these factors here as they may become significant in other experimental configurations.

Compton scattering contributes a large portion to the total interaction cross section of 10–20 keV x rays with matter. A material of uniform thickness and mass density as is the case for our samples causes x rays to scatter uniformly across the image. As a result, ignoring the effects of Compton scatter can underestimate $B S$⁠. For our study, we employed MCNP to model our experimental setup and simulate the signal from Compton scatter recorded on the detector.⁶⁹ We found that <1% of the detector signal were attributed to Compton scattered x rays (Table S2 in the supplementary material). Many of the Compton scattered x rays missed the detector due to the large sample-to-detector distance. Therefore, adopting a large sample-to-detector distance is a useful strategy to minimize Compton scatter. There are cases where this is not possible, however; for instance, imaging at x ray energies >20 keV or Compton scatter from materials outside the x ray beam falling on the detector. For these cases, a post hoc Monte Carlo-based simulation of the scatter map may be a viable option to correct for Compton scatter.⁷⁰ 

Scintillators often exhibit both short and long decay time components that can last from nanoseconds to hours. They can unfortunately produce unwanted ghosting or background signal in the images called afterglow. While LSO scintillators are known to only have a single decay time of 43 ns,⁵⁷ which is less than the time of 153.4 ns between x ray pulses, studies have shown evidence of afterglow that possibly stem from thermoluminescence and stray electrons between electron bunches.^64,71,72 Also, in our calculation of $B S$⁠, we assumed $ζ i(E)$ was approximately constant between the XPCI image of the sample and its corresponding flat field image because they were recorded immediately after one another. While this was proven to be a good approximation for our case because of the agreement in $B S$ values calculated from the XPCI images and hydrodynamic simulations, $ζ i(E)$ may significantly vary from x ray pulse-to-pulse under brighter x ray sources or for different scintillators. These findings warrant further investigation on the performance of scintillators particularly after long term exposures to intense x ray pulses.

The MPCI system was modeled to perfectly convert photons outputted from the scintillator into an electrical signal. In reality, each detector is a cascaded system uniquely coupling the scintillator, objective lens, image intensifier and photodiodes. While these differences in coupling between cameras are mostly removed with flat field correction, Fig. 10 shows evidence of systematic biases in the computed $B S$ between cameras and possibly spatially within cameras. One source of these systematic biases could be from radiation-induced damage to the optical components. Though precision of our current quantitative methods is limited by stochastic fluctuations in the pulse-to-pulse x ray beam intensity, x ray beam stability are expected to improve over time such that cross-calibration of the cameras is needed to further increase measurement accuracy. This is particularly important when comparing dynamic events captured sequentially by different cameras [e.g., Figs. 8(c) and 8(e)].

As mentioned, phase contrast was not included in Eq. (1) because shocked regions that are well away from the material interface and shock fronts/release waves are dominated by x ray attenuation. Still, Fig. 12 (bottom row) shows that phase contrast can add 1%–2% to $B S$ in regions well away from the material interface and shock fronts/release waves. Also, for some segmented regions, there is only a small area from which to compute $B S$ after excluding areas with significant phase contrast. Some possible solutions to reducing phase contrast includes shortening the sample-to-detector distance or increasing the x ray energy. However, it is important to keep in mind that this will also increase Compton scatter and reduce visibility of the shock front and sensitivity to $B S$⁠.

All the imaging factors mentioned above are important to consider when designing XPCI shock experiments for quantitative measurements. Accurately simulating XPCI images ahead of imaging may be an important addition to this work to maximize both shock front visibility and area dominated by x ray attenuation.⁷³ 

Free-space propagation-induced phase contrast translates spatial jumps in $B S$ into bright and/or dark Fresnel fringes on XPCI images. We exploited this to locate the material interface, incident shock, reflected/release wave and transmitted shocked fronts, in order to segment the regions between them and to measure $θ$⁠. Fresnel fringes, however, provide more than boundaries for segmenting, they encode information about the sample response to the propagating shock wave. Here, we discuss what we have inferred about the geometry and behavior of the shocked samples from their XPCI image.

TPX/PMMA [Fig. 7(a)], Teflon/PMMA [Fig. 8(a)], and PMMA/PMMA [Fig. 9(a)] material interfaces appear vastly different in their XPCI image. In Figs. 7(a) and 9(a), there is a prominent dark band with bright fringes on both its sides along the material interface. In contrast, Fig. 8(a) shows two dark bands at the material interface. To identify the physical origin of these features, we performed 2D XPCI simulations of the material interface using the angular spectrum method (Fig. S4 in supplementary material).⁴⁸ We found that two dark Fresnel fringes occur when the epoxy is sufficiently thick and the angle between the material interface and x ray beam direction (or tilt for short) is small. As the tilt increases and/or the epoxy thickness decreases, the two dark Fresnel fringes combine into a single dark Fresnel fringe. Furthermore, spatial separation between the bright and dark fringes increases with tilt. Consequently, in the Teflon/PMMA case, the material interface is angled almost parallel to the x ray beam direction with an epoxy thickness $≳10μ$m. While in the TPX/PMMA and PMMA/PMMA cases, the material interface is also angled almost parallel to the x ray beam direction but with an epoxy thickness $≲10μ$m.

Keeping in mind the epoxy thickness determined for each case, we looked at the behavior of the material interface as it is traversed by the incident shock wave. In the Teflon/PMMA case, Fig. 8(c) shows two dark fringes disappearing behind the incident shock front. This is evidence of the epoxy being compressed. In the TPX/PMMA and PMMA/PMMA cases, their single dark fringe alters slightly in appearance but remains single as the incident shock wave propagation across the material interface. It is difficult to determine whether the epoxy layer expanded or compressed but we know at least that it did not expand beyond $∼$10  $μ$m since two dark fringes would appear. In any case, it is possible to calculate and track from XPCI images changes in the epoxy layer thickness during interactions with a shock wave, but this is beyond the scope of this work.

In addition, we observed a faint dark fringe along the deflected material interface, as indicated by a black arrow in Figs. 7(c), 8(e), and 9(c). The source of these fringes can be explained by looking at the hydrodynamic simulations in Fig. 12. The simulations show that the incident shock front is curved by edge release waves. As a result, the x ray beam projected through parts of the material interface that have and have not been deflected. The faint dark fringes represents the latter.

We investigated the shape of the incident shock front by modeling it as a logistic function and simulating XPCI images using the angular spectrum method.⁴⁸ The incident shock front appeared as bright and dark Fresnel fringes in the simulated XPCI images, as was similarly observed in the recorded XPCI images. Like the material interface, the appearance of the Frensel fringes was sensitive to the angle between shock front and x ray beam direction (not shown). At angles close to zero, the incident shock front appears as a single dark fringe. This single dark fringe splits into a bright and a dark fringe that become increasingly separated as the angle increases. Applying this knowledge to Fig. 7(c), the large separation between bright and dark fringe representing the incident shock front and the single dark fringe representing the material interface indicates that they are not parallel. Similar conclusions can be drawn for Fig. 9(c), whereas for Fig. 8(c), the incident shock front and material interface appear to be parallel. While variation in the angle between the incident shock front and material interface would not significantly affect $B S$ and $θ$⁠, it would be important to incorporate the variation in the angle in a fully three-dimensional (3D) hydrodynamic simulation to accurately simulate and compare the interaction of the shock wave and material interface with the XPCI images.

In all three cases, multiple (secondary) Fresnel fringes was seen trailing behind the incident shock front (primary Fresnel fringes). These secondary fringes are marked by a white arrow in Figs. 7(c), 8(e), and 9(c). It is likely that these secondary fringes represent the material jetting from the impactor. This is supported by the hydrodynamic simulations in Fig. 12. The jetting material causes a sharp increase in $B S$⁠, which, therefore, translates to a pair of bright and dark fringes in the XPCI image. But, there is more than one pair, which may be indicative of perturbations in the jetting material that evolve into Richtmyer–Meshkov instabilities.^74,75 This was not captured by our hydrodynamic simulations due possibly to exclusion of the material strength, which inhibits the magnitude of jetting. It may also explain why in the XPCI images the jetted material traveled slower than that predicted by our hydrodynamic simulations.

As mentioned in Sec. III B 1, a faint line is seen in Fig. 9(c) that resembles a reflected wave. This is surprising given that the target and window are both PMMA. However, Marsh⁷⁶ showed that ambient mass densities of PMMA can range from 1.180 to 1.191 g/cm³. Thus, we performed a hydrodynamic simulation of a shock wave propagating across a $30 °$ oblique material interface between a 1.191 g/cm³ target PMMA and a 1.180 g/cm³ window PMMA to produce a release wave. We then simulated its XPCI image using the angular spectrum method⁴⁸ and added Gaussian noise with the standard deviation for each pixel calculated from Fig. 5(b). The release wave was not visible in the presence of noise as shown in Fig. 13(a). The same XPCI simulation was performed, but the mass density for the target and window PMMA reversed to produce a reflected shock wave. The reflected shock wave was also not visible in the presence of noise [Fig. 13(b)].

Alternatively, the faint bright line could originate from the epoxy between the target and window material. We performed similar hydrodynamic simulations but with a 5  $μ$m thick layer of EPON 828 epoxy with a density of 1.15 g/cm³ sandwiched between a 1.186 g/cm³ target and window PMMA. EOS data for AngstromBond AB9110 epoxy was not available; instead, EPON 828 was chosen for having similar mass density and mechanical properties. Simulation of the XPCI image showed a faint bright and dark fringe in Fig. 13(c) as observed in Fig. 9(c). To explain this phenomenon, when the incident shock front propagates across the material interface, some of it reflects off the target/epoxy interface to shock the target to a higher mass density shock state. The remaining transmits across the epoxy and reflects off the epoxy/window interface and eventually releases the target to a lower mass density shocked state. Consequently, there exist a thin region of elevated mass density that produces the faint line in the XPCI image [Fig. 9(c)]. The simulated faint line in Fig. 13(c) appears brighter than that seen in Fig. 9(c), which suggests that the epoxy thickness may be less than 5  $μ$m and/or the epoxy density may be closer to that of PMMA.

To confirm which of or if both the small impedance mismatch and epoxy are responsible for the faint line is possible by measuring $B S$ around the faint line. Unfortunately, our Monte Carlo-based method needs to resolve changes in $B S$ smaller than its current precision of 3%–5%.

Methods for direct, quantitative measurements of $B S$ and $θ$ using XPCI were developed. These methods were tested on gas-gun-induced oblique shock wave polymer interface interaction experiments. Computation of $B S$ in the shocked regions of the polymers were <0.1% accurate and precise to within 3%–5%, while $θ$ was measured with an uncertainty of $0.2 °$⁠. These measurements were largely limited by the pulse-to-pulse variation in the x ray beam intensity and Poisson noise. Our methods are expected to improve with planned upgrades in facilities such as APS and Linac Coherent Light Source to enhance x ray beam stability, brilliance, and spatial coherence. Based on these results, we anticipate using XPCI in shock polar configurations to accurately determine and constrain EOS of materials with redundant measurements.

Our work demonstrated that XPCI can become a highly accurate quantitative tool for developing and testing hydrodynamic codes on the three-shock and two-shock and PM fan configurations but it can easily be extended to study other configurations that include bow shocks and mach stems, for example. In materials that exhibit a mixture of phases, e.g., polysulphone or plastic bonded explosives, measuring $B S$ across these regions can help constrain multi-phase EOS and reaction rate models. Measurement of $θ$ can further constrain these models to ensure shock wave behavior is consistent with material response.

Our Monte Carlo method could also directly measure mass density by using cylindrical rather than cuboidal targets. Similar to Rigg et al.,⁴⁴ a detector with a sufficiently large FOV would be used to image the shock wave propagation and ejected material. By treating the shock compressed target as symmetric around the cylindrical axis, the release material can be removed via Abel transform to measure only the mass density of the shocked region. This can be combined with the measured $θ$ to develop EOS models.

XPCI showed high contrast dark bands created by material jetting from the sample boundary. We found that the traveling speed of the jetting material and its 3D structure are appreciably different from our hydrodynamic simulations. This is indicative of mechanical properties beginning to influence material response, whereas initially, they behaved as a hydrodynamic flow. This observation suggests that there is a possible extension to this work to deduce the dynamic strength of materials by looking at other shock-induced behaviors, e.g., spallation, void collapse, and fracture.^56,75,77

Our hydrodynamic simulations and SPA show when an oblique shock wave travels from a higher (Teflon) to lower impedance (PMMA) solid material, a PM fan is released into Teflon. While the PM fan is not obviously visible in Fig. 8 or in Fig. S1 in the supplementary material, our Hough transform-based method was able to detect it. This was achieved by adjusting the image threshold to include a large number of noisy pixels and restricting our search for the PM fan to within $θ= 2 °$ and $r=7μ$m (where $r$ and $θ$ describe a line $r=xcos⁡θ+ysin⁡θ$ in (⁠ $x$⁠, $y$⁠) Cartesian space) of that predicted by our hydrodynamic simulations and SPA. Future work is needed to improve the detection of PM fans and weak shock waves such as the reflected shock wave, either through improved imaging conditions or image post-processing, particularly for materials with poorly defined EOS where we do not know where to narrow our search.

Finally, another avenue of future work is the inclusion of phase contrast in our Monte Carlo-based method. This would enable reconstruction of $B S$ across the shock fronts, release waves, and material interface. Thus, it provides a more detailed comparison between shock wave experiments and computation models. The challenge is knowing beforehand what material each pixel contains. This is complicated by having some pixels enclose a mixture of materials (partial volume artifact). But, an even greater challenge is incorporating the effects of phase contrast into the statistical properties of the image intensity and, therefore, the recovered $B S$⁠, as has been done for a class of linearized phase retrieval techniques.⁷⁸ 

See the supplementary material for (I) a detailed derivation of C i j ( r → m ; B S ), (II) unprocessed XPCI images, (III) tabulation of B S and θ values, (IV) MCNP simulated effects of Compton Scatter, (V) description and suppression of discrete sampling effects, (VI) hydrodynamic simulation of oblique shock front traversing a Teflon/PMMA and PMMA/PMMA interface, and (VII) XPCI simulation of material interface.

This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). This publication is based upon work performed at the Dynamic Compression Sector, which is operated by Washington State University under the U.S. Department of Energy (DOE)/National Nuclear Security Administration Award No. DE-NA0003957. This research used resources of the Advanced Photon Source, a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. A.F.T.L. acknowledges the support of a Laboratory Directed Research and Development Director’s Fellowship (20200744PRD1). T.D.A. was supported by the Advanced Simulation Computing-Physics and Engineering Models (ASC-PEM) program at Los Alamos National Laboratory. The iHMX and Conventional High Explosives Grand Challenge Programs supported performing and analyzing the experiments. We gratefully acknowledge Tim Pierce, Tate Hamilton, Claudine Armenta, and Adam Golder for assistance in preparing hardware and samples and Bryan Zuanetti for his insightful feedback on the manuscript.

The authors have no conflicts to disclose.

Andrew F. T. Leong: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Christopher M. Romick: Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Cynthia A. Bolme: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Writing – review & editing (equal). Tariq D. Aslam: Methodology (equal); Writing – review & editing (equal). Nicholas W. Sinclair: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Pawel M. Kozlowski: Supervision (equal); Writing – review & editing (equal). David S. Montgomery: Data curation (equal); Formal analysis (equal); Supervision (equal); Writing – review & editing (equal). Kyle J. Ramos: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.