Ejecta microjets offer an experimental methodology to study high-speed particle laden-flow interactions, as microjets consist of millions of particulates traveling at velocities of several kilometers per second and are easily generated by most common shock drives. Previous experiments on the OMEGA Extended Performance laser found that collisions between two counter-propagating laser-driven tin ejecta microjets varied as a function of drive pressure; jets generated near shock pressures of 10 GPa passed through each other without interacting, whereas jets generated at shock pressures of over 100 GPa interacted strongly, forming a cloud around the center interaction point. In this paper, we present a more systematic scan of tin ejecta microjet collisions over intermediate pressure regimes to identify how and at what shock pressure interaction behavior onsets. Radiographs of interacting microjets at five different laser drive energies qualitatively demonstrate that interaction behavior onsets slowly as a function of laser drive energy. Quantitative mass and density metrics from each radiograph provide trends on jet characteristics and collisional mass dispersion. It is observed that jetting mass, jet densities, and mass dispersion increase with increasing drive pressures and that the increased jet density at the higher drive energies may account for the increased mass dispersion. This work provides an important step in the understanding of tin ejecta microjet collisions and paves the way for future studies on the physics dominating high-speed particle-laden flow interactions.

The study of high-velocity particle-laden flow interactions is broadly relevant to fields ranging from planetesimal evolution to meteorite impact behavior.1–7 Such flows are comprised of numerous micrometer-scale particles that collide with other particles in motion, altering mass–velocity distributions and density profiles. The specific behavior of any single collision varies with parameters such as impact velocity, particle material, and angle of impact and can be often predicted through focused experiments or large-scale simulations.8–10 Predicting the results of colliding streams of particles is more difficult, as the simultaneous and subsequent interactions of thousands or millions of particles leave large-scale simulations untenable and experimental data sparser. As such, more direct data are needed to understand how particle-laden flows interact across a variety of conditions.

Ejecta microjets offer one experimental methodology to study flow interactions, as jets consist of many micrometer-scale particles that travel at velocities above several kilometers per second.11–13 Microjets are generated when a shock wave reaches the rear surface of a material and interacts with a surface feature, such as a groove.14 The feature inverts as a limiting case of the Richtmyer Meshkov instability and generates a propagating jet of particulates.15–20 As jets are easily generated via most commonly used shock drives and jet parameters can be tuned via sample and drive choices, experiments can leverage microjetting to study specific interaction cases of particle-laden flows.

Recent studies on the interactions of counter-propagating planar tin ejecta microjets observed for the first time a difference in microjet interaction behavior as a function of shock pressure; low-density jets generated by shock pressures near 10 GPa passed through each other unattenuated, while higher density jets generated by shocks of over 100 GPa interacted strongly, dispersing mass into a cloud around the center point.21 Simulations of the jet interactions that assumed elastic collisions between jet particulates suggested that the observed difference in interaction behavior could be attributed to the differences in jet density alone, as the collision probability in particle-laden flows is known to increase with increased particulate density.22,23 However, the simulations failed to capture the complete shape and spread of the experimentally observed interaction cloud, suggesting more work is needed to understand the physics dominating jet interaction behavior.

Tin is a material that undergoes phase changes over the pressures explored in the previous experiment, releasing into the solid phase at pressures below 19.5 GPa, a mixed phase between 19.5 and 33 GPa, a liquid between 33 and 50 GPa, and shocking and releasing into liquid at pressures above 50 GPa.24,25 It is known that the phase of the bulk shocked material affects the resulting jet characteristics, such as particulate phase, mass–velocity distributions, particle size distributions, and total mass.14,26–29 But unknown is how and whether the phase of the particulates within the jets affects the collective particle interaction behavior. Further studies on jet interactions over such transitional pressures could provide key insights into how tin phase affects the bulk collisional properties.

In this paper, we highlight recent laser-driven ejecta experiments that image the interactions of tin ejecta microjets at intermediate drive pressures, exploring the full spectrum of tin release behavior. By observing intermediate pressures, we seek to understand how jet characteristics and the resulting interaction behavior evolve with increasing drive pressure, as a way to assess the relative contributions of density vs phase to interactions. We develop metrics to assess jet characteristics and mass dispersion and observe quantifiable trends as a function of laser drive energy (and, thus, shock pressure). This work finds for the first time how interaction behavior onsets, providing an important step in probing the physics governing microjet interaction behavior.

The remainder of the paper is structured as follows. Section II describes the experimental platform and setup. Section III shows the imaging results from the experiments, and Sec. IV highlights the methodologies used in data analysis. Section V presents the results and discussion, and the paper concludes in Sec. VI.

To study the interactions of tin ejecta microjets across a wide range of pressures, we performed experiments at the OMEGA Extended Performance (EP) laser facility in Rochester, New York.30 The platform, similar to that described in previous work,21,31 is shown in Fig. 1(a). The target consists of two tin foils mounted together on a stalk with target normals intersecting at 130 °. The foils are 100  μm thick, 3.2  × 3.2 mm 2 squares. A 30  μm plastic ablator rests on the outer surfaces of the tin samples to increase the laser shock-drive efficiency. The tin samples have 60 ° opening angle triangular grooves carved 45  μm deep across their inner surfaces, and the grooves are oriented such that imaging occurs down the jet axes.

FIG. 1.

The laser-driven jet interaction experiments were performed on a platform designed for OMEGA EP. (a) The imaging diagnostic views the interactions of two planar ejecta microjets, driven by two long pulse lasers. Copper shields reduce the background signal on the imaging plate detector and masks on tin inner surfaces limit the jetting material that propagates to the center such that observations are made through more uniform jet conditions. (b) The masks are perpendicularly oriented to the groove on each 3.2 × 3.2 mm 2 tin inner surface, creating a 1 mm opening and allowing a 1 mm region of the jet to propagate forward. (c) The point projection radiography scheme uses a backlighter foil over a pinhole on a thick substrate to generate a spherical He- α x-ray source, which projects an image of the interacting ejecta on an image plate diagnostic. (d) A typical raw radiograph shows target features and the jet interactions in the center of the image. Tin calibration steps on the target allow a direct mapping from x-ray intensity on the image plate to the areal density of tin.

FIG. 1.

The laser-driven jet interaction experiments were performed on a platform designed for OMEGA EP. (a) The imaging diagnostic views the interactions of two planar ejecta microjets, driven by two long pulse lasers. Copper shields reduce the background signal on the imaging plate detector and masks on tin inner surfaces limit the jetting material that propagates to the center such that observations are made through more uniform jet conditions. (b) The masks are perpendicularly oriented to the groove on each 3.2 × 3.2 mm 2 tin inner surface, creating a 1 mm opening and allowing a 1 mm region of the jet to propagate forward. (c) The point projection radiography scheme uses a backlighter foil over a pinhole on a thick substrate to generate a spherical He- α x-ray source, which projects an image of the interacting ejecta on an image plate diagnostic. (d) A typical raw radiograph shows target features and the jet interactions in the center of the image. Tin calibration steps on the target allow a direct mapping from x-ray intensity on the image plate to the areal density of tin.

Close modal

Two ultraviolet ( λ = 351 nm) beams (EP beam numbers 3 and 4) deliver energy to the tin foils in 8 ns square drives with variable energies at experiment time t = 0 ns, sending shock waves into the tin, inverting the grooves, and generating planar jets that propagate toward the center. The beams are used with 1800  μm super-gaussian phase plates to ensure regions of planar drives in the centers of the targets. Tantalum masks, offset by 400  μm from the tin inner surfaces, limit the material reaching the interaction point to a 1 mm region at the center of 3.2 mm tin foil in the most uniform region of the laser drive; the masks ensure imaging through the most uniform jet conditions. The mask configuration is shown in Fig. 1(b) and has been shown to have minimal edge effects on the jetting material.

At a later time, a third ultraviolet beam (EP Beam 2) heats a backlighter foil in a 1 ns square pulse in a 300  μm spot size over a pinhole aperture to generate a flash of spherical x rays for radiography. The heater laser energy is specified to maximize the conversion efficiency of laser energy to He- α line emission for the specific foil material type.32 Backlighter foil types in these experiments included iron (Fe) and zinc (Zn), which generate atomic He- α line emission at 6.7 and 9.0 keV, respectively. The backlighter foil is mounted on a 75  μm thick tantalum (Ta) substrate with a 20 μm pinhole laser drilled in the center at an angle of 20 ° to the substrate normal, as indicated in Fig. 1(c), to reduce probability of detector damage from driven target debris.33 The laser is pointed to the location of the pinhole projected to the back surface of the Ta substrate to ensure generation of a bright, spherical x-ray source for imaging.

A shielded image plate detector collects x rays attenuated by the jetting sample material. The pinhole x-ray source is located 23 mm away from the ejecta target and the image plate can be located 288–648 mm away from the ejecta target, varying the image field of view and magnification (14x to 29x). An example radiograph of jet interactions at the 14x magnification is shown in Fig. 1(d), highlighting all target features. Tin steps of 3, 6, 9, 12, and 15  μm located on the bottom of the image correlate the x-ray intensity on the image plate to areal density; some targets had only three calibration steps.

Across three separate shot days, we collected radiographs of interacting tin ejecta microjets at five different nominal laser drive energies (200, 300, 400, 600, and 800 J) with two different x-ray backlighter energies, resulting in ten total radiographs. The laser energies were selected to span between the 70 and 1200 J drives used in the previous bounding experiment, which generated shock release pressures near 10 and 100 GPa, respectively.21 The radiographs can be seen in Fig. 2, with the Fe (Zn) backlighter data on the top (bottom) row. The average laser drive energies specific to each shot are cited below the images; individual as-delivered laser drive energies varied by up to 13%, with an additional 5% facility uncertainty in the cited values. Each image represents a 1500  μm wide  × 1000  μm tall field of view in the image plane with the center of the image selected to be the center of the interaction point. Interacting planar jets appear in all images, as well as the releasing tin free surface in the upper lefthand and righthand corners. The planar jets extend into the imaging plane by 1 mm, as restricted by the masks described in Sec. II.

FIG. 2.

Raw radiographs of interacting planar microjets for different laser drive energies, with energy increasing to the right. The images show qualitative differences in jet appearance and interaction behavior as a function of drive pressure, with the strongest interaction clouds occurring above laser drives of 600 J. The images on the top row were generated via an iron (6.7 KeV He- α line emission) backlighter and the bottom-row images with a zinc (9.0 KeV He- α line emission) backlighter. The different x-ray energies provide better imaging contrast over different density regimes, with the iron data most clearly demonstrating the interaction cloud formation but the zinc data providing quantitative areal density of the jets and surrounding regions.

FIG. 2.

Raw radiographs of interacting planar microjets for different laser drive energies, with energy increasing to the right. The images show qualitative differences in jet appearance and interaction behavior as a function of drive pressure, with the strongest interaction clouds occurring above laser drives of 600 J. The images on the top row were generated via an iron (6.7 KeV He- α line emission) backlighter and the bottom-row images with a zinc (9.0 KeV He- α line emission) backlighter. The different x-ray energies provide better imaging contrast over different density regimes, with the iron data most clearly demonstrating the interaction cloud formation but the zinc data providing quantitative areal density of the jets and surrounding regions.

Close modal

The Fe and Zn backlighter radiographs provide two complementary sets of data and both show similar trends in interaction as a function of laser drive energy but appear different for several reasons. The Fe images feature higher magnification, increasing the imaging resolution. However, the higher magnification results in more complex background subtraction and less certainty in the precise image scale, which motivated the transition to a lower magnification for subsequent shots. In addition, while the relatively low x-ray energy of the Fe backlighter most clearly shows the formation of the interaction cloud, the dynamic range within the cloud and jets is low. The jets and portions of the interaction clouds attenuate all x rays, effectively rendering the Fe backlighter radiographs as shadowgraphs and incapable of density analysis. The lack of quantitative information available from the Fe data motivated the collection of complimentary data and the transition to a higher-energy Zn backlighter for subsequent shots.

While the Fe backlighter images present limited quantitative information, the high-resolution and optical-like properties provide valuable qualitative information on jet characteristics and collisional mass dispersion. First, the shape and quality of the ejecta jets vary with laser drive energy. At the lowest drive energy, the jets clearly display structure, which is more typically observed for jets releasing near the solid phase.14 As the laser drive energy increases, the appearance of structure reduces until the jets appear more uniform along their directions of propagation. There is some indication of interaction occurring even in the lowest drive images, but at higher energy drives, large clouds appear around the center point of interaction. These clouds extend both downward and upward from the center, indicating increased collisions between jet particulates as the drive energy increases.

The Zn backlighter images show similar qualitative characteristics, although they are less sensitive to attenuation by the lowest density regions. Again, the jet structure appears at the lower drives and an interaction cloud forms at the highest drive. The Zn backlighter images provide good contrast through the jetting region and, as such, are the focus of later quantitative analysis in this paper. Both sets of images show signs of asymmetry in the drive on the left vs right side of the image. This is an inherent limitation of the platform due to small facility timing and laser energy uncertainties.

While direct measurements of the shock pressure before release were not made in this experiment, the radiographs enable measurement of the releasing free surface velocity via a time-of-flight assumption, which can be used to infer the approximate pressure regimes. We assume the free surface to be ballistic upon release, as the experiments are performed in vacuum. For each radiograph, we measure the distance of propagation of the bulk free surface and divide by the propagation time to estimate the bulk free surface velocity. The propagation time is the time at which the image is taken, subtracted by the approximate shock breakout time, which is bounded by an understanding of the sample thickness and range of reasonably possible shock speeds. The free surface velocity, u f s, can then be assumed to be approximately twice the particle velocity, u p ( u f s = 2 u p), along previously published values of the tin shock Hugoniot, resulting in estimates for the shock pressures.34,35

It should be noted that this method of pressure inference comes with significant uncertainties. One portion of these uncertainties arises from measurement uncertainties, including shock release time ( ±10 ns), image time ( ±10 ns), backlighter motion blur (1 ns), samples thickness ( ±5  μm, accounted for in shock breakout time uncertainty), surface position measurement at the image time due to the rough appearance of the releasing free surface ( ±25  μm), and surface position measurement due to differences in image magnification into the plane of the image ( ±50  μm). Added in quadrature, these uncertainties result in a 9% uncertainty in the velocity and an associated 10% in the inferred pressure.

However, the cited 10% uncertainty does not account for the uncertainty associated with using such a method to ascertain pressure regimes. It is possible that the free surface velocity as measured could be slower than the free surface velocity immediately upon release due to energy dissipation. In addition, the diamond-turned surfaces of the tin samples result in slight jetting features, as evident by the rough surface appearances at the time of radiograph, which could result in an over-estimation of the free surface velocity. Due to the difficulty of ascribing precise uncertainty quantification for these factors, we conservatively estimate a possible ±25% systematic error by use of this methodology.

Figure 3 presents the inferred pressures for all 10 shots and 20 different drives (EP beam numbers 3 and 4 for each of the 10 radiographs) plotted along with a power-law fit, with exponent found to be 0.68; the 0.68 exponent tracks closely to empirical formulas describing shock pressures as a function of laser-driven ablation laser intensity, which scale pressure with laser intensity to the 2/3 power.36 The error bars on each point convey the 10% error for each piece of analysis. Also plotted in dashed lines above and below the fit to the data are power law fits to the data at the ±25% uncertainty bounds. As shown in Fig. 3, the pressures may range from as low as 20 GPa to as high as 90 GPa from the lowest to the highest drive energy. While the pressure associated with any specific laser drive energy is uncertain, it is expected that the pressure increases systematically according to the power law fit as a function of laser drive energy. Thus, comparisons between radiographs from lower to higher laser drives correspond to comparisons between lower vs higher tin release pressures.

FIG. 3.

Inferred pressure as a function of laser drive energy is plotted along with a power-law fit (exponent of 0.68). The pressures are determined by assuming that the free surface velocity is twice the particle velocity in the shock front before release. The error bars represent measurement uncertainties on each shot, while the dashed contours indicate reasonable bounds of uncertainty on the method used to ascertain pressure. Even within the bounds of uncertainty, in this experiment, the data are seen to span transitional regions of tin release phase behavior.

FIG. 3.

Inferred pressure as a function of laser drive energy is plotted along with a power-law fit (exponent of 0.68). The pressures are determined by assuming that the free surface velocity is twice the particle velocity in the shock front before release. The error bars represent measurement uncertainties on each shot, while the dashed contours indicate reasonable bounds of uncertainty on the method used to ascertain pressure. Even within the bounds of uncertainty, in this experiment, the data are seen to span transitional regions of tin release phase behavior.

Close modal

The horizontal colored bands on Fig. 3 demarcate the different shock release behaviors of tin, ranging from solid β-phase release to liquid shock and release. Figure 3 highlights that these experiments explored jet interactions through pressure drives that span transitional regimes of tin phase upon release, despite the wide uncertainty bounds on any specific point. It is worth noting that the phase of the jet likely differed slightly from that of the bulk free-surface material, as has been demonstrated by previous work;27 the shock wave decayed as it traversed the tin sample, and as such, the shock pressure was higher when the shock wave initialized the jet at the vertex of the groove compared to when the shock wave subsequently reached the tin rear surface; this pressure difference is expected to be on the order of 10%. Given that ejecta literature often describes jet generation with reference to the shock breakout pressure of the free surface, we maintain this convention, acknowledging that the phase of the jet may be skewed slightly higher toward melt when comparing to the phase of the free surface.

Table I catalogues the parameters of all shots and associated uncertainties, similarly highlighting the broad range of tin and jet conditions explored in this experiment. The pressure values are cited with the 10% uncertainty bounds, but we again note that the systematic uncertainties allow for all values to be up to 25% higher or lower. The “Left Drive” refers to the tin jet on the left side of the radiograph (driven by EP beam number 4), and “Right Drive” correlates with the tin jet on the right (driven by EP beam number 3). The “Image Time” column refers to the time between tin drive laser turn on at t = 0 ns and x-ray backlighter heater beam turn on. The imaging times were selected to ensure jet collisions would be observed. Because of the long delay times between the tin drive beams and backlighter heater beam, atypical for normal EP operations, the facility cited an uncertainty of ±10 ns to the as-delivered image time. The facility cites a ±5% uncertainty to the as-reported laser energy values.

TABLE I.

Summary of experimental parameters on the ten shots highlighting the wide range of pressures explored in the experiments as well as the shot-to-shot variations.

Left-side drive parametersRight-side drive parameters
Laser shot parameters(OMEGA EP Beam 4)(OMEGA EP Beam 3)
Shot numberBacklighter foil materialNominal laser energy (J)Image time (ns)Delivered laser energy (J)Jet velocity (km s−1)Inferred shock pressure (GPa)Delivered laser energy (J)Jet velocity (km s−1)Inferred shock pressure (GPa)
37 542 Zn 200 500 ± 10 226.9 ± 11.3 4.0 ± 0.6 28 ± 3 200.7 ± 10.0 4.1 ± 0.6 29 ± 3 
37 543 Zn 300 450 ± 10 320.9 ± 16.0 4.2 ± 0.6 36 ± 4 318.9 ± 15.9 4.2 ± 0.6 37 ± 4 
37 544 Zn 400 375 ± 10 402.6 ± 20.13 5.4 ± 0.8 45 ± 5 408.1 ± 20.4 5.3 ± 0.8 43 ± 4 
38 454 Zn 600 330 ± 10 623.0 ± 31.1 6.1 ± 0.9 60 ± 6 606.2 ± 30.3 5.9 ± 0.9 55 ± 6 
38 455 Zn 800 315 ± 10 826.6 ± 41.33 6.8 ± 1.0 72 ± 7 814.4 ± 40.7 6.6 ± 1.0 69 ± 7 
36 378 Fe 200 500 ± 10 183.2 ± 9.2 4.0 ± 0.6 25 ± 3 195.2 ± 9.8 3.8 ± 0.6 24 ± 2 
36 379 Fe 300 450 ± 10 302.8 ± 15.1 4.7 ± 0.7 36 ± 4 304.5 ± 15.2 4.1 ± 0.6 32 ± 3 
36 377 Fe 400 375 ± 10 396.3 ± 19.8 4.9 ± 0.7 44 ± 4 410.2 ± 20.5 5.1 ± 0.8 42 ± 4 
36 376 Fe 600 330 ± 10 616.3 ± 30.8 5.6 ± 0.8 60 ± 6 617.2 ± 30.9 5.2 ± 0.8 54 ± 5 
36 374 Fe 800 315 ± 10 771.0 ± 38.6 5.8 ± 0.9 63 ± 6 801.4 ± 40.1 5.9 ± 0.9 71 ± 7 
Left-side drive parametersRight-side drive parameters
Laser shot parameters(OMEGA EP Beam 4)(OMEGA EP Beam 3)
Shot numberBacklighter foil materialNominal laser energy (J)Image time (ns)Delivered laser energy (J)Jet velocity (km s−1)Inferred shock pressure (GPa)Delivered laser energy (J)Jet velocity (km s−1)Inferred shock pressure (GPa)
37 542 Zn 200 500 ± 10 226.9 ± 11.3 4.0 ± 0.6 28 ± 3 200.7 ± 10.0 4.1 ± 0.6 29 ± 3 
37 543 Zn 300 450 ± 10 320.9 ± 16.0 4.2 ± 0.6 36 ± 4 318.9 ± 15.9 4.2 ± 0.6 37 ± 4 
37 544 Zn 400 375 ± 10 402.6 ± 20.13 5.4 ± 0.8 45 ± 5 408.1 ± 20.4 5.3 ± 0.8 43 ± 4 
38 454 Zn 600 330 ± 10 623.0 ± 31.1 6.1 ± 0.9 60 ± 6 606.2 ± 30.3 5.9 ± 0.9 55 ± 6 
38 455 Zn 800 315 ± 10 826.6 ± 41.33 6.8 ± 1.0 72 ± 7 814.4 ± 40.7 6.6 ± 1.0 69 ± 7 
36 378 Fe 200 500 ± 10 183.2 ± 9.2 4.0 ± 0.6 25 ± 3 195.2 ± 9.8 3.8 ± 0.6 24 ± 2 
36 379 Fe 300 450 ± 10 302.8 ± 15.1 4.7 ± 0.7 36 ± 4 304.5 ± 15.2 4.1 ± 0.6 32 ± 3 
36 377 Fe 400 375 ± 10 396.3 ± 19.8 4.9 ± 0.7 44 ± 4 410.2 ± 20.5 5.1 ± 0.8 42 ± 4 
36 376 Fe 600 330 ± 10 616.3 ± 30.8 5.6 ± 0.8 60 ± 6 617.2 ± 30.9 5.2 ± 0.8 54 ± 5 
36 374 Fe 800 315 ± 10 771.0 ± 38.6 5.8 ± 0.9 63 ± 6 801.4 ± 40.1 5.9 ± 0.9 71 ± 7 

While the radiography images provide valuable qualitative information about the onset of interaction behavior, more quantitative analysis can be used to understand how jet characteristics and interaction dynamics change as a function of laser drive energy and tin shock pressure. Specifically, we seek to develop metrics that can be used as quantitative points of comparison between each radiograph to assess whether the observed collisional behavior onsets slowly over the drive energies explored or appears to onset over a threshold energy. As this work explores tin over transitional phases, a rapid change in collisional behavior may suggest that phase plays a dominant role in collisional behavior, whereas a slower change may reduce the importance of phase compared to other effects, such as jet density.

For radiographs, the measured intensity of the x-ray signal on the image plate detector relates to the attenuating material areal density and, thus, to the mass. We can define multiple metrics that quantify density and mass for each image to gain a more comprehensive understanding of how the relative effects of density and phase play a role in the observed jet interactions by comparing the metrics as functions of laser drive energy. The metrics used include the following: total jetting mass, average jet areal density, and the ratio of mass dispersed by jet collisions. To obtain the total mass, the areal density of each pixel multiplied by the pixel area is summed. Similarly, the jet average density is measured by averaging areal density over a region of the jet as a way to understand particle packing density. While these two metrics provide insights into the trends of jet characteristics as a function of laser drive energy, another metric is needed to quantify how mass is distributed by collisions and assess how collisional behavior evolves.

To that end, we define both the radial sum of mass into angular bins and the related ratio of dispersed mass. The radial sum of mass into angular bins uses the center point of interaction as the origin and sums mass radially in angular bins around the center point. The resulting curve of total mass vs angle, particularly for the angles in the lower half of the image, provides an assessment of how much mass is dispersed by collisions; in the case of limited collisions between jet particulates, two prominent peaks would appear in the negative angle portion, whereas in the case of significant collisions, two less-prominent peaks would appear with mass dispersed between them. The mass ratio is then defined as the ratio of dispersed mass vs the ballistic jet mass (mass in the jet area); a ratio of one amounts to equal mass dispersed to that in the two post-collision jets and a ratio of zero suggests no mass has been dispersed. Since collisions dominate mass dispersion, lower mass ratios correspond to less interaction between particles in the jets and higher mass ratios to more interaction.

The remainder of this section describes in more detail how the Zn radiograph data are processed and demonstrates how the quantitative metrics are assessed on one example radiograph. The first processing step involves subtracting a linear background from all pixels in the image by averaging the intensity value in a region known to have zero x-ray transmission and subtracting the average value from all pixel intensities. Then, the free space in the background image is used generate a two-dimensional Gaussian fit to the radiation background shape to flatten the image. The Gaussian function is chosen as it most accurately represents the intensity distribution of the spherical x-ray source on the image plate detector.

Tin calibration steps of known thickness then convert the measured x-ray intensity of each pixel to areal density. Figure 4(a) shows a flattened radiograph for one Zn radiograph (average drive energy of 213 J). The green rectangles on the calibration steps indicate the regions over which the intensity values are averaged for intensity to areal density fitting and fitting statistics. Figure 4(b) shows the fit of intensity to areal density along with uncertainties, with the fit and fit uncertainty bounds generated using the SciPy orthogonal distance regression library. As indicated on the graph, the fit of pixel intensity to areal density is assumed to follow the function I = a ( e b x + e c x ), where I is the intensity as measured on the image plate, x is the areal density, and a, b, and c are free fitting parameters, for optimal density reconstruction, as has been performed in the previous work.37 The intensity of each pixel then maps to an areal density value to obtain Fig. 4(c), the radiograph in areal density color scale. The color scale maximum of the image as displayed optimizes contrast within the jetting region for this specific image as a demonstration; the maximum areal density observed in the jets is approximately 5 mg  cm 2.

FIG. 4.

The radiographs are processed to return quantitative metrics on density and mass by correlating the x-ray intensity on the image plate to tin areal density. (a) A flattened radiograph, with the green boxes indicating six different regions over which the pixel intensities are averaged to correlate the tin steps of known areal density to pixel intensity. (b) X-ray intensity is fit to areal density via an exponential function. (c) Intensity is converted to areal density, resulting in the radiograph with areal density color scale.

FIG. 4.

The radiographs are processed to return quantitative metrics on density and mass by correlating the x-ray intensity on the image plate to tin areal density. (a) A flattened radiograph, with the green boxes indicating six different regions over which the pixel intensities are averaged to correlate the tin steps of known areal density to pixel intensity. (b) X-ray intensity is fit to areal density via an exponential function. (c) Intensity is converted to areal density, resulting in the radiograph with areal density color scale.

Close modal

Mass summation is performed over a trapezoidal region, as shown in Fig. 5(a). An areal density threshold away from the free surface defines the limits of the trapezoidal boundary, ensuring that summation does not over-estimate the jetting mass by including free surface mass. In addition, pixel values are only summed if their areal density exceeds one standard deviation above the background noise, slightly under-counting the total mass. This method also slightly under-counts total jet mass by excluding the stemming region of the jet closest to the free surface. However, this method provides a standard point of comparison between all images and, as such, is chosen to provide data on the trend of mass in the jetting region as a function of laser drive energy.

FIG. 5.

(a) The trapezoidal region over which the mass is summed. Trapezoid boundaries are defined using a cutoff areal density value from the free surface, ensuring uniform comparisons across all radiographs. (b) The region used to determine average areal density as a further point of comparison between images. The areal density provides a measure of particle packing fraction in the jets.

FIG. 5.

(a) The trapezoidal region over which the mass is summed. Trapezoid boundaries are defined using a cutoff areal density value from the free surface, ensuring uniform comparisons across all radiographs. (b) The region used to determine average areal density as a further point of comparison between images. The areal density provides a measure of particle packing fraction in the jets.

Close modal

Figure 5(b) shows the same image, slightly zoomed-in, to illustrate the region averaged to obtain average jet areal density. Areal density values are averaged over a rectangular region that is 35  μm wide × 400  μm long in order to capture the average density of the bulk of the pre-interaction jet material. The 35  μm width was selected to provide the most consistent metric for comparison across radiographs. Due to facility limitations, target alignment in the chamber often occurs with up to a 1 ° error. As such, the 1 mm dimension of the planar microjet into the plane may be radiographed at 1 ° off axis, causing the jet to appear artificially wide. In addition, the two Zn radiographs at the highest drive energies were taken at slightly different image magnifications than at the lower energies, resulting in slightly different resolutions and apparent jet widths. Given the image-to-image variations of target alignment and magnification, the 35  μm wide region along the center of the jet can capture more reliably the trends of jet density across radiographs.

Such measurements of density in the pre-interaction region of the jets are inherently limited in that they do not measure the actual density of the region that has undergone interactions. The density and velocity distributions of microjets vary as a function of jetting time and position along the jet axis, and as such, this form of density quantification only provides a point of comparison between images as a way to identify trends. Individual areal density values may over- or under-predict the density of the interacting portion of the jet. Other methods to assess jet density were assessed, such as point-wise identification of the maximum jet density, and were found to return similar trends between images as a function of laser drive energy.

Finally, Fig. 6(a) shows the example image with the origin at the center point of interaction and axes in dashed lines. Angle θ is defined from the right horizontal axis, as indicated. Positive angles represent the mass in the top half of the image and negative angles represent mass in the bottom. An example of the summation is shown in Fig. 6(b), with the two large peaks on the right showing total mass on the top half of the image (“pre-interaction jets”) and the two peaks on the left showing the mass in the bottom (“post-interaction jets”). The sums exclude a small circular region at the center of interactions such that all mass outside the main peaks can be assumed to be dispersed by collisions.

FIG. 6.

Data processing included the summation of mass along radii from the center point of interaction into angular bins to quantify if mass spread away from the jetting region due to particle collisions. (a) A schematic of how axes are defined within the trapezoidal region for radial summation. All mass within an angular bin summed to generate the mass per degree. The dotted circle in the center shows the region that is not included in summation to characterize more clearly interacted mass. (b) The result of radial summation. Negative angles represent the jets post-interaction, with the post-interaction jet peaks indicated on the image. Positive angles represent the pre-interacted jet mass. Mass between the jetting peaks is assumed to be mass that was dispersed as a result of jet collisions.

FIG. 6.

Data processing included the summation of mass along radii from the center point of interaction into angular bins to quantify if mass spread away from the jetting region due to particle collisions. (a) A schematic of how axes are defined within the trapezoidal region for radial summation. All mass within an angular bin summed to generate the mass per degree. The dotted circle in the center shows the region that is not included in summation to characterize more clearly interacted mass. (b) The result of radial summation. Negative angles represent the jets post-interaction, with the post-interaction jet peaks indicated on the image. Positive angles represent the pre-interacted jet mass. Mass between the jetting peaks is assumed to be mass that was dispersed as a result of jet collisions.

Close modal

Gaussian fitting of the radial sums on the lower half of each image to extract the mass ratio is demonstrated in Fig. 7. The lower half of the image is used for mass dispersion analysis, as the lower half contains the majority of the post-interaction jet mass. The mass scale for each image is normalized by the total amount of post-interacted mass to compare more consistently across images. The fitting function consists of two Gaussians summed with one super-Gaussian of order seven; the order was selected as the best fit to the data but ultimately did not affect data interpretation within the error bars. The two Gaussian peaks, representing mass in the jet area, are centered at the peaks observed in the fitting region. The Gaussian widths are similarly constrained by the width of the data peaks and the observed widths of the pre-interaction jets. The super-Gaussian is fit in the middle to obtain the best possible fit of the total region, again limiting the width of the function to lie reasonably between the two other peaks. The areas under the two gaussian peaks are taken to be the mass in the jet area and the area under the super-Gaussian in the center is taken to be the mass in the middle region, or the dispersed mass. Figures 7(a) and 7(b) show fitting examples for the lowest drive and highest drive images, respectively. Due to the potential variability in selection of Gaussian fitting parameters, the mass numbers derived from fitting carry a 20% uncertainty.

FIG. 7.

Gaussians fit to the radial sums of the lower half of the images provide a lower bound on mass dispersed by collisions and allow for comparisons between radiographs to understand when mass begins to be dispersed. This figure shows Gaussian fitting for two different bounding radiographs, (a) at the lowest nominal drive energy of 200 J and (b) at the highest nominal drive energy of 800 J. Two Gaussians define the mass in the jet area, and a super Gaussian demarcates mass in the center region. The mass in the center region is assumed to have been dispersed by collisions.

FIG. 7.

Gaussians fit to the radial sums of the lower half of the images provide a lower bound on mass dispersed by collisions and allow for comparisons between radiographs to understand when mass begins to be dispersed. This figure shows Gaussian fitting for two different bounding radiographs, (a) at the lowest nominal drive energy of 200 J and (b) at the highest nominal drive energy of 800 J. Two Gaussians define the mass in the jet area, and a super Gaussian demarcates mass in the center region. The mass in the center region is assumed to have been dispersed by collisions.

Close modal

There are several key points of uncertainty worth noting in this analysis. The first is that in the higher energy drive case, as seen in Fig. 7(b), some dispersed mass at the lower and higher angles below and above the jet areas is unaccounted for in the Gaussian fits. The second is that experimental laser drive asymmetries may result in one jet arriving earlier to the center point than the other, as noted in the earlier qualitative description of the images. Thus, some of the mass in the post-collision region in the lower half of the image will be mass that passed through the center without having collided with a counter-propagating jet. In addition, by only considering the lower half of the image in this analysis, we neglect to consider any mass that is dispersed upward in the image as a result of collisions. As such, the mass ratio does not provide an absolute measurement of the total amount of dispersed mass. Rather, the mass ratio provides a lower bound to the amount of dispersed mass and acts as a metric by which to understand at what laser drive energy mass begins to be dispersed in a significant fraction.

The different analysis metrics can now be used to assess trends of jet and collision characteristics over the laser drives explored in this experiment. Because the as-delivered laser energy is a direct measurable of the shot day with relatively low uncertainty, the quantitative metrics are plotted as functions of energy, instead of as functions of the inferred pressure. The colored regions shading each plot indicate the possible tin phases of each laser drive energy and are slanted to indicate the possible pressure regions with regard to possible ±25% uncertainties in obtaining the tin release pressure from the bulk free surface velocity. Each data point is plotted on top of a thin, dashed vertical line to highlight the possible range of tin phase behavior explored in each shot within those uncertainty bounds.

Figures 8(a) and 8(b) show the trends in total mass and the average jet density, respectively. The total mass increases monotonically as a function of laser drive energy, with the most mass released at the highest drives, in what are likely liquid release phases; the increase in mass post-melt is consistent with previous jetting and ejecta studies.14,38 Similarly, the average jet density increases with laser drive energy, with the notable exception being the lowest laser drive. This anomaly results from the fact that jet density is measured in the back region of the jet that has yet to undergo interactions. It is a known phenomenon that jets generated in the solid phase have a bimodal density distribution due to a secondary release of jet material following spallation;14,31 this results in a lower density front region and a higher density region at the back of the jet. Because the lowest laser drive energy in these most recent experiments likely corresponds with a mixed- or solid-phase release, density is measured in the higher density back jet region, making this value appear artificially high. Also evident in this plot is that the two highest-energy laser drive shots have the largest error bars; these two shots were performed with targets with only three tin calibration steps, resulting in greater uncertainties in the fitting to calibrate image plate intensity to areal density.

FIG. 8.

Trends of total mass and average jet density as functions of laser energy are assessed across the five radiographs. (a) Mass increases monotonically with drive pressure, while (b) the average jet density shows some non-linearity in response due to jet structure complexities. The colored regions in each plot identify possible tin release phases explored in the experiment as a function of laser drive energy. The vertical dashed lines behind the data points from the top to the bottom of the plot span the possible phases explored at each individual laser drive energy within the bounds of pressure inference uncertainty.

FIG. 8.

Trends of total mass and average jet density as functions of laser energy are assessed across the five radiographs. (a) Mass increases monotonically with drive pressure, while (b) the average jet density shows some non-linearity in response due to jet structure complexities. The colored regions in each plot identify possible tin release phases explored in the experiment as a function of laser drive energy. The vertical dashed lines behind the data points from the top to the bottom of the plot span the possible phases explored at each individual laser drive energy within the bounds of pressure inference uncertainty.

Close modal

The jet characterization metrics of mass and density show clear and connected trends across the laser drive energies; higher laser drive energies generate higher density jets and more mass. As we ultimately aim to understand how such jet characteristics relate to collisional metrics, we turn to the mass ratio, which provides a metric by which to understand how mass dispersal changes. Figure 9(a) shows the mass ratio as a function of laser drive energy and highlights that mass dispersion into the center region similarly increases as a function of drive energy. These consistent trends of slowly increasing total mass, jet density, and dispersion suggest that collisional behavior onsets gradually over the laser drive energies explored. Because these experiments span tin transitional drive pressures, the lack of a threshold over which one metric suddenly increases suggests that phase may not be a primary driver in the observed collisional behavior. However, because the data spans multiple tin phases, we still aim to understand whether the increased mass dispersion we observe at the higher drives can be accounted for by increased jet density or whether liquid particles seem to interact differently than solid.

FIG. 9.

(a) The ratio of the mass in the middle area to the mass in the jet area provides a quantitative metric for the amount of mass spread into the center region by collisions. The mass ratio increases monotonically with laser energy, implying that collisional frequency similarly increases. (b) The normalized collision index, which relates to the particle packing density in the jets as well as particle velocity, assesses the relative frequency of particle collisions. The predicted frequency of particle collisions is observed to increase greatly at the higher energy laser drives.

FIG. 9.

(a) The ratio of the mass in the middle area to the mass in the jet area provides a quantitative metric for the amount of mass spread into the center region by collisions. The mass ratio increases monotonically with laser energy, implying that collisional frequency similarly increases. (b) The normalized collision index, which relates to the particle packing density in the jets as well as particle velocity, assesses the relative frequency of particle collisions. The predicted frequency of particle collisions is observed to increase greatly at the higher energy laser drives.

Close modal

To address the question of how phase may affect jet interactions, we generate a phase-independent metric to understand the relative frequency of particle collisions as functions of measurable jet qualities: density and velocity. The frequency of collisions between particles in particle-laden flows scales with a collision index, demonstrated in previous work and defined as ρ 1 ρ 2 ( v 1 + v 2 ) / 2, where ρ 1 and ρ 2 are the average jet densities of the left and right jets and v 1 and v 2 are the jet velocities.22 For the case of this work, the collision index is normalized by the index at the lowest laser drive to use the index as a point of comparison between all radiographs. Two key assumptions appear in this calculation: (1) the particle size distributions are identical in all drive cases, and (2) the uncorrelated kinetic energy of the particles (a required metric for the collisional index) scales with the jet tip velocity. Jet velocity is quantified via time-of-flight, similarly to the methodology used to obtain free surface velocity. Velocities range from 4.0 to 6.8 km  s 1 from the lowest to highest laser drive energies, as seen in Table I, with errors of up of ±15% due to limitations of the time-of-flight measurement of a post-interaction jet.

The normalized collision index as a function of laser drive energy is plotted in Fig. 9(b). The collision index, a function of jet density and velocity alone, stays relatively constant at the lower laser drives and changes most drastically at the highest drives, suggesting that particles within the denser and faster jets undergo collisions with 5 to nearly 15 times the frequency of those particles in the lower-density jets. The normalized collision index can also be plotted against the mass ratio as a way to assess dispersed mass as a function of the collision frequency. As shown in Fig. 10, data from the three lowest laser drives cluster at the bottom left of the plot with low mass ratios and low collision indices, while the mass ratios increase sharply above a certain collision index threshold. The strong trend of increased collisional frequency with increased mass dispersion suggest that the jet density and velocity are key drivers in how collisions affect mass dispersion and that the increased mass dispersion observed in these experiments may be due to the increase in density alone. While the bounds of uncertainty on the precise shock pressure explored in this experiment are large, the largest collision indices occur certainly above tin melt-upon-release threshold, thus suggesting that phase may play a role in collisions, if only in how the bulk phase affects jet properties such as particulate density. However, we conclude that jet density appears to be the strongest observed influence on the observed mass dispersion.

FIG. 10.

The mass ratio as a function of the normalized collision index suggests that mass spread by collisions greatly increases once the jets reach a certain velocity and density threshold. For the case of tin jets, the increased collision ratio and mass spread occurs over a certain density threshold achieved in this experiment at the highest energy laser drives.

FIG. 10.

The mass ratio as a function of the normalized collision index suggests that mass spread by collisions greatly increases once the jets reach a certain velocity and density threshold. For the case of tin jets, the increased collision ratio and mass spread occurs over a certain density threshold achieved in this experiment at the highest energy laser drives.

Close modal

In this work, we presented two complimentary sets of x-ray radiographs of tin ejecta microjet interactions occurring over a range of different laser drive energies, expanding upon previous work, which only observed interactions at two bounding pressure extremes. We inferred that the pressures explored in these experiments span the different release phase regions of tin, and we offered, for the first time, a complete picture of how and when increased mass dispersion onsets in tin jet interactions. Qualitative analysis of the radiographs, as well as detailed quantitative comparisons between images, suggested that jet densities and mass increase gradually with laser drive energy. These trends correlate with increased mass dispersion and an increased normalized collision frequency, suggesting that a higher probability of collisions in the higher density jets may cause the observed increase in mass dispersion. It still remains to be understood whether liquid particles interact differently than do solid under these extreme velocity collision conditions, but this work suggests that density is a primary driver of mass dispersion and that phase of the releasing material should be taken into account if more mass is known to be released above a phase transition boundary.

Future work will continue to reduce experimental uncertainties to explore the relative contributions of jet density and material phase on mass dispersion. More imaging frames per shot are necessary to reduce error bars on measurements of velocities in the system, including jet velocities, and thus on shock pressures. In addition, with further shot time or platform modifications, independent or simultaneous measurements of shock pressures before release at the different laser drive energies would provide a more robust understanding of the phases explored in this experiment and the resulting jet qualities. Similarly, independent radiographs of jets that do not undergo interactions would provide more robust measurements of jet densities and total interacted mass, as a way to describe collisional behavior outside of metrics meant only as points of comparison.

In addition, we plan to continue experiments on different jetting materials in order to study interactions at similar jet densities but different material phases. Complimentary experiments could be performed to assess some of the other unknowns, such as jet particle size distributions, which are thought to vary as a function of material and shock pressures. As such, the liquid jet particle sizes may differ greatly from those in the solid or mixed-phase jets, potentially affecting interaction probabilities. While difficult to perform, further analysis of particle characteristics as a function of shock drive would help to increase our physical understanding of the mechanisms at play. Likewise, the study of individual collisions of solid particulates vs liquid droplets could illustrate how the interactions differ and if collisional conditions reach complex material regimes, such as the material vapor dome, in either case.

And finally, finer scans of laser energy, specifically in between 400 and 600 J, stand to resolve the trends further. The trends we observed in the images show slow onsets of collisional behavior that appear as near-linear functions of laser drive energy. The results of interactions in intermediate drives could either confirm the observed near-linear increases in mass and mass dispersion or could show a stronger thresholding effect. Either way, a finer scan of laser drive energies with lower uncertainties on drive pressures stands to answer more certainly on the physics driving tin microjet interactions.

While uncertainties remain, this work broadens our understanding of how microjet interactions evolve across a wide range of tin release phases and provides an important step on the path to understand the physics governing microjet collisional behavior. The results from these experiments clarify the scope of interaction possibility for tin microjects and pave the way for future jetting interaction experimentation, allowing further studies on particle-laden flow collisional behavior for a wide variety of broadly relevant materials.

LLNL-JRNL-862372. The authors would like the thank Dr. Alan Wan for the initial support required to begin this project. They would also like to thank Dr. Hans Rinderknecht and Dr. Channing Huntington for their initial leadership in the work and to Livermore target fabrication and General Atomics for their continued support of high-precision millimeter-scale targets. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.

The authors have no conflicts to disclose.

Alison M. Saunders: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Yuchen Sun: Data curation (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – review & editing (equal). Jeremy A. K. Horwitz: Conceptualization (supporting); Data curation (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (equal). Suzanne J. Ali: Data curation (supporting); Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Jon H. Eggert: Conceptualization (equal); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Supervision (supporting). Kyle K. Mackay: Conceptualization (supporting); Investigation (supporting); Writing – review & editing (equal). Brandon E. Morgan: Conceptualization (equal); Formal analysis (supporting); Methodology (supporting); Writing – review & editing (equal). Fady M. Najjar: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (supporting); Writing – review & editing (equal). Hye-Sook Park: Conceptualization (equal); Investigation (supporting); Methodology (supporting); Project administration (supporting); Supervision (supporting). Yuan Ping: Conceptualization (equal); Project administration (supporting). Jesse Pino: Conceptualization (equal); Funding acquisition (lead); Investigation (supporting); Methodology (supporting); Project administration (equal); Supervision (supporting); Writing – review & editing (equal).

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

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