Experiments were performed on grooved Sn and Cu samples to study the temporal evolution of microjets. Jets were generated by the impact of gun-launched flyer plates against the back of grooved targets made from either Cu or Sn (groove depth of ∼250 μm). The Hugoniot states in the various Sn targets encompassed conditions where solid phases are maintained throughout (7 and 16 GPa) and also conditions where melting occurs upon the release of compression (25 and 34 GPa); the transition occurs near a Hugoniot pressure of 23 GPa. Cu targets at 27 and 56 GPa provide comparisons in which the jets move at similar speeds but remain solid. In all cases, the spatial distribution of mass within the microjets was measured using high-speed synchrotron radiography. The result is a time history of the jet thickness profile from which quantities like total jet mass and jet velocity can be derived. In both the solid and liquid states, we generally observe that an increase in the shock strength leads to an increase in jet mass. However, this trend breaks down for Hugoniot states near the transition from continuously solid to melted-on-release. This is evidenced by the observation that there was no difference in the rate of mass flow in Sn jets at 16 and 25 GPa, while similar pressure jumps on either side of this range caused substantial changes in the jet mass. This contrasts with the behavior of smaller polishing defects that were present on the same samples (∼1 μm deep). From these, no ejecta mass was detected below the melt boundary, but obvious microjets were generated once melting occurred. This indicates that crossing the bulk melt-on-release threshold can alternately promote or inhibit the flow of mass into microjets based on the amplitude of the initial perturbation.

When a compressive shock wave releases at a material's free surface, a local structure, such as grooves or tool marks, can invert and launch jets of material that travel substantially faster than the main surface. This process is understood to function as a Richtmyer–Meshkov instability (RMI), in which the growth of surface perturbations is driven by the misalignment of pressure and density gradients at the locally non-planar interface.1,2 If the process continues unchecked, the RMI spikes will form jets and disintegrate into a cloud of ejecta. The mass of ejecta produced by this process is influenced by the initial surface profile, the strength and profile of the shock, the material's density, and its resistance to shear and cavitation.2,3 These phenomena are of direct scientific interest to fields like planetary impact4 and indirectly in other shock-related work where they can disrupt measurements,5 damage equipment,6 or even present safety hazards.7 

It is especially interesting to study what happens to this process when a melt boundary is crossed because material strength and cohesion undergo abrupt changes. Changes to the strength of the material in the jet will affect its disintegration,8 while changes to the base material can influence the flow of material into the jet and, hence, quantities like the peak velocity.9 In this respect, theoretical and experimental works have both described a sudden increase in the mass of ejecta and microjets as the base material is driven above the melt boundary.5,10–12 In the well-studied case of Sn, this means exceeding a Hugoniot pressure of ∼23 GPa, above which the internal energy is high enough that melting occurs during isentropic release.13 This tends to agree with an intuitive sense of the process, whereby the strengthless molten material is more prone to RMI growth. However, recent hydrodynamics14 and molecular dynamics calculations15 predict that the mass of material released by a single microjet reaches a plateau for pressure conditions where melting begins. Other calculations by Durand et al.16–18 have examined liquid and solid jets using molecular dynamics to reveal length scale and phase transition effects. Together this suggests a nuanced role of melting in the production of ejecta as a feature that often, but not always, leads to an increase in the mass of ejected material.

The objective of the current study is to test these melt-related predictions, emphasizing the effect of melting on jet mass, but also on jet velocity, morphology, and breakup. To accomplish this, synchrotron-based radiography was used to image microjets during their formation. The methods are like prior work in which the attenuation of transmitted radiation has been used to estimate the mass and velocity of high-speed surface ejecta.12,19–23 Using Sn samples, the initial shock pressure was varied to span the ∼23 GPa Hugoniot state above which the material will melt upon release, sometimes termed the melt-on-release pressure.13,24 Cu samples provide a control case in which the same high velocities could be reached without melting.25 The results support the conclusion that the material phase is a key parameter for understanding jet formation14 but also suggest that phase effects are inextricably linked to the length scale of the originating surface features.10 It will be concluded that, for a given material and loading scenario, crossing the bulk melt threshold can either reduce or promote jetting based on the initial length scale of the surface profile.

Polished Sn and Cu samples (15 × 16 × 1–2 mm) were engraved with centrally located and nominally identical V-grooves (62 ± 1° root). The Sn samples were cut from commercial purity (99.9%) rolled sheet (McMaster-Carr), while the Cu samples consisted of half-hard OFHC bar stock (Thyssenkrupp). The Cu and Sn had approximate grain sizes of 150 and 450 μm, respectively. An example groove profile is shown in Fig. 1(a), having been measured with a laser scanning confocal microscope (Keyence VK-X200). The radius at the root was at most 10 μm. The individual groove widths and sample thicknesses are listed in Table I. Several of the grooves were terminated part way across the sample such that the experimental field of view would show any jetting/ejecta not attributable to the V-groove, i.e., a control region to show surface ejecta. The samples' planar top surfaces were polished with diamond slurry, which produced scratch depths of <0.1 μm. The Sn samples also had occasional steps of ∼1 μm due to grain relief and an overall RMS <1 μm. An example Sn surface profile is shown in Fig. 2, which was measured using an interferometer having a nominal lateral resolution of 1.1 μm and depth resolution of <1 nm (Zygo NewView 7300). The black arrow shown in Fig. 2(a) indicates the location of the digital cross section shown in part (b). The Cu was polished to a similar or better degree, though as will be seen later, its surface finish is less critical than that of Sn for the purpose of minimizing RMI ejecta. The inside faces of the Sn grooves were smooth within 1 μm, with tool chatter the likely cause of observed variations, as shown in Fig. 3. In this figure, the groove axis is aligned along the direction of Slice 1. Note that due to the angle required to collect interferometry data inside the groove, the Slice 2 axis is slightly foreshortened and only the length scale in the Slice 1 direction is accurate.

FIG. 1.

(a) Example of a measured groove profile in a Sn substrate, with the defining dimensions labeled. The root radius was <10 μm and the angle was 62 ± 1°. The width of each groove is given in Table I. (b) Schematic of the grooved sample, with flyer and imaging directions marked.

FIG. 1.

(a) Example of a measured groove profile in a Sn substrate, with the defining dimensions labeled. The root radius was <10 μm and the angle was 62 ± 1°. The width of each groove is given in Table I. (b) Schematic of the grooved sample, with flyer and imaging directions marked.

Close modal
FIG. 2.

Example surface profile for a planar region of Sn, showing the sub-micrometer scratches and micrometer-scale grain relief. The black arrow in (a) shows the axis of the cross-sectional slice in part (b).

FIG. 2.

Example surface profile for a planar region of Sn, showing the sub-micrometer scratches and micrometer-scale grain relief. The black arrow in (a) shows the axis of the cross-sectional slice in part (b).

Close modal
FIG. 3.

Surface profile taken from within the groove showing the tool chatter.

FIG. 3.

Surface profile taken from within the groove showing the tool chatter.

Close modal
TABLE I.

Geometry, materials, and impact conditions for each shot. PC is polycarbonate.

Shot material/IDGroove width (μm)Sample thickness (μm)Flyer materialFlyer velocity (km/s) ±0.5%Expected pressure (GPa)Bulk phase on releaseBase velocity (km/s) ±0.02Jet velocity (km/s) ±0.02
Sn1 301 ± 1 1523 ± 3 PC 1.62 6.9 Solid 0.61 1.87 
Sn2 297 ± 1 1635 ± 7 6061 Al 1.52 15.9 Solid 1.28 3.20 
Sn3 293 ± 2 1530 ± 8 6061 Al 2.16 24.6 Liquid 1.80 4.02 
Sn4 286 ± 3 1635 ± 5 OFHC Cu 1.86 34.2 Liquid 2.32 4.79 
Cu1 270 ± 4 1109 ± 21 OFHC Cu 1.24 26.8 Solid 1.24 3.31 
Cu2 291 ± 0 1021 ± 11 OFHC Cu 2.25 55.9 Solid 2.28 5.32 
Shot material/IDGroove width (μm)Sample thickness (μm)Flyer materialFlyer velocity (km/s) ±0.5%Expected pressure (GPa)Bulk phase on releaseBase velocity (km/s) ±0.02Jet velocity (km/s) ±0.02
Sn1 301 ± 1 1523 ± 3 PC 1.62 6.9 Solid 0.61 1.87 
Sn2 297 ± 1 1635 ± 7 6061 Al 1.52 15.9 Solid 1.28 3.20 
Sn3 293 ± 2 1530 ± 8 6061 Al 2.16 24.6 Liquid 1.80 4.02 
Sn4 286 ± 3 1635 ± 5 OFHC Cu 1.86 34.2 Liquid 2.32 4.79 
Cu1 270 ± 4 1109 ± 21 OFHC Cu 1.24 26.8 Solid 1.24 3.31 
Cu2 291 ± 0 1021 ± 11 OFHC Cu 2.25 55.9 Solid 2.28 5.32 

Samples were shock compressed via parallel plate impact loading at the Dynamic Compression Sector at the Advanced Photon Source at Argonne National Laboratory (Sector 35-ID), schematically shown in Fig. 1(b). Flyer materials, velocities, and sample pressures for each shot are listed in Table I. All flyers were at least 4 mm thick and 10.2 mm in diameter, dimensions that prevent the rear surface release from arriving at the free surface during the time of the experiment. The pressures listed in Table I are calculated from the measured impact speed and equation of state data compiled by Steinberg.26 The Hugoniot pressures for shots Sn1 and Sn2 were below the melt-on-release pressure (∼23 GPa), while Sn3 and Sn4 were above it.13,27 All four Sn shots were below the shock melt pressure of ∼49 GPa.13 For easy reference, the expected state of the material upon release is also listed in Table I. These reflect the bulk state, not possible localized melting within the jet due to plastic heating.14,28

Jetting behavior was observed via high-speed radiography using the detector system described by Jensen et al.,29 which was placed 1 m from the sample. This system allows unblurred images to be collected at intervals matching the electron bunch timing of 153.4 ns. Using the U17 undulator, the x-ray spectrum has a narrow peak at 23.5 keV, with half-maximum values of 22.9 and 23.7 keV, and a highly repeatable spatial intensity profile. Samples were aligned within ±3.5 mrad of the beam, meaning at most 30 μm of the center is shadowed by the upstream edge of the sample. Parallax effects from the beam are negligible because the divergence is only a few tens of μrad. This permits material thickness to be accurately calculated from the measured attenuation. Each sample was imaged perpendicularly to the groove axis, i.e., through the thin dimension of the jet sheet or “sheet view.” This orientation was chosen because it provides more information on the jet morphology than would be available from a view along the jet axis, which would have been more akin to a shadowgraph. This perspective also causes edge effects to be confined to isolated regions that can be easily excluded from the analysis. Each field of view also contained offset stacked foils of the sample material as quantitative absorption references to check bias and phase-contrast effects. The Sn foils were 60 ± 2 μm thick and the Cu foils were 19 ± 2 μm thick.

Figure 4 presents a representative dynamic radiograph for one of the Sn targets, with the important features labeled. All radiographs have been transformed so that the sample surface is true to the vertical axis and the shockwave moves from left to right. Furthermore, the images have also undergone a standard flat-field correction. The bright-field images were captured after translating the specimen clear of the field of view, resulting in some minor artifacts due to changes in the position of window defects. Figure 4 shows that the leading edges of the jet and parent slab are clearly visible, along with distinct structures within the jet. Subsequent sections will discuss these features in detail for each shot, as well as extract quantitative trends.

FIG. 4.

Representative dynamic image showing the base sample, emerging jet, no-groove control region, and static calibration foils.

FIG. 4.

Representative dynamic image showing the base sample, emerging jet, no-groove control region, and static calibration foils.

Close modal

Figures 5 and 6 show the dynamic radiographs for Sn and Cu samples, respectively. Each column represents an individual shot, with the shot name and Hugoniot pressure labeled at the top. The rows contain sequential time series of radiographs, for which 153.4 ns elapsed between frames. For a review of the main features in each image, please refer to Fig. 4. As a reminder, only Sn3 and Sn4 are shocked above the melt-on-release pressure.

FIG. 5.

Dynamic images for Sn1–4, arranged in columns. The frames in each row are separated by 153.4 ns. For brevity, neither preceding static nor later trailing frames are shown.

FIG. 5.

Dynamic images for Sn1–4, arranged in columns. The frames in each row are separated by 153.4 ns. For brevity, neither preceding static nor later trailing frames are shown.

Close modal
FIG. 6.

Dynamic images for Cu1–2, arranged in columns. The frames in each row are separated by 153.4 ns. For brevity, neither preceding static nor later trailing frames are shown.

FIG. 6.

Dynamic images for Cu1–2, arranged in columns. The frames in each row are separated by 153.4 ns. For brevity, neither preceding static nor later trailing frames are shown.

Close modal

Considering the jet breakup, it was observed that Sn1, Sn2, and Cu1 (solid) show wavy bands breaking up into a net-like field. The principal difference in these three cases is that the jet front in Sn1 is more jagged than in Sn2 or Cu1. These breakup patterns presumably originate from the combination of plastic instability and elastic release waves which govern the breakup of other high velocity (solid) metal jets.30 In contrast, Sn3 and Sn4 (melted) show a different pattern, where striations are observed perpendicular to the leading edge of the jet, and there is no evidence of breakup. Cu2 (solid) shows neither striations nor breakup, instead appearing as a nearly smooth sheet at each observation time.

The greater jaggedness in the jet front of Sn1 relative to the other cases is intriguing as no substantial difference in sample geometry or initial material state is known to account for it. One possible origin is heterogeneity in shear strength caused by the grain structure. Such an effect would be imparted by the orientation dependence of crystal plasticity. Because strength effects diminish in proportion to inertial effects as velocity gradients increase, it makes sense that a crystal plasticity mechanism would be more apparent in Sn1 than Sn2. While it is difficult to quantify a wavelength for the Sn1 jet front, it appears not very dissimilar to the grain scale shown in Fig. 2. A complimentary explanation is the formation of a multi-wave shock structure. At the 7 GPa pressure experienced by Sn1, neither the elastic precursor nor the β-BCT phase transition waves are overdriven.13 Using molecular dynamics, Durand et al.17 showed how the reflection of these waves from the free surface can introduce structure into the jet breakup. It is hoped that future simulations can explain the intertwined effects of crystal plasticity, phase change, and elastic precursor waves.

It is tempting to credit melting for the lack of breakup in Sn3 and Sn4 (melted). This would accord with expected changes in jet stability for a decreasing yield strength30 and an increasing strain rate sensitivity,8 both of which would accompany melting. However, Cu2 (solid) shows the same prolonged intact elongation. So, while melting may certainly contribute, it does not seem to be a necessary condition for the delayed breakup seen in Sn3 and Sn4. Instead, the effect may be due to the fact that increasing jet velocity causes an increase in stable elongation independent of material property changes.30,31 Further supporting this notion is the fact that Cu2 shares a nearly identical particle velocity as Sn4 (Table I).

The molten jets (Sn3 and Sn4) show longitudinal striations along the direction of shock propagation which are not observed in any of the solid jets. Similarly, only Sn3 and Sn4 show lateral striations in the leading part of the jet, i.e., parallel to the shock front. Therefore, these features do indeed seem related to the melt transition experienced by the Sn samples. Similar features have been seen in prior reports on jets believed to be in the molten state.12 While the striation origin has yet to be conclusively understood, the melt-induced drop in shear strength presumably means that secondary instabilities can grow when the melt conditions are met. Though eliminated by melting upon release, the grain structure of the material could have theoretically seeded such an instability during the loading portion of the wave when the material remained solid. However, the grain scale (Fig. 2) was far larger than the wavelength of these striations, so this cause seems unlikely. Nor is an elastoplastic or two-phase wave structure likely the cause, as these are overdriven at the relevant pressures. Instead, the striation wavelength is a closer match to that of the tool marks seen within the grooves (∼10 μm), though no definite conclusion is drawn here. Nor is it clear why the lateral striations are apparently confined to the jet tip, while the longitudinal ones extend the length of the growing jet.

One additional difference between Sn1 and Sn2 vs Sn3 and Sn4 bears notice, and this is the behavior in the no-groove control regions. Here, Sn1 and Sn2 show no observable ejecta, whereas a series of microjets is present in both Sn3 and Sn4, with the effect most pronounced in Sn4 (see the spike-like structures in the bottom of frames 2 and 3 in Fig. 5). This behavior is like that previously observed by others (Refs. 5, 32, and 33), in which surface melting produces a sharp rise in ejecta mass. The supported pressure waves used in this study imply that the source of this ejecta is most likely small-scale surface imperfections, not micro-spall/cavitation.34 The relative sparsity and uniform coverage of these microjets suggest that the originating imperfections were few and scattered across the surface, for example, the ∼1 μm surface steps due to grain relief, or perhaps other isolated defects of similar size. Defects of substantially larger size can be ruled out from the surface inspections.

The position of both the jet tip and the parent slab can be identified from the images shown in Figs. 5 and 6. These positions have been plotted in Figs. 7(a)7(f) where the jet positions are shown by open markers and those of the slab surface shown by filled markers. Using an intensity-based edge extraction algorithm, these interfaces can be repeatably located with ∼1-pixel precision. However, there is some ambiguity as to the proper intensity, i.e., apparent thickness, at which to fix this point. This leads to an estimated uncertainty of ±2 pixels (3.3 μm). The resulting error bars have not been included because they would be smaller than the markers. Extrapolating the flat surface position back to zero displacement provides a means to determine the shock breakout time independent of any other diagnostic, avoiding possible cross timing issues. All times reported are relative to this reference.

FIG. 7.

The position of the flat sample face and jet tip for the Sn1, Sn2, Sn3, Sn4, Cu1, and Cu2 experiments are shown in parts (a)–(f), respectively. The origin is taken as the position of the sample face at the time of shock breakout. Error bars have not been included because they would be smaller than the markers.

FIG. 7.

The position of the flat sample face and jet tip for the Sn1, Sn2, Sn3, Sn4, Cu1, and Cu2 experiments are shown in parts (a)–(f), respectively. The origin is taken as the position of the sample face at the time of shock breakout. Error bars have not been included because they would be smaller than the markers.

Close modal

Applying a linear fit to the positions shown in Fig. 7, the velocities for the jet tip and base surface have been plotted relative to one another in Fig. 8. Of note is the gradual decrease in the slope between Sn points as the planar velocity increases. A similar prediction was made by Mackay et al.14 and previously measured by de Rességuier et al.9 With only two copper data points, such a trend cannot currently be evaluated for that material. The position uncertainty has been propagated into the velocity, yielding error bars which are again not shown because they are nearly equal to the marker size. The timing uncertainty between frames adds no appreciable error.

FIG. 8.

The speed of the jet tip as a function of free surface speed for the Sn and Cu samples. Error bars have not been included because they would be of comparable size to the markers.

FIG. 8.

The speed of the jet tip as a function of free surface speed for the Sn and Cu samples. Error bars have not been included because they would be of comparable size to the markers.

Close modal

To determine the mass of the jet, the attenuation at each pixel was first calculated using reference light and dark field values from within each image. These were provided by the thick sample base and free space far ahead of it. By integrating along the vertical image axis, a line-out of the mean attenuations within the jet was found. A 96% confidence interval on each point was calculated using the standard deviation and number of pixels within each of the bright field, dark field, and jet regions. Applying Beer's law, the attenuation was then converted to a thickness times density (ρd) along the jet's cross section.35 Mass attenuation coefficients of 12.69 and 21.13 cm2/g have been used for the Cu and Sn, respectively, based on the peak x-ray energy of 23.5 keV.36 While the value of ρd is the most directly inferred quantity in this analysis, it can be converted into jet thickness (d) by approximating the jet to be at ambient density. To first order, this approximation is reasonable because of the thinness of the jet in comparison to the time scale and wave speed, which imply an ample opportunity for the jet center to equilibrate with the zero-pressure free surface. Still, both values will be reported because ρd is more rigorous, while d is likely to be more easily understood.

The effects of phase contrast, as well as other systematic errors, e.g., the assumed monochromatic beam, were evaluated from regions of the dynamic images where the stationary foils were located. For Sn, the mean thickness calculated from these x-ray attenuations was 93% of that measured mechanically before the shot. The effect of phase contrast can be seen by the light and dark bands around the edges of the static Sn foils shown in Figs. 5 and 6. Calculated thickness profiles for one such foil are shown for several dynamic frames in Fig. 9. This shows that the peaks of the fringes extend more than 10 μm on either side of the foil edge, with deviations in the apparent thicknesses that in many cases exceed 10–30%. While that makes phase-contrast effects the single largest source of uncertainty in the thickness measurement, this trade-off is necessary to resolve fine structure within the jets.

FIG. 9.

Thickness of the Sn foil during three sequential dynamic frames, showing the line-out variation, frame to frame repeatability, and phase contrast effects. The solid black lines represent the thickness as measured mechanically prior to the shot, with dashed lines showing the uncertainty in that measurement.

FIG. 9.

Thickness of the Sn foil during three sequential dynamic frames, showing the line-out variation, frame to frame repeatability, and phase contrast effects. The solid black lines represent the thickness as measured mechanically prior to the shot, with dashed lines showing the uncertainty in that measurement.

Close modal

The calculated thickness profiles of the jets are shown in Figs. 10–15, with each subplot corresponding to the observation time labeled in their upper right. The spatial origin is taken at the sample's planar surface position at shock breakout. The dotted blue lines represent the 96% confidence interval, calculated as described above. Qualitatively, the main difference in jet shapes appears to be the ratio of the thickness at the tip and waist, which was generally greater for higher velocity cases. At later observation times, the Sn thicknesses ranged from 5 to 10 and 13 to 47 μm at the waist and tip, respectively. Similar plots for the no-groove control regions are not presented because the mass was too low for a meaningful measurement. The data for thickness vs position are included in tabular form in the supplemental material.

FIG. 10.

Jet thickness (d) vs position for the Sn1 jet for each frame in which it could be calculated for the entire jet. The paired right axis shows the more rigorous value of thickness times density (ρd). Dotted lines bracketing the solid blue line indicate the 96% confidence interval. Vertical dashed lines indicate the identified positions of the jet tip and base.

FIG. 10.

Jet thickness (d) vs position for the Sn1 jet for each frame in which it could be calculated for the entire jet. The paired right axis shows the more rigorous value of thickness times density (ρd). Dotted lines bracketing the solid blue line indicate the 96% confidence interval. Vertical dashed lines indicate the identified positions of the jet tip and base.

Close modal
FIG. 11.

Jet thickness vs position for Sn2, otherwise like Fig. 10.

FIG. 11.

Jet thickness vs position for Sn2, otherwise like Fig. 10.

Close modal
FIG. 12.

Jet thickness vs position for Sn3, otherwise like Fig. 10.

FIG. 12.

Jet thickness vs position for Sn3, otherwise like Fig. 10.

Close modal
FIG. 13.

Jet thickness vs position for Sn4, otherwise like Fig. 10.

FIG. 13.

Jet thickness vs position for Sn4, otherwise like Fig. 10.

Close modal
FIG. 14.

Jet thickness vs position for Cu1, otherwise like Fig. 10.

FIG. 14.

Jet thickness vs position for Cu1, otherwise like Fig. 10.

Close modal
FIG. 15.

Jet thickness vs position for Cu2, otherwise like Fig. 10.

FIG. 15.

Jet thickness vs position for Cu2, otherwise like Fig. 10.

Close modal

While the spatial profiles of the jets may be of interest for validating simulations, an integral measure is more useful for revealing the overall trends. As such, Fig. 16 shows the total linear mass of the jet vs time, found by simply integrating each curve in Figs. 10–13. This provides the mass per unit length of the jet. This is not expressed as mass per unit area, as is typical for surface ejecta measurements because the grooves have no periodicity and hence no well-defined area. As the integration of jet thickness is somewhat sensitive to where the base of the jet is measured, the base and tip positions have been marked by the dashed vertical lines in Figs. 10–13. The base was taken as that point where the Sn reached a thickness of 120.5 μm (the 1/e attenuation length). The error bars represent the difference between the upper and lower confidence intervals in Figs. 10–13. This provides an upper bound on the error, since uncorrelated thickness errors would add in quadrature, leading to a substantially smaller estimate of the uncertainty. However, the larger value has been plotted in recognition of the apparent correlation between neighboring values of the thickness.

FIG. 16.

Mass per unit length of the jet as a function of time since shock breakout.

FIG. 16.

Mass per unit length of the jet as a function of time since shock breakout.

Close modal

The most interesting feature of Fig. 16 is that Sn2 (solid) and Sn3 (melted) accumulated mass at nearly identical rates for the entire observation time, despite having been shocked to pressures differing by 8.7 GPa and spanning the melt-on-release threshold. This stands in contrast to the trend seen between Sn1 and Sn2, where increasing the pressure by 9.0 GPa led to a substantial increase in the accumulation of jet mass. The same was true for Sn3 and Sn4 (ΔP=9.6GPa). So, increasing the shock pressure led to increasing jet mass accumulation rates both above and below the melt-on-release pressure but not in its immediate vicinity. This finding is not an artifact of unwanted background ejecta, from which the bias acts in the opposite direction, i.e., a minor overestimation of the jet mass for Sn3 and Sn4.

This observation may surprise our intuition of how fluid jets should behave vis-a-vis solid jets. It also may appear to contradict what was seen in the no-groove control regions, not to mention multiple prior works on Sn ejecta (see, for example, Refs. 21, 32, and 33). However, these cases all involve differences in length scale. In those instances where melting was accompanied by an increase in ejected mass, the initial perturbations were relatively small, on the order of 1 μm or less. For comparatively large grooves (45 μm deep), Mackay et al.14 recently predicted that crossing the melt-on-release condition of Sn would produce a plateau in the mass of microjets like has been observed here at a somewhat larger length scale.

This size effect may be attributable to the role of shear strength in RMI growth. Consider that the relative asymptotic amplitude of an RMI in an elastoplastic medium is directly proportional to the amplitude of the initial perturbation and indirectly proportional to the yield strength.37 This scaling, according to Piriz et al.37 can be given as follows:

ξξiξiρkξiup2Y,

where ξi and ξ are, respectively, the initial and asymptotic amplitudes of the perturbation; k is the perturbation wave number; ρ is density; up is the shock wave's particle velocity; and Y is the yield strength. Once ξξiξi grows large enough, jets and subsequent ejecta form. This implies that even the low strength of Sn can be sufficient to prevent jetting if the initial perturbations are “small enough,” which in the present experiments apparently means of order ∼1 μm. When that strength is removed by melting, then the growth becomes unbounded at much smaller length scales. For much deeper initial grooves, the effect of strength can be swamped long before melting occurs. In that instance, a drop in strength upon melting would be expected to have little impact, since a jet is bound to form in any case.

However, the preceding discussion does not provide an explanation for the observed stagnation in the jet mass near the melt-on-release threshold. While the relevant physics appears to be contained in the simulations from Mackay et al.,14 an understanding of the most relevant aspects remains unknown. At present, we can only speculate that the effect is related to partial localized melting. According to simulations,14 the material near the groove can be driven to melt at pressures below the bulk melt-on-release threshold because of the accumulation of plastic work. Since the deformation is concentrated near the base of the groove, so too is the localized melting, which leads to molten pockets within a solid bulk. This creates a heterogeneous material in which the flow is apparently different than when either constituent phase exists in the same geometry in isolation. In any case, additional work is required to reach a more confident and precise answer.

The effects of release melting have been experimentally investigated for a series of Sn microjets, along with comparisons to Cu jets of similar velocity and solid conditions in the base sample material. This revealed that for large grooves (∼250 μm deep, 62° angle), there was a stagnation in the rate at which the jet mass increased as the melt-on-release condition was exceeded. The opposite was seen for the mass of ejecta originating from smaller defects (∼1 μm deep steps) in the control regions, as has already been well established by others.5,10–12 The conclusion is that the effect of melting on jet formation is size-dependent. To better understand the origins of this phenomenon, the full position and thickness data for each jet have been made available, along with error bars, in the hopes that it may aid future computational studies.

See the supplementary material for thickness vs position data for each jet shown in Figs. 10–15 to facilitate future comparisons. In these tables, the position (μm) data are in column 1 and span from the jet base to the tip. The thickness (μm) values are in column 2.

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. 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 U.S. Department of Energy (DOE) Office of Science User Facility, operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357. This work has been assigned an LLNL information management release number of LLNL-JRNL-821093.

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

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