## Abstract

A metallic microparticle impacting a metallic substrate with sufficiently high velocity will adhere, assisted by the emergence of jetting—the splash-like extrusion of solid matter at the periphery of the impact. In this work, we compare real-time observations of high-velocity single-microparticle impacts to an elastic–plastic model to develop a more thorough understanding of the transition between the regimes of rebound and bonding. We first extract an effective dynamic yield strength for copper from prior experiments impacting alumina spheres onto copper substrates. We then use this dynamic yield strength to analyze impacts of copper particles on copper substrates. We find that up to moderate impact velocities, impacts and rebound velocities follow a power-law behavior well-predicted on the basis of elastic-perfectly plastic analysis and can be captured well with a single value for the dynamic strength that subsumes many details not explicitly modeled (rate and hardening effects and adiabatic heating). However, the rebound behavior diverges from the power-law at higher impact velocities approaching bonding, where jetting sets on. This divergence is associated with additional lost kinetic energy, which goes into the ejection of the material associated with jetting and into breaking incipient bonds between the particle and substrate. These results further support and develop the idea that jetting facilitates bonding where a critical amount of bond formation is required to effect permanent particle deposition and prevent the particle from rebounding.

## Introduction

While the mechanics of elastic–plastic impacts have been extensively modeled [16], experimental comparisons are generally limited to subsonic velocities or large impactors [5,79]. However, high-velocity microscale impacts are relevant to some contemporary applications, including erosion, micrometeorite capture, and particle-spray processes. For example, cold spray coating is a technique where particle–substrate bonding is achieved with ∼1–100 µm particles impacted above a minimum “critical velocity” that is typically supersonic and below which particles rebound [1013]. An outward extrusion of the material from the interface between the impacting particle and substrate, known as jetting, has been considered closely related to bonding [1417]. Numerous mechanisms have been proposed for jetting [14,1820] although there is limited direct observational support. Experimental approaches to study mechanics at the relevant high impact velocities and small impactor sizes are vital for a thorough understanding of such mechanistic questions.

The recent development of an all-optical microballistic test platform has provided direct real-time observations of high-velocity single-microparticle impacts in a number of systems [2125]. Impact behaviors of metallic particles have been directly observed across the rebound and bonding regimes, providing more precise measurements of critical velocity for bonding as well as real-time observations of bonding-associated jetting and ejection [17]. While many of these investigations are centered around bonding at and above the critical velocity [17,26,27], the elastic–plastic mechanics for high-speed impacts in the lower-velocity rebound regime also involve extreme conditions worthy of study. Further, the measurement of both inbound and outbound velocities on such non-bonding impacts provides rich data from which to extract mechanical properties.

Therefore, in this work, we focus on the elastic–plastic rebound behavior of microparticles and compare experimental data with plastic impact models to extract some effective elastic–plastic properties. We first examine impacts of alumina particles on copper, originally conducted to measure metal hardness at very high strain rates [28]. By comparing to a simple model for the impact of an elastic sphere with an elastic-perfectly plastic (EPP) half-space [2], we extract an effective dynamic yield strength for copper. We then use this model and extracted strength to discuss the energy dissipation mechanisms present in a copper–copper impact system relevant to cold spray. Our goal is to analyze rebound behavior to develop a more comprehensive understanding of the impact and bonding behavior of metallic microparticles in the context of cold spray and to better appreciate the energy dissipation associated with the process of jetting.

## High-Velocity Micro-Impact Experiments

We use an all-optical platform, the laser-induced particle impact test (LIPIT), schematically shown in Fig. 1(a), to conduct impact experiments [17,21,22]. Particles are dispersed onto and launched from a “launching pad,” consisting of a glass substrate (210-µm thick), an ablative gold layer (60-nm thick), and a polyurea polymer layer (30-µm thick). The launching pad surface is imaged with an optical microscope, and a single particle is selected and measured immediately before being launched with a high-power laser pulse (Nd-YAG, 532-nm wavelength, 10-ns duration). This pulse is focused to ablate the gold layer, which in turn expands the polymer film and accelerates the particle toward the target. A second laser pulse (640-nm wavelength, 30-μs duration) illuminates the field of view, and 16 images of the impact event are captured by an ultra-high-speed camera (Specialised Imaging, SIMX16) with 5-ns exposure and variable interframe time. From these images, the impact and rebound velocities are measured with an uncertainty of ±2%. Particles impact normal to the substrate surface within ±3°. More details on the experimental setup and launch pad fabrication can be found in previous papers [17,22]. Figure 1(b) shows eight frames of a 12.2-µm diameter copper particle impacting a copper substrate at 400 m/s and then rebounding at 30 m/s. The coefficient of restitution (CoR) in this experiment is thus ∼30/400 = 0.075.

Fig. 1
Fig. 1
Close modal

For a portion of the analysis below, we consider impacts of 14 µm alumina particles, purchased from Inframat Advanced Materials LLC (Amherst, NY). Those experiments were already described in a previous study [28], and the data are reproduced here for impacts on a copper substrate of 3.175-mm thickness purchased from OnlineMetals (Seattle, WA). For this work, additional new experiments were conducted at lower rates using the same materials and procedures. In addition, new experiments were conducted on the same copper substrates, using atomized spherical copper particles with a nominal size of 10 µm purchased from Alfa Aesar (Ward Hill, MA). Substrate surfaces were ground and polished to a nominal 0.04 µm finish prior to impact experiments.

## Elastic–Plastic Impact Analysis

Our primary goal in this work is to understand the impact response of copper microparticles on copper substrates. We therefore aim to quantitatively evaluate impact and rebound velocity information, such as those shown in Fig. 1, in a way that permits the extraction of material properties such as the dynamic strength. Material behavior at high rates is complex, involving strain hardening, strain-rate hardening, adiabatic heat generation and transport, thermal softening, etc. Here, we are interested in a simple analysis with broad applicability to a number of materials and conditions where we will tend to aggregate these complex physics into some “effective” elastic–plastic properties. For this purpose, we consider the work of Wu et al. [2,3], who presented a series of detailed finite element computations in which they modeled the rebound behavior of both an elastic sphere impacting an EPP half-space and an EPP sphere impacting a rigid wall. They found that in either case, the coefficient of restitution could be empirically fitted with a simple non-dimensionalized power-law
$CoR=VrVi=α(VyVi⋅E*Yd)1/2$
(1)
In this expression, Vi is the impact velocity, Vr is the rebound velocity, Yd is the dynamic yield strength that controls plasticity at the rates in question, and E* is the reduced elastic modulus defined by elastic moduli E and Poisson’s ratios ν of the two materials
$1E*=1−ν12E1+1−ν22E2$
(2)
The velocity at which plastic deformation initiates is termed Vy, and is defined by Johnson [1] as
$Vy=(26Yd5ρE*4)1/2$
(3)
with ρ the density of the impacting material. Finally, α is a prefactor determined from fitting simulated impacts of an elastic sphere on an EPP substrate and an EPP sphere on a rigid wall, yielding values of 0.78 and 0.62, respectively. The power-law behaviors of Eq. (1), including the specific exponent of −1/2 with respect to Vi, are valid beyond the cases established by Wu et al. [2,3], as established for example in the more elaborate finite element simulations of impacts of aluminum, copper, and stainless steel microspheres onto matched substrates [6].
In the simulations of Wu et al., only simple elastic and plastic deformations were considered with no strain or strain-rate hardening and no adiabatic heating considered. As such, their analysis is formally limited to velocities below
$ρVi2Yd=0.1$
(4)
where heating effects are negligible, as established by Johnson [1]. For many metals, this velocity limit is below 100 m/s, and applying this analysis at higher velocities involves greater approximation. The present study includes high-velocity impacts well above this threshold, ranging from 100 to nearly 900 m/s, which involves quantities of ρVi2/Yd up to and beyond 1. Productive use of this analysis, therefore, requires that we acknowledge the production of heat and account for it in the changes to the dynamic yield strength. We achieve this by first extracting an “average” or “effective” plastic yield stress Yd for copper which essentially averages over all of the strains, rates, and temperatures that may prevail in the contact plasticity problem; we do this by analyzing impacts of alumina impactors on copper to extract Yd with Eq. (1). We then subsequently use this extracted yield stress for copper with Eq. (1) to model our system of interest, copper impacting copper, in the same velocity range. As we shall see, this approach appears effective although it certainly subsumes a great deal of deformation physics into the simple characteristic strength parameter Yd.

## Dynamic Yield Strength of Copper

To first develop insight into the effective dynamic strength of copper under the relevant experimental conditions (scale, strain, rate, etc.), we revisit a prior study done by our group where nominally rigid alumina particles (14.0 ± 0.5 µm diameter) were used as impactors to measure the dynamic hardness of copper [28]. The CoR data from those experiments are reproduced in Fig. 2(a), along with new data from the present work on the same system at lower velocities.

Fig. 2
Fig. 2
Close modal

Taking the alumina sphere as elastic and unyielding in the impacts against the softer copper (α = 0.78), we fit the data corresponding to impacts between 100 and 800 m/s with Eq. (1) to extract an “effective” or average dynamic yield strength of the copper substrate. The additional, low-velocity data are disregarded in this analysis, as they likely represent a low-strain-rate regime of rebound behavior, described by Johnson [1] where CoR ∝ Vi−1/4. We take values of density, Young’s modulus, and Poisson’s ratio to be independent of impact velocity, and based on the fit in Fig. 2(b) on double-logarithmic axes, we extract a value for the effective dynamic yield strength Yd = 450 MPa.

Interestingly, a single, constant Yd value appears to present a reasonable approximation that fits all of our experimental data, despite the many assumptions in the analysis. We further provide values of Yd corresponding to each individual impact (Fig. 2(c)). These Yd values are found to lie in a tight range across all our experimental velocities, with a standard deviation of just ∼45 MPa. We use this standard deviation subsequently to establish a range of expected behavior.

## Copper on Copper Impacts

With an effective dynamic yield strength for copper under the conditions of interest, we now proceed to consider the copper–copper system. Data for copper particles (12.5 ± 1 µm diameter) impacted on copper substrates between velocities of 100 and 900 m/s are shown in Fig. 3(a) and show a typical trend of CoR decreasing up to an apparent critical velocity above which particles adhere rather than rebounding. The critical velocity is calculated by finding the smallest single group of impacts with an equal number of rebound and bonding events and taking the average between the highest and lowest velocities of that group. This is analogous to finding the “ballistic limit” [29] for bonding. In this case, the critical velocity is the average between two impacts, yielding 580 m/s.

Fig. 3
Fig. 3
Close modal

We note that the critical bonding velocity is certainly a system-specific quantity and changes with details of the investigated material, including particle size [12] and chemical content, particularly oxygen content [30]. In copper, these variables have effects that spread the measured critical velocity over a broad range from ∼300 to ∼600 m/s or more [31]. The present value (580 m/s) as well as the trend of the data in Fig. 3(a) are in good agreement with data for similar single-particle impacts published previously [17] for slightly larger particles (14 ± 2 µm diameter) in the copper–copper system with the same commercial source. It is also in agreement with other reported values based on cold spray experiments with copper powders of oxygen contents standard for atomized commercial powders, around 0.1–0.15wt% [12,31]. In our subsequent analyses, we use only the new data from the present study with an average particle diameter of 12.5 µm and a critical velocity of 580 m/s.

Three scanning electron microscopy (SEM) images of impact sites are shown in Figs. 3(b)3(d) for Vi = 400, 550, and 770 m/s, respectively; these are denoted by arrows in Fig. 3(a) as well. At 770 m/s, the impacted particle appears bonded to the substrate with substantial signs of jetting around the edges of the particle where “lips” of prior jets can be seen originating from both the substrate and the particle (denoted by arrows). Similar signs of jetting are present around the edge of the crater formed at 550 m/s as well (Fig. 3(c)) although perhaps with smaller lips and less contiguously around the perimeter. At the lowest of the three velocities, no jet lips are observed in Fig. 3(b).

In the prior work from our group [17,26,27] and others [1416], jet formation has been closely linked to impact bonding under the premise that jet formation involves severe plastic deformation right at the interface between the particle and substrate where the adhesive bond forms. Such severe jetting deformation promotes flattening of microscopic surface roughness/asperities, and spreading and removal of contaminating surface films like native oxide and generally assists in the formation of clean metal-on-metal contacts that permit adhesive welding to occur. The observations in Fig. 3(d) support this general line of argument with the presence of substantial jets associated with the bonded state. However, Fig. 3(c) also suggests that it is possible to initiate some jets around the impact site without necessarily producing enough of a bond to result in permanent particle adhesion. Permanent particle adhesion apparently requires some critical amount of interfacial plasticity provided by jetting. Impacts like the one shown at 550 m/s are not sufficient to achieve that condition. Thus, jetting emerges at sufficiently high impact velocities but sets on below the critical velocity at which there is sufficient jetting to achieve bonding.

Closer inspection of the CoR data also leads us to a similar view—the transition between rebounding and bonding is not just a discontinuity; it also involves a divergence from the power-law scaling before adhesion occurs. This can be best seen in Fig. 4(a) by examining the copper–copper data in a double-logarithmic fashion. For lower velocities, we see a reasonably convincing conformity of these data to the characteristic scaling law of Eq. (1), CoR ∝ Vi−1/2. While fitting various ranges of the data can give slightly different values of the power-law exponent (0.6–0.7), the theoretical value of −1/2 is supported both by the EPP model [2,3] and more advanced mechanical models that incorporate hardening, rate effects, etc. [6]. Taking the dynamic yield strength of copper extracted independently earlier (Yd = 450 MPa), we present fitting-parameter-free predictions of Eq. (1) both for an elastic sphere impacting an EPP substrate (α = 0.78) and an EPP sphere impacting a rigid substrate (α = 0.62). The data lie rather close to the latter prediction, and in fact, a least-squares fitting to a single parameter (using only the data below 550 m/s) gives a best-fit α = 0.57, with a standard deviation as shown by the shaded band in Fig. 4(a) to account for uncertainty in Yd. The fact that the fitted prefactor is close to that expected for a plastic sphere impacting a rigid wall is intuitively reasonable: in matched material impacts, the substrate experiences a shallow spherical indentation which might have a characteristic strain on the order of 0.1, whereas the impacted particle is severely flattened by a factor of as much as ∼1.5, implying a far higher characteristic strain of ∼0.4. To first order, then, the particle bears most of the deformation, as contemplated in the EPP particle-on-rigid-wall model, explaining the conformity of the data to that model.

Fig. 4
Fig. 4
Close modal
As impact velocities rise above about 400 m/s and approach the critical velocity for bonding, the experimental CoR increasingly deviates from the power-law of Eq. (1) to values lower than expected for simple elastic–plastic behavior. The divergent CoR behavior in this range suggests that simple elastic–plastic impact mechanics, even when calibrated to account for high rates (and adiabatic heating), are not sufficient to describe the physics of these impacts. Viewed another way, there is an additional loss of kinetic energy in this range of velocities that apparently does not result simply from the plastic formation of a crater and the compression of the impactor. We can quantitatively assess this excess lost energy by first determining for each impact the rebound kinetic energy
$Er=mVr22$
(5)
where the mass of each particle m is calculated individually based on measured diameter. The difference between the actual rebound kinetic energy and that predicted by Eq. (1) (with α = 0.57) is presented in Fig. 4(b) as the excess lost energy, which rises as the impact velocity approaches the critical velocity.

The largest observed quantity of excess lost energy is ∼5.7 nJ and that at the critical velocity is 7.1 nJ. For reference, the impact kinetic energies are on the order of microjoules, three orders of magnitude greater, meaning that simple elastic–plastic mechanics still accounts for the vast majority of impact energy dissipation. Nevertheless, this last small proportion of energy loss is needed to slow and stop the particle and, therefore, plays a key role in the bonding behavior.

## Deviation From the Power-Law

Interestingly, the same kind of power-law deviation in the run-up to adhesion is seen in other published single-particle impact experiments on other materials. This is shown in Figs. 5(a)5(c) by replotting the published data for Ni, Al, and Zn, respectively [17]. In each case, the power-law of Eq. (1) is shown with α = 0.57 fixed and fitted to the lower-velocity data with a single parameter, Yd, as shown in Table 1.

Fig. 5
Fig. 5
Close modal
Table 1

Critical velocities, dynamic yield strengths, and divergence velocities for the present data on Cu as well as previously published data [17]

MaterialCuNiAlZn
Critical velocity (m/s)580 ± 12655 ± 5810 ± 14540 ± 25
Dynamic yield strength (MPa)450 ± 45800 ± 32290 ± 10470 ± 21
Divergence velocity (m/s)510 ± 36540 ± 25620 ± 42420 ± 17
MaterialCuNiAlZn
Critical velocity (m/s)580 ± 12655 ± 5810 ± 14540 ± 25
Dynamic yield strength (MPa)450 ± 45800 ± 32290 ± 10470 ± 21
Divergence velocity (m/s)510 ± 36540 ± 25620 ± 42420 ± 17

Whereas it is common to tabulate critical bonding velocities for different materials systems, we propose that the velocity at which impacts diverge from power-law behavior also represents a characteristic quantity worth tabulating and understanding. We determine the divergence velocity, termed Vd, in the same manner as was described above for the critical bonding velocity. First, a velocity range containing an equal number of impacts conforming and not conforming to the power-law is found such that the number of impacts in this range is the minimum number required to ensure that only one such range exists for a given data set. The velocity is taken as the average of the two outermost points, while the error is half their difference. Performing this analysis on our Cu data and the literature data from Fig. 5 returns the values in Table 1.

Hassani et al. proposed a hydrodynamic argument for jetting where the shock formed upon impact detaches from the particle–substrate edge leading to local tension that can initiate a jet. If the local tension in the jet exceeds the material spall strength, the jet extends and fragments into small ejecta [26]. For a matched material system in the limit where the shock speed, Cs, is independent of impact velocity, their result can be written as follows:
$Vspall∝KρCs$
(6)
where K is the bulk modulus and ρ is the density.

In their work, Hassani et al. showed a strong conformity of measured critical adhesion velocities to this expected scaling, as shown in Fig. 6. The adhesion condition involves considerable jetting, concomitant considerable interface straining, clean metal-on-metal contact, and bonding to a degree that the particle remains permanently attached. However, as illustrated in Fig. 3(c), there is some degree of jetting that is apparently not enough to achieve full irreversible particle adhesion at lower velocities, setting in at the point of power-law divergence. We hypothesize that the scaling of Eq. (6) should thus also be seen in the divergence velocity as a marker of the onset of such spall. This is tested in Fig. 6 where Vd is also plotted against K/ρ/Cs. The proportionality observed here suggests that the power-law divergence we observe originates from hydrodynamic phenomena.

Fig. 6
Fig. 6
Close modal

Taken together, all of these observations and analysis in Figs. 36 speak to the emergence of jetting as the source of power-law divergence in metal-on-metal impacts near the bonding critical velocity. The extra lost energy in these impacts is, therefore, most likely attributable to jetting (and its consequences), and we turn our attention to a detailed discussion of that in what follows.

## Jetting-Associated Energy Dissipation Mechanisms

As noted earlier, the maximum amount of excess energy loss that we are able to measure by virtue of the power-law divergence of our experimental data on copper is Ed = 7.1 nJ. This value corresponds to the energy predicted by the rebound power-law (Eq. (1)) at the critical velocity, and it is of similar magnitude when evaluated for the other metals in Fig. 5: Ed = 5.2, 6.2, and 5.1 nJ for Ni, Al, and Zn, respectively. This excess energy loss is associated with jetting, and we suggest two possible major contributions to Ed.

First, there is a clear source of energy loss associated with material ejection upon jetting. As the ejected material travels away from the interface at high speeds, the kinetic energy carried by it, Ed(K), is
$Ed(K)=12mjVj2$
(7)
with mj and Vj, respectively, the total mass and average velocity of the ejected material. Although we expect the ejected material to be some combination of both pure metal and oxide [27], we approximate ρj as the density of copper, 8930 kg/m3, and based on observations of material ejection in the prior work on Al [17], we suggest average ejection velocity Vj to be ∼1 km/s. With these values, we would require an ejected mass of the material of mj = 1.4 × 10−14 kg at the critical velocity to fully account for the lost particle kinetic energy Ed by only this mechanism. For our experiments, this would correspond to 0.16% of the initial particle mass. A molecular dynamics study observed, in impacts of copper spheres on rigid substrates under comparable conditions, material ejection on the same order of magnitude relative to the initial impactor mass [32]. Although there are extensive experimental efforts to measure impact-induced ejection mass in the context of planetary bodies, these experiments generally consider nonmetallic and porous targets [3335]. We are not aware of any experimental efforts to assess this mass directly in the present context of metallic microscale impacts, and we encourage experimentation to evaluate it in future work. In any event, 0.16% of the particle mass could be a plausible magnitude for the jetting-associated ejecta in experiments near the critical velocity, meaning that the kinetic energy of the jets themselves may account for a significant amount of the power-law deviation.
Second, because the jetting process facilitates bonding by providing intimate and pristine metallic surfaces [1417], we expect metallic bonds to form whenever there is some amount of jetting, even if the particle subsequently detaches from the substrate and rebounds. The excess lost energy Ed would then be associated with the energy of refracturing those metallic bonds. This view is in line with simulations contemplating the effects of temporary bonding on impact and rebound behavior [36]. Simplistically, we can estimate this debonding energy, Ed(D), as that dissipated in a mode I fracture event in a plane stress condition
$Ed(D)≈KIC2EAD$
(8)
where KIC is the mode I fracture toughness, E is the elastic modulus, and AD is the area that bonds and then must refracture to permit particle rebound. The modulus of copper is 110 GPa, but the other parameters in this expression are not known exactly. The fracture toughness can be bounded between that for perfectly brittle fracture (KIC ∼1 MPa m1/2) and that for ideally coherent bulk copper (KIC ∼60 MPa m1/2) [37]. However, these bounds are too far apart to be practically useful and intuitively do not correspond to the expected physical situation of evaluating a transient metallic bond interface in copper.

A better approximation can be made by directly using the bulk fracture toughness of unannealed cold-sprayed copper deposits, which are on the order of ∼9 MPa m1/2 [37] and which better reflect the nature of the bonded regions in the present experiments. With this value, Eq. (8) suggests that a bonded contact area of AD ≈ 10 µm2 would be required to fully account for the excess energy dissipation at the critical velocity corresponding to ∼5% the total contact area of the flattened particles in our experiments. This value seems reasonable when compared with cross-sectional images of particles deposited via cold spray [15,3840] where, prior to annealing and other post-processing techniques, the particle–substrate interface is extremely imperfect. The actual amount of the bonded interface area can vary between 15% and 95% depending on how far above the critical velocity the impact occurs [41]. In our calculation, we are concerned with impact precisely at the critical velocity, so our value of the 5% bonded area being even below that of 15% in the above experiments seems reasonable. Our value of 5% is also reasonable in comparison to observations like that of Fig. 3(c), below the critical velocity, where incipient jets are seen but represent a truly miniscule amount of the surface area for bonding.

Thus, starting at the onset of jetting and continuing above the critical velocity, it seems that the amount of interfacial area associated with jetting and bonding increases, corresponding to an increasing Ed until and beyond particle arrest and adhesion. After particle adhesion, we anticipate a further increase in the bonded area with impact velocity. It would be highly desirable in future work to quantitatively evaluate the dependence of the transiently and permanently bonded areas on impact velocity both above and below the critical adhesion velocity.

In conclusion, both of the above two effects would thus seem to be relevant in the discussion of energy dissipation in jetting and adhesion near the critical velocity. Other factors, such as the energy lost in breaking up jets into droplets, increases in the surface area, fracturing of oxides, etc., are also at play, but we have estimated these to be of significantly lower magnitude in their contributions to Ed compared with the kinetic energy of the ejected material and the debonding energy.

Further experimental quantification of both material ejection and incipient bonding is required to ascertain which of the two major energy dissipation mechanisms is more significant. Nonetheless, our present observations and calculations suggest there is a threshold level of jetting required to effect permanent bonding. Below this threshold, we observe substrate jetting and divergent rebound behavior. It is only when a sufficient extent of jetting is achieved, with material ejection and incipient bonding, that the particle permanently deposits onto the substrate.

## Conclusion

This work presents a new quantitative view of microparticle impacts over a range of velocities that span from rebound to those where solid-state material ejection and particle bonding occur. First, by comparing single-particle experiments of alumina particles impacting a copper substrate with a model for elastic particle impacts on an elastic-perfectly plastic substrate, we were able to determine an “effective” dynamic yield strength of copper, spanning a large range of velocities, and subsuming a great deal of deformation physics, including hardening and adiabatic heating effects. This dynamic yield strength, in turn, can be used to predict with very high accuracy the rebound behavior for impacts of copper on copper. What is more, such analysis reveals that in the metal-on-metal situation, rebound behavior increasingly diverges at higher impact velocities approaching the bonding transition. This power-law divergence is also seen in three other matched material systems in literature data we have reevaluated. Together, an analysis of all four materials suggests that the onset of this power-law divergence is linked to jetting and the hydrodynamic phenomenon of spall. Microscopic observations align with this view where craters left behind near—but below—the critical adhesion velocity show incipient jet structures on the surface.

We discuss two jetting-associated energy dissipation mechanisms—the kinetic energy transfer to ejected material and the fracture of incipiently formed bonds. We quantitatively estimate whether these mechanisms can effect the divergent energy loss observed in the present copper impacts and found both to be plausible sources of the excess lost energy in the run-up to particle adhesion. Our results demonstrate that jetting is not just a necessary condition for bonding but that there is additionally a threshold of jetting-induced energy dissipation that is required to prevent rebound and create permanent bonds. By analyzing rebound behavior, we have expanded our understanding of jetting in metallic microparticle impacts to build toward a more comprehensive understanding of the impact-induced metallic bonding that is fundamental to cold spray.

## Acknowledgment

This work was primarily supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, and Division of Materials Sciences and Engineering under Award DE-SC0018091. The work was performed in facilities supported by the U.S. Army Research Office and CCDC Army Research Laboratory through the Institute for Soldier Nanotechnologies, under Cooperative Agreement No. W911NF-18-2-0048. The key equipment (the multi-frame camera) was provided through the Office of Naval Research DURIP (Grant No. N00014-13-1-0676).

## Conflict of Interest

There are no conflicts of interest.

## References

1.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
2.
Wu
,
C.
,
Li
,
L.
, and
Thornton
,
C.
,
2003
, “
Rebound Behaviour of Spheres for Plastic Impacts
,”
Int. J. Impact Eng.
,
28
(
9
), pp.
929
946
. 10.1016/S0734-743X(03)00014-9
3.
Wu
,
C.-Y.
,
Li
,
L.-Y.
, and
Thornton
,
C.
,
2005
, “
Energy Dissipation During Normal Impact of Elastic and Elastic–Plastic Spheres
,”
Int. J. Impact Eng.
,
32
(
1–4
), pp.
593
604
. 10.1016/j.ijimpeng.2005.08.007
4.
Brach
,
R. M.
, and
Dunn
,
P. F.
,
1998
, “
Models of Rebound and Capture for Oblique Microparticle Impacts
,”
Aerosol Sci. Technol.
,
29
(
5
), pp.
379
388
. 10.1080/02786829808965577
5.
Brach
,
R. M.
,
Dunn
,
P. F.
, and
Li
,
X.
,
2000
, “
Experiments and Engineering Models of Microparticle Impact and Deposition
,”
,
74
(
1–4
), pp.
227
282
. 10.1080/00218460008034531
6.
Yildirim
,
B.
,
Yang
,
H.
,
Gouldstone
,
A.
, and
Müftü
,
S.
,
2017
, “
Rebound Mechanics of Micrometre-Scale, Spherical Particles in High-Velocity Impacts
,”
Proc. R. Soc. A Math. Phys. Eng. Sci.
,
473
(
2204
), p.
20160936
. 10.1098/rspa.2016.0936
7.
Kharaz
,
A. H.
,
Gorham
,
D. A.
, and
Salman
,
A. D.
,
1999
, “
Accurate Measurement of Particle Impact Parameters
,”
Meas. Sci. Technol.
,
10
(
1
), pp.
31
35
. 10.1088/0957-0233/10/1/009
8.
Gorham
,
D. A.
, and
Kharaz
,
A. H.
,
2000
, “
The Measurement of Particle Rebound Characteristics
,”
Powder Technol.
,
112
(
3
), pp.
193
202
. 10.1016/S0032-5910(00)00293-X
9.
Labous
,
L.
,
Rosato
,
A. D.
, and
Dave
,
R. N.
,
1997
, “
Measurements of Collisional Properties of Spheres Using High-Speed Video Analysis
,”
Phys. Rev. E—Stat. Physics, Plasmas, Fluids, Relat. Interdiscip. Top.
,
56
(
5
), pp.
5717
5725
. 10.1103/PhysRevE.56.5717
10.
,
H.
,
Kreye
,
H.
,
Gärtner
,
F.
, and
Klassen
,
T.
,
2016
, “
Cold Spraying—A Materials Perspective
,”
Acta Mater.
,
116
, pp.
382
407
. 10.1016/j.actamat.2016.06.034
11.
Moridi
,
A.
,
Hassani-Gangaraj
,
S. M.
,
Guagliano
,
M.
, and
Dao
,
M.
,
2014
, “
Cold Spray Coating: Review of Material Systems and Future Perspectives
,”
Surf. Eng.
,
30
(
6
), pp.
369
395
. 10.1179/1743294414Y.0000000270
12.
Schmidt
,
T.
,
Gärtner
,
F.
,
,
H.
, and
Kreye
,
H.
,
2006
, “
Development of a Generalized Parameter Window for Cold Spray Deposition
,”
Acta Mater.
,
54
(
3
), pp.
729
742
. 10.1016/j.actamat.2005.10.005
13.
Papyrin
,
A.
,
Kosarev
,
V.
,
,
S.
,
Alkimov
,
A.
, and
Fomin
,
V.
,
2006
,
Cold Spray Technology
,
Elsevier Science
,
Oxford, UK
.
14.
,
H.
,
Gärtner
,
F.
,
Stoltenhoff
,
T.
, and
Kreye
,
H.
,
2003
, “
Bonding Mechanism in Cold Gas Spraying
,”
Acta Mater.
,
51
(
15
), pp.
4379
4394
. 10.1016/S1359-6454(03)00274-X
15.
King
,
P. C.
,
Busch
,
C.
,
Kittel-Sherri
,
T.
,
Jahedi
,
M.
, and
Gulizia
,
S.
,
2014
, “
Interface Melding in Cold Spray Titanium Particle Impact
,”
Surf. Coatings Technol.
,
239
, pp.
191
199
. 10.1016/j.surfcoat.2013.11.039
16.
Vidaller
,
M. V.
,
List
,
A.
,
Gaertner
,
F.
,
Klassen
,
T.
,
Dosta
,
S.
, and
Guilemany
,
J. M.
,
2015
, “
Single Impact Bonding of Cold Sprayed Ti-6Al-4V Powders on Different Substrates
,”
J. Therm. Spray Technol.
,
24
(
4
), pp.
644
658
. 10.1007/s11666-014-0200-4
17.
Hassani-Gangaraj
,
M.
,
Veysset
,
D.
,
Nelson
,
K. A.
, and
Schuh
,
C. A.
,
2018
, “
In-Situ Observations of Single Micro-Particle Impact Bonding
,”
Scr. Mater.
,
145
, pp.
9
13
. 10.1016/j.scriptamat.2017.09.042
18.
Bae
,
G.
,
Kumar
,
S.
,
Yoon
,
S.
,
Kang
,
K.
,
Na
,
H.
,
Kim
,
H.-J.
, and
Lee
,
C.
,
2009
, “
Bonding Features and Associated Mechanisms in Kinetic Sprayed Titanium Coatings
,”
Acta Mater.
,
57
(
19
), pp.
5654
5666
. 10.1016/j.actamat.2009.07.061
19.
Grujicic
,
M.
,
Zhao
,
C. L.
,
DeRosset
,
W. S.
, and
Helfritch
,
D.
,
2004
, “
Adiabatic Shear Instability Based Mechanism for Particles/Substrate Bonding in the Cold-Gas Dynamic-Spray Process
,”
Mater. Des.
,
25
(
8
), pp.
681
688
. 10.1016/j.matdes.2004.03.008
20.
Li
,
W.-Y.
,
Zhang
,
C.
,
Li
,
C.-J.
, and
Liao
,
H.
,
2009
, “
Modeling Aspects of High Velocity Impact of Particles in Cold Spraying by Explicit Finite Element Analysis
,”
J. Therm. Spray Technol.
,
18
(
5–6
), pp.
921
933
. 10.1007/s11666-009-9325-2
21.
Lee
,
J.-H.
,
Veysset
,
D.
,
Singer
,
J. P.
,
Retsch
,
M.
,
Saini
,
G.
,
Pezeril
,
T.
,
Nelson
,
K. A.
, and
Thomas
,
E. L.
,
2012
, “
High Strain Rate Deformation of Layered Nanocomposites
,”
Nat. Commun.
,
3:1164
. 10.1038/ncomms2166
22.
Veysset
,
D.
,
Hsieh
,
A. J.
,
Kooi
,
S.
,
Maznev
,
A. A.
,
Masser
,
K. A.
, and
Nelson
,
K. A.
,
2016
, “
Dynamics of Supersonic Microparticle Impact on Elastomers Revealed by Real-Time Multi-Frame Imaging
,”
Sci. Rep.
,
6
(
25577
), pp.
1
7
. 10.1038/srep25577
23.
Veysset
,
D.
,
Kooi
,
S. E.
,
Мaznev
,
A. A.
,
Tang
,
S.
,
Mijailovic
,
A. S.
,
Yang
,
Y. J.
,
Geiser
,
K.
,
Van Vliet
,
K. J.
,
Olsen
,
B. D.
, and
Nelson
,
K. A.
,
2018
, “
High-Velocity Micro-Particle Impact on Gelatin and Synthetic Hydrogel
,”
J. Mech. Behav. Biomed. Mater.
,
86
, pp.
71
76
. 10.1016/j.jmbbm.2018.06.016
24.
Wu
,
Y.-C. M. C. M.
,
Hu
,
W.
,
Sun
,
Y.
,
Veysset
,
D.
,
Kooi
,
S. E.
,
Nelson
,
K. A.
,
Swager
,
T. M.
, and
Hsieh
,
A. J.
,
2019
, “
Unraveling the High Strain-Rate Dynamic Stiffening in Select Model Polyurethanes—The Role of Intermolecular Hydrogen Bonding
,”
Polymer (Guildf)
,
168
, pp.
218
227
. 10.1016/j.polymer.2019.02.038
25.
Sun
,
Y.
,
Wu
,
Y.-C. M.
,
Veysset
,
D.
,
Kooi
,
S. E.
,
Hu
,
W.
,
Swager
,
T. M.
,
Nelson
,
K. A.
, and
Hsieh
,
A. J.
,
2019
, “
Molecular Dependencies of Dynamic Stiffening and Strengthening Through High Strain Rate Microparticle Impact of Polyurethane and Polyurea Elastomers
,”
Appl. Phys. Lett.
,
115
(
9
), p.
093701
. 10.1063/1.5111964
26.
Hassani-Gangaraj
,
M.
,
Veysset
,
D.
,
Champagne
,
V. K.
,
Nelson
,
K. A.
, and
Schuh
,
C. A.
,
2018
, “
,”
Acta Mater.
,
158
, pp.
430
439
. 10.1016/j.actamat.2018.07.065
27.
Hassani-Gangaraj
,
M.
,
Veysset
,
D.
,
Nelson
,
K. A.
, and
Schuh
,
C. A.
,
2019
, “
Impact-Bonding With Aluminum, Silver, and Gold Microparticles: Toward Understanding the Role of Native Oxide Layer
,”
Appl. Surf. Sci.
,
476
, pp.
528
532
. 10.1016/j.apsusc.2019.01.111
28.
Hassani
,
M.
,
Veysset
,
D.
,
Nelson
,
K. A.
, and
Schuh
,
C. A.
,
2020
, “
Material Hardness at Strain Rates Beyond 106 S−1 Via High Velocity Microparticle Impact Indentation
,”
Scr. Mater.
,
177
, pp.
198
202
. 10.1016/j.scriptamat.2019.10.032
29.
Carlucci
,
D. E.
, and
Jacobson
,
S. S.
,
2008
,
Ballistics: Theory and Design of Guns and Ammunition
,
CRC Press
,
Boca Raton, FL
.
30.
Gilmore
,
D. L.
,
Dykhuizen
,
R. C.
,
Neiser
,
R. A.
,
Roemer
,
T. J.
, and
Smith
,
M. F.
,
1999
, “
Particle Velocity and Deposition Efficiency in the Cold Spray Process
,”
J. Therm. Spray Technol.
,
8
(
4
), pp.
576
582
. 10.1361/105996399770350278
31.
Li
,
C. J.
,
Li
,
W. Y.
, and
Liao
,
H.
,
2006
, “
Examination of the Critical Velocity for Deposition of Particles in Cold Spraying
,”
J. Therm. Spray Technol.
,
15
(
2
), pp.
212
222
. 10.1361/105996306X108093
32.
Germann
,
T. C.
,
2006
, “
Large-Scale Molecular Dynamics Simulations of Hyperthermal Cluster Impact
,”
Int. J. Impact Eng.
,
33
(
1–12
), pp.
285
293
. 10.1016/j.ijimpeng.2006.09.049
33.
Housen
,
K. R.
, and
Holsapple
,
K. A.
,
2011
, “
Ejecta From Impact Craters
,”
Icarus
,
211
(
1
), pp.
856
875
. 10.1016/j.icarus.2010.09.017
34.
Johnson
,
B. C.
,
Bowling
,
T. J.
, and
Melosh
,
H. J.
,
2014
, “
Jetting During Vertical Impacts of Spherical Projectiles
,”
Icarus
,
238
, pp.
13
22
. 10.1016/j.icarus.2014.05.003
35.
Kurosawa
,
K.
, and
,
S.
,
2019
, “
Impact Cratering Mechanics: A Forward Approach to Predicting Ejecta Velocity Distribution and Transient Crater Radii
,”
Icarus
,
317
, pp.
135
147
. 10.1016/j.icarus.2018.06.021
36.
Rahmati
,
S.
, and
Jodoin
,
B.
,
2020
, “
Physically Based Finite Element Modeling Method to Predict Metallic Bonding in Cold Spray
,”
J. Therm. Spray Technol.
,
29
(
4
), pp.
611
629
. 10.1007/s11666-020-01000-1
37.
Kovarik
,
O.
,
Siegl
,
J.
,
Cizek
,
J.
,
,
T.
, and
Kondas
,
J.
,
2020
, “
Fracture Toughness of Cold Sprayed Pure Metals
,”
J. Therm. Spray Technol.
,
29
(
1–2
), pp.
147
157
. 10.1007/s11666-019-00956-z
38.
Li
,
W.-Y.
,
Yin
,
S.
, and
Wang
,
X.-F.
,
2010
, “
Numerical Investigations of the Effect of Oblique Impact on Particle Deformation in Cold Spraying by the SPH Method
,”
Appl. Surf. Sci.
,
256
(
12
), pp.
3725
3734
. 10.1016/j.apsusc.2010.01.014
39.
Goldbaum
,
D.
,
Chromik
,
R. R.
,
Yue
,
S.
,
Irissou
,
E.
, and
Legoux
,
J. G.
,
2011
, “
Mechanical Property Mapping of Cold Sprayed Ti Splats and Coatings
,”
J. Therm. Spray Technol.
,
20
(
3
), pp.
486
496
. 10.1007/s11666-010-9546-4
40.
Guetta
,
S.
,
Berger
,
M. H.
,
Borit
,
F.
,
Guipont
,
V.
,
Jeandin
,
M.
,
Boustie
,
M.
,
Ichikawa
,
Y.
,
Sakaguchi
,
K.
, and
Ogawa
,
K.
,
2009
, “
Influence of Particle Velocity on Adhesion of Cold-Sprayed Splats
,”
J. Therm. Spray Technol.
,
18
(
3
), pp.
331
342
. 10.1007/s11666-009-9327-0
41.
Schmidt
,
T.
,
,
H.
,
Gärtner
,
F.
,
Richter
,
H.
,
Stoltenhoff
,
T.
,
Kreye
,
H.
, and
Klassen
,
T.
,
2009
, “
From Particle Acceleration to Impact and Bonding in Cold Spraying
,”
J. Therm. Spray Technol.
,
18
(
5–6
), pp.
794
808
. 10.1007/s11666-009-9357-7