The Laser-Induced Particle Impact Test (LIPIT) can be used to probe projectile, target, and synergistic projectile-target microscale-level responses under high strain rate deformation. LIPIT’s advantages over other microscale techniques include the ability to controllably launch a single microparticle and precisely characterize the projectile momentum and kinetic energy before and after target impact. In addition, a LIPIT apparatus possesses a small laboratory footprint and is suitable for extension to high-throughput testing. Hence, LIPIT experiments have been used to study the dynamic response of many polymers, gels, and metals in different structural forms with various target thickness to projectile diameter ratios. These microscopic high-strain-rate ( >106s−1) deformation behavior and impact energy absorption studies were used to infer deformation mechanisms, as well as to suggest promising materials for macroscopic applications. Geometric scale, however, can significantly influence dynamic material behavior through scale-induced changes in event time, strain rate, projectile/target material homogeneity, and more. In this study, such geometric-scale ef- fects are probed. Noncrystalline alumina spheres ranging five orders of magnitude in diameter ( dp= 3µm to 10 mm) were launched into scaled amorphous polycarbonate targets at normal incidence using either LIPIT or a gas gun, depending on the scale. The projectile impact velocity and the projectile diameter to target thickness ratio were held constant in all experiments ( vi= 550 m/s and ht/dp= 0.25, respectively). Impact energies spanned over 11 orders of magnitude, from hundreds of Joules down to nanoJoules, representing the broadest range ever addressed in a single experimental impact study. Nominal target perforation times and strain rates varied by roughly three orders of magnitude. Length scale reduction resulted in a remarkable ∼230% amplification in specific energy absorption ( E∗ p) and an ∼240% increase in relative impact deforma- ∗Corresponding author. J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, Texas, 77843, United States. Email Address :jacob rogers@tamu.edu ∗∗Corresponding author. J. Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, College Station, Texas, 77843, United States. Email Address :telacyjr@tamu.edu Preprint submitted to TBD September 19, 2024 tion area. The fraction of target material engaged during impact more than doubled at the smallest scale. Normalized estimated target ballistic limit velocities were roughly three times higher in the LIPIT experi- ments. The improved performance of targets at smaller scales likely stemmed from rate- and size-induced enhancements of yield and failure stress values, shifting the bulk failure mode from plugging to dishing. These findings demonstrate that lower length scale dynamic material behavior may be profoundly distinct from that at the macroscale, necessitating a more rigorous assessment of scale effects when developing new materials and structures for use in extreme environments. Keywords: Laser induced particle impact test (LIPIT), Single-stage gas gun, Scanning electron microscopy (SEM), Optical microscopy, Specific energy absorption, Length scale, Strain rate, Polycarbonate, Alumina, Thin films, Profilometry, Laser confocal microscopy, Microspheres, Impact scaling, Geometric scaling, Elastic Plastic Impact Computation (EPIC) code 1. Introduction and Motivation Nature has many types of spectacular impacts, including collisions of galaxies transcending eons [1, 2], 1 asteroid-planetary impacts triggering mass extinctions [3], biological predator events like the potent strikes 2 of mantis shrimp against clam shells [4], and star-powered fusion of atoms lasting only zeptoseconds [5]. 3 Similarly, human-engineered collisions play vital roles in asteroid/meteoroid redirection [6], construction and 4 fabrication processes [7], hypersonic missile defense [8], kinetic energy weapon development [9], cold spray 5 applications [10], innovative drug delivery methods [11], and fusion power generation [12]. Investigations of 6 impact dynamics fuel the development of ballistic armor [13], automobile collision passenger safeguards [14], 7 and spacecraft micro-meteoroid/orbital debris shielding [15]. The impact energies can range from yotta- 8 Joules (1024J) to attoJoules (10−18J) and can transform into thermal, chemical, potential, and mechanical 9 energy, activating material elastic or inelastic deformation, fracture, fragmentation, melting, vaporization, 10 sublimation, ionization, fission, and fusion. Moreover, the mechanisms and processes at play can change 11 dramatically with spatial and temporal scales. 12 Many impacts can be characterized by the projectile and target material properties, projectile diameter 13 (dp), target thickness ( ht), and impact velocity ( vi), provided that the target’s lateral dimensions are suffi- 14 ciently large to minimize boundary effects. The following discussion focuses on a set pair of projectile and 15 target materials. For a given target thickness to projectile diameter ratio ( ht/dp), increasing vishortens the 16 primary event duration, which in turn raises the strain and heating rates. These rates are closely linked 17 to the instantaneous material properties and phase [16]. Enlarging the spatial scale alone effectively lowers 18 strain and heating rates through a corresponding increase in event duration. Decreasing target scale has 19 the opposite effect. For a given combination of ht,dp,ht/dp, there exists a critical velocity that defines the 20 2 transition between target perforation and penetration, commonly termed the ballistic limit velocity ( v50) 21 [17]. Conversely, for a given vi, there is a critical geometric ratio, ( ht/dp)crit, that defines the upper bound 22 for target perforation. An identical spherical impact to a thin target, i.e.,ht/dp≪(ht/dp)crit, typically 23 yields thin film perforation, which resembles axisymmetric membrane stretching and puncturing [18]. As 24 ht/dpapproaches ( ht/dp)crit, ballistic limit perforation occurs, characterized by target cratering, cracking, 25 shear banding, spalling, bulging, dishing, petalling, and/or shear plugging [18, 19]. As the target becomes 26 semi-infinite, ht/dp≫(ht/dp)crit, extensive target penetration and cratering can occur. Material hierarchi- 27 cal inhomogeneities, from macroscale aggregates to microscale crystal grains, result in spatial variations and 28 anisotropies in material properties and failure mechanisms that complicate matters further [20]. 29 Historically, scaled impact research has predominantly focused on ballistic impact and planetary science 30 applications, with most attention given to composites, granular materials, and lightweight metals (see, e.g., 31 [21–25]). However, these studies often rely on certain simplifying assumptions (fully hydrodynamic behavior, 32 axisymmetry, etc.) and are only applicable to a relatively narrow range of potential impact scenarios [ e.g., 33 cratering with ht/dp≫(ht/dp)crit]. Despite the prevalence of natural and artificial impacts, the knowledge of 34 how energy transformation processes and dissipation mechanisms vary with length scale remains insufficient, 35 particularly when transitioning from macro ( ∼10−2m) to micro ( ∼10−6m) material length scales. Such a 36 gap in understanding must be addressed as material characterization techniques transition to increasingly 37 smaller length scales [26–29]. 38 At the microscale, executing and characterizing controlled impacts is fraught with challenges in experi- 39 mental repeatability, measurement uncertainties, sample preparation, and stochastic variations in material 40 properties. The Laser-Induced Particle Impact Test (LIPIT) has become an attractive method for prob- 41 ing microscale projectile, target, and synergistic projectile-target responses to high strain rate deformation 42 [30–35]. LIPIT’s advantages over other microscale launching techniques include the ability to controllably 43 launch a single microparticle with relatively low kinetic energy ( ∼nanoJoules) and precisely characterize the 44 projectile momentum and kinetic energy before and after target impact. A LIPIT apparatus also has a small 45 laboratory footprint ( ∼1×1 m2) and is suitable to employ in high-throughput testing. A comparison of the 46 achievable impact velocities obtained viaa variety of launching techniques is shown in Fig. 1a (adapted from 47 [36]). LIPIT experiments have been applied to investigate impact responses of polymers, gels, and metals, 48 in various structural forms [35–42]. These prior studies have leveraged microscopic high-rate deformation, 49 failure, and relative impact energy absorption results to identify promising materials for macroscopic appli- 50 cations. However, the specific energy absorbed in these microscopic events often exceeds that observed at 51 the macroscale by at least an order of magnitude. For reference, Fig. 1b displays a comparison between 52 the specific energy absorption ( E∗ p) values obtained from microscopic and macroscopic experiments, as re- 53 3 Figure 1: An overview of relevant launching techniques and experimental results from the literature: (a) representative launch capabilities as a function of impactor scale (adapted from [36]) and (b) specific energy absorption as a function of impact velocity for macro- and micro-scales (adapted from [40]). Apparatuses include the single-stage gas gun (SSGG), single-stage powder gun (SSPG), two-stage light gas gun (2SLGG), three-stage light gas gun (3SLGG), Van de Graaff accelerators (VDF), laser-driven flyer (LDF), laser-induced particle impact test (LIPIT), rocket sled, rail gun, and plasma gun. Materials impacted with similar ht/dpratios include multi-walled carbon nanotubes (MWCNT) of thicknesses h106andh205nm, multi-layer graphene (MLG), polystyrene (PS), polycarbonate (PC), poly(vinylidene fluoride-co-trifluoroethylene) P(VDF-TrFE), as well as macroscopic target aluminum, steel, Kevlar, and Dyneema. ported in a previous review [40]. Some microscopically determined E∗ pvalues [ e.g., for multi-walled carbon 54 nanotube (MWCNT) sheets] exceed macroscopic values for metals by over 2,300%! This stark difference un- 55 derscores the need to exercise extreme caution when extrapolating microscopic high-rate material behavior 56 to successively higher spatial scales. 57 Scale-induced changes in impact phenomena are a strong function of the projectile and target materials 58 and geometries, the impact velocity, and other factors. This is particularly true for the dynamic behavior 59 of polymers, where large deformations and phase transformations can occur along with temporally and 60 spatially varying gradients in temperature, stress, strain, and strain rate. For instance, at the microscale, 61 target thickness reduction alone has been shown to enhance E∗ p[43]. One systematic approach to isolating 62 length scale influences involves the selection of macroscopically and microscopically homogeneous target and 63 projectile materials while holding the impact velocity and relative geometries of the target and projectile 64 constant. Amorphous polycarbonate (PC) is one such target material. PC is a tough and transparent 65 thermoplastic with a broad range of applications in electronics, automobiles, construction, medicine, space 66 exploration, and ballistic protection [44–50]. Moreover, PC inherently has just two material length scales— 67 the root-mean-square distance between its chain ends (proportional to its weight average molecular weight, 68 Mw) and the average distance between its chain entanglements (inversely proportional to its entanglement 69 molecular weight, Me). In general, an increase in Mwfor fixed Meincreases the number of entanglements 70 per chain ( Ne∼Mw/Me) but does not necessarily change the bulk volume density of entanglements ( νe). 71 4 For amorphous thermoplastic polymers such as PC, polystyrene (PS), and poly(methyl methacrylate) 72 (PMMA), their material yield stress is generally proportional to strain rate ( ˙ ε) and inversely proportional 73 to temperature ( T) [51]. These effects have been widely studied for glassy thermoplastics using universal 74 axial tension/compression testing ( ˙ ε∼10−4–101s−1), rheometry ( ˙ ε∼10−3–103s−1), Split-Hopkinson 75 tension/compression/torsion bar testing ( ˙ ε∼103–104s−1), and Taylor impacts ( ˙ ε∼105s−1) over a range 76 of temperatures ( −150⪅T⪅200◦C) [51–53]. The rate and temperature dependent mechanical behavior 77 of glassy polymers can be attributed to their αandβtransitions occurring at temperatures TαandTβ, 78 respectively [52]. For these polymers, Tαdefines the glass transition temperature ( Tg). For PC, Tα≈147◦C 79 andTβ≈ −100◦C. In its glassy range ( T < T α), PC’s tensile, compressive, and torsional yield stress 80 all exhibit a linear increase with log ˙ εat constant Tand a linear decrease with Tat a constant ˙ ε(i.e., PC 81 behavior is described by the Eyring theory) [52, 53]. These relationships deviate from linearity at T≈ −50◦C 82 due to reduced backbone chain motion, leading to the observed βmechanical damping peak at Tβ. Polymers 83 are quite sensitive to hydrostatic pressure, which typically raises their yield strength, bulk modulus, and 84 Tg. Post-yielding behavior in PC ( Tg∼147◦C) results from a balance between adiabatic thermal softening 85 and strain (and strain rate) hardening. At and above the yield stress, PC displays extensive visco-plastic 86 flow, while other glassy polymers, like PMMA ( Tg∼136◦C) and PS ( Tg∼100◦C), undergo a ductile to 87 brittle transition that is exacerbated by increasing strain rate [54]. PC maintains this ductility over a broad 88 range of Tand ˙εvalues due to its unusually low Tβ, contrasting with most thermoplastics that are brittle 89 forT < T g. PC, however, can embrittle when subjected to ultra-high strain rates, ultra-low temperatures, 90 or annealing. 91 ForT < T gand moderate to low strain rates, PC generally yields, flows, converts approximately 50– 92 60% of plastic work to heat, and exhibits damage localization in the form of shear banding and crazing 93 [55–59]. Above the glass transition temperature, PC deforms homogeneously. Fracture occurs as crazes 94 nucleate and break down into tensile cracks, which propagate as new crazes form at their tips. Despite 95 its low thermal conductivity, PC’s enhanced toughness through deformation-induced molecular orientation 96 generally prevents its fracture from heat-induced shear instabilities. When subjected to macroscale ballistic 97 impact ( ˙ ε≲105s−1), finite PC targets (0 .1< h t/dp<1.8) exhibit a variety of impact-driven failure 98 modes, including dishing, petalling, cratering, cone cracking, and plugging depending on the ht/dpratio 99 [47]. The combined adiabatic heating from shocks and plastic work generates high temperatures, and this 100 localized heating is enhanced by PC’s low thermal conductivity. Elevated temperatures and ensuing visco- 101 plastic flow during impact can potentially suppress or even “erase” evidence of prior, low temperature shear 102 banding, crazing, and other failure mechanisms. Even so, under macroscale hypervelocity impact conditions 103 ( ˙ε∼106–107s−1), thicker PC targets (0 .8< ht/dp<3.2) appear to fail more brittly, hinting at a possible 104 5 ductile-to-brittle transition with increasing strain rate [44]. Such a transition (if present) is influenced by 105 projectile/target shapes, impact velocity, event duration, length scale, and more. 106 Although the macroscale behavior of PC has been extensively studied, it has been the subject of only 107 one LIPIT study [50]. These LIPIT experiments demonstrated that an increase in PC entanglement 108 density ( νe) led to a notable increase in E∗ p. PC typically has a relatively high entanglement density 109 (νe∼1026entanglements/m3) compared to that of other glassy polymers like polystyrene (PS) ( νe∼ 110 1024entanglements/m3). A higher entanglement density generally enhances material toughness and defor- 111 mation resistance by forming an energy-absorbing network that restricts chain mobility, evenly distributes 112 stresses, enhances elasticity, and hinders crack growth [60–64]. Moreover, unlike other glassy polymers, 113 PC’s high-rate mechanical behavior is less susceptible to adiabatic heating, potentially explaining its 81% 114 higher LIPIT E∗ pvalue at vi= 500 m/s compared to PS ( cf.Fig. 1b) [65]. Such material behavior could 115 be responsible for the nominal entanglement-driven increases in E∗ preported in [50]. This previous study, 116 however, did not compare LIPIT E∗ presults with corresponding values from macroscale tests. In fact, a 117 systematic experimental analysis of how length scale influences ballistic impact phenomena from macroscale 118 to microscale has yet to be conducted for any material system . 119 In this study, rigid noncrystalline alumina spheres ranging five orders of magnitude in diameter ( dp= 120 3µm–10 mm) were launched into PC targets of thickness htat normal incidence using either LIPIT or a 121 single stage gas gun, depending on the scale. The projectile impact velocity ( vi≈550 m/s) and the ratio of 122 target thickness to projectile diameter ( ht/dp≈0.25) were held fixed for all experiments. Impact energies 123 span from hundreds of Joules down to nanoJoules, significantly expanding on previous relevant investigations 124 [50, 66, 67]. The specific energy absorption ( E∗ p), local plastic deformation, and deformation microstructures 125 are compared across all considered scales. The decrease in length scale results in a significant increase in 126 specific energy absorption and relative deformation area. Numerical predictions of PC impact behavior for 127 short times are compared with the experimental results to show limitations in current continuum-based 128 material modeling approaches. These preliminary observations show that length scale influences cannot be 129 ignored when developing and employing emerging experimental techniques that promise accelerated material 130 discovery ( e.g., LIPIT, laser-driven flyers, and nanoindentation). Moreover, macroscopically observed impact 131 phenomena and accompanying theory/models may not translate well to the microscale, as a decrease in 132 spatial scale for fixed ht/dpresults in higher average strain and heating rates. In light of these implications, 133 this study seeks to explore the influence of geometric scale on impact phenomena. 134 6 2. Methodology: Scaling the Impact 135 Films of PC were perforated at vi≈550 m/s by noncrystalline rigid alumina spheres with diameters 3, 10, 136 100, 500, 1,000, 4,000, and 10,000 µm. Regardless of length scale, the ratio of target thickness to projectile 137 diameter was held fixed, i.e.,ht/dp= 0.25. These viandht/dpvalues guaranteed target perforation while 138 ensuring the projectile remained intact, thus maintaining consistent impact scenarios across scales. This 139 effectively isolated the influence of key scale-dependent factors (strain rates, heating rates, projectile impact 140 energy, target areal density, etc.) on PC’s deformation, failure, and specific energy absorption. 141 2.1. The Projectile Launching Techniques 142 The breadth of geometric scales probed in this study necessitated the use of two distinct projectile 143 launching techniques. For the 3 µm and 10 µm diameter projectiles, a LIPIT apparatus in the Materials 144 Microstructures and Properties Laboratory at Texas A&M University (TAMU) was employed [41, 42].1The 145 LIPIT technique involves using a laser pulse to ablate a gold film sandwiched between a glass substrate and 146 a crosslinked polydimethylsiloxane (PDMS) film [32, 36, 40, 43, 68]. Before launch, alumina microparticles 147 were distributed on the elastomer on the side of the “launch pad” that faces away from the laser’s point of 148 incidence (Fig. 2a). The launch pads were prepared as described in Ref. [42]. The laser pulse generates a 149 plasma that expands and rapidly deforms the PDMS layer, propelling a projectile through lab air at high 150 velocity towards a target (Fig. 2b). The laser power can be adjusted to vary the projectile’s launch velocity. 151 In general, microparticles with diameters ranging 1–50 µm can be readily accelerated to velocities surpassing 152 1 km/s. A dedicated study to probe how length scale andvelocity influence impact phenomena together is 153 ongoing. 154 The larger projectiles were launched using a single-stage gas gun (SSGG) and accompanying aeroballistic 155 range located within the TAMU Hypervelocity Impact Laboratory (HVIL) [69].2The SSGG launching 156 mechanism operates using helium gas, initially contained in a high-pressure reservoir at pressures up to 157 34 MPa (Fig. 2c). Upon remote activation of a fast-acting valve, the He gas rapidly expands down the 158 launch tube (barrel), accelerating a projectile towards its target (Fig. 2d). The SSGG can launch single 159 projectiles ranging from 2.0–12.7 mm in diameter or clusters of particles down to 100 µm in diameter using 160 a simultaneously launched distributed particle (SLDP) technique, to velocities ranging 0.1–1.5 km/s [70]. As 161 an aside, launch velocities up to 8 km/s can be achieved using the HVIL two-stage light gas gun [69]. For the 162 SSGG technique, a single projectile or collections of projectiles were loaded into a four-piece spherical cavity 163 1Material Microstructures and Properties Laboratory Website: https://elt.engr.tamu.edu/ . 2TAMU Hypervelocity Impact Laboratory Website: https://telacyjr.engr.tamu.edu/facility/ hypervelocity-impact-laboratory-hvil/. 7 Figure 2: Projectile launching devices employed in this study: (a, b) a laser-induced particle impact test (LIPIT) apparatus and (c, d) a single-stage light gas gun. Both devices are shown in (a, c) pre- and (b, d) post-launch configurations. The LIPIT schematic was adapted from Ref. [36]. sabot. During free-flight through a tank containing lab air, the sabot segments radially separated from the 164 projectile(s). The projectile(s) then passed through an annular steel plate, which halted the sabot. Single 165 4 mm and 10 mm spheres were each launched in an appropriately sized sabot, while the SLDP technique was 166 used to launch the 100, 500, and 1,000 µm diameter particles to increase the likelihood of observable target 167 impacts. After launch, the SLDPs would naturally disperse radially due to aerodynamic forces, resulting 168 in a well-distributed impact site pattern on the corresponding PC thin film, with individual sites being far 169 enough apart to be considered as separate impacts (nearest neighbor impact site spacing ≫10dp) [70]. 170 2.2. Materials and Sample Preparation 171 The macroscopically isotropic and homogeneous PC target materials were sourced from Plaskolite, LLC. 172 [TUFFAK®;ht: 1,016 µm (PC11016) and 2,380 µm (PC12380)] [71] and Rowland Advanced Polymer Films 173 8 [RowTec®;ht: 127 µm (PC2127) and 254 µm (PC2254)] [72]. Differential scanning calorimetry (DSC) 174 curves for both as-received PC1 and PC2 samples show the materials have very similar glass transitions 175 (Tg= 148–153◦C) (Fig. 3a). Both materials had a mass density of ρt= 1.20 g/cc and key quasi-static 176 mechanical properties that differed by less than 3% (see Supplementary Information, Sect. SI.3). The 177 polymers’ molecular weight averages ( Mn= number average, Mw= weight average, Mz= Z-average) and 178 molecular weight distribution ( PDI =Mw/Mn, polydispersity) were measured using a TOSOH Ambient 179 Temperature gel permeation chromatography (GPC) instrument with tetrahydrofuran (THF) as the solvent 180 and polystyrene as the calibration standard. Both commercial materials had similar chromatographs. For 181 PC1, Mn= 27 ,737 g/mol, Mw= 55 ,711 g/mol, Mz= 87 ,908 g/mol, PDI = 2.01. For PC2, Mn= 182 28,927 g/mol, Mw= 55 ,876 g/mol, Mz= 87 ,749 g/mol, PDI = 1.93 (see Table S3 in Supplementary 183 Information, Sect. SI.4). 184 The sample preparation technique was dependent on the film thickness. Targets with ht= 127 ,254,1,016 , 185 and 2,380 µm were cut directly from the as-received PC1 or PC2 material and placed between two aper- 186 tured steel fixture plates (Fig. 3b and 3c). The 30 µm thick samples (PC230) were created by heated 187 compression of the PC2127material. Given the small in-plane dimensions of the resulting targets (around 188 25.4 mm), a custom fixture with a grid pattern was created to hold multiple square thin films (Fig. 3d). 189 This arrangement enhanced the likelihood of target impact during a given SLDP launch, thereby facilitating 190 more extensive post-impact analysis. For LIPIT samples, the PC2127material was dissolved using 50/50 191 dichloromethane/toluene, then deposited on a silicon wafer attached to a spin coater. High-rate rotation 192 and solvent evaporation yielded roughly 0.75 µm (PC20.75) and 2.25 µm (PC22.25) thick films, which were 193 placed on a transmission electron microscopy (TEM) grids serving as LIPIT target fixtures (Fig. 3e). For 194 all macroscopic SSGG experiments, the target assembly (target plus fixture) was centered on the launch 195 tube (impact) axis. Translation stages were used as part of the LIPIT apparatus to identify a suitable 196 projectile on the launch pad and impact site on the target inside a TEM grid square. In all experiments, 197 target boundaries were far enough from the impact point to prevent in-plane reflected waves from affecting 198 penetration/perforation dynamics ( i.e., perforation times were less than ∼10% of wave travel times; see 199 Supplementary Information, Sect. SI.2). 200 Alumina (Al 2O3) was chosen as the projectile material due to its relative homogeneity across the given 201 length scales. The noncrystalline spheres were sourced from various commercial vendors depending on 202 their diameter ( dp): Huake Scientific Research Materials Co., Ltd. ( dp= 3µm and 10 µm), Corpuscular 203 Microspheres-Nanospheres ( dp= 100 µm), Goodfellow Cambridge Ltd. ( dp= 500 µm and 1,000 µm), and 204 Kyocera Corporation ( dp= 4,000µm and 10,000 µm). For reference, Fig. 4 shows representative images 205 of the projectiles at each length scale captured with either an optical camera (Figs. 4a and 4b) or Thermo 206 9 Figure 3: An overview of the polycarbonate (PC) targets: (a) a representative differential scanning calorimetry curve for each as-received target material and (b) the target preparation process for each film thickness, ht. Two materials were used in target preparation: Plaskolite, LLC. TUFFAK®(PC1; ht= 1.02, 2.38 mm) and Rowland Advanced Polymer Films RowTec®(PC2; ht= 0.75, 2.25, 30, 127, 250 µm). All material had similar densities ( ρt) and glass transition temperature ( Tg) values. Targets were fixed (c) between apertured plates ( ht= 127 ,254,1,016 ,2,380 µm), (d) in an array on a custom grid ( ht= 30 µm), or (e) on a 200 mesh TEM grid ( ht= 0.75,2.25µm). Fisher Helios NanoLab 660 dual-focused ion beam-scanning electron beam microscope operating with an 207 incident beam energy and working distance of 1 kV and 4 mm, respectively (Figs. 4c–4g). In this figure, the 208 projectile diameter ( dp) decreases from left to right and top to bottom: (a) 10 mm, (b) 4 mm, (c) 1 mm, 209 (d) 500 µm, (e) 100 µm, (f) 10 µm, and (g) 3 µm. The projectiles displayed consistent sphericities and 210 diameters even at the finest length scale, maintaining relative uniformity across the impact experiments. 211 With a density of ρp= 3.95 g/cc, the projectiles varied in mass from mp= 5.6×10−11g (dp= 3µm) to 212 mp= 2.1×100g (dp= 10 mm). To underscore the extensive range of scale, each projectile’s impact kinetic 213 energy ( Ei=1/2mpv2 i) was calculated, assuming vi= 550 m/s and neglecting rotational kinetic energy 214 (Fig. 4h). The ballistic impacts presented in this work span eleven orders of magnitude in kinetic energy 215 (∼10−9–102J), representing the broadest range ever addressed in a single experimental impact study. 216 Prior to each experiment, the PC target thickness was measured using either a digital caliper ( ht> 217 100µm) or a Keyence VK-X3000 Three-Dimensional (3D) Surface Profiler laser confocal microscope ( ht< 218 10 Figure 4: Micrographs showcase the alumina projectiles used in the impact experiments with the following diameters ( dp decreases from left to right): (a) 10 mm, (b) 4 mm, (c) 1 mm, (d) 500 µm, (e) 100 µm, (f) 10 µm, and (g) 3 µm. For reference of scale, the (h) impact kinetic energy ( Ei=1/2mpv2 i) for each of these dpvalues is plotted, assuming impact velocities of vi= 550 m/s and projectile masses of mp=1/6πd2 pρp, where ρp= 3.95 g/cc is the projectile density. 100µm).3In addition, the Keyence microscope was used to inspect the films for any obvious defects 219 (impurities, cracks, etc.), significant thickness variations, and excessive surface roughness. Similarly, the 220 diameter of each projectile was measured using either calipers, optical microscopy, or scanning electron 221 microscopy (SEM), depending on the diameter. These steps were essential to ensure that the ht/dp≈0.25 222 ratio was largely maintained and that target inhomogeneities were minimized. 223 The LIPIT thin films varied slightly in thickness due to inherent variability in the spin coating and/or 224 compression molding process, affecting the ht/dpratios. However, they remained near the intended value 225 of 0.25. Residual internal stresses in the PC targets caused by material manufacturing and processing 226 might affect impact energy absorption and deformation, particularly at smaller scales. The spin-coating and 227 compression-molding sample preparation techniques, however, did not introduce any noticeable material 228 anisotropy, as evidenced by cross-polarizing microscopy (refer to Supplementary Information, Sect. SI.5). 229 Similarly, the characteristic length of the repeating molecular unit was much smaller than the minimum 230 LIPIT sample thickness ( ht= 0.75µm), suggesting that the local arrangement of PC molecules did not have 231 a significant effect on target energy absorption. For instance, the projected volume of the ht= 0.75µm 232 target material underneath the projectile (π/4d2 pht) was∼100,000 times greater than the volume of a sphere 233 3Laser confocal and scanning electron microscopy were performed at the TAMU Small Scale Mechanical Behavior Laboratory. 11 Figure 5: Schematic overview of the experimental methodology showcasing consistent diagnostics and impact scenarios across all scales, with the projectile diameter ( dp) provided in bold to highlight the scale. The projectile launch apparatus/technique (SSGG, SSGG + SLDP, or LIPIT) is also shown for reference. enclosing the average PC chain as defined by its root-mean-square end-to-end distance (see Supplementary 234 Information, Sect. SI.1) [73]. Furthermore, while surface effects can depress Tgandνevalues for very thin 235 films ( ht<100µm), such effects were negligible in this study ( ht≥750 nm) [74, 75]. 236 2.3. Quantifying Specific Energy Absorption and Normalized Deformation Area 237 For the SSGG experiments, a high-speed Shimadzu HPV-X2 camera (250k fps, exposure times 1,000– 238 1,500 ns) [76] captured high-contrast, shadowgraphic images of each impact event. Light from high-intensity 239 LED arrays, passed through a diffuser box, was used to illuminate the event [77]. A 50 mm lens was fitted 240 onto the Shimadzu camera for the 1, 4, and 10 mm diameter particle impact experiments, while an Infinity 241 K2 Distamax long distance microscope with a CF2 objective was used for the 100 µm and 500 µm SLDP 242 tests. The LIPIT experiments were captured at a frame rate of 1 GHz and exposure time of 5 ns using 243 a Specialized Imaging SIMX camera [78], where a collimated laser was used for illumination. High-speed 244 images, as well as OM and SEM micrographs, were used to verify projectile diameters. These diagnostic 245 setups ensured consistent, comparable, and high-contrast shadowgraphic images across all scales. Figure 5 246 includes an idealized schematic for each set of experiments (arranged in descending order by projectile size), 247 including details of the projectile, target, launcher, lighting, and imaging system. 248 The consistent impact conditions ( vi≈550 m/s, ht/dp≈0.25) and relative projectile/target material 249 properties ensured negligible projectile deformation and no projectile fragmentation occurred for all exper- 250 iments. Each alumina projectile remained intact during PC target perforation and exited with a residual 251 12 Figure 6: Simplified diagrams illustrate the projectile: (a) approaching the target at time t0with velocity viand (b) after perforating the target at time tfwith a residual velocity vr. Diagram (c) represents the method for normalizing the energy absorption of the target, Ep, using the mass of a conceptualized “ejected” material plug (highlighted with crosshatching); the plug mass is defined as mt=π/4ρthtd2 p. The final diagram, (d), depicts the normalized deformation area concept, expressed asAd/Ap=D2/d2 p, where Dis the diameter of the circle defined by Ad. velocity vr(Fig. 6a and 6b). Hence, the in-situ videographic images also facilitated the measurement of vi 252 andvrusing open-source motion tracking softwares, such as Tracker [79] and ImageJ [80]. These velocity 253 measurements were then used to estimate the energy absorption of the target material, given by 254 Ep=1 2mpv2 i−1 2mpv2 r−Edrag, (1) where mpis the mass of the projectile and Edragis projectile kinetic energy loss due to drag, which was non- 255 negligible for the microsphere impacts and calculated using methods described in the supporting information 256 of Refs. [43, 81]. The energy absorption can be normalized by the mass of an ideal plug of target material 257 “ejected” by the projectile during target perforation (Fig. 6c) to facilitate ballistic performance comparisons 258 across various materials and geometric scales [40, 42, 50]. The nominal specific energy absorption may be 259 defined as 260 E∗ p=Ep mplug, (2) where mplug=π/4d2 pρthtandρt= 1.20 g/cc is the mass density of the PC target material. 261 After each experiment, optical micrographs of the perforations and out-of-plane deformation on both 262 the impact and exit sides of the target were captured using a Keyence VK-X3000 3D Surface Profiler. The 263 through thickness coordinate of the impacted target surface, zt(r), was measured as a function of radial 264 distance ( r) from the center of impact (Fig. 6d). An impact-damaged region was defined for each target in 265 terms of an in-plane circular region of diameter D= 2rD, where rDis the maximum radial distance beyond 266 13 Figure 7: A summary of the experimental workflow employed in this study. Key steps include (1) sample preparation, (2) pre-impact sample inspection with the Keyence VK-X3000 3D Surface Profiler, (3) scaled impact experiment using either the LIPIT apparatus or SSGG, (4) optical microscopy and laser confocal microscopy of impact damage with the Keyence instrument, and (5) scanning electron microscopy (SEM) of impact damage using the ThermoFisher Helios NanoLab 660 dual-focused ion beam-scanning electron beam microscope. which the transverse deflection of the impact surface is less than 5% of the unimpacted plate thickeness,4267 i.e., 268 D= 2rD, such that 1 −2 ht|zt(r)|<0.05. (3) This diameter was used to characterize the normalized deformation area , 269 Ad Ap=D2 d2p, (4) consistent with definitions used in the literature [50]. One-dimensional (1D) deformation profiles ( zt(r)) 270 collected at each scale were normalized by the corresponding projectile diameter allowing for direct compar- 271 isons of the relative deformation for each test. SEM imaging of the LIPIT perforations on both the target 272 impact and exit surfaces was performed using the ThermoFisher Helios NanoLab 660 dual-focused ion beam- 273 scanning electron beam microscope (1 kV operating voltage, 4 mm working distance). Hence, the variations 274 in deformation behavior and bulk failure mechanisms across length scales were probed. An overview of the 275 experimental workflow is summarized in Fig. 7. 276 3. Results and Discussion for the Scaled ht/dpImpacts 277 This section highlights essential findings from the scaled impact experiments, including specific energy 278 absorption ( E∗ p) and normalized deformation area ( Ad/Ap) measurements, as well as representative optical 279 microscopy images and profilometry scans of each perforation on both the front and back surfaces of each 280 4A deflection value of approximately 5% was chosen as it loosely marks the threshold for detectable deflection without considerable influence from noise in the profilometry data. 14 target. SEM micrographs of the LIPIT perforations are also presented. E∗ pandAd/Apare also compared to 281 calculated values from the Elastic Plastic Impact Computation code (EPIC) simulations across all considered 282 length scales (see Supplementary Information, Sect. SI.7). The constant ht/dpimpact experiments were 283 performed at seven distinct length scales, which can be described by the diameter of the alumina projectiles: 284 dp= 3, 10, 100, 500, 1,000, 4,000, and 10,000 µm. For each scale, at least three tests were run to establish 285 average data points and corresponding standard deviations, which tended to increase at smaller scales. 286 3.1. Spatial Scale Dependency of Specific Energy Absorption and Normalized Deformation Area 287 Regardless of length scale and launch technique (SSGG, SSGG + SLDP, or LIPIT), the projectile’s motion 288 was captured using high-speed shadowgraphy. Figure 8 shows in descending order representative high-speed 289 images captured before (Figs. 8a, 8c, 8e, 8g, 8i, 8k, 8m) and after (Figs. 8b, 8d, 8f, 8h, 8j, 8l, 8n) each scaled 290 impact event by either the HPV-X2 ( dp= 100, 500, 1,000, 4,000, 10,000 µm) or SIMX ( dp= 3 and 10 µm) 291 camera. Yellow arrows and red dashed lines superimposed on the images highlight the direction of projectile 292 motion and relative target thickness inside/on the corresponding fixture, respectively. The projectile impact 293 velocity ( vi) and residual velocity ( vr) were determined for all experiments by tracking its horizontal motion 294 across multiple shadowgraphs. The launching technique, relative length scale, and average impact and 295 residual velocities are also shown in Fig. 8 for reference. Variations in viat a given length scale were 296 relatively minor ( <10%). The key experimental parameters ( dp, vi, vr, ht, ht/dp, E∗ p,andAd/Ap), as well 297 as the launch apparatus used at each length scale, are summarized in descending order in Table 1. 298 The specific energy absorption [ E∗ p, Eq. (2)] of each scaled PC target was calculated for all experiments 299 using the measured viandvrvalues provided in Table 1. The variations in experimental htandvivalues, 300 coupled with measureable deviations in dp, likely explain the standard deviations in E∗ pandAd/Apat a given 301 scale, particularly the notable increase in standard deviations with decreasing dp. A heightened sensitivity 302 of the target material to velocity changes occurs as htdecreased due to increasing strain and heating rates. 303 Table 1: A summary of the scaled impact experiments performed in this study. A minimum of three experiments were performed at each length scale to quantify the effects of impact velocity and target thickness variations on specific energy absorption ( E∗ p). No. dp(µm)vi (m/s)vr (m/s)ht(µm) ht/dpLaunch ApparatusE∗ p (MJ/kg)Ad/Ap 1 10,000 ±2.5 561 ±29 482 ±29 2,380 ±130 0.24 ±0.01 SSGG 0 .38±0.02 1.43 ±0.00 2 4,000 ±2.5 537 ±20 437 ±20 1,016 ±25 0.25 ±0.01 SSGG 0.42 ±0.02 1.70 ±0.00 3 1,000 ±2.5 539 ±4 434 ±4 254 ±25 0.25 ±0.03 SSGG SLDP 0.45 ±0.01 1.82 ±0.01 4 500 ±2.5 529 ±4 385 ±4 127 ±13 0.25 ±0.03 SSGG SLDP 0.57 ±0.01 2.15 ±0.02 5 100 ±2.5 523 ±47 251 ±47 30 ±5 0.30 ±0.05 SSGG SLDP 0.77 ±0.16 2.66 ±0.13 6 10 ±0.80 522 ±42 207 ±42 2.25 ±0.25 0.23 ±0.04 LIPIT 1.12 ±0.15 3.92 ±0.62 7 3 ±0.32 535 ±34 38 ±34 0.75 ±0.05 0.25 ±0.04 LIPIT 1.25 ±0.24 4.81 ±1.01 dp, projectile diameter; vi, projectile impact velocity; vr, residual velocity; ht, target thickness; ht/dp, target-thickness- projectile-diameter ratio; E∗ p, specific energy absorption [Eq. (2)]; Ad/Ap, normalized deformation area [Eq. (4)]. 15 Figure 8: High-speed images capture the impact events: the alumina projectiles in images (a, c, e, g, i, k, m) are depicted prior to impact, and those in images (b, d, f, h, j, l, n) are shown after target perforation. Each row in the image array corresponds to a specific projectile diameter ( dp), which decreases sequentially from the top to the bottom. The target films are highlighted using red dotted lines for enhanced visibility (the dark areas around the films indicate target fixtures). Yellow arrows show the direction of projectile motion. Information provided alongside each series details the launching method (SSGG, SSGG + SLDP, or LIPIT), the impact velocity ( vi), residual velocity ( vr), and an accompanying scale. This could also partially explain the observed higher standard deviations. Under the assumption of constant 304 projectile-target homogeneous materials, relative geometries, and impact conditions, E∗ pwould be expected 305 16 Figure 9: Key impact experiment results as a function of length scale (projectile diameter): (a) specific energy absorption, Eq. (2), and (b) normalized deformation area, Eq. (4). Both metrics follow similar power law trends, decreasing dramatically with length scale. Also included are deformation-area-normalized E∗ pvalues [ E∗∗ p= (Ap/Ad)E∗ p], as well as EPIC code predictions forE∗ pandAd/Ap. to remain constant in the absence of any scaling effects . Contrarily, E∗ pexhibited a dramatic increase with 306 a decrease in scale, ranging from approximately 0.4 MJ/kg for the ht= 2.38 mm (thickest) target to as 307 high as ∼1.25 MJ/kg for the ht= 0.75µm (thinnest) target (a roughly 230% increase; cf. Table 1). The 308 specific energy absorption is plotted in Fig. 9a for each geometric scale, demonstrating an inverse power 309 law relationship with projectile diameter ( E∗ p∝d−0.16 p). The error bars on each data point represent the 310 standard deviation in E∗ pprovided in Table 1. As an aside, the LIPIT E∗ pmeasurements align well with 311 previously reported values for vi≈500 m/s PC film impacts, though with reduced ht/dp≈0.04 ratios and 312 26,000 < M w<59,000 g/mol [50]. The notable rise in E∗ pwith decreasing scale shown in Fig. 9a emphasizes 313 that without proper physical understanding and scaling laws, microscopic impact phenomena cannot be 314 directly extended to the macroscale or visa versa . 315 Post-impact characterization and measurements often help unravel in-situ material behavior and failure. 316 The effective deformation area ( Ad;cf. Fig. 6d) provides a simple estimation of the extent to which 317 the target material was engaged and subsequently affected (through permanent deformation or failure) 318 by the impact event. To establish a comparison baseline that spans various length scales, this area was 319 normalized by the projected area of the projectile [ Ad/Ap; Eq. (4)]. Assuming no scaling effects, the 320 17 normalized deformation area would remain largely unchanged. Interestingly, however, measurements taken 321 from optical microscopy and laser confocal micrographs indicate a similar trend between the Ad/ApandE∗ p, 322 both increasing dramatically with decreasing length scale. In fact, the normalized deformation area rises by 323 a factor of three from the largest ( dp= 10 mm, ht= 2.38 mm) to the smallest ( dp= 3µm,ht= 0.75µm) 324 scale ( cf. Table 1). Similar to E∗ p, the LIPIT Ad/Apvalues are consistent with the results of Chan et al. 325 [50], who found that at the microscale, PC’s E∗ prises with entanglement density and deformation area. As 326 shown in Fig. 9b, Ad/Apfollows an inverse power law trend with projectile diameter ( Ad/Ap∝d−0.16 p). 327 Hence, as the length scale decreases, a larger proportion of the target material undergoes deformation and 328 failure. Notably, the scaling exponents for Ad/ApandE∗ pare virtually the same. 329 An alternative approach to normalizing impact energy absorption ( Ep) is by employing the deformation 330 area rather than the projectile’s projected area [ i.e.,E∗∗ p= (Ap/Ad)E∗ p]. Interestingly, if 5% is chosen 331 as the deflection threshold (see Eq. 3), the E∗∗ pvalues remain largely constant across the different scales 332 (see Fig. 9a; dash-dot line). These findings reveal that ( i) relative target features resulting from impact- 333 induced deformation and failure lack some consistency across length scale, even when the projectile/target 334 materials and geometries and the impact velocity are unchanged, and ( ii) the added volume of target material 335 undergoing deformation as length scale decreases [( Ad−Ap)ht] is responsible (through plastic work, heating, 336 etc.) for the rise in E∗ p. 337 3.2. Micrographs and Normalized Cross-Sectional Profiles of the Scaled Perforations 338 Post-experiment, the impacted PC samples were immediately sealed at room temperature to preserve 339 the deformation and minimize contamination. Subsequent examinations of the impact and exit surfaces 340 were conducted using OM ( dp≥100µm) and SEM ( dp≤10µm), depending on the length scale ( cf. 341 Fig. 7). Figure 10 offers a visual comparison of the damage from alumina sphere impacts of diameters 342 dp= 10,4,1,0.5 mm, arranged in descending order. Figures 10a, 10c, 10e, and 10g illustrate the impact 343 sides of the perforated films, and Figs. 10b, 10d, 10f, and 10h depict the exit sides. Each image includes a 344 superimposed yellow dashed circle representing the projectile’s projected diameter, and a scale bar positioned 345 in the bottom right corner. The microscopy technique is provided in the bottom left corner of each image. 346 Micrographs of the remaining perforations ( dp= 100 ,10,3µm) are displayed in Fig. 11 using the same 347 layout. 348 In all experiments, the deformed target area exceeded the projectile’s projected area ( Ad/Ap>1;cf. 349 Fig. 9b). The effective diameter of the perforation hole/opening was consistently smaller than that of 350 the projectile, suggesting a degree of hole closure post-perforation, akin to material “self-healing” [82, 83]. 351 Notably, the perforations from projectiles of 4 mm (Figs. 10a and 10b) and 100 µm (Figs. 11a and 11b) 352 18 diameters seemed completely closed, possibly due to slightly larger ht/dpratios. Minimal out-of-plane 353 deformation was observed on the impact face, in contrast to the significant deformation on the exit face 354 across all considered length scales. Signs of jetting, the ejection of material from the impactor-target contact 355 point, were present (Figs. 10e–10h), and there was little to no evidence of large-scale crazing or cracking. 356 The apparent absence of these features might be due to adiabatic heating elevating temperatures above 357 Tg, leading to substantial visco-plastic flow that smoothed out such localized deformation features. In fact, 358 evidence of material flow was present in all micrographs. A slightly higher degree of surface roughness was 359 present in the ht= 30 µm target sample ( dp= 100 µm) due to the compression molding process (Figs. 11a 360 and 11b); however, with nominal peak-to-valley distances being less than 0 .01dp, the experimental results 361 were likely unaffected. Although the geometry and regularity of the perforations appear to vary with scale, 362 the micrographs do not show any obvious changes in primary failure mechanisms or material behavior, such 363 as ductile to brittle transitions. Figures 10 and 11, however, do visually demonstrate the growth of the 364 deformation area ( Ad) relative to the projectile’s projected area ( Ap) as the length scale decreases. 365 Isometric SEM images at a 48◦angle from the film plane were also taken of the same dp= 10 µm (Figs. 366 S5c and S5d) and dp= 3µm (Figs. S5g and S5h) projectile perforations to better highlight the perforation 367 geometry and deformation features not evident in the normal SEM images. The relatively large conical 368 perforation geometries seen in the isometric scans were characteristic of the LIPIT samples (more later) and 369 are in general agreement with previous LIPIT results for PC thin films [50]. 370 Another objective of this study was to probe scale-induced changes in perforation geometry. The OM 371 and SEM images in Figs. 10 and 11 offer only limited quantitative data on the cross-sectional perforation 372 geometries, so laser confocal microscopy was employed to collect profilometry data on both the front and back 373 surfaces of the perforations at each length scale. To facilitate comparison and visualization, the front and back 374 side 1D profiles were normalized using the projectile diameter ( dp) and shifted by the appropriate average 375 ht/dpratio ( cf. Table 1). This normalized representative two-dimensional (2D) cross-sectional profiles, as 376 shown in descending order in Fig. 12. These cross sections are annotated with corresponding dpandhtvalues 377 and highlighted with distinct colors. Matching microscopy images ( cf.Figs. 10 and 11) are provided for 378 reference on the right, with the imaging method noted at the top left of each micrograph. A scaled projectile 379 profile ( dp/dp= 1) moving downward at velocity vris depicted at the bottom of the plot, illustrating 380 that the normalized effective deformation area increases as the length scale decreases ( Ad/Ap∝d−0.16 p). 381 The figure also clearly shows that the perforation openings are consistently smaller than the projectile. 382 Interestingly, the thicker specimens exhibit signs of plugging failure behavior, while the thinner specimens 383 seem to undergo more bulk bending or “membrane-like” deformation (dishing) [47]. This apparent transition 384 in predominant target failure mode is consistent with the observed rise in E∗ p(cf.Fig. 9a). Dishing involves 385 19 Figure 10: Orthogonal micrographs (along impact axis) display perforations in the PC targets caused by impacts from alumina spheres with diameters of (a, b) dp= 10 mm, (c, d) dp= 4 mm, (e, f) dp= 1 mm, and (g, h) dp= 0.5 mm. Within the array of images, the left column (a, c, e, g) represents the impact face of the target, whereas the right column (b, d, f, h) depicts the exit face. On each micrograph, a 2D projection of the projectile is overlaid, indicated by a dashed yellow line, to serve as a point of reference. The microscopy technique is provided in the bottom left of each image. 20 Figure 11: Orthogonal micrographs (along impact axis) display perforations in the PC targets caused by impacts from alumina spheres with diameters of (a, b) dp= 100 µm, (c, d) dp= 10 µm, and (e, f) dp= 3µm. Within the array of images, the left column (a, c, e) represents the impact face of the target, whereas the right column (b, d, f) depicts the exit face. On each micrograph, a 2D projection of the projectile is overlaid, indicated by a dashed yellow line, to serve as a point of reference. The microscopy technique is provided in the bottom left of each image. proportionally more target material in plastic deformation than plugging, which is typically more localized 386 and less deformative (more later). 387 3.3. The Effect of Size on Target Perforation Time, Strain Rate, and v50388 In the impact experiments, reducing the spatial scale alone dramatically increased the nominal target 389 strain and heating rates due to a corresponding decrease in the duration of the penetration/perforation event. 390 Specifically, the nominal target perforation time, tp≈2ht/(vi+vr), decreased from about 10 µs to 10 ns 391 21 Figure 12: Representative experimental one-dimensional cross-sectional profiles of the perforated PC target samples (hatched region, figure inset). All profile data was captured using laser confocal microscopy and normalized by the respective projectile diameter ( dp) for comparison across length scales. At each scale, impact and exit side profiles (dark solid lines) are displaced by the corresponding ht/dpvalues provided in Table 1 for ease of visualization. An impact side micrograph of the target perforation is shown for each cross-sectional profile, with the microscopy technique noted in the top left corner: optical microscopy (OM) or scanning electron microscopy (SEM). Length scale decreases from top to bottom. Vertical dashed lines show the projectile diameter ( dp/dp= 1) relative to the perforation regions: normalized effective deformation area [ Ad/Ap; Eq. (4)] grows with decreasing length scale. 22 (by∼1,000 times!) when transitioning from the macro- to microscale (Fig. 13a). This temporal reduction 392 potentially inhibited some failure mechanisms from contributing to smaller-scale energy absorption, while 393 activating or enhancing the contribution of others (through added strain/strain rate hardening, thermal 394 softening, etc). One simple approach to estimating the nominal strain rate within the thin targets involves 395 using an analytical expression, such as that derived by Lee et al. [32]: 396 ˙˜ε=tp 2vi Rc2 , (5) where Rc≈vctpis the cone radius, vc≈1.23c||[vi/(√ 2c||)]2/3is the cone velocity [84], and c||≈1,400 m/s 397 is the in-plane wave speed. Calculated ˙˜εvalues, also plotted against projectile diameter in Fig. 13a, show 398 a roughly 1,000-fold rise ( ∼105to∼108s−1) from the largest to smallest length scale. As an aside, for 399 dp⪅500µm, the ˙˜εvalues exceeded the strain rate commonly used to define the transition from macroscale 400 terminal ballistic to hypervelocity impact regimes ( ∼106s−1).5These extreme strain rates undoubtedly 401 affected the target’s local, instantaneous yield stress ( σy,t) and failure stress ( σf,t). Previous studies have 402 shown that PC’s yield stress exhibits a bilinear relationship with log ( ˙˜ε), roughly doubling as the strain rate 403 is increased from ∼10−4s−1to∼104s−1(see, e.g., [85–88]).6This hardening is generally the result of a 404 competition between loading rate and rate of polymer chain motion/alignment and entanglement dynamics. 405 Although strain rates tested in these studies fall short of those shown in Fig. 13a, time-temperature superpo- 406 sition extrapolations of stress-strain data suggest PC’s σy,tandσf,tvalues, as well as its modulus, continue 407 increasing proportional to strain rate [88]. Such dynamic material behavior likely resulted in higher specific 408 target impact resistance ( σf,t/ρtor∼σy,t/ρt) at the smaller scales. Enhancement of σy,tandσf,tmay have 409 been partially offset by added thermal softening from increased adiabatic heating rates at the microscale: 410 ˙q∼βpEp/(mplugtp),βpbeing the fraction of Epconverted to heat ( ∼0.5–0.6 for PC). 411 Other aspects of the scaled impacts also showed size dependency. For example, the time required for the 412 projectile to perforate the target ( tp), relative to some scale-specific characteristic time ( t∗), was notably 413 larger for the LIPIT experiments. Here, t∗can be defined as the duration for a compression wave, traveling 414 at velocity ct(PC’s bulk sound speed), to traverse a distance ht. From the largest to smallest scale, tpct/ht 415 roughly doubled (Fig. 13b). Interestingly, this increase coincides with a seven-order magnitude reduction 416 5The shift from the terminal ballistic regime to the hypervelocity impact regime can be defined by criteria like sonic thresholds, initial to complete melting conditions, or characteristic strain and heating rates of the projectile or target. At the macroscale, a common nominal impact-induced target strain rate used to characterize the transition to the hypervelocity impact regime is ∼106s−1(i.e.,˙˜εcrit∼106s−1). The impact velocity corresponding to ˙˜εcrit(vcrit) decreases with spatial scale, assuming the projectile-target materials and geometries are fixed [ i.e.,vcrit=f(dp, ht, ht/dp, . . .); see Fig. S8]. 6Experimental techniques used to determine σy,tandσf,tin these studies depended on the desired loading rate but in- cluded screw-driven and hydraulic tensile testing, dynamic mechanical analysis, and split-Hopkinson pressure bar (SHPB) experimentation. 23 Figure 13: Key parameters that vary with length scale, dp, for fixed projectile-target materials and relative geometries: (a) perforation time, tp, decreases and nominal strain rate, ˙˜ε, increases, respectively, by a factor of ∼103from the largest to smallest length scale [32, 84]; (b) the fraction of target material engaged during impact, ∼tpct/ht, roughly doubles, while the ratio of projectile kinetic energy to target area density, Ei/AD t, drops by seven orders of magnitude, for the microscale LIPIT experiments; (c) the normalized projectile residual velocity, vr/vi, scales proportionally with dp; and (d) the normalized target ballistic limit velocity, v50/vi, is enhanced by 200% at the microscale. A two-term power law was fitted through the vr/vi measurements in (c); this vr/vi∝a+bd−0.25 p fit was used to generate the solid red lines in (a), (b), and (d). in the ratio of projectile kinetic energy ( Ei) to target area density ( ADt=htρt), as illustrated in Fig. 417 13b. tpct/htis also proportional to the fraction of target material subjected to rapid deformation before 418 perforation occurs.7Hence, as dpdecreased, proportionally more of the target ( e.g.,∼πc2 tt2 phtρt/mplug) was 419 7Here, tp≈2ht/(vi+vr) is used for simplicity. The ratio tpct/htwould be more accurately estimated using, tp≈2δ/(vi+vr), assuming δ, the distance over which the projectile and target interact until complete perforation occurs, could be determined. 24 involved in projectile energy dissipation. These findings are consistent with the observed inverse scaling 420 of normalized deformation area with dp(Ad/Ap∝d−0.16 p;cf. Fig. 9b). Furthermore, they align with the 421 transition in bulk target failure mode from predominantly plugging to dishing as the spatial scale decreased 422 (cf. Fig. 12). Dishing engages a relatively larger volume of target material, resulting in more extensive visco- 423 plastic deformation (radial stretching and thinning) before it fails primarily in tension [18]. Consequently, 424 the “relative time to material failure” ( ∼tpct/ht) for dishing can significantly exceed that of plugging, which 425 is more abrupt and localized. These factors make dishing more efficient at absorbing impact energy. 426 Length scale reduction also resulted in a ∼90% decrease in the projectile’s residual velocity ( vr; Fig. 427 13c), indicating that the target’s ballistic limit velocity ( v50) increased with decreasing dp. For a fixed 428 projectile-target material combination and ht/dpratio, a transition from bulk plugging to dishing failure of 429 a ductile target plate has been associated with decreasing projectile residual velocity ( i.e.,viapproaching 430 v50) [18, 47]. Hence, the coupled increase of the target’s v50and degree of dishing with decreasing spatial 431 scale is expected. This consistency, however, does not address whythe target’s v50rose with decreasing dp. 432 A simple method to estimate v50for chunky projectiles (length to diameter ratio near unity) impacting thin 433 plates involves using the Recht-Ipson model [89, 90], derived from momentum and energy considerations: 434 v50 vi=s 1−v2r v2 i 1 +mplug mp2 . (6) Average values for dp,ht,vi, and vr(cf. Table 1) were used in Eq. 6 to approximate the PC targets’ 435 normalized v50values, which saw a ∼200% increase from the largest to smallest scale (Fig. 13d). A two- 436 term power law fitted through the vr/vimeasurements ( vr/vi≈0.98−0.04d−0.25 p; Fig. 13c) was used to 437 generate empirical predictions (solid red lines) for all calculated parameters shown in Figs. 13a, 13b, and 438 13d. Notably, the fitted v50estimates scale with d−0.25 p, a trend likely attributable to rate and/or size effects 439 on the target’s yield and failure stress. Evans et al. [91] recently showed that σf,t∝d−0.5 pfor a number 440 of polymers (including PC) across LIPIT and gas gun length scales ( dp∼10−6–10−2m). This relationship 441 was linked to the dependency of a material’s failure stress on the −1/2 power of flaw size, as described 442 in linear elastic fracture mechanics (LEFM) ( i.e.,dpis analogous to flaw size) [91–93]. A Buckingham Π 443 analysis was also employed to develop the expression v50= 1.15(σf,t/ρp)0.5(ht/dp)0.5, which, for fixed ht/dp444 andρp, suggests that v50∝d−0.25 p if the σf,t∝d−0.5 pscaling holds [91]. A similar scaling of v50with dpcan 445 be demonstrated using simple analytical expressions, such as that developed by Phoenix and Porwal [84] 446 for a two-dimensional thin target: v50= (1 + mplug/mp)(2Aphtσf,t/mplug)0.5ε0.25 f, where εfis the target’s 447 failure strain. Assuming fixed εf,ht/dp, and ρt/ρp, this analytical expression also predicts that v50scales 448 with d−0.25 p ifσf,t∝d−0.5 p. The consistency of these predicted v50-dprelationships with that shown in Fig. 449 25 13d strongly suggests that the observed enhancements of the PC target’s E∗ p,Ad/Ap,vr, and v50at smaller 450 scales are linked to an increase in its σy,tandσf,tvalues, resulting from strain rate hardening, actual size 451 effects, or both. 452 In essence, the decrease in length scale alone drastically shortened the duration of the impact event. 453 This temporal reduction amplified impact-induced strain rates in the target, which in turn increased its 454 instantaneous, local yield and failure stress. This strength enhancement allowed the target to better resist 455 abrupt failure by plugging, facilitating a transition to bulk failure viadishing. Consequently, at smaller 456 scales, a larger fraction of the target material was involved in deformation and failure and relatively more 457 time was available for energy dissipation mechanisms to operate. This combined effect ultimately led to the 458 observed rise in the target’s specific energy absorption ( cf. Fig. 9a). These insights offer a path forward for 459 material modeling efforts focused on capturing the observed size dependency of ballistic impact performance. 460 3.4. Predicted Specific Energy Absorption and Normalized Deformation Area Using EPIC 461 Complementary numerical impact simulations in EPIC ( vi≡550 m/s and ht/dp≡0.25) were used to 462 approximate the projectile residual velocity ( vr) at each length scale. The simulation methodology, material 463 models, and model parameters are provided in Supplementary Information, Sect. SI.7. While the models 464 account for strain and strain rate hardening, as well as thermal softening, of the target material, they are 465 calibrated only for strain rates up to roughly 104s−1due to limited experimental data available in the 466 literature. In addition, these simulations do not account for the effect of microstructural factors (molecular 467 weight, number of entanglements per chain, etc.) on polymer dynamic behavior. The simulated viandvr 468 values were then used with Eq. (2) to arrive at predicted specific energy absorption ( E∗ p) values (see Fig. 9a, 469 dashed line). The simulations fail to reflect the experimental rise in E∗ pwith decreasing projectile diameter 470 (dp). The EPIC simulations were also used to predict changes in normalized deformation area with scale, 471 as shown in Fig. 9b. These predictions are based on the diameter Dof a circular region encompassing all 472 out-of-plane target deformation quantified using the same method described in Fig. 6d and Eq. (4). Similar 473 to the E∗ pcalculations, EPIC predicts negligible change in Ad/Ap. 474 Despite the material models used [Eqs. (S5), (S6), and (S7)] being sensitive to strain rates and temper- 475 ature, they do not accurately replicate the experimental target response. This disagreement exists although 476 simulated strain rates are nominally consistent with expected experimental values ( i.e., they grow by orders of 477 magnitude). Most high-rate material models are calibrated using experimental results from Split-Hopkinson 478 Bar compression tests at strain rates up to 104s−1—but this is over three orders of magnitude less than 479 nominal LIPIT values [94]. Moreover, initial sample temperatures typically do not exceed roughly 200◦C 480 for polymers. Therefore, a likely explanation for the discrepancy between the simulated and experimen- 481 26 talE∗ pvalues could be that the material models and EOS employed are not designed and/or calibrated 482 for the specific impact conditions under investigation. Even so, calibrating existing established material 483 models will likely be an insufficient approach: closing the gap in understanding may require the develop- 484 ment of entirely new material models or computational approaches that accurately capture scale-induced 485 changes in predominant material deformation behavior, phase transformations, and failure mechanisms that 486 are negligible/overwhelmed at larger scales. These issues are discussed in more detail in the Supplementary 487 Information, Sect. SI.8. 488 4. Conclusions 489 When the length scale decreases from centimeters to microns, the dynamic behavior of materials can dra- 490 matically change due to corresponding modifications in event duration, strain and heating rates, temperature, 491 state of stress, material inhomogeneities, among other factors. Addressing this issue is vital to the success 492 of emerging high-throughput material characterization techniques. The Laser-Induced Particle Impact Test 493 (LIPIT) has proven to be a valuable method for studying high strain rate deformation of various materials at 494 the microscale, thanks to its precision, minimal lab space requirement, and high-throughput testing capacity. 495 Initial observations from LIPIT indicate that microscale energy absorption can exceed macroscale results by 496 over a decade. This current study intentionally probes geometric scaling effects on dynamic material behav- 497 ior by using a LIPIT apparatus and a single-stage light gas gun to launch noncrystalline alumina spheres 498 of 3, 10, 100, 500, 1,000, 4,000, and 10,000 µm in diameter into scaled ht/dp= 0.25 polycarbonate (PC) 499 targets. Impact energies ranged from hundreds of Joules to nanoJoules, covering eleven decades, the widest 500 range probed in a single impact study. Perforation times decreased and strain rates increased, respectively, 501 by about three orders of magnitude from the largest to smallest scale. Reducing the length scale increased 502 target specific energy absorption ( E∗ p) by approximately 230% and the impact deformation area ( Ad/Ap) 503 by about 240%. Interestingly, ( Ap/Ad)E∗ pvalues exhibited no apparent size dependency. The fraction of 504 the target material engaged during impact more than doubled at the smallest scale. Target ballistic limit 505 velocities scaled with d−0.25 p and were about three times higher in the LIPIT experiments. The relative in- 506 crease in target performance at smaller length scales corresponded to a transition in its predominant failure 507 mode from plugging to dishing. This apparent transition was likely caused by rate- and/or size-induced 508 enhancements in PC’s yield and failure stress values. 509 Complementary numerical simulations of these impacts, however, do not show any increase in specific 510 energy absorption or normalized deformation area even though predicted strain rates reflect expected values. 511 Such discrepancies underscore the limitations of current material models when scaling down impacts from 512 27 macroscale (10−2m) to microscale (10−6m). Experimental and simulation design, implementation, and 513 analysis would be complicated by the introduction of material hierarchical inhomogeneities, such as those 514 present in metals, carbon nanotube mats, etc. The findings of this study strongly suggest that length 515 scale cannot be ignored in both developing and applying established and innovative microscale material 516 characterization techniques, particularly for advanced material discovery. For these reasons, one key goal 517 of this study is to stimulate scientific dialogue and begin developing a framework for investigating how 518 geometric scale affects impact phenomena. 519 Acknowledgments 520 This research is based upon work partially supported by the National Science Foundation (NSF) Graduate 521 Research Fellowship under Grant No. 1746932. A portion of this research was also sponsored by the Army 522 Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-22-2-0106. The 523 views and conclusions contained in this document are those of the authors and should not be interpreted 524 as representing the official policies, either expressed or implied, of the Army Research Laboratory or the 525 U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government 526 Purposes notwithstanding any copyright notation herein. 527 Use of the Texas A&M University Soft Matter Facility (RRID:SCR 022482) and contribution of Dr. 528 Peiran Wei are acknowledged. Use of the Texas A&M University Microscopy and Imaging Center Core 529 Facility (RRID:SCR 022128) and contribution of Dr. Stanislav Vitha are also acknowledged. Gavin Lukasik 530 assisted with the set up and alignment of the Infinity K2 Distamax long distance microscope with a CF2 531 objective used for the dp= 100 µm and dp= 500 µm SSGG experiments. Zhen Sang performed a subset of 532 the SEM imaging of the PC target perforations following the LIPIT experiments. 533