A combined numerical and experimental study to elucidate primary breakup dynamics in liquid metal droplet-on-demand printing

Additive manufacturing (AM) of metals is a powerful technology that enables rapid production of parts with unmatched design freedom.¹ With the advantages of increased performance and reduced cost over conventional manufacturing methods, metal AM has transitioned from merely a prototyping technique to manufacturing of high-value components.^1–3 Various methods have been employed in metal AM: selective laser melting (SLM) being the most popular method,⁴ selective laser sintering (SLS),⁵ directed energy deposition,⁶ binder jetting,⁷ sheet lamination,⁸ extrusion,^9,10 and liquid metal jetting (LMJ).^11–19 

Liquid metal jetting technology is attractive because it is closely related to the well-developed ink jet technology and provides various advantages among the other metal AM methods: low feedstock processing requirements, fast build times of dense mechanically robust parts with negligible waste, minimal post-processing, and the potential to support a broad range of metals or alloys. In Droplet on Demand (DoD) LMJ, a momentum pulse pushes molten metal out of an orifice in controlled, discrete droplets successively to form structures of arbitrary shape. This freeform casting process produces droplets with diameters in the range of ∼0.1 mm to 1 mm that quickly solidify upon contact with previous layers, resulting in dense near-net shapes. We have previously shown that this process has negligible effects on the mechanical properties of tin specimens when compared to input feedstock.¹⁸ 

Various actuation methods have been employed to create momentum pulses to dispense discrete metal droplets: combining laser melting and pneumatic pressure to deliver CuSn12 braze droplets,¹¹ using electromagnetic (Lorentz) force to print solder,¹⁷ aluminum,¹³ tin,²⁰ and silver²⁰ droplets, implementing piezoelectric force to deposit solder droplets,^14,21 impacting a solid rod to eject solder droplets,²² and using solely pneumatic force to generate droplets of pure Cu¹⁵ and Al alloys.^12,16,23 This work focuses on pneumatic actuation because of its simplicity, compatibility with nearly all metals, and versatility in its ability to print in DoD (pulsed pressure) or jetting (constant pressure) modes.

The dynamics of DoD printing of Newtonian non-metallic liquids have been extensively studied.^24–32 In addition to experiments, several modeling techniques have been employed to characterize droplet dynamics, including early work using Marker-and-Cell methods,³³ finite element simulation,^26,29 volume of fluid (VoF) methods,^19,34,35 and, more recently, the lattice Boltzmann method.³¹ Both the experimental and simulation studies typically focus on high viscosity, low surface tension Newtonian liquids (i.e., high Ohnesorge number, Oh = 1/Z = $μ/σρR$ ≳ 0.1, where μ is the droplet viscosity, ρ is the droplet density, and R is the ejection nozzle inner radius) and seek to characterize the drop morphology and trajectory during the ejection process.

Successful printing requires optimizing the droplet length at breakup to suppress satellite droplet formation and ensure sufficient momentum in the print direction. This leads to the identification of a fluid printability window or a “printability phase diagram” initially requiring 0.1 ≲ Oh ≲ 1,^24,36,37 although more recent work has recognized that the Oh limits change as a function of the strength of the ejection pulse (characterized by the Weber number, We = ρRU²/σ, where U characterizes the average velocity in the nozzle during the ejection).³⁰ Printing at low Oh < 0.1 is indeed possible, but the range of acceptable We must also decrease. This is especially relevant as the high density and surface tension of liquid metals often lead to DoD printing at extremely low Oh. Several authors have observed the controlled ejection of liquid metal droplets with Oh < 0.1.^15,20,22 Relatively little work, however, has focused on studying the detailed drop morphology and dynamics in the low Oh regime.^{30,32,38–40} There have been even fewer combined experimental and computational studies applied directly to pneumatically actuated liquid metal printing.¹⁹ Despite the importance of LMJ technologies, a detailed characterization of droplet formation and breakup dynamics in the physically relevant, low Oh parameter regime is still missing.

In this study, we seek to address this gap by focusing on elucidating the DoD process from ejection to primary breakup for pneumatically actuated, molten tin droplets. Using a combined experimental and computational approach, we follow the droplet dynamics and morphology from initial ejection to pinch-off. We perform high-speed videography to extract the temporal dynamics of the droplet shape and compare them to VoF simulations across parameter space with a focus on the approach to small Oh.

Figure 1 illustrates the liquid metal printing apparatus. Three fast-switching, high flow valves (Festo Corp. 525193) are used in parallel to pressurize a crucible (20 ml 316 SS syringe, KD Scientific 780816) with Ar gas that is metered via an electronic regulator (Kelly Pneumatics HFR-100TK-0505-R-CT). Valve switching is controlled by a MOSFET-amplified signal from a waveform generator (Keysight Tech. 33512B), and the crucible pressure is observed using a static pressure transducer (Omega PX409-030G5V-EH). The crucible is heated with a custom holder (not shown) containing four 60 W cartridge heaters (Omega HDC19109), each with its own temperature controller and thermocouple feedback. The crucible holder is attached to a stage (Thorlabs LTS150) oriented to provide translation along the vertical axis. A custom 303 SS nozzle with 100 µm inner diameter (TECDIA, Inc.) is used to generate molten tin droplets. The print head is surrounded by a 2 in. diameter quartz tube that is purged with Ar gas to create an oxygen-free environment, as surface oxidation drastically changes the droplet morphology and behavior (see Fig. S1 of the supplementary material). Oxygen levels inside the tube are monitored with an optical oxygen sensor periodically to avoid heat damage to the probe (FSO2-4 with TROXROB3 probe, Pyroscience GmbH). An X–Y stage (Thorlabs MLS203) is used to center the tube with the print head. Droplet generation at the nozzle is recorded with a high-speed camera (Photron FASTCAM Mini AX200 with an Infinity K2 DistaMax lens) at 80 kHz frame rate using a backlighting method with an LED illuminator (SugarCUBE Ultra White LED, Ushio America, Inc.) equipped with a light guide and a focusing lens. A custom Matlab (MathWorks) program is used to control valve operation (frequency and duty cycle), set drive pressure, and provide real-time feedback of the pressure in the vessel.

Approximately 60 g of 99.95% pure tin feedstock (8886K11, McMaster-Carr) is loaded into the crucible. The valves are opened with constant pressure to purge the crucible of oxygen with Ar gas. The crucible is then heated to 380 ± 2 °C to melt the tin and ensure the nozzle tip is at ∼265 °C, which is well above tin’s melting point of 232 °C. The print head is lowered into a glass tube that is filled with Ar gas to limit oxidation. Droplets are generated by actuating the valves together at 50 Hz with an 11.5% duty cycle (on time 2.3 ± 0.2 ms) and a 10.5 psi pressure head. High-speed videography was used to capture the formation of 75 droplet events using an 80 kHz frame rate for 128 × 320 pixel² frames (≈1 × ≈2.6 mm² and 8 µm/pixel).

Image processing routines in the Mathematica software package (version 12.0.0)⁴¹ are used to identify and calculate each droplet’s volume, length, centroid position, and leading tip position for each 12.5 µs time step. Droplets are identified via contrast thresholding based on a globally determined threshold. Each droplet is analyzed from the time it first starts to extrude from the base of the nozzle until it reaches the bottom of the frame. The volume of each droplet is determined using a numerical integration approximation assuming cylindrical symmetry where the droplet is sliced into discrete cylinders one pixel in height denoted by a single row in the image and each cylinder’s diameter is equal to the width of the droplet at the respective row in the image (see Fig. S2 of the supplementary material).

A computational model is developed to elucidate the droplet ejection and primary breakup process. This simulation is principally focused on the droplet dynamics in the run-up to primary breakup. The droplet and gas dynamics are governed by the momentum conservation equation and the continuity equation,⁴² 

where length scales are made dimensionless with the nozzle radius, R, and timescales are made dimensionless with the capillary time scale, $tc=ρR3/σ$⁠, with the fluid density ρ and the surface tension σ. In Eq. (1), λ_ρ = 1 in the liquid and λ_ρ = ρ/ρ_gas in the gas phase. Gravity acts in the z-direction and is parallel to the droplet ejection direction. At the liquid–gas interface, the interface momentum balance relates the jump in the stress across the drop surface, S, to the surface tension force via

where the dimensionless stress is T = −pI + λ_μ(⁠$∇$v + $∇$v^T), $κ^$ is the dimensionless mean local curvature, and n is the surface normal, with λ_μ = 1 in the liquid phase and λ_μ = μ_gas/μ in the gas phase.

The droplet dynamics are described by three dimensionless groups: We, Oh, and Bo. In pneumatic droplet ejection, stronger pressure pulses lead to a larger Weber number (We), while increasing the surface tension decreases We. The Ohnesorge number (Oh) is the ratio of the capillary time scale to the viscous time scale. Low values of Oh are expected when surface tension forces are stronger than viscous dissipation forces. Finally, the Bond number quantifies the impact of gravitational forces on droplet deformation and is defined as

where g is the value of gravitational acceleration. In this work, we note that Bo ≪ 1 and, although still modeled, gravity effects will have a negligible impact on drop deformation dynamics.

Together with the appropriate boundary conditions, these dimensionless groups fully prescribe the problem. We perform droplet breakup simulations using the volume-of-fluid solver implemented in the commercial software STAR-CCM+ v12.04.010 (Siemens). The governing equations [Eqs. (1) and (2)] are discretized using a whole domain formulation, and the continuum surface force model is applied to model surface tension forces in Eq. (3).⁴³ The liquid and gas dynamics are solved simultaneously and we assume incompressibility for both fluids (ρ_gas = 1.184 15 kg/m³ and μ_gas = 1.855 08 × 10⁻⁵ Pa s). The 2D axisymmetric simulation domain is presented in Fig. 2(j). The nozzle dimensions match those of the experiment, as detailed above, and we include a 1.5 mm inlet region for the nozzle. To prevent edge effects, the domain extends 15 mm (150 nozzle diameter lengths) from the nozzle exit, parallel to the axis-of-revolution, and 3 mm (30 nozzle diameter lengths) perpendicular to the axis. Variable mesh resolution is used. A fine mesh with mesh size s_mesh = 64 cells/nozzle radius (∼0.78 µm cell size) is used in the rectangular area defined by extending from the nozzle inlet to 5 mm below the nozzle exit and 1.5 mm out perpendicularly from the axis-of-revolution. Outside of this region, the mesh gradually coarsens to an ultimate cell size of 50 µm. We use outlet boundaries at the extreme edges of our simulation. The exterior nozzle surfaces are non-wetting walls (180° contact angle), while the interior surfaces are wetting walls (0° contact angle).

A time dependent plug-flow is imposed at the inlet. This is in contrast to the pulse employed for the benchmarking studies where we use the time dependent profiles from the works of Castrejon-Pita et al.²⁶ and Xu and Basaran.²⁹ The image analysis suggests that the volumetric flowrate through the nozzle is approximately constant during the ejection [see Sec. III and text associated with Fig. 2(k)]. Thus, to compare to experiments, we assume a constant input velocity over a given pulse duration t_p. In this work, simulations are performed over a range of We, Oh, and t_p. To ensure adequate time resolution and satisfaction of the convective Courant number criterion, Co = U_CellΔt_sim/s_mesh < 0.3, the time step is chosen such that Δt_sim < 10⁻⁴ $ρR3/σ$⁠. The region from the nozzle exit to the nozzle entrance is completely filled with liquid at the beginning of the simulation, and both fluids are assumed initially stagnant. The simulation is allowed to progress until the droplet first breaks up, i.e., primary droplet breakup.

Figure 2 illustrates the information that is extracted for each experimentally generated droplet. Figures 2(a)–2(i) show an image sequence of a typical droplet generation event that progresses from the beginning of extrusion [(a)–(d)] up to immediately before (e) and after (f) primary droplet breakup at t_B = 875 µs. After breakup, surface tension forces collapse the extruded tin to minimize surface area, resulting in a series of oscillations between oblate (g) and prolate (h) spheroidal shapes about the spherical equilibrium (i). No satellite droplet formation is observed over the 75 droplet generation events. Figure 2(j) illustrates the equivalent simulation domain showing liquid tin ejection.

Figure 2(k) illustrates how video analysis is used to quantify droplet generation into parameters that help build and provide direct feedback to the model. The linear increase in volume during droplet extrusion extracted from videography is fit to provide the volumetric flow rate (Q), extrusion velocity of tin out of the nozzle, and We. Droplet volume (also the resulting calculated droplet diameter) is taken when the droplet first breaks up (primary breakup), where cylindrical symmetry and accuracy of volumetric calculations are highest. The subsequent fluctuations in volume after breakup are an artifact of droplet oscillations into and out of the focal plane. The droplet centroid position is mapped to determine the ejected droplet velocity, U_eject. The axial length of the droplet is tracked to determine the length at breakup, L_B, which is the distance between the nozzle tip and the leading edge of the drop at breakup. Table I summarizes the parameter extraction of 75 droplet generation events, showing high reproducibility of ∼300 µm diameter droplets with no satellite formation (coefficient of variation, CV, of 3.1%).

The VoF method has been used previously to simulate droplet breakup dynamics in DoD processes.^19,34,35 To our knowledge, no simulations of droplet-on-demand printing using STAR-CCM+ have been reported in the literature. The computations are thus first validated against two previous studies and are in excellent agreement with the reported breakup times and breakup lengths (see Fig. S3 of the supplementary material).^26,29 In these validation studies, Oh = 0.36²⁶ and Oh = 0.1.²⁹ The surface tension of liquid metals is an order of magnitude larger than typical aqueous-based systems, and for our experimental system, Oh_exp = 0.0044, which is far lower than either of these benchmarks. Simulation at this extremely low Oh was not possible using the methods in this manuscript, as we found that the computation was unstable—the droplet would artificially disintegrate during ejection owing to the unstable simulation of the droplet surface. Reducing the time step size did not improve the computational performance. Our approach is to use our simulations to understand the behavior of the ejected droplets as we approach Oh → 0 by systematically decreasing Oh to the lowest values we can simulate and characterizing the droplet dynamics and morphology within this limit. In our work, the minimum simulated Oh is 0.031.

To bracket the experimental data and understand the droplet formation process, simulations are performed over a broad range of the three variable parameters in the model: We, Oh, and the pulse time, t_p (equivalently, the ratio of the ejected droplet volume to the cylinder volume, R_Drop: R, as described below). For a fixed nozzle radius, Bo is constant and is thus not varied. Previous literature studies suggest a fluid printability window defined by the range We ≳ 1 and 0.1 ≲ Oh ≲ 1.^24,35 More recently, it has been recognized that lower Oh printing is possible as long as We also decreases.³⁰ Specifically, by extrapolating data points obtained at less extreme values, Liu and Derby suggest that when We = 1, the range is 0.0035 < Oh < 0.035 (note that we use the radius as the characteristic length scale). The experimentally determined Oh_exp is indeed inside this range, and as shown above, we are able to repeatedly form well-controlled, discrete droplets without the formation of satellites.

In the experimental work, a discrete pressure pulse is used to eject single droplets. During the pulse, the pressure rapidly increases and then returns to ambient (Fig. S4). Without a detailed model of the internal system gas dynamics and hydraulics, the pressure pulse cannot be directly mapped to the velocity response at the nozzle inlet. As noted above, a constant velocity is applied over a fixed velocity pulse duration t_p, as implied by the constant volumetric growth rate from the image analysis in Fig. 2(i). This is an assumption, as the droplet volume is an integrated quantity and rapid changes in the pulse cannot be easily extracted. By varying t_p at fixed We, the volume of liquid introduced in the domain is also varied. Assuming that all of the injected volume becomes a spherical droplet yields $tp=4RDrop3/3R2U$⁠, where R_Drop is the equilibrium radius of a drop formed from the injected fluid. The dimensionless pulse duration or push time, τ_p ≡ t_p/t_c, is then

The dimensionless push time is a function of both the We and the injected fluid volume.

Using the constant extrusion velocity and the size of the ejected drop, the amount of liquid that exits the nozzle provides an approximate value for the velocity pulse of t_p ≈ 0.9 ms (R_Drop: R = 3). From Table I, we note that We_exp = 2.3. Using a surface tension of 593 mN/m from droplet oscillation analysis (see the supplementary material) and literature values at the relevant print temperature for μ = 2 mPa s⁴⁴ and ρ = 6885 kg/m³,⁴⁵ Oh_exp = 0.0044 (Re = We^1/2/Oh = 344). Accurate high resolution simulations at low Oh are a long-standing challenge;⁴² to our knowledge, there are no examples of high resolution VoF simulations at these Oh values. The lowest reported value of the high resolution VoF simulation is consistent with the range in our simulations, Oh = 0.019.³⁸ 

The breakup time and length as a function of We, Oh, and R_Drop: R for actuation with a fixed, constant velocity over a fixed pulse duration are presented in Fig. 3. Several simulation trends, consistent with previous work using oscillatory actuation pulses, are immediately evident.²⁹ Across all ejected droplet sizes, the droplet length at breakup (L_B) increases and the breakup time (t_B) decreases with increasing We. We also see that at constant We, the drop breakup time increases with increasing Oh, as viscous forces slow the break-up process. The breakup length generally also increases, but the dependence on Oh varies and is particularly weak for the intermediate pulse duration [Fig. 3(c)]. In contrast to the previously reported work and as a consequence of our choice of pulse actuation, at fixed Oh and We, droplet breakup lengths and times increase as the injected fluid volume, R_Drop: R, is increased.

In Fig. 3, the experimentally determined breakup times and lengths are overlaid to show that several simulation parameters lead to reasonable agreement with the experimental data. The experimental data are binned onto a grid with a bin size of 50 µm × 50 µs to create the probability distribution function shown in Fig. 3(a). As the injected volume (R_Drop: R) is varied in Figs. 3(b)–3(d), a number of combinations of Oh and We lead to nearly equivalent breakup times and distances. Interestingly, the We over which we see agreement between the experiment and simulation, We = 1.5–2.0, is fairly constant and in agreement with the experimentally determined value of We = 2.3. However, a wide range of Oh values appear to agree with the experimental data. For intermediate sized drops, R_Drop: R = 2.6, simulations across all Oh and at fixed We = 1.5–2.0 are in near agreement with the experimental data. Smaller volumes (R_Drop: R = 2.2) are in agreement only at higher Oh. Simulations at larger injected volumes, coincident with the experimentally determined volume (R_Drop: R = 3.0), show improved agreement in the breakup time as we approach low Oh but, generally, exceed the breakup length.

A more local and detailed comparison between the simulation and experiment can be made by following the bottom-most point of the drop as a function of time.²⁶ The droplet tip trajectory is defined as the distance from the nozzle exit to the bottom tip of the droplet along the nozzle axis of revolution. The droplet tip trajectory coincides with the breakup length at the breakup time, t_B. In Fig. 4, this trajectory is plotted for all 75 experimentally measured droplets and the simulation data, which are in closest agreement with the experimentally determined breakup times and distances from Fig. 3.

At fixed We, an ideally extruded column of liquid corresponds to a straight line on this plot with the slope equal to the ejection velocity. For an actual evolving droplet, the tip position is highly dependent on the drop shape during the drop ejection process. In general, increasing slope in the tip position vs time trajectory implies that the fluid droplet is stretching in the ejection direction and thinning in the perpendicular direction because of acceleration. Decreasing slope implies the opposite: the tip is decelerating in the axial direction as surface tension forces cause the droplet curvature to increase and thus grow laterally (i.e., perpendicular to the droplet ejection path). For all of the simulated low momentum (We < 4), low Oh (Oh < 0.3) drop tip trajectories in Fig. 4, we see an initial deceleration, implying a laterally growing liquid mass, followed by a rapid acceleration due to pinch-off of the accumulated volume. Since Bo is small in these simulations, this acceleration is attributed to surface tension forces squeezing fluid toward the droplet leading edge. Physically, this appears as a nearly pendant drop that suddenly accelerates during pinch-off. The lag before acceleration is seen across ejected droplet sizes but is most pronounced for larger droplets, R_Drop: R = 3.0. Notably, this trend is not observable in the experimental data, as these trajectories more closely track the expected trajectory from high Oh ejection and the ideally extruded cylindrical column of liquid ejected at constant velocity. Indeed, across R_Drop: R, the low Oh, low We trajectories are in least agreement, both quantitatively and qualitatively.

As Oh is increased, the droplet paths become steeper and the slope becomes more constant. Contrary to the expectation from the experimentally determined Oh_Exp = 0.0044, the simulations at short (R_Drop: R = 2.2) and intermediate (R_Drop: R = 2.6) pulse durations with higher Oh are in much better agreement. Specifically, the trajectory is of nearly constant slope but then shows a noticeable deceleration as the droplet approaches breakup. This leads to a linear trajectory followed by a distinct “hook” shape immediately before breakup, in agreement with the experimental trajectories. Notably, this hook is absent in the low Oh simulations and we thus attribute the deceleration at high Oh to the stronger viscous forces.

In Figs. 4(a) and 4(c), we also include low Oh = 0.1 trajectories ejected with a stronger pulse, We = 4. Although these simulations do not match the breakup times and distances for the experiment, the tip trajectories, nearly straight lines, are now in better qualitative agreement with the experimental data. However, as expected from the low viscosity ejection, the characteristic deceleration “hook” at breakup is absent.

These observations contrast with expectations: As the simulations more closely match the experimental parameters, the agreement should improve. However, for the low Oh, We = 2 simulations, the droplet trajectory is quantitatively inaccurate and qualitatively inconsistent. It is only as We increases beyond We = 2 at low Oh or by increasing Oh at We = 2 that the simulations begin to replicate the characteristics of the experimental trajectories. The experiments thus appear to include elements of breakup from both high and low Oh parameter regimes.

Figure 5 compares the droplet shapes at breakup for experiments (a) and simulations (b). The experimental droplet shape was generated by averaging the pixel grayscale intensities for all 75 droplet images before normalizing them to unity, creating an intensity map where the pixel value denotes the likelihood of droplet presence. The simulation droplet shapes are snapshots taken immediately after breakup. The top of the frame is directly below the nozzle tip for both experiments and simulations.

As shown in Fig. 5, there is a drastic difference between the experimental and simulated droplet shapes at breakup. The experimental intensity map confirms that this liquid metal jetting system generates consistent droplets at breakup with a “pill-shaped” morphology with a diameter of ∼200 µm and a length of ∼600 µm. Simulations with Oh ≥ 0.5 show roughly spherical drops with diameters of ∼200 μm to 300 µm with nearly the entirety of the mass concentrated far away from the nozzle (∼350 μm to 550 µm). In all simulation cases, any discontinuities in the fluid stream between the nozzle and droplet are an artifact of low image resolution; thin filaments indeed span the apparent gaps just before breakup. These thin filaments are predominant in high viscosity droplet ejection, Oh ≥ 0.5, but are notably absent in the experimental results. Increasing the pulse duration (larger R_Drop: R) not only increases the droplet size but also adds a tail with increasing radius (measured perpendicularly from the ejection axis) for lower Oh. As the viscosity decreases (Oh ≤ 0.2), the droplets develop noticeably thicker tails that grow into secondary masses, which will result in satellite droplets. The lowest simulated Oh (i.e., all Oh ≤ 0.2 in the figure) provided the closest match to the experimental data with regards to the droplet length and pinch point close to the nozzle (∼50 µm), although we do note that the breakup time is not in agreement and the morphology is clearly different. Of particular note, the experimental droplet formation is also distinguished from the simulations in how the extruded tin immediately splays out into a column of fluid that is twice the inner diameter of the nozzle, as shown in Fig. 2(b). In contrast, simulations show a fluid column that is no larger than the internal diameter of the nozzle.

The simulations are used to systematically probe the behavior of the droplets as Oh decreases. Figure 6 shows the shape at breakup at fixed We = 2 to We = 4, bracketing the experimental value, and R_Drop: R = 2.2 while Oh = 0.978 to 0.031. As we follow the progression to lower Oh for We = 2, the droplet position at breakup approaches the nozzle. This low momentum ejection leads to a discrete spherical droplet at breakup with a long, thread-like tail (note that the tail is below the captured image resolution and is thus not visible). Moreover, as Oh is decreased, the drop becomes generally more spherical, a qualitative departure from the experimental expectation. For values less than Oh = 0.130, the droplet will strongly couple to the exterior of the nozzle and will not produce a discrete drop. We thus exclude these simulations.

At We = 3 and We = 4, there is a similar decreasing trend with the breakup length as Oh decreases and surface tension more effectively decelerates the droplet. However, in contrast to the lower momentum ejection, the droplet tail becomes thicker and develops a distinct and consistent pinch-point near the nozzle exit. These two observations are qualitatively consistent with the experimental results even as we are operating at larger We = 3 = 1.29 We_exp. The shape of the droplet, although much closer to the experiment, still departs from the observed “pill” morphology.

We have experimentally demonstrated the repeatable ejection of extremely low Oh droplets with no satellites. Recent work on the liquid printability operational phase diagram has recognized that the printable Oh is a function of the We.³⁰ The bounds shift to lower Oh as We decreases, with a minimum printability window at We = 1 and 0.0035 < Oh < 0.035. Our experimental work, with We_exp = 2.3 and Oh_exp = 0.0044, falls near these bounds and within the expanded printable region. It thus serves as an experimental verification of this extreme corner of printability.³⁰ Furthermore, in contrast to more complicated push–pull strategies,⁴⁶ our results imply that a very simple ejection pulse, an apparently constant velocity impulse over fixed time, is effective. To better understand the behavior in this extreme corner of the printability phase diagram, our experiment and model probe the droplet dynamics as the simulation approaches Oh_exp = 0.0044.

The experimental droplet dynamics in the run-up to breakup appearing in Figs. 2(a)–2(e) reveal an extruded “pill” that is larger than the nozzle radius and has no discernible tail formation, nor internal pinch-off, and thus no satellite drop formation. No simulation parameters can replicate this behavior. Across the data, it appears that the experimental droplets most closely behave like an ideally extruded cylinder with a deceleration just before breakup, leading to linear trajectories with a “hook.” This is a surprising result. Such behavior implies that viscous forces are dominating the droplet behavior during the droplet ejection process (i.e., high Oh). In Fig. 4, the experimental droplet tip trajectories are indeed in close agreement with the simulation results at high Oh, but the droplet shape at breakup images in Fig. 5 shows that the droplet morphology at high Oh is distinctly different from the experimental results. Important signatures of high Oh breakup including the emergence of a long thread-like tail are absent in the experimental system.

Alternatively, the trajectories of the droplet tip at low Oh and low We deviate even more strongly from the experiment. For example, simulations with We = 2 show that the droplet should eject as a single nearly spherical droplet. Simulations of low Oh droplet ejection at higher We = 3 and 4 [Figs. 6(b) and 6(c)] show a more uniform mass distribution from the droplet tip to the nozzle exit. As Oh decreases, the drop becomes more elongated, the mass is centered near the leading edge, and the curvature of the elongated droplet increases. This trend in mass distribution more closely aligns with the experimental results, but the experimental droplets show much less curvature variation in the axial direction in the tail than would be expected at such low values of Oh. The shapes predicted from the simulations are consistent with experimental observation in aqueous systems and with other VoF computations.^{30,32,38–40} In the previously published work, as in the simulations presented here, the droplet mass is concentrated at the leading-end of the droplet, with pronounced curvature near breakup, and a long, thick tail that tapers toward the nozzle exit. In contrast, the experimental drop shapes at breakup presented here are far more cylindrical. We do note strong agreement between the experiment and simulation in the location of the pinch-point for the We = 3 ejection as we decrease Oh [cf. Figs. 6(b) and 5(a)].

The simplest explanation for deviations between our measured and simulated droplet shapes is that we are not accounting for an essential process. The image analysis in Fig. 5 shows that the droplet does not wet the exterior nozzle walls, yet the liquid tin shows a radius during extrusion which is twice the nozzle radius. Thus, it is possible that internal, structural nozzle geometry aberrations (e.g., clogging) could cause the fluid to “splay” out from the nozzle exit. This seems unlikely, as the droplet shape is consistent throughout the course of the experiment and between different experiments that employ unused instances of the nozzle. Alternatively, recent work by He et al.³¹ has demonstrated that wettability in the interior of a completely clean nozzle can have a dramatic impact on the breakup length. In our simulations, we assumed an internal contact angle of 0°, but it is possible that in the experimental system, internal aberrations or partial wetting/non-wetting could dramatically impact the exit velocity profile.

A strong change in the velocity profile at the nozzle exit could direct the fluid laterally, causing the droplet to splay during ejection. Additionally, this effect, coupled with the internal fluid dynamics of the nozzle, could lead to internal acceleration/deceleration of the flow and cause an error in the measured We_exp. Recall that by using the flow rate to determine the velocity in the nozzle, we are implicitly assuming a bulk, uniform flow. More appropriate accounting of wetting effects would alter the characteristic We_exp. Indeed, given the improved qualitative agreement between the experimental data and the We = 3 and low Oh simulations, it is possible that the differences are due principally to error in We and an ejection velocity from the nozzle exit which is not along the ejection direction.

The nearly cylindrical shape and constant velocity trajectory of the experimentally measured droplets imply strong viscous forces. This ejection behavior is inconsistent with the measured physical parameters (i.e., Oh_exp = 0.0044). However, we rule out the values of our physical experimental parameters as a source of error. Our droplet oscillation measurements described in the supplementary material are consistent with the reported value of surface tension, and we expect negligible error in the measurement of tin density and the nozzle inner diameter. Previous work has shown that oxide formation on liquid tin can strongly impact the measurement of viscosity,⁴⁴ but that would merely lead to a higher Oh value and, as we discussed above, the droplet shape at high Oh breakup deviates strongly from model expectations. It thus appears that the experimentally measured droplet dynamics are inconsistent with a constant property model of viscous droplet breakup. We rule out the possibility that we are introducing modeling error, as we have validated our simulation against non-metals.

Instead, it appears that our model is missing an essential description of the droplet breakup physics. In addition to the impact of wetting described above, another possibility is the role of oxide in temporarily altering the surface properties of the droplet. A previous literature study on metal droplet oscillation measurements has shown that the presence of oxide strongly impacts droplet dynamics and requires the introduction of surface viscosity.⁴⁷ We constantly purge Ar through the printing chamber, but enough oxygen could still be present to form a thin oxide film. Although measured oxygen content was below the O₂ sensor’s 50 ppm detection limit, oxide has been shown to form on molten tin in environments of <0.25 ppm O₂.⁴⁴ The measurements of droplet oscillations post-breakup imply that the surface tension is constant and consistent with literature values for molten tin, but the high surface area-to-volume ratio during the ejection process, coupled with the temporarily localized droplet, may allow for the transient introduction of a surface film. This film could “shatter” during the violent droplet ejection/detachment and free-fall oscillation would thus not impact oscillation frequency measurements while still impacting the droplet shape at breakup.

Finally, we note that the heated liquid metal is injected into an environment with uncontrolled ambient gas temperature. The high metal conductivity implies that the droplet is isothermal. Indeed, the Biot number for the droplet, characterizing the external convective time scale to the internal conductive times scale, is nearly zero (Bi ≈ 10⁻⁴) during the majority of the droplet formation process. However, near the breakup event, the perimeter to area ratio will tend to infinity and thermal gradients will briefly appear. Locally, the heat flux to the environment becomes important and the droplet temperature can transiently approach the ambient temperature. Temperature dependent parameters may help explain deviation from the expected behavior.

We have built a custom liquid metal printer that was verified with high-speed video to reliably dispense ∼300 µm diameter tin droplets. This represents an experimental realization of extremely low Oh, satellite-free drop-on-demand printing and provides verification of an extreme corner of the printability phase diagram. High speed videography was used to track the detailed droplet dynamics during the ejection process which revealed that the drop behaves like an extruded “pill” with no discernible tail formation nor internal pinch-off. Video analysis was used to extract parameters necessary to build a computational VoF model that was used to simulate over a wide range of We, Oh, and R_Drop: R. Simulations were performed to understand the droplet dynamics and morphology as Oh is decreased and compared to the experimental data. No parameter combinations completely described the droplet dynamics in the run-up to breakup. Instead, the modeling results enabled the identification of both high and low Oh breakup signatures in the experimental system. The tip trajectories appeared to agree with high Oh breakup, but the observed large tails during the breakup process were a hallmark of low Oh breakup. Indeed, the closest model of the droplet appeared to be an ideally extruded cylinder which decelerated near breakup. Several possibilities for the discrepancy between the experiment and simulation were presented with a focus on the fluid ejection at the nozzle exit. The experiments showed a strong splaying flow at the nozzle exit, which may be driven by wetting in the nozzle interior.

Understanding low Oh DoD printing is critical to the continued improvement of liquid metal printing technologies, and here, we have identified a droplet morphology during ejection, which leads to satellite-free printing. Future work will explore droplet ejection across a broader range of process parameters and seek to further elucidate the factors impacting droplet shape, breakup, and satellite formation, including thermal effects, wettability, and the role of surface oxides.

See the supplementary material for information on the effects of the presence of oxygen on droplet morphology, detailed information on the methods used to analyze the high speed video of droplet ejection to determine breakup dynamics and material properties as well as a sample video, and details on the benchmarking of the model to the previously published works.

V.A.B. and N.N.W. contributed equally to this work.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07-NA27344 and was supported by the LLNL-LDRD program under Project No. 18-SI-001 (Grant No. LLNL-JRNL-814299).

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