Nanodroplet Flight Control in Electrohydrodynamic Redox 3D Printing

Abstract

Electrohydrodynamic 3D printing is an additive manufacturing technique with enormous potential in plasmonics, microelectronics, and sensing applications thanks to its broad material palette, high voxel deposition rate, and compatibility with various substrates. However, the electric field used to deposit material is concentrated at the depositing structure, resulting in the focusing of the charged droplets and geometry-dependent landing positions, which complicates the fabrication of complex 3D shapes. The low level of concordance between the design and printout seriously impedes the development of electrohydrodynamic 3D printing and rationalizes the simplicity of the designs reported so far. In this work, we break the electric field centrosymmetry to study the resulting deviation in the flight trajectory of the droplets. Comparison of experimental outcomes with predictions of an FEM model provides new insights into the droplet characteristics and unveils how the product of droplet size and charge uniquely governs its kinematics. From these insights, we develop reliable predictions of the jet trajectory and allow the computation of optimized printing paths counterbalancing the electric field distortion, thereby enabling the fabrication of geometries with unprecedented complexity.

Introduction

The autofocusing effect is a double-edged sword for electrohydrodynamic (EHD) 3D printing. On the one hand, this phenomenon, describing the attraction of charged droplets toward a printed object in response to the electric field concentration it creates, helps the fabrication of high-aspect-ratio structures and ensures sub-100 nm resolution.¹ It also allows for the precise positioning of quantum dots on plasmonically active structures for photonics applications² and improves stitching reliability in multimaterial printing using both a multinozzle³ and a single nozzle with a double-channel⁴ approach. On the other hand, the nontrivial droplet deflection involved detrimentally complicates the fabrication of 3D architectures. Compared to localized electrodeposition^5,6 or focused electron/ion beam-induced deposition (FEBID/FIBID),^7,8 EHD printing is relatively limited in its geometric freedom due to this autofocusing effect. Despite the technique’s unique and promising advantages, compatibility with various types of substrates, large material palette, and high deposition rate⁹, even the fabrication of seemingly simple geometries is impacted. For instance, printing a wall that intersects a previously printed one leads to undesirable discontinuities in the structure,¹⁰ and this is only partially improved by keeping a low aspect ratio (wall height over thickness) in combination with concomitant printing of the two walls.¹¹ As further breakthroughs in EHD 3D printing will require an approach for the fabrication of arbitrary shapes, overcoming the challenges of autofocusing is critical to pushing beyond pillars, spirals,³ bridges,³ and walls¹² and unleashing the potential of the technique. Reaching this crucial milestone necessitates, first, a better understanding of the droplet dynamics in the EHD process and, second, a versatile strategy for creating printing paths that compensate for the autofocusing effect.

To that end, efforts were made to understand how perturbations in the electric field by the deposited voxels (the 3D equivalent of a pixel) influence the droplet dynamics and, thus, the landing position of the subsequent voxels. Yudistira et al. systematically studied the deflection and the breakup of ejected droplets in response to the accumulation of charged solvent on a dielectric substrate.¹³ In the case of electrospun fibers, another application of EHD printing, the charge carried by the collected fibers strongly influences the deposition of the following ones.¹⁴ Through conscientious design, these electrostatic interactions can be harnessed to allow for the precise positioning of EHD fibers and droplets. By introducing external deflection electrodes, charged droplets can be deflected in a controlled way to create macroscale 2D patterns.¹⁵ Extending this approach, Liashenko et al. demonstrated the electrostatic steering of electrospun fibers to create well-defined 2D patterns and out-of-plane stacks.¹⁶ The deflection speed enabled by the external electrodes outperforms the capabilities of the mechanical stages and thus allows for fiber alignment without buckling.

In parallel, efforts have been made to alleviate the negative impact of the electrostatic deflection. O’Connell et al. derived an empirical model to compute printing paths counteracting the effect of electric field distortion around hemispheres of different sizes.¹⁷ Though successful for this particular geometry, the printing path correction for an arbitrary shape, which continuously evolves during printing, requires a more agile and comprehensive approach, such as numerical simulation by the finite element method (FEM). Other nanoscale printing techniques, such as aerosol jet printing¹⁸ or FEBID/FIBID,^19,20 readily benefited from FEM modeling for geometrical optimization, but work on EHD printing is still emerging. The modeling of EHD jets assisted in the design of a multinozzle array print head that addresses the crosstalk issues between individual nozzles.²¹ Richner et al. provided a more fundamental study by observing the deflection of EHD droplets from a vertical path in response to electric field modification by thin-film electrodes.²² Their FEM model determined the droplet trajectory and provided, by comparison with experimental results, an estimation for the charge and the size of the ejected droplets. Despite the increasing interest in numerical methods for understanding the EHD process and improving jet straightness from a multinozzle array, they have not yet been employed to optimize geometrical conformity between design and print outcome.

To tackle this challenge of high geometric fidelity in EHD printing, we here develop a combined experimental and modeling approach to investigate the flight dynamics of charged droplets using the electrohydrodynamic redox 3D printing (EHD-RP) process as a model system. EHD-RP has recently been introduced as an electrochemical alternative to EHD,^4,23 in which the nanoparticle suspension is replaced with metal ions in solution. This technique advantageously combines both the high deposition rate of EHD and the various benefits of EC-based nanoscale 3D printing,²⁴ including superior as-deposited material quality,²⁵ tunable microstructures,²⁶⁻²⁸ and no requirement for postprinting annealing steps. Notably, EHD-RP distinguishes itself from other nanoscale additive techniques by its potential for multimaterial printing⁴ and its unique ability to synthesize multicomponent alloys with precise compositions.²⁹

Here, the electrostatic deflection of charged droplets using thin-film electrodes allows for their precise positioning to create various geometries. We also examine the flight path alteration resulting from the presence of a nanopillar printed on the substrate to provide a physical basis for the autofocusing effect in EHD-RP. In comparing these experimental observations with predictions from the developed FEM simulations, this work provides new insights into the droplet characteristics and their role in the deflection of the droplet’s path. Finally, our development of an FEM-based optimization tool highlights key insights into the autofocusing phenomena in EHD-RP and enables printing path correction for electrostatic autofocusing. Through these tools, EHD-RP and other EHD techniques can reach greater geometric complexity and open the door for the promise of fast, tunable microstructural design on the nanoscale.

Results and Discussion

Electrostatic Deflection by External Electrodes

Electrohydrodynamic redox 3D printing (Figure 1a) relies on the polarization of an immersed sacrificial anode with a ≈100 V electric potential, leading to (i) the electrodissolution of the anode, thereby generating solvated metal ions, and (ii) the electrohydrodynamic ejection of ion-loaded solvent droplets from a quartz capillary nozzle. Upon landing on a grounded substrate, the solvent evaporates, and (iii) the ions are reduced, giving rise to a metallic voxel. The droplet formed in this process carries a charge; thus, its flight is impacted by the orientation of the electric field. In a standard EHD configuration where the autofocusing effect is not considered, the electric field arises solely from the potential difference between the nozzle and the counter electrode, e.g., a grounded flat substrate, leading to a spatial configuration with circular symmetry and thus a straight droplet flight path. In order to study the droplet deflection, we break the circular symmetry of the electric field by introducing polarizable thin-film electrodes in the vicinity of the printing nozzle. Using only one electrode (1D) with nozzle translation parallel to the electrode (Figure 1b), the shape of the deposited copper lines strictly follows the triangular waveform of the electrical signal. Also, the magnitude of the deflection appears to increase linearly with the signal amplitude (Figure S1a).

Figure 1.

EHD-RP with electrostatic jet deflection. (a) Schematic of the printing process. A sacrificial copper anode inserted in an acetonitrile-filled quartz capillary is polarized with a ≈100 V potential. The strongly positive voltage leads to (i) generation of solvated metal ions, while the electric field triggers (ii) electrohydrodynamic ejection of ion-loaded droplets. Upon landing (iii), the ions are reduced and the solvent evaporates, leading to the creation of a metallic voxel. (b) SE micrographs showing 1D deflection of the EHD droplets by an electrode parallel to the nozzle movement and polarized with a 30, 60, and 90 V voltage amplitude. (c) SE micrographs showing electrostatic deflection of the EHD droplets in all in-plane directions using two perpendicular and independently polarized electrodes. (d,e) Out-of-plane cylinders, each of which was deposited with a constant nozzle position and a sinusoidal signal applied to both deflection electrodes. Imaging tilt: (b,d) 0° and (c,e) 55°.

Electrostatic deflection offers a unique approach for the precise positioning of voxels with lateral jet speeds and accelerations unattainable by nanopositioning systems. The introduction of a second electrode perpendicular to the first unlocks the complete in-plane positioning of the droplet and enables the deposition of 2.5D structures (Figure 1c). In this approach, a sinusoidal signal applied to both electrodes with a π/2 phase shift between the two leads to the deposition of a cylinder (Figure 1e). Interestingly, the 9 cylinders visible here elongate and/or shrink along the two principal axes (Figure 1d). Given that each cylinder was fabricated with the same signal amplitude but with 9 different nozzle positions, the strength of the electric field produced by the deflection electrode rapidly decays with distance. For the most elongated cylinders [top-left and bottom-right in (d)], the longer residence time of the EHD jet at the extremities even causes a splitting of the structure in two. Also, with the volume of each cylinder being the same, changes in the lateral dimensions lead to variations in height. Accordingly, the thinnest cylinder on the bottom-left of Figure 1c,d is also the tallest.

The postprinting deflection amplitude is derived from SE micrographs in the 1D configuration (Figure 2a) as a function of the distance from the steering electrode edge and compared to the deflection amplitude predicted by a 3D FEM model (Figure 2b). The model, built in the COMSOL Multiphysics environment, uses the Electrostatics module to predict the electric field distribution and, thereafter, computes the trajectory of a spherical droplet within this field by using the Charged Particle Tracing module. The assumption of droplet sphericity is discussed in Section S1 (Supporting Information). In accordance with the experimental measurement, the amplitude predicted by the model decreases with the distance from the steering electrode (Figure 2b). A cross-sectional map of the simulated electric potential distribution is displayed in Figure 2c,d for a −60 and +60 V deflection potentials. In these two cases, the influence of the electrode voltage on both the electric field lines between the nozzle and substrate (light yellow) and the droplet path (light blue) is visible. The model also shows that, for the 7 μm nozzle distance used here, electrode potential has little to no impact on the electric field lines in the vicinity of the nozzle opening. Thus, the droplet trajectory does not deviate from a vertical path right after ejection from the nozzle but rather from a later stage.

Figure 2.

Electrostatic deflection and comparison to FEM modeling. (a) Copper lines showing waviness as a result of electrostatic deflection with a triangular waveform. The amplitude of the deflection directly follows the signal amplitude (30, 60, and 90 V). (b) Deflection amplitude as a function of the distance from the steering electrode for a 60 V signal amplitude, as obtained experimentally and derived from FEM modeling for droplets with various radius·charge (r·q) products. The error bars are an estimation of the confidence interval for the deflection measurement from SE micrographs. (c,d) 2D maps of the electric potential distribution predicted by the 3D FEM COMSOL Multiphysics model showing the electric field lines and the simulated droplet path for a deflection voltage of (c) −60 and (d) +60 V. The droplet in these maps has a 90 nm radius and carries a charge of 167e. (e) Model predictions for the deflection amplitude, taken as half the peak-to-peak value between −60 and +60 V deflection voltage, and (f) coefficient of determination (r²) showing the degree of agreement between the model predictions and experimental data for various droplet radii and number of elementary charges. The results unveil the relationship that links the droplet deflection to its r·q product.

The amplitude of the deflection from the vertical path appears to be closely linked to the characteristics of the droplet, for instance, its radius and the number of elementary charges that it carries (Figure 2e). The model indicates a greater degree of deflection in lighter or more weakly charged droplets. More specifically, all droplets with matching radius-charge (r·q) products undergo a similar degree of deflection. Accordingly, the best agreement between the FEM model and the experiments (Figure 2f) is found for all couples of r and q whose product lies around 12–15 μm e (or between 7–30 μm e to stay conservative with respect to the uncertainty in the experimental data).

Droplet Deflection in Response to the Autofocusing Effect

The autofocusing effect, i.e., the attraction of charged droplets to readily printed structures, represents both an opportunity and a challenge for the EHD-RP process. On the one hand, it improves reliability in the fabrication of out-of-plane wire-like structures, such as pillars, by attracting charged droplets to the apex; it attenuates any imprecision in the positioning of the printing nozzle. On the other hand, the resulting droplet deviation from a straight trajectory complicates the fabrication of complex geometries, for which the use of nontrivial printing paths needs to be developed. Here, we highlight this phenomenon by studying the droplet deflection in response to the presence of a 2 μm-tall pillar as a function of the offset, i.e., the lateral nozzle displacement between the first pillar and the secondary branch. The flight path, whose end point is revealed by the branch attachment point in the secondary electron (SE) micrographs, as discussed in Section S2 (Supporting Information), appears strongly impacted by the presence of the first pillar (Figure 3a). This effect is amplified for smaller offsets between pillars. For offsets of 0.7 μm or smaller, the deflection is large enough that the droplet lands on the side of the first pillar. Considering the pillar width (190 nm), this observation implies a sideways deflection up to 0.6 μm from the vertical path. In Section S3 (Supporting Information), we discuss the origin of the small volume that does not deflect and thus deposits below the branch.

Figure 3.

Autofocusing effect and predictions of the FEM model. (a) SE micrographs revealing how the presence of a nanopillar influences the deposition of a subsequent branch. The blue arrows are a guide to the eye in visualizing the deflection. Imaging tilt: 55°. (b) 2D cross-section of the electric potential distribution as predicted by the 3D FEM model and the resulting droplet (90 nm radius, 380e charge) flight path for a 0.7 μm branch offset. (c) Lateral (x) and vertical (z) landing positions of the droplet derived experimentally from the attachment point of the secondary branch, as measured in SE micrographs, and comparison to predictions of the FEM model for different r·q products. Each experimental data point is averaged from 18 data sets obtained from 2 independent experiments (9 data sets each, see Figure S2) and the error bars denote the resulting standard deviation. (d) Coefficient of determination (r²) that reflects the fitting quality of the FEM model to the experimental data for different droplet radii and number of charges. The results highlight how the r·q product governs the droplet trajectory.

The map of the electric potential predicted by the FEM model shows how the small radius of curvature associated with the body and the apex of the pillar locally increases the magnitude of the electric field and bends the field lines toward it (Figure 3b), thereby leading to substantial droplet deflection. The FEM model can replicate the deflection behavior closely for all offsets investigated (Figure 3c), and the best fit is obtained for an r·q product of ≈34.2 μm e [(34 ± 4) μm e accounting for the error estimate in the experimental data]. In line with the computational results for droplet deflection by thin film electrodes, the r·q relationship holds here (Figure 3d) despite the fundamentally different configurations of droplet deflection caused by a protruding structure on the substrate.

Relevance of the r·q Product to the Flight Dynamics of EHD Droplets

Predictions of the two aforementioned FEM models, electrostatic deflection by a thin-film electrode and the autofocusing effect, reveal how the droplet flight path is closely bound to the product of its radius and charge: r and q. In both of these configurations, droplets with identical r·q (or, in terms of mass, m^1/3·q) follow similar trajectories (Figure S3). Quantitatively, the deviation from this relationship is small enough that, for the electrostatic deflection of droplets characterized by r·q = 15.0 μm e, increasing the radius by a factor 4 (from 45 to 180 nm, i.e., 64 × larger mass) leads to less than 20 nm difference in the landing position (Figure 2f). In comparison, a much smaller increase in the droplet size from 45 to 57 nm, while keeping the number of charges constant, already leads to a larger shift of 30 nm.

Additionally, the trajectory deflection scales inversely with the r·q product. A heavy or strongly charged droplet follows a straighter path (i.e., less impacted by the bending of the electric field lines) than a light or weakly charged droplet. This result follows intuition with respect to the radius; a heavier droplet has more momentum and thus deflects less, but is more nuanced with respect to the charge. While the lateral acceleration during the flight scales with a stronger charge (Figure S3c), a stronger charge also leads to an increase in the vertical acceleration (Figure S3b) and, ultimately, less time allowed for lateral deflection. Thus, for a more strongly charged droplet, the reduced flight time allows for less lateral deflection, despite the increase in lateral electrostatic force.

This result is particularly interesting, given that the deflection of charges in response to an electric or magnetic field typically follows the mass-to-charge m/q ratio and not m^1/3·q. In mass spectrometry measurements, m/q is the ratio discriminating between species and is independent of the type of mass selection (sector field, time-of-flight, and quadrupole filter). Here, the observed difference in behavior arises from the aerodynamic drag force acting on the droplet in the model, which is negligible in mass spectrometry (see the comparison in the absence of drag force, Figure S4). Consequently, droplet trajectories are sensitive to the dynamic viscosity of the surrounding medium, which is determined by factors including gas composition, temperature, and pressure. However, the r·q relationship holds independently of these conditions. As the magnitude of the drag force tends toward zero (Figure S4b), for instance, by reducing the gas pressure, all flight trajectories converge regardless of the r·q product. Our findings are taken and discussed in the context of the experimental conditions reported: argon gas at atmospheric pressure and room temperature.

To support and understand the model’s predictions with regards to the r·q relationship, a simple analytical model can be derived with the following assumptions:

• 1.The space between the nozzle and the substrate is divided between two zones (acceleration and deflection) at a distance h from the substrate.
• 2.In the first zone, the droplet accelerates vertically due to the electric field, while the drag is neglected. The total voltage drop in this zone is denoted by ΔV.
• 3.In the second zone, the droplet decelerates due to the drag force and is deflected laterally by a constant electric field E_x, while the vertical electric field is neglected.
• 4.In the second zone, the lateral velocity is small compared to the vertical one (Figure S3b,c), such that the lateral component of the drag force is neglected.
• 5.The drag follows Stoke’s law, F = 6πμrv, with μ as the dynamic viscosity, r as the radius of the droplet, and v as the droplet speed.

With these assumption, the resulting deflection is calculated as , with , and ρ as the solvent density. In this relationship, r and q are found only in the form of their product r·q. Therefore, in accordance with the predictions of the FEM model, all droplets with identical r·q values are equally impacted by the lateral electric field. Moreover, the comparison between the experimental results and the predictions of the models (both numerical and analytical) allows for the determination of only its r·q product, not the individual values of r and q. In that regard, the predictions of the analytical and FEM models for the droplet deflection differ appreciably for the investigated range of size and charge. For instance, the analytical model is not valid for r·q > B² (≈80 μm e, for larger r·q, the argument of the logarithm is negative), which is not negligibly far from the value predicted by FEM. Therefore, while the analytical model supports our phenomenological understanding and validates the FEM model, we will rely on numerical methods for a quantitative assessment of the droplet deflection.

The r·q products obtained from the two above-mentioned experimental approaches do not perfectly correspond: (34 ± 4) μm e from the autofocusing experiment vs 7–30 μm e from the experiment with a deflection electrode. The origin of the discrepancy may arise from a neglect of the role of solvent evaporation. In experiments, the droplet mass decreases in flight due to evaporation and changes the r·q product. Thus, for longer flight times, the r·q product is on average smaller. Since the nozzle distance was larger for the experiments with the deflection electrode (9 vs 7.5 μm), it may explain the smaller r·q derived from this approach. Another source of error could be the absence of a dielectrophoretic force in the model. Since this force is proportional to the gradient of the squared magnitude of the electrical field, which is different for the two configurations, it would alter the droplet path differently. While the significance of the dielectrophoretic force in the droplet dynamics is still unclear, it is most likely negligible, as discussed in Section S4 (Supporting Information).

The value derived from the autofocus configuration appears to be more reliable for the two following reasons. First, the estimated error in the SEM measurement is significantly smaller for this experiment (see error bars, Figure 3c). Second, the FEM model of this configuration appears to be more sensitive to the r·q product, as small changes in r·q lead to substantial variations in the prediction of the landing position. The latter is particularly true around the transition between where the droplet lands on the pillar vs the substrate (see Figure 3c, branch offset between 0.6 and 1 μm). Thus, in the following analysis, we will assume an r·q value of (34 ± 4) μm e.

One can further narrow down this estimation of r·q by considering the Rayleigh limit,³⁰, which sets the maximal number (n) of elementary charges (e) a droplet of radius (r) and surface tension (γ) can sustain without a Coulomb explosion. Taking 34 μm e for r·q and 28.66 mN m^–1 for γ,³¹ the Rayleigh limit is reached for a 45 nm radius, which corresponds to a charge of 755 e. Therefore, the droplet characteristics have to satisfy both r·q = 34 μm e and r > 45 nm. Finally, for the EHD process, it is generally assumed that the droplet size is on the order of the feature size, which, for a pillar of 190 nm, would imply a radius of ≈95 nm and thus a charge of ≈358 e. In future investigations, the droplet charge could be derived in an alternative way by dividing the overall reduction current by the droplet frequency obtained with an optical method (by measuring the variation in intensity when a droplet crosses a laser beam).

Although the exact droplet size and charge remain uncertain, the results of the previous sections showed that the r·q product is sufficient to predict the flight behavior.

Printing Paths Compensating for the Autofocusing Effect

Using the r·q product calculated from the autofocusing experiments, we transform the FEM model into an optimization tool in which the lateral nozzle position is iteratively adapted to reduce the absolute distance between the calculated and target landing positions. This enables a corrected nozzle path that compensates for the autofocusing effect. Note that we chose the r·q derived from the autofocusing experiments, 34 μm e, first for the reasons discussed in the previous paragraph, and second since the model should correct for the droplet deflections that arise from this exact phenomenon. As a proof of principle, we derive the optimal printing path for a simple geometry, which, despite its apparent ease, represents a challenge in the absence of print corrections: two closely spaced nanopillars. These are designed to be 2 μm in height and spaced by 300 nm, resulting in a gap of half the pillar width (assuming 200 nm-thick cylinders). As visible in the left-most micrograph of Figure 3a, this geometry represents a challenge that cannot be achieved by printing the first pillar and, afterward, the second. Without correction, the second pillar grows from the tip of the first. One simplistic approach that decreases the autofocusing effect is layer-by-layer printing. Thus, to provide a fair comparison between this printing strategy and the FEM-derived path corrections, the two pillars are printed in a layer-by-layer fashion, with each pillar segment being either 400 or 800 nm in height. Moreover, we further reduce the autofocusing effect by halving the length of the first deposited segment, thereby introducing a shift that decreases the maximal height difference between the pillars during fabrication to only 1/2× the segment length instead of 1×.

The nozzle positions optimized by the FEM model and corresponding droplet flight path are shown for a subset of growth steps in Figure 4a–c. The correction is stronger, and therefore more crucial, the taller the nongrowing pillar is compared to the growing one. The magnitude of the correction also increases with the overall height of the two nanopillars (Figure 4c), which stresses the importance of the correction for more complex structures. The results also highlight the limitations of the layer-by-layer approach in reducing the autofocusing effect. Even in the extreme case of infinitely thin layers, i.e., the two pillars have the same height during the whole growth process, a significant correction still applies: 19 nm for step i + 2in Figure 4b,c and even 24 nm when the 2 pillars reach 2 μm in height (last growth step). This correction seems small, but without it, the error in the pillar apex position would increase for each new layer, and the final outcome would not be correct. This illustrates the limited potential for a layer-by-layer printing scheme to decrease the autofocusing effect and emphasizes the necessity of FEM-calculated printing path corrections.

Figure 4.

Printing path optimization for compensation of the autofocusing effect. (a) Overview of the geometrical configuration of the FEM model. (b) Flight path (blue) after optimization of the nozzle positioning for a selection of growth steps. All trajectory end points are correctly situated at the apex of the growing pillar. (c) Nozzle positions with and without correction as a function of printing time for 800 nm-long pillar segments, assuming a 3 μm s^–1 vertical growth rate. SE micrographs displaying the experimental outcomes for (d–f) 800 nm and (g–i) 400 nm segment length. The corrected paths enable the deposition of nanopillars with a high degree of geometrical conformity (e,h), while appreciable divergence is observed in the absence of correction (d,g). Imaging tilt: 80°.

The experimental results further corroborate the benefits of this approach. Pillars fabricated using a corrected path (Figure 4e,h) are considerably closer to the design geometry than the uncorrected ones (d,g) for both 800 nm segment steps (f) and 400 nm segment steps (i) (see also similar results from an independent experiment in Figure S5). The 100 nm gap presented here is the only parameter we investigated and does not correspond to the limit of the method. From the perspective of the model, this gap may be reduced willingly to the point where the two pillars meet. In practice, however, the higher droplet attraction to the neighboring pillar would decrease the chances of print success. In the predictions of the model, this would translate into an increased sensitivity of the droplet landing position to the printing parameters (e.g., the nozzle position). In other words, the differences between the model and the real conditions would have a greater impact on the print outcome. In line with these considerations, the small error visible at the top of the corrected pillars with large segment steps (e) may have two possible origins. First, the pillars shown are slightly higher than designed: 2.3 μm instead of 2 μm (growth rate was ≈3.45 μm s^–1 instead of the expected 3 μm s^–1). Consequently, the antenna effect was stronger while printing than in the simulation, which may explain the increased deflection observed. Second, the top of the pillar is thinner than the bottom, and it thus grew faster (assuming a constant volumetric deposition rate, resulting in an increased droplet attraction). In any case, the defect remains minor in comparison to the uncorrected geometry. Thus, the results demonstrate the potential of FEM to bridge the gap between basic geometries achieved by trial and error, the current state of the art in EHD printing, and the fabrication of arbitrarily complex 3D shapes.

Toward Arbitrary Geometries

To generalize this approach to the fabrication of arbitrary shapes, the 3D models are discretized into printing steps, each corresponding to an added spherical voxel. For each printing step, i.e., building block, the FEM model constructs the geometry as the union of the previous spheres and then optimizes the nozzle position accordingly. The experimental outcomes for a double helix, a bicycle, and a tree-like network with and without corrected printing paths are visible in Figure 5. In the absence of correction, the topology of the fabricated structures is not guaranteed: the two strands of the double-helix (a) collapse on each other, and the four main branches of the nanotrees (g) do not emanate from a unique point. Conversely, the corrected nano-objects display excellent geometrical conformity with the model and good reproducibility, as shown in Figure S6. Thus, the use of autofocusing-corrected printing paths unlocks the fabrication of complex 3D shapes by EHD printing and opens new prospects for this additive manufacturing technique in various fields of nanotechnologies.

Figure 5.

Unlocking the fabrication of arbitrary shapes. Nanoscale 3D geometries deposited by EHD-RP (a,d,g) without and (b,e,h) with autofocusing-corrected printing paths. (c,f,i) 3D models, (c) double helix, (f) bicycle, and (i) tree-like networks, are discretized into spherical building blocks. From red to blue, the spheres’ color denotes the order in which each segment was printed. Imaging tilt: (a,b) 85°, (d,e) 86.5°, and (g,h) 89.5°.

These results illustrate both the success of model-based corrections and some of the fabrication challenges linked to EHD 3D printing. First, the voxels sitting directly on the substrate require additional residence time for an equivalent volume of deposited material compared to subsequent voxels. This leads to shorter pedestals for the spiral’s strands (Figure 5a–c) and larger cropping of the lower parts of the bicycle wheels (Figure 5d–f). This effect is likely related to the spreading of the copper nuclei on the substrate for the first landing droplets and may be corrected by investigating the delayed time for the pillar nucleation. Second, the bicycle frame illustrates the difficulty of fabricating overhanging structures with a beam angle approaching 0° from the horizontal. In this example, the top tube at 25° above the horizontal (between saddle and handlebars) was fabricated successfully, while the 9° down tube (which has been simplified here into a tube connecting the wheels) was not. Although 0° beams have been reported,⁴ their fabrication requires a precise match between the growth rate and the nozzle movement, and thus, any imprecision in the assumed growth rate can be detrimental. Third, bridges, such as bicycle wheels, can be printed successfully but show limited reproducibility. This topology is particularly challenging because the two sides of the wheel need to have the exact same height in order to close the gap; otherwise, the droplets are attracted by the higher side, and the gap remains. Finally, the shape of the print tends to increasingly diverge from the model during fabrication; a positioning error of a voxel decreases the prediction reliability for the next. Thus, model accuracy is crucial for expanding the presented strategies toward larger structures. Considering the currently achieved precision despite the rudimentary data set used to derive the model parameters, i.e., the droplet attraction to a nanopillar in Figure 3, there is a large potential for refining the model.

Conclusions

We present two experimental approaches to study the deflection of charged droplets involved in the EHD-RP process. First, a voltage signal applied to external deflection electrodes directly controls the trajectory of EHD droplets and allows for their precise positioning in the fabrication of 2.5D structures. Second, we quantify the alteration to the droplet’s landing position due to a nearby nanopillar (i.e., the autofocusing effect), which further highlights the challenges in achieving high geometrical complexity by EHD printing. The reported FEM simulations agree with both experimental approaches and unveil that the product of droplet size and charge predominantly governs its flight dynamics. In the current study, this quantity (r·q) lies around 34 μm e and likely corresponds to a droplet radius of ≈95 nm and a charge of ≈358e. Finally, we show how the printing paths can be optimized by an FEM model to compensate for the electrostatic deflection of the charged droplets. This strategy unlocks the deposition of complex 3D nano-objects that are not achievable otherwise and unleashes the full potential of EHD 3D printing in the field of microfabrication.

Methods

Printing Procedure and Setup

The printing nozzles were prepared with a pipette puller (P-2000, Sutter Instrument) from quartz capillaries (QF100-70-15, Sutter Instrument). The pulling parameters were as follows: Cycle 1: Heat 810, Filament 5, Delay 128, Pull 50, Velocity 30; Cycle 2: Heat 700, Filament 4, Delay 130, Pull 75, Velocity 50, leading to aperture diameters of 150–220 nm. The nanopipettes were filled with acetonitrile (Optima, Fisher Chemical) using a glass syringe (Gastight #1010, Hamilton) equipped with a 20 nm syringe filter (Anotop 10, GE Whatman). Cu wires used as sacrificial anodes (99.9999%, 0.25 mm-thick, Alfa Aesar) were etched in 65% concentrated nitric acid (Sigma-Aldrich) for 10 s, rinsed in water (Optima LC/MS grade, Fisher Chemical), stored in ethanol (EMSURE, absolute for analysis, Supelco, Merck), and finally inserted into the nanopipettes. The printing process was conducted in an inert Ar atmosphere with <100 ppm of O₂, as monitored by a Module ISM-3 oxygen sensor (PBI Dansensor), to reduce the oxidation of the deposited copper. Substrates used for printing were Si wafers [(100), SiMat, with 50 nm diffusion barriers of SiO₂ and Si₃N₄] coated by DC magnetron sputtering (PVD Products Inc.) with a 3 nm Ti film and an 80 nm Au film. The thin-film electrodes were made by a standard lithography process, with a patterning step performed by laser writing (DWL66+, Heidelberg). The steering setup with electrical wiring is visible in Figure S7. The nanopositioning stage (QNP piezo stage with Ensemble controller, Aerotech) and the printing voltage provided by a power source (B2962, Keysight) were controlled with a custom Matlab script. The printing voltage was set to 110 V in all experiments. The ramping profiles of the piezo stage between the autofocusing-corrected waypoints were performed by a script written in the AeroBasic programming language of Aerotech and running on the stage controller.

Microscopy

The samples were imaged with a FEG-SEM (Magellan 400, FEI, USA) and with a FIB-SEM station (NVision40, Zeiss, Germany), both operating in the in-lens mode with an acceleration voltage of 5 kV for the electron source.

FEM Model

All 3D numerical models were built with COMSOL Multiphysics software. The AC/DC module computed the electric potential distribution for a printing potential applied at the capillary surface and a grounded substrate. The droplet trajectory was calculated thereafter using the Charged Particle Tracing module, comprising an electrostatic force and a drag force as defined by Feng et al. for the motion of a viscous sphere subjected to interfacial slip.³² The correction for a viscous sphere, accounting for the existence of flows within the droplet, is preferable when the particle viscosity is not substantially larger than the surrounding fluid (here, the viscosity ratio is only ≈15), while the interfacial slip correction is justified by the large Knudsen number in the present study (≈1, defined as the mean free path of gas molecules over particle radius). The nozzle positions that best compensate for the autofocusing effect were determined using the derivative-free BOBYQA (bound optimization by quadratic approximation) method, which iteratively adapts the model geometry to reduce the deviation between the computed and desired landing positions. The double-pillar geometry in Figure 4 was modeled in COMSOL by two cylinders with ellipsoid ends and variable total heights. The arbitrary shapes in Figure 5 are wire-like structures modeled and discretized in Blender software. The file conversion from Blender to COMSOL Multiphysics and ultimately to the 3D-printer software was managed by a custom Wolfram Mathematica script.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsami.3c10829.

• Assumption of droplet sphericity, equivalence between branch attachment and droplet landing position, explanation for the nondeflection of some droplets in Figure 3a, influence of dielectrophoretic force, influence of the deflection electrode potential and the printing voltage, micrographs used to derive the data in Figure 3c, droplet trajectory, speed, and flight time, calculated droplet deflection with and without drag force, additional data to Figure 4, success rate in printing bicycle geometry, setup for electrostatic deflection, and force magnitude comparison between dielectrophoretic and electrostatic force (PDF)

The authors declare no competing financial interest.