⚡ electrical engineering

Predicting droplet landing in angle-controlled surface acoustic wave jetting

This paper demonstrates a practical framework for predicting droplet landing positions in nozzle-free, angle-controlled surface acoustic wave jetting by linking ejection angles to landing coordinates through a geometric model, achieving high placement precision and cross-day predictability while defining operational limits for single-droplet ejection.

Original authors: Md Aminul Islam, Sara Adibi, James Friend

Published 2026-10-23

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CC BY 4.0

Original authors: Md Aminul Islam, Sara Adibi, James Friend

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where printing delicate biological samples, like living cells or soft gels, requires no tiny nozzles to push the material through. Traditional inkjet printers rely on these microscopic openings, but they are prone to clogging when the liquid is thick or filled with particles. To solve this, scientists have turned to sound waves. By using a specific type of vibration called a surface acoustic wave, they can shake a hanging drop of liquid until it pinches off and flies into the air. This method is clean and clog-free, but for years, a major question remained unanswered: once the drop is launched, where exactly will it land? Without a way to steer the drop or predict its destination, this technology could not be used for precise tasks like drawing circuits or assembling tissue.

A team of researchers has now mapped out the rules for steering these sound-driven drops and predicting where they will settle. They built a system where a hanging drop of liquid sits above a flat surface. By applying sound waves from two different directions, they can create an imbalance that pushes the drop off-center, sending it flying at a specific angle. The researchers discovered that the direction the drop travels depends entirely on the difference in the push from the two sides. If the push is equal, the drop goes straight down. If one side pushes harder, the drop tilts toward that side. They also found a specific range of power settings that ensures only one drop is launched cleanly, avoiding the messy spray of smaller droplets that often follows a main drop.

The team then tested how accurately they could place these drops. They set up a gap of 5.2 millimeters between the hanging drop and the collection surface. In a series of experiments, they fired the drops at various angles and measured where they hit. Crucially, these measurements were not made against a fixed, external target on the surface. Instead, the researchers defined the "target" for each experiment based on where a drop landed when the sound waves were perfectly balanced. This means the study demonstrates how reliably the drops land relative to that specific balanced shot, rather than predicting their absolute position on the substrate. The results showed a distinct pattern in the errors. When the drops were sent straight down, they landed very close to the balanced reference point. However, as the angle of the shot increased, the landing spot became less predictable, but only in the direction of the shot. The drops would wander back and forth along the line of travel, but they stayed remarkably steady side-to-side. This means the error is not random; it stretches out like a line rather than a circle. The researchers found that the further the drop had to travel and the sharper the angle, the more the landing spot would drift, but this drift was consistent enough to be measured and predicted relative to the balanced shot.

To see if this prediction held up over time, the researchers ran tests on different days, sometimes weeks apart. They used data from the first six days to build a simple guide for where the drops should land relative to the balanced shot, and then tested that guide on the final two days without making any adjustments. The guide worked well, predicting the landing spots with an average error of less than one millimeter. This proved that the relationship between the sound settings and the landing spot is stable, even as the equipment or environment changed slightly from day to day. They also demonstrated that by firing drops in quick succession, they could draw continuous lines on the surface. While the lines were not perfectly straight and varied slightly in width, the fact that they could be drawn at all showed that this method could create complex shapes, not just single dots.

The study concludes that while the technology is not yet perfect enough for the ultra-precise absolute placement required in high-end manufacturing, it has moved from a chaotic spray to a predictable tool for relative positioning. The researchers identified that the main source of error comes from the angle of the shot and the distance the drop must travel. They suggest that by bringing the target surface closer to the drop, perhaps to just a few millimeters away, the accuracy could be significantly improved. This work provides a clear framework for understanding how sound waves can be used to move tiny amounts of liquid with intention, paving the way for future devices that can print with materials that would clog any traditional nozzle.

Technical Summary: Predicting Droplet Landing in Angle-Controlled Surface Acoustic Wave Jetting

Problem Statement
While nozzle-free droplet generation via Surface Acoustic Waves (SAW) offers significant advantages for printing sensitive materials (e.g., cells, bioinks, and functional inks) by avoiding clogging and shear stress, a critical gap remains in controlling where these droplets land. Conventional nozzle-based inkjet printing achieves high spatial resolution but struggles with material compatibility. Existing nozzle-free methods, including laser-induced transfer and acoustic jetting, often lack precise control over deposition location or require complex alignment. This work addresses the specific challenge of predicting droplet landing positions in an angle-controlled SAW jetting system, where droplets are steered from a pendant parent drop toward a collection substrate without moving the source or the printhead.

Methodology
The authors utilized a downward-facing SAW jetting platform equipped with opposing focused interdigital transducers (FIDTs). By creating a time-integrated acoustic acceleration imbalance between the transducers, they steered droplets ejected from a pendant parent drop at specific angles ( $\theta$ ) toward a substrate.

Key methodological components included:

Regime Identification: The study employed an overall acceleration ratio ( $\Upsilon$ ) to define operating windows, distinguishing between no-ejection, single-droplet, satellite-droplet, and multiple-droplet regimes.
Geometric Deposition Model: A geometric model was developed to link the ejection angle to the landing position. The model references all measurements to a "balanced shot" (where transducer inputs are equal, resulting in near-vertical ejection) within each experiment, rather than an absolute external coordinate system.
Experimental Design: Experiments were conducted over 20 trials across eight collection dates with a parent-drop-to-substrate spacing ( $L$ ) of 5.2 mm. The study analyzed 80 steered shots across four velocity ratios ( $r = \dot{\xi}_1/\dot{\xi}_2 \in \{0.2, 0.4, 2.5, 5.0\}$ ) and a reference ratio ( $r=1$ ).
Predictive Testing: A held-out predictive test was performed where a mapping of command ratios to landing positions, calibrated on six earlier dates ( $n=60$ ), was applied to two later dates ( $n=20$ ) without recalibration.
Line Deposition Analysis: The internal geometry of one-dimensional lines formed by sequential deposition was characterized using centerline offset and local width metrics on hydrophilic substrates.

Key Results

Anisotropic Placement Scatter: The study found that placement variability is strongly anisotropic. In-plane error (steering direction) had a standard deviation of 0.59 mm, while transverse error was only 0.06 mm (a ~10:1 ratio). The in-plane error scales with the steering angle according to an angular jitter model ( $\sigma_\theta \approx 4.9^\circ$ ) amplified by a $\sec^2\theta$ factor.
Operating Windows: The acceleration ratio $\Upsilon$ successfully identified the single-droplet ejection regime ( $1 \le \Upsilon < 1.5$ ), separating it from regimes producing satellites or multiple droplets.
Predictive Accuracy: The held-out predictive model achieved a Root-Mean-Square Error (RMSE) of 0.71 mm (95% confidence interval: 0.41–1.01 mm) when predicting landing positions on new dates based on prior calibration. The largest errors occurred at the strongest steering condition ( $r=0.2$ ), where day-to-day drift is amplified by the geometric projection.
Geometric Discrepancy: The measured landing positions exhibited an "overshoot" relative to the physical separation distance ( $L=5.2$ mm), corresponding to an empirical lever arm ( $L_{eff}$ ) of approximately 8.3 mm. The authors attribute this to a combination of trajectory-angle measurement timing, per-experiment origin placement bias, and post-impact centroid displacement on the PDMS substrate, noting that ballistic curvature alone cannot explain the outward shift.
Line Fidelity: Proof-of-concept line deposition demonstrated that internal line geometry (straightness and width uniformity) can be quantified, though width variability was heavily influenced by wetting and coalescence on the hydrophilic substrate.

Significance and Claims
The paper establishes a practical framework for predicting relative droplet landing in angle-controlled SAW jetting. The authors explicitly state that their contribution is the characterization of relative placement repeatability and calibrated relative prediction, not absolute target placement. They emphasize that the reported sub-millimeter precision is modest compared to established inkjet or acoustophoretic methods but represents a critical step for a clog-resistant, nozzle-free method.

The work identifies the limits imposed by steering angle, separation distance, and actuation regimes. It concludes that reducing the separation distance ( $L$ ) toward the millimeter scale is a primary route to improving placement accuracy by suppressing geometric amplification of errors and post-impact displacement effects. The study provides the foundational metrics necessary for future work aiming at absolute placement control in SAW-based additive manufacturing.

Technical Summary: Predicting Droplet Landing in Angle-Controlled Surface Acoustic Wave Jetting

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