# Droplet and jet morphologies using solenoid microvalves for controlled fluid dispensing

Maxime Facon, Yahya Rharbi, Pierre Batocchi, Albert Magnin, Nadia El Kissi

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Maxime Facon, Yahya Rharbi, Pierre Batocchi, Albert Magnin, Nadia El Kissi. Droplet and jet morphologies using solenoid microvalves for controlled fluid dispensing. Chemical Engineering Science, 2025, 318, pp.122118. 10.1016/j.ces.2025.122118. hal-05145610

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# Droplet and jet morphologies using solenoid microvalves for controlled fluid dispensing

Maxime Facon<sup>a,b</sup>, Yahya Rharbi<sup>a,*</sup>, Pierre Batocchi<sup>b</sup>, Albert Magnin<sup>a</sup>, Nadia El Kissi<sup>a,*</sup>

## ARTICLE INFO

**Keywords:**

Solenoid microvalve

Dispensing valve

Drop formation

Jet breakup

Time-frequency operational diagrams

## ABSTRACT

This study explores the controlled generation of droplets and jets through magnetic solenoid microvalves, a key process for achieving tailored fluid morphologies that are essential for applications in bioprinting, printed electronics, and automotive painting. High-speed imaging of droplet and jet formation using aqueous Newtonian solutions of alkali-swellable emulsion (ASE) was conducted across varying actuation frequencies, valve opening times, and applied pressures. The imaging data enabled the establishment of operational diagrams by defining key parameters, including valve opening time, actuation frequency, and applied pressure – to map the conditions required to produce diverse morphologies, including single droplets, droplets with satellites, multi-droplet formations, and continuous jets. Quantitative analyses reveal the transitions between regimes, driven by Rayleigh-Plateau instabilities and mechanical perturbations, shaping droplet elongation, jet fragmentation, and satellite formation. The microvalve operation is modeled by integrating electrical behavior, temporal magnetic field evolution, mechanical forces, and viscous effects. This predictive framework accurately maps flow rates and identifies the transition zones between different ejection regimes. By correlating droplet and jet morphologies with governing instabilities, this work enhances the understanding of solenoid microvalve dynamics, offering valuable insights for optimizing fluid dispensing in precision-driven technologies.

## 1. Introduction

The controlled generation of droplets or jets, ranging from nanoliter to microliter volumes at high frequencies, is a fundamental requirement in advanced printing technologies.(Carou-Senra et al., 2024; Dong et al., 2006; Hoath, 2016; Lefebvre and McDonell, 2017; Li et al., 2015; Lin and Reitz, 1998; Lohse, 2022; Ng and Shkolnikov, 2024; Ozbolat et al., 2017; Tripathi et al., 2025) This level of precision is vital for applications where the accurate and consistent placement of fluids in two or three dimensions determines both the performance and functionality of the final product. Technologies such as inkjet, 3D bioprinting, multi-tone automotive painting, and printed electronics exemplify the increasing need for robust methods capable of delivering fluids with tailored properties under stringent conditions. For instance, 3D bioprinting requires the deposition of bioactive materials with minimal stress to maintain cellular viability, (Chen et al., 2022; Forget et al., 2017; Weygant et al., 2024a) pharmaceutical applications demand precise droplet control for accurate dosage and effective drug delivery,

(Bonhoeffer et al., 2017; Carou-Senra et al., 2024; Tam et al., 2021) whereas printed electronics depend on uniform droplet formation to ensure circuit integrity.(Islam et al., 2024; Khan et al., 2020) Similarly, automotive painting involves the injection of fluids with high solid content to achieve optimal coverage and gloss.(Pendar et al., 2022).

Several techniques have been developed to address this need, including continuous inkjet (CIJ) and drop-on-demand (DOD) methods. (Cao et al., 2024; Dong et al., 2006; Gudapati et al., 2016; Li et al., 2015; Mishra et al., 2010; Nasehi et al., 2023; Phung and Kwon, 2019; Shah et al., 2021) DOD technology, in particular, ejects droplets only when required, providing precise control over droplet volume and placement. (Hoath, 2016; Tam et al., 2021) In contrast, CIJ operates under constant pressure, enabling higher flow rates and continuous operation. Droplet formation in CIJ systems is achieved through forced perturbations, typically induced by piezoelectric actuation. Among the emerging solutions, dispensing microvalves have demonstrated significant potential, meeting several key demands in these applications.(Bonhoeffer et al., 2017; Tam et al., 2021; Chen et al., 2022; Weygant et al., 2024b; Nasehi

<footnote><sup>a</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, F-38000 Grenoble, France</footnote>

<footnote><sup>b</sup> Nippon Paint Automotive France, ZA Champ 7 Laux, 38190 Le Champ-Près-Froges, France</footnote>

<footnote>* Corresponding authors at: Univ. Grenoble Alpes, CNRS, Grenoble INP, LRP, F-38000 Grenoble, France.</footnote>

<footnote>E-mail addresses: yahya.rharbi@univ-grenoble-alpes.fr (Y. Rharbi), albert.magnin@univ-grenoble-alpes.fr (A. Magnin), nadia.elkissi@univ-grenoble-alpes.fr (N. El Kissi).</footnote>
et al., 2023; Mutschler et al., 2014; Ng et al., 2017; Kang et al., 2024; Chen et al., 2023) These dispensing valves can be actuated either by piezoelectric(Bonhoeffer et al., 2017) or magnetic mechanisms. (Mutschler et al., 2014; Nasehi et al., 2023) The magnetically actuated valves, use a solenoid to drive a plunger that opens and closes the nozzle at high frequencies, typically in the microsecond to millisecond range. (Mutschler et al., 2014; Nasehi et al., 2023) This design allows for the controlled ejection of droplets or jets under constant pressure, offering flexibility in handling a wide range of fluid properties, including high-viscosity and rheologically complex fluids. The resulting morphologies, from isolated droplets to continuous jets and from satellite formation to jet fragmentation, are governed by parameters such as nozzle geometry, applied pressure, and actuation frequency.

Although the utility of these technologies is well established, the diversity of the morphologies they generate remains incompletely understood.(Chen et al., 2023; Kang et al., 2024; Mutschler et al., 2014; Zhao et al., 2017) This gap stems from a lack of comprehensive studies that systematically examine the underlying mechanisms and operational parameters. This is particularly important given that the morphologies produced by solenoid microvalves exhibit complex dynamics influenced by multiple factors.(Castrejón-Pita et al., 2012) Among these, Rayleigh-Plateau instabilities, driven by capillary forces, play a dominant role in controlling the jet fragmentation and the size and spacing of the resulting droplets.(Rayleigh, 1878) Additional perturbations, such as mechanical vibrations induced by actuator motion, pressure fluctuations at the nozzle exit, and disturbances propagating along the jet, further complicate the dynamics.(Kalaaji et al., 2003; Lopez et al., 1999; Maîtrejean et al., 2021) These perturbations can amplify, resulting in the formation of satellite droplets or irregular morphologies. To control the produced morphologies it is crucial to appreciate these behaviors, through the monitoring and analysis of the interplay between fluid properties, fluid dynamics and process parameters, that remains incompletely understood.(Nasehi et al., 2023) Gaining such insights is essential for optimizing the performance of solenoid microvalves and achieving precise and reproducible control over calibrated droplet formation for target applications, particularly for fluids with complex viscosities and rheological properties.(Chen et al., 2023) For instance, in the case of bioprinting, some studies have focused on solenoid-type microvalves, examining valve dynamics and predicting the flow rate in specific configurations in the steady state regime.(Mutschler et al., 2014) Other investigations have centered on the droplet generation mechanisms in valves similar to those studied in this work, aiming to visualize and simulate the formation of individual droplets under the shortest opening times achievable by these devices (on the order of a few hundred microseconds).(Nasehi et al., 2023) However, these studies do not fully capture the complex dynamics associated with valve opening and closure. In particular, they overlook the influence of the magnetic field, valve response, and fluid viscosity, key factors that impact not only overall jet formation but also, more specifically, the generation of individual droplets.(Chen et al., 2023; Nasehi et al., 2023; Mutschler et al., 2014) Even more critical is the ability to predict and control single-droplet formation using this valve, as this process occurs exclusively within the transient regime, where opening and closing dynamics dominate. Therefore, a deeper understanding of the relationship between valve dynamics and droplet formation is essential. Moreover, the transition to stable jetting regimes remains poorly understood, particularly regarding how instabilities dissipate during jet formation in solenoid microvalves. A more comprehensive understanding of these transient phenomena is crucial to achieve precise, stable, and predictable performance, effectively bridging the gap between droplet-based and jet-based applications.

This study systematically investigates the mechanisms governing droplet and jet formation in solenoid microvalves, with a particular focus on the interplay between fluid properties, valve dynamics, and operating parameters. High-speed imaging is used to capture the morphologies of ejected fluids under various actuation conditions, allowing

for the identification of distinct ejection regimes and their transition boundaries. These observations are synthesized into operational diagrams that map regime transitions as functions of valve opening time, actuation frequency, and applied pressure.

To deepen the understanding of these regimes, quantitative analyses are conducted to determine droplet volume, jet velocity, and effective flow rates, based on image analysis and gravimetric measurements. The dynamics of jet breakup and satellite formation are further interpreted using linear stability theory, incorporating relevant dimensionless parameters such as the Weber and Ohnesorge numbers. Additionally, a comprehensive model of the microvalve operation is developed, integrating electromagnetic actuation, valve mechanics, and fluid flow across the different internal geometries. This model accurately predicts the flow behavior in both steady-state and transient regimes, providing valuable insights into valve response time and actuation dynamics. These coupled approaches form a unified framework for describing and controlling fluid ejection in solenoid microvalves. The findings define critical parameters for optimizing droplet and jet morphologies in precision dispensing applications including bioprinting, printed electronics, and industrial coatings.

## 2. Material and methods

### 2.1. Materials

The ASE (Rhéovis AS 1130, supplied by BASF) used in this study is an alkali-swellable emulsion, composed of a polyacrylic acid-based copolymer provided as a 100 nm particle emulsion with an acidic pH of 4 and a concentration of 30 %. This was used as supplied. A 2-dimethylaminoethanol (DMAE) solution from Sigma-Aldrich was used for pH adjustment, and double-deionized (DDI) water was used to dilute the ASE.

### 2.2. Solution preparation

The ASE emulsion was diluted with DDI water to a concentration of 0.17 %. Ionization of polyacrylic acid was achieved by dropwise addition of a 10 % DMAE solution under continuous stirring using an IKA C-MAG HS 4 magnetic stirrer. The pH of the solution was adjusted to 9, exceeding the pKa of the polyacrylic acid (pKa = 6.5), to ensure complete ionization of the acrylic acid groups. The density of the mixture is very close to $ρ = 1000 \text{ kg/m}^3$ at 23 °C. The 0.17 % ASE solution was selected as a model Newtonian fluid to allow for a simplified analysis of microvalve behavior. At concentrations above ~ 0.3 %, ASE exhibits pronounced shear-thinning and yield-stress characteristics. To avoid the complexity introduced by such nonlinear rheological behavior, the concentration was kept low to ensure Newtonian behavior within the operational shear range.

### 2.3. Rheometry

Two rheometric techniques were employed. Rotational rheometry was used to determine the behavior of the fluids at low shear rates and to characterize the transient response of fluids in correlation with nozzle dynamics and the rheometer characteristic response time. Capillary rheometry was used to investigate fluid behavior at high shear rates, representative of process conditions.

Rotational rheometry measurements were performed using two rheometers, the DHR3 and the ARES G2, both from TA Instruments. The ARES G2 was equipped with a cone-plate geometry, using a 25 mm diameter cone with an angle of 1°59'15'' and a truncation gap of 0.042 mm. For the DHR3, a Couette geometry was used, consisting of a rotor with a diameter of 28 mm and a stator with a diameter of 30 mm and a height of 42 mm, resulting in a 1 mm gap between the rotor and the stator. The DHR3 rheometer was used to measure the viscosity under steady-state conditions at constant shear rates ranging from $10^{-3}$ to $10^3$
$s^{-1}$, allowing sufficient time at each shear rate to reach a steady-state regime before progressing to the next rate. Viscosity was calculated from the measured shear stress $\tau$ and the applied shear rate $\dot{\gamma}$ as $\eta = \dot{\tau}/\dot{\gamma}$. All measurements were performed at a controlled temperature of 23 °C, maintained using a Peltier effect thermoregulation system.

Stress relaxation tests were conducted using the ARES-G2 rheometer following a three-step measurement procedure: an initial shear rate step at 0.01 $s^{-1}$ followed by a high shear step at 1000 $s^{-1}$, and a final measurement step at 0.01 $s^{-1}$. The temporal evolution of stress was monitored to determine the fluid's relaxation time during the third step. A Newtonian oil with similar viscosity was used to evaluate the rheometer's response time, which was found to be approximately 40 ms. Relaxation measurements on ASE 0.17 % indicate that the fluid's relaxation time is shorter than the intrinsic response time of the rheometer (40 ms).

**Capillary rheometry.** Capillary rheometry was performed using a custom-built setup, consisting of a 0.6 l thermostated reservoir, a pressure control chamber, and interchangeable capillary dies with specific diameter D and length L. Nitrogen gas was used to apply a controlled pressure measured via a Prisma Instrument PI105 pressure sensor (0.1 to 50 bar, ±0.6 bar accuracy). The mass flow rate Q was determined by weighing the extruded fluid over time with a SCAIME AQ C3 load cell (accuracy ± 0.85 g), allowing calculation of flow rate in steady state regime for each applied pressure. Under laminar, incompressible, and isothermal flow conditions, the apparent shear rate and wall shear stress are calculated as follows:

$$ \dot{\gamma}_{app} = \frac{32Q}{\pi D^3} \tau = \frac{\Delta P_c \cdot D}{4L} \quad (1) $$

where, $\Delta P_c$ is the pressure drop across the capillary, corrected from entrance effect using the Couette-Bagley correction. (Macosko, 1994) The effective shear rate $\dot{\gamma}$ is obtained from the apparent shear rate using the Rabinowitch-Mooney correction. (Macosko, 1994)

Measurements were conducted using a 250 µm diameter capillary (±10 µm) with an L/D ratio of 200, achieving apparent shear rates from 15,000 to 500,000 $s^{-1}$. Pressures were kept within ranges to maintain low inertial effects, and measurements were valid for Reynolds numbers (Re) below 500, ensuring no transition to turbulent flow.

The results of rotational and capillary measurements are summarized in Fig. S1, which shows the evolution of shear stress as a function of shear rate over five decades. A slight shear-thinning behavior is observed at low to intermediate shear rates, while Newtonian behavior dominates at high shear rates, with a viscosity of 3.2 ± 0.32 mPa·s. It will be shown in the following that the estimated shear rate in the nozzle during valve operation, given by $\dot{\gamma} \approx 8$ V/D (where V is the flow velocity and D the nozzle diameter), is at least 420,000 $s^{-1}$ for $V > 8$ m/s and $D = 150$ µm, Newtonian behavior can be safely assumed throughout the study.

## 2.4. Surface tension measurement

The surface tension of ASE solutions was measured using the pendant drop method on a Digidrop GBX Scientific device. A droplet (~5 µl) was formed on a 0.73 mm diameter needle, and surface tension ($\gamma$) was calculated using the Young-Laplace equation from the droplet's geometry. Surface tension measurements were conducted in a temperature-controlled room maintained at 23 °C. To minimize the influence of evaporation, all measurements were performed promptly and under stable ambient conditions. Each measurement was repeated more than 10 times, and the reported values correspond to the mean values calculated from these repetitions. The error bars represent the standard deviation. The surface tension was found to be $\gamma = 70 \pm 5$ mN/m.

## 2.5. Solenoid microvalve

The study utilizes SMLD 300G electromagnetic microvalves, designed by Fritz Gyger (Thun, Switzerland) for medium to high-viscosity fluids. A schematic of the SMLD 300G microvalve is shown in Fig. 1 (left) with the typical dimensions of each component given in Table S1 in the Supporting Information. The microvalve features a conically shaped sapphire seat (a) with a 150 µm diameter, 1 mm long nozzle, serving as the closure point for a 1 mm ruby ball (b) attached to the end of a slender rod. The valve remains normally closed due to a compressed spring (c), ensuring a secure seal when inactive. At the core of the system is a flat, mobile plunger (d), measuring 1.39 mm × 2.2 mm with a length of 8 mm, which extends into the thinner rod supporting the ruby ball. This plunger moves within a cylindrical housing (e), 2.25 mm diameter and 33 mm length. A flat metallic stopper (f) measuring 2.2 mm × 1.39 mm and 10 mm in length, is positioned behind the plunger to limit its backward motion, restricting the valve's maximum opening displacement to 60 µm. The valve is actuated by a solenoid coil (g), which surrounds the channel with an inner diameter of 2.65 mm, an outer diameter of 5.65 mm, and a length of 14 mm.

Fig. 1 (right) illustrates the electrical signal characteristics and definition of representative times. The actuation of the microvalve follows a controlled cyclic sequence of opening and closing, driven by the application of a predefined electric current profile. Each cycle begins with a peak current of 1000 mA applied for 400 µs, defined as the peak time, $t_p$, to ensure rapid and efficient valve opening. After this peak period, the current is reduced to 200 mA and maintained for a holding time, $t_h$, that can be varied between 80 µs and 10,000 µs. The microvalve closes when the current is cut off, marking the end of the cycle and preparing the system for the next activation. The solenoid has a resistance $R = 6 \ \Omega$ and an inductance $L = 0.8$ mH, resulting in an electrical time constant of $L/R = 133$ µs (see Table S1).

One of the main parameters of the microvalve in this work is the opening time, $t_o$, which will be arbitrarily defined as the total duration of the applied current, expressed as $t_o = t_p + t_h$. It should be noted that the dynamic processes governing valve opening and closing may not correspond exactly to $t_o$, as these mechanical responses involve several additional factors that will be discussed later. Nonetheless, the total electrical current duration, $t_o$, will be used as a representative indicator characterizing the microvalve's opening. The pulse train frequency was systematically varied between 50 Hz and a maximum of $1/t_o$, with an upper limit of 2000 Hz under our experimental conditions.

## 2.6. Printing bench

The system includes a 1 L reservoir for the printing fluid, maintained at a controlled pressure between 0.5 and 2.5 bar. The fluid passes through a 20 µm particle filter to remove large particles and then through a heat exchanger that regulates its temperature at 23 ± 0.1 °C. A WIKA S-11 pressure sensor, with a sensitivity of ± 3 mbar, monitors the fluid pressure before it enters the electromagnetic microvalve. To account for pressure losses along various sections of the channel, the pressure used in calculations corresponds to the value measured at the sensor.

For each set of experimental conditions, pressure and opening time in particular, the ejected fluid was collected on a Sartorius BP 3100 S balance (precision 0.01 g) over a 30-second period, and the mass collected per unit time was used to quantify the printing mass flow rate, $Q_{print}$, under steady state regime.

The ejected fluid was visualized using a backlighting configuration and captured with a Phantom VEO 440 high-speed camera (4 Megapixels, 2560 x 1600 resolution), operating at 1100 frames per second at maximum resolution. A strip window was selected to achieve the best spatial resolution and a frame rate approaching 11,000 fps. This configuration provided an image every 90.9 µs, which was maintained throughout the experiment. The camera, equipped with an Irix 150 mm
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** (Left) Schematic of the solenoid microvalve of the LSMD Gyger type, depicting (a) the flat valve seat containing the nozzle, (b) the ruby valve ball, (c) the closing spring, (d) the mobile plunger, (e) cylindrical housing (f) the stationary anchor, (g) the magnetic coil and (h) the electrical cable controlling the coil. The arrow indicates the flow direction. (Right) Schematic representation of the temporal diagram illustrating the current vs. time applied during the operating cycle of the microvalve.</figcaption></figure>

Macro 1:1f/2.8 Dragonfly lens and positioned 29.5 cm from the nozzle, provides optimal focus and high-resolution imaging. Uniform illumination was achieved with a PHLOX® HSC BACKLIGHTS 10x10 cm LED panel, enhancing contrast for clear observation of fluid morphology. The captured images are in 16-bit grayscale format. The objective and camera configuration established a defined visualization field for droplet and jet observation. This field extended approximately 2 cm from the microvalve nozzle exit, capturing the jet’s emergence and progression until it left the visualization area.

## 2.7. Image analysis

It enables to characterize the dimensions of the ejected morphologies, length and width in particular, as well as their aspect ratio, volume, velocity, etc. To achieve this, the images were calibrated using the known dimensions of the microvalve, which served as a reference scale and two software programs were used for image analysis. The first analysis method involved the use of ImageJ (National Institutes of Health) to determine the primary dimensional characteristics of jets and droplets. Measurements included length and width at various positions along the jet or droplet, with a primary focus on the maximum values. The aspect ratio of these objects (length/width) was systematically calculated and analyzed. Furthermore, the morphology of the jet tail and satellite droplets was analyzed separately using the same procedure. This method also facilitated statistical counting of both primary and satellite droplets which can form depending on flow conditions and fluid properties, providing quantitative data on droplet formation and distribution. The second program, developed in Python using OpenCV, applied grayscale thresholding to distinguish droplets from the bright background, with a predefined threshold based on the average grayscale intensity across all images. After thresholding, images were binarized to detect droplet contours and calculate their volume, positions, and profiles along the jet. As with ImageJ, this second image analysis technique also enabled us to determine the aspect ratio and the count of droplets and satellites droplets.

## 2.8. Ejection protocol

For each pressure setting, ranging from 0.5 to 2.5 bar, multiple opening times ($t_o$) were selected, varying between 480 µs and 10400 µs. Each combination of pressure and opening time was tested at frequencies ranging from a lower limit of 50 Hz to an upper limit of $1/t_o$. Each visualization at the microvalve exit, started at least 5 s after the beginning of fluid ejection, to allow the system to reach a steady state. For each set of experimental conditions (pressure, frequency, opening

time), the printing mass flow rate, $Q_{print}$ was determined and several ejection cycles were recorded. At least five cycles were analyzed, thus providing an average representation of the phenomena observed.

# 3. Experimental results and discussions

## 3.1. Qualitative observations of the fluid ejection

In the following, high-speed imaging will be first used to describe the images captured during printing cycles at a fixed pressure and frequency, while varying the opening time, allowing for the identification of different ejection regimes. Next, the influence of frequency on the observed morphologies and ejection regimes will be examined, followed by an analysis of the pressure effect. Finally, an "opening time"-"Frequency" diagram will be established for each pressure level to highlight the conditions leading to different morphologies and ejection regimes.

## 3.2. Time evolution of drops and jet during the valve opening cycle

High-speed imaging was employed to capture the sequence of events in drop ejection dynamics, with frames recorded every 90.9 µs. This visualization tracks the droplet’s progression from its initial emergence at the nozzle to its movement across the observation field. Fig. 2 highlights the time evolution of droplet and jet morphologies at 0.5 bar for opening durations $t_o$ ranging from 480 µs to 2400 µs, at a frequency of 50 Hz. Additional cases for other $t_o$ values (from 480 µs to 5400 µs) can be found in Supporting Information, Fig. S2. Fig. 2a to 2e illustrate the detailed morphological changes at the nozzle exit of an ASE solution at 0.17 % concentration, under a pressure of 0.5 bar and a frequency of 50 Hz. These images provide a clear view of each stage in the ejection process for valve opening times of 480, 500, 650, 1400, and 2400 µs, with frames taken at 90.9 µs intervals. The onset of valve opening (marked as "o") is identified by the first appearance of fluid at the nozzle outlet, while valve closure ("c") corresponds to the visible thinning of the jet.

First, at short valve opening times ($t_o$), below 480 µs (not shown), no fluid is ejected, especially at a pressure of 0.5 bar. However, at 480 µs, a droplet forms at the nozzle exit, emerging in a slightly elongated shape (Image 2 on Fig. 2a). As it detaches from the nozzle, the droplet may either form a single droplet (not shown) or be followed by one or two small satellite droplets (Image 9). With a slight increase to 500 µs (Fig. 2b), the main droplet becomes more elongated (Image 4). After the microvalve closes (Image 5), the main droplet continues to progress, leaving behind a trailing filament that thinns as it extends over a few millimeters (Images 6 to 10). The trailing filament eventually detaches
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Series of images showing the ejection of ASE solution at 0.17 % concentration during valve opening cycle using a solenoid microvalve (SMLD 300G) at a pressure of 0.5 bar and a frequency of 50 Hz. The image sequence is separated by 90.9 µs intervals. The valve opening durations, from top to bottom, are (a) 480 µs, (b) 500 µs, (c) 650 µs, (d) 1400 µs, (e) 2400 µs part 1, before closing and (f) 2400 µs part 2 post closing. "o" and "c" indicate respectively the opening and closing of the micro-valve.</figcaption></figure>

from both the nozzle and the main droplet (Image 11, Fig. 2b), fragmenting into several small satellite droplets as can be seen on Image 14.

This regime, corresponding to valve opening times up to 500 µs is identified as the “single drop regime”. It can be associated to the “drop-on-demand technology”.

For intermediate $t_o$ values, higher than 650 µs and up to approximately 1400 µs, (Fig. 2c and 2d) and additional images provided in the

Supporting Information (Fig. S2), the ejection dynamics display distinct behaviors. A small, elongated jet emerges from the nozzle, exhibiting perturbations and multiple morphological heterogeneities (Images 1 to 12, Fig. 2c and 2d). Near the nozzle exit, within a region extending up to roughly 1 cm, the jet undergoes several breaking events at the leading edge, forming multiple droplets (Images 10 to 12, Fig. 2c). Following valve closure, (Image 9, Fig. 2c), the jet continues to progress, leaving a
<figure><img src="image_5.png" /><figcaption>Fig. 2. (continued).</figcaption></figure>

thin, stretched filament behind. The jet then fragments into several droplets (Images 12 to 26, Fig. 2c). The trailing filament also produces small satellite droplets as it detaches from the main jet, with all events occurring within the initial centimeter from the nozzle exit (Images 12 to 26, Fig. 2c). Similar observations can be made in Fig. 2d, the number of droplets resulting from the main jet and trailing filament fragmentation, increasing with the opening time. This can be accounted for by comparing the final images obtained at each opening time, as illustrated in the last frames of Fig. 2c and 2d and Fig. S2.

This regime, corresponding to intermediate valve opening times, from 650 µs and up to about 1400 µs, is identified as the “intermediate regime”.

Finally, at higher valve opening times ($t_0 \ge 1400$ µs), the ejection process transitions into a continuous jetting regime (Fig. 2e and Fig. S2). Spatial perturbations observed at the nozzle exit dissipate along the jet’s length, resulting in a more uniform morphology dominated by capillary-like instabilities (Images 12 to 26, Fig. 2e). Droplet breakups near the nozzle occur rapidly as the droplet emerges but become less frequent as
the jet extends further from the nozzle. However, once the jet stabilizes, at a length on the order of a couple of centimeters, the breakup frequency rises again. Following valve closure (Image 26, Fig. 2e), the thin filament near the nozzle, fragments into small satellite droplets (Images 26 to 50, Fig. 2f), while the primary jet itself breaks into larger droplets.

This regime, observed for valve opening times of 1400 µs and beyond, is identified as the ‘‘continuous jet regime’’. It is analogous to the operating principle known as ‘‘continuous ink jet technology’’.

The impact of varying valve opening times on droplet morphology is summarized in Fig. 3, capturing images just before valve closure from Fig. 2 and Fig. S2. The transition from single droplet and ultimately, to a continuous jet is clearly illustrated as the opening time increases, with individual droplets and occasional satellites at short $t_0$ (Images 1 to 3, Fig. 3), successive multiple droplets at intermediate $t_0$ (Images 4 to 6, Fig. 3), and a stable jet at high $t_0$ (Images 7 and 8, Fig. 3). This tunable control over ejection modes, from drop-on-demand to continuous jetting, highlights the system’s versatility making it suitable for various applications spanning both DOD and continuous jet printing technologies.

The same experiments were conducted over a range of frequencies from 50 Hz up to the maximum achievable frequency for each valve opening time, i.e. around $1/t_0$. Using high speed imaging, the observed ejection regimes were analysed to determine the effect of frequency at a given pressure of 0.5 bar with valve opening times ranging from 480 µs to 10,000 µs. The results are detailed in the following paragraph.

### 3.3. Opening Time-Frequency diagram for drop and jet formation

A preliminary analysis of the images obtained at different frequencies and presented in Supporting Information, Fig. S3, reveals that, for frequencies up to approximately 1000 Hz, droplet shapes and

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Series of images capturing the jet morphology at the moment of valve closure during the ejection of a 0.17 wt% aqueous ASE solution through the SMLD 300G microvalve at a pressure of 0.5 bar. The jet is shown for valve opening durations of 480, 500, 550, 650, 900, 1400, 2400, and 5400 µs (from left to right).</figcaption></figure>

ejection patterns within each regime remain largely unaffected, maintaining consistent behavior with only minor fluctuations. As shown in Fig. S3(c) and S3(d), when the ejection frequency exceeds 1000 Hz, the distance between successive jets, originating from consecutive actuation cycles, decreases to the point where dynamic interactions between them becomes unavoidable. In this regime, the trailing end of a jet, which decelerates as it moves away from the nozzle, can be overtaken by the faster leading front of the following jet, resulting in jet-jet collisions. These interactions lead to instabilities such as droplet coalescence, trajectory deviation, fragmentation, or splashing. This behavior is referred to as correlated jets. The emergence of correlated jets marks a transition from isolated jetting events to a regime where the dynamics of each ejection are influenced by those that precede and follow it.

These findings are summarized in the ‘‘opening time’’–‘‘frequency’’ diagram shown in Fig. 4 corresponding to a pressure of 0.5 bar. In this diagram, *uncorrelated jets* are represented by dark symbols, while *correlated jets*, typically occurring at frequencies above 1000 Hz, are indicated by open symbols. Notably, the ejection regimes identified in the previous section remain evident and are largely independent of frequency, further highlighting the robustness of the observed regimes classification, i.e.

- *no ejection regime*, observed for opening time below 480 µs at 0.5 bar, likely due to insufficient valve opening or inadequate fluid flow establishment time;

- *single drop ejection regime*, observed for 480 µs and 500 µs opening times, characterized by the ejection of one or, at most, two drops, often accompanied by several satellite droplets;

- *intermediate regime*, up to 2000 µs, where a highly perturbed jet rapidly breaks into multiple droplets, with the jet length increasing as the holding time increases;

- *continuous jetting regime*, marked by a long, homogeneous filament that exhibits capillary-like instabilities, resulting in periodic breakage into droplets.

### 3.4. Effect of pressure on the observed morphologies and ejection regimes

The same set of experiments was performed at higher pressures and

<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Operational diagram of the solenoid microvalve for ASE solution at 0.17 % and 0.5 bar. The points represent different tests conducted at specific opening times ($t_0$) and frequencies. Solid dots correspond to uncorrelated ejections, while open circles indicate correlated jets. The dashed lines distinguish various regimes: the black line marks the boundary of the no ejection region, the area between the black and red lines defines the single droplet regime, the region between the red and green lines indicates the intermediate zone where the jet transitions to a structurally homogeneous state, and beyond the green line, the continuous jetting regime is observed with a stable, uniform jet.</figcaption></figure>
the morphologies observed at 0.5 bar, including single drop formation, the intermediate regime and continuous jetting, remain consistently present at 1 and 2 bar as shown in Supporting Information Figs. S4(a) and S4(b). As a result, the “opening time”-“frequency” diagrams appear generally similar across all pressures from 0.5 to 2 bar (Fig. 4 and Fig. S4).

However, the valve opening time marking the transition between different regimes shift decreases with increasing pressure. Specifically, the transition from the single drop regime to the intermediate regime decreases from approximately 550 µs at 0.5 bar to below 480 µs at 2 bar. Similarly, the transition from the intermediate regime to the continuous jetting regime shifts from around 2000 µs at 0.5 bar to approximately 650 µs at 2 bar. These observations highlight the strong influence of pressure on the dynamics of jet formation and regime boundaries.

The effect of pressure is further illustrated in Fig. S5, which presents images captured at various pressures and valve opening times. For instance, at a fixed opening time of $t_0 = 480$ µs, the ejected fluid volume clearly increases with pressure. As the pressure rises from 0.5 bar to 2 bar, a visible transition from the single drop regime to the intermediate regime is observed confirming that higher pressures promote earlier regime transitions by enhancing fluid delivery and jet development.

These observations underscore the critical role of pressure in determining ejection regimes, complementing the previously established influences of valve opening time and frequency. By fine-tuning these parameters, it becomes possible to control fluid dynamics within the nozzle and tailor the system for specific printing applications. More specifically, our results demonstrate that, at a given pressure and frequency, valve opening time predominantly governs the ejection regime, defining the transition from single-drop formation to continuous jetting. In contrast, frequency exerts only a limited influence on the observed morphologies and does not affect the occurrence of ejection regimes. However, at frequencies exceeding approximately 1000 Hz, disturbances such as droplet interactions, merging, and trajectories interference were observed, phenomena collectively referred to as correlated jets. Operating microvalves at such high frequencies inherently involves fluid relaxation dynamics, particularly as the frequency approaches the inverse of the fluid’s characteristic relaxation time. Nonetheless, morphological analysis in this study did not reveal any definitive signatures of polymer relaxation influencing the frequency-dependent behavior. This suggests that relaxation effects may be either negligible or masked under the current experimental conditions. A direct measurement of relaxation times corresponding to this frequency range is inaccessible with the rheometrical setup, which restricts relaxation measurements to a minimum time of 40 ms, corresponding to a maximum frequency of 25 Hz (see Materials and Methods).

## 4. Quantification of the fluid ejection

These observations emphasize the significance of process control parameters, particularly the dynamics of microvalve operation (opening and closing), and their direct influence on fluid flow behavior and ejection velocity. To quantitatively assess these effects, droplet and jet morphologies, specifically their volume and displacement during valve actuation, were analyzed through image processing techniques, enabling a time-resolved study of their evolution throughout the process. To perform this analysis, custom scripts were developed using Python and OpenCV. These tools were employed to detect and characterize droplet shapes, allowing the determination of droplet diameters at various distances from the nozzle exit. Assuming cylindrical symmetry, these measurements enabled the calculation of surface area and volume for each droplet. The program also distinguishes between primary droplets, satellite droplets, and jets, providing a comprehensive morphological classification of the ejected fluid structures.

In the following sections, we first present the results of the ejected volume calculations. We then analyze and discuss the corresponding flow rates and ejection velocities associated with each actuation cycle.

Finally, we quantify the droplet aspect ratio within the single drop regime and characterize jet fragmentation behavior observed during the continuous jet regime.

### 4.1. Ejected volume during microvalve operation

Fig. 5a presents the calculated volume of fluid ejected during a single printing cycle as a function of time, for a 0.17 % ASE solution at a constant pressure of 0.5 bar, across various valve opening times $t_o$. The results show a gradual increase in ejected volume over time, reflecting the period during which the valve is open and fluid is actively flowing. This is followed by a maximum, after which the curve either plateaus or decreases, indicating that fluid ejection has ceased due to valve closure. From these curves, two key performance indicators of the valve are extracted: the maximum ejected volume $v_{max}$ and the corresponding time at which it is reached, denoted $t_o^e$.

In Fig. 5b, $v_{max}$ is plotted as a function of the imposed valve opening time ($t_o$), with the evolution of $t_o^e$ shown in the inset. The data reveal that both $v_{max}$ and $t_o^e$ increase almost linearly for $t_o$ values above 600 µs.

<figure><img src="image_8.png" /></figure>

Fig. 5. (a) The ejected volume, derived from image analysis, as a function of time starting from the fluid exit at the nozzle. The data correspond to the ejection of an ASE solution at 0.17 % concentration under a pressure of 0.5 bar. Each curve represents a different valve opening time ($t_o$), as specified in the legend. (b) The maximum ejected volume $v_{max}$, defined as the volume at which the curves in Fig. 5a reaches a plateau and/or no longer increase, is plotted against $t_o$. The insert shows the time $t_o^e$ at which $v_{max}$ is reached (determined in Fig. 5a), plotted as a function of $t_o$. The dashed line is a guide for the eyes. The solid line corresponds to the simulated ejected volume as a function of $t_o$, based on Equations (3) and (4) (see below). The simulation parameters include a viscosity $\eta = 3.2$ mPa.s and the known electrical and mechanical properties of the microvalve (see Materials and Methods).
However, for shorter $t_o$ values, around 500 µs or less, both metrics drop sharply. This important and unexpected result indicates that, for valve opening times of the order of 500 µs or less, the maximum volume is ejected before the nominal valve closure time. In other words, the effective valve opening time ($t_o^e$) is shorter than the imposed $t_o$, suggesting delayed fluid response or incomplete valve actuation at $t_o$ values around 500 µs or less. Consequently, when operating in this regime, $t_o^e$ should be considered as a more accurate indicator of valve behavior than the programmed $t_o$.

Fig. 5b also shows that both $v_{max}$ and $t_o^e$ approach zero around $t_o = 450$ µs, marking the threshold below which the valve no longer opens sufficiently to sustain fluid ejection. This behavior suggests that the actual valve dynamics deviate from the programmed electrical signal (illustrated in Fig. 1b), particularly for $t_o$ values below 600 µs. The discrepancy highlights that the valve's response is not solely governed by the applied current, but is also influenced by a combination of electrical and mechanical factors, including coil inductance and resistance, spring stiffness, and the fluid's properties. Importantly, this deviation cannot be attributed to the inductive–resistive time constant of the solenoid alone (133 µs, see Materials and Methods), nor to fluid viscoelasticity, as rheological measurements confirm that the 0.17 % ASE solution behaves in a Newtonian-like manner. Furthermore, the shape of the $v_{max}$ vs. $t_o$ curve offers valuable insights into the valve's dynamic behavior. In particular, the sharp onset of ejection around 450 µs suggests that delayed closure, in addition to delayed opening, may contribute significantly to the observed trends. These results underscore the complexity of valve-fluid interactions and the importance of considering mechanical lag and fluid inertia in high-speed microvalve operation.

## 4.2. Flow velocity during ejection process

The flow velocity, measured from image analysis during an opening cycle, is evaluated in the fully developed jet regime i.e. continuous jet. During each valve opening cycle, the instantaneous flow velocity is determined as the speed of the moving droplet or jet, calculated based on the distance traveled between two consecutive frames (90.9 µs). Measurements are taken from the moment the valve opens until the droplet exits the field of view. For individual droplets, the velocity is measured directly; for multi-droplet formations, it is based on the leading droplet; and for jets, it is measured at the head of the jet. The results for the ASE solution at 0.17 % at high $t_o$ values are presented in Fig. 6a. It is noted that the velocity level is relatively high.

## 4.3. Determining the effective flow rate $Q_{eff}$ during ejection process

Linear trend of $v_{max}$ vs. $t_o$ for $t_o > 600$ µs (Fig. 5b) likely indicates the

establishment of a steady-state flow regime between the microvalve and the ejected fluid, making this threshold suitable for quantifying the flow rate by dividing $v_{max}$ by $t_o^e$. In contrast, the pronounced nonlinearity indicated by a sharp discontinuity in the curve $v_{max}$ vs. $t_o$ for $t_o < 600$ µs suggests that the system remains in a transient regime, complicating direct flow rate quantification.

At a given pressure, the flow rate $Q_{print}$ accounting for the volume ejected during the opening cycles is determined by weighing the mass of fluid ejected over a period of 30 s, with the valve opening frequency arbitrarily set at frequency f (see Materials and Methods). Consequently, the effective flow rate during a printing cycle, $Q_{eff}$, is calculated by the relation:

$$Q_{eff} = \frac{Q_{print}/f}{t_o^e} \qquad (2)$$

$t_o^e$ being the fluid ejection time during valve opening (see §2.1 and Fig. 5). Let's recall that for opening times over 600 µs, $t_o^e$ approaches $t_o$ (see §2.1 and Fig. 5).

$Q_{eff}$ is represented in Fig. 6b as a function of pressure. As expected, it is in complete agreement with results on velocity discussed earlier (Fig. 6a). Throughout the investigated pressure range, the evolution of $Q_{eff}$ does not follow a strictly linear Poiseuille-like behavior. Instead, the flow exhibits a continuously curved trend with a downward concavity, suggesting a deviation from purely viscous behavior. At higher pressures, the data suggest a tendency toward a quadratic evolution ($Q_{eff} \propto \sqrt{P}$), which is characteristic of inertial-dominated flow as described by Bernoulli's theorem. (White, 2011) This behavior reflects a gradual transition in which both viscous and inertial effects coexist and contribute to the overall flow dynamics in the microvalve dispensing process.

## 4.4. Analysis of single drop regime

Fig. S5(a) shows the images obtained during single drop regime for the 0.17 % ASE solution at 480 µs and different pressures. At 0.5 bar, the ejected fluid predominantly results in the formation of a single droplet, with a length-to-diameter ratio close to 1 (Fig. 7). As pressure increases, we can observe on Fig. S5(a) that the length of the ejected fluid increases. The aspect ratio derived from the image analysis, monotonically increases with increasing pressure with values reaching up to 10 at 2.5 bar (Fig. 7).

After valve closure, the fluid near the nozzle undergoes stretching and thinning, forming a trailing filament before eventually detaching from the nozzle (Fig. S5(a)). The length of this filament increases with pressure. Meanwhile, the primary ejected fluid breaks into several droplets within the first centimeter from the nozzle, with the droplet

<figure><img src="image_9.png" /><figcaption>Fig. 6. (a) Flow velocity derived from image analysis as a function of pressure for the ASE solution at 0.17 % at high $t_o$ values. (b) Flow rate $Q_{eff}$ measured using equation (1), as a function of pressure for the ASE solution at 0.17 %, calculated in the steady-state regime at high $t_o$ values. The dashed lines represent fits to Equation (3) (see below) for a fully opened nozzle. The calculation incorporates the detailed characteristics of the microvalve, (see Table S1) assuming a viscosity ($\eta$) of 3.2 mPa·s.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Aspect ratio of the main droplet as a function of pressure for the ASE solution at 0.17 % and an opening time of 480 µs. The aspect ratio is defined as the ratio of droplet length to diameter. The inset shows the number of satellite droplets following the main droplet as a function of pressure.</figcaption></figure>

count scaling with the length of the ejected fluid and thus increasing with pressure (Fig. S5(a)). As a result, the regime allowing the generation of a single droplet is restricted to a narrow range of opening time to (~480 µs) and low pressures (~0.5 bar) in the case of 0.17 % ASE. At the same time, the trailing filament fragments into smaller satellite droplets, with their number increasing as pressure rises. At 0.5 bar, only one or two satellites form, whereas at 2.5 bar, more than 15 satellites are generated at 480 µs (Fig. 7 and Fig. S5(a)).

According to reference(Castrejón-Pita et al., 2012), the breakup conditions of the trailing filament can be assessed using its aspect ratio (Fig. 7 and Fig. S5(a)) and the Ohnesorge number, which characterizes the balance between viscous and capillary forces Oh = $\frac{\eta}{\sqrt{\rho v R}}$, with R radius of the filament and $\gamma$ the surface tension. For the present system, the Oh number is approximately 0.041, and the aspect ratio of the trailing filament is estimated to exceed 50, as observed in Fig. S5(a). These values place the filament firmly within the breakup regime described in the diagram from reference.(Castrejón-Pita et al., 2012) However, this analysis is less straightforward when considering the breakup of the main droplet at high pressures (above 1.5 bar). In this regime, the droplet exhibits an aspect ratio between 5.8 and 10, positioning it within the transition zone between the breakup and collapse regimes in the diagram from reference.(Castrejón-Pita et al., 2012) This transitional behavior makes its breakup dynamics less predictable within this framework.

### 4.5. Analysis of the jetting regime

At higher $t_0$ values (Fig. S5), during the valve opening stage, the fluid gradually transitions into a stable jetting regime, generating jets extending beyond 20 mm in length. Fig. 8 illustrates the jet length ($L_{jet}$) from its emergence at the nozzle up to the end of the valve opening cycle for the 0.17 wt% aqueous ASE solution flowing at 0.5 bar for an opening time of 5400 µs, with 90.9 µs between frames. The jet length increases progressively over time, reaching around 20 mm at 5000 µs (Image 8b), beyond which the length shows only minor variations. This behavior suggests a characteristic breaking length, potentially influenced by Rayleigh-Plateau instabilities that govern jet stability under these

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** (a) Series of images showing the ejection sequence of a 0.17 wt% aqueous ASE solution at 0.5 bar using a solenoid microvalve (SMLD 300G). The images capture the progression from the start of ejection to valve closure for an opening duration of 5400 µs, with 90.9 µs between frames. (b) Jet length (●) as a function of time measured through image analysis starting at the jet's exit from the nozzle. The count of jet breakup events over time is represented by (○).</figcaption></figure>

conditions.

During the jet's progression, breakups are consistently observed, primarily initiating from the jet front (head) and following two distinct regimes, as evidenced by the data presented in Fig. 8a. The number of breakup events ($n_{break}$) is determined by analyzing successive images and counting the instances where a droplet detaches from the continuous jet. Fig. 8b shows how the number of jet breakup events ($n_{break}$) evolves over time. In the initial phase, up to 5000 µs, breakups occur at a relatively steady rate, with a breakup frequency ($f$) of approximately 2.63 events per millisecond, as indicated by the slope of Fig. 8. This steady behavior shifts beyond 5000 µs, where $f$ increases significantly to about 9 events per millisecond, corresponding to the jet reaching a more stable length. In this second regime, the breakup rate closely matches the jet's forward velocity, sustaining a semi-steady length. In contrast, the earlier phase with a lower breakup rate allows the jet to continue extending.

The perturbation wavelength $\lambda$ is estimated directly from image analysis by measuring the distance between consecutive necking points (troughs) along the progressing jet. Alternatively, a more precise value of lambda is determined by finding the best linear fit for the plot of $\lambda \cdot n_{break} + L_{jet}$ vs. time, optimizing for the value of $\lambda$ that minimizes the $R^2$. These analysis leads to $\lambda = 780 \ \mu m \pm 70 \ \mu m$ for ASE 0.17 % and 0.5 bar. The corresponding dimensionless wavenumber $k = 2\pi R/\lambda = 0.64 \pm 0.06$, using $R$ as the nozzle radius 75 µm.

Linear stability analysis of capillary instabilities in free jets relies on solving the dispersion equation, which describes the exponential growth of surface perturbations. This equation provides the perturbation growth rate, $\beta$, as a function of the wavenumber, $k$, enabling the identification of the optimal wavenumber, $k^*$, corresponding to the maximum growth rate, $\beta^*$ (Supporting Information Section S1).(Levich and Tobias, 1963;
Weber, 1931) Fluid viscosity and inertial effects are introduced through dimensionless parameters, specifically the Ohnesorge number and the Weber number (We = $\frac{\rho v^2 R}{\gamma}$), respectively (Section S1). Using the fluid viscosity at high shear rates (3.2 mPa·s), surface tension (70 mN/m), and air density $\rho_a$ at 20 °C (1.2 kg/m³), the Ohnesorge number is estimated to be Oh = 0.041, while the Weber number is calculated as We = 68.5. These parameters lead to an optimal wavenumber $k^*=0.68$, (Section S1) which closely aligns with the experimentally determined value of k = 0.64.

The breakup process of the stable jet significantly changes following valve closure, as illustrated in Fig. 2f, for the 0.17 wt% ASE solution at 0.5 bar with $t_0$ of 2400 µs. Several stages are observed: the trailing filament progressively thins (Images 27 to 30) and eventually breaks at the nozzle exit (Image 30), allowing the jet to advance further. As the jet advances, it undergoes fragmentation into multiple droplets, a process consistent with free jet breakup mechanisms. (Castrejón-Pita et al., 2012; Hoath et al., 2013; Planchette et al., 2019) Each primary droplet is often accompanied by small satellite droplets (Images 30 to 50). Similarly, the trailing filament fragments into multiple smaller satellite droplets (Images 34 to 50). The breakup dynamics of both the main jet and the trailing filament are assessed through the aspect ratio vs. Ohnesorge number diagram from references. (Castrejón-Pita et al., 2012; Hoath et al., 2013) With a calculated Oh of 0.041 and an aspect ratio exceeding 50, both fall within the breakup regime, consistent with the observed fragmentation process. (Castrejón-Pita et al., 2012; Hoath et al., 2013).

## 5. Numerical predictions of the flow rate and velocity

A detailed understanding of the flow behavior within the microvalve requires analyzing both the valve opening dynamics and the fluid flow through its various sections, as illustrated in Fig. 1 and Table S1 in supplementary information. This analysis enables numerical predictions of the time-dependent flow rate, Q(t), and velocity, which can then be compared with experimentally measured values, namely the effective flow rate $Q_{eff}$ and velocity under steady-state conditions (Fig. 6). Such comparisons serve as a reference for evaluating the microvalve's performance. The opening dynamics are characterized by the gap $x(t)$ between the ruby ball and its seat, which ranges from 0 µm when the valve is closed to $x_0$ = 60 µm when fully open (see Materials and Methods, Solenoid microvalve).

### 5.1. Steady-State analysis: Contributions to pressure drop and flow velocity

To begin assessing the microvalve's performance, the steady-state regime is considered, where the gap is fixed at $x_0$ and both the flow rate and velocity remain constant. In this configuration, numerical predictions of flow rate and velocity are compared to the experimentally measured flow rates $Q_{eff}$ and corresponding velocity under steady-state conditions (Fig. 6), providing a benchmark for evaluating the microvalve's operational characteristics.

The volumetric flow rate $Q(t)$ is related to the pressure drop $\Delta P$ between the inlet and outlet through energy conservation principles. Under laminar flow conditions, this relationship is described by Equation (3). The equation accounts for pressure losses due to viscous dissipation (scaling linearly with $Q$) throughout the valve's geometry, as well as kinetic energy contributions at the inlet and outlet (scaling with $Q^2$). (Planchette et al., 2019).

$$ \frac{8\eta L_{plu} Q(t)}{\pi \left( R_{cyl}^4 - R_{plu}^4 + \frac{R_{cyl}^2 - R_{plu}^2}{\ln(R_{cyl}/R_{plu})} \right) (1 - \alpha)} + \frac{6\eta Q(t)}{\pi x(t)^3} \ln\left(\frac{R_{disc}}{R_{noz}}\right) + \frac{8\eta Q(t) L_{noz}}{\pi R_{noz}^4} \\ + \frac{\rho Q(t)^2}{2\pi^2 R_{noz}^4} = \Delta P_{total} \quad (3) $$

In the steady-state analysis, both the gap $x(t)$ and the flow rate $Q(t)$ are held constant at $x_0$ = 60 µm and $Q$, respectively.

The first term in Equation (3) corresponds to the pressure drop due to flow through the flat section formed between the plunger (d in Fig. 1, Table S1) and the stopper (f in Fig. 1). This region consists of a 1.2 mm wide flat section within a cylindrical channel of 2.25 mm diameter (e in Fig. 1) spanning a total length $L_{plu}$ of 18 mm (comprising 8 mm from the plunger and 10 mm from the stationary anchor, see Fig. 1 and Table S1). The flat portion effectively reduces the flowable cross-section to approximately 1 - $\alpha$, where $\alpha \approx 0.2$, meaning about 80 % of the full circular area is available for flow. Despite the flat geometry, this section is approximated as annular flow between two concentric cylinders with inner radius $R_{plu}$ = 0.6 mm and outer radius $R_{cyl}$ = 1.125 mm. Due to the relatively large gap in this section compared to other flow regions, the associated pressure drop is negligible and can be safely omitted without compromising the overall analysis.

The second term in Equation (3) represents the pressure drop through the narrow flow region between the ruby ball (b, Fig. 1) and the valve seat (a, Fig. 1). The converging flow in this region is modeled as flow between two parallel discs of radius $R_{disc}$ separated by a gap distance $x_0$ (a and b in Fig. 1) corresponding to a fully open valve. The radius $R_{disc}$ is estimated to lie between the nozzle radius (75 µm) and the ruby ball radius (500 µm) with 250 µm used as a representative value in the analysis. Evaluating the model with different values of $R_{disc}$ provides insight into its sensitivity and helps establish a confidence range for the predictions.

The third term in Equation (3) addresses the pressure drop in the final cylindrical section of the nozzle (a, Fig. 1), characterized by a radius $R_{noz}$ = 75 µm and length $L_{noz} \approx 1$ mm.

The fourth term in Equation (3) captures the kinetic energy contributions at the inlet and outlet of the microvalve. However, due to the 15-fold difference in diameters between these sections, the kinetic energy at the inlet is roughly 50,000 times smaller than at the outlet. Consequently, the inlet term is negligible, and only the outlet contribution is retained in the calculations.

The numerical evaluation of Equation (3) indicates that the dominant contributions to the pressure drop come from the 2nd, 3rd and 4th terms in Equation (3),) that is, the Poiseuille flow resistance in the converging disc region and nozzle, and the kinetic energy contribution. This outcome aligns well with experimental trends and supports the model's accuracy under the given test conditions. As shown in Fig. 6b, this analysis yields flow rate predictions that align closely with the experimentally measured effective flow rates in the steady-state regime across the full tested pressure range (0.5 to 2.5 bar). The agreement is particularly strong when using a viscosity of 3.2 mPa·s and setting $R_{disc}$ = 250 µm.

Similarly, the flow velocity computed as $(Q/4\pi)R_{noz}^2$ offers a reasonable estimate of the measured flow velocity in the steady-state regime (Fig. 6a) using the same parameters (viscosity 3.2 mPa·s and $R_{disc}$ = 250 µm). While the fit for velocity is slightly less precise than for the flow rate, the model remains a good approximation. The observed discrepancies likely stem from an underestimation of the jet velocity, resulting from jet breakup during its progression, or obtain a velocity value that accurately reflects the actual flow rate (as determined by weighing), the measured velocity should be corrected by adding $2 R_{drop} \times$ break-up frequency to the measured velocity, where $R_{drop}$ is the

<footnote n="1">¹ Under the flow conditions considered in this study, for example at 0.5 bar, with an estimated flow velocity of 7.8 m/s (Fig. 6a) and the fluid viscosity being of 3.2 mPa·s, Reynolds number is approximately 365 and 146 for the cylindrical nozzle and for the annular flow respectively. Both values are well below the critical threshold of 2300, confirming that the flow remains laminar in both regions and thereby validating the applicability of Equation (3).</footnote>
droplet radius and $f_{breakup}$ is the breakup frequency. Although jet breakup is not explicitly modeled here, the results remain consistent with experimental data, further confirming the robustness of the analysis.

## 5.2. Time resolved evaluation of flow during valve opening

A complete characterization of fluid flow through the microvalve requires not only analysis under steady-state conditions but also during the transient opening and closing phases. Capturing this dynamic behavior involves understanding the valve's response to the applied magnetic field, the mechanical dynamics of the actuation system, and the resulting time-dependent flow rates. These phenomena can be explored through numerical simulations for the prediction of the temporal evolution of the ejected flow rate and the maximum ejected volume.

The microvalve opening is defined by the time-dependent gap width $x(t)$ between the ruby ball and its seat. This gap evolves according to the balance of forces acting on the system: an electromagnetic driving force opposed by both elastic restoring forces and frictional resistance. Accurately predicting $x(t)$ enables the calculation of the time-dependent flow rate $Q(t)$, thereby capturing the full transient behavior during both the opening and closing sequences.

Equation (3) serves as the foundation for estimating the flow rate $Q(t)$ throughout the valve's actuation cycle, provided that the evolution of $x(t)$ is known. This function is derived taking into account the system's kinematics, considering the electromagnetic force, the mass and geometry of the moving plunger, spring restoring forces, and other relevant mechanical and geometric parameters. The governing equation for the plunger's motion, and thus for $x(t)$, is a second-order differential equation:

$$m \frac{d^2 x(t)}{dt^2} = \frac{1}{2} \frac{dL_{\text{eff}}}{dx} I(t)^2 - F_{\text{spring}} - F_r - F_p + Fe + Fd \quad (4)$$

In this equation, the right-hand side represents mechanical inertia. The first term on the left-hand side corresponds to the electromagnetic force. Based on the solenoid's resistance $R$ and inductance $L$, the electrical time constant is estimated as $\tau_e = L/R = 133$ µs (see Materials and Methods). This time constant determines the current response to a step voltage, resulting in exponential growth according to $I(t) = I_{\text{max}}(1 - e^{-t/\tau_e})$. This relationship describes the typical behavior of an RL circuit under sudden excitation. (Irwin and Nelms, 2010) The resulting electromagnetic force is modeled as $\frac{1}{2} dL_{\text{eff}}/dx I(t)^2$ where $dL_{\text{eff}}/dx$ is the gradient of effective inductance with respect to plunger displacement. (Fitzgerald et al., 2009) This magnetic force retracts the plunger and the attached ruby ball, producing a displacement $x(t)$ that varies between 0 µm (valve closed) to 60 µm (valve fully open).

The second term in Equation (4) accounts for the opposing elastic force from the return spring, estimated between 0.1 and 0.2 N, based on the spring's stiffness and compression. Given the limited travel range of the plunger (up to 60 µm), both electromagnetic and elastic forces can be treated as approximately constant during actuation of the valve. This assumption simplifies the dynamic modeling of the system while maintaining acceptable accuracy.

The third term represents the wall shear force acting on the plunger ($\tau_e$), as it moves within its cylindrical housing. This frictional force, arising from viscous shear in the annular gap, is described in Equation (5):

$$F_r = 2\pi R_{\text{plu},c} L_{\text{plu}} \eta \frac{\partial V_z(r)}{\partial r} \alpha \approx 4\pi \eta \frac{\dot{x}(t) R_{\text{plu},c}^2 L_{\text{plu}}}{R_{\text{cyl}}^2 - R_{\text{plu},c}^2} \alpha \quad (5)$$

Parameters include the plunger radius $R_{plu,c}$, plunger length $L_{plu}$, cylinder radius $R_{cyl}$, active circular fraction $\alpha \sim 0,2$ and the plunger velocity $\dot{x}(t)$. Within the narrow annular gap of 250 µm, even at high retraction speed (e.g. 1 m/s), the wall shear force is estimated at $\sim 30$ mN, that is negligible compared to the dominant actuation forces (e.g., electromagnetic and spring forces), and can thus be reasonably ignored in the dynamic analysis.

The fourth term accounts for pressure forces ($F_p$) acting on the front and back faces of the plunger. Assuming pressure symmetry across these surfaces, the net pressure-induced force is zero, and this term can also be neglected.

The final two terms in Equation (4) model the impact dynamics when the plunger reaches the travel limits (i.e., contact with the anchor or valve seat). An elastic response is introduced as $F_e = k_s \Delta x(t)$, along with a damping force $F_d = -k_d \dot{x}(t)$. These forces, activated upon contact, account for energy dissipation and rebound behavior during impact. A smooth transition is ensured by using a sharp hyperbolic tangent function, enabling a localized yet continuous application of stopping forces as the plunger approaches its endpoints. To ensure this behavior and obtain the smallest residual oscillations post impact, values used were $k_s = 10^7$ N/m and $k_d = 200$ N·s/m.

Equation (4) enables the time-resolved evaluation of flow during valve opening, and specifically allows for the calculation of the temporal evolution of the fluid flow rate $Q(t)$. At each time step following the initiation of electric current, Equation (4) is solved to compute the instantaneous plunger displacement $x(t)$. This displacement is then used as input to Equation (3), which is solved iteratively to determine the time-dependent flow rate $Q(t)$. The total ejected volume $v_{\text{max}}$, is subsequently obtained by integrating $Q(t)$ over the duration of a single valve opening cycle. All simulations are performed under the assumption of a constant applied current of 1000 mA throughout the opening phase.

A representative simulation of $Q(t)$ as a function of time following microvalve actuation is shown in Fig. 9 for various total opening times $t_o$. After electrical excitation, $Q(t)$ remains nearly zero and exhibits a noticeable delay before it begins to increase. This delay results from the exponential rise of the electric current in the solenoid coil, which governs the gradual buildup of magnetic force. This force must first overcome the plunger's inertial response and the counteracting spring force before the plunger begins to move, initiating fluid flow and leading to the subsequent increase in $Q(t)$. For $t > 500$ µs, a steady-state flow regime is established, during which $Q(t)$ reaches a plateau, indicating that the valve is fully open. Upon microvalve deactivation, $Q(t)$ gradually decreases until it returns to zero. This decline is primarily attributed to the exponential decrease of the electric current, which leads to a progressive reduction in magnetic force. Combined with the mechanical inertia of the plunger, this results in a delayed closure of the valve and a correspondingly gradual decrease in $Q(t)$. This transient effect during the opening and closure of the microvalve is likely responsible for the

<figure><img src="image_12.png" /><figcaption>Fig. 9. Simulated flow rate as a function of time, obtained using the protocol described by Equations (1) and (2), for different valve opening durations ($t_o = 300$ µs, 800 µs, 1300 µs, and 2200 µs). The simulation incorporates fluid viscosity as well as the electrical and mechanical properties of the microvalve.</figcaption></figure>
non-linear increase of $v_{max}$ observed at short $t_0$ values in Fig. 5b.

To determine the parameters that best capture the evolution of $v_{max}$ as a function of the opening time $t_0$, simulations were performed by varying both the electromagnetic force coefficient $\frac{1}{2} \frac{d\text{eff}}{dx}$ and the spring force $F_{\text{spring}}$. The calculated $v_{max}$ values were then compared with experimental measurements using a least-squares method. The best fit was obtained for $\frac{1}{2} \frac{d\text{eff}}{dx} = 0.5 \text{ N/A}^2$ and $F_{\text{spring}} = 0.1 \text{ N}$.

As shown in Fig. 5b, the simulation (solid line) closely matches the experimental data (solid black dots) for the variation of $v_{max}$ with $t_0$. In the range 600–2500 µs, both exhibit an approximately linear increase in $v_{max}$, indicating a quasi-steady regime where the valve is fully open and the flow is stabilized.

For $t_0 < 500$ µs, the simulation also captures the sharp decrease in $v_{max}$ observed experimentally. This regime reflects the transient dynamics of valve actuation, involving electromagnetic force buildup, mechanical acceleration, and fluid inertia. During this short time span, the valve does not open sufficiently to establish steady flow, leading to a significant drop in ejected velocity. This behavior supports the existence of a dynamic threshold in $t_0$, below which effective flow cannot be established due to the delayed electromechanical-fluidic response.

Although the model captures the general trend well, particularly at short to, further refinement is needed to accurately represent the underlying physics. The main challenge lies in modeling the coupled electromagnetic–mechanical response and the effects of fluid acceleration, captured by the $\partial Vz(r)/\partial t$ term in the Navier-Stokes equations. For instance, the characteristic time for steady-state flow to develop, estimated as $\tau = \rho \times R_{\text{noz}}^2 / 8\eta$, is about 150 µs under test conditions, comparable to the valve’s transient response. This indicates that unsteady flow effects are as influential as electrical dynamics in the early stages of microvalve operation.

## 6. Conclusion

This study provides a detailed understanding of the dynamics governing droplet and jet formation in solenoid microvalves, establishing operational diagrams that map the transition between various ejection regimes, including single droplets, droplets with satellites, multi-droplet formations, and continuous jets. High-speed imaging was essential to capturing these transitions, revealing how valve opening time, actuation frequency, and applied pressure influence the resulting morphologies. The imaging data, combined with morphological analyses, highlighted the critical role of Rayleigh-Plateau instabilities and mechanical perturbations in shaping droplet elongation, jet breakup, and satellite formation. A comprehensive model was developed to describe the microvalve’s operational process by integrating electrical behavior, magnetic field evolution, mechanical forces, and viscous effects. This predictive framework accurately characterizes flow rates and delineates the boundaries between different ejection regimes, providing insights into the interplay between fluid rheology and valve dynamics. The operational diagrams established in this work define the key parameters for controlling fluid morphology – namely, the valve opening time, actuation frequency, and applied pressure – allowing precise control over the transition between droplet and jet formation. The findings emphasize the importance of adjusting these parameters to optimize the performance of solenoid microvalves for precision dispensing of ASE solutions. These insights are essential for advancing technologies in bioprinting, printed electronics, and industrial coating, where reproducibility and control over fluid morphology are critical to performance. Future efforts will focus on refining the understanding of how different fluid rheologies influence droplet and jet morphologies and the corresponding utilization diagrams. Special attention will be given to rheologies of particular interest for industrial applications, including rheology modifiers, high-viscosity fluids, and yield-stress fluids. This will enable further expansion of solenoid microvalve capabilities, addressing complex formulation needs and enhancing precision in

demanding operational environments.

### CRediT authorship contribution statement

**Maxime Façon:** Visualization, Investigation, Data curation. **Yahya Rharbi:** Writing – original draft, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. **Pierre Batocchi:** Supervision, Funding acquisition. **Albert Magnin:** Supervision, Methodology. **Nadia El Kissi:** Writing – review & editing, Supervision, Resources, Project administration.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The Rhéologie et Procédés Laboratory is affiliated with the LabEx Tec 21 (Investissements d'Avenir - Grant agreement no ANR-11-LABX-0030). Additionally, the laboratory is associated with the PolyNat Carnot Institute (Investissements d'Avenir - Grant agreement no ANR-11-CARN-030-01). We thank the National Association for Research and Technology (ANRT) for awarding a doctoral grant (CIFRE agreement) to one of us (M.F.). The author expresses gratitude to Mohamed Karrouch, Didier Bleses and Vincent Verdoot for their invaluable assistance in conducting the experiments, and to Frédéric Hugenell for the insightful discussions.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.ces.2025.122118.

### Data availability

Data will be made available on request.

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