Motion Control of Gallium-Based Liquid Metal Droplets in Abrasive Suspensions Within a Flow Channel

Abstract

Gallium-based room-temperature liquid metal is a promising multifunctional material for microfluidics and precision machining due to its high mobility and deformability. However, precise motion control of gallium-based liquid metal droplets, especially in abrasive particle-laden fluids, remains challenging. This study presents a hybrid control framework for regulating droplet motion in a one-dimensional PMMA channel filled with NaOH-based SiC abrasive suspensions. A dynamic model incorporating particle size and concentration effects on the damping coefficient was established. The system combines a setpoint controller, high-resolution voltage source, and vision feedback to guide droplets to target positions with high accuracy. Experimental validation and MATLAB simulations confirm that the proposed dynamic damping control strategy ensures stable, rapid, and precise positioning of droplets, minimizing motion fluctuations. This approach offers new insights into the manipulation of gallium-based liquid metal droplets for targeted material removal in micro-manufacturing, with potential applications in microelectronics and high-precision surface finishing.

1. Introduction

Gallium-based liquid-metal (LM) alloys [1] such as eutectic gallium indium (EGaIn, mp $15.5 °\mathrm{C}$) and gallium indium tin (Galinstan, m.pt. $15.5 °\mathrm{C}$) have been attracting more attention from researchers in recent years. The unique physical/chemical properties of LM, such as high thermal/electrical conductivity($3.4\times {10}^{-6} \mathrm{S}\cdot {\mathrm{m}}^{-1}$), extreme flexibility, a large surface tension ($>400 \mathrm{m}\mathrm{N} {\mathrm{m}}^{-1}$), an extremely low vapor pressure (<10⁻⁶ Pa at 500 °C), and relatively low toxicity compared to mercury [2] and their ability to form a thin oxide shell when exposed to oxygen [3] has many applications. These exceptional characteristics make room-temperature liquid metals promising for a wide range of applications, such as soft and stretchable electronic components [4,5,6,7], devices [8,9,10], small-scale vehicles and robots [11,12], soft sensors [13,14], tunable antennas and apertures [15,16], as well as fluidic optical components and displays [17,18]. Furthermore, liquid metals hold significant potential for the development of soft robots [19], a long-standing ambition in engineering. Despite these advances, precise control of the surface tension of liquid metals, a critical factor for achieving desired motion and deformation, remains a key challenge. Addressing this issue is essential for unlocking the full potential of these materials in next-generation applications.

In the application environment of liquid metal droplets, NaOH solution is commonly employed as the electrolyte, not only to facilitate the dissolution of the native oxide layer on the liquid metal surface but also to promote the formation of a stable electrical double layer, thereby enabling electrocapillary actuation and maintaining surface activity during operation [20]. When abrasive particles, such as silicon carbide, are introduced into the NaOH solution at varying particle sizes and concentrations, the rheological properties of the solution undergo significant changes, resulting in the formation of a distinct “rheological fluid.” The presence of these abrasive particles alters the viscosity and flow characteristics of the solution, which, under the influence of an applied electric field, can affect the motion of the liquid metal. Specifically, the particle size and concentration may directly influence factors such as the viscosity, surface tension, and shear resistance of the solution.

Despite the vast potential of liquid metal droplets in various applications, precise control of their motion in abrasive particle-laden fluids presents significant challenges. The random distribution of abrasive particles within the fluid introduces additional resistance to the motion of liquid metal droplets, thereby influencing their trajectory and velocity [21]. Particularly under the influence of an electric field, the interaction mechanisms between the liquid metal and the abrasive particles become more complex. Effectively controlling the motion and velocity of liquid metal droplets through the electric field, therefore, remains a critical issue that must be addressed. Consequently, the development of new control strategies and mathematical models to tackle the challenges of liquid metal motion control in abrasive particle-laden fluids is of paramount importance.

Various methods have been developed to manipulate liquid metal droplets, which can be broadly categorized into several approaches: electrical fields [22,23], magnetic fields [24,25], electrochemical reactions [26,27], photochemical processes [28], and ionic imbalance [29]. Among these, the application of an external electric field in an ionic NaOH solution to actively control the surface tension of liquid metal droplets is the most widely employed, due to its practicality and ease of implementation. According to Lippmann’s equation, the external electric field can disrupt the charge distribution within the electrical double layer (EDL) at the liquid metal- NaOH solution interface, thereby generating a surface tension gradient along the liquid metal droplet’s surface. This gradient induces motion and deformation of the droplet [30].

Numerous studies have explored the use of electric fields to drive the movement of liquid metal droplets. For instance, a direct current (DC) electric field has been applied to drive a mercury slug within a microfluidic channel, with the average speed of the mercury theoretically estimated [31]. Similarly, gallium-based liquid metal droplets have been shown to exhibit linear and reciprocating motion under DC and alternating current (AC) electric fields, respectively, as reported in [23,30,32]. More recently, a low-power, high-efficiency micropump was developed by applying an AC electric field with a DC offset to a Galinstan (68.5 wt% gallium, $21.5 \mathrm{w}\mathrm{t}\mathrm{\%}$ indium, and $10 \mathrm{w}\mathrm{t}\mathrm{\%} \mathrm{t}\mathrm{i}\mathrm{n}$) droplet confined within a cylindrical chamber embedded in fluidic chips [33]. As shown in the work by Xie et al. [34], liquid metal droplet motion control is critical in a variety of applications such as soft robotics and microfluidics. Despite these advancements, precise control of liquid metal droplet motion remains a significant challenge, with many aspects still to be fully understood. However, existing research has predominantly focused on the motion control of liquid metals in homogeneous fluids, such as pure NaOH solutions, with relatively little attention given to the motion characteristics of liquid metals in abrasive particle-laden fluids.

In abrasive particle-laden fluids, the presence of solid particles markedly alters the rheological properties of the medium—including viscosity, density, and surface tension—which in turn significantly affects the dynamics of liquid metal droplets subjected to external electric fields [35]. Despite this, the specific roles of particle size and concentration in modulating these effects remain insufficiently explored in the literature. Addressing this knowledge gap, the present study systematically investigates the motion behavior of liquid metal droplets in NaOH-based abrasive suspensions with varied SiC particle sizes and concentrations under applied electric fields. By quantitatively analyzing how these particle characteristics influence the viscosity of the mixed solution and the damping term in the liquid metal dynamic equation, we establish a comprehensive control strategy for achieving precise manipulation of liquid metal droplets in complex fluid environments. The findings offer new theoretical and practical insights for optimizing liquid metal motion and surface processing in multiphase microfluidic systems.

To achieve the aforementioned objectives, we have established an advanced experimental platform, which integrates a programmable power supply, an industrial camera imaging system, and a comprehensive data acquisition and feedback control unit. Initially, a dynamic model was constructed to elucidate the motion behavior of liquid metal droplets subjected to an external electric field, with particular emphasis on the roles of surface tension-driven forces, viscous resistance, and channel wall friction. Building upon this model, we designed a setpoint controller that leverages real-time visual feedback to compute the optimal voltage input, thereby enabling precise manipulation of droplet motion via PID control.

The validity and robustness of the proposed control methodology were systematically demonstrated through experiments conducted in PMMA-based microfluidic channels. Overall, this hybrid control framework is expected to enable highly accurate regulation of liquid metal droplet dynamics, while also offering new perspectives for the application of liquid metals in complex fluidic environments. Furthermore, our results highlight the significant potential of this approach for high-precision material removal, providing valuable insights for advanced micro-manufacturing processes where stringent surface quality control is required.

2. Materials and Methods

2.1. Experimental Setup and Materials

In the experiment, as illustrated in Figure 1, EGaln ($75 \mathrm{w}\mathrm{t}\mathrm{\%}$ gallium, $25 \mathrm{w}\mathrm{t}\mathrm{\%}$ indium) was employed as the droplet material. The droplets were dispensed into open-top PMMA channels with widths of $3,4,5,6$, and 7 mm and a fixed length of 80 mm. Using a syringe, EGaln droplets with radii ranging from 0.15 cm to 0.35 cm were generated and introduced into the channels. A series of abrasive suspensions with varying particle sizes ($100 \mathsf{\mu }\mathrm{m}$ to $500 \mathsf{\mu }\mathrm{m}$) and volume concentrations ($2\mathrm{\%}$ to $10\mathrm{\%}$) were prepared. To ensure uniform dispersion of SiC particles in the NaOH solution, a magnetic stirrer was used. All abrasive particles were classified using standard sieves: 170 mesh (90–106 $\mathsf{\mu }\mathrm{m}$), 120 mesh (125–150 $\mathsf{\mu }\mathrm{m}$), $80$ mesh (180–212 $\mathsf{\mu }\mathrm{m}$), $60$ mesh (250–270 μm), and 50 mesh (270–300 $\mathsf{\mu }\mathrm{m}$). Both the NaOH solution and silicon carbide (SiC) were supplied by Dongguan Zhongwei Abrasive Technology Co., Ltd., Dongguan, China. The viscosity of each suspension was measured using a viscometer under different SiC particle size and concentration conditions to evaluate their effects on the solution’s viscosity. The input voltage was applied via a programmable AC/DC power supply (ITECH IT7221, ITECH Electronic Co., Shenzhen, China). Additionally, an MV-CU060-10GC camera (Hikvision, Hangzhou, China) was employed to record the experiments and acquire real-time position data of the liquid metal, which was transmitted to the controller to enable real-time tracking of droplet position, velocity, and acceleration. The precise control framework for liquid metal droplet dynamics is schematically presented in Figure 2. The actual experimental channel are shown in Figure 3.

2.2. Experimental Procedure and Closed-Loop Control

The acquired position data was transmitted to a MATLAB-based controller for closed-loop feedback regulation. In particular, MATLAB was utilized to integrate and coordinate the experimental system, making extensive use of the Image Acquisition Toolbox, Image Processing Toolbox, and Instrument Control Toolbox for real-time image acquisition, object recognition, system control, and data recording. The overall hybrid control framework, enabling precise manipulation of liquid metal droplet dynamics under varying abrasive conditions, is schematically illustrated in Figure 2.

The electrolyte was a $0.5 \mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}\mathrm{N}\mathrm{a}\mathrm{O}\mathrm{H}$ alkaline solution that filled all channels. The electrode was graphite electrode whose size is $\varphi 10\times 20 \mathrm{m}\mathrm{m}$. We obtained the accurate displacement measurement of the liquid metal droplet every 0.1 s from the video using MATLAB R2022a (MathWorks, Natick, MA, USA) software. Next, based on the displacement data, we calculated the velocity and acceleration of liquid metal droplet using matlab software. From the definition in (A18), we obtain that $K_{2,0}=0.045 \mathrm{m}\mathrm{P}\mathrm{a}·\mathrm{s}$ and $K_3=0.0001$. All open-top PMMA channel samples used in the experiment had a rectangular cross-section with a width and depth of 5 mm, and lengths of 40, 50, 60, 70, or 80 mm. The height of the solution must enable the solution to cover the liquid metal droplet [34].

The liquid metal droplet is highly sensitive to circuit voltage due to its extremely high surface tension, with even a small voltage fluctuation in the circuit sufficient to induce a displacement in the static liquid metal droplet. Although the voltage output module of the AC/DC power supply (ITECH IT7221, ITECH Electronic Co., Shenzhen, China) has a precision exceeding 0.1 V, it does not provide adequate voltage accuracy. Therefore, we propose a voltage output strategy that enables high-resolution voltage output to the circuit, as shown in Figure 4. The AC/DC power supply’s voltage output module has an accuracy of 0.1 V, used for providing output voltages to the two electrodes. Additionally, the voltage measurement module of the AC/DC power supply (ITECH IT7221, ITECH Electronic Co., Shenzhen, China) has a measurement accuracy of 0.1 V and is employed to measure the input voltage to the control circuit, feeding back any deviations to the PID controller, formulated as follows:

$$u(k)=K_p\cdot e(k)+K_i\cdot \sum _{i=0}^k e(i)\cdot dt+K_d\cdot {\displaystyle \frac{e\left(k\right)-e(k-1)}{dt}}$$

(1)

where $e\left(k\right)$ represents the error at the $k$-th step, $\sum _{i=0}^k e(i)\cdot dt$ is the cumulative integral error, and ${\displaystyle \frac{e\left(k\right)-e(k-1)}{dt}}$ denotes the differential error. $u\left(k\right)$ represents the output voltage error. The control gains are $k_i=0.1,k_d=1$, and $k_p=6$.

Thanks to the above strategy, the enhanced power supply ensures that the voltage output accuracy meets the control requirements, with a response time of less than 300 ms. These parameters generally require iterative tuning during the experimental process, based on factors such as droplet mass, channel resistance, response speed requirements, and the intensity of external disturbances.

As illustrated in Figure 4, in the proposed control system, position feedback plays a critical role in achieving precise motion control of the droplet. The position feedback is obtained through real-time image processing, where the droplet’s centroid is extracted from the captured images. This feedback signal is subsequently used to compute the position error, which is defined as the difference between the target position ($g_q$) and the actual position of the droplet. The position error, denoted as $e(t)=g_q-y(t)$, serves as the primary input to the PID controller, enabling continuous adjustment of the control signal.

The closed-loop system continuously compares the desired position with the actual position, thus allowing the PID controller to dynamically adjust the voltage output. This voltage, in turn, influences the droplet’s movement, ensuring it follows the target trajectory with minimal deviation. The real-time feedback mechanism effectively mitigates steady-state errors and improves the overall system stability, ensuring the droplet reaches the target position with high precision.

By incorporating position feedback into the control loop, the system compensates for any disturbances or inaccuracies in the motion, making it adaptive to changes in the environment or system parameters. This feedback-driven approach is crucial for maintaining high-accuracy control in systems where precise positioning is essential. The detailed theoretical analysis of the liquid metal droplet dynamics modeling and controller design is provided in Appendix A.

To evaluate the proposed strategy, simulations are conducted under three key scenarios: varying target positions, droplet sizes, and damping coefficients modulated by particle size and concentration.

The experimental design encompassed three scenarios: (1) varying the target positions (20–40 mm) to evaluate the system’s positional accuracy at different distances; (2) altering the droplet radii to assess the robustness of the control system under different droplet properties; and (3) adjusting the particle size and concentration of the abrasive fluid to test the system’s adaptability to environmental changes. Each experimental configuration was repeated more than 50 times to ensure statistical reliability and minimize random errors. The processed experimental data were compared with simulation results to analyze trends in position, velocity, and acceleration.

3. Results and Discussion

The motion control of liquid metal droplets in abrasive suspensions is a complex task influenced by factors such as viscous resistance, friction, and external driving forces. This study proposes a closed-loop control strategy based on a PID controller, which dynamically adjusts the applied voltage in response to real-time feedback of the droplet’s position and velocity. By employing this approach, the droplet achieves rapid response and high-precision localization, even under varying environmental conditions. The theoretical foundation of the control system is validated through Lyapunov stability analysis, ensuring robust convergence and stability.

To evaluate the proposed strategy, simulations are conducted under three key scenarios: varying target positions, droplet sizes, and damping coefficients modulated by particle size and concentration. These simulations demonstrate the adaptability and effectiveness of the control system in achieving precise motion control. The findings not only provide theoretical insights into the dynamics of liquid metal droplets but also offer practical guidance for their integration into advanced microfluidic devices and flexible electronics, addressing critical challenges in precision engineering.

3.1. Motion Response of Liquid Metal Droplets at Different Target Positions

Figure 5 illustrates the dynamic response of a liquid metal droplet under varying target positions ($20 \mathrm{m}\mathrm{m}, 25 \mathrm{m}\mathrm{m}, 30 \mathrm{m}\mathrm{m}, 35 \mathrm{m}\mathrm{m}$, and 40 mm), highlighting the temporal evolution of position, velocity, and acceleration. The analysis reveals that the droplet exhibits rapid response and high-precision control across all target positions. The position curves demonstrate that the droplet swiftly approaches the target position and stabilizes, with longer response times observed for larger target distances. However, the droplet consistently reaches the desired position with remarkable accuracy. The velocity and acceleration curves further elucidate the droplet’s dynamic behavior: the droplet accelerates rapidly at the initial stage, decelerates promptly as it nears the target, and eventually stabilizes with zero velocity and negligible acceleration. This dynamic trajectory underscores the control system’s ability to effectively regulate the droplet’s motion, ensuring precise localization.

As the target position increases, the peak velocity and acceleration also rise, reflecting the greater force required for the droplet to achieve rapid response over longer distances. This trend highlights the adaptability and stability of the control system. The results demonstrate the system’s superior performance in both dynamic and steady-state control, offering theoretical insights and practical guidance for the motion control of liquid metal droplets in abrasive fluids. These findings are particularly significant for applications aimed at achieving precision machining and advancing technologies in microfluidics and flexible electronics.

Figure 6 presents the experimental results for the motion of gallium-based liquid metal droplets under various target positions (20, 25, 30, 35, and 40 mm). The experimental trajectories clearly demonstrate that the droplets achieve their designated target positions with high accuracy and minimal overshoot. For direct comparison and validation, these results are juxtaposed with the simulation data (see Figure 5). Notably, both experimental and simulated trajectories exhibit rapid convergence during the initial phase (0–0.5 s), followed by stable and nearly oscillation-free settling. The overall agreement in trend between the experimental data and model predictions further confirms the effectiveness and reliability of the proposed control strategy for achieving precise multi-target positioning.

It should be noted that, compared to the simulation, the experimental velocity and acceleration profiles show minor fluctuations. These small oscillations are primarily attributable to unavoidable experimental noise, physical limitations of the actuators, and delays in image acquisition and feedback control. Such minor deviations are typical in practical systems and do not compromise the overall stability or positioning accuracy.

3.2. Motion Response of Liquid Metal Droplets at Different Particle Sizes and Concentrations

Figure 7 illustrates the dynamic behavior of liquid metal droplets under varying particle sizes (${\mathrm{d}}_{\mathrm{p}}=100 \mathsf{\mu }\mathrm{m}$ to $300 \mathsf{\mu }\mathrm{m}$) and concentrations ($C_p=2\mathrm{\%}$ to $10\mathrm{\%}$). Across all parameter combinations, the droplet accurately reaches the target position $(20 \mathrm{m}\mathrm{m})$ with minimal overshoot and consistent stabilization. The position profiles indicate robust control system performance, as variations in particle size and concentration have negligible impact on the steady-state behavior.

The velocity and acceleration curves highlight the droplet’s rapid response and controlled deceleration. Slight increases in peak velocity and acceleration are observed with larger particle sizes and higher concentrations, reflecting the need to overcome increased damping effects. However, the overall response trends remain consistent, demonstrating the system’s stability and adaptability to changes in fluid properties.

These findings confirm the control system’s ability to maintain precise and stable droplet manipulation in abrasive fluid environments with varying particle size and concentration. This robustness is critical for applications such as precision machining and microfluidic technologies, where consistent performance is required despite complex fluid conditions.

As depicted in Figure 8, the experiments with varying abrasive fluid parameters (particle sizes of 100–300 μm and concentrations of 2–10%) confirmed the control system’s adaptability to environmental variations. As shown in Figure 11, the velocity and acceleration plots, larger particle sizes and higher concentrations led to marginally increased peak velocities and accelerations. However, the overall motion patterns and stabilization times remained consistent, indicating that the control system effectively mitigates the impact of fluid parameter variations.

3.3. Motion Response of Liquid Metal Droplets at Different Droplet Radii

Figure 9 demonstrates the dynamic behavior of liquid metal droplets with varying radii ($\mathrm{r}=0.15 \mathrm{c}\mathrm{m},0.20 \mathrm{c}\mathrm{m},0.25 \mathrm{c}\mathrm{m},0.30 \mathrm{c}\mathrm{m}$, and 0.35 cm), highlighting the evolution of position, velocity, and acceleration over time. The results show that all droplets successfully converge to the target position (20 mm) with high accuracy, regardless of size. Smaller droplets exhibit slightly longer response times and higher peak velocities and accelerations due to increased mass, but the system maintains consistent stabilization with minimal overshoot.

The velocity and acceleration curves indicate that larger droplets require higher momentum and forces to overcome inertia, reflected in slightly elevated peaks. However, the overall trends across all droplet sizes remain consistent, demonstrating the robustness and adaptability of the control system. The system effectively regulates acceleration and deceleration, ensuring smooth and precise droplet positioning.

These findings highlight the control system’s ability to handle droplets of varying sizes without compromising precision or stability. This robustness is essential for applications requiring accurate droplet manipulation, such as in abrasive fluid environments or microfluidic technologies, where size variability is inevitable. The study offers valuable insights into the scalability and reliability of the control process for practical applications.

As shown in Figure 10, for varying droplet radii ($0.15 \mathrm{c}\mathrm{m}, 0.20 \mathrm{c}\mathrm{m}, 0.25 \mathrm{c}\mathrm{m}, 0.3 \mathrm{c}\mathrm{m}$, and 0.35 cm), experimental results closely matched the simulation (see Figure 7) trends. Smaller droplets exhibited slightly higher velocities and accelerations due to reduced mass, while larger droplets required more time to stabilize at the target position. Despite these differences, the control system maintained stable performance across all conditions, validating its robustness in adapting to different droplet dimensions.

Figure 11 illustrates the droplet manipulation process as captured from experimental video footage. The manipulation sequence depicted in Figure 11a–e was conducted within a rectangular channel. In these images, the liquid metal droplet appears as a spherical object, while its target destination is indicated by a small red circle. Figure 11a presents the initial configuration, and Figure 11b–d display the dynamic manipulation at various time points. Figure 11e demonstrates that the liquid metal droplet successfully reached its designated target. The complete manipulation process can be viewed in the Supplementary Videos S1–S5 uploaded.

The experimental velocity and acceleration profiles aligned with simulation predictions, exhibiting a rapid initial response followed by smooth deceleration. The observed peak values in velocity and acceleration were consistent across experiments and simulations, further validating the control system’s dynamic modeling accuracy. Additionally, the time required for droplets to stabilize at the target position showed minimal deviation between experiments and simulations, supporting the system’s precision and reliability.

4. Conclusions

This study presents a robust control strategy for the precise manipulation of liquid metal droplets in abrasive suspensions. By combining a closed-loop PID controller with real-time feedback, the proposed method dynamically adjusts the applied voltage to achieve high-precision droplet positioning in complex fluid environments. The integration of particle size and concentration into the damping coefficient model enhances the understanding of liquid metal dynamics, offering novel insights into motion control within non-Newtonian fluids.

Simulation and experimental results demonstrate the effectiveness of the control system across varying conditions, including target position, droplet size, and damping properties. The strong agreement between simulation and experimental data further validates the reliability and accuracy of the proposed strategy, confirming its suitability for applications in flexible electronics, microfluidics, and precision machining.

Unlike previous studies, this research systematically addresses the impact of abrasive particle properties on the dynamics of liquid metal droplets, providing a comprehensive approach to motion control. This novel perspective opens new avenues for fine-tuning damping dynamics in complex fluidic systems, enabling enhanced control flexibility for industrial and biomedical applications.

Future work will focus on extending this control framework to three-dimensional fluidic systems and exploring its adaptability to dynamically changing external fields. Additionally, integrating machine learning algorithms could improve real-time optimization and further enhance the system’s responsiveness, broadening its industrial applicability. These advancements have the potential to revolutionize microfluidic devices, biomedical diagnostics, and precision manufacturing systems.