Stroking through Electrolyte: Liquid Metal Droplet Propulsion through Pulse Time Modulation

Active droplets play important roles in microfluidics, robotics, and micro‐electromechanical systems. As a special class of active droplets that are conductive, reactive, and of high surface tension, liquid metal droplets (LMDs) can be driven by electric‐field‐induced surface (Marangoni) flows to function as reconfigurable components in actuators, sensors, catalytic reactors, and antennas. Stimulating LMDs using an electric field induces concurrent electro‐hydrodynamic flows and electrochemical surface oxidation (passivation). It is however difficult to decouple these two effects which brings complexity in controlling LMD motions. To address this challenge, pulse time modulation (PTM) signals are used. PTM enables controlled LMD displacement by propelling the droplets forward during the voltage‐on phases and facilitating surface recovery from oxidation during the voltage‐off phases. Counterintuitively, by taking such intermittent “rests”, the LMDs effectively inhibit the unfavorable impact of oxidation, granting high motion controllability. Combining high‐speed imaging, motion tracking, machine learning, and electrochemical analysis, the study reveals how electro‐hydrodynamic flows and surface oxide formation/dissolution interplay to generate well‐defined motion regimes. The study further develops a quasi‐analytical model to describe droplet motions and designs a rotary LMD motor to showcase the versatility of the approach. This work provides the fundamental framework and viable strategy for designing innovative liquid metal‐based systems.


Introduction
3][4][5][6][7][8][9][10][11] In many of such systems, Marangoni flows (or surface-tension-driven flows) provide the mechanism for droplet propulsion. [12,13]In these systems, when the droplets feature sizes comparable to or below the liquid's capillary scale the Marangoni flow dominates, and also the motion is strongly influenced by surface tension.[16][17][18][19][20][21][22][23] It has been proven challenging to create an evolving concentration or temperature gradient to guide moving droplets in real time.This is because both mass and thermal diffusion are slow transport processes, which render their fast on-demand control difficult.This is especially true when it comes to small-scale and highly dynamic systems.On the other hand, chemical controls of droplets have also raised many questions including limited energy reserves and external control over the reactions.Altogether, the overarching goal of achieving programable and highly controlled motion of droplets has so far been limited.
Liquid metals, specifically those made of gallium and its alloys, have been shown in reconfigurable and soft systems, biomedical applications, motorized devices, pumps, and soft robotics.Particularly, conductive liquid-state droplets, at room or nearroom temperature, have been well investigated in such settings.[26] When immersed in an electrolyte medium, applying an electric field gradient can generate Marangoni flow on the surface of these liquid metal droplets (LMDs), creating an electro-hydrodynamic flow. [12,27,28][31][32] However, to date in practice, the precise motions of electric-field-driven LMDs could not be fully harnessed by considering hydrodynamics alone.The motion of electrically driven LMDs still lacks adequate understanding and proper control even for simple actions such as displacement and on-demand halts.This is likely due to the fact that electrohydrodynamic flow of gallium-based droplet-electrolyte systems are influenced by a thin metal (hydro)oxide layer which is electrochemically formed on the droplet surface. [33]The interplays between surface convection, viscous effect, oxide formation, and dissolution bring a high-level complexity to the LMD system.The route through which the viscous effect and surface oxidation affects the LMD propulsion remains unanswered.So far, the most commonly used DC driving strategies have been demonstrated to lead to undesirable deceleration in many settings, which is unwanted for systems relying on long-term high-speed droplet propulsion. [27]The mechanism that underlies the deceleration of the LMDs remains elusive.
Highly effective propulsion is often seen in biosystems.For instance, fish and jellyfish are good swimmers, and they both use undulated body locomotion to propel themselves through water at high speeds and low energy consumption. [34,35]Inspired by such undulating bio locomotion, here we introduce a pulse time modulation (PTM) strategy, in an electrochemical setting, to improve LMD driving efficiency and precision of motion control.We show that turning off the driving voltage for suitable time intervals using PTM, instead of using continuous driving voltages, is able to offer great controls for droplet propulsion.That is, the LMDs took intermittent rests to obtain higher velocity and better long-term stability.We investigate the interplay between Marangoni flow and surface electrochemical oxidation of LMDs in determining droplet motions.We find that the proposed strategy can fine-tune the driving surface flow, oxide formation, and dissolution to realize different well-defined motion regimes.We apply a number of techniques including high-speed imaging, motion tracking, machine learning, and electrochemical measurements to analyze the motion behavior, flow patterns, and surface activities, which allowed the understanding of the coupled hydrodynamics and electrochemistry in the system.The developed quasi-analytical model reproduces the droplet motions under different driving conditions.We further build a LMD motor and use PTM driving signals to achieve long-term stable propulsion of the droplet at high velocities, revealing device-level viability and robustness of our strategy.

Results and Discussion
In the first stage we show that, in an electrolyte, pulsed timemodulated voltages can enable controlled, high-speed, and stable motions of the LMDs.This is compared to the displacement of droplets of the same liquid metal under applied constant DC voltages.In these experiments, the modulation of the PTM signal was carried out by varying the durations for which the voltage pulses are turned on (t on ) or turned off (t off ).As shown in Figure 1a and Figure S1 (Supporting Information), the experimental setup consisted of a channel (length L = 100 mm, width W = 15 mm) filled with an electrolyte of sodium hydroxide (NaOH, concentration c = 0.5 mol L −1 ).Eutectic gallium indium (EGaIn, melting point: 15.5 °C) was selected as the liquid metal so that the droplet could remain liquid at room temperature.The electrical driving signal was applied using two graphite electrodes which were placed at the two ends of the channel, under which the LMD advanced to the anodic electrode.The droplet movement was captured by a camera for motion tracking and analysis (Figure S2, Supporting Information). [36,37]he LMD movement in an electric field is known as a result of the surface Marangoni flows induced by an applied potential gradient following the Lippman equation  =  0 − 1 2 CΔU 2 EDL , where  is the surface tension, C is the capacitance of the electric double layer (EDL), ΔU EDL is the potential difference across the EDL and  0 is the surface tension at ΔU EDL = 0 (Figure 1b). [38]t should be noted that after applying the voltage, a drop in potential first occurs across the EDL of the electrodes, followed by a linear change across the electrolyte (Figure S3, Supporting Information).In addition, the side of the LMD facing and opposing the anodic electrode become negatively and positively polarized, respectively. [39]Such a polarization is responsible for the asymmetrical oxidation of the LMD, which will be discussed in detail later.
Distinct LMD motions were observed when PTM signals (Figure 1c-f) and DC signals (Figure 1g-j) were applied (Movie S1, Supporting Information).The magnitude of the applied signal (U S = 15 V) and droplet diameter (D = 3 mm) were fixed for both the PTM and DC driven cases for comparison and consistency.The DC-driven LMD first accelerated to a peak velocity of v ∼ 45 mm s −1 within 150 ms but quickly experienced a gradual deceleration to v < 10 mm s −1 (Figure 1g-i).The PTM-driven LMD displayed different types of droplet motions by regulating t off (t on = 10 ms fixed) (Figure 1d).We note that the power consumption of the PTM signal (P PTM ) is less than that of DC signals of the same U S (P DC ), and their ratio can be estimated by P PTM /P DC = t on /(t on + t off ).When t off was small (e.g., t off = 3 ms), a fast increase followed by a gradual decrease in velocity was observed (Figure 1e), which was similar to the DC-driven droplet motion (Figure 1i).When t off was increased to 10 ms, an increased velocity to ≈70 mm s −1 was obtained, which was overall stable until the LMD reached the end of the channel.Further increasing t off (e.g., t off = 30 and 80 ms) led to stable but continually decreasing velocity profiles.At sufficiently large t off (e.g., t off = 80 ms), the discernible oscillation of the LMD movement became evident due to significant deceleration during prolonged t off .
The distinct droplet motions actuated by DC and PTM were reflected by the electrolyte flow patterns induced at the dropletelectrolyte interface.These flow patterns were revealed by flow visualization (Experimental Section).In the case of the PTM-driven LMD, dissipating pulsed wakes (downstream wave patterns) in the electrolyte were observed, indicating the pulsed motion of the droplets surface (Figure 1f).By contrast, when subjected to a DC signal, the LMD created continuous streamlines within the surrounding electrolyte as it moved through it (Figure 1j).As such, the PTM-and DC-driven LMDs adopted a "stroking" and "gliding" mechanism for traversing the electrolyte, respectively.
We then examined the influence of t off on the time-dependent velocity profiles of the LMD (with U S = 15 V and D = 3 mm fixed) for three t on values, namely, t on = 5 ms (Figure 2a; Figure S4a, Supporting Information), t on = 10 ms (Figure 2b; Figure S4b, Supporting Information) and t on = 30 ms (Figure 2c; Figure S4c, Supporting Information).These velocity profiles were classified into three regimes: 1) DC-like velocity profiles that showed an initial acceleration followed by a fast deceleration under small t off , that were assigned as regime 1 (R-1) profiles.Such a velocity profile was also observed at high-frequency (>500 Hz) PTM signals (Figure S5, Supporting Information); 2) Fast and steady velocity profiles of the LMD at intermediate t off, that were categorized as regime 2 (R-2) profiles; 3) Decreased and oscillating velocity profiles at further increased t off , that were defined as regime 3 (R-3) profiles.It's worth noting that all the PTMdriven LMDs exhibited oscillating motions, but with short t on and t off the oscillations were hardly noticeable (Figure S6, Supporting Information).The same tripartite velocity profile categorization was applied to other t on while t off was fixed (Figure S7, Supporting Information).Such a categorization of the velocity profiles only became evident when PTM signals were applied to regulate t on and t off individually.Similar velocity profile regimes could not be reproduced using frequency modulation, because both t on and t off depend on f which makes the categorization less apparent (Figure S5, Supporting Information). [6,30]he Reynolds number was defined as Re = vD/μ with  the density and μ the dynamic viscosity of the electrolyte, v the displacement velocity, and D the diameter of the liquid metal droplet, respectively.The equation implies that for given droplet diameter and velocity, Re should have similar dependence to the motion regime of the LMDs.It should be noted that Re alone is not able to define the flow pattern since the LMD motion shows transient and complex changes and the faster LMD surface movement relative to the total droplet velocity cannot be taken into account.
Based on the aforementioned classification of a large experimental data set of (t on , t off , and D), a phase diagram of the LMD velocity profile regimes was constructed using a neural network machine learning model (see Supporting Information, Table S1, Figure S8, Supporting Information).The model was trained with experimental data (scatters in Figure 2e) to reproduce the three regimes matching the three types of velocity profiles.The experimental data points for different regimes of velocity profiles fitted reasonably well into their respective regions for a fixed droplet size (D = 3 mm, Figure 2e), indicating a good accuracy of the machine learning model (see validation in Supporting Information).We extended the machine learning model to different droplet sizes, which allowed us to predict their velocity profile regimes with a considerably reduced number of experiments (Table S2 and Figure S9, Supporting Information).As shown in Figure 2f, increasing the droplet size D shifted the boundary of the R-1 and R-2 regions toward smaller t off , resulting in a contraction of the former and an expansion of the latter, respectively.In comparison, the R-3 region was less influenced by D.
We further compared the velocity profiles of a 3 mm LMD driven by DC and PTM (R-2) signals with varied U S (from 5 to 19 V, Figure 2d).Both signal types showed that increasing U S generally led to higher velocities due to an increased surface tension gradient, as predicted by the Lippman equation.The change in U S did not alter the velocity profile characterization of the DC-and PTM-driven LMD.Importantly, the PTM velocity profiles always showed a higher velocity and better stability than the DC velocity profiles for all the tested voltages.It should be noted that, when compared to a DC-driven voltage of the same magnitude, the PTM signals had a lower effective (time-average) voltage input, which was able to improve the driving efficiency with reduced power consumption.
The acceleration profiles derived from the velocity curves from Figure 1e,i showed the more stabilized motions of the PTMdriven LMD compared to the DC-driven LMD (Figure S10, Supporting Information).Note that the maximum acceleration was reached by the DC-driven LMD.This was because the DC signal provided a constant driving voltage, while that of the PTM signal only presented during t on .The reason for the DC signals to be less efficient than the PTM counterparts is the formation and accumulation of surface oxide on the LMD, which is to be discussed in detail in the next section.
We further investigated the LMD motions by heating up the system to 40°C (Figure S11, Supporting Information).The results are compared to the results from Figure 1e at 20 °C.The outcome shows that temperature greatly influences the droplet motion, and that the droplet velocity increases when the system temperature is increased from 20 to 40 °C.We note that increasing the temperature of the electrolyte not only changes its viscosity, but also alters the interfacial chemical and electrochemical processes (including oxidation and oxide dissolution).The latter can have a stronger influence on the droplet's motion.
The motion of the LMDs was driven by the applied voltage through the electrocapillary effect.However, the contribution of the applied voltage to the droplet propulsion is not always beneficial.In the presence of a directional external electric field, the LMD surface electrochemically oxidizes.The resulting surface oxide layer negatively impacts the motion.Previous works on solid metallic spheres under similar conditions have suggested that the oxidation is asymmetrical. [40]To explain the electrochemical processes occurring at the LMD surfaceelectrolyte interface, its time-dependent interfacial potential (U-t, Figure 3b,e,h,k) and time-dependent current (i-t, Figure 3c,f,i,l) induced by the applied DC and PTM signals (Figure 3a,d,g,j) were measured and compared by using the LMDs as the working electrode (Experimental Section, Figures S12-S15, Supporting Information).Before applying any driving signal on the two external graphite electrodes, the liquid metal was biased with its open circuit potential (OCP = −1.54V) so that there was zero current.
When a DC voltage was applied to the graphite electrodes (Figure 3a), an abrupt change (within 25 ms) in the potential and current across the LMD-electrolyte interface was observed (Figure 3b,c).Such a transient response was attributed to the fast oxidation as well as the capacitance change of the EDL.Both the interfacial potential U and current I gradually decreased and eventually became constant.However, U and I never returned to their initial value, which was indicative of the change of the interface due to the presence of surface oxide.In addition, the persistent current (Figure 3c) meant that oxidation of the LMD surface always took place under the DC polarization to compensate for the counteracting oxide dissolution process and to reach a steady state.
Different from the DC polarization, the oxidation and oxide dissolution processes dominated during the t on and t off periods of each PTM cycle.Therefore, the PTM driving signal allowed the LMD to remove the surface oxide during t off and reduced the negative impacts of oxidation on the droplet motion.As shown in Figure 3e,h,k,f,i,l, the LMDs were able to respond to the PTM signals by showing cyclic changes in U and I.During t on , the LMD went through part of the DC process and was driven to move forward while accumulating surface oxide on the backside.The oxide is expected to be of nanoscale thickness and thus invisible. [41]ontinuous surface oxide accumulation and thickening were prevented during t off in which the oxide was dissolved by the NaOH electrolyte.For each fixed t on , there existed an optimal t off for achieving droplet driving efficiency, which was expected to be the time required for complete oxide removal (t c , Figure 3l).When the external potential was terminated, a counter current (opposite direction to that during t on ) was generated, which declined gradually.This current was attributed to both the fast change in EDL capacitance (see reference measurement with graphite as the WE in Figure S14, Supporting Information) and the relatively slow change in surface chemistry due to oxide dissolution.Therefore, a zero current was expected upon complete surface oxide removal in the i-t curves.This explanation matched well with our experimental observations.
In the case of t on = 5 ms, t c was determined to be 12 ms.When t off was significantly smaller than t c (e.g., 3 ms), oxide removal during t off was ineffective, leading to DC-like (R-1) droplet velocity profiles (Figure 3f).When t off was close to t c (e.g., 10 ms), effective oxide removal was achieved, and fast yet stable (R-2) droplet motion was observed (Figure 3i).Further increasing t off to higher than t c didn't contribute to improving droplet motion through the oxide removal mechanism (Figure 3l).On the contrary, it decelerated the droplets as viscous drag became dominant when no driving voltage was applied during t off .Apparent oscillations in velocity profiles were observed for large t off due to the significant velocity drop when the driving voltage was terminated.Similar trends were also observed for other t on (e.g., t on = 10 ms, t on = 30 ms, see Figure S16, Supporting Information).The results indicated that longer t on led to a thicker accumulation of the oxide layer, which in turn increased t c (Figure S17, Supporting Information).Therefore, it was the surface oxidation that decelerated the DC-driven LMDs.Using pulsed driving signals, i.e., intermittently cutting off the driving signal, can be beneficial for improving droplet motion provided that efficient oxide removal can be achieved at minimal t off (t off ≈ t c ).We visualized the flow patterns of the surrounding electrolyte created by the moving LMDs under both DC and PTM driving conditions (Figure 3m-p).It revealed interesting hydrodynamics underlying the different types of droplet motions.In the current system, the LMDs gained the forward net propulsion force mainly from conveying the upstream electrolyte backward via stroking.For the DC-driven case, continuous streamlines were formed in the surrounding electrolyte.The separation point between the liquid metal surface flow and the electrolyte flow was seen not on the equator region (middle point) but shifted to the front side of the LMD, generating an outwardflowing electrolyte stream (Figure 3m; Movie S2, Supporting Information), even when the droplet completely stopped (Movies S2 and S3, Supporting Information).Based on the aforementioned characterizations, we attributed such a shift of the separation point and the divergent electrolyte flow to the oxidation of the opposite (backside) part of the droplet and the suppression of the surface flow therein.In this case, the net force acting on the LMD, generated by pushing the electrolyte, also formed an angle with the droplet moving direction.Consequently, only its projection to the droplet moving direction became the effective propulsion.
The R-1 PTM-driven LMDs showed DC-like flow patterns, apart from that pulsed streamlines were induced by the pulsed driving signal (Figure 3n; Movie S3, Supporting Information).The similarity in the flow patterns between the DC-driven and R-1 PTM-driven LMDs explains the similar velocity profiles of the two.The presence of surface oxide on the backside of the LMDs was believed to be the reason shared by both cases.In the R-2 and R-3 PTM-driven droplet motions, the increase in t off allowed the surface oxide of LMDs to be sufficiently removed, rendering its influence negligible.In these cases, the flow separation point was observed to be at around the equator of the droplets and the electrolyte streams to be aligned with the droplet moving direction (Figure 3o; Movie S4, Supporting Information; Figure 3p; Movie S5, Supporting Information).As a result, the propulsion acting on the LMDs was maximized.The flow patterns remained unchanged over the whole duration (until the LMDs reached the channel end), showing the stability of the droplet motion after effective surface oxide removal.This rationalized the observed high droplet velocity in the R-2 and R-3 droplet motion.The decrease in droplet velocity in R-3 droplet motion was mainly due to the deceleration by viscous drag during prolonged t off .
Our results indicate that viscous effects and surface oxide formation were the two critical phenomena that affected the LMD motions under both DC-driven and PTM-driven conditions.As shown in Figure 4a, during the t on phase, the surface Marangoni flow of the LMDs pushes the adjacent electrolyte backward through the viscous effect (assuming nonslip boundary condition), which generates a counter force that drives the forward motion of the LMD.In other words, the viscous effect is responsible for the generation of the driving force F Prop for the LMDs during t on .The presence of a solid oxide layer can greatly affect the surface tension driven flow of the LMDs, and here the influence Ψ of the oxide layer on F Prop is expressed with the Gompertz function: [42] where (t) is the oxide layer thickness which is determined by oxide growth during t on and the oxide dissolution during t off .The parameters c 1 , c 2 , c 3 and c 4 are coefficients related to the applied driving conditions and system electrochemistry and were experimentally determined.Ψ = 1 for (t off t on −1 << 1) and Ψ = 0 for (t off t on −1 >> 1) indicate the high (positive) and low (negative) impact of oxidation on F Prop , respectively (Figure 4b).
When the potential is off during the t off phase, the LMDs decelerate due to viscous drag described by the following equation: where  is the LMD density, C D,off is the drag coefficient and A = (D/2) 2 is the cross-sectional area of the droplet.For a given t on , increasing t off favors minimizing the impact of oxidation but increasing that of viscous drag (Figure 4b).The maximum driving force occurs when t off is sufficiently long for effective oxide removal but reasonably short for avoiding prolonged viscous deceleration (t off ≈ t c ).By considering the net propulsion (F Prop ), influence of oxidation () and viscous drag (F D ) for t on and t off separately, we build the following equations to describe droplet motion over n (n = 1, 2, 3, …) PTM cycles with the signal period T (T = t on + t off , see SI for more details): Equation ( 3) allowed the determination of droplet velocity at any time points and thus the velocity profiles of the LMDs.As shown in Figure 4c, our quasi-analytical model was able to generate all regimes of experimentally observed velocity profiles by tuning t off (fixed t on ).The successful implementation of the model confirms that surface oxide formation and viscous drag are the two critical phenomena affecting the droplet motion.It also evidences that using PTM is a viable strategy to fine tune both phenomena and achieve efficient LMD actuation.
With a detailed understanding of the motion of the PTMdriven LMDs, we further proposed an offset PTM (O-PTM) strategy to improve the driving efficiency for certain types of droplet motions.Considering that for the R-1 motion, the noneffective surface oxide removal caused the LMDs to quickly slow down after applying the PTM signal, we introduced high frequency (15 kHz) and short (13 μs) offset pulses (−15 V) during the t off phases (Figure 5a; Figure S18, Supporting Information) with an effective voltage of −3 V.These short-offset pulses provided a counter electric field which hydrodynamically and electrochemically facilitated oxide removal in addition to the oxide dissolution by NaOH.As shown in Figure 5b, applying the O-PTM signal switched the R-1 PTM-driven droplet motion (v < 20 mm s −1 ) to a high-velocity and stabilized R-2 motion (v ≈ 50 mm s −1 ).The O-PTM strategy was less effective for R-2 motions when the influence of oxide was less significant (Figure 5c).When applying the O-PTM signal to R-3 motion LMDs (Figure 5d), where the impact of the oxide is minimal because of prolonged t off , the droplet velocity decreased due to the reversed Marangoni flows induced by each offset pulse.Leveraging the high-speed and stable droplet motion driven by PTM, we developed a miniature metal rotary motor (Figure 5e-g).A LMD (D = 3 mm) immersed in a NaOH solution (c = 0.5 mol L −1 ) was used as the rotor which was driven by three electrodes arranged symmetrically in a circular chamber.R-2 PTM (t on = t off = 10 ms) signals were applied across each two electrodes when the LMD was located in between.To obtain continuous and stable rotary motion, three-phase PTM signals were applied (Figure 5h).The switching of the signals between the electrodes was controlled by an inductive sensor underneath the bottom of the chamber, which was able to detect the real-time position of the LMD.The LMD exhibited stable average velocity and continuous circular motion (Figure 5i; Movie S6, Supporting Information).It was able to work across a driving voltage range from 9 V to 25 V and obtain a maximum average velocity of 126 mm s −1 under 21 V driving voltage (Figure 5j; Figure S19, Supporting Information).This corresponds to a displacement of 42 droplet body lengths (diameter) per second, which is the highest continuous speed achieved controllably with a single LMD.Further increasing the voltage did not lead to higher droplet velocity, which can be attributed to the observed strong electrolysis (as accompanying bubble generation was seen) on the graphite electrodes negatively affecting the driving electric-field.The motor device was tested for long-term operation, and it demonstrated an overall stable velocity profile over 20 min continuous operation (Figure 5k).The slight variations in the velocity were thought to be caused by the change in ion concentrations (e.g., Ga oxide dissolution) in the solution.
In this work, we did not change the viscosity of the media during the experiments and we did not experiment with the influence of added mechanical energy. [43,44]It is highly likely that both viscosity and mechanical forces will have significant impacts on the LMDs motions.We suggest that studies of these effects should be conducted in the future works.

Conclusion
The introduction of the PTM strategy enabled us to successfully control Marangoni flow, surface oxidation, and viscous force in controlling the motions of LMDs.To comprehensively investigate the application of PTM, we designed a series of experiments for studying the motion of LMDs driven by external electric fields that were pulse time modulated.By regulating the on-state off-state times of the PTM signals individually, the contributions of the critical factors were fine-tuned for obtaining distinct droplet motion regimes.We demonstrated that effectively inhibiting the adverse impact of the surface oxide of the LMDs could lead to highly controlled droplets, at high speed and stability, as well as a reduced power consumption.The flow visualization/analysis and surface electrochemical characterization supported our interpretation regarding the mutual interplays between these critical factors.Subsequently, a quasi-analytic model developed accounting for the Marangoni-flow-induced driving force, surface oxidation, and viscous effect that enabled us to define experimental observations of droplet motion regimes.Additionally, machine learning was applied to extend our prediction of droplet motion regimes to a wide droplet size range.Our findings depicted a clear mechanistic picture for the electric-fielddriving motion of LMDs using voltage pulses of different on/off states and provided a readily applicable non-contact strategy for the programmable actuation of LMDs.Eventually, by showcasing a rotary LMD-based motor operated and controlled by PTM, we evidenced that this work could be implemented in practical settings and empower next-generation droplet-based applications.

Experimental Section
Droplet Motion Tracking: The experimental setup consisted of a weighed eutectic gallium indium (EGaIn, prepared by melting 75 wt% Ga and 25 wt% In) droplet placed on a horizontally aligned, curved glass channel within an acrylic container filled with a 0.5 mol L −1 sodium hydroxide (NaOH) solution (Figure 1a; Figure S1, Supporting Information).The desired LMD diameter was achieved by measuring its weight with a milligram balance (ENTRIS641-1S, Sartorius Lab Instruments).Two cylindrical graphite electrodes were positioned 100 mm apart on both ends of the channel.The electrodes were connected to the dual full-bridge driver (L293B), the external power supply (Keysight E36311A), and the Arduino microcontroller, which defined the signals (DC, PTM, O-PTM) to generate the electric field.The applied supply voltage Us experienced a voltage drop of ≈0.6 V due to the driver.To improve the repeatability of the experiments, each run started with bringing the LMD shortly into contact with the graphite electrode to achieve reproducible electrochemical states. [45]he movement of the LMD was recorded by a smartphone camera (framerate: 60 Hz, resolution: 1080p).The videos were postprocessed into image sequences using Python.A Matlab program identified the LMD position for each image to obtain the absolute position, velocity, and acceleration over time.
High-Speed Camera Imaging: The experimental setup was identical to the setup used for the position tracking except that the standard camera was replaced by the highspeed camera Phantom VEO-610-L-18G-M (lens: 100 mm F.28 CA-Dreamer Macro 2x).The electrolyte flow was visualized with food dye.The high-speed camera works with a grayscale imaging sensor, so the color information was not restored in the images.
Electrochemical Measurements: The experimental setup consisted of a three-electrode cell with the LMD as the working electrode, the reference electrode (CHI 150), and a gold wire as the counter electrode (Figure S12, Supporting Information).The electrochemical measurements were performed with an electrochemical workstation (CH Instruments, Model 760E).The EGaIn droplet was held by a 3D-printed platform (Polylactide) with a depression within the same acrylic container from the previous experiments and immersed within a 0.5 mol L −1 NaOH solution.The threeelectrode cell was aligned perpendicular to the external electric field generated by the graphite electrodes to minimize the influence of the potential gradient across the electrolyte on the measurements.OCP and i-t (initial voltage = −1.56V) were conducted while applying the external electric field.Each run was performed twice, but with the signal electrodes reversed and the results averaged to compensate for small misalignments (Figure S13, Supporting Information).All experiments were conducted at room temperature.

Figure 1 .
Figure 1.Liquid metal droplet (LMD) driven by pulse time modulated (PTM) voltages and its comparison to DC driven LMDs.a) Schematic of the experimental setup.b) Schematic of the potential gradient profile across the channel and polarization of the LMD when an external voltage is applied.cf) PTM-driven LMD motion: c) Time-lapse images of the LMD movement (time between frames: 200 ms, t on = 10 ms, t off = 10 ms); d) The applied PTM signal; e) Droplet velocity as a function of time.For t off = 10 and 30 ms, the velocity curves terminate upon the droplets reaching the end of the channel; f) Flow-field visualization of the electrolyte surrounding the LMD.g-j) DC-driven LMD motion: g) Time-lapse images of the LMD movement (time between frames: 200 ms); h) The applied DC signal; i) Droplet velocity as a function of time; j) Flow-field visualization of the electrolyte surrounding the LMD.The upper part images of (f and j) are high-speed camera snapshots, and the lines in the lower parts are schematics illustrating the flow patterns (not necessarily the streamlines).

Figure 2 .
Figure 2. Characterization and categorization of LMD velocity profiles.a-c) Velocity profiles for PTM-signal driven LMD (D = 3 mm) at varied t off and fixed t on = 5 ms (a), 10 ms (b), 30 ms (c) and their respective Reynolds number (Re).d) Comparison of velocity profiles for DC-driven (dashed lines) and PTM-driven (solid lines) LMDs under different voltages U S .In insets (a-d), the velocity curves terminate upon the droplets reaching the end of the channel.e) Phase diagram of velocity profiles based on t on and t off .The scatters represent experimental data points (R-1: diamonds; R-2: triangles; R-3: circles).The color-filled regions were drawn based on the described machine learning classification.The inset shows a magnified region with low t on and t off .f) Phase diagrams of velocity profile regimes for different droplet sizes D as indicated.

Figure 3 .
Figure 3. Electrochemical and hydrodynamic analysis of electric-field-driven LMDs.a-l) Electrochemical characterizations of the LMDs under the external polarization of U S .a,d,g,j) Voltage signals applied between the two graphite electrodes.a) DC signal and PTM signals with fixed t on = 5 ms and d) t off = 3 ms, g) t off = 10 ms, and j) t off = 30 ms. b,e,h,k) Time-dependent surface potential (OCP) and c,f,i,l) current of the LMDs measured using a threeelectrode electrochemical system.m-p) Visualization of electrolyte flow patterns under different types of droplet motions after the initial acceleration phase.The inset cyan arrows and red dots indicate the flow direction of the electrolyte and flow separation points, respectively.The black arrows indicate the moving direction of the LMDs.

Figure 4 .
Figure 4. Force analysis of electric field driven LMDs.a) Schematic force analysis of the driving force F Prop and the drag force F D and schematic LMD motion under the influence of asymmetrical surface oxidation.b) Relationship between the oxide layer impact  the viscous drag F D specifically during the t off -phase and the driving force F Prop in dependency of each velocity profile regime.c) Three types of velocity profiles generated using the quasianalytical model developed in this study.The same PTM conditions as in Figure 1e were used to generate the curves.

Figure 5 .
Figure 5. Demonstration of an extended PTM strategy and the application of the PTM strategy in a droplet-based system.a) An example of the offset PTM (O-PTM) signal with a counter electric field during t off to facilitate oxide removal.b-d) Comparison of velocity profile types for the same LMDs driven by PTM and O-PTM signals.e-g) Image sequence for the liquid metal-based motor device controlled by a three-phase PTM signal.h) The applied three-phase PTM signal with t on = t off = 10 ms and U S = 20 V. i) The distance-time plot of the PTM-driven LMD at the beginning of its motion.The inset figure shows the x-y position (circular motion) of the LMD over time.j) Average velocity of the liquid metal rotor under different PTM driving voltages (t on = t off = 10 ms).k) Velocity-time of the liquid metal rotor during long-term continuous operation.