Vibration‐Enabled Mobility of Liquid Metal

Directed liquid metal (gallium‐based) manipulation and actuation are paramount for copious applications, including soft robotics, soft electronics, and targeted drug delivery. Although there are several strategies available to achieve mobility of liquid metals in a “wet” environment. Strategies to achieve and improve mobility of liquid metal droplets and puddles in a “dry” environment are scarce and rely on metallophobic surface design or liquid metal marbles. Here, high mobility of Galinstan is elucidated by combining metallophobic surface design and vertical vibrations. Vibration frequencies between 20 and 30 Hz are conducive to droplet movement and threshold inclination angles of 0.5° to 1° are observed upon actuation by these vibrations. The method itself is applicable for a wide range of droplet sizes (30 and 2000 µL) and very robust. The droplet movement typically comprises of periodic receding and advancing of the droplet and commences via a rolling mechanism rather than a gliding mechanism. Finally, it is shown that small (0.5 mm height) obstacles can be traversed by this method, indicating that it can be used in concert with other strategies, such as surface structuring strategies, which open up pathways for mobility and controlled actuation of liquid metal droplets in air.


Introduction
Control and manipulation of liquids play fundamental roles in microfluidics, water harvesting, chemical reactions, and biomedical analysis.In recent years, droplet-based motion has garnered Both general means necessitate a reason for directional movement, which arises due to asymmetry.Passive driving forces are the aforementioned chemical gradients and topographical gradients. [1,2]Active mechanisms for movement rely on external energy input, such as electric, magnetic, light, heat (i.e., Marangoni effect-based), and vibrations, [3] albeit the driving force may be derived from an asymmetry in the surface texture/structure [3a,c] or the external energy. [4]These kinds of water transport methods are already long in use by nature, for instance, the feeding mechanism of capillary feeding shorebirds relies on ratcheting of tiny droplets by time dependent changes in beak geometry, a combination of a passive (geometry) and active (change in geometry of the beak). [5]Vibrations have shown interesting features, such as the ability to move liquid upward a slope at specific drive frequencies and amplitudes. [6]3c] Similarly, liquid infused ratchets in combination with horizontal vibrations were recently used to drive water droplet motion for droplet volumes from 0.05 to 2000 μL, [7] showing significant progress in directed transport of liquids.While there are many methods to move and manipulate water and organic liquid droplets, manipulation of liquid metals is more demanding.
Liquid metals, metals, and alloys liquid at or near room temperature, have garnered attention in the scientific community due to an interesting combination of properties.Granted the long-used liquid metal mercury has been phased out due to safety concerns, see Minamata convention. [8]Nowadays, mercury has been substituted by gallium-based liquid metals.These metals and alloys feature electrical and thermal conductivity, low viscosity, low toxicity, and a reactive and regenerating surface. [9]Therefore, their potential for application as chemical reaction environments, [10] in soft electronics, [11] and in soft robotics is scrutinized. [12]For many applications, especially, for soft electronics (i.e., patterning methods), biomedical applications (i.e., controlled drug delivery), and soft robotics, reliable and controlled movement or manipulation (i.e., deformation) of the liquid metal is essential.
Mobility and directed transport of gallium-based liquid metals are more challenging than for "conventional" liquids.This difficulty arises from the reactivity of gallium toward oxygen, as these liquid metals form a self-limiting oxide skin.This oxide skin is sticky toward most surfaces while featuring yield stress. [13]Because of the sticky nature of the liquid metal oxide, liquid metals are often actuated in liquids, where one can exploit the slip layer. [13,14]Liquid metals do not stick to smooth surfaces when a thin layer of water or organic solvent is deposited on the surface first.Interestingly, slip layers can also be used to achieve rebound of liquid metals.Here, the slip layer thickness was found to impact the rebound of liquid metal droplets while helping to avoid the adhesion of the liquid metal. [15]Rebound of liquid metals can also be observed on structured surfaces, an example with enhanced rebound is the article by Wang et al. who exploited micro/nano needles for storing the impact energy and releasing a portion of it for rebound. [16]Generally, actuation of liquid metals is easier in liquids than in air.Actuation means in liquid or on a liquid layer rely on magnetic fields, [12a,17] electric fields, [18] chemical gradients (chemotaxis), [19] fuels (i.e., bubble propulsion), [20] and ultrasound (low μm-scale object). [21]Though the latter three methods have only been shown for small droplets.Without an exterior liquid phase, controlled actuation of liquid metals becomes arduous, as the liquid sticks to other materials.Therefore, non-stick, so-called metallophobic, surfaces have been established, which rely on hierarchical roughness of the surface minimizing the contact area between the liquid metal oxide and the substrate. [22]Another method involves decorating small liquid metal droplets with micro-or nanoparticles, generating liquid metal marbles.Both methods can be used to achieve limited mobility of the liquid metal. [23]The latter is impacted by particle detachment and generation of pristine surface areas under stresses for larger droplets (ca.>100 μL).Movement and deformation of liquid metal droplets "in air" have been generally obtained by gravity (tilting the sample) or by magnetic actuation.In our experience, both methods are rather unreliable, as the shape of the droplet dictates the necessary force (i.e., tilting angle).Further, actuation of magnetic liquid metal droplets by a magnetic field can deform the liquid metal droplet substantially, and perhaps rendering it unresponsive due to adhesion induced by the forces acting on the liquid metal, that is, by forced wetting. [13]erein, we showcase a new paradigm to actuate liquid metal droplets (Galinstan, a eutectic alloy of gallium, indium, and tin; denoted EGaInSn) in air (without another liquid, i.e., without slip layer).The method encompasses the combined use of a metallophobic substrate and vertical vibrations.The method features great repeatability, control of the linear translational speed by control of the tilting angle, and a large transport volume range, from 30 to 2000 μL and potentially even greater volumes.Further, we show that only minute inclination angles can be employed to actuate the liquid metal droplets.There are many possible combinations of vibration frequencies and amplitudes, in-clination angles, and droplet sizes.Therefore, initially, the vertical vibration was optimized for each droplet size, namely, the frequency and amplitude.Then, the threshold inclination angle and the droplet linear translational speed at a fixed vibration were analyzed.Finally, the movement mechanism was analyzed, and the ability of liquid metal droplets to cross obstacles was investigated.Due to the high mobility induced by the vertical vibrations, this method can be used in conjunction with other actuation methods, that is, magnetic actuation, [12a] or with textures to enable controlled and on demand directed movement of liquid metals.

Materials
The eutectic alloy of gallium, indium, and tin (EGaInSn) with a composition of Ga 68.5 wt%, In 21.5 wt%, Sn 10 wt%, denoted in literature as Galinstan, was used in all experiments.The properties of the liquid metal are as follows: density () = 6440 kg m −3 , pristine surface tension () = 605 mN m −1 -oxidized surface tension ca.9a,24] The alloy was purchased from Dongguan Wochang Metal Products Co. (Dongguan, China).90 mm standard polystyrene petri dishes were used as substrates (Biosharp, Hefei, China), and were coated by the liquid metal phobic (metallophobic) coating Neverwet (Rust-Oleum, Vernon Hills, United States). [25]A structured and rough diamond coating was generated with specific liquid metal phobic morphology on a Si-sample according to a procedure outlined earlier. [24,26]

Characterization
Scanning electron microscopy (SEM) micrographs and energydispersive X-ray spectroscopy (EDS) mappings were taken with the APREO S (Thermo Fisher Scientific) at an acceleration voltage of 5 kV.Micro-Raman scattering was measured with the HORIBA LabRAM HR800 Evolution Raman spectrometer employing an excitation wavelength of 532 nm.

Fabrication of Liquid Metal Phobic Samples via Neverwet Coating
Polystyrene petri dishes were cut (the outer rim was trimmed) to obtain flat PS disks from single use petri dishes and glued to cardboard.Then, the base coat (Neverwet) was applied twice and left to dry for at least 30 min in the fume hood (denoted as step 1 in the instructions).Subsequently, the top coat (step 2) was applied two times and the sample was left to dry overnight (12 h).The coating was applied from an approx.distance of 20 cm.More information on the coating procedure and some key points to keep in mind can be found in the instructions manual of the commercial coating and the article by the Dickey group. [25]ww.afm-journal.de

Setup and Workflow of Vibration-Induced Movement of the Liquid Metal
A Frederiksen Vibration generator no 218 500 with a circular Chladni plate was used to generate precise and defined vibrations (Frederiksen Scientific, Drammen, Norway).The Chladni plate was outfitted with a paper coating to protect it from liquid metal attachment and corrosion.On this protective paper coating the liquid metal phobic samples were placed (glued).A Keysight 33210A Function/Arbitrary Waveform Generator (Keysight Technologies, Santa Rosa, United States) was employed to generate square-shaped waves with frequencies ranging from 1 to 60 Hz at amplitudes ranging between 0.1 and 10 Volt (later denoted V).The function generator was directly connected to the vibration generator.Furthermore, the vibration generator was placed on a manual tilting stage, which can measure the tilting angle between −20°and 20°.For liquid metal droplet deposition on the sample, the tilting stage was set to 0°.Then, the sample (attached to the vibration generator) was placed below a self-built dispensing unit, comprising a fluorocarbon tip, a 20 cm long silicone rubber peristalsis tube, and the Leadfluid TYD01 syringe pump (Baoding Lead Fluid Technology Co., Ltd., Hebei, China) outfitted with a 10 mL syringe.The 10 mL syringe was initially filled with EGaInSn and the pump was used to control the liquid metal droplet volume.We estimate the deviation of the droplet volume to ≈15 μL (see Supporting Information).A certain volume, between 30 and 2000 μL, was dispensed by the pump and placed carefully on the sample at a distance <1 cm between the nozzle and the sample.Then, the horizontal level of the Chladni plate was assessed and aligned by use of a spirit level (in both directions, x and y).Subsequently, the tilting stage was set to the desired tilting angle and vibrations were applied to move the droplets.The vertical vibrations are schematically shown in Figure 1.As blank experiments, droplets were moved by only tilting the stage or by tilting the stage after flattening the drops by using vibrations (i.e., strong vibrations: 4-6 V at 20 to 30 Hz).Note that the real amplitude (in mm) of the vibration is for this setup dependent on the drive frequency and amplitude in volt.The amplitude in mm at 27.5 Hz was 1.1 and 3.3 mm for 2 and 6 V and, respectively.These values were determined from measurements via a high speed camera (see below).After the experiment, the horizontal level of the Chladni plate was ascertained a second time.Movement of the droplets was captured in a topview (or rarely: side view) setup by a Nikon D7100 camera (Nikon Corporation, Tokyo, Japan) at an acquisition rate of 29 Hz (29 pictures per second) with a Nikon AF-S DX Nikkor 35 mm f1.8G lens (Nikon Corporation, Tokyo, Japan) or in a side-view configuration by a high-speed camera (Phantom Miro Lab 110, Vision Research, Wayne, United States) at an acquisition rate of 400 Hz, a resolu- tion of 768 pixels x 768 pixels, and an exposure time of 2400 μs.Movement was determined from the advancing front edge of the droplet.

Metallophobic Nature of the Coatings
Although this work does not focus on the study of liquid metal phobic or better liquid metal oxide phobic (metallophobic) surfaces, a short overview of the two different liquid metal phobic surfaces and the origin of their liquid metal repellency are given here briefly.
Gallium-based liquid metals react with oxygen rapidly and form gallium oxide (mixed oxides of Ga 2 O 3 and Ga 2 O). [27]This oxide skin is considered to be sticky on many substrates. [13]Many methods have been established to avoid the adhesion of liquid metals, but most of them are only applicable in liquid surroundings.One rationale to inhibiting the adhesion of liquid metal (with oxide skin) in air without the use of surrounding liquids is the use of roughness combined with hydrophobic or even superhydrophobic surfaces. [13]A high roughness reduces the contact area between the liquid metal and the surface substrate drastically, which is ascribed to the yield stress of the oxide skin.The liquid metal oxide is in the case of a rough substrate only in contact with surface protrusion and does not infiltrate pits between protrusion as long as the pit and protrusion dimensions are favorable. [28]Notably, wettability also plays a role, but a small change in wettability is not that impactful. [13]Consequently, several rough surfaces feature liquid metal repellency.Here, the Neverwet coating and structured diamond coatings were used.The former was used due to the ease of access and fabrication, while the latter was chosen due to the low friction coefficient of diamond.Furthermore, during experiments leading to a previous report on the surface tension of the liquid metal oxide skin, [24] we found that liquid metal droplets, especially larger ones, were very mobile on structured diamond coatings upon exposition to vibrations.
In Figure 2, the surface morphology of the Neverwet and diamond coatings are shown.The Neverwet coating appears to be homogeneous (Figure 2a) and comprises of two layers of roughness.The microscale roughness originates from clusters or aggregates spaced out hundreds of micrometers.These clusters have a height of tens of micrometers, as discussed by the Dickey group. [25]The second layer of roughness originates from hills of nanoparticles with 100 nm or less and comprise of silica nanoparticles (ca.20 nm).Further characterization of such a Neverwet coating can be found in the article by the Dickey group. [25]It should be noted that this kind of coating is F-terminated.The F-termination, in combination with the rough and porous structure of Neverwet, leads to hydrophobic or superhydrophobic surfaces.After the experiments were concluded (without cleaning), the water contact angle was 134°± 5°.
Similarly, the diamond coating features hierarchical roughness.The coating comprises of diamond hemispheres of ≈2 μm height and 2.2 ± 0.5 μm width, the coating itself is continuous and nanocrystalline (50 to 250 nm grain size).The hemisphere density on the sample is an important parameter for liquid metal repellency and was chosen to be close to that employed for the superhydrophobic diamond coating in our previous article. [24]n comparison to the previously published coating, the density of the hemispheres is slightly higher, leading to somewhat coalesced diamond hemispheres, as seen in Figure 2d.The observed surface structure arises from the two-step deposition process and the prior intricate seeding procedures, as outlined previously. [26]riefly, in the first seeding step, nanodiamond seeds are stabilized by oxalic acid, leading to controlled low seeding densities on the substrate.Then, the diamond is grown by hot filament chemical vapor deposition (HFCVD), resulting in the growth of hemispheres due to the growth mechanism of diamond (Vollmer-Weber; also known as island growth). [29]Subsequently, a second seeding procedure with a highly stable nanodiamond colloid (and conducive surface charge, that is positive Zeta potential of the nanoparticles) was executed, yielding a high seeding density on the sample.Finally, a second HFCVD growth process was executed, resulting in the growth of a coalesced diamond film on the sample.To verify the identity of the coating, Raman spectra were measured (Figure S1, Supporting Information).The peak at 1333 cm −1 verifies the presence of diamond.Further, the shoulder at 1350 cm −1 and the peak at 1582 cm −1 signify the presence of graphite (or graphitic material) and correspond to the D and G bands, respectively, [30] a common by-product during the diamond synthesis.Though the diamond coating features a structured surface and diamond films are initially hydrophobic, the diamond sample was hydrophilic with a water contact angle of 35°± 11°.This hydrophilicity arises from slow oxidation of the diamond surface (the sample used was ≥ 6 months old).Representative images of the contact angles are shown in Figure S2 (Supporting Information).
Both coatings feature hierarchical roughness and are good candidates for liquid metal-repellent surfaces.The surface wettability (contact angle) of liquid metals on these materials is low and the contact angles are high, with angles typically >160°, [24,25] and increase upon vibrating to 170°on the rough diamond sample, as reported earlier. [24]As a note of caution, contact angles of liquid metals are highly subjective due to the yield stress of the oxide skin. [31]Further, liquid metal droplets are considered to exist in the Cassie-state on rough surfaces, [28a] stabilized by the high surface tension and yield stress of the oxide skin, and the liquid metal is not anticipated to transition to the Wenzel state upon exposition to vibration (a transition to the Wenzel state would likely lead to adhesion of the liquid metal to the surface, see forced wetting). [13]To signify the excellent liquid metal repellency of the Neverwet coating, an SEM micrograph in combination with maps were taken after the movement experiments below were concluded.Notably, no cleaning of the samples was conducted before SEM and EDS measurements.Neither in the SEM micrograph nor in the EDS maps or the EDS spectrum the presence of liquid metal (gallium, indium, or tin) could be ascertained (see Figure S3, Supporting Information), signifying the excellent liquid metal repellency of the Neverwet coating.For the diamond coating a similar behavior is expected as long as the coating is Hterminated (hydrophobic).Besides the advantage of low friction coefficient of the diamond coating it also possesses a high chemical resistance and can be easily cleaned by acid or base if contaminated by liquid metal (oxide) adhesion. [32]Notably, these two structured surfaces only serve as examples of usable materials and were chosen due to their availability rather than their performance.We anticipate that other liquid metal repellent surfaces (likely based on hierarchical roughness [13,22a,33] ) can be used in a similar fashion and might even show better performance than the coatings featured here.

Liquid Metal Movement on Inclined Surfaces by Vibrations
In general, it is difficult to move liquid metal in air, as it is outfitted with the sticky oxide skin and features yield stress.Further liquid metal droplet movement is commencing via rolling, and the issue of the yield stress of the oxide is that non-equilibrium shapes, for instance, ovaloid or flattened shapes, are often obtained due to an impact of the droplet on the surface.Similarly, flattening can be caused by vibrations, and this flattening has an impact on the roll-off angle (mobility).This makes reliable and repeatable measurement of roll-off angles even for very small droplets challenging.To set the stage, the necessary in-clination angles for liquid metal droplets rolling off the Neverwet coated polystyrene were determined, as shown in Figure 3a.Small droplets (30 μL) roll-off the sample at an inclination angle of ≈13.5°± 0.5°.However, flattened droplets show an increased roll-off angle of ≈19.7°± 1.0°.Relatively low roll-off angles have been published for round liquid metal droplets deposited very gently (or by improving their roundness) on metallophobic surfaces and liquid metal marbles, that is, ca.23a,b,34] Maybe overlooked in this context is the fact that the roll-off angle is dependent on the droplet volume and droplets and marbles deform upon deposition.Small non-circular objects generally need a higher inclination angle in order to roll off (see Figure 3a).23a] The necessary inclination angle increases for droplets of 100 μL and then, decreases for bigger droplets to ≈6.8°± 0.2°(2000 μL).The higher necessary inclination angle for 100 μL stems from the non-circularity of the droplet, as it appears ovaloid (higher contact area and lower center of mass, which are both detrimental to droplet movement), and the low overall mass.The low mass is also the reason for the high inclination angle observed for the 30 μL droplet, as the contact line pinning or adhesion to the substrate has to be overcome.Bigger droplets are more flattened but necessitate a much lower inclination angle due to their high mass.This is due to the fact that the liquid in the oxide skin shifts the weight to the lower region of the droplet and this mass is able to move the oxide skin in the process.It should be noted that this motion is still a "rolling" motion and not a sliding motion (see Section "Motion mechanism").One can see this for "semi-metallophobic" surfaces, such as paper, where larger liquid metal droplets roll down while leaving a liquid metal oxide trail behind them on the paper. [35]In this case, the adhesion of the liquid metal oxide toward the paper overcomes the cohesion of the liquid metal.
Shifting to vibration-enabled movement.In Figure 3b, the minimum inclination angle necessary to actuate droplets with a volume between 30 and 2000 μL is plotted versus the droplet volume for the Neverwet and the diamond coating when the droplet is actuated by vibrations.Smaller droplets were not characterized as they tended to jump erratically.For this measurement, the vibration frequency and amplitude were optimized, as shown in Figure 3c.As frequency, 27.5 Hz was used for the measurement, while the amplitude and inclination angle were varied.Generally, the amplitude was varied between 2 and 6 V (amplitude 1.1 to 3.3 mm), but most of the time, 2 to 3 V (amplitude 1.1 to 1.65 mm) would move droplets once the inclination threshold for movement was reached.One should note that during vibrationenabled movement the droplets deform (are flattened). [24]The minimum inclination angle for the liquid metal to move on Neverwet appears with ca.1°± 0.3°across all drop sizes slightly higher than that on the diamond coating (0.5°± 0.3°), perhaps due to the low friction coefficient of diamond. [29,36]However, scatter in these small values is observed and thus no conclusive comment on this is given.An example of the movement of liquid metal on the diamond sample at a low inclination angle is given in Figure S4 (Supporting Information).It shows video stills from the droplet movement at an inclination angle of 0.5°: droplet volume 600 μL, vibration 27.5 Hz, and 3 V (1.65 mm amplitude).Due to the low inclination angle, the droplet movement is very slow.Interestingly, the threshold inclination angle is rather independent of the droplet volume, suggesting a somewhat different mode of motion than "normal rolling" (see Section "Motion mechanism").An exception to this may be very small droplets with volumes (30 μL or less, Bond number <5), as they appear to necessitate a lower inclination angle (Neverwet coating) than the bigger droplets.For each droplet volume, several parameters need to be optimized, namely, the drive amplitude (between 0.5 and 10 V), the frequency (between 5 and 60 Hz), and the inclination angle.In Figure 3c, the inclination angle was set to 3°and the drive amplitude, frequency, and the droplet volume varied.For droplets (≤ 650 μL, Bond number between 5 and 15), an optimal frequency range between 20 and 30 Hz is observed, and the necessary drive amplitudes are ca. 2 V.For lower frequencies, the necessary drive amplitude increases rapidly.Similarly, for higher frequencies, the threshold amplitude increases, yet the droplets remain relative mobile.For instance, movement can still be obtained by a higher inclination angle or manually tilting.Though, the motion itself remains relatively slow compared to the linear translation speed observed in the optimal frequency range, and it does not appear to depend on the drive amplitude at these higher frequencies.For bigger droplets (or puddles; ≥1000 μL, Bond number > 15), the liquid metal moves even at low (i.e., 5 Hz) and high frequencies (i.e., 60 Hz) at driving amplitudes of 4 V or lower.However, the optimal frequency range remains between 20 and 30 Hz, unaltered from the optimal range for the smaller droplets, and driving voltages ≈1 V are only necessary to induce movement of the liquid metal on the Neverwet coating.3c,7,37] Sessile droplet oscillation can be either rocking, resulting from center of mass oscillations or Rayleigh oscillations.4b] In this equation, n denotes the mode number, an integer of 2 or higher,  denotes the surface (or interfacial) tension, R is the liquid drop radius, and  is the density of the liquid.Similarly, the rocking resonance frequency can be calculated by Equation (2).
Here, h(Θ) is a dimensionless geometric function that has been determined numerically dependent on the contact angle of the liquid on the surface by Celestini and Kofman. [38]The Rayleigh frequency (n = 2) of the liquid metal droplets and puddles was determined under the assumption that the surface tension is 365 mN m −1 (oxidized) and that the oxide skin behaves liquid-like to 251 Hz (30 μL) to 30 Hz (2000 μL).The rocking frequency was calculated by using a h (170°) of 0.024. [38]The resonance rocking frequency was estimated to 18 Hz (30 μL) to 2.2 Hz (2000 μL).However, one should note that the calculation of the rocking frequency heavily relies on the knowledge of the contact angle (Θ).The contact angle (170°) used here was taken from our previous article on the estimation of the surface tension of the liquid metal oxide skin, [24] though one should note the untrustworthiness of contact angle measurements in liquid metal research stated earlier. [31]The forced frequencies show good actuation performance and are for the biggest droplet close to the Rayleigh frequency and for the smallest droplet volume reasonably close to the rocking resonance.However, we assume that neither of the resonances contribute much (except for the largest droplet volume: 2000 μL) to the mobility of the liquid metal, rather we suggest that the shape of the curves in Figure 3c stems from the transferred impulse.The transferred impulse is affected by the "real" amplitude (in mm) and the frequency (in Hz).Yet, the real amplitude of the vibration dwindles with increasing frequency of the vibration even though the driving voltage remains unaltered (see manual of the vibration generator).An increase in frequency by itself increases the transferred impulse.However, the reduction in real vibration amplitude reduces the transferred impulse.The competition of these effects results in the parabola shape of the curves observed in Figure 3c.
Notably, the threshold driving voltage necessary to induce movement on the inclined sample is lower the higher the vol-ume of the droplet is, perhaps related to the higher driving force (due to higher mass of the droplet).3c] Another argument is the fact that a competition or balance between the pushing effect caused by large droplet mass upon vibration (compare Figure 7) and the hindering effect caused by large contact area and contact line pinning exists.
As an example of droplet movement, Figure 4 shows the top view of droplet motion of 30 and 1000 μL liquid metal droplets on Neverwet coated polystyrene.In Movies S1 and S2 (Supporting Information), droplet motion are shown, which correspond to the stills shown in Figure 4. Figure 4a shows the linear translation of a 30 μL droplet down an incline of 2°.The vibration used here is defined by a frequency of 27.5 Hz and an amplitude of 6 V (3.3 mm).The droplet crossed the distance of the sample (ca.7 cm) in <1 s.The stills of the taken video suggest that the droplet detaches from the surface (3rd and 6th stills), as the droplets become smaller (due to the high surface tension of ca.365 mN m −1 ). [24]Furthermore, one can observe a flattened droplet with higher surface area than the original one in the third still, suggesting impact of the droplet.However, this is difficult to ascertain from top view videos.Further, vibration patterns can be observed on the other stills.For the big droplet with a volume of 1000 μL, linear translation of the droplet appears to be a bit slower, and the video as well as the stills do not suggest that the droplet detaches from the surface.Again, vibration patterns can be observed.Similar vibration patterns have been described by Zhao et al. for liquid metal in alkaline aqueous solution [39] and by Steen et al. for water droplets. [40]Beside these patterns, one can observe liquid metal scattering and dispersal for larger droplets at high amplitudes (≥ 6 V) when the frequency is between 20 and 30 Hz.For instance, as shown in Movie S3 (Supporting Information) for a 1000 μL droplet excited with a vibration of 27.5 Hz and 8 V (4.4 mm, Weber number (We) ca.760).Droplets may be ejected in the middle or at the side of the liquid metal puddle.Therefore, the drive amplitude needs to be controlled rigorously.One should note that the Ohnesorge number (Oh) is far below 1 (dependent on volume but <0.001, see Supporting Information).Therefore, the aerodynamic Weber number, a number relating the disruptive forces with the restorative forces (surface tension), is the determining factor for the breakup of droplets.A high Weber number is indicative for a tendency of droplet fragmentation.Indeed, high Weber numbers are obtained (see Table S1, Supporting information).For an amplitude of 1.1 mm (2 V, 27.5 Hz), the Weber numbers were between 2.5 and 10.0, increasing with droplet volume.These values are still low and the droplets do not fragment.In contrast, for an amplitude of 3.3 mm (6 V, 27.5 Hz), the Weber numbers are much higher, between 22.1 and 89.7.These values are far in the range where droplet fragmentation is observed in secondary atomization. [41]However, droplet impact (with water at Weber numbers <230) [42] and vibration of water on a substrate at a Weber number of 69 may not yield in droplet ejection or catastrophic breakup (that is dependent on surface structure). [43]In the case of liquid metals, bouncing of liquid metal on a thin water layer without droplet ejection was found for Weber numbers of 10 and greater. [15]In this case the droplets are further stabilized by the oxide skin, which features altered mechanical properties, that is, higher viscosity and elasticity, [27,44] and presumably the viscosity is able to dissipate energy while the high elasticity of the oxide helps holding the droplet together.In addition to the Weber and Ohnesorge numbers, the Reynolds and Bond numbers are given in the Supporting Information.

Dependence of the Liquid Metal Linear Translational Speed on Inclination Angle and Droplet Volume
Figure 5 has a closer look at the liquid metal movement on the inclined Neverwet coating upon actuation with vibrations.Here, the vibration frequency was set to 27.5 Hz and the vibration amplitude was generally set to 2 V (1.1 mm), except for the smallest droplet size of 30 μL (6 V, 3.3 mm), as this droplet size did not reliably move upon use of a drive amplitude of 1.1 mm presumably due to reduced inertia.In Figure 5a,c, the time-displacement curves for droplets with volumes of 30 and 1000 μL are shown while Figure 5b,d depict the corresponding velocities.Upon excitation with the specific vibration, the 30 μL droplet moves downward.Initial displacement is slow and acceleration is observed in the time-displacement curve in the first 0.2 s.Then, the velocity (displacement versus time) remains constant, which can be ascertained by the linear region in the graphs (Figure 5a).The impact of the inclination angle is directly visible in the time-displacement curves and in the average velocity graph in Figure 5b.At minute inclinations (i.e., 0.5°), the movement is very slow with a velocity of 2.5 ± 0.5 cm s −1 .The velocity increases with inclination linearly up to a velocity of 14.3 ± 1.4 cm s −1 at an inclination angle of 5°.Higher inclination angles were not measured, as for bigger droplets movement might arise at angles >5°due to jitter stemming from manual adjustment of the tilting stage.At first glance, the droplet movement of 1000 μL droplets appears similar to that of the 30 μL droplets.However, a longer run in time of ca.0.3 s is needed and the velocity is lower with 0.7 ± 0.3 cm s −1 (1°) and 7.7 ± 0.8 cm s −1 (5°), as seen in Figure 5c,d.Figure 5e,f suggests that this is a general trend.The droplet movement is for the 30 μL droplet much faster than for the bigger ones, yet one should note that a higher driving amplitude was used for the 30 μL droplets.34d] The similar velocity irrespective of the droplet volume might stem from a common movement mode.We assume that the movement is either sliding/gliding or "rolling" (see Section Movement mechanism).Bigger droplets have a higher mass, which increases the driving force due to the tilted surface and the exerted vibrations, but they are also encumbered with higher fiction/adhesion -retardation forces -counteracting the movement due to the bigger contact area (or better?pinned contact line length) and mass of the liquid metal.The higher velocity observed for the small droplet is ascribed to the different movement mode, "jumping movement", arising from the different vibration amplitude used.During jumping movement, the droplet is not in contact with the surface, which alleviates friction and adhesion forces with the surface counteracting the driving force of droplet motion and the droplet is able to move faster.Magnetic actuation of liquid metal droplets without a liquid surrounding the liquid metal have been shown by He et al. featuring a velocity of 4.5 cm s −1 with a response time of 0.58 s. [34d] Both, linear translational speed and response time, are better for our approach while the linear translational speed itself can be controlled by the inclination angle.Notably, vibration frequency and vibration amplitude may also influence the linear transversal speed of the liquid metal.

Movement Mechanism
In this article, we suggested that the mechanism for movement depends on the actuation method, either inclined surface or the vibration-induced movement.From experience with liquid metal marbles, we know that liquid metal marbles roll down an inclined surface, even at rather low inclination angles.However, rolling off is impeded by a low roundness of the droplets and large droplets are difficult to move, as they might leave behind their liquid metal oxide patina. [35]In Figure 3a, it was shown that a considerable tilting angle is needed to actuate liquid metal down a slope, even when the surface is metallophobic.The reason for the relatively high roll off angles measured above are the movement mechanism, which is a rolling motion (see also Figure 6 and; Movies S4 and S5, Supporting Information), and the fact that the roundness of the droplets was not improved by a prior rolling procedure (compare to liquid metal marbles).The rolling motion is also observed for the bigger droplets.For instance, the 1000 μL liquid metal droplet shows this "rolling" motion.Upon inclining the tilting stage, the liquid inside the oxide skin is moved toward the lower part of the droplet, thereby pushing the oxide skin toward the ground.Concurrently, less bulk liquid metal is located at the higher section of the liquid metal droplet.Due to the tilted surface, gravity exerts a driving force toward the lower part of the sample.As the liquid metal now moves downward, the oxide skin at the higher part of the droplet is lifted (detaches from the surface).This lifting results from the high yield stress of the oxide skin, operating in a similar way as rubber encapsulation or liquid marbles.For liquid puddles (or liquid marbles) movement has been shown to work via rolling in a mixture of free surface and caterpillar motion (depending on the volume of the droplet). [45]iquid metal movement induced by vibrations either commences via jumping and "rolling" or via "rolling".In Figure 7 (Movies S6 and S7, Supporting Information), the movement of 30 and 1000 μL droplets is shown.The former was excited by 27.5 Hz, 3.3 mm vibration, the latter by 27.5 Hz, 1.1 mm vibration while in both cases the inclination angle was set to 3°.For the smaller droplet, one can see in the image at t = 0 s that the droplet is compressed and flattened during upward movement of the vibrating plate.Once the plate moves downward, the liquid metal droplet is expelled from the surface, as seen in the images at t = 0.0075 and 0.0275 s.During the time the liquid metal droplet is airborne, it travels a fair distance downward the inclined sample and impacts the substrate after ca. 2 oscillations of the sample (see image: t = 0.0650 s).A slight droplet movement downward the sample is also observed during contact with the sample.In an additional experiment, the liquid metal oxide skin was marked with silica beads and rolling of the whole droplet (30 μL) was observed while airborne and in contact with the surface.The large droplet (1000 μL) does not detach from the surface, rather it appears to slide on the surface, as shown in Figure 7.At the beginning (t = 0 s), the vibrating plate moves upward, and the droplet (or puddle) is compressed.Once the plate moves downward, the liquid metal maintains some momentum, and the center of mass is lifted.Concurrently, the elevated (receding) contact line recedes while the lower contact line (advancing contact line) is maintained.At 0.0275 s, the plate moves upward and the puddle is compressed by the substrate surface, resulting in a bigger puddle area.This time, the lower (advancing) contact line advances, while the receding contact line is maintained.The process can be summarized as periodic receding (uphill) and advancing (downhill) of the droplet.The vibration thus enables movement by alternately depinning the droplet-surface contact lines.We assumed that larger liquid metal droplets slide down a slope once actuated with vibrations rather than rolling down.To validate or disprove this hypothesis, the 1000 μL liquid metal puddle was marked with silica beads, as shown in Figure S5 (Supporting Information).Beads remaining stationary on the liquid metal puddle would indicate that the liquid metal puddle moves via a sliding mechanism.However, the silica beads moved in the same direction as the liquid metal droplet.The silica bead located at the bottom (middle) moved much faster than the beads located at the right and left side.All these facts indicate that the droplet "rolls" down by periodically lifting the oxide skin (detaching from the substrate) at the receding end of the puddle and lowering of the liquid metal oxide skin at the advancing section of the puddle, where it would contact the substrate.As the liquid metal droplets behave (in our experience) similar to liquid marbles, room-temperature liquid metals might be interesting model systems due to the robustness of the approach for liquid marble systems, that is, for directed movement by vibrations.
Notably, droplets were also able to cross small obstacles.For instance, the 650 μL droplet could cross an edge with a height of 0.5 mm, as shown in Figure 8.The droplet was actuated for this traversal with a 27.5 Hz and 1.1 mm vibration at an inclination angle of 5°.The stills in Figure 8 (and Movie S8, Supporting Information) show that the droplet moves from the Neverwetcoated sample (white) to the diamond-coated sample (black) and crosses the obstacle very rapidly.This traversal is related to projection of some mass of the liquid metal across the edge (0.1 s) and the tilting angle combined with gravity resulting in a net downward force.We also tested an obstacle height of 0.8 mm.However, traversal of such a high obstacle by droplets was unreliable.Still, these results indicate that with surface design and surface structuring techniques one may be able to establish motion without employing a general tilting angle, but by using structures, [46] such as (texture) ratchets. [7,37]4b,7,47] Furthermore, one may consider a combination of metallophobic surfaces and vibrations with magnetic actuation or even motion in a "wet" environment. [48]e also observed a few times liquid metal droplets climbing upward (against an inclined surface) similar to results shown for water droplets and other more conventional liquids. [49]However, at this point we could not obtain repeatable results on this matter (maybe more prevalent at frequency ≥ 30 Hz) and thus, we cautiously point to the fact that this effect might be exploitable for directed liquid metal motion in the future.

Conclusion
In this work, we strived to enable high mobility of liquid metals without the use of a surrounding liquid.3c,47c] As metallophobic surfaces, hierarchical structured Neverwet and diamond coatings were used.In addition, vertical vibrations at an optimal frequency of 27.5 Hz were chosen to improve mobility across all droplet sizes.Notably, with this method, we were able to actuate a broad range of droplet sizes with the aid of only minute inclination angles, as the threshold inclination angles were between 0.5°and 1°.34d] Furthermore, the linear translational speed was homogeneous and high (i.e., 5.5 cm s −1 at an inclination angle of 4°) and rather independent of the droplet volume (≥100 μL), which is related to the common movement mechanism of the larger droplets.The movement mechanism was found to be based on rolling of the oxide skin, stemming from periodic receding (uphill) and advancing (downhill) of the droplet's oxide skin actuated by the two different directions of the vertical vibration.The vibration thus enables linear translation by alternately depinning the droplet-surface contact lines.Finally, we showed that liquid metal puddles are able to traverse small (height 0.5 mm) obstacles, indicating that this method can be combined with surface structuring to achieve more directed motion.In regard to directed linear translation, other vibrations, such as horizontal vibrations, asymmetric vibrations, and combination of these, and their combination with surface structures are anticipated to further improve liquid metal manipulation and directed liquid metal transport.Another future path is that directed liquid metal movement may be achieved by a combination of the metallophobic surface design and vibration with the known magnetic actuation, [48a,b] a common technique used in soft robotics.

Figure 1 .
Figure 1.Schematic showing the movement of a liquid metal droplet due to vertical vibrations when the sample is inclined at an inclination angle ().

Figure 2 .
Figure 2. Morphology of the liquid metal repellent surfaces.a) SEM micrograph of the Neverwet coating on polystyrene and its magnification (b).c) SEM micrograph of the structured diamond coating on Si and its magnification (d).

Figure 3 .
Figure 3. a) Roll-off angle of liquid metal droplets dependent on the droplet volume for both as-deposited deposited and flattened droplets, the liquid metal phobic sample was the Neverwet coated polystyrene.b) Minimum inclination angle necessary to move liquid metal droplets on Neverwet or diamond when exposed to vibrations (frequency 27.5 Hz, 2 V (1.1 mm) to 6 V (3.3 mm) amplitude).c) Minimum amplitude (in V) necessary to move a liquid metal droplet dependent on the frequency and droplet volume.The inclination angle of the sample (Neverwet coating) was set to 3°.The size of the symbols reflects the droplet size for sake of clarity.

Figure 4 .
Figure 4. Sequence of photographs illustrating droplet motion induced by vibrations.a) Motion of a 30 μL droplet actuated by a vibration at 27.5 Hz and 6 V (3.3 mm).b) Motion of a 1000 μL droplet (or puddle) actuated by a vibration at 27.5 Hz and 2 V (1.1 mm).Both movements were recorded at an inclination angle of 2°.

Figure 5 .
Figure 5. Movement of EGaInSn on the Neverwet coating upon application of vibrations.a) Time-displacement curve of EGaInSn droplets with a volume of 30 μL dependent on the inclination angle of the sample upon actuation with vibrations (27.5 Hz, 3.3 mm amplitude) and b) the corresponding velocity versus inclination angle diagram.c) Time-displacement curve of EGaInSn droplets with a volume of 1000 μL dependent on the inclination angle of the sample upon actuation with vibrations (27.5 Hz, 1.1 mm amplitude) and d) the corresponding velocity versus inclination angle diagram.e) Comparison of time-displacement curves of liquid metal droplets; the droplet volume was varied between 30 and 2000 μL.The inclination angle was maintained at 4°.The vibration used was 27.5 Hz, 1.1 mm amplitude, except for 30 μL droplet (27.5 Hz, 3.3 mm amplitude).f) Velocity of droplet movement versus droplet size at fixed inclination angle of 4°.The vibration used was 27.5 Hz, 1.1 mm amplitude, except for 30 μL droplet (27.5 Hz, 3.3 mm amplitude).

Figure 6 .
Figure 6.Droplets moving down a slope on a metallophobic diamond coated Si sample.The red arrows indicate that the movement is commencing via a rolling motion.

Figure 7 .
Figure 7. Movement of liquid metal droplets on Neverwet-coated PS upon exposition to vibrations.The 30 μL droplet was exposed to 27.5 Hz at 3.3 mm and the 1000 μL droplet to 27.5 Hz at 1.1 mm.The inclination angle was set to 3°.The arrows depict the movement direction of the substrate while the numbers in the image depict the elapsed time.Further, the advancing and receding contact line is indicated in the image with a red dotted line.

Figure 8 .
Figure 8. Droplet movement across an obstacle (0.5 mm edge height).For this experiment, a 650 μL EGaInSn droplet was used, an inclination angle of 5°was set, and the vibration used was 27.5 Hz at 1.1 mm.The height of the obstacle (black substrate) is 0.5 mm and the obstacle material is a diamond-coated Si sample.The white sample is the Neverwet-coated PS.