Multimodal Locomotion of Magnetic Droplet Robots Using Orthogonal Pairs of Coils

Recently, several fluidlike, shape‐adaptable magnetic microrobots have been developed to overcome the drawbacks posed by the rigid structure of traditional magnetic microrobots. However, most of their control systems rely on permanent magnet‐based systems, which are unfeasible for large‐scale environments, such as in vivo applications. Herein, an electromagnetic system comprising orthogonal pairs of coils is used to generate global magnetic fields, that is, magnetic fields that are applied equally to large volumes, for the multimodal locomotion of ferrofluidic robots composed of magnetic micro‐/nanoparticles. Three main magnetic field configurations, with their respective locomotion mechanisms, are explored: uniform rotating fields for droplet fission–fusion motions and locomotion; magnetic trapping point‐based locomotion for single‐ and collective‐droplet manipulation; and field‐free region‐based dragging locomotion and selective droplet control. The effectiveness of the proposed multimodal locomotion mechanisms and potential applications are experimentally demonstrated in minichannels through selective mixing, targeted delivery, and optimized magnetic heating applications. These locomotion mechanisms using global fields enable the use of fluidlike magnetic microrobots in large‐volume applications.


Introduction
The primary limitation of traditional magnetic milli/microrobots is their lack of adaptability to their working environment, a result of their permanent rigid structure and limited locomotion mechanisms.While various locomotion mechanisms for magnetic robots have been developed, each robot is typically limited to work with only one or, at most, a couple of mechanisms. [1]Examples of these include Janus microdimer walkers propelled by oscillating magnetic fields [2] and helicoidal robots driven by planar rotating magnetic fields (RMFs), [3] each possessing a single locomotion mechanism.Alternatively, trimer-like microrobots can employ up to two locomotion mechanisms, depending on whether planar rotating or conical RMFs are applied. [4]Although, the development of soft body magnetic robots has bolstered their adaptability by permitting a broader range of locomotion mechanisms, [5,6] their morphological adaptability remains limited, especially within channels smaller than the size of the robot.
As a result, several magnetic elastomers [7][8][9] and fluidlike magnetic robots [10][11][12][13] have been developed in recent years.For instance, a novel amoeba-inspired magnetic droplet robot has been developed, which can perform applications such as in situ chemical reactions, thrombolysis, and phagocytosis. [14]hese magnetic elastomers and fluid-like magnetic robots (hereafter referred to as droplet robots) exhibit high deformability, enabling them to modify their shape to match the geometry of the surrounding environment, thereby facilitating their movement through tight channels.They also possess the capability to divide into smaller robots (fission) or amalgamate into larger robots (fusion).Furthermore, certain models are capable of transitioning between solid and fluid states, thus broadening their potential applications. [8,15]However, in most research related to the development of these robots, their applicability is only demonstrated using magnets controlled by hand [8][9][10][11][12] or, in some cases, through mechanical stages. [13,16]Moreover, some studies that explore the locomotion of these robots in 3D environments rely solely on manually controlled magnets. [17,18]lectromagnets are essential for precisely controlling the locomotion of droplet robots.Typical configurations rely on controlling planar arrays of small electromagnets with iron cores, [19][20][21] position controlling a main magnet with electromagnets, [22] or combining a magnet, electromagnets, and a mechanical platform. [23]These systems create large magnetic gradients that can easily perform fusion-fission tasks, collective control, and selective actuation of droplets.However, these systems are intrinsically restricted to small-scale (except when combined with mechanical stages) [23] and 2D applications.Current electromagnetic actuation systems (EMA) for 3D locomotion of droplet robots consist of large ferromagnetic core solenoid electromagnets with a small working volume (a few centimeters) that exerts magnetic forces capable of controlling a single-droplet robot. [24,25]These 3D systems cannot create uniform magnetic fields, perform the fusion-fission of droplets, nor conduct swarm or selective locomotion.
The easiest way to increase the scale of an EMA system is by implementing orthogonal pairs of coils, such as Helmholtz and Maxwell coils, to generate global magnetic fields throughout the entire working space.Recent research has unveiled several multimodal locomotion mechanisms for ferrofluid robots, including jumping, fission-fusion, and stretching-based translation, all achieved using global uniform magnetic fields produced by three orthogonal pairs of Helmholtz coils. [26]However, the limited magnetic force has restricted the size of channels through which a droplet can pass, necessitating the splitting of droplets into smaller units to navigate narrow channels.Furthermore, neither swarm control nor selective locomotion were possible.
Previous studies have demonstrated the collective locomotion of droplet robots using a Helmholtz coil system.To achieve this, four additional electromagnets were added to the system to exert magnetic forces that changed the position of a swarm of magnetic droplets. [27]The swarm was assembled by applying RMFs to droplets located at the water-air interface.In another study, a magnetic needle was added to assemble droplets around the needle as a swarm and control their position. [28]However, this method is impractical for enclosed environments and requires a robotic platform to control the needle.A promising alternative method to control the collective locomotion of magnetic droplet robots is the use of a magnetic "trapping point" (TP), which is a special magnetic field distribution characterized by a region of maximum magnetic field toward which magnetic robots are attracted. [29]Although collective locomotion using TP locomotion generated by Helmholtz coils has been demonstrated in a swarm of traditional magnetic robots, [30] it has not yet been demonstrated in fluid-like robots.
Currently, there are no control mechanisms for selective manipulation of magnetic droplets using global magnetic fields.However, selective 1D actuation of rigid magnetic helical actuators constrained to threaded channels has been achieved through the use of global fields. [31]This was accomplished utilizing a special field distribution known as the "selection field (SF)," characterized by a region of zero magnetic field at its center.This magnetic field configuration is common in magnetic particle imaging systems, a novel experimental imaging modality. [32]n summary, there are currently no mechanisms for controlling the collective swarm locomotion and individual selectively locomotion of droplet robots using only global fields, fully exploiting their morphological adaptability for their adaptive magnetic control.This is crucial for expanding their applicability to larger scales and enabling 3D control.In this work, several multimodal locomotion mechanisms were developed for the control of droplet robots using global fields.The global fields are produced by a system (Figure 1a) composed of three orthogonal pairs of Helmholtz coils (H x , H y , and H z with their respective axis in the x-, y-, and z-axis), a pair of Maxwell coils (Mc, with their center in the z-axis), and a high-frequency solenoid coil (Hc) for inductive heating.Three main magnetic field distributions, as shown in Figure 1b, are generated: uniform RMFs, a magnetic TP, and a field-free region (FFR).
As shown in Figure 1c, the rotating fields are used for both fission-fusion and locomotion of droplet robots, while TP and FFR are primarily used for collective and selective locomotion, respectively.This is the first demonstration of collective and selective locomotion of droplet robots using global fields, as far as we know.Additionally, both TP and FFR can be utilized for dragging locomotion of magnetic droplets.Magnetic field distributions are analyzed through computer simulations, and locomotion mechanisms are experimentally demonstrated using ferrofluid magnetic robots composed of magnetic nanoparticles.Figure 1d illustrates potential applications of these locomotion mechanisms, including targeted delivery of cargoes, selective pumping control, and optimized magnetic heating by controlling the amount of magnetic fluid at the target region.These three applications are experimentally demonstrated in minichannels.The results of this study suggest that global magnetic fields can be used for the multimodal control of fluidlike magnetic robots, making them suitable for large-scale applications, such as in vivo applications.

Magnetic Field Distributions
In this work, global fields are generated by an EMA system that consists of three orthogonal pairs of Helmholtz coils, a pair of Maxwell coils, and a high-frequency solenoid (see Figure S1, Supporting Information).The system is described in more detail in a previous work. [30]Unlike traditional systems based on Helmholtz coils, all coils in this EMA system can be independently controlled, rather than in pairs.The smallest pair of Helmholtz coils, H z , possess a radius of 140 mm.Theoretically, they can produce a uniform field within a distance of 140 mm.However, the pair of Maxwell coils have a radius of 70 mm.To avoid significant inhomogeneities in the gradient distribution (below 10%), the working volume of the system for the proposed control strategy was limited to 60 Â 60 Â 50 mm 3 (respectively in the x-, y-, and z-axis).Further details on the properties of the coils can be found in Table S1, Supporting Information.The inductive heating coil produces an alternating sinusoidal magnetic field of 20 mT at a frequency of 200 kHz.

Uniform Time Static and Rotating Magnetic Fields
The three pairs of Helmholtz coils are used to generate uniform magnetic fields along the working space.In this system each coil is controlled independently, resulting in six distinct control currents for the Helmholtz coils: I x1 , I x2 , I y1 , I y2 , I z1 , and I z2 , which respectively flow through the pairs of coils H x , H y, and H z. However, in the experiments conducted in this study, the pairs of Helmholtz coils produced only a uniform magnetic field.Thus, the control currents can be simplified as (1) And their respective magnetic fields B x , B y , B z are where k x , k y , and k z are constants related to the geometric properties of the coils.Two different magnetic field distributions are generated: a time-static uniform magnetic field, hereafter referred to as static magnetic field (SMF), which is where θ 1 and ϕ 1 are the azimuthal (angle between the xand y-axis) and polar angles (angle between the z-axis and XY plane) of the applied SMF, respectively, and B SMF is the magnitude.
where θ 2 and ϕ 2 are the azimuthal and polar angles of the rotation axis of RMF, respectively, and B RMF and ω are the magnitude and angular frequency, respectively.For clarity, a diagram illustrating the orientations of the SMF and RMF is provided in Supporting Information (see Figure S2, Supporting Information).

Magnetic Trapping Point
A TP is a magnetic field distribution that features a single region of maximum magnetic field, which attracts any magnetized object toward it.This study used the lower Maxwell coil (Mc 1 ) to generate the TP distribution.As shown in Figure 2a, Mc 1 generates a magnetic gradient that pushes any magnetized object in -z direction.However, depending on the vertical distance from the coil, it also generates a radial gradient (G r ) that either pushes a magbot toward to or away from the axis of the coil.The point at which the radial gradient changes direction is known as the critical distance.The field distributions shown in Figure 2a for the main XY plane are above the critical distance.As shown, even though the magnetic field points outward from the origin, the magnetic gradient pushes toward its center.This magnetic field distribution is further discussed in previous works. [29,33]

Field-Free Region
Flowing currents with the same magnitude but opposite directions in a pair of Maxwell coils generate a special magnetic field distribution referred to as a SF, which includes a region of zero magnetic field at its center, known as FFR.The SF is where p the given position, and G is the magnetic gradient tensor, which for a pair of maxwell coils aligned with the z-axis is given by where diag(x) denotes a diagonal matrix with the elements x on its diagonal.The relation between g z and the current I m flowing through the Maxwell coils is where k m is a constant dependent on the geometrical properties of the Maxwell coils.The magnetic field and magnetic gradient distributions generated in the main XY and XZ planes are shown in Figure 2b.In both planes, the magnetic field is zero at the origin and increases linearly in all directions, whereas the magnetic gradient pushes outward from the origin.Additionally, in the XY plane, the radial gradient (G r ) is uniform.
Traditionally, for systems comprised of orthogonal pairs of coils, the magnitude of the gradient differs significantly between axes due to the increasing size of the coils along each axis.Figure 2c compares the magnetic gradient produced by a single coil of each pair of Helmholtz coils along their working axis and the gradients G z and G r produced by Mc, with a current flow of 10 A. The results show that the magnitude of the gradient produced by Mc is uniform and almost three times higher in the z-axis and over four times higher than in any radial direction when compared to the other coils.6]

Results
All experiments were conducted using %100-300 μL of ferrofluid immersed in ionized water.Due to the properties of the purchased magnetic fluid, it was not possible to perform experiments in a different liquid or without it.

RMF-Based Control: Locomotion and Fission-Fusion Mechanisms
The magnetic fluid used in this work is composed of Fe 3 O 4 magnetic nanoparticles, which exhibit nearly superparamagnetic behavior.Therefore, its magnetic moment (m ¼ VM, where V is the volume of the droplet and M its magnetization) depends on the applied magnetic field, and it vanishes in the absence of a magnetic field.The magnetization of a ferrofluid is , where H ext is the applied control magnetic field strength, H f the internally produced magnetic field strength of the fluid, and χ is the magnetic susceptibility.The first multimodal locomotion mechanisms explored in this work rely solely on the application of RMFs, as described in Figure 3 and Movie SM1, Supporting Information.
These RMFs are generated exclusively by the three pairs of Helmholtz coils.An RMF with a rotation axis perpendicular to the z-axis induces a rolling locomotion of a magnetic droplet along the rotation direction, as depicted in Figure 3a.The locomotion was tested on a droplet along a double-spiral trajectory (Figure 3a, Movie SM1a, Supporting Information) from the upper-right portion of the trajectory (t 0 ) to the lower left-side part of the trajectory (t f ), by applying an RMF of 3.5 mT at 3.6 Hz.This locomotion mechanism is straightforward and widely reported. [23,26]ne advantage of droplet robots is their fusion-fission mechanisms.In this study, these mechanisms are achieved by applying an RMF rotating along the z-axis, with varying frequencies.As described in Figure 3b, the application of the RMF causes the MNPs within the droplet to chain together, resulting in an enlargement of the droplet.While the enlarged droplet rotates in response to the RMF, a dragging force (F D ) is applied in opposite directions at the two ends of the droplet.These forces produce a shear stress at the center of the droplet, causing it to divide into two. [37]The timelapse images of Figure 3b (see Movie SM1b, Supporting Information) show the fission process of a single droplet into four droplets, achieved by applying an RMF of 14 mT at 5 Hz.As soon as the field is applied (t 1 ), the droplet enlarges itself and starts rotating, ultimately getting sliced by its center (t 2 -t 4 ), resulting in two smaller droplets (t 5 ), which repeat the same mechanism (t 6 -t 8 ) and further divide into a total of four droplets.
While the droplets exhibit an ellipsoidal shape, they generate rotating flows that also repel each other, maintaining a distance at which their magnetic interactions remain negligible, thereby preventing them from merging.However, at higher frequencies, the droplets lose their ability to follow the RMF, resulting in their collapse into a round shape that no longer generates repulsive flows between droplets.When the droplets are close enough for their magnetic dipolar interactions to become significant, an attractive magnetic force (F m ¼ ∇ðm ⋅ BÞ) occurs between them.As the droplets approach each other, viscous dragging can induce a planetary motion without coalescence. [38]By minimizing the droplet's shape, the drag forces diminish and the droplets' thin liquid films rupture, leading to droplet fusion in an RMF.This mechanism is shown in Figure 3c (see Movie SM1c, Supporting Information).Timelapse images in Figure 3c show the fusion mechanism for three droplets subjected to an RMF of 8 mT at 20 Hz.When the RMF is applied, the three droplets begin rotating and assume a thin disk shape (prolate motion) (t 1 ); then droplets m 2 and m 3 get attracted to each other, causing them to clash and rotate together until they merge into a single larger droplet (t 2 -t 4 ).Finally, the remaining small droplet m 1 is dragged toward the large droplet and merges into a single one (t 5 -t 7 ).
An attempt was made to achieve collective locomotion of various droplets under an RMF, as shown in Figure 3d (see Movie SM1d, Supporting Information).At the start of the experiment, four droplets (m 1 -m 4 ) were placed equidistantly, with m 1 located at the beginning of the trajectory (t 0 ).The droplets were controlled so that the m 1 followed the trajectory, by applying an RMF of 3.5 mT at 3.6 Hz.However, as the droplets move, the distances between them decreased until they collided with each other and merged (m 1 with m 2 at t 2 , m 3 with m 4 at t 4 ), ending the experiment with two droplets instead of the initial four.This is due to the fact that uniform fields uniformly magnetized all the droplets (at these low frequencies), resulting in attractive forces between the droplets during their rotation.This highlights the need of a gradient field that induces different magnetizations on the droplets, allowing them to exert repulsive forces that prevent merging.
Another important characteristic of fluid-like magnetic robots is their environmental shape adaptability.For this purpose, usually large gradients produced by magnets are required.However, we decided to test the droplet's adaptability through its locomotion within channels that decreased in width from 5 to 1 mm, using only uniform RMFs of 3.5 mT at 3.6 Hz.As shown in Figure 3e (see Movie SM1e, Supporting Information), the droplet successfully passed through the 5 mm (t 0 -t 1 ), 4 mm (t 1 -t 2 ), and 3 mm (t 2 -t 3 ) channels.However, its propulsion was not strong enough to pass through the 2 mm channel.To pass through the channel using only uniform RMFs, the droplet must be split so that it can pass through. [26]However, in channels as small as this one, fission is not feasible through uniform RMFs, and a gradient field is required.

Trapping Point-Based Locomotion: Single and Collective Dragging Locomotion
When a current is passed through Mc 1 , a TP field distribution is generated.As illustrated in Figure 4a, if a magnetic droplet is placed within such distribution, the magnetic gradient will push it toward the TP which coincides with the axis of the coil in the z-axis.Then, by applying an SMF, the position of TP changes and the droplet is dragged along.This locomotion mechanism was experimentally demonstrated by controlling the locomotion of a droplet along a complex trajectory consisting of semicircles, as shown in Figure 4a (see Movie SM2a, Supporting Information).The timelapse image reveals a change in the shape of the droplet, transforming from circular to ellipsoidal, during its trajectory through the XY plane.To further scrutinize this phenomenon, the shape of the droplet in the ZY plane was compared at varying I m , as shown in Figure 4b.With I m at zero, the droplet assumes a semispherical shape that begins to elongate along the Z-axis, evoking the image of a semiellipsoid.This semiellipsoid progressively enlarges its major diameter as I m increases.The cause of this transformation is the magnetic field applied, which incites the creation of MNP chains within the droplet.These chains expand proportionally with increasing magnetic field strength.As evidenced by the magnetic field distribution images in Figure 4b (which exclude the magnetic field attributable to the droplet's magnetization), the shape of the droplet ceases to change for applied magnetic fields exceeding 11 mT (I m > 10 A), indicating the attainment of its magnetic saturation state.
Subsequently, the position of TP was controlled along the Y-axis by applying a constant I m = 40 A and a variable SMF to analyze the shape of the droplet, as shown in Figure 4c.The value of SMF was varied along the Y-axis, from 6 to À6 mT in increments of 2 mT, with each condition labeled in sequence from t 0 to t 6 .This manipulation resulted in displacement of the droplet from y = 18 mm in -Y direction to y = À18 mm, as well as a change in the orientation of the droplet.These changes in orientation and position are respectively attributable to the applied magnetic field and magnetic gradient distributions, as demonstrated in Figure 4c.The applied SMF imposes a magnetic torque on the MNP chains within the droplet, effecting an immediate alteration in the orientation, σ, of the droplet upon application of SMF.This orientation subsequently changes from 0°to 6°, 22°, and 30°for SMF magnitudes of 2, 4, and 6 mT, respectively.
Due to the low viscosity of water, the shape of the droplet is unaffected by viscous forces and is exclusively dependent only on the applied magnetic field (see Movie SM2b, Supporting Information).The images on the right side of Figure 4c display the fluidic flow generated by the dragging locomotion of the droplet at its various orientations.As it can be observed, for conditions t 0 -t 3 , the droplet has an inclination (positive σ) opposite to the droplet's dragging motion (in -y direction).This generates a fluidic flow that concentrates on the droplet's right side.However, when the droplet's inclination (negative σ) aligns with the direction of the dragging motion (t 4 -t 6 ), its flow resistance escalates, thus producing flows that concentrate in the direction of the drag.
If several droplets are placed inside a TP distribution, they will assemble with their center at TP.This phenomenon was demonstrated by successively splitting the initial droplet (same as Figure 4) into 3, 6, and 13 droplets, which were then subjected to a TP with I m = 40 A, as shown in Figure 5a.In all cases, the droplets assembled with their center coinciding with the TP.Each droplet equally separated from its neighboring droplets due to a magnetic repulsive force between them.Furthermore, it is observed that the distance between droplets decreases as their size decreases, due to the decline in their repulsive force product of their volume reduction.
Similar to the locomotion of a single droplet, the position of the magnetic swarm can also be manipulated by controlling the position of the FFR.To demonstrate this, a magnetic swarm composed of six droplets was made to perform collective locomotion along a trajectory composed of straight lines, as shown in Figure 5b (see Movie SM3a, Supporting Information).After applying the TP distribution (t 0 ), the droplets assembled with a "lead droplet" at its center, and the remaining five droplets were equidistantly separated around the lead droplet.The position of the lead droplet matched the position of the TP.As observed in the timelapse images, unlike attempted collective locomotion using RMFs, as the controlled swarm moved the distance between the droplets was maintained.This is because the TP distribution induces different magnetizations in the radial direction (following the magnetic distribution shown in Figure 5a) that generate repelling forces between the droplets.
The ability of the swarm to adapt its shape to the shape of the transited channel when controlled by TP-based locomotion was verified by controlling a swarm of three robots through a minichannel with varying channel widths and complex geometries, as shown in Figure 5c (see Movie SM3b, Supporting Information).The swarm began its locomotion in a 20 mm-wide squared region (t 0 ) and was forced through a 5 mm-wide channel (t 1 -t 3 ).Along its trajectory through the narrow channel, only one droplet could pass at a time, so the geometry of the swarm adapted to successfully pass through.After that, the swarm regained its original shape, but the distance between each other decreased in the 10 mm channel (t 3 -t 5 ) compared to its original state (t 0 ).A constant change in the separation distance of the droplets can be observed through the locomotion of the swarm along the zigzag trajectory (t 5 -t 6 ).When the swarm reached the end of the trajectory, a circle with radius of 5 mm, the swarm once again regained its original shape (t 8 ).

FFR-Based Locomotion: Enhanced Dragging Locomotion and Selective Locomotion
The gradient distribution generated by the SF, characterized by its FFR, can be used to exert a magnetic force on magnetic droplets, dragging them away from the FFR.The distributions shown in the superior section of Figure 6a depict the magnetic field and magnetic gradient distributions when only the Maxwell coils are energized with a current of 10 A (G r = 0.35 mT mm À1 ), showing the FFR at the center.Although the magnetic field lines point toward the FFR, the gradient exerts a magnetic force on the droplets that pushes them in a radial direction away from the FFR.From the magnetic force equation, it is deducted that the force exerted on a droplet will be zero when the FFR is at the center of the droplet and will increase as the separation distance between the FFR and the droplet increases.If an SMF is applied, the position of the FFR can be changed to match that of a targeted droplet so that the surrounding droplets are pushed away from the targeted droplet, as shown in the images of Figure 6a.Furthermore, the magnitude of the gradient is constant but can be controlled by the applied current; as observed in the distribution images, the gradient duplicated when I m was increased from 10 A (G r = 0.35 mT mm À1 ) to 20 A (G r = 0.7 mT mm À1 ).Using this principle, the locomotion of a magnetic droplet was retested in the same minichannel used for rolling locomotion, as shown in Figure 3e.As the timelapse images in Figure 6b (see Movie SM4a, Supporting Information) demonstrate, besides passing successfully the 5 mm (t 0 -t 1 ), 4 mm (t 1 -t 2 ), and 3 mm (t 2 -t 3 ) channels by applying an G r = 0.1 mT mm À1 , the magnetic force exerted by the SF allowed the droplet to pass successfully through the 2 mm (t 3 -t 4 ) and 1 mm (t 4 -t 6 ) channels by further increasing G r to 0.2 and 0.3 mT mm À1 .This was impossible when using rolling locomotion.Furthermore, it can be observed that for the droplet to successfully go through the 1 mm channel, the droplet splits into two droplets (t 5 ), which later merge into one.
Another important locomotion mechanism that can be implemented with the FFR is the selective locomotion of the droplets, as shown in Figure 6c (see Movie SM4b, Supporting Information).
In this experiment, an SF with G r = 0.2 mT mm À1 and an RMF of 3.5 mT with a frequency of 3.6 Hz were applied.First, the FFR was applied at the center of four droplets (m 1 -m 4 ), pushing them toward the corners of the glass container (t 0 -t 2 ). Figure 6d shows how the FFR changes when an RMF rotating along the y-axis is applied, with the FFR matched to the position of a droplet.In the XY plane, the FFR alternates along the x-axis, which would exert a magnetic force on the droplet.However, due to the fast and symmetrical alternating gradient, the overall force is 0. In the XZ plane, an offset is induced in the FFR, causing it to rotate with its axis at the droplet.This rotation induces a RMF at the location of the droplet, which causes the droplet to also rotate.Furthermore, droplets located outside of the FFR rotation trajectory cannot locomote, as the magnetic field outside of the trajectory does not rotate completely.
Using the described mechanism, m 1 was selectively controlled along a star trajectory.First, the FFR was positioned at the center of m 1 and an RMF was applied, propelling m 1 through rolling locomotion from the upper-right corner to the initial position of the starshaped trajectory (t 3 -t 5 ).Then, m 1 followed the star trajectory until it returned to its original position (t 6 -t 12 ).After that, m 1 is moved back to the upper right corner (t 13 -t 14 ) and the position of the FFR was adjusted to match m 2 .Next, m 2 was moved toward the initial position of the star trajectory (t 15 -t 16 ) and controlled along the star trajectory (t 17 -t 21 ).While m 1 and m 2 were being controlled, the nontargeted droplets remained at their respective corners, unable to rotate.

Demonstration of Potential Applications
To demonstrate the potential applications of the proposed multimodal locomotion mechanisms, we conducted three experiments: selective pumping, selective targeted delivery, and optimized magnetic heating.For the selective pumping experiment, we utilized two separated minichannels, each consisting of three circular chambers with a radius of 10 mm and interconnected by 2 mm-width channels.Red dye was added at the superior right end of the upper channel and green dye was added at the superior right end of the lower channel.We first performed simultaneous pumping in both channels, as shown in the timelapse images in Figure 7a (see Movie SM5, Supporting Information).A magnetic droplet (m 1 and m 2 ) was then placed in the middle chamber of each minichannel (t 0 ) and moved toward the chambers containing the dye (t 1 -t 2 ) using FFR-based dragging locomotion.Then, an RMF of 6.6 mT with a frequency of 9 Hz was applied, rotating along the z-axis.This resulted in a rotating motion of the droplets, which pumped the dyes toward the other two respective chambers of each minichannel (t 3 -t 4 ).It took about 104 and 152 s for the green and red dies, respectively, to be completely distributed along the channels.Then selective pumping was performed as shown in Figure 7b (see Movie SM5, Supporting Information).In this case, the droplets were positioned in the lower-left sided chamber of each minichannel (t 0 ).Then, m 1 was moved toward the chamber containing the green dye, and an SF with G r = 0.2 mT mm À1 was applied with its FFR positioned at droplet's location while applying an RMF of 5.6 mT and frequency of 6 Hz (t 1 ).Along with the rotation induced in the droplet by the SMF, the SF generated magnetic forces that pushed the droplet toward the chamber, resulting in a more efficient pumping that finished distributing the dye in only 44 s (t 2 ).Next, m 2 was moved toward the chamber containing the red dye (t 3 ).Selective pumping was performed following the previously described procedure (t 4 -t 5 ).The dye was again distributed in the channel, this time within 32 s.Besides proving being a successful mechanism for selective pumping, the proposed mechanism also resulted in an enhanced pumping performance.
An experiment was conducted to demonstrate the ability to selectively control magnetic droplets for targeted delivery applications.The experiment was performed inside a minichannel with an overall size of 46 Â 46 mm, as shown in Figure 8a (see Movie SM6, Supporting Information).Initially, a droplet was placed in the initial chamber, which had a radius of 5 mm.The droplet was then moved to a larger chamber (Figure 8b) and split into smaller droplets.One of the resulting droplets was selected and transferred to the cargo chamber, which contained photocurable resin balls with an average radius of 250 μm (Figure 8c) (see Experimental Section/Methods).Due to their hydrophobic nature, the resin balls exhibited no interaction with the surrounding water and were capable of adhering to the surface of the hydrophobic ferrofluid droplets.
The selected droplet was loaded with four cargoes (time-lapse images in Figure 8d) and transported through 3 mm channels toward target area 1, as observed in the timelapse images of Figure 8e.The resin balls remained on the surface of the droplet without penetrating it due to the droplet's surface tension.During control of the targeted droplet, the remaining droplets were unable to synchronize with the RMF and propel themselves.The above procedure was repeated, where the main large droplet was split, and one of the resulting smaller droplets was transferred to the chamber containing the cargo, as observed in Figure 8f.The targeted droplet was then loaded with the remaining cargoes (Figure 8g) and moved through the narrow channel toward the target area 2 (Figure 8h).Overall, the experiment demonstrated the potential of using magnetic droplets for targeted delivery applications in microfluidic systems.The selective control of individual droplets using RMF allows for precise delivery of cargoes to specific locations, which can be useful in various biomedical and chemical applications. [39]hile these cargo delivery experiments were only demonstrative, and the cargoes could not be separated from the droplet after reaching the targeted regions, the solid cargoes can be substituted with hydrophilic liquid cargoes.These liquid cargoes can then be injected into the droplet, enabling their separation from the droplets after delivery. [20,26]nother use of the proposed locomotion mechanisms is the ability to regulate the amount of magnetic fluid during magnetic heating applications to control the temperature rise of the droplets.To demonstrate this, some magnetic fluid was placed within a minichannel that diverged into three paths ending in circular chambers (P1-P3), as shown in Figure 9a (see Movie SM7, Supporting Information).In each of the circular channels, an optical fiber thermal probe was placed to measure the temperature of each chamber during the experiment, as shown in Figure 9b.
First, the initial droplet (t 0 ) was split (t 1 ) and some of the resulting droplets were selectively controlled toward chambers P1 (t 2 ) and P2 (t 3 ), as observed in Figure 9c.Then, for the first heating cycle, the heating coil was turned on (t 4 ), and the droplets were heated for 60 s.The respective thermal images showed that more heat was produced by the droplet in P1, whereas the heat produced by the droplet in P2 barely showed an increase in the thermal images.In comparison, the temperatures measured by the temperature probes show that the temperature rose from about 14 °C to 41.9°and 35.4 °C in P1 and P2, respectively.Since no fluid was distributed to P3, no temperature increase was observed in that chamber.The temperatures shown in the thermal images are significantly lower than those measured by the temperature probes because the thermal images show the temperature at the surface of the water in which the droplets are immersed, whereas the temperature probes were in direct contact with the droplets.
After the heating coil was turned off, the droplet at P1 was removed (t 6 ) and merged with the remaining droplet in the initial chamber (magenta circle) (t 7 ), as observed in Figure 9d.The large droplet was then split (t 8 ) and some droplets were transferred toward P2 (t 9 ).The heating coil was turned back on (t 10 ), and the droplets were heated for 60 s (t 11 ), causing the temperature in P2 and P1 to increase to 54.9°and 19.7 °C, respectively.The higher and lower temperatures in P2 and P1, compared to the first heating cycle, are due to the respective addition and removal of magnetic fluid from those chambers.Hence, these mechanisms can be used to control the temperature rise during magnetic heating.

Conclusion
This study introduced various magnetic field distributions generated by global fields for the multimodal locomotion of fluidlike magnetic robots.First, we demonstrated the rolling locomotion and fission-fusion mechanisms of magnetic droplets using RMFs.However, we found two main drawbacks with the RMFs: droplets could not pass through channels smaller than 2 mm, and when attempting collective droplet control, the droplets merged.These limitations highlight the need for magnetic gradient field distributions to control droplet robots in global field systems.Utilizing the TP magnetic distribution, which generates magnetic gradients, we were able to demonstrate single and collective dragging locomotion control of magnetic droplets.While the RMF mechanisms magnetized the droplets of the magnetic swarm equally, the TP distribution exerts different magnetizations to each droplet, resulting in a repulsive force that prevents them merging.This was further demonstrated by controlling a magnetic swarm inside a minichannel with varying width, causing the separation distance between the droplets and their assembled shape to constantly adapt to the shape of the minichannel, ultimately preventing them from merging into larger droplets.
By applying a SF gradient field distribution, two FFR-based locomotion mechanisms were also proposed.First, it was demonstrated that by controlling the position of the FFR with respect to a targeted droplet, the magnetic force exerted on the droplet could be controlled.As a result, using FFR-based dragging locomotion the droplet was able to successfully pass through 2 and 1 mm-width channels, which was impossible using RMF-based locomotion.Second, a selective locomotion strategy was developed and demonstrated by matching the FFR with a targeted droplet.The FFR exerted a magnetic force on the nontargeted droplets, pushing them toward the walls of the container.After an RMF was applied, only the targeted droplet was able to synchronize with the RMF and follow a star-shaped trajectory.
The potential applications of the proposed mechanism were experimentally demonstrated through the selective pumping control of dye and the selective targeted delivery of cargoes using the selective control mechanisms.Furthermore, the regulation of the temperature during inductive magnetic heating in targeted areas through the control of the quantity of droplet contained within the target area was also demonstrated.The applicability of these control mechanisms in 3D environments for droplet robots is yet to be explored and will be addressed in future works.The main significance of mechanisms demonstrated in this work is that they allow for extension of the application of fluidlike magnetic robots to larger working volumes (e.g., animal or even human scale).This has previously been limited to small working volumes (up to a few cm) due to the existing droplet control systems that rely on permanent magnets or small coil arrays.

Experimental Section
Materials: The magnetic fluid, etching, and coating agents were purchased from Magron, Korea.3D printing zdental model sand resin was purchased from UNIZ, USA.Basic 3D printing UV-sensible resin was purchased from Shenzhen Anycubic Technology Co, China.
Manufacturing of Test Minichannels: Experiments in this study were performed either in a glass box or 3D-printed minichannels.The glass box was made of 3 mm-thick glass and had a working volume of 50 mm Â 50 mm Â 45 mm.Minichannels were created using a Slash 2 Plus 3D printer (UNIZ, USA).An etching agent was applied to both the glass box and the minichannels for 30 min, after which they were rinsed with water and dried.Subsequently, a hydrophobic coating agent was applied to both the box and the minichannels.
Manufacturing of Photocurable Resin Balls: Basic 3D printing UV resin was mixed with oil-based black paint.Then droplets of about 70 nl were precipitated into 100 000 cs silicone oil while exposed to UV light.
Fluidic Flow Simulations: Simulations for the fluidic flow induced by the dragging locomotion of the droplets were performed using the laminar flow and moving mesh interfaces of the CFD module of COMSOL.

Figure 1 .
Figure 1.Ferrofluid droplet multimodal locomotion concept.a) EMA system comprising three pairs of Helmholtz coils, a pair of Maxwell coils, and a heating coil.b) Main magnetic field distributions produced by the system for multimodal locomotion of magnetic droplets: RMFs, magnetic TP, and FFR.c) Multimodal locomotion mechanisms of a ferrofluid droplet: rolling locomotion, fission, fusion, and selective and swarm locomotion mechanisms.d) Potential applications of the multimodal locomotion mechanisms: targeted delivery, selective flow control, and optimized magnetic heating.

Figure 2 .
Figure 2. Comparison of magnetic gradient distributions.a,b) Magnetic field and magnetic gradient field distributions along the main XY and XZ planes produced by the single coil Mc 1 and the pair of Maxwell coils with a current of 10 A, generating a TP and FFR, respectively.c) Graph comparing the magnetic gradient produced by the different coils of the EMA system along their respective axis when a current of 10 A is flowing.

Figure 3 .
Figure 3. RMF-based multimodal control mechanisms.a) Rolling locomotion mechanism: an RMF with rotating axis perpendicular to the XY plane propels a magnetic droplet along the direction of rotation.The rolling direction coincides with the radial axis (r) of the plane (Zr) in which the RMF rotates.The mechanism is used to control a droplet along a double-spiral trajectory.b) Droplet fission mechanism: a low-frequency RMF ( f < 10 Hz) with rotation along the z-axis produces splitting of a droplet due to the shear stress at its centroid resulting from the dragging forces.c) Droplet fusion mechanism: a high-frequency RMF ( f > 20 Hz) with rotation along the z-axis diminishes dragging forces and allows a magnetic force, due to dipolar magnetic interactions, to attract the droplets, one to each other, and merge.d) Failure of swarming locomotion along a double-spiral trajectory using rolling locomotion due to uncontrolled droplet fusion.e) Rolling locomotion of a droplet along channels with successive shorter widths ranging from 5 to 2 mm.

Figure 4 .
Figure 4. TP-based single locomotion.a) Single-locomotion mechanism: the magnetic gradient distribution drags a magnetic droplet toward the TP located at the center of Mc 1 .By controlling the position of TP with the superposition of an SMF, locomotion of a droplet through a complex trajectory is achieved.b) Change in the shape of a droplet and magnetic field distribution produced by Mc 1 for different values of I m .c) Magnetic field and horizontal magnetic gradient distributions produced by the superposition of the TP generated by Mc 1 and different values of B x that change the position of TP; timelapse images of the change in position and orientation of a droplet due to the different applied fields, and simulated fluidic flow fields induced by the dragging locomotion of the droplet for the different applied magnetic fields.

Figure 5 .
Figure 5. TP-based collective locomotion.a) Images of the assembly of 3, 6, and 13 magnetic droplets induced by the application of a TP.b) Timelapse images of the collective locomotion of the assembly of six droplets, achieved by applying an SMF to control the position of TP. c) Timelapse images of the locomotion of a magnetic droplet swarm through a minichannel to test the swarm's environmental adaptability.

Figure 6 .
Figure 6.FFR-based control mechanisms.a) Magnetic field and magnetic gradient field distributions at the main XY plane for different positions of FFR exerting a magnetic force diverting from the FFR.b) FFR-based dragging locomotion of a magnetic droplet through channels with decreasing width from 5 to 1 mm.c) Timelapse images of the selective control of magnetic droplets along a star-shaped trajectory in order: separation, selective positioning of m 1 , selective locomotion of m 1 , removal of m 1 and selective positioning of m 2 , and selective locomotion of m 2 .d) Magnetic field and magnetic gradient field distributions at the main XY and XZ planes showing the change in the field distributions when an RMF is applied.

Figure 7 .
Figure 7. Selective pumping control.a) Timelapse images of the simultaneous pumping of red and green dye within two separate minichannels using magnetic droplets.b) Timelapse images of the selective pumping of dye within two separate minichannels using magnetic droplets in order: inferior channel, green dye; and superior channel red dye.

Figure 8 .
Figure 8. Selective cargo loading and delivery using selective droplet control.a) Experimental set: a minichannel containing a magnetic droplet and several cargo (250 μm-radius photocurable resin balls).b) Locomotion of the magnetic droplet toward a wide region.c) Droplet fission and selective locomotion of a smaller droplet toward the cargo region.d,e) Timelapse images of the selective loading (d) and delivery (e) of cargo.f ) Fission of large droplet and selective locomotion of a smaller droplet toward the cargo region.g,h) Timelapse images of the selective loading (g) and delivery (h) of cargo.

Figure 9 .
Figure 9. Optimized magnetic heating by droplet amount at targeted regions through droplet selective control.a) Experimental set comprising a minichannel with a magnetic droplet and three temperature probes placed at three targeted regions (P1-P3) within a heating coil.b) Graph of the temperature measured by the three temperature probes at the targeted regions during the experiment.c) Timelapse images of the droplet fission and its distribution toward the targeted regions P2 and P1; and thermal images at the beginning and end of the first thermal cycle performed after droplet distribution.d) Time-lapse images of droplet removal from P1 (t 6 -t 7 ) and droplet addition to P2 (t 9 ) and thermal images at the beginning and end of the second thermal cycle after droplet redistribution.