Coupling Magnetic Torque and Force for Colloidal Microbot Assembly and Manipulation

For targeted transport in the body, biomedical microbots (μbots) must move effectively in three‐dimensional (3D) microenvironments. Swimming μbots translate via asymmetric or screw‐like motions while rolling ones use friction with available surfaces to generate propulsive forces. Previously the authors have shown that planar rotating magnetic fields assemble μm‐scale superparamagnetic beads into circular μbots that roll along surfaces. In this, gravity is required to pull μbots near the surface; however, this is not necessarily practical in complex geometries. Here, the authors show that rotating magnetic fields, in tandem with directional magnetic gradient forces, can be used to roll μbots on surfaces regardless of orientation. Simplifying implementation, a spinning permanent magnet is used to generate differing ratios of rotating and gradient fields, optimizing control for different environments. This use of a single magnetic actuator sidesteps the need for complex electromagnet or tandem field setups, removes requisite gravitational load forces, and enables μbot targeting in complex 3D biomimetic microenvironments.

Magnetic μbots convert an external-fieldinduced torque into movement.Swimming μbots [31,32] locomote in the bulk fluid with nonreciprocal motion while rolling μbots use shear gradients near surfaces to move.Here, we focus on μwheels, a type of magnetic rolling μbot that reversibly assembles in situ from colloidal superparamagnetic beads into 2D disks of varying size upon application of a planar rotating magnetic field.μWheels are particularly suited for applications in the vasculature due to their ability to actively reconfigure.They share the speed advantages of larger systems and the access advantages of small systems, as they can assemble into large swarms to increase power and motility, then disassemble into their colloidal building blocks when the field is removed.We have shown that this reconfigurability enables unique actuation modes that enhance climbing, drug delivery, [3] spreading, and velocity. [14]hen brought near a surface with a load force, typically gravity but other forces have been demonstrated, [33,34] rolling μbots use a shear gradient near a surface boundary to move with velocity V ∝ ω b R where ω b is the μbot angular velocity and R the boundingcircle radius. [35]Better control over the directionality and strength of the load force would enhance the ability of rolling μbots to traverse microenvironments that are tortuous and surfaces non-normal to gravity.In this work, we demonstrate a novel μbot actuation approach where the load is created through magnetic field gradients that complement the torque and translationinducing rotating magnetic field.
Most magnetic μbots are driven by changing the direction of the bulk magnetic field in time which induces a torque.Field direction is simple to control with three independent electromagnet axes, often in a six-coil Helmholtz arrangement. [36]In a Helmholtz setup, the gradient of the field is negligible so no magnetophoresis is induced.μBots also have low field strength requirements (< 10 mT), so coils can be small, low power, and in a simple arrangement for small working volumes [37] as the field strength B scales with the inverse of the distance from the magnet cubed, B ∝ 1=r 3 .[40][41] These techniques sometimes rely on closed loop microbot position tracking, an approach that is difficult to implement with microbot swarms.In general, the gradient component of the field induces a magnetophoretic force in magnetic material that is proportional to the inverse of the distance from the magnet to the fourth power, F ∝ 1=r 4 .Together, precise control of magnetophoresis for μbots requires larger magnetic actuation equipment with significant power and infrastructure requirements for human-scale use. [42]o translate, μwheels require a planar rotating field and an orthogonal load force to roll along a surface.We have previously shown that, with gravity alone, μwheels can roll up surfaces at angles up to 80°, demonstrating that only a small component of the load force needs to be in the direction of the surface, [14] reducing the requirements for effective 3D μwheel translation.In effect, one requires only a rotating magnetic field and one directional variable force to oppose gravity.A single rotating permanent magnet above the working volume can be used to simultaneously induce magnetophoretic forces to counteract gravity while still retaining the rotating field. [43]As the torque and force from a single magnet are coupled and differ in scaling, the ratio of the magnitude of the torque and force can be readily altered by changing the distance between the magnet and sample.The effects on microbot actuation of the torque [44] and force [45] from a rotating permanent magnet have been previously studied.In this work, we combine these approaches and demonstrate that by spinning and altering the distance of a nearby permanent magnet we can induce the assembly of superparamagnetic particles into μwheels with three distinct modes.At large magnet distances, magnetophoresis is low, gravity dominates, and μwheels roll along bottom surfaces (Figure 1a).At medium distances, magnetophoresis can overcome weight allowing μbots to roll on inverted surfaces (Figure 1b).At small distances, magnetophoresis dominates and μwheels can be collected and localized into a specified area.Once the magnet is removed, μwheels disassemble back into their constituent beads (Figure 1c).With this simple actuation scheme, μwheels can move on all surfaces regardless of orientation with respect to gravity and navigate complex 3D biomimetic microenvironments.

Coupled Magnetic Torque and Force from a Single Spherical Permanent Magnet
The field generated from a spherical permanent magnet (Figure 2a) is described by the dipole equation where B is the magnetic flux density vector or magnetic field strength, μ 0 the magnetic permeability of free space, r the distance, r the norm of r, and m m the magnetic moment of the permanent magnet.Inspection of this equation reveals that the magnitude of the field scales with the inverse distance cubed, B ∝ 1=r 3 (Figure 2b).Also, when the magnet moment m m is in line with r, the field is greatest as m m ⋅ r is maximized.When m m is orthogonal to r, the field drops by half (Figure 2c); therefore, at a fixed r and as m m is rotated around the angular velocity vector ω m (Figure 2a), an elliptical rotating field is generated (Figure S1, Supporting Information).Note that for this study ω m is set to be orthogonal to m m and r xy ¼ 0. A torque is induced when a magnetic dipole m b is misaligned with an external field B and is generally calculated via τ ¼ m b Â B. Since the colloidal beads that comprise μwheels are superparamagnetic, their moment is proportional to the local field strength, m b ¼ V bead χB=μ 0 .For simplicity across all μwheel shapes and sizes, the simple dipolar model is used here [46] where the bead moments are independent and are calculated only once based on the external field calculated from Equation (1).When the magnet rotates with angular velocity ω m , μwheels assemble into disks and rotate from the induced torque.The magnetization of the μwheel lags behind the magnetic field resulting in a magnetic memory torque [47] τ ¼ V bead nχ 00 B 2 b ω b =μ 0 where χ 00 is the imaginary magnetic susceptibility, n the number of beads, and b ω b the field rotation unit vector at the μwheel.It follows that the torque on a μwheel is then proportional to the inverse distance from the magnet to the sixth power, τ ∝ B 2 ∝ 1=r 6 , a consequence of the bead moment being dependent on the local field strength.Note here that the rotation of the magnet ω m above the sample creates an opposite rotation direction at the sample ω b , and thus an opposite μwheel rotation.In this study, the rotation rate is only 5 Hz enabling μwheels to rotate at the same rate as the field.When near a surface (< 2 μm), [33] μwheels roll with a head- The bead also experiences a magnetophoretic force F ¼ ∇ðB ⋅ BÞV bead χ=μ 0 .This force scales with the inverse of the distance to the seventh power F ∝ 1=r 7 as the force is dependent on the shape of the field as well as the bead moment m b .We define a nondimensional average force, F ¼ F=W where F is the magnetophoretic force on a single bead and W the buoyant weight force on a single bead.With this, F < 1 when gravity dominates and F > 1 when magnetophoresis is greater than the gravitational force.With a single magnetic dipole source, the field and force are coupled with the magnet distance r.The ratio between the field and force can be varied by changing the magnet distance r where the z, or against-gravity component, of the average normalized force Fz is shown (Figure 1b).In this study, the magnet is placed above the sample, so that the largest component of the force is opposite the direction of gravity, allowing locomotive freedom and movement on inverted surfaces.
At small magnet distances (r=r m < 3), F ≫ 1 and magnetophoresis dominates, resulting in an MP-dominated mode (Figure 2c) where locomotion due to the induced μwheel torque is insignificant and movement is based primarily on magnetophoresis.In confined systems, MP-dominated mode can be used to create a trapping area which concentrates magnetic material where the field is greatest.At medium distances (3 < r=r m < 4), Fz > 1 and magnetophoresis is greater than the weight force.In this MP-assisted mode, μwheels lift off and roll on inverted surfaces.Upon rotation of the field, μwheels translate and use these new accessible surfaces to break symmetry and roll (Supplementary Video 1, Supporting Information).The velocity of the μwheels due to the induced torque is much greater than the velocity due to the x-y component of the MP force, Fxy (Figure S2, Supporting Information) and μwheels retain their maneuverability.Also, since μwheels move in the microenvironment, the variability of the field perpendicular to the magnet is less relevant as the working volume is smaller.Across a 16 cm 2 region, Fz remains above 2 at r z ¼ 8.3 without having to reposition the magnet.At large distances (r=r m > 4), Fz and gravity dominates.μWheels use gravity as the requisite load force in this MP-low mode and retain their locomotion capabilities as previously reported with electromagnet actuation. [48]Here, the rotating permanent magnet creates an elliptical rotating field with a slower field frequency (5 Hz), contrasting with previous studies with a circular field and higher frequency (40 Hz). [14,35]

μWheel Manipulation
Due to the superparamagnetic nature of the constituent beads that form μwheels, the three actuation modes can be readily switched (Figure 3a-d, Supplemental Video 2, Supporting Information).In the MP-dominated mode, beads collect in a trapping region against a wall where ∇B is the lowest.At a Fxy ¼ 6, bead clusters initially travel toward the trapping point with a minimum velocity of %40 μm s À1 .By rotating the magnet around the z axis, the direction of the moment vector can be changed, selecting the collection region without moving the magnet position in the xy plane (Figure 3a,b,e,f ).This is possible due to the fixed 20°c amber angle of the magnet (θ m ) which means m xy 6 ¼ 0 in all orientations.When the field is removed, μwheels disassemble into their constituent beads (Figure 3c).In this system, this occurs at r > 30 cm.Note that disassembly is not required between actuation modes, but does affect μwheel size formation, as studied previously. [14]Additionally, collecting magnetic material in this way has been previously studied in detail due to its many applications in localized drug delivery and particle fractionization. [49,50]Wheel trajectories were imaged, analyzed, and averaged over their entire length to characterize manipulation in the MPassisted and MP-low modes in a horizontal glass parallel plate sample chamber.At high magnet separations, here, r z ¼ 11.3, 13.5 cm, μwheels use gravity to roll on the bottom surface with positive velocity in the MP-low mode.In the MP-assisted mode, here r z ¼ 8.3, 8.9 cm, μwheels move to the top inverted surface and roll in the opposite direction with negative velocity (Figure 3h, Supplementary Video 1, Supporting Information).Consistent with previous experimental results with electromagnets, [48] μwheels with bounding-circle radius R roll with velocity V ∝ ω b R. Here, all μwheels rotate in step with the field, leading to the linear V ∝ R behavior across all R due to the single field frequency reported here.Due to the shape of the field from the permanent magnet, the camber angle of the rotating field θ f is not constant at every point in the working volume (Figure 3g).However, μ wheels are capable of rolling at high camber angles [48] (< 80°) and are minimally affected by the %20°change in θ f across 4 cm 2 .Additionally, the variable load force generated by the magnet across various r z does not strongly influence the traction under these experimental conditions (Figure 3i).
The shape of the formed μwheels is determined by a balance between the drag torque due to the fluid and the magnetic torque.At lower field strengths, fluid drag overcomes these dipoledipole attractions within the μwheel and the structure collapses into round, disk-like shapes.As the field strength increases, more linear μwheel conformations are preferred due to the alignment of the dipoles within the bead.To characterize this, we calculate a roundness according to A = sinðθ f ÞπR 2 where A is the measured projected area of the μwheel.With μwheels assuming 2D configurations due to the planar field, [48] roundness characterizes disk shape.Generally, roundness decreases as the field strength increases (Figure 3j).The low rotation frequency combined with high field strengths generated by the permanent magnet enables low-roundness chains to form.
At high field strengths, these chains can get too large and exhibit instability and breakup, as reported elsewhere. [51,52]he elliptical nature of the field used here likely increases this phenomenon, due to the dipole-dipole interaction strength halving twice every rotation.Due to the difficult nature of tracking these spurious trajectories, these unstable chains are not included in the data analysis.At low field strengths and a low field frequency of 5 Hz, we observe a rolling instability in large, round μwheels (R > 30 μm), consistent with other experimental observations. [53]This suggests that 5 Hz is in the transition region between an end-over-end and edgewise rolling mode.

Magnetophoresis-Assisted Movement Enables Full 3D Rolling
To demonstrate complex μwheel maneuvers in 3D by switching actuation modes, a glass capillary is placed horizontally.Here, μwheels can use gravity as the requisite load force and move up the edges of the capillary where the slope approaches 90°a s observed in previous work. [14]When switching to MP-assisted mode however, the top half of a cylinder is now available for μwheel translation.Here, Fz > 1 and magnetophoresis are used as the requisite load force.To demonstrate switching between modes and the capability it provides, a helical maneuver is performed inside the capillary (Figure 4a,b; Supplemental Video 3, Supporting Information).When the μwheel reaches the edge of its locomotion capability in the MP-low mode, the mode is switched to the MP-assisted mode by decreasing r z to move across the top half of the capillary.Additionally, this experiment is conducted in a fluid twice the viscosity of water, demonstrating that higher fields enable μwheel translation in viscous fluids.
To demonstrate that complex 3D biological networks are now traversable by μwheels, a 3D-printed capillary network model was fabricated.In this, a bolus of beads was injected into the model and trapped at a designated start region with a small permanent magnet.In separate experiments, a μwheel swarm was driven into the four distinct capillary branches (Figure 4c-i; Supplemental Video 4, Supporting Information).The left and bottom branches, located in plane and in the direction of gravity, are accessible solely using MP-low mode.When targeting branches like these, gravity is used as the requisite load force and the bottom capillary surfaces are used to move.The top branch, however, is directly above the main thoroughfare channel and requires a nongravitational load force to access.Using the MP-assisted mode, the μwheel swarm is attracted to the top surface of the channel and rolls up and into the target branch.Lastly, the upper-right branch is accessible with either the MPlow or MP-assisted mode.Here, the MP-low mode was first used to steer into the branch followed by the MP-assisted mode to switch to the top surface and quickly climb the steeply inclined channel.Across all experiments, >90% of the swarm was localized in the desired capillary branch.Swarm manipulation was also enhanced by using the previously studied "switchback" field pattern in which the μwheel size distribution was decreased and climbing ability increased. [14]After targeting a certain branch, the MP-dominated mode can be used to collect and concentrate the beads.This is particularly effective when the branch has a component in the direction toward the magnet.Otherwise, branches like the bottom branch have a natural collection behavior due to gravity collecting the beads.

Discussion
By combining a rotating bulk magnetic field with a controllable gradient force, the manipulation of colloidal μwheels in 3D geometries is greatly extended.This approach takes advantage of the superparamagnetic properties of μwheels, enabling reversible assembly and switching between modes.This property fits well with field-gradient techniques and allows prevention of large agglomerations which are especially important to avoid in biomedical applications.As a result, this approach is applicable to many of the complex 3D networks in the body including the pulmonary, circulatory, lymphatic, and excretory systems.
High field strengths directly increase the torque available to drive μbots.For paramagnetic μbots, τ ∝ B 2 , further increasing the effect of field strength on μbot manipulation.This could prove useful for increasing μbot translation rates by increasing the maximum rotation rate of μbots before they step-out of the field's rotation rate and enable movement through viscous fluids in the body.For μwheels, higher fields favor eccentric shapes over perfect disks.These elliptical μwheels are more efficient; for a given number of beads, a higher bounding-circle radius μbot can be constructed.Since V ∝ ωR, the larger the radius, the faster the μwheel.Additionally, recent research has shown that eccentric, or slender, microrollers reduce out-ofplane rotational flows and perform better in confined geometries. [54]ue to the simple fields required for μwheel assembly and movement, a single rotating permanent magnet can readily supply the necessary fields.Since μwheels only require a small load force component into a nearby surface and can climb extreme slopes, [14] the magnet only needs to be placed above the working volume to counteract gravity while simultaneously applying a bulk rotating field.While corkscrew-shaped magnetic swimming μbots would still rotate in the presence of the rotating field, the locomotion direction for swimming μbots is typically perpendicular to their rotation.Consequently, they are more sensitive to the orientation of the local field than μwheels, which keep rolling as they tilt. [48,55]Additionally, magnetophoresis would only increase μbot translation when the force vector is in the same direction as locomotion and likely would hinder the intended manipulation.
The permanent magnet actuator in this study is simple and does not require additional scaling up for human use, unlike electromagnetic systems typically used for μbot research.The 5 cm diameter spherical magnet used here is suited for MP-assisted use 7-9.5 cm (up to %25 mT) and MP-low use up to 15 cm (2 mT).This working distance allows penetration into most deep regions of the body.If needed for specific applications, a 10 cm magnet would project its field twice as far but, at a given field magnitude, the gradient would be decreased due to the homothetic property of magnetic objects. [56]While the use of a permanent magnet for μbot manipulation has been previously studied, [55,57,58] its application for assembly and full 3D manipulation of rolling μbots has not been reported.These previous studies however have demonstrated how such magnets can be affixed to a robotic arm for positioning over the entire body [57] or can be spun in fluid for >100 Hz rotation rates. [58]

Conclusion
Simultaneous use of the torque and force components of a magnetic field enables μwheels to move on previously inaccessible inverted surfaces.Microwheel swarms can switch modes between collection, inverted translation, and gravity-assisted translation by simply changing the balance between magnetic bulk and gradient fields.Such coupled force and torque actuation is particularly fitted to rolling μbots as the requisite load force is low and perpendicular to the rotation direction.With this approach, full μwheel navigation is now possible through 3D networks regardless of the orientation with respect to gravity.Implementation of this scheme was significantly simplified using a single rotating permanent magnet with three degrees of freedom, sidestepping the need for prohibitively large human-scale electromagnet systems.

Experimental Section
Magnetic Fields: The magnetic field generated by a spherical permanent magnet is perfectly described by the dipole equation (Equation ( 1)) outside the magnet body. [56]The moment of a hard permanent magnet is calculated by m m ¼ B r V=μ 0 with B r the residual flux density, V the magnet volume, and μ 0 the vacuum magnetic permeability.The moment m m , by convention, points from the magnet center through its North pole.The magnet used in this study is a neodymium sphere of diameter 50.8 mm and B r ¼14 800 Gauss (N52 sphere, Applied Magnets, Plano TX, USA).
The moment of a superparamagnetic bead is assumed to be a soft magnet below saturation and is calculated by m b ¼ V bead χH 0 where V bead is the bead volume, χ the magnetic susceptibility, and H 0 the external magnetic induction field.In this study, the magnetization of the medium (water) is assumed to be negligible so B ¼ μ 0 H.The beads used in this study are M-450 Dynabeads (Thermo Fisher) with a reported magnetic susceptibility [59] of χ m ¼ 1.63.
3D-Printed Magnet Actuator: A custom robotic actuator (Figure 5) was designed with three degrees of freedom (DoF) to create the necessary fields for gradient-assisted microbot rolling (software Fusion 360, Autodesk).The body of the device was 3D-printed (Prusa Mk3sþ, Prusa Research, Prague, Czech Republic) in polyethylene terephthalate glycol plastic (Hatchbox 3D, Pomona, CA) with specialty parts purchased from McMaster-Carr (Douglasville, GA).The use of metal in the device, especially close to the magnet, was minimized to prevent eddy currents.The motors used were brushless dual shaft motors (D5065 270KV, ODrive Robotics) with an encoder (CUI AMT102-V, Figure 5. Rotating permanent magnet apparatus.The top construction contains a 2" N52 spherical permanent magnet capable of rotation rates up to 20 Hz and heading angles 0°to 360°.The bottom construction moves the sample and microscopy train on a linear axis toward and away from the rotating magnet. ODrive Robotics) mounted on each secondary shaft.Control was performed with an ODrive v3.6 board that interfaces with Python and allows the motors to be precisely controlled with position and velocity feedback control.The device consists of an upper portion that rotates and directs the rolling direction of the magnet (1st and 2nd DoF) and a lower portion that moves the sample and microscopy train to a prescribed distance from the magnet (3rd DoF).
Only three DoF are required for the necessary magnet movements.First, the magnet can spin around a given angular velocity vector b ω m up to 20 Hz.The spherical permanent magnet is securely fixed in a 3D-printed holder with a shaft in line with b ω m .For this study, the magnet moment b m m was arranged perpendicular with b ω m by aligning the magnet poles orthogonal to the shaft.This shaft is rotated using a pulley and belt system with a nearby motor.Second, b ω m can be moved 360°in the xy plane to set the rolling direction.This is performed by affixing the magnet rotator to a large 3D-printed gear assembly which can rotate to a specific position with software input.Optionally, the z component of b ω m , which sets the magnet camber angle, can be changed by varying the mounting angle of the rotator to this 3D-printed gear with a 3D-printed insert.For this study, the magnet camber angle was fixed at 20°for easier viewing of μwheel shape, size, and rotation rate.Third, the sample distance to the magnet can be varied to change the magnetic field and force strength.This is performed by moving the lower portion with the sample and microscopy train toward or away from the upper portion with the magnet with a motor-driven belt.
The lower portion is modular and can be switched between a microscope mode for microscale imaging or a macromode that supports a smartphone.The microscope consists of a 10x objective (Nikon) mounted to a high-speed camera (pco.panda4.2, Excelitas PCO GmbH) that is moved and focused with a 3-axis linear translation stage (460-XYZ, Newport).An LED light source with a collimator (LEDD1B with COP1-A, ThorLabs) is mounted to the left of the sample, and light is directed through the sample to the microscope using a mirror mounted at 45°.For macromode, the microscope is replaced with a 3D-printed holder for a smartphone (S22þ, Samsung).In this mode, the LED light source is removed and the 45°mirrors are replaced with a white backsplash for enhanced imaging contrast.
Macroscale μWheel Swarm Demonstration: A sample well was constructed using double-sided tape (3M VHB Tape GPH-060GF) placed around the edges of a 1 Â 3" glass slide.Approximately 2 μL of 4.5 μm beads with initial concentration of 4 Â 10 8 beads mL À1 (Dynabeads M-450 Epoxy, Thermo Fisher) were diluted 100Â in 0.2 wt% sodium dodecyl sulfate (SDS) (Sigma-Aldrich).This was added to the reservoir and the remainder of the volume was filled with 0.2 wt% SDS solution.Videos were recorded using the macroimaging mode from below.
Microwheel Velocities: A sample chamber was constructed by cutting an %1 Â 1.5 cm rectangle in double-sided tape (3M VHB Tape GPH-060GF) and placing it on a 1 Â 3" glass slide.Approximately, 60 μL of 500x diluted 4.5 μm beads (Dynabeads M-450 Epoxy, Thermo Fisher) in 0.2 wt% sodium dodecyl sulfate (SDS) (Sigma-Aldrich) was added to the reservoir and covered with a square 22 mm glass cover slip.The sample was placed in the holder and imaged using a custom microscope (see 3D-printed magnetic actuator methods).Videos of μwheels rolling at each experimental condition were recorded and then analyzed using custom μbot tracking software written in Julia. [60]elical Motion in a Cylindrical Capillary: A glass capillary (OD 1000 μm and ID 580 μm) was suspended over a glass slide with a 3D-printed enclosure filled with dimethyl sulfoxide (DMSO) (Sigma-Aldrich).The capillary was filled with %1 μL of 500x diluted 4.5 μm Dynabeads in 0.2 wt% SDS in DMSO.The DMSO (refractive index = 1.479) enabled clear viewing through the round capillary with minimal refraction at the curved edges.The custom microscope (see Methods) was used to record μwheel movements with the focus fixed to show out-of-plane movements.
Capillary Network Targeting: A 3D biomimetic capillary network model was designed in CAD software Fusion 360 with four distinct branches off a thoroughfare channel.The model was 3D printed using stereolithography (Form 3, FormLabs) in clear plastic.After removing support material and washing thoroughly with isopropyl alcohol (IPA) (Sigma-Aldrich), the inlets were plumbed with 0.01 0 0 ID by 0.030 0 0 OD clear Tygon tubing (Cole Parmer) and fixed in place with two-part epoxy.The outside surfaces of the devices were sprayed with three coats of glossy clear paint (249 117 Gloss Clear, Rust-Oleum) to increase transparency.After curing, the internal channels were subsequently rinsed with 1 mL of IPA, 1 mL of deionized water, and 1 mL of 0.2 wt% SDS.Then, 0.3 mL of 30x diluted 4.5 um Dynabeads was loaded while a permanent magnet was simultaneously placed near the inlet junction.The device was placed in the sample holder of the magnetic actuator in macrophotography mode.Videos were recorded in the macroimaging mode from below, showing the progress of the microswarm throughout magnetic targeting.

Figure 1 .
Figure 1.A single rotating permanent magnet can be used to create the simple fields required to a) assemble and spin μwheels while b) simultaneously supplying a load force through magnetophoresis (MP) opposite the direction of gravity.c) When the magnet is removed μwheels disassemble.

Figure 2 .
Figure 2. Magnetic field near a spherical permanent magnet.a) A spherical permanent magnet (r m = 2.54 cm) has two poles with the moment m m pointing through the North pole.The magnet is rotated around the angular velocity vector ω m which, for this study, is set perpendicular to m m .The magnet camber angle θ m measures the angle between the xy plane and ω m and is kept constant in this study at θ m ¼ 20°.b) Left axis: The average field strength B a distance r away from m m .Right axis: The average normalized magnetophoretic force Fz ¼ F z =W where F z is the magnetophoretic force on a single bead in the z, or b r direction and W the buoyant weight force on a single bead.B ∝ 1=r 3 and Fz ∝ 1=r 7 .B inset) During rotation, B is at a maximum when m and r are parallel and drops by half when they are orthogonal.The magnetophoretic load drops by 4x when m and r are orthogonal (r z ¼ 8.9cmÞ.c) Modes: Magnetophoresis (MP) low, MP-assisted, and MP-dominated.

Figure 3 .
Figure 3. Permanent magnet μwheel manipulation.a,b) Beads are collected through magnetophoresis in the MP-dominated mode.c) When the field is removed (r > 30 cm) μwheels reversibly disassemble.d) μWheels roll in the MP-low mode.Scale for (a)-(d) = 0.55 cm.e,f ) Force uniformity in the x-y plane at r z ¼ 7.0 cm, corresponding to images in (a) and (b).Beads accumulate in the blue region where Fxy is the lowest.For e), b m m ¼ ½0, sinðπ=9Þ, À cosðπ=9Þ and for f ), b m m ¼ ½0, À sinðπ=9Þ, À cosðπ=9Þ.g) Uniformity of the rotational field in the x-y plane.Arrows correspond to b ω xy of the field at each test point, while the color corresponds to the field camber angle θ f ¼ sin À1 ðb ω z Þ. h) Measured velocity and bounding-circle radius of μwheels in MP-assisted and MP-low modes at distances r z ¼ 8.3, 8.9, 11.3, 13.5 cm, corresponding to B ¼ 21.1, 17.1, 8.4, 4.9 mT.i) Measured traction of μwheels across all field conditions and μwheel shapes.j) Measured roundness of μwheels where a value of 1 corresponds to a circular μwheel and a roundness of %0.2 corresponds to a single bead wide chain.Points outside the box plots represent outliers.

Figure 4 .
Figure 4. Magnetophoresis assistance enables gravity-independent μwheel rolling.a) Individual μwheel performing a helical maneuver in a cylindrical capillary.The overlaid trajectory color is the height from the bottom of the cylinder in the normal, gravity direction.The MP-low mode is used for the beginning and end, while the MP-assisted mode is used in the middle to move on the top half of the capillary.Scale = 300 μm.b) Illustration demonstrating the programmed helical trajectory.c) Rendered CAD model of the bifurcating capillary network.d) Fabricated 3D-printed device.Scale = 1 cm.e) Snapshot of bead bolus at the start location.Scale = 0.75 cm.f ) After targeting to the left capillary branch.g) After targeting to the bottom branch.h) After targeting to the upper-right branch.i) After targeting to the top branch.