## Abstract

Microbot locomotion is challenging because of the reversible nature of microscale fluid flow, a limitation that can be overcome by breaking flowfield symmetry with a nearby surface. We have used this strategy with rotating wheel-shaped microbots, microwheels (μwheels), that roll on surfaces leading to enhanced propulsion and fast translation speeds. Despite this, studies to date on flat surfaces show that μwheels roll inefficiently with substantial slip. Taking inspiration from the mathematics of roads and wheels, we demonstrate that μwheel velocities can be significantly enhanced by changing microroad topography. Here, we observe that periodic bumps in the road can be used to enhance the traction between μwheels and nearby walls. Whereas continuous μwheel rotation with slip is observed on flat surfaces, a combination of rotation with slip and nonslip flip occurs when μwheels roll on surfaces with periodic features, resulting in up to fourfold enhancement in translation velocity. The unexpectedly fast rolling speed of μwheels on bumpy roads can be attributed to the hydrodynamic coupling between μwheels and road surface features, allowing nonslip rotation of entire wheels along one of their stationary edges. This road-wheel coupling can also be used to enhance μwheel sorting and separation where the gravitational potential energy barrier induced by topographic surfaces can lead to motion in only one direction and to different rolling speeds between isomeric wheels, allowing one to separate them not based on size but on symmetry.

## INTRODUCTION

Microscale propulsion has attracted considerable interest in recent years because of the desire for self-directing in vivo devices (*1*, *2*) and the challenging nature of low Reynolds number flows (*3*). External field-based approaches, especially those using magnetic fields, are among the most promising because they are inherently noncontact and require no chemical fuel. To enable propulsion induced by magnetic torque (*4*, *5*) and without addition of external forces or field gradients, microbot symmetry must be broken either with appropriately designed shape (*6*–*9*) or with proximity to walls (*10*–*20*). However, fabricating asymmetric microdevices of suitable geometry can be challenging, motivating wall-based methods such as microwalkers (*10*), artificial cilia (*11*), microworms (*12*), microcarpets (*13*), microwheels (μwheels) (*14*, *17*), microlassos (*18*), and microswimmer swarms (*15*, *16*, *19*, *20*).

Smooth surfaces, however, are associated with substantial slip with inefficiencies that set the upper limit on translation velocity. Here, we identify surface topographies that register with μwheel structure, taking inspiration from the mathematics of roads and wheels where it can be shown that, for any given wheel shape, there is a complementary road for optimal translation (*21*, *22*). For example, smooth-riding bicycles can be made with square-shaped wheels on roads constructed from a series of truncated catenaries (Fig. 1). Here, we investigate the translation of μwheels of varying size and shape on surfaces designed to mimic their corresponding ideal roads with the goal of enhancing translation velocity. In doing so, we develop methods where wheels of different structure interact differently with different roads, providing opportunities for not only faster devices but also for μwheel separations and sorting, paving the path toward designing micro- and nanodevices capable of enhanced translation, control, and movement within real topographically complex environments.

## RESULTS

### μWheel translation on flat surfaces

Building on previous work where we have demonstrated that colloidal devices such as pumps and valves can be assembled in two dimensions with external fields (*23*, *24*), we have recently demonstrated that uniform three-dimensional (3D) magnetic fields can create colloid-based μwheels capable of rapid translation on flat surfaces (*14*, *17*). In this, negatively charged superparamagnetic particles of radius *a*, denser than their surrounding solvent, assembled into close-packed μwheels via isotropic interactions induced by an in-plane rotating magnetic field, *B _{xy}* is the field strength, ω

*/2π is the applied field frequency, and*

_{M}*n*and the degree of rotational symmetry ξ.

In the presence of an in-plane rotating magnetic field, fabricated μwheels were laid flat and spun on a surface without net translation; however, upon application of an additional AC field along the *z* direction, *z* and *xy* components, μwheels stood up and rolled (Fig. 2A and movie S1). Specifically, they inclined relative to the surface and spun in the same plane as the field at a camber angle θ* _{c}* = tan

^{−1}(

*B*/

_{xy}*B*). The angular rotational frequency ω of a μwheel is related to the applied angular field frequency ω

_{z}*as*

_{M}*is the critical frequency related to the overall applied field strength (*

_{c}*25*). In this work, we typically ran experiments with 0.628 ≤ ω

*≤ 314 rad/s. For 2D disc-like μwheels, the time-averaged translation velocity*

_{M}*R*, with radius

_{n}*14*,

*17*).

Because μwheels were assembled from individual spherical colloidal building blocks, their perimeter was not strictly circular. As a result, μwheel translation on flat surfaces was influenced by geometry and the specific manner in which μwheels interacted with the surface. This is seen in Fig. 2C, where the instantaneous velocities of μwheels over one period of rotation demonstrated oscillations inversely proportional with their degree of rotational symmetry. To describe this observation, we modeled the rolling of μwheels of different size and symmetry on flat surfaces by extending the analysis of Tierno *et al.* (*26*, *27*). Although greater detail can be found in the Supplementary Materials, here, we focus on dimers (*n* = 2, ξ = 2), whose motion is shown in movie S2 and fig. S2B. In this case, the instantaneous velocities parallel ν(*t*) and perpendicular ν* _{z}*(

*t*) to the substrate can be written as

*x*and

*h*are the displacements of a dimer parallel and perpendicular to the substrate based on the center of mass.

*V*

_{0}= 4ω

*af*/3 is a characteristic velocity, where

*f*(

*a/h*) is a correction factor that accounts for the rotation of a sphere near a wall (

*28*–

*30*)(see “Modeling for μwheel translation on flat and textured surfaces” section in the Supplementary Materials). β

*and α*

_{i}*are the hydrodynamic mobilities of lobe*

_{i}*i*moving parallel and perpendicular to the substrate and are functions of the lobe-wall separation

*h*. In our work, because μwheels were very close to the substrate, we used expressions (eq. S3) based on asymptotic lubrication theory (

_{i}*31*), which provides the correct limits that α

*, β*

_{i}*→ 0 when the contact distance*

_{i}*a*/

*h*→ 1.

_{i}We solved Eq. 2 for dimers and plotted the instantaneous velocities in fig. S2 (B and C). In our calculations, we set the initial separation between the μwheel edge and the substrate to 0.2*a* = 450 nm determined by balancing the buoyancy force on the μwheel and the electrostatic repulsion between the μwheel and the substrate. We see that ν(*t*) varied with the dimer orientation during rolling and reached a maximum when the dimer stands up and a minimum when it is parallel to the surface. This behavior arises from the difference in the hydrodynamic mobilities (β_{1} − β_{2}) between lobes, which depends on lobe-wall separations *h*_{1} and *h*_{2} (eq. S3). When a dimer is suspended in bulk fluid far away from the substrate, no net translation is expected because their mobilities are equal, β_{1} = β_{2} = 1; similarly, when a dimer is very close to the substrate but lies parallel with both lobes equally separated from the substrate, ν equals 0 because, once again, the mobilities are equal and β_{1} = β_{2} = 0. When the dimer stands up, however, the difference in mobilities β* _{i}* is largest, and the dimer has the greatest speed. Matching μwheel symmetry, the dimer orientation changes twice per rotation as does the velocity. We note that during rotation the μwheel center of mass did not remain a constant distance from the substrate (fig. S2C), rather it oscillated with the same period as its lateral velocity because of differences in mobilities α

*between lobes. Only perfectly circular wheels such as monomers roll with their center-of-mass constant elevation above a flat substrate. Note that over the course of rotation, our modeling predicts only ~0.1*

_{i}*a*oscillation from the average center of mass to substrate separation, a short distance that explains relatively smooth rolling during dimer translation. This can also be observed by tracking the displacement of the centers of mass of both lobes, which exhibit a characteristic 180° phase difference (fig. S2D). By solving eq. S6 for μwheels with different degrees of symmetry (2 ≤ ξ ≤ 6), we observed similar features in ν and ν

*. As shown in Fig. 2C, predictions for the lateral velocities*

_{z}*v*correspond to our experimental data.

### μWheel translation on periodic topographic surfaces

On flat surfaces, μwheels demonstrate substantial slip (~90%) because the μwheel-wall hydrodynamic interaction weakly contributes to net translation. We improved coupling and increased translation velocities by introducing periodic features on the substrate. In our case, as wheel shape varied, substrate structure could be modified most simply by changing road periodicity and spacing to match wheel features. Ideal substrates for rolling without slip for μwheels of different symmetry can be predicted numerically (Fig. 3A and see Supplementary Materials for details).

To mimic continuous surfaces with repeating catenary features, we fabricated topographic surfaces using polydimethylsiloxane (PDMS) replicas of linear diffraction gratings (Fig. 3B). Experimental measurement by atomic force microscopy (AFM) shows that the spacing between ridges was 9.9 μm, maximum height was 3.83 μm, and blaze angle was γ = 26°. One of the advantages of this surface is that effective spacings between ridges can be varied by simply changing μwheel rolling angle θ (Fig. 3B). We began with dimers where, by tracking the center of mass of each lobe, we identified two distinct translation modes during rolling (Fig. 3C and movie S3). Mode I occurs when dimers translate between two peaks, during which they rotate along their center of mass, evidenced by the characteristic 180° phase difference between lobes. This mode is the same as observed on flat substrates (fig. S2D) and is associated with weak wheel-surface interaction, substantial slip, and a relatively low net translation velocity. With addition of surface features, however, a second mode, mode II, appears where μwheels approach and flip over peaks. In mode II, the lobe adjacent to the peak remains almost motionless, whereas the other rotates about a full dimer length; essentially, the dimer flips without slip. This is reflected by the almost constant lateral position for the red lobe during mode II in Fig. 3C. As dimers rolled on these periodic surfaces, these two modes alternated effectively via “slip and flip,” leading to a faster net translation velocity on patterned surfaces than on flat ones.

To better understand the slip and flip translation of μwheels on textured surfaces, we again used Eq. 2 but then incorporated surface topography *h _{s}*(

*x*) into the mobilities α

*and β*

_{i}*in eq. S3 via*

_{i}*h*

_{1}=

*h*+

*a*cos ω

*t*−

*h*(

_{s}*x*+

*a*sin ω

*t*) and

*h*

_{2}=

*h*−

*a*cos ω

*t*−

*h*(

_{s}*x*−

*a*sin ω

*t*), where

*h*

_{1}(

*h*

_{2}) is the separation between the center of lobe 1 (lobe 2) and the patterned substrate (fig. S3A). We note that this method is an approximation; the calculation of exact sphere mobilities near nonflat surfaces is nontrivial, and only the influence of walls with small-amplitude, sinusoidal deformations has to date been reported (

*32*). As a result and instead of modeling the rolling of dimers along surfaces that exactly match our experiments, we used small trapezoidal bumps to represent microroad surface features. As shown by

*h*(

_{s}*x*) in fig. S3B, the spacing between bumps

*d*was 4.4

*a*, the same as that between gratings in our experiments. The height and length of the bumps were then tuned to be 0.38

*a*and 0.2

*a*, respectively, so that the dimer rolled at the same time-averaged speed as observed in our experiments. The instantaneous velocities parallel (ν

*) and perpendicular (*

^{p}*11*,

*30*). Therefore, lobe 2 remained stationary, whereas lobe 1 (black) rotated along lobe 2’s center of mass, leading to a flip. Because α

_{2}, β

_{2}~0, the difference in mobilities between the two lobes, and hence the translation velocity, is at a maximum during flip. This model reveals that noncircular μwheels could roll faster on appropriate bumpy microroads due to the hydrodynamic coupling and the resulting nonslip flip. Although the sawtooth-like surface topography in our experiments was more complicated, the simplified trapezoidal bumps in our modeling captured the underlying physics. When we set a very small initial separation of 0.05

*a*between the dimer edge and flat substrate in simulations, we observed sequential lobe flips instead of normal rolling (movie S5). This again is due to the marked mobility drop of one lobe that was very close to the substrate, whereas the other lobe simply rotated around it.

The slip and flip modes were also apparent via Fourier transform of the instantaneous μwheel translation velocity ν* ^{p}* (Fig. 3, C and D, insets), where two frequency peaks arose. The first frequency, the slip frequency ω

*, is related to the angular rotational frequency and μwheel symmetry, ω*

_{s}*= ξω. The second frequency, the flip frequency ω*

_{s}*, is related to how often a μwheel encounters a surface peak, governed by the surface feature spacing*

_{f}*d*and μwheel rolling angle θ:

With a combination of slip and flip, *ν _{s}* and

*ν*, as well as τ

_{f}*and τ*

_{s}*, are the instantaneous translation velocities and periods for the slip and flip motion with τ = τ*

_{f}*+ τ*

_{s}*. Note that τ*

_{f}*and τ are related to ω*

_{f}*and ω*

_{s}*by τ*

_{f}*= 2π/ζω = 2π/ω*

_{f}*and τ = 2π/ω*

_{s}*. If we define*

_{f}*C*and C

_{s}*as the propulsion coefficients for slip and flip, respectively, then*

_{f}*/ω*

_{f}*as*

_{s}Following Eq. 3, we measured * _{s}*, and ω

*for rolling μwheels of different sizes and symmetries at different rolling angles θ, both along and against the blaze direction (Fig. 4A). We fitted experimental data with Eq. 3 to find*

_{f}*C*/

_{f}*C*= 4.07,

*C*/

_{s}*C*= 0.22. The ratio

*C*/

_{s}*C*< 1 indicates some deceleration in the flip mode along θ = 0° due to channel edges. The ratio

*C*/

_{f}*C*> 1 indicates that efficient flipping led to roughly fourfold faster net translation velocity on topographic surfaces. By adjusting the spacing between bumps on the microroad, we could completely eliminate the slip mode, allowing the dimer to flip continously as shown in movie S6. Via substitution of Eq. 3, the ratio of ω

*/ω*

_{s}*becomes*

_{f}*/ω*

_{s}*and*

_{f}### A μwheel “rectifier”

One advantage of constructing surfaces from diffraction gratings is that the surface feature slope is steeper in one direction of travel, the blaze direction, than the other (Fig. 3B), allowing us to investigate the impact of energy barrier on rolling. If the magnetic field frequency is below a critical value, then dimers will be trapped between surface features when rolling against the blaze direction (Fig. 5 and movie S7). If the direction is switched, however, then dimers will translate readily at all field rotation frequencies, a rectifying effect that can be understood by considering the relevant forces (*33*, *34*). For a μwheel rolling along a surface slope with angle γ (Fig. 5A, inset), μwheels experience (i) a translation-rotation coupling hydrodynamic force *F*_{d,tr} ∼ 6πηω*R _{n}*

^{2},where η is the solvent viscosity; (ii) a gravitational force

*F*

_{g}=

*mg*sinγ, where

*F*

_{f}= μ

*cosγ,where μ*

_{k}mg*is the kinetic friction coefficient. The balance between these three forces determines whether a μwheel can overcome the geometric barrier to reach the next surface peak or, if*

_{k}*F*+

_{f}*F*

_{d,tr}<

*F*, be trapped. Because

_{g}*F*

_{d,tr}is proportional to the angular rotational frequency ω, there exists a critical frequency ω* below which the dimer does not translate. By letting

*F*+

_{f}*F*

_{d, tr}∼

*F*, we estimated that ω* ~ 3.73 rad/s for dimers rolling along θ = 90° against the blaze direction, close to the experimentally determined value (ω* ~ 3.77 rad/s). In addition, aiding translation here is that the slope length along the blaze direction is shorter than the dimer long axis, allowing it to easily roll over surface peaks without having to overcome the larger gravitational energy barrier (Fig. 5B). For μwheels larger than the slope length in either direction—diamond-shaped tetramers, for example—rectifying disappears because they can translate in either direction at all frequencies. On the other hand, the square-shaped tetramer can only move with the blaze direction at all frequencies.

_{g}### Separation of isomeric μwheels by symmetry

For a given rotation rate and on flat surfaces, μwheel size is the most important factor in determining velocity, and isomeric μwheels roll at about the same speed; for example, diamond- and square-shaped tetramers translated at similar velocities (Figs. 2A and 6A), and they could not be easily separated from each other. Topographic surfaces, however, could be exploited to separate isomeric μwheels. For example, diamond wheels interacted more effectively with the textured surfaces than squares with (ω* _{f}*/ω

*)*

_{s}_{diamond}> (ω

*/ω*

_{f}*)*

_{s}_{square}(Fig. 4A). With a larger velocity enhancement for diamond μwheels (Fig. 4A), one could separate them from square μwheels by rolling them simultaneously on the same textured surface (Fig. 6B and movie S8). An alternative approach to separate isomers is to take advantage of the rectifier effect and manipulate the applied field frequency because there is a frequency range within which the difference in the change of gravitational energy during rolling between diamond and square wheels is large enough so that diamond wheels translate continuously, whereas square wheels remain stationary (Fig. 6C and movie S9). Our observations are reminiscent of physical methods that rely on geometric constraints, such as arrays of posts or obstacles, to bias the net transport direction of objects of different sizes, including cells and colloids for separations (

*35*–

*37*). Instead of size, here, we used the coupling between translation speed and symmetry of objects induced by patterned surfaces, offering a new way to separate microscopic objects.

## CONCLUSION

We have investigated the rolling of superparamagnetic μwheels of different sizes and symmetries driven by a rotating magnetic field on flat and textured surfaces. On flat surfaces, the overall translation speed of μwheels is governed by their size with substantial slip during rotation. With addition of periodic features on the surface, however, we observe two alternating translation modes, slip and flip, which are tied to surface topography. The flip mode arises because of the hydrodynamic coupling between the textured surface and the μwheel, which allows simple rotation of the whole wheel along one of its stationary edges. Because the flip motion shows no slip, the overall μwheel translation velocity is significantly enhanced. We further demonstrate that the unique propulsion behavior of μwheels on topographic surfaces can be used for separation of isomeric μwheels of different symmetry. The hydrodynamic coupling between textured surfaces and μwheels revealed in our study may lead to more efficient propulsion of microdevices in topographically complex environments.

## MATERIALS AND METHODS

### Materials

Dynabeads M-450 Epoxy (4.5 μm in diameter) were obtained from Thermo Fisher Scientific Inc. Poly(diallyldimethylammonium chloride) [PDADMAC; molecular weight (*M*_{w}) = 40,000 to 50,000] was purchased from Sigma-Aldrich Co. LLC. and used as received. PDMS (SYLGARD 184 Kit) was purchased from Dow Corning.

### μWheel fabrication

To assemble and permanently link μwheel constituent particles, we placed a mixture of 20 μl of Dynabeads M-450 Epoxy (10 mg/ml) and 200 μl of 1% PDADMAC (*M*_{w} = 40,000 to 50,000) under an *x-y* planar rotating magnetic field (*B _{xy}* = 5 to 6 mT) at room temperature for 20 to 30 min. In this process, μwheels of differing size and structure were formed, each irreversibly bound due to bridging of the positively charged polyelectrolytes across negatively charged Dynabeads composing each wheel.

### Topographic surface fabrication

We fabricated PDMS replicas of diffraction gratings (Thorlabs) of known spacing (100 lines/mm; no. GR2550-10106). In this, a commercially available PDMS two-component kit was used where a mixture of elastomer and curing agent (10:1) was poured over the grating and cured under vacuum for 60 min. Once cured, PDMS replicas were peeled from the masters and placed on a cleaned glass slide for use.

### Magnetic field control

A 3D magnetic field was created with five air-cored copper solenoid coils (50 mm in inner diameter, 51 mm in length, and 400 turns with current capacity of 3.5 A) as shown in fig. S1A. Current signals controlled by MATLAB (MathWorks Inc.) and an output card (National Instruments, NI-9263) were amplified (KEPCO, BOP-40-5m) and then passed through those coils to generate the magnetic field. In-time signal monitoring was performed via a data card (National Instruments, NI-USB-6009) and Gaussmeter (VGM Gaussmeter, AlphaLab Inc.). μWheel propulsion was captured at a frame rate of 200 frames per second via charge-coupled device camera (EPIX Inc., SV643M) mounted on an inverted microscope (Olympus, IX 71).

## SUPPLEMENTARY MATERIALS

robotics.sciencemag.org/cgi/content/full/4/32/eaaw9525/DC1

Materials and Methods

Fig. S1. Experimental setup and the rolling of a μwheel under a 3D magnetic field.

Fig. S2. μWheels translating on flat surfaces.

Fig. S3. Schematics for modeling the rolling of a dimer on a textured surface and textured surface with trapezoidal bumps used in the simulations.

Fig. S4. Calculation of an ideal road for a square (4,4) μwheel translating without slip.

Movie S1. Translation of a 7-mer and a 5-mer under a 3D magnetic field on a flat surface.

Movie S2. Translation of a dimer on a flat surface.

Movie S3. Translation of a dimer under a 3D magnetic field on a topographic surface along the blaze direction.

Movie S4. The simulated translation (slip and flip) of a dimer on a topographic surface with evenly spaced trapezoidal bumps.

Movie S5. The simulated translation (sequential flip of two lobes) of a dimer on a flat substrate.

Movie S6. The simulated translation (continuous flip) of dimer on a topographic surface with trapezoidal bumps spaced by *d* = 1.39*a*.

Movie S7. The rectifier effect for the translation of a dimer under a 3D magnetic field on a topographic surface.

Movie S8. Comparison of a diamond and square μwheel translation on the flat (top) versus topographic surface against the blaze direction (bottom).

Movie S9. Translation of a diamond and square against the blaze direction.

This is an article distributed under the terms of the Science Journals Default License.

## REFERENCES AND NOTES

**Funding:**T.Y., N.W., and D.W.M.M. acknowledge financial support from the National Aeronautics and Space Administration (grant no. NNX13AQ54G). K.B.N. and D.W.M.M. also thank the National Institutes of Health under grants R21NS082933 and R01NS102465.

**Author contributions:**T.Y. and A.T. performed the experiments. T.O.T. designed the μwheel binding and wrote custom codes. T.Y. and T.O.T. built the experimental setups. K.B.N., N.W., and D.W.M.M. conceived the project. T.Y., K.B.N., N.W., and D.W.M.M. analyzed the experimental results and wrote the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to support the conclusions of this manuscript are included in the main text or Supplementary Materials.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works