Microwheels on microroads: Enhanced translation on topographic surfaces

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, Bxycos(ωMt)x̂+Bxysin(ωMt)ŷ, where B_xy is the field strength, ω_M/2π is the applied field frequency, and x̂ and ŷ are directional unit vectors (fig. S1). In the studies here, we irreversibly bound μwheel constituent particles together with the addition of positively charged polyelectrolytes [poly(diallyldimethylammonium chloride)] during assembly, yielding a mixture of μwheels of varying size and morphology (Fig. 2A, inset). We characterized these μwheels with the number of constituent particles n and the degree of rotational symmetry ξ.

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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, Bzcos(ωMt+φ)ẑ, where φ is the phase lag between the 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_z). The angular rotational frequency ω of a μwheel is related to the applied angular field frequency ω_M asω=ωM,ωM≤ωcω=ωM−ωM2−ωc2,ωM>ωc(1)where ω_c is the critical frequency related to the overall applied field strength (25). In this work, we typically ran experiments with 0.628 ≤ ω_M ≤ 314 rad/s. For 2D disc-like μwheels, the time-averaged translation velocity ν¯ is proportional to ωR_n, with radius Rn∝na, leading to ν¯/ωa∝n. Supporting this scaling, experiments show that larger μwheels rolled faster on flat substrates (Fig. 2B), and μwheels of similar size, such as diamond- and square-shaped tetramers, rolled with similar velocity. We note that the slope of the line in Fig. 2B, ν¯/ωRn, is far from unity (~0.1), indicating substantial slip during rolling, consistent with earlier studies (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 asν(t)V0=dxdt=−12(β2−β1)cos(ωt)νz(t)V0=dhdt=12(α2−α1)sin(ωt)(2)where x and h are the displacements of a dimer parallel and perpendicular to the substrate based on the center of mass. V₀ = 4ωaf/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_i and α_i are the hydrodynamic mobilities of lobe i moving parallel and perpendicular to the substrate and are functions of the lobe-wall separation h_i. In our work, because μwheels were very close to the substrate, we used expressions (eq. S3) based on asymptotic lubrication theory (31), which provides the correct limits that α_i, β_i → 0 when the contact distance a/h_i → 1.

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 0.2a = 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 (β₁ − β₂) between lobes, which depends on lobe-wall separations h₁ and h₂ (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, β₁ = β₂ = 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 β₁ = β₂ = 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_i 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 ~0.1a 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_z. As shown in Fig. 2C, predictions for the lateral velocities 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).

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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_i and β_i in eq. S3 via h₁ = h + a cos ωt − h_s(x + a sin ωt) and h₂ = h − a cos ωt − h_s(x − a sin ωt), where h₁ (h₂) 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 4.4a, the same as that between gratings in our experiments. The height and length of the bumps were then tuned to be 0.38a and 0.2a, respectively, so that the dimer rolled at the same time-averaged speed as observed in our experiments. The instantaneous velocities parallel (ν^p) and perpendicular (νzp) to the topographic substrate could then be solved numerically (Eq. 2 and eqs. S3 and S8), and their motion could be shown (movie S4). In Fig. 3D, we plot the lateral displacement of the center of mass of each dimer lobe. We also observed the appearance of the translation mode I slip characterized by a 180° phase difference and mode II flip with almost zero displacement of one lobe (the red line in Fig. 3D). By further examining movie S4, we found that the existence of bumps on the road made lobe 2 (red) contact the bump closely. As a result, its parallel and perpendicular hydrodynamic mobilities were almost zero following lubrication theory (11, 30). Therefore, lobe 2 remained stationary, whereas lobe 1 (black) rotated along lobe 2’s center of mass, leading to a flip. Because α₂, β₂~0, 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 0.05a 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_s, is related to the angular rotational frequency and μwheel symmetry, ω_s = ξω. The second frequency, the flip frequency ω_f, is related to how often a μwheel encounters a surface peak, governed by the surface feature spacing d and μwheel rolling angle θ: ωf=2πν¯psinθ/d, where ν¯p is the average μwheel lateral translation velocity on a topographic surface.

With a combination of slip and flip, ν¯p can be expressed as ν¯p=∫0τsνsdt+∫0τfνfdt∫0τdt=τsτν¯s+τfτν¯f, where ν_s and ν_f, as well as τ_s and τ_f, are the instantaneous translation velocities and periods for the slip and flip motion with τ = τ_s + τ_f. Note that τ_f and τ are related to ω_s and ω_f by τ_f = 2π/ζω = 2π/ω_s and τ = 2π/ω_f. If we define C_s and C_f as the propulsion coefficients for slip and flip, respectively, then ν¯s=CsωRn, ν¯f=CfωRn, and ν¯p=ωRn[(1−ωfωs)Cs+ωfωsCf]. Recalling that the average μwheel translation velocity on flat surfaces ν¯=CωRn (Fig. 2B), the ratio of average translation velocities becomes proportional to ω_f/ω_s asν¯pν¯=(CfC−CsC)ωfωs+CsC(3)

Following Eq. 3, we measured ν¯p, ω_s, and ω_f 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 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_s/ω_f becomesωsωf=ξωd2πν¯psinθ=(d2πaCs)⋅ξnsinθ+(CfCs−1)(4)suggesting a linear relationship between ω_s/ω_f and ξ/nsinθ. This was observed experimentally (Fig. 4B), where deviations for 7-mers could be attributed to effectively smaller symmetry degrees. In addition, the bigger slope associated with rolling along the blaze indicates a longer time required to climb inclined surfaces between peaks, a feature not included in our simple model.

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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 ∼ 6πηωR_n² ,where η is the solvent viscosity; (ii) a gravitational force F_g = mgsinγ, where m=nΔρ(43πa3) is the effective μwheel mass; and (iii) a wet friction force F_f = μ_kmgcosγ,where μ_k 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 F_f + F_d,tr < F_g, be trapped. Because 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_g, 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.

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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_f/ω_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.

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