Controlled assembly of microscale objects can be achieved by exploiting interactions that dominate at that length scale. Capillary interactions are an excellent candidate for this purpose; microparticles trapped at fluid interfaces disturb the interface shape, migrate, and assemble to minimize the interfacial area. These interactions are independent of microparticle material properties and so can be used to assemble objects of arbitrary materials. By using a magnetic robot as a mobile distortion source, additional control over assembly can be achieved. For example, millimeter-scale magnetic robots that are heavy enough to distort the interface have been used to generate long range capillary attractions and collect passive particles that are hundreds of micrometers in diameter. However, for smaller robots and particles, gravity is less important, and capillary interactions rely on interface distortions from undulated contact lines. We use a magnetic microrobot to manipulate passive microparticles at the water/hexadecane interface via an interplay of hydrodynamic and capillary interactions. Furthermore, we demonstrate preferred docking at corners of a square microrobot without the need for high resolution position control. We modulate the strength of docking interactions, allowing structure assembly and release. Finally, we design undulated docking stations with multiple stable sites for cargo delivery. The ability to dynamically manipulate microparticles and their structures at fluid interfaces creates new possibilities for manufacturing of complex microstructures.
The need to assemble microscale elements into structures with high resolution is of broad interest in materials science and micromanufacturing. This interest drives research at the intersection of microrobotics and colloidal science to manipulate and assemble particle building blocks and to control their structures via external electric1–3 and magnetic fields.4,5 However, such field-driven strategies are limited to colloids with specific material properties and are ineffective for passive colloids that do not respond to the applied fields. In order to manipulate passive colloids, robotic strategies have been applied, including contact6,7 and noncontact manipulation.8,9 However, these schemes typically rely on complex geometry and high resolution control. Capillary interactions at fluid interfaces hold tremendous promise for robotic directed assembly. These interactions are a powerful tool to assemble passive colloids10 and to integrate responsive and passive particles11–13 into robust structures that can be driven by external fields. In the robotics arena, the manipulation and assembly of passive particles have been achieved by exploiting capillary interactions between submillimeter particles and a millimeter scale robot.14 While this hybrid approach of using a magnetically driven robot to interact via capillarity with passive particles should apply to smaller, microscale robots and to colloids of negligible weight, the origin, range, and strength of capillary interactions differ significantly from the heavy, larger counterparts. Far from contact, capillary interactions are dominated by a ubiquitous interaction that occurs for all microparticles at fluid interfaces. However, near contact, interactions depend strongly on microrobot geometry, which can be designed to achieve particular tasks. These include site specific attachment with tunable interaction strengths, which can be harnessed to transfer microparticle cargo to docking sites. This programmable assembly via geometry opens new opportunities for microrobotics and programmable structures.
In this Letter, we harness capillary interactions between magnetic microrobots and passive polystyrene particles trapped at fluid interfaces. To exploit recent findings of capillary forces that attract microparticles to sites with a high interface curvature,15,16 we demonstrate directed assembly and micromanipulation of these particles. In particular, we design microrobots with different shapes and use visual feedback to control their position using an electromagnetic control system as shown in Figs. 1(a) and 1(b). We start with a circular microrobot and demonstrate attraction via a ubiquitous mode of capillary interaction between a moving micro-object and a passive particle trapped at the fluid interface. The interactions are complex as both the microrobot's motion and capillary interactions contribute significantly to the dynamics. Thereafter, directed assembly of assembly of passive particles toward high curvature sites at the interface generated by the microrobot is achieved using the sharp corners and associated strong capillary interactions on a square microrobot. We then modulate the curvature of the robot's features by making a flower-shaped robot in order to reduce the strength of the capillary interactions, allowing weaker binding of the particle. Finally, we design a docking station with out-of-plane undulations and strong capillary interactions and demonstrate transfer of passive cargo from a microrobot to this station.
When a particle attaches to an interface between immiscible fluids, it deforms the interface shape. This deformation is determined by the particle's weight and by the shape of the contact line or the contour where the interface and particle meet. The shape of this contour is typically highly undulated owing to contact line pinning at sharp edges or on surface features like roughness sites or chemical heterogeneities.17 The capillary energy, Ecap, associated with the interface deformation is given by , where γ is the surface tension and is the change in the interface area because of the interface deformation.
The importance of the particle weight relative to surface tension forces is determined by the magnitude of the nondimensional Bond number, , where g is the gravitational acceleration, a is the characteristic length of the particle, and is the density difference between the two phases. With , surface tension forces dominate over gravitational forces and motions of microparticles are driven by capillary interactions to minimize the interfacial area.
Passive polystyrene particles used in this study have small . Typically, these particles have an undulated, pinned contact line where the fluid interface intersects the particle.18 This results in a distorted interface, with changes in the interface height far from the passive particle having a quadrupolar symmetry with amplitude , as is required for particles with weak body forces and torques.19 The deviatoric curvature of the interface , defined as the difference between principal curvatures of the host interface, is an important quantity in determining forces experienced by microparticles at the interface, as the changes in the interface area owing to the interactions of the particle and the microrobot can be recast in terms of this quantity.
The capillary energy of interaction for a microparticle trapped in an interface with deviatoric curvature is given by . The associated capillary force on the microparticle is oriented along the deviatoric curvature gradient of the interface20 with the expression , where and its gradients are evaluated at the center of the passive particle. The pose of the microrobot and its shape can be used to locally tune the interface shape and associated curvature fields. This is shown schematically in Fig. 1(c) for the circular disk, which also excites a quadrupolar distortion at the interface. This treatment suffices for far-field interactions. Near contact, the details of the interface shape determine capillary interactions.
At static curved fluid interfaces, microparticles migrate under the action of this force toward the high curvature gradient sites.10 Here, microrobots driven by external magnetic fields are used to deform fluid interfaces, generating capillary interactions that guide microparticle assembly. The magnetic microrobots and passive docking sites used in this study are shown in Fig. 1(d) and fabricated through photolithography as described in Fig. S1 in the supplementary material and have . At the interface, the undulated contact line pins along the upper edge of the microrobot due to the ridgelike roughness created by the sputtering process (Fig. S2). In this study, interface deformations from the microrobot's undulated contact line dominate over those from its weight, which is therefore neglected in the analysis of microrobot-particle interactions.
The magnetic manipulation platform consists of four electromagnetic coils (APW Company) mounted on an aluminum supporting structure arranged around the workspace as shown in Fig. 1(b); visual feedback is provided by a CCD camera (Point Grey Grasshopper3 Monochrome) mounted on a Zeiss inverted microscope (ZEISS Axio Vert.A1). A DAQ (USB-3104, Measurement Computing) is used to control the current via a custom power electronics circuit. Each coil is connected with a DC power supply (XG 850W, Sorensen) to generate the desired magnetic field. Only two coils, one of each pair, are energized at a time depending on the trajectory. The direction of motion can be tuned by changing the ratio of currents on powered coils.
The water-oil interface studied in this Letter is formed by gently pipetting hexadecane (Sigma-Aldrich) to fully cover the interface of de-ionized water in a Petri dish. The oil-water interface is selected rather than a free surface to minimize any disturbances from air currents in the laboratory. Magnetic microrobots are introduced to the interface with a syringe needle or pipette tip. The movements of the microrobots are controlled by the system described above or a permanent magnets (K&J Magnetics). The passive polystyrene microspheres with diameters and (Spherotech) are cleaned with water and isopropyl alcohol (IPA) and stored in hexadecane. They are introduced to the interface by pipetting the suspension gently into the oil phase and allowing them to sediment until they encounter the oil-water interface, where they adsorb and become trapped.
Actuation of the microrobot enables the interface deformation field, or, alternatively, to be dynamically positioned, allowing passive particles in the vicinity of the microrobot to be assembled via capillary interaction. When the interface distortions from the microrobot and from the microsphere overlap, the distance between the two will decrease to minimize the deformation area, hence minimizing the total capillary energy through this attractive interaction. Figure 2(a) shows a time-stamped image of polystyrene particles being assembled by a circular magnetic microrobot. The microrobot is moving toward the particle at the interface and finally attracts the particle to its edge; their trajectories are indicated in Fig. 2(b). The inset in Fig. 2(b) shows the interferometric image of the interface height around a circular robot, which reveals a weak downward monopole owing to the robot's weight and a quadrupolar deformation in the far field, as shown in Fig. 2(c), and irregular distortions owing to the undulated contact line in the near field as shown in Fig. 2(d). The average magnitudes of the quadrupoles are and as determined from multiple interferometric measurements (for details, see the supplementary material). As we are neglecting the weak monopolar deformation from its weight, the deviatoric curvature at the center of mass of the particle is expressed as18 , where L is the center-to-center distance between the passive particle and the microrobot. Therefore, the dominating capillary force is expressed as
where γ is the average surface tension of the two phases.
Hydrodynamic interactions also play an important role in this process as the two objects approach on the fluid interfaces. Dani et al.21 studied pair interactions for identical microspheres trapped at the oil/water interface and they demonstrated both theoretically and experimentally a velocity reduction due to the hydrodynamic interaction between the pair. We observed similar behaviors between microrobots and passive particles in the experiment indicated by the red dots in Fig. 2(e).
We approximate the flow created by the microrobot moving at the interface under the external field as a Stokeslet plus a potential dipole at its center.22 Passive microparticles in the vicinity of the microrobot can sense both the flow field and the capillary field. The corresponding approach velocity between a moving microrobot and the passive microparticle when moving along line of centers, from Faxn's law, is expressed as
where U and are the velocities of the microrobot and passive particle, μ is the average viscosity of the two phases, and is given by (1).
To examine the validity of (2), we substitute the relevant parameters in this expression, including . The velocity U and separation distance L are extracted from the videos. We plot the approach velocity as a function of separation distance in Fig. 2(e) where the theory agrees qualitatively with experimental results. As the robot is driven toward its target from afar, the approach velocity decreases due to long-ranged hydrodynamic interaction. However, once within a critical distance, , capillary attraction rapidly pulls the pair together. The discrepancy between theory and experiment may come from several sources. First, there are higher order modes for capillary interactions between the objects near contact. Second, the hydrodynamic model adopted is strictly valid for far field interactions between spheres, while our robot is disk-shaped. A third possible source of error is that we have not accounted for the effect of interface deformation on hydrodynamic drag. However, the theory captures the main features of the interaction and agrees qualitatively with experiment.
While the circular microrobot demonstrates the concept of capillary directed assembly, we exploit a square microrobot with similar Bo to show the importance of high curvature sites generated in the interface near corners. Because of its weight, the robot rests slightly below the plane of the interface (). This causes strong distortions and associated pronounced deviatoric curvature gradients near the corners. Instead of randomly assembling anywhere as on the circular microrobot, passive particles near the square microrobot now migrate to its corner, as shown in Fig. 3(a). The deviatoric curvature field and curvature gradient around the square microrobot with a pinned contact line and weak vertical displacement are simulated via the finite element method, performed using COMSOL Multiphysics, as shown in Fig. 3(b). The curvature gradient, as shown by white arrows in Fig. 3(b), and the magnitude of the capillary force along the diagonal near the corner, as shown in Fig. 3(c), indicate the strong local capillary interactions near the corner. These interactions result in a strong capillary bond that is hard to break. Tracking the approach velocity of the pair [Fig. 3(d)] reveals the behavior similar to the circular microrobot far from contact. However, in the near field, the rapid approach and strong local attraction to corners are apparent. For smooth heavy robots, corner attraction is predicted to occur over distances comparable to the half-side of the robot. Roughness generates competing interactions; corner assembly can be preserved by increasing the microrobot weight (see the supplementary material).
Strong capillary interaction is desirable for assembly but not for release. We can prescribe intermediate strength interactions by designing a planar flower-shaped microrobot of with a periodic wavy structure of wavelength and amplitude . We also make this robot lighter, and so the weak weight results in a weaker interface depression. This results in a deviatoric curvature field near each protrusion with gradients that are gentler than those at sharp corners, and the resulting capillary force is one order of magnitude weaker as shown in Fig. S3. The “petals” on the flowers define docking sites, with colloids closest to the microrobot attaching to either one or two protrusions. The energy from these docking sites is quite local, and so colloids that approach the microrobot chain with each other due to particle-particle interaction. This chain can be transported together with the microrobot or broken under external torque. As the chain breaks under external torque, assemblies of passive particles (i.e., the dimer labeled as 1 and 2 in Fig. 4) will be released from the microrobot. Through such a process, various structures of particles or passive cargo can be assembled, transported, and released at the fluid interface.
In addition to assembly and release, delivery of passive particles to desirable locations is another important micromanipulation process toward building microstructures. To do this, we introduce a passive docking station with out-of-plane undulations to the fluid interface as shown schematically in Fig. 5(a). This docking station is fabricated via lithography from the negative resist SU-8 developed under UV light through an undulated mask that defines the wavy form and length of the docking station. The thickness of the resist film defines the width of the docking station (see Fig. S4). Contact line pinning on the top wavy edge gives an abrupt height change close to the docking station that decays over distances comparable to the wavelength of the feature. Because of this local interface distortion, the deviatoric curvature field, shown in Fig. 5(b) using the open-source software Surface Evolver,23 is orders of magnitude stronger than that around the planar microrobots, as is the associated capillary force. As a result, passive particles can be transferred as cargo from a circular microrobot to this docking station. After delivering the passive particle to one of the out-of-plane protrusions, the capillary bond between the circular microrobot and passive particle can be broken under external torque as shown in Fig. 5(c). The microrobot is then driven away by the external field to complete the transfer process.
In this Letter, directed assembly and micromanipulation of passive particles at the fluid interface are achieved via the interplay of hydrodynamic and capillary interactions using magnetic microrobots. The microrobot acts as a mobile fluid interface deformation source and generates a flow field as it is maneuvered through the workspace to interact with passive microparticles. Hydrodynamic interaction is more important than capillary attraction in the far field and pushes passive particles away, whereas capillary attraction dominates in the near field and assembles passive particles at high curvature gradient sites. By strategically designing the geometry of the microrobot, preferred assembly sites, such as corners, for passive particles can be created. Furthermore, the interaction strength can be tuned by modulating the curvature of the microrobots or out-of-plane shape of docking stations, allowing passive particles cargo to be delivered to desired sites. The strategy we have shown in this Letter will have potential applications in single cell manipulation and micromanufacturing.
See the supplementary material for supplementary information and videos.
We gratefully acknowledge the support of NSF Grant No. DMR-1607878.