Noncontact orientation of objects in three-dimensional space using magnetic levitation

The Orientations of Nonspherical Objects Levitating in a MagLev Device.

As a preliminary study, we levitated a Nylon screw (9 mm in length) in the MagLev device. Finite element simulations based on the parameters (dimensions, strength of the magnetic field, magnetic susceptibility of the solution) of this device show that, to a good approximation, the gradient of the magnetic field is linear, with a constant slope between the surface of the top magnet to the surface of the bottom magnet, and the magnetic field is zero at the center of the device (27). We rested the MagLev device on a flat laboratory bench, and hence the x′-, y′-, z′ axes coincided with the x-, y-, z axes of the laboratory frame of reference. The concentration of MnCl₂ was 1.50 M, yielding a solution of density ρ_s ∼ 1.15 g/cm³ (measured with a pycnometer). The density of the solution was similar to that of the screw, ρ_o = 1.15 g/cm³ (manufacturer’s data). We chose the direction vector p→ to point along the long axis of the screw. The screw levitated at the center of the device and adopted an orientation with p→ parallel to the surface of the magnets (Fig. 2A). We next modified the shape of the screw by cutting the shaft to a length of 2.5 mm. The shortened screw, while still levitating at the center of the device, adopted an orientation with p→ pointing perpendicular to the surface of the magnets (Fig. 2B). Because we only changed the geometry of the screw, albeit substantially, we inferred that geometry played a role in determining the orientation of objects in this MagLev device.

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Fig. 2.

Equilibrium orientations of nonspherical objects in MagLev. (A and B) A Nylon screw orients differently when the length of the shaft was reduced from 9.5 to 2.5 mm. (C–F) Plots of the orientation of the objects (angle α) versus their aspect ratios A_R = T/l (schematic). Each data point is an average of seven replicate objects. The error bars represent the SD. The x-error bars are smaller than the data point. The dashed vertical line is the value of the critical aspect ratio A_R^*, predicted by theory. (Insets) Representative images of objects levitating in the MagLev device in each plot. The black arrow indicates the direction of p→. The cross in the background is for reference, and the horizontal line in the cross measured 30 mm.

We designed a series of experiments with model objects to explore the role of geometry for orientation in the MagLev device. We machined objects out of organic polymers with circular, annular, square, and triangular 2D cross-sections, each with a constant “characteristic” length, l. Depending on the object, l was the diameter of the circle, the outer diameter of the annulus, the side width of the square, or the length of the sides of the equilateral triangle (Fig. 2 C–F). We varied the thickness T of the objects in the third dimension to produce cylinders, annular cylinders, square prisms, and triangular prisms. Levitating these relatively simple, symmetric 3D objects with MagLev allowed us to obtain a theoretical understanding of the governing physics. To classify the shapes, we defined a nondimensional aspect ratio parameter A_R to be the ratio between the thickness of the object and its characteristic length (i.e., A_R = T/l; Fig. 2). We set p→ to be aligned along the thickness axis of the object.

We started by levitating objects of small A_R, and progressively levitated objects with larger A_R. We captured images of the objects along the y′–z′ plane and measured the angle α that p→ subtended with respect to the z′ axis of the device. We defined α to be zero when p→ was parallel to the z′ axis. The value of α was clustered either around 0° or 90°—p→ was parallel or perpendicular to the surface of the magnets—with at most a 10° (typically <5°) variation observed between replicate objects. Plots of α versus A_R for each of the shapes revealed that α jumped abruptly from 0° to 90° at what appeared to be a critical value of A_R, which we denote as A_R^*. The value of A_R^* appeared to be different for the different shapes. For the cylinder we observed 0.84 ≤ A_R^* ≤ 0.88, for the annular cylinder we observed 1.04 ≤ A_R^* ≤1.12, for the square prism we observed 0.90 ≤ A_R^* ≤1.10, and for the triangular prism we observed 0.65 ≤ A_R^* ≤ 0.73. The value of A_R^* and the orientation of the objects in the y′–z′ plane did not depend on the shape of the magnets, the levitation height of the objects (Fig. S1), or the distance d between the two magnets (varied from 45 to 65 mm) (Fig. S2), suggesting that the observed effects are purely a function of the shape of the objects.

The orientation of the object in the x′–y′ plane, as expected, did depend on the shape of the magnets. For square magnets, the objects centered in the magnetic field and aligned along the diagonals (Fig. S3). For disk-shaped magnets, the final orientation of the object in the x′–y′ plane was dependent on the history of sample. Shaking the container, or removing the container and replacing it in the MagLev device, caused the orientation of the object in this plane to change (data not shown). The orientation of the object in the y′–z′ plane, however, was still fixed and determined only by A_R.

Modeling the Height and Orientation of Nonspherical Objects in MagLev.

The dependence of the height and orientation of objects on shape was one for which we wished to have an analytical treatment. We consider the potential energy of an arbitrary object located in a region with superimposed magnetic and gravitational fields (a MagLev system). Eq. 1 gives the energy density (energy per unit volume) of the MagLev system.u=umag+ugrav=−12μoΔχ(r→)B→2−Δρ(r→)g→⋅h→.[1]In this equation, u_mag is the magnetic contribution and u_grav is the gravitational contribution to the total potential energy density, Δχ(r→)=χo(r→)−χs is the magnetic susceptibility of the object relative to a homogeneous medium, Δρ(r→)=ρo(r→)−ρs is the density of the object relative to a homogeneous medium, and h→=(0,0,h) is the height of the object. In general, the object can be heterogeneous in both density and magnetic susceptibility such that these functions depend on the position coordinate r→.

At static equilibrium, the potential energy U=∫Vu dV, where V is the volume of the object, has to be minimized. Finding the equilibrium configuration involves minimizing simultaneously the energy associated with the levitation height and orientation of the object. Parameterizing the object, and numerically solving the resulting set of multivariable equations (minimization has to be performed over the spatial coordinates and the distributions of density and susceptibility), provides the levitation height and equilibrium orientation for arbitrary objects in arbitrary magnetic fields.

Simplifications of Eq. 1 allow analytical closed-form solutions that provide physical insight. The equilibrium levitation height h0 will occur where dU/dh=0. For a linearly varying magnetic field, the levitation height of the centroid of the object in the MagLev depends only on the average susceptibility χ¯o=1/V∫Vχo(r→)dV and the average density ρ¯o=1/V∫Vρo(r→)dV of the object, regardless of the shape and the distribution of the heterogeneities within the object.

The equilibrium orientation(s) at angle α will occur at the local minima of U, where dU/dα=0 and d2U/dα2>0. We choose a body-fixed coordinate system p→(x″,y″,z″) aligned with the principal axes of the object, and fix the x′′ axis to remain parallel to the x′ axis of the MagLev reference frame (we include the full derivation and a procedure to find this preferred reference frame in SI Appendix). We proceed to analyze the rotation of the object around the x′ axis with the same convention as in the experiments and parameterize orientation as the angle α that p→ subtends with respect to the z′ axis. Eq. 2 gives this energy for an object that is homogeneous in susceptibility and density.U(α)=βΔχVλz2(1−R)sin2⁡α.[2]In this equation, β=2B02/μ0d2, λz2 is the principal second moment of area along the z″ axis, and R is the ratio of the principal second moments of area along the y″- and z″ axes.

Fig. 3A shows a plot of Eq. 2 at representative values of R. For values of R < 1, U(α)∝sin2⁡α, and the potential minima occur at α = 0° and 180°. For values of R > 1, U(α)∝−sin2⁡α∝cos2⁡α, and the potential minima occur at α = 90° and 270°. All other values of α result in energies that lie within these extrema and are not stable. Thus, objects with uniform density will only orient with α = 0° or α = 90°. This result rationalizes the experimental observations in Fig. 2. When R approaches 1, the linear theory predicts a flat energy landscape. Adapting the analysis that led to Eq. 2 for nonlinear magnetic fields by retaining higher order terms in the expression for B→ provides solutions for the orientation of these objects (SI Appendix).

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Fig. 3.

Energy and orientation of objects in MagLev. (A) Plot of the potential energy as a function of α (the angle that p→ makes with respect to the z′ axis) (Eq. 2). R is the ratio of the second moment of area of the object. For R < 1, continuous black line, the two (degenerate) minima in potential energy occur at α = 90° and 270^o. For R > 1, dashed and dotted black line, the two (degenerate) minima in potential energy occur at α = 0° and 180°. When R approaches 1, the linear theory predicts a flat energy landscape. The schematic at the top of the plot shows the orientation of the object with respect to z′. (B) Plot of α versus R for the experimental objects in Fig. 2. All of the data collapse onto a master curve with the transition in orientation at R = 1.

We calculate the value of A_R^* at which it is energetically favorable for the objects to switch orientation from α = 0° or α = 90°, and plot the results in Fig. 2 C–F as a dashed line (see SI Appendix for full calculation). Our calculations match our experimental results excellently. Furthermore, plotting α versus R results in the collapse of our data for all of the shapes onto a master curve where the transition between orientations occurs at R = 1 (Fig. 3B).

Manipulating the Orientation of Objects by Rotating the MagLev Device.

We used a Nylon screw (8.5 mm in length) to illustrate the process involved in manipulating—without contact with a solid surface—the orientation of an object suspended inside an entirely closed container of paramagnetic liquid. We controlled the orientation of the screw by rotating the MagLev device together with the container of paramagnetic liquid. The concentration of MnCl₂ was again 1.50 M, and thus the screw levitated at the center of the device. Fig. 4 shows the orientation of the screw in the y–z plane when the device was rotated 360° counterclockwise about the x axis (the z′ axis rotated relative to the z axis). For reference, we used a 30 × 22-mm cross as a background, keeping the cross fixed with respect to the laboratory frame of reference. The screw, suspended in solution, rotated in the laboratory frame of reference and tracked the angle of rotation of the magnets (Fig. 4B). Rotations about the other two axes resulted in similar outcomes (data not shown). We conclude that rotating (and translating) about the x-, y-, and z axes allows arbitrary orienting and positioning of objects in 3D with respect to the laboratory frame of reference. Fig. S4 demonstrates that the orientation of the objects can also be manipulated by moving only the magnets, while keeping the container stationary—a procedure that might be useful in certain situations: for example, when access to the oriented objects from the top of the container is desired.

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Fig. 4.

Controlling the orientation of a levitating object in laboratory space by rotating the MagLev device. (A) Schematic of the experimental setup. θ is the angle that the z′ axis makes relative to the z axis. (B) Experimental images taken along the y–z plane of a Nylon screw (8.5 mm in length) in the MagLev. We kept the cross in the background fixed relative to the laboratory. The screw tracks the position of the magnets, rotating a full 360° with respect to the laboratory frame of reference. The white double-headed arrows indicate the orientation of the axis of the magnetic field gradient. (C) Similar rotations caused the screw to translate and contact the wall of the container when the density of the screw was greater than the density of the solution. Further rotations caused the screw to flip orientation. For scale, the horizontal line in the cross is 30 mm.

Choosing a solution that has the same density as the object is important for contactless manipulation of objects by rotating the device. Fig. 4C shows the results when the density of the solution is lower than the density of the object (e.g., the same screw used in Fig. 4B). Rotating the device to 45° caused the screw to translate toward the walls of the container, and eventually to contact the wall. Further rotation of the device to 90° caused the screw, which was touching the container, to flip, and prevented its further manipulation. Although not shown here, it is rational to speculate that normal forces on an object, due to contact with a hard wall, might damage or deform soft, sticky, or fragile objects.

Why does using a solution of lower density cause the screw to contact the wall of the container? When the density of the solution is equal to the density of the object, the gravitational force acting on the object is zero, and the center of volume of the object levitates at the center of the device (SI Appendix). When the density of the solution is less than that of the object, force balance requires that the object equilibrate at a smaller levitation height (the example shown in Fig. 4C), due to the nonzero gravitational force. Reversing the direction of the force vectors describes the situation for objects with a density higher than the solution, and the object equilibrates at a larger levitation height. Rotating the direction of the magnetic force (always acting along the z′ axis) with respect to the direction of the gravitational force (always acting along the z axis) produces a component of the net force that acts perpendicular to the z axis. The perpendicular component of the force, which increases in magnitude with increasing angles of rotation and reaches a maximum at θ = 90°, causes the object to translate toward the wall to maintain static equilibrium.

It is clear that when the gravitational force is zero, the object remains fixed at the center of the device. This configuration allows arbitrary rotations of the device without the object contacting the walls of the container. A practical means of matching the density of the liquid to an object of unknown density is to start with a concentrated solution of paramagnetic salt and progressively dilute the solution until the object levitates at the center of the device.

Manipulating the Orientation of Objects with External Magnets.

Another method of controlling the orientation of objects, without contact with a solid surface, is by using external magnets to modify the magnetic field generated by the fixed coaxial magnets in the MagLev device. It is energetically favorable for the paramagnetic liquid in the container to respond to changes in the magnetic field by redistributing volume to occupy regions of locally high field strength. This movement of liquid will indirectly cause the displacement of levitating diamagnetic objects in the MagLev device.

We demonstrate this method by manipulating the orientation of a Nylon screw (2 cm in length) in the x′–y′ plane of a MagLev device equipped with disk-shaped magnets (Fig. 5). We used circular magnets because this geometry resulted in a circularly symmetric field in the x′–y′ plane. Thus, the screw does not have any preferred orientation in this plane. Magnets with shapes of lower symmetry, for example square and rectangular magnets, favor the orientation of the object along specific planes of symmetry, such as along the diagonals (Fig. S4) (27, 47). Fig. 5B shows an image of the screw viewed along the x′–y′ plane of the device. A cross pattern affixed to the bottom magnet is provided as a guide to the eye.

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Fig. 5.

Manipulating the orientation of an object in the x′–y′ plane of a MagLev device with an external magnet. (A) Schematic of the experimental setup. Due to the cylindrical symmetry of the magnetic field, the long axis of the screw does not have a preferred orientation in the x′–y′ plane. The image in B shows one of the orientations the screw adopts when placed in the device. (C) We moved an external magnet close to the screw to align the screw head along the red lines of the pattern. The brown square indicates the approximate position of the external magnet. Scale bar, 5 mm. Also see Fig. S5 for images taken along the z′–y′ plane of a screw being manipulated with external magnets.

We used a small cubic magnet (0.64 × 0.64 × 0.64 cm, magnetic field strength at the surface ∼0.4 T) to generate, externally, a localized region of high magnetic field strength to manipulate the orientation of the head of the screw. We brought the small magnet to a distance of about 2 cm from the head of the levitating screw (the walls of the container prevented a closer approach of the magnet). The head moved away from the small magnet and came to rest after rotating ∼45° away from the surface of the external magnet. By moving the magnet around the exterior of the container, we oriented the head of the screw along the four principal axes of the cross (Fig. 5C). At each position, the screw remained at its new orientation even after the small magnet was moved away from the device. Furthermore, combinations of several external magnets allowed finer control of the orientation of levitating objects (Fig. S5).

Orienting Soft and Sticky Objects in MagLev.

MagLev shows particular promise for manipulating and controlling the orientation of soft materials. As proof or principle, we used MagLev to manipulate objects fabricated out of hydrogels, elastomers, and colloids—three classes of materials with diverse technological applications.

Hydrogels have been used to fabricate actuators (15), soft robots (5, 9), and artificial tissues (6⇓–8). For example, directed assembly of hydrogel strips and blocks laden with different cell types is a promising method for engineering artificial tissues (6⇓–8). Hydrogels used in biomimetic applications are soft (6⇓–8), and tend to stick to surfaces due to the capillary action of the liquid film on the hydrogel surfaces. In Fig. 6A, we used MagLev in combination with an external magnet to orient a slab of poly(N-isopropylacrylamide) hydrogel; ordinarily, this material would deform or break when handled by a hard gripper (Young’s modulus, E, of the hydrogel ∼ 1,000 Pa). We inserted the object into the MagLev device by gently releasing the hydrogel from the mold into the paramagnetic liquid. In the MagLev container, the object was able to assume its natural shape, and the competing magnetic and gravitational forces acting on it determined its orientation. An external magnet allowed the control of the orientation of the sharp end of the hydrogel relative to the four principal axes of the cross.

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Fig. 6.

Manipulation of soft, sticky, and easily deformable objects. (A) Photographs along the x′–y′ plane showing control of the orientation of a poly(N-isopropylacrylamide) hydrogel using external magnets. We used the same experimental setup as in Fig. 4. The sharp end of the hydrogel was made to point in the four principal axes of the cross pattern. The brown square indicates the approximate position of the external magnet. The hydrogel levitated stably in each position after we withdrew the external magnet. (B) Images along the z′–y′ plane of a soft-gripper component made out of EcoFlex 0030. The orientation of the gripping face was changed with respect to the laboratory frame of reference by rotating the magnets. The black double-headed arrows indicate the orientation of the axis of the magnetic field gradient, and θ is the angle of rotation of the magnets with respect to the z axis. (C) Schematic and picture of an armored droplet. The droplet adopts a stable peanut shape due to the jamming of the polystyrene particles on its interface. (D) We controlled the orientation of the armored droplet with respect to the cross pattern by using an external magnet. The manipulation of the position of the object with MagLev did not deform this soft solid. Scale bars: (A) 5 mm, (B) the horizontal line of the cross is 30 mm, (C) 2 mm, and (D) 5 mm.

In Fig. 6B, we controlled the orientation of a pneumatically actuated soft gripper made out of the silicone elastomer EcoFlex. The gripper, a modular part of a larger soft robot assembly (48), deforms easily when subject to moderate forces and tends to adhere to surfaces due to the low surface energy of cured EcoFlex. Rotating the magnets allowed control over the orientation of the gripping face with respect to the laboratory frame of reference. We envision that MagLev, with further development, could extend modular strategies for the assembly (48) of robots to materials that are softer than elastomers [e.g., hydrogels (9)].

Self-assembled granular and colloidal materials, held together by relatively weak physical bonds, are a class of soft or fragile condensed matter that shows promise as stimuli-responsive materials and containers (11, 13, 14, 49⇓–51). These materials, despite being composed predominantly of fluid, can adopt nonspherical shapes (i.e., they can demonstrate solid-like properties) due to the jamming of the colloidal particles on their surfaces (12, 51, 52). The capillary bonds that confer their solid-like properties are weak, and hence these solids have yield strengths on the order of γ/R ∼ tens of Pa (53). The surface tension of the liquid is γ (N/m), and the radius of the object is R. Although such low yield strengths are sufficient to maintain the shape of the objects against gravity and thermal agitation, once fabricated, these objects cannot be manipulated with hard grippers without causing irreversible plastic deformation due to localized shear melting of the jammed colloidal monolayer (53, 54). Fig. 6C shows control over the orientation of a nonspherical perfluorodecalin droplet covered with a jammed monolayer of 10-µm-diameter polystyrene particles. We obtained the stable nonspherical peanut-like object by forcing two spherical particle-covered droplets to fuse by squeezing them mechanically. MagLev thus allows active manipulation of self-assembled diamagnetic granular structures without requiring the use of magnetic or paramagnetic particles (11). All of the manipulations demonstrated here can also be performed on composite objects with metallic components (Fig. S6).