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# Elastic Torque and the Levitation of Metal Wires by a Nematic Liquid Crystal

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Science  30 Jan 2004:
Vol. 303, Issue 5658, pp. 652-655
DOI: 10.1126/science.1092608

## Abstract

Anisotropic particles suspended in a nematic liquid crystal disturb the alignment of the liquid crystal molecules and experience small forces that depend on the particles' orientation. We have measured these forces using magnetic nanowires. The torque on a wire and its orientation-dependent repulsion from a flat surface are quantitatively consistent with theoretical predictions based on the elastic properties of the liquid crystal. These forces can also be used to manipulate submicrometer-scale particles. We show that controlled spatial variations in the liquid crystal's alignment convert the torque on a wire to a translational force that levitates the wire to a specified height.

Complex fluids are soft materials that have liquid-like properties but differ from ordinary liquids because of internal structure on the nanometer or micrometer scale. The classic example of a complex fluid is a polymer solution containing long chain-like molecules whose presence alters the flow behavior. Complex fluids are ubiquitous; they occur, for example, in living organisms and in many important technologies. When particles are suspended in a complex fluid, effects from entropy, interfacial phenomena, or other sources can generate forces on the particles that are not present in ordinary liquids (1). Understanding and controlling these forces are important scientific challenges. For example, the emergent technique of microrheology uses the motion of small particles in a complex fluid to uncover characteristics of the fluid at length scales and time scales that are inaccessible to bulk probes (2). The relations between the forces that give rise to this motion and the intrinsic properties of the fluid are key to interpreting microrheology experiments. Alternatively, the forces imposed on particles in a complex fluid can be used to manipulate the particles and arrange them into unique and useful structures (36).

Nematic liquid crystals are complex fluids possessing anisotropy that introduces both forces and torques on suspended particles (717). A nematic is composed of rod-like molecules and is characterized by an alignment of the long axes of the molecules along a particular direction. At an interface between a nematic and a surface, interactions between the surface and the liquid crystal molecules can create an energetically preferred orientation for the nematic alignment. For surfaces that are not flat, such as those of particles suspended in the nematic, these preferences can distort the otherwise uniform alignment, with a corresponding cost to the nematic's elastic energy. The tendency to minimize this energy can lead to forces on the particles. For example, recent studies on spherical particles that orient the nematic perpendicular to their surface have shown that the resulting distortions mediate interactions between the particles (712). However, few probes exist that can directly access such forces. Hence, despite their significance, few geometries exist for which quantitative information about them has been extracted (8, 12, 18), and the exploitation of these forces for manipulating particles remains largely unexplored. We have developed methods to manipulate anisotropic particles in nematic liquid crystals. The particles' shape introduces orientational degrees of freedom that lead to fluid-imposed forces and torques, which we have measured quantitatively. Specifically, by balancing the fluid-imposed forces on magnetic nanowires with either applied magnetic fields or gravity, we have determined the elastic torque on a wire and have measured a repulsion between a wire and a flat surface that is mediated by the nematic. As a demonstration of how these interactions can be controlled, we showed that a spatial variation in the nematic's alignment converts the torque into a translational force that can levitate a wire to a desired height.

In our experiments, isolated, highly cylindrical nickel wires 175 nm in radius (19) were contained in the room-temperature nematic liquid crystal pentylcyanobiphenol (5CB). The wire lengths ranged from 14 to 35 μm. The 5CB with nanowires was held between parallel glass plates, which were separated by 100 μm. The plates were coated with rubbed polyimide to create a uniform orientation for the nematic alignment parallel to the plates. Typically, one specifies the local alignment direction for a nematic by a vector n, called the director. Nematics are invariant under inversion of the director, n → –n, and their elastic energy is minimized when the orientation of n is spatially uniform. Polarization microscopy observations showed that the director at our wires' surfaces oriented parallel to the wires' long axis, a configuration known as longitudinal anchoring. Therefore, in equilibrium a wire orients with its long axis parallel to the director, as shown schematically in Fig. 1A. For a wire at some other orientation, as in Fig. 1B, the director must distort to satisfy this boundary condition. Such distortions increase the elastic energy U of the liquid crystal, and they consequently introduce a restoring torque that attempts to realign the wire with the director.

As a first step toward understanding the forces acting on anisotropic particles in nematics, we measured this elastic torque through microscopy observations. The samples were positioned on the stage of a Nikon TE2000 inverted microscope within a set of magnetic coils that produced magnetic fields of precisely controlled magnitude and direction within the microscope's focal plane. Images of the wires in the 5CB were made with a long-working-distance ×40 objective. Because the nickel wires possess a magnetic dipole moment μ parallel to their long axis, a small magnetic field β (in the range from 1 to 10 G) applied at an angle ϕ to the director rotates the wire to an angle Θ with respect to the director, as in Fig. 1B (20). The wire comes to rest when the magnetic torque, μBsin(ϕ – Θ), is balanced by the torque from the gradient in elastic energy $Math$(1) Figure 1C shows results from measurements of Bsin(ϕ – Θ) versus Θ for an isolated wire 14 μm long. The observed linear relation over an extended range of Θ implies that the elastic energy grows quadratically with the angle of the wire; that is, U ∼ Θ2.

This result supports theoretical predictions based on the elastic energy of the nematic. In considering the behavior of an elongated inclusion in a nematic, Brochard and de Gennes (15) calculated this energy as a function of the inclusion's orientation. For the case of longitudinal anchoring, they exploited an analogy between the elastic energy of the nematic and the electrostatic field energy of an object at fixed potential to predict that the energy varies with particle orientation as $Math$(2) where K is the Frank elastic constant of the nematic (21) within the approximation K11 = K22 = K33 = K and CL/2ln(L/2R) is the capacitance of a wire with length L and radius R (22). The Θ2 dependence in Eq. 2 is not derived from a small angle approximation but is analogous to the V 2 voltage dependence of the stored energy in a capacitor. The slope of the line in Fig. 1C, approximately 2.9 ± 0.1 G/radian, matches the expected value, 4πCK/μ ≈ 3 G/radians, where C ≈ 2 × 10–4 cm, K ≈ 5 × 10–7 dynes (23), and μ/L ≈ 3 × 10–11 emu/μm (24), thus confirming quantitatively the validity of Eq. 2. This slope corresponds to a torque that grows with angle as 1 × 10–10 dyne cm/radian.

Although the symmetry of the nematic indicates that elastic distortions should be symmetric across Θ = π/2, the linear growth in the torque with Θ extends to Θ approaching π, as shown in Fig. 2A. However, applying a magnetic field that rotates the wire past Θ = π causes an abrupt switch in the sign of the torque, so that Θ goes from lagging ϕ to leading ϕ. After this switch, the torque again varies linearly with Θ, with its magnitude crossing zero at Θ = 2π. A schematic diagram in Fig. 2B for the elastic energy illustrates this process. As the wire rotates past Θ = π/2, the director enters a metastable configuration as it remains on the energy branch, labeled I, with a minimum at Θ = 0. Transitions to the equilibrium director configuration corresponding to branch II are suppressed. At Θ = π, the director switches from the configuration of branch I to a different metastable configuration corresponding to branch III, ultimately reaching a distortion-free configuration at Θ = 2π.

Movies tracking the change from branch I to branch III show that during the transition, the wire spins about its long axis by 180°. (This effect can be seen very clearly for wires that are slightly bent.) Schematic pictures of the nematic distortions, shown in Fig. 2C, indicate how flipping the wire in this way can facilitate the transition. The director field in the rotation plane remains unchanged with respect to the wire before and after the flip, suggesting that no intermediate configurations with prohibitively large energy must be traversed to move between branches I and III. In contrast, switching between branches I and II requires the nucleation of a disclination line (15) whose energy makes such transitions generally unfeasible. Although the diagrams in Fig. 2C provide a sense of the transition from branch I to III, an interesting question remains as to how the nematic imparts angular momentum to the wire parallel to its long axis to cause the flip. A mapping of the full three-dimensional director field would be needed to clarify this issue.

Beyond the torques characterized in Figs. 1 and 2, orientation-dependent nematic distortions can generate interactions between a wire and other objects. For example, because of the longitudinal anchoring at the wire surface and the parallel orientation of the director at the glass plates resulting from the rubbed polyimide, a wire that rotates away from Θ = 0 near a plate will introduce a twist in the director concentrated between the underside of the wire and the plate. Following the electrostatic analogy introduced by Brochard and de Gennes, this geometry maps to the problem of a wire at fixed voltage lying parallel to a plane held at a different voltage. The capacitance per unit of length in this electrostatic problem decreases with increasing separation of wire and plane, leading to a repulsive force between them (16). We have observed that sedimented wires rotated by a magnetic field rise off the bottom glass surface, demonstrating the existence of a corresponding repulsion in the liquid crystal system. Figure 3 displays results for the equilibrium height above the bottom plate reached by a wire when the elastic forces on it are balanced by gravity. In the measurement, the wire, 24 μm long, is held between glass surfaces with a separation of 80 μm. The dashed line in the figure is the relation between Θ and the wire's height z expected based on a balance of gravity and the repulsive force from a single surface (16) $Math$(3) where g is the gravitational constant and δM = πR2LNi – ρn) is the effective mass of the nickel wire with density ρNi in 5CB with density ρn. With increasing height, repulsion between the wire and the top glass plate becomes increasingly important. To approximate this effect, we model the balance of the gravitational force on the wire, δMg = 2 × 10–8 dynes, with a superposition of the repulsive forces from top and bottom surfaces as determined separately using Eq. 3. The solid line in Fig. 3 is the result of a fit to this form (25).

We note that the interaction between a wire and a plate implies not only a repulsive force when Θ > 0, but a torque that tries to restore Θ to 0. We have found that the magnitude of this torque grows quickly as a wire approaches within roughly 5 μm of a plate. At larger heights, the torque becomes independent of height to within the precision of our measurements. The results of torque on a wire presented in Figs. 1 and 2 were obtained on wires lifted by the repulsive force to large heights, so that the experimental conditions best approximated those of a wire in a bulk nematic, as considered by Brochard and de Gennes (15).

The orientation-dependent elastic energy of an anisotropic particle suspended in a nematic can be exploited to manipulate the particles via judiciously designed spatial variations in the director orientation. Figure 4A shows a schematic of a twisted nematic in which a nickel nanowire is made neutrally buoyant. The upper and lower plates of the cell are treated so that the director aligns parallel to each surface but at orientations that are offset from each other by π/2. The director within the cell thus twists through an angle of π/2 with a uniform pitch to match both boundary conditions. When a large magnetic field (B ≈ 50 G) is applied so that a wire aligns parallel with the field, the elastic energy becomes a function of the wire's height. Following Eq. 2, a wire near the bottom of the cell feels an “elastic lift” $Math$(4) where z0 is the height at which the field is parallel to the local director and ∂Θ/∂z is the magnitude of the twist. This lift can easily reverse the sedimentation of wires by gravity. Figure 4B shows results for the height of a wire 35 μm long as a function of time as it rises in a twisted cell with total thickness of 100 μm for three different values of z0. Assuming the lift is countered by gravity and a viscous drag, the equation of motion for the wire is $Math$(5) where Γ =4πηL/[ln(L/2R) + 0.84] is the Stokes drag coefficient for a cylinder (26), and η is the effective viscosity of the nematic. The solid lines in Fig. 4B are the results of fits to the solution of this equation $Math$(6) with the time constant, τ = Γ/[4πKC(∂Θ/∂z)2], constrained to have the same value, τ = 270 s, in all three fits. These results give clear qualitative support for Eqs. 4 to 6 in describing the levitation. Quantitative comparisons are made difficult by the complicated anisotropy of the nematic's viscoelasticity (21). However, selecting characteristic values for 5CB of η ∼ 1 Poise (27) and K ≈ 5 × 10–7 dynes leads to τ ≈ 140 s, which is in reasonable agreement with the measured value.

This manipulation of wires with director gradients suggests methods for self-assembly using more elaborate spatially dependent director fields. For example, as a straightforward extension of the levitation, a liquid crystal that twists through many periods (such as a cholesteric liquid crystal) could serve as host to the creation of a periodic array of planes of oriented nanowires. Another possible application of these forces involves wire sorting through the repulsion from a flat surface. As Eq. 3 indicates, gravity and the repulsive force balance at a height that varies to first order with R–2 and is independent of L when Θ is fixed; for example, by a large external field. Hence this repulsion can be used to spatially separate wires of different radii.

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