Anisotropic Stokes Drag and Dynamic Lift on Spheres Sedimenting in

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Anisotropic Stokes Drag and Dynamic Lift on Spheres Sedimenting in a Nematic Liquid Crystal Joel B. Rovner, Daniel H. Reich, and Robert L. Leheny* Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, United States ABSTRACT: The motion of silica spheres with homeotropic anchoring sedimenting within nematic liquid crystal 4-cyano-4′-pentylbiphenyl (5CB) has been studied at low Ericksen number. The magnitude of the spheres’ velocity depends on the angle θ between the far-field nematic director and the gravitational force, indicating an anisotropic Stokes drag. When the director is oriented at an oblique angle to the gravitational force, the velocity also acquires a component normal to the force, demonstrating the existence of a lift force generated by the fluid. The magnitude and direction of the velocity as functions of θ quantitatively obey theoretically predicted forms. Ruhwandl and Terentjev,14,15 in an experiment on colloidal (magnetic) cylinders translating in a nematic fluid.21 However, because of complications arising from the cylindrical shape of the particles in that experiment, we were unable to make a direct comparison between the observed lift and the predictions of Ruhwandl and Terentjev. In this Letter, we report the results of an experiment measuring the translational motion of spherical colloids sedimenting through a nematic in which we again observe dynamic lift when the applied (gravitational) force is at an oblique angle to the nematic director. The spherical particle shape in this case facilitates comparison with the theory, and we find good quantitative agreement between the observed magnitude and direction dependence of the lift and that predicted theoretically.

I. INTRODUCTION When colloids are suspended in structured fluids such as thermotropic liquid crystals, extraordinary phenomena unseen in simple liquids can emerge. One well-studied example is the distortion of the nematic director and the formation of topological defects around colloids in a nematic fluid as a result of the boundary conditions set by the anchoring of the director at the particle surfaces. These distortions and the effective interactions between colloids that they engender have been investigated for a variety of colloidal shapes and anchoring conditions,1−8 and the ability to manipulate and order colloids through such liquid-crystal-mediated forces has been advanced as a method for colloidal self-assembly.9−11 In addition to such effects that result from the static properties of colloids in liquid crystals, the complicated hydrodynamics of liquid crystals12 can similarly lead to unusual behavior in colloidal dynamics. For instance, owing to a nematic’s anisotropic viscous properties, the mobility of a colloid becomes dependent on its direction of motion, leading to an effective drag viscosity that is different when the colloid moves parallel versus perpendicular to the far-field nematic director n̂ . This difference in mobilities parallel and perpendicular to n̂ has been investigated in several contexts experimentally and theoretically through both direct determination of the Stokes drag and through measurements of anisotropic diffusivity.13−24 A theoretical description of this behavior invokes a tensorial form of the Stokes formula for the viscous drag force F⃗D on a particle moving with velocity v,⃗ 14,15 ⃡ ⃗ FD⃗ = −Mv

II. EXPERIMENTAL PROCEDURES The experiments involved tracking the motion of silica spheres translating under the force of gravity through the nematic 4-cyano-4′pentylbiphenyl (5CB, Kingston Chemicals, purity >99.8%). Polydisperse spheres with radii ranging from R = 8 to 14 μm were treated with the silane N-octadecyldimethyl[3-(trimethoxysilyl)propyl]ammonium chloride (DMOAP, UCT Specialties) to induce homeotropic surface anchoring.25 5CB containing a dilute suspension of spheres was introduced via capillary action into liquid-crystal cells composed of parallel glass slides separated by 80 μm and coated with rubbed polyimide (PI-2555, HD MicroSystems) for uniform planar anchoring. The cells were heated to above the nematic−isotropic transition temperature, TNI = 34 °C, and cooled slowly to create large regions of uniform, defect-free nematic order. Polarization microscopy images of the spheres in nematic 5CB showed that each sphere was accompanied by a hedgehog defect near its surface, indicating strong homeotropic anchoring.26,27 The liquid-crystal cells were mounted on the stage of an inverted optical microscope (Nikon Eclipse TS100) on a level table, and the entire microscope was tilted by an angle β. Because of the density

(1)

where M⃡ is the resistance tensor. Beyond the drag anisotropy, an additional and even more striking consequence of eq 1 is the creation of a dynamic “nematic lift” force wherein an object moving through a nematic at an oblique angle to the director acquires a velocity component perpendicular to the applied force. Recently, we observed the existence of this lift force, originally predicted by © 2013 American Chemical Society

Received: December 19, 2012 Revised: February 1, 2013 Published: February 2, 2013 2104

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mismatch between silica and 5CB, the spheres experienced a gravitational force, Fg = 4πR3(ρs − ρ5CB)g/3, where ρs = 2.0 g/cm3 and ρ5CB = 1.0 g/cm3 are the densities of silica and 5CB, respectively, and g is the acceleration due to gravity. Tilting the microscope introduced a component of the force parallel to the cell substrates, F = Fg sin β, as illustrated schematically in Figure 1a, causing the spheres to

that we ascribed to contact between the spheres and substrate, leading us to restrict the experiments only to spheres with strong homeotropic anchoring. All measurements reported here were conducted with β ≈ 20.5°. Typical velocities at this tilt angle were in the range of 0.3 < v < 0.5 μm/s, which was sufficiently large that the sedimentation dominated the diffusive motion but was still small enough to be at low Ericksen number Er. Specifically, Er = ηRv/K ≈ 0.05, where η ≈ 70 mPa·s is an average viscosity30 and K ≈ 5 pN is the average Frank elastic constant of 5CB.31 This small value of Er indicates that the spheres’ motion did not appreciably alter the director field around the spheres from its static configuration, such as causing the hedgehog defect to reposition significantly with respect to the sphere surface as seen in calculations and simulations when Er ≳ 1.32,33 Indeed, an analysis of polarization images of the nematic texture when the spheres were in motion versus when they were stationary revealed no observable changes. The spheres’ motion was also at low Reynolds number (Re ≈ 10−7) and hence well in the regime of Stokes flow.

III. RESULTS AND DISCUSSION Figures 2 and 3 display the main results for the velocity of spheres sedimenting through nematic 5CB as a function of the

Figure 1. Schematic diagrams illustrating the experimental geometry of a sphere falling under the influence of gravity. (a) Side view of the liquid-crystal cell on the microscope tilted by an angle β so that suspended spheres experience a gravitational force, F⃗ = F⃗g sin β, parallel to the cell substrates. (b) Top view of a sphere showing the angle θ between the far-field director n̂ and the force as well as the angle of deflection ϕ between the velocity v⃗ and the force. translate. The moving spheres were imaged with either a 4×/0.10 objective or a long-working-distance 20×/0.40 objective along with a single-lens reflex camera (Nikon D3100) with a digital remote timer to obtain the spheres’ velocities. Only spheres that were at least several diameters from any neighboring sphere were monitored to ensure that sphere−sphere interactions did not influence the motion. To characterize the anisotropy in the drag and possible lift force, the magnitudes and directions of the spheres’ velocities were measured as a function of the angle θ between the applied gravitational force and the nematic director. We specify the direction of the velocity and hence the degree of lift by the angle ϕ between the velocity and the applied force, as shown schematically in Figure 1b. Changes in θ were achieved by rotating the cells about the microscope’s optical axis while keeping the tilt of the microscope fixed. In this way, the direction of the gravitational force with respect to the axes of the camera images remained unchanged while the far-field director n̂ was rotated. Thus, any changes in the direction of the spheres’ motion could be associated unambiguously with changes in the relative directions of F⃗ and n̂. For each value of θ, the motion of each sphere was tracked for at least 100 μm. On the basis of the resolution with which we could determine the position of the spheres in each video frame, we estimate that we could determine ϕ with a precision of ±0.3°. All velocity measurements were performed at room temperature. Measurements of the spheres’ radii were performed by heating 5CB to the isotropic phase to obtain a sharp image of the spheres’ surfaces. Because of the sedimentation of the spheres, a concern in the experiment was the possible contribution to the drag on the spheres from their proximity to the bottom substrate. As discussed below, we suspect that the proximity of the spheres and substrate indeed affected the hydrodynamic drag. However, direct contact between the spheres and the substrate was avoided by the incompatibility of the homeotropic anchoring at the surface of the spheres and the planar anchoring at the substrate, which created a repulsive force that acted to levitate the spheres several micrometers away from the substrate, as observed previously.5,28,29 Nevertheless, to test that the spheres responded to the gravitational force as expected, their velocity was determined for a range of microscope tilt angles β. The velocity varied linearly with sin β, as expected for Stokes drag, indicating that no additional forces, such as contact friction between the sphere and substrate, affected the motion. In contrast, measurements on spheres without strong homeotropic anchoring revealed anomalous mobilities

Figure 2. Magnitude of the spheres’ velocity as a function of the angle θ between the applied force and the far-field director normalized by the velocity at θ = 0. The solid circles show the experimental results. The solid line shows the predicted form from eq 6 using the experimental value of the friction ratio, Rf = 1.22, and the dashed lines show the bounds on the prediction based on the uncertainty in Rf.

Figure 3. Angle of deflection ϕ between the applied gravitational force and the resulting spheres’ velocity as a function of the angle between the applied force and the far-field director. The solid circles show the experimental results. The solid blue line shows the predicted form from eq 5 using the experimental value of the friction ratio, Rf = 1.22, and the black dashed lines show the bounds on the prediction based on the uncertainty in Rf. The dashed−dotted blue line displays the result of a best fit to the data for the friction ratio Rf = 1.26.

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angle θ between the far-field director n̂ and the gravitational force. Figure 2 shows the magnitude of the velocity normalized by the magnitude when the director is aligned with gravity (θ = 0), and Figure 3 shows the angle ϕ separating the directions of the gravitational force and the velocity. In each case, the quoted results are the mean values from measurements on six spheres with radii ranging from 8 to 14 μm, and the error bars are the standard deviations of the mean. The values showed no systematic variation with radius within this range. As Figure 2 illustrates, the spheres’ velocity decreases as the angle between the director and the applied force increases from zero, illustrating the anisotropy in the Stokes drag, and reaches a minimum when the force and director are perpendicular. Meanwhile, as seen in Figure 3, the direction of the velocity diverges from that of the applied force as the angle between the director and the force increases from zero, demonstrating the presence of a lift force. This lift force is largest when θ ≈ 45°, where ϕ reaches a maximum at ϕ ≈ 7°. As expected, the lift force approaches zero when the applied force is along the highsymmetry directions parallel and perpendicular to the director (θ = 0 and 90°). The trends in Figures 2 and 3 can be understood quantitatively as a direct consequence of the tensorial form of the Stokes formula in eq 1. Following Ruhwandl and Terentjev, we take advantage of the symmetry of the nematic to diagonalize the resistance tensor M⃡ by choosing one axis parallel to n̂. In this case, eq 1 leads to F cos θ = M v cos(θ − ϕ)

(2)

F sin θ = M⊥v sin(θ − ϕ)

(3)

interface that were also anomalously small, despite the fact that 5CB dominates the drag as a result of its much larger viscosity compared to that of water.24 They suggested that the drag was altered from the bulk behavior because of differences in the nematic ordering around the particle near the interface compared to that in the bulk, an effect that similarly could be occurring in the present experiments because of the proximity of the substrate. Determining conclusively whether modifications to the director field and flow field from the nearby substrate conspire to reduce the drag anisotropy as compared to that of a sphere far from any boundaries would require the formidable task of solving the Leslie−Ericksen equations12 under these specific geometric and anchoring conditions. A potentially interesting direction for future experiments would be to explore the changes in colloid mobility in the nematic as the substrate spacing is decreased because in isotropic liquids confinement between two walls is known to affect the hydrodynamic drag strongly.34 Nevertheless, using only the experimentally determined ratio Rf as input, one can, through eqs 2 and 3, arrive at a predicted form for the strength of the lift force, as expressed through the angle ϕ, as a function of θ, ⎛ tan θ ⎞ ⎟⎟ ϕ = θ − tan−1⎜⎜ ⎝ Rf ⎠

The solid line in Figure 3 displays this form with Rf = 1.22, and the dashed lines show the bounds on this prediction based on the uncertainty in Rf. As the figure illustrates, the experimental results and eq 5, which has no free parameters, agree closely. If instead we consider Rf to be a free parameter and fit eq 5 to the results for ϕ(θ), we find that the best fit, which is shown by the dashed−dotted line in Figure 3, gives Rf = 1.26, in close agreement with the measured value. Furthermore, eqs 2 and 3 can be manipulated to obtain a predicted form for the normalized velocity at all θ, again with only Rf as input,

where M∥ and M⊥ are the Stokes frictions for motion parallel and perpendicular to n̂, respectively, and F is the applied force (in this case, the component of gravity parallel to the substrates). The ratio of the frictions Rf is obtained from the velocities at θ = 0 and 90°: Rf ≡

M⊥ v(0) = = 1.22 ± 0.03 M v(90)

(5)

v (θ ) = v(0)

(4)

−1/2 ⎡ ⎛ ⎛ ⎞⎞⎤ ⎛ ⎞⎞ ⎛ ⎢cos2⎜⎜tan−1⎜ tan θ ⎟⎟⎟ + R f 2 sin 2⎜⎜tan−1⎜ tan θ ⎟⎟⎟⎥ ⎢⎣ ⎝ R f ⎠⎠⎥⎦ ⎝ R f ⎠⎠ ⎝ ⎝

We note that this ratio is somewhat smaller than that found in other studies of colloidal dynamics in 5CB. For instance, experiments monitoring the diffusion of colloids with homeotropic anchoring in 5CB have obtained a ratio of diffusion coefficients parallel and perpendicular to n̂ in the range of 1.4−1.6.17,18,22 We do not have a definitive explanation for the smaller ratio obtained in the present experiment compared with these previous experiments. One difference with previous experiments is that our colloids were accompanied by hedgehog defects in the director, whereas the earlier experiments involved colloids with Saturn-ring defects. Hedgehog defects are signatures of stronger anchoring,27 which suppresses the drag anisotropy.14,15,17 Another possible source of difference with earlier studies is the proximity of the spheres to the bottom substrate in the present experiment. This proximity should alter both the director-field configuration around the spheres, due to the nearby boundary condition on the director set by the substrate anchoring, and the hydrodynamic flow field past the spheres, as compared to that of spheres remote from any boundaries. We note that a similar effect from a nearby boundary appears to influence the mobility of colloids at the 5CB−water interface. Specifically, Abras et al. recently reported values of the drag anisotropy of colloids at the 5CB−water

(6)

The solid line in Figure 2 displays this form with Rf = 1.22, and the dashed lines again show the bounds on the prediction based on the uncertainty in Rf. The good agreement between the theoretical and measured variation in velocity with θ seen in Figure 2, combined with the close agreement for ϕ illustrated in Figure 3, demonstrates that the full directional dependence of the Stokes drag, including the generation of a lift force, can be understood quantitatively through the simple assumption of a tensorial form for the Stokes formula with a resistance tensor that is diagonalized along the director.

IV. CONCLUSIONS These experiments have demonstrated the presence of a nematic-generated lift force on spherical colloids and its quantitative connection to anisotropy in the Stokes drag. As shown previously, the degree of anisotropy in the drag on a colloid in a nematic is sensitive to the anchoring conditions at the colloid surface,14,15,17,21 thus implying a direct correlation 2106

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(14) Ruhwandl, R. W.; Terentjev, E. M. Friction drag on a cylinder moving in a nematic liquid crystal. Z. Naturforsch., A 1995, 50, 1023− 1030. (15) Ruhwandl, R. W.; Terentjev, E. M. Friction drag on a particle moving in a nematic liquid crystal. Phys. Rev. E 1996, 54, 5204−5210. (16) Billeter, J. L.; Pelcovits, R. A. Defect configurations and dynamical behavior in a Gay-Berne nematic emulsion. Phys. Rev. E 2000, 62, 711−717. (17) Stark, H.; Ventzki, D. Stokes drag of spherical particles in a nematic environment at low Ericksen numbers. Phys. Rev. E 2001, 64, 031711. (18) Loudet, J. C.; Hanusse, P.; Poulin, P. Stokes drag on a sphere in a nematic liquid crystal. Science 2004, 306, 1525−1525. (19) Gleeson, H. F.; Wood, T. A.; Dickinson, M. Laser manipulation in liquid crystals: an approach to microfluidics and micromachines. Philos. Trans. R. Soc. London, Ser. A 2006, 364, 2789−2805. (20) Verhoeff, A. A.; van Rijssel, J.; de Villeneuve, V. W. A.; Lekkerkerker, H. N. W. Orientation dependent Stokes drag in a colloidal liquid crystal. Soft Matter 2008, 4, 1602−1604. (21) Rovner, J. B.; Lapointe, C. P.; Reich, D. H.; Leheny, R. L. Anisotropic Stokes drag and dynamic lift on cylindrical colloids in a nematic liquid crystal. Phys. Rev. Lett. 2010, 105, 228301. (22) Škarabot, M.; Muševič, I. Direct observation of interaction of nanoparticles in a nematic liquid crystal. Soft Matter 2010, 6, 5476− 5481. (23) Mondiot, F.; Loudet, J. C.; Mondain-Monval, O.; Snabre, P.; Vilquin, A.; Würger, A. Stokes-Einstein diffusion of colloids in nematics. Phys. Rev. E 2012, 86, 010401. (24) Abras, D.; Pranami, G.; Abbott, N. L. The mobilities of microand nano-particles at interfaces of nematic liquid crystals. Soft Matter 2012, 8, 2026−2035. (25) Noel, C. M.; Giulieri, F.; Combarieu, R.; Bossis, G.; Chaze, A. M. Control of the orientation of nematic liquid crystal on iron surfaces: application to the self-alignment of iron particles in anisotropic matrices. Colloids Surf., A 2007, 295, 246−257. (26) Lubensky, T. C.; Pettey, D.; Currier, N.; Stark, H. Topological defects and interactions in nematic emulsions. Phys. Rev. E 1998, 57, 610−625. (27) Mondain-Monval, O.; Dedieu, J. C.; Gulik-Krzywicki, T.; Poulin, P. Weak surface energy in nematic dispersions: Saturn ring defects and quadrupolar interactions. Eur. Phys. J. B 1999, 12, 167−170. (28) Lapointe, C.; Cappallo, N.; Reich, D. H.; Leheny, R. L. Static and dynamic properties of magnetic nanowires in nematic fluids. J. Appl. Phys. 2005, 97, 10Q304. (29) Pishnyak, O. P.; Tang, S.; Kelly, J. R.; Shiyanovskii, S. V.; Lavrentovich, O. D. Levitation, lift, and bidirectional motion of colloidal particles in an electrically driven nematic liquid crystal. Phys. Rev. Lett. 2007, 99, 127802. (30) Chmielewski, A. G. Viscosity coefficients of some nematic liquid crystals. Mol. Cryst. Liq. Cryst. 1986, 132, 339−352. (31) Madhusudana, N. V.; Pratibha, R. Elasticity and orientational order in some cyanobiphenyls: part 4. Reanalysis of the data. Mol. Cryst. Liq. Cryst. 1982, 89, 249−257. (32) Stark, H.; Ventzki, D. Non-linear Stokes drag of spherical particles in a nematic solvent. Europhys. Lett. 2002, 57, 60−66. (33) Zhou, C.; Yue, P.; Feng, J. J. The rise of Newtonian drops in a nematic liquid crystal. J. Fluid Mech. 2007, 593, 385−404. (34) Faucheux, L. P.; Libchaber, A. J. Confined Brownian motion. Phys. Rev. E 1994, 49, 5158−5163.

between a particle’s surface properties and the magnitude of the resulting lift force that could find applications such as in separation technologies. In some cases, for instance in some lyotropic liquid crystals, this anisotropy can be remarkably large (D∥/D⊥ ≈ 4),23 thus one can expect that the effects of the lift force, although somewhat modest in the present experiment, could be made quite strong with appropriate choices of the structured fluid. Recently, Mondiot et al. presented a perturbative approach to the Leslie−Ericksen equations that enabled them to relate the anisotropy in colloid mobility in a nematic to the fluid’s Miesowicz viscosity parameters.23 This capability further opens the door to engineering colloidal dynamics in liquid crystals as part of the broader efforts to exploit the unique properties of colloid/liquid-crystal composites in developing new materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: (410) 516-6442. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank G. Drazer and R. Devendra for helpful discussions. Funding was provided by the NSF (DMR-1207117).

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REFERENCES

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