Direct Measurement of Colloidal Particle Rotation and Field

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Direct Measurement of Colloidal Particle Rotation and Field Dependence in Alternating Current Electrohydrodynamic Flows Jesu´s Santana-Solano,† David T. Wu,‡ and David W. M. Marr*,† Chemistry and Chemical Engineering Departments, Colorado School of Mines, Golden, Colorado 80401 ReceiVed January 27, 2006. In Final Form: April 11, 2006 We have measured the influence of both applied alternating current (AC) field strength and frequency on the electrohydrodynamic (EH) flows present in colloidal systems near an electrode surface. The effect of the flows is visualized by the rotation of the colloids, fluorescently labeled by a novel technique involving EH-driven aggregation of much smaller tracer colloids to the surface of the larger colloids. Our results show an E2 dependence of these flows, consistent with an induced charge mechanism for effective colloidal interactions. We have also observed a crossover in frequency that suggests a change in the origin of the induced charge, consistent with predictions from available theory. The EH flows appear to be hydrodynamically screened inside clusters, as evidenced by the lack of rotation of interior colloids and the cluster-size independent rotation rate of colloids on the boundary.

Introduction The controlled assembly of colloidal particles has received significant attention in recent years because of the potential application of nano- and microstructured materials in many fields. Ordered colloidal systems have lattice spacings ranging from nanometers to micrometers and can therefore diffract ultraviolet, visible, and near-infrared light. One can take advantage of this for a variety of applications, including sensors,1 narrow-band optical filters,2 optical switches, photonic band gap materials, waveguides,3 and other types of optical and electrooptical devices.4 Photonic crystals could allow significant advances in the miniaturization and high-speed performance of integrated circuits5 and have profound applications for telecommunications, lasers, fiber optics, data processing, and display technologies.5,6 Currently, an important limitation in the development of such colloid-based technologies is in the control of crystal symmetries and in the slow times associated with crystal formation. Most efforts to create three-dimensional colloidal crystals rely either on the use of gravity to allow colloids to slowly settle and form dense ordered phases or the use of strong charge-induced colloidal repulsions to induce colloidal crystallization. However, both approaches have significant drawbacks. Gravity-induced ordering requires highly monodisperse colloids and careful control of the density mismatch to limit defect density. Electrostatic-repulsioninduced ordering can require long periods of time and very careful control of the colloidal interactions. Recently, we have developed techniques using applied alternating current (AC) electric fields7 combined with light8 on confined colloids that allow for rapid * Corresponding author. E-mail: [email protected]. † Chemical Engineering Department. ‡ Chemistry and Chemical Engineering Departments. (1) Asher, S. A.; Holtz, J.; Weissman, J.; Pan, G. Mesoscopically periodic photonic-crystal materials for linear and nonlinear optics and chemical sensing. MRS Bull. 1998, 23, 44-50. (2) Sunkara, H. B.; Jethmalani, J. M.; Ford, W. T. Composite of colloidal crystals of silica in poly(methyl methacrylate). Chem. Mater. 1994, 6, 362-364. (3) Joannopoulos, J. D.; Villeneuve, P. R.; Fan, S. Photonic crystals. Solid State Commun. 1997, 102 (2-3), 165-173. (4) Xia, Y.; Gates, B.; Yin, Y.; Lu, Y. Monodispersed colloidal spheres: old materials with new applications. AdV. Mater. 2000, 12 (10), 693-713. (5) Fairley, P. The microphotonics revolution. Technol. ReV. 2000, 103 (4), 38-44. (6) Joannopoulos, J. D.; Villeneuve, P. R.; Fan, S. Photonic crystals: putting a new twist on light. Nature 1997, 386, 143-149. (7) Gong, T.; Wu, D. T.; Marr, D. W. M. Electric field reversible threedimensional colloidal crystals (cover article). Langmuir 2003, 19, 5967.

and reversible formation of colloidal crystals with tunable interactions that overcome many of these limitations. A detailed understanding of the mechanisms in the ordering of these systems remains elusive, however, and has motivated the detailed examination of simpler two-dimensional (2D) analogues. Significant previous work has focused on field-directed colloidal ordering in 2D systems that exhibit novel phase behavior9-11 and unique optical properties.12-14 As first reported by Richetti et al.,15 electric fields15-20 can induce a “lateral attraction” on electrode surfaces that can be used to create colloidal crystallites. This attraction was at first surprising since the induced dipoles of the colloids should strongly repel. The fact that individual colloids travel toward each other, even across very long distances, strongly suggested that these apparent attractions are driven by hydrodynamic flows. Trau et al.,16 Yeh et al.,17 and (8) Gong, T.; Marr, D. W. M. Photon-directed colloidal crystallization. Appl. Phys. Lett. 2004, 85, 3760. (9) Marcus, A. H.; Rice, S. A. Phase transitions in a confined quasi-twodimensional colloid suspension. Phys. ReV. E 1997, 55 (1), 637-656. (10) Murray, C. A.; van Winkle, D. H. Experimental observation of two-stage melting in a classical two-dimensional screened Coulomb system. Phys. ReV. Lett. 1987, 58, 1200-1203. (11) Murray, C. A.; Sprenger, W. O.; Wenk, R. A. Comparison of melting in three and two dimensions: microscopy of colloidal spheres. Phys. ReV. B 1990, 42, 688-703. (12) Lin, S. Y.; Chow, E.; Hietala, V.; Villeneuve, P. R.; Joannopoulos, J. D. Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal. Science 1998, 282, 274-276. (13) Leonard, S. W.; Mondia, J. P.; Driel, H. M. v.; Toader, O.; John, S.; Busch, K.; Birner, A.; Go¨sele, U.; Lehmann, V. Tunable two-dimensional photonic crystals using liquid-crystal infiltration. Phys. ReV. B 2000, 61 (4), R2389R2392. (14) Chow, E.; Lin, S. Y.; Johnson, S. G.; Villeneuve, P. R.; Joannopoulos, J. D.; Wendt, J. R.; Vawter, G. A.; Zubrzycki, W.; Hou, H.; Alleman, A. Threedimensional control of light in a two-dimensional photonic crystal slab. Nature 2000, 407, 983-986. (15) Richetti, P.; Prost, J.; Barois, P. Two-dimensional aggregation and crystallization of a colloidal suspension of latex spheres. J. Phys. Lett. 1984, 45, L1137-L1143. (16) Trau, M.; Saville, D. A.; Aksay, I. A. Field-induced layering of colloidal crystals. Science 1996, 272, 706-709. (17) Yeh, S.-R.; Seul, M.; Shraiman, B. I. Assembly of ordered colloidal aggregates by electric-field-induced fluid flow. Nature 1997, 386, 57-59. (18) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Electrophoretic assembly of colloidal crystals with optically tunable micropatterns. Nature 2000, 404, 5659. (19) Bo¨hmer, M. In situ observation of 2-dimensional clustering during electrophoretic deposition. Langmuir 1996, 12 (24), 5747-5750. (20) Solomentsev, T.; Bo¨hmer, M.; Anderson, J. L. Particle clustering and pattern formation during electrophoretic deposition: a hydrodynamic model. Langmuir 1997, 13, 6058-6068.

10.1021/la060270g CCC: $33.50 © 2006 American Chemical Society Published on Web 05/19/2006

Colloidal Particle Rotation in AC EH Flows

Bo¨hmer19 have all proposed mechanisms where this lateral attraction results from electrohydrodynamic (EH) effects arising from charge accumulation near the electrodes due to the passage of ionic current. In AC fields, Saville21-23 and Nadal et al.24 have proposed analogous mechanisms where lateral variations in concentration polarization induce a spatially varying free charge that induces electroosmotic fluid motion in the presence of an electric field, causing the particles to move together. Fagan, et al.25-27 have elaborated on this significantly and have investigated the electrolyte and frequency dependence of these field-induced flows. These specific mechanisms proposed for the electric-field driven interactions between colloids vary depending on the regime: for example, whether the field is AC or direct current (DC), or whether the frequency is low enough to allow for significant Faradaic reaction or formation of a concentration polarization layer. Other factors that have been accounted for include the effects of specific counterion identities, image charges, and whether the free charges that respond to the field are localized around the electrodes or the colloids. The dominance of a particular factor depends on the regime in which the assembly takes place. However, under applied AC fields, the mechanisms are electrohydrodynamic in that they essentially propose that the force exerted on the fluid is proportional to the product of the unbalanced charge and the local field. Since the unbalanced charged is, to lowest order, linear in the applied field, the magnitude of the force imparted on the fluid scales as the electric field squared. The frequency dependence of the force then appears in the quantity of unbalanced charge and the electric field acting upon it. Because of the axial symmetry of the colloid in conjunction with that of the bounding surface, the flows have the general geometry of a lateral flow toward the particle that is expelled normal to the plane (or the reverse). While these theories describe mechanisms where flow is driven by the action of an electric field on the unbalanced charge, the experimental observations corroborating these theories have focused on characterizing the translational motion of the more easily tracked colloids.24 Although this provides useful information, measurement of colloid rotation provides a more direct method of characterizing the influence of the applied field on fluid flow and the apparent colloid interactions. Because the rotation reflects the local flow field near the colloid surface, it may help validate or distinguish a mechanism, such as whether the unbalanced charge is localized near the colloid or the electrode. In this article, we describe a method to directly observe colloid rotation, and report on the electric field dependence of this rotation. As such, the approach reported in this paper can be used to gain a more detailed understanding of electric-field-driven colloid (21) Trau, M.; Saville, D. A.; Aksay, I. A., Assembly of colloidal crystals at electrode interfaces. Langmuir 1997, 13, 6375-6381. (22) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Assembly of colloidal aggregates by electrohydrodynamic flow: kinetic experiments and scaling analysis. Phys. ReV. E 2004, 69, 021405. (23) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Electrically guided assembly of planar superlattices in binary colloidal suspensions. Phys. ReV. Lett. 2003, 90 (12), 128303. (24) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B.; Ajdari, A. Electrically induced interactions between colloidal particles in the vicinity of a conducting plane. Phys. ReV. E 2002, 65, 061409. (25) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Evidence of multiple electrohydrodynamic forces acting on a colloidal particle near an electrode due to an alternating current electric field. Langmuir 2005, 21, 1784-1794. (26) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Vertical motion of a charged colloidal particle near an AC polarized electrode with a nonuniform potential distribution: theory and experimental evidence. Langmuir 2004, 20, 48234834. (27) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Vertical oscillatory motion of a single colloidal particle adjacent to an electrode in an AC electric field. Langmuir 2002, 18, 7810-7820.

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assembly, both in dilute conditions and in the technologically relevant cases of dense assembled phases or clusters. Experimental Methods Uniform polystyrene particles 4.2 µm in diameter (surfactantfree white carboxylate-modified latex (CML), Interfacial Dynamic Corporation, Oregon, OR) were employed as the large colloidal species, and surfactant-free fluorescent Nile Red CML polystyrene latex particles 340 nm in diameter (chosen to minimize any surface charge differences between the large and small particles) were used as labels. The deionized water used was purified by a Millipore Milli-Q system (resistivity: ∼18 MΩ‚cm). Our microfluidic cells were fabricated with indium tin oxide-coated glass slides (Delta Technologies, Ltd., Stillwater, MN) that were first coated with 0.5 wt % Triton X-100 (Aldrich Chemical Co., Inc., Milwaukee, WI). These were sandwiched together using a 50 µm Kapton spacer (DuPont High Performance Materials, Circleville, OH) and sealed with poly(dimethylsiloxane) (PDMS) (Sylgard Brand, Dow Corning Corporation, Midland, MI). These cells were used for both the labeling process and for measuring particle rotation. AC electric fields were applied to the cell using a function generator (DS360, Stanford Research System, Sunnyvale, CA), and rotation was observed using a Zeiss Axioplan 2 with a 60× objective in dark-field and fluorescence microscopy. Labeling. An aqueous colloidal mixture was prepared at an electrolyte concentration of 1.0 mM KCl. The larger colloids were deposited on the lower electrode by gravity, covering an area fraction of 0.29, while the tracer colloids with a volume fraction of 10-4 remained dispersed in the volume. To initiate the labeling process, we applied an AC electric field of 0.14 V/µm at a frequency of 100 Hz. Under these conditions, the EH flow translates the large particles toward one another, producing tight 2D crystals, while the small colloidal labels are dragged toward these crystals by the induced flows. At the center of the clusters, the converging flows move upward, confining the tracers to the top surface of the larger colloids, to which they are attracted at these field conditions. After ∼2 min, most of the fluorescent particles are trapped in a region around the cluster center. In this process, the tracers aggregate on the larger particles with a concentration depending primarily on the time the electric field is applied and its intensity, with a larger intensity leading to faster aggregation. The tracer particles are also attracted to each other under these field conditions, leading to recognizable fluorescent domains. Once the central particles are covered by fluorescent labels, irreversibility is demonstrated through subsequent centrifugation and separation of the larger particles to remove the excess tracers in the dispersion. Rotation Measurement. Labeling the spheres allows characterization of the EH flows when an AC field is applied. These flows not only translate colloids toward one other but also induce rotation. The rotational axis depends on the coupling of the flows associated with each particle and its neighbors and directly indicates the direction and strength of the net flow around each colloid. Under AC electric fields, particles separated by more than 5 diameters showed no rotation, providing a rough indication of the range of the EH interactions under the field strengths employed here. Typically, upon application of the field, particles began rotating and translating toward one other until clusters were formed. After the particles assembled into stable clusters, only those remaining on the cluster borders rotated, as illustrated and shown in Figure 1. To quantify microparticle rotation, several experimental simplifications were made. The area fraction of 0.11 was low enough that the average distance between neighboring clusters was greater than 5 diameters, leading to negligible interaction. In addition, only particles with a single fluorescent tracer region were tracked to simplify analysis and provide rotation about a fixed axis. The rotational frequency of these particles was then measured by monitoring the integrated brightness of the particle image (IDL 6.2, RSI, Boulder, CO). Figure 2 provides an example of the measured integrated brightness in which the flow drives the labeled particle to rotate with an average frequency of 0.3 Hz (with an applied field at 100 Hz and 0.08 V/µm). In this figure,

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Santana-Solano et al. intensity from the colloid being tracked. This method was previously used by Shelby et al.29 to study the rotational motion of single microparticles confined in microvortices. The duration of the “bright” pulse is variable, depending on the location of the tracer region relative to the rotation axis and on the tilt of the rotation axis relative to the electrode surface. Additionally, because of the microscope optics and the angle of illumination, the field of vision corresponds to less than half the surface of the sphere, and so the intensity signal is “dark” for a longer interval than it is “light”. These effects do not alter the rotational period, so we are able to extract a rotational velocity with good precision from intensity curves such as those shown in Figure 2. As can be seen, the periodicity is very regular over many cycles.

Results and Discussion

Figure 1. (a) Illustration of experimental setup where the actual electrode spacing is 50 µm. Under applied AC fields, nonsymmetric hydrodynamic flows induce rotation for colloidal particles on the 2D aggregate boundaries: (b,c) initially and (d,e) after 10 s (scale bar ) 10 µm). Rotation direction is indicated with arrows (see movie provided as Supporting Information).

Figure 2. Regular intensity periods measured upon application of the electric field due to the rotating labeled colloid. For quantification, significantly lower label concentrations were employed (inset). the primary peaks correspond to the configuration in which the tracer region is highest. This rotational motion is quite stable and can be maintained in a steady state for significant periods of time; typically, we follow the rotation for ∼1.5 h in generating the data provided here. In these measurements, we employ voltages where convection clearly dominates over rotational Brownian motion. With rotational diffusion times on the order of tens of minutes, we estimate a Peclet number (the ratio of convective to diffusive transport) of approximately 30, even at our lowest applied voltages and rotational frequencies. While it is possible to apply a more complex and thorough approach of image processing and mapping onto a deduced rotational motion based on observed projections in video frames,28 we adopted a simpler but more direct strategy of monitoring the periodicity of the integrated (28) Makadia, A.; Daniilidis, K. Direct 3D Rotation Estimation from Spherical Images via a Generalized Shift Theorem. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Madison, WI, 2003; IEEE Computer Society: Piscataway, NJ, 2003; pp 217-224.

The translational and rotational motion of the colloid results from the sum of any direct forces, F, and torques, L, on the colloid and the forces and torques exerted by the integrated stress of the fluid on the colloid surface. Steady-state motion occurs when the forces and torques from the fluid, at a terminal velocity and angular velocity, balance all other forces and torques. Together with the knowledge of any direct forces on the colloids, a measurement of the translational and angular velocities of the colloid thus provides a measure of the flow field surrounding a colloid. Here, we present a novel analysis relating the apparent translation per rotation to the ratio of direct to EH forces. For an isolated colloid near a wall, the axially symmetric EH flow exerts a normal force but no torque. Other colloids entrained in this flow, however, can rotate and translate, as illustrated in Figure 1. By symmetry, the far-field flow near a wall at z ) 0 is a simple shear u(r) ∼ κzGˆ with shear rate κ ∼ F-4, where F ) xx2+y2. The prefactor measures the net driving force exerted on the fluid and depends on the height and size of the colloid as well as the distribution and location of the unbalanced charge that is responding to the electric fields. Let us first consider the translation per rotation of a free colloid in this far-field shear flow. Far from the wall, Faxen’s Law states that a sphere at a distance l from the wall translates with velocity U equal to the local fluid velocity, κl, and rotates with angular velocity Ω equal to the local angular velocity of the fluid, κ/2. Since the rotation rate is independent of colloid radius a or distance from the wall, while the velocity increases with distance from the wall, l > a, the distance traveled per full colloid rotation, 2πU/Ω is always greater than 4πa. Thus, viewed from above, the colloid will appear to be “slipping” forward at least twice as quickly compared to rolling on a surface, where the distance traveled per rotation is 2πa. For closer distances, the hydrodynamic effect of the wall is to reduce both the velocity and the angular velocity relative to the unbounded case. Chaoui et al.30 recently used a recurrence relation for a spherical harmonic expansion in bipolar coordinates to calculate flow solutions for a free sphere in a plane-bounded shear flow, accurate to nine significant digits, even down to separations of a millionth of a particle diameter, which makes it accurate enough to improve upon existing lubrication expressions. While the distance traveled per rotation does decrease, it only does so very slowly: using the expressions in Chaoui et al.,30 even for a gap size of ∼10% of the colloid radius, while the velocity and angular velocity are reduced by approximately 25%, the distance traveled per rotation is only reduced by 2%

(29) Shelby, J. P.; Mutch, S. A.; Chiu, D. T. Direct manipulation and observation of the rotational motion of single optically trapped microparticles and biological cells in microvortices. Anal. Chem. 2004, 76, 2492-2496. (30) Chaoui, M.; Feuillebois, F., Creeping flow around a sphere in a shear flow close to a wall. Q. J. Mech. Appl. Math. 2003, 56 (3), 381-410.

Colloidal Particle Rotation in AC EH Flows

to ∼(0.98)4πa. For gap sizes of 1% of a colloid radius, the velocity and angular velocity are reduced by approximately half, and the distanced traveled per rotation is reduced by about 5% relative to the no-wall case. Thus, even near the wall, free colloids in a shear flow will appear to be slipping forward compared to rolling. In fact, our observations show that typical distances traveled per colloid rotation are much less than that for rolling. While this does not appear to be possible for free colloids from the pure hydrodynamic considerations above in the far-field regime, it is possible for colloids that experience a repulsive force from each other. Morphological observations of colloidal assemblies under quasi-2D confinement and EH flows31 have indeed suggested that the effective attractions mediated by EH flows are, to varying degrees, countered by repulsive induced dipole-induced dipole interactions that result from the applied AC fields. While these repulsions are not strong enough to overcome the EH attractions in the unconfined case here, they may be strong enough to noticeably slow the translations of the colloids toward each other. In a straightforward extension of the classical approach employed by Chaoui et al., we solve for the creeping flow motion of a forced sphere in a shear flow near a wall as a linear superposition of the solutions of three cases: (i) a nonrotating sphere translating with velocity Ux in the x-direction in a quiescent fluid parallel to a wall, which produces a force Ftx ) -6πaµftxxUx and torque Cty ) 8πa2µctyxUx; (ii) a nontranslating sphere rotating with angular velocity Ωy in a quiescent fluid near a wall, which produces a force Frx ) 6πa2µfrxyΩy and torque Cry ) -8πa3µcryyΩy; and (iii) a stationary sphere in a shear flow with shear rate κ near a wall, which produces a force Fκx ) 6πaµfκxxκl and torque Cκy ) 4πa3µcκyxκ. In the above, we follow the notation in Chaoui et al., and the expressions for force and torque define the corresponding height-dependent friction factors, f and c, above. The fluid viscosity is µ. The conditions of force and torque balance determine U and Ω as a linear combination of the free colloid solution, proportional to κ, and the solution for a forced colloid without shear, proportional to the applied force and torque. In the absence of torques, as is the case for induced dipole-induced dipole forces, the velocity and angular velocity are then r

U ) Ufree +

cyy F 6πµa ft cr - fr ct xx yy xy yx

Ω ) Ωfree +

cyx F 2 t r 6πµa fxx cyy - frxy ctyx

t

where Ufree and Ωfree are the unforced colloid velocity and angular velocity. The linear structure is analogous to Faxen’s equations for a forced colloid far from a wall, but with friction factors reflecting the hydrodynamic influence of the wall. We note that the distance traveled per rotation of a forced colloid in a quiescent fluid is thus 2πa(cryy/ctyx). At a gap of 10% of the radius, the ratio cryy/ctyx is approximately 16, and increases with gap size. At a gap of about 1% of a colloid radius, the ratio cryy/ctyx is about 8. Since these are larger than the values for a free sphere in a shear flow, a repulsive force will act to slow the translational approach of a colloid more than the rotation, leading to an apparent reduction of the distance traveled per rotation. Since the induced dipole-induced dipole forces do not produce torque, the rotation rate can be used to measure the strength of (31) Gong, T.; Wu, D. T.; Marr, D. W. M. Two-dimensional electrohydrodynamically induced colloidal phases. Langmuir 2002, 18, 10064-10067.

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the force due to EH flow. For a known value of height from the surface, for instance, as done in the total internal reflection microscopy (TIRM) measurements by Fagan et al.,26 the friction factors can be calculated, and the ratio of the direct force to the EH force, and hence their magnitudes, can be calculated from the equations above. The rotation rate can also be used to test the distance dependence of the flow, namely, the expected 1/F4 dependence at large distances. Analysis of the rotation dependence on the EH flow field becomes more complicated as the colloids come into closer proximity of each other and sense the detailed shape of the flow, which may also depend on factors such as distribution and location of the induced charges responding to the electric field. Since the different theoretical mechanisms proposed give predictions for the velocity field as a function of position about a given colloid, it may be possible to directly relate the observed rotation of a colloid with the surrounding velocity field that also serves to drive colloids together or apart. In practice, however, several other factors influence how surrounding fluid flow couples to colloid rotation. First, for particles with a large number of labels, there is the apparent influence of friction between the colloid and the confining electrode surfaces, manifest in stick-slip rotation, and perhaps responsible for a drift in the axis of rotation sometimes observed over the course of a few minutes. Our measurements were thus carried out on selected colloids with small label numbers that exhibited rotation about a fixed axis without apparent stickslip. Upon application of electric fields, clusters of the labeled particles formed on the order of a minute. Pairs of free colloids that attracted each other were observed to start rotating toward each other as they were pairing up. Since the cluster formation was relatively transient, we focused our observations on established clusters of different sizes at steady state. Here, it was observed that the only colloids that exhibited noticeable rotation were those at the outer border (perimeter) of the cluster. There was visible motion of free small tracer colloids concentrating as a shifting cloud about the center of clusters, but no detectable rotation of the interior colloids themselves. Furthermore, the rotation of the colloids on the perimeter of the cluster was invariably in the direction toward the center of the cluster, as depicted in Figure 1. This observation is qualitatively consistent with the general idea that each colloid is setting up a flow field that laterally draws fluid toward it and expels it vertically. A simplistic superposition of these individual flow fields, which neglects hydrodynamic interactions between colloids, suggests that the radial component of the flow field is small in the center and largest at the perimeter. The dense arrangement of colloids inside a cluster, however, limits this superposition view, and acts to hydrodynamically screen out flow in the interior, leaving the dominant flow to occur at the cluster exterior. A similar conclusion is reached if we consider, for instance, in a mechanism based on electrode concentration polarization, partial cancellations of the EH driving force due to unbalanced charged near each colloid. This hydrodynamic screening and damping of the EH driving force in the interior of a cluster is an important observation for the purpose of understanding and controlling colloid assemblies. Our first measurements concern the variation of the rotation rate of the colloid with the amplitude of the electric field. Although the flow fields cannot be linearly superimposed, we expect, in the laminar regime, the velocities to scale linearly with the applied forces. For the reasons stated in the Introduction, we generically expect the force to be quadratic in the field strength, as is predicted in a variety of mechanisms and consistent with prior observations

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Figure 3. Boundary colloid rotation rate fr as a function of applied AC field potential ∆φ where E2 dependence is observed. For lines from bottom to top, Np ) 53, Np ) 45, Np ) 75, Np ) 17, and Np ) 27, and the slope of the line fits are independent of the number of particles Np in the cluster (inset).

of colloid translational motion. This E2 dependence of the velocity field should thus produce an E2 dependence in the rotation rate, as seen in Figure 3 for five representative clusters of differing size and shape. The quality of the E2 behavior strongly supports the EH (as opposed to electrokinetic) nature of the flow fields under all conditions presented in this paper. Because of hydrodynamic screening, as described above, we expect that the flow velocity approaches a constant value as the cluster size increases, that is, the flow approaches a finite value for a straight edge of boundary colloids. This independence of the rotation speed with the cluster size (at a given amplitude and frequency of the electric field) was indeed observed for a range of cluster sizes, as seen in the inset of Figure 3 plotting the slope of the E2 dependence of the rotation rate versus the cluster size. If the slope is not controlled by the size of the cluster, this leaves the question of what is responsible for the variation in slope seen in Figure 3. From observations of comparable variations in rotation rates of relatively isolated colloids, we conclude that these variations are not due to collective effects, but are due to local inheterogeneities, including particle environment or differential friction due to differing coverage by the tracer particles. While quantitative extraction of flow profiles is limited by these uncertainties, the scaling behavior seen for individual colloids, including influences by surrounding colloids, appears to be robust. Turning now to the dependence of the colloid rotation rate on the electric-field AC frequency, Figure 4 shows that there appears to be a rough crossover from a relatively constant rotation rate at low frequencies to a regime at higher field frequency, where the rotation rate decreases inversely with the field frequency, consistent with the scaling presented by Ristenpart et al.22 The apparent crossover frequency range of ∼500 Hz is also consistent with that seen in other experiments22,25 corresponding to the suppression of Faradaic reactions by the double-layer capacitance.

Santana-Solano et al.

Figure 4. Boundary colloid rotation rate vs applied AC field frequency with an electric field intensity of 0.08 V/µm.

The data does not extend to sufficiently low field frequencies to make a conclusion about the low frequency limit. However, we do point out that a constant rotation rate at low frequencies is possible with an EH mechanism (and the data are clearly still showing E2 field dependence), for instance, by an induced-charge electroosmotic mechanism (ICEO) as proposed by Bazant et al.32 Since the induced charge is in phase with the electric field in this EH mechanism, the net force on the fluid does not cancel out over an AC cycle.

Conclusion Using a fluorescence-based labeling approach, we have measured the influence of both applied AC field strength and frequency on the EH flows present in colloidal systems. Upon field application, we have observed that particles on the induced colloidal aggregate perimeter rotate with a rate independent of cluster size, consistent with hydrodynamic screening and the partial cancellation of EH forces inside the cluster. These results clearly show an E2 dependence of the EH flows and thus demonstrate that the effective colloidal interactions arise from an induced charge mechanism. In addition, we have observed a crossover in frequency that suggests a change in the origin of the induced charge at approximately 500 Hz, consistent with predictions from available theory. These observations are expected to be relevant for the EH-driven assembly of colloidal crystals and structures. Acknowledgment. This work was supported by NASA under grant #NAG9-1364. The authors would like to thank Armand Ajdari for stimulating discussions on EH flows. Supporting Information Available: A brief movie (0.6 MB) of colloid rotation is available in quicktime format. This material is available free of charge via the Internet at http://pubs.acs.org. LA060270G (32) Bazant, M. Z.; Squires, T. M. Induced-charge electrokinetic phenomena: theory and microfluidic applications. Phys. ReV. Lett. 2004, 92, 066101.