Anomalous Particle Rotation and Resulting Microstructure of Colloids

Oct 25, 2008 - Anomalous Particle Rotation and Resulting Microstructure of Colloids in AC Electric Fields. Pushkar P. Lele, Manish Mittal, and Eric M...
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Langmuir 2008, 24, 12842-12848

Anomalous Particle Rotation and Resulting Microstructure of Colloids in AC Electric Fields Pushkar P. Lele, Manish Mittal, and Eric M. Furst* Department of Chemical Engineering and Center for Molecular and Engineering Thermodynamics, UniVersity of Delaware, 150 Academy Street, Newark, Delaware ReceiVed July 11, 2008. ReVised Manuscript ReceiVed September 2, 2008 嘷 w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/Langmuir. We study the transition of ordered structures to disordered bands and vortices in colloidal suspensions subjected to AC electric fields. We map the critical frequencies and field biases at which particles form disordered bands and vortices. These results are interpreted based on the trajectory dynamics of particle pairs using blinking optical tweezers. Under conditions that vortices are observed, individual particle pairs rotate out of alignment with the field. The direction and magnitude of these interactions determine the orientation and average angular velocity of the band revolution. Increasing the frequency of the electric field reduces the anomalous rotation of the particles pairs, consistent with the frequency dependence of the suspension order-to-disorder transition. This anomalous rotation is consistent with a torque on doublets generated by the mutual polarization of particles and phase lag of the induced dipoles.

I. Introduction The use of electric fields to drive colloidal particles into selfassembled structures has generated significant interest in recent years.1-4 When colloidal particles of low (high) dielectric constants are suspended in a fluid of high (low) dielectric constant and subjected to an electric field, they become polarized. Dipole moments are induced in the particles by the release of free or bound polarizational charges at the interphase boundary, which is a consequence of the differences in electrical conductivities and the dielectric permittivities between the dispersed particles and the medium.5 In addition, the perturbation of the double layer from its equilibrium state by the external fields at low frequencies leads to a larger dipole moment.6,7 Previous work has shown that AC electric fields induce the assembly of microspheres into two-dimensional colloidal crystals and other patterns.1,8-11 Similar phenomena are observed using DC electric fields.12-14 Here, we are specifically interested in studying the order-disorder transition in parallel electrode geometries, which exploits both dielectrophoresis and polarization-induced particle interactions to assemble colloidal crystals and other novel structures.4,15-17 * Corresponding author. E-mail: [email protected]. (1) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706–709. (2) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature 1997, 386, 57–59. (3) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Nature 2000, 404, 56–59. (4) Lumsdon, S. O.; Kaler, E. W.; Velev, O. D. Langmuir 2004, 20, 2108– 2116. (5) Pohl, H. A. Dielectrophoresis; Cambridge University Press: Cambridge, UK, 1978. (6) Mittal, M.; Lele, P. P.; Kaler, E. W.; Furst, E. M. J. Chem. Phys. 2008, 129, 064513–7. (7) Basuray, S.; Chang, H. C. Phys. ReV. E 2007, 75, 060501–4. (8) Yethiraj, A.; van Blaaderen, A. Nature 2003, 421, 513. (9) Gong, T.; Marr, D. W. M. Langmuir 2001, 17, 2301. (10) Ristenpart, W.; Aksay, I.; Saville, D. Phys. ReV. Lett. 2003, 90, 128303. (11) Hoggard, J. D.; Sides, P. J.; Prieve, D. C. Langmuir 2008, 24, 2977–2982. (12) Bohmer, M. R. Langmuir 1996, 12, 5747–5750. (13) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408–3413. (14) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334–6336. (15) Hermanson, K. D.; Lumsdon, S. O.; Williams, J. P.; Kaler, E. W.; Velev, O. D. Science 2001, 294, 1082. (16) Lumsdon, S. O.; Kaler, E. W. Appl. Phys. Lett. 2003, 82, 949–951. (17) Millman, J. R.; Bhatt, K. H.; Prevo, B. G.; Velev, O. D. Nat. Mater. 2005, 4, 98–102.

Electric fields also lead to the formation of structures and flows in colloidal suspensions which disrupt or complicate the aim of creating ordered 2D and 3D arrays of colloidal particles. These structures include tumbling clouds and toroidal rings in DC fields18 and the formation of dense bands and vortices of colloidal particles in AC fields.19-26 Bands and vortices of colloidal particles in AC electric fields were reported by Stauff19 and were also discussed by Gamayunov and co-workers in their work in thinly spaced optical slides.20 Isherwood et al.21 showed in their experiments that band formation is a time dependent phenomenon and occurs only in AC fields. This observation was confirmed by Jennings et al.,22 who theorized that the bands are a direct result of developing instabilities in the presence of AC electric fields. Prost and co-workers instead suggested that macroscopic gradients in the electrolyte concentration, which were thought to arise due to the field-induced concentration gradients near the particle surfaces, lead to instabilities which cause the bands to appear.23,24 Two other groups attribute the presence of bands to the mutual polarization of particles,25,26 causing them to rotate. This latter idea is supported by experiments using biological cells. Adjacent cells have been observed to rotate due to the presence of neighboring cells in AC electric fields.27 For instance, cells aligned as a chain in the field direction rotate in the axis perpendicular to the electric field. The angular velocity of the rotation is greatest when the chain’s angle of inclination is (45° to the electric field. Electrophoretic rotation of polystyrene doublets has also been observed due to differences (18) Han, Y.; Grier, D. G. J. Chem. Phys. 2005, 122, 164701. (19) Stauff, J. Kolloidzeitschrift 1995, 143, 162–171. (20) Gamayunov, N. I.; Murtsovkin, V. A. Kolloidn. Zh. 1983, 45, 760–763. (21) Isherwood, R.; Jennings, B. R.; Stankiewicz, M. Chem. Eng. Sci. 1987, 42, 913–914. (22) Jennings, B. R.; Stankiewicz, M. Proc. R. Soc. London, Ser. A 1990, 427, 321–330. (23) Isambert, H.; Ajdari, A.; Viovy, J. L.; Prost, J. Phys. ReV. E 1997, 56, 5688–5704. (24) Isambert, H.; Ajdari, A.; Viovy, J. L.; Prost, J. Phys. ReV. Lett. 1997, 78, 971–974. (25) Kiriyama, T.; Ozawa, T.; Akimoto, T.; Yoshimura, H.; Mitsui, T. Jpn. J. Appl. Phys 1997, 36, 7282–7288. (26) Hu, Y.; Glass, J. L.; Griffith, A. E.; Fraden, S. J. Chem. Phys. 1994, 100, 4674–4682. (27) Holzapfel, C.; Vienken, J.; Zimmermann, U. J. Membr. Biol. 1982, 67, 13–26.

10.1021/la802225u CCC: $40.75  2008 American Chemical Society Published on Web 10/25/2008

Colloid Particle Rotation and Microstructure in AC

in the zeta potentials between particles in doublets.28-30 In their work, Hu and co-workers26 suggest that the mutual polarization induces neighboring particles to rotate. In this work, we study the microstructure and underlying microscopic interactions between colloidal particles in AC fields in frequency regimes where anomalous rotation occurs. Using blinking optical tweezers, we show that particle pairs in an AC electric field rotate relative to each other and fall out of alignment to the electric field lines. The underlying mechanism is consistent with the mutual polarization mechanism postulated by Hu et al.26 As we will discuss, these anomalous microscopic interactions are also present in microstructures, such as particle chains and clusters; it is this characteristic doublet rotation that leads to the formation of band structures. Before discussing our results, we first describe the experimental methods employed in this work.

II. Experimental Details A. Materials and Sample Preparation. We use suspensions of monodisperse polystyrene latex particles (Polysciences, Inc.) with an average diameter 2a ) 3.04 ( 0.12 µm. Particles are washed prior to use by centrifugation in ultrapure water (conductivity 18.2 MΩ · cm) before redispersing them in a 10 µM solution of KCl. The density mismatch of polystyrene (FPS ) 1050 kg/m3) causes particles to sediment to the bottom slide. When investigating macroscopic colloidal structures, we use φ ) 0.01 solids volume fraction which leads to an area coverage on the lower sample surface of approximately 0.5. For particle interactions measurements, we use a dilute suspension of φ e 10-4 to reduce hydrodynamic interactions with neighboring particles and to ensure stable optical trapping. The electric field is generated using coplanar gold electrodes. The electrodes are fabricated on glass slides (25 × 75 cm, plain microscope slide, Fischer Scientific). The glass slides are first cleaned in a freshly prepared Nocrhomix solution (cat. no. 19-010, Godax Laboratories). Next, we use vapor deposition to create parallel electrodes using an approximately 10 nm thick layer of chromium followed by a 100 nm gold layer. The gap between the planar electrodes is 2 mm. The electrodes are plasma cleaned (PDC-32G, Harrick Plasma) before use in experiments. The fluid cell is constructed using an adhesive spacer to create a gap of approximately 100 µm between a microscope coverslip (1.5, 22 × 22 mm, Corning) and the microscope slide. Colloidal solution is introduced by capillary action with care to avoid the introduction of bubbles. The ends of the cell are then sealed using a fast UV curing epoxy (Norland optical adhesive, P/N 8101) to prevent evaporation and convection. AC electric fields are applied using a high voltage ramp amplifier (Exfo RG-91) driven by a function generator (33220A, Function/arbitrary waveform generator, Agilent Technologies). To minimize the DC component of the signal, we use a 2 µF capacitor. The sample is illuminated using a halogen lamp, and images are captured on a charge-coupled device (ccd) camera (KPM1-AN, Hitachi) and stored on a digital video recorder at a rate of 30 frames/s. Individual frames are digitally transferred to a workstation for further image processing. B. Blinking Optical Tweezers. Our optical tweezer setup is shown in Figure 1. We position polystyrene particles by focusing a Nd:YAG laser beam (λ ) 1064 nm, Coherent Compass 1064 400 M) to a diffraction limited spot at the sample plane using a water immersion microscope objective (63×, 1.2 numerical aperture Zeiss C-apochromat). Multiple traps are created with a time-sharing technique that uses repeated, discontinuous scanning of a laser beam at discrete locations.31 This is achieved by a pair of perpendicular acoustooptic deflectors (AOD, AA Optoelectronics AA) which controls the angle of the beam as it enters the back aperture of the microscope objective. A computer software interface provides digital control (28) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. (29) Velegol, D.; Catana, S.; Anderson, J. L.; Garoff, S. Phys. ReV. Lett. 1999, 83, 1243. (30) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1991, 145, 362. (31) Pantina, J. P.; Furst, E. M. Langmuir 2004, 20, 3940–3946.

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Figure 1. Blinking optical tweezer setup.

over the AOD positioning of the laser. A custom software generates a digital value that is converted to a voltage to be applied across the AOD crystal by a 16-bit data acquisition (DAQ) board (National Instruments BNC-2110). The sample is positioned in the x-y plane by controlling the microscope stage (MS-2000-XY, Applied Scientific Instrumentation) with an X-Y joystick. We generate blinking optical traps by periodically shuttering the laser beam using a signal from a separate function generator with a 1:3 duty cycle (DS-360, Stanford Research Systems). Particle trajectories are obtained from the digital images using standard particle tracking algorithms.32 Since out of plane fluctuations contribute to the 2D tracking error, the effect is minimized by shuttering the beam for no more than 0.6 s, or approximately 0.05 times the characteristic particle diffusion time τ0 ) 6πη a3/kBT where kBT is the thermal energy and η is the solvent (water) viscosity. We estimate the out of plane movement does not exceed approximately 0.2 µm on average. For blinking tweezer experiments, the solids volume fraction in our samples is kept to a minimum (φ ) 10-3-10-4) to study particle pairs in isolation. Particles are held far from the sample walls (≈50 µm).

III. Results A. Order-Disorder Phase Diagram. We characterize the suspension structure over a range of frequencies and electric field biases for a fixed colloidal volume fraction and salt concentration. Three general structures are observed: (i) isotropic disordered suspensions; (ii) crystalline, partially crystalline, or chains of particles aligned in the field direction; (iii) large bands or vortices. Examples of these structures are shown in Figure 2. We define a band to be a distinct zigzag cluster of colloidal particles in 2D or 3D with definite boundaries separating the high and low density areas in the sample (see Figure 2d). A band may or may not have any chaining and each limb within the band makes an average orientation angle with respect to the field lines, which usually falls between 30-80° with respect to the field. The band structure is observed to rotate with time, with particles on the edges of the aggregate moving with a higher velocity than at the center. Bands are typically organized with angles that alternate in the direction orthogonal to the field, with particle rotation also occurring in opposite directions. Because of this rotation, we also refer to these dynamic, disordered structures as vortices. In each experiment, we apply the field strength and frequency to a well dispersed suspension and wait for 10 min for the structure to evolve before making observations. Each point represents an average of four separate regions sampled within our planar electrode cell. For each subsequent experiment, we completely redisperse the colloidal suspension and again apply the field at a set frequency in a stepwise manner, thus ensuring that we start from a random, isotropic suspension. Fourier (32) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298–310.

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Figure 2. Structures formed at various electric field strengths at the frequency ν )20 kHz. These show Fourier transforms of raw images. The scale bar is 50 µm.

Random Dispersion. Below a critical field strength, the suspension maintains a disordered, random state. This critical field strength is weakly dependent on the frequency. Because the polarizability of the particle decreases with increasing frequency, the field strength required for chain formation is expected to increase.4,6 Chain Structures and 2D Crystals. At higher frequencies (ν > 2 kHz) and above the critical field strength λ ∼ 10, where λ ) πoβ2a3E2/kBT, a transition to ordered structures occurs as chains of particles form and coalesce into 2D arrays. The factor λ is a ratio of polarization-induced interparticle forces to Brownian forces, where the interaction energy between induced dipoles is

( ar ) (3 cos θ - 1)

Uclip ) -2πa3β2E02 Figure 3. Order-disorder phase diagram for polystyrene particles in AC electric fields. At low applied field strengths, the suspension exhibits a transition from random, isotropic to dipolar chains and crystals, similar to that reported in ref 4. At high field strengths, a second disordered phase of disordered bands and vortices appears. Each point represents the observed structure 10 min after applying the field from a welldispersed state.

transforms of the bright field images (Figure 2) are analyzed to calculate the alignment factors,33

An )

∫2π 0 I(φ) cos(nφ) dφ ∫2π 0 I(φ) dφ

(1)

where I(φ) is the intensity azimuthal average at the wavevector q ≈ 2 µm-1. Among the alignment factors, n ) 6 is of particular interest, because it captures the presence of hexagonal order in the images. To identify ordered phases, we use a threshold value of A6 g 0.2. A similar method is used to determine the presence of band structures, using A2 at q ∼ 0.15 µm-1. The phase identification is confirmed by visual observation of the bright field images. The alignment factors at wavevectors corresponding to nearcontact of the particles and far field length scales provide quantitative information about the presence of bands and chains in the suspension allowing us to construct a phase diagram, shown in Figure 3. The order-disorder phase diagram for particles in AC electric fields can be summarized by three regimes: (1) random dispersion; (2) chain structures and 2D crystals; (3) disordered bands. (33) Walker, L. M.; Wagner, N. J. Macromolecules 1996, 29, 2298–2301.

3

2

(2)

Here, a is the particle radius, β is equivalent to the ClausiusMosotti factor,34 E0 is the applied field strength, and θ is the angle the pair makes with the electric field lines. Crystalline arrays that completely fill the two-dimensional area, previously reported by Lumsdon et al.,4 are not observed due to the lower area fraction of the suspension in this study. As the frequency increases, the range of field strengths where chaining is observed expands (cf. Figure 3). However, at lower frequencies (ν < 2 kHz), particles tend to rotate out of alignment with the field, which leads to a reentrance to disordered structure through the formation of particle bands, described below. Disordered Bands. At moderate to high field strengths, again depending on the frequency, a second disordered regime emerges in the form of bands of particles that develop throughout the suspension. At lower frequencies (i.e., ν )1 kHz), such bands or vortices develop even at moderate field strengths (E ∼ 4 kV). However at higher frequencies (ν ) 12 kHz), these bands and vortices form only at high field strengths (E ∼ 10 kV). Thus, the ordered regime is bounded by two disordered regions: one where the particles interactions are dominated by thermal energy (random dispersion) and a second where the interactions are driven by the rotation of particles (bands and vortices). We also performed experiments with density matched suspensions using 50/50 mixtures of deuterium oxide (D2O, 99.9% pure, Sigma Aldrich) and water, as well as experiments with the electrodes on the bottom of the sample cell and observed the same particle band structures (data not shown). In section III.C, we discuss measurements of the characteristic doublet rotation measured using blinking optical tweezers to understand this second order-to-disorder transition with increasing field strength. Interestingly, a small coexistence region on the (34) Jackson, J. D. Classical Electrodynamics: Wiley: New York, 1975.

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Figure 4. Kinetics of band formation. (A) Suspension within a minute of application of electric fields. (B) Phase separation at 500 V/m, 1 kHz at time t )3 min. (C) Effect of increasing field strength to 2232 V/m (time elapsed since step change ) 45 s). (D) Development of band and transition in to vortices, total time elapsed ∼11 min. Scale bar ) 50 µm.

phase diagram indicates a simultaneous presence of chains or 2D ordered arrays and bands. In this region, weak, vortexlike flows of particles exist in all bands and isolated crystals at all frequencies, even those that contain significant order. At higher frequencies, these flows are often not obvious because the band circulates very slowly, preserving the crystalline structure. B. Time Dependence of Band Formation. The structure of the suspension was also observed after applying a field strength at a fixed frequency and then changing the field after around 2-5 min without redispersing the suspension. We observe that structures evolve more quickly than experiments in which we start from a random dispersion, as described previously. As seen in Figure 4, at ν ) 1 kHz and field strength E ) 500 V/m, the particles in a random dispersion coalesce but do not form chains. Increasing the electric field strength by a step change without redispersing the particles results in greater attraction between the particles, but order is not visible because of the rotation on pairs of particles, which disrupts the formation of dipolar chains. Eventually regions with higher concentration of particles densify, forming compact bands. The particles along the periphery of the bands begin to move along the structure, “hopping” from one particle to another and setting the assembly into vortex flow. At higher frequencies however, the contribution from anomalous rotation is weaker, and hence chain break up and vortex flows are slower, thereby enabling the particles to form ordered structures. Figure 5 demonstrates this phenomenon at ν ) 8 kHz and E ) 5000 V/m. Within two minutes, an isolated crystal is observed (Figure 5A). After 15 minutes, the crystal has very slowly devolved and a bandlike structure can be seen developing in Figure 5B. C. Short Range Pair Interactions. To study the near-contact particle interactions (r ∼ 2a, where a is the particle radius), we use optical traps to position two particles in the bulk away from the glass surfaces, in a dilute solution. As described in section II, we repeatedly trap and release them from a fixed initial separation and initial orientation, using blinking optical tweezers. We study two orientations of the pair with respect to the electric field lines, θ0 ) 0° and 30°. Approximately 1000 particle trajectories were obtained for each particle pair, with each trajectory sampling 0.6 s. Experiments were carried out over a range of frequencies from 600 Hz to 6 kHz. In the experiments, the field strength was maintained constant at 7500 V/m.

Figure 5. (A) Crystalline domains for ν ) 8kHz, E0 ) 5000 V/m at time t ) 2 min after application of the electric field. (B) Crystal structure devolves into a bandlike formation after 15 min. The scale bar is 40 µm.

Images of one such trajectory are shown in Figure 6 over a total time of 0.6 s at a field frequency ν ) 600 Hz and field strength E ) 7500 V/m. After being released by the optical traps, the particles approach each other along the field direction, consistent with the expected polarization-induced dipolar interaction. Surprisingly, however, as particles come closer, they rotate out of alignment with the electric field. The rotation continues until the particles reach almost 60° with respect to the field, after which they appear to repel each other, moving to larger separations. The rotation is unexpected, since we anticipate that particles should experience forces that align in the field direction based on the induced dipolar interactions.35,36 Results over many particle trajectories are summarized in Figures 7-9. In Figure 7, we show the distribution of final angles that particle pairs make with respect to the field field lines at the end of the trajectory (0.6 s) for the initial angle θ0 ) 0°. The plot shows that the distribution is significantly wider at the lowest frequency (600 Hz), indicating that the doublet rotation is stronger at 600 Hz compared to higher frequencies at the same field strength. The data at ν ) 2 and 6 kHz are fit using a single Gaussian, while the data for ν ) 600 Hz is fit by the sum of symmetric Gaussian curves at θ ) (20°. The latter is consistent with diffusion of the particles superimposed on a finite drift velocity. Figures 8 and 9 show the average separation between the particles as it evolves with the average absolute angles the pair makes with the electric field lines, for two initial orientations of the doublet with respect to the field and two field frequencies. When the pair is oriented parallel to the field lines (θ0 ) 0°), the particles approach each other rapidly and attain a separation of around 200 nm, which remains constant over the lag time as the particles begin to rotate. In case of the 30° orientation, the particles approach, rotate and move away without reaching a stable separation. A common feature in these experiments is that the (35) Jones, T. B. Electromechanics of Particles; Cambridge University Press: Cambridge, UK, 1995. (36) Hu, C.-I. J.; Barnes, F. S. Radiat. EnViron. Biophys. 1975, 12, 71–76.

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Figure 6. Two particles are held and released by optical tweezers at field frequency ν ) 600 Hz and field strength E ) 7500 V/m. The particles approach when the traps are turned off and rotate around each other to an orientation perpendicular to the electric field lines. The scalebar is 3 µm.

Figure 7. Pair angle with respect to field lines 0.6 s after the traps are turned off. The two particles held in blinking optical tweezers parallel to the electric field lines. The solid lines are fits to the data. Electric field strength ) 7500 V/m.

Figure 9. Mean separation with respect to the average angle between particles. The observations are averaged over 1000 trajectories, each lasting 0.6 s. The electric field strength E0 ) 7500 V/m, and the initial orientation angle is 30°.

Figure 10 also confirms that the self-organization into bandor vortexlike structures are not a result of large scale streaming flows, which were reported earlier by Morgan and Green.37 In their work, Morgan and Green reported flows in which the rotation occurs out of the plane of the electrodes, whereas the particle vortices studied here exhibit rotation in the plane of the electrodes. In the current work, strong streaming flows have been eliminated by spacing the electrodes far apart (2 mm versus the values of 0.025-0.5 mm used in ref 37) and working at the midpoint between the electrodes.

IV. Discussion

Figure 8. Mean separation with respect to the average angle between particles. The observations are averaged over 1000 trajectories, each lasting 0.6 s. The electric field strength E0 ) 7500 V/m, and the initial orientation angle is 0°.

polarization-induced interactions are weaker at 2 kHz,6 and therefore, the particles approach each other more slowly. The rotation, as indicated by the angle distribution, is weaker as well. D. Chain Collapse. A dramatic example of structural transitions driven by the anomalous rotation is shown in Figure 8. At a field strength E ) 7500 V/m, we form chains of particles at a high frequency (ν ) 12 kHz), then rapidly change the frequency to a lower value, passing through the expected order-disorder transition. The chains of particles are observed to suddenly collapse along the field direction as particles rotate around each other when the frequency is changed stepwise to ν ) 1 kHz. Smaller aggregates of triplets and doublets, as well as single particles then repel each other and migrate in the direction orthogonal to the applied field, forming patterns reminiscent of the early stage of particle band formation.

The above experiments demonstrate the connection between the microscopic rotation of particle pairs in AC electric fields at low frequencies, which we study directly using blinking optical traps, and the appearance of a disordered microstructure that disrupts the otherwise ordered structures of 2D dipolar chains and crystals. To explain the rotation of particles out of alignment with the applied electric field, we follow the theory of Hu and co-workers.26 Their mechanism of band formation results from the mutual polarization of particles near the relaxation frequency of the dominant polarization mechanism. Hu et al. observed band or vortex formations similar to those observed here, but in a frequency regime where the double layer polarization is not important (ν ∼ 100 kHz). Nonetheless, the model is applicable in any regime where mutual induction lags the polarization due to the applied electric field, regardless of the polarization mechanism, which we show below. First, consider two spherical particles in an AC electric field, as shown in Figure 11. For simplicity, the polarized double-layer particle entity is treated as a point dipole with a dipole moment (37) Morgan, H.; Green, N. G. AC Electrokinetics: Colloids and Nanoparticles; Research Studies Press, Ltd: Baldock, Hertfordshire, England, 2003.

Colloid Particle Rotation and Microstructure in AC

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µi ) 4π0a3β(ω)Ei

(5)

The torque L on particle 2 is calculated from the cross-product of the real components of the total dipole moment and electric field27,35

L ) R{µtot} × R{Etot}

(6)

where Etot ) E + Ei and µtot ) µ + µi. Since E and µ are parallel, as are Ei and µi,

L ) R{µ} × R{Ei} + R{µi} × R{E}

(7)

Each term is evaluated, with the resulting expression for the total torque on the particle given by

( ar ) sin 2θ(1 - cos 2ψ)δ 3

L ) 6π0R2E02a3

(8)

φ

As seen from eq 8, the lag in polarization causes a torque on each particle due to mutual induction, which is maximum for a phase lag between the particle polarization and applied AC field of Ψ ) 90°. Although R{µ} × R{Ei} and R{µi} × R{E} result in opposing torques, the moment induced by the neighboring particle µi lags the moment induced by the applied field µ, resulting in a net torque on the particles. For force-free particles, doublets will rotate in response to this torque. From Faxe´n’s second law, the torque on particle i is related to its angular velocity Ωi and the velocity field generated by particle j, uj ) a3Ωj × r/r3, Figure 10. Image sequence showing the breakup of chains due to doublet rotation when the field frequency is changed. Chains are formed at 12 kHz, 7500 V/m. A step change in frequency from 12 kHz to 1 kHz occurs at time t ) 0 s. Boxes highlight the nature of the aggregate collapse for three chains. The scale bar is 100 µm. Enhanced movie file depicting change in the field frequency.

1 Li ) 8πa3µ (∇ × uj)i - Ωi 2

[

]

(9)

Solving this coupled set of equations for L1 ) L2 ) L shows that Ω1 ) Ω2 ) Ω, and thus,

Ω)

L 8πa µ(1 + a3/2r3)

(10)

3

In response to the fluid motion induced by the rotation of the particles, the particles will rotate around each other, similar to the behavior shown in Figure 6. From Faxe´n’s first law, the effective force is

Fθ )

6πµa4Ω r2

(11)

Substituting the magnitude of the induced torque leads to

Fθ ) Figure 11. Mutual polarization due to neighboring particles. The field due to dipole 1 induces µi at position 2 in addition to the already present µ.

µ induced by the electric field E ) Eoe-iωt, where ω is the field frequency. The dipole moment µ is given by

µ ) 4π0a3β(ω)E

(3)

where the polarizability is β(ω) ) R(ω)eiΨ. The factor eiΨ accounts for the polarization lagging the electric field by a phase angle Ψ. The dipole moment µ of particle 1 induces a field Ei at the position of particle 2, leading to an additional contribution to the dipole moment µi for the second particle. The field Ei is given by the dipole approximation as35

Ei )

(

1 3(µ · r)r µ - 3 4π0 r5 r

)

and thus, the induced moment due to this field is

(4)

9π0R2E02a2 a 5 sin 2θ(1 - cos 2ψ) 2(1 + a3/2r3) r

()

(12)

To inhibit this rotation, the dipolar force along δθ restoring the particle alignment in the field direction must be greater than Fθ. From eq 2, the force of the dipole-dipole interaction is

( ar ) sin 2θ cos ψ

Fθclip ) 6π0R2E02a2

4

2

(13)

Taking the ratio of the two forces yields

Fθ

) clip

Fθ

3 tan2 ψ a 2(1 + a3/2r3) r

()

(14)

As eq 14 shows, the force due to mutual polarization can become on the same order of magnitude as the dipolar force as the phase lag increases. The double layer polarization phase lag can be calculated by numerically solving the standard electrokinetic model.38-41 Using a recent solution due to Hill et al.,40 we obtain the polarizability β(ω) and its corresponding phase angle Ψ depending on the

12848 Langmuir, Vol. 24, No. 22, 2008

Figure 12. Numerical solutions to the standard electrokinetic model showing the influence of double layer polarization on the phase lag. κa ∼ 16.5, a ) 1.5 × 10-6 m.

double layer thickness, particle surface charge and the field frequency. Figure 12 shows the dependence of phase lag on the frequency range of our microscopic interactions experiment and the zeta potentials. For the limiting case of ζ f 0, the double layer polarization is insignificant and the phase lag is zero. For ζ ) -137.5 mV, the phase lag is large in the range of field frequencies ν ) 100 - 600 Hz, which is approximately the region where the observed doublet rotation is also strong. As the frequency increases, the phase lag decreases, requiring a larger applied field strength to generate rotation of the particles. Furthermore, as the frequency increases, the total polarizability also decreases, leading to a greater decrease in the interaction.6 Notably, increasing the ionic strength also weakens the polarization and should also inhibit doublet rotation. In an assembly of particles, these interactions are more complex because of the mutual polarization by multiple neighbors. Qualitatively, the mutual polarization model agrees with the average trajectories for doublets that are initially at an angle θ0 ) 30°, shown in Figure 8. Particles attract predominantly along the radial direction, with little of the tangential motion that would be expected from the dipolar interactions. Quantitatively, the rotation out of the field direction is stronger than expected, and becomes dominant when the particles are at close separations (r - 2a < 0.2a). This suggests a more complicated interaction in the near-field than the point dipolar theory can capture, perhaps as a result of overlap of the diffuse double layers. Overall, the results of the model and blinking laser tweezer experiments demonstrate the origin of the banded microstructure. (38) Hunter, R. J. Foundations of Colloid Science; Oxford University Press Inc.: New York, 2001. (39) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 258, 56–74. (40) Hill, R. J.; Saville, D. A.; Russel, W. B. J. Colloid Interface Sci. 2003, 263, 478–497. (41) Hill, R. J.; Saville, D. A. Colloids Surf., A 2005, 267, 31–49.

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At high frequencies or ionic strengths, we expect that the torque on particles is not strong enough to prevent particles from chaining and forming ordered structures. However, as we have shown, weak rotation can still persist, leading to a slow migration of particles over long times. As a result, the order may still be disrupted, as shown in Figure 6. At higher field strengths, depending on the field frequency, particle rotation can be strong, causing a disruption of the otherwise ordered structure of dipolar chains and crystals. This suggests that conditions in which the rotation-induced torque exceeds the force restoring particles to alignment in the field direction leads to the formation of the band or vortex microstructure over time, corresponding to the upper order-to-disorder transition in Figure 3.

V. Conclusions We investigated the microstructure of polarizable particles in AC electric fields as a function of field strength and frequency. We reported three key findings: (1) The order-disorder transition as a function of field strength and frequency exhibits an ordered regime consisting of dipolar chains and crystals between two disordered regimes at low and high field strengths. While the transition to order at low field strengths is consistent with the polarization-induced interactions becoming larger than the thermal energy of the particles, the re-entrant disordered regime at high field strengths is marked by large-scale rotating structures. (2) Using blinking laser tweezers to characterize the interaction of particle pairs, we demonstrated that particle pair rotation opposite of alignment in the field direction is the microscopic phenomenon driving the observed suspension behavior. The pair particle rotation was observed to decrease with increasing frequency, thus requiring higher field strengths, in agreement with the frequency and field strength dependence of the second orderto-disorder transition. (3) The particle rotation is consistent with a mechanism based on the mutual polarization of particles and is an effect of the phase lag between the polarization and the applied and mutually induced field. The connection we made between particle interactions and suspension microstructure answers a long-standing debate regarding the mechanism underlying the band structures in experiments employing parallel electrode geometries.23-26 Acknowledgment. We acknowledge helpful discussions with E. Kaler, O. Velev, S. Gangwal, T. Squires, N. Wagner, J. Brady, J. Swan, J. McMullan, and A. Grillet. We thank Prof. R. J. Hill for lending us his codes for numerical solutions to the standard electrokinetic model. Funding of this work was provided by the National Science Foundation (NIRT CBET-0506701) and Sandia National Laboratories (contract no. 678286). LA802225U