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Evidence of Multiple Electrohydrodynamic Forces Acting on a Colloidal Particle near an Electrode Due to an. Alternating Current Electric Field. Jeffrey A. Fagan,* ...
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Evidence of Multiple Electrohydrodynamic Forces Acting on a Colloidal Particle near an Electrode Due to an Alternating Current Electric Field Jeffrey A. Fagan,* Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received July 29, 2004. In Final Form: November 23, 2004 Total internal reflection microscopy was used to monitor the elevation of 4-7.5 µm diameter particles near an electrode in response to an oscillating electric field with amplitude up to 8.5 kV/m. The media were 0.15 mM electrolyte solutions of HNO3, NaHCO3, and KOH, and the frequency band was 40 Hz to 10 kHz. Polystyrene-sulfonate particles were used in bicarbonate and KOH solutions, while polystyrene-amine particles were used in nitric acid. At frequencies less than 500 Hz, large oscillations in elevation at the driving frequency with small superimposed Brownian excursions were observed. At frequencies above 1 kHz, deterministic oscillations in elevation were negligible compared to Brownian fluctuations, which allowed transformation of histograms of elevations into potential energy profiles. The ac field drew the particle closer on average to the electrode in KOH solutions (compared to the no-field average elevation) and the field pushed the particle farther from the electrode in NaHCO3. In HNO3 a reversal of average height was observed at a frequency of 300 Hz at 1.7 kV/m with the particle being drawn closer to the electrode at low frequencies and being pushed away at higher frequencies. The reversal reflects two different electrohydrodynamic mechanisms. Analysis of the data at a high frequency (10 kHz) revealed a net force that was attractive in KOH and repulsive in HNO3. This net force scaled with E2ω-1, where E is the amplitude and ω is the frequency.

Introduction The forces that an alternating electric field exerts on a colloidal particle near an electrode are of interest for use in microfluidic devices, displays, and other optical applications or biosensors.1-4 The mechanism(s) producing these forces are not completely understood. Observations of both aggregation and disaggregation of particle doublets and ensembles of particles at different frequencies, and in different electrolytes, indicate that the phenomena are complex.5-14 Theories advanced to explain the observed behavior depend on the frequency, particle size, electrolyte conductivity, electric field distribution, and whether the current passes through the capacitance of the electrode’s double layer or through an electrochemical reaction.8,9,13-19 * To whom correspondence should be addressed. (1) Gong, T.; Wu, D. T.; Marr, D. M. Langmuir 2002, 18, 1006410067. (2) Joanopoulos, J. D. Nature 2001, 414, 257-258. (3) Gleason, N. J.; Nodes, C. J.; Higman, E. M.; Guckert, N.; Askay, I. A.; Schwarzbauer, J. E.; Carbeck, J. D. Langmuir 2003, 19, 513-518. (4) Brisson, V.; Tilton, R. D. Biotechnol. Bioeng. 2002, 77, 290-295. (5) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408-3413. (6) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334-6336. (7) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706-708. (8) Trau, M.; Saville, D. A.; Askay, I. A. Langmuir 1997, 13, 63756381. (9) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature (London) 1997, 386, 57-59. (10) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Nature 2000, 404, 56-59. (11) Kim, J.; Garoff, S.; Anderson, J. A.; Sides, P. J. Langmuir 2002, 18, 5387-5391. (12) Nadal, F.; Argoul, F.; Kestener, P.; Pouligny, B.; Ybert, C.; Adjari, A. Eur. Phys. J. E 2002, 9, 387-399. (13) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B.; Adjari, A. Phys. Rev. E 2002, 061409, 1-7. (14) Ristenpart, W. D.; Askay, I. A.; Saville, D. A. Phys. Rev. E 2004, 021405, 1-8. (15) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2002, 18, 78107820. (16) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2004, 20, 48234834.

Investigators extending at least back to Yeh et al.9 have suggested that an electroosmotic mechanism due to the interaction of lateral components of the electric field around a particle with space charge on the electrode is responsible for this motion. Recently Ristenpart et al.14 reported a scaling analysis of this mechanism applicable above approximately 500 Hz, predicting dependence of the induced force on field strength and frequency. The source of momentum was the product of diffuse layer charge q(t) (due to polarization of the electrode’s double layer) with lateral electric field components Er of a dipole (due to the obstruction of current by the dielectric particle). This mechanism requires no electrode reaction and accounts for both a power of 2 on the applied field and a power of -1 on frequency. This model, however, overestimates the strength of the lateral components of the electric field because the dipole approximation of the field around the particle ignores the electrode’s termination of the disturbance in the electric field.20 Nevertheless, the model provides testable predictions of the dependence of force on field and frequency above 500 Hz. We have been exploring the vertical response of a single particle in an ac field near an electrode in order to study the forces without the complications of multiparticle effects.15-18,21 The arrangement appears in Figure 1a. We apply an ac field E(t) normal to the electrode and record data that allows determination of h(t) where h(t) is of order 100 nm. At zero applied field, the particle bobs up and down within a potential well defined by gravitation and electrostatic repulsion between the particle and the (17) Sides, P. J. Langmuir 2001, 17, 5791-5800. (18) Sides, P. J. Langmuir 2003, 19, 2745-2751. (19) Bonnefont, A.; Argoul, F.; Bazant, M. Z. J. Electroanal. Chem. 2001, 500, 52-61. (20) Sides, P. J.; Tobias, C. W. J. Electrochem. Soc. 1980, 127, 288291. (21) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2003, 19, 66276632.

10.1021/la048076m CCC: $30.25 © 2005 American Chemical Society Published on Web 01/26/2005

Forces on a Colloidal Particle

Figure 1. Schematic of the apparatus: (A) A colloidal particle levitates over the bottom electrode. The figure shows the overall direction of the applied field E and the definition of h(t). (B) A flow cell is formed from parallel planar electrodes separated by a nonconducting spacer and optically connected to a dovetail prism. ac voltage is applied across the two electrodes. The linearly polarized beam from a 17 mW HeNe laser is totally internally reflected at the bottom electrode-solution interface, producing an evanescent wave within the fluid phase. A colloidal particle approaching the electrode scatters the evanescent wave. The photomultiplier tube records the intensity. In the diagram the particle is not to scale; the electrode spacing is approximately 200+ times the average particle diameter.

electrode. This investigation showed that the average gap between a single spherical colloidal particle and a planar ITO electrode changed when the field was applied, thus a net force acted on the particle.15,16 At frequencies less than a few hundred hertz, the force was electrolyte dependent and correlated with the longitudinal (i.e., parallel to the overall field) motion observed for particle doublets and swarms. When the average particle height was depressed, particles separated, and when the average particle height increased, the particles aggregated.11,15 Comparison of experimental data and numerical calculations for frequencies around 100 Hz indicated the presence of an additional E2 force on the particle, and not merely the combined effects of colloidal forces, Brownian motion, and an O(E) interaction between the normal component of the electric field and charge on the particle itself.16,21 This work16 led to a hypothesis that the additional force depends on faradaic reaction(s) at the electrode for frequencies less than a few hundred hertz. Electrode reactions alter the distribution of the current underneath the colloidal particles from a primary current distribution20 (characterized by a uniform electrical potential on the electrode) to a current distribution under the particle that is more uniform.22 An electrode participating in a faradaic reaction does not terminate Er on its surface. Under this condition, the lateral electric field at the electrode surface interacts with the electrode’s diffuse layer charge and drives an electroosmotic flow underneath the particle. The mechanism depended on a theoretically expected and experimentally found phase angle (between the current and the apparent zeta potential of the electrode) that was different from the π/2 phase angle associated with electrolyte resistance in series with electrode capacitance. This electroosmotic flow accounted for the observed behavior at frequencies of approximately 200 Hz and lower.16 This mechanism, however, cannot contribute to particle ordering at frequencies >1 kHz where the double layer capacitance short circuits the faradaic impedance and where others have shown the existence of attractive forces.1,7-9,12,14 This context prompted two inquiries that yielded the results reported here. First, we explored frequencies from 40 Hz to 10 kHz in single-particle

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experiments to capture effects both where faradaic reactions can occur and where faradaic reactions are unimportant. Second, three different electrolytes (KOH, NaHCO3, and HNO3) at the same concentration were used to test for electrolyte dependence. This work documents the existence of net ac electric-field-driven forces on a single particle at frequencies both above and below a kilohertz, and the dependence of those forces on frequency, voltage, and, for the first time, electrolyte. Terms must be defined at the outset of describing such complex phenomena. The acronym “EK” (i.e., electrokinetic) will refer to the interaction of the imposed electric field E with the charge around the particle itself. This interaction is proportional to E and is most important at the lowest frequencies. The term “electrohydrodynamic” will mean any flow where the source of momentum is proportional to E2. The term FCEO (faradaically coupled electroosmosis) will refer to an electroosmotic interaction between Er(t) and the diffuse layer charge of the electrode, q(t), where the phase angle between the apparent zeta potential of the electrode and current is different from π/2. This mechanism will be of primary importance when the frequency is greater than a few tens of hertz and less than about 200 Hz. We will use ICEO (induced charge electroosmosis), a term introduced by Bazant and Squires23 for an electrohydrodynamic flow due to induced charge, to refer to flow resulting from the interaction of Er with q(t), where the electrode is purely capacitive. This mechanism is of primary importance above approximately 500 Hz. Experimental Section TIRM of Particles in Equilibrium. Using total internal reflection microscopy24 (TIRM), one can measure the timedependent height of a colloidal particle bobbing stochastically in equilibrium above a solid surface. A diagram of the instrument appears in Figure 1b. A HeNe laser beam totally reflects from the electrode-solution interface of the flow cell. The colloidal particle near the electrode scatters evanescent radiation present in the solution adjacent to a solid boundary. A photomultiplier tube captures the scattered light digitally in photon counting mode. The intensity of the scattered light for a spherical particle decreases exponentially with height above the electrode-solution interface

I(t) ) IB + I0e-βh

(1)

where I(t) is the scattered intensity, IB is the background scattering, β-1 is the decay length set by the physics, and I0 + IB is the scattered intensity of the sphere at h ) 0. Both IB and I0 are independently measurable, which allows eq 1 to be solved for the time-dependent height of the particle, h(t). Given a large set of independent intensity measurements, one calculates the probability distribution p(h) of the particle’s sampled elevations. Once p(h) is known for a Brownian particle, Boltzmann’s equation

p(h) ) Ae-φ(h)/kBT

(2)

(where φ(h) is the potential energy (PE), kB is Boltzmann’s constant, T is temperature, and A is a normalizing constant) can be applied to determine the PE as a function of height. Since the PE is related to the force on a particle by F(h) ) -∂φ/∂h, TIRM can be used either to track the time-dependent height of a particle or, for Brownian particles, to measure the forces acting on it. TIRM of Electrically Driven Particle Motion. The data recorded for electrically driven nonequilibrium experiments was the same except that an analog photocurrent signal from the photomultiplier tube (PMT) also was recorded by an oscilloscope to obtain information about phase angles among the applied (23) Bazant, M. Z.; Squires, T. M. Phys. Rev. Lett. 2004, 92, 066101-

(22) Dukovic, J.; Tobias, C. W. J. Electrochem. Soc. 1987, 134, 33143.

4. (24) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93-125.

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Figure 2. (A) Experimental probability histogram for a polystyrene-aliphatic amine colloidal particle in 0.15 mM nitric acid. The most likely height, which represents the elevation of mechanical equilibrium, is hm ) 170.6 nm. The particle samples elevations above and below hm due to Brownian motion. The probability histogram represents 196 608 data points. (B) The calculated PE profile for the probability histogram presented in Figure 2A. The potential energy of the particle increases on either side of hm due to electrostatic repulsion from the electrode and gravity pulling toward the electrode. As the slope of gravitation is less than the slope due to electrostatic repulsion, the probability distribution and thus the average height, 〈h〉 ) 236.9 nm, is skewed away from the electrode. Also included is a three-parameter theoretical curve where the forces due to electrostatic repulsion and gravity were fit from the data assuming linear superposition. potential, induced current, and particle motion. Phase angles between the current and the particle response were calculated by measuring the time difference between the maximum in the voltage drop across a resistor in series with the cell and the maximum in recorded intensity from the PMT, which corresponds to the minimum particle height.25 Several different methods were employed to interpret the scattered intensities recorded from the PMT as a function of time. At frequencies ∼40 Hz. The net downward force on the particle decayed as the frequency rose beyond 200 Hz but was still appreciable up to 1000 Hz. In NaHCO3 a net force pushed the particle away from the electrode at all frequencies, to the point where the particle spent most of the cycle out of the evanescent wave and no meaningful data could be taken below 200 Hz at this field strength. Both of these behaviors were different from the functionality observed in Figures 4 and 5 for HNO3, where the net force on the particle was downward between ∼40 and 300 Hz but upward above 300 Hz. Although not shown in Figure 5 because the data were not taken at the same applied voltage, experiments in nitric acid at all applied voltages revealed that 〈h〉 approached the equilibrium value as the frequency approached 10 kHz. The net force on the particle also depended on the magnitude of the electric field, which is demonstrated for KOH in Figure 6. Below 100 Hz, 〈h〉 at 2580 V/m was higher than 〈h〉 at 1610 V/m, but at higher frequencies the average particle heights were reversed. This reversal as a function of field strength was also observed in the other two electrolytes. Figure 7 contains PE profiles for two separate particles in a 10 kHz electric field with the superficial electric field being the varied parameter. For KOH (Figure 7A), increasing the field strength steepened the outer arm of the PE profile, which means that a net force pulled the particle closer to the electrode. The PE profiles for a particle in HNO3 (Figure 7B) reveal that increasing the amplitude of the electric field in nitric acid pushed the particle away from the electrode at 10 kHz, opposite from the behavior in KOH. The corresponding 〈h〉 in KOH and HNO3 at these frequencies is shown in Figure 8; 〈h〉 was depressed in KOH but increased in HNO3 as the electric field increased. The results shown in Figures 7 and 8 were reproducible; similar experiments with bicarbonate as the electrolyte were not reproducible at 10 kHz. At frequencies below 8 kHz increasing the amplitude of the electric field always pushed the particle away, but by 10 kHz the force was sufficiently small that other as yet uncontrolled effects were present in bicarbonate. The effect of frequency on the measured apparent PE profile is shown in Figure 9 for a particle in NaHCO3 with the oscillating electrical potential kept constant. As the frequency of oscillation increased, the PE profiles approached the equilibrium profile, which means that the net force levitating the particle above its equilibrium height decreased. Thus the apparent additional long-range net repulsive force acting on the particle weakened as the frequency increased. The phase angle between the particle motion and the electric field in the solution appears in Figure 10. In each of the three electrolytes, KOH, NaHCO3, and HNO3, the difference between the measured phase angle and the expected 90° phase angle becomes statistically zero above a few hundred hertz. Discussion Forces and Frequency. Figures 4, 5, and 6 are strong evidence of force acting with net direction on particles near surfaces during ac polarization. The magnitude and sign of the net force depend on frequency, field strength, and electrolyte. Figure 4 shows that a net force on the particle exists both at relatively low frequency, where the amplitude of the EK motion is discernible, and at high

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Figure 4. Particle trajectories in nitric acid at a 4 Vpp (peak to peak) applied potential and various frequencies. The trajectories are shown in phase time to allow for easy comparison. The average height of the particle is smaller than its equilibrium value at 104 and 210 Hz but is increased relative to the equilibrium height for 275 Hz and higher despite the substantially larger amplitude of oscillation at the lower frequencies. The trend with frequency is not perfectly monotonic but the overall reversal of height with respect to the equilibrium height is unmistakable.

Figure 5. Relative average height as a function of frequency for a particle in each of the three electrolytes for an apparent electric field of 2 kV/m. In KOH, application of the electric field at frequencies above 40 Hz pulls the particle toward the electrode, whereas in NaHCO3 the 〈h〉 is always above the equilibrium height, and in HNO3 the 〈h〉 reverses at ∼350 Hz. The clear electrolyte dependence at frequencies above 500 Hz is not accounted for in the ICEO mechanism as formulated by Ristenpart et al.14

frequency where the amplitude of the particle motion is barely perceptible. Figure 5 demonstrates a variety of behavior at constant field strength. The particle is biased below its zero field average height in KOH over most of the frequency range, while the particle in bicarbonate is biased in the opposite direction. Figures 4 and 5 show a surprising effect of HNO3, in which the particle reverses its position relative to its zero field height when the frequency is raised from 100 to 1000 Hz. This dependence on frequency suggests that the mechanism generating the force changes when the frequency is increased from low frequency to high frequency. In prior

work we described two mechanisms that exist at low frequencies. First, the lower mobility of a particle for motion toward a surface and its higher mobility for motion away from the surface cause any particle to drift positively from its equilibrium height during ac excitation. Called the EK effect, it controls the average height of the particle at sufficiently low frequency because in that limit the particle goes far from the surface on its outward swing. Second, a faradaically coupled electroosmotic force (FCEO) at frequencies below about 200 Hz accounts for the increased average 〈h〉 in bicarbonate and the decreased 〈h〉 in the other two electrolytes at low frequency. This force, which reflects an interaction between the charge of the diffuse layer and lateral electric field components arising from redistribution of current under the particle, was the first mechanism that explained the decrease of 〈h〉 observed in KOH.16 FCEO, however, does not account for the existence of a force at higher frequencies where the double layer capacitance short circuits the faradaic channel. Others have suggested that an induced charge electroosmotic flow (ICEO) mechanism exists at frequencies higher than about 500 Hz.14 The superposition of low-, intermediate-, and highfrequency mechanisms can explain the results of Figure 5. The data for KOH cover the complete range of effects. The EK effect raises the particle above its equilibrium height below 60 Hz; FCEO pulls down on the particle below 200 Hz and the ICEO force pulls down on the particle above 500 Hz. The data for bicarbonate likewise contain all the effects. Both the EK effect and FCEO force raise the particle at low frequencies, and the ICEO force sustains the positive deviation from the zero field height when the frequency is above 500 Hz. HNO3 provides the clearest example of a transition between intermediate frequency effects and high-frequency effects. A strong downward force from the FCEO mechanism dominates the particle at frequencies at least as high as 40 Hz and below 200 Hz. As increasing frequency short circuits the faradaic chan-

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Figure 6. Relative average height of a particle in KOH as a function of frequency at two different applied apparent electric fields. The increased field leads to both increased order E effects at frequencies below 200 Hz, as well as increasing the order E2 depression in elevation observed at frequencies above 200 Hz. The magnitude and direction of the change with field strength are not uniform with frequency. Similar increases in the magnitude of the change from equilibrium are found in both HNO3 and NaHCO3. The lines are guides for the eye.

Figure 7. (A) Potential energy profiles for an ∼6.7 µm polystyrene-sulfonate particle in 0.15 mM KOH for four different applied 10 kHz electrical potentials and equilibrium. Stronger fields result in an increase in the apparent additional downward force on the particle. The displacement of the PE minimum along the x-axis as the field increases is a direct consequence of a force pulling down on the particle. The fanning out of the electrostatic arm of the PE curve for the various field strengths is an artifact of forcing the PE to be zero at its minimum in all cases. (B) Potential energy profiles for an approximately 4.4 µm diameter polystyrene sphere in 0.15 mM nitric acid (pH ∼ 3.6) and exposed to a 10 kHz applied alternating electric potential. As the magnitude of the applied potential increases, the particle experiences additional upwardly directed force. This particle effectively became neutraly buoyant at an apparent maximum field of ∼7800 V/m.

nel, the ICEO mechanism takes over and suspends the particle on average over its equilibrium height. Experiments in nitric acid at other field strengths, not shown, verified that the particle rose out of the evanescent wave at 25 Hz, in accord with the EK effect, and approached the equilibrium value at high frequency. These arguments remain speculative because no available quantitative theory reveals dependence of ICEO on electrolyte, but two facts deserve attention. In all three electrolytes, a transition of behavior occurs between 200 and 500 Hz, precisely where the capacitive channel short circuits the faradaic channel. Second, in nitric acid and bicarbonate, where ICEO leads to a positive deviation of

〈h〉 from the zero field case, the cation mobility exceeds the anion mobility while the opposite is true for KOH. Sides18 pointed out this connection between the direction of electrohydrodynamic flow and ion mobilities, but for a mechanism different from ICEO. Thus a transition occurs at the frequency range where one expects it and there are differences of physical properties related to the motion of ions in the solution. Figure 6 shows that conclusions about the frequency ranges where the three different forces dominate the behavior are contingent on field strength. In KOH, raising the field strength extends the transition between the EK and FCEO effects from 50 Hz to approximately 100 Hz.

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Figure 8. Relative average heights of the particles shown in Figure 7 as a function of the apparent maximum electric field at 10 kHz. The average height decreases in KOH and increases in HNO3 with increasing field magnitude. An equal change in relative height does not indicate an equal change in the additional net force on the particle. A larger net force is required to pull the particle closer than to push it away by an equal distance.

Increasing the electric field also extends appreciable effects of the net force on the particle from 1 to 2 kHz. This pattern was also observed in the two other electrolytes. Dependence of the High-Frequency Force on the Electric Field and Frequency. We analyze the apparent PE profiles for two different particles exposed to a 10 kHz electric field in KOH and HNO3 given previously in Figure 7. Assuming linear superposition of the forces acting on the particle, we removed the contribution due to gravity from the measured profiles. The result of subtracting the PE contribution due to gravity is shown in Figure 11 for the 10 kHz KOH data shown in Figure 7A. The equilibrium PE is zero at large gaps when gravity is subtracted, as expected; however the remaining PE curves all have a positive slope, and the apparent additional force on the

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particle is approximately linear with particle height over a substantial range of elevations. The results of subtracting the gravity contribution to the HNO3 profiles (not shown) from Figure 7B are similar; however the remaining linear slope to the apparent PE profiles is negative. The residual “unknown” force on the particle in either electrolyte is the slope of the PE curve, due to the definition of F(h) ) - ∂φ/∂h. Using this relation as a basis for plotting the apparent additional force due to the electric field as a function of the apparent maximum field, one obtains Figure 12. Best-fit power law curves to the data of both electrolytes yield similar values for the exponents n on En, 1.77 for KOH and 1.68 for HNO3. Both of these values are in reasonable agreement with the expected value of n ) 2 for an order E2 mechanism. This residual force is substantially smaller in HNO3 than in KOH. Additional experiments have indicated that particle size is an important parameter in determining the magnitude of the induced force, and the diameter of the particle in the HNO3 experiment was measurably smaller than the particle used in the hydroxide solution, but this does not account for the factor of 10 difference in magnitude at any value of E. Additional experimental data (not shown) at other frequencies (primarily 1 and 2 kHz) yielded best-fit values for the exponent similar in magnitude to those observed at 10 kHz. The average value and standard deviation for the exponent, for the six series in which at least five different voltages were applied, was 2.01 ( 0.45. Analogously to the above, the gravity contribution was also removed from the PE profiles presented in Figure 9, in which the varied parameter was the frequency of the electric field. The slopes of the remaining apparent force, from the approximately linear section of the remaining data, are plotted for NaHCO3 over a frequency range of 800 Hz to 10 kHz in Figure 13. The best-fit exponent to the NaHCO3 data versus the reciprocal of frequency is 0.81. Data points are also shown for particles in HNO3, the best-fit exponent for this particle was 1.02. Data for KOH also showed a dependence on the reciprocal of frequency.

Figure 9. Apparent potential energy profiles for a ∼7.5 µm polystyrene-sulfonate particle in 0.15 mM NaHCO3 as a function of frequency. As the frequency increases, the profiles approach the equilibrium profile.

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Figure 10. Phase angles between particle motion and the measured current in an in-series resistor. As the frequency increases above several hundred hertz, the phase angle becomes immeasurably different from the 90° difference expected between forces in phase with the electric field in the solution and the particle motion. In a previous publication,16 we demonstrated that at low frequencies, phase angles differing from 90° are likely indicators of an electrode reaction driven E2 force. These data were taken for an applied potential difference of 5 Vpp, or ∼2 kV/m maximum apparent electric field.

Figure 11. Potential energy profiles from Figure 7A with the gravitational contribution removed. The lines are best-fit curves by a four-parameter function, y ) Ae-Bh + Ch + D. This combination of an exponential and linear functionality is a common feature of equilibrium potential energy profiles due to DLVO type forces. The induced force causes the particle to appear heavier with increasing field in KOH. The listed V/m is the superficial maximum electric field.

These results for dependence of the ICEO on E2 (Figure 12) agree with the findings of Ristenpart et al.14 Likewise our measured functionalities of 1/f 0.81 and 1/f 1.02 in NaHCO3 and HNO3 (Figure 13) respectively agree with their predicted frequency dependence. Ristenpart et al. however did not experiment with different electrolytes and their scaling analysis does not account for possible electrolyte dependence of the force. Also there might be

effects still to be uncovered, such as slight dependence of the surface charge on the solid as protons or hydroxyl ions are moved in and out of the double layer. The next step in understanding these complex phenomena is to undertake a detailed numerical calculation of the forces arising from the interaction of an imposed electric field with an electrode on which a dielectric particle rests. These calculations are currently being performed.

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Figure 12. log-log plot of the apparent vertical force due to application of the 10 kHz electric field versus the apparent field strength for the two particles in Figure 7. In the 0.15 mM KOH solution (from Figure 7A) the apparent force is directed toward the plate and is substantially larger in magnitude than the repulsive force measured in the HNO3 experiment (Figure 7B). The best power law fit (minimization of squared error) curves indicate that the change in slopes is approximately proportional to E2; the optimal fit for KOH was F ∝ E1.77 and for HNO3 was F ∝ E1.68. The approximate diameters of the two particles were 6.7 and 4.4 µm for KOH and HNO3 respectively.

Figure 13. Apparent electric field induced force on a particle in 0.15 mM NaHCO3 and 0.15 mM HNO3 as a function of frequency for an apparent ∼2.5 kV/m max applied electric field. The data points for NaHCO3 are the slope values from four-parameter (y ) Ae-Bh + Ch + D) best fits to the apparent PE profiles presented in Figure 9 with the gravitational force subtracted. The HNO3 apparent PE profiles were similarly analyzed. The solid lines are the best power law fits to the apparent force in each electrolyte; the exponent of the line is 0.81 in NaHCO3 and 1.02 in HNO3. Both are approximately linear with inverse frequency.

Conclusion New experimental evidence shows that single colloidal particles near an electrode and exposed to an ac electric field at frequencies between 500 Hz and 10 kHz experience an O(E2) force generated by the application of the electric

field. The force does not require faradaic reactions. The dependence on the square of the applied field and the dependence on the reciprocal of frequency above 1 kHz are in agreement with a recently published scaling analysis. The force was shown to be electrolyte dependent in its effect, which was not part of the analysis. This force

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Figure 14. Amplitude of EK-induced oscillation as a function of the maximum electric field strength for a 6 µm diameter particle with a zeta potential of -100 mV at a frequency of 1 kHz. Also shown is the average diffusion length for the same size particle in a 1 ms interval, assuming that the initial height of the particle is 300 nm. The ratio of the particles’ excursion amplitude to the diffusion length is unity when the applied field amplitude is between 600 and 700 V/m in this case.

is different in cause from a previously observed E2 force present at lower frequencies. We suggest that a natural transition between the two forces could occur around ∼200-500 Hz, as the extinction of the faradaic current path through the electrode with increasing frequency changes the distribution of current on the electrode in the vicinity of the particle. Acknowledgment. The National Science Foundation Grants CTS-00089875, and CTS-0338089 supported this work. Appendix Validity of the Quasi-Equilibrium Analysis at Frequencies >1 kHz. A quasi-equilibrium approach was used to investigate the net forces on the particle induced by the ac field for frequencies greater than a few hundred hertz; i.e., the intensity signal from TIRM was processed as equilibrium data. Two conditions must be imposed on this method. First, the stochastic component must be primarily responsible for the instantaneous particle motion for the elevation sampling to be essentially Brownian. Second, the oscillation of the particle must not significantly distort the measured value for 〈h〉 over each measurement interval from its true value. To demonstrate when the first constraint is satisfied, we first estimate the total amplitude of oscillation due to the electric field at any given field strength, particle height, and frequency. Multiplying the height-corrected electrophoretic force on the particle with the appropriate mobility and time period

Amp Osc ∼

1 F (h,〈|E(t)|〉)(R(h)/6πηa) (A.1) 2f EK

gives a reasonable estimate. In eq A.1, f is the frequency of the field, R(h) is a height dependent correction term to the infinite fluid mobility, 1/6πηa, and FEK(h,〈E(t)〉) is the electrophoretic force caused by a static electric field whose magnitude equals the root mean square (rms) electric field,

〈|E(t)|〉. The O(E) force, assuming both a thin double layer, and a static electrophoretic mobility, is defined as

FEK(h,t) ) σ(h) 6π0aζsphereE(t)

(A.2)

where ζsphere is the zeta potential of the particle and σ(h) is a correction factor due to the differences in flow patterns between sedimenting and electrophoretically driven spheres. Both R(h) and σ(h) are defined in refs 15 and 22. In Figure 14, the amplitude of particle motion from eq A.1 is plotted versus field strength for three average particle heights at a frequency of 1 kHz, for a particle with a zeta potential of -100 mV. The amplitude of particle motion is 5 nm when 〈h〉 is 200 nm, f ) 1 kHz, and ζ ) -100 mV. An intuitive first requirement for use of the equilibrium analysis is that the amplitude of driven oscillation should be less than the diffusion length over the sampling interval. The diffusion length for the particle is Ldiffusion ∼(4D(h)τ)1/2, where D(h) ) R(h)D∞ and τ is the time interval, given the Stokes-Einstein value for the particle diffusion coefficient, D∞ ) kBT/6πηa. At 1 kHz, for an 1865 V/m electric field and a sampling interval, τ, of 1 ms the ratio of the O(E) oscillation to the diffusion length, R ) Amp Osc/(4Dτ)1/2, is ∼2.75. This value is greater than unity, but it is not clear where the line between too large and small enough lies. To determine whether the analyses used in this report were valid, numerical calculations were performed where the average motion of the particle was calculated for the addition of only the oscillatory O(E) force to the equilibrium DLVO forces due to electrostatic repulsion, van der Waals force, and gravity.20 The calculated PE profile for a 6 µm diameter particle exposed to a case where R equals 5 (800 Hz, 2985 V/m electric field) is compared to the equilibrium PE profile in Figure 15. The two profiles are in reasonable agreement, indicating that the EK oscillations, in this case of ∼60 nm, do not substantially alter the measured PE profile at a ratio of 5. The typical value for this ratio for the data presented in this paper is