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Langmuir 2006, 22, 9846-9852
Mechanism of Rectified Lateral Motion of Particles near Electrodes in Alternating Electric Fields below 1 kHz Jeffrey A. Fagan,*,† Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213 ReceiVed April 4, 2006. In Final Form: July 11, 2006 A rectified electroosmotic flow mechanism and its expression in a quantitative model account for the net lateral motion of colloidal particles above a uniform planar electrode in an alternating electric field that drives a faradaic reaction on the electrode surface. Specific comparison to published particle doublet trajectories at 100 Hz in sodium hydroxide and sodium bicarbonate electrolytes demonstrates that the model quantitatively agrees with the experimental doublet trajectories when only independently measurable parameters are employed. This model reproduces the experimental signatures of the published particle pair motion at 100 Hertz: dependence of the direction of motion on the electrolyte, order of magnitude of the interparticle velocity, invariance of the lateral motion to changes in the particle zeta potential, and observed steady separation between particles that otherwise tend to aggregate. The model is expected to apply up to approximately 1 kHz, at which essentially all of the alternating current flows through the double-layer capacitance and not the faradaic reaction.
Introduction The electrically driven motion of colloidal particles along the surface of a planar electrode has attracted substantial scientific interest as a “gentle” method for particle manipulation. The mechanisms responsible for the net motion have been difficult to discover; theories and models have not explained all observations of frequency and electrolyte dependence reported in the literature.1-14 Recent work15,16 suggests that no single mechanism is responsible for the net motion of particles over all frequencies; instead, different phenomena are present in each frequency range. At zero frequency, Solomentsev et al.17-19 demonstrated that electroosmotic flow due to the interaction of the electric field with the diffuse charge in the particle’s double layer drives the separation or aggregation of particles dependent upon the sign of the electric field and the zeta potential of the particles. * Corresponding author. E-mail:
[email protected]. † Current address: National Institute of Standards and Technology: Polymers Division, Gaithersburg, Maryland 20899. (1) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408-3413. (2) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334-6336. (3) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706-708. (4) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 6375-6381. (5) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature (London) 1997, 386, 57-59. (6) Gong, T.; Marr, D. W. M. Langmuir 2001, 17, 2301-2304. (7) Sides, P. J. Langmuir 2001, 17, 5791-5800. (8) Kim, J.; Guelcher, S. A.; Garoff, S.; Anderson, J. A. AdV. Colloid Interface Sci. 2002, 96, 131-142. (9) Kim, J.; Garoff, S.; Anderson, J. A.; Sides, P. J. Langmuir 2002, 18, 53875391. (10) Brisson, V.; Tilton, R. D. Biotechnol. Bioeng. 2002, 77, 290-295. (11) Zhou, H.; Preston, M. A.; Tilton, R. D.; White, L. R. J. Colloid Interface Sci. 2005, 285, 845-856. (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 2004, 20, 4823-4834. (16) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2005, 21, 1784-1794. (17) Solomentsev, Y.; Bohmer, M.; Anderson, J. L. Langmuir 1997, 13, 60586068. (18) Guelcher, S. A.; Solomentsev, Y. E.; Anderson, J. L. Powder Technol. 2000, 110, 90-97. (19) Solomentsev, Y. U.; Guelcher, S. A.; Bevan, M.; Anderson, J. L. Langmuir 2000, 16, 9208-9216.
Figure 1. Schematic of the simulated two-particle geometry. The two identical particles are levitated an equal time-dependent distance, h(t), from the surface and are separated by the distance R.
Alternatively, for frequencies above 1 kHz the induced charge electroosmotic flow proposed by Yeh et al.5 and developed by Saville and co-workers4,14 based on the time-dependent polarization of the electrode’s double layer can account for the experimental observations of lateral motion at those frequencies. However, this model predicts aggregation under all circumstances. In this contribution, we focus on the intermediate frequency band between dc and 1 kHz where particles aggregate or separate depending on the electrolyte, which is not explained by either the dc electroosmosis mechanism or by the high-frequency induced-charge mechanism. We demonstrate that the combination of two hydrodynamic mechanisms and one induced electrostatic mechanism, acting simultaneously, explains the experimental observations. The combination of the two mechanisms rectifies the otherwise reversible motion from each of the two hydrodynamic mechanisms; in the absence of this collaborative effect, no net lateral motion of particle pairs due to hydrodynamic forces is produced. First Hydrodynamic Mechanism: Electrokinetic (EK) Flow. Consider the particles of Figure 1. An electric field applied normal to a planar electrode acts on the diffuse cloud of counterions surrounding any given particle to create electroosmotic flow along the particle’s surface. Depending on the direction of the applied electric field and the sign of charge borne by the particle’s cloud, this electroosmotic flow is directed either away from or toward the electrode. This flow pumps fluid along the surface of the electrode toward or away from the particle, respectively. Other particles become entrained in this flow and either aggregate or separate, depending on the direction of flow along the surface of the electrode. We refer to this as the electrokinetic (EK) mechanism of lateral motion; it explains the
10.1021/la060899j CCC: $33.50 © 2006 American Chemical Society Published on Web 10/25/2006
Mechanism of Rectified Lateral Motion of Particles
observations of lateral motion made with dc electric fields.17-20 For ac electric fields, the EK flow about the particle is modulated in phase with the applied field, with neighboring particles experiencing both attraction and repulsion over each cycle of the electric field. However, this O(E) modulation of the EK flow is symmetric, thus no net aggregation or separation occurs. Second Hydrodynamic Mechanism: Faradaically Coupled Electroosmotic Flow (FCEO). The second mechanism collaboratively responsible for the net lateral motion is called Faradaically coupled electroosmosis.15 When electrode reactions are present, finite rate electrode kinetics lead to the presence of a nonzero lateral electric field component within the electrode’s diffuse double layer beneath the sphere. The action of the lateral electric field on the electrode’s diffuse-layer charge cloud drives a second electroosmotic flow with both lateral and vertical components. We refer to this O(E2) phenomenon as the Faradaically coupled electroosmotic (FCEO) force mechanism.15 By itself, however, the lateral component of FCEO flow is too weak an effect and does not extend far enough in the lateral direction to explain the experimental observations of aggregation and separation for particle pairs. Thus, neither mechanism (EK or FCEO) alone can explain observations of lateral motion reported for ac fields. The vertical force exerted by the FCEO flow on a particle is, however, sufficiently strong to affect both the average height and range of vertical motion dramatically. In addition, the vertical force is experimentally observed to perturb the phase angle between the instantaneous height of a single particle and the electric field at that instant away from the expected value of 90°. Thus, although the primary effect of the FCEO flow is orthogonal to the EK flow, the FCEO mechanism breaks the symmetry of the EK flow field. EK and FCEO Acting in Concert. The core hypothesis of this contribution is that the EK and FCEO mechanisms act together to aggregate or separate particles for alternating applied fields at frequencies below 1 kHz. The key concept is that the vertical component of the FCEO mechanism shifts the phase angle between the particle height and the electric field away from the expected 90°.16 This phase shift between the particle height and the electric field breaks the symmetry of the EK flow around two adjacent particles. In this hypothesis, a sum of vertical forces leading to a phase angle of less than 90° separates two particles, a phase angle greater than 90° brings them together, and a phase angle of 90° induces no motion because the symmetry is unbroken. The collaborative effect of these two interactions can directly account for the electrolyte-dependent motion of particle pairs observed in experiments at frequencies below 1 kHz. Electrostatic Mechanism: Induced Dipole Repulsion. Although the two hydrodynamic mechanisms described above are sufficient to account for the aggregation and separation of particles, the induced dipole repulsion described by Gong and Marr21 also occurs in this system. In this mechanism, the alternating electric field induces an effective separation of charge between the upper pole of each particle and the lower pole; because the induced charge separation is always in sync for the two particles, the particles experience electrostatic repulsion proportional to the square of the field strength. The aim of this contribution is to present a model that comprises all of these effects. The resulting model successfully accounts for observed particle behavior. (20) Bo¨hmer, M. Langmuir 1996, 12, 5747-5750. (21) Gong, T.; Marr, D. W. M. Langmuir 2001, 18, 2301-2304.
Langmuir, Vol. 22, No. 24, 2006 9847
Theory We describe a model that accounts for the observations of Kim et al.,8,9 who monitored the relative separation R(t) between the centers of two adjacent particles in the absence of nearby particles. Figure 1 is a schematic of Kim’s geometry in which two identical dielectric spheres near an electrode are influenced by an alternating field applied normal to the electrode. To simulate the results of Kim et al.,8,9 each sphere is treated both as a “target” of the other and as a “tracer” sphere entrained in the flow of a target. Expression for Instantaneous Lateral Motion due to EK Flow. Calculation of the net radial speed of entrainment dR/dt begins with calculation of Ur, the instantaneous lateral velocity of a tracer sphere relative to its target sphere. One can calculate Ur as the product of the local entrainment velocity and a hindrance factor19
Ur ) qw(h) Vr(R)
(1)
where Vr is the radial speed of entrainment calculated from Faxen’s law plus the component of electrophoresis calculated using Smoluchowski’s equation and the local radial electric field due to the presence of the target sphere. Faxen’s law and the Smoluchowski equation give the velocity of entrainment in bulk viscous fluid, so qw(h) is applied to account for the hindrance due to the proximity of the particle to the electrode. Solomentsev et al.19 evaluated Vr numerically, assuming that both the target sphere and the tracer sphere are at the same elevation h above the planar electrode. Their result is relatively insensitive to the elevation h of the spheres over the range of elevations of interest here.15,22 We estimated Vr using the following polynomial fit of their numerical data for h/a ) 0.05
Vr(r) )
ζE∞
6
() r
∑ An a n)0
η
n
(2)
where a is the radius of either sphere, E∞ is the unperturbed electric field generating the electroosmotic flow around the target sphere, ζ is the zeta potential of the target sphere, and and η are the electric permittivity and viscosity of the fluid. Equation 2 is valid only for 2 e R/a < 5. Similarly, the hindrance factor qw was estimated from 7
qw(h) )
() h
∑ Bn a n)0
n
(3)
The values of An and Bn appear in Table 1. Equation 3 is valid for 0.01 < h/a < 0.08. For larger h/a, eq 4 provides a better fit to the exact calculation.19
qw(h) )
15 3 + h h -1.7931 ln + 5.0310 -32.0 ln + 57.528 a a (4)
()
()
Solomentsev et al. originally applied eqs 1-4 to the dc case. We adapt them to describe the oscillating field case by specifying
E(t) ) E0 cos(ωt)
(5)
in which t is time and ω is 2π/f. Phase Angle and Form of Vertical Motion. Experimentally, the vertical motion of a single particle in response to an ac electric (22) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2002, 18, 7810-7820.
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Fagan et al.
Table 1. Values of Coefficients An and Bn in Equations 2 and 3 as Well as the Parameter Values Used in the Simulation to Match the Experimental Conditions of Kim et al.8,9 a Fitting Parameters n
An
Bn
n
An
Bn
0 1 2 3
-0.215 0.4 -0.19 0.0417
0.2331 7.988 -265.5 6841
4 5 6 7
-4.76 × 10-3 2.76 × 10-4 -6.42 × 10-6
-1.1230 × 105 1.121 × 106 -6.101 × 106 1.395 × 107
f s a
Model Parameters 78.5 η 2.56 E∞ 4.85 µm
ζ θ
KOH Parameters -0.058 V 〈h〉 80° 28 ∆h
200 nm 100 nm
ζ θ
NaHCO3 Parameters -0.0725 V 〈h〉 98° 28 ∆h
450 nm 225 nm
0.001 kg/ms 5 kV/m
R ) 4πa30f
a The values for the phase angle in each electrolyte were taken from previous measurements reported in refs 15 and 22. Both 〈h〉 and ∆h were chosen as reasonable representations of the behaviors of actual particles in the proper electrolyte exposed to the specified electric field.
field is a complicated function of frequency, field strength, particle size, electrode material, and electrolyte composition. Knowledge is insufficient at this time to calculate the phase angle and average instantaneous height from these elementary parameters. We know from experiments, however,15,16,22 that h(t) varies approximately sinusoidally around an average height, 〈h〉, at the frequency of the applied electric field, with a phase angle between the local electric field and the particle height, θ, of approximately 70110°. Thus, we simulate the time-dependent height of each (negatively charged) particle with the following expression:
h(t) ) 〈h〉 - ∆h cos(ωt - θ)
where ∆qw ) -∆h(dqw/dh|〈h〉) is the amplitude of oscillations in qw. Note that the drift velocity vanishes for θ ) 90° and has opposite signs for θ on either side of 90°. Induced Dipole Repulsion. The disturbance to a uniform electric field E∞ in an unbounded fluid of dielectric constant f caused by a dielectric sphere of radius a and dielectric constant s is the same as the disturbance caused by a point dipole of strength RE∞ aligned with the electric field. The apparent polarizability is
(6)
We have attributed the phase offset in particle height from 90° to the FCEO flow as detailed in the Introduction and Discussion sections. However, whatever its source, the fact that the phase angle θ is not 90° and that its value is system-dependent is sufficient to account for the observed motion. Simple Model for Drift. To understand how the phase angle affects the direction of drift in lateral separation between two particles, consider a linearized form of the equations. For small excursions in elevation, the linearized form of eq 3 is
(
s - f s + 2f
)
(10)
When the fluid contains ions and the sphere is charged, the counterion cloud surrounding the sphere can also become polarized, which was recently considered by Zhou et al.11 For the present case of thin double layers, the contribution from the polarization of the counterion cloud can be neglected. The electrostatic repulsion between two identical point dipoles oriented normal to the line connecting their centers is
FDD )
3(RE∞)2 4π0fR4
(11)
where 0 is the electric permittivity in vacuo and f is the dielectric constant of the fluid. The additional separating velocity due to the dipole-dipole interaction is thus
UDD )
qw F 6πηa DD
(12)
Relative Lateral Motion of Two Particles. We obtain a master expression for the rate of aggregation or separation of two particles by summing the contributions of the hydrodynamic and electrostatic forces described in previous sections. Assuming that Re , 1, the rate of mutual entrainment by two identical non-Brownian spheres is twice the electroosmotically driven velocity Ur (eq 1) plus the velocity due to the induced dipole repulsion (eq 10):
dR ) 2Ur + UDD dt
(13)
where ∆Vr equals the coefficient of cos(ωt) after eq 5 is substituted into eq 2 and also represents the amplitude of the oscillations in Vr. Using these approximations in eq 1 and then averaging over one period τ ) 2π/ω, we obtain the net drift velocity
Equation 13 is the equation by which the velocity of approach or separation of two particles was calculated. We have neglected hindrance related to the flow of fluid between the two spheres. Imposing a realistic model by Dufrense et al.23 of the two-body effects on the interparticle mobility did not qualitatively affect the comparison of calculated and experimental doublet trajectories. To summarize, the trajectories [h(t), R(t)] describing the relative motion between isolated pairs of identical particles on the electrode surface were predicted as follows. We prescribe an initial value for the separation distance R(0), usually 2.1a or 4.5a. Time is discretized into intervals ∆t ) 2π/500ω, so there are 500 steps per cycle of the electric field. At any given time t, R(t) is known, E∞(t) and h(t) can be calculated from eqs 5 and 6 and the parameters in Table 1, and Vr and qw can be calculated from eqs 2 and 3 using the values of An and Bn specified in Table 1. Finally, Ur and UDD are calculated from eqs 1 and 10, and the separation for the next time step can be calculated from
〈Ur〉 ) ∆Vr∆qw cos(θ)
R(t + ∆t) ) R(t) + (2Ur + UDD)∆t
qw(h) ) qw(〈h〉) +
dqw | (h - 〈h〉) ) dh 〈h〉 dqw qw(〈h〉) | ∆h cos(ωt - θ) (7) dh 〈h〉
where the second equality was obtained by substituting eq 6. Ignoring the weak r dependence of Vr in eq 2 compared to the strong dependence on E∞, we can estimate oscillations in Vr using
Vr ) ∆Vr cos(ωt)
(8)
(9)
(14)
Mechanism of Rectified Lateral Motion of Particles
Figure 2. (Line) Calculated interparticle spacing between two ∼10µm-diameter polystyrene particles in KOH under the same electric field conditions as in the experiment. (Symbols) Data taken from Kim et al.,8,9 showing experimental observations of particle pair repulsion. The data shown is for five experimental trajectories involving pairs of ∼9.7-µm-diameter polystyrene particles in 0.1 mM potassium hydroxide electrolyte exposed to a 100 Hz, 5 kV/m maximum alternating electric field. The calculated trajectory based on the rectified flow model closely approximates the experimental measurements.
R(t + ∆t) becomes R(t) for the next time step, and the procedure is then repeated.24
Langmuir, Vol. 22, No. 24, 2006 9849
Figure 3. (Line) Calculated interparticle spacing between two ∼9.7µm-diameter polystyrene particles in sodium bicarbonate under the same electric field conditions as in the experiment. (Symbols) Data taken from Kim et al.8,9 showing experimental observations of particle pair aggregation. The experimental data shown is for ∼9.7-µmdiameter polystyrene particles in 0.1 mM sodium bicarbonate electrolyte exposed to a 100 Hz, 5 kV/m max sinusoidal alternating electric field. The calculated trajectory correctly predicts the direction of motion and also displays the termination of aggregation at a separation distance larger than contact.
Comparison of Calculated and Experimentally Measured Trajectories. The calculated relative motion of two identical particles in KOH electrolyte, from eq 14 using the parameters of Table 1, appears in Figure 2. The two particles repel from an initial separation distance of 2 radii to ∼5.5 radii in a little over 2 min of real time. The net force on the particles decreases as the separation distance increases, leading to a monotonic decrease in speed as the particles separate. We compare the calculations with experimental data from Kim et al.8,9 The experimental conditions of refs 8 and 9, which reported the interparticle separation of particle doublets in applied ac fields, closely match the conditions of Fagan et al.15,16,22 under which the vertical motion of single particles was measured, with the exception that the particle used for the vertical motion measurements was on average 6 to 7 µm in diameter rather than the 9.7 µm used by Kim et al. In particular, the particle doublet trajectories presented here are reproduced from ref 9. The parameters reported in ref 9 and those used in these simulations are presented in Table 1. The calculations and experimental data in Figure 2 are obviously similar. Figure 3 compares the experimental data and results of the model in sodium bicarbonate electrolyte. The model calculations in NaHCO3 match the direction, the order of magnitude, and the steady-state gap of the experimental data. The steady centerto-center separation distances of ∼2.5 radii reflect the induced dipole repulsion; the aggregative flow otherwise would bring the particles into contact. Thus, the steady gap is the distance at which the aggregative and repulsive forces are balanced; in both experiment and simulation, particles maintain this distance indefinitely and return to the same distance after any perturbation. In Figure 3, the quantitative difference between the simulated and experimental trajectories may be due to the fact that the vertical motion measured in single-particle experiments was measured for particles smaller than those used in the doublet motion experiments.9,15,22 This difference is more likely to
quantitatively affect the predicted behavior in bicarbonate solutions because the net upward force on the particle increases approximately with ∼a15,25 whereas the mass of the particle increases with a3. Thus, the average elevation of the particles in bicarbonate was lower than the elevation used in these calculations, which would have the effect of increasing the aggregation rate. In contrast, in KOH, the particle is pushed toward the electrode regardless of the particle size, and the observed average elevation is nearly insensitive to a. The most impressive feature of Figures 2 and 3 is the correct prediction of the opposite direction of lateral interparticle motion in KOH and NaHCO3; in both the simulations and the experiments, particles on the electrode come together in NaHCO3 (Figure 3) but move apart in KOH (Figure 2). The observed direction is a direct consequence of the particular electrolyte. The sole important difference between the parameter sets used for the two simulations (Table 1) is the phase angle θ. Recall that the particular values in Table 1 are based on our observations of the vertical motion of single particles. Experiments showed that the phase angle is always less than 90°26 in hydroxide and nitric acid solutions on ITO whereas in NaHCO3 it is greater than 90°. These phase angles are directly attributable to the presence of the multiple forces acting on the vertical motion of each particle, but values for these phase angles and the effect of choosing a specific electrolyte are not yet predictable from first principles. Taken together, the trajectories shown in Figures 2 and 3 are the first simulations to demonstrate both electrolyte dependence and the correct order of magnitude of interparticle velocity in this frequency range. One might also ask if it would be possible to visually record the stepwise motion of the particles toward each other to verify the rectification mechanism because an oscillatory trajectory has not been recorded with video microscopy above a few hertz. However, at 100 Hz, the amplitude of oscillation over each cycle of the field is less than 1% of the particle radius (i.e.
