Langmuir 2007, 23, 6983-6990
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Electrolyte-Dependent Pairwise Particle Motion near Electrodes at Frequencies below 1 kHz James D. Hoggard,* Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon UniVersity, 5000 Forbes AVenue, Pittsburgh, PennsylVania 15213 ReceiVed January 8, 2007. In Final Form: April 3, 2007 A model incorporating a phase angle between an applied electric field and the motion of particles driven by it explains electrolyte-dependent pairwise particle motion near electrodes. The model, predicting that two particles aggregate when this phase angle is greater than 90° but separate when the phase angle is less than 90°, was based largely on contrasting behavior in two electrolytes (KOH and NaHCO3) used with indium tin oxide (ITO) electrodes. The present contribution expands the experimental evidence for this model to KOH, NaHCO3, NaOH, NH4OH, KCl, and H2CO3 solutions with Pt, as well as ITO electrodes. The phase angle correlation was verified in all cases. Comparisons of the model predictions to experimental data show that the sign and order of magnitude of rates of change in the separation distances between particle pairs are correctly predicted.
Introduction Controlling the aggregative behavior of colloidal particles is important in display technologies, biosensors, microfluidic devices, and other applications.1 Experimental investigations2-24 have shown that colloidal particles near an electrode and close to each other move parallel to the electrode surface when an electric field is applied normal to the surface. Particles in proximity to each other and near electrodes aggregate or separate depending on experimental conditions. Understanding the mechanisms underlying this motion would be useful for design of directed-assembly particulate systems. * To whom correspondence should be addressed. E-mail: jhoggard@ andrew.cmu.edu. (1) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2006, 22, 9846-9852. (2) Kim, J.; Anderson, J.; Garoff, S.; Sides, P. Langmuir 2002, 18, 53875391. (3) Solomentsev, Y.; Bohmer, M.; Anderson, J. L. Langmuir 1997, 13, 60586068. (4) Solomentsev, Y. E.; Guelcher, S. A.; Bevan, M.; Anderson, J. L. Langmuir 2000, 16, 9208-9216. (5) Guelcher, S. A. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1999. (6) Guelcher, S. A.; Solomentsev, Y. E.; Anderson, J. L. Powder Technol. 2000, 110, 90-97. (7) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706-708. (8) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 6375-6381. (9) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature (London) 1997, 386, 57-59. (10) Gonzalez, A.; Ramos, A.; Green, N. G.; Castellanos, A.; Morgan, H. Phys. ReV. E. 2000, 61, 4019-4028. (11) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. ReV. Lett. 2003, 90, 128303-1-128303-4. (12) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. ReV. E 2004, 69, 021405-1-021405-8. (13) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B.; Ajdari, A. Phys. ReV. E 2002, 65, 161509-1-061409-8. (14) Liu, Y.; Narayanan, J.; Liu, X. J. Chem. Phys. 2006, 124, 124906-1124906-8. (15) Gong, T.; Marr, D. W. M. Langmuir 2001, 17, 2301-2304. (16) Gong, T.; Wu, D. T.; Marr, D. W. M. Langmuir 2002, 18, 10064-10067. (17) Santana-Solano, J.; Wu, D. T.; Marr, D. W. M. Langmuir 2006, 22, 5932-5936. (18) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B. Phys. ReV. E 2002, 65, 061409-1-061409-8. (19) Bazant, M. Z.; Squires, T. M. Phys. ReV. Lett. 2004, 92, 066101-1066101-4. (20) Fagan, J. A. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 2005. (21) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2002, 18, 7810-7820. (22) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2003, 19, 6627-6632. (23) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2004, 20, 4823-4834. (24) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2005, 21, 1784-1794.
A hierarchy of mechanisms explaining the observations and organized along an axis of frequency is emerging. Under dc polarization, Anderson et al.2-6 showed that electroosmotic flow associated with each particle determines whether the particles advance toward each other or retreat and that the effect is proportional to the electric field; Fagan et al.1,20,23 call this effect, which is first order in electric field, “EK” flow. This dc model by itself, however, cannot account for particle aggregation or separation under ac polarization because the distance the particles advance toward each other over one-half of the electric field cycle is equal and opposite to the distance they retreat over the other half cycle. Fagan et al.1 proposed that electrochemical reactions break this symmetry in the EK flow so that a net change in the separation between two particles occurs over each cycle, resulting in net velocity (Ud) of aggregation or separation. A phase angle (θ ) between the cell current (I) and the elevation of the particle above the electrode (h) correlates with observations of the electrolyte-dependent drift velocity, Ud, which is proportional to the square of the electric field (O(E2)) during ac polarization at frequencies up to 1 kHz. In KOH, θ was less than 90°, and in NaHCO3, θ was greater than 90°, which correlated with the disaggregation observed in KOH and the aggregation observed in NaHCO3. The approximate upper bound of 1 kHz arises because above this frequency all the current is consumed in charging the electrodes and very little participates in electrochemical reactions; since the rate of charging is not kinetically limited, θ approaches 90°. Above 1 kHz induced charge on the electrode, an O(E2) effect originally proposed by Yeh et al.9 and subsequently elaborated by Saville and co-workers11-12 better predicts the response observed. Induced-dipole repulsion, another O(E2) force described by Gong et al.,15-16 also exerts an influence, both above and below 1 kHz. The focus of the present contribution is the intermediate frequency range below 1 kHz where Fagan et al.1 modeled electrolyte-dependent center-to-center particle separation distance (R) as a function of time for an isolated pair of identical particles when faradaic reactions are occurring (this model will hereafter be referred to as “Fagan’s model”). Fagan’s model comprises several effects: (1) EK flow tugs the particles together and pushes them apart in each cycle. (2) The second effect is another
10.1021/la070049j CCC: $37.00 © 2007 American Chemical Society Published on Web 05/24/2007
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the electric field lines around the sphere results in a component of the electric field tangent to the electrode, which causes electrophoresis of nearby particles. Solomentsev et al.4 calculated this contribution from Smoluchowski’s equation and added it to the entrainment speed. Fagan’s model neglects the weak dependence of this speed on h and employs a polynomial fit of their numerical results as a function of the center-to-center distance R for h/a ) 0.05 Figure 1. Schematic representation of an isolated pair of particles. Both particles are assumed to be spherical and of equal radius, a. When an ac electric field, E, is applied normal to electrode surface, the particle height, h, oscillates out of phase with the electric field. Electroosmotic flow around the particle also oscillates leading to a fluctuation in the center-to-center particle separation distance, R. If the phase angle between E and h deviates from 90°, Fagan’s model and experiments suggest that the particles either separate or aggregate depending on the phase angle.
electroosmotic flow caused by a tangential component of electric field beneath the particle acting on the charge in the double layer of the electrode, which we believe results in the measured phase angle, θ. (3) The elevation, h, oscillates while the particles experience height-dependent hindrance to particle motion parallel to the plane of the electrode.1,20-25 (4) The final component of the model is induced-dipole repulsion.15,16,26 Combining the primary features described above, Fagan’s model confirmed that particle pairs disaggregate for θ less than 90°, while particle pairs aggregate when θ exceeds 90°. The model accounted for the experimental observations of Kim et al.2 and predicted separation in KOH and aggregation in NaHCO3. Fagan’s model was formulated using evidence from two electrolytes (KOH and NaHCO3) interacting with one electrode material (ITO). The evidence, while clear, was narrowly based. Furthermore, much data in the literature11-12,15-18 were acquired with KCl solution and Pt electrodes, as well as ITO. Thus, there was a need to show that Fagan’s model is generally valid for more electrolytes and a metallic electrode. The purpose of this investigation was to test the correlation between θ measured on single particles and R measured as a function of time on isolated pairs of particles at low frequencies in several different electrolytes above both Pt and ITO electrodes.
Theory Lateral Motion of Isolated Pairs. A potentiostat controlled by a function generator applies a voltage between the two electrodes which varies sinusoidally with time at some frequency, ω. If the amplitude of the voltage is not too large, the resulting current also varies sinusoidally with time. Dividing the measured current by the specific conductance and the electrode area yields the electric field, normal to the electrode, which each particle sees. Denote that quantity as
E(t) ) ∆E cos(ωt)
(1)
This electric field produces electroosmotic flow around each particle which entrains nearby particles. The component of the velocity of entrainment tangent to the electrode for a second particle at the same elevation h (see Figure 1) was estimated by Solomentsev et al.4 using Faxen’s law. In addition, bending of (25) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 637651. (26) Zhou, H.; Preston, M. A.; Tilton, R. D.; White, L. R. J. Colloid Interface Sci. 2005, 285, 845-856. (27) Einstein, A. InVestigations on the Theory of Brownian Motion; Fu¨rth, R., Ed.; translated by Cowper, A. D. (1926, reprinted 1956); Einstein, Collected Papers, vol. 2, pp 170-82, 206-22.
V(R,t) ) qp(R/a)
0fζp E(t) η
(2)
where 6
qp(F) )
A nF n ∑ n)0
(3)
where 0 is the permittivity of free space, f is the dielectric constant of the fluid, ζp is the zeta potential of the particles, η is viscosity of the fluid, a is the particle radius and An are the coefficients used as a polynomial fit. These are from n ) 0-6: 0.215, -0.4, 0.19, -0.0417, 4.76 × 10-3, -2.76 × 10-4, and 6.42 × 10-6. This polynomial approximation to qp is valid for 2 < F < 5. Faxen’s law and Smoluchowski’s equation apply in unbounded fluids. The presence of the nearby electrode introduces additional fluid drag, which reduces the response of the particle. The extent of reduction (for a freely rotating sphere translating parallel to a plane wall) depends on h/a and was estimated in Fagan’s model from
UEK(R,h,t) ) qw(h/a)V(R,t)
(4)
where 7
qw(H) ) qw(H) )
BnHn ∑ n)0
for H e 0.045
(5a)
15 3 + -1.7931 ln H + 5.031 -32 ln H + 57.528 for H > 0.045 (5b)
UEK is essentially the net speed of hindered entrainment calculated by Solomentsev et al.4 which was found to correctly model most of their observations of lateral motion induced by dc electric fields; Fagan calls this the “EK model”. When the two particles are close to one another, they experience repulsion between their instantaneous dipoles, which also contributes to lateral motion. Fagan estimated this contribution from Stokes’ law modified by the hindrance factor qw:
UDD(R,h,t) ) qw(h/a)
FDD(R,t) 6πηa
(6)
where FDD is the repulsion between the two identical dipoles of magnitude RE(t):
FDD(R,t) )
3[RE(t)]2 4π0fR4
where R is the apparent polarizability of the particles:26
(7)
Electrolyte-Dependent Pairwise Particle Motion
R ) 4πa30f
p - f p + 2f
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(8)
and where p is the dielectric constant of the particle. Adding the contribution from dipole repulsion to that from the electroosmotic flow around each sphere gives the net rate of change of separation:
dR ) 2UEK + UDD dt
(9)
The EK contribution is doubled because each of the two particles in the pair generates an electroosmotic contribution to flow which entrains the other particle. Fagan et al.21 made careful observations of h(t) of single isolated particles as a function of frequency and amplitude of the applied voltage. Although the oscillations in h(t) are periodic in time, they are not purely sinusoidal: the half of the cycle corresponding to small elevations is more blunted that the other half. Varying the frequency and amplitude of the electric field causes the mean elevation, the amplitude of the oscillations in h(t), and the shape of the waveform to change in a complex fashion. Explaining those changes was the subject of several of their papers. Moreover, the oscillations in h(t) are out of phase with oscillations of the applied electric field (E(t)). Assuming the electrophoretic response of the particle is quasi-steady, one would expect dh/dt to be proportional to E(t) with no phase lag; this implies that h(t) is expected to be 90° out of phase with E(t). Fagan et al.1 reported measurements of the phase angle which vary between 50° and 100°, depending on the frequency and electrolyte. For the purposes of the present paper, we approximate the shape of oscillations in h(t) to be purely sinusoidal:
h(t) ) havg - ∆h cos(ωt - θ)
(10)
Values of the parameters havg, ∆h, and θ are those measured on single isolated particles. Effect of Phase Angle. The effect of the phase angle θ can be clearly seen if we assume the oscillations in E are small so we can linearize qw(H) and neglect oscillations in qp(F). Then UEK has one contribution which is proportional to cos(ωt) and a second contribution which is proportional to cos(ωt) cos(ωt - θ). When averaged over any full cycle, the first contribution vanishes whereas the second contribution leads to net drift as a speed1
UEK ) ∆V∆qw cosθ
(11)
where
∆V ) qp(R/a)
0fζp ∆E η
(12)
represents the amplitude of oscillations in V and where
∆qw ) ∆h
|
dqw dh
havg
(13)
represents the amplitude of the oscillations in qw. From eqs 11-13, oscillations in elevation h and in the associated hindrance factor qw are essential to achieving a net drift. Moreover, the oscillations in h and qw must be out of phase with the oscillations in the driving electric field by an angle θ which differs from 90°.
Contribution from Brownian Motion. For one-dimensional Brownian motion along the x axis, the probability of finding a particle between x and x + dx at time t, given that it was at x0 at t ) 0 is given by p(x,x0,t) dx where
p(x,x0,t) )
1
x4πDt
[
exp -
]
(x - x0)2 4Dt
(14)
is the well-known probability density function and represents a Gaussian distribution centered at x0 whose variance grows linearly with time. The first three moments of this distribution are given by
{
1 for n ) 0 x for n)1 x ) -∞ x p(x,x0,t) dx ) 0 2 x0 + 2Dt for n ) 2 n
∫
∞
n
}
(15)
If there are two identical Brownian particles, with initial locations x10 and x20, their mean-square, relative displacement also grows linearly with time:
(x2 - x1)2 ) x22 - 2(x2)(x1) + x21 ) (x20 - x10)2 + 4Dt (16) This assumes that p(x1,x10,t) and p(x2,x20,t) are both given by eq 14 which is independent of the position of the other particle, although that means that the two particles might overlap. If the particles are also free to move along the y axis, then their y coordinates evolve at the same rate:
(y2 - y1)2 ) (y20 - y10)2 + 4Dt
(17)
This assumes the p(y1,y10,t) and p(y2,y20,t) are both given by eq 14 which are independent of the x-coordinates. The centerto-center distance is then given by
R2 ) (x2 - x1)2 + (y2 - y1)2 ) R20 + 8Dt
(18)
Finally, any hindrance in the hydrodynamic mobility of the particles owing to proximity to the wall also hinders their diffusion coefficient
R2 ) R20 + 8qw(havg/a)Dt
(19)
Experimental Section Figure 1 shows a graphical depiction of some of the variables described previously, and the experimental apparatus appears in Figure 2. The flow cell consists of two 2.54 × 7.62 cm2 electrodes separated by a 1.40 mm polycarbonate spacer. A 100 nm thick ITO layer on a coated glass slide (Bioptics Corp.) was the upper electrode. The bottom electrode was either an ITO or 100 nm thick Pt (Radient Technologies Inc.) coated glass slide. The upper ITO electrode had an inlet and outlet for moving fluid through the cell. The two electrodes and spacer were optically coupled to a prism providing an angle of incidence equal to 68° for the total internal reflection microscopy (TIRM) apparatus.28 Experiments were conducted in 0.15 mM solutions of KOH, KCl, NaHCO3, NH4OH, and NaOH. Deionized water was sparged with nitrogen gas for at least 1.5 h to remove dissolved CO2 which can lead to carbonic acid formation in solution. Salt was added and the solution was capped under nitrogen in order to prevent uncontrolled equilibration with CO2 in the atmosphere. H2CO3 solutions were purchased from Fisher Scientific. These had a pH of 5.7 and were not diluted. The electrodes were each immersed in ∼10 mM KOH (28) Prieve, D. C. AdV. Colloid Interface Sci. 1999, 82, 93-125.
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Figure 2. Experimental apparatus. Thick solid lines represent current passed from the function generator and potentiostat through the flow cell, thin sold lines represent voltages sent to the computer for data collection, and dotted lines represent voltages monitored on the oscilloscope. This experimental setup allows the measurement of θ between E and the oscillatory vertical motion of a single particle. h of single particles were recorded by measuring the intensity of HeNe laser light scattered by particles in an evanescent wave near the electrode surface using TIRM. θ measurements were made by monitoring the TIRM signal and the applied current on an oscilloscope. The separation distance of two isolated particles was measured with the aid of images captured by the CCD camera. solution for at least an hour followed by washing with deionized water and drying with nitrogen; this procedure established surface charge that was vital for preventing the particles from depositing irreversibly on the electrode. After assembly, the cell was filled with electrolyte and allowed to equilibrate for an hour. Surfactant-free, 5.7 µm diameter polystyrene-sulfonate particles from Interfacial Dynamics Corporation were then injected and allowed to sediment to the bottom electrode. Sinusoidally modulated electric fields were generated by an Agilent 33120A function generator controlling an EG&G PAR Model 173 potentiostat-galvanostat operating in potentiostat mode with a Model 179 digital coulometer. The working electrode was connected to the upper ITO electrode and the reference and counter electrodes were shorted and attached to the bottom ITO or Pt electrode. To avoid any contamination of the electrodes by different electrolytes, the electrodes were replaced when each new electrolyte was tested. The electrodes were deemed to be stable if there were no visible gaps in the Pt or ITO films, the electrode color was normal (electrodes can “burn out” if the applied voltages are too high), and the currentvoltage relationship of the electrode was sinusoidal and consistent with previous measurements. Occasionally, a dc electric field was applied to attempt to rest particles directly on the electrode surface in order to make a known scattered light intensity measurement using TIRM. The top ITO electrode would become discolored at voltages greater than ∼3 V. If this occurred, that electrode was immediately discarded. After cleaning a Pt electrode multiple times, visible gaps in the Pt film could sometimes be observed under the microscope. These electrodes were also discarded. The current and voltage versus time curves were always consistent. The zeta potentials of the particles (ζp) were measured in each electrolyte prepared as described above (0.15 mM except H2CO3) using a Malvern Instruments Zetasizer 3000HSA. This instrument measures electrophoretic mobilities using light scattering, from which ζp are determined from the Henry equation.29 The Zetasizer cell consists of a flow cell between two gold electrodes. The values of ζp were -105 ( 1, -78 ( 2, -96 ( 3, -95 ( 1, -104 ( 2, and (29) Shaw, D. J. Introduction to Colloid and Surface Chemistry; ButterworthHeinemann Ltd: Oxford, 1989. (30) Kasumi, H.; Sides, P. J.; Anderson, J. L. J. Colloid Interface Sci. 2004, 276, 239-247.
Hoggard et al.
Figure 3. Experimentally measured Vh signal and its conversion to elevation. (a) A raw PMT signal, Vh, as a function of time. The thin line represents a single unaveraged measurement of Vh and the thick line represents the average of 1024 of these 20 ms samples. In this paper, Vh signals used to make θ measurements were averaged over 1024-4096 cycles. This was sufficient to generate reproducible waveforms. (b) The actual h(t) is calculated from the averaged Vh signal in (a).28 The intensity of the scattered light is exponentially dependent on h. Note that the maximum in Vh corresponds to the minimum in h because the intensity of the scattered light increases with decreasing h. The solution in this case was 0.15 mM KOH above a Pt electrode at 100 Hz. -55 ( 2 mV in KOH, KCl, NaHCO3, NH4OH, NaOH, and H2CO3, respectively. The pH of each solution was measured using an Accumet AR60 pH/conductivity meter. These were 9.8, 5.8, 9.4, 9.2, 9.7, and 5.6, respectively. Single Particle Measurements. The quantities h, h0, havg, and θ were measured for single particles using TIRM. A HeNe laser beam was totally internally reflected at the bottom electrode-electrolyte interface resulting in an evanescent wave. The angle of incidence of the laser beam was 68°, and the penetration depth was 113.7 nm. A particle near the bottom electrode scatters the evanescent photons with an intensity exponentially dependent on h. The intensity of the scattered light was measured using a Photon Technology International Model 814 photomultiplier tube (PMT). When used in digital mode, this apparatus precisely measures h of a single particle above an electrode with nanometer resolution. When in analog mode, the PMT reported a voltage proportional to the scattering intensity (Vh) to an Agilent 54624A oscilloscope. Since the applied current (I) was monitored on the same oscilloscope via an analog signal from the potentiostat, θ between the two signals was readily determined. Particles chosen for single-particle experiments had to meet the following criteria: they were freely mobile and exhibited Brownian motion; they remained in the evanescent wave for at least the majority of the cycle; and the current was symmetric and approximately sinusoidal. To test if the particles were mobile, fluid was briefly pumped through the cell before and after the experiment with zero applied electric field. A particle was mobile if it moved freely in response to this flow. To observe if the particle underwent obvious Brownian motion, the HeNe laser was turned on with zero applied electric field. If the intensity of the scattered light showed stochastic changes in Vh, and thus h, the particle was accepted for experimentation. The cell current waveform was acceptable as long as the
Electrolyte-Dependent Pairwise Particle Motion
Figure 4. Calculation of the time lag between the applied current and h. (a) I (dashed line) and Vh (solid line) are depicted. (b) θ between I and h was calculated by measuring the difference in time between the maxima of the two waveforms and correcting for a frequency-dependent time lag in the PMT. These waveforms were obtained in 0.15 mM KOH above an ITO electrode at 100 Hz, and E0 was 1786 V/m. In this instance, the phase difference was 2.31 ms. From these data and after the time lag correction, θ was calculated to be 81.4°. The measured θ was generally reproducible to within several degrees depending on the amount of noise in the Vh peak. waveform was approximately sinusoidal and symmetric, typically valid above 50 Hz; below this frequency, nonlinearities in the current-voltage relation degraded the symmetry. h0 vanished as the frequency neared 1 kHz. Averaging of the scattered light over many cycles eliminated Brownian noise in the height data. The number of averaged cycles varied from 1024 to 4096 depending on the frequency and the amount of noise. At low frequencies, fewer cycles were needed than at high frequencies to obtain reproducible data. The time of the maximum in the PMT signal was measured by fitting a parabola to the waveform. It was observed that a phase lag exists in the PMT and that it depended on the frequency of the applied field. In order to obtain accurate measurements of θ, this time lag had to be measured and subtracted from Vh. No correction was needed for I. The method and results for this calibration are described in Appendix 1. Particle Pair Experiments. To measure R(t) in isolated particle pair systems, the flow cell was backlit with white light and the HeNe laser was blocked. Images were acquired using a Dage MTI CCD100 camera connected to a monitor. Images were recorded onto a computer on which R was measured at 5 s intervals. Pairs of particles were chosen using the same criteria established in the single particle experiments, with the addition that the particles were mobile with respect to each other. R0 was slightly greater than two radii for separating systems and approximately four radii for aggregating systems. Both the small and large initial standoffs were initially tried for each particle pair with each electrolyte-electrode combination as a control. R(t) was measured on at least four particle pairs for each electrolyte-electrode system in order to mitigate effects of random particle motion. All particle pair experiments in this paper were conducted at 100 Hz with E0 equal to 1786 V/m (5 Vpp) unless otherwise noted.
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Figure 5. θ and R(t) results for electrolytes-electrodes that aggregated: NaHCO3-Pt (solid circles) NaHCO3-ITO (open circles), KCl-Pt (solid triangles), and KCl-ITO (open triangles). (a) Values of θ for several electrolytes-electrodes were all greater than 90°. (b) R(t) shows that isolated particle pairs aggregated. In all cases, particle pairs aggregated in each electrolyte-electrode when the phase angle was greater than 90°.
Results and Discussion When an electric field is applied at a low frequency (
