Vertical Oscillatory Motion of a Single Colloidal Particle Adjacent to an

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Vertical Oscillatory Motion of a Single Colloidal Particle Adjacent to an Electrode in an ac Electric Field Jeffrey A. Fagan,* Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received March 11, 2002. In Final Form: July 1, 2002 The vertical motion of a levitated colloidal particle immersed in aqueous solution, located near an electrode, and subjected to an alternating electric field normal to the electrode was recorded using total internal reflection microscopy. The particle’s height was sinusoidal in time during the portion of the cycle above its average height; during the remainder of the cycle, however, hydrodynamic hindrance arising from the particle’s proximity to the electrode did not allow it to descend below the average height with the same amplitude as above. The particle’s height averaged over a full cycle exhibited a minimum in its frequency and voltage dependences in potassium hydroxide but increased monotonically with frequency in sodium bicarbonate. The phase between the response of the particle and the applied electric field varied significantly with frequency. Both the profoundly different response in potassium hydroxide and sodium bicarbonate and the significant variation in phase angle with frequency are not expected on the basis of colloidal and electrophoretic forces alone; another force is suggested. The electrohydrodynamic model of Sides (2002) is qualitatively consistent with these observations.

Introduction Electrophoretic deposition is often used to bring colloidal particles to a surface.1 While exploring the behavior of loosely deposited particles, many investigators2-10 have observed aggregation in both ac and dc electric fields. Identifying the mechanisms responsible for the motion is important because the ability to control particle motion at an electrode is potentially significant in novel display technologies, among other applications. Two theories explain lateral motion of particles in response to normally directed fields during electrophoresis: the electrokinetic model (EK)7,8 and the electrohydrodynamic model (EH).4,5,11 According to the EK model, the electric field normal to the electrode acts on the diffuse cloud of counterions next to the particle to cause electroosmotic flow around the particle toward or away from the electrode depending on the sign of the particle’s zeta potential and the sign of the electric field. This flow in turn moves fluid along the surface of the electrode either toward or away from the particle depending on the direction of the electroosmotic flow. Particles near the electrode and located within a few radii of each other are mutually entrained in the flow along the surface and are * Corresponding author: tel 412 268-2261; e-mail jfagan@ andrew.cmu.edu. (1) Sarkar, P.; Nicholson, P. S. J. Am. Chem. Soc. 1996, 79, 19872002. (2) Bo¨hmer, M. Langmuir 1996, 12, 5747-5750. (3) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408-3413. (4) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334-6336. (5) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 63756381. (6) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature (London) 1997, 386, 57-59. (7) Solomentsev, Y. U.; Bo¨hmer, M.; Anderson, J. L. Langmuir 1997, 13, 6058-6068. (8) Solomentsev, Y. U.; Guelcher, S. A.; Bevan, M.; Anderson, J. L. Langmuir 2000, 16, 9208-9216. (9) Kim, J.; Guelcher, S. A.; Garoff, S.; Anderson, J. A. Adv. Colloid Interface Sci. 2002, 96, 131-142. (10) Kim, J.; Garoff, S.; Anderson, J. A.; Sides, P. J. Langmuir 2002, 18, 5387-5391. (11) Sides, P. J. Langmuir 2001, 17, 5791-5800.

thereby brought together or driven apart. This is the mechanism of aggregation or disaggregation in the EK model. The EH model likewise holds that circulation imposed on the fluid is a source of lateral motion; however, it is based on electrical body forces appearing wherever there are electrolyte concentration gradients and not just near the particle’s surface. The normal electric field itself, acting through faradaic processes at the electrode, spawns electrolyte concentration gradients that produce unbalanced charge in the fluid adjacent to the electrode’s surface; the sign and magnitude of this charge depend on the electric field, the sign of the reacting species, and the mobilities of the ions.11 In the absence of any particles, no component of the electric field acts along the surface to cause flow. The presence of a dielectric particle near the electrode, however, bends the field lines around the particle. The lateral field component of the bent field lines forces the unbalanced charge to move parallel to the electrode’s surface. Other particles near the electrode become entrained in this flow along the surface and are thereby brought together or driven apart. This is the mechanism of aggregation or disaggregation in the EH model. Let En denote the normal component of the electric field in the absence of any particles and ζ denote the zeta potential of the particles. Motion of particles along the surface of the electrode can be expected to scale like Enζ in the EK model and like En2 in the EH model (since both the induced charge and the surface component of the electric field are proportional to En). In particular, ζ is absent from the EH model. In an ac electric field, the movement of particles toward one another during onehalf of the cycle is virtually canceled out in the EK model by movement away from one another during the second half of the cycle. In the EH model, the direction of lateral motion of particles is in the same direction during both halves of the cycle. Anderson and co-workers have qualitatively and quantitatively shown that in dc electric fields the lateral velocity of a particle is proportional to the zeta potential of the

10.1021/la025721l CCC: $22.00 © 2002 American Chemical Society Published on Web 09/12/2002

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Langmuir, Vol. 18, No. 21, 2002 7811

particle, and the direction of the particle interaction is determined by the direction of the electric field.7,8 Both observations are fundamental to the EK model. When ac electric fields were applied, however, the investigators observed that either aggregation or disaggregation is possible, and the direction can be reversed at the same frequency and electric field strength by changing the electrolyte.9,10 Furthermore, Kim et al.10 demonstrated that convective motion of particles toward each other is independent of particle zeta potential. Electrokinetic effects therefore do not appear to be a strong factor in ac aggregation. Many observations of two-particle motion in ac fields, however, are consistent with the EH mechanism proposed by Trau et al.5 as articulated by Sides11 for a single particle. Previous research has focused on lateral motion of particles under ac electric fields as a test for the suggested mechanisms; to our knowledge, no one has examined the details of particle motion in the direction normal to the electrode. This contribution, therefore, is intended to explore the longitudinal oscillatory motion of particles near an electrode as a possible indicator of the mechanisms responsible for the aggregation or dispersion of particles. The motion of a single particle in the direction normal to the electrode was tracked with the aid of total internal reflection microscopy (TIRM12). This work comprises measurements of the colloidal particle’s motion, a comparison with the motion expected from EK theory, and a comparison of observed behavior with predictions of the EH model. The most surprising observation is that the ac field draws the particle closer to electrode in KOH but pushes the particle farther from the electrode in bicarbonate. This qualitatively different behavior in the two electrolytes is not expected from the EK model and suggests another force must be acting, such as the force in the EH model. Theory No Applied Field. A colloidal particle near a horizontal surface samples different elevations by Brownian motion in the absence of an externally applied electric field. Boltzmann’s equation relates the probability of finding the sphere at any instantaneous elevation h to the potential energy (PE) φ(h) of the sphere,

p(h) ) Aeφ(h)/(kT)

(1)

where h is the instantaneous thickness of the gap between the planar surface and the closest point on the particle, and A is a normalization constant chosen so that ∫∞0 p(h) dh ) 1. We expect double-layer repulsion to be well modeled with the aid of linear superposition and Derjaguin’s approximations when the separation distance is several Debye lengths. Then, for a 1:1 electrolyte, the total PE profile obeys

φ(h) ) B exp(-κh) + φvdw(h) + Gh

(2)

where B ) 64πa(kT/e)2 tanh(eψ1/4kT) tanh(eψ2/4kT), is the dielectric permittivity of water (using rationalized units), a is the radius of the sphere, e is the elemental charge, ψ1 and ψ2 are the Stern potentials of the sphere and the plate, κ ) (2Ce2/kT)1/2 is the Debye parameter, C is electrolyte concentration, and G ) 4/3πa3(Fs - Ff)g is the net weight of the sphere. The van der Waals attraction is severely retarded and screened at the separations sampled by particles in our (12) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93-125.

experiments. We followed the method of Bevan and Prieve13 for polystyrene-glass to measure the van der Waals attraction for polystyrene-ITO. The empirical relation for 6 µm particles is

φvdw(h) ) -6.8kBT exp

-h (46nm )

(3)

Both the preexponential and the decay length shown are larger for the ITO/polystyrene system than for glass/ polystyrene, so the force in this case is twice as large and the range is about 15% greater. The PE profile represented by eq 2 has a single minimum at an elevation we denote as hm, which represents the most likely elevation in the absence of an externally applied electric field. The average elevation under this condition is given by

1 t ∫0∞h|E)0e-φ(h)/k T dh ) tmax ∫0

〈h0〉 ) A

B

max

h|E)0 dt

(4)

where “E ) 0” indicates no applied electric field. The average elevation 〈h0〉 is expressed as a probabilityweighted elevation in the first equality of eq 4. Experimentally, 〈h0〉 can be obtained equivalently as the time average expressed by the last term on the right-hand side of eq 4. A non-Brownian particle eventually settles to the minimum in the potential energy profile and remains there indefinitely. Thus, the average elevation of a nonBrownian particle must be hm. In contrast, a Brownian particle continuously samples different elevations according to the Boltzmann distribution given by (1). Under the conditions relevant to this study, the probability density p(h) resembles a skewed Gaussian, with a maximum at hm but skewed in the direction of h > hm; this leads to an average elevation 〈h0〉 calculated from (4), which is about 20% higher than hm for the particles of this study. Applied Field. Under the influence of a sinusoidal electric field strong enough to exert a force on the charged particle comparable to the colloidal forces, the particle oscillates up and down with Brownian fluctuations superimposed. Here we attempt to predict the deterministic part of the stationary oscillations by performing a force balance on a non-Brownian particle. We acknowledge at the outset that this analysis is only partially correct because Brownian motion affects even the average elevation of the particle, as described above for the no-field case. The effect of Brownian motion on the average elevation of a particle in an oscillating electric field is less easy to calculate, requiring numerical solution of a diffusion equation with time- and position-dependent coefficients. In this paper, we use only a primitive, empirical correction for Brownian motion; that is, we add the difference between hm and 〈h0〉 to the average elevation computed for a non-Brownian particle in order that the model behave correctly at the limits of voltage and frequency. The small particle Reynolds number means that inertia is negligible so the flow around the particle is well described by Stokes equation. Even in an infinite fluid, the relaxation time for a 6 µm sedimenting particle to reach terminal velocity is on the order of a2/ν or 10-5 s, which is very short compared to the period of oscillations. When the particle is very close to the electrode, the time to reach terminal velocity is probably much less, owing to the much shorter distance that momentum must diffuse. Hence, at each instant we expect the particle to undergo (13) Bevan, M.; Prieve, D. C. Langmuir 1999, 15, 7925-7936.

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Fagan et al.

translation at its terminal velocity driven by the local colloidal and electrokinetic forces. A balance of colloidal, electrokinetic, and viscous drag forces requires

dh

∑i Fi ) 0 ) FC(h) + FEK(h,t) - R(h)6πµa dt

(5)

where a is the particle radius, h is the particle “height” defined as in the previous subsection, µ is the fluid viscosity, and R(h) is a correction to the drag force for sedimenting spheres very close to a wall. An approximate equation for R(h), fit to the results of Brenner et al.,14,15 is

R(h) )

6h2 + 9ah + 2a2 6h2 + 2ah

(6)

The colloidal force in eq 5 comprises electrostatic, gravitational, and van der Waals components and can be calculated by combining eq 2 with

FC ) -

dφ dh

(7)

We represent the electrokinetic force as the product of a field-dependent term and a sinusoid.

FEK ) σ(h)6πζaE∞ sin(ωt)

(8)

where ζ is the zeta potential of the particle and E∞ is the amplitude of the applied electric field. σ(h) is an additional correction factor for a sphere undergoing electrophoresis; it is associated with the differences in flow patterns between sedimenting vs electrophoretically driven spheres. An equation for σ(h) proposed by Solomentsev et al.7 as an empirical fit to the numerical result of Keh and Lien16 is

σ(h) )

h + 1.554a h + 0.3a

(9)

Equation 5 can be solved for the instantaneous particle velocity, ready for integration.

dh FC(h) + FEK ) dt 6πµaR(h)

(10)

In the previous subsection, we defined the elevations hm and 〈h0〉 as the most likely and average heights with no applied field; we now define additional important elevations: (i) h h (τ): “phase-dependent averaged height” refers to a time-dependent height obtained by averaging over many cycles the height observed at one particular phase in the oscillation of the applied electric field. This signal averaging reduces the effect of Brownian fluctuations and other N random noise, i.e., h h (τ) ≡ (1/N)∑n)1 h[τ + 2π(n - 1)/ω], where n is the index of the nth cycle acquired during an experiment, N is the total number of cycles, and τ is a value in seconds such that 0 e τ e 2π/ω. The measured h’s are observed at sufficiently long times after starting the ac field that the random initial conditions have been forgotten. (14) Brenner, H. Chem. Eng. Sci. 1961, 16, 242-251. (15) Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 1753-1777. (16) Keh, H. J.; Lien, L. C. J. Chin. Inst. Chem. Eng. 1989, 20, 283290.

(ii) 〈h〉 “experimental applied field average height” refers to the simple time average of the measured time-varying particle height h(t) over many full cycles when an electric field is applied, i.e., 〈h〉 ≡ (1/2π)∫2π h (τ) d(ωτ). 0 h (iii) 〈hEK〉, 〈hEK/D〉, or 〈hEK/BC〉 “theoretical applied field average height” refer to the time average of the theoretical time-varying particle height h over a full cycle when an electric field is applied, i.e., 〈hEK〉 ≡ (1/2π)∫2π 0 h(τ) d(ωτ), where h(τ) is obtained from the integration of eq 10 at sufficiently long times after starting the ac field that random initial conditions have been forgotten. D and BC in the subscript refer to deterministic or Browniancompensated values, as described below. 〈hEK〉 with no modifier refers to either quantity. Integration of the deterministic height derivative, eq 10, and insertion into the definition of the theoretical applied field average height yields the convective limit prediction for 〈hEK〉; this is eq 11.

〈hEK/D〉 )

[

∫02π ∫tt

1 2π

FC + FEK

final

6πµaR(h)

initial

]

du d(ωt)

(11)

Equation 11 represents the height behavior expected in the limit of dominant convective motion because Brownian motion was neglected in the force balance, eq 5, from which eq 11 was generated. The D in the subscript denotes the purely deterministic nature of the calculation. Since Brownian motion was not included in eq 11, 〈hEK/D〉 does not predict the equilibrium limit of 〈h0〉 at zero applied electric field, where diffusive motion is dominant. Instead, 〈hEK/D〉 predicts hm, the height at which the colloidal forces sum to zero, as the limit with no applied field. We can “correct” the electrokinetic model prediction calculated from eq 10 by adding the difference between 〈h0〉 and hm to the calculated 〈hEK/D〉. By adding the correction term, ∆BM ) 〈h0〉 - hm, the predicted particle behavior becomes indistinguishable from the expected behavior at very low field strength or very high frequencies. This compensation for Brownian motion is purely empirical and is included such that the model matches the known limit when no electric field is applied. The added height also assumes that the effects of Brownian motion on 〈hEK/D〉 are unaffected by the electric field. (This ad hoc correction would not be necessary if the convective diffusion equation were solved as mentioned at the outset of the description of this model.) The equation used for this model is

〈hEK/BC〉 )

1 2π

[

final

FC + FEK

initial

6πµaR(h)

∫02π ∫tt

]

du d(ωt) + ∆BM (12)

where the subscript BC denotes “Brownian compensated.” Both 〈hEK/D〉 and 〈hEK/BC〉 results will be displayed along with the collected data. Experiments TIRM12 was employed to detect the motion of charged particles exposed to an oscillating normally directed electric field near an electrode. The intensity of light scattered from an evanescent wave by a single particle is measured and converted to particle height via

I(h) ) I0e-βh + Ib

(13)

which relates the intensity of scattered light I(h) to particle height h. Ib is the intensity reading when no particle is in view, I0 + Ib is the intensity of light scattered by the particle when in contact with the wall (h ) 0), and β-1 is the penetration depth of the evanescent wave. As h f ∞, the intensity exponentially decays


Figure 1. (a, top) Schematic of the experimental apparatus; two ITO electrodes face each other across an electrolyte, either potassium hydroxide or sodium bicarbonate. The light scattered by the particle, from the evanescent wave created at the interface, is collected to measure the particle height. The particle is not to scale in this figure. (b, bottom) Larger view of inset from (a). Note the scattered light form the evanescent wave and the small size of h(t) relative to the particle size. to Ib regardless of the precise h; thus, particle height can only be accurately measured when it is less than several β-1. While large penetration depths allow height to be measured over a larger range, small penetration depths lead to greater sensitivity; a primary penetration depth of 145.7 nm was chosen as a reasonable compromise. The accuracy of independent particle height measurements depends on the statistical properties of Ib, which has both a mean and a standard deviation. The error bars on any height measurement were calculated from eq 13 by adding and subtracting the standard deviation in Ib to each measured intensity. The optical apparatus appears in Figure 1. A helium-neon laser (λ ) 632.8 nm, Melles Griot model #05 LHP 927) provided photons for the evanescent wave. A Zeiss Universal microscope with a total magnification of 400× and a photomultiplier tube (Photon Technology International 814 module) captured the scattered light. The photomultiplier tube typically counted photons digitally; an analog signal was recorded on an oscilloscope (Agilent Technologies model #54624A) for phase angle measurements. In TIRM measurements, digital photon counting is not always an accurate measurement of the instantaneous particle height. Each individual intensity reading, I, represents the total number of photons scattered during the previous sampling period ∆t. If the particle’s elevation changed during this period, then I represents an average over the elevations sampled during this period. Owing to the nonlinear nature of (13), the average

Langmuir, Vol. 18, No. 21, 2002 7813 elevation deduced from the average intensity by (14) is heavily weighted toward smaller elevations. In order for I to lead to a true instantaneous elevation, any fluctuations in elevation during ∆t must be small compared to the penetration depth β-1 of the evanescent wave. If a particle is moving at a high enough velocity, U(t), such that U(t)∆tβ is not small, then the averaging that occurs during digital sampling causes the measured height of the particle to be distinguishably lower than the true height. In practice, this limitation was only significant for combinations of low frequencies and strong applied fields that caused U(t)∆tβ to exceed five. Nominal 6.2 µm diameter sulfonated polystyrene particles from Interfacial Dynamics Corp. were suspended in 0.13 mM solutions of potassium hydroxide or sodium bicarbonate. The electrolyte solutions were prepared with deionized water (R > 16 MΩ) immediately prior to each set of experiments. All initial solution pH measurements were consistent with both the Debye length measured from the PE profile and the expected value. The particle solution also was made just prior to use by combining the electrolyte solution with a much smaller volume of concentrated particle dispersion. The dispersion was injected into the experimental cell appearing in Figure 1 and comprising two indium tin oxide (ITO)coated glass slides (Bioptics Corp.) separated by a 1.34 mm polycarbonate spacer and optically connected to a prism using matching fluid (R.P. Cargille Laboratories Inc., Cedar Grove, NJ). The particles sedimented to the electrode from which the majority were then flushed by injection of additional solution to increase the average nearest-neighbor distance. Particles that displayed unchanging scattered intensities or that obviously differed from the nominal 6.2 µm diameter were rejected. A total of 100 000 observations of scattering intensity recorded at 10 ms intervals were taken from each new particle in order to accurately determine 〈h0〉 and its field-free PE profile from eq 1. TIRM measurements in the absence of an electric field, 100 000 points at 1 ms intervals, were also interleaved between electric field experiments to check for drift and to equilibrate the system. The interleaved field-free measurements were consistent with the initial PE profile over long time periods in KOH; however, in bicarbonate the behavior of the particles often changed, including attraction to the electrode followed by adhesion after several hours. When a particle’s behavior changed in bicarbonate, all of the data collected since the last consistent field-free measurement were discarded, and the cell was disassembled. A typical ac electric field experiment consisted of three parts. First, the field was applied for 2 min for the particle to reach a stationary but oscillating behavior. Then the intensity was measured at 1 ms intervals for between 100 and 500 s, and finally the field was disconnected for at least 3 min between trials. The time between experiments was maintained to allow the ion concentration profiles to relax to equilibrium and to allow any heat generated via ohmic heating to dissipate. The room temperature was maintained within (1 °C. Integer relationships between sampling rates and the period of the electric field at certain frequencies were exploited during experiments to allow for phase consistent measurements. For example, at a digital sampling rate of 1000 Hz, the period of 40 Hz (25 ms), 50 Hz (20 ms), 62.5 Hz (16 ms), 100 Hz (10 ms), 125 Hz (8 ms), etc., waves contain an integer number of samples per cycle. The data acquired in each cycle was superimposed to eliminate random noise such as Brownian motion and thereby determine h h (τ) as described in the definition of this quantity.

Results and Discussion In the absence of an applied electric field, a colloidal particle samples a distribution of elevations around the hm due to Brownian motion; representative experimental height data appear in Figure 2. Potential energy profiles constructed from the distribution of particle heights via eq 1 for a 6.6 µm sulfonated polystyrene particle above an ITO electrode at several salt concentrations and zero applied electric field appear in Figure 3. This manner of

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Figure 2. Particle height h(t) recorded as a function of time for a 6 µm polystyrene particle above an ITO electrode with no applied electric field in 0.13 mM potassium hydroxide. The changes in elevation of the particle are caused by Brownian motion.

Figure 3. Measured potential energy profiles for a 6.6 µm polystyrene sphere above an ITO electrode without an applied electric field as a function of salt concentration at constant pH. Each profile was recorded at a different salt concentration: from left to right the concentrations were 0.715, 0.503, 0.305, 0.168, and 0.077 mM potassium hydroxide. Salt concentrations above 0.077 mM consisted of 0.077 mM potassium hydroxide and potassium chloride. At higher salt concentrations, the particle samples a smaller range of heights closer to the surface. As concentration is increased, more of the van der Waals attraction is evident.

displaying the potential energy profile17,18 adjusts the reference state for potential energy at each salt concentration so that the linear portions (where gravity dominates) of each profile overlap. The hm for each salt concentration is the point of lowest potential energy on its curve in Figure 3. 〈h0〉, calculated from experimental data such as in Figure 3, was typically 25-30 nm above hm. Figure 4 shows the relative positions of 〈h0〉 and hm for a representative potential energy profile. Dynamics of Particles Subjected to an Alternating Electric Field. Initial Response to an Alternating Voltage and the Approach to Steady Oscillation. A particle undergoing Brownian motion in an equilibrium potential well, as in Figure 2, behaves very differently when an alternating electric field is applied. Figure 5 shows four (17) von Grunberg, H. H.; Helden, L.; Leiderer, P.; Bechinger, C. J. Chem. Phys. 2001, 114, 22. (18) Haughey, D.; Earnshaw, J. C. Colloids Surf. A: Physicochem. Eng. Aspects 1998, 136, 217-230.

Fagan et al.

Figure 4. Schematic of the difference in average, 〈h0〉, and most likely, hm, heights at equilibrium given an asymmetric potential energy profile. The difference in the two quantities is caused by Brownian motion; in this case Brownian motion causes the particle to spend more time at the elevations greater than hm because the potential energy slope due to gravity is less steep than the slope due to electrostatic repulsion; this results in 〈h0〉 being greater than hm.

Figure 5. Four transient trajectories of particle height from the application of a 62.5 Hz, 3 Vpp electric field. In each case, independent of starting height, the initial electric field shot the particle up to a maximum height; as all the forces on the particle came into play, the moving average decreased toward the 〈h〉 denoted by the line at 100 nm. The heavy black line is the average trajectory over the coming cycle of the electric field. The line at approximately 140 nm is the average before the electric field is applied.

individual responses of the same particle to a suddenly applied electric field alternating at 62.5 Hz. The initial motion of the particle is shown as a function of the logarithm of time so the response of each particle can be distinguished. The initial height of the particle was different in each of the four trajectories in Figure 5, yet within 250 ms the four trajectories overlap. Thus, despite different initial particle heights, all the trajectories became nearly indistinguishable after a brief transition time. The total vertical distance traveled by each particle in the first 16 ms (first cycle of the field) was greater than the distance traveled during any later cycle of the electric field. Figure 6, which displays the average of five transient responses to a 100 Hz 11.2 V/cm electric field in KOH, shows that the characteristic behavior continues unchanged over a longer time period. The moving average of the particle height in Figure 6 reaches the stationary average value in less than half a second. In both Figures


Langmuir, Vol. 18, No. 21, 2002 7815

Figure 6. A moving average height of five transient particle height responses to the application of a 3 Vpp, 100 Hz external electric field. The trajectories show the initial maximum of Figure 5, a rapid fall through 〈h0〉 (long dashes) to steady oscillatory behavior about the 〈h〉 (short dashes). The dashed line is the average height before application of the oscillating field. The moving average of the trajectory over one cycle displays only small deviations from the 〈h〉 once the steady oscillatory behavior is reached.

5 and 6, the new 〈h〉 that the particle adopts over several hundred milliseconds equaled the average height measured after several minutes. Waiting 2 min for the stationary state was more than sufficient to record steady behavior. Particle Trajectories during Steady Oscillation. We now compare the behavior of a particle without an applied field to the steady oscillation obtained after the transition period. Parts a and b of Figure 7 show sequentially measured particle heights with no applied electric field and with field applied, respectively, over the time interval corresponding to one ac cycle; 〈h0〉 is also plotted in Figure 7a as a heavy horizontal line. The force due to the electric field dominates the particle motion at electric fields above a few V/cm; Brownian dynamics are superimposed on the driven motion. Of note in Figure 7b is that the Brownian excursions from the average value (indicated by the solid curve) are apparent when the particle is far from the surface and suppressed when the particle is close. The amplitude of motion is approximately 200 nm, which is about 3% of the particle’s diameter. TIRM sensitively measures the driven dynamics of particles near surfaces. Phase-consistent averaging, eliminating the effect of Brownian excursions, resolves the correlated particle trajectories from the stochastic height data. Figure 8 shows h h (τ) in KOH at several applied field strengths and 50 Hz. Both oscillatory and nonoscillatory components are present in h h (τ). For example, the particle height at 1.4Vpp is always lower than the height at 1Vpp, but the amplitude of its motion is greater, which indicates an offset. The offset can be removed by normalizing each particle trajectory by 〈h〉; thus, Figure 9 shows normalized particle trajectories in KOH at 50 Hz varying applied field strengths. Figure 9 emphasizes both the monotonic increase in amplitude of the particle excursions and the increasing distortion from sinusoidal behavior of the particle trajectories with increased applied field strength. The distortion is highlighted in Figure 10 where a single particle trajectory from Figure 9 is shown with a sinusoid fit to the highest 10 points; also shown for comparison are a reflection of the trajectory about its average height and another reflection about the half period. Both reflections emphasize the asymmetry of the particle motion and the deviation from purely sinusoidal behavior.

Figure 7. (a) Absolute height of a particle observed at 1 ms intervals over approximately 40 ms elapsed time, with no applied field. The solid line is 〈h0〉 obtained after averaging 150 repetitions of such data. (b) The corresponding heights observed with a 50 Hz, 5 Vpp external applied field. The individual points in (b) show stochastic Brownian fluctuations superimposed on the driven motion. Averaging 150 cycles produces the smooth line, h h (τ). Although the elevations for single pass data show some variation, particularly at higher elevations, individual trajectories are closely correlated to h h (τ). An important note is that the error bars on the individual points are height dependent due to the height dependent ratio of the constant background scatter with the exponentially related scattering due to the particle.

Discussion of the Dynamic Studies. The measured h(t) of a particle exposed to a 50 Hz ac electric field (Figure 7b) demonstrates that a charged particle moves perceptibly in response to a low-frequency ac electric field. The amplitude of motion was a few percent of the particle radius. The response of a particle at low fields is sinusoidal because the particle experiences only constant or weakly changing forces and mobility as it oscillates a small distance up and down due to the electric field. As the electric field strength is increased, however, the amplitude of the oscillations increases (see Figure 8), and the forces on the particle change substantially over the range of elevations sampled. The particle’s trajectory becomes an imperfect sinusoid as shown in Figure 10. The amplitude of the particle’s excursion below 〈h〉 becomes less than the amplitude above and cannot be fit with same sine function as the upper half of the trajectory. This distortion of the sinusoid below 〈h〉 is an effect expected in the EK model due to the asymmetry of colloidal forces and the reduced particle mobility at small separations.14,15 As the particle approaches the surface, the mobility of the particle decreases with height (see eq 6). This effect, independent

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Figure 8. Effect of applied field strength on the 5000 cycle h h (τ) of a polystyrene particle above an ITO electrode subjected to an applied electric field of 50 Hz. Note that the trajectory at 1.4 Vpp is at all points below the trajectory for 1 Vpp, despite the larger amplitude of motion at 1.4 Vpp.

Figure 9. Normalized height comparison of trajectory shapes at 50 Hz as a function of applied electric field strength. The trajectories become less sinusoidal with increasing field strength, which increases the amount of time the particle is at an elevation smaller than its average.

of direction if inertial effects are small, directly reduces the distance the particle travels for a constant force as h(t) f 0. The hindered mobility consequently reduces the amplitude of the particle’s excursion below 〈h〉, which increases the fraction of the cycle that the particle spends beneath its average, which accounts for the forward shift (with increasing voltage) of the time at which the particle crosses its average value as per Figure 8. Averaged Response of Particles to an Alternating Electric Field. Average Particle Heights. We now focus on 〈h〉, the time average height, as an important source of information about the forces acting on the particle. Figures 11 and 12 show 〈h〉/〈h0〉 in KOH as a function of voltage at constant frequency and as a function of frequency at constant applied voltage, respectively. The first remarkable result is that both Figures 11 and 12 exhibit minima in 〈h〉/〈h0〉 for KOH. The particle is pushed, on average, toward the electrode in KOH as voltage is increased from zero in Figure 11. At higher voltage amplitudes, the average height 〈h〉 in KOH returns to the

Fagan et al.

Figure 10. Comparison of averaged height h h (τ) at 5 Vpp, 50 Hz (solid line) to a perfect 50 Hz sinusoid fit to the highest nine points of the trajectory (dashed line). The trajectory clearly deviates from the sinusoid below 〈h〉. A reflection of the bottom half of the trajectory is included to emphasize the difference in amplitude of motion.

Figure 11. Measured (symbols) and EK model predicted (solid line, eq 12; dashed line, eq 11) dependence of the quotient 〈h〉/ 〈h0〉 of a polystyrene particle above an electrode as a function of applied field strength for a 100 Hz electric field. The measured 〈h〉/〈h0〉 in potassium hydroxide (closed circles) goes through a minimum. In contrast, the 〈h〉/〈h0〉 measured in sodium bicarbonate (open triangles) is monotonic and opposite in initial direction relative to the potassium hydroxide data. The reversal of the measured behavior is evidence that Brownian motion does not cause the minimum in 〈h〉/〈h0〉 and that a force absent from the EK model must be present. The Brownian adjusted EK model never predicts an 〈hEK/BC〉/〈h0〉 less than unity.

equilibrium average. The average height 〈h〉 in KOH is greater than the equilibrium average height at the lowest frequencies (Figure 12), plunges below the equilibrium height at intermediate frequencies, and returns toward the equilibrium height at the highest frequencies sampled. The 〈h〉/〈h0〉 of a particle in bicarbonate also appears in Figure 11 (open triangles) as a function of voltage at 100 Hz. The second remarkable result of this work is that the particle in bicarbonate (Figure 11) is pushed on average away from the electrode as voltage is increased from zero, in stark contrast to its behavior in KOH. In fact, the particle moves so far away from the electrode in bicarbonate, at least half a micron, that TIRM is unable to track it except for the low voltages shown in Figure 11. The observation that the behavior of 〈h〉/〈h0〉 in bicarbonate is qualitatively different from the behavior in KOH is significant.


Langmuir, Vol. 18, No. 21, 2002 7817

Figure 12. Measured (points) and EK model predicted (solid line, eq 12; dashed line, eq 11) dependence of the normalized average height of a polystyrene particle above an electrode. The applied voltage was 5 Vpp in KOH. The EK model predictions are monotonic, while 〈h〉/〈h0〉 clearly goes through a minimum.

Figures 11 and 12 include the two theoretically calculated 〈hEK〉 predictions of the electrokinetic model also divided by 〈h0〉. 〈hEK/D〉 and 〈hEK/BC〉 for a polystyrene particle calculated at a frequency of 100 Hz and normalized by 〈h0〉 appear in Figure 11. With or without the compensation for Brownian motion, the model predicts a monotonic increase of average height with applied voltage in Figure 11. Also, both the deterministic and Brownian-corrected EK models predict that 〈hEK〉 exceeds 〈h0〉 at low frequencies (Figure 12) as a response to the standard colloidal forces and the dependence of mobility on the gap. 〈hEK〉/〈h0〉 is greater than unity at low frequencies because both the reduction in mobility and PE profile shape favor an increase in 〈hEK〉 as frequency decreases. The diminishing mobility of the particle with decreasing height and the impenetrability of the wall shift 〈hEK〉 outward from 〈h0〉, since, starting at the same initial position, the particle cannot move as great a distance toward the surface as it can move away from the surface for the same applied force. Additionally at lower frequencies, the particle has time to travel farther, which increases the amplitude of oscillation causing the particle to spend a greater proportion of time at higher elevations. Thus, 〈hEK〉 exceeds 〈h0〉 at low frequencies according to both model and experiment. An important detail to remember is that an electrokinetic theory of particle motion does not distinguish between bicarbonate and KOH at the same concentration and ζ; thus, the predictions for 〈hEK/D〉 and 〈hEK/BC〉 are identical for both electrolytes even though Figure 11 reveals that the particle behaves quite differently in the two electrolytes. Phase Angle Measurements. We measured the phase angle between the internal electric field (i.e., current) and the particle motion as well as the angle between the internal electric field and the external electric field. Figure 13 shows representative data from which these phase angles were measured. The current in Figure 13a is nearly a pure sinusoid, but in Figure 13b, it deviates from pure sinusoidal behavior; there are inflections indicating nonlinear current-voltage behavior. Likewise, the intensity of scattered light from the particle loses symmetry at lower frequency as well. Figure 14 summarizes the two measured phase angles over a range of frequencies. The phase angle between the current and the scattered light intensity is π/2 at high frequency but declines from this value at frequencies below 100 Hz. The phase angle

Figure 13. (a) Current (dashed curve), applied voltage (solid curve), and intensity (dotted curve) signals recorded at 50 Hz for a 5 Vpp electric field. The current and intensity values have been scaled for visual comparison to the voltage. The phase angle between the current and the light intensity is marked as θ. The light intensity is at maximum when the particle is closest to the surface. (b) Same information as in (a), but the frequency is 5 Hz. The phase angle between the current and the external voltage is labeled as φ. Note the nonlinear current-voltage relation at 5 Hz, evidence that a faradaic reaction is occurring.

between the current and the voltage is zero at high frequencies but increases from zero below 100 Hz. Discussion of Averaged Results: Evidence of an Additional Force. The key experimental result of this paper is the profoundly different trends in average particle height with frequency and amplitude observed with KOH and NaHCO3. In KOH at 100 Hz, Figure 11 shows that a particle’s 〈h〉 is approximately unaffected at low voltages, decreases to go through a minimum (below hm) at larger voltages, and then increases at higher voltages. In NaHCO3 at 100 Hz, a particle’s 〈h〉 (also Figure 11) increases strongly and monotonically as the voltage is increased from zero. These data indicate the presence of a force that acts differently in different solutions. While electrokinetic forces clearly move the particle up and down as in Figure 7b, neither the double layer force nor the electrokinetic force depends on the choice of 1:1 electrolyte (zeta potentials are practically unchanged at our conditions). This invariance with electrolyte is illustrated by the theoretical EK model averages provided in Figure 11. There is only one average curve and its Brownian-adjusted twin; there is not a curve for KOH and a different curve for bicarbonate. Coupling of Brownian motion with the ac electrokinetic force might lead to a minimum in the dependence of the average height on frequency or voltage,

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Figure 14. Summary of current-voltage (circles) and currentintensity (triangles) phase angles as functions of frequency in 0.15 mM potassium hydroxide. The highest frequency data were measured with a faster acquisition setting of the photomultiplier electronics to ensure that enough samples were acquired. We overlapped measurements to demonstrate the effect of any artifacts due to this change; they were minimal. Most significant is the current-intensity phase angle, which does not deviate measurably from 90° in the EK model, but deviates significantly from 90° in a nonmonotonic manner in the experiment. This discrepancy is a strong indicator that another force on the particle, possibly an EH effect, is present. Additionally, model predictions for the current-voltage phase angle are plotted for the conditions {Rohmic ) 612 Ω, RCT ) 709 Ω, and Cdl ) 8.5 µF/cm2} (solid line, eq 16) and {Rohmic ) 612 Ω, Cdl ) 8.5 µF/cm2, RCT ) ∞} (dashed line, eq 16).

but this dependence is not likely to depend on the choice of electrolyte. Furthermore, neither the deterministic model nor the Brownian-compensated model (eq 12) matches the observed behavior in either electrolyte, although the slope of the KOH data at large applied fields (strong EK convection) does agree with the model predictions. In all cases, the results of the semiempirical model are monotonic. The fact that different behavior is observed in two systems that are indistinguishable to the electrokinetic and colloidal forces is the primary result in this paper and the main argument for the existence of an additional force in the system. Further evidence of an additional force is the observed 〈h〉 in KOH with a 5 Vpp electric field (Figure 12). 〈h〉 is higher than 〈h0〉 at low frequencies for a constant voltage in KOH; it decreases below 〈h0〉 in the midrange of frequencies and approaches 〈h0〉 at frequencies greater than a kilohertz. The solid line in Figure 12 shows that both models for 〈hEK〉 decrease from a large value at low frequencies and asymptotically approach constant values, 〈h0〉 and hm, respectively, at high frequencies. Both model predictions are monotonic. In Figure 12, 〈h〉 resembles the calculated lines at low frequencies, is lower than either prediction at intermediate frequencies, and asymptotically approaches the Brownian-adjusted model at high frequencies. At high frequencies, the particle does not move far enough for the average height to be significantly affected by the electrophoretic motion. If there is a force that is present at frequencies lower than a kilohertz, it vanishes at frequencies above that level. For the average height of a particle to decrease with the application of an ac electric field, as in KOH, either the particle must experience an additional force or one of the colloidal forces must decrease. Additionally, the change or additional force must be strong enough to counteract both the electrokinetic effects and the asymmetry of the

Fagan et al.

Figure 15. A representation of how EH forces would affect a colloidal particle near an electrode in potassium hydroxide. The fluid is imagined to flow outward from the particle, requiring fluid to flow down on top of the particle to maintain continuity. The flow of the fluid exerts a hydrodynamic drag on the particle, which lowers its 〈h〉, and affects the phase of the particle motion relative to the current.

colloidal forces. A first hypothesis was that increases in the ionic strength near the electrode, due to either the electric field or a reaction, shorten the Debye screening length in a manner that couples with the particle motion to decrease the average height. However, simulations of particle motion that included the expected effect on the Debye screening length indicated ionic strength effects were much smaller than the electrokinetic effect and did not change the monotonic nature of the electrokinetic response; therefore, other possible mechanisms must be explored to explain the additional force on the particle. The EH Model. Another phenomenon that is expected near an electrode in an ac electric field is electrohydrodynamically driven flow. As such, a force on the particle caused by the EH mechanism might be responsible for the observed behavior. Sides11 showed that the interactions between a lateral secondary electric field and unbalanced charge arising from concentration gradients produce a spatially varying body force within the fluid near the particle. Radial convection of the fluid near the particle driven by the EH body force requires fluid to move past the particle, either upward or downward, to satisfy mass continuity. The drag on the particle caused by the fluid convection could theoretically cause an additional effective force on the particle. Figure 15 shows a schematic representation of the possible fluid flow arising from this outwardly directed force density. In the EH model, the lateral force at a given position for a binary electrolyte with a reacting anion, averaged over one cycle, is estimated by11

〈F h r(r*, ζ)〉 ≈

[

] (

)

-RTtˆ ∂φ* ∂2θ 1 Re itot* 2 4Fa2κ∞ ∂r* ∂ζ2

(14)

where is the electric permittivity, R is the gas constant, T is temperature, F is Faraday’s constant, κ∞ is the bulk electrolyte conductivity, θ is the normalized electrolyte concentration, and itot is the total current. In this equation only, ζ represents the distance from the electrode normalized by the particle diameter. The equation shows that the force is proportional to the total current and the second derivative of the concentration. Nonzero concentration gradients are a necessary condition for the existence of a force according to this model, and concentration gradients only exist when at least part of the total current is faradaic; an EH force only arises, therefore, when an electrochemical


Langmuir, Vol. 18, No. 21, 2002 7819

reaction is occurring. The most interesting features of eq 14 are the lack of dependence on particle charge and the proportionality to a pure transport parameter, given by

ˆt ≡

t + t+ z+ z-

(15)

where ti are the transference numbers; in an electrolyte with no concentration gradients, they represent the fraction of current carried by a particular ion. The quantity ˆt can be either positive or negative depending on the mobilities of the ions of the binary electrolyte. For a binary system with a known reacting anion, a change in the sign of ˆt would reverse the direction of longitudinal force on the particle. In KOH, for example, the parameter ˆt is negative, and the average force density as a whole is therefore positive. To satisfy mass continuity in KOH, electrolyte from above the particle must replace the fluid moving away from the particle; the replacement fluid flows down on top of the particle and flows outward away from the particle, the direction of convection which should cause a downward drag force on the particle. According to the theory, the flow is low at low frequencies, goes through a maximum at intermediate frequencies, and decreases at high frequencies, all in the range from 0 to 1000 Hz. Comparison to Figure 12 shows that 〈h〉 of a particle in KOH is unaffected at high frequencies, where the EH flow is expected to be weak, goes through a minimum, near in frequency to maximum in predicted EH flow, and increases at low frequency. Thus, the observations in KOH are consistent with an EH mechanism. As a side note, Kim et al.10 have confirmed that particle pairs move apart in KOH and that the effect decreases with frequency. Kim et al.10 have also shown that particle pairs near an electrode in an ac electric field move toward each other in solutions of NaHCO3. For particles to aggregate in NaHCO3, the direction of the flow around a particle in NaHCO3 must be the reverse of that in KOH. If the flow passing the particle in bicarbonate is upward, the particle should be levitated upward away from 〈h0〉. Thus, in contrast to the depressed 〈h〉 in KOH, an increased 〈h〉 is expected for bicarbonate. The predictions of the EH mechanism appear to agree with the observed reversal in direction between KOH and NaHCO3; ˆt is negative in KOH and is positive for NaHCO3. However, Sides’s11 EH model is for a binary electrolyte with a known reacting anion; since it is unlikely that either the cation or the bicarbonate ion participate directly in the electrode reaction, the model is not directly applicable without further investigation. The data for NaHCO3 (open triangles) in Figure 11 nevertheless demonstrate that 〈h〉 in bicarbonate increases strongly and monotonically with electric field strength, as predicted by the reversal in direction of EH flow, at the same conditions that 〈h〉 was depressed in KOH. Comparison with the same theoretical EK predictions also indicates that the increase in height is much stronger than the predicted 〈hEK/BC〉/〈h0〉 due to the EK model, which makes it unlikely that an additional force is present in KOH but not in NaHCO3. In fact, the levitation effect in bicarbonate is strong enough that 〈h〉 becomes difficult to measure using TIRM; the particle rises out of the range where scattering due to the particle exceeds the background noise. Evidence of an EH Force from Phase Measurements. Phase angle measurements also support the hypothesis that EH forces affect the vertical oscillatory motion of single particles. Two phase angles are of interest: one is

Figure 16. Equivalent linear circuit element model of the electrochemical cell. The presence of capacitors in the circuit causes the phase angle between the current and external electric field to be nonzero.

the phase angle between the internal electric field (current) and voltage that arises because an electrochemical circuit contains both resistances and capacitance. The second is the phase angle between the internal electric field and the output of the photomultiplier (i.e., the height of the particle). An electrochemical cell is often depicted as the equivalent circuit shown in Figure 16, which contains two faradaic reaction resistances in parallel with double-layer capacitances. These elements, modeling the electrode and its faradaic reaction, are connected by an ohmic solution resistance. In the absence of any mass-transfer related impedance, the phase between the current and the voltage can be deduced from the equivalent circuit,

Φ ) arctan

(

2ωCdlRF2

)

RΩ(1 + ω2Cdl2RF2) + 2RF

(16)

where Φ is the phase angle, RΩ is the solution resistance, and Cdl and RF are the double-layer capacitance and faradaic resistances, respectively. Depending on the combination of Cdl, RF, and RΩ, the phase angle can vary from zero at all frequencies to leading the external field by 90° at zero frequency. Finite values of the faradaic resistances lead to a phase angle that is a nonmonotonic function of frequency and approaches zero at both frequency limits. While the current/voltage phase angle in Figure 14 does not clearly decline toward zero at low frequency, the phase angle also does not appear to be heading toward 90°, as it would if there were no faradaic resistance. Additionally, Figure 13b shows that at low frequency the current through the cell is no longer linearly dependent on overpotential. The inflections in the curve suggest nonlinear current-voltage behavior. Both of these observations are evidence that a finite faradaic resistance (denoting the presence of a reaction) is present in the system. This is important because an electrode reaction is a necessary component of the EH theory. The phase angle between the current and the particle height also is an important quantity. If the quasi-steadystate assumption holds, the observed maximum in the output of the photomultiplier (minimum in the particle height) should lag the maximum of the electric field across the electrolyte solution by exactly 90°. The fact that this phase angle is neither 90° in Figure 14 nor a monotonic function of frequency in the range of frequency where EH effects are pronounced provides strong evidence that forces other than those caused by electrokinetic effect are present in the system. The EK mechanism does not predict a significant phase shift from the current at any frequency. Conclusions The motion of a colloidal particle normal to an electrode when exposed to an alternating electric field was a complex

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function of the applied field strength, the frequency of the field, the charge on the particles, and the choice of electrolyte. The electric field profoundly affected the particle dynamics in the direction normal to the electrode. The motion up and down was not sinusoidal because the hydrodynamic hindrance to motion and colloidal forces varied strongly as a function of height. In bicarbonate solutions, particles are substantially pushed away from the electrode during ac polarization, on average perhaps as much as a micron. In stark contrast, in KOH solutions, particles are drawn closer to the electrode. The average behavior of the particle with an ac field was not consistent

Fagan et al.

with a deterministic superposition of electrokinetic and colloidal forces in the convective limit. The observed behavior was consistent, however, with the presence of an additional time varying force on the particle. In particular, the behavior of this force was consistent with an electrohydrodynamic model for particle-pair motion in ac electric fields. Acknowledgment. This work was supported by a grant from the National Science Foundation, CTS0089875. LA025721L

Vertical Oscillatory Motion of a Single Colloidal Particle Adjacent to an

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