Single and Pairwise Motion of Particles near an Ideally Polarizable

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Single and Pairwise Motion of Particles near an Ideally Polarizable Electrode Christopher L. Wirth,* Reza M. Rock, Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States ABSTRACT: Single-particle longitudinal motion and pairwise lateral motion was investigated while the particles were excited by an oscillating electric field directed normally to an electrode proximate to the particles. The electrode was polarized over a range of potential insufficient to drive electrochemical reactions, a range called the “ideally polarizable region”. The particles’ motion was qualitatively dependent on the choice of electrolyte despite the absence of electrochemical reactions. As when electrochemical reactions were not explicitly excluded, the phase angle θ between particle height and electric field correlated with the particles' separation or aggregation during excitation. A simple harmonic oscillator model of the particles’ response, including colloidal and hydrodynamic forces and including the Basset force not previously cited in this context, showed how θ can increase from 0° at low frequencies, cross 90° at ∼100 Hz, and then increase to 180° as frequency was increased. The model captured the essence of experimental observations discussed here and in earlier works. This is the first a priori prediction of θ for this problem.

1. INTRODUCTION Control of nanometer- to micrometer-scale particles benefits engineering applications ranging from the preparation of photonic materials1 to the separation of colloids.2,3 An electric field is a common tool for manipulating particles through electrophoresis, dielectrophoresis, and electroosmosis. One approach based on electrokinetic phenomena produces colloidal crystals on an electrode by application of a direct or alternating current (dc or ac) electric field.4
34 Although the mechanism that drives crystallization in dc fields is well established, the mechanism responsible for particle aggregation in ac fields is much more subtle.35 Particle assembly in ac fields depends on particle confinement, electric field strength, and frequency. Perhaps the most dramatic dependence of particle assembly on system parameters is the effect of electrolyte. Negatively charged particles dispersed in KOH, NaOH, or NH4OH separated upon application of a 100 Hz electric field, but the same particles dispersed in NaHCO3 or KCl aggregated.29 This behavior was observed both on tin-doped indium oxide (ITO) and on platinum thin-film electrodes. Studying the relative motion between two particles is a first step for developing an understanding of multibody colloidal crystals. Consider two colloidal particles near a working electrode (WE) as shown in Figure 1. Particles move laterally (xy plane in Figure 1) upon application of an ac electric field. Most investigators monitor the time-dependent separation distance r(t) in the xy plane because this information is accessible with video microscopy (VM). Unfortunately, ordinary VM typically has low temporal resolution (∼30 ms) when compared with the period of oscillations of electric field (1
100 ms). Consequently, any effect occurring on the time scale of the electric field’s period eludes ordinary VM. High-speed VM is available, but a second problem arises when trying to resolve the microscopic amplitudes of lateral motion in any given cycle. At r 2011 American Chemical Society

Figure 1. Isolated pair of colloidal particles proximate to a working electrode (WE). The particles each have radius a, height h, and particle
particle separation distance r. Video microscopy (VM) is used to measure r but can only measure steady motion because of low spatial and temporal resolution of typical VM experiments. Total internal reflection microscopy (TIRM) is used to measure h. TIRM measurements reveal the particles’ h oscillates at the primary driving frequency of the electric field.

100 Hz, the lateral amplitude of oscillations is approximately 1% of the particle’s radius, which is difficult to resolve.27 Total internal reflection microscopy (TIRM), however, measures WE
particle separation distance h(t) with both nanometer and microsecond resolution. These measurements reveal that the particle oscillates longitudinally (z axis in Figure 1) at the primary frequency of the electric field with amplitude up to 20% of the particle radius a. While observing a single particle’s vertical motion with TIRM in a search for clues about the mechanism of lateral motion, Fagan et al.17 discovered a phase angle between the oscillating particle height and the electric field driving the particles’ motion. Under pseudosteady electrophoresis, the instantaneous particle velocity normal to the electrode is directly proportional to the instantaneous electric field. Received: May 8, 2011 Revised: June 30, 2011 Published: July 21, 2011 9781

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Langmuir A sinusoidal electric field should engender a sinusoidally varying velocity, the integral of which [i.e., h(t)] is 90° out of phase with the electric field. Thus, the authors expected to measure the phase angle θ = 90°. However, while dispersed in KOH, NaOH, or NH4OH, the particles separated and θ < 90°, but when dispersed in NaHCO3 or KCl, the particles aggregated and θ > 90°. The sign of cos θ strongly correlated with the direction of pairwise particle motion. This correlation between single-particle vertical motion (i.e., h) and pairwise lateral motion (i.e., r) was embedded in a drift velocity model.28 The essence of the authors’ model was that quasi-steady electroosmotic flow driven along a target particle’s surface entrains neighboring particles; θ ¼ 6 90° breaks the antisymmetry of the sinusoidally varying flow. Hoggard et al.29 provided substantial experimental evidence supporting the model. The theory was successful but required measurement of the phase angle; the apparatus necessary for the measurement is available in only a few laboratories. A model that allows prediction of the phase angle from more elementary properties is desired. Fagan et al.23 speculated that faradaic reactions were responsible both for the phase angle, and hence for lateral motion, and for its electrolyte dependence at low frequencies. In other words, the phase angle in general and the opposite behavior in different electrolytes in particular, which resembles chemical-like selectivity, might have ensued from the chemistry of electrochemical reactions at the electrode. Different chemical reactions and different phase relationships between the current and electrode potential could have been occurring in different electrolytes. Faradaic current was thought to perturb the working electrode’s equilibrium ζ potential, which is possible for dc currents.36 One observation supporting faradaic reactions as the cause was that lateral particle motion was electrolytedependent in low-frequency (less than 100 Hz) electric fields but not in high-frequency (greater than 1000 Hz) electric fields; the importance of faradaic current relative to double-layer charging current decreases with increasing frequency.35 These two fundamental points, the relevance of faradaic reactions and the origin of the phase angle, are the topics of this paper. We have directly tested the importance of faradaic reactions by conducting experiments in a region of electrode potential where electrochemical reactions were not thermodynamically possible. Even though no electrochemical reactions occurred, we found that phase angles and strong electrolyte dependence of the relative motion (between particles in an isolated pair) persisted. The knowledge that the phase angle must arise from purely physiochemical effects spurred new inquiry into its origin. The phase angle itself, the missing component of the model developed by Fagan et al.,28 is the second focus of this contribution. We present a dynamic model that describes the longitudinal motion of a charged particle in the presence of an ac electric field normal to an electrode. Incorporating double layer repulsion, fluid friction, the electrophoretic force, inertia, and a force not previously recognized in this context, the Basset force, the model demonstrates a purely physical basis for the existence of the phase angle. The model, although it provides a convincing basis for the phase angle, does not yet predict the electrolyte dependence observed experimentally; we suggest possible refinements involving ion mobility, which might lead to phase angles having different signs of cos θ in different electrolytes.

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oscillatory force on the particle. As a first step in predicting this phase angle, let us ignore the corrections associated with the particle’s close proximity to an electrode. Consider a rigid sphere, immersed in an infinite sea of fluid, responding to an oscillatory force. The equation governing one particle’s longitudinal motion along the z-axis is MP

The phase angle between oscillations in the elevation of a single isolated particle and the electric field results from the action of an

ð1Þ

where Mp is the mass of the particle, Fd is the dynamic drag force on the particle, Fc is the summation of colloidal forces acting on the particle, ω is the angular frequency of the electric field, and t is time. QE∞eiωt is the electrophoretic force acting on the particle for an electric field oscillating as eiωt, E∞ is the electric field at infinity, and Q is the particle’s apparent electrokinetic charge. Q is much smaller than the true charge borne by a particle with a radius much larger than the solution’s Debye screening length, k
1: see Appendix A. The dynamic drag force comprises quasi-steady Stokes drag, inertia of the displaced fluid, and the Basset force. Fd is given by37 Fd ¼


f

dh 1 d2 h þ Mf 2 þ 6ηa2 dt 2 dt

! rffiffiffiffiffiffiffiZ t 2 0 πFf d hðt Þ dt 0 p ffiffiffiffiffiffiffiffiffiffi η
∞ dt 02 t
t0

ð2Þ where f is the usual Stokes-drag coefficient (f = 6πηa), Mf is the mass of the fluid displaced by the particle, t0 is a dummy integration variable representing past values of the time t, and η and Ff are the fluid’s viscosity and density, respectively. The first term on the righthand side of eq 2 is commonly found in colloid science; the remaining two terms might be less familiar. The second term is the inertia of the fluid displaced by the particle. This term arises because any acceleration of the particle also causes some fluid to accelerate. The third term is the Basset, or “history”, force arising from the transient development of the momentum boundary layer next to the oscillating particle for an arbitrary history of past position h(t). The present model is the first inclusion of the Basset force in this context. Both the fluid inertia as well as the Basset force are corrections that account for unsteady drag on the particle. Both terms are derived by solving the unsteady form of the Stokes equation in which the nonlinear inertial terms (in the more general Navier
Stokes equation) are dropped (requires Reynolds number , 1) but the linear inertial term Ff dv/dt is retained. More information on eq 2 can be found in Kim and Karrila.37 The colloidal forces acting on the particle are given by the derivative of the potential energy, ϕ(h), between the particle and wall [i.e., ∑Fc =
dϕ(h)/dh]. When both ka . 1 and kh . 1, the total potential energy as a consequence of colloidal forces between the particle and wall is ϕðhÞ ¼ Be
kh þ Gh  2 kT tanh B ¼ 64πεa e

k¼

2. THEORY 2.1. Simple Harmonic Oscillator Model of Particle Motion.

d2 h ¼ Fd þ Fc þ QE∞ eiωt dt 2

2e2 C∞ εkT

ð3Þ eζp 4kT

!



eζe tanh 4kT

 ð4Þ

!1=2 ð5Þ

4 G ¼ πa3 ðFp
Ff Þg 3

ð6Þ

where eq 5 is for a symmetric 1:1 electrolyte, ε is water’s dielectric permittivity, k is Boltzmann’s constant, T is temperature, e is the 9782

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elementary charge, C∞ is the bulk number concentration of electrolyte, and ζp and ζe are the particle’s and electrode’s Stern potential. Equation 3 has a single minimum at h = hm. The charge parameter B can be expressed in terms of hm, leaving38 ϕðhÞ
ϕðhm Þ G
kðh
hm Þ G ¼ ½e ðh
hm Þ
1 þ kT kkT kT

ð7Þ

which describes the potential energy between a sphere and wall at the relative separation distance h
hm. In the absence of an applied electric field, the particle relaxes to its equilibrium position h = hm and remains there forever. In the limit of a very weak oscillatory electric field, the particle undergoes steady oscillations around h = hm with the amplitude of the oscillations in h(t) proportional to the amplitude of oscillations in E(t). In this limit, Fc becomes linear in h. The leading terms of a Taylor-series expansion of eq 7 around the most probable separation distance, hm, are ϕðhÞ = ϕðhm Þ þ

Gk ðh
hm Þ2 2

ð8Þ

Finally, the total colloidal force acting on the particle is obtained by differentiating Fc ¼


dϕ ¼
Gkðh
hm Þ dh

ð9Þ

While use of ϕ(h) given by eq 3 or 7 to calculate Fc makes eq 1 nonlinear in the general case, in the limit of a very smallamplitude electric field, eq 1 becomes linear in h. Except for the Basset force, this linearized form of eq 1 corresponds to the classic damped harmonic oscillator. After definition of X = (h
hm) and M = (Mp + 1/2Mf), eq 1 is rewritten as M

d2 X dx þ f þ 6ηa2 dt 2 dt

rffiffiffiffiffiffiffiZ t 2 0 πFf d Xðt Þ dt 0 pffiffiffiffiffiffiffiffiffiffi þ GkX ¼ QE∞ eiωt η
∞ dt 02 t
t0

ð10Þ A particular solution is XðtÞ ¼

QE∞    eiωt ¼ Λeiθ eiωt pffiffiffiffi 1 þ i 2 Gk
Mω þ iωf 1 þ N pffiffiffi 2

ð11Þ where N is a dimensionless group that is the product of the Reynolds (Re) and Strouhal (St) numbers     a2 ωFf UaFf ωa ¼ ð12Þ N ¼ η η Re U St and U is the amplitude of oscillations in dh/dt. When written in polar form as Λeiθ, the complex coefficient in eq 11 has an argument θ and amplitude Λ. The angle θ represents the phase angle between oscillations in height and electric field. Each term on the left-hand side of eq 10 contributes to the phase angle in eq 11. The inertial term gives a phase angle of π, the quasi-steady Stokes term gives a phase angle of π/2, the Basset force term gives a phase angle of 3π/4, and the colloidal force term gives no phase angle. The authors stress that eq 11 is a preliminary model; any dependence of hydrodynamic wall interactions on separation distance was neglected and it assumes both ka . 1 and kh . 1. A numerical solution to the full nonlinear problem will be the subject of future work.

Figure 2. (a) Expanded and collapsed views of the ETIRM fluid cell. (b) Experimental apparatus. (1) Potential signal from function generator to potentiostat. (2) Potential signal from function generator to oscilloscope to be recorded. (3) Measured current from potentiostat to oscilloscope to be recorded. (4) Scattered light intensity from photomultiplier tube to oscilloscope to be recorded. (5) Working electrode lead. (6) Reference electrode lead. (7) Counterelectrode lead. (8) Optics stack including microscope, photomultiplier tube (PMT), and charge-coupled device (CCD) camera. The PMT is used for single-particle phase-angle measurements, and the CCD camera is used for pairwise measurements. (9) Dove tail optical prism. (10) Incident ray.

2.2. Ideally Polarizable Electrode. To capture important low-frequency behavior while still avoiding faradaic reactions, we employ the working electrode’s ideal polarization window as a means to eliminate faradaic current during experiments. During a cyclic voltammogram (CV), the potential of a working electrode is changed relative to a reference electrode while recording current. The dependence of the recorded current on the electrode potential helps determine whether the current is only capacitive (ideally polarizable electrode, IPE) or a combination of faradaic and capacitive (non-IPE). Faradaic current depends on the magnitude of potential, in addition to scan speed of the potential, and other kinetic parameters of the electrochemical reaction. Capacitive current primarily depends on the rate at which potential is being scanned and the total capacitance of the electric double layer. In the absence of specific adsorption, capacitive current arising from charging of the double layer is given by  
1 1 1 þ ¼ V_ Ctot ð13Þ ic ¼ V_ Cc Cdl where V_ is the rate at which the electrode potential—defined as the potential of the working electrode minus the potential of a 9783

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reference electrode of the same kind located just outside the diffuse layer—is changing. [Note that while such precise positioning of the reference electrode is impractical in a real cell, the error due to any additional ohmic drop is negligible at slow scan rates.] The total differential capacitance of the electric double layer (Ctot) is the quotient of the current within the ideally polarizable region (IPR) and V_ . A CV scans the electrode potential both positively and negatively, resulting in positive (ic+) and negative capacitive current (ic
). The difference between these two (Δi) is used to measure Ctot. V_ þ ¼
V_
¼ V_ Δi ¼ icþ
ic
¼ V_ þ Ctot
V_
Ctot ¼ 2V_ Ctot

ð14Þ

Equation 14 will be used to identify and characterize the IPR region. The electrode will be considered ideally polarizable if the relationship between Δi and V_ is linear.

3. EXPERIMENTAL SECTION 3.1. Experimental Apparatus. Single and pairwise particle measurements were conducted with an enhanced TIRM apparatus we call the “electrochemical total internal reflection microscope” (ETIRM) and is illustrated in Figure 2. As described below, this instrument couples electrochemical analysis with a standard TIRM.38 The complete experimental apparatus includes the ETIRM cell, an Agilent model 33120 function generator, an Agilent model 54620 oscilloscope, a Zeiss Universal microscope, a Photon Technologies Inc. photomultiplier tube, a Dage-MTI CCD100 camera, a 1100 series Uniphase 632 nm wavelength HeNe 10 mW cylindrical tube laser, a potentiostat, and a personal computer (not shown). Two potentiostats were used in this work. A Princeton Applied Research VersaSTAT3 was used to identify and characterize the IPR (section 4.1), and a model 276 potentiostat from the same company was used to conduct single and pairwise particle measurements (section 4.2). The important electrical components of the ETIRM cell comprise a working electrode (WE), reference electrode (RE), and counterelectrode (CE). Implementation of a three-electrode cell was necessary to isolate the WE and rigorously identify the ideally polarizable region (IPR). The WE potential was defined as the potential drop between the WE and RE. Since no significant current flows to the RE during a voltage scan and no electrochemical reactions are expected on it, the RE provides a convenient and constant reference state for the measurement of potential on the WE. The CE supplied sufficient current to maintain the potential drop between the WE and RE. The WE and CE were ITO thin films (30
60 Ω 3 square
1, 30
60 nm thick) on a glass substrate obtained from Sigma Aldrich, each with an exposed surface area equal to 10.25 cm2. The RE was a 0.040 cm thick silver/silver chloride (Ag/ AgCl) electrode with an exposed surface area equal to 0.58 cm2; the RE was prepared by a well-established procedure.39 All potentials cited in this work are relative to RE as the reference state (ref). The gaps between WE
RE and RE
CE were maintained by 0.038 cm thick Teflon spacers. All electrodes were connected to the external circuit via conductive silver epoxy obtained from Allied Electronics. Potassium hydroxide (KOH) and sodium bicarbonate (NaHCO3) were used in this work at concentrations of 0.15 mM for all experiments. The electrolyte solutions were prepared from ultrapure (resistivity∼18.2 MΩ 3 cm) water that was vigorously deaerated for 2 h with 99.995% pure nitrogen gas. The authors stress that careful deaeration is crucial. Carbon dioxide readily absorbs into alkaline solution, subsequently forming carbonate and lowering the solution’s pH. Across all the experiments, the average pH of 0.15 mM KOH and 0.15 mM NaHCO3 was measured to be 10.0 and 7.4, respectively. The particles used were nominally 5.7 μm

Figure 3. Particle scattering light from evanescent wave. The intensity of light scattered, I(h), exponentially depends on the separation distance h through eq 15. diameter sulfonated polystyrene spheres obtained from Interfacial Dynamics Corporation. Particles were centrifuged down and redispersed 10 times to remove contaminants. Dynamic light scattering with a Malvern Zetasizer 3000HSa was used to measure the electrophoretic mobility of the particles dispersed in each electrolyte. The measured mobilities were
7.96 ((0.21) and
8.29 ((0.85) μm 3 cm 3 V
1 3 s
1 in 0.15 mM NaHCO3 and 0.15 mM KOH, respectively.

3.2. Electrochemical Measurements To Identify the Ideally Polarizable Region. A CV was used to identify the IPR. Electric

potential was scanned between
0.5 and 1.2 V or between
0.5 and 1.3 V in the same solution at 100 mV 3 s
1 for 20 cycles in 0.15 mM KOH or 0.15 mM NaHCO3, respectively, to identify the IPR. Once a steady measurement was obtained at the 10th cycle, the flat region of the CV was identified. Total capacitance of the electric double layer was measured within the IPR by scanning between
0.2 and 0.8 V at different scan speeds in both electrolytes. The scan speeds chosen were 10, 25, 50, 75, 100, 250, and 500 mV 3 s
1. Ten cycles at each potential scan speed were conducted. The difference in current between the forward and reverse scan (see eq 14) was plotted versus scan speed to determine total doublelayer capacitance. 3.3. Measuring the Single-Particle Phase Angle. ETIRM was used to determine the instantaneous separation distance between a particle and the working electrode. Scattered light intensity (see Figure 3) was collected by a PMT and subsequently analyzed. Intensity, I(h), is related to h by IðhÞ ¼ I0 e
βh þ Ib β¼

4π λ

ð15Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðne sin RÞ2
ðns Þ2

ð16Þ

where I is the instantaneous intensity of light scattered, I0 is the intensity of light scattered from a particle at h = 0, β
1 is the decay length of the evanescent wave, I b is the background intensity of light, λ is the wavelength of the incident laser, ns is the refractive index of the dispersing electrolyte, ne is the refractive index of the thin-film electrode support, and R is the angle of incidence of the laser. Analog signals for potential, current, and scattered light intensity were recorded and 1024 cycles were averaged by the oscilloscope. Discrete Fourier transforms of current and scattered light intensity were compared to determine θ (see Appendix B). The phase angle between scattered light intensity and current density is reported here because that was the phase angle reported in an earlier paper.29

3.4. Particle Tracking for Pairwise Particle Measurements. Particle tracking experiments were performed at 16 magnification with a Dage-MTI CCD100 camera with a final output of 640  480 pixels and 30 frames 3 s
1, giving a resolution of ∼0.4 μm 3 pixel
1. Image analysis routines for particle identification and tracking were adapted from open source code written by Blair and Dufresne,40 based on the particle tracking algorithm of Crocker and Grier41 . The particle centers were identified to subpixel accuracy by first applying a bandpass filter to each 9784

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Figure 4. Cyclic voltammogram (CV) of indium tin oxide working electrode scanned at 100 mV 3 s
1 while submerged in (—) 0.15 mM KOH or (---) 0.15 mM NaHCO3. The CV displays regions with (hydrogen and oxygen evolution) and without (ideally polarizable region) faradaic current. The thermodynamic minimum of the size of the ideally polarizable region is 1.23 V for the electrolysis reaction.

Figure 6. Difference in current density between the forward scan and reverse scan (Δi/2) plotted as a function of scan speed for (a) 0.15 mM KOH and (b) 0.15 mM NaHCO3. The slope of a regression line is equal to the electric double layer capacitance. Data at four potentials are shown: (b) 0, (O) 0.2, ([) 0.4, and (]) 0.6 V vs ref. The linearity of each set of points demonstrates that the interface is acting a pure capacitor. A line with a slope of 10 μF 3 cm
2 is shown as a reference.

Figure 5. Cyclic voltammetery of an indium tin oxide working electrode in (a) 0.15 mM KOH and (b) 0.15 mM NaHCO3 between
0.2 and 0.8 V. Each graph displays current density for eight scan rates, 10, 25, 50, 75, 100, 250, and 500 mV 3 s
1; the larger scan rates have larger absolute values of current density. The difference in current density between the forward scan and backward scan (Δi) was regressed with scan speed to give the electric double layer’s capacitance (see eq 14). frame and locating the brightness-weighted centroid of the pixels for each particle. Characteristic intensity and radius of gyration were calculated for each particle, which were used to reject false particle identifications. By this method, the center-to-center distance r was determined to an accuracy of less than 0.5 pixel for one frame 3 s
1 of video over the course of each experiment.

4. RESULTS AND DISCUSSION 4.1. Identifying the Ideally Polarizable Region. Figure 4 shows measured current density as a function of potential for 0.15 mM KOH and 0.15 mM NaHCO3 at a potential scan rate of

100 mV 3 s
1. The CVs in Figure 4 feature windows of flat current density with zero slope between two regions of current density with nonzero slopes at positive and negative potentials, which are characteristic of oxygen and hydrogen evolution, respectively. The CV for each electrolyte featured a range of ideally polarizable potential whose width is approximately 1.3 V. This range of potential corresponds to the range of stability of water with respect to electrolysis (1.23 V).42 The two electrolytes shared an IPR between
0.2 and 0.8 V. Subsequently, eight CVs were conducted between
0.2 and 0.8 V at systematically varied scan rates to determine the capacitance of the electric double layer (see Figure 5). As expected from eq 14, the measured current density increased with increased scan speed (i.e., the CV shows a capacitive hysteresis of current between forward current and reverse current). Figure 6 shows the difference in current density (Δi/2) plotted as a function of potential scan speed for an ITO electrode submerged in 0.15 mM KOH or 0.15 mM NaHCO3 at four potentials along the CV. The linearity of the relationship between current density and scan speed confirms that the double layer was behaving as a capacitor and the region being scanned was ideally polarizable. Further, the y-intercept—which represents the faradaic contribution to current—is negligibly small, once again confirming that the current is purely capacitive. 4.2. Single-Particle Longitudinal and Pairwise Lateral Measurements. The phase angle between scattered light intensity and current density was measured for a variety of frequencies from 10 to 500 Hz while the particles were dispersed in 0.15 mM KOH or 0.15 mM NaHCO3 and the electrode was polarized from
0.2 to 0.8 V. Recall that the experiments described in section 4.1 were conducted at far smaller frequencies (∼0.05 Hz) than used here. Because the impedance of a capacitor is inversely proportional to frequency, higher frequency is expected to favor capacitive current over faradaic current;35 thus 9785

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Figure 8. Micrographs of pairwise particle motion under the influence of an ac electric field turned on at time = 0 s. Particles dispersed in 0.15 mM KOH separated, but particles dispersed in 0.15 mM NaHCO3 aggregated. The working electrode was polarized between
0.2 and 0.8 V at 50 Hz. The small dot of light visible at one pole of the particles is light scattered from the evanescent wave. Notice the scattering is more intense for particles dispersed in KOH; this indicates the particles are closer to the working electrode. Figure 7. Phase angle between scattered light intensity from a particle and current density for (b) 0.15 mM KOH and (O) 0.15 mM NaHCO3. The solid line is a calculation of the phase angle between scattered light intensity and current based on the model presented in section 2.1. The working electrode was polarized between
0.2 and 0.8 V. The phase angle was electrolyte-dependent despite removal of any possibility of electrochemical reaction.

the electrode remains an IPE. Results from all phase-angle measurements are summarized in Figure 7. The solid curve in Figure 7 is a calculation that will be discussed in the next section. Phase angles in 0.15 mM KOH decreased and diverged from 90° as frequency decreased, while phase angles in 0.15 mM NaHCO3 increased and diverged from 90° as frequency decreased. The phase angles measured in 0.15 mM NaHCO3 appeared to reach a maximum at the lowest frequencies, perhaps presaging a decrease in phase angle at lower frequencies. Phase angles at frequencies less than 40 Hz for particles dispersed in NaHCO3 could not be measured because the particle migrated beyond the range of the evanescent wave in this frequency regime for the entire cycle. Because the particles dispersed in KOH remained in the evanescent wave, this result indicated that the average height of the particle was greater in NaHCO3 than in KOH, which was consistent with previous work not restricted to the IPR.23 One way to avoid this problematic behavior and probe lower frequencies in both electrolytes is to use heavier particles, which would have smaller fluctuations in height when exposed to an ac electric field. Pairwise particle motion in an ac electric field was recorded with video microscopy and subsequently analyzed by use of a particle-tracking algorithm. The electrolyte and electrode polarization window were identical to the single-particle measurements described above. Two sets of experiments were conducted: one set at 50 Hz and one at 100 Hz. Micrographs of a representative experiment at 50 Hz for each electrolyte are shown in Figure 8. The images show that when the electric field was turned on, particles initially equal distances apart separated when dispersed in 0.15 mM KOH but aggregated when dispersed in 0.15 mM NaHCO3. Results from application of the particle tracking algorithm are shown in Figures 9 and 10. Each data set represents the average of at least seven experiments, except for 0.15 mM KOH at 100 Hz, which was conducted twice; the increased scatter in the average is a result of running the experiment fewer times. The full particle tracking analysis supported the observations illustrated by Figure 8 that particles dispersed in 0.15 mM KOH separated faster than predicted from diffusion and dipole
dipole repulsion, but particles dispersed in 0.15 mM NaHCO3 aggregated. This observation is consistent with previous work not restricted to the IPR.

Figure 9. Separation distance r between two particles dispersed in 0.15 mM NaHCO3, plotted as a function of time. The working electrode was polarized between
0.2 and 0.8 V at either (a, b) 50 Hz or (b, O) 100 Hz. Each point represents the average of multiple trials that all approximately began when the particles had a separation equal to 2a. The error bars represent 1 standard deviation from the average. These data show that two particles dispersed in NaHCO3 aggregated upon application of an ac field (consistent with previous work) even while there was no possibility of faradaic reactions.

Previous work hypothesized that faradaic reactions were responsible for electrolyte-dependent motion at low frequencies.28 These authors suggested that θ deviated from 90° depending on the nature of the reaction occurring at the nearby electrode. Work presented here clearly shows that longitudinal and lateral motion depend qualitatively on dispersing electrolyte in the absence of electrochemical reactions. In other words, electrolyte-dependent motion is not a consequence of electrochemical reactions occurring at the nearby working electrode’s surface; the chemical-like selectivity remains without the chemistry. 4.3. Simple Harmonic Oscillator Model. The origin of a phase angle between the particle height and the electric field 9786

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Figure 11. Calculation of the relative amplitudes or contributions of each term from the model presented in section 2.1.

Figure 10. Separation distance r between two particles dispersed in 0.15 mM KOH, plotted as a function of time. The working electrode was polarized between
0.2 and 0.8 V at either (a, b) 50 Hz or (b, O) 100 Hz. Each point represents the average of multiple trials that all approximately began when the particles had a separation equal to 2a. The error bars represent 1 standard deviation from the average. The solid line is predicted separation based on diffusion and dipole
dipole repulsion. These data show that two particles dispersed in KOH separated upon application of an ac field (consistent with previous work) even while there was no possibility of faradaic reactions.

has remained an open question since it was first measured in 2002;17 we suggest an origin herein. While developing imaging amperometry,43,44 we discovered that θ approached 0° at very low frequencies (∼0.01 Hz). We reasoned that the particle was at its equilibrium height (i.e., h at which ∑F = 0) at every instant in time while slowly changing the electric field, causing the particle’s motion to be in unison with the electric field (i.e., θ = 0°), which is quite different from the original assumption of pseudosteady electrophoresis that leads to θ = 90°. Clearly quasi-steady electrophoresis does not apply at all frequencies. This realization led us to revisit the dynamics of the single-particle motion. The solid curve in Figure 7 shows θ calculated for a 6 μm diameter particle on the basis of the model developed in section 2.1. The phase angle increased from 0° as frequency increased from 0 Hz, crossed 90° at a frequency of 118 Hz, and then continued to increase to 180° at much higher frequencies. The key additions to the model were the inclusion of inertial effects, which were previously thought to be insignificant, and the Basset force. In fact, phase angles greater than 90° are possible only if inertia is included.

Phase angles above or below 90° are predicted with the appropriate inclusion of these effects. The curve shown in Figure 7 is surprisingly close to experimental measurements when it is considered that this model has no adjustable parameters and was developed without taking into account the influence of the nearby boundary on hydrodynamic interactions. The relative amplitudes of each component in eq 11 were identified by dividing the left-hand side the equation by QE∞ and subsequently taking the modulus of each term (see Figure 11). The y-axis of Figure 11 is the scaled amplitude of colloidal, quasisteady Stokes drag, Basset, or inertia (including the fluid inertia from eq 2). The shaded region in Figure 11 denotes the frequency regime where phase-angle measurements have been conducted. At frequency f < 1 Hz, colloidal forces (double layer repulsion plus gravity) were larger than all other contributions and are solely responsible for balancing the electrokinetic force. This is the regime where the particle was at its quasi-equilibrium height because the electric field is changing very slowly. The phase angle in this regime approached 0° (see Figure 7). Stokes drag began to be important at f = 1 Hz. The Basset force became important at f = 100 Hz, about the frequency that the phase angle crossed 90°. Inertial effects began to dominate at f > 1000 Hz. Three of the four effects are important within the shaded region; this combination produced complicated phase-angle behavior shown in Figure 7. Figure 11 shows that the Basset force contributed to the particle’s motion over a band of frequencies that included the regime where experiments were conducted, which is especially notable because this force was not considered in this context previously. 4.4. Limitations of the Model in Its Present Form. Two primary assumptions make the results of this contribution more illustrative than predictive. The first assumption, that the potential energy of the particle is a parabolic function of distance, was discussed previously in section 2.1. The second assumption is neglect of certain influences of the proximate wall. All three components of the frictional force (see eq 2) may be influenced by proximity to the electrode. Stokes drag for a particle approaching a boundary increases by a factor proportional to a/h, but the full effect of the nearby boundary on the virtual inertia and Basset force of an oscillating particle is still unclear. The experimental technique presented herein may serve as a way to measure these forces, as high-resolution signals of both the driving force (i.e., current) and response (i.e., scattered light intensity) can easily be measured. The apparent electrokinetic charge, Q, also depends on proximity to the electrode. Previous models for quasi-steady electrophoresis near an electrode revealed that the distance 9787

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Figure 12. Scaled electrophoretic mobility of a particle plotted as a function of scaled particle ζ potential. (b, O) Particle dispersed in 0.15 mM KOH and 0.15 mM NaHCO3, respectively, calculated via the algorithm of O’Brien and White.47 The solid line is the relationship between mobility and ζ potential in the limit of ka . 1 (Smoluchowski). The electrolyte-dependent curves differ by a few percent when the scaled ζ potential is larger than about 4. The difference arises from the dependence on ion diffusion coefficient.

dependence of the electrophoretic mobility (or the apparent charge Q) was different when the electrostatic potential was assumed to be spatially uniform on the electrode45 or when the current density was assumed to be spatially uniform on the electrode.46 Finally, the approximation of colloidal forces by the harmonic potential given by eq 8 should be relaxed; the colloidal forces are more generally given by eq 7. While eq 8 represents a linear Hookean spring, the force corresponding to eq 7 is not linear in the displacement from equilibrium (h
hm). For a linear spring, the phase angle is independent of the magnitude of the driving force QE∞; for a nonlinear spring, the phase angle will depend on the driving force. All of the corrections suggested above make eq 2 nonlinear or make the coefficients dependent on h such that a simple analytic solution for h(t) is no longer evident. These refinements will require a numerical solution to eq 2. Despite the simplifications, the model reveals how a phase angle vital to symmetry breaking can arise, and it predicts phase angles reasonably near values found experimentally. 4.5. Origin of Electrolyte Dependence. One possible explanation for the origin of electrolyte dependence is modification of the particle’s apparent electrokinetic charge by the dispersing electrolyte. To illustrate how this might occur, consider the special case of quasi-steady (neglect inertia) electrophoresis far from the electrode (Fc = 0). When the fluid inertia and the Basset force in eq 2 are also neglected, eq 1 becomes dh Q ¼ EðtÞ ¼ mEðtÞ dt 6πηa

ð17Þ

The coefficient of E is the familiar electrophoretic mobility m. The second equality above serves as the definition for the apparent electrokinetic charge Q. In the asymptotic limit of very thin double layers (i.e., ka f ∞), the electrophoretic mobility is given by the well-known result first derived by Smoluchowski m¼

εζp η

ð18Þ

In this limit, neither the mobility (nor the electrokinetic charge) depend on the choice of electrolyte. This state of affairs arises

because the relatively weak, externally applied electric field does not significantly perturb the ion concentration profiles away from their (equilibrium) Boltzmann distributions established by the much stronger electric field inside the counterion cloud. More generally (for finite ka), the electrophoretic mobility does depend on the ion mobilities. Figure 12 shows how the electrophoretic mobility depends on ζ potential for the two electrolytes used in this study; even for our relatively large particles (ka = 115), a few percent difference can be seen between the electrolytes. This difference might be substantially larger when the particle resides close to the electrode (h , a). The ionic mobilities or diffusion coefficients enter the model through the ionic continuity equations, which are needed to replace the equilibrium Boltzmann distributions when ka is finite. However, merely changing the magnitude of the electrophoretic mobility or charge Q does not alter the phase angle predicted by eq 1 if that equation remains linear in h. The response of a (linear) damped harmonic oscillator corresponds to the amplitude of oscillations in h(t) being directly proportional to the amplitude of the driving force QE(t); the charge Q does not affect the phase angle. Of course, some of the refinements suggested in the previous section (e.g., replacing eq 8 by eq 7) make the model nonlinear; then the magnitude of Q might affect the phase angle. A direct effect of electrolyte choice on the phase angle is provided by the model for dynamic mobility.48 In this model, the dynamic electrophoretic mobility m is a complex number. In other words, the electrophoretic velocity mE(t) is not in phase with the electric field E(t). This implies that our electrophoretic force QE(t) is also out of phase with the electric field E(t). Even the grossly oversimplified model of eq 18 would then predict that an electrolyte-dependent phase angle between the electrophoretic force and the electric field will lead to an electrolytedependent phase angle between the elevation h(t) and the electric field E(t). Despite the simplicity of the model presented in section 2.1, this is the first report of a frequency-dependent phase angle predicted from first principles in this context. Refinement of this model to include wall hindrance to longitudinal motion and the effect of ionic transport are the next steps.

5. CONCLUSION We have presented results for single and pairwise motion of particles within an electrode’s ideally polarizable region. The phase angle between scattered light intensity and current density for a single particle oscillating in an electric field remains electrolyte-dependent despite the absence of electrochemical reactions. Particles dispersed in KOH have θ < 90°, while particles dispersed in NaHCO3 have θ > 90°. Furthermore, pairwise particle motion also remains electrolyte-dependent. Particles dispersed in KOH separated and those dispersed in NaHCO3 aggregated upon application of identical polarization and frequency. These results are consistent with those presented in earlier papers that did not specifically operate in a region where electrochemical reactions were impossible. Thus we can now rule out electrochemical (redox) reactions as the source of electrolyte dependence. New modeling suggested that the phase angle arises from the balance of particle and fluid inertia, colloidal forces, quasi-steady Stokes drag, and the Basset force. Updates to the model are required to appropriately account for hydrodynamic interactions between the particle 9788

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and nearby electrode, as well as the appropriate expression for the particle’s apparent electrokinetic charge when neither ka . 1 nor kh . 1.

’ APPENDIX A: APPARENT ELECTROKINETIC CHARGE OF A PARTICLE A charged colloidal particle feels a force upon application of an electric field. If it is assumed that the particle behaves as a point charge, the basic equation for the force is Fel ¼ qE

ðA-1Þ

where q is charge and E is the electric field. A quasi-steady viscous drag force is generated that points in the direction opposite particle motion (see Figure A-1). The Stokes drag on a particle of radius a, moving with speed U in the direction of Fel, is Fdrag ¼
6πηaU

ðA-2Þ

Starting from an initial condition of U = 0, the particle’s speed will continue to increase until the two forces are balanced and the particle reaches a terminal velocity, U∞. Solving Fel + Fdrag = 0 gives q E ðA-3Þ U∞ ¼ 6πηa Only in the asymptotic limit of ka f 0 does a particle behave as a point charge. In the opposite limit of ka f ∞ (more relevant for our experiments), the electrophoretic velocity is given by the Smoluchowski equation U∞ ¼

εζp E η

ðA-4Þ

Invoking the equivalent assumption of ka f ∞, we can relate ζp to the surface charge density σ by the Gouy
Chapmann equation for a binary, univalent electrolyte ! eζp kT σ ¼ 2 kε sinh ðA-5Þ e 2kT Multiplying σ by the surface area of the particle gives the true charge of the particle, Q ! eζp 2 2 kT ðA-6Þ Q ¼ σ4πa ¼ 8πa kε sinh e 2kT Although eq A-3 is not valid in the limit of ka f ∞, we can equate it with eq A-4 to define an apparent electrokinetic charge Qapp εζp Qapp ¼ 6πηa η

ðA-7aÞ

Qapp ¼ 6πaεζp

ðA-7bÞ

Taking the ratio of Qapp to Q Qapp 3 ¼ 4 Q

eζp kT ka sinh

Figure A-1. Illustration of the balance between the force (Fel) and viscous drag (Fdrag). A positive particle will move in the direction of the electric field.

Finally, in the limit of small ζ potential, ζp f 0 Qapp 3 ¼ ðkaÞ
1 4 Q

Notice that the apparent electrokinetic charge is always the same sign but (when ka . 1) is much smaller than the true charge borne by the colloidal particle. A simple explanation is that much of the true charge borne by the particle is effectively neutralized by the counterion cloud surrounding it.

’ APPENDIX B: USE OF A DISCRETE FOURIER TRANSFORM TO DETERMINE PHASE ANGLE Potential, current, and scattered light intensity signals were collected with an oscilloscope during each experiment described in section 4.2. The signals used for the discrete Fourier transform were an average of 1024 cycles (computed by the oscilloscope). Each average signal comprised a total of N = 2000 points in time distributed uniformly over one cycle. The potential, current, and scattered light intensity signal arrays were processed with a discrete Fourier transform given by the algorithm For n ¼ 0 , ..., N
1 XðnÞ ¼

N
1

∑ xðkÞeð2πi=NÞnk k¼0

!

ðA-8Þ

ðB-1Þ

where x(k) is the time domain signal value at k = 0, ..., N
1. X(n) is a sequence of complex numbers in the frequency domain with an argument (i.e. angle, Ω) and amplitude, the latter of which is the quantity that is plotted on the y-axis in a frequency spectrum. X(n) plotted as a function of frequency is more useful than x(k) for identifying the primary driving frequency by a peak in the spectrum. Thus, the frequency domain was generated by multiplying the sampling frequency (1/Δt) by n. An example frequency spectrum from an experiment in 0.15 mM NaHCO3 at 100 Hz is shown in Figure B-1. The highest peak in the frequency spectrum occurs at the fundamental driving frequency of applied field. Ω was calculated at the primary peak of the frequency spectrum of X(n/Δt). The difference in angles at each peak of the signal was calculated to produce the desired phase angle. For example, the phase angle between scattered light intensity (SLI) and current (C), θ, is calculated by θ ¼ ΩCpeak
ΩSLI peak

eζp 2kT

ðA-9Þ

ðB-2Þ

A previously published correction to account for a time delay in the photomultiplier tube29 was then applied to θ to give the final values published herein. 9789

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Figure B-1. Frequency spectra of (a) potential signal (units of volts), (b) current signal (units of 10 mA), and (c) scattered light intensity signal (units of volts) for a particle dispersed in 0.15 mM NaHCO3 and polarization between
0.2 and 0.8 V vs ref at 100 Hz. X(n/Δt) is plotted on the y-axis. Each signal displays a strong peak at the fundamental driving frequency of the electric field. The scattered light intensity (SLI) displays periodic peaks because of the inherent nonlinearity arising from both the particle’s motion and logarithmic dependence of SLI on particle height.

’ AUTHOR INFORMATION Corresponding Author

*E-mail [email protected]; phone 412-268-3282; fax 412268-7139.

’ ACKNOWLEDGMENT We thank our colleague Professor Aditya Khair, who drew our attention to ref 37, in which the Basset force is described. The National Science Foundation supported this research under CBET 0730391. The PPG Foundation, through the Achievement Rewards for College Scientists (ARCS) program, provided support for C.L.W. The Carnegie Institute of Technology also provided support to C.L.W. with a Bertucci graduate fellowship. ’ REFERENCES (1) Joannopoulos, J. D.; Villeneuve, P. R.; Fan, S. H. Photonic crystals: Putting a new twist on light. Nature 1997, 386 (6621), 143–149. (2) Zhou, H.; White, L. R.; Tilton, R. D. Lateral separation of colloids or cells by dielectrophoresis augmented by AC electroosmosis. J. Colloid Interface Sci. 2005, 285 (1), 179–191. (3) Lecuyer, S.; Ristenpart, W. D.; Vincent, O.; Stone, H. A. Electrohydrodynamic size stratification and flow separation of giant vesicles. Appl. Phys. Lett. 2008, 92 (10), No. 104105. (4) Richetti, P.; Prost, J.; Barois, P. Two-dimensional aggregation and crystallization of a colloidal suspension of latex spheres. J. Stat. Phys. 1985, 39 (1
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