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Research Papers

Using Shear and Direct Current Electric Fields to Manipulate and Self-Assemble Dielectric Particles on Microchannel Walls OPEN ACCESS

[+] Author and Article Information
Necmettin Cevheri

G. W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405
e-mail: ncevheri@gatech.edu

Minami Yoda

G. W. Woodruff School
of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332-0405
e-mail: minami@gatech.edu

Manuscript received January 2, 2015; final manuscript received January 16, 2015; published online February 12, 2015. Editor: Boris Khusid.

J. Nanotechnol. Eng. Med 5(3), 031009 (Aug 01, 2014) (8 pages) Paper No: NANO-15-1001; doi: 10.1115/1.4029628 History: Received January 02, 2015; Revised January 16, 2015; Online February 12, 2015

Manipulating suspended neutrally buoyant colloidal particles of radii a = O (0.1–1 μm) near solid surfaces, or walls, is a key technology in various microfluidics devices. These particles, suspended in an aqueous solution at rest near a solid surface, or wall, are subject to wall-normal “lift” forces described by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory of colloid science. The particles experience additional lift forces, however, when suspended in a flowing solution. A fundamental understanding of such lift forces could therefore lead to new methods for the transport and self-assembly of particles near and on solid surfaces. Various studies have reported repulsive electroviscous and hydrodynamic lift forces on colloidal particles in Poiseuille flow (with a constant shear rate γ· near the wall) driven by a pressure gradient. A few studies have also observed repulsive dielectrophoretic-like lift forces in electroosmotic (EO) flows driven by electric fields. Recently, evanescent-wave particle tracking has been used to quantify near-wall lift forces on a = 125–245 nm polystyrene (PS) particles suspended in a monovalent electrolyte solution in EO flow, Poiseuille flow, and combined Poiseuille and EO flow through ∼30 μm deep fused-silica channels. In Poiseuille flow, the repulsive lift force appears to be proportional to γ·, a scaling consistent with hydrodynamic, versus electroviscous, lift. In combined Poiseuille and EO flow, the lift forces can be repulsive or attractive, depending upon whether the EO flow is in the same or opposite direction as the Poiseuille flow, respectively. The magnitude of the force appears to be proportional to the electric field magnitude. Moreover, the force in combined flow exceeds the sum of the forces observed in EO flow for the same electric field and in Poiseuille flow for the same γ·. Initial results also imply that this force, when repulsive, scales as γ·1/2. These results suggest that the lift force in combined flow is fundamentally different from electroviscous, hydrodynamic, or dielectrophoretic-like lift. Moreover, for the case when the EO flow opposes the Poiseuille flow, the particles self-assemble into dense stable periodic streamwise bands with an average width of ∼6 μm and a spacing of 2–4 times the band width when the electric field magnitude exceeds a threshold value. These results are described and reviewed here.

Recent interest in microfluidics involving transport through microchannels, i.e., channels with hydraulic diameters ranging from a few μm to a few hundred μm [1], has inspired significant interest in the near-wall dynamics of particles in a flowing solution. For example, many microfluidics-based immunoassays and nucleic acid assays use “beads” functionalized with “probes” (e.g., antibodies) to preconcentrate “targets” (e.g., proteins, nucleic acid fragments, and viral particles), then transport the beads to the wall of the device for subsequent analysis by surface-mounted sensors [2].

The dynamics of dielectric colloidal particles of average radius a = O (0.1–1 μm) near a planar dielectric wall suspended in an aqueous electrolyte solution at rest is a classic problem of colloid science. In a dilute suspension where interparticle interactions are negligible, the lift forces, i.e., the forces normal to the wall, acting on a near-wall particle are from the DLVO theory [3], electrostatic forces due to the interaction of the particle and wall electric double layers (EDLs), attractive van der Waals forces, and the gravitational force, which are negligible for neutrally buoyant particles. For particles and walls with ζ-potentials of the same sign, which is the usual case, the electrostatic interactions are of course repulsive.

Moreover, colloidal particles are also subject to additional lift forces in these microchannel flows, which are typically driven by a pressure gradient or a voltage gradient (i.e., electric field) [4]. When the flow is driven by a pressure gradient Δp/L, the resultant fully developed Poiseuille flow through microchannels is essentially linear with constant shear rate γ· near the wall. In such a shear flow, neutrally buoyant a = 4–9 μm particles are repelled from the wall by shear-induced electrokinetic, or electroviscous, lift [5,6] due to the polarization of the particle EDL associated with convection of mobile counterions by the flow. Theoretical and modeling studies of electroviscous lift, which evaluate the Maxwell stress tensor [7] and incorporate hydrodynamic effects including particle rotation [8,9], suggest that the magnitude of shear-induced electrokinetic lift is proportional to γ·2. The experimental estimates of the electroviscous lift force are, however, typically at least an order of magnitude greater than theoretical predictions [9].

Shear-induced lift forces have also been observed in field-flow fractionation (FFF). Williams et al. [10] showed in their FFF experiments that the total lift force on a = 5–20 μm near-wall PS (latex) particles suspended in a fluid with viscosities μ < 2 cP in a shear flow exceeded the expected inertial lift [11], and concluded that there was an additional lift force, “hydrodynamic lift,” that had a magnitude proportional to γ·.

Voltage gradients, i.e., electric fields, are also commonly used to “pump” fluids in microchannels, in part because the resultant EO flow has a (nearly) uniform velocity profile that minimizes convective dispersion. In a flow driven by an applied electric field of magnitude |E|, suspended particles can experience (in addition to electrophoretic and dielectrophoretic forces) a repulsive “dielectrophoretic-like” lift force due to the breakdown of the symmetry of the Maxwell stress tensor in the gap between the dielectric particle and the dielectric wall [12]. The magnitude of this force is proportional to E2 and a2, based on both theoretical [12] and experimental studies [13,14], but the actual force magnitudes reported in the experiments are significantly greater than the values predicted by the theory.

To our knowledge, there are no experimental studies of the dynamics of particles suspended in a flow driven by both a pressure gradient and an electric field. Numerical and experimental studies have shown, however, that the velocity field in such a flow is simply the superposition of the Poiseuille and EO velocity fields, at least for the creeping flows typical of microchannels [15-17].

Over the last decade, our group has used evanescent wave-based particle tracking to study both flow and particle dynamics in both EO and Poiseuille flow through fused-silica microchannels. Because the intensity of evanescent waves decays exponentially normal to the wall with a length scale, the intensity-based penetration depth zp, which is typically ∼100 nm, only particles within the first few hundred nm next to the wall are visualized. Moreover, the three-dimensional position of the particles can be determined from their position in the image and the brightness (i.e., intensity) of the particle images.

Evanescent-wave particle tracking, therefore, simultaneously measures both the wall-normal distribution and the displacements/velocities of near-wall particle tracers. The remainder of this paper reviews our previous studies of the lift forces acting upon a = 245 nm suspended fluorescent carboxylate-terminated PS in EO flow, Poiseuille flow, and describes recent results in combined Poiseuille and EO flow.

This section summarizes the experimental procedures; further details can be found in Refs. [18] and [19]. The working fluid was an aqueous sodium tetraborate (Na2B4O7) solution prepared by dissolving Na2B4O7·10 H2O salt (Acros Organics 419450010) in double distilled de-ionized water (with an initial resistivity > 16 MΩ cm), at a molar concentration of 1 mM, giving a Debye length κ-1 ≈ 7 nm. This fluid was seeded with a = 245 nm fluorescent carboxylate-terminated PS particles (Life Technologies F8812), with a polydispersity of 7.5 nm according to the manufacturer, at a bulk particle number density c = 2.7 × 1016 m−3, corresponding to volume fraction ϕ = 1.7 × 10−3. In all cases, the particle suspensions were sonicated, filtered through syringe filters to remove aggregated particles, and degassed. The ζ-potential of the suspended particles ζp ≈ −50 mV, as measured by laser-Doppler microelectrophoresis (Malvern Zetasizer).

The dilute suspension was driven through fused-silica microchannels of nominal depth H ≈ 30 μm and width W ≈ 300 μm by pressure gradients Δp/L < 1.1 bar/m and electric fields E=Ei (where i is the direction of the Poiseuille flow) with |E| < 95 V/cm. The ζ-potential of the fused-silica wall ζw ≈ −125 mV, based on EO mobility measurements and the Helmholtz–Smoluchowski equation assuming thin EDLs.

The flow in the center of the microchannel (i.e., away from the side walls) was illuminated from below with evanescent waves with zp = 110–120 nm generated by the total internal reflection of an argon-ion laser beam at a wavelength λ = 488 nm coupled into the channel using an isosceles right triangle prism at the bottom wall (with a manufacturer-quoted root mean square roughness of 3 nm). The laser beam was shuttered to produce 0.5 ms pulses (and hence exposure times).

The fluorescence from the particles was imaged by an electron multiplying charge-coupled device camera (Hamamatsu C9100-13) through a 525 ± 25 nm bandpass emission filter using a magnification 63, numerical aperture 0.7 microscope objective (Leica PL Fluotar L) and recorded on the hard drive of a personal computer as 512 × 144 pixels images, with physical dimensions of 130 μm × 37 μm. At least 750 image pairs (separated by time interval Δt = 2 ms and spaced 0.2 s apart) were acquired within a single experiment over a total image acquisition time of about 150 s.

The location of the center of each particle was determined by cross-correlation with a two-dimensional Gaussian function, and used to identify and remove images of overlapping or aggregated particle tracers. Assuming that the particle image intensity Ip has the same exponential decay as the evanescent-wave illumination, the particle-wall separation h (i.e., the distance between the edge of the particle and the wall)Display Formula

(1)Ip(h)=Ip0exp{-hzp}

Here, Ip is defined to be the area-averaged integral of the grayscale values in the particle image and Ip0 is the average of Ip values obtained from particles that are attached to the bottom wall at the end of each experimental run. An ensemble of h values over at least (in most cases) 104 particle images are used to calculate the particle number density profile c(h) over 20 nm bins.

The distribution of particles near the wall is nonuniform, even in the absence of flow, due to DLVO interactions as discussed previously. The magnitude of the net lift force can be estimated, assuming that the particles have a Boltzmann distribution, from the slope of the particle potential ϕ. The normalized particle-wall interaction potential energy profileDisplay Formula

(2)φ(h)kT=-ln{c(h)c}

where k is the Boltzmann constant and T is the absolute temperature of the fluid. The potential energy profiles are smoothed by lowpass filtering over a 60 nm wide window. To “isolate” the potential energy due to the flow ΔφF, the potential energy for the “no flow” case φnf (which should correspond to the potential predicted by DLVO theory) is subtracted from ϕDisplay Formula

(3)ΔφF(h)kT=φ(h)kT-φnf(h)kT

The magnitude of the corresponding lift force due to flow F is then the slope of ΔφF(h), which is estimated by linear regression for 100 nm ≤ h ≤ 280 nm (the data at h < 100 nm were excluded from the estimate because of their large uncertainties). The result is effectively the lift force averaged over this range of h for particles subject to a constant γ·.

Evanescent-wave particle tracking was used to study colloidal particle dynamics at h ≤ 300 nm and estimate lift forces in EO flow, Poiseuille flow, and combined Poiseuille and EO flow. The results for combined flow are separated into the case where E > 0 and the Poiseuille and EO flows are in the same direction, and the case where E < 0 and the Poiseuille and EO flows are in opposite directions.

EO Flow.

Figure 1 shows the particle number density profiles c(h) for EO flow (in the absence of shear) at E = 0 V/cm (×), 4.7 V/cm (), 9.5 V/cm (), 16.5 V/cm (), 23.6 V/cm (), and 33.1 V/cm (). The E = 0 case here is taken to be the no flow case, although there is still a very weak flow with speeds less than 1 μm/s (versus typical flow speeds of at least 102μm/s), because of difficulties in ensuring that the free surface of the upstream and downstream reservoirs are at exactly the same height. The flow Reynolds number (based on H and the average speed) Re < 10−4.

For E ≤ 9.5 V/cm, c increases with h until the number density reaches a maximum at h ≈ 100 nm; c then appears to decrease very slowly as h increases. For larger values of E, c increases monotonically with h, at least for h ≤ 300 nm, i.e., the near-wall region illuminated by the evanescent waves. Presumably, c recovers to c (denoted by the dashed line) farther from the wall. For E ≥ 9.5 V/cm, the number density at a given h decreases as E increases, suggesting that applying a steady electric field parallel to the channel axis leads to a repulsive lift force that increases with electric field magnitude.

The standard deviation in c at a given value of h over four independent experiments is less than 10% (15% for the largest value of E) for h > 100 nm, but can be as great as 70–80% for h < 80 nm, in part because there are very few particles that close to the wall. The number density profiles for E = 0 and 4.7 V/cm are hence identical within experimental uncertainty.

Figure 2 shows the normalized particle-wall interaction potential energy profiles φ(h)/(kT) estimated from the data shown in Fig. 1 (with the same legend). As expected, the h value at the potential energy minimum corresponds to the peak of the particle number density profile in Fig. 1. Figure 3 shows the normalized particle potential energy profiles due to flow ΔφF(h)/(kT) at E = 4.7 V/cm (), 9.5 V/cm (), 16.5 V/cm (), 23.6 V/cm (), and 33.1 V/cm (), where the potential energy for the E = 0 case was subtracted from the profiles shown in Fig. 2. Given the large standard deviations in c near the wall, the particle potentials due to flow are shown only for h = 100 nm and 280 nm.

Figure 4 shows a log–log plot of the repulsive lift force normalized by its maximum value F/Fmax estimated from the slope of the potential energy profiles shown in Fig. 3 as a function of normalized electric field E/Emax. The error bars denote the standard deviation over four experiments, while the gray solid and black dashed trend lines have slopes of 1 and 2, respectively. The force appears to be proportional to E2, in agreement with previous theoretical predictions [12] and experimental observations [13,14] of the dielectrophoretic-like lift force discounting the result at the lowest electric field, which has a standard deviation greater than the actual force. Note, however, that these EO flow results span less than a decade variation in E (from 4.7 V/cm to 33.1 V/cm).

Poiseuille Flow.

Poiseuille flow was studied at Δp/L = 0 bar/m, 0.1 bar/m, 0.43 bar/m, 0.64 bar/m, 0.83 bar/m, and 1.04 bar/m, corresponding to near-wall shear rates γ· = 0 s−1, 150 s−1, 650 s−1, 960 s−1, 1250 s−1, and 1580 s−1 [19]. The flow Reynolds number (based on H and the average speed) Re < 1, and E = 0 in all cases. Figure 5 shows the particle number density as a function of particle-wall separation c(h) for γ· = 0 s−1 (×), 150 s−1 (), 650 s−1 (), 960 s−1 (), 1250 s−1 (), and 1580 s−1 (). The profile at γ· = 0 s−1 is the no flow case also shown in Fig. 1. Although not shown here, the velocity data obtained from these particle images are in reasonable agreement with the velocity profiles predicted by theory [19].

In all cases, the number density profiles are nonuniform and qualitatively similar: c increases with h until the number density reaches its maximum at h ≈ 100 nm, and then c is essentially constant (or decreases very slowly) as h increases. The decrease in c at a given h as γ· increases suggests that there is a repulsive lift force in this shear flow, like the case for EO flow, and that the magnitude of this force increases with γ·.

Figure 6 shows the corresponding normalized particle potential energy profiles due to flow ΔφF(h)/(kT) with the same legend as Fig. 5. Figure 7 shows the magnitude of the repulsive lift force F estimated from the slope of these potential energy profiles as a function of γ·. The force magnitude appears to be proportional to shear rate, although the data barely span an order of magnitude in γ·. The scaling of the lift force in Poiseuille (i.e., shear) flow is therefore consistent with hydrodynamic, versus electroviscous, lift [10]. Ranchon and Bancaud [20] have also observed a force that repels a = 100–150 nm particles in with a magnitude proportional to γ· in their microscopy studies of the Poiseuille flow of a water–glycerin electrolyte solution through 0.91.9 μm deep channels, validated by Brownian simulations.

Combined Poiseuille and EO Flow: E > 0.

We first consider the case where E > 0 and the Poiseuille and EO flows are in the same direction [19]. Figure 8 shows the particle number density profiles c(h) measured for combined Poiseuille and EO flow at (a) γ· = 650 s−1 and (b) 1580 s−1, and E = 0 V/cm (×), 0.9 V/cm (○), 2.4 V/cm (▽), 4.7 V/cm (◻), 7.1 V/cm (◇), and 9.5 V/cm (△). The number density decreases as E increases at a given h, and variations in c are at most indiscernible at the highest value of E on this scale. Indeed, no data were obtained at E > 9.5 V/cm because there were too few particles in the region illuminated by the evanescent wave to obtain good statistics, and the c(h) profiles shown at E = 9.5 V/cm are for only about 3 × 103 particle images. The number density also decreases as γ· increases at a given E. Finally, the decrease in c (at a given h) appears to be much larger than that for EO flow at γ· = 0 at the same values of E (cf. the changes in c in Fig. 1 between E = 0 and E = 9.5 V/cm). This enhancement could of course be due to the additional lift force due to shear, but Fig. 5 suggests that this effect is too small to account for this large a decrease. Figure 9 then shows the particle potentials due to combined flow estimated from these c(h) data at (a) γ· = 650 s−1 and (b) 1580 s−1 with the same legend as Fig. 8.

Figure 10 shows the magnitude of the lift force due to combined flow F that repels particles from the wall (and hence reduces c) as a function of the applied electric field E at γ· = 650 s−1 (●) and 1580 s−1 (▲). The force magnitude appears to be proportional to E, since linear curve-fits to these data (dashed lines) have R2 > 0.997. The scaling of this force with electric field suggests that the lift force on colloidal particles observed in combined flow is not dielectrophoretic-like.

Furthermore, the estimates of F for combined flow are much greater than the sum of the dielectrophoretic-like lift force observed in EO flow at the same E and the hydrodynamic lift force observed in Poiseuille flow at the same γ·. For example, F ≈ 14 fN at the largest value of E = 9.5 V/cm and γ· = 650 s−1, versus a dielectrophoretic-like lift force at the same E of ∼1.6 fN and a hydrodynamic lift force at the same γ· of 2.5 fN.

Figure 11 presents a log–log plot of the lift force magnitude F as a function of the shear rate γ· for combined flow at E = 4.7 (▼) and 9.5 V/cm (◼). The error bars denote the standard deviation, again over four experiments. Interestingly, the magnitude of the lift force appears to be proportional to (γ·)N, where N = 0.45 and 0.46 based on power-law curve-fits (dashed lines) to the data at E = 4.7 V/cm and 9.5 V/cm, respectively. So, the lift force in combined flow does not appear to have the scaling of the hydrodynamic lift observed in Poiseuille flow, where N = 1. These data, like those for EO and Poiseuille flow are, however, over a limited range of experimental parameters, barely spanning an order of magnitude in E and γ·.

Combined Poiseuille and EO Flow: E < 0.

Changing the polarity of the electric field so that E < 0, and the Poiseuille and EO flows are therefore in opposite directions, has some surprising effects on the particle dynamics. First, the lift force on the particles changes direction, and the particles are attracted to (versus repelled from) the wall. Figure 12 shows particle number density profiles c(h) for combined Poiseuille and EO flow at (a) γ· = 650 s−1 and (b) 1580 s−1, and E = 0 V/cm (×) (again, the no flow case), −0.9 V/cm (○), −2.4 V/cm (▽), −4.7 V/cm (◻), −7.1 V/cm (◇), and −9.5 V/cm (△). The number density increases rapidly as h increases (note that the scale for c is an order of magnitude greater than the other number density profiles), reaching a maximum value (about ten times the bulk number density at E = −9.5 V/cm) at h 100 nm, then decreasing as h increases to a value significantly greater than c for |E| > 0. Although the profiles are qualitatively very similar for these two different shear rates, the actual values of c are consistently lower at a given h and |E| at the higher γ·.

Figure 13 shows the corresponding particle potential profiles due to combined flow at (a) γ· = 650 s−1 and (b) 1580 s−1 for E = −0.9 V/cm (○), −2.4 V/cm (▽), −4.7 V/cm (◻), −7.1 V/cm (◇), and −9.5 V/cm (△). The potentials are for the most part negative, corresponding to an attractive interaction. Figure 14 then plots this lift force due to combined Poiseuille and EO flow as a function of |E| for γ· = 650 s−1 () and 1580 s−1 (). Here, negative and positive values of F correspond to attractive and repulsive lift forces, respectively, to be consistent with the lift forces shown here for the other flow cases. The magnitude of the force seems to be proportional to E, as was the case for E > 0, with R2 > 0.99 for linear curve-fits to these data (dashed lines). Clearly, the lift force on colloidal particles observed in combined flow for both positive and negative E is not dielectrophoretic-like, and may instead be electrostatic in nature.

When |E| exceeds a critical “threshold” value, the particles self-assemble into bands that are aligned with the flow [18], and this threshold value appears to increase with γ·. Figure 15 shows typical evanescent-wave particle visualizations over a field of view ∼130 μm square in combined flow for γ· = 650 s−1 and E = 0 V/cm (a), −28 V/cm (b), −35 V/cm (c), and −70.7 V/cm (d). The attractive force in combined flow increases the number of near-wall particles (cf. Figs. 15(a) and 15(b)). For |E| ≥ 33 V/cm (at γ· = 650 s−1), the particles self-assemble into nearly periodic bands parallel to the flow (Figs. 15(c) and 15(d)).

As |E| increases (at a given γ· and for values of |E| above this threshold), the bands, which are 5–6 μm in width, become more “defined,” with fewer particles visible between the bands, and the average spacing of the bands increases, from ∼19 μm at E = −35 V/cm to ∼26 μm at E = −94.5 V/cm [18]. The “height,” or wall-normal dimension, of these bands remains undetermined, however, since only particles at h < about 360 nm are visible in these images.

Moreover, the density of the particles in the bands appears to be quite high—so high that that it is difficult to distinguish diffraction-limited images of individual particles in these images. Preliminary visualizations of several of these particle bands where ∼10% of the particles are fluorescently labeled suggest that the particles are in a liquid (versus ordered crystalline) state, and move in the same direction as, but with a speed less than, that of the flow.

The band formation is reversible, with bands forming and disappearing O (1 min) after the electric field is turned on and off, respectively. Once formed, the bands are stable over space and time: They are observed over the entire experimental run time, which is typically 10 min in this steady flow, and over most of the ∼25 mm length of the channel, though the bands become more defined farther downstream. Initial observations also show that they are deflected by debris attached to the bottom wall of the fused-silica microchannel (Fig. 16), which suggests that it may be possible to control the locations where these bands form by patterning this surface.

Evanescent-wave particle tracking was used to visualize particle dynamics and quantify lift forces on colloidal PS particles within ∼300 nm of the wall in EO and Poiseuille flow through fused-silica microchannels. The results in combined EO and Poiseuille flow show that the combination of a (steady) electric field parallel to the channel axis and shear flow have unexpected effects on particle dynamics. The lift force can be either attractive or repulsive, depending upon the direction of the electric field. Moreover, the magnitude and scaling of this lift force with electric field and shear rate are fundamentally different from previous observations of such (e.g., dielectrophoretic-like, electroviscous, and hydrodynamic) lift forces. In combined flow where the EO flow is in the direction opposite from the Poiseuille flow, the particles, after being attracted to the wall, self-assemble into nearly periodic dense bands aligned with the flow direction above a critical electric field magnitude.

There appears to be little theoretical understanding of these phenomena at present, so understanding these initial results is the focus of current and future work. Nevertheless, these observations suggest that there may be new ways to manipulate and self-assemble particles near solid surfaces using a combination of electric field and flow shear in microchannels, which may be useful in microfluidic-based immunoassays and nanomaterials synthesis, among other applications.

This work is supported by the NSF Fluid Dynamics Program by Grant No. CBET-1235799.

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References

Reyes, D. R., Iossifidis, D., Auroux, P. A., and Manz, A., 2002, “Micro Total Analysis Systems. 1. Introduction, Theory, and Technology,” Anal. Chem., 74(12), pp. 2623–2636. [CrossRef] [PubMed]
Lim, C. T., and Zhang, Y., 2007, “Bead-Based Microfluidic Immunoassays: The Next Generation,” Biosens. Bioelectron., 22(7), pp. 1197–1204. [CrossRef] [PubMed]
Probstein, R. F., 2003, Physicochemical Hydrodynamics: An Introduction, 2nd ed., Wiley, Oxford, UK, Chap. 8.
Stone, H. A., and Kim, S., 2001, “Microfluidics: Basic Issues, Applications, and Challenges,” AIChE J., 47(6), pp. 1250–1254. [CrossRef]
Alexander, B. M., and Prieve, D. C., 1987, “A Hydrodynamic Technique for Measurement of Colloidal Forces,” Langmuir, 3(5), pp. 788–795. [CrossRef]
Bike, S. G., Lazarro, L., and Prieve, D. C., 1995, “Electrokinetic Lift of a Sphere Moving in Slow Shear Flow Parallel to a Wall I. Experiment,” J. Colloid Interface Sci., 175(2), pp. 411–421. [CrossRef]
Bike, S. G., and Prieve, D. C., 1992, “Electrohydrodynamics of Thin Double Layers: A Model for the Streaming Potential Profile,” J. Colloid Interface Sci., 154(1), pp. 87–96. [CrossRef]
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Warszyński, P., Wu, X., and van de Ven, T. G. M., 1998, “Electrokinetic Lift Force for a Charged Particle Moving Near a Charged Wall—A Modified Theory and Experiment,” Colloids Surf. A, 140(1–3), pp. 183–198. [CrossRef]
Williams, P. S., Moon, M. H., Xu, Y., and Giddings, J. C., 1996, “Effect of Viscosity on Retention Time and Hydrodynamic Lift Forces in Sedimentation/Steric Field-Flow Fractionation,” Chem. Eng. Sci., 51(19), pp. 4477–4488. [CrossRef]
Cox, R. G., and Brenner, H., 1968, “The Lateral Migration of Solid Particles in Poiseuille Flow—I. Theory,” Chem. Eng. Sci., 23(2), pp. 147–173. [CrossRef]
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Liang, L., Ai, Y., Zhu, J., Qian, S., and Xuan, S., 2010, “Wall-Induced Lateral Migration in Particle Electrophoresis Through a Rectangular Microchannel,” J. Colloid Interface Sci., 347(1), pp. 142–146. [CrossRef] [PubMed]
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Figures

Grahic Jump Location
Fig. 11

Log–log plot of repulsive lift force magnitude due to combined flow as a function of shear rate at E = 4.7 (▼) and 9.5 (◼) V/cm. The dashed lines are power-law curve-fits to these data, and have slopes of 0.45 and 0.46 (with R2= 0.92 and 0.96), respectively [19].

Grahic Jump Location
Fig. 10

Repulsive lift force magnitude due to combined Poiseuille and EO flow as a function of electric field E at γ· = 650 (●) and 1580 s−1 (▲) [19]

Grahic Jump Location
Fig. 9

Normalized particle potential profiles due to combined Poiseuille and EO flow ΔφF(h)/(kT) at γ· = 650 (a) and 1580 s−1 (b). The legend is the same as that in Figure. 8.

Grahic Jump Location
Fig. 8

Particle number density profiles c(h) measured in combined Poiseuille and EO flow at γ·= 650 (a) and 1580 s−1 (b), and E = 0 (×), 0.9 (○), 2.4 (▽), 4.7 (◻), 7.1 (◇), and 9.5 (△) V/cm. Again, the dashed line represents c∞.

Grahic Jump Location
Fig. 7

Repulsive lift force magnitude due to Poiseuille (shear) flow as a function of near-wall shear rate. The dashed line is a linear curve-fit to these data, with a correlation coefficient R2 = 0.996 [19].

Grahic Jump Location
Fig. 6

Normalized particle potential due to Poiseuille flow ΔφF/(kT) as a function of h for γ· = 150 (◼), 650 (▼), 960 (●), 1250 (▲), and 1580 (◆) s−1

Grahic Jump Location
Fig. 5

Near-wall particle number density profiles c(h) in Poiseuille flow at γ· = 0 (×), 150 (◼), 650 (▼), 960 (●), 1250 (▲), and 1580 (◆) s−1. The dashed line denotes the bulk number density c∞ = 2.7 × 1016 m−3.

Grahic Jump Location
Fig. 4

Log–log plot of the repulsive lift force magnitude due to EO flow as a function of electric field magnitude, both normalized by their maximum values. The solid trend line has a slope of 1, while the dashed trend line has a slope of 2.

Grahic Jump Location
Fig. 3

Normalized particle potential due to EO flow ΔφF/(kT) as a function of h for E = 4.7 (), 9.5 (), 16.5 (), 23.6 (), and 33.1 () V/cm

Grahic Jump Location
Fig. 2

Normalized potential φ/(kT) as a function of h at E = 0 (×), 4.7 (), 9.5 (), 16.5 (), 23.6 (), and 33.1 () V/cm

Grahic Jump Location
Fig. 1

Near-wall particle number density profiles c(h) for EO flow at E = 0 (×), 4.7 (), 9.5 (), 16.5 (), 23.6 (), and 33.1 () V/cm. The dashed line denotes the bulk number density c∞ = 2.7 × 1016 m−3.

Grahic Jump Location
Fig. 12

Particle number density profiles c(h) measured in combined Poiseuille and EO flow at γ· = 650 (a) and 1580 s−1 (b), and E = 0 (×), −0.9 (○), −2.4 (▽), −4.7 (◻), −7.1 (◇), and −9.5 (△) V/cm.

Grahic Jump Location
Fig. 13

Normalized particle potential profiles due to combined Poiseuille and EO flow ΔφF(h)/(kT) at γ· = 650 (a) and 1580 s−1 (b), and E = −0.9 (○), −2.4 (▽), −4.7 (◻), −7.1 (◇), and −9.5 (△) V/cm

Grahic Jump Location
Fig. 14

Lift force due to combined Poiseuille and EO flow as a function of electric field magnitude at γ· = 650 () and 1580 () s−1

Grahic Jump Location
Fig. 15

Evanescent-wave visualizations of particles in Poiseuille flow at γ· = 650 s−1 (a) and in combined EO and Poiseuille flow at E = −28 (b), −35 (c), and −70.7 (d) −95 V/cm [18]. The direction of the Poiseuille flow is from left to right in the images.

Grahic Jump Location
Fig. 16

Image showing the entire ∼350 μm width of the channel with the bands (observed at γ· = 650 s−1 and E = −47 V/cm) deflecting around a piece of dust (with a diameter of ∼30 μm)

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