Article pubs.acs.org/Langmuir
Lift Forces on Colloidal Particles in Combined Electroosmotic and Poiseuille Flow Necmettin Cevheri† and Minami Yoda*,† †
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0405 United States S Supporting Information *
ABSTRACT: Colloidal particles suspended in aqueous electrolyte solutions flowing through microchannels are subject to lift forces that repel the particles from the wall due to the voltage and pressure gradients commonly used to drive flows in microfluidic devices. There are very few studies that have considered particles subject to both an electric field and a pressure gradient, however. Evanescent-wave particle tracking velocimetry was therefore used to investigate the near-wall dynamics of a dilute suspension of 245 nm radius polystyrene particles in a monovalent electrolyte solution in Poiseuille and combined electroosmotic (EO) and Poiseuille flow through 30μm-deep fused-silica channels. The lift force observed in Poiseuille flow, which is estimated from the near-wall particle distribution, appears to be proportional to the shear rate, a scaling consistent with hydrodynamic lift forces previously reported in field-flow fractionation studies. The estimates of the lift force observed in combined flow suggest that the force magnitude exceeds the sum of the lift forces observed in EO flow at the same electric field or in Poiseuille flow at the same shear rate. Moreover, the force magnitude appears to be proportional to the electric field magnitude and have a power law dependence on the shear rate with an exponent between 0.4 and 0.5. This unexpected scaling suggests that the repulsive lift force observed in combined electroosmotic and Poiseuille flow is a new phenomenon, distinct from previously reported electroviscous, hydrodynamic lift, or dielectrophoretic-like forces, and warrants further study.
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INTRODUCTION The dynamics of dielectric colloidal particles of radii a = O(0.1−1 μm) near a planar dielectric wall suspended in a conducting medium, such as a quiescent aqueous electrolyte solution, is a classic problem of colloid science. More recently, the near-wall dynamics of suspended particles in a flowing electrolyte solution has become a problem of interest because of its applications in microfluidics. Transport in the microfluidic devices known as laboratories-on-a-chip (LoC) or micrototal analysis systems (μTAS)1,2 commonly entails incompressible creeping flows through microchannels, defined here to be channels with hydraulic diameters ranging from a few micrometers to a few hundred micrometers. The rapid diffusion and reduced sample and reagent volumes associated with shrinking devices down to the micrometer scale should increase the throughput and sensitivity, as well decrease the cost, of these devices.3 At such small length scales, surface forces also become significant as a result of relatively large surface areas and small volumes. Given that surface forces are usually short-ranged, that is, significant only within ∼0.5 μm of the surface, such forces should affect mainly the transport of particles near the microchannel walls. In the absence of flow, the forces acting on a near-wall particle are, based upon Derjaguin−Landau− Verwey−Overbeek (DLVO) theory,4 electrostatic forces due to the interaction of the particle and wall electric double layers (EDLs), attractive van der Waals forces, and the gravitational © 2014 American Chemical Society
force, which is negligible if the particle density is equal to that of the solution. For particles and walls with surface charges of the same sign, which is the usual case, the electrostatic interactions are repulsive. In the presence of a flow, however, colloidal particles are often subject to additional forces due to the presence of pressure or voltage gradients that are commonly used to drive microchannel flows.5 In fully developed Poiseuille flow through microchannels driven by a pressure gradient Δp/L, for example, the parabolic velocity profile is essentially simple shear flow near the wall, and it has been known for more than 20 years that large a = 4−9 μm polystyrene (PS) particles suspended in simple shear of a water−glycerol mixtures are repelled from the wall due to what was later termed a “shear-induced electrokinetic lift force”.6 This lift force is due to electroviscous effects, that is, polarization of the EDL due to convection of the mobile counterions within the EDL by the flow. In their studies using lubrication theory, Bike and Prieve7 studied this lift force theoretically, considering only the Maxwell stress tensor. They then developed a more general theory,8 relaxing the lubrication approximation. The analysis by Warszyński et al.9 also included hydrodynamic effects as well as the Maxwell stress tensor. Perhaps the most complete analysis of this problem is the Received: June 30, 2014 Revised: September 29, 2014 Published: October 24, 2014 13771
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EDLs. The flow was driven by dc electric fields with E < 33 V/cm and pressure gradients Δp/L < 1.1 bar/m through fusedsilica channels with a nominal depth of 30 μm. The images of these particles (with density ρp = 1.05 g/cm3), when illuminated with evanescent waves generated by the total internal reflection (TIR) of light at the interface between the flow and the fused-silica wall, can be processed to yield their 3D position measured with respect to the wall. Given that the wallnormal component of the velocity is effectively zero this close to the wall because of the no-flux boundary condition, only the velocity components parallel to the wall are estimated by matching the particle images in successive exposures to their nearest neighbors. The velocity profiles obtained for h ≤ 300 nm are in good agreement with theoretical predictions for the expected Poiseuille flow velocities. The brightness of the particle image can then, with appropriate calibration, be used to determine the wall-normal position of the particle, usually characterized in terms of the particle−wall separation, h, based on the assumption that this brightness has an exponential decay with wall-normal distance with a length scale identical to that of the illumination. The near-wall particle distribution and the forces acting on the particles can then be determined from ensemble of particle−wall separations from O(104) particle images in most cases.
analysis by the method of matched asymptotic expansions by Cox,10 who also considered hydrodynamic effects, including particle rotation as well as the Maxwell stress tensor. Shear-induced lift forces have also been observed in field-flow fractionation (FFF) studies. Williams et al.11 showed 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 could not be completely due to standard inertial lift forces12 and therefore hypothesized that there was an additional lift force mechanism, which they termed “hydrodynamic lift.” Voltage gradients, that is, electric fields, are also commonly used to “pump” fluids in microchannels, in part because the electroosmotic (EO) flow driven by an electric field parallel to the channel axis has an essentially uniform velocity profile that minimizes convective dispersion. Particles in such an electric field are also subject to electrophoretic (EP) and dielectrophoretic (DEP) forces.13 Furthermore, an electric field applied parallel to the channel axis, which is the usual case for EO flows, can lead to a repulsive force normal to the wall, which is known as a “dielectrophoretic-like lift force”.14,15 Yariv’s analysis16 showed that a nonzero lift force can be generated due to the breakdown of the symmetry of the Maxwell stress tensor in the gap between the dielectric particle and the dielectric wall and that this force has a magnitude that scales with E2, where E is the electric field magnitude, and a2. A recent experimental study17 of steady EO flows through 30-μm-deep fused-silica channels driven by dc electric fields showed that a = 240−463 nm PS, and silica particles with particle-wall gap h ≤ 300 nm were repelled from the wall by a force with a magnitude of O(1−10 fN) that was proportional to E2 and a2. Although the scaling of the force was consistent with dielectrophoretic-like lift, estimates of the actual magnitude of the force was significantly (up to 40 times) larger than that predicted by Yariv,16 possibly because this analysis assumes a particle−wall gap much greater than a. Near-wall transport of colloidal PS particles suspended in a flowing aqueous electrolyte solution is of importance in a variety of microfluidics applications. Microfluidics-based immunoassays and nucleic acid assays, for example, use PS or magnetic “beads” functionalized with “probes” (e.g., antibodies) to preconcentrate “targets” (e.g., proteins, nucleic acid fragments, viral particles).18 In many cases, the beads are then brought near the wall so that surface-mounted sensors can determine if the beads have bound target. An improved understanding of the forces acting on colloidal particles as they approach a wall due to applied pressure gradients and electric fields could therefore lead to new approaches for manipulating and sorting functionalized beads. Both numerical and experimental studies19−21 have shown that driving a flow by imposing both a pressure gradient and an electric field will give a flow with a velocity field that is simply the superposition of the Poiseuille and EO flows for these creeping flows. Although Johann and Renaud22 used a combination of Poiseuille and EO flows, albeit in distinct regions of their microfluidic device, to sort yeast cells almost a decade ago, we know of no studies that have looked at how the forces due to the combination of a pressure gradient and an electric field affect the near-wall dynamics of rigid particles, much less cells, in such flows. Evanescent wave-based particletracking velocimetry23 was therefore used to study the dynamics of a = 125 and 245 nm fluorescent PS particles suspended at number densities of O(1016 m−3) in the flow of a dilute monovalent electrolyte solution for the case of thin
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EXPERIMENTAL DETAILS
Materials and Chemicals. The working fluid in these experiments was an aqueous 1 mmol/L sodium tetraborate (Na2B4O7) solution consisting of sodium tetraborate decahydrate salt (Acros Organics 419450010) dissolved in double distilled deionized (DDI) water from a water purification system (Barnstead E-pure Ultrapure D4641) with an initial resistivity of ∼16 MΩ·cm. The solution was seeded with fluorescent carboxylate-terminated PS particles with radii a = 125 nm ± 4.5 nm (mean ± standard deviation) or a = 245 nm ± 7.5 nm (Life Technologies F8811, F8813), based on the manufacturer’s specifications, at nominal bulk number densities c∞ = 2.7 × 1016 m−3, corresponding to volume fractions, ϕ, of 2.2 × 10−4 and 1.7 × 10−3, respectively. The particles had absorption and emission peaks at 490 and 513 nm, respectively. The Debye length of the solution was λD = 6.8 nm, so λD ≪ a in all cases. The a = 125 nm and a = 245 nm particle solutions were sonicated for 15 min and filtered through syringe filters with pore sizes of 0.45 or 0.8 μm (Millipore SLHV033RS, SLAA033SS), respectively, to remove aggregated particles. Filtering these solutions introduced minor variations in the actual value of the bulk number density in these experiments. The solutions were then degassed under vacuum at an absolute pressure of O(10 kPa) for another 15 min to minimize pH changes due to absorption of CO2. Finally, the ζ potentials of the particles ζp were measured in these dilute suspensions using laser−Doppler microelectrophoresis with a Malvern Zetasizer; ζp = −67.8 ± 1.0 mV (average ± standard deviation) for the a = 125 nm particles and ζp = −47.3 ± 1.42 mV for the a = 245 nm particles. Procedure. These experiments studied combined EO and Poiseuille flow of dilute colloidal suspensions through isotropically wet-etched fused-silica microchannels with nominally trapezoidal cross sections of depth 30 μm and width 300 μm. The EO flow was driven by dc electric fields of magnitude E < 33 V/cm along the channel axis, generated by a low- or high-voltage dc power supply (Instek GPR3510HD, Stanford Research Systems PS325) attached to platinum electrodes, where the upstream electrode at the inlet was the cathode, and the downstream electrode at the outlet was the anode. The electrodes (Figure 1) were sealed to four-way tubing connectors (Fisher Scientific 15-315-32B) attached to inlet and outlet reservoirs (consisting of a ∼5 mm segment of i.d. 4 mm borosilicate glass tubing) of the channel. In all cases, E was small enough that Joule heating was negligible; the temperature of the fluid, measured by a thermocouple 13772
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identifying and removing images of overlapping or aggregated tracers, the displacements of the remaining particles in the plane parallel to the wall were determined by matching particles in the first images of the pair to their nearest neighbor in the second image of the pair, since the average interparticle distance of 3.3 μm was significantly greater than the particle displacements due to convection, which were typically less than 1.5 μm. The particle−wall separation, h, or the distance between the particle edge and the wall was then determined from the particle image intensity, Ip, which was defined to be the sum of the grayscale values (of each pixel in the particle image) divided by the number of pixels. Assuming that the particle image intensity has the same exponential decay as the evanescent-wave illumination where the length scale of the decay is zp,
Figure 1. Sketch of the combined flow setup showing the tubing connectors at the inlet and exit reservoirs of the microchannel. The flow and wall-normal directions are the x and z directions, respectively.
⎧ h⎫ ⎬ Ip(h) = Ip0exp⎨ − ⎪ ⎪ ⎩ zp ⎭ ⎪
at the exit, was within 1 °C of the ambient temperature, which varied between 19 and 21 °C. The Poiseuille flow, which was in the same direction as the EO flow, was driven by pressure gradients Δp/L < 1.1 bar/m generated hydrostatically by adjusting the height of a reservoir of DDI water that was attached to another port on the four-way connector at the inlet reservoir. A piece of tubing with a horizontal section at the exit was attached to the corresponding port of the four-way connector at the outlet reservoir to ensure that the exit pressure was constant over the course each experiment, and the remaining “top” ports of the four-way connectors were sealed. At the start of each experimental run, the channel was cleaned, then filled with the dilute suspension, and mounted on the stage of an inverted epi-fluorescence microscope (Leica DMIRE2). Each experimental run consists of several sets of experiments in which Δp/L increases from 0 to 1.1 bar/m; within each set of experiments at a given value of Δp/L, E increases from zero to its maximum value. For the experiments with a = 245 nm particles, the maximum value of E was 9.5 V/cm for Δp/L > 0, and 33 V/cm in all other cases. The bottom wall of the channel (with a manufacturer-quoted rms surface roughness of 3 nm) was illuminated with evanescent waves generated by the TIR of a CW argon ion laser beam at a wavelength λ = 488 nm shuttered by an acousto-optic modulator and coupled into the substrate with a fused-silica isosceles right triangle prism (Supporting Information Figure S1). The intensity-based penetration depth of the illumination was zp = 110 nm−120 nm, on the basis of measurements of the distance between successive TIR spots. The fluorescence from the particles in each experiment (i.e., at a given value of Δp/L and E) was imaged through a 525 ± 25 nm bandpass emission filter to isolate the longer-wavelength fluorescence from the illumination over a sequence of 750 image pairs, for which the image exposure time was τ = 0.5 ms. The time interval within the image pair, Δt = 2 ms, and the time between image pairs was 0.2 s (thereby ensuring that “new” particles were imaged in each pair), resulting in a total image acquisition time of ∼150 s for each measurement. The images of the flow in the center of the channel (to minimize any effects from the side walls) were acquired by an electron multiplying charge-coupled device (EMCCD) camera (Hamamatsu C9100-13) through a 63× magnification, 0.7 numerical aperture microscope objective (Leica PL Fluotar L) and recorded on the HD of a PC as 512 × 144 pixels images, with physical dimensions of 130 μm × 37 μm. As discussed in the Results and Discussion section, to ensure steady-state flow, images are acquired only at least 120 s after the start of a new experiment at a different value of E within a set at a given Δp/L, and at least 15 min after the start of a new set at a different value of Δp/L. At the end of each experimental run, a magnesium chloride (MgCl2) solution (nominal molar concentration of 7 mM) seeded with particles was injected into the channel to attach the particles to the fused-silica walls. Calibration images of these particles attached to the wall were acquired with the same illumination and imaging setup used in the experiments. The location of each particle center in a given image was determined by cross-correlation with a 2D Gaussian function. After
⎪
(1)
I0p
where is the average of the Ip values from the image of the wallattached particles acquired at the end of the corresponding experimental run. The ensemble of particle−wall separations was determined from at least 104 particle images (except for the combined flow cases at E = 9.5 V/cm, where at least 3 × 103 images were analyzed), and used to calculate the particle number density profile, c(h), over 20-nm-wide bins in h. The displacements of the a = 125 nm and a = 245 nm particles were then divided into three layers, each containing a similar number of samples, based on h: (1) 0 ≤ h ≤ 100 nm,(2) 100 nm ≤ h ≤ 200 nm, and (3) 200 nm ≤ h ≤ 300 nm. The particle velocity components parallel to the wall were then simply the particle displacement in each layer divided by Δt. The particle velocities, averaged over all the particles in each layer, are then placed at the average distance from the wall sampled by the particles, based upon c(h). As noted earlier, suspended particles will be convected by the flow and subject to electrophoresis. The particle velocity measured in these experiments is therefore
u p = U + uep = U +
εζpE μ
(2)
where U is the flow velocity, and uep is the electrophoretic velocity. Here, we assume that the Helmholtz−Smoluchowski relation for the electrophoretic velocity is valid (where ε and μ are the permittivity and absolute viscosity, respectively, of the fluid) because the hindrance of the electrophoretic mobility due to particle−wall interactions should be negligible in these studies.24
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RESULTS AND DISCUSSION Poiseuille Flow Velocity Profiles. Figure 2 shows the flow velocity profile U(z), where z is the wall-normal position of the centers of the a = 245 nm particles, for Poiseuille flow (E = 0) at Δp/L = 0.1−1.04 bar/m. These velocities are averaged over four independent experiments; in all cases, the standard deviations in U are smaller than the symbols. The dotted lines, which denote the analytical solution for Poiseuille flow (in the absence of particle−wall interactions), show that the flow near the wall is simple shear, with a shear rate, γ̇, proportional to Δp/L (Table 1 gives the near-wall shear rates, γ̇, corresponding to the values of Δp/L studied here). The dashed lines represent the analytical solution by Goldman et al.25 for the velocity of a neutrally buoyant near-wall particle in simple shear flow, including the hydrodynamic drag due to the presence of the wall, which causes the particle to “lag” the flow. The velocity data are in reasonable agreement with Goldman’s solution, except for the results obtained farthest from the wall (at 200 nm ≤ h ≤ 300 nm) at Δp/L = 0.83 and 1.04 bar/m, which significantly underestimate the expected velocity. These discrepancies may be due to difficulties in distinguishing the 13773
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Figure 2. Flow velocity, U, as a function of z (a particle at z = a touches the wall) in Poiseuille flow. The dotted lines represent the corresponding simple shear flow profiles; the dashed lines represent the particle velocities corrected for the additional hydrodynamic drag due to the wall.25
Figure 3. Particle number density profiles c(h) for a = 245 nm particles in Poiseuille flow.
1.04 bar/m, c also appears to decrease as Δp/L increases at a given separation, suggesting that there is a lift force that drives the particles away from the wall with a magnitude that increases as Δp/L increases. The average uncertainty (i.e., standard deviation over four experiments) in c is 6% for h ≥ 100 nm. It should be noted that the uncertainties in c are likely overstated due to minor variations in c∞ (cf. Experimental Details). Unfortunately, it is impractical to accurately measure c∞ in the bulk of the flow during the experiments. Figure 4 shows c(h) for a = 245 nm particles for EO flow (Δp/L = 0) at E = 0−33.1 V/cm. Note that the E = 0 case here
Table 1. Near-Wall Shear Rates, γ̇, Corresponding to the Pressure Gradients Δp/L Δp/L [bar/m] γ̇ [s−1]
0.1 150
0.43 650
0.64 960
0.83 1250
1.04 1580
signal from the noise this far from the wall. Although results are not shown here, the velocity results obtained with the a = 125 nm particles were also in reasonable agreement with the analytical solution of Goldman et al.,25 albeit with some underestimation of the velocity for the results far from the wall. Particle Number Densities near the Wall. The agreement between the data and the analytical solution for the particle velocity in this shear flow suggests that the technique can estimate the wall-normal position of the particles with reasonable accuracy. Evanescent-wave particle tracking was therefore used to estimate the particle number density, c(h), following the procedure detailed in the previous section. Figure 3 shows c(h) for the a = 245 nm particles, where h is the particle−wall separation (so z = h + a) obtained from the same four experiments in Poiseuille flow (E = 0) at Δp/L = 0−1.04 bar/m. Unless stated otherwise, the error bars in this and subsequent figures represent the standard deviation in the results over four independent experiments. Note that there may still be a very weak flow at the lowest value of Δp/L (i.e., “0” Bar/m), albeit with speeds below 1 μm/s, or 2−3 orders of magnitude less than the speeds shown in Figure 2. The number-density profiles are all qualitatively similar, with a monotonic increase in c with h until the number density reaches a peak value at h ≈ 100 nm and then very little change (or perhaps a small decrease) in c farther away from the wall, with what appears to be a slow recovery to c∞ (denoted by the dashed line) at h > 300 nm (i.e., beyond the region accessible with evanescent-wave particle tracking). As predicted by DLVO theory, the peak in c, corresponding to the most probable value of h, occurs several Debye lengths from the wall, due to van der Waals and particle−wall EDL interactions. Although the profiles appear to “cluster” in two different groups, namely Δp/L = 0 and 0.1 bar/m and Δp/L = 0.43−
Figure 4. Similar to the previous figure, but for electroosmotic flow. The error bars are smaller than the symbols in some cases.
is the “no flow,” or Δp/L = 0 bar/m case in the previous figure. The number density profiles for the two lowest values of E are nearly identical, suggesting that increasing E has at most a weak effect on c. For E ≥ 9.5 V/cm, however, c decreases as E increases at a given h, with the maximum in c at h ≈ 100 nm completely disappearing for E ≥ 16.5 V/cm. These results 13774
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Figure 5. Similar to Figures 3 and 4, but for combined EO and Poiseuille flow at (a) Δp/L = 0.43 bar/m and (b) 1.04 bar/m. Both plots have the same vertical axis and legend.
where k is the Boltzmann constant and T is the absolute temperature of the fluid, subject to the boundary condition that ϕ → 0 as h → ∞. The potential profiles were smoothed by an unweighted lowpass filter over 60 nm windows. Figure 6 shows the particle potential profiles ϕ(h) calculated from the c(h) shown in Figure 3 for Poiseuille flow (E = 0) at
suggest that there is a lift force acting on the tracer particles at larger values of E, presumably the dielectrophoretic-like lift force observed previously in EO flows under similar conditions.14,15 Although results are not shown, these changes in c(h) with increases in Δp/L or E were not observed for smaller a = 125 nm particles, and only results for the a = 245 nm particles are presented in this section. Figure 5 shows the particle number density profiles c(h) measured for combined EO flow at E = 0−9.5 V/cm and Poiseuille flow at (a) Δp/L = 0.43 bar/m and (b) Δp/L = 1.04 bar/m. Again, the number density decreases as E increases at a given h, with c < 0.2c∞ for E ≥ 4.7 V/cm. 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. Moreover, a comparison of Figure 5a and b shows that the decrease in c increases with Δp/L, or the shear rate γ̇ at a given E. It also appears that the decrease for combined flow is much larger than that for EO flow at Δp/L = 0 (Figure 4) at the same E. This enhancement could, of course, be due to the additional shear-induced electrokinetic lift force due to γ̇, but the results of Figure 3 suggest that this effect is too weak to account for such a large effect. These observations suggest that the lift force for combined EO and Poiseuille flow that repels a = 245 nm particles away from the wall may exceed the sum of the shear-induced electrokinetic lift force due to γ̇ and the dielectrophoretic-like lift force due to E. The magnitude of this repulsive force can be estimated from the slope of the potential energy profile ϕ(h) of the particle−wall interaction, which can, in turn, be estimated from c(h) if this is the steady-state particle, and hence a Boltzmann, distribution. On the basis of calibrations (Figure S2 in the Supporting Information), a minimum of 750 image pairs were obtained for each case 15 min after changing Δp/L to ensure steady-state conditions. Particle−Wall Interaction Potential Energy for Poiseuille Flow. For a Boltzmann distribution, the normalized particle−wall interaction potential energy is ⎧ c(h) ⎫ ϕ(h) ⎬ = −ln⎨ kT ⎩ c∞ ⎭
Figure 6. Normalized particle−wall interaction potential energy profile ϕ(h)/(kT) for Poiseuille flow.
Δp/L = 0−1.04 bar/m. The profiles are similar for the two lowest values of Δp/L, with ϕ decreasing as h increases until it reaches a minimum at h ≈ 110 nm, then increasing as h increases. As expected, the h value at the potential energy minimum corresponds to the peak of the particle density profile, that is, most probable particle position (cf. Figure 3). At larger Δp/L, however, the location of the maximum shifts, and there is, at best, a weak decrease in ϕ at larger h. Note that |ϕ| < kT in all cases, which suggests that shear has a weak effect on
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the particle−wall interaction potential energy, at least for Δp/L ≤ 1.04 bar/m. Although results are not shown here, DLVO theory for the “no flow” (Δp/L = 0 bar/m) case gives a particle−wall potential profile in which the most probable separation distance is at h ≈ 70−80 nm. In Figure 6, this separation distance is hence larger and the gradient in the potential energy is less steep than those for the profile predicted by DLVO theory, but the profiles derived from these experimental data and predicted by DLVO theory are in reasonable agreement for h > 100 nm. These trends are similar to those observed in our previous studies of EO flow17 and have also been observed in similar total internal reflection microscopy (TIRM) studies.26,27 Although some of this discrepancy may be due to surface roughness,27 we suspect that these discrepancies are also due to image noise and hindered Brownian diffusion normal to the wall; the TIRM studies cited here considered larger particles with a > 1 μm where Brownian effects are negligible. In these experiments, the rms displacement due to unconfined Brownian diffusion during the exposure time is ∼30 nm. Moreover, the diffusion of these particles is hindered by the wall, so the particles have a greater probability of diffusing away from (vs toward) the wall, suggesting that hindered Brownian effects, which are most significant for smaller values of h, will tend to shift the minimum to higher values of h and broaden the minimum. As an initial attempt to isolate and estimate the potential energy change due to the presence of flow (and hence nonzero γ̇ and E), the increase in the potential above that for the Δp/L = 0 bar/m (and E = 0) case, which should correspond to the potential energy for a particle suspended in a quiescent fluid, was subtracted from the profiles shown in Figure 6 for Δp/L > 0: ΔϕF(h) kT
=
ϕ(h) ϕ(h) − kT kT
Δp / L = 0
Figure 7. Additional particle potential energy due to flow ΔϕF(h)/(kT) calculated from the data shown in Figure 6.
(4)
Figure 7 shows ΔϕF(h)/(kT) for Poiseuille flow at Δp/L = 0.1−1.04 bar/m. The positive values of ΔϕF show that the particle−wall potential increases due to the presence of shear. Lift Force Due to Shear Flow. The magnitude of this repulsive lift force - due to simple shear flow is then taken to be the slope of ΔϕF(h)/(kT), which is estimated by linear regression of the data shown in Figure 7 over 100 nm ≤ h ≤ 280 nm. Only the data farther from the wall were considered because of the agreement between the experimentally obtained particle−wall potentials and the predictions from DLVO theory (for the “no flow” case) over this range of h, and because the standard deviations in the data are smaller in this region. Note that these data should all be at the same shear rate because the flow this close to the wall is essentially simple shear (cf. Figure 2). Figure 8 shows -(γ )̇ ; here, the pressure gradients Δp/L = 0.1, 0.43 0.64, 0.83, and 1.04 bar/m correspond to shear rates γ̇ = 150, 650, 960, 1250, and 1580 s−1, respectively. The error bars denote the standard deviation over three independent experiments, and the dashed line represents the curve fit by linear regression. The lift force magnitude, - , appears to be proportional to γ̇ (and hence, Δp/L); a linear curve fit of these data (dashed line) gives a correlation coefficient R2 = 0.977. As mentioned earlier, various types of lift forces acting on near-wall particles subject to a shear flow have been studied by a number of groups. Many of these studies attributed what they termed electroviscous lift to the distortion, more specifically the
Figure 8. Magnitude of repulsive lift force due to shear (Poiseuille) flow as a function of shear rate. The dashed line represents a linear curve fit to these data, which have a maximum standard deviation of 1.1 fN.
breakdown of symmetry, of the electric and hydrodynamic stress fields in the region between the particle and the wall by the flow. Wu et al.28 compared the predictions by Bike and Prieve,7,8 Warszyński et al.,9 and Cox.10 In subsequent work, Warszyński and van de Ven29 mention that their “refined” theory9 for the 2D case of a cylinder near a wall (extended to the 3D case of a sphere near a wall using the Derjaguin approximation) corrects an error in Cox’s model.10 The electroviscous force magnitude predicted by all of these theories is proportional to the square of the particle velocity, u, and, hence, the square of γ̇ in simple shear flow. However, the forces in these Poiseuille flow experiments are proportional to γ̇, as shown in Figure 8. Hence, the scaling observed here is similar to that for the hydrodynamic lift force reported by Williams et al.11 from their FFF measurements, with a magnitude 13776
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Figure 9. Normalized particle−wall interaction potential profiles due to flow for combined EO flow at (a) Δp/L = 0.43 and (b) 1.04 bar/m and for EO flow (c) (Δp/L = 0). The profiles in parts a and b have the same vertical axis and legend. The error bars are smaller than the symbols in some cases.
-∝
a3γ ̇ h
deep channels, validated by Brownian dynamics simulations, to show that there are almost no particles within 60 nm of the wall. They attribute their observations to the existence of a lift force of ∼0.5 pN that repels particles from the wall. Their results suggest that the magnitude of this force is proportional to the shear rate, in agreement with the scaling reported here, although particle confinement is significant in these studies (unlike the experiments described here), with the particles occupying as much as a third of the channel depth. Their results also imply that the magnitude of the force increases with the ionic strength of the solution, given that they observe much less repulsion for particles suspended in the same sodium tetraborate solution studied here. Furthermore, Kazoe et al.30 have also recently used evanescent wave-based particle tracking velocimetry to track a = 32 nm particles in Poiseuille flow through 410-nm-deep channels and reported that their velocity profiles, while in agreement with the parabolic profile predicted by theory in the bulk, were larger than the theoretical predictions near the wall. Although they suggested that this behavior could be due to
(5)
Moreover, the magnitudes of the observed forcesabout 4.6 fN at the maximum value of γ̇ = 1580 s−1 (cf Figure 8)is much greater than that predicted by the various electroviscous force models: for γ̇ = 1580 s−1, the magnitude of the electroviscous force is ∼1.7 × 10−3 fN based on the Bike and Prieve model8 (eq 3 in Wu et al.28), ∼3.5 fN based on the (possibly flawed) Cox model10 (eqs 6 and 7 in Wu et al.28), and ∼0.52 fN based on the “refined” model of Warszyński and van de Ven.29 Given that neither the scaling with shear rate nor the actual value of the force are what would be expected for electroviscous lift, it seems highly unlikely that this lift force is due to electroviscous effects. Very recently, Ranchon and Bancaud [private communication] have used bright-field microscopy studies of the dynamics of a = 100 nm−150 nm particles (suspended in a solution consisting of 160 mM Tris−HCl, 160 mM boric acid, 5 mM EDTA and glycerin) in Poiseuille flow through 0.9−1.9-μm13777
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apparent slip caused by “molecular behavior in the fluid near the wall,” their observations are also consistent with a repulsive force driving these nanoparticles away from the wall. Potentials and Lift Force for Combined EO and Poiseuille Flow. A similar procedure was used to determine the potential energy profiles for the combined flow, as well as for EO flow (Δp/L = 0). The potential energy due to flow ΔϕF(h)/(kT) was then calculated for both cases. Figure 9 shows ΔϕF(h)/(kT) for combined flow at (a) Δp/L = 0.43 bar/m and (b) 1.04 bar/m, and E = 0−9.5 V/cm and for EO flow (Δp/L = 0) (c) at E = 4.7−33.1 V/cm. The potential difference for combined flow is clearly much greater than that for EO flow in the absence of shear at, for example, E = 4.7 V/cm (△) and 9.5 V/cm (□).
over the range of E shown in Figure 10) suggest that the magnitude of the dielectrophoretic-like lift force observed for EO flow in the absence of shear is proportional to E2, this lift force appears to be distinct from dielectrophoretic-like lift force observed in EO flow with no shear. Figure 11 shows a log−log plot of the lift force magnitude due to flow as a function of the shear rate for combined flow at
Figure 11. Graph of the lift force magnitude, - [log scale], as a function of shear rate, γ̇ [log scale], for combined flow at E = 4.7 (▽) and 9.5 V/cm (□). The lines represent power law curve fits to these data. In a few cases, the error bars are smaller than the symbols.
E = 4.7 and 9.5 V/cm. The lines are power law curve fits to these data with the exponent of the curve fit to the E = 4.7 V/ cm data (solid line) 0.492 (R2 = 0.973) and the exponent for the E = 9.5 V/cm (dashed line) 0.444 (R2 = 0.965). Although these data are over a limited range of shear rates, the magnitude of the lift force appears to be proportional to (γ̇)N, where N ≈ 0.4−0.5. The lift force observed in combined electroosomotic and Poiseuille flow also appears to be distinct from the electroviscous force6−10,28,29 and hydrodynamic lift force11 observed in shear flow in the absence of an electric field, which scale with γ̇2 and γ̇, respectively. The scaling of - with E, as seen in Figure 10 suggests that the lift force may have characteristics similar to electrostatic forces, which are, of course, proportional to E. In the absence of shear, a steady electric field creates a dipole along the flow direction due to the redistribution of the mobile counterions in the particle EDL, and the effect of this dipole on the Maxwell stress tensor generates wall-normal electroviscous lift forces.29 Although the lift forces observed in combined flow are not, based on their scaling, electroviscous in nature, we speculate that the combination of the steady electric field along the channel axis and the shear flow may create a dipole due to the mobile counterions surrounding the particle that is “inclined” with respect to the flow direction and that the wall-normal force that we observe here may be the wall-normal component of the electrostatic repulsion between this dipole, which is no longer parallel to the flow direction, and the counterions in the wall EDL. However, we know of no lift forces that are proportional to (γ̇)N, where N ≈ 0.4−0.5 in similar situations. Indeed, the only similar scaling that we are aware of is that observed in nonNewtonian power law fluids, where the viscous stress is proportional to (γ̇)N. However, non-Newtonian behavior seems implausible in these dilute (nominal bulk volume fraction
Figure 10. Lift force magnitude - as a function of electric field magnitude, E, for combined EO and Poiseuille flow at Δp/L = 0−1.04 bar/m. The dashed lines represent linear curve fits to the combinedflow results.
Figure 10 shows the magnitude of this repulsive lift force due to flow, - , estimated again from the slope of the ΔϕF(h)/(kT) results shown in Figure 9 over 100 nm ≤ h ≤ 280 nm as a function of the electric field magnitude E for combined flow at Δp/L = 0.43 bar/m and 1.04 bar/m, as well as for EO flow (i.e., Δp/L = 0). The lift forces for combined EO and Poiseuille flow are much greater than those for EO flow in the absence of shear; indeed, the minimum value of - of about 4 fN for combined flow exceeds the maximum value for EO flow, which is 1.6 fN. This large enhancement in the lift force observed in combined flow cannot, however, be due to the additional shearinduced electrokinetic lift force due to γ̇, since these forces are 1.9 fN and 4.6 fN for Δp/L = 0.43 bar/m and 1.04 bar/m, respectively (Figure 8). So the values of - obtained in these combined flow studies are much larger than the sum of the shear-induced electrokinetic lift force observed in Poiseuille flow at the same γ̇ (or Δp/L) and the dielectrophoretic-like lift force observed in EO flow at the same E. Moreover, - appears to scale with E in combined flow; the dashed lines in Figure 10 represent linear curve fits to these estimates of the lift force, with R2 = 0.997 for Δp/L = 0.43 bar/m and 1.04 bar/m. Given that previous studies16,17 and our results for EO flow (not shown, and admittedly not evident 13778
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0.17%) colloidal suspensions. Clarifying this scaling and its origins is the focus of current research.
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ASSOCIATED CONTENT
* Supporting Information
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Experimental setup and steady-state calibrations. This material is available free of charge via the Internet at http://pubs.acs. org/.
SUMMARY We report here studies of the dynamics of near-wall a = 245 nm fluorescent PS particles suspended in a monovalent aqueous electrolyte solution at a volume fraction of 0.17% flowing through 30-μm-deep fused-silica channels driven by pressure gradients as great as 1.1 bar/m and voltage gradients (i.e., electric fields) as great as 33 V/cm. The molar concentration of the aqueous solution was 1 mM, giving a Debye length λD ≪ a. Evanescent-wave particle tracking was used to determine both the flow velocity parallel to the wall and the distribution of the particles along the wall-normal direction for particle−wall separations h ≤ 300 nm. This close to the wall, the fluid velocity is essentially simple shear flow with a linearly varying velocity profile for Poiseuille flow with no electric field. The velocity results obtained for this case are, as expected, in reasonable agreement with previous analytical predictions25 for the velocity of a neutrally buoyant particle in simple shear, including the effect of the hydrodynamic drag on the particle due to the wall. The near-wall particle distribution, characterized by the particle number density profile, and the average force estimated from these profiles suggest that shear leads to a lift force that drives the a = 245 nm particles away from the wall. Although similar phenomena have been observed in other studies of near-wall particle dynamics in simple shear and dubbed electroviscous lift6−10 or hydrodynamic lift,11 the estimates of the lift forces observed in this study suggest that the scaling of the magnitude of this repulsive force, which is proportional to the shear rate, is consistent with hydrodynamic (vs electroviscous) lift. Although results are not shown here, the observations for EO flow with no pressure gradient are consistent with previous results reporting the existence of a dielectrophoretic-like lift force that also repels the particles from the wall.16,17 The most surprising observations are, however, for combined EO and Poiseuille flow driven by both a pressure gradient and an electric field. Again, the near-wall particle distributions suggest that there is a lift force that repels the particles from the wall. However, this force is much stronger than that observed for either Poiseuille flow or EO flow alone: our estimates suggest that the magnitude of this repulsive lift force significantly exceeds the sum of the (possibly hydrodynamic) lift force at the same shear rate and the dielectrophoretic-like lift force at the same electric field. Furthermore, the force magnitude scales with E, vs the E2 expected for the dielectrophoretic-like lift force, suggesting that this is an electrostatic repulsion, which we speculate may be due to the polarization of the particle EDL along a direction that is not parallel to the flow. Most surprising, these data suggest that the force is proportional to (γ̇)N, where N appears to range from 0.4 to 0.5, vs N = 2 for an electroviscous force, or N = 1 for the hydrodynamic lift force. These results suggest that the lift force observed in combined Poiseuille and EO flow is a new and, at present, unexplained phenomenon and that the combination of an applied electric field and shear flow affects the electric field in the particle−wall gap and the counterion distribution and, hence, the polarization of the EDL around the particle. Verifying these phenomena over a wider range of parameters and testing our conjectures about the mechanisms underlying this lift force are the focus of current work.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +1-404-894-6838. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by National Science Foundation award CBET-1235799. We thank J. P. Alarie and J. M. Ramsey in the Department of Chemistry at the University of North Carolina for providing the microchannels.
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REFERENCES
(1) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Micro total analysis systems. 1. Introduction, theory, and technology. Anal. Chem. 2002, 74, 2623−2636. (2) Auroux, P. A.; Iossifidis, D.; Reyes, D. R.; Manz, A. Micro total analysis systems. 2. Analytical standard operations and applications. Anal. Chem. 2002, 74, 2637−2652. (3) Whitesides, G. M. The origins and the future of microfluidics. Nature 2006, 442, 368−373. (4) Probstein, R. F. Physicochemical Hydrodynamics: An Introduction, 2nd ed.; John Wiley and Sons: Oxford, UK, 2003. (5) Stone, H. A.; Kim, S. Microfluidics: Basic issues, applications, and challenges. AIChE J. 2001, 47, 1250−1254. (6) Alexander, B. M.; Prieve, D. C. A hydrodynamic technique for measurement of colloidal forces. Langmuir 1987, 3, 788−795. (7) Bike, S. G.; Prieve, D. C. Electrohydrodynamic lubrication with thin double layers. J. Colloid Interface Sci. 1990, 136, 95−112. (8) Bike, S. G.; Prieve, D. C. Electrohydrodynamics of thin double layers: A model for the streaming potential profile. J. Colloid Interface Sci. 1992, 154, 87−96. (9) Warszyński, P.; Wu, X.; van de Ven, T. G. M. Electrokinetic lift force for a charged particle moving near a charged wall − a modified theory and experiment. Colloids Surf., A 1998, 140, 183−198. (10) Cox, R. G. Electroviscous forces on a charged particle suspended in a flowing liquid. J. Fluid Mech. 1997, 338, 1−34. (11) Williams, P. S.; Moon, M. H.; Xu, Y.; Giddings, J. C. Effect of viscosity on retention time and hydrodynamic lift forces in sedimentation/steric field-flow fractionation. Chem. Eng. Sci. 1996, 51, 4477−4488. (12) Cox, R. G.; Brenner, H. The lateral migration of solid particles in Poiseuille flow − I. Theory. Chem. Eng. Sci. 1968, 23, 147−173. (13) Velev, O. D.; Bhatt, K. H. On-chip manipulation and assembly of colloidal particles by electric fields. Soft Matter 2006, 2, 738−750. (14) Saiki, K.; Sato, Y. Nanoscale structure of electrokinetically driven flow obtained from large-area evanescent wave excitation. R. Soc. Chem. 2004, 296, 255−257. (15) Liang, L.; Ai, Y.; Zhu, J.; Qian, S.; Xuan, X. Wall-induced lateral migration in particle electrophoresis through a rectangular microchannel. J. Colloid Interface Sci. 2010, 347, 142−146. (16) Yariv, E. “Force-free” electrophoresis? Phys. Fluids 2006, 18, No. 031702. (17) Kazoe, Y.; Yoda, M. Experimental study of the effect of external electric fields on interfacial dynamics of colloidal particles. Langmuir 2011, 27, 11481−11488. (18) Lim, C. T.; Zhang, Y. Bead-based microfluidic immunoassays: the next generation. Biosens. Bioelectron. 2007, 22, 1197−1204.
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(19) Dutta, P.; Beskok, A. Analytical solution of combined electroosmotic/pressure driven flows in two-dimensional straight channels: finite Debye layer effects. Anal. Chem. 2001, 73, 1979−1986. (20) Monazami, R.; Manzari, M. T. Analysis of combined pressuredriven electroosmotic flow through square microchannels. Microfluid. Nanofluid. 2007, 3, 123−126. (21) Barz, D. P. J.; Zadeh, H. F.; Ehrhard, P. Measurements and simulations of time-dependent flow fields within an electrokinetic micromixer. J. Fluid Mech. 2011, 676, 265−293. (22) Johann, R.; Renaud, P. A simple mechanism for reliable particle sorting in a microdevice with combined electroosmotic and pressuredriven flow. Electrophoresis 2004, 25, 3720−3729. (23) Li, H. F.; Yoda, M. Multilayer nano-particle image velocimetry (MnPIV) in microscale Poiseuille flows. Meas. Sci. Technol. 2008, 19, 075402. (24) Keh, H. J.; Anderson, J. L. Boundary effects on electrophoretic motion of colloidal spheres. J. Fluid Mech. 1985, 153, 417−439. (25) Goldman, A. J.; Cox, R. G.; Brenner, H. Slow viscous motion of a sphere parallel to a plane wall − II. Couette flow. Chem. Eng. Sci. 1967, 22, 653−660. (26) Bevan, M. A.; Prieve, D. C. Direct measurement of retarded van der Waals attraction. Langmuir 1999, 15, 7925−7936. (27) Suresh, L.; Walz, J. Y. Direct measurement of the effect of surface roughness on the colloidal forces between a particle and a flat plate. J. Colloid Interface Sci. 1997, 196, 177−190. (28) Wu, X.; Warszyński, P.; van de Ven, T. G. M. Electrokinetic lift: observations and comparisons with theories. J. Colloid Interface Sci. 1996, 180, 61−69. (29) Warszyński, P.; van de Ven, T. G. M. Electroviscous forces on a charged cylinder moving near a charged wall. J. Colloid Interface Sci. 2000, 223, 1−15. (30) Kazoe, Y.; Iseki, K.; Mawatari, K.; Kitamori, T. Evanescent wavebased particle tracking velocimetry for nanochannel flows. Anal. Chem. 2013, 85, 10780−10786.
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Lift Forces on Colloidal Particles in Combined Electroosmotic and Poiseuille Flow Necmettin Cevheri† and Minami Yoda*,† †
G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0405 United States S Supporting Information *
ABSTRACT: Colloidal particles suspended in aqueous electrolyte solutions flowing through microchannels are subject to lift forces that repel the particles from the wall due to the voltage and pressure gradients commonly used to drive flows in microfluidic devices. There are very few studies that have considered particles subject to both an electric field and a pressure gradient, however. Evanescent-wave particle tracking velocimetry was therefore used to investigate the near-wall dynamics of a dilute suspension of 245 nm radius polystyrene particles in a monovalent electrolyte solution in Poiseuille and combined electroosmotic (EO) and Poiseuille flow through 30μm-deep fused-silica channels. The lift force observed in Poiseuille flow, which is estimated from the near-wall particle distribution, appears to be proportional to the shear rate, a scaling consistent with hydrodynamic lift forces previously reported in field-flow fractionation studies. The estimates of the lift force observed in combined flow suggest that the force magnitude exceeds the sum of the lift forces observed in EO flow at the same electric field or in Poiseuille flow at the same shear rate. Moreover, the force magnitude appears to be proportional to the electric field magnitude and have a power law dependence on the shear rate with an exponent between 0.4 and 0.5. This unexpected scaling suggests that the repulsive lift force observed in combined electroosmotic and Poiseuille flow is a new phenomenon, distinct from previously reported electroviscous, hydrodynamic lift, or dielectrophoretic-like forces, and warrants further study.
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INTRODUCTION The dynamics of dielectric colloidal particles of radii a = O(0.1−1 μm) near a planar dielectric wall suspended in a conducting medium, such as a quiescent aqueous electrolyte solution, is a classic problem of colloid science. More recently, the near-wall dynamics of suspended particles in a flowing electrolyte solution has become a problem of interest because of its applications in microfluidics. Transport in the microfluidic devices known as laboratories-on-a-chip (LoC) or micrototal analysis systems (μTAS)1,2 commonly entails incompressible creeping flows through microchannels, defined here to be channels with hydraulic diameters ranging from a few micrometers to a few hundred micrometers. The rapid diffusion and reduced sample and reagent volumes associated with shrinking devices down to the micrometer scale should increase the throughput and sensitivity, as well decrease the cost, of these devices.3 At such small length scales, surface forces also become significant as a result of relatively large surface areas and small volumes. Given that surface forces are usually short-ranged, that is, significant only within ∼0.5 μm of the surface, such forces should affect mainly the transport of particles near the microchannel walls. In the absence of flow, the forces acting on a near-wall particle are, based upon Derjaguin−Landau− Verwey−Overbeek (DLVO) theory,4 electrostatic forces due to the interaction of the particle and wall electric double layers (EDLs), attractive van der Waals forces, and the gravitational © 2014 American Chemical Society
force, which is negligible if the particle density is equal to that of the solution. For particles and walls with surface charges of the same sign, which is the usual case, the electrostatic interactions are repulsive. In the presence of a flow, however, colloidal particles are often subject to additional forces due to the presence of pressure or voltage gradients that are commonly used to drive microchannel flows.5 In fully developed Poiseuille flow through microchannels driven by a pressure gradient Δp/L, for example, the parabolic velocity profile is essentially simple shear flow near the wall, and it has been known for more than 20 years that large a = 4−9 μm polystyrene (PS) particles suspended in simple shear of a water−glycerol mixtures are repelled from the wall due to what was later termed a “shear-induced electrokinetic lift force”.6 This lift force is due to electroviscous effects, that is, polarization of the EDL due to convection of the mobile counterions within the EDL by the flow. In their studies using lubrication theory, Bike and Prieve7 studied this lift force theoretically, considering only the Maxwell stress tensor. They then developed a more general theory,8 relaxing the lubrication approximation. The analysis by Warszyński et al.9 also included hydrodynamic effects as well as the Maxwell stress tensor. Perhaps the most complete analysis of this problem is the Received: June 30, 2014 Revised: September 29, 2014 Published: October 24, 2014 13771
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EDLs. The flow was driven by dc electric fields with E < 33 V/cm and pressure gradients Δp/L < 1.1 bar/m through fusedsilica channels with a nominal depth of 30 μm. The images of these particles (with density ρp = 1.05 g/cm3), when illuminated with evanescent waves generated by the total internal reflection (TIR) of light at the interface between the flow and the fused-silica wall, can be processed to yield their 3D position measured with respect to the wall. Given that the wallnormal component of the velocity is effectively zero this close to the wall because of the no-flux boundary condition, only the velocity components parallel to the wall are estimated by matching the particle images in successive exposures to their nearest neighbors. The velocity profiles obtained for h ≤ 300 nm are in good agreement with theoretical predictions for the expected Poiseuille flow velocities. The brightness of the particle image can then, with appropriate calibration, be used to determine the wall-normal position of the particle, usually characterized in terms of the particle−wall separation, h, based on the assumption that this brightness has an exponential decay with wall-normal distance with a length scale identical to that of the illumination. The near-wall particle distribution and the forces acting on the particles can then be determined from ensemble of particle−wall separations from O(104) particle images in most cases.
analysis by the method of matched asymptotic expansions by Cox,10 who also considered hydrodynamic effects, including particle rotation as well as the Maxwell stress tensor. Shear-induced lift forces have also been observed in field-flow fractionation (FFF) studies. Williams et al.11 showed 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 could not be completely due to standard inertial lift forces12 and therefore hypothesized that there was an additional lift force mechanism, which they termed “hydrodynamic lift.” Voltage gradients, that is, electric fields, are also commonly used to “pump” fluids in microchannels, in part because the electroosmotic (EO) flow driven by an electric field parallel to the channel axis has an essentially uniform velocity profile that minimizes convective dispersion. Particles in such an electric field are also subject to electrophoretic (EP) and dielectrophoretic (DEP) forces.13 Furthermore, an electric field applied parallel to the channel axis, which is the usual case for EO flows, can lead to a repulsive force normal to the wall, which is known as a “dielectrophoretic-like lift force”.14,15 Yariv’s analysis16 showed that a nonzero lift force can be generated due to the breakdown of the symmetry of the Maxwell stress tensor in the gap between the dielectric particle and the dielectric wall and that this force has a magnitude that scales with E2, where E is the electric field magnitude, and a2. A recent experimental study17 of steady EO flows through 30-μm-deep fused-silica channels driven by dc electric fields showed that a = 240−463 nm PS, and silica particles with particle-wall gap h ≤ 300 nm were repelled from the wall by a force with a magnitude of O(1−10 fN) that was proportional to E2 and a2. Although the scaling of the force was consistent with dielectrophoretic-like lift, estimates of the actual magnitude of the force was significantly (up to 40 times) larger than that predicted by Yariv,16 possibly because this analysis assumes a particle−wall gap much greater than a. Near-wall transport of colloidal PS particles suspended in a flowing aqueous electrolyte solution is of importance in a variety of microfluidics applications. Microfluidics-based immunoassays and nucleic acid assays, for example, use PS or magnetic “beads” functionalized with “probes” (e.g., antibodies) to preconcentrate “targets” (e.g., proteins, nucleic acid fragments, viral particles).18 In many cases, the beads are then brought near the wall so that surface-mounted sensors can determine if the beads have bound target. An improved understanding of the forces acting on colloidal particles as they approach a wall due to applied pressure gradients and electric fields could therefore lead to new approaches for manipulating and sorting functionalized beads. Both numerical and experimental studies19−21 have shown that driving a flow by imposing both a pressure gradient and an electric field will give a flow with a velocity field that is simply the superposition of the Poiseuille and EO flows for these creeping flows. Although Johann and Renaud22 used a combination of Poiseuille and EO flows, albeit in distinct regions of their microfluidic device, to sort yeast cells almost a decade ago, we know of no studies that have looked at how the forces due to the combination of a pressure gradient and an electric field affect the near-wall dynamics of rigid particles, much less cells, in such flows. Evanescent wave-based particletracking velocimetry23 was therefore used to study the dynamics of a = 125 and 245 nm fluorescent PS particles suspended at number densities of O(1016 m−3) in the flow of a dilute monovalent electrolyte solution for the case of thin
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EXPERIMENTAL DETAILS
Materials and Chemicals. The working fluid in these experiments was an aqueous 1 mmol/L sodium tetraborate (Na2B4O7) solution consisting of sodium tetraborate decahydrate salt (Acros Organics 419450010) dissolved in double distilled deionized (DDI) water from a water purification system (Barnstead E-pure Ultrapure D4641) with an initial resistivity of ∼16 MΩ·cm. The solution was seeded with fluorescent carboxylate-terminated PS particles with radii a = 125 nm ± 4.5 nm (mean ± standard deviation) or a = 245 nm ± 7.5 nm (Life Technologies F8811, F8813), based on the manufacturer’s specifications, at nominal bulk number densities c∞ = 2.7 × 1016 m−3, corresponding to volume fractions, ϕ, of 2.2 × 10−4 and 1.7 × 10−3, respectively. The particles had absorption and emission peaks at 490 and 513 nm, respectively. The Debye length of the solution was λD = 6.8 nm, so λD ≪ a in all cases. The a = 125 nm and a = 245 nm particle solutions were sonicated for 15 min and filtered through syringe filters with pore sizes of 0.45 or 0.8 μm (Millipore SLHV033RS, SLAA033SS), respectively, to remove aggregated particles. Filtering these solutions introduced minor variations in the actual value of the bulk number density in these experiments. The solutions were then degassed under vacuum at an absolute pressure of O(10 kPa) for another 15 min to minimize pH changes due to absorption of CO2. Finally, the ζ potentials of the particles ζp were measured in these dilute suspensions using laser−Doppler microelectrophoresis with a Malvern Zetasizer; ζp = −67.8 ± 1.0 mV (average ± standard deviation) for the a = 125 nm particles and ζp = −47.3 ± 1.42 mV for the a = 245 nm particles. Procedure. These experiments studied combined EO and Poiseuille flow of dilute colloidal suspensions through isotropically wet-etched fused-silica microchannels with nominally trapezoidal cross sections of depth 30 μm and width 300 μm. The EO flow was driven by dc electric fields of magnitude E < 33 V/cm along the channel axis, generated by a low- or high-voltage dc power supply (Instek GPR3510HD, Stanford Research Systems PS325) attached to platinum electrodes, where the upstream electrode at the inlet was the cathode, and the downstream electrode at the outlet was the anode. The electrodes (Figure 1) were sealed to four-way tubing connectors (Fisher Scientific 15-315-32B) attached to inlet and outlet reservoirs (consisting of a ∼5 mm segment of i.d. 4 mm borosilicate glass tubing) of the channel. In all cases, E was small enough that Joule heating was negligible; the temperature of the fluid, measured by a thermocouple 13772
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identifying and removing images of overlapping or aggregated tracers, the displacements of the remaining particles in the plane parallel to the wall were determined by matching particles in the first images of the pair to their nearest neighbor in the second image of the pair, since the average interparticle distance of 3.3 μm was significantly greater than the particle displacements due to convection, which were typically less than 1.5 μm. The particle−wall separation, h, or the distance between the particle edge and the wall was then determined from the particle image intensity, Ip, which was defined to be the sum of the grayscale values (of each pixel in the particle image) divided by the number of pixels. Assuming that the particle image intensity has the same exponential decay as the evanescent-wave illumination where the length scale of the decay is zp,
Figure 1. Sketch of the combined flow setup showing the tubing connectors at the inlet and exit reservoirs of the microchannel. The flow and wall-normal directions are the x and z directions, respectively.
⎧ h⎫ ⎬ Ip(h) = Ip0exp⎨ − ⎪ ⎪ ⎩ zp ⎭ ⎪
at the exit, was within 1 °C of the ambient temperature, which varied between 19 and 21 °C. The Poiseuille flow, which was in the same direction as the EO flow, was driven by pressure gradients Δp/L < 1.1 bar/m generated hydrostatically by adjusting the height of a reservoir of DDI water that was attached to another port on the four-way connector at the inlet reservoir. A piece of tubing with a horizontal section at the exit was attached to the corresponding port of the four-way connector at the outlet reservoir to ensure that the exit pressure was constant over the course each experiment, and the remaining “top” ports of the four-way connectors were sealed. At the start of each experimental run, the channel was cleaned, then filled with the dilute suspension, and mounted on the stage of an inverted epi-fluorescence microscope (Leica DMIRE2). Each experimental run consists of several sets of experiments in which Δp/L increases from 0 to 1.1 bar/m; within each set of experiments at a given value of Δp/L, E increases from zero to its maximum value. For the experiments with a = 245 nm particles, the maximum value of E was 9.5 V/cm for Δp/L > 0, and 33 V/cm in all other cases. The bottom wall of the channel (with a manufacturer-quoted rms surface roughness of 3 nm) was illuminated with evanescent waves generated by the TIR of a CW argon ion laser beam at a wavelength λ = 488 nm shuttered by an acousto-optic modulator and coupled into the substrate with a fused-silica isosceles right triangle prism (Supporting Information Figure S1). The intensity-based penetration depth of the illumination was zp = 110 nm−120 nm, on the basis of measurements of the distance between successive TIR spots. The fluorescence from the particles in each experiment (i.e., at a given value of Δp/L and E) was imaged through a 525 ± 25 nm bandpass emission filter to isolate the longer-wavelength fluorescence from the illumination over a sequence of 750 image pairs, for which the image exposure time was τ = 0.5 ms. The time interval within the image pair, Δt = 2 ms, and the time between image pairs was 0.2 s (thereby ensuring that “new” particles were imaged in each pair), resulting in a total image acquisition time of ∼150 s for each measurement. The images of the flow in the center of the channel (to minimize any effects from the side walls) were acquired by an electron multiplying charge-coupled device (EMCCD) camera (Hamamatsu C9100-13) through a 63× magnification, 0.7 numerical aperture microscope objective (Leica PL Fluotar L) and recorded on the HD of a PC as 512 × 144 pixels images, with physical dimensions of 130 μm × 37 μm. As discussed in the Results and Discussion section, to ensure steady-state flow, images are acquired only at least 120 s after the start of a new experiment at a different value of E within a set at a given Δp/L, and at least 15 min after the start of a new set at a different value of Δp/L. At the end of each experimental run, a magnesium chloride (MgCl2) solution (nominal molar concentration of 7 mM) seeded with particles was injected into the channel to attach the particles to the fused-silica walls. Calibration images of these particles attached to the wall were acquired with the same illumination and imaging setup used in the experiments. The location of each particle center in a given image was determined by cross-correlation with a 2D Gaussian function. After
⎪
(1)
I0p
where is the average of the Ip values from the image of the wallattached particles acquired at the end of the corresponding experimental run. The ensemble of particle−wall separations was determined from at least 104 particle images (except for the combined flow cases at E = 9.5 V/cm, where at least 3 × 103 images were analyzed), and used to calculate the particle number density profile, c(h), over 20-nm-wide bins in h. The displacements of the a = 125 nm and a = 245 nm particles were then divided into three layers, each containing a similar number of samples, based on h: (1) 0 ≤ h ≤ 100 nm,(2) 100 nm ≤ h ≤ 200 nm, and (3) 200 nm ≤ h ≤ 300 nm. The particle velocity components parallel to the wall were then simply the particle displacement in each layer divided by Δt. The particle velocities, averaged over all the particles in each layer, are then placed at the average distance from the wall sampled by the particles, based upon c(h). As noted earlier, suspended particles will be convected by the flow and subject to electrophoresis. The particle velocity measured in these experiments is therefore
u p = U + uep = U +
εζpE μ
(2)
where U is the flow velocity, and uep is the electrophoretic velocity. Here, we assume that the Helmholtz−Smoluchowski relation for the electrophoretic velocity is valid (where ε and μ are the permittivity and absolute viscosity, respectively, of the fluid) because the hindrance of the electrophoretic mobility due to particle−wall interactions should be negligible in these studies.24
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RESULTS AND DISCUSSION Poiseuille Flow Velocity Profiles. Figure 2 shows the flow velocity profile U(z), where z is the wall-normal position of the centers of the a = 245 nm particles, for Poiseuille flow (E = 0) at Δp/L = 0.1−1.04 bar/m. These velocities are averaged over four independent experiments; in all cases, the standard deviations in U are smaller than the symbols. The dotted lines, which denote the analytical solution for Poiseuille flow (in the absence of particle−wall interactions), show that the flow near the wall is simple shear, with a shear rate, γ̇, proportional to Δp/L (Table 1 gives the near-wall shear rates, γ̇, corresponding to the values of Δp/L studied here). The dashed lines represent the analytical solution by Goldman et al.25 for the velocity of a neutrally buoyant near-wall particle in simple shear flow, including the hydrodynamic drag due to the presence of the wall, which causes the particle to “lag” the flow. The velocity data are in reasonable agreement with Goldman’s solution, except for the results obtained farthest from the wall (at 200 nm ≤ h ≤ 300 nm) at Δp/L = 0.83 and 1.04 bar/m, which significantly underestimate the expected velocity. These discrepancies may be due to difficulties in distinguishing the 13773
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Figure 2. Flow velocity, U, as a function of z (a particle at z = a touches the wall) in Poiseuille flow. The dotted lines represent the corresponding simple shear flow profiles; the dashed lines represent the particle velocities corrected for the additional hydrodynamic drag due to the wall.25
Figure 3. Particle number density profiles c(h) for a = 245 nm particles in Poiseuille flow.
1.04 bar/m, c also appears to decrease as Δp/L increases at a given separation, suggesting that there is a lift force that drives the particles away from the wall with a magnitude that increases as Δp/L increases. The average uncertainty (i.e., standard deviation over four experiments) in c is 6% for h ≥ 100 nm. It should be noted that the uncertainties in c are likely overstated due to minor variations in c∞ (cf. Experimental Details). Unfortunately, it is impractical to accurately measure c∞ in the bulk of the flow during the experiments. Figure 4 shows c(h) for a = 245 nm particles for EO flow (Δp/L = 0) at E = 0−33.1 V/cm. Note that the E = 0 case here
Table 1. Near-Wall Shear Rates, γ̇, Corresponding to the Pressure Gradients Δp/L Δp/L [bar/m] γ̇ [s−1]
0.1 150
0.43 650
0.64 960
0.83 1250
1.04 1580
signal from the noise this far from the wall. Although results are not shown here, the velocity results obtained with the a = 125 nm particles were also in reasonable agreement with the analytical solution of Goldman et al.,25 albeit with some underestimation of the velocity for the results far from the wall. Particle Number Densities near the Wall. The agreement between the data and the analytical solution for the particle velocity in this shear flow suggests that the technique can estimate the wall-normal position of the particles with reasonable accuracy. Evanescent-wave particle tracking was therefore used to estimate the particle number density, c(h), following the procedure detailed in the previous section. Figure 3 shows c(h) for the a = 245 nm particles, where h is the particle−wall separation (so z = h + a) obtained from the same four experiments in Poiseuille flow (E = 0) at Δp/L = 0−1.04 bar/m. Unless stated otherwise, the error bars in this and subsequent figures represent the standard deviation in the results over four independent experiments. Note that there may still be a very weak flow at the lowest value of Δp/L (i.e., “0” Bar/m), albeit with speeds below 1 μm/s, or 2−3 orders of magnitude less than the speeds shown in Figure 2. The number-density profiles are all qualitatively similar, with a monotonic increase in c with h until the number density reaches a peak value at h ≈ 100 nm and then very little change (or perhaps a small decrease) in c farther away from the wall, with what appears to be a slow recovery to c∞ (denoted by the dashed line) at h > 300 nm (i.e., beyond the region accessible with evanescent-wave particle tracking). As predicted by DLVO theory, the peak in c, corresponding to the most probable value of h, occurs several Debye lengths from the wall, due to van der Waals and particle−wall EDL interactions. Although the profiles appear to “cluster” in two different groups, namely Δp/L = 0 and 0.1 bar/m and Δp/L = 0.43−
Figure 4. Similar to the previous figure, but for electroosmotic flow. The error bars are smaller than the symbols in some cases.
is the “no flow,” or Δp/L = 0 bar/m case in the previous figure. The number density profiles for the two lowest values of E are nearly identical, suggesting that increasing E has at most a weak effect on c. For E ≥ 9.5 V/cm, however, c decreases as E increases at a given h, with the maximum in c at h ≈ 100 nm completely disappearing for E ≥ 16.5 V/cm. These results 13774
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Figure 5. Similar to Figures 3 and 4, but for combined EO and Poiseuille flow at (a) Δp/L = 0.43 bar/m and (b) 1.04 bar/m. Both plots have the same vertical axis and legend.
where k is the Boltzmann constant and T is the absolute temperature of the fluid, subject to the boundary condition that ϕ → 0 as h → ∞. The potential profiles were smoothed by an unweighted lowpass filter over 60 nm windows. Figure 6 shows the particle potential profiles ϕ(h) calculated from the c(h) shown in Figure 3 for Poiseuille flow (E = 0) at
suggest that there is a lift force acting on the tracer particles at larger values of E, presumably the dielectrophoretic-like lift force observed previously in EO flows under similar conditions.14,15 Although results are not shown, these changes in c(h) with increases in Δp/L or E were not observed for smaller a = 125 nm particles, and only results for the a = 245 nm particles are presented in this section. Figure 5 shows the particle number density profiles c(h) measured for combined EO flow at E = 0−9.5 V/cm and Poiseuille flow at (a) Δp/L = 0.43 bar/m and (b) Δp/L = 1.04 bar/m. Again, the number density decreases as E increases at a given h, with c < 0.2c∞ for E ≥ 4.7 V/cm. 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. Moreover, a comparison of Figure 5a and b shows that the decrease in c increases with Δp/L, or the shear rate γ̇ at a given E. It also appears that the decrease for combined flow is much larger than that for EO flow at Δp/L = 0 (Figure 4) at the same E. This enhancement could, of course, be due to the additional shear-induced electrokinetic lift force due to γ̇, but the results of Figure 3 suggest that this effect is too weak to account for such a large effect. These observations suggest that the lift force for combined EO and Poiseuille flow that repels a = 245 nm particles away from the wall may exceed the sum of the shear-induced electrokinetic lift force due to γ̇ and the dielectrophoretic-like lift force due to E. The magnitude of this repulsive force can be estimated from the slope of the potential energy profile ϕ(h) of the particle−wall interaction, which can, in turn, be estimated from c(h) if this is the steady-state particle, and hence a Boltzmann, distribution. On the basis of calibrations (Figure S2 in the Supporting Information), a minimum of 750 image pairs were obtained for each case 15 min after changing Δp/L to ensure steady-state conditions. Particle−Wall Interaction Potential Energy for Poiseuille Flow. For a Boltzmann distribution, the normalized particle−wall interaction potential energy is ⎧ c(h) ⎫ ϕ(h) ⎬ = −ln⎨ kT ⎩ c∞ ⎭
Figure 6. Normalized particle−wall interaction potential energy profile ϕ(h)/(kT) for Poiseuille flow.
Δp/L = 0−1.04 bar/m. The profiles are similar for the two lowest values of Δp/L, with ϕ decreasing as h increases until it reaches a minimum at h ≈ 110 nm, then increasing as h increases. As expected, the h value at the potential energy minimum corresponds to the peak of the particle density profile, that is, most probable particle position (cf. Figure 3). At larger Δp/L, however, the location of the maximum shifts, and there is, at best, a weak decrease in ϕ at larger h. Note that |ϕ| < kT in all cases, which suggests that shear has a weak effect on
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the particle−wall interaction potential energy, at least for Δp/L ≤ 1.04 bar/m. Although results are not shown here, DLVO theory for the “no flow” (Δp/L = 0 bar/m) case gives a particle−wall potential profile in which the most probable separation distance is at h ≈ 70−80 nm. In Figure 6, this separation distance is hence larger and the gradient in the potential energy is less steep than those for the profile predicted by DLVO theory, but the profiles derived from these experimental data and predicted by DLVO theory are in reasonable agreement for h > 100 nm. These trends are similar to those observed in our previous studies of EO flow17 and have also been observed in similar total internal reflection microscopy (TIRM) studies.26,27 Although some of this discrepancy may be due to surface roughness,27 we suspect that these discrepancies are also due to image noise and hindered Brownian diffusion normal to the wall; the TIRM studies cited here considered larger particles with a > 1 μm where Brownian effects are negligible. In these experiments, the rms displacement due to unconfined Brownian diffusion during the exposure time is ∼30 nm. Moreover, the diffusion of these particles is hindered by the wall, so the particles have a greater probability of diffusing away from (vs toward) the wall, suggesting that hindered Brownian effects, which are most significant for smaller values of h, will tend to shift the minimum to higher values of h and broaden the minimum. As an initial attempt to isolate and estimate the potential energy change due to the presence of flow (and hence nonzero γ̇ and E), the increase in the potential above that for the Δp/L = 0 bar/m (and E = 0) case, which should correspond to the potential energy for a particle suspended in a quiescent fluid, was subtracted from the profiles shown in Figure 6 for Δp/L > 0: ΔϕF(h) kT
=
ϕ(h) ϕ(h) − kT kT
Δp / L = 0
Figure 7. Additional particle potential energy due to flow ΔϕF(h)/(kT) calculated from the data shown in Figure 6.
(4)
Figure 7 shows ΔϕF(h)/(kT) for Poiseuille flow at Δp/L = 0.1−1.04 bar/m. The positive values of ΔϕF show that the particle−wall potential increases due to the presence of shear. Lift Force Due to Shear Flow. The magnitude of this repulsive lift force - due to simple shear flow is then taken to be the slope of ΔϕF(h)/(kT), which is estimated by linear regression of the data shown in Figure 7 over 100 nm ≤ h ≤ 280 nm. Only the data farther from the wall were considered because of the agreement between the experimentally obtained particle−wall potentials and the predictions from DLVO theory (for the “no flow” case) over this range of h, and because the standard deviations in the data are smaller in this region. Note that these data should all be at the same shear rate because the flow this close to the wall is essentially simple shear (cf. Figure 2). Figure 8 shows -(γ )̇ ; here, the pressure gradients Δp/L = 0.1, 0.43 0.64, 0.83, and 1.04 bar/m correspond to shear rates γ̇ = 150, 650, 960, 1250, and 1580 s−1, respectively. The error bars denote the standard deviation over three independent experiments, and the dashed line represents the curve fit by linear regression. The lift force magnitude, - , appears to be proportional to γ̇ (and hence, Δp/L); a linear curve fit of these data (dashed line) gives a correlation coefficient R2 = 0.977. As mentioned earlier, various types of lift forces acting on near-wall particles subject to a shear flow have been studied by a number of groups. Many of these studies attributed what they termed electroviscous lift to the distortion, more specifically the
Figure 8. Magnitude of repulsive lift force due to shear (Poiseuille) flow as a function of shear rate. The dashed line represents a linear curve fit to these data, which have a maximum standard deviation of 1.1 fN.
breakdown of symmetry, of the electric and hydrodynamic stress fields in the region between the particle and the wall by the flow. Wu et al.28 compared the predictions by Bike and Prieve,7,8 Warszyński et al.,9 and Cox.10 In subsequent work, Warszyński and van de Ven29 mention that their “refined” theory9 for the 2D case of a cylinder near a wall (extended to the 3D case of a sphere near a wall using the Derjaguin approximation) corrects an error in Cox’s model.10 The electroviscous force magnitude predicted by all of these theories is proportional to the square of the particle velocity, u, and, hence, the square of γ̇ in simple shear flow. However, the forces in these Poiseuille flow experiments are proportional to γ̇, as shown in Figure 8. Hence, the scaling observed here is similar to that for the hydrodynamic lift force reported by Williams et al.11 from their FFF measurements, with a magnitude 13776
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Figure 9. Normalized particle−wall interaction potential profiles due to flow for combined EO flow at (a) Δp/L = 0.43 and (b) 1.04 bar/m and for EO flow (c) (Δp/L = 0). The profiles in parts a and b have the same vertical axis and legend. The error bars are smaller than the symbols in some cases.
-∝
a3γ ̇ h
deep channels, validated by Brownian dynamics simulations, to show that there are almost no particles within 60 nm of the wall. They attribute their observations to the existence of a lift force of ∼0.5 pN that repels particles from the wall. Their results suggest that the magnitude of this force is proportional to the shear rate, in agreement with the scaling reported here, although particle confinement is significant in these studies (unlike the experiments described here), with the particles occupying as much as a third of the channel depth. Their results also imply that the magnitude of the force increases with the ionic strength of the solution, given that they observe much less repulsion for particles suspended in the same sodium tetraborate solution studied here. Furthermore, Kazoe et al.30 have also recently used evanescent wave-based particle tracking velocimetry to track a = 32 nm particles in Poiseuille flow through 410-nm-deep channels and reported that their velocity profiles, while in agreement with the parabolic profile predicted by theory in the bulk, were larger than the theoretical predictions near the wall. Although they suggested that this behavior could be due to
(5)
Moreover, the magnitudes of the observed forcesabout 4.6 fN at the maximum value of γ̇ = 1580 s−1 (cf Figure 8)is much greater than that predicted by the various electroviscous force models: for γ̇ = 1580 s−1, the magnitude of the electroviscous force is ∼1.7 × 10−3 fN based on the Bike and Prieve model8 (eq 3 in Wu et al.28), ∼3.5 fN based on the (possibly flawed) Cox model10 (eqs 6 and 7 in Wu et al.28), and ∼0.52 fN based on the “refined” model of Warszyński and van de Ven.29 Given that neither the scaling with shear rate nor the actual value of the force are what would be expected for electroviscous lift, it seems highly unlikely that this lift force is due to electroviscous effects. Very recently, Ranchon and Bancaud [private communication] have used bright-field microscopy studies of the dynamics of a = 100 nm−150 nm particles (suspended in a solution consisting of 160 mM Tris−HCl, 160 mM boric acid, 5 mM EDTA and glycerin) in Poiseuille flow through 0.9−1.9-μm13777
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apparent slip caused by “molecular behavior in the fluid near the wall,” their observations are also consistent with a repulsive force driving these nanoparticles away from the wall. Potentials and Lift Force for Combined EO and Poiseuille Flow. A similar procedure was used to determine the potential energy profiles for the combined flow, as well as for EO flow (Δp/L = 0). The potential energy due to flow ΔϕF(h)/(kT) was then calculated for both cases. Figure 9 shows ΔϕF(h)/(kT) for combined flow at (a) Δp/L = 0.43 bar/m and (b) 1.04 bar/m, and E = 0−9.5 V/cm and for EO flow (Δp/L = 0) (c) at E = 4.7−33.1 V/cm. The potential difference for combined flow is clearly much greater than that for EO flow in the absence of shear at, for example, E = 4.7 V/cm (△) and 9.5 V/cm (□).
over the range of E shown in Figure 10) suggest that the magnitude of the dielectrophoretic-like lift force observed for EO flow in the absence of shear is proportional to E2, this lift force appears to be distinct from dielectrophoretic-like lift force observed in EO flow with no shear. Figure 11 shows a log−log plot of the lift force magnitude due to flow as a function of the shear rate for combined flow at
Figure 11. Graph of the lift force magnitude, - [log scale], as a function of shear rate, γ̇ [log scale], for combined flow at E = 4.7 (▽) and 9.5 V/cm (□). The lines represent power law curve fits to these data. In a few cases, the error bars are smaller than the symbols.
E = 4.7 and 9.5 V/cm. The lines are power law curve fits to these data with the exponent of the curve fit to the E = 4.7 V/ cm data (solid line) 0.492 (R2 = 0.973) and the exponent for the E = 9.5 V/cm (dashed line) 0.444 (R2 = 0.965). Although these data are over a limited range of shear rates, the magnitude of the lift force appears to be proportional to (γ̇)N, where N ≈ 0.4−0.5. The lift force observed in combined electroosomotic and Poiseuille flow also appears to be distinct from the electroviscous force6−10,28,29 and hydrodynamic lift force11 observed in shear flow in the absence of an electric field, which scale with γ̇2 and γ̇, respectively. The scaling of - with E, as seen in Figure 10 suggests that the lift force may have characteristics similar to electrostatic forces, which are, of course, proportional to E. In the absence of shear, a steady electric field creates a dipole along the flow direction due to the redistribution of the mobile counterions in the particle EDL, and the effect of this dipole on the Maxwell stress tensor generates wall-normal electroviscous lift forces.29 Although the lift forces observed in combined flow are not, based on their scaling, electroviscous in nature, we speculate that the combination of the steady electric field along the channel axis and the shear flow may create a dipole due to the mobile counterions surrounding the particle that is “inclined” with respect to the flow direction and that the wall-normal force that we observe here may be the wall-normal component of the electrostatic repulsion between this dipole, which is no longer parallel to the flow direction, and the counterions in the wall EDL. However, we know of no lift forces that are proportional to (γ̇)N, where N ≈ 0.4−0.5 in similar situations. Indeed, the only similar scaling that we are aware of is that observed in nonNewtonian power law fluids, where the viscous stress is proportional to (γ̇)N. However, non-Newtonian behavior seems implausible in these dilute (nominal bulk volume fraction
Figure 10. Lift force magnitude - as a function of electric field magnitude, E, for combined EO and Poiseuille flow at Δp/L = 0−1.04 bar/m. The dashed lines represent linear curve fits to the combinedflow results.
Figure 10 shows the magnitude of this repulsive lift force due to flow, - , estimated again from the slope of the ΔϕF(h)/(kT) results shown in Figure 9 over 100 nm ≤ h ≤ 280 nm as a function of the electric field magnitude E for combined flow at Δp/L = 0.43 bar/m and 1.04 bar/m, as well as for EO flow (i.e., Δp/L = 0). The lift forces for combined EO and Poiseuille flow are much greater than those for EO flow in the absence of shear; indeed, the minimum value of - of about 4 fN for combined flow exceeds the maximum value for EO flow, which is 1.6 fN. This large enhancement in the lift force observed in combined flow cannot, however, be due to the additional shearinduced electrokinetic lift force due to γ̇, since these forces are 1.9 fN and 4.6 fN for Δp/L = 0.43 bar/m and 1.04 bar/m, respectively (Figure 8). So the values of - obtained in these combined flow studies are much larger than the sum of the shear-induced electrokinetic lift force observed in Poiseuille flow at the same γ̇ (or Δp/L) and the dielectrophoretic-like lift force observed in EO flow at the same E. Moreover, - appears to scale with E in combined flow; the dashed lines in Figure 10 represent linear curve fits to these estimates of the lift force, with R2 = 0.997 for Δp/L = 0.43 bar/m and 1.04 bar/m. Given that previous studies16,17 and our results for EO flow (not shown, and admittedly not evident 13778
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0.17%) colloidal suspensions. Clarifying this scaling and its origins is the focus of current research.
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ASSOCIATED CONTENT
* Supporting Information
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S
Experimental setup and steady-state calibrations. This material is available free of charge via the Internet at http://pubs.acs. org/.
SUMMARY We report here studies of the dynamics of near-wall a = 245 nm fluorescent PS particles suspended in a monovalent aqueous electrolyte solution at a volume fraction of 0.17% flowing through 30-μm-deep fused-silica channels driven by pressure gradients as great as 1.1 bar/m and voltage gradients (i.e., electric fields) as great as 33 V/cm. The molar concentration of the aqueous solution was 1 mM, giving a Debye length λD ≪ a. Evanescent-wave particle tracking was used to determine both the flow velocity parallel to the wall and the distribution of the particles along the wall-normal direction for particle−wall separations h ≤ 300 nm. This close to the wall, the fluid velocity is essentially simple shear flow with a linearly varying velocity profile for Poiseuille flow with no electric field. The velocity results obtained for this case are, as expected, in reasonable agreement with previous analytical predictions25 for the velocity of a neutrally buoyant particle in simple shear, including the effect of the hydrodynamic drag on the particle due to the wall. The near-wall particle distribution, characterized by the particle number density profile, and the average force estimated from these profiles suggest that shear leads to a lift force that drives the a = 245 nm particles away from the wall. Although similar phenomena have been observed in other studies of near-wall particle dynamics in simple shear and dubbed electroviscous lift6−10 or hydrodynamic lift,11 the estimates of the lift forces observed in this study suggest that the scaling of the magnitude of this repulsive force, which is proportional to the shear rate, is consistent with hydrodynamic (vs electroviscous) lift. Although results are not shown here, the observations for EO flow with no pressure gradient are consistent with previous results reporting the existence of a dielectrophoretic-like lift force that also repels the particles from the wall.16,17 The most surprising observations are, however, for combined EO and Poiseuille flow driven by both a pressure gradient and an electric field. Again, the near-wall particle distributions suggest that there is a lift force that repels the particles from the wall. However, this force is much stronger than that observed for either Poiseuille flow or EO flow alone: our estimates suggest that the magnitude of this repulsive lift force significantly exceeds the sum of the (possibly hydrodynamic) lift force at the same shear rate and the dielectrophoretic-like lift force at the same electric field. Furthermore, the force magnitude scales with E, vs the E2 expected for the dielectrophoretic-like lift force, suggesting that this is an electrostatic repulsion, which we speculate may be due to the polarization of the particle EDL along a direction that is not parallel to the flow. Most surprising, these data suggest that the force is proportional to (γ̇)N, where N appears to range from 0.4 to 0.5, vs N = 2 for an electroviscous force, or N = 1 for the hydrodynamic lift force. These results suggest that the lift force observed in combined Poiseuille and EO flow is a new and, at present, unexplained phenomenon and that the combination of an applied electric field and shear flow affects the electric field in the particle−wall gap and the counterion distribution and, hence, the polarization of the EDL around the particle. Verifying these phenomena over a wider range of parameters and testing our conjectures about the mechanisms underlying this lift force are the focus of current work.
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AUTHOR INFORMATION
Corresponding Author
*Phone: +1-404-894-6838. E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by National Science Foundation award CBET-1235799. We thank J. P. Alarie and J. M. Ramsey in the Department of Chemistry at the University of North Carolina for providing the microchannels.
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REFERENCES
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