Langmuir 2005, 21, 10941-10947
10941
Movement of Colloidal Particles in Two-Dimensional Electric Fields Junhyung Kim,†,‡,§ Stephen Garoff,*,†,‡,£ John L. Anderson,†,‡,# and Luc J. M. Schlangen| Department of Chemical Engineering, Center for Complex Fluid Engineering, and Physics Department, Carnegie Mellon University, Pittsburgh, Pennsylvania, Display Device Group, LCD R&D Center, Samsung Electronics, Yongin, Korea, Department of Chemical Engineering, Case Western University, Cleveland, Ohio, and Philips Research Laboratories, Eindhove, The Netherlands Received May 9, 2005. In Final Form: August 23, 2005 We characterize the movement of carbon black particles in inhomogeneous, two-dimensional dc electric fields. Motivated by display applications, the particles are suspended in a nonpolar solvent doped with a charge control agent. The two-dimensional fields are generated between strip electrodes on a glass slide spaced 120 µm apart with field strengths up to 104 V/m. Such fields are insufficient to drive either electrohydrodynamic instabilities or natural convection due to ohmic heating, but they move the particles between the electrodes in about 30 s. In the center region between the strip electrodes, the particles move by electrophoresis; that is, the particle velocity is proportional to the electric field. However, when imposing a constant-potential or constant-current boundary condition at the electrodes to derive the electrical field, the electrophoretic mobility calculated from the measured particle velocities is outside the range of mobilities predicted from the theory of O’Brien and White. Near the electrodes the particles either speed up or slow down, depending on the polarity of the electrode, and these changes in velocity cannot be explained simply by electrophoresis in a spatially varying electric field. We suggest that this anomalous motion arises from electrohydrodynamic flows originating from the interaction between the space charge of the polarized layers above the electrodes and the electric field. Approximate calculations indicate such flows could be sufficiently strong to explain the anomalous trajectories near the edges of the electrodes.
Introduction Electrophoretically driven particle motions are employed in display technologies in which particles are rotated1 or translated within pixel-size compartments.2-6 Two-dimensional electric fields are employed to move particles to specific locations within the compartments, as illustrated in Figure 1. Nonpolar solvents are often used to inhibit electrochemistry and allow fields to be sustained long enough to move particles over distances on the order of 100 µm. In this paper, we explore the motion of particles dispersed in a nonpolar liquid in an electric field between strip-patterned electrodes that create twodimensional, spatially nonuniform electric fields. We use dodecane, doped with a charge control agent (CCA), as the suspending medium and carbon black as the suspended particles. * To whom correspondence should be addressed. † Department of Chemical Engineering, Carnegie Mellon University. ‡ Center for Complex Fluid Engineering, Carnegie Mellon University. § Samsung Electronics. £ Physics Department, Carnegie Mellon University. # Case Western University. | Philips Research Laboratories. (1) Gyricon Media, http://www.gyriconmedia.com. (2) Comiskey, B.; Albert, J. D.; Yoshizawa, H.; Jacobson, J. An Electrophoretic Ink for All-Printed Reflective Electronic Displays. Nature 1998, 394, 253-255. (3) Murau, P.; Singer, B. The Understanding and Elimination of Some Suspension Instabilities in an Electrophoretic Display. J. App. Phys. 1978, 49, 4820-4829. (4) Dalisa, A. L. Electrophoretic Display Technology. IEEE Trans. Electron Devices 1977, ED-24, 827-834. (5) Elliott, G. Recent Developments in Liquid Crystals and Other New Display Techniques. Radio Electron. Eng. 1976, 46, 281-295. (6) Sipix, http://www.sipix.com.
Figure 1. Use of two-dimensional electric fields to move particles laterally in thin liquid films. One application is a colloidal filter overlay on a color array. The particles block light at the pixel level.
A system such as that described above poses a number of scientific questions. The conduction mechanism in such nonpolar fluids has been previously studied.7-18 While electrophoresis might be expected to dominate particle motion in one-dimensional electric fields, the forces applied (7) Denat, A., Gosse, B. and Gosse, J. P. Electrical Conduction of Solutions of an Ionic Surfactant in Hydrocarbons. J. Electrostatics 1982, 12, 197-205. (8) Denat, A.; Gosse, B.; Gosse, J. P. High Field dc and ac Conductivity of Electrolyte Solutions in Hydrocarbons. J. Electrostatics 1982, 11, 179-194. (9) Denat, A.; Gosse, B.; Gosse, J. P. Ion Injection in Hydrocarbons. J. Electrostatics 1979, 7, 205-225. (10) Douwes, C.; Waarden, M. v. Current Decay during Measurement of dc Electric Conductivity in Solutions in Hydrocarbons. J. Inst. Pet. 1967, 53, 237-250. (11) Klinkenberg, A.; Minne, J. L. Electrostatics in the Petroleum Industry; Elsevier Publishing Company: the Hague, The Netherlands, 1958. (12) Novotny, V.; Hopper, M. A. Transient Conduction of Weakly Dissociating Species in Dielectric Fluids. J. Electrochem. Soc: Electrochem. Sci. Technol. 1979, 126, 925-929. (13) Lewis, T. J. The Basic Processes of Conduction in Dielectric Liquids. In IEEE 11th International Conference on Conduction and Breakdown in Dielectric Liquids; IEEE: New York, 1993; pp 32-41. (14) Felici, N.; Gosse, B.; Gosse, J. P. Factors Controlling Ion Injection by Metallic Electrodes in Dielectric Liquids. In Proceedings of International Conference on Conduction and Breakdown in Dielectric Liquids; IEEE: New York, 1975; pp 103-106.
10.1021/la051233c CCC: $30.25 © 2005 American Chemical Society Published on Web 10/05/2005
10942
Langmuir, Vol. 21, No. 24, 2005
Kim et al.
Figure 2. Strip electrode cell. The three strips are tin-doped indium oxide layers 0.11 µm thick on a glass slide. The strips are 30 µm wide (w) and 2.4 cm long (L). The center-to-center spacing between the strips is 150 µm, and the fluid gap (h) is 30 ( 10 µm. The x and y coordinates have an origin at the center of the middle strip.
to the particle in an inhomogeneous, two-dimensional electric field are not as well characterized. In this paper, we find regions in the two-dimensional field where electrophoresis does dominate the particle motion. However, we find that another force, perhaps arising from electrohydrodynamic flows, has a very significant impact on particle movement. Design of display devices will need to take such forces into account. Experimental Section Details of the experiments are presented elsewhere.19 Dodecane (Aldrich, 99+%) was used as the hydrocarbon solvent, and OLOA 371 (Chevron) was the charge control agent at a concentration in the range 3-4 wt%. Carbon black particles (Black Pearl 130, Cabot Co.) were suspended in the solutions. Ten to twenty microliters of a 10 wt% batch carbon black/ dodecane/OLOA dispersion was introduced into glass bottle filled with another OLOA/dodecane solution using a micropipet. The dispersion was then mixed for 10-15 min in a PowerGen 125 homogenizer (Fisher Scientific, PA). The conductivity of the solutions, measured with a Scientifica model 627 conductivity meter (Scientifica, NJ), was 27.2 ((3.4) nS/m, and the concentration of the poly-disperse carbon black was 45 ((10) ppm. The suspended particles had an average diameter of 1 µm, as determined by optical microscopy. The three electrodes were thin strips of tin-doped indium oxide (ITO) films on a glass slide (Figure 2) and were obtained from the Philips Research Laboratories (Eindhoven, The Netherlands). The strips were 30 µm wide, spaced 150 µm center-to-center, and 0.11 µm high. We introduced a drop (∼20 µL) of the dispersion on the glass substrate with the strip electrodes and then placed a glass microscope cover slip (Fisher Scientific, 2.4 cm × 4.0 cm) on top of the dispersion drop. The weight of the cover slip and surface tension of the suspension determined thicknesses of the liquid layer above the electrodes. The average layer thickness was 30.2 ( 9.8 µm. After the particles sedimented onto the slide containing the strip electrodes, video microscopy analysis showed that the particles were uniformly distributed across the surface. A dc potential (V0) was applied to the center strip using a Keithley 2400 sourcemeter, while the two outer strips were grounded. Transient currents were measured with a Keithley 6514 electrometer, and a typical transient is shown in Figure 3. The transient current decayed rapidly at first, due to electrode (15) Mechlia, T., Gosse, B., Denat, A., and Gosse, J. P. Electrophoretic Determination of the Electric Charge at the Liquid-Solid Interface: Relation with Conduction Phenomena. In 8th International Conference on Conduction and Breakdown in Dielectric Liquids; IEEE: New York, 1984; pp 293-297. (16) Dikarev, B. N.; Karasev, G. G.; Bolshakov, V. I.; Romanets, R. G.; Potapov, I. V. Electrization and Electrical Conduction of Dielectric Liquids. J. Electrostatics 1997, 40-41, 147-151 (17) Felici, N. J. Conduction and Electrification in Dielectric Liquids: Two Related Phenomena of the Same Electrochemical Nature. J. Electrostatics 1984, 15, 291-297. (18) Kim, J.; Anderson, J. L.; Garoff, J.; Schlangen, L. J. M. Ionic Conduction and Electrode Polarization in a Nonpolar Liquid containing a Charge Control Agent. Langmuir 2005, in press. (19) Kim, J., Dynamics of Particles in Spatially and Temporally Varying Electric Fields near Electrodes, PhD Thesis, Carnegie Mellon University, Pittsburgh, PA, 2004.
Figure 3. An example of transient current for V0 ) 5 V applied to the center strip. polarization, but remained nearly constant after about five seconds. The “residual” current (Ires) persisted for minutes. This current was used to determine the electric field between the electrodes, as discussed later. Typically, Ires was about 10% of the current at time zero (when V0 was first applied). The trajectories of particles were measured when the current had decayed to its residual value. Particle motions were tracked with an Olympus IMT-2 inverted microscope with a long working distance objective (10× objective lens and 1× intermediate lens). Trajectories were recorded using a Sony 77 CCD camera and a Panasonic V-AG 7355 SVHS video recorder with a Panasonic WJ-810 time-date generator to provide timing of events. Frames from the videotape were digitized at a rate of 10 frames per second with a Scion Image LG-3 frame grabber installed in a Macintosh G3 computer. Positions of particles as a function of time were determined. We only analyzed relatively isolated particles to minimize the effects of particleparticle interactions. To maintain the conditions of isolated particles, we only tracked particles in the cell for a single application of the potential. To perform another experiment, the cell was disassembled, cleaned, and reassembled before applying the potential again. We used sufficiently low voltages in our experiments so neither convection due to ohmic heating nor electrohydrodynamic instabilities affected the observed particle motions. Even applying an ac potential of 20 V peak-to-peak at 100 Hz (creating ohmic heating well above that in our experiments) did not induce any observable convective motion of the particles. Application of a dc field to a low-conductivity liquid can create an electrohydrodynamic instability.3 Our voltages are below the critical values needed to induce such instabilities. When we applied dc fields much larger than those used in the experiments reported here, we saw irregular motion of the particles due to these electrohydrodynamic instabilities.
Results The transient current, depicted in Figure 3, indicates that the electrodes polarized with time due to accumulation of charge and possibly some free energy expended in charge transfer across the metal surface. After 5 s, the current reached a nearly constant value, which we call the ‘residual’ current, and at that time, we began tracking particles. Panels a and b of Figure 4 show trajectories of particles when dc potentials of 5 and -5 V, respectively, were applied to the center strip electrodes, with the two outer strips grounded. In both cases, the particles moved toward the higher-potential electrode, indicating they were negatively charged and the motion was electrophoretic in nature. In the center region between the strips (at distances greater than ≈30 µm from the electrode edge), the particles moved at nearly constant velocity (Ux) directed toward the electrode of higher potential. In Figure 4, the crossing trajectories might reflect the fact that the
Movement of Colloidal Particles in Electric Fields
Langmuir, Vol. 21, No. 24, 2005 10943
fields do show some asymmetry in the magnitude of the fields between the central and outer electrodes. We know that the particles above the glass surface did not significantly change elevation (their y value) because they remained in focus while moving in the x direction. (Our resolution limit for determining vertical motion is ≈3 µm.) While we report here the motion of particles that have settled against the glass surface between the electrode strips, some particles remained suspended well above the glass surface. By sweeping the focal plane of the microscope through the cell gap, we confirmed that the x motion of particles was similar at all elevations in the cell, i.e., the particles suspended well above the glass surface moved in the same direction as those near the electrode, albeit with slower speeds. We note that none of the behaviors discussed depend on time for t > 5 s, i.e., a particle passing a particular position between the electrodes exhibits a similar trajectory as another particle passing the same point at another time. Thus, the forces driving particle motion are steady in time for t > 5 s in the region between the electrodes. Discussion
Figure 4. (a) Sample trajectories of particles when V0 ) +5 V on the center strip. (b) Sample trajectories of particles when V0 ) -5 V. Table 1. Particle Velocity Far from the Edges of the Electrode Strips (Ux, µm/s)), Residual Current Divided by Initial Current (Ires/I0), Residual Electric Field at y ) 0.0166 Midway between the Strips (Eres,x, V/cm)), and Electrophoretic Mobility (µe ) Ux/Eres,x, (µm/s)/(V/cm)) versus the Electrical Potential Applied to the Center Strip (V0, V)a Eres,x V0
Ux
Ires/I0
CP
CC
µe CP
CC
+3.0 -1.64 ( 0.36 0.13 ( 0.02 24.7 14.3 -0.066 -0.115 +5.0 -2.54 ( 0.75 0.11 ( 0.02 34.8 20.1 -0.073 -0.126 -5.0 2.42 ( 0.56 0.11 ( 0.01 -34.8 -20.1 -0.069 -0.120 a
CP refers to constant potential (eq 7a), and CC refers to constant current (eq 7b).
carbon black dispersions are not homogeneous and have a significant variation in size and probably charge. Leastsquares fitting of the linear portion of the trajectories of several particles for each value of the potential on the center strip gives the results listed in Table 1. As the particles moved within about 30 µm of the higherpotential electrode, their speed increased and the particles stuck to the electrode edge. Particles initially located very close to or above the lower potential electrodes migrated to a position some small distance from the electrode edge and then ceased to move. The increase in velocity of particles at the higher-potential electrode began at larger distances from the electrode and was greater when the center strip was at the high potential. Similarly, the position where the particles ceased motion near the lowerpotential electrode was greater when that electrode was at the center (≈30 µm from the electrode edge) compared to when the lower potential was at the outer electrode (≈3-5 µm). As shown later, calculations of the electric
Determination of the Electric Field. The electrophoretic mobility (µe) of a particle is defined as Ux/Ex. The field is not known accurately for the period of time we observed the particle motion because the electrodes were significantly polarized, as indicated by the large drop in current shown in Figure 3. Because a comprehensive model for the electrode polarization is not available, we must estimate Ex from the measured current. The primary current in a conducting liquid, that is, the current outside the space charge of the electrode polarization layers, is governed by Laplace’s equation in the absence of ion concentration gradients:20
∇2Φ ) 0
(1)
When the potential V0 is initially applied to the center strip (and the outer strips are grounded), the potential in the liquid in contact with the electrode equals the potential of the metal. In this ‘constant-potential’ case, the boundary conditions for (eq 1) are
center strip: Φ ) V0, outer strips: Φ ) 0
(2)
We also assume an insulating condition on the glass surfaces and at the outer boundary of the cell:
glass:
∂Φ ∂Φ ) 0; x f (∞: )0 ∂y ∂x
(3)
Solution of (eq 1) with the above boundary conditions leads to the determination of the field. Equation 1 was numerically solved with the appropriate boundary conditions using commercial CFD packages.19,21 In the numerical solution, the second boundary condition in eq 3 was specified at x equal to three times the center-to-center distance between the strips, since the results did not change in the calculations when this boundary was moved to larger x. In terms of dimensionless parameters (Φ scaled by V0, x and y scaled by the width, w, of the strip electrodes), the field (Ex ) -∂Φ/∂x) near the glass surface is shown in Figure 5a. Note that, away from the edges of the electrodes (20) Newman, J. S. Electrochemical Systems; Prentice Hall: New York, 1991. (21) Luis Rojas performed the calculations using ANSYS-CFX5.6, which is a CFD code based on finite volume-based finite element method with coupled multigrid solver using hexaedral elements.
10944
Langmuir, Vol. 21, No. 24, 2005
Kim et al.
of the liquid, while in the constant-current scenario the scaling is I/KL. These two parameters are connected through the cell constant B for the constant-potential case, which is defined by:
I0 )
K V B 0
(5)
where B is given by
1
B)-
∫0
2L
1/2
(6)
∂Φ dx ∂y
For our cell, the constant-potential solution gives B ) 2.6/L. Using the above relationships, we obtain the following for the dimensional electric field for the residual current, Ires, in either the constant-potential or constantcurrent scenarios:
( )( )( ) ( ) ( ) ( )( )
constant potential: Eres,x ) constant current: Eres,x )
Figure 5. The x component of the electric field (dimensionless) at one particle radius from the glass surface (y ) 0.0166 ) a/w) as determined by solving eq 1 for the potential in the fluid. The geometry of the electrode cell is shown in Figure 2. x is made dimensionless by the width of the strips (w). Note that the fields at y ) 0.0166 are essentially the same as for y ) 0. (a) Constant-potential boundary condition on the strips; Φ is scaled by V0, the potential applied to the center strip. (b) Constantcurrent boundary condition on the strips; Φ is scaled by I/KL where I is the total current, K is the solution conductivity, and L is the length of the strips.
(40 < x < 110 µm), Ex is approximately constant; Ex ) 0.19 at the midpoint between the strips, which is slightly below 0.25, the value obtained from the ratio of the difference in potential (eq 1) between the two strips divided by the distance (4 in dimensionless terms) between the edges. As expected, the magnitude of the field increases dramatically near the edges of the strips. The assumption of constant potential on the electrode surfaces is valid immediately upon application of the potential V0 to the center strip; however, as charge builds up on the electrodes, the potential in the liquid just outside the space charge of the double layer changes. The polarization is greatest near the edges of the strips where the field is most intense, thereby leading to a condition that is modeled in the extreme with a constant-current density across the electrodes. In terms of dimensionless potential (Φ scaled by I/KL where L is the length of the strips, I is the total current, and K is the solution conductivity), the ‘constant-current’ boundary condition is
center strip:
∂Φ 1 ∂Φ ) -1; outer strips: ) (4) ∂y ∂y 2
Figure 5b shows the field between the strips that was obtained by numerically solving21 eqs 1, 3, and 4. The field is essentially constant and equal to 0.285 at the midpoint between the strips. In the constant-potential scenario, the potential of the center strip (V0) is the scaling parameter for the potential
Ires V0 ∂Φ I0 w ∂x
Ires ∂Φ ) wLK ∂x cc Ires 1 V0 ∂Φ I0 LB w ∂x
(7a)
cp
(7b)
cc
where (-∂Φ/∂x)cp and (-∂Φ/∂x)cc are plotted in panels a and b of Figure 5, respectively, and I0 is the current immediately upon application of potential V0 to the center electrode. To compare the constant-current field with the constant-potential field, (-∂Φ/∂x)cc should be divided by LB. Thus, for the same total current, Eres,x on the glass surface at the midpoint between the strips is lower for the constant-current case, the ratio being (0.285/0.190)/2.6 ) 0.58. We also note that the asymmetry in the field between the center and outer strips is greater for the constantcurrent case. The validity of the above expressions for our experiments over the range 5-100 s depends on the correctness of boundary condition eqs 2 or 4 and requires the conductivity of the solution to remain constant during this time. The constant-potential boundary condition in eq 2 neglects any field changes that result from accumulation of charged species near the electrode surfaces during double-layer formation. The constant-current boundary condition expressed by eq 4 would be applicable when the resistance to charge transfer at the electrode surface controls the local current, thus leveling the current across the electrode. If the system begins with a constant-potential condition, then the high fields (Ey) at the edge of the strips would produce more countercharges, thereby reducing the effective potential for driving the charged species in the bulk solution. The actual case in our experiments, after the current begins to decay, is probably not modeled by either the constant-potential or constant-current boundary condition, rather it most likely somewhere between these two extremes. The assumption of constant conductivity can be checked by comparing the charge conducted over the 100 s period of observation with the total charge initially in solution. If the strips were perfectly polarized, i.e., there was no charge exchange between the metal and the solution, then the current would lead to build up of the charge carriers at the electrode surfaces with possible depletion of the charge carriers in the bulk solution. Integration of the current over 100 s indicates that less than 2% of the charge
Movement of Colloidal Particles in Electric Fields
Langmuir, Vol. 21, No. 24, 2005 10945
carriers initially present in the solution was transferred to the electrodes; thus, the conductivity of the solution can be considered time independent. We make a final point about the difference between the constant-potential and constant-current fields: in the latter case, Ex is nonzero on the strips. This is important because this field drives a flow of the liquid from the glass surface toward each strip, which might explain the anomalous trajectories of the particles near the edges of the electrodes as discussed later. For the constantpotential assumption Ex ) 0 on the electrodes, so that electrohydrodynamic flow of the liquid would not exist. Electrophoretic Mobility of the Particles. The electrophoretic mobility of the carbon black particles (µe ) Ux/Ex) can be determined using the values for the measured particle velocity (Ux) and residual current. Ex is estimated from either eq 7a or eq 7b, using the values of the field in Figure 5a and b at the midpoint between the strips.. The results are shown in Table 1. The constancy of the mobility with respect to direction and magnitude of the field, whichever electrode boundary condition is used, is good evidence that the particle motion is indeed due to classical electrophoresis between the strips. Kornbrekke et al.22 obtained a mobility of -0.04 (µm/s)/(V/ cm) with similar carbon black particles suspended in a lower concentration (1 wt%) of OLOA in dodecane. The negative mobilities mean the particles are negatively charged in these solutions. By considering the solutions to be ‘simple’ electrolytes, we can apply the classical theory for electrophoresis to relate the experimental mobility to the zeta potential of the particle. We assume the carbon black particles are essentially dense with negligible conduction or liquid flow in the interior. The mobility is a function of the zeta potential of the particle’s surface (ζ) and the product κa. Smoluchowski’s equation applies for κa f ∞:
µe,s )
ζ η
(8)
The viscosity (η) and permittivity () of dodecane at 25 °C are 1.37 × 10-3 kg/m‚s and 2.0 times the permittivity of vacuum. The mobility is smaller than the Smoluchowski prediction at finite κa and large magnitudes of ζ. To determine the zeta potential of the particles from the measured mobility, we need to determine κ for our solutions. Assuming the ions in the dodecane are carried by micelles of the PIBS and the valence of each charged micelle is (1, the Debye parameter is given by
κ)
(DK )
1/2
(9)
where K is the conductivity of the solution and D is the diffusion coefficient of the micelles. While the particles contribute about 10-20% to the conductivity to the solution, we assume all the conductivity is due to the PIBS micelles. We have determined the micelle diffusion coefficient to be 1.6 × 10-11 m2/s.18,19 The measured value of K is 27 nS/m; thus, κ ) 9.8 × 106 m-1, and κa ) 4.9. Numerical calculations of the electrophoretic mobility, obtained19 using the method of O’Brien and White,23 are plotted in Figure 6. We selected three values of κa; note that the effect of κa over this range is small. According to (22) Kornbrekke, R. E.; Morrison, I. D.; Oja, T. Electrophoretic Mobility Measurements in Low Conductivity Media. Langmuir 1992, 8, 1211-1217. (23) O’Brien, R. W.; White, L. W. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.
Figure 6. Theoretical relationship between electrophoretic mobility and particle zeta potential as calculated from the theory of O’Brien and White23 for a symmetrically charged electrolyte. Triangles, κa ) 3; circles, κa ) 4; inverted triangles, κa ) 5. The mean experimental mobilities, assuming either constant potential or constant current on the electrodes to calculate the field, are marked with broken lines.
the O’Brien and White theory, the magnitude of the mobility reaches a maximum value as ζ increases, and the mobility actually decreases as ζ increases in magnitude. The average mobility determined from the constantpotential solution to the field, -0.069 (µm/s)/(V/cm), is close to but just outside the maximum predicted from the theory (-0.065). The experimental mobility based on the electric field calculated using the constant-current assumption (-0.120) is significantly greater than the theoretical maximum. Note that Smoluchowski’s equation, though not expected to apply to the conditions of our experiments, predicts ζ ) - 93 mV from the experimental mobility. We expect our estimates of the field Ex made with the two assumptions of constant potential and constant current on the electrodes would bracket the real field, but we cannot exclude the fact that the inconsistency between experiment and theory might be related to errors in estimating the field. Another possible explanation is that the estimate of κa is too low. We assumed all the conductivity is due to the charge control agent (PIBS), and neglected contributions by the charged particles themselves, which would underestimate κ. Furthermore, we chose a mean particle size of 0.5 µm radius, whereas some of the particles might have been larger. If κa ) 11 or 15, instead of 5 as estimated from the conductivity, the mobility predicted from O’Brien and White’s theory would be -0.071 and -0.078, respectively. These values lie between the experimental mobilities determined from the constant-potential and constant-current fields. There are two other complications that cloud the comparison of our particle mobilities with theory. The first is a possible boundary effect on the mobility, arising because the particles had settled onto the glass surface between the strip electrodes. Boundary effects on electrophoresis are generally weak, especially at large κa, where κ is the Debye parameter and a is the particle’s radius (0.5 µm).24 The calculations of Keh and Chen25 for spheres with large κa show that particles moving parallel to a rigid planar surface experience at most a 10% decrease or increase when the particle is very close to the surface. The wall effect is stronger when κa is below 10,26 but it (24) Anderson, J. L. Ann. Rev. Fluid Mech. 1989, 21, 61. (25) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (26) Ennis, J., and Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497.
10946
Langmuir, Vol. 21, No. 24, 2005
Kim et al.
Figure 7. Comparison between an experimental trajectory (symbols) and the trajectory predicted from eq 10 (line).
is still much weaker than for movement of particles by other forces (such as diffusion or gravity). The second uncertainty is electro-osmosis of the liquid resulting from the charge on the glass. We have no independent measurement of electro-osmosis, so we cannot rule out its effect on particle motion. Anomalous Motion Near the Edges of the Strip Electrodes. As seen in Figure 4, the particles moved at higher speeds as they approached the higher-potential electrode. To see if this could be due to higher fields near the edge of the strip, we assume eq 7a holds for all x along the lower surface of the cell and integrate the following equation:
dx ) µeEres,x(x) dt
(10)
where µe is the constant mobility determined from the central region far from the strips where Eres,x is essentially constant. As seen for a typical particle in Figure 7, the higher velocity near the more positively charged strip cannot be accounted for by simple electrophoresis. Note that were we to use the constant-current calculation of Eres,x, this conclusion would not change. To consider options for explaining the higher particle velocities near the higher-potential electrode and the stalling of particle near the lower-potential electrode, we calculate the equivalent force that would be needed to account for the discrepancy between the observed particle trajectory and that given by eq 10. Since the time scale for reaching terminal velocity is of order microseconds for this system, a quasi-steady hydrodynamic analysis is valid. Let Ux(x) be the observed particle velocity in the x direction. The equivalent force needed to explain the discrepancy between Ux and the predicted electrophoretic velocity, µeEres,x, is given by
Feq,x ) 6πηa(Ux - µeEres,x)
(11)
Figure 8 shows the equivalent force on typical particles, using eq 7a for Eres,x. Note that using eq 7b for the field produces results comparable to those obtained using eq 7a. As a reference for the magnitude of Feq,x, the magnitude of the equivalent electrophoretic force, Fep,x ) 6πηaµeEres,x, varies from 0.15 pN near the higher-potential electrode edge to about 0.05 pN in the center between the electrodes. At the point where a particle stalls near the lower potential electrode, Feq,x ) -Fep,x. When the higher-potential electrode is at the center, the equivalent force is as much as seven times larger than the electrophoretic force even 10 µm from the electrode edge. When the outer strip has
Figure 8. Additional equivalent force on a typical particle. (a) The higher potential (V0 ) +5 V) is at the center strip. (b) The higher potential (0 V) is at the outer strips (V0 ) -5 V applied to the center strip).
the higher potential, the equivalent force is comparable to the electrophoretic force. The fact that the direction of the additional equivalent force does not point toward the lower potential surface across the entire range of x between the strips suggests that an electro-osmotic flow arising from the negatively charged double layer on the glass cannot account for the additional force accelerating particles toward the higher-potential electrode.27 The fact that the additional equivalent force always points toward electrode edges, where electric field gradients are higher, suggests it might arise from a dielectrophoretic force. This force is determined by the difference in the dielectric constant of the liquid and the particle, the volume of the particle, and gradient of square of electric fields:
p - 3 a ∇(E2) Fdi ) 2π p + 2
(12)
where p is the permittivity of the particle and E is the magnitude of the electric field. Approximating E by eq 7a, the maximum dielectrophoretic force near the edges of electrodes is of order 10-3 pN, which is 2 orders of magnitude smaller than the electrophoretic force and 3 orders of magnitude smaller than additional equivalent force acting on the particle. The result is basically the same when we use eq 7b for the field. Hence, dielectrophoresis is not a plausible explanation for the observed (27) The sign of the electrophoretic mobility requires that the carbon black particles have a negative zeta potential. Since the particles do not stick to the glass surface between the electrodes, the zeta potential of the glass in this system is probably negative. Electro-osmotic flow arising from this negative potential would oppose the electrophoretic motion of the particles, which are also negatively charged.
Movement of Colloidal Particles in Electric Fields
Langmuir, Vol. 21, No. 24, 2005 10947
Figure 9. Sketch of the electrohydrodynamic flow caused by the electric field (Ex) acting on the polarized charge over each strip for the case of constant current (Figure 5b) with a positive potential applied to the center strip. Only one-half of the cell is shown.
deviations from simple electrophoresis near the edges of the electrode strips. Electrohydrodynamic Flow and Its Effect on Particle Motion. If the constant-current boundary condition on the electrodes is valid, then there is a lateral electric field above each electrode, as shown in Figure 5b. The fluid velocity above each electrode is given by (dimensional variables are used now)
y f λ+: v f veh ∼
λσ E η x
electrophoretic velocity of the particle is given by eq 10; from Table 1, we have Ux ) -0.12Ex. Because the electric field is the same as for electrophoresis, the convective velocity of the particle at the edge of the center strip should be 1-2 orders of magnitude greater than the electrophoretic velocity if the constant-current assumption were valid on the electrode. Even though the determination of the electrophoretic mobility indicates the actual electrodes were probably not exactly at a constant-current density, any condition between constant potential and constant current would produce an electrohydrodynamic flow qualitatively consistent with that sketched in Figure 9. Note that a finite Ex is required on each electrode to produce the electrohydrodynamic flow described above. If we assume a constant-potential condition, then Ex is zero and there would be no fluid convection generated by this mechanism. Therefore, the details of the charge transfer at the surfaces of the electrodes determine the strength of the flow and must be known to predict the trajectory of particles with reasonable accuracy.
(13)
where λ is the thickness of the space-charge region, which should be comparable to the Debye screening length (κ-1) of the solution if the solution is not significantly depleted of charge carriers. σ is the polarized charge per area of electrode surface and is independent of x in the constantcurrent case, though it does increase in magnitude with time. Ex is a function of x on each strip (see Figure 5b). Equation 13 applies at the edge of the space-charge region (hence the superscript +) and provides a boundary condition that could be used to solve for the velocity field throughout the cell (with zero fluid velocity on the glass surfaces, neglecting any electroosmotic flow due to the charge on the glass). The fluid convection described above might explain the speeding up of the particles as the approached the more positive strip and the stalling at the more negative strip. Consider the case when the center strip is more positive (Figure 9). Ex is positive everywhere between the midline of the strips along y ≈ 0. Because σ is negative along the center strip, eq 13 indicates the flow is toward the center strip; this flow increases the particle’s speed as it approaches the strip. Conversely, σ is positive on the outer strip and hence ve is positive near the outer strip, thereby opposing the electrophoretic motion. Note also that this electrohydrodynamic velocity should decay with the length scale ∼w. An analysis28 of similar flows generated near a no-slip surface (such as the glass surface between the strips) shows the velocity field decays approximately as w-3. If this is true for the flow described here, then it would explain our observation that the departure from electrophoretic motion only occurs when the particles are within a distance of order w from the edge of the strips. While the flow sketched in Figure 9 explains the qualitative nature of the particle motion, the case is more compelling if the magnitude of the electrohydrodynamic velocity is greater than the electrophoretic velocity. The polarization charge is found by integrating the current; σ ≈ -10-3 µC/m2 on the center strip after 10 s when it is at 5 V (see Figure 3). Setting λ ) κ-1 ≈ 100 nm, eq 13 gives veh ≈ -7.3Ex µm/s where the field is in V/cm. The (28) Solomentsev, Y., Guelcher, S. A., Bevan, M., and Anderson, J. L. Langmuir 2000, 16, 9208.
Summary and Conclusions We have studied particle motion near a glass surface in a nonpolar liquid in two-dimensional electric fields. When the particles are in the central region between two strip electrodes and not too close to the edges, they move by electrophoresis. The electrophoretic mobility is determined from the particle trajectories and estimates of the electric field that are based on constant-potential or constant-current boundary conditions on the strip electrodes. The particle mobility is negative with a magnitude that is constant over the electric fields studied (V0 ) -5 f 5 V).; however, the magnitude of the mobility lies outside the range allowed by the classical theory of O’Brien and White.23 When the particles are close to the strips, the particles undergo anomalous motion. The particles move much faster than predicted from electrophoresis alone as they approach the higher-potential electrode strip, but they stall near the edge of the lower potential strip. Mechanisms such as dielectrophoresis and electro-osmotic flow driven by charge on the glass surfaces cannot account for this motion. Electrohydrodynamic flow created by interactions between the polarized layer of charge above each electrode and the spatially varying electric field near each electrode strip is a possible explanation of the anomalous motion. The direction of the flow would enhance the electrophoretic motion of the particles as they approach the more positive strip and oppose the particle motion near the more negative strip, consistent with the experimental observations. An estimate of the fluid velocity above center electrode, assuming the extreme case of constant-current density on the electrode, produces a value that is 1-2 orders of magnitude greater than the electrophoretic velocity. Acknowledgment. We thank Dr. Luis Rojas, Visiting Faculty at the Institute for Complex Engineered Systems at Carnegie Mellon University and Professor, Department of Energy Conversion, Universidad Simo´n Bolı´var, Caracas, Venezuela, for obtaining the numerical solutions to eq 1 and providing the results for the fields reported in Figure 5a and b. Financial support for this work was obtained from Carnegie Mellon University and the Philips Research Laboratories in Eindhoven, The Netherlands. LA051233C
