Langmuir 2004, 20, 4823-4834
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Vertical Motion of a Charged Colloidal Particle near an AC Polarized Electrode with a Nonuniform Potential Distribution: Theory and Experimental Evidence Jeffrey A. Fagan,* Paul J. Sides, and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received October 28, 2003. In Final Form: March 15, 2004 Electroosmotic flow in the vicinity of a colloidal particle suspended over an electrode accounts for observed changes in the average height of the particle when the electrode passes alternating current at 100 Hz. The main findings are (1) electroosmotic flow provides sufficient force to move the particle and (2) a phase shift between the purely electrical force on the particle and the particle’s motion provides evidence of an E2 force acting on the particle. The electroosmotic force in this case arises from the boundary condition applied when faradaic reactions occur on the electrode. The presence of a potential-dependent electrode reaction moves the likely distribution of electrical current at the electrode surface toward uniform current density around the particle. In the presence of a particle the uniform current density is associated with a nonuniform potential; thus, the electric field around the particle has a nonzero radial component along the electrode surface, which interacts with unbalanced charge in the diffuse double layer on the electrode to create a flow pattern and impose an electroosmotic-flow-based force on the particle. Numerical solutions are presented for these additional height-dependent forces on the particle as a function of the current distribution on the electrode and for the time-dependent probability density of a charged colloidal particle near a planar electrode with a nonuniform electrical potential boundary condition. The electrical potential distribution on the electrode, combined with a phase difference between the electric field in solution and the electrode potential, can account for the experimentally observed motion of particles in ac electric fields in the frequency range from approximately 10 to 200 Hz.
Introduction The lateral motion of colloidal particles near an electrode under normally directed ac electric fields is currently being investigated for applications in mesoscale surface patterning, novel colloidal phase formation,1 optical devices,2 and cell manipulation.3,4 Most research has focused on the aggregation of doublets and multiparticle clusters to determine the driving mechanism for this motion.5-17Fagan et al.,18 however, investigated the response of the particle height to normally directed alternating electric fields; the (1) Gong, T.; Wu, D. T.; Marr, D. M. Langmuir 2002, 18, 1006410067. (2) Joanopoulos, J. D. Nature (London) 2001, 414, 257-258. (3) Gleason, N. J.; Nodes, C. J.; Higman, E. M.; Guckert, N.; Askay, I. A.; Schwarzbauer, J. E.; Carbeck, J. D. Langmuir 2003, 19, 513-518. (4) Brisson, V.; Tilton, R. D. Biotechnol. Bioeng. 2002, 77, 290-295. (5) Kim, J.; Garoff, S.; Anderson, J. A.; Sides, P. J. Langmuir 2002, 18, 5387-5391. (6) Trau, M.; Saville, D. A.; Askay, I. A. Langmuir 1997, 13, 63756381. (7) Yeh, S. R.; Seul, M.; Shraiman, B. I. Nature (London) 1997, 386, 57-59. (8) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408-3413. (9) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334-6336. (10) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706708. (11) Bo¨hmer, M. Langmuir 1996, 12, 5747-5750. (12) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Nature (London) 2000, 404, 56-59. (13) Solomentsev, Y.; Bohmer, M.; Anderson, J. L. Langmuir 1997, 13, 6058-6068. (14) Guelcher, S. A.; Solomentsev, Y. E.; Anderson, J. L. Powder Technol. 2000, 110, 90-97. (15) Solomentsev, Y. E.; Guelcher, S. A.; Bevan, M.; Anderson, J. L. Langmuir 2000, 16, 9208-9216. (16) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B. Phys. Rev. E 2002, 65, 061409. (17) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. Rev. E 2002, 69, 021405-021412. (18) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2002, 18, 78107820.
change in average particle height depended on the electrolyte18 and correlated with the direction of lateral motion observed for particle clusters at the same conditions.19 Neither an electrokinetic (EK) mechanism without Brownian effects18 nor the combination of an EK mechanism with Brownian motion19 explained the observed motion. Here, we explore two effects related to the case where the frequency of oscillation is low enough that a faradaic reaction occurs at the electrode. This constraint limits the practical frequency values from roughly 10 to a few hundred Hz. At lower frequencies than order 10 Hz, the nonlinear current/voltage dependence becomes important, while beyond a few hundred Hz most of the ac current is capacitive and bypasses the faradaic reaction. The two effects accompanying the inclusion of an electrode reaction, an electrohydrodynamic (EH) mechanism6,20,21 for particle motion and an electroosmotic (EO) mechanism, are evaluated. The EH mechanism for lateral particle motion, which Trau et al.6 proposed and Sides20,21 elaborated into a quantitative model, is qualitatively consistent with both lateral5 and vertical18 observations. Concentration polarization adjacent to the electrode produces free (unbalanced) charge in the diffusion layer; the charge interacts with lateral components of the electric field to produce a distributed body force on the liquid within the region of concentration gradients. This body force drives fluid flow that imparts force to the particle. Yeh et al.7 hypothesized that the lateral electric field component interacts with the free charge in the diffuse part of the electrode’s double layer to generate force on (19) Fagan, J. A.; Sides, P. J.; Prieve, D. C. Langmuir 2003, 19, 66276632. (20) Sides, P. J. Langmuir 2001, 17, 5791-5800. (21) Sides, P. J. Langmuir 2003, 19, 2745-2751.
10.1021/la036022r CCC: $27.50 © 2004 American Chemical Society Published on Web 05/06/2004
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the particle; we call this the EO mechanism. If the electrode reaction is reversible, the overpotential required to drive the reaction is small, and the potential of an electrode in the vicinity of a sphere is uniform. Thus, the lateral electric field within the electrode’s double layer is negligible, and insignificant EO force is generated. However, measurements18 indicated the presence of a potential-dependent electrode reaction. Reflecting these considerations, Sides20,21 implemented a linear relationship between the current and the electrode potential to account for the semiinsulating effect of an electrode reaction on the electrode surface. The boundary condition on the electrode potential also affects the vertical forces acting on single particles, increasing the EK force, and adding contributions from both the EO and EH mechanisms at frequencies where faradaic reactions occur. Kinetic limitations on the rate of an electrode reaction redistribute the electric field in the region beneath the particle, imposing a significant lateral electric field on the charge in the diffuse double layer of the electrode. This interaction drives electroosmotic flow along the electrode and thereby generates EO force on the particle. Finally, the new field distribution enhances the EK force because the field intensity is increased along the surface of the particle relative to the case where the electrode potential is uniform.22 The effects of changing the boundary condition from a uniform potential distribution to a nonuniform potential distribution on the electrode, and new data on the vertical motion of particles subjected to oscillating electric fields, are the focus of this contribution. We explore the behavior in the relatively low-frequency range where faradaic effects are present. The EH, EO, and enhanced EK forces on the particle are calculated numerically for a colloidal particle above an electrode, and the forces are inserted into a deterministic model for calculating colloidal particle heights in an alternating electric field. The theoretical results are related to new measurements of the behavior of particles near electrodes. The inclusion of electrode kinetics in the boundary condition, and a voltage-dependent zeta potential, can explain the experimentally measured behavior of a single colloidal particle in ac electric fields at frequencies below ∼0.2 kHz. Origin of the EK, EH, and EO Forces Application of an ac or dc electric field across an aqueous electrochemical cell containing charged colloidal particles suspended over one of the electrodes generates forces that supplement the forces acting on the particles under equilibrium conditions. The added forces reflect the Coulombic interaction of the electric field with either fixed charge on the particle or free charge in the electrolyte. The free charge in the geometry of a particle above a planar electrode resides in the double layers of both the charged particle and the electrode and in a region of free charge associated with a diffusion layer near the electrode.20,21 Both the interaction of the free charge with the electric field and the distribution of the electric field itself must be included to determine the total force on the particle. Electrode Boundary Conditions, Current Distribution, and the Electric Field. Laplace’s equation and appropriate boundary conditions govern the electric field distribution around the particle appearing in Figure 1. Faradaic reactions can occur when the frequency of the applied voltage is sufficiently small that not all of the current is capacitive, usually less than a few hundred Hz. (22) Keh, H. J.; Lien, L. C. J. Chin. Inst. Chem. Eng. 1989, 20, 283290.
Fagan et al.
Figure 1. Geometry of the axisymmetric system used for the calculation of the EO, enhanced EK, and EH forces to represent a spherical dielectric particle near an electrode. The two electrodes are located on the z ) 0 and z ) 50 µm planes. The axis of symmetry is at r ) 0. Inset: a sketch of the electric field lines when the current density is uniform. The large type numbers enumerate the boundaries referenced in Table 1.
When the electrode kinetics are fast and concentration gradients are small, the electrode surface has a uniform potential and the current distribution is nonuniform. The current flowing to the electrode in this case is zero at the axis of symmetry, increases to a maximum approximately two radii from the axis, and then decreases toward the value expected between two plane parallel electrodes.23 Additionally, because the electrode potential is uniform on the electrode, the lateral electric field is zero at the electrode surface. This is often termed the “primary current distribution”.24 When the electrode reaction rate depends on potential and concentration gradients are small, the electrode boundary condition becomes an equality between the current flowing to the electrode according to Ohm’s law from solution and the current calculated according to the Butler-Volmer current/voltage relation.24 The current distribution in this case, often termed a “secondary current distribution”,24 can vary from nonuniform to uniform depending on the kinetic parameters and the current level. A secondary distribution is more realistic, especially when the particles are small. In this case, the current is distributed more uniformly on the electrode near the particle, and the potential at the inner edge of the diffuse layer varies with radial position. The important consequence of a secondary current distribution is a nonzero lateral component of the electric field at the electrode. The local electric field lines around the particle for both the primary and secondary current distributions appear in Figure 2A,B. The electric field lines intersect the plane of the electrode at 90° for a uniform potential (case A) and at less than 90° for case B. Thus, a lateral electric field acts on the diffuse layer charge in case B. Relationship between Choice of Boundary Conditions and Forces. The interaction of the electric field with a thin layer of charge near a surface generates a slip velocity at the boundary; the velocity is proportional to both the electric field and the zeta potential of the surface. (23) Sides, P. J.; Tobias, C. W. J. Electrochem. Soc. 1980, 127, 288291. (24) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1991.
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The Smoluchowski equation governs the fluid velocity according to
∇2φ ) 0
B 0rζsurfaceE νfluid ) η
where φ is the electrical potential. The electric potential boundary conditions are a specified oscillating potential at the top electrode, φ ) V0 cos(ωt) (boundary 3), and no penetration of the electric field into the particle or the radial boundaries, En ) 0 (boundaries 2, 4, 5), where En is the component of the electric field normal to the particle or boundary surface. Additionally, the Butler-Volmer equation
(1)
where 0 is the permittivity of free space, r is the relative permittivity of the medium, ζsurface is the zeta potential of the surface, E B is the electric field, and η is the fluid viscosity. Equation 1 can be applied either to a particle surface, as in the case of electrophoresis, or to an electrode surface as in the case of electroosmosis, once the electric field is known. The combined specification of uniform potential on the electrode and a uniform electrolyte concentration produces only the EK force due to the interaction of the electric field with the double layer of the particle. No EO flow exists because the lateral component of the electric field is zero in the double layer. Likewise, if concentration gradients are neglected, no EH body force acts on the fluid. Relaxing the assumption of uniform concentration allows the appearance of a concentration gradient of electrolyte due to the introduction and removal of ions at the electrode surface. Since a small amount of unbalanced charge is associated with concentration gradients,24 this causes a layer of free charge to exist outside of both the particle’s and the electrode’s double layers. This diffusion layer is the region of free charge mentioned earlier; its penetration depth into the solution is directly related to the square root of the ambipolar diffusivity divided by the frequency of the electric field. The free charges present in this diffusion layer interact with the nonuniform electric field due to the presence of the dielectric particle to generate the EH body force in the fluid. A transport parameter tˆ20,21 determines the sign of the free charge. This parameter can change sign in different electrolytes, accounting for the reversal in particle motion observed in different electrolytes. The EH force on the particle is due to the integrated stress and pressure acting on the particle surface due to the body force in the fluid. Additionally, changing the uniform potential BC to a secondary current distribution due to a potential-dependent electrode reaction both produces a substantial EO force and enhances the EK force. A nearly uniform current distribution requires a nonuniform potential at the electrode underneath the particle; thus, a nonzero in-plane electric field component exists in that region and interacts with the ions of the electrode’s diffuse layer to generate EO flow. A nearly uniform current distribution also concentrates the electric field beneath the particle relative to the primary current distribution. This has the effect of intensifying the current density flowing along the particle’s surface, which substantially strengthens the EK force. Theory Calculation of the Forces. Two equations must be solved to determine the force on the particle due to the electric field. These are Stokes’s equation
0 ) -∇p + µ∇2ν - Fe∇φ
(2)
with continuity
∇‚ν ) 0
(3)
where p is pressure, µ is the viscosity, ν is the fluid velocity, Fe is the local unbalanced charge, and φ is the electric potential, and Laplace’s equation for the electrical potential
[ (
i ) i0 exp
) (
RaF (V (t) - φ|z)0) RT 0 -RcF (V0(t) - φ|z)0) exp RT
(4)
)]
(5)
was linearized24 and used to provide a relation for the electrode beneath the particle (boundary 1) that describes the kinetic relationship between the potential and its gradient given a linear faradaic reaction on the electrode
i
|
z)0
) -κcond
|
(Rc + Ra)i0F ∂φ (V0(t) - φ|z)0) ) ∂z z)0 RT
(6)
where κcond is the electrolyte conductivity at the initial electrolyte concentration, Ri are kinetic symmetry parameters, i0 is the exchange current density, and V0(t) is the oscillating electric potential of the electrode. The electrical potential of the electrode itself, φ|z)0, was fixed at 0 V. Scaling of eq 8 produces a parameter J24 defined as J ≡ [(Rc + Ra)ai0F]/RT, which was set to a value representative of a potassium hydroxide electrolyte and an ITO electrode, 0.1.21 Large values are characteristic of a uniform potential distribution and fast reaction kinetics, while low values of J mean a more uniform current distribution. Once φ(r,z) is known, the Stokes equation can be solved for the fluid velocity using the appropriate boundary conditions. For the EK and EO forces, the Fe∇φ term in the Stokes equation due to distributed unbalanced charge in the medium is identically zero; the velocity field propagates from the boundary condition of a slip velocity on the particle or electrode surface, respectively. The boundary conditions on the Stokes equation for the EO and EK forces are presented in Table 1. In all calculations the pressure was fixed to Preference ) 0 at one point in the domain. For the EH calculation, the Fe∇φ term in the Stokes equation is nonzero due to the electrolyte concentration gradient near the electrode. This would appear to require the simultaneous solution of both the Navier-Stokes equation and the concentration profile. However, if mass convection is neglected,21 the electrolyte concentration can be decoupled from the fluid flow equations and calculated independently. The concentration of a binary electrolyte when mass convection is negligible is described by the diffusion equation
∂celectrolyte ) Delectrolyte∇2celectrolyte ∂t
(7)
where celectrolyte is the concentration of the electrolyte and Delectrolyte is the ambipolar diffusivity of a symmetric electrolyte; Delectrolyte ) 2D-D+/(D- + D+), where D+,- are the individual salt diffusivities. The unbalanced charge that appears in Stokes’s equation is calculated from the concentration gradient via
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Table 1. Hydrodynamic Boundary Conditions for the EK, EO and EH Force Calculations boundarya
EK model
1
ν)0
2 3 4 5
∂ν/∂r ) 0 ν)0 n‚T ) 0 νfluid-ith direction ) (0rζsurfaceE B ith direction)/η
EO model
EH model
νfluid-rdirection ) (0rζsurfaceE B r/η) ∂ν/∂r ) 0 ν)0 n‚T ) 0 ν)0
ν)0 ∂ν/∂r ) 0 ν)0 n‚T ) 0 ν)0
a The listed boundary numbers are taken from Figure 1, the diagram of the calculation space.
2 RTtˆ ∇ celectrolyte Fe ) F celectrolyte
(8)
where F is Faraday’s constant, R is the gas constant, is the permittivity of the solution, and ˆt, defined as ˆt ) t+/z + + t-/z-, where t+ ≡ z+D+/(z+D+ - z-D-) for a symmetric electrolyte.21 The boundary conditions on the electrolyte concentration at the bottom electrode is
-Delectrolyte
|
∂celectrolyte ∂z
) z)0
|
t+ i z-υ-F
(9)
z)0
Numerical Solution
The flux of mass is zero into the remaining surfaces. Although the diffusion equation is time-dependent, the initial distribution of any given average concentration does not affect the final stationary state. The hydrodynamic boundary conditions for the EH calculation are presented in Table 1. As in the EO and EK calculations, the pressure was additionally specified at one point in the domain. Particle Motion in an Applied AC Field. For comparison to experimental results, the forces described in previous sections must be used to calculate particle trajectories or an average particle height. Given relations for the vertical forces on a particle due to the EK, EO, and EO mechanisms, numerical simulations of average particle height (such as described in ref 19) can be performed. In ref 19, Femlab was used to numerically solve the convective-diffusion equation for a statistical ensemble of noninteracting particles
∂c/∂t + ∇‚(-D(h)∇c + cU(h,t)) ) 0
(10)
where c(h,t) is the particle concentration, D(h) is the position-dependent diffusion coefficient for the particle, and U(h,t) is the migration velocity caused by the local force field. In refs 18 and 19 and in these calculations, the migration velocity is always assumed to be at its terminal value. The migration velocity can be modeled as the sum of the forces acting on the particle: colloidal, electrokinetic, electrohydrodynamic, and electroosmotic, multiplied by the position-dependent mobility, m(h).
U(h,t) ) m(h)
To calculate a result comparable to the experimental results in ref 18, the concentration distribution that results from the solution of eq 10 can be normalized by the total number of particles, ∫∞0 c(h) dh, to generate a probability density profile, p(h) ) c(h)/∫∞0 c(h) dh, where p(h) is the probability density of finding a particle at a particular height. The probability density profile can be used to easily calculate both the average and time-dependent behavior of a single particle. For a typical colloidal particle at equilibrium, the probability density resembles a skewed Gaussian with a single maximum, the height at which a particle is most likely to be found, hm, and an average height, 〈h〉, 10-20% larger than hm. When a time-dependent alternating electric field is present, the probability density converges to an oscillatory stationary state characterized by instantaneous most-likely and average heights that vary continuously with time. The instantaneous most likely height is denoted as hm(t), and the instantaneous average height is havg(t) ) ∫∞0 p(h,t)h dh. Averaging havg(t), acquired at discrete intervals ∆t apart, over one full cycle of the ac field yields the time-averaged height, 〈h〉 ) (1/N) N ∑i)1 havg(i∆t), in which N∆t is equal to the period of oscillation. This is the quantity to compare to the experimental results at 100 Hz.25
∑Fi ) m(h)[Fcolloidal(h) + FEK(h,t) +
FEH(h,t) + FEO(h,t)] (11)
where Fcolloidal, FEK, FEO, and FEH are the colloidal, electrokinetic, electroosmotic, and electrohydrodynamic forces, respectively. The equations for Fcolloidal and m(h) are identical to those in ref 18; the relations for FEK and FEO are presented later in this contribution. The boundary conditions on eq 10 are no flux at both the electrode surface and infinity (1 µm).
Calculation of the EH Force Function. To calculate the EH force on a single particle equivalent to those used in the experiments of ref 18, several modified versions of the model presented in ref 21 were created. In all cases, the particle was made smaller, to the size used in ref 18, the far lateral boundary was moved from 50 to 40 µm, and the numbers of both total and boundary elements were increased. These changes increased the overall accuracy of the calculation while decreasing the computational cost. The reduction in the width of the geometry is justified, as the ratio of the new width to the particle radius is greater in these calculations than in ref 21. The geometry used for the EH force calculations appears in Figure 1. This axisymmetric domain, with the particle of radius a located a distance h above the electrode, is expansive enough such that the lateral gradients at the boundaries far from the particle are negligible. Also, the number of boundary elements was systematically increased to ensure that the calculated force did not depend on either the number of boundary or total elements. The values for the parameters used in the calculation of the EH force are collected in Table 2. All calculations of the EH force were performed for parameters matching those of a potassium hydroxide electrolyte above an indiumtin oxide electrode. The EH body force induces fluid flow in the regions above and around the particle. Directly underneath the particle, however, the fluid is too constricted for flow to occur due to the EH mechanism, and instead a pressure gradient is generated. z-directed force on the particle comes from both the pressure gradient and the fluid flow generated by the EH body force in the electrolyte. The total z-directed force due to the EH effect can be calculated by numerical integration. The vertical force on the particle as a function of time was calculated via (25) The determination of 〈h〉 from eq 1, where the heights of an infinite number of noninteracting particles were predicted over one cycle, is equivalent to averaging the measured height of a single particle over a large number of cycles; the experiments typically gathered data over more than 10 000 cycles. This is the basis for comparing the numerical solution to the experimental results.
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Table 2. Numerical Values for Important Parameters Used in the EO Force Calculations parameter
symbol
particle radius electrolyte concentration viscosity fluid density J factor number of elements
3 × 10-6 m 0.15 mM
µ F J
ζparticle
0.001 kg/(m s) 1000 kg/m3 0.1 ∼6000 elements (EO and EK) ∼3000 elements (EH) ∼250 elements (EO and EK) ∼200 elements (EH) -60 mV
ζparticle
-10 mV
ζ0
-80 mV
ζ0
-10 mV
number of boundary elements particle zeta potential (KOH) particle zeta potential (NaHCO3) electrode zeta potential (KOH) electrode zeta potential (NaHCO3)
value
a celectrolyte
boundary integration within Femlab.26 The quantities integrated to calculate the force on the particle were the pressure and the z-directed components of the viscous shear stress. In equation form, the integral was
FEH z-direction ) IAn‚T‚ez da
(12)
where ez is unit vector in the z-direction, n is the normal vector to the sphere, and T is the total stress tensor, integrated over the surface of the sphere in the axisymmetric geometry. As a test of the boundary integration utility, Femlab was used to calculate the z-directed force on a sedimenting sphere enclosed in an infinitely long cylinder with a finite radius. The analytical solution to the low Reynolds number version of this problem is well-known:27
q)
6πaµV∞ Ffixed cylinder
) 1 - 2.104(a/rcylinder) + 2.087(a/rcylinder)3 (13)
where q is the ratio of the drag force on a sedimenting sphere in an infinite fluid, given a velocity V∞, divided by the force on a sphere sedimenting with the same velocity in a cylinder with a finite radius, rcylinder is the radius of the infinitely long cylinder, and Ffixed cylinder is the force on the sedimenting particle necessary to maintain the velocity, V∞, in the finite radius cylinder. The values of q calculated via Femlab and the analytical expression appear in Figure 3; the numerically calculated force on the particle agrees with the mathematical solution. For all simulations, four cycles of the electric field were sufficient for the time-dependent force on the particle to converge within 1% of the final value. The convergence of the calculated force is demonstrated in Figure 4. The decreasing amplitude of the percent change of the EH force from the same phase in the previous cycle over time demonstrates that the system was well converged when the result was accepted. Constraints, addressed in detail in refs 20 and 21, apply to the EH calculations in this paper. Those assumptions (26) Note: as of Femlab V2.3a there is an error in the axisymmetric boundary integration code; the correct function for integrating the force on the boundary is -2*pi*Ky_cns+2*pi*r*p*ny. (27) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics, 2nd ed.; Noordhoff International Publishing: Leyden, The Netherlands, 1973; p 317.
limit the range of possible calculations to e200 Hz and applied fields e4 kV/m in a 0.15 mM KOH solution due to the increasing importance of transient terms and larger deviations from uniform electrolyte concentration distributions. Values for the applied field were chosen to match those applied experimentally in ref 18. Thus, the peakto-peak voltages reported throughout this paper represent the equivalent voltage that would be applied to generate the appropriate electric field if the electrode spacing were 1.34 mm, as in ref 18, not the 50 µm spacing used in the EH calculation.28 Calculation of the EO and Additional EK Forces. The EO force and the additional EK force were calculated using the same geometry, shown in Figure 1, as for the EH force calculation. However, the calculation of the EO and EK forces were time-independent, so a greater number of nodes and elements were available within the limits of our computational resources. Typical values for the parameters used to calculate the EO and EK forces are reported in Table 2. To calculate the EO and EK forces, the zeta potentials of the two surfaces were alternately set to 0 or -100 mV to determine the individual magnitude of each force. For both mechanisms, the force on the particle was found to be linearly related to the zeta potential within the entire range of electric fields explored, (5000 V/m. The integration of the force on the particle was carried out as in the EH calculation. As a test of the accuracy of this method, the electrode’s boundary condition was reset to a uniform potential distribution, and the force on the particle was calculated for comparison with the analytical solution of Keh and Lien.22 The results of this comparison are shown in Figure 5; the clear match between the calculated force and the theory is further evidence of the accuracy of our methods for calculating force on the particle. Calculation of AC Field Particle Probability Density Curves and Average Heights. Two methods were used to calculate the average height of a particle when an ac electric field was applied. For calculations involving only the EH, colloidal, and EK forces, a modified version of the calculation performed in ref 19 was used to calculate the particle probability density curves and average particle heights presented in this paper. Specifically the modifications were the following: the upper boundary was moved inward from 1.5 to 1.0 µm, and the EH force was added to the sum of forces, eq 3. These modifications are justifiable; the particle concentration in the eliminated region was negligible at all times in the calculations of ref 19, and the addition of the solely downward directed EH force was expected to further reduce the particle concentration near the upper boundary. All parameters, except the inclusion of EH force terms, were identical to those used in ref 19. Convergence times to stationary state were equivalent to those in ref 19. For average particle height calculations that involved the EO and enhanced EK forces the method of ref 18 was used, which eliminated the Brownian motion term, D(h)∇c, from eq 10, and substituted a fixed compensation to the calculated average height to account for the difference between hm and 〈h〉. This method was chosen because the parameters used in the electroosmotic force calculation are only estimates, and the calculations of ref 19 showed that rigorously accounting for Brownian motion results in less than a 10% correction to the expected average height (28) The actual voltage entered into the simulation was therefore significantly smaller than the quoted 5 Vpp value; however, this additional complexity is justified by the ability to directly compare these results to the data of refs 18 and 19.
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Figure 2. The local electric field lines around a 6 µm diameter dielectric particle located 250 nm above the electrode surface. In (A), the boundary condition on the electrode is a uniform electrical potential with a fast reaction. In (B), the boundary condition reflects the consequences of a sluggish electrode reaction. Twenty streamlines are shown in each plot. The lateral component of the electric field is nonzero at the surface in the secondary current distribution. The J factor value used to calculate (B) was 0.1.
Figure 3. The theoretical ratio of force necessary to move a sedimenting sphere in an infinite medium at the same velocity as sphere sedimenting in an infinitely long cylinder of finite radius as a function of the ratio of the sphere radius to the cylinder radius. The numerical methods used to calculate the individual points shown in the figure accurately reproduce the analytical27 results. The cylinder used in the calculation typically contained ∼160 boundary elements and ∼2400 total elements.
vs the method used in ref 18. The deterministic nature of the average height calculations including the EO force does not qualitatively influence the results of these simulations. Experiments TIRM29 was the basis of the experiments. A totally internally reflected helium-neon laser generates an evanescent wave at the electrode surface; when a particle is close to the surface, the evanescent wave is scattered and the intensity of the scattered light is exponentially related to the height of the particle. A photomultiplier tube senses the amount of scattered light. A dovetail prism was optically connected to the bottom electrode of the fluid cell that consisted of two parallel, tin-doped indium oxide electrodes purchased from Bioptechs Inc. A (29) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93-125.
Figure 4. Percent change in the EH force on the particle from the same phase in the previous cycle for the last three cycles simulated. The four curves are for 1, 2, 4, and 10 Vpp with the particle-plate gap fixed at 200 nm. The percent change decreases with time for any given phase, and the majority of the values were changing less than 1% (horizontal dashed lines) by the end of the fourth total cycle. The main point is that 40 ms (four cycles) was enough time to establish a stationary state. nonconducting spacer separated the 10 cm2 electrodes by 1.34 mm. Fluid connections are made through the upper electrode so that fluid can be injected into the cell when necessary without opening the otherwise closed system. An Agilent 33120A arbitrary waveform generator generated the sinusoidal potential that was applied to the electrochemical cell by an EG&G PAR 273A potentiostat/galvanostat. An Agilent 54624A oscilloscope monitored the applied potential, the photocurrent from the photomultiplier tube, and the voltage drop across a 10.1 ohm resistor in series with the electrochemical cell. A typical value for the total measured resistance was ∼400 ohm for a KOH electrolyte. The solutions, 0.15 mM solutions of potassium hydroxide and sodium bicarbonate, were prepared with deionized water sparged with N2 for at least 1.5 h. All solutions were prepared, diluted, and injected into the cell under a nitrogen atmosphere. The electrodes that comprise the fluid flow cell were soaked for a minimum of 1 h in the working solution and were additionally flushed with working solution prior to particle injection. The particles were 6.2 µm nominal diameter, surfactant-free, poly-
Vertical Motion of a Charged Colloidal Particle
Figure 5. Comparison of Femlab simulated and theoretical solution for the electrokinetic force on a sphere near a planar electrode with the uniform potential boundary condition.22 The electric field was 1 V/m, the particle zeta potential was -100 mV, and the particle diameter was 6 µm. The numerical methods accurately capture the behavior of this known case.
Figure 6. The z-direction EH force per unit area, ∂Fz/∂area, along the surface of a 6 µm particle 400 nm above the electrode exposed to a 5 Vpp oscillating electric field. The force is shown at different phases of one cycle. The primary contribution to the force/area is derived from the pressure gradient beneath the particle, which accounts for the maximum magnitude of the force/area at the bottom pole of the particle. The midplane of the particle is located at approximately 4.7 × 10-6 m. Eight distinct phases are shown; however, the two phases when the electric field is zero are indistinguishable from the line of zero force density. The two maxima in the force/area magnitude are roughly coincident with the times of maximum electric field. (styrenesulfonate) particles from Interfacial Dynamics Corp. Additional details of the TIRM apparatus can be found in refs 18 and 29.
Results and Discussion The effect of using the potential-dependent electrode boundary condition is best addressed in two parts: first attending to the EH force and then concentrating on the EO and EK forces as a pair. The EH Calculation. The vertical EH force on the particle is not skin drag from the EH driven fluid convection but stems primarily from increased pressure beneath the particle. This EH pressure (Figure 6) is highest near the pole of the particle nearest the electrode, where almost no fluid convection occurs, and is near zero about the particle’s equator, which is at the same height as the
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Figure 7. The z-directed EH force on the particle as a function of time for a 100 Hz, 5 Vpp equivalent (18.7 V/cm) ac electric field at particle elevations of 200 and 400 nm. The points are calculated. At 200 nm the force on both the upper hemisphere and the lower hemisphere are shown, and the total force is shown for both the 200 and 400 nm cases. The fitting curve virtually overlaps the net EH force over the entire cycle. To distinguish the relative strength of the EH force, the data were normalized by the magnitude of the force due to gravity, G, on the particle, 6.1 × 10-14 N.
approximate center of the EH driven convection cell from the electrode. Integrating the calculated EH force per unit area (Figure 6) over the surface of the particle yields the net force on the particle due to the EH mechanism at any point in the cycle (Figure 7). The points shown are calculated forces on the upper and lower hemispheres at a particle/electrode gap of 200 nm, the sum (and hence net force) at 200 nm, and the net force for a gap of 400 nm. The E2 nature of the EH force is apparent; the primary EH force is always in the downward direction, despite the reversal in the sign of the electric field in half the cycle. The other indicator of the E2 nature of the force is the doubling of the frequency according to the trigonometric identity between the square of the cosine and the cosine of twice the angle. The EH force is positive briefly for KOH parameters; positive EH force is caused by the retardation of the diffusion layer as it progress outward from the electrode, relative to the forcing electric potential at the electrode surface. Although the previous figures demonstrated that the net EH force on the particle is directed downward for the hydroxide solution parameters, as is the 〈h〉 in the hydroxide solution experiments, the net force on the particle due to the EH mechanism is small compared to the other forces acting on the particle. This is shown in Figure 8, where the net force on the particle, rationalized by the force due to gravity, and averaged over one cycle is shown as a function of the applied field and particle height. The rationalized force is about 1/10 the gravity force throughout the range of likely particle heights at the applied fields utilized in ref 18. The average height of the particle, 〈h〉, including the EH force, was calculated in the manner of ref 19 to demonstrate the quantitative effects of the EH force. The results of the calculation, which included the EH force, Brownian motion, colloidal forces, and the constant potential boundary condition EK force, appear in Figure 9. Including the EH force along with the effects of diffusion and the EK excitation moves calculations in the right direction (toward the experimental data), but not by much. Although qualitatively consistent with experimental observations, the EH force due to an
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Figure 8. Average z-directed EH force on the particle scaled by the magnitude of the buoyancy force. The curves are from top to bottom, 400, 300, 200, and 100 nm particle heights; the curves are in identical order in the inset figure. The symbols are values calculated from the simulation, and the lines are power E2 splines. The term Vpp refers to volts peak-to-peak and is related to the electric field by the relation Vpp ) 2 × 1.34 mm ×E B ∞. The magnitude of the EH force is on the order of 1/10 of gravitation.
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functions of the particle height. The forces are particularly nonlinear when the particle height is small but approach their values for the uniform electrical potential boundary condition when the particle is high above the electrode. The crucial result shown in Figure 11 is that EO and EK forces are nearly equally important in determining the particle motion when the particle is near the surface. Both forces exhibit similar trends in magnitude with respect to the choice of the J parameter as shown in Figure 12. At large J, the EO force decreases to zero because the lateral electric field vanishes at J ) ∞, and the EK force goes to the Keh and Lien22 value. The circle points and triangular points of Figure 12 overlap at high J because the electrode reaction is not consuming much of the applied potential. At small J, both the EO and EK forces (open and filled triangles) vanish if the cell voltage is held constant because the applied potential is spent running the reaction, and thus a lower electric field is available in the solution. However, if the parameter held constant is the overall electric field in solution, both the EO and EK forces approach maximum values as J decreases, and the current distribution becomes uniform (J ) 0). For a 6 µm diameter dielectric particle, the EO force as a function of particle height for a J factor of 0.1 (KOH value9) is approximated by eq 14.
FEO(h,E B ∞) )
B∞ ζplateE [5.197 + 128.3e-h/47.15nm + η × 108 64.17e-h/182.6nm] (14)
The additional EK force is well modeled by a correction term to the constant potential boundary condition solution of Keh and Lien.22
[
B ∞,J) ) FEK-constpot(h,E B ∞) 1 + FEK-j(h,E -τ1h
ψ1e
+ ψ2e-τ2h
]
1 + ψ3(J - 0.1) + ψ4(J - 0.1)2
Figure 9. Simulated average height curves and experimental data comparison. The four cases shown are for EK force with a particle in Brownian motion,19 EK force alone,18 EH added to EK, and all three effects included. The calculated average height curve 〈hEH(Vpp)〉 including the EH force is little different from the average height curve, 〈hEK(Vpp)〉, presented in ref 18 and improved upon in ref 19. Neither curve resembles the experimental average height measurements in either potassium hydroxide or sodium bicarbonate (inset: filled circles and open triangles, respectively).
electrode reaction is insufficient to account for the observed changes of average height of the particle. The EK/EO Calculation. The streamlines for the fluid flows induced by both the EK and EO forces appear in Figure 10A,B for a J value of 0.1. The streamlines emphasize the flow near the equator for the EK effect and near the pole adjacent to the electrode for the EO effect. The calculated values for the EK and EO forces as a function of height, relative to the buoyancy force, are shown in Figure 11. At a J value of 0.1, both the EO and the EK forces are of the same order and are substantial in magnitude. In contrast to the EH force, which is an E2 effect, both the EK and EO forces are proportional to the applied electric field through eq 1. However, because of the geometry of the system, both forces are nonlinear
(15)
where τi and Ψi are fitting parameters whose values are given in Table 3. The EK force produced by a specified uniform electrode potential was discounted as an explanation for the experimentally observed behavior of ref 18 in a prior publication;19 the question remains whether the enhanced magnitude of the EK force causes any qualitative changes in the expected average height of the particle, 〈h〉, in the absence of an additional EO force. The answer to this question is no. The enhanced magnitude of the EK force in the absence of the EO force only leads to a faster increase in 〈h〉 with voltage due to the increased amplitude of oscillation, but neither causes any qualitative change in the functionality of 〈h〉 nor brings about quantitative agreement with the experimental observations of ref 18 in sodium bicarbonate solutions. This leaves the EO force as the root cause for the observed particle behavior. An EO force alone, however, is insufficient to account for the experimentally observed changes of average height of the particle even though the magnitude of the force is correct. If the EO force is symmetric and in phase with the electric field, the integrated EO force over one cycle is quite small even though the amplitude of particle oscillation increases. An EO force that is out of phase with the electric field or that displays hysteresis, however, would not integrate to zero over a cycle of the electric field. One indication of a phase angle or hysteresis in the
Vertical Motion of a Charged Colloidal Particle
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Figure 10. Streamlines induced by a 1000 V/m electric field for the EK model boundary conditions (A) and the EO model boundary conditions (B). In the EK model, the fluid flow is due to the interaction of the electric field with the particle’s double layer, while in the EO model (B) the interaction is with the electrode’s double layer. It is the location of the interaction that leads to the difference in the structure of the circulation patterns. The fluid velocity is ∼30% larger in the EK model. In both calculations the 6 µm diameter particle was 250 nm above the electrode, and the value of the J was 0.1. Note that the streamlines do not correspond to equally spaced values of the stream function.
Figure 11. EO and EK force for an 18.7 V/m electric field as a function of height with a J factor of 0.1 for a 6 µm diameter particle. The EO force on the particle is of the same order as the EK force and is substantially larger than the force due to gravity. A uniform potential boundary condition at the electrode substantially reduces the EK force, and the EO force vanishes as expected.
EO force would be a phase angle different from π/2 in the motion of the particle relative to the driving electric field. Our prior work18 in fact indicated that the phase angle between the electric field and the particle motion in KOH is less than the π/2 expected from a simple relationship between the force on the particle and the electric field. We conducted additional experiments to reconfirm the phase difference between the electric field and motion of the particle and to test whether the phase angle depended on the electrolyte under otherwise equivalent operating conditions. The results are presented in Figures 13 and 14. Figure 13 shows the experimentally measured evolution of the particle position from equilibrium, when power is applied to the cell at t ) 0, until the time when the particle motion is in the stationary state. The average height of the particles in KOH decreases while the average height in bicarbonate increases. In Figure 14, the mea-
Figure 12. The dependence of the EK (filled symbols) and EO (open symbols) forces as a function of the J factor depending on the electrical quantity that is held constant with the change in J, the applied potential (triangles), or the electric field strength (circles). The calculations were done for a 6 µm diameter particle 250 nm above an electrode, assuming a zeta potential of -100 mV on each surface. If the electric field is held constant, the two forces are maximum at J ) 0 where the current distribution below the particle is uniform. Table 3. Numerical Values for the EK Force Fitting Parameters of Eq 18 ψ1
ψ2
ψ3
ψ4
τ1
τ1
1.576
3.651
0.0189
1.0022
2.5407 × 10-3
1.384 × 10-2
sured electrical current and TIRM photocurrent are recorded for identical applied potentials as a function of time. The output in these experiments, photocurrent, is directly proportional to the intensity of the scattered light. Thus, the minimum height of the particle occurs at the time of maximum photocurrent. In Figure 14, if the particle experienced only electrophoretic and colloidal forces, the maximum in photocurrent for both electrolytes would be almost exactly 90° out of phase from the maximum electrical current. The results show that for KOH, as in
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Figure 13. Typical experimental particle 62.5 Hz trajectories in KOH (triangles) and in NaHCO3 (circles) recorded when the electric field was applied at time ) 0. The initial direction of the electric field was reversed in the two electrolytes for clarity. The thick lines are the moving average of the point data over the next cycle, and the dashed lines are the respective long time average heights. In KOH, the average height decreases, and in NaHCO3, the average height increases.
Figure 14. Analog applied potential, current, and TIRM photocurrent waveforms. In TIRM, the photocurrent is inversely related to the height of the particle above the electrode; e.g., at the point of maximum photocurrent, the particle is actually at its minimum elevation. Black lines indicate KOH and the gray lines NaHCO3. The vertical dotted line indicates a point in the cycle that is 90° from the maximum of the electric current. If the forces on the particle were in phase with the electric field in solution, the maxima for both solutions would occur at the time mark of the vertical dotted line. In KOH, the motion of the particle is accelerated relative to the expected 90° phase angle, and in NaHCO3, the motion of the particle is retarded relative to the 90° phase angle.
our prior work, the phase angle is less than π/2. The interesting new result is that the phase angle in bicarbonate exceeds π/2. How could a phase angle between the EO force and the electric field occur? Consider the equivalent circuit of Figure 15. The series resistor Rohm represents the electrolyte and the parallel resistor Rf and capacitor Cdl represent the faradaic resistance and the capacity of the double layer, respectively. If an alternating current is applied to this circuit, the electric field in the resistor is in phase with the current. The electrode potential, however, defined as the potential drop between the electrode and the solution located just outside the double layer, is out of phase with the current by the phase angle Θ ) tan-1(ωCdlRf) for linear components. Several groups30-32 (30) Hu, K.; Fan, F.; Bard. A. J.; Hillier, A. J. Phys. Chem. B 1997, 101, 8298-8303. (31) Barten, D.; Kleijn, J. M.; Duval, J.; v. Leeuwen, H. P.; Lyklema, J.; Stuart, M. A. Langmuir 2003, 19, 1133-1139.
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Figure 15. Equivalent circuit diagram for one ITO electrode and the electrolyte solution. Cdl is the double-layer capacitance, Rf is the faradaic resistance, and Rohm represents the electrolyte. If an alternating current is applied to this circuit, the electrode potential, i.e., the potential drop across the parallel components, will be out of phase with the current and hence the electric field in the series resistor Rohm.
have observed or calculated changes to the apparent zeta potential of an electrode as a function of electrode potential, which would consequently introduce a phase angle in the EO force relative to the electric field in solution. We conclude that ac polarization of an electrode can cause the zeta potential of the electrode to oscillate at the frequency of the ac polarization and that the phase difference between the zeta potential and the current can be nonzero. The formula for Θ also predicts that the phase angle should rise toward π/2 as frequency increases, as long as the circuit elements behave linearly, which was also observed.18 The experimental evidence is also consistent with the hypothesis that differences in the charging of the electrode’s double layer are responsible for the divergent directions of particle motion observed in KOH and NaHCO3. On the basis of these experimental results and the conclusion that only the EO model supplies the magnitude of force required to explain our observations, we suggest that the zeta potential of the electrode oscillates at the frequency of the applied potential and causes the integrated EO force to be nonzero. As a first approximation, we suggest that ζ oscillates sinusoidally about a base value ζ0, with a phase angle δ between the zeta potential and the applied electric field, with a magnitude of oscillation ζ1.
ζelectode ) ζ0 + ζ1 cos(ωt + δ)
(16)
The term ζ1 was chosen to be
ζ1 ) [ζ0/ζslope]Vpp
(17)
where Vpp is the applied voltage and ζslope is a fitting parameter. ζ0 was taken as the equilibrium zeta potential of ITO in the experiment. The chosen fitting parameter was 2Vpp, as this value yields good agreement with experimental results, while also limiting the change in zeta potential to within a reasonable range.18,19 The average heights that result from choosing this formulation for the change in the zeta potential are shown in Figure 16. When the phase angle is less than 90°, the particle is depressed, on average, toward to the electrode. When the phase angle exceeds 90°, the particle is pushed away from the electrode. Thus, depending on the phase (32) Bonnefont, A.; Argoul, F.; Bazant, M. Z. J. Electroanal. Chem. 2001, 500, 52-61.
Vertical Motion of a Charged Colloidal Particle
Figure 16. Average height of the particle as a function of δ, the phase angle of the electrode zeta potential change with respect to the electric field, at four voltages and 100 Hz. All of the curves show a change from depressed 〈h〉 to increased 〈h〉 near the 90° phase angle. This qualitatively agrees with the observed trends in potassium hydroxide (phase angle 90°, increased 〈h〉). These lines were calculated via the deterministic model of ref 18 but with corrections made for the EO and EK forces. The values used in eq 19 for these calculations appear in Table 2.
angle, the particle can either be depressed or substantially elevated. The measured phase angle between the electric field and the force on the particle was neither 90° nor the same in the two electrolytes, which implies that different average heights 〈h〉 should be observed, as demonstrated in calculations also displayed in Figure 16. However, to generate different behavior in 〈h〉 for different electrolytes, the charging of the zeta potential must be electrolyte dependent. This is likely to be the case because different ionic species clearly interact with the electrode differently; refs 32 and 33 by Bazant et al. and Prieve demonstrate different curves for the effective zeta potential as a function of the current density, which would depend on the electrolyte. To demonstrate that the phase angles in the particle motion shown in Figure 14 could be the result of a phase angle in a dynamic zeta potential, a plot of the observed phase angle in the particle motion as a function of the phase angle in the dynamic zeta potential is shown in Figure 17. The observed phase angle in particle motion is predicted by the deterministic simulation to change clearly with respect to the chosen phase angle in the zeta potential. Additionally, the response in the particle motion phase angle is similar to the phase angles previously observed in Figure 14 and ref 18. Calculations of the average height curves in bicarbonate and hydroxide solutions, using the experimentally observed phase angles between the particle motion and the electric field in the solution from Figure 14 as δ and assuming that eq 17 holds, yield relations for 〈h〉 that appear in Figure 18. The calculated curves are qualitatively consistent with the experimental data of ref 18. These are the first calculations to demonstrate significant depression of the average height in potassium hydroxide. The opposite direction and quantitative correlation between theory and experiment in the bicarbonate solution is also novel. These results are evidence that the induced EO flow due to an electrode reaction is the key mechanism controlling net single particle motion. (33) Prieve, D. C. Colloids Surf.: A Phys. Chem. Eng. Asp., submitted for publication.
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Figure 17. The observed phase angle in the particle motion relative to the electric field in the solution as a function of the phase angle in the dynamic zeta potential, δ. The phase angle in the particle motion clearly changes when δ is not equal to 90°. This line was calculated via the deterministic model of ref 18 but with corrections made for the EO and EK forces. The values used in eq 19 for this calculation appear in Table 2.
Figure 18. Comparison of experimental relative average height data (KOH f circles, NaHCO3 f triangles) with the predictions of the EO inclusive model for KOH (dot-dash line) and NaHCO3 (dashed line). The phase angles used in the calculation were taken from the experimental results in Figure 12 and were 77° and 98.1°, respectively. The solid black line is the calculated 〈h〉/〈h0〉 for a 90° phase angle. These lines were calculated via the deterministic model of ref 18. The values used in eq 16 for these calculations appear in Table 2.
This conclusion has important implications for multiparticle lateral motion; phase angles between the particle motion and the electric field other than 90° lead to a net lateral flow over each cycle of the electric field. The reason for this is that the product of the lateral particle mobility and the lateral force on the particle due to the radially directed fluid flow from the electrophoretic motion of the particle does not integrate to zero over a cycle if the phase angle between the particle motion and the electric field is other than 90°. Phase angles 90° lead to net aggregation, although the magnitude of the lateral force is yet to be calculated. This observation is qualitatively consistent with the behavior observed for particle doublets by Kim et al.5 Conclusions The change in average height of a colloidal particle in an ac field was shown to be qualitatively and quantitatively
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consistent with a theoretical model that includes a nonuniform electrical potential boundary condition. Specification of an electrode reaction with finite rate kinetics necessitates the choice of a boundary condition that is intermediate to both the uniform electrical potential and uniform current distribution boundary conditions. The choice of this boundary condition engenders additional forces on a particle near the electrode. The key conclusion is that the electrolyte-dependent interaction of the electric field with the free charge in the double layer of the electrode is the root cause of the experimental results presented in ref 18. Additionally, this result yields a key insight into the lateral motion of particle pairs or multiples. The inclusion of the EO force leads to an asymmetry in the particle’s vertical motion, which causes the lateral motion
Fagan et al.
of the particle, coupled through the lateral mobility to have a net component. The direction of net motion is determined by the phase angle in the EO mechanism. The evidence presented herein suggests that the cause of multiparticle aggregation/dispersion at low frequencies on uniform electrodes is due to a net EK driven flow. Work is ongoing to directly measure a material’s zeta potential under the application of current and on calculating the projected lateral particle motion due to the phase angle in particle motion. Acknowledgment. This work was supported by the National Science Foundation Grant CTS-00089875. LA036022R
