Langmuir 2003, 19, 2745-2751
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Calculation of Electrohydrodynamic Flow around a Single Particle on an Electrode Paul J. Sides* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received September 4, 2002. In Final Form: January 9, 2003 A numerical solution of the equations describing electrohydrodynamic flow around a single particle next to an electrode during passage of alternating current is presented. The Stokes equations, the diffusion equation, and Laplace’s equation are solved simultaneously. The results confirm earlier approximate calculations showing that electrohydrodynamic flow is a significant factor in aggregative and deaggregative behavior of colloidal particles near electrodes. Both electrode kinetics and the ratio of particle size to diffusion layer thickness affect not only the strength of the flow around the particle but also its direction. Sluggish electrode kinetics causes the lateral flow around a particle in KOH solution to move away from it at 100 Hz, but fast electrode kinetics causes the flow to reverse at the same frequency. Increasing the frequency can cause flow reversal, which might explain experimental observations of the existence of a critical frequency above which particles separate and below which they aggregate in alkaline solution. The critical frequency is inversely proportional to the square of the particle radius.
Introduction Particles can not only be deposited on electrodes by the action of fields applied normally to the electrode1 but also can be made to aggregate laterally and self-order in some cases.2-13 Particles have moved in a variety of circumstances including dc and ac polarization, frequency variation from zero to megahertz, particle size variation from nanometric to micrometric, and particle composition variation from dielectric to metallic. Theory and suggestions have appeared in the literature to explain this behavior. Anderson and co-workers9-12 generated a quantitative direct current model based on electrokinetic flow stimulated by the interaction of the applied electric field’s surface normal component with the charge in solution adjacent to colloidal particles. In an electrohydrodynamic flow model proposed by Trau et al.4,5 and Yeh et al.,7 concentration polarization adjacent to the electrode produces a finite free charge in the diffusion layer; the charge interacts with any lateral electric field arising from nonuniformities to produce a distributed body force on the liquid within the region of concentration gradients. * Phone: 412 268 3846. E-mail:
[email protected]. (1) Sarkar, P.; Nicholson, P. S. J. Am. Ceram. Soc. 1996, 79, 19872002. (2) Giersig, M.; Mulvaney, P. Langmuir 1993, 9, 3408-3413. (3) Giersig, M.; Mulvaney, P. J. Phys. Chem. 1993, 97, 6334-6336. (4) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706-708. (5) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 63756381. (6) Bo¨hmer, M. Langmuir 1996, 12, 5747-5750. (7) Yeh, S.-R.; Seul, M.; Shraiman, B. I. Nature 1997, 386, 57-59. (8) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Nature 2000, 404, 56-59. (9) Solomentsev, Y.; Bohmer, M.; Anderson, J. L. Langmuir 1997, 13, 6058-6068. (10) Guelcher, S. A. Investigating the Mechanism of Aggregation of Colloidal Particles during Electrophoretic Deposition. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1999. (11) Guelcher, S. A.; Solomentsev, Y. E.; Anderson, J. L. Powder Technol. 2000, 110, 90-97. (12) Solomentsev, Y. E.; Guelcher, S. A.; Bevan, M.; Anderson, J. L. Langmuir 2000, 16, 9208-9216. (13) Matsui, T.; Lee, S.; Nakayama, K.; Ozaki, M.; Yoshino, K. Mol. Cryst. Liq. Cryst. 2001, 367, 2825-2831.
The theory based on concentration polarization, however, had not been elaborated for the case of particles until Sides14 assembled diffusion, electrochemical phenomena, and hydrodynamics into a quantitative model of electrohydrodynamic flow in the vicinity of an individual particle. This work showed how the electrode kinetics and capacitance, the diffusion of electrolyte, and the frequency of excitation combine to exhibit phenomena observed experimentally. Sides14 identified three dimensionless groups, the ratio of the reaction resistance to the capacitive impedance, the ratio of the diffusion layer thickness to the particle diameter, and a difference of ion transference numbers, that govern the overall problem. In its most striking prediction, the model showed that the transference parameter could be used to predict the direction of flow toward or away from the particle. Experiments have confirmed this to be true for particles in alkaline solution, a frequency of 100 Hz, and particle radii of 5 µm.15 The model, however, was at best a first approximation because it compounded a one-dimensional solution of the diffusion equation with a simplistic treatment of the hydrodynamics. Nadal et al.16 recently provided new evidence of the behavior of particles both in particle pairs and en masse on electrodes in alkaline solution. They identified a critical frequency below which large groupings of particles aggregate and above which they “explode apart” from tightly organized formations. They explained the transition as a tradeoff of dominance of attractive hydrodynamic forces and repulsive dipole forces. At frequencies below the critical frequency, the Stokes forces dominate and aggregate the particles, while above the critical frequency the dipole forces dominate and the particles repel each other. They found that the critical frequency depended on the particle radius with an exponent of -2.3. The purpose of the present work was to revisit the electrohydrodynamic model of Sides14 with a simultaneous (14) Sides, P. Langmuir 2001, 17, 5791-5800. (15) Kim, J.; Anderson, J.; Garoff, S.; Sides, P. Langmuir 2002, 18, 5387-5391. (16) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B.; Ajdari, A. Phys. Rev. E 2002, 65, 061409-1-061409-8.
10.1021/la026509k CCC: $25.00 © 2003 American Chemical Society Published on Web 02/08/2003
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Sides
0 ) -∇p + µ∇2v - Fe∇φ
(1)
Equation 1 is the Stokes equation with a source term due to the electrohydrodynamic effect. v is vector velocity, p is pressure, µ is viscosity, Fe is the local unbalanced charge density, and φ is the electric potential. The companion to eq 1 is the equation of continuity of the fluid,
∇‚v ) 0 Figure 1. A sketch of the setting of the problem in a semiinfinite domain. A dielectric sphere resides near an electrode. A counter electrode (not shown) provides an oscillating field E∞. This electrochemical system comprises ohmic resistance in the electrolyte, with charge-transfer resistance and double layer capacitance at the electrode. The electrochemical reaction is the reduction of a neutral species N to anion A- and back.
treatment of the diffusion and electrohydrodynamic flow in the vicinity of a particle on an electrode. The results confirm the conclusions of the previous work and provide new insight, even suggesting an alternative explanation of the observations of Nadal et al.16 Theory The geometry of the problem appears in Figure 1. The axisymmetric domain is infinite in r and semi-infinite in z. An electrode bounds the domain at z ) 0. A dielectric sphere of radius a resides on the axis of symmetry, and a gap h separates the particle from the electrode. Several assumptions restrict the scope of the solution: (1) binary electrolyte; (2) small deviations from uniform concentration; (3) linear electrode kinetics; (4) negligible convection of mass and momentum; (5) frequencies low enough that the transient term of the motion equation is negligible, and quasi-steady flow in response to the excitation; (6) an impedance of the electrode reaction less than the double layer impedance; (7) sinusoidally varying voltage; (8) electroneutrality. The restriction to binary electrolyte allows solution of a single diffusion equation for the salt. The maximum concentration variation calculated for any reported result was less than 10% of the bulk concentration, so Laplace’s equation could be solved for the potential. The use of linear kinetics allows exploration of the effects of kinetics on the current distribution without the complexity of the full Butler-Volmer equation.17 Assumptions 4 and 5 are good for the small particles and low velocities and frequencies of the problem; the order of the convective terms was less than 1/1000 of that of the diffusive terms in all of the equations. Both the electrical and flow aspects of the problem responded sufficiently quickly that quasi-steady behavior was a good approximation. Assumption 6 restricts the problem to relatively low frequencies, less than about 100 Hz depending on the charge transfer resistance and the double layer capacitance. The significance of this assumption is that the current is purely faradaic and in phase with the voltage. In regard to assumption 8, Sides14 argued that there is no inconsistency between the calculation of unbalanced charge and the assumption of electroneutrality at frequencies below approximately a kilohertz. One must simultaneously solve fluid flow, mass diffusion, and electrical equations to obtain the electrohydrodynamic velocity. (17) Newman, J. S. Electrochemical Systems, 1st ed.; Prentice Hall: Englewood Cliffs, NJ, 1973.
(2)
The boundary conditions on v are no-slip on the particle and electrode, symmetry on the axis, and zero velocity at infinity. The pressure is specified at one point in the domain. The next equation is Laplace’s equation for the electric potential, φ. The potential oscillates in a quasi-steady fashion with an applied oscillating field at infinity in the z direction. All derivatives in r vanish at the axis and at infinity.
∇2φ ) 0
(3)
The boundary condition on the electrode describes the kinetic relationship between the potential and its gradient. The faradaic reaction is assumed to be linear with potential, following the equation17
(Rc + Ra)ioF ∂φ | ) (Vo(t) - φ|z)0) ∂z z)0 RT
i|z)0 ) -κ
(4)
Here κ is electrolyte conductivity; Ri are kinetic symmetry parameters; io is the exchange current density which governs the rate of reaction; F is Faraday’s constant; Vo(t) is the oscillating electric potential of the electrode; and R and T are the gas law constant and temperature, respectively. The third equation is the diffusion equation for the concentration of binary electrolyte, c, where D is the ambipolar diffusivity of the binary electrolyte.
∂c ) D∇2c ∂t
(5)
The boundary conditions on the diffusion equation are symmetry in the radial direction at the axis and at infinity. The salt concentration is constant at infinity in the axial direction, and the flux of electrolyte at the electrode is proportional to the faradaic current flowing into the electrode.17
t+ ∂c i| -D |z)0 ) ∂z z-ν-F z)0
(6)
Here t+ is the transference number of the cation, z- is the anion’s charge number, and ν- is the anion’s stoichiometric coefficient of dissolution. Solution of the diffusion equation allows calculation of the charge density appearing in eq 1. For a binary electrolyte, the unbalanced charge density in a region of a nonzero concentration gradient is given by17
Fe ) -
RTt( 2 i‚∇c ∇ ln c + 2 F z+ν+λc
(7)
is the electrolyte permittivity; i is vector current density; z+ and ν+ are the charge number and stoichiometric dissociation coefficient of the salt; λ is the equivalent conductance of the salt; and t( is a pure transport
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transfer resistance. The charge density was scaled by RTt(/4Fa2. The boundary conditions on the hydrodynamic equations were taken to be no-slip on the electrode and the particle and zero stress on all other boundaries. Laplace’s equation for the potential becomes
∇2φ* ) 0
Figure 2. A sketch of the domain for the numerical solution. The sphere was placed at 5% of its radius from the electrode in this axisymmetric geometry.
parameter defined as (t+/z+ + t-/z-) where t- is the transference number of the anion. Sides14 argued that two of the three terms resulting from the product of this equation and the electric field cancel out over one period in a linear sinusoidal model for the oscillating potential and concentration, so the only term contributing to a net force averaged over one period of oscillation is
Fe )
2
RTt( ∇ c F c
(8)
Equations 1-6 and 8 are the essential relationships that control the net electrohydrodynamic velocity in the vicinity of a particle. There is no analytical solution for this set of equations, but they can be solved numerically. The problem described above for an infinite domain is solved with the finite element code FEMLAB in the domain of Figure 2. The length scale of the domain is taken as 5 times the diameter of the particle, sufficiently large not to profoundly affect the motion in the vicinity of the particle. As a check on the suitability of the domain size and the mesh which concentrated points in the vicinity of the sphere, the mesh used in the calculations was employed to solve Laplace’s equation in the vicinity of a dielectric sphere on an electrode for which there are analytical solutions. The result of the finite element calculation followed the analytical result closely. Scaling the equations, one obtains their dimensionless form containing the essential parameters. Motion and continuity (cylindrical coordinates):
-
(
)
∂2u ∂2u u 1 ∂u ∂φ* ∂Π + )0 + - + - F/e ∂η ∂η ∂η2 ∂σ2 η2 η ∂η -
(
)
∂Π ∂2v ∂2v 1 ∂v ∂φ* - F/e + )0 + 2+ 2 ∂σ η ∂η ∂σ ∂η ∂σ
∂φ* | ) 2J(1 - φ*|σ)0) ∂σ σ)0
(12)
The parameter J, defined as17
J≡
(Rc + Ra)aioF κRT
(13)
expresses a ratio of relative resistance to flow of current; it is a ratio of the ohmic resistance along a path of length a to the charge-transfer resistance presented by the reaction. Low values of J characterize a sluggish reaction and therefore a tendency toward uniform current distribution. High values of J indicate that ohmic resistance in the electrolyte controls the current distribution. In the present case, low values of J mean that the current density, even under the sphere, is uniform along the electrode. High values of J mean that the current density is small under the sphere, goes through a maximum of about 3% about 2 radii from the axis, and then decreases toward its uniform value at infinity of radius. The diffusion equation in dimensionless form is
(
)
∂2θ 1 ∂θ ∂2θ ∂θ ) δ* 2 + + ∂τ η ∂η ∂σ2 ∂η
(14)
where θ is dimensionless concentration c/c∞ and δ* ≡ [D/(4a2ω)]1/2 is another important parameter of the system of equations;14 δ* governs the extent to which concentration gradients are located mainly below the plane defined by the equator of the particle or are distributed both above and below the equator. The boundary conditions on eq 14 are symmetry at the radial endpoints and a fixed concentration at the upper wall. The flux and the current are related by eq 6 which in dimensionless form is
∂φ* ∂θ | )E | h ∂σ σ)0 ∂σ σ)0
(15)
where
where u is dimensionless radial velocity and v is dimensionless axial velocity; ζ and η are axial distance and radial distance, respectively; Π is dimensionless pressure. All distances were scaled by the particle diameter, and velocities were scaled by
RTt( Vo Fµ (L + a/J)
The boundary conditions are symmetry in the radial direction at the axis and the wall, an imposed voltage Vo sin ωt at the wall opposite the electrode, and a dimensionless form of eq 4 at the electrode.
(9)
u ∂u ∂v + + )0 η ∂η ∂σ
vo ≡
(11)
(10)
The electric potential was scaled by Vo, the amplitude of the exciting voltage. The parameter J (defined below) is a dimensionless ratio of the ohmic resistance to the charge-
E h ≡
Vo κt(L 2z-ν-FDc∞ L + a/J
The equations described above govern the variation of potential, fluid flow, concentration, and pressure in the domain of Figure 2. Three important parameters characterize the problem: (1) δ* is the ratio of the diffusion penetration depth to the particle diameter. (2) The parameter J governs the current distribution around the particle, as described above. (3) The parameter E h is a dimensionless electric field that contains the transference parameter t( whose sign can vary depending on the relative mobility of the two ions.
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Figure 3. Results of the simultaneous simulation of current flow, diffusion, and electrohydrodynamics around a sphere on an electrode in the domain of Figure 2. J ) 0.1, δ* ) 0.2 in 0.1 mM KOH at 100 Hz. The graphs are a snapshot of the solution at π/2 in the cycle: (a) equipotential lines; (b) contours of concentration of electrolyte; (c) flow magnitude and direction. The flow is away from the particle in this case of slow kinetics and “high” frequency.
Figure 4. Results of the calculation under the same conditions of Figure 3 but at 3π/2 in the cycle: (a) concentration contours; (b) flow magnitude and direction. The concentration contours are still quite near the surface, but the gradient is reversed because hydroxyl is being removed from solution. The extra contour in (a) is due to the wave effect of a sinusoidally varying source at the lower edge of the domain. The flow is nevertheless in the same direction as in Figure 3c.
Results The calculations simulated conditions where Sides14 predicted that separation of particles was possible and Kim et al.15 confirmed it experimentally. The diameter of the particle was 10 µm, and it was positioned 0.5 µm above the electrode. The electrolyte was KOH at a concentration of 0.1 mM. Since the electrochemical reaction was suspected to involve oxygen, for which the reaction kinetics is slow, the J parameter was taken as 0.1 to reflect a more uniform current distribution. The frequency of oscillation was 100 Hz where experimental evidence exists.15 The potential distribution, concentration distribution, and flow
field at π/2 in the cycle appear in Figure 3a-c. The equipotential lines of Figure 3a show the expected bending around the sphere. The current, which flows perpendicularly to the lines of equal potential, flows back under the sphere, which reflects the nearly uniform current distribution. In the first quadrant of the cycle, the anion is injected into the electrolyte by the oxidation of the neutral species. The anion attracts the cation, and a diffusion boundary layer forms. Thus the concentration (Figure 3b) is locally increased near the surface as reflected in the gathering of the lines of equal concentration near the surface. Mass is diffusing out from under the sphere normal to the contours. The most interesting graph is Figure 3c, which shows that at this point in the cycle the lateral flow adjacent to the particle is positive, which means that two particles near each other would repel each other under these circumstances, as predicted by Sides.14 The same set of graphs at 3π/2 in the cycle appear in Figure 4a,b. Now the polarity of the electrode is reversed and the concentration at the electrode is a minimum. The extra contour in Figure 4b means that there are maxima and minima in the concentration profile as a function of distance normal to the electrode, as expected from a sinusoidally varying source at the lower edge of the domain. Figure 4b shows that the flow is nevertheless again positive, meaning that two adjacent particles repel each other during both halves of the cycle in these circumstances. Thus there is a net tendency for two particles to move away from each other during each cycle under these conditions. Figure 5a-c is the same set of graphs at π/2 where the only difference from Figure 3 is that the electrode kinetics is more facile, J ) 10. The consequence of this difference
Figure 5. Results of the simultaneous simulation of current flow, diffusion, and electrohydrodynamics around a sphere on an electrode in the domain of Figure 2. J ) 10, δ* ) 0.2 in 0.1 mM KOH at 100 Hz. The graphs are a snapshot of the solution at π/2 in the cycle: (a) equipotential lines; (b) contours of concentration of electrolyte; (c) flow magnitude and direction. This is a case of facile electrode kinetics. The flow is toward the particle in this case of fast kinetics and “high” frequency.
Electrohydrodynamic Flow around a Single Particle
Figure 6. Results of the simultaneous simulation of current flow, diffusion, and electrohydrodynamics around a sphere on an electrode in the domain of Figure 2. J ) 0.1, δ* ) 0.5 in 0.1 mM KOH at 100 Hz. The graphs are a snapshot of the solution at π/2 in the cycle: (a) contours of concentration of electrolyte; (b) flow magnitude and direction. This is a case of sluggish electrode kinetics as in Figure 3 but at a frequency low enough that the diffusion layer thickness equals the particle radius. The flow is toward the particle in this case of sluggish kinetics but “low” frequency.
is apparent in all three graphs of Figure 5. The equipotential lines show that much less current is flowing in under the particle, which substantially changes the concentration distribution near the particle. This effect is even more obvious in Figure 5b where the concentration contours bend away from the particle’s base, indicating that the ionic flux underneath the particle is small. The net effect on the flow is to cause the direction to be now negative instead of positive (Figure 5c). Thus fast electrode kinetics in this case causes two particles near each other to aggregate. Figure 6 is the same set of graphs at π/2 as shown in Figure 3 where the only difference is that the diffusion layer is now approximately 1 radius thick. The concentration contours extend farther into the domain. Even with the same slow kinetics of Figure 3 and relatively uniform current distribution, the lateral flow is toward the particle, not away from it. Figure 7 is a plot of the velocities at a particular point in the domain adjacent to the particle ((1.75, 0.55) in dimensionless coordinates) as a function of phase in the cycle for the circumstances of Figures 3, 5, and 6. The flow at the selected point in the domain is representative of the lateral flow in the vicinity of the particle and is consistent with the radial separation at which Kim et al.15 reported particle velocities. The bias in velocity either toward or away from the particle can be positive or negative depending on the value of the kinetic parameter J or δ*. According to Figure 7, the velocity for the case of a thin diffusion layer (δ* ) 0.2) and sluggish electrode kinetics (J ) 0.1) is almost always positive during the cycle. The velocity for the case of a thin diffusion layer (δ* ) 0.2) and facile kinetics (J ) 10) reverses itself during the cycle but is overall more negative than positive. The velocity for the case of sluggish kinetics (J ) 0.1) and a relatively thick boundary layer (δ* ) 0.5) is almost always negative. Figure 8 summarizes the effects of the three parameters of the problem on these velocities; it reports the average dimensionless velocity that a tracer particle would experience at the location used in Figure 7. The data points are calculated results, and the lines are linear fits to the calculated points. The velocity is positive for the case J ) 0.1 and δ* ) 0.2, and the velocity is negative for the other two cases as shown in Figure 8. The results in Figure 8 indicated that the velocity can be represented by a linear equation when E h is the
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Figure 7. A plot of the instantaneous velocities that prevail at a point in the domain near the particle (1.75, 0.55 in dimensionless coordinates scaled by particle diameter). The graphs reflect the indications in Figures 3, 5, and 6 that the overall flow is away from the particle for sluggish electrode kinetics and thin diffusion layers, but otherwise toward the particle.
Figure 8. Velocities at the position described in Figure 7 were averaged over one cycle, scaled by eq 10, and plotted against the dimensionless electric field defined in eq 15. The results plotted in this manner are the data points, and the lines are linear fits to the results, which demonstrate that the scaled velocity is proportional to the electric field. Since the reference velocity itself is proportional to the electric field, the dimensional velocity is proportional to the square of the electric field.
dependent variable and where the slope is a function of both J and δ*.
〈v〉 ) ψo(J,δ*)E h vo
(16)
Thus the trends for varying J and δ* can be summarized conveniently as plots of ψo as a function of these two parameters. For example, Figure 9 shows the variation of the coefficient ψo with the parameter J for a fixed δ*. As J increases, the velocity coefficient ψo becomes less negative and changes sign when J is approximately 3. This is the value of J where the velocity reverses. Likewise,
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Figure 9. The coefficient ψo defined in eq 16 is plotted against the kinetic parameter J at constant δ* ) 0.2. For small values of J, the flow coefficient is negative, which means a positive velocity. The important point is that the sign of the coefficient reverses at J ≈ 3.
Figure 10. The velocity coefficient ψo defined in eq 16 is plotted against the dimensionless boundary layer thickness δ* at constant J ) 0.1. For high frequencies where δ* is low, the flow coefficient is negative, which means that the particles separate; for low frequencies, the opposite occurs. The corresponding results from previous calculations based on an approximate model are included. Note that the velocity reverses direction in the new calculation.
one can plot the velocity coefficient ψo as a function of δ* at constant J, as in Figure 10. This plot also shows that the coefficient can be either positive or negative depending on the value of δ*. The flow reverses at a δ* value of approximately 0.3. Discussion Figures 3-6 demonstrate the physical changes occurring during one cycle of the voltage. The concentration of electrolyte near the electrode is enriched during the anodic part of the cycle and depleted during the cathodic part. Unbalanced charge appears and vanishes as the cycle proceeds. The charge and the lateral electric field around the particle interact and thereby engender electrohydrodynamic flow that causes adjacent particles to aggregate or repel each other. The original purpose of this contribution was to investigate in more detail the prediction that colloidal particles near an electrode should repel each other in alkaline solution when subjected to an alternating electric field.14 This prediction was based on the fact that the parameter t( is negative for KOH solution. The results of the present calculations confirmed that result, which was also demonstrated experimentally.15 The new insight is that a particular set of circumstances even in alkaline solution is required to make particles repel each other electrohydrodynamically. This is the case illustrated in
Sides
Figure 3. The current distribution on the electrode must be uniform enough that significant current flows under the particle. In addition, the frequency must be high enough (or the particle large enough, or the diffusivity small enough) that the concentration gradients are weak above the plane of the particle’s equator. The electric field associated with the current flowing along the electrode toward the particle interacts with the concentration gradients near the electrode to produce a flow outward away from the particle. Facile kinetics, which means ohmic considerations dominate the current distribution, can cause flow reversal as shown in Figure 5 even in alkaline solution. When the electrode kinetics is fast, the current density beneath the particle decreases severely because the particle shields the electrode. Thus current does not flow laterally along the electrode to enter the region beneath the particle. The gradients of concentration beneath the plane of the particle’s equator interact with weak lateral electric fields, as per Figure 5, and this interaction is not sufficient to drive a net lateral flow away from the particle. If the diffusion layer is thick enough so that significant concentration gradients exist above the plane of the particle’s equator, the flow can reverse in alkaline solution even though the current distribution is uniform, as per Figure 6c. The motion due to the outward flowing current above the plane of the equator dominates the flow in this case. The linearity of the calculated results in Figure 8 is entirely expected for this set of equations. The unbalanced charge density is proportional to the electric field through the passage of faradaic current through the electrode. This implicit proportionality and the explicit dependence of the driving force for flow on electric field mean that the velocity should be proportional to the square of the electric field strength. The absolute calculated velocities are scaled by the reference velocity of eq 10 which is itself proportional to the electric field. Thus the scaled velocities of Figure 8 are linear with the dimensionless electric field. This linearity of the velocities when plotted in the manner of Figure 8 was used to advantage in summarizing the dependence of the scaled velocities on the parameters J and δ* as in Figures 9 and 10. The flipping of the sign of the coefficient ψo with each of these parameters demonstrates the complexity of effects possible in this system. Figure 10 especially indicates possibilities for experimental verification. In a system where the diffusion coefficient is fixed, as for a particular electrolyte, one can change the value of the parameter δ* by changing either the particle size or the frequency. In fact, these results indicate that for KOH at 100 Hz, particles of 5 µm in radius should separate but particles whose radius is 1 µm should aggregate. Comparison with Prior Work. The results confirm essential findings of the approximate model of Sides,14 and they also exhibit new behavior. The calculations show that the direction of flow depends on properties of the electrolyte, frequency, and particle size. The new results are demonstrations that both the intrinsic electrode kinetics and the ratio of particle size to diffusion layer thickness affect not only the strength of the flow around the particle but also its direction. In particular, sluggish electrode kinetics leading to a more uniform current distribution causes the flow around a particle in KOH solution to move away from it for relatively high frequencies but fast electrode kinetics causes the flow to reverse at these same frequencies. Likewise, for constant electrode kinetics, an increase of frequency above a certain critical value can cause flow reversal. The relationship between
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and they separated when the frequency was switched to a value above that frequency. Figure 11 is a plot of data from Nadal et al.16 showing the dependence of critical frequency. They estimated that the slope of the data was -2.3. The transition from an aggregative mode to a separative mode where frequency is involved is a striking feature of Figure 10 of the present work. For J ) 0.1, the critical value of δc* at which the flow in the vicinity of a single particle changed from toward the particle to away from it was about 0.3. Further investigation revealed that the crossover value of δc* was insensitive to values of J below 0.5, that is, in cases of relatively slow kinetics and small particles such as for micron size particles on indiumtin oxide. For example, variation of J from 0.05 to 0.5 caused the crossover value δc* to change from 0.25 to 0.3. The critical frequency predicted by the theory and calculations of this work then becomes
Figure 11. Lines of critical frequency calculated from eq 17a,b appear on a plot of the data of Nadal et al. (ref 16). The experiments were conducted in alkaline solution. The approximate agreement of the slope and the values of the predicted critical frequency with the experimental results of Nadal et al. indicates that one can explain their observed reversal of behavior without invoking physical effects other than electrohydrodynamics.
the approximate calculations and the new results appears in Figure 10 where calculated values from the method of Sides14 are compared to results from the present work. The dependence of the velocity coefficient on the diffusion layer parameter is similar in both cases, but the new calculations reveal that a change of direction is possible at lower frequencies, below which the δ* parameter exceeds approximately 0.3. These theoretical results illuminate experiments recently reported in the literature. For example, the calculated maximum separation velocity according to this work was 56 nm/s for conditions comparable to experiments where Kim et al.15 found 150 ( 60 nm/s. Thus the model correctly predicts the direction of motion but the value is somewhat lower than experimentally measured. At the present time, the reason for this difference is not known, but the calculations involved only one particle and used linear kinetics, which would certainly affect the quantitative results. The quantitative agreement between experiment and calculated velocities requires more investigation. Nadal et al.16 recently reported that particles in alkaline solution behaved differently above and below a critical frequency. Below the critical frequency they aggregated,
D νc = 0.44 2 a
(17a)
D νc = 0.64 2 a
(17b)
depending on whether one uses the upper value of 0.3 or the lower value of 0.25. These two equations are plotted in Figure 11 along with the data of Nadal et al.16 The diffusivity was 2.13 × 10-9 m2/s for the sodium hydroxide electrolyte. First, without assuming any dipole interaction, the electrohydrodynamic model predicts a slope of -2 which agrees well with the slope of their results below 1 kHz. Second, the predicted value of the critical frequencies, obtained with no adjustable parameters, is not far from their experimentally determined result, which indicates that the electrohydrodynamic model can account for the observed results without invoking any other force. Conclusions A simultaneous solution of Laplace’s equation, the diffusion equation, and Stokes equations confirmed and extended the results of earlier work on electrohydrodynamic flow in the vicinity of a particle. The current work showed that facile electrode kinetics changed the current distribution around a particle sufficiently to reverse the flow direction. Also, the calculations showed that there is a critical frequency above which particles separated and below which they aggregated in alkaline solutions. The predictions of the model agree with recently reported observations of such a critical frequency. Acknowledgment. This work was supported by the National Science Foundation, Grant CTS-0089875. LA026509K
