Electrohydrodynamic Particle Aggregation on an ... - ACS Publications

Langmuir 2001, 17, 5791-5800

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Electrohydrodynamic Particle Aggregation on an Electrode Driven by an Alternating Electric Field Normal to It Paul J. Sides* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

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Received April 10, 2001. In Final Form: July 12, 2001 An approximate model for the velocity due to electrohydrodynamic flow of electrolyte in the vicinity of a dielectric sphere near an electrode is described. The model considers the interaction of a lateral electric field, caused by the presence of the sphere, with free charge produced in the diffusion layer near an electrode when alternating current is passed through it. An equation based on the model predicts that adjacent dielectric particles aggregate or separate depending on the relative magnitude of the individual ionic conductivities and the frequency of oscillation in the case of a binary electrolyte. If the anion reacts and its conductance exceeds the cation conductance, the particles repel each other; the particles aggregate if the reverse is true. The equation also predicts that separation or aggregation depends on frequency of oscillation of the current. The model accounts for observed effects of frequency and particle size found in the literature. Finally, a dimensionless group that places a constraint on the frequency at which electroneutrality remains a good assumption for calculating electrohydrodynamic flow in oscillating systems is derived.

Introduction Electrophoretic deposition of particles on surfaces is an important technology.1 In the previous decade, investigators of electrophoretic deposition have noticed that particles not only are deposited but also aggregate laterally and self-order in some cases.2-12 Particles have moved in a variety of circumstances including direct current (dc) and alternating current (ac) polarization, frequency variation from zero to megahertz, particle size variation from nanometric to micrometric, and particle composition variation from dielectric to metallic. Experimental parameters associated with reported phenomena are collected in Table 1 (“entries” refers to column 1 of Table 1); some features are summarized as follows. (1) Micron-size particles move in fields of order 0.1 kV/m under dc polarization (entries 1 and 11) but aggregate at fields >1 kV/m in ac polarization (entries 3, 8, and 12). (2) Aggregation in ac fields occurred at higher frequencies for smaller particles (compare entry 4 to entries 6-8). (3) Particles stopped or became repulsive as the frequency was increased (entries 4, 9, and 13). (4) Particles moved * Phone: 412-268-3846. Fax: 412-268-2183. E-mail: ps7r@ andrew.cmu.edu. (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.

Table 1. Phenomena All Observed on Polystyrene Particles in Separate Experiments in Three Laboratoriesa a E entry ref (µm) (kV/m) 1 2 3 4 5 6 7 8 9 10 11 12 13

6, 9 6, 9 10 10 7 7 7 7 7 5 5 5 5

5 5 5 5 1 1 1 1 1 1 1 1 1

0.25 -0.25 2.5 2.8 0 16 22 25 >40 0.25 >0.25 10 5-10

ω (kHz)

interaction

0 0 0.03-0.5 1 1 1 1 1 1-103 0 0 106

aggregation, ordering repulsion aggregation repulsion no motion aggregation, loose order aggregation, medium order aggregation, crystalline repelling clusters no aggregation aggregation aggregation no motion

a The experiments in ac polarization used much higher fields than the dc experiments. a ) particle radius; E ) electric field perpendicular to the electrode in the absence of the particles; ω ) frequency of the electric field.

more quickly as they approached each other in dc polarizationbut moved more slowly as they approached in ac polarization (ref 10). Theory and suggestions have appeared in the literature to explain this behavior. Anderson and co-workers9-12 generated a quantitative dc model based on electrokinetic (EK) flow stimulated by the interaction of the applied electric field’s surface normal component with the charge in solution adjacent to colloidal particles. A charged particle near an electrode causes liquid in its vicinity to move when a dc field is applied, creating a circulating axisymmetric flow around the particle. Liquid adjacent to the particle flows along the particle surface away from the electrode for appropriate electrode polarization and particle charge, and it flows in along the electrode toward the particle to maintain mass continuity; the net effect is a sweeping flow along the electrode. A tracer particle in the vicinity of the original particle is entrained in the flow and moves toward the latter. Two particles of similar size

10.1021/la0105376 CCC: $20.00 © 2001 American Chemical Society Published on Web 08/15/2001

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Langmuir, Vol. 17, No. 19, 2001

Sides

are mutually entrained. An equation allowing estimation of the lateral fluid velocity, and hence the maximum particle velocity, is

ζpot UEK ) 0.1 E η 0

(1)

which is the electrophoretic velocity the particle would adopt in the bulk solution multiplied by 0.1. UEK is both a characteristic fluid velocity and the maximum possible velocity of an adjacent tracer particle entrained in the flow. is the permittivity of the electrolyte; ζpot is the zeta potential of the particle; η is the liquid viscosity; E0 is a constant overall electric field applied normal to the electrode. The coefficient 0.1 corrects for the secondary nature of the flow, about 10% of the electrophoretic velocity. (The primary flow of electrolyte is in the normal direction away from the electrode; the flow along the electrode, providing continuity of mass, is approximately an order of magnitude weaker, hence the factor of 0.1 multiplying the electrophoretic velocity.) This equation gives an estimate of achievable particle velocities: the velocity is positive, and hence two adjacent particles separate, when the signs of zeta potential and field are the same; when the signs are different, the velocity is negative and they aggregate. The EK model qualitatively and quantitatively agrees with experimental data on aggregation of particle pairs during dc polarization. Linearity of the effect with applied field seems to rule out a net contribution of the EK effect to lateral motion during alternating polarization of the electrode, but there might be consequences of the breaking of hydrodynamic symmetry by the electrode in the vicinity of the particles. A model that explores these consequences does not yet exist. Dielectrophoresis might account for other observations such as a change of behavior with frequency. Yeh et al.7 mentioned dielectrophoresis for its potential as a source of normally directed force on the particles but did not examine it as a potential cause of aggregation. The dielectrophoretic (DE) force arises from the interaction of a nonuniform electric field and the dipole moment induced in the particle by a field. The particle velocity UDE according to this mechanism can be approximated by15,16

UDE )

aR/p 2 a2R/p ∇|E ˜ |2 ≈ 10-2 E 6η 6η 0

(2)

where E is the local electric field in the vicinity of two adjacent particles, a is the particle radius, and R/p is the real part of the complex polarizability. Lateral electric field gradients arise because the dielectric particles deflect current around them, which produces gradients in the plane of the electrode; the strength of the lateral gradient is about 1/10 of the magnitude of the overall field imposed normal to the electrode. Since the dielectrophoretic interaction of two adjacent particles in a field applied normally to the electrode depends only on the lateral field, the factor of 0.1 must be applied twice to give an estimate of the field strength. The complex polarizability depends on the permittivities and conductivities of the particles and the medium; it can change sign as a function of (13) Newman, J. S. Electrochemical Systems, 1st ed.; Prentice Hall: Englewood Cliffs, NJ, 1973. (14) Sides, P. J.; Tobias, C. W. J. Electrochem. Soc. 1980, 127, 288291. (15) Tsukahara; Sakamoto, S. T.; Watarai, H. Langmuir 2000, 16, 3866-3872. (16) Green, N. G.; Ramos, A.; Morgan, H. J. Phys. D: Appl. Phys. 2000, 33, 632-641.

Figure 1. (a) The geometry of the sphere adjacent to the electrode. h is taken to be 5% of the diameter for the calculations. (b) The model electrochemical reaction of a neutral to an anionic state, along with the equivalent circuit for the reaction. No Warburg impedance appears because the departure of the solution concentration from its bulk value everywhere in the domain is assumed to be small.

frequency, often above 100 kHz, and has a value between +1 and -1/2.16 The third principle explaining lateral motion due to normally directed fields is the electrohydrodynamic flow 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. This theory, however, has not been elaborated for the case of particles to the point where predictive calculations based on it can be compared to experiments. It should be stressed that all three of the effects in the foregoing discussion can be active when particles are near an electrode during polarization. In fact, this complexity partially accounts for the variety of experimental observations. The real need is to probe what effects are important for a given set of process parameters such as electric field, frequency, and type of polarization. The particle velocities in the dc polarization case and the dielectrophoretic case are characterized by the equations presented, but the electrohydrodynamic case until now had no equivalent. Investigation of the strength of the electrohydrodynamic flow in the vicinity of a single particle during passage of alternating current and deduction of an equation that is explicit about the role of various physical properties were the goals of this work. Theory An electrically driven body force in the equation of motion of a liquid is the basis of the electrohydrodynamic explanation of particle motion proposed by Trau et al.4,5 The task is to develop an estimate of the strength of the electrohydrodynamic flow created by a single particle on a surface. Consider a single dielectric particle of radius a on a working electrode as shown in Figure 1a; the counter electrode is at infinity The electrolyte is a solution of binary salt having ions charged z+ and z- and dissociating into ν+ and ν- ions, respectively. An oscillating voltage applied to this system oxidizes the anion to form a neutral species during the anodic part of the cycle and regenerates the anion during the cathodic part of the cycle, as shown in Figure 1b. Consumption and production of charged species at the electrode create nonzero concentration gradients

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in the electrolyte adjacent to the electrode, which causes a small imbalance of charge in the diffusion layer. Poisson’s equation combined with an expression for the electric field in an electrochemical system of a binary salt governs the charge density Fe.

Fe ) ∇‚E

(3)

where13

E)

F i + (z ν D + z-ν-D-) ∇c (4) z+ν+Λc z+ν+Λc + + +

Here, i is the total current density, c is the neutral salt concentration, and Λ is its equivalent conductance. Di are the individual ionic diffusivities, zi are the ionic charges, νi are the stoichiometric coefficients of dissociation. F is Faraday’s constant. The diffusion equation for binary electrolyte and the equation for conservation of charge complete the suite of differential relationships necessary for calculation of the free charge.

∂c ) D ∇2c ∂t

(5)

∇‚i ) 0

(6)

D is the ambipolar diffusion coefficient of the dissolved salt. Equations 4-6 contain an assumption of electroneutrality, which makes their use with eq 3 paradoxical as Newman has pointed out.13 Analysis included in the Appendix leads to a dimensionless group that quantifies the severity of the inconsistency for oscillating systems. In the present case, the inconsistency is not significant below 1 kHz. The product FeE is the force per unit volume exerted on the liquid. Neglecting convection of momentum, one can write the complete differential description of electrohydrodynamic flow in this system,

-∇p + ∇2u + FeE ) ∇‚u ) 0

∂Fu ∂t

(7) (8)

subject to conditions of no slip at the particle and electrode surface, and zero velocity at infinity. p is pressure, u is the vector local fluid velocity, F is the fluid density, and t is time. The six eqs 3-8 and six unknowns i, c, E, u, Fe, and p must be solved in the vicinity of the particle to calculate the flow pattern and hence the resulting particle motion. The problem is axisymmetric for a single particle, but coupling between the flow and the electrochemical aspects of the problem complicates it. This set of equations applies to an electrode with zero capacitance and zero charge-transfer resistance; one must include the double layer and finite reaction kinetics for real electrodes. Since the purpose of this analysis is to estimate the strength of the effect, the following decomposition of the problem is proposed as a foundation for further experimental and theoretical inquiry. The transverse electric field caused by the presence of the dielectric sphere is imposed on the one-dimensional solution to the electrochemical problem in the absence of the sphere. In short, the charge density calculated from a one-dimensional solution of the mass transfer aspects of the problem is multiplied by the radial electric field calculated from a solution of Laplace’s equation for a dielectric sphere near an electrode. This approach neglects the charge arising

from radial concentration gradients, but these gradients are secondary in nature and hence small with respect to the primary gradients normal to the electrode. The electric field normal to the electrode is the primary contributor to the free charge, while the radial electric field that interacts with the free charge is a consequence of the disturbance in the potential around the sphere. Free Charge Density. The analysis of the free charge density begins by substituting eq 4 into eq 3 in order to deduce an expression for the free charge to be found in any region of space where ionic current is flowing and/or concentration gradients exist.13

(

)

Fe i‚∇c RT t+ t- 2 )+ + ∇ ln c 2 F z+ zz+ν+Λc

(9)

where ti is the transference number of species i. Thus, the free charge density for a one-dimensional problem is

(

)

∂c itot Fe RT t+ t- ∂2 ln c ∂z )+ + F z+ z- ∂z2 z+ν+Λc2

(10)

itot ) ic + if

(11)

where

itot is the total current density, if is the current associated with the faradaic reaction, and ic is the capacitive current through the electrode’s double layer. One must solve the diffusion equation to obtain the concentration function to be inserted in eq 10. The solution should reveal the concentration’s dependence on position, time, applied voltage V, and electrochemical parameters. Consider the electrochemical reaction involving reduction of a neutral species to produce an anion at an electrode as shown in Figure 1b. The equivalent circuit of this electrochemical reaction, also in Figure 1b, is taken as an ohmic solution resistance RΩ in series with a parallel arrangement of a double layer capacitor Cdl and a frequency-independent linear charge-transfer resistance Rct. The Warburg impedance is neglected because the model is restricted to small fractions of the limiting current. The solution of eq 5 for alternating current applied to the electrochemical system appearing in Figure 1b is

θ)

[x

t+(1 - j)if c )1+ exp c∞ z-ν-Fx2ωDc∞

]

2ωa2 ζ(1 + j) D (12)

where c∞ is the bulk salt concentration, ω is the frequency of oscillation, ζ is dimensionless distance (z/2a), and j is the square root of -1. Analysis of the circuit of Figure 1b leads to the following equation for if where Z is the complex impedance of the circuit of Figure 1b and V is the applied sinusoidally varying voltage.

if )

(

)

RΩ V 1Rct Z

(13)

The complex impedance Z includes contributions of the ohmic resistance, the charge-transfer resistance, and the electrical double layer modeled as the capacitor in Figure 1b. The concentration in eq 12 decays exponentially from the surface value to the bulk value over a distance roughly equal to (2D/ω)1/2. Increasing the frequency of oscillation

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both reduces the amplitude of the oscillation of concentration, because more current goes through the double layer capacitance, and thins the diffusion layer. Combining eqs 10 and 12, one obtains an equation for the charge imbalance in a one-dimensional diffusion layer with alternating polarization.

( )

()

itot 1 ∂θ RTtˆ 1 ∂2θ RTtˆ 1 ∂θ Fe ) - + 2aκ∞ θ2 ∂ζ 4Fa2 θ ∂ζ2 4Fa2 θ2 ∂ζ

2

(14)

where ˆt ≡ t+/z+ + t-/z-. Thus, the charge in the region of concentration gradients is a complex function of time and distance from the electrode. The inconsistency of using the assumption of electroneutrality in the derivation of eq 14 leads to neglect of the derivative of the free charge density on the right-hand side, as mentioned previously. The argument presented in the Appendix shows that the term is negligible below a kilohertz. Radial Electric Field. The dielectric particle of Figure 1a forces the current to flow around it, which engenders radial electric field gradients. If the fraction of limiting current is small, and hence departures from uniform concentration are small, solution of Laplace’s equation gives an estimate of the radial electric field at every position. Sides and Tobias14 derived an equation for the ohmically limited potential and current distribution around a dielectric sphere on an electrode, but the finite reaction rate on a real electrode, coupled with the small size of the particles, produces a uniform current distribution on the electrode. The justification of this assumption is that the ohmic penalty for uniform distribution of current around a small obstruction is small compared to the kinetic penalty for nonuniformity. This comparison is expressed by the dimensionless quantity a/(κ∞Rct) for linear polarization.13 This relation weighs the ohmic penalty for redistribution of current against the resistance presented by the reaction kinetics at the electrode. As this quantity becomes less than unity and smaller, the current density distribution becomes more uniform. For particles of less than 10 µm radius and conductivities corresponding to solutions of a millimolar or less, this parameter is less than unity for charge-transfer resistances as low as 0.001 Ω m2. The thin film semiconductor electrodes such as ITO have charge-transfer resistances of order 1 Ω m2, which indicates that the assumption of uniform current density is probably quite good. One must solve therefore for the potential and current distribution around the particle for a uniform flux to the electrode surface. Assuming no concentration gradients and solving Laplace’s equation for a spherical particle located a distance h from an electrode, with conditions of a linear field at infinity, uniform flux on the electrode, and zero flux into the particle, one obtains ∞

∑ Bn cosh ξ Pn(µ) - E0z n)0

φ ) γE0xcosh ξ - µ

(15)

for the electric potential in the vicinity of the sphere. Here, γ is the bispherical length parameter; µ and ξ are the bispherical coordinates,17 and z is distance perpendicular to the electrode. This equation clearly shows the meaning of E0 as the overall electric field applied normally to the electrode. Use of the generating function for Legendre Polynomials18 allows one to deduce the coefficients of eq (17) Moon, P.; Spencer, D. E. Field Theory Handbook; SpringerVerlag: Berlin 1961. (18) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1970, 33, 88-93.

15 in order to satisfy the no-flux condition on the particle surface. The alternating radial electric field is given by

Er )

itot ∂φ* κ∞ ∂r*

(

)

(16)

where r* ≡ r/2a and φ* ) φ/(-2aE0). Electrohydrodynamic Force. To this point in the derivation, the equations are exact within the limits imposed by electroneutrality both for an alternating current applied to a linearized electrode reaction in a onedimensional infinite domain and for a uniform current distribution around a spherical particle on an electrode in a medium with no concentration gradients. For the purposes of obtaining an estimate of the expected lateral force on the electrolyte at small fractions of the limiting current, we take the product of the free charge deduced from the concentration equation and the radial electric field from the solution of Laplace’s equation. If the departure of the concentration near the electrode from its bulk value is relatively small, the error in using Laplace’s equation for calculating the electric field is likewise small. Thus, an estimate of the force per unit volume on the liquid is

(

( )

itot 1 ∂θ RTtˆ 1 ∂2θ + F h r ≈ FeEr ) - 2aκ∞ θ2 ∂ζ 4Fa2 θ ∂ζ2 RTtˆ 1 ∂θ 2 itot ∂φ* (17) ∂r* 4Fa2 θ2 ∂ζ κ∞

( ))

( )

κ∞ is the bulk electrolyte conductivity. Multiplication through the charge expression by the electric field produces three terms contributing to the force density. The first and third terms, involving the concentration gradient and the gradient squared, respectively, are the product of three sinusoids oscillating about zero; these products vanish when averaged over one cycle. Thus, the force density reduces to

F hr ≈

( ) ( )

RTtˆ 1 ∂2θ itot ∂φ* ∂r* 4Fa2 θ ∂ζ2 κ∞

(18)

Equation 18 expresses the local electrohydrodynamic driving force for flow as a function of position and time. The time-averaged force density at any distance r* from the axis of the bubble and ζ from the electrode can be found from

〈F h r(r*,ζ)〉 ≈

[

] ( )

-RTtˆ ∂φ* ∂2θ / 1 Re itot 2 4Fa2κ∞ ∂r* ∂ζ2

(19)

where i/tot is the complex conjugate of the total current density and θ is taken as constant at unity consistent with the assumption of a small departure from limiting current on the anodic half of the cycle. (A complete evaluation of eq 18 without this assumption confirmed that the two methods agreed within a tenth of a percent.) The remaining task is to convert the time-averaged force density calculated from eq 19 to a fluid velocity. For the purposes of estimating the fluid velocities to be expected and for reasons that will become clear in the results section, we integrate

d2vr

-η

dz2

≈ FeEr ) 〈F h r(r*,ζ)〉

(20)


Langmuir, Vol. 17, No. 19, 2001 5795

twice to obtain the desired estimate of vr with the result appearing in eq 21. This equation estimates the timeaveraged local velocity vr expected if the spatial force distribution deduced in eq 19 is applied in fluid adjacent to an infinitely long wall in contact with an infinite medium where z is distance from the wall, as before.

4a2 [ζ η

v(ζ,r*) ) K

∫0∞ 〈Fh r(r*,ζ)〉 dζ ∫0ζ∫0w 〈Fh r(r*,u)〉 du dw]

(21)

(In this equation, u and w are dummy variables of integration.) The multiplicative constant K has been inserted to reflect the approximate nature of the calculation. If the model is approximately correct, its value should be between 0.1 and 10. A crucial point is that the velocity estimated in eq 21 reflects not only the average force density in the particle’s vicinity but also the force distribution in the direction normal to the electrode. Observability Index. The particle Peclet number introduced by Anderson and co-workers12 can be considered an “observability index” in the problem. The significant distance in this group is the particle radius, and the diffusion coefficient is the particle’s diffusion coefficient. This dimensionless group compares the velocity of the fluid to the particle’s Brownian mobility; it functions essentially as a convenient dimensionless parameter for weighing convective motion versus diffusive motion of the particles. When the observability index is large with respect to unity, the motion of the particles should be observable against the background of Brownian motion.

Figure 2. The dimensionless concentration of electrolyte (eq 12) as a function of phase angle at various distances. Transport properties are those of Table 2 for potassium hydroxide. ω/2π ) 100 Hz; Vo ) -6 V; c∞ ) 0.1 mM; Rct ) 0.01 Ω m2; Cdl ) 0.05 F/m2. The concentration oscillates as the faradaic reaction switches from anodic to cathodic. The amplitude of oscillation decreases exponentially from the surface according to eq 12. The oscillation at any distance from the surface is phase shifted from the value at ζ ) 0.

aUEH 6πa2ηUEH 6πa2η vr(0.5, 0.5) ) ) (22) Dp kT kT

electrode. The charge-transfer resistance was treated as a parameter of the calculation. All other values were physical constants or specified values such as the particle diameter. Alternating current has implications for the behavior of electrolyte concentration near the electrode, for the relationship between the region of concentration variation, and for the free charge density. The concentration of electrolyte oscillates about the bulk value in a region near the electrode as the faradaic current changes between cathodic and anodic polarization, as shown in Figure 2. The concentration at any distance is phase shifted from its value at the surface. The maximum 15% deviation of the concentration from the bulk value shows that the system is well away from limiting current during the cycle. The distance from the electrode over which concentration gradients exist is given approximately by (2D/ω)1/2; when normalized by the particle diameter, this becomes (D/2ωa2)1/2. This important relationship appears in Figure 3 for 1 and 5 µm particles. A 5 µm particle protrudes from the diffusion layer even at frequencies as low as 10 Hz as shown in Figure 3, but a 1 µm particle is comparable in scale to the diffusion layer only above 100 Hz. Figure 4 is a representative calculation of the total free charge density during one cycle of oscillation, for various distances from the electrode. The oscillation of the concentration and of its second derivative means that the charge density calculated according to eq 14 likewise varies throughout each cycle and its amplitude decreases with distance from the electrode as shown in Figure 4. The presence of the dielectric particle disturbs the flow of current to the electrode. The disturbance in the potential is illustrated in Figure 5 which shows normalized equipotentials in the region adjacent to the particle. The equipotentials are parallel to the electrode surface (ζ ) 0) at large distances from it and curve in toward the

PeEH ≡

Here, Dp is the individual particle diffusion coefficient and UEH is a characteristic velocity for the electrohydrodynamic flow (and hence the maximum particle velocity) taken as equal to the local lateral flow velocity calculated from eq 21 at a height and radius corresponding to the radius of the particle. Expressions for the EK and DE models can be derived similarly from eqs 1 and 2:

PeEK )

0.6πζpotE0a2 kT

(23)

PeDE )

10-2πR/pa3 2 E0 kT

(24)

Equation 21 allows an estimate of the expected electrohydrodynamic velocity, while eqs 22-24 are the desired comparative relationships; they allow calculation of the individual contributions of electrohydrodynamic, electrokinetic, and dielectrophoretic based particle motion, each referred to the same basis. Results The basis for the calculations that follow is an aqueous solution of a 1:1 salt having the properties of 10-4 molar potassium hydroxide or sodium bicarbonate shown in Table 2. The dimensionless distance of the particle (relative to its diameter) from the electrode is 0.05. An oscillating voltage of amplitude 6 V was applied to the equivalent circuit of Figure 1b where the ohmic resistance was taken as 2.18 Ω m2 and the double layer capacitance was 0.05 F/m2 as measured by Guelcher10 for an indium tin oxide

Table 2. Electrical and Transport Properties Used in the Calculations property λ+ × 104 λ- × 104 t+ tD z+ zCdl ˆt

m2/ohm equiv m2/ohm equiv none none m2/s none none F/m2 none

NaHCO3

KOH

50.11 41.5 0.547 0.453 1.21 × 10-9 1 -1 0.05 0.094

73.5 197.6 0.271 0.729 2.85 × 10-9 1 -1 0.05 -0.458

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Figure 3. The ratio of the diffusion layer thickness to the particle diameter. D ) 1.2 × 10-9 m2/s. The 5 µm particle protrudes from the diffusion layer even at 10 Hz, as shown by the value of this ratio being always less than unity on this graph. The 1 µm particle is well within the region of concentration variation at 10 Hz and begins to protrude from it above 100 Hz.

Figure 4. The total free charge density (C/m3) as a function of phase angle (eq 14) for various dimensionless distances from the electrode. The calculation parameters are the same as in Figure 2. The charge density changes sign as a function of the phase, and it decreases with distance from the electrode.

spherical particle surface as a consequence of the zeroflux condition there. The amplitude of the electric field distribution in the z direction at three radii appears in Figure 6. The electric field amplitude changes sign at a distance ζ of approximately 0.5 as the current reverses direction to flow back under the particle, as required by the uniform current distribution. The disturbance due to the particle decays by a factor of about 3 even at 1 diameter away from the axis of the particle. The field strength is greater at distances less than 0.5; the requirement of uniform current distribution pulls current through the constricted space between the spherical surface and the electrode and hence concentrates the field in that region. The oscillating force on the liquid at various distances from the electrode appears in Figure 7. This figure represents a case (hydroxide solution) where the forces acting on the liquid are primarily positive and therefore drive liquid away from the particle. The distribution of the time-averaged force per unit volume for a chargetransfer resistance of 0.01 Ω m2 appears in Figure 8 as a function of distance from the electrode normalized by the diameter of the particle. This is a case with electrolyte properties of sodium bicarbonate where the flow is primarily toward the particle, in contrast to Figure 7. At 30 Hz, the force density is negative, has its maximum magnitude at the electrode (ζ ) 0), and decays with

Sides

Figure 5. Equipotentials around a dielectric sphere on an electrode for uniform current density. h* ) 0.05. The curvature of the equipotentials is nonzero at the surface. A uniform current density requires current to flow in from infinity along the electrode.

Figure 6. Radial electric field amplitude (V/m) as a function of ζ at various radii. The normal field is approximately 3000 V/m. The field magnitude is larger in the region below the equatorial plane than above it because the requirement of uniform current density draws current into the constricted space between the sphere and the electrode. The strength of the radial field decreases strongly with distance from the axis of the sphere.

distance. Negative force densities generate negative flow toward the particle, while positive values drive fluid away from the particle. Thus, two adjacent particles move toward each other when force densities are negative. The force density magnitude at 30 Hz decreases strongly at distances from the electrode greater than 0.5 with this set of parameters because the effective diffusion layer is about 0.38 on this scale. Obeying an exponential decay, force density is noticeable to about 2-3 times this value. The force density magnitude at the surface becomes more negative as frequency increases due to the shrinking diffusion layer but decreases strongly with distance, and a positive hump appears. This change of sign of the force density means that the electrolyte on average is being pushed in opposite directions at various distances from the surface. The flow velocities calculated from eq 21 at the height of the center of the particle, dimensionless radius r* ) 0.5 and K ) 1, appear in Figure 9. Calculations are shown for the conditions of Table 2 (bicarbonate) with chargetransfer resistances from 10-4 to 10-1 Ω m2. (An electrode of 1 cm2 would have charge-transfer resistances from 1 Ω


Langmuir, Vol. 17, No. 19, 2001 5797

Figure 7. Lateral force density (N/m3) as a function of phase angle during one cycle. The calculation parameters are the same as in Figure 2 with the additional parameter particle radius a ) 5 µm. The force density varies during one cycle but overall is positive in this case for KOH. This means that fluid tends to flow away from a particle on the surface in these circumstances. Figure 9. Calculations based on eq 21 for the conditions of Figure 8 (bicarbonate), K ) 1.0, and the shown Rct. The velocities are negative at low frequencies, which means particles should aggregate. The velocity magnitude goes through a maximum (approximately 50 Hz), and its sign changes at frequencies depending on the charge-transfer resistance.

Figure 8. Force density distribution near the electrode at zero phase angle (eq 18) during one cycle. Cdl ) 0.05 F/m2; c∞ ) 10-4 M. Calculation parameters are for bicarbonate-like properties shown in Table 2. Rct ) 0.01 Ω m2. Note that the force density is now predominantly negative at lower frequencies upon switching to an electrolyte with different transport properties. The force density distribution changes as more current passes through the capacitance of Figure 1b and the diffusion layer also shrinks. As the current goes out of phase with the driving voltage, repulsive force develops at the higher frequencies (>200 Hz).

to 1 kΩ, respectively, with these values.) The effect of increasing frequency from low values is to produce higher velocity magnitudes that go through a magnitude maximum at relatively low frequency (see for example the curve marked “0.001” in Figure 9 at 50 Hz) and decrease as the frequency increases. The velocity at r* ) 0.5 reverses direction at the two lower values of the charge-transfer resistance. Particle size plays a crucial role in the electrohydrodynamic mechanism because the diffusion layer thickness is inversely proportional to the square root of the frequency of oscillation of the current. The particle can be well within or almost all outside of the diffusion layer at a particular frequency depending on its radius. An example of the effect appears in Figure 10, where the same calculation of velocity appears for particles of radius 1 and 5 µm at equal field strength over the same range of frequency. The magnitude maximum appearing at 50 Hz for the 5 µm particle is pushed out to 400 Hz, and the overall effect is much smaller for the smaller particle. Much higher fields are necessary to move smaller particles. If the flow is negative, particles generating this type of flow are mutually entrained. The sign of the force exerted on the liquid can be positive (repulsive), however, for a

Figure 10. The effect of particle radius on the behavior with frequency (eq 21) with K ) 1.0. Conditions are the same as in Figure 2, but for particles 5 and 1 µm in radius. The maximum in magnitude occurs at higher frequencies for smaller particles.

different electrolyte. The sign of the coefficient multiplying the second derivative in eq 19 depends critically on the relationship of the ionic diffusivities. The individual ionic conductance for sodium is larger than the conductance for bicarbonate; hence, the parameter ˆt is positive for sodium bicarbonate. The coefficient is negative in aqueous KOH, however, because the hydroxyl ion has a much higher ionic conductance than the potassium ion (see Table 2). The reversal of the overall force appears in Figure 11 where all the calculation parameters are identical except that the transport properties correspond to those for potassium hydroxide rather than sodium bicarbonate. The last of the calculated results is Figure 12 which shows the observability index of the EH model compared to the DE model as a function of frequency. The index can be positive or negative depending on the direction of flow; its sign gives the direction of flow, and the magnitude

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Figure 11. The effect of a change in electrolyte from sodium bicarbonate to KOH assuming the same reaction as in Figures 9 and 10. All conditions are the same as in Figure 2 except that in one case the solution is bicarbonate and in the other is KOH. See Table 2 for properties. The direction of velocity is opposite in the two cases, and the magnitude is in general greater in KOH.

Figure 12. A comparison of the observability of particle motion by dielectrophoresis and by electrohydrodynamic forces for 5 µm (radius) particles in bicarbonate solution. The electrohydrodynamic effect is larger than the dielectrophoretic effect at lower frequencies.

indicates the strength of the effect on the particle compared to background Brownian motion. The EH model yields observabilities greater than 10 which means the EH effect should be measurable in two-particle experiments. The DE effect is constant because experiments have shown that reversal from positive dielectrophoresis to negative dielectrophoresis usually occurs at frequencies above 100 kHz. The DE contribution is weaker than the EH effect at low frequencies. Discussion Interpretation of the Calculations. The results of the previous section should be considered indicators of potential particle behavior. The assumption of linear

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kinetics is a substantial simplification particularly at low frequencies where currents can exceed the maximum for linear behavior in real systems. Introduction of nonlinear current voltage behavior requires a different analysis because the behavior is not sinusoidal. The model also does not include the contributions of lateral concentration gradients interacting with the lateral electrical fields; these have been assumed to be small. Despite these shortcomings, the model provides insight and predictions of behavior worth checking with experiments. Figures 2-8 give an indication of the complexities of the temporal and spatial phenomena. The electrohydrodynamic response of electrolyte in the vicinity of a dielectric particle on an electrode is a complex phenomenon with strong frequency-dependent effects. Frequency governs the relationship between the diffusion layer thickness and the particle size as expressed by the dimensionless ratio δ* ) (D/ωa2)1/2. Values of δ* . 1 mean that the particle is well within the diffusion layer where the electric field surrounding the particle can interact with the Laplacian of the concentration. δ* , 1 means that most of the particle is outside the region of concentration variation. Furthermore, frequency governs the distribution of current between the double layer capacitance and the faradaic reaction, depending on Γ* ) ωRctCdl. If Γ* . 1, most of the current goes through the capacitance rather than through the faradaic reaction, and vice versa. Comprehending the interplay between these two parameters is the key to understanding Figure 9: If δ* . 1 and Γ* , 1 at a particular frequency, the direction of flow is in the primary direction set, as discussed below, by the parameter ˆt. In this case, the magnitude of the charge imbalance increases with the square root of frequency because the diffusion layer shrinks toward the particle diameter as the frequency is increased. The force density distribution in the vicinity of the particle is relatively constant under these conditions. This situation corresponds to the lowest frequency end of the curve for Rct ) 10-4 Ω m2 of Figure 9. If δ* , 1 and Γ* , 1, the direction of flow remains the same, but the velocity is reduced because the diffusion layer is confined to a range much smaller than the particle diameter and the free charge is proportional to the inverse square root of frequency. This corresponds to the highfrequency end of the lowest resistance curve of Figure 9. If δ* . 1 and Γ* . 1, much of the total current goes through the capacitance and hence the concentration effects are reduced so the velocities are lower. The overall flow is still negative. This situation corresponds to the low-frequency end of the high resistance curve of Figure 9. If δ* , 1 and Γ* . 1, the combination of a small diffusion layer and a shift of current to the capacitance of the double layer causes substantial positive force density to appear, enough so that the overall flow reverses, as shown at about 100 Hz on the high resistance curve of Figure 9. The second insight is the effect of particle size on the observed behavior. Figure 10 makes clear that higher electric fields are necessary for smaller particles and that two observers could come to different conclusions about aggregation and repulsion of particles at low frequency if one is using relatively large or small particles. At frequencies below 100 Hz, under the conditions of Figure 10, particles 1 µm in radius separate while particles 5 µm in radius strongly aggregate. This effect again points out the complex interaction of the particle size, the electrical behavior, and the mass transfer. The third insight provided by the model is an indication of the primary direction of flow. As discussed above,


increasing frequency by itself can reverse the flow, but the combination of an electrode reaction with the transport properties of the particular electrolyte sets the lowfrequency flow direction. In particular, if the anion is the reacting species and if the ionic diffusivity of the anion is less than the ionic diffusivity of the cation, the flow is negative and particles should aggregate. If the ionic diffusivity of the cation is less than the diffusivity of the anion, the flow is opposite, as shown in Figure 11. For example, the particles should not aggregate in ac fields in a binary electrolyte involving a base where the hydroxyl ion is the main ionic reactant and the field strength is such that the reaction is at a small fraction of the limiting current. The situation is reversed, however, for bicarbonate as the electrolyte; the transference parameter is positive, so the force acts in the negative direction, as shown by Figure 11. The observability index calculations indicate that the EH effect should be experimentally accessible for twoparticle measurements. At higher frequencies, the DE effect can influence the results. Comparison with Experiment. The observations of Table 1 can be revisited in light of the results of this investigation. 1. Micron-size particles move in fields of order 0.1 kV/m under dc polarization but aggregate at fields >1 kV/m in ac polarization. The electrokinetic interaction in dc polarization is inherently a stronger effect because the primary driving field is the normal field, which is typically 10 times the lateral gradients. 2. Aggregation in ac fields occurred at higher frequencies for smaller particles. Since the correspondence of the diffusion layer thickness with the particle size is a critical feature of this electrohydrodynamic model, it contains the necessary elements to explain this observation. Specifically, the thickness of the diffusion layer corresponds to the particle diameter only at much higher frequencies for much smaller particles. For example, the frequency ratio required to reduce the diffusion layer thickness from 10 to 2 µm is 25 due to the inverse square root dependence of diffusion layer thickness on frequency. 3. Particles stopped or became repulsive as the frequency was increased. The effects of high frequency on the mass transfer aspects and on the distribution of current between the faradaic reaction and the double-layer capacitance act to reduce the EH flow or even to reverse the direction as the diffusion becomes thin with respect to the particle diameter and/or current is shunted from the faradaic resistance to the double-layer capacitance. At very low frequency, when the diffusion layer is thicker than the particle diameter and if the ωRctCdl product is less than unity, the model predicts that increasing frequency increases the particle velocity, but this remains a prediction not yet observed. 4. Particles moved more quickly as they approached each other in dc polarization but moved more slowly as they approached in ac polarization. This behavior dramatically demonstrates the difference between EH and EK behavior. The EK flow is driven by normally directed electric field gradients that are intensified when the particles approach closely. The EH flow, driven by lateral gradients, is diminished because lateral gradients are naturally smaller when two equivalent particles approach each other. Conclusions A model for electrohydrodynamic flow near a dielectric sphere on an electrode undergoing ac polarization accounts for several disparate observations in the literature. The

Langmuir, Vol. 17, No. 19, 2001 5799

decrease of velocity and even flow reversal with increasing frequency are a result of the complicated interplay of the thickness of the diffusion layer with the particle size and the electrodics of the system. The primary direction of flow depends on the transport properties of the electrolyte and the details of the electrode reaction. Specific issues to be investigated are as follows: 1. The predictions should be independent of particle charge. 2. The strength of the effects should be proportional to the square of the electric field. 3. One should observe reversal of flow upon a suitable change of counterion if the reacting species remains the same. 4. Particle size is a key parameter. Perhaps the best chance to observe the maximum in velocity magnitude predicted at low frequencies is to use particles that are less than 5 microns in diameter. Use of small particles pushes the position of the velocity maximum out to larger frequencies where the electrode kinetics is more linear, in keeping with the assumptions of the model. 5. There should be a search for different electrode/ electrolyte combinations that allow confirmation of the effects of charge-transfer resistance and the transference parameter. Acknowledgment. This work was supported by the NASA microgravity program Grant NAG3-2159 and National Science Foundation Grant CTS-0089875. Appendix The analysis relies on the paradoxical use of both electroneutrality and Poisson’s equation, which requires some explanation. First, an estimate of the departure from electroneutrality must be obtained. Following Newman,13 one begins with the flux expression for the nonreacting ion. In the present transient case, the flux of the cation is only zero at the electrode, but this is where the expected departures from electroneutrality are the largest, so little significance is lost.

Fc+Ez ) RT

∂c+ ∂z

(A.1)

Solving for Ez and introducing the result into Poisson’s equation (electroneutrality not assumed), one obtains

z+c+ + z-c- )

2 RT ∂ ln c+ F ∂z2

(A.2)

Scaling this equation for a 1:1 electrolyte with z ) (2D/ ω)1/2z*, one obtains an estimate for the departure from electroneutrality.

RTω ∆c ≈ c∞ 2F2c D ∞

(A.3)

The departure from electroneutrality is proportional to frequency because the second derivative in eq A.2 increases with frequency; diffusion tends to relax differences in concentration. Using values from Table 2, one estimates the departure from electroneutrality to be 1 part in 103 at 1 kHz. This departure is insignificant for most aspects of the problem, and use of electroneutrality is clearly justified. The nonzero charge excess nevertheless makes the simultaneous use of electroneutrality and Poisson’s equation (see eq 9) paradoxical.

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One resolves the inconsistency by suspending the assumption of electroneutrality and examining the consequences. The total free charge density contributing a nonzero average electrohydrodynamic force density is given by

Fe RT ∂2 ln c 1 ∂iz RT 1 ∂2c 1 ∂Fe ) ˆt ) ˆt + F ∂z2 κ ∂z F c ∂z2 κ ∂t

(A.4)

The first term on the right-hand side of the final expression is the primary source of electrohydrodynamic flow at small fractions of the limiting current, as derived in the main text; the second term involving the derivative of the charge density with time is a consequence of the suspension of the assumption of electroneutrality. Scaling the latter term

to the former provides an estimate of their relative importance. This process yields a dimensionless group ω/κ∞ˆt that determines the impact of the assumption of electroneutrality on this aspect of the problem. At low frequencies or high conductivity, the departure from electroneutrality is inconsequential. If, however, the individual ionic conductivities are quite close, this parameter becomes large and electroneutrality cannot be safely neglected in the electrical aspects of the problem. This parameter is less than 0.1 for the cases and results presented in this work; hence, use of electroneutrality and Poisson’s equation is consistent with the stated purpose to estimate the strength of the electrohydrodynamic flow. LA0105376

Electrohydrodynamic Particle Aggregation on an ... - ACS Publications

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