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Langmuir 1997, 13, 6058-6068
Particle Clustering and Pattern Formation during Electrophoretic Deposition: A Hydrodynamic Model Yuri Solomentsev,*,† Marcel Bo¨hmer,‡ and John L. Anderson† Colloids, Polymers and Surfaces Program, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, and Philips Research Laboratories, Eindhoven, Professor Holstlaan 4, 5656 AA Eindhoven, The Netherlands Received March 19, 1997. In Final Form: August 19, 1997X Clustering of latex particles 4-10 µm in diameter during and after electrophoretic deposition of the particles onto flat electrodes has been reported by Bo¨hmer (Langmuir 1996, 12, 5747). The particles interacted over length scales comparable to their size in the formative stages of the clusters. Combinations of two or more clusters already deposited approached each other to form larger agglomerates. A model based on electroosmotic flow about charged particles near surfaces is developed here to explain these observations. A charged, nonconducting particle near or on a flat conducting surface creates flow in the adjacent fluid due to electroosmosis about the particle’s surface. Fluid is drawn laterally toward the particle near the electrode and pushed outward from the particle farther away from the electrode above the particle. Another particle near the electrode will be drawn toward the central particle by this convection. We first solve for the flow field about a single particle and then compute the rearrangement of neighboring particles in response to the flows. The clustering times for different initial configurations of sets of particles (e.g., regular versus irregular spacing) are calculated. The average clustering times for irregular configurations are greater than those for regular arrays. The qualitative and quantitative features of the experimental observations are captured by this model if the hindrance effect of the solid wall is taken into account. For example, the model correctly predicts the observed declustering (separation) of particles when the polarity of the electric field is reversed as well as the observed cluster-to-cluster motion.
Introduction Electrophoretic deposition of colloidal particles onto an electrode can result in an ordered arrangement of particles, as shown in Figure 1. Long range attractions between particles close to conducting surfaces have been observed in both ac and dc electric fields.1-3 In recent experiments Bo¨hmer4 observed that particles of size 4 and 10 µm diameter formed large ordered clusters on the electrode surface even when the coverage was far below that of a close-packed monolayer. Several observations are particularly interesting: (a) The aggregation occurred after the particles were close to the surface or deposited. (b) The aggregation could be reversed, even without removing the particles from the electrode, by reversing the polarity of the field. (c) Two or more clusters that had formed on the surface often aggregated into a larger cluster. Bo¨hmer concluded that the aggregation phenomenon is not attributable to short range colloidal forces because the particles and clusters interacted over length scales comparable to their size and the tendency for aggregation decreased when the ionic strength was increased. Bo¨hmer suggested that the aggregation of deposited particles was due to hydrodynamic effects resulting from electroosmosis about each particle. Most models of electrophoretic deposition focus on the velocity of the particle as it approaches the electrode.5 The formation of aggregates is assumed to arise because * To whom correspondence should be addressed. Electronic mail:
[email protected]. † Colloids, Polymers and Surfaces Program, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. ‡ Philips Research Laboratories, Eindhoven, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. X Abstract published in Advance ACS Abstracts, October 15, 1997. (1) Trau, M.; Saville, D. A.; Aksay, I. A. Science 1996, 272, 706. (2) Richetti, P.; Prost, J.; Barois, P. J. Phys. Lett. 1984, 45, L-1137. (3) Crocker, J. C.; Grier D. G. Phys. Rev. Lett. 1996, 77, 1897. (4) Bo¨hmer, M. Langmuir, 1996, 12, 5747. (5) Koelmans, H.; Overbeek, J. Th. G. Discuss. Faraday Soc. 1954, 18, 52.
S0743-7463(97)00294-1 CCC: $14.00
Figure 1. Example of a monolayer of 4 µm polystyrene latex particles on an ITO-coated glass slide. The layer was prepared following the procedure described by Bo¨hmer.4 The original latex suspension provided by the manufacturer was diluted 10 times with 10-5 M KCl solution. The particles were deposited at 2.0 V for 30 min followed by a fixation at 6.0 V for 30 s. Then the sample was taken out of the solution and dried. As discussed by Bo¨hmer,4 some additional fusion of domains may have taken place in the process of drying.
of coagulation just prior to deposition, which results from a higher local ionic strength caused by the electrode reactions, and because of dipole interactions between the particles as one approaches the other at the surface.6,7 Trau et al.1 suggested that the clustering of particles observed in ac and dc fields1-3 is due to electrohydrodynamic flows caused by concentration gradients arising from the electrode reactions. No model to date has considered the effects of the flow generated about each particle by the electric field. When the double layer is (6) Estrelia-Lopez, V. R.; Ul’berg, Z. R.; Konashvili, S. A. Kolloidn. Zh. 1982, 44, 74. (7) Lavrov, I. S.; Smirnov, O. V. J. Appl. Chem. USSR 1969, 42, 1459.
© 1997 American Chemical Society
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Figure 3. Single particle held stationary near an electrode. The orientation of this diagram is 90° relative to Figure 2.
computing the electrophoretic velocity of a nonconducting sphere toward an electrode, but because they were interested only in the particle’s velocity as a function of position from the electrode, they did not analyze the velocity field of the fluid. We follow the analysis of Keh and Lien9 to obtain a semianalytical solution for the velocity field. The velocity field is then used to calculate the dynamics of groups of particles, assuming that hydrodynamic and electrostatic interactions between particles can be neglected. The calculations demonstrate that electroosmosis is sufficiently strong to explain the observations by Bo¨hmer4. Electroosmotic Velocity Field
Figure 2. Streamlines for electroosmotic flow about a negatively charged particle of radius a deposited at a separation distance h* ) 0.1a from a positive electrode. If the particle and electrode were to have the same charge polarity, then the streamlines would remain the same but the direction of fluid flow would be reversed from that shown in the figure.
thin relative to the particle diameter, interactions among a group of particles having the same ζ potential and suspended (without touching each other) in an infinite liquid phase cancel, such that each particle in the group moves at the velocity it would have alone.8 Thus, in a group of particles suspended in a fluid there is no tendency for the particles to move toward or away from each other. However, the situation is quite different for two or more particles constrained near a conducting surface. Figure 2 shows the streamlines of flow about a single charged, nonconducting particle held fixed near a flat electrode. The particle is assumed to be negative, and the electric field points into the liquid (i.e., the electrode is positive). The electric field interacts with the positive charge in the diffuse part of the double layer at the particle’s surface, causing electroosmotic flow. The flow is directed away from the electrode near the particle and pulls fluid toward the particle near the electrode surface. A second particle that has deposited in the vicinity of the first particle would be convected toward the first particle, and vice versa. Electroosmotic flows about deposited particles thus promote aggregation, as suggested by Bo¨hmer;4 the question is whether or not these flows are sufficiently strong to explain the aggregation he observed. In this paper we compute the fluid velocity field due to electroosmosis about a deposited particle. Keh and Lien9 and Keh and Anderson10 considered this problem when (8) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61. (9) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (10) Keh, H. J.; Lien, L. C. J. Chin. Inst. Chem. Eng. 1989, 20, 5, 283.
A spherical particle of radius a is held stationary near a conducting wall (the electrode). We use a cylindrical coordinate system (r*, θ, z*) with the z* axis perpendicular to the wall, as shown in Figure 3. The coordinates of the center of the sphere are {0, 0, a + h*}, where h* is the gap between the sphere and the wall. It is convenient to nondimensionalize the coordinates using the radius of the particle; the dimensionless cylindrical coordinates are denoted by (r, θ, z), and the dimensionless gap is denoted by h. The particle is large compared with the thickness of the double layer (κa . 1, where κ is the Debye screening parameter). The electrical potential Φ in the fluid outside the double layer is described by Laplace’s equation:
∇2Φ ) 0
(1a)
with the following boundary conditions:
when z ) 0: Φ ) 0 as z f ∞: Φ f -aE∞z
(1b)
on the sphere’s surface: n‚∇Φ ) 0
(1c)
where E∞ ) constant is the electric field strength far from the particle. The fluid velocity u and pressure p are described by the following equations and boundary conditions
η∇2u - ∇p ) 0
(2a)
∇‚u ) 0
(2b)
when z ) 0: u ) 0 as z f ∞: u f 0, p f constant
(2c)
on the sphere’s surface: u ) (ζ/η)∇sΦ
(2d)
where and η are the permittivity and viscosity of the fluid, ζ is the electrostatic potential of the surface of the particle, and ∇s is the surface gradient operator.
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The boundary condition (eq 2d) on the sphere’s surface really applies at the outer edge of the double layer, but because the Debye length 1/κ is small compared to the particle’s radius (κa . 1) this surface is essentially the same mathematically as the true surface of the particle.8 The fluid velocity field is found by solving eqs 1-2. The force required to hold the center of the particle at the distance z ) 1 + h can be determined by integrating the hydrodynamic stress over the surface of the particle (i.e., the outer edge of the double layer). The problem is solved by using bispherical coordinates (ξ, ψ, θ):
r)
c sin ψ cosh ξ - cos ψ
(3a)
z)
c sinh ξ cosh ξ - cos ψ
(3b)
c ) xh(2 + h)
(3c)
where c is the bispherical length parameter. Note that ξ ) 0 corresponds to the surface of the electrode, while ξ ) ξ0 ) arccosh(1 + h) corresponds to the surface of the sphere. Since the problem has cylindrical symmetry, it is useful to introduce a stream function Ψ, which is related to the components of the velocity in bispherical coordinates by 2
uξ )
1 (cosh ξ - cos ψ) ∂Ψ ∂ψ a2 c2 sin ψ
(4a)
2
uψ ) -
1 (cosh ξ - cos ψ) ∂Ψ ∂ξ a2 c2 sin ψ
(4b)
This definition of Ψ automatically satisfies eq 2b. The electric potential Φ and stream function Ψ can be found as expansions in bispherical harmonics: ∞
Φ ) cE∞(cosh ξ - cos ψ)1/2
∑ Rn ×
( ) n)0
sinh n +
1
2
ξPn(cos ψ) - E∞az (5)
∞
Wn(ξ)G-1/2 ∑ n+1 (cos ψ) n)1
Ψ ) (cosh ξ - cos ψ)-3/2
(
(6)
1 1 ξ + Bn sinh n - ξ + 2 2 3 3 Cn cosh n + ξ + Dn sinh n + ξ 2 2
Wn(ξ) ) An cosh n -
)
(
(
)
)
(
)
where Pn and Gn-1/2 are Legendre and Gegenbauer polynomials, respectively, and Rn, An, Bn, Cn, and Dn are the constants which are determined from the boundary conditions on the sphere’s surface and on the electrode. Note that the functions in eqs 5 and 6 automatically satisfy the governing eqs (eqs 1a and 2a,b) and the far field (z f ∞) conditions. The above infinite series are exact solutions for Φ and Ψ, but an approximation is introduced because the series are truncated at finite values of n. In our calculations we truncated the series in eqs 5 and 6 at n ) N (typically N was between 30 and 150). A set of 5N + 1 linear equations for the unknown coefficients were obtained from the boundary conditions on the sphere and electrode. This set of equations was solved by standard methods, and the stream function Ψ was
Figure 4. (a) Dimensionless lateral velocity of fluid ur/(ζE∞/ η) at r ) 3. The solid line is for h ) 0.1; the dotted line is for h ) 0.05. (b) Dimensionless lateral velocity of fluid ur/(ζE∞/η) for h ) 0.1. The solid line is for z ) 3; the dashed line is for z ) 1.1.
calculated; the velocity at any point in the fluid could then be calculated from eq 4. Figure 2 shows streamlines for h ) 0.1. We have calculated the streamlines for h ) 0.05 and 0.10 and found them to be indistinguishable near the electrode. The fluid velocity component ur at fixed r is plotted versus z in Figure 4a. There are three interesting features of Figures 2 and 4. First, the fluid velocity is always toward the deposited particle near the electrode (z , 1). Second, far from the electrode the velocity is toward the deposited particle at large r and away from the particle at small r. And third, there is a zone of circulation. Thus, the effect of this flow on influencing the trajectory of another particle is different depending on its relative position; the flow could be attractive to the deposited particle or repulsive. We consider this z-position dependent behavior later. The important point is that when a particle is near the electrode, say h e 0.1, then it will always be convected toward a second deposited particle due to the electroosmotic flow caused by the second particle. Previous works have been concerned with the effect of the wall on the electrophoretic velocity.9-11 This is determined by using eqs 4-6 to compute the hydrodynamic force on the particle and then balancing this force with the hydrodynamic force on a sphere moving at velocity U perpendicular to the wall. The result is plotted in Figure 5 and compared with the result for sedimentation (or motion by any body force acting on the particle). Note that in this graph the electrophoretic velocity is normalized by its value far from the wall, (ζ/η)E∞, while the sedimentation velocity is normalized by its far field value, 2a2∆Fg/(9η), where ∆F is the difference in densities (11) Feng, J. J.; Wu, W. Y. J. Fluid Mech. 1994, 264, 41.
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for each mechanism are comparable far from the surface. This situation has been noted by Barton and Subramanian12 and Lowenberg and Davis13 for the thermophoretic movement of bubbles and drops near boundaries. We emphasize that colloidal forces between the particle and the wall are neglected in the determination of f; such forces could prevent detachment of a particle even when E∞ > Ec. Clustering of Deposited Particles by Electroosmotic Flow
Figure 5. Electrophoretic (dashed line) and sedimentation (solid line) velocities of a particle moving perpendicular to a conducting wall as a function of the separation distance (h). The sedimentation velocity is normalized by the far field value (2a2∆Fg/9η), while the electrophoretic velocity is normalized by its far field value (ζ/η)E∞.
Figure 6. Function f(h), defined by eq 7a, versus separation (h) between the particle and the wall.
between the particle and fluid and g is the gravitational acceleration. As noted by others, the wall effect on sedimentation is much greater than that on electrophoresis.9-11 The different wall effect on electrophoresis versus sedimentation produces an interesting situation. We ask the question what field is required to lift a particle off the surface, assuming there are no significant colloidal forces between the particle and the surface? This critical field, Ec, is determined by setting the electrophoretic velocity equal to minus the sedimentation velocity and defining a function f(h) to account for the wall:
Ec )
2 a2∆Fg f(h) 9 ζ
v0 ) u0 +
(7b)
Note that by definition f f 1 as h f ∞. Moreover, it can be shown that the function f(h) is the ratio of net forces exerted on the particle by gravitational and electric fields; consequently f(h) is finite at h ) 0, since both of these forces are finite. The critical field decreases with decreasing separation distance because the wall effect is greater on sedimentation. Thus electrophoresis is a more efficient mechanism for driving particles toward or away from a surface than is sedimentation when the velocities
a2 2 (∇ u)0 6
(8)
where u0 is the undisturbed (in the absence of the particle) velocity of the fluid evaluated at the center of the particle. Note that eq 8 partially describes the wall effect, since the second term on the right side is influenced by the wall. Our calculations show that the second term produces about a 30% reduction in the velocity of the particle in the r direction as compared to ur0. The second effect is a direct hindrance of the wall. A correction factor q is defined to account for this effect:
(7a)
f(h) is the ratio of the values given by the two curves in Figure 5; this function is plotted in Figure 6. A good representation (within 5%) of f(h) is given by the following empirical equation:
h + 0.300 f(h) ) h + 1.554
The electroosmotic flow caused by a deposited particle (i.e., a particle for which h is small and essentially constant) affects the trajectory of its neighbors. Far from the electrode the dominant motion of the particles is in the z direction, caused by electrophoresis and sedimentation, and the influence of the electroosmotic flow is negligible. Close to the electrode, h e 0.1, the interesting motion is in the plane parallel to the electrode because there would be no such velocity component in the absence of deposited particles. In this section we consider the dynamics of groups of deposited particles and assume that there is a small but finite gap (0 < h , 1) between the deposited particles and the electrode surface. Thus, we are investigating the clustering process after particles have been brought to the surface by electrophoresis and sedimentation. The lateral velocity of the fluid (ur) near a single deposited particle at r ) 3 is shown in Figure 4a. Figure 4b presents variations of ur with r at fixed z. Note that in general the convective velocity of a second particle (“test particle”) centered at the point with coordinates {r, z} relative to the first (deposited) particle (see Figure 2) would be smaller than the value of ur due to the influence of the wall. There are two hydrodynamic effects responsible for this. First, in the absence of boundaries the velocity of a sphere in a convective flow can be calculated from Faxen’s law:
vr ) qv0r 0 < q < 1
(9)
v0r ) v0‚ir
(10)
where
Goldman et al.14 have shown that for 0.001 < h < 0.1 the translational velocity of a sphere in a linear shear flow near a solid wall would be 30-60% less than the fluid velocity. Consequently we expect that q ) O(1) for 0.01 < h < 0.1. The precise relation between the fluid velocity (u) and the particle velocity (v) can only be determined by solving the electrostatic and hydrodynamic equations (12) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1990, 141, 146. (13) Lowenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1993, 160, 265. (14) Goldman, A. J.; Cox, R. G.; Brenner, H. Chem. Eng. Sci. 1967, 22, 653.
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Figure 8. Initial and final configuration of a five-particle group arranged in a face-centered square. b is the initial center-tocenter distance between the particles at the corners. Figure 7. Relative convective velocity (vr) of a second particle caused by the electroosmotic flow generated by the first particle (assumed fixed). The velocity is scaled by the electrophoretic velocity of the particles far from the electrode. h ) 0.1 (solid line) and 0.05 (dotted line). q ) 1 in eq 9.
for two spherical particles near a conducting wall. In the calculations presented here we have used the approximation described by eqs 8-9 with q ) 1 to calculate particle velocities and then adjusted q to fit the calculations with the experimental data. The analysis is for equal size particles, and because one value of q has been assumed for all particles, we have implicitly assumed that the gap h is about the same for all particles. A second approximation in our simulations is to assume that the fluid velocity u in eq 8 equals the sum of the electroosmotic velocities caused by all the deposited particles without allowing for interactions among them. That is, we neglect particle interactions as they would affect the local electric and velocity fields. We first consider the relative motion between two deposited particles. The line of centers defines the x axis, and one of the particles is fixed (i.e., it is not allowed to move). The relative velocity of the second particle toward the first (fixed) particle equals the magnitude of the convective velocity vr caused by the first particle, assuming q )1. Figure 7 is a plot of vr versus the center-to-center distance between the particles. Note that if the first particle were allowed to move because of the electroosmosis generated by the second particle, then the relative velocity between the two particles in the absence of other particles would be 2|vr|. The dynamics of a group of deposited particles is determined by the collective electroosmotic flows from each. We first consider the velocity of particle i caused by electroosmosis from particle n, which is expressed generally in Cartesian coordinates (x, y, z) as
vi,n ) vr(x(xi - xn)2 + (yi - yn)2)
[(xi - xn)i + (yi - yn)j]
x(xi - xn)2 + (yi - yn)2 (11)
where vr(d) is the relative velocity defined by eq 9 and plotted in Figure 7 and (xi, yi) are the coordinates of the center of particle i. Because particle interactions are neglected with respect to their effects on the electric and velocity fields, the net electroosmotic velocity is the sum of the velocities caused by each particle. Thus, the velocity of particle i in the group is N
vi )
∑ vi,n
n)1 n*i
The position of particle i at time t is given by
xi(t) ) xi(t)i + yi(t)j
(13)
Combining the above two equations gives 2N scalar equations for the trajectories of the particles of the ensemble near the electrode. In vector form these equations can be written as follows:
dxi(t) ) vi(t), i ) 1, ..., N dt
(14)
Time has been made dimensionless by t0:
t0 )
η a ζE∞
(15)
To close the problem, the initial positions of the particles must be specified:
xi(0) ) x0i yi(0) ) y0i
(16)
To avoid overlap of the particles when they approach each other, we set vi,j ) vj,i ) 0 when contact is first made between particles i and j. Simulations of particle clustering were performed with two basic initial configurations of the group of particles: a regular array and an irregular distribution. The facecentered square array is pictured in Figure 8. The time (T) required for five particles in Figure 8 to come together was calculated by solving eq 14 subject to the initial conditions characterized by the array spacing b. The mean velocity of collapse, Va, is defined as
b 2( ) - 2 x 2 V (b) ) 2
a
T(b)
(17)
The results of these calculations are plotted in Figure 9. The average velocity of clustering appears to be exponential in b; an exponential fit of the data is shown in Figure 9b:
Va ) 0.7322e-0.2766b
(18)
Equation 18 approximates the calculated values of Va within 5%. The length b (normalized by the particle radius) is related to the area fraction (R) of deposited particles by
(12) b)
x5πR - 2
(19)
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Figure 11. Clustering time of a three-particle group arranged in (a) an equilateral triangle (open circles), (b) a straight line (open squares), and (c) a random group (filled triangles). The broken line is eq 22a. The center-to-center distance (d) is defined by eq 20. The vertical bars are the standard deviations of MonteCarlo simulations of the random group.
Figure 9. (a) Clustering time as a function of initial separation for five particles arranged as shown in Figure 7. q )1. (b) Average velocity of clustering (see eq 14) versus initial spacing. q )1. The line is eq 18.
Figure 12. Clustering velocity of a three-particle group arranged in (a) an equilateral triangle (open circles), (b) a straight line (open squares), and (c) a random group (filled triangles). The broken line is eq 22b.
(without overlap) in the area. The mean interparticle distance at t ) 0 is defined as follows for all three initial configurations:
1 d ) (d12 + d13 + d32) 3
(20)
where dij is the center-to-center distance between particles i and j. The clustering time T is defined as the time when one of the particles touches two others, and the clustering velocity VA is defined by Figure 10. Initial configurations of a three-particle group.
VA(d) ) Taking, for example, a 20% area fraction, we have b ≈ 6.9; from Figure 9b we see that Va is about 15% of the electrophoretic velocity of the particles in suspension if q(h) is equal to unity. Models based on regular placement of particles do not always adequately describe random systems, especially at low area fractions. To estimate effects of a random nature of the clustering process, we performed the following calculations. The three configurations depicted in Figure 10 were considered: (a) three particles placed at the corners of an equilateral triangle; (b) three particles placed in a line; (c) three particles placed randomly
d-2 T(d)
(21)
T(d) and VA(d) were calculated from eqs 14 and 16 as functions of initial interparticle distance assuming q ) 1 in eq 9. In the case of random initial placement, 10-15 initial configurations were chosen. Note that the positions were chosen randomly except for not allowing overlap. T and VA for this case were calculated as averages over all initial configurations. The calculations for all three initial configurations are presented in Figures 11 and 12. The vertical bars in these graphs represent the standard deviations from Monte Carlo simulations for random placement of the particles. The following equations are empirical fits of the mean
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Figure 13. Calculations showing formation of two small clusters which move toward each other to form a larger cluster.
values of the corresponding quantities for the case of the random initial placement:
T(d) ) 2.19(d - 2) exp(0.277d)
(22a)
VA(d) ) 0.456 exp(-0.277d)
(22b)
Equation 22a approximates the calculated values for T(d) within 6% for 2 < d < 7. Equation 22b is obtained from eqs 21 and 22a. As in the case of the set of five particles considered in Figure 8, VA and T display exponential dependence on the interparticle distance. For d ) 6 we find VA to be about 10% of the electrophoretic velocity of the particles in suspension if q is assumed to be equal to 1. Note from Figure 7 that the magnitude of vr reaches its maximum at r ) 2 (when two particles touch each other). This means that the tendency to attract is strongest at the point of contact. Consequently, the aggregates formed by secondary electroosmotic flow are stable and can be destroyed only if the field is reversed or turned off. This means that if an aggregate (doublet, triplet, etc.) is formed, it must move as a whole toward another cluster of particles. To illustrate this, we consider the initial configuration shown in Figure 13. Here the six particles are placed in two subgroups of three each. At first the particles of each subgroup form a small cluster, and then the two clusters move toward each other. Note that time is scaled by t0, which is defined in eq 15. Comparison with Experimental Observations Measurements were performed on a glass slide with a 270 nm indium/tin oxide (ITO) layer with a sheet
resistance of 8.2 Ω per square. The slide was placed in a cylindrical cell with an inner diameter of 4 mm. On top of the cylinder a Pt counter electrode was placed. The distance between the ITO slide and the counter electrode was 6 mm. The particles were viewed using a Leitz Metallovert Microscope with a 20× or 50× dark field objective and a Philips LDH0703 CCD camera. Images were stored on a videotape, and selected images were digitized. Measurements were performed on 10 mm surfactant-stabilized polystyrene spheres (Duke Scientific) dispersed in 10-4 M KNO3 aqueous solutions. The particles had an electrophoretic mobility of 6.0 × 10-8 m2/(s V), which was measured with a Zetasizer 4 (Malvern). Constant voltages of 1 and 1.5 V, followed by -1 and -1.5 V, respectively, were applied. The current in the system was monitored in the course of the experiment. The electric field was calculated from the measured current and specific conductivity of the solution above the electrode, which was 40 µS/cm (this value is higher than expected because of the excess of surfactant present in the system). At applied voltages of 1 and 1.5 V, the particles retained their mobility even when they appeared to be on the surface. A small but clearly visible Brownian motion was observed for the deposited particles, thus indicating the presence of a finite film of a liquid between the deposited particles and the electrode. At applied potentials of 1 and 1.5 V, long range attractive forces between the particles deposited on the electrode were observed. The range of the observed interparticle attraction was on the order of several particle diameters. In the process of deposition the particles formed ordered clusters as shown in Figure
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Figure 14. Time series for the formation of clusters of 10 µm PS particles starting from a semiregular array at 1.5 V.
14. Figure 14 is a time series of cluster formation for an applied voltage of 1.5 V. Before the photographs were taken, the particles were first deposited at 1.5 V followed by -1.5 V for 10 min. Due to gravitational forces the particles remained on the surface upon application of the negative voltage but were well dispersed (i.e., no clusters). The positive potential was then applied again, and the particles began to form clusters which eventually led to fairly well ordered arrays. After 60 s the first doublets formed (Figure 14b). Subsequently, these doublets became triplets and the two doublets on top of the figure moved toward each other (Figure 14b,c). The clusters became more ordered (see for instance the triplet in Figure 14c and d). Clustering of single particles and cluster-cluster aggregation occurred until, as seen in the last picture of
the series taken after 12 min, only one single particle was left and several large ordered clusters were formed (Figure 14f). This series of photographs shows both the long range nature of the interactions and the cluster-cluster aggregation phenomenon. Reorientation of particles in a cluster can be seen as well; for example, in Figure 14d and e a triplet deformed upon approaching the big cluster at the top of the picture and reoriented itself to fit in the lattice. Larger clusters also reoriented; for example, see the configuration of the cluster at the very top of Figure 14e and f. If the microscope was focused about 15 µm away from the surface, the particles were mainly seen on the periphery of the existing clusters. No suspended clusters were observed in the bulk, suggesting that there was no
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Figure 15. Declustering upon reversal of the field at -1.5 V.
Solomentsev et al.
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Figure 16. Absolute value of electric field as a function of time: filled squares, applied voltage of -1.0 V; filled circles, applied voltage of -1.5 V.
Figure 17. Experimental values for the mean interparticle distance (see eq 20) as a function of time for six triplets: open symbols, applied voltage of -1 V (three triplets); filled symbols, applied voltage of -1.5 V (three triplets).
Figure 18. Declustering time for triplets TE(d) as a function of the mean center-to-center distance defined by eq 20. TE(d) is the experimental value in the form of eq 23: open symbols, applied voltage of -1.0 V; filled symbols, applied voltage of -1.5 V. Symbols correspond to the data plotted in Figure 17. The curve is the theoretical prediction calculated from eq 24 with q ) 0.50. The vertical bars are the standard deviations of Monte-Carlo simulations for the range 2 < d < 5 (10-15 trials). Horizontal bars show the standard deviations of the distance measurements (experiment).
interparticle interaction far from the plate. It was observed that the particles coming to a cluster move around it until they find an equilibrium position on the border of the crystalline structure. Quite often an approaching particle was visible as a vague, large gray structure, which became smaller as it moved closer to the
Langmuir, Vol. 13, No. 23, 1997 6067
electrode. This particle would well fit on the top of the aggregate if it had continued to move in the same direction. However, it was usually observed that the aggregate closed itself, and the incoming particle drifted to the edge of the aggregate and then down to the surface. Such behavior was always seen if the clusters were smaller than 20 particles. This observation is in line with the idea that fluid is transported upward along the particles. Incoming particles are led to the periphery of the existing clusters. The particles under experimental conditions tend to form a monolayer, and the second layer is only formed when the average size of the clusters is larger than approximately 20 particles. This fact suggests that the particles are convected by an electroosmotic flow toward the perimeter of clusters already formed on the surface. Note also that the particles remained mobile while on the surface of the electrode. Some Brownian motion, especially for the smaller (4 µm) particles, was observed.4 Upon reversal of the field, the particles drifted apart from each other (Figure 15). In order to quantify the aggregation rates, a fixed initial arrangement of particles was desirable. Since the effect considered here is reversible with electric field, instead of agglomeration rates the declustering rates can be measured. For this observation a set of triplets was used. Quantitative analysis of particle migration was performed for deposition at 1 and 1.5 V followed by declustering at -1 and -1.5 V, respectively. The negative voltages were low enough so the particles stayed on the surface due to the gravitational force (E < Ec). The electric field was calculated from the measured current and fluid conductivity using an electrode area of 14.9 mm2, and a typical plot of the electric field as a function of time is given in Figure 16. It can be seen that initially the field strength decreased rapidly and became approximately constant after about 100 s. The time required for double-layer charging is much shorter (on the order of milliseconds). The changes in the electric field (i.e., current density at fixed electrode voltage) on the long time scale observed in our experiments were probably due to the kinetics of the electrode reactions and polarization of the electrode. In Figure 17 the experimentally measured average interparticle distance is given for six different triplets drifting apart as a function of time for -1.0 V (three triplets) and -1.5 V (three triplets). Without an applied field, the particles in the triplets also declustered due to Brownian motion, indicating that the clustering was caused by a field-induced attraction. We expect the effects of Brownian motion to be small for the larger particles (10 µm) and higher fields (-1.5 V electrode potential). The particle velocity along the electrode at -1.5 V is initially about twice as high as that for the velocity for -1.0 V, which agrees with the difference in electric field strength between these applied voltages, indicating the linear character of the phenomenon. The measured time series of mean interparticle distances during triplet declustering allows for a comparison between our theory and the experiment. In order to eliminate the time dependence of the field and to make a direct comparison with the theoretical predictions, the parameter TE(d) is defined as a dimensionless time:
∫0tE(τ) dτ
ζ T (d) ) t E
atη
(23)
TE(d) was plotted as a function of the experimentally measured mean interparticle distance d. A linear interpolation of the experimental data (current density) was used for integration in eq 23. Since the theory developed
6068 Langmuir, Vol. 13, No. 23, 1997
Solomentsev et al.
above is linear in the applied electric field E, the plot in Figure 11 can be used to calculate the average time T(d) required for a cluster to separate to the state when the mean interparticle distance reaches the value d. T(d) can then be compared to the experimentally observed time TE(d). Equation 22a was obtained assuming that the parameter q ) 1. To account for the hindrance effect of the wall on clustering/declustering time, eq 22a has to be modified. Since vr is proportional to q, eq 22a becomes
T(d) )
2.19 (d - 2) exp(0.277d) q
(24)
Allowing q to be the only adjustable constant, eq 24 was fitted to averaged experimental data for TE(d). Such a fit is reasonable, since the distance between the particle and the electrode was not measured in the experiment and there is no theory to predict q(h). As discussed above (see eq 9), we expect the fitted value of q to be O(1) but less than 1. A comparison of the values of T(d) calculated from eq 24 to experimental data through the parameter TE(d) is presented in Figure 18. Note that the vertical bars shown in the plot are standard deviations of Monte-Carlo simulations (10-15 trials for randomly placed triplets). The data for the applied voltages of -1.0 and -1.5 V in Figure 18 are superimposable, as expected from the linear dependence on the electric field. By fitting eq 24 to the experimental data, we obtained q ) 0.50. The calculations for linear shear flow by Goldman et al.14 show q ) 0.5 and
0.6 for h ) 0.01 and 0.03, respectively; thus, our best-fit value of q is reasonable given the expected range of h. Conclusions The analysis presented here shows that the effects observed during the electrophoretic deposition of colloidal particles on electrodes under dc conditions4 can be explained in terms of electroosmotic flows around the particles near the electrode. These effects include clustering of single particles in the vicinity of the surface only, formation of patterns on the surface, long range interactions, reversible declustering of aggregates, the twodimensional structure of the deposited particles, and cluster-to-cluster agglomeration. A hydrodynamic theory accounting for these flows is developed. There is good quantitative agreement between the predictions of this theory and the experimental data for the clustering of single particles into triplets, if the hindrance effect of the wall is taken into account. We conclude that, near the electrodes, convection caused by electroosmosis about deposited particles leads to clustering of the particles and ultimately to formation of patterns. The observed destruction of the patterns and clusters is also the result of electroosmotic flows when the polarity of the electric field is reversed. Acknowledgment. This work was partially supported by the National Science Foundation under Grant CTS-9420780. LA970294A
