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Aggregation Dynamics for Two Particles during Electrophoretic Deposition under Steady Fields Yuri Solomentsev,† Scott A.Guelcher,‡ Michael Bevan,§ and John L. Anderson*,| Colloids, Polymers and Surfaces Program, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, Advanced Products Research and Development Laboratory, Semiconductor Products Sector, Motorola, 3501 Ed Bluestein Boulevard, Austin, Texas 78721, Bayer Corporation, Polyurethanes Division, South Charleston Technical Center, Building 740, South Charleston, West Virginia 25303, and Particulate Fluid Processing Centre, School of Chemistry & Department of Chemical Engineering, University of Melbourne, Victoria, 3010, Australia Received April 5, 2000. In Final Form: September 5, 2000 The motion of particles deposited on an electrode by electrophoresis is governed by electrokinetics, electrohydrodynamics, and Brownian diffusion. Under a dc electric field, the particles attract each other through their electroosmotic flows, but Brownian diffusion tends to randomize the distribution. Here, we develop a mathematical model for the time evolution of the probability of separation between two deposited particles. Predictions from the model for the mean separation versus time and the standard deviation about the mean separation are compared with experimental data for pairs of polystyrene latex particles deposited on thin-film metallic electrodes. The good agreement in the absence of adjustable parameters indicates that the convective-diffusion analysis based on electrokinetics is the mechanism behind particle aggregation during electrophoretic deposition under dc field conditions, that is, electroosmotic convection drives the particles together and the Brownian motion of the particles tends to separate them.
Introduction Colloidal particles deposited near an electrode have been observed to aggregate and form large, two-dimensional, ordered clusters in the presence of dc electric fields.1-3 Bo¨hmer1 reported several interesting observations regarding the self-ordering process: the aggregation was reversible when the direction of the field was reversed; the clusters were stable only when the field was applied, and they dispersed when the field was turned off; and the apparent interaction distance between particles was several particle radii (20-30 µm), which is significantly larger than the range of colloidal forces. Solomentsev et al.4 proposed an electrokinetic model for particle aggregation based on convection caused by electroosmotic flow about the deposited particles. They compared particle trajectories calculated from their model with experimental trajectories for the aggregation of sets of three particles and found good quantitative agreement using only one adjustable parameter that accounts for the hindrance effect of the electrode surface on the mobility of the deposited particles. Guelcher et al.5 experimentally studied the timedependent aggregation and separation of pairs of particles for different dc electric fields, particle sizes, and zeta potentials. The aggregation trajectories of pairs of de* To whom correspondence should be addressed. † Motorola. ‡ Bayer Corporation. § University of Melbourne. | Carnegie Mellon University. (1) Bohmer, M. Langmuir 1996, 12, 5747. (2) Trau, M.; Saville, D. A.; Aksay I. A. Science 1996, 272, 706. (3) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 6375. (4) Solomentsev, Y.; Bohmer, M.; Anderson, J. L. Langmuir 1997, 13, 6058. (5) Guelcher, S. A.; Solomentsev, Y.; Anderson, J. L Powder Technol. 2000, 110, 90.
posited particles from an initial center-to-center separation of 3.5 particle radii were recorded using optical microscopy, and the video frames were analyzed to compute values of separation versus time for each set of particles. Their data show that, on average, the relative velocity between the pairs of particles is proportional to the electric field and the zeta potential of the particles and scales with the particle size, as predicted by the electrokinetic theory. In this paper, we present an analysis of two-particle dynamics on the surface of an electrode when a dc electric field is applied. The relative movement of the particles is determined by convective entrainment of each particle by the other (electroosmotic flow), secondary electrophoresis caused by the disturbance to the electric field by the particles, and random Brownian motion (two-dimensional diffusion). The objective of the analysis is to find the statistical mean separation and the standard deviation about the mean for a pair of particles as a function of time. In the next section, we develop the convectivediffusion model for a pair of deposited particles. The important dimensionless parameter is a Peclet number based on the particle size and the strength of the electroosmotic flow. Then, the model is used to predict the mean separation between the particles and the standard deviation versus time. Finally, predictions from the theory in the absence of adjustable parameters are compared with experimental data for ensembles of trajectories.5 Our studies of the Brownian motion of deposited particles are summarized in the Appendix. Convective-Diffusion Model for Two Particles The geometry of the two particles and the electrode is shown in Figure 1. The gap h between each particle and the surface is assumed to be the same and constant with time; the value of h is determined by external forces (electrophoretic, gravitational) and colloidal forces (repulsive). The particle size (a ) radius) is typically of order
10.1021/la0005199 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/03/2000
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Figure 1. Two equal size particles deposited on an electrode. Both particles are mobile with a two-dimensional Brownian diffusion coefficient D ) qD∞, where D∞ is the diffusion coefficient of each particle in bulk solution. The gap h is assumed to be the same for both particles.
Figure 3. Vr as a function of the dimensionless separation (r) from particle 1. h ) 0.05, solid curve; h ) 0.03, dashed curve.
caused by the electroosmotic flow (superscript H) and secondary electrophoresis (superscript E) about particle 1, is given by
U2r ) q(h)Vr; Vr ) VrH + VrE Figure 2. Electroosmotic flow around a spherical particle held stationary near an electrode. If the electrode is positive and the particle is negative, then liquid is drawn toward the particle near the electrode and pushed upward by electroosmosis caused by the electric field acting on the positive diffuse charge of the double layer about the particle. If the direction of the electric field is reversed, the streamlines are the same, but the direction of flow is reversed.
micrometers, and the particles have two-dimensional Brownian mobility characterized by a diffusion coefficient. Our objective is to derive and solve the equation that yields the mean value of the center-to-center separation (r) versus time (t). All variables are made dimensionless by scaling factors; length by a, velocity by U0, and time by a/U0, where the velocity scale is the electrophoretic velocity of the particle far from the electrode.
U0 )
| | ζ E η 0
(1)
In the above equation, E0 is the electric field in the absence of the particle, and η are the permittivity and viscosity of the liquid, resepctively, and ζ is the zeta potential of the particles. The particles attract each other because of the electroosmotic flow about each particle, as shown in Figure 2. The flow is driven by the electrokinetic slip velocity on the surface of each particle, which is caused by the electric field acting on the double layer of the particle. Solomentsev et al.4 solved Laplace’s equation for the electrical potential and hence the electric field (E(1)) about a single particle (particle 1) on the electrode and then used this result to solve for the electroosmotic velocity field (v(1)) represented by the streamlines in Figure 2; particle 2 was ignored in this analysis. The velocity v(1) entrains particle 2 and draws it toward particle 1. In addition, there is a component to the field E(1) that moves particle 2 with respect to 1; we call this “secondary electrophoresis”. The resulting velocity of particle 2 (U2r) along the line of centers (unit vector ir),
(
VrH ) v(1) + VrE )
(2)
)
a2 2 (1) ∇v ‚ir 6 2
(3)
(ζηE ) ‚i (1)
2
(4)
r
where q is a wall hindrance coefficient that is a weak function of h. O’Neill6 derived an exact solution for q versus h for a freely rotating sphere translating parallel to a wall; this relationship is approximated by the following equation:7
[
h < 0.025: q(h) ) -
8 ln(h) + 0.9588 15
-1
]
0.01< h < 0.10: q(h) ) 0.2331 + 7.988h 265.5h2 + 6841.0h3 - 1.123 × 105h4 + 1.121 × 106h5 - 6.101 × 106h6 + 1.395 × 107h7
(5)
The subscript 2 in eqs 3 and 4 means that the field variables are evaluated at the position of the center of particle 2 (z ) 1 + h) as if that particle were not there. The expression for VrH is Faxen’s law, which should be viewed as an approximation because it applies to a sphere in an unbounded fluid.8 Note that VrH brings the particles together, and VrE moves them apart; the magnitude of VrH is about 7 times that of VrE when r ≈ 3. Figure 3 shows the function Vr for two values of h; this function is relatively insensitive to h for h < 0.1. Figure 4 shows the migration velocity of particle 2 (Ur2 ) qVr). An empirical fit to Vr for h ) 0.05 that is accurate over the range 2 < r < 11 is given by (6) O’Neill, M. E. MathematikaI 1964, 11, 67. (7) Guelcher, S. A. Investigating the Mechanism of Aggregation of Colloidal Particles during Electrophoretic Deposition. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 1999. (8) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: The Hague, The Netherlands, 1983.
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Solomentsev et al.
Pe )
aU0 D∞
(9)
where D∞ is the diffusion coefficient of a single particle in an unbounded fluid (D∞ ) kT/6πηa). The first term in brackets in eq 8 results from diffusion of the particles, and the second from electroosmotic flow and secondary electrophoresis. Note that the same hindrance coefficient q multiplies both the diffusive (D ) qD∞) and the electrokinetic (Ur2 ) qVr) terms, and the factor of 2 is needed to account for the effects of each particle on the other. Consistent with the approximation adopted for calculating the convective particle velocity, we have neglected hydrodynamic interactions between the particles when expressing the diffusion coefficient but account for the interactions between each particle and the electrode (coefficient q). The probability function must be normalized. Figure 4. Hindered velocity of particle 2 (U2r ) qVr) as a function of the dimensionless separation (r) from particle 1. h ) 0.05, solid curve, q ) 0.38; h ) 0.03, dashed curve, q ) 0.35. H
Vr ) -0.17019r
-3
-4
- 44.669r
+
-5
167.60r
- 162.44r-6
VrE ) -0.01590r-3 + 3.9480r-4 -
Because the electroosmotic flow is approximated at large distances from the particle by a point force directed perpendicular to the electrode surface, we expect vr(1) (and hence VrH) to decay as r-3 when r f ∞.9 Log-log plots over the range 5 < r < 11 show that VrH ≈ -6.863r-3.33 and VrE ≈ 0.8987r-3.54. We have assumed that the electrode surface produces no electroosmotic flow. This assumption is valid for an ideal electrode of constant potential, because the electric field along the surface of the electrode (z ) 0) is zero. However, nonuniform current distributions caused by defects and adsorbed layers of charged molecules (e.g., contamination) could produce an electric field along the surface. Because any model of this effect would require some details of the kinetics of charge transfer at the electrode surface, we do not consider this effect. In addition to electroosmotic convection and secondary electrophoresis, the particles on the electrode experience Brownian diffusion. To account for coupling between convective and diffusive transport, we define a pair probability function F(r,t) as the probability that the center-to-center distance between the two particles is between r and r + dr. The distribution function must satisfy the conservation relation
(7)
∫2∞Fr dr ) 1
(10)
As r f ∞, the electroosmotic convection becomes negligible, leaving Brownian motion to randomize the distribution. Thus, at large separations F must vanish.
(6)
12.0092r-5 + 9.8817r-6
1 ∂ ∂F )(rJ ) ∂t r ∂r r
2π
r f ∞: F f 0
(11)
When the particles are in contact, a no-flux boundary condition is appropriate because the particles do not coagulate and cannot overlap.
r ) 2: Jr ) 0
(12)
The initial condition is determined by defining t ) 0 for each aggregation cycle as the time when the probability function is known.
t ) 0: F ) F0(r)
(13)
The model above is a statistical description of the dynamics of two particles. If the time evolution of the probability density function F(t,r) is known, then measurable quantities such as the mean separation µ(t) and the standard deviation σ(t) over many trials at the same conditions (Pe) can be calculated. The quantities µ(t) and σ(t) are defined as follows:
∫2∞rFr dr
(14)
∫2∞r2Fr dr - µ2(t)]1/2
(15)
µ(t) ) 2π σ(t) ) [2π
These quantities can be calculated from experimental data5,7 and compared to predictions based on the solution of eqs 7-13. Predictions from the Model
where Jr is the flux of probability
[
Jr ) 2q -
1 ∂F + VrF Pe ∂r
]
(8)
and the Peclet number is defined as (9) Pozrikidis, C. Boundary Integral and Singularity Methods for Linearized Viscous Flow; Cambridge University Press: new York, 1992.
Electroosmosis, electrophoresis, and Brownian motion are competing forces, the former tending to bring the particles together and the latter tending to randomize and hence separate them on average. By multiplying eq 7 by r2 and integrating over the range 2 < r < ∞, we obtain the following equation for the mean separation:
∫2∞r-1Fr dr + 2π∫2∞VrFr dr]
2π dµ ) 2q dt Pe
[
(16)
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In deriving the above, we set F ) 0 at r ) 2; this is justified at early times. Given the initial separation r0 as a known quantity, we have
t ) 0:
[
]
dµ 1 ) 2q + Vr(r0) dt r0Pe
(17)
A critical initial radius (rc) can be defined such that, when r0 > rc, the mean separation grows with time, meaning that, on average, the pair of particles will separate rather than aggregate.
-rcVr(rc) )
1 Pe
(18)
For example, when Pe ) 10, then rc ) 6. Indeed, there were occasions in the experiments when two particles slowly moved away from each other when r0 was large.7 We solved eqs 7-13 numerically subject to the following initial condition:
t ) 0: F0(r) )
[
]
(r - r0)2 2 exp π 2σ 2
x
{ [ ( )]
σ0 r0 1 + erf
r0
x2σ0
0
+
( )}
r02 2 σ exp π 0 2σ02
x
(19)
Figure 5. Scaled mean separation (µ) of aggregating pairs of particles as a function of dimensionless time (t), as predicted from the theory. Dashed curve, Pe ) 100; dotted curve, Pe ) 300; solid curve, Pe ) 500. Note that, for Pe > 100, the mean separation is essentially unaffected by Brownian motion and eq 22 applies.
Also, the boundary condition at infinity was approximated as
r ) 11: F ) 0, Jr ) 0
(20)
The initial condition is a Gaussian distribution; we used a small standard deviation (σ0 ) 0.01). Trials with different (small) values of σ0 show that the statistical results are essentially independent of this parameter. The standard Crank-Nicholson algorithm10 was used to solve the convective-diffusion problem. The initial mean (r0) was always taken as 3.5 times the particle radius. Figure 5 shows the predicted mean separation versus time. The trajectories are essentially independent of Pe for Pe > 100 and are given simply by integration of the mutual approach velocity 2Ur2. Figure 6 shows the standard deviation versus time for three values of Pe; these plots are essentially independent of Pe at values of 100 or greater. The following expression is a good fit to the curves in Figure 6, except near t ) 0:
Pe1/2σ ) 2.15(qt)0.526
(21)
Experiments 5,7
Guelcher et al. have reported experimental results for the trajectories of pairs of polystyrene latex spheres deposited on thin-film metal electrodes, either gold or tindoped indium oxide (ITO). The particles were polystyrene latex spheres in aqueous surfactant-free solutions; the particle surfaces were modified by the supplier with carboxylate groups to impart a negative charge. Particles with diameters in the range of 2.5-10 µm were studied with applied dc electric fields in the range of 20-100 V/m. The electrophoretic mobilities of the particles were measured in a microelectrophoresis apparatus;7 U0 was (10) Press, W. H.; Flannery, B. P.; Teukolsky S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: New York, 1989.
Figure 6. Scaled standard deviation about the mean trajectory as a function of dimensionless time, as predicted from the theory. Three values of Pe (100, 300, and 500) were used. Equation 21 fits the curves, which are almost indistinguishable on this graph.
calculated from the mobility and the electric field of the aggregation experiment. The separation versus time for each pair was determined by video microscopy and automated image analysis. An example of photomicrographs of a pair of particles aggregating to form a doublet is shown in Figure 7. For each set of experimental conditions, a total of 10 aggregation trajectories were measured. The complete set of data of Guelcher et al. show that the relative velocity between the particles was proportional to the electric field and zeta potential of the particles, and the length scale of the mean trajectories was proportional to the particle size. However, there was considerable scatter among the trajectories; an example of an ensemble of 10 trajectories of 10 pairs of particles at identical conditions is shown in Figure 8. Here, we quantitatively test our convective-diffusion model for two-particle aggregation against the data of
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Solomentsev et al. Table 1. Comparison of q Calculated from TIRM Measurements of h (see eq 5) and Brownian Diffusion (BD) Experimentsa systemb
BD q j ( σq
TIRM q j ( σq
h (TIRM)
6.3 µm diameter/0.1 mM 0.342 ( 0.108 0.378 ( 0.016 0.0458 NaHCO3/ITO electrode (146 nm) 6.3 µm diameter/1.0 mM 0.377 ( 0.064 NaHCO3/gold electrode 9.7 µm diameter/0.1 mM 0.365 ( 0.010 0.0376 NaHCO3/ITO electrode (182 nm) a q j and σq are the mean value and the standard deviation of q over the set of the experiments. b The particles were carboxylatemodified polystyrene latex. See ref 7 for details.
Figure 7. Photomicrographs of a pair of particles aggregating. The particles are 9.7-µm-diameter carboxylate-modified polystyrene latex spheres (ζ ) -65 mV) on a gold electrode in contact with 1.0 mM NaHCO3. The electrode was positive, and the electric field was 80 V/m directed into the solution.
Table 2. Results from the Lateral Diffusion Experiments with 6.3-µm-Diameter Latex Particles (carboxylated surface) on a Gold Electrode in Contact with 1.0 mM NaHCO3 particle
Dx µm2 s-1
Dy µm2 s-1
qx
qy
NΞ
1 2 3 4 5 6 7
0.0258 0.0371 0.0260 0.0354 0.0249 0.0289 0.0054
0.0267 0.0306 0.0232 0.0376 0.0250 0.0291 0.0114
0.333 0.478 0.335 0.457 0.321 0.373 0.0694
0.345 0.395 0.300 0.485 0.323 0.375 0.147
790 790 790 790 790 790 627
Table 3. Summary of Results from Table 2a x direction y direction all
mean q
standard deviation
0.383 0.370 0.377
0.068 0.066 0.064
a The result for particle 7 was not included in the averages as it was obviously stuck to the surface.
Figure 8. Separation between pairs of particles aggregating to form a doublet as a function of time for 9.7-µm-diameter particles (ζ ) -65 mV) deposited on a gold thin-film electrode in contact with 1.0 mM NaHCO3. Each symbol shape represents one trajectory, and the data are for the same conditions (particle type, electric field, solution ionic strength and electrolyte, and electrode material). The electrode was positive, and the electric field was 60 V/m. Note that each of the 10 trajectories is for a different pair of nominally identical particles.
Guelcher and co-workers.5,7 One major issue is whether the scatter in the trajectories, as shown in Figure 8, can be quantitatively explained by Brownian motion of the particles or whether must one invoke other semi-random events such as convection due to concentration polarization of the electrodes3 or defects on the surface of the electrode. We focus on one complete set of data for 9.7-µm carboxylated latex spheres on a gold electrode in 1.0 M NaHCO3. Data for other particles and electrode surfaces are available;5,7 although not as complete as the data examined here, these other data are quantitatively consistent with the theory and support the conclusions reached in this paper. Determination of The Hindrance Parameter (q). The particle-electrode separation distance h is an important parameter because it determines the value of q. It is important to determine q independently of the aggregation experiments to test the theory and to ascertain that h is sufficiently small ( 3.525 were neglected in the calculations. This modification of the trajectories is justified because the history of the separation should not be important; that is, we are not concerned about how the pair of particles reached a separation of 3.5 particle radii. The mean and standard deviation of the separation were calculated for the ensemble of pairs at each value of time. As seen in Figure 5, diffusion is predicted to have a negligible role in determining the mean separation versus time when Pe > 100. Therefore, we expect, under this condition, that the scaled mean trajectories measured at different electric fields for a given system should be determined by integrating the following equation:
dµ ) 2qVr(µ) dt
(22)
The unscaled experimental mean trajectories for 9.7µm-diameter particles deposited on a gold electrode in contact with 1.0 mM NaHCO3 are presented in Figure 9. In the experiments, the electric field ranged from 20 to 100 V/m. The same data are plotted as scaled variables in Figure 10. The solid line was calculated from the theory (eq 22) using q ) 0.377, the mean value for 6.3-µm particles on gold determined from the lateral diffusion measurements (see Table 1). The dotted lines were calculated using the mean value plus or minus one standard deviation of q (σq ) 0.064). For separations greater than about 3 particle radii, the agreement between the experimental and calculated trajectories is good given that there are no adjustable parameters in the theory. For smaller separations, the pairs of particles approach each other faster than predicted by our convective-diffusion theory. This is probably due to neglect of direct two-particle electrostatic and hydrodynamic interactions (i.e., due to the approximation of linear superposition of single-particle
Figure 11. Scaled standard deviation (σ) for the mean trajectories shown in Figure 9. The solid line is eq 21 with q ) 0.377, while the dotted lines are for q ) 0.377 ( 0.064.
fields), because the trajectories still scale5 with a and U0 even though the experimental trajectories deviate from the model. Comparison of Standard Deviations. The scaled experimental standard deviations of the mean trajectories shown in Figure 9 are plotted in Figure 11. The experimental standard deviation at t ) 0 is not zero because of errors associated with controlling the initial positions of the particles; that is, at t ) 0, the separation ranged from 3.475 to 3.525. The electric field varied from 20 to 100 V/m, corresponding to Peclet numbers in the range of 98488. The theoretical predictions (solid curve) were made from the calculations plotted in Figure 6 (see eq 21) using the mean experimental value q ) 0.377. The dotted lines were calculated using q ) 0.377 ( 0.064 (one standard deviation). For the fields of 20-60 V/m (Pe ) 98-261),
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the experimental and calculated standard deviations are comparable, confirming our hypothesis that Brownian motion is the most important source of the observed scatter in the individual pair trajectories. However, for fields greater than 60 V/m, the experimental standard deviations are greater than the theoretical predictions. Similar results were observed with smaller (6.3-µmdiameter) particles.7 In general, for Pe e 300, the experimental and calculated standard deviations are comparable, indicating that Brownian motion is the most important source of the scatter in trajectories. For Pe > 300, the experimental and calculated standard deviations are comparable for dimensionless times t < 0.7-3.5, depending on the magnitude of Pe (and hence the electric field), but at longer times, the experimental standard deviations consistently became larger than the values predicted by the model. Moreover these discrepancies occurred earlier for higher Peclet numbers (higher current densities). It is possible that the increase in the experimental standard deviation at higher Peclet numbers and longer times can be attributed to electrohydrodynamic convection3, which becomes stronger as the field strength increases and ion concentration gradients become significant. Concentration gradients of charge-transfer species associated with the electrode reaction should become more significant with increasing current density (which is proportional to the field) and time. Another potential cause of disagreement between theory and experiment is the standard deviation of the particle size (7%). To minimize this effect, only pairs of particles that appeared to be the same size, as resolved optically under the microscope, were chosen for the experiments, so the uncertainty due to polydispersity within any one pair is on the order of the uncertainty in the initial separation. However, variations in size between pairs of nominally the same particles could have been more significant and might account for some of the variation of the data in Figure 11 for the lower values of Pe. Guelcher7 discusses more thoroughly the role of the particle size distribution. Conclusions The convective-diffusion model developed here for the evolution of the probability function for the separation of a pair of equal spherical particles on an electrode, given by eqs 7-13, fits the experimental data of Guelcher and co-workers5,7 in terms of the ensemble-averaged separation (µ) and standard deviation (σ) versus time. All of the parameters in the modelsparticle size and zeta potential, the convective field Vr, and the hindrance parameter qs were measured or calculated independently. This work, along with the scaling analysis by Guelcher et al.,5 provides strong evidence that particle aggregation during electrophoretic deposition under dc fields is driven by electrokinetics, primarily electroosmotic convection. Agreement between theory and experiment for the standard deviation of separation versus time (Figure 11) also demonstrates that Brownian motion is the dominant factor in the scatter observed between trajectories under the same conditions (see Figure 8). However, the fact that the experimental standard deviations of the separation are greater than the theoretical predictions at longer times implies that there is another driving force for scatter in addition to Brownian motion, possibly electrohydrodynamic convection caused by concentration polarization at the electrodes3. Acknowledgment. This work was partially supported NSF Grant CTS-9814064 and the Philips Research
Solomentsev et al.
Laboratories, Eindhoven, The Netherlands. S.G. acknowledges financial support through NASA Training Grant NGT5-50054. The authors thank Darrell Velegol of Pennsylvania State University for supplying the Scion Image macro used to measure the particle separation distances. S.G. thanks Esther de Beer, Marcel Bo¨hmer, Bart Fokkink, and Eric Meulenkamp for their assistance with electrochemistry and pair aggregation experiments during his internship at Philips Research Laboratories in Eindhoven, The Netherlands. Appendix: Determination of the Gap (h) and the Hindrance Coefficient (q) for a Deposited Particle An important parameter of the theory for particle aggregation is the hindrance coefficient q, which is a measure of the mobility of a particle in the plane parallel to the electrode’s surface. Here, we describe two methods for determining this coefficient: direct measurement of the gap (h) by TIRM, from which q can be computed using published hydrodynamic results, and direct measurement of q by study of the two-dimensional Brownian motion of isolated particles. The particles were surfactant-free polystyrene latex spheres of diameter 6.3 or 9.7 µm with carboxylated (negative) surfaces, the same as used in the two-particle aggregation experiments. Total Internal Reflection Microscopy. Total internal reflection microscopy (TIRM)11 experiments were performed to determine the gap between a deposited particle and the ITO-coated glass electrode. The deposited particle was illuminated by an evanescent wave produced at the electrode/solution interface with the same geometry shown in Figure 1. By measuring the intensity of the light scattered from the particle in the evanescent wave, the instantaneous gap between the particle and the electrode can be measured with exponential sensitivity using the equation12
I(h) ) I0 exp(-βh)
(23)
where β is defined as
β)
4π (n sinθ)2 - n22 λx 1
(24)
and I0 is the scattered intensity for contact between the particle and the electrode (h ) 0), n1 is the refractive index of the glass substrate of the ITO electrode, n2 is the refractive index of the electrolyte solution, θ is the angle of incidence, and λ is the wavelength of the incident laser. TIRM was used to measure the Brownian excursions of deposited particles normal to the electrode surface. These height excursions occur relative to the most probable separation, hm, which is the height at which the sum of the surface and gravitational forces on the particle are equal to zero (condition for mechanical equilibrium or thermodynamic energy minimum). In the absence of an applied field, electrostatic forces between the particle and the wall act to to push the particle away from the electrode, while a gravitational force pushes it toward the electrode. No van der Waals interactions were detected in these measurements. The most probable separation, hm, is determined from the maximum in the measured distribution of heights, p(h), which is related to the minimum in the particle/electrode potential energy profile, Φ(h), by Boltzmann’s equation (12) Chew, H.; Wang, D. S.; Kerker, M. Appl. Opt. 1979, 18, 2679.
Aggregation Dynamics during Electrophoretic Deposition
p(h) ) A exp
(
)
-Φ(h) kT
Langmuir, Vol. 16, No. 24, 2000 9215
(25)
where A is a normalization constant and k is Boltzmann’s constant. Experimentally, p(h) is constructed from a histogram of intensity measurements and hm is determined using eqs 23 and 24 and I(hm), which is the intensity associated with the maximum of the product N(I) × I(h), where N(I) is the number of times each intensity is sampled in the intensity histogram. The values for hm were then used to calculate values of the hindrance parameter q. The value of hm for 6.3-µm particles in 0.1 mM NaHCO3 was 0.0458 (or equivalently, 146 nm in dimensional terms). The value of the hindrance parameter at the potential minimum, q(hm) ) 0.378, was calculated from the solution developed by O’Neill.6 We also calculated the mean value of q defined by
〈q〉 )
∫0∞q(h) p(h) dh
(26)
The differences between 〈q〉 and q(hm) are insignificant here. Lateral Diffusion Experiments. The hindrance parameter was measured directly by observing the Brownian motion of single particles in the plane of the electrode. From these data, the two-dimensional diffusion coefficient (D) of the particles was determined. The hindrance parameter was then calculated using
q ) D/D∞
tE - tI tF
(∆x)i+1 ) xi+γ - xi for i ) 0 to N∆x - 1
where the subscript denotes the frame number and γ ) tI/tF. The mean displacements were calculated from
∆x )
1
N∆x
∑
N∆xk)1
(∆x)k
∆y )
1
N∆y
∑ (∆y)k
N∆yk)1
(30)
If Brownian motion were the only contribution to particle motion and the surface were isotropic, then the mean displacements would be zero. However, because it was difficult to level the cell, the particles were often observed to “roll” along the electrode under the influence of gravity. If the nonzero mean displacement is assumed to result from rolling, then the velocities of the particle in the x and y directions are given by
Ux )
∆x tI
Uy )
∆y tI
(31)
To obtain the Brownian component of the displacements, we define the relative displacements as
Ξk ≡ ∆xk - ∆x
Πk ≡ ∆yk - ∆y
(32)
The sample variances sΞ2 and sΠ2 of the relative displacements in the x and y directions are given by
(28)
For a 20-min experiment (tE ) 20 min) with frames grabbed every tF ) 1.5 s and a time interval of tI ) 22.5 s, N∆x ) N∆y ) 785 displacements. From the image analysis of the videotapes, a set of data was obtained for the location of the particle center versus time. The displacements in the x and y directions were calculated using the following equations:
(29)
(∆y)i+1 ) yi+γ - yi for i ) 0 to N∆y - 1
(27)
where D∞ is particle’s diffusion coefficient in the bulk solution, which is given by the Stokes-Einstein equation (D∞ ) kT/6πηa). The concept of the lateral diffusion experiments is illustrated in Figure 12. These experiments were performed in the cell used to measure the two-particle aggregation trajectories. The cell was filled with a dilute particle suspension and the particles were allowed to settle for 10-20 min until a sufficient number had deposited on the electrode. One particle distanced at least 10 radii from other particles was located optically, and its motion was recorded on a videotape for 20 min. This procedure was repeated as necessary to obtain tapes of the motion of five or more particles. From the videotapes, frames were “grabbed” every tF seconds, and the location of the center of the particle was determined for each frame. The displacements in the x and y directions (∆x, ∆y) over successive time intervals of duration tI were determined. For example, frame 0 was grabbed at time 0 s, frame 1 at tF s, ..., frame 9 at 9tF s, frame 10 at 10tF s, etc. The table of displacements in the x direction was generated as follows: (∆x)1 ) x(tI) - x(0), (∆x)2 ) x(tI + tF) - x(tF), (∆x)3 ) x(tI + 2tF) - x(2tF), etc. The total number of displacements N∆x ) N∆y were calculated from the following equation:
N∆x )
Figure 12. Measurement of displacements due to Brownian motion in the plane of the electrode: (a) side view, (b) bottom view. The displacement of the center of the particle in the x and y directions were measured over successive time intervals of duration tI (22.5 s).
sΞ2 )
1
∑ k)1
N -1 Ξ
NΞ
(Ξk - Ξ h )2
sΠ2 )
1
(Πk - Π h )2 ∑ k)1
N -1 Π
NΠ
(33)
The mean relative displacements Ξ h and Π h are zero according to eq 32. In the limit of a large number of observations (frames), the sample standard deviations sΞ and sΠ approximate the population standard deviations; therefore
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Langmuir, Vol. 16, No. 24, 2000
Solomentsev et al.
Figure 13. Histogram for the system 6.3-µm CML/1.0 mM NHCO3/gold. Key to symbols: circles, statistical relative frequency calculated from the experimental data; triangles, probability calculated assuming a normal distribution; line, fit to the calculated probabilities.
Dx )
sΞ2 2tI
Dy )
sΠ2 2tI
(34)
Most of the displacement distributions followed a Gaussian distribution, as expected. A typical distribution is shown in Figure 13. More details of the statistical analysis are given by Guelcher.7 The average value of the hindrance parameter q is defined as
q(h) )
(
)
1 1 Dx Dy + ) (qx + qy) 2 D∞ D∞ 2
(35)
The experimental results for 6.3-µm particles on a gold electrode in contact with 1.0 mM NaHCO3 are provided in Tables 2 and 3. The mean mobility of 0.377 reported
for our particles is comparable to the values of 0.505 and 0.224 reported by Dabros et al.13 for 3.4- and 5.1-µm polystyrene latex spheres, respectively, deposited on a glass slide. Note that particle 7 had an anomalously low value of the diffusion coefficient. Such sluggish particles, some totally stationary, indicating that they were “stuck” to the electrode, were occasionally observed. A comparison between the values of q determined from TIRM and the lateral diffusion experiments is made in Table 1. The agreement is very good. The following conclusions can be drawn from the results of these two methods in the absence of an applied electric field: (1) the mobile particles deposited on the electrode experienced random Brownian motion; (2) the two-dimensional diffusion coefficient was hindered by the electrode such that it was 30-40% of the bulk diffusion coefficient, corresponding to a gap h e 0.05; and (3) there was no statistically significant difference between the mean values of q in orthogonal directions, indicating that the electrode surface was locally isotropic. Lateral diffusion coefficients were also determined when an electric field was applied.7 First, q was determined without the field, and then the electrode was polarized either positive (corresponding to our two-particle aggregation studies) or negative (corresponding to separation of pairs of particles).5 The cell was allowed to relax for about 3-5 min between electrode polarization steps so that concentration gradients established by the electrochemical reactions would dissipate. A statistical analysis of the results indicates that the electric field had an insignificant effect on the value of q when the electrode was positive but did have a small effect when it was negative. Because we report here only two-particle trajectories when the electrode was positive, we conclude that the electric field driving the particle aggregation had a negligible effect on q. LA0005199 (13) Dabros, T.; Warzynski, P.; van de Ven, T. G. M. J. Colloid Interface Sci. 1994, 162, 254.
