Langmuir 1999, 15, 2865-2870
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A Study of Silica Nanoparticle Adsorption Using Optical Reflectometry and Streaming Potential Techniques Robert A. Hayes,*,† Marcel R. Bo¨hmer, and Lambertus G. J. Fokkink Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands Received June 8, 1998. In Final Form: December 31, 1998 The kinetics of adsorption of monodisperse nanosized silica particles (32-190 nm diameter) on silicon wafers modified with an aminopropylsilane have been studied by optical reflectometry under stagnantflow conditions. The adsorption of negatively charged particles on a positively charged wafer surface is driven by electrical double layer attraction and was not found to be measurably reversible. The initial rate of particle adsorption decreases with increasing particle size while the equilibrium amount adsorbed increases with electrolyte concentration. The effect of particle adsorption on the ζ potential of similarly modified glass slides was measured using streaming potential apparatus. The coverage of particles was varied using electrolyte as indicated by the reflectometry measurements. The results suggest that displacement of the plane of shear is not localized to particle covered areas.
Introduction In a number of industrial processes, particle adsorption plays a key role. In mineral processing and wafer cleaning particle attachment is deleterious and must be reversed. In the fabrication of television screens, the controlled adsorption of particles is desirable. Small silica particles are used as coatings of the phosphor particles which are applied on the inside of the screen. The coatings greatly improve the handling of the phosphor. However particle adsorption processes have been relatively understudied in comparison to molecular adsorption. Matijevic and coworkers have used a packed column method to study the adsorption1 and subsequent desorption2-4 of monodisperse metal oxide colloids on glass and steel surfaces. Healy and co-workers have applied their heterocoagulation model5 to the adsorption of mixed oxide systems.6 A stagnant-flow method for the study of particle adsorption in the absence of hydrodynamic factors has been proposed by a number of authors.7,8 This geometry is generally combined with optical microscopy to obtain quantitative information regarding the kinetics of particle adsorption.9 The adsorption of nanosized particles, which are beyond the resolution of optical microscopy, has been monitored by reflectometry.10-12 The effect of adsorption of charged particles on an * Current address of corresponding author: Philips Research, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. E-mail:
[email protected]. † Former address: Ian Wark Research Institute, University of South Australia, Mawson Lakes, SA 5095, Australia. (1) Kallay, N.; Nelligan, J. D.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1983, 79, 65. (2) Kolakowski, J. E.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1979, 75, 65. (3) Kuo, R. J.; Matijevic, E. J. Chem. Soc., Faraday Trans. 1979, 75, 2014. (4) Kuo, R. J.; Matijevic, E. J. Colloid Interface Sci. 1980, 78, 407. (5) Hogg, R.; Healy, T.W.; Fuerstenau, D. W. J. Chem. Soc., Faraday Trans. 1966, 62, 1638. (6) Healy, T.W.; Wiese, G. R.; Yates, D. E.; Kavanagh, B. V. J. Colloid Interface Sci. 1973, 42, 647. (7) Dabros, T.; Van de Ven, T. G. M. Colloid Polym. Sci. 1983, 261, 694. (8) Adamczyk, Z.; Warszynski, P. Adv. Colloid Interface Sci. 1996, 63, 41. (9) Marston, N.; Vincent, B. Langmuir 1997, 13, 14. (10) Boonekamp, E. P. Ph.D. Thesis, University of Utrecht, 1994. (11) Bo¨hmer, M. R. J. Colloid Interface Sci. 1998, 197, 251. (12) Bo¨hmer, M. R.; van der Zeeuw, E. A.; Koper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242.
oppositely charged surface may be assessed by electrokinetic measurements. It is generally assumed that the shear plane, at which the ζ potential is sampled is displaced from the solid - liquid interface by an adsorbed ion diameter or a distance not exceeding a nanometer.13 While this is logical for surfaces that have subnanometer roughness, one may question whether it is the case for more physically heterogeneous, or morphologically complex, surfaces. The systematic effect of roughness on electrokinetic behavior has, to the best of our knowledge, not been reported. In this work, the adsorption of nanosized monodisperse silica particles on an aminopropylamine modified silicon wafer immersed in aqueous electrolyte is the focus. The wafer is maintained at pH 5.6 where the surface is positively charged. The adsorption behavior is mitigated by the attractive double layer forces between oppositely charged particles and substrate and double layer repulsion between adjacent adsorbing particles. An elegant demonstration of this effect can be found in the centrifugation14,15 and, more recently, scanning electron microscopy16 (SEM) studies of Vincent and co-workers. In the current investigation the effect of particle size and electrolyte concentration on the kinetics of adsorption are studied by reflectometry. The effect of particle adsorption on the electrokinetic behavior is studied by streaming potential measurements. If the electrolyte concentration is varied during deposition, the coverage of particles can be controlled. After free particles are flushed away, the streaming potential was measured at different ionic strengths. Experimental Section Materials. The silica samples used in this work were Ludox AS40 (Dupont) and Monospher 25, 100, and 200 (Merck). The Ludox sample was dialyzed before use. Particle sizes were obtained by photon correlation spectroscopy using a PCS system with an ALV5000 correlator. The particle radii (rp) were 16, 20, 45, and 95 nm, respectively. All electrolyte and particle/electrolyte (13) Hunter, R. J. ζ potential in colloid science, Academic Press: London, 1988. (14) Vincent, B.; Young, C. A. J. Chem. Soc., Faraday Trans. 1 1980, 76, 665. (15) Vincent, B.; Jagiello, J.; Luckham, P. F.; Tadros, T. F. J. Chem. Soc., Faraday Trans. 1 1980, 76, 674. (16) Harley, S.; Thompson, D. W.; Vincent, B. Colloids Surf. 1992, 62, 163.
10.1021/la980668f CCC: $18.00 © 1999 American Chemical Society Published on Web 03/23/1999
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Figure 1. Homogeneous slab model17 used to calculate the reflection coefficients, Rp and Rs, using tabulated values of the optical constants.12 The calculations were made at a wavelength of 632.8 nm and an incident angle of 70°. solutions were made up in reagent grade KNO3 (Merck) in high purity water. Particle concentrations were 100 mg/L unless otherwise stated. For reflectometry the substrates were silicon wafers with a 100 nm thick SiO2 film. For streaming potential measurements glass microscope slides were used. In either case substrates were cleaned by rinsing sequentially in detergent (Extran) solution, water, ethanol and heptane followed by drying in a stream of filtered nitrogen prior to 15 min UV/ozone treatment. The surfaces were then modified by immersion in a solution of bis(aminopropyl)trimethoxysilane (0.15 wt. %) in 2-propanol/water (50/50% by wt.) for 15 min. On removal they were thoroughly rinsed in 2-propanol, dried in a nitrogen stream and stored in aluminum foil prior to reflectometry or streaming potential measurements. Both reflectometry and streaming potential measurements were conducted at pH 5.6 ( 0.2 in a constant temperature room (20 ( 1 °C) in which all solutions were equilibrated overnight. At this pH the surface is positively charged as the isoelectric point (iep) of the aminosilane surface occurs at pH 9.2. Reflectometry. In reflectometry the change in polarization of a linearly polarized laser beam is related to the amount of material adsorbed at the reflecting interface. The intensity ratio, S, of parallel (p) and perpendicularly (s) polarized light
S)
Ip Is
(1)
is measured. The procedure for converting the raw signal to surface coverage of particles is described in detail elsewhere.12 Only the esssential features and modifications to the procedure are described here. The adsorbed amount (Γ) can be expressed as
Γ)
1 ∆S A s So
(2)
where So is the intensity ratio prior to adsorption and ∆S is the change in intensity ratio. The sensitivity factor As is defined by
As )
d(Rp/Rs) 1 dΓ (Rp/Rs)o
dn 1 Γ dc dp
Table 1. Comparison of Constant and Coverage-Dependent Values of the Reflectometry Sensitivity Factor, Asa As ) aΓ+ b (m2/g) dp (nm)
As const (m2/g)
102 a
b
32 40 90 190
9.48 ( 0.03 9.35 ( 0.04 8.1 ( 0.1 3.06 ( 0.07
3.7 ( 0.3 4.8 ( 0.2 5.55 ( 0.08 1.68 ( 0.05
9.18 ( 0.02 8.88 ( 0.02 6.85 ( 0.02 2.28 ( 0.02
a The fitting was done in the refractive index range 1.333-1.370. The intercept of the normalized reflectivity ratio was constrained to 1.0 (Figure 2).
g/cm3 and refractive index of 1.449. A linear dependence of refractive index on the amount of adsorbed silica is assumed. A plot of normalized reflectivity ratio (eq 3) as a function of coverage is readily constructed. Only a limited range of refractive index, and therefore adsorbed amounts, is required because the volume fraction of silica particles will not exceed 0.3. A typical plot, for dp ) 90 nm is reproduced in Figure 2. The reflectivity ratio-Γ relationship is approximately linear and in work to date linearity has been assumed. However deviations from linearity, which highlight limitations of the simple optical model, become more pronounced for large adsorbed amounts. Therefore in this work fitting with a second order polynomial was also investigated (Figure 2). Differentiation of this polynomial yielded a coverage dependent As
As ) aΓ + b
(3)
In this work the reflection coefficients, Rp and Rs, have been calculated assuming a homogeneous slab model (Figure 1). The Abeles’ matrix method,17 applicable to stratified planar systems, has been used. The aminosilane layer, which provides the positive charge to attract and adhere the adsorbing silica particles, is assumed to have negligible thickness. The thickness of the particle layer is set to a single particle diameter, dp. The reflectivity ratio, Rp/Rs, is then calculated as the refractive index is incremented. The amount adsorbed, Γ, is related to the refractive indices of the particle layer, neff, and water, nw, via
neff ) nw +
Figure 2. Normalized reflectivity ratio (eq 3) as a function of adsorbed amount for a silica particle diameter of 90 nm. The symbols (+) correspond to values calculated using a homogeneous slab model and the line to the fitted second-order polynomial. The intercept was constrained to the value of 1. The sensitivity factor, As, is simply obtained by differentiation of Y(Γ).
(4)
where dn/dc ) 0.058 cm3/g for a silica particle density of 2.0 (17) Born, M.; Wolf, E. Principles of optics; Pergamon Press: Oxford, England, 1980.
(5)
A comparison of the As values obtained by the two fitting procedures is made in Table 1. While the differences are minor for small dp the major deviations for larger dp and higher coverages led to the general use of the more refined fitting procedure in the current work. Substitution of As into eq 2 resulted in a quadratic equation from which the adsorbed amount could be directly obtained
Γ)
-b + xb2 + 4a∆S/So 2a
(6)
The particle coverage, θ, simply equals Γ/Γmax where Γmax ) 2Fdp/ 3. In a typical experiment, electrolyte was channelled through the reflectometer cell, the modified wafer was positioned, and the ratio of parallel to perpendicular intensities (S) was monitored to ensure that the baseline, from which So (typically 0.9) is obtained, was steady. So can be adjusted simply by rotating the incident laser beam. Switching of a valve immediately adjacent
Silica Nanoparticle Adsorption
Figure 3. Normalized reflectivity ratio as a function of time in 10-3 M KNO3 at pH 5.6. The inset is an expansion of the initial adsorption phase. to the reflectometer cell then allowed the introduction of particles (the electrolyte concentration was unchanged) with a consequent increase in S. The reflectometer signal was monitored until no further increase was evident, which took between 10 min and 1 h, depending on particle size. At this point particle-free electrolyte solution was then redirected through the cell. No evidence of particle desorption was observed in this work. Flow rates were maintained at 1.25 cm3/min, which corresponded to a Reynolds number of 15.25.7 The angle of incidence of the He/Ne laser with the wafer was set to 70°. Streaming Potential. An Anton Paar EKA, which readily accommodated the surface modified glass microscope slides, was used. The dimensions of the flow channel were 76 × 10 × 0.3 mm3. The reflectometry measurements were used as a guide in choosing the appropriate electrolyte concentration so as to obtain the desired coverage. Of course the difference in flow conditions precludes a strictly quantitative correlation between the two flow geometries. Particle (either 40 or 90 nm diameter)/electrolyte solutions were introduced to the streaming potential chamber and allowed to equilibrate with the modified surfaces for 1 h. Free particles were then flushed from the streaming potential cell with electrolyte. The streaming potential was then measured at a range of electrolyte concentrations. The usual protocol was to move from low ionic strength to high followed by repeating the initial solution measurement so as to check whether the continual flushing of the cell had led to any particle removal. Again no evidence of particle removal was observed. The ζ potential was calculated from the streaming potential in the usual manner.13 The ζ potentials were corrected for surface conductance using the cell constant calculated from conductivity and conductance measurements at high ionic strength (3 × 10-2 M KNO3) where surface conduction is negligible.18 In 10-2 and 10-3 M KNO3 the corrections were negligible. However in 10-4 M KNO3 the surface conduction correction was more significant.
Results and Discussion Reflectometry. The raw data for the adsorption of silica particles from 10-3 and 10-2 M electrolyte on the aminosilane modified wafer are reproduced in Figures 3 and 4, respectively. The rate of adsorption is highest for the smallest silica particles. The amount adsorbed increases with electrolyte concentration. Both of these trends, and indeed the magnitude of the change in reflectometer signal, (18) Scales, P. J.; Grieser, F.; Healy, T. W.; White, L. R.; Chan, D. Y. C. Langmuir 1992, 8, 965.
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Figure 4. Normalized reflectivity ratio as a function of time in 10-2 M KNO3 at pH 5.6. The inset is an expansion of the initial adsorption phase.
Figure 5. Comparison of the coverages obtained using the constant (- - -) and coverage dependent (s) values of the sensitivity factor, As. The KNO3 concentration was 10-2 M and the pH 5.6.
are entirely consistent with the results obtained for silica particles adsorbing on surfaces modified by adsorption of cationic polymer.11,12 Conversion of the reflectometric data to coverage (θ) was done according to the procedure described in the Experimental Section. In Figure 5 a comparison is made between the coverages obtained using both constant and coverage dependent values of the sensitivity factor, As. It is clear that at coverages exceeding 0.2 there is a significant deviation between the two methods. The discrepancy exceeds 20% in the case of the largest particles (190 nm) at maximum coverage. It was for this reason that the coverage dependent values of As were used here. The effect of electrolyte concentration on coverage for the larger silica particles is shown in Figure 6. While there is a major effect of electrolyte concentration on coverage the maximum values obtained are significantly less than the random sequential adsorption (RSA)8 jamming limit of 0.55. Clearly the particles are not behaving as hard spheres as the repulsion between diffuse layers of adjacent particles increases the interparticle separation and therefore decreases the coverage. If one
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Figure 6. Effect of KNO3 concentration on the coverage of 90 and 190 nm silica at pH 5.6. The wafer with 190 nm particles adsorbed from 10-3 M KNO3 was also analyzed by SEM. The coverage obtained was 0.285 ( 0.007.
Hayes et al.
Figure 8. ζ potential of surfaces with 90 nm silica particles adsorbed. The coverage is varied by altering the deposition electrolyte concentration, with higher concentrations corresponding to higher coverages (the lowest coverage was obtained by lowering the particle concentration from 100 to 10 mg/L and reducing the contact time to 15 min). After equilibration, the free particles were flushed away with electrolyte, and the dependence of the ζ potential on ionic strength was observed. The particle coverages were obtained by SEM analysis of glass slides at the completion of experiments (Figure 9). Table 2. ζ Potentials of Aminopropylsilane Surfaces Calculated from Streaming Potential Measurements in KNO3 Solutions
Figure 7. Effect of KNO3 concentration on the coverage of 32 nm (- - -) and 40 nm (s) silica at pH 5.6.
compares the coverages obtained with the RSA limiting coverage, then it is clear that the effective particle size (particle plus double layer) generally includes more than three Debye lengths (κ-1). This issue is explored in more detail later. The rate of adsorption of the 90 nm particles is independent of electrolyte concentration as expected.12 The reason for the slight difference in initial rate for the 190 nm particles is unknown. SEM analysis of the surface that had 190 nm particles adsorbed from 10-3 M KNO3 gave a somewhat higher value of coverage than was obtained by reflectometry. The reason for this discrepancy, observed elsewhere,12 is simply that the He/Ne laser beam has a finite size, and the lower coverage region outside the stagnant flow zone makes some contribution to the reflectometer signal. The use of a more focused beam would presumably minimize this problem. The initial rate of adsorption of 32 and 40 nm particles is essentially coincident (Figure 7) and unaffected by electrolyte concentration. The maximum coverage did not exceed 0.25 in either case. ζ Potential. The ζ potential of the bare aminosilane surface (Table 2) was dependent on electrolyte concentration, with its magnitude increasing with decreasing concentration. Particles were introduced to the aminosilane surfaces in differing ionic strength solutions so as to vary the coverage, with lower KNO3 concentrations corresponding to lower coverages as demonstrated by the
I (M)
ζ (mV)
10-4 10-3 10-2
39.4 ( 0.3 27.8 ( 0.4, 32.4 ( 1.2, 28.6 ( 0.9 15.6 ( 3.7, 15.9 ( 2.1
reflectometry measurements. The ζ potential of the surface at different coverages of 90 nm silica is shown in Figure 8. Not surprisingly the presence of negatively charged silica particles on the surface displaced ζ in a negative direction with the sign of the potential reversing at coverages greater than 0.2. At maximum coverage, ζ plateaued at about -15mV. The ζ potential of the 90 nm particles was independently determined with a Malvern Zetasizer IV19 to be -40 (-42) mV in 10-3 (10-4) M KNO3. One might have expected for particles of this size adsorbed at maximum coverage that ζ would correspond to that of the individual particles. Clearly this is not the case. The coverage of 90 nm particles was measured with SEM at the completion of a number of streaming potential measurements (Figure 9). At maximum coverage (corresponding to 3 × 10-2 M KNO3), θSEM was 0.36. It appears then that the measured ζ for the particle adsorbed surface scales approximately with the coverage. At lower coverages the underlying surface makes a more significant contribution to the measured ζ. The measured ζ for the particle adsorbed surface remained constant at its minimium value for a considerable range of deposition KNO3 concentration. After deposition from 10-5 M KNO3 there was a slight positive shift in ζ (θSEM ) 0.26). To reduce the coverage further, a nonequilibrium deposition was conducted with the concentration of silica particles reduced by an order of magnitude (to 10 ppm) and the contact time reduced to 15 min. In this case ζ was positive and between 7 and 18 mV, depending on the ionic strength during ζ-potential measurement. The effect of adsorbing the smaller 40 nm silica particles on the electrokinetic properties was also investigated (19) Minor, M.; van der Linde, A. J.; van Leeuwen, H. P.; Lyklema, J. J. Colloid Interface Sci. 1997, 189, 370.
Silica Nanoparticle Adsorption
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Figure 10. ζ potential of surfaces with 40 nm silica particles adsorbed. The coverage is varied by altering the deposition electrolyte concentration (horizontal axis), with higher concentrations corresponding to higher coverages (the lowest coverages were obtained by lowering the particle concentration from 100 to 10 mg/L and then also by reducing the contact time). After equilibration, the free particles were flushed away with electrolyte, and the dependence of ζ potential on ionic strength was observed. Table 3. Particle Coverages Determined by Reflectometry and SEM with Corresponding Debye Length Parameters techniquea
rp (nm)
κ-1 (nm)
θp
nc
nκ-1 (nm)
SEM/refl reflectom reflectom reflectom reflectom SEM/str potl SEM/str potl SEM/str potlb reflectom reflectom reflectom reflectom
95 95 95 45 45 45 45 45 20 20 16 16
9.6 9.6 3.0 9.6 3.0 1.75 96 96 9.6 3.0 9.6 3.0
0.285 0.237 0.347 0.185 0.272 0.360 0.263 0.143 0.170 0.246 0.164 0.235
3.80 5.12 8.06 3.37 6.25 5.95 0.206 0.447 1.65 3.27 1.37 2.80
36.5 49.2 24.2 32.4 18.8 10.4 19.8 42.9 15.8 9.81 13.2 8.40
a Technique by which coverage was determined. b Nonequilibrium experiment. c Determined from eq A5, Appendix.
Figure 9. Ex situ SEM images of particles adsorbed in the streaming potential chamber. The coverages were determined to be (from top to bottom) 0.143 ( 0.013, 0.263 ( 0.003, and 0.360 ( 0.007. The streaming potential plates were horizontal during deposition. However particle coverages on both the upper and lower plates were found to be concordant by SEM indicating that sedimentation was not an issue.
(Figure 10). The general behavior was consistent with that observed for the larger particles with the minimum value of ζ potential obtained at higher coverages in the range -3 to -12 mV depending on the measurement ionic strength. For the smaller particles there was a consistent trend in ζ potential with measurement ionic strength. The magnitude of the ζ potential increased as the measurement ionic strength was decreased. The ζ potential of the particles was independently determined19 to be -26.5 (-33) mV in 10-3 (10-4) M KNO3. The trend in ζ potential with measurement ionic strength may simply reflect the similar trend for the particles alone, in contrast to the measurements for the larger 90 nm particles. Unfortunately the particles were too small to enable SEM
to be used to independently determine the particle coverages. For the larger particles the Debye length (κ-1) was always considerably less than the particle diameter whereas for the smaller particles the particle size was closer to the range of Debye lengths studied. As a result the underlying surface may be expected to play a greater electrokinetic role, particularly at low ionic strengths in the presence of smaller particles. This effect is exacerbated by the lower coverages obtained with smaller particles at a given ionic strength. If one compares the coverages obtained in the streaming potential cell (determined by SEM) with those obtained under stagnant flow conditions (determined by reflectometry and SEM) there is good agreement at high coverages (Table 3). At lower coverages the streaming potential coverages are higher than those obtained in stagnant flow. This presumably reflects the much greater time that particles spend adjacent to the substrate in the former case, which increases the probability of finding a vacant site for adsorption. Controlling Particle Coverages. In this study the equilibrium coverages of particles of various sizes adsorbed on a surface from electrolytes of varying concentration have been measured. The RSA model8 predicts a maximum coverage of hard spheres of 0.546. The measured coverages
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a minor contribution to the ζ potential when the interparticle separation is less than 40 nm. At particle separations corresponding to a particle diameter the contribution of the substrate to the measured ζ potential is much more significant. The shear plane does not therefore closely follow the undulations in the particleadsorbed surface. The particle-coverage depends on particle size as well as on the electrolyte concentration dependent Debye length.
Figure 11. Diffuse layer thickness as a function of particle size, dp. The reduction in coverage compared to the RSA limit has been assumed to be due to the diffuse layer thickness (see Appendix). Only equilibrium data are included. Coverages obtained by reflectometry are denoted by symbols (+, 10-2 M; 0, 10-3 M) with the lines corresponding to linear fits through the low particle size data. The value denoted by n was obtained by SEM for the reflectometer sample with dp ) 190 nm and I ) 10-3 M. The values denoted by triangles were obtained by SEM analysis of streaming potential data after deposition from 3 × 10-2 M (open) and 10-5 M (filled) KNO3.
(Table 3) are less than this value due to the double layer repulsion between adjacent particles. One can define an effective particle size which combines the diffuse layer component with the particle size. By comparing the measured coverage with the RSA limit (see Appendix for details) one can obtain a value for the diffuse layer thickness in units of κ-1 or nm (Figure 11). The diffuse layer thickness is found to monotonically increase with both particle size and electrolyte concentration. If the coverage were reduced simply by double layer repulsion then the particle separation might be expected to reduce to a fairly consistent number of Debye lengths. While the reason for this systematic variation is unclear the implications for particle adsorption are useful. Large particles exert more repulsive force on adjacent particles at the same number of Debye lengths than do smaller particles. Conclusions The adsorption of silica particles on aminosilanemodified surfaces with respect to particle size, electrolyte concentration, and kinetics is entirely consistent with parallel observations for cationic polymer-treated surfaces.11 The adsorption of silica particles reverse the potential of the aminosilane surface at higher coverages. At coverages of 90 nm silica particles exceeding 0.3, the electrokinetic contribution of the aminosilane surface is negligible and the ζ potential is simply reduced from the value of the individual particles by the particle area fraction. At lower coverages the aminosilane surface makes
Acknowledgment. We are grateful to the staff, in particular Ton de Laat, of the Centre for Fabrication Technology (CFT) at Philips for performing the PCS measurements and Merck for making available a sample of noncommercial silica spheres. The generously offered advice and time of Arjen Boogard and Gerald Belder regarding reflectometry and streaming potential measurements is warmly acknowledged. Appendix: Calculation of Average Particle Separation in Units of K-1 Ai correspond to occupied areas and the diffuse layer thickness is in units (n) of κ-1.
Ap ) π rp2
(A1)
Adl ) π [2 rpnκ-1 + n2 (κ-1)2]
(A2)
Adl θdl 0.546 - θp ) ) Ap θp θp
(A3)
Combining these equations
(
θp ) 0 (A4) 0.546
(2 rp κ-1)2 - 4 (κ-1)2 rp2 1 -
θp 1 (A5) 0.546 2(κ-1)2
(κ-1)2n2 + (2 rp κ-1)n + rp2 1 -
)
then
n ) -(2rpκ-1) +
x
(
)
Only the positive root of eq A4 returns a positive, and therefore physically realistic, value of n. LA980668F
