Stability Ratio in Binary Hard Sphere Suspensions, Measured via

Langmuir 2003, 19, 4127-4137

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Stability Ratio in Binary Hard Sphere Suspensions, Measured via Time-Resolved Microscopy V. A. Tolpekin,* M. H. G. Duits, D. van den Ende, and J. Mellema University of Twente, Faculty of Science and Technology, Physics of Complex Fluids group Associated with the J.M. Burgerscentrum for Fluid Mechanics, and Institute of Mechanics, Processes and ControlsTwente (IMPACT), P.O. Box 217, 7500 AE Enschede., The Netherlands Received December 20, 2002. In Final Form: March 7, 2003 Concentrated suspensions of hard spheres with a bimodal size distribution provide a system with a very rich behavior, which largely still has to be explored. We present here an experimental study, focused at the kinetics of aggregation between the large spheres, as caused by the presence of the small spheres. At high concentrations of the small particles, the pair potential between the large spheres is considered to contain not only an attractive well in the contact region but also a repulsive barrier at larger separations. This barrier was confirmed to be present in our system and was seen to have a profound influence on the initial aggregation rate. To test the adequacy of the binary hard sphere (BHS) model to describe our system, we have performed quantitative measurements and modeling of the rate of aggregate formation. Three-dimensional video microscopy, with a spatial resolution at the level of single (large) particles, allowed us to determine the time evolutions for monomer, dimer, and trimer number densities. By fitting these to theoretical model expressions, characteristic times were extracted and translated into stability ratios. The experiments were performed with colloidal silica spheres, having respective diameters of 920 and 50 nm for the large and small particles. Particle volume fractions ranged from 0.01 to 0.35% for the large particles and from 30 to 40% for the small particles. Besides experiments under quiescent conditions, we also performed measurements in flow, for Peclet numbers up to 20. The time evolutions of the various aggregation number densities were found to correspond well with Smoluchowski theory. The stability ratio W was found to increase sharply with small-particle concentration, in correspondence with the predictions. The magnitudes of the experimental W were found to be 4-11 (2-3) times larger than those predicted for a BHS in quiescent fluid (under shear).

1. Introduction Colloidal particles interacting through a pair potential that contains an attractive well at small separations and a repulsive barrier at larger distances provide a rather interesting case when one considers the aggregation of such particles. Especially when one can achieve control of the height of the barrier, a variety of properties or behaviors can be obtained. The most sensitive property is the colloidal stability (more quantitatively, the characteristic time scale of the aggregation). Also, different aggregate structures can be obtained by controlling the height of the potential barrier. A well-known example is that of fractal aggregation, where the fractal dimension is determined by the extent to which the aggregation is “diffusion”- or “reaction” (barrier)-limited. After the initial aggregate formation, the presence of a barrier can also play a role in restructuring (e.g. compaction) of the aggregate, which usually takes place over longer time scales. A classical example of a colloidal system where a repulsion has to be overcome before the particles can go into associative contact is a charge-stabilized aqueous system in which also a significant van der Waals attraction is present. In such systems, a barrier can be present, depending on the electrolyte concentration(s). A less frequently studied case in experimental colloid science is the binary hard sphere (BHS) system. In this system, one can also obtain a pair potential with an attraction preceded by a repulsion. Such a potential can be induced between the large particles, provided that the asymmetry between the two sizes is large enough and that the concentration of the small particles is high enough. The origin of this (effective) potential lies in the entropy of the system where a pair of large particles is embedded in a “sea of small

particles”. When the large particles are separated by a small surface distance (comparable to the small-particle diameter), the presence of small particles in the gap is entropically disfavored, which leads to a lower osmotic pressure in the gap (effective attraction). However, before two large particles can come so close together, small particles will have to be removed from the shells surrounding the large particles, which is also entropically disfavored (repulsion). Besides the different origin of the BHS depletion potential, also its properties are clearly distinguished from the classical (DLVO-like) potential. Both the depth of the attractive well and the height of the repulsive barrier can be set (though not independently) via the small-particle concentration. At sufficiently low small-particle concentration, the potential will become negligibly small (i.e. interaction energies < kT), resulting in a hard sphere system. This situation is easily obtained, even when starting from a concentrated and flocculated system (by simple dilution). This degree of reversibility cannot be matched by DLVO-type systems. At higher small-particle concentrations, the flocculation of the large particles can be tuned from weak to strong, while also the time scale of the aggregation can be manipulated. This level of control makes the binary hard sphere system an interesting case. This point has also been recognized by several research groups that have performed simulations on the structure, correlations, and phase behavior of binary hard sphere systems. Since there are three degrees of freedom in this system (two concentrations and a size ratio), a large area is still unexplored, but already a rich phase behavior has been found (see, e.g., refs 1-6). Of interest for the present paper is the “corner” of the parameter space where the large-particle concentration is low and the small-particle

10.1021/la027045t CCC: $25.00 © 2003 American Chemical Society Published on Web 04/04/2003

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concentration is high. In this case, one can define an effective potential between two large particles, caused by the presence of the small particles. The properties of this potential were recently addressed in calculations and in Monte Carlo simulations.7,8 An approximate expression for the effective pair potential was presented, which gave a satisfactory description for the phase behavior in the case of a sufficient size asymmetry. On the experimental side, fewer studies have been performed on BHS dispersions. Perhaps one of the reasons for this is the difficulty in making a colloidal hard sphere mixture in practice. Despite the success in mapping the behavior of different kinds of particle suspensions to that of hard spheres in the late eighties (and after), several examples can be found in the literature where mixing two systems of particles that behaved as hard spheres “between themselves” resulted in markedly different behavior than that predicted for binary hard spheres (see, e.g., refs 9 and 10). Such deviations can be due to various kinds of interactions between the different types of particles, like adsorption9,10 or halo formation.11 In the few experimental studies where the observed phase behavior was supposed to correspond well to that of BHS (see, e.g., refs 12 and 13), still some comments were made on the possibility of slight deviations at the level of the direct particle interactions. It was only recently that experimental evidence could be found for an effective pair potential similar to the predictions for BHS. In experiments by Crocker et al,14 large colloidal PMMA spheres were mixed with small PS latex spheres in an aqueous medium containing also NaCl and SDS. Using an optical line tweezer allowed them to confine the motion of two large particles to one dimension. By measuring the pair distribution function P(r) (at various concentrations of small particles), it could be demonstrated that a pair of large particles can experience a depletion attraction near contact and a depletion repulsion at larger separation. In the present study we describe experiments with binary sphere dispersions in a nonaqueous medium. Both the large and the small particles consist of (nondeformable) silica cores, coated with a grafted layer of octadecyl chains to make them dispersible in oil. The van der Waals attraction is suppressed by using a (near) refractive index matching solvent. A confocal scanning laser microscope was used to study the structure (formation) of the large particles in real time. We will first corroborate the existence of effective attractions between the large particles and then shift focus to the (initial) rate of (1) Dijkstra, M. Curr. Opin. Colloid Interface Sci. 2001, 6, 372-382. (2) Dijkstra, M.; van Roij, R.; Evans, R. J. Chem. Phys. 2000, 113, 4799-4807. (3) Dijkstra, M.; van Roij, R.; Evans, R. Phys. Rev. E 1999, 59, 57445771. (4) Dijkstra, M.; van Roij, R.; Evans, E. Phys. Rev. Lett. 1998, 81, 2268-2271. (5) Malherbe, J. G.; Amokrane, S. Mol. Phys. 2001, 99, 355-361. (6) Lue, L.; Woodcock, L. V. Mol. Phys. 1999, 96, 1435-1443. (7) Dijkstra, M.; van Roij, R.; Evans, R. Phys. Rev. Lett. 1999, 82, 117-120. (8) Gotzelmann, B.; Roth, R.; Dietrich, S.; Dijkstra, M.; Evans, R. Europhys. Lett. 1999, 47, 398-404. (9) Duits, M. H. G.; May, R. P.; Vrij, A.; de Kruif, C. G. J. Chem. Phys. 1991, 94, 4521-4531. (10) van Ewijk, G. A.; Philipse, A. P. Langmuir 2001, 17, 72047209. (11) Tohver, V.; Smay, J. E.; Braem, A.; Braun, P. V.; Lewis, J. A. Proc. Natl. Acad. Sci. U. S. A. 2001, 98, 8950-8954. (12) Steiner, U.; Meller, A.; Stavans, J. Phys. Rev. Lett. 1995, 74, 4750-4753. (13) Imhof, A.; Dhont, J. K. G. Phys. Rev. Lett. 1995, 75, 1662-1665. (14) Crocker, J. C.; Matteo, J. A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352-4355.

Tolpekin et al.

Figure 1. Schematic overview of the paper.

aggregation. The latter quantity was measured both under quiescent conditions and in shear flow. In our analysis we will compare these experimental data to calculated aggregation rates. The latter quantities were obtained by modeling the transport mechanism (Brownian motion, shear flow) for bringing the particles close together, the effective particle interactions, and the suspension viscosity. The most critical test will be on the adequacy of the BHS model to describe the effective interactions between the large particles. The rate of aggregation should be very sensitive to the presence of a potential barrier, as predicted. This paper is further organized as follows: In section 2 we will present the theoretical background and equations relevant to the aggregation kinetics of binary hard spheres. In section 3 the preparation of the fluids and the instrumentation will be described. In sections 4 and 5 the results will be presented and discussed, while in section 6 the conclusions will be summarized. To further facilitate reading the paper, we have also included a schematic overview, illustrating the connections between the various sections (Figure 1). 2. Theory 2.1. Population Balance and Operational Definition of the Stability Ratio. When aggregates are characterized only via the number k of particles they contain, the aggregation kinetics can be expressed via the time-dependence of the corresponding number densities nk. The conservation of mass sets the boundary condition ∞

np )

∑ knk k)1

(1)

while the initial distribution is given by

nk(t)0) ) nk(0) for k g 1

(2)

For t > 0, the time-dependence of the number density nk is governed by collisions between aggregates, leading to new and bigger aggregates (neglecting break-up). Assuming that new aggregates can only be created via binary

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Langmuir, Vol. 19, No. 10, 2003 4129

encounters, the following system of coupled differential equations is obtained:

dnk dt

)

1k-1

∑ 2 i)1

dimensionless quantities νk ) nk/np and τ ) t/tBr proc, we obtain

∞

∑ i)1

B(i,k-i)nink-i - nk

∞

B(i,k)ni

νtot(τ) )

(3)

where the kernel B(i,k) represents the “aggregation rate” between aggregates of type i and k. These reaction kernels are of key importance to this paper, since they depend on the mechanism of aggregation (Brownian or shear flow) and on the (repulsive part of the) pair potential. In the case of Brownian aggregation15 (“perikinetic flocculation”)

BBr(i,k) )

(

)

2kT 1 1 (Ri + Rk) + Ri Rk 3ηWBr ik

(4)

while in case of shear-induced aggregation (“orthokinetic flocculation”)

4γ˘ (Ri + Rk)3 B (i,k) ) sh 3Wik sh

1 σiσk

(5)

(6)

where σk is the area fraction in the outer shell of an aggregate that is occupied by particles; σk increases with the size of the aggregate, and we estimate it as

σk ) k(1-3/D)

(7)

Brownian Aggregation. Simplifying the kernel for the Brownian case (for equal sized particles) to

BBr(i,k) )

8kT 3ηWBr

(9)

totτ

µ1 (1 + µtotτ)2

µ2 + (µ12 + µtotµ2)τ (1 + µtotτ)3

(10)

(11)

ν3(τ) ) µ3 + 2(µ1µ2 + µtotµ3)τ + (µ13 + 2µtotµ1µ2 + µtot2µ3)τ2 (1 + µtotτ)4 (12)

with Rk the collision radius of an aggregate containing k particles and γ˘ the rate of shear. Wik is a stability ratio and depends on the (effective) interaction potential, the hydrodynamic interactions between aggregates, and the size of the aggregates. We will write it in the form Wik ) WW ˜ ik, where W ) W11 is the stability ratio for a pair of monomers and W ˜ ik incorporates the dependence of Wik on Sm aggregate size. W is defined by W ) BSm 11 /B11, where B11 3 equals BSmB ) 8kT/(3η) or BSmS ) 32a γ˘ /3, respectively, for Brownian or shear-controlled aggregation, in the absense of a repulsive potential barrier and two-particle hydrodynamic interactions. B11 is the “real” kernel for dimer formation, where these effects are incorporated. These effects will be modeled in sections 2.3 and 2.4 to obtain theoretical predictions for W. The radius of the aggregate Rk in eqs 4 and 5 can be expressed as a function of k, assuming fractal-like aggregates with a fractal dimension D: k ) (Rk/a)D, with a the radius of the monomer particle. Considering the probability that two primary particles belonging to different aggregates touch each other when those aggregates slightly overlap, one may estimate the relative stability ratio as

W ˜ ik )

ν1(τ) )

ν2(τ) )

µtot

∑ νk(τ) ) 1 + µ k)1

(8)

with η the medium viscosity, makes it possible to solve the population balance equations (eq 3) analytically, as shown by Smoluchowski.16 Rewriting the result for (15) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: 1989. (16) von Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129.

where µk ) νk(τ)0) and µtot ) νtot(τ)0) and

tBr proc )

πηa3WBr φkT

(13)

with φ ) 4/3πa3np the particle volume fraction. Equation 13 will be used further on in this paper to calculate WBr from the experimental results. Shear-Controlled Aggregation. Simplifying the kernel function for the shear-controlled aggregation makes it possible to obtain again an analytical solution:

Bsh(i,k) )

16γ˘ a3 (i + k) 3Wsh

(14)

Defining τ ) t/tsh proc in this case, the solution can be written as

νtot(τ) ) µtote-τ

(15)

ν1(τ) ) µ1 exp(-τ - µtot(1 - e-τ))

(16)

ν2(τ) ) (µ2 + µ12(1 - e-τ)) exp(-τ - 2µtot(1 - e-τ)) (17) 1 ν3(τ) ) µ3 + 3µ1µ2(1 - e-τ) + 3µ13 1 - e-τ - (1 2

(

(

e

-2τ

))

) exp(-τ - 3µtot(1 - e-τ)) (18)

Here we have identified tsh proc with

tsh proc )

πηa3 sh W φkT

(19)

Equation 19 will be used further on to calculate Wsh from the experimental results. We also solved the population balance equations (eq 3) with nonsimplified aggregation kernels (eqs 4-7) numerically, for several values for the fractal dimension D between 1.5 and 3. For τ < 0.2, deviations from the approximate analytical solution (eqs 9-12 and refs 15 and 19) were less than 1%. So, in the following we will use the analytical expressions for fitting the experimental data. 2.2. Effective Interaction in Binary Hard Sphere Systems. The statistical interaction between two large

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Tolpekin et al.

distribution function. Under these conditions, this enhancement can be estimated as

W ) W0/(1 + 0.257Pe1/2/W0)

Figure 2. Depletion interaction potential calculated with eq 20. φSP values: solid line, 30%; dashed line, 35%; dash-dotted line, 40%.

hard spheres in the presence of small hard spheres is described with an average interaction potential. An approximate expression for U(r) was taken from4

Udepletion(λ) 1+q )[9φ2(1 + 4φ)λ + 3φ(1 + 4φ + kT 2q 10φ2)λ2] (20) where

λ)

1 r - -1 2aq q

for -1 < λ < 0, q ) asp/a is the ratio between the small and large sphere radii, and r is center to center distance. In principle, φ in eq 20 should be the reservoir packing density of the small particles. In our case, the volume fraction of big particles was never more than 1%, so we set φ ) φs, where φs is the volume fraction of small particles. A graph of the potential given by eq 20 is presented in Figure 2. 2.3. Modeling the Stability Ratio in a Quiescent Fluid. The stability ratio W for interacting monodisperse particles of radius a in a quiescent fluid can be calculated with15

∫

WBr ) 2a

∞

U(r)/kT

e dr 2a 2 r G(r/a)

(21)

where r is center to center distance, U(r) is the interaction potential, and G(r/a) is a hydrodynamic interaction function. We will use an approximate expression for G(x) which has the proper asymptotic behavior for x f 2 and x f ∞. It also represents the intermediate data points tabulated by Batchelor17 quite well:

G(x) ) 1 -

1 15 3 + - 0.140625 2x x3 4x4 exp(7.34(2 - x)) (22)

Equation 21 was integrated numerically in order to calculate the value for WBr. 2.4. Modeling the Stability Ratio in Shear Flow. According to Russel et al.,15 the flux on a particle is enhanced by the shear flow. For small Peclet numbers (Pe) one can consider this enhancement as a perturbation on the Brownian flux still assuming an isotropic pair (17) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1-29.

(23)

where W0 is the stability ratio for the suspension in the case Pe ) 0, as given by eq 21. However, in our experiments the Pe values ranged from 5 to 20, so one is neither in the Pe , 1 limit nor in the Pe . 1 limit. A full analysis of this case, taking into account the full anisotropic pair distribution, is beyond the scope of this paper. To simplify things, we consider two stages in the aggregation process. In the first stage the flux of partner particles on a tagged particle outside their interaction range (r > b), where b ) 2a(1 + q), is considered; in the second stage the flux inside the interaction shell (2a < r < b) is considered. Due to the barrier in the interaction potential, the concentration gradient for r > b is very small, and in this region diffusion can be neglected. Hence, the concentration of partner particles, p(r), at the boundary r ) b is determined by the flow field. From trajectory calculations, that were done using the method described by Feke and Schowalter,18 we obtained pb/n ) 1.598 for b/a ) 2.1086, independent of the position at the boundary. Inside the interaction shell (2a < r < b) the radial concentration gradient will be strong and diffusion will dominate the convection, if the Peclet number is not too large. In this regime we neglect the radial convection and solve the isotropic diffusion problem:15

{ (

dU/kT dp 1 d 2 FGp + dF dF F2 dF

)} ) 0

(24)

with the boundary conditions p(2) ) 0 and p(2(1 + q)) ) pb. Here F ) r/a. Solving this ordinary differential equation and calculating the flux

Jr )

2kT dU/kT dp Gp + aζ dF dF

(

)

(25)

gives the doublet formation rate:

1 4kT I ) n2 2 3η

pb/n 2+2qexp(U(s)/kT)

∫2

s2G(s)

(26) ds

This should be compared to the Brownian rate at Pe ) 0:

1 4kT I0 ) n2 2 3η

1 ∞exp(U(s)/kT)

∫2

s2G(s)

(27) ds

from which eq 21 follows. Hence, for our case the shear flow enhances the aggregation rate with

I ) I0

∫2∞

pb

exp(U(s)/kT)

ds s2G(s) 2+2qexp(U(s)/kT) ds n 2 s2G(s)

∫

(28)

The stability ratio under shear is given by

W/W0 ) I0/I

(29)

where W has been defined as the ratio between the pair formation rate under shear and the Smoluchowski rate


for Brownian aggregation, instead of shear-induced aggregation: W ) BSmB/B11 (see section 2.1). To obtain the range of Peclet numbers for which eqs 26-29 are valid, we estimate the lower limit for Pe from the condition that the influx at the interaction boundary, r ) b, should be larger than the diffusive flux. Equation 25 gives the diffusive flux while the convective influx was calculated from the trajectory calculations: Jr ) [p(r) dr/ dt]r)b. This resulted in the lower limit Pelow(φsp), which is a function of the interaction potential and so of φsp: Pelow ranges between 5 for φsp ) 0.20 and 2 for φsp ) 0.40. The upper limit, Peup, was also found from the trajectory calculations:

Peup )

aFmax a dU/kT ) | 2kT 2 dr r)b

Above this value for Pe the shear flow is able to push a particle over the potential barrier; below it is not. Peup ranges between 30 for φsp ) 0.20 and 180 for φsp ) 0.40. Hence, in the range 5 < Pe < 30, eqs 28 and 29 can be applied for our systems. 3. Experimental Section 3.1. Synthesis of the Silica Particles. Two kinds of colloidal silica particles were synthesized. Common to both preparations was that the method of Sto¨ber et al.19 was used to synthesize primary silica cores from tetraethyl orthosilicate (TEOS) in the presence of ammonia, water, and ethanol. To make the large particles visible with the fluorescence confocal microscope, a relatively small amount of dye was added to the TEOS in the first preparation step. The dye was fluorescein isothiocyanate (FITC), coupled to aminopropyltriethoxysilane (APTES) as described by Verhaegh et al.20 The radius of the thus obtained fluorescent cores was 250 nm. Directly after their formation, these organosilica particles were overgrown with a few nanometer thin layer of “pure Sto¨ber” silica to stabilize them. Also the 2.0 M high ammonia concentration needed to obtain the particles affects the colloidal stability somewhat. Therefore, a 1:1 dilution with ethanol was done as the next step. Primary particles were grown out to their target radius via seeded growth, as described by Coenen et al.21 For the large particles this growth was allowed to take place only via small steps, thus mimicking the continuous addition method as described by Giesche,22 which minimizes secondary nucleation. By adding TEOS (and water) three times a day, the outgrowth could be completed in 1 week. A relatively small amount of secondary nuclei was seen to be formed in the last growth steps. These particles could be separated off using differential sedimentation. The final particle radii were 25 nm for the small particles and 460 nm for the large particles. These silica cores were coated with a dense layer of octadecyl chains via the method described by van Helden et al.23,24 This surface coating renders the particles hydrophobic and provides a steric stabilization against flocculation. Sedimentation (using an ultracentrifuge at 10 000g for the small particles) and resuspension in pure solvent were applied to remove unreacted octadecyl chains and to transfer the particles. The small particles were initially transferred to pure cyclohexane, to facilitate their sedimentation, and subsequently dried and redispersed into chloroform. The large particles were directly transferred to chloroform. (18) Feke, D. L.; Schowalter, W. R. J. Fluid Mech 1983, 133, 17-35. (19) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (20) Verhaegh, N. A. M.; van Blaaderen, A. Langmuir 1994, 10, 14271438. (21) Coenen, S.; de Kruif, C. G. J. Colloid Interface Sci. 1988, 124, 104-110. (22) Giesche, H. J. Eur. Ceram. Soc. 1994, 14, 205-214. (23) Van Helden, A. K. J. Colloid Interface Sci. 1980, 76, 418. (24) Van Helden, A. K. J. Colloid Interface Sci. 1981, 81, 354.

Langmuir, Vol. 19, No. 10, 2003 4131 Table 1. Characterization of Small and Large Particles fluorescent core radius total particle radius mass density

small particle

large particle

25 nm 1.72 g/mL

250 nm 460 nm 1.755 g/mL

3.2. Particle Characterizations. Particle size and shape (distributions) were measured with transmission electron microscopy (TEM). The (460 nm) large particles were showing a polydispersity (standard deviation) of 8 ( 2% and good sphericity. The (25 nm) small particles showed some irregularities in the shape but were still fairly spherical. Here the polydispersity was 11 ( 2%. The mass densities of the particle systems in chloroform were determined by weighing an accurately known volume of stock dispersion, drying this amount in an oven, and measuring the weight of the remaining particulate material. Assuming no excess mixing volumes, the particle density Fp was calculated from

1/〈F〉 ≡ 〈v〉 ) wpvp + (1 - wp)vs with 〈F〉 the measured density of the dispersion, wp the measured weight fraction of the dried particles, and vs ≡ 1/Fs the specific volume of the solvent. The latter quantity was measured in the same vessel prior to the dispersion, and it corresponded well with the literature value of 1.485 g/mL at 25 °C. In accordance with the previous equation, the particle weight concentration cp and volume fraction φp can be calculated from

1/cp ) (vp - vs) + vs/wp

and

φp ) vPcp

The results of the mentioned characterizations are summarized in Table 1. 3.3. Preparation of the Mixtures. Mixtures of small and large particles in chloroform were prepared from stock dispersions of the separate particle systems. In calculating the needed amounts, again the assumption of “no excess mixing volumes” was used. Volume fractions of the small- and large-particle stock dispersions amounted approximately to 0.45 and 0.07, respectively; their precise values were calibrated from time to time to avoid composition errors due to solvent evaporation. For each experiment, 1.0 mL of mixture was prepared in a 2 mL capped vial by first adding the small-particle stock, then the chloroform, and finally the large-particle stock. After each addition, the weight was measured, to enable an a posteriori calculation of the precise concentrations. In all cases, the mixtures were freshly prepared, that is, (typically 10) min before the start of the experiment. Homogenization was achieved by shaking the vial shortly and gently with the hand. We chose not to use vigorous mixing methods such as stirring or whirlmixing (not even for short times), to minimize flow-induced aggregation. The strong yellow color of the large particles was helpful in verifying that indeed a homogeneous mixture was obtained. In addition to the ingredients mentioned, also 10 µL of a 0.25% (by weight) solution of calcium 2-ethylhexanoate (ABCR) in chloroform was added, to prevent the buildup of static charges within the fluid. 3.4. Confocal Scanning Laser Microscope (CSLM). An UltraView CSLM system (Perkin-Elmer) was used for imaging the fluorescent large particles. This system comprises (among others) an Nikon Eclipse TE200 inverted microscope, supplied with a 100×, N.A. 1.30 immersion oil objective. Using a Physik Instrumente P-721.17 piezotube with a PZ 73E high-speed Z axis controller, the vertical displacement of the objective (and, with it, the Z-location in the sample) can be set with submicron resolution. Excitation of the FITC fluorescence is done with the 488 nm line of a Melles-Griot 643-PEYB-A01 krypton/argon laser. The image from the Yokogawa CSU10 confocal head is recorded with a Hamamatsu IEEE 1394 C4742-95-12ERG camera. This digital CCD camera has 1344 pixels × 1024 pixels. Pixel binning ×2 allowed scanning rates up to 65 ms per image. The corresponding pixel size amounted to (2 × 0.0645) µm. Images from the camera were directly stored on a PC hard disk. 3.5. Shear Cell for the CSLM. A home-built shear cell with plate-plate geometry was mounted on top of the inverted microscope of the CSLM. The bottom of the shear cell was made

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of 0.17 mm thick glass to allow the observations. All other parts were made of stainless steel. Steady rotation of the upper plate established the flow of the fluid in the cell. Using a MATTKE 2842/T505/PG30 motor, the rotational speed could be gradually varied between 0 and 2 rpm with steps of 0.001 rpm. The stage with the shear cell could be moved relative to the objective of the microscope in the horizontal direction, allowing observations at different radial positions in the shear cell. The actual radial position was varied between 0 and 9 mm and was measured with an accuracy of 0.5 mm. The gap width was measured with a micrometer and varied between 200 and 300 µm ((25 µm) for different experiments. The diameter of the sample compartment was 30 mm. To prevent evaporation of the chloroform, a vapor lock filled with glycerol was used. About 1 mL of dispersion was prepared for each experiment. The shear cell was first filled and then closed by lowering the upper plate with the vapor lock mounted on it. A small hole in the vapor lock allowed for pressure equalization. This hole was closed with a PTFE stopper directly hereafter. Next the microscope was focused, and the recording was started. 3.6. Acquiring Images. Complete 3D information on the fluorescent particle configuration (positions) at rest and under shear has been obtained by performing XY-scans at different Z-positions of the sample. Each 3D scan consists of (typically) 60 XY-scans (images) spaced by 1 µm in the Z-direction. Each image, in turn, was a two-dimensional 87 × 66 µm2 CSLM image captured with an exposure time of 65 ms. The time interval between successive 3D scans was gradually changed during the experiment. In the beginning of the experiment, where the quickest changes were expected, it was set to zero. Later the delay was increased up to 120 s in some cases, thus optimizing image storage efficiency and minimizing particle photobleaching. The latter was of major importance in the case of experiments in quiescent fluid, because the same set of particles was exposed for a long time. In the case of experiments in shear flow, the problem of photobleaching was less important because the flow continuously refreshes the set of particles in the observation window. 3.7. Image Analysis: Classification of Aggregates. In our analysis of the first stages of the flocculation kinetics, the distribution of single particles (monomers) and small aggregates (oligomers) was measured for several time points. Each time point corresponded to one 3D scan for experiments in quiescent fluid and three to five consecutive 3D scans for experiments done in shear flow. The total duration of 3D scans identified with one “time point” was always much shorter than the duration of the experiment. In a typical analysis, 300 to 400 entities were considered for each time-point. This was sufficient to obtain monomer, dimer, and trimer fractions with an acceptable accuracy. The analysis was done in a semiautomated way. First of all, intensity contributions not coming from particles had to be eliminated. For each image, local intensity maxima were identified, after which a 2D (X, Y) Gaussian intensity profile was fitted for each local maximum. Considering the intensity and the half-width of the profiles allowed an unambiguous detection of all particles and rejection of all other local intensity maxima. This was done with in-house developed software. The next step was to classify the detected entities into monomers, dimers, trimers, and so on. Here a problem had to be addressed: aggregates extending in the vertical (Z) direction can only be partially observed in (X, Y) optical sections. However, in measuring the aggregation number, each particle in the aggregate has to be counted once. We have solved this problem by defining new “optical sections” (from now on termed Z-slices) from the data and subsequently analyzing these sections manually. Considering the vertical (Z) intensity profile of a single particle, it shows that a peak width of 1 µm (i.e. equal to the step size of the Z-scan) is found at an intensity of approximately half times the maximum value (see Figure 3). This means that particles which are more than 0.5 µm away from the focal plane can be rejected by applying a relative intensity threshold of 0.5 (i.e. 0.5 times the maximum intensity observed in that Z-slice, corresponding to a particle in focus). This correspondence between the step size and the rejection criterion allows us to make visible only particles in a narrow

Tolpekin et al.

Figure 3. Intensity profile of a particle along the Z-direction.

Figure 4. Illustration of how aggregates appear in optical (XY) sections taken successively at different (Z) heights. Black objects are in the plane of focus, while gray objects are slightly out of focus but not rejected by the threshold. The (here: downward) displacements between successive images represent motions. The aggregate in the lower right corner is a dimer in the XY plane, while the aggregate in the upper left corner is a trimer. Z-slice, while ensuring that all particles in the volume covered by the whole Z-scan will be detected. To cover for slight (if any) intensity variations between particles in the same layer, or at different optical depths or at different times, we have used in most cases a slightly lower intensity threshold. As a consequence hereof, it was ensured that all particles were detected at least in one slice. However, it also meant that most particles were found in two consecutive Z-slices. This made it necessary to perform the counting of entities and aggregation numbers manually, by comparing images at neighboring Z-positions. The advantage hereof was that also slight displacements of entities due to thermal and/or convective motion between two subsequently recorded images could be taken into account. This procedure is illustrated Figure 4. Examples of obtained CSLM images are presented in Figure 5. Particles on the images obtained in shear flow look elongated in the direction of the flow. The degree of elongation depends on the particle velocity. This reduces the quality of the images. However, it provides additional information about the position of the particles relative (above or below) to the focal plane, thus simplifying the classification of the aggregates. 3.8. Local Measurements. 3.8.1. Large-Particle Concentration. A potentially complicating factor is the gravity settling of single large particles (and in later stages also their aggregates). This leads to an increase in concentration near the bottom of the cell, which in turn causes an increase in the (local) aggregation rate. The particle counting method described in section 3.7 allowed us to measure the concentration as a function of the Z-position and to exclude the affected Z-slices near the bottom of the cell. This is also illustrated in Figure 6. The profiles in Figure 6 correspond to transient states, since the height of the sediment layer grows with time. The growth rate of this layer can also be estimated a priori, in which case it can be used as a design tool for the experiment. In our calculations we have considered only the monomer particles and assumed that sedimentation leads to a complete removal of particles from the top of the cell, leading to a depleted top layer with a thickness vsedt. The corresponding number of sedimented particles (per unit area) after time t

Nsed )

3φvsedt 4πa3


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Figure 5. Representative CSLM image: (a) in a quiescent fluid; (b) in shear flow. Table 2. Relative Viscosity of a Small-Particle Dispersions as a Function of Volume Fractiona φs (%) ηr

31.2 5.57

33.8 7.69

36.1 12.3

36.9 14.3

37.4 12.0

38.8 19.4

39.6 21.4

aThe solvent viscosity used for normalizing the data is 5.8 × 10-4 Pa‚s.

Figure 6. Concentration of large particles as a function of (Z) height above the bottom of the cell. Different sets of data correspond to different time points: t ) 25 (b), 51 (9), 177 (1), and 400 (2) min. The curves represent our model calculations. is assumed to be distributed over the whole liquid column H according to a Boltzmann distribution:

nexcess(z) )

4. Results

( )

Nsed z exp z0 z0

z0 )

with

3kT 4πa3∆Fg

where we have used H . z0. Note that nexcess is a number of particles per volume. It is supposed to be added to the number density of the homogeneous dispersion:

nbulk )

3φ 4πa3

so that

(

n(z,t) ) nbulk 1 +

vsedt -z/z0 e z0

)

Now defining (somewhat arbitrarily) the height of the concentrated layer Hcl as the location where nexcess < 0.1nbulk, we obtain

(

)

vsedt Hcl(t) ) z0 ln 10 z0

This simple (quasi-equilibrium, assuming fast diffusion compared to the sedimentation rate) model was found to describe our experimental (transient) data (as given, for example, in Figure 6) well. The calculated value for z0 was 5-6 µm in our case. For each experiment, images at different times t were analyzed only at z-values satisfying z > Hcl(t). For the time ranges covered by our experiments, the values for Hcl(t) were found between 5 and 30 µm. This is well below the maximum height zmax ) 60 µm as determined by the working distance of the microscope objective. Inserting zmax in the previous equation yields a maximum time for the experiments:

tmax )

3.8.2. Local Shear Rate. Owing to the sparseness of the large fluorescent particles, also the local shear rate could be measured, by tracking particles at different Z-positions. This allows us to compare the locally measured γ˘ local ≡ ∆vx/∆z with the global value γ˘ global ≡ ωr/h. A number of test measurements were done to verify the correspondence. In most cases the measured local shear rate turned out to be between 80% and 100% of the calculated global value. The local shear rate was in no case found to depend on height (up to 60 µm) or on time. We, therefore, ascribe the small deviations to errors in calibrating the gap thickness and the radial position.

( )

z0 zmax exp 10vsed z0

4.1. Aggregation of Large Particles due to Small Particles. As a first characterization of the effective interactions induced by the small particles, we have measured the sedimentation velocity of the large particles (aggregates) as a function of small-particle concentration. These experiments were done using capped glass tubes of 10 cm height (Schott GL14) and containing 4 mL of the fluid mixture. The strong yellow color of the large particles, the transparency of the small particles, and the sharp boundary between sedimenting dispersion and continuous phase made it easy to follow the process with the naked eye. Neglecting interactions between the sedimenting particles, and assuming that the small-particle dispersion through which they move can be represented by its macroscopic mass density F(φs) and viscosity η(φs), the steady settling velocity for a single large particle is given by

v°sed )

2∆F(φs)ga2 9η(φs)

This equation was used to normalize the measured sedimentation velocities. To that purpose, we calculated ∆F(φs) using the particle densities from Table 1, and assumed no excess mixing volumes. The low shear viscosity η(φs) was measured with a Contraves LS40 rheometer; see Table 2 for the results. To interpolate or extrapolate to the concentrations of the sedimentation experiments, these data were fitted

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Tolpekin et al. Table 3. Description of Experiments and Fit Results for the Stability Ratio exp code name Q1 Q2 Q3 F1 F2 F3 F4

Figure 7. Sedimentation velocity of large particle aggregates, normalized by the steady “Stokesian” value ν°sed for a monomer.

with Quemada’s expression:25

ηr ) (1 - φ/φm)-2 resulting in the value φm ) 0.503. In the case that large-particle aggregates are formed, the sedimentation speed will increase tremendously, which makes the measurement of vsed/v°sed a sensitive way of detecting aggregation. The results are shown in Figure 7. The large values (i.e. .1) for the normalized sedimentation speed for φs g 0.27 provide clear evidence that aggregation takes place. It is interesting to note that the normalized sedimentation velocity at φs ) 0.40 amounts only to a value of 3. This is ascribed to a slow onset of the aggregation process: it takes an appreciable time before the particles start to form aggregates. During this “lag time” the particles are still present as monomers and, hence, sediment with a normalized velocity equal to 1. Once the aggregates are formed, they sediment much more quickly, but this only makes a modest contribution to the timeaveraged sedimentation velocity. A slowing down of the aggregation process at large φs is in accordance with the expectations for binary hard spheres. The formation of aggregates was also confirmed by inspection of the tube contents with CSLM. Dilution of the aggregated systems with pure chloroform to low small-particle concentrations lead to disaggregation of the large particles into (colloidally stable) monomers. This corroborates the absence of significant attractions other than the depletion force. 4.2. Measurements of the Initial Aggregation Rate. Initial aggregation rates were measured by keeping track of the distribution of small aggregates (predominantly monomers, dimers, and trimers) as a function of time. Recordings of the CSLM images started between 5 and 15 min after the samples had been mixed: in all cases this turned out (a posteriori, see also Table 3) to be a relatively short time span compared to the measured characteristic aggregation time tprocess. Image acquisition was prolonged, either up to the point where large aggregates were observed (for short experiments) or overnight (for experiments with a long duration). The analysis of the images was done for times below the tmax defined in section 3.8.1. A summary of the experiments performed in the quiescent state (code names Q1, Q2, and Q3) and in flow (F1, F2, F3, and F4) is given in Table 3. For each experiment, the characteristic aggregation time tprocess was extracted by fitting the time evolution of (25) Quemada, D. In Lecture notes in Physics: Stability of Thermodynamic Systems; Cases-Vasquez, J., Lebon, J., Eds.; Springer: Berlin, 1982; p 210.

SP

φ, % LP

29.8 35.3 39.7 30.3 34.6 38.8 39.1

0.0204 0.0577 0.224 0.0147 0.0116 0.350 0.141

γ˘ , s-1

Pe

tprocess, h

Wexp

Wpred

0 0 0 8.7 4.6 3.3 7.6

0 0 0 7.0 6.0 8.0 20

5.1 ( 0.9 11 ( 2.3 95 ( 14 2.0 ( 1.2 5.4 ( 1.6 1.2 ( 0.2 6.1 ( 1.2

15 ( 3 47 ( 13 791 ( 152 4(2 5(2 20 ( 4 38 ( 8

3.67 12.4 70.5 2.01 9.15 63.3 73.7

the various number densities (monomers, dimers, trimers) in a simultaneous manner, with the model presented in section 2. A few issues had to be addressed here. The first issue concerns the state of the fluid at t ) 0, that is, at the start of the recording. Inspection of the first observed aggregate distributions revealed that, in all experiments, a small fraction of dimers (and also some trimers) was already present at that time. This could be ascribed to two different origins. First, a small fraction of permanent dimers (and some trimers) may have been present already in the stock dispersion of the large particles; such structures can be formed in the synthesis stage where the particles have not yet been coated with a hydrophobic layer. Second, and more importantly, the inevitable mixing of the small and large particles itself provides an opportunity for the large particles to meet and form (small) aggregates. They are enabled to do so by the time lapsed before the start of the experiment and by the flow of the mixing. This (small) complication has been accommodated for in the fitting, by considering not only the aggregation time but also the initial (i.e., at the start of the recording) number fractions of the dimers (µ2) and trimers (µ3) as freely adjustable parameters for each experiment. The second issue concerns the inaccuracies in the various number densities (monomers, dimers, trimers, total). A distinction can be made between different kinds of uncertainties. First there were some “ambiguous cases”, where even with the manual procedure elaborated in section 3.7 it could not be stated for sure what was observed. Two types occurred: (a) “did the considered entity contain either i or i + 1 monomers?”, or (b) “were particles connected, or just in close proximity?” To cover for this problem, minimum-count and maximum-count numbers were introduced for each class i. In the case of ambiguity, the maximum-count numbers for the involved classes were increased with the appropriate number, while the minimum-count numbers were left unaltered. This provided a first set of upper and lower bounds for the number fractions νi, calculated from νi ≡ Ni/N0. Here Ni is the number of i-mers counted in the image volume V, while N0 is calculated from N0 ≡ ∑iiNi, the total number of particles in V. A second source of inaccuracy in the determination of the several νi’s is the inherent statistical fluctuation in counted numbers {Nk}. We calculated errors in νi’s taking into account that each νi depends on the whole set of {Nk} (via N0). These errors turned out to be much more important (2-10 times larger) than the errors due to classification ambiguities. The fitting of the data proceeded as follows. First the νtot(t) data were fitted for the adjustable parameters µtot and tprocess. Then, keeping these parameters fixed, the ν1(t) data were fitted for µ1. Next, keeping µtot, tprocess, and µ1 fixed, the ν2(t) were fitted for µ2, and in an analogous fashion the ν3(t) data were fitted for µ3. Since tprocess is extracted only in the first fit, the model fits of ν1, ν2, and ν3(t) serve to provide a consistency check. From the tprocess


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Figure 8. Aggregation kinetics at rest: (a) Q1; (b) Q2; (c) Q3. Squares correspond to data for ν1, open squares for ν2, triangles for ν3, and circles for νtot. The solid lines correspond to the model described in section 2.1.

Figure 9. Aggregation kinetics in shear flow: (a) F1; (b) F2; (c) F3; (d) F4. Squares correspond to data for ν1, open squares for ν2, triangles for ν3, and circles for νtot. The solid lines correspond to the model described in section 2.1.

found, the stability ratio W was calculated with the help of eqs 13 (for the quiescent state) and 19 (for shear flow). The results of fitting the aggregation model (eqs 9-12 for experiments in quiescent fluid and eqs 15-18 in shear flow) to the experimental data are shown in Figures 8 and 9, for the cases of aggregation at rest and in flow, respectively. The description of the experimental data for νtot and νk by the model turns out to be satisfactory in all cases. We remark here that making the allowance for an aggregate distribution at t ) 0 gave a significant improvement of the quality of the fit. The corresponding numerical values obtained for the stability ratio and the initial aggregate distribution are listed in Tables 3 and 4. From the values in Table 3 it can be seen that the characteristic aggregation time scales tprocess are much longer than the 5-15 min lapsed between the mixing and the recording. Also, the number fractions µ (at t ) 0) in Table 4 are not clearly correlated to a normalized time

Table 4. Fit Results for the Aggregate Number Fractions at the Start of the Recordings exp code name

µ

µ1

µ2

µ3

Q1 Q2 Q3 F1 F2 F3 F4

0.88 ( 0.02 0.91 ( 0.01 0.903 ( 0.009 0.89 ( 0.02 0.898 ( 0.007 0.88 ( 0.01 0.883 ( 0.007

0.78 ( 0.02 0.82 ( 0.01 0.773 ( 0.008 0.77 ( 0.03 0.792 ( 0.008 0.78 ( 0.01 0.767 ( 0.009

0.103 ( 0.009 0.077 ( 0.005 0.112 ( 0.009 0.12 ( 0.02 0.11 ( 0.01 0.102 ( 0.006 0.110 ( 0.008

0.006 ( 0.003 0.004 ( 0.002 0.014 ( 0.003 0.003 ( 0.002 0.005 ( 0.002 0.006 ( 0.004 0.005 ( 0.003

lapse (i.e. the time lapsed before the recording was started, divided by tprocess). In all cases, about 20% of the particles turned out to be incorporated in a small aggregate (an oligomer) at t ) 0. Although the large-particle dispersions before mixing were not entirely free of oligomers, a 20% fraction is unrealistically high. A more likely explanation is that even the gentle manual shaking, as could not be avoided, promotes the aggregation in a significant manner.

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Figure 10. Stability ratio. The circles correspond to experimental data in a quiescent fluid; the triangles correspond to experimental data in a sheared fluid. The solid (dashed) lines indicate a lower and upper limit for the prediction of the BHS model in a quiescent (sheared) fluid.

As can be seen from the data on tprocess in Table 3, all our measurements lie in the time range τ < 0.2. Thus, the approximate analytical solutions eqs 14-18 are applicable for the data description. The stability ratios obtained at rest and in flow are plotted in Figure 10, to allow a comparison with the calculated W(φsp) curves for binary hard spheres. Because the particle size ratio q cannot be more precisely defined than ∆q/q ≈ 0.1, W(φs) was calculated for both q + ∆q (lower curve) and q - ∆q (upper curve). These curves were used as upper and lower limits for the predicted stability ratio. The presented accuracies in the fitted value for tprocess and the calculated values of stability ratio are determined by the errors in the ν values and calculated using the standard techniques for error propagation. Figure 10 shows that the experimental dependences on φs are fairly well reproduced by the model curves. The experimental values at rest differ from the prediction with a factor of 4-11. Under shear the experimental values are in the range of the predictions. Since the stability ratio is the result of an integration over the pair potential, the deviations could be due to the magnitude and/or the width of the barrier. Assuming that only the height of the barrier is underestimated by the BHS model potential, the difference would be about 2kT. Another reason for the deviations can be due to hydrodynamics: when two large particles are close together, the small particle suspension cannot be considered as a homogeneous continuum anymore. The free energy required for pushing small particles out of the gap between the large particles may lead to a reduction of the effective G(r). On the other hand, at even smaller distances between the large particles, no small particles can penetrate the gap anymore, resulting in a lower effective viscosity in that region. Without rigorous calculations (simulations), it cannot be said what the net effect of these counteracting effects will be on the hydrodynamics and, hence, the stability ratio. 5. Discussion In the qualitative sense, our bidisperse sphere mixtures show a good correspondence with the predicted behavior for binary hard spheres. Quantitative characterization of the effective potential between the large particles is a more difficult task. This may be due to model assumptions that were made, as well as to slight deviations from true hard sphere (direct) interactions between the different

Tolpekin et al.

kinds of particles. In the following we shall comment on these issues. 5.1. Effective Depletion Potential. The expression (eq 20) for the depletion potential used in calculating the stability ratio was in fact a simplification of the real potential, in that it is a third-order expression in the smallparticle volume fraction, which was shown to provide an equally accurate account of Monte Carlo simulation data for the phase diagram.4 As far as we know, there was no accurate quantitative experimental confirmation of eq 20. It should also be noted that eq 20 assumes monodispersity for the small and for the large particles. Theoretical studies26,27 have pointed out that polydispersity for the small particles tends to lower the entropic barrier in the large-particle potential. Most drastic effects were found for polydispersities above 20%,26 which is higher than the 12% for our system. The only other experimental study (up till now) of the effective pair potential in BHSs was that of Crocker et al.14 It is interesting to compare the results of both studies (keeping in mind that different particle system were used). A common finding is that, on increasing φs, the aggregation rate goes down tremendously. A difference, however, is that, in the case of Crocker et al., the height of the repulsive barrier in U(r) was more or less constant in the range φs ) 30-40 %, whereas in our case the stability ratio (which depends on the height of the barrier) was seen to grow with φs, which is at least qualitatively in line with predictions for BHS. 5.2. Direct Particle Interactions. It is difficult to exclude that slight deviations from truly hard sphere direct interactions (between the small and/or the large particles) are responsible for the quantitative differences between the measured and predicted stability ratios. A recent overview28 shows the (computer simulated) effects of superimposing “small-small”, “large-large”, or “smalllarge” potentials onto the hard sphere potential. Looking for the superpositions that were supposed to cause an increased effective repulsion between the large particles, three possibilities came into the picture, but they all seem to be unlikely: (1) attractions between small and large particles (i.e. adsorption), for which no indications could be found in electron microscopy experiments, (2) longranged additional repulsions between the large particles, which were not indicated by the high packing fraction (close to 0.6) of sediments formed under normal gravity, and (3) additional repulsions between the small particles, which were not indicated by the volume fraction dependence of the viscosity. Comparing the viscosity data in Table 2 to literature values for hard spheres29 gives a good correspondence, after setting the hydrodynamic specific volume to 0.725 mL/g, which is close to the 0.71 mL/g used in ref 29 for very similar (hard sphere) silica particles in cyclohexane. A more direct characterization of the direct particle interactions should be possible with, for example, small angle neutron scattering, but is beyond the scope of the present paper. 5.3. Model for W in Shear Flow. The experiments under shear can be compared with our “first-order” calculations of the stability ratio in section 2.4. The two main assumptions made in its derivation were (a) the particle flux on a tagged particle outside the interaction (26) Goulding, D.; Hansen, J. P. Mol. Phys. 2001, 99, 865-874. (27) Piech, M.; Walz, J. Y. J. Colloid Interface Sci. 2000, 225, 134146. (28) Louis, A. A.; Allahyarov, E.; Lo¨wen, H.; Roth, R. Phys. Rev. E 2002, 65, 061407. (29) Van Der Werff, J. C.; De Kruif, C. G. J. Rheol. 1989, 33, 421454.


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6. Conclusions

Figure 11. Critical Peclet value (dots), below which trajectories never lead to doublet formation, and the maximum repulsive force (dashed line) as calculated from eq 20, both as a function of φs. Also indicated are the (Pe, φs) parameters for the experiments under shear (crosses) and the lower limit for Pe (dashed dotted line) for which the model of section 2.4 is valid.

shell is fully deterministic and controlled by the flow field, while inside the interaction shell the particle flux is determined by the Brownian motion due to the gradient in particle concentration, and (b) the diffusion process inside the interaction shell is isotropic. Assumption a is valid as long as the shear flow is too weak to push a particle over the potential barrier but strong enough to provide sufficient particles at the interaction boundary for the diffusion process. This leads to a range of Peclet numbers for which the model is valid; from our calculations, we concluded that the upper limit is determined by the force with which the flow can push a particle against the repulsive force, as indicated in Figure 11. The lower limit is estimated from the radial influx of particles at the interaction boundary, which is roughly Jin ) 0.4naγ˘ , and the outflux Jout at the boundary, given by eq 25. As stated before, this limit ranges between 5 for φs ) 0.20 and 2 for φs ) 0.40. This limit is also indicated in Figure 11. The validity of assumption b is harder to estimate. One can only say that the concentration gradient in the radial direction is expected to be much larger than that in the tangential directions because in the diffusion model the gradient should balance the counteracting potential force and this force has only a radial component.

We presented a new method for studying the process of colloidal particle aggregation, and for the first time, to the best of our knowledge, measurements on initial aggregation under shear are presented. The advantage of our method is the direct observation of the aggregates/ structures both in quiescent and in flowing fluids. Unlike other methods, such as light scattering or a particle sizer,30,31 this method does not depend on a model for interpretation of the results in the first case, and it does not need a transportation of parts of the sample outside the shear cell for analysis. Another advantage is the possibility to measure locally, and thus to check, the homogeneity of the sample. The disadvantage of our method is a laborious procedure to classify aggregates and, more importantly, the necessity to accumulate much information in order to reduce the statistical fluctuations of the measured quantities. It is clear from the present study that our bidisperse sphere mixtures behave almost quantitatively, as one would expect for binary hard spheres. By controlling the concentration of the small particles φs, we were able to manipulate the (effective) pair potential between the large particles, from steeply repulsive (hard sphere like), via weakly attractive, to strongly attractive and with a repulsive barrier. These findings are in line with expectations, since the particles are essentially undeformable spheres, van der Waals attractions are suppressed via the refractive index matching, and no evidence was found for long-ranged repulsive interactions (after the antistatic agent was added). Still, it is one of only a few experimental studies where the presence of a repulsive-attractive potential in binary sphere mixtures is demonstrated. Acknowledgment. This work has been supported by the Foundation for Fundamental Research on Matter (FOM), which is financially supported by The Netherlands Organization for Scientific Research (NWO). LA027045T (30) Oles, V. J. Colloid Interface Sci. 1992, 154, 351-358. (31) Serra, T.; Colomer, J.; Casamitjana, X. J. Colloid Interface Sci. 1997, 187, 466-473.

Stability Ratio in Binary Hard Sphere Suspensions, Measured via

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