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Shear-induced structure in a colloidal suspension
Shear-Induced Structure in a Colloidal Suspension II. Light Scattering C. MATHIS,* G. BOSSIS,* AND J. F. B R A D Y t *Laboratoire de Physique de la MatiOre Condens#e, Universit£ de Nice, Parc Valrose, 06034 Nice Cedex, France; and t Department of Chemical Engineering, California Institute of Technology, Pasadena, California 91125 R e c e i v e d S e p t e m b e r 3, 1987; accepted D e c e m b e r l, 1987
We study, by light diffusion, the deformation of the pair distribution functionof a colloidalsuspension in a cylindrical Couette flow. The suspension is made of monodisperse latex particles (do = 750 A) at a volume fraction • = 1.3%. At equilibrium, the suspension has a liquid-like structure. The structure factor S(k) is recorded for different orientations of the diffusion wave vector in the plane of shear for Peclet numbers (Pc = 3,a2/Do) between 10-2 and 10-1. In order to get rid of different spurious causes of scattering the spectra are normalized by the same suspension in a gas-like state, obtained by the addition of salt. The structure is shown to deform asymmetricallywhen we increase the shear rate: it increases slowlywith a small linear shift of the maximum on the compression side whereas the intensity falls rapidly on the dilatation side. These experimental results are discussed with the help of numerical simulations on a similar system. © 1988AcademicPress,Inc. the structure of suspension. Light scattering (12, 16-20) and neutron scattering (21) are well-adapted technics (convenient wavelength) to study the structural changes of these systems. The effect of a shear on the structure was first demonstrated by Ackerson and Clark (16, 22 ) using a Couette cell and their results were compared to molecular dynamics computer simulations (23). This paper is the experimental part of a whole study devoted to the investigation of the structure deformation in sheared colloidal suspensions. The first part (referred to as I) reviews the theoretical approaches and compares them to the numerical simulation results. Using light diffusion technics we have studied the deformation of the structure factor in a large angular range, allowing us to compare significantly our results with the structure factors given by a linear approximation (cf. I).
INTRODUCTION
A colloidal s u s p e n s i o n can be defined as a diphasic system made of submicron solid particles in a liquid solvent. It follows that paint, blood, clay, and so on belong to that category. The study of physics of such materials is made difficult by the number of the constituents and by their complex interactions. So it is convenient to use a simple model suspension of polystyrene spheres (latex), with a sharp distribution of sizes, in a simple and neutral solvent. The aim of studies devoted to such systems is the understanding of the different interact i o n s - p a r t i c l e / particle, particle/solvent-and the correlation between microscopic structure and macroscopic properties (viscosity, electrical conductivity, mass transport, etc.). Numerous papers have been published and concerned with: (i) the growth and structure of the colloidal crystals in a bi- or tridimensional domain (1-5); (ii) the coagulation phenomena ( 6 - 9 ) ; and (iii) the effects of temperature (10-12), ionic force of the solvent (13), and sphere concentration (14, 15 ) on
MATERIALS AND METHODS
The particles are polystyrene spheres negatively charged ( - S O 4 groups bound to the surface) and supplied in aqueous solution by 16
002 1-9797/88 $3.00 Copyright© 1988 by AcademicPress, Inc. All rights of reproduction in any form reserved.
Journalof Colloidand InterfaceScience,Vol. 126, No. 1, November 1988
SHEAR-INDUCED STRUCTURE, II
Dow Chemical Co. The indicated mean diameter is do = 2a = 910 A with a dispersion = +29 A; the volumic fraction is q~ = 0.1. We have determined • by weighing samples before and after evaporating the water: = 0.106 + 0.003. The mean diameter has been measured by electronic microscopy, taking care to neglect the very few particularly small particles; the results are reported in Table I. Other measurements, made in our laboratory by photon correlation spectroscopy, give a hydrodynamic diameter of 860 A but the real diameter is usually 10% lower. We must be careful in announcing a value of do for many reasons: (i) measurements by PCS are not very precise, (ii) the hydrodynamic diameter is often different from the real diameter, and (iii) measurements by electronic microscopy are made with a relatively weak number of spheres (---250). In any event it seems reasonable to use do = 750 A as a mean value in our calculations, a value smaller by nearly 20% than the one reported by Dow Chemical Co. We must note that a smaller particle diameter could be an alternative reason to explain the systematic difference between the latex concentration calculated after scattering experiments and that determined from a dilution factor in a paper of Grtiner and Lehmann (24). We have kept the electric charge value proposed by Grtiner and Lehmann (24) and that is compatible with other studies (16): 1000e- per particle. The suspending fluid is a mixture of glycerol and water carefully distilled and deionized TABLE I Determination of the Mean Diameter of the Particles by Electronic Microscopy Measurements Sample
(~)(~)
o(A)
1 2 3 4
742 714 747 699
19 20 32 45
Note. The enlarging ratio of the electronic microscope has been verified.
17
(53% glycerol). This choice results from a compromise imposed by several necessities: (i) to match the suspension optical index with that of the cell's quartz walls (/'/quartz = 1.458, nsolvent= 1.407, and nsuso~ns.= 1.407 + 0.193/b), (ii) to reduce the multiple scattering in the suspension, and (iii) to avoid the Taylor instability under shear. The density of this fluid is Psolvent = 1.136 g cm -3 (at 20°C) and the relative permittivity er, measured at 2 MHz, is Er = 68. The experimentally determined viscosity is nso~vent= 9.14 cP. So, taking Ppnlyst. 1.049 g cm -3 , the spheres' sedimentation velocity (buoyant) is 2.9 × 10-~ m s -~ and can be neglected. The suspension is carefully deionized by a slow agitation with a mixed-bed resin (Amberlite MB-3) for 24 h, and every surface (quartz or glass) in contact with the suspension is previously washed using deionized water and resin. After such a treatment the suspension, entirely crystallized, is transferred in the measuring cell where it shows the characteristic iridescence of a longrange ordered suspension in the liquid state. The colloidal suspension is investigated with quasi-elastic light scattering (QELS), by recording the scattered intensity S(k) with k = Iki - l ~ l = 47rn/kosin(Od/2) where Oa is the scattering angle (see Fig. 1). The same diagnostic is used, in another wave-length range in crystallography. The suspension of spherical latex particles experiences a shear stress in a cylindrical Couette flow. The Couette cell is made of transparent fused silica (quartz): this material is preferred to glass because of its low refractive index and also because of the absence of ionic impurities on the quartz walls. The cell is described in Fig. l a. It allows a rotating Couette flow with a maximum shear rate: 7max = 80 Hz. The velocity profile is linear with a theoretical variation less than 5% in the central part of the flow where the measurements are made. The use of glycerol-water solvent coupled with the rather low shear rates prevents the apparition of the Taylor instability. Moreover, to reduce the quantity of suspension and to avoid the onset of secondary flow, the cell's bottom is filled with C6F6E, a =
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
18
MATHIS, BOSSIS, AND BRADY
a
b Y
FIG. 1. (a) The rotating Couette cell; R~ = 3.975 cm, R2 = 4.440 cm. (b) The angular convention; the above position of ki is kept constant during the experiments.For - 180 < 0d < 0, k is in the compressional range.
liquid which exhibits a perfect neutrality with respect to the suspension (p = 1.70 g c m -3, ~/ = 7.25 cP). To avoid a contamination of the suspension by the carbon dioxide of the air we maintain a permanent atmosphere of Ar above the suspension's free surface. The indexmatching vessel (Fig. l) contains eucalyptol or a mixture of glycerol and water in order to minimize the refraction of either incident or emerging beams. However, to bypass various opto-geometrical problems (multiple scattering and absorption of light, residual error in optical alignment during the detector's rotation, lack of the cylinder's circularity, etc.), we have used an original normalization method: we change the state of the suspension by adding some salt (CNaa = 10 -4 mole lit e r - 1). In doing so, we obtain a reference gaseous-like state, which is used to normalize all the other spectra. It must be noted that this method includes the division by F(0), the form factor of an isolated particle. Another apparatus, made of a small cylindrical quartz cell at the center of a glass vessel filled with eucalyptol, has also been used to obtain the form factor or static scattering spectra. Journal of Colloid and Interface Science, VoL !26, No. 1, November 1988
The light from a 35-mW H e - N e laser (~ = 6328 A) is attenuated ( T = 10%) and powermonitored. The incident beam, properly polarized and spatially filtered, is chopped at 30 Hz; scattered light falls on a photodiode which is rotating around a scattering region positioned at the center of the interval between the two cylinders. The signal is treated with a synchronous amplifier P.A.R. 124 A. Standard measurements taken many times during the experiment ensure that there is not more than 2% variation in intensity. The suspension does not significantly evolve during the course of an experiment. RESULTS
(1) Form Factor of the Particles In order to test the apparatus, a very dilute and nondeionized suspension has been studied to obtain the form factor of the particles. Presented in Fig. 2 are the theoretical form factors for spherical particles of 700, 750, and 910 in diameter and the experimental one for our particles: it appears that the average particle diameter is approximately 700 A which is
SHEAR-INDUCED
STRUCTURE,
19
II
1,
0,9,
O,B,
0,7,
0,6,
Theta.(°) 0,5
I 20
I 40
I 50
I 80
I 100
I 120
I ~40
I 160
I I80
FIG. 2. F o r m factors o f particles. (. • • ) Experimental points; ( - - - ) theoretical curves.
consistent with our previous determination o f do.
(3) Sheared Colloidal Suspensions Measurements with a shear stress have been done with a suspension at a concentration @ = 0.0132. The interparticular distance dexp = 2~-/kmax is found to be 2460 & at zero shear. The D e b y e - H f i c k e l length K-1 = (F 2. ZCiZ~/RTEoEr)-1/2 is 282 & in the case of a completely deionized suspension. The suspension is subjected to shear rates reported in Table III with corresponding Peclet numbers Pe = 67rna33,/kTand Taylor numbers Ta = 4¢oZd4/p 2 × (R2/(R 2 - R2)) where d = R2
(2) Static Structure Factors In the same manner, we have recorded static structure factors for many concentrations of suspended particles. In Fig. 3 we see the difference between structure factors obtained in a crystallized and a liquid state. The completely deionized suspensions, crystallized or in liquid state, show a strong iridescence. Values o f the m a x i m a 0ma., kmax, and the corresponding interparticle distance doxp = 27r//~ax are given in Table II.
-- RI.
All the measurements are made in the geometry indicated in Fig. lb with a fixed k i . In
2O0 180
S (Thetad)(a. °)
c60 14o 120
/
100 80 5o, 40
The'cad(°)
20, o
I 20
I 40
I 60
I 80
I 100
I 120
FIG. 3. Static structure factors o f a • = 0.0033 suspension; ( - - ) solid state (crystal); (- • • ) liquid state. Journal of Colloid and Interface Science, Vol, 126, No. 1, November 1988
20
MATHIS, BOSSIS, AND BRADY TABLE II
Mean Interparticular Distances in Function of the Volumic Concentration, Determined with the Help of Recorded Static Structure Factors ~b × 103
1 2 -
-
3.3 -
-
5 -
-
8 10
State
Liquid --Crystal Liquid Crystal Liquid ---
Om~ (dg)
km~ X 10-5 (cm -~)
d~p (A)
46 58 61 75 74 88 89 102 112
1.03 1.28 1.34 1.61 1.59 1.84 1.86 2.06 2.19
6100 4910 4690 3900 3950 3420 3380 3050 2870
this case the wave vector k = ki - ko is rotating in the shear plane but is also changing in modulus. Recordings of S~(k) (where y corresponds to the shear rate) with 3" > 0 and 3, < 0 (reversed rotation of inner cylinder) show the symmetrical (with respect to 3') deformation of the spectra. (Figs. 4a and 4b). In the following the shear rate is positive as defined in Fig. 1. We can see in Fig. 4a that for 0d > 0 (the wave vector k is in the "dilatational range") the intensity at the m a x i m u m of SV(k) is decreasing with the shear rate; meanwhile, the intensity at the m a x i m u m is increasing for 0d < 0 (k is in the "compressional range"). Moreover, a displacement of 0max toward smaller values and a broadening of the scattering spectrum are visible with increasing shear rate. It is important to note that the angular displacement is not identical for 0d > 0 or 0d < 0.
Following the previous work of Ackerson and Clark (16), at low shear rates we can assume an elliptical deformation of the structure, expressed as g~(r) = g ° ( r [ l -
g~(r) =
g°(r) -- xy/r2rM3"rdg°/dr.
[2]
Equation [2 ] is a particular form of the more general expansion (cf. I) g~(r) = g°(r)(1 -
Pe/2f(r)xy/r2).
[3]
The Fourier transform of Eq. [2 ] allows us to obtain the structure factor S~(k), S~(k) =
S°(k) + kxkJk2"cM3"kdS°/dk,
[4]
or equivalently SV(k) = S°[k(1 +
kxky/k2zM3")].
[5]
Experimentally, we can determine S°(k), and S'~(k). Following Ackerson and Clark (16), we can obtain a value for ~M, from Eq. [5] and the positions of the m a x i m a of S'~(k). With the angular convention described in Fig. lb, Eq. [5] becomes
dS°(k)/dk,
(4) Comparison with the Low Shear Rate Linear Theory
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
[1]
The deformation is proportional to the shear rate and its relative amplitude Ar/r can be estimated by equating the elastic stress GAr[ r (where G is the modulus of rigidity) and the viscous stress ~3". With this hypothesis the normalizing time r is equal to the Maxwell relaxation time rM = 7 / G . On the other hand, in the case of a small shear rate, we can do a linear expansion of g ( r ) as a function of rMT,
S~(k)
The pair-distribution function g0(r), as well as the structure factor S ° ( k ) , exhibits a spherical symmetry in the case of a suspension of spheres at rest. These two functions are modified when the suspension is experiencing a shear stress, which reduces the symmetry of the system.
xy/r2r3,]).
= S°[k(1 +
TABLE
[ sin 0d ~
~)rM3")l.
..
[6]
III
Shear Rates and Dimensionless Numbers of the Flow (Ta~ = 3416) 3' (Hz)
Pe
Ta
6.39 12.79 25.58
1.43 × 10-2 2.86 × 10-2 5.72 × 10-2
65 261 1044
SHEAR-INDUCED STRUCTURE, II
21
a 260
240
S (Theta}(~. o)
220 200 180
180 t40 t20 100 80 I -180
......
d~ t
I
I
I
I
I
I
I
-135
-90
-45
45
90
135
180
b 240 220
S (Theta~(a...)
200 lflO 160 140 t20 100 80 60 I -180
I
I
I
I
I
I
l
%35
-90
-45
45
90
t35
t80
FIG. 4. Structure factorswith (a) positive applied shear rates; (b) negative applied shear rates. ( 1 ) Hz; (2) 3" = 6.39 Hz; (3) 3" = 12.79 Hz; (4) 3" = 25.58 Hz. Denoting kYmaxas the value o f k corresponding at the m a x i m u m of S~(k), o
krnax = kTmax(1 +
TM3`)
[7]
or rM3` = ( 2 / s i n 0%x) × (sin(0%x/2)/sin(0~max/2) - 1).
[8]
The results reported in Fig. 5 concern only the part of spectra showing an increasing intensity with 3' (the wave vectors k are on the compressional range). We get rM = 8.4 + 0.2 ms. Ackerson and Clark, exploring the same part of the spectra (16), had found approximately the same value, but with a suspension at ¢ = 0.0016 and do = 1090 ~. However, the Maxwell relaxation times obtained in the "di-
3' =
0
latational part" of each spectrum are strongly dependent on the shear rate. Anyway we keep ZM = 8.4 + 0.2 ms as a working value in the following. We can note that the values of rM3` seem small enough to justify the expansion [2]. Indeed, with the knowledge of rM, we are able to compare experimental curves and S~(k) as given by [41. We can write from Eq. [ 4]
$7(0) = S°(O) 4- TrM X {sin 0 sin(O/2)/cos(O/2)}dS°/dO.
[9]
Equation [ 9 ] gives theoretical spectra (Figs. 6b and 7) in which the evolution of the position of maxima is quite correct for the lower shear rate but where the intensity change is qualitatively correct only for the "compresJournal of Colloid and Interface Science, Vol. 126, No. 1, N o v e m b e r 1988
22
MATHIS, BOSSIS, AND BRADY
0,3 Tau~Gamma
0o25,
. ] "
0,2, Ooi5, O,l, O,05, ..............
0
Gamma(H,) I
I
5
I
t0
15
I
20
I
25
I
30
FIG. 5. Experimental zM'y a function of 3,: the resulting slope is r ~ = 8.4 ms.
a 285 S (Theta.) (....)
235
~.85
-15o
-IBO
b
Thetad(°)
-~2o
-90
a85 S (Theta~ (~...)
235
185
-~a0
/
~ /
-150
Thetad(°)
-tao
-90
FIG. 6. Structure factor S(0) in the neighborhood of 0%x: (a) experimental curves; (b) theoretical curves. (1) 3' = 0 Hz; (2) 3, = 6.39 Hz; (3) 3' = 12.79 Hz; (4) 3' = 25.58 Hz. Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
SHEAR-INDUCED STRUCTURE, II
23
950
S (Theta~(~. °.)
3O0
250
200
4
150
100
50
Ol -JBO
,\ -135
The t.ad (':')
I
I
-90
-45
I
45
I
90
I
135
I
t80
FIG. 7. Structure factor S(O) obtained by using Eq. [9] with rM = 8.4 ms. (1) 3" = 0 Hz; (2) 3" = 6.39 Hz; (3) 3' = 12.79 Hz; (4) 3" = 25.58 Hz.
sional range." The agreement is only qualitative; in particular, the experimental curves do not cross themselves at the same point (Figs. 6a and 6b) for each side of the spectra (the m a x i m a of S°(0)) as predicted by Eq. [4]. A more significative test of the linear theory can be made by comparing S°(O) with [S~(0) + S - ~ ( 0 ) ] / 2 . Indeed these two quantities should be identical if the structure deformation were linear in % whatever the deformation function f ( r ) m a y be. The difference, shown in Fig. 8, shows the discrepancy between theoretical and experimental results, which can reach the same order of magnitude as the difference S ~ - S o itself.
DISCUSSION The comparison between experiments and theoretical predictions based on a linear theory shows clearly that this theory does not describe completely the rearranging p h e n o m e n a of the particles for Peclet numbers of a few percent. In particular, this theory does not give a correct prediction about the observed dissymmetry of evolution of the two parts, (compression and dilatation) of the same spectrum. Actually, this dissymmetry is evident in Fig. 4, where the intensity is falling rapidly with the shear rate on the extensional side, whereas we note only a small increase with the shear
4O
30
A
(s'+~)/2-s ° (~...,
/~
Thetaa(°) -20 -180
-90
90
180
FIG. 8. The difference [S~(0) + S-~(0)]/2 - S°(O). (1) 3, = 6.39 Hz; (2) 3" = 12.79 Hz; (3) 3' = 25.58 Hz. Journal of Colloid and Interface Science, Vol. 126, No. 1, November I988
24
MATHIS, BOSSIS, A N D B R A D Y
rate on the compressional side. This behavior is very similar to the one we have observed by numerical simulation (cf. I, Fig. 7 ), where we can see the same dissymmetry in the angular pair-distribution function when the Peclet number changes from 0.01 to 0.1 with the softsphere system. We have emphasized in I that the condition of validity of the linear expansion in the shear rate was Pe r* 2 < 1 rather than Pe < 1, where r* is the normalized effective radius of the particles corresponding to the half of the nearest-neighbor equilibrium separation. In the simulation we have found that the upper limit of the linear regime was roughly Pe r* 2 = 0.5. In the experiments Pe r* 2 ranges from 0.15 to 0.60. For the lowest shear rate the agreement with the linear theory is not too bad (compare the curves 1 and 2 of Fig. 7 with those of Fig. 4, or also compare Fig. 6a to Fig. 6b for the curves 1 and 2), except near the maximum of S(k). Thus, the limiting value of Pc r* 2 for a linear behavior, found by simulation, is at least three times larger than that obtained by experiment. This is very likely due to a difference in structure a n d / o r interparticle potential between the simulated and the real system. For instance, our simulations are for a monolayer and the equivalent 3D volume fraction: cI, 0.001 (which is representative of the lowest density compatible with the existence of a liquid or a crystalline state) is well below the experimental one (0.013). Nevertheless, the scaling factor should account for this difference. Another possible explanation comes from the presence of a large deformable ionic cloud in these systems. When the particles are pushed toward each other by the shear force, the ionic cloud of each particle is deformed, and, in first approximation, this deformation gives rise to opposing dipoles aligned along the interparticle axes. This dipolar potential is long ranged and repulsive; it will act in the same direction as the shearing force helping to repel the particles from each other on the extensional side. This supplementary interaction could explain why the structure is easier Journal of Colloid and Interface Science, Vol. 126,No. 1, November1988
to deform in these experiments than that predicted by simulation. CONCLUSION
We have shown that for Peclet numbers as low as a few percent the deformation of the structure was not linear in the shear rate. In particular, whereas the linear theory can apply to the compressional side and be useful in predicting the order of magnitude of the Maxwell relaxation time, it fails completely on the extensional side. These observations apply to colloidal systems whose principal characteristic is the long range of the repulsive DebyeHtickel potential. A possible explanation of the difference in behavior as a function of the Peclet number between the experimental and the simulated system could come from the deformation of the ionic cloud during the rotation of particles around each other. Other experiments with differing concentrations (0.02 > ,I~ > 0.001), and therefore differing values of the effective radius r*, are needed in order to confirm the behavior of the structure factor as a function of Pe r* 2. Furthermore we plan to use birefringence and dichroism to obtain more information on the angular structure induced by the flow. A new cell has been designed that will at the same time permit us to cover a larger range of angles (0d E [--180, + 180]) and also to explore the different angles while keeping constant the modulus of k. On the other hand, a more complete comparison with numerical simulations would need 3D runs with a few hundred particles. This would be too costly in computer time with full Stokesian dynamics but for particles interacting through a long-range repulsive potential, some simplification of the hydrodynamic interactions can be allowed and we are currently testing the validity of these approximations. ACKNOWLEDGMENTS We thank Mr. P. Durual of Atochem who has kindly furnished us with the CrF6E. We are also indebted to Gaston Maruejouls and Robert Clapier for their technical
SHEAR-INDUCED STRUCTURE, II support. This work is supported by the CNRS (ATP MIAM).
REFERENCES 1. Pieranski, P., Contemp. Phys. 24, 25 (1983). 2. Williams, R., and Crandall, R. S., Phys. Lett. A 48(3), 225 (1974). 3. Hiltner, P. A., and Krieger, I. M., J. Phys. Chem. 73(7), 2386 (1969). 4. Chaikin, P. M., Di Meglio, J. M., Dozier, W. D., Lindsay, H. M., and Weitz, D. A., in press. 5. Mitaku, S., Ohtsuki, T., Katsumi, E., Kishimoto, A., and Okano, K., Japan. J. Appl. Phys. 17(2), 305 (1978). 6. Zeichner, G. R., and Schowalter, W. R., J. Colloid Interface Sci. 71, 237 (1979). 7. Bensley, C. N., and Hunter, R. J., J. Colloidlnterface Sci. 92, 436 (1983). 8. Bensley, C. N., and Hunter, R. J., J. Colloidlnterface Sci. 92, 448 (1983). 9. De Rooy, N., De Bruyn, P. L., and Overbeek, J. T., J. Colloid Interface Sci. 75, 542 (1980). 10. Schaefer, D. W., J. Chem. Phys. 66, 3980 (1977). 11. Williams, R., Crandall, R. S., and Wojtowicz, P. J., Phys. Rev. Lett. 37(6), 348 (1976).
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12. Hartl, W., and Versmold, H., J. Chem. Phys. 81(5), 2507 (1984). 13. Hartl, W., and Versmold, H., Z. Phys. Chem. Neue F. 139, 247 (1984). 14. Pusey, P. N., and Van Megen, W., in "Complex and Supermolecular Fluids" (S. A. Safran, Ed.). WileyInterscience, New York, 1986. 15. Fujita, H., and Ametani, K., Japan. J. AppL Phys. 16(7), 1091 (1977). 16. Ackerson, B. J., and Clark, N. A., Physica A 118, 221 (1983). 17. Kops-Werkhoven, M. M., and Fijnaut, H. M., J. Chem. Phys. 77(5), 2242 (1982). 18. Nieuwenhuis, E. A., and Vrij, A., J. Colloidlnterface Sci. 72, 321 (1979). 19. Tomita, M., Takano, K., and Van de Ven, T. G. M., J. Colloid Interface Sci. 92, 367 (1983). 20. Brown, J. C., Pusey, P. N., Goodwin, J. W., and Ottewill, R. H., J. Phys. A 8(5), 664 (1975). 21. Ackerson, B. J., Hayter, J. B., Clark, N. A., and Cotter, L., J. Chem. Phys. 84(4), 2344 (1986). 22. Clark, N. A., and Ackerson, B. J., Phys. Rev. Lett. 44(15), 1005 (1980). 23. Hanley, M. J. H., Rainwater, J. C., Clark, N. A., and Ackerson, B. J., J. Chem. Phys. 79(9), 4448 (1983). 24. Griiner, F., and Lehmann, W. P., J. Phys. A 15, 2847 (1982).
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
We study, by light diffusion, the deformation of the pair distribution functionof a colloidalsuspension in a cylindrical Couette flow. The suspension is made of monodisperse latex particles (do = 750 A) at a volume fraction • = 1.3%. At equilibrium, the suspension has a liquid-like structure. The structure factor S(k) is recorded for different orientations of the diffusion wave vector in the plane of shear for Peclet numbers (Pc = 3,a2/Do) between 10-2 and 10-1. In order to get rid of different spurious causes of scattering the spectra are normalized by the same suspension in a gas-like state, obtained by the addition of salt. The structure is shown to deform asymmetricallywhen we increase the shear rate: it increases slowlywith a small linear shift of the maximum on the compression side whereas the intensity falls rapidly on the dilatation side. These experimental results are discussed with the help of numerical simulations on a similar system. © 1988AcademicPress,Inc. the structure of suspension. Light scattering (12, 16-20) and neutron scattering (21) are well-adapted technics (convenient wavelength) to study the structural changes of these systems. The effect of a shear on the structure was first demonstrated by Ackerson and Clark (16, 22 ) using a Couette cell and their results were compared to molecular dynamics computer simulations (23). This paper is the experimental part of a whole study devoted to the investigation of the structure deformation in sheared colloidal suspensions. The first part (referred to as I) reviews the theoretical approaches and compares them to the numerical simulation results. Using light diffusion technics we have studied the deformation of the structure factor in a large angular range, allowing us to compare significantly our results with the structure factors given by a linear approximation (cf. I).
INTRODUCTION
A colloidal s u s p e n s i o n can be defined as a diphasic system made of submicron solid particles in a liquid solvent. It follows that paint, blood, clay, and so on belong to that category. The study of physics of such materials is made difficult by the number of the constituents and by their complex interactions. So it is convenient to use a simple model suspension of polystyrene spheres (latex), with a sharp distribution of sizes, in a simple and neutral solvent. The aim of studies devoted to such systems is the understanding of the different interact i o n s - p a r t i c l e / particle, particle/solvent-and the correlation between microscopic structure and macroscopic properties (viscosity, electrical conductivity, mass transport, etc.). Numerous papers have been published and concerned with: (i) the growth and structure of the colloidal crystals in a bi- or tridimensional domain (1-5); (ii) the coagulation phenomena ( 6 - 9 ) ; and (iii) the effects of temperature (10-12), ionic force of the solvent (13), and sphere concentration (14, 15 ) on
MATERIALS AND METHODS
The particles are polystyrene spheres negatively charged ( - S O 4 groups bound to the surface) and supplied in aqueous solution by 16
002 1-9797/88 $3.00 Copyright© 1988 by AcademicPress, Inc. All rights of reproduction in any form reserved.
Journalof Colloidand InterfaceScience,Vol. 126, No. 1, November 1988
SHEAR-INDUCED STRUCTURE, II
Dow Chemical Co. The indicated mean diameter is do = 2a = 910 A with a dispersion = +29 A; the volumic fraction is q~ = 0.1. We have determined • by weighing samples before and after evaporating the water: = 0.106 + 0.003. The mean diameter has been measured by electronic microscopy, taking care to neglect the very few particularly small particles; the results are reported in Table I. Other measurements, made in our laboratory by photon correlation spectroscopy, give a hydrodynamic diameter of 860 A but the real diameter is usually 10% lower. We must be careful in announcing a value of do for many reasons: (i) measurements by PCS are not very precise, (ii) the hydrodynamic diameter is often different from the real diameter, and (iii) measurements by electronic microscopy are made with a relatively weak number of spheres (---250). In any event it seems reasonable to use do = 750 A as a mean value in our calculations, a value smaller by nearly 20% than the one reported by Dow Chemical Co. We must note that a smaller particle diameter could be an alternative reason to explain the systematic difference between the latex concentration calculated after scattering experiments and that determined from a dilution factor in a paper of Grtiner and Lehmann (24). We have kept the electric charge value proposed by Grtiner and Lehmann (24) and that is compatible with other studies (16): 1000e- per particle. The suspending fluid is a mixture of glycerol and water carefully distilled and deionized TABLE I Determination of the Mean Diameter of the Particles by Electronic Microscopy Measurements Sample
(~)(~)
o(A)
1 2 3 4
742 714 747 699
19 20 32 45
Note. The enlarging ratio of the electronic microscope has been verified.
17
(53% glycerol). This choice results from a compromise imposed by several necessities: (i) to match the suspension optical index with that of the cell's quartz walls (/'/quartz = 1.458, nsolvent= 1.407, and nsuso~ns.= 1.407 + 0.193/b), (ii) to reduce the multiple scattering in the suspension, and (iii) to avoid the Taylor instability under shear. The density of this fluid is Psolvent = 1.136 g cm -3 (at 20°C) and the relative permittivity er, measured at 2 MHz, is Er = 68. The experimentally determined viscosity is nso~vent= 9.14 cP. So, taking Ppnlyst. 1.049 g cm -3 , the spheres' sedimentation velocity (buoyant) is 2.9 × 10-~ m s -~ and can be neglected. The suspension is carefully deionized by a slow agitation with a mixed-bed resin (Amberlite MB-3) for 24 h, and every surface (quartz or glass) in contact with the suspension is previously washed using deionized water and resin. After such a treatment the suspension, entirely crystallized, is transferred in the measuring cell where it shows the characteristic iridescence of a longrange ordered suspension in the liquid state. The colloidal suspension is investigated with quasi-elastic light scattering (QELS), by recording the scattered intensity S(k) with k = Iki - l ~ l = 47rn/kosin(Od/2) where Oa is the scattering angle (see Fig. 1). The same diagnostic is used, in another wave-length range in crystallography. The suspension of spherical latex particles experiences a shear stress in a cylindrical Couette flow. The Couette cell is made of transparent fused silica (quartz): this material is preferred to glass because of its low refractive index and also because of the absence of ionic impurities on the quartz walls. The cell is described in Fig. l a. It allows a rotating Couette flow with a maximum shear rate: 7max = 80 Hz. The velocity profile is linear with a theoretical variation less than 5% in the central part of the flow where the measurements are made. The use of glycerol-water solvent coupled with the rather low shear rates prevents the apparition of the Taylor instability. Moreover, to reduce the quantity of suspension and to avoid the onset of secondary flow, the cell's bottom is filled with C6F6E, a =
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
18
MATHIS, BOSSIS, AND BRADY
a
b Y
FIG. 1. (a) The rotating Couette cell; R~ = 3.975 cm, R2 = 4.440 cm. (b) The angular convention; the above position of ki is kept constant during the experiments.For - 180 < 0d < 0, k is in the compressional range.
liquid which exhibits a perfect neutrality with respect to the suspension (p = 1.70 g c m -3, ~/ = 7.25 cP). To avoid a contamination of the suspension by the carbon dioxide of the air we maintain a permanent atmosphere of Ar above the suspension's free surface. The indexmatching vessel (Fig. l) contains eucalyptol or a mixture of glycerol and water in order to minimize the refraction of either incident or emerging beams. However, to bypass various opto-geometrical problems (multiple scattering and absorption of light, residual error in optical alignment during the detector's rotation, lack of the cylinder's circularity, etc.), we have used an original normalization method: we change the state of the suspension by adding some salt (CNaa = 10 -4 mole lit e r - 1). In doing so, we obtain a reference gaseous-like state, which is used to normalize all the other spectra. It must be noted that this method includes the division by F(0), the form factor of an isolated particle. Another apparatus, made of a small cylindrical quartz cell at the center of a glass vessel filled with eucalyptol, has also been used to obtain the form factor or static scattering spectra. Journal of Colloid and Interface Science, VoL !26, No. 1, November 1988
The light from a 35-mW H e - N e laser (~ = 6328 A) is attenuated ( T = 10%) and powermonitored. The incident beam, properly polarized and spatially filtered, is chopped at 30 Hz; scattered light falls on a photodiode which is rotating around a scattering region positioned at the center of the interval between the two cylinders. The signal is treated with a synchronous amplifier P.A.R. 124 A. Standard measurements taken many times during the experiment ensure that there is not more than 2% variation in intensity. The suspension does not significantly evolve during the course of an experiment. RESULTS
(1) Form Factor of the Particles In order to test the apparatus, a very dilute and nondeionized suspension has been studied to obtain the form factor of the particles. Presented in Fig. 2 are the theoretical form factors for spherical particles of 700, 750, and 910 in diameter and the experimental one for our particles: it appears that the average particle diameter is approximately 700 A which is
SHEAR-INDUCED
STRUCTURE,
19
II
1,
0,9,
O,B,
0,7,
0,6,
Theta.(°) 0,5
I 20
I 40
I 50
I 80
I 100
I 120
I ~40
I 160
I I80
FIG. 2. F o r m factors o f particles. (. • • ) Experimental points; ( - - - ) theoretical curves.
consistent with our previous determination o f do.
(3) Sheared Colloidal Suspensions Measurements with a shear stress have been done with a suspension at a concentration @ = 0.0132. The interparticular distance dexp = 2~-/kmax is found to be 2460 & at zero shear. The D e b y e - H f i c k e l length K-1 = (F 2. ZCiZ~/RTEoEr)-1/2 is 282 & in the case of a completely deionized suspension. The suspension is subjected to shear rates reported in Table III with corresponding Peclet numbers Pe = 67rna33,/kTand Taylor numbers Ta = 4¢oZd4/p 2 × (R2/(R 2 - R2)) where d = R2
(2) Static Structure Factors In the same manner, we have recorded static structure factors for many concentrations of suspended particles. In Fig. 3 we see the difference between structure factors obtained in a crystallized and a liquid state. The completely deionized suspensions, crystallized or in liquid state, show a strong iridescence. Values o f the m a x i m a 0ma., kmax, and the corresponding interparticle distance doxp = 27r//~ax are given in Table II.
-- RI.
All the measurements are made in the geometry indicated in Fig. lb with a fixed k i . In
2O0 180
S (Thetad)(a. °)
c60 14o 120
/
100 80 5o, 40
The'cad(°)
20, o
I 20
I 40
I 60
I 80
I 100
I 120
FIG. 3. Static structure factors o f a • = 0.0033 suspension; ( - - ) solid state (crystal); (- • • ) liquid state. Journal of Colloid and Interface Science, Vol, 126, No. 1, November 1988
20
MATHIS, BOSSIS, AND BRADY TABLE II
Mean Interparticular Distances in Function of the Volumic Concentration, Determined with the Help of Recorded Static Structure Factors ~b × 103
1 2 -
-
3.3 -
-
5 -
-
8 10
State
Liquid --Crystal Liquid Crystal Liquid ---
Om~ (dg)
km~ X 10-5 (cm -~)
d~p (A)
46 58 61 75 74 88 89 102 112
1.03 1.28 1.34 1.61 1.59 1.84 1.86 2.06 2.19
6100 4910 4690 3900 3950 3420 3380 3050 2870
this case the wave vector k = ki - ko is rotating in the shear plane but is also changing in modulus. Recordings of S~(k) (where y corresponds to the shear rate) with 3" > 0 and 3, < 0 (reversed rotation of inner cylinder) show the symmetrical (with respect to 3') deformation of the spectra. (Figs. 4a and 4b). In the following the shear rate is positive as defined in Fig. 1. We can see in Fig. 4a that for 0d > 0 (the wave vector k is in the "dilatational range") the intensity at the m a x i m u m of SV(k) is decreasing with the shear rate; meanwhile, the intensity at the m a x i m u m is increasing for 0d < 0 (k is in the "compressional range"). Moreover, a displacement of 0max toward smaller values and a broadening of the scattering spectrum are visible with increasing shear rate. It is important to note that the angular displacement is not identical for 0d > 0 or 0d < 0.
Following the previous work of Ackerson and Clark (16), at low shear rates we can assume an elliptical deformation of the structure, expressed as g~(r) = g ° ( r [ l -
g~(r) =
g°(r) -- xy/r2rM3"rdg°/dr.
[2]
Equation [2 ] is a particular form of the more general expansion (cf. I) g~(r) = g°(r)(1 -
Pe/2f(r)xy/r2).
[3]
The Fourier transform of Eq. [2 ] allows us to obtain the structure factor S~(k), S~(k) =
S°(k) + kxkJk2"cM3"kdS°/dk,
[4]
or equivalently SV(k) = S°[k(1 +
kxky/k2zM3")].
[5]
Experimentally, we can determine S°(k), and S'~(k). Following Ackerson and Clark (16), we can obtain a value for ~M, from Eq. [5] and the positions of the m a x i m a of S'~(k). With the angular convention described in Fig. lb, Eq. [5] becomes
dS°(k)/dk,
(4) Comparison with the Low Shear Rate Linear Theory
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
[1]
The deformation is proportional to the shear rate and its relative amplitude Ar/r can be estimated by equating the elastic stress GAr[ r (where G is the modulus of rigidity) and the viscous stress ~3". With this hypothesis the normalizing time r is equal to the Maxwell relaxation time rM = 7 / G . On the other hand, in the case of a small shear rate, we can do a linear expansion of g ( r ) as a function of rMT,
S~(k)
The pair-distribution function g0(r), as well as the structure factor S ° ( k ) , exhibits a spherical symmetry in the case of a suspension of spheres at rest. These two functions are modified when the suspension is experiencing a shear stress, which reduces the symmetry of the system.
xy/r2r3,]).
= S°[k(1 +
TABLE
[ sin 0d ~
~)rM3")l.
..
[6]
III
Shear Rates and Dimensionless Numbers of the Flow (Ta~ = 3416) 3' (Hz)
Pe
Ta
6.39 12.79 25.58
1.43 × 10-2 2.86 × 10-2 5.72 × 10-2
65 261 1044
SHEAR-INDUCED STRUCTURE, II
21
a 260
240
S (Theta}(~. o)
220 200 180
180 t40 t20 100 80 I -180
......
d~ t
I
I
I
I
I
I
I
-135
-90
-45
45
90
135
180
b 240 220
S (Theta~(a...)
200 lflO 160 140 t20 100 80 60 I -180
I
I
I
I
I
I
l
%35
-90
-45
45
90
t35
t80
FIG. 4. Structure factorswith (a) positive applied shear rates; (b) negative applied shear rates. ( 1 ) Hz; (2) 3" = 6.39 Hz; (3) 3" = 12.79 Hz; (4) 3" = 25.58 Hz. Denoting kYmaxas the value o f k corresponding at the m a x i m u m of S~(k), o
krnax = kTmax(1 +
TM3`)
[7]
or rM3` = ( 2 / s i n 0%x) × (sin(0%x/2)/sin(0~max/2) - 1).
[8]
The results reported in Fig. 5 concern only the part of spectra showing an increasing intensity with 3' (the wave vectors k are on the compressional range). We get rM = 8.4 + 0.2 ms. Ackerson and Clark, exploring the same part of the spectra (16), had found approximately the same value, but with a suspension at ¢ = 0.0016 and do = 1090 ~. However, the Maxwell relaxation times obtained in the "di-
3' =
0
latational part" of each spectrum are strongly dependent on the shear rate. Anyway we keep ZM = 8.4 + 0.2 ms as a working value in the following. We can note that the values of rM3` seem small enough to justify the expansion [2]. Indeed, with the knowledge of rM, we are able to compare experimental curves and S~(k) as given by [41. We can write from Eq. [ 4]
$7(0) = S°(O) 4- TrM X {sin 0 sin(O/2)/cos(O/2)}dS°/dO.
[9]
Equation [ 9 ] gives theoretical spectra (Figs. 6b and 7) in which the evolution of the position of maxima is quite correct for the lower shear rate but where the intensity change is qualitatively correct only for the "compresJournal of Colloid and Interface Science, Vol. 126, No. 1, N o v e m b e r 1988
22
MATHIS, BOSSIS, AND BRADY
0,3 Tau~Gamma
0o25,
. ] "
0,2, Ooi5, O,l, O,05, ..............
0
Gamma(H,) I
I
5
I
t0
15
I
20
I
25
I
30
FIG. 5. Experimental zM'y a function of 3,: the resulting slope is r ~ = 8.4 ms.
a 285 S (Theta.) (....)
235
~.85
-15o
-IBO
b
Thetad(°)
-~2o
-90
a85 S (Theta~ (~...)
235
185
-~a0
/
~ /
-150
Thetad(°)
-tao
-90
FIG. 6. Structure factor S(0) in the neighborhood of 0%x: (a) experimental curves; (b) theoretical curves. (1) 3' = 0 Hz; (2) 3, = 6.39 Hz; (3) 3' = 12.79 Hz; (4) 3' = 25.58 Hz. Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988
SHEAR-INDUCED STRUCTURE, II
23
950
S (Theta~(~. °.)
3O0
250
200
4
150
100
50
Ol -JBO
,\ -135
The t.ad (':')
I
I
-90
-45
I
45
I
90
I
135
I
t80
FIG. 7. Structure factor S(O) obtained by using Eq. [9] with rM = 8.4 ms. (1) 3" = 0 Hz; (2) 3" = 6.39 Hz; (3) 3' = 12.79 Hz; (4) 3" = 25.58 Hz.
sional range." The agreement is only qualitative; in particular, the experimental curves do not cross themselves at the same point (Figs. 6a and 6b) for each side of the spectra (the m a x i m a of S°(0)) as predicted by Eq. [4]. A more significative test of the linear theory can be made by comparing S°(O) with [S~(0) + S - ~ ( 0 ) ] / 2 . Indeed these two quantities should be identical if the structure deformation were linear in % whatever the deformation function f ( r ) m a y be. The difference, shown in Fig. 8, shows the discrepancy between theoretical and experimental results, which can reach the same order of magnitude as the difference S ~ - S o itself.
DISCUSSION The comparison between experiments and theoretical predictions based on a linear theory shows clearly that this theory does not describe completely the rearranging p h e n o m e n a of the particles for Peclet numbers of a few percent. In particular, this theory does not give a correct prediction about the observed dissymmetry of evolution of the two parts, (compression and dilatation) of the same spectrum. Actually, this dissymmetry is evident in Fig. 4, where the intensity is falling rapidly with the shear rate on the extensional side, whereas we note only a small increase with the shear
4O
30
A
(s'+~)/2-s ° (~...,
/~
Thetaa(°) -20 -180
-90
90
180
FIG. 8. The difference [S~(0) + S-~(0)]/2 - S°(O). (1) 3, = 6.39 Hz; (2) 3" = 12.79 Hz; (3) 3' = 25.58 Hz. Journal of Colloid and Interface Science, Vol. 126, No. 1, November I988
24
MATHIS, BOSSIS, A N D B R A D Y
rate on the compressional side. This behavior is very similar to the one we have observed by numerical simulation (cf. I, Fig. 7 ), where we can see the same dissymmetry in the angular pair-distribution function when the Peclet number changes from 0.01 to 0.1 with the softsphere system. We have emphasized in I that the condition of validity of the linear expansion in the shear rate was Pe r* 2 < 1 rather than Pe < 1, where r* is the normalized effective radius of the particles corresponding to the half of the nearest-neighbor equilibrium separation. In the simulation we have found that the upper limit of the linear regime was roughly Pe r* 2 = 0.5. In the experiments Pe r* 2 ranges from 0.15 to 0.60. For the lowest shear rate the agreement with the linear theory is not too bad (compare the curves 1 and 2 of Fig. 7 with those of Fig. 4, or also compare Fig. 6a to Fig. 6b for the curves 1 and 2), except near the maximum of S(k). Thus, the limiting value of Pc r* 2 for a linear behavior, found by simulation, is at least three times larger than that obtained by experiment. This is very likely due to a difference in structure a n d / o r interparticle potential between the simulated and the real system. For instance, our simulations are for a monolayer and the equivalent 3D volume fraction: cI, 0.001 (which is representative of the lowest density compatible with the existence of a liquid or a crystalline state) is well below the experimental one (0.013). Nevertheless, the scaling factor should account for this difference. Another possible explanation comes from the presence of a large deformable ionic cloud in these systems. When the particles are pushed toward each other by the shear force, the ionic cloud of each particle is deformed, and, in first approximation, this deformation gives rise to opposing dipoles aligned along the interparticle axes. This dipolar potential is long ranged and repulsive; it will act in the same direction as the shearing force helping to repel the particles from each other on the extensional side. This supplementary interaction could explain why the structure is easier Journal of Colloid and Interface Science, Vol. 126,No. 1, November1988
to deform in these experiments than that predicted by simulation. CONCLUSION
We have shown that for Peclet numbers as low as a few percent the deformation of the structure was not linear in the shear rate. In particular, whereas the linear theory can apply to the compressional side and be useful in predicting the order of magnitude of the Maxwell relaxation time, it fails completely on the extensional side. These observations apply to colloidal systems whose principal characteristic is the long range of the repulsive DebyeHtickel potential. A possible explanation of the difference in behavior as a function of the Peclet number between the experimental and the simulated system could come from the deformation of the ionic cloud during the rotation of particles around each other. Other experiments with differing concentrations (0.02 > ,I~ > 0.001), and therefore differing values of the effective radius r*, are needed in order to confirm the behavior of the structure factor as a function of Pe r* 2. Furthermore we plan to use birefringence and dichroism to obtain more information on the angular structure induced by the flow. A new cell has been designed that will at the same time permit us to cover a larger range of angles (0d E [--180, + 180]) and also to explore the different angles while keeping constant the modulus of k. On the other hand, a more complete comparison with numerical simulations would need 3D runs with a few hundred particles. This would be too costly in computer time with full Stokesian dynamics but for particles interacting through a long-range repulsive potential, some simplification of the hydrodynamic interactions can be allowed and we are currently testing the validity of these approximations. ACKNOWLEDGMENTS We thank Mr. P. Durual of Atochem who has kindly furnished us with the CrF6E. We are also indebted to Gaston Maruejouls and Robert Clapier for their technical
SHEAR-INDUCED STRUCTURE, II support. This work is supported by the CNRS (ATP MIAM).
REFERENCES 1. Pieranski, P., Contemp. Phys. 24, 25 (1983). 2. Williams, R., and Crandall, R. S., Phys. Lett. A 48(3), 225 (1974). 3. Hiltner, P. A., and Krieger, I. M., J. Phys. Chem. 73(7), 2386 (1969). 4. Chaikin, P. M., Di Meglio, J. M., Dozier, W. D., Lindsay, H. M., and Weitz, D. A., in press. 5. Mitaku, S., Ohtsuki, T., Katsumi, E., Kishimoto, A., and Okano, K., Japan. J. Appl. Phys. 17(2), 305 (1978). 6. Zeichner, G. R., and Schowalter, W. R., J. Colloid Interface Sci. 71, 237 (1979). 7. Bensley, C. N., and Hunter, R. J., J. Colloidlnterface Sci. 92, 436 (1983). 8. Bensley, C. N., and Hunter, R. J., J. Colloidlnterface Sci. 92, 448 (1983). 9. De Rooy, N., De Bruyn, P. L., and Overbeek, J. T., J. Colloid Interface Sci. 75, 542 (1980). 10. Schaefer, D. W., J. Chem. Phys. 66, 3980 (1977). 11. Williams, R., Crandall, R. S., and Wojtowicz, P. J., Phys. Rev. Lett. 37(6), 348 (1976).
25
12. Hartl, W., and Versmold, H., J. Chem. Phys. 81(5), 2507 (1984). 13. Hartl, W., and Versmold, H., Z. Phys. Chem. Neue F. 139, 247 (1984). 14. Pusey, P. N., and Van Megen, W., in "Complex and Supermolecular Fluids" (S. A. Safran, Ed.). WileyInterscience, New York, 1986. 15. Fujita, H., and Ametani, K., Japan. J. AppL Phys. 16(7), 1091 (1977). 16. Ackerson, B. J., and Clark, N. A., Physica A 118, 221 (1983). 17. Kops-Werkhoven, M. M., and Fijnaut, H. M., J. Chem. Phys. 77(5), 2242 (1982). 18. Nieuwenhuis, E. A., and Vrij, A., J. Colloidlnterface Sci. 72, 321 (1979). 19. Tomita, M., Takano, K., and Van de Ven, T. G. M., J. Colloid Interface Sci. 92, 367 (1983). 20. Brown, J. C., Pusey, P. N., Goodwin, J. W., and Ottewill, R. H., J. Phys. A 8(5), 664 (1975). 21. Ackerson, B. J., Hayter, J. B., Clark, N. A., and Cotter, L., J. Chem. Phys. 84(4), 2344 (1986). 22. Clark, N. A., and Ackerson, B. J., Phys. Rev. Lett. 44(15), 1005 (1980). 23. Hanley, M. J. H., Rainwater, J. C., Clark, N. A., and Ackerson, B. J., J. Chem. Phys. 79(9), 4448 (1983). 24. Griiner, F., and Lehmann, W. P., J. Phys. A 15, 2847 (1982).
Journal of Colloid and Interface Science, Vol. 126, No. 1, November 1988