J . Phys. Chem. 1990, 94, 1164-1171
1164
The data presented in Figure 9 and 10 also suggest that the ring and the whole molecule are undergoing anisotropic reorientation, and the aliphatic chain is undergoing multiple internal rotations about each C-C bond. Figure 11 gives a good graphical illustration of the mobility gradient along the chain with the chain mobility increasing as one moves away from the methine carbon (carbon 9, Figure 1). A plot of N T , vs carbon number, starting with the methine carbon 9 shows a linear dependence. According to the model for multiple internal rotation^,'^ this observation is indicative of equal rotational diffusion constants for all bonds in the chain except for the terminal methyl group. An increase in slope indicates that the rotational diffusion constant increases for all carbons, and this behavior persists even at the high pressure of 5000 bar. Preliminary theoretical analysis of the Tiand NOE data shows that both simple models involving overall and internal rotation and models involving anisotropic reorientations with multiple internal rotations do not reproduce the experimental data. It appears that one has to use a Cole-Davidson d i ~ t r i b u t i o nof~ ~ correlation times. This approach has successfully been used in our laboratory to interpret the temperature and pressure dependence of deuterium relaxation times in selectively deuterated glycerols.38 Availability of T , and NOE values for fluidity changes of 5 orders of magnitude offers a unique opportunity not only to test rigorously various theoretical models for dynamics of complex liquids but, for the first time, these experiments will allow one to characterize the density and temperature effects on the molecular dynamics in complex liquids, Theoretical analysis of the experimental T , and NOE data is in p r o g e ~ sin~ our ~ laboratory.
10 0
8 0
-5 -
6 0
* L
10
20
0 0
9
10 14
12
11
CARBON NUMBER
Figure 11. Dependence of N T , for chain carbons in EHB and EHC at 80 OC for 1 and 5000 bar: m, EHB, 80 OC, 1 bar; 0,EHB, 80 “C, 5000 bar; A, EHC, 80 OC, 1 bar; A, EHC, 80 OC, 5000 bar.
along the ethYlhexYl chain, the bwer the Parameter K . This signifies that its mobility is higher. In agreement with this approximative evaluation Of relative mobilities are the values Of R = T ~ ~ ~ / where T D , f m l n is the correlation time, calculated35from , 0.61 58 and the T , minimum at lower temperature, where W T ~ , = T D is the correlation time calculated from the Debye equation. For example, at -20 OC for EHB the R values for individual carbons exhibit the following progression R ( 5 ) > R(3,7) R(4,6) > ~ ( 1 0 , 1 4 )> R(11,12,13,15), The detailed analysis of K and R values will be given in detail elsewhere.36
Acknowledgment. This research was partially supported by the Air Force Office for Scientific Research under Grant AFOSR 89-0099 and the National Science Foundation under Grant NSF C H E 85-09870 and Grant NSF DMR 86-12860 and Shell Research (U.K.).
(35) Jonas, J.; Amdt, E. R. J . Magn. Reson. 1978, 32, 297. (36) Adamy, S.; Grandinetti, P. J.; Masuda, Y.;Campbell, D. Manuscript
in
(37) Davidson, D. W.; Cole, R. M. J . Chem. Phys. 1951, 19, 1484. (38) Wolfe, M.; Jonas, J. J . Chem. Phys. 1979, 71, 3252.
preparation.
Experimental Studies on the Rheology of Hard-Sphere Suspensions near the Glass Transition Louise Marshall and Charles F. Zukoski IV* Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801 (Received: July 17. 1989)
We have investigated the rheological behavior of sterically stabilized colloidal silica particles of three different sizes at volume fractions above 0.5. Despite a small surface charge, which elevated the intrinsic viscosity from the Einstein value of 2.5, the particles were found to behave essentially as hard spheres in the concentrated suspensions and to have properties highly reminiscent of molecular glasses. The zero shear rate viscosity, characteristic of disordered suspensions and present at all volume fractions, diverges as 6 0.6 and is well-described by the Doolittle equation for glassy flow. For suspensions with a relative zero shear rate viscosity greater than 5 X lo2,shear thickening was observed. Characteristic time scales for particle rearrangement determined from critical shear rates for shear thinning and shear thickening were found to follow trends predicted for molecular glasses. A transition from a liquidlike linear relaxation response to glassy stretched exponential behavior was observed as volume fraction was increased. The onset of the glassy relaxation response, indicative of nondecaying correlations, occurred near a volume fraction of 0.52.
-
I. Introduction Computer simulation and analytical studies of hard-sphere liquids have provided many insights into the origins and manifestations of the glass transition. 1-3 These studies suggest that ( 1 ) Woodcock, L. V. Ann. N.Y.Acad. Sci. 1981, 37, 274. (2) Jackle. J. Rep. Prog. Phys. 1986, 49, 171.
0022-3654/90/2094-1164$02.50/0
hard-sphere liquids fall out of equilibrium with the liquid state into a metastable vitreous phase (i.e., undergo a phase transition) at volume fractions between 0.50 and 0.64. The broad range of transition volume fractions arises from the different methods of (3) Angell, C. A.; Clarke, J. H. R.; Woodcock, L. V. Ado. Chem. Phys. 1981, 48, 397.
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990 1165
Rheology of Hard-Sphere Suspensions TABLE I: Particle Properties of Silica Sols TEM diameter, sol nm
hydrodynamic diameter,onm
A". g
Kb
lnlc
2.000 f 0.005 2.003 f 0.005 2.090 f 0.005
3.69 f 0.01 3.50 f 0.01 3.85 f 0.01
cm-'
~~
LMS20E LMS2 1E LMS22E
288 f 11 81.7 f 12 205 f 7
286 f 12 89 f 5 210f 10
1.846 f 0.001 1.747 f 0.001 1.843 f 0.001
a Particle diameter determined by dynamic light scattering in Decalin at 18 "C. bConstant relating suspension concentration to suspension viscosity as defined in eq 1 . Intrinsic viscosity defined by conversion of concentration to volume fraction, eq 2.
treating thermodynamic properties of the glass, which is characterized variously as having a random close packed (Bernal) structure, structural rigid it^,^ a heat capacity close to that of an equivalent crystalline solid at the same pressure,' and a vanishing self-diff~sivity.~ Confirmation of predictions of the equilibrium and dynamic behavior of the hard-sphere glassy state is hindered by a lack of suitable molecular liquids sufficiently hard-sphere-like to act as model systems. Recent studies of several classes of colloidal suspensions have shown that their equilibrium phase behavior and transport properties are well-described in terms of hard-sphere-like particle interaction~.~-l~ Pertinent to this discussion are observations of order-disorder phase transitions in colloidal suspensions with coexisting phases having volume fractions of 0.49 and 0.55, close to those predicted for molecular hard spheres.6J0 In addition, amorphous suspensions, readily distinguishable from those possessing long-range order by the lack of iridescence produced by Bragg scattering, are easily prepared at volume fractions greater than 0.5 and above a volume fraction of 0.55 show a limited tendency to crystallize over extended periods of time (months to years9). Pusey and van Megen9 have recently reported studies on the self-diffusivity in hard-sphere suspensions at volume fractions above 0.5 and find the development of structural relaxation times longer than can be measured experimentally at volume fractions of 0.56. This behavior is similar to that observed in molecular dynamics simulations of hard- and soft-sphere molecular liquids near the glass tran~itionl-~ and measurements of the time dependence of the density correlation function of an ionic glass." In this paper we investigate the flow behavior of hard-sphere-like suspensions at volume fractions in the glass transition region. We compare our results with the flow properties observed for molecular liquids, and those predicted for hard spheres near the glass transition, and find remarkable similarities. In these experiments we work at constant suspension volume and temperature. Thus, by changing the particle volume fraction, we are, in effect, changing the (osmotic) pressure on the system. We therefore feel privileged that this work is included in a Festschrift for Harry Drickamer, who has contributed so much to our understanding of the chemical and physical properties of matter through pressure changes. The remainder of this paper is structured as follows. In section I1 we describe our experimental system and the rheological studies performed, followed, in section 111, by a discussion of our results in light of data published on similar colloidal suspensions and on the hard-sphere-like nature of the particle interactions. In order to reconcile our zero stress viscosities over the entire volume fraction range, we argue that the particles carry a small surface charge, large enough to provide an intrinsic viscosity greater than the Einstein hard-sphere value of 2.5 but small enough that the (4) Stoessel, J. P.; Wolynes, W. P. J . Chem. Phys. 1984, 80, 4502. (5) Ottewill, R. H. In Colloidal Dispersions; Goodwin, J. W., Ed.; Royal Society: London, 1982; Chapter 9. (6) Vrij, A.; Jansen, J. W.; Dhont, J. K. G.; Pathmamanoharan, C.; Kopswerkhoven, M. M.; Fijnaut, H. M. Faraday Discuss. Chem. SOC.1983, No. 76, 19. (7) van Megen, W.; Underwood, S. M.; Ottewill, R. H.; Williams, N. St. J.; Pusey, P. N. Faraday Discuss. Chem. Soc. 1987, No. 83, 47. ( 8 ) Jansen, J. W.; de Kruif. C. G.; Vrij, A. J . Colloid Interface Sci. 1986, 114, 471. (9) Pusey, P. N.; van Megen, W. Phys. Reu. Lett. 1987,59, 2083. (IO) Hachisu, S.; Kobayashi, Y.; Kose, A. J . Colloid Interface Sci. 1973, 42, 342. ( 1 1 ) Ullo, J. J:; Yip, S. Phys. Reu. Let?. 1985, 54, 1509.
equilibrium thermodynamic properties of the suspensions at elevated volume fractions are not substantially altered from those of hard spheres. In section IV the zero stress creep compliance and steady-state viscosity of our suspensions are compared with similar studies on molecular liquids near the glass transition and with the predictions made for hard spheres. Finally, in section V we discuss our results and present our conclusions. 11. Experimental Section
System Preparation. The suspensions used in this work are composed of esterified silica sols pioneered by Van Helden et a1.I2 Silica particles were synthesized by ammonia-catalyzed hydrolysis of tetraethyl orthosilicate (TEOS, Fischer Scientific ACS certified reagent grade) at 50 O C , followed by a seeded growth reaction to grow particles to a final desired size and increase solids yield from the preparations. As described by Bogush et aI.,I3 additions of TEOS and water at a mole ratio of 1:2 were made to the reaction vessel at 8-h intervals, until the desired size was achieved. The particles grow according to the formula d,, = do(Vfd/Vo)1/3, where do and dfinalare the initial and final particle sizes and Vo and Vfinalare the initial and total volumes of TEOS added in the growth reaction. These sols were then esterified, by heating the particles in octadecanol to ca. 200 O C at atmospheric pressure for 4 h. Excess octadecanol was then removed by centrifugation in chloroform and cyclohexane. The particles were finally redispersed in cyclohexane for storage. For the rheological measurements the particles were suspended in Decalin (cisltrans-decahydronaphthalene, 99%, Aldrich). The particles were centrifuged out of the cyclohexane and then dried under vacuum on a rotary evaporator and left under vacuum overnight. The dry powder was then resuspended in Decalin with sonication. Concentrated samples were obtained by further rotary evaporation. Volume fractions were calculated from mass fractions determined by drying suspensions to constant mass at temperatures below 70 O C , using the formula 4 = [(mass of silica)/p:]/[(mass of silica)/p,' (mass of Decalin)/pdv],where p,' and pdvare the densities of the silica and Decalin, respectively. As discussed below, we feel that this method of determining volume fraction provides a lower bound on the volume occupied by the particles in the suspension. Particle density was determined gravimetrically (Table I). Suspensions of known masses of silica and Decalin were made up in a volumetric flask in triplicate at 18 O C . Particle densities were determined from the mass of the solids, the Decalin mass, and the density of Decalin at 18 OC (determined to be 0.885 f 0.001 g ~ m - ~ )These . values are comparable with those quoted in the l i t e r a t ~ r e . ' ~ *We ' ~ attribute the slightly higher values to the particle synthesis at 50 O C and esterification at atmospheric pressure. The particles were sized by using transmission electron microscopy on a Philips 400 TEM and by dynamic light scattering with a Brookhaven Instruments 2030 PCS. The particle size data on the three silica sols used in this study are summarized in Table I. While the particles are all of narrow size distribution, the suspension LMS21 E, which contained the smallest particles, was
+
(1 2) van Helden, A.; Jansen, J. W.; Vrij, A. J . Colloid Interface Sci. 1981, 81. - - ,354. --
(13) Bogush, G. H.; Tracey, M. A,; Zukoski, C. F. J . Non-Crysl. Solids 1988, 104, 95. (14) de Kruif, C. G.; van Iersel, E. M. F.; Vrij, A,; Russel, W. B. J . Chem. Phys. 1986, 83, 47 I I.
1166 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
found to contain a sizable fraction of irregularly shaped particles. Rheological Measurements. Rheological measurements were made using a Bohlin Reologi constant stress rheometer, fitted with a cup and bob geometry. Most measurements were made with the C25 cup and bob (bob diameter of 25 mm and a gap width of 1.2 mm), which gave a stress range of 0.025-235 Pa. However, occasional use of the C14 cup and bob (bob diameter of 14 mm and gap width of 0.6 mm) allowed measurements up to 1250 Pa. Creep and recovery times for the experiments varied from 250 s to I O ks to allow the sample to achieve steady-state flow and recover equilibrium structures prior to applying a new stress. The approach to steady state was determined by an algorithm in the controlling software, which calculated the fit of J = Bt", where J is the creep compliance, B is a constant, and t is the time after the stress is applied. When n equals 1, the system is at steady state. Except for the most viscous samples (>lo4 Pa s), n was within 5% of unity. For the highest viscosities the value was significantly lower, and a nonlinear fitting routine, discussed in section IV, was used to extrapolate the viscosity of the sample. With sterically stabilized suspensions, care must be taken that van der Waals forces do not dominate particle interactions. By working with particles below 400 nm in diameter, and at temperatures well above the critical flocculation temperature, attractive interactions can be minimized. In addition, by choosing a solvent that closely matches the particles in index of refraction, van der Waals forces can be significantly reduced. However, a balance must be struck between minimizing interactions and choosing a solvent with a low vapor pressure as the effects of evaporation are critical at high volume fractions. Some initial studies on a suspension of 1 10-nm-diameter particles (not used in the present work) and samples of LMS2OE and LMS22E, suspended in Decalin at a volume fraction of 0.5, showed that no changes in shear viscosity or high-frequency modulus occurred until the suspensions were cooled below 0-5 "C. This is in agreement with the studies of van Helden et al.,ls who report a phase transition occurring at 4 OC for a similar system of 61nm-diameter particles in Decalin, as observed by turbidity measurements and visual phase separation. In addition, during light scattering studies no signs of aggregation were observed for the suspensions studied, unlike particles with a diameter of 700 nm which slowly aggregated under the same conditions. We feel that these results show that van der Waals forces are small but are unable to show that attractive interactions are completely absent. In all the rheological studies, suspension temperatures were held at 18 OC and a solvent trap filled with Decalin was used to minimize evaporation. This allowed measurements on a sample to be made over several days, with no noticeable changes in transport properties due to drying. Measurements were made on suspensions containing the three particles over a wide range in concentration. Reconcentrating a sample under vacuum had no effect on suspension viscosity; however, care was taken not to heat the particles above 70 OC.'* The viscosity of Decalin and the intrinsic viscosities of the suspensions were determined by using a Couette double concentric cylinder geometry (stress range 0.0035-35 Pa). The rheometer was calibrated with standard silicone oils for all geometries, and suspension viscosity measurements were found to be geometry independent. Samples of LMS22E at volume fractions of 0.590 and 0.592 were studied in two different concentric cylinder geometries. While the viscosities at a given stress do not superimpose due to the sensitivity of viscosity to volume fraction in this region, the samples show very similar viscosity-stress curves, with the onset of shear thickening and shear thinning occurring at very similar stresses. These results suggest that wall slip is not a problem in these systems. 111. Results and Comparison with Other Hard-Sphere Systems
Families of shear viscosityshear stress curves over a wide range in volume fraction are displayed for the three different particles ( 1 5 ) van Helden, A. K.; Jansen, J. W.: Vrij, A. J . Colloid Interface Sci. 1986, 114, 47 1 .
Marshall and Zukoski 10 10
; ;10 m a v h lo
2
Figure 1. Shear viscosity as a function of shear stress for several volume fractions of suspension LMS20E.
D
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'
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0.592 0.590 0.584 D 0.571 0 0.543
8 4
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'
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Figure 3. Shear viscosity as a function of shear stress for several volume fractions of suspension LMS22E.
in Figures 1-3. All the suspensions show similar behavior. At low volume fraction no shear thinning is observed. As the volume fraction is increased above 0.2, shear thinning develops at a characteristic stress dependent on both particle size and volume
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990 1I67
Rheology of Hard-Sphere Suspensions
.r
10
I
.
1.25
1.20
; 3 3
10
lo 0.95 0.00 ~
0 00
0 20
0 40
0 60
Figure 4. Relative zero shear stress viscosity as a function of volume fraction. Present work: ( 0 )LMSZOE, (D) LMSZlE, (A) LMS22E. Previous studies: ( 0 )de Kruif et al.I4with 6 recalculated, (a) Papir and Krieger,2' (-) Doolittle equation2' with @,,, = 0.638, and (---) Quemada equation2)with &, = 0.638. Error bars indicate 1 standard deviation.
fraction. At very high volume fractions the drop in viscosity with increasing shear stress becomes large (1-3 orders of magnitude), and shear thickening is observed. At all particle concentrations studied, a zero-stress Newtonian plateau viscosity was observed. A plot of the relative zero-stress viscosity, qo/qc,as a function of volume fraction is presented in Figure 4. Here qc is the viscosity of the continuous phase, Decalin, at 18 "C (2.8 X Pa s). At the lowest stresses, and highest viscosities, the strain measured is on the order of the resolution of the instrument. Even with long time measurements of 10 ks, and the nonlinear least-squares fitting routine described in section IV, there can be considerable variation in estimates of the zero stress viscosity. As a result, the error bars increase sizably as the viscosity increases, especially above 1O4 Pa s. Comparison of our studies with theoretical predictions of suspension behavior is hindered by the need to convert from a measurable quantity such as particle concentration [(mass of particles)/(volume of suspension)], c, or weight fraction [(mass of particles)/(mass of suspension)], wf, to volume fraction [(volume of particles)/(volume of suspension)], 4. As discussed by de Kruif et al.,I4who studied the steady-state viscosity of similar suspensions, conversion of these measurable quantities to volume fraction requires knowledge of the particle density or specific volume. Rather than use the volumetrically determined particle density, pSy, for this purpose, de Kruif et al. suggest a specific volume determined from the slope of the relative viscosity-particle concentration curve in the limit as c 0. In this limit the viscosity can be written as q / q c = 1 + K c = 1 + [q]4 (1) where K is the measured slope and [ q ] is the suspension intrinsic viscosity. In hard-sphere suspensions, [q] takes the Einstein value of 2.5. By modeling the particles as hard spheres, the particle specific volume, q ( $ / c ) , is written as
-
q = K/2.5 (2) and q is used to convert concentrations to volume fractions over the entire concentration range. Figure 5 shows typical data for the relative viscosity of suspension LMS22E as a function of particle concentration in the dilute limit. In these experiments c was determined from a known mass of particles brought to a known volume with Decalin at 18 OC. Suspension viscosities were measured by using the Couette double concentric cylinder (double gap) geometry over a wide stress range. Values of K extracted from similar plots for the different particles are given in Table I . If we follow the procedure of relating c to r$ through the
0.02
0.04
0.06
0.08
0.10
0 2
c (g~m-~)
'p
Figure 5. Relative viscosity as a function of weight concentration for LMS22E.
prescription of eq 2, we find that the highest concentration we studied corresponds to a volume fraction of 0.69, well above the dense random close packing limit of 0.64. None of the suspensions studied showed signs of ordering, and all had zero shear stress plateau viscosities characteristic of disordered suspensions. Particle specific volumes, calculated from eq 2, would suggest that the particles uptake as much as 50%of their dry mass in solvent, yet no significant differences were observed between the hydrodynamic diameters (as determined by PCS) and diameters measured by electron microscopy. While there is evidence that the particles are capable of absorbing solvent (i.e., they are porous),I2 we feel that the pore volume available to Decalin is a very small fraction of the particle volume. As a consequence, we seek alternative explanations for the elevated intrinsic viscosities, [ q ] ,as calculated from measured values of K using q = l/p,V in eq 2. Factors other than solvation contributing to elevate intrinsic viscosities are particle shape, the presence of aggregates, or particle interactions. Transmission electron micrographs indicate that the particles are spherical and have narrow size distributions. Dynamic light scattering shows that the particles are not aggregated, suggesting that attractive interaction potentials are not contributing to elevate [ q ] . It would seem unlikely that the esterified particles suspended in a nonpolar solvent like Decalin would support a charge. However, the early work of van Helden et al.,'* characterizing the structure factor of very dilute suspensions of particles of similar sizes, prepared in a similar manner suggeststhe presence of electrostatic repulsion. While most evident in polar solvents, the effects of particle charge were also found for particles suspended in cyclohexane and n-octane. To confirm the presence of particle charge in our suspensions, two qualitative experiments were performed. First, the particles were found to move toward the positive electrode under an applied field and to reverse direction when the polarity of the field was reversed. Second, a comparison of the current density from suspensions of the particles, and that of Decalin, indicates a substantial increase in conductivity associated with the particles. While the field strengths used in these two qualitative experiments are large, and nonlinear effects are observed, the presence of a small particle surface charge is confirmed. As a result, the use of intrinsic viscosity to calculate particle specific volumes was abandoned, and in further discussions we report volume fractions determined from weight fractions using p z . The presence of a surface charge suggests the particles do not interact like hard spheres, and as shown by the present work, and that of van Helden et al., at low volume fractions this is apparent. A simple calculation of the primary electroviscous effectI6indicates (16) Booth,
F. Proc. R.Soc. London
1950, A203, 533.
1168 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
that in these nonaqueous media, where double layers are well extended, a very low particle charge (on the order of 5 mV or less) would give rise to the elevation of intrinsic viscosity. As we are working principally at elevated volume fractions, we must consider the interaction of the double layers, Le., the secondary electroviscous effect and its consequences. A measure of this can be determined by calculation of the Huggins coefficient, which represents the order (@) term in the viscosity equation and takes on well-established values for hard-sphere systems, showing substantial elevation when particle interactions are large. For suspensions LMS21E and LMS22E sufficient data were collected to calculate Huggins coefficients of 6 and 4. Our intrinsic viscosity measurements were made over a limited range of volume fraction, so uncertainties in the order (42)term are large. That these values are close to the theoretically predicted values for hard spheres of Batchelor17 (6.2) and BeenakeP (5.2) shows that the particle interactions are unaffected by the small charge present and gives us confidence in treating the suspensions as hard spheres, at elevated volume fraction. In addition, calculation of the ratio of electrostatic to Brownian forces acting on the particle^,^^^^^ for the magnitudes suggested by calculation of the primary electroviscous effect, gives values in the range 3 x 10-9-3 X indicating that thermal forces are far more important than electrostatic forces in controlling the suspension structure and rheology. In this work we are concerned with the glass-forming nature of the silica sols. Consequently, we are interested in the location of the order-disorder phase transition boundary and the volume fraction at which a disordered, liquidlike phase becomes metastable. When dilute samples ,of LMS22E were allowed to settle, crystallization was observed after several months, but as previously noted, the concentrated suspensions showed no signs of ordering. The value of the Huggins coefficient, and the small magnitude of electrostatic repulsive forces relative to thermal forces, suggests that the phase behavior of the silica sols will be dominated by the particle core rather than long-range soft repulsions. From these observations, we conclude that the disordered suspensions studied here become metastable with respect to an ordered phase at a volume fraction with a upper bound of 0.5. In Figure 4 a comparison is made between the zero stress viscosities reported here and those of de Kruif et al.,I4 with their concentrations converted to volume fractions by using the reported particle density. The observed differences can be attributed to variation in particle preparation technique and suspending medium, as well as uncertainties in converting reported concentrations to volume fractions. The results of Papir and Krieges' on suspensions of cross-linked polystyrene in benzyl alcohol are also reported without adjustments to volume fraction. It should be noted that elevated intrinsic viscosities were also reported for this system. At high volume fractions, the lack of superposition of the viscosities of suspensions composed of different sized particles reported here reflects differences in the maximum packing fractions and, perhaps, subtle variations in particle interaction potentials which are magnified as volume fraction is increased. As the smaller particles (LMS2I E) have the broadest particle size distribution and contain a fair number of nonspherical particles (as is typical of sub-100-nm silica particles prepared from T E O P ) , particle packing is expected to be less efficient, and as a consequence their maximum packing fraction is expected to be lower than that of the other two particle types. The differences become observable at volume fractions >0.5 where viscosities are very sensitive to the free volume in the suspension. The lack of superposition in the data sets shown in Figure 4 indicates that the suspension transport properties are very sensitive to small variations in particle type, and, due to the dramatic increase in 70 for 4 > 0.5, superposition of data sets will also rely on very accurate knowledge of volume fraction. We acknowledge real differences in divergence from the data of Papir and Krieger and can, at present, offer no explanation other than (17) Batchelor, G.K.J. Fluid Mech. 1977, 83, 97. (18) Beenaker, C. W. J. Physica 1984, A128, 48. (19) Russel, W. B. J . Colloid Inrerface Sci. 1976, 55, 590. (20) Russel, W. B. J . Fluid Mech. 1978 85. 209. (21) Papir, Y. S.; Krieger. 1. M . J . Colloid Interface Sci. 1970, 34, 126.
Marshall and Zukoski details of particle chemistry and the resulting variations in particle interaction potentials. IV. Colloidal Glasses As the thermal properties of colloidal suspensions will be dominated by the suspending medium, anomalies in heat capacity are unlikely to provide insight into colloidal glass transitions. Instead, we rely on the response of the suspension to an applied stress to probe this transition. The divergence of suspension viscosity at elevated volume fractions is commonly associated with the approach of a limiting packing density above which flow is no longer possible. These concepts are the core of the Krieger-DoughertyZ2 and QuemadaZ3 expressions relating viscosity to volume fraction. Recently, the extensive calculations of Wagner and Russelz4 indicate that, at elevated volume fraction, suspension flow is dominated by thermodynamic stresses which are internally coupled with many-body hydrodynamic stresses through particle distribution functions. For hard-sphere suspensions, the maximum packing density is found experimentally to lie near the random close packed limit of 0.64. As it is possible to construct packing configurations, allowing for free slippage of individual particles, or zones of particles, past one another up to crystalline close packing at 4 = 0.74, the rapid rise in viscosity at 4 > 0.5 reflects a predisposition of hard-sphere suspensions toward a random close packed structure. This behavior suggests that metastable amorphous structures of hardsphere colloidal suspensions are maintained, as these configurations place the suspension in a relative free energy minimum. As suggested by Woodcock,' Ackerson and P ~ s e y ?and ~ Pusey and van Megen? the structural and dynamic response of hard-sphere colloidal suspensions at elevated volume fractions is reminiscent of supercooled molecular liquids approaching the glass transition. Following this approach, we have analyzed our data in light of experimentally observed and theoretically predicted behavior of molecular liquids near the glass transition. Three major points of comparison are described below. The scaling of zero stress viscosity on volume fraction and the time scales characterizing structural relaxation are shown to follow the behavior observed for molecular liquids and predicted for hardsphere glasses. In addition, the dynamic response of the suspension to the application of a stress is found to be typical of behavior expected for glasses and shows a transition from a liquidlike to a glassylike response at a volume fraction near 0.52. Viscosity Divergence at the Glass Transition. The free volume model of Cohen and T u r n b u P predicts the fluidity, I-', of hard-sphere liquids in accordance with the Doolittle equationz7 (eq 3), which relates the free volume associated with molecules to the critical free volume required for diffusive motion. When the system is densified, the free volume shrinks and approaches the critical free volume. When the excess free volume in the system is zero, the particles are "frozen" in position. Originally shown to describe the temperature dependence of molecular liquids approaching the glass transition,27the Doolittle equation may be written V / % = c exP(ANTur*/vr) (3) where uf is the free volume available to molecules in the system, equal to [( 1 - 4) - (1 - 4,,,)] VT, or ($,,, - 4) VT, and uf* is the free volume needed for diffusion to occur, which is on the order of the molecular volume. VT is the total volume, and NTis the number of molecules. Thus, the right-hand side of eq 3 can be written as C exp(Aq4/(4,,, - 4)). Woodcock and Ange1Iz8studied the self-diffusivity of hard-sphere liquids and found that Do/D follows the free volume model predictions, with d, = 0.6377 (the volume fraction at random close packing) and A a constant close to unity,
(22) Krieger, I. M.; Dougherty, T.J. Trans. SOC.Rheol. 1959, 1 1 1 , 137. (23) Quemada, D. Aduances in Rheology; Universidad Nacionai Autonama de Mexico: 1984; Vol. 2. (24) Wagner, N . J.; Russel, W. B. Physica 1989, A155, 475. (25) Ackerson, B. J.; Pusey, P. N . Phys. Reu. Len. 1988, 61, 1033. (26) Cohen, M.; Turnbull, D. J . Chem. Phys. 1959, 31, 1164. (27) Doolittle, A. K. J . Appl. Phys. 1951, 22, 417. (28) Woodcock, L. V.; Angell, C. A. Phys. Rev. Left. 1981, 47, 1129.
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990 1169
Rheology of Hard-Sphere Suspensions lo* 10
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1)
Figure 6. Relative zero shear stress viscosity fit to the Doolittle equation:" (a) LMSZOE, (M) LMSZIE, (A)LMS22E. Error bars indicate 1 standard deviation.
( 2 9 ) Rallison, J. M.; Hinch, E. J. J .
Fluid Meeh. 1986, 167, 131.
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Figure 7. Magnitude of shear thinning (qo - qJqo) as a function of volume fraction: (a) LMSZOE, (a) LMSZlE, (A)LMS22E.
in accordance with the experimental findings of Doolittle and the Cohen and Turnbull model. This exponential scaling is somewhat different from the Krieger-DoughertyZ2 equation, the expression of Q ~ e m a d a , and * ~ the predictions of RallisonZ9for the coefficient of diffusion in dense hard-sphere suspensions, which takes the form
Do/D = t / q c = (1 - 4/4m)-* (4) As shown in Figure 6, the Doolittle equation is found to linearize the zero stress viscosity. The data show scatter, but a distinct linearity is present, with a gradient of 1.65 indicating that uf* is greater than a particle volume. For particles LMS2OE and LMS22E, 4,,, was taken as 0.638, while for the smallest particles G,, was taken as 0.61. The lower value of maximum packing fraction for the smallest particles is attributed to this system's broader particle size distribution and greater shape irregularity. The small change in maximum packing fraction gives rise to several orders of magnitude change in viscosity as close packing is approached (Figure 4), indicating the sensitivity of suspension properties to the approaching limiting packing fraction. Figure 4 also shows the fits of the Quemada expression and the Doolittle equation to the measured zero shear viscosity, with 4, = 0.638 for both, C equal to 1.20, and A equal to 1.65 in the Doolittle equation. As can be seen, the Quemada expression describes the data best at low volume fractions but underestimates the viscosity considerably as $, is approached. We believe that the Quemada expression underestimates the high volume fraction viscosity because it is an extrapolation of the liquid line into the metastable region where the suspensions undergo a transition into the glassy state. By contrast, the Doolittle equation fits the divergence extremely well but, as is expected for a glass model, fails to describe the low volume fraction behavior of the suspension where it is out of the metastable regime. Shear thinning in colloidal suspensions is attributed to the transition from thermal to hydrodynamic control of particle interactions and, consequently, of suspension structure. The degree of shear thinning (qo - qm) is a measure of the thermodynamic stresses on the particles in the equilibrium s t r u ~ t u r e . 'Here ~~~~ qm is the high shear rate limiting Newtonian viscosity, which, for our systems, is obscured at elevated volume fractions by the existence of shear thickening. However, a lower bound on the magnitude of the shear thinning can be made by estimating qm from the lowest viscosity measured before the onset of shear thickening. This estimate proves to be quite accurate as the degree of shear thinning is several orders of magnitude at volume fractions >0.54. Figure 7 shows the volume fraction dependence of (so qm)/qo,a function that changes between zero and unity as volume
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Figure 8. Critical Peclet numbers as a function of volume fraction (see text for definitions). Open symbols: Pel,2. Closed symbols: Pe, (a) LMSZOE, (M) LMSZIE, (A) LMS22E.
fraction is increased. The rapid increase in zero stress viscosity is translated into the rapid approach of this function to unity a t a volume fraction near 0.52. The ability of the Doolittle equation to correlate the zero stress viscosities and the rapid increase in (qo - vm) indicate that the particle motion is hindered due to a reduction in free volume and that thermodynamic stresses increase rapidly at a volume fraction above 0.5. These results match qualitatively with the behavior expected for materials undergoing a glass transition. Below more evidence for such a transition arising from the suspension's dynamic properties is presented. Time Scales for Relaxation Processes. Time scales for structural rearrangement increase dramatically for molecular liquids near the glass transition. Two characteristic time scales can be extracted from the steady-state viscosity data presented in Figures 1-3. Following previous work in this field, the shear rate where the viscosity drops to halfway between the low and high shear stress limiting values, yll2,is taken as a characteristic inverse time scale for structural relaxation due to thermal forces acting on the particles. In Figure 8 a plot of yl/z in terms of the characteristic Peclet number for shear thinning, Pel/z = qca3yl,2/kBT, is presented as a function of volume fraction. In agreement with previous work on other systems,30 Pel/z is approximately constant up to a volume fraction near 0.5 and then (30)van de Werff, J. C.;de Kruif, C.G. J . Rheol. 1989, 33, 421.
1170 The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
Marshall and Zukoski
drops precipitously, indicating an extreme lengthening in relaxation TABLE II: Rheological Data for Suspension LMSZOE times. The close grouping of yl12(within the large experimental wt uncertainties associated with determining these shear rates) for fraction Q vo/vca vJv? Y1/2c YmC the three particle sizes when scaled on a3 adds additional support 0.7401 0.578 6 X lo7 1.1 X lo-' 3.31 X lo4 6 X lo-' to our contention that the suspensions are composed of hard0.7380 0.574 3.8 X lo6 1.07 x 104 1 x 10-4 2.9 X lo-' sphere-like particles and agrees with the results of van der Werff 0.7048 0.534 1.9 x 105 1.72 X lo2 6 X 4.5 x 10' and de K r ~ i f . ~ ~ 3 x 10' 0.6890 0.515 8.94 X lo2 1.21 x 102 2 x 10-1 A second characteristic relaxation time can be extracted from 0.6874 3.6 X 10' 0.510 7.58 x 103 2.20 x 102 1.5 X the shear thickening behavior. The shear rate where the suspension 0.6853 5.73 x 10' 0.509 5.55 x 102 6.37 X 10' 3 x lo-' viscosity passes through a minimum as the applied stress is in1.7 X lo2 0.6632 0.486 8.17 X 10' 2.14 X 10' I 6 2 x 102 0.6361 0.459 1.50 X 10' 9.30 creased, ym, is taken to indicate the onset of shear thickening. d d 4.77 0.4660 0.275 7.30 Shear thickening is associated with a transition from a suspension structure composed of planes of particles which slip over one a Uncertainties in viscosity range from 1-2% at low volume fractions another in the direction of flow to a disordered s t r u c t ~ r e . ~ ~ . ~ 'to
