Viscosity of Hard-Sphere Suspensions: Can We Go Lower?

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Viscosity of Hard-Sphere Suspensions: Can We Go Lower? Vijay Gopalakrishnan and Charles F. Zukoski* Department of Chemical and Biomolecular Engineering, UniVersity of Illinois, Urbana, Illinois 61801

To date, approaches seeking a definitive link between colloidal suspension microstructure and the corresponding viscosity predict that for dense suspensions, hard-sphere systems possess the lowest-attainable viscosity. Interparticle interactions are thought to increase suspension viscosity because of the additional energy that is required to overcome these interactions to sustain flow. Here we report our recent findings on the ability of weak depletion attractions to generate additional free volume within dense suspensions that then enhance particle diffusivities and result in a viscosity that is lower than the hard-sphere value. The minimum achievable viscosity relative to the hard-sphere value grows with increasing volume fraction in a continuous fashion as the glass-transition volume fractions are approached. Calculations using a recently developed model that incorporates the ability of thermal motions to relax particle localizations in cages within a mode coupling framework qualitatively capture our experimental findings. The study highlights the presence of localization barriers in dense suspensions above volume fractions of ∼0.40-0.45 that grow stronger with volume fractions up to and beyond the glass-transition volume fractions and challenges long-held notions that hard spheres possess the lowest-attainable viscosities at these dense volume fractions. I. Introduction Control of suspension flow properties is critical to the design and production of a number of industrial products including ceramics, foods, paints and coatings, and consumer and medical products. The ability to make accurate predictions of suspension viscosity and modulus from knowledge of the governing parameters, such as the concentration of solids and the composition of additives and the continuous phase, remains a significant challenge requiring a comprehensive understanding of the impact of microstructure on the particle dynamics, which ultimately governs suspension-flow behavior. Of particular interest to this work is the effect of attractions on suspension microstructure and resulting shear viscosity. In particular, we explore the influence of weak attractions in colloidal systems that with an increasing volume fraction eventually lead to a gelled or nonergodic state. This paper builds on Russel’s contributions to hard-sphere rheology,1,2 depletion interactions,3 and links between statistical mechanics and the mechanics of suspensions.4,5 As suspensions are concentrated, their viscosities become an increasingly sensitive function of the volume fraction, φ, of particles. The viscosity is intimately correlated with the longtime diffusive dynamics of the particles that constitute the suspension and the increased work required to shear a suspension at elevated loading is directly correlated with the decreased long-time diffusivity. In a significant development, Cohen and co-workers6 derived a predictive link between the suspension microstructure, described via the equilibrium structure factor, S(q), and the zero shear viscosity, η0, of suspensions. The model is defined as the binary collisions in a mean field (BCMF) model. Through the BCMF model, Cohen and co-workers demonstrated that hard-sphere suspension viscosities are most sensitive to the microstructure and dynamics on the length scale of the order of a particle diameter. On this length scale, a reference particle in a concentrated suspension experiences the presence of a cage of its local neighbors that presents additional resistance to particle diffusion that dominates the relaxation of * To whom correspondence should be addressed. E-mail: czukoski@ uiuc.edu. Tel.: 217-333-0034. Fax: 217-333-5052.

correlations in density fluctuations in these systems. As the suspension volume fraction increases (i.e., φ g 0.30-0.35), the caging effect significantly slows diffusion and results in a dramatic increase in the zero shear viscosity. Above a φ value of ∼0.56, Pusey and Van Megen7,8 observed glasslike long-time diffusion of hard spheres where particles are trapped in cages of neighbors that for time scales of the experiment can amount to several months. Van Megen and Underwood9 performed a detailed set of dynamic light-scattering experiments by going up in small steps in volume fraction in a well-characterized hard-sphere system. By observing the long time behavior of the intermediate scattering functions that are dominated by cooperative dynamics, they determined the glass transition volume fraction, φg, to be the lowest volume fraction at which the intermediate scattering function reaches a longtime plateau over 10 orders of magnitude in delay time. From these experiments, φg was determined to be around 0.57-0.58. Cooperative phenomena leading to glassy dynamics has been treated in the ideal mode coupling theory (IMCT)10-13 by providing a self-consistent microscopic theory that describes the coupling of particle dynamics at one length scale with structural correlations on all length scales. IMCT predicts an abrupt transition from a system with long-range diffusion to one where this diffusion no longer occurs, indicative of a glass or a gel. As in the BCMF model, IMCT makes predictions through integrals over the equilibrium structure factor. Beyond the glass transition, long-range particle motion is arrested, and the particles are indefinitely localized. For a system of hard spheres, the IMCT predicts a glass transition at φ of 0.52. By applying a rescaling argument, van Megen and Underwood demonstrated that beyond the short-time scales, the IMCT predictions capture the long-time relaxation of the intermediate scattering function, thereby successfully predicting a vanishing long-time diffusion as the glass transition is approached. A qualitatively similar argument can apply to systems in which interparticle attractions are systematically turned on. For a fixed volume fraction, over a small range of increase in the strength of attraction, the long-time self-diffusivity dramatically drops indicating the formation of gels. Analogous to cages in hard-sphere glasses, interparticle attractions of sufficient strength

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localize particles with respect to their neighbors. As the suspension approaches the gel point, the suspension viscosity increases dramatically. Brady14 related the viscosity of a suspension to the short-time diffusivity of a particle that was dependent on the radial distribution function at contact (g(D), D being the particle diameter), thereby accounting for the influence of suspension microstructure on macroscopic flow properties. He obtained the following relationship: η0 ) η∞ + (12/5)ηsφ2g(D)(D0/Dss(φ)), where η∞ is the infinite frequency viscosity, D0 is the diffusivity of a single Brownian particle at infinite dilution, and Dss(φ) is the short-time self-diffusivity at the particular φ. Here, η0 is the zero shear viscosity when the microstructure within the suspension is isotropic. When extended to systems with attractions, it is expected that, as particles come closer to one another, both g(D) and Dss(φ) will increase monotonically thereby leading to a proportional increase in η0. On the basis of these models, the effect of attractions is similar to that of increasing volume fraction in that increasing both parameters always drives the system closer to an irreversibly localized state. Thus, one could conclude that for a given volume fraction at zero shear rate, an isotropic microstructure in suspensions that results solely from volume-exclusion effects would produce the lowest-attainable suspension viscosity. Attractive or repulsive interactions would only serve to increase suspension viscosity. However, Poon and co-workers15,16 recently identified a “glass-melting” phenomenon in which, when a small amount of polymer was added to a glassy hard sphere suspension (φ ≈ 0.60), particles that are arrested are able to escape their localized states by diffusing out of their cages. This was seen by a complete relaxation of the intermediate scattering function. The glass-melting phenomenon was accompanied by a reduction in the magnitude of the first peak in the structure factor which is a measure of the coherence of the first cage around a reference particle. The reduction in coherence of the first cage allows the central particle to diffuse out of an otherwise permanently localized state, thereby leading to complete decay in the intermediate scattering function. Working with a similar system, Shah et. al. measured a similar reduction in the first peak in S(q) at volume fractions (φ ≈ 0.40-0.45) substantially lower than φg.17 Previous work18 has shown an accelerated decay in the density fluctuations at these volume fractions upon the addition of polymer that is conceptually similar to observations in glass-melting systems.16,19 In addition, the suspension viscosity was shown to pass through a minimum with increasing polymer concentrations at volume fractions far from the glass transition suggesting that particle localization similar to that seen in glasses occurs at volume fractions well below those predicted by IMCT. Indeed, using structure factors that have been shown to capture subtle details of changes in S(q) as nonadsorbing polymer is added to suspensions of hard spheres, both the BCMF model and the recently proposed Schweizer and Saltzman (SS) approach20,21 predict this minimum in viscosity. These results challenge the long-held notion that hard-sphere-like interactions hold the lowest-attainable viscosity at a given volume fraction. Here we extend our preliminary studies to provide more detailed studies of the effects of weak attractions on the zero shear viscosity of dense near-hard-sphere suspensions, both as a function of the polymer concentration at a constant volume fraction and as a function of the colloid volume fraction at a constant polymer concentration as φg is approached. We make comparisons of experimental observations from a wellcharacterized near-hard-sphere system with predictions from the SS model to test the ability of the model to capture the

qualitative and quantitative features of the experimental trends. The following section describes the experimental system and the experimental setup employed in our study. In section III, we present our experimental results and a discussion that is followed by section IV which involves comparing model predictions with trends from the experiment. In section V, we draw two major conclusions. First, the localization that ultimately results in glass formation is initiated at volume fractions well below those typically described as glassy. The smooth transition of mechanical properties up to and through the glass transition suggests that the abrupt ergodic/nonergodic transition predicted by IMCT does not capture the dynamics of the suspension mechanics well. Our results suggest that the model of Schweizer and Saltzman that incorporates an activated relaxation mechanism into a “naı¨ve” mode-coupling formalism22 provides a world view more consistent with the experimental data. Second, we demonstrate that over a wide volume-fraction range, weak attractions can decrease suspension viscosity II. Experimental Section Silica particles were synthesized using the Stober method.23 The recipe involves the hydrolysis of tetraethyl ortho-silicate in an ethanol medium in the presence of a base (ammonium hydroxide). The reactants are added at specific concentrations to obtain particles with a diameter (D) of 48 nm as measured by dynamic light scattering (λ ) 614 nm and θ ) 150°). The size measurement was also confirmed via scanning electron microscopy (SEM) images and a size estimate from the sidebounce ultra small-angle x-ray scattering (SB-USAXS) experiments. Since a seeded growth technique24 was not employed, precise control over the polydispersity could not be established. As discussed in the next section, the spread in our particle size distribution has a nontrivial impact on the structure factor obtained from SB-USAXS experiments. Once the particles are synthesized, the ethanol solvent is boiled off, and the particles are dispersed in a medium of liquid octadecanol at 210 °C for 6 hours. A steric barrier of octadecyl hairs on the silica surface formed by the reaction of octadecanol with the surface hydroxide bonds of the silica particles prevents the aggregation of particles that results from van der Waals attractions. The excess octadecanol is removed by washing the particles with chloroform and centrifuging the particles out of the wash solution. This process is repeated 4 times to ensure that all traces of unreacted octadecanol are removed. After the wash, the particles are dried in a vacuum oven for 24 h to remove all traces of chloroform. The polymer used in the study is polystyrene of molecular weight, Mw ) 2460 g (Mw/Mn ) 1.07), purchased from the Sigma Aldrich company. Polymer concentration (cp) was measured relative to the overlap concentration (c* p): c* p ) 3Mw/ 4πRg3NA, where NA is the Avogadro number and Rg is the radius of gyration. The solvent used to disperse the silica particles and polymer in this study is cis-decalin, which was also purchased from the Sigma Aldrich company. At room temperature, polystyrene is in a near-ideal state in decalin. The characterization of the polymer in decalin has been performed by Shah et al.;25 using their results, Rg ) 1.31 nm, and therefore 2Rg/D ) 0.055. Suspensions were prepared by adding specific amounts of stock hard-sphere suspension and polymer solution and diluting the mixture with the requisite amount of decalin to bring the suspension to the desired volume fraction, φ and polymer concentration, cp/c*p. Volume fractions were measured by making dry weight measurements and employing a particle density of ∼1.6 g/mL in the calculations. This value is consistent with detailed measurements made on octadecyl-coated silica particles in the past.26

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Rheology is performed on a Bohlin CVOR rheometer which is operated in a constant stress mode. All experiments were performed in a cup and bob geometry, where the bob diameter was 14 mm and the gap size was 0.7 mm. This geometry permitted accurate measurements of viscosity, η, to stresses, σ, as low as 0.1 Pa for our studies and, thereby, allowed the measurement of the zero shear plateau in viscosities up to a decade in stress magnitude. A solvent trap was employed in all runs to prevent evaporation of the solvent from the suspension. This was especially important at the highest volume fractions of ∼0.54, where a small amount of solvent loss can substantially alter flow properties. A common protocol was employed to ensure consistency in measurements. Once 3 mL of the sample is loaded, it is presheared at 100 s-1 for 30 s, followed by a 10 s equilibration time. Then measurement is commenced by performing an upward and downward stress sweep. At each value of stress, a delay time of 60 s was set to achieve steady state, followed by a measurement time of 20 s, during which the measured data is averaged. Prior creep studies showed that a 60 s period was sufficient to achieve a steady state for all the suspensions over the entire window of stresses. For the highest volume fractions of 0.51 and 0.54, a roughened bob was used to prevent any slip at the bob-suspension interface. All relevant figures display viscosity relative to that of the solvent, ηr, and stress is nondimensionalized as (σD3/8kBT), where kB is the Boltzmann’s constant and T is the temperature (T ) 25 °C). Side-bounce ultra small-angle x-ray scattering (SB-USAXS) was performed at the Argonne photon source (APS) at the Argonne National Laboratories (ANL) in Argonne, IL. The optics setup consists of two sets of channel-cut Si 111 crystals that define the beam and eliminate slit-smearing effects. This obviates the need for a desmearing routine of the extracted scattering data. An advantage of the setup is that the static structure factor, S(q), from these experiments directly provides an accurate measure of the correlations in density fluctuations in the suspension up to the first peak in the structure factor. III. Results and Discussion Correlations in density fluctuations on length scales of a particle diameter are strongly associated with the mechanics of hard-sphere suspensions. In this work, we explore the relationship between these correlations and the phenomenon of glass melting in glassy colloidal suspensions. We investigate changes in shear viscosity and microstructure in suspensions of hard spheres to which we add a nonadsorbing polymer. The polymer induces an effective attraction between the particles, the strength of which scales on cp/c*p and the range of which is controlled by the polymer radius of gyration, Rg. In these systems, extensive studies indicate that for 2Rg/D < 0.1, upon increase of the polymer concentration, suspensions gel prior to undergoing an equilibrium phase transition. In suspensions with volume fractions above the glass transition, enhancements of particle diffusivities have been reported to occur at polymer concentrations below that required to induce gelation.10,15,16,18,19,27 This phenomenon is associated with weak depletion attractions disrupting the cage of nearest neighbors around a reference particle, thus increasing particle mobility. Enhanced mobilities occur even though the depletion attractions drive the particles closer together. Experimental evidence for the disruption of the cage structure is seen by a reduction in the first peak of the S(q), labeled in this paper as S(q*), where q* is the position of the first peak. At volume fractions substantially below the glass transition, S(q*) is also seen to pass through a minimum upon increasing cp/c*p.17 The suppression in S(q*) with the addition of polymer

is quantitatively captured by the polymer reference interaction site model (PRISM) developed by Fuchs and Schweizer.17,28 In our previous studies we demonstrated that the decrease in S(q*) with increasing cp/c*p is associated with increases in diffusivity and decreases in suspension viscosity. In this paper, we expand on our preliminary studies with two sets of experiments. First, suspension microstructures and corresponding viscosities are measured over a wide volume fraction range (0.38 e φ e 0.54) at a fixed polymer concentration, cp/c*p ) 0.03, chosen to be near the viscosity minimum, obtained from our preliminary work.18 This window of volume fractions encompasses the range of φ where we expect caging forces to play a significant role in particle dynamics. The second set of experiments are performed at φ ) 0.54 with increasing polymer concentration. For all suspensions, we were able to extract an accurate measure of zero shear viscosities by measuring the low-shear plateau in viscosity for nearly a decade in magnitude of applied stress. As Figure 5 indicates, the zero shear viscosity of suspensions in the absence of any polymer nearly replicate measurements made on other experimental hardsphere systems over the entire range of volume fractions. However, we recognize that the small mismatch could be the result of residual van der Waals attractions that results from the minor refractive index mismatch.29 As indicated below, these attractions are very weak. As a result, we refer to suspensions in the absence of any polymer as hard-sphere systems. Below we first discuss the microstructures of the suspensions and then turn to mechanical properties. Suspension Microstructure. Figure 1a shows a plot of the scattered intensity against the normalized wavevector, qD, at a low φ ) 0.04. At this volume fraction, interparticle correlations are nonexistent and hence, the scattered intensity reflects the form factor, P(q), of the particle. As mentioned in section II, particles synthesized by the Stober method intrinsically have broader particle size distributions as the average particle size decreases. Accounting for such a size distribution is important when investigating the detailed scattering properties of the suspensions. For this purpose, we have employed the model developed by Vrij30,31 that describes an analytical method for predicting the angular dependent scattering properties of hardsphere suspensions containing an arbitrary particle size distribution. The scattering properties (scattering intensity, form factor, scattering amplitude, etc.) are described by functions composed of terms involving particle size distribution (PSD) weighted averages. The structure factor from Vrij’s treatment is thus an h (q), where R(q) is the average structure factor, Sh(q) ) R(q)/Ffh2P normalized scattering intensity, F is the number density of all particles, hf is the averaged scattering amplitude, and P h (q) is an average form factor. For details of the calculations, we encourage readers to refer to the original references.30, 31 The dotted line in Figure 1a is the prediction of the form factor for spherical particles having a Gaussian distribution in particle size with a mean diameter of 48 nm and a 12% standard deviation, scaled to match the scattered intensity. However, as can be seen in Figure 1a, the drop-off in experimental intensity from the low qD Guinier plateau to the first minimum is more gradual than that predicted by the model suggesting that the scattering includes a contribution from an additional particle size distribution. When an additional broader distribution of particles with a mean diameter of 96 nm and standard deviation of 30%, comprising just 1% of the total particle number density, is incorporated into the original PSD, the model predictions of the form factor (solid line in Figure 1a) capture the experimental scattered intensity for all qD.


Figure 2. S(q) against qD at φ ) 0.54 with increasing polymer concentration.

Figure 1. (a) Scattering intenstiy at low volume fraction (φ ≈ 0.04) against dimensionless wavevector, qD, with predictions from the Vrij model30,31 for scattering from particles with polydispersity. Open circles are experimental data points. The dashed line is the prediction for particles with a diameter of 48 nm and a 12% standard deviation in size distribution. The solid line is the prediction for particles with a diameter of 48 nm and a 12% standard deviation in size distribution and another broad distribution of particles with ∼30% polydispersity and a mean size of 96 nm. The large particles contribute to only 1% of the total number density of particles. (b) Static structure factor, S(q), against the dimensionless wavevector, qD, at different volume fractions. The data points are experimental structure factors, whereas the solid lines are fits from Vrij’s model30,31 for an average structure factor for polydisperse suspensions. The composition of particles is as mentioned in the caption for Figure 1a.

One possible reason for the presence of small quantities of a broad distribution of particles in the background of the original distribution could lie in the technique of particle preparation. Each synthesis produces limited quantities of particles. As a result, we prepared several batches to produce a sufficient quantity of particles for the two sets of experiments. It is possible that repeated usage of the same reaction vessel could have resulted in a minute fraction of particles from a previous batch of synthesis acting as seed particles for a growth process in a subsequent synthesis run. Considering we had to prepare several batches of the particles, it is possible that the impurities had a broad distribution in size centered at a mean size close to twice the intended particle size. We are confident that since the number density of the large particles is significantly small (∼1% of the total), their role in

determining the microstructure of the suspension is negligible. However, the 12% standard deviation in the size distribution of 48 nm particles has a significant effect on S(q). Figure 1b plots the S(q) against qD for the hard sphere suspensions obtained from SB-USAXS experiments. One notable feature is the decrease in magnitude of S(q*) for all volume fractions when compared to the predictions for hard-sphere systems using the Percus Yevick closure (i.e., S(q*) at φ ) 0.38, 0.43, 0.48, 0.51, and 0.54 for monodisperse hard spheres using the PercusYevick closure are ∼1.91, 2.3, 2.9, 3.43, and 4.14, respectively). Experimentally, we observe that the corresponding values are ∼1.57, 1.76, 2.08, 2.34, and 2.53. When comparisons of experimental data are instead made with predictions for the average S(q) from Vrij’s model,30,31 where the particle size distribution is the same as that used to fit the form factor data, there is a satisfactory fit to the experimental S(q) of the hardsphere suspensions, as shown in Figure 1b. Thus, a spread in particle size distribution can explain the differences between measured and predicted monodisperse hard-sphere structure factors. This suggests that the spread in particle size distribution in our system plays a nontrivial role in modulating the structural correlations on the length scale of a particle diameter. As emphasized in the following discussion, the purpose of this study is to compare structural and viscosity modifications relative to the hard-sphere suspension and therefore the moderate particle size distribution does not complicate our conclusions. The agreement of the Vrij model with our data suggests that equilibrium density fluctuations are those of particles with a modest spread in particle size distribution that experience solely volume exclusion interactions. Figure 2 shows modifications to the experimentally determined S(q) upon addition of polymer at a constant volume fraction. At a φ ) 0.54, the addition of a small amount of polymer brings about a reduction in the magnitude of the first peak in the structure factor, S(q*). As the polymer concentration is increased further, S(q*) goes through a minimum before growing in magnitude as the gel boundary is approached. Over the measurable range of qD, the presence of polymer affects the correlations of density fluctuations to the greatest extent at the position of the first peak, q* (Figure 2). The reduction in S(q*) is indicative of modification to the microstructure on the length scale of a particle diameter and is interpreted as a decrease in the coherence of the first cage surrounding any reference particle. An interesting feature of these data is the shift in the position of the first peak to higher qD values with increasing cp/c*p. As discussed in previous studies, increasing polymer concentration increases the strength of attraction and results in the particles composing the first cage to come closer to a reference particle. These trends are better illustrated in Figure

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Figure 3. Static structure factor at the first peak, S(q*) vs the nondimensionalized polymer concentration, cp/c* p, at φ ) 0.54. The inset shows the plot of the position of the first peak, q*D vs cp/c* p.

Figure 4. S(q*) vs φ for hard spheres (open circles) and at a cp/c* p ) 0.03 (half solid squares). The lines are drawn to guide the eye.

3 where the main figure is a plot of S(q*) against cp/c*p and the inset plots q*D against cp/c*p. At cp/c*p ) 0, S(q*) ≈ 2.53. At a cp/c*p ) 0.02, S(q*) reaches its minimum value of ∼2 before it again increases with increasing polymer concentration. We note that, despite a moderate particle size distribution which modulates the absolute value of S(q*) when compared to that of a monodisperse suspension, the depression in S(q*) with the addition of polymer is of a similar magnitude to that observed for monodisperse near-hard-sphere suspensions studied in the past.17,18 Poon and co-workers15,16,32 report similar trends in poly(methyl)methacrylate hard-sphere suspensions with the addition of nonadsorbing polymer at a φ ≈ 0.60. Dynamic light-scattering experiments performed over a wide range of wave vectors showed that the decrease in the structural coherence of the cage in the glassy regime is accompanied by a complete decay of the intermediate scattering function. This phenomenon is defined as glass melting. A subsequent increase in polymer concentration eventually leads to another nonergodic state, sometimes defined as a gel, which is brought about by the relocalization of particles in attractive cages. Attractive cages result from particles being effectively “stuck” to their neighbors because of strong attractive forces at high polymer concentrations. In structurally arrested systems, the long-time plateau value of the intermediate scattering function, f(t f ∞), measured at q*D, is nonzero, and the fraction of decay from the initial value of 1 is a measure of the localization length of the particle. In hard-sphere glasses, f(t f ∞) has decayed to a value of ∼0.6-0.7 suggesting that localization lengths are on the order of the particle diameter. However, in attractive glasses, f(t f ∞) decays only by a few percentage points indicating a localization length on the order of a few percent of the particle diameter. Schweizer and coworkers33 have shown that, in attractive glasses formed from the addition of polymer, the localization length is on the order of the radius of gyration of the polymer. Despite being below the accepted hard-sphere glass-transition volume fraction of ∼0.57-0.58, the results shown in Figures 2 and 3 replicate behavior seen in glassy systems. The liquidlike nature of the suspension in the absence of polymer and the shortened relaxation times upon the addition of polymer are confirmed by Figure 6 which displays well-defined plateau zero shear viscosities for a range of polymer concentrations. The presence of structural modifications in liquid suspensions that are identical to those in glassy systems leading to similar enhancements in particle dynamics (as discussed later) suggests that the phenomenon of “cage melting” commences in dense suspensions and gradually gains significance with increasing φ up to and beyond the glass transition, at which point long-range

motion is finally arrested over experimental time scales, thereby defining a glassy state. This conclusion has been previously surmised in the theory developed by Saltzman and Schweizer21 where the appearance of a “dynamic free-energy” barrier to diffusion, the signature for particle localization, occurs at φ ) 0.432 for hard-sphere suspensions, much below the experimental φg. To explore the volume fraction range over which cage melting can be experimentally measured, we fixed cp/c*p at a value of 0.03 and increased the volume fraction from 0.38 to 0.54. The choice for the lower boundary on φ was governed by that volume fraction at which caging is expected to have a limited impact on particle dynamics. We anticipate that this becomes significant at φ ≈ 0.35, where shear thinning becomes significant. Experiments were performed at a constant polymer concentration, cp/c*p ) 0.03, chosen because the minimum in S(q*) is found in the vicinity of this concentration. Figure 4 shows changes in S(q*) for suspensions in the absence of polymer and at a cp/c*p ) 0.03 for a range of volume fractions. Over the entire range of volume fractions, there is a suppression in the value of S(q*, cp/c*p ) 0.03) relative to the hard-sphere value. The nonuniform suppression over this range in φ can be attributed to the fact that the window of polymer concentrations in which we observe liquidlike suspensions before the transition to a nonergodic state (gel) becomes increasingly narrow with increasing φ. Since the minimum in S(q*) is approximately at the center of this window, the cp/c*p corresponding to the minimum in S(q*) shifts to smaller values with increasing φ. Thus the minimum in S(q*) is not always observed at a cp/c*p ) 0.03. The data in Figure 4 confirm that the coherence of the first cage around a reference particle is disrupted upon addition of polymer over the entire range of volume fractions studied. Suspension Rheology. Figure 5 shows a comparison of the relative zero shear viscosity (i.e., zero shear viscosity normalized to that of the solvent, ηr0), for suspensions in the absence of polymer and compares these measurements with those measured on well-characterized monodisperse hard-sphere suspensions. The solid line is the prediction from the recent model of Schweizer and Saltzman20 that quantitatively captures the increase in ηr0 with increasing φ for previously well-characterized experimental hard-sphere suspensions discussed in the work of Russel and co-workers.2 The values reported here for ηr0 also correspond with those measured on similar systems.34 Figure 6 shows a plot of the relative viscosity, ηr, of the suspension against the applied external stress at φ ) 0.54 and increasing cp/c*p. At the highest stresses, the high shear viscosity monotonically increases with increasing polymer concentration. This trend points to a progressively increasing contribution


Figure 7. Relative zero shear viscosity, ηr0, vs φ for hard-sphere suspensions (b) and with polymer cp/c*p ) 0.03 (9). With increasing φ, the decrease in ηr0 becomes increasingly significant as is shown by the inset where the viscosity relative to the hard-sphere value is plotted vs φ. Figure 5. Comparison of the relative zero shear viscosity, ηr0, as a function of the volume fraction, φ. The solid line is the prediction of the Schweizer and Saltzman model20 which passes through the scatter of data points from previous well-characterized experimental systems. The open circles are experimental data points for our system.

Figure 6. Relative viscosity, ηr, vs the dimensionless stress at φ ) 0.54 with increasing polymer concentration. The zero shear plateau, observed for a decade in stress magnitude, goes through a minimum in magnitude as the polymer concentration is increased. The inset, in a log-linear plot, shows the evolution of the zero shear viscosity relative to the hard-sphere value as a function of the polymer concentration.

from hydrodynamic stresses which is consistent with previous work on attractive suspensions.35 However, ηr0 evolves in a nonmonotonic fashion. The inset shows a plot of η0/η0HS (viscosity relative to the value in the absence of polymer (or hard sphere)) with increasing cp/c*p. As discussed above, the magnitude of S(q*) is a measure of the coherence of the nearestneighbor cage. In dense suspensions, the coherence of this cage governs the resistance to long-time particle diffusion.16 A comparison with Figure 3 shows a strong correlation of η0/η0HS with S(q*) indicating the intimate coupling of dynamical time scales that are correlated with density fluctuations on the length scale of a particle diameter and the suspension viscosity in the limit of zero shear where the equilibrium microstructure is not distorted by the applied stress. Figure 7 shows a plot of the ηr0 of suspensions at cp/c*p ) 0.03 and the corresponding suspensions in the absence of polymer as a function of volume fraction. One striking feature in the plot is the impact of the presence of polymer on ηr0 with increasing volume fraction. While small at φ ) 0.38, the reduction in zero shear viscosity becomes increasingly significant with increasing volume fraction. At φ ) 0.54, zero shear viscosity at cp/c*p ) 0.03 is only ∼25% of the hard-sphere

viscosity. The inset plot in Figure 7 shows the zero shear viscosity relative to the hard-sphere value. A comparison with Figure 4 shows that the magnitude of reduction in S(q*) at a particular cp/c*p is weakly linked to the magnitude of the reduction in ηr0 over the window of explored volume fractions. By performing two independent sets of experiments, we are thus able to delineate the dominating contributions to the evolution of the zero shear viscosity relative to the hard-sphere value (i.e., the polymer concentration and the colloid volume fraction). The first set of experiments indicates that, at any given φ, changes in viscosity are dependent on the polymer concentration, cp/c*p and that this is linked directly to the impact of polymer concentration on the magnitude of S(q*), which in turn controls the magnitude of caging resistance. In the second set of experiments, the polymer concentration is held constant thereby approximately fixing the polymer-induced cage disruption. As particle localization gets stronger with increasing φ, small modifications in the nearest-neighbor cage that lead to a decrease in S(q*) have an increasingly larger impact on the flow behavior of the suspension, such that the dense glassy suspensions are reduced to liquidlike behavior. IV. Comparisons with Model Predictions Schweizer and Saltzman21 recently proposed a model that integrates the naı¨ve mode coupling theory (NMCT) of Kirkpatrick and Wolynes22 with the concept of single-particleactivated hopping mechanisms. The full-blown ideal mode coupling theory (IMCT) predicts the glass transition at φ ≈ 0.52. Above this volume fraction, particles are localized and are unable to diffuse out of their cages. Thus, the IMCT predicts that, above the glass transition, the intermediate scattering functions are unable to decay to zero and possess a nonzero long-time plateau value (i.e., f(t f ∞) * 0). Experimental studies8,9 have shown that ergodic systems can be obtained at volume fractions above that predicted by the IMCT. Brownian dynamics simulations36 show that as the glass transition is approached, the dominant mechanism for relaxation of the density fluctuations is the presence of single-particle-activated hopping mechanisms over localization barriers. Constructing a dynamic free-energy landscape from knowledge of the structure factor, S(q), Schweizer and Saltzman relax the assumption of permanent localization in glassy suspensions by predicting relaxation times for the “hopping” of a particle over a freeenergy barrier to escape localization. For hard-sphere suspensions, the NMCT predicts a glass transition at φ ) 0.432. By using the dynamic free-energy concept to predict single-particle

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Figure 8. Predictions of the SS model. The figure plots the viscosity relative to that of the hard-sphere suspension at the corresponding φ against the polymer concentration, cp/c* p. The model predicts the onset of a significant decrease in viscosity followed by a drastic increase on approaching the gel boundary. The decrease in viscosity is magnified in the inset plot.

relaxation times above this volume fraction and incorporating the hopping time as an additional diffusion time into Cohen’s model6 to predict viscosities, Schweizer and Saltzman can capture experimental measurements of the zero shear viscosity for hard-sphere systems over the entire range of experimentally accessible volume fractions. Predictions of the original Cohen theory, the BCMF model, severely underestimate the zero shear viscosity of hard-sphere suspensions above φ ≈∼ 0.45-0.50. However, incorporation of the hopping time into the calculations corrects for this underestimation and provides a near-quantitative match with existing experimental data. For a detailed discussion on the model, we direct interested readers to the aforementioned references. With knowledge of the structure factor of the suspension, the model offers an avenue to predict the viscosity of colloidal suspensions. Since the PRISM model,28 developed by Schweizer and co-workers, has been shown to accurately capture the trends in S(q) with the addition of polymer in dense suspensions on monodisperse spheres,17 we use this model to provide S(q) as input into the Schweizer and Saltzman model to predict zero shear viscosities. Below, the discussion focuses on the ability of the Schweizer and Saltzman model to capture our experimental observations. A discussion of the model can be found in our previous work,18 as well as the original references of the model.20 We would like to emphasize that the PRISM model does not incorporate a distribution in particle size as is present in our experimental system. Thus the predictions of the PRISM and hence the SS model are restricted to suspensions of monodisperse particles. However, our observations on the evolution of S(q*) and ηr0 from experiment essentially replicate those for systems with nearly monodisperse distributions in particle sizes.17,18 The magnitude of reduction in S(q*) with increasing cp/c*p, relative to the hard-sphere value, is nearly identical to that observed in the aforementioned studies on nearmonodisperse systems. As a result, all of our comparisons with predictions from the SS model are made relative to the hardsphere suspension at the corresponding φ. Predictions of the SS model for the zero shear viscosity normalized to the hard-sphere value (absence of polymer) are shown in Figure 8. The calculations for the structure factor were performed on an athermal code developed by Fuchs and Schweizer.28 At φ ) 0.38 and 0.43, which are volume fractions lower than the NMCT-predicted glass-transition volume fraction

Figure 9. Minimum viscosity obtained with the addition of polymer relative to that of the hard-sphere suspension vs φ. The circles are values from the experiment and the squares are predictions from the SS model at the respective φ.

(φ ) 0.432), since the particles are not localized, the hopping time is zero, and hence the predictions are given by the original BCMF model.6 At φ ) 0.48, 0.51 and 0.54, the model predicts a reduction in η0 with increasing polymer concentration. Subsequent increases in polymer concentration lead to an increase in viscosity, and η0 increases rapidly as the gel boundary is approached. This is clearly seen in the case of φ ) 0.54. The inset plot in Figure 8 magnifies the region of the main plot where a reduction in viscosity is observed. A number of interesting features are observed in the inset plot. With increasing φ, the minimum in η0 relative to the hard-sphere value becomes more significant. This is consistent with experimental observations, where as the volume fraction increases, the decrease in viscosity with the addition of polymer becomes more substantial. Another interesting feature is the transition of the position of the minimum in η0/η0HS to smaller polymer concentrations as the volume fraction increases. Since we did not perform experiments over a range of cp/c*p for every φ, we are unable to compare this prediction with experimental trends. However, an important distinction between the model prediction and experiment is the position of the minimum in zero shear viscosity at φ ) 0.54. This minimum is observed at cp/c*p ≈ 0.03 in the experiment, whereas the model predicts the minimum in η0 to exist at cp/c*p ≈ 0.15. The model thus underestimates the window of polymer concentrations within which the zero shear viscosity is less than the hard-sphere value. In Figure 9, we compare the maximum reduction in η0 obtained at each φ as determined from the SS model with experimental results for cp/c*p ) 0.03. The squares are predictions from the model, and the circles are values from experiment. As discussed earlier, the decrease in η0 relative to the hardsphere value becomes more significant with increasing φ. The model qualitatively captures this trend. However, at the highest φ, the model underestimates the magnitude of the decrease in η0. The comparison of model predictions with experimental values indicates that the SS model qualitatively captures the evolution of η0 with increasing cp/c*p. Quantitative prediction of the position and magnitude of the minimum in η0 does not match observations from experiment. This suggests that the model underestimates the impact of the depression in S(q*) on the zero shear viscosity at the highest volume fractions. It is also possible that weak attractions between particles could result in a deviation from the model calculations. Krishnamurthy and Wagner29 use a semiempirical approach to demonstrate that the presence of very small attractions can generate additional free volume in the suspensions thereby reducing viscosities. Thus


the observed viscosity reduction may be enhanced by the presence of weak residual attractions. The correlation of the experimental trends with model predictions strongly suggests that viscosity enhancements in hard-sphere suspensions at volume fractions above ∼0.38 result from temporary particle localization. Although numerous studies6,37 highlight the significant role that caging forces play with increasing φ, beginning at φ ≈ 0.25-0.30, Cohen’s model is unable to satisfactorily predict the nonmonotonic trends in viscosity upon the addition of polymer.18 The presence of a resistance in the form of a dynamic free-energy barrier that temporarily localizes the particle in the cage, in addition to a simple diffusional resistance from the cage, plays a significant role in enhancing the ability of the BCMF model to make predictions that approach experimental values. Our present study therefore supports the conclusion that localization processes, although dominant in glassy suspension mechanics, gradually grow into prominence in dense suspensions at volume fractions that are significantly below the glass-transition volume fractions. V. Conclusion The focus of this study is to present an investigation of the effect of weak depletion attractions on the zero shear viscosity of dense colloidal suspensions. To enable accurate measurements of the zero shear viscosity on the rheometer, we are limited to small particle sizes at which distributions in particle sizes from existing synthesis techniques are nontrivial. With a standard deviation in particle size of ∼12%, SB-USAXS experiments show that a moderate polydispersity does affect the structure factor and especially S(q*). This makes absolute comparisons with model predictions difficult. However, the polydispersity in particle size does not swamp out the nonmonotonic variations in S(q*) with the addition of polymer, and in fact, the magnitude of reduction in S(q*) is nearly identical to that observed for monodisperse systems. Thus we were able to successfully compare model predictions with experimental trends when the modifications to η0 were normalized to the corresponding hardsphere value. Our study builds on Cohen’s earlier work6 to emphasize the dominant role of particle caging at φ ≈ 0.35 and greater that strongly influences suspension zero shear viscosities. Our first set of experiments at φ ) 0.54 highlight the strong correlation of η0 with density fluctuations on the length scale of the nearestneighbor cage. If this cage can be disrupted, as is shown with weak depletion attractions, zero shear viscosities can be reduced. Krishnamurthy and Wagner also recently demonstrated this via a simplified model29 that captures the ability of weak depletion attractions to generate free volume thereby enhancing particle diffusion that overwhelms the attractive interactions between the particles that favors localization. The second set of experiments in which cage disruption and subsequent viscosity reduction with depletion attractions was explored over a wide range in colloid volume fractions reveals that structural arrest in glassy systems is better described by the smooth buildup in caging and particle localization that slow long-range particle motion with increasing φ until the system appears arrested over experimental time windows. Despite being localized, particles remain thermally activated suggesting that the presence of a glass transition is associated with rapidly changing relaxation times as the transition boundary is approached and the inability of experimental techniques to capture relaxation times that are arbitrarily large. Finally, the study highlights the unique ability of weak depletion attractions to perturb suspension microstructure in

dense ergodic suspensions in such a manner as to substantially reduce viscosities. The general view has been that for a suspension to have the lowest-possible viscosity, its microstructure should be that which is determined solely by hard-spherelike interactions. However, weak depletion attractions open a whole new set of possibilities to design suspensions with viscosities lower than that of hard sphere suspensions that then begs the question “how low can you go?” Acknowledgment This work was submitted to celebrate Bill Russel’s 60th Birthday and builds on his contributions to hard sphere rheology, depletion interactions, and links between statistical mechanics and the mechanics of suspensions. What is reported here could not have been done with out what was done by Bill and his colleagues in the past. The authors would like to thank Prof. K. S. Schweizer and Prof. M. Fuchs for access to their code for predicting S(q) in depletion systems. The SEM studies were carried out in the Center for Microanalysis of Materials, University of Illinois, which is partially supported by the U.S. Department of Energy under Grant DEFG02-91-ER45439. The USAXS experiments were conducted at the UNICAT facility at Argonne National Laboratories in Chicago, IL. The UNICAT facility at the Advanced Photon Source (APS) is supported by the U.S. DOE under Award DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, the Oak Ridge National Laboratory (U.S. DOE contract DE-AC05-00OR22725 with UTBattelle, LLC), the National Institute of Standards and Technology (U.S. Department of Commerce), and UOP LLC. The APS is supported by the U.S. DOE, Basic Energy Sciences, Office of Science under Contract W-31-109-ENG-38. Literature Cited (1) Dekruif, C. G.; Vanlersel, E. M. F.; Vrij, A.; Russel, W. B. Hardsphere colloidal dispersions-Viscosity as a function of shear rate and volume fraction. J. Chem. Phys. 1985, 83, 4717. (2) Cheng, Z. D.; Zhu, J. X.; Chaikin, P. M.; Phan, S. E.; Russel, W. B. Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions. Phys. ReV. E 2002, 65, 041405. (3) Gast, A. P.; Hall, C. K.; Russel, W. B. Polymer-induced phase separations in nonaqueous colloidal suspensions. J. Colloid Interface Sci. 1983, 96, 251. (4) Russel, W. B.; Gast, A. P. Nonequilibrium statistical-mechanics of concentrated colloidal dispersions-Hard-spheres in weak flows. J. Chem. Phys. 1986, 84, 1815. (5) Wagner, N. J.; Russel, W. B. Nonequilibrium statistical-mechanics of concentrated colloidal dispersions-Hard-spheres in weak flows with many-body thermodynamic interactions. Phys. A 1989, 155, 475. (6) Cohen, E. G. D.; Verberg, R.; de Schepper, I. M. Viscosity and diffusion in hard-sphere-like colloidal suspensions. Phys. A 1998, 251, 251. (7) Pusey, P. N.; Vanmegen, W. Phase-behavior of concentrated suspensions of nearly hard colloidal spheres. Nature 1986, 320, 340. (8) Pusey, P. N.; Vanmegen, W. Observation of a glass-transition in suspensions of spherical colloidal particles. Phys. ReV. Lett. 1987, 59, 2083. (9) Vanmegen, W.; Underwood, S. M. Glass-transition in colloidal harde spheres-Measurement and mode-coupling-theory analysis of the coherent intermediate scattering function. Phys. ReV. E 1994, 49, 4206. (10) Fuchs, M. MCT results for a simple liquid at the glass transition. Transp. Theory Stat. Phys. 1995, 24, 855. (11) Fuchs, M.; Gotze, W.; Mayr, M. R. Asymptotic laws for taggedparticle motion in glassy systems. Phys. ReV. E 1998, 58, 3384. (12) Gotze, W. Recent tests of the mode-coupling theory for glassy dynamics. J. Phys.: Condens. Matter 1999, 11, A1. (13) Gotze, W.; Sjogren, L. Relaxation processes in supercooled liquids. Rep. Prog. Phys. 1992, 55, 241. (14) Brady, J. F. The rheological behavior of concentrated colloidal dispersions. J. Chem. Phys. 1993, 99, 567. (15) Pham, K. N.; Puertas, A. M.; Bergenholtz, J.; Egelhaaf, S. U.; Moussaid, A.; Pusey, P. N.; Schofield, A. B.; Cates, M. E.; Fuchs, M.;

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ReceiVed for reView November 13, 2005 ReVised manuscript receiVed March 29, 2006 Accepted April 4, 2006 IE051255U

Viscosity of Hard-Sphere Suspensions: Can We Go Lower?

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