Rheological Behavior of Polymerically Stabilized Suspensions: Two

Langmuir 2001, 17, 5757-5767

5757

Rheological Behavior of Polymerically Stabilized Suspensions: Two Different Polymer Layers Compared P. A. Nommensen, D. van den Ende,* M. H. G. Duits, and J. Mellema Twente Institute of Mechanics, J.M. Burgers Centrum, Rheology Group, Department of Applied Physics, University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands Received July 10, 2000. In Final Form: May 21, 2001 Linear and nonlinear rheological behavior of a suspension of polymer-coated colloidal spheres was experimentally investigated for systems with a polymer layer thickness comparable to the core size of the particles. The low shear plateau of the flow curves increases 5 orders of magnitude with increasing concentration in a very narrow range around a critical concentration. Below this critical concentration, the dependence of the low shear viscosity on concentration differs significantly from hard sphere behavior. Above the critical concentration, low shear viscosity plateaus were observed, too, followed by an extreme shear thinning in which the shear stress was virtually constant. In this concentration range, hysteresis was observed. The behavior at high shear rates was captured with lubrication-based modeling. Viscoelastic behavior could only be measured at concentrations above the transition. The observed storage moduli were virtually frequency independent. Their concentration dependence was satisfactorily described with a model based on the work of Elliott and Russel [see: J. Rheol. 1998, 42, 361.]. An essential ingredient of this model is the radial pair distribution function, g(r12). Using Monte Carlo simulations to calculate g(r12), both ordered and disordered structures were found above a concentration close to the critical concentration found from the flow curves. These structure differences caused only a marginal difference in calculated values for the storage modulus.

1. Introduction Polymerically stabilized suspensions can largely be classified by the thickness of the stabilizing polymer layer L relative to the radius of the solid core Rc. This perhaps intuitive classification applies especially well to the rheological behavior. Suspensions of particles with a very thin layer, that is, L/Rc , 1, behave (almost) hard-spherelike.1,2 If the layer thickness is much larger than the core radius, L/Rc . 1, the particles behave like multiarm star polymers (see for example Kapnistos et al.3). When the layer thickness is comparable to the core radius, an interesting intermediate regime is found. There, a mixture of hard sphere and polymer-like behavior can be expected. In experimental investigations of suspensions of particles with Rc/L g 1.4,4-6 the hard sphere behavior was found to dominate at low to intermediate concentrations. Only at very high concentrations did the properties of the polymer layer manifest themselves. These two concentration regimes can be demonstrated with the viscoelastic behavior in which a liquid to solid transition is observed. Below this transition, the suspension behaves like a system of Brownian hard spheres.6 Above the transition concentration, the storage modulus has become (almost) frequency independent. Due to the deformable polymer layer, it is possible to reach effective volume fractions φeff ) (4π/ 3)(Rc + L)3n, with n being the number density, well above the maximum close packing of hard spheres. Attempts to model the rheological behavior have been scarce so far. They have been focused mainly on two (1) Van der Werff, J. C.; de Kruif, C. G. J. Rheol. 1989, 33, 421. (2) Phan, S.-E.; Russel, W. B.; Cheng, Z. D.; Zhu, J. X.; Chaikin, P. M.; Dunsmuir, J. H.; Ottewil, R. H. Phys. Rev. E 1996, 54, 6633. (3) Kapnistos, M.; Semenov, A. N.; Vlassopoulos, D.; Roovers, J. J. Chem. Phys. 1999, 111, 1753. (4) D’Haene, P. Rheology of polymerically stabilized suspensions. Ph.D. Thesis, Katholieke Unversiteit Leuven, Belgium, 1992. (5) Neuha¨usler, S.; Richtering, W. Colloids Surf., A 1995, 97, 39. (6) Nommensen, P. A.; Duits, M. H. G.; Lopulissa, J. S.; Van Den Ende, D.; Mellema, J. Prog. Colloid Polym. Sci. 1832, 110, 144.

evident features of the polymer layer: the permeability and the deformability. When the solvent can flow through the polymer layer, this will affect the hydrodynamic interactions between the particles. In rheological experiments, this is reflected most strongly at high concentrations and high deformation rates, where lubrication effects dominate. Potanin and Russel7 incorporated the permeability into calculations for the lubrication force between two polymer-coated particles. With small adaptations to this model, we have obtained a good description of the high shear viscosity of several experimental systems with Rc/L in the range of 1.4-7.2.8 When two particles are pushed together, the polymer layers have to deform. This deformation raises the Helmholtz potential of the polymer layers, causing a repulsive interaction between the particles. Like in charged stabilized suspensions, this repulsive interaction influences the rheological behavior enormously. Perhaps the most closely related rheological property is the highfrequency storage modulus G′∞. Elliott and Russel9 modeled G′∞ in a similar fashion as Wagner10 did for charged stabilized suspensions, taking also into account the permeability of the polymer layer. Recently, we were able to obtain a quantitative description of the experimental G′∞ of a silica/poly(dimethylsiloxane) system with a Rc/L of 1.4 with this model.11 These results form an encouragement to further explore this new and interesting field of rheology. Especially, a further step toward star polymers would be interesting. For particles with Rc/L g 1.4, descriptions which take only the permeability and the deformability into account resulted in satisfactory agreement. This relatively simple (7) Potanin, A. A.; Russel, W. B. Phys. Rev. E 1995, 52, 730. (8) Nommensen, P. A.; Duits, M. H. G.; Van Den Ende, D.; Mellema, J. Phys. Rev. E 1999, 59, 3147. (9) Elliott, S. L.; Russel, W. B. J. Rheol. 1998, 42, 361. (10) Wagner, N. J. J. Colloid Interface Sci. 1993, 161, 169. (11) Nommensen, P. A.; Duits, M. H. G.; Van Den Ende, D.; Mellema, J. Langmuir 2000, 16, 1902.

10.1021/la000972k CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001

5758

Langmuir, Vol. 17, No. 19, 2001

Table 1. Properties of the PDMS Used to Synthesize Both SMD65 and SJL23U Mn Mw/Mn segment length a χ in heptane a

8 × 104 g/mol 1.4 0.84 nm 0.33-0.45a

See Nommensen et al. (ref 11) for a detailed discussion.

description can be expected to break down for particles with smaller Rc/L. The polymer character of these particles will be more pronounced. This opens the possibility to detect new phenomena, like entanglements between polymers grafted to different particles. In this paper, we present a study on a polymerically stabilized suspension with a smaller Rc/L. The properties of the polymer chains prior to the grafting onto the cores were identical to those of the previously mentioned suspension with Rc/L of 1.4. The core radius was however synthesized to be half the size. With this suspension, we can investigate the influence of curvature by comparing the experimental results with those of the previous system. Another challenging possibility is to test our modeling of G′∞ and η∞. Here, we have the advantage of already knowing some parameters for the new system including those that in the previous study were used as fit parameters. This allows a more rigorous test case for the modeling. The paper is organized as follows: In the next section, we present the suspension characterization and details of the rheological measurements. This is followed by a first comparison of the experimental data using empirical relations that are typically used for dispersions. The modeling section starts with a description of the polymer layer and the pair potential. The latter is used to obtain the radial pair distribution functions for several concentrations using Monte Carlo simulations. Using the description of the polymer layer, G′∞ and η∞ are calculated. In the last part of this paper, newly discovered features of this particle system are discussed.

Nommensen et al. Table 2. Summary of the Characterization Results SJL23U Measured Quantities Rc 40 ( 3 nm polydispersity cores 10% Rh 85 ( 5 nm density core Fc 1.9 g/mL specific volume dry particle vp 1.8 g/mL PDMS-content by weight 17% L ) Rh - Rc f σ

Derived Quantities 45 nm 8.3 × 102 0.040 nm-2 ) 5.4 mg/m2

SMD65 82 ( 5 nm 8% 140( 5 nm 2.2 g/mL 1.9 g/mL 12% 58 nm 5.3 × 103 0.063 nm-2 ) 8.4 mg/m2

2.1. Materials and Sample Preparation. The measurements presented in this paper were performed on suspensions of spherical silica cores onto which the polymer poly(dimethylsiloxane) (PDMS) has been endgrafted. The solvent is heptane which is a good solvent for PDMS at the temperature of the measurements, 25 °C. The synthesis of the particles was done following the two-step recipe published earlier by us.6 In a nutshell: First, the bare silica cores were prepared according to the method of Sto¨ber et al.12 PDMS molecules were grafted in a separate procedure to the bare particles using the method of Auroy et al.13 An advantage of this two-step method is the possibility to characterize the silica cores before the PDMS is grafted to them. In previous studies,6,8,11 we analyzed the rheological behavior of suspensions with the particle core radius 1.4 times the polymer layer thickness (SMD65). Here, we mainly present data of suspensions of particles with a smaller Rc/L value (SJL23U). To obtain this system, we synthesized cores with a smaller radius and grafted PDMS chains on their surface. The PDMS chains had, before grafting, the same properties as for SMD65 (see Table 1). A comparison between the two particle systems is given in Table 2. The grafting density σ of SJL23U turned out to be different. It was hard to get good control over the precise magnitude of σ with the grafting method of Auroy. Like Castaing et al.,14 we have used diisopropyl ethylamine as a catalyst to compensate for the lower surface reactivity of the smaller silica

cores. Nevertheless, σ is lower than for the previously studied system. But the stability of the suspension turned out to be good. The characterization of SJL23U was done using similar techniques as used for SMD65.6 It was found that L of SJL23U is much larger than the averaged lateral distance between the polymer chains (σ-1/2 ≈ 5.0 nm giving Lσ1/2 ≈ 9). So the polymer chains are strongly stretched and form brushlike structures. A similar result was obtained for SMD65 which had a Lσ1/2 of 17. All sample suspensions were made from a single stock of known concentration. Concentrated samples were made by centrifuging a weighed amount of stock, pipetting off the calculated weight of heptane, and subsequently shaking the suspension vigorously. The speed during centrifugation was set to 1000 rpm (g ≈ 2 × 103 m/s2) during a period of 150 h. We used this low acceleration to prevent irreversible aggregation between the particles, which has been seen to occur (by ourselves and others14) during prolonged interparticle contacts at very high osmotic pressures. This phenomenon is ascribed to a polymer bridging similar to that described by Castaing et al.15 For our particles, it occurred only for φc > 0.22, which corresponds to interparticle distances that are much smaller than the distances involved in our experiments and modeling. All samples that were prepared via the mild centrifugation mentioned above could be easily redispersed afterward. Pipetting or shearing the suspensions was never seen to cause aggregation. Light scattering experiments on samples diluted after the mild centrifugation confirmed the absence of aggregates. From the dry particle weight fractions w, weight concentrations c were calculated according to 1/c ) (vp - vs) + vs/w with vp and vs being the gravimetrical specific volumes of the dry particles and the solvent, respectively. The core volume fraction is calculated from the weight concentration using the relation φc ) cvc(1 - wp) where vc denotes the gravimetrical specific volume of the core and wp denotes the weight fraction of polymer per particle. 2.2. Rheological Measurements on SJL23U. Various rheological properties of SJL23U were measured as a function of concentration with a Contraves LS40 and a Haake RS150. Both instruments were equipped with a homemade vapor lock to prevent changes in concentration during the measurements. Ethylene glycol was used as the sealing liquid. Using the vapor lock, negligible changes in concentration were detected after the sample had been inserted, over a period of approximately 12 h. However, during sample insertion a small amount of the suspending liquid could evaporate, inducing a small increase in the concentration. While being small, this increase will have been different for each sample. At high concentrations, this can still have a significant effect; this is evidenced by the sample prepared with φc ) 7.9% which has a slightly higher (instead of lower) yield stress and viscosity than the 8.0% sample, as can be seen in Figure 1. For the lower concentrations, the Contraves LS40 was used to measure the flow curves in the range of shear rates 1 < γ˘ < 80 s-1. A Couette geometry was used with inner and outer radii of 5.5 and 6.0 mm, respectively. The steady-state condition was

(12) Sto¨ber, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62. (13) Auroy, P.; Auvray, L.; Leger, L. J. Colloid Interface Sci. 1992, 150, 187.

(14) Castaing, J. C.; Allain, C.; Auroy, P.; Auvray, L.; Pouchelon, A. Europhys. Lett. 1996, 36, 153. (15) Castaing, J. C.; Allain, C.; Auroy, P.; Auvray, L. Eur. Phys. J. B 1999, 10, 61.

2. Experimental Section

Rheology of Polymerically Stabilized Suspensions

Figure 1. Flow curves of SJL23U for several concentrations. The solid lines represent the fits to (1). Flow curves at high concentrations show different behavior. The dashed lines help to guide the eye. Low shear viscosity plateaus are followed by an extreme shear thinning region. Note also the hysteresis loops. Especially the flow curve at 8.0% shows a dramatic shear history dependence. The inset shows the stress as a function of the shear rate for this sample. Viscosities measured with increasing shear rate were in all cases found to be larger than the curves with decreasing shear rate. checked by monitoring the viscosity at constant shear rate as a function of time. For the more concentrated samples, a Haake RS150 has been used to measure flow curves and the viscoelastic behavior. The RS150 was equipped with a cone-plate geometry with a diameter of 60 mm and a 2° angle. It is a controlled stress apparatus in contrast to the LS40, which is a controlled rate instrument. We used a controlled stress apparatus for the more concentrated samples because we expected (dynamic) yielding behavior, which was also observed for SMD65.8 The RS150 can also be used in controlled rate mode. In this mode, the stress is adapted using a feedback loop until the desired shear rate is obtained. The time necessary for this usually amounts a few seconds. For measuring the linear viscoelasticity, it is important to have a well-defined structure in the beginning of the experiment. In previous studies (see for example ref 4), hairy particle suspensions were found to show a clear shear history dependence in their viscoelastic behavior. Therefore, samples were given a long time (3-7 days) of rest after preparation. No sedimentation was observed during this time. Subsequently, they were brought into the rheometer using a Finntip pipet and subjected to the following protocol: The elastic modulus G′ was measured at a frequency of 1 Hz and the smallest accessible stress amplitude τ ) 0.014 Pa during 600 s. This measurement allows an investigation of the influence of the (following) series of preshears. A series of preshears was applied over a period of 120 s with γ˘ equal to 1, -1, 2, -2, 4, and -4 s-1. By change of the direction of the applied shear rate, the sample is homogenized inside the measurement geometry without excessive deformation. In this fashion, the induction of a nonequilibrium particle configuration is avoided as much as possible. One hour of rest was given to

Langmuir, Vol. 17, No. 19, 2001 5759 allow the configuration of particles and polymers to relax. G′ was measured at a frequency of 1 Hz and the smallest accessible stress amplitude τ ) 0.014 Pa during 60 s. In most cases, this gave a reproduction of the measurement directly after sample injection. G′ was measured as a function of the stress amplitude to determine the limit of the linear regime. G′ and G′′ were measured as a function of frequency. Here, the measurement at 1 Hz indicated in all cases that the rest structure was still intact. In a creep experiment, γ(t) was measured for several applied (small) stresses τ. From these curves, γ˘ (τ) was obtained as well as the viscosity η ) τ/γ˘ . Thus, Newtonian viscosity plateaus for 3 × 10-6 s-1 < γ˘ < 1 × 10-2 s-1 could be obtained. More points of the flow curve were determined using the “controlled rate mode” in the γ˘ -range 3.0 × 10-2 to 1.0 × 103 s-1 starting from low to high γ˘ and back. After each change in γ˘ , the sample was allowed to adjust for some time (typically 60 s) after which the stress was measured. Unless stated otherwise, all data presented in this paper were acquired using this protocol. 2.2.1. Flow Curves. The relative viscosity ηr ) η/ηs, with ηs ) 0.386 mPa s being the solvent viscosity, is shown in Figure 1 as a function of shear stress for core volume fractions 1.5% e φc e 9.7%. The concentration of the samples has a profound influence on the viscosity. Besides a tremendous increase of its magnitude with increasing concentration, the qualitative behavior also changes drastically. Dilute suspensions show only Newtonian behavior. Concentrated suspensions show a strong shear stress dependence of the viscosity. There is a clear transition visible between φc ) 7.5% and 8.0%. Above this transition, the magnitude of the viscosity changes drastically over the full τ-range. Also, the shape of the flow curve changes. There is still a low shear viscosity plateau visible, but it is followed by an extreme shear thinning region in which the viscosity drops 4 orders of magnitude in a very narrow stress range. The flow curves at concentrations below the transition can be described quite accurately as a function of γ˘ with the expression given by Cross:16

ηr ) ηr,∞ +

ηr,0 -ηr,∞ γ˘ m 1+ γ˘ crit

( )

(1)

By fitting the Cross relation to ηr(γ˘ ), we obtained relative values for the low and high shear viscosity, ηr,0 and ηr,∞, and the critical shear rate γ˘ crit. The results of these fits are plotted in Figure 1 as solid lines (note that we calculated τ by multiplying γ˘ with the fitted result for η(γ˘ )). The concentration dependence of ηr,0 and γ˘ crit will be discussed in section 3, and that of ηr,∞ will be discussed in section 4.3. The description with the Cross relation breaks down for volume fractions above the transition. Here, two different stress regimes could be distinguished. For stresses between 0.1 and 1.0 Pa, plastic behavior was observed with a dynamic yield stress τy as also has been reported for other polymerically stabilized suspensions.4,8,17,18 At stresses below τy, small but steady shear rates could be detected. These shear rates were proportional to the applied shear stress, indicating a low shear viscosity plateau. To check the steady state of the shear rate in these creep experiments, stress jump experiments were performed. This type of experiment is a sequence of three applied stresses each during a time span that is long enough (typically 10 min) to ensure that the sample has reached a steady shear rate. The sequence starts by applying a stress τ0 < τy to the sample, after which τ is increased in a single step to τ1, also smaller than τy. In the last part, the applied stress is reduced again in a single step to τ0. During this experiment, the shear amplitude and shear rate are monitored. The third part of the experiment is the one of interest. Directly after the stress was reduced to τ0, the sample started to rotate backward, thus releasing part of the elastic contribution to the strain. After some time, the sample began to rotate again in the forward direction. The shear rate leveled off to a value close to (16) Cross, M. M. J. Colloid Sci. 1965, 20, 417. (17) Jones, D. A. R.; Leary, B.; Boger, D. V. J. Colloid Interface Sci. 1992, 150, 84. (18) Vermant, J.; Raynaud, L.; Mewis, J.; Ernst, B.; Fuller, G. G. J. Colloid Interface Sci. 1999, 211, 221.

5760


the steady-state value in the first part of the experiment. Hence, the measured shear rate is due to dissipative processes and not due to slow elastic relaxations. A second point to check is the extreme shear thinning, which is sometimes addressed to wall slippage (see for example Buscall et al.19). To test the wall slippage hypothesis, measurements were also performed with a plate-plate geometry. Here, the extreme shear thinning behavior was again observed. Changing the distance between the plates resulted in variations of the viscosity, smaller than the typical experimental error, so no gap effects were observed, as would be expected in case of wall slippage. Another peculiarity of the flow curves just above the transition concentration is the hysteresis effect. The shape of the flow curve depends on the direction in which the shear rate was changed. We investigated this phenomenon by measuring the flow curve using the controlled rate mode where we altered γ˘ stepwise. The time between two steps was typically 60 s, which is long enough for the Haake RS150 to adjust itself. We started with small γ˘ and measured up to γ˘ ) 1.0 × 103 s-1. From here on, the shear rate was decreased. At high γ˘ , the “down going” curve reproduced the “up going” curve quite well. But for smaller γ˘ , the down going curve resulted in smaller magnitudes for the stress. These deviations are the most pronounced in the region where the shear thinning is the strongest. For the sample with φc ) 8.0%, the down going curve seems to behave as a member of the family of flow curves at low concentrations, down to a certain point. Here, the fluid seems to (re)discover its stable state, giving rise to a dramatic viscosity increase and a multiple valued γ˘ (τ) in the measured curve. At higher concentrations, this was also seen although less pronounced. We also measured the flow curve of φc ) 8.0% in the controlled stress mode. Again, hysteresis was observed. 2.2.2. Viscoelastic Behavior. The viscoelastic behavior of SJL23U measured with harmonic oscillations showed also a sharp transition between φc ) 7.5% and 8.0%. Below the transition, it was not possible to apply small enough stresses to observe linear behavior. Even at the smallest accessible stress amplitude, we observed a strain response with an amplitude larger than 10%, which is much larger than the typical value for the critical strain observed for SMD65 (γcrit = 2%). Above the transition, we were able to do linear measurements. This was checked by measuring the storage modulus G′ at ω ) 0.6 and 6.0 rad/s as a function of τ. G′ is roughly constant at small stresses and drops suddenly to (almost) zero at stresses larger than τcrit. The magnitude of τcrit is, within the experimental error, identical to that of the dynamic yield stress τy observed in the flow curves. When G′ is plotted versus the strain, the end of the linear regime is not marked with an abrupt collapse of G′. At γ < γcrit, G′ is almost independent of γ, but at γ > γcrit, G′ decreases gradually with increasing γ. The magnitude of γcrit was for all concentrations approximately 2%. The storage modulus G′ is almost independent of the frequency as can be seen in Figure 2. The loss modulus G′′ shows in all cases a minimum. We have checked that this minimum is not due to inertia effects. This type of behavior was also seen in our previous study at high concentrations. To investigate the consistency between the linear viscoelastic and the flow curve measurements, we have compared the linear viscosity η′ and the steady shear viscosity η. In the limit of slow deformations, these quantities should become equal. In Figure 3, we plotted the steady shear viscosity together with η′ ) G′′/ω for the sample with φc ) 8.8%. At low frequencies, the magnitude of η′ approached that of the viscosity plateau of the flow curve. The expected limiting level for η′(ω f 0) implies that G′′(ω) has a maximum near ω ≈ 10-3 rad/s, indicating a relaxation time of approximately 103 s.

Nommensen et al.

Figure 2. The elastic moduli as a function of the frequency for concentrations above the transition (b, 8.0%; 9, 7.9%; [, 8.5%; 1, 8.8%; 2, 9.7%). The filled symbols present the G′ data, and the open symbols represent the G′′ data.

Figure 3. Comparison of steady shear viscosity with G′′/ω ) η′r: O, η′r(ω); b, ηF(γ˘ ).

applicability of these relations to SJL23U as a diagnostic tool, that is, to reveal its characteristic behavior compared to other soft sphere systems. Deviations from these relations would not be unexpected considering the relatively small ratio Rc/L of SJL23U. First, we attempted to describe the low shear viscosity for samples below the transition concentration. In a number of previous studies (see for example refs 4, 17, and 20), the volume fraction dependence of η0 was excellently described by the relation of Quemada,21

(

ηr,0 ) 1 -

)

φv φmax

-2

(2)

For polymerically stabilized suspensions, a number of empirical relations involving rheological quantities have been reported in the literature. Here, we investigate the

with φv being the volume fraction of the sample based on either the core or the outer radius of the particle at low concentration (the latter is often denoted as the effective volume fraction) and φmax being the corresponding volume fraction where the viscosity diverges. Since Quemada’s equation has been proven successful in describing the low shear viscosity of hard spheres, a good fit to eq 2 indicates effective hard sphere behavior of the particles: the softness of the particles can be taken into account by using an effective hard sphere radius. As shown in Figure 4, the volume fraction dependence of ηr,0 of SJL23U below the transition is poorly described by eq 2. We demonstrate this by plotting the experimental data together with calculations of eq 2 with a φmax of 7.5% and 8.0%. The curve for φmax ) 7.5% describes the data quite well up to φc ≈ 6%. At higher volume fractions up to the transition ηr,0 cannot be described satisfactorily

(19) Buscall, R.; McGowan, J. I.; Morton-Jones, A. J. J. Rheol. 1993, 37, 621.

(20) Buitenhuis, J.; Fo¨rster, S. J. Chem. Phys. 1997, 107, 262. (21) Quemada, D. Rheol. Acta 1977, 16, 82.

3. Empirical Relations


Langmuir, Vol. 17, No. 19, 2001 5761

10-2. This slope was estimated theoretically by Buscall22 to be 0.02 for suspensions of soft repulsive spheres where both τy and G′∞ are dominated by the pair potential. The observed correlation between τy and G′∞ indicates that also for this system not only G′∞ but also τ is related to the particle pair potential. 4. Modeling G′∞(Oc) and η∞(Oc)

Figure 4. The low shear viscosity vs core volume fraction. It is not possible to describe the data using a Quemada expression as can be seen by the solid lines that represent calculations with a φmax of 0.06 and 0.078.

Figure 5. Reciprocal critical shear rate versus the low shear viscosity for SJL23U (b) and SMD65 (4). The solid lines are power law fits; the dashed line represents a linear fit to data of SJL23U.

anymore with eq 2. The position where ηr,0 diverges is described well by the curve with φmax of 8.0%, but this curve gives a poor description of all data below φc ) 8.0%. The absence of a good fit to eq 2 indicates that the particles do not behave as effective hard spheres at volume fractions below the transition. Also, the location of the shear thinning transition was investigated in previous studies on soft spheres below the transition concentration. As a characteristic value, one can take the critical shear rate γ˘ crit, defined as the shear rate where the viscosity equals 1/2(η0 + η∞). D’Haene4 found that the data of several systems with different particle radii could be collapsed onto a master curve by multiplying γ˘ crit with η0a3/kBT, where η0 is the low shear viscosity of the suspension. When scaling in a similar fashion, our data for both SJL23U and SMD65 fall also on this master curve. This scaling behavior corresponds to a linear relation between γ˘ crit-1 and η0. Looking more precisely, the data of SMD65 indicated a slightly stronger power behavior. These data, that covered 5 orders of magnitude, were better described by a power law function with 1.23 ( 0.04 as exponent. Fitting a power law function to the data of SJL23U results in an exponent of 0.8. The range of η0 is too small (3 × 10-3 to 4 × 10-2 Pa s) to obtain this exponent with high accuracy. In Figure 5, also the fit of the function γ˘ crit-1 ) c0η0 to the experimental data has been plotted. When τy is plotted versus G′∞, a linear dependence is obtained with a slope of (1.6 ( 0.1) × 10-2. This linear dependence was also seen in our previous work, in other polymerically stabilized suspensions, and in charge stabilized suspensions all with a slope in the range 2-4 ×

4.1. The Polymer Layer. Before we can start modeling the rheological behavior, we first have to obtain a description of the polymer layer on the particles, anticipating that the properties of the entire suspension are dominated by those of the polymer layer. In a previous study, we used two different methods from polymer physics to calculate the layer thickness under noncompressed conditions and to calculate the change in the Helmholtz potential of the polymer layer under a uniform compression.11 This layer thickness is an important parameter in the modeling of both η∞(φ) and G′∞(φ). With the assumption that during compression of a particle pair no lateral relaxations take place in the layer, the pair interaction potential Φ(r12), that is, the change in the Helmholtz potential under a local deformation, can be obtained. This Φ(r12) is of key importance in the modeling of G′∞(φ). Both models for the polymer layer were capable of describing the observed concentration dependence of G′∞ for the SMD65 particles within the experimental uncertainties. Here, SJL23U is confronted with the same two models to investigate their applicability for particles with a smaller Rc/L. The first model is a mean field approach for a polymer brush on a curved surface using an analytical description (hereafter referred to as AA). The second model is a self-consistent field lattice model which uses spherical symmetry (hereafter referred to as SCF). AA is based on a variation technique that was developed by Li and Witten23 for strongly stretched, multiarm star polymers in a marginal solvent. In this mean field description, the Helmholtz potential F is written as the sum of an elastic, entropic term and an excluded volume interaction term. The latter term represents the osmotic pressure contribution to F which is written as a virial expansion, truncated after the quadratic term. F is a functional of the segment volume fraction n(r). Li and Witten obtained n(r) by minimizing the Helmholtz potential. We extended this model11 by incorporating an impenetrable core with radius Rc. The SCF model is an extension of the polymer adsorption theory of Scheutjens and Fleer24,25 and has been published by Wijmans and Zhulina.26 The polymer layer is described by a spherical lattice where each site is occupied with either a polymer segment or solvent molecule. The equilibrium segment distribution is calculated by taking into account all possible conformations, each weighted by its Boltzmann factor. The Helmholtz potential of the polymer layer can be calculated directly without the need for a virial expansion. In the calculations of compressed polymer layers, special attention must be paid to the outer bounding surface that confines the volume accessible to the polymer chains. If this wall is chosen to be impenetrable, a depletion zone for the segments occurs near the wall confining the polymer layer to a smaller volume than (22) Buscall, R. J. Chem. Soc., Faraday Trans. 1991, 87, 1365. (23) Li, H.; Witten, T. A. Macromolecules 1994, 27, 449. (24) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (25) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (26) Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214.

5762


Figure 6. The segment volume fraction profile of SJL23U calculated by AA and SCF together with the segment volume fraction of SMD65 calculated by SCF, all for noncompressed states.

was intended. We modeled the wall as a mirror to make this depletion zone disappear. The parameters used in the calculations for both AA and SCF are listed in Tables 1 and 2. In the modeling of G′∞ of SMD65, we used the solvent quality χ as fit parameter. The best results were obtained for the SCF calculations with χ ) 0.45 and for the AA calculations with χ ) 0.36. The discrepancy in the value for χ was explained by the neglect of the higher order terms in the virial expansion for F in AA. A smaller value for χ gives a larger excluded volume interaction that covers for the absence of the higher order terms. In the AA calculations, the best-fit value found for χ can thus be expected to depend on the average segment volume fraction of the layer. We used χ again as a fit parameter in the AA calculations for describing η∞ and G′∞ (to be discussed in sections 4.3 and 4.4). A reasonable description of both quantities is obtained when χ is set to 0.40. In the SCF modeling, we just kept χ equal to 0.45. The corresponding segment volume fraction profiles are shown in Figure 6. The calculated segment volume fraction in SJL23U is considerably lower than that in SMD65, making higher order terms in the virial expansion in F less important. Recalling that the fitted value of χ may absorb errors due to neglect of these higher order terms in AA, the higher value for χ in the AA description of SJL23U is in accordance with the expectations. The calculated segment volume fraction profiles n(r) from both models agree quite well for SJL23U despite some qualitative differences, which were already discussed in our previous work.11 The good correspondence between both model calculations for n(r) justifies the approximation in the AA model that the polymer chains are strongly stretched. Note that this assumption is not made in the SCF model. At the periphery of the polymer layer, the strong stretching approximation is not applicable since the segment volume fraction becomes too low there. In that region, the SCF curve becomes concave resulting in a (Gaussian) tail that extends over more than 100 nm. Both models predict a layer thickness of comparable magnitude, 41 and 40 nm by AA and SCF (ignoring the contribution of the tail), respectively, which compares well with the experimental result for the layer thickness of 45 ( 6 nm. Results of the pair potential calculations can be seen in Figure 7. The potential Φ(r12) calculated with SCF has a larger value over the full r12 range than Φ(r12) calculated with AA. But at intermediate and strong compression, the agreement is fairly good. For particle pairs with r12 - 2Rc > 80 nm, AA predicts no compression of the polymer layers, in contrast to SCF. According to SCF, particle pairs

Nommensen et al.

Figure 7. Pair potentials calculated with either SCF or AA.

start feeling each other at larger separations resulting in a nonzero Φ(r12) for r12 - 2Rc > 80 nm. 4.1.1. Interpenetrations. The modeling presented so far can also be used to consider the possibility of other processes occurring when the polymer brushes belonging to different particles are brought into interaction. A conceivable process would be the penetration of “guest chains” into “host brushes”. The driving force for these interbrush penetrations is the increase of the mixing entropy which lowers the Helmholtz potential. Brush interpenetrations, if/when they occur, can influence the rheology of hairy particle suspensions in different ways. For rheological quantities associated with equilibrium properties such as G′∞, it is sufficient to consider the effect of penetrations on the particle pair potential. For rheological properties that depend on time scales, it is of interest to know whether temporary entanglements between polymers belonging to different particles should be taken into account. In the strong stretching regime, that is, for polymer brushes, chain penetration is expected to have a negligible effect on the rheology if the number of segments per chain is high. Martin and Wang27 investigated the effect of interbrush penetration for brushes on flat surfaces under compression, using a self-consistent field lattice model. In their calculations, both brushes were incorporated, which makes it possible to directly observe penetrations in the segment volume fraction profiles. For brushes with 1000 segments per chain, the extent of penetration is negligible compared to the layer thickness. The effect is more pronounced for chains with 200 segments per chain, but the extent of penetration is still small. We recall here that the PDMS grafted on SJL23U has approximately 400 segments per chain. Martin and Wang also calculated the force profiles for compression of the two brushes. Only at very slight compression did interbrush penetrations influence the magnitude of the interaction. Introducing curvature for the grafting surface influences these conclusions somewhat; however, the segment volume fraction profiles for spherical particles were very similar to those obtained by Martin and Wang. This implies that also for our spherical brushes, interpenetrations can be expected to be of no importance to the rheological behavior. However, at the periphery of the polymer layer the strong stretching approximation will break down for our particles. In that region, interpenetration is more likely. The AA model does not account for this breakdown of the strong stretching approximation, but the SCF model does. We used the shape of the segment volume fraction profile to estimate where the strong stretching approximation breaks down. For strong stretching, the profile has, like (27) Martin, J. I.; Wang, Z.-G. J. Phys. Chem. 1995, 99, 2833.


Langmuir, Vol. 17, No. 19, 2001 5763

Figure 8. The segment volume fraction profile n(h) under uniform compression as calculated with SCF with the sealing mirror at h ) 50, 45, 40, 35, and 30 nm.

the AA model predicts, a convex shape. In absence of strong stretching, the chains follow Gaussian statistics, which results in a concave shape. Hence, the transition from strong stretching to Gaussian behavior is expected to occur near the inflection point in the volume fraction profile as calculated by SCF. We use the position of the inflection point again to mark the regions where interpenetration cannot be excluded for compressed polymer layers. Segment volume fraction profiles were calculated with SCF with the polymer layer compressed by a mirror wall. These profiles are plotted in Figure 8. The regions where interpenetrations can occur are typically a few nm thick for low to moderate compressions. Comparing this distance with the Kuhn length indicates that the interpenetration region is only a few Kuhn lengths deep. 4.2. The Particle Configuration. The rheological behavior of a suspension depends strongly on the particle configuration. But this configuration changes at the moment the suspension is forced to flow. For weak flows such as in small amplitude harmonic oscillations, these changes are small so a series expansion (i.e., a perturbation) around the equilibrium particle configuration is a suitable choice for modeling the rheological behavior. In section 4.4, such a series expansion is used in the calculation of G′∞. This section focuses on the results of the Monte Carlo simulations that were performed to obtain the radial pair distribution function at rest as a function of the particle concentration. The simulation followed the Metropolis scheme using the pair potential Φ(r12) as calculated in section 4.1 for the particle-particle interaction. Most runs were performed with 216-256 particles in the simulation box. A few test runs were done with 1024 particles. All runs consisted of at least 30 × 103 steps, but some runs were continued over more than 2.5 × 105 steps. During the runs, the order parameter Q6 was calculated to monitor the structure. Q6 is defined as28

Q6 )

(

4π 13

6

∑

m)-6

| |)

2Leff ) r1kT - 2Rc

1/2

Q h 6m

2

within the first shell of g(r12) with respect to a certain particle are considered to be nearest neighbors of that particle. The Ylm are spherical harmonics which depend on the polar and azimuth angles of r. Due to its quadratic form, Q6 is invariant under rotations of the coordinate system. In simulations with a finite number of particles, a random configuration results in a Q6 smaller than 0.1. For crystalline structures, Q6 becomes larger, 0.45 and 0.5 for face-centered cubic (fcc) and body-centered cubic (bcc), respectively. At the start of a simulation run, Q6 was allowed to reach a steady-state value after which the radial pair distribution g(r12) was determined by averaging over the remaining part of the simulation. The shape of g(r12) and the resulting value of Q6 were used to characterize the particle configuration. The position and the height of the maxima in g(r12) were determined to facilitate this. For each core volume fraction, separate runs were performed, each with a different starting configuration. The results of the simulations showed a freezing transition at φfr where the exact location of φfr depends on the details of Φ(r12). At φc < φfr, the resulting g(r12) is insensitive to the starting configuration. This changes drastically for φc > φfr. In this regime, the starting configuration has a striking influence on the outcome of the simulation. Depending on the starting configuration, both ordered and disordered structures were obtained for simulation runs with the same φc. In some runs, crystallization was observed to a fcc lattice with stacking faults. More details about this can be found in ref 29. The volume fraction of the freezing transition φfr was determined using the results from a series of simulation runs starting with fcc lattices with decreasing volume fractions. In our opinion, this is the most reliable way of determining φfr since the kinetic barrier for melting is smaller than for crystallization. Below a certain volume fraction, the fcc structures disappeared completely. When plotting Q6 versus φc (see Figure 9), a sudden jump in Q6 marks φfr. The magnitude of φfr depends on the details of the particle system as can be seen in Table 3 where we collected the results for SJL23U and SMD65. The layer thickness has a profound influence on φfr. So it might be instructive to transform φfr to an effective volume fraction. This is done by multiplying φfr with (1 + L/Rc)3. A problem in such a calculation is the determination of L, which is not unambiguous. In AA, L is defined as the distance from the surface where the segment volume fraction becomes zero. The magnitude for L in the SCF calculations is determined by extrapolating the segment volume fraction from the inflection point using a quadratic function. Further, it can be expected that the layers are already deformed to some extent before the repulsion due to deformation becomes noticeable. For that reason, we use an effective layer thickness

with

Q h lm )

1 Nb

/

Ylm(θ(rij), φ(rij)) ∑ i,j

(3)

/ where the latter summation ∑i,j runs over the total number of nearest neighbor pairs, Nb, in the sample volume, with position ri,j relative to each other. All particles

(28) Steinhardt, P. J.; Nelson, D. R.; Ronchetti, M. Phys. Rev. B 1983, 28, 784.

(4)

to calculate φfr,eff. Here, r1kT is the separation between the particle centers for Φ(r1kT) equals 1kT. For all the systems, φfr,eff has a comparable magnitude: 48-49%, independent of the value for Rc/L. Hence, the freezing transition occurs well before random close packing. Following Russel et al. [ref 30, p 343] who predicted the freezing transition for charged colloidal systems, we can (29) Duits, M. H. G.; Nommensen, P. A.; Van Den Ende, D.; Mellema, J. Colloids Surf., A 2001, 183-185, 335. (30) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.

5764


Nommensen et al.

Figure 9. The order parameter Q6 (a) and the average nearest neighbor position (b) are plotted versus φc. The curves represent different starting configurations based on (×) fcc, (0) bcc, and (4 and O) both liquidlike.

arguments:

Table 3. Location of the Freezing Transition φfr(r1kT/2Rc)3

system

method

φfr

SJL23U SJL23U SMD65

AA SCF SCF

7.5% ( 0.2% 7.1% ( 0.1% 11.5% ( 0.5%

Hav ) 2Rc

48% ( 2 48% ( 1 49% ( 2

pinpoint an effective hard sphere radius on a pair potential of 1kT. The transition should then occur at φfr,eff ) 50%, which is in nice agreement with the results of our Monte Carlo simulations. 4.3. Modeling the High Shear Viscosity. In the regime of high frequencies or high shear rates, hydrodynamic interactions form the dominant contribution to the viscosity. In previous work,8 we presented a model that takes only these hydrodynamic interactions into account. The model predicted the high shear viscosity of SMD65 rather well. In this approach, the relative viscosity was the summation of the solvent viscosity µ, the single particle contribution to the viscosity, and the viscosity due to lubrication µlub:

ηr,∞ ) 1 + [η]cφc +

µlub µ

(5)

φmax φc

1/3

-1

(7)

In this approach, the particle configuration is represented only via the maximum packing fraction φmax. At high shear rates, the lubrication modeling gave the best description when setting φmax to 0.638 which can be identified with random close packing. For the calculation of the highfrequency viscosity (needed to calculate G′∞), the lubrication model can also be used. Obviously, one then has to recognize that the microstructure will be different for high frequencies compared to high shear rates. This is represented by a different φmax, which was obtained from the results of section 4.2. The intrinsic viscosity [η]c is, just as Flub, influenced by the polymer layer. At low concentrations, particles with a dense and/or thick polymer layer can be modeled well by approximating them as uniform permeable spheres with radius Rc + L and permeability δ02. An expression for the intrinsic viscosity was given by Wiegel:32

[η]c )

with [η]c the intrinsic viscosity. The lubrication contribution µlub can be related to the lubrication force Flub by the expression31

2Rc Flub(Hav,δ0) µlub )9 µ 2Rc + Hav 6πµRcV

[( ) ]

5 2

(

)

J1(∆) L 1+ Rc 10 1 + 2J1(∆) ∆

3

(8)

with

(6)

Flub is proportional to the velocity V with which the particles approach each other. Here, we use the same expression for Flub as in our previous work.8 It was developed for particles that are coated with a polymer layer.7,8 The polymer layer hinders the solvent to flow inside the gap between the particles thus increasing the hydrodynamic interaction between them. By modeling the polymer layer as a permeable medium with permeability δ02, the consequences of its presence for the viscosity can be calculated. In their derivation of µlub, Frankel and Acrivos assumed that the motion of the particles is affine and that the pair distribution function is sharply peaked so it can be characterized by a δ-peak at the averaged nearest neighbor position 〈r12〉. The calculation of µlub is then carried out for particles with a core to core surface separation Hav equal to 〈r12〉 - 2Rc. Hav can be related to φc using geometrical (31) Frankel, N. A.; Acrivos, A. Chem. Eng. Sci. 1967, 22, 847.

∆)

3 tanh(∆) R 3 and J1 ) 1 + 2 δ0 ∆ ∆

(9)

The term in brackets in eq 8 is a correction term that transforms the intrinsic viscosity based on the volume fraction of the uniform permeable particles to [η]c, based on the core volume fraction. The magnitude of δ0 can be identified with the characteristic “mesh size” of the polymers7 which can be related to the grafting density σ in the case where the polymers are stretched:33

δo ≈ σ-1/2

(10)

With the determination of δ0 and the experimental value of L, we have quantified all parameters necessary to calculate the high shear viscosity: Rc, L, δ0, µ, φmax, and (32) Wiegel, F. W., Fluid Flow Through Porous Macromolecular Systems; Springer-Verlag: Berlin, 1980. (33) Alexander, S. J. Phys. 1977, 38, 977.


Langmuir, Vol. 17, No. 19, 2001 5765

Figure 10. Comparison of the experimental high shear viscosity with the model calculations with δ0 ) 5.0 nm (according to eq 10), 2.5 nm, and 0 nm (impenetrable polymer layer). The magnitude for the layer thickness (45 nm) was set to the experimental result. The larger error bars at the highest concentrations are associated with the extrapolations needed to obtain η∞ (see Figure 1).

φc (see Table 2). In Figure 10, the theoretical predictions for SJL23U are compared with the experimental results. The model calculations underestimate η∞(φc) a little. By decreasing δ0 compared to the prediction of eq 10, we improve the description considerably. The magnitude of δ0 that results in a good description is relatively small. From a comparison with SMD65, one should expect a value larger than 5 nm. The SMD65 particles have a much denser polymer layer than the particles of SJL23U, and the model calculations of SMD65 indicate a δ0 of 5 nm. 4.4. Modeling the High-Frequency Elastic Modulus. The model calculations presented in this section are based on the work of Elliott and Russel.9 They have derived an expression for G′ in the limit of small amplitude oscillatory shear at high frequency,

3φc2 Rc3 G′∞ ) 3 kT Rc 5π

gs r 2 ∫(W - (1 - A)r12dΦ/kT dr12 ) i 12

Figure 11. Comparison between the model calculations (3, SCF; O, AA) and the experimental G′∞ (0).

The Monte Carlo simulation results were not unambiguous. At concentrations above φfr, both ordered and disordered structures showed up (see section 4.2). G′∞ was calculated with both the ordered and disordered structures.29 The calculated values for G′∞ were identical within the accuracy of the calculation. In Figure 11, we compare the model calculations with the experimental G′∞. The error in the AA calculations was estimated by performing calculations in which the magnitudes of Rc and Nn were varied within their range of experimental error. The error in the SCF calculation will have a comparable magnitude. The calculations result in a satisfactory agreement with the experimental data. Although the calculated values for G′∞ are larger than the experimental values, the experimental values are within the accuracy range of the AA calculations. Comparison of the AA and SCF results shows that the model calculations are very sensitive to Φ, so G′∞ can be estimated only within a factor of 2-3. Still, this agreement is satisfactory, considering the mismatch of more than 1 order of magnitude in earlier studies.9

dr12

5. Discussion

(11)

5.1. Experimental Results. In this paper, flow curve and linear viscoelastic measurements on a suspension of particles coated with a thick polymer layer over a wide range of particle concentrations are presented. Below the critical concentration (i.e., the concentration were the particles start to overlap each other), flow curves could be measured using a conventional controlled rate instrument (LS40) without any problems with respect to measurable ranges. Above the critical concentration, the suspensions behaved quite differently. Because of the high values of the low shear plateau, η(τ) and γ˘ (τ) could be determined only in creep experiments using a controlled stress instrument (RS150), resulting in shear rates as small as 3 × 10-6 s-1. Linear behavior in the plateau regime was checked with shear step experiments. The observed strong shear thinning for stresses of 0.1-1 Pa raised the question of whether it could be due to possible wall slippage. To check this, measurements were performed in different geometries. Wall slippage should influence the results, but no differences were observed. So it was concluded that wall slippage did not occur. The hysteresis effect might have to do with the use of the controlled rate mode of the RS150. This was checked by measuring the shear thinning part of the 8% curve also in controlled stress mode. The two modes gave the same results, so the hysteresis effect shown by the samples must be considered as a material property.

with A and W being hydrodynamic functions depending on η′∞ and δ0. The product gsi reflects changes in the particle configuration due to the applied oscillatory shear. This model incorporates hydrodynamic, Brownian, and direct particle pair interactions. The particle configuration under shear is written as a series expansion around the equilibrium radial distribution function g, which was obtained with Monte Carlo simulations. The specific modeling of the interactions by Elliott and Russel was developed for particles with thin layers (L , Rc). We adapted the pair interaction taking into account the thick polymer layer.11 This resulted in a better description of the measured G′∞(φc) behavior of SMD65. In our extended modeling, no assumptions were made that would preclude a priori its application also to SJL23U. Most ingredients such as Φ(r12) were already calculated in previous sections. Here, we briefly summarize the parameters that were used in the calculations. We start with the hydrodynamic interactions where we set the permeability according to eq 10 and we calculated η∞ using eq 5 where we set Hav to rnn - 2L instead of using eq 7. By taking the position of the first maximum in the radial pair distribution function rnn, we used results of the Monte Carlo simulations. The pair potential Φ was calculated with the parameters as listed in Tables 1 and 2 where χ was set to 0.40 and 0.45 for AA and SCF, respectively.

5766


5.2. η(τ,Oc). New features of the flow curves were observed at low to intermediate stresses. The concentration dependence of the low shear viscosity up to the transition concentration is significantly different from (effective) hard sphere behavior. This is evidenced especially by the abrupt change in the magnitude of η0 near the transition. In the φc region of 7.5% to 8.0%, η0 increases more than 5 orders of magnitude. Expressions which are successful in describing η0 of nearly hard sphere systems as a function of concentration predict a more gradual increase of η0. Departure from the effective hard sphere behavior was also reported by Buitenhuis and Fo¨rster20 for micelle systems. They performed a systematic study on a series of block copolymer micelles. Most of their samples showed a volume fraction dependence following eq 2. But two of their samples which they characterized as “very soft” and “polymeric” could not be fitted with this equation. A remarkable difference between these samples and SJL23U is that the two micelle systems exhibit a steady increase of ηr,0 up to the highest measured concentrations. We observed η0 plateaus above the transition. In our previous study on particle suspensions with a larger Rc/L, we did not observe these plateaus, but then we used controlled rate measuring devices with a lowest accessible shear rate of 2 × 10-4 s-1. So we cannot exclude the possibility that the plateaus would have been observable at γ˘ outside the range of the devices. The abruptness of the change in characteristics of the flow curve might suggest a phase transition. Results of the Monte Carlo simulations corroborate this since the position of the freezing transition corresponds well with the volume fraction where the transition in the rheological behavior was observed. An alternative explanation is the occurrence of interbrush penetration which might induce the formation of aggregates, held together via entanglements. At a critical volume fraction, such aggregates would span the measuring gap, hence causing a steep increasing of the viscosity. Another interesting feature is the hysteresis of the flow curves. The time between the measurement of subsequent data points was long enough to rule out artifacts caused by the transient response of the Haake RS150, so the behavior can be attributed to sample effects. In other hairy particle suspensions, hysteresis was also observed, like in suspensions of poly(butylacrylate-styrene) particles in water with Rc/L of 6.4 and 7.518 and in our previous model suspensions (SMD65). However, a hysteresis as extreme as that observed in the flow curve at φc ) 8.0% was to our knowledge not reported earlier for a hairy particle suspension. In a general sense, the hysteresis can be understand as follows. Up to a critical stress τy the particle configuration is only slightly deformed, resulting in a linear response to the applied stress. If the stress exceeds τy, the “low stress structure” breaks up and a new one builds up. The “high stress structure” has a much lower viscosity. In a stress ramp experiment, with increasing stress a low shear viscosity plateau will be observed followed by an abrupt shear thinning. If the stress is decreased, the high stress structure can persist even at stresses below τy. The rebuilding of the low stress structure takes some time tb. If γ˘ is larger than 1/tb, the flow field prevents the rebuilding of the low stress structure. We propose two mechanisms that might explain the hysteresis. In the just-mentioned paper,18 Vermant used smallangle light scattering to measure the structure of their suspensions under simple shear flow. At low stresses, a

Nommensen et al.

liquidlike structure was observed. At high stresses, bundlelike ordering appeared. The second mechanism that might explain the hysteresis assumes that the rest structure is characterized by a lattice. At low stress or shear rate, the particles move along a zigzag trajectory. Increasing the stress or shear rate can result in a transition to straight trajectories. The structure is then characterized by sliding layers. This transition was investigated by Gray and Bonnecaze35 who performed simulations on the shearing of a lattice of strongly repulsive colloidal particles for several lattice orientations. When they sheared a fcc lattice parallel to the (111) planes and along the 〈211〉 direction, they observed an hysteresis-like viscosity curve which has a remarkable resemblance with our experimental curve. 5.3. G′∞. The model of Elliott and Russel for describing the concentration dependence of G′∞, which was successfully applied previously for SMD65, also gave a satisfactory description for the SJL23U suspensions. In light of the difference in relative layer thickness and the substantially different segment volume fractions, this can be regarded as a corroboration of the model. This conclusion is supported by the solvent quality parameter χ which was found to be plausible and consistent with earlier findings. The lubrication model for calculating the concentration dependence of η∞ gave a fairly good description of the experimental data with a permeability having the expected order of magnitude. The precise value obtained from the model fit was lower than expected on the basis of the grafting density. To be able to make a more pronounced statement, one would have to reduce uncertainties in model parameters such as the layer thickness, for which the predicted η∞(φc) is very sensitive. On the experimental side, an extension to higher shear rates would also be desirable to reduce extrapolation errors. Also, in a more general sense we would like to stress the importance of accurate characterization of the particles. The modeling of G′∞ for example depends heavily on this. Here, small experimental errors in the parameters can result in significant uncertainties in the calculations for G′∞. These could be reduced if the pair potential can be probed directly. With the development of atomic force microscopy and optical tweezers, nowadays these experiments are within reach. Any influence of polydispersity of the polymer chains was neglected in our modeling despite the magnitude for Mw/Mn of 1.4. Introducing polydispersity alters the segment volume fraction profile of the polymer layer.34 The smaller chains push the larger chains away from the surface, thus increasing the layer thickness. In section 4.3, we already argued that a larger layer thickness improves the description of η∞. A larger layer thickness also increases the reach of the pair interaction. But the segment volume fraction at the periphery will be lower than for monodisperse polymers. This softens the repulsion. Hence, it is difficult to predict the effect of polydispersity on G′∞ beforehand. Summarizing, the discrepancies in the modeling of η∞ and G′∞ might to some extent be due to polydispersity. A better correspondence between the particle system and its modeling could be obtained by modeling the polydispersity or fractionating the polymer batch before grafting. Both remedies require a lot of effort. 6. Conclusions We have measured the rheological behavior of suspensions of silica particles stabilized by endgrafted PDMS (34) Levicky, R.; Koneripally, N.; Tirell, M. Macromolecules 1998, 31, 2616. (35) Gray, J. J.; Bonnecaze, R. T. J. Rheol. 1998, 45, 1121.


chains. The radius of the silica core and the thickness of the polymer layer were equal for these particles, which made it interesting to make a comparison with previous results for particles with larger cores. For the new particles, the polymer layer had both a stronger curvature and a lower segment density. Yet with the same models from polymer physics as used before, it was still possible to obtain a satisfactory description of the experimental G′∞ (and η∞) data. Together with the consistent values found for the fit parameters, this result corroborates our modeling approach for the polymer layer. A salient detail in our modeling of G′∞ was that in the Monte Carlo calculations of the radial pair distribution function g(r12), both ordered and disordered structures were found above a certain particle concentration. These different structures gave however only a marginal difference in the calculation of G′∞. Besides scaling models,36,37 hardly any modeling exists for the rheological behavior of polymerically stabilized (36) Genz, U.; D’Aguanno, B.; Mewis, J.; Klein, R. Langmuir 1994, 10, 2206.

Langmuir, Vol. 17, No. 19, 2001 5767

suspensions at low shear rates. Yet a very interesting behavior was observed in this regime. Above a critical particle concentration, the flow curves changed in shape, showing very high low-shear viscosity plateaus followed by an extreme shear thinning. Also, a remarkable hysteresis was observed. Possible qualitative explanations for this behavior have been given. The critical concentration was found to correspond well with the crystallization threshold as found in Monte Carlo simulations and in experiments on other soft repulsive particles. Acknowledgment. This work is part of the research program of the Foundation for Chemical Research (CW) with financial support from The Netherlands Organization for Scientific Research (NWO). We thank J. van Male for performing the SCF calculations, B. W. M. Kuipers for measuring the hydrodynamic radius, and J. S. Lopulissa for synthesizing the silica/PDMS particles. LA000972K (37) Maranzano, B. J.; Wagner N. J. Rheol. Acta 2000, 39, 483.

Rheological Behavior of Polymerically Stabilized Suspensions: Two

Recommend Documents