Shear-Induced Microstructural Evolution of a Thermoreversible

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Shear-Induced Microstructural Evolution of a Thermoreversible Colloidal Gel Priya Varadan and Michael J. Solomon* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 Received October 26, 2000. In Final Form: March 14, 2001 We report a study of the effect of shear deformation on the static structure factor, S(q), of a thermoreversible gel of organophilic colloidal silica (a ) 40 nm) in the solvent hexadecane. Small- and wide-angle light scattering measurements show that the quiescent structure of these gels is consistent with that of fractal clusters with dimension d ) 2.4 (independent of volume fraction, φ) and finite radius (ξ), which is a function of φ for the range 0.01 < φ < 0.1. Upon application of low shear rate deformation (γ˘ e 30 s-1), we observe an increase in ξ and d, relative to the quiescent conditions. For this to be the case, mass conservation requires that the number density of clusters be dramatically reduced upon shearing. The increase of d and ξ and the concomitant decrease in the number density of clusters point to the profound effect of shear on the long-range structure of colloidal gels. At high shear rates (γ˘ > 30 s-1) we observe anisotropy of S(q) in the flow-vorticity plane. The observed two-lobe butterfly patterns are oriented in the flow direction for all φ studied. The anisotropy persists after cessation of shear, although some partial relaxation is observed at the highest shear rate studied (γ˘ ) 120 s-1). Start-up of steady shear experiments performed for φ ) 0.035 reveals a monotonic increase of S(q) at low q (aq ) 0.032), which is consistent with an increase in both the fractal dimension, d, and cluster radius, ξ. Comparison of the time evolution of S(aq)0.032) with transient rheological measurements performed under the same conditions reveals that the monotonic increase in S(aq ) 0.032) occurs on a time scale identical to that required for the stress response to attain steady state.

1. Introduction and Background Colloidal gels are commonly encountered as intermediates during the manufacture of ceramics, magnetic storage media, coatings for antireflective and automotive applications, inks, and paints.1 They exhibit unusual rheological properties such as solidlike linear viscoelasticity2,3 and an apparent yield stress.4,5 While the rheology of colloidal gels is an indicator of the interaction potential between the colloidal particles,6,7 it is also a function of the gel structure. However, this relationship between gel structure and rheology remains poorly understood. Characterization of the structural changes in colloidal gels as they undergo shear deformation would help elicit a better understanding of the origins of their unusual rheology. The process of gelation has been described as the growth of clusters of particles. These clusters then associate (or percolate) to form a space-filling network.7-9 This process succeeds if the particles are present above a critical concentration or if the attractive interparticle interactions are of sufficient strength. Theoretical10-12 and experi(1) Brinker, J. C.; Scherer, G. W. Sol Gel Science: The physics and chemistry of sol-gel processing; Academic Press: New York, 1990. (2) Buscall, R.; McGowan, I. J.; Mills, P. D. A.; Stewart, R. F.; Sutton, D.; White, L. R.; Yates, G. E. J. Non-Newtonian Fluid Mech. 1987, 24, 183. (3) Chen, M.; Russel, W. B. J. Colloid Interface Sci. 1991, 141, 565577. (4) Barnes, H. A.; Walters, K. Rheol. Acta 1985, 24, 323-326. (5) Nguyen, Q. D.; Boger, D. V. Annu. Rev. Fluid Mech. 1992, 24, 47-88. (6) Firth, B. A.; Hunter, R. J. J. Colloid Interface Sci. 1976, 57, 266275. (7) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (8) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1992, 68, 3327-3330. (9) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41, 197-217. (10) Witten, T. A.; Sander, L. M. Phys. Rev. Lett. 1981, 47, 14001403. (11) Meakin, P. J. Colloid Interface Sci. 1983, 96, 415-424.

mental13,14 studies have shown that in dilute aggregating dispersions the clusters are highly branched and resemble fractals. The fractal dimension of these clusters has been shown to depend on the mechanism of aggregation. These mechanisms include diffusion-limited aggregation, diffusion-limited cluster-cluster aggregation, and chemically limited cluster-cluster aggregation.10,12,15 Experimental studies of dilute gels formed under quiescent conditions have shown that the structure is consistent with fractal clusters of finite extent.8,16 That is, aggregation occurs until the fractal clusters fill space, at which time the clusters form a space-filling network or gel. However, the fractal cluster description is not applicable concentrated colloidal gels,9 and in this case analogies with percolation transition have been used to understand gelation.17 Recent attempts to link gelation to the glass transition may also prove fruitful for colloidal gels.18,19 The microstructures of colloidal gels have been successfully probed by scattering techniques such as X-ray, neutron, and light scattering. Recently, the quiescent local structure of a colloidal gel has also been directly visualized by confocal laser scanning microscopy.20 The microstruc(12) Jullien, R.; Kolb, M.; Botet, R. J. Phys., Lett. 1984, 45, L977981. (13) Schaefer, D. W.; Martin, J. E.; Wiltzius, P.; Cannell, D. S. Phys. Rev. Lett. 1984, 52, 2371-2374. (14) Weitz, D. A.; Lin, M. Y.; Huang, J. S. In Complex and Supramolecular Fluids; Safron, S. A., Clark, N. A., Eds.; WileyInterscience: New York, 1986; pp 509-549. (15) Jullien, R.; Botet, R.; Mors, P. M. Faraday Discuss. Chem. Soc. 1987, 83, 125. (16) Dietler, G.; Aubert, C.; Cannell, D. S.; Wiltzius, P. Phys. Rev. Lett. 1986, 57, 3117-3120. (17) Stauffer, D. Introduction to Percolation Transition; Taylor and Francis: London, 1985. (18) Bergenholtz, J.; Fuchs, M. Phys. Rev. E 1999, 47, 5706. (19) Fabbian, L.; Gotze, W.; Sciortino, F.; et al. Phys. Rev. E 1999, 59, R1347. (20) Verhaegh, N. A. M.; Asnaghi, D.; Lekkerkerker, H. N. W. Physica A 1999, 264, 64-74.

10.1021/la001504d CCC: $20.00 © 2001 American Chemical Society Published on Web 04/21/2001

Thermoreversible Colloidal Gel

ture of dilute, aqueous, density-matched (in a mixture of H2O/ D2O) suspensions of polystyrene colloids, aggregated by the introduction of a divalent electrolyte, has been studied by light scattering.21 Light scattering studies of aqueous colloidal gels are limited to low volume fractions because of the large refractive index contrast between the particle and the solvent. This large contrast causes multiple scattering, even at dilute volume fractions. The microstructure of concentrated aqueous gels of silica has been more often studied by neutron scattering, which is insensitive to the refractive index variation.9,22,23 One class of model systems for studying gelation by light scattering are suspensions of sterically stabililized silica in organic solvents such as benzene,24 decalin,9 dodecane,25 and hexadecane,3 which are marginal for the grafted chains. They are suitable for light scattering studies since the particles and the solvent are approximately refractive index matched. The small refractive index contrast allows the study of even high volume fraction gel structure. Colloidal dispersions of silica spheres grafted with octadecyl chains show a rich phase behavior, which is a function of both the nature of the solvent and the temperature. Although the origin of the interparticle potential in these systems is poorly understood, its range is believed to be short (radius of gyration of grafted steric layer ∼ 1-3 nm), and its strength is sensitive to temperature. As the temperature is decreased, the strength of the attractive potential increases, and at some critical temperature, gelation and/or phase separation may occur. As opposed to the silica-benzene system,26 in the silicahexadecane system there is no evidence of phase separation. Only gelation is observed as temperature or volume fraction is varied.3 Measurements of the static structure factor of adhesive sphere suspensions have been found to be consistent with the Baxter potential given by

where the Baxter interaction parameter, τ, is a function of temperature.27 The origin of the attractive potential is thought to involve conformational changes of the grafted layer, which are temperature-dependent because of the marginal quality of the solvent. While the role of residual van der Waals forces in mediating the short-range interaction is not fully understood, these interactions are minimized because of the approximate refractive index matching of the colloid and solvent. The phase diagram for adhesive spheres has been shown to exhibit a firstorder gas-liquid transition with the associated spinodal and critical point at φ ) 0.1213 and τ ) 0.0976 and a (21) Carpineti, M.; Ferri, F.; Giglio, M.; Paganini, E.; Perini, U. Phys. Rev. A 1990, 42, 7347-7354. (22) Hanley, H. J. M.; Butler, B. D.; Straty, G. C.; Bartlett, J.; Drabarek, E. J. Phys.: Condens. Matter 1999, 11, 1369-1380. (23) Sinha, S. K.; Freltoft, T.; Kjems, J. In Kinetics of Aggregation and Gelation; Family, F., Landau, D. P., Eds.; Elsevier Science Publishers B. V.: Amsterdam, 1984; pp 87-90. (24) Verduin, H.; de Gans, B. J.; Dhont, J. K. G. Langmuir 1996, 12, 2947-2955. (25) Rouw, P. W.; de Kruif, C. G. Phys. Rev. A 1989, 39, 5399-5408. (26) Verduin, H.; Dhont, J. K. G. J. Colloid Interface Sci. 1995, 172, 425-437. (27) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770.

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solid-liquid transition.28 Various hypotheses for the mechanism of gelation in dispersions of adhesive spheres have been advanced. It has been speculated that the gelation transition is a consequence of percolation, a frustrated gas/solid phase transition, or freezing of the organophilic chains on the surface of the particles.3,25,28 Nevertheless, the exact mechanism of gelation of adhesive spheres remains unclear. Application of shear flow induces changes in microscopic structure, which in turn affect rheological behavior. While many studies have reported the quiescent structure of aggregated and gelled colloidal suspensions, there have been few studies on the effect of an applied shear flow on these systems. Yet, it is known that the viscoelastic properties of colloidal gels are highly shear history dependent. Thus, the effect of flow on structure is profound and often irreversible. Rheological studies on octadecyl grafted silica in hexadecane, decalin, and tetradecane have shown that these gels possess measurable yield stresses and low-frequency elastic moduli, and the steady-state shear viscosity is shear thinning.3,9 Neutron scattering studies on sheared organophilic silica-tetradecane gels showed that the fractal dimension increased with shear rate, suggesting the densification of floc structure with shear.9 Even at high shear rates, where significant shear thinning was observed, the cluster size was still larger than 10 particle diameters. Small-angle neutron scattering studies on aqueous concentrated silica suspensions gelled at pH ) 8 and a constant shear rate have also revealed an increase in the measured fractal dimension upon gelation during shear when compared to a suspension gelled under quiescent conditions.22 Theories have been proposed to relate gel structure to rheological properties. Models based on the fractal structure of clusters have been developed to describe the linear viscoelastic modulus, limit strain, γM (maximum strain the gel can sustain and remain in the linear viscoelastic limit), and steady shear viscosity of dilute colloidal gels exhibiting fractal scaling.29-32 These theories are in reasonable agreement with experimental rheological measurements on different flocculated suspensions.3,28,30,31 Application of the model of Potanin and co-workers to the steady shear rheological measurements of a weakly aggregating dispersion of polystyrene latex particles suggested that the fractal dimension of the cluster during shear flow was higher than the quiescent condition.31 While the above models have been applicable for dilute colloidal gels, alternative models based on percolated networks, which describe the role of microstructure at long length scales corresponding to the intercluster regime, have been utilized to understand the rheology of concentrated colloidal gels.17,33 Percolation theory offers the possibility that the removal of only a few network connections can destroy the percolated network and thereby significantly affect the elastic properties of the gel. The volume fraction dependence of the elastic modulus of organophilic silica-hexadecane gels has been shown to be consistent with percolation theory.28 While these scaling theories based on fractal aggregation and percolation describe the linear viscoelastic and steady shear behavior (28) Grant, M. C.; Russel, W. B. Phys. Rev. E 1993, 47, 2607-2613. (29) Buscall, R.; Mills, P. D. A.; Yates, G. E. Colloids Surf. 1986, 18, 341-358. (30) de Rooij, R. D.; Ende, v. d.; Duits, M. H. G.; Mellema, J. Phys. Rev. E 1994, 49, 3038. (31) Potanin, A. A.; de Rooij, R.; Ende, V. d.; Mellema, J. J. Chem. Phys. 1995, 102, 5845-5853. (32) Shih, W.-H.; Shih, W. Y.; Kim, S.-I.; Liu, J.; Aksay, I. A. Phys. Rev. A 1990, 42, 4772-4779. (33) Mall, S.; Russel, W. B. J. Rheol. 1987, 31, 651-681.

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of gels, they do not describe the transient evolution of the microstructure during shear, particularly for nonlinear deformation. Recent computer simulations, however, of colloidal gels provide insight into such behavior. Doi and Chen performed 2-D and 3-D simulations of aggregating colloids in shear flow.34 Interparticle interactions were modeled with a sticking probability, and hydrodynamic interactions were accounted for with the free draining approximation; Brownian motion was neglected. The simulations revealed that at low volume fraction compact clusters were formed, while at high volume fractions loose networks were formed. West and co-workers studied the breakup of colloidal gels under applied strain by means of 2-D simulation.35 This study explicitly considered the effect of angular rigidity of particle-particle bonds on microstructural evolution. The time evolution of stress obeyed a power law scaling prior to yielding. Hydrodynamic interactions were neglected in these simulations. Simulation results to date report isotropic structure during shear deformation; however, recent experimental light scattering studies of gelled suspensions of organophilic silica in benzene have revealed that anisotropic structures form during shear and persist after cessation of flow.24 While the effect of steady shear on the microstructure of colloidal gels has been studied, to our knowledge sufficient quantitative modeling is lacking, particularly on the long length scales typical of cluster dimensions. Furthermore, there have been no detailed experimental studies of the microstructural evolution in colloidal gels during the start-up of steady shear. Start-up of steady shear experiments would provide insight into the microstructural change that occurs as the gel is deformed from the quiescent state. In this paper we present small- and wide-angle light scattering results of a thermoreversible gel of colloidal stearyl silica spheres in hexadecane. The quiescent structure of the gel is characterized, extending the scattering vector range of earlier studies to the intercluster regime.3 The effect of the shear rate in steady shear and start-up of steady shear on the microstructure of the quiescently gelled materials is studied by flow smallangle light scattering. The experimental results are discussed in light of the recent theories and simulations for flow-induced microstructure in aggregated suspensions. The observed structural anisotropy in the flowvorticity plane is related to enhanced concentration fluctuations. 2. Experimental Section 2.1. Synthesis and Characterization of Colloidal Silica. The silica particles used for this study were synthesized according to the procedure of Stober et al.36 and Bogush et al.37 The particles were then grafted with octadecyl chains by the method of Van Helden and co-workers.38 The particles were dried under a continuous stream of nitrogen at 60 °C for 3 days prior to use. The particle size was determined by dynamic light scattering (DLS, ALV compact goniometer, ALV-5000/E correlator). To determine the particle size by DLS, the colloids were dispersed in cyclohexane at concentrations φ < 0.0005, and the intensity autocorrelation function g2(τ) was collected at T ) 25 °C at q ) 10.9 µm-1 where q is the scattering vector. The average hydrodynamic radius was determined by a constrained regu(34) Doi, M.; Chen, D. J. Chem. Phys. 1989, 90, 5271-5279. (35) West, A. H. L.; Melrose, J. R.; Ball, R. C. Phys. Rev. E 1994, 49, 4237. (36) Stober, W.; Fink, A.; Bohn, E. J. Colloid Interface Sci. 1968, 26, 62-69. (37) Bogush, G. H.; Tracy, M. A.; Zukoski IV, M. A. J. Non-Cryst. Solids 1988, 104, 95-106. (38) Van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1980, 81, 354-368.

Varadan and Solomon Table 1. Gelation Temperatures of the Quiescent Gels (Measured by DLS) as a Function of Volume Fraction φ

Tg (°C)

φ

Tg (°C)

0.01 0.025

23.5 26.2

0.055 0.1

27.8 29.4

larization method (CONTIN) fit to g2(τ).39 The average hydrodynamic radius determined by this method was 40 nm. The standard deviation of the particle size as measured from scanning electron microscopy (SEM) was 4 nm. 2.2. Determination of Gelation Temperatures. Suspensions were prepared by dispersing known amounts of the dried silica (F ) 1.8 g/cm3 9) in twice-filtered (0.1 µm, Anotop, Whatman Inc., England) hexadecane. The gelation temperatures were determined with an accuracy of (0.1 °C by dynamic light scattering according to the method described by Solomon and Varadan.40 Briefly, the dynamic structure factor, f(q,t), was extracted from measurements of the time-average normalized intensity autocorrelation function gt2(q,t) by the method of Pusey and van Megen.41 The gel point was determined as the temperature at which nonergodic behavior was first observed. This temperature coincides with significant retardation of the decay of f(q,t). Table 1 shows the gelation temperature (Tgel) of different volume fraction suspensions. 2.3. Wide-Angle Light Scattering. Wide-angle light scattering was conducted on an ALV compact light scattering goniometer. The incident wavelength of light was λ0 ) 0.488 µm, and the scattering vector varied from 3.8 µm-1 < q < 35.6 µm-1. Temperature was controlled to (0.02 °C. The scattering volume was estimated to be 0.002 mm3 at θ ) 90°. For colloidal gels that exhibit nonergodic behavior, the time-averaged intensity of scattered light need not equal the ensemble-averaged scattered intensity. To collect ensemble average data, scattered intensity was measured for numerous scattering volumes (n > 100). 2.4. Small-Angle Light Scattering. 2.4.1. Optical System. A schematic of the optical setup used for the small-angle light scattering apparatus is shown in Figure 1a. The design of this apparatus was based on that of Cumming and co-workers.42 Polarized light from a 10 mW He-Ne laser (Melles Griot Inc., Irvine, CA) was attenuated by a quarter-wave plate and a polarizing beam splitter. The vertically polarized light was then passed through the sample. The sample was placed in a flat optical cuvette of path length 2 mm (Helma Inc., NY). The cuvette was positioned in a temperature-controlled refractive index matched vat (described below). An off-axis paraboloidal mirror (f ) 100 mm, Melles Griot, Irvine, CA) was used to collimate the scattered light from the sample that was placed at the focus of the mirror. The collection angle was approximately 20° from incidence. A circular beam stop cut from a neutral density filter of optical density 5.0 (Newport Inc., Irvine, CA) was positioned beyond the mirror to prevent saturation of the CCD. The collimated light from the mirror was then passed onto a pair of lenses (planoconvex lens, diameter ) 76 mm, f ) 300 mm; achromatic lens, diameter ) 12.7 mm, f ) 25 mm; Newport Inc., Irvine, CA) that are separated by a distance equal to the sum of focal lengths. A CCD detector collected the resulting collimated beam, reduced in diameter by a factor of 12. The detection optics was positioned at an angle (11°) such that the incident beam was incident at one end of the mirror, thereby maximizing the angular range of the device. The use of the off-axis paraboloidal mirror in the detection setup enabled measurement from 2° < θ < 20°. This apparatus was used for the quiescent experiments. Different detection optics was used to detect scattering in the full flow-vorticity plane in the flow small-angle light scattering experiments, as shown in Figure 1b. The off-axis paraboloidal mirror was replaced by a planoconvex lens (diameter ) 80 mm, f ) 100 mm, Melles Griot Inc., Irvine, CA). The shear cell was positioned in the refractive index matched vat, such that the sample in the gap at the front end of the shear cell (with respect to the detector) was at the focus of the lens. This lens collimated (39) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229-242. (40) Solomon, M. J.; Varadan, P. Phys. Rev. E, in press. (41) Pusey, P. N.; van Megen, W. Physica A 1989, 157, 705-741. (42) Cumming, A.; Wiltzius, P.; Bates, F. S.; Rosedale, J. H. Phys. Rev. A 1992, 45, 885-897.


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Figure 1. (a) Schematic of the small-angle light scattering apparatus. (b) Schematic of the flow small-angle light scattering apparatus. the scattered rays from the sample. The detection optics was positioned coincident with the incident beam, so that the beam passed through the center of the lens. The remainder of the detection optics was unchanged from above. The range of scattering angles accessible for the flow experiments was 2° < θ < 12.5°. Note, since the laser beam passes through the entire shear cell, the angularly dependent scattering may be affected by additional scattering from the sample in the gap at the rear end of the shear cell. Geometric analysis of the scattering showed that this spurious scattering (at angles >3°) is rejected by the second detector lens. That is, in effect, the detector lens (diameter ) 12.7 mm) functions as a pinhole in the device for these rays. Since this analysis leads us to anticipate some possible effect due to the scattering at angles less than 3° from the front end of the shear cell, an additional flow experiment was performed to determine its extent. An iris (diameter ∼ 2 mm) was added to the optical train to block any additional scattering. Comparison of the data showed that its inclusion had no significant effect on the angularly dependent scattering. Consequently, we conclude

that for our cell design and detector format the effect of additional scattering from the rear end of the shear cell does not significantly affect the data analysis. The quiescent small-angle light scattering experiments were performed with an 8-bit video CCD camera (Cohu 4915, Cohu Inc., San Diego, CA), with a spatial resolution of 796 × 494 and pixel size of 8.4 µm (h) × 9.8 µm (v). Quiescent images were acquired and digitized by means of a framegrabber (Scion LG-3, Scion Corp., Frederick, MD) in a Power Macintosh G3, using NIH-image software (a public domain software program developed by the U.S. National Institutes of Health and is available at http://rsb.info.nih.gov/nih-image), and analyzed using MATLAB software. The CCD detector used in the flow small-angle light scattering experiments was a 12-bit cooled digital CCD camera (Sensicam VGA, Cooke Corporation, Auburn Hills, MI) with a maximum readout rate of 30 kHz, a spatial resolution of 640 × 480 pixels, and pixel size of 9.9 µm × 9.9 µm. Images were acquired by means of an image acquisition board, using the acquisition software Sensicontrol (Cooke Corporation, Auburn

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Hills, MI), in a personal computer (Optiplex GX100, Dell Computers), and analyzed using MATLAB software. 2.4.2. Temperature Control and Refractive Index Matching. A cylindrical glass vat with two plane parallel windows of optical quality glass was used to provide the necessary temperature control and refractive index matching for the quiescent and flow experiments. For the flow experiments, it was necessary to match the refractive index of the curved surface of the shear cell with an exterior vat fluid to eliminate scattering and refraction at the glass interface. Hexadecane (refractive index ) 1.434) was used as the refractive index matching fluid. The temperature of the vat was controlled by external water circulation to within (0.5 °C. The optical vat provided the required temperature control for the quiescent and shear experiments and prevented the distortion of scattering patterns due to the optical surfaces of the shear cell. 2.4.3. Taylor-Couette Shear Cell. The shear cell used for flow-SALS experiments was a cylindrical Couette cell, consisting of an inner solid glass cylinder (diameter ) 30 mm) and an outer hollow glass cylinder (inner diameter ) 33 mm). The inner solid cylinder and the outer hollow cylinder were made of Corning 7740 glass (refractive index ) 1.47), whose surfaces were ground and polished. The inner solid cylinder was rotated at a constant rate, by means of a microstepper motor (25000 steps/rev, Zeta 6104 series, Compumotor, Parker Hannifin Corp., Troy, MI). The range of shear rates investigated for the steady shear experiments was 0.3 s-1 e γ˘ e 120 s-1. While conducting flow experiments at low shear rates (γ˘ < 1 s-1), the motor was connected to a gearbox (Bayside Motors, Accutrol Inc., MI) with a 100:1 ratio, to ensure that a continuous deformation was applied. The whole device was placed in the refractive index matching vat. The shear cell was positioned in the vat such that the incident beam passed through its center. A calculation of the Taylor number for a Newtonian fluid with solvent viscosity comparable to the fluids studied leads us to conclude that the Taylor number for all shear rates investigated is below the critical Taylor number of flow instability.43 As a further test for flow instability (whether of an inertial or viscoelastic nature), the time dependence of the stress for a particular sample was monitored in a Taylor-Couette cell and in cone and plate flow (ARES rheometer, Rheometrics Scientific, NJ). The near equivalence of the measured stresses in the two geometries confirms the absence of the flow instabilities. 2.4.4. Experimental Procedure for Quiescent and Flow Light Scattering Measurements. Samples were prepared by dispersing dried silica particles of a known weight in filtered hexadecane (0.1 µm, Anotop, Whatman Inc., England). The sample was heated to 60 °C and vortex mixed to disperse the particles. Subsequently, the optical cell with the sample was placed in the temperature-controlled vat at 60 °C, and the temperature was reduced to the desired value. No consolidation of the quiescent samples was observed over the experimental time scales. In the quiescent experiments, the images were captured at a rate of 30 frames/s (using 8-bit CCD) and averaged over six frames to reduce any noise. The procedure followed for the flow experiments was as follows. Prior to each experiment, the sample was reheated in the shear cell to 60 °C and presheared at a rate of 120 s-1 for 120 s to remove any residual structure. The temperature was then lowered to the desired value, after which the sample was allowed to thermally equilibriate for 15 min. This procedure was followed to ensure reproducible microstructure. The sample was then sheared at the desired rate for 60 s, and images were captured with the 12-bit digital camera using an exposure time of 1-2 ms (depending on the sample). No significant change in the scattered intensity was observed when the sample was sheared for 120 s when compared to that for 60 s. Shearing the sample for longer time periods (>2 min) resulted in some consolidation (sedimentation) of the gel for φ ) 0.035 and φ ) 0.055 but not for φ ) 0.10. Hence, 60 s was chosen as the duration of flow for the steady shear experiments. For the transient study, the structural evolution was monitored for 60 s after the inception of flow. Images (43) Drazin, P. G.; Reid, W. H. Hydrodynamic Stability; Cambridge University Press: New York, 1985.

Varadan and Solomon were captured at a rate of 25 frames/s to enable the detection of microstructural changes that occur during the start-up of steady shear. 2.4.5. Data Analysis. For samples that scatter isotropically, the isointensity lines on the CCD correspond to constant scattering angles (θ°) and constant scattering vectors (q). Determination of the scattering vector, in the case of the quiescent experiments using the off-axis paraboloidal mirror, involved a geometric transformation. Once the lines of constant scattering vector were determined, the average intensity corresponding to a scattering vector was measured as described in the next paragraph. In the case of the flow experiments a planoconvex lens was used instead of the mirror. The flow and vorticity axis corresponded to φ ) 0° and 90°, respectively, where φ is the azimuthal angle. In the case where anisotropic scattering patterns were observed, the scattering along the flow and vorticity direction was analyzed. Because of the orientation of the anisotropic scattering patterns and the presence of the beam stop, the scattering intensities in the flow direction and vorticity direction were measured by averaging the intensities in a region corresponding to φ ) 0° ( 20° and φ ) 90° ( 10°, respectively. The average intensity value corresponding to each scattering angle or q-vector was determined by averaging the intensities within the segment θ - 0.125° e θ < θ +0.125° so as to provide equally spaced scattering intensity data points over the entire scattering vector range of the device. The CCD dark current was subtracted from the measured intensities. The scattered intensities from the sample were background corrected by subtracting the scattering intensity of the solvent measured in the optical cell. In the shear experiments the scattering intensity of the suspension at 60 °C was used as the background, for reasons described by Grant and Russel.28 The device performance and data analysis procedure was tested for a dilute sample of polystyrene spheres (radius ) 1.53 µm, Polysciences, Warrington, PA). The measured form factor showed good agreement with Mie theory. The scattered intensity I(q) was converted to the structure factor S(q) according to the proportionality Is(q) ) FP(q) S(q). Here, F is number density of particles and P(q) is the particle form factor given by the expression P(q) ) [3(sin(aq) aq cos(aq))/(aq)3]2 for a sphere of radius a. Here, q, the scattering vector is given by q ) (4πn/λ0) sin(θ/2) where θ is the scattering angle, λ0 is the wavelength of incident light in a vacuum, and n is the refractive index of the medium. In the quiescent experiments, the scattered intensities were normalized by the value at aq ) 1.0 to account for the concentration dependence.

3. Experimental Results 3.1. Quiescent Microstructure. The quiescent microstructure of the organophilic silica-hexadecane gels was characterized for 0.5 µm-1 < q < 35.6 µm-1 by means of small- and wide-angle light scattering. These studies were performed at T) 25 °C for φ g 0.025 and T ) 21 °C for φ < 0.025. The temperatures were chosen to be below the gelation temperature of each volume fraction. The maximum attenuation of the beam was 20% for φ ) 0.1 at T ) 25 °C for a path length of 2 mm. Hence, the effect of multiple scattering is expected to be negligible.44,45 The effect of gel volume fraction on the structure factor, S(q), is shown in Figure 2. For volume fractions less than φ ) 0.075, S(q) shows an extended region of power law scaling at high scattering vector. At low q, the intensity tends to a q-independent plateau. The lower limit of the power law scaling extends to lower q as the volume fraction is decreased. The power law scaling of the structure factor, S(q), is characteristic of a fractal object, where the power law exponent corresponds to the fractal dimension d. The high volume fraction gels (φ ) 0.075 and φ ) 0.1) show only a limited regime of power law behavior at high q. (44) Cipelletti, L. Phys. Rev. E 1997, 55, 7733-7740. (45) Verhaegh, N. A. M.; Asnaghi, D.; Lekkerkerker, H. N. W.; Giglio, M.; Cipelletti, L. Physica A 1997, 242, 104-118.


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Figure 2. S(q) as a function of the dimensionless scattering wavevector for organophilic silica-hexadecane gels of different concentrations, measured by both small- and wide-angle light scattering. The measurements were performed at 25 °C for all volume fractions except for φ ) 0.015 and φ ) 0.01 where the temperature was 21 °C. The solid lines are fits to the model of Sinha and co-workers.23 Table 2. Dimensionless Cluster Radius (ξ/a) of the Quiescent Gels As Determined from the Model of Sinha et al.23 for the Range of Volume Fractions Investigated φ

ξ/a

φ

ξ/a

0.01 0.015 0.025 0.035

27.0 19.4 9.8 8.1

0.055 0.075 0.10

4.1 2.7 2.0

The curves in Figure 2 are fits to the fractal cluster model of Sinha et al.,23,46 where the gel structure is taken to be one of randomly distributed fractal clusters. In this case, g(r) ∼ Ard-3 exp(-r/ξ), where ξ is the cluster radius and d is the fractal dimension. Since S(q) and g(r) are Fourier transform pairs, it has been shown that

S(q) )

(

S(0)

)

[1 + (qξ)2](d-1)/2

sin(d - 1) tan-1(qξ) (2) (d - 1)qξ

From Figure 2, the measured S(q) is adequately modeled by eq 2, except at the highest volume fractions. For the thermoreversible gels, d ) 2.4, independent of volume fraction. The error in d as estimated by analysis of fit sensitivity and residuals is (10%. Figure 2 apparently indicates that the small-angle scattering device underreports the scattering at the highest angles probed by the instrument, since in the region of overlap of the two devices (0.15 < aq < 0.2) there is some difference in the data. Nevertheless, the assignment of fit parameters is not sensitive to this error, since exclusion of these data at the highest angles from the fits to eq 2 changed d and ξ by less than 5%. The model parameter ξ/a is volume fraction dependent: results are listed in Table 2. The results in Table 2 corroborate the failure of the model at the highest volume fractions, since the values of ξ/a tend to unrealistically small values in this limit. (46) Ferri, F.; Frisken, B. J.; Cannell, D. S. Phys. Rev. Lett. 1991, 67, 3626-3629.

Figure 3. Effect of temperature on the S(q) for φ ) 0.055 measured by both small- and wide-angle light scattering.

Figure 3 shows the effect of temperature on the quiescent structure (φ ) 0.055). As the temperature is lowered, S(q) dramatically increases for all q. At the gel point it was observed that temporal fluctuations in the scattered intensity in the q range 0.02 < aq < 0.2 were arrested, since the speckles on the CCD appeared stationary. Note that below the gel point little change is observed in the scattered intensity as the temperature is further decreased. A note regarding the high-temperature structure of the dispersion is required: At high temperature, a dispersed fluid structure is expected, and S(q) should differ only slightly from unity in the volume fraction range studied. Moreover, for a particle size of 40 nm and at aq < 1, the form factor is nearly unity and independent of scattering vector. Thus, the scattered intensity at high temperatures should be essentially independent of q. However, our data at 30 °C and at higher temperatures show a q dependence even at low aq. Chen and Russel3 also reported persistent deviations from the calculated form factor at low q for colloidal silica particles synthesized in the same way. They suggested that the anomalous scattering could be due to the presence of permanent aggregates. Analogous behavior was also seen in cyclohexane. We have considered this contribution to the scattering and concluded that it does not significantly impact our results since they are exclusively in the gel regime, where the scattering intensity is always at least an order of magnitude larger than the anomalous contribution. Figure 3 supports this conclusion. 3.2. Shear-Induced Microstructure. The effect of shear on the silica-hexadecane system was studied for three different volume fractions, φ ) 0.035, 0.055, and 0.10, using flow small-angle light scattering (0.02 < aq < 0.125). Because microstructural evolution was of interest, no preshear was applied to the gels. Rather, the initial condition for all the studies was the quiescently gelled state. Figure 4 shows the effect of steady-state shear (γ˘ ) 120 s-1) on S(q) for a gel (φ ) 0.055) at different temperatures. At a temperature near the gel point of the sample (Tgel ) 28.4 °C), the effect of shear on the dispersion was minimal. At temperatures significantly lower than the gel temperature there was a dramatic increase in S(q) upon the application of shear. Note that no evolution in S(q) was observed at any temperature, for many hours, upon cessation of shear. Thus, the structural changes are

2924


Varadan and Solomon Table 3. Effect of Steady Shear on the Fractal Dimension and Dimensionless Cluster Size at γ3 e 30 s-1 for O ) 0.035, 0.055, and 0.10 shear rate (s-1)

Figure 4. Effect of shear on S(q) measured in the flow direction at different temperatures for φ ) 0.055 measured by flow smallangle light scattering. Intensities along the flow direction for all the flow experiments were measured by averaging over a 20° sector (φ ) 20°) along the flow axis.

Figure 5. S(q) at low shear rates (γ˘ e 30 s-1) for φ ) 0.1 at T ) 25 °C. S(q) is isotropic at these shear rates, in the q range probed by SALS. The data are offset for clarity. The solid lines are fits to the model of Sinha and co-workers.23 The inset shows the measured data. The subsequent flow experiments were performed at T ) 25 °C.

long-lived, even after the deformation is stopped. The subsequent shear experiments were performed at temperatures lower than the gel point for each volume fraction, where the effect of shear was the greatest, as shown for the case of φ ) 0.055 in Figure 4. At low shear rates (γ˘ e 30 s-1) steady-state SALS patterns were isotropic (i.e., S(q) did not depend on the azimuthal angle φ in the flow-vorticity plane). The application of shear resulted in the isotropic increase of scattered intensities over the entire q range probed by the flow-SALS device. Figure 5 shows the effect of γ˘ on the measured structure factor of a gel (φ ) 0.10). The data are offset in the y-axis for clarity to reveal the shape of the scattering curves. The shape of the scattering curves

d

ξ/a

quiescent 0.3 0.6 3 12

φ ) 0.035 2.4 2.58 2.47 2.84 2.98

8.1 a a a 44.2

quiescent 3 12

φ ) 0.055 2.4 2.86 3

4.1 45.2 31.5

quiescent 1.2 3 12 30

φ ) 0.10 2.4 2.7 2.77 2.7 2.3

2.0 48.0 31.7 26.0 44.0

a Note: for φ ) 0.035 at low shear rates, the deviation from the fractal scaling was not significant enough for the cluster size to be extracted with reasonable uncertainty using the model of Sinha et al.23

is of interest because it reflects structural information such as the cluster size and fractal dimension. While the quiescent S(q) of this sample did not show a significant q dependence in the scattering vector range probed by flow-SALS, the structure factor measured during flow showed a near power law scaling, especially at high q. The deviation from fractal scaling, which is indicative of cluster effects, was shifted to low q. Hence, qualitatively the effect of shear is to create variations in the structure at low q that are absent in the quiescent material. Similar trends were observed for the other two volume fractions studied, φ ) 0.035 and 0.055. To quantitatively assess the effect of steady shear flow on the gel structure, we applied the fractal cluster model for S(q), given by eq 2. The solid curves in Figure 5 show the fits from which d and ξ/a were determined. The results obtained for φ ) 0.035, 0.055, and 0.1 are listed in Table 3. It is evident from Table 3 that both the fractal dimension, d, and dimensionless cluster radius, ξ/a, of the gels change upon shear. For the gels with φ ) 0.035 and 0.055 the fractal dimension tends to increase with shear rate (γ˘ < 30 s-1). However, the fractal dimension for φ ) 0.1 shows a modest increase. It is interesting to note that the fractal dimension of the low volume fraction (φ ) 0.035, 0.055) at the highest shear rates was nearly three, suggestive of a dense suspension microstructure in the q range probed. Although the effect of steady shear on the fractal dimension has been studied in other gels, its effect on the cluster radius has not previously been experimentally quantified.9,22 The results in Table 3 for the dimensionless cluster size (ξ/a) show that the application of shear causes a significant increase in the cluster size relative to the quiescent condition. It was difficult to establish a trend for the dependence of cluster size on shear rate for the lowest volume fraction gel (φ ) 0.035), because the deviation from the fractal scaling was shifted to very low scattering vectors. Therefore, the deviation from the fractal scaling of S(q) was not significant enough in the q range studied to enable ξ/a to be extracted with reasonable uncertainty. The data for φ ) 0.10 are most illustrative, generally showing a trend toward decreased cluster size with increased shear rate. Note that the model parameter ξ/a is always larger for the sheared than for the quiescent material. The implications of the model parameters


Figure 6. Isointensity contours showing the evolution of the anisotropy in the flow-vorticity plane at high shear rates for φ ) 0.1. The intensity values of the contours are shown in each image.

extracted from the steady shear measurements for γ˘ e 30 s-1 will be considered more completely in the discussion. While at low shear rates (γ˘ e 30 s-1) the scattering patterns were isotropic, at higher shear rates anisotropy of the structure factor was observed in the flow-vorticity plane. The contour plots of the captured images are shown in Figure 6. Only a few isointensity contours are shown for clarity. The contour plots show that the anisotropy is in the form of a two-lobe butterfly pattern, with a greater increase of S(q) in the flow direction than in the vorticity direction. The two-lobe butterfly patterns are oriented in the flow direction, independent of shear rate (γ˘ > 30 s-1). Quantitative details of the anisotropy in S(q) change as the shear rate is increased. In particular, the magnitude of S(q) at low q is suppressed in both the flow and vorticity directions as γ˘ increases. This trend can be seen in Figure 7, where S(q) in the flow (φ ) 0°) and vorticity (φ ) 90°) directions has been extracted from Figure 6 data. In addition to the suppression of the magnitude of S(q) with increasing γ˘ , the onset of deviation from the fractal scaling which was observed at the lowest scattering vectors for low shear rates (cf. Figure 5) was shifted to higher scattering vectors. This difference in the low- and high-γ˘ behavior suggests, qualitatively, that the structures formed at high shear rates are smaller than those formed at low shear rates. Note that while S(q) in the vorticity direction is lower than in the flow direction, it is higher than the S(q) of the quiescent gel for all shear rates investigated. Scattering anisotropy was also observed for lower volume fraction gels (φ ) 0.055 and φ ) 0.035) at high shear rates. Figure 8 shows the q dependence of the structure factor for the three volume fractions at a shear rate 120 s-1. The data for each volume fraction are offset in the y-axis for clarity. The φ ) 0.1 gel shows more pronounced anisotropy than the lower volume fraction gels, because the difference between the magnitude of S(q) in the flow and vorticity direction is larger and extends

Langmuir, Vol. 17, No. 10, 2001 2925

Figure 7. Structure factor of φ ) 0.1 measured along the flow and vorticity directions for different shear rates. The intensities in the vorticity direction were measured by considering a 10° sector (φ ) 10°) about the vorticity axis. The open symbols correspond to data measured in the flow direction, and the solid symbols correspond to those measured in the vorticity direction.

Figure 8. Comparison of the structure factor along both flow and vorticity directions for the three volume fractions at γ˘ ) 120 s-1. The intensity data are offset for clarity.

to larger scattering vectors (aq > 0.04) than for the φ ) 0.035 and 0.55 gels. The power-law exponent for the low volume fraction (φ ) 0.035, 0.055) gels is approximately 3.5. The large magnitude of the scaling exponent and other features of the anisotropy in S(q) at high rates will be discussed further in the next section. For γ˘ < 120 s-1, the microstructural changes induced by shear were permanent, since no evolution of S(q) was observed after the cessation of flow. However, at the highest shear rate studied γ˘ ) 120 s-1, S(q) increased significantly at low q in both the flow and vorticity directions upon cessation of flow. This partial relaxation of structure was very rapid (t < 1 s). The total change observed is shown in Figure 9 for φ ) 0.055. Similar behavior was observed for the other volume factions studied. Even though partial relaxation was observed,

2926


Figure 9. Structure factor during steady-shear flow at γ˘ ) 120 s-1 and after cessation of shear for φ ) 0.055. The open symbols correspond to data measured in the flow direction, and the filled symbols correspond to those measured in the vorticity direction.

Varadan and Solomon

Figure 11. Evolution of S(q) in the flow direction with time during start-up of steady shear at γ˘ ) 0.6 s-1 for φ ) 0.035. The power law exponent of the dependence of S(q) on q increases with time.

Figure 12. Evolution of S(q) at low q (aq ) 0.032) with time at different shear rates for φ ) 0.035.

Figure 10. Isointensity contours showing the transient evolution of scattered intensities for φ ) 0.035 at a shear rate of 0.6 s-1. The intensity values of the contours are shown in each image.

residual anisotropy in S(q) still persisted long after cessation of flow. To understand the evolution of the microstructure from the quiescent state upon application of shear, transient studies for the gel φ ) 0.035 were conducted at different shear rates. Figure 10 shows contour plots of the temporal evolution of S(q) in the flow-vorticity plane at γ˘ ) 0.6 s-1. A few isointensity contours are shown for clarity. Although a certain amount of scattering anisotropy is observed at short times, S(q) is isotropic at long times (consistent with steady-state studies reported earlier), as evident from the circular intensity contours shown in Figure 10. Figure 11 shows the evolution of S(q) (in the flow direction) with time for the same conditions. S(q) increases monotonically

with the time. As time increases, S(q) assumes a powerlaw scaling, the exponent of which appears to also increase with time. At long times, the deviation from the fractal scaling is no longer observable even at the lowest q probed. The increase in S(q) at low q is rapid at early times. At later times, the q dependence of the structure factor converges to its steady-state behavior. Beyond 40 s little further evolution of S(q) is observed. Similar monotonic increases in S(q) were observed for other shear rates. The fractal dimension determined by the fractal cluster model in eq 2 was found to increase monotonically to d ∼ 3, in the q range probed (data not shown). To represent the dependence of the temporal evolution of S(q) on γ˘ , S(q,t) at low q (aq ) 0.032) is plotted for 0.3 s-1 e γ˘ e 30 s-1 in Figure 12. Phenomenologically, S(q,t) increases rapidly with time and converges toward a steadystate value (for γ˘ > 3 s-1), while at lower shear rates (γ˘ e 3 s-1), S(q,t) shows a more gradual increase with time. The trend of increasing fractal dimension (d(t)) was consistent with the observed trend in S(q,t) at aq ) 0.032.


Langmuir, Vol. 17, No. 10, 2001 2927

The implications of these observed changes in S(q) upon start-up of steady shear will be discussed in the following section. 4. Discussion This study has characterized the quiescent and shearinduced structures of organophilic silica-hexadecane gels in the volume fraction range 0.01 < φ < 0.1. We first summarize the key observations made in this study. The quiescent structure of these gels was modeled as fractal clusters with an intracluster fractal dimension of 2.4 (independent of volume fraction) and a volume fraction dependent cluster size. Application of shear resulted in an increase in the cluster size and fractal dimension of these gels. While at low shear rates the scattering patterns were isotropic, at high shear rates anisotropy in the form of two-lobe butterfly scattering patterns was observed. The effect of shear on the structure factor was irreversible, and the measured structure factor showed no further change after the shear was stopped, except for γ˘ ) 120 s-1, where some partial structural relaxation was observed upon cessation. Start-up of steady shear experiments revealed a monotonic increase in S(q,t) at low q (aq ) 0.032) with time for 0.3 s-1 e γ˘ e 30 s-1. In Figure 3 the quiescent structure of the organophilic silica-hexadecane gels was found to be consistent with that of fractal clusters of a finite size. The fractal dimension, d, of the thermoreversible gels is high relative to dilute aqueous polystyrene gels formed by the addition of divalent electrolyte for which diffusion-limited clustercluster aggregation has been reported (d ) 1.8); nevertheless, it is consistent with results for other systems.23,46,47 d ) 2.5 has been reported for 3-D simulations of diffusionlimited aggregation processes.10 Chen and Russel reported a fractal dimension of d ) 2.08 ( 0.29 for a system similar to ours, but with a slightly different particle radius.3 The large difference in the fractal dimension between the thermoreversible gels in our study and the aqueous gels could be due to differences in the aggregation mechanisms of the systems. The aggregation of adhesive spheres is weak (∆Umin/kT < 10) and reversible, which may lead to a gel structure with a high value of d. The model of Sinha et al. describes the data for the low volume fractions well; however, for φ ) 0.075 and φ ) 0.1 there is some discrepancy between the fit and the data, especially at low scattering vectors. This might be due to the gels being of too high a volume fraction to be described by a fractal cluster model (note: ξ/a < 3 for φ ) 0.075 and 0.1). Alternatively, long-range structure of the fractal clusters or small contribution to the scattering by spurious aggregates could play a role. Gelation is often thought to occur when the fractal clusters of aggregating particles grow until the cluster radii fill space, in a manner that is independent of volume fraction. The clusters formed under such conditions obey the scaling:

ξ ∝ φ1/(d-3) a

(3)

Alternatively, an effective cluster volume fraction may be defined. In this case

φcluster ξ ) φ a

3-d

()

(4)

(47) Vacher, R.; Woignier, T.; Pelous, J.; Courtens, P. Phys. Rev. B 1988, 37, 6500.

Figure 13. Volume fraction dependence of the dimensionless cluster radius (ξ/a). The symbols are from fits to experimental data. The solid line shows the predicted scaling for fractal clusters with d ) 2.4, which are distributed independent of volume fraction.

The scaling of the dimensionless cluster size (ξ/a) with the volume fraction is plotted in Figure 13. The solid line shown is the scaling of eq 3, given d ) 2.4. According to this prediction, the clusters fill space in a manner that is independent of volume fraction (i.e., φcluster is a constant). Dilute gels of aqueous polystyrene have been shown to obey this prediction.8 Figure 13 suggests that the simple picture of the gel structure offered by eq 3 may not hold for the thermoreversible colloidal gels. Since the fractal dimension d ) 2.4 is independent of volume fraction, φcluster, the effective volume fraction of clusters must not be constant for all samples. Indeed, applying eq 4 indicates that the volume fraction of clusters in the gel with colloid volume fraction of 0.01 is approximately half that in the gel with colloid volume fraction of 0.10. This may be due to the fact that these gels are of a significantly higher volume fraction than the aqueous polystyrene gels, which obey eqs 3 and 4 with φcluster independent of φ. In addition, the possibility of nonrandom percolation leading to a volume fraction dependence of the cluster packing would account for the observations. Nonrandom percolation could possibly apply to thermoreversible gels, since conditions are known to be not far removed from the spinodal curve.24,40 The long-range correlations present in the vicinity of the spinodal could promote directed percolation that would be consistent with the observations in Figure 13. It should be noted that the (10% error in the characterization of d could also be relevant, since the slope in Figure 13 is a sensitive function of d. Alternatively, the observations in Figure 13 might signify real differences between the gelation mechanisms of the silica-hexadecane and aqueous polystyrene systems. Below the gelation temperature, shear had a significant effect on the organophilic silica-hexadecane system. In contrast, no measurable effect of shear was observed in the fluid phase. The changes in the q dependence of the structure factor were quantified by applying the model of Sinha et al. Shear flow caused an increase in the fractal dimension, d, and dimensionless cluster radius (ξ/a) relative to their quiescent values. For d < 3, the increase in d implies the densification of the cluster on length scales corresponding to the scattering vector range over which

2928


Varadan and Solomon

Table 4. Effect of Steady Shear on the Number of Particles Per Cluster and the Number Density of Clusters (Gcluster) Relative to the Quiescent Condition for O ) 0.035, 0.055, and 0.1 γ˘ (3 s-1) γ˘ (12 s-1)

φ

d

ξ/a

Npa

Fclusterb

0.035 0.055 0.1 0.035 0.055 0.1

2.84 2.86 2.77 2.98 3 2.7

11.0 15.9 5.5 7.7 13.0

1800 2700 500 1000 1200

5.56 × 10-4 3.86 × 10-4 1.71 × 10-3 9.69 × 10-4 7.93 × 10-4

a Number of particles per cluster (N ) ) (ξ/a)d. b Number density p of clusters (Fcluster) ) φ/[(ξ/a)da3]. Note: the entries for ξ/a, Np, and Fcluster are relative to those of the quiescent conditions.

the power-law scaling of S(q) was observed. Shear-induced densification has been previously reported for other kinds of colloidal gels.9,22 Cluster densification due to shear is plausible: Shear deformation of a gel network inevitably leads to the rupture of particle-particle bonds. Concurrently, particleparticle contacts that can result in reaggregation occur readily as particles or clusters move along streamlines in shear flow. Particularly if the system is thermodynamically unstable, as has been suggested for thermoreversible gels,24,40 this shear-induced disaggregation/reaggregation process will tend to yield a more spatially inhomogeneous structure. The result is a structure that is locally dense but in which mass conservation is maintained by the presence of large void regions. Because for certain volume fractions and shear rates, d ∼ 3 (cf. Table 3), we conclude that this process occurs nearly completely in some instances. Recent 2-D simulations of aggregating colloidal spheres corroborate this picture.35 In addition to the increase in the fractal dimension, we also observe an increase in the dimensionless cluster size. Clusters convected by shear flow may aggregate upon contact, due to the attractive interactions of particles on the periphery of the clusters. Thus, according to this view, clustercluster collisions that occur along the streamlines of shear flow lead to the formation of larger clusters. Because of the requirement of mass conservation, the shear-induced increase in ξ/a and d relative to quiescent conditions requires a concomitant decrease in the number of clusters per unit volume. A simple calculation illustrates this effect: For fractal clusters of radius ξ, consisting of particles of radius a, the number of particles per cluster is given by Np ) (ξ/a)d. The number density of clusters (Fcluster) is thus φξ-dad-3, where φ is the colloid volume fraction. Results for different volume fractions and shear rates are listed in Table 4. The results illustrate a profound reduction in Fcluster upon the application of shear. The effect of shear on this quantity is more striking than either the changes in ξ/a or d and is due to the fact that both of these quantities increased relative to the quiescent case. The decrease in Fcluster is consistent with the picture of shearinduced spatial inhomogeneity discussed earlier. Although both d and ξ increased relative to the quiescent condition for γ˘ > 30 s-1, anisotropy in S(q) in the flowvorticity plane was also noted in this shear rate range. The onset of anisotropy appeared to occur at a critical shear rate; however, the possibility of its occurrence at lower shear rates at a lower q than could be accessed by our experimental setup cannot be ruled out. Because S(q) is the Fourier transform of spatial concentration fluctuations, increased concentration gradients result in the enhancement of S(q) in the flow direction. This implies the growth of domains of high concentration, the axes of which are oriented in the vorticity direction. The anisot-

ropy in S(q) for φ ) 0.035 and 0.055 is observed at length scales larger than the cluster size (ξ/a ∼ 25). Thus, while the clusters themselves may be isotropic, they may organize in a manner that is anisotropic, leading to an increased concentration gradient along the flow direction. These flow-induced compositional inhomogeneities are retained in the gel structure after cessation of shear. They indicate that flow induces the formation of a highly nonequilibrium structure. In viscoelastic polymeric systems two-lobe and fourlobe patterns have been reported in shear and extensional flows.48,49 More recently, the presence of two-lobe scattering patterns in the flow-vorticity plane upon shear deformation have been reported for a dispersion of silica spheres (a ) 100 nm) in a polymer melt,50 an associating polymer solution with latex particles,51 and an organophilic silica-benzene gel.24 Observations in the viscoelastic systems have been explained by means of the HelflandFredrickson theory of stress-induced enhancement of concentration fluctuations.52 However, a simple version of this theory that neglects the effect of convective backflow and uses the second-order fluid constitutive equation fails to predict the behavior observed in the flow-vorticity plane for our system.53 The above theory predicts elliptic patterns and not the observed two-lobe butterfly patterns. A full explanation of the observations will await a more complete understanding of the constitutive model most appropriate to describe the viscoelasticity of colloidal gels. In addition to anisotropic S(q) for φ ) 0.035 and 0.055, S(q) in both the flow and vorticity directions obeyed power law scaling with exponent 3.5 for the approximate range 0.04 < aq < 0.12. A power law scaling greater than three is unusual for aq < 1; however, the structure factor in Porod scattering, observed for systems with sharp interfaces, exhibits a q-4 dependence.54 Unsheared gels of colloid-polymer mixtures that undergo spinodal decomposition also exhibit similar large exponents.45 It is interesting that this scattering behavior is observed in our system even though the suspension is macroscopically stable on the experimental time scale. Measurements of the time dependence of S(q) during the start-up of steady shear provide information about structural evolution due to an applied deformation. For φ ) 0.035, the change in shape of S(q) upon deformation is consistent with an increase in the cluster radius, ξ, and the cluster fractal dimension, d. The monotonic increase in d (data not shown) agrees with simulations of Chen and Doi.55 Although, as shown in Figure 11, after approximately 60 s S(q) has reached steady state across nearly the complete range of q studied, a slight evolution is still observed at the smallest q accessible. This trend is more clearly observable in Figure 12, where at the lowest shear rates studied (0.3 and 0.6 s-1) S(q) at aq ) 0.032 is still evolving for t ∼ 60 s. The continued evolution at long times and low shear rates might be due to some sedimentation of the large, locally dense fractal clusters formed (48) van Egmond, J. W.; Werner, D. E.; Fuller, G. G. J. Chem. Phys. 1992, 96, 7742-7757. (49) Wu, X. L.; Pine, D. J.; Dixon, P. K. Phys. Rev. Lett. 1991, 66, 2408-2411. (50) DeGroot, J. J. V.; Macosko, C. W.; Kume, T.; Hashimoto, T. J. Colloid Interface Sci. 1994, 166, 404. (51) Belzung, B.; Lequeux, F.; Vermant, J.; Mewis, J. J. Colloid Interface Sci. 2000, 224, 179-187. (52) Helfland, E.; Fredrickson, G. Phys. Rev. Lett. 1989, 62, 24682471. (53) Fuller, G. G. Optical Rheometry of Complex Fluids; Oxford University Press: New York, 1995. (54) Porod, G. In Small-Angle X-Ray Scattering; Glatter, O., Kratky, O., Eds.; Academic Press: London, 1982. (55) Chen, D.; Doi, M. J. Chem. Phys. 1989, 91, 2656-2663.


Figure 14. Comparison of the evolution of S(q) at low q with the stress evolution of φ ) 0.035 at γ˘ ) 3 s-1. The transient rheology was performed in a Taylor-Couette shear cell on a constant strain rate rheometer.

during the shear deformation. Alternatively, the trend is consistent with the long times often required to attain steady-state behavior in sheared colloidal suspensions. The latter case, if correct, would be evidence of a real correspondence between S(q) at low q (structure on long length scales) and the long-time rheological response of colloidal suspensions. To probe this possible correspondence further, the transient rheology of a thermoreversible colloidal gel (φ ) 0.035) was studied under conditions of the S(q) measurements. Results for γ˘ ) 3.0 s-1 are reported in Figure 14. S(q) at aq ) 0.032 (in the flow direction) is plotted for comparison. The stress response is as expected for a fluid with an apparent yield stress. In particular, the maximum of the stress overshoot observed at short times has been often used as a measure of the static yield stress.56 The stress decay after the overshoot first occurs rapidly and is later followed by a slow evolution toward an eventual steady state. By comparing the transient rheology with the time evolution of S(aq ) 0.032), a number of interesting features concerning the relationship between the two are revealed. First, the monotonic increase in S(aq ) 0.032) occurs on the same time scale required for the stress response to attain steady state. Since the increase in S(aq ) 0.032) is due to an increase in both the cluster radius and fractal dimension, the evolution of these two structural quantities is certainly an important determinant of the rheological response, especially in the region after the stress overshoot. It thus appears necessary that both ξ and d be included (56) Liddell, P. V.; Boger, D. V. J. Non-Newtonian Fluid Mech. 1996, 63, 235-261.

Langmuir, Vol. 17, No. 10, 2001 2929

in rheological modeling. Although the evolution in fractal dimension is typically considered (cf. Potanin et al.31 for example) in modeling, this is less often the case for the cluster radius, particularly for models of nonlinear rheological response. (However, see Krall and Weitz57 for a recent fractal cluster model of the linear viscoelastic response.) Second, the phenomenon of yielding is not too sensitively probed by measurements of S(q) at low q, since the overshoot in the stress response that is known to be related to yielding has little correspondence with S(aq ) 0.032) in the flow direction, which increases monotonically. Thus, either yielding is associated with structural changes that occur for q less than is accessible by our apparatus or it is caused by subtle changes that are not well quantified by measurement of a spatially averaged quantity such as S(q). Third, a slightly different picture is obtained, however, if the anisotropy in S(q) is considered. As shown in Figure 10, all the colloidal gels show an initial transient region of anisotropy of S(q) in the flow-vorticity plane, even at low shear rates. At longer times a return to an isotropic S(q) is observed for γ˘ e 30 s-1. The region of transient anisotropy in S(q) has been noted in Figure 14. It is interesting that this region roughly corresponds to the interval over which the stress overshoot is observed. Although the relationship among flow-induced structural anisotropy, yielding, and the stress overshoot is unclear, Figure 14 suggests that future investigations in this area are warranted. This study has shown that the effect of shear on the microstructure of thermoreversible colloidal gels involves a complex phenomenology of cluster densification, expansion, and anisotropy due to stress-induced enhanced concentration fluctuations at high shear rates and that the transient evolution of S(q) is consistent with timedependent rheology. However, a number of issues remain to be more completely explored, including the exact relationship between the gel constitutive model and the observed anisotropy and the small strain changes in S(q) that precipitate yielding. Nevertheless, the observation of the significant effect of shear on long length scales, particularly involving cluster expansion, densification, and anisotropy, indicates that future progress toward understanding gel rheology and dynamics should involve explanations of these interesting structural features. Acknowledgment. This work was supported in part by NSF (CTS-9813824), Dupont, and the donors of the Petroleum Research Fund (PRF), administered by the American Chemical Society. We acknowledge helpful discussions with Dr. R. Butera, Prof. G. Fuller, Prof. R. Larson, and Prof. R. Ziff. LA001504D (57) Krall, A. H.; Weitz, D. A. Phys. Rev. Lett. 1998, 80, 778.

Shear-Induced Microstructural Evolution of a Thermoreversible

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