Multi Length Scale Analysis of the Microstructure in Sticky Sphere

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Multi Length Scale Analysis of the Microstructure in Sticky Sphere Dispersions during Shear Flow H. Hoekstra,† J. Mewis,† T. Narayanan,‡ and J. Vermant*,† Department of Chemical Engineering, Katholieke Universiteit Leuven, W. de Croylaan 46, B-3001 Leuven, Belgium, and European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France Received June 6, 2005. In Final Form: August 31, 2005 The effect of shear flow on the microstructure in a weakly aggregated suspension is investigated. Monodisperse small silica particles with a grafted layer of 1-octadecanol are dispersed in n-tetradecane, yielding a thermoreversible sticky sphere model suspension. A combination of small angle light scattering and ultra small and small-angle X-ray scattering techniques have been used, in situ and time resolved, to study the flow-induced anisotropy of the microstructure. In this manner, the length scales from the single particle size to that of the spatial organization of the aggregates can be covered. Harmonic expansion of the structure factor demonstrates that anisotropy develops in the microstructure on all relevant length scales. Possible real space interpretations of the scattering information are discussed in conjunction with implications for the nonlinear rheological behavior.

1. Introduction

n ∝ rD

Suspensions can form colloidal gels when attractive particles are present at a sufficiently large volume fraction to produce a space filling network. As for attractive colloidal glasses, the transition from the fluid to the solid state in colloidal gels has been explained by jamming of the particle dynamics.1 Many technological applications are based on the fact that this colloidal gel structure can be reversibly broken down by flow. As a result, these materials display complex rheological behavior which includes phenomena such as a yield stress, thixotropy, and aging.2 Microstructural theories have been developed to describe and predict the flow behavior of these materials. To obtain the rheological properties, the changes in spatial organization of the particles due to the effects of flow have to be calculated.3-6 The available models of this nature assume that the aggregate structure is fractal and that flow mainly affects the aggregate size. Unfortunately, only fragmentary data are available on the microstructure in reversibly aggregated suspensions during flow, and a complete analysis of the full hierarchical microstructure during flow is still lacking. The structure of aggregated suspensions at rest has been analyzed with various scattering techniques. X-ray, neutron, and light scattering measurements have shown that aggregates are often hierarchical self-similar structures7,8 that can be characterized by a single fractal dimension7,9

where D is the fractal dimension and n is the number of particles inside a spherical volume with radius r. Experiments have confirmed the theoretical values for D, ranging from 1.75 for diffusion-limited cluster aggregation (DLCA) to 2.1 for reaction limited cluster aggregation (RLCA) (see e.g. ref 8 and references therein). Not all suspensions however fall in this range; for example, for thermoreversible aggregation, the values found for D are close to 2.410,11 which is well above the RLCA limit. More recently, imaging techniques have also been used to directly probe the microstructure in a quantitative manner. Varadan and Solomon12 used confocal scanning laser microscopy to quantify the full 3D long-range structure in dense colloidal gels of varying volume fraction, formed by short-range attractive interparticle interactions. More traditional microscopic methods often require the colloidal system to be confined to two dimensions in order to study its microstructure in sufficient detail.13,14 Yet, only under special flow conditions can confocal and conventional microscopy be applied to study the flow induced changes in aggregated networks.15,16 Scattering techniques have proven to be more useful in the quantitative analysis of the microstructure in more concentrated systems during flow;10,17-19 therefore, these techniques will be used here.

* To whom correspondence should be addressed. E-mail: [email protected]. † Katholieke Universiteit Leuven. ‡ European Synchrotron Radiation Facility. (1) Prasad, V.; Trappe, V.; Dismore, A. D.; Segre, P. N.; Cipelletti, L.; Weitz, D. A. Faraday Discuss. 2003, 123, 1-12. (2) Dullaert, K.; Mewis, J. Rheol. Acta 2005, 45 (1), 23-32. (3) de Rooij, R.; Potanin, A. A.; van den Ende, D.; Mellema, J. J. Chem. Phys. 1993, 99 (11), 9213-9223. (4) de Rooij, R.; Potanin, A. A.; van den Ende, D.; Mellema, J. J. Chem. Phys. 1994, 100 (7), 5353-5360. (5) Potanin, A. A.; Derooij, R.; van den Ende, D.; Mellema, J. J. Chem. Phys. 1995, 102 (14), 5845-5853. (6) Bartehelmes, G.; Pratsinis, S. E.; Buggisch, H. Chem. Eng. Sci. 2003, 58 (13), 2893-2902. (7) Schaefer, D. W.; Martin, J. E.; Wiltzius, P.; Cannell, D. S. Phys. Rev. Lett. 1984, 52 (26), 2371-2374.

(1)

(8) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. Adv. Colloid Interface 2002, 95 (1), 1-50. (9) Weitz, D. A.; Oliveria, M. Phys. Rev. Lett. 1984, 52 (16), 14331436. (10) Varadan, P.; Solomon, M. J. Langmuir 2001, 17 (10), 29182929. (11) Solomon, M. J.; Varadan, P. Phys. Rev. E 2001, 63, 051402 1-9. (12) Varadan, P.; Solomon, M. J. Langmuir 2003, 19 (3), 509-512. (13) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46 (4), 20452054. (14) Murray, C. A.; Grier, D. G. Annu. Rev. Phys. Chem. 1996, 47, 421-462. (15) Mohraz, A.; Solomon, M. J. J. Rheol. 2005, 49 (3), 657-681. (16) Hoekstra, H.; Vermant, J.; Mewis, J. Langmuir 2003, 19 (22), 9134-9141. (17) de Groot, J. V.; Macosko, C. W.; Kume, T.; Hashimoto, T. J. Colloid Interface Sci. 1994, 166 (2), 404-413. (18) Verduin, H.; deGans, B. J.; Dhont, J. K. G. Langmuir 1996, 12 (12), 2947-2955. (19) Pignon, F.; Magnin, A.; Piau, J. M. Phys. Rev. Lett. 1997, 79 (23), 4689-4692.

10.1021/la051488q CCC: $30.25 © 2005 American Chemical Society Published on Web 10/14/2005

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A generic observation for weakly aggregated dispersions is that they display large scale inhomogeneities during flow. Typically, small angle light scattering produces a characteristic two-lobe “butterfly” pattern.10,17-21 The length scale associated with the underlying anisotropic organization is much larger than that of the primary particle. Formation of roller-like structures has been proposed as an explanation for these scattering patterns.17,19 Whereas direct microscopic evidence for such structures is unavailable for aggregated colloidal dispersions, aggregated confined emulsions in shear flow reveal the formation of such roller-like structures oriented in the vorticity direction.22 In the latter experiments, the size of the attractive droplets is comparable to the gap size; hence, it is not certain that these observations can be extrapolated to attractive dispersions in general. Butterfly patterns are not unique to aggregated suspensions in shear flow. Stable mixtures of latex particles and associating polymer,23 polymer solutions that display phase separation during shear flow,24 and immiscible polymer blends with a viscoelastic polymer dispersed phase in a Newtonian matrix25 also show similar light scattering patterns. Hobbie et al.26 argued that butterfly scattering patterns could arise as a consequence of elastic effects. Experiments on two-dimensional suspensions suggest, that in the case of weakly aggregated systems, the development of butterfly patterns is due to the directional dependence of break-up and aggregation.16 To discriminate between the different mechanisms, probing structure development in the plane of shear, i.e., the vorticity plane, (we use the convention that the projection planes in shear flow are referred to by the direction of their normal) should be required. Unfortunately, in most scattering experiments, information is obtained through projection onto the velocity gradient plane. Whereas butterfly patterns are commonly generated in SALS experiments, evidence for local scale microstructural anisotropy in reversibly aggregated suspension is less general. Small angle neutron scattering experiments indicated anisotropy for sticky sphere systems,27 whereas no small length scale anisotropy was observed in clay dispersions.19 Apart from possibly creating an anisotropic microstructure, X-ray and light scattering studies have shown that shear flow also tends to densify the aggregates in concentrated suspensions.10,28 A full microstructural picture of reversibly aggregated suspensions submitted to flow is still lacking. Here, the flow-induced microstructure in such a material is studied over length scales, ranging from the primary particle size to the aggregate size and the spatial organization of the aggregates. To achieve this, a suitable model colloid and corresponding techniques have to be selected. Additional constraints are imposed by the requirement that all shear (20) Vermant, J. Curr. Opin. Colloid Interface 2001, 6 (5-6), 489495. (21) Vermant, J.; Solomon, M. J. J. Phys.-Condens. Mater. 2005, 17, R187-R216. (22) Montesi, A.; Pena, A. A.; Pasquali, M. Phys. Rev. Lett. 2004, 92 (5), 058303 1-4. (23) Belzung, B.; Lequeux, F.; Vermant, J.; Mewis, J. J. Colloid Interface Sci. 2000, 224 (1), 179-187. (24) van Egmond, J. W.; Werner, D. E.; Fuller, G. G. J. Chem. Phys. 1992, 96 (10), 7742-7757. (25) Hobbie, E. K.; Jeon, H. S.; Wang, H.; Kim, H.; Stout, D. J.; Han, C. C. J. Chem. Phys. 2002, 117, 6350-6359. (26) Hobbie, E. K.; Lin-Gibson, S.; Wang, H.; Pathak, J. A.; Kim, H. Phys. Rev. E, 2004, 69 (6), 061503. (27) Woutersen, A. T. M.; May, R. P.; de Kruif, C. G. J. Rheol. 1993, 37 (1), 71-88. (28) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41 (2), 197-218.

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history and sample loading effects should be eliminated. Therefore, a thermoreversible suspension is chosen that consists of amorphous silica particles protected by a grafted oligomer layer and dispersed in n-tetradecane. This system yields colloidal stability at high temperatures, but aggregation can be induced by lowering the temperature. The structure-probing techniques used include twodimensional SAXS and SALS, scattering dichroism, and USAXS. With different flow geometries and optical arrangements, we attempt to elucidate the effect of flow on the microstructure of sticky sphere dispersions at multiple length scales. 2. Materials and Methods 2.1. Preparation of the Sticky Sphere Suspension. Charge stabilized silica particles were obtained as an aqueous suspension (Ludox TM-50, weight fraction φw) 0.499 from DuPont, currently obtainable from Grace Davison). These particles were coated with a layer of 1-octadecanol (Fluka, purum grade), following the procedure used by van Helden et al.29 for Sto¨ber silica. First, the particles had to be transferred from the aqueous medium into ethanol. For that purpose, 20 droplets of sulfuric acid (BDH, 98%) were mixed with 200 g of Ludox dispersion under vigorous stirring, and subsequently, 630 g of technical grade ethanol was added. Upon addition of the ethanol, some coagulum collected at the bottom of the flask. The clear supernatant was transferred to a 6 L flask, after which an additional amount of 2700 g of technical grade ethanol was added, reducing the water content of the mixture below the azeotropic concentration. Finally, 602 g of 1-octadecanol was added. The ethanol/water mixture was distilled using standard equipment at atmospheric pressure. Upon reaching a vapor temperature of 80 °C, the distillation was stopped. The flask was maintained in an oven at a constant temperature of 230 °C for 4.5 h. After cooling, the excess of 1-octadecanol was removed by vacuum distillation in an oil bath at 210 °C at a column pressure of 1 mbar, resulting in a 1-octadecanol vapor temperature of approximately 160 °C. The remaining colloidal particles were allowed to cool while maintaining a vacuum. Upon reaching room temperature, the vacuum was released, and the particles were redispersed in cyclohexane, resulting in a yellowish suspension. The thus prepared suspension remained stable in cyclohexane for several months, and even centrifugation up to 4000 RCF did not cause the particles to sediment. The average particle diameter was 26.8 nm with a polydispersity of 12%, as derived from X-ray scattering measurements on dilute suspensions in cyclohexane. The polydispersity was obtained by assuming a Schultz type size-distribution.30 To prepare the actual sticky sphere suspensions, the particles were dried by passing a dry nitrogen stream over the sample, followed by overnight drying in vacuo. The dispersion was then formulated by adding the dried silica powder to n-tetradecane (Sigma-Aldrich, 99%). Upon heating, a stable transparent suspension formed that gelled when it was cooled to room temperature. Most experiments were performed on a suspension containing 5% volume of particles in n-tetradecane, which provided the best compromise between the requirements of rheological and scattering measurements. Lower volume fractions suffer from sedimentation, especially during shear flow. However, in X-ray scattering measurements, concentrated aggregated dispersions can suffer from multiple scattering at ultra small angles due to the high contrast between the silica particles and the dispersing medium. For the light scattering measurements from the vorticity plane, some experiments were also performed on a 10% dispersion. 2.2. Flow-Scattering Setups. Combined rheological and light scattering experiments during flow were conducted using a PaarPhysica MCR300 rheometer, equipped with a specially designed parallel plate flow cell in borosilicate glass. For all experiments (29) van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81 (2), 354-368. (30) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79 (5), 24612469.

Microstructure in Sticky Sphere Dispersions a 10 mW He-Ne laser was used as the light source. The light beam was sent through the sample along the velocity gradient direction by means of a set of prisms. The 2D scattering patterns were collected on a semi-transparent screen and recorded with a 10-bit progressive scan digital CCD-camera (Pulnix TM-1300). Images were stored on a computer using a digital frame grabber (Coreco Tci-Digital SE). Subsequently, they were analyzed with in-house developed software (SalsSoftware). To characterize the orientation angle of the scattering entities in the vorticity plane, some light scattering experiments were conducted using a concentric cylinder geometry. Orientation was also investigated by measuring the scattering dichroism. For this purpose, laser light was modulated at 50 kHz using an optical train consisting of a Glan Thomson polarizer (Newport), a photoelastic modulator (PEM, Beaglehole instuments), and a zero order quarter wave plate (Newport).31 The transmitted light intensity was detected with a photodiode, and harmonic variations were analyzed with two DSP lock-in amplifiers (SR830 from Stanford Research Systems). For the dichroism measurements a Couette flow cell with a sample height of 16.5 mm was used. A quartz rod was used to guide the light in the gap in order to reduce the optical path length from 16.5 to 5.5 mm. In this manner, sufficient light was transmitted for the measurements. SAXS experiments were performed at the high brilliance beam line (ID02) of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. 2D-Scattering images were collected using a Thomson X-ray image intensifier coupled to a 14-bit Frelon CCD camera. The distance between scattering volume and detector could be varied from 1 to 10 m, with the entire beam flight path in a vacuum (apart from the sample). Scattering images were normalized and corrected for background scattering and detector characteristics using standard ESRF procedures.32 Flow was generated in a polycarbonate Couette cell with a gap of 1 mm, mounted on a Haake Rotovisco RV30 stress-controlled rheometer. The Couette cell could be translated horizontally, resulting in the beam travelling through the sample in either the gradient or the velocity direction. Hence, this experiment probes scattering from the velocity gradient plane and the velocity plane, respectively. For all experiments, a sample to detector distance of 10 m was chosen, accessing a (2D) q range of 0.010.5 nm-1 at a wavelength of 0.99 Å. USAXS measurements with a Bonse-Hart camera enabled extension of the q range toward the light scattering regime. During these measurements, the flow cell containing the sample is placed between two identical channel-cut Si crystals. The crystal in front of the sample acts as a monochromator, and the second crystal behind the sample serves as the analyzer. A third channel-cut crystal is used as a crossed analyzer to obtain scattering curves in point geometry.33 Scattering from different angles of the sample is obtained by rotating the first analyzer crystal using small angular steps. This provides a so-called “rocking curve”. The scattering curves of the sample can be determined by subtracting the rocking curve of the cell with solvent from the sample rocking curve. An example of the background and sample data is shown in Figure 1. A polycarbonate Couette flow cell attached to a stepper motor was used for the flow experiments on the USAXS line. By moving the Couette, line scans could be made along the velocity direction in the gradient plane and along the gradient direction in the velocity plane. With the sample and setup used, a q range of 0.0012-0.63 nm-1 could be covered. Above q ) 0.002 nm-1 the ratio between background and sample signal was at least a factor 4. For a sample of 10% particles in tetradecane, a broadening of the sample rocking curve from gradient scans at temperatures in the aggregated state was observed. This is indicative of multiple scattering occurring at these ultra small angles, and therefore, we restricted the USAXS measurements to a concentration of 5% particles. (31) Fuller, G. G. Optical rheometry of complex fluids; Oxford University Press: New York, 1995. (32) Narayanan, T.; Diat, O.; Bosecke, P. Nucl. Instrum. Methods A 2001, 467, 1005-1009. (33) Pontoni, D.; Narayanan, T. J. Appl. Crystallogr. 2003, 36, 787790.

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Figure 1. Rocking curve for a suspension of 5 vol % particles in n-tetradecane at 38.0 °C at a shear rate of 100 s-1 (circles). The full line is the rocking curve of the cell with solvent. The resulting scattering curve is shown in the inset.

3. Results and Discussion 3.1. Rheology. To ensure good reproducibility of the rheological and structural properties, a specific protocol was followed. The sample was first heated to 60 °C in order to melt the gel and to produce a stable suspension, this was followed by slow cooling to 38.0 °C. The final temperature was 3° lower than the gel point, which was identified as the temperature where the speckle pattern freezes in a light scattering experiment.34 The heating/ cooling cycle was repeated before each measurement to fully erase the mechanical history, including sedimentation effects. In most free surface experiments the surface was covered by a layer of ethylene glycol. This in combination with the large difference in the temperatures of the thermal protocol and the boiling point of tetradecane (253 °C) prevented evaporation of the solvent. Before starting a rheological or structural measurement, the suspension was allowed to gel and equilibrate at the final temperature for 15 min. The combination of temperature and time proved adequate for the storage and the loss moduli to reach a constant value. Subsequently, the sample was pre-sheared at a rate of 100 s-1 until a steady state was reached, yielding a reproducible initial condition for further experiments. Lowering the shear rate to the final value and awaiting steady-state resulted in the viscosity curve shown in Figure 2. Steady state values are typically reached after 500 s. For longer measurement times, sedimentation became apparent, resulting in a continuous decrease of the stress level over time. The stress and viscosity data in Figure 2 are corrected for the variation of shear rate in the parallel plate flow cell. With Couette measurements that are less prone to the effects of sedimentation, it was verified that the same steady state viscosities were obtained when flow was immediately started up at the final shear rate without a preshear at 100 s-1. The times for reaching steady state were much longer in this case however. Hence, using this protocol without preshear in combination with parallel plate geometries would result in significant sedimentation effects due to the times required to reach steady state. Therefore, samples were always presheared. The occurrence of slip, which can severely disturb both the rheological and the scattering measurements, can be ruled out because reproducible steady-state results were obtained for different geometries and different mechanical (34) Verduin, H.; Dhont, J. K. G. J. Colloid Interface Sci. 1995, 172 (2), 425-437.

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Figure 2. Flow curve of a suspension of sticky silica spheres in n-tetradecane at 38.0 °C at a volume fraction of 0.05. The error bars are within the size of the symbols. The insets are the corresponding SALS patterns for the indicated shear rates. The solid line is the viscosity predicted by Batchelor’s equation.

Figure 3. USAXS data for the quiescent structure of a dispersion of silica in n-teradecane (φ ) 0.05) at different temperatures. The curves are shifted along the vertical axis for clarity.

histories. Figure 2 also includes the SALS patterns, which were obtained at the same time as the rheological measurements, and are discussed below. The flow curve can be described by the Herschel-Bulkley model (τ ) τy + kγ˘ n) with an apparent yield stress τy of 0.12 Pa and a power law index n of 0.87. The high shear viscosity expected for a stable suspension obtained from Batchelor’s equation35 is marked by a full line. 3.2. USAXS. 3.2.1. Quiescent Structure. USAXS measurements were used to characterize both the quiescent and sheared microstructure. Figure 3 shows the scattering curves for a dispersion with φ ) 0.05 in n-tetradecane at three different temperatures. The curves are shifted along the vertical axis for clarity. At a temperature of 60 °C, corresponding to the highest temperature used in the preheating protocol, the scattering curve confirms the absence of structures with large length scales. The scattering curves at small scattering vector, however, lie above the measured form factor, suggesting either some residual attraction between the particles or the presence of a small fraction of particles that are permanently aggregated in doublets. When the sample is cooled to 38 °C, the increase in scattering at q < 0.02 nm-1

Figure 4. (a) USAXS line scans in the gradient plane along the velocity direction for the sticky sphere suspension (φ ) 0.05, T ) 38.0 °C, γ˘ ) 0 s-1 and 100 s-1). The arrow on the right corresponds to the particle diameter. The full line is a fit to the data using eq 2; (b) Difference between scans along the flow direction in the vorticity plane and along the gradient direction in the velocity plane. In both panels a and b, the scattering curves for different shear rates are shifted vertically for clarity.

shows that structures develop at large length scales. The constant slope of the scattering curve at low q suggest fractal behavior. When eq 2 is fitted to the scattering data, a fractal dimension of 2.4 ( 0.1 is obtained. This value agrees well with what has been previously measured for other sticky sphere systems.10,28 Further cooling to a temperature of 13 °C results in a gel that no longer flows under the action of its own weight when sample vials are turned upside down. Whereas the rheological properties are strongly affected, the scattering curves in the sticky regime seem independent of temperature. This suggests that the structure might be kinetically arrested during the temperature quench. 3.2.2. Structure during Flow. Microstructural models assume isotropic aggregates during flow, and they suggest that the aggregate size is a strong function of the applied shear rate.3-6 These structural features can be tested in situ with USAXS measurements, because of the high resolving power of X-rays. First, line scan measurements were performed in the velocity gradient plane along the velocity direction. The same thermal preconditioning protocol as for the rheological measurements was employed at each shear rate. Figure 4a shows the resulting scattering curves for 0 and 100 s-1. As can be seen from a comparison of the curves obtained during flow with that under quiescent conditions (shifted for clarity), the effect

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of flow is limited. To asses the possible directional dependency of the structure, the USAXS setup was modified to enable scanning in two perpendicular directions. Line scans in the velocity gradient plane along the velocity direction were complemented with scans in the velocity plane along the velocity gradient direction as shown in Figure 4b for small values of the scattering vector. To quantify the aggregate size, the scattering data were fitted with the following expression:

I(q) ) NS(q)P(q) ) N[S(q)f + S(q)p]P(q)

(2)

with S(q) the structure factor, P(q) the form factor, and N a proportionality constant. For P(q), the form factor of homogeneous spherical particles with a Schultz type of size distribution was used. S(q)p is the part of the structure factor that accounts for direct particle interactions. For S(q)p, we used the analytic expression for a square well potential, in the mean spherical approximation using the Percus-Yevick closure. The specific expression used for P(q) and S(q)p can be found in refs 30, 46, and 47. S(q)f is the structure factor of the fractal clusters, which is given by36-38

S(q)f )

Figure 5. Crossover length from eq 3 as a function of shear rate for the sticky sphere suspension φ ) 0.05, T ) 38.0 °C, (O) velocity scans in the velocity gradient plane, (0) velocity gradient scans in the velocity plane.

sin[(D - 1) tan-1(qζ)] (3) (D - 1)qζ [1 + q2ζ2](D-1)/2 S(0)

D is the fractal dimension and ζ is a crossover length, which can be used as a measure for the aggregate size.10 The solid line in Figure 4a is a fit to the scattering data using eq 2, with the volume fraction fixed to 0.05. For all shear rates, the R2 values of the fit were above 0.99. A fractal structure implies isotropy, and therefore, the fit based on eq 2 should give parameter values that are identical for the two different scanning directions. The evolution of the crossover length, ζ, with applied shear rate is shown in Figure 5. The error bars indicated correspond to the standard error of the estimate of ζ. The values for ζ translate to an apparent aggregate size in the micrometer size range (aapp ) [(D + 2)/D]1/2Rg ) [(D + 2)(D + 1)/2]1/2ζ).37 The slope of the regression lines displayed in Figure 5 has a value of approximately -0.16 for the velocity scans and -0.18 for the velocity gradient scans, indicating an unexpectedly weak dependence of the apparent aggregate size on shear rate. Furthermore, the apparent size is different for the two scanning directions, indicating an anisotropic shape of the aggregates. (35) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97-117. (36) Ferri, F.; Frisken, B. J.; Cannell, D. S. Phys. Rev. Lett. 1991, 67 (25), 3626-3629. (37) Khlebtsov, N. G.; Melnikov, A. G. J. Colloid Interface Sci. 1994, 163 (1), 145-151. (38) Freltoft, T.; Kjems, J. K.; Sinha, S. K. Phys. Rev. B 1986, 33 (1), 269-275. (39) Johnson, S. J.; Fuller, G. G. J. Colloid Interface Sci. 1988, 124 (2), 441-451. (40) Vermant, J.; Van Puyvelde, P.; Moldenaers, P.; Mewis, J.; Fuller, G. G. Langmuir 1998, 14 (7), 1612-1617. (41) van Egmond, J. W. Macromolecules 1997, 30 (25), 8045-8057. (42) Wagner, N. J.; Ackerson, B. J. J. Chem. Phys. 1992, 97 (2), 14731483. (43) Maranzano, B. J.; Wagner, N. J. J. Chem. Phys. 2002, 117 (22), 10291-10302. (44) Silbert, L. E.; Farr, R. S.; Melrose, J. R.; Ball, R. C. J. Chem. Phys. 1999, 111 (10), 4780-4789. (45) Silbert, L. E.; Melrose, J. R.; Ball, R. C. Mol. Phys. 1999, 96 (11), 1667-1675. (46) Pontoni, D.; Finet, S.; Narayanan, T.; Rennie, A. G. J. Chem. Phys. 2003, 119 (12), 6157-6165. (47) Klein, R.; D’Aguanno, B. Light scattering: principles and development; Clarendon: Oxford, U.K., 1996.

Figure 6. Anisotropy between flow and gradient directions from USAXS data for a 5% (v) dispersion in n-tetradecane at 38.0 °C, as calculated from eq 4.

To investigate this anisotropy in more detail a simple anisotropy factor was calculated from the data in Figure 4b, it is defined as

AUSAXS )

x∑

[log(Ivelocity(q)) - log(Igradient(q))]2

(4)

q

The intensities used in this expression are absolute intensities. The summation is carried out over the entire range of scattering vectors probed in the experiments, to produce an integrated measure. As the scattering curves span many decades in scattered intensity, the squared difference of the logarithm of the scattering intensity is chosen as a measure for the anisotropy. The log prevents weighing the difference in plateau value at low q in favor of differences at higher q where the scattered intensity is much lower. In Figure 6, the anisotropy is plotted as a function of shear rate. It increases monotonically with the applied shear rate. Over the probed q range, the scans in the velocity direction only reflect a change in apparent size. In the velocity gradient scans, however, there is also a change in slope of the scattering curve at intermediate q values. Figure 4b shows this effect, where the increased slope in the gradient scan as compared with the velocity scan is observed for the shear rate of 100 s-1 in a small interval of q values centered around q ≈ 0.01 nm-1. Similar data were obtained at other shear rates but are not shown

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in Figure 4b. The slope increases from the quiescent value of 2.4 ( 0.1 to 2.76 ( 0.1 at a shear rate of 250 s-1. Because the increase only occurs over a limited q range, and as the structure is anisotropic, these values do not provide a “fractal dimension”. The increase in slope can be attributed to densification of the aggregates in the velocity gradient direction. Varadan et al.10 and Rueb et al.28 also observed shear-induced densification of thermoreversible aggregated suspensions. The effect observed in the present work is less pronounced than in previous work, but more importantly, it is shown to be direction dependent. Summarizing, the shear-induced anisotropy as displayed in Figure 6 is the result of a directional dependence of apparent aggregate size (Figure 5) and of a direction dependent densification of the aggregates in shear flow. These data already prove that large-scale anisotropy develops in the microstructure, especially at the higher shear rates probed in our experiments. 3.3. SALS. To investigate the nature of the anisotropy in more detail and to extend the data to even smaller values of the scattering vector, 2D-SALS measurements were used. The insets given in Figure 2 are the steady state SALS patterns projected on the velocity gradient plane, obtained simultaneously with the rheological measurements. Clearly, the pattern becomes more anisotropic with increasing shear rate: a pronounced butterfly shape develops progressively, which does not relax when the flow is stopped. To quantify the anisotropy, a method is used that is free from assumptions regarding the specific symmetries of the scattering patterns. This method is based on the eigenvalues of the second-order moment tensor of the intensity distribution and calculates the anisotropy as39

A)

x[∑I(x, y)xx - ∑I(x, y)yy]2 + 4[∑I(x, y)xy]2 I

(5)

In eq 5 (x, y) are position coordinates representing a location on the scattering pattern (with x and y the two principal directions); I(x, y) is the intensity at that point, and I is the total intensity summed over all pixels on the CCD array. In this manner, an invariant characterizing the anisotropy is obtained. This measure is not sensitive to minor changes in anisotropy. Furthermore, it amplifies contributions far from the center of the scattering pattern; hence, it is biassed towards the smaller length scales. Nonetheless, the method has been used extensively to characterize systems that develop anisotropic scattering patterns during flow.24,39,40 In the case of anisotropic scattering from concentration fluctuations in phase separating polymer solutions, there is a direct link between the second-order moment of the structure factor and the stress tensor.41 Figure 7 displays the anisotropy as calculated from eq 5 for the SALS patterns as a function of the applied shear rate. In the range of shear rates covered by the experiments, the anisotropy increases monotonically with the applied shear rate. When the anisotropy is divided by the corresponding stress level, it becomes clear that it increases less than the stress. Typically, scattering angles in the light scattering experiments lie between 1° and 10°. In real space, this corresponds to sizes of approximately 4-40 µm (λ ) 632.8 nm). Hence, the anisotropy derived from the light scattering experiments refers to the micrometer size range in real space. This is a much larger length scale than that of the single particles and should be related to the internal structure and the spatial organization of the aggregates.

Figure 7. Anisotropy of the SALS images (velocity gradient plane) calculated from eq 5 and anisotropy divided by stress (φ ) 0.05, T ) 38.0 °C.)

Figure 8. (a) Flow cell used for the SALS experiments from the vorticity plane. (b) Scattering pattern from the vorticity plane for a suspension of 10% (v) particles in n-tetradecane at 39.0 °C (γ˘ ) 35 s-1 and sample height = 500 µm).

To access the scattering information from the vorticity plane, issues concerning multiple scattering need to be resolved. To avoid multiple scattering in a 2D SALS experiment, the optical path length through the sample has to be reduced (see Figure 8) and scattering from a stagnant layer of fluid at the bottom of the Couette has to be prevented. To achieve this, a layer of ethylene glycol was added at the bottom and the gap of the Couette was only partially filled with the suspension. To minimize the subsequent distortions of the flow field, the viscosity of the layer was adjusted to closely match that of the shear thinning dispersion at the applied shear rate. Scattering from the vorticity plane results in an anisotropic pattern with an average orientation of the major axis between the flow direction (0°) and the extensional direction of the flow field (-45°), as is shown in Figure 8 for a 10% dispersion. At lower and higher volume fractions patterns with orientations deviating from the velocity direction were also observed, but the anisotropic scattering was most pronounced for the 10% dispersion at the applied shear rate. When the flow is stopped the anisotropy remains present and the orientation of the pattern does not change. Comparable results were obtained by Mohraz and Solomon15 in flow-SALS experiments from the vorticity plane on aggregated polystyrene suspensions. Due to the use of the glycol layer, the effects of shear rate on orientation could not be investigated using SALS. Therefore, the average orientation of the scattering pattern has been measured with scattering dichroism on a 5% dispersion. These measurements suffer less from multiple scattering as only the properties of the transmitted beam are analyzed. To reduce and vary the optical path length, a quartz rod was inserted into the gap. Figure 9 shows the measured evolution of the average orientation angle with applied shear rate. At low shear rates, the orientation of the dichroism approaches the -45° direction, whereas at higher shear rates it evolves toward the velocity direction.

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Figure 9. Orientation angle in the vorticity plane as determined from scattering dichroism measurements for a suspension of 5% (v) particles in n-tetradecane at 38.0 °C (θ ) 0 corresponds to the velocity direction).

Figure 10. Anisotropy with 2-fold symmetry from SAXS patterns projected onto the velocity gradient plane for the 5% (v) suspension at 38.0 °C, the position of the arrow corresponds to a length of one particle diameter.

3.4. SAXS. From the previous sections a picture emerges of a complex microstructural organization at the aggregate level. To investigate the presence of anisotropy on a more local scale, 2D SAXS measurements are used. Local length scales from about 20 particle diameters to below the particle size could be probed. At these length scales, the anisotropy is inevitably less pronounced than in the SALS experiments, because the organization of only a limited number of particles is studied. Therefore, a method different from that used to analyze the SALS patterns has to be used here. A suitable method has been proposed by Wagner and Ackerson.42,43 In this method a perturbation to the isotropic structure factor is expanded in spherical harmonics of specific symmetry. This method is more sensitive to small levels of anisotropy. For suspensions of Brownian hard spheres in shear flow, the expansion can be related to the stress.42 For weakly aggregated, nonergodic, suspensions as used in this work, a theoretical link is not yet established. The anisotropy can be calculated from43

Figure 11. Anisotropy with 4-fold symmetry from SAXS patterns in the velocity gradient plane for the 5% (v) suspension at 38.0 °C. The position of the arrow corresponds to a length of one particle diameter.

An )

∫02π[I(q,θ,γ˘ ) - I(q,θ,γ˘ ) 0)] cos(nθ) dθ

1 2π

(6)

In eq 6, n is an integer, the value of which determines the symmetry of the spherical harmonic. No differences were observed when rotating clockwise and counter-clockwise. Hence, the vorticity axis is an axis of symmetry, and therefore, the only nonzero components of the expansion are those with n being even numbers. To use eq 6, the isotropic scattering pattern at rest is subtracted from the scattering pattern obtained during shear flow. The resulting image is then weighed by a cosine function with appropriate symmetry and integrated over the azimuthal angle θ to obtain An. The flow direction is taken as zero angle. The same thermal protocol as used in the SALS and USAXS measurements was applied before each measurement. The resulting A2 values for the 5% suspension are displayed in Figure 10. With increasing shear rate, the anisotropy at T ) 38.0 °C extends to smaller length scales. At shear rates above 100 s-1, A2 differs significantly from zero at length scales of approximately 3 particle diameters on. The negative sign of the A2 contribution indicates that the major axis of the scattering pattern is oriented in the vorticity direction. With respect to the scattering patterns at rest, there is a reduction of scattering in the velocity direction and an increase of scattering in the vorticity

direction as can be deduced from the increasing magnitude of A2. Based on the USAXS, SALS, and SAXS information, it can therefore be concluded that the isotropic fractal concept, generally used to model the rheological behavior of this type of materials, fails at all length scales down to a few particle diameters. This is at least the case at higher shear rates where the stresses in the interior of the aggregates are sufficiently large to distort even the local aggregate microstructure. The level of the 2-fold contribution to the anisotropy A2 seems to saturate at higher shear rates as similar values are obtained at 100 and 1000 s-1. At the higher shear rates probed in the SAXS experiments, the A4 component for 4-fold symmetry also emerges. Hence, the second-order perturbation of the scattering pattern (n ) 4) also becomes important, although its magnitude is smaller than A2 at the same q values. Figure 11 shows the anisotropy term for A4, for scattering probed from the velocity gradient plane. No significant contributions of higher order symmetry (n > 4) were observed in the range of shear rates covered in the present experiments. In addition to the velocity gradient plane, the scattering onto the velocity plane is also accessible with Couette geometry (see section 2). Due to the much longer path

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Figure 12. Anisotropy with 2-fold symmetry from SAXS patterns in the velocity plane for the 5% (v) suspension at 38.0 °C. The position of the arrow corresponds to a length of one particle diameter.

Figure 13. Overview of the structural information from the different scattering techniques during shear flow for a sticky sphere suspension. In the scattering planes marked with an X no information is available for the given scattering technique.

lengths through the sample the data now suffer from multiple scattering, which renders the scattering more isotropic in nature. The results of the harmonic analysis for the A2 term are shown in Figure 12. The presence of anisotropy is obvious from the nonzero values for A2 at low q values. As for the SAXS results from the velocity gradient plane, the orientation of the scattering pattern is independent of the applied shear rate and the direction of shear. Hence, the vorticity direction is again an axis of symmetry in the scattering pattern. At fixed q the values of A2 increase monotonically with increasing shear rate and the values are all positive, indicating that the major axis of the scattering pattern is oriented along the velocity gradient direction in this plane. 4. Discussion The main results of the scattering experiments are summarized in the cartoons of Figure 13. Three relevant structural levels are probed, corresponding to the size and

Hoekstra et al.

organization of the aggregates, their internal structure and the local organization of the individual colloidal particles. From the combined information of the scattering at different length scales a consistent picture of the microstructure emerges. 4.1. Large Length ScalesSALS. The SALS results in Figure 2 correspond to the most common type of flowSALS experiments in which light is sent along the velocity gradient direction. An anisotropic butterfly shaped pattern oriented in the velocity direction is observed, in agreement with previous results for different types of aggregated colloidal suspensions in shear flow.10,17,19 The scattering patterns represent a large scale inhomogeneity, possibly associated with the formation of roller-like structures.17,19 Yet, for such roll cells, an isotropic shape of the scattering pattern in the vorticity plane is expected. The experimental scattering patterns of this type, shown in Figure 8, however reveal a clear anisotropic shape. Recently, Hoekstra et al.16 reported an anisotropic microstructure in the vorticity plane for aggregated 2D suspensions, and it was attributed to a hydrodynamic mechanism based on directional dependence of aggregation and breakup. In such systems, large structures are formed in the compressional direction of the flow field, and their subsequent rotation and breakup when oriented in the extensional direction causes the observed anisotropy. This was used to explain an orientation of the scattering pattern in the vorticity plane with the major axis in the velocity direction.16 For the 3D suspensions investigated here, the orientation of the major axis of the scattering pattern from the vorticity plane (Figure 8) shows a shear rate dependence. When increasing the shear rate, it gradually evolves from an orientation along the compressional axis of the flow field (-45°), toward the flow direction (Figure 9). This might be caused by a number of factors. At low shear rates, just after the yielding of the network, the large aggregate structures can be expected to hinder each other while rotating. They may even be temporarily arrested when oriented along the compressional direction causing the average correlation length to be higher along a direction approaching -45°. Such arrested structures along the compressional direction have been predicted in simulations of concentrated aggregated suspensions.44 When the aggregates become smaller, they no longer interact as strongly and can freely rotate resulting in patterns that gradually become oriented in the flow direction. 4.2. Intermediate Length ScalesUSAXS. USAXS measurements were mainly carried out to investigate the dependence of the aggregate size on shear rate. When scans are taken in the velocity gradient plane along the flow direction (v b) or in the velocity plane along the velocity b˘ ), a weak dependence on shear rate gradient direction (γ is obtained as shown in Figure 5. Yet, as inferred from Figures 4 and 5, not only the aggregate shape but also the internal aggregate structure is anisotropic. A difference in crossover length, shown in Figure 5, indicates that the aggregates are larger in the velocity direction than in the velocity gradient direction, resulting in a stronger scattering in the velocity direction. Figure 4, however, indicates that on a more local scale (larger q) the situation is reversed. The scattering is now more pronounced in the gradient direction than in the velocity direction, which suggests a greater internal density in the aggregates when scanning along the velocity gradient direction. Such a densification in the velocity gradient direction is consistent with the formation of locally densely packed particles, which form when aggregates collide along the compressional axis of the flow field.

Microstructure in Sticky Sphere Dispersions

4.3. Small Length ScalesSAXS. The local scale anisotropy was further analyzed by means of 2D-SAXS patterns, using an expansion in spherical harmonics (Figures 10-12). The negative values of A2 in Figure 10 mean that the intensity of the scattering patterns from the velocity gradient plane decreases more rapidly in the velocity direction than in the vorticity direction. Hence, the iso-intensity contour plot of scattered intensity from this plane has its major axis along the vorticity direction (see also Figure 13). Simulations of concentrated aggregated suspensions predict such behavior.45 The volume fractions in the simulation studies were however much higher than in the experiments, and Bragg-like peaks were obtained in the calculated scattering pattern. Nevertheless, local scale compression leads to denser (or stronger correlated) structures in the compressional direction. The result of a projection of such a structure in the velocity gradient plane is denser in the velocity than in the vorticity direction. This effectively results in iso-contours of the scattering pattern that give an orientation in the vorticity direction and negative values for the A2 coefficient, as is observed in Figure 10. At the highest shear rates, the anisotropy (A2) saturates, suggesting the aggregate compressibility is limited. Somewhat surprisingly, the local scale anisotropy shows biaxiality. The positive values of A2 from the velocity plane (Figure 12) mean that the iso-intensity SAXS plots in this plane are oriented in the velocity gradient direction (see also Figure 13) and not in the vorticity direction. The nature and causes of the small length scale anisotropy in the velocity plane are not understood. Experimental data in the literature on the local organization of particles in concentrated aggregated suspensions in shear flow are scarce. A similar adhesive sphere system (stearyl silica in benzene) has been studied by Woutersen et al.27 with neutron scattering. Although the range of length scales, as compared to the primary particle diameter, was less extensive than in the present experiments, these authors find a local scale shear-induced anisotropy in the microstructure. The orientation of the scattering patterns is, however, at odds with ours.27 This might be explained by the fact that their system is much more concentrated than ours (φ ) 0.38). At such concentrations and in the q range used in their work, the scattering is dominated by the liquid peak of the structure factor of interparticle interactions and the power law region of the aggregate structure is not probed. 4.4. Multi Length Scale MicrostructuresOverview. The results shown schematically in Figure 13 for all length scales lead to the following conceptual picture of the flow induced microstructure. The apparent aggregate size during flow, measured by USAXS using line scans in the flow and velocity gradient directions, reveals a weak dependence on shear rate and is shown to be direction dependent. This means that considering only changes in aggregate size to explain the rheological behavior is an oversimplification. Scattering measurements suggest a much more complex hierarchy. A large scale structural inhomogeneity is induced by flow which leads to an anistropic scattering patterns as probed by SALS in both the velocity gradient plane (butterfly patterns in Figure 2) and the vorticity plane (Figure 8). These patterns are representative of a large scale anisotropic organization of the aggregates in these materials during shear flow. The magnitude of the anisotropy increases with increasing

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shear rate (Figure 7). From scattering dichroism measurements, it can be concluded that the average orientation in the vorticity plane evolves from an orientation close to -45° toward the flow direction with increasing shear rate (Figure 9). The large length scale anisotropy and the anisotropic aggregate shape can be explained by invoking directional dependence of aggregation and break-up processes, as was observed in 2D suspensions.16 The direction dependent assembly and subsequent rupture also result in aggregates which are anisotropic in size, larger structures existing along the compression axis of the flow field, as shown by the USAXS measurements. Finally the observed densification of the aggregates on a more local scale combined with the reversal of the local scale anisotropy observed in SAXS measurements, are consistent with the mechanism of structure formation. The aggregates are internally compressed when these larger scale structures are formed by compressing aggregates together. 5. Conclusions Subjecting weakly aggregated suspensions to a shear flow results in a complex microstructure. Hence, we investigated shear-induced microstructures on a single model system over a wide range of scattering vectors and this from different scattering planes. By using a combination of light and X-ray scattering during flow, it is shown that microstructural anisotropy is significant on all length scales above the particle size. The concept of isotropic fractals, often used to model the rheological behavior of aggregated suspensions, cannot be used for these materials. Instead, the microstructure during shear flow is more complex and depends on length scale and direction. The overall microstructural picture that could emerge and reconciles most of the scattering information is that of compressible aggregates which form large anisotropic structures by aggregation and break-up events. The aggregates combine and separate at very specific orientations of their backbones, i.e., along the compression and extension directions of the flow field, respectively. SALS measurements from the vorticity plane reveal an anisotropic butterfly pattern, with a shear rate dependent orientation angle. It can be explained by considering that, at low shear rates, the rotation of the large structures formed along the compressional direction is hindered, resulting in arrested orientations in that direction as suggested by earlier simulation results. At higher shear rates, this hindering diminishes, as the aggregate size is reduced. The microstructural organization of the aggregates is then consistent with what was observed in 2D experiments16 where an orientation of the scattering patterns in the flow direction was observed. SALS measurements indicate that the building blocks of these large structures are anisotropically shaped aggregates. These aggregates have an increasingly more dense, anisotropic, internal organization as the shear rate is increased. Acknowledgment. We acknowledge the support of the European Synchrotron Radiation Facility (ESRF) in providing the facilities and financial support for the X-ray scattering experiments (experiment SC-1054). We acknowledge the Fund for Scientific Research-Flanders (FWO-Vlaanderen) for funding. LA051488Q