Rheol Acta (2009) 48:871–881DOI 10.1007/s00397-008-0341-9

ORIGINAL CONTRIBUTION

New insights on fumed colloidal rheology—shearthickening and vorticity-aligned structuresin flocculating dispersions

Ajay Singh Negi · Chinedum O. Osuji

Received: 15 June 2008 / Accepted: 16 November 2008 / Published online: 24 December 2008© Springer-Verlag 2008

Abstract We investigate the rheology of dilute dis-persions of fumed colloidal particles with attractiveinteractions in hydrocarbon liquids. Surprisingly, thesesystems display shear thickening due to the break-down of densified flocs and a concomitant increase inthe effective volume fraction of the fractal particlesin the fluid. We show that this shear thickening iscontrolled by a critical stress and accompanied by apositive increase in the first normal stress difference,N1, at the shear thickening transition. This is in contrastto the well-known hydrocluster mechanism of shearthickening in concentrated hard-sphere and repulsivesystems. Gel elasticity depends strongly on the stressapplied to suspensions in preshear, scaling roughly asG′ ∼ σ 2

preshear. We propose a simple model to accountfor these results in terms of the cluster number den-sity determined by the preshear stress. At low shearrates, vorticity-aligned aggregates are present at γ̇ ≈100 s−1. In this regime, the system displays a smallbut noticeable increase in viscosity on increasing shearrate. We investigate the effect of tool roughness andfind that wall slip is not responsible for the observedphenomena. Instead, the increase in the apparent vis-cosity results from increased flow resistance due to thepresence of gap-spanning log-like flocs in rolling flow.

Paper presented at Workshop on Rheophysics on January 7–8,2008 in Paris, France.

A. S. Negi · C. O. Osuji (B)Department of Chemical Engineering,Yale University, New Haven, CT 06511, USAe-mail: chinedum.osuji@yale.edu

A. S. Negie-mail: ajay.negi@yale.edu

Keywords Carbon black · Colloidal gels ·Shear thickening · Stress relaxation · Flocculation ·Shear-induced structures

Introduction

The rheology of colloidal dispersions may be high-lighted by the display of shear-thickening behaviorin which the viscosity of the system grows with in-creasing shear rate (Barnes 1989). Such flows areof immense practical concern, as they are relevantin the fluid handling of particulate suspensions in awide variety of processes, ranging from cement mixingto the manufacture of cosmetics and filled polymers(Bergstrom 1998; Chadwick et al. 2002). Experimental(Chow and Zukoski 1995; Bender and Wagner 1996;Maranzano and Wagner 2001) and simulation studies(Brady and Bossis 1985; Melrose 2003; Melrose et al.1996) have considerably advanced the understandingof this phenomenon, and shear thickening is now wellunderstood, generally, to arise due to the formationof shear-induced hydroclusters. Briefly, particles areforced towards contact along the compressive axis ofthe shear flow, resulting in enhanced viscous dissipa-tion by fluid flow between particle surfaces with smallseparations. The critical condition for the onset of shearthickening may be rationalized in terms of a balance be-tween the shear stress forcing particles towards contactand any stabilizing forces that oppose close approach,as in Eq. 1 for Brownian spheres

3πηoγ̇ca3

2h= −kBT

∂ ln g(r)∂r

+ Frep (1)

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where a is the particle radius, h = r − 2a is the distancebetween two particles, g(r) is the particle distributionfunction, and ηo is the viscosity of the suspending fluid,replaceable by the suspension viscosity, η, as a wayof accommodating many body effects. Frep is the non-Brownian repulsive force between particles that mayarise from electrostatic contributions or the presence ofstabilizing polymer brushes on the particle.

In hard sphere and repulsively interacting colloidalsuspensions, hydrocluster formation leading to shearthickening is commonly observed at high loadings, ϕ �40%. The increase of viscosity with shear rate maybe smooth and continuous or discontinuous, in whichcase a dramatic jamming of the system can occur withthe viscosity diverging as the particles are forced intocontact in clusters with system-size dimensions. By con-trast, shear thickening of this nature is not known norpredicted for flocculating colloidal systems, i.e., systemswith substantial attractive interactions (Barnes 1989;Gopalakrishnan and Zukoski 2004; Raghavan andKhan 1997). In such gel-forming materials, flow curvesexhibit monotonic shear thinning behavior. Viewedfrom the perspective provided by Eq. 1, this is sur-prising, as the critical stress for hydrocluster formationshould be expected to decrease with increasing strengthof attraction between particles or decreasing Frep, thatis, as the barrier to hydrocluster formation decreases.A systematic study of the effect of interaction strengthwas conducted in depletion-induced gels where finecontrol over the interaction strength is provided bythe polymer depletant concentration (Gopalakrishnanand Zukoski 2004). Here, the critical shear stress atthe onset of shear thickening is seen, in fact, to de-crease as a function of interaction strength, but cru-cially, the extent of shear thickening also decreasesuntil it completely vanishes for systems with substantialinteractions sufficiently strong to induce flocculation.An important emerging concept is that hydroclusterformation does in fact occur in aggregating systems, butthe resulting increase in the hydrodynamic contributionto the viscosity does not offset the large decrease inthe thermodynamic contribution, and so, the overallobservable is a decrease in viscosity on increasing shearrate, i.e., monotonic shear thinning (Gopalakrishnanand Zukoski 2004).

Fumed colloidal particles are commodity materialsthat find application in a great variety of fields. Forexample, carbon black is used as a pigment in inksand paints and a filler in rubber tires. Fumed silicais widely used as a filler and rheology modifier ingels and rubbers. Fumed alumina finds application incoatings, adhesives, and personal care products. Many

of these applications fall under the rubric of “pastes,”the rheophysics of which have been well reviewed(Coussot 2007). All are produced via flame processingof an appropriate feedstock (heavy oil, SiCl4, Al2Cl6)under specific conditions that result in the productionof nanoscale primary particles, ca. 1–10 nm, that collideand sinter in the flame to form larger aggregates. Itis these aggregates that are considered the colloidal“particles,” with particle sizes ranging generally fromO(10−1) to O(10+1)μm. They are characterized by afractal dimension, dp

f , which describes the size depen-

dence of the particle mass as M ∼ Rdpf . The smaller the

fractal dimension, the more open or porous the particlestructure is. Typical dimensions are anywhere from2 to 2.8 as characterized by X-ray scattering (Smithand Zukoski 2004; Helsen and Teixeira 1986). Unsur-prisingly, the suspension rheology of these structuredparticles is of considerable importance in the handlingand processing of the particles themselves and of themany complex fluids that incorporate them in theirformulation.

Considerable attention has been devoted to thistopic, as well as to the rheological modification, pri-marily mechanical reinforcement, of fumed particlefilled polymers. Slurries of these particles in purelyviscous or inelastic media display qualitatively simi-lar rheology, as do their “three-dimensionally full” or“fully solid” counterparts. Under attractive conditions,they aggregate and form gels composed of networksof particle flocs, with an elastic modulus that scaleswith composition as G′ ∼ ϕα (Khan and Zoeller 1993;Grant and Russel 1993). The scaling exponent α is afunction of the interaction strength between particles,with typical values of 3 ≤ α ≤ 5. Under excluded vol-ume or repulsive interactions, no flocculation occurs.The steady flow response shows a Newtonian plateaufollowed by shear thinning above a critical shear rate.At sufficiently high volume fractions, they exhibit shearthickening due to the formation of hydroclusters asdiscussed above for solid particles. Notably, the effec-tive volume fraction of the fumed colloidal particles asdeduced from the zero-shear viscosity via the Batchelorexpression is several factors larger than the nominalvolume fraction, calculated on a mass density basis.This is attributed to the large hydrodynamic radius ofthe fractal aggregates and the entrapment of fluid in theporous particles. Thus, shear thickening attributed tohydrocluster formation has been observed at nominalvolume fractions as low as a few percent (Raghavan andKhan 1997; Fischer et al. 2007). In parallel to the obser-vation of hydrocluster-driven shear thickening in hard-sphere and repulsive fumed particulate systems, shear

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thickening is not expected under flocculating conditions(Raghavan and Khan 1997).

Here, we report on the observation of modest shearthickening in flocculating fumed colloidal dispersionsof carbon black and alumina in hydrocarbon media.Shear thickening in these systems is driven not byhydrocluster formation but by breakup of dense fractalclusters resulting in an increase in the effective volumefraction of the system, as detailed in our previous pub-lication regarding carbon black in tetradecane solvent(Osuji et al. 2008). We consider the effect of varyingsolvent viscosity and present results illustrating that thebreakup of clusters is controlled by a critical stress, andnot a critical shear rate. We find a strong correlationbetween the elastic modulus of gels and the preshearstress to which they are subjected, underpinned by thedifference in suspension microstructure in the shearthinning vs shear thickening regimes. These systemsdisplay measurable residual stresses on cessation offlow, and we characterize their dynamics as dependenton the preshear to which the system is subjected. Themicrostructure of the shear-thickened gels displays adramatic instability, resulting in the formation of highlyanisotropic vorticity-aligned aggregates on shear flow,studied here in alumina dispersions. These results shednew light overall on the rheology of fumed colloidalmaterials and help explain some curious phenomenaobserved in these materials in past reports (Raghavanand Khan 1995; Kawaguchi et al. 2001).

Experimental

Carbon black was obtained from Cabot. The particulargrade used, Vulcan X72R, exists as ≈ 0.5-μm-diameterparticles with a particle fractal dimension dp

f = 2.2

and bulk density ρcarbon = 1.8 g/cm3. Fumed alumina(SpectrAl 100) was also obtained from Cabot withbulk density ρalumina = 3.7 g/cm3. Tetradecane (AldrichChemical, η = 2.8 cPs, ρ = 0.76 g/cm3) and a medium-bodied mineral oil (Aldrich Chemical, η = 45 cPs, ρ =0.85 g/cm3) were blended in various proportions to pro-vide suspending media of tunable viscosity, accordingto a logarithmic mixing rule. Under these conditions,i.e., in nonpolar solvents, there is no charge on thesurface of the particles, and so, they interact solely viaan attractive Van der Waals potential. A concentratedstock of the fumed colloidal particles in the relevantfluid was produced by vortex mixing for about 2 min,sonication for another 10–30 min, and vortexing againfor 2 min. Samples for measurement were then pre-

pared from the well-mixed stock by dilution, followedby vortexing and sonication.

Rheological characterizations were performed usinga combination of unaltered and roughened tool sur-faces in cone-plate, parallel-plate, and Couette geome-tries. Steady flow characterizations were performed onan ARES LS-1 (Rheometrics) strain controlled instru-ment, as well as with an MCR301 (Anton-Paar USA)stress-controlled rheometer run in strain control mode.Dynamic data were recorded on the MCR301. The fasthardware-based feedback of this instrument was usedto monitor residual stresses on shear rate quenches byobserving the applied stress required to keep the sam-ple under constant zero strain. For these experiments,the rheometer was installed on an air-floated table toeliminate vibration of the sample by mechanical roomnoise. Optical characterization was conducted in straincontrol in parallel-plate and cone-plate geometries us-ing the MCR301 instrument. A fiber-guided ring lightsource was positioned together with a CCD camera be-low the glass plate of the rheometer to permit imagingvia reflection off the top tool of the instrument.

All measurements were conducted at 25◦C, withtemperature control provided by Peltier elements. Forsteady flow measurements, two different methods wereapplied. The first approach involved preshearing sam-ples at the highest shear rate to which they would subse-quently be exposed, ≈ 1, 000 s−1, followed by repeateddescending and ascending shear rate sweeps. Alterna-tively, repeated shear sweeps were performed, but thefirst ascending sweep data were discarded due to the in-consistent mechanical history involved in handling andloading the sample into the rheometer. Both methodswere found to produce consistent, reproducible flowcurves in good agreement, and data for steady flowis presented here without additional specification ofthe preshear routine. Equilibration at each data pointwas ensured by using ≈ 60-s stabilization time and≈ 10–15 s integration time, with a minimum data den-sity of 10 pts./decade. In the case of measurementsconducted on the ARES, the data collection time wasdoubled as two directions of rotation were used at eachdata point in order to avoid artifacts due to instrumentoffsets of residual stresses in the samples. It is impor-tant to note that the mixing performed on samples,although thorough, served only to macroscopically ho-mogenize the materials. The settling and heterogeneityintroduced by the handling and loading of the samplenecessitates an appropriate preshear after the sample isloaded to ensure consistent results. All samples werechecked to verify that edge-fracture, possible in theshear flow of sticky systems, did not occur.

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For dynamic measurements, samples were pres-heared at a fixed rate of 50 s−1 for 30 min and thenpresheared at the particular shear rate of interest againfor 30 min before finally being allowed to sit quiescentlyfor 15 min. Following this gelation time, the storageand loss moduli were characterized using a frequency ofω = 1 rad/s and strain γ = 0.1%, which was within thelinear regime as confirmed by dynamic strain sweeps,conducted at ω = 1 rad/s. Frequency sweeps were con-ducted in the linear regime using the same strain,γ = 0.1%.

Results and discussion

Steady flow behavior

The flow curves for carbon black and alumina exhibitqualitatively similar features. The samples exhibit two

Fig. 1 Steady-state flow curves on ascending rate sweeps oncarbon black (top) and alumina (bottom) suspensions in a 3:1(by weight) mixture of mineral oil with tetradecane. Data weremeasured on ARES-LS1 instrument using a 2◦, 50-mm cone.For the alumina samples, the tool surfaces were covered with agritted paper and confirmed the absence of roughness effects inthis regime. Filled symbols are stress, open symbols are viscosity

regimes of increasing viscosity with increasing shearrate, with shear thinning behavior at all other points.The first regime, at lower shear rates, ∼ 100 s−1, fea-tures a somewhat irregular, slight increase in viscosityor, at times, what appears to be more of a plateau in theflow curve. This will be referred to as the structuringregime. At higher shear rates, in the second regime, thesamples shear thicken modestly, with a 2–5× increase inviscosity, as shown for data taken using the cone-plategeometry in Fig. 1. This will be referred to as the shear-thickening regime. Within the shear-thinning regions,both materials exhibit the thixotropy that is commonlyencountered in flocculated gels—stepped increases ofshear rate result in time-dependent decreases in theviscosity as the material is broken down in the flow(Barnes 1997). In the shear thickening regime, sus-pensions display a negative thixotropy as the viscosityincreases with time in response to a stepped increase ofshear rate.

Correspondingly, samples display thixotropic loopswith the viscosity on descending rate sweeps above thatat the corresponding shear rate on the ascending arm ofthe flow curve, Fig. 2. The departure of the descendingsweep from the ascending curve is, notably, at the peakof the shear-thickening regime on the ascending curve.

From our data, we extract the critical shear rateson ascending and descending sweeps, γ̇ a

c , γ̇ dc and cor-

responding critical shear stresses, σ ac , σ d

c that denotethe transitions to and from shear thickening flow. Wefind qualitatively different composition scaling of thecritical stress and rate on the ascending and descendingcurves, Fig. 3. In particular, γ̇ a

c ∼ ϕ−1, whereas γ̇ dc ∼ ϕ0.

Correspondingly, σ ac ∼ ϕ1 and σ d

c ∼ ϕ2. For the peakviscosity, γ̇ a

p ∼ ϕ−1, whereas γ̇ dp ∼ ϕ0. Additionally,

Fig. 2 Thixotropic loops for carbon black dispersion in a 1:1tetradecane–mineral oil fluid. Filled symbols are for ascendingsweeps, open symbols for descending runs

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Fig. 3 Composition dependence of critical stress and shear rateat the ascending and descending shear thickening transitions(top) and peaks (bottom). The lines in the graphs serve as guides,with slopes as indicated by the value of the exponents in the text

σ ap ∼ ϕ1.5 and σ d

p ∼ ϕ2.5. With only one-third of adecade in composition range, the actual value of theexponents cannot be robustly determined. However, itis noteworthy that the sum of the absolute values ofthe exponents is roughly constant for the ascending anddescending critical transitions and peaks. This suggestsa consistent variation of the critical viscosity and peakviscosities with composition both on the ascending anddescending sweeps.

In order to determine whether the transition to shearthickening flow is rate- or stress-controlled, we exam-ined the flow curves across a number of fluid viscosities.The critical shear rate varied inversely with the viscosityof the solvent, and the data can be rescaled onto a sin-gle curve using the hydrodynamic stress, σh = γ̇ ηsolvent,indicating that shear thickening in these systems iscontrolled by a critical stress, as seen from the data ofFig. 4.

Fig. 4 Ascending and descending sweep flow curves for 4 wt.%carbon black in oil–tetradecane (C14) mixtures of 0:4, 1:3, 1:1,and 3:1 compositions. The shear rate is reduced to the hydro-dynamic stress by multiplying by the viscosity of the solvent,producing superposition of the relative viscosities, indicating thatshear thickening is controlled by a critical stress

Along with the viscosity characterization, steady flowexperiments provided measurements of the normalstress behavior. It should be noted that normal stressdifferences in colloidal suspensions are notoriously dif-ficult to measure accurately (Kolli et al. 2002; Zarragaet al. 2000; Singh and Nott 2003), and more so for dilutesystems. Full-scale deflection of the strain gauge mech-anism on most commercial instruments is achieved atO(20N), which, for a typical 50-mm cone, correspondsto a stress of O(104) Pa, whereas the stresses generatedby these dilute dispersions are at least two orders ofmagnitude smaller than that. Further compounding theproblem is the quadratically increasing inertial contri-bution to the normal force at high shear rates, Finertial =−(3/20)ρ(Rω)2, where ρ is density, R is the radiusof the tool, and ω is the rotational velocity (Kulickeet al. 1977). Nonetheless, by making careful correctionsto the normal stress signal measured using the ARESLS-1 instrument and averaging over several runs, wewere able to accurately and reproducibly resolve thefirst normal stress difference (N1) behavior in the shearthickening regime of tetradecane dispersions of carbonblack, Fig. 5. We find that N1 shows a marked upturn atthe shear thickening transition, followed by a peak andeventual downturn. The marked upturn, however, wasonly visible for more concentrated samples, ϕ >4 wt.%.This behavior is in contrast to the negative first normalstress differences that develop on hydrocluster-driventhickening in concentrated hard-sphere systems (Singhand Nott 2000, 2003). The downturn in the normal

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Fig. 5 Normal stress data on ascending rate sweeps of carbonblack dispersions in tetradecane, taken as the difference withrespect to the value at the lowest shear rate, �N1(γ̇ ) = N1(γ̇ ) −N1(γ̇ = 10). Data are averaged over four separate runs, and theerror bar represents the standard deviation in the normal stressat each data point. Inset shows the viscosity and normal stressbehavior the 6 wt.% sample, where the colocation of the shearthickening transition and the sharp upturn in N1 is clearly visible

stresses at high shear rates does not appear to be re-lated to any improper inertial correction of the data.It appears to be coincident with the peak of the shearthickening response in the more concentrated samples,but its origin is currently unexplained.

In order to understand the origins of the observedrheology, we imaged the microstructures that werepresent at different points of the steady flow curve for adilute alumina dispersion, which permitted reasonablygood optical contrast, as shown in Fig. 6. At low shearrates, in the shear-thinning regime, the system consistsof large “chunks” of the broken gel. In the structur-ing regime, we observe the appearance of vorticity-aligned flocs that adopt a log-rolling motion. These flocsgradually break down in the shear-thinning regime thatfollows the structuring regime, eventually forming acollection of densified clusters. Finally, in the shear-thickening regime, these dense clusters are broken upand the system consists of small, finely dispersed aggre-gates that dramatically increase the optical density ofthe sample. Shear thickening in the dispersion materialsis caused by the increase in the effective fraction ofthe particles when large clusters are broken up anddispersed. These results are in excellent agreement withthe observed structures in carbon black systems, pub-lished previously (Osuji et al. 2008), and are analogousto a “cluster dilation” model advanced to explain a

Fig. 6 Microstructure at different shear rates for a 2 wt.% alu-mina dispersion in 3:1 oil/tetradecane fluid using a 1◦ 50-mmcone. Field of view is 1 × 1 cm

negative thixotropy in concentrated ferric oxide sus-pensions (Kanai and Amari 1995). This mechanism ismarkedly different than the hydrocluster formation,which drives thickening in nonflocculating suspensions.It is expressly dictated by the fractal nature of the parti-cles, which allows them to form compact aggregates dueto interpenetration, as it were, of the highly structuredparticle surfaces and should be a general mechanismacross many types of fumed particles.

While flow curves measured at higher shear rates,γ̇ > 100 s−1, are smooth and of excellent reproducibil-ity, data collected at lower shear rates were less consis-tent from run to run, and rather noisy (Osuji and Weitz2008). In some cases, for instance, the viscosity wasobserved to decrease monotonically with decreasingshear rate. One possibility is that wall slip occurs atshear rates far below 100 s−1, as it is well known to occurin the yielding of sticky colloidal systems such as these(Walls et al. 2003). Transient stress measurements onstep-wise reduction of shear rates at different gaps inthe parallel-plate geometry, however, suggest that wallslip is not a large factor (Yoshimura and Prud’homme1988; Dullaert and Mewis 2005) and that homogeneousflow is achieved at the shear rates considered, Fig. 7.Further, the use of tools covered by roughened surfaceswas found to preserve features of the flow curve and didnot eliminate the unusually low and decreasing viscosi-ties sometimes observed on decreasing shear rates inthe low-shear-rate regime.

It appears instead that sedimentation of the parti-cles at low shear rates is responsible for the lack of

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Fig. 7 Transient stress measurements on 4 wt.% carbon black intetradecane. Data are for step-wise reductions in shear rate from100 to 2.5 and 0.5 s−1 gaps of 400, 600, and 800 μm in the parallelplate geometry

consistency of the data. The use of sequential ascend-ing and descending runs with long equilibration delaysallows substantial time for sedimentation to occur, andis exacerbated at low shear rates where shear-inducedresuspension cannot occur. Indeed, consistent smoothflow curve data were only obtained when the Couettegeometry was used, as shown in Fig. 8. In this geometry,the effects of sedimentation are mitigated by the largeheight (34 mm) of the tool. We observe good agreementbetween ascending and descending flow curves in thelow-shear-rate limit, all the way down to 10−3 s−1, andthe small increases in viscosity in the structuring regime

Fig. 8 Ascending rate flow curves for carbon black dispersionsin 3:1 oil/tetradecane fluid measured in the Couette geometry onARES-LS1 instrument

are consistently present. This is in contrast to datameasured in the cone-plate geometry where the effectsof sedimentation are disproportionately amplified. Itshould be stressed that these data in the Couette weremeasured on the ARES-LS1 instrument using two di-rections per data point, with equilibration and torque-averaging times of 60 and 10 s per direction, resulting ina high degree of confidence that these are equilibriumflow curves.

Vorticity aligned structure formation

Vorticity alignment of structures in complex fluids ap-pears to be a quite general phenomenon, having beenobserved in several associating systems under flow, suchas thixotropic clay gels (Pignon et al. 1997), nanotubesuspensions (Lin-Gibson et al. 2004), and attractiveemulsion droplets (Montesi et al. 2004). Their appear-ance is usually associated with a negative first normalstress difference. The observation of these structuresin steady-state flow in carbon black and alumina dis-persions is, thus, not surprising, although we did notobserve consistently measurable negative normal stressdifferences in the structuring regime of our system.However, the rheology associated with their formationhere appears to be unique, as their appearance is con-current with a slight increase in the apparent viscosityof the system, though Pasquali et al. have observed asmall Newtonian plateau coincident with the formationof vorticity-aligned emulsion droplet aggregates. Whilethe exact mechanism of the viscosity effect remainsunknown, it may stem from the significant occlusion ofthe tool cross-section by the cylindrical flocs. Our mea-surements show that the floc width is approximatelyequal to the gap of the tool, which, assuming a circularcross section, implies that the flocs span the tool gap.This would give rise to lubrication stresses in the flowof fluid in the small spaces between the tool and flocsurfaces and may be at the origin of the observedviscosity behavior.

The stability of the vorticity-aligned structures isstrongly dependent on gap size. As seen in the data ofFig. 6, the structures are not present beyond a certainradial distance from the center of the cone. While theyare present as steady-state structures, they also formstrikingly and then gradually breakup as a transientresponse in the deformation of shear-thickened sus-pensions at a fixed shear rate or on quenches fromhigh shear rate thickening flow into the shear thinningregime of the flow curve (Osuji and Weitz 2008). Thisis shown in Fig. 9 for a 2% alumina suspension. Here,

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Fig. 9 Vorticity aligned flocs in rolling flow in 2% aluminain 50 mm, 1◦ cone-plate. The sample was quenched fromγ̇ = 1,000 s−1 to 10 s−1. Images are taken from a through d at0, 4, 18, and 78 s after the rate quench. Field of view is 1 × 1 cm

as also reported for a flocculated nanotube system(Lin-Gibson et al. 2004), the structure formation isstrongly parameterized by the gap size. In the currentsystem, their formation may be governed by the rapidaggregation of the small finely dispersed clusters pro-duced in the shear thickening flow. Given the R3

dependence of torque on radial distance in the cone-plate geometry, (Macosko 1994), the contribution fromthe log-like flocs to the apparent viscosity is small asthe structures are destabilized at larger radial distancesfrom the cone center. The use of the parallel plategeometry suffers from the radial dependence of theshear rate, and so, the rheology of these systems wouldbest be studied as a function of gap size in the Couettegeometry.

Gel elasticity

Given the differences observed in suspension mi-crostructure between regimes delineated by γ̇c, the crit-ical shear thickening transition, it can be expected thatthe mechanical properties of the gels formed after ces-sation of preshear should differ as well. Dynamic mea-surements were carried out in the cone-plate geometryon the MCR instrument on carbon black suspensions in3:1 blends of mineral oil with tetradecane. The sampleswere found to exhibit either a very small or immea-surable frequency dependence of the elastic modulus,Fig. 10. The strain dependence of the moduli is char-acteristic of soft glassy materials, with a pronouncedupturn in the loss modulus, G′′, on yielding of thenetwork, Fig. 11. The yield strains appear to increasefor samples produced at higher shear rates, but the

Fig. 10 Frequency sweeps on samples subjected to differentpreshear flows. Five-percent carbon black in 3:1 oil/tetradecanefluid

uncertainty in accurately extracting a yield strain makesit difficult to asses this quantitatively. At very largestrains, γ ∼ 100–1,000%, there are small increases inthe two moduli. This corresponds with the structuringregime observed in steady flow experiments, and thedynamic shear rate, γω, is similar to the steady flow ratein the steady measurements for these samples. Agingwas not observed in these samples and the modulusremained steady over ≈ the period of observation (1 h),including a 15-min sweep performed to check the timeevolution of the modulus, Fig. 12.

When considered as a function of the preshear stress,the elastic moduli display a marked power-law depen-dence for samples sheared above the shear-thickeningtransition, scaling roughly as G′ ∼ σ 2, Fig. 13. The

Fig. 11 Strain sweeps on samples subjected to different preshearflows. Five-percent carbon black in 3:1 oil/tetradecane fluid

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Fig. 12 Storage and loss moduli of gels remain constant duringmeasurement after 30-min quiescent gelation. Five-percent car-bon black in 3:1 oil/tetradecane fluid

strong dependence of the modulus on the prestress canbe understood in terms of the gel structure that resultsupon cessation of shear. Gelation in these fluids is veryrapid and the system presents an elastic modulus atthe shortest measurable times after cessation of shear,t ≤ 5 s. The gel modulus can then be viewed as simply acohesive energy density (Larson 1999) that results fromthe contact between clusters, the dimensions of whichare set by the magnitude of the preshear stress.

The shear force on a cluster of radius Rc is dis-tributed over Ns particles in the plane of shear of thecluster. The cluster, a fractal object, has Ns ∼ R

d f −1c ,

where d f is the fractal dimension of the cluster. The at-tractive interaction, U , between particles of dimension

Fig. 13 Elastic modulus of gels as function of preshear stress.Inset: data rescaled according to Eq. 3

a balances this force such that the dependence of thecluster size on the shear stress is as given in Eq. 2.

Rd f −3c ∼ 4πσa

U(2)

The elastic modulus is then given as G′ ∼ ν U , whereν is the number density of clusters, ν ∼ ϕ/R

d fc . Thus,

G′ ∼ φσ

d f3−d f (3)

Assuming a value of 2 for the fractal dimension, thissuggests that G′ ∼ σ 2φ. Replotting the data in thesereduced terms produces a good agreement between oursimple scaling model and the data, Fig. 13. The dataare in rough agreement with experimental data thatdemonstrate that the number of particles in a floc scalesroughly as Nc ∼ σ−1 (Sonntag and Russel 1987). Thechoice of fractal dimension for the gel is reasonable,though somewhat arbitrary. Previous data on carbonblack gels in 100% tetradecane fluid were better fit,assuming a fractal dimension of 1.8. Curiously, themaximum elastic moduli of those gels were a factor of≈ 3× higher than that achieved in the more viscous 3:1oil/tetradecane fluid.

The results reported here shed some light on thework of Raghavan et al., in which the structural recov-ery of flocculated silica gels was studied as a functionof time after large-amplitude oscillatory shear at dif-ferent strains. They observed that the recovered mod-ulus was strongly dependent on the strain applied, andthat higher moduli were obtained if the samples weresubjected to larger strains. They hypothesize that thedifference is due to changes in the network structure,with smaller strains leading to rearrangement of thenetwork into larger flocs, leading to a reduction in thenumber of “crosslinks” in the systems. This reason-ing is in line with our findings. Our experiments haveprovided direct imaging that supports these assertions,and our scaling model accounts well for the data. Thestrong dependence of the modulus on the preshearstress calls into question the utility of the compositionscaling of the modulus, G′ ∼ ϕα as a metric for theinteraction strength in fumed colloidal systems. Thestandard procedure in experimental investigations is toapply a high rate preshear to “initialize” the system,and then to measure the modulus, but our data suggestthat a more accurate approach would involve applyinga composition scaled stress as the preshear.

The gelation that occurs on cessation of shear flowcan be thought of as a “mechanical quench” of thedispersion from a freely flowing gel state into an ar-rested gel. In the alumina and carbon black materi-als studied here, for the volume fractions considered,

880 Rheol Acta (2009) 48:871–881

gelation was very rapid, and samples display a finitemodulus at the earliest measurable times (∼ 5 s) aftercessation of shear. Such near-immediate establishmentof a percolated network after stopping flow has alsobeen observed in silica gels (Raghavan and Khan 1995),as well as suspensions of ferric-oxide (Kanai and Amari1993). This sol–gel transition can be expected to involvethe display of residual or internal stresses as the gelnetwork is deformed with respect to an equilibriumrelaxed configuration. Such a stress, σi, should be pro-portional to the modulus of the system, through theamount of the out-of-equilibrium strain, γd, and shouldrelax gradually in time (Ramos and Cipelletti 2005).The concept of such a “trapped stress” has been in-voked to account for the strange dynamics observedin the aging of some glassy systems (Bandyopadhyayet al. 2004; Bellour et al. 2003). Measurements of stressdecays after stress-jumps enable a decoupling of theviscous and elastic stress components and have beenmade on a variety of systems in the short time limit(≤ 100 s) (Dullaert and Mewis 2005; Kaffashi et al.1997). Using the fast-feedback control loop of ourMCR instrument, we successfully measured thesestresses after a fast quench (0.1 s duration) to thestationary state out to 103 s after cessation of shear,Fig. 14. Measurements are conducted on carbon blackdispersions in 3:1 oil/tetradecane fluids that have beensubjected to different preshears. We find a weak powerlaw decay, with σi ∼ t−0.1 across all shear rates, with theinitial value of the shear stress, σi(t = 0.1), on the orderof 1–10% of the preshear stress. Reversing the directionof the preshear correspondingly switches the sign ofthe residual stress. The data are in excellent agreementwith the time-dependence observed for suspensions

Fig. 14 Residual stresses measured on cessation of shear fromdifferent shear rates

Fig. 15 Modulus of gels as a function of the magnitude of theresidual stress, measured just after cessation of shear

of carbon black in tetradecane measured on an AR-G2 instrument (Osuji et al. 2008), suggesting that thisapproach can be satisfactorily used if very short timedata are not required. The direct proportionality ex-pected between the residual stress and the modulus isconfirmed by these measurements, Fig. 15.

A standard stress relaxation measurement in thelinear regime of course yields such a direct correspon-dence between σ and G′ as well, but in the currentexperiment, the stresses that are relaxing are not mea-sured in response to a step-strain on an equilibriumstructure, but arise solely on the rapid rate quench ofthe system. These stresses are of nontrivial magnitude.For example, for gels prepared at 700 s−1, the resid-ual stress just after cessation of shear is about 3 Pa,Fig. 14, whereas the yield stress measured in the strainsweep, Fig. 11, is only about 2.5 Pa. The presence ofresidual stresses in these systems should be taken intoaccount in designing stress relaxation experiments asthe stress resulting from the applied step-strain in theexperiment is superposed on the existing residual stress.This can lead to artifacts such as negative stresses onstrain-controlled instruments such as the ARES, whichautomatically offset the transducer at the start of thestep-strain. At present, we are unable to reconcile thepresence of internal stresses that are significant in com-parison to measured yield stresses in these materials.More detailed studies are required. Here, the MCRinstrument is particularly well suited to these studies,as it permits specification of the braking time overwhich the sample is quenched from the flowing to thestationary state.

Rheol Acta (2009) 48:871–881 881

Conclusion

The observation of shear thickening in flocculatingsystems is a rather surprising result. It is a modesteffect relative to the shear thickening and jammingthat occur in dense, hard-sphere systems. The data arehighly reproducible and indicate that the transition iscontrolled by a critical stress and is concurrent withan upturn in the first normal stress difference. Shearthickening here can be well understood in terms ofthe breakdown of clusters and the resultant increase inthe effective volume fraction of particles in the fluid.This effect is exactly as a result of the unique fractalmorphology of the colloidal particle concerned, and so,the mechanism is not relevant for Euclidean solid parti-cles. The strong dependence of the gel modulus on thepreshear stress is properly rationalized in terms of thecluster number density that results from the preshearflow. This has important implications in applicationswhere fumed colloidal particles are used as fillers influids, including polymers, as the elasticity of the fluidcan be strongly influenced by the mixing conditions.Surfactants can be used to sterically modulate the at-tractive interaction between particles, and this would beexpected to alter the shear thickening behavior. In pre-liminary experiments, we have observed that the criticalshear thickening transition shifts to lower stresses, aswould be expected for decreased floc strength due tothe presence of the surfactant. Finally, the dynamics ofthe rolling motion of the log-like flocs is of interest.A careful analysis may yield insight into rheology ob-served in the structuring regime.

Acknowledgements The authors would like to acknowledgeDr. Chanjoong Kim and Prof. David Weitz of Harvard Universityfor fruitful discussions and funding from the National ScienceFoundation under grant number CBET-0828905.

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