Rheologica Acta manuscript No. (will be inserted by the editor) New Insights on Fumed Colloidal Rheology - Shear Thickening and Vorticity Aligned Structures in Flocculating Dispersions Ajay Singh Negi · Chinedum O. Osuji Received: date / Accepted: date Abstract We investigate the rheology of dilute disper- sions of fumed colloidal particles with attractive inter- actions in hydrocarbon liquids. Surprisingly, these sys- tems display shear thickening due to the breakdown of densified flocs and a concomitant increase in the effec- tive volume fraction of the fractal particles in the fluid. We show that this shear thickening is controlled by a critical stress and accompanied by a positive increase in the first normal stress difference, N 1 at the shear thick- ening transition. This is in contrast to the well-known hydrocluster mechanism of shear thickening in concen- trated hard-sphere and repulsive systems. Gel elasticity depends strongly on the stress applied to suspensions in pre-shear, scaling roughly as G ∼ σ 2 pre-shear . We propose a simple model to account for these results in terms of the cluster number density determined by the pre-shear stress. At low shear rates, vorticity aligned aggregates are present at ˙ γ ≈ 10 0 s -1 . In this regime the system displays a small but noticeable increase in viscosity on increasing shear rate. We investigate the effect of tool roughness and find that wall slip is not responsible for the observed phenomena. Instead, the increase in the apparent viscosity results from increased flow resistance due to the presence of gap-spanning log- like flocs in rolling flow. A. Negi Department of Chemical Engineering, Yale University, New Haven, CT 06511 Tel.: 203-436-4059 E-mail: [email protected]C. Osuji Department of Chemical Engineering, Yale University, New Haven, CT 06511 Tel.: 203-432-4357 Fax: 203-432-4387 E-mail: [email protected]Keywords carbon black · colloidal gels · shear thickening · stress relaxation · flocculation · shear induced structures PACS 82.70.Dd · 82.70.Gg · 83.60.Rs 1 Introduction The rheology of colloidal dispersions may be highlighted by the display of shear thickening behavior in which the viscosity of the system grows with increasing shear rate [1]. Such flows are of immense practical concern as they are relevant in the fluid handling of particu- late suspensions in a wide variety of processes, ranging from cement mixing to the manufacture of cosmetics and filled polymers [2,3]. Experimental [4–6] and sim- ulation studies [7–9] have considerably advanced the understanding of this phenomenon and shear thicken- ing is now well understood, generally, to arise due to the formation of shear-induced hydroclusters. Briefly, particles are forced towards contact along the compres- sive axis of the shear flow resulting in enhanced viscous dissipation by fluid flow between particle surfaces with small separations. The critical condition for the onset of shear thickening may be rationalized in terms of a balance between the shear stress forcing particles to- wards contact and any stabilizing forces which oppose close approach, as in Equation 1 for Brownian spheres 3πη o ˙ γ c a 3 2h = -k B T ∂ ln g(r) ∂r + F rep (1) where a is the particle radius, h = r - 2a the dis- tance between two particles, g(r) the particle distri- bution function and η o the viscosity of the suspend- ing fluid, replaceable by the suspension viscosity, η,
Transcript
Rheologica Acta manuscript No.(will be inserted by the editor)
New Insights on Fumed Colloidal Rheology - ShearThickening and Vorticity Aligned Structures inFlocculating Dispersions
Ajay Singh Negi · Chinedum O. Osuji
Received: date / Accepted: date
Abstract We investigate the rheology of dilute disper-sions of fumed colloidal particles with attractive inter-actions in hydrocarbon liquids. Surprisingly, these sys-tems display shear thickening due to the breakdown ofdensified flocs and a concomitant increase in the effec-tive volume fraction of the fractal particles in the fluid.We show that this shear thickening is controlled by acritical stress and accompanied by a positive increase inthe first normal stress difference, N1 at the shear thick-ening transition. This is in contrast to the well-knownhydrocluster mechanism of shear thickening in concen-trated hard-sphere and repulsive systems. Gel elasticitydepends strongly on the stress applied to suspensionsin pre-shear, scaling roughly as G′ ∼ σ2
pre−shear. Wepropose a simple model to account for these results interms of the cluster number density determined by thepre-shear stress. At low shear rates, vorticity alignedaggregates are present at γ ≈ 100 s−1. In this regimethe system displays a small but noticeable increase inviscosity on increasing shear rate. We investigate theeffect of tool roughness and find that wall slip is notresponsible for the observed phenomena. Instead, theincrease in the apparent viscosity results from increasedflow resistance due to the presence of gap-spanning log-like flocs in rolling flow.
A. Negi
Department of Chemical Engineering, Yale University, NewHaven, CT 06511Tel.: 203-436-4059E-mail: [email protected]
C. Osuji
Department of Chemical Engineering, Yale University, NewHaven, CT 06511
The rheology of colloidal dispersions may be highlightedby the display of shear thickening behavior in whichthe viscosity of the system grows with increasing shearrate [1]. Such flows are of immense practical concernas they are relevant in the fluid handling of particu-late suspensions in a wide variety of processes, rangingfrom cement mixing to the manufacture of cosmeticsand filled polymers [2,3]. Experimental [4–6] and sim-ulation studies [7–9] have considerably advanced theunderstanding of this phenomenon and shear thicken-ing is now well understood, generally, to arise due tothe formation of shear-induced hydroclusters. Briefly,particles are forced towards contact along the compres-sive axis of the shear flow resulting in enhanced viscousdissipation by fluid flow between particle surfaces withsmall separations. The critical condition for the onsetof shear thickening may be rationalized in terms of abalance between the shear stress forcing particles to-wards contact and any stabilizing forces which opposeclose approach, as in Equation 1 for Brownian spheres
3πηoγca3
2h= −kBT
∂ ln g(r)∂r
+ Frep (1)
where a is the particle radius, h = r − 2a the dis-tance between two particles, g(r) the particle distri-bution function and ηo the viscosity of the suspend-ing fluid, replaceable by the suspension viscosity, η,
2
as a way of accommodating many body effects. Frep
is the non-Brownian repulsive force between particlesthat may arise from electrostatic contributions, or thepresence of stabilizing 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 may besmooth and continuous, or discontinuous in which casea dramatic jamming of the system can occur with theviscosity diverging as the particles are forced into con-tact 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 [1,10,11]. In suchgel forming materials, flow curves exhibit monotonicshear thinning behavior. Viewed from the perspectiveprovided by Equation 1, this is surprising as the criticalstress for hydrocluster formation should be expected todecrease with increasing strength of attraction betweenparticles or decreasing Frep, that is, as the barrier tohydrocluster formation decreases. A systematic studyof the effect of interaction strength was conducted indepletion induced gels where fine control over the in-teraction strength is provided by the polymer deple-tant concentration [10]. Here, the critical shear stressat the 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 decreases un-til 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,but the resulting increase in the hydrodynamic contri-bution to the viscosity does not offset the large decreasein the thermodynamic contribution, and so the overallobservable is a decrease in viscosity on increasing shearrate, i.e. monotonic shear thinning [10].
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 silica iswidely used as a filler and rheology modifier in gels andrubbers. Fumed alumina finds application in coatings,adhesives and personal care products. Many of theseapplications fall under the rubric of “pastes,” the rheo-physics of which have been well reviewed [12]. All areproduced via flame processing of an appropriate feed-stock (heavy oil, SiCl4, Al2Cl6) under specific condi-tions that results in the production of nano-scale pri-mary particles, ca. 1-10 nm, that collide and sinterin the flame to form larger aggregates. It is these ag-gregates which are considered the colloidal “particles”,
with particle sizes ranging generally from O(10−1) toO(10+1)µm. They are characterized by a fractal di-mension, dp
f , which describes the size dependence of
the particle mass as M ∼ Rdpf . The smaller the frac-
tal dimension, the more open or porous is the particlestructure. Typical dimensions are anywhere from 2 to2.8 as characterized by x-ray scattering [13,14]. 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 their for-mulation.
Considerable attention has been devoted to this topic,as well as to the rheological modification, primarily me-chanical reinforcement, of fumed particle filled poly-mers. Slurries of these particles in purely viscous orinelastic media display qualitatively similar rheologyas do their “three-dimensionally full” or “fully solid”counterparts. Under attractive conditions, they aggre-gate and form gels composed of networks of particleflocs, with an elastic modulus that scales with compo-sition as G′ ∼ ϕα [15,16]. 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 as dis-cussed above for solid particles. Notably, the effectivevolume fraction of the fumed colloidal particles as de-duced from the zero-shear viscosity via the Batchelorexpression is several factors larger than the nominal vol-ume fraction, calculated on a mass density basis. Thisis attributed to the large hydrodynamic radius of thefractal 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 [11,17]. Inparallel to the observation of hydrocluster driven shearthickening in hard-sphere and repulsive fumed partic-ulate systems, shear thickening is not expected underflocculating conditions [11].
Here, we report on the observation of modest shearthickening in flocculating fumed colloidal dispersions ofcarbon black and alumina in hydrocarbon media. Shearthickening in these systems is driven not by hydroclus-ter formation but by breakup of dense fractal clustersresulting in an increase in the effective volume frac-tion of the system, as detailed in our previous publica-tion regarding carbon black in tetradecane solvent [18].We consider the effect of varying solvent viscosity andpresent results illustrating that the breakup of clusters
3
is controlled by a critical stress, and not a critical shearrate. We find a strong correlation between the elasticmodulus of gels and the pre-shear stress to which theyare subjected, underpinned by the difference in sus-pension microstructure in the shear thinning vs. shearthickening regimes. These systems display measurableresidual stresses on cessation of flow, and we charac-terize their dynamics as dependent on the pre-shear towhich the system is subjected. The microstructure ofthe shear thickened gels displays a dramatic instability,resulting in the formation of highly anisotropic vortic-ity aligned aggregates on shear flow, studied here inalumina dispersions. These results shed new light over-all on the rheology of fumed colloidal materials and helpexplain some curious phenomena observed in these ma-terials in past reports [19,20].
2 Experimental
Carbon black was obtained from Cabot Corporation.The particular grade used, Vulcan X72R, exists as ≈0.5µm diameter particles with a particle fractal dimen-sion dp
f = 2.2 and bulk density ρcarbon = 1.8 g/cm3.Fumed alumina (SpectrAl 100) was also obtained fromCabot with bulk density ρalumina = 3.7 g/cm3. Tetrade-cane (Aldrich Chemical Co., η = 2.8 cPs, ρ = 0.76g/cm3) and a medium-bodied mineral oil (Aldrich Chem-ical Co., η = 45 cPs, ρ = 0.85 g/cm3) were blended invarious proportions to provide suspending media of tun-able viscosity, according to a logarithmic mixing rule.Under these conditions, i.e. in non-polar solvents, thereis no charge on the surface of the particles and so theyinteract solely via an attractive Van der Waals poten-tial. A concentrated stock of the fumed colloidal parti-cles in the relevant fluid was produced by vortex mixingfor about 2 minutes, sonication for another 10-30 min-utes and vortexing again for 2 minutes. Samples formeasurement were then prepared from the well-mixedstock by dilution, followed by 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 room
noise. 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 pre-shearing sam-ples at the highest shear rate to which they would sub-sequently be exposed, ≈ 1000s−1, followed by repeateddescending and ascending shear rate sweeps. Alterna-tively, repeated shear sweeps were performed, but thefirst ascending sweep data was discarded due to theinconsistent mechanical history involved in handlingand loading the sample into the rheometer. Both meth-ods were found to produce consistent, reproducible flowcurves in good agreement, and data for steady flow ispresented here without additional specification of thepre-shear routine. Equilibration at each data point wasensured by using ≈ 60 s stabilization time, and ≈ 10-15s integration time, with a minimum data density of 10pts./decade. In the case of measurements conducted onthe ARES the data collection time was doubled as twodirections of rotation were used at each data point in or-der to avoid artifacts due to instrument offsets of resid-ual stresses in the samples. It is important to note thatthe mixing performed on samples, although thorough,served only to macroscopically homogenize the materi-als. The settling and heterogeneity introduced by thehandling and loading of the sample necessitates an ap-propriate pre-shear after the sample is loaded to ensureconsistent results. All samples were checked to verifythat edge-fracture, possible in the shear flow of stickysystems, did not occur.
For dynamic measurements, samples were pre-shearedat a fixed rate of 50 s−1 for 30 minutes and then pre-sheared at the particular shear rate of interest againfor 30 minutes before finally being allowed to sit qui-escently for 15 minutes. Following this gelation time,the storage and loss moduli were characterized using afrequency of ω =1 rad/s and strain γ = 0.1%, whichwas within the linear regime as confirmed by dynamicstrain sweeps, conducted at ω = 1 rad/s. Frequencysweeps were conducted in the linear regime using thesame strain, γ = 0.1%.
4
3 Results and Discussion
3.1 Steady Flow Behavior
The flow curves for carbon black and alumina exhibitqualitatively similar features. The samples exhibit tworegimes 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 viscosity,or 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 2-5X increasein viscosity, as shown for data taken using the cone-plate geometry in Figure 1. This will be referred to asthe shear thickening regime. Within the shear thinningregions, both materials exhibit the thixotropy that iscommonly encountered in flocculated gels - stepped in-creases of shear rate result in time-dependent decreasesin the viscosity as the material is broken down in theflow [21]. In the shear thickening regime, suspensionsdisplay a negative thixotropy as the viscosity increaseswith time in response to a stepped increase of shearrate.
Correspondingly, samples display thixotropic loopswith the viscosity on descending rate sweeps above thatat the corresponding shear rate on the ascending arm ofthe flow curve, Figure 2. The departure of the descend-ing sweep from the ascending curve is, notably, at thepeak of the shear thickening regime on the ascendingcurve.
From our data, we extract the critical shear rateson ascending and descending sweeps, γa
c , γdc and cor-
responding critical shear stresses, σac , σd
c that denotethe transitions to and from shear thickening flow. Wefind qualitatively different composition scaling of thecritical stress and rate on the ascending and descend-ing curves, Figure 3. In particular, γa
c ∼ ϕ−1, whereasγd
c ∼ ϕ0. Correspondingly, σac ∼ ϕ1 and σd
c ∼ ϕ2. Forthe peak viscosity, γa
p ∼ ϕ−1, whereas γdp ∼ ϕ0. Ad-
ditionally, σap ∼ ϕ1.5 and σd
p ∼ ϕ2.5. With only 1/3 ofa decade in composition range, the actual value of theexponents cannot be robustly determined. However, itis noteworthy that the sum of the absolute values of theexponents is roughly constant for the ascending and de-scending critical transitions and peaks. This suggests aconsistent 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 examinedthe flow curves across a number of fluid viscosities. The
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Fig. 1 Steady-state flow curves on ascending rate sweeps on car-
bon black (top) and alumina (bottom) suspensions in a 3:1 (byweight) mixture of mineral oil with tetradecane. Data were mea-
sured on ARES-LS1 instrument using a 2◦, 50 mm cone. For
the Alumina samples, the tool surfaces were covered with a grit-ted paper and confirmed the absence of roughness effects in this
regime. Filled symbols are stress, open symbols are viscosity.
critical shear rate varied inversely with the viscosity ofthe solvent, and the data can be rescaled onto a singlecurve using the hydrodynamic stress, σh = γηsolvent,indicating that shear thickening in these systems is con-trolled by a critical stress, as seen from the data of Fig-ure 4.
Along with the viscosity characterization, steadyflow experiments provided measurements of the normalstress behavior. It should be noted that normal stressdifferences in colloidal suspensions are notoriously dif-ficult to measure accurately [22–24], and moreso for di-ulte systems. Full scale deflection of the strain gaugemechanism on most commercial instruments is achievedat O(20N), which for a typical 50 mm cone correspondsto a stress of O(104) Pa whereas the stresses gener-ated by these dilute dispersions are at least 2 ordersof magnitude smaller than that. Further compound-ing the problem is the quadratically increasing inertialcontribution to the normal force at high shear rates,Finertial = −3/20ρ(Rω)2 where ρ is density, R is theradius of the tool and ω is the rotational velocity [25].
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osity
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Fig. 2 Thixotropic loops for carbon black dispersion in a 1:1
tetradecane-mineral oil fluid. Filled symbols are for ascendingsweeps, open symbols for descending runs.
Nonetheless, by making careful corrections to the nor-mal stress signal measured using the ARES LS-1 in-strument and averaging over several runs, we were ableto accurately and reproducibly resolve the first normalstress difference (N1) behavior in the shear thickeningregime of tetradecane dispersions of carbon black, Fig-ure 5. We find that N1 shows a marked upturn at theshear thickening transition followed by a peak and even-tual downturn. The marked upturn, however, was onlyvisible for more concentrated samples, ϕ > 4 wt.%. Thisbehavior is in contrast to the negative first normal stressdifferences that develop on hydrocluster driven thick-ening in concentrated hard sphere systems [26,24]. Thedownturn in the normal stresses at high shear rates doesnot appear to be related to any improper inertial cor-rection of the data. It appears to be coincident withthe peak of the shear thickening response in the moreconcentrated samples, but its origin is currently unex-plained.
In order to understand the origins of the observedrheology, we imaged the microstructures that were presentat different points of the steady flow curve for a dilutealumina dispersion, which permitted reasonably goodoptical contrast, as shown in Figure 6. At low shearrates, in the shear thinning regime, the system con-sists of large “chunks” of the broken gel. In the struc-turing regime, we observe the appearance of vorticityaligned flocs which adopt a log-rolling motion. Theseflocs gradually break down in the shear thinning regimethat follows the structuring regime, eventually forminga collection of densified clusters. Finally, in the shearthickening regime, these dense clusters are broken upand the system consists of small, finely dispersed ag-gregates that dramatically increase the optical density
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Fig. 3 Composition dependence of critical stress and shear rate
at 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.
of the sample. Shear thickening in the dispersion ma-terials is caused by the increase in the effective frac-tion of the particles when large clusters are brokenup and dispersed. These results are in excellent agree-ment with the observed structures in carbon black sys-tems, published previously [18] and are analogous toa “cluster dilation” model advanced to explain a neg-ative thixotropy in concentrated ferric oxide suspen-sions [27]. This mechanism is markedly different thanthe hydrocluster formation which drives thickening innon-flocculating suspensions. It is expressly dictated bythe fractal nature of the particles which allows themto form compact aggregates due to interpenetration asit were, of the highly structured particle surfaces andshould be a general mechanism across many types offumed particles.
While flow curves measured at higher shear rates,γ > 100 s−1, are smooth and of excellent reproducibil-ity, data collected at lower shear rates were less con-
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 hydrody-
namic stress by multiplying by the viscosity of the solvent, pro-ducing superposition of the relative-viscosities, indicating that
shear thickening is controlled by a critical stress.
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Fig. 5 Normal stress data on ascending rate sweeps of carbon
black dispersions in Tetradecane, taken as the difference withrespect to the value at the lowest shear rate, ∆N1(γ) = N1(γ)−N1(γ = 10). Data are averaged over 4 separate runs, and the errorbar represents the standard deviation in the normal stress at eachdata point. Inset shows the viscosity and normal stress behavior
the 6 wt.% sample where the co-location of the shear thickeningtransition and the sharp upturn in N1 is clearly visible.
sistent from run to run, and rather noisy [28]. In somecases for instance, the viscosity was observed to de-crease monotonically with decreasing shear rate. Onepossibility is that wall slip occurs at shear rates far be-low 100 s−1, as it is well known to occur in the yieldingof sticky colloidal systems such as these [29]. Transientstress measurements on step-wise reduction of shearrates at different gaps in the parallel-plate geometry
Fig. 6 Microstructure at different shear rates for a 2 wt.% Alu-mina dispersion in 3:1 oil:tetradecane fluid using a 1◦ 50 mm
cone. Field of view is 1 cm x 1 cm
however, suggest that wall slip is not a large factor[30,31] and that homogeneous flow is achieved at theshear rates considered, Figure 7. Further, the use oftools covered by roughened surfaces was found to pre-serve features of the flow curve, and did not eliminatethe unusually low and decreasing viscosities sometimesobserved on decreasing shear rates in the low shear rateregime.
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Fig. 7 Transient stress measurements on 4 wt.% carbon black
in tetradecane. Data are for step-wise reductions in shear ratefrom 100 to 2.5 and 0.5 s−1 gaps of 400, 600 and 800 µm in the
parallel plate geometry.
It appears instead that sedimentation of the par-ticles at low shear rates is responsible for the lack ofconsistency of the data. The use of sequential ascend-ing and descending runs with long equilibration de-lays allows substantial time for sedimentation to oc-cur, and is exacerbated at low shear rates where shear-
7
induced re-suspension cannot occur. Indeed, consistentsmooth flow curve data were only obtained when theCouette geometry was used, as shown in Figure 8. Inthis geometry, the effects of sedimentation are mitigatedby the large height (34 mm) of the tool. We observegood agreement between ascending and descending flowcurves in the low shear rate limit, all the way down to10−3 s−1 and the small increases in viscosity in thestructuring regime are consistently present. This is incontrast to data measured in the cone-plate geometrywhere the effects of sedimentation are disproportion-ately amplified. It should be stressed that these data inthe Couette were measured on the ARES-LS1 instru-ment using 2 directions per data point, with equilibra-tion and torque-averaging times of 60 and 10 secondsper direction, resulting in a high degree of confidencethat these are equilibrium flow curves.
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Fig. 8 Ascending rate flow curves for carbon black dispersionsin 3:1 oil:tetradecane fluid measured in the Couette geometry onARES-LS1 instrument.
3.2 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 [32], nanotube suspensions [33]and attractive emulsion droplets [34]. Their appear-ance is usually associated with a negative first normalstress difference. The observation of these structures insteady-state flow in carbon black and alumina disper-sions is thus not surprising, although we did not observeconsistently measurable negative normal stress differ-ences in the structuring regime of our system. How-ever the rheology associated with their formation here
appears to be unique as their appearance is concur-rent 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 remains un-known, 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 circu-lar cross section, implies that the flocs span the toolgap. This would give rise to lubrication stresses in theflow of fluid in the small spaces between the tool andfloc surfaces 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 ofFigure 6, the structures are not present beyond a cer-tain radial distance from the center of the cone. Whilethey are present as steady state structures, they alsoform strikingly and then gradually breakup as a tran-sient response in the deformation of shear thickenedsuspensions of at a fixed shear rate or on quenches fromhigh shear rate thickening flow into the shear thinningregime of the flow curve [28]. This is shown in Figure9 for a 2% alumina suspension. Here, as also reportedfor a flocculated nanotube system [33], the structureformation is strongly parameterized by the gap size. Inthe current system, their formation may be governed bythe rapid aggregation of the small finely dispersed clus-ters produced in the shear thickening flow. Given theR3 dependence of torque on radial distance in the cone-plate geometry, [35], the contribution from the log-likeflocs to the apparent viscosity is small as the structuresare destabilized at larger radial distances from the conecenter. The use of the parallel plate geometry suffersfrom the radial dependence of the shear rate and so therheology of these systems would best be studied as afunction of gap size in the Couette geometry.
3.3 Gel Elasticity
Given the differences observed in suspension microstruc-ture between regimes delineated by γc, the critical shearthickening transition, it can be expected that the me-chanical properties of the gels formed after cessationof pre-shear should differ as well. Dynamic measure-ments were carried out in the cone-plate geometry onthe MCR instrument on carbon black suspensions in3:1 blends of mineral oil with tetradecane. The sam-ples were found to exhibit either a very small or im-measurable frequency dependence of the elastic mod-ulus, Figure 10. The strain dependence of the moduli
8
Fig. 9 Vorticity aligned flocs in rolling flow in 2% alumina in 50
mm, 1◦ cone-plate. Sample was quenched from from γ = 1000s−1
to 10 s−1. Images are taken from (a) through (d) at 0, 4, 18 and78 seconds after the rate quench. Field of view is 1 cm x 1 cm.
is characteristic of soft glassy materials, with a pro-nounced upturn in the loss modulus, G′′, on yielding ofthe network, Figure 11. The yield strains appear to in-crease for samples produced at higher shear rates, butthe uncertainty in accurately extracting a yield strainmakes it difficult to asses this quantitatively. At verylarge strains, γ ∼ 100-1000%, there are small increasesin the two moduli. This corresponds with the structur-ing regime observed in steady flow experiments and thedynamic shear rate, γω, is similar to the steady flowrate in the steady measurements for these samples. Ag-ing was not observed in these samples and the modulusremained steady over ≈ the period of observation (1hour), including a 15 minute sweep performed to checkthe time evolution of the modulus, Figure 12.
Fig. 12 Storage and loss moduli of gels remain constant duringmeasurement after 30 minute quiescent gelation. 5% carbon black
in 3:1 oil:tetradecane fluid.
When considered as a function of the pre-shear stress,the elastic moduli display a marked power-law depen-dence for samples sheared above the shear thickeningtransition, scaling roughly as G′ ∼ σ2, Figure 13. Thestrong dependence of the modulus on the pre-stress 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 at theshortest measurable times after cessation of shear, t ≤5s. The gel modulus can then be viewed as simply acohesive energy density [36] that results from the con-tact between clusters, the dimensions of which are setby the magnitude of the pre-shear 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
df−1c
where df is the fractal dimension of the cluster. The at-
9
tractive interaction, U , between particles of dimensiona balances this force such that the dependence of thecluster size on the shear stress is as given in Equation2.
Rdf−3c ∼ 4πσa
U(2)
The elastic modulus is then given as G′ ∼ ν U whereν is the number density of clusters, ν ∼ ϕ/R
dfc . Thus,
G′ ∼ φσdf
3−df (3)
Assuming a value of 2 for the fractal dimension, thissuggests that G′ ∼ σ2φ. Replotting the data in thesereduced terms produces a good agreement between oursimple scaling model and the data, Figure 13. The datais in rough agreement with experimental data on thatdemonstrates that the number of particles in a flocscales roughly as Nc ∼ σ−1 [37]. The choice of fractaldimension for the gel is reasonable, though somewhatarbitrary. Previous data on carbon black gels in 100%tetradecane fluid were better fit assuming a fractal di-mension of 1.8. Curiously, the maximum elastic moduliof those gels were a factor of ≈ 3X higher than thatachieved in the more viscous 3:1 oil: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 reasoningis in line with our findings. Our experiments have pro-vided direct imaging that supports these assertions, andour scaling model accounts well for the data. The strongdependence of the modulus on the pre-shear stress callsinto question the utility of the composition scaling ofthe modulus, G′ ∼ ϕα as a metric for the interactionstrength in fumed colloidal systems. The standard pro-cedure in experimental investigations is to apply a highrate pre-shear to ‘initialize’ the system, and then tomeasure the modulus but our data suggest that a moreaccurate approach would involve applying a composi-tion scaled stress as the pre-shear.
The gelation that occurs on cessation of shear flowcan be thought of as a “mechanical quench” of the
1 10 1001
10
100
1000
1 10 1001
10
100
1000
1 10 1001
10
100
1000
Preshear Stress (Pa)
5%
0.1 1 101
10
100
1000
0.1 1 101
10
100
1000
0.1 1 101
10
100
1000
5%
3%
G' (
Pa)
2
4%
3%
G' (
Pa)
4%
Fig. 13 Elastic modulus of gels as function of pre-shear stress.
Inset: data re-scaled according to Equation 3
dispersion 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,gelation was very rapid, and samples display a finitemodulus at the earliest measurable times (∼ 5 s) af-ter cessation of shear. Such near-immediate establish-ment of a percolated network after stopping flow hasalso been observed in silica gels [19] as well as suspen-sions of ferric-oxide [38]. This sol-gel transition can beexpected to involve the display of residual or internalstresses as the gel network is deformed with respect toan equilibrium relaxed configuration. Such a stress, σi,should be proportional to the modulus of the system,through the amount of the out-of-equilibrium strain, γd,and should relax gradually in time [39]. The concept ofsuch a “trapped stress” has been invoked to accountfor the strange dynamics observed in the aging of someglassy systems [40,41]. Measurement of stress decays af-ter stress-jumps enable a decoupling of the viscous andelastic stress components and have been made on a va-riety of systems in the short time limit (≤ 100 s) [31,42]. Using the fast-feedback control loop of our MCRinstrument, we successfully measured these stresses af-ter a fast quench (0.1 s duration) to the stationary stateout to 103 s after cessation of shear, Figure 14. Mea-surements are conducted on carbon black dispersionsin 3:1 oil:tetradecane fluids that have been subjectedto different pre-shears. We find a weak power law de-cay, 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 pre-shear stress. Reversing the direc-tion of the pre-shear correspondingly switches the signof the residual stress. The data are in excellent agree-ment with the time-dependence observed for suspen-sions of carbon black in tetradecane measured on an
10
AR-G2 instrument [18], suggesting that this approachcan be satisfactorily used if very short time data is notrequired. The direct proportionality expected betweenthe residual stress and the modulus is confirmed bythese measurements, Figure 15.
A standard stress relaxation measurement in the lin-ear regime of course yields such a direct correspondencebetween σ and G′ as well, but in the current experi-ment, the stresses that are relaxing are not measuredin response to a step-strain on an equilibrium structure,but arise solely on the rapid rate quench of the system.These stresses are of non-trivial magnitude. For exam-ple, for gels prepared at 700 s−1, the residual stressjust after cessation of shear is about 3 Pa, Figure 14whereas the yield stress measured in the strain sweep,Figure 11 is only about 2.5 Pa. The presence of resid-ual stresses in these systems should be taken into ac-count in designing stress relaxation experiments as thestress resulting from the applied step-strain in the ex-periment 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 MCR in-strument is particularly well suited to these studies as itpermits specification of the braking time over which thesample is quenched from the flowing to the stationarystate.
Fig. 14 Residual stresses measured on cessation of shear fromdifferent shear rates.
1E-3 0.01 0.1 1 101
10
100
1000
1E-3 0.01 0.1 1 101
10
100
1000
1E-3 0.01 0.1 1 101
10
100
1000
G' (
Pa)
Residual Stress (Pa)
4% 5%
3%
Fig. 15 Modulus of gels as a function of the magnitude of the
residual stress, measured just after cessation of shear.
4 Conclusion
The observation of shear thickening in flocculating sys-tems is a rather surprising result. It is a modest effectrelative to the shear thickening and jamming that occurin dense hard sphere systems. The data are highly re-producible and indicate that the transition is controlledby a critical stress and is concurrent with an upturnin the first normal stress difference. Shear thickeninghere can be well understood in terms of the breakdownof clusters and the resultant increase in the effectivevolume fraction of particles in the fluid. This effect isexactly as a result of the unique fractal morphology ofthe colloidal particle concerned and so the mechanismis not relevant for Euclidean solid particles. The strongdependence of the gel modulus on the pre-shear stressis properly rationalized in terms of the cluster num-ber density that results from the pre-shear flow. Thishas important implications in applications where fumedcolloidal particles are used as fillers in fluids, includingpolymers, as the elasticity of the fluid can be stronglyinfluenced by the mixing conditions. Surfactants can beused to sterically modulate the attractive interactionbetween particles, and this would be expected to alterthe shear thickening behavior. In preliminary experi-ments, we have observed that the critical shear thick-ening transition shifts to lower stresses, as would be ex-pected for decreased floc strength due to the presence ofthe surfactant. Finally, the dynamics of the rolling mo-tion of the log-like flocs is of interest. A careful analysismay yield insight into rheology observed in the struc-turing regime.
Acknowledgements The authors would like to acknowledge
Dr. Chanjoong Kim and Prof. David Weitz of Harvard Univer-
11
sity for fruitful discussions, and funding from the National ScienceFoundation under grant number CBET-0828905.
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