Shear thickening of cornstarch suspensions

Transcription

1 Shear thikening of ornstarh suspensions Abdoulaye Fall, François Bertrand, Guillaume Ovarlez, Daniel Bonn To ite this version: Abdoulaye Fall, François Bertrand, Guillaume Ovarlez, Daniel Bonn. Shear thikening of ornstarh suspensions. Journal of rheology, Amerian Institute of Physis, 202, 56, pp <0.22/ >. <hal > HAL Id: hal Submitted on 8 Jun 202 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Shear Thikening of Cornstarh Suspensions Abdoulaye Fall,2, François Bertrand 2, Guillaume Ovarlez 2 and Daniel Bonn,3 Van der Waals Zeeman Institute, University of Amsterdam, Valkenierstraat 65, 08XE Amsterdam, The Netherlands 2 Université Paris Est, Laboratoire Navier, UMR 8205 CNRS-ENPC-IFSTTAR, 2 Allée Kepler Champs sur Marne, Frane 3 Laboratoire de Physique Statistique, Eole Normale Supérieure, 24 Rue Lhomond 7523Paris Cedex 05, Frane We study the rheology of ornstarh suspensions, a non Brownian partile system that exhibits disontinuous shear thikening. Using Magneti Resonane Imaging (MRI), the loal properties of the flow are obtained by the determination of loal veloity profiles and onentrations in a Couette ell. For low rotational rates, we observe shear loalization harateristi of yield stress fluids. When the overall shear rate is inreased, the width of the sheared region inreases. The disontinuous shear thikening is found to set in at the end of this shear loalization regime when all of the fluid is sheared: the existene of a non-flowing region thus seems to prevent or delay shear thikening. Marosopi observations using different measurement geometries show that the smaller the gap of the shear ell, the lower the shear rate at whih shear thikening sets in. We thus propose that the disontinuous shear thikening of ornstarh suspensions is a onsequene of dilatany: the system under flow attempts to dilate but instead undergoes a jamming transition beause it is onfined. This proposition is onfirmed by an independent measurement of the dilation of the suspension as a funtion of the shear rate. It is also explains the MRI observations: when flow is loalized, the non-flowing region plays the role of a dilatany reservoir whih allows the material to be sheared without jamming. I. Introdution: Shear thikening has been observed for a wide variety of suspensions [Barnes, (989)]. The phenomenon is frequently enountered during the proessing of onentrated dispersions in various industries where it has a strong impat on energy onsumption. Shear thikening an be defined as an inrease in the steady-state shear visosity η of a fluid with the shear rate γ when the latter exeeds some ritial value γ. The detailed mehanism of shearthikening is still under debate [see e.g. Boersma et al. (990), Franks et al. (2000), Hoffman (972), Maias et al. (2003), Laun et al (992), Hoffman (974), Foss and Brady (2000), Akerson (990) and Chen et al (994). The majority of investigations of shear thikening were onduted on olloidal suspensions (Boersma et al. (990), Franks et al. (2000), Hoffman (972)]. Bender and Wagner (996) and Maranzano and Wagner (2002) attribute the phenomenon to the shear-indued formation of hydrodynami lusters transient onentration flutuations that are driven and sustained by the applied shear field. The visosity rise is ontinuous at low volume frations, but an also be disontinuous at higher ones [Maias et al. (2003), Laun et al (992), O'Brien and Makay (2000), Bertrand et al. (2002)], probably beause of aggregation of hydrolusters reating a jammed network [Hoffman (974), Foss and Brady (2000)]. In the latter ase, the lustered shear thikened state may be a long lived metastable state haraterized by a large yield stress, as shown by Cates et al. (2005). In the piture of Bender and Wagner (996), the formation of flow indued hydrolusters results in an inreased dissipation of energy and, onsequently, the visosity inreases. The ommonly aepted piture for shear thikening in Brownian suspensions is now the formation of shear-indued hydrolusters [see e.g. the reent summary by Wagner and Brady (2009)]. This results in large hydrodynami stresses in rapid flowing suspensions, and there is simulation and other evidene that these are the dominant stresses in the shear-thikened regime [Bender and Wagner (996), Phung et al. (996)]. However, some of these simulations are limited to relatively modest volume frations ( ϕ ); experiments at muh higher onentrations suggest that instead a thermodynami mehanism may take plae [O Brien and Makay (2000)]. In any ase it seems highly unlikely that the stati jamming observed by Bertrand et al. (2002) is due to hydrodynami interations alone, sine in the persisting solid phase there is no marosopi flow to provide those interations. Also, as far as we know, hydrodynami models for shear thikening offer no immediate explanation of nonmonotoni regions of the flow urve [Holmes et al. (2005)]. Thus it seems lear that mehanisms other than pure hydrodynamis are at work in suffiiently dense shear-thikening suspensions [O Brien and Makay (2000), Bertrand et al. (2002)]. Indeed, it has already been emphasised [Ball and Melrose (995), Melrose and Ball (995; 2004a,b)] that deviations from pure lubriation fores an dominate the physis of any hydrodynamially lustered state [Holmes et al. (2005)]. Shear thikening is also observed in non-brownian suspensions with larger partile sizes [Williamson and Hekert (93); Fall et al. (2008; 200); van der Werff and de Kruif (989); Sellitto and Kurhan (2005) Berthier, et al (2000); Brown and Jaeger (2009; 200)]; here the mehanisms at play are less lear. Reently, Brown and Jaeger (2009) have shown a transition between a shear thinning and a shear thikening regime where the shear thikening

3 / n behavior is haraterized by σ γ. For their measurements n 0. 5 far from jamming and n ontinuously dereases towards 0 upon approahing jamming (that is, the volume fration ϕ ϕ ). However Fall et al. (200), in max a non-brownian partile suspensions similar that those of Brown and Jaeger show that the intrinsi behavior (from loal MRI measurements) shows only visous (n = ) or granular saling (with n = 0.5) and that shear thikening simply orresponds to the transition between the two regimes. Note that, we all this regime granular; however it is important to realize that the saling law with n = 0.5 an orrespond to any inertial flow, inluding that of Newtonian fluids, and is not neessarily a granular saling. The MRI data showed that in steady-state suh systems are heterogeneous due to partile migration, and that onsequently the marosopi stress strain rate relationship annot be diretly related to the loal onstitutive behavior and thus in partiular to the shear thikening. Here, we ompare loal and global measurements for what is perhaps the best known example of a shear thikening suspension: ornstarh partiles suspended in water. We show that the shear thikening an in fat be viewed as a re entrant solid transition in this system: (i) at rest the material is solid beause it has a (small) yield stress; (ii) for low shear rates, shear banding (loalization) ours, and the flowing shear band grows with inreasing shear rate, the shear thus liquefies the material; (iii) shear thikening happens at the end of the loalization regime, where all of the material flows, subsequently it suddenly beomes solid again. In addition, (iv) we find a pronouned dependene of the ritial shear rate for the onset of shear thikening on the gap of the measurement geometry, whih an be explained by the tendeny of the sheared system to dilate. This is onfirmed by an independent measurement of the dilation of the suspension as a funtion of the shear rate. It also explains the MRI observations: when flow is loalized, the non-flowing region plays the role of a dilatany reservoir whih allows the material to be sheared without jamming. This paper follows up on our earlier work on shear thikening of ornstarh [Fall et al., (2008)], but is muh more detailed in that here we present also the MRI measurements of the onentration, more detailed measurements of the veloity profiles, plate-plate measurements, osillation measurements and more detailed measurements of the variation of the gap of the plate-plate ell under an imposed normal stress. In order for these new data to be omprehensible we do have to repeat some of the earlier data and disussion. In this way we obtain a more omplete piture of the shear thikening behavior. II. Materials and methods The ornstarh partiles (from Sigma Aldrih) are relatively monodisperse partiles with, however, irregular shapes [Figure ]. Suspensions are prepared by mixing the ornstarh with a 55 wt% solution of CsCl in demineralized water. The CsCl allows one to perfetly math the solvent and partile densities [Merkt et al. (2004)]. We study suspensions of volume fration ranging between 38% and 46%, and fous here mainly on the behavior of a 44% ornstarh suspension that is representative of the rest. The effet of hanging the volume fration will be disussed in detail in Se.III.6. Figure : Mirograph of the ornstarh partiles. Experiments are arried out with a vane-in-up (inner ylinder radius Ri = 2.5 mm, outer ylinder radius Re = 8.5 mm, height H = 45 mm) or parallel plate geometry on a ommerial rheometer (Bohlin C-VOR 200) that imposes either the torque or the rotational veloity (with a torque feedbak). The vane geometry is equivalent to a ylinder with a rough lateral surfae whih redues wall slip [Larson (999)]. The inside of the up is also overed with the granular partiles using double sided adhesive tape. For the parallel plate geometry, the upper plate is of 40 mm diameter; both plates are roughened. Veloity profiles in the flowing sample were obtained with a veloity ontrolled magneti resonane imaging (MRI) rheometer from whih we diretly get the loal veloity distribution in a Couette geometry with a gap of.85 m. We investigated the stationary flows for inner ylinder rotational veloity Ω ranging between 0.2 and 0 rpm, orresponding to overall shear rates between 0.04 and 2.35 s - [Raynaud et al. (2002), Rodts et al. 2

4 (2005), Bonn et al.(2008)]. All the measurements were done in the same laboratory with a ontrolled humidity of 40%; ornstarh is likely to take up some water from the atmosphere. III. Experimental results. Typial marosopi behavior Let us first present the typial behavior observed when shearing a ornstarh suspension (Figure 2) Visosity (Pa.s) 00 0 E Shear rate (s - ) Figure 2: Apparent visosity vs. shear rate when performing a step stress test in a vane in up geometry. η are similar for all measured volume frations: at low stresses, η dereases with inreasing applied stress, refleting shearthinning behavior. As the stress inreases further, η inreases, refleting shear-thikening behavior, and then reahes a plateau. This abrupt inrease in visosity observed is harateristi of disontinuous shear thikening. To better understand this behavior, we performed MRI measurements in a Couette geometry. The visous properties at higher stresses were measured with step stress tests. The basi trends in ( γ ) 2. Loal rheology: veloity and onentration profile measurements In Figure 3, we plot the dimensionless veloity profiles for the steady flows of a ornstarh suspension, for various rotational veloities ranging from 0.2 to 9 rpm. V(r)/V i,0 0,8 0,6 0,4 0,2 0,0 Veloity (m/s) 0,0020 0,005 0,000 0,0005 0,0000 γ yield 4,4 4,8 5,2 5,6 Radius (m) 4,4 4,8 5,2 5,6 6,0 Radius (m) Ω (rpm) ,8 0,6 0,4 0,2 Figure 3: Dimensionless veloity profile in the gap obtained by MRI measurements. Insert: the shear rate at the interfae between sheared and unsheared regions is given by the slope of the veloity profile at that point, taken of ourse in the moving part of the material. 3

5 The MRI measurements of the veloity profiles first, show that there is no slip to within the experimental unertainty. The data shown is normalized on the speed of the rotating inner ylinder, and extrapolates to unity, showing that slip is negligible. Seond, the flowing part of the sample oupies only a small fration of the gap at low rotation veloities: we observe shear loalization. The veloity profiles are omposed of two regions: the part lose to the inner ylinder is moving, and the rest is not. For the lower rotation speeds, sine the part of the material that does not move is subjeted to a stress, this implies that the suspension has a yield stress. The yield stress an be determined from the ritial radius r at whih the flow stops: the shear stress at a given radius r as a funtion of the applied torque T and fluid height H follows from momentum balane, and thus the yield stress at r follows immediately as 2 σ y = T / 2πHr. The yield stress turns out to be on the order of 0.3 Pa. Although it seems obvious that onentrated suspensions that show shear thikening also have a yield stress, we have not found literature omparing the prethikening flow behavior to a Bingham model as we do here, with the exeption of the reent work of Brown and Jaeger (Brown and Jaeger, 200) where the pre-thikening behavior was ompared to a Hershel-Bulkley model. This is probably due to the fat that the yield stress is low. We an detet it relatively easily here beause we use the MRI data. A striking observation is that the shear rate at the interfae between sheared and unsheared regions is different from zero, in ontrast with what is observed in simple yield stress fluids [Bonn and Denn (2009), Moller et al. (2009), Ovarlez et al., 2008, 200]. We indeed observe that the slope of the veloity profile at that point [Figure 3, Insert] is equal to 0.2s -. This implies that below γ yield 0.2s there is no stable flow. The existene of a ritial shear rate for yield stress materials has been disussed previously in detail, and requires that the system be (slightly) thixotropi. The ritial shear rate assoiated with the yielding of thixotropi materials has been disussed in detail elsewhere [Moller et al. (2006), (2008), (2009)]. In our ase, this is likely to be due to ompetition between slight sedimentation or reaming, i.e., a density mathing that is not perfet and shear-indued resuspension [Coussot et al., (2002a, b); Fall et al. (2009)]. Upon inreasing the rotation rate, a larger part of the fluid is sheared, and for the highest rotation speeds the sheared region oupies the entire gap. We are unable to go to higher rotation rates in the MRI sine the shear thikening sets in immediately when the shear band oupies the entire gap of the Couette ell, and when it does the motor of the rheometer is no longer suffiiently strong to rotate the inner ylinder: shear thikening is observed as an abrupt inrease of the measured torque on the rotation axis. We therefore see the onset of shear thikening at the first shear rate for whih all of the material is sheared; we will show below that this is likely to happen beause the presene of a non-flowing region delays and attenuates the shear thikening. Shear stress (Pa) Ω (rpm) , 0 Shear rate (s - ) Thikening Conentration (-) 0,50 0,45 0,40 0,35 0,30 0,25 4,4 4,8 5,2 5,6 6,0 Radius (m) Ω (rpm) Figure 4: (a) Loal shear stress as a funtion the loal shear rate. The line is a fit to the Bingham model: σ=σ + kγ y with σ y 0. 35Pa, k = 0. 30Pa. s. b) Loal onentration profiles in the Couette gap geometry for a ornstarh suspension sheared at various rotational veloities ranging from 3 to 9 rpm. From the veloity profiles v (r), we an determine the onstitutive behavior. In the flowing part, the loal shear rate within the Couette gap an be given as v v γ () r =. By ombining γ () r with the loal shear stress r r σ r = T / 2πHr at eah radial position r, for various rotational veloities Ω, one obtains the onstitutive equation () 2 4

6 σ ( γ ) of the fluid. The loally measured onstitutive equation of the ornstarh suspension is shown in Figure 4 (a). It is onsistent with the observation that σ 0. Pa and γ 0.2s. y 3 yield The MRI also allows us to measure the partile onentration; to within the experimental unertainty of ±0.2% in volume fration the partile onentration is homogeneous throughout the gap. Thus, during the flow and at the onset of shear thikening, the suspension remains homogeneous within the gap. As a result, no migration of partiles is observed to the limit of the transition to the shear thikening regime. This is perhaps surprising in the light of reent measurements of shear thikening in suspensions of spherial partiles, where shear thikening is always aompanied by partile migration (Fall et al, 200). In Couette flows, the onsequene of migration is an exess of partiles near the outer ylinder [Leighton and Arivos (987a, b), Ovarlez et al. (2006), Huang and Bonn (2007)]. However our measurements do not ompletely rule out partile migration; we estimate, from the MRI data that the maximum gradient in partile onentration, if any, is around 0.%. This is indeed very small when ompared to migration for suspensions of spherial partiles, where gradients of several perents are observed in the same MRI Couette ell. The ritial shear rate is due to the existene of a yield stress and is in priniple deoupled from the onset of shear thikening. However, in the Couette geometry used for the MRI experiments, the existene of a yield stress makes that part of the material flows, and another part does not beause the stress it is subjeted to is smaller than the yield stress. As shown by the plate-plate presented below, the existene of a non-flowing region influenes the onset of shear thikening, and so the ritial shear rate is indiretly oupled to the shear thikening phenomenon. To the ontrary, for the ritial stress both the dilation and the measurements as a funtion of volume fration show that the onset of shear thikening is diretly dependent on the stress. The emergene of shear thikening at the end of the loalization regime then suggests that the non-flowing part plays an important role in the shear thikening. To assess what this role is, lassial rheology measurements were onduted. 3. Role of a dead zone In order to investigate in detail the influene of the non-flowing region in the Couette ell on the observed shear thikening, parallel plate geometry is used. This geometry has the additional advantage that there is no reservoir of partiles present, as is the ase of the Couette. If we need to, a non-flowing region an be reated in this geometry by simply leaving a few milliliters of paste around the gap and in ontat with the sample between the plates [Figure 5]. In addition this geometry allows us to measure the normal stresses. Figure 5: example of a non flowing region or dead zone in parallel plate geometry. The rheometer measures a torque T and a rotation rate ω, whih are related to the stress and shear rate at the 3 edge r = R of the sample by σ = 3T / 2πr and γ = 2πr ω / b, with R the plate radius and b the gap size. In this ase the shear rate being up and shear thikening is favored. The visosity and normal stresses are measured with a gap size of 0.8 mm. 5

7 Normal stress σ N (Pa) ,0 0, 0 00 Shear rate (s - ) Visosity (Pa.s) 000 Without surplus With surplus E-3 0,0 0, 0 00 Shear rate (s - ) Figure 6: Role of surplus of paste on the shear thikening transition: Evolution of normal stress and visosity as a funtion of the applied shear rate. Figure 6 shows the evolution of visosity and normal stress with the applied shear rate. For low shear rates, a typial behavior σ / γ of a shear thinning fluid is observed. At a ertain shear rate, a very abrupt inrease in visosity is observed. However, this abrupt inrease in visosity is only observed when the surplus of paste around the plates is arefully removed. If a few milliliters of suspension are left on the bottom plate in ontat with the paste between the two plates, the shear thikening is muh attenuated: there is no abrupt inrease in visosity. Defining the ritial shear rate as the first shear rate for whih the apparent visosity goes up, the surplus of paste inreases the ritial shear rate very signifiantly. We also observe that the shear thikening is aompanied by the emergene of large normal stresses. For low shear rates, the normal stresses are very small. However, from the ritial shear rate on, grows and beomes very important. The ritial shear rate for whih normal stresses appear is very similar to that for whih the visosity inreases. We note also that if a surplus of paste is left around, the shear thikening is aompanied by muh lower normal stresses. These results provide a possible explanation for the MRI experiments in whih shear thikening only happens when all of the material flows in the Couette geometry. When only part of the material flows, the dead zone plays a role analogous to the surplus in the parallel plate experiments, whih delays and weakens the shear thikening, as is observed here in the parallel plate geometry. At the end of the shear loalization regime, the Couette system is suddenly analogous to the parallel plate geometry without surplus; the ritial shear rate is thus suddenly lowered and the material jams. In onlusion, it seems evident that, indeed, the presene of a non-flowing region delays and attenuates the shear thikening. We an note that these findings are onsistent with a piture that hydrolusters formation leads to disontinuous shear thikening and jamming when the hydrolusters perolate the struture [Wagner and Brady, (2009); Cates et al (2005)]. 4. Dilation effet These data suggest that, in both measurement geometries, the dead zone plays the role of a reservoir of dilation whih helps the material to flow without jamming. Indeed, the prinipal information obtained from the normal stress measurement is their on-off behavior, whih is quantitatively linked with the onset shear rate of shear thikening, as was verified by studying different volume frations [Fall et al. (2008)]. The normal stresses are reminisent of the shear indued dilatany of dry granular matter: when sheared, it will dilate in the normal diretion of the veloity gradient. Dilatany is a diret onsequene of ollisions between the grains: to aommodate the flow, the grains have to roll over eah other in the gradient diretion, and hene the material will tend to dilate in this diretion. However, in our system without a surplus, the grains are onfined, both between the plates and in the solvent. The latter provides a onfining pressure that is mainly due to the surfae tension of the solvent, making it impossible to remove grains from the suspension. As suggested by Cates et al (2005) and Fall et al. (2008), the onfinement pressure assoiated with this should be on the order of the surfae tension over the grain size, P = γ / R 7000Pa. This onfinement pressure is in the same order of magnitude as the typial normal stresses measured near the onset of shear thikening [Figure 6]. In addition, this gives a maximal dilation that is on the order of partile diameter ( 20μm ); ompared to the radius of the parallel plate ell this gives a maximum dilation of about 0.%, too small to be deteted by our MRI density measurements. 6

8 In dilatany [Reynolds (885)], the volume of a olletion of partiles must inrease upon shearing to enable flow. This has been suggested as a possible mehanism for jamming in onentrated olloidal suspensions [Cates et al. (2005)]. Dilation within a fixed volume of suspending liquid involves the formation of fore transmitting hydrolusters, whose growth eventually auses partiles to enounter the air-liquid interfae. This generates large apillary fores at the free surfae, whih an then balane the normal inter-partile fores and resist further motion. The partiles an thus form spanning hydrolusters in lose ontat, jamming the sample. This may then frature into millimetre-sale granules [Cates et al. (2005)]. Jamming of olloids is also seen in pipe and hannel flows [Isa et al. (2009); Haw (2004)]; here free surfaes are not present. As the upper plate is retrated, a filament forms whih narrows and eventually breaks. An elongational flow, in ontrast to more onventional shear and pipe geometries, therefore implies an inrease of the interfaial area during flow. Although purely elongational flow an be ahieved by exponential plate separation, a onstant separation speed is loser to fiber-spinning and other industrial proesses. In these, a purely tensile loading evokes a mixed flow ombining elongation and shear. Reent studies [Bishoff White et al. (200)] have demonstrated that extensional rheometry an suessfully be performed on olloidal suspensions. Bishoff White et al measured the tensile stresses of a (φ ~ 0.355) ornstarh solution. They observed a flowable filament at low extension rates but beyond this found a transition to brittle frature. The interations in this system are poorly haraterized but learly attrative (see Figure of [Bishoff White et al. (200)]), presumably due to strong van der Waals fores. Suh interations ould strongly influene the flow behavior as they do in strongly aggregating olloids at lower densities [Bishoff White et al. (200)]. We further investigate the role of dilation by doing osillatory rheology (Figure 7). We find that, the material shows a linear elasti behaviorg 0 G at low shear stresses, i.e. it behaves as a "solid" material. Inreasing the stress, a solid-liquid transition is found haraterized by a yield stress at whih G' G" ; then the suspension begins to flow and G ' < G". These observations are onsistent with the behavior observed in steady shear (see above). Inreasing the applied stress even more, the liquid regime quikly ends by the shear thikening of the system (haraterized by an abrupt inrease of G and G ), for a ritial shear strain γ G'; G" (Pa) 00 "Solid" "Liquid" "Solid" 0 0, 0,0 E-3 G' G" E Shear strain γ (-) (a) Critial shear strain γ (-),6,4,2,0 0,8 0,6 (b) 0,4 0,8,2,6 2,0 Gap (mm) f (Hz) Figure 7: (a) Elasti modulus (G') and loss modulus (G") in the 44% volume fration suspension as a funtion of shear strain for an imposed shear stress (0.00 to 00 Pa) at Hz in a vane geometry. (b) Critial strain vs. gap and frequeny It is lear from Figure 7 that the nonlinearity between stress amplitude and strain amplitude beomes more and more signifiant as the stress amplitude inreases. Moreover, the appearane of nonlinear behavior is aompanied by non-sinusoidal responses in the osillatory shear experiments. As the higher harmoni signals emerge, the material funtions suh as G and G lose their original physial meaning. Considerable efforts [Yu et al. 2009; Cho et al. 2005; Ewoldt et al. 2008; Klein et al. 2007] have been made towards obtaining useful and desirable material information from suh LAOS (Large Amplitude Osillatory) experiments. For our purposes, the nonlinear osillatory experiments are useful for haraterizing the onset of shear thikening as well for determining the time sales required to generate the shear thikening response. Indeed, in the osillatory experiments, the first natural interpretation of the observation of shear thikening would be that the shear rate γ = 2πγf applied during the osillations is equal to the ritial shear rate γ observed in the ontinuous shear experiments when γ γ. With a frequeny f =Hz, the ritial osillatory shear rate at the onset of thikening is 6.5 s -, a value 3 times higher than the ritial shear rate observed during ontinuous shear. To understand this differene, we have performed the same experiments for several frequenies. We observe, in Figure 7(b), that the ritial shear strain γ at the onset of thikening is onstant; it is 7

9 also independent of the gap sized, in ontrast with γ (see below). The onlusion is that γ is indeed the relevant physial quantity in the osillatory experiments. This observation means that, even for high shear rates, thikening annot take plae if a two neighboring grains did not experiene a relative motion of order of partile diameter; this is onsistent with the major role of dilation evidened above. 5. Confinement effet It is therefore tempting to see whether the shear thikening phenomenon itself an be attributed entirely to the onfinement: if the ornstarh is onfined in suh a way that the grains annot roll over eah other, this ould in priniple lead to an abrupt jamming of the system. Experimentally, instead of setting the gap size in the rheometer for a given experiment, we an impose the normal stress and vary the gap size in order to reah a target value of the normal stress. If this is done for different shear rates, and the target value for the normal stress is taken to be zero, we an obtain the dependene of the gap variation on shear rate. A typial measurement is shown in Figure 8 (a), where we impose a onstant shear rate and measure the gap and visosity as a funtion of time. This shear rate and initial gap ombination are beyond the shearthikening transition in shear rate, and thus the visosity (shear stress) starts to strongly inrease, as do the normal stresses. The latter then leads, through a feedbak loop, to an inrease in the gap, allowing the system to dilate, until the shear thikening disappears altogether: the shear stress is bak to low values (less than 0 Pa). This unambiguously demonstrates that the shear thikening is a dilation effet, and that taking away a onfining fator makes the thikening disappear altogether. Normal stress (Pa) Normal stress Gap size Measurement time (s) Gap size (mm) Shear Stress (Pa) Measurement time (s) 0000 Figure 8: Time evolution for a γ =.6s applied shear rate of: (a) the gap size and the normal stress; (b) Normal (irles) and shear stresses (squares). The rheometer is set to hange the rotation rate of the tool through a feedbak loop to make sure the shear rate is onstant, even though the gap is hanging Normal Stress (Pa) More quantitatively, repeating this experiment for different shear rates [Figure 8(b)], one an obtain the gap hange as a funtion of the shear rate that allows the suspension to flow freely, i.e., without developing normal stresses due to partile ollisions. The linear evolution of Δ h with the shear rate Δh = βγ with (0.273±0.03) mm.s Data Linear fit β Δh (mm) Applied shear rate (s - ) Figure 9: Variation of the gap aording to the shear rate. 8

10 This an be ompared to the parallel-plate experiments in whih the gap was systematially varied. Figure 0(a) shows the measured apparent visosity as a funtion of shear rate for different gaps. For low shear rate, again a shear thinning behavior is observed. At the ritial shear rate, a very abrupt inrease in visosity is observed; the main point here is that this ritial shear rate inreases with inreasing gap. Comparison between parallel plate, one and plate, and Couette ells showed idential ritial shear rates to within the experimental unertainty showing that the shear rate gradient present in our parallel plate geometry does not strongly affet our results. Visosity (Pa.s) Normal Stress (Pa) Shear rate (s - ) 0,0 0, 0 Shear rate (s - ) Critial shrear rate (s - ) α 2 With surplus Without surplus Linear fit Gap (mm) Figure 0: (a) Apparent visosity and normal stress as a funtion of shear rate for different gaps in mm. Measurements were made with a parallel plate rheometer (Bohlin C-VOR 200) with radius R =20 mm. (b) Evolution of the ritial shear rate a funtion of the gap. The error bars orrespond to the unertainty (reproduibility) of the experiments. We then define the ritial shear rate as the shear rate for whih both the apparent visosity inreases and for whih non-zero normal stresses are first observed: both values are very similar and, more importantly, inrease linearly with the gap [Figure 0]. Again, if a few milliliters of suspension are left on the bottom plate in ontat with the paste between the two plates, the ritial shear rate strongly inreases and beomes roughly independent of the gap size [Figure 0]. The ritial shear rate with a surplus of paste present is, in addition, the same as that found in the large gap Couette ell, in whih there is also a reservoir of partiles present in the non-flowing region. The flow urve of Figure 3 shows that in the MRI experiments the ritial shear rate is ~ 4 s - ; as soon as this shear rate is exeeded, the system shear thikens. We therefore onlude that the ritial shear rate for thikening obeys γ = γ αh for h < h M I and onstant above; here h is the gap size, α = (4.95±0.44) (mm.s) -, γ M the ritial shear rate when the suspension is suffiiently onfined and γ I the intrinsi ritial shear rate. Moreover, this linear evolution of a ritial shear rate with the gap size presents a striking similarity with the dilation results shown in Figure 9. Indeed, we an observe that the value of α = 4.95 is roughly onsistent with the value found in Figure 9 ( β = ), providing a quantitative hek that indeed the dilatany is responsible for the shear thikening. 6. Shear thikening as a Visous/Granular transition Another parameter that has a large effet on the ritial shear rate is the volume fration. At low volume frations: ϕ < 0. 4, shear thikening is either less dramati or absent [van der Werff and de Kruif (989)]. In Figure, we show the typial evolution of the visosity as a funtion the shear rate of ornstarh suspension with different onentrations. 9

11 Visosity (Pas) , Φ (%) , 45,5 0,0 0, 0 00 Shear rate (s - ) σ (Pa) Critial shear rate (s - ) Conentration (%) Conentration (%) Rough Vane Rough PP40 Smooth Vane CP40-4 Smooth PP40 Figure : Visosity as a funtion a shear rate for different onentrations. (b) Critial shear stress of shear thikening vs. onentration in different measurement geometries. Inset: Critial shear rate of shear thikening vs. onentration. γ dereases roughly linearly with inreasing onentration and seondly that the ritial shear stress of shear thikening remains roughly onstant at σ ( ϕ) 20Pa onsistent with the work of [Fall et al. (200)]. In this piture, for low stresses, the visosity of the interstitial fluid lubriates the ontats and one reovers a visous behavior; this agrees with the observed Bingham behavior provided the stress on the system is muh larger than the yield stress so that the latter an be negleted. However for higher stresses, a Bagnold-type dry granular rheology would be expeted. The hallmark of this behavior is that the stress in the Bagnold regime, i.e., beyond shear thikening sales inertially [Fall et al. (200), Lemaître et al. (2009), Mills and Snabre (2009)], We found that the ritial shear rate ( ϕ) 2 σ γ implying. However, for the ornstarh, this annot be verified diretly beause the system beyond shear thikening has a very high visosity making measurement over a large range of shear rates impossible. In addition, for the highest shear rates one observes instabilities in both the parallel plate ell and the Couette ell that are probably due to the normal stresses, and that make that the fluid is expelled from the gap. However remaining around the onset shear rate, we an verify that in the seond, shear thikening regime, the system behaves similarly to a dry granular material, whih is exatly what is at the basis of the lubriated to inertial transition. If the system is a fritional one (as a dry granular material should be), due to the steri interation between the partiles, the loal shear stress σ indues a loal normal stress σ = μσ n where μ is the marosopi frition oeffiient. Shear stress (Pa) μ = 0, Normal stress (Pa) Figure 2: Proportionality between normal stress and shear stress in the shear thikening regime for the 44% volume fration suspension 0

12 Figure 2 shows the evolution of shear stress as a funtion a normal stress for different gap sizes of parallel plate geometry. We found that the normal stresses hange linearly with the shear stresses. This linearity an be used to define the marosopi frition oeffiient of the suspension; we find a value 0.62±0.0: this value is similar to that found in dry granular materials [da Cruz et al, (2005)] and other shear-thikening fluids [Lootens et al. (2003, 2005); Brown and Jaeger, (20)]. This suggests that in the shear thikening regime, the interstitial fluid plays no more role as lubriation fores beome negligible: ontats behave like "dry" ontats. In this ase, shear thikening an therefore again be onsidered as a transition from lubriated regime to a fritional regime under flow. The shear thikening transition is then due to diret ontats between partiles indued by the shear. 7. Shear thikening as a re-entrant jamming transition In our situation, the yield stress is smaller than the ritial stress and onsequently the sample yields and flows before thikening. If the ritial stress is the smallest one, the thikening behavior may even disappear altogether [Gopalakrishnan and Zukoski, (2004), Brown and Jaeger (2009)]. Our data show that for the ornstarh suspensions there are two ritial stresses for whih the apparent visosity beomes infinite: first, upon approahing the yield stress from above, the visosity diverges, in agreement with the MRI observations that the flow behavior is lose to that of a Bingham fluid. Seond, at the ritial stress for thikening, an almost disontinuous jump of the visosity is observed. These observations suggest a solid liquid solid transition. This is in addition qualitatively the same piture as that obtained from by the osillatory shear experiments (Figure 7). Visosity (Pa.s) 00 slope 0 'solid' 'liquid' 'solid' Applied Stress (Pa) Figure 3: Solid liquid solid transition; apparent visosity vs. stress when performing a step stress test. This is a typial example of disontinuous shear thikening, in whih the region with positive slope of (γ ) ours in a stress range that is nearly independent of paking fration as mentioned above in Figure. This slope inreases with paking fration, approahing η σ orresponding to a disontinuous stress/shear-rate relation. These results are similar to theory that suggests that shear thikening is due to a re-entrant jamming transition [Sellito and Kurhan (2005)]. It has been suggested for glassy systems that applying a shear is equivalent to inreasing the effetive temperature with whih the system attempts to overome energy barriers [Berthier et al. (2000)]. If now a system has a re-entrant solid transition as a funtion of temperature, the solid phase may also be indued by the shear, leading to shear thikening [Sellito and Kurhan (2005); Fall et al. (2008)], as is also observed here (Figure 3). Indeed, the sample is solid in the sense that G >>G (Figure 7) and the torque beomes too large for the MRI rheometer to turn. In the onvential rheology, there are instabilities due to the large elastiity that expel the starh from the gap of the measurement geometry, making determination of an apparent visosity diffiult. σ IV. Conlusion We have studied the flow behavior of dense suspensions of non-olloidal partiles, ornstarh partiles in water, by oupling loal veloity and onentration measurements through MRI tehniques, and marosopi rheometri experiments in Couette and parallel plate geometries. The MRI data reveals that the flow exhibits shear-banding at low rotation veloities of the inner ylinder. The MRI also showed that the material is homogeneous and no migration is observed, ontrary to what happens for non-brownian suspensions of spherial partiles [Fall et al., (200)].

13 Classial rheology then shows that the ritial shear rate for the onset of shear thikening depends on the onfinement: it evolves linearly with the gap size. This linear evolution is a diret onsequene of the dilatany observed in this suspension under shear. Indeed, from dilation measurements, we have shown that the appliation of a shear rate higher than the ritial shear rate for thikening diretly leads to a dilation of the suspension. Conversely, taking away the onfinement attenuates the shear thikening behavior, or even makes it disappear altogether. For the MRI observations this implies that when flow is loalized, the nonflowing region plays the role of a dilatany reservoir whih allows the material to be sheared without undergoing a jamming transition. This onviningly shows that the dilation is at the origin of the disontinuous thikening. In addition, in osillatory experiments we found a ritial shear strain for shear thikening that appears independent of the frequeny. The value of the ritial shear strain ( γ ) also supports the idea that disontinuous shear thikening is a diret onsequene of shear indued dilatany that is hindered by the onfinement. The ritial shear rate assoiated with the yield stress is due to the existene of a yield stress and is in priniple deoupled from the onset of shear thikening. However, in the Couette geometry used for the MRI experiments, the existene of a yield stress makes that part of the material flows, and another part does not beause the stress it is subjeted to is smaller than the yield stress. As also shown by the plate-plate measurements (Figure 6), the existene of a non-flowing region influenes the onset of shear thikening, and so the ritial shear rate is indiretly oupled to the shear thikening phenomenon. However both the dilation and the measurements as a funtion of volume fration show that the ritial stress for the onset of shear thikening is diretly oupled to the onset behavior. Referenes Akerson, B. J. Shear indued order and shear proessing of model hard sphere suspensions. J. Rheol., 34, (990). Ball R. C. and J. R. Melrose, Lubriation breakdown in hydrodynami simulations of onentrated olloids, Adv Colloid Interfae Si 59, 9 30 (995). Barnes H.A. Shear-Thikening ('Dilatany') in Suspensions of Non aggregating Solid Partiles Dispersed in Newtonian Liquids. J. Rheol. 33, (989). Bender, J.W. and N.J. Wagner, Reversible shear thikening in monodisperse and bidisperse olloidal dispersion. J. Rheol. 40, (996). Berthier L., J. L. Barrat, and J. Kurhan, Phys. Rev. E 6, 5464 (2000). Bertrand E. et al. From shear thikening to shear-indued jamming, Phys. Rev. E 66, (2002) Bishoff White, E.E. Chellamuthu, M. & Rothstein, J.P. Extensional rheology of a shear-thikening ornstarh and water suspension. Rheol. Ata 49, 9-29 (200). Boersma, W. H., J. Laven, and H. N. Stein, Computer simulations of shear thikening of onentrated dispersions, J. Rheol. 39, (995) Boersma, W. H., J. Laven, and H. N. Stein, Shear thikening (dilatany) in onentrated dispersions, AIChE. J. 36, (990). Bonn D. et al., Some appliations of magneti resonane imaging in fluid mehanis: Complex flows and omplex fluids Annu. Rev. Fluid Meh. 40, 209 (2008). Bonn D. and M. Denn., Yield stress fluids slowly yield to analysis, Siene 324, 40 (2009). Brown E. and H. M. Jaeger, Dynami jamming point for shear thikening suspensions, Phys. Rev. Lett. 03, (2009). Brown E., Niole A. Forman, Carlos S. Orellana, Hanjun Zhang, Ben Maynor, Douglas Betts, Joseph M. DeSimone, and Heinrih M. Jaeger, On the generality of shear thikening in suspensions, Nature Materials 9, (200) Brown E. and H. M. Jaeger, The role of dilation and onfining stresses in shear thikening of dense suspensions (20) Cates, M.E., M. D. Haw, and C. B. Holmes, Dilatany, jamming, and the physis of granulation J. Phys. Condens. Matter 7, S257 (2005). Chen, L.B., Akerson, B.J., & Zukoski, C.F. Rheologial onsequenes of mirostrutural transitions in olloidal rystals. J. Rheol., 38, -25, (994). 2

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