Macroscopic Discontinuous Shear Thickening vs Local Shear Jamming in Cornstarch

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by  A. Fall

Macroscopic Discontinuous Shear Thickening vs Local Shear Jamming in Cornstarch
A. Fall,1 F. Bertrand,1 D. Hautemayou,1 C. Mezi`ere,1 P. Moucheront,1 A. Lemaˆıtre,1 and G. Ovarlez1,2
1Laboratoire Navier (UMR CNRS 8205), Universit´e Paris Est, Champs-sur-Marne, France
2University of Bordeaux, CNRS, Solvay, LOF, UMR 5258, 33608 Pessac, France
(Dated: January 23, 2015)
We study the emergence of discontinuous shear-thickening (DST) in cornstarch, by combining
macroscopic rheometry with local Magnetic Resonance Imaging (MRI) measurements. We bring
evidencethatmacroscopic DSTisobservedonly whentheflowseparatesintoalow-density flowing
and a high-density jammed region. In the shear-thickened steady state, the local rheology in the
5
flowingregion,isnotDSTbut,strikingly,isoftenshear-thinning. Ourdatathusshowthatthestress
1
jumpmeasuredduringDST,incornstach,doesnotcaptureasecondary,high-viscositybranchofthe
0
local steady rheology, but results from the existence of a shear jamming limit at volume fractions
2
quitesignificantly below random close packing.

n
a
J Granular suspensions are seemingly simple systems starch [16–18], the paradigmatic system for this phe-

2 composed of non-colloidal rigid particles immersed in nomenon,byaccessingexperimentallythelocalresponse.
2 Newtonian fluids. Although the only relevant interac- To this aim, we have designed a new velocity-controlled
tions are hydrodynamic and contact forces, they present rheometer that can exert steady torques up to 10 N.m
]
t arichrheology[1],whichincludesshear-thinningorshear onawide-gapCouettecellinsertedinourMagneticRes-
f
o thickening, normal stress differences, shear banding and onance Imaging (MRI) scanner. It enables us to access
s yieldstressbehaviour. Withincreasingpackingfractions, localvelocity and particle volume fraction profiles in the
t. continuous shear thickening (CST) appears at lower and flowing sample [13, 19], for a wide range of strain rates
a lower strain rates and becomes more and more abrupt. (γ˙) covering CST and DST. We thus find that macro-
m
This eventually leads to the emergence of a spectacu- scopic DST is observed when the flow localizes, i.e. the
- lar phenomenon: discontinuous shear thickening (DST), system separates into a low-density flowing and a high-
d
an order-of-magnitude jump of the stationary (macro- density jammed region. In the shear-thickened steady
n
o scopic) stress when strain rate crosses some threshold state,the localrheologyinthe flowingregionisnotDST
c value. DSTisobservedinanarrowrangeofpackingfrac- but, strikingly, is most often shear-thinning. Our data
[ tionsnearrandomclosepackingandrecentworkssuggest support that DST should be attributed to the existence

that it is a dynamic or shear jamming transition. This of a shear jamming limit γ˙ (φ) at volume fractions φ
1 DST
v motivates a considerable attention from experiments [2– significantly smaller than random close packing.
6 4], numerical simulation [5–7] and theory [8, 9]. The cornstarch (Sigma) is the same as in [11, 16]: it
1 The origin of shear thickening is essentially unknown. presents slightly polydisperse particles of size ∼ 20µm
6
Proposed mechanisms include: the crossover to Bagnold with irregular shapes. We have measured the random
5
0 scaling due to particle inertia [10]; the coupling between close packing fraction φRCP ≃ 55% by tapping the pow-
. normal and shear stresses associated with material dila- der for long time in a container [20]. We have also mea-
01 tancy [11, 12]. In numerical simulations DST was ob- sured the random loose packing φRLP ≃ 35% as the
5 served in systems which are forced to remain homoge- density achieved after sedimentation of a dilute suspen-
1 neous by periodic boundary conditions [5]. It was thus sion [21, 22]. The suspensions are prepared by mixing
: suggested that it might arise from a jump of the local cornstarchin demineralized water at initial volume frac-
v
response, which occurs at fixed packing fraction, and is tions ranging from 33.5% to 43.9%.
i
X caused by the proliferation of frictional contacts [5, 9]. Let us start with macroscopic rheometry measure-
r Examples abound, however, in granular suspension ments using a Malvern Kinexus Pro stress-controlled
a
rheology where the macroscopic response is sharply dif- Couette rheometer. The inner and outer cylinders radii
ferent from the local behaviour: the apparent yield are: Ri = 12.5mm and Ro = 18mm; the inner cylin-
stress [13, 14] or a transient DST behavior [10], for ex- der height is h = 37.5mm; the gap is large enough to
ample, were both shown to result from the emergence avoidconfinementeffects [11, 12, 15]. Bothcylinders are
of flow inhomogeneities due respectively to density dif- roughenedtoavoidwallslip. WereportonFig.1-(a)the
ferences and migration. The observation of finite size torque T measured on the inner cylinder during a 3 min
effects in DST measurements [11, 12, 15], at sizes up to logarithmic ramp of Ω (the inner cylinder rotation rate)
100particles,thusquestionstheideathatDSTisapurely from 5.10−2rpm to 2.103rpm. Near the inner cylinder,
local phenomenon. In this context, it is unclear how a the local stress is
phasediagramforthelocalresponsecanbededucedfrom
macroscopic experimental data and compared with nu- τ(Ri,Ω)=T(Ω)/(2πRi2h) (1)
merical [5], and theoretical [9] works.
This situation motivates us to study DST in corn- the localstrainrateisaprioriunknown,butisclassicaly
2
estimated as:
γ˙(Ri,Ω)≃2ΩRo2/(Ro2−Ri2) (2)
Combining Eq. (1) and (2) yields the apparent viscosity
η ≡ τ/γ˙ vs γ˙ relation reported on Fig. 1-(b). Note that
at low η macroscopic inertial effects may arise (Taylor-
Couette instabilities), which limit the accessible γ˙ range
at low φ’s.
At low γ˙, the response is clearly shear-thinning. It
crosses over to CST around a characteristic strain rate
γ˙ defined at that where η reaches a minimum. Vis-
CST
cosityjumpsatsomehigherstrainrateγ˙ whichchar-
DST
acterizes the onset of DST.
The classical estimate of γ˙(Ri) as Eq. (2), which is
basedontheNewtoniansolution,remainsunsatisfactory.
An exact expression exists so long as the material is ho-
mogeneous in the studied torque range:
∞
dΩ
γ˙(Ri;T0)=2 T (3)
nX=0 (cid:18) dT(cid:19)(cid:12)(cid:12)(cid:12)T=T0 RRoi 2n
(cid:12) (cid:0) (cid:1) FIG. 1. (Color online). Macroscopic rheometry data. (a)
It cannot be used around the DST transition, however, Torque T vs rotation rate Ω during a logarithmic ramp at
becausethe apparentsingularityofdT/dΩ cannotbe re- various packing fractions φ. (b) Apparent viscosity vs ap-
solvedexperimentally. Neverthelesswehavecheckedthat parent shear rate extracted from (a). (c) Critical shear rates
usingthefirsttwotermsof(3)givesthesamequalitative γ˙CST and γ˙DST vs φ. (d) Reversibility test at φ = 43.9%:
behaviour as in Fig. 1-(b) up to DST. We will show be- succession of up,down, and up ramps.
low that the flow does remain homogeneous below DST,
which allows us to use (3) to accurately compute the
γ˙CST and γ˙DST values that are reported on Fig. 1-(c). up to the unknow prefactor τ(Ri,Ω).
Interestingly, (i) at any volume fraction CST is always InFig2,weplotthesteadyvelocityandconcentration
observed before DST; (ii) both γ˙ and γ˙ seem to profiles thus measured for a few Ωs. Velocity is normal-
CST DST
vanish aroundthe same volume fraction φc which we es- ized by its value at the inner cylinder.
timate to be ≈45%≪φ by linear extrapolation. For small Ω’s (5 and 7 rpm), we find that the density
RCP
Now we turn to velocity-controlled MRI rheometry. profile is homogeneous while the flow extends through-
OurCouettecellhasinnerandouterradiiRi =3cmand out the gap. It is known that non-Browniansuspensions
Ro = 5cm (resp.) and inner cylinder height h = 11cm. mayslowlybecomeinhomogeneousdue toshear-induced
Both cylinders are roughened to avoid slip, which we migration and sedimentation [13, 14]. We checked that
checkedfromvelocityprofiles. Allexperimentsdiscussed thisdoesnotoccurbeforestrainslargerthanafewthou-
below are performed by preparing a homogeneous mate- sands, which is much larger than the strain range over
rialwithmeanvolumefractionφ =43.9%(experiments which we collect data. Since density is uniform, we can
0
at 40%, 41% and 42.5% show similar features). Note access the local rheologyη(γ˙,φ0) as follows. For each Ω,
thatourMRIcelldimensionsdifferfromthoseofthecell matching the single parameter τ(Ri,Ω) in Eq. (4) pro-
used to obtain the macroscopic data of Fig 1. Hence, to videsthecompleteη(r,Ω)profile,whichcanthusbeplot-
captureDST withthis newcell,therangeofrotationve- ted vs γ˙(r,Ω). These values are compared on Fig. 3-(a)
locitiesmustbeadapted: atφ =43.9%,weuseΩvalues with the local rheology data obtained from macroscopic
0
between5rpmand100rpm. In eachMRI experiment,Ω rheometry using Eqs. (1) and (3) to evaluate the local
is held fixed until steady state is reached. stressand strainrate near the inner cylinder. Clearly,in
OurMRI[13,23]providesthestationarypackingfrac- the shear-thinning regime, the stationary local response
tion φ(r,Ω) and azimuthal velocity v(r,Ω) at any radial within the gap matches our macroscopic measurements
position r. From the latter we extract the local strain This local homogeneous response is observed so long
rate γ˙(r,Ω) = v/r − ∂v/∂r. We do not have access as the maximal local strain rate, which is reachedat the
to torque measurement. But, since the local stress is inner cylinder, lies below the γ˙DST(φ0) value identified
τ(r,Ω)=τ(Ri,Ω)Ri2/r2 intheCouettegeometry,wecan in macroscopic rheometry [Fig. 1-(c)]. Homogeneity and
estimate the local viscosity profile as: locality then enable us to estimate the critical rotation
rate at which DST is expected as: Ω (φ )≃8rpm.
DST 0
τ(Ri,Ω) Ri2 A sudden transition occurs as soon as Ω crosses
η(r,Ω)= (4)
γ˙(r,Ω) r2 ΩDST(φ0). AsshownonFig.2-(a),theflowthenabruptly
3
FIG. 3. (a) Comparison of local rheometry data obtained
from MRI measurements (open symbols) and near the inner
cylinder in macroscopic rheometry (filled symbols) in homo-
geneous conditions. Upper data: φ = 43.9%; lower data:
φ ≈ 33.5%. (b) From the velocity profiles [Fig. 2-(a)]: frac-
tion of thegap which is jammed vsΩ.
homogeneities. Namely,the volume fractiondecreasesin
theflowinglayer,whileitincreasesinthejammedregion,
as required by the conservation of particle number. As
Ω increases beyond Ω (φ ), the progressive extension
DST 0
of the flowing layer is accompanied by a broadening of
the low-density region. At high strain rates, the den-
sity saturates, in the flowing layer, at a packing fraction
φmin ≃ 33% <∼ 35% ≃ φRLP and, in the jammed region,
atφ∼φ . Itisnoteworthythatthedensityprofilecan
RCP
achieve multiple forms depending on shear history [24].
Letusemphasizethatthechangeofdensitycreatedby
theDSTtransitionisirreversible. Indeed,onceastation-
ary profile φ(r,Ω ) has been produced by ramping Ω up
1
FIG.2. SteadyMRIdataforaφ0 =43.9%cornstarchsuspen- to some arbitraryΩ1 >ΩDST, we found that the density
sionanddifferentrotationalvelocitiesΩ. (a)velocityprofiles. profile remains the same under any subsequent lowering
(b) density profiles; solid lines indicate φRLP and φRCP. of Ω. This irreversibility shows up in our macroscopic
rheometry setup (the small Couette cell) as illustrated
on Fig. 1-(d) where we plot the torque T vs Ω during:
stopsinalargeregion. Namely,thevelocityprofilejumps (i) an initial up-ramp that drives the system through
fromoneoftherightmostcurves,correspondingtohomo- the DST transition; followed by (ii) a down-ramp. The
geneousflows,to the leftmost one,i.e. the moststrongly torquesmeasuredduringtheup-anddown-rampsclearly
localizedflow. Notethatmeasuringasinglevelocitypro- lie on different branches. However, if we subsequently
file requires accumulating MRI data over ∼ 30s, which (iii) re-increase Ω, torque T(Ω) tracks the data previ-
corresponds here to a strain of order 50. Upon cross- ouslyobtainedonthe down-ramp. Hence we reasonthat
ing ΩDST(φ0), the first measurable velocity profile is al- on the down (ii) and up (iii) ramps, the system explores
ready localized. The flow subsequently remains steady quasi-reversibly a family of states associated with some
overthousands units of strains. DST is thus clearly con- inhomogeneous density profile set by Ω , the maximum
1
comitant with shear localization. value of the rotation rate achieved during shear history.
As Ω increases further, the velocity profiles progres- We now probe the local rheology within the shear-
sively extend to the right (i.e. towards the outer cylin- thickened flow. Three of our datasets, for Ω = 40, 60,
der). In all cases, the system remains separated into a and100rpm,presentafinite-widthregionofroughlyuni-
flowing layer near the inner cylinder and a jammed re- form packing fraction φ ≃ 33%. Using only the data
min
gion near the outer one. The fraction of the gap that is collected in these regions, we can hence implement the
jammed is reported on Fig. 3-(b): it jumps at ΩDST(φ0) same approach as in the fully homogeneous case, to ac-
and then slowly decreases. cess the local rheology η(γ˙,φ) at φ = φ . For each Ω,
min
Comparing these velocity profiles [Fig. 2-(a)] with lo- wematchtheprefactorτ(Ri,Ω)andplotinFig.3-(a)the
cal density data [Fig. 2-(b)], we find that, quite strik- resultingη(r,Ω)vsγ˙(r,Ω)datapoints. Clearly,thelocal
ingly, the flow localization at Ω = 10rpm >∼ ΩDST(φ0) η(γ˙,φmin) rheology is shear-thinning, and extrapolates
is associated with the sudden emergence of density in- the φ=33.5% rheometry measurements.
4
vanish. As φc is quite distinct from the jamming point
φ ≃55%,acontinuumofjammedstatesisaccessible
RCP
beween φc and φRCP.
In the literature, experimental observations of macro-
scopicDSTareusuallyanalyzedwhileassumingthatthe
flow remains homogeneous: the DST stress jump hence
is interpreted as picking up a secondary, high-viscosity,
branch of the steady local response. As we have shown,
however,homogeneity does not hold. The stress jump is
apropertyoftheglobalresponse,andisassociatedwitha
globalreorganizationoftheflow,whilethelocalresponse
alwayslieswithintheshear-thinningorCSTregimes. Of
course,thisisshownhereforcornstarch,butwenotethat
today,thereisnotintheliterature,asingleexperimental
data point that can be trusted to pick up the presumed
secondarybranchofs-shapedrheologies,sincethereisno
other local observation of the flow response.
Moreover,theDSTscenariothatweareunveilinghere
is rather unexpected in view of recent models based on
FIG. 4. Phase diagram: scatter plot of the local
s-shape flow curves [9]. Indeed, at the onset of macro-
(γ˙(r,Ω),φ(r,Ω))valuescomparedwiththeγ˙CST(φ)(red)and scopic DST, the part of the flow that reaches locally a
γ˙DST(φ) (blue) lines obtained from macroscopic rheometry
limit of stability, i.e. the γ˙ line, is the region near
[Fig. 1-(c)]. DST
theinnercylinder,wherethelocalshearrateismaximal.
From s-shape models, in their current formulation, we
would expect this locally unstable region to jump to a
ThesedatademonstratethatmacroscopicDSTcannot
high-viscositystate. Instarkcontrast,wefindthatwhen
bestraightforwardlyinterpretedascapturingacompara-
DSToccurs,thisregionjumpstowardalow-density,low-
ble changeofthe localresponseatfixedvolumefraction.
viscosity shear-thinning state. Meanwhile jamming oc-
Instead,itresultsfromtheseparationofthesystemintoa
curs in the region near the outer cylinder which, at the
low-density-flowing and a high-density-jammed regions.
onset of DST, was well within its stability limit.
Somewhat paradoxically, the local response was found
It finally appears that DST is determined by two ma-
above to be shear-thinning in the flowing layer of the
jor factors: (i) the existence of a limit γ˙ (φ) beyond
DST
macroscopic shear-thickenedstate.
which no steady flow can be accessed; (ii) the fact that
In the flowing layer, the density plateau only devel- as φc ≪ φRCP, a continuum of jammed states exists.
ops around φmin. It is hence at this packing fraction Indeed, due to (i) the inner cylinder region eventually
only that we can extract the local rheology η(γ˙,φ) from reaches the γ˙ (φ) shear-jamming limit, that it can
DST
MRI data. To further qualify the local response, we only escape by decreasing its local volume fraction; it
now construct a phase diagram for all the local flow and thus drives migration throughout the gap, because (ii)
jammed states. To this end, we collect the operating thegrowingjammedlayercanstorematerialsignificantly
points(γ˙(r,Ω),φ(r,Ω))obtainedwiththeMRIforallΩ’s above φc. Both (i) and (ii) are direct consequences of a
andallr’sandreportthemonFig.4,wheretheyarecom- unique physical phenomenon, shear jamming.
paredwith the curves γ˙CST(φ) and γ˙DST(φ) that delimit It is crucial here that the γ˙DST(φ) line is not vertical,
the flow regimes previously identified using macroscopic hence can be crossed by increasing γ˙ at given φ. It is
rheometry. OurMRIdataincludepointsobtainedinthe so because the asymptotic flow regimes (high and low γ˙,
flowinglayeraswellasinjammedregions: inthe scatter resp.) havetwodifferentlimit volumefractions(≃φRLP
plot, they are respectively found on the left (low φ, with and φc, resp.). This contrasts sharply with [10] where
non-vanishing γ˙) and right (high φ, γ˙ ≃0) hand sides. jamming is rate-independent and DST purely dynami-
Local flow point fall in either the shear-thinning or cal. It remains open to identify the microscopic mecha-
CST domains, which supports that both types of lo- nismsthat accountfor the splitting ofthe jamminglimit
cal rheologies are found. Moreover, they all lie below observed here. Several may be envisionned, such as the
the γ˙ (φ) line, which thus appears to be an intrinsic increasedmobilizationoffrictionalcontactswithincreas-
DST
material limit beyond which no steady flow can be sus- ing shear rate [5], or the angularity of particles, which is
tained. Noflowisfoundinparticularatpackingfractions significantforcornstarch,orfinallyparticleinertia,which
above the value φc ≃ 45% where γ˙CST(φ) and γ˙DST(φ) mightbecausingtheCST[10]thatalwayspreceedsDST.
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