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Microscopic mechanism for the shear-thickening of non-Brownian
suspensions
Nicolas Fernandez,1 Roman Mani,2 David Rinaldi,3 Dirk Kadau,2
Martin Mosquet,3 HélèneLombois-Burger,3 Juliette Cayer-Barrioz,4
Hans J. Herrmann,2 Nicholas D. Spencer,1 and Lucio Isa1, ∗
1Laboratory for Surface Science and Technology, Department of
Materials, ETH Zurich, Switzerland
2Computational Physics for Engineering Materials, Department of
Civil,
Environmental and Geomatic Engineering, ETH Zurich,
Switzerland3Lafarge LCR, Saint Quentin-Fallavier, France
4Laboratoire de Tribologie et Dynamique des Systèmes - UMR 5513
CNRS, École Centrale de Lyon, France
We propose a simple model, supported by contact-dynamics
simulations as well as rheology andfriction measurements, that
links the transition from continuous to discontinuous
shear-thickeningin dense granular pastes to distinct lubrication
regimes in the particle contacts. We identify a localSommerfeld
number that determines the transition from Newtonian to
shear-thickening flows, andthen show that the suspension’s volume
fraction and the boundary lubrication friction coefficientcontrol
the nature of the shear-thickening transition, both in simulations
and experiments.
Flow non-linearities attract fundamental interest andhave major
consequences in a host of practical applica-tions [1, 2]. In
particular, shear-thickening (ST), a viscos-ity increase from a
constant value (Newtonian flow-Nw)upon increasing shear stress (or
rate) at high volumefraction φ, can lead to large-scale processing
problemsof dense pastes, including cement slurries [3].
Severalapproaches have been proposed to describe the micro-scopic
origin of shear-thickening [4–7]. The most com-mon explanation
invokes the formation of ”hydroclus-ters”, which are responsible
for the observed continuousviscosity increase [6, 8, 9] and which
have been observedfor Brownian suspensions of moderate volume
fractions[10, 11]. However, this description no longer holds
forbigger particles and denser pastes, where contact net-works can
develop and transmit positive normal stresses[12]. Moreover, the
link between hydroclusters and CSTfor non-Brownian suspensions is
still a matter of debate[13]. Additionally, dense, non-Brownian
suspensions canalso show sudden viscosity divergence under flow
[14–17] with catastrophic effects, such as pumping failures.In
contrast to a continuous viscosity increase at any ap-plied rate,
defined as continuous shear-thickening (CST),the appearance of an
upper limit of the shear rate de-fines discontinuous
shear-thickening (DST). This CST toDST transition is observed when
the volume fraction ofthe flowing suspension is increased above a
critical value,which depends on the system properties, e.g.
polydis-persity or shape, and on the flow geometry [3, 18].
Anexplanation for its microscopic origin is still lacking
[19].Moreover, experiments have demonstrated that the fea-tures of
the viscosity increase (slope, critical stress) canbe controlled by
tuning particle surface properties suchas roughness [20] and/or by
adsorbing polymers [21, 22].These findings suggest that
inter-particle contacts play acrucial role in the macroscopic flow
at high volume frac-tions. A more precise description of these
contacts istherefore essential to interpret the rheological
behavior.
In this paper, we present a unified theoretical frame-work,
supported by both numerical simulations and ex-
perimental data, which describes the three flow regimesof rough,
frictional, non-Brownian particle suspensions(Nw,CST,DST) and links
the Nw-ST (in terms of shear)and the CST-DST transitions (in terms
of volume frac-tion) to the local friction. Our microscopic
particle-contact based description, as opposed to
macroscopicscaling, explains both the occurrence of DST and
recov-ers Bagnold’s analysis [5] for CST, respectively above
andbelow a critical volume fraction.
The lubricated contact between two solid surfaceshas been widely
studied in the past [23]. It is nowcommonly accepted that different
lubrication regimesoccur as a function of a characteristic number,
theSommerfeld number s. For two identical spheres,s = ηfvRp/N ,
where ηf is the fluid viscosity, v is thesliding speed between the
two solid surfaces, Rp isthe radius of the spheres and N is the
normal load.At high s (”hydrodynamic regime”-HD), a fluid filmfully
separates the two sliding surfaces and the frictioncoefficient µ
depends on s. For low s, below a criticalvalue sc, the lubrication
film breaks down and contactsbetween the microscopic asperities on
each surfacesupport most of the load. This ”boundary
lubrication”regime (BL) exhibits friction coefficients that only
veryweakly depend on s. For intermediate values of s thesystem is
in a ”mixed regime”, where the sharpness ofthe transition depends
on the system properties (e.g.contact roughness, rheology of the
fluid. See Fig.1a)[23].
Both experiments and models show that Nw flow isstable below a
critical shear rate γ̇c where the contactsbetween particles are HD
lubricated. On the other hand,a particle-contact-dominated flow
requires, by definition,that s < sc and it is equivalent to a
dense dry gran-ular flow (i.e. no suspending fluid lubrication
effects).Dense granular flows follow a quadratic scaling of
thenormal and shear stress P and τ with the shear rate γ̇
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FIG. 1. (color online) a) Schematic Stribeck curve. Evolutionof
the friction coefficient, µ, versus the Sommerfeld number, s,for a
lubricated contact. b) Apparent viscosity, η, versus theshear rate,
γ̇, from the numerical simulations. c) Numericalsimulations
friction law (black line) and probability distribu-tions of s, P
(s), for all contacts at several shear stresses asdefined in b. d)
Frequencies of BL contacts, PBL, and HDcontacts, PHD = 1 − PBL, as
a function of γ̇ for the stressesdefined in b. The simulations data
in b-c-d have φ = 0.58,µ0 = 0.1 and sc = 5× 10−5.
(Bagnold scaling) through a volume-fraction-dependentfactor [5];
this implies that the apparent viscosity riseslinearly with γ̇ and
that the system shear thickens con-tinuously (see Fig.1b). This
scaling can be expressed interms of a dimensionless parameter, the
inertial number
I = γ̇Rp�
ρp
P, only depending on φ and µ for rigid par-
ticles with density ρp [24].
Given the definition of s, this leads tos ∝ ηfI2/γ̇ρpR2p. This
Bagnold (CST) regime ispossible as long as γ̇ is larger than γ̇c ∝
ηfI2/scρpR2p,showing the link between γ̇c and sc when particle
con-tacts dominate. This transition was partially proposed,with
macroscopic arguments, by Bagnold [5, 25, 26].Nevertheless, our
microscopic analysis also accounts forvolume fraction effetcs.
In our model, the existence of two lubrication mech-anisms
(boundary and hydrodynamic) implies twodifferent jamming volume
fractions φmax, above whichflow is not possible. If the system is
hydrodynamicallylubricated, the jamming volume fraction φHD
maxis at
random close packing φRCP , regardless of the boundaryfriction
coefficient [27]. Conversely, when the system isin a
boundary-lubricated Bagnold regime, the jammingvolume fraction
φBL
maxdecreases with µ [28, 29]. Both
φHDmax
and φBLmax
are independent of γ̇ for non-Brownianparticles. It follows that
φRCP = φHDmax ≥ φBLmax(µ).When φ ≤ φBL
max≤ φHD
max, the transition from hydro-
dynamic to boundary-dominated flow is possible and
the suspension exhibits CST, as reported above andpredicted by
Bagnold. When φBL
max< φ ≤ φHD
max, the
transition to a Bagnold regime is forbidden, and theshear rate
cannot exceed γ̇c: the system undergoes DST.As a consequence,
φBL
maxis the critical volume fraction for
DST and therefore it can be tuned by changing the par-ticle
friction coefficient. Both numerical simulations andexperiments
fully and independently support our model.
In concentrated systems most of the dissipation arisesfrom
particles that are in, or close to, contact and notfrom Stokesian
drag [25, 30]. This motivates using Con-tact Dynamics [31–35] to
simulate dense suspensions ofhard, spherical, frictional particles
using a simplifiedStribeck curve (no mixed regime) as friction law
(seeFig.1c and Eq.1). Only one dissipative mechanism, ei-ther BL or
HD, is taken into account in each contact.This constitutes the
simplest physical description of alubricated contact.The boundary
lubrication between two rough particles
is described using Amontons-Coulomb friction, i.e.
thecoefficient of friction µ0 being independent of the load,the
speed and the apparent contact area [23].In the HD regime, the
hydrodynamic interactions be-
tween two neighboring particles are long-lived and can
bedescribed by standard, low-Reynolds-number fluid me-chanics with
a lubrication hypothesis [36], from which afriction coefficient can
be calculated as a function of theSommerfeld number µ = 2πsln( 56πs
) (see SupplementalMaterial for full derivation). The lubrication
hypothe-sis breaks down when the particles are too far apart
(i.ewhen s is large) and therefore we consider only a rangeof γ̇
where s of almost all the contacts is smaller than alimit value,
slim = 10−1.The friction law used for the simulations is then:
µ(s) =
�µ0 if s < sc2πs ln( 56πs ) if sc < s < slim
(1)
In our Contact Dynamics simulations the normal forcesare
calculated based on perfect volume exclusion, us-ing zero normal
restitution coefficient, and we simulatestress-controlled (τ)
simple shear between moving andfixed rough walls (obtained by
randomly glued parti-cles) at a constant volume fraction [37, 38].
The rect-angular simulation box dimensions are (Lx, Ly, Lz) =(25R,
10R, 27R), where Lz is the distance between thetwo walls and R the
radius of the largest particle inthe simulations. We use periodic
boundary conditionsin both x and y directions. The presence of hard
confine-ment mimics experimental conditions, and simulationswith
Lees-Edwards boundary conditions that are peri-odic in the three
directions show the same qualitativebehavior (see Supplemental
Material). The particle radiiare uniformly distributed between 0.8R
and R to pre-vent crystallization. When fixing φ, µ0, R, ρp and
sc,the physics of the system is characterized by a single
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dimensionless number: λ =√τρR/ηf . λ can be under-
stood as the ratio between the microscopic time scale ofthe
lubricating fluid, ηf/τ , and of the granular medium,R�ρ/τ [24].
Increasing the shear stress τ quadratically
is equivalent to decreasing ηf linearly. In our simula-tions, we
vary ηf and keep τ fixed. After the system hasreached its steady
state with a linear velocity profile (noshear band), we measure the
time averaged velocity ofthe moving wall �vwall�, thus γ̇ =
�vwall�/Lz and the ap-parent viscosity of the suspension η is given
by τ/γ̇. Thequantities γ̇, τ and η are measured in units of
ηf/ρpR2,η2f/ρpR2 and ηf (see Supplemental Material for
details).
The simulations (see Figs.1b and 2) reproduce a tran-sition
between a Newtonian regime at low shear rates(independent of µ0 and
dominated by HD-lubricated con-tacts) to a ST regime with
increasing γ̇, for which bound-ary lubricated contacts are
dominating. In the absenceof hydrodynamics in the friction law,
such a transitionis lost (see Supplemental Material). Indeed, in
Fig.1cfor increasing applied stress, the distributions of s inall
the particle contacts shift toward the BL regime inthe Stribeck
curve. In our simulations, the system shearthickens when at least ≈
20% of the contacts are belowsc. In Fig.1d, the percentage of
particles in BL and HDcontacts is plotted against γ̇ for the
stresses defined inFig.1b. For low µ0, this ST regime is continuous
and fitswith a Bagnoldian scaling (η ∝ γ̇). Here, the
viscosityincreases with µ0, as in a dry granular medium [24].
Thisscenario changes as µ0 goes beyond a critical value, here0.35
for φ = 0.58 (Fig.2). Then, the system cannot besheared above a
critical shear rate for any shear stress:the system shear-thickens
discontinuously.
FIG. 2. (color online)Apparent viscosity versus γ̇ for
differentµ0 and sc = 5 ·10−5 in simulations. In the Newtonian
regime,the viscosity does not depend on µ0 but on φ. At φ =
0.58,for µ0 ≤ 0.3, the system experiences CST, where the
viscositydepends on the friction coefficient. For µ0 ≥ 0.35, the
systemjams at sufficiently large γ̇. Data points for φ = 0.59
showDST for µ0 = 0.3. Inset: Zoom of the transition zone.
The transition from CST to DST does not only oc-cur when
increasing µ0 but also when increasing φ: thesystem experiences CST
at φ = 0.58, µ0 = 0.3 butexperiences DST for φ = 0.59 and the same
µ0 (seeFig2). Moreover, as predicted in our theoretical model,the
CST-DST transition occurs when φ is increased abovea φBL
max(µ0=0.3), compatible with [28].
In brief, the numerical simulations confirm that ourtheoretical
framework sets the sufficient conditions toexplain Nw-ST and
CST-DST transitions.
Our model is also independently supported by ex-periments where
the link between local friction andmacroscopic rheology is
established using quartz sur-faces. We first show experimentally
that the volumefraction of the CST-DST transition is indeed φBL
max
and then that it can be tuned by modifying µ0. Thisis
demonstrated by using four different comb poly-mers, i.e.
poly(methacrylic acid) (PMAA) graftedwith poly(ethylene glycol)
(PEG) side chains, whichare dissolved in a Ca(OH)2 saturated
aqueous buffersolution with 20 mmol/L K2SO4. The co-polymers
weresynthesized by radical polymerization in water accordingto [39,
40]. Their specifications, obtained from aqueousgel permeation
chromatography (GPC) are (backbonesize in kDa, number of carboxylic
acids per side chainand side chain size in kDa): Polymer A:
PMAA(4.3)-g(4)-PEG(2), Polymer B: PMAA(3.4)-g(2.3)-PEG(2),Polymer
C: PMAA(4.3)-g(4)-PEG(0.5) and Polymer D:PMAA(5)-g(1.5)-PEG(2).
Once in the buffer solution,these comb polymers are readily
adsorbed onto a neg-atively charged surface, such as quartz, by
calcium-ionbridging, and create a stable and highly solvated
PEGcoating on the solid surface [41] that is known to modifythe BL
coefficient of friction [42]. The conclusions of theexperiments are
not dependent on the choice of system,which is a model material for
industrial applications(e.g. cement slurry), for which the friction
coefficientcan be easily tuned.
The rheological analysis was performed on suspensionsof ground
quartz (Silbelco France C400, D50 = 12µm)with Φ between 0.47 and
0.57 in the alkali polymer solu-tions (see Supplemental Material
for details). We initiallymeasured φBL
maxvia compressive rheology by high-speed
centrifugation (acceleration ≈ 2000g) of a fairly low
con-centration suspension (φ = 0.47) in a 10mL measuringflask and
calculating the average sediment volume frac-tion for the various
polymers. During sedimentation athigh speed, particles come into
contact and jam, pro-ducing a looser sediment compared to
frictionless ob-jects. After 20 minutes of centrifugation, no
further evo-lution is observed and we measured φBL
max(A) = 0.578,
φBLmax
(B) = 0.560, φBLmax
(C) = 0.555, φBLmax
(D) = 0.545(see Fig.3a).The CST-DST transition was then measured
by shear
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FIG. 3. (color online) a) Sediment heights for the differ-ent
polymers after centrifugation. b) Viscosity vs shear ratewith
adsorbed polymer A for various φ of quartz micropar-ticle
suspensions. c) Viscosity vs shear rate for the four ad-sorbed
polymers at analogous φ (φ(A) = 0.537, φ(B) = 0.537,φ(C) = 0.538,
φ(D) = 0.535). d) Oswald-De Waele exponentn vs the reduced volume
fraction (same symbols as in c). In-set: Same data in log-log plot.
The solid line is a power-lawfit for (n− 1) vs reduced volume
fraction.
rheometry in a helicoidal paddle geometry (Anton Paar301
rheometer, see [21] Fig.4 for geometry description)with a
descending logarithmic stress ramp after pre-shear(from 700 to 0.01
Pa in 100s). The viscosity curvesare divided into two main parts:
at low shear rate, thefluid shows a Newtonian behavior with a
viscosity thatdepends on volume fraction [43] (Fig.3b) but not
onthe polymer coating (Fig.3c). For high shear rates, thefluid
shear-thickens. At moderate volume fractions, thesystem undergoes
CST with τ ∝ γ̇2 (Bagnoldian regime)as observed by [44], while for
the higher volume fractionsin our experiment, the abruptness of ST
increasesquickly at a critical Φ (see Fig.3b for Polymer A).
Abovethis threshold, the suspensions display DST. In orderto
quantify this critical volume fraction, the flow curvesfor the
various φ in the ST regime are fitted by anOswald-De Waele power
law: η ∝ γ̇n. In Fig3d, we showthat n(φ) diverges exactly at the
polymer-dependentφBLmax
that we measured independently by centrifugation,as predicted by
our model. Moreover, the data fromthe different polymer coatings
collapse onto a singlemaster curve as a function of a reduced
volume fraction(φBL
max− φ)/φBL
maxthat does not depend on surface
properties. A similar collapse was observed for particles
FIG. 4. (color online)φBLmax as a function of the coefficient
offriction in boundary regime µ0 for the four polymers (samesymbols
as in Fig.3). The CST and DST regions are high-lighted in the
graph.
of different shapes [45].
To complete our analysis we measured the BL fric-tion
coefficients µ0 between a polished rose quartz stonesurface
(Cristaux Suisses, Switzerland) and a 2 mm di-ameter borosilicate
sphere (Sigma-Aldrich, USA) coatedwith the four different polymers,
using a nanotribome-ter (CSM instruments, Switzerland). The contact
wasimmersed into a drop of polymer solution. The experi-ments were
realized in an N2 atmosphere at sliding ve-locities between 10−5
and 10−3 m/s (see SupplementalMaterial for a protocol). The
measured values of µ0 re-ported Fig.4in are speed independent, as
expected in theBL regime. The differences in the friction for the
differentpolymers have been previously ascribed to a variation
ofthe PEG unit density on the surfaces [46], stemming froman
equilibrium between entropic side chain repulsion
andbackbone/surface electrostatic attraction (through cal-cium
bridging).
Finally, Fig.4 shows the direct correlation between theBL
coefficients of friction and the measured maximumvolume fraction
φBL
maxthat separates CST and DST, as
included in our model. φBLmax
is a decreasing function ofthe particle friction coefficient in
the boundary regime,as predicted by simulations [28, 29].
Using a simple theoretical framework, independentlybacked up by
simulations and experiments, we have iden-tified the microscopic
origin of both continuous and dis-continuous shear-thickening of
dense non-Brownian sus-pensions as the consequence of the
transition from hydro-dynamically lubricated to boundary lubricated
contacts.The central role played by friction introduces the lo-cal
Sommerfeld number as the controlling parameter forthe transition
between Newtonian and shear-thickeningregimes, as demonstrated by
our numerical simulations.The presence of two distinct lubrication
regimes as a
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5
function of the Sommerfeld number is furthermore at theorigin of
the Nw-ST transition. In particular, the frictioncoefficient in the
boundary regime, which we tuned ex-perimentally by polymer
adsorption on the particle sur-face, governs the nature of the ST
transition. Distinctlubrication regimes imply that the jamming
volume frac-tions in the viscous regime φHD
maxand in the Bagnoldian
regime φBLmax
are not the same in general, given that onlythe latter depends
on the friction coefficient. ThereforeCST is found when φHD
max≥ φBL
max≥ φ, while the sus-
pension exhibits DST when the transition to the inertialregime
is impossible because φHD
max≥ φ > φBL
max. Thus,
in the absence of transient migration effects [44], the lo-cal
volume fraction and friction coefficient determine thestable
microscopic flow mechanism, which is either CSTor DST [44, 47, 48].
Moreover, our model does not re-quire any confinement at the
boundaries, but only thatlocally φ > φBL
max. This condition is fulfilled by prevent-
ing particle migration out of the shear zone, either
byconfinement during steady-state shear [18] or by keepingthe shear
duration short enough [49].
The generality and consistency of our data and of theproposed
model sets a global framework in which the tri-bological (friction)
and rheological properties of densenon-colloidal systems are
intimately connected. Thisconcept is expected to have an impact on
a host of prac-tical applications and relates fundamental issues
such asflow localization [50] and the solid-liquid-solid
transitionof granular pastes [14].
Acknowledgments - The authors thank Fabrice Tou-ssaint for
scientific discussions during preliminary workand Cédric Juge,
Abdelaziz Labyad and Serge Ghi-lardi for technical support.
The authors acknowledge financial support by LafargeLCR.
Furthermore, this work was supported by the FP7-319968 grant of the
European Research Council, the Am-bizione grant PZ00P2 142532/1 of
the Swiss NationalScience Foundation and the HE 2732/11-1 grant of
theGerman Research Foundation.
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-
Microscopic mechanism for the shear-thickening of non-Brownian
suspensionsSUPPLEMENTAL MATERIAL
Nicolas Fernandez,1Roman Mani,
2David Rinaldi,
3Dirk Kadau,
2Martin Mosquet,
3Hélène
Lombois-Burger,3Juliette Cayer-Barrioz,
4Hans J. Herrmann,
2Nicholas D. Spencer,
1and Lucio Isa
1, ∗
1Laboratory for Surface Science and Technology, Department of
Materials, ETH Zurich, Switzerland
2Computational Physics for Engineering Materials, Department of
Civil,
Environmental and Geomatic Engineering, ETH Zurich,
Switzerland3Lafarge LCR, Saint Quentin-Fallavier, France
4Laboratoire de Tribologie et Dynamique des Systèmes - UMR 5513
CNRS, École Centrale de Lyon, France
NUMERICAL SIMULATIONS
Friction law calculation
In order to simulate our dense paste we need to model
the behavior of the particles and the fluid. We assume
that the fluid layers between particles are very thin com-
pared to the particle radius and of the same order of
magnitude as the amplitude of their surface roughness.
This assumption allows us to avoid solving explicit equa-
tions for the suspending fluid, which are extremely com-
putationally expensive even for a single contact. It also
permits the use of a local friction law that takes into ac-
count the lubrication effect of the suspending fluid in the
form of the well-known Stribeck curve. The system can
then be easily simulated with contact dynamics.
A typical Stribeck curve shows two main lubrication
regimes separated by a ”mixed” zone: a boundary lubri-
cation (BL) regime where the asperities on the two slid-
ing surfaces are in contact and a hydrodynamically lubri-
cated (HD) regime where the shear of the fluid film is re-
sponsible for energy dissipation. Implementing boundary
lubrication is straightforward in simulations because the
friction coefficient in this regime is usually independent
of the speed and the load. The novelty in our simulations
is the implementation of the hydrodynamic regime in the
particle-particle contact friction law.
As already reported in the main body of the article, the
hydrodynamic interactions between two similar neighbor-
ing spheres in the HD regime are described by standard
low-Reynolds-number fluid mechanics with a lubrication
hypothesis (i.e. inter-surface distance small compared to
the radius of the particle). The drag force on one par-
ticle due to the other is given in the canonical reference
frame of the contact by (for overall formula [1] and for
detailed calculations: diagonal terms [2] and the non-
diagonal term [3]) :
FHD1→2 =π
10ηfRp
�−15h−1 12h− 12
0 −10ln(h−1)
� �vNvT
�(1)
where h is the surface-to-surface distance normalizedby Rp, ηf
is the fluid viscosity and vN and vT are thenormal and tangential
components of the local relative
speed respectively. By definition of the reference frame:
vT ≥ 0. Please note that spinning around the normaldirection is
neglected because other components of the
drag dominate for small h, i.e. the normal spinning dragdoes not
diverge when h −→ 0 [4].Additionally, the typical Reynolds numbers
of the
particle Rep =ρpR
2pγ̇
ηf(with ρp being the density of
the particle) are small for the range of shear rates
that we investigate experimentally. Inertia effects are
thus negligible and only the steady-state lubrication is
relevant. Moreover, the particles cannot interpenetrate
each other and the existence of a long-lived contact
itself implies that vN = 0, given that for vN > 0
theparticles leave each other. This hypothesis is justified by
prior simulation studies, in which the contact duration
between particles in dense granular media was long
compared to γ̇−1 [5]. As an additional consequence ofthis
hypothesis, the shear thickening that we observe
cannot be due to some hydroclusters that are arising
from attractive viscous forces in opening contacts.
With these hypotheses, Newton’s second law on one of
the spheres, projected on the normal and tangential axis
of the contact, becomes:
6πηfRp
5√h
vT + Fext
N= 0 (2)
− πηfRpln(h−1)vT + F extT = 0 (3)
with F extN
and F extT
being the sums of external forces
applied on the sphere by means of other contacts pro-
jected on the normal and tangential axes.
At this stage, following standard hydrodynamic lubri-
cation analysis [6], the minimum thickness of the fluid
layer between the contacting particles is given by equa-
tion (2):
h = (6π
5s)2 (4)
with s, the Sommerfeld number as defined in the paper.The ratio
between the tangential viscous drag and the
-
2
normal load gives the coefficient of friction µ reported inthe
main body of the paper:
µ =�F ext
T�
�F extN
� = 2πsln(5
6πs) (5)
As already explained in the main article, this formula
is valid only for small s because the lubrication hypoth-esis is
valid only for h ∝ s2 slim. In any case we have demonstrated in
thenumerical simulations that there are less than 0.5% ofthe
contacts with s > 10−1 in the considered range ofshear stresses
and that the resulting forces are weak.
Finally, as an additional consequence of the long-lived
contact hypothesis, the normal restitution coefficient, eN ,is
zero for both lubrication regimes. It can be seen phys-
ically as the result of the intense damping of any normal
speed when two particles are near each other, as seen
in Eq.1. Moreover, the normal restitution coefficient is
known to have a minor impact on the behavior of dense
granular media [7].
Variation of the parameters
In the simulations, the stress τ , the shear rate γ̇ andthe
apparent viscosity η can be tuned by varying thedimensionless
number
λ =
√τρR
ηf(6)
The time scale of the problem is set by [T ] =�ρ/τR
and thus, we can introduce the dimensionless shear rate
γ̇, velocity ṽ and load Ñ via
γ̇ = ˜̇γ
�τ/ρ
R= ˜̇γλ
ηfρR2
v = ṽ
�τ
ρ(7)
N = ÑτR2
The Sommerfeld number can be expressed as
s = ηfvr
N= ηf
ṽr
Ñ√ρτR2
=ṽr
ÑRλ= r̃
ṽ
Ñλ−1 (8)
where r = r̃R is the normalized particle radius. Forunequal
spheres, we assume that we can replace r by theaverage radius
2r−1
c= r−11 + r
−12 such that
s = r̃cṽ
Ñλ−1 (9)
Here, we readily see that for fixed φ, µ0, sc the only con-trol
parameter is λ. In our simulations, we varied λ to
obtain ˜̇γ as the simulation output. From Eq. (6) thestress τ is
obtained via τ = λ2η2
f/(ρR2) such that the
apparent viscosity is given by
η =τ
γ̇=
λ˜̇γηf (10)
Role of the lubricating fluid
As outlined in the main article, the observation of a
transition from a Newtonian to a shear thickening regime
relies on the presence of a lubricating fluid. Now we show
that, also in the numerical model, it is essential to in-
clude a lubricating contact law in order to observe this
kind of transition. Fig. 1 shows data from the main ar-
ticle superimposed to simulations where the lubrication
is disregarded, i.e. µ = const, or in other words, allcontacts
are always in the boundary lubrication regime.
Here, we see that the curve µ = 0.2 = const exhibitspure
Bagnoldian scaling for any shear rate γ̇ and coin-cides at large γ̇
with the data where lubrication is takeninto account (curve µ0 =
0.2). For µ = const, there isno transition to a Newtonian regime.
Furthermore, the
vertical solid line indicates that the system is jammed at
any applied stress for µ = 0.4 = const as opposed to lu-bricated
contacts where flow is possible at sufficiently low
γ̇ (see µ0 = 0.4). Note that as we consider infinitely
hardparticles in our simulations, the jamming point does not
depend on the applied stress as opposed to soft sphere
simulations as in Ref. [8].
−1 0 1 2 3 4−2
0
2
4
6
8
10
log 10(γ̇)
log10
(τ)
-∞
µ0 = 0.4µ0 = 0.2µ = 0.2 = constµ = 0.4 = const
FIG. 1. Shear stress τ as a function of γ̇ for two
differentfriction coefficients. Stars and squares correspond to
datawhere the lubrication is taken into account in the
numericalmodel, whereas filled circles correspond to a constant µ,
i.e.lubrication is disregarded. The solid line schematically
showsthe jammed state, where no flow is possible at any
appliedstress when lubrication is disregarded.
-
3
Effect of confining walls
In order to investigate the influence of the confin-
ing walls, we also modified the boundary conditions in
the vertical z-direction by removing the walls and us-ing
Lees-Edwards [9] boundary conditions instead. Shear
in x-direction is induced by considering moving (in x-direction)
mirror images of the simulation box. Shortly
speaking, particles at the bottom of the simulation box
interact with particles at the top of the box, which have
an x coordinate displaced by an amount δx = tvshearwhere vshear
is the shearing velocity and t is time.When particles cross the
boundaries in z-direction, thex-velocities are corrected by an
amount δv = vshear. Asin the main article, we simulate
stress-controlled shear-
ing at mean stress τ0 where the shearing velocity vshear
ismeasured. In order to keep a constant mean stress, vshearis
adjusted via the equation of motion v̇shear = (τ0 − τ)[10] where τ
= −1/V
�iF ixrizis the actual shear stress
of the sample. The sum runs over all contacts i havingcontact
force Fi and distance vector ri connecting thetwo centers of the
spheres. The effective viscosity of the
suspension is given by η = τ0/γ̇ where γ̇ = vshear/Lz andLz is
the system size in z-direction.
−1 0 1 2 3
1.5
2
2.5
3
3.5
4
4.5
5
5.5
log 10(γ̇)
log10
(η)
µ0 = 0.4µ0 = 0.3µ0 = 0.2µ0 = 0.1
FIG. 2. Effective viscosity as a function of the shear rate
fordifferent µ0 at φ = 0.595 for a system with periodic
boundaryconditions in all three directions.
Fig. 2 shows η as a function of γ̇ for different bound-ary
lubrication friction coefficients µ0 and volume frac-tion φ =
0.595. As in the main article, we observe twotransitions, from
Newtonian to Bagnold as γ̇ is increasedand from continuous to
discontinuous shear thickening as
µ0 is increased. These calculations suggest indeed thatfriction
law is responsible for both the different observed
flow regimes and the transitions between them. In par-
ticular the DST transition is not due to the presence
of confining walls. The critical volume fraction and µ0
at which the system experiences DST are slightly larger
compared to the values in the main article, but no qual-
itative differences are found. This is due to the fact that
the presence of hard walls has the effect of reducing the
accessible volume to the particles in the system [8].
POLYMER ADDITION AND RHEOLOGY
In shear rheology, any addition of small quantities of
adsorbing polymers decreases the viscosity, both in the
low-(deflocculation) and high-shear-rate (ST) regimes.
Above a certain mass of polymer per unit mass of quartz,
further addition of polymer no longer changes the suspen-
sion viscosity. This saturation takes place at polymer
concentrations in the solution below 2 %mass, thus far
below levels that could change the viscosity of the sus-
pending fluid. All the rheology experiments were thus
carried out with excess polymer relative to the satura-
tion mass ratio.
The working polymer mass ratios were r(A) = 3.1mg/g
SiO2, r(B) = 3.1 mg/g
SiO2, r(C) = 1.5 mg/g
SiO2
and r(D) = 4.0 mg/gSiO2
. These working concentrations
have not been precisely optimized, nevertheless the small
value of r(C) can be explained by polymer C shorter sidechains
that create a thinner brush layer and thus a lower
adsorbed mass per unit surface [11]. In addition to 20
mmol/L of K2SO4, an excess of Ca(OH)2 (6 mg/gSiO2 )was added to
maintain saturation of the buffer solution
even after the reaction of OH− with the surface of thequartz
grains and the adsorption of calcium ions. In simi-
lar conditions, the ζ-potential of silica surfaces have
beenreported to be around 6mV [12]. A drop of Surfynol MD-
20 (antifoaming Gemini surfactant, from Air Products,
USA) was also added to prevent air trapping during the
mixing.
Even if the adsorption of the polymers seemed imme-
diate, the experiments were performed at least 10 min
after the initial mixing. During this period the suspen-
sion can settle a bit, the suspension is then homogenize
by mixing and by the pre-shear (increasing logarithmic
stress ramp from 0.01 to 700 Pa in 100 s for the shearrheology,
and 5 min of whirly mixing)
FRICTION MEASUREMENTS
The friction between quartz grains cannot be directly
and accurately measured due to their size and their com-
plex shape. In order to establish the difference between
the lubrication ability of the four tested polymers, the
friction force was measured on a model tribosystem, a
borosilicate glass sphere with a radius of 2 mm (from
Sigma-Aldrich) and a polished quartz plane. The stone
was polished using a set of decreasing SiC polishing
papers and diamond pastes down to 1µm-grade (from
-
4
FIG. 3. Friction force for the 3 sliding speeds and a
constantload of 100 mN on one cycle of amplitude 2mm for polymerC.
Each value was recorded over a 8µm sliding distance (dueto time
resolution of the apparatus) but only 50 values perspeed (equally
spaced) are plotted
Struers, Denmark). Then the polished stone and the
spheres were cleaned for 20 min using Piranha solution
(7:3 concentrated H2SO4/30% H2O2). The sliding sur-faces were
immersed without contact between each other
in the drop of the polymer alkali solution (100 µL, poly-mer
concentration of 5 %mass) for 10 min before the be-
ginning of the measurement. The polymers are adsorbed
on both surfaces via calcium bridging under alkali con-
ditions. In addition, in order to prevent acidification of
the solution by CO2 dissolution, the measurements wereperformed
under a N2 atmosphere at room pressure andtemperature. Neither
modification of pH nor CaCO3precipitation was observed during the
experiments. The
experiments were carried out at a sliding velocity of 640,
70 and 17 µm/s under a constant load of 100 mN over a2 mm long
track under reciprocating conditions. Under
these conditions, according to Hertz’ theory [6], the av-
erage local pressure is around 0.1GPa and the expectedcontact
radius is 16µm, which remains small compared
to the sliding distance.
Between two measurements, the sphere was renewed
and the quartz surface was washed using a large volume
of neutral pH buffer solution and then pure water (Milli-
Q system from Millipore, 18.2MΩ.cm) in order to desorband remove
the polymer without modifying the quartz
surfaces by aggressive chemical and thermal cleaning pro-
cedures. As stated in the paper and shown in Fig.3, the
friction force does not depend on the sliding velocity and
is stable with time. The average friction coefficient is
then calculated over a distance of 1.2 mm.
∗ Corresponding author: [email protected][1] P. Coussot and
C. Ancey, Rhéophysique des pâtes et des
suspensions (EDP Sciences, 1999).
[2] S. Kim and S. J. Karrila, Butterworth-Heinemann(1991).
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W. Snidle and J. F. Archard, Proceedings of the In-
stitution of Mechanical Engineers 184, 839 (1969).[5] Y.
Forterre and O. Pouliquen, Annual Review of Fluid
Mechanics , 1 (2008).[6] G. W. Stachowiak and A. W. Batchelor,
Engineering tri-
bology (Butterworth-Heinemann, 2005).[7] F. Da Cruz, F. Chevoir,
J. N. Roux, and I. Iordanoff,
Tribology Series , 53 (2003).[8] M. P. Ciamarra, R. Pastore, M.
Nicodemi, and
A. Coniglio, Physical Review E 84 (2011),
10.1103/Phys-RevE.84.041308.
[9] M. P. Allen and D. J. Tildesley, Computer Simulation
ofLiquids, Oxford Science Publications (Oxford UniversityPress,
USA, 1989).
[10] M. Otsuki and H. Hayakawa, Physical Review E 83,051301
(2011).
[11] S. S. Perry, X. Yan, F. T. Limpoco, S. Lee, M. Müller,and
N. D. Spencer, ACS Applied Materials & Interfaces1, 1224
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mailto:[email protected]://onlinelibrary.wiley.com/doi/10.1002/aic.690400418/abstracthttp://onlinelibrary.wiley.com/doi/10.1002/aic.690400418/abstracthttp://dx.doi.org/10.1063/1.1707699http://www.annualreviews.org/doi/abs/10.1146/annurev.fluid.40.111406.102142http://www.annualreviews.org/doi/abs/10.1146/annurev.fluid.40.111406.102142http://store.elsevier.com/product.jsp?isbn=9780750678360&pagename=searchhttp://store.elsevier.com/product.jsp?isbn=9780750678360&pagename=searchhttp://dx.doi.org/10.1016/S0167-8922(03)80034-2http://dx.doi.org/10.1103/PhysRevE.84.041308http://dx.doi.org/10.1103/PhysRevE.84.041308http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0198556454http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0198556454http://dx.doi.org/10.1103/PhysRevE.83.051301http://dx.doi.org/10.1103/PhysRevE.83.051301http://pubs.acs.org/doi/abs/10.1021/am900101mhttp://pubs.acs.org/doi/abs/10.1021/am900101mhttps://www.jstage.jst.go.jp/article/jact/7/1/7_1_5/_articlehttps://www.jstage.jst.go.jp/article/jact/7/1/7_1_5/_article
Microscopic mechanism for the shear-thickening of non-Brownian
suspensionsAbstractAcknowledgmentsReferences