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

2

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

3

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

4

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

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.

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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).

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[12] C. Schroefl, M. Gruber, M. Lesti, and R. Sieber, Journalof Advanced Concrete Technology 7, 5 (2009).

mailto:lucio.isa@mat.ethz.chhttp://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