Rheology of dense granular suspensions under extensional flow

Oliver Cheal1 and Christopher Ness2

1Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom2Department of Chemical Engineering and Biotechnology,

University of Cambridge, Cambridge CB3 0AS, United Kingdom(Dated: April 24, 2018)

We study granular suspensions under a variety of extensional deformations and simple shear usingnumerical simulations. The viscosity and Trouton’s ratio (the ratio of extensional to shear viscosity)are computed as functions of solids volume fraction φ close to the limit of zero inertia. Suspensionsof frictionless particles follow a Newtonian Trouton’s ratio for φ all the way up to φ0, a universaljamming point that is independent of deformation type. In contrast, frictional particles lead toa deformation-type-dependent jamming fraction φm, which is largest for shear flows. Trouton’sratio consequently starts off Newtonian but diverges as φ → φm. We explain this discrepancyin suspensions of frictional particles by considering the particle arrangements at jamming. Whilefrictionless particle suspensions have a nearly isotropic microstructure at jamming, friction permitsmore anisotropic contact chains that allow jamming at lower φ but introduce protocol dependence.Finally, we provide evidence that viscous number rheology can be extended from shear to exten-sional deformations, with a particularly successful collapse for frictionless particles. Extensionaldeformations are an important class of rheometric flow in suspensions, relevant to paste processing,granulation and high performance materials.

I. INTRODUCTION

Industrial and geophysical processes that involve densesuspensions in motion invariably exhibit combinationsof shear and extensional flow [1, 2]. To achieve a use-ful description of their rheological properties, one musttherefore start with a sound knowledge of the materialresponse to both types of deformation. Despite this clearrequirement, most of the recent influential developmentsin the understanding of suspension rheology (both ex-perimental [3, 4] and numerical [5–7]), and indeed drygranular material rheology (see for example Refs [8, 9]),have focussed exclusively on shear flows. This shortcom-ing is understandable in part due to the relative diffi-culty of achieving purely extensional flows experimen-tally. Extensional deformations are, however, more se-vere than shearing in the sense that material elementsmove apart exponentially, rather than linearly, with time(or strain) [10], so in practical applications they may wellprove to dominate the overall rheological phenomenology.The relative importance of extensional to shear rheologi-cal properties is traditionally quantified using Trouton’sratio, the ratio of extensional to shear viscosity.

Extensional rheology is better understood in polymers,with many successful experimental approaches havingbeen developed over the past four decades. Classicaltechniques include melt stretching [11], filament stretch-ing [12, 13], flows through a contraction [14–19] and lu-bricated squeezing [20, 21]. Led by experimental insightsfrom such techniques, constitutive models for polymerrheology have long benefitted from understanding bothshear and extensional flows [22, 23]. For the continuedprogression of the field of suspension rheology, it is es-sential that the understanding of arbitrary deformationscan be brought up to speed with that of polymers.

In recent years, there have been a number of experi-

mental studies of the extensional rheology of dense sus-pensions using similar techniques to those above. Therehas been notable emphasis on shear thickening systems,a particular class of suspension that sits close to thecolloidal-granular interface [4]. A popular route hasbeen to use a filament stretching device to probe thehigh deformation rate uniaxial extension regime. Insuch experiments, Rothstein and coworkers observedstrain hardening in nano- and micrometre particle sus-pensions, with light scattering results suggesting particleself-organisation as the origin [24, 25]. The approach isrobust enough to detect changes in particle concentrationand solvent properties [26] and to examine properties rel-evant to printing and other applications [27, 28]. Devicesof this type have the added complexity of a liquid-airinterface, the shape of which distorts under rapid exten-sional flows, leading naturally to a connection betweenstrain hardening and granule formation [29]. Another se-ries of experiments placed a tensile load on a cornstarchsuspension [30], leading to the unexpected result thatshear jamming and shear thickening, both purportedlymanifestations of stress-induced particle friction [31], canbe independently inhibited using chemical modifiers [32].

A simulation model predicting shear thickening underextensional flow has emerged recently [33], and founda Trouton’s ratio of 4 for planar extension (both aboveand below shear thickening), surprisingly consistent withthe prediction for a Newtonian fluid. The analysis fo-cussed on the effect of polydispersity on shear thickeningat a small number of volume fractions. It is has not yetbeen explored whether there is solids volume fraction de-pendence or deformation-type dependence beyond planarextension. Experimental measurements of Trouton’s ra-tio have been reported for suspensions and found to beO(10) in the Brownian regime [34] and slightly above theexpected Newtonian values in the granular regime [35].

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Furthermore, particle roughness was found to enhancethe extensional viscosity, demonstrating the importanceof including particle-particle friction in numerical modelsand constitutive descriptions of extensional flow.

The focus on shear thickening is understandable givenits ubiquity in applications, but it has led to an overlook-ing of the underlying rheological behaviour of granular(by which we mean athermal) suspensions under exten-sional flow. This underlying behaviour is typically de-scribed under shear flow by the much-used viscous num-ber model, the so-called µ(Iv)-rheology [3]. It is a robustframework for rheological modelling of granular parti-cles of arbitrary particle-particle friction, and takes asits basis the assumption that for sufficiently hard spheresthe only relevant stress scale is the hydrodynamic one.This leads naturally to apparent Newtonian rheology inwhich all stresses scale linearly with shear rates, butremain highly sensitive to the solid volume fraction φ.The presence of a particle pressure under shear flow [3],complicated by an ambiguity in measured values of re-ported normal stress differences [36, 37], however, leadscrucially to the denomination ‘quasi’-Newtonian for anydense granular suspension described by µ(Iv)-rheology.Consequently, it is not clear whether this framework canbe generalised to extensional flows, and in particularwhether ratios of normal to shear stresses (expressed asTrouton’s ratio) should be truly Newtonian in such cir-cumstances, in spite of the direct proportionality betweenstresses σ and shear rates γ. In this respect, extensionalflows are an important class of rheometric flow for study-ing suspensions close to jamming.

In this article, we use numerical simulations to studydense granular suspensions under extensional flow. Weimplement a minimal discrete element-type numericalmodel that keeps track of the trajectories and forceson overdamped, neutrally buoyant suspended spheres,which are updated in a deterministic way according toNewtonian dynamics. The force terms comprise hy-drodynamic lubrication and harmonic contact potentialswith friction. The model operates in the athermal, non-inertial regime. We consider planar, uniaxial and biax-ial deformations and compute the Trouton’s ratios as afunction of the solid volume fraction, using shear flowas a reference. The distinction identified recently be-tween sliding and rolling contacts for suspended parti-cles [38, 39] and, by extension, the role of frictional forcesin suspensions of large particles, ought still to be validfor extensional flows, so it is essential to consider ex-plicitly the role of static friction between particles. Forthis reason, our model allows hydrodynamic lubricationforces to break down on some surface roughness length-scale, and we thereafter consider direct particle-particlecontacts with static friction coefficient µ.

We first describe our numerical simulation methodol-ogy (Section II), before describing the deformation typesstudied and the method of imposing them (Section III).We then describe the response of the material during thestraining period (Section IV), and go on to demonstrate

the divergence of the shear and extensional viscositieswith volume fraction (Section V). We find a discrepancyin the critical volume fractions for suspensions of fric-tional particles that can be explained by considering themicrostructural configurations at jamming (Section VI).Finally, we discuss to what extent the results for exten-sional flow can be mapped on to viscous number rheology(Section VII).

II. NUMERICAL MODEL

Our model considers athermal, noninertial, neutrallybouyant particles that represent a suspension corre-sponding to that used in the seminal experiment of Boyeret al. [3]. We consider a periodic domain containing bidis-perse spheres with solids volume fraction φ. The parti-cles have density ρ and radii a and 1.4a, mixed in equalnumbers. The simulation box is initialised with 12,000(shear) or 15,000 (extension) particles (we explore theimportance of system size in the Appendix) placed ran-domly before being relaxed to achieve minimally overlap-ping states. In what follows, we report ensemble averagesover five realisations obtained by changing the initial con-figurations using a random seed. The simulation box isdeformed according to a velocity gradient tensor U∞.Suspended particles are thus subjected to a rate of straintensor with symmetric and antisymmetric parts E∞ andΩ∞ respectively, where the background fluid flow at xfollows U∞(x) = E∞x+ Ω∞ × x.a. Hydrodynamic forces Hydrodynamic interactions

between particles are based upon the resistance matrixformalism described by Refs [40–42]. Following Ball andMelrose [43] we consider short-ranged, frame-invariant,pairwise interactions. For neighbouring particles 1 and2, translating with velocities U1, U2 and rotating at Ω1,Ω2, and with centre-centre vector r (and n = r/|r|)pointing from particle 2 to particle 1, it can be shown [44]that the force F h and torque Γh on particle 1 are givenby:

F h/ηf =(XA11n⊗ n+ Y A11(I − n⊗ n))(U2 −U1)

+ Y B11(Ω1 × n) + Y B21(Ω2 × n),(1a)

Γh/ηf =Y B11(U2 −U1)× n− (I − n⊗ n)(Y C11Ω1 + Y C12Ω2),

(1b)

where ηf is the Newtonian viscosity of the suspendingliquid. For particle radii a1 and a2, the surface-surfaceseparation is given by h = |r|−(a1+a2), which we nondi-mensionalise as ξ = 2h/(a1 + a2). The scalar resistancesXA

11, Y A11, Y B11 , Y B21 , Y C11 and Y C12 comprise short rangecontributions that diverge as 1/ξ and ln(1/ξ) and aregiven in Appendix A. We neglect interactions that haveh > 0.05a (with a the smaller particle radius). The per-force hydrodynamic stresslet is Shij = − 1

2

(Fhi rj + Fhj ri

).

3

A drag force and torque act on particle 1 at position x1,given by

F d = −6πηfa1(U1 −U∞(x1)), (2a)

Γd = −8πηfa31(Ω1 −Ω∞(x1)), (2b)

leading to per-particle contributions to the stresslet givenby Sd = − 20π

3 ηfa31E∞.

b. Contact forces Following experimental evidencethat lubrication layers break down in suspensions underlarge stress [38], and, equivalently, for large particles [4],we use a minimum h defined as hmin = 0.001a, belowwhich hydrodynamic forces are regularised and particlesmay come into mechanical contact. For a particle pairwith contact overlap δ = ((a1+a2)−|r|)Θ((a1+a2)−|r|)and centre-centre unit vector n, we compute the contactforce and torque on particle 1 according to [45]:

F c = knδn− ktu (3a)

Γc = a1kt(n× u) (3b)

where u represents the incremental tangential displace-ment, reset at the initiation of each contact. kn and ktare stiffnesses, with kt = (2/7)kn. The tangential forcecomponent is restricted by a Coulomb friction coefficientµ such that |ktu| ≤ µknδ. For larger values of |ktu|,contacts enter a sliding regime. We take the stresslet asScij = −F ci rj .

Trajectories are computed from the above forces. Con-tact and hydrodynamic forces and torques are summedon each particle and the trajectory is updated accord-ing to a Velocity-Verlet algorithm. The dynamics arecontrolled by three dimensionless quantities: the vol-ume fraction φ, the Stokes number St and the stiffness-scaled shear rate ˆγ. We ensure that the Stokes numberSt = ργa2/ηf remains 1 throughout to approximateover-damped conditions. We found O(10−3) to be suffi-ciently small in practice and achieved this by setting par-ticle radius a = 0.5 [length], density ρ = 1 [mass/length3],suspending fluid viscosity ηf = 0.1 [mass/(time× length)]and shear rate γ = 0.001 [1/time]. In this limit we ex-pect rate-independent rheology in which all stresses scalelinearly with deformation rates. The extent to which theparticles may be considered hard spheres is set by theshear rate rescaled with particle stiffness, as given byˆγ = 2γa/

√kn/(2ρa) [8]. We set ˆγ < 10−5 through-

out by setting kn = 50000 [mass/time2]. The modelis implemented in LAMMPS [46]. The overall stress ten-sor is computed by summing over all of the stresslets

σ = −2ηfE∞ + 1

V

(∑i S

d +∑phSh +

∑pcSc)

where

the sums are over individual particles i, hydrodynami-cally interacting pairs ph and contacting pairs pc.

III. DESCRIPTION OF APPLIEDDEFORMATIONS

Rate ofdeformation E∞

Magnitude|E∞|

Viscosity

Simpleshear

0 12γ 0

12γ 0 0

0 0 0

γ σ12/γ = η†

Planarextension

−γ 0 00 γ 00 0 0

2γ (σ22−σ11)/γ = 4η†

Uniaxialextension

− 1

2γ 0 0

0 − 12γ 0

0 0 γ

√3γ (σ33−σ11)/γ = 3η†

Biaxialextension

γ 0 00 γ 00 0 −2γ

2

√3γ (σ11−σ33)/γ = 6η†

TABLE I. Rate of deformation tensors E∞, their magnitudes,and the viscosity definitions for each type of flow explored inthis work.

We consider shear, planar, uniaxial and biaxial flows asillustrated schematically in Figures 1(a)-(d) respectively.Consider the general velocity gradient tensorU∞ in threedimensions which has components ∂vi/∂xj . From this,we obtain the components of the symmetric rate of defor-

mation tensor as E∞ij = 12

(∂vi∂xj

+∂vj∂xi

). Shear flows have

a corresponding rotational part Ω∞ij = 12

(∂vi∂xj− ∂vj

∂xi

),

while extensional flows are irrotational. For the case of aNewtonian fluid, the stress tensor is then given simply byσij = −pδij+2η†E∞ij where η† is the Newtonian viscosity.In Table I we present the rate of deformation tensors E∞

corresponding to each of the flow types explored in thiswork, as well as the magnitudes |E∞| =

√2E∞ : E∞ and

the associated viscosity definitions, which follow Ref [2].It is noted that the framework of Jones et al. [47] dictatesthat for the uniaxial Trouton’s ratio we should comparethe shear viscosity at γ with the uniaxial extensional vis-cosity at

√3γ. Comparing the reported deformation rate

magnitudes in Table I, we see that this requirement is sat-isfied. We take the Trouton’s ratios (Tr) as the ratios ofeach of the extensional viscosities to the shear viscosity.These lead to values of the Newtonian Trouton’s ratio of4, 3 and 6 for planar, uniaxial and biaxial flows, respec-tively. We will use these values as a basis for comparisonfor the extensional flows modelled in this work.

Volume-conserving deformations are applied to thesimulated suspension by incrementally changing the di-mensions of the periodic box according to the relevantrate of deformation tensor. To simulate simple shear, weuse a triclinic periodic box with a tilt length Lxy (seeFigure 1(a)) that is incrementally increased linearly intime as Lxy(t) = Lxy(t0) + Lyγt, giving a deformation

4

γt0 0.5 1 1.5 2

Troutonratio

01

34

6

10

16

y

zx

a

y

zx

b

c

d

h

i

shearuniaxialplanarbiaxial

Lxy2

shearuniaxialplanarbiaxial

e

f

g

γt = 2 γt = 1.5 γt = 1 γt = 0.5

.

.

.

.strain γt

strain γt

rela

tive

visc

osity

η/η

f Tr

outo

n’s ra

tio

.

.

FIG. 1. Schematics of the deformations applied in this work. Shown are (a) simple shear; (b) planar extension; (c) uniaxialextension; (d) biaxial extension. In each case the wireframe box illustrates the box dimensions at an earlier time and the redarrows indicate the directions of the applied deformation. The upper coordinate diagram refers to (a) while the lower one refersto (c)-(d). The box deformations lead to uniform velocity gradients. Shown in (e)-(g) are examples of the velocity gradientsobtained during uniaxial extension at increasing strain increments. (h) Plot of viscosity as a function of strain for each flowtype (at φ = 0.45 and friction µ = 1) showing start-up period (shaded) and the steady flow period (unshaded). Black arrowindicates region from which viscosities are used for averaging. (i) Plot of Trouton’s ratio as a function of strain for each flowtype. Dashed lines represent the corresponding Newtonian values. In each case the grey shaded area represents the maximumand minimum values obtained during five independent simulation runs. Coloured bars next to figure labels (a)-(d) correspondto the colours in (h) and (i).

that is entirely equivalent to that obtained using a Lees-Edwards boundary condition. For extensional flows, theleading box dimension is increased with time accordingto L(t) = L(t0)eγt to give a constant true strain rate,that we quantify as γ. The other box dimensions arevaried accordingly to conserve the volume. We verifiedthat neglecting particle-particle interactions and impos-ing simply the deformation protocol described here andthe Stokes drag forces described above leads to parti-cle trajectories that follow precisely the affine deforma-tion of the simulation box. The velocity of any particlethat crosses a periodic boundary is remapped accordingto the velocity gradient across the box perpendicular tothat boundary. The velocity gradient at any point in thesimulation box at any time represents the overall appliedbox deformation and thus the particles are subjected touniform velocity gradients as illustrated in Figure 1(e)-(g).

Whereas the shear deformation can be continuallyremapped to permit arbitrarily large deformations, theextensional deformations are constrained in magnitudesince our simulation approach involves ‘shrinking’ one of

the box dimensions with time. Taking the uniaxial de-formation as an illustrative example, we initiate the sim-ulation box with 15,000 particles of radii a and 1.4a andwith a cuboidal box of dimensions 171.4a× 171.4a× 10a(giving φ = 0.4 in this case). During the period for whichwe observe strain-independence of the viscosity (see be-low), the box dimensions remain O(10)a in x, y and z.There is uniform straining throughout the sample duringthis period, with steady state locally acting velocity gra-dients that match the overall box deformation (see Fig-ure 1 and Supplementary Video). We verified that thereis no system size dependence by simulating a smaller sam-ple and achieving a comparable (though shorter) steady-state period. Our numerical model breaks down at largeextensional strains as the contracting dimensions of thesimulation box reach O(1) particle radii and particles‘see’ themselves through periodic images (see Appendix).Notwithstanding the difficulty in achieving large defor-mations, the approach we describe here to achieve steadyvelocity gradients during various extensional deforma-tions has been discussed and applied previously in severalworks across glassy and polymeric systems (see, for ex-

5

ample, Refs [48–54]). To reach larger strains, it is neces-sary to implement remappings such as those described byKraynik and Reinelt [33, 55, 56] for planar deformations.For materials involving long time/length scales (polymermelts for instance), these boundaries are essential. Fordense suspensions, one the other hand, that can reacha steady state within strains of 1 or 2 [57], they maybe useful for some studies but are not crucial to studysteady phenomena. The planar deformation used in thiswork is equivalent to that acting between remappings ofthe Kraynik-Reinelt scheme.

IV. EVOLUTION OF SUSPENSION VISCOSITYWITH STRAIN

Starting from a quiescent state with minimal particle-particle contacts, we begin the constant-rate deforma-tion. The viscosity for each is computed from the sim-ulation data according to the definitions in Table I. Forexample, the suspension viscosity under uniaxial exten-sion is given by η = (σzz − σxx)/γ. This result is fur-ther rescaled by the suspending fluid viscosity ηf and wethus present reduced viscosities as η/ηf . Viscosity ver-sus strain plots for each deformation type are given inFigure 1(h), at volume fraction φ = 0.45 and frictioncoefficient µ = 1. For small strains γt < 1 we iden-tify start-up regimes in which the viscosity increases withstrain. During this time, the number of direct particle-particle contacts increases with strain and a flow-inducedmicrostructure establishes [57, 58]. As can be seen, weare able to achieve a strain-independent region with astrain magnitude γt = O(1). It is noted that the biaxialextension simulation is conducted using the output fromthe uniaxial extension as the initial condition. Thus theinitial period presented for biaxial flow corresponds toa flow reversal rather than to a start-up from a quies-cent state. Interestingly, a familiar characteristic surgein stresses [57] (hydrodynamic in origin) is observed atvery small strains, indicative of placing closed particlecontacts under tension as discussed by Refs [59, 60].

In Figure 1(i) we give the evolution of Trouton’s ratiowith strain, evaluated by rescaling the extensional vis-cosities by the shear viscosity at each strain increment.There are two interesting features to note. The moststriking is that, for γt > 1, the results are remarkablyclose to the Newtonian values. This suggests that thequasi-Newtonian character of overdamped suspensionsextends beyond the linear scaling of shear stresses withshear rates. The result for planar extension matches thatpredicted by an independent simulation model [33]. Thesecond interesting feature is the surge in Trouton’s ra-tio for each of the extensional flows, with a maximumat around γt = 0.5. This indicates a faster microstruc-tural evolution for extensional flows compared to shearflows. Such a finding is consistent with the form of theapplied deformations, which see fluid elements move to-gether/apart exponentially for extensional flows but lin-

early for shear flows. The shaded regions in Figure 1(i)represent the maximal and minimal values obtained overfive independent simulation runs, indicating a very weakdependence on the initial configuration. Error bars arethus omitted from the following results and discussion.

V. EVOLUTION OF THE VISCOSITY ANDTROUTON’S RATIO WITH VOLUME

FRACTION

Presented in Figure 2 is the evolution of viscosity forfrictionless (a) and frictional (b) particles and the evolu-tion of Trouton’s ratio Tr for frictionless (c) and frictional(d) particles, with volume fraction φ for each deforma-tion type. In general, the viscosities for all flow typesfollow the form η/ηf = α(1−φ/φc)−β , with φc a generic‘critical’ volume fraction and β a scaling parameter muchdiscussed in the literature [61] and reported to be ≈ 2 inshear flow experiments (see, for example, [4]). Follow-ing conventional nomenclature [31] we drop φc and labelthe frictionless and frictional jamming points φ0 and φm,respectively, where φm depends on µ. We fitted such aform to our biaxial extension data and found α = 1.25,β = 1.6 and φ0 = 0.644 for frictionless and α = 1.1, β = 2and φm = 0.575 for frictional particles. Also shown in (a)and (b) are the hydrodynamic and contact contributionsto the shear viscosity which, when summed and comple-mented by the Stokes term 2ηfE

∞ lead to the total shearviscosity. We find that for frictional particles, contactsdominate even for φ < 0.4, while for frictionless particlescontacts only become dominant at φ > 0.54. Comparablebehaviour of the contact and hydrodynamic viscosities isobtained for all of the flow types.

For φ > (φ0, φm), the suspension enters a jammed stateas indicated by the shaded red region in Figures 2(a)-(d).Here, flow is only possible through particle deformationsand thus for strictly hard spheres, jamming representsflow arrest. For the nearly-hard spheres considered inthis work, we enter a high stress flowing regime in whichan elasticity emerges corresponding to the stiffness of theparticle-particle repulsion. Such a region can only be ob-served experimentally when the particles are sufficientlysoft, for example in emulsions [62]. In any case, the flowin this region is not strictly viscous and thus is not ex-pected to obey Newtonian Trouton’s ratios.

The values of Tr presented in Figures 2(c)-(d) demon-strate a remarkably broad range of volume fractions forwhich the flow appears to be approximately Newtonian,persisting up to φ ≈ 0.62 for frictionless and φ ≈ 0.54for frictional particles. Tr reaches between 7 and 8 underbiaxial extension, but given the logarithmic scale overwhich the overall stresses are varying, we consider a dis-crepancy of ≈ 25% from the Newtonian value to be in-substantial. Interestingly, our simulation result impliesthat there is very weak dependence of Trouton’s ratio onparticle-particle friction for φ < 0.54, despite the impor-tance of the contact stress contribution and the dominant

6

0.4 0.45 0.5 0.55 0.6 0.651

34

6

10

0.4 0.45 0.5 0.55 0.6 0.65 0.71

34

6

10

0.4 0.45 0.5 0.55 0.6 0.65 0.7100

102

104

0.62 0.64 0.6610-2

10-1

100

101

0.4 0.45 0.5 0.55 0.6 0.65100

102

104

106a b

c d

shearuniaxialplanarbiaxial

shear hydroshear contactbiaxial fit

quasi-Newtonian jammed jammedquasi-Newtonian

approaching jamming

approaching jamming

frictionless frictional

frictionalfrictionless

uniaxialplanarbiaxial

uniaxialplanarbiaxial

φ0 φm

volume fraction φ

volume fraction φ

volume fraction φ

volume fraction φ

rela

tive

visc

osity

η/ηf

rela

tive

visc

osity

η/ηf

Trou

ton’s

ratio

Trou

ton’s

ratio

η/ηφ

=φ0biaxial

0.55 0.56 0.57 0.58 0.5910-2

100

φ0biaxial η/ηφ

=φmbiaxial

φmbiaxial

FIG. 2. Top: Divergence of the reduced viscosity η/ηf as a function of volume fraction φ for shear, planar, uniaxial and biaxialflow with frictionless (a) and frictional (b) particles. Also shown are the relative hydrodynamic and contact contributions tothe shear viscosity. Qualitatively equivalent results are obtained for the extensional flows (not shown). Highlighted in thered shaded region in (a) and (b) are the ‘jammed’ regions where the rheology is no longer expected to be viscous and thusa Newtonian Trouton’s ratio is not expected. Insets (a) and (b): Same data as (a) and (b) but focussing on the region nearjamming where there is a discrepancy in jamming volume fraction. Plotted on the y-axes are the viscosities rescaled by theirvalues at jamming (as measured under biaxial extension). Highlighted in Inset (b) in the red circle is the anomalous point forshear flow, which enters the jammed region at higher volume fractions than extensional flows. Bottom: Evolution of Trouton’sratio with volume fraction φ for planar, uniaxial and biaxial flow with frictionless (c) and frictional (d) particles. Highlightedare regions where the ratio matches that of a Newtonian fluid (blue), where it deviates on the approach to jamming (green),and where it is fully jammed (red).

role of friction in setting the viscosity at these volumefractions.

An interesting disparity between frictionless and fric-tional particles emerges in the ‘approaching jamming’region, for volume fractions 0.54 < φ < 0.65. Inthe frictionless scenario, a narrow transition window of∆φ ≈ 0.02 exists in which Tr rapidly and monotoni-cally switches from its low (φ < φ0) plateau to a high(φ > φ0) plateau. The monotonicity suggests that each ofthe flowing states approach a common, deformation-type-independent, value of φ0. The width of this ‘approachingjamming’ region decreases with increasing particle stiff-ness as the transition to jamming becomes sharper (seeAppendix). By contrast, Tr for frictional particles beginsto exceed its Newtonian values around ∆φ ≈ 0.05 belowjamming φm. This suggests there is a window in whichthe extensional viscosity of suspended frictional particles

exceeds the shear viscosity by up to an order of mag-nitude. In fact, we find (see Appendix) that this spikein the Trouton’s ratio for frictional particles scales withthe stiffness of the particles, strongly suggesting that atvolume fractions in this region, Tr actually represents aratio between jammed and flowing states (rather thantwo stiffness-independent flowing states as is the case forfrictionless particles, which show no such scaling) thusimplying a discrepancy in φm for different flow types. Re-turning to the viscosity divergence plotted in Figure 2(b)Inset, we verify that the surge in Tr corresponds to a mis-match in the frictional jamming volume fraction φm fordifferent flow types, as highlighted by the red circle thatindicates the entry to jamming for shear flow is shiftedto the right with respect to extensional flows. The ex-tensional viscosities tend to diverge at a common volumefraction that is approximately 0.005 below that for shear

7

flow. Based on this monotonicity and nonmonotonicityin Tr for frictionless and frictional particles respectively,we thus conclude that φm depends subtly upon the defor-mation type whereas φ0 does not. (Though there appearsto be a visual mismatch between φ0 values for differentdeformation types in Figure 2a, the monotonicity of Tr(Figure 2c) proves that the shift is only in the y−axisand not in x.)

It is also noted that there is weak φ dependence ofTr above φ0 (φm) for frictionless (frictional) particles. Ifwe crudely take the rheology here to be quasistatic [8],and thus dependent on the ‘shape’ of the deformationtensor but not the relative magnitude of the deformationrate, we can obtain a reasonable prediction of Tr abovejamming. Specifically, for planar, uniaxial and biaxialflows we obtain, respectively, Tr ∼ 4/2, ∼ 3/

√3 and

∼ 6/(2√

3) above jamming, regardless of particle-particlefriction, corresponding to the representative viscositiesrescaled by the magnitude of E∞ (see Table I).

The deformation type dependence of φm suggests aclear route to intermittent jamming through changes indeformation type. For example, at a volume fraction ofφ ≈ 0.575, a suspension of frictional particles is quasi-Newtonian under shear flow, but jammed under exten-sional flow. This poses a direct challenge to industrialprocesses that involve mixed flow, suggesting that a fluidelement at fixed volume fraction might transiently jamand unjam dependent upon the instantaneous flow typeto which it is subjected. Such an effect is not predictedfor frictionless particles.

VI. MICROSTRUCTURAL BEHAVIOUR CLOSETO JAMMING

We observed a deformation-type-independent criti-cal volume fraction φ0 for frictionless particles, but adeformation-type-dependent φm for frictional particles.This is consistent with earlier observations that frictionaljamming, which occurs at φm, shows protocol depen-dence and hysteresis. Flow arrest in frictional particles isthus often described as a fragile or shear-jamming tran-sition that masks an underlying true jamming transitionwhich occurs at φ0 (with φ0 > φm) [63].

When frictional forces are large, percolating chains ofstable but fragile particle-particle contacts can permitjamming with considerable anisotropy at volume frac-tions below φ0 (see, for example, Refs [63–65]). In suchsystems, experiments show hysteretic effects whereby thematerial initially jams at some low packing fraction (sim-ilar to our φm here) but upon further perturbations itconsolidates and approaches φ0 [66]. No such hysteresisis observed at frictionless jamming [67], which thus oc-curs when the material reaches an isotropic (or, at least,more isotropic, see Baity-Jesi et al. [68]) packed state.Our suspension of frictionless particles might thus reachjamming at an isotropic configuration that is not protocol(i.e. deformation type) dependent, whereas the frictional

particles reach jamming when their dynamically evolvingforce chains are able to percolate the system and permitan anisotropic jammed state, which is necessarily proto-col dependent.

To test whether this description is suitable for explain-ing our observed divergence of Tr for frictional (but notfrictionless) particles, we consider the microstructuralanisotropy at the critical volume fraction. To do this,we consult a familiar form of fabric tensor defined asAij = 〈ninj〉 − 1

3δij [69], where ni is a particle-particleunit vector and angular brackets represent an averageover all particles that are in mechanical contact (definedwhen |r| < (a1 + a2)). For a large, isotropic sample, oneobtains Aij → 0. We use scalar representations of thefabric, A, corresponding to the viscosity definitions givenin Table I, for shear A := Axy, planar A := Ayy − Axx,uniaxial A := Azz − Axx and biaxial A := Axx − Azzdeformations.

Fabric data are presented for frictionless and frictionalparticles in Figures 3(a) and (b) respectively. In all casesnegative A corresponds to a preferential orientation ofcontacts along the compressive flow axis, which we il-lustrate schematically in Figures 3(c) and (d) for shearand planar extension deformations, respectively. In Fig-ures 3(e) and (f) we plot the distribution of the vectorni projected onto the xy-plane for shear and planar flow.There is always an alignment of contacts along the com-pressive axes, regardless of φ, friction and deformationtype. We find that A→ 0 as φ increases, indicating thatthe microstructure generally becomes more isotropic asjamming is approached. Crucially, it is observed thatthere is a strong disparity in the values of A measuredat the jamming point when comparing frictionless andfrictional particles. Frictionless particles jam when A iscloser to 0 (indicative examples are A = −0.028 for shearand A = −0.029 for planar deformations at φ0, indicatedin Figure 3a) indicating that flow-arrest is achieved witha more isotropic microstructure than in frictional flows,which have A = −0.063 and A = −0.113 respectively attheir respective φm (Figure 3b). This finding is also ap-parent in the radial distributions shown in Figures 3(e)-(f). These show a more anisotropic distribution of con-tact forces at the jamming volume fraction for frictionalcompared to frictionless particles, with a surplus of par-ticle contacts along the NW-SE axis under shear flowand the E-W axis under planar flow. In contrast, theprofiles for frictionless particles are, while not perfectlycircular, rather more uniform. Moreover, there is littledeformation type dependence in the value of A at φ0 forfrictionless particles, suggesting that, although the def-initions of A vary with each case, jamming occurs witha similarly isotropic structure in each case. By contrast,there is quite some variation in A at φm for frictionalparticles, again emphasising the dependence upon defor-mation type.

Frictionless particles only jam when their arrangementis nearly isotropic, so it doesn’t matter what type of de-formation we apply; frictional particles can jam in an

8

0.48 0.5 0.52 0.54 0.56 0.58 0.6-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.5 0.55 0.6 0.65-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

b

A = -0.063

A = -0.113

a frictionless

shearuniaxialplanarbiaxial A = -0.028

A = -0.029

y

zx

shear

planar

frictional

frictionless φ = 0.645

frictional φ = 0.58

shear

frictionless φ = 0.645

frictional φ = 0.575

planar

y

z x

y

z x

d

c

f

e

volume fraction φ

volume fraction φ

scal

ar fa

bric

Asc

alar

fabr

ic A

FIG. 3. Behaviour of the microstructural fabric close to the critical volume fraction for each deformation type exploredin this work. Shown are the scalar fabric A for (a) frictionless particles and (b) frictional particles. Dashed arrows showexemplary values of A at jamming. In (c) and (d) we draw schematic illustrations of microstructural fabric for shear flow andplanar extension, respectively. Dark shading corresponds to load-bearing contacts while light shading corresponds to ‘spectator’particles [70]. Straight blue arrows indicate compressive axes; curved blue arrows indicate rotational component; grey arrowsindicate streamlines corresponding to affine flow. Shown in (e) and (f) are radial distributions of particle-particle contacts(defined when |r| < (a1 + a2)) at the critical volume fraction, projected onto xy for shear and planar flows. The shape of thedistribution reflects the values of A (dashed black lines show the result when our algorithm is run using 5× 105 random pointsdistributed uniformly over a spherical surface). Suspensions of frictionless particles jam with A less than half the frictionalvalue, and thus the distribution is more circular.

anisotropic state, so it matters how we deform them upto this point. We thus conclude that frictionless particleshave constant Tr all the way to φ0 because different defor-mation types share this critical volume fraction; frictionalparticles have a deformation type dependent φm whichis highest for shear flows, meaning Tr diverges betweene.g. φuniaxialm and φshearm .

VII. MAPPING THE EXTENSIONALDEFORMATIONS ONTO VISCOUS NUMBER

RHEOLOGY

We finally verify that the numerical model describedherein predicts flow behaviour under shear and extensionthat qualitatively follows the viscous number rheologyframework proposed by Boyer et al. [3] very well. Theviscous number is defined as Iv = ηf γ/P (for suspend-ing fluid viscosity ηf , deformation rate γ and pressureP ) and works as an analogue of the inertial number usedin dry granular material modelling [9]. In the athermal,non-inertial limit, the rheological state of a suspension

can be uniquely defined using two functions that relatethe volume fraction φ and the stress ratio τ = σ/P to theviscous number Iv. The stress ratio, which is ordinarilytaken as the ratio between the shear stress and mean nor-mal stress (i.e. the pressure P ), is defined in this workaccording to the viscosity definitions given in Table I.Specifically, we replace the shear stress with a genericstress σ given by σ := σxy for shear, σ := σyy − σxx forplanar, σ := σzz − σxx for uniaxial and σ := σxx − σzzfor biaxial deformations. For each flow type, the pressureis taken simply as P = − 1

3

∑i=x,y,z σii. The functions

φ(Iv) and τ(Iv) are presented in Figure 4. Crucially,qualitatively consistent behaviour is observed for bothshear and extensional flows and for both frictionless andfrictional particles. Comparing frictionless and frictionalcases quantitatively, we find discrepancies in the criti-cal φ, as discussed above, as well as discrepancies in thelimiting τ at low Iv, which has been discussed earlierby Da Cruz et al. [71]. We also show in Figure 4(b)and (d) the predictions based on the model proposedby Boyer et al. [3], for which they give parameters ap-

9

propriate for frictional particles (we use their parame-ters here). Sources of discrepancy between the presentresult and the model prediction are variations in poly-dispersity (which alter the numerical value of the criticalvolume fraction φ measured when Iv → 0), variations inparticle-particle friction coefficient (which alter the nu-merical value of the limiting stress ratio σ/P measuredas Iv → 0) and variations in particle hardness (which al-ter the critical viscous number at which volume fractionsmay exceed the critical volume fraction).

Considering φ(Iv) in Figures 4(a) and (b), we find somediscrepancy in the quantitative results for different flowtypes below the critical volume fraction. Interestingly, wefind that a convincing collapse of the data in this region isobtained if we redefine Iv based on the magnitude of thedeformation rate tensor, that is replacing γ with |E∞| togive I ′v = ηf |E∞|/P , Figure 4(a) and (b) [Inset]. Thisresult implies that an alternative Trouton’s ratio (Trp)may be defined for the mean normal stresses, taking val-ues that correspond approximately to |E∞| for each flowtype. Comparing τ(Iv) in Figures 4(c) and (d), we sim-ilarly find a qualitative match for all flow types, but aquantitative discrepancy. Since we have demonstratedsatisfactory correspondence to Newtonian Tr for a broadrange of φ, as well as a convincing collapse of φ(Iv) with|E∞| that implies an equivalent Trp, we crudely definea rescaled stress ratio as τ ′ = (σ/Tr)/(P/Trp). Using τ ′

and I ′v as defined above, we again are able to collapse thedata, Figure 4(c) and (d) [Inset]. The collapse is partic-ularly convincing for frictionless particles and still rathergood for frictional particles.

This result demonstrates that, provided the stressesare rescaled appropriately by their Trouton’s ratios(which we have shown can be considered as Newtonianfor a broad range of φ), the viscous number rheologyframework proposed by Boyer et al. [3] can predict therheology for all of the deformations considered in thiswork with a single set of parameters. We provide ex-amples of such a fitting for the frictionless case (Fig-

ures 4(a) and (c) (Inset)), using φ(I ′v) = φ0/(1 + I′1/2v )

and τ ′(I ′v) = µ1 + (µ2−µ1)/(1 + I0/I′v) + I ′v + 2.5φ0I

′1/2v

with φ0 = 0.644, µ1 = 0.1, and the parameters µ2 = 0.7and I0 = 0.005 following Boyer et al. [3]. Similarly forthe frictional case (Figures 4(b) and (d) (Inset)), with

φ(I ′v) = φm/(1 + I′1/2v ) and τ ′(I ′v) = µ1 + (µ2 − µ1)/(1 +

I0/I′v) + I ′v + 2.5φmI

′1/2v with φm = 0.575 and the pa-

rameters µ1 = 0.32, µ2 = 0.7 and I0 = 0.005 follow-ing Boyer et al. [3]. Since the necessary stress rescal-ings derive directly from the relationships between therate of strain tensors defined above, and according toNewtonian rheology (at least for volume fractions up toslightly below jamming), we can characterise the rheol-ogy of athermal, noninertial particle suspensions in anyof the studied flows based on the rate-independent for-mulation [3]. Interestingly, the log-linear axes in Fig-ures 4(c) and (d) Inset reveal a potential mismatch inthe functional form of τ ′ for frictionless and frictional

particles on the approach to jamming. We expect thatthis does not derive from the effects of polydispersityor particle hardness mentioned above, but rather repre-sents a qualitative difference in the nature of the stressesat flow arrest when contacts are sliding or rolling. Theasymptotic behaviour of τ ′(I ′v) for frictionless particleshas been previously demonstrated in the absence of hy-drodynamic interactions [7], and further analyses basedon the current model are deferred to future work.

VIII. CONCLUSION

We have thus shown that for a broad range of volumefractions the underlying extensional rheology of densesuspensions can be described simply by a NewtonianTrouton’s ratio. This leads to a good agreement with vis-cous number rheology, provided the stresses are rescaledappropriately by the Trouton’s ratio, which is availablea priori from the known deformation tensor. For sus-pensions of frictionless particles, our model predicts noflow-type dependence on the critical volume fraction forjamming φ0 and consequently the Trouton’s ratios arefixed up to φ ≈ 0.63. This result is relevant for ather-mal suspensions with normal repulsive interactions be-tween particles, for example emulsions and silica suspen-sions below shear thickening. By contrast, a disparity injamming volume fractions φm for different deformationsemerges for frictional particles, suggesting that mixedflows with shear and extensional components might jamand unjam at fixed volume fraction and stress, simplydue to changes in the deformation. This is relevant forsuspensions of large granular particles of the type de-scribed under the framework of Boyer et al. [3], and alsofor silica suspensions above the onset of shear thickening.

It would be interesting to determine whether, in prac-tice, chaotic flow or even oscillating flows of the typedescribed by Pine et al. [72] that can eliminate particle-particle contacts might serve to inhibit the role of load-bearing force chains and thus extend the range of volumefractions that exhibit Newtonian Trouton’s ratios evenfor frictional particles. Achieving a general descriptionof extensional rheology is relevant to numerous applica-tions that involve mixed flows of dense suspensions, no-tably in footstuffs [73], ceramic paste extrusion [1, 74] andcalcium phosphate injections for bone replacement treat-ments [75]. In addition, dense suspensions are emergingas a useful material for energy dissipation during impacts,for which both biaxial [76] and uniaxial [77] configura-tions are relevant.

IX. ACKNOWLEDGEMENT

CN acknowledges the Maudslay-Butler Research Fel-lowship at Pembroke College, Cambridge for financialsupport. We thank Jin Sun and Zeynep Karatza for use-ful discussions and Ranga Radhakrishnan for sharing his

10

10-6 10-4 10-2 1000.4

0.45

0.5

0.55

0.6

0.65

0.7

I ′v = ηf |E∞|/P10-6 10-4 10-2 100

φ0.4

0.5

0.6

0.7

10-6 10-4 10-2 1000

2

4

6

8

I ′v = ηf |E∞|/P10-6 10-4 10-2 100

τ′=

(σ/T

r)/(P/T

r p)

10-1

100

10-6 10-4 10-2 1000.4

0.45

0.5

0.55

0.6

0.65

10-6 10-4 10-2 1000

1

2

3

4

5

volu

me

fract

ion φ

volu

me

fract

ion φ

Boyer (2011)

Boyer (2011)

a b

c d

frictionless frictional

frictionless frictional

viscous number Iv = ηfγ/P viscous number Iv = ηfγ/P

viscous number Iv = ηfγ/P viscous number Iv = ηfγ/P

stre

ss ra

tio τ

= σ

/P

stre

ss ra

tio τ

= σ

/P

shearuniaxialplanarbiaxial

. .

..

φ(Iv’)

I ′v = ηf |E∞|/P10-6 10-4 10-2 100

φ

0.4

0.5

0.6

I ′v = ηf |E∞|/P10-6 10-4 10-2 100

τ′=

(σ/T

r)/(P/T

r p)

0.5

1

2

φ

φ

(I 0v) =0

1 + I01/2v

(I 0v) =m

1 + I01/2v

φ(Iv’)

τ’(Iv’) τ’(Iv’)

FIG. 4. Viscous number rheology for shear and extensional flows. Shown are the volume fraction as a function of viscousnumber for frictionless (a) and frictional (b) particles, and the stress ratio as a function of viscous number for frictionless (c)and frictional (d) particles. We redefined the viscous number replacing γ with |E∞| and redefined the stress ratio by rescalingσ with a Newtonian Trouton’s ratio for each flow type. We thus arrive at the collapsed plots of volume fraction as a function ofviscous number for frictionless ((a), Inset) and frictional particles ((b), Inset) and stress ratio as a function of viscous numberfor frictionless ((c), Inset) and frictional particles ((d), Inset). We provide fits to the Inset data according to the expressionsgiven therein, with the parameters given in the main text. Also shown in (b) and (d) are the predictions given by Boyer et al.[3] based on shear flow experiments.

derivation of the lubrication forces [44]. A video showinga typical deformation and the scripts needed to reproduceit are given at https://doi.org/10.17863/CAM.13415.

Appendix A: Scalar resistances for hydrodynamiclubrication forces

The scalar resistances used in the hydrodynamic forcemodel described in Section II follow those presented byKim and Karrila [42] and are given (for β = a2/a1) by

XA11 = 6πa1

(2β2

(1 + β)31

ξ+β(1 + 7β + β2)

(5(1 + β)3)ln

(1

ξ

)),

(A1a)

Y A11 = 6πa1

(4β(2 + β + 2β2)

15(1 + β)3ln

(1

ξ

)), (A1b)

Y B11 = −4πa21

(β(4 + β)

5(1 + β)2ln

(1

ξ

)), (A1c)

Y B21 = −4πa22

(β−1(4 + β−1)

5(1 + β−1)2ln

(1

ξ

)), (A1d)

Y C11 = 8πa31

(2β

5(1 + β)ln

(1

ξ

)), (A1e)

Y C12 = 8πa31

(β

10(1 + β)ln

(1

ξ

)). (A1f)

Appendix B: Check that the simulation result isn’taffected by finite-size effects

Using simple shear as a test case, we simulate vari-ous periodic box sizes (i.e. particle numbers) to checkthat there are no finite size effects. For simulations withN > 3000, we find rather convincing system size inde-pendence. Thus we conclude that the results presentedin this work, which all have N> 10000, are not influencedby system size.

11

Appendix C: Demonstration of simulation breakingdown for large uniaxial strains

As discussed in the main text, our extensional flowsimulations do not allow arbitrarily large deformations,but rather are limited by the shrinking length of the com-pressive axes. We tested the maximal strain that can bereasonably achieved under uniaxial extension by deform-ing the box until the measured viscosity shows unphysicalbehaviour, Figure A2. For both system sizes considered,we are able to obtain a strain independent viscosity in thestrain window 1→ 4.5. We thus constrain the averagingwindow for all extensional flow simulations considered inthis work to that range of strains.

Appendix D: Role of particle stiffness

To confirm that the spike in Trouton’s ratio observedfor frictional particles does indeed represent a ratio be-tween a flowing and a jammed state, we repeated thesimulations using particles with increased stiffness. Since

the stresses in the flowing states are roughly independentof particle stiffness (since we are already near the hardparticle limit) while the jammed state stresses scale withkn/a [8], we find that the magnitude of the spike in Trou-ton’s ratio for frictional particles at φ = 0.575 scales withthe particle stiffness, Figure A3.

0.5 0.52 0.54 0.56 0.58 0.6

102

103

104

105

volume fraction φ

rela

tive

visc

osity

η/ηf

FIG. A1. Divergence of simple shear viscosity for frictionalparticles (µ = 1) with volume fraction for several differentsystem sizes, measured in terms of total particle number N.

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Volume fraction φ0.55 0.6 0.65 0.7

Trouton’s

ratio

1

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3

4

Volume fraction φ0.52 0.56 0.6 0.64

Trouton’s

ratio

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outo

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volume fraction φ

frictional

ˆ = 2a/p

kn/(2a) = 4.5 x 10-6

3.2 x 10-6

2.2 x 10-6

1.5 x 10-6

FIG. A3. (a) Trouton’s ratio versus volume fraction foruniaxial deformation and fricitonless particles with increasingparticle stiffness kn, kt. (b) Trouton’s ratio versus volumefraction for uniaxial deformation and fricitonal particles withincreasing particle stiffness kn, kt.

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