Temporally heterogeneous dynamics in granular flows

0.52 MB·English

by  L. E. Silbert

Temporally heterogeneous dynamics in granular flows
∗
Leonardo E. Silbert
James Franck Institute, University of Chicago, Chicago, Illinois 60637 and
Department of Chemistry and Biochemistry, University of California Los Angeles, Los Angeles, California 90095
(Dated: February 2, 2008)
Granular simulations are used to probe the particle scale dynamics at short, intermediate, and
5 longtimescalesforgravitydriven,densegranularflowsdownaninclinedplane. Onapproachtothe
0 angle of repose, where motion ceases, the dynamics become intermittent over intermediate times,
0 with strong temporal correlations between particle motions – temporally heterogeneous dynamics.
2 Thisintermittencyischaracterisedthroughlargescalestructuraleventswherebythecontactnetwork
periodically spansthe system. A characteristic time scale associated with these processes increases
n
as the stopped state is approached. These features are discussed in the context of the dynamics of
a
J supercooled liquids near theglass transition.

6
PACSnumbers: 81.05.Rm,45.70.Mg,64.70.Pf,61.20Lc
2

]
t Granular materials persist at the forefront of contem- Granularsimulationsofchuteflowshavecapturedfea-
f
o porary research due to the extreme richness in their dy- tures of the rheology with remarkable accuracy [10, 11].
s namics,eitherundershear[1],flowingoutofahopper[2], Threeprincipalflowregimesareobservedforchuteswith
.
t or driven by gravity [3]. Because thermodynamic tem- roughbases(seeFig.1). Ifθistheinclinationangleofthe
a
m perature plays little role in determining these features, chute: i) no flow occurs below the angle of repose θr, ii)
granularmaterialsarewidelyrecognizedasamacroscopic steadystateflowexistsintheregionθ <θ <θ ,andiii)
- r u
d analogue of athermal systems far from equilibrium [4]. unstable flow occurs for θ >θu. For all θ <θu, the den-
n The notion of jamming [5] provides a unifying frame- sity is constant throughout the bulk, away from the free
o work to describe the behaviour of a wide range of sys- top surface and lower boundary, with smoothly varying
c
tems, from the molecular level (supercooled liquids and depth profiles of the flow velocity and the principle and
[
glasses), microscale (colloidal glasses), through to the shearstresses[10]. Thelocationofθ alsodependsonthe
r
1
macroscale(granularmatter),neartheirjammingtransi- heightofthe flowingpile h [7, 11]. Pouliquenintroduced
v
tion. The motivation for this work comes from the view thequantityh (θ)thatencodesallthe(undetermined)
7 stop
1 that granular flows can be used as a macroscopically ac- informationabout the roughness and effective friction of
6 cessible analogue through which we can gain a better the base to relate the dependency of θr on h. In the
1 understandingofthecriticalslowingdownofthedynam- language of glasses and colloids, h (θ) represents the
stop
0 ics in systems near their point of jamming. One of the glass transition or yield stress line, respectively.
5
features associated with the approach of the glass tran-
0
sitionisthe conceptofspatiallyheterogeneousdynamics
/
t
a [6]. This workpresents features ofa drivengranularma-
m terialflowingdownaninclinedplane–chuteflows–that
- exhibitlargescale,cooperativeeventsonapproachtothe
d angle of repose where flow stops – its jamming point.
n FIG. 1: Phase behaviour of granular chute flow (schematic).
o Gravitydrivenflowscontinuetobestudiedextensively θr istheangleofreposeatwhichflowceases. θm isthemaxi-
c mumangleofstabilityatwhichflowisinitiatedfromrest. θu
duetotheirubiquitythroughoutnature–avalanches,de-
: is themaximum angle for which stable flow is observed. The
v
i bris flows, and sand dunes – and also because of their angle θi at which intermittentdynamicsemerges, theshaded
X importance in materials handling throughout industry. region, coincides with θi ≈θm. For the simulations reported
r More recently, there has been a focus of attention to- here, θr =19.30◦±0.01◦, θm =20.3◦±0.2◦, and θu ≈26◦.
a wards understanding the properties of chute flows be-
cause of the ability of obtaining well-defined and repro-
ducible steady state flow regimes. Despite the fact that Thisworkcharacterisesthelarge-scalefluctuationsand
recentexperimentshavebeenabletodetermine,indetail, emerging spatial and temporally heterogeneous dynam-
macroscopicinformationofgranularflows [7, 8], probing ics for a flowing state in the vicinity of θ . This is in
r
the internal dynamics remains problematic due to the contrast to discrete avalanches in heap flows [14]. θ is
i
opaqueness of the particles. Still, such flows offer a rela- the angle, for θ <θ ≤θ , such that, over time scales in-
r i
tivelycleansysteminwhichtodeveloptheoreticalmodels termediatebetweenballisticpropagation,atshorttimes,
forconstitutiverelationsaswellasdescribingthe flowat and diffusive motion, at long times, the particle dynam-
the level of the grain size [9]. This is where the role of icsbecomestronglyintermittent. Thisintermittencyisa
simulation has proved extremely useful. consequenceoflargespatialandtemporalcorrelationsin
2
particlemotions,throughtherelaxationandreformation sured normal to the inclined plane in the z direction:
ofmechanicallystableclustersthatperiodicallyspanthe ∆z2 ≡< (z(t) − z(0))2 >. During continuous flow,
system. A characteristic time scale τ , associated with thereisaclearcrossoverbetweentheballisticregime,for
c
these correlations is found to increase as (θ−θ )→0. t/τ <10−1,anddiffusiveregime,t/τ >10. Closertothe
r
Thesimulationswereperformedusingthe discreteele- jammingpoint, the MSDoscillatesovertimes intermedi-
ment technique with interactions appropriate for granu- ate between them. The appearanceof noveldynamics in
lar materials [10]. Monodisperse (soft) spheres of diame-
ter d and mass m flow downa roughened, inclined plane 101 24o
due to gravity g. Initially, θ is set to θu to remove any 100 20o
preparation history, then incrementally reduced in steps 10−1 19.5o
of 0.01◦ ≤ ∆θ ≤ 0.5◦. Particles interact only on con- 10−2 19.35o
tact (cohesionless), along directions normal and tangen- 10−3 19.32o
tial(viastaticfriction)tothevectorjoiningtheircentres. 2z −4
∆ 10
Contacts are defined as two overlapping neighbours. All
−5
10
runs were performed with a particle friction coefficient
−6
10
µ = 0.5 and coefficient of restitution ǫ = 0.88. Most of −7 (a)
the results presented are for N = 8000 particles flowing 10
−8
down a chute that is L =20d long, L =10d wide (the 10
x y −2 −1 0 1 2
10 10 10 10 10
xy-plane employs periodic boundary conditions), and an
averageflowheighth≈40d(withafreetopsurface). See t/τ
Fig. 5. Massively, large-scale, parallel simulations were
used for N = 160000, with (L ,L ,h) = (100,40,40)d,
x y
to demonstrate that the observed features are not a sys- FIG.3: Themeansquareddisplacementinthedirectionnor-
tem size artifact. Length scales are non-dimensionalised maltotheplane∆z2,withdecreasinganglefrom toptobot-
tom. Allcurves havebeen shifted for clarity.
bytheparticlediameterdandtimeinunitsofτ =pd/g.
The intermittent regime was first identified through
measurements of the kinetic energy as shown in Fig. 1. the vicinity of θ is a consequence of strong correlations
r
In the continuous-flow, steady state regime the energy emergingintime-time quantitiessuchasthevelocityau-
is approximately constant with small fluctuations about tocorrelation function, C(t) ≡< v (t)v (0) >, where v
z z z
themeanvalue. Astheinclinationangleisincrementally is the velocity component in the direction normal to the
decreased towards θr, changes in the time profile of the plane. See Fig. 4(b). These fluctuating quantities are a
energy first become apparent for θi ≈20.2◦. signature of temporally heterogeneous dynamics. To in-
vestigate this further, the average coordination number
0.04
o z, Fig. 4(a), is shown over the intermediate time win-
) 19.5
u. 20o dow, demonstrating that the structural processes asso-
a. ciated with these correlations involve large-scalecooper-
( ative events. During the flowing phase of the intermit-
y 0.03
g tentregime,zremainslowerthanthemechanicallystable
r limitforfrictionalspheresz =4[15]. Inthestaticphase,
e c
n the systems attains a value z ≈ z , thus almost coming
c
E
to rest. (There is always a residual creeping motion as
0.02
c flow does not completely stop until θ .)
i r
t
e To facilitate such processes, the contact force network
n
i 24o undergoes temporal transitions between static and flow-
K
ing phases. This evolution is captured in the (distinct)
0.01
0 2 4 6 8 10 particle-particlecontactforce-forcetimecorrelationfunc-
t/τ tion F(t), as shown in Fig.4(c). The physical picture of
the intermittent regime is thus: over some characteris-
tic time scale the system oscillates between almost total
mechanical stability and fluid-like flow. This involves
FIG.2: Averagekineticenergyperparticleoveranintermedi-
atetimewindowforN =8000anddifferentinclinationangles time-dependent, system-spanning, structural relaxation
as indicated. The symbols are for N =160000 at θ=19.5◦. events through the break-up and reformation of particle
contacts. This process is best depicted in the simulation
snapshots of Fig. 5.
Becauseoftheconvectivenatureoftheflow,themean- Theintermittentregimecanthereforebeconsideredas
squared displacement (MSD), shown in Fig. 3, is mea- temporally biphasic. During one phase, z ≈ z , so the
c
3
4 depending on the time window of observation.
z 3 100
2
1 (a) 10−1
) 1 10−2
0
0.5
(
C ) −3
0 f 10
/ (
) P
t −0.5
(
C −1 (b) 10−4
1 10−5
)
(t −6
F 0.5 10
(c) 0 2 4 6 8 10
0
f
0 2 4 6 8 10
τ
t/
FIG.6: Probability distribution function P(f) of thenormal
contact forces (normalised so f¯=1) for θ =20◦; total (solid
FIG.4: Intermediatetimewindowassociatedwiththeoscilla- line), the high-z phase (dashed), and the low-z phase (dot-
tionsintheMSD,for19.35◦ (dashedline),19.5◦ (thick),and ted). In the high-z phase, the contact forces are binned into
24◦ (dotted): (a) coordination number z, (b) velocity auto- a histogram if z > 3.5 for that configuration, and similarly
correlation functionC(t),and(c)force-force timecorrelation if z < 2.0 in the low-z curve. The full P(f) for θ = 22◦
function F(t).
(symbols) is shown for comparison.
Theintermittentregimeispeculiarduetothecomplex
interplay between several different characteristic length
and time scales. Identifying the structural entities of
characteristic size ℓ, remains a matter of debate [9, 16].
However, the general picture is as follows: one expects
ℓ to increase with decreasing θ, or correspondingly, the
shearrateγ˙. Then,θ istheanglewhereℓ≈h,resulting
r
in the transition from a dynamic state to a static one.
The development of an intermittent regime suggests an-
other competing time scale, whereby ℓ ≈ h can occur,
but other mechanisms drive the system away from sta-
FIG. 5: The contact force network at θ = 20◦ for three suc- bilitybeforethesystempermanentlyjams. Thisseemsto
cessive times. (a) Snapshot corresponds to a peak in the co-
coincidewiththeregimewherethescalingoftheflowve-
ordination number,(b)atrough,and(c)thenextpeak. The
locitydiffersfromthatofthecontinuousflowregime[11].
shade and thickness of the lines represent the magnitudes of
Presently, it is not clear how to approach this problem
the normal forces between two particles in contact. The in-
clination has been removed in the figures and the frame is a without a more sophisticated particle scale analysis.
guide to theeye. The simulation axes are indicated. Progress can still be made by identifying the charac-
teristictimescalesoftheintermittentregime. Oneisthe
periodicity in the kinetic energy, correlation functions,
andz. Thistimescaleisratherinsensitivetoθ inthein-
system is close to mechanical stability, while the flowing termittentphase. However,C(t)hasanenvelopeofdecay
phasehasaconsiderablylowercoordination. Asignature that corresponds to the extent of the intermittency. A
of this is seen in the probability distribution function of time scale τ , that characterisesthe initial decay in C(t)
c
the normal contact forces P(f). The partial distribu- is shown in Fig 7. Interestingly, a power-law divergence:
tions shown in Fig. 6 are for the high-z phase (defined τ ∼ (φ−φ )−γ, with γ ≈ 0.65, captures the behaviour
c r
as configurations with z > 3.5) which resembles that of quite well. As a comparison, a modified Vogel-Fulcher
a static packing–exponentialdecayathighforces– and formisalsoshowninFig.7,whichissatisfactorycloseto
separately the low-z phase (z < 2), with a high-f tail thejammingpoint,butdeviatesfurtheraway. Theavail-
that decays more slowly. This suggests that it may be able range for analysis is somewhat restricted through
possible to distinguish between different types of flows this particulardefinition ofτ , soit remains unclear how
c
4
thistimescalemightcorrespondtothemorefamiliardef- time scales associated with the onset of dynamical het-
initions of relaxation times in supercooled fluids [17]. A erogeneity and cooperativity are experimentally accessi-
more appropriate definition of τ is being investigated. ble, granular flows offer a convenient system to address
c
questions often associated with glasses at the molecular
2
10
and colloidal level. This emerging picture thus begs the
question, can existing theories that are currently being
appliedtotraditionalglassesandcolloidsbe extendedto
granular systems?
I thank Bulbul Chakraborty for insightful conversa-
τ
1 tions and Gary S. Grest for critiquing the manuscript.
10
c
GrantsupportfromGrantNos.NSF-DMR-0087349,DE-
FG02-03ER46087, NSF-DMR-0089081, and DE-FG02-
03ER46088is gratefully acknowledged.
0
10
−2 −1 0
10 10 10
∗ Electronic address: [email protected]
θ − θ
[1] J.-C. Tsai, G. A. Voth, and J. P. Gollub, Phys. Rev.
r Lett. 91, 064301 (2003); M. Toiya, J. Stambaugh, and
W. Losert, Phys. Rev.Lett. 93, 088001 (2004).
FIG. 7: Characteristic time scale τc, with error bars, asso- [2] E89.,L0o4n5g5h0i1, N(2.0E02a)s;wJar.,Cahnodi,NA..MKeundorno,llPi,hRy.s.RR.eRvo.sLaeletst,.
ciated with the intermittent structural events at intermedi-
and M. Z. Bazant, Phys.Rev.Lett. 92, 174301 (2004).
ate times as a function of distance from the jamming point
(θ−θr). The solid line is a power law fit: τ ∼ (θ−θr)−γ, [3] N. Taberlet, P. Richard, A. Valance, W. Losert, J. M.
with γ ≈ 0.65; whereas the dashed line is of the modified Pasini, J.T. Jenkins,andR.Delannay,Phys.Rev.Lett.
aVnodgepl-≈Fu0lc.3h9e.r form: τc ∼ exp{A(θ−θr)−p}, with A ≈ 0.66 9R1e,v.26L4e3t0t.19(320,0036)8;0A01.V(2.0O0r4p).eandD.V.Khakhar,Phys.
[4] L. P. Kadanoff, Rev.Modern Phys. 71, 435 (1999).
[5] A. J. Liu and S. R.Nagel, Nature 396, 21 (1998).
In conclusion, gravity-driven, dense, granular flows [6] M.D.Ediger,Ann.Rev.Phys.Chem.51,99(2000),and
down an inclined plane exhibit intermittent dynamics references therein.
[7] O. Pouliquen, Phys.Fluids 11, 542 (1999).
over intermediate time scales in the vicinity of the an-
[8] E. Azanza, F. Chevoir, and P. Moucheront, J. Fluid
gle of repose. The intermittent regime signals the onset
Mech. 400, 199 (1999); C. Ancey, Phys. Rev. E 65,
of temporal heterogeneous dynamics, whereby the short
011304 (2001).
time motion is ballistic, at long times diffusive, but at [9] P. Mills, D. Loggia, and M. Tixier, Europhys. Lett. 45,
intermediate times a combination of creeping flow and 733(1999);I.S.AransonandL.S.Tsimring,Phys.Rev.
trap-like dynamics exists. The system becomes trapped E 64, 020301 (2001); A. Lemaitre, Phys. Rev. Lett. 89,
in temporally metastable states [18] through large scale 064303(2002);D.Erta¸sandT.C.Halsey,60,931(2002);
M. Y.Louge, Phys.Rev. E 67, 061303 (2003).
cooperative events. Correlations between these events
[10] D.Erta¸s,G.S.Grest,T.C.Halsey,D.Levine,andL.E.
decay,butdosomoreslowlyasthe jammingpointisap-
Silbert, Europhys. Lett. 56, 214 (2001); L. E. Silbert,
proached. The behaviour of the characteristictime scale
D.Erta¸s, G.S.Grest, T.C.Halsey,D.Levine,andS.J.
τc, shares similarities with the properties of supercooled Plimpton, Phys.Rev. E 64, 051302 (2001).
liquids in the vicinity of the glass transition. Such dy- [11] L.E.Silbert,J.W.Landry,andG.S.Grest,Phys.Fluids
namical similarities were recently probed experimentally 15, 1 (2003).
in vibrated bead packs [20]. Therefore, in a typically [12] L. E. Silbert, D. Erta¸s, G. S. Grest, T. C. Halsey, and
D. Levine, Phys.Rev.E 65, 051307 (2002).
heuristic fashion, an analogy with glassy dynamics the-
[13] C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel,
orycanbe made. θ marksthe pointwhere alldynamics
r Phys. Rev.Lett. 86, 111 (2001).
cease and in this sense plays the role of the glass transi- [14] P.-A. Lemieux and D. J. Durian, Phys. Rev. Lett. 85,
tion temperature Tg. Indeed, earlier work [12] showed 4273 (2000).
that the flowing-static transition resembles the way a [15] S. F. Edwards, Physica A 249, 226 (1998).
model liquid approaches the glassy phase [13]. However, [16] L. E. Silbert, unpublished.
the way the dynamics change at θ , is similar is spirit to [17] C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMil-
i lan, and S. W. Martin, J. App.Phys. 88, 3113 (2000).
the way the dynamics in a supercooled fluid changes at
[18] J.-P. Bouchaud, A. Comtet, and C. Monthus, J. Phys. I
themodecouplingtemperatureT [19]. Withthisviewin
c France 5, 1521 (1995).
mind, one can associate θi with Tc. These results there- [19] W. G¨otze and L. Sj¨ogren, Rep. Prog. Phys. 55, 241
fore hint towards a unifying picture of dynamical het- (1992).
erogeneities in dense, amorphous systems. Because the [20] G. D’Annaand G. Gremaud, Nature 413, 407 (2001).