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Dynamics of Grains in Driven Granular Media

Narayanan Menon and Douglas J. Durian

MRS Proceedings / Volume 463 / 1996DOI: 10.1557/PROC­463­313

Link to this article: http://journals.cambridge.org/abstract_S1946427400214048

How to cite this article:Narayanan Menon and Douglas J. Durian (1996). Dynamics of Grains in Driven Granular Media. MRS Proceedings,463, 313 doi:10.1557/PROC­463­313

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DYNAMICS OF GRAINS IN DRIVEN GRANULAR MEDIA

NARAYANAN MENON, DOUGLAS J. DURIANDepartment of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547

ABSTRACT

We use diffusing-wave spectroscopy (DWS) to study microscopic dynamics in the interior of 3-dimensional granular systems. We study two granular systems where particle motions are excitedby different driving mechanisms -- gravity-driven channel flow and a gas-fluidized bed. In bothinstances we obtain detailed information about short-time collisional dynamics such as rmsvelocity fluctuations, mean free paths, and collision frequencies. We also observe a slowcrossover from short-time ballistic motions to long-time, grain-scale diffusive motions.

INTRODUCTION

In the absence of external driving forces, a granular system comes to rest because ofdissipative interactions such as friction and inelastic collisions between individual grains.However, when subject to geophysical processes, human handling or manipulation in industrialsituations, granular systems are driven into motion 1. We do not, however, have an established'fluid mechanics' for sand to describe macroscopic response to external sources of energy. Themajor effort toward such a continuum description are in the form of kinetic theories 2 based on theintuition that sand may be described as a gas of inelastic particles. In contrast to molecular fluids,the scale of velocity fluctuations is non-thermal and in these models is introduced as a temperature(referred to as the 'granular temperature') which is determined by the state of flow. While thisqualitative picture appears reasonable (at least in the limit of a dilute, highly-driven system), it hasnever been established by experiment or been quantitatively probed in real 3-dimensional flows.

In this article we present a direct probe of local, short-time dynamics of grains in dense,fully three-dimensional systems with Diffusing Wave Spectroscopy (DWS) 3 a multiple lightscattering technique which measures mean-square particle displacements as a function of timedown to 108 secs with a resolution of 1A. For two different driving mechanisms -- gravity-drivenflow and gas-fluidization -- we deduce important microscopic inputs to a description of transport :the rms velocity fluctuations, the mean free path and mean collision frequencies. Since granularsystems are opaque due to scattering at grain surfaces, most previous experimental work has beenconfined to quasi-two-dimensional flows. MRI techniques [4] and (less directly) video imaging oftracer beads [5] can yield information about 3-dimensional flows. However, as we shall showhere, the short-time collisional dynamics are at too short length (.01 - 1 gm) and time (10-_6104 s)scales to be directly accessible to these techniques. Furthermore, we shall also show that therelationship between the short-time collisional motions and long-time diffusive and convectiveprocesses is more complex than previously believed, rendering it impossible to infer the collisionalparameters from measurements at scales greater than the particle size.

EXPERIMENT

MaterialsWe use cohesionless, smooth, spherical glass beads with diameters, d = 49±5, 95+15,

97±11, and 194+17 jtm. The beads are prepared by baking them and cooling in a dry, inertatmosphere before use. In channel flow, we monitor the effect of humidity on the beads and

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Mat. Res. Soc. Symp. Proc. Vol. 463 0 1997 Materials Research Society

repeat this procedure as necessary. In the fluidized bed, a continuous flow of N2 gas preventssuch effects.

Gravity-driven channel flow:Sand flows from a large reservoir into a vertical, rectangular channel (30 cm high, 10 cm

wide and 0.3 - 1 cm thick), and then flows out through a grid of finely-spaced holes at the bottom.The flow velocity in the channel is varied between .03 and 3 cm/s by controlling the size andspacing of the grid. The flow is stationary, and if a uniform grid is used, may be characterized by asingle average density and flow velocity, V,-, everywhere in the channel. Spatial gradients arefound to be small in all three directions by long-range video microscopy, light transmission, andDWS. We have also studied non-uniform flows, where we create macroscopic velocity gradientsusing a non-uniform grid at the outlet. The arrangement of beads in flow shows no evidence ofdensity inhomogeneities or crystalline clusters. The effect of the ambient air is small: the Bagnoldnumber, given by the ratio of grain inertia to the Stokes drag, is greater than 104.

Fluidized bedThe bed is contained in a glass cell of square cross-section (5.9 cm x 5.9 cm) and

supported on a frit which also functions as a gas distributor. The fluidizing gas is N2, meteredinto a chamber below the distributor at a regulated pressure through a needle valve. A flowmetermeasures the volume flow rate of gas. The pressure drop across the frit is measured by adifferential pressure transducer as a function of gas velocity and is subtracted from the totalpressure drop to obtain the drop across the bed.

Light scatteringWe illuminate the sample using an Ar+ laser with wavelength, X=488 or 514 nm and 3mm

beam waist. Incident photons perform random walks through the sample due to multiplescattering by the beads, and interfere producing a speckled interference pattern. A description ofdiffusive photon transport 3 through the medium involves two length scales: a transport mean freepath, 1*, and an absorption length 1A. These lengths are determined by measuring the fraction offight transmitted through the sample, T, as a function of its thickness and fitting to the solution ofthe diffusion equation treating the sample as a slab of infinite extent and thickness, L:

T sinh(G4y) + fy'cosh(/-y)

(I + y) sinh( L - +2 (-Ycoshr L )2

(1) *,-0cosh 3,

where y 31*/lA (we assume here and in equation -2 that the extrapolation length ratio 3, ze=l). In .Fig. I we show the results of such a fit applied totransmission through our samples. 10"'

DWS involves measuring intensity 10.2

fluctuations of speckles due to grain motions.From the measured intensity autocorrelation 10 -3

function, we compute the normalized electric- Sample thickness. L (cm)field autocorrelation function gi(T) using theSiegert relation 3: g,(T) = [<I(t)I(0)>/ <1(E)> 2- Fig. I Transmission vs. thickness, L, in cm forI ]/p, where p is a constant related to the 194jim beads (open circles) and 95jtm beads (solid

collection optics. Using diffusion theory with the circles). The lines are fits to eqn. I

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1.0Exposure to air

0.5

10-6 10 4 (s) 10-2

Fig.2 Electric field autocorrelation function,gi(Q) vs. T in secs. Circles (open and solid)represent 95gim beads, prepared dry, the plus (+)and cross (x) symbols represent increasingexposure to atmosphere.

1.0

"-'-0.5

'~~12

A 10-

V10-12

10-6 10-5 1 4 ( 0-3z (s)

Fig. 3 a) Electric field autocorrelation fItvs t, in transmission (+) and backscatte(Ar2(T)) vs r. The solid line i(8V'r) 2/(l+(C/,r) 2), showing ballistic mirandom velocity, 8V, in a cage of size 8Vshows <Ar 2(T)> vs T for 194 grn sanIA/I*= 5 ). The open symbols are from Isolid circles are video measuremexperpendicular to the flow. In both the infigure, there is sub-diffusive motionindicated by dashed lines) over 3-4 decadt > t .

appropriate boundary conditions for the specificsample and scattering geometry in use andincluding absorption effects 3, gi(t) may then beinverted to obtain the mean-squareddisplacement <Ar2(r)> of the scattering sites. Therelevant expressions are: gl ('r) = f(x + y) / f(y)where in transmission,

sinh(vx-) + -cosh(i-)f(x) = N (2)

(1+ x) sin L, x +2 coshn IjX

and in back-scattering, f(x)=exp(-24x)where x=x(t)-k2<Ar 2(t)>, k=2ir/ X, and y= 31*/IA.

RESULTS

Gravity-driven channel flow:- 0.32cm/s Typical data for normalized electric-field:95 [tm autocorrelation functions, gi(G), are shown in

Fig. 2 for light transmitted through the sample.scattering- The agreement of the solid and open circles,

which show different experimental runs onidentically prepared beads, demonstratesreproducibility. The shape of gl(t) is verydifferent for sand that is aged by exposure to air-. presumably by adsorbing moisture -- in spiteof the fact that the flow velocity, Vf, is not

-.028 pm appreciably affected by aging. Aside frompresenting a useful diagnostic to test thepreparation of beads, this suggests that thefeatures in gl() are determined by relativemotions of beads rather than their average flow(which decorrelates g1 (t) at times of 0.1- I Os)

10 10- In Fig. 3a we show two more examplesof the electric field autocorrelation function gi(r)measured in transmission and backscattering on95 ptm sand with a flow velocity Vf = 0.32 cm/s

rnction, gixr), along with the grain displacement, <Ar 2(,r),ring (o). (b) obtained by the prescription of eqns 2 a and b.xs a fit toson with a The dynamics inferred from very different gi(T)

. The inset in Fig. 3a coincide, showing the robustness ofd ( the analysis. This also implies that the dynamicsnd (1"/d=5.3,

)WS and the of grains are uniform across the thickness of theits in 1-d, channel, since photons transmitted through theset and main sample are scattered by sand grains through the(diffusion is bulk of the cell whereas photons backscatteredes in time for from the sample are scattered mainly by sand

315

-.3" .10-o4 .o.. 102, .nA .. to .... °, .... 0-1,

grains within a few 1* of the walls. Theshort-time motion of sand grains isballistic, i.e. (Ar2(T)>=(8V) 2r2, where 6V isa randomly directed velocity. The meanfree time tc represents the duration of thisballistic flight, which is terminated by acollision with a neighbouring grain. Themean free path, s=5Vto, corresponds tothe mean distance between surfaces ofneighbouring beads. Further changes inthe relative positions of grains of grainsrequires many collisions. The slowincrease in <Ar 2(T)> at times longer than T.

is interpreted as a gradual distortion ofthe cage of neighbours. The solid line inFig. 3b shows ballistic motion of grainswith a randomly directed velocity 5Vwithin a fixed cage of size s.

In Fig. 4 we display thedependencies of the mean velocityfluctuation (8V), the mean free path (s)and mean free time (r,) on the flowvelocity, Vf. Transmission measurementstaken on cells of different thicknesses aswell as in back-scattering have the samedependence on Vf showing that velocitygradients perpendicular to the walls of thechannel are negligibly small. (Gradients inthe other two directions are directlyconfirmed to be small by scanning thebeam over the sample). Fig. 4a showsagain that 6V is comparable in magnitudeto Vf and that 5V o) V f 2 over this range.The mean free time tc, (Fig. 4b) has only aweak dependence on Vf. The collision

70

to0

10-Vf (cm/s)l1 10

Fig.4 Microscopic scales of motion vs. macroscopic flowvelocity Vf. The open symbols are for 95 jim beads(squares and circles are transmission in 0.32 and 0.625 cmthick cells; diamonds are back-scattering). The solidsymbols are for 194 jtm beads (circles and squares aretransmission in 0.625 and 0.92 cm thick cells; diamondsare back-scattering). (a) 5V vs Vf, both in cm/s. The linesare power-law fits with exponent 2/3 to the 95 ýtm (dashed)and 194 pim (solid) data. (b) Mean collision time, T, (sec)vs. Vf. (c) Mean free path, s (=-Vr,), scaled to the particlediameter, d. The lines show (5V)

2/2gd, the fraction of itsown diameter a particle must fall to attain a speed 8V

frequency ranges from about 500 kHz (for d=95 pim) to 10 kHz (for d=194 pnm). The mean freepath, s, is small compared to the particle diameter, d. Thus, the dilation is tiny (-4). 1 um) rangingfrom 0.01 to 0.1% of d. An optical or MRI image therefore cannot even identify the flow asbeing in a collisional regime (as opposed to particles being in constant contact). The physicalpicture of collisional dynamics represented by the data in Fig. 4 are consistent with energy andmomentum balance at the level of orders of magnitude as is illustrated by the lines in Fig. 4cwhich show the dilation required to produce a free fall velocity of 8lV, scaled to the particlediameter. Naive energy and momentum budgets do not, however, recover the dependence of themeasured parameters on the driving velocity Vf. The rate of inelastic losses, 1/2m(8V)2(I-e 2)/ T, iscomparable to the rate of gravitational work, mgVf, but balancing these two terms with theassumption that T is independent of Vf(as suggested by Fig. 4b) gives an incorrect scaling of SVCc Vf•1 2. Likewise, assuming that velocities of neighbouring grains are uncorrelated and that acollision completely decorrelates the velocity autocorrelations yields an estimate me8V- mgtc.

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This also leads to an incorrect scaling 8V -c V½. A treatment of the particle motions asuncorrelated thus seems insufficient, suggesting that effects such as velocity correlations betweenparticles or slowly decaying velocity autocorrelations, may be important. Long time motions,which we discuss next, give further indication of such complex dynamics.

At times longer than r,, the relative displacement of particles is characterized by sub-diffusive motion over several decades in time, as displayed in Fig. 3b and its inset. Indeed, we donot obtain the diffusive limit, <Ar 2(r)) cc T, even at the end of the range available to DWS. Thefour data points (solid circles) at the extreme long time end of the inset in Fig. 3b show data fromlong-range video microscopy of self-diffusion of sand grains in the direction transverse to Vf andin the plane of the channel. Our data indicate that the parameters of collisional dynamics such as8V and s are not simply related to long-time diffusion and therefore cannot be deduced frommeasurements at long time- and length-scales.

The data of Fig. 4a also demonstrate that velocity fluctuations can be large even in theabsence of a macroscopic velocity gradient, contrary to the simplest models of granularhydrodynamics. In Fig. 5 we plot <Ar 2(r))taken at various points in a cell withspatially varying velocity gradients (dashed 10lines). For comparison, we also plot (solid i---

lines) data taken at two different values ofVf in a uniform flow. It is quite clear that A 10-1

2

the imposition of a macroscopic velocitygradient alters the transition fromcollisional dynamics discussed above, but v 14 . 10has little influence on the ballistic regime 10I6 10-5 " (s) 10-4 10-3itself. In particular, this shows that 8V is(sinselfInsiartivela, to aiets, a that 8Vis Fig. 5. Effect of velocity gradients. The solid lines are in aquite insensitive to gradients, a fact that is uniform flow with Vf=0.5 and 1.1 cm/s, the dashed lines arenot captured in kinetic theories of flow. in velocity gradients, the local flow velocity is 0.4 - 0.5 cm/s.

Fluidized bedsWe have taken DWS data for particle motions in beds composed of 49 and 97 Pim glass

beads. Beads of this size and density fall in the Geldart A category, in which increasing gasvelocity causes the static bed first to fluidize uniformly and then admit macroscopic bubbles at ahigher flow rates. The hysteretic nature of these transitions has led to the belief that particles inthe bed are able to attain less compact packings in the uniformly fluidized state and do notactually lose contact with neighbouring particles. For a narrow regime of gas flow rates abovethe minimum velocity for fluidization, theDWS signal indicates that the particles .

are static. This, for the first time, 0-10"0. T2,7

directly confirms the hypothesis that Einterparticle contacts may be sustainedabove minimum fluidization. As A 10-12

bubbling commences, we visually %. , ,observe large-scale convective motions vin the bead pack. Microscopic dynamics 10-" 7induced by this overall convection are 10"6 10-4 (s) 10-2 100obtained in the DWS signal as shown g Fig. 6. <Ar 2(T)> vs. t at several gas flow rates (increasing fromFig 6 where we plot <Ar 2(t)> for a range right to left) for 49pam beads fluidized by N 2 gas. The dashedof flow rates from just above the uniform lines represent diffusion (,

1) and ballistic motion (E2

)

317

fluidization threshold to flow rates well in excess of the bubbling threshold (flow rates increasefrom right to left). Just above the bubbling threshold we observe diffusive motions (as indicatedby the dashed line with a slope T). As the gas flow is further increased, the collisional regime isrevealed (ballistic motion is indicated by the dashed line with slope r2.) and we can obtain the'granular temperature' 6, (8V) 2 , the collision frequency, and the mean free path just as in the caseof channel flow. 6V grows rapidly with increasing gas flow just above the bubbling threshold butappears to turn over to a slower dependence on gas velocity at higher rates of gas flow as may beseen in the crowding of the curves at the extreme left of Fig. 6

CONCLUSIONS

We have created a simple granular flow and described a technique which allows a fullcharacterization of the macroscopic and microscopic behaviour of the system. This should serveas an experimental bench-mark for any theory of flow, which we feel is vital in a field wheretheory has been advanced well beyond experiment in the absence of such constraints. What havewe directly learnt that is new? We believe that for the first time, we have been able to study areal, three dimensional flow and present both a qualitative and quantitative picture of thecollisional regime. While hitherto unproven, the fact that collisional dynamics do occur will notsurprise many. What is surprising and new are the scales of motion, the fact that a 'temperature'occurs in the absence of any shear gradient (and indeed, seems to be only weakly affected by agradient), and a well-defined puzzle regarding the dependence of this 'temperature' and the otherscales of the collisional dynamics on the average driving velocity. Yet another new andunexpected fact regards the relationship between the short-time collisional dynamics and the long-time behaviour. The large separation in time scale is reminiscent of dynamics in viscous liquidsand dense colloids and suggest that there may be analogous cooperative dynamics underlying theproblem of flow. The interplay between macroscopic velocity gradients and the nature of thecross-over from ballistic to diffusive regimes is also not captured in existing models of grain flow.

We have also demonstrated the utility of DWS techniques in characterizing particlemotions in gas-fluidized beds. We are able to show directly that particle contacts persist in theuniformly fluidized regime. At higher flow rates we are able to track both diffusional andcollisional dynamics. DWS thus is a versatile, non-intrusive and sensitive probe of particlemotions in granular systems at length scales much smaller than the grain size and is potentially adiagnostic of wide utility even in real industrial settings.

REFERENCES

1. H. M. Jaeger, S.R. Nagel, R.P. Behringer, Phys. Today, 49, 32 (1996).2. R.A. Bagnold, Proc. R. Soc. Lond. A 225, 49 (1954);; J.T. Jenkins, S.B. Savage, J. FluidMech, 130, 187 (1983); P.K. Haff, ibid., 134, 401 (1983)3. D.J. Pine, D.A. Weitz, Dynamic Light Scattering: The method and some applications, ed. WynBrown, pg. 652 (Clarendon, Oxford, 1993)4. M. Nakagawa et al., Exp. Fluids 16, 54 (1993); E.E. Ehrichs et al., Science, 267, 1632, (1995)5. V.V.R. Natarajan, M.L. Hunt, E.D. Taylor, J. Fluid Mech., 304, 1 (1995); 0. Zik, J. Stavans,Europhys. Lett., 16, 255 (1991).6. G.D. Cody et al. Powd. Tech. 87, 211 (1996) infer 8V from the acoustic response of thecontainer to discrete collisions of particles.

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