PHYSICAL REVIEW A VOLUME 45, NUMBER 6

Vibrated powders: Structure, correlations, and dynamics

15 MARCH 1992

G. C. BarkerTheory and Computational Science Group, AFRC Institute ofFood Research, Colney Lane, Norwich NR4 7UA,

United Kingdom

Anita MehtaTheory of Condensed Matter Group, Cavendish Laboratory, Madingiey Road, Cambridge CB3 OHE,

United Kingdomand Interdisciplinary Research Center in Materials for High Perfo-rmance Applications, University ofBirmingham, Edgbaston,

Birmingham B152TT, United Kingdom*(Received 13 September 1991)

The structure and dynamics of powders subjected to vibration are investigated by a nonsequential andcooperative computer-simulation approach in three dimensions. Starting from a microscopic model ofthe physics, we are able to probe independent and collective effects in the dynamics of vibrated powders,as well as in the resulting structures. In particular, we analyze the role of cooperative structures such asbridges, which are always present in reality and which cannot be formed by purely sequential processes.We look in depth at the behavior of the volume fraction and coordination number as a function of theintensity of vibration, as well as at correlation functions describing contacts between neighboring grains,also as a function of intensity. Satisfying agreement with the qualitative predictions of earlier analyticwork is obtained, and a framework is laid for future investigations.

PACS number(s): 05.40.+j, 05.60.+w, 81.90.+c, 82.70.—yI. INTRODUCTION AND AN OUTLINE

OF THE MODEL

Powders are materials that are composed of dense col-lections of solid grains. They vary in their composition,ranging from coarse-grained aggregates to fine-gradepowders; in their packing, ranging from loosely to close-packed states; and in their states of motion, ranging fromstationary piles to continuous flowing masses. They havebeen of interest to engineers [1,2] for a long time, but it isonly recently that they have become an important and ex-citing area of theoretical [3—9] and experimental [10—13]physics.

Powders exhibit behavior that is neither completelysolidlike nor completely liquidlike, but intermediate be-tween the two. In addition to phenomena exhibited byother amorphous systems, their randomness of shape andtexture strongly influences their static and dynamic prop-erties. They are highly nonlinear and hysteretic, as aconsequence of which they show complexity, so that theoccurrence and relative stability of a large number ofmetastable configurations govern their behavior. Finally,a unique feature of granular materials is that they showdilatancy [1], which is the ability to sustain different de-grees of packing.

However, the subjects of this paper are those featuresof the static and the dynamic properties of granular ma-terials that are universal, i.e., that do not depend on thedetails of the particle sizes or on the material propertiesof the individual grains. Examples of such properties arethe existence of a fixed maximum random-packing frac-tion or the size segregation induced by shaking. In orderto investigate such characteristic granular behavior wehave investigated a model powder that is made frommonodisperse, hard spherical grains so as to highlight the

(generic) microscopic behavior of the grains, and the waythat this influences the macroscopic physics of granularmaterials. We have used a three-dimensional computer-simulation method to obtain microscopic details of thegrain configurations and to probe the independent-particle and the collective effects that occur within a vi-brated bed of grains. We present here an extended ac-count of previous work [9], in which we investigate phe-nomena that contribute to the behavior of dry powders.

Thermal agitation in a powder takes place on an atom-ic rather than a particulate scale; therefore it is externalvibrations that play an essential role in the behavior ofpowders. In the absence of external agitation, the grainsare frozen into one configuration (since their thermal en-ergy is insufFicient to generate the equivalent of Brownianmotion), which represents one of the many possible meta-stable states of the system —we note in passing that thisis one of the reasons why powders show complexity. Inthe microscopic model [3], on which this work was based(a quantitative version of which is presented elsewhere[8]), a granular pile is represented by an assembly of po-tential wells, each representing a local cluster of grains,while the effect of vibration applied to the pile is modeledas being an effective noise H. If H is greater than thebinding energy of the particles to their clusters, then thegrains are ejected, and move into neighboring clusters; interms of the real powder, this means that grains are eject-ed individually (independent particle relaxa-tion) fromtheir clusters. Conversely, if H is small relative to thebinding energies of the particles, they are not ejected; thisenergy goes into the reorganization of the grains (collec-tive relaxation) within their clusters to minimize voids.The claim is [3,8] that for high intensities of vibration,the dominant process is single-particle relaxation,whereas collective relaxation dominates at low intensities.

45 3435 1992 The American Physical Society

3436 G. C. BARKER AND ANITA MEHTA

It will be realized that while single-particle relaxationleads to a rapid decay of the slope, it will lead to a lowpacking fraction and a rough surface. Equally, when col-lective relaxation dominates, the slope will relax slowly ornot at all; on the other hand, slow collective reorganiza-tion of particles will lead to efficient void filling, i.e., to ahigh packing fraction and a smooth surface. We have inearlier work [4] investigated the phenomenon of granularrelaxation in relation to the decay of the slope of a sand-pile subjected to vibration, and will focus here on theeffect of the relaxational dynamics on the structure of thepowder and the correlations within it.

II. A SURVEY OF REORGANIZATION SCHEMES:THE PHILOSOPHY UNDERLYING OUR OWN

The static powder is only characteristic of the methodof preparation —thus demonstrating hysteresis; and anensemble of configurations, built from independent reali-zations of the whole powder using the same method, isrepresentative of a particular aggregation method. Manyaggregation schemes have been investigated in this way,including the deposition model of Void [14], ballisticdeposition [15], close packing with surface restructuring[16], and diffusion-limited aggregation [17]. Of theseschemes, the simplest is the Void model, in which parti-cles stick instantaneously on impact. In ballistic deposi-tion, no trajectories are computed, and aggregation sitesare chosen from a list, while in diffusion-limited aggrega-tion, random-walk trajectories precede the aggregationphase. In all these cases, the initial choice of a site ter-minates the process, i.e., particles stick on impact. Themodel of Ref. [16] goes further, in that particles are al-lowed to roll around after impact on a stationary aggre-gate until they find a local minimum of potential energy,but this is still a sequential process. In contrast, theschemes developed here and elsewhere [4,9] contain col-lective restructuring, where the aggregate restructuressimultaneously with the incoming particles, thus makingour process nonsequential and cooperative, and thereforecapable of incorporating realistic reorganization process-es. This essential ingredient makes our methods muchmore reflective of many-particle events in a movinggranular system.

When mechanical energy is supplied to a powder, inthe form of stirring, shaking, or conveying operations,periods of release are introduced. During the periods ofrelease, the grains have some freedom to rearrange theirpositions relative to their neighbors, and the powder"jumps" [3] between different, but related, grainconfigurations. In this case a series of grainconfigurations represents the dynamic response of thepowder to forcing excitations. In general, this responsehas both transient and steady-state components. Thus ashaken powder follows a path through the phase space ofpowder configurations, which depends on both the dy-namics of the individual grains and on the intensities andfrequencies of the component vibrations of the drivingforce.

In practice, qualitatively similar driving forces may re-sult in rather different behavior in the powder. Thus vi-brations are frequently used to enhance powder mixing,

whereas, in contrast, they may also be identified as asource of size-segregation effects [18—20]. Similarly, slowshaking, or "tapping, "may be used as a means of powdercompaction, especially after a pouring process, but agitat-ed powders can also be significantly more fluid than theirunshaken counterparts. Our aim is to distinguish, interms of individual and collective relaxations [9], thedifferent microscopic responses that underlie the macro-scopic response of a powder subject to vertical vibrationsat different intensities. In our simulation model [4,9], thedriving force is periodic, and leads to clearly definedperiods of dilation of the powder assembly, betweenwhich we have static configurations of grains. The driv-ing force is applied uniaxially and is coupled homogene-ously to the powder, so that free volume is introduceduniformly. During the periods of dilation the grainmotion is dominated by a strong uniaxial gravitationalfield and by hard-core interactions with neighboring par-ticles and the container base.

For a noncohesive powder, it is clear that stirring,shaking, and pouring are a11 many-particle operations.During these processes the particles follow complicatedtrajectories, composed of free-fall segments punctuatedby hard, inelastic collisions with the other particles be-fore they reach stable positions in a static assembly.These trajectories are fundamentally nonsequential, thatis, the route of one particle to its stable position cannotbe computed without simultaneously computing theroutes of many other particles. Stable configurations arethose in which each particle rests in a potential-energyminimum, and therefore cannot lower its potential energyany further by local or nonlocal motion. In practice, thismeans that each particle is in contact with at least threeothers.

The static configurations of grains that result fromshaking reflect the essentially nonsequential nature of theprocess. These configurations contain particle bridges[4,9] and a wide variety of void shapes and sizes that donot occur in sequentially deposited aggregates. In thiscontext, a bridge is a stable arrangement of particles inwhich at least two of the particles depend on each otherfor their stability. Bridges cannot be formed by thesequential placement of particles into stable positions butare a natural consequence of the simultaneous settlingmotion of closely neighboring particles. In our simula-tions [9] we have approximated the precise particle tra-jectories by using a low-temperature Monte Carlomethod supplemented by a nonsequential random-close-packing algorithm. This is a compromise. At the ex-pense of losing information concerning the granular dy-namics, we can efhciently produce static structures thatcorrespond to a nonsequentia1 deposition process. Previ-ous simulations [15,16] have failed to build in this aspectof shaking.

Our method falls between authentic granular-dynamicssimulations [21,22] and previous shaking simulations[15], which combine sequential deposition with a searchfor global minima of the potential energy. Visscher andBolsterli [15] have performed computer simulations of vi-brated beds of hard spheres; they have, however, inter-preted the shaking process only in terms of its outcome,

45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND . . ~ 3437

i.e., in terms of the final, static configurations of thegrains. Their shaken grain configurations were built, us-ing an adapted random-close-packing procedure; that is,by adding grains one by one at sites of minimum poten-tial energy chosen from a set of trial random-close-packing sites. The resulting packings, which remain fullysequential, have volume fractions P—=0.60, which aregreater than those for unshaken configurations but stillsignificantly below the maximum volume fraction for ran-dom close packing of /=0. 64 [23]. More recently, Rosa-to et al. [24] introduced a two-dimensional Monte Carlomethod to study the size segregation that is induced byshaking. Their method includes important nonsequentialfeatures but does not include a criterion for the stabilityof the packing, and hence cannot be used, directly, to fol-low the changes in volume fraction or particle coordina-tions that occur as a result of applied vibrations. In athree-dimensional simulation, Soppe [25] produced non-sequential consolidated packings by combining a MonteCarlo compression with ballistic deposition. This methodalso omits an explicit stability criterion, but, by using aparticular prescription for annealing the packing, it leadsto unstable beds of particles with volume fractionsP=—0.60. Stable packings that contain features due tononsequential reorganization were used by Duke, Barker,and Mehta [4] to study the steady relaxation of the slopeof a two-dimensional pile of hard particles that is causedby vertical vibrations. These simulations reproducedqualitative features of the relaxation and indicated thatcollective particle motions, over length scales comparablewith the nonsequential structural components, were im-portant. This method forms an integral part of the shak-ing simulations presented below.

Granular-dynamics simulations are usually performedin one of two distinct regimes. First there is a grain-inertial regime [21,22], in which instantaneous, inelastictwo-particle collisions dominate the motion. These simu-lations model powders under highly energetic (kinetic)flow conditions, and they are most efficient at moderateparticle densities of P—=0.3 —0.4. In may ways, the im-plementation of granular dynamics in the grain-inertialregime follows the standard methods established for themolecular dynamics of complex fluids using collections ofrough hard spheres. However, one important distinctionarises because the collisions between particles, unlikethose between molecules, are inelastic. The secondgranular-dynamics regime, called the quasistatic regime,is used to model [26] the slow, collective motion of close-packed ($~0.55) collections of particles. In this case,the contact forces between two particles are most impor-tant, and the organization of computer simulations re-volves around the efficient solution of many simultaneousequations of motion for interacting particles. For mostreal materials, the precise nature of the contact forces isunclear —the so-caIied principles of limiting friction andindeterminacy of stress [2], which are familiar to chemi-cal engineers [27], say that the internal stress in a granu-lar assembly is indeterminate, because the friction be-tween two grains in contact can lie anywhere betweenzero and a limiting value. Therefore the applications ofgranular dynamics in the quasistatic regime [26] are re-

stricted by (ad hoc) estimates of contact forces usuallyconstructed from viscous and harmonic elements. Mostshaking processes take place in a series of regimes thattraverse the spectrum from grain inertial to quasistatic,and therefore "shaking" is difficult to simulate with acontinuously tuned granular-dynamics prescription.Thus, during a cycle of a shaking process, the grains mayexperience local particle densities that vary fromP=—0.3 —0.6 and may go through periods of rapid motionas well as through periods of slow relaxation. We hope toreport granular-dynamics simulations of shaking else-where, but here we shall introduce a hybrid technique.

Before describing the details of our method, we wouldlike to comment on a few other approaches. Cellular au-tomata [28] are being used increasingly to model granularflow; while these are powerful tools, both because of theirflexibility and their relative speed, they are limited bytheir lattice-based formulation. They are thus good forqualitative descriptions of powder flow, but cannot probedetailed particulate structure during and after flow. Thekinetic-theory-type approaches of Haff [29] and Jenkinsand Savage [30] and the hydrodynamic approaches ofJackson [31] are appropriate for the situation of rapidshear, where the grains are in constant motion, and theassembly is assigned a "granular temperature" deter-mined by the average mean-square velocity of the grains.These methods are, ho~ever, inappropriate for the situa-tion where the grains are in slow, or no, motion withrespect to each other —the continuum approach fails, be-cause the discreteness of the grains and the effects of fric-tion and restitution at individual collisions become in-creasingly important. Our method [9], on the otherhand, is able to probe details of particulate structure; inaddition, we do not assume a continuum basis or a singlegranular temperature, and are therefore able to copebetter with the quasistatic regime of slow shear.

III. DETAILS OF OURSIMULATION TECHNIQUE

In our simulations we have a bed (Fig. 1) of mono-disperse, hard spheres above a hard, impenetrable, plane

FIG. 1. Schematic diagram of the geometry of the simula-tion. Hard particles form a periodic bed above an impenetrablebase: the diagram shows the primary simulation cell, which isrepeated in two perpendicular directions.

3438 G. C. BARKER AND ANITA MEHTA 45

base that is at z =0. The particle bed is periodic, with arepeat distance of I. sphere diameters, in two perpendicu-lar directions, x and y, in the plane. Each primary simu-lation cell contains N spheres. A unidirectional gravita-tional field acts downwards, i.e., along the negative-zdirection.

Initially, the spheres are placed in the cell using asequential random-close-packing procedure [32]. Thespheres are introduced, one at a time, from large z and atrandom lateral positions, and follow complex paths,which are composed of vertical line segments and circu-lar arcs, until they reach stable positions in contact eitherwith three other spheres or with the hard base. Thesesphere trajectories correspond to rolling motions separat-ed by periods of free fall. In this sequential deposition,the moving sphere rolls over spheres that are already lo-cated in stable positions; that is, incoming spheres cannotdisrupt the stable packing, and they cannot interfere withother aggregating particles. In this sense, the aggregationis slow, and the gravitational field is strong. Many au-thors have analyzed the sequential, close-packed arrange-ments of spheres [15,23,33] that are obtained using thisprocedure. For monodisperse spheres there are boundarylayers that extend for approximately five sphere diame-ters both above the hard base and below the free surface.These layers contain quasiordered arrangements ofspheres. Apart from this, the packing is homogeneouswith a mean volume fraction $0=0.581+.001 and amean sphere coordination co =6.00+0.02 [34]. Thesevalues are not altered substantially by introducing a smallamount (-5%) of polydispersity, and they adequatelydescribe the packings that are used as initialconfigurations for our shaking simulations.

In our simulations, the packing is subject to a series ofnonsequential, X-particle reorganizations. Each reorgan-ization is performed in three distinct parts: first, a verti-cal expansion or dilation: second, a Monte Carlo consoli-dation; and finally a nonsequential close-packing pro-cedure. We shall call each full reorganization a shake cy-cle or, simply, a shake. The duration of our model shak-ing processes and the lengths of other time intervals areconveniently measured in units of the shake cycle.

The first part of the shake cycle [4,9] is a uniform verti-cal expansion of the sphere packing, accompanied by ran-dom, horizontal shifts of the sphere positions. Sphere i,at height z, , is raised to a new height z,'=(I+a)z, . Foreach sphere, new lateral coordinates are assigned, accord-ing to the transformation x'=x +g„,y'=y +g, provid-ing they do not lead to an overlapping sphereconfiguration; here g„and g are Gaussian random vari-ables with zero mean and variance e . The expansion in-troduces a free volume of size e between the spheres andfacilitates their cooperative rearrangement during phases2 and 3 of the shake cycle. This expansion is uirtual: weseek merely to introduce a free volume, not to model aphysical expansion. The parameter e is a measure of theintensity of vibration; although we do not know the exactfunctional relationship between these two quantities, weexpect them to vary monotonically for reasonably smalle. We have assumed that the freedom of motion of theparticles in the interior of the packing increases with the

intensity of the applied vibrations.In the second phase of the cycle, the whole system is

compressed by a series of displacements of individualspheres. Spheres are chose at random and displaced ac-cording to a very-low-temperature, hard-sphere, MonteCarlo algorithm. A trial position for sphere i is given byr,

' =r, +ad, where a is a random vector with components—1 a,a, a, ~1, and d defines the size of a neighbor-hood for the spheres. The move is accepted if it reducesthe height of sphere i without causing any overlaps. Allthe successful moves reduce the overall potential energyof the system. The process continues until the efficiencywith which moves are accepted, measured by batch sam-pling, falls below a threshold value e. Here d and e arefree parameters that are chosen to optimize the computa-tional method. It is shown below that there is a regime inwhich the static results are not strongly dependent onthis choice.

Finally, the sphere packing is stabilized using an exten-sion of the random-close-packing method describedabove. The spheres are chosen in order of increasingheight and, in turn, are allowed to roll and fall into stablepositions. In this part of the shake cycle spheres may rollover, and rest on, any other sphere in the assembly. Thisincludes those spheres that are still to be stabilized andthat may, in turn, undergo further rolls and falls. In thisway, touching particles can be continually moved untilno further rolling is possible. This procedure allows theformation of complex, stable, structural components, likebridges and arches, which cannot be constructed by pure-ly sequential processes [4,9].

The outcome of a shake cycle is to replace one stableclose-packed configuration by another. In theseconfigurations, each particle occupies a cluster that isformed by its neighbors, and a "shake" is thus a reorgani-zation scheme for a set of clusters. The role of the indivi-dual parts of the shake cycle is clear. Expansionrepresents a challenge of variable degree to the integrityof the clusters. The Monte Carlo compression reinstatesthose clusters that were deformed and, when necessary,creates new clusters where the previous ones were des-troyed. Finally, the stabilization phase positions the par-ticles inside the set of clusters established in phase 2. Inphase 2, the Monte Carlo procedure generates a time-ordered sequence of states that culminates with a statethat has an isolated potential well for each particle. Al-though this does not replicate at every instant the actualdynamical processes that lead to the static configurationof spheres, and the dynamical information that it con-tains will depend on the choices for d and e, it seemsreasonable that the set of clusters that is produced is nottoo sensitive to these details.

In practice, during phase one of the nth shake cycle,the mean volume fraction of the assembly falls from P„to P„,/( I+a). In phase 2 the volume fraction steadilyincreases to P„—=P„,, and in phase 3 it remains approxi-mately constant. In contrast, the mean coordinationnumber is reduced from c„&to zero in the expansionphase of the nth shake and remains zero throughout theMonte Carlo compression, but, during stabilization, it in-creases steadily to c„=—c„

45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3439

IV. SOME COMMENTS ON THE TRANSIENTREGIME, AND STEADY-STATE RESULTS

The continuous evolution of particle positions and ve-locities that occurs during a physical shaking process isreplaced, in our simulations, by a time-ordered, discreteset of static N-particle configurations. The members ofeach set are the nonsequentially reorganized close pack-ings that are obtained after integral numbers t of comp-leted shake cycles starting from a sequential randomclose packing. Each set of configurations may be labeledby three parameters e, d, and e. For each member ofthese sets we have evaluated "material" properties, suchas the volume fraction and the mean coordination num-ber, from the central portion of the packing in order tominimize the surface effects. In all cases, this volumecontains more than 50% of the spheres in the simulationcell.

A. Volume fractions and coordination numbers

Figure 2 shows the variation of the volume fraction Pwith the number of shakes t for two different shaking in-tensities e=0 05 and. 0.5. In both cases e =d =0.01,N =1300, and L =8 particle diameters. At low intensity,the volume fraction increases slowly for t & 50 and Quctu-ates around a steady value, /=0. 598+.003, at largertimes. This corresponds to a slow compaction towards avibrational steady state. In this state nonsequential reor-ganizations of.intensity a=0.05 leave the volume fractionof the packing substantially unaltered. The steady statedoes not depend on the particular choice of startingconfiguration or on particular sets of pseudorandomnumbers used during the shake cycles. For @=0.5, theevolution of the volume fraction of the packing is morecomplicated. The first two shake cycles significantlyreduce the volume fraction of the packing from that ofthe sequential deposit to /=0. 562+0.002. Followingthis there is another transient period, t (30, in which thevolume fraction partially recovers. Finally, for t )40another vibrational steady state is achieved, with

0=0.569+0.002.Thus, over a range of shaking intensity, repeated non-

sequential reorganization leads to packings with bulkproperties that are insensitive to further vibrations. Theproperties of the vibrational steady states will be dis-cussed, in detail, below. Results have been obtained bytaking averages from sets of m consecutive configurationsin the steady-state shaking regime with m =—50.Throughout, we have used simulations with N =-1300 andL = 8 particle diameters, for which the mean depth of thepacking is approximately 20 particle diameters. For shal-lower packings, the measured volume fractions are toolarge. This is because the hard base at z =0 causes someordered, denser regions to occur in the lowest layers ofthe packing. These layers are unimportant when measur-ing the volume fractions of packings with depths greaterthan ten particle diameters. We have also tested thedependence of the volume fraction on the cell size L forfixed bed depths, and conclude that serious size depen-dence is absent for L ~ 8.

Monte Carlo consolidation is, structurally, the mostinfluential, and computationally the most intensive partof each shake cycle. The duration of this phase, whichcan be measured in terms of the number of Monte Carlosteps per particle NMc/N, can be increased either by de-creasing e (the terminating efficiency of the Monte Carlosequence) or by decreasing d (the maximum size of eachMonte Carlo step). However, the results of hybrid simu-lations are not related trivially to the details of the MonteCarlo component alone. In Fig. 3 we have plotted, forseveral values of d, the steady-state volume fraction Pagainst the length of the Monte Carlo consolidation.Each data point in Fig. 3 has been obtained from aseparate simulation, with @=0.5, N—= 1300, and L =8particle diameters, by averaging the volume fraction over20 consecutive steady-state shaking configurations. Mostimportantly, for long Monte Carlo consolidations, i.e.,for sufficiently small values of e, the volume fraction datacollapses onto a single, constant value that is independentof d.

Figure 3 shows that, in the absence of the Monte Carlo

0.6

0.6 +%sV+~e~g zPjhsv

C0

~0.58—".V-

+

+0.56—

I

50 100

o0.58—

0.56—

~ ~

0X

,+

2000

NMc/N

X +X

X

o $o + +

+o ~

I

4000

FIG. 2. Volume fraction of monodisperse hard spheres plot-ted against the number of cycles t, of a computer-simulatedshaking process. The initial sta~e is a sequential random closepacking, with volume fraction 0.581, and the shaking intensityis a=0.05 (~ ) and 0.5 (+).

FIG. 3. Steady-state volume fraction of monodisperse hardspheres plotted against the length of Monte Carlo consolidation(measured in Monte Carlo steps per particle NMc/N). TheMonte Carlo consolidation is the second phase of a three-phasecomputer-simulated shaking process with shaking intensity@=0.5. The maximum Monte Carlo step lengths are d =0.004(~ ), 0.01(+),0.05 (o), and 0.2 (X).

3440 G. C. BARKER AND ANITA MEHTA 45

0.6

C0o0.580

E~0.56—0

0.540.0

I

0.5I

1.0 1.5

Shake Intensity

FIG. 4. Steady-state volume fraction of monodisperse hardspheres plotted against the shaking intensity.

phase, the steady-state volume fraction, for shaking in-tensity @=0.5, is /=0. 589+0.001. This is greater thanthe volume fraction for a sequential deposition process,which itself could be viewed as a shaking process,without a Monte Carlo phase, of intensity e= ao. For thesmallest values of d, the Monte Carlo consolidation haslittle effect for NMC/N & 10 . This indicates that, in thefirst part of the Monte Carlo dynamics, the particles ex-perience a period of free diffusion. For longer consolida-tions, and for larger values of d, the downward motion ofthe particles is a collective process. The regions of Fig. 3in which P decreases with NMc/N correspond to in-creased numbers of cooperative features that are trappedinto the stable, close-packed structure by extending theperiod of Monte Carlo consolidation. The minima in Fig.3 indicate that premature termination of the Monte Carloconsolidation can cause too many large bridges and voidsto be trapped in the stable packing. The mean coordina-tion number of the spheres e remains weakly dependenton d for long Monte Carlo consolidations, but, in testswith e =0.05 and 0.5, the values of e obtained by extrapo-lation to d =0 do not differ substantially from those ob-tained with d =0.01.

After the above comments on the transient regime, wenow discuss the steady state: the results presented in thefollowing are mean values taken from m -=50 consecutivecycles in the steady-state regime of the shaking process.Figure 4 shows the variation of the steady-state volumefraction P with the intensity of vibration e For e). 1.0,the volume fraction is only weakly dependent on e withP—=0.550+0.003. However, P rises sharply as e is re-duced below a=1.0 and, for e 0.2, the shaken assemblyadopts configurations that are more compact than thosefor sequentially deposited spheres. This is a clear mani-festation of the collective nature of the structures that areintroduced by a shaking process.

Figure 5 shows the variation of the steady-state meancoordination number of the spheres c with the intensityof vibration e. For e 0.25, the mean coordination de-creases as e increases, and it is approximately constant atc —=4.48+0.03 for larger intensities. The mean coordina-tion number in a shaken assembly is substantially belowthat for a sequential deposit (c =—6.0) refiecting the pres-ence of bridges and other void-generating structures.

4.650.4

o 460CD

0o4.55-(3C:U~ 45-

oz-0

0.0

4.450.0

I

0.5I

1.0 1.5

Also shown in Fig. 5 are the mean fractions, P(n) forn =3—9, of spheres that are n-fold coordinated in pack-ings subjected to steady-state shaking vibrations at inten-sities @=0.05 and 0.5. Most spheres touch four or five oftheir neighbors. For larger values of e, the proportion offourfold-coordinated spheres is increased, largely at theexpense of sixfold-coordinated spheres; i.e., the peak ofthe distribution moves towards lower coordination num-bers.

From Figs. 4 and 5 we note that for 0.25 ~ e~ 1.0, thesteady-state volume fraction steadily decreases with e,while the mean coordination number remains constant.This is consistent with the interpretation that, on the onehand, the density of bridges is independent of e, but that,on the other, the shapes of the bridges become in generalmore eccentric (and therefore more wasteful of space) ase is increased.

B. Network analysis

Each stable configuration of spheres has associatedwith it a network, called the contact network, which canbe formed by drawing line segments between the centersof all pairs of touching spheres. We have studied the evo-lution of the contact network in order to follow localsphere correlations during shaking. For each sphere i attime t we define an (N —1)-dimensional vector b;(t), suchthat the jth element of b;(t) is unity if sphere i is touch-ing sphere j at time t, and zero otherwise. Figure 6 showsthe variation with time of the average autocorrelationfunction

z(t) = (b, (t').b, (t +t') lib, (t')i ib;(t +t')i ),for spheres in the interior of the packing, at two shakingintensities @=0.05 and 0.5. In both instances the initialrate of breaking of contacts is greatest, and for largertimes t ~10 the rate becomes approximately constant.The main conclusion from the figure is that contactcorrelations disappear relatively slowly for low intensitiesof vibration: more quantitatively, we find that a singlenonsequential reorganization with e =0.5 is approximate-

Shake Intensity

FIG. 5. Mean coordination number of monodisperse hardspheres plotted against the shaking intensity. The inset showsthe mean fractions P(n) of spheres that are n-fold coordinatedin the steady-state regime of the shaking process that has shak-ing intensity @=0.05 (solid lines) and 0.5 (dashed lines).

45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3441

1.0

0.8—

0.6—

0.4—

~ ~ ~ ~~ ~ ~ ~ ~ ~ y ~ 0 ~ ~ ~ ~ ~ ~, ,

++

++++++++

+++++++m++g

0.2 I10

I

20 30

FIG. 6. Autocorrelation function z(t) of the contact networkplotted against the number of shake cycles t for rnonodispersehard spheres in the steady-state regime. The shaking intensityis a=0.05 (~ ) and 0.5 (+).

gains a new one, sphere G. In this case, the overall im-pression is one of network disruption. The network con-nectivity is altered significantly in this case, and a com-parison of Figs. 7(b) and 7(d) shows many examples ofbond creation and annihilation.

A notable insight to be gained from Figs. 7(b)—7(d) isthat bridge collapse occurs more frequently for large vi-brations. In Fig. 7(b), spheres H and I rest on, and sup-port, each other and therefore form part of a bridge; thisfeature is retained in Fig. 7(c) (contact network aftersmall vibrations), but not in Fig. 7(d) (contact networkafter large vibrations), where sphere H gains additionalsupport by contacting sphere A from above. We see froma comparison of Figs. 7(b) and 7(d) that the bridge incor-porating spheres H and I has collapsed after a single(large-intensity) shake.

C. Correlation functions

ly twice as eScient at disrupting the contact network asone with @=0.05.

The behavior of z(t) is consistent with snapshot obser-vations of consecutive contact network configurations.Figure 7 highlights the responses of a smal1 group ofneighboring spheres, which are in the interior of a muchlarger packing, to vibrations of two different intensities.Figure 7(a) shows the spheres that are instantaneouslywithin a spherical capture volume, and Fig. 7(b) showsthe contacts between them. We note that contacts be-tween spheres at the periphery of the capture volume andspheres that are outside it are not represented in Fig. 7.The capture volume is centered on sphere A and has a ra-dius of approximately two sphere diameters. In Figs.7(b) —7(d) small balls mark the positions of centers of theclose-packed spheres, and rods represent the sphere con-tacts. The initial configuration of packed spheres, whichis a configuration obtained at the end of one particularshake cycle in the steady-state shaking regime witha=0.05, is shown in Fig. 7(a), and its associated contactnetwork is shown in Fig. 7(b). Figures 7(c) and 7(d) arethe contact networks for configurations that are obtainedafter the application of one further complete shake cycle,with intensity a=0.05 and 0.5, respectively.

In the initial configuration sphere A rests on spheres B,C, and D, and is touched by one other sphere, labeled E,which rests on it. After an additional shake cycle witha=0.05, the sphere A remains stabilized in the same waybut has gained a further contact, with sphere F, fromabove. There are many differences between the networksin Figs. 7(b) and 7(c) but they are mainly small changes,of the sphere positions and the rod orientations, which donot grossly alter the network connectivity. During theextra shake cycle, one sphere has left the capture volumeand another has entered it, so that the numbers of centersin Figs. 7(b) and 7(c) are identical. The overall impres-sion is one of network deformation.

In contrast, the network in Fig. 7(d) does not closelyresemble the one in Fig. 7(b). There is a net loss of twospheres from the capture volume during the additionalshake cycle with a=0.5. After this extra shake, sphere Aretains only two of its original supporting neighbors and

The pair distribution functions of particle positions,It (r) for separations in a horizontal plane and g(z) forseparations in the vertical direction, are illustrated in Fig.8 for @=0.05 and 0.5. The data sets for these functionswere collected, over m =-25 cycles, from horizontal slabswith a thickness of one sphere diameter and from verticalcylinders with cross sections equal to that of one sphere.In both directions, the structure is similar to that expect-ed for dense, hard-sphere fluids. The short-range order ismost pronounced in the horizontal direction, while thepair distribution function in the z direction, g (z), is rela-tively insensitive to variations of the shaking intensity.Both functions indicate the presence of a second shell ofneighbors at a separation of approximately two particlediameters: we conclude from these figures that theshort-range order decreases with increasing intensity ofvibration, in accord with intuition.

During a shake cycle, each particle i is shifted in posi-tion by Ar;=Ax;i+Ay;j+hz, k, where i, j, and k areunit vectors in the x, y, and z directions. We have plot-ted, in Fig. 9, correlation functions of the vertical com-ponents of displacement hz; for e=0 05 and 0.5.. H(r)measures the correlations in a horizontal plane and G(z)measure the correlations in the vertical direction accord-ing to

H(r) =(bz;bz, 5(It;, I —r)8(lz;, I ——,') & I& Ihz; I &

G( )=(&;&,&(I;, I —)8(It;, I ——,') &/(I&;I &',

where t, =(x, —x ) +(y, —"y ), z,"=z;—z, and 8(x) isthe complement of the Heaviside step function. Theaverages are taken over all pairs of spheres i and j andover m =—25 shake cycles. We note that, over the rangeof shaking intensities we have studied, the mean size ofvertical displacements during a shake cycle, ( Ib,z, I &, is amonotonic, increasing function of the intensity. Figure9(a) shows that H(r) decreases rapidly to zero with in-creasing r, and that there is a small decrease in the mag-nitude of the longitudinal displacement correlations,measured in the transverse direction, as the shaking in-tensity is increased. The data in Fig. 9(a) give an esti-mate for the horizontal range over which the spheres

3442 G. C. BARKER AND ANITA MEHTA 45

L

(b)

Pg

plI

.]i"

(c)

FI~. 7. (a) Three-dimensional representation of a cluster of 35 spheres. The cluster is part of a large assembly of spheres that havebeen subjected to shaking vibrations with intensity a=0.05. (b) The contact network that corresponds to the cluster of spheresshown in (a). Small balls represent the centers of the packed spheres and rods represent the contacts between them. The centers ofthe spheres B, C, D, and E, which contact the central sphere A, have been colored red. Contacts with the spheres that are outside ofthe cluster have not been shown. (c) The contact network that corresponds to the cluster of spheres that is obtained after the clusterin (a) is subjected to a further shake cycle with intensity a=0.05. (d) The contact network that corresponds to the cluster of spheresthat is obtained after the cluster in (a) is subjected to a further shake cycle with intensity a=0.5.

45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3443

1.5 1.5 model [3], which says that collective (independent-particle) motions predominate for lower (higher) intensi-ties of vibration.

++

++

+ ~+

+a+~+yg+

N 10~ +

~ + ~ ~ ~++ + . g+++

~ p +~+ ~ + ++++I

0.5 0.5

FIG. 8. Pair distribution functions of particle positions h (r)and g (z) for monodisperse hard spheres in the steady-state re-gime plotted against horizontal displacement r and vertical dis-placement z. The shaking intensity is @=0.05 (~ ) and 0.5 (+ ).The peak heights, which are not shown, have been estimated ash (1)=6.35(6.20) and g (1)=4.40(4.25) for a=0.05(0.5).

0.4=„- 1.0—

L

0.2-+' ~+

0.0~ ++ +,g+~ ++ +

I~ ~ T

2r

N

~ 0.5— +

0.0—

~~ +++ ~

~ +g+qy + + g~--

~ ~ + + +++"

FIG. 9. Correlation functions H(r) and G(z) for the verticaldisplacements of spheres during a single cycle of the steady-state shaking process plotted against horizontal displacement r,and vertical displacement z. The shaking intensity is a=0.05(~ ) 0.5 (+).

move collectively during a shake cycle, and thus providea measure of the typical "cluster size" in the transversedirection.

Clearly, during vertical shaking, the motion of a parti-cle is more sensitive to the positions and the motion ofthose neighbors that are above or below than it is to thosethat are alongside. Figure 9(b) shows that the correla-tions of the longitudinal displacements measured in thelongitudinal direction are stronger than those measuredin the transverse direction, that is, G(z) has a large firstpeak and, at large displacements, it decreases more slow-ly than 8 (r). Also, G (z) depends strongly on the intensi-ty of the vibrations, and, for small e, it has a distinct(negative) minimum at approximately z =1.3 sphere di-ameters. This implies that at these separations, whichare typical of vertical particle separations in shallowbridges, many sphere displacements are not stronglycorrelated, and several of them move in opposite direc-tions; hence this feature is consistent with the slowcompression or collapse of shallow bridges. The correla-tion functions of the transverse components of the spheredisplacements are negative at small separations, which isconsistent with spheres sliding past each other as they aredisplaced in the x and y directions.

We conclude from all the above that the size of a typi-cal dynamical cluster, in both longitudinal and transversedirections, decreases with increasing intensity of vibra-tion. This verifies the predictions of the microscopic

D. The "hole" space

We have concentrated on the static properties and thepair correlations of spheres that form a random-close-packed structure. Equally fundamental, and intimatelyrelated, problems concern the nature of the continuousnetwork of empty space, consisting of pores, necks, andvoids, etc. , which complement the physical structure. Inorder to investigate the pore space of shaken packings,we have constructed the complex structures formed fromoverlapping holes. For a close-packed bed of spheres, theoverlapping holes are another species of spheres, each ofwhich touches four of the packed spheres. The holes mayoverlap each other but cannot intersect any of the packedspheres. For a monodisperse close packing, the max-imum hole size is approximately the same as the spheresize, and the minimum hole diameter is 0.224 times thatof the spheres, corresponding to the hole at the center ofa regular tetrahedron formed from four spheres.

Figure 10 shows a small section of the overlapping holestructures for vibrated packings with a=0.05, 0.5, and1.5, and Fig. 11 shows the corresponding distributionfunctions for the hole radii. From Fig. 11, it is clear thatsmall-intensity shaking is an efficient method of removinglarger holes from the overlapping hole structure, and,therefore, a method for removing large voids from apacking without producing a regular structure. We alsonote, from Fig. 10, that low-intensity shaking leads tolarge numbers of isolated holes, and isolated hole pairs,whereas the larger-intensity vibrations create clearlydefined strings of connected, overlapping holes. This im-portant feature has clear implications for the transportproperties of vibrated beds of particles, and we hope toreport on these in more detail at a later date.

E. The surface

In addition to furnishing data on the bulk properties,our simulations can be used to obtain information aboutthe surface of shaken particulate assemblies. Surface mea-surements are subject to larger uncertainties than bulkmeasurements because they involve only a fraction of theparticles contained in the simulation cell, and also be-cause they are generally more susceptible to system sizedependence. For simulations with L =8 particle diame-ters and N=-1300, we have measured the mean-squaresurface width o =L g, (z; —zo) defined by the spheresi, which have heights z, and which are the highestspheres in each L vertical columns that have cross sec-tions of one square-sphere diameter. zo is the meanheight of the bed. All the surfaces that we have exam-ined are smoother than the surface of a sequentially de-posited aggregate that has o. =0.44+0.02. For e &0.5the surface width is approximately independent of e ando. =—0. 16+0.02. For larger shaking intensity, cr in-creases with e, and for @=1.5, o- -=0.23+0.02. This is inkeeping with the qualitative predictions I3] of our model,which state that greater surface roughening arises as aconsequence of violent vibrations. We have not been able

3AAA G. C. BARKER AND ANITA MEHTA 45

to establish the scaling properties of o. ,' however, wehave confirmed that these trends are followed by othermeasures of the surface irregularity. Most notably wehave used a Monte Carlo method to investigate the reac-tion surface for ballistic aggregation with small test parti-cles, which have a diameter of 0.001 sphere diameters-in this case as well, the mean-square width of the reac-tion surface follows the behavior outlined above.

Other studies of surface roughening I35 —37] have con-centrated on the scaling regime appropriate to sequentialdeposition in the presence of noise. Since our current ap-proach is restricted to sizes below the scaling regime, be-cause we incorporate complex and nonsequential reor-ganization processes, we are unable to compare our re-sults on surfaces with those presented in Refs. I35 —37].

0.2

0.15—

0.1—CL

~ ~+ +0

~ ++ 0 0~ +0

0.05—

0 0 T T T T T)

(

0

~ +0 ~ + 0

0++ 0

0+ 0 (I

a +4I I

0.0 0.1 0.2 0.3 0.4

FIG. 11. Distribution functions I'(r) for the radii r of theoverlapping holes presented in Fig. 10. The shaking intensity is@=0.05 (~ ), @=0.5 (+), and @=1.5 (0).

Id

4.

I""'%I

S'de

('"(Td» Ij. IRhlHI~,

'((

r

''d0 (

However, the results we have presented do provide aquantification of surface roughening in cooperatively res-tructured packings. We hope to report further on the sur-face properties of shaken particulate assemblies, as wellas the surface-penetration effects, at a later stage.

V. DISCUSSION

(c)T p»»

* P

'dd»d

FIG. 10. Sections of the overlapping hole structures that aretopologically complementary to the structures formed by thespheres. The shaking intensity is (a) @=0.05, (b) @=0.5, and (c)@=1.5.

In the preceding sections we have shown that oursimulations provide direct and meaningful microscopicobservations of nonsequentially reorganized granularstructures. The simulation technique allows us to ob-serve the packing both internally and nondestructively(which is outside the scope of current experimental tech-niques), and it therefore provides a unique opportunity tolearn about the bulk behavior and the transient responsesof granular solids subjected to vibration by focusing onthe relaxation mechanisms at the particulate level. Thusour simulation method provides a description that is su-perior to a continuum description and that provides thebasis for a fundamental understanding of realistic (nonse-quential) particle dynamics.

The hybrid Monte Carlo method used above allows usto construct, both efficiently and consistently, nonsequen-tial reorganizations of random close packings, but, in sodoing, it sacrifices detailed knowledge of the granular dy-namics and is unable to look at the effects of the qualityand the frequency of the applied vibrations. In this sense,we have not built a model of one particular shaking pro-cess from which quantitative data will result, but havedesigned a working tool to study the qualitative featuresof shaking and nonsequential processes, in general.

All our simulations have been performed using rnono-disperse collections of spheres in open systems. It is un-likely that the introduction of a small amount of po-lydispersity, in either the sizes or the shapes of the parti-cles, would seriously alter our conclusions. However, it iscertain that, ultimately, the introduction of variations inthe sizes and the shapes of the particles would cause newshaking-induced effects, such as size segregation and theappearance of particularly favorable close packings, to in-teract with, and probably cloud, the effects that we haveobserved. Also, we have employed periodic boundary

VIBRATED PO%DERS: STRUCTURE, CORRELATIONS, AND. . . 3445

conditions throughout in order to compensate, at least inpart, for the size restrictions that are imposed by ourcomputational limits. It must be emphasized thatconfining walls and the particle wall interactions play amajor role in most powder-handling applications. Theextension of our scheme to include the effects of po-lydispersity, confining walls, as well as other interparticleinteractions, represents a primary goal of our futurework.

We note that the algorithm, as employed here, does notensure the homogeneity of bridge nucleation. This is be-cause, in the ordered consolidation in phase 3 of ourshake cycle, the packing gradually produces a gap be-tween consolidated and unconsolidated particles. Thesize of our simulations makes this effect relatively unim-portant, but, in larger simulations, the effect can be over-come by incorporating into the consolidation phase athree-dimensional extension of the local shifts that wereintroduced by Duke et al. [4] to ensure homogeneousdistribution of bridges in a two-dimensional shaken pile.

The results we have presented establish links betweenthe observed changes of the material properties, whichoccur as the shaking intensity varies, and the underlyingmicroscopic correlations of the particle positions and dis-placements. From these results we identify competingroles for the independent-particle and the collective re-laxation mechanisms that occur in nonsequentially reor-ganized random close packings —and verify earlier pre-dictions [3] that independent-particle (collective) effectsdominate at high (low) vibration intensity. Contact net-work measurements show that at high intensities, indivi-dual particles are regularly ejected out of their local envi-ronments, and, hence, one particle may sample manydifferent environments over a short period of time. Incontrast, at low shaking intensities, each particle experi-ences a slow deformation of its environment during shak-ing, and the identity of the particles that form its closeneighbors remains relatively constant; thus, at theselower shaking intensities it is rare for a particle to make atransition into a totally new environment.

In our packings, the particles that form parts ofcooperative structures, such as bridges and arches, aresubject to the different rates of change of their local envi-ronment caused by different shaking intensities. Thisleads to nonsequential reorganization behavior which de-pends, qualitatively, on the shaking intensity. Duringhigh-intensity shaking, cooperative structures form anddisappear rapidly, so that most of the bridges, etc. , thatare present in one particular configuration are only oneor two generations old. These "immature" bridges haveshapes that are those most favored at their formation,and these are, in general, wasteful of space. Thus thepacking fraction takes a low value. In the low-intensityregime, the cooperative structures form and then deform,along with their local environment, over several furthercycles before they become too tenuous to survive. In thiscase, a packing may contain bridges that are many gen-erations old ("mature") and that have shapes that arefavored by their stability against disruption. This includesshapes that have relaxed downwards and are therefore"Batter"; the result is a higher packing fraction, i.e., a

minimization of the void space, and a shift of the holesize distribution to smaller sizes.

According to our definition of a "bridge, " several ofthe spheres that form part of the structure will have adeficit of neighbors, particularly in the downward direc-tion, and, therefore will have low coordination numbers.Thus the mean coordination number of a packing will de-pend on the density of bridge contacts, and in turn thisdensity will depend on the number density of the bridgesand on the mean size of a bridge (i.e., the mean number ofspheres that are required to construct one bridge). Wecan determine the latter from our measurements of H(r);these indicate the lateral extent over which the verticaldisplacements of the spheres are positively correlatedduring one shake cycle, and our results show (Fig. 9) thatthe mean bridge width is quite insensitive to the shakingintensity. Nonzero correlations extend over approxi-mately 1.5 sphere diameters, indicating an average bridgewidth in the region of 3.0 sphere diameters, for both largeand small intensities of vibration.

Given the observed independence of the mean coordi-nation number on e, for @~0.25, we infer that the num-ber density of bridges is approximately constant in thisregime. For @&0.25, the mean coordination numberrises, which indicates (since the bridge size stays approxi-mately constant) that the mean number density of bridgesfalls.

We now show the effect of the nature of the collapsedbridges on the resultant packing. For the lowest shakingintensities, the packing includes regions that result fromslowly collapsed, mature, and Hatter bridges, i.e., bridgesthat deformed considerably before their contact networkwas disrupted. We suggest that these are regions of par-ticularly efficient random close packing and thereforecause the mean volume fraction to rise above the valuethat can be obtained by sequential packing processes.The enhanced short-range correlations of the particle po-sitions (cf. Fig. 8), which we observed for a=0.05, areconsistent with this interpretation. In the high-intensityregime, it is the "immature" and angular bridges that col-lapse, and the aftermath of such collapses is not distin-guishable from a sequentially deposited structure, withsmaller short-range correlations in particle positions andlower packing fractions. The above thus illustrates (viathe specific mechanism of bridge collapse) the point [3]that, at low intensities of vibration, collective reorganiza-tion of particles (and the consequent slow rearrangementof particle bridges) will lead to the efficient filling ofvoids, and the converse.

In conclusion, then, we have presented a detailed studyof the microscopic processes at work in the interior of avibrated granular pile. We have investigated the bulkstructure, by analyzing the behavior of the volume frac-tion and the distribution of coordination numbers as afunction of the shaking intensity. We have also presenteda detailed study of the contact networks and their auto-correlation functions before and after vibration, and haveshown that earlier predictions [3,8] regarding the roles ofindependent-particle and collective relaxation mecha-nisms are verified. The spatial correlations in the pileafter vibration have been investigated by examining the

3446 G. C. BARKER AND ANITA MEHTA

pair-correlation functions of particle positions, and thedistributions of voids, as a function of intensity, whichprovide valuable clues to the static as well as to the trans-port properties, with particular reference to the impor-tant issue of bridge formation and collapse. Finally, wehave shown directly the dynamical behavior of grainssubmitted to vibration, by examining the displacementcorrelation functions as a function of intensity anddemonstrating that the slow motion of clusters predom-inates at lower intensities, relative to the motion of in-

dependent particles, and the converse. We look forwardgreatly to experimental verification of this rich array oftheoretical [3,4,8,9] results, which our present work in itsdetail has brought within reach of current experimentaltechniques [38].

ACKNOWLEDGMENTS

We thank Richard Needs, Mike Cates, and MalcolmGrimson for useful comments on the manuscript.

*Present address.[1]R. A. Bagnold, Proc. R. Soc. London Ser. A 225, 49

(1954); 295, 219 (1966).[2] R. L. Brown and J. C. Richards, Principles of Powder

Mechanics (Pergamon, Oxford, 1966).[3] Anita Mehta, in Correlations and Connectivity; Geometri

cal Aspects of Physics, Chemistry and Biology, edited by H.E. Stanley and N. Ostrowsky (Kluwer, Dordrecht, 1990),pp. 88-108.

[4] T. A. J. Duke, G. C. Barker, and Anita Mehta, Europhys.Lett. 13, 19 (1990).

[5] Anita Mehta and S. F. Edwards, Physica A 168, 714(1990).

[6] L. P. Kadanoff' et al. , Phys. Rev. A 39, 6524 (1989).[7] H. M. Jaeger et al. , Europhys. Lett. 11,619 (1990).[8] Anita Mehta, R. J. Needs, and S. Dattagupta (unpub-

lished).[9] Anita Mehta and G. C. Barker, Phys. Rev. Lett. 67, 394

(1991).[10]G. W. Baxter et al. , Phys. Rev. Lett. 62, 2825 (1989).[ll] H. M. Jaeger, C. Liu, and S. R. Nagel, Phys. Rev. Lett. 62,

40 (1989).[12] G. A. Held et al. , Phys. Rev. Lett. 65, 1120 (1990).[13]Ory Zik and Joel Stavans (unpublished).[14] M. J. Void, J. Colloid Interface Sci. 14, 158 (1959).[15] W. M. Visscher and M. Bolsterli, Nature 239, 504 (1972).[16] R. Jullien and P. Meakin, Europhys. Lett. 4, 1385 (1987).[17]T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400

(1981).[18]J. Bridgwater, Powder Technol. 15, 215 (1976).

[19]D. S. Parsons, Powder Technol. 13, 269 (1977).[20] J. C. Williams and G. Shields, Powder Technol. 1, 134

(1967).[21]C. S. Campbell and C. E. Brennan, J. Fluid Mech. 151, 167

(1985).[22] O. R. Walton, Int. J. Eng. Sci. 22, 1097 (1984).[23] J. D. Bernal, Proc. R. Soc. London, Ser. A 280, 299 (1964).[24] A. Rosato et al. , Phys. Rev. Lett. 58, 1038 (1987).[25] W. Soppe, Powder Technol. 62, 189 (1990).[26] P. A. Cundall and O. D. L. Strack, Geotechnique 29, 47

(1979).[27] See, e.g. , Tribology in Particulate Technology, edited by M.

J. Adams and B.J. Briscoe (Hilger, Bristol, 1987).[28] G. W. Baxter and R. P. Behringer, Phys. Rev. A 42, 1017

(1990).[29] P. K. Haff, J. Fluid Mech. 134, 401 (1983).[30] J. T. Jenkins and S. B. Savage, J. Fluid Mech. 130, 187

(1983).[31]R. Jackson, in Theories ofDispersed Multiphase Flow, edit-

ed by R. Meyer (Academic, New York, 1983).[32] G. C. Barker and M. J. Grimson, J. Phys. Condens.

Matter 1, 2779 (1990).[33]J. G. Berryman, Phys. Rev. A 27, 1053 (1983).[34] G. C. Barker (unpublished).[35] S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. Lon-

don, Ser. A 381, 17 (1982).[36] M. Kardar, G. Parisi, and Y. C. Zhang, Phys. Rev. Lett.

56, 889 (1986).[37] R. Jullien and R. Botet, Phys. Rev. Lett. 54, 2055 (1985).[38] S. K. Sinha (private communication).