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Granular Packing

Under VibrationEric Clément, Laurent Labous and Loic Vanel, Université Pierre et Marie Curie, Paris

When vibrated, assemblies of grains exhibit a very rich phenomenology.The

material – a versatile state of matter itself – can behave in ways reminiscent of

solids, of gases or of fluids.There are even signs of unexpected motion

What the lazy physicist has for breakfast isinstant coffee, made from grains, and acereal like cornflakes. He or she buys thecoffee as a dry powder, and the cereal as adry grain. Both are processed in cleverways by their manufacturer, almost cer-tainly involving, at some stage, vibration.

In fact many practical industrial situa-tions involve the vibration of powders andgranulates as an important part of a fabri-cation process. Mechanical agitation isused to transport grains along conveyorbelts or to sort granular assemblies viadevices using sifting or spontaneous size-segregation effects. This can be seen as anatural occurrence in mixtures of com-mercial cocktail nuts shaken during trans-portation: you always see the larger nutson top. But this spontaneous size segrega-tion due to shaking, or any other mechani-cal action like shearing, is a nuisancewhen the aim is to obtain properly mixedparticulate materials. For example, themanufacturing of homogeneous pharma-ceutical tablets out of various powderedchemicals is a headache for engineers.Segregation must also be mastered to pro-duce the new highly resistant concretematerials, made of solid particles ranging

from 100 Angstroms to a few millimetresin size.

For some applications, vibration is usedto obtain a fluid-like matter, but interest-ingly, vibration is also used to obtain amore solid-like matter such as compactedpackings. We will describe a process ofiron casting that uses this effect. Also, inthe making of all ceramics the compactionphase, usually achieved by vibration, is anessential step that ensures the futuremechanical strength of the material. Thistype of technology, developed by engi-neers through the years, uses the extreme-ly versatile and ubiquitous properties ofgranular assemblies whose apparentbehaviour may depend critically on thenature of the grains, the presence of a sur-rounding gas, the shape of the containerboundaries and the modes of vibration.

Recently, these issues were addressed ata more fundamental level by physiciststrying to understand the basic mecha-nism. How can mass, momentum, andenergy transfer from a vibrating bound-ary, explain the rich variety of behaviourfound? Our aim is to give an overview ofvarious phenomena that may take placewhen an assembly of grains is shaken. We

shall show that some minimal conceptsand a very crude model of local granularinteractions demonstrate many of the keyfeatures.

Since what happens is so diverse, weshall only discuss granular material sub-ject to a vertical vibration, with no otherinteractions, other than solid on solid con-tact forces. In particular, we ignore theeffect of a surrounding gas, even though itcan be very important in practice, espe-cially when the size of the grain is small,of the order of 100 micrometers or less.We shall also ignore adhesion forces,whether from the van der Waals forces orfrom the liquid bridges created by humidi-ty. This means that we focus on situationsinvolving millimetre-sized dry grains,rather than micrometric-sized powders.

First we must discuss the problem oflocal energy dissipation and its implica-tions for a collection of grains. We canthen assess situations where the grains aresubject to a vertical vibration. It willemerge that, in different conditions, thegrain behave as a solid block, as a gas, oras a condensed fluid. As a fluid there canbe behaviours such as convection rolls,spontaneous heaping of the surface, theslow dynamics of glassy materials, andstanding wave patterns analogous to thesurface Faraday instability of fluids. Ourlevel of description of the interactions isnot derived from any firm fundamentalbasis. Unlike molecular physics the inter-actions will not come from fundamentallaws like quantum theory or Maxwell’sequations. In practice, we know onlycrudely the relevant physical interactionsoccurring during the contact of twomacroscopic bodies. Usually the collision-al interaction between spherical grainsmay be described using two basic quanti-ties, such as a normal and a tangentialrestitution coefficient for the relativevelocities before and after an impact. Thissimple approach could conceal the factthat these coefficients incorporate a seriesof very complex mechanical phenomena,involving elasticity, and also irreversibleprocesses such as plastic or visco-elasticdeformation. Restitution coefficients arenot fundamental material properties. Theymay vary with strength and angle ofimpact. There are many different models,but the major feature is the local nature of

Fig 1 Irreversible collision of two dissipative (ie energy

losing) particles. Due to the dissipative nature of the

collision between real granular particles, the time

reversibility of the interaction is broken. This peculiar

feature of granular materials is the origin of much

complex behaviour

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dissipation. Already one has a situationradically different from the standard mol-ecular dynamic vision of kinetic theory. Inthe classical kinetic approach, timereversibility and energy conservation ofthe local interactions are fundamentalingredients from which one can derivenice properties such as the H-theorem,and establish a rigorous route to macro-scopic thermodynamic laws. These fea-tures disappear with local dissipativeinteractions.

When two macroscopic bodies collide

(figure 1) it is always possible to decidewhich is the past and which is the futureof the collision, from knowledge of thetrajectories. Roughly, one may say thatlocal dissipation tends to guide the trajec-tories of two colliding bodies to approacheach other after each impact. This impliesa new situation (figure 2), which shows theresults of a numerical simulation using asimplified dissipation model. The normalrestitution coefficient is e = 0.6 (e is thenormal relative velocity after the collision,divided by the normal relative velocitybefore the collision). The simulation startswith the initial ‘big-bang’ situation shown

in figure 2a – particles distributed on asurface with random positions and veloci-ties. After several collisions per particle aninstability is observed which is directlylinked to the local dissipation mechanism.Figure 2b shows that the ‘cooling’ of thegranular assembly is not homogeneous.Regions of higher densities are regions oflower pressure and lower ‘temperature’;they have the tendency to attract moreparticles. In this situation the only avail-able energy sources are the spontaneousshearing fluctuations generally present inany fluid-like system. The soft-clusteringmode shown is the result of a subtle com-petition between cooling and heating. Itwas noticed first by a team at theUniversity of Tel Aviv (I. Goldhirsh and G.Zanetti Phys. Rev. Lett. 70 (1993) 1619).

Another important characteristic ofassemblies of grains is the existence of afinite dissipation length. When a movingboundary gives energy to the grain assem-bly, the collision wave caused is dampedexponentially with a length proportionalto the size of the grains. The numericalpre-factor which sets this length scale,increases when the dissipation decreases.For granular assemblies which are biggerthan this dissipation length, what happensbears some analogy with a phase transi-tion. Starting from a packing where thegrains are separated, and then after thepassage of the collision wave, there aredense zones characterized by binary colli-

Fig 2 Cooling of a ‘granular gas’. Numerical simulation of a hard sphere dissipative gas. Fig 2a left An initial

homogeneous configuration with a random velocity distribution. Fig 2b right After about 20 collisions per particle

dense clusters form due to local energy dissipation.These clusters grow because they are low pressure regions

Fig 3 Vibrofluidization of a granular packing

Fig 3a left Submitted to a vertical vibration a thin layer of weakly dissipative grains will expand when the input

energy is large. From left to right, the pictures are the result of simulations for increasing vibration amplitude. In the

last picture the granular column reaches a steady state resembling a kinetic gas; this is the so-called ‘fluidized state’

Fig 3b above Thicker layers of grains dissipate so much energy that they cannot be entirely fluidized.Taken from an

experiment in a bidimensional cell this snapshot shows that only a few grains on the top surface are fluidized while

the remaining grains move as one block

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sions whose number increases tremen-dously. In numerical simulation this num-ber of collisions would diverge in a finitetime. In practice this means that the sys-tem will reach a state where the particlesundergo long-lasting and multiple con-tacts: they cannot be described by inde-pendent binary collisions any more. Thegranular assembly behaves more or lesslike a compact solid or a disordered glassymaterial.

When rate of energy input is largeenough to prevent excessive ‘cooling’ andwhen the height of the layer is small withrespect to the dissipation length, a steadystate is reached. The assembly looks like agaseous column, the ‘fluidized state’, whichhas been studied in parallel by manyteams in Paris, in Germany and inCambridge. Figure 3a, shows the results ofa numerical simulation of a weakly dissi-pative and shallow granular assembly,excited by a sinusoidally driven vibratingplate. As the acceleration is increased, afluidized state develops, and the assemblyresembles a gas column. If the number ofbead layers is increased, a novel state isreached. The fluidization wave arisingfrom collisions with the bottom plate isalmost damped on a time scale compara-ble to the period of vibration. The systemseparates into two phases: at the top is adense fluid (figure 3b); at the bottom, asolid-like structure appears. The regulartriangular packing of the solid structureof figure 3b is due to two-dimensionalconfinement of the metal beads used inthe experiment. In a three-dimensionalsituation, the appearance would be morelike amorphous packing.

Finally, when the height of the layer ismuch larger than the dissipation length,all the input energy is dissipated in thebulk, and the system behaves like a solidinelastic block on a vibrating plate. To avery good approximation all the energyreceived from the impact is immediatelydissipated in the bulk and the granularassembly velocity adjusts to the platevelocity. In this case, the granular layermay move more or less like a distortingsnake. This is the so-called sub-harmonicinstability found by a team at the EcoleNormale Supérieure in Lyon, France. If thedriving acceleration is very large, the solidblock is launched with such a high veloci-ty after each collision that it undergoes afree flight lasting at least one full period ofvibration. Due to the initial disorder of thegranular assembly and friction with thewall, different parts of the system mayrespond with different phases. When one

part touches the plate another part canstill be in the middle of a free flight. Thisis the case for the top part of the snapshotdisplayed in figure 4 where the left andright parts of the layer touch the bottomplate and where the middle part is still inthe air. One period later the opposite situ-ation is observed (see bottom part of fig-ure 4). The packing responds like aninelastic block with the different partsseparated by a phase defect whichdecreases with time. Ultimately the systemwill respond as a single inelastic solidblock.

The dense state is the most complexand versatile situation. We have men-tioned already the sub-harmonic instabili-ty which happens for rather large horizon-tal layers. When the aspect ratio is closerto one (more like a square), if the frictionbetween the grains and the lateral walls isimportant, another phenomenon predom-inates. For sinusoidal vibrations with anacceleration larger than gravity themotion of a solid inelastic block on a plate

separates into two distinct phases. One is afree flight phase where the solid blockleaves the plate and performs a free flight.The other is a contact phase, initiated by acollision between the packing and theplate. At the beginning of the free flightthere is a moment when the velocity of thegranular layer is directed upwards but thelateral boundary goes downwards, and theballistic motion of the layer is damped dueto friction. Actually, this contrary motionis the origin of a singular boundary layereffect which results in two symmetricalconvection rolls. This effect was found andstudied by a group at the University ofParis, and also later at the University ofChicago, US. For example, when an hori-zontal layer of sand is vibrated in a con-tainer with vertical boundaries, a surfaceinstability is observed. The instability isdirectly caused by two convection rolls oneach side, which bring grains downwardsalong the walls and upwards in the centreof the bulk (figure 5). This difference offluxes creates heaps on the free surface,

Fig 4 During a vibration cycle, a granular layer may detach from the floor and perform a free flight. When the time

of free flight becomes larger than the vibration period, an horizontal thick layer may become unstable.This is the so-

called ‘sub-harmonic instability’. The top picture displays an experimental snapshot of the instability. The bottom

picture is a snapshot one period of vibration later

Fig 5 Boundaries such as the lateral walls are responsible for the existence of convection rolls which bring

materials upwards at the centre of the layer and downwards at the side. As shown by this experimental picture,

the surface may become unstable and form a heap where the mass transfer mechanism is a continuous

avalanche; this contributes to the flow

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which show that the state is not really fluid(a heap could not exist) nor really solid(convection rolls would be absent).

At higher vibrational energy, the localagitation of the grains increases. The heapstructure is mechanically unstable and,eventually, an horizontal free surface isrestored. In this case, the flow of the grainsreally is akin to a fluid. This property isused in the industrial process of iron cast-ing. When a fluid-like state has beenachieved by mechanical agitation of asand container, a polystyrene replica ofthe piece to be moulded is plunged in thesand bed in such a way that when thevibration stops the piece is surroundedhomogeneously by the sand. The aim ofthe next stage is to cast a hot liquid metal,which burns out and replaces the poly-styrene piece. Any vapour escapes easilythrough the porous sand. A successful pro-cedure needs a sand matrix of highmechanical strength, able to resist largestress gradients during the cooling phaseof the metal. The fluidized bed, which set-tles in its rather loose packing once vibra-tion has stopped, does not have this prop-erty. So, before casting the liquid metal thegranular assembly is again vibrated, but ina different parameter range. The systemcompacts itself forming a dense and solidstructure, very resistant to compression,but still very easy to remove by topplingthe container, after the metal piece hascooled.

The compaction process has recentlyshown links to more fundamental behav-iour. Physicists at the University ofChicago, US, studied the density variationin a long vertical column of grains sub-mitted to a series of vertical taps. Therewas an initial fast compaction stage. Aftera short time slow dynamical behaviouremerged, characterized by a logarithmicincrease of the density (figure 6). This log-arithmic variation could last several daysbefore reaching a steady state. The under-lying idea is: to get an additional increasein density when the system is already verypacked it is necessary to have a collectivereorganization of large parts of the pack-ing. This collective phenomenon is expo-nentially costly in time for particles withindependent and random motions. Suchbehaviour is reminiscent of many process-es of ageing in other physical contexts,such as the glass transition. Several mod-els have been proposed which followanalogies with the statistical physics ofspin glasses.

Another surprising feature of vertically-vibrated layers of grains is the existence of

Fig 6 above By choosing suitable vibration parameters, it is possible to make a granular column more and more

compact. In a given volume, the percentage of granular solids increases with time, at first rapidly, but thereafter it

takes a very long time to reach a final density. Mainly, the compaction process is logarithmic in time

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resonant standing waves. These were firstdiscovered by physicists in Lyon, France,and observed more recently in a differentgeometry at the University of Texas, US.The Texas experiment shows that, whensubject to vertical vibration, a thin layer ofgrains on an horizontal plate may producevarious resonant patterns. These includesquares, stripes and even hexagons. Wereproduce (figure 7) the results of a com-puter simulation of this behaviour, for alayer of 6 grains vibrated sinusoidally atan acceleration amplitude of 3.2 times theacceleration of gravity. These views, fromabove, show two successive phases sepa-rated by a period equal to the period ofexcitation. The system displays a squarestructure. It is strikingly reminiscent ofthe surface instabilities observed last cen-tury by Michael Faraday for vibrated lay-ers of fluids. The phenomenon resultsfrom parametric excitation of gravitywaves or capillary surface waves, and thepatterns seen result from non-linear cou-plings between waves travelling in differ-ent directions. The Texas group hasrecently reported the formation of anoriginal localized structure which theyhave baptized the ‘oscillon’. It occurs justabove the threshold of instability. This sin-gular oscillating structure is formed alter-nately from a peak and a hole.Surprisingly, it is possible to create situa-tions where several of these ‘oscillon’structures couple to form dimers, trimersor even higher order ‘molecules’. In figure8a we show a snapshot of an experimenton a confined bidimensional layer of alu-minum beads. The top picture shows asnapshot of a resonant wave; this is a low-dimensional analog of the Faraday insta-bility, and corresponds to a vertical cut ofa picture like figure 7. The lower picture,figure 8b, shows an oscillon localized atthe centre of the cell. A full understandingof these waves is still lacking. There areopen problems and challenging questionson the many fundamental and unveiledaspects of energy and momentum trans-port in these complex states of matterwhich can be formed with granular mate-rials.

Fig 7 left and above The Faraday instability in granular assemblies. A numerical simulation of square patterns as

observed from the top of a vertically vibrated container. Snapshots left and above are separated in time by one

period of the excitation showing the exchange of peaks and cavities from one period to the next.This phenomenon

is well know in fluids and was discovered by Michael Faraday in 1831

Fig 8 below The Faraday standing wave pattern in a bidimensional cell.

Fig 8a is an experimental snapshot of a standing wave for a layer of aluminium beads confined in a bidimensional

cell. A vertical cut of a pattern (as in fig 7) would display such a structure. Fig 8b is the snapshot of an ‘oscillon’, which

is a solitary peak pattern occurring at the edge of the Faraday instability

Further readingH.M. Jaeger, S.R. Nagel and R.P.Behringer Rev. Mod. Phys. 68 (1996)1259 a review of recent developmentsin the physics of granular matter I. Gutman Industrial Use ofMechanical Vibrations (BusinessBooks, London, 1968) applications