Jamming in granular matter

Jamming in granular matter

A. Garcimartín*, I. Zuriguel*, D. Maza*, J.M. Pastor* and L.A. Pugnaloni†

* Departamento de Física y Matemática Aplicada, Facultad de Ciencias, Universidad de Navarra,31080 Pamplona, Spain

† Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, United Kingdom

Abstract. An array of systems can get jammed: traffic jams, compaction of granular materials,the glass transition, colloidal gels, panic and stampedes of people, foams, … The question thatarises is whether all these systems share their features in a more fundamental way, enabling us tothink of a jammed system as a new state of matter. Jamming could then be amenable to bedescribed as a phase transition. A particular experiment aimed at understanding thisphenomenon is presented here. It consists of a silo filled with grains, with an orifice in the base.Grains begin to pour freely, and then – if the orifice is not much larger than several graindiameters – the flow stops because an arch is formed: it jams. By measuring the number ofgrains between successive jammings (an avalanche), a measurement of the jamming probabilityis obtained. From the features of the statistical distribution of avalanches, some interestingcharacteristics of this process can be grasped. In particular, the probability density function ofavalanches shows a clear exponential tail, therefore the system is not critical. It can be rescaledwith only one parameter, namely the mean avalanche size (or, alternatively, the jammingprobability of a single grain). We pay particular attention to the study of some variables thatcould, in principle, influence the process in order to ascertain which are relevant to thephenomenon of jamming, and we show the existence of a critical radius for the orifice abovewhich no jamming can occur.

INTRODUCTION

We often experience that things get jammed. We shake the saltcellar because saltstops pouring, we get caught in slow traffic and we exasperate waiting to leave thecinema through a door that cannot take a sudden swell of people. We can easily recallmany situations where a flow of particles is arrested, and this we call – in a broadsense – a jam. The notion that these and other phenomena can be rigorously described,in physical terms, as sharing fundamental resemblances, has recently been put forwardby Liu and Nagel [1] . Thus, the glass transition can be thought of – and particularly inthe context of mode coupling theory – as molecules that stop exploring an intricateenergy landscape when cooled, i. e., they get jammed in one energy valley. Colloids,foams, granular flows, traffic and stampedes are also included among the systemsdisplaying jamming.

It has even been suggested that the jammed state is a phase of matter, and thatjamming is amenable to be described as a phase transition in the thermodynamicsense. With a sensible choice of variables, Liu and Nagel propose a phase diagram thatcould be pertinent to all the above mentioned systems [2]. Other authors have coined

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the term “fragile matter” to describe the jammed state [3], and they give a qualitativedescription of it, that can be rigorously applied to some cases: it is mechanically stableregarding the so-called compatible stresses, but unstable against incompatible stresses.Maybe it is more intuitive to consider that jamming is the response of a system to theapplied external stresses by developing mechanical structures that block the flow.These structures, that we shall call arches, are stable against the stresses that originatedthem, but unstable against other stresses: they can be easily broken when submitted toforces different from those that create and sustain them. Domes in architecture are agood example; just by removing the keystone the vault can be demolished.

Attractive as the notion can appear for the physicist, it may be difficult to actuallydescribe jamming as a phase transition in any of those systems. This typically involvesdefining an order parameter that is zero for one of the phases and different from zeroin the other. This change in the order parameter is seen when changing the controlparameter [4]. But there is no prescription for the choice of these variables, andpicking the right one can turn out to be complicated [5]. It is therefore of great interestto make available experimental results that can be framed within these general ideas inorder to provide explicit instances on which to support the theoretical notions.

Of all the systems that get jammed, granular matter is probably the bestparadigm [6]. In fact, if one considers the bare essentials of jamming, it involvesdiscrete particles that – through friction or other forms of dissipation – arrangethemselves in mechanically stable structures (arches) that stop some flow. Thesimplest example is the flow of a large number of particles through an obstacle (apipe, an orifice, etc.). This is just the type of experiment that we are presenting here.

We have chosen a cylindrical container (a silo) much larger than the grains, with anorifice in the base from where the particles fall freely. If the orifice is not much largerthan the grains, the flow will eventually stop due to jamming. This arrangement is alsointeresting from the industrial point of view, because it is widely used in the handlingof granular materials. The flow from such a silo has indeed been studied in thiscontext [7], but the main interest is usually to keep it flowing without blockage andlittle attention has been paid to the regime where the stream is arrested. It is worth tonotice that it is sometimes mentioned the existence of a “critical radius” of the orifice,above which the grains will not jam. The use of the word critical in this context,however, is not related to the language of phase transitions, in the sense that a powerlaw divergence of some variable when approaching the critical radius has not beendescribed. The assessment of such a feature is one of the goals of our work.

In selecting the configuration of a vertical silo we have restricted ourselves to amotion due to gravity, but it nevertheless allows us to consider a range of variablesand to ponder their influence on the jamming events. As stated above, this is mostinteresting when trying to investigate which are the parameters relevant to the phasetransition, if indeed there is one. It is not known why do systems get jammed, in thesense that we have no explanation from first principles. Therefore a phenomenologicaldescription must rely on the knowledge of the facts that do affect jamming. Hence theimportance of exploring experimentally the influence of the variables that could inprinciple have some effect in the process. These variables can be broadly classifiedinto two groups. First, we can think of all the features of the granular medium thathave to do with friction and elasticity. The type of material from which the grains are

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made, the roughness and other surface properties determine the friction and therestitution coefficient, and hence the dissipation when two particles collide. Theinvolved processes can be considered as dynamic in nature. On the other hand, thereare several properties, such as the shape of grains, the size, the size dispersion, and soon, which could affect the geometry of the bridges and therefore the mechanicalstability of the arches. These properties could then be labelled as static. One of the firstquestions that arises is whether jamming is a static or a dynamic process. Theinvestigation of the influence of material properties, on one hand, and geometricalproperties, on the other, can shed light on the subject. Thus, this experiment goes wellbeyond a simple taxonomy or a mere description of observations, becausefundamental knowledge of general application can be gained from the results.

This paper is structured as follows. First, we give a detailed description of theexperimental system. We then present the data obtained in a typical run and theinformation that can be procured. The existence of a critical radius – in the sense of apower law divergence – is studied, as well as the influence of material and geometricproperties. Finally, we draw some conclusions.

EXPERIMENTAL SETUP

We have used many different grains in our experiment, changing both the material(glass beads, lead shots, steel balls and plastic pellets) and the shape (usuallyspherical, but we also use rice, lentils and soup grains). For some of these materials,such as for glass beads, a range of diameters has also been explored. The size of thebeads was always big enough so that their weight surpasses any possible cohesiveforces (due to electrostatic charge or humidity, for example). Practical considerationsalso pose a limit to the maximum size of the beads. The diameter of the different beadsspans from half a millimeter to six millimeters.

The grains were poured in a cylindrical silo following always the same procedure inorder to obtain a repeatable volume fraction (the ratio between the volume of thegrains and the total volume of the grains plus the interstitial air). The so-calleddistributed filling [8], which can be accomplished by quickly pouring the granularmaterial into the silo, gave in our case a volume fraction of 0.59 ± 0.01 formonodisperse spherical grains. As we will explain below, the volume fraction remainsconstant along the experimental run.

Several different silos were used in the experiment, most of them made of stainlesssteel, but some were made of glass in order to allow direct observation of the grainsand assess any possible effect due to the particularities of the container. We usedseveral silo diameters to make sure that the bin can be considered infinite in the radialdirection, i. e., that the phenomenon is not affected by the boundaries. We havechecked the reported fact that this happens when the silo diameter is larger than 30particle diameters [9]. In all the experiments reported here the silo diameter is alwayslarger than 30 grain diameters. The height of the silos is 50 cm; typically, severalhundred thousands grains are needed to fill the silo.

The base of the silo is a changeable circular piece of metal in which a circularorifice was bored at the centre with a milling machine or a lathe. The pressure of the

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granular material at the base of the silo saturates quickly due to the redirection of thestresses to the lateral walls, a phenomenon known as the Janssen effect [10]. Thishappens whenever the level of the granular material in the silo exceeds approximately1.5 times the silo diameter. Care was taken that this condition was amply satisfied byregularly refilling the silo.

Grains that pour freely from the orifice fall on an electronic scales that measure thenumber of grains fallen between two consecutive jammings (an avalanche). Thisbalance is connected to a computer where this measurement is stored (see Fig. 1). Amicrophone near the outlet registers the noise of the falling beads; the duration of theavalanche is obtained in a digital oscilloscope and stored in the computer.

As we want to study a large number of avalanches, a device has been devised toautomatically unblock the orifice without changing the volume fraction of the granularmaterial. This is accomplished by means of an air jet aimed at the outlet of the silo.Pressurised air does not change the volume fraction, thus ensuring that it remainsconstant along the experimental run. The valve that releases the air jet is also operatedfrom the computer. The air is usually pressurised to 3 bars.

The computer program governing the set-up works as follows. An avalanche istriggered by opening the valve during 0.4 seconds. When the avalanche ends, ameasurement from the scale is taken, converted into number of grains and stored forfuture analysis along with the duration of the avalanche, obtained from theoscilloscope. After a waiting time of a few seconds, the procedure is repeated. The run

FIGURE 1. Sketch of the experimental set-up. S: silo; O: oscilloscope; M: microphone; V: electricvalve; A: air jet; W: electronic scales; PC: computer.

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is automatically stopped whenever the level of the granular material inside the siloapproaches a height equal to 1.5 times the silo diameter, for the reasons explainedabove. The silo can then be refilled and the measurements are resumed. Other data(such as the temperature and relative humidity of the room) are stamped in the filealong with the measurements; they have been used to discard external influences. In atypical run, several thousands of avalanches are registered. More details on theexperimental setup and the procedures are explained elsewhere [11].

RESULTS

Our first task was to test whether the size of the outlet orifice can be rescaled inorder to assemble all the measurements with a single control variable. This has beenfound to be the case [6]: the adimensional radius R (the ratio between the diameter ofthe orifice and the diameter of the beads) is the relevant parameter, in the sense thatthe absolute sizes of the beads and the orifice give the same statistics for theavalanches whenever the ratio R is the same. It has also been checked that theavalanches are uncorrelated, i. e. that the size of one avalanche does not depend on thesize of the previous avalanche [6]. The histogram of the avalanche size s – where s isthe number of beads – is a good way to describe the results.

Avalanche Size Distribution

Let us first consider the case of monodisperse spherical beads. The histogram of theavalanche size s for R=3 is shown in Fig. 2. It shows a peak that divides the histograminto two parts: a power law growth to the left, and an exponential tail to the right. Thisis revealed when both parts are separately plotted in logarithmic and semilogarithmicscale respectively (Fig. 3).

0 50 100 150 200

0,00

0,02

0,04

n R(s

)

sFIGURE 2. Histogram for the avalanche size s for R=3. The dashed line marks the peak, where thesize of the avalanche with largest probability is found.

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It is interesting to note that the shape of the histogram is similar to the forcedistribution among grains that results from the Q-model [1].

The exponential tail of the histogram can be explained if we assume that eachparticle has a probability p of passing through the outlet orifice which is independentof its neighbours. Then, if p is the probability that one grain gets past through theoutlet without blocking it, and taking into account that the probability of blocking istherefore (1-p), the probability of an avalanche of s grains is just

( ) ( ) sR ppsn −= 1 (1)

When taking the logarithm of Eq. 1, we see that the slope of the histogram insemilogarithmic scale is just the logarithm of p, which provides a way to obtain itexperimentally. Equation (1) is reminiscent of the percolation model in one dimension[12], which considers a chain of independent events.

On the other hand, no simple argument has come to us for the power law thatappears for s smaller than the peak. Moreover, as the peak usually corresponds to arather small value of s, the power law is uncertain in the sense that it barely spansthrough a decade (see Fig. 3). Direct observation suggests that this region has to dowith the fact that when an arch is destroyed, there is a minimum preferred size for theavalanche, namely, the number of beads that formed the arch: as they were mutuallystabilised, when taking out one of them all the others will fall. Then an avalanche ofvery small size is not probable, and the exponential breaks down before reaching verysmall values of s.

The exponential nature of the histogram (except for very small avalanches) impliesthat there is a characteristic size in the system. We can take, for example, the meanavalanche size <s>. All the histograms for different R can be rescaled and theycollapse in the same curve [11]. This precludes the interpretation of this phenomenonin the framework of self-organised criticality [13], since there is a characteristic size inthe system.

25 50 75 100 125

10-3

10-2

1 10

10-3

10-2

n S(R)

sFIGURE 3. The same histogram than in Fig. 2 is plotted in two parts: the tail is plotted insemilogarighmic scale, and in the inset, the growth to the left of the peak is shown in logarithmic scale,revealing a power law. The lines are linear fits.

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The existence of a critical radius

Let us now turn to the question of whether exists a radius Rc beyond which nojamming occurs. It is not intuitive that such a radius should exist. In principle, nothingprevents the formation of an extremely big arch that blocks the outlet orifice, even if itis very large; it would be a matter of waiting a sufficiently long time for the arch toform. In order to investigate this, we have obtained the mean avalanche size as afunction of R. The results are shown in Fig. 4 for spherical glass beads. A divergenceof <s> is clearly seen.

In order to call this a phase transition (between a regime where the flow will stopdue to jamming and another where it will never stop) it would be interesting to see if<s> diverges near the critical radius following a power law (the mean avalanche size isanalogous to the mean cluster size in the percolation theory). The complication is thatthis should happen near the critical radius, which is very difficult to approach. Wehave carried out several experimental runs where the mean avalanche was bigger thana million beads. This requires a constant replenishment of the silo and brings aboutvery long duration of avalanches. It is impractical to attempt further approach to Rc

with our setup. Moreover, the range of R explored barely covers a decade.Nevertheless, we have tried to fit the data with a power law like the following:

γ−=

cRR

As (2)

where A and γ are constants. This fit is provided in Fig. 4. The values obtained areA=9900±100, Rc=4.94±0.03 and γ=6.9±0.2. The value of A has no particularsignificance. The critical exponent γ is very large compared to typical values of phasetransitions. It is remarkable that the power law fit holds even far from Rc. This is not

1 2 3 4 510-1

101

103

105

107

2 4

<s>

R

FIGURE 4. The mean avalanche size <s> as a function of R. The data correspond to spherical glassbeads. A linear plot is shown in the inset, while a semilogarithmic plot is shown in the main graph. Thesolid line is the fit explained in the text.

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usual in phase transitions, although there are some cases – like in Rayleigh-Bénardconvection – where the same has been observed [14].

Influence of material properties and shape

As stated above, the study of the phenomenon with changing material propertiesand grain shape can provide interesting clues to the process. Among the materialproperties, those affecting friction and elasticity and the surface roughness are themost immediately attractive to explore. We have carried out several runs with beads ofdifferent materials, apart from glass: lead shots, which are very soft; stainless steelbeads; and plastic pellets (made of delrin), with a high restitution coefficient. In orderto study the influence of surface roughness we have immersed glass beads in an acidsolution. It is difficult to measure the surface roughness, but the effectiveness of themethod is checked by observing the beads roll. Initially they followed straight paths;after the treatment, the path is irregular. We have also checked that the surface wasirregular with a microscope.

The surprising result is that all these changes do not produce any measurable effectin jamming. Both p (as obtained from the histogram) and the behaviour of <s> (Fig. 5)remain unchanged. This means that jamming is not directly related to the details offriction among grains or the elasticity of the material, nor to the surface properties ofthe particles.

A moderate size dispersion of the beads (5% in radius) has also revealed itself tocause no effect whatsoever in the statistics. It is known that beads with a small sizedispersion prevent the formation of geometrically ordered clusters. This result impliesthat local order of the grains was either absent with monodisperse beads (the mostprobable assumption, as will be explained shortly) or that it does not influence the

1 2 3 4 510-1

101

103

105

107

<s>

R

FIGURE 5 The mean avalanche <s> plotted in semilogarithmic scale versus R. Compare with Fig. 4.Data include: glass beads (circles); lead shots (stars); steel balls (down triangles); delrin pellets (uptriangles); glass beads with 5% radius dispersion (diamonds); rough glass spheres (squares).

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jamming phenomenon.The next step involves changing the shape of the particles, departing from the

spherical form. To this end, we have carried out several runs with various grains: rice,lentils, and pasta soup grains (cylindrical grains with conical ends, with a diameter ofabout 2 mm). Rice and lentils can be considered prolate balls and oblate ballsrespectively, while pasta grains are more complicated in shape. The results are shownin Fig. 6, where the behaviour of <s> as a function of R is displayed. It can be seenthat the shape drastically affects the jamming phenomenon. It is noteworthy that therestill is a critical radius for grains that are not spherical. The size of Rc changesdepending on the shape, but the divergence can always be fitted with Eq. (2). Evenmore, the critical exponent γ remains approximately the same.

Visual inspection of the arches formed at the orifice show that the non-sphericalgrains orient themselves in peculiar arrangements. A more detailed study of this topicand the changes that occur in the histogram of avalanches will be found in [11].

The implications of these results are substantial. As explained in the introduction,they hint that jamming is an affair of the geometrical arrangement of arches and theirmechanical stability, and that dynamic effects (the peculiarities of the collisionsamong grains) have no influence on it.

CONCLUSIONS

We have shown that jamming can be described by a characteristic avalanche size,that we have taken to be the mean avalanche size <s>. This is a consequence of theexponential tail found in the histograms of avalanches. Except for the case of verysmall avalanches, the statistics for a given R are completely described with <s>.

2 3 4 5

101

103

105

107

<s>

R

FIGURE 6. The mean avalanche <s> plotted in semilogarithmic scale versus R for different grainshapes. This figure is to be compared with Fig. (4). Data correspond to rice (squares), pasta grains(circles) and glass spheres (up triangles). Solid lines are fits according to Eq. (2).

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The behaviour of <s> when R is changed reveals the existence of a critical radiusRc, in the sense that there is divergence that can be fitted by a power law. Even thoughthe critical point is difficult to approach, the sheer increase in the mean avalanche sizeis well established. This means that beyond Rc no jamming would occur.

By changing the granular material used in the experiments, we have been able toexplore the influence of the material properties (shape, roughness, elasticity, and soon). We have shown that the relevant factors are the geometric properties (such as sizeand shape). The implication is that jamming is a phenomenon closely related to themechanical stability of arches, and dynamic processes such as collisions do not havean influence on it.

A more detailed work is needed to properly describe the phase transition; maybethe flow (number of grains per unit time that fall from the outlet) or a related quantitycan be put forward as the order parameter. Research is currently under way in thisdirection.

ACKNOWLEDGMENTS

Financial support from the Spanish Ministry of Science and Technology throughprojects BFM2002-00414 and HF2002-0015, from the Government of Navarre, andfrom the Universidad de Navarra through a PIUNA project, is acknowledged. I.Z. andJ.M.P. are grateful to Asociación de Amigos de la Universidad de Navarra for a grant.L.A.P. is a member of CONICET (Argentina). We are indebted to Jean-FrançoisBoudet, Yacine Amarouchene and Hamid Kellay (CPMOH, University of Bordeaux)for their suggestions and comments.

REFERENCES

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1992.13.Bak, P., Tang, C., and Wiesenfeld, K., Phys. Rev. Lett. 59, 381-384 (1987).14.Platten, J.K., and Legros, J.C, Convection in Liquids, Berlin: Springer-Verlag, 1984, p. 449.

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