113Physics of granular matter: pattern formation and applications

© 2009 Advanced Study Center Co. Ltd.

Rev.Adv.Mater.Sci. 20(2009) 113-124

Corresponding author: Christof A. Kruelle, e-mail: christof.kruelle@hs-karlsruhe.de

PHYSICS OF GRANULAR MATTER: PATTERNFORMATION AND APPLICATIONS

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Received: February 03, 2008

Abstract. A summary of results is presented from experiments in granular systems, which areexcited by vertical and/or horizontal vibrations. The transitions between different dynamic statesdepend on internal properties of the granular system like the density of particles and on externalparameters of the driving shakers. Characteristic for granular systems are such counterintuitivephenomena as the crystallization by increasing the vibration amplitude and thereby the energyinput, or the rise of large particles in a sea of smaller ones (Brazil-nut effect). For horizontal shak-ing of a binary system the demixing of small and large particles is found to occur at the samecritical particle density as the liquid-solid transition. The combined action of both vertical and hori-zontal vibrations for the controlled transport of bulk solids is utilized already for a long time inindustrial applications. In a certain driving range it is known that standing surface waves occur athalf the forcing frequency. The three experiments presented here indicate that the four majorconcepts describing the complex behavior of a vibrated granular system, namely phase transition,segregation, pattern formation, and transport are closely related and yield a rewarding field forfuture research.

1. INTRODUCTION

An accumulation of macroscopic grains, set inmotion by an external driving force, can show sur-prising behavior (for examples, see Fig. 1). A pe-culiar phenomenon - called segregation - occursas soon as heterogeneous particles are implied, ingeophysical rock avalanches as well as in pro-cessed powders in the food or chemical industry[1,2]. This de-mixing of grains which differ in size,density, or surface properties, has been intensivelystudied since the 1990s in laboratory experiments,e.g. in rotating drums [3] and under vertical [4] or,more recently, horizontal linear [5] excitation. An-other phenomenon frequently encountered whenhandling granular material is the transition from adisordered phase to a more organized state, whenthe density of grains is increased beyond a criticalvalue. This can be observed, for instance, in mono-

disperse particles under vertical vibration withoutgravity [6] or under horizontal translational excita-tion [7].

In this brief review we report on experimentalinvestigations of both phase transitions and seg-regation phenomena in granular systems, whichare agitated by three different vibration exciters.When the particles are externally forced to performstochastic movements a “granular temperature”can be defined as mean kinetic energy in the cen-ter-of-mass system. For describing the physicalproperties of this ensemble one can ask, in anal-ogy to thermodynamic phase transitions:· Are there critical temperatures at which the inter-nal structure undergoes qualitative changes?

· Which order parameters characterize the dynam-ics of the transition?

· What are the consequences for granular mix-tures?

114 C.A. Kruelle

Fig. 1. (a) Sand dune decorated with aeolian ripples, Great Sand Dunes National Monument, Colorado,USA, (b) Oscillating granular surface wave on a vibratory conveyor.

Fig. 2. (a) Sketchy representation of the experimental device for the vertical vibration of a binary mixtureof spheres. The container diameter is 9.4 cm. Temporal evolution of (b) initially 8 mm glass beads on topof 15 mm polypropylene, which show the classical Brazil-nut effect and (c) 10 mm bronze spheres on 4mm glass beads showing the reverse Brazil-nut effect.

2. VERTICAL VIBRATION

Granular media consisting of small and large par-ticles tend to de-mix when shaken vigorouslyenough. The “Brazil-nut effect” - if a particle mix-ture is shaken vertically, the larger particles will endup on top of the smaller ones, like the nuts in amuesli package - became some kind of Drosophilamelanogaster of granular media research [4,8-11].Numerical simulations could validate the rise oflarger particles [12,13]. Proposed mechanisms are,for example, convective motion of the smaller par-ticles, which drag the larger ones to the top [9], orthe filling of voids by the small particles, therebylifting the larger ones [4,12,13]. Contrary to thesecommon observations Shinbrot et al. [14], however,noticed that a large particle, depending on its den-sity, could also sink to the bottom of the container.

Recent theoretical investigations [15-17] explainedthat either effects, the rise or descend of the largerparticles, may occur. The latter case has becomeknown as “reverse Brazil-nut effect”. The border-line between both effects has been predicted byHong, Quinn, and Luding [15] in a simple relationbetween the size d

l/d

s and mass ratios m

l/m

s of the

large and small particles.One approach to describe an externally driven

granular medium is to consider the individual par-ticles as hard spheres. The driving could be - as inthe case of the Brazil-nut effect - a vertical vibra-tion of the container, which confines the granules.For strong driving, where all particles are in con-tinuous motion, the ideas of the kinetic theory ofgases can be applied. Then it is possible to definea granular temperature T in analogy to a gas, us-ing the mean kinetic energy:

115Physics of granular matter: pattern formation and applications

Fig. 3. Phase space for particle properties. The plot shows the regimes where reversed and classicalBrazil-nut effects occur depending on the particle properties. Each small symbol represents one of 178experiments. The solid line separating both areas is given by Eq. (2). The large symbols indicate theprediction of 3D molecular-dynamics simulations performed by Hong et al. [15] with up to 3600 particles.

N

mv v

vN

v

i

i

i

i

∝ −

=

∑

∑

1

2

1

1 2� � , with

, (1)

where N is the total number of particles, and mi

and vi are the mass and the velocity of the i-th par-

ticle, respectively. Hong [18] calculated, on the basisof this model, a critical granular temperature T

c =

mgdµ/µ0, below which a system of monodisperse

spheres condensates. This critical temperaturedepends on the diameter d of the particles and theinitial filling height µ, measured in units of d. Hereg denotes the gravitational acceleration and µ

0 is a

constant, which depends on the spatial dimensionand the underlying packing structure [18].

For a binary particle mixture different criticaltemperatures do exist, as pointed out by Hong etal. [15]. If a binary mixture of spheres is agitatedby an external shaker, such that the granular tem-perature is in between the two critical values, one

type of particle condensates while the other remainsfluidized. It depends on the size and mass of theparticles, which particle species will condensateand therefore sink to the bottom of the container.Following Rosato et al. [12], Hong et al. claimedthat the cross-over condition is given when the ra-tio of the critical temperatures is equal to the vol-ume ratio of the two particle species, which leads,in D dimensions, to the simple relation

d

d

m

ms s

Dl l=������

−

1

1

. (2)

If, in 3D, the diameter ratio is larger than the squareroot of the mass ratio, the particle mixture shouldshow Brazil-nut effect and vice versa.

The prediction of Eq. (2) was put to the test [19]by preparing an instable layering, followed by acontrolled shaking of the container. Our experimen-tal device (see Fig. 2a) operates at frequencies fbetween 0 and 100 Hz and normalized accelera-

116 C.A. Kruelle

Fig. 4. Sketchy representation of the experimentaldevice for the horizontal circular vibration of amonolayer of glass beads. The container diameteris 29 cm. The bottom glass plate is coated with anelectrically conducting indium-tin oxide layer tosuppress static electricity.

tions Γ = A(2πf)2/g up to 40, where A is the shakingamplitude and g the gravitational acceleration. Theacceleration Γ is measured with an accelerometerattached to the base plate of the Perspex cylinder.The granular materials used were spherical par-ticles of various densities and diameters.

In order to check whether a binary mixture wouldshow Brazil-nut behavior or its reverse, we pre-pared a presumed instable configuration, i.e., weput several layers of the type, which were predictedby Eq. (2) to rise during shaking on the bottom ofthe Perspex vessel. On top of these we placed oneor more layers of the second type. If the predictionwas correct, we would observe that the particleson top started to move through the layers of theother particle type, ending up at the bottom of thevessel (see Figs. 2b and 2c). On the other hand, ifthe prediction turned out to be wrong the initial lay-ering would be stable.

Most experiments have shown clearly either theBrazil-nut effect (Fig. 2b) or the reverse form (Fig.2c). For some particle combinations, a mixed statewas stable. An overview of the experimental re-sults as well as numerical simulations by Hong,Quinn, and Luding [15] together with their theoreti-cal borderline are shown in Fig. 3. For 81% of thetested combinations (145 out of 178) the predic-tion of Eq. (2) was correct. The prediction failedwhen one particle type was made of aluminum orpolyurethane. We assume that for these materialsthe condition of hard spheres, which is one of themain propositions in the theory, was not met. Wealso noticed that, in the case of the reverse Brazil-nut effect, it is crucial to choose an appropriate fill-ing height. It turned out that the effect is completelydestroyed if the initially lower layer is too large.

We currently focus on experimental verificationsof the premise of this segregation mechanism viacondensation. If, at a fixed external driving, a mono-disperse set of glass beads exceeds a critical num-ber of particles a phase transition can be observedfrom a fluidized “gas-like” state to a condensed“crystalline” state. Systematic studies of the depen-dence of the condensation temperature T

c on the

internal parameters (number, size, and materialdensity of the spheres) as well as the external con-ditions (amplitude and frequency of the shaker) arepromising [20].

3. HORIZONTAL CIRCULARVIBRATION

In this section we present a model system, con-sisting of two species of glass beads with different

size rolling in a horizontal shaker, where both phe-nomena, segregation and phase transition, arefound to be closely related since they occur at aboutthe same granular density [21].

A sketch of our experimental device is shownin Fig. 4. The particles in the dish are excited in ahorizontal circular vibration, i.e. a circular move-ment of the entire platform due to the superposi-tion of two sinusoidal vibrations in perpendiculardirections. The frequency f of the table motion canbe tuned from 0.5 to 2.0 Hz, with a preset ampli-tude A = n/8.2.54 cm, where n=2,…8. The granu-lar system is composed of a variable number, N,of spheres with diameters d= 0.4 or 1.0 cm. Toobtain a size-independent control parameter thefilling fraction m is defined as the total cross-sec-tional area N.π(d/2)2 of all spheres divided by thesurface π(D/2)2 of the cell with diameter D = 29cm. During the motion, the positions of all particlesare captured with a charge-coupled-device (CCD)camera fixed to the moving table.

During the experiments we noticed globalchanges in the dynamics of the system while cross-ing a critical threshold of the filling fraction µ. Inparticular, the spheres became arranged in regu-lar, triangular patches. To study this phenomenon,

117Physics of granular matter: pattern formation and applications

Fig. 5. (a) Image of a monolayer of 410 spheres (filling fraction µ = 0.49) rolling on a table shaken with anamplitude A= 2.22 cm and a frequency f = 1.67 Hz. (b) Contour plot of the corresponding 2D powerspectrum. (c) Image of a monolayer of 580 spheres (µ= 0.69) under the same excitation, and (d) its 2Dpower spectrum. The snapshots are taken in a square of 15 cm at the center of the cell.

a monolayer of monodisperse glass spheres isplaced inside the dish. After an early stage, duringwhich the system loses all traces of the arbitraryinitial configuration, an image is taken. To under-line the developed structured state a 2D Fouriertransformation (FFT) is performed. Results of thesemeasurements are shown in Fig. 5, for low andhigh filling fractions of spheres.

At low density, the configuration of grains doesnot show any structures. Its 2D power spectrumdisplays a continuous intensity distribution within acircle of radius k

d = 2π/d. In contrast, at high den-

sity, the small spheres arrange in a triangular lat-tice. In the 2D power spectrum six peaks appearat a wave number k d

02 3 2= π / ( / ) , evenly

spaced by an angle of π/3.

The dynamics of the particles is obviously dif-ferent in these two regimes. At low density eachparticle is free to follow its own trajectory until itcollides with its neighbor, like in a fluid. On the otherhand, at high density, the particles are forced intoa collective motion inside a 2D crystal. The cross-over between these two regimes is reminiscent ofa liquid-solid transition. To specify the transitionpoint, a characteristic order parameter is extractedfrom the power spectra. The spectral intensity isintegrated radially in an annulus 0.98k

0<k<1.02k

0

around the expected peaks. The resulting averagedintensity defines a function of the azimuthal angleϕ. In the structured state this function is supposedto present six equidistant peaks.

118 C.A. Kruelle

Fig. 6. Phase diagram of the critical filling fraction µc vs. shaker amplitude A showing the regime µ > µc,seg

(squares), where segregation occurs. The liquid-solid transition line µ > µc,ls

(circles) is obtained from thenormalized Fourier intensity data for two different particle sizes. The inset shows Johannes Kepler’soriginal drawing of a 2D hexagonal close packing [40].

To obtain a single characteristic number, whichquantifies the order of the structured state, theangular space [0,2π] is subdivided into six equalparts. The intensities in each interval are summedup yielding a singly peaked function I(ϕ) in the re-duced angular range ϕ ∈ [0,π/3]. Dividing I(ϕ) byits arithmetic mean yields a normalized intensity

II

iI

ii

i

n

i

i

i

i

i

ϕϕ

ϕ

ϕπ

� �� �

� �=

= ⋅ =

=

∑1

312

1max

max

max

max

,

, , (3)

from which an order parameter α = In,max

-1 is de-rived. The dependence of this quantity a on thefilling fraction m is highly nonlinear. A liquid-solidlike transition can be observed around a critical fill-ing fraction µ

c,ls.

For the segregation experiments, a binary mix-ture of glass beads consisting of 19 large particlesof diameter d

l = 1.0 cm imbedded in a variable num-

ber of small spheres with diameter ds = 0.4 cm are

prepared. Initially the large spheres are placed ona regular triangular lattice with 3 cm spacing be-tween the centers of two nearest neighbors. Thisdistance L

nn(t), averaged over all the 19 large

spheres, is measured in real time [22]. For a clearcharacterization of the segregation the mean valueof the distance between nearest neighbors in theasymptotic regime, L∞, is measured as a functionof the filling fraction µ. Here, a transition betweena non-segregated state at low density and a seg-regated state at high density can be defined. Sinceboth transitions depend on the shaker amplitudewe explored this dependence further. Fig. 6 showsthe resulting phase diagram of the critical filling frac-tions µ

c,seg

and µc,ls, respectively, vs. shaker ampli-

tude A. These critical filling fractions seem to be

119Physics of granular matter: pattern formation and applications

Fig. 7. (a) Annular conveyor: (1) Torus-shaped vibration channel, (2) Adjustable support, (3) Elastic band,(4) Vibration module with unbalanced masses, (5) Electric motor with integrated frequency inverter. (b)Driving module with four unbalanced masses. (c) Principle modes of oscillation: linear, elliptic, and circu-lar.

independent on the driving frequency but decreasewith increasing shaker amplitude. The regimewhere segregation occurs is always slightly abovethe liquid-solid transition line. This suggests thatthe granular phase transition is a precondition forsegregation.

The transition is in the regime from 0.55 to 0.75and depends on the amplitude of the driving shakerin a counterintuitive way: the liquid-solid transitionline can be crossed by increasing the amplitudeand thereby the energy input. A similar phenom-enon was coined “freezing by heating” in the con-text of driven mesoscopic systems [23].

We offer the following thoughts to elucidate theconnection between the segregation in the binarymixture and the phase transition in the monodis-perse layer of spheres. It seems that the granularmaterial does not stay uniformly distributed butchooses the configuration which minimizes theenergy input for a given number of particles andexcitation parameters, in order to reduce theamount of energy to be dissipated. For µ ≤ µ

c all

particles move randomly with a uniform distribu-tion, i.e., they prefer not to collide with the lateral

wall. Indeed at low densities only few particle-wallcollisions can be observed. We have measuredthat, in this case, large and small particles roll in-side the dish in almost the same fashion. No seg-regation occurs. On the other hand, if the fillingfraction exceeds µ

c, a boundary-hitting regime be-

gins. This change can be seen and even heard inthe experiment. In this regime the granules are con-tinuously hit by the lateral wall. This process tendsto decrease the extension of the monolayer whichorganizes itself in a more and more condensedstate, leading to a triangular lattice, when the exci-tation is increased even further. Following the ideasof Edwards et al. [24], this is also the reason whysegregation occurs: beyond the critical filling frac-tion the collisions with the lateral wall become thedominant driving mechanism. The granular mate-rial is continuously compressed by the collisionswith the lateral wall and therefore tends to increaseits compactness by organizing itself in a triangularlattice. Since the larger particles disturb this order-ing process the system pushes these intruders intothe central region of the dish thereby reducing thenumber of holes and defects [25,26].

120 C.A. Kruelle

Fig. 8. Phase diagram of the transport behavior at circular forcing. The alternating arrows correspond tothe transport direction with clockwise circular forcing.

We conclude that the main mechanism for sizesegregation in our binary system is the compres-sion force exerted by the lateral boundary andmediated by the developing lattice of the smallerspheres.

4. VIBRATORY CONVEYOR

The combined action of vertical and horizontal vi-bration of the support has been utilized already fora long time for the controlled transport of bulk car-goes in a whole variety of industrial processes [27-29]. So-called “vibratory conveyors” are well es-tablished in routine industrial production for con-trolled transport of bulk solids. The transportedgoods can thereby be moved through three differ-ent mechanisms in the desired direction:· sliding of the particles by asymmetric horizontalvibration (“gliding principle”),· ballistic trajectories of the particles caused by in-clined linear oscillation (“throw principle”), and· horizontal transport through vertical vibration of asupport with an asymmetric saw tooth profile ofthe base (“ratchet principle”).

Because of the complicated interactions betweenthe vibrating trough and the bulk solids both glideand throw movements of the particles frequentlyappear within one oscillation cycle. Apart from theamplitude and frequency, the trajectory form of theconveyor’s motion also exerts an influence. There-fore it would be desirable to acquire a thoroughknowledge of the dependence of the transport be-havior on the three principle oscillation forms: lin-ear, circular and elliptic.

First results from circular oscillations showgranular surface waves in a certain range of fre-quencies. Such pattern formation and segregationphenomena have add in the past years to the in-terest of both engineers and physicists. It makesup the core problem of the discipline “Physics ofGranular Material”. Using the vibratory conveyor,the question can be addressed: “How do structureformation and transport properties of granular ma-terial on vibrated surfaces influence each other?”

Through the construction of a ring-shaped vi-brating channel, the long-time dynamics of aclosed, mass conserving system devoid of distur-

121Physics of granular matter: pattern formation and applications

Fig. 9. (a) Experimental setup with transparent trough and conical mirror placed in the center of the ring.The reflected image of the surface profile is captured with a high-speed CCD camera on top of a mirror.(b) Anamorphotic image reflected from a conical mirror. (c) Section of a reconstructed granular surface.

bances from the influx and outpouring of grainscan be studied. The main piece of the vibratoryconveyor (c.f. Fig. 7a) is a torus-shaped vibratingchannel of light-weight construction (carbon fiberstrengthened epoxy) with an average diameter of450 mm and a channel width of 50 mm. Firmlyconnected to the channel are four symmetricallyarranged rotating vibrators. The complete oscilla-tion system, the channel and drivers, are sus-pended with elastic bands in a highly adjustableframe.

The driving of the vibratory conveyor is done byan electric motor with integrated frequency inverter(SIEMENS Combimaster 1UA7). The above is con-nected via compensation clutches to the vibrators(see Fig. 7b). The excitation results from the si-multaneous counter rotation of unbalancedmasses, which are coupled with a gear. If two syn-chronously acting linear vibrators are oriented per-pendicularly, then - through appropriate choice ofthe phase shift between the two - the shape of thevibration can be chosen and numerous Lissajousfigures (linear, circular, elliptical) can be produced(see Fig. 7c). The phase can easily be adjustedthrough the choice of angle at which the unbalancedmasses are mounted (for example 90 degrees forcircular vibration).

For the first trial, the channel was loaded with450 g of carefully sieved glass beads with meandiameter 1.1 mm. The layer thickness of the grainswas about five particle diameters. The averagetransport speed was determined from the time ofcirculation of a colored tracer particle that the bulkcarried along. This occurs automatically through aPC supported image processing system that de-tects the signal changes from the passing of the

tracer through a row of the image and records theassociated transit time.

Surprisingly, the mean transport direction de-pends, for circular agitation, on the amplitude Aand frequency f in a characteristic manner (seeFig. 8). Below a critical value Γ = 0.44, the trans-port is impeded through static friction. Above thisthreshold, the bulk velocity initially increases be-fore reaching its first maximum at Γ = 1.1, with halfthe circulation speed Aω of the circular motion. In-creasing the excitation further leads to a decelera-tion of the granular flow, actually reversing its di-rection of motion in the range 2.70 < Γ < 4.01. Thisalternating behavior repeats itself at still higher Γ.

Through application of five different unbalancedmasses, the oscillation amplitude A can be variedbetween 0.53 mm and 2.35 mm. It is found thatthe conveyor speed increases linearly with theamplitude. The critical machine numbers Γ, atwhich the transport behavior changes qualitatively,are independent of the oscillation amplitude, at leastfor Γ < 5. In the frequency-amplitude parameterspace (see Fig. 8) the threshold values lie on f-2

hyperbola of constant acceleration (“isoepitachs”).For industrial applications, this observed rever-

sal effect is relevant as the direction of a granularflow can be selected through the frequency of theexcitation alone. One can employ such two-wayconveyors for example in larger cascading trans-port systems as control elements to convey thematerial to different processes as needed.

5. GRANULAR SURFACE WAVES

The first appearance of a pattern in the grains canbe observed at Γ = 2.4. The surface of the par-ticles is then no longer flat, but rather shows peri-

122 C.A. Kruelle

Fig. 10. (a) Scaled velocity v* of surface waves and granular flow. (b) Circles represent the measuredwavelength λ plotted over the normalized acceleration Γ. Throughout the measurement the driving ampli-tude was kept constant at 1.47 mm. The solid line is the graph of the function λ = 1.3cm + 20cm.Γ-1.9, thebest fit to the data.

odic maxima and minima, quickly oscillating up anddown (see Fig. 1b). Similar phenomena, namelystanding waves in purely vertically shaken granu-lar material, have been already observed [30,31].In the system presented here, because of the sym-metry breaking through the circular driving of thevibratory conveyor, these surface waves are trav-eling and therefore influence the bulk velocity. Forthe intended studies, the periodic boundary condi-tions of the channel geometry make it easier toanalyze the topology of the surface waves bymeans of a Fourier spectrum of waves.

For comparison with theoretical models it isnecessary to determine the dynamic surface pro-file h(ϕ,t) during the transport process with highspatial and temporal resolution. This task has beensolved by Pak and Behringer [32] only for a smallsection of an annular trough. Our container con-sists of a 2 cm wide annular channel with opentop, 7 cm high Plexiglas walls, and a radius ofR = 22.5 cm giving a circumference of L

0 =141 cm

(see Fig. 9a). The granular system is observed fromthe top via a conical mirror placed in the center ofthe ring, similar to Ref. [32]. Thus a side view ofthe whole channel is captured with a single high-speed digital camera (resolution: 1280

×

1024 pix-

els at rates up to 500 images per second). Fig. 9b

shows an anamorphotic image reflected from theconical mirror. The wavy granular surface is seenas a jagged ring around the tip of the cone. Forreconstructing the true shape of the profile h(ϕ,t)digital image processing is performed which deliv-ers 360° panoramic side views of the granular pro-file in the channel. A small section of such a recon-structed granular surface is presented in Fig. 9c.The spatial resolution is sufficient for detectingsingle particles of 2 mm size. The channel is litfrom outside through diffusive parchment paperwrapped around the outer wall, hence particlesappear dark in front of a bright background.

If Γ exceeds 1, the vertical component of thecircular acceleration will cause the grains to de-tach from the bulk followed by a flight on a ballisticparabola. This results in a much less dense-packed, fluidized, state with highly mobile constitu-ents. In a certain driving range between Γ

1 ≈ 3.3

and Γ2 ≈ 3.7 a locking of the time-of-flight between

successive bounces and the period of the circularvibration occurs [34]. The initially flat bed becomesdestabilized, and undulations of the granular sur-face occur in the range 2.4 < Γ < 4.5 [35,36]. High-speed CCD imaging showed that they oscillate withhalf the excitation frequency (“f/2 waves”). As men-tioned, in contrast to previous work [32, 37-39] done

123Physics of granular matter: pattern formation and applications

at purely vertical vibration the present waves arenot stationary but are transported along the annu-lar trough. The drift velocity of the wave patterncan be measured by applying a phase-locked im-aging technique with fixed time delay ∆t = 2T. Thisis done with a camera which is triggered by a pick-up signal taken from the rotating unbalancedmasses. From these images space-time diagramscan be constructed. The deviation of the stripedpattern from the vertical is taken as a measure forthe wave speed.

A comparison of the wave speed with the bulkvelocity of the transported particles shows that bothvelocities are identical (Fig. 10a). In particular, areversal of the wave velocity is also possible byadjusting the vibration frequency alone. In the rangeΓ ≈ 4.5, when the granular flow is reversed a sec-ond time, surface waves disappear. The granularsurface becomes flat but still oscillates at twice thevibration frequency. Above Γ ≈ 5.5 standing wavesare observed again but this time repeating theirpatterns after four shaker periods (“f/4 waves”). Thisscenario is reminiscent to the wave patterns foundby Bizon et al. [39] for vertical vibration of a later-ally extended system.

A more detailed analysis of the Γ dependenceof the wave length λ shows

λ λ αΓ Λ Γ� � = + ⋅min , (4)

see Fig. 10b. For the minimum wavelength λmin weobtain a value of 1.3 ± 0.3 cm, which is approxi-mately the depth of the granular layer. The valuesfor the other parameters are Λ = 20 ± 4 cm andα = -1.9 ± 0.3. This is consistent with the results ofMetcalf et al. [30] who examined the dependenceof the wavelength on the peak acceleration Γ atconstant frequency.

6. CONCLUSIONS

To summarize, the vibratory systems presentedhere have demonstrated their potential for the in-vestigation of the pattern formation and transportproperties of granular materials in a systematic way.Considering the granular pattern formation, thedescribed annular apparatus is unrivaled for itscapability to study solid-liquid transitions since, dueto the permanent transport of all particles at peri-odic boundary conditions, the influence of spatialinhomogeneities is excluded:· propagating patterns which persist along the whole

system cannot be caused by local inhomogene-ities of the container and

· one can wait until any transient due to coarseningprocesses of the developing patterns have dis-appeared.

For a further understanding of the general be-havior of granular material on vibratory conveyorsthe next steps are· the development of a better understanding of seg-regation phenomena in multidisperse systems,

· the investigation of clustering patterns insubmonolayers, and

· the qualitative modelling of the spatiotemporalsurface structure.

ACKNOWLEDGEMENTS

We would like to thank A. Aumaître, A.P.J. Breu,R. Brito, H. El hor, H.-M. Ensner, A. Götzendorfer,R. Grochowski, F. Landwehr, S.J. Linz, J. Kreft, I.Rehberg, M. Rouijaa, T. Schnautz, M. Schröter, S.Strugholtz, D. Svenšek, and P. Walzel for valuablediscussions.

Support by Deutsche Forschungsgemeinschaft(DFG-Sonderprogramm “Verhalten granularerMedien”, Kr1877/2) is gratefully acknowledged.

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