A bi-disperse granular gas in a compartmentalized system is experimentally
found to cluster competitively: Depending on the shaking strength, the
clustering can be directed either towards the compartment initially containing
mainly small particles, or to the one containing mainly large particles. The
experimental observations are quantitatively explained within a flux model.Comment: 4 pages, 4 figures, Phys. Rev. Lett., in pres

Lohse, Detlef

Mikkelsen, René

van der Meer, Devaraj

van der Weele, Ko

English

arXiv

A bidisperse granular gas in a compartmentalized system is experimentally found to cluster competitively: Depending on the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the one containing mainly large particles. The experimental observations are quantitatively explained within a flux mod

University of Twente Research Information

VOLUME 89, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 18 NOVEMBER 2002Competitive Clustering in a Bidisperse Granular GasRené Mikkelsen, Devaraj van der Meer, Ko van der Weele, and Detlef LohseDepartment of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands(Received 27 May 2002; published 1 November 2002)214301-1A bidisperse granular gas in a compartmentalized system is experimentally found to clustercompetitively: Depending on the shaking strength, the clustering can be directed either towards thecompartment initially containing mainly small particles or to the one containing mainly largeparticles. The experimental observations are quantitatively explained within a flux model.DOI: 10.1103/PhysRevLett.89.214301 PACS numbers: 45.70.MgFIG. 1 (color online). Images from two experiments with amixture of P1  300 large steel beads (r1  2:50 mm, m1 0:500 g) and P2  600 smaller ones (r2  1:25 mm, m2 0:063 g). The initial distribution of the beads is the same inboth experiments: f35P1;13P2g in compartment A and f25P1;23P2gin B. For moderate shaking strength (f  60:0 Hz, a 2:0 mm, top row) a cluster is formed in the compartmentinitially dominated by large particles (A). For milder shaking(f  37:5 Hz, a  2:0 mm, bottom row) the clustering takesof the experiment (see Fig. 1, right) depends on theshaking strength.place in the compartment initially containing most of the smallparticles (B).Clustering is one of the important features of granulargases, arising from the inelastic collisions between theparticles. Energy is dissipated in each collision such that adense region will dissipate more energy, becoming evendenser, resulting in the formation of a cluster of slowparticles [1]. It is a prime example of structure formationin a system far from equilibrium.Thus far, most attention has been given to clustering inmonodisperse systems where all particles are identical. Inthat case, the clustering simply occurs in a region thatoriginally is a bit denser than the others. In the presentpaper we concentrate on a bidisperse granular gas, con-sisting of large and small particles. Here the clusteringturns out to be competitive: it can occur either in a regionwhich originally is populated mainly by small particlesor in a region with mainly large particles.The experimental setup (see Fig. 1) consists of a cylin-drical perspex tube with inner diameter 11.2 cm andheight 42.2 cm, divided into two equal compartmentsby a wall of height 6.0 cm. This compartmentalizatonmakes it possible to get a clear-cut picture of the cluster-ing process. The tube is mounted on a shaker with adjust-able amplitude a and frequency f. The inverse shakingstrength D / 1=af2[defined more precisely in Eq. (8)below] is one of the crucial parameters for the clusteringbehavior.To minimize the effect of statistical fluctuations in theexperiments, we take a sufficiently large number of beadsof each species: P1  300 large and P2  600 smallones. The small and large beads are both made of steel(so they have the same coefficient of normal restitution,e  0:85), with radius ratio   r1=r2  2.Clustering behavior.—The clustering behavior isstrongly influenced by the initial distribution of the twospecies over the compartments. As an example we takean initial distribution in which compartment A has themajority of the large particles and compartment B mostof the small ones. This situation is depicted in Fig. 1, left,with f35P1;13P2g  f180; 200g in compartment A andf25P1;23P2g  f120; 400g in compartment B. The outcome0031-9007=02=89(21)=214301(4)$20.00 The full bifurcation diagram is given in Fig. 2: Forsufficiently strong shaking (small D) a uniform distribu-tion is found. This is denoted as regime 0. For moderateshaking strength (moderate D, regime I) the particlescluster in compartment A, the one initially dominatedby large particles. The reason is as follows: Many of thesmall beads quickly cluster in compartment A, where thedissipation is highest due to the greater number of largebeads, which (with their larger mass and surface area) actas ‘‘coolers.’’ The remaining beads in compartment Bjump higher than before, since there are fewer collisions,and also large ones now start to make it over the wall intocompartment A. After about 20 seconds to a minute(depending on the value of D) the final state is reached:a dynamical equilibrium with practically all large par-ticles and most of the small ones in compartment A and2002 The American Physical Society 214301-1VOLUME 89, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 18 NOVEMBER 2002only a few rapid small particles in compartment B (seeFig. 1, top row).For very mild shaking (high D, regime II) the cluster-ing process is much slower and, more importantly, it goesin the opposite direction. The particles now cluster incompartment B, i.e., in the compartment initially domi-nated by small particles (see Fig. 1, bottom row). Theseries of events is as follows: First, the larger particleshardly jump at all. They stay close to the bottom, trans-ferring energy from the vibrating bottom to the smallerparticles, which thereby gain comparatively large veloci-ties (like tennis balls jumping higher on top of a basket-ball than on the plain floor). So it is easier for the smallbeads to leave compartment A (with more large beads).The remaining particles become more mobile, and after acouple of minutes the first large beads start to make itacross the wall into B, where they are immediately swal-lowed by the developing cluster. With every particle thatleaves compartment A, the process progressively speedsup. The total clustering time is on the order of 5 to20 minutes, steeply increasing with the value of D.Flux model.—The above behavior can be explained interms of a dynamical flux model. It is a generalization ofthe model derived by Eggers for clustering in the mono-disperse case [2–4]. First we consider the compartmentsseparately and from this derive the flux of particles leav-ing each compartment.The gas within a compartment is assumed to be inthermal equilibrium with respect to the granular tem-perature T  12mihv2i i (the mean kinetic energy of theparticles), which is taken to be the same for both speciesi  1; 2. This is an approximation: Two different granulartemperatures may in fact coexist in binary mixtures [5](and at higher densities the species are known to segre-gate [6]), but taking this into account would add anunknown and possibly not constant parameter to ourmodel. As in Ref. [2], and in fair agreement with mo-lecular dynamics simulations [7], we assume that thegranular temperature T is approximately constantthroughout the compartment.The same simulations [7] show that both species obey abarometric height distribution (as expected for dilutegranular gases, where one can use the standard equationof state and momentum balance [2]). That is, their numberdensities show an exponential decay with the height z:niz  ni0 expmigz=T: (1)The density at ground level follows from R10 nizdz Ni as ni0  migNi=T. Here Ni is the number of par-ticles (of species i) in the compartment under consider-ation and  is its ground area. Equation (1) says that thelarger particles, with larger mi, become dilute faster withgrowing z than the small particles.The temperature T is determined by the balance be-tween energy input J0 (through collisions with the bot-tom) and dissipation Q (through the collisions between214301-2particles). To determine the energy input, we assume forsimplicity a small-amplitude sawtooth motion of thebottom, such that colliding particles always find it mov-ing upward with velocity vb  af. This means that aparticle coming down with vertical velocity componentvzi is bounced back with vzi  2vb, and the energy gainper collision is 2mivbvzi  vb. Multiplying this by thenumber of collisions [/ ni0vzi per species], and assum-ing an isotropic Maxwellian velocity distribution (hv2zii 13 hv2i i), this gives the rate of energy input:J0  Xi1;2ni043vbT 2~~2p~~3p v2b~~~~~~~~miTp : (2)As typically vb  vi, the first term is much larger thanthe second, which we therefore neglect. Then J0 43n10  n20vbT 43gvbm1N1 m2N2.This must be balanced by the rate of energy loss Qthrough the inelastic collisions between the particles. It isequal to the product of the number of collisions per unitvolume and the energy loss per collision / 1 e2 mimj= mi mjvi  vj2, averaged over the velocitydistribution, and integrated over the whole compartment.It consists of three terms, representing collisions betweentwo particles of type 1, type 2, and types 1 and 2,respectively [8]:Q 16~~~~pg1 e2T1=2r21~~~~~~m1pN21  r22~~~~~~m2pN2247r1  r22~~~~~~~~m12pN1N2;(3)with~~~~~~~~m12pm1m2m1~~~~m2p  m2~~~~m1p  34  ~~~~~~m1p  ~~~~~~m2p m1 m22 : (4)Equating J0 andQ yields the granular temperature T ofthe compartment:T af21441 e22; (5)where the effective mass  is given byN1; N2m1N1 m2N2r21~~~~~~m1pN21  r22~~~~~~m2pN22 47 r1  r22 ~~~~~~~~m12p N1N22:(6)It is through this quantity that the populations of the twospecies within one compartment influence each other.The central quantity of the model is the flux functionFi, i.e., the number of particles (of species i) that leave thecompartment per unit time. It is the product of the densityniz and the horizontal velocity (~~~~~~~~~~~~~~~~2T=mip), inte-grated over the space above the wall (width b) fromz  h to some cutoff height hH [9]:214301-2FIG. 2 (color online). Bifurcation diagram showing the threedifferent clustering regimes O, I, and II. The particle numbersNi (i  1; 2) are given relative to the symmetric solution: Ni 12Pi. The squares and diamonds are experimental data, corre-sponding to the numbers of large and small particles, respec-tively, in compartment A (solid symbols) and B (open symbols).The curves represent the equilibrium situation according to thetheoretical flux model. The initial condition is always the sameas in Fig. 1.VOLUME 89, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 18 NOVEMBER 2002FiN1; N2 ~~~~~~~~2TmisbZ Hhnizdz~~~~~~~~~~2TmisbNiemigh=T1 emigH=T ANi~~~~mireDmi=: (7)In the last step we have linearized expmigH=T,implying that Hhv2i i=g. The prefactor determin-ing the absolute rate of the flux is given by A12~~2p1e2gbH=af. The dimensionless parameterD, which governs the clustering behavior, has the formD144ghaf21e22: (8)FIG. 3 (color online). Evolution of the bidiperse system, calculate200. The top rows show the number of large and small particles acurves represent compartment A, and the dashed ones compartmentevolve: N1B (large) and N2B (small). The cross indicates the initial ccomes about through a bifurcation, whereas the transition from (b214301-3The evolution of the number of particles NiA in com-partment A (i  1; 2) is given by the net balance betweenthe (outgoing) flux from A to B and the (incoming) fluxfrom B to A:dNiAdt FiN1A; N2A  FiN1B; N2B FiN1A; N2A  FiP1  N1A; P2  N2A; (9)where we have used particle conservation, NiANiB  Pi. The evolution of the (complementary) particlenumbers in compartment B is governed by the sameequation with A and B interchanged.Flux model versus experiment.—In Fig. 2 the resultsfrom the flux model are shown together with the experi-mental data, using the same initial distribution as in Fig. 1.The agreement between theory and experiment is verygood, and the same three regimes are found: For vigorousshaking (regime 0, D< 10) the system settles into theunclustered homogeneous state. For moderate shaking(regime I, 10<D< 150) we find clustering in compart-ment A, and for mild shaking (D > 150) the clusteringtakes place in compartment B. The transition fromregime I to II is quite abrupt.The experimental observation that the small particlesalways (in both regimes I and II) are the first to cluster,followed later by the large ones, is also reflected in theflux model. This can be seen in Fig. 3, where the evolutionof the various particle numbers [obtained from Eq. (9)] isshown. The same plots also illustrate the different timescales for type-I and type-II clustering: The time it takesthe system to reach its equilibrium situation grows rapidlywith growing D (decreasing shaking strength), just as inexperiment.In the lower part of Fig. 3 the corresponding flowdiagrams are depicted, showing how the particle numbersN1B and N2B in compartment B evolve, for any initialcondition. The cross denotes the initial condition used inthe experiments (fN1B; N2Bg  f120; 400g), and the ar-rows show the evolution of the system. For very strongd from the flux model, for (a) D  1, (b) D  100, and (c) D s a function of time (note the different time scales): the solidB. The flow diagrams (bottom row) show how the contents of Bondition fN1B; N2Bg  f120; 400g. The transition from (a) to (b)) to (c) is seen to be the result of a shifting basin boundary.214301-3FIG. 4 (color online). Phase diagram, showing the threeclustering regimes as a function of the inverse shaking strengthD and the size ratio   r1=r2. The shaded areas are calculatedfrom the flux model, and the open circles (no clustering), stars(type I clustering), and triangles (type II clustering) correspondto experiments. The experimental results on the vertical dashedline   2 also feature in Fig. 2. The initial distribution ofparticles is always taken to be the same as in Figs. 1–3.VOLUME 89, NUMBER 21 P H Y S I C A L R E V I E W L E T T E R S 18 NOVEMBER 2002shaking (left plot) only one stable fixed point exists: theuniform distribution f150; 300g.In the middle plot, for moderate shaking strength, thehomogeneous state has become unstable and has givenway to two new stable fixed points. These correspond tocompartment B being either nearly empty (fixed point inthe lower left corner) or well filled (upper right). Theirbasins of attraction are indicated by the shading. Ourstarting point happens to lie in the basin of the lowerleft point, and so we end up with compartment B beingnearly empty. The arrows show that the small particlescluster first, and only when these have attained their finaldistribution do the large ones follow. Indeed, the flux oflarge particles is appreciable only when the number ofsmall particles in the compartment has become very low.Finally, in the rightmost plot (for mild shaking) we seethat the basin boundary has shifted, which explains theswitch from regime I to regime II: Our initial conditionnow lies within the basin of attraction of the fixed point inthe upper right corner. So we end up with a full compart-ment B. The same plot shows that the fixed points movefurther into their corners as D grows, i.e., the clusteringbecomes more pronounced for decreasing shakingstrength, just as in the monodisperse case [2,3].Further parameters.—Until now we have taken theratio of the radii   r1=r2  2, but how does the phe-nomenon depend on  in general? This ratio has a markedeffect on the critical D values where the transitionsbetween the regimes 0, I, and II take place. In Fig. 4we show the position of these regimes as a function of  ,from the flux model and from experiments. The verticaldashed line corresponds to   2 studied in Figs. 1–3. Itis seen that for  < 1:5 the transition to regime II isimmediate: here the larger beads are not sufficiently bigto compensate for the fact that they are a minority. It isthe larger number of beads that decides where the clustergoes, just as for the monodisperse case (  1). On the214301-4other hand, for high values of  , the dominant size of thelarge beads always makes them the decisive factor (onlyregime I survives). It is precisely the intermediate region1:5< & 2:3 in which the competition takes place. For  1:6 we witness the particularly interesting sequence0-II-I-II, both in the model and in experiment.Another parameter of interest is the ratio between thetotal particle numbers of each species, "  P1=P2 (up tonow we have always taken "  1=2). Also this ratioobviously has a big influence on the clustering behavior:a larger value of " means that the large beads become amore important minority (or even a majority for " > 1)and hence type-I clustering will gain ground.Conclusion.—The clustering behavior of a bidispersegranular gas is much richer than the monodisperse caseand shows a surprising new feature: The clustering can bedirected either towards the compartment initially con-taining mainly large particles (type-I clustering) or to theone containing mainly small particles (type-II), simplyby adjusting the shaking strength. All the experimentalobservations can be explained quantitatively by a bidis-perse extension of the Eggers flux model.This work is part of the research program of theStichting FOM, which is financially supported by NWO.[1] I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619(1993); H. Jaeger, S. Nagel, and R. Behringer, Rev. Mod.Phys. 68, 1259 (1996); L. Kadanoff, Rev. Mod. Phys. 71,435 (1999).[2] J. Eggers, Phys. Rev. Lett. 83, 5322 (1999).[3] K. van der Weele, D. van der Meer, M. Versluis, andD. Lohse, Europhys. Lett. 53, 328 (2001); D. van derMeer, K. van der Weele, and D. Lohse, Phys. Rev. E 63,061304 (2001).[4] D. van der Meer, K. van der Weele, and D. Lohse, Phys.Rev. Lett. 88, 174302 (2002).[5] W. Losert, D. G.W. Cooper, J. Delour, A. Kudrolli, andJ. P. Gollub, Chaos 9, 682 (1999); R. D. Wildman and D. J.Parker, Phys. Rev. Lett. 88, 064301 (2002); K. Feitosa andN. Menon, Phys. Rev. Lett. 88, 198301 (2002).[6] A. Rosato, K. J. Strandburg, F. Prinz, and R. H.Swendsen, Phys. Rev. Lett. 58, 1038 (1987); J. B.Knight, H. M. Jaeger, and S. R. Nagel, Phys. Rev. Lett.70, 3728 (1993); D. C. Hong, P.V. Quinn, and S. Luding,Phys. Rev. Lett. 86, 3423 (2001).[7] R. Mikkelsen et al. (to be published).[8] We neglect the dissipation resulting from collisions withthe wall, treating them as being completely elastic.[9] Above the cutoff height, the state variables of the twocompartments are in equilibrium and hence no net fluxoccurs. In principle, H will depend on the mean free pathof the particles, but here we take it to be constant.Naively integrating up to h! 1 would lead to an (un-physical) nonzero flux for Ni  0. To see this, consider acompartment without any type 2 particles: then T /1=N21 [from Eqs. (5) and (6)] and hence F1 loses its N1dependence in the dilute limit [cf. second line of Eq. (7)],making it nonzero for N1  0.214301-4

Competitive Clustering in a Bi-disperse Granular Gas

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