Temperature Oscillations in a Compartmentalized Bidisperse Granular Gas

Meiying Hou,* Hongen Tu, Rui Liu, Yinchang Li, and Kunquan LuBeijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences,

Beijing, 100080

Pik-Yin Lai† and C. K. Chan‡

Institute of Physics, Academia Sinica, NanKang Taipei and Department of Physics and Center for Complex Systems,National Central University, Chungli

(Received 16 May 2007; published 12 February 2008)

A granular clock is observed in a vertically vibrated compartmentalized granular gas composed of twotypes of grains with the same size. The dynamics of the clock is studied in terms of an unstableevaporation or condensation model for the granular gas. In this model, the temperatures of the two types ofgrains are considered to be different, and they are functions of the composition of the gas. Oscillations inthe system are driven by the asymmetric collisions properties between the two types of grains. Both ourexperiments and model show that the transition of the system from a homogeneous state to an oscillatorystate is via a Hopf bifurcation.

DOI: 10.1103/PhysRevLett.100.068001 PACS numbers: 45.70.Mg, 05.20.Dd, 05.70.Ln

At first sight, granular gases [1–3] share many similari-ties with their molecular counterparts. However, for com-partmentalized systems, phenomena which are impossiblefor equilibrium molecular gases such as Maxwell’s demon[4] or clustering [5] can occur in granular gases. Thesephenomena are possible because differences in granulartemperatures [6] can be stably sustained between the com-partments. Recently, an even more intriguing phenomenonknown as the granular clock [7,8] has been reported. Forthe clock, the clustering of the grains between the compart-ments becomes periodic (Fig. 1) when the monodispersegas is replaced by a bidisperse one.

The mechanism of granular clocks is still unclear. Onepossible origin is the Brazil nut effect (BNE) or the reverseBNE (RBNE) when there is vertical segregation [7,8].Since the clustering in monodisperse gas is driven by thedifference in temperatures between the compartments, it isplausible that the oscillation in temperatures in a bidispersesystem is also a driving force. For a bidisperse system, it isstill not clear if there is only one single temperature [9] forthe whole system or if the two types of grains have theirown temperatures [10,11]. Intuitively, the scenario of asingle temperature in a bidisperse gas would not lead toclustering oscillation because it is similar to the case ofmonodisperse gas. We are then left with the possibility ofdual temperatures in the system. Presumably, the clock isdriven by the oscillations of the dual temperatures.

In this Letter, we report results of an experiment de-signed to remove the effect of BNE in a compartmentalizedsystem by using a bidisperse system with grains of thesame size. We find that a granular clock can still beproduced. Our results can be understood by a model de-veloped to understand the mechanism of the granular clockin terms of dual temperatures. No BNE or RBNE assump-tions are needed. Therefore, even grains of the same size ina bidisperse system can produce granular oscillations.

Predictions of our model are verified by experiments.Both our experiments and model show that the transitionfrom a homogeneous state to an oscillatory state in such asystem is via a Hopf bifurcation.

Our experimental setup consisted of a rectangular glasscontainer 2.6 cm wide and 5.4 cm long with a height of13.3 cm, divided into two equal compartments by analuminum barrier 0.2 cm thick and with a height (h) of1.5 cm from the bottom. Steel balls (SBs) and glass beads(GBs) of the same size, 0.5 mm in radius, are used. In theexperiments, the total number of grains is fixed at 960,

FIG. 1 (color online). The phase diagram for the homogenous(HS), oscillatory (OS), and segregated (SS) states. See thecaptions of Figs. 3 and 4 for the uncertainties in vc and vf.Also shown are the photographs of the configurations of thesystem with glass beads being darker (red) and steel balls beinglighter (silver). Snapshots (a)–(e) are taken during one cycle ofthe clock. Note that f � 60 Hz and the period of the clock(Fig. 4) can range from tens to hundreds of seconds.

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while the ratio �0 � NGB=NSB varies from 1:1 to 7:1,where NGB and NSB are the number of glass beads andsteel balls, respectively. Vertical sinusoidal vibrations withfrequency f and amplitude a are applied to the containerto create granular gases with a shaking velocity v withv � 2�af.

Results of the experiments are summarized as a phasediagram in terms of v and �0 as shown in Fig. 1. Similar tothe findings of Ref. [8], three states [homogenous (HS),oscillatory (OS), and segregated (SS)] in the distribution ofgrains can be observed. These phases are separated by twotransition velocities vf and vc which are functions of �0.Experiments with different sets of a and f have shown thatthe boundaries of these three regimes are sensitive only tothe combination of af / v. For a fixed value of �0, as the vis being lowered, the system is first in HS when v > vc andthen turns into OS when vc > v > vf. The system be-comes segregated when v is lowered below vf. The fea-tures of the transition points vc and vf are that theamplitude of the oscillations (�) increases with (vc � v)while the period of the oscillation (�) decreases with (v�vf). Note that � is close to saturation around vf, and it isimpossible to determine � close to vc because of strongfluctuations in the system. Obviously a new type of granu-lar clock is produced without the BNE or RBNE as shownin the snapshots for the OS in Fig. 1.

Let us first consider a two-compartment system which ismade up of a right compartment (RC) and a left compart-ment (LC). Hereafter, quantities in the LC and RC will bedenoted by the L and R subscripts, respectively. If thegranular clock experiments were performed with molecu-lar gases, one would have needed to change the tempera-tures (T) of the two compartments in a periodic mannerexternally. For example, a molecular gas can be made toevaporate from LC and condense into RC if TL is raisedhigher than TR and vice versa. We show below that thisperiodic temperature change can be generated endoge-nously by an instability in the dynamics of a bidispersegranular gas.

For a monodisperse granular gas, there is a single granu-lar temperature for the system which is approximately T /�uN�

2, with N being the number of grains and u is somecharacteristic velocity which is related to the shakingvelocity (v) of the system [4]. The approximation of asingle height-independent temperature has been shown tobe rather satisfactory by molecular dynamics simulations[4,5]. T is inversely proportional to N2 because most of thedissipation of the system comes from the collisions of thegrains among themselves, and higher density means morecollisions and therefore a lower T. When v is large, onewould expect that the grains are free to move between LCand RC. Therefore, the grains are distributed equally(NR � NL � N=2) in LC and RC with TR � TL.However, when v is lowered to a point that most of thegrains in RC and LC cannot be exchanged, the systemcan be unstable. Consider a fluctuation of �N in the

number of grains in the RC, i.e., NR � N=2� �N, NL �N=2� �N. In this case, TL will be higher than TR, andmore grains will jump from LC to RC than those from RCto LC. During this process, TL is raised further by theevaporation of grains and vice versa for the condensationin RC. Therefore this process is unstable, and it will notstop until the difference in grain number between the twocompartments is so large that the smaller probability forgrains to jump from RC to LC is compensated by a largenumber of NR. This is the scenario in Ref. [4].

Next, consider the temperature in a bidisperse granulargas in a single compartment with a total number of N �NA � NB grains and a number ratio �0 � NA=NB betweenthe two types of grains A and B. Although a single T issometimes assigned for such a system [5], we argue that TAand TB are in general different. For such a system, there arethree types of binary collisions: namely, AA, AB, and BB.The most important one here is the AB collision which isthe source of energy exchange between the two types ofgrains. Let us further assume that during AB collisions, B isslowed down by Awhile A is accelerated by B. That is, A isgetting kinetic energy from B. For such a system, TA andTB will be a function of �0. For example, TA will be higherwhen �0 is smaller and vice versa for TB. In other words,the temperature of B grains is lowered by the presence of Awhile the presence of B grains increases the temperature ofA grains. Therefore, when � changes, the temperatures fora particular kind of grain will also be changed.

For such a bidisperse gas in a two-compartment system,relevant parameters are NA � NAL � NAR, NB �NBL � NBR, �L �

NALNBL

, and �R �NARNBR

. For high v, wewill have NAL=NA � NBL=NB �

12 (i.e., �L � �R � �0),

TAL � TAR, and TBL � TBR with TB < TA. When thestrength of shaking is lowered, we can always get to apoint at which only type A grains are free to exchangebetween the two compartments. When this happens, thesystem will be similar to that of a single type of A. In such acase, our arguments about clustering will be applicable.That is, there will be an unstable evaporation and conden-sation of A. However, the situation here is more compli-cated because there will be changes in �L and �R duringthis process.

When grains A evaporate from RC and condense intoLC, both TAL and TBL are lowered while those in RC willbe raised. When enough of grains A are evaporated fromRC, the TBR will be raised, and it can be so high that grainsB start to evaporate from RC and condense in LC, too.Since this is an unstable situation, once it happens, more ofgrains B will jump from RC to LC. However, when grainsB start to jump from RC to LC, TAL starts to rise, too. Whenenough B has jumped from RC to LC, TAL will be so highthat grains A start to jump from LC back to RC. Again, thisis an unstable situation which will lead most of grain A tojump from LC to RC. That is, there is an oscillation ingrains A. Similar arguments will show that there will alsobe oscillations in grains B, too.

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Let us denote the masses and radii of grains A and B bymA;mB and rA; rB, respectively, the restitution coefficientsof AA, BB, and AB collisions by eA, eB, and eAB, respec-tively, and the bottom area of the compartment by �. Withthese notations, we have TAL � �

vADANAL

�2 and TBL �

� vBDBNBL

�2, where DA � 4����

�p

r2A�1� e

2A�=� [12] and simi-

larly for DB. The two velocities vA and vB are related to vas vA � v=p��� and vB � v=q��� for some general func-tions p and q, where @p=@NBL < 0 and @q=@NAL > 0. Therequirements on @p=@NBL and @q=@NAL are used to imple-ment our assumption that the temperature of grains A israised by the presence of B, while the presence of grains Alowers the temperature of grains B. It can be shown [13]theoretically that the situation of grains A gaining energydue to A-B collisions occurs when eABmB > mA. With thisformulation, there are two equations for the balance of thefluxes of A and B between RC and LC as

1

KA_NAL � �

N2AL

vA��L�e�aA�N

2AL=v

2A��L��

��NA � NAL�

2

vA��R�e�aA��NA�NAL�

2=v2A��R��

1

KB_NBL � �

N2BL

vB��L�e�aB�N

2BL=v

2B��L��

��NB � NBL�2

vB��R�e�aB��NB�NBL�

2=v2B��R��;

(1)

where aA � ghD2A (aB is similarly defined) with g being

the acceleration due to gravity and h is the height of thewall separating LC and RC; KA (KB) are constants thatdepend on the properties of A (B). Note that the forms ofthe above equations are similar to those used in Refs. [7,8].However, the physical meanings of the above equations aredifferent from those of Refs. [7,8]. Here the dynamicalvariables are the number of grains A and grains B in one ofthe compartments, whereas those of Refs. [7,8] are thefractions of grains A and B in one of the compartments.For grains of a single type [4], the fraction completelydetermines the properties of the system. But for a binarymixture, one also needs � to specify the effects of one typeof grain on the other.

Obviously, NAL � NA=2 and NBL � NB=2 is a fixedpoint, but it will be stable only when v is large. When vis smaller than a threshold vc, it will become unstable andturn into a limit cycle. To be specific, we use reasonablefunctional forms p��� � 1

2 �2�1

1��� and q � 2p (form I)for actual calculations. Note that vA � vwhen �! 1 andvB � v when �! 0 as expected. Furthermore, it can beshown that [14] the fixed point loses its stability at vcvia a supercritical Hopf bifurcation giving rise to a stablelimit cycle. We have also checked that the same behaviorholds for other reasonable forms such as p��� �1� 1

2 e��=�0 , q��� � 1� �=�0 (form II), and p��� � 1�

12 e����=�0�

2=2�, q��� � 1� ��=�0�2 (form III).

Figure 2 shows the null clines of Eq. (1) for differentvalues of v together with the trajectory of an arbitraryinitial point. In Fig. 2, we have chosen the values of aAand aB in Eq. (1) such that vc is close to unity for a systemwith KA � KB � 1, N � 2000, and �0 � 2. Note that thedifference in aA and aB can originate from either thedifference in the restitution coefficients or in their radii.It can be seen that the fixed point �NA2 ;

NB2 � [Fig. 2(a)] turns

into a limit cycle when v is lowered [Fig. 2(b)]. As v isfurther reduced, both the amplitude (�) and period (�) ofthe limit cycle increases [Fig. 2(c)]. When � is largeenough, new stable fixed points are created [Fig. 2(d)].The main difference between our model and that of Ref. [8]is that in Ref. [8] BNE is needed while only asymmetry incollision properties is needed here, be it mass or size.

In the experiments, it can be observed that the SB isgiving energy to the GB during collisions and therefore theGB is playing the role of grains A. To compare the modelwith the experimental results quantitatively, we examine �and � of the oscillation close to the fixed points vc and vf.In the experiments, � is measured by a stopwatch and � ismeasured by counting the number of steel SBs in a com-partment by stopping the experiment when the SBs areseen to be least populated in one of the compartments.Figure 3 is the measured dependence of � on v� vctogether with a fit of the form � �vc � v�1=2. It can be

0.0 0.2 0.4 0.6 0.8 1.00.0

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0.8

1.0

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1.0(a) (b)

(c) (d)

FIG. 2 (color online). Simulated phase trajectories as a func-tion of shaking velocity v showing the three different states ofthe systems. (a) Stable fixed point for the homogeneous state atNAL=NA � NBL=NB � 1=2 with v � 1:01, (b) the oscillatorystate as a limit cycle when v � 0:99, (c) the amplitude � of thelimit cycle increases when v is further lowered to 0.97, and(d) finally another fixed point is reached where there is a stablesegregation of grains with v � 0:927.

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seen that our data are consistent with a supercritical Hopfbifurcation only for small vc � v. Presumably, this is dueto the fact that linear stability analysis will fail when vc �v is large. Figure 4 shows both the �v� vf� dependence of� obtained from experiments and from our model fromEq. (1) (inset). It can be seen that all of the data can be putin the form � �v� vf��� with� ’ 1 for the experimentsand different values of � for different forms of p and q.

This last finding suggests that our flux model gives a goodquantitative description of the physics of the system onlyclose to the unstable point, independent of the forms of pand q used. However, detailed dynamics of the system willbe needed to determine the value of �.

It is clear that our model captures the essential featuresof the clock phenomenon. In fact, our argument can beapplied even to the heaping phenomenon [15]. For such acase, in the language of our model, grains from the ma-jority parts of the compartment are evaporated and thencondensed at one of the corners of the compartment toform the heap. There will be large temperature differencesbetween the grains in the heap and those outside the heap.Finally, it is known that the RBNE can also be driven bydensity differences [16]. However, our system is a dilutegas; the effects considered in Ref. [16] are probably notapplicable.

This work was supported in part by CNSF GrantNo. A0402-10474124 and by CAS Grant No. KACX2-SW-02-06. P. Y. L. and C. K. C. acknowledge the supportby NSC under Grants No. 95-2112-M-008-009 and No. 95-2112-M-001-003 and by NCTS.

*mayhou@aphy.iphy.ac.cn†pylai@phy.ncu.edu.tw‡ckchan@gate.sinica.edu.tw

[1] H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod.Phys. 68, 1259 (1996).

[2] L. P. Kadanoff, Rev. Mod. Phys. 71, 435 (1999).[3] P. G. de Gennes, Rev. Mod. Phys. 71, S374 (1999).[4] J. Eggers, Phys. Rev. Lett. 83, 5322 (1999).[5] R. Mikkelsen, D. van der Meer, Ko van der Weele, and

D. Lohse, Phys. Rev. Lett. 89, 214301 (2002); Phys.Rev. E 70, 061307 (2004).

[6] A. Mehta and S. F. Edwards, Physica (Amsterdam) 157A,1091 (1989).

[7] R. Lambiotte, J. M. Salazar, and L. Brenig, Phys. Lett. A343, 224 (2005).

[8] S. Viridi, M. Schmick, and M. Markus, Phys. Rev. E 74,041301 (2006).

[9] R. D. Wildman and D. J. Parker, Phys. Rev. Lett. 88,064301 (2002).

[10] Klebert Feitosa and Narayanan Menon, Phys. Rev. Lett.88, 198301 (2002).

[11] G. W. Baxter and J. S. Olafsen, Granular Matter 9, Nos. 1–2, 135 (2007).

[12] DA and DB can be interpreted as dissipation factorsbecause they vanish in the limit of elastic collisions.However, such a limit is not relevant in our phenomeno-logical model in which dissipations are always important.

[13] M. Hou, P. Y. Lai, and C. K. Chan (unpublished).[14] J. Guckenheimer and P. Holmes, Nonlinear Oscillations,

Dynamical Systems, and Bifurcations of Vector Fields(Springer, New York 1983).

[15] P. Evesque and J. Rajchenbach, Phys. Rev. Lett. 62, 44(1989).

[16] T. Shinbrot, Nature (London) 429, 352 (2004).

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.50.0

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1.0

1.1

vC

- v (cm/sec)

= 7:1

= 6:1

= 5:1

= 4:1

= 3:1

= 2:1

FIG. 3 (color online). Variation of oscillation amplitude � as afunction of distance away from the critical velocity (vc � v) forvarious �0. The solid line is a fit of the data to the form ��vc � v�

1=2 for small vc � v with vc being adjusted within 20%of the experimental observation to give the best fit. Uncertaintiesof the data are smaller than the size of the symbols.

0 1 2 3 4 5 6 7 80

100

200

300

400

0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98

0.01

0.02

0.03

0.04

0.05

0.06

Form III= -0.59

Form II= -0.50

(Arb

itra

ryU

nit)

v (Arbitrary Unit)

Form I= -0.32

(sec

)

v - vf

(cm/sec)

FIG. 4 (color online). Variation of � as a function of v� vfmeasured for various �0. The symbols are the same as those inFig. 3. The solid lines are fits of the data to the functional form� �v� vf��1 with vf being a fitting parameter. The insetshows the results from the calculations of Eq. (1) with the threedifferent forms of p and q. Note that the calculated results can allbe fit to the form � �v� vf�

�� but with different values of �.Uncertainties of the data are smaller than the size of the symbols.

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