On cyclic oscillation in granular gas

ARTICLES20. Sato, N, Meijer, L, Skaltsounis, L et al., Maintenance of pluripo- Chinese Science Bulletin 2005 Vol 50 No. 8726-730tency in human and mouse embryonic stem cells through activationof Wnt signaling by a pharmacological GSK-3-specific inhibitorOn cyclic oscillation inNat.Med.,2004,10(1):55-63granular gasES cells, Cell Struct Funct., 2001, 26(3): 137-14MIAO Tiande, LIU Yuan",2, MIAO Fuxingescu.S, Stemness, fusion and renewal of hematopoMU Qingsongand embryonic stem cells, J Cell Mol. Med., 2003, 7(2): 103-112. 1. Deparment of Mechanics, Lanzhou University, Lanzhou 730000,23, Haegele, L, Ingold, B, Naumann, H. et al., Wnt signalling inhibitsneural differentiation of embryonic stem cells by controlling bone2. School of Civil Engineering, Lanzhou Jiaotong Ur730070. Chinamorphogenetic protein expression, Mol. Cell Neurosci., 2003, Correspondence should be addressed to Miao Tiande (email: miaotd@24(3):696-708Izu.edu.cn)24. Rattis, F M. Voermans, C. Reya, T, Wnt signaling in the stem cell Abstract In a Maxwells demon experiment on a diluteniche, Curr Opin. Hematol, 2004, 11(2): 88-94granular gas made up of two species of particles placed in25. Humphrey, R.K. Beattie, G.M., Lopez, A.D. et al., Maintenance two closely connected containers, a cyclic oscillation similarof pluripotency in human embryonic stem cells is StAT3 indeto the chemical oscillations is recognized that the two speciesof particles cluster alternately in the two containers. Thenpendent, Stem Cells, 2004, 22(4): 522-530the nonlinear behavior is qualitatively explained based on a26. Smith, A G, Embryo-derived stem cells: Of mice and men, Annu. flux model.Rev. Cell Dev Biol. 2001.17: 435-462Keywords: granular gas, cyclic oscillation.DOI:10.1360/982004-612Stat3 in IL-6-induced regulation of growth and differentiation inMI leukemia cells, EMBO., 1996, 15(14): 3651-36.One of the key differences between ordinary molecular28. Palmieri,SL, Peter, W Hess, H et al., Oct-4 transcription factor gas and granular gas is the tendency of the latter to sponis differentially expressed in the mouse embryo during establish- taneously form highly concentrated regions or clusters[ment of the first two extraembryonic cell lineages involved in im- Recently, a simple experiment with a vibrofluidizedplantation, Dev. BioL., 1994, 166(1): 259--267granular gas has arrested attention, where an especialNichols, J. Zevnik. B. Anastassiadiclustering phenomenon has been observed4.5. In this ex-pluripotent stem cells in the mammalian embryo depends on the periment, a number of steel beads are initially distributedPOU transcription factor Oct4, Cell, 1998, 95(3): 379--391uniformly in a box divided into two identical compartments by a wall with a narrow horizontal hole near thepresson or box's bottom Shaking the box verticaOct-3/4 defines differentiation dedifferentiation or self-renewala gaseous state and are free to travel between the comEScl,Nat. Genet.,2000,24(4):372-376partments through the hole. After a short time, most of the31. Becskei, A. Serrano, L. Engineering stability in gene networks by beads migrate spontaneously to one of the two compart-autoregulation, Nature, 2000, 405(6786): 590-593ments. The compartment contains a lot of beads, while the32. Mitsui, K, Tokuzawa, Y, Itoh, H et al, The homeoprotein Nanog other contains only a few. The beads in the experiment actis required for maintenance of pluripotency in mouse epiblast and as Maxwell,s demon in classical thermodynamics, whichES cells,cell,2003,113(5):631-642preferentially makes the gas molecules pass from leftright or vice versa. Therefore the above experiment is33. Chambers, I, Colby, D,,Robertson, M. et al., Functional expres- vividly called the Maxwells demon experiment. Somesion cloning of Nanog, a pluripotency sustaining factor in embrexperimental and theoretical work about this phenomenononic stem cells, Cell, 2003, 113(5): 643--655has been carried out 5-8134. Cavaleri, F, Scholer, H R, Nanog: A new recruit to the embryonicVan der meer et al. [9I studied a bidisstem cell orchestra, Cell, 2003, 113(5): 551-552.composed of some large and small beads. Here a competi35. Fujikura, J, Y amato, E, Yonemura, S et al, Differentiation of em- tive cluster is found: depending on the shaking strengthbryonic stem cells is induced by GATA factors, Genes Dev, 2002, the clustering can be directed either towards the compartment initiallother containi中国煤化工 beads or to the36. Yoshida-Koide U. Matsuda. T. Saikawa K. et al.. Involvement ofCNMHRas in extraembryonic endoderm differentiation of embryonic stemmixture composed of particles of different species andcells,biocheM.biophys.res.commun2004,313(3):475-481.diametersHereadilutegranulargascomposedoftw(Received October 10, 2004; accepted March 9, 2005) species of particles placed in a compartmentalized system7方数据Chinese Science Bulletin Vol 50 No.8 April 2005ARTICLESis experimentally investigated. An interesting phenome- process. We turn on the shaker. After about 204 s the mil-non is observed: the granular gas is found to undergo a lets in compartment a quickly cluster into compartment Bcyclic oscillation that the two species of particles cluster through the hole while the number of the mungs in comalternately in the two compartments under vertical excita- partment A is almost unchanged, as shown in Fig. 1(b)After about 290 S, there are a few millets in compartment1 Experimental method and model predictionA, and the mungs start to pass through the wall into comThe experimental setup(Fig. 1), similar to that of ref. partment B(Fig. I(c). After the majority of the two spe[5], consists of a rectangle perspex box with a base of 5 cles get into compartment b(t= 364 s, Fig. 1(d), the mil-cmx7 cm and a height of 30 cm, divided into two equal lets start to leave compartment B. About 250 s later,thecompartments(labeled as A and B respectively) by a wall mungs follow the millets until most of them return to Awith a narrow horizontal slit of dimension 6 mmx cm at These processes are shown in Fig. 1(e)-1(f). The whole2cm from the base The box is mounted on a shaker with process lasts 700 s or so. Then this process periodicallyadjustable amplitude a and frequency ffollows with the same period. We call this phenomenon aThe experimental material is a binary mixture of millets cyclic oscillation in granular gas(1.0-1.2 mm in diameter, e,=0.67 in restitution coefThen, to investigate the effect of the initial distributionof the two species in each compartment on the cyclicficients,2.5x10 g in mass)and mungs(2.0-2.5 mm in oscillation, we consider four kinds of situationsdiameter, e2=0.4 in restitution coefficients, 500x10 g N, N2A=(1/2P. 1/2P2),(1/5R.4/5B)in mass). The number of millets(P, is 4000 and that ofmungs(P2) is 400(P, R. Here Nia and n2a denote the number ofThe initial distribution of the two species is depicted in millets and mungs in compartment A at initial time re-ig. 1(a),with (4/5R. 4/5P2)= 3200,) in com- spectively. The total number of each species(P1, P2) ispartment A and (1/5P, 1/5P2)=1800, 80] in compart- granular gas is observed in all of these situations, imply.ment B. The shaker's amplitude and frequency are 2 mm ing that this phenomenon is independent of the initial disand 20 Hz. a digital camera is used to record the whole tribution of particles and it is a stable limit cycle(b)t=204s(c)t=290s中国煤化工(e)=594sCNMHGFig. 1.(a)-(f) Six images from a cyclic oscillation experiment. The number of millespectively. The initial distribution of two species in compartment A and B are 4/5P. 4/5P, and 15P. 1/ 5P,. The shaking amplitude and frequency are a=2 mm and f= 20 Hz. These images show the initial periodic transporting process of two species in the cyclicoscillation experiment灼9数鸦 ence Bulletin Vo.50No.8Apm2005727ARTICLESIt is found that the oscillation cycle is strongly influ- wards movement. If the parameters of the two kinds ofnced by the shaking amplitude. Table 1 gives the de- particles, such as diameters, masses, restitution coeffipendence of the oscillation cycle I on the applied shaking cients, and the ratio of the number of them, are properlyamplitude when the excited frequency keeps constant. The selected, the granular system will adjust itself to make thecycle I decreases obviously with the shaking amplitude. number of millets in one compartment larger than that ofhere is no cyclic oscillation found after the amplitude mungs during the vibration process. Here the SRBNP andexceeds 3 mm. This shows that when the excited fre- the cluster will occur, which results in the cyclic oscillaquency is held constant, and that there is an amplitude tion. For example, sometime after the experiment starts,region in which the cyclic oscillation in granular gas can and if the majority of millets and mungs cluster in com-be observedpartment A, while the number of millets is much largerIn Table 2, the ratio of the total particle number of each than that of mungs(Fig. 1(a), the movement status ofpecies,P: P, is changed and the other parameters are these particles is similar to the rBNP Mungs stay close toconstant. The oscillation will not appear when the ratio is the bottom and hardly jump, transferring energy from theless than a certain value. Different kinds of binary mixture vibrating bottom to millets and activate them while theyare found to influence the oscillating phenomenon, too. become in idleness relatively. So it is easier for the milletsWe take some samples for experiments several timesto leave compartment A. When more and more milletsmillets and sticky rices, beads and beads, mungs and escape to compartment B(Fig. 1(b)), the mung beans inbeads, sticky rice and beads. The two latter mixtures show compartment a become more mobile so that they start tothe cyclic oscillation while the others do notmove across the wall into B gradually, where they areimmediately swallowed by the millets and lose movabilitTable 1 The oscillating cycle varing with the applied shaking amplitude in B(Fig. I(c)). At the same time, the millets in compartShaking amplitude a1752.053.0ment b become more mobile(the mungs transfer energyOscillating cycle //min 28to millets). when the number of the mung beans increasesThe shaking frequency f is 20 Hz. When the shaking anto a certain extent(most mungs and millets stay in B), theexceeds 3.0 mm there is no cyclic oscillation foundmillets start to cluster to compartment A. At last the millets and the mung beans show a cyclic oscillationFurther experiments show that the cyclic oscillation inTable 2 The effect of the ratio of the total number of the two species on granular gas coexists with the SrBnP and the cluster. Ifthe cyclic oscillation5:1the srBnp and the cluster do not happen, the cyclic osCyclic oscillationillation will not occur. So there exist important intrinsWhen the ratio is less than 5 there is no cyclic oscillation foundrelations between the SrBNp and the cluster and the cydenotes existence of the oscillation and "n" denotes non -existenceclic oscillation. However. the existence of the srbnp andthe cluster during the vibration depends on many factors2 Analysis of experimental resultssuch as the material parameters of particlesIt has been pointed out that inelastic collisions between 3 A flux modelthe particles play an important role in driven granular sys-tems and can cause size segregation and clustering.InTo explain the existence of the asymmetric steady stateour experiments millets and mungs are free to travel bethe maxwells demonproposedtween compartments A and B, and the number of them in analytical theory based on the kinetic theory for diluteeach compartment varying all the time. When millets in granular gas. The gas within a compartment is assumed toone compartment are much more than mungs, most mungsbe in thermal equilibrium with respect to the granularcluster at the center of the compartment and stay close to temperature T- v2 ND(D is special dimension; v is thethe bottom. Millets move around the mungs and the ma- velocity of a particle). This definition of temperature isjority of them jump on top of mungs. This phenomenon is customary for granular media. He considered the inter-similar to the reverse Brazil nut problem(rBnp) o(In change of particles between the two compartments as anour experiments this phenomenon and the cluster disap- effusion process and gave a flux model depicting the depear gradually with a decrease in the ratio between the pendence of the number density (n)on the timeparticles numbers of each species(P: P2). Similar rednverse Brazil nut problem (SRBNP) and the cluster between grains affect the distribution of them and the trans中国煤化工port of energy. They make mung beans transport energy where the ceC N MH Gnts depending orfrom the vibrating bottom to millets, which thereby gain the total number ot particles and their properties(masscomparatively large velocities. At the same time, millets restitution coefficient e, diameter d, etc. ) on the geometryjump above the large grains, which resist the mungs up- of the system(height h of the hole, ground area S of each72方数据Chinese Science Bulletin Vol 50 No. 8 April 2005ARTICLEScompartment), and on the frequency f and amplitude a ofthe shaking. The theoretical prediction for the bifurcationdiagram compares fairly well with the results from simulations using the molecule dynamics method as well aswith those from experiments 5,7,8t06A mixture system composed of two kinds of particleshas more complicated interactions than the granular system consisting of a species of particles. Function (1)can-not give a satisfactory explain to the cyclic oscillation inour experiments. Considering the coupling of the two spe-0.1cies, which comes from the damp between the fluxes, weintroduce some coupling terms fi n2,f(1-2), f2/050I00150200250300350400450500t/sf2(1-n). The controlling equations are shown as fol- Fig. 2. The time evolution of n and n2. Here coupling coefficients arelowsgiven as: f,=0.5, 2=-0.5, 6=4.5 and b,=5.0. The initialmost ofb=4(+1n2)nevalues of two species are no=0.8, n2=0.8, whichparticles are placed in compartment A at the initial time. The numericalresult shows an obvious cyclic oscillation, which agree with experimen+A1(1+f1n2)(1-n1)etal observation qualitativen2=-42+n)n用The time evolution of the ratio n, for four differentinitial distributions of the two species in compartment A is+A2(1+fn11-n2)2e-b(h-n)2given in Fig. 3. Although there are different initial distrbutions in compartment A, the oscillation cycle is alwayshere n, and n, denote the ratio of the numberof mil- a constant (=152 S). It implies that the oscillation ofts and mung beans in compartment a to their total num- millets and mungs is independent of the initial distributionber respectively. fi, f2, b, and b2 are coupling coef- of two species (n1o,n2o),which iso consistent withficients depending on material parameters(suchexperimental resultscoefficients of restitution, diameters and masses. the totalnumber of the two species, etc. ) The values of thosequantities should be determined through furthern12920.m20-08-n10-08.m0-0.8lents. Here, our aim is to give a qualitative discussiontherefore, we will simply take some specific values forthose quantities to solve the differential equations by thefourth-order Runge-Kutta method0.5The time evolutions of n, and n2 are shown in Fig. 20.4he values of the coupling coefficients f,, f2, b,b, are taken as 0.5, -0.5, 4.5 and 5.0 respectively. At the0.1b1=45,b2=5,t=152sstart of the simulation viz. t=o s. the number of milletsand mungs in compartment A are n10=0.8, /20=0.8 respectively; in other words, most of both species are incompartment A. After 10 s or so, ni decreases to 0. 17 Fig 3. The time evolution of the ratio n, for different initial distribu-sharply and n2 becomes slightly less, which means most of tions of millets and mungs in compartment A Here coupling coefficientsmillets cluster in compartment B and the majority of are the same as those in Fig. 2. The oscillation cycles in these evolutionsare always about 152 s, being consistent with our experimental resultmungs are still in compartment A. Hereafter, nI begin to qualitativelyincrease slowly and the decreasing of n2 becomes rapid inA. At about t=110 s nI has the same value as n2 in A,For different coupling coefficients, we obtain differentlal to 0.3, that is, 70 percent of both species are in final distributompartment B Subsequently, n2 follows ni to increase to distribution ar中国煤化工Fig.4: Given tworesume their initial clustering status in compartmentoupling coeCNMHG, two species ofgradually. The numerical results qualitatively agree with particles symmeuncany aisuidute ll compartment a and Bthe experimental observations. The time evolutions of n, with b1= 1.0 and b2= 2.0, cluster in one compartmentand n, show an obvious cyclic oscillation.with b1=9.0 and b2=6.0, and separately cluster in eachChiaese seience Bulletin Vol 50 No8 April 2005729ARTICLEScompartment with10.0 and b2=15.0. These experiments, this work is only consistent with the experinumerical results have been observed in our experimentsments qualitatively, but not quantitatively. a possible reason lies in the inapplicability of the Boltzmann distribu1.2tion to a dilute granular gas in a gravitational field, whichH1(b-1.b2=2)=n(b1=10b2=15)6)n2(b1=1,b2=2)n2(b1=10b2=15)Ignoresments(Fig. 1(a)-(f)) showed that the distributions ofbinary mixtures, which always vary with the mixture ratioare different from boltzmann distribution when thenumber of large particles is far less than that of small paricles, the former always cluster at the center of the bottomand are surrounded by the latter. The distribution of largef1=0.5,f=0.5particles is far away from Boltzmann distribution. Whenthe number of small particles becomes less, the distribution of large particles is close to Boltzmann distribution.0100200300400500600700Thus it is important to estimate the density distribution ofparticle mixtures for further research in granular mixtureFig 4. The time evolution for different coupling coefficients For given dynamics. Presently, experimental researches are still thef1=0.5andf2=-0.5ecies of particles symmetrically distribute imainstream for this subject, and a satisfactory kinetic the-A with b, =9.0 and b: =6.0, and separately cluster in each compartments ory has not been presented yet. We expect to extend ourwith b,= 10.0 and b,=150. These numerical results have been observed work to multi-mixtures to interpret abundant properties ofthe granular systemThConclusions and discussionsScience Foundation of China( Grant Nos. 10172041 and 10402012)The mixture of two different species of particles dis- Referencesplays more abundant dynamical phenomena than the systems composed of one single kind of particles under ver-Jaeger, H. M, Nagel S. R, Behringer, R. P,, Granular solids, liq-tical excitation. In addition to the clustering behavior,uids, and gases, Reviews of Modern Physics, 1996, 68(4): 1259there emerges a surprising new feature: the two species of1273particles cluster alternately in compartment A or B and 2. Herrmann, H J, Granular matter, Physica A,2002,313: 188--210hdergo a cyclic oscillation. The existence of this beha3. Herrmann, H. J, Luding, S, Cafiero, R. Dynamics of granularior similar to the classical chemical oscillations dependson particle parameters(mass m, restitution coefficient e,systems, Physica A, 2001, 295: 93-100diameter d, etc )and the mixture ratio of these particles4. Troy, S, Fernando, J. M., Noise to order, Nature, 2001, 410: 251and is independent of the initial distribution of the particles in the compartments. Given two species of particles, 5. Eggers, J, Sand as Maxwells Demon, Physical Review Letterthe oscillation cycle is also independent of the initial dis-999,83(25):5322-5325tribution of the particles in two compartments, while there 6. Brey, J. J, Moreno, F, Garcia-Rojo, R, Ruiz-Montero, M.J., Hyare some intrinsic relationships between the oscillationdrodynamic Maxwell demon in granular systems, Physical Reviewcycle and the shaking strength.E,2001,65(1):011305The dynamical behavior of complex granular gas is a 7. van der Weele, K, van der Meer, D, Versluis, M, Lohse, D,Hys-nificant phenomenon. It enriches phenomena ofself-organization and suggests a potential way for underteretic clustering in granular gas, Europhysics Letters, 2001, 53standing the intrinsic rules in the field of granular media.328—334The experimental data in this paper shows that the intrin-8. van der meer. d. van der Weele. K. Lohse. d. Bifurcation dia-sic factor, which makes the phenomenon come into beinggram forcompartmentalized granular gases,Phys.Rev. E,2001,63:is coexistent with similar reverse Brazil nut problem and061304.the cluster caused by inelastic collisions in mixtures. By 9. Mikkelsen. R. van der Meer, D, van der Weele, K. Lohse. Dexpanding Eggers flux model, we developed a theoreticalCompetitive clustering in a bidisperse granular gas, Phys. Revnodel to consider the coupling of granular mixtures. Just中国煤化工as shown in Figs 2-4, our flux model is consistent withItal data qualitatively10. Hong, DCNMHG brazi nut problemHowever, in this paper only some preliminary work hasen percolation anu condensation, Phys. Rev.been done. Although a flux model for binary mixtures is1,86:3423-3434presented to explain the cyclic oscillation observed in our(Received December 12, 2004; accepted February 22, 2005)73方数据Chinese Science Bulletin Vol 50 No. 8 April 2005