CorrelationbetweenVoronoivolumesindiscpackings

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1 February EPL, 97()344 doi:.9/95-575/97/344 CorrelationbetweenVoronoivolumesindiscpackings Song-ChuanZhao (a),stacysidle,harryl.swinney andmatthiasschröter (b) Max-Planck-InstitutfürDynamikundSelbstorganisation-Bunsenstr.,D-3773Göttingen,Germany,EU CenterforNonlinearDynamicsandDepartmentofPhysics,UniversityofTexasatAustin Austin, TX 787, USA received 9 September ; accepted in final form December published online 3 January PACS n Granular systems PACS 45.7.Cc Static sandpiles; granular compaction PACS j Disordered solids Abstract We measure the two-point correlation of free Voronoi volumes in binary disc packings, wherethepackingfractionφ avgrangesfrom.875to.838.weobserveshort-rangedcorrelations overthewholerangeofφ avgandanticorrelationsforφ avg>.877.thespatialextentoftheanticorrelationincreaseswithφ avgwhilethepositionofthemaximumoftheanticorrelationandthe extentofthepositivecorrelationshrinkwithφ avg.weconjecturethattheonsetofanticorrelation corresponds to dilatancy onset in this system. Copyright c EPLA, Introduction. Dry granular matter interacts only via elastic and frictional forces, which require particles to be in contact; spatially extended interactions like Van der Waals forces typically play no role. However, granular particles form networks of force chains [], which implies the existence of local correlations. Lechenault etal. have shown that the logarithm of the distribution of free volumes in granular matter scales in a non-extensive way with the cluster size, which implies the existence of correlations between Voronoi cells[]. A similar scaling was also observed in monodisperse sphere packings[3]. These experimental observations raise the question how the spatial extent of such correlations changes with packing fraction. This question is especially important for granular systems with glassy behavior, where a number of groups have studied the length scale related to spatially heterogeneous dynamics[4 7]. In this paper we demonstrate the existence of both correlations and anticorrelationinthefreevoronoivolumes,andwemeasuretheir spatial extent as a function of volume fraction. Experiment. Experiments are performed in a two-dimensional air fluidized bed, as sketched in fig. (a). TheparticlesareabinarymixtureofTeflondiscswith diametersofd s =6mmandd l =9mm.Theyareconfined (a) songchuan.zhao@ds.mpg.de (b) matthias.schroeter@ds.mpg.de ThefreeVoronoivolumeisthedifferencebetweentheactual Voronoivolumeofacellandthevolumeofacellinahexagonal packing. between two vertical glass plates (thickness mm) separated by a distance slightly larger than the thickness of the discs(3.86 mm). The bed contains approximately 75discsofeachsize. Thediscpackingistappedfrombelowbyairpulses flowing through a distributor of open-porous foam(duocel 4 PPI aluminium foam). Electrostatic charging is minimized by grounding the distributor. The duration and strength of air pulses are controlled by two Waston Smith 6B464 mechanical pressure regulators and two pairs of Jefferson 6 series electronic valves. Three sensors below the distributor are used to measure air pressure (Validyne DP5), humidity(honeywell HIH-36) and temperature(ysi 4433). Typical humidities and temperatures are3.8±.3%and4.8±.4 C. Theaveragepackingfractionφ avg valueiscontrolled bythetypeanddurationoftheairpulses.theuseof different tapping modes(cf. fig. (b)) enables us to vary φ avg from.875 to.838, as shown in table. The mostcompactbedsareobtainedbyfirstdrivingthebed toanewconfigurationusingaprimarypulse,andthen following that by shorter secondary pulses to compactify the bed. The duration of the primary pulse and the strengthofairflowarefixed,butthenumberandduration of the secondary pulses are adjustable. Looser packings are obtained using two airflow pathways with different flow rates; when the primary pulse stops, the secondary pathway still provides some small flow, which slows down the settling discs. 344-p

2 Song-Chuan Zhao etal. (a) (b) (c).8 primary pulse secondary pulses Pressure (bar).6.4. valve valve distributor time (s) extended pulse 5mm Reg Reg Pressure (bar).6.4. air source..6. time (s) 9mm Fig. :(Color online)(a) Sketch of the experimental setup(particles not drawn to scale).(b) Examples of the different tapping modes. The pressure is measured below the distributor.(c) Image of the experimental packing. Table : Parameters of the air pulses. All the experiments start with the same primary pulse(3. bar measured at the regulator, ms), either followed by several secondary pulses with the same pressure or accompanied by an extended pulse. φ avg Extendedpulse Secondarypulses bar ms bar ms ms ms.856 5ms ms ms ms ms ms+5 5ms ms+3 5ms Aftereachtap,thepackingisallowedtorelaxforfour seconds. Then an image of the packing is taken by a CCDcamerawithaNikkor5mmlens.Onlythecentral region of the packing (5 9mm ), five small disc diameters away from boundaries, is captured. The spatial resolution is.7 mm per pixel. For each experiment 8 configurations are imaged. Image processing. To analyze the configuration we compute the Voronoi tessellation of the packing. Because we consider a D system, volume and area are used interchangeably in the following. In a first step the centers andsizesofthediscsarecalculatedwithanaccuracyof. pixel using a template correlation technique. TherearetwomethodstoidentifyVoronoicellsina bidisperse system, radical tessellation[8] and navigation map [9]. Radical tessellation takes the boundaries of the cells as the collection of points whose tangents to neighboring particles are equal length. The navigation maptakesthecellsasthecollectionofpointscloserto the surface of the corresponding particles than others in the system. In this work the navigation map is computed numerically on a grid of /6 pixel resolution. An example ofthenavigationmapisshowninfig.(a). Our packing is prepared under gravity. Sidewalls introduce slow convection rolls during an air flow pulse, and distributor inhomogeneities introduce gradients in the airflow.itisthereforenotsurprisingtofindgradientsof thepackingfractioninthesystem.theanalysisisdone for the central spatially most homogeneous region. The spatial variance of the local packing fraction (averaged over the whole 8 taps) is calculated for circles of diameter8d s (seethewhitedashedcircleinfig.(a)).thenthe region with the smallest variance(in all cases smaller than.43) is chosen as the analysis region for the experiment. Additionally, the evolution of the global packing fraction and the geometrical contact number in that region are examined to make sure that no segregation occurs during thecourseoftheexperiment. Results. Individual Voronoi volumes and free volume distribution. The distribution of individual Voronoi volumes v Thespatialvarianceofthewholesystemis p

3 Voronoi volume correlation (a) (b) (c) log(p(v/v s )) Voronoi volume v/v s (d) P(v/v s ) 8 x 3 6 small φ=.875 both 4 large (e) P(v/v s ) 8 x large small both φ= Voronoi volume v/v s Voronoi volume v/v s Fig. :(Color online)(a) Voronoi cells computed using the navigation map for the packing in fig. (c) and labeled with different colors. The white dashed circle indicates the size of the region over which we perform further analysis for over 8 individual configurations.(b)theprobabilitydistributionofthevolumeofindividualvoronoicells;thetwopeakscorrespondtothetwo sizesofdiscs.(c)anillustrationoftheconstructionofpairsbyahexagonalwheel.thewheelvertices(darkdots)definepairs of points. The free volume of the Voronoi cells to which the vertices belong is used for the two-point correlation measurement. (d), (e) Free volume distributions of small discs, large discs, and both for two different packing fractions. The volume is normalizedbythevolumeofsmallparticlesv s=πd s/4.theprobabilitiesarenormalizedbythenumberofsmall,large,and both discs, respectively. can be computed directly from the results of the navigationmap.anexampleisshowninfig.(b).becausewe do not want to distinguish between small and large discs, we follow Lechenault etal.[] and use in the subsequent analysisthefreevoronoivolumev f =v v min.theminimumvolumev min isthevolumethatagrainwouldoccupy inahexagonalpackingofidenticaldiscs.itequals 3 d, wheredisthediameterofthecorrespondinglarge(d l ) discsorsmall(d s )discs.whiletherespectivemeanfree volumesofsmallandlargediscsstilldiffer( v s=.35 f and v f l =.58inunitsofv s=πd s/4forφ avg =.838), the success of the subsequent analysis justifies this step by hindsight. The free volume distributions of small and large discs and both types together are presented for two packing fractionsinfig.(d)and(e).forlargevolumesthedecay is exponential but besides this feature none of the free volume distributions and volume distributions in fig. (b), (d) and (e) could be fitted reasonably with a gamma distribution. This result differs from D and 3D monodisperse packings[ ]. Therelativeheightofthetwopeaksinthefreevolume distributions changes with φ avg. A recent study of the probability distributions of quadron volumes (an alternative tessellation introduced in [3]) showed that the positionandheightofthesepeakscanbetracedtoconditional probabilities of cell volumes at given coordination numbers[4].sothechangesvisibleinfig.(d)and(e) mayalsoberelatedtoachangeofcontactnumber. Two-point correlation of free Voronoi volume. To quantify the correlations between Voronoi volumes we use the two-point correlation function: C ij (L)= (v i v i )(v j v j ) σ, () where i, j correspond to two points in a distance L belongingtotwovoronoivolumes,andv i,v j arethefree volumes of these Voronoi cells.... indicates averages over all the 8 different packings created by flow pulses. v i and v j are the mean free volumes at these points (computed separately to remedy the effect of remainingsmallgradients),andσ =(σi +σ j )/isthe corresponding variance. To obtain better statistics, 4 pairs of points with the samelareselected.foreachpairc ij (L)iscomputed using eq. (). In practice the pairs are selected in the following way: We construct a hexagonal wheel centered intheanalysisregion.thelengthofeachedgeofthewheel issettobel(cf.fig.(c)).thesixouterpointsandthe centerformpairsthatareseparatedbyadistancel. Foreachsuchpair,thefreevolumeofthetwoVoronoi cells to which the two vertices belong are taken as v i andv j ineq.().thenthehexagonistranslatedtofive different positions and rotated to four different angles for each position. However, the whole wheel stays inside the whitecircleinfig.(a).thenthe4pairsareaveraged: corr(l)= 4 C ij (L). () i,j 344-p3

4 Song-Chuan Zhao etal. corr(l).. φ =.875 φ =.838 corr(l).. vertical horizontal L/d s L/d s Fig. 3: (Color online) Depending on φ, corr(l) exhibits both correlations and anticorrelations. Inset: the anisotropy measurementforφ avg=.838.thetwopointcorrelationof pairsplacedhorizontally(darkgreen)andvertically(brown), rather than constructed from the hexagon. corr(l).8.4 L C Aneg.. A neg φ avg L min exp(-l/l AC ) L/d s Fig. 4: (Color online) Three characteristic lengths: L C, obtainedfromalinearfit(dashedblackline)tocorr(l)for small L; L min, corresponding to the minimum of corr(l); and L AC, obtained from an exponential fit (green line) to corr(l)forl>l min.theinsetshowsthenegativeareaa neg ofcorr(l)asafunctionofφ avg. The correlation function corr(l) is shown for two values of φ avg in fig. 3. For low φ avg, corr(l) decays to zero and then fluctuates around it. Positive values of corr(l) indicate that the two free Voronoi volumes deviate from theaverageinthesamedirection.forhighφ avg,corr(l) decreases to a negative minimum and then increases towards. Negative values of corr(l) characterize anticorrelations: Voronoi cells at this distance deviate in opposite directions from the average free volume. We define three characteristic lengths to describe corr(l)(seefig.4).firstwedoalinearfittocorr(l)for arangecenteredathalfofthemaximumofcorr(l)with width±/d s.thepointwherethisfitcrosseszeroyields the length L C. For measurements of corr(l) showing anticorrelationwedefineasecondlength,l min,whichis extracted from a local parabolic fit around the minimum characteristic length/d avg 3.5 φ AC L AC L min L C φ avg Fig. 5: (Color online) Packing fraction dependence of the characteristic lengths of correlation L C and anticorrelation L minandl AC.Thelengthsarenormalizedusingtheaverage discdiameterd avg,whichcorrespondsto.3.8timesd s. ofcorr(l).thethirdlength,l AC,isobtainedfroman exponentialfitrangingfroml min totheendofcorr(l) (weallowforasmalloffsetinthisfit,butthemagnitude ofthisoffsetislessthan.6inallcases). Tofindtheonsetoftheanticorrelations,theareaA neg wherecorr(l)<(cf.fig.4)isplottedasafunctionof φ avg,asshownintheinsetoffig.4 3.Anextrapolation of a linear fit to A neg to zero defines the threshold foranticorrelations,φ AC =.877±.5.Forφ>φ AC, corr(l) can be both positive and negative; therefore, it cannotbedescribedbyapowerlaw[]. ThedependenceofL C,L min,andl AC onφ avg isshown infig.5:l C andl min slowlydecreasemonotonicallywith φ avg,whilel AC growsapproximatelylinearly.thefinite extent of correlations and anticorrelations can also be seen infig.6,whichshowshowthevarianceofthefreevolume of a cluster scales with the number N of particles in thecluster.inthelargen limitalinearrelationshipis recovered as predicted by the central limit theorem in the absence of correlations. Gravity breaks the isotropy in our experimental setup; this anisotropy is visible in the correlations plotted in the inset of fig. 3, where pairs of horizontal and vertical points are averaged separately. While the qualitative features of corr(l) are independent of direction, the lower statistics of this analysis does not allow us to extract the corresponding characteristic lengths. We have also performed our analysis with the radical tessellation and have found that all features stay qualitatively the same. Quantitatively there are slight differences: φ AC =.88±.;L C,L min andl AC are6%,%and 3% larger on average. 3 WhencomputingA neg wetakeintoaccounttheoffsetofthe exponentialfitdeterminingl AC. 344-p4

5 Voronoi volume correlation ) log (σ N f L 8d s cluster 3 log (N).76 Fig. 6: (Color online) The correlations have finite spatial extent,asshownbythevarianceσ f N oftheaveragefreevolume ofacompactclusterasafunctionofthenumbernofgrains included (open blue circles). In the absence of correlations the central-limit theorem predicts a slope of one. While the slopeforsmallclusters(n<)is.76,inthelargenlimita slopeofoneisrecovered.alsotheslopeofahexagonalwheel cluster withsidelengthl 8d s(opentriangles)equalsone. Hereφ avg=.838;datasetsareshiftedforclarity. Discussion. This is the first observation of anticorrelation between Voronoi volumes in a granular system. Onereasonforthisisthatearlierstudiesofvolumefluctuations focused on the cluster composed of neighboring particles[,3]. There the variance of average free volume pergrainσn asafunctionofthenumberofgrainsincluded N wasmeasured,andthescalingσn N α withα< indicated the existence of correlations. However, the scalingbetweenσn andnisameasureofthecorrelationintegratedoverthewholeclustersize.inthelargenlimit,the relationbetweenn α andcorr(l)is N α = C N L(N) corr(l)ldl, (3) where C is a constant proportional to the density of the cluster (πl (N) would be the size of the cluster). The relatively small contribution of anticorrelations to the integral in eq. (3) would be hard to distinguish from experimental noise(cf. fig. 6). Therefore, a direct measurement of two-point correlation is necessary to find anticorrelation. In recent years a statistical mechanics approach for granular systems has been developed where the Hamilton function is replaced by a volume function[5]. Voronoi cells have been used to construct this function[,6,7]. While the discovery of anticorrelation has consequences for such an approach, some properties like the distribution of the local packing fraction have been shown to not depend on correlations[8]. It is possible that not only the extent but even the existence of positive correlation is an artifact of Voronoi tessellation. By definition the Voronoi tessellation assigns spacetoeachparticleinacertain equal way.inthe caseofthenavigationmap,theedgeofavoronoicell isthecollectionofpointsthathaveequaldistancerto the surface of the neighboring particles. The increase in avoronoivolumecouldroughlybeseenastheincrease in r, therefore ther of its neighboring Voronoi cell. Consequently the Voronoi tessellation itself gives rise to a positive correlation between neighboring Voronoi volumes. To verify whether this effect dominates the evolution of L C,weestimatetheaveragesized vor ofvoronoicellsin the system in the following way. In a homogeneous system, φ avg couldbewrittenastheratiooftheaverageparticle volumetotheaveragevoronoivolume,φ avg =d avg/d vor. Giventhebidispersenatureofoursystemd avg isinthe rangeof.3.8d s.thisestimationleavesusd vor = d avg / φ avg.rescalingl C withd vor showsthatl C decays slightly faster than explained by simple compaction. This could stem from the appearance of the anticorrelation beyondφ AC.Itwillbeinterestingtotestthishypothesis with another way of tessellation, such as quadrons[3]. Aboveφ AC thefluctuationofthevolumeofonegrain causes more and more grains to be anticorrelated, which isindicatedbytheincreaseofl AC andthedecreaseof L min.concerningthephysicalinterpretationoftheseanticorrelations,weconjecturethatφ AC mightcorrespondto theonsetofdilatancy.in3dsystemsitiswellestablished that loose granular material collapses under shear, while granular material denser than dilatancy onset expands [9,]. This expansion can be understood as competition between the grains for free volume. Our results show that correlations between the volumes of subunits depend on the specific system under consideration. For example, in froths there exist anticorrelations of the number of faces between neighboring cells[ 4]. At thesametimethevolumeofafoamcellisproportional to the number of faces[,5]. These features combined indicate that the volume of neighboring cells should be anticorrelated. ItwouldbeinterestingtoknowhowL AC woulddevelop at even higher density. However, we can not compactify the system above φ =.838 without partial segregation. In this case no anticorrelations occur within the segregated patches.theonlysurvivinglengthisl C,whichcharacterizes the size of the segregated patches. Conclusion. Our measurements of the two-point correlation function for binary disc packings yield a shortranged positive correlation over the whole range of packing fractions Above a volume fraction of.877 ±.5 we observe anticorrelation in the free Voronoi volumes. These anticorrelations reach a maximum at a distance of about 3.5 small particle diameters. They then decay exponentially with distance, with an exponent growing linearly with packing fraction. 344-p5

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