The effect of boundaries on the force distributions in granular media is
illustrated by simulations of 2D packings of frictionless, Hertzian spheres. To
elucidate discrepancies between experimental observations and theoretical
predictions, we distinguish between the weight distribution {\cal P} (w)
measured in experiments and analyzed in the q-model, and the distribution of
interparticle forces P(f). The latter one is robust, while {\cal P}(w) can be
obtained once the local packing geometry and P(f) are known. By manipulating
the (boundary) geometry, we show that {\cal P}(w) can be varied drastically.Comment: 4 pages, 4 figure

A.J. Liu

A.V. Tkachenko

C. Liu

C.S. O’Hern

D. Howell

D.L. Blair

D.M. Mueth

Ellák Somfai

F. Radjai

G. Løvoll

H.A. Makse

H.M. Jaeger

I.K. Ono

J.M. Erikson

Jacco H. Snoeijer

L.E. Silbert

L.P. Kadanoff

Martin van Hecke

P. Claudin

P.G. de Gennes

S. Luding

S.J. Antony

S.N. Coppersmith

Wim van Saarloos

English

arXiv

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Saarloos, W. van

Snoeijer, J.H.

Hecke, M. van

Somfai, E.

Leiden University Scholary Publications

Effect of boundaries on the force distributions in granular media

Crossref

Force and weight distributions in granular media: Effects of contact geometry

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The influence of the local contact network on interparticle forces and effective particle weights is studied in simulations of two-dimensional packings of frictionless, Hertzian spheres. The weight distribution P(w) changes qualitatively when approaching a boundary and differs for regular and irregular packings, while the interparticle force distribution P(f) is robust. We provide examples where P(w) at the boundary, which is the quantity probed experimentally, deviates substantially from P(f) in the bulk. Discrepancies between the P(w)’s predicted by the q model and measured in experiments are due to differences in the contact geometry.Biological and Soft Matter Physic

Hecke, M.L. van

RAPID COMMUNICATIONSPHYSICAL REVIEW E 67, 030302~R! ~2003!Force and weight distributions in granular media: Effects of contact geometryJacco H. Snoeijer,1 Martin van Hecke,2 Ellák Somfai,1,* and Wim van Saarloos11Instituut-Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands2Kamerlingh Onnes Lab, Universiteit Leiden, Postbus 9504, 2300 RA Leiden, The Netherlands~Received 24 October 2002; published 25 March 2003!The influence of the local contact network on interparticle forces and effective particle weights is studied insimulations of two-dimensional packings of frictionless, Hertzian spheres. The weight distributionP(w)changesqualitativelywhen approaching a boundary and differs for regular and irregular packings, while theinterparticle force distributionP( f ) is robust. We provide examples whereP(w) at the boundary, which is thequantity probed experimentally, deviates substantially fromP( f ) in the bulk. Discrepancies between theP(w)’s predicted by theq model and measured in experiments are due to differences in the contact geometry.DOI: 10.1103/PhysRevE.67.030302 PACS number~s!: 45.70.Cc, 05.40.2a, 46.65.1geleotiorregeiodin-meszeroeled-a-dtresly-k-.errti-veleledkenorkicek isforceA crucial property of granular materials is their heterogneity @1#. In particular, the strong fluctuations of interparticforces and the organization of the largest of these in tenuforce networks have recently attracted considerable atten@2–9#. The most basic characterization of the force netwis the probability distribution of the contact forces. Measuments@2–6#, numerical simulations@7,8#, and theory@9# allfind that force distributions decay exponentially for larforce @10#, albeit with nonuniversal exponents. The behavfor small forces is less well settled but relevant: it may incate arching@11# or jamming@12#.In this paper, we will unravel the effect of the local cotact geometry on the distributions of interparticleforce Fandeffective particleweight W; the weight is defined as the suof the vertical components of all downward pointing forcon a particle@see Fig. 1~b!#. While the distribution of forcesF is the primary object one ultimately wishes to characteriit is difficult to access experimentally. In experiments wheparticle-wall forces are measured by means of imprintscarbon paper@2–4# or force sensors@5# one probes theweightof the bottom particles. The weight is also the primquantity described by theq model, which is a lattice modein which weights are randomly redistributed over a fixnumber of supporting grains@9#.Our central point is that while the distribution ofF isrobust, the distribution ofW is profoundly influenced by thecontact geometry, in particular bythe number of downwardpointing contact forces nc . This has the following consequences.~i! The bulk and bottom weight distributions are qualittively different due to geometrical effects: on an averagencis smaller for bottom than for bulk particles@ ee Fig. 1~a!#.~ii ! For small weights, the weight distribution of theqmodel is qualitatively different from distributions observeexperimentally; this is due to the fixed contact geome~fixed nc) of the q model.~iii ! The distributions of bottom weights and bulk forcare in generaldifferent; regular contact geometries or highcompressed packings@4# exhibit substantial differences be*Present address: Department of Physics, University of WarwCoventry CV4 7AL, UK.1063-651X/2003/67~3!/030302~4!/$20.00 67 0303-usonk-r-,enytween bulk force and bottom weight distributions. For pacings in which many bottom particles havenc51, however,the bulk force and bottom weight distributions are similarWe will focus on frictionless, Hertzian spheres undgravity. The particle-boundary forces are then purely vecal, and we will refer to their magnitude as the effectiweightW of a particle. To allow comparison to theq model,we define the effective weight also for bulk particles@seeFig. 1~b!#:Wj[mjg1(̂i &~FW i j !z . ~1!Here mj denotes mass,g denotes gravity,FW i j are the inter-particle forces, and the sum runs overnc ‘‘up’’ contacts,associated with the particles that exert a force on particjfrom above. In the following, we rescaleW and F to theiraverage values at a certain height; we write the rescaweights and forces asw and f with distributionsP(w) andP( f ).We will probe P(w) and P( f ) by examining boundaryeffects in an irregular packing~Figs. 2 and 3!, and by study-ing a regular packing where the packing order can be broby curving the boundary~Fig. 4!. Our framework can now besummarized as follows. The geometry of the contact netwhas a strong effect onP(w), but P( f ) is very robust. Theweight distribution for particles with a givenc , Pnc(w), isrobust and behaves aswnc21 for small w. P(w) can be de-k,FIG. 1. ~a! Detail of a typical packing in our simulations; thheighth denotes the distance from the bottom. The force networrepresented by the black lines whose thickness represents themagnitude.~b! Definition of interparticle forcesF and weightW fora particle withnc52.©2003 The American Physical Society02-1r-nrgeresuchthendii.aryon-perch.mfortom-toi-es-RAPID COMMUNICATIONSSNOEIJERet al. PHYSICAL REVIEW E 67, 030302~R! ~2003!FIG. 2. ~a! The weight distributionP(w) at various heightsbetween 10 and 30 in the bulk~open circles! for 2,h,3 ~fullcurve! and at the bottom~dots!. Inset: ^w2& which quantifies thewidth of P(w) as a function of heighth. The sharp transition ofP(w) near the bottom is clearly visible.~b! P( f ) in the bulk~opencircles! and for the layer-to-layer forces near the bottom~dots!; theinset showsP( f ) for bulk forces on a log-lin scale.~c! Detail of atypical packing showing dominance near the bottom of layerlayer forces~black lines! to intralayer forces~white lines! in deter-mining w. The numbers show the values ofnc for the respectivebottom particles.~d!,~e! Scatter plot of (f i j ,w i j ) for ~d! the bulkforces and~e! the layer-to-layer forces near the bottom.FIG. 3. ~a,b! Decomposition ofP(w) according to Eq.~3! ~a! inthe bulk ~open circles! and ~b! at the bottom~dots!; the measuredbulk values for the fractions$r0 ,r1 ,r2 ,r3% in Eq. ~3! are $0.01,0.11, 0.52, 0.36% and the bottom values are$0.08, 0.46, 0.44, 0.02%;as explained after Eq.~3!, we excluded the intralayer, almost horzontal forces at the bottom.~c!,~d! When rescaled to the averagvalue for each distribution function,P1(w) ~c! andP2(w) ~d! areessentially the same in the bulk~open circles! and at the bottom~dots!.03030composed asP(w)5(ncrncPnc(w), wherernc are the frac-tions of particles that havenc50,1,2, . . . ‘‘up’’ contacts.Differences ofrnc between boundary particles and bulk paticles, and between irregular and regular (q model-like!packings explain the differentP(w)’s for these cases. Wher0 andr1 are large, the total weight distributionsP(w) ex-hibits a plateau at small weights and a slow decay at laweights; whenr2 and r3 become large,P(w) becomessharply peaked. In this way,P(w)’s small weight behavior aswell as its exponential decay rate for large weights@10# re-flect the packing geometry.(a) Numerical method.Our two-dimensional~2D! pack-ings consist of frictionless spheres~3D! under gravity; theparticles interact through normal Hertzian forces, whef }d3/2 andd denotes the overlap distance@13#. Unless notedotherwise, the material constants and gravity are chosenthat a particle deforms 0.1% under its own weight, andparticle radii are drawn from a flat distribution betwee0.4,r ,0.6. Masses are proportional to the cube of the raThe container has a width of 24, employs periodic boundconditions in the horizontal direction, and has a bottom csisting of a fixed hard support. The data shown in this pawere obtained from 1100 realizations of 1180 particles eaWe construct our packings by letting the particles relax froa gaslike state.(b) Boundary effects.In Figs. 2~a! and 2~b!, the distribu-tions for the weight and interparticle forces are shownbottom~dots! and bulk~open circles! particles. ForP(w) weobserve a substantial difference between bulk and bot-FIG. 4. ~a!,~b! Packing and force networks in a weakly polydiperse packing near a flat bottom~a! and a curved bottom~two circlesegments of radius 20 glued together! ~b!. ~c! Distribution ofweightsP(w) on the flat bottom~solid line!, convex curved bottom~dashed line!, and the concave curved bottom~dotted line!. Thevarious shapes originate from the corresponding$r0 ,r1 ,r2%: $0.00,0.10, 0.90%, $0.02, 0.39, 0.58%, and $0.04, 0.46, 0.50%, respectively.~d! P( f ) in the bulk (h ranging from 10 to 20! for the flat, convex,and concave bottoms; the inset shows theP( f )’s on a log-lin scale~the curves are offset for clarity!.2-2othdcallr--teeher-ufigarththscmedngioehegstefar0°thethel foro--al-e-ht-st.er,yinrkir-or ari-les.be-b-ddRAPID COMMUNICATIONSFORCE AND WEIGHT DISTRIBUTIONS IN GRANULAR . . . PHYSICAL REVIEW E67, 030302~R! ~2003!particles, and the transition between the bulk and bottbehaviors is remarkably sharp: in the slice 2,h,3 ~fullcurve!, the weight distribution is already bulklike@Fig. 2~a!and inset#. The force distributionP( f ) is more robust; theonly difference between bulk and bottom distributions ischange of the small peak aroundf 50.7 for bulk forces to aplateau forP( f ) near the bottom@Fig. 2~b!#. It is intriguingto note that this change is reminiscent of what is proposean identification of thejamming transition @12#. The goodagreement betweenP( f )bulk and Pbottom(w) for w*0.3 isfortuitous and, as we will show below, due to the large fration of bottom particles that havenc50 or 1.Figure 2~c! illustrates that due to particular geometry ofpacking near the bottom, the interparticle forces naturadivide up into almost horizontal intralayer forces and ‘‘layeto-layer’’ forces. The distributions of the contact anglesw i jare shown in Figs. 2~d! and 2~e! for bulk and boundary contacts. In the bulk the contact angles are uniformly distribuand independent ofu fW u: our packing is isotropic. Near thboundary, however, this isotropy is strongly broken: tangles of the ‘‘layer-to-layer’’ forces between bottom paticles and those in the layer above are concentrated arop/3 and 2p/3.Decomposition ofP(w). To understand the change oP(w) near the bottom, consider the typical packing of F2~c!. The forces between neighboring bottom particlesalmost purely horizontal, so these hardly contribute tototal force on the particle from above@the ‘‘weight’’ in Eq.~1!#. Thus, the average value ofnc , the number of particlesthat contribute to its weight, is on an average lower atbottom than in the bulk. Let us now argue whythe probabil-ity of finding a small value of w increases with smaller nc.Consider Eq.~1! for fixed nc and analyzePnc(w), the weightprobabilities restricted to particles of givennc . As long asthe joint probability distribution of the interparticle forceremains finite for small forces, it follows from a phase-spaargument thatPnc~w!}wnc21 for w→0, ~2!for all nc>1 @14#. The particles that do not feel a force froabove havenc50 and give ad-like contribution at W5mg; for deep layers this occurs forw!1.To check the validity of this idea, we have determinPnc(w) both in the bulk and near the bottom by determininc for each particle and decomposing the weight distributP(w) into thePnc(w)’s,P~w!5(ncrncPnc~w!. ~3!In Figs. 3~a! and 3~b! we show this decomposition for thbulk and bottomP(w)’s, while P1(w) andP2(w) are shownin Figs. 3~c! and 3~d!. In determining the value ofnc weexplicitly exclude the intralayer bottom contacts; for tstrongly polydisperse packings, these correspond to anthat deviate less than 13° from the horizontal. For our emates of the bulkrnc’s we do not exclude any forces, i.e., th03030meas-ydnd.eeeenlesi-‘‘cutoff angle’’ is strictly 0°. Since the bulk is isotropic, acutoff angle of 13° leads to a change of the order of 15%,too little to explain the change ofr1 by a factor of more than4. For simplicity, we therefore have kept a cutoff angle ofin the bulk.The crucial difference between bottom and bulk arefractions rnc of particles that experiencenc other particlespressing on them from above; thePnc(w)’s obey Eq.~2! andare very similar, except for a small difference betweenbulk and bottomP1(w)’s for small w @15#.The behavior ofP(w) for smallw is dominated byr0 andr1 , but these fractions also affect the steepness of the taiP(w) via the normalization conditions*dw P(w)51 and*dw wP(w)51. A small value of limw↓0P(w) leads to asharply peaked distribution that falls off with a large expnent@Fig. 2~a! for the bulk and Fig. 3~d!#, while a plateau atsmall w leads to a slower exponential decay at largew @Fig.2~a! for the bottom and Fig. 3~c#. The contact geometry affects the small weight distribution and hence, via the normization constraints, the exponential decay rate of the largweight tail [10].(c) Regular packings.We have seen above that the weigdistributionP(w) is very sensitive to the local packing geometry, while the distribution of interparticle forces is robuFor the particular example shown in Figs. 2 and 3, howevPbulk( f ) and Pbottom(w) look rather similar due to a largeamount of particles withnc51. In order to provide an ex-ample whereP( f ) andP(w) are radically different, and totest the robustness ofP( f ), we consider a packing of weaklpolydisperse particles, 0.49,r ,0.51. Above a flat bottom,the particles form an almost perfect hexagonal packing@Fi .4~a!#, while for a curvedbottom this order is broken@Fig.4~b!#. The flat bottom leads to particles with mostlync52,and indeedP(w);w for small w in agreement with Eq.~2!@see Fig. 4~c!#. For the curved bottom a dramatic changethe fractionsrnc and correspondinglyP(w) occurs @Fig.4~c!#. P( f ) in the bulk are all indistinguishable@see Fig.4~d!# and very similar toP( f ) in the strongly polydispersecase@Fig. 2~b!#: even major changes in the contact netwodo not affect the bulk P( f ).In experiments, the differences between regular andregular packings were found to be minute@3#. However, theexperimental data for regular packings shows evidence fd peak atw50, which indicates that there arenc50, andlikely also nc51 particles present. Even though the expementalpackingis regular, thecontact networkis apparentlynot, possibly due to friction and the hardness of the particThe regular packing also clarifies the discrepancytween theP(w)’s in measurements andq model@9#. In sucha lattice model, the weight of a particle is randomly distriuted overnc neighbors. In the simplest casenc52; the re-sulting P(w) is then qualitatively similar to what we founfor the flat bottom; in particular,P(w);w for small w @9#.This is due to the absence of particles withnc51: in simu-lations of theq model with random connectivities, we founa variety ofP(w)’s with finite limw↓0P(w) oncer1.0 @16#.2-3inurtgr-calaeresof-a-ingoreeetestofener,r-vese ofrgu-enac-RAPID COMMUNICATIONSSNOEIJERet al. PHYSICAL REVIEW E 67, 030302~R! ~2003!Finally, note thatf q[qw in the q model is more like a con-tact force, and thatP( f q) remains finite for smallf q @9,16#.(d) Perspective.Our message is clear: in experimentswhich the forces on a boundary are probed, one measeffectively the weight distributionP(w) that is sensitive tothe local packing near the bottom and whicha priori is dif-ferent from the robust force distributionP( f ). Why P( f ) isso robust remains an intriguing open issue.From the experimentally obtainedP(w), one can estimatethe fractionsr0 and r1 which dominate the small weighdistribution. The height of the peak atw50, clearly presentin Fig. 5 of Ref. @3# and in Ref.@4# is given by r0 /Dw,whereDw is the bin width. Ignoring this peak and estimatinlimw↓0P(w) gives the value ofr1P1(0); direct measure-ments ofr1 would yield P1(0). Based on these consideations, it seems that for most experimentally observed paings, r0 and r1 are substantial at the boundary, so thPbottom(w) is similar to P( f )bulk ~apart from ad peak atw50). The same argument probably holds for recent simutions by Silbertet al. @17#.However, in recent experimental data on highly deformpackings of rubber beads@4#, a clear decrease in thed peakand limw↓0P(w) is visible. We think that these trends aentirely consistent with a decrease ofr0 andr1 and increaseof r2 and r3 , as expected for highly deformed packingSuch changes in ther ’s would also lead to a steepeninghyEev03030esk-t-d.the exponential tail ofP(w); this steepening is clearly visible in the data of Ref.@4#.To further test our framework in experiments, direct mesurements of the fractionsr i would be very welcome. Alter-natively, one can directly influence these fractions by placa layer of relatively small or large beads at the bottom. Fsmall beads,r0 andr1 will be enhanced, leading to a largP(w) for smallw, and a slow exponential decay for largew.Relatively large bottom beads will suppressr0 and r1 andincreaser2 , etc.; the limw↓0P(w) is then suppressed, thexponential decay becomes steep andP(w) appears stronglypeaked. SinceP( f ) in the bulk is not influenced by a singllayer of boundary grains, this may be the simplest way toour framework.An important issue for future study is clearly the rolefriction and dimensionality. Our numerical study has bedone in two dimensions with frictionless spheres; howevrecent studies indicate@8# that the coordination number fo3D packings with friction is similar to those of 2D frictionless packings. Qualitatively, the picture with which we haadvanced is therefore expected to capture the realistic cathree dimensions with friction because our phase-space aments are independent of dimension.We are grateful to Martin Howard and Hans van Leeuwfor numerous illuminating discussions. J.H.S. and E.S.knowledge support from the FOM foundation.redl-,@1# H.M. Jaeger, S.R. Nagel, and R.P. Behringer, Rev. Mod. P68, 1259 ~1996!; P.G. de Gennes,ibid. 71, 374 ~1999!; L.P.Kadanoff, ibid. 71, 435 ~1999!.@2# D.M. Mueth, H.M. Jaeger, and S.R. Nagel, Phys. Rev. E57,3164 ~1998!.@3# D.L. Blair et al., Phys. Rev. E63, 041304~2001!.@4# J.M. Eriksonet al., Phys. Rev. E66, 040301~2002!.@5# G. Lo”voll, K.J. Målo”y, and E.G. Flekko”y, Phys. Rev. E60,5872 ~1999!.@6# D. Howell, R.P. Behringer, and C. Veje, Phys. Rev. Lett.82,5241 ~1999!.@7# F. Radjaiet al., Phys. Rev. Lett.77, 274 ~1996!; S. Luding,Phys. Rev. E55, 4720~1997!; F. Radjaiet al., Phys. Rev. Lett.80, 61 ~1998!; A.V. Tkachenko and T.A. Witten, Phys. Rev.62, 2510 ~2000!; S.J. Antony,ibid. 63, 011302~2000!; C.S.O’Hern et al., Phys. Rev. Lett.88, 075507~2002!.@8# H.A. Makse, D.L. Johnson, and L.M. Schwartz, Phys. RLett. 84, 4160~2000!.@9# C. Liu et al., Science269, 513 ~1995!; S.N. Coppersmithet al., Phys. Rev. E53, 4673~1996!.s..@10# This tail is only exponential if the particle deformations asufficiently small@8#. Our maximal deformations are 2% anour tails are not perfectly exponential.@11# P. Claudinet al., Phys. Rev. E57, 4441~1998!.@12# A.J. Liu and S.R. Nagel, Nature~London! 396, 21 ~1998!; C.S.O’Hern et al., Phys. Rev. Lett.86, 111~2001!; I.K. Ono et al.,ibid. 89, 095703~2002!.@13# Contact Mechanics, edited by K.L. Johnson~Cambridge Uni-versity Press, Cambridge, England 1985!.@14# Such scaling is also implicit in theq model, although therenc>2 so thatP(0)50.@15# Bottom particles withnc50 or 1 are typically smaller thanaverage@see Fig. 2~c!#. For these particles, the intralayer ‘‘amost horizontal’’ forcesdo contribute slightly to the weightswhich enhances the probability for small~but nonzero! weightsat the expense ofP(0); this effect is visible forP0,bottom(w)andP1,bottom(w) in Fig. 3~b!.@16# J.H. Snoeijeret al. ~unpublished!.@17# L.E. Silbert, G.S. Grest, and J.W. Landry, Phys. Rev. E66,061303~2002!.2-4

Effect of boundaries on the force distributions in granular media

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