Using sedimentation to obtain precisely controlled packings of noncohesive
spheres, we find that the volume fraction $\phi_{\rm RLP}$ of the loosest
mechanically stable packing is in an operational sense well defined by a limit
process. This random loose packing volume fraction decreases with decreasing
pressure $p$ and increasing interparticle friction coefficient $\mu$. Using
X-ray tomography to correct for a container boundary effect that depends on
particle size, we find for rough particles in the limit $p \to 0$ a new lower
bound, $\phi_{\rm RLP} = 0.550 \pm 0.001$.Comment: significantly revised, published versio

Harry L. Swinney

Matthias Schröter

Melissa Jerkins

Mohammad Saadatfar

S. Edwards

Tim J. Senden

Tomaso Aste

English

arXiv

Jerkins, M.

Schröter, M.

Swinney, H.

Senden, T.

Saadatfar, M.

Aste, T.

MPG.PuRe

Onset of mechanical stability in random packings of frictional spheres

Swinney, H.-L.

Senden, T.-J.

Max Planck Society eDoc Server

Crossref

Onset of Mechanical Stability in Random Packings of Frictional Spheres

Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction ?RLP of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure p and increasing interparticle friction coefficient ?. Using x-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit p?0 a new lower bound, ?RLP=0.550±0.001

Jerkins, Melissa

Schröter, Matthias

Swinney, Harry

Senden, Tim

Saadatfar, Mohammad

Aste, Tomaso

Kent Academic Repository

Onset of Mechanical Stability in Random Packings of Frictional SpheresMelissa Jerkins, Matthias Schro¨ter,* and Harry L. Swinney†Center for Nonlinear Dynamics and Department of Physics, University of Texas at Austin, Austin, Texas 78712, USATim J. Senden, Mohammad Saadatfar, and Tomaso AsteDepartment of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University,0200 Canberra, ACT, Australia(Received 15 November 2007; revised manuscript received 28 April 2008; published 2 July 2008)Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that thevolume fraction RLP of the loosest mechanically stable packing is in an operational sense well defined bya limit process. This random loose packing volume fraction decreases with decreasing pressure p andincreasing interparticle friction coefficient . Using x-ray tomography to correct for a container boundaryeffect that depends on particle size, we find for rough particles in the limit p ! 0 a new lower bound,RLP  0:550 0:001.DOI: 10.1103/PhysRevLett.101.018301 PACS numbers: 45.70.Cc, 46.65.+g, 47.57.efIntroduction.—If granular materials such as sand, sugar,or snow are excited strongly (e.g., by shaking or shearing),they exhibit fluidlike behavior. However, after the excita-tion stops, dissipation quickly produces a static packingthat is mechanically stable under its own weight.Experiments [1–8] and simulations [9–13] have shownthat the volume fraction has a well-defined lower limit,RLP, called random loose packing (RLP).The value of RLP depends on the particle-particle in-teractions. Packings of cohesive particles like fine powdersare stable under their own weight for values of RLP as lowas 0.15 [5–8]. However, many granular materials do notexhibit cohesive forces. Simulations of frictionless elasticnoncohesive spheres have found the onset of a finite bulkmodulus at the jamming point, J  0:64 [9–11]. Realspheres have friction and then it has been suggested thatRLP depends on the density difference between the parti-cles and the surrounding fluid [1– 4]. The lowest volumefraction reported thus far, RLP  0:555, was observed forslowly sedimenting spheres in a liquid of nearly the samedensity [3].Here we demonstrate a limit process that yields well-defined values of RLP that depend on pressure and coef-ficient of friction. The results are discussed in the contextof a statistical mechanics approach based on the ensembleof all mechanically stable configurations [14].Experiment.—Mechanically stable packings of glassspheres were prepared by allowing the particles to sedi-ment following flow pulses in a water fluidized bed. Thefluidized bed was contained in a vertical polycarbonatetube with an inner diameter D of 12.8 mm and a lengthof 230 mm. The tube’s bottom end was closed by a dis-tributor consisting of a porous bronze disk (height, 8 mm;nominal pore size, 25 m). A programmable syringepump (Harvard Apparatus) created pulses of constantflow rate Q. During a flow pulse of 2 min length thegranular medium fluidized and expanded until it reacheda steady state height. After each flow pulse, the particlessedimented onto the distributor and formed a mechanicallystable packing whose volume fraction depended on Q, asshown in Fig. 1. A higher value of Q resulted in a moreexpanded fluidized bed, longer sedimentation time, andlower  of the sediment. Packings created in this wayare independent of the state of the sample prior to thelast flow pulse [15]. This property is important for anystatistical mechanics approach [16]. The volume fractionsin Fig. 1 are averaged over the whole sample:  m=Ahsed, where  is the particle density, m is the totalmass of all the spheres, A is the cross sectional area of thetube, and hsed is the height of the sedimented sampledetermined from images. 0.55 0.56 0.57 0.58 0.59 0  10  20  30  40volume fraction φflow rate (ml/min)322 µm (D)277 µm (C)167 µm (B)96 µm (A)0.550.56 0  100  200  300  400φ RLPdiameter (µm)FIG. 1 (color online). The approach to random loose packingin a limit process is achieved using flow pulses in a liquidfluidized bed. Data for different particle diameters were fit to(1) to obtain RLP. Letters in parentheses refer to the particlesamples in Table I. Inset: Diameter dependence of RLP withoutcontainer size correction (see text). Sample height was 97 mm atRLP; five flow pulses were averaged for each flow rate.PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 20080031-9007=08=101(1)=018301(4) 018301-1 © 2008 The American Physical SocietyThe properties of the different samples of particles aregiven in Table I. The density  of the particles in eachsample was measured with an accuracy of 0.06% using aGay-Lussac specific gravity bottle and a MicromeriticsAccuPyc 1330 gas pycnometer; the average  was2:48 g=cm3. To characterize the frictional properties ofthe samples we measured the angle of repose under water:a beaker containing a layer of particles about 5 mm highwas tilted until the particles started to move. To obtain anespecially rough sample (F) we soaked part of sample Cfor 3 h in hydrofluoric acid. Sample E consists of spheresthat were smoothed by exposure to more than 45 000 flowpulses in a fluidized bed [15].RLP is defined by a limit process.—The main improve-ment over earlier studies using sedimenting particles [3,7]is that our control of Q allows us to change the sedimen-tation time independent of the liquid density. This proce-dure reveals the convergence of  to RLP. Theobservation that the slowest relaxing preparation yieldsthe loosest packings agrees with simulations of frictionaldisks and spheres [11,12]. Figure 1 shows that Q is welldescribed by the fit function used in [15], Q  RLP  aQ b ; (1)which we use to determine RLP.Dependence on particle diameter.—The inset of Fig. 1indicates that RLP decreases with particle diameter; how-ever, this decrease is due to lower volume fraction near acontainer wall, an effect known since the earliest studies[1]. This effect is explained in Fig. 2(a): since particlescannot penetrate the container wall, voids are larger thereand the volume fraction of the layer adjacent to the bound-ary is lower than bulk measured in the core of the sample.The difference between apparent averaged over the wholecontainer and bulk increases with the ratio d=D andproduces the trend displayed in the inset of Fig. 1.We examined the finite size effect using x-ray tomog-raphy [17], Fig. 2. In each run positions of 1:5 105spheres were measured with a resolution of better than0.1% of a sphere diameter [18]. Figure 2(c) shows thedifference between the apparent volume fraction using allparticles and the bulk volume fraction (measured using theVoronoi volumes [18] of all particles that are at least 4daway from the container walls). For Q< 10 ml=min ,apparent values from the tomographic measurements agreewith the results (dotted curve) for the 277 m particles inFig. 1, but for Q> 10 ml=min the tomographic values arelarger due to unavoidable vibrations during the recordingof the tomogram.The inset of Fig. 2(c) shows the difference between thebulk and apparent  as a function of apparent. A linear fityields 0.56 0.57 0.58 0.590.60 0  10  20  30  40  50volume fractionflow rate (ml/min)apparent volume fractionbulk volume fraction 0.005 0.007 0.009 0.011 0.56  0.57  0.58  0.59φ bulk - φ apparentφapparentc)b)a)container wallbulk particlesboundary layerFIG. 2. (a) Illustration of the necessity of a finite size correction of the volume fraction determined from the total sample volume.Particles near the boundary have a lower volume fraction than those in the bulk. (b) Cross section of a 3-dimensional x-ray tomogramof the fluidized bed; the particles have a diameter of 255 m (sample G). The inner diameter of the polycarbonate tube (black circle) is12.8 mm. (c) Bulk volume fraction for particles that are at least 4d away from boundary, and the apparent volume fraction apparent forall particles. The dotted line corresponds to the fit to the 277 m particles in Fig. 1. The inset shows a linear fit (2) to the differencebetween bulk and apparent volume fractions. Sample height was 39 mm at RLP.TABLE I. Properties of the different samples of glass spheres.Particle diameters d and standard deviations  were measuredwith a Camsizer (Retsch Technology). Angles of repose underwater were averaged over 10 measurements.Sample d (m)  (%) Supplier Angle of reposeA 96 15.6 Cataphote 24:8 1:0B 167 16.1 Cataphote 26:1 0:7C 277 7.6 Cataphote 25:3 0:8D 322 9.3 Cataphote 25:5 0:7E 261 5.0 MoSci 24:0 0:8F 257 7.8 Cataphote 27:7 1:3G 255 2.7 MoSci 26:6 0:7PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 2008018301-2 bulk  apparent  0:122apparent  0:505: (2)For all further experiments we used only spheres withdiameter 261 or 257 m (sample E or F), and we cor-rected for the effect of finite container size using (2).Influence of pressure.—The stress inside a column ofgrains differs from the hydrostatic case in two ways:(i) anisotropy—the horizontal stress xx in the columndiffers from the vertical stress zz; (ii) wall friction—thepart of the load carried by the frictional sidewalls increaseswith depth z below the surface. Consequently, xx and zzsaturate with z. In our analysis we use a pressure depen-dence on height given by the Janssen model [19], whichassumes a constant stress ratio K  xx=zz everywhere ina sample. Experiments show that this model is a fairapproximation in the absence of external loads [20,21].The model gives a saturation of pressure (p  zz) withdepth, pz  psat1 ez=l; (3)to a constant value psat  gD=4KW , where W is thecoefficient of particle-wall friction, g is the gravitationalacceleration,  is the density difference between spheresand surrounding liquid, and l  D=4KW . Equation (3)indicates two ways of controlling the pressure distributioninside the column: (I) Increasing the sample height, whichincreases the fraction of the sample at psat. If RLP in-creases with p, then the average RLP measured by ourmethod should increase with sample height. This behavioris confirmed in Fig. 3. (II) Decreasing the density differ-ence , which decreases psat but keeps the pressureprofile unchanged. We increased the fluid density to ashigh as 2:39 g=cm3, close to the 2:48 g=cm3 particle den-sity, by adding sodium polytungstate to the water. Resultsfor different  (Fig. 4) again confirm that RLP decreaseswith decreasing p.A limit 0RLP would be given by matching the fluid andparticle densities, but in this limit there would be nosedimentation and no connected granular packings wouldform. Therefore, we extrapolate to determine 0RLP: In theabsence of theory we follow [5] and use the pressuredependence close to the jamming point known for friction-less static soft spheres [9–11,22] and frictionless thermalhard spheres [23]: RLP  0RLP a; (4)where we identify 	 psat. A fit of RLP for smoothparticles (sample E) in Fig. 4 yields 0RLP  0:5550:006. The value of   0:51 0:25 is approximate be-cause our derivation of (4) did not take into account the dependence of K [20].Influence of frictional properties.—Figure 4 shows thatRLP for the rough spheres was lower than for the smoothspheres. For the rough spheres a fit to (4) yields 0RLP 0:550 0:001 and   0:89 0:16. The decrease ofRLP and 0RLP with increasing friction agrees with anotherexperiment [24], model [25], and simulations [11–13,26].Discussion.—Our experimental results and numericalsimulations [11,12] both show that RLP is well definedin an operational sense: in the limit of infinitesimally slowpreparation, the volume fraction of a sample converges toRLP, independent of the details of preparation. Careshould be taken in comparing theory for frictionless hardspheres with the experimental results, in part because ofthe different possible ways of defining mechanical stability[27,28].0.60 0.59 0.58 0.57 30 20 10volume fraction φ realflow rate  (ml/min)21 mm73 mm144 mm 0.58 0.57 0.56 16080 0φ RLPsample height (mm)FIG. 3. Dependence of  on Q for different sample heights(261 m spheres, sample E). Values of  were corrected forfinite size effect using (2). Solid lines are fits to (1). Inset: Theresultant RLP values as a function of sample height. 0.55 0.56 0.57 0  0.4  0.8  1.2  1.6Random Loose Packing  φ RLPdensity difference: ρparticle - ρliquid   (g/cm3)smooth spheres (sample E)rough spheres (sample F)FIG. 4. RLP as a function of the density difference betweenparticles and fluid, for smooth (261 m) and rough (257 m)particles. Solid lines are fits to (4). Data have been corrected forthe finite size effect using (2). Sample height was 64 mm, and thepoints are averages obtained at the highest possible flow rate,since for small density differences the flow rate range was toosmall for a meaningful fit with (1).PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 2008018301-3The observation of a well-defined RLP can be consid-ered within the framework of a statistical mechanics ofstatic granular material [14], where a configurational en-tropy S is defined as the logarithm of the number ofmechanically stable configurations for a given , p, andfriction coefficient. Two different approaches can explainRLP using two different assumptions of how S depends on. The first approach assumes that RLP is the smallest where S becomes larger than zero. This is compatible withthe existence of looser, highly ordered configurations [27],as their number seems not to grow exponentially withsystem size, so S  0.The second approach is supported by numerical resultson the number of stable configurations of frictional disks[26], where S has a maximum at RLP. This idea agrees withslow sedimentation leading to RLP; it is simply the mostprobable configuration. If the sedimentation speed is in-creased, the additional kinetic energy allows the system toexplore the local energy landscape and find rarer but lowerpotential energy (denser) configurations. Further, the maxi-mum of S and therefore RLP moves to higher values of for decreasing friction [26]. This agrees with our resultsand with simulations of frictionless disks that have amaximum of S at random close packing [29]. Our resultsindicate also that increasing p shifts the maximum of S in asimilar way.Conclusions.—Mechanically stable packings of spheresprepared with increasing sedimentation time display alower bound of their volume fraction, RLP, which de-pends on the pressure and the coefficient of friction butnot on the diameter of the spheres. In the limit of zeropressure we have found a new lowest value of RLP,0:550 0:001.We thank Massimo Pica Ciamarra and Antonio Conigliofor sharing their unpublished results, and BrandonMcElroy for his assistance with the Camsizer. Further wethank W. D. McCormick, Narayanan Menon, CharlesRadin, Leo Silbert, Jack Swift, and participants of theAspen Jamming workshop for helpful discussions. Thiswork was supported by Robert A. Welch FoundationGrant No. F-0805.*matthias.schroeter@ds.mpg.de†swinney@chaos.utexas.edu[1] G. Scott, Nature (London) 188, 908 (1960).[2] R. Rutgers, Nature (London) 193, 465 (1962).[3] G. Y. Onoda and E. G. Liniger, Phys. Rev. Lett. 64, 2727(1990).[4] R. Ojha, N. Menon, and D. J. Durian, Phys. Rev. E 62,4442 (2000).[5] J. M. Valverde, M. A. S. Quintanilla, and A. Castellanos,Phys. Rev. Lett. 92, 258303 (2004).[6] J. M. Valverde and A. Castellanos, Europhys. Lett. 75, 985(2006).[7] K. J. Dong, R. Y. Yang, R. P. Zou, and A. B. Yu, Phys. Rev.Lett. 96, 145505 (2006).[8] P. B. Umbanhowar and D. I. Goldman, Am. J. Phys. 74,720 (2006).[9] H. A. Makse, D. L. Johnson, and L. M. Schwartz, Phys.Rev. Lett. 84, 4160 (2000).[10] C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel,Phys. Rev. Lett. 88, 075507 (2002); C. S. O’Hern, L. E.Silbert, A. J. Liu, and S. R. Nagel, Phys. Rev. E 68, 011306(2003).[11] H. P. Zhang and H. A. Makse, Phys. Rev. E 72, 011301(2005).[12] K. Shundyak, M. van Hecke, and W. van Saarloos, Phys.Rev. E 75, 010301(R) (2007).[13] Leonardo Silbert (private communication).[14] S. Edwards and R. Blumenfeld, in Granular Physics,edited by A. Mehta (Cambridge University Press,Cambridge, England, 2007).[15] M. Schro¨ter, D. I. Goldman, and H. L. Swinney, Phys. Rev.E 71, 030301(R) (2005).[16] T. Aste and T. Di Matteo, Phys. Rev. E 77, 021309 (2008).[17] A. Sakellariou, T. J. Sawkins, T. J. Senden, and A. Limaye,Physica (Amsterdam) 339A, 152 (2004); T. Aste, M.Saadatfar, A. Sakellariou, and T. J. Senden, Physica(Amsterdam) 339A, 16 (2004); T. Aste, M. Saadatfar,and T. J. Senden, Phys. Rev. E 71, 061302 (2005).[18] T. Aste, T. D. Matteo, M. Saadatfar, T. J. Senden, M.Schro¨ter, and H. L. Swinney, Europhys. Lett. 79, 24003(2007).[19] M. Sperl, Granular Matter 8, 59 (2006).[20] L. Vanel and E. Cle´ment, Eur. Phys. J. B 11, 525 (1999).[21] L. Vanel, P. Claudin, J.-P. Bouchaud, M. E. Cates, E.Cle´ment, and J. P. Wittmer, Phys. Rev. Lett. 84, 1439(2000).[22] T. S. Majmudar, M. Sperl, S. Luding, and R. P. Behringer,Phys. Rev. Lett. 98, 058001 (2007).[23] C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006).[24] Narayanan Menon (private communication).[25] Y. Srebro and D. Levine, Phys. Rev. E 68, 061301 (2003).[26] Massimo Pica Ciamarra and Antonio Coniglio,arXiv:0805.0220.[27] S. Torquato and F. H. Stillinger, J. Appl. Phys. 102,093511 (2007).[28] S. Torquato and F. H. Stillinger, J. Phys. Chem. B 105,11 849 (2001).[29] G.-J. Gao, J. Bławzdziewicz, and C. S. O’Hern, Phys. Rev.E 74, 061304 (2006).PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R S week ending4 JULY 2008018301-4

Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction phiRLP of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure p and increasing interparticle friction coefficient mu. Using x-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit p-->0 a new lower bound, phiRLP=0.550+/-0.001

Jerkins, M

Schröter, M

Swinney, HL

Senden, TJ

Saadatfar, M

Aste, T

UCL Discovery

Onset of mechanical stability in random packings of frictional spheres.

Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction ϕRLP of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure p and increasing interparticle friction coefficient μ. Using x-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit p→0 a new lower bound, ϕRLP=0.550±0.001

Swinney, Harry L.

Senden, Tim J.

The Australian National University

Onset of Mechanical Stability in Random Packings of Frictional SpheresMelissa Jerkins, Matthias Schröter,* and Harry L. Swinney†Center for Nonlinear Dynamics and Department of Physics, University of Texas at Austin, Austin, Texas 78712, USATim J. Senden, Mohammad Saadatfar, and Tomaso AsteDepartment of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University,0200 Canberra, ACT, Australia(Received 15 November 2007; revised manuscript received 28 April 2008; published 2 July 2008)Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that thevolume fraction RLP of the loosest mechanically stable packing is in an operational sense well defined bya limit process. This random loose packing volume fraction decreases with decreasing pressure p andincreasing interparticle friction coefficient . Using x-ray tomography to correct for a container boundaryeffect that depends on particle size, we find for rough particles in the limit p! 0 a new lower bound,RLP  0:550 0:001.DOI: 10.1103/PhysRevLett.101.018301 PACS numbers: 45.70.Cc, 46.65.+g, 47.57.efIntroduction.—If granular materials such as sand, sugar,or snow are excited strongly (e.g., by shaking or shearing),they exhibit fluidlike behavior. However, after the excita-tion stops, dissipation quickly produces a static packingthat is mechanically stable under its own weight.Experiments [1–8] and simulations [9–13] have shownthat the volume fraction has a well-defined lower limit,RLP, called random loose packing (RLP).The value of RLP depends on the particle-particle in-teractions. Packings of cohesive particles like fine powdersare stable under their own weight for values ofRLP as lowas 0.15 [5–8]. However, many granular materials do notexhibit cohesive forces. Simulations of frictionless elasticnoncohesive spheres have found the onset of a finite bulkmodulus at the jamming point, J  0:64 [9–11]. Realspheres have friction and then it has been suggested thatRLP depends on the density difference between the parti-cles and the surrounding fluid [1– 4]. The lowest volumefraction reported thus far, RLP  0:555, was observed forslowly sedimenting spheres in a liquid of nearly the samedensity [3].Here we demonstrate a limit process that yields well-defined values of RLP that depend on pressure and coef-ficient of friction. The results are discussed in the contextof a statistical mechanics approach based on the ensembleof all mechanically stable configurations [14].Experiment.—Mechanically stable packings of glassspheres were prepared by allowing the particles to sedi-ment following flow pulses in a water fluidized bed. Thefluidized bed was contained in a vertical polycarbonatetube with an inner diameter D of 12.8 mm and a lengthof 230 mm. The tube’s bottom end was closed by a dis-tributor consisting of a porous bronze disk (height, 8 mm;nominal pore size, 25 m). A programmable syringepump (Harvard Apparatus) created pulses of constantflow rate Q. During a flow pulse of 2 min length thegranular medium fluidized and expanded until it reacheda steady state height. After each flow pulse, the particlessedimented onto the distributor and formed a mechanicallystable packing whose volume fraction depended on Q, asshown in Fig. 1. A higher value of Q resulted in a moreexpanded fluidized bed, longer sedimentation time, andlower  of the sediment. Packings created in this wayare independent of the state of the sample prior to thelast flow pulse [15]. This property is important for anystatistical mechanics approach [16]. The volume fractionsin Fig. 1 are averaged over the whole sample:  m=Ahsed, where  is the particle density, m is the totalmass of all the spheres, A is the cross sectional area of thetube, and hsed is the height of the sedimented sampledetermined from images. 0.55 0.56 0.57 0.58 0.59 0  10  20  30  40volume fraction φflow rate (ml/min)322 µm (D)277 µm (C)167 µm (B)96 µm (A)0.550.56 0  100  200  300  400φ RLPdiameter (µm)FIG. 1 (color online). The approach to random loose packingin a limit process is achieved using flow pulses in a liquidfluidized bed. Data for different particle diameters were fit to(1) to obtain RLP. Letters in parentheses refer to the particlesamples in Table I. Inset: Diameter dependence of RLP withoutcontainer size correction (see text). Sample height was 97 mm atRLP; five flow pulses were averaged for each flow rate.PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 20080031-9007=08=101(1)=018301(4) 018301-1 © 2008 The American Physical SocietyThe properties of the different samples of particles aregiven in Table I. The density  of the particles in eachsample was measured with an accuracy of 0.06% using aGay-Lussac specific gravity bottle and a MicromeriticsAccuPyc 1330 gas pycnometer; the average  was2:48 g=cm3. To characterize the frictional properties ofthe samples we measured the angle of repose under water:a beaker containing a layer of particles about 5 mm highwas tilted until the particles started to move. To obtain anespecially rough sample (F) we soaked part of sample Cfor 3 h in hydrofluoric acid. Sample E consists of spheresthat were smoothed by exposure to more than 45 000 flowpulses in a fluidized bed [15].RLP is defined by a limit process.—The main improve-ment over earlier studies using sedimenting particles [3,7]is that our control of Q allows us to change the sedimen-tation time independent of the liquid density. This proce-dure reveals the convergence of  to RLP. Theobservation that the slowest relaxing preparation yieldsthe loosest packings agrees with simulations of frictionaldisks and spheres [11,12]. Figure 1 shows thatQ is welldescribed by the fit function used in [15], Q  RLP aQ b; (1)which we use to determine RLP.Dependence on particle diameter.—The inset of Fig. 1indicates that RLP decreases with particle diameter; how-ever, this decrease is due to lower volume fraction near acontainer wall, an effect known since the earliest studies[1]. This effect is explained in Fig. 2(a): since particlescannot penetrate the container wall, voids are larger thereand the volume fraction of the layer adjacent to the bound-ary is lower than bulk measured in the core of the sample.The difference between apparent averaged over the wholecontainer and bulk increases with the ratio d=D andproduces the trend displayed in the inset of Fig. 1.We examined the finite size effect using x-ray tomog-raphy [17], Fig. 2. In each run positions of 1:5 105spheres were measured with a resolution of better than0.1% of a sphere diameter [18]. Figure 2(c) shows thedifference between the apparent volume fraction using allparticles and the bulk volume fraction (measured using theVoronoi volumes [18] of all particles that are at least 4daway from the container walls). For Q< 10 ml=min ,apparent values from the tomographic measurements agreewith the results (dotted curve) for the 277 m particles inFig. 1, but for Q> 10 ml=min the tomographic values arelarger due to unavoidable vibrations during the recordingof the tomogram.The inset of Fig. 2(c) shows the difference between thebulk and apparent  as a function of apparent. A linear fityields 0.56 0.57 0.58 0.590.60 0  10  20  30  40  50volume fractionflow rate (ml/min)apparent volume fractionbulk volume fraction 0.005 0.007 0.009 0.011 0.56  0.57  0.58  0.59φ bulk - φ apparentφapparentc)b)a)container wallbulk particlesboundary layerFIG. 2. (a) Illustration of the necessity of a finite size correction of the volume fraction determined from the total sample volume.Particles near the boundary have a lower volume fraction than those in the bulk. (b) Cross section of a 3-dimensional x-ray tomogramof the fluidized bed; the particles have a diameter of 255 m (sample G). The inner diameter of the polycarbonate tube (black circle) is12.8 mm. (c) Bulk volume fraction for particles that are at least 4d away from boundary, and the apparent volume fraction apparent forall particles. The dotted line corresponds to the fit to the 277 m particles in Fig. 1. The inset shows a linear fit (2) to the differencebetween bulk and apparent volume fractions. Sample height was 39 mm at RLP.TABLE I. Properties of the different samples of glass spheres.Particle diameters d and standard deviations  were measuredwith a Camsizer (Retsch Technology). Angles of repose underwater were averaged over 10 measurements.Sample d (m)  (%) Supplier Angle of reposeA 96 15.6 Cataphote 24:8 1:0B 167 16.1 Cataphote 26:1 0:7C 277 7.6 Cataphote 25:3 0:8D 322 9.3 Cataphote 25:5 0:7E 261 5.0 MoSci 24:0 0:8F 257 7.8 Cataphote 27:7 1:3G 255 2.7 MoSci 26:6 0:7PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008018301-2 bulk  apparent  0:122apparent  0:505: (2)For all further experiments we used only spheres withdiameter 261 or 257 m (sample E or F), and we cor-rected for the effect of finite container size using (2).Influence of pressure.—The stress inside a column ofgrains differs from the hydrostatic case in two ways:(i) anisotropy—the horizontal stress xx in the columndiffers from the vertical stress zz; (ii) wall friction—thepart of the load carried by the frictional sidewalls increaseswith depth z below the surface. Consequently, xx and zzsaturate with z. In our analysis we use a pressure depen-dence on height given by the Janssen model [19], whichassumes a constant stress ratioK  xx=zz everywhere ina sample. Experiments show that this model is a fairapproximation in the absence of external loads [20,21].The model gives a saturation of pressure (p  zz) withdepth, pz  psat1 ez=l; (3)to a constant value psat  gD=4KW , where W is thecoefficient of particle-wall friction, g is the gravitationalacceleration,  is the density difference between spheresand surrounding liquid, and l  D=4KW . Equation (3)indicates two ways of controlling the pressure distributioninside the column: (I) Increasing the sample height, whichincreases the fraction of the sample at psat. If RLP in-creases with p, then the average RLP measured by ourmethod should increase with sample height. This behavioris confirmed in Fig. 3. (II) Decreasing the density differ-ence , which decreases psat but keeps the pressureprofile unchanged. We increased the fluid density to ashigh as 2:39 g=cm3, close to the 2:48 g=cm3 particle den-sity, by adding sodium polytungstate to the water. Resultsfor different  (Fig. 4) again confirm that RLP decreaseswith decreasing p.A limit 0RLP would be given by matching the fluid andparticle densities, but in this limit there would be nosedimentation and no connected granular packings wouldform. Therefore, we extrapolate to determine 0RLP: In theabsence of theory we follow [5] and use the pressuredependence close to the jamming point known for friction-less static soft spheres [9–11,22] and frictionless thermalhard spheres [23]: RLP  0RLP a; (4)where we identify 	 psat. A fit of RLP for smoothparticles (sample E) in Fig. 4 yields 0RLP  0:5550:006. The value of   0:51 0:25 is approximate be-cause our derivation of (4) did not take into account the dependence of K [20].Influence of frictional properties.—Figure 4 shows thatRLP for the rough spheres was lower than for the smoothspheres. For the rough spheres a fit to (4) yields 0RLP 0:550 0:001 and   0:89 0:16. The decrease ofRLP and0RLP with increasing friction agrees with anotherexperiment [24], model [25], and simulations [11–13,26].Discussion.—Our experimental results and numericalsimulations [11,12] both show that RLP is well definedin an operational sense: in the limit of infinitesimally slowpreparation, the volume fraction of a sample converges toRLP, independent of the details of preparation. Careshould be taken in comparing theory for frictionless hardspheres with the experimental results, in part because ofthe different possible ways of defining mechanical stability[27,28].0.60 0.59 0.58 0.57 30 20 10volume fraction φ realflow rate  (ml/min)21 mm73 mm144 mm 0.58 0.57 0.5616080 0φ RLPsample height (mm)FIG. 3. Dependence of  on Q for different sample heights(261 m spheres, sample E). Values of  were corrected forfinite size effect using (2). Solid lines are fits to (1). Inset: Theresultant RLP values as a function of sample height. 0.55 0.56 0.57 0  0.4  0.8  1.2  1.6Random Loose Packing  φRLPdensity difference: ρparticle - ρliquid   (g/cm3)smooth spheres (sample E)rough spheres (sample F)FIG. 4. RLP as a function of the density difference betweenparticles and fluid, for smooth (261 m) and rough (257 m)particles. Solid lines are fits to (4). Data have been corrected forthe finite size effect using (2). Sample height was 64 mm, and thepoints are averages obtained at the highest possible flow rate,since for small density differences the flow rate range was toosmall for a meaningful fit with (1).PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008018301-3The observation of a well-defined RLP can be consid-ered within the framework of a statistical mechanics ofstatic granular material [14], where a configurational en-tropy S is defined as the logarithm of the number ofmechanically stable configurations for a given , p, andfriction coefficient. Two different approaches can explainRLP using two different assumptions of how S depends on. The first approach assumes that RLP is the smallest where S becomes larger than zero. This is compatible withthe existence of looser, highly ordered configurations [27],as their number seems not to grow exponentially withsystem size, so S  0.The second approach is supported by numerical resultson the number of stable configurations of frictional disks[26], where S has a maximum at RLP. This idea agrees withslow sedimentation leading to RLP; it is simply the mostprobable configuration. If the sedimentation speed is in-creased, the additional kinetic energy allows the system toexplore the local energy landscape and find rarer but lowerpotential energy (denser) configurations. Further, the maxi-mum of S and therefore RLP moves to higher values of for decreasing friction [26]. This agrees with our resultsand with simulations of frictionless disks that have amaximum of S at random close packing [29]. Our resultsindicate also that increasing p shifts the maximum of S in asimilar way.Conclusions.—Mechanically stable packings of spheresprepared with increasing sedimentation time display alower bound of their volume fraction, RLP, which de-pends on the pressure and the coefficient of friction butnot on the diameter of the spheres. In the limit of zeropressure we have found a new lowest value of RLP,0:550 0:001.We thank Massimo Pica Ciamarra and Antonio Conigliofor sharing their unpublished results, and BrandonMcElroy for his assistance with the Camsizer. Further wethank W. D. McCormick, Narayanan Menon, CharlesRadin, Leo Silbert, Jack Swift, and participants of theAspen Jamming workshop for helpful discussions. Thiswork was supported by Robert A. Welch FoundationGrant No. F-0805.*matthias.schroeter@ds.mpg.de†swinney@chaos.utexas.edu[1] G. Scott, Nature (London) 188, 908 (1960).[2] R. Rutgers, Nature (London) 193, 465 (1962).[3] G. Y. Onoda and E. G. Liniger, Phys. Rev. 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Lett. 76, 149 (2006).[24] Narayanan Menon (private communication).[25] Y. Srebro and D. Levine, Phys. Rev. E 68, 061301 (2003).[26] Massimo Pica Ciamarra and Antonio Coniglio,arXiv:0805.0220.[27] S. Torquato and F. H. Stillinger, J. Appl. Phys. 102,093511 (2007).[28] S. Torquato and F. H. Stillinger, J. Phys. Chem. B 105,11 849 (2001).[29] G.-J. Gao, J. Bławzdziewicz, and C. S. O’Hern, Phys. Rev.E 74, 061304 (2006).PRL 101, 018301 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending4 JULY 2008018301-4

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Onset of mechanical stability in random packings of frictional spheres

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