Jamming of frictional spheres and random loose packing

Transcription

1 PAPER Soft Matter Jaing of frictional spheres and rando loose packing Leonardo E. Silbert* Received 29th January 2010, Accepted 16th March 2010 First published as an Advance Article on the web 7th May 2010 DOI: /c001973a The role of the friction coefficient,, on the jaing properties of disordered, particle packings is studied using coputer siulations. Copressed, soft-sphere packings are brought towards the jaing transition the point where a packing loses echanical stability by decreasing the packing fraction. The values of the packing fraction at the jaing transition, f c, gradually decrease fro the rando close packing point for zero friction, to a value coincident with rando loose packing as the friction coefficient is increased over several orders of agnitude. This is accopanied by a decrease in the coordination nuber at the jaing transition, z c, which varies fro approxiately six to four with increasing friction. Universal power law scaling is observed in the pressure and coordination nuber as a function of distance fro the generalised, friction-dependent jaing point. Various power laws are also reported between the f c, z c, and. Dependence on preparation history of the packings is also investigated. Introduction Granular packings can exist over a range of densities generally depending on the generation protocol and the nature of the grain grain interactions. For frictionless spheres, where the particle friction coefficient ¼ 0, rando close packing (rcp), or the axially rando jaed state, describes the densest possible packing of dry, cohesionless, spheres whose structure contains no long-range order. 1 8 Rando close packing is welldefined in the sense that it represents a reproducible packing state as observed in nuerous experients and siulations of frictionless sphere packings of hard particles that interact priarily through excluded volue effects. 9 All such studies agree that rando close packing occurs at a packing fraction, f rcp z This is in contrast to frictional packings, > 0, which can exist over a substantial range in packing fraction fro f rcp all the way down to f rlp z 0.55, the value often quoted as rando loose packing. 10,11 Although, in reality, it is actually quite difficult to generate loose packed states. Consequently, rando loose packing is a uch less well-developed concept than rando close packing. The strong history dependence and characterisation of frictional packings persists as an experiental issue thus opening the door for siulations to shed light on the nature of rando loose packing of frictional particles. Frictionless sphere-packings and the rcp state have received uch focus in ters of jaing the transition fro a jaed, rigid, solid-like state, with finite shear and bulk oduli, to an unjaed, liquid-like state. 6,12 14 The siplest syste exhibiting such a transition occurs in a packing of over-copressed, purelyrepulsive, soft, frictionless spheres, at zero teperature in the absence of gravity. As the packing fraction, f, is decreased, the packing undergoes a transition at f c, between jaed and Departent of Physics, Southern Illinois University, Carbondale, IL, USA. E-ail: lsilbert@physics.siu.edu This paper is part of a Soft Matter theed issue on Granular and jaed aterials. Guest editors: Andrea Liu and Sidney Nagel. unjaed phases that occurs abruptly at f c ¼ f rcp. Approaching the jaing transition fro above, f/f + c, the average nuber of contacting neighbours per particle the coordination nuber z approaches the inial value required for echanical stability z iso, also known as isostaticity. For frictionless spheres, z ¼ 0 iso ¼ 6 (in 3D, ¼ 4in2D) Thus, the interplay between echanical stability and axially rando, 4 in soe sense, provides a eaningful operational definition of the rcp state. Although it is worthwhile to note that the particle positions are disordered no long range order exists but they are not copletely rando as they ust satisfy echanical equilibriu. The question now arises, to what extent do these developing ideas of jaing apply ore generally in systes with noncentral force laws, such as frictional, granular packings? The identification of the jaing threshold in frictionless sphere packings to the well-known rcp state begs the question, can these sae ideas lead to a ore concrete definition of the rando loose packing state for frictional spheres? These concerns are investigated here by studying how interparticle friction affects the jaing properties of onodisperse spheres. Here it is shown that the extrapolated values of the packing fraction f c, and the coordination nuber z c, at the jaing transition depend on the friction coefficient, as presented in Table 1 and Fig. 1, in agreeent with conclusions of recent experients 11 and theory, 5,18 22 and that f c / f rlp and z c / z iso ¼ N ¼ 4 (in 3D, ¼ 3 in 2D) in the liit of large friction. Here, ziso ¼ N represents the isostatic state for frictional spheres, corresponding to the inially required coordination nuber for infinitely hard, frictional spheres. 23 Although the ain focus of this study is three diensional packings, for copleteness, soe inforation for 2D packings is also provided in Fig. 1 and Table 1. To date, the influence of friction has been studied in the context of three diensional granular packings, 18,24 26 and, to soe extent, their jaing properties. 5,19,22 Makse and co-workers 22 recently ade significant contributions to our understanding of frictional packings fro a theoretical point of view and the results presented here are fully consistent with these 2918 Soft Matter, 2010, 6, This journal is ª The Royal Society of Cheistry 2010

2 Table 1 Extrapolated values of the packing fraction f c, and coordination nuber z c, at the jaing transition for different friction coefficient for 3D onodisperse spheres (top) and 2D bidisperse discs (botto). The associated errors on these estiates range fro less than 1% for sall to 4% at larger values of 3D f c z c D f c z c Siulation odel Fig. 1 Dependence of the critical values of the packing fraction f c (filled circles), and coordination nuber z c (open squares), on the particle friction coefficient, for onodisperse spheres in 3D (upper panel) and bidisperse discs in 2D (lower panel). The insets are paraetric plots of f c against z c. Sybol size is representative of saple-to-saple fluctuations and error bars. previous studies. In saying that, however, ost earlier studies have not accurately addressed the issue as to whether frictional packings ja in the sae way as frictionless ones nor has the issue of history dependence been seriously considered. These are the two principal thees investigated in this study. What is apparent is that the coputational odels eployed are fully relevant to address realistic systes. In particular, experients 27,28 and siulations 21,29,30 of pseudo-2d, jaed, disc packings show good agreeent suggesting that the phenoenon of jaing sees to be applicable to real, frictional aterials. Here, it is shown that friction does indeed play an essential role in deterining the jaing transition for 3D sphere-packings as is the case in two diensional siulations of the Leiden group. 21,30 The principal result is the contention that the concept of the rando loose packing state becoes a friction-dependent property. The coonly quoted value for f rlp z 0.55, is only realised in the liit of large friction. Moreover, power law scaling is observed in the pressure and excess coordination relative to the friction-dependent jaing transition and that these critical values also exhibit power law behaviour with friction coefficient. The siulation technique used here eploys N ¼ 1024 onodisperse, inelastic, soft-spheres of diaeter d ¼ 1 and ass ¼ 1, within a cubic siulation box with periodic boundary conditions without gravity. Particles interact only on contact when they overlap, at which point they are considered to be contacting neighbours, through a repulsive, linear spring-dashpot. The force is characterised by the particle stiffness k n, t, and inelasticity by the coefficient of restitution e n, t, in the noral (n) and tangential (t) directions with respect to the contact surfaces. Unless otherwise stated, for > 0, a static friction law tracks the history of the friction forces over the lifetie of a contact that satisfies the Coulob yield criterion. 18,31 33 In this work, the particle Poisson ratio, n ¼ 1 l 4 þ l ¼ 0,34 where the ratio of the tangential to the noral particle stiffness is denoted by, lh k t ¼ 1, for this study. k n The initial configurations were generated by taking a dilute assebly of particles in a disordered, liquid-like configuration, then instantaneously quenching these configurations into overcopressed, jaed packed states at f i ¼ After this rapid copression the configurations were then allowed to relax into a echanically stable state. Unless otherwise stated, during this initial copression friction was switched off. In an alternative protocol to be discussed later, friction was switched on during the initial quench to the over-copressed state. The packing fraction of these echanically stable packings were then increentally decreased towards the jaing threshold with the friction coefficient set at the desired value. After each increental change in the packing fraction, the packing was rapidly quenched back into a echanically stable state by setting e n, t z 0. This procedure was continued until the difference in the potential energy between successive increents ( 10 16, at which point the siulation run terinates. To iprove statistical uncertainty, all results are averaged over at least five independent realisations. Results and discussion The jaing transition is identified as the point at which the pressure p, coputed fro the contact stresses, goes to zero. The jaing transition packing fraction f c, is obtained as a fitting paraeter by extrapolating the f p curve to p ¼ 0. Likewise, the coordination nuber at the transition, z c, is siilarly obtained by fitting the (p, z) data to the functional for used for frictionless spheres 14 and bubbles: 35 p f (z z c ) 1/2. This sae power This journal is ª The Royal Society of Cheistry 2010 Soft Matter, 2010, 6,

3 law exponent has also been used to fit experiental 28 and nuerical data 19,21 of frictional systes. The transition values, f c and z c, for different, are shown in Table 1 and plotted in Fig. 1. The insets in Fig. 1 represent the boundary between stable and unstable states. For 3D, onodisperse spheres with ¼ 0, f 0 ¼ ¼ c ¼ and z 0 c ¼ , are indistinguishable fro previous studies that used siilar and different algoriths. 7,14,18,19,36 These results are ¼ consistent with f rcp and z 0 iso. As increases, {f c, z c } decrease. In the liit of large friction, $ 1, these values saturate ¼ at about, f N ¼ c z 0.55 and z N c z 4, which coincide with the ¼ values of f rlp and z N iso. To check that these values do indeed correspond to the hard-particle liit, the particle stiffnesses were varied over five orders of agnitude keeping their ratio l ¼ 1. The resulting critical values obtained were all indistinguishable within statistical error. For copleteness data for two diensional, bidisperse disc packings (with particle size ratio 1 : 1.4) is also shown in Fig. 1 and Table 1. The 2D data are consistent with other jaing results, 21 siulations of sheared granular aterials, 37 and studies on force indeterinacy. 20 This behaviour can be reasoned by the following arguents. For sall friction coefficients the tangential forces do not contribute significantly to the stability of the packing and hence the packings are not that different fro frictionless systes. This behaviour persists, as shown in Fig. 1, up to friction coefficients approaching order unity where the typical tangential force first becoes coparable to the noral force. Hence, the friction forces start to play a significant role in stabilising the packing. The precise value of the friction coefficient where this transition occurs is likely to depend on the particle Poisson ratio n. In the work presented here n ¼ 0. Preliinary data with both negative and positive values of n indicate a possible dependence of the critical values on n, but these differences are practically within statistical uncertainty. Other granular siulations 18 with n ¼ 1 6 find siilar behaviour in the packing fraction and coordination nuber indicating that the data presented here is a general stability property of frictional systes. Associated changes in the structural properties of the packings with friction are highlighted in Fig. 2 and Fig. 3, where force network inforation is presented. Fig. 2 shows the noral force networks where the noral forces between particles in contact are represented by lines connecting the centres of the particles. Thicker, darker line shading indicate forces with increasing agnitude. As the critical packing fractions and coordination nubers decrease with increasing friction the Fig. 3 Distributions, P(f n, t ), of the, (a) noral F n, and (b) tangential F t, contact forces, noralised by their respective ean values. Data at approxiately the sae Df z 10 4 for four different friction coefficients are shown: ¼ 0.001, 0.1, 0.25, 1. The data at ¼ 0.25 is ephasised. density of contacts likewise decreases as is seen in these snapshots. It is also worth pointing out that although the global force network by necessity pervades the syste, there is little indication of long-ranged, correlated, force chain structures in the noral forces. Fig. 3 shows the distributions of the agnitudes of the noral and tangential forces for packings close to the critical states for different friction coefficients. In all cases, the distributions are characterised by exponential-like tails at larger forces. However, there are soe subtle changes occurring at the sall force plateau-peak region. Notably, on careful inspection of the data at ¼ 0.25, which has been ephasised in the figures and corresponds to a value of the friction coefficient the sallest forces only for friction coefficients where the critical values are decreasing with friction in Fig. 1, both the noral and tangential distributions exhibit an upturn at the sallest forces. Thus indicating that the decrease in the critical values indicated in Fig. 1 is correlated with the fraction of sall forces in the syste. Scaling Motivated by earlier studies on frictionless systes, it is possible to rescale all the data for different, by using the easure Df ^ f f c, as the distance for each syste for a given, fro their respective jaing transitions, i.e. it is now not appropriate to easure the distance to the jaing transition using just the zero-friction value f c ¼ 0, for different. Fig. 4 shows the power-law scaling, data collapse for the pressure and the excess coordination nuber Dz ^ z z c, for all friction coefficients: Fig. 2 Snapshots of the noral force networks for three different friction coefficients at approxiately the sae pressures or equivalently the sae Df z 10 4 ; (a) ¼ 0, (b) ¼ 0.1, and (c) ¼ 1. Lines represent the noral forces between particles in contact. Thicker, darker lines indicate larger forces. Particles not shown for clarity Soft Matter, 2010, 6, This journal is ª The Royal Society of Cheistry 2010

4 Fig. 4 Power law scaling of the pressure p, and excess coordination nuber Dz, as a function of the excess packing fraction, Df, over and above the extrapolated values at the jaing threshold {z c, f c}. Different sybols represent different friction coefficients: ¼ 0(>), (B), 0.01 (,), 0.1 (+), 0.5 (*), 1 (O), and 10 (grey B). The solid lines represent power laws with exponents 0.5 and 1 as indicated. P f Df; Dz fðdfþ 1=2 : This result shows that jaing frictional spheres look like jaing frictionless spheres provided one identifies the frictionspecific jaing threshold values f c and z c. Furtherore, the jaing thresholds for the packing fractions and coordination nubers exhibit several power-law relationships as a function of the friction coefficient, as shown in Fig. 5 and Fig. 6. Up to the large-friction liit, where both f c and z c practically saturate, the transition values relative to the ¼ 0 values, scale as, Df c;0 h f ¼0 c f c 0:74½9Š ; Dz c;0 h z ¼0 c z c 0:64½4Š : The corresponding easures, Df c, N ^ (f c fc ¼ N ) and Dz c, N ^ (z c z c ¼ N ), relative to the infinite friction liit do not exhibit siilar relations, however, the following power laws are also observed: Df c;0 ðdz c;0 Þ 1:23½10Š ; Df c;n ðdz c;n Þ 0:65½8Š ; (1) (2) (3) Fig. 6 Power laws between; (a) Df c,0 ^ (f c ¼ 0 f c ), and Dz c,0 ^ z c ), the jaing values relative to the zero-friction state, and (b) (z ¼ 0 c Df c, N ^ (f c f c ¼ N ) and Dz c, N ^ (z c z c ¼ N ), the jaing values relative to the large-friction state. Solid lines are power law fits to data described by eqn (3), with power law exponents and in (a) and (b) respectively. It is not clear as to the origin of eqn (2) and eqn (3), although the first relation in eqn (3) is consistent with eqn (1). These results are siilar to two diensional systes, 21 where the corresponding four power law exponents were obtained: 0.77, 0.70, 1.1, and 0.59, and thus hints at soe possible underlying universal properties of frictional packings irrespective of diensionality. The results presented thus far have iplications regarding the statistical echanical enseble forulation of granular (rando) packings espoused by Edwards 38 and has been extensively explored by Makse and co-workers 22 and others. 39 Edwards revision of statistical echanics proposes that the properties of a granular packing can be coputed fro a statistical average over equally probable configurations. For infinitely hard and frictional particles the partition function entering the analogue of the canonical enseble is sued over the full range of all possible states copatible with echanical stability, fro rando loose packing, f rlp ¼ 0.55, up to f rcp ¼ However, the results presented here suggest that generalising to finite friction requires one to take into account friction-dependent constraints: the su-over-states should only include those echanically stable states copatible with the value of, i.e. project onto the su of states fro all possible configurations only the echanically stable ones. 41 Practically, this ay be achieved by cutting-off the su-over-states at the appropriate value of f c: Ð 42 f frlp df 0 / Ð f f c df 0 ; or in the icrocanonical foralis, states with z < z c should be given zero weight. This is precisely the approach taken by Makse and co-workers who deduced a frictional packing phase diagra 22 which is consistent with the results presented in Fig. 1. Fig. 5 Power laws for, Df c, 0 ^ (f c ¼ 0 f c )(), the packing fraction, and Dz c,0^ (z c ¼ 0 z c )(-), the coordination nuber, at jaing relative to the zero friction values, over several orders of agnitude in. Solid lines are power law fits to data described in eqn (2), with power law exponents and in (a) and (b) respectively. Protocol dependence The siulation protocol ipleented here generates sphere packings that are statistically identical to other works, 6,19 22 resulting in siilar values for the transition packing fraction f c, and coordination nuber z c, as presented in Fig. 1 and Table 1. These values correspond to the hard sphere liit where the particles are just touching at the point of jaing. This can be seen by coparing the various interaction and siulation odels (linear-spring or Hertzian interaction forces, olecular dynaics or contact dynaics siulations). 18,20 22,36 However, the path This journal is ª The Royal Society of Cheistry 2010 Soft Matter, 2010, 6,

5 taken to arrive at the jaing transition ay differ, particularly when friction is present. Because of the hysteretic nature of the frictional forces the precise force configurations explored during this path will depend on the preparation history of the packings. The ajor factor influencing the frictional forces are the fraction of slipping contacts n s. The plasticity index is defined by, 18 z ^ F t /F n, where F n, t are the agnitudes of the {noral,tangential} forces. Here, slipping contacts are identified as those contacts at or on the verge of yielding: z $ The qualitative nature of the results are not sensitive to the precise definition of slipping contacts. The data of Fig. 7 shows how the probability distributions P(z) depend of friction coefficient for packings prepared very close to the jaing threshold. For sall friction coefficients < 0.2, the distributions are doinated by a large fraction of contacts that are close to yielding, z > 0.9. These systes correspond to points in Fig. 1 just before the rapid decrease in f c with. However, for larger friction coefficients, > 0.5, where the transition values begin to saturate, the weight of the distribution shifts to saller z. Hence, a ajority of contacts becoe stabilised as seen by the growth of a hup at z z 0.25 for ¼ 1 in Fig. 7. Siilar changes in P(z) have been seen in siulations of confined packings 43 and two diensional systes. 21 Although the qualitative features of the distributions shown in Fig. 7 are robust over different protocols, the actual nubers of slipping contacts depends on the preparation history of the packings. To illustrate this point Fig. 8 shows the fraction, n s,of slipping contacts over a wide range of friction coefficients for packings that are close to the jaing threshold. The two sets of data correspond to slightly different packing preparation histories. The open sybols are for the protocol used throughout this work to generate the previously shown data: a dilute packing was over-copressed with friction initially set to zero prior to the decopression procedure. The filled sybols, on the other hand, are for packings where friction was set to the desired value during the initial over-copression stage. The differences between the two protocols clearly show up at the lower friction coefficients where the newly odified protocol exhibits a uch larger fraction of slipping contacts. The rapid copression with friction leads to a build up of the frictional forces that quickly causes contacts to reach the Coulob yield criterion as the syste is decopressed towards the jaing threshold. However, these differences diinish for larger friction coefficients. The exact nature of the frictional build up and relaxation is currently under Fig. 8 Fraction of slipping contacts n s at Df() z over a large range in friction coefficients. By definition, n s ( ¼ 0) ¼ 1. The two sets of data represent two slightly different protocols as discussed in the text. Differences at sall friction values diinish at larger friction coefficients for the two protocols. investigation. It is also worth pointing out that the extrapolated jaing thresholds are identical within statistical uncertainty between the two protocols. Thus, the protocol dependence does not see to affect the geoetrical properties of the packing, but will likely have an effect on echanical properties, such as yielding under flow or the application of external forces. To probe the nature of protocol dependence further, the algorith used to generate the packings was also odified in the following ways: In one case a dynaic friction odel was used that depends only on the instantaneous friction force. For this syste the jaing threshold lies uch closer to the frictionless liit. In the other data, the history of the contacts was reset at different intervals during the siulation procedure thereby suppressing the build-up of frictional forces. The corresponding evolution of the coordination nuber with packing fraction is shown in Fig. 9 for packings these packings where the contact history and frictional forces were treated differently. Thus, erasing the history of the frictional forces between particles in contact can have a profound effect on the resulting evolution of the packing. What Fig. 9 deonstrates is that friction strongly influences the jaing thresholds and hence the possible range of echanically stable states for a given friction coefficient. This ay therefore explain, in part, the observation why real, frictional granular aterials, ore often than not, tend to for packings that are interediate between the rando close and rando loose liits: even inor rearrangeents that cause the particles to oentarily coe out of contact will erase the history of frictional contacts and thus the syste will appear to be coposed of particles of a lower friction coefficient than their actual values. This aspect of the history dependence of the frictional forces has been utilised recently in experients on real, frictional, granular particles ade to iic frictionless systes. 44 Fig. 7 Distributions, P(z), of the plasticity index, z ^ F t /F n, where F n, t are the agnitudes of the noral and tangential forces, for different friction coefficients, within 10 4 of the jaing threshold. Conclusions In suary, cohesionless, frictional spheres that interact only on contact, exhibit a jaing transition that shares any siilarities with that for frictionless systes, provided one identifies the appropriate friction-dependent jaing transition packing fraction f c, and coordination nuber z c. Using these 2922 Soft Matter, 2010, 6, This journal is ª The Royal Society of Cheistry 2010

6 Fig. 9 Evolution of the coordination nuber, z, with decreasing packing fraction, f. Line styles represent different friction odels: frictionless ( ¼ 0, solid line); history-dependent static friction odel (dotted), static-friction odel reset at different rates (dot-dash and dash), and dynaic instantaneous friction odel with no contact history (long-dash), ¼ 1.0. Inset shows the rando close packing region. friction-dependent quantities rather than just the frictionless and frictional isostatic values ay lead to a better understanding of existing scaling results. 30,45 In saying that, the role of packing generation protocol and friction-induced history dependence persists as a topic that has received relatively little attention to date. Understanding how packings traverse the packing stability diagra of Fig. 1 will likely help resolve soe of these issues. Rando loose packing, f rlp z 0.55, appears to be a state that can only be achieved in the large-friction liit. These results, however, do not preclude the fact that even lower values of f rlp ay be obtained using different protocols, such as sedienting particles in a fluid, 46 or tuning the particle interactions to include additional forces, such as cohesion. 47,48 The power-law dependence on of both f c and z c, relative to the frictionless syste hints at a ore universal behaviour. The origin of these scalings reains unclear although one possible explanation for the results of Fig. 5 and Fig. 6 ight lie in spatial correlations between the frictional forces. As shown in Fig. 8, there is a systeatic decrease in the fraction of slipping contacts with increasing friction. One ight therefore expect that for sall friction coefficients there are large spatial regions of contacts that are slipping, whereas for large friction, these are replaced by non-obilised contacts. Hence, fluctuations are sall in either case. However, interediate between these two extree states, large-scale, fluctuating regions ight occur. Preliinary data suggests that there is indeed an increase in correlations that follow these arguents and such ideas are currently being pursued. Applications of the results presented here could potentially pave the way towards a generalised forulation of an equation of state for granular aterials. Progress along these lines has been ade recently. 22 However, the hysteretic nature of the frictional forces can have a significant effect on the properties of the packing. It is interesting to note that the friction coefficients of any aterials lie in the range 0.01 # # 1. Coincidentally, these values of correspond to the region of echanically stable states where sall changes in can lead to large changes in the packing fraction and coordination nuber of the packing. Hence, any processes that cause changes in the surface friction of the constituent particles, such as roughening or polishing, can result in packings with draatically different stability properties. Alternatively, aterials could be designed with varying echanical properties based on their frictional properties. Acknowledgeents It is a pleasure to thank M. van Hecke and M. Schr oter for several insightful discussions related to this work. Support fro an SIU ORDA faculty seed grant and the National Science Foundation CBET is greatly appreciated. References 1 J. D. Bernal and J. Mason, Nature, 1960, 188, G. D. Scott, Nature, 1960, 188, J. G. Berryan, Phys. Rev. A: At., Mol., Opt. Phys., 1983, 27, S. Torquato, T. M. Truskett and P. G. Debenedetti, Phys. Rev. Lett., 2000, 84, H. A. Makse, D. L. Johnson and L. M. Schwartz, Phys. Rev. Lett., 2000, 84, C. S. O Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 68, A. Donev, S. Torquato and F. H. Stillinger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 71, L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 73, R. D. Kaien and A. J. Liu, Phys. Rev. Lett., 2007, 99, G. Y. Onoda and E. G. Liniger, Phys. Rev. Lett., 1990, 64, M. Jerkins, M. Schr oter, H. L. Swinney, T. J. Senden, M. Saadatfar and T. Aste, Phys. Rev. Lett., 2008, 101, Jaing and Rheology, ed. A. J. Liu and S. R. Nagel, Francis & Taylor, New York, Unifying Concepts in Granular Media and Glasses, ed. A. Coniglio, A. Fierro, H. J. Herrann and M. Nicodei, Elsevier, Asterda, C. S. O Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett., 2002, 88, S. Alexander, Phys. Rep., 1998, 296, 65. This journal is ª The Royal Society of Cheistry 2010 Soft Matter, 2010, 6,

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