The Spot Model for random-packing dynamics

Transcription

1 Mechanic of Material 38 (2006) The Sot Model for random-acking dynamic Martin. Bazant Deartment of Mathematic, Maachuett Intitute of Technology, 77 Maachuett Avenue, Cambridge, MA 02139, United State Dedicated to Prager Medalit, Salvatore Torquato Abtract The diffuion and flow of amorhou material, uch a glae and granular material, ha reited a imle microcoic decrition, analogou to defect theorie for crytal. Early model were baed on either ga-like inelatic colliion or crytal-like vacancy diffuion, but here we rooe a cooerative mechanim for dene random-acking dynamic, baed on diffuing ot of intertitial free volume. Simulation with the Sot Model can efficiently generate realitic flowing acking, and yet the model i imle enough for mathematical analyi. Starting from a non-local tochatic differential equation, we derive continuum equation for tracer diffuion, given the dynamic of free volume (ot). Throughout the aer, we aly the model to granular drainage in a ilo, and we alo briefly dicu glay relaxation. We conclude by dicuing the roect of ot-baed multicale modeling and imulation of amorhou material. Ó 2005 Elevier Ltd. All right reerved. 1. Introduction addre: bazant@mit.edu Profeor Torquato ha made ioneering contribution to the characterization of random acking and their relation to roertie of heterogeneou material (Torquato, 2002). Hi recent work reject the claical notion of random cloe acking of hard here and relace it with the more recie concet of a maximally random jammed tate (Torquato et al., 2000; Torquato and Stillinger, 2001; Kanal et al., 2002; Donev et al., 2004b). In thee tudie and other invetigating the jamming tranition (OÕHern et al., 2002, 2003), however, the focu i on the tatitical geometry of tatic acking, and not on the dynamic of nearly jammed acking in flowing amorhou material. Dene random-acking dynamic i at the heart of condened matter hyic, and yet it remain not fully undertood at the microcoic level. Thi i in contrat to dilute random ytem (gae), where BoltzmannÕ kinetic theory rovide a ucceful tatitical decrition, baed on the hyothei of randomizing colliion for individual article. Similar ingle-article theorie can alo be alied to molecular liquid at tyical exerimental time and length cale, where kinetic energy i able to fully dirut local acking (Hanen and McDonald, 1986). The difficulty arie in decribing liquid at very mall (atomic) length and time cale, over which the trajectorie of neighboring article are trongly correlated the o-called cage effect /$ - ee front matter Ó 2005 Elevier Ltd. All right reerved. doi: /j.mechmat

2 718 M.. Bazant / Mechanic of Material 38 (2006) Fig. 1. EyringÕ mechanim for flow in vicou liquid: a ingle article jum from one available cage to another by exchanging with a void moving in the ooite direction. Thi difficulty i extended to much larger length and time cale in uercooled liquid (Götze and Sjögren, 1992), glae (Angell et al., 2000), and dene granular material (Jaeger et al., 1996), where kinetic energy i inufficient to eaily tear a article away from it cage of neighbor. A a reult, one mut omehow decribe the cooerative motion of all article at once. In dene ordered material (crytal), cooerative relaxation and latic flow are mediated by defect, uch a intertitial, vacancie, and dilocation, but it i not clear how to define defect for homogeneou diordered material. The challenge in decribing random-acking dynamic i related to the concet of hyer-uniform oint ditribution, recently introduced by Torquato and Stillinger (2003). In a dilute ga, article undergo indeendent random walk and thu have the uniform ditribution of a Poion roce (Hadjicontantinou et al., 2003), where the variance of the number of article, N, cale with the volume, V: Var(N) =hni = qv, where q i the mean denity. In a condened hae, the article ditribution mut be hyer-uniform, with much maller fluctuation, roortional to the urface area: Var(N) / V (d 1)/d, where d = 3 i the atial dimenion, o it i clear that article cannot fluctuate indeendently. In a crytal, hyer-uniformity i a roerty of the ideal lattice, which i reerved during diffuion and latic flow by the motion of iolated defect. For dene diordered material, however, no imle flow mechanim ha been identified, which reerve hyer-uniformity. Eyring (1936) wa erha the firt to ugget a microcoic mechanim for vicou flow in liquid, analogou to vacancy diffuion in crytal. He rooed that the acking evolve when individual article jum into available cage, thu dilacing reexiting void, a hown in Fig. 1. Much later, the ame hyothei wa ut forth indeendently in theorie of the gla tranition (Cohen and Turnbull, 1959; Turnbull and Cohen, 1970), hear flow in metallic glae (Saeen, 1977), granular drainage from a ilo (Litwinizyn, 1958; Mullin, 1972), and comaction in vibrated granular material (Boutreux and de Genne, 1997), although it i not clear that all of thee author intended for the model to be taken literally at the microcoic level. Since article and void imly witch lace, it eem the Void Model can only be imulated on a ingle, fixed configuration of article, but real flow are clearly not contrained in thi way. Thi difficulty i aarent in the work of Caram and Hong (1991), who neglected random acking and imulated the Void Model on a lattice in an attemt to decribe granular drainage. By now, free volume theorie of amorhou material (baed on void) have fallen from favor, and exeriment on glay relaxation (Week et al., 2000) and granular drainage (Choi et al., 2004) have demontrated that acking rearrangement are highly cooerative and not due to ingle-article ho. In a recent theory of granular chute flow down inclined lane, Ertaß and Haley (2002) have otulated the exitence of coherent rotation, called granular eddie to motivate a continuum theory of the mean flow. Although the theory uccefully redict Bagnold rheology and the critical layer thickne for flow, Landry and Gret (2003) have failed to find any evidence for granular eddie in dicrete-element imulation of chute flow (Landry et al., 2003). In gla theory, Adam and Gibb (1965) introduced the concet of region of cooerative relaxation, whoe ize diverge at the gla tranition. Modern tatitical mechanical aroache are baed on mode-couling theory (Götze and Sjögren, 1992), which accurately redict denity correlation function in imle liquid (Kob, 1997), although a clear microcoic mechanim, which could be ued in a article imulation, ha not really emerged. Cooerative rearrangement have alo long been recognized in the literature on heared glae. Orowan (1952) wa erha the firt to otulate localized hear tranformation in region of enhanced atomic diorder. Argon (1979) later develoed the idea of intene hear tranformation at low temerature, which underlie the tochatic model of localized inelatic tranformation (Bulatov and

3 M.. Bazant / Mechanic of Material 38 (2006) Argon, 1994). A imilar notion of hear tranformation zone (ST) ha alo been invoked by Falk and Langer (1998) in a continuum theory of hear reone, which ha recently been extended to account for free-volume creation and annihilation in glae (Lemaître, 2002b) and granular material (Lemaître, 2002a). Thi henomenology eem to cature many univeral feature of amorhou material, although the microcoic icture of + and ST tate remain vague. In thi aer, we rooe a imle model for the kinematic of dene random acking. In Section 2, we introduce a general mechanim for tructural rearrangement baed on the concet of a diffuing ot of free volume. In Section 3, we aly the Sot Model to granular drainage. In Section 4, we analyze the diffuion of a tracer article via a nonlocal, nonlinear tochatic differential equation, in the limit of an ideal ga of ot. In Section 5, we derive equation for tracer diffuion in granular drainage, which deend on the denity, drift, and diffuivity of ot (or free volume). We cloe by dicuing oible alication to glae in Section 6 and ot-baed multicale modeling and imulation of amorhou material in Section The Sot Model 2.1. Motivation Our intuition tell u that a article in a dene random acking mut move together with it nearet neighbor over hort ditance, followed by gradual cage breaking at longer ditance. In imle liquid, thi tranition occur at the molecular cale (<nm) over very hort time (<) comared to tyical exerimental cale. In uercooled liquid and glae, the time cale for tructural relaxation effectively diverge and i relaced by low, ower-law decay (Angell et al., 2000; Kob, 1997; Hanen and McDonald, 1986), although the length cale for cooerative motion remain relatively mall (Week et al., 2000). In granular drainage, cage breaking occur lowly, over time cale comarable to the exit time from the ilo, o that cooerative motion i imortant throughout the ytem at the macrocoic cale (Choi et al., 2004). Another curiou feature of granular drainage i the imortance of geometry: all fluctuation in a dene flow eem to have a univeral deendence on the ditance droed for a wide range of flow rate (Choi et al., 2004). In a ene, therefore, increaing the flow eed in thi regime i like fatforwarding a film, aing through the ame equence of configuration, only more quickly. The only exiting theory to redict thi roerty (a well a the mean flow rofile in ilo drainage) i the Void Model, ince increaing the flow eed imly increae the injection rate of void, but not their geometrical trajectorie. However, the model incorrectly redict cage breaking and mixing at the cale of individual article. Thi aradox of granular diffuion i a fruitful tarting oint for a new model of random-acking dynamic General formulation Let u uoe that the cage effect give rie to atial correlation in article velocitie, with correlation coefficient, C(r), for two article earated by r. More generally, when there i broken ymmetry, e.g. due to gravity in granular drainage, there i a correlation coefficient, C ab ðr 1; r 2 Þ, for the a velocity comonent of a article at r 1 and the b velocity comonent of a article at r 2. Perha the imlet way to encode thi information in a microcoic model i to imagine that article move cooeratively in reone to ome extended entity a ot which caue a article at r to move by DR ¼ wðr ; r ÞDR ð1þ when it move by DR near r. Although the ot i not a defect er e, like a dilocation in an ordered acking, it i a collective excitation which allow a random acking to rearrange. In rincile, the ot influence, w, could be a matrix cauing a mooth ditribution of local Fig. 2. The ot mechanim for cooerative diffuion: a grou of neighboring article make mall correlated dilacement in reone to a diffuing ot of exce intertitial volume.

4 720 M.. Bazant / Mechanic of Material 38 (2006) tranlation and rotation about the ot center r, but a reaonable firt aroximation i that it i imly a collective tranlation the ooite direction from the ot dilacement, a illutrated in Fig. 2. In thi cae, w i a calar function, whoe hae i roughly that of the velocity correlation function. More reciely, under ome imle aumtion, we how below that C(r) i the overla integral of two ot influence function earated by r. Since the ot influence i related to the cage effect, we exect that w and C will decay quickly with ditance, for earation larger than a few article diameter. Due to the local tatitical regularity of dene random acking, we might exect a ot to retain it hae a it move, in which cae w deend only on the earation vector, r ðiþ rðjþ, although thi aumtion might need to be relaxed in region of large gradient in denity or velocity. Phyically, what i a ot? Since article move collectively in one direction, a ot mut correond to ome amount of free intertitial volume (or miing article ) moving in the other direction. If article are ditributed with number denity q (r )and a ot at r carrie a tyical volume, V (r ), then an aroximate tatement of volume conervation i V DR ¼ dr qðr Þwðr ; r ÞDR ðr Þ. ð2þ For article ditributed uniformly at volume fraction, /, thi reduce to a imle exreion for the local volume carried by a ot, V ðr Þ¼/ dr wðr ; r Þ; ð3þ which can thu be interreted a a meaure of the otõ total influence. A very imle, ot-baed Monte Carlo imulation roceed a follow. Given a ditribution of (aive) article and (active) ot, the random dilacement, DR ðjþ, of the jth ot centered at r ðjþ, of the ith induce a random dilacement, DR ðiþ article centered at r ðiþ DR ðiþ ¼ w rðiþ ; rðjþ þ DR ðjþ DR ðjþ ð4þ Fig. 3. The trail of ot correond to tranient, retating chain of article. Each ot undergoe an indeendent random walk, with an aroriate drift and diffuivity for free volume, which leave in it trail a thick chain of article retating in the ooite direction, a hown in Fig. 3. In Eq. (4), we chooe to center the ot influence on the end of it mall dilacement, but it i alo reaonable to ue the midoint of the dilacement (Rycroft et al., 2005) or it beginning. In the infiniteimal limit, dicued below, thee choice are analogou to different (Stratonovich v. Itô) definition of tochatic differential (Riken, 1996). In rincile, the drift velocity, diffuivity, and influence function of ot could deend on local variable, uch a tre, and temerature (or rather, ome uitable microcoic quantitie related to contact force and velocitie, reectively). Sot could alo interact with each other, undergo creation and annihilation, and oe a tatitical ditribution of ize (or influence function). The imlet kinematic aumtion, however, which cature the baic hyic of the cage effect, i that ot are identical and maintain their roertie while undergoing indeendent (non-interacting) random walk. In articular, the contant influence function, w(jr r j), i choen to be tranlationally invariant in ace and time. It turn out that thi model allow rather realitic multicale imulation, while remaining analytically tractable Multicale model The imle ot mechanim above give a reaonable decrition of tracer diffuion and low cage breaking in random acking, but it doe not trictly enforce acking contraint (or, more generally, inter-article force). A uch, article erform indeendent random walk in the long-time limit, which eventually lead to uniform denity fluctuation with Poion tatitic. For a comlete microcoic model, we mut omehow reerve hyer-uniform acking. Thi may be accomlihed by adding a relaxation te to the ot-induced dilacement, a hown in

5 M.. Bazant / Mechanic of Material 38 (2006) (a) (b) (c) Fig. 4. Multicale imulation of denely acked (nearly) hard here with the Sot Model: (a) a block of neighboring article tranlate ooite to the dilacement of a ot of free volume; (b) the block and a hell of neighbor are allowed to relax under oft-core reulive force; (c) the net cooerative rearrangement combine thee two te. (Particle dilacement are greatly exaggerated for clarity.) Fig. 4. Firt in (a), a ot dilacement caue a imle correlated dilacement, a decribed above, e.g. uing Eq. (4) with ome hort-ranged choice of w(r) with a finite cutoff. Next in (b), the affected article and a hell of their nearet neighbor are allowed to relax under aroriate inter-article force, with more ditant article held fixed. For imulation of (nearly) hard grain, the mot imortant force come from a oft-core reulion, which uhe article aart only if they begin to overla. Although it i not obviou a riori, the net ot-induced cooerative dilacement, hown in Fig. 4(c), eaily roduce very realitic flowing acking, while reerving the hyical icture of the model (Rycroft et al., 2005). In ractice, the correlated nature and mall ize of the ot-induced block dilacement reult in very mall and infrequent article overla, only near the edge of the ot, where ome hear occur with the background acking. A a reult, it eem the detail of the relaxation are not very imortant, although thi iue merit further invetigation. In any cae, the algorithm i intereting in it own right a a method of multicale modeling, ince it combine a macrocoic imulation of imle extended object (ot) with localized, microcoic imulation of article. 3. Alication to granular drainage 3.1. Exerimental calibration and teting The claical Kinematic Model for the mean velocity in granular drainage (Nedderman, 1991), which comare fairly well with exeriment (Tüzün and Nedderman, 1979; Samadani et al., 1999; Choi et al., 2005), otulate that the mean downward velocity, v, atifie a linear diffuion equation, ov oz ¼ b$2? v; ð5þ where the vertical ditance z lay the role of time and the horizontal dimenion (with gradient, $? ) lay the role of ace. The microcoic jutification for Eq. (5) i the Void Model of Litwinizyn (1958, 1963) and Mullin (1972, 1974), where article-ized void erform directed random walk uward from the orifice. A dicued above, thi microcoic mechanim i firmly contradicted by article-tracking exeriment (Choi et al., 2004), but, a hown below, a imilar macrocoic flow equation can be derived from the Sot Model, where ot diffue uward with a (horizontal) diffuion length, b ¼ VarðDx Þ ; ð6þ 2d h Dz where Dx i the random horizontal dilacement of a ot a it rie by Dz and d h = 2 i the horizontal dimenion. A tyical value for 3 mm gla bead i b 1.3d, where d i the article diameter. The hae of the ot influence function can be inferred from meaurement of atial velocity correlation in exeriment (Bazant et al., 2005) or imulation by the dicrete-element method (DEM) with frictional, vico-elatic here (Rycroft et al., 2005). The imlet aumtion i a uniform herical influence with a finite cutoff, wðrþ ¼ w r < d =2; ð7þ 0 r > d =2; where exeriment and imulation find d 5d. Thi i conitent with our interretation of the ot mechanim in term of the cage effect, where a article move with it nearet neighbor. The tyical number of article affected by a ot,

6 722 M.. Bazant / Mechanic of Material 38 (2006) N ¼ / d 3 ; ð8þ d i thu N 72, for / For a uniform ot, the condition of volume conervation, Eq. (2), read V ðdx ; Dz Þ¼ NV ðdx ; Dz Þ. ð9þ Uing Eq. (1), thi rovide an exreion for the ot influence, w ¼ V D/ ð10þ NV / 2 in term of D/, the change in local volume fraction due to the reence of a ingle ot. In DEM imulation of granular here in ilo drainage (Rycroft et al., 2005), the local volume fraction varie in the range, / = , within the rough bound of jamming, / = 0.63 (Torquato et al., 2000; Kanal et al., 2002; OÕHern et al., 2002, 2003), and random looe acking, / = 0.55 (Onoda and Liniger, 1990). If we attribute D/// = 1% to a ingle ot, then we find w 0.017, but, if many ot, ay N = 10, can overla, then thi etimate i reduced by 1/N.We thu exect, w = Thi rediction can be teted againt exeriment and DEM imulation. We can ue Eq. (9) and (10) to infer w from a meaurement of the horizontal article diffuion length b ¼ VarðDx Þ 2d h j Dz j ¼ w2 VarðDx Þ 2d h wdz ¼ wb. ð11þ DEM imulation (Rycroft et al., 2005) and article-tracking exeriment (Choi et al., 2004) for imilar flow yield w = b /b = d/1.14d = and w = d/bpe x = d/(1.3d)(321) = , reectively (where Pe x i a Péclet number). Thee value are conitent with the model, thu roviding ome uort for it microcoic hyothei. A otõ influence i related to the free volume it carrie by Eq. (3). In the cae of a uniform ot of diameter, d, it total free volume i given by V ¼ /wd3 ; ð12þ 6 which i related to the article volume, V,by 3 ¼ wn. ð13þ V ¼ w/ d V d For dene granular drainage, the tyical value N =72andw = imly V = 0.18V. In ummary, by comaring the model to exeriment and DEM imulation, we reach a quantitative decrition of ot in granular drainage: a ot carrie around one fifth of a article volume, read out over a region of roughly five article diameter. Thi delocalized icture of free volume diffuion i rather different from that of the claical Void Model, which i the (unhyical) limit, V = V and N = w = Simle Monte Carlo imulation We have een that the Sot Model can be uccefully calibrated for granular drainage, o it rovide a reaonable tarting oint for Monte Carlo imulation. Firt, we dicu imle imulation by Bazant et al. (2005) uing the baic mechanim in Fig. 2 and Eq. (4) alied to granular drainage in a quai-two-dimenional ilo, a in the exeriment of Samadani et al. (1999) and Choi et al. (2004). The ilo ha a narrow orifice (not much bigger the a ot) and i very wide, with little influence from the ide wall. For imlicity, we imulate the model in two dimenion (d h = 1) uing uniform, circular ot with the arameter inferred from exeriment above: w = , d =5d, b = 1.3d. (Very imilar reult may be obtained with a Gauian influence function.) We make no attemt to decribe the dynamic of the orifice which control the rate of introduction of ot and (for volume conervation) their vertical drift velocity. Intead, we eek to decribe the teady velocity ditribution, u to a contant roortional to the flow rate, a well a the random trajectorie of tracer article. Since ot do not interact, we allow ot to a through the ytem one at a time, following directed random walk with vertical dilacement, Dz = 0.1d, and random horizontal dilacement, Dx ¼ ffiffiffiffiffiffiffiffiffiffi 2bDz ¼0:51d. After each ot dilacement, (Dx,Dz ), all article within a ditance 0.5d = 2.5d of the ot center undergo a tiny block dilacement, (Dx,Dz )= w(dx,dz )= ( d,0.0012d). Within the block, the acking i reerved during motion, but at the edge there i ome hear, uncontrained by inter-article force. The imulation begin with a random acking of identical dik, colored with horizontal trie (10d thick) to aid in viualizing the ubequent evolution. A nahot of the ot imulation at a later time i

7 M.. Bazant / Mechanic of Material 38 (2006) Fig. 5. Simulation of granular drainage in a quai-two-dimenional ilo. To: two-dimenional imulation uing (a) the Sot Model without acking contraint and (b) the Void Model (Bazant et al., 2005). Bottom: three-dimenional imulation uing (c) the Sot Model with multicale relaxation and (d) the Dicrete Element Method for frictional, vico-elatic here (Rycroft et al., 2005). Particle are colored according to their initial oition in horizontal trie, 10d thick.

8 724 M.. Bazant / Mechanic of Material 38 (2006) hown in Fig. 5(a). For comarion, a imulation of the ame ituation with the Void Model on a twodimenional lattice, following Caram and Hong (1991), i hown in Fig. 5(b), along with the central lice of a DEM imulation by Rycroft et al. (2005) in three dimenion (15d thick) in Fig. 5(d), which i imilar to the exeriment. All nahot are taken after roughly the ame amount of drainage ha occurred. Although the mean flow rofile i imilar in the ot and void imulation and reaonably cloe to exeriment, the void imulation dilay far too much diffuion and mixing, ince the initial horizontal trie are comletely mixed down to the ingle article level inide the flow region. In contrat, the interface between the colored layer remain fairly har in the ot imulation, a in exeriment and the DEM imulation. The diffuion of individual tracer article i alo decribed fairly well, thu jutifying the mathematical analyi below. Without acking contraint, however, hyer-uniformity i gradually lot, a article begin to overla and oen ga, which i mot aarent in the region of highet hear near the orifice Multicale ot imulation Next, we dicu imulation by Rycroft et al. (2005) uing the multicale algorithm in Fig. 4 (with an internal relaxation te) alied to a three-dimenional drainage imulation, tarting from the ame initial condition and geometry a the DEM imulation in Fig. 5(d). The baic otinduced article dynamic i the ame a before, only in three dimenion (with uniform herical ot) in order to avoid the trong tendency for hard dik to crytalize in two dimenion. A minor change i that the ot influence i centered on the midoint of each ot dilacement (rather than the end). We emloy the imlet oible relaxation cheme, where each air of overlaing article in a relaxation zone (a here of diameter, d +4d) i uhed aart by a dilacement, a(d r) roortional to the overla, d r, while keeing article fixed outide a here of diameter, d +2d. It turn out that the article dilacement in the relaxation te (Fig. 4(b)) are tyically at leat four time maller than thoe in the baic ot te (Fig. 4(a)). Therefore, the mean cooerative motion (Fig. 4(c)) i quite conitent with the imle icture of original model. A ot imulation with imilar arameter a above, including uch a relaxation te with a = 0.8, i hown in Fig. 5(c). The rate of introducing ot at the orifice and their uward drift velocity ha alo been calibrated for comarion to the DEM imulation, at the ame intant in time. Clearly, the imle relaxation i able to reerve realitic random acking, and in many way the ot imulation in Fig. 5(c) i inditinguihable from the much more comutationally demanding DEM imulation in Fig. 5(d). Not only are the mean velocity rofile and diffuion length reroduced, but o are variou microcoic tatitic of the acking geometry, uch a the two-body and three-body correlation function (Rycroft et al., 2005). Thee urriing reult eem quite inenitive to the detail of the relaxation te, although thi iue require further tudy. The robut behavior of the multicale algorithm eem due to the cooerative (block-like) nature of ot-induced dilacement, which caue article overla to be extremely mall and infrequent. Another, deeer reaon may be that the geometry of dene flowing random acking ha univeral feature, which may be achieved by the totally different dynamic of ot and DEM imulation. In any cae, the ability to imulate flowing dene random acking by an efficient algorithm (at leat 100 time fater than DEM) could have broad alicability for diordered material, not jut granular flow. 4. Mathematical analyi of diffuion 4.1. A non-local, nonlinear SDE In thi ection, we return to the general formulation of the Sot Model and analyze tracer diffuion in the continuum limit. It i clear from the imulation in Fig. 5 that the baic model in Fig. 2 give a reaonable decrition of the dynamic of a ingle article tracer, even though the multicale relaxation te in Fig. 4 i needed to reerve realitic acking. The relatively mall ize of the relaxation dilacement make it reaonable to regard them a mall additional noie in a mathematical analyi of tracer diffuion. Here, we will neglect thi mall (but comlicated) noie and view it average effect a incororated tatitically into the ot influence function, w(r,r ), in Eq. (4). We begin by artitioning ace a hown Fig. 6, where the nth volume element, DV ðnþ, centered at r ðnþ

9 M.. Bazant / Mechanic of Material 38 (2006) contain a random number, DN ðnþ, of ot at time t (tyically one or zero). In a time interval, Dt, uoe that the jth ot in the nth volume element make a random dilacement, DR ðj;nþ (which could be zero). According to Eq. (4), the total dilacement, DR, of a article at r in time Dt i then given by a um of all the random dilacement induced by nearby moving ot, DR ¼ X n XDN ðnþ j¼1 wðr ; r ðnþ þ DR ðj;nþ ÞDR ðj;nþ. ð14þ Note that the atio-temoral ditribution of ot, DN ðnþ, i another ource of randomne, in addition to the individual ot dilacement, DR ðj;nþ, o that each article dilacement i given by a random um of random variable. In the limit of infiniteimal dilacement, we arrive at a non-local, nonlinear tochatic differential equation (SDE) dr ðtþ ¼ dn ðr ; tþwðr ðtþ; r þ dr ðr ; tþþdr ðr ; tþ; ð15þ where the tochatic integral i defined by the uual limit (infinitely refined artition of ace) of the random Riemann um in Eq. (14). Thi equation differ from tandard nonlinear SDE (Riken, 1996) in two baic way: (i) the tracer trajectory, r ðtþ ¼ t dn = 0 ¼0 d R( r ) dr ðþ d R( r ) dn = 1 d R ( r ) Fig. 6. Sketch of a article interacting with aing ot, howing ome the quantitie in Eq. (15). ð16þ i aively driven by a tochatic ditribution of moving influence (ot), dn (r,t), which evolve in time and ace, rather than by ome internal ource of indeendent noie, and (ii) the tochatic differential, dr (t), i given by a non-local integral over other tochatic differential, dr (r,t), aociated with thee moving influence, which lie at oition, r, at finite ditance away from the article at r The continuum limit In general, the variou tochatic differential in Eq. (15) are correlated, which ignificantly comlicate analyi. Here, we will make the reaonable firt aroximation of an ideal ga of ot, where the tracer article ee an indeendent, random configuration of non-interacting ot at each infiniteimal time te. A in an ideal ga (Hadjicontantinou et al., 2003), ot are thu ditributed according to a Poion roce with a given mean denity, q (r,t). In addition to neglecting correlation caued by interaction between ot, we diregard the following fact: (i) the ditribution of ot in ace, {dn (r,t)}, at time t deend exlicitly on the ditribution and dilacement at the reviou time, t dt, via the ot random walk; (ii) each ot, due to it finite range of influence, affect the ame article for a finite eriod of time, o any eritence (autocorrelation) in the ot trajectory i tranferred to the article, in a nonlinear fahion controlled by w(r,r ). In the ot-ga aroximation, the tracer article erform a random walk with indeendent (but non-identically ditributed) dilacement, which deend non-locally on a Poion roce for finding ot. Therefore, the roagator, P (r,tjr 0,t 0 ), which give the robability denity of finding the article at r at time t after being at r 0 at time t 0, atifie a following Fokker Planck equation (Riken, 1996), which take the following form: op þ $ ðu P Þ¼$$ : ðd P Þ ot with drift velocity, u ðr; tþ ¼ hdr ðr; tþi dt and diffuivity tenor, D ab ¼ lim Dt!0 hdr ðr; tþi Dt ð17þ ð18þ ðr; tþ ¼hdRa drb i. ð19þ 2dt

10 726 M.. Bazant / Mechanic of Material 38 (2006) (Here $$: A denote P a Pb o2 A ab ox a ox b.) The Fokker Planck coefficient may be calculated by taking the aroriate exectation uing Eq. (14) in the limit DV ðnþ! 0 and Dt! 0 (in that order), which i traightforward ince we aume that ot do not interact. Here, the ot dilacement, DR ðjþ ðr ðnþ Þ, and the local number of ot, DN ðnþ, are indeendent random variable in each time interval, and they are indeendent of the ame variable at earlier time. In order to calculate the drift velocity, we need only the mean ot denity, q (r,t), defined by hdn ðnþ i¼q ðr ; tþdv ðnþ. The reult, u ðr ; tþ ¼ dv wðr ; r Þ½q ðr ; tþu ðr ; tþ 2D ðr ; tþ$q ðr ; tþš ð20þ exhibit two ource of drift. The firt term in the integrand i a article drift velocity, which ooe the ot drift velocity, u ðr; tþ ¼ hdr ðr; tþi ð21þ dt a in Eq. (9). The econd term, which deend on the ot diffuion tenor, D ði;jþ ðr; tþ ¼ hdrðiþ 2dt dr ðjþ i ; ð22þ contain ome noie-induced drift, tyical of nonlinear SDE (Riken, 1996), which caue article to climb gradient in the ot denity. Thi extra drift i crucial to enure that article eventually move toward the ource of ot, e.g. the orifice in granular drainage. Both contribution to the drift velocity in Eq. (20) are averaged non-locally over a finite region, weighted by the ot influence function, w(r,r ). In order to calculate the diffuivity tenor, we alo need information about fluctuation in the ot denity. From the ot-ga aroximation, we have hdn ðnþ DN ðmþ i¼d m;n hðdn ðnþ Þ 2 i¼oððdv ðnþ Þ m Þ; where m = 1 for a Poion roce and m < 1 for a hyer-uniform roce (Torquato and Stillinger, 2003). It turn out that uch fluctuation do not contribute to the diffuion tenor (in more than one dimenion), and the reult i D ðr ; tþ ¼ dv wðr ; r Þ 2 q ðr ; tþd ðr ; tþ. ð23þ Note that the influence function, w, aear quared in Eq. (23) and linearly in Eq. (20), which caue the Péclet number for tracer article to be of order w maller than that of ot (or free volume), a in Eq. (11). Higher-order term a Kramer Moyall exanion generalizing Eq. (17) for finite indeendent dilacement, which do deend on fluctuation in the ot denity, are traightforward to calculate, but beyond the coe of thi aer. Such term are uually ignored becaue, in ite of imroving the aroximation, they tend to roduce mall negative robabilitie in the tail of ditribution (Riken, 1996). In granular material, however, velocity gradient can be highly localized, o the correction term could be ueful Satial velocity correlation tenor For any tochatic roce rereenting the motion of a ingle article, it i well-known that tranort coefficient can be exreed in term of temoral correlation function via the Green Kubo relation (Riken, 1996). For examle, the diffuivity tenor in a uniform flow i given by the time integral of the velocity auto-correlation tenor, 1 D ab ¼ dthu a ðtþu b ð0þi; ð24þ 0 where U ðtþ ¼fU a g¼dr =dt i the tochatic velocity of a article. (A imilar relation hold for ot.) In the Sot Model, nearby article move cooeratively, o the tranort roertie of the collective ytem alo deend on the two-oint atial velocity correlation tenor, C ab ðr 1; r 2 Þ¼ hu a ðr 1ÞU b ðr 2Þi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð25þ hu a ðr 1Þ 2 ihu b ðr 2Þ 2 i which i normalized o that C ab ðr; rþ ¼1. We emhaize that the exectation above i conditional on finding two article at r 1 and r 2 at a given moment in time and include averaging over all oible ot ditribution and dilacement. Subtituting the SDE (15) into Eq. (25) yield C ab ðr 1; r 2 Þ¼ dv q ðr Þwðr 1 ; r Þwðr 2 ; r ÞD ab ðr Þ. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D ab ðr 1ÞD ab ðr 2Þ ð26þ auming indeendent ot dilacement. Eq. (26) i an integral relation for cooerative diffuion, which relate the atial velocity correlation

11 M.. Bazant / Mechanic of Material 38 (2006) tenor to the ot (or free volume) diffuivity tenor via integral of the ot influence function, w(r,r ). If the tatitical dynamic of ot i homogeneou (in articular, if D i contant), then the relation imlifie R dv C ab ðr q 1;r 2 Þ¼ ðr Þwðr 1 ;r Þwðr 2 ;r Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R dv q ðr Þwðr 1 ;r Þ R 2 dv 0 q ðr 0 Þwðr 2;r 0 Þ2 ð27þ if alo the tenor i diagonal, D ab / d a;b. If the tatitical dynamic of article i alo homogeneou, a in a uniform flow (q = contant), then it imlifie even further R dv C ab ðrþ ¼ wðr r Þwð r Þ R ; ð28þ dv wðr Þ 2 where we have aumed that the ot influence function, and thu the correlation tenor, i tranlationally invariant (r = r 1 r 2 ). In thi limit, a mentioned above, the velocity correlation function i imly given by the (normalized) overla integral for ot influence earated by r Relative diffuion of two tracer The atial velocity correlation function affect many-body tranort roertie. For examle, the relative dilacement of two tracer article, r = r 1 r 2, ha an aociated diffuivity tenor given by D ab ðr 1 ; r 2 Þ¼D ab ðr 1ÞþD ab ðr 2Þ 2C ab ðr 1; r 2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D aa ðr 1ÞD bb ðr 2Þ. ð29þ In a uniform flow, the diagonal comonent take the imle form D aa ðrþ ¼2D aa ð1 Caa ðrþþ; ð30þ which may be ued above to etimate the cagebreaking time, a the exected time for two article diffue aart by more than one article diameter. A more detailed calculation of the relative roagator, P(r,tjr 0,t 0 ), neglecting temoral correlation (a above) would tart from the aociated Fokker Planck equation, opðr; tþ ot ¼ $$ : ðdðrþpðr; tþþ ð31þ with a delta-function initial condition. (In a nonuniform flow, one mut alo account for noieinduced drift and motion of the center of ma.) Thi analyi doe not enforce acking contraint, o it allow for two article to be earated by le than one diameter. A hard-here reulion may be aroximated by a reflecting boundary condition at jrj = d when olving equation uch a (31), but there doe not eem to be any imle way to enforce inter-article force exactly in the analyi. 5. Tracer diffuion in granular drainage 5.1. Statitical dynamic of ot The analyi in the reviou ection make no aumtion about ot, other than the exitence of well-defined local mean denity, mean velocity, and diffuion tenor, which may deend on time and ace. A uch, the reult may have relevance for a variety of dene diordered ytem exhibiting cooerative diffuion (ee below). In thi ection, we aly the analyi to the ecific cae of granular drainage, in which ot diffue uward from a ilo orifice, a in Fig. 5. Our goal here i imly to how how to derive continuum equation from the Sot Model in a articular cae, but not to tudy any olution in detail. For imlicity, let u aume that each ot undergoe mathematical Brownian motion with a vertical drift velocity, u ¼ v ^z, and a diagonal diffuion tenor, 0 1 D ¼ D? D? D k C A; ð32þ which allow for a different diffuivity in the horizontal (?) and vertical (k) direction due to ymmetry breaking by gravity. In that cae, the roagator for a ingle ot tracer, P (x,z,tjx 0,z 0,t), atifie another Fokker Planck equation, op ot þ o oz v ð P Þ ¼ $ 2? D? P o 2 þ D k oz 2 P. ð33þ The coefficient may deend on ace (e.g. larger velocity above the orifice than near the tagnant region), a uggeted by the hae of ome exerimental denity wave (Baxter et al., 1989). The geometrical ot roagator, P ðx j z; x 0 ; z 0 Þ, i the conditional robability of finding a ot at horizontal oition x once it ha rien to a height z from an initial oition (x 0,z 0 ). For contant v and D, the geometrical roagator atifie the diffuion equation,

12 728 M.. Bazant / Mechanic of Material 38 (2006) op oz ¼ b$2? P ; ð34þ where b ¼ D? v i the kinematic arameter. If ot move indeendently, thi equation i alo atified by the teady-tate mean ot denity, q (x,z), analogou to Eq. (5) of the Kinematic Model. However, the mean article velocity in the Sot Model, Eq. (20), i omewhat different, a it involve non-local effect (ee below). The time-deendent mean denity of ot, q (x,z,t), deend on the mean ot injection rate, Q(x 0,z 0,t) (number/area time), which may vary in time and ace due to comlicated effect uch a arching and jamming near the orifice. It i natural to aume that ot are injected at random oint along the orifice (where they fit) according to a ace time Poion roce with mean rate, Q. In that cae, if ot do not interact, the atial ditribution of ot within the ilo at time t i alo a Poion roce with mean denity, q ðx; z; tþ ¼ dx 0 dz 0 dt 0 Qðx 0 ; z 0 ; tþ t 0 <t P ðx; z; t j x 0 ; z 0 ; t 0 Þ. ð35þ For a oint-ource of ot (i.e. an orifice roughly one ot wide) at the origin with flow rate, Q 0 (t) (number/time), thi reduce to q ðx; z; tþ ¼ dt 0 Q 0 ðt 0 ÞP ðx; z; t j 0; 0; t 0 Þ; ð36þ t 0 <t where P i the uual Gauian roagator for Eq. (33) in the cae of contant u and D. In reality, ot hould weakly interact, but the reaonable decrition by the Kinematic Model ugget that ot diffue indeendently a a firt aroximation in granular drainage (Choi et al., 2005) Statitical dynamic of article Integral formulae for the drift velocity and diffuivity tenor of a tracer article may be obtained by ubtituting the ot denity which olve Eq. (35) into the general exreion (20) and (23), reectively. For examle, if ot only diffue horizontally ðd k ¼ 0Þ, then the mean downward velocity of article i given by v ðr; tþ ¼ dv wðr ; r Þq ðr ; tþv ðr ; tþ. ð37þ Note that the mean article velocity i a non-local average of nearby ot drift velocitie. For imlicity, let u conider a bulk region where the ot denity varie on cale much larger than the ot ize. In thi limit, the integral over the ot influence function reduce to the following interaction volume : V k ðrþ ¼ dr wðr; r Þ k ð38þ for k = 1,2. (Note that V 1 = V above.) The equation for tracer-article dynamic (17) then take the form, op ot ¼ o v q oz 2D k oq oz V 1 P 2$? D? ð$ o 2?q ÞV 1 P D k oz 2 q V 2 P þ $ 2? D? q V 2 P. ð39þ Again, it i clear that recaling the ot denity i equivalent to recaling time. When the ot dynamic i homogeneou (i.e. u and D are contant), Eq. (39) imlifie further 1 op ¼ v V ot o oz þ b? $2 þ b k o 2 ðq oz 2 P Þ 2b? $ ðp $q Þ 2b k o oz P oq oz ; ð40þ where b? ¼ b ¼ D? =v and b k ¼ D k =v are the ot diffuion length and b? ¼ b V 2 =V 1 and b k ¼ b k V 2 =V 1 are the article diffuion length. In thi aroximation, the latter are given by the imle formula, b? R b ¼ bk dv? b ¼ wðr; r Þ 2 R k dv wðr; r Þ ; ð41þ which generalize Eq. (11) for a uniform ot with a har cutoff. The hyical meaning of the diffuion length become more clear in the limit of uniform flow, q = contant. In term of the oition in a frame moving with the mean flow, f = v t z, where v = v V q, we arrive at a imle diffuion equation, op of ¼ b? $2? þ o 2 bk P oz 2 ; ð42þ where f, the mean ditance droed, act like time, conitent with the exerimental finding of Choi et al. (2004).

13 M.. Bazant / Mechanic of Material 38 (2006) Poible alication to glae We have een that the Sot Model, in it imlet form, accurately reroduce the kinematic of bulk granular drainage, o it i temting to eculate that it might be extended to flow in other amorhou material. In thi ection, we briefly conider evidence for ot-like dynamic in glae, but we leave further extenion of the Sot Model for future work. Exeriment have revealed amle ign of dynamical heterogeneity in uercooled liquid and glae (Hanen and McDonald, 1986; Kob, 1997; Angell et al., 2000), but the direct obervation of cooerative motion ha been achieved only recently. Rather than comact region of relaxation, Donati et al. (1998) have oberved tring-like relaxation in molecular dynamic imulation of a Lennard-Jone model gla. The trength and length cale of correlation increae with decreaing temerature, conitent with the Adam Gibb hyothei. Such cooerative motion would be difficult to oberve exerimentally in a molecular gla, but Week et al. (2000) have ued confocal microcoy to reveal three-dimenional cluter of fater-moving article in a dene colloid. In the uercooled liquid hae, cluter of cooerative relaxation have widely varying ize, which grow a the gla tranition i aroached. In the gla hae, the cluter are much maller, on the order of ten article, and do not roduce ignificant rearrangement on exerimental time cale. Thee obervation ugget that the Sot Model may have relevance for tructural rearrangement in imle glae. String-like relaxation i reminicent of the trail of a ot, in Fig. 3. An atomically thin chain might reult from the random walk of a ot, roughly one article in ize, but carrying le than one article of free volume. Larger region of correlated motion might involve larger ot and/ or collection of interacting ot. Some key feature of the exerimental data of Week et al. (2000) eem to uort thi idea: (i) correlation take the form of neighboring article moving in arallel direction, a in Fig. 2; and (ii) the large cluter of correlated motion tend to be fractal of dimenion two, a would be exected for the random-walk trail of a ot, a in Fig. 2(b). For a comlete theory of the gla tranition, however, one would reumably have to conider interaction between ot and thermal activation of their creation, motion, and annihilation. 7. Concluion In thi aer, we have introduced a mechanim for tructural rearrangement of dene random acking, due to diffuing ot of free volume. Even without inter-article force, the Sot Model give a reaonable decrition of tracer dynamic, which i trivial to imulate and amenable to mathematical analyi, tarting from a non-local tochatic differential equation. With a imle relaxation te to enforce acking contraint, the Sot Model can efficiently roduce very realitic flowing acking, a demontrated by the cae of granular drainage from a ilo. The ot mechanim may alo have relevance for glay relaxation and other henomena in amorhou material. Regardle of variou material-ecific alication, the ability to eaily roduce three-dimenional dene random acking i intereting in and of itelf. Current tate-of-the-art algorithm to generate dene random acking are artificial and comutationally exenive, eecially near jamming (Torquato et al., 2000; Kanal et al., 2002; OÕHern et al., 2002, 2003). A oular examle i the molecular dynamic algorithm of Lubachevky and Stillinger, 1990, which imulate a dilute ytem of interacting article, whoe ize grow linearly in time until jamming occur. For each random acking generated, however, a earate molecular dynamic imulation mut be erformed. In contrat, the Sot Model roduce a multitude of dene random acking (albeit with ome correlation between amle) from a ingle imulation, which i more efficient than molecular dynamic, ince it doe not require the mechanical relaxation of all article at once. It would be intereting to characterize the tye of dene acking generated by the Sot Model and comare with the reult of other algorithm. The enitivity of the reult to the choice of ot influence function, relaxation rocedure, and arameter hould alo be tudied ytematically. The Sot Model (with relaxation) alo rovide a convenient aradigm for multicale modeling and imulation of amorhou material, analogou to defect-baed modeling of crytal. The iteration between global meocoic imulation of ot and local microcoic imulation of article lead to a tremendou aving in comutational effort, a long a the ot dynamic i hyically realitic for a given ytem. For examle, a imle extenion of the multicale imulation of granular

14 730 M.. Bazant / Mechanic of Material 38 (2006) here by Rycroft et al. (2005) would be to different article hae, uch a ellioid, which have been hown to ack more efficiently than here (Donev et al., 2004a). The only change in the imulation would be to modify the inter-article force in the relaxation te for a oft-core reulion with a different hae. A more challenging and fruitful extenion would be to incororate mechanic into the multicale imulation, beyond geometrical acking contraint. One way to do thi may be ue the information about inter-article force in the ot relaxation te to etimate local tree, which could then affect the dynamic of ot. Perha dicrete ot dynamic could alo be connected to claical continuum model of granular material, uch a Mohr Coulomb laticity theory (Nedderman, 1991). It may alo be neceary to move article directly in reone to mechanical force, in addition to the random cooerative dilacement caued by ot. Such extenion eem neceary to decribe forced hear flow in granular and glay material. For now, at leat we have a reaonable model for the kinematic of random acking. Acknowledgment Thi work wa uorted by the US Deartment of Energy (Grant DE-FG02-02ER25530) and the Norbert Wiener Reearch Fund and NEC Fund at MIT. The author i grateful to J. Choi, A. Kudrolli, R.R. Roale, C.H. Rycroft for many timulating dicuion and to A.S. Argon, L. Bocquet, M. Demkowicz, R. Raghavan for reference to the gla literature. Reference Adam, G., Gibb, J.H., On the temerature deendence of cooerative relaxation roertie in gla-forming liquid. J. Chem. Phy. 43, Angell, C.A., Ngai, K.L., McKenna, G.B., McMillan, P.F., Martin, S.W., Relaxation in glaforming liquid and amorhou olid. J. Al. Phy. 88, Argon, A.S., Platic deformation in metallic glae. Acta Metall. 27, Baxter, G.W., Behringer, R.P., Fagert, T., Johnon, G.A., Pattern formation in flowing and. Phy. Rev. Lett. 62, Bazant, M.., Choi, J., Rycroft, C.H., Roale, R.R., Kudrolli, A., A theory of cooerative diffuion in dene granular flow, unublihed. Boutreux, T., de Genne, P.G., Comaction of granular mixture: a free volume model. Phyica A 244, Bulatov, V.V., Argon, A.S., A tochatic model for continuum elato-latic behavior. Modell. Simul. Mater. Sci. Eng. 2, Caram, H., Hong, D.C., Random-walk aroach to granular flow. Phy. Rev. Lett. 67, Choi, J., Kudrolli, A., Roale, R.R., Bazant, M.., Diffuion and mixing in gravity driven dene granular flow. Phy. Rev. Lett. 92, Choi, J., Kudrolli, A., Bazant, M.., Velocity rofile of granular flow inide ilo and hoer. J. Phy.: Conden. Matter 17, S2533 S2548. Cohen, M.H., Turnbull, D., Molecular tranort in liquid and glae. J. Chem. Phy. 31, Donati, C., Dougla, J.F., Kob, W., Plimton, S.J., Poole, P.H., Glotzer, S.C., Stringlike cooerative motion in a uercooled liquid. Phy. Rev. Lett. 80, Donev, A., Cie, I., Sach, D., Variano, E.A., Stillinger, F.H., Connelly, R., Torquato, S., Chaikin, P.M., 2004a. Imroving the denity of jammed diordered acking uing ellioid. Science 303, Donev, A., Torquato, S., Stillinger, F.H., Connelly, R., 2004b. Jamming in hard here and dik acking. J. Al. Phy. 95, Ertaß, D., Haley, T.C., Granular gravitational collae and chute flow. Eurohy. Lett. 60, Eyring, H., Vicoity, laticity, and diffuion a examle of abolute reaction rate. J. Chem. Phy. 4, Falk, M.L., Langer, J.S., Dynamic of vicolatic deformation in amorhou olid. Phy. Rev. E 57, Götze, W., Sjögren, L., Relaxation rocee in uercooled liquid. Re. Prog. Phy. 55, Hadjicontantinou, N.J., Garcia, A.L., Bazant, M.., He, G., Statitical error in article imulation of hydrodynamic henomena. J. Comut. Phy. 187, 274. Hanen, J.-P., McDonald, I.R., Theory of Simle Liquid. Academic, London. Jaeger, H.M., Nagel, S.R., Behringer, R.P., Granular olid, liquid, and gae. Rev. Mod. Phy. 68, Kanal, A.R., Torquato, S., Stillinger, F.H., Diverity of order and denitie in jammed hard-article acking. Phy. Rev. E 66, Kob, W., The mode-couling theory of the gla tranition. In: Exerimental Aroache to Suercooled Liquid: Advance and Novel Alication. ACS Book, Wahington,. 28. Landry, J.W., Gret, G.S., rivate communication. Landry, J.W., Gret, G.S., Silbert, L.E., Plimton, S.J., Confined granular acking: tructure, tre, and force. Phy. Rev. E 67, Lemaître, A., 2002a. Origin of a reoe angle: kinetic of rearrangement for granular material. Phy. Rev. Lett. 89, Lemaître, A., 2002b. Rearrangement and dilatency for heared dene material. Phy. Rev. Lett. 89, Litwinizyn, J., Statitical method in the mechanic of granular bodie. Rheol. Acta 2 3, 146. Litwinizyn, J., The model of a random walk of article adated to reearche on roblem of mechanic of looe media. Bull. Acad. Pol. Sci. 11, 593. Lubachevky, B.D., Stillinger, F.H., Geometric roertie of random dik acking. J. Stat. Phy. 60,