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Developments in Modern Physics

by: Reece Metz

Developments in Modern Physics PHYS 110

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Reece Metz
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Date Created: 10/29/15
PHYSICAL REVIEW E 71 061306 2005 Random close packing revisited Ways to pack frictionless disks Ning Xu1 Jerzy Blawzdziewicz1 and Corey S O Henil392 1Department of Mechanical Engineering Yale University New Haven Connecticut 065208284 USA Department of Physics Yale University New Haven Connecticut 065208120 USA Received 20 March 2005 published 28 June 2005 publisher error corrected 6 July 2005 We create collectively jammed CJ packings of 5050 bidisperse mixtures of smooth disks in two dimen sions 2D using an algorithm in which we successively compress or expand soft particles and minimize the total energy at each step until the particles are just at contact We focus on small systems in 2D and thus are able to nd nearly all of the collectively jammed states at each system size We decompose the probability P for obtaining a collectively jammed state at a particular packing fraction 1 into two composite functions 1 the density of C packing fractions p which only depends on geometry and 2 the frequency distri bution B which depends on the particular algorithm used to create them We nd that the function p is sharply peaked and that B depends exponentially on d We predict that in the in nitesystem size limit the behavior of P in these systems is controlled by the density of C packing fractionsinot the frequency distribution These results suggest that the location of the peak in P when Ngtoo can be used as a protocolindependent de nition of random close packing DOI 101103PhysReVE71061306 I INTRODUCTION Developing a statisticalmechanical description of dense granular materials structural and colloidal glasses and other jammed systems I composed of discrete macroscopic grains is a dif cult longstanding problem These amor phous systems possess an enormously large number of pos sible jammed con gurations however it is not known with what probabilities these con gurations occur since these sys tems are not in thermal equilibrium The possible jammed con gurations do not occur with equal probabilityiin fact some are extremely rare and others are highly probable Moreover the likelihood that a given jammed con guration occurs depends on the protocol that was used to generate it Despite dif cult theoretical challenges there have been a number of experimental and computational studies that have investigated jammed con gurations in a variety of systems e experiments include studies of static packings of ball bearings 23 slowly shaken granular materials 45 sedi menting colloidal suspensions 6 and compressed colloidal glasses 7 The numerical studies include early Monte Carlo simulations of dense liquids 8 collision dynamics of grow ing hard spheres 9 serial deposition of granular materials under gravity 10712 various geometrical algorithms 13715 compression and expansion of soft particles fol lowed by energy minimization 16 and other relaxation methods 17 The early experimental and computational studies found that dense amorphous packings of smooth hard particles fre quently possess packing fractions near random close packing pm which is approximately 064 in threedimensional 3D monodisperse systems 18 and 084 in the 2D bidisperse systems discussed in this work 1519 However more re cent studies have emphasized that the packing fraction at tained in jammed systems can depend on the process used to create them Different protocols select particular con gura tions from a distribution of jammed states with varying de grees of positional and orientational order 20 15393755200571606130692300 0613061 PACS numbers 8105Rm 8270y 8380Fg Recent studies of hardparticle systems have also shown that different classes of jammed states exist with different properties 21 For example in locally jammed LI states each particle is unable to move provided all other particles are held xed however groups of particles can still move collectively In contrast in collectively jammed CI states neither single particles nor groups of particles are free to move excluding oater particles that do not have any con tacts Thus C states are more jammed than L states In this article we focus exclusively on the properties of collectively jammed states These states are created using an energy minimization procedure 1619 for systems com posed of particles that interact Via soft niterange purely repulsive and spherically symmetric potentials Energy minimization is combined with successive compressions and decompressions of the system to nd states that cannot be further compressed without producing an overlap of the par ticles As explained in Sec II this procedure yields collec tively jammed states of the equivalent hardparticle system In previous studies of collectively jammed states created using the energyminimization method we showed that the probability distribution of collectively jammed packing frac tions narrows as the system size increases and becomes a 8 function located at 10 in the in nitesystemsize limit 1619 We found that 10 was similar to values quoted pre viously for random close packing 18 The narrowing of the distribution of C packing fractions as the system size in creases is shown in Fig l for 2D bidisperse systems How ever it is still not clear why this happens Why is it so dif cult to obtain a collectively jammed state with lt1 at 10 in the largesystem limit One possibility is that very few collec tively jammed states exist with lt1 at 10 Another possibility is that collectively jammed states do exist over a range of pack ing fractions but only those with packing fractions near 10 are very highly probable Below we will address this question and other related problems by studying the distributions of collectively jammed states in small bidisperse systems in 2D For such 2005 The American Physical Society XU BLAWZDZIEWICZ AND O HERN 150 120 A90 60 30 0 077 083 1 FIG 1 The probability distribution PltS to obtain a collectively jammed state at packing fraction qS in 2D bidisperse systems with N 18 dotted line 32 dashed line 64 dot dashed line and 256 solid line systems we we will be able to generate nearly all of the collectively jammed states Enumeration of nearly all CJ states will allow us to decompose the probability density P to obtain a collectively jammed state at a particular packing fraction into two contributions P p 8 1 The factor p in the above equation represents the density of collectively jammed states ie p d measures how many distinct collectively jammed states exist within in a small range of packing fractions d The factor 8 denotes the effective frequency ie the counts averaged over a small region of with which these states occur We note that the density of states p is determined solely by the topological features of con gurational space it is thus independent of the the protocol used to generate these states In contrast the quantity 8 is protocol dependent because it records the average frequency with which a CI state at occurs for a given protocol For example for al gorithms that allow partial thermal equilibration during com pression and expansion the frequency distributions are shifted to larger compared to those that do not involve such equilibration The decomposition 1 will allow us to determine which contribution p or 8 controls the shape of the prob ability distribution P in the largesystem limit Others have studied the inherent structures of hardsphere liquids and glasses but have not addressed this speci c question 2223 We will show below that p controls the width of the distribution of CI states in the in nite systemsize limit We also have some evidence that the location of the peak in P in the largeN limit is also determined by the largeN behavior of p We will also argue that for many proce dures the protocol dependence of the frequency distribution 8 is too weak to substantially shift the peak in P for large systems Thus our results suggest that for a large class of algorithms the location of the peak in P can be used as a protocolindependent de nition of random close packing in the in nitesystemsize limit PHYSICAL REVIEW E 71 061306 2005 II METHODS Our goal is to enumerate the collectively jammed con gu rations in 2D bidisperse systems composed of smooth repul sive disks We will focus on bidisperse mixtures composed of N 2 large and N 2 small particles with a diameter ratio 7714 because it has been shown that these systems do not easily crystallize or phase separate 1516 We consider sys tem sizes in the range N 4 256 particles For N 10 we were able to nd nearly all of the collectively jammed states For Nl2 14 we found more than 90 60 of the total number Since the number of collectively jammed states grows so rapidly with N we are not able to calculate a large fraction of the CI states for N gt 14 but as we will show below we can still make strong conclusions about the shape of the distribution of CI states in large systems We utilize an energyminimization procedure to create collectively jammed states 16 We assume that the particles interact via the purely repulsive linear spring potential 6 Vrij 51 rijdij2 dijrij17 2 where e is the characteristic energy scale rij is the separation of particles 139 and j dijddj 2 is their average diameter and x is the Heaviside step function The potential 2 is nonzero only for rijltdij ie when the particles overlap Jammed states are obtained by successively growing or shrinking particles followed by relaxation via potential en ergy minimization until all particles excluding oaters in the system are just at contact In these prior studies we showed that the distribution of collectively jammed states does not depend sensitively on the shape of the repulsive potential Vrj Note that our process for creating jammed states differs from the xedvolume energyminimization procedure implemented in Ref 16 In the description be low the energies and lengths are measured in units of e and the diameter of the smaller particle d1 For each independent trial the procedure begins by choosing a random con guration of N particles at an initial packing fraction A in a square box with unit length and periodic boundary conditions The positions of the centers of the particles are uncorrelated and distributed uniformly in the box We have found that the results do not depend on the initial volume fraction A as long it is signi cantly below the peak in p We chose i060 for most system sizes After initializing the systems we nd the nearest local potential energy minimum using the conjugate gradient algo rithm 24 We terminate the energyminimization procedure when either of the following two conditions is satis ed 1 two successive conjugate gradient steps n and n1 yield nearly the same total potential energy per particle VH1 V Vlt 510 16 or 2 the total potential energy per par ticle is extremely small V1lt me 10 16 Following the potential energy minimization we decide whether the system should be compressed or expanded to nd the jamming threshold If Vn1gtVmaX2gtlt10 16 par ticles have nonzero overlap and thus small and large particles are reduced in size by Ad1d1A l 2 and Adz nAdl re spectively If on the other hand Vn1Vmim the system is 061306 2 RANDOM CLOSE PACKING REVISITED WAYS TO below the jamming threshold and all particles are thus in creased in size After the system has been expanded or com pressed it is relaxed using potential energy minimization and the process is repeated Each time the procedure switches from expansion to contraction or vice versa the packing fraction increment A is reduced by a factor of 2 The initial expansion rate was A i10 4 When the total potential energy per particle falls within the range Vmaxgt Vgt me the process is terminated and the jammed packing fraction is recorded If the nal state con tains oater particles with two or fewer contacts we remove them minimize the total potential energy and slightly com press or expand the remaining particles to nd the jamming threshold Note that the nal con gurations are slightly com pressed with overlaps in the range 10 9lt1 rjdljlt 10 8 We have veri ed that our results do not depend strongly on the parameters me Vmax and Arbi For each system size N this process is repeated using n independent random initial conditions and the resulting jammed con gurations are analyzed to determine whether they are collectively jammed and unique III ANALYSIS OF JAMMED STATES To verify if a given nal con guration is collectively jammed we analyze the eigenvalue spectra of the dynamical or rigidity matrix 16 MiWB where the indices 139 and j refer to the particles and a8x y represent the Cartesian coordinates For a system with Nf oaters and N N Nf particles forming a connected network the indices 139 and j range from 1 to N Thus the dynamical matrix has dN rows and columns where d2 is the spatial dimension By differ entiating the interparticle potential we nd that the elements of the dynamical matrix with 139 9E j are given by 25 lquot A A A A Mime ffan rijarij Cijrijarij v 3 I where tij6Vlarij and cij62Vl art2 while those with 139 j are given by Mime Mime 4 J The dynamical matrix 3 and 4 has dN real eigenval ues 5 d of which are zero due to translational invariance of the system In a collectively jammed state no set of par ticle displacements is possible without creating an overlap ping con guration therefore the dynamical matrix has ex actly dN d nonzero eigenvalues In our simulations we use the criterion igt 5mm for nonzero eigenvalues where 6mm 106 is the noise threshold for our eigenvalue calculations We note that our energyminimization algorithm for cre ating jammed states does occasionally yield a con guration that is not collectively jammed These states however are not considered in the current study The number of trials that yield collectively jammed states out of the original n trials is denoted m The fraction of trials that give locally but not collectively jammed states nl I LI L decreases with in creasing system size from 5 at N6 to less than 1 for N gt 12 PHYSICAL REVIEW E 71 061306 2005 FIG 2 Five distinct collectively jammed states that exist at the same packing fraction O81O 73 for a 2D bidisperse system with N 12 particles In a d unshaded particles are in the same po sitions while the shaded particles are in different locations from panel to panel Particles labeled 1 2 and 3 are in different positions in panels 1 and e while the other particles are in the same positions We determine whether two collectively jammed states are distinct by comparing the sorted lists of the nonzero eigen values of their respective dynamical matrices If the relative difference between two corresponding eigenvalues differs by more than Ediff10 3 the con gurations are treated as dis tinct By comparing the topology of the network of particle contacts in a representative sample of CI states we have found that this criterion is suf cient to reliably determine whether two states are distinct or identical This procedure allows us to determine the number us of distinct collectively jammed states at each xed number of independent trials m As expected if two CJ states have dif ferent packing fractions they are distinct with different con tact networks and dynamical modes This property holds with very high numerical precision the packingfraction difference of 10 13 already assures that the two states are distinct However it is not true that all collectively jammed states with the same packing fraction are identical For example the CJ states shown in Fig 2 have the same packing fraction but they possess different contact networks and eigenvalue spectra This is a clear demonstration that two collectively jammed con gurations at the same packing fraction can have very different structural properties We have also calculated the total number of contacts be tween particles NC ie the number of bonds that satisfy rijltdij in our slightly compressed jammed con gurations We nd that the number of contacts in the collectively jammed states satis es the relation 1726 chNlcmn2dN d 1 5 The minimum number of contacts required for mechanical stability of the system NE can be calculated by equating the number of degrees of freedom to the number of con straints Note that an extra constraint is required to prevent 061306 3 XU BLAWZDZIEWICZ AND O HERN 0 E 1 D qlt392 g D B 3 3 A r 4 v A 5 V 1 0 1 2 3 4 A FIG 3 Fraction fA of distinct collectively jammed states with an excess number of contacts ANc NHill over a range of system sizes N6 circles 8 squares 10 diamonds 12 upward tri angles and 14 downward triangles For N 6 all C states have A20 particle expansion We have found that nearly all of the col lectively jammed states have NCNICmn fewer than 1 of these states have NCgtNICI1111 as shown in Fig 3 All con gu rations that are not collectively jammed have fewer contacts than NE IV RESULTS In the preceding two sections we described our methods for generating and counting distinct collectively jammed states We will now present the results from these analyses We will rst discuss how the number of CI states depends on parameters such as the number of trials and system size We then decompose the probability density of obtaining a CI state at a given packing fraction Eq 1 into the density p of CI packing fractions and their frequency distribution 8 We also consider under what conditions all of the pos sible CI states can be enumerated and determine whether strong conclusions can be made about the distributions of CI states in large systems even though complete enumeration is not possible Our studies of the number of distinct CJ states nS versus the number of independent trials n led to several surprising observations First we nd that these systems possess a sig ni cant fraction of rare CJ states and thus an exponentially large number of trials are required to obtain nearly all states Second a master curve appears to describe nSn for systems with N 2 10 as shown in Fig 4 Each data point in this gure was obtained by averaging over at least 100 distinct permutations of the n trials Our numerical results indicate that when us is more than about 20 of the total number of distinct CJ states n3 the curve nSn can be accurately ap proximated by 2 n n E 1 A10g10lt gt 7 6 where A z 005 Our direct computations for small systems N 6 8 and 10 and numerical ts to the master curve 6 for N 12 and 14 indicate that both n and n3 increase exponentially with system size as shown in Fig 5 However both these quanti PHYSICAL REVIEW E 71 061306 2005 06 01 nsns 04 8 6 4 2 log n n5quot FIG 4 Fraction of C states nsn0t versus the ratio of the num ber of trials nt to the total number of trials nEot required to nd all C states n for several system sizes N6 dot dashed line 8 dashed line 10 thick solid line 12 long dashed line and 14 dotted line The curves for N 10 12 and 14 collapse The thin solid line is a least squares t to Eq The diamond upward triangle and downward triangle symbols give the maximum num ber of trials attempted for N 10 12 and 14 respectively ties remain nite for any nite system In particular for the smallest system sizes we increased the total number of trials by at least a factor of 10 and did not nd any new collec tively jammed states We also used several different algo rithms for generating CJ states eg compression and ex pansion of particles followed by relaxation using molecular dynamics with dissipative forces along fij and frictional forces perpendicular to fly and these did not lead to any new CJ states that were not already found using the protocol described in Sec II The maximum number of trials and fraction of CI states obtained are provided in Table I As indicated in Eq 1 the probability distribution P for obtaining a collectively jammed state at a particular pack ing fraction can be factorized into two composite func tions the density of CI states p and the frequency 8 with which these states occur In our simulations the distri bution P is calculated from the relation 14 d gt P n1 P d 7 Here np is the total number of CI states counting all repetitions of the same state with packing fractions below A 25 v FIG 5 The total number of distinct CJ states n circles and the number of trials required to nd them squares versus system size N The solid and dotted lines have slopes equal to 12 and 17 respectively 061306 4 RANDOM CLOSE PACKING REVISITED WAYS TO TABLE I Maximum number of trials performed min and frac tion of C states obtained nSngotmax versus system size N N n ns n2 max 6 106 10 8 106 10 10 29 x 106 10 12 28 x 106 090 14 26 x 106 060 The density of CI states is evaluated using an analogous relation p d ns n5 7 where ns is the number of distinct CJ states that have been detected in the packingfraction range below lt0 In fact we have used the number of distinct packing fractions to de ne p in place of the number of distinct CJ states ns However this does not affect our results because dis tinct states with the same are rare in 2D bidisperse sys tems We note that both the probability density 7 and the density of CI states 8 are normalized to 1 The frequency distribution 8 P p is normalized accordingly Below we show how P p and 8 depend on the fraction of CI states nSn0t and system size N To plot these distributions we used ten bins with the endpoint of the nal bin located at the largest CI packing fraction lm for each N We recall that the distribution of CI packing frac tions p does not depend on the protocol used to generate the CJ states The protocol dependence of the distribution P is captured by the frequency distribution 8 The probability distribution P of CI states is shown in Fig 6 for two small systems N 10 and 14 The results indi cate that P depends very weakly on the fraction nSn0t of CI states obtained only 5 of the CJ states are required to capture accurately the shape of P for these systems This result holds for all system sizes we studied which implies that the distribution of CI states can be measured reliably 07 0774 078 082 H 086 FIG 6 Probability distribution P for obtaining a C state at 05 for N 10 solid line and N 14 dotted line at nSngotmax The distributions at nsn t005 squares for N210 and triangles for tot N214 overlap those with larger nSns PHYSICAL REVIEW E 71 061306 2005 20 15gt E10 207 315 Q10 FIG 7 Density of collectively jammed packing fractions p for a N10 b 12 and c 14 at nSn t02 solid lines 04 dotted lines 06 dot dashed lines 08 long dashed lines and 10 dashed lines even in large systems 1619 Note that the width and loca tion of the peak in P do not change markedly over the narrow range of N shown in Fig 6 To see signi cant changes in P the system size must be varied over a larger range P for N18 32 64 and 256 is shown in Fig 1 at xed number of trials n104 The width of the distribution narrows and the peak position shifts to larger as the system size increases In Ref 16 we found that P for this 2D bidisperse system becomes a 5 function located at 00842 in the in nitesystemsize limit What causes P to narrow to a 5 function located at 00 when N gt 00 Is the shape of the distribution P deter mined primarily by the density of states p or does the frequency distribution 8 play a signi cant role in deter mining the width and location of the peak We will shed light on these questions below We rst show results for p and 8 as functions of the fraction rLSn0t of distinct CJ states obtained In Fig 7 p is shown for several small systems In contrast to the total distribution P the density of states p depends on rLSn0t signi cantly For N10 a system for which we can calculate nearly all of the CJ states the curve p reaches its nal height and width when nSn tO5 However its shape still slowly evolves as rLSn0t increases above 05 the lowlt0 part of the curve increases while the highlt0 side de creases This implies that the rare CJ states are not uniformly distributed in lt0 but are more likely to occur at low packing fractions below the peak in p Similar results for p as functions of rLSn0t are found for N 12 and 14 By compar 061306 5 XU BLAWZDZIEWICZ AND O HERN FIG 8 Density of collectively jammed packing fractions p for N 8 solid 10 dotted 12 dot dashed and 14 long dashed at msniotmax ing p at xed nSng we also nd that p narrows with increasing N To further demonstrate that p narrows the density of states is plotted in Fig 8 for several system sizes at rLSng max listed in Table I The dependence of the frequency distribution 8 on the system size N and the fraction rLSn0t of CI states obtained is illustrated in Fig 9 The results show that in contrast to the functions P and p the distribution 8 achieves its maximal value at the highest packing fraction for which CI states exist lm By comparing 8 for different system sizes at xed rLSn0t we nd that max increases with increas ing N The frequency distribution 8 becomes more strongly peaked at lm as rLSn0t increases The evolution of 8 with rLSn0t can be explained by noting that 8 EP p and that P does not depend on rLSn0t for 20 a 15 60 b 20 200 150 3 E100 07 074 FIG 9 Frequency distribution 8 for a N 10 b 12 and c 14 for nSn t02 solid lines 04 dotted lines 06 dot dashed lines 08 long dashed lines and 10 dashed lines PHYSICAL REVIEW E 71 061306 2005 log 84 10910 Bn Iogm Mb FIG 10 Frequency distribution 8n normalized by the peak value for a N210 b N212 and c N 14 at nSn t02 solid lines 04 dotted lines 06 dot dashed lines 08 long dashed lines and 10 dashed lines Least squares ts to exponential curves thin solid lines are also shown for the largest nSn0t at each N nSngotBODS according to the results shown in Fig 6 The density of states p and the frequency distribution 8 must therefore behave in opposite ways to maintain constant P As shown earlier in Fig 7 the peak in p widens for nSn tlt05 and shifts to lower packing fractions as rLSn0t increases Thus the distribution 8 must decrease at low packing fractions and build up at large packing frac tions with increasing nSngot In Fig 10 we show the frequency distribution 8 8 8max which is normalized by the peak value Bmax The results are plotted on a logarithmic scale The frequency distribution varies strongly with CJ states with small packing fractions are rare and those with large packing frac tions 083 occur frequently We nd that 8 is ex ponential over an expanding range of as rLSn0t increases For N 10 8 increases exponentially over nearly the en tire range of at nSn t1 We see similar behavior for N 12 and 14 in panels b and c of Fig 10 thus we expect 8 to be exponential as NSNEOt for N gt 10 We have calculated leastsquares ts to 8 A3 Xp33 9 for the largest rLSn0t at each system size As pointed out above the frequency distribution becomes steeper with in creasing N we nd that B 3 increases by a factor of 35 as N increases from 10 to 18 not shown Note that reasonable estimates of B 3 can be obtained even at fairly low values of nSng We showed in Fig 10 that the frequency distribution is not uniform in in contrast it increases exponentially with A Figure 11 shows another striking result the frequency distribution is also highly nonuniform within a narrow range of A In this gure we plot the cumulative distribution F h of 061306 6 RANDOM CLOSE PACKING REVISITED WAYS TO 4 I l 1 n 0 02 04 06 08 1 I 1 ins 6 1 L 1 I 0 02 04 06 08 1 in S FIG 11 Cumulative probability distribution F h of C states in a narrow range of packing fractions dq for N 12 The index 139 de notes the position of the state in a list ordered by the frequency of occurrence and as is the total number of states in the given interval The solid lines correspond to bins centered on 2073 075 077 and 07939 the dashed lines labeled 1 2 3 and 4 correspond to bins centered on 2065 081 083 and 069 respectively The width of each bin is A 002 The inset shows that the data are well described by Eq 10 where AF24 and a varies from 03 to 04 the probabilities of jammed states in a narrow interval d versus the index i in a list of all distinct states in d ordered by the value of the probability of each state The data for several different intervals appear to collapse onto a stretched exponential form Fhexp AF1 ins 10 where as is the number of distinct CJ states within d and the exponent a varies from 03 to 04 These results clearly demonstrate that C states can occur with very different fre quencies even if they have similar packing fractions From our studies of small systems we nd that both the density p of CI packing fractions and the frequency dis tribution 8 narrow and shift to larger packing fractions as the system size increases See Figs 7 and 10 How do these changes in p and 8 affect the total distribution P and can we determine which changes dominate in the large system limit To shed some light on these questions we consider the position of the peak in P with respect to the maximal packing fraction of CI states lm for several sys tem sizes In the absence of changes in p as a function of lm the maximum of P should shift toward max with increasing system size because the frequency distribution 8 becomes more sharply peaked at max accord ing to the results in Fig 10 However as shown in Fig 12 we nd the opposite behavior over the range of system sizes we considered the peak of P shifts away from lm This suggests that the density of states not the frequency distri bution plays a larger role in determining the location of the peak in P in these systems Additional conclusions about the relative roles of the the density of states and the frequency distribution on the posi tion and width of P can be drawn from our observation that the frequency distribution 8 is an exponential function PHYSICAL REVIEW E 71 061306 2005 PW 1349quot N U FIG 12 P thick lines and Bn thin lines for N10 solid line 12 dotted line and N 14 dot dashed line at nsngotmax where A PltS and Bn for N 10 12 and 14 have been shifted by A 0013 0005 and 0 respectively so that Bmax for the three system sizes coincide 8n for each N has also been ampli ed by a factor of z20 of cf the discussion of results in Fig 10 and that P is Gaussian for suf ciently large systems as shown in 16 and illustrated in Fig 13 If we assume that the exponential form of the frequency distribution 9 remains valid in the large system limit the density of states p P 8 is also Gaussian with the identical width 0N The location of the peak in P is 12ltNgt ltNgt B3N02N 11 where N is the location of the peak in p In previous studies 16 we found that the width of P scaled as a N Q with 0055 We have also some indication that 30 i AZO 10 o 0 I I I 40 39 39 150 d 39 39 39 100 gt 078 08 085 084 086 FIG 13 The distribution P of C states for N a 18 b 32 c 64 and d 256 are depicted using circles The solid lines are least squares ts of the large QS side of PltS to Gaussian distributions 061306 7 XU BLAWZDZIEWICZ AND O HERN BBO39Z decreases with increasing system size 350220017 at N212 compared to 0012 at N218 However we are not currently able to estimate BB in the largeN limit If the systemsize dependence of B B is weaker than N2 the quantity B 502 will tend to zero and the frequency distri bution will not in uence the location of the peak in Plt1 In this case Plt1 becomes independent of the frequency distri bution in the limit New for a class of protocols that are characterized by a similar frequency distribution 3 as our present protocol Thus as our preliminary results suggest random close packing can be de ned as the location of the peak in plt1 when N ace and this de nition is completely independent of on the algorithm used to generate the C states In the opposite case where the systemsize depen dence of BB is stronger than N2 the position of the peak in Plt1 results from a subtle interplay between the density of states and the frequency distribution However even in this case one can argue that the dependence of the position of the peak only weakly depends on the protocol a shift of the peak position requires an exponential change in the frequency dis tribution 31 V CONCLUSIONS We have studied the possible collectively jammed con gurations that occur in small 2 periodic systems com posed of smooth purely repulsive bidisperse disks The C states were created by successively compressing or expand ing soft particles and minimizing the total energy at each step until the particles were just at contact By studying small 2D systems we were able to enumerate nearly all of the collec tively jammed states at each system size and therefore de compose the probability distribution P lt1 for obtaining a C state at a particular packing fraction lt1 into the density plt1 of C packing fractions and their frequency distribution 31 The distribution 31 depends on the particular proto col used to generate the C con gurations while plt1 does not This decomposition allowed us to study how the protocolindependent plt1 and protocoldependent 31 in uence the shape of Plt1 These studies yielded many important and novel results First the probability distribution Plt1 of C states is nearly independent of 113112 and thus it can be measured reliably even in large systems This nding validates several previous measurements of Plt1 1619 Second the number of dis tinct C states grows exponentially with system size In ad dition a large fraction of these con gurations are extremely rare and thus an exponentially large number of trials are required to nd all of the C states Third the frequency distribution 31 is nonuniform and increases exponentially with lt1 We also found that even over a narrow range of lt1 the frequency with which particular C states occur is strongly nonuniform and involves a large number of expo nentially rare states Finally we have shown that Plt1 be comes Gaussian in the largeN limit Since Plt1 plt1Blt1 and 31 is exponential we expect that plt1 is also Gaussian and controls the width of Plt1 for large N We also have preliminary resulm that suggest that the contribu PHYSICAL REVIEW E 71 061306 2005 tion from 31 to the shift of the peak in Plt1 decreases with increasing N We expect that plt1 will determine the location of the peak in Plt1 in the largeN limit1 and thus it is a robust protocolindependent de nition of random close packing in this system VI FUTURE DIRECTIONS Several interesting questions have arisen from this work that will be addressed in our future studies First we have shown that the frequency with which C states occur is highly nonuniform It is important to ask whether the rare states can be neglected in analyses of static and dynamic properties of jammed and nearly jammed systems For ex ample we have shown that Plt1 is insensitive to the fraction Itsn0t of C states obtained and thus Plt1 is not in uenced by the rare C states However rare C states may be impor tant in determining the dynamical properties of jammed and glassy systems if these states are associated with passages or channels from one frequently occurring state to another Moreover an analysis of the density of states and the fre quency distribution both as a function of lt1 and locally in lt1 may shed light on the phasespace evolution of glassy sys tems during the aging process A closely related question is what topological or geo metrical features of con gurational phase space give rise to the exponentially varying frequency distribution Can one for example uniquely assign a speci c volume in con gura tional space to each jammed state A candidate for such a quantity is the volume QC of con guration space in which each point is connected by a continuous path without particle overlap to a particular C state It is likely that those C states with large QC will occur frequently for a typical com paction algorithm while those with small QC will be rare Another important question is whether the results for plt1 31 and Plt1 found in 2D bidisperse systems also hold for other systems such as monodisperse systems in 2D and 3D Does plt1 still control the behavior of Plt1 or does the frequency distribution play a more dominant role in de termining Plt1 To begin to address these questions we have enumerated nearly all of the distinct collectively jammed states and calculated Plt1 plt1 and 31 in small 2D periodic cells containing N 24732 equalsized particles In our preliminary studies we have found several signi cant differences between 2D monodisperse and bidisperse systems which largely stem from the fact that partially or dered states occur frequently in the monodisperse systems First in 2D monodisperse systems there is an abundance of distinct C states that exist at the same packing fraction For example in a monodisperse systems with N224 multiple distinct states occur at 19 of the C packing fractions com pared to less than 1 in bidisperse systems with N 14 Sec ond for the system sizes studied quantitative features of the distributions of C states depend on whether N is even or odd Third Plt1 can possess two strong peaks For example two peaks in Plt1 occur at lt110805 and lt120844 for N 24 as shown in Fig 14 Moreover the largelt1 peak that corresponds to partially ordered con gurations is a factor of 0613068 RANDOM CLOSE PACKING REVISITED WAYS TO 40 30 O 07 FIG 14 PltS solid line p dotted line and 8 dot dashed line for a 2D monodisperse system with N 24 3 taller than the smallt peak that corresponds to amorphous con gurations Finally the maximum in 8 coincides with the largelt0 peak in P and 8 decays very rapidly as decreases As shown in Fig 14 the rapid decay of 8 signi cantly suppresses the contribution of the peak in p t0 the total distribution P Thus 8 which depends on PHYSICAL REVIEW E 71 061306 2005 the protocol used to generate the CJ states may strongly in uence the total distribution P even in moderately sized 2D monodisperse systems Many open questions concerning monodisperse systems in 2D will be answered in a forthcoming article 27 We will measure the shape of P as a function of system size and predict whether p or 8 controls the width and location of the peak or peaks in the largeN limit The fact that 8 strongly in uences P at small and moderate system sizes explains why it has been so dif cult to determine random close packing in 2D monodisperse systems 18 different protocols have yielded different values for prep 2028 ACKNOWLEDGMENTS Financial support from NSF Grant Nos CTS0348175 J B and DMR0448838 NXCSO is gratefully ac knowledged We also thank Yale s High Performance Com puting Center for generous amounts of computer time 1 Jamming and Rheology edited by A J Liu and S R Nagel Taylor amp Francis New York 2001 2 J D Bernal Nature London 188 910 1960 3 G D Scott Nature London 188 908 1960 4 J B Knight C G Fandrich C N Lau H M Jaeger and S R Nagel Phys Rev E 51 3957 1995 5 P Phillippe and D Bideau Europhys Lett 60 677 2002 6 R P A Dullens and W K Kegel Phys Rev Lett 92 195702 2004 7 J X Zhu M Li R Rogers W Meyer R H Ottewill W B Russell and P M Chaikin Nature London 387 883 1997 8 J L Finney Proc R Soc London Ser A 319 495 1970 9 B D Lubachevsky F H 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Press New York 1986 25 A Tanguy J P Wittmer F Leonforte and J L Barrat Phys ReV B 66 174205 2002 26 A Donev S Torquato and F H Stillinger Phys Rev E 71 011105 2005 27 N Xu J Blawzdziewicz and C S O Hern unpublished 28 A Donev S Torquato F H Stillinger and R Connelly J Appl Phys 95 989 2004 061306 9


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