Role of Jammed States & Frequency Distributions in Granular Packing, Study notes of Physics

Download Role of Jammed States & Frequency Distributions in Granular Packing and more Study notes Physics in PDF only on Docsity! Random close packing revisited: Ways to pack frictionless disks Ning Xu,1 Jerzy Blawzdziewicz,1 and Corey S. O’Hern1,2 1Department of Mechanical Engineering, Yale University, New Haven, Connecticut 06520-8284, USA 2Department of Physics, Yale University, New Haven, Connecticut 06520-8120, USA
Received 20 March 2005; published 28 June 2005; publisher error corrected 6 July 2005 We create collectively jammed
CJ packings of 50-50 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 find 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
into two composite functions:
1 the density of CJ packing fractions 

, which only depends on geometry, and
2 the frequency distri- bution 

, which depends on the particular algorithm used to create them. We find that the function 

 is sharply peaked and that 

 depends exponentially on
. We predict that in the infinite-system-size limit the behavior of P

 in these systems is controlled by the density of CJ packing fractions—not the frequency distribution. These results suggest that the location of the peak in P

 when N→ can be used as a protocol-independent definition of random close packing. DOI: 10.1103/PhysRevE.71.061306 PACS number
s: 81.05.Rm, 82.70.y, 83.80.Fg I. INTRODUCTION Developing a statistical-mechanical description of dense granular materials, structural and colloidal glasses, and other jammed systems 1 composed of discrete macroscopic grains is a difficult, long-standing problem. These amor- phous systems possess an enormously large number of pos- sible jammed configurations; however, it is not known with what probabilities these configurations occur since these sys- tems are not in thermal equilibrium. The possible jammed configurations do not occur with equal probability—in fact, some are extremely rare and others are highly probable. Moreover, the likelihood that a given jammed configuration occurs depends on the protocol that was used to generate it. Despite difficult theoretical challenges, there have been a number of experimental and computational studies that have investigated jammed configurations in a variety of systems. The experiments include studies of static packings of ball bearings 2,3, slowly shaken granular materials 4,5, 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 10–12, various geometrical algorithms 13–15, 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
rcp, which is approximately 0.64 in three-dimensional
3D monodisperse systems 18 and 0.84 in the 2D bidisperse systems discussed in this work 15,19. 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 configura- tions from a distribution of jammed states with varying de- grees of positional and orientational order 20. Recent studies of hard-particle systems have also shown that different classes of jammed states exist with different properties 21. For example, in locally jammed
LJ states, each particle is unable to move provided all other particles are held fixed; however, groups of particles can still move collectively. In contrast, in collectively jammed
CJ states neither single particles nor groups of particles are free to move
excluding “floater” particles that do not have any con- tacts. Thus, CJ states are more “jammed” than LJ states. In this article we focus exclusively on the properties of collectively jammed states. These states are created using an energy minimization procedure 16,19 for systems com- posed of particles that interact via soft, finite-range, purely repulsive, and spherically symmetric potentials. Energy minimization is combined with successive compressions and decompressions of the system to find 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 hard-particle system. In previous studies of collectively jammed states created using the energy-minimization method, we showed that the probability distribution of collectively jammed packing frac- tions narrows as the system size increases and becomes a  function located at
0 in the infinite-system-size limit 16,19. We found that
0 was similar to values quoted pre- viously for random close packing 18. The narrowing of the distribution of CJ packing fractions as the system size in- creases is shown in Fig. 1 for 2D bidisperse systems. How- ever, it is still not clear why this happens. Why is it so dif- ficult to obtain a collectively jammed state with


0 in the large-system limit? One possibility is that very few collec- tively jammed states exist with


0. Another possibility is that collectively jammed states do exist over a range of pack- ing fractions, but only those with packing fractions near
0 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 PHYSICAL REVIEW E 71, 061306
2005 1539-3755/2005/71
6/061306
9/$23.00 ©2005 The American Physical Society061306-1 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

 = 



 .
1 The factor 

 in the above equation represents the density of collectively jammed states i.e., 

d
measures how many distinct collectively jammed states exist within in a small range of packing fractions d
. The factor  denotes the effective frequency
i.e., the counts averaged over a small region of
 with which these states occur. We note that the density of states 

 is determined solely by the topological features of configurational space; it is thus independent of the the protocol used to generate these states. In contrast, the quantity 

 is protocol dependent, because it records the average frequency with which a CJ 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, 

 or 

, controls the shape of the prob- ability distribution P

 in the large-system limit. Others have studied the inherent structures of hard-sphere liquids and glasses, but have not addressed this specific question 22,23. We will show below that 

 controls the width of the distribution of CJ states in the infinite system-size limit. We also have some evidence that the location of the peak in P

 in the large-N limit is also determined by the large-N behavior of 

. We will also argue that for many proce- dures the protocol dependence of the frequency distribution 

 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 protocol-independent definition of random close packing in the infinite-system-size limit. II. METHODS Our goal is to enumerate the collectively jammed configu- 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 =1.4 because it has been shown that these systems do not easily crystallize or phase separate 15,16. We consider sys- tem sizes in the range N=4–256 particles. For N10, we were able to find nearly all of the collectively jammed states. For N=12
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 CJ states for N 14, but as we will show below, we can still make strong conclusions about the shape of the distribution of CJ states in large systems. We utilize an energy-minimization procedure to create collectively jammed states 16. We assume that the particles interact via the purely repulsive linear spring potential V
rij = 2
1 − rij/dij2
dij/rij − 1 ,
2 where is the characteristic energy scale, rij is the separation of particles i and j, dij =
di+dj /2 is their average diameter, and
x is the Heaviside step function. The potential
2 is nonzero only for rij dij—i.e., 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 floaters 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 V
rij. Note that our process for creating jammed states differs from the fixed-volume energy-minimization procedure implemented in Ref. 16. In the description be- low, the energies and lengths are measured in units of and the diameter of the smaller particle d1. For each independent trial, the procedure begins by choosing a random configuration of N particles at an initial packing fraction
i 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
i as long it is significantly below the peak in 

. We chose
i=0.60 for most system sizes. After initializing the systems, we find the nearest local potential energy minimum using the conjugate gradient algo- rithm 24. We terminate the energy-minimization procedure when either of the following two conditions is satisfied:
1 two successive conjugate gradient steps n and n+1 yield nearly the same total potential energy per particle,
Vn+1 −Vn /Vn =10−16, or
2 the total potential energy per par- ticle is extremely small, Vn+1 Vmin=10 −16. Following the potential energy minimization, we decide whether the system should be compressed or expanded to find the jamming threshold. If Vn+1 Vmax=2 10 −16, par- ticles have nonzero overlap and thus small and large particles are reduced in size by d1=d1
/
2
 and d2=d1, re- spectively. If, on the other hand, Vn+1Vmin, the system is FIG. 1. The probability distribution P

 to obtain a collectively jammed state at packing fraction
in 2D bidisperse systems with N=18
dotted line, 32
dashed line, 64
dot-dashed line, and 256
solid line. XU, BLAWZDZIEWICZ, AND O’HERN PHYSICAL REVIEW E 71, 061306
2005 061306-2 The density of CJ states is evaluated using an analogous relation 

d
= ns

+ d
 − ns

 ns ,
8 where ns

 is the number of distinct CJ states that have been detected in the packing-fraction range below
. In fact, we have used the number of distinct packing fractions to define 

 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 CJ states
8 are normalized to 1. The frequency distribution 

= P

 /

 is normalized accordingly. Below, we show how P

 , 

, and 

 depend on the fraction of CJ states ns /ns tot and system size N. To plot these distributions, we used ten bins with the endpoint of the final bin located at the largest CJ packing fraction
max for each N. We recall that the distribution of CJ packing frac- tions 

 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 

. The probability distribution P

 of CJ 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 ns /ns tot of CJ 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 CJ states can be measured reliably even in large systems 16,19. 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 significant changes in P

, the system size must be varied over a larger range. P

 for N=18, 32, 64, and 256 is shown in Fig. 1 at fixed number of trials nt=10 4. 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  function located at
0=0.842 in the infinite-system-size limit. What causes P

 to narrow to a  function located at
0 when N→? Is the shape of the distribution P

 deter- mined primarily by the density of states 

, or does the frequency distribution 

 play a significant role in deter- mining the width and location of the peak? We will shed light on these questions below. We first show results for 

 and 

 as functions of the fraction ns /ns tot of distinct CJ states obtained. In Fig. 7, 

 is shown for several small systems. In contrast to the total distribution P

, the density of states 

 depends on ns /ns tot significantly. For N=10, a system for which we can calculate nearly all of the CJ states, the curve 

 reaches its final height and width when ns /ns tot 0.5. However, its shape still slowly evolves as ns /ns tot increases above 0.5; the low-
part of the curve increases while the high-
side de- creases. This implies that the rare CJ states are not uniformly distributed in
, but are more likely to occur at low packing fractions below the peak in 

. Similar results for 

 as functions of ns /ns tot are found for N=12 and 14. By compar- TABLE I. Maximum number of trials performed, nt max, and frac- tion of CJ states obtained
ns /ns totmax versus system size N. N nt max
ns /ns totmax 6 106 1.0 8 106 1.0 10 29 106 1.0 12 28 106 0.90 14 26 106 0.60 FIG. 6. Probability distribution P

 for obtaining a CJ state at
for N=10
solid line and N=14
dotted line at
ns /ns totmax. The distributions at ns /ns tot=0.05
squares for N=10 and triangles for N=14 overlap those with larger ns /ns tot. FIG. 7. Density of collectively jammed packing fractions 

 for
a N=10,
b 12, and
c 14 at ns /ns tot=0.2
solid lines, 0.4
dotted lines, 0.6
dot-dashed lines, 0.8
long-dashed lines, and 1.0
dashed lines. RANDOM CLOSE PACKING REVISITED: WAYS TO … PHYSICAL REVIEW E 71, 061306
2005 061306-5 ing 

 at fixed ns /ns tot, we also find that 

 narrows with increasing N. To further demonstrate that 

 narrows, the density of states is plotted in Fig. 8 for several system sizes at
ns /ns totmax listed in Table I. The dependence of the frequency distribution 

 on the system size N and the fraction ns /ns tot of CJ states obtained is illustrated in Fig. 9. The results show that in contrast to the functions P

 and 

, the distribution 

 achieves its maximal value at the highest packing fraction for which CJ states exist,
max. By comparing 

 for different system sizes at fixed ns /ns tot we find that
max increases with increas- ing N. The frequency distribution 

 becomes more strongly peaked at
max as ns /ns tot increases. The evolution of 

 with ns /ns tot can be explained by noting that 

  P

 /

 and that P

 does not depend on ns /ns tot for ns /ns tot0.05 according to the results shown in Fig. 6. The density of states 

 and the frequency distribution 

 must therefore behave in opposite ways to maintain constant P

. As shown earlier in Fig. 7, the peak in 

 widens
for ns /ns tot 0.5 and shifts to lower packing fractions as ns /ns tot increases. Thus, the distribution 

 must decrease at low packing fractions and build up at large packing frac- tions with increasing ns /ns tot. In Fig. 10, we show the frequency distribution n

 =

 /max, which is normalized by the peak value max. 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

0.83 occur frequently. We find that n

 is ex- ponential over an expanding range of
as ns /ns tot increases. For N=10, n

 increases exponentially over nearly the en- tire range of
at ns /ns tot=1. We see similar behavior for N =12 and 14 in panels
b and
c of Fig. 10; thus we expect n

 to be exponential as Ns /Ns tot→1 for N 10. We have calculated least-squares fits to n = A exp
B

9 for the largest ns /ns tot at each system size. As pointed out above, the frequency distribution becomes steeper with in- creasing N; we find that B increases by a factor of 3.5 as N increases from 10 to 18
not shown. Note that reasonable estimates of B can be obtained even at fairly low values of ns /ns tot. We showed in Fig. 10 that the frequency distribution is not uniform in
; in contrast, it increases exponentially with
. Figure 11 shows another striking result; the frequency distribution is also highly nonuniform within a narrow range of
. In this figure, we plot the cumulative distribution Fh of FIG. 8. Density of collectively jammed packing fractions 

 for N=8
solid, 10
dotted, 12
dot-dashed, and 14
long-dashed at
ns /ns totmax. FIG. 9. Frequency distribution 

 for
a N=10,
b 12, and
c 14 for ns /ns tot=0.2
solid lines, 0.4
dotted lines, 0.6
dot- dashed lines, 0.8
long-dashed lines, and 1.0
dashed lines. FIG. 10. Frequency distribution n

 normalized by the peak value for
a N=10,
b N=12, and
c N=14 at ns /ns tot=0.2
solid lines, 0.4
dotted lines, 0.6
dot-dashed lines, 0.8
long-dashed lines, and 1.0
dashed lines. Least-squares fits to exponential curves
thin solid lines are also shown for the largest ns /ns tot at each N. XU, BLAWZDZIEWICZ, AND O’HERN PHYSICAL REVIEW E 71, 061306
2005 061306-6 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 Fh = exp− AF
1 − i/ns ,
10 where ns is the number of distinct CJ states within d
and the exponent  varies from 0.3 to 0.4. These results clearly demonstrate that CJ states can occur with very different fre- quencies even if they have similar packing fractions. From our studies of small systems, we find that both the density 

 of CJ packing fractions and the frequency dis- tribution 

 narrow and shift to larger packing fractions as the system size increases.
See Figs. 7 and 10. How do these changes in 

 and 

 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 CJ states
max for several sys- tem sizes. In the absence of changes in 

 as a function of
−
max, the maximum of P

 should shift toward
=
max with increasing system size, because the frequency distribution  becomes more sharply peaked at
max accord- ing to the results in Fig. 10. However, as shown in Fig. 12, we find the opposite behavior over the range of system sizes we considered: the peak of P

 shifts away from
max. 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  is an exponential function of

cf. the discussion of results in Fig. 10 and that P

 is Gaussian for sufficiently 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

 /

 is also Gaussian with the identical width 
N. The location of the peak in P

 is
P *
N =
 *
N + B
N2
N ,
11 where
 *
N is the location of the peak in 

. In previous studies 16, we found that the width of P

 scaled as  N−, with  0.55. We have also some indication that FIG. 11. Cumulative probability distribution Fh of CJ states in a narrow range of packing fractions d
for N=12. The index i de- notes the position of the state in a list ordered by the frequency of occurrence and ns is the total number of states in the given interval. The solid lines correspond to bins centered on
=0.73, 0.75, 0.77, and 0.79; the dashed lines labeled 1, 2, 3, and 4 correspond to bins centered on
=0.65, 0.81, 0.83, and 0.69, respectively. The width of each bin is 
=0.02. The inset shows that the data are well described by Eq.
10, where AF 2.4 and  varies from 0.3 to 0.4. FIG. 12. P



thick lines and n



thin lines for N=10
solid line, 12
dotted line, and N=14
dot-dashed line at
ns /ns totmax, where

=
+
. P

 and n

 for N=10, 12, and 14 have been shifted by 
=0.013, 0.005, and 0 respectively, so that max for the three system sizes coincide. n

 for each N has also been amplified by a factor of 20. FIG. 13. The distribution P

 of CJ states for N
a 18,
b 32,
c 64, and
d 256 are depicted using circles. The solid lines are least-squares fits of the large-
side of P

 to Gaussian distributions. RANDOM CLOSE PACKING REVISITED: WAYS TO … PHYSICAL REVIEW E 71, 061306
2005 061306-7