Is Random Clo Packing of Spheres Well Defined?
S.Torquato,1,*T.M.Truskett,2and P.G.Debenedetti2
1Department of Chemistry and Princeton Materials Institute,Princeton University,Princeton,New Jery08544 2Department of Chemical Engineering,Princeton University,Princeton,New Jery08544
(Received1November1999)
Despite its long history,there are many fundamental issues concerning random packings of spheres that remain elusive,including a preci definition of random clo packing(RCP).We argue that the current picture of RCP cannot be made mathematically preci and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm.We suggest that this impas can be broken by introducing the new concept of a maximally random jammed state, which can be made preci.
PACS numbers:05.20.Jj,61.20.–p
Random packings of identical spheres have been stud-
ied by biologists,materials scientists,engineers,chemists,
付款申请单and physicists to understand the structure of living cells,
武术大全liquids,granular media,glass,and amorphous solids,to
mention but a few examples.The prevailing notion of ran-
方兴未艾什么意思dom clo packing(RCP)is that it is the maximum den-
sity that a large,random collection of spheres can attain
and that this density is a universal quantity.This tradi-
tional view can be summarized as follows:“ball bearings
and similar objects have been shaken,ttled in oil,stuck
with paint,kneaded inside rubber balloons—and all with
no better result than(a packing fraction of)...0.636”[1].
One aim of this paper is to reasss this commonly held
view.First,we obrve that there exists ample evidence
in the literature(in the form of actual and computer ex-
periments)to suggest strongly that the RCP state is ill
defined and,unfortunately,dependent on the protocol em-
ployed to produce the random packing as well as other
system characteristics.In a classic experiment,Scott and
Kilgour[2]obtained the RCP value f cഠ0.637by pour-ing ball bearings into a large container,vertically vibrating
the system for sufficiently long times to achieve maximum
densification,and extrapolating the measured volume frac-
tions to eliminate finite-size effects.Important dynami-
cal parameters for this experiment include the pouring rate
and both the amplitude and frequency of vibration.The
key interactions are interparticle forces,including(ide-
ally)repulsive hard-sphere interactions,friction between
the particles(which inhibits densification),and gravity.It
is clear that the final volume fraction can depend nsi-
tively on the system characteristics.Indeed,in a re-
cent experimental study[3],it was shown that one can
achieve denr(partially crystalline)packings when the
particles are poured at low rates into horizontally shaken
containers.
Computer algorithms can be ud to generate and
study idealized random packings,but the final states
are clearly protocol dependent.For example,a popular
rate-dependent densification algorithm[4]achieves f c
between0.642and0.649,a Monte Carlo scheme[5]gives
f cഠ0.68,and a“drop and roll”algorithm[6]yields f cഠ0.60.It is noteworthy that,in contrast to the last algorithm,the first two algorithms produce configurations in which either the majority or all of the particles are not in contact with one another.We are not aware of any algorithms that truly account for friction between the spheres.
However,we suggest that the aforementioned incon-sistencies and deficiencies of RCP ari becau it is an ill-defined state,explaining why,to this day,there is no theoretical determination of the RCP density.This is to be contrasted with the rigor that has been ud very re-cently to prove that the denst possible packing fraction f for identical spheres is p͞
p
18ഠ0.7405,corresponding to the clo-packed face-centered cubic(fcc)lattice or its stacking variants[7].
The term“clo packed”implies that the spheres are in contact with one another with the highest possible co-ordination number on average.This is consistent with the view that RCP is the highest possible density that a random packing of clo-packed spheres can posss.However,the terms“random”and“clo packed”are at odds with one another.Increasing the degree of coordination,and thus, the bulk system density,comes at the expen of disorder. The preci proportion of each of the competing effects is arbitrary and therein lies the problem.In what follows, we supply quantitative evidence of the ill-defined nature of RCP via computer simulations,and we propo a new notion,that of a maximally random jammed state.
A preci mathematical definition that supplants the RCP state should apply to any statistically homogeneous and isotropic system of identical spheres(with specified interactions)in any space dimension d.Although we discard the term“clo packed,”we must retain the idea that the particles are in contact with one another,while maintaining the greatest generality.We say that a particle (or a t of contacting particles)is jammed if it cannot be translated while fixing the positions of all of the other particles in the system.The system itlf is jammed if each particle(and each t of contacting
particles)is jammed [8].This definition eliminates systems with“rattlers”(freely roaming caged particles)in the infinite-volume
20640031-9007͞00͞84(10)͞2064(4)$15.00©2000The American Physical Society
limit.We recognize that jammed structures created via
公公再爱我一次computer algorithms[9]or actual experiments will contain
a very small concentration of such rattler particles,the
preci concentration of which is protocol dependent.
Thus,in practice,one may wish to accommodate this type
of a jammed structure,although the ideal limit described
北京限行规定above is the preci mathematical definition of a jammed
state that we have in mind.Nevertheless,it should
be emphasized that it is the overwhelming majority of音乐的的英文
spheres that compo the underlying“jammed”network
that confers rigidity to the particle packing.
Our definition of the maximally random jammed(MRJ)
state is bad on the minimization of an order parame-
ter described below.The most challenging problem is
quantifying randomness or its antithesis:order.A many-
particle system is completely characterized statistically by
the N-body probability density function P͑r N͒associated with finding the system with configuration r N.Such com-
plete information is never available and,in practice,one
must ttle for reduced information.From this reduced in-
formation,one can extract a t of scalar order parameters
c1,c2,...,c n,such that0#c i#1,;i,where0cor-
responds to the abnce of order(maximum disorder)and
1corresponds to maximum order(abnce of disorder).
The t of order parameters that one lects is unavoidably
subjective,given that there is no single and complete scalar
measure of order in the system.
However,within the necessary limitations,there is
a systematic way to choo the best order parameters to
be ud in the objective function(the quantity to be mini-
mized).The most general objective function consists
of weighted combinations of order parameters.The t
of all jammed states will define a certain region in the
n-dimensional space of order parameters.In this region of jammed structures,the order parameters can be divided up into two categories:tho that share a common minimum and tho that do not.The strategy is clear:retain tho order parameters that share a common minimum and discard tho that do not since they are conflicting measures of order.Moreover,since all of the parameters sharing a common minimum are esntially equivalent measures of order(there exists a jammed state in which all order parameters are minimized),choo from among the the one that is the most nsitive measure,which we will simply denote by c.From a practical point of view, two order parameters that are positively correlated will share a common minimum.
Consider all possible configurations of a d-dimensional
system of identical spheres,with specified interactions,at a
sphere volume fraction f in the infinite-volume limit.For
every f,there will be a minimum and maximum value of
the order parameter c.By varying f between zero and
its maximum value(triangular lattice for d2and fcc
lattice for d3),the locus of such extrema define upper
and lower bounds within which all structures of identical
spheres must lie.Figure1shows a schematic(not quanti-tative)plot of the order parameter versus volume fraction. Note that at f0the most disordered(c0)configu-rations of sphere centers can be realized.As the packing fraction is incread,the hard-core interaction prevents ac-cess to the most random configurations of sphere centers (gray region).Thus the lower boundary of c,reprenting the most disordered configurations,increas monotoni-cally with f.The upper boundary of c corresponds to the most ordered structures at each ,perfect open lat-tice structures(c1).Of cour,the details of the lower boundary will depend on the particular choice of c.Nev-ertheless,the salient features of this plot are as follows: (i)all sphere structures must lie within the bounds and (ii)the jammed structures are a special subt of the al-lowable structures[10].We define the MRJ state to be the one that minimizes c among all statistically homogeneous and isotropic jammed structures.
To support the aforementioned arguments,we have car-ried out molecular dynamics simulations using systems of 500identical hard spheres with periodic boundary con-ditions.Starting from an equil
ibrium liquid configura-tion at a volume fraction of f0.3,we compresd the system to a jammed state by the well-known method of Lubachevsky and Stillinger[9]which allows the diame-ter of the particles to grow linearly in time with a dimen-sionless rate G.Figure2a shows that the volume fraction of the final jammed states is inverly proportional to the compression rate G.A linear extrapolation of the data to the infinite compression rate limit yields fഠ0.64,
which
FIG.1.A schematic plot of the order parameter c versus volume fraction f for a system of identical spheres with pre-scribed interactions.All structures at a given value of f must lie between the upper and lower bounds(white region);gray region is inaccessible.The boundary containing the subt of jammed structures is shown.The jammed structures are shown to be one connected t,although,in general,they may exist as multiply disconnected.Point A reprents the jammed struc-ture with the lowest density and point B reprents the denst ordered jammed ,clo-packed fcc or hexagonal lattice for d3,depending on the choice for c).The jammed structure which minimizes the order parameter c is the maxi-mally random jammed state.
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φ
0.0
0.5
1.0
1.5
2.0
(Γ x 103)−1
Q
0.00.2
0.4
0.6
0.8
1.0
T
FIG.2.Molecular dynamics simulation results for the hard-sphere system.(a)The reciprocal compression rate G 21ver-sus the volume fraction f of the final jammed state of hard spheres using the molecular dynamics compression protocol of Lubachevsky and Stillinger [9].The jammed state occurs when the diameters can no longer increa in time,the sphere colli-sion rate diverges,and no further compression can be achieved after relaxing the configuration at the jammed volume fraction.Each point reprents the average of 27compressions,and the dashed line is a linear fit to th
e data,which yields f ഠ0.64when G 210.(b)The Q -T plane for the hard-sphere system,where T and Q are translational and orientational order parame-ters,respectively.Shown are the average values for the jammed states of (a)(circles),as well as states along the equilibrium liq-uid (dotted line)and crystal (dashed line)branches.
is clo to the suppod RCP value reported by Scott and Kilgour.
To quantify the order (disorder)in our jammed struc-tures,we have chon to examine two important measures of order:bond-orientational order and translational order [11].The first is obtainable in part from the parameter Q 6and the cond is obtainable in part from the radial distri-bution function g ͑r ͒(e.g.,from a scattering experiment).To each nearest-neighbor bond emanating from a sphere,one can associate the spherical harmonics Y lm ͑u ,w ͒,using the bond angles as arguments.Then Q 6is defined by [12]
Q 6ϵ
√4p 136
X
m 26j Y 6m j 2!1͞2
,(1)where Y 6m denotes an average over all bonds.For a com-pletely disordered system in the infinite-volume limit,Q 6equals zero,whereas Q 6attains its maximum value for
space-filling structures (Q fcc
亭子简笔画
6
ഠ0.575)in the perfect fcc crystal.Thus,Q 6provides a measure of fcc crystallite for-mation in the system.For convenience we normalize the
orientational order parameter Q Q 6͞Q fcc
6by its value in the perfect fcc crystal.
Scalar measures of translational order have not been well studied.For our purpos,we introduce a translational or-der parameter T which measures the degree of spatial or-dering,relative to the perfect fcc lattice at the same volume fraction.Specifically,
T ÉP N C i 1͑n i 2n ideal
i
͒P N C i 1͑n fcc
i 2n ideal i ͒É
,(2)where n i (for the system of interest)indicates the average
occupation number for the shell of width a d centered at a distance from a reference sphere that equals the i th nearest-neighbor paration for the open fcc lattice at that den-sity,a is the first nearest-neighbor distance for that fcc lattice,and N C is the total number of shells (here we
choo d 0.196and N C 7).Similarly,n ideal i and n fcc
i are the corresponding shell occupation numbers for an ideal gas (spatially uncorrelated spheres)and the open fcc crystal lattice.Obrve that T 0for an ideal gas (perfect randomness)and T 1for perfect fcc spatial ordering.
The relationship between translational and bond-orientational ordering has heretofore not been character-ized.We have measured both T and Q for the jammed structures generated by the Lubach
evsky-Stillinger algo-rithm and have plotted the results in the Q -T plane in Fig.2b [13].This order plot reveals veral key points.First,we obrve that T and Q are positively correlated and therefore are esntially equivalent measures of order for the jammed structures.Therefore,in eking to determine the MRJ state using T and Q ,one would arch for jammed structures that minimize Q ,the more nsitive of the two measures.Our preliminary results indicate that the MRJ packing fraction f MRJ ഠ0.64for 500spheres using the Lubachevsky-Stillinger protocol.It should be noted,however,that a systematic study of other protocols may indeed find jammed states with a lower degree of order as measured by Q .Moreover,we notice that the degree of order increas monotonically with the jammed packing fraction [11].The results demonstrate that the notion of RCP as the highest possible density that a random sphere packing can attain is ill defined since one can achieve packings with arbitrarily small increas in volume fraction at the expen of small increas in order.For purpos of comparison,we have included in the or-der plot of Fig.2b results for the equilibrium hard-sphere system for densities along the liquid branch and densi-ties along the crystal branch,ending at the maximum
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clo-packed fcc state[14].Interestingly,the equilibrium structures exhibit the same monotonicity prop
erties as the jammed ,T increas with increasing Q and the degree of order increas with the packing fraction. Note that neither Q nor T are equal to unity along the equilibrium crystal branch becau of thermal motion. To summarize,we have shown that the notion of RCP is not well defined mathematically.To replace this idea, we have introduced a new concept:the maximally ran-dom jammed state,which can be defined precily once an order parameter c is chon.This lays the mathemati-cal groundwork for studying randomness in den pack-ings of spheres and initiates the arch for the MRJ state in a quantitative way not possible before.Nevertheless, significant challenges remain.First,new and efficient pro-tocols(both experimental and computational)that generate jammed states must be developed.Second,since the char-acterization of randomness is in its infancy,the systematic investigation of better order parameters is crucial.
We thank F.H.Stillinger,T.Spencer,J.H.Conway,and M.Utz for many valuable discussions.S.T.was supported by the Engineering Rearch Program of the Office of Basic Energy Sciences at the U.S.Department of Energy (DE-FG02-92ER14275)and the Guggenheim Foundation. He also thanks the Institute for Advanced Study in Prince-ton for the hospitality extended to him during his stay there. P.G.D.was supported by the Chemical Sciences Division of the Office of Basic Energy Sciences at the U.S.De-partment of Energy(DE-FG02-87ER13714).T.M.T.was supported by
NSF.
*Corresponding author.
Electronic address:torquato@matter.princeton.edu.
[1]See remarks published in Nature(London)239,488(1972).
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(1969).
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(1972).
[7]T.J.Hales(to be published).
[8]A fascinating open question is:How many jammed con-
figurations exist at each packing ,what is the density of states for jammed configurations?
[9]B.D.Lubachevsky and F.H.Stillinger,J.Stat.Phys.60,
561(1990).
[10]The jammed structures with the lowest density for d.
演讲的艺术1have yet to be identified.Examples of low-density jammed structures for d2have been noted by B.D.
Lubachevsky,F.H.Stillinger,and E.N.Pinson,J.Stat.
Phys.64,504(1991).For d3,the clo-packed simple cubic lattice(contained by rigid boundaries)with a packing fraction of fp͞6ഠ0.52is an obvious example,but this is most likely not the lowest-density jammed structure.
[11]Other reasonable choices for order parameters were tested,
including an information-theoretic entropy,and resulted in the same qualitative behavior as the order parameters described here(Q and T).All of the results,as well as the utility of such order parameters for general many-particle systems(including glass)will be reported in a longer paper.
[12]P.J.Steinhardt,D.R.Nelson,and M.Ronchetti,Phys.Rev.
B28,784(1983).
[13]Our preliminary tests indicate that jammed structures with
packing fractions in the range0.68,f,0.74can be produced when very low compression rates(G,1023)are employed.However,the structures are consistently more disordered(as determined by T and Q)than the equilibrium fcc crystal structures at the same packing fraction. [14]We also found that clo-packed crystals compod of ran-
dom quences of fcc and hcp layer placements cannot be considered to be“random packings”by our chon cri-teria.Both Q and T are not only higher than the mini-mum indicated in Fig.2b for the jammed structures,but Q lies in the range0.84,Q,1,depending on the stacking arrangement.
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