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PHYSICAL REVIEW A VOLUME 45, NUMBER 6
Vibrated powders: Structure, correlations, and dynamics
15 MARCH 1992
G. C. BarkerTheory and Computational Science Group, AFRC
Institute ofFood Research, Colney Lane, Norwich NR4 7UA,
United Kingdom
Anita MehtaTheory of Condensed Matter Group, Cavendish
Laboratory, Madingiey Road, Cambridge CB3 OHE,
United Kingdomand Interdisciplinary Research Center in Materials
for High Perfo-rmance Applications, University ofBirmingham,
Edgbaston,
Birmingham B152TT, United Kingdom*(Received 13 September
1991)
The structure and dynamics of powders subjected to vibration are
investigated by a nonsequential andcooperative computer-simulation
approach in three dimensions. Starting from a microscopic model
ofthe physics, we are able to probe independent and collective
effects in the dynamics of vibrated powders,as well as in the
resulting structures. In particular, we analyze the role of
cooperative structures such asbridges, which are always present in
reality and which cannot be formed by purely sequential
processes.We look in depth at the behavior of the volume fraction
and coordination number as a function of theintensity of vibration,
as well as at correlation functions describing contacts between
neighboring grains,also as a function of intensity. Satisfying
agreement with the qualitative predictions of earlier analyticwork
is obtained, and a framework is laid for future investigations.
PACS number(s): 05.40.+j, 05.60.+w, 81.90.+c, 82.70.—yI.
INTRODUCTION AND AN OUTLINE
OF THE MODEL
Powders are materials that are composed of dense col-lections of
solid grains. They vary in their composition,ranging from
coarse-grained aggregates to fine-gradepowders; in their packing,
ranging from loosely to close-packed states; and in their states of
motion, ranging fromstationary piles to continuous flowing masses.
They havebeen of interest to engineers [1,2] for a long time, but
it isonly recently that they have become an important and ex-citing
area of theoretical [3—9] and experimental [10—13]physics.
Powders exhibit behavior that is neither completelysolidlike nor
completely liquidlike, but intermediate be-tween the two. In
addition to phenomena exhibited byother amorphous systems, their
randomness of shape andtexture strongly influences their static and
dynamic prop-erties. They are highly nonlinear and hysteretic, as
aconsequence of which they show complexity, so that theoccurrence
and relative stability of a large number ofmetastable
configurations govern their behavior. Finally,a unique feature of
granular materials is that they showdilatancy [1], which is the
ability to sustain different de-grees of packing.
However, the subjects of this paper are those featuresof the
static and the dynamic properties of granular ma-terials that are
universal, i.e., that do not depend on thedetails of the particle
sizes or on the material propertiesof the individual grains.
Examples of such properties arethe existence of a fixed maximum
random-packing frac-tion or the size segregation induced by
shaking. In orderto investigate such characteristic granular
behavior wehave investigated a model powder that is made
frommonodisperse, hard spherical grains so as to highlight the
(generic) microscopic behavior of the grains, and the waythat
this influences the macroscopic physics of granularmaterials. We
have used a three-dimensional computer-simulation method to obtain
microscopic details of thegrain configurations and to probe the
independent-particle and the collective effects that occur within a
vi-brated bed of grains. We present here an extended ac-count of
previous work [9], in which we investigate phe-nomena that
contribute to the behavior of dry powders.
Thermal agitation in a powder takes place on an atom-ic rather
than a particulate scale; therefore it is externalvibrations that
play an essential role in the behavior ofpowders. In the absence of
external agitation, the grainsare frozen into one configuration
(since their thermal en-ergy is insufFicient to generate the
equivalent of Brownianmotion), which represents one of the many
possible meta-stable states of the system —we note in passing that
thisis one of the reasons why powders show complexity. Inthe
microscopic model [3], on which this work was based(a quantitative
version of which is presented elsewhere[8]), a granular pile is
represented by an assembly of po-tential wells, each representing a
local cluster of grains,while the effect of vibration applied to
the pile is modeledas being an effective noise H. If H is greater
than thebinding energy of the particles to their clusters, then
thegrains are ejected, and move into neighboring clusters; interms
of the real powder, this means that grains are eject-ed
individually (independent particle relaxa-tion) fromtheir clusters.
Conversely, if H is small relative to thebinding energies of the
particles, they are not ejected; thisenergy goes into the
reorganization of the grains (collec-tive relaxation) within their
clusters to minimize voids.The claim is [3,8] that for high
intensities of vibration,the dominant process is single-particle
relaxation,whereas collective relaxation dominates at low
intensities.
45 3435 1992 The American Physical Society
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3436 G. C. BARKER AND ANITA MEHTA
It will be realized that while single-particle relaxationleads
to a rapid decay of the slope, it will lead to a lowpacking
fraction and a rough surface. Equally, when col-lective relaxation
dominates, the slope will relax slowly ornot at all; on the other
hand, slow collective reorganiza-tion of particles will lead to
efficient void filling, i.e., to ahigh packing fraction and a
smooth surface. We have inearlier work [4] investigated the
phenomenon of granularrelaxation in relation to the decay of the
slope of a sand-pile subjected to vibration, and will focus here on
theeffect of the relaxational dynamics on the structure of
thepowder and the correlations within it.
II. A SURVEY OF REORGANIZATION SCHEMES:THE PHILOSOPHY UNDERLYING
OUR OWN
The static powder is only characteristic of the methodof
preparation —thus demonstrating hysteresis; and anensemble of
configurations, built from independent reali-zations of the whole
powder using the same method, isrepresentative of a particular
aggregation method. Manyaggregation schemes have been investigated
in this way,including the deposition model of Void [14],
ballisticdeposition [15], close packing with surface
restructuring[16], and diffusion-limited aggregation [17]. Of
theseschemes, the simplest is the Void model, in which parti-cles
stick instantaneously on impact. In ballistic deposi-tion, no
trajectories are computed, and aggregation sitesare chosen from a
list, while in diffusion-limited aggrega-tion, random-walk
trajectories precede the aggregationphase. In all these cases, the
initial choice of a site ter-minates the process, i.e., particles
stick on impact. Themodel of Ref. [16] goes further, in that
particles are al-lowed to roll around after impact on a stationary
aggre-gate until they find a local minimum of potential energy,but
this is still a sequential process. In contrast, theschemes
developed here and elsewhere [4,9] contain col-lective
restructuring, where the aggregate restructuressimultaneously with
the incoming particles, thus makingour process nonsequential and
cooperative, and thereforecapable of incorporating realistic
reorganization process-es. This essential ingredient makes our
methods muchmore reflective of many-particle events in a
movinggranular system.
When mechanical energy is supplied to a powder, inthe form of
stirring, shaking, or conveying operations,periods of release are
introduced. During the periods ofrelease, the grains have some
freedom to rearrange theirpositions relative to their neighbors,
and the powder"jumps" [3] between different, but related,
grainconfigurations. In this case a series of grainconfigurations
represents the dynamic response of thepowder to forcing
excitations. In general, this responsehas both transient and
steady-state components. Thus ashaken powder follows a path through
the phase space ofpowder configurations, which depends on both the
dy-namics of the individual grains and on the intensities
andfrequencies of the component vibrations of the drivingforce.
In practice, qualitatively similar driving forces may re-sult in
rather different behavior in the powder. Thus vi-brations are
frequently used to enhance powder mixing,
whereas, in contrast, they may also be identified as asource of
size-segregation effects [18—20]. Similarly, slowshaking, or
"tapping, "may be used as a means of powdercompaction, especially
after a pouring process, but agitat-ed powders can also be
significantly more fluid than theirunshaken counterparts. Our aim
is to distinguish, interms of individual and collective relaxations
[9], thedifferent microscopic responses that underlie the
macro-scopic response of a powder subject to vertical vibrationsat
different intensities. In our simulation model [4,9], thedriving
force is periodic, and leads to clearly definedperiods of dilation
of the powder assembly, betweenwhich we have static configurations
of grains. The driv-ing force is applied uniaxially and is coupled
homogene-ously to the powder, so that free volume is
introduceduniformly. During the periods of dilation the grainmotion
is dominated by a strong uniaxial gravitationalfield and by
hard-core interactions with neighboring par-ticles and the
container base.
For a noncohesive powder, it is clear that stirring,shaking, and
pouring are a11 many-particle operations.During these processes the
particles follow complicatedtrajectories, composed of free-fall
segments punctuatedby hard, inelastic collisions with the other
particles be-fore they reach stable positions in a static
assembly.These trajectories are fundamentally nonsequential,
thatis, the route of one particle to its stable position cannotbe
computed without simultaneously computing theroutes of many other
particles. Stable configurations arethose in which each particle
rests in a potential-energyminimum, and therefore cannot lower its
potential energyany further by local or nonlocal motion. In
practice, thismeans that each particle is in contact with at least
threeothers.
The static configurations of grains that result fromshaking
reflect the essentially nonsequential nature of theprocess. These
configurations contain particle bridges[4,9] and a wide variety of
void shapes and sizes that donot occur in sequentially deposited
aggregates. In thiscontext, a bridge is a stable arrangement of
particles inwhich at least two of the particles depend on each
otherfor their stability. Bridges cannot be formed by thesequential
placement of particles into stable positions butare a natural
consequence of the simultaneous settlingmotion of closely
neighboring particles. In our simula-tions [9] we have approximated
the precise particle tra-jectories by using a low-temperature Monte
Carlomethod supplemented by a nonsequential random-close-packing
algorithm. This is a compromise. At the ex-pense of losing
information concerning the granular dy-namics, we can efhciently
produce static structures thatcorrespond to a nonsequentia1
deposition process. Previ-ous simulations [15,16] have failed to
build in this aspectof shaking.
Our method falls between authentic granular-dynamicssimulations
[21,22] and previous shaking simulations[15], which combine
sequential deposition with a searchfor global minima of the
potential energy. Visscher andBolsterli [15] have performed
computer simulations of vi-brated beds of hard spheres; they have,
however, inter-preted the shaking process only in terms of its
outcome,
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45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND . . ~ 3437
i.e., in terms of the final, static configurations of thegrains.
Their shaken grain configurations were built, us-ing an adapted
random-close-packing procedure; that is,by adding grains one by one
at sites of minimum poten-tial energy chosen from a set of trial
random-close-packing sites. The resulting packings, which remain
fullysequential, have volume fractions P—=0.60, which aregreater
than those for unshaken configurations but stillsignificantly below
the maximum volume fraction for ran-dom close packing of /=0. 64
[23]. More recently, Rosa-to et al. [24] introduced a
two-dimensional Monte Carlomethod to study the size segregation
that is induced byshaking. Their method includes important
nonsequentialfeatures but does not include a criterion for the
stabilityof the packing, and hence cannot be used, directly, to
fol-low the changes in volume fraction or particle coordina-tions
that occur as a result of applied vibrations. In athree-dimensional
simulation, Soppe [25] produced non-sequential consolidated
packings by combining a MonteCarlo compression with ballistic
deposition. This methodalso omits an explicit stability criterion,
but, by using aparticular prescription for annealing the packing,
it leadsto unstable beds of particles with volume fractionsP=—0.60.
Stable packings that contain features due tononsequential
reorganization were used by Duke, Barker,and Mehta [4] to study the
steady relaxation of the slopeof a two-dimensional pile of hard
particles that is causedby vertical vibrations. These simulations
reproducedqualitative features of the relaxation and indicated
thatcollective particle motions, over length scales comparablewith
the nonsequential structural components, were im-portant. This
method forms an integral part of the shak-ing simulations presented
below.
Granular-dynamics simulations are usually performedin one of two
distinct regimes. First there is a grain-inertial regime [21,22],
in which instantaneous, inelastictwo-particle collisions dominate
the motion. These simu-lations model powders under highly energetic
(kinetic)flow conditions, and they are most efficient at
moderateparticle densities of P—=0.3 —0.4. In may ways, the
im-plementation of granular dynamics in the grain-inertialregime
follows the standard methods established for themolecular dynamics
of complex fluids using collections ofrough hard spheres. However,
one important distinctionarises because the collisions between
particles, unlikethose between molecules, are inelastic. The
secondgranular-dynamics regime, called the quasistatic regime,is
used to model [26] the slow, collective motion of close-packed
($~0.55) collections of particles. In this case,the contact forces
between two particles are most impor-tant, and the organization of
computer simulations re-volves around the efficient solution of
many simultaneousequations of motion for interacting particles. For
mostreal materials, the precise nature of the contact forces
isunclear —the so-caIied principles of limiting friction
andindeterminacy of stress [2], which are familiar to chemi-cal
engineers [27], say that the internal stress in a granu-lar
assembly is indeterminate, because the friction be-tween two grains
in contact can lie anywhere betweenzero and a limiting value.
Therefore the applications ofgranular dynamics in the quasistatic
regime [26] are re-
stricted by (ad hoc) estimates of contact forces
usuallyconstructed from viscous and harmonic elements. Mostshaking
processes take place in a series of regimes thattraverse the
spectrum from grain inertial to quasistatic,and therefore "shaking"
is difficult to simulate with acontinuously tuned granular-dynamics
prescription.Thus, during a cycle of a shaking process, the grains
mayexperience local particle densities that vary fromP=—0.3 —0.6
and may go through periods of rapid motionas well as through
periods of slow relaxation. We hope toreport granular-dynamics
simulations of shaking else-where, but here we shall introduce a
hybrid technique.
Before describing the details of our method, we wouldlike to
comment on a few other approaches. Cellular au-tomata [28] are
being used increasingly to model granularflow; while these are
powerful tools, both because of theirflexibility and their relative
speed, they are limited bytheir lattice-based formulation. They are
thus good forqualitative descriptions of powder flow, but cannot
probedetailed particulate structure during and after flow.
Thekinetic-theory-type approaches of Haff [29] and Jenkinsand
Savage [30] and the hydrodynamic approaches ofJackson [31] are
appropriate for the situation of rapidshear, where the grains are
in constant motion, and theassembly is assigned a "granular
temperature" deter-mined by the average mean-square velocity of the
grains.These methods are, ho~ever, inappropriate for the situa-tion
where the grains are in slow, or no, motion withrespect to each
other —the continuum approach fails, be-cause the discreteness of
the grains and the effects of fric-tion and restitution at
individual collisions become in-creasingly important. Our method
[9], on the otherhand, is able to probe details of particulate
structure; inaddition, we do not assume a continuum basis or a
singlegranular temperature, and are therefore able to copebetter
with the quasistatic regime of slow shear.
III. DETAILS OF OURSIMULATION TECHNIQUE
In our simulations we have a bed (Fig. 1) of mono-disperse, hard
spheres above a hard, impenetrable, plane
FIG. 1. Schematic diagram of the geometry of the simula-tion.
Hard particles form a periodic bed above an impenetrablebase: the
diagram shows the primary simulation cell, which isrepeated in two
perpendicular directions.
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3438 G. C. BARKER AND ANITA MEHTA 45
base that is at z =0. The particle bed is periodic, with arepeat
distance of I. sphere diameters, in two perpendicu-lar directions,
x and y, in the plane. Each primary simu-lation cell contains N
spheres. A unidirectional gravita-tional field acts downwards,
i.e., along the negative-zdirection.
Initially, the spheres are placed in the cell using asequential
random-close-packing procedure [32]. Thespheres are introduced, one
at a time, from large z and atrandom lateral positions, and follow
complex paths,which are composed of vertical line segments and
circu-lar arcs, until they reach stable positions in contact
eitherwith three other spheres or with the hard base. Thesesphere
trajectories correspond to rolling motions separat-ed by periods of
free fall. In this sequential deposition,the moving sphere rolls
over spheres that are already lo-cated in stable positions; that
is, incoming spheres cannotdisrupt the stable packing, and they
cannot interfere withother aggregating particles. In this sense,
the aggregationis slow, and the gravitational field is strong. Many
au-thors have analyzed the sequential, close-packed arrange-ments
of spheres [15,23,33] that are obtained using thisprocedure. For
monodisperse spheres there are boundarylayers that extend for
approximately five sphere diame-ters both above the hard base and
below the free surface.These layers contain quasiordered
arrangements ofspheres. Apart from this, the packing is
homogeneouswith a mean volume fraction $0=0.581+.001 and amean
sphere coordination co =6.00+0.02 [34]. Thesevalues are not altered
substantially by introducing a smallamount (-5%) of polydispersity,
and they adequatelydescribe the packings that are used as
initialconfigurations for our shaking simulations.
In our simulations, the packing is subject to a series
ofnonsequential, X-particle reorganizations. Each reorgan-ization
is performed in three distinct parts: first, a verti-cal expansion
or dilation: second, a Monte Carlo consoli-dation; and finally a
nonsequential close-packing pro-cedure. We shall call each full
reorganization a shake cy-cle or, simply, a shake. The duration of
our model shak-ing processes and the lengths of other time
intervals areconveniently measured in units of the shake cycle.
The first part of the shake cycle [4,9] is a uniform verti-cal
expansion of the sphere packing, accompanied by ran-dom, horizontal
shifts of the sphere positions. Sphere i,at height z, , is raised
to a new height z,'=(I+a)z, . Foreach sphere, new lateral
coordinates are assigned, accord-ing to the transformation x'=x
+g„,y'=y +g, provid-ing they do not lead to an overlapping
sphereconfiguration; here g„and g are Gaussian random vari-ables
with zero mean and variance e . The expansion in-troduces a free
volume of size e between the spheres andfacilitates their
cooperative rearrangement during phases2 and 3 of the shake cycle.
This expansion is uirtual: weseek merely to introduce a free
volume, not to model aphysical expansion. The parameter e is a
measure of theintensity of vibration; although we do not know the
exactfunctional relationship between these two quantities, weexpect
them to vary monotonically for reasonably smalle. We have assumed
that the freedom of motion of theparticles in the interior of the
packing increases with the
intensity of the applied vibrations.In the second phase of the
cycle, the whole system is
compressed by a series of displacements of individualspheres.
Spheres are chose at random and displaced ac-cording to a
very-low-temperature, hard-sphere, MonteCarlo algorithm. A trial
position for sphere i is given byr,
' =r, +ad, where a is a random vector with components—1 a,a, a,
~1, and d defines the size of a neighbor-hood for the spheres. The
move is accepted if it reducesthe height of sphere i without
causing any overlaps. Allthe successful moves reduce the overall
potential energyof the system. The process continues until the
efficiencywith which moves are accepted, measured by batch
sam-pling, falls below a threshold value e. Here d and e arefree
parameters that are chosen to optimize the computa-tional method.
It is shown below that there is a regime inwhich the static results
are not strongly dependent onthis choice.
Finally, the sphere packing is stabilized using an exten-sion of
the random-close-packing method describedabove. The spheres are
chosen in order of increasingheight and, in turn, are allowed to
roll and fall into stablepositions. In this part of the shake cycle
spheres may rollover, and rest on, any other sphere in the
assembly. Thisincludes those spheres that are still to be
stabilized andthat may, in turn, undergo further rolls and falls.
In thisway, touching particles can be continually moved untilno
further rolling is possible. This procedure allows theformation of
complex, stable, structural components, likebridges and arches,
which cannot be constructed by pure-ly sequential processes
[4,9].
The outcome of a shake cycle is to replace one
stableclose-packed configuration by another. In
theseconfigurations, each particle occupies a cluster that isformed
by its neighbors, and a "shake" is thus a reorgani-zation scheme
for a set of clusters. The role of the indivi-dual parts of the
shake cycle is clear. Expansionrepresents a challenge of variable
degree to the integrityof the clusters. The Monte Carlo compression
reinstatesthose clusters that were deformed and, when
necessary,creates new clusters where the previous ones were
des-troyed. Finally, the stabilization phase positions the
par-ticles inside the set of clusters established in phase 2.
Inphase 2, the Monte Carlo procedure generates a time-ordered
sequence of states that culminates with a statethat has an isolated
potential well for each particle. Al-though this does not replicate
at every instant the actualdynamical processes that lead to the
static configurationof spheres, and the dynamical information that
it con-tains will depend on the choices for d and e, it
seemsreasonable that the set of clusters that is produced is nottoo
sensitive to these details.
In practice, during phase one of the nth shake cycle,the mean
volume fraction of the assembly falls from P„to P„,/( I+a). In
phase 2 the volume fraction steadilyincreases to P„—=P„,, and in
phase 3 it remains approxi-mately constant. In contrast, the mean
coordinationnumber is reduced from c„&to zero in the
expansionphase of the nth shake and remains zero throughout
theMonte Carlo compression, but, during stabilization, it
in-creases steadily to c„=—c„
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45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3439
IV. SOME COMMENTS ON THE TRANSIENTREGIME, AND STEADY-STATE
RESULTS
The continuous evolution of particle positions and ve-locities
that occurs during a physical shaking process isreplaced, in our
simulations, by a time-ordered, discreteset of static N-particle
configurations. The members ofeach set are the nonsequentially
reorganized close pack-ings that are obtained after integral
numbers t of comp-leted shake cycles starting from a sequential
randomclose packing. Each set of configurations may be labeledby
three parameters e, d, and e. For each member ofthese sets we have
evaluated "material" properties, suchas the volume fraction and the
mean coordination num-ber, from the central portion of the packing
in order tominimize the surface effects. In all cases, this
volumecontains more than 50% of the spheres in the
simulationcell.
A. Volume fractions and coordination numbers
Figure 2 shows the variation of the volume fraction Pwith the
number of shakes t for two different shaking in-tensities e=0 05
and. 0.5. In both cases e =d =0.01,N =1300, and L =8 particle
diameters. At low intensity,the volume fraction increases slowly
for t & 50 and Quctu-ates around a steady value, /=0. 598+.003,
at largertimes. This corresponds to a slow compaction towards
avibrational steady state. In this state nonsequential
reor-ganizations of.intensity a=0.05 leave the volume fractionof
the packing substantially unaltered. The steady statedoes not
depend on the particular choice of startingconfiguration or on
particular sets of pseudorandomnumbers used during the shake
cycles. For @=0.5, theevolution of the volume fraction of the
packing is morecomplicated. The first two shake cycles
significantlyreduce the volume fraction of the packing from that
ofthe sequential deposit to /=0. 562+0.002. Followingthis there is
another transient period, t (30, in which thevolume fraction
partially recovers. Finally, for t )40another vibrational steady
state is achieved, with
0=0.569+0.002.Thus, over a range of shaking intensity, repeated
non-
sequential reorganization leads to packings with bulkproperties
that are insensitive to further vibrations. Theproperties of the
vibrational steady states will be dis-cussed, in detail, below.
Results have been obtained bytaking averages from sets of m
consecutive configurationsin the steady-state shaking regime with m
=—50.Throughout, we have used simulations with N =-1300 andL = 8
particle diameters, for which the mean depth of thepacking is
approximately 20 particle diameters. For shal-lower packings, the
measured volume fractions are toolarge. This is because the hard
base at z =0 causes someordered, denser regions to occur in the
lowest layers ofthe packing. These layers are unimportant when
measur-ing the volume fractions of packings with depths greaterthan
ten particle diameters. We have also tested thedependence of the
volume fraction on the cell size L forfixed bed depths, and
conclude that serious size depen-dence is absent for L ~ 8.
Monte Carlo consolidation is, structurally, the mostinfluential,
and computationally the most intensive partof each shake cycle. The
duration of this phase, whichcan be measured in terms of the number
of Monte Carlosteps per particle NMc/N, can be increased either by
de-creasing e (the terminating efficiency of the Monte
Carlosequence) or by decreasing d (the maximum size of eachMonte
Carlo step). However, the results of hybrid simu-lations are not
related trivially to the details of the MonteCarlo component alone.
In Fig. 3 we have plotted, forseveral values of d, the steady-state
volume fraction Pagainst the length of the Monte Carlo
consolidation.Each data point in Fig. 3 has been obtained from
aseparate simulation, with @=0.5, N—= 1300, and L =8particle
diameters, by averaging the volume fraction over20 consecutive
steady-state shaking configurations. Mostimportantly, for long
Monte Carlo consolidations, i.e.,for sufficiently small values of
e, the volume fraction datacollapses onto a single, constant value
that is independentof d.
Figure 3 shows that, in the absence of the Monte Carlo
0.6
0.6 +%sV+~e~g zPjhsv
C0
~0.58—".V-
+
+0.56—
I
50 100
o0.58—
0.56—
~ ~
0X
,+
2000
NMc/N
X +X
X
o $o + +
+o ~
I
4000
FIG. 2. Volume fraction of monodisperse hard spheres plot-ted
against the number of cycles t, of a computer-simulatedshaking
process. The initial sta~e is a sequential random closepacking,
with volume fraction 0.581, and the shaking intensityis a=0.05 (~ )
and 0.5 (+).
FIG. 3. Steady-state volume fraction of monodisperse hardspheres
plotted against the length of Monte Carlo consolidation(measured in
Monte Carlo steps per particle NMc/N). TheMonte Carlo consolidation
is the second phase of a three-phasecomputer-simulated shaking
process with shaking intensity@=0.5. The maximum Monte Carlo step
lengths are d =0.004(~ ), 0.01(+),0.05 (o), and 0.2 (X).
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3440 G. C. BARKER AND ANITA MEHTA 45
0.6
C0o0.580
E~0.56—0
0.540.0
I
0.5I
1.0 1.5
Shake Intensity
FIG. 4. Steady-state volume fraction of monodisperse hardspheres
plotted against the shaking intensity.
phase, the steady-state volume fraction, for shaking in-tensity
@=0.5, is /=0. 589+0.001. This is greater thanthe volume fraction
for a sequential deposition process,which itself could be viewed as
a shaking process,without a Monte Carlo phase, of intensity e= ao.
For thesmallest values of d, the Monte Carlo consolidation
haslittle effect for NMC/N & 10 . This indicates that, in
thefirst part of the Monte Carlo dynamics, the particles
ex-perience a period of free diffusion. For longer consolida-tions,
and for larger values of d, the downward motion ofthe particles is
a collective process. The regions of Fig. 3in which P decreases
with NMc/N correspond to in-creased numbers of cooperative features
that are trappedinto the stable, close-packed structure by
extending theperiod of Monte Carlo consolidation. The minima in
Fig.3 indicate that premature termination of the Monte
Carloconsolidation can cause too many large bridges and voidsto be
trapped in the stable packing. The mean coordina-tion number of the
spheres e remains weakly dependenton d for long Monte Carlo
consolidations, but, in testswith e =0.05 and 0.5, the values of e
obtained by extrapo-lation to d =0 do not differ substantially from
those ob-tained with d =0.01.
After the above comments on the transient regime, wenow discuss
the steady state: the results presented in thefollowing are mean
values taken from m -=50 consecutivecycles in the steady-state
regime of the shaking process.Figure 4 shows the variation of the
steady-state volumefraction P with the intensity of vibration e For
e). 1.0,the volume fraction is only weakly dependent on e
withP—=0.550+0.003. However, P rises sharply as e is re-duced below
a=1.0 and, for e 0.2, the shaken assemblyadopts configurations that
are more compact than thosefor sequentially deposited spheres. This
is a clear mani-festation of the collective nature of the
structures that areintroduced by a shaking process.
Figure 5 shows the variation of the steady-state
meancoordination number of the spheres c with the intensityof
vibration e. For e 0.25, the mean coordination de-creases as e
increases, and it is approximately constant atc —=4.48+0.03 for
larger intensities. The mean coordina-tion number in a shaken
assembly is substantially belowthat for a sequential deposit (c
=—6.0) refiecting the pres-ence of bridges and other
void-generating structures.
4.650.4
o 460CD
0o4.55-(3C:U~ 45-
oz-0
0.0
4.450.0
I
0.5I
1.0 1.5
Also shown in Fig. 5 are the mean fractions, P(n) forn =3—9, of
spheres that are n-fold coordinated in pack-ings subjected to
steady-state shaking vibrations at inten-sities @=0.05 and 0.5.
Most spheres touch four or five oftheir neighbors. For larger
values of e, the proportion offourfold-coordinated spheres is
increased, largely at theexpense of sixfold-coordinated spheres;
i.e., the peak ofthe distribution moves towards lower coordination
num-bers.
From Figs. 4 and 5 we note that for 0.25 ~ e~ 1.0,
thesteady-state volume fraction steadily decreases with e,while the
mean coordination number remains constant.This is consistent with
the interpretation that, on the onehand, the density of bridges is
independent of e, but that,on the other, the shapes of the bridges
become in generalmore eccentric (and therefore more wasteful of
space) ase is increased.
B. Network analysis
Each stable configuration of spheres has associatedwith it a
network, called the contact network, which canbe formed by drawing
line segments between the centersof all pairs of touching spheres.
We have studied the evo-lution of the contact network in order to
follow localsphere correlations during shaking. For each sphere i
attime t we define an (N —1)-dimensional vector b;(t), suchthat the
jth element of b;(t) is unity if sphere i is touch-ing sphere j at
time t, and zero otherwise. Figure 6 showsthe variation with time
of the average autocorrelationfunction
z(t) = (b, (t').b, (t +t') lib, (t')i ib;(t +t')i ),for spheres
in the interior of the packing, at two shakingintensities @=0.05
and 0.5. In both instances the initialrate of breaking of contacts
is greatest, and for largertimes t ~10 the rate becomes
approximately constant.The main conclusion from the figure is that
contactcorrelations disappear relatively slowly for low
intensitiesof vibration: more quantitatively, we find that a
singlenonsequential reorganization with e =0.5 is approximate-
Shake Intensity
FIG. 5. Mean coordination number of monodisperse hardspheres
plotted against the shaking intensity. The inset showsthe mean
fractions P(n) of spheres that are n-fold coordinatedin the
steady-state regime of the shaking process that has shak-ing
intensity @=0.05 (solid lines) and 0.5 (dashed lines).
-
45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3441
1.0
0.8—
0.6—
0.4—
~ ~ ~ ~~ ~ ~ ~ ~ ~ y ~ 0 ~ ~ ~ ~ ~ ~, ,
++
++++++++
+++++++m++g
0.2 I10
I
20 30
FIG. 6. Autocorrelation function z(t) of the contact
networkplotted against the number of shake cycles t for
rnonodispersehard spheres in the steady-state regime. The shaking
intensityis a=0.05 (~ ) and 0.5 (+).
gains a new one, sphere G. In this case, the overall im-pression
is one of network disruption. The network con-nectivity is altered
significantly in this case, and a com-parison of Figs. 7(b) and
7(d) shows many examples ofbond creation and annihilation.
A notable insight to be gained from Figs. 7(b)—7(d) isthat
bridge collapse occurs more frequently for large vi-brations. In
Fig. 7(b), spheres H and I rest on, and sup-port, each other and
therefore form part of a bridge; thisfeature is retained in Fig.
7(c) (contact network aftersmall vibrations), but not in Fig. 7(d)
(contact networkafter large vibrations), where sphere H gains
additionalsupport by contacting sphere A from above. We see froma
comparison of Figs. 7(b) and 7(d) that the bridge incor-porating
spheres H and I has collapsed after a single(large-intensity)
shake.
C. Correlation functions
ly twice as eScient at disrupting the contact network asone with
@=0.05.
The behavior of z(t) is consistent with snapshot obser-vations
of consecutive contact network configurations.Figure 7 highlights
the responses of a smal1 group ofneighboring spheres, which are in
the interior of a muchlarger packing, to vibrations of two
different intensities.Figure 7(a) shows the spheres that are
instantaneouslywithin a spherical capture volume, and Fig. 7(b)
showsthe contacts between them. We note that contacts be-tween
spheres at the periphery of the capture volume andspheres that are
outside it are not represented in Fig. 7.The capture volume is
centered on sphere A and has a ra-dius of approximately two sphere
diameters. In Figs.7(b) —7(d) small balls mark the positions of
centers of theclose-packed spheres, and rods represent the sphere
con-tacts. The initial configuration of packed spheres, whichis a
configuration obtained at the end of one particularshake cycle in
the steady-state shaking regime witha=0.05, is shown in Fig. 7(a),
and its associated contactnetwork is shown in Fig. 7(b). Figures
7(c) and 7(d) arethe contact networks for configurations that are
obtainedafter the application of one further complete shake
cycle,with intensity a=0.05 and 0.5, respectively.
In the initial configuration sphere A rests on spheres B,C, and
D, and is touched by one other sphere, labeled E,which rests on it.
After an additional shake cycle witha=0.05, the sphere A remains
stabilized in the same waybut has gained a further contact, with
sphere F, fromabove. There are many differences between the
networksin Figs. 7(b) and 7(c) but they are mainly small changes,of
the sphere positions and the rod orientations, which donot grossly
alter the network connectivity. During theextra shake cycle, one
sphere has left the capture volumeand another has entered it, so
that the numbers of centersin Figs. 7(b) and 7(c) are identical.
The overall impres-sion is one of network deformation.
In contrast, the network in Fig. 7(d) does not closelyresemble
the one in Fig. 7(b). There is a net loss of twospheres from the
capture volume during the additionalshake cycle with a=0.5. After
this extra shake, sphere Aretains only two of its original
supporting neighbors and
The pair distribution functions of particle positions,It (r) for
separations in a horizontal plane and g(z) forseparations in the
vertical direction, are illustrated in Fig.8 for @=0.05 and 0.5.
The data sets for these functionswere collected, over m =-25
cycles, from horizontal slabswith a thickness of one sphere
diameter and from verticalcylinders with cross sections equal to
that of one sphere.In both directions, the structure is similar to
that expect-ed for dense, hard-sphere fluids. The short-range order
ismost pronounced in the horizontal direction, while thepair
distribution function in the z direction, g (z), is rela-tively
insensitive to variations of the shaking intensity.Both functions
indicate the presence of a second shell ofneighbors at a separation
of approximately two particlediameters: we conclude from these
figures that theshort-range order decreases with increasing
intensity ofvibration, in accord with intuition.
During a shake cycle, each particle i is shifted in posi-tion by
Ar;=Ax;i+Ay;j+hz, k, where i, j, and k areunit vectors in the x, y,
and z directions. We have plot-ted, in Fig. 9, correlation
functions of the vertical com-ponents of displacement hz; for e=0
05 and 0.5.. H(r)measures the correlations in a horizontal plane
and G(z)measure the correlations in the vertical direction
accord-ing to
H(r) =(bz;bz, 5(It;, I —r)8(lz;, I ——,') & I& Ihz; I
&
G( )=(&;&,&(I;, I —)8(It;, I ——,') &/(I&;I
&',
where t, =(x, —x ) +(y, —"y ), z,"=z;—z, and 8(x) isthe
complement of the Heaviside step function. Theaverages are taken
over all pairs of spheres i and j andover m =—25 shake cycles. We
note that, over the rangeof shaking intensities we have studied,
the mean size ofvertical displacements during a shake cycle, (
Ib,z, I &, is amonotonic, increasing function of the intensity.
Figure9(a) shows that H(r) decreases rapidly to zero with
in-creasing r, and that there is a small decrease in the mag-nitude
of the longitudinal displacement correlations,measured in the
transverse direction, as the shaking in-tensity is increased. The
data in Fig. 9(a) give an esti-mate for the horizontal range over
which the spheres
-
3442 G. C. BARKER AND ANITA MEHTA 45
L
(b)
Pg
plI
.]i"
(c)
FI~. 7. (a) Three-dimensional representation of a cluster of 35
spheres. The cluster is part of a large assembly of spheres that
havebeen subjected to shaking vibrations with intensity a=0.05. (b)
The contact network that corresponds to the cluster of spheresshown
in (a). Small balls represent the centers of the packed spheres and
rods represent the contacts between them. The centers ofthe spheres
B, C, D, and E, which contact the central sphere A, have been
colored red. Contacts with the spheres that are outside ofthe
cluster have not been shown. (c) The contact network that
corresponds to the cluster of spheres that is obtained after the
clusterin (a) is subjected to a further shake cycle with intensity
a=0.05. (d) The contact network that corresponds to the cluster of
spheresthat is obtained after the cluster in (a) is subjected to a
further shake cycle with intensity a=0.5.
-
45 VIBRATED POWDERS: STRUCTURE, CORRELATIONS, AND. . . 3443
1.5 1.5 model [3], which says that collective
(independent-particle) motions predominate for lower (higher)
intensi-ties of vibration.
++
++
+ ~+
+a+~+yg+
N 10~ +
~ + ~ ~ ~++ + . g+++
~ p +~+ ~ + ++++I
0.5 0.5
FIG. 8. Pair distribution functions of particle positions h
(r)and g (z) for monodisperse hard spheres in the steady-state
re-gime plotted against horizontal displacement r and vertical
dis-placement z. The shaking intensity is @=0.05 (~ ) and 0.5 (+
).The peak heights, which are not shown, have been estimated ash
(1)=6.35(6.20) and g (1)=4.40(4.25) for a=0.05(0.5).
0.4=„- 1.0—
L
0.2-+' ~+
0.0~ ++ +,g+~ ++ +
I~ ~ T
2r
N
~ 0.5— +
0.0—
~~ +++ ~
~ +g+qy + + g~--
~ ~ + + +++"
FIG. 9. Correlation functions H(r) and G(z) for the
verticaldisplacements of spheres during a single cycle of the
steady-state shaking process plotted against horizontal
displacement r,and vertical displacement z. The shaking intensity
is a=0.05(~ ) 0.5 (+).
move collectively during a shake cycle, and thus providea
measure of the typical "cluster size" in the
transversedirection.
Clearly, during vertical shaking, the motion of a parti-cle is
more sensitive to the positions and the motion ofthose neighbors
that are above or below than it is to thosethat are alongside.
Figure 9(b) shows that the correla-tions of the longitudinal
displacements measured in thelongitudinal direction are stronger
than those measuredin the transverse direction, that is, G(z) has a
large firstpeak and, at large displacements, it decreases more
slow-ly than 8 (r). Also, G (z) depends strongly on the intensi-ty
of the vibrations, and, for small e, it has a distinct(negative)
minimum at approximately z =1.3 sphere di-ameters. This implies
that at these separations, whichare typical of vertical particle
separations in shallowbridges, many sphere displacements are not
stronglycorrelated, and several of them move in opposite
direc-tions; hence this feature is consistent with the
slowcompression or collapse of shallow bridges. The correla-tion
functions of the transverse components of the spheredisplacements
are negative at small separations, which isconsistent with spheres
sliding past each other as they aredisplaced in the x and y
directions.
We conclude from all the above that the size of a typi-cal
dynamical cluster, in both longitudinal and transversedirections,
decreases with increasing intensity of vibra-tion. This verifies
the predictions of the microscopic
D. The "hole" space
We have concentrated on the static properties and thepair
correlations of spheres that form a random-close-packed structure.
Equally fundamental, and intimatelyrelated, problems concern the
nature of the continuousnetwork of empty space, consisting of
pores, necks, andvoids, etc. , which complement the physical
structure. Inorder to investigate the pore space of shaken
packings,we have constructed the complex structures formed
fromoverlapping holes. For a close-packed bed of spheres,
theoverlapping holes are another species of spheres, each ofwhich
touches four of the packed spheres. The holes mayoverlap each other
but cannot intersect any of the packedspheres. For a monodisperse
close packing, the max-imum hole size is approximately the same as
the spheresize, and the minimum hole diameter is 0.224 times thatof
the spheres, corresponding to the hole at the center ofa regular
tetrahedron formed from four spheres.
Figure 10 shows a small section of the overlapping
holestructures for vibrated packings with a=0.05, 0.5, and1.5, and
Fig. 11 shows the corresponding distributionfunctions for the hole
radii. From Fig. 11, it is clear thatsmall-intensity shaking is an
efficient method of removinglarger holes from the overlapping hole
structure, and,therefore, a method for removing large voids from
apacking without producing a regular structure. We alsonote, from
Fig. 10, that low-intensity shaking leads tolarge numbers of
isolated holes, and isolated hole pairs,whereas the
larger-intensity vibrations create clearlydefined strings of
connected, overlapping holes. This im-portant feature has clear
implications for the transportproperties of vibrated beds of
particles, and we hope toreport on these in more detail at a later
date.
E. The surface
In addition to furnishing data on the bulk properties,our
simulations can be used to obtain information aboutthe surface of
shaken particulate assemblies. Surface mea-surements are subject to
larger uncertainties than bulkmeasurements because they involve
only a fraction of theparticles contained in the simulation cell,
and also be-cause they are generally more susceptible to system
sizedependence. For simulations with L =8 particle diame-ters and
N=-1300, we have measured the mean-squaresurface width o =L g, (z;
—zo) defined by the spheresi, which have heights z, and which are
the highestspheres in each L vertical columns that have cross
sec-tions of one square-sphere diameter. zo is the meanheight of
the bed. All the surfaces that we have exam-ined are smoother than
the surface of a sequentially de-posited aggregate that has o.
=0.44+0.02. For e &0.5the surface width is approximately
independent of e ando. =—0. 16+0.02. For larger shaking intensity,
cr in-creases with e, and for @=1.5, o- -=0.23+0.02. This is
inkeeping with the qualitative predictions I3] of our model,which
state that greater surface roughening arises as aconsequence of
violent vibrations. We have not been able
-
3AAA G. C. BARKER AND ANITA MEHTA 45
to establish the scaling properties of o. ,' however, wehave
confirmed that these trends are followed by othermeasures of the
surface irregularity. Most notably wehave used a Monte Carlo method
to investigate the reac-tion surface for ballistic aggregation with
small test parti-cles, which have a diameter of 0.001 sphere
diameters-in this case as well, the mean-square width of the
reac-tion surface follows the behavior outlined above.
Other studies of surface roughening I35 —37] have con-centrated
on the scaling regime appropriate to sequentialdeposition in the
presence of noise. Since our current ap-proach is restricted to
sizes below the scaling regime, be-cause we incorporate complex and
nonsequential reor-ganization processes, we are unable to compare
our re-sults on surfaces with those presented in Refs. I35
—37].
0.2
0.15—
0.1—CL
~ ~+ +0
~ ++ 0 0~ +0
0.05—
0 0 T T T T T)
(
0
~ +0 ~ + 0
0++ 0
0+ 0 (I
a +4I I
0.0 0.1 0.2 0.3 0.4
FIG. 11. Distribution functions I'(r) for the radii r of
theoverlapping holes presented in Fig. 10. The shaking intensity
is@=0.05 (~ ), @=0.5 (+), and @=1.5 (0).
Id
4.
I""'%I
S'de
('"(Td» Ij. IRhlHI~,
'((
r
''d0 (
However, the results we have presented do provide
aquantification of surface roughening in cooperatively
res-tructured packings. We hope to report further on the sur-face
properties of shaken particulate assemblies, as wellas the
surface-penetration effects, at a later stage.
V. DISCUSSION
(c)T p»»
* P
'dd»d
FIG. 10. Sections of the overlapping hole structures that
aretopologically complementary to the structures formed by
thespheres. The shaking intensity is (a) @=0.05, (b) @=0.5, and
(c)@=1.5.
In the preceding sections we have shown that oursimulations
provide direct and meaningful microscopicobservations of
nonsequentially reorganized granularstructures. The simulation
technique allows us to ob-serve the packing both internally and
nondestructively(which is outside the scope of current experimental
tech-niques), and it therefore provides a unique opportunity
tolearn about the bulk behavior and the transient responsesof
granular solids subjected to vibration by focusing onthe relaxation
mechanisms at the particulate level. Thusour simulation method
provides a description that is su-perior to a continuum description
and that provides thebasis for a fundamental understanding of
realistic (nonse-quential) particle dynamics.
The hybrid Monte Carlo method used above allows usto construct,
both efficiently and consistently, nonsequen-tial reorganizations
of random close packings, but, in sodoing, it sacrifices detailed
knowledge of the granular dy-namics and is unable to look at the
effects of the qualityand the frequency of the applied vibrations.
In this sense,we have not built a model of one particular shaking
pro-cess from which quantitative data will result, but havedesigned
a working tool to study the qualitative featuresof shaking and
nonsequential processes, in general.
All our simulations have been performed using rnono-disperse
collections of spheres in open systems. It is un-likely that the
introduction of a small amount of po-lydispersity, in either the
sizes or the shapes of the parti-cles, would seriously alter our
conclusions. However, it iscertain that, ultimately, the
introduction of variations inthe sizes and the shapes of the
particles would cause newshaking-induced effects, such as size
segregation and theappearance of particularly favorable close
packings, to in-teract with, and probably cloud, the effects that
we haveobserved. Also, we have employed periodic boundary
-
VIBRATED PO%DERS: STRUCTURE, CORRELATIONS, AND. . . 3445
conditions throughout in order to compensate, at least inpart,
for the size restrictions that are imposed by ourcomputational
limits. It must be emphasized thatconfining walls and the particle
wall interactions play amajor role in most powder-handling
applications. Theextension of our scheme to include the effects of
po-lydispersity, confining walls, as well as other
interparticleinteractions, represents a primary goal of our
futurework.
We note that the algorithm, as employed here, does notensure the
homogeneity of bridge nucleation. This is be-cause, in the ordered
consolidation in phase 3 of ourshake cycle, the packing gradually
produces a gap be-tween consolidated and unconsolidated particles.
Thesize of our simulations makes this effect relatively
unim-portant, but, in larger simulations, the effect can be
over-come by incorporating into the consolidation phase
athree-dimensional extension of the local shifts that
wereintroduced by Duke et al. [4] to ensure homogeneousdistribution
of bridges in a two-dimensional shaken pile.
The results we have presented establish links betweenthe
observed changes of the material properties, whichoccur as the
shaking intensity varies, and the underlyingmicroscopic
correlations of the particle positions and dis-placements. From
these results we identify competingroles for the
independent-particle and the collective re-laxation mechanisms that
occur in nonsequentially reor-ganized random close packings —and
verify earlier pre-dictions [3] that independent-particle
(collective) effectsdominate at high (low) vibration intensity.
Contact net-work measurements show that at high intensities,
indivi-dual particles are regularly ejected out of their local
envi-ronments, and, hence, one particle may sample manydifferent
environments over a short period of time. Incontrast, at low
shaking intensities, each particle experi-ences a slow deformation
of its environment during shak-ing, and the identity of the
particles that form its closeneighbors remains relatively constant;
thus, at theselower shaking intensities it is rare for a particle
to make atransition into a totally new environment.
In our packings, the particles that form parts ofcooperative
structures, such as bridges and arches, aresubject to the different
rates of change of their local envi-ronment caused by different
shaking intensities. Thisleads to nonsequential reorganization
behavior which de-pends, qualitatively, on the shaking intensity.
Duringhigh-intensity shaking, cooperative structures form
anddisappear rapidly, so that most of the bridges, etc. , thatare
present in one particular configuration are only oneor two
generations old. These "immature" bridges haveshapes that are those
most favored at their formation,and these are, in general, wasteful
of space. Thus thepacking fraction takes a low value. In the
low-intensityregime, the cooperative structures form and then
deform,along with their local environment, over several
furthercycles before they become too tenuous to survive. In
thiscase, a packing may contain bridges that are many gen-erations
old ("mature") and that have shapes that arefavored by their
stability against disruption. This includesshapes that have relaxed
downwards and are therefore"Batter"; the result is a higher packing
fraction, i.e., a
minimization of the void space, and a shift of the holesize
distribution to smaller sizes.
According to our definition of a "bridge, " several ofthe
spheres that form part of the structure will have adeficit of
neighbors, particularly in the downward direc-tion, and, therefore
will have low coordination numbers.Thus the mean coordination
number of a packing will de-pend on the density of bridge contacts,
and in turn thisdensity will depend on the number density of the
bridgesand on the mean size of a bridge (i.e., the mean number
ofspheres that are required to construct one bridge). Wecan
determine the latter from our measurements of H(r);these indicate
the lateral extent over which the verticaldisplacements of the
spheres are positively correlatedduring one shake cycle, and our
results show (Fig. 9) thatthe mean bridge width is quite
insensitive to the shakingintensity. Nonzero correlations extend
over approxi-mately 1.5 sphere diameters, indicating an average
bridgewidth in the region of 3.0 sphere diameters, for both
largeand small intensities of vibration.
Given the observed independence of the mean coordi-nation number
on e, for @~0.25, we infer that the num-ber density of bridges is
approximately constant in thisregime. For @&0.25, the mean
coordination numberrises, which indicates (since the bridge size
stays approxi-mately constant) that the mean number density of
bridgesfalls.
We now show the effect of the nature of the collapsedbridges on
the resultant packing. For the lowest shakingintensities, the
packing includes regions that result fromslowly collapsed, mature,
and Hatter bridges, i.e., bridgesthat deformed considerably before
their contact networkwas disrupted. We suggest that these are
regions of par-ticularly efficient random close packing and
thereforecause the mean volume fraction to rise above the valuethat
can be obtained by sequential packing processes.The enhanced
short-range correlations of the particle po-sitions (cf. Fig. 8),
which we observed for a=0.05, areconsistent with this
interpretation. In the high-intensityregime, it is the "immature"
and angular bridges that col-lapse, and the aftermath of such
collapses is not distin-guishable from a sequentially deposited
structure, withsmaller short-range correlations in particle
positions andlower packing fractions. The above thus illustrates
(viathe specific mechanism of bridge collapse) the point [3]that,
at low intensities of vibration, collective reorganiza-tion of
particles (and the consequent slow rearrangementof particle
bridges) will lead to the efficient filling ofvoids, and the
converse.
In conclusion, then, we have presented a detailed studyof the
microscopic processes at work in the interior of avibrated granular
pile. We have investigated the bulkstructure, by analyzing the
behavior of the volume frac-tion and the distribution of
coordination numbers as afunction of the shaking intensity. We have
also presenteda detailed study of the contact networks and their
auto-correlation functions before and after vibration, and
haveshown that earlier predictions [3,8] regarding the roles
ofindependent-particle and collective relaxation mecha-nisms are
verified. The spatial correlations in the pileafter vibration have
been investigated by examining the
-
3446 G. C. BARKER AND ANITA MEHTA
pair-correlation functions of particle positions, and
thedistributions of voids, as a function of intensity, whichprovide
valuable clues to the static as well as to the trans-port
properties, with particular reference to the impor-tant issue of
bridge formation and collapse. Finally, wehave shown directly the
dynamical behavior of grainssubmitted to vibration, by examining
the displacementcorrelation functions as a function of intensity
anddemonstrating that the slow motion of clusters predom-inates at
lower intensities, relative to the motion of in-
dependent particles, and the converse. We look forwardgreatly to
experimental verification of this rich array oftheoretical
[3,4,8,9] results, which our present work in itsdetail has brought
within reach of current experimentaltechniques [38].
ACKNOWLEDGMENTS
We thank Richard Needs, Mike Cates, and MalcolmGrimson for
useful comments on the manuscript.
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