Physica A 249 (1998) 232–238
Subharmonic surface waves in vibrated granular
media 1
D.C. Rapaport ∗
Physics Department, Bar-Ilan University, Ramat-Gan 52900, Israel
Abstract
Using a simplied model of a dissipative discrete-particle uid we study the formation of surface waves under vertical vibration. The only dissipative forces included in this two-dimensional
molecular dynamics simulation act normally to the line of contact during collisions; transverse
frictional forces commonly used in granular models are not required. Depending on the parameters, surface waves are observed to form with a half or a quarter of the driving frequency.
c 1998 Elsevier Science B.V. All rights reserved.
Keywords: Granular media; Surface waves; Molecular dynamics
1. Introduction
Granular materials [1] continue to provide a theoretical challenge, with simulation
playing an important role in attempting to understand the mechanisms responsible for
the complex behavior. Among the many fascinating properties exhibited by these systems are the surface waves that appear when a thin granular layer is vertically vibrated
[ 2 – 6]. Novel aspects of this phenomenon described recently include wave patterns having the form of stripes and hexagons [3,4], period doubling [4], and localized excitations [5]. Several simulations of vertically vibrated granular systems have been reported;
the goal of the earlier work was, e.g., to examine the convective ows that develop
[7], and the ability of vibration to segregate grains according to size [8]. Following the
latest surface-wave experiments, two simulation studies have focused on the nature of
the waves themselves [9,10]; both describe observations of standing-wave patterns at
half the driving frequency in 2-D systems with rigid side walls. The former employed
a soft-disk model; the transverse damping forces commonly used in modeling granular
ows are stated to be essential for observing stable wave patterns. The latter considered
1
Partially supported by the Israel Science Foundation.
author. E-mail: [email protected].
∗ Corresponding
c 1998 Elsevier Science B.V. All rights reserved
0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 4 7 0 - 6
D.C. Rapaport / Physica A 249 (1998) 232–238
233
a hard-disk system, with collisions governed by a velocity-dependent restitution coecient; to avoid inelastic collapse – an artifact of event-driven simulation in which the
collision interval tends to zero due to energy dissipation – an articial time cuto was
introduced to restrict the dissipation rate. A very recent report [11] demonstrated the
comparatively good agreement that exists between the surface wave dispersion relations
obtained from simulation and experiment.
It is plausible that surface waves represent a more general phenomenon associated
with the excitations of a dissipative many-particle system, one that is not strongly
dependent on a particular granular model. Motivated by this idea, we have carried out
a series of molecular dynamics (MD) simulations of soft-disk particles subject only to
a (relative-) velocity-dependent damping force that acts in a direction normal to the line
of contact. This force is the only means of energy dissipation; other frictional forces
used in studying granular ows are omitted from the model. Furthermore, to avoid
possible spurious eects due to vertical walls, these have been replaced by periodic
boundaries; while horizontal periodicity will tend to select wavelengths commensurate
with the system width, it can be argued that this is preferable to the unknown eects of
rigid walls. The claim [10] that wavelike eects produced in a soft-disk system might
be due to collective excitations in a network of densely packed disks is countered by
the observations that the oscillations are of suciently large amplitude that particles
tend to lose contact with their neighbors for part of each cycle, and that the material
rapidly comes to rest when the driving force is removed; the advantage of a “soft”
interaction is the avoidance of the inelastic collapse endemic to the hard-disk model.
2. Computational details
The particles interact with a force based on the Lennard-Jones potential: for particle
separation r the force is f r = (48=r 2 )[(=r)12 − 1=2(=r)6 ]r, with a cuto at rc =
21=6 ; this produces a strong excluded-volume repulsion acting over a narrow range,
so that in a typical collision only relatively small overlap occurs. A damping force that
depends on the relative velocity, C, acts between particles with r¡rc ; this has the form
f n = −n (C · r)r=r 2 , and a value n = 5 is sucient to ensure rapid energy dissipation.
These two elements, or their equivalents, are included in all granular simulations, but
the other forces that are generally introduced to mimic friction (involving transverse
damping or restoring forces, as well as rotational motion) are omitted.
The particle sizes are chosen randomly within a limited range to reduce the tendency
of the material to pack as a lattice. Reduced units are dened in which the value of
(and the mass) of the largest particles, and , are set to unity; such a choice is
standard in MD simulations and establishes the time scale. A gravitational force g is
also present. Conventional MD simulation techniques are used [12], and the equations
of motion are solved by the leapfrog method. The container has a specied width
and periodic lateral boundaries, and the base consists of a vertically oscillating row of
overlapping disks whose position is y(t) = A sin(2ft). The force law given above also
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D.C. Rapaport / Physica A 249 (1998) 232–238
applies to interactions with the base. A dimensionless acceleration, = (2f)2 A=g, can
be dened; for particles unable to store elastic energy the particle layer cannot separate
from the base unless ¿1.
Measurements of the wave prole are based on the mean particle height in a series
of (typically 64) vertical columns covering the entire width of the system. Mean height
is less susceptible to noise than maximum height, and because the waveforms involve
the entire bulk of the material, and not just the upper surface, little information about
the waveform is lost; the tendency for a few of the topmost particles to take ight
makes it dicult to devise an alternative automated scheme for characterizing the wave
prole. Once the system has had sucient time to reach a “steady” state, a series of
observations of the wave prole are carried out. A computational “stroboscope” is
employed that captures the waveform at a xed phase in each base vibration cycle –
choosing the instant at which the base is at its lowest position adequately resolves the
behavior for the parameter values used here. To allow for the period doublings, four
separate averages are evaluated, each covering every fourth cycle.
Because the behavior appears to depend on all the parameters of the problem, we
consider just a single vibration frequency f = 0:4; the number of particles is 10 × W
(where W is the container width) – this ensures a layer thickness of approximately 10
particles. Measurements begin only after waiting 100 vibration cycles, and the following
40 cycles are then used to produce the wave prole averages. The results shown
here are limited to cases where clearly resolved stationary waveforms develop; many
parameter combinations produce time-varying waveforms that might possibly yield to
more detailed analysis.
3. Results
In some cases the waveform develops fairly quickly, within about 30 vibration cycles;
in others the process is slower, with the possible appearance of relatively long-lived
transient states having a dierent number of peaks; in yet other cases the surface waveform fails to settle down over the course of the simulation and the averaged waveform
conceals these short-lived peaks. Transient states can dier between similar runs using
dierent initial random velocities, even though the same kind of waveform eventually
appears. Contrary to the observation of [9], the absence of transverse damping forces
does not prevent the formation of stable waveforms.
Fig. 1 shows examples of how the waveform (as dened above) varies with
for
a xed value of A, and the dependence on A for xed . In both cases W = 130, and
a common vertical scale (extending from 2 to 12) is used. The upper graphs exhibit
very dierent kinds of behavior. No signicant wave pattern develops for = 1. At
= 1:4 there are ve peaks located at alternating positions, an indication that the
system has passed its rst period-doubling bifurcation; the surface waveform oscillates
with frequency f=2. For = 2:2 there are three peaks that appear every second cycle in
alternating positions; the pattern repeats every fourth cycle, so the frequency is f=4 –
D.C. Rapaport / Physica A 249 (1998) 232–238
235
Fig. 1. The -dependence of the vertical waveform (based on mean particle height) for constant A, and
the A-dependence for constant ; each of the four curves is an average over every fourth cycle, the same
vertical scale is used in all cases (it is unrelated to the horizontal scale), and W = 130.
the system has experienced a further period-doubling. Thus, not only are surface waves
corresponding to the rst subharmonic excitation (f=2) observed, as in previous simulations [9,10], but waveforms due to the second subharmonic excitation (f=4) also
appear. The
values of the bifurcations are less than those obtained experimentally
[4,6] because the disks store elastic energy during their extended collisions. The lower
graphs in Fig. 1 show four alternating peaks at frequency f=4 for A = 1, ve peaks
at f=2 for A = 2, and four peaks at f=2 for A = 2:5. To demonstrate that the present
results are in quantitative agreement with previous work, Fig. 1 shows that the wavelength for A = 2, = 2 is 26 (reduced units), a value midway between experiment and
other simulation results [11].
Fig. 2 shows how the number of peaks varies with W, for both f=2 and f=4
excitations. Periodic boundaries naturally favor patterns whose wavelength is an exact
fraction of the system width; there is a limited ability for adjustment, but defects of
various kinds (such as one of the peaks being poorly formed) can also appear in some
instances. Certain parameter combinations yield a superposition of waves with dierent
wavelengths, whereas transient states can show dierent numbers of evenly spaced
peaks.
One fascinating aspect of the phenomenon is the actual motion of the particles. Only
a fully animated view of the self-organization that develops can convey the details of
the behavior, but for the purposes of this article a few screen snapshots will have to
suce. Fig. 3 shows images made at the lowest base position over four consecutive
cycles for W = 70 (A = 2, = 1:6); this is an example of an f=2 excitation, with
the waveform repeating in alternate cycles. Examination of the motion over several
vibration cycles reveals the presence of horizontal, stationary density waves, and that
the peaks of the vertical waveform are located above the density maxima. There is
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D.C. Rapaport / Physica A 249 (1998) 232–238
Fig. 2. Waveform dependence on system width; a common vertical scale is used, and the width is scaled to
t the gures.
Fig. 3. Particle congurations at the lowest base position during four consecutive cycles of an f=2 excitation.
often considerable noise, so that even with well-formed patterns some cycles tend to
have “cleaner” waveforms than others. At certain stages in the cycle, voids of various
sizes can appear between the base and the lowest particles, as seen experimentally.
Further details of the collective motion can be extracted by “painting” the particles
dierent colors, depending on their horizontal positions at a particular instant, and
then watching how these vertical color bands evolve. The results suggest that there is
gradual mixing, but the short-term behavior is dominated by a to-and-fro motion in
which (for the f=2 case) particles ow down both sides of each peak to be swept
up to form the adjacent peaks. A quantitative estimate of this motion is obtained
D.C. Rapaport / Physica A 249 (1998) 232–238
237
Fig. 4. Horizontal displacement (rms and absolute mean) as a function of time.
from the cumulative horizontal (x) grain displacement. Fig. 4 shows both the rms
and absolute mean displacements for two cases (taken from Fig. 1), one in which an
f=2 surface waves occur, the other in which no signicant surface excitation develops.
The behavior in the former case is indicative of diusive motion (∝ t 1=2 ) on which
the lateral oscillations are superimposed (with the absolute displacements appearing to
provide closer agreement than the rms, although this detail requires further study); in
the latter case the behavior is essentially diusive.
4. Conclusion
The results presented here establish that surface excitations are a general feature of
dissipative particulate systems, independent of many of the features usually incorporated
into granular models. In reality, such systems are of course granular media, or synthetic
variations such as metal balls, but this highly simplied model shares their behavior.
Surface waves are the preferred mechanism for dissipating energy, and in this respect
the simulation has succeeded in capturing the essence of the phenomenon, including
the multiple period doublings observed experimentally. Future 3-D simulations should
provide a glimpse of the much richer variety of behavior that can arise, but even for
the 2-D case there is a large parameter space that we have only just begun to explore.
The challenge ahead is to utilize the wealth of detail available within the simulational
framework to determine why the system behaves precisely as it does.
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