Experiments with Vertically Vibrated Granular Materials
Kathryn Brothers, Patricia Levon, Daniel Summerhays & Mark Vondracek,
Evanston Township High School, 1600 Dodge Avenue, Evanston, IL 60204;
vondracekm@eths.k12.il.us
Abstract
Several experiments were designed by high school students to investigate
different properties of vertically vibrated granular material. Several layers of materials
such as fine sand and small bronze particles were vibrated at different frequencies and
amplitudes, and unusual, localized patterns called oscillons were observed. The patterns
that form can be examined in detail as part of an inquiry project in high school or college
classes or as an independent science research project since the behavior of vibrated
granular materials is not fully understood theoretically. This article also encourages other
high school students and teachers to become involved in independent science research.
Many common, everyday materials like sand, salt, and sugar are granular
materials. Though they are common, they have truly fascinating and unexpected
properties under certain conditions. For instance, piles of sand can be stable up to a
critical slope angle, when the pile breaks down and a resulting avalanche appears. But
rather than having the whole pile collapse, only a thin layer of sand near the surface
collapses and ‘flows’ down hill.1 On the other hand, when a dish of sand is shaken
vertically, properties associated more with fluids are observed. In this article we describe
some relatively simple and inexpensive experiments that can be done in high school or
college physics labs to investigate properties of vertically vibrated granular materials.
The waves, ripples, and varieties of interference patterns commonly observed and
associated with liquids have also been found in vertically vibrated granular materials.
Researchers have discovered that when thin layers of granular materials like sand, small
copper balls, or small glass beads are vertically vibrated, they form patterns and waves
that are sustained for long periods of time.2 In addition, at low frequencies (around 20
Hz) small mounds of particles called oscillons can appear. When the peaks of the
oscillons are connected, honeycomb and checkerboard-like patterns can be seen, and at
other frequencies the oscillons form patterns of stripes and hexagons.3 As the system is
vibrated up and down, the particles move in vertical, circular convection currents and are
constantly circulated through the formation of an oscillon. At higher frequencies (near 35
Hz), Paul Umbanhowar, a leading researcher at Northwestern University, has observed
the formation of lines and waves.4
When trying to develop theoretical models that describe the fascinating motion of
vertically vibrated granular materials, we can begin to represent the vertical position of
the surface that pushes the granular material by
Y = A cos (t) .
(1).
Here Y is the vertical displacement from equilibrium, A is the amplitude of motion,  is
angular frequency (and is related to the driving frequency by  = 2f, where f is in hertz),
and t is time. In order to describe the resulting motion of the system, the velocity and
acceleration can be found by differentiating Eq. (1) to get
v = (dY/dt) = -Asin (t) (2)
and
a = (dv/dt) = -A2 cos (t) . (3)
The magnitude of the maximum upward acceleration occurs when cos ( t) is equal to
one, or amax = A2. We are interested in the acceleration because this is directly related
to some of the forces that individual particles of the material experience. In terms of the
driving frequency, the maximum acceleration can be represented as amax = A(2f )2 =
A(42f 2). From this, an equation to represent the acceleration of the particles over the
acceleration due to gravity can be written as
= (a /g) = [A(42f 2)] /g ,
(4)
where  is referred to as the acceleration amplitude. The acceleration amplitude is a
dimensionless, scaled variable that has been used in earlier analyses of the phenomena
studied in this investigation. The values of amplitude (A) and frequency (f) can be
determined experimentally, and  can be calculated. We present the derivation of the
acceleration amplitude here for the benefit of those readers who go on to read other
research articles in this field, where  is used extensively.
At present, no accepted theoretical model exists that can accurately predict or
fully explain all the patterns that are observed in vertically vibrated granular materials.
One may think it should not be too difficult since we have expressions such as pV = nRT
for gases containing billions of molecules treated as solid particles. But gas molecules
are far apart with respect to the size of the molecules, meaning we can ignore interactions
between molecules, whereas samples of granular materials with far fewer particles are
very close together and the constant interactions between adjacent particles are quite
difficult to calculate for the system. This is unfortunate because the overall behavior of
granular materials is an area of interest for various industrial engineers as well as geoscientists, where, for example, the avalanching of a large system of particles is important.
In addition, this phenomenon is interesting for the many-body problem that it is
associated with.
Previous studies of vertically vibrated granular materials have focused on finding
appropriate frequencies and amplitudes that will lead to various patterns in the material.
Other experiments have looked at the effects of size of the particles and the number of
layers of particles in a container on the patterns that result. But many aspects of the
motion have not been studied experimentally in any depth. These would include the
vertical and horizontal mixing properties of the layers of granular material when these
patterns are present. Also, high frequency (>50 Hz) has largely been overlooked in
previous studies. Because experiments to test these phenomena are relatively simple,
numerous experiments in an interesting area of science (normally completely ignored in
physics courses) are possible to carry out in class or by interested students.
The Experiment
In order to study the properties of a granular material that was vibrating against
gravity, a support structure for the sample must be built and calibrated. We constructed
two different systems for different experimental runs. One support structure consisted of
a square wooden board that was raised on wooden legs so it rested above a PASCO
mechanical vibrator (Model SF-9324). A hole was cut out of the wooden board in order
to place a four-legged aluminum strut that was clamped to the board. The aluminum strut
had a central section that was large enough for a 90-mm diameter plastic petri dish to rest
on it. This central section was large enough to keep the petri dish stable and level, and to
prevent it from tipping to one side and affecting the formations in the sand. Plastic petri
dishes were used instead of glass dishes because they are lighter and would not interfere
with the vibrating as much. In addition, we found that the plastic dishes created minimal
electrostatic effects with the sand. The petri dish was secured to the aluminum strut and
provided a volume in which to pour fine sand. The sand had an average diameter of
approximately 0.42 mm. The mechanical vibrator was attached to the central section of
the aluminum strut with a banana plug connector that fit into the vibrator and strong
epoxy. A PASCO digital function generator – amplifier (Model PI-9587) was used to
drive the vibrator and shake the sand vertically. Two rulers were attached on the board
near the petri dish so that distances between oscillons could be measured. Figure 1 is a
picture of the complete experimental apparatus. The second structure consisted of a
number of cardboard pieces glued to the actual vertical vibrating rod of the mechanical
vibrator. A petri dish was then attached to the top piece after being leveled.
Figure 1: Picture of the support stand to which the petri dish was attached.
The petri dishes can be filled with a granular material of choice. One of our
experiments used seven layers of fine white sand while another experiment used bronze
powder. We found that bronze powder, due to its superior spherical uniformity and
smaller size, produced effects that were much clearer and more well defined than sand,
which consists of larger particles that have jagged surfaces. Part of the experimental
process will be to find an appropriate number of layers of the material that will make
oscillons and patterns. If too much material was used, the patterns became asymmetric or
disappeared altogether.
The frequency could be adjusted over large ranges of values using the digital
generator. The amplitude varied inversely with frequency and was measured with a pen
fixed to the petri dish that drew on a piece of fixed cardboard. We found the optimal
amplitudes for frequencies that produced patterns in the sand.
We would like to outline one novel experiment that was done that produced
preliminary results that, to our knowledge, have not been made before, in order to
demonstrate how this type of experiment can lead to original research and inquiry
projects. Once several sets of patterns were identified for our experimental apparatus, we
proceeded to investigate how layers of sand mixed in the vertical direction as well as the
horizontal direction. Some theoretical models predict there should be significant vertical
mixing but little horizontal mixing,5 but to the best of our knowledge there have been few
experimental investigations into the validity of these predictions. We decided to take
some of the white sand we were using and add food coloring to make different colored
samples of sand. The idea was to use different colored layers of sand to track the mixing
of the layers when the sand was vibrated. The white sand absorbs food coloring and
keeps its original shape and collision properties. The dyed sand was baked for several
minutes in an oven at 350°F in order to evaporate any absorbed moisture. The dyed sand
was tested by itself to ensure the same patterns appeared and were identical to the
original white sand at the same amplitudes and frequencies. To determine the time
progression of mixing, we used a Sony Digital 8 camcorder to record the shaking petri
dish of sand, and looked at the surface patterns frame-by-frame. The time between
frames was 1/30th of a second, and the vibration frequency was at 17 Hz. Figure 2 shows
waves that are formed at higher frequencies with sand, and Figure 3 shows an oscillon
pattern in the bronze powder at around 19 Hz.
Figure 2: Time-ordered sequence of wave patterns formed in sand at 40 Hz. There is about
0.5 seconds between shots. The sequence goes from top left to top right, then bottom left to
bottom right.
Figure 3: Photo of oscillons at 19 Hz. The material being used is fine bronze powder.
Horizontal mixing was tested using two different methods. The first method
consisted of filling half of the petri dish cross-sectional area with white sand and the
other half with blue sand, with each half covering a half-circle when looking at the dish
from above. The system was vibrated vertically and the two halves mixed very little. In
some runs there was some amount of mixing (over a period of 60 seconds), and this was
almost entirely due to the fact that the dish became more unbalanced and non-level over
such long time intervals. Some individual particles drifted slightly away from their local
convection currents and bounced around the dish randomly. This was also apparent in
plain white sand when small impurities (individual grains of black sand) could be seen
remaining in local, circular and elliptical paths at some locations while small numbers of
individual grains moved around the dish in random paths over much greater distances.
The individual grains that moved around much greater areas of the petri dish can perhaps
be best described as looking like small particles that undergo Brownian motion on the
surface of some liquids. Normally these grains were observed to follow convection
current patterns, but we were not able to measure or document the paths of those that
moved throughout larger areas.
The second method used to test for horizontal mixing was to use petri dishes with
walls cutting across the petri dish, splitting it into four equal sections (a quarter of the full
circle). We looked for any changes of the normal pattern formation at the same
frequencies and amplitudes stated earlier. The walled petri dishes appeared to have no
effect whatsoever on the formations of oscillons, their patterns, or their vertical mixing.
Any waves that were generated appeared to go through the walls at higher frequencies
and the barriers did not affect the oscillon formations. This was as expected since we
observed little horizontal motion and mixing.
These data and observations suggest there are complicated forces acting on
individual grains of sand as well as layers of sand, and that where they end up vertically
is random. The global, periodic patterns that are seen on the surface, which is an effect of
larger collections of particles, are not indicative of chaotic motions individual grains are
going through locally. One can imagine that individual grains of sand collide quickly
with other grains, with most of the collisions happening off-center. After colliding, there
would be horizontal components of momentum, but because of the large numbers of
particles in the system there would be only short horizontal paths before colliding with
other particles. This set of collisions would result in very little net horizontal
displacement for the majority of grains, but there would be larger vertical displacements
due to the vertical forces involved (gravity and the driving force of the petri dish on all
grains of sand). The resultant motion of most grains of sand would be something along
the lines of an ellipse, and any model that eventually works and describes vertically
vibrating granular systems will need to account for this random motion, perhaps by
statistical means.
Conclusions and Further Study
We have been able to make some new observations and measurements of the
behavior of vertically vibrated granular materials. It is clear from the experiments there is
little horizontal mixing, and that the small amount that does occur is mostly due to the
fact that the system is not completely level over long time periods of vibration. The
motion of the vast majority of particles is vertical, and such motion manifests itself in a
wide array of patterns depending on the combination of frequency and amplitude of the
shaker.
These experiments were carried out by Kathryn, Patricia, and Dan between their
junior and senior years of high school. Their papers6 were recognized locally and
nationally in several competitions, including the Intel Science Talent Search, the Junior
Science and Humanities Symposium, and local and state (Illinois) science fairs. We want
to encourage other high school teachers to work with interested students and sponsor
independent and original science research.7 It is possible to do good, novel research with
relatively simple and inexpensive equipment, particularly in physics. There was never a
need for university lab facilities or equipment, and the students designed and built their
own experiments. Below we list several other investigations that could be done in this
area of study to give some further ideas for others to try.
Future investigations could involve tracking the movement and progression of a
single grain of material through the dish by using a single, different-colored grain. This
would allow observing and measuring local effects rather than global effects. Note that
the number of layers could be varied to see how the mixing changes. Other objectives
could be the investigation of the differences in patterns and formations of different
materials and different sized particles, as well as different sized containers. Particles of
glass and metal, which would be more spherically symmetric and uniform in size than
sand, could be vibrated and compared/contrasted with sand, especially the timing data.
Also, similarities and differences in the behaviors of granular materials that are vertically
vibrated could be compared to the behaviors of these materials in other environments,
such as rotated systems and avalanching.
Many people may consider something like vibrating sand or powders to be a
simple system at first sight, but clearly it is more complicated when one begins working
with it. The work presented in this paper shows there are aspects of this phenomenon
that have largely been ignored, and by making more precise and detailed measurements,
more sophisticated models will one day lead to a better theoretical understanding of these
fascinating granular systems. It could also allow teachers to develop sets of inquiry
projects for students, which fit in nicely with most local, state, and federal science
standards, since this area of physics is likely never introduced in any introductory physics
course.
References
1. H.M. Jaeger and Sidney Nagel, “Physics of the granular state,” Science 255, 15231530 (March 20, 1992).
2. Fransisco Melo, Harry L. Swinney, and Paul Umbanhowar, “Transition to
parametric wave patterns in a vertically oscillated granular layer,” Phys. Rev. Let. 72
(1), 172-175 (January 3, 1994).
3. Fransisco Melo, Harry L. Swinney, and Paul Umbanhowar, “Periodic, aperiodic,
and transient patterns in vibrated granular layer,” Physica A 249, 1-9 (1998).
4. Paul Umbanhowar, “Patterns in the sand.” Nature 389, 541-542 (October 9,
1997).
5. Paul Umbanhowar, Professor of Physics at Northwestern University. Interview
by the authors of this paper, October 24, 2001, Evanston, IL.
6. To view copies of research papers written by Evanston students, see
http://facweb.eths.k12.il.us/chemphys/science_research_papers.htm.
7. Robert Horton and Mark Vondracek, “Creating and maintaining a high school
physics research program,” Phys. Teach. 42, 334-338(Sept. 2004).