Figure 5
Patterns - Figure 6
Matthew Evan Thrasher
Bartlesville High School
Bartlesville, Oklahoma 74006
Do you miss the time before you started school, when you could wile away your days creating your own empire of castles and kingdoms in your sandbox? Some of us have trouble leaving this age behind, but as we mature, we create more complicated patterns. Since the behavior of granular material is not well understood, yet it is used in countless industries, I studied the patterns created and the general behavior of a vertically oscillated granular layer in my 1997 Science project. To experiment, a shaker system was designed and created from a speaker driven by an amplified computer-controlled sine wave. At the core of the experiment, I tested five variables; the frequnecy of vibration, the amplitude of vibration, the depyh of the granular layer, the granular material, and the sample container. I observed the granular layer behave in intriguing manners. First, at low amplitudes, I observed movement phenomenon ranging from donut-convection to gravity-defying leveling. At higher amplitudes, I observed extraordinary patterns from hexagons to stripes. As to the most intriguing occurrence, oscillons, which are isolated peaks and craters of material, they behave in the most mysterious of fashions. Overall, the patterns are dictated by the density of the oscillons and geometry, not by sample container, granular material, or depth. From my data and results, I formulated seven theories. As the next step in my project, I plan to research application of my theories to industries and the real world. Hypothecial applications could range from matter distribution in the universe to pharmaceutical mixing.
Remember playing in your childhood sandbox? Most children noticed a few things about the nature of sand: it can easily be shaped, it was fun to play with, and it made cleaning clothes a nightmare. But non-linear dynamic scientists, what adults who play in sandboxes are called, notice a multitude of other intriguing and unexplainable phenomena in the behavior of sand and other granular materials: granular materials expand instead of contract when pressure is applied. The pressure in a silo is independent of depth after a certain point (Which is why an hourglass is filled with sand instead of water). My research explores extraordinary patterns and phenomenon which occur spontaneously when layers of granular material are shaken vertically (1).
The behavior of granular materials is not just of aesthetic interest. Countless industries, such as agriculture, pharmaceutics, plastics, and chemical, use granular materials in both raw and finished products (2). However, amazingly little is known about the true behavior, interaction, and intrinsic, potential worth of granular materials and the properties they add to normal substances. Thus, the area of research for this science fair project is granular materials: their behavior, their interactions, and their application in industry.
Research pertaining to granular materials is limited. Fundamental behavior, such as internal stresses and critical angle has been explored in the past. But only recently has work been initiated on the topic of vertically vibrated granular materials. The idea for this project came from an article in the August 31, 1996 issue of Science News magazine (3). I hypothesize that vibrated granular materials behave as a collective group, much like a fluid. Through this project's experimentation, this relation to a fluid was proven at the base level, yet granular materials behave in completely unique ways. This is evident in rogue oscillons, which are peaks of granular materials over twenty-five times the height of the granular layer.[fig. 11]
As in every experiment, the equipment and materials used have to be designed in such a way to truly isolate and test the variables to insure accurate results. Over a period of two months, the equipment was designed, materials were gathered, and a shaker system [fig. 1] to vertically oscillate a sample of grains was built.
A sine wave, the purest subharmonic waveform, oscillates the sample (4). Since the frequency of the sine wave, which is the cycles per second, and the amplitude, the total displacement of the wave, needs to be controlled with exacting accuracy and precision, the computer shareware WaveGen version 2.1c by David D. Dight was employed on a Gateway 2000 computer. The frequencies were tested between 1-300 hertz. The signal output from the computer was not powerful enough to drive the sample, so a JVC RX-250 FM/AM Computer Controlled Receiver amplified the signal. The amplitude was controlled varying the amplitude [volume] on the receiver with a supercritical step-volume controller. From the receiver, an 11-inch woofer speaker was adapted to drive the sample by attaching a circular piece of 1/8 inch Plexi-glass with a 6-inch carriage bolt attached upright on the center of the Plexi-glass was epoxied to the speaker cone. In order to protect the voice coil from any abrasive grains and to stabilize the sample to keep oscillation strictly vertical, Plexi-glass was placed over the speaker. A hole was drilled into the center of the Plexi-glass sheet and a washer with a vertical extension was greased and affixed around the hole. To further stabilize the speaker and allow leveling, the speaker was housed in a speaker box. [Fig. 1] Door stops on each corner allowed leveling. A pair of tweeters was employed to sound in excessive driving amplitudes which distort the waveform and could potentially overload the equipment.
To visualize the behavior of the samples of granular materials, two halogen lamps illuminated the sample at a low angle to show depth, contours, and the dynamics. Since the patterns observed are going in and out of phases at half the driving frequency [from five to 150 times a second], the human eye simply combines the images into one. A Sharp Viewcam Model Vl-H450 was employed to take images of the granular materials while being oscillated. This allowed not only real-time visualization, but also the capabilities to analyze frame by frame the behavior of granular material on a computer.
Since the speaker has limited driving force, three containers were designed to minimize mass and maximize flat, level area. A shortened glass beaker, a circular aluminum bowl, and a octagonal glass dish were employed to contain the various samples. Different shapes of containers tested the side-wall effects on the patterns. The bottoms of each container were basically flat, except the glass beaker, whose bottom had two circular, concentric ribs. This variable tested the effects of the driving surface's shape and contours on the granular material's behavior. Nuts were epoxied onto the bottom of each container in the center to allow attachment to the carriage bolt for vibration.
Another important variable is the depth of the granular materials. Research resulted in two methods being developed and utilized. The depth of the material was calculated by the volume of granular material divided by the area of the container's bottom surface divided by the average diameter of the granular material. The depth was controlled directly or by tilting the container to one side via the door stop leveling. This caused a steady decline of particle depth to the uphill side; thus, providing a range of particle depths.
Five separate variables were tested in this experiment. All experiments in references were done in a vacuum; however, these experiments were completed in air at 700 feet above sea level to study the effects of air on the material's behavior (5). Frequencies were tested at various frequencies from 1 to 300 hertz. The amplitude was varied from 0 gamma units [dimensionless acceleration amplitude=4pi2f2Ag-1] (6) to 3.5 gamma units. Three different containers were each tested. Fourteen different granular materials with differing size [dust to 2.5 mm], shape [circular or irregular], density [1.6-7.9 g/cm3], composition, and coefficient of restitution were analyzed with the shaker system (fig. 10). Each granular material was tested separately and mixed. Finally, the depth of the media was varied. All in all, this experiment was designed for five variables: the granular material, depth, sample container, frequency, and amplitude of vibration.
Basically, only one major difficulty occurred during this experimentation. The nuts that attach each container to the carriage bolt was not in the exact center; thus creating torque and causing perfect leveling to be very difficult. This was then used as an advantage to test a multitude of particle depths simultaneously.
Generally, granular media behave in most ways like a liquid or gas, in a few ways like a solid, and in some ways like nothing else. These experiments show this through the classification of the phenomenon observed into the two categories: patterns and movement. These observed behaviors are applied to the world.
Movement, as I define it, is a phenomenon that does not have a recursive visual pattern. At this stage, the granular materials act as a fluid. Movement occurs while the granular level is in a flat state (around 1 g), during the transition from a flat state to a pattern (around 2 g's), at extreme amplitudes (around 3 or more g's), or in all cases. Seventeen different movement behaviors occur during these three stages. These first five behaviors occur at all frequencies and amplitudes. Most of the behaviors observed have no names yet, thus I labeled them according to their behavior and italicized them in this report.
First of all, when a layer of granular material vibrates, the different horizontal layers separate, because of the same transfer of energy that is exhibited when one pool ball solidly hits another. This behavior is called dilation. (fig. 9) In order for either layer to move, the layers must separate from each other because of the compacting of round balls. Industry could use granular materials as a stable foundation, because of the lack of dilation when pressure is applied. A granular layer when vibrated undergoes a repetitive sequence of unpacking (dilation), during free-flight, and packing (compacting), during impact with the sample container. The settling of plastic, pharmaceutics, and other granular products during shipping leaves spaces of air that can either erode, harm the product, or consume shipping space that could be used more efficiently. Thus, using dilation and the packing sequence together could increase efficiency and shelf-life dramatically through pre-compacting of granular materials.
One interesting behavior of grains is that when pressure is applied from the top, the granular layer creates small "arches" of stress that distribute the weight so that only a small fraction of the media supports the entire load, while regions of media feel no stress at all (7). This is similar to the modern workplace and the distribution of stars and heavenly bodies throughout the universe(8).
If the dish is not perfectly level, exaggerated leveling occurs at low amplitudes. Exaggerated leveling is the building up of a mound on the lower side of an unleveled container. One possible explanation is that because of the tilted bottom each grain is attracted to the lowest point by gravity. This behavior could be useful in industry to level machinery, belts, and equipment or products. At higher amplitudes, gravity-defying leveling occurs. This kind of leveling flattens the granular layer according to the angle of the container. This shows that extreme amplitudes cancel out gravity's influence. When the amplitude increases dramatically, a gush of material to the uphill side collides with the opposite side; thus, a shock wave is sent back and forth. This causes sloshing tides.
As a common phenomenon, critical angle, usually around 35 degrees, but a function of the diameter of the media, is the angle in which the material will flow. After critical angle is exceeded, only the top few layers, slide or avalanche down until the side of the mound reaches a stable angle below critical angle. This behavior can be used to efficiently hold stockpiles of material in containing units.
When the sides of the mound exceed the critical angle, the top few layers avalanche down the side of the mound until a stable angle is reached. Then, because of exaggerated leveling, these grains are forced underneath the mound, due to of gravity's influence and other grains sliding. Once the grains reach the lowest point at the corner of the container, the other grains push the lowest grains up and out of the way. This set of events initiates the circular flow of material. The three forms of this convection are heap, horizontal, and donut convection. Heap, which is convection in heaps, and circular convection, which is convection around the plate, occur because of this behavior.
When a handful of sand is dropped, it forms a crater and disperses, meaning to spread. When dispersion occurs in vibrating containers, the granular material moves outward. But the side walls force the granular material on top in the same action as heap convection; thus, a donut-shaped convection is initiated. It is conjectured that the middle layers have a complex vortex movement, like the cloud layers of Jupiter. One use of convection is to keep the material in raw stockpiles mixed to insure the material is fresh and prevents a substance to becoming outdated. It could also be used for mixing purposes.
When more than one size of media is vibrated, the different kinds of grains sort into layers by size. This is called size segregation. When the media achieves free-flight, the smaller grains fit in gaps between the larger grains; thus, over time, the smaller grains work their way to the bottom, while the larger grains are forced to the top. One possible use for size segregation is separation. An automatic separator can be constructed to siphon off different layers of different sized media.
When vibrating fine materials at high amplitudes, the air resistance of the particles becomes large enough to have a significant effect, called air heaping. (fig. 8)(9). In the case of air heaping, mounds are formed because each particle wants to fall faster than air resistance will allow, so like a flock of geese cutting through the air, the particles form a group; thus, leaving void spaces. During high amplitudes in air, bubbles occur when chambers of air are caught beneath the layer during free-flight (fig. 9). As the air is forced from beneath during collision, air collects into large cavities, like geese. These "bubbles" are expelled with some media at the surface. I call these geysers. When geysers occur within heaps, bubbles form only at one specific particle depth that varies with the frequency and particle diameter. So, sets of bubbles form in this geyser region which creates an entire line of geysers. This could be applied to create clouds of dust that increase efficiency of reactions and other processes in all industries, including oil and gas production.
When solidified chunks of granular media are vibrated, the "rock" is broken down into smaller granules, because of cracks and fissures produced by the collisions. Perhaps, this could provide insight into geology, plate tectonics, and slow crack growth. Some materials break down further and produce a cloud of dust. Again, this cloud of dust applies to many industries and increases the efficiency of reaction and processing.
When the granular material is vibrated at about 2 gamma units, patterns spontaneously arise out of the flat layer. Patterns are defined in this project as formations with repetitive and phasing visual phenomena. Most movement qualities have no significant influence on the layers at this point because of the extreme amplitude.
Imagine shaking a flat, granular layer up and down. Now, if the shaking increases past the acceleration due to gravity, the grains start to leave the plate(fig. 9). If everything were perfect, the granules would simply bounce up and down, but imperfections, chaos, complex interactions, and dynamics complicate models. Introducing a second layer causes the two layers of grains to be knocked off free-flight because of mid-air collisions, the granules now begin to behave differently than in the single layer model.
It is exactly these added layers that cause patterns of granular material to arise out of a flat layer. In the multiple layer model, a random ball is sent off of its original trajectory. When this ball is set off course, it eventually hits another ball. That ball hits another, and so on. The effect of these collisions is a grouping of material. This group hits the plate, causing a sloshing consisting of a mountain-phase and a valley-phase that are localized, called oscillons. Since every action must have an equal and opposite reaction, the peak smoothly transforms into a crater. At the critical amplitude of oscillons (about 2.2 gamma units), I think that a random ball has enough energy to displace enough balls to get an oscillon started. I call it the activation energy. The oscillon is then randomly and spontaneously created from the flat layer. The sloshing oscillon then disperses to fill the entire sample container with oscillons, just like a drop of water in a still pond fills the entire pond with ripples. This exact same phenomenon occurs with oscillons except the spreading patterns consist of the two alternating phases because of continual excitation. With this single ball, an entire set of events are initiated that end with a beautiful pattern. Patterns are composed of many fused oscillons. Once an oscillon pattern initiates, it is possible to sustain it at amplitudes below that of the original amplitude. This is amplitude hysteresis (from the word "history"(10). I believe amplitude hysteresis exists because the sloshing motion has to get started, but once created is easy to maintain.
Oscillons usually exist fused with other oscillons that compose a pattern. But in the transitions between patterns, there lies an oscillon region where oscillons can be isolated.(fig. 3) These isolated oscillons behave in a manner completely different than that of patterns. In patterns, frequency and amplitude dictate what the pattern will be. In oscillons, the only aspect that changes is the size, which is directly and linearly related to the frequency and amplitude. The average oscillon height is usually four times the depth of the granular layer. Again, since the behavior of oscillons is unknown, I could only name them accordingly.
Oscillons behave like matter at the nuclear level. Oscillons exhibit a magnetic behavior, like a nuclear attraction. When two oscillons of opposite phases come close or are generated close to each, they fuse into a diatomic crystal. When two oscillons of the same are close, they repel each other or an oscillon of the opposite phase is created in between. Once combined in a crystal, oscillons have a very strong bond, like a nuclear bond. Never in these experiments has a crystal been separated.
Oscillons also form patterns. Oscillons can combine in crystals in numerous configurations: a diatomic pair of oscillons, like diatomic gases; strings of oscillons, like a polymer(fig. 2); and a trimer, a group of four with one central oscillon and one oscillon every 60 degrees. Observed in patterns, but never isolated, I believe square, triangle, and hexagon patterns exist. Another new phenomenon observed is that of rogue oscillons which are oscillons over twenty-five times the height of the granular layer that spew material, and may be many oscillons in-phase with each other while a resonance within the ball is reached. Rogues compare to geysers, but rogues occur in patterns.(fig. 11)
When two oscillons are initially created at the same time, they can be of different phases. The relative time each is created at and has existed can be figured by how much of the sample container is occupied by oscillons of that phase. When two patterns of opposite phase meet, or a pattern meets with a side wall, a generation pool is created (fig. 7). A generation pool is a transition between two different phases. In the case of the transition between the side wall and pattern, this generation pool forms with new oscillons that merge into the pattern. This flow is in the direction of the uphill side in an unbalanced container. In the case where the transition is between two patterns of opposite phase, the two patterns generate new oscillons flowing into the pattern of the opposite phase. Thus, this motion creates the visual effect of horizontal bubbling.
Varying the material produced the same basic, qualitative patterns, but the quantitative aspects changed. This is simple to understand using simple logic. Because balls are different sizes, the amplitude and frequency change relative to the particle size, all of which depend on a standard of length. Size, frequency, and amplitude are relative to particle size. So, in the results and conclusions, a dimensionless unit is used in accordance with a natural unit, a unit occurring in nature, such as the speed of light or sound. With differing coefficients of restitution and density, these variables simply needed relative amplitudes. Amazingly, the shape of the material did not matter. I think this is because of the net effect of the collisions and has to do with the universal nature of granular materials, not the insignificant shape. In conclusion, granular materials that differ in size, shape, density, coefficient of restitution, and composition have the same universal behavior. I even believe that any material, if scaled properly, would behave in the same way. This proves that these patterns are a universal behavior of granular materials. The depth of the material has numerous effects on the pattern. The thicker the layer the more stable the pattern and the higher amplitude is needed to excite the pattern (11). In the transition areas between patterns, thicker layers take the pattern of higher frequencies, while thinner layers take the lower frequency pattern.
The patterns were the same, independent of the container. This proves that the behavior of granular media is not dependent on the composition, shape, or size of the container or particles. The only attributes needed are a hard sturdy bottom, with sides and something light enough for driving. Depending on the frequency and amplitude, the pattern created varies. Amplitude and frequency have two different, yet coherent effects on the pattern. Increased amplitude increases the size of the oscillons that compose the patterns, while higher frequencies decrease the size of the oscillons that compose the patterns. Increased amplitude and frequency increases the density and the dispersive tendencies of the patterns, respectively. This behavior of patterns is ruled by one unifying nature. Although recondite at first glance, this nature is simply a set of repetitions and ordered periodicty. This nature is ordered by geometry and the packing of circles in more efficient ways.
With increasing frequencies, the pattern undergoes numerous changes. These pattern transformations are ruled by the density of the single oscillons that compose the entire pattern and geometry. (fig. 5, 6) The density of the oscillons is simply defined as the relative number of oscillons in the confines of the sample container. This density, oscillons per square unit, is increased by higher frequencies, the number of cycles per second.
At lower frequencies and amplitudes above the critical parameters for the existence of patterns, matrices of squares occur (fig. 4). With higher frequencies, the pattern changes into a triangle pattern, then to a stripe pattern. After this, the new patterns are interactions between stripes. There are four degrees of stripes: uniform, rolling, ribbing, and streaming stripes. The patterns change into another pattern by an oscillon of the next density forming somewhere, usually the side of the dish, and spreading throughout the dish. In the transition areas between two different patterns, the two different patterns can coexist for considerable periods.
If the amplitude is increased from any pattern, another degree of patterns arise. These pattern conversions are ruled not by density, but by the dispersive degree of the pattern. Higher amplitudes mean that the granular material hits the dish with more force, so the media disperses more and creates larger craters. By increasing amplitude, the patterns go through three degrees of dispersion. The first is the normal, base pattern; squares, triangles, or stripe. The second degree is a hexagon pattern in all cases. The third is nebulae fingers.
As the second degree, hexagons (fig. 12) are formed because they are the most sided polygon that can fit together in a matrix; thus, taking the form of looking more like a circle, an infinite-sided shape. It could then be conjectured that at an infinite amplitude, the shape of the oscillons would be perfectly circular. To account for the problem of circles not fitting to together to form a lattice, I suppose the space in between each circle would be filled by increasingly smaller circular oscillons, assuming the granular material was infinitely small. Hexagons are always formed, but at the higher degrees of stripes and amplitudes, it is harder to visualize. The third degree of dispersion is what I call nebulae fingers because of their visual similarities to star nebulae formations seen in the sky. These nebulae fingers are the chaotic in-phase and out-phase combination of the single oscillons that compose the patterns at extreme amplitudes.
The first degree of stripes is a uniform, in-line striped pattern. (fig. 7) This pattern is formed by the fusing of an entire matrix of oscillons into single rows. The peaks of the fused lines of oscillons compose the mountain or peaks of the stripe. The crater phase of oscillons form the valleys of the striped pattern. When frequency is increased further, rolling stripes, are formed. Rolling stripes occur when two or more sets of stripes that are aligned mix. This creates a pattern of stripes that phase in and out. If the sample container is unleveled, then the striped pattern rolls or travels across the plate; hence, the name. At higher frequencies, ribbing stripes, as I call them, occur when different orientations of stripes mix which creates a Moiré pattern. The net effect of this interaction is a pattern of stripes that is segmented. Finally, the last degree of stripes is the streaming stripes. Streaming stripes occur when two or more set of stripes are flowing against each other and are out of phase relative to each other. These streaming stripes form a terraced semi-flat layer. The layer is actually covered by many stripes except that are so small, they are virtually impossible to see. The only visual sign of them are terraces. I believe these terraces are formed because the fused oscillons that compose a striped pattern are separated or torn apart. This happens in order for stripes to go around each other then refuse beyond the terrace.
With this pattern of occurrence set forth, the pattern at all frequencies and all amplitudes can then be predicted. If the amplitude is greater than that of this first set of patterns, the plate will start to throw the granular layers so high that the sample container will have oscillated another time while the granular media was in free-flight. This creates patterns that are identical of the previous patterns and are called f/x waves, x being the number of times the plate has to vibrate for the granular material to complete an entire cycle. Only F/4 patterns have been observed (13). I believe there are patterns above those observed already. In the periods between the patterns and the next higher order of patterns, such as f/2 and f/4 patterns, there lies a region where the terraces dominate. Terraces at the point are called kinks which can now take the form of spirals, jagged lines, and other bizarre line patterns. In conclusion, geometry dictates the pattern which will form at which frequency and amplitude because of the density of oscillons and the degree of dispersion.
Overall, I have observed the behavior of granular material. From there, I have formulated, discussed, and explained the universal behavior of granular materials in seven theories. In this paper, seven theories have been explained. First, The Theory of Universality and Intrinsically proves that the behavior of granular material is universal and an inherent property. The Theory of Normality explains that the fluid, movement phenomenon at low amplitudes exists because of the inadequate driving energy. Another theory includes The Theory of Oscillon Packing Density, pertaining to the oscillons' density as related to the patterns. Also, The Theory of Periodicity accounts for repetition in the pattern scheme with increasing amplitude. The Theory of Nuclear Behavior compares oscillons as representative of matter at the nuclear level. The Theory of the High Collective describes granular flow at high amplitudes as many independent balls acting collectively to form the whole. Finally, The Theory of Normality explains that granular behavior at low amplitudes is like a fluid.
As mentioned, research done before these experiments is limited. However, the results that were available were congruent to my own and were used as the control in making sure everything was working correctly. To further the impact of this project, I applied these findings to the real world. Through these experiments, insight into the behavior of granular materials, vibrated or not, will hopefully provide understanding when dealing with these materials.
Since the behavior of granular material observed is universal, only a few sources of error existed. The effects of the environment and the air slightly degraded the stability and sharpness of the patterns. The distortion caused by the speaker, amplifier, computer, and wires on the sine wave signal is uncontrollable. The impossibility of perfectly vertical vibration could cause undetermined effects. Although existent, none of these sources deter the universal behavior significantly.
This subject is most theories and not much solid laws. One would be searching new phenomenon. The amazing aspect is that the subject is simple balls. These fluid and nonlinear dynamics perplex and stretch our minds to further our horizons. In doing experiments, one should make sure the equipment and material is adequate before starting, because this is the only major obstacle in most research. Since granular material's behavior is still open for exploration, a few ideas on further future research include studying the effects of driving at resonant frequencies; relationship of patterns, quasicrystals, and extreme amplitudes, performing computer simulations, investigating higher degrees of waves (f/8, etc.) and multi-frequency forcing.
Overall, I hope to provide some insights into the complex behavior of granular materials through my experimentation, research, and ideas.
First and more foremost, I would like to thank Mr. Paul Umbanhowar at the University of Texas at Austin for all of his guidance, advice, and information. As always, my parents have supported me in all my endeavors. I must also recognize all my teachers do for me. Also, Nuclear Metals, Inc. provided me with free research grade material which let me truly investigate the behavior of nature. Finally, I would like to recognize the impact and convenience of the Internet for information retrieval and communication.
