Results and DiscussionIt would seem that the degree of crystallinity, which is defined as the number of grains in a

crystalline state divided by the total number of grains, ranges from 0.63 to 0.71. Binary ratios which are closer to 50%-50% exhibit a larger degree of crystallinity than ratios at the two extremes. This suggests that the exact composition of binary distributions is of some significance.

Refer to the histograms plotted below. It can be seen that these histograms exhibit behavior which is characteristic of a bimodal distribution. This is significantly different from a monodisperse crystallite distribution which has only one peak. Note that the bimodal distribution becomes less defined around the 50%-50% ratio. Also notice that small crystallites are far more prevalent than larger crystallites. The data suggests that the influence of crystallite binding energy may be non-uniform depending on the binary ratio of a specific metallic alloy. This is an important first step in uncovering the significance of crystallite formation in randomly-close packed, granular media.

AbstractThe goal of this project is to quantitatively study the degree of crystalinity and the crystalite size

distrobution in a binary distribution of small spherical grains via statistical methods. This degree of crystalinity shall be measured as a function of the binary ratio of small to large spheres. The configuration of granular spheres will be used as a model system which represents a close analogue to a binary distribution of atoms within a metallic alloy. Some physicists theorize that the degree of crystalinity within metallic alloys has a direct influence upon the exact thermophysical properties which that alloy exhibits.

Andrew Abraham, Advisor: Dr. K. KriebleDepartment of Physics and Earth Science, Moravian College

In Cooperation with: Dr. Y.W. Kim, Lehigh University

For years, physicists have been studying the methods and mechanisms related to the packing of granular material in an enclosed volume (2). Recently, some physicists (6) have been investigating the possibility of extending the concepts of granular physics to areas of thermal physics. Specifically, the exact atomic arrangement of atoms within a given solid is of particular interest. Some solids are crystalline (morphous) in nature. These solids are characterized by a long-range ordering of the positions of the atoms within the solid and typically exhibit a repetitious pattern in their atomic structure (2). In contrast, amorphous solids (glass) do not exhibit any long-range order in their atomic structure. A “pure” amorphous solid would have a 100% randomized distribution of atoms within its atomic structure. In practice, however, most amorphous solids have small pockets of organized “crystallites,” or mini-crystals (6), which are distributed amongst areas of atoms which have purely randomized positions. These crystallites, and the atoms surrounding them, may be molded using a configuration of randomly-close packed, spherical grains. Each grain will approximate the relative size and shape of an atom within a solid.

This study is specifically interested in modeling systems of metallic alloys. A metallic alloy is comprised of a minimum of two different metallic species. In this study, the simplest case of a metallic alloy is molded using a binary distribution of steel ball bearings. The larger bearing has a diameter of 3.23mm ± 5% while the smaller bearing has a diameter of 2.51mm ± 5%. The ratio of the large to small ball bearing diameters is roughly 1.29. To put these values in perspective, note that the alloy bronze is comprised of copper (Cu 29) and tin (Sn 50) (5). Tin, being a larger atom, a has an average atomic diameter of 3.164Ǻ, while copper, the smaller atom, has an atomic diameter of 2.551Ǻ (5). The ratio of these two atom diameters is 1.240, which is only 3.6% different from the binary distribution of the steel ball bearings being used in this study.

The scientific community has long understood the significance of potential and kinetic energy in regards to thermal physics (6). For example, the temperature of an object is defined as the root-mean-square of the kinetic energies of the constituent atoms within that object (6). In its simplest form, temperature is a statistical measure of the kinetic energy of atoms which comprise an object. Obviously, the temperature (linear kinetic energy) of an object will dictate the way in which that object interacts with the physical world. Likewise, the crystallites within amorphous solids represent a certain amount of “binding energy” which holds the atoms of the crystallite together (6). The small amount of potential energy in each member of the crystallite may add up to an energy significant enough to have a large impact upon the thermophysical properties of that particular metallic alloy. It is the goal of this project to measure the average crystallite size, as well as the size distribution, of a randomly-close packed , 2-D configuration of spherical, granular media.

Introduction

To acquire good statistical data, a method was developed which could produce large numbers of crystallites within a (relatively) short period of time. A chamber was constructed which was capable of holding approximately 600 spherical ball bearings. The dimensions of the chamber are 8.0x11.25x0.327cm. Note that the interior thickness of the chamber is just large enough to allow the large ball bearings to freely move about, but does not allow any bearings to rest on top of one another. The upper and lower sides of the chamber are constructed of 8.0x11.25cm sheets of transparent, electrically conductive (prevents coulomb forces) glass. A computer controlled a stepping motor was attached to this glass chamber. The computer would rotate the chamber 4-5 times such that the changing direction of gravity would randomize the distribution of spheres within the enclosure. Once finished, the computer would then trigger circuitry which activated a solenoid which, in turn, activated the shutter on a digital camera. This camera would capture a photograph of the configuration of spheres and store it for later analysis. Once the camera had captured the image, the process would repeat. A total of 270-340 images would be taken for each ratio of binary media. Binary ratios ranging from 90% small– 10% large, to 10% small– 90% large were taken at 10% intervals, each acquiring ~300 images.

In order to extract meaningful statistical data from the images, it became necessary to program an image-processing algorithm which could automatically identify the exact position of each sphere within an image. This program was written using the C programming language for the sake of runtime efficiency. This program has the ability to accurately locate roughly 97% of the spheres within the image, which is good enough for the purpose of this investigation. After identification, a second program was developed using the Python programming language. This program determined which spheres were in contact with one another by comparing their relative positions and accounting for their respective radii. Next, this program used a series of rules and set operations to “build” up each crystallite within the image. The number and size of the crystallites within each image were recorded, compiled, and finally graphed once all images from each binary ratio had been processed.

Experimental Method

I wish to specifically thank three influential people who greatly helped me during my honors research.1) Dr. Krieble -Honors Advisor2) Dr. Roeder -Honors Leazon3) Dr. Kim -Project Coordinator

Each has been more than gracious towards me and has undeniably earned my respect and admiration.

Acknowledgements

References1. Polytetrahedral Nature of the Dense Disordered Packings of Hard Spheres. Anikeenko, A. V. and Medvedev, N. N. June 8, 2007, Physical Review Letters.2. Is Random Close Packing of Spheres Well Defined? Torquato, S., Truskett, T. M. and Debenedetti, P. G. 10, March 6, 2000, Physical Review Letters, Vol. 84.3. Packing in the Spheres. Weitz, David A. February 13, 2004, Science Magazine, Vol. 303, pp. 968-969. www.sciencemag.org.4. Random Close Packing of Hard Spheres and Disks. Berryman, James G. 2, February 1983, Physical Review A, Vol. 27, pp. 1053-1061.5. American Institute of Physics Handbook. 3rd Edition. s.l. : McGraw-Hill, 1972.6. Kim, Dr. Y.W. Bethlehem, Winter 2008-2009. Mulltiple Interviews .7. FERREZ, Jean-Albert. Dynamic Triangulations for Efficent 3D Simulation of Granular Materials. s.l. : École Polytechnique Fedetale de Lausanne, 2001.

Figure 1: Original Data Figure 2: Original, Processed by Image Recognition Program

Figure 3: Crystallites which have been Identified

Figure 4: Experimental Apparatus

Figure 9-13: Data

Poster By: Andrew Abraham