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Statistical characteristics of particles on a liquid surface Backer, Lara; Moreno, Sofia Jan 5, 2012

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2  STATISTICAL CHARACTERISTICS OF PARTICLES ON A LIQUID SURFACE               Lara Backer Sofia Moreno   Project Sponsors: Dr. Bud Homsy Dr. Harish Dixit   Project 1259 ENPH 479 The University of British Columbia January 5, 2012  3  EXECUTIVE SUMMARY  The purpose of this project was to measure statistical properties of a multi-particle system of rod-like particles, approximately 10mm in length and 1mm in diameter, placed on the surface of a fluid in a funnel as it drains, serving to hasten the surface tension effects due to the ?Cheerios Effect?. Top-view images of the system were automatically captured as the experiment progressed and the fluid drained. Approximately five images were selected from fifteen experiments. The particles in the images were identified using MATLAB and Image J, and the data was run through MATLAB scripts that generated radial, orientation, and void area (areas of fluid enclosed by particles) distribution plots that denoted the packing efficiency of the particles.  The radial distribution plots, denoting the probability density of particles encountered at different distances from a reference particle were found to be more or less independent of initial configuration and the rate of the fluid draining from the funnel; in all experiments, the probability of encountering a particle at a distance of one particle width increased over time.  Similarly, for orientation distribution plots, the initial configuration and rate of fluid draining did not seem to have a large impact on the final configuration of particles, but it was noted that at the end of experiments, the particles tended to be oriented in the same direction, particularly for particles at one particle width from each other, as expected. The void area distributions indicated that the cylindrical particles tend to pack efficiently because the number of smaller void areas increased with time for all experiments, found by comparing the initial void area distribution to the final distribution.  Running experiments with different initial particle concentrations (numbers of particles present) and measuring the radial, orientation, and void area distributions would further support or contradict results of this project, which indicate that the final configuration?s distribution functions would always be the same. Furthermore, the method of detecting void areas can be improved to distinguish void areas which are almost enclosed by particles, but due to a small connection to a larger void area are not considered isolated. This could provide a better gauge of the packing efficiency of the cylindrical particles.   4  TABLE OF CONTENTS  Executive Summary ....................................................................................................................................................... 3 List of Figures ................................................................................................................................................................. 5 1.0 Introduction ...................................................................................................................................................... 6 1.1 Background and Significance of Project ....................................................................................................... 6 1.2  Statement of Problem and Project Objectives ............................................................................................. 6 1.3  Scope and Limitations .................................................................................................................................. 7 2.0 Discussion ......................................................................................................................................................... 8 2. 1 Theory .......................................................................................................................................................... 8 2.2  Methods ...................................................................................................................................................... 9 2.2.1 Experimental Methods ............................................................................................................................ 9 2.2.2 Image Processing ..................................................................................................................................... 9 2.2.3 Data Analysis .......................................................................................................................................... 10 2.3 Experiment ................................................................................................................................................. 12 2.3.1 Experimental Apparatus ........................................................................................................................ 12 2.3.2 Analysis and Flowchart .......................................................................................................................... 12 2.4  Results ........................................................................................................................................................ 13 2.4.1 Radial Distribution Results ..................................................................................................................... 14 2.4.2 Orientation Distribution Results ............................................................................................................ 15 2.4.3 Efficiency Distribution Results ............................................................................................................... 16 2.5  Discussion and Sources of Error ................................................................................................................. 18 3.0 Conclusions ..................................................................................................................................................... 19 4.0 Project Deliverables ........................................................................................................................................ 20 5.0 Recommendations .......................................................................................................................................... 21 Appendix A - Code ....................................................................................................................................................... 22 Image Analysis Code ................................................................................................................................................ 22 ImageJ Macros ........................................................................................................................................................ 22 Complete Data Code ............................................................................................................................................... 22 Radial and Orientation Distribution Codes ............................................................................................................. 22 Void Area Code ........................................................................................................................................................ 22 Appendix B ? User Manual .......................................................................................................................................... 23 References ................................................................................................................................................................... 24  5  LIST OF FIGURES Figure 1: Bubbles aggregating on the walls of a petri-dish (7) ...................................................................................... 8 Figure 2: Schematic of an example of the ?Cheerios Effect? (7) ................................................................................... 8 Figure 3: Example of Selected Images for a given experiment ...................................................................................... 9 Figure 4: Normalization area in domain ...................................................................................................................... 10 Figure 5: Example result for radial distribution function ............................................................................................. 11 Figure 6: Stokely et al. results of void area distribution of research in two-dimensional packing of prolate granular material (6) .................................................................................................................................................................. 11 Figure 7: Particles seeded on a glycerin surface .......................................................................................................... 12 Figure 8: experimental setup ....................................................................................................................................... 12 Figure 10: Particle centroids and experiment domain ................................................................................................ 14 Figure 9: Experiment Flow Chart ................................................................................................................................. 13 Figure 11: Radial distribution functions for initial configuration. From left, rapid-intermediate-slow ....................... 15 Figure 12: Radial distribution functions for final configuration. From left, rapid-intermediate-slow ......................... 15 Figure 13: Orientation distribution functions for initial configuration. From left, rapid-intermediate-slow .............. 16 Figure 14: Orientation distribution functions for final configuration. From left, rapid-intermediate-slow ................ 16 Figure 15: Initial void area distribution for rapid drip rate .......................................................................................... 17 Figure 16: Final void area distribution for rapid drip rate ........................................................................................... 17 Figure 17: Initial void area distribution for an experiment with intermediate drip rate ............................................. 17 Figure 18: Final void area distribution for an experiment with intermediate drip rate .............................................. 17 Figure 19: Initial void area distribution for an experiment with slow drip rate .......................................................... 17 Figure 20: Final void area distribution for an experiment with slow drip rate ............................................................ 17 Figure 21: Three particles enclosing a void area ......................................................................................................... 21 Figure 22: Three particles almost enclosing a void area.............................................................................................. 21      6  1.0 INTRODUCTION This section outlines the significance of the research, as well as the scope and limitations of the project. 1.1 BACKGROUND AND SIGNIFICANCE OF PROJECT In the 1940?s, Nicolson studied the interactions between particles on a fluid surface 4. His research focused on the use of ?bubble rafts? as models of imperfections of crystal and polycrystalline lattices, like those found in metals. His research proved that grain boundaries, grain growth, and the behavior of other material imperfections could be observed by considering lattices composed of bubbles on a fluid surface as solid-like.  One area of microfluidics gaining interest today is that of self-assembly, discrete objects forming ordered and disordered structures. Most recently, deeper knowledge and understanding of lateral attractive forces and surface tension effects have been used to construct hollow, elastic shells made of self-assembling colloidal particles 2. These shells, called ?colloidosomes?, act as shields for emulsion droplets or bubbles. They are of particular interest due to their potential for cellular immunoisolation used for cell encapsulation because their composition is variable, using different solvents, particles, and contents.  Technological applications are interested in the behavior of the attractive forces between particles on a fluid interface, as seen in the above-mentioned studies 3. Research and an understanding of the attractive forces and dynamics of particles on an interface have made the creation of a controlled structure of aggregate particles possible. Therefore, this research can also lead to improved manufacturing of micro-elecromechanical systems? components.  However, few studies have considered multi-particle systems. Our project focused on a multi-particle system of rod-like particles, approximately 10mm in length and 1mm in diameter, and the effects of capillary attraction on the particles? respective position and orientation angle in the system?s final configuration. Therefore, the majority of the results generated in this project were compared to each other.  Blackman observed the behavior of 2 cylindrical and spherical particles on a liquid-gas interface. The study allowed him to determine the effect of viscosity on the interfacial drag coefficient 1. These results led Paria Karimi, a summer co-op student of Dr. Homsy, to investigate the capillary attractions in a multi-particle system, using rod-like particles of the average size previously mentioned. Our project continued this investigation, focusing on measuring and interpreting statistical properties of the particles such as the radial, orientation, and void area distribution functions for given particles on a fluid surface.  1.2  STATEMENT OF PROBLEM AND PROJECT OBJECTIVES  The goal of this project was to measure the statistical properties of cylindrical particles on the surface of a fluid as described below. By the end of the project, the properties that were investigated were the probability density functions of another particle existing at a distance of r+dr from an arbitrary particle, the orientation angle of another particle at an angle of ?+d? with respect to an arbitrary particle, as well as the distribution of enclosed ?void areas? of fluid between particles to measure the efficiency of particle packing.  7  1.3  SCOPE AND LIMITATIONS In this project images taken from Dr. Homsy?s previous co-op student and images from experiments conducted by the group members were analyzed and the resulting radial, orientation, and probability distributions were compared. Given the pre-existing data and that collected over the last semester, approximately fifteen experiments were selected to be analyzed and of those fifteen, five images were selected from each.  The experimental set up was altered, but the resulting images were improved. Only one camera was used, as opposed to two, and it was mounted higher above the experiment. The particles were placed on the surface of the fluid in a funnel, and the duration of each experiment (i.e. drip rate) was controlled by a valve.  The final selected images were then run through a MATLAB script, mostly written by the previous student, to identify the particles; however, any remaining, unidentified particles had to be manually identified using Image J, a free software. Finally, the data was run through MATLAB functions that generated radial, orientation, and void area distribution graphs. These results were compared with other results from experiments of similar duration.  The results look similar to the outcome expected. However, it should be noted that the project is limited by the quality of the images. Due to the constantly changing quality of each image, the image processing code was not written to run unaltered for each image. Ensuring the parameters in the code are optimized to identify particles of a certain image time consuming, limiting the scope of this project with respect to the number of experiments conducted during the project term, as well as the improvements made to experimental set up.    8  2.0 DISCUSSION This section contains the theory relevant to the project, the methods used to conduct experiments and analyze data, and a flowchart of the system. Results will also be presented and discussed. 2. 1 THEORY Particles floating on a liquid surface experience forces due to surface tension and buoyancy, among others. These forces cause capillary attraction, affecting the particles to self-assemble in ordered and disordered structures. An example of this can be seen by North Americans almost every morning in their cereal bowl. The cereal pieces tend to form clusters on the liquid-milk surface, due to this everyday example, the phenomenon has been labeled by many as, the ?Cheerios effect? 7. Other common examples are: the bubbles of carbonated drinks aggregating on the walls of a glass, as seen below in Figure 1; dry seasoning sprinkled on soup; and single strands of hair adhering to the periphery of a bathtub 5.   FIGURE 1: BUBBLES AGGREGATING ON THE WALLS OF A PETRI-DISH (7)  When evaluating the behavior of particles on a liquid surface at equilibrium, one must consider the balance of linear and angular momentum both on the surface plane and out of plane 7. For example, buoyancy, a vertical force, is an important factor in determining the behavior of particle interaction.   Consider the simple case of one bubble on a water-air interface, near a glass wall, as shown in Figure 2; this model is also used to explain the behavior between multiple bubbles 7. The geometry at the wall of the interface between the fluids illustrates the fundamental reasons these attractive forces are observed. At this point the surface of the water is significantly altered, as is that of the air, also known as the meniscus effect, and because the bubble is buoyant, gravity exerts an upward force on the bubble, but it cannot separate from the liquid-gas interface. Therefore, it moves upward along the meniscus, simultaneously moving closer to the wall, so there appears to be an attractive force between the bubble and the wall.   FIGURE 2: SCHEMATIC OF AN EXAMPLE OF THE ?CHEERIOS EFFECT? (7)  9  As was stated in the introduction, few studies consider multi-particle systems, which is why the results of our experiments and those of Dr. Homsy?s co-op student were compared to each other. The performance metric is not to compare our results to those of another research project; rather, the performance metric is to look for similarities in the results of this project?s experiments. 2.2  METHODS 2.2.1 EXPERIMENTAL METHODS The images tracking the movements of the particles were captured by a camera mounted above the funnel. An image was captured every one to two minutes, depending on the duration of the experiment. Draining of the fluid was controlled by a valve, resulting in experiments of different durations. The following types of experiments were chosen to be analyzed: 1. Rapid drip rate (1-2 hours to complete the experiment) 2. Intermediate drip rate (3-4 hours) 3. Slow drip rate (>5 hours) A detailed description of the apparatus and other equipment is provided in the following section.  To analyze the different types of experiments, four or five images were taken from each experiment at a given time interval. For example, if the experiment lasted four hours to completion, the first image was chosen in addition to one image taken every hour following the first image, totaling five selected images, as seen in Figure 3. This allows for statistical analysis at different stages of each experiment.  For the purpose of improving image quality, only one camera was used, so only one experiment ran at a given time. The previous student ran two experiments simultaneously because she wanted equal dripping rates. Lighting distribution was slightly improved from that in the images of the previous student, and the camera was mounted farther from the funnel to allow for a larger viewing frame.  t=0 t=1hr t=2hr t=3hr t=4hr FIGURE 3: EXAMPLE OF SELECTED IMAGES FOR A GIVEN EXPERIMENT  2.2.2 IMAGE PROCESSING MATLAB and Image J were both used for image processing, to identify the particles. MATLAB code described in the Appendix A was written to generate radial and orientation distributions, as well as to generate a void area distribution for a given image.  Another method suggested to identify particles was to initially identify all the particles in the first image. Given that each image is captured every one to two minutes would allow for memory tracking. Therefore, given the initial data of the system the sequential particle data could be predicted and then checked. This method was not 10  pursued due to time and software limitations. MATLAB, although extremely powerful, could not do this on its own, so we decided to continue with identifying particles for each image separately.  2.2.3 DATA ANALYSIS Each selected image was analyzed and the resulting radial, orientation, and void area distributions were found. This was achieved by identifying the domain, the circular fluid-funnel interface, as well as the centroids and orientations for all of the particles.  The radial distribution and orientation distribution functions depicted the probability densities of the particles. The graphs were generated by using methods of binning; every time a particle was encountered, the distribution function bin was increased at the radius that it was encountered at, with respect to a reference particle. Every particle was used as a reference particle, comparing all other particles to it, so the bin encompassed the entire experimental domain.   The values added to the bins were normalized by the overall density of the particles in the experiment and the area of the radial strip at r+dr from the reference particle encompassed in the domain. Multiple methods were attempted to determine this area. Initially, the area of the strip was found by determining the angle of the radial strip that was included within a square domain from the image, but determining the quadrant that the strip intersected the square domain proved to be difficult. Another method was to determine the angle of the radial strip inside a circular domain, by equating equations of circles with desired radii to find intersection points. Figure 4 shows this method, with the area at r+dr from the white reference particle only shown where it is within the blue experimental domain. This method was chosen to be used in the final code, as it was the most efficient method and the easiest to implement. This code is referenced further in Appendix A.     The resulting plots of the radial distribution function for the final particle configuration in the experiments were expected to look similar to that in Figure 5. A peak was expected at the distance of one ?radius?, or width of a particle; this is assuming most particles will be packed side by side. Similarly, a peak was expected at one radius for the orientation distribution because as the particles begin to pack it is most probable to find parallel particles next to a given particle.   FIGURE 4: NORMALIZATION AREA IN DOMAIN 11               FIGURE 5: EXAMPLE RESULT FOR RADIAL DISTRIBUTION FUNCTION  The resulting void area distribution plots were expected to be downward sloping, particularly for the images taken in the later stages of an experiment. An example of such a graph from Stokely et al.?s research is presented in Figure 6. This is due the expectation that as the fluid drains and the area covered by the particles increases in relation to the surface are of the fluid, the number of small, enclosed void areas should increase and the number of larger void areas should decrease 6.    FIGURE 6: STOKELY ET AL. RESULTS OF VOID AREA DISTRIBUTION OF RESEARCH IN TWO-DIMENSIONAL PACKING OF PROLATE GRANULAR MATERIAL (6)  The proposal for this project, ?Measuring the Statistical Properties for Experiments with Different Initial Conditions?, suggested measuring statistical properties such as initial concentrations of particles as well as different initial particle orientations and placements and drip rates, but this was not pursued due to time restrictions.   Therefore, the main focus of the project was to improve the image processing procedure and to generate a function that could measure the statistical properties of a given state in an experiment; the remaining time was spent reproducing rapid, intermediate, and slow experiments to support previous research.   12  2.3 EXPERIMENT 2.3.1 EXPERIMENTAL APPARATUS 162 Nylon 6-6 cylinders were used as the particles in the experiment. They were chosen because they produced a sharper contrast which improved image analysis compared to the contrast provided by acrylic and glass type particles. The cylinders were 10mmx1mm, and were placed on the surface of a solution of 80% glycerin and 20% water in a glass funnel, as shown in Figure 7. This solution was selected based on the density, which was higher than the density of the particles, allowing them to float, but low enough to allow the solution to drain from the funnel at a reasonable rate. The funnel was mounted on a metal stand. Plastic tubes ran from the bottom of the funnel to empty the water by a height difference, and the rate of water emptied was controlled by a ball valve at the end of the tube. This forced the particles to group faster than normal, in a quasi-static method.  FIGURE 7: PARTICLES SEEDED ON A GLYCERIN SURFACE Figure 8 shows the entire experimental apparatus, which consisted of a 1280x1024 pixel camera with a 96dpi and 8-bit depth, was mounted on a horizontal beam, on a stand above the funnel apparatus, and was connected directly to a laptop to record images. Three IKEA halogen lamps were placed around the funnel to illuminate the particles, making them easy to recognize in images.  FIGURE 8: EXPERIMENTAL SETUP  2.3.2 ANALYSIS AND FLOWCHART  The images were taken from the laptop, and analyzed using a combination of Matlab 2012 and ImageJ (an open source software). A flowchart of the experimental process is shown in Figure 9, and a step by step procedure for 13  running scripts for image analysis and calculating statistical properties of the particles using Matlab and ImageJ is included in Appendix B. Scripts were written to process the images and extract the particle centroids and orientations, and were able to locate 100% of the particles in the experiment. Scripts were also written to use this data in order to calculate the radial and orientation distributions, and to determine the efficiency of particle packing by calculating the void areas, and are referenced further in Appendix A.   2.4  RESULTS As mentioned previously, three flow rates were used to compare the data from the experiments: slow experiments, which took approximately 10 hours to drain; intermediate experiments that ran approximately 4 hours, and rapid experiments that took place in approximately one hour. Images of these experiments are shown in Figure 10 and 11 for initial and final configurations, respectively.  Place particles on fluid surface Start to drain fluid from funnel by opening valve to desired flow rate Record images of the experiment Use ImageJ and Matlab to obtain centroids and angles of particles from images Use Void Area and Radial Distribution codes to generate graphs of the statistical properties of particles FIGURE 9: EXPERIMENT FLOW CHART 14      From each of these experiments, the centroids of the particles were located inside the funnel domain as shown in Figure 12. In this figure, the axes merely depict the pixel values of the image.  FIGURE 12: PARTICLE CENTROIDS AND EXPERIMENT DOMAIN 2.4.1 RADIAL DISTRIBUTION RESULTS This gave rise to plots of the radial distribution functions for each of the particles, shown in Figure 13 for the initial particle configuration directly after particles were placed on the surface, as well as the plots shown in Figure 14 for the final particle configuration when the particles were the most tightly packed without becoming caught on the funnel walls. FIGURE 10: IMAGE OF INITIAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW FIGURE 11 : IMAGE OF FINAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW 15        The vertical axis depicts the how density of the particle system varies by distance from a reference particle. This radial distribution function, called g(r), was normalized so that in a homogeneous mixture at a large distance from a reference particle, the density of particles found would become uniform. The horizontal axis is the radius of the image in pixels, divided by the particle width in pixels. Thus, a peak of particles on the radial distribution graphs at one ?radius? means that the probability is high to find particles lined up next to each other. 2.4.2 ORIENTATION DISTRIBUTION RESULTS Additionally, the orientation distribution graphs shown in Figure 15 and Figure 16 were generated for the same rapid, intermediate, and slow drain time at the final and initial particle configurations. The functions displayed in these graphs, called Q(r), describe the orientation of particles located at certain distances, with respect to a reference particle.  FIGURE 13: RADIAL DISTRIBUTION FUNCTIONS FOR INITIAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW FIGURE 14: RADIAL DISTRIBUTION FUNCTIONS FOR FINAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW 16       The x- axis in these figures is the same as for the radial distribution function graph, and the normalization for the probability distribution function is such that there will be a larger peak for particles that are aligned and have no difference in orientations; an orientation distribution of zero signifies a random distribution of particle orientations. 2.4.3 EFFICIENCY DISTRIBUTION RESULTS The graphs of the resulting initial and final void area distributions are presented as follows in Figures 17-22. It should be noted that each void area was normalized over the average area of a particle, and the axes are log-log scale.  FIGURE 15: ORIENTATION DISTRIBUTION FUNCTIONS FOR INITIAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW FIGURE 16: ORIENTATION DISTRIBUTION FUNCTIONS FOR FINAL CONFIGURATION. FROM LEFT, RAPID-INTERMEDIATE-SLOW 17   FIGURE 17: INITIAL VOID AREA DISTRIBUTION FOR RAPID DRIP RATE   FIGURE 18: FINAL VOID AREA DISTRIBUTION FOR RAPID DRIP RATE   FIGURE 19: INITIAL VOID AREA DISTRIBUTION FOR AN EXPERIMENT WITH INTERMEDIATE DRIP RATE   FIGURE 20: FINAL VOID AREA DISTRIBUTION FOR AN EXPERIMENT WITH INTERMEDIATE DRIP RATE   FIGURE 21: INITIAL VOID AREA DISTRIBUTION FOR AN EXPERIMENT WITH SLOW DRIP RATE   FIGURE 22: FINAL VOID AREA DISTRIBUTION FOR AN EXPERIMENT WITH SLOW DRIP RATE   The initial void area graphs on the left, display a greater number of large than small void areas. In the case of the slow drip rate, this is not as obvious; however, when compared other initial graphs, the peaks occur at approximately the same values for void area. In the first rapid drip rate experiment, comparing the final to the initial state, the vertical axis has a smaller scale in the figure to the left. Therefore, it can be stated that the number 18  of smaller areas is increasing. Similarly, a change in scale occurs for the following results and the same observation is made, that the number of smaller void areas increases between initial and final distributions.  2.5  DISCUSSION AND SOURCES OF ERROR From Figures 13 and 14 in the previous section, it is clear that the radial distribution function peaks at 1 radius, that is to say, at 1 particle width separation. Therefore, particles were more likely to be present at closer distances to a reference particle.  Furthermore, between the final and initial configurations, the peaks of the radial distribution functions increased for small radii. After letting the particles settle, they were more prone to be in ?rafts?, lined up next to each other. Additionally, the number of particles in rafts was seen to increase between initial and final configurations for all drip rates, so the final configuration was not dependent the rate that the funnel emptied. For the orientation distribution in Figures 15 and 16, the orientation peaks at 1 radius as well. Thus, particles that were closer together were more likely to be aligned with the same orientation, whereas further from the radius the particle alignments were more random.  Similarly to the radial distribution, the orientation graph peaks at 1 radius increased at the final configuration, and the orientation distribution was not found to be dependent on drip rate. When comparing the final distribution plot of void area sizes to number of void areas, it is apparent that at least two from each type of experiment resemble the predicted behavior of a downward sloping line. However, most of the final graphs also have a decrease in smallest areas, but please note that the scales on the final graphs are not the same as those of the initial graphs. After interpreting the change in scale, it is observed that the number of smaller areas, in fact, increases and the number of larger areas decreases. These results imply that the cylindrical particles have efficiency in their packing formation, as the surface are of the fluid decreases. Packing efficiency describes the number of contacts a particle can have; for example, the maximum number of contacts would result in 100% packing efficiency. However, given the data obtained through these experiments, the exact value of their packing efficiency cannot be determined at this time; a mathematical derivation of packing efficiency is necessary, like that of spherical particles. If a theoretical value can be obtained, it should be used as the reference from which to measure the packing efficiency of the particles. The main source of error in the results is due to significant differences in image quality. The lighting is not consistent between experiments and images, resulting in the need to change parameters in the image processing code for every selected image, which often cannot identify all of the particles. Image J allows the user to manually identify the particles by drawing a line over each particle, then clicking the F1 key. The images taken from the previous student?s experiments sometimes exclude particles because the camera is too close to the funnel. Therefore, some information is omitted from the first images selected in a given experiment. Finally, the method used to distribute the particles on the fluid surface can have an impact on the final state of the system. The particles were sprinkled by hand onto the fluid surface, and tweezers were used to separate particles that had fallen on each other. This method was consistent enough for the scope of this project; however, an even more random and consistent method for distributing particles should be attempted, if desired, to observe if it impacts the final configuration of the particles. 19  3.0 CONCLUSIONS Code was written to analyze the statistical properties of particles that are affected by surface tension under the ?Cheerio Effect?. The properties that were chosen to be analyzed were the radial and orientation distribution functions, and code to compute the packing efficiency of particles based on the sizes of ?void areas?. Experiments were run under three conditions: rapid, intermediate, and slow drip rates, and the graphs of the statistical properties for both the final and initial configurations were presented. Radial and orientation distributions were found to be independent of drip rate, and it was found that at the final, most drained configuration, the probability density distribution peaks increased at short distances compared to the initial configuration. Thus, particles were more inclined to be close to each other and aligned at a similar orientation angle, regardless of their initial configuration and the drip rate. For the void area distributions, the packing efficiency was also found to increase at the final configuration compared to the initial configuration as there was an increase of small void areas present in the final configuration.     20  4.0 PROJECT DELIVERABLES List of Deliverables: 1. Final Recommendation Report 2. Matlab script files for radial distribution function and orientation distribution function, image processing, and calculating void areas 3. ImageJ macros files to locate particle centroids and orientations 4. Documentation on the use of the scripts and macros  There was no need for anything to be purchased for the experiment, as all equipment was already purchased and present in the UBC Complex Fluids lab.  21  5.0 RECOMMENDATIONS The scope of this project did not allow for some further development of the results and analysis, mainly due to time restrictions. The following are recommendations of alternate techniques and additional data to be collected and analyzed. 1. Radial, orientation, and void area distributions should be measured for different initial concentrations of particles. This involves running the same types of experiments, rapid, intermediate and slow drip rates, with more particles and less particles than used during this research project. The results would prove whether the initial concentration of particles has an effect on the final radial, orientation, and void area distribution functions.  2. Improving void area analysis to detect void areas that are not enclosed, but are almost surrounded by particles. For example, three particles can surround a void area to make a triangle, as seen in Figure 23. However, the MATLAB script to find void areas cannot detect the triangular shape like that of Figure 24, so the triangular area is considered part of a larger void are that extends outside the triangular perimeter.   FIGURE 23: THREE PARTICLES ENCLOSING A VOID AREA   FIGURE 24: THREE PARTICLES ALMOST ENCLOSING A VOID AREA   The resulting void area distribution would be more informative if it were possible to identify the second triangular area as a separate void area, not part of any connected area, because it would give a better indication of the packing of the particles and the structures formed by the particles in the final configuration. Setting a threshold for the minimum distance between particles that would result in a void area would allow areas such as the triangular area in Figure 23 to be analyzed. The exact value of the threshold is unknown at present, but can be estimated and tested.  3. Even though the void area distributions indicated that the cylindrical particles had some packing efficiency, the exact value of that efficiency cannot be measured from the results obtained in this project. A theoretical analysis of packing efficiency for cylindrical particles could be conducted to obtain a reference from which to compare the already existing data. For example, spherical particles have a maximum contact number of 6, for any two dimensional lattice; if all of the particles were to have a contact number of 6 then the packing efficiency of spherical particles is 100%. Studies have been done to estimate the contact numbers of rod-like particles for three dimensional packing using simulations, such as Wouterse et al, and found that for ?spherocylinders?, rod-like particles, with high aspect ratios the contact number was much higher than that of spheres for a given volume concentration 9.  22  APPENDIX A - CODE IMAGE ANALYSIS CODE This code, entitled ?InitialParticleFinding.m? was written in Matlab primarily by a previous co-op student. By altering threshold parameters, it was used to filter the images and detect particles. It inserted the detected centroids and orientations into a file, and produced a final image with the remaining particles that were not detected. IMAGEJ MACROS Two macros were written: one created files for to store the data from the unidentified particles, and one enabled the user to draw a line across a particle, which allowed the user to easily store the resulting centroid and angle. While all the particles could be detected from this code, the previous image analysis allowed the user to more efficiently locate the majority of the particles. COMPLETE DATA CODE This Matlab code, entitled ?CompleteData.m? combined the particle data found in the original image analysis and the ImageJ files. RADIAL AND ORIENTATION DISTRIBUTION CODES  The radial and orientation distribution code was written in Matlab and titled ?rad_dist_circle.m?. It was used to create the radial and orientation distribution graphs, and referenced another code, ?FindAlpha.m? to determine the area to use to normalize the probability distributions. VOID AREA CODE Additional code was written in Matlab, entitled ?Voidarea.m?, which calculated the void areas present in the image, and provided the user with a graph of the void area distribution.    23  APPENDIX B ? USER MANUAL In order to process an image, follow the steps below: 1. Use InitialParticleFinding.m in Matlab: a. Follow instructions at the top of function to adjust parameters (particularly image file name call, other parameters mentioned depend on brightness of image; if particles are deleted between figure 1 and 2, threshold is too low) b. Figure 1: Initial image c. Figure 2: converts to black and white d. Figure 3: Thresholds below 1000 currently e. Figure 4: Deletes edges from particles to help with analysis; gradient method f. Figure 5: Further erosion of non particulate shapes g. Figure 6: Deletes items less than the number of given pixels (600 currently) h. Figure 7: Deletes first round of particles, adds centroids and angles to file i. Figure 8: Further erodes and filters j. Figure 9: Deletes particles below threshold number (500 currently) k. Figure 10: Last round of particle deletion and addition to file; additional particles not detected are saved as ?unidentified particles? l. Figure 11: Displays identified particles 2. Navigate to ImageJ program a. File->open->?unidentified particles? b. Plugins->macros->edit (both Initialize Data File_Macro and Centroid_Angle_Macro) c. In Initialize Data File_Macro, edit the file to open to be in desired folder; name it ??/unidC.txt?, at the top of the file, go to macros->run macro; file should now be created d. In Centroid_Angle_Macro, go to macros->install macro e. In image and ImageJ main window, select the *Straight* tool, click and drag from one end of particle to opposite end along the long way, click F1 after particle is selected, the centroid and angle resulting should now be stored. Repeat for all particles, they will now be numbered. f. In pop up results window, file->save as ??/unidA.txt? g. In saved file, delete the first line of centroid and angle so that it can be read in by Matlab h. Close imageJ 3. In Matlab, open CompleteData m-file a. Ensure that w and h are the correct pixel width and height (find by opening an image and using the data cursor on a particle) b. Change line 21 to the name of the original image (want to plot the rectangles on this) c. Change line 29 and 30 to the correct min and max radius (domain radius should be between these); difference between min and max should not exceed 100 d. Change line 33 sensitivity to finding ?domain? to 0-1 e. If multiple circles are found, choose only one (alter lines 35-39 for this) f. Run CompleteData.m (will only work for Matlab 2012) 4. Open VoidArea.m a. Change filename, line 20, to original image name b. Run code for void area distribution  5. For radial and orientation probability distributions, run rad_dist_circle.m 24  REFERENCES  1 J. Blackman, ?Capillary attraction of parallel cylinders and spheres", USRA Summer report, Dept. of Mathematics, Univ. British Columbia, Vancouver, Canada (2011).  2 D. Dinsmore, M F. Hsu, M. G. Nikolaides, M. Marquez, A. R. Bausch, and D. A. Weitz, ?Colloidosomes: Selectively Permeable Capsules Composed of Colloidal Particles", Science 298, 1006-1009 (2002).  3 H. N. Dixit and G. M. Homsy, ?Capillary effects on floating cylindrical particles?, Preprint, 2012.  4 M. M. Nicolson, ?The interaction between floating particles?, Proc. Cam. Phil. Soc., 45, 288295 (1949).  5 Phil Schewe and Ben Stein, Number 745 #2, September 15, American Institute of Physics (2005).  6 K. Stokely, A. Daicou, and Scott V. Franklin, ?Two-Dimensional packing in prolate granular material?, Physical Review E 67, 051302 (2003).  7 D. Vella and L. Mahadevan, ?The Cheerios effect", Am. J. Phys. 73, 819-825 (2005).  8 Weeks, E. (n.d.). Pair Correlation Function. Retrieved 2012, from Emory Physics Website: http://www.physics.emory.edu/~weeks/idl/gofr.html 9 A. Wouterse, S. Luding, and A. P. Philipse, ?On contact numbers in random rod packing?, Granular matter, Vol 1 pp. 169-177 (2009). 

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