MEASUREMENT OF THE THERMAL PROPERTIES OF A WEAKLY-COUPLED COMPLEX (DUSTY) PLASMA Except where reference is made to the work of others, the work described in this disertation is my own or was done in collaboration with my advisory commite. This disertation does not include proprietary or clasified information. Jeremiah Wiliams Certificate of Approval: Edward Thomas, Jr. Stephen F. Knowlton Asociate Profesor, Chair Profesor Physics Physics Yu Lin Alen Landers Profesor Asistant Profesor Physics Physics Joe F. Pitman Interim Dean Graduate School MEASUREMENT OF THE THERMAL PROPERTIES OF A WEAKLY-COUPLED COMPLEX (DUSTY) PLASMA Jeremiah Wiliams A Disertation Submited to the Graduate Faculty of Auburn University in Partial Fulfilment of the Requirements for the Degre of Doctor of Philosophy Auburn, Alabama December 15, 2006 ii MEASUREMENT OF THE THERMAL PROPERTIES OF A WEAKLY-COUPLED COMPLEX (DUSTY) PLASMA Permision is granted to Auburn University to make copies of this disertation at its discretion, upon request of individuals or institutions and at their expense. The author reserves al publication rights. Signature of Author December 15, 2006 Date of Graduation iv VITA Jeremiah Wiliams, son of Douglas and Mary Wiliams, was born on August 30, 1976 in Binghamton, New York. He graduated from John F. Kennedy High School in 1994. He atended Dickinson College and graduated with a Bachelor of Science degre in Physics and Mathematics in 1998. He entered Graduate School at the University of California, Los Angeles and graduated with a Masters of Science degre in Physics in 2000. After spending thre years as a faculty member in the Department of Physics at Ilinois Wesleyan University, he entered Graduate School at Auburn University to pursue a Doctoral degre in Physics in August 2003. He was maried to Anne-Evan (Kale) Wiliams, daughter of W. Wilford Kale Jr. and Louise Lambert Kale, on December 20, 2003. Their son, Franklin Douglas Wiliams, IV, was born on July 6, 2005. v DISERTATION ABSTRACT MEASUREMENT OF THE THERMAL PROPERTIES OF A WEAKLY-COUPLED DUSTY PLASMA Jeremiah Wiliams Doctor of Philosophy, December 15, 2006 (M.S., University of California, Los Angeles 2000) (B.S., Dickinson College, 1998) 315 Typed Pages Directed by Edward Thomas, Jr. Knowledge of the velocity space distribution function facilitates the characterization of the thermal and dynamical properties of a system. In this work, a comprehensive series of experiments and numerical simulations are used to measure the velocity space distribution function of the microparticle component of a dusty plasma. In particular, this work introduces the use of stereoscopic particle image velocimetry techniques in the area of dusty plasma. This disertation presents the results of extensive simulations of the PIV measurement technique, as applied to dusty plasmas, and the vi discusses the first measurements of the thre-dimensional velocity space distribution function of a weakly-coupled dusty plasma. This work demonstrates that it is posible to simultaneously measure al thre velocity components over an iluminated slice of a dust cloud, alowing for a more acurate measure of the transport and energy properties of the dust cloud. From these measurements, the velocity space distribution function of a stable weakly-coupled complex dusty plasmas has been constructed for a wide range of experimental conditions. From the simulations, it is found that there is a unique mapping function that relates measured PIV distribution to the underlying particle velocity space distribution function provided that the PIV analysis is applied to dust particles that have a size distribution that is nearly monodisperse. This mapping function is strongly dependent on the number density of the microparticle component that is being measured. Experiments are performed using dust clouds composed of either 1.2 ? 0.5 ?m alumina particles, 6.22 ?m diameter melamine microspheres or 3.02 ?m diameter silica microspheres. From the measured distribution functions, the bulk kinetic temperature of the microparticle component was extracted as a function of the neutral gas presure. It was found that the bulk kinetic temperature of the microparticle component was anisotropic and significantly larger than the other plasma species. The heating mechanism responsible for these large temperatures is a function of the neutral presure and appears to be more eficient with higher dust number densities. vii ACKNOWLEDGEMENTS I would like to thank my advisor, Dr. Edward Thomas, Jr., for his support, advice and insight throughout this proces. They were invaluable. In addition to the few tricks up his sleve, I would also like to thank him for his encouragement and for keeping Lord Elroy at bay. Beyond my advisor, there are many people who I would like to thank for fruitful discussions. An incomplete list includes Dr. Steve Anderson for his asistance with some of the technical aspects of the PIV software and technique, Dr. John Gore for his insight on measuring the number density of the dust clouds. Dr. Andrew Post-Zwicker for general discussions and providing the SEM images of the dust particles, Dr. Alen Landers for his insight on the statistical aspects of the analysis, Dr. Jianjun Dong for his insight on distribution functions and thermodynamics and Mr. Wiliam Wiliams for his insight in the area of optics. Finaly, I would like to thank my former instructors, my colleagues, the members of the Plasma Sciences Laboratory, and al of the people whose names currently elude me for their support and encouragement over the years. Finaly, I would like to thank my family for their unending support and encouragement. Without their support and sacrifice, I would not have made it to this point. And of course, I must acknowledge my wife, Anne-Evan, for her support, patience and understanding over the last few years. In particular, thank you for picking up the slack over the last few months when I needed to spend time on this research. vii Style manual or journal used AIP Style Manual Computer software used Microsoft Word ix TABLE OF CONTENTS LIST OF FIGURES....................................................xii LIST OF TABLES...................................................xxv CHAPTER 1: INTRODUCTION..........................................1 1.1 HISTORICAL OVERVIEW............................................1 1.2 BASICS OF DUSTY PLASMA..........................................3 1.2.1 What is a dusty plasma?........................................3 1.2.2 Examples of man-made dusty plasmas.............................11 1.2.3 Examples of naturaly-occurring dusty plasmas......................15 1.3 PREVIOUS WORK................................................20 1.3.1 Thermal studies involving phase transitions.........................21 1.3.2 Thermal studies involving laser heating............................32 1.3.3 Other thermal studies..........................................34 1.4 SCOPE OF THIS THESIS............................................37 CHAPTER 2: APARATUS.............................................38 2.1 EXPERIMENTAL DEVICE...........................................38 2.1.1 Vacuum Vesel..............................................38 2.1.2 Vacuum and Gas Systems......................................40 2.1.3 Plasma and Dusty Plasma Generation.............................41 2.2 PARTICLE IMAGE VELOCIMETRY.....................................45 2.2.1 Principles of Particle Image Velocimetry...........................50 2.2.2 Two-Dimensional Particle Image Velocimetry......................52 2.2.3 Stereoscopic Particle Image Velocimetry...........................62 2.2.3.1 Image Dewarping and Calibration.............................71 2.2.3.2 Reconstructing thre-dimensional velocity information.............75 2.2.4 Stereoscopic Particle Image Velocimetry System at Auburn University..................................................77 2.2.4.1 Description of the system....................................77 2.2.4.2 Verification of the system...................................79 2.2.4.3 Measurement of the resolution limt of the system..................86 2.3 PARTICLE DENSITY MEASUREMENT..................................90 2.3.1 Density Measurement by Light Scatering Technique.................92 2.3.2 Monodisperse vs. polydisperse distributions of the microparticle component..................................................94 2.3.3 Determination of the scatering eficiency...........................95 x CHAPTER 3: SIMULATION OF THE PIV MEASUREMENT..................97 3.1 MOTIVATION...................................................97 3.2 COMPUTATIONAL METHODOLOGY...................................99 3.3 RESULTS.....................................................103 3.3.1 Uniform Mas..............................................104 3.3.2 Mas Distribution...........................................110 3.3.2.1 Fixed width of particle size distribution........................115 3.3.2.2 Efect of width of particle size distribution......................118 3.3.2.3 Weighted PIV calculation..................................122 3.4 DISCUSION OF SIMULATION RESULTS...............................125 CHAPTER 4: EXPERIMENTAL RESULTS...............................128 4.1 EXPERIMENTAL METHODOLOGY....................................128 4.2 RESULTS WITH MICROPARTICLES HAVING A POLYDISPERSE SIZE DISTRIBUTION..................................................139 4.3 RESULTS WITH MICROPARTICLES HAVING A MONODISPERSE SIZE DISTRIBUTION..................................................150 4.4 DISCUSION OF RESULTS.........................................161 CHAPTER 5:CONCLUSIONS..........................................176 5.1 SUMARY OF SIMULATIONS OF THE PIV MEASUREMENT ECHNIQUE........176 5.2 SUMARY OF THE XPERIMENTAL MEASUREMENTS.....................178 5.3 SUMARY OF RESULTS...........................................180 5.4 FUTURE DIRECTIONS............................................181 REFERENCES......................................................184 APENDIX 1: MATHEMATICAL BACKGROUND........................189 A.1.1: MATHEMATICAL FOUNDATION OF THE 2D PIV TECHNIQUE..............189 A.1.1.1 Mathematical description of image formation.....................189 A.1.1.2 athematical description of the Cross Correlation Analysis Technique................................................194 A1.2: PROOF OF THE CORRELATION THEOREM.............................201 A1.3: MATHEMATICS OF THE PINHOLE CALIBRATION MODEL..................203 APENDIX 2: C+ CODE TO SIMULATE THE PIV MEASUREMENT.........207 A2.1: CONFIGURATION FILE FOR THE PIV SIMULATIONS......................207 A2.2: C+ CODE FOR THE PIV SIMULATION...............................209 A2.3: ADITIONAL IBRARIES.........................................221 APENDIX 3: IGOR MACROS.........................................229 A3.1: DESCRIPTION OF IGOR..........................................229 A3.2: ANALYSIS CODE FOR SIMULATION DATA WITH MONODISPERSE PARTICLES....230 A3.3: ANALYSIS CODE FOR SIMULATION DATA WITH POLYDISPERSE PARTICLES.....238 A3.4: ANALYSIS CODE FOR EXPERIMENTAL DATA...........................248 A3.5: 1D DRIFTING MAXWELIAN FUNCTION FIT CODE......................253 xi APENDIX 4: LABVIEW CODE........................................254 A4.1: DESCRIPTION OF LABVIEW......................................254 A4.2: MASTER LABVIEW CODE.......................................255 A4.3: RUN_ME_FOR_VECTOR_PREPROCESING_USE.VI......................257 A4.3.1: Convert_Data_to_Matrix_for_Vector_Preprocesing.vi.............259 A4.3.2: Select_ROI_3_for_Vector_Preprocesing_AUTO.vi................261 A4.3.2.1: Predefined_Thresholding.vi...............................263 A4.3.2.2: Extract_Data_ROI.vi....................................264 A4.3.2.3: Count_Zeros.vi.........................................265 A4.3.2.4: Valid_vectors_method_1.vi...............................266 A4.3.3: Get_File_List_for_Vector_Preprocesing.vi......................268 A4.3.4: Generate_File_Names_for_Vector_Preprocesing.vi................269 A4.3.5: Select_ROI_3_for_Vector_Preprocesing.vi......................270 A4.3.5.1: Manual Thresholding.vi..................................271 A4.3.6: Read_Single_Data_File_m_to_n_measurements.vi.................273 A4.4: RUN_ME_TO COMPILE_Z_LOCATIONS_INTO_CLOUD.VI..................274 A4.4.1: Get_File_List_to Compile_z_locations_into_cloud.vi...............276 A4.5: RUN_ME_FOR_SCATERING_EFICEINCY.VI..........................277 A4.5.1: Manual_Particle_Counting_for_Scatering_eficeincy.vi.............278 A4.5.2: Read_in_Images_(directory structure)_for_Scatering_eficeincy.vi....280 A4.5.2.1: Get_directory_file_list_for_Scatering_eficeincy.vi.............282 A4.5.2.2: Read_in_Image_data_for_particle_density.vi..................284 A4.5.3: Automatic_Particle_Counting_for_Scatering_eficeincy.vi..........285 A4.5.3.1: Adjust Image Display Range (fixed value).vi..................287 A4.6: RUN_ME_FOR_PIV_FACTOR.VI...................................289 A4.6.1: Get_File_List_PIV_factor.vi..................................291 A4.6.2: Compute_PIV_factor_to_file.vi...............................292 A4.6.3: Density_Range.vi..........................................292 A4.7: RUN_ME_TO_MEASURE_PARTICLE_DENSITY.VI.......................293 A4.7.1: Generate_File_List_for_particle_density.vi.......................297 A4.7.2: User_defines_ROI_for_particle_density.vi.......................298 A4.7.3: Average n images (float, single set of images).vi...................300 A4.7.4: Compute Particle Density (known factor).vi......................301 A4.7.4.1: Length.vi.............................................303 A4.7.5: Save 2D Data.vi...........................................304 A4.7.5.1: Generate 2D Aray to Save.vi..............................305 A4.7.6: Save 3D Data.vi...........................................307 A4.7.6.1: Generate 3D Aray to Save.vi..............................308 A4.8: RUN_ME_M_TO_N_MEASUREMENTS.VI.............................309 A4.8.1: Select_ROI_m_to_n_measurements.vi..........................311 A4.8.2: Extract_ROI_m_to_n_measurements.vi.........................312 A4.8.3: Convert_Data_to_Matrix_m_to_n_measurements.vi................314 xii LIST OF FIGURES Figure 1.1: Image of particles iluminated by a laser in a plasma reactor. The bluish region is a region of dust particles iluminated using laser light scatering. Under this region, a silicon wafer is visible. The inset shows an SEM image of one of these dust particles, which is grown inside the reactor.....................................12 Figure 1.2: SEM images of microparticle formation from the ASDEX-U tokamak [25]..............................................13 Figure 1.3: Depicting where dust clouds are suspended in (a) an rf discharge and (b) ? (d) a dc discharge...................................14 Figure 1.4: Image of suspended dust particles in an argon plasma. The fluctuation of the particle density indicates the presence of a dust- acoustic wave.............................................14 Figure 1.5: Showing examples of dusty plasmas near earth: (a) images of a flame and soot particles suspended in the flame (b) lightening near the launch pad prior to the launch of STS-8, (c) noctilucent clouds and (d) exhaust from the Apollo 10 space shuttle launched on May 18, 1969. (Figures courtesy of NASA)......................16 Figure 1.6: Showing examples of dusty plasmas in the solar system: (a) an image taken by the Hubble Space Telescope of Comet LINEAR after breaking into several ?mini-comets? and (b) images of the spokes in Saturn?s B rings taken by Voyager 2. The spokes are highlighted in the inset. (Figures courtesy of NASA)...............17 Figure 1.7: Showing examples of dusty plasmas in the interstelar medium: (a) Sharples 140 (Image courtesy of NASA/JPL-Caltech), (b) a piece of cosmic dust (Image courtesy of NASA) and (c) NGC 1999 (Image courtesy of NASA and The Hubble Heritage Team (STScI).................................................19 Figure 1.8: Showing the observed phases of a plasma crystal: (a) the crystaline state, (b) the floe and flow state, (c) the vibrational state and (d) the fluid-like state. (Images courtesy of H. Thomas, Max Planck Institute for Exterestrial Physics, Garching, Germany)........23 xii Figure 2.1: A schematic of the 3DPX device showing (a) the top view of the experimental setup and (b) a detailed drawing of Section 1 of the experimental device where the experiments are conducted. The coordinate system shown in this figure wil be used throughout this document. The x-y plane is defined by the plane of the laser sheet, while the z-direction is perpendicular to the laser sheet. It is noted that gravity acts in the y-direction..............................39 Figure 2.2: A photograph of the 3DPX device and the stereoscopic particle image velocimetry diagnostic system...........................40 Figure 2.3: A photograph of the gas and vacuum subsystems on the 3DPX device. The tube leaving the bottom of the photograph is connected to a roughing pump.................................41 Figure 2.4: A photograph of the flange and electrode asembly used to generate a plasma in the 3DPX device..........................42 Figure 2.5: Photographs of the dust trays used in the experiments involving (a) polydisperse and (b) monodisperse microparticles. In al of the experiments presented in this disertation, the dust tray is electricaly floating.........................................43 Figure 2.6: Schematic drawing showing the locations where dust is typicaly trapped in the 3DPX device. In (a), the dust is suspended below the anode in an anode sheath trap. In (b), the dust is suspended above the dust tray, in a sheath trap.............................44 Figure 2.7: Images of a dust cloud suspended in (a) the anode sheath trap and (b) the sheath trap. The dust cloud is iluminated using (a) a Nd:YAG laser (? = 532 nm) and (b) a diode laser (? = 632 nm). The purple glow that is visible in the background is an argon plasma..................................................45 Figure 2.8: Video images of a dust cloud in the presence of a Langmuir probe. As a bias is applied to the probe, a void is formed inside the dust cloud....................................................46 Figure 2.9: A schematic showing the typical setup for a two-dimensional quantitative imaging system..................................48 Figure 2.10: Characteristic images taken when using (a) particle streak velocimetry (Image courtesy of H. Thomas, Max Planck Institute for Exterestrial Physics, Garching, Germany), (b) particle tracking velocimetry, (c) particle image velocimetry and (d) laser speckle velocimetry.........................................49 Figure 2.11: A cartoon depicting a 2D PIV system (Figure used with permision of LaVision, Inc.)..........................................51 xiv Figure 2.12: Orientation of a stereo-PIV system in the (a) forward-backward (b) forward-forward and (c) backward-backward scatering configuration. In the forward-backward setup, the cameras are located on the same side of the flow of interest, while the cameras observe the flow of interest on opposite sides of flow in the backward-backward and forward-forward setup. (Figures used with permision of LaVision, Inc.).............................52 Figure 2.13: A cartoon depicting the PIV analysis proces (Figure courtesy of LaVision, Inc.)............................................53 Figure 2.14: Flow diagram for the PIV analysis proces.......................54 Figure 2.15: A lens system consisting of a spherical lens (L 2 ) with focal length f 2 and a cylindrical lens (L 1 ) with focal length f 1 is used to generate a laser sheet of height L D , for ilumination of the tracer particles. Incident on L 1 is a laser beam have a beam diameter L d ..............54 Figure 2.16: Diagram showing the imaging of a particle in the laser sheet on the CD camera..............................................56 Figure 2.17: Implementation of the cross correlation technique using fast Fourier Transforms.........................................59 Figure 2.18: Depicting how sub-pixel resolution is achieved in the PIV analysis. The bars denote the computed values of the cross-correlation function, while the curve depicts a best fit Gaussian to the computed values in the (a) x-direction and (b) y-direction. The peak from the curve fit yields the desired displacement with far greater acuracy than can be found from the raw cross correlation data. The actual displacement in the images used in this analysis was 8 pixels in the x-direction and 8.5 pixels in the y-direction. The displacement that is extracted from the fit was found to be 7.92 pixels in the x-direction and 8.55 pixels in the y-direction. It is noted that the cros correlation analysis performed in the analysis generating this figure was far simpler than what is done in the DaVis software.........................................61 Figure 2.19: Showing the motion observed by the PIV system, ! r v measured , as opposed to the actual motion, ! r v actual , of the particle. The true motion of the particle is depicted by the solid arow, while the dashed arow depicts the measured motion.......................63 Figure 2.20: Plot of the ratio of the velocity measured by the PIV system, v measured , to the true velocity, v actual , as a function of the angle ?. For ? > 25?, the eror exceds ten percent........................64 xv Figure 2.21: Depicting the orientation of the cameras in the two standard stereoscopic particle image velocimetry orientation: (a) translational method and (b) angular displacement method. (Figures used with permision of LaVision, Inc.)..................65 Figure 2.22: Depicting how the out-of-plane component is extracted using the stereoscopic set-up. A diference in the measured displacements is observed if there is motion in the out-of-plane direction and is related to the displacement in the out-of-plane motion. (Figure used with permision of LaVision, Inc.)...................66 Figure 2.23: Depicting the limited depth of field that arises due to the angular displacement method of stereoscopy. It is noted that only the dots on the right hand side of the image are in focus....................67 Figure 2.24: Depicting the Scheimpflug criteria, where the image, object and lens planes al intersect at a single point.........................67 Figure 2.25: Showing the possible lens tilts that can be encountered when enforcing the Scheimpflug criteria. In (a) the lens is not tilted, (b) the lens has not been tilted enough, (c) the lens is titled to the Scheimpflug angle and (d) has the lens has tilted too far.............68 Figure 2.26: Depicting the image sen in Figure 2.23, once Scheimpflug criteria has been enforced..........................................69 Figure 2.27: Depicting the image sen in Figure 2.24 once the image has been dewarped into the global coordinate system. The red grid depicts the anticipated location of the dots on one plane of the calibration target. It is noted that the distortion observed in Figure 2.26 is gone and the magnification is uniform across the field of view........70 Figure 2.28: Showing two 3D calibration targets used on the 3DPX device. The large target, shown in (a), consist of an aray of Maltese crosses, while the smaler target, shown in (b), consist of an aray of dots......72 Figure 2.29: Showing the eror that is introduced when the laser sheet and calibration plate are not properly aligned.........................73 Figure 2.30: Flow diagram of the self-calibration proces......................74 Figure 2.31: Depicting the geometry of the stereo-PIV system as viewed from (a) above and (b) the side. It is noted that the angles ? and ? do not have to be the same for both camera. In this example, the laser sheet propagates defines the x-y plane...........................76 Figure 2.32: Schematic depicting the top view of the experimental setup and the orientation of the stereo-PIV hardware..........................79 Figure 2.33: Showing the PIV analysis proces used by the DaVis software........80 xvi Figure 2.34: SEM image of the silica microspheres used in experiments verifying the functionality of the stereo-PIV system. {SEM image courtesy of Andrew Post-Zwicker, Princeton Plasma Physics Laboratory}..............................................81 Figure 2.35: Orientation of the electrodes and dust tray used to verify the functionality of the stereo-PIV system..........................82 Figure 2.36: Schematic showing the orientation of the stereo-PIV and LS imaging systems used to verify the functionality of the stereo-PIV system...................................................83 Figure 2.37: A video image sequence acquired using the second camera system confirming the displacement in the z-direction. For increased clarity, the cloud is outlined in red. The yelow rectangle in the first image denotes the location of the laser sheet for the stereo- PIV system. The dust tray that holds the source powder can be sen in the bottom, center of the images, while the perturbation electrode is visible in the lower right of the images.................84 Figure 2.38: Four images of the reconstructed velocity field from the stereo-PIV system for times t = 0.8, 1.4, 1.8, and 2.4 seconds. The red-gren shading at the center of each of the four panels represents the z- component of the velocity of the microparticle cloud. The image labeled t = 0.8 sec, is before the application of the pulse and there is no net z-motion. At t = 1.4 sec, the pulse is applied and the cloud is moving in the +z-direction as indicated by the bright red contours. At t = 1.8 sec, the pulse has reversed and the particle cloud is moving in the ?z-direction as indicated by the gren contours. By the last image, t = 2.4 sec, the perturbation pulse has ended and the cloud has returned to a state of no net z-motion.........85 Figure 2.39: Distribution of the z-component of velocity for the four cases shown in Figure 2.37........................................86 Figure 2.40: Showing the eror in the stereo-PIV system in the (a) x-, (b) y- and (c) z-direction. The optical setup for this measurement corresponds to the one that was used in taking the data in Section 4.2 of this disertation. It is noted that the eror is comparable to what one observes in a two dimensional PIV system................89 Figure 2.41: An inverted image of a strongly-coupled dusty plasma. (Figure courtesy of H. Thomas, Max Plank Institute for Extreterestrial Physics, Garching, Germany).....................91 Figure 2.42: An inverted image of a dust cloud showing a region of known density, highlighted by the box. The image has been inverted to beter se the dust particles...................................93 xvii Figure 2.43: Video image of dust clouds composed of multiple particle sizes. The original particle distributions included spheriglas particles having a mean size of 10 ?m and 325 mesh (? 45 ?m) silica. The banding structure that is observed is a characteristic feature of dust clouds containing a size distribution............................95 Figure 2.44: Showing the fluctuation level in the scatering eficiency, ?, as a function of image number used in the calibration procedure. The fluctuation level is five percent................................96 Figure 3.1: Showing the efect of the PIV measurement on the measured distribution. In (a), one ses the velocity distribution that is being measured, while (b) shows the distribution that is measured using the PIV technique. It is noted that the measured distribution has a narower width, which is a result of an averaging that is intrinsic to the measurement technique. Physicaly, the result of this averaging would be the measurement of a smaler temperature, i.e. T PIV < T dust ................................................98 Figure 3.2: A cartoon ilustrating the PIV measurement. The dots represent the particles that sed the flow of interest and the red and blue boxes denote interogation regions over which the PIV analysis is performed. Here, the interogation region would be a 2 ? 2 bin region. For the experiments presented in Chapter 4, a bin corresponds to a 6 ? 6 pixel region of the acquired images. The PIV technique returns a vector that describes the average motion of al of the particles in the interogation region. Two examples are sen in red and blue. It is noted that there is a 50% overlap in the interogation region........................................100 Figure 3.3: The result for a simulation of an ideal PIV measurement is depicted. The crosses depict the distribution of velocities that is input into the PIV simulation, while the solid line depicts the distribution of velocities that is returned by the simulated PIV measurement. The expected narowing of the distribution of velocities is observed.......................................103 Figure 3.4: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a uniform size. Here, the results for alumina particles (r = 0.6 ?m, ? = 3800 kg/m 3 ) are shown. The blue symbol represents the mean value of ? out /? in for 24 simulations over a wide range of kinetic temperatures, while the eror bars represents the extreme values observed for the ratio ? out /? in .................................................105 xvii Figure 3.5: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a uniform size. Here, the result of a PIV simulation comparing two types of tracer particles is shown. The crosses represent alumina particles (r = 0. 6 ?m, ? = 3800 kg/m 3 ) and circles represent silica particles (r = 1.45 ?m, ? = 2600 kg/m 3 )....107 Figure 3.6: Showing the results of a simulated PIV measurement when the tracer particles have a non-zero drift velocity. Here, the red curve shows the underlying velocity space distribution that is being measured, while the blue curve depicts the distribution of velocities that is measured using the PIV technique. It is noted that the narowing of the measured distribution is observed, but there is no observed efect of the value of the drift velocity...............108 Figure 3.7: Showing the efect that the drift velocity has on the PIV measurement. In (a), the ratio of widths is plotted as a function of the efective particle density. Here the crosses denote the result when there is no drift velocity and the circles represent the result when there is a non-zero drift velocity. The drift velocities used in the simulation are sen in (b)................................109 Figure 3.8: A plot showing the overlay of the velocity distribution for monodisperse silica particles, r = 1.45 ?m, in red and polydisperse silica particles, r = 1.45 ? 0.5 ?m, in blue for the same kinetic temperature, T dust = 15 eV. It is noted that the velocity distribution for polydisperse particles has a larger width.....................111 Figure 3.9: Showing how the weighted average was used to find the efective width of the velocity distribution when the particles had a size distribution in the PIV simulations............................112 Figure 3.10: Showing the number of iterations needed to compute a meaningful value for the weighted average...............................114 Figure 3.11: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a size distribution. Here, the results for alumina particles (r = 0.6 ? 0.25 ?m, ? = 3800 kg/m 3 ) are shown. The blue symbol represents the mean value of ? out /? in for 24 simulations over a wide range of kinetic temperatures, while the eror bars represent the extreme values observed for the ratio ? out /? in .................................................116 xix Figure 3.12: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a size distribution. Here, the results from a simulation involving alumina particles having a size distribution (r = 0.6 ? 0.25 ?m, ? = 3800 kg/m 3 ), depicted by crosses, are compared with the results from a simulation involving monodisperse alumina particles (r = 0.6 ?m, ? =3800 kg/m 3 ), depicted by circles........................................117 Figure 3.13: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of particle density for tracer particles having a size distribution with a varying width, ?r, relative to the mean radius, ? r ,. In al thre plots, the crosses depict a size distribution with ? r , = 1.5 ?m, the triangles depict a size distribution with ? r , = 3.0 ?m and the dots depict a size distribution having ? r , = 50 ?m. In (a) the width of the size distribution relative to the mean value, ?r/? r , of 0.25, in (b) ?r/? r = 0.5 and is (c) ?r/? r = 0.75..........................119 Figure 3.14: Plot of the ratio of the width of the output to the input velocity distribution as a function of mean particle size for simulations with ?r/? r = 0.75. Here, crosses represent an efective particle density, n d,efective , of approximately 4.9, circles represent n d,efective ? 3 and triangles represent n d,efective ? 2. The dashed curve depicts the portion of the distribution included due to the truncation of the size distribution (e.g. r ? 0.25 ?m)................................120 Figure 3.15: Simulation results showing the efect on the measured distribution when on includes the weighting due to particle size in the PIV technique. The solid black curve depicts the underlying velocity distribution, the dotted red curve depicts what is measured by the PIV technique when al particle contribute equaly and the dashed blue curve depicts what is measured by the PIV technique when the particle size is a factor in the cross correlation analysis. The strong peak at zero velocity that occurs when including the weighting due to particle size is not observed experimentaly........123 Figure 3.16: A curve relating the efective particle density that is used in the simulations to particle densities that would be measured in an experiment. It is important to note that the slope of this curve wil depend on the particular calibration of the PIV system.............126 xx Figure 3.17: An inverted image of a typical dust cloud. Superimposed on this image is a grid that represents the interogation regions that are used in the PIV calculation. It is noted that the particle density is not uniform of the interogation region, which corresponds to a 2 ? 2 box bin (12 ? 12 pixels)................................127 Figure 4.1: A photograph of (a) the electrodes used in the experiments described in this disertation and (b) the location of the dust cloud when the electrodes are aranged in the cathode-anode-tray configuration. In (b), the lower electrode (anode) is visible. A dust cloud composed of 1.51. ?m diameter silica microspheres is iluminated by a laser diode (? = 632 nm) and is visible in red. The purplish background is an argon plasma.....................129 Figure 4.2: A typical video image of a weakly-coupled dusty plasma for (a) 1.2 ? 0.5 ?m diameter alumina microparticles, (b) 6.22 ?m diameter melamine microspheres and (c) 3.02 ?m diameter silica microspheres.............................................130 Figure 4.3: A plot depicting the oscilation of the PIV camera when the linear translation stage comes to rest. Here, the (a) vertical and (b) horizontal motion of a fixed location inside of the 3DPX device is tracked over the first ten images in a measurement sequence.......131 Figure 4.4: Showing the proces by which the underlying velocity space distribution function is obtained. In (a), the measured velocity distribution, sen in red, and the resolution eror in the stereo-PIV system, shown in blue, are overlaid. In (b), the measured velocity distribution, sen in red, and the resolution eror in the stereo-PIV system, shown in blue, are added to yield an intermediate distribution, shown in gren. In (c), the mapping function corrects the width of the intermediate function, shown in gren, to acount for the intrinsic averaging, which yields the velocity space distribution, sen in black. The velocity space distribution function shown in (d), again in black, can then be used to study the thermal properties of the microparticle component of the dust cloud...................................................135 Figure 4.5: Showing the kinetic dust temperature of 6.22 ?m diameter melamine microspheres as a function of the number of PIV measurements included in the distribution function. The symbol represents the mean temperature from the 45 - n measured temperatures, while the eror bars depict the range of temperatures that are measured. The values for n used in this calculation are n = 1, 2, 5, 10, 25 and 45. It is observed that the average temperature is relatively constant. However, the level of fluctuation decreases as more measurements are included in the velocity distribution........................................137 xxi Figure 4.6: Showing the measured velocity distributions for diferent numbers of included measurements. The red curve depicts the velocity in the x-direction, the gren curve depicts the velocity in the y- direction and the blue curve depicts the velocity in the z-direction. It is observed that there is les fluctuation in the shape of the velocity distribution as the number of measurements included increases................................................138 Figure 4.7: Showing (a) the size distribution and (b) an SEM image of the microparticles used in the experiments presented in Section 4.2. In (a) the red circles depict measured values for the particle size, while the blue curve is a Gausian fit. (SEM image courtesy of Andrew Post-Zwicker, Princeton Plasma Physics Laboratory).......139 Figure 4.8: Typical velocity distributions in the (a) x-, (b) y- and (c) z- direction. The crosses depict the measured distribution of velocities, while the solid curve represents the fit to a drifting Maxwelian velocity distribution. The experimental parameters for this data are p = 120 mTorr, V anode = 231 V and V cathode = -91 V......142 Figure 4.9: Multiple measurements of the kinetic temperature of the microparticle component taken over a period of an hour with constant experimental conditions (p = 110 mTorr, V anode = 230 V, V cathode = -89 V)..........................................144 Figure 4.10: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of neutral gas presure. The diferent colors indicate data from diferent experimental runs. It is noted that the eror bars are statistical and do not acount for fluctuations in the background plasma.........................146 Figure 4.11: A plot of the kinetic temperature for dust clouds composed of polydisperse alumina microparticles (r = 0.6 ? 0.25 ?m. ? = 3800 kg/m 3 ), normalized using Equation 4.3, as a function of neutral gas presure.......................................148 Figure 4.12: A plot of (a) the number density of the microparticle component as a function of the neutral gas presure and (b) the kinetic temperature of the microparticle component as a function of the number density of the microparticle component. It is noted that the color coding is consistent with Figure 4.10, with each color representing data taken using a single dust cloud..................149 Figure 4.13: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of neutral gas presure. The diferent colors indicate data from diferent experimental runs. It is noted that the eror bars are statistical and do not acount for fluctuations in the background plasma.........................151 Figure 4.14: Schematic showing a side view of the electrode orientation.........153 xxii Figure 4.15: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of the neutral gas presure. The data from Cloud 3 is shown in gren, while the red data indicates data from Cloud 4..................................155 Figure 4.16: Normalized temperature as a function of neutral gas presure for dust clouds composed of (a) melamine and (b) silica microspheres. In (a), the red data corresponds to the cloud that was generated at low presure, while the blue data corresponds to the cloud that was generated at high presure. In (b), the data in black, purple, gren and red corresponds to Clouds 1 ? 4 respectively..............................................157 Figure 4.17: Normalized temperature in the (a) x- (b) y- and (c) z-direction as a function of the neutral gas presure for the dust clouds composed of silica microspheres. The data in black, purple, gren and red corresponds to Clouds 1 ? 4 respectively........................159 Figure 4.18: A plot of the kinetic temperature as a function of the number density of the microparticle component for dust clouds composed of (a) melamine and (b) silica dust. In (b), the gren and blue data corresponds to Cloud 3, while the red data corresponds to Cloud 4....160 Figure 4.19: An image depicting the location of a cloud that is suspended in the (a) tray sheath and (b) anode sheath. In (a) the lower electrode is not visible, but the dust tray can be sen in the lower central region of the image. In (b), the upper electrode is visible, but the dust tray is not. In (c), the relative location of the equilibrium position for a dust cloud in the anode and tray sheath are shown.................164 Figure 4.20: Distribution of velocities in a single slice from the central region of a dust cloud suspended in the (a) tray sheath and (b) anode sheath. It is observed that the asymmetry sen in the anode region is significantly reduced when the dust cloud is suspended in the tray sheath..................................................165 Figure 4.21: A plot of the width of the measured velocity distribution as a function of the time betwen the laser pulses. It is observed that the width of the distribution gets smaler as the time betwen laser pulses increases. The lines indicate the estimated resolution limit of the stereo-PIV system....................................167 Figure 4.22: A plot of the ratio of the widths: ? y :? x in red and ? z :? x , in blue, as function of time betwen laser pulses. It is noted that the asymmetry is preserved regardles of the time betwen laser pulses..................................................169 xxii Figure 4.23: Distribution of velocities throughout the volume of a dust cloud suspended in the anode spot. Each plot represents the distribution of velocities measured using the stereo-PIV system at a z-location given in the upper right hand corner of the plot. It is observed that the fluctuations in the measured distribution is smalest in the central region of the cloud, where the clouds are bigger and more vectors are reconstructed. Additionaly, it is observed that the same asymmetry is observed throughout the cloud and in the distribution for the entire cloud sen in the lower right tile..........170 Figure 4.24: Contour plots of the velocity (left column) and speed (right column) in the x- (top row) y- (middle row) and z-direction (bottom row) for a single slice of a weakly-coupled dusty plasma. The lines depict contours of constant velocity/speed, while the shaded background more clearly ilustrate the spatial variation of the motion. It is observed that there is a great deal of spatial structure, particularly in the z-direction.........................172 Figure 4.25: Showing the spatial variation in the z-direction of (a) the number density, (b) T z , (c) T y and (d) T z ...............................175 Figure A.1.1: Imaging of a particle in the laser sheet on the CD camera. The fluid flow is iluminated by a laser shet at the object plane and imaged on the CD aray at the image plane. In this figure, lower case leters denote quantities measured by the CD camera, while capital leters denote quantities iluminated by the laser sheet. The position of a particles at time t 0 are denoted by x 1 and t 1 by x 1 ?.......190 Figure A.1.2: Depicting the correlation function. The displacement used to generate this correlation plane was 8 pixels in both x and y..........197 Figure A.1.3: Depicting the proces used to construct a PIV vector. Identical n ? n evaluation regions from Image 1, taken at time t 0 , and Image 2, taken at time t 1 , are extracted. A cross correlation analysis of these two image subsets is done and the peak in the correlation plane yields the desired in-plane displacement, ! r d .................198 Figure A.1.4: Showing the formation of a correlation plane. A square region of the two acquired images is extracted. The subset of Image 2 is shifted around the subset of Image 1 and the quantity in Equation A.1.17 is computed................................199 Figure A.1.5: Cross correlation function for simulated images. In (a) and (b), the generated images used in this analysis are sen. Each image contains a tracer particle and random noise that defines the noise floor. The defined shift in the tracer particle was 3 pixels in x and y. In (c), a top view of the correlation plane is sen, while (d) shows a topographic view of the correlation plane................200 Figure A.1.6: Showing the basic idea of the pinhole model.....................203 xxiv Figure A.1.7: Top view of the 3D pinhole model. Particles are located in the world coordinate system and are iluminated by a laser sheet........205 xxv LIST OF TABLES Table 1.1: The results from thermal studies made during the phase transitions observed in a plasma crystal. It is observed that the experiments were performed over a wide range of experimental conditions, but they al exhibited the same key result: a notably larger temperature in the fluid-like regime......................................29 Table 1.2: Showing the results from thermal studies where a phase transition was induced by heating the plasma crystal using an external means (laser heating). As observed in Section 1.3.1, there is a notably larger temperature in the fluid-like regime.......................32 Table 1.3: Showing other thermal studies involving a dusty plasma.............35 1 CHAPTER 1: INTRODUCTION In Section 1.1, a short historical overview plasmas and dusty plasmas is presented. In Section 1.2, a basic introduction to dusty plasmas, their formation and examples of their existence in nature, the laboratory and the industrial seting is given. In Section 1.3, the work that has been done to date studying the thermal properties of a dusty plasma is summarized. Finaly, in Section 1.4, the purpose of this research, the basic structure of this disertation and a summary of the major findings wil be presented. 1.1 HISTORICAL OVERVIEW In nature, mater naturaly exists in one of four states: solid, liquid, gas and plasma. In a solid, the atoms (or molecules) that make up the material are closely packed together and locked in a rigid latice structure. Additionaly, the atoms (or molecules) interact strongly. As energy is added to the system, the atoms (or molecules) gain energy and are able to break the bonds that hold them in the rigid latice structure. This proces denotes a phase transition from a solid state to a liquid state. In the liquid state, the atoms (or molecules) that make up the mater are closely spaced but interact weakly. Consequently, the liquid state does not hold a fixed shape, which is a characteristic of the solid state. As more energy is added, another phase transition is observed into the gas 2 state. In this state, the atoms (or molecules) that make up the mater are spread far apart and interact very weakly. Typicaly, the only interactions betwen the atoms (or molecules) in the gas state are due to collisions with other atoms (or molecules). As more energy is added, electrons within the atoms gain enough energy to break fre from the atoms forming an ionized gas. This quasi-neutral collection of ions, electrons and neutral particles makes up the final, and most common state of mater, plasma [1]. Sir Wiliam Crooke first observed the plasma state in the laboratory in 1879 [2]. In these early studies, a discharge tube, known now as a Crookes Tube, was used and the plasma state was refered to as ?radiant mater.? The field was pioneered by Irving Langmuir, who coined the term ?plasma? in 1928 based on the similarity that he observed betwen the way that blood plasma caries red and white blood cels and the way that he imagined that plasma as an electrified fluid that caried ions and electrons [3]. Among the observations made by Langmuir was perhaps the first observation of a dusty plasma [4]. In these experiments, an argon plasma was generated in a Pyrex glas tube by a heated tungsten filament (cathode) and disc shaped anode. By temporarily lowering the temperature of the cathode; tungsten was sputtered from the filament. These bits of tungsten were observed to have a profound impact on the discharge and under certain conditions, were observed to streak through the discharge at incredibly large velocities, on the order of 10 to 30 cm/s. The existence of these streamers led to this type of discharge being refered to as a ?streamer discharge.? [4] After these initial observations, litle work was done examining plasmas that included particulate mater. However, interest in such plasmas was revived in 1982 with 3 the observation of spokes in Saturn?s B ring [5]. This was followed a number of other discoveries that have lead to an explosive growth in this area of research [6]. Among these discoveries were the realization that the observed contamination of semiconductor materials in plasma procesing facilities was due to the growth of particles in the plasma [7, 8], the plasma crystal [9-13] and a variety of new plasma modes [14-18]. Additionaly, the field of dusty plasmas has led to an increase in the basic understanding of many fundamental aspects of plasma physics. These include research in the areas of charge particle collection by Langmuir probes [19, 20] and the electrical structure of plasma sheaths [21, 22]. Recently, dusty plasmas have proven to be a model system for studying the properties of fluids at the kinetic level [23, 24] and there has been an increased interest in understanding the generation and transport of dust as applied to applications involving fusion plasmas [25- 28]. Furthermore, because the microparticle component of this type of plasma can be visualized directly, their properties can be studied at the kinetic level and it is possible, in principle, to measure the full distribution function of the microparticle component. 1.2 BASICS OF DUSTY PLASMA 1.2.1 WHAT IS A DUSTY PLASMA? Dusty plasmas (also refered to as complex plasmas, colloidal plasmas and fine particulate plasmas) are a four-component system composed of ions, electrons, neutral particles and charged particulate mater (i.e. the dust component) [29, 30]. The size of the particulate mater in this state of mater covers many orders of magnitude. In the 4 laboratory, the dust component consists of particles ranging in size from the nanometer to the micrometer scale. In space, the particulate mater can be a few meters in size. As a result, the dust component can be considered both macroscopic, when compared to the other plasma components, and at the same time microscopic, when compared to the length scales of the plasma environment that they are a part of. In both the laboratory environment and nature, the particulate mater becomes charged and acts as a third charged species. The presence of this third charged species significantly increases the complexity of the plasma. Hence, these systems are often refered to as complex plasmas. Once the particles are in the plasma, they wil become charged through a number of possible mechanisms [29-31]. These mechanisms include the direct collection of plasma particles (i.e. ion and electron currents), thermionic emision, ionizing radiation (i.e. nuclear particles) and photoionization (i.e. ultraviolet, UV, radiation). Nuclear efects have great relevance in space environment, in particular in stelar regions. UV efects play a large role in the dusty plasmas in space and were studied in early experiments that examined the charging of dust grains. In the discharge plasmas that are investigated in this work, the relevant charging mechanisms are the electron and ion flux to the particles from the surrounding plasma. In this environment, one can model the dust grains similar to an electricaly floating probe. It is noted that new insight has ben gained into the understanding of the standard probe diagnostics of plasma physics due to research in this field [19, 20]. In a typical discharge plasma, the negative species (electrons) are much more mobile than the positively charged species (ions). As a result, the dust grains reach a 5 negative floating potential with respect to the plasma. The electrostatic potential of the dust grain can be found by balancing the electron repulsion and the ion collection given in Equation 1.1. ! dq d dt =I e +I i (1.1) where q d is the charge acumulated on the particulate mater, I e is the electron current and I i is the ion current. There are two key results that come from Equation 1.1. First, the charge on the dust grain is a dynamic quantity. As a result, the charge on the dust grain must be solved for self-consistently with other plasma parameters, such as the ion and electron densities. Secondly, for a given set of plasma conditions, Equation 1.1 implies that there wil be a finite charging time for the grain to reach its equilibrium charge. For typical lab conditions, this is on the order of tens of microseconds. Because of the short charging time relative to the typical experimental time scales (~seconds) considered here, it is asumed that the dust grains have a constant equilibrium charge that wil be estimated using the following steps. The ion and electrons are typicaly approximated using orbit-limited charging currents [31], where it is asumed that the ions and the electrons obey a Maxwelian distribution at temperatures T i and T e , respectively. For systems where the dust grains have a negative surface potential with respect to the plasma, ? d , the orbit-limited electron and ion currents are given by 6 ! I e ="4#r d 2 n e k B T e 2#m e $ % & ? ( ) exp e* d k B T e + , - . / 0 (1.2) ! I i =4"r d 2 n i e k B T i 2"m i # $ % & ? ( 1) Z i e* d k B T i + , - . / 0 (1.3) where e is the electron charge, k B is Boltzmann?s constant, ! Z i is the ion charge, n e and n ? are respectively the electron and ion number density, m e and m ? are respectively the electron and ion mas and r d is the dust radius. The ! exp e" d k B T e # $ % & ? ( term is Equation 1.2 represents the electron repulsion, while the ! 1" Z i e# d k B T i $ % & ? ( ) term in Equation 1.3 represents the ion collection. It is noted that the model for the charging currents given in Equations 1.2 and 1.3 are derived asuming the ?dust in plasma? limit. In this low density limit, r d < ? D < a, where r d is the radius of the dust grain, ? D is the efective Debye length and a is the interparticle spacing. In the dusty plasma limit, where r d < ? D ~ a, the scatering of ions and electrons by the dust component is more complicated. When this more complicated, and realistic, scatering is considered, it is found that the currents given in Equations 1.2 and 1.3 are slightly larger than the actual currents to the dust grain. As a result, the value of the grain charge computed using this model for the charging currents wil provide an upper bound [29]. 7 The efect of this third charged species on the plasma is significant. First, the amount of negative charge that is tied to the dust grain can be quite substantial. As a result, there can be a significant reduction is the local fre electron or ion populations in the vicinity of dust grains, depending on whether the dust grain acquires a net positive or negative charge. For the typical discharge plasma that contains a relatively high concentration of dust particles, it is possible that Z d n d > n e . Similarly, in plasmas where photoionization is a dominant charging mechanism, photoemision is responsible for the ejection of electrons from the dust grains surface. In this case, it is possible that Z d n d > n i . As a result, the standard condition of quasi-neutrality in a plasma must be modified. ! n i =n e +Z d n d (1.4) where Z d is the number of electron charge on the dust particle, while n i , n e and n d are the number density of the ion, electron and dust populations respectively. Using this condition of quasineutrality, it is possible to solve for the potential of the of the dust particle when the charging currents have reached an equilibrium value, i.e. I i +I e = 0. By combining Equations 1.1 ? 1.4 and treating the dust particles as spherical capacitors (i.e. ! q d =4"# o r d $ d ), it is found that ! 1+P e" d k B T e # $ % & ? ( ) * , - . m i T e ei exp e" d k B T e / 0 1 2 3 4 =15 e" d k B T i # $ % & ? ( (1.5) 8 where the Havnes parameter, ! P=4"n d r # o k B T e en i $ % & ? ( ) [32], is introduced. The Havnes parameter is a dimensionles quantity that describes an efective screning in a dusty plasma. By numericaly solving Equation 1.5, the surface potential of the dust grain is determined. Using typical discharge conditions for the plasma studied in this disertation (e.g. an argon plasma with alumina microparticles where T i = 0.025 eV, T e = 3 eV, n d = 1 ? 10 10 m -3 , n e = n i = 5 ? 10 14 m -3 , r d = 0.6 ? 10 -6 m), a dust potential of -6.7 V and a dust charge of approximately 2800 electrons is found. It is noted that for these conditions under the condition of quasineutrality, Z d n d / n i ~ 0.056; i.e. the dust has collected just over 5% of the electrons from the local plasma. The second efect that this third charged species has on a plasma involves modifying the screning length. As a result, the screning length inside a dusty plasma can be quite diferent than what is observed in the standard ion-electron plasma. The standard screning (Debye) length, ? D , in a dusty plasma is given by a combination of the ion and electron Debye lengths [33] as ! 1 " D 2 = 1 " D,i 2 + 1 " D,e 2 (1.6) where ! " D,# = $ o k B T # Z # n # e 2 is the standard Debye length for species ? (ions and electrons) and ? o is the permitivity of fre space. For the plasma parameters noted above, ? D,e = 5.76 ? 10 -4 m, ? D,i = 5.26 ? 10 -5 m and ? D = 5.23 ? 10 -5 m ~ ? D,i . At the present 9 time, experiments appear to give diferent results as to the appropriatenes of ? D,e or ? D as the scaling length and this is an active topic of research in the dusty plasma community. A third efect of the third charged species is an increase, when compared with those in a traditional two-component plasma, in the time scales for dusty plasma phenomena, i.e. the plasma reacts more slowly. In large part, this is due to the very large mas of the dust grains, when compared to the ion and electron mases, i.e. m i /m d , m e /m d < 1. This new time scale is parameterized by the dust-plasma frequency, ? pd . ! " pd = 4#n d Z d 2 e 2 m d (1.7) For the plasma parameters noted above, ? pd = 171.45 rad/s. This new time scale is of particular benefit to the experimentalist, as it alows a great variety of optical imaging technologies ? some as simple as basic CD cameras ? to be used to explore plasma physics phenomena. This is a powerful advantage for the study of dusty plasma and one that is exploited in this work. A fourth efect arises from the fact that the dust has a very smal charge to mas ratio, when compared to the other plasma species (e.g. the ions and the electrons). Consequently, the dynamic response of the dust gives rise to many new wave phenomena. A number of these, including the dust-acoustic [17, 34, 35] and dust 10 latice [18] wave, have been extensively explored both in theory and in experiment and have proved to be a valuable tool for testing the non-linear properties of a dusty plasma. Finaly, because of the high charge, the electrostatic interaction betwen dust particles and betwen the dust particles and the surrounding plasma can become quite large. In some cases, the electrostatic energy can exced the kinetic energy, in which case the system becomes highly coupled. In this state, a wide range of new phenomena, such as the crystaline plasma state, can be observed [9-13]. To quantify this, the Coulomb coupling parameter, which is a ratio of the electrostatic potential energy to the thermal energy, is used ! "= Z d 2 e 4#$ o ak B T d % & ? ( ) * exp+ r d , D - . / 0 1 2 (1.8) where ? D is the Debye length given in Equation 1.6, and T d and Z d are the temperature and dust charge respectively. It is noted that the term in brackets represents the ratio of electrostatic to thermal energy, while the exponential term acounts for screning efects. For ? < 1, the system is in a gas like state. For ? ~ 1, the system is in a liquid-like state and for ? > 1, the system is in a solid, or crystaline, state. For the plasma crystals to occur, ? ? 170 [36]. 11 1.2.2 EXAMPLES OF MAN-MADE DUSTY PLASMAS Dusty plasmas can be generated in research or industrial plasma devices. In some cases, the presence of the dust particles can be quite detrimental to the desired plasma environment. In other cases, the presence of the dust is exploited as a valuable commodity. This section gives a few examples of these types of dusty plasmas applications. An early example of a dusty plasma was first observed by Gary Selwyn at IBM in the fabrication of microelectronics, as sen in Figure 1.1. Here, dust particles are produced during plasma etching of silicon wafers. The bluish glow is from laser light that is scatered from the dust particles that form during the etching proces [7]. Dust particle formation can also occur in thin film deposition using reactive gases [8, 37, 38]. In these cases, the dust is an undesirable contaminant that needs be removed from or controlled in the plasma environment [39-41]. On the other hand, the presence of dust can also be quite beneficial in some industrial applications. A few examples where the presence of dust can be beneficial include the growth of diamond films, the fabrication of particles of known size and shape [42] or of particle with desired surface and optical properties [43, 44], and the implantation of nanoparticles to enhance surface properties (i.e. amorphous silicon solar cels) [45, 46]. 12 Figure 1.1: Image of particles iluminated by a laser in a plasma reactor. The bluish region is a region of dust particles iluminated using laser light scatering. Under this region, a silicon wafer is visible. The inset shows an SEM image of one of these dust particles, which is grown inside the reactor. Recently, the presence of dust in fusion devices has become a topic of great interest [25-28]. When the fusion plasma comes into contact with the wal of the device or with other plasma facing components, material can be vaporized or sputtered off of the surface. The results of these plasma-wal interactions is ilustrated in Figure 1.2. Here, SEM images of material that acumulates on the floor of a fusion device are shown. This dust is not confined by the magnetic fields and if these particles travel from the wal into the core plasma, they can become a major energy loss mechanism. Additionaly, the dust can transport tritium in the plasma, as wel as become radioactive themselves, making the dust also an additional safety concern. 13 Figure 1.2: SEM images of microparticle formation from the ASDEX-U tokamak [25]. For the majority of laboratory studies, dusty plasma are generated in either radio frequency (rf) and direct current (dc) discharge plasmas. Additionaly, the rf discharge plasma is also quite common in the industrial seting. In both types of discharges, the dust particles are suspended in regions of high electric field, where the electric force balances the gravitational force, i.e. m d g = q d E, where m d is the mas of the dust grain, g is the aceleration due to gravity, q d = Z d e is the charge of the dust grain and E is the local electric field. Typical suspension locations can be sen in Figure 1.3. 14 Figure 1.3: Depicting where dust clouds are suspended in (a) an rf discharge and (b) ? (d) a dc discharge. Figure 1.4 shows an example of dusty plasma formed in the laboratory using a dc discharge device. Figure 1.4: Image of suspended dust particles in an argon plasma. The fluctuation of the particle density indicates the presence of a dust-acoustic wave. 15 1.2.3 EXAMPLES OF NATURALY-OCCURRING DUSTY PLASMAS Dusty plasmas are somewhat ubiquitous in nature. Near the earth, examples of dusty plasma include flames, lightning and noctilucent clouds. Flames, as sen in Figure 1.5(a) are an example of a weakly-coupled ionized plasma. The glow that is observed is due to blackbody radiation of soot particles that are suspended in the plasma [29]. When lightning occurs, a plasma is created in an atmosphere that is full of water droplets and pollutants. These bits of particulate mater become charged, forming a dusty plasma, Figure 1.5(b). Another example of lightning, known as bal lightning, is possibly due to the oxidation of nano-sized silicon particles that are suspended in the air after a lightning strike occurs [47]. Further from the earth?s surface, ~85 km above the surface, noctilucent clouds [48] are observed, as sen in Figure 1.5(c). Here micron sized ice-coated particles are suspended in a plasma. A final example, Figure 1.5(d), is sen in the exhaust of a spacecraft as it is launched into space. 16 (a) (b) (c) (d) Figure 1.5: Showing examples of dusty plasmas near earth: (a) images of a flame and sot particles suspended in the flame (b) lightening near the launch pad prior to the launch of STS-8, (c) noctilucent clouds and (d) exhaust from the Apolo 10 space shutle launched on May 18, 1969. (Figures courtesy of NASA) As one moves away from the earth and into space, the abundance of dust is observed by the existence of planets and stelar objects. As a result, there are many examples of dusty plasmas in space. In our imediate solar system, two examples of dusty plasmas include comets and planetary rings. Comets [49], Figure 1.6(a), can be thought of as ?dirty? snowbals consisting of ice and other particular mater on the order of a few microns in size. Saturn?s rings are another example, one that led to a resurgence 17 in this area of study. In particular, a great succes of dusty plasmas is the description of the spoke features that have been observed in Saturn?s rings [50], Figure 1.6(b). (a) (b) Figure 1.6: Showing examples of dusty plasmas in the solar system: (a) an image taken by the Huble Space Telescope of Comet LINEAR after breaking into several ?mini-comets? and (b) images of the spokes in Saturn?s B rings taken by Voyager 2. The spokes are highlighted in the inset. (Figures courtesy of NASA) 18 As one moves into the interstelar media, there are numerous examples of dusty plasmas consisting of ice, silicates, graphite and so on. Some examples include the interstelar clouds [51, 52], space nebulae [53], and zodiacal light clouds [54]. In fact, the formation of objects in the universe come from first being in the dusty state and it is not surprising that dusty plasmas are the most common structural form of mater in the universe. Some of these structures can be sen in Figure 1.7. An image in the infrared of Sharples 140 taken with the Spitzer Space Telescope is sen in Figure 1.7(a). Here, thre stars are surrounded by dense dust, which prevents them from being observed in the visible region of the spectrum. The bright red bowl that is visible is the dust cloud. Figure 1.7(b) shows an SEM image of a piece of cosmic dust. In Figure 1.7(c), an image of the NGC 1999 nebula taken with the Hubble Space Telescope is sen. This reflection nebula is visible because a young star (V380 Orionis), located to the left of center iluminates the surrounding dust. In the darken region to the right of V380 Orionis is a Bok globule, a region of gas and cosmic dust so dense that it block al light. It is noted that the presence of this dust can be an important factor in the behavior of the system. For instance, the dust may provide a dominant cooling mechanisms. Additionaly, the dust can be an important plasma source or an important recombination mechanism. Dust can also provide coupling betwen the electromagnetic and gravitational fields [55-58]. Further, the presence of dust in the interstelar medium can be an important aspect of planet and stelar formation [59-64]. Indeed, given the abundance of dust in the plasma environment, one might even ask if there is such a thing as a non-dusty plasma [6]. However, over the last twenty years, much work has gone 19 into this field [6], leading to a more fundamental understanding of a wide range of physics phenomena. (a) (b) (c) Figure 1.7: Showing examples of dusty plasmas in the interstelar medium: (a) Sharples 140 (Image courtesy of NASA/JPL-Caltech), (b) a piece of cosmic dust (Image courtesy of NASA) and (c) NGC 199 (Image courtesy of NASA and The Huble Heritage Team (STScI). 20 1.3 PREVIOUS WORK The measurement of the thermodynamic properties of a dusty plasma is experimentaly chalenging problem because of the dificulties asociated with measuring the thre-dimensional velocities of the microparticles. In spite of this, there have been a number of measurements made of the kinetic temperature of the dust component using the plasma crystal [10, 65, 66] and, more recently, the dust cluster [67, 68]. In particular, there has been a great deal of interest in studying the evolution of the dust kinetic temperature during the observed melting transition that occurs in the plasma crystal. More recently, there have been a number of experiments that have measured the temperature of the dust component while simulating random heating of the microparticle component using lasers [69, 70]. For the most part, al of these experiments have been performed using microparticles suspended in an rf discharge plasmas. In what follows, a review of measurements of the thermal properties of a dusty plasma is presented. Section 1.3.1 presents results from thermal studies on systems during observed phase transitions from a strongly-coupled, crystaline state to weakly-coupled fluid like state. Section 1.3.2 presents the results of thermal studies done while actively heating a dusty plasma using lasers. Section 1.3.3 presents other studies involving the thermal properties of a dusty plasma. 21 1.3.1 THERMAL STUDIES INVOLVING PHASE TRANSITIONS Early measurements of the thermal properties of a dusty plasma were performed on strongly-coupled systems and focused on measuring the temperature of the dust component as a strongly coupled, two dimensional plasma crystal underwent a phase transition to a fluid-like state. Once a strongly-coupled plasma crystal was generated, a phase transition was induced by varying experimental parameters. Typicaly, there are thre mechanisms that can be used to initiate a phase transition [71]. 1. Increasing the applied rf power (in a dc system, this corresponds to increasing the current). As the power (current) increases, the plasma density increases. This results in a decrease in the Debye shielding length and the dust particles move closer together. This leads to a decrease in the interaction energy with surrounding particles and a transition from an ordered to a disordered state occurs. 2. Decreasing the neutral gas presure. This mechanism induces a phase transition through two mechanisms. First, a decrease in the neutral gas presure changes the viscosity of the background gas (i.e. collisions betwen neutral and dust particles) and the rate at which energy is disipated. Second, a change in the presure wil change the background plasma parameters. Together, these two efects can initiate a phase transition from an ordered to a disordered state. 3. Adding particles to the system that are more masive than those in the plasma crystal. In this case, the more masive particles wil move to an equilibrium position below the plasma crystal. The interaction of the dust particles and the 22 confining field generates an instability that leads to a phase transition. If a confining field is not present, this method cannot be used. Of these thre methods, varying the neutral presure is an experimentaly convenient method of controlling the phase transition. During the observed phase transition, four distinct states were identified [71]: crystaline, ?flow and floe,? vibrational and fluid. In the crystaline state, the dust particles are aligned in a rigid latice structure. It is possible that the dust grain exists in multiple vertical planes. The dust particles tend to interact strongly with surrounding particles in the same plane. However the vertical spacing of the planes is such that the particles in diferent layers tend to not interact strongly. As a result, these systems, even if they are thre dimensional, can be treated as a two dimensional system. The ?flow and floe? state is characterized by regions where the dust is aligned in a crystaline state (floes) and regions of directed transient flows (flows). In this state, the thermal velocities of the microparticles are close to the thermal speed of the background neutral gas and the transient flows have magnitudes that are about half of the thermal sped of the background neutral gas. Additionaly, in this state, one occasionaly observes vertical particle migration to other other layers. In the vibrational state, the orientational ordering increases and there is a decrease in the flow regions. The vibrational and thermal energies increase, as does the vertical migration. Additionaly, there is a decrease in the translational order. In the fluid state, there is no longer any observed crystaline structure. These states were identified by examining orientational correlation functions, motion of the particles, viscosity, interaction cross sections and self-difusion. Each of these states can be sen in Figure 1.8. 23 (a) (b) (c) (d) Figure 1.8: Showing the observed phases of a plasma crystal: (a) the crystaline state, (b) the floe and flow state, (c) the vibrational state and (d) the fluid-like state. (Images courtesy of H. Thomas, Max Planck Institute for Exterestrial Physics, Garching, Germany) To extract a kinetic temperature, the motion of the particles was observed by scatering laser light from the microparticles and recording video images of the scatered light. From the video images that were acquired a velocity for each of the microparticles was obtained from the measured displacements and the time betwen acquired video images. This is possible because the lower particle number density of the plasma crystals permits each particle to be clearly identified and tracked over time. Using these 24 velocities, one of four techniques were used to extract a kinetic temperature of the microparticle component, T j . It is noted that the index, j, refers to a symbol (j = M, MD, MKE, NM) denoting the method used to obtain the temperature and that this temperature T j refers to the quantity k B T j , where k B is Boltzmann?s constant. It is noted that thre of the four methods described below (Maxwelian, Mean Kinetic Energy and Normal Modes) measure the same temperature. In the first method, which wil be denoted by the symbol M, the distribution of measured velocities was fit to a Maxwelian velocity distribution, Equation 1.9. ! f= m 2"T M exp# mv#v d () 2 2T M $ % & ? & ( & * & (1.9) where v is the velocity of a particle, v d is the drift velocity, m is the mas of the dust grain and T M is the kinetic temperature of the microparticle component. From the fit, a temperature is then extracted. In the second method, which wil be denoted by the symbol MD, a temperature was extracted by measuring a velocity from the maximum observed displacement, ?x max . By knowing the time, t, over which this displacement was measured, Equation 1.10 can be used to extract a temperature, T MD . It is noted that this is a measure of the maximum energy, not a bulk temperature measurement. ! T MD =m "x max t # $ % & 2 (1.10) 25 In the third method, which wil be denoted by the symbol MKE, the mean kinetic energy of al of the dust particles is calculated. This energy is then equated to a temperature using Equation 1.11. ! T MKE =mv"v () 2 (1.11) It is noted that this method may overestimate the temperature, for a number of reasons [66]. If Equation 1.11 is expanded, the temperature is defined as the diference betwen the total kinetic energy and the mean drift energy. The kinetic energy has contributions from the thermal motion and correlated wave motion. In measuring the kinetic temperature, one is interested only in the contribution from the thermal motion. However, it is dificult to compute the kinetic energy stored in coherent oscilations from experimental data. This dificulty lies in the way in which the data is acquired. Identification of coherent oscilations requires observing the motion of many particles over large distances. However, an acurate measurement of the microparticle velocity requires visualizing a smal number of particles. In a number of the earlier experiments, only a smal region of the plasma crystal was imaged. As a result, the energy stored in coherent oscilations may be underestimated. In a similar vein, the drift velocity can be dificult to measure. If the sampling time is large, compared to the time for drift velocities to change direction, then the drift velocity is averaged to zero. With care, it is possible to addres these concerns [66]. However, it is likely that the temperatures reported in measurements 1, 2, 3 and 5 in Table 1.1 may be overestimated. It is unclear what eror is introduced in these measurements due to these concerns. Nonetheles, it is 26 also the case that the true temperature wil be strongly correlated with the reported values. The final method used to determine the kinetic temperature of the microparticle component involves examining the normal modes of the plasma crystal [67]. In this approach, the normal modes are computed from the dynamical matrix given in Equation 1.12. ! E ",#,ij = $ 2 E r ",i r #,j (1.12) where r ? ,i denotes the x or ycoordinate of the i th particle with (?, ? = x, y) and E is the total energy of the system, given by Equation 1.13. ! E= 1 2 m" o 2 r i 2 i=0 N # + Z 2 e 4$% o 2 exp& r ij ? D ( ) * + , - r iji>j N # (1.13) Here, the first term denotes the potential energy of the radial confinement with resonance frequency ? o and the second term denotes the screned Coulomb interaction betwen particles with screning length ? D . It is noted that the radial position of the i th particle, r i , is with respected to the center of mas and r ij = |r i - r j |. The eigenvectors of the matrix defined by Equation 1.12 are the normal modes. One then projects the measured 27 velocities, v i , are projected onto the normal modes vectors, e i,l as defined in Equation 1.14. ! v l t()= r v i t()" r e i,l i=1 N # (1.14) where the normal modes aredenoted by the index l = 1, . . ., 2N. From this, the normal modes are calculated in the form of a spectral density ! S l "()= 2 T v l t()exp#i"t()d # T 2 T 2 $ 2 (1.15) Each mode wil have an efective temperature given by Equation 1.16. ! T NM = 1 2N mS l "()d 0 # $ l=0 2N % (1.16) where ! v l 2 =S l "()d 0 # $ . Summing the contribution of al the modes wil give the mode temperature, T NM . The measurement of the kinetic temperature that were made during the observed phase transition from an ordered to a disordered state are summarized in Table 1.1. Al of the reported measurements, with the exception of Measurement 4, were performed using either a plasma crystal or a dust cluster. In each of these measurements, the motion of the microparticles was observed by iluminating the microparticles with a laser sheet. 28 As a result, there is an implicit asumption that the motion in these experiments is two dimensional, i.e. the motion is confined in the plane iluminated by the laser shet. For these plasma crystal or dust clusters, this is a reasonable approximation. This asumption may not be as valid for Measurement 4, resulting in an underestimation in the kinetic dust temperature. In al of these experiments, there is a common observation: a significant increase in the kinetic temperature of the microparticle component is observed as the system moves from the crystaline state to the fluid state. Further, the kinetic temperature in the fluid-like state is found to be larger, and in most cases significantly larger, than the temperature of any of the plasma components (e.g. the electrons, ions and background neutrals). Finaly, it is noted that the kinetic temperature of the dust particle is very diferent than the surface temperature of the dust particle. Indeed, it has been observed that the surface temperature of the dust particles is close to room temperature [72]. In Experiment 1 [65], a plasma crystal composed of monodisperse melamin- formaldehyde microspheres having a diameter of 9.4 ?m was generated at a neutral gas presure of 640 mTorr in an rf discharge plasma. A phase transition was induced using two mechanisms: reduction of the neutral gas presure and increasing the applied rf power. The motion of the microparticles was observed using a 50 fps video camera and a kinetic temperature was computed using the MKE method described above. Further, the temperature that was extracted using the MKE method was used to generate a Maxwelian velocity distribution that fit the experimentaly observed velocity distribution with excelent agrement. It is noted that the temperature of 0.7 eV represents the 29 Table 1.1: The results from thermal studies made during the phase transitions observed in a plasma crystal. It is observed that the experiments were performed over a wide range of experimental conditions, but they al exhibited the same key result: a notably larger temperature in the fluid-like regime. 30 resolution limit of the video camera used. As a result, it is likely that the kinetic temperature in the crystaline state was lower than what is reported. In Experiment 2 [10], a plasma crystal composed of monodisperse melamin- formaldehyde microspheres having a diameter of 6.9 ?m was generated at a neutral gas presure of 315 mTorr in an rf discharge plasma. A phase transition was induced by reducing the neutral gas presure. The motion of the microparticles was observed using a CD camera and a kinetic temperature was computed using the MKE and M methods described above. It was observed that these two methods yielded the same results. In Experiment 3 [66], a plasma crystal composed of monodisperse melamin- formaldehyde microspheres having a diameter of 9.4 ?m was generated at a neutral gas presure of 90 mTorr in an rf discharge plasma. A phase transition was induced by reducing the neutral gas presure. The motion of the microparticles was observed using a 30 fps CD camera with a shutter speed of 1/250 s. The kinetic temperature was computed using the MKE method described above. In Experiment 4 [67], a plasma crystal composed of monodisperse melamin- formaldehyde microspheres having a diameter of 9.55 ?m was generated at a neutral gas presure of 100 mTorr in an rf discharge plasma. A phase transition was induced by decreasing the neutral gas presure. The motion of the microparticles was observed using a 50 fps video camera and a kinetic temperature was computed using the MKE and NM methods described above. The temperatures computed using these two methods were in agrement. In Experiment 5 [73], a plasma crystal composed of monodisperse melamin- formaldehyde microspheres having a diameter of 10.24 ?m was generated at a neutral gas 31 presure of 0.35 mTor in a dc discharge plasma. In this experiment, the suspended dust cloud exhibited two distinct regions. In the upper region of the cloud, the microparticles existed in strongly-coupled state, while the microparticles in the lower region of the cloud were in a fluid-like state. Unlike the other experiments, the motion of the microparticles may not haven been limited to the two-dimensional plane of the laser shet. However, it is also noted that a detailed study of the temperature was not preformed and that the reported temperatures represent a maximum energy in the iluminated plane of the suspended dust cloud, not a bulk measurement as is the case in the other experiments discussed here. In Experiment 6 [68], a dust cluster composed of monodisperse melamin- formaldehyde microspheres having a diameter of 7.7 ?m, 9.5 ?m or 12.26 ?m was generated at a neutral gas presure of 135 mTor in an rf discharge plasma. A phase transition was induced using two mechanisms: reduction of the neutral gas presure and increasing the applied rf power. The motion of the microparticles was observed using a 25 - 100 fps video camera, depending on the state of the dust cluster. The kinetic temperature was computed using the MKE and NM methods described above. It is noted that each method yielded the same kinetic temperature for the microparticle component. Finaly, it was observed that smaler particle obtained a higher kinetic temperature at al values of neutral gas presure. In total, the results of experiments 1-4 and 6 in Table 1.1 suggests two significant results that wil be evaluated in this work. First, in al cases, the distribution of the microparticle velocities was determined to obey a Maxwelian distribution for these strongly-coupled dusty plasmas. It wil be shown that this is also valid for the weakly- 32 coupled systems studied in this work. Second, as discussed at the outset of this section, increasing the neutral gas presure is an efective method of controlling the microparticle temperature. In particular, increases in the neutral presure corresponds to a lowering of the dust temperature. This wil be evaluated for weakly-coupled dusty plasmas composed of both uniform and non-uniform particle size distributions in this work. 1.3.2 THERMAL STUDIES INVOLVING LASER HEATING While the experiments in Section 1.3.1 focused on measuring the kinetic temperature of the microparticle component that naturaly occurred at a given set of experimental conditions, there have also been a number of studies that involved the active modification of the dust particle temperature [69, 70]. In these experiments, a plasma crystal or dust cluster was generated and heated using a laser. To simulate an equilibrium heating proces, the laser excitation was randomly applied to various locations in the plasma crystal. The results of these experiments are summarized in Table 1.2 and are discussed briefly below. State Presure [mTorr] T dust [eV] Method Reference 1 Crystaline Liquid 37.5 or 75 0.1 8 KE Wolter et al., Phys Rev E, 71, 036414 (2005) 2 Crystaline Liquid 5 ~0.1 ~20 MKE Nosenko et al., Phys Plasmas, 13, 032106 (2006) Table 1.2: Showing the results from thermal studies where a phase transition was induced by heating the plasma crystal using an external means (laser heating). As observed in Section 1.3.1, there is a notably larger temperature in the fluid-like regime. 33 In the first of these experiments [69], a dust cluster consisting of 10 ? 200 monodisperse melamine formaldehyde microspheres having a diameter of 9.55 ?m or 12.25 ?m was generated at a neutral gas presure of 37.5 or 75 mTorr in an rf discharge plasma. A phase transition was induced by heating the dust cluster using a variable power (0-200 mW) Nd:YAG laser. The motion of the microparticles was observed using a 50 fps CMOS camera and a kinetic temperature was computed using the MKE method previously described. Further, it was observed that the distribution of velocities was Maxwelian and excelent agrement betwen the width of the velocity distribution and the kinetic temperature computed using the MKE method was observed. As was observed in the experiments discused in Section 1.3.1, there was a significant increase in the kinetic temperature of the microparticle component. It is noted that the experiment from [69] highlighted in Table 1.2 was performed on a dust cluster consisting of 44 microparticles. A final observation from this experiment was that the heating was much stronger, under identical experimental conditions, for smaler particles. In the second of these experiments [70], a plasma crystal consisting of approximately 6700 melamine formaldehyde microspheres having a diameter of 8.09 ?m was generated at a neutral gas presure of 5 mTorr in an rf discharge plasma. A phase transition was induced by heating the plasma crystal using two laser beams (? = 532 nm) from opposite sides of the plasmas crystal with a total heating power from 0 to 10.55 W. A region of the plasma crystal consisting of 1100 microspheres was observed using a 29.97 fps video camera and a kinetic temperature was computed using the MKE method described above. Further, it was observed that the distribution of velocities was 34 Maxwelian. As a result of the heating, a phase transition from the crystal to disordered state was observed. As was observed in the experiments discused in Section 1.3.1, there was a signifigant increase in the kinetic temperature of the microparticle component, in this case from ~0.1 eV in the crystal state to ~20 eV in the disordered state 1.3.3 OTHER THERMAL STUDIES Beyond the experiments described in the previous sections, there have been four other experiments that have measured the kinetic dust temperature [74-76]. One of these experiments (Experiment 1 in Table 1.3) focused on examining wave modes in a dusty plasma in the various states of a plasma crystal, two of these experiments (Experiments 2 and 3 in Table 1.3) involved looking at shocks in a complex dusty plasma, while the fourth (Experiment 4 in Table 1.3) measured the kinetic temperature of a dust cloud trapped in an rf discharge over a wide range of experimental conditions. Unlike the experiments previously discussed, there was no observed phase transition in Experiment 4 in Table 1.3. The results of these experiments are summarized in Table 1.3 and are discussed briefly below. 35 State Presure [mTorr] T dust [eV] Method Reference 1 Crystaline Partialy Melted Liquid 7.5 0.037 14.5 54.9 S. Nunomura, et al., Phys Rev Let, 94, 045001 (2005) 2 Before Shock After Shock 13.5 0.5 ~300 MKE D. Samsonov, et al., Phys Rev Let, 92, 255004 (2004) 3 Ahead, H Ahead, V Behind, H Behind, V 375 1.7 9.2 2.0 3.1 M V.E. Fortov et. al., Phys Rev E, 71, 036413 (2005) 4 Axial Azimuthal 10-50 ? 500 ? 1000 MKE M. Schabel et al., J. Appl . Phys. 86, 1834 (1999). Table 1.3: Showing other thermal studies involving a dusty plasma. In Experiment 1 [74], a plasma crystal composed of ~5000 monodisperse plastic microspheres having a diameter of 8.9 ?m was generated at a neutral gas presure of 7.5 mTorr in an rf discharge plasma. A phase transition was induced by adding a smal amount of 12.7 ?m onodisperse plastic microspheres. The motion of the microparticles was observed using a video camera operating at 22 ? 154 fps and a kinetic temperature was computed using the M method described above. It is noted that the velocity distributions were observed to be very close to Maxwelian. In Experiment 2 [75], a plasma crystal composed of monodisperse plastic microspheres having a diameter of 8.9 ?m was generated at a neutral gas presure of 7.5 mTorr in an rf discharge plasma. A 0.1 m diameter tungsten wire located 4 m below the plasma crystal was pulsed to -100 V for 50 ms, inducing a compresional shock in the plasma crystal. This shock caused the plasma crystal to melt imediately behind the 36 propagating shock wave. The motion of the microparticles was observed using a video camera and a kinetic temperature was computed using the MKE method described above. In Experiment 3 [76], a dust cloud composed of monodisperse plastic microspheres having a diameter of 1.87 ?m was generated at a neutral gas presure of 375 mTorr in a dc discharge plasma. To induce a shock, a batery of 1.2 mF capacitors was charged to 1.2 kV and discharged through a 16 turn copper coil using a 14 ? resistor (~90 A of current through the coil). This pulse causes the plasma to move upward for a very short period of time. During this time, the dust particles are no longer in mechanical equilibrium and begin to fal under the influence of gravity. When the plasma returns to its original configuration after the pulse has ended, the particles are pulled back to their original location. The particles that are located at a lower position in the vertical direction have falen a greater distance and as a result, a region of high dust density is created. The motion of the microparticles in the horizontal (H) and vertical (V) directions was observed using a video camera operating at 1000 fps and a kinetic temperature was computed using the M method described above. In Experiment 4 [77], a dust cloud composed of monodisperse polystyrene latex microspheres having a diameter of 10.2 ?m was generated over a range of neutral gas presure from 10 to 50 mTorr in an rf discharge plasma. The motion of the microparticles was measured using laser doppler velocimetry, LDV, and confirmed using LS techniques. From the velocities measured using the LDV technique, a kinetic temperature was computed using the MKE method described above for the motion paralel (azimuthal), ! T || , and perpendicular (axial), ! T " , to the electrode. For the particle 37 concentrations observed in this experiment, the motion of the microparticle component was observed to be random and uncorrelated. Additionaly, the velocity distributions measured were independent of the filtering conditions for the LDV measurement. There are two key results that were observed in this experiment that are relevant to the work presented in this disertation. First, the temperature increased with decreasing neutral gas presure and increasing power. Second, the observed temperatures were anisotropic, with ! T " :T || #2 across a wide range of experimental conditions. 1.4 SCOPE OF THIS THESIS In this disertation, two new results are presented. First, the use of a new diagnostic, stereoscopic particle image velocimetry (stereo-PIV), in the study of dusty plasmas is introduced. Using this new diagnostic, the first detailed measurements of the thermal properties of a dc glow discharge dusty plasma are then presented. In Chapter 2, the experimental apparatus and procedures used in the experiments presented in this disertation are discused. Chapter 3 wil discuss simulations of the PIV measurement technique, while Chapter 4 wil present the first experimental measurements of the thermal properties of a weakly-coupled dusty plasma in a dc glow discharge. Finaly, conclusions and directions for future work wil be discussed in Chapter 5. 38 CHAPTER 2: APARATUS The work described in this disertation was performed using the 3D-Dusty Plasma Experiment (3DPX) device [78]. As is the case with al plasma devices, 3DPX consists of a number of subsystems, including the vacuum vesel, vacuum and gas systems, plasma and dusty plasma generation systems, and diagnostic systems; each of which wil be discussed in the following chapter. Section 2.1 describes the 3DPX device [78], as wel as the techniques employed for the plasma and dusty plasma generation. Section 2.2 wil describe the PIV technique and the stereoscopic particle image velocimetry diagnostic [79], which is both the centerpiece of the 3DPX device and the primary diagnostic for the experiments described in this disertation. Section 2.3 describes how measurements of the particle density are performed [80]. 2.1 EXPERIMENTAL DEVICE 2.1.1 VACUM VESEL The 3DPX vacuum vesel [78] is composed of two 10 cm (4?) inner diameter stainles stel six-way crosses. One of these crosses, which wil be refered to as Section 1, is a modified six-way cross having a length of 47.5 cm. The most prominent feature of Section 1 is a large rectangular window, 7 cm tal by 24 cm wide, on the front 39 of the chamber. Experiments are performed in Section 1 of the device, where the large rectangular window alows for optimal optical aces. Section 2 of the 3DPX vacuum vesel is a standard ISO100 stainles stel six-way crosses. This section holds al of the vacuum pumps and gas fed systems. In this configuration, the experimental region of the system is separated from the gas flows that are generated by the gas and vacuum subsystems. The overal length of the chamber is 71 cm (28?). A schematic of the 3DPX device is shown in Figure 2.1. Figure 2.1: A schematic of the 3DPX device showing (a) the top view of the experimental setup and (b) a detailed drawing of Section 1 of the experimental device where the experiments are conducted. The cordinate system shown in this figure wil be used throughout this document. The x-y plane is defined by the plane of the laser shet, while the z-direction is perpendicular to the laser shet. It is noted that gravity acts in the y-direction. 40 A photograph of the 3DPX device can be sen in Figure 2.2. Figure 2.2: A photograph of the 3DPX device and the stereoscopic particle image velocimetry diagnostic system. 2.1.2 VACUM AND GAS YSTEMS A dynamic vacuum is maintained in 3DPX using a 1.0 l/s ULVAC GLD-051 roughing pump. An ultimate base presure of p < 10 mTorr can be maintained in the 3DPX device. For most dusty plasma experiments performed in 3DPX, experiments are done at neutral gas presures ranging from p = 90 to 300 mTorr. Gas, typicaly argon, enters the 3DPX vacuum vesel through a two-stage system. First, the gas flows through a ?? diameter NUPRO cutoff valve which maintains a hard seal for the system. Next, the gas is sent through an Edwards LV10K fine control leak 41 valve. Both the gas feds and vacuum pumps are located in section 2 of 3DPX to minimize neutral gas flows that may efect the confinement and behavior of the dust particle suspended in the plasma. An photograph of the gas and vacuum subsystems can be sen in Figure 2.3. Figure 2.3: A photograph of the gas and vacum subsystems on the 3DPX device. The tube leaving the botom of the photograph is conected to a roughing pump. 2.1.3 PLASMA ND USTY PLASMA GENERATION A dc glow discharge plasma is generated in 3DPX using a pair of electrodes that are biased relative the grounded chamber wal, i.e. an anode and cathode. The electrodes are square stainles stel plates, each having a length of 2.5 cm. The electrodes are 42 soldered to 18 gauge wire and enter the 3DPX device through ?? and ?? coupling asemblies that are located on the back side of the chamber. The coupling asembly consists of two, ?? and one, ?? coupling equaly spaced fedthroughs on an ISO100 flange. Each electrode can be moved in the z-direction. A side view of this setup can be sen in Figure 2.4. Figure 2.4: A photograph of the flange and electrode asembly used to generate a plasma in the 3DPX device. To generate a plasma, argon gas is introduced into the chamber until the desired dynamic presure is achieved (typicaly, p ~ 100 mTorr). The anode is biased at V anode ~ 250 V and the cathode is biased at V cathode ~ -100 V, relative to the grounded chamber wal. Once the plasma is generated, the presure and bias voltages are adjusted until the desired experimental conditions are reached. 43 In these experiments, the dust is located on a smal tray located below the electrodes. For experiments using alumina microparticles (Al 2 O 3 , r d = 0.6 ? 0.25 ?m), the dust tray is sen in Figure 2.5(a) was used. For experiments involving monodisperse dust, the dust tray sen in Figure 2.5(b) is used. (a) (b) Figure 2.5: Photographs of the dust trays used in the experiments involving (a) polydisperse and (b) monodisperse microparticles. In al of the experiments presented in this disertation, the dust tray is electricaly floating. The dust tray rests on a hollow, ?? diameter cylindrical stainles stel tube that enters the chamber through a linear fed through mounted an ISO100 flange located at the bottom of Section 1 of the 3DPX device. The tray can be moved verticaly (y- direction) in the vacuum vesel and is electricaly floating. Once the plasma has formed, a flux of ions and electrons from the plasma charges dust particles acording to Equations 1.1 ? 1.5. If enough charge has acumulated on the dust particle to alow the electric force on these dust particles to exced the gravitational and adhesion forces that 44 hold the dust particles to the surface of the chamber, the dust particles enter the plasma and become levitated. Typicaly, these particles become trapped in the sheath of the anode, imediately below the anode, or the sheath of the dust tray, imediately above the dust tray. This is depicted in Figure 2.6. For the experiments in this disertation, the dust clouds were primarily suspended in the anode spot trap. Figure 2.6: Schematic drawing showing the locations where dust is typicaly traped in the 3DPX device. In (a), the dust is suspended below the anode in an anode sheath trap. In (b), the dust is suspended above the dust tray, in a sheath trap. Photographs of dust clouds suspended in these regions can be sen in Figure 2.7. As is the case in other experiments involving a dc glow discharge [81-83], the dust clouds in 3DPX are characterized by sharply defined boundaries betwen the dust cloud and surrounding plasma. 45 (a) (b) Figure 2.7: Images of a dust cloud suspended in (a) the anode sheath trap and (b) the sheath trap. The dust cloud is iluminated using (a) a Nd:YAG laser (? = 532 nm) and (b) a diode laser (? = 632 nm). The purple glow that is visible in the background is an argon plasma 2.2 PARTICLE IMAGE VELOCIMETRY Much like other plasmas, dusty plasmas are an example of a complex, self- organized non-linear system. However, dusty plasmas have the added advantage of alowing for the direct visualization of one component of the system on the kinetic level. Nonetheles, diagnostic studies of dusty plasmas often presents a non-trivial problem. The standard diagnostic probe techniques of plasma physics, e.g. in-situ probes such as Langmuir or emisive probes, perturb the suspended particle cloud as sen in Figure 2.8. As a result, it is necesary to study these systems using non-invasive, typicaly optical, techniques. The most common of the optical techniques is laser light scatering (LS) [8, 84], depicted in Figure 2.9. Generaly, a laser, whose wavelength is shorter than the size of the microparticles, is used to iluminate the particle cloud. The light that is scatered by 46 the particles is then recorded, typicaly using a CD, CMOS or stil digital camera. In practice, a CW red (? ~ 632 nm) or gren (? ~ 532 nm) laser is used for these types of measurements. Figure 2.8: Video images of a dust cloud in the presence of a Langmuir probe. As a bias is aplied to the probe, a void is formed inside the dust cloud. Many of the LS techniques were developed in the fluid mechanics community and are collectively known as particle-based quantitative imaging techniques [84]. The simplest of these LS techniques are most efective for low density dusty plasmas were the motion of the individual particles can be resolved. For higher density cases, 47 e.g. when particle overlap can become problematic, more advanced LS techniques becomes necesary. When one is interested in measuring the velocity field in a flow, there are a number of quantitative imaging techniques that can be used. In a typical fluid measurement, these techniques involve injecting discrete particles into the flow field, a proces known as ?seding?, and then observing the motion of these tracer particles over a large region of the flow. For these techniques to yield meaningful information, there are two of conditions that these tracer particles must satisfy. First, they must be large enough to be observed opticaly. Second, the tracer particles must be neutraly buoyant with respect to the flow, i.e. they must be sufficiently smal such that they travel with the fluid flow without disturbing the flow. The most common of these particle-based quantitative imaging techniques [84] are particle streak velocimetry (PSV), particle tracking velocimetry (PTV), particle image velocimetry (PIV) and laser speckle velocimetry (LSV). The setup for each of these techniques is esentialy the same. Light, from any number of sources, is used to iluminate the flow of interest. Typicaly, a laser beam is expanded into a two- dimensional light sheet, which is oriented in the direction of the flow of interest. Light from the laser sheet then scaters off of the tracer particles located in the flow of interest and is recorded by a camera that is oriented perpendicular to the laser sheet. It is noted that if digital imaging (i.e. a CD camera) is used in the measurement, then the above techniques are denoted as digital, i.e. digital particle image velocimetry (DPIV) [85]. A typical setup for measuring two-dimensional flows is shown in Figure 2.9. 48 Figure 2.9: A schematic showing the typical setup for a two-dimensional quantitative imaging system. There are, however, very important diferences in these techniques. In particle streak velocimetry, PSV, the image exposure time is long relative to the time that a tracer particle occupies a given position. The result is a streak in the recorded image, the length of which can be used to extract a velocity using the exposure time. In particle tracking velocimetry, PTV, the seding density of the tracer particles is sufficiently low that each particle image is clearly observable without overlap or interference with other imaged particles. Images are acquired in a short period of time so that the particles are observed as discrete dots and acquired with frequency sufficiently large to alow the motion of each individual particle to be tracked. In laser speckle velocimetry, LSV, the seding density of tracer particles is sufficiently high that individual particles cannot be detected due to overlap with neighboring particles. As a result, one observes a speckle patern. In particle image velocimetry, PIV, the seding density of the tracer particles lies in an intermediate range betwen what is used in the PTV and LSV technique. Here, the individual tracer particles can be sen, but there is some overlap with neighboring particles. Figure 2.10 depicts images from each of these diagnostic techniques. 49 (a) (b) (c) (d) Figure 2.10: Characteristic images taken when using (a) particle streak velocimetry (Image courtesy of H. Thomas, Max Planck Institute for Exterestrial Physics, Garching, Germany), (b) particle tracking velocimetry, (c) particle image velocimetry and (d) laser speckle velocimetry. For the experiments presented in this disertation, the PIV measurement technique was employed. The details of the PIV measurement technique are presented in the following sections. Section 2.2.1 discusses the basic features of the PIV measurement technique. Section 2.2.2 discusses the implementation of two-dimensional PIV technique. Section 2.2.3 discusses an extension of two-dimensional PIV technique to 50 measure thre-dimensional velocities. Finaly, the system that was used in making the measurements described in this disertation wil be discussed in Section 2.2.4. 2.2.1 PRINCIPLES OF PARTICLE IMAGE VELOCIMETRY Digital particle image velocimetry is a whole-field, non-invasive optical technique that alows instantaneous fluid flow measurements [85]. In the typical PIV experimental configuration [79, 86], a flow is seded with micron-sized particles that are then caried by the fluid flow of interest. These tracer particles are sufficiently smal that they travel with the flow of interest without disturbing it and are then iluminated by a light source. Typicaly, a pair of laser pulses, that are aligned to folow the same optical path, iluminate the tracer particles at a user-defined time interval, ?t laser . The duration of the laser pulses is sufficiently smal (on the order of nanoseconds) that the particles do not move more than 10% of their diameter. This rule of thumb value prevents streaking of particle images, which significantly reduces the quality of the computed velocities. These laser pulses are normaly shaped into a two-dimensional light sheet using an asortment of optical components. Light from the laser sheet wil scater off of the tracer particles and is then recorded using either a CD or film camera that is, typicaly, oriented perpendicular to the laser sheet. These images, once in digital form, are divided into a series of interogation regions of size n ? n pixels. It is then possible to extract the mean displacement for al of the particles that are located in each interogation region using the PIV technique. Using this displacement and the time betwen subsequent iluminations of the tracer particles, ?t laser , one can compute a velocity. A sketch of a ?clasic? PIV system, 2D PIV, is sen in Figure 2.11. Using the 2D PIV setup, one is able to recover the projection of the motion in the plane of the laser 51 sheet, i.e. a single PIV measurement provides a ?snapshot? of the flow velocities over the whole field. Figure 2.1: A carton depicting a 2D PIV system (Figure used with permision of LaVision, Inc.) In the experiments that are discussed in this disertation, an extension of the two- dimensional PIV technique, known as stereoscopic particle image velocimetry or stereo- PIV, is used [87]. Stereo-PIV techniques alow one to measure the thre-dimensional velocities (v x , v y and v z ) of the tracer particles. In this approach, two cameras are oriented obliquely to the laser sheet, as sen in Figure 2.12. Here, each camera functions as a two- dimensional PIV system. Using the velocity profiles that are generated from each camera system (v x,1 , v y,1 , v x,2 , v y,2 ) and the orientation of the cameras relative to the laser sheet, one can compute the thre-dimensional velocity profile. 52 (a) (b) (c ) Figure 2.12: Orientation of a stereo-PIV system in the (a) forward-backward (b) forward-forward and (c) backward-backward scatering configuration. In the forward-backward setup, the cameras are located on the same side of the flow of interest, while the cameras observe the flow of interest on oposite sides of flow in the backward-backward and forward-forward setup. (Figures used with permision of LaVision, Inc.) It is noted that the application of the PIV technique in dusty plasmas is somewhat novel, when compared to the typical applications of this technique in the fluid dynamics community. Unlike the typical use of these techniques in fluid mechanics [79], where the tracer particles sed the flow of interest, in dusty plasmas, the tracer particles are the suspended microparticles (i.e. the system of interest). Despite this diference, the PIV technique has been succesfully applied to many types of systems in dusty plasmas [78, 88-91]. 2.2.2 TWO-DIMENSIONAL PARTICLE IMAGE VELOCIMETRY In this section, the ?clasical?, two-dimensional PIV technique is described. It is noted that there are numerous algorithms and recording techniques that have been developed for computing the velocity vectors [79]. In what follows, the discussion is restricted to the techniques that are used in procesing the data that is acquired on the 3DPX device. 53 The images from which velocities are computed are recorded using the double- frame/single exposure PIV recording technique. Using this approach, a laser fires at times t 0 and t 1 = t 0 + ?t laser . The light that scaters off of the tracer particles from the first laser pulse, t 0 , is recorded in the first frame, while the light from the second laser pulse, t 1 , is stored in the second frame. The image is broken into evaluation windows of size n ? n pixels and a cross-correlation betwen identical evaluation regions in these two frames is performed to compute the average displacement of al the tracer particles located in the evaluation region. This is depicted in Figure 2. 13. Figure 2.13: A carton depicting the PIV analysis proces (Figure courtesy of LaVision, Inc.) The cross-correlation is done using the ?standard? two-dimensional Fourier cross- correlation technique, which is the most commonly used algorithm in the fluid dynamics community. A flow diagram for this proces is sen in Figure 2.14. 54 Figure 2.14: Flow diagram for the PIV analysis proces. To visualize the seded flow, a laser beam is shaped into a laser sheet using a combination of spherical (L 2 ) and cylindrical (L 1 ) lenses to iluminates a slice of this fluid flow, as depicted in Figure 2.15. Figure 2.15: A lens system consisting of a spherical lens (L 2 ) with focal length f 2 and a cylindrical lens (L 1 ) with focal length f 1 is used to generate a laser shet of height L D , for ilumination of the tracer particles. Incident on L 1 is a laser beam have a beam diameter L d . The height and width of the laser sheet can be computed using geometric optics. For the system shown in Figure 2.15, these quantities are given by Equation 2.1 55 ! L D = f 2 f 1 L d w=2 2.44"f 2 d (2.1) where L D is the height of the laser sheet, L d is the diameter of the beam, f i are the focal lengths asociated with the lenses as numbered in Figure 2.15 and w is the waist (thicknes) of the beam. In a typical PIV measurement [79], the nomenclature used has the tracer particles located in the object plane and the CD aray is the image plane. At time t 0 , the particles at a location ! r X i are iluminated by the laser and imaged on the CD aray at a location ! r x i . At a later time, t 1 , the particles have moved to a location ! r X i ' and are imaged on the CD aray at a location ! r x i ' x i ?. As a result, the real displacement, ! r D , appears as displacement ! r d on the CD aray. This is depicted in Figure 2.16. 56 Figure 2.16: Diagram showing the imaging of a particle in the laser shet on the CD camera. Asuming that there is no distortion in the imaging optics and that there is a uniform magnification across the field of the field of view, the displacement in the recorded image, ! r d = r x i '" r x i , is related to the real displacement, ! r D = r X i '" r X i , through Equation 2.2. ! d x =x i '"x i ="M(D x +D z x i ' o ) d y =y i '"y i ="M(D y +D z y i ' o ) (2.2) 57 A key asumption in the application of 2D PIV is the displacement is predominately in the plane of the laser sheet (i.e. the x-y plane), which is to say that ! D z "0. In this limit, the observed motion is described by Equation 2.3. ! d x d y " # $ % & ? =M D x y " # $ % & ? (2.3) The mathematical details of the imaging proces can be found in Appendix A.1.1.1. To calculate the motion of the particles, a cros correlation analysis [92] is performed betwen a subset of an image taken at time t 0 , which is described by an intensity function I 0 (x, ?), and a subset of an image taken at time t 1 , which is described an by intensity function I 1 (x, ?). In practice, one does not analyticaly atempt to estimate the displacement function, ! r d . Instead, one statisticaly finds the best match betwen the two images, t 0 and t 1 , using the discrete cross correlation function given by Equation 2.4. ! Rk,l()=Ix,y()" I x+k,y+l( ) l=#L L $ k=#K K (2.4) where I and I? are subsets of images I 0 (x, ?) and I 1 (x, ?), with I? being larger than I, centered at location (x, y) of the CD aray. Physicaly, the image element I is shifted by an amount (k, l) over the image element I?. This shifting operation is done over a range 58 of values (-M ? k ? M, -N ? l ? N). At each location, the quantity in Equation 2.4 is computed, forming a correlation plane of size 2M+1 by 2N+1. The mathematical details of this are given in Appendix A.1.1.2. There are two points that bear mentioning on the application of the cross correlation technique. First, and as noted previously, it is only possible to recover linear displacements using this technique. To acount for this in practice, one must use a sufficiently smal ?t laser and an interogation region that is smal enough that there are no second order displacements. Second, this is a computationaly intensive proces, requiring O[N 2 ] computations to form a correlation plane of spatial dimension N ? N. However, there is a more computationaly eficient means of computing the correlation function in Equation 2.4. This technique makes use of the correlation theorem, which is derived in Appendix A.1.2. The correlation theorem states that cross correlation of two functions, I and I? is equivalent to a complex conjugate multiplication of their Fourier transforms, or ! Rx,y()" ? I # ? I ' * (2.5) where ! ? I and ! ? I ' * are the Fourier transform of I and I?, respectively. The Fourier Transforms provide a computationaly eficient approach to computing the cross correlation function. The flow diagram of the PIV analysis is sen in Figure 2.17. 59 Figure 2.17: Implementation of the cros corelation technique using fast Fourier Transforms. Once the cross correlation plane has been computed, the largest peak in the correlation plane is fited using a Gausian curve. In the DaVis software, thre adjoining points from the correlation plane are fit using a Gaussian peak fit in both the x and y direction. To do this, the correlation plane is scanned for the maximum value, R (i, j) . The adjoining four correlation points in the correlation plane, R (i-1, j) , R (i+1, j) , R (i, j-1) and R (i, j+1) , are extracted. A Gaussian curve, defined in Equation 2.6, is then fit in both the x- and y- directions. From these fits, the location of the correlation peak can be identified to within 0.05 pixels, depending on the quality of the images used. ! fr()=Cexp" r o "r() 2 k # $ % & % ? ( % ) % (2.6) where r denotes a coordinate position (here, it can denote either the x- or the y-position), C and k are constants, and the r o denotes an offset in either the x- or y-direction defined in Equation 2.7. 60 ! x o =i+ lnR i"1,j() "lnR i+1,j() 2lnR i"1,j() 4ln i,j() 2lnR i+1,j() y o =j+ lnR i,j"1() lnR i,j+1() 2lnR i,j"1() 4ln i,j() 2lnR i,j+1() (2. 7) where i and j denote discrete locations in the correlation plane and R (i, j) denotes the value of the correlation value at location (i, j). This proces is depicted in Figure 2.18. Although the PIV algorithm described above yields generaly good results, there are additional refinements that can be used to improve the quality of the results generated in the PIV analysis. The first technique involves overlapping evaluation regions. In this work, a 50% overlap is used. This more than doubles the number of independent vectors that are computed and more acurately maps the velocity field. It is noted that a 50% overlap is the maximum overlap that can be used and stil obtain independent PIV vectors. The second technique is multi-pas correlation. In this proces, the correlation analysis is repeated m times. In the first pas, a displacement map is generated. In the next pas, when the correlation analysis is repeated, the subregion extracted from the second image used in the correlation analysis is offset in the direction of displacements that were computed in the first correlation analysis. This proces is then repeated until the m iterations have been completed. For each iteration, the displacement field that was generated in the previous correlation analysis is used to generate the image shift for the sub region in the second image. 61 100x10 3 80 60 40 20 0 Cross Correlation [AU] 12111098765 y-displacement [pixels] Cross Correlation Gaussian Fit (a) (b) 160x10 3 140 120 100 80 60 40 20 0 Cross Correlation [AU] 12111098765 x-displacement [pixels] Cross Correlation Gaussian Fit Figure 2.18: Depicting how sub-pixel resolution is achieved in the PIV analysis. The bars denote the computed values of the cros-corelation function, while the curve depicts a best fit Gausian to the computed values in the (a) x-direction and (b) y-direction. The peak from the curve fit yields the desired displacement with far greater acuracy than can be found from the raw cros corelation data. The actual displacement in the images used in this analysis was 8 pixels in the x-direction and 8.5 pixels in the y- direction. The displacement that is extracted from the fit was found to be 7.92 pixels in the x-direction and 8.5 pixels in the y-direction. It is noted that the cros corelation analysis performed in the analysis generating this figure was far simpler than what is done in the DaVis software. The multi-pas correlation analysis yields a more acurate result for four reasons. First, this approach minimizes the eror in the measured displacement by keeping the correlation peak near the origin of the correlation plane. Second, the image shift that is 62 used is localy determined. As a result, a shift that more closely resembles the flow is used than what would otherwise be obtained using a global offset. Third, this approach alows one to use a smaler evaluation region, which wil generate a higher spatial resolution. Finaly, this approach results in a more acurate velocity field, as fewer particles are likely to leave the evaluation region [93]. In the work in this disertation, a two pas correlation analysis is performed. On the first pas, a 16 ? 16 pixel evaluation region is used and a 12 ? 12 pixel evaluation region is used in the second pas. 2.2.3 STEREOSCOPIC PARTICLE IMAGE VELOCIMETRY While the PIV approach is a powerful measurement technique, it does have limitations. Foremost among these limitations is that the ?clasical? PIV measurement can only measure the projection of the velocity vector in the plane of the laser sheet. This eror can be sen in Figure 2.19 and can lead to a potentialy significant source of eror: any motion that is transverse to the plane of the laser sheet is lost. Further, this lack of information regarding this third component of velocity can introduce an unrecoverable eror in the in-plane motion. To quantify this, consider the motion that is shown in Figure 2.19. 63 Figure 2.19: Showing the motion observed by the PIV system, ! r v measured , as oposed to the actual motion, ! r v actual , of the particle. The true motion of the particle is depicted by the solid arow, while the dashed arow depicts the measured motion. In Figure 2.19, the laser is directed along the x-axis, while the PIV camera is oriented along the direction of the z-axis, alowing for visualization of motion in the x-y plane. Suppose that the particle is moving purely in the x-z plane with a speed v actual at an angle ? with respect to the x-axis. The velocity that is measured by the PIV system wil be ! r v measured = r v actual cos" ? i . The eror is negligible for motion that is esentialy restricted to the x-axis, i.e. along the direction of the ilumination sheet of light. As ? increases, the eror becomes progresively larger as indicated in Figure 2.20. It is clear that there can be a strong need for knowledge of this third component of the motion. 64 1.0 0.8 0.6 0.4 0.2 v measured : v actual 806040200 ! (?) Figure 2.20: Plot of the ratio of the velocity measured by the PIV system, v measured , to the true velocity, v actual , as a function of the angle ?. For ? > 25?, the eror exceds ten percent. There are numerous methods that can be implemented to extract information about the third velocity component [94]. The most straightforward of these methods involves the addition of a second 2D PIV measurement from a diferent viewing axis [87]. This approach is generaly refered to as stereoscopic particle imaging velocimetry. In practice, there are two typical orientations for the two camera stereo-PIV system, refered to as the lens translational and the angular displacement method. The orientation of the cameras in each of these approaches is shown in Figure 2.21. For the work that is presented in this disertation, the angular displacement method is employed. 65 (a) (b) Figure 2.21: Depicting the orientation of the cameras in the two standard stereoscopic particle image velocimetry orientation: (a) translational method and (b) angular displacement method. (Figures used with permision of LaVision, Inc.) Figure 2.22 provides a qualitative ilustration of the ?out of plane? velocity reconstruction using two cameras. Because of the diferent orientations, each camera wil measure a diferent particle displacement. However, if the orientation of each camera, relative to the light sheet, is known, it is possible to reconstruct the out-of-plane velocity component of the motion. This proces is quantitatively described in Section 2.2.3.2. 66 Figure 2.2: Depicting how the out-of-plane component is extracted using the stereoscopic set-up. A diference in the measured displacements is observed if there is motion in the out-of-plane direction and is related to the displacement in the out-of-plane motion. (Figure used with permision of LaVision, Inc.) For a stereoscopic system using the angular displacement method, the measurement precision of the out-of-plane component increases as the angle betwen the cameras increases. However, this oblique viewing results in a limited depth of field. As a result, it is dificult to obtain an image that is focused over the entire field of view, as sen in Figure 2.23. 67 Figure 2.23: Depicting the limited depth of field that arises due to the angular displacement method of stereoscopy. It is noted that only the dots on the right hand side of the image are in focus. This limited depth of field can only be acommodated for by tilting the image plane such that the image, lens and object plane intersect at a common point. This is known as the Scheimpflug criteria (or the ?Keystone Correction?) [95] and is ilustrated in Figure 2.24. Figure 2.24: Depicting the Scheimpflug criteria, where the image, object and lens planes al intersect at a single point. 68 To enforce the Scheimpflug criteria, the lens for the CD camera is mounted on a Scheimpflug adapter. This alows for adjustment of the lens plane until the thre planes are properly aligned. When the thre planes are properly aligned, the entire field of view wil be in focus. Figure 2.25 shows how the Scheimpflug criteria is enforced. Figure 2.25: Showing the posible lens tilts that can be encountered when enforcing the Scheimpflug criteria. In (a) the lens is not tilted, (b) the lens has not ben tilted enough, (c) the lens is titled to the Scheimpflug angle and (d) has the lens has tilted to far. 69 In practice, one must manualy adjust the Scheimpflug adapter until the image appears to be in focus over the entire field of view. In Figure 2.26, one ses the image from Figure 2.23 once the Scheimpflug criteria has been enforced. Figure 2.26: Depicting the image sen in Figure 2.23, once Scheimpflug criteria has ben enforced. Enforcement of the Scheimpflug criteria introduces a strong perspective distortion, which can be sen in Figure 2.26. As a result, a rectangular image appears as an isosceles trapezoid. This is due to two types of distortion: anamorphic and shearing distortion. The anamorphic distortion generates a foreshortening of the x-dimension (horizontal direction) by the cosine of the object tilt angle. The shearing distortion results in a non-uniform magnification in the y-dimension (vertical direction) as one moves across the x-dimension. To compensate for these two forms of distortion, the PIV camera must be empiricaly calibrated. This is done with the aid of a 3-D ?target? sen in Figure 2.27. This calibration proces [96], which is more completely discussed in Section 2.2.3.1, wil generate a mapping function for each PIV camera which wil 70 transform, or ?dewarp,? the acquired images from the image plane (i.e. what is acquired by the cameras) to the global coordinate system (i.e. where the actual displacements are computed). In Figure 2.27, one ses the image from Figure 2.26 dewarped into the gloabal coordinate system for the camera orientation used to acquire the image sen in Figure 2.26. The region of the image that is black are lost in the dewarping proces, while the red cross depicts the expected location of the dots on one plane of the calibration target. Figure 2.27: Depicting the image sen in Figure 2.24 once the image has ben dewarped into the global cordinate system. The red grid depicts the anticipated location of the dots on one plane of the calibration target. It is noted that the distortion observed in Figure 2.26 is gone and the magnification is uniform acros the field of view. Once the acquired images have been dewarped into the global coordinate system using the mapping functions that were generated in the empirical calibration proces, two sets of 2D velocity vectors, one for each camera, are generated. It is noted that the requirement of uniform image magnification that is asumed in the derivation of the imaging proces used in the 2D PIV analysis has been satisfied once the images have 71 been dewarped into the global coordinate system. Additionaly, from the calibration proces, one gains knowledge of the orientation of the cameras relative to each other and the laser sheet. Using this geometry, one can extract the thre-dimensional velocity field. It is noted that the system is over specified, i.e. the thre velocity vector components are computed from the two known two dimensional velocity vectors. As a result, the velocities that are computed using the stereo-PIV technique tend to be more acurate that those found using the clasical 2D PIV approach. Section 2.2.3.1 describes the calibration techniques, which are used on the 3DPX experiment to compute the image dewarping parameters and to determine the orientation of the two cameras relative to each other. Section 2.2.3.2 describes how to extract the thre-dimensional velocity information. 2.2.3.1 IMAGE DEWARPING AND CALIBRATION In order to compute the thre-dimensional velocities acquired using this stereoscopic setup, the two-dimensional vectors from each PIV setup must be computed in the same coordinate system, i.e. the global coordinate system. To determine the mapping function from the image plane to the global coordinate system, an empirical calibration proces is performed using Tsai?s pinhole model [97]. In this approach, each camera images a thre-dimensional, two-level calibration plate containing a defined grid of marks. Two calibration plates that are used on the 3DPX device are sen in Figure 2.28. Using these images, a mapping function for each PIV camera is generated using Tsai?s pinhole. The pinhole model is based on the theorem of intersecting lines and 72 maps a particle, which exist in the ?global? 3-D coordinate system (X W , Y W , Z W ), to a location on the CD aray, the camera coordinate system (x, y). The mathematical details of this model can be found in Appendix A.1.3. Figure 2.28: Showing two 3D calibration targets used on the 3DPX device. The large target, shown in (a), consist of an aray of Maltese croses, while the smaler target, shown in (b), consist of an aray of dots. One critical aspect in the calibration proces is the need for a perfect alignment of the laser sheet and the calibration target [96]. Any misalignment wil result in an eror in the mapping function. As a result, the images wil not be dewarped into the same world coordinate system, leading to an eror in the computed velocity field. This is sen in Figure 2.29. Here, the laser is not exactly aligned with the global coordinate system. As a result, each camera wil detect that the particle appears in slightly diferent positions in the global coordinate systems. The particle wil appear shifted to the left by camera 1 and to the right by camera 2. The diference in these positions defines a disparity vector and can be a major source of eror in the resulting velocity field. 73 Figure 2.29: Showing the eror that is introduced when the laser shet and calibration plate are not properly aligned. It is possible, however to tweak the calibration to acount for possible misalignments betwen the laser sheet and calibration plate that existed during the initial calibration. This proces, known as self-calibration [96], simplifies the calibration proces and results in significantly smaler eror. A flow diagram for the self-calibration proces is sen in Figure 2.30 and described below. 74 Figure 2.30: Flow diagram of the self-calibration proces. After the system has been calibrated using a calibration plate and the pinhole model, a sequence of ~50 images of a particle patern is acquired. By cross correlating the acquired images of the particle patern, it is posible to determine if there is deviation betwen the two camera?s field of views. If the dewarped images are exactly the same, then there wil be no displacement field generated. If there is a deviation, the resulting displacement field, known as a disparity vector map, can be used to correct the camera calibration. Using the disparity map, one can determine the corresponding world points in the measurement plane using a standard triangulation method. These world points are then fited to a thre dimensional plane, which is used to correct the mapping function of each camera. It is noted that the triangulation method requires a volume mapping function, which is why a pinhole model is used in the calibration proces. This proces 75 can be repeated to improve the results. However, the fit tends to converge after a few (~3) pases. 2.2.3.2 RECONSTRUCTING THREE-DIMENSIONAL VELOCITY INFORMATION Using geometric optics, the particle image displacement for the system sen in Figure 2.16 was found in Section 2.2.2 to be ! d x =x i '"x i ="M(D x +D z x i ' o ) d y =y i '"y i ="M(D y +D z y i ' o ) (2.8) In the stereoscopic system, each camera wil generate a two-dimensional velocity field, ! r V =U W,i V W,i ( ), given by Equation 2.9. ! U W,i =" # x i "x i M$t V W,i =" # y i "y i M$t (2.9) where i is an index denoting the camera (right or left). It is noted that the displacements and velocities given by Equations 2.8 and 2.9 respectively have been computed in the global coordinate system. Consequently, it is asumed that the acquired images have 76 been dewarped as described in Section 2.2.3.1 prior to the PIV calculation being performed. In practice, this is the case. To extract the thre-dimensional velocities, the following geometry is asumed for the two PIV cameras: ? i denotes the angle in the x-z plane betwen the z-axis and the ray that connects the tracer particle and the recording plane through the lens center and ? i denote the corresponding angle in the y-z plane. This is ilustrated in Figure 2.31. Figure 2.31: Depicting the geometry of the stereo-PIV system as viewed from (a) above and (b) the side. It is noted that the angles ? and ? do not have to be the same for both camera. In this example, the laser shet propagates defines the x-y plane. For the geometry shown in Figure 2.31, the angles ? i and ? i are defined as ! tan" i = # x i z o tan$ i = # y i z o (2.10) 77 where i is again an index denoting the camera (right or left). Using these angles and the velocities that are computed from each PIV camera, the velocities in the global coordinate system, ! r V =V x , y ,V z ( ) , are given by Equation 2.11. ! V x = U W,r tan" l +U W,l tan" r tan r tan" l V y = V W,r tan# l +V W,l tan# r tan r tan# l = V W,r +V W,l 2 + V z 2 tan# l $tan# r ( ) V z = U W,r $U W,l tan" r +tan" l (2.11) 2.2.4 STEREOSCOPIC PARTICLE IMAGE VELOCIMETRY SYSTEM AT AUBURN UNIVERSITY This section describes the hardware of stereo-PIV system that was used in the studies presented in this disertation. Section 2.2.4.1 describes the hardware and software components. Section 2.2.4.2 describes initial measurements that were made to verify the functionality of this system. Finaly, Section 2.2.4.3 describes how the uncertainty in the system is measured. 2.2.4.1 DESCRIPTION OF THE SYSTEM The stereo-PIV system used in the Plasma Sciences Laboratory at Auburn University was purchased from LaVision Inc. in 2003. It consists of a pair of frequency doubled neodymium-doped yttrium aluminium garnet, Nd:YAG, lasers (? = 532 nm) and 78 a pair of cross-correlation CD cameras oriented in the backscatering configuration. The laser used in this system is a Solo PIV II, manufactured by New Wave Research. This dual cavity pulsed laser has a repetition rate of 15 Hz and an output energy of 50 mJ per pulse. The laser pulses are shaped into a laser sheet using a -10 m focal length cylindrical lens. The CD cameras used in this system are Imager Intense cross correlation CD cameras. This 12-bit progresive scan CD camera has a 1376 ? 1040 pixel aray that acquires images at a frame rate of 10 frames per second. To improve the collection eficiency of the CD elements of the camera, the cameras are cooled to an operating temperature of -10?C using Peltier elements. In the stereoscopic arangement, the cameras operate at a maximum frame rate of 5 Hz. Each camera is configured with a Scheimpflug lens adapter and a 60 m Nikon MicroNikkor lens. To reduce extraneous light signals, each camera is fited with a narow band pas filter, ? = 532 ? 10 nm. Both the laser and the camera systems are mounted on a single axis linear translation stage that has 500 m of travel, alowing for measurements of the velocity vectors anywhere in the volume of the 3DPX device. A sketch of the experimental setup is sen in Figure 2.32. 79 Figure 2.32: Schematic depicting the top view of the experimental setup and the orientation of the stereo- PIV hardware. A software system from LaVision Inc., DaVis (Data Visualization), is used to control the PIV hardware (laser, cameras and translation stage) and to compute the thre- dimensional velocity vectors. A flow diagram of the full PIV analysis proces used by the DaVis software sen in Figure 2.33. 2.2.4.2 VERIFICATION OF THE SYSTEM Shortly after acquiring the stereo-PIV system, an experiment was performed to verify the system?s thre-dimensional measurement capabilities [78]. In this investigation, a dusty plasma was generated and a perturbation was applied to induce an out-of-plane (i.e., z-direction) displacement of the dust cloud. Two independent imaging systems (the stereo-PIV system and a separate LS system) were used to confirm the motion of the cloud. 80 Figure 2.3: Showing the PIV analysis proces used by the DaVis software. 81 In this experiment, an argon dc glow discharge plasma is generated using a pair of square, stainles stel electrodes that are 2.5 cm on a side. The electrodes are biased with respect to the electricaly grounded vacuum vesel wals. The upper electrode functions as the anode and is biased to V anode = 200 V. The lower electrode functions as the cathode and is biased to V cathode = -90 V. The chamber is filed with argon gas to a presure p = 114 mTorr. In this configuration, the dust cloud formed in the sheath of the dust tray. The microparticles used in this experiment are silica microspheres having a size distribution defined by r = 1.5 ? 0.5 ?m. The source of the particles for this experiment was located on a square tray that is ~ 2.5 cm below the cathode. An SEM image of the microparticles used in this experiment is shown in Figure 2.34. Figure 2.34: SEM image of the silica microspheres used in experiments verifying the functionality of the stereo-PIV system. {SEM image courtesy of Andrew Post-Zwicker, Princeton Plasma Physics Laboratory} 82 To induce the perturbation in the z-direction, a third electrode is used. This perturbation electrode is rectangular copper plate and is located below and behind the anode and cathode. The orientation of the electrodes and dust tray are shown in Figure 2.35. Figure 2.35: Orientation of the electrodes and dust tray used to verify the functionality of the stereo-PIV system. To verify that a shift in the z-direction occurred, a second video system, oriented perpendicular to the PIV laser sheet, is used. This second system consisted of smal laser diode (? = 635 nm, red), a third, 30 frame per second CD video camera, and a video recorder. The orientation of the PIV system and this secondary video system are sen in Figure 2.36. 83 Figure 2.36: Schematic showing the orientation of the stereo-PIV and LS imaging systems used to verify the functionality of the stereo-PIV system. A triangular pulse of duration 650 ms raises the voltage on the perturbation electrode from V p = 50 V to V p = 100 V. As a result of this pulse, the particle cloud experiences both a vertical (y-direction) and an out-of-plane (z-direction) shift. Images from the second camera system that verify the motion in the z-direction are sen in Figure 2.37. 84 Figure 2.37: A video image sequence acquired using the second camera system confirming the displacement in the z-direction. For increased clarity, the cloud is outlined in red. The yelow rectangle in the first image denotes the location of the laser shet for the stereo-PIV system. The dust tray that holds the source powder can be sen in the botom, center of the images, while the perturbation electrode is visible in the lower right of the images. A twenty image pair sequence (40 images per camera) was acquired using the stereoscopic PIV system. The lasers are fired with a time separation of ?t laser = 2.5 ms and the time separation betwen succesive image pairs is ?t image = 0.2 sec. Four of the reconstructed velocity measurements from the 20-image sequence are shown in Figure 2.38. Here, the arows represent the two-dimensional velocity of the microparticles in the plane of the laser sheet, while the contours represent the z- component of velocity (i.e. the out-of-plane motion). The measurements indicate that in the first (t = 0.8 sec) and fourth (t = 2.4 sec) images shown, the microparticles, on average, have no directed z-motion. However, at times t = 1.4 sec and t = 1.8 sec, there is a significant out-of-plane motion. The particle cloud is first pushed in the +z-direction 85 with the application of the pulse, and then restored to its original position by returning in the -z-direction. t = 0.8 sec t = 1.8 sec t = 1.4 sec t = 2.4 sec Figure 2.38: Four images of the reconstructed velocity field from the stereo-PIV system for times t = 0.8, 1.4, 1.8, and 2.4 seconds. The red-gren shading at the center of each of the four panels represents the z- component of the velocity of the microparticle cloud. The image labeled t = 0.8 sec, is before the aplication of the pulse and there is no net z-motion. At t = 1.4 sec, the pulse is aplied and the cloud is moving in the +z-direction as indicated by the bright red contours. At t = 1.8 sec, the pulse has reversed and the particle cloud is moving in the ?z-direction as indicated by the gren contours. By the last image, t = 2.4 sec, the perturbation pulse has ended and the cloud has returned to a state of no net z-motion. This out-of-plane movement is further confirmed by plotting the distribution of the particle velocities in the z-direction as shown in Figure 2.39. Here, the distribution of the z-component of velocity, v z , is plotted for each of the four cases shown in Figure 2.37. Again, it is noted that for first (t = 0.8 sec) and fourth (t = 2.4 sec) images shown, the velocity distributions are centered at v z = 0 m/s. The t = 1.4 sec case is centered about v z = 12 m/s and the t = 1.8 sec case is centered about v z = -10 m/s. 86 60 50 40 30 20 10 0 number (a.u.) -30 -20 -10 0 10 20 30 velocity (x10 -3 m/s) (5) t = 0.8 sec (8) t = 1.4 sec (10) t = 1.8 sec (13) t = 2.4 sec Figure 2.39: Distribution of the z-component of velocity for the four cases shown in Figure 2.37. 2.2.4.3 MEASUREMENT OF THE RESOLUTION LIMT OF THE SYSTEM As discussed in Section 2.2.3, the stereo-PIV system is a complicated optical diagnostic. As a result, it is not surprising that it is dificult to directly calculate the measurement uncertainty for this system. In the past, there have been theoretical [98] and experimental [99] studies that have atempted to quantify the uncertainty of the stereo- PIV technique. In the theoretical study, a simple geometric eror model for stereoscopic particle image velocimetry technique was developed. This model provides insight into the diferences betwen the translational and angular configurations. However, the results are limited as a number of erors are not included in the eror model developed. These sources of erors include lens aberations, diferences in the laser power used for measuring each PIV image and erors in the algorithm used in the cross-correlation analysis. Not surprisingly, there were observed diferences betwen the theoretical 87 results and what was measured experimentaly. Further, one cannot necesarily apply the results from the experimental studies to other systems because there wil be diferent sources erors and diferent quantities of erors in al of the elements of the system. As a result, the experimental and theoretical studies provide a guideline for uncertainty but they cannot necesarily be directly applied to other experimental setups. Beyond the sources of erors that exist in the hardware components of the stereo- PIV system, there is an additional eror that is introduced by the software algorithm that computes the velocity vectors. This numerical eror can, and has been, quantified for various analysis schemes by using standardized, ?ideal? images from the PIV Chalenge (http:/ww.pivchalenge.org). It is noted that in this case, the images that are analyzed represent what one might cal an ?ideal? PIV measurement, where there are no erors in the system (e.g. eror due to optical distortion) and the exact calibration is known. Using these images, the DaVis software used in these studies was found to have an eror of 0.03 pixels [100]. It is, however, possible to experimentaly measure the total uncertainty of a stereo-PIV system, using a ?zero-displacement? test. To do this, two images are taken with a sufficiently smal time increment betwen them that there wil be efectively no motion betwen the two images. By running these images through the PIV analysis software, one can determine the uncertainty in the PIV measurement. Such a measurement wil include al erors due to hardware and software sources. By plotting the distribution of vectors that are reported in this measurement, one can extract the uncertainty in the stereo-PIV measurement. This result depends on the calibration and 88 optical setup. Consequently, this zero-displacement test needs to be repeated whenever the system is recalibrated. The results of one such measurement can be sen in Figure 2.40. In this measurement the optical setup that was used in the experiments presented in Section 4.2 was used and the time betwen images, ?t laser , was set to 1 ?s. It is noted that there was no motion observed in the images taken in the zero-displacement test. Further, under the asumption that the particles having a peak velocity of 10 m/s and an image resolution of 35 ?m/pixel; a ?t laser , of 1 ?s corresponds to a displacement of 0.0002 pixels. These values, which are typical for the experimental systems used in this disertation, yield a displacement that is wel below the limitation of the software algorithm used. 89 1200 800 400 0 Counts [A.U.] 0.40.20.0-0.2-0.4 Position [pixels] 1200 800 400 0 Counts [A.U.] 0.40.20.0-0.2-0.4 Position [pixels] 1200 800 400 0 Counts [A.U.] 0.80.60.40.20.0-0.2-0.4-0.6-0.8 Position [pixels] ! x [pixel] Fit width = 0.105 ? 0.001 [pixels] ! y [pixel] Fit width = 0.092 ? 0.001 [pixels] ! z [pixel] Fit width = 0.199 ? 0.002 [pixels] (a) (b) (c) Figure 2.40: Showing the eror in the stereo-PIV system in the (a) x-, (b) y- and (c) z-direction. The optical setup for this measurement coresponds to the one that was used in taking the data in Section 4.2 of this disertation. It is noted that the eror is comparable to what one observes in a two dimensional PIV system. 90 2.3 PARTICLE DENSITY MEASUREMENT While the PIV technique is particularly wel suited for studying the dusty plasmas that exist in the weakly-coupled regime, there are a few drawbacks to this technique. Unlike the particle tracking techniques that were used in the previous studies of the thermal properties of a strongly-coupled dusty plasma, the PIV technique does not return a velocity for each of the particles in the system being studied. Instead, what is obtained is a velocity that represents the average motion of a group of particles. The result of this is an intrinsic averaging in the velocities that are measured and is a wel-known feature of the PIV technique [85, 101, 102] in the fluid dynamics community. In traditional PIV measurements, this efect is negligible and as a result, there have been no studies to quantify this. In the next chapter, the results of a computational study of this phenomenon are presented. However, at this time, two key results from these computational studies are noted. First, the distribution of velocities that are measured using PIV techniques are narower that the velocity distribution that is being measured. There is, however, a unique mapping function that directly relates what is measured to the underlying velocity space distribution function. This mapping function depends strongly on two features of the dust cloud: the number density of the suspended microparticles and the size distribution of microparticles that are suspended. Consequently, it is critical to know what the number density (i.e. particles/volume) of the microparticle component. Additionaly, it is necesary to have knowledge of what is suspended in the particle cloud (i.e. are the suspended particles monodisperse or do they have a size distribution). 91 To addres the isue of the number density of the suspended dust particles, thre techniques have been used in previous experiments. The simplest approach involves directly counting the total number of particles from the acquired images. This technique requires each particle to occupy multiple pixels in the image of the dust cloud and very low number densities, such as what is common in the strongly-coupled systems sen in Figure 2.41. Figure 2.41: An inverted image of a strongly-coupled dusty plasma. (Figure courtesy of H. Thomas, Max Plank Institute for Extreterestrial Physics, Garching, Germany) The second technique involves measuring the optical extinction and then, using Mie Theory, use this value to determine the number density [103-105]. Recently, a third method, which is an extension of the first technique, has been used [80]. In this 92 approach, the amount of light scatered by a known number of particles is determined and used to compute the particle density, even in regions where the particle density is sufficiently high that individual particles can not be resolved. In the systems studied here, the number densities are sufficiently high that the first technique cannot be used. Additionaly, the second technique requires, for the experiments presented in this disertation, measuring extinction ratios are too smal to be reliably measured. In Section 2.3.1, the third technique, which is used to determine the particle density, is discussed. 2.3.1 DENSITY MEASUREMENT BY LIGHT SCATERING TECHNIQUE In a recent study by the Piel group at Kiel University in Germany, a new technique for estimating the number density directly from video images was introduced [80]. In this method, the density of the microparticle component can be determined from the video images using Equation 2.12 ! n dust = I total V cloud " (2.12) where n dust is the number density of the microparticle component, I total is the amount of light scatered by the dust cloud, V cloud is the volume of the dust cloud iluminated by the laser sheet and ? is a scatering eficiency. The amount of light scatered by the dust cloud and the volume of the dust cloud are measured directly from the video images. The 93 scatering eficiency is a measure of the amount of light scatered by a single dust particle and is measured from the video images in regions of known particle density, i.e. in a region where one can manualy count the number of particles such as what is sen in Figure 2.42. Figure 2.42: An inverted image of a dust cloud showing a region of known density, highlighted by the box. The image has ben inverted to beter se the dust particles. There are a few caveats to this technique that bear mentioning. First, the images of the dust cloud analyzed cannot have any saturated pixels. For the data presented here, the PIV cameras used have sufficient depth, 12 bits of resolution or 4096 grayscale levels compared to 256 levels for a typical 8 bit camera, that this was not an isue. Secondly, the suspended microparticles must be esentialy monodisperse. This result is supported 94 by observations described in Section 2.3.2, and by simulation results, to be discussed in Section 3.3.2.3. Finaly, the amount of light scatered per particle, i.e. the scatering efeciency ?, needs to constant. In regions where one can measure the particle density from the video images, this was found to be the case to within 5% and is described in Section 2.3.5.2. 2.3.2 MONODISPERSE VS. POLYDISPERSE DISTRIBUTIONS OF THE MICROPARTICLE COMPONENT If a dusty plasma is composed of microparticles that have a size distribution, there is an outstanding question regarding what subset of the total distribution is suspended in the plasma. Although there have been atempts to measure the suspended distribution in past studies [106], this remains a dificult and unresolved isue. However, it is possible to determine if the suspended microparticles have a size distribution or if the suspended microparticles are monodisperse. There are two methods of determining this. The first method comes from examining the shape of the velocity distribution that is measured. Simulations of the PIV measurement have revealed that there is an anamolous peak that appears at zero in the velocity distribution measured using the PIV technique. This wil be discussed in detail in Section 3.3.2.3. The second method is to examine the images of the suspended dust particles. Based on studies by McKe et al. [107] and Annaratone et al. [108], the suspension of diferent sized particles gives rise to unique structures in the observed dust clouds. In particular, the dust clouds tend to orient 95 themselves into distinct band or regions of diferent particle size. In Figure 2.43, an image of a dust cloud with multiple particle sizes is sen. Here, the suspended cloud contained two distinct particle sizes, spheriglas having a mean size of 10 ?m and 325 mesh (? 45 ?m) silica. While it is unclear what is being suspended from size distribution of the dust used in these experiment, the banding structure indicates that particles having multiple sizes are suspended. Figure 2.43: Video image of dust clouds composed of multiple particle sizes. The original particle distributions included spheriglas particles having a mean size of 10 ?m and 325 mesh (? 45 ?m) silica. The banding structure that is observed is a characteristic feature of dust clouds containing a size distribution. 2.3.3 DETERMINATION OF THE SCATERING EFICIENCY. A key parameter in the method described in Section 2.3.1 to determine the number density of the microparticle component is amount of light scatered per particle, ?. This parameter is measured using a series of images where the number density is sufficiently low that it is possible to count the number of particles in the video 96 images. The total pixel intensity is computed in this region and the ratio of the number of particles to total pixel intensity yields the quantity ?. To do this, a series of 45 - 60 images were analyzed. After a background was subtracted, the image sequence was procesed using the code found in Appendix A.4.5. In this code, each image was opened and the user selected a region of countable particle density. If the number density was sufficiently low that the particles could be uniquely identified, an automatic routine was used. If the particle density was too high for this to be done, i.e. if there was overlap of particles such that the software could not distinguish al of the particles in the defined region, the number of particles was manualy counted. The software then computed the total pixel intensity in the region of interest. From these quantities, the intensity per particle, ?, is computed. In the regions where the user had to manualy count the number of particles, there was a fluctuation level of approximately 5%. Figure 2.44 shows the typical fluctuation level in this parameter. 2000 1500 1000 500 0 Scattering Factor 4530150 Image Number Figure 2.4: Showing the fluctuation level in the scatering eficiency, ?, as a function of image number used in the calibration procedure. The fluctuation level is five percent. 97 CHAPTER 3: SIMULATION OF THE PIV MEASUREMENT 3.1 MOTIVATION Knowledge of the kinetic velocity space distribution function alows one to fully describe the thermodynamic state of a given system. Through the use of stereoscopic PIV techniques, it is possible to measure the thre-dimensional distribution of kinetic velocities for the microparticles component of a dusty plasma. In these experiments, the stereo-PIV system measures the thre-dimensional velocities of the particles iluminated using a one milimeter thick laser sheet. By scanning the stereo-PIV system through the entire cloud, a reconstruction the full thre-dimensional distribution of velocities is determined. This measurement is described in detail in Chapter 4. An important concern that has been demonstrated in studies within the fluid mechanics community is that PIV measurements have an inherent bias toward smaler values of velocity [85, 101, 102]. This arises from the fact that in the PIV measurement technique, each velocity vector represents the average motion of a cluster of tracer particles. In this averaging proces, the motion of the higher speed particles tends to be suppresed. As a result, a distribution of velocities that is measured using PIV techniques is generaly narower than the underlying velocity distribution function, as ilustrated in Figure 3.1. Consequently, the distribution of velocities that is measured using PIV 98 techniques is not the true velocity space distribution function of the system being studied, but rather something that is closely related. Figure 3.1: Showing the efect of the PIV measurement on the measured distribution. In (a), one ses the velocity distribution that is being measured, while (b) shows the distribution that is measured using the PIV technique. It is noted that the measured distribution has a narower width, which is a result of an averaging that is intrinsic to the measurement technique. Physicaly, the result of this averaging would be the measurement of a smaler temperature, i.e. T PIV < T dust . This chapter describes a series of simulations that atempt to relate the distribution of velocities measured using the PIV technique to the underlying velocity space distribution. With such a mapping function, it wil be possible to estimate the temperature of the dust component of a dusty plasma. It wil be shown that this mapping function is influenced by the size distribution and by the particle density of the suspended mircoparticles. The existence of a mapping function and the efect that these experimental parameters have on that mapping function are examined by numericaly simulating the PIV measurement. In Section 3.2, the simulation methodology is described. Section 3.3 presents the results of these simulations. Section 3.4 contains a discussion of these results and how these simulations are connected to real experiments. 99 3.2 COMPUTATIONAL METHODOLOGY When a PIV measurement of a dusty plasma is made, two video images, separated in time by ?t laser , are acquired. Each video image is decomposed into n ? n pixel interogation regions, where n is typicaly betwen 12 and 32. Using sequential pairs of images, the average displacement, d, for al of the particles located in an interogation region is computed. Using these two quantities, a spatialy averaged velocity vector is constructed for each interogation region. Experimentaly, velocities are computed on a grid with a 50% overlap to maximize the acuracy of the reconstructed velocity field. This proces is ilustrated in Figure 3.2. The simulations described in this chapter sek to reproduce this proces. A simulation of the PIV measurement has been developed to compare the distribution that is produced by the PIV measurement to the underlying distribution function. In the simulation code, an analysis grid of 400 ? 400 bins is created. Each bin is asigned a random number of particles up to ten. 100 Figure 3.2: A carton ilustrating the PIV measurement. The dots represent the particles that sed the flow of interest and the red and blue boxes denote interogation regions over which the PIV analysis is performed. Here, the interogation region would be a 2 ? 2 bin region. For the experiments presented in Chapter 4, a bin coresponds to a 6 ? 6 pixel region of the acquired images. The PIV technique returns a vector that describes the average motion of al of the particles in the interogation region. Two examples are sen in red and blue. It is noted that there is a 50% overlap in the interogation region. Each particle in the simulation is asigned thre physical atributes: a radius, a mas density and a thre-dimensional velocity vector. In asigning these values, the following asumptions are made. Al of the particles are asumed to be spherical. Therefore, the selection of a mas density, which represents the material composition of the particles, e.g. silica, alumina or melamine formaldehyde, and a radius simulataneously selects the particle mas. Additionaly, al of particles are asumed to have radii large enough to be detectable by the PIV imaging system. In the experiments performed in this disertation, the lasers used have a wavelength of 532 nm. As such, the simulation particles are constrained to have a radii r ? 0.25 ?m. The asignment of the velocity atribute depends on the type of particle size distribution, monodisperse or polydisperse, being studied. For simulations having monodisperse (uniform size) particles, the mas is computed directly from the particle 101 radius and mas density. For each vector direction, the particles are asumed to follow a 1D Maxwelian velocity distribution that is parameterized by the particle mas, m, and a kinetic temperature, T d , as defined in Equation 3.1. ! fv j () = m 2"k B T d,j exp# mv j #v d,j ( ) 2 2k B T d,j $ % & ? & ( & * & (3.1) where, v j is the velocity of a particle, v d,j is the drift velocity, k B is Boltzmann?s constant and the subscript j refers to the vector direction (j = x, y, z). The velocity of the tracer particle is then extracted at random from the velocity distribution. It is noted that, for most of the simulations presented, the drift velocity, v d,j was set to zero. Further, the velocities are selected for each vector direction (x, y and z) from their respective 1D Maxwelian distribution. In the simulations involving a polydisperse size distribution, there is an additional step because the particles have diferent sizes. First, the particle radius, r d,j was extracted at random from a normal distribution of radii, defined by the mean radius, ? r , and the width, ?r, of the size distribution in Equation 3.2. ! r d,j (? r ,")= 1 2#"r exp$ r$? r () 2 2" % & ? ( * ? + r,0.25 ?m (3.2) The mas, m d,j of the j th particle is then computed from this radius. It is noted that the truncation of particle sizes in Equation 3.2 corresponds to the difraction limit that is 102 observed experimentaly. Under the asumption that al of the particles have the same kinetic temperature in a given vector direction, this mas and the kinetic temperature are used to generate a Maxwelian velocity distribution from which the velocities, v d,j are extracted at random in the same fashion described previously To simulate the PIV measurement, the average velocity of al of the particles in an interogation region of size 2 ? 2 bins is computed. For reference, a bin is represented by a single box in Figure 3.2. This is repeated over the entire simulation grid using a 50% overlap in the evaluation region, which is consistent with what is done in the PIV velocity reconstruction algorithm used in the experimental work presented in this disertation. A single bin corresponds to a 6 ? 6 pixel region of the image used in the PIV analysis. To verify the acuracy of the code, a simulation of an ideal PIV measurement is made. In this simulation, particles having a uniform size and temperature (i.e. T x = T y = T z = T) are uniformly distributed over the simulation grid. The results of this simulation is sen in Figure 3.3,where the input velocity distribution, represented by crosses, and the velocity distribution that is generated by the simulated PIV measurement, represented by the solid curve, are plotted. The simulated PIV measurement clearly shows a narower velocity distribution than the input distribution. This is consistent with the expected suppresion of larger particle velocities and gives confidence in the performance of the code. 103 Figure 3.3: The result for a simulation of an ideal PIV measurement is depicted. The croses depict the distribution of velocities that is input into the PIV simulation, while the solid line depicts the distribution of velocities that is returned by the simulated PIV measurement. The expected narowing of the distribution of velocities is observed. 3.3 RESULTS In Section 3.2, it was demonstrated that the simulation code yields the anticipated results for an ideal PIV measurement. In this section, simulation results are presented covering a wide range of parameters that sek to characterize the relationship betwen the input and output velocity distributions. In these simulations, several important parameters are considered for their potential influence on the output velocity distribution: radius (mas) of the tracer (dust) particle, kinetic temperature of the tracer (dust) particles, number density of the tracer (dust) particles and, in the case of particles having a size distribution, the width of the size distribution. It wil be shown in this section that 104 the number density of the particles and the width of the size distribution play a dominant role in determining the relationship betwen the input and output velocity distributions. For most of the simulation results presented, the data is presented in the format of the ratio of the width of the output (PIV measurement) to the input (velocity space distribution) velocity distribution, ? out /? in , as a function of the efective number density of the tracer (dust) particles, which is computed directly over the simulation grid. It is noted that the value of ? is a function of the particle mas and kinetic temperature. In these results, the mas is known and as a result, the value of ? can be regarded as equivalent to the kinetic temperature. 3.3.1 UNIFORM MAS The first series of simulation results involve particles having a uniform radius/mas. Here, alumina particles with a mas density of ? = 3800 kg/m 3 and a radius r = 0.6 ?m are considered. A series of simulations are performed considering efective number densities ranging 0.01 to 5 particles/bin. For each value of the efective number density, 24 simulations are run in which the kinetic dust temperature that defines the width of the input distribution is varied betwen 1 eV ? T d ? 300 eV. The results of these simulations are shown in Figure 3.4, where the blue bar represents the mean value for ? out /? in for al of the temperature cases and the eror bars in red represent the maximum and minimum values for ? out /? in for each efective particle density. 105 Figure 3.4: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a uniform size. Here, the results for alumina particles (r = 0.6 ?m, ? = 380 kg/m 3 ) are shown. The blue symbol represents the mean value of ? out /? in for 24 simulations over a wide range of kinetic temperatures, while the eror bars represents the extreme values observed for the ratio ? out /? in . It is observed that the temperature of the particles has no significant impact on the width of the output velocity distribution at any given density. However, the number density of particles clearly has a strong influence on the simulation results. It is believed that this is due to the averaging that occurs in the PIV measurement. Since a PIV measurement yields the average motion of al of the particles located in an evaluation region, the particles whose velocities are toward the tails of the distribution function (i.e. generaly those with higher velocities) tend to be suppresed in the averaging proces. As the number density of particles increases, the simulation result suggests that this averaging proces becomes increasingly important and the width of the measured distribution of velocities becomes smaler. 106 By contrast, as the particle density decreases, the ratio of widths approaches unity. Experimentaly, this corresponds to the PIV system functioning as a particle tracking system. In this low-density limit, each individual particle is measured directly and there is minimal averaging. In the limit of the efective particle density going to zero, the distribution of velocities that is measured is the underlying distribution function and one would expect that the ratio of widths would go to one. This result is a further validation of the correct functioning of the simulation code. In Figure 3.5, two separate simulations are shown comparing two diferent particle mases with a fixed kinetic temperature, T dust = 15 eV. Here, alumina particles (r = 0.6 ?m, ? = 3800 kg/m 3 ) are depicted by crosses, while silica particles (r = 1.45 ?m, ? = 2600 kg/m 3 ) are depicted by circles. For these two particle parameters, the mas of the silica particles is larger than the mas of the alumina particles by a factor of approximately ten. This simulation result suggests that the particle mas has no impact on the results of the PIV measurement. This result is not surprising, considering that the kinetic temperature was observed to have no efect on the PIV measurement. Both the mas and the kinetic temperature of the particles wil define the width of the velocity distribution function that is being measured (i.e. the input into the simulations being discussed here). However, this does not influence the measurement proces. As a result, the ratio of the width of the output to the input velocity distribution is unchanged. 107 1.0 0.8 0.6 0.4 0.2 0.0 Ratio Of Widths 5.04.54.03.53.02.52.01.51.00.50.0 Particle Density [particles / bin] Figure 3.5: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a uniform size. Here, the result of a PIV simulation comparing two types of tracer particles is shown. The croses represent alumina particles (r = 0. 6 ?m, ? = 380 kg/m 3 ) and circles represent silica particles (r = 1.45 ?m, ? = 260 kg/m 3 ). In the simulations presented above the drift velocity of the particles was set to zero. When including a drift velocity, there is no observed efect on the PIV measurement. This is sen in Figure 3.6, where the input and output velocity distribution for an efective number density of 3.5 particles per bin are overlaid. As was sen previously, a narowing in width of the distribution returned by the PIV measurement is observed. It is also noted that there is no efect on the value of the drift velocity, v d = 0.2 m/s. 108 1.0 0.8 0.6 0.4 0.2 Counts 0.20020.20010.20000.19990.1998 Velocity [m/s] Figure 3.6: Showing the results of a simulated PIV measurement when the tracer particles have a non-zero drift velocity. Here, the red curve shows the underlying velocity space distribution that is being measured, while the blue curve depicts the distribution of velocities that is measured using the PIV technique. It is noted that the narowing of the measured distribution is observed, but there is no observed efect of the value of the drift velocity In Figure 3.7(a), the ratio of widths is plotted as a function of efective number density for simulations dust particles having diferent drift velocities. Here, the circles represent the simulation where the drift velocity was nominaly 0.2 m/s, while the crosses represent the results when v d = 0 m/s. In Figure 3.7(b), the computed drift velocities for the results sen are plotted as a function of the efective number density. These results again show that a net drift velocity has no impact on the resulting PIV distributions. 109 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 Drift Velocity 543210 Particle Density [particles/bin] v drift = 0 mm/s v drift = 20 mm/s 1.0 0.8 0.6 0.4 0.2 0.0 Ratio of Width 543210 Particle Density [particles/bin] v drift = 0 mm/s v drift = 20 mm/s (a) (b) Figure 3.7: Showing the efect that the drift velocity has on the PIV measurement. In (a), the ratio of widths is ploted as a function of the efective particle density. Here the croses denote the result when there is no drift velocity and the circles represent the result when there is a non-zero drift velocity. The drift velocities used in the simulation are sen in (b). 110 3.3.2 MAS DISTRIBUTION In most of the early experiments, the dust particles used were polydisperse; i.e. there was a distribution of particle sizes. Therefore it is necesary to determine the efect of a size (mas) distribution has on the velocity distributions that are measured using the PIV technique. This section presents the results from simulations where the particles measured using the PIV technique had a size distribution. Because of the size distribution, care must be taken in the simulation proces and in the analysis of the data. In particular, there are thre isues that need to be considered: the asignment of particle velocities, the determination of the width of the velocity distribution and the way in which the PIV calculation is performed. The proces by which the velocities are asigned is complex, when compared to the work that was discused in Section 3.3.1. A particle size is extracted at random from a distribution of particle sizes defined by the mean radius, ? r , and the width of the size distribution ?r, defined in Equation 3.2. This size (mas) is then used to generate a velocity distribution for a defined kinetic temperature of the dust, under the asumption that al of the particles have the same temperature. Velocities are then extracted at random from this distribution. This proces is repeated to form the velocity distribution that the simulation wil measure. The resulting velocity distribution wil have an efective width that is defined by an input temperature and a mas distribution defined by the size distribution, ? r ? ?r. As sen in Figure 3.8, the velocity distributions that are 111 generated when there is a size distribution appears Maxwelian with a larger width than those that are generated for a monodisperse particle size for the same kinetic temperature. 1.0 0.8 0.6 0.4 0.2 0.0 Normalized Counts -0.050 -0.025 0.000 0.025 0.050 velocity [m/s] r = 1.45 ?m r = 1.45 ? 0.5 ?m Figure 3.8: A plot showing the overlay of the velocity distribution for monodisperse silica particles, r = 1.45 ?m, in red and polydisperse silica particles, r = 1.45 ? 0.5 ?m, in blue for the same kinetic temperature, T dust = 15 eV. It is noted that the velocity distribution for polydisperse particles has a larger width. It is no longer possible to simply fit the input (i.e. the underlying velocity space distribution that is being ?measured? in the simulation) and output velocity distribution (i.e. what is ?measured? in the simulation) to a Maxwelian. The reason for this lies in the multiple possible value of the mas. When there is a distribution of particle sizes, there is not a single mas to use in the fit and as a result a range of mases can be used in the fit. This leads to a distribution of possible temperatures. As a result, an alternative method is needed to compute an efective width of the velocity distributions. In this method, the velocity distribution is fit to a Maxwelian asuming that al of the particles have a known radius, r i . The resulting width is then 112 weighted to the probability, ? i , that a particle has that given size. The proces is depicted in Figure 3.9. Figure 3.9: Showing how the weighted average was used to find the efective width of the velocity distribution when the particles had a size distribution in the PIV simulations. Here, the distribution of particle radii were used in the simulations are fit to a normal distribution. From this fit, the probability, ? i , that a particle has a particular radius, r i. , is extracted as sen in Figure 3.9. Starting at the minimum detectable particle size, r = 0.25 ?m, the distribution of velocities that is generated in the simulated PIV measurement is fit to a Maxwelian, under the asumption that al of the particles have this radius. This proces is then repeated by moving the asumed radius through the size distribution N times using equaly spaced increments for the particle size. At each size, r i , a width, ? i , is extracted from the fit of the velocity distribution and probability that the 113 particle has that radius, ? i , is extracted from the fit of the size distribution. The efective width of the velocity distribution, ? efective , is computed using Equation 3.3. ! " effective = # i i $ " i # i i (3.3) where the sum runs from 1 (i.e. a particle of having radius r = 0.25 ?m) to N (i.e. a particle of having radius r = ? r + 9.5?, where ? is the width of the size distribution). In is noted that in the limit of monodisperse dust particles (i.e. of ?r ? 0), the efective width defined in Equation 3.3 reduces the width that was used in Section 3.3.1. The value of N was selected such that it was large enough that the efective width that was computed no longer fluctuated, as sen in Figure 3.10; typicaly, N ? 100. The same proces is used to find an efective width of the velocity distribution that is input into the PIV simulation. Finaly, the proces by which the PIV measurement is simulated needs to be addresed. In the previous section involving monodisperse particles, the PIV measurement is simulated by computing the average velocity of al of the particles in the interogation region using Equation 3.4. ! v j = 1 N j v i i " (3.4) where N j is the number of particles in the j th interogation region and v i is the velocity of the i th particle, which is located in the j th interogation region. 114 (a) (b) (c) (d) 20 15 10 5 0 T dust [eV] 140120100806040200 Number of Iterations T actual T dust 0.14196 0.14195 0.14194 0.14193 0.14192 0.14191 0.14190 Ratio of widths 140120100806040200 Number of Iterations Ratio of Width, ! ! 3.38 particles/bin 16 12 8 4 0 T dust [eV] 140120100806040200 Number of Iterations T actual T dust 0.96810 0.96800 0.96790 0.96780 0.96770 Ratio of widths 140120100806040200 Number of Iterations Ratio of Width, ! ! 0.05 particles/bin Figure 3.10: Showing the number of iterations neded to compute a meaningful value for the weighted average. When the particles are monodisperse, this provides an acurate model of the measurement, as al of the particles wil contribute equaly in the cross correlation analysis that generates the PIV velocities. However, when the particles are polydisperse, al of the particles wil not contribute equaly in the cross correlation analysis. Asuming that the particles have a diameter larger than the wavelength of light used, it is known from Rayleigh Scatering that the amount of light that scaters off of the a particles scales with the square of the particle size. As a result, the contribution to the cross correlation wil go like the size of the particle to the fourth power. Consequently, larger particles wil contribute more in the calculation of the PIV vectors. Experimentaly, this can be acounted for by using multi-intensity layer PIV [109]. In this approach, one thresholds 115 the image and performs the PIV calculation using a smal range of pixel intensities. This analysis is repeated over the range of pixel intensities that occur in the image. Once this has been done a number of times spanning the intensity in the acquired images, one combines the results of each evaluation into a single vector field. Experimentaly, this technique is not used in the work presented in this thesis and it is necesary to understand what efect this would have on the measured distribution. Nevertheles, there is insight that can be gained into the PIV technique by asuming that polydisperse particles contribute equaly in the PIV calculation. As a result, the results of simulations involving polydisperse particles using Equation 3.4 for the PIV calculation wil be presented in Sections 3.3.2.1 and 3.3.2.2. In Section 3.3.2.3, simulations that acount for the weighting due to particle size wil be presented. 3.3.2.1 FIXED WIDTH OF PARTICLE SIZE DISTRIBUTION The next series of simulation results focus on simulations involving tracer particles having a size (mas) distribution defined by r = ? r ? ?r. In Figure 3.11, the results for simulations involving particles having a distribution given by r = 0.6 ? 0.25 ?m and a mas density of ? = 3800 kg/m 3 (alumina) are shown. 116 Figure 3.1: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a size distribution. Here, the results for alumina particles (r = 0.6 ? 0.25 ?m, ? = 380 kg/m 3 ) are shown. The blue symbol represents the mean value of ? out /? in for 24 simulations over a wide range of kinetic temperatures, while the eror bars represent the extreme values observed for the ratio ? out /? in . As before, a series of simulations are performed over a wide range of particle densities and temperatures. In this case, 24 simulations were run in which the temperature of the input distribution is varied betwen 1 eV < T < 50 eV. The blue symbol represents the mean value for ? out /? in and the eror bars in red represent the maximum and minimum values for ? out /? in for each efective particle density. It is again observed that the kinetic temperature of the particles has no significant impact on the width of the output velocity distribution at any given density. Additionaly, the density of particles continues to have a strong influence on the simulation results. 117 1.0 0.8 0.6 0.4 0.2 0.0 Ratio of Widths 5.04.54.03.53.02.52.01.51.00.50.0 Particle Density [particles / bin] Figure 3.12: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of the efective particle density for tracer particles having a size distribution. Here, the results from a simulation involving alumina particles having a size distribution (r = 0.6 ? 0.25 ?m, ? = 380 kg/m 3 ), depicted by croses, are compared with the results from a simulation involving monodisperse alumina particles (r = 0.6 ?m, ? =380 kg/m 3 ), depicted by circles. Figure 3.12 compares the results of simulations for particles having a mas distribution (r = 0.6 ? 0.25 ?m, ? = 3800 kg/m 3 ), depicted by crosses, and a uniform mas (r = 0.6 ?m, ? = 3800 kg/m 3 ), depicted by circles, at a fixed dust temperature of T dust = 15 eV. While it is observed that the efective particle density has a significant impact on the width of the output velocity distribution at any given density, the efect is notably stronger when monodisperse particles are used. These simulation results suggest that the mas distribution broadens the width of the velocity distribution that is measured using the PIV technique, which is consistent with what was observed in Figure 3.8. 118 3.3.2.2 EFECT OF WIDTH OF PARTICLE SIZE DISTRIBUTION The next set of simulations seks to characterize the influence of the width of the size distribution on the width of the output velocity distribution. That is, if the particle size distribution is specified by r = ? r ? ?r, does the size of the quantity ?r/? r influence the width of the output velocity distribution? To examine this, thre cases are considered for a fixed temperature: ?r/? r = 0.25, 0.5 and 0.75 for a range of mean particle sizes ranging from 1.5 ?m to 50 ?m. For reference, the simulation results presented in Figure 3.11 had ?r/? r ? 0.42. The results for these simulations are sen in Figure 3.13, where (a) shows the results for ?r/? r = 0.25, (b) shows the results for ?r/? r = 0.5 and (c) shows the results for ?r/? r = 0.75. In al thre plots, the crosses represent the simulations where ? r = 1.5 ?m, the triangles represent the simulations where ? r = 3.0 ?m and the dots represent the simulations where ? r = 50 ?m. There are thre observations of note here. First, as has been the case in al other simulations, the density of particles has a strong influence on the width of the velocity distribution that is measured using the PIV technique. Second, as the ratio, ?r/? r , increases, the efect that the PIV measurement has on the width of the 119 1.0 0.8 0.6 0.4 0.2 0.0 Ratio of Widths 5.04.54.03.53.02.52.01.51.00.50.0 Particle Density [particles / bin] ? = 1.5 x 10 -6 m ? = 3.0 x 10 -6 m ? = 50 x 10 -6 m (a) (b) (c) 1.0 0.8 0.6 0.4 0.2 0.0 Ratio of Widths 5.04.54.03.53.02.52.01.51.00.50.0 Particle Density [particles / bin] ? = 1.5 x 10 -6 m ? = 3.0 x 10 -6 m ? = 50 x 10 -6 m 1.0 0.8 0.6 0.4 0.2 0.0 Ratio of Widths 5.04.54.03.53.02.52.01.51.00.50.0 Particle Density [particles / bin] ? = 1.5 x 10 -6 m ? = 3.0 x 10 -6 m ? = 50 x 10 -6 m Figure 3.13: Plot of the ratio of the width of the output to the input velocity distribution, ? out /? in , as a function of particle density for tracer particles having a size distribution with a varying width, ?r, relative to the mean radius, ? r ,. In al thre plots, the croses depict a size distribution with ? r , = 1.5 ?m, the triangles depict a size distribution with ? r , = 3.0 ?m and the dots depict a size distribution having ? r , = 50 ?m. In (a) the width of the size distribution relative to the mean value, ?r/? r , of 0.25, in (b) ?r/? r = 0.5 and is (c) ?r/? r = 0.75. 120 output velocity distribution becomes les severe. This is consistent with previous observations that the size distribution broadens the output velocity distribution. Further, it suggests that this broadening of the output velocity distribution increases as the width of the size distribution (?r/? r ) increases. Finaly, there is an interesting result that is particularly clear in Figure 3.13. For a fixed particle density, the mean size of the particles, ? r , increases, the efect that the PIV measurement has on the width of the output velocity distribution appears to approach a constant value. In an atempt to characterize this result, the ratio of the output to input velocity distribution widths, ? out /? in , is plotted as a function of the mean particle size, ? r , for thre diferent efective particle densities with ?r/? r = 0.75 in Figure 3.14. 0.60 0.55 0.50 0.45 0.40 0.35 Ratio of Widths 5040302010 Mean Radius [microns] 0.909 0.900 0.891 0.882 0.874 0.865 % Included Figure 3.14: Plot of the ratio of the width of the output to the input velocity distribution as a function of mean particle size for simulations with ?r/? r = 0.75. Here, croses represent an efective particle density, n d,effective , of aproximately 4.9, circles represent n d,effective ? 3 and triangles represent n d,effective ? 2. The dashed curve depicts the portion of the distribution included due to the truncation of the size distribution (e.g. r ? 0.25 ?m). 121 Here, the crosses represent an efective particle density, n d,efective , of approximately 4.9 particles per bin, the circles represent an efective particle density of approximately 3 particles per bin and the triangles represent an efective particle density of approximately 2 particles per bin. It is observed that as the mean particle size increases, the ratio of widths has an asymptotic behavior. This is a common for each of the particle densities, suggesting that it is intrinsic to the PIV measurement and not an efect of the particle density. A similar result is observed when ?r/? r = 0.5. However, in this case, the trend much les pronounced. This trend was not sen in any of the other the previously discussed simulations. Consequently, it appears that if the quantity ?r/? r is roughly 0.5 or larger, this efect is observed. This efect is a result of the fraction of the size distribution that is included in the PIV simulation. As previously noted, the size distribution of the tracer particles is truncated such that particles having a radius les than 0.25 ?m are not included in the simulation. This truncation in the size distribution has an indirect efect on the values of velocity that can be included in the simulation. The velocities are selected from a Maxwelian velocity distribution whose width is proportional to the temperature of the particles and inversely proportional to the mas of the tracer particles. As ?r/? r increases, the fraction of the size distribution that is lost by this truncation increases as the mean value, ? r , decreases. This is sen in the dashed curve in Figure 3.14, where the percent of 122 the distribution that is included for ?r/? r = 0.75 is plotted as a function of the mean particle size. By not including the smaler sized particles at a fixed temperature, the larger values of velocity wil not be included in the simulation. Consequently, the velocity distribution that the PIV measurement acts on wil be narower and the averaging that takes place in the PIV measurement wil not be as severe. As a result, the narowing efect that the PIV measurement has on the width of the output velocity distribution, for fixed values of ?r/? r , would be smaler as the mean size of the particle distribution decreases. In the majority of the simulations that have been presented, the width of the size distribution has been relatively smal compared to the mean value and this truncation of the size distribution has had litle efect. However, as this truncation becomes more severe, the fraction of the particle size distribution that is included in the simulation impacts the shape of the mapping function that relates the underlying velocity distribution function to what is measured by the PIV technique. 3.3.2.3 WEIGHTED PIV CALCULATION In the previous two sections, an underlying asumption in the simulations presented was that the particles al contribute equaly in the PIV measurement. However, under the asumption of Rayleigh scatering, the particles wil scater light that is proportional to r d 2 . This wil lead to an r d 4 weighting in the cross correlation analysis that 123 generates the PIV velocities. To acount for this, the computation of the average velocity of the particles in an interogation region was modified to Equation 3.5. ! v j = 1 N j r i 4 v i ? r 2 i " (3.5) When one acounts for the weighting that occurs due to the amount of light that is scatered by the microparticles (i.e. the microparticles size), one observes a notably diferent looking distribution of velocities, as sen in Figure 3.15. 1.0 0.8 0.6 0.4 0.2 0.0 Normalized Counts -0.10 -0.05 0.00 0.05 0.10 Velocity [m/s] Velocity Distibution Unweighted PIV distribution Weighted PIV distribution Figure 3.15: Simulation results showing the efect on the measured distribution when on includes the weighting due to particle size in the PIV technique. The solid black curve depicts the underlying velocity distribution, the doted red curve depicts what is measured by the PIV technique when al particle contribute equaly and the dashed blue curve depicts what is measured by the PIV technique when the particle size is a factor in the cros corelation analysis. The strong peak at zero velocity that ocurs when including the weighting due to particle size is not observed experimentaly. 124 Here, the underlying velocity distribution is depicted by the solid black curve, what is measured asuming a uniform contribution of particles sizes (i.e. Equation 3.4) is depicted by the red doted curve and what is measured asuming a that the particle size is a factor in the cross correlation analysis is depicted by the dashed blue curve (i.e. Equation 3.5). While it is observed that the expected narowing in the measured distribution occurs, the narowing is much more severe. In particular, there is a very strong peak at zero velocity. This peak was characteristic in al of the simulations performed using Equation 3.5. Further, it is noted that the distributions are no longer Maxwelian and, in this case, there is no longer a mapping function that relates the measured distribution to the underlying velocity space distribution function. It is noted, however, that this strong peak at zero velocity was not observed experimentaly. As a result, detailed simulations of this phenomenon were not performed. It is noted however, that this result is significant in that it suggest many of the dust clouds studied in this work, even for cases where the original powder was polydisperse, led to particle suspensions that were fairly monodisperse. If this were not the case, the measured distributions would have a strong peak at zero velocity. Additionaly, the light intensity recorded by the PIV camera remains generaly constant over the particles, supporting the asertion that the suspended particles may be nearly monodisperse. 125 3.4 DISCUSION OF SIMULATION RESULTS Simulations of the PIV measurement have shown that there is a unique mapping function that relates the widths of the input and output velocity distribution provided that the PIV technique is applied to dust particles that are esentialy monodisperse. In order generate a mapping function that wil connect the distribution of velocities that is measured using the PIV technique to the underlying velocity space distribution, what remains is to relate the efective particle density that is used in the simulations and the particle density that appears in the experiment, i.e. it is necesary to identify a relationship betwen the units of particle density used in the simulation (particles/bin) and what can be measured in an experiment (e.g. particles/cm 3 ). In the simulations, a 2 ? 2 bin region corresponds to the volume of the evaluation region in a PIV experiment. The volume of an evaluation region is given by ? 2 n 2 t, where n is half the side of the interogation region, t is the thicknes of the laser sheet and ? is a conversion factor from pixels to meters. The value of ? wil depend on the calibration of the camera system used in making the PIV measurement and n wil depend on the size of the interogation region used in the PIV analysis (i.e. for a 12 ? 12 pixel interogation region, n = 6). For one such calibration on the 3DPX device, n = 6 pixels, t = 1 m and ? ? 43 ?m/pixels. Using these values, the calibration connecting the efective particle density used in the simulations and the particle density that would be measured in an experiment is sen in Figure 3.16. 126 Typicaly, the density of the tracer particles is not uniform over the whole field that is measured using a PIV system. This is certainly the case in studies of the weakly- coupled dusty plasmas that occur in a dc discharge plasma, as can be sen by inspection of the inverted image of a dust cloud in Figure 3.17. It is noted that simulations where the particle density was uniform over the simulation grid yielded the same results for a wide variety of functional forms for the density, provided that the efective particle density was the same. Consequently, the particle density of interest would be the particle density over the volume that the PIV measurement is made. Figure 3.16: A curve relating the efective particle density that is used in the simulations to particle densities that would be measured in an experiment. It is important to note that the slope of this curve wil depend on the particular calibration of the PIV system. 127 Figure 3.17: An inverted image of a typical dust cloud. Superimposed on this image is a grid that represents the interogation regions that are used in the PIV calculation. It is noted that the particle density is not uniform of the interogation region, which coresponds to a 2 ? 2 box bin (12 ? 12 pixels). 128 CHAPTER 4: EXPERIMENTAL RESULTS In this chapter, the first extensive measurements of the thermal properties of a weakly-coupled dusty plasma in a dc glow discharge are presented. In particular, the velocity space distribution function of the microparticle component is measured and from this, a kinetic temperature is extracted. The experimental methodology is described in Section 4.1. Results for microparticles having a polydisperse distribution are presented in Section 4.2 and results obtained using monodisperse microparticles are presented in Section 4.3. Section 4.4 contains a discussion of these results. 4.1 EXPERIMENTAL METHODOLOGY The primary diagnostic technique used in the experiments presented in this chapter is stereoscopic particle image velocimetry (stereo-PIV). In these experiments, a thin slice (~1 m) of the dusty plasma is iluminated by a pair of laser pulses, shaped into ?two-dimensional? shets using a cylindrical lens. The time betwen the laser pulses for the experiments described in this chapter, unles otherwise noted, is held fixed at 250 ?s to facilitate measurement of the kinetic motion of the microparticles. The electrodes used to generate the plasma are sen in Figure 4. 1. 129 There were two electrode configurations that were used in the experiments in this chapter: the cathode-anode-tray and the anode-cathode-tray configuration. The bulk of the measurements were made using the cathode-anode-tray configuration. In this configuration, the upper electrode was biased negative relative to the grounded chamber wal (cathode) and the lower electrode was biased positive with respect to the chamber wal (anode). In the anode-cathode-tray configuration, the electrode biases are switched. In both configurations, the dust tray was located below both of the electrodes and was electricaly floating. (a) (b) Figure 4.1: A photograph of (a) the electrodes used in the experiments described in this disertation and (b) the location of the dust cloud when the electrodes are aranged in the cathode-anode-tray configuration. In (b), the lower electrode (anode) is visible. A dust cloud composed of 1.51. ?m diameter silica microspheres is iluminated by a laser diode (? = 632 nm) and is visible in red. The purplish background is an argon plasma. As noted in Section 1.3, previous studies of the thermal properties of dusty plasmas were restricted to strongly-coupled systems, mostly in rf discharge systems. In 130 those experiments, the microparticles were organized into ?rigid? uniform structures with a fixed inter-particle spacing that is comparable to the electron Debye length, ~100 ? 300 ?m. Consequently, for most strongly-coupled experiments, tracking of individual particles is sufficient to determine the particle velocities. By contrast, the dust clouds in these dc glow discharge experiments, as sen in Figure 4.2, contain no obvious spatial ordering. Further, the interparticle spacing can vary across the cloud from tens of microns (~? Di ) to a few hundred microns (~? De ). (a) (b) (c) Figure 4.2: A typical video image of a weakly-coupled dusty plasma for (a) 1.2 ? 0.5 ?m diameter alumina microparticles, (b) 6.2 ?m diameter melamine microspheres and (c) 3.02 ?m diameter silica microspheres. 131 Using the stereo-PIV system, measurements of the thre-dimensional velocity vectors of the microparticles are made over the entire volume of the suspended particle cloud. This is acomplished by stepping the laser sheet through the entire cloud volume in 1 m increments in the z-direction, the direction perpendicular to the plane of the laser sheet. At each z-location, a sequence of n, typicaly 60, PIV measurements are made. Since the efective PIV measurement rate is 5 Hz, the recording of these 60 images requires ~12 seconds. However, after moving the laser/camera asembly to a new z- location, the camera oscilates for a short period of time, ~0.8 to 1.0 seconds as sen in Figure 4.3. -12.0 -11.8 -11.6 -11.4 Horizontal position [mm] 1.81.51.20.90.60.30.0 Time [s] 10.8 10.6 10.4 10.2 10.0 Vertical Pposition [mm] (a) (b) Figure 4.3: A plot depicting the oscilation of the PIV camera when the linear translation stage comes to rest. Here, the (a) vertical and (b) horizontal motion of a fixed location inside of the 3DPX device is tracked over the first ten images in a measurement sequence. 132 To acount for the oscilation of the cameras that is observed, the first fiften images (~3 seconds) of each measurement sequence are excluded from the analysis. Al of the subsequent velocities that were measured over the entire cloud volume are combined to obtain an experimentaly measured distribution of velocities for each velocity component (v x , v y and v z ) for the entire cloud. In the experiments presented in this chapter, a dc glow argon discharge plasma is generated in the 3DPX device [78] using a pair of biased electrodes. For these experiments, the cathode-anode-tray configuration was used. Once the plasma has ben established, dust particles are introduced and a dust cloud forms in the anode sheath below the lower electrode (anode), as sen in Figure 4.1(b). The experiments described in the following sections were performed over a range of presures. It is noted that while these experiments were performed over a wide range of experimental conditions, care was taken to ensure that there was no visible global or collective transport of the microparticles in the dust cloud. Specificaly, there was no observed change in the phase of the dust cloud, as was observed in the aforementioned rf experiments. Further, as the presure was varied, the anode and cathode voltages were adjusted to maintain stable dust clouds with no observable collective efects (i.e. waves, oscilations or other large scale transport structures). Once a stable dust cloud has been formed, a distribution of velocities is measured and the presure is varied. Once a new set of experimental conditions has ben established, the system is not disturbed for a period of time (~3 ? 5 minutes) before taking the next measurement. The dust clouds that were studied in this experiment are a closed system in the sense that there are no dust particles entering or leaving the dust cloud over the duration 133 of the experiment. At the same time, these systems are open in the sense that energy flows frely betwen the background plasma and the suspended dust cloud. As a result, the dust clouds studied here represent a non-equilibrium thermodynamic system where the dust clouds are in local equilibrium with the surrounding plasma. Consequently, one can use some ideas from equilibrium thermodynamics to study these systems. To interpret the experimentaly measured velocity distributions, there is one asumption that is used in the analysis for both the polydisperse and monodisperse microparticles: it is asumed that the microparticles have reached a local equilibrium with the background plasma. As a result of this asumption, it is asumed that the velocity distribution for each vector directions can be modeled as a drifting 1D Maxwelian, as given in Equation 4.1. ! f= m 2"k B T d exp# mv#v d () 2 2k B T d $ % & ? ( & * (4.1) where v is the velocity, m is the mas of the dust particle, k B is Boltzmann?s constant, T d is the dust temperature and v d is the drift velocity. There are thre points that bear mentioning. First, the temperatures that are measured here represent the bulk temperature of the entire cloud. Second, it is noted that a drifting Maxwelian was used for completenes and there was no observed drift in the data that is presented in this chapter. Finaly, from this fit it is possible to extract the underlying distribution function after acounting for the eror in the PIV system and the averaging that is intrinsic to the PIV measurement. 134 Once the velocity distribution has been measured, the first step is to acount for the uncertainty in the stereo-PIV system. For clarity, the uncertainty here refers to the resolution limit of the stereo-PIV system that was discussed in Section 2.2.4.3, i.e. the eror that is due to hardware (e.g. lens eror) and software (e.g. limits of the PIV algorithm) sources. As sen in Section 2.2.4.3, the resolution eror in the stereo-PIV system is Maxwelian. Since we are measuring the thermal properties of a dusty plasma by measuring the distribution of particle velocities, which are also Maxwelian in nature, the eror sen in Figure 2.39 can be directly acounted for using Equation 4.2. ! " true 2 =" measured 2 #" resolution 2 (4.2) where ! " measured is the dispersion that is measured, ! " resolution is the dispersion that comes from the eror measurement discussed in Section 2.2.4.3 and ! " true is the actual dispersion. To acount for the averaging that is intrinsic to the PIV technique, as described in Chapter 3, the density of the microparticle component is determined using the technique described in Section 2.3. With this value, the mapping function developed in Section 3.3.1 is used to extract the underlying velocity space distribution function, from which the thermal properties can be extracted. This analysis proces is sen in Figure 4.4. 135 (a) (b) (c) (d) 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] -0.4 -0.2 0.0 0.2 0.4 Velocity [m/s] -0.4 -0.2 0.0 0.2 0.4 Velocity [m/s] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] Figure 4.4: Showing the proces by which the underlying velocity space distribution function is obtained. In (a), the measured velocity distribution, sen in red, and the resolution eror in the stereo-PIV system, shown in blue, are overlaid. In (b), the measured velocity distribution, sen in red, and the resolution eror in the stereo-PIV system, shown in blue, are aded to yield an intermediate distribution, shown in gren. In (c), the maping function corects the width of the intermediate function, shown in gren, to acount for the intrinsic averaging, which yields the velocity space distribution, sen in black. The velocity space distribution function shown in (d), again in black, can then be used to study the thermal properties of the microparticle component of the dust cloud. At this point, it is important to emphasize that information is not lost due to the time that is required to complete the measurement of a distribution of velocities. When a single measurement is made, the duration of this measurement is 250 ?s. At each z- 136 location, 60 measurements are made with a frequency of 5 Hz. For a cloud that is 9 m thick, the distribution function for the stable dust cloud that is fit to Equation 4.1 includes approximately 405 measurements taken over a period of roughly thre minutes. Given this extended time, it is important to verify that information is not lost due to the sampling rate. To verify that this is not the case, a series of 45 stereo-PIV measurements were made at a single z-location. In this experiment, a dust cloud composed of monodisperse (i.e. uniform) melamine microspheres having a radius of 3.11 ?m was generated in an argon dc glow discharge with p = 160 mTorr, V anode = 208 V and V cathode = -73 V. The data was then analyzed in the following fashion: the vectors from n measurements were used to generate a distribution of velocities. This was done 60 - n times; e.g. for n = 3, the data from easurements 1 - 3, 2 - 4, 3 - 5, . . ., 58 - 60 would be analyzed and a set of temperatures (T 1-3 , T 2-4 , T 3-5 , . . ., T 58-60 ) was extracted. Using this technique, we analyzed this series of data using a variable number of measurements. The results from this analysis is sen in Figure 4.5. It is noted that the temperatures that are reported here come directly from the fit of the measured distribution. The mapping functions that were discussed in Chapter 3 were not used. However, over the duration of the measurement, the appearance and structure (e.g. the size and density) of the dust cloud did not change. Consequently, the PIV correction factor would be the same for al of the data and the net result would be a constant vertical shift in the data shown. Consequently, the esential physics is preserved. 137 500 400 300 200 100 0 T x [eV] 4530150 Number of Measurements Included 500 400 300 200 100 0 T y [eV] 4530150 Number of Measurements Included 2000 1500 1000 500 0 T z [eV] 4530150 Number of Measurements Included Figure 4.5: Showing the kinetic dust temperature of 6.2 ?m diameter melamine microspheres as a function of the number of PIV measurements included in the distribution function. The symbol represents the mean temperature from the 45 - n measured temperatures, while the eror bars depict the range of temperatures that are measured. The values for n used in this calculation are n = 1, 2, 5, 10, 25 and 45. It is observed that the average temperature is relatively constant. However, the level of fluctuation decreases as more measurements are included in the velocity distribution. 138 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] -0.2 -0.1 0.0 0.1 0.2 velocity [m/s] -0.2 -0.1 0.0 0.1 0.2 velocity [m/s] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] v x v y v z n = 1 n = 2 n = 5 n = 10 n = 25 n = 45 Figure 4.6: Showing the measured velocity distributions for diferent numbers of included measurements. The red curve depicts the velocity in the x-direction, the gren curve depicts the velocity in the y-direction and the blue curve depicts the velocity in the z-direction. It is observed that there is les fluctuation in the shape of the velocity distribution as the number of measurements included increases. 139 It is observed that as the number of measurements included in the distribution increased, the level of fluctuation in the measured temperature decreased. Consequently, the use of a large number of images results in beter statistics, but not a loss of physical information. This is supported in the observed distribution of velocities, as sen in Figure 4.6. It is noted that the same experimental test was run using the 1.2 ? 0.5 ?m diameter alumina microparticles and the same results were observed. 4.2 RESULTS WITH MICROPARTICLES HAVING A POLYDISPERSE SIZE DISTRIBUTION In the experiments described in this section, dust clouds composed of 1.2 ? 0.5 ?m diameter alumina microparticles were suspended in an argon plasma over a range of neutral gas presures from 100 to 260 mTorr. The size distribution for the dust particles and an SEM image of the dust particles used can be sen in Figure 4.7. 40 30 20 10 0 Wt% 543210 Particle Size [?m] (a) (b) Figure 4.7: Showing (a) the size distribution and (b) an SEM image of the microparticles used in the experiments presented in Section 4.2. In (a) the red circles depict measured values for the particle size, while the blue curve is a Gausian fit. (SEM image courtesy of Andrew Post-Zwicker, Princeton Plasma Physics Laboratory) 140 In order to analyze the measured velocity distributions, there are two additional asumptions: the subset of microparticles that are actualy suspended in the plasma are both spherical and monodisperse. Based on what is sen in Figure 4.7, the microparticles are clearly not spherical. However, this asumption is only used to compute a mas using the mean particle size of 0.6 ?m (m d = 3.44 ? 10 -15 kg). If the second asumption, that the suspended microparticles are monodisperse, is valid, then the mas that is computed using the asumption of spherical particles may be incorrect. However, the key results wil be preserved, as the eror wil result in a shift in the numerical value of the temperatures measured. Consequently, the second asumption, that the suspended particles are monodisperse, is important but its validity is not imediately obvious. One of the chalenges of this experiment is knowing precisely which particles from the source distribution of particle sizes becomes suspended in the plasma. This uncertainty wil clearly have an impact on the evaluation of the measured velocity distributions. Nonetheles, there is experimental evidence that the size distribution of the suspended particles is significantly more narow, possibly even monodisperse, than the source powder. Evidence for this is thre-fold. First, the intensity of the scatered light imaged by the CD cameras does not vary in any significant way across the image of the dust cloud. If the suspended particles had a size distribution, the intensity would fluctuate in a fashion that is not consistent with observations. Further, the fluctuation in the intensity level that was observed for dust clouds whose source powder had a polydisperse size distribution was comparable to what was observed for dust clouds whose source powder were monodisperse microspheres used in the experiments presented in Section 4.3. Second, there are several observations that when particles of 141 diferent sizes are suspended in a dusty plasma, the particle become segregated acording to size, often with sharp boundaries betwen the diferent particle sizes [107, 108]. None of these characteristic structures are observed in the dust clouds used in the experiments presented in this section. Finaly, the measured velocity distributions do not exhibit a strong peak at zero velocity, which is suggested by the simulation results discussed in Section 3.3.2.3. As a result, there is sufficient evidence to proced with the asumption that an efectively monodisperse population of microparticles are suspended in the plasma. Provided that al of the aforementioned conditions are satisfied, it is possible to use the velocity distributions to extract a bulk kinetic temperature for the entire cloud. It is noted that the measured distribution, as sen in Figure 4.8, is in reasonably good agrement with the asumption of a Maxwelian distribution of microparticle velocities, which supports the underlying asumptions that the microparticles have achieved a local thermodynamic equilibrium. It is noted that the widths of the velocity distributions (e.g. temperature) sen in Figure 4.8 are highly anisotropic. As sen in Measurement 2 in Table 1.1 and Measurement 4 in Table 1.3, there was an observed anisotropy in the temperatures measured in the previous experiments. However, unlike many of the rf dusty plasma experiments in which the dust particles remain confined in highly wel-ordered 2D structures, the particle clouds in this experiment often have extended 3D structures. Furthermore, as shown in Figure 4.1, the orientation of the electrodes and trays do not alow any of the vector directions to be symmetric to another. In the y-direction, gravity 142 1700 1275 850 425 0 Counts -0.10 -0.05 0.00 0.05 0.10 v x [m/s] 2000 1500 1000 500 0 Counts -0.10 -0.05 0.00 0.05 0.10 v y [m/s] 1700 1275 850 425 0 Counts -0.2 -0.1 0.0 0.1 0.2 v z [m/s] Measured Distribution Maxwellian Fit (a) (b) (c) Figure 4.8: Typical velocity distributions in the (a) x-, (b) y- and (c) z-direction. The croses depict the measured distribution of velocities, while the solid curve represents the fit to a drifting Maxwelian velocity distribution. The experimental parameters for this data are p = 120 mTor, V anode = 231 V and V cathode = -91 V. 143 breaks the symmetry with the x- and z-directions. The symmetry in the x- and z- direction is broken by the electrodes being fed into the chamber in the z-direction and that the electrodes are not located directly above each other. Therefore, some degre of asymmetry may be expected among the distributions of the thre velocity components. A full discussion of the interpretation of the temperature asymmetry in the experiment wil be given in Section 4.4. With the confirmation that the stereo-PIV system can measure a velocity distribution, it is now possible to evaluate the thermal properties of the dusty plasma over a variety of plasma conditions. An initial experiment tests the reproducibility of the temperature measurement for a stable cloud in an argon dc glow discharge plasma. Here, six measurements are made of a single dust cloud over a period of an hour. During these measurement, the experimental parameters were held constant with p = 110 mTorr, V anode = 230 V, V cathode = -89 V and there was no observable change in the dust cloud or in the background plasma over the duration of the experiment. In the absence of external perturbations, it is expected that the temperature of the microparticle component should remain constant. The result of this measurement is sen in Figure 4.9. The eror bars are due to uncertainty in the number density of the microparticle component, which is then propagated through the mapping function developed in Section 3.3.1. As was noted previously, the temperature is observed to be anisotropic with the largest value in the z- direction. 144 100 2 3 4 5 6 1000 Temperature [eV] 654321 Measurement T z T y T x Figure 4.9: Multiple measurements of the kinetic temperature of the microparticle component taken over a period of an hour with constant experimental conditions (p = 10 mTor, V anode = 230 V, V cathode = -89 V). It is observed that the measured temperatures are significantly larger than the other plasma components. For the 3DPX device, the electron, ion and neutral temperatures are typicaly 2-3 eV, ? 0.1 eV and 0.037 eV respectively. However, the values of the kinetic dust temperature that are reported in Figure 4.9 are not unexpected. As was sen in Table 1.1, the temperatures that were measured in previous work involving the strongly-coupled dusty plasmas exhibited a temperature in the fluid-like state that was far in exces of the other plasma components. By comparison, the experimental conditions of this study are further into the fluid regime than the previous experiments. This observation, at least qualitatively, is imediately obvious upon examining the images of the dust clouds that are studied in this disertation. On the other hand, it is much more dificult to quantitatively justify this claim. 145 Typicaly, the coupling parameter ?, defined in Equation 1.8, is used to quantify the state of the system in the strongly-coupled dusty plasmas. Using the parameters for the systems in this disertation, the values of ? ranged from ? ~ 1 in the x- and y- directions to ? ~ 0.1 in the z-direction. However, the definition of ?, which asumes an isotropic dust temperature and spatialy uniform dust densities, is dificult to apply to the thermaly anisotropic, spatialy extended thre-dimensional dust cloud structures that are observed in this experiment. Nonetheles, the estimated values for the coupling parameter place this system clearly in the weakly-coupled regime (? ? 1), therefore it is not unexpected that this system has a higher temperature than was reported in the earlier studies. Further, of the studies presented in Table 1.1, only measurement 5, was done in a DC glow discharge. The measurements reported in that work represent the maximum energy observed and are in exces of the values reported here. In a second experiment, the objective is to investigate the scaling of the kinetic dust temperature with changes in the neutral gas presure. The result of this measurement is sen in Figure 4.10, where the kinetic temperature of thre dust cloud are shown as a function of neutral gas presure. In taking this data, a dust cloud was generated at a low neutral presure. As the neutral presure was increased (100 ? p ? 260 mTorr), the electrode biases were adjusted to maintain a stable dust cloud (227 ? V anode ? 263 V, -80 ? V cathode ? -93 V). 146 900 600 300 0 T z [eV] 260240220200180160140120100 Neutral Pressure [mTorr] 150 100 50 0 T y [eV] 250 200 150 100 50 0 T x [eV] (a) (b) (c) Figure 4.10: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of neutral gas presure. The diferent colors indicate data from diferent experimental runs. It is noted that the eror bars are statistical and do not acount for fluctuations in the background plasma. 147 First, it is noted that the data for al thre dust clouds exhibit the same qualitative behavior as a function of neutral presure. Initialy, at lower neutral presures, there is a higher temperature in each vector direction. As the neutral presure increases, the temperature suddenly increased at 110 mTorr. Beyond this presure, there is a decrease in the kinetic temperature of the dust component betwen 110 and 140 mTorr. At presures above p = 140 mTorr, the temperature of the dust cloud remains constant and smaler than what was observed at lower neutral presures. While the kinetic temperatures reported here are anisotropic and are on very diferent scales, the temperatures variation is the same for al thre vector directions. This is shown in Figure 4.11. Here the measured temperatures have been normalized in each direction using Equation 4.3. ! T normalized, i = T" min, i max, i T min, i (4.3) where T i is the temperature being normalized, T min,i and T max,i is the minimum and maximum temperature, respectively, in a given vector direction denoted by the index i = x, y, and z. This suggests that the mechanism responsible for heating the microparticle to the observed kinetic temperatures acts in al thre components, though not equaly, e.g. there is preferential heating. 148 1.0 0.8 0.6 0.4 0.2 0.0 T normalized 260240220200180160140120100 Neutral Pressure [mTorr] T x, normalized T y, normalized T z, normalized Figure 4.1: A plot of the kinetic temperature for dust clouds composed of polydisperse alumina microparticles (r = 0.6 ? 0.25 ?m. ? = 380 kg/m 3 ), normalized using Equation 4.3, as a function of neutral gas presure Two interesting observations are noted during the decrease in temperature betwen 110 and 140 mTorr. First, there was a change in the structure of the dust cloud that corresponds to the observed change in the temperature. At the lower presures, the dust clouds are physicaly larger, have a higher density and are highly disordered. By comparison, al of the dust clouds at a neutral gas presure greater than 140 mTorr are observed to have the same size and structure, which is notably smaler, significantly lower number density and are more structured than the clouds observed at lower presures. Despite this change in the physical structure of the cloud, the dust clouds in the experiments presented here remained solidly in a fluid-like regime. Using the definition of ? = (? x , ? y , ? z ) given in Equation 1.8, a value of ? ~ (0.25, 0.35, 0.05) was observed at presures below 110 mTorr and a value of ? ~ (1.15, 1.40, 0.25) at presures 149 greater than 140 mTorr. Secondly, there appears to be a correlation betwen the dust density and the temperature, which is sen in Figure 4.12. 75 50 25 0 Particle Density [mm -3 ] 27522517512575 Neutral Pressure [mTorr] 200 150 100 50 0 T x [eV] 7550250 Number Density [mm -3 ] (a) (b) Figure 4.12: A plot of (a) the number density of the microparticle component as a function of the neutral gas presure and (b) the kinetic temperature of the microparticle component as a function of the number density of the microparticle component. It is noted that the color coding is consistent with Figure 4.10, with each color representing data taken using a single dust cloud. In Figure 4.12(a), there is a sharp drop in the number density of the microparticle component that is observed as the neutral gas presure increases. This drop in density corresponds to the observed drop in temperature. This is more clearly sen in Figure 4.12(b), where the kinetic temperature in the x-direction is plotted as a function of the 150 number density of the microparticle component. It is noted that the same result is sen in both the y- and z-direction. Based upon this, it appears that the heating mechanism is more eficient with higher dust density. 4.3 RESULTS WITH MICROPARTICLES HAVING A MONODISPERSE SIZE DISTRIBUTION The objective of the experiments described in this section is to investigate the scaling of the kinetic dust temperature with changes in the neutral gas presure for dust clouds composed of monodisperse microspheres. To acomplish this, separate experiments were performed using dust clouds composed of either 6.22 ?m diameter melamine (? = 1510 kg/m 3 ) or 3.02 ?m diameter silica (? = 2000 kg/m 3 ) microspheres. The particles were suspended in an argon plasma over a range of neutral gas presures from 98 to 215 mTorr. The measurement and analysis techniques developed for the studies with polydisperse particles are used to analyze these two systems. In Figure 4.13, the kinetic temperature of two dust clouds composed of the melamine microspheres is shown as a function of the neutral gas presure. For the data shown in red, a dust cloud was generated at a low presure, p = 102 mTorr. As the presure was increased (102 ? p ? 169 mTorr), the electrode biases were adjusted to maintain a stable dust cloud (208 ? V anode ? 211 V, -76 ? V cathode ? -73 V). For the data shown in blue, a dust cloud was generated at a high presure, p = 172 mTorr. As the presure was decreased (172 ? p ? 134 mTorr), the electrode biases were held constant (V anode = 209 V, V cathode = -172 V) maintaining a stable dust cloud. 151 15x10 3 12 9 6 3 0 T z [eV] 175150125100 Neutral Pressure [mTorr] 3000 2000 1000 0 T y [eV] 3000 2000 1000 0 T x [eV] (a) (b) (c) Figure 4.13: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of neutral gas presure. The diferent colors indicate data from diferent experimental runs. It is noted that the eror bars are statistical and do not acount for fluctuations in the background plasma. 152 First, it is noted that the data for both dust clouds exhibit the same qualitative behavior observed in the dust clouds whose source particles was polydisperse. Initialy, at lower neutral gas presures, there is a higher temperature in each vector direction. Additionaly, with the exception of 108 mTorr, the temperature appears to be relatively constant in this region. At 108 mTorr, the measured temperature is significantly larger than the other measurements. As the neutral presure increases, there is a decrease in the kinetic temperature of the dust component betwen 110 and 130 mTor. Beyond p = 130 mTorr, the temperature of the dust cloud remains relatively constant. Further, over al of the presures examined, the measured temperatures exhibit a strong anisotropy with the temperature being significantly larger in z-direction (transverse to he plane of the laser sheet). To quantify this anisotropy, the ratio of the width of the distribution function in the y-direction to the width of the distribution function in the x- direction, ? y :? x , and the corresponding ratio betwen the z- and x-direction, ? z :? x , is computed. These ratios provide an efective measure of the asymmetry in the system. In both the alumina and melamine data, this ratio was found to be constant. For the alumina dust clouds, ? z :? x = 4.49 ? 0.05 and ? y :? x = 0.83 ? 0.02. For the melamine dust clouds, ? z :? x = 4.41 ? 0.11 and ? y :? x = 0.90 ? 0.03. Given the large anisotropy in the z-direction and that this anisotropy appears to be independent of the particle mas or charge, it is important to confirm whether this observation is an experimental artifact caused by the measurement technique or a true measurement of the behavior of the dusty plasma. This is investigated in detail in the next section. However, to determine if it was posible to control this anisotropy, additional experiments were performed on dust clouds compose 153 of 3.02 ?m diameter monodisperse silica microspheres. In an atempt to change the observed anisotropy, the orientation of the electrodes was slightly changed. As sen in Figure 4.14, the electrodes are not verticaly aligned. There is a smal shift in the electrode position in the z-direction, by an amount ?z, to facilitates confinement of the dust clouds. For the measurements described below involving the silica microspheres, the shift, ?z, was decreased, relative to the shift that was used in the measurements involving the melamine microspheres and alumina dust. Figure 4.14: Schematic showing a side view of the electrode orientation. In this new configuration, the temperature of four dust clouds was measured over a wide range of neutral gas presures. For Cloud 1, a dust cloud was generated at a low presure, p = 91 mTorr. As the presure was increased (91 ? p ? 145 mTorr), the electrode biases were held constant (V anode = 207 V, V cathode = -90 V) maintaining a stable dust cloud. For Cloud 2, a dust cloud was generated at a low presure, p = 98 mTorr. As the presure was increased (98 ? p ? 126 mTorr), the electrode biases were adjusted to 154 maintain a stable dust cloud (199 ? V anode ? 200 V, -95.8 ? V cathode ? -92.5 V). For Cloud 3, a dust cloud was generated at a low presure, p = 100 mTorr. As the presure was increased (100 ? p ? 215 mTorr), the electrode biases were adjusted to maintain a stable dust cloud (209 ? V anode ? 215 V, -101.0 ? V cathode ? -89.5 V). For Cloud 4 , a dust cloud was generated at a high presure, p = 189 mTorr. As the presure was decreased (189 ? p ? 100 mTorr), the electrode biases were adjusted to maintain a stable dust cloud (208 ? V anode ? 209 V, -89.9 ? V cathode ? -76V). Of these four clouds, Clouds 3 and 4 were generated in background plasmas that are comparable to what was observed with the melamine and alumina dust clouds. The remaining clouds were les dense. As a result, Clouds 1 and 2 can be quantitatively compared with the other data in a normalized fashion only. In Figure 4.15, the kinetic temperature of Clouds 3 and 4 are shown as a function of presure. It is noted that the data for both dust clouds exhibit the same qualitative behavior that was observed using alumina dust and melamine microspheres. Initialy, at lower neutral presures, there is a higher temperature in each vector direction. There is a sharp increase in temperature at neutral gas presures near 110 mTorr, which is consistent with previous measurements involving dust clouds composed of melamine microspheres. As the neutral gas presure increases, there is a decrease in the kinetic temperature of the dust component until p ~ 125 mTorr. Beyond p = 125 mTorr, the temperature of the dust cloud remains constant. It is interesting to note that this sharp increase in temperature appears to be independent on the neutral gas presure, as it occurs with increasing and decreasing neutral gas presure. 155 1200 800 400 0 T z [eV] 21519016514011590 Neutral Pressure [mTorr] 300 200 100 0 T y [eV] 300 200 100 0 T x [eV] (a) (b) (c) Figure 4.15: Kinetic temperature of the microparticle component in the (a) x-, (b) y- and (c) z-direction as a function of the neutral gas presure. The data from Cloud 3 is shown in gren, while the red data indicates data from Cloud 4. 156 One of the motivating factors in these experiments with the silica microspheres was examining if it was possible to alter the anisotropy in temperature that was observed. As was previously observed, the anisotropy was constant over al of the data that was taken. By shifting the position of the electrodes in the z-direction, but leaving their orientation otherwise unchanged, it was expected that this would cause a change in the anisotropy in the z-direction. Here, it was observed that the anisotropy in the z-direction is smaler. For these dust clouds, ? z :? x = 4.09 ? 0.04 and ? y :? x = 0.86 ? 0.01. Despite the anisotropic temperatures, the variation in the kinetic temperature for both the melamine and silica dust, is the same in al thre vector directions. This is sen in Figure 4.16, where the kinetic temperatures are normalized acording to Equation 4.3, which is repeat below. ! T normalized, i = T" min, i max, i T min, i In Figure 4.16(a), the data for the melamine microspheres is plotted, while the silica microspheres is sen in Figure 4.16(b). For the silica data, the data for Clouds 1-4 are plotted. Again, this is suggestive that the mechanism responsible for heating the microparticle component to the observed kinetic temperatures acts in al thre components, though not equaly. 157 1.0 0.8 0.6 0.4 0.2 0.0 T normalized 180170160150140130120110100 Neutral Pressure [mTorr] T z, normalized T y, normalized T x, normalized (a) (b) 1.0 0.8 0.6 0.4 0.2 0.0 T normalized 22519817114411790 Neutral Pressure [mTorr] Tx Ty Tz Figure 4.16: Normalized temperature as a function of neutral gas presure for dust clouds composed of (a) melamine and (b) silica microspheres. In (a), the red data coresponds to the cloud that was generated at low presure, while the blue data coresponds to the cloud that was generated at high presure. In (b), the data in black, purple, gren and red coresponds to Clouds 1 ? 4 respectively. In both sets of data, a sharp decrease in the temperature is observed. For the melamine microspheres, this decrease occurs betwen 110 and 130 mTorr. For the silica microspheres, this decrease occurs betwen 105 and 125 mTor. In both cases, the 158 observed decrease in temperature is comparable to the range of neutral presures where a strong decrease in the kinetic temperature was observed for the polydisperse data. By more closely examining the silica data, it is possible to more clearly se the structure of the sharp increase in temperature that is observed in around p = 110 mTorr. This is sen in Figure 4.17, where the normalized temperature for dust clouds composed of silica microspheres is plotted as a function of presure over the range of presures where the kinetic temperature is observed to suddenly increase. In order to beter se the structure, the kinetic temperature was normalized acording to Equation 4. 4. ! T normalized, i = T i max, i (4.4) where T i is the temperaturebeing normalized, T max,i is the maximum temperature for a given cloud in a given vector direction denoted by the index i = x, y, and z. While the presure at which this increase in temperature occurs varies in the four cases sen here, al of the data exhibits the same type of structure with neutral gas presure. 159 1.0 0.8 0.6 0.4 0.2 0.0 T z, normalized 14013012011010090 Neutral Pressure [mTorr] 1.0 0.8 0.6 0.4 0.2 0.0 T y, normalized 1.0 0.8 0.6 0.4 0.2 0.0 T x, normalized (a) (b) (c ) Figure 4.17: Normalized temperature in the (a) x- (b) y- and (c) z-direction as a function of the neutral gas presure for the dust clouds composed of silica microspheres. The data in black, purple, gren and red coresponds to Clouds 1 ? 4 respectively. 160 Additionaly, the dependence of the kinetic temperature on the number density of the microparticle component and the kinetic temperature are sen in Figure 4.18 for (a) melamine and (b) silica dust. (a) (b) 300 250 200 150 100 50 0 T x 3530252015105 Number Density [mm -3 ] 3000 2500 2000 1500 1000 500 0 T x [eV] 765432 Number Density [mm -3 ] Figure 4.18: A plot of the kinetic temperature as a function of the number density of the microparticle component for dust clouds composed of (a) melamine and (b) silica dust. In (b), the gren and blue data coresponds to Cloud 3, while the red data coresponds to Cloud 4. 161 In Figure 4.18(a), it is clear that the kinetic temperature of the melamine dust exhibits the same dependence on dust number density as sen in the dust clouds composed of alumina microparticles. In the case of the silica data, however, a very diferent result is observed. It is observed that, for increasing presure, the number density increased and the kinetic temperature decreased, which is exactly opposite what was observed in the alumina and melamine data. Further, the data plotted in blue in Figure 4.18(b) corresponds to the region where the kinetic temperature was observed to increase at lower presures. On the other hand, with decreasing presure, the silica data behaved in the same fashion as the alumina and melamine data. Consequently, this data suggests that a smal change in the experimental geometry (i.e. a shift in the z-position of the electrodes) can lead to a significant change in the behavior of the dust. 4.4 DISCUSION OF RESULTS In the experiments described in the previous sections, there were two results that were consistently observed: the measured temperatures are highly anisotropic and significantly larger than the temperature of the background plasma. While it is presently unclear what the heating mechanism is responsible for the observed dust temperature, it is clear that the heating mechanism is more eficient for higher particle number densities. These results are consistent with previous observations made using the plasma crystal [65-68, 73, 77, 110]. Within the dusty plasma literature, there are a couple of theoretical models that may explain the observed heating of the microparticles. In papers by Meltzer, et al. [65] 162 and Joyce, et al. [111], it is proposed that the streaming of ions past the dust particles can trigger an instability that leads to dust particle heating, notably, as a function of the neutral gas presure. Although this work is primarily focused on microparticles that are trapped in the sheath region of rf discharges, the microparticles in this experiment may be subject to a similar mechanism due to the ion flow. Alternatively, fluctuations in the spatial location or the microparticle charge have been proposed as mechanisms that can lead to microparticle heating [112]. The observations of this study, notably Figures 4.12 and 4.18 give an indication that increasing particle density, or conversely, reduced inter-particle spacing, corresponds to an increase in the microparticle temperature. With reduced interparticle spacing, the microparticles can more closely interact with their neighboring particles giving rise to an enhanced heating of the microparticles. It is noted, however, that this mechanism would have to compete with the reduction in charge due to screning efects from the increased dust density. In either case, it is clear that the heating of the microparticles is significant in this experiment. Additionaly, the heating mechanism occurs simultaneously, though asymmetricaly, in al thre vector directions. By contrast, the observation of highly asymmetric temperatures is dramatic. It is important to determine if the observed asymmetry is an artifact of the stereo-PIV measurement technique or an indication of the underlying proceses that are occurring in the experiment. Recal that the orientation of the electrodes and dust tray in the interior of the 3DPX device, as sen in Figure 4.14, does not suggest that there is any reason for symmetry in 3DPX. 163 In particular, it is noted that the experimental setup does not have a prefered direction of symmetry. The force of gravity acts in the y-direction, breaking the symmetry in the x-y plane (e.g. the plane of the laser sheet). Similarly, there is no reason to expect symmetry betwen in the x-z plane. First, the electrodes are fed into the chamber from the back, e.g. in the z-direction. Further, for improved confinement, the upper electrode is shifted behind the lower electrode, i.e. the anode and cathode are not verticaly aligned. As a result, the plasma sheath extends around the anode, forming the region where the dust clouds are confined. Additionaly, the dust clouds described in the previous sections are suspended in the anode sheath, imediately below the lower electrode, where the behavior of the plasma is quite complex. In order to explore the asymmetric behavior of the kinetic dust temperature measurements in 3DPX, results from thre diferent experiments are compared. In al of the experiments discussed to this point in this disertation, the electrodes have been oriented the cathode-anode-tray configuration and the dust is confined in the region imediately below the anode, as sen in Figure 4.19(b). However, this is not the only experimental orientation for which dust can become trapped in the 3DPX device. There is an alternative configuration, the anode-cathode-tray or A-C-T configuration, that can be used to confine the dust particles [113]. In this arangement, the dust clouds become confined in the region imediately above the dust tray, i.e. in the sheath region of the dust tray, as sen in Figure 4.19(a). For comparison, the equilibrium location for the dust clouds in both configurations are schematicaly depicted in Figure 4.19(c). 164 Figure 4.19: An image depicting the location of a cloud that is suspended in the (a) tray sheath and (b) anode sheath. In (a) the lower electrode is not visible, but the dust tray can be sen in the lower central region of the image. In (b), the uper electrode is visible, but the dust tray is not. In (c), the relative location of the equilibrium position for a dust cloud in the anode and tray sheath are shown. The dust clouds in the A-C-T configuration are significantly further from the electrodes that generate the plasma. Given that the equilibrium location of the dust cloud in this configuration is further from the plasma source, it is possible that the dust cloud may be les susceptible to any asymmetry asociated with the plasma source. A comparison is made betwen the distribution of velocities in a single slice of a dust cloud suspended in the tray and anode sheath shown in Figure 4.20 165 (a) (b) 1.0 0.8 0.6 0.4 0.2 0.0 N o r m a l i z e d C o u n t s 0.20.10.0-0.1-0.2 velocity [m/s] Cathode-Anode-Tray Configuration v x vy vz Tx ~ Ty < Tz 1.0 0.8 0.6 0.4 0.2 0.0 N o r m a l i z e d C o u n t s 0.20.10.0-0.1-0.2 velocity [m/s] Anode-Cathode-Tray Configuration v x v y v z Tx ~ Ty ~ Tz Figure 4.20: Distribution of velocities in a single slice from the central region of a dust cloud suspended in the (a) tray sheath and (b) anode sheath. It is observed that the asymetry sen in the anode region is significantly reduced when the dust cloud is suspended in the tray sheath. In Figure 4.20(a), a dust cloud was suspended in an argon plasma (p = 102 mTorr, V anode = 173 V, V cathode = -140 V) with the electrodes configured in the A-C-T configuration. The time betwen laser pulses (i.e. the duration of a single measurement) was set to ?t laser = 1.0 ms. In Figure 4.20(b), a dust cloud was suspended in an argon plasma (p = 120 mTorr, V anode = 215 V, V cathode = -97.5 V) with the electrodes configured in the C-A-T configuration. The time betwen laser pulses was set to ?t laser = 0.5 ms. It is noted that diferent times betwen laser pulses were used to extract slightly diferent information about the type of motion (i.e. kinetic or fluid-like) of interest. As has been discussed previously, a crucial part of this project is determining the validity of the measurements of the distribution function made using the stereo-PIV system. Typicaly, this can be asesed by comparing the width of the distribution that is measured to the width of the distribution measured using zero-displacement test described in Section 2.2.4.3. This was done for al of the data that has been presented in the previous sections of this chapter, however this measurement was not made for the 166 optical configuration used in the experiments described in this section. However, the zero-displacement test measures the eror in the software algorithm and the erors in the optical components of the stereo-PIV system. Given that sources of eror have remained relatively unchanged over the course of al of the experiments that have been presented in this disertation, it is posible to make an asesment as to whether or not the asymmetric distributions are valid. Typicaly, the measurement eror in the z-direction is on the order of 0.2 pixels, while the eror in the x- and y-directions are on the order of 0.1 pixels. Using the time betwen laser pulses, ?t laser , and the camera calibration (i.e. the number of microns per pixel), it is possible to estimate the resolution limit. The widths of the distributions sen in Figure 4.20 are larger than the resolution eror giving confidence in the validity of measured data. The data presented in Figure 4.20 also suggests that the asymmetries observed in the velocity distributions can vary significantly depending on the trapping location of the cloud. While this does not eliminate the possibility that there may be a contribution due to the PIV diagnostic, the observed asymmetries in the temperature measurements are, in part, a ?real? feature of the experiment. Nonetheles, there remain two unresolved questions. First, it is not clear if the smaler asymmetry observed in the tray sheath is due to the larger time interval betwen laser pulses. Second, the distributions shown in Figure 4.20 are from a single slice of the dust cloud. As a result, it is not clear if the asymmetry is due to the location where we are looking or if the observed asymmetry is a global feature. 167 To addres the first question, a stable dust cloud was generated using the anode- cathode-tray configuration. In this configuration, the dust cloud was suspended in the tray sheath using the following experimental conditions: p = 106 mTor, V anode = 197.2 V and V cathode = -117.3 V. In this experiment, a series of 40 measurements were made at a single slice of the dust cloud and the time betwen the laser pulses was varied from 250 ?s to 2.5 ms. For al values of ?t laser used, the final interogation region was a 12 ? 12 pixel region of the acquired PIV images. The resulting distributions were fit and the width of the distribution in each vector direction is plotted as a function of time betwen laser pulses in Figure 4.21 50 40 30 20 10 ! measured [mm/s] 2500200015001000500 !t laser ["s] x-direction y-direction z-direction Figure 4.21: A plot of the width of the measured velocity distribution as a function of the time betwen the laser pulses. It is observed that the width of the distribution gets smaler as the time betwen laser pulses increases. The lines indicate the estimated resolution limit of the stereo-PIV system. As was the case for the data in Figure 4.18, the zero displacement test was not performed for the optical setup used in this measurement. However, based on the 168 previously measured values for the resolution eror in the stereo-PIV system, the measurement limit can be estimated and is found to be smaler than what is measured. The estimated resolution limit is indicated by the dashed lines in Figure 4.21. Additionaly, it is observed that the measured width becomes smaler as the time betwen laser pulses increases. It is thought that this may be due to more particles leaving the interogation region, held constant at a 12 ? 12 pixel, as the time betwen laser pulses increases. This shows that the size of the interogation region used is crucial and depends strongly on the time betwen laser pulses. As described previously, the interogation size of 12 ? 12 pixel works very wel for a time betwen laser pulses of 250 ?s, e.g. the majority of the microparticles do not leave the interogation region betwen laser pulses. To gain insight into the nature of the asymmetry that is being measured, the ratio of the width in the y-direction to the width in the x-direction, ? y :? x , and the ratio of the width in the z-direction to the width in the x-direction, ? z :? x , is computed. These ratios, which are an efective measure of the asymmetry in the system, are plotted as a function of time betwen laser pulses in Figure 4.22 It is observed that the ratio of widths, ? y :? x and ? z :? x , is maintained regardles of the time betwen the laser pulses. For comparison, the ratios in Figure 4.18(a) are ? z :? x = 1.74 and ? y :? x = 0.68. The ratios in Figure 4.18(b) are ? y :? x = 1.00 and ? z :? x = 2.59. As a result, the data sen in Figure 4.19 can be compared directly using this ratio of widths, even though the time betwen laser pulses is diferent. 169 These measurements show that the observed asymmetry in temperature is, in large part, due to the microparticles themselves and is not simply a function of the PIV diagnostic. At the present time, it is unclear why the asymmetry varies as it is observed to and it remains an open question as to why the asymmetry is so much stronger when the 2.0 1.5 1.0 0.5 0.0 Ratio 2500200015001000500 !t laser ["s] ! y : ! x ! z : ! x Figure 4.2: A plot of the ratio of the widths: ? y :? x in red and ? z :? x , in blue, as function of time betwen laser pulses. It is noted that the asymetry is preserved regardles of the time betwen laser pulses. dust cloud is suspended in the anode spot. Nonetheles, the observed asymmetry is real and not a systematic eror in the stereo-PIV system. Finaly, it is noted that the asymmetry that is observed in the anode spot, while present in the bulk distribution of the dust cloud, is also present throughout the cloud. Figure 4.23 shows the distribution of velocities at each z-location through the entire volume of a dust cloud suspended in the anode spot. This suggests that if an asymmetry is present in a single slice of the cloud, then it wil be present throughout the cloud and wil be observed in the bulk cloud. 170 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] -0.2-0.10.0 0.1 0.2 Velocity [m/s] -0.2-0.10.0 0.1 0.2 Velocity [m/s] -0.2-0.10.0 0.1 0.2 Velocity [m/s] -0.2-0.10.0 0.1 0.2 Velocity [m/s] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] 1.0 0.8 0.6 0.4 0.2 0.0 Counts [AU] z = 86 [mm] z = 87 [mm] z = 88 [mm] z = 89 [mm] z = 90 [mm] z = 91 [mm] z = 92 [mm] z = 93 [mm] z = 94 [mm] z = 95 [mm] z = 96 [mm] z = 97 [mm] z = 98 [mm] z = 99 [mm] z = 100 [mm] z = 101 [mm] z = 102 [mm] z = 103 [mm] z = 104 [mm] Cloud Figure 4.23: Distribution of velocities throughout the volume of a dust cloud suspended in the anode spot. Each plot represents the distribution of velocities measured using the stereo-PIV system at a z-location given in the uper right hand corner of the plot. It is observed that the fluctuations in the measured distribution is smalest in the central region of the cloud, where the clouds are biger and more vectors are reconstructed. Aditionaly, it is observed that the same asymetry is observed throughout the cloud and in the distribution for the entire cloud sen in the lower right tile. 171 Finaly, it is noted that the temperatures that are reported here describe the bulk temperature of the entire dust cloud. While this measurement is consistent with the previous experiments involving the rf systems, it is not entirely clear that this definition of temperature is entirely acurate or necesarily appropriate for the weakly-coupled dusty plasmas studied in this disertation. This is hinted at in Figure 4.23, where it appears that the width of the distributions varies throughout the cloud. Indeed, a detailed examination of the velocity measurements reveals that there is a great deal of structure in the motion of the dust particles and this observed spatial variation in the motion is significantly larger than what was observed in the experiments involving the rf systems. This is sen in Figure 4.24, where contours of velocity in the x-, y- and z-direction are plotted for a single slice of a dust cloud. A cursory inspection of Figure 4.24 shows that not only is there a large spatial variation in the observed motion, there is much more structure in the z-direction. While this justifies the need for the stereo-PIV diagnostic for these studies, it raises an interesting question about the appropriatenes of the usual Maxwelian definition of temperature based upon the width of a distribution function. In the kinetic description, it is expected that the kinetic temperature for a system should be the same, regardles of whether one examines the entire systems or a smal subset of the system. While this applicable for the experimental studies in the rf systems, it appears that this may not be the case for the weakly-coupled dusty plasmas studied here 172 -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -16 -15 -14 -13 -12 -11 -10 y [mm] 2116116 x [mm] -0.10 -0.05 0.00 0.05 0.10 v x [mm/s] 100 80 60 40 20 0 |v x | [mm/s] -0.10 -0.05 0.00 0.05 0.10 v y [mm/s] -0.10 -0.05 0.00 0.05 0.10 v z [mm/s] 100 80 60 40 20 0 |v y | [mm/s] 100 80 60 40 20 0 |v z | [mm/s] Figure 4.24: Contour plots of the velocity (left column) and sped (right column) in the x- (top row) y- (midle row) and z-direction (botom row) for a single slice of a weakly-coupled dusty plasma. The lines depict contours of constant velocity/sped, while the shaded background more clearly ilustrate the spatial variation of the motion. It is observed that there is a great deal of spatial structure, particularly in the z- direction. To quantify this, the temperature is measured as a function of the z-location for a dust cloud composed of 6.2 ?m diameter melamine microspheres in an argon plasma generated with p = 122 mTorr, V anode = 209 V and V cathode = -76 V. The result of this measurement is sen in Figure 4.25. It is noted that the large uncertainty in the reported temperature at z = 86 m is due to the poor statistics that occur at the front of the cloud. 173 There are thre comments that bear mentioning on what is observed. First, the number of measurements at the front of this cloud was relatively smal, which acounts for the larger uncertainty in the measurement. Second, these measurements stil involve averaging over the x-y plane and as a result, there is already a loss of spatial information. Nonetheles, it is observed that there is a significant variation in the temperature throughout the dust cloud. However, it does appear that a weighted average of the temperature throughout the cloud is related to the bulk temperature, T bulk , of the cloud using Equation 4.5. ! T bulk = T i " i i # " i i (4.5) where T i and ? i are the temperature and number density of the microparticle component in the i th slice and the index i extends over the extent of the cloud. The values for the temperature that is extracted from the entire velocity distribution of the cloud, T avg , and temperature found using the weighted average defined in Equation 4.5, T bulk , are reported in Figure 4.25. It is noted that the temperature that is measured using the PIV measurements over the entire dust cloud is consistent to the average temperature at each z-location, when weighted by the number density. The diferences in temperature are likely due to the large uncertainty in the front of the dust cloud. Consequently, the temperatures that are reported here are consistent with previous measurements, it is not clear if this necesarily the best measurement for the energy of the dust cloud. Clearly, it 174 is necesary to examine the spatial structure of the temperature for these systems in more detail. 175 10 3 2 3 4 5 6 10 4 T z [eV] 105100959085 z-location [mm] 10 2 2 4 10 3 2 4 10 4 T y [eV] 10 2 2 4 10 3 2 4 10 4 T x [eV] 10 8 6 4 2 0 Dust Density [mm -3 ] T Bulk,x = 778? 35 [eV] T avg,x = 951.4 ? 49.1 [eV] T Bulk,z = 3185.7 ? 136.2 [eV] T avg,z = 3513.5 ? 148.6 [eV] T Bulk,y = 693.2 ? 33.6 [eV] T avg,y = 886.6 ? 64.1 [eV] Figure 4.25: Showing the spatial variation in the z-direction of (a) the number density, (b) T z , (c) T y and (d) T z . 176 CHAPTER 5:CONCLUSIONS The work in this disertation presents the first detailed measurements of the velocity space distribution function of a weakly-coupled dusty plasma. Using the 3DPX device, the velocity space distribution of a dust cloud was measured over a wide range of experimental conditions. Using these velocity distributions and the results of extensive simulations of the PIV measurement technique, a bulk kinetic temperature for the entire dust cloud was determined. A summary of the work presented in this disertation and directions for future research are given in this chapter. In Section 5.1, a summary of the simulations of the PIV measurement technique is presented. In Section 5.2, a summary of the experimental measurements is presented. Finaly, directions for future research are discussed in Section 5.3 5.1 SUMARY OF SIMULATIONS OF THE PIV MEASUREMENT ECHNIQUE The measurements in this disertation were made using the particle image velocimetry technique. This technique is a valuable tool in the dusty plasma community, as it provides a whole-field, non-invasive instantaneous measurement of the microparticle transport over the entire iluminated volume of the suspended dust cloud. Unlike particle tracking techniques, this technique returns the average velocity of a number of particles. 177 While this approach is particularly useful for studying flows and higher number density dusty plasmas where individual particles can not be easily resolved, it does present a chalenge when atempting to measure the velocity space distribution function. The reason for this lies in the underlying averaging that occurs in this measurement technique, which results in the suppresion of the higher speed particles and an inherent bias towards smaler values of velocity. While this is a wel-known feature of the PIV technique in the fluid mechanics community, litle work has been done to determine if it is possible to relate what is measured using PIV techniques to the underlying velocity distribution function. Extensive simulations of the PIV measurement technique were performed to relate the measured distribution of velocities to the underlying velocity distribution function. Asuming that the underlying distribution is Maxwelian, it was found that when the PIV analysis is restricted to an esentialy monodisperse distribution of tracer particles, the measured distribution is also Maxwelian and has a width that is equal to or smaler than the width (temperature) of the distribution being measured. These simulations demonstrate that there is a unique mapping function that relates the measured and underlying velocity distribution and depends on the size distribution and the number density of the suspended microparticle component. At relatively low particle number densities, the simulations showed that the PIV system acts as a particle tracking system, which is wel known experimentaly. As the particle number density increases, the width of the measured distribution becomes smaler. Indeed, the factor that relates the width of the measured distribution to the 178 underlying velocity distribution exhibits a clear functional dependence on the number density of the microparticle component. It was also shown that for a distribution of particle sizes, e.g. a polydisperse distribution, the curvature of this mapping function is influenced by the width of the particle size distribution. The curvature becomes more extreme as the width of the suspended size distribution becomes smaler. It is noted that in order to observe this efect, the PIV technique must be applied to an esentialy monodisperse distribution of tracer particles, which can be acomplished using the multi-intensity PIV technique. If the suspended microparticles have a size distribution and the multi-intensity PIV technique is not used, the measured distributions are not Maxwelian and exhibit a strong peak at zero velocity. In this case, there is no mapping function to relate the measured velocity distribution to the underlying velocity distribution. 5.2 SUMARY OF THE XPERIMENTAL MEASUREMENTS Knowledge of the kinetic velocity space distribution function is an important first step in the description of the thermodynamic state of a given system. In this disertation, it was shown that through the use of stereoscopic PIV techniques, it is possible to measure the thre-dimensional distribution of kinetic velocities of the microparticle component of a dusty plasma. The first new result presented was an experimental study verifying the applicability of the stereo-PIV technique to study of dusty plasmas. In particular, a perturbation was applied to induce a displacement out of the plane of the laser sheet. 179 This motion was then observed using the stereo-PIV system. As a result, a new diagnostic technique was added to the field of dusty plasmas. Using stereoscopic particle image velocimetry, the kinetic velocity space distribution function of a weakly-coupled dusty plasmas was measured for dust clouds composed of polydisperse 1.2 ? 0.5 ?m diameter alumina dust, monodisperse 6.22 ?m diameter melamine microspheres or monodisperse 3.02 ?m diameter silica microspheres. This was acomplished by scanning the stereo-PIV system through the entire dust cloud, making measurements in 1 m increments and reconstructing the full thre-dimensional distribution of velocities. From these measured distributions, a bulk kinetic temperature in each spatial direction of the suspended dust clouds was determined. These experiments were performed in the 3DPX device using dust clouds suspended in an argon dc glow discharge plasma over a wide range of experimental conditions. The electrode configuration for these experiments was a cathode-anode-tray arangement. From these measurements, there were thre key results observed. First, it was observed that the kinetic temperatures of the dust clouds are substantialy larger than the other plasma components. This result is consistent with and larger than previous measurements that were made using strongly-coupled dusty plasmas. Secondly, it was observed that the temperatures are asymmetric. This asymmetry was shown to be a real efect, and not a consequence of the PIV measurement technique. While it is not clear what mechanism is responsible for heating the dust particles, there are a few conclusions that can be drawn about the behavior of the mechanism that is responsible. The heating mechanism is preferential, generating a significantly larger temperature in the z-direction of the experimental geometry. Additionaly, the heating mechanism is more eficient at 180 lower neutral gas presures and appears to be more eficient with higher number densities of the microparticle component. From the literature, there are two mechanisms that may be responsible for the observed heating. However, further work is needed to determine if either candidate can explain the observed heating. 5.3 SUMARY OF RESULTS In this disertation, a new diagnostic technique, stereoscopic particle image velocimetry, was introduced in the study of dusty plasmas. Using this technique, the velocity space distribution function of a stable weakly-coupled dust cloud composed of a variety of particle sizes was measured as a function of the neutral gas presure. It was observed that the distributions were anisotropic. Further, it was shown that this anisotropy is real and was varied significantly. Using the measured velocity space distribution function, a bulk kinetic temperature of the microparticle component was determined. It was found that the kinetic temperature was significantly larger than the other plasma components, but it is presently unclear what is mechanism is responsible for the preferential heating that was observed. In the proces of completing this work, there were a number of isues that have been uncovered that require further investigation. These wil be discused in the following section. 181 5.4 FUTURE DIRECTIONS The introduction of the stereo-PIV diagnostic technique opens a wide range of possible studies for weakly-coupled dusty plasmas, including more acurate measurements of transport phenomena, studies of the thermodynamic properties of a dusty plasma, and the visualization of these systems using a phase-space like construction. Additionaly, there are a number of isues that have been uncovered in this disertation that require further investigation. The simulations that have been presented here yield a great deal of insight into the behavior of the PIV measurement technique and the mapping functions that have been developed are of great use. It may be possible to develop an analytic framework to generate these mapping functions. Additionaly, it would be of great value to experimentaly measure such a mapping function. This would most easily be done using a two-dimensional plasma crystal, where it is far easier to control the particle density. Further, there is an outstanding question on the size of the anisotropy, particularly in the z-direction. While it was shown in this disertation that this anisotropy is a real efect caused by the underlying heating measurement, it is unclear if a portion of this anisotropy is due to PIV technique. By using a system that is symmetric, such as the Coulomb or Yukawa bals that have recently been observed in an rf discharge [88, 114, 115], one can measure the existence of an anisotropy in the PIV technique directly. Additionaly, using the Coulomb bals, one can measure directly the thre-dimensional velocity space distribution function [114] alowing one to again directly test the dependence of the mapping functions that have been developed through simulation. 182 Beyond the outstanding isues involving the stereo-PIV technique, there are a number of physics questions that have ben raised in this disertation that need closer examination. The most obvious entails identifying the heating mechanism that is responsible for the preferential heating that is observed. One possible solution that has been proposed is a two-stream instability betwen the ions and dust particles. As a result, it is necesary to measure the ion flow in the vicinity of the equilibrium location of the dust cloud. Further, there are a number of interesting details that could be further examined. For instance, there was an anomalously large temperature at a particular set of experimental conditions (6.22 ?m diameter melamine microspheres, p = 108 mTorr, V anode = 209 V and V cathode = -76 V). One possible explanation of would be a resonance in the heating mechanism. This could be tested by replacing the manual leak valve with a mas flow controller, which would alow one to carefully and systematicaly measure the bulk temperature of the dust cloud as a function of presure. Further insight into the heating mechanism could be gained by examining if the temperature of the microparticle temperature is a function of the equilibrium charge on the dust grain. An initial study of this dependence would involve generating a dust cloud under similar experimental parameters using diferent sized dust grains. 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Section A.1. 3 presents the mathematical details of the pinhole model. A.1.1: MATHEMATICAL FOUNDATION OF THE 2D PIV TECHNIQUE The following description follows the presentation of Adrian [79]. A.1.1.1 MATHEMATICAL DESCRIPTION OF IMAGE FORMATION Consider a fluid flow iluminated by a laser sheet and depicted in Figure A.1.1. 190 Figure A.1.1: Imaging of a particle in the laser shet on the CD camera. The fluid flow is iluminated by a laser shet at the object plane and imaged on the CD aray at the image plane. In this figure, lower case leters denote quantities measured by the CD camera, while capital leters denote quantities iluminated by the laser shet. The position of a particles at time t 0 are denoted by x 1 and t 1 by x 1 ?. In a given region of this iluminated slice, i.e. the object plane, there are N tracer particles. At time t 0 , the position, ! r X i , of the tracer particles, ? 0 , in this interogation volume can be described by Equation A.1.1. ! " 0 = r X 1 r X 2 M r X N # $ % % & ? ( ( with r X i = X i Y i Z i # $ % & ? ( (A.1.1) 191 where the position of the i th particle in coordinate space is give by X i , Y i and Z i . These particles wil be imaged on the CD element at location ! r x i = x i y i " # $ % & ? , where x i and y i denote a position on the CD aray. Using a similar notation, the location of these particles can be described by Equation A.1.2. ! r " = r x 1 r x 2 M r x N # $ % % & ? ( ( with r x i = x i y i # $ % & ? ( (A.1.2) Here, we note the use of the convention of capital leters to denote objects in the object plane and lower case leters denote objects in the image plane. Further, it is noted that positions in the image and object plane are related to each other by Equation A.1.3. ! x i =MX i y i =MY i (A.1.3) where M is the magnification. Implicit in Equation A.1.3 is the asumption that the magnification is uniform across the interogation volume. This asumption is reasonable in the application of 2D PIV, where the camera is oriented perpendicular to the laser sheet. 192 At time t 1 = t o + ?t, the N particles in the interogation volume have moved to locations ! r X i '= r X i + r D , where ! r X i ' and ! r D i = D i,x i,y D i,z " # $ % & ? denote the new position and the displacement of the i th particle respectively. Regardles of how complicated the motion, the displacement of the particles can be described as a linear translation, ! r D , provided that the time betwen succesive images, ?t, is smal enough. Under this asuming, the position of the original N particles is given by Equation A.1.4. ! " 1 = r X 1 ' r X 2 ' M r X N ' # $ % % & ? ( ( = r X 1 + r D r X 2 + r D M r X N + r D # $ % % & ? ( ( with r X 1 '= X i +D x Y i + y Z i +D z # $ % & ? ( (A.1.4) The displacement which is described by Equation A.1.4 is depicted in Figure A.1.1. Asuming that there are no distortions in the imaging optics, the location of the particle in the second image, ! r " x i , can be related to the location of the particle in the first image, ! r x i , using the matrix formulation of optics, defined in Equation A.1.5. ! r " x i = t P o t D o t P #1 o r x i (A.1.5) 193 where ! r " x defines a location in the plane of the (virtual) light sheet, ! r x i defines a location in the fluid flow, ! t P "1 defines the transformation of a point in fluid flow to a point on the image plane (i.e. the CD aray), ! t D defines the displacement of the particles in the fluid flow, ! t P is the transformation of a point on the image plane onto the plane of the (virtual) light sheet. These quantities, in their matrix form are given below. ! t P = 1000 0100 0000 00" 1 z o 1 # $ % % % & ? ( ( ( (A.1.6) ! t D = 100D X 010 Y 001D Z 0001 " # $ $ % & ? ? (A.1.7) ! t P "1 = 1000 0100 000"z o (1+M) 000" # $ % % & ? ( ( (A.1.8) From this, one can se that the displacement in the recorded image, ! r d = r x i '" r x i , is related to the real displacement, ! r D , through Equation A.1.9. 194 ! d x =x i '"x i ="M(D x +D z x i ' o ) d y =y i '"y i ="M(D y +D z y i ' o ) (A.1.9) A key asumption in the application of 2D PIV is that the displacement is predominately in the plane of the laser sheet (i.e. the x-y plane), which is to say that ! D z "0. In this limit, the observed motion is described by Equation A.1.10. ! d x d y " # $ % & ? =M D x y " # $ % & ? (A.1.10) A.1.1.2 MATHEMATICAL DESCRIPTION OF THE CROS CORELATION ANALYSIS TECHNIQUE The intensity profile, I, of an image of the tracer particles that is formed on the CD aray can be described using Equation A.1.1. ! I 0 =I 0 r x ," () =V r X i () # r x i ()$ r x % r x i () i=1 N & =V r X i () # r x % r x i () i=1 N & (A.1.11) where ! V r X i () is a function that describes the light that is scatered off of a particle located at ! r X i and ! " r x i () is a function that describes the spread of light scatered off of a particle 195 located at ! r X i measured at ! r x i due to al optical elements. Similarly, the image formed on the CD aray at time t 1 would then be described by Equation A.1.12. ! I 1 =I 1 r x ," () =V r X j + r D ( ) j=1 N # $ r x % r x j ( ) & r x % r x j % r d ( ) (A.1.12) It is noted that a diferent index, j, is used here when summing over the particles in the image to highlight that this describes the image at t 1 (i.e. the second image). The next step in the proces is to determine the displacements. This is typicaly done by localy cross correlating the two images, t 0 and t 1 . The cross correlation of these two images, describe by Equations A.1.11 and A.1.12 respectively, is given by Equation A.1.13. ! R r s ," , r D () = 1 A I 0 r x ," () I 1 r x ," () # = 1 A V r X i () V r X j + r D ( ) $ r x % r x i ()$ r x % r x j + v s % r d ( ) A # i,j & =V r X i () V r X j + r D ( ) R $ r x i % r x j + v s % r d ( ) i,j (A.1.13) where ! r s is the separation vector in the correlation plane, A is the area of the interogation region, ! R " r x i # r x j + v s # r d ( ) $ 1 A " r x # r x i ()" r x # r x j + v s # r d ( ) A % and the indices are used to denote particles in diferent frames (i.e. t 0 and t 1 ). Equation (A.1.13) can be clarified by breaking the sum into the i = j and the i ? j terms. 196 ! R r s ," , r D () =R # v s $ r d () V r X i () V r X i + r D ( ) i=1 N % + V r X i () V r X j + r D ( ) R # r x i $ r x j + v s $ r d ( ) i&j (A.1.14) where, the first term in the summation contains from the i = j terms, while the second term contains the i ? j terms. Physicaly, the first term contains the correlation information betwen identical particles (i.e. the desired displacement information). The second term is a convolution of the mean intensities of the images and the correlations betwen random particles. The second term defines the noise level in the correlation plane. Using the notation of Adrian [79] , the correlation function given in Equation A.1.15 can be writen as ! R r s ," , r D () =R D r s ," , r D () +R F r s ," , r D () +R C r s ," , r D () (A.1.15) where ! R D r s ," , r D () represents the correlation information betwen identical particles (i.e. the first terms in Equation A.1.14), ! R F r s ," , r D () represents the correlations betwen random particles and ! R C r s ," , r D () represents the convolution of the mean intensities. Figure A.1.2 depicts a typical cross correlation function. 197 ? R D ? R F +R C Figure A.1.2: Depicting the corelation function. The displacement used to generate this corelation plane was 8 pixels in both x and y. Let us now more closely examine the term of interest, ! R D r s ," , r D () . ! R D r s ," , r D () =R # v s $ r d () V r X i () V r X i + r D ( ) i=1 N % (A.1.16) This function is maximum when ! v s = r d . Provided that the signal of interest ! R D r s ," , r D () exceds the noise floor, one needs to find the location of the peak for this portion of correlation function. It is noted that when this analysis is done, one specifies how much larger this portion of correlation function must be, relative to the noise floor defined by the quantity ! R F r s ," , r D () +R C r s ," , r D () , i.e. a signal to noise ratio. In the experiments presented in this thesis, this value is set to 1.2. The location of this peak directly yields the desired average in-plane displacement, ! r d . Typicaly, the correlation function is fited and the peak is found from the fited function, alowing for sub-pixel resolution in the measured displacement. 198 The proces to implement the cross-correlation technique is relatively straightforward and can be sen in Figure A.1.3. Figure A.1.3: Depicting the proces used to construct a PIV vector. Identical n ? n evaluation regions from Image 1, taken at time t 0 , and Image 2, taken at time t 1 , are extracted. A cros corelation analysis of these two image subsets is done and the peak in the corelation plane yields the desired in-plane displacement, ! r d . In practice, one does not analyticaly atempt to estimate the displacement function, ! r d . Instead, one statisticaly finds the best match betwen the two images, t 0 and t 1 , using the discrete cross correlation function, ! Rk,l()=I 0 x,y()I 1 x+k,y+l( ) l="L L # k="K K (A.1.17) where I and I? are subsets of images I 0 (x, ?) and I 1 (x, ?), with I? being larger than I, centered at location (x, y) of the CD aray. Physicaly, the image element I is shifted by 199 an amount (k, l) over the image element I?. This shifting operation is done over a range of values (-M ? k ? M, -N ? l ? N). At each shift location, the quantity in Equation A.1.17 is computed to form a correlation plane of size 2M+1 by 2N+1. This is sen in Figure A.1.4. Figure A.1.4: Showing the formation of a corelation plane. A square region of the two acquired images is extracted. The subset of Image 2 is shifted around the subset of Image 1 and the quantity in Equation A.1.17 is computed For a particularly simple case, the results of this operation is sen in Figure A.1.5. 200 (a) (b) (c) (d) Figure A.1.5: Cros corelation function for simulated images. In (a) and (b), the generated images used in this analysis are sen. Each image contains a tracer particle and random noise that defines the noise flor. The defined shift in the tracer particle was 3 pixels in x and y. In (c), a top view of the corelation plane is sen, while (d) shows a topographic view of the corelation plane. There are two points that bear mentioning on the application of the cross correlation technique. First, and as noted previously, it is only possible to recover linear displacements using this technique. To acount for this in practice, one must use a sufficiently smal ?t and a smal enough interogation region that there are no second order displacements, such as a displacement gradient. Second, this proces is computationaly intensive proces, requiring O[N 2 ] computations to form a correlation plane of spatial dimension N ? N. However, there is a computationaly more eficient means of computing the correlation function in Equation A.1.17. This technique makes 201 use of the correlation theorem, which states that cross correlation of two functions, I and I? is equivalent to a complex conjugate multiplication of their Fourier transforms, or ! Rx,y()" ? I # ? I ' * (A.1.18) where ! ? I and ! ? I ' * are the Fourier transform of I and I?, respectively. As a result, the cross correlation analysis can be done using Fourier Transforms, which is how the PIV vectors are computed in practice. The proof of Equation A.1.18 is given in the next section. A1.2: PROOF OF THE CORRELATION THEOREM In the folowing, consider two functions: ! fx,y() and ! gx,y(). By definition, the cross correlation of these two functions, ! h"x,y(), is given by Equation A.1.19. ! h"x,y()=gx,y()#fx,y()=gx,y()$f*x%"x,y%"y( )dxdy %& & ? (A.1.19) The Fourier Transform of f(x, y) and g(x, y) are, by definition, given in Equation A.1.20. ! Gp,q()= 1 4" 2 gx,y()exp#ipx+qy( ){ }dxdy #$ $ % Fp,q()= 1 4" 2 fx,y()exp#ipx+qy( ){ }dxdy #$ $ % (A.1.20) 202 We now consider the convolution of ! fx,y() and ! gx,y() as the inverse Fourier transform of the product of ! Gp,q() and ! F*p,q() in Equation A.1. 21. ! j("x,y)=g#f=Gp,q()F*p,q()expip"x+q"y( ){ }dpdq $% % & (A.1.21) Substituting the definition of ! Gp,q() from Equation A.1.20, we find ! j("x,y)= 1 4# 2 gx,y()exp$ipx+qy( ){ }dxdy $% % & ? ( ) * + , Fp,q()expip"x+q"y( ){ }dpdq $% % & (A.1.22) Changing the order of integration, ! j("x,y)=gx,y() 1 4# 2 F*p,q()exp$ip"x$x()+q"x$x()( { } dpdq $% % & ? ( * + , dxdy $% % & (A.1.23) But, ! 1 4" 2 F*p,q()exp#ip$x#x()+q$x#x()( { } dpdq #% % & is a phase shifted Fourier Transform of ! Fp,q(). Thus, ! j("x,y)=gx,y()fx#"x,y#"y( )dxdy #$ $ % (A.1.24) 203 Consequently, ! h("x,y) and ! j("x,y) are the same function and the cross correlation of two functions is equivalent to complex conjugate multiplication of the same two functions. This is the correlation theorem. A1.3: MATHEMATICS OF THE PINHOLE CALIBRATION MODEL The pinhole model is a mathematical model that describes a camera setup and is defined by eleven parameters. It is based on the theorem of intersecting lines and provides a complete mapping of volume of interest. The basic idea, in two dimensions, of this model is shown in Figure A.1.6. Figure A.1.6: Showing the basic idea of the pinhole model. 204 Here, a point in the real world at P(y, z), located in the object plane, is projected onto the image plane at location P?(y?, z?). Based on the geometry of Figure A.1.6, a relationship betwen the position P(y, z) and P?(y?, z?) is given by A.1.25. ! " y " z = y z # " y =f y z (A.1.25) Extending this into thre dimensions, one can map a thre dimensional point in the global coordinate system, (X W , Y W , Z W ) to image coordinates on the CD aray (x, y). For applications to a thre dimensional system, an additional transformation is needed. First, the iluminated object, which exist in ?global? 3D coordinates (X W , Y W , Z W ), must be projected onto a 2D ?object plane,? (X C , Y C , Z C ), which is then imaged by the CD aray in the camera. In order to map a location in the global coordinate system (i.e. what is iluminated by the laser sheet) onto the camera coordinate system (i.e. what is imaged by the camera), as sen in Figure A.1.7, a rotation and a translation are performed. This is described mathematicaly in Equation A.1.26. ! r X C = t R r X W + t T (A.1. 26) where ! r X C =X C ,Y C ,Z C ( ) is a position in the camera coordinates, ! r X W =X W ,Y W ,Z W ( ) is a position in the global coordinate system, and ! t R and ! t T describe the rotation and translation necesary to move from the global coordinate system and the camera system. It is noted that the rotation and translation here are the six external parameters in the pinhole model used here. 205 Figure A.1.7: Top view of the 3D pinhole model. Particles are located in the world cordinate system and are iluminated by a laser shet. Once the iluminated particles have been projected into camera coordinate system, a generalized version of Equation A.1.25 relates a position in the camera system to where it is projected, asuming that there are no sources of distortion in the system. ! x u =f X C Z C y u =f Y C Z C (A.1. 26) where ! f is related to the focal length of the lens and the distance from the image to the object plane, ! T z . 206 ! f=f lens 1+ f lens T z +f lens " # $ % & ? (A.1.27) It is noted that the distortion in the system ay cause ! x u , y u ( ) to not be the exact location where the image forms on the CD aray. These location are, however, related. The location where the image actualy forms on the CD aray is given by ! x=S x d pixel +x o y= y d S pixel +y o (A.1.28) where S x is a distortion parameter, S pixel is the size of a pixel, (x o , y o ) is the principle point (typicaly near the center of the CD aray) and (x d , y d ) are the distorted camera position given by ! x d =x u 1+k 1 r+k 2 r 2 ( ) y d =y u 1+k 1 r+k 2 r 2 ( ) (A.1.29) where k 1 and k 2 are radial distortion terms and ! r 2 =x d 2 +y d 2 . Of these, ( ! f, S x , k 1 , x o , y o ) define the five internal camera parameters of the pinhole model used. 207 APENDIX 2: C+ CODE TO SIMULATE THE PIV MEASUREMENT Simulations of the PIV measurement were presented in Chapter 3. In this appendix, the basic code that was writen in C+ to simulate this measurement technique is presented. Appendix A.2.1 contains a copy of the configuration that was used, while the PIV simulation code is found in Appendix A.2.3 and the code for non-standard libraries used are included in Appendix A.2.4. A2.1: CONFIGURATION FILE FOR THE PIV SIMULATIONS 208 209 A2.2: C+ CODE FOR THE PIV SIMULATION 210 211 212 213 214 215 216 217 218 219 220 221 A2.3: ADITIONAL IBRARIES 222 223 224 225 226 227 228 229 APENDIX 3: IGOR MACROS A3.1: DESCRIPTION OF IGOR IGOR Pro is an application for the presentation and analysis of data. This application, published by WaveMetrics, Inc., has a powerful scripting language. In this appendix, we present the code that was writen to analyze both the experimental and simulation data. Appendix A.3.2, we present the code used to analyze the simulation data having a uniform size, while the code for analyzing the simulation results involving a distribution of microparticle sizes is found in Appendix A.3.3. Appendix A.3.4 contains the code that was used to analyze the experimental data, while Appendix A3.5 contains the code that was used to fit data to a 1D drifting Maxwelian velocity distribution function. 230 A3.2: ANALYSIS CODE FOR SIMULATION DATA WITH MONODISPERSE PARTICLES 231 232 233 234 235 236 237 238 A3.3: ANALYSIS CODE FOR SIMULATION DATA WITH POLYDISPERSE PARTICLES 239 240 241 242 243 244 245 246 247 248 A3.4: ANALYSIS CODE FOR EXPERIMENTAL DATA 249 250 251 252 253 A3.5: 1D DRIFTING MAXWELIAN FUNCTION FIT CODE 254 APENDIX 4: LABVIEW CODE A4.1: DESCRIPTION OF LABVIEW LabVIEW is a general-purpose graphical programing environment that includes al of the standard features of text-based programing languages. However, unlike these text based programing languages, LabVIEW uses a graphical programing language, known as G, to create programs known as ?virtual instruments?, or ?VIs.? It is published by National Instruments. A VI consists of two parts, the front panel and the block diagram. The front panel provides a customizable interactive user interface, while the block diagram provides a graphical representation of the program code. National Instruments provides numerous libraries of functions and sub-routines for the rapid development of code for data acquisition, analysis and presentation. The VI uses a hierarchical and modular structure. This means that the top-level VI may contain a number of sub-VI?s, smal applications that perform a specific function not initialy provided by National Instruments. There were many applications that were writen in LabVIEW for use in this disertation. The code found in Appendix A4.2 alows the user to aces and run the code used in this disertation to reduce the PIV vectors generated by DaVis and analyze 255 the video data to determine the particle density of the microparticle component. The code used to reduce and the raw PIV vectors at each z-location is found Appendix A4.3, while the code used to compile the PIV vectors at each z-location into a single cloud is found in Appendix A4.4. The code used to determine the scatering eficiency, ?, is found Appendix A4.5, while the code that is used to extract the factor relating the distribution of velocities measured using the PIV technique to the underlying velocity space distribution function can be found in Appendix A4.6. The code used to measure the number density of the dust clouds can be found Appendix A4.7 and the code used to generate a velocity distribution using only a portion of the acquired data is found in Appendix 4.8. A4.2: MASTER LABVIEW CODE This vi wil launch al of the code that is related to the data procesing that is done for the measurements of the distribution function for a dusty plasma. Connector Pane 256 Front Panel Block Diagram 257 List of SubVIs and Expres VIs with Configuration Information Run_Me_for_Vector_Preprocesing_Use.vi Appendix 4.3 Run_Me_to Compile_z_locations_into_cloud.vi Appendix 4.4 Run_Me_for_Scatering_eficeincy.vi Appendix 4.5 Run_Me_for_PIV_factor.vi Appendix 4.6 Run_Me_to_measure_particle_density.vi Appendix 4.7 Run_Me_m_to_n_measurements.vi Appendix 4.8 A4.3: RUN_ME_FOR_VECTOR_PREPROCESING_USE.VI This code extracts the non-zero vectors that are generated in the PIV analysis. 258 Connector Pane Front Panel Block Diagram 259 List of SubVIs and Expres VIs with Configuration Information Convert_Data_to_Matrix_for_Vector_Preprocesing.vi Appendix 4.3.1 Select_ROI_3_for_Vector_Preprocesing_AUTO.vi Appendix 4.3.2 Get_File_List_for_Vector_Preprocesing.vi Appendix 4.3.3 Generate_File_Names_for_Vector_Preprocesing.vi Appendix 4.3.4 Select_ROI_3_for_Vector_Preprocesing.vi Appendix 4.3.5 Read_ Single_Data_File_m_to_n_measurements.vi Appendix 4.3.6 A4.3.1: CONVERT_DATA_TO_MATRIX_FOR_VECTOR_PREPROCESSING.VI Converts the vector file (exported from DaVis in .txt format) into a grid form and returns and cluster aray that holds al of the data. Connector Pane 260 Front Panel Block Diagram 261 A4.3.2: SELECT_ROI_3_FOR_VECTOR_PREPROCESSING_AUTO.VI The code automaticaly identifies the location of the cloud. Images are generated to check for acuracy of results. Connector Pane Front Panel 262 Block Diagram List of SubVIs and Expres VIs with Configuration Information Predefined_Thresholding.vi Appendix 4.3.2.1 Extract_Data_ROI.vi Appendix 4.3.2.2 Count_Zeros.vi Appendix 4.3.2.3 Valid_vectors_method_1.vi Appendix 4.3.2.4 263 A4.3.2.1: PREDEFINED_THRESHOLDING.VI The user generates a Mask that defines the location of the dust cloud. Connector Pane Front Panel 264 Block Diagram A4.3.2.2: EXTRACT_DATA_ROI.VI Converts a user-defined ROI in an image to aray form. Connector Pane Front Panel 265 Block Diagram A4.3.2.3: COUNT_ZEROS.VI Counts the number of zeros that occur in the "Input Aray" and returns this value, "Number of zeros" Connector Pane Front Panel 266 Block Diagram A4.3.2.4: VALID_VECTORS_METHOD_1.VI Kep only the valid vectors Connector Pane 267 Front Panel Block Diagram 268 A4.3.3: GET_FILE_LIST_FOR_VECTOR_PREPROCESING.VI This code generates a list of files to be opened. Connector Pane Front Panel 269 Block Diagram A4.3.4: GENERATE_FILE_NAMES_FOR_VECTOR_PREPROCESING.VI Generates list of filenames to save results to disk. Connector Pane Front Panel 270 Block Diagram A4.3.5: SELECT_ROI_3_FOR_VECTOR_PREPROCESING.VI The valid vectors from a defined ROI are kept, while the others are discarded. Connector Pane Front Panel 271 Block Diagram List of SubVIs and Expres VIs with Configuration Information Manual Thresholding.vi Appendix 4.3.5.1 Extract_Data_ROI.vi Appendix 4.3.5.2 Count_Zeros.vi Appendix 4.3.5.3 Valid_vectors_method_1.vi Appendix 4.3.5.4 A4.3.5.1: MANUAL THRESHOLDING.VI The user generates a Mask that defines the location of the dust cloud. 272 Connector Pane Front Panel Block Diagram 273 A4.3.6: READ_SINGLE_DATA_FILE_M_TO_N_MEASUREMENTS.VI Reads in a single vector file exported from DaVis in .txt format. Connector Pane Front Panel 274 Block Diagram A4.4: RUN_ME_TO COMPILE_Z_LOCATIONS_INTO_CLOUD.VI This code compiles the files at each z-location into a single file, i.e. al of the vectors across the cloud. Connector Pane 275 Front Panel Block Diagram List of SubVIs and Expres VIs with Configuration Information Get_File_List_to Compile_z_locations_into_cloud.vi Appendix 4.4.1 276 A4.4.1: GET_FILE_LIST_TO COMPILE_Z_LOCATIONS_INTO_CLOUD.VI Generates an aray, "File Names", that list al of the files in a folder, "Directory." The file structure is coded for the Windows directory structure. Connector Pane Front Panel Block Diagram 277 A4.5: RUN_ME_FOR_SCATERING_EFICEINCY.VI This code is used to determine the pixel intensity per particle for images from DaVis. Connector Pane Front Panel 278 Block Diagram List of SubVIs and Expres VIs with Configuration Information Manual_Particle_Counting_for_Scatering_eficeincy.vi Appendix 4.5.1 Read _in_Images_(directory structure)_for_Scatering_eficeincy.vi Appendix 4.5.2 Automatic_Particle_Counting_for_Scatering_eficeincy.vi Appendix 4.5.3 A4.5.1: MANUAL_PARTICLE_COUNTING_FOR_SCATERING_EFICEINCY.VI User selects a region of known density and enters the number of particles in that region. 279 Connector Pane Front Panel Block Diagram 280 A4.5.2: READ_IN_IMAGES_(DIRECTORY STRUCTURE)_FOR_SCATERING_EFICEINCY.VI Reads in IMX, IM7 (LaVision image format) images stored in "Directory Location". The user specifies which frame to read in using "Select Frame" and what region of the image is wanted using ROI. This portion of the image is stored in an aray, "Image Aray". 281 Connector Pane Front Panel 282 Block Diagram List of SubVIs and Expres VIs with Configuration Information Get_directory_file_list_for_Scatering_eficeincy.vi Appendix 4.5.2.1 Read_in_Image_data_for_particle_density.vi Appendix 4.5.2.2 A4.5.2.1: GET_DIRECTORY_FILE_LIST_FOR_SCATERING_EFICEINCY.VI Generates an aray, "File Names", that list al of the files in a folder, "Directory Location". The file structure is coded for the Windows directory structure. 283 Connector Pane Front Panel Block Diagram 284 A4.5.2.2: READ_IN_IMAGE_DATA_FOR_PARTICLE_DENSITY.VI Reads in IMX, IM7 (LaVision image format) images and returns the image data in an aray. Connector Pane Front Panel 285 Block Diagram A4.5.3: AUTOMATIC_PARTICLE_COUNTING_FOR_SCATERING_EFICEINCY.VI Automaticaly counts the number of particles in a region. Connector Pane 286 Front Panel Block Diagram 287 List of SubVIs and Expres VIs with Configuration Information Adjust Image Display Range (fixed value).vi Appendix 4.5.3.1 A4.5.3.1: ADJUST IMAGE DISPLAY RANGE (FIXED VALUE).VI Image values are capped at a User Selected value. Intensities greater than this value are set to zero. Connector Pane 288 Front Panel Block Diagram 289 A4.6: RUN_ME_FOR_PIV_FACTOR.VI This code wil determine the scatering efeciency that is needed to extract a particle denisty from the images acquired using the stereo-PIV system Connector Pane Front Panel 290 Block Diagram List of SubVIs and Expres VIs with Configuration Information Get_File_List_PIV_factor.vi Appendix 4.6.1 Compute_PIV_factor_to_file.vi Appendix 4.6.2 Density _Range.vi Appendix 4.6.3 291 A4.6.1: GET_FILE_LIST_PIV_FACTOR.VI Generates a list of text files to open. It is asumed that the the file to open, "Summary.txt," is stored in a series of folders located in the folder defined by "Directory." Returns in an aray containing the file names. Connector Pane Front Panel Block Diagram 292 A4.6.2: COMPUTE_PIV_FACTOR_TO_FILE.VI Interpolates the PIV Factor from the mapping function generated by the simulations of the PIV measurement. Connector Pane Front Panel Block Diagram A4.6.3: DENSITY_RANGE.VI Computes the range of densities, based on the experimental uncertainty. 293 Connector Pane Front Panel Block Diagram A4.7: RUN_ME_TO_MEASURE_PARTICLE_DENSITY.VI This code wil read in images taken by DaVis and then determine the paticle density of a cloud. Connector Pane 294 Front Panel Block Diagram 295 296 List of SubVIs and Expres VIs with Configuration Information Generate_File_List_for_particle_density.vi Appendix 4.7.1 Read_in_Image_data_for_particle_density.vi Appendix 4.5.5.2 User_defines_ROI_for_particle_density.vi Appendix 4.7.2 Average n images (float, single set of images).vi Appendix 4.7.3 297 Compute Particle Density (known factor).vi Appendix 4.7.4 Manual Thresholding.vi Appendix 4.3.5.1 Save 2D Data.vi Appendix 4.7.5 Save 3D Data.vi Appendix 4.7.6 A4.7.1: GENERATE_FILE_LIST_FOR_PARTICLE_DENSITY.VI Generates a list of image files to open. It is asumed that the camera images are stored in "Base Directory" and the images to be used are stored in the folowing location: Base Directory: Image@ z-location:Image - Background: Raw -> World Image Returned in an aray of number that stores the number of images at each z-location, "Number of images per z-location: and an aray of the file names. Connector Pane 298 Front Panel Block Diagram A4.7.2: USER_DEFINES_ROI_FOR_PARTICLE_DENSITY.VI The user defines a region of interest in "Image". Connector Pane 299 Front Panel Block Diagram List of SubVIs and Expres VIs with Configuration Information Adjust Image Display Range (fixed value).vi Appendix 4.5.3.1 300 A4.7.3: AVERAGE N IMAGES (FLOAT, SINGLE SET OF IMAGES).VI Returns the average of the n-images stored in the "Input aray" Connector Pane Front Panel 301 Block Diagram A4.7.4: COMPUTE PARTICLE DENSITY (KNOWN FACTOR).VI Compute the size and particle density of a dust cloud. Connector Pane 302 Front Panel Block Diagram 303 List of SubVIs and Expres VIs with Configuration Information Length.vi Appendix 4.7.4.1 A4.7.4.1: LENGTH.VI Determines the length of a cloud. It is asumed that the cloud is solid and only counts values that exced the threshold intensity. Connector Pane Front Panel Block Diagram 304 A4.7.5: SAVE 2D DATA.VI Saves 2D Aray to a tab-delimited text file. Connector Pane Front Panel 305 Block Diagram List of SubVIs and Expres VIs with Configuration Information Generate 2D Aray to Save.vi Appendix 4.7.5.1 A4.7.5.1: GENERATE 2D ARRAY TO SAVE.VI Generates the header information for the 2D arays to be saved to disk. 306 Connector Pane Front Panel Block Diagram 307 A4.7.6: SAVE 3D DATA.VI Saves a 3D aray to a series of-tab delimited text files. Each page of the aray is writen to a file. Connector Pane Front Panel 308 Block Diagram List of SubVIs and Expres VIs with Configuration Information Generate 3D Aray to Save.vi Appendix 4.7.5.1 A4.7.6.1: GENERATE 3D ARRAY TO SAVE.VI Generates the header information for the 3D arays to be saved to disk. 309 Connector Pane Front Panel Block Diagram A4.8: RUN_ME_M_TO_N_MEASUREMENTS.VI This code generate m sets of data containing n measurements. 310 Connector Pane Front Panel Block Diagram 311 List of SubVIs and Expres VIs with Configuration Information Read_ Single_Data_File_m_to_n_measurements.vi Appendix 4.3.6 Select_ROI_m_to_n_measurements.vi Appendix 4.8. Extract_ROI_m_to_n_measurements.vi Appendix 4.8.2 Get_directory_file_list_for_Scatering_eficeincy.vi Appendix 4.5.2.1 Convert_Data_to_Matrix_m_to_n_measurements.vi Appendix 4.8.3 A4.8.1: SELECT_ROI_M_TO_N_MEASUREMENTS.VI User defines a mask of the cloud. Connector Pane 312 Front Panel Block Diagram List of SubVIs and Expres VIs with Configuration Information Manual Thresholding.vi Appendix 4.3.5.1 A4.8.2: EXTRACT_ROI_M_TO_N_MEASUREMENTS.VI Extracts valid vectors. 313 Connector Pane Front Panel Block Diagram 314 A4.8.3: CONVERT_DATA_TO_MATRIX_M_TO_N_MEASUREMENTS.VI Converts the vector file (exported from DaVis in .txt format) into a grid form. Connector Pane Front Panel 315 Block Diagram