Structure formation and wave phenomena in moderately coupled dusty plasmas

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Fall 2011 Structure formation and wave phenomena in moderately coupled dusty plasmas Jonathon Robert Heinrich University of Iowa Copyright 2011 Jonathon Heinrich This dissertation is available at Iowa Research Online: Recommended Citation Heinrich, Jonathon Robert. "Structure formation and wave phenomena in moderately coupled dusty plasmas." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Physics Commons

2 STRUCTURE FORMATION AND WAVE PHENOMENA IN MODERATELY COUPLED DUSTY PLASMAS by Jonathon Robert Heinrich An Abstract Of a thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa December 2011 Thesis Supervisor: Professor Robert L. Merlino

3 1 ABSTRACT Dusty plasmas, defined as plasmas of ions, electrons, neutrals, and charged micron to sub-micron dust particles, support a rich diversity of physical states. These states (ranging from solids to liquids to gas) are determined by the ratio of the Coulomb potential energy between dust particles to the particles kinetic energy and allow for a broad range of phenomena, from crystallization to dust acoustic waves. Due to various dusty plasma interactions, dust acoustic waves can be nonlinear and exhibit various wave phenomena, from topological wave defects to shock waves to structure formations. In this thesis, I investigate a spectrum of plasma and wave interactions in liquid-like dusty plasmas and focus on a range of dust acoustic wave phenomena observed experimentally in a dc discharge dusty plasma. By developing various experimental techniques, dust acoustic wave di raction and topological wave defects, dust acoustic shock waves, temporal dust acoustic wave growth, and structure forming dust acoustic waves were observed. I begin in Chapter 2 with the di raction of dust acoustic waves, which I investigated by introducing a glass rod into the dusty plasma. The resulting di raction pattern was compared to acoustic wave di raction in a neutral gas. In addition to the di raction pattern, topological wave defects were observed to form. I continue Chapter 2 with a preliminary investigation into topological wave defects in dust acoustic waves. Chapter 3 follows with three nonlinear dust acoustic wave experiments. I created ashocktubelikeprofilefordustacousticwavesusingasingleslit.theshock-tube like potential resulted in two sets of nonlinear dust acoustic waves, coalescing high and low amplitude waves and dust acoustic waves that developed into dust acoustic shock waves. The self-excited dust acoustic shock waves were compared to theoretical models. The third nonlinear dust acoustic wave phenomenon that I investigated was areversedriftmodethatappearsinhighamplitudedustacousticwaves.iproposea

4 2 wave process based on dust particle dynamics in high amplitude dust acoustic waves to explain the observations. In Chapter 4, I describe an experimental technique that I developed to create a quiescent drifting dusty plasma. The drifting dusty plasma was used to observe dust acoustic wave growth from thermal density fluctuations. The observed growth rate and frequency were compared kinetic and fluid models. In Chapter 5, I describe experimental observations of a structure forming instability in dusty plasmas. By increasing the discharge current, transient and aperiodic dust density striations formed. I characterized the transient and stationary modes and compared the stationary mode to an ionization/ion-drag instability and a polarization instability. Abstract Approved: Thesis Supervisor Title and Department Date

5 STRUCTURE FORMATION AND WAVE PHENOMENA IN MODERATELY COUPLED DUSTY PLASMAS by Jonathon Robert Heinrich A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Physics in the Graduate College of The University of Iowa December 2011 Thesis Supervisor: Professor Robert L. Merlino

6 Copyright by JONATHON ROBERT HEINRICH 2011 All Rights Reserved

7 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Jonathon Robert Heinrich has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the December 2011 graduation. Thesis committee: Robert L. Merlino, Thesis Supervisor John A. Goree Gregory G. Howes Frederick N. Ski Robert E. Yager

8 To Bob and Janie Heinrich ii

9 ACKNOWLEDGEMENTS I extend my sincere gratitude to my advisor Robert Merlino for his invaluable guidance, support, and discussions through out my graduate education. I would like to thank Su-Hyun Kim for his assistance, mentoring, and discussions. I would also like to thank Michael Miller for his technical knowledge and support. The work in this thesis was supported by the DOE and NSF. iii

10 TABLE OF CONTENTS LIST OF TABLES vi LIST OF FIGURES vii LIST OF SYMBOLS xiii CHAPTER 1 INTRODUCTION Dusty Plasmas: In Space, the Ionosphere, and Fusion Reactors Charging of Dust Particles Laboratory Dusty Plasmas Waves in Dusty Plasmas Goals for this Dissertation SCATTERING AND TOPOLOGICAL DEFECTS IN DUST ACOUS- TIC WAVES Dusty Plasmas as a Tool to Study Basic Fluid Dynamics The interaction of dusty plasma with an obstacle Experimental Design and Apparatus Dust Acoustic Wave Di raction about a Cylinder Topological Defects Summary NONLINEAR DUST ACOUSTIC WAVES Dust Acoustic Shock Waves, Solitons, and Other Nonlinear Waves Experimental Design and Apparatus Observation of Dust Acoustic Shock Waves Comparison with Theoretical Work Backward Drifting Nonlinear Dust Acoustic Waves Summary TEMPORAL DUST ACOUSTIC WAVE GROWTH Wave Growth Experimental Design and Apparatus Observations and Discussion Comparison with Fluid and Kinetic Models Summary iv

11 5 STRUCTURE FORMATION IN DUSTY PLAMAS APPENDIX 5.1 Structure Forming Processes Ionization instability Polarization instability Experimental Design and Apparatus Experimental Observations and Theoretical Comparison of Structure Formations Transitional mode Stationary mode Pressure dependence in the ionization instability Summary A LIST OF PUBLICATIONS A.1 Papers Based on Thesis A.2 Papers Not Based on Thesis BIBLIOGRAPHY v

12 LIST OF TABLES Table 2.1 Experimental Parameters for Di raction, Scattering, and Topological Defects Experimental Results: Iron Dust Experimental Results: Silica Dust Experimental Parameters for Dust Acoustic Wave Growth Experimental Parameters for Dusty Plasma Structure Formations.. 67 vi

13 LIST OF FIGURES Figure 1.1 Detailed evolution of the spoke structure in Saturn s B ring. The smaller images on the left were taken at 10 minute intervals by Voyager 2, The evolution of the spoke structure is attributed to dusty plasma interactions (Image courtesy of Calvin Hamilton.) DC discharge experimental apparatus. The direction of gravity and the magnetic field are indicated. A postive bias is applied, as shown, between the anode and chamber wall Image of dust acoustic waves. The dust acoustic waves were recorded at 500 frames per second in the dc discharge dusty plasma machine detailed in Fig Taken with kaolin dust. The anode is located on the right of the image with the dust cloud to the left. The dust acoustic waves are propagating from the right of the image to the left The linear response of the high-speed camera to scattered light intensity. The image brightness is in arbitrary units. To measure the light sensitivity of the high-speed camera the illuminating laser intensity was varied and the resulting scattered light was recorded Experimental configuration with a glass rod. Top view Di raction of dust acoustic waves about a circular rod. The anode is on the left of image, with the dust cloud to the right. The circular glass rod is located in the center of the image surrounded by a dark dust void. Dust acoustic waves are propagating from the left of the image to the right. Primary waves begin to the left of the rod. Secondary waves can be seen originating to the right (downstream) of the rod. Taken with kaolin dust. Recorded at 500 frames per second Di raction of pressure waves by a cylinder. (a) Two-dimensional acoustic pressure contours for di raction by a cylinder. (b) Scaled image of dust acoustic wave di raction by a cylinder. The di racting cylinder located downstream at (0,0). x and y are in units of wavelength. =1 and a= vii

14 2.5 Topological wave defects due to scattering by a rod. The bifurcation of wavefronts are circled in red and the spatial defect line is denoted by the dashed yellow line. Images (a)-(f) are taken at 0.01 sec intervals. The bifurcated defects propagate up the dust acoustic wavefronts Double slit di raction and wave-defect evolution. The wave-defects are circled in red. In (a) and (f) the waves that make up final wavefronts are colored for clarity. Images (a)-(f) are taken at sec intervals. The green wavefronts in (a) break o of the blue and yellow wavefronts and join together (f) Wavefront breaking of bifurcated waves due to a wave scattering from aglasssheet. Images(a)-(f)aretakenat0.002secintervals. The topological defects are circled. The glass plate is labeled in (a). Density striations perpendicular to k are visible in the wavefronts in (a). As the waves evolve with time the density depletions between the striations grow (b)-(e) until the bifurcated wave vanishes (f) The e ect of a single slit on dust acoustic waves. (a) Unobstructed dust acoustic waves, (b) slit located 11.5 mm from the anode, (c) slit located 15.0 mm from the anode. A pair of high and low amplitude dust acoustic waves are marked by a bracket in (b), with the high amplitude wave to the left and the low amplitude wave to the right. A dust acoustic shock wave is indicated in (c) The average dust density with a single slit. The shock-tube like system is apparent in the intensity profile of the single-slit configuration, (a). The averaged dust cloud is pictured in (b). The region the dust density profile was taken over is marked in red. Large gradients in the intensity profile correspond to large gradients in the dust density Confluence of high and low amplitude dust acoustic waves. The wave peaks locations are plotted with respect to time. The fast (high amplitude) wave and the slow (low amplitude) wave are indicated with the blue and red lines, respectively. The coalesced wave is marked in green. Also plotted are the amplitudes (in arbitrary units) of the waves. The plots clearly show the coalesced wave traveling at the fast wave s speed as well as no significant amplitude di erence between the fast and coalesced waves viii

15 3.4 The observed temporal dust acoustic wave density profiles. Wave amplitudes were taken 2.68, 2.82, 2.95, and 3.10 cm from the anode over a 0.16 s period. The image intensity saturated for the wave peaks 2.68 cm from the anode, creating a false flattening of the wave crests. The dust acoustic waves develop into shocks as they propagate away from the anode. Two wavefronts are identified and connected between observation locations with a dashed line. Taken over multiple wave cycles to show regularity of phenomena Shock amplitude and width,,ofasingledustacousticshockwave plotted with respect to position Evolution of dust acoustic shock waves. (a) Numerical solution for the evolution of non-dispersive dust acoustic shock waves. Calculated following Eliasson and Shukla s derivation [45].A cubic spine fitting of the solution is plotted for 30, 40, and 60 ms. (b) Shock wave development of a single observed normalized dust acoustic wave. The di erence in trough dust densities is due to limitations of the numerical model Fine structure in high amplitude dust acoustic waves. The density profile of the high amplitude waves is shown in (a) in units of N d = n d /n d0. A sample image of high amplitude dust acoustic waves with the reverse drift mode is shown in (b). The non-propagating waves are circled in yellow. The wave spectrum in Fig. 3.8 was sampled across the white rectangle. The region corresponding to the density profile in (a) is outlined in red The wave spectrum of fine structure in high amplitude dust acoustic waves over the highlighted region in Fig. 3.7(b). The acoustic and fine structure modes are labeled. The wave spectrum was sampled over 1024 frames recorded at 500 frames per second Experimental apparatus designed to trap a secondary dust cloud. (a) Schematics of the experimental apparatus with a 12 cm diameter mesh. The inter-wire spacing of the mesh allowed laser light to pass through unobstructed. The mesh was designed with a variable bias with respect to the chamber, permitting for a secondary dust cloud to be trapped and later ejected when the bias was removed. The distance between the mesh and the anode was adjustable. (b) Image of the dusty plasma suspension with the biased mesh and a trapped secondary dust cloud. The primary and secondary clouds as well the anode, mesh, and coordinate system are labeled. Here the mesh is 15.5 cmfromtheanode ix

16 4.2 Floating potential taken with an emissive probe in the absence of dust. The mesh was located 14.2 cm from the anode. The potential well created by the biased mesh that traps the secondary dust cloud was located from 9 to 13 cm from the anode Axial plasma density in the absence of dust for 5 and 7 ma discharge currents, corresponding to the silica and iron dust experiments, respectively. The plasma density falls o exponentially from the anode Images of the drifting dust cloud with spontaneously excited dust acoustic waves taken at 0.08 second intervals. Taking t =0sfrom the first observable traces of dust acoustic waves, the streaming dust cloud is shown in (a) at t = 0.05 s. Early dust acoustic wave growth is shown in (b) at t =0.03 s. Fully developed dust acoustic waves are shown in (c) at t =0.11 s. A curvature in the wavefronts of can be seen in (c) with a sample wavefront highlighted in yellow. Plots (d)-(f) show the corresponding dust density profiles taken across the dotted line in (a). The spatial slice in (a) is also the line that the space-time plot in Fig. 4.5(a) was taken over. Taken with iron dust Space-time diagram of the spontaneous excitation of dust acoustic waves in a streaming dust cloud. The primary dust cloud is towards the right and the secondary dust cloud is seen towards the top left of the image (note the absence of waves). The anode is the line on the right of the image and the circular mesh is on the left. The region of wave growth is marked and the direction of ion-flow is indicated. The time is taken from when the bias was removed from the mesh. The spatial slice that the space-time plot in (a) was taken over is indicated in Fig. 4.4(a). Average dust cloud drift speeds for (a) and (b) from 0 to 0.5 s are 6.1 and 9.8 cm/s, respectively. The smooth transition between drifting DAWs and non-drifting DAWs seen in (a) from 0.9 to 2 s is due to lower dust drift speed. The wave collisions seen in (b) from 0.5 to 1.4 s are due to a larger dust drift speed. Taken with iron dust x

17 4.6 Examples of the observed dust acoustic wave growth. In (a),(c) the amplitude measured between the peaks and troughs of single dust acoustic waves (boxed in (b) and (d)) are plotted vs. time for iron and silica dust, respectively. The portions of the observed amplitude growth used to calculate the growth rate are indicated and the exponential fits used to calculate the growth rates are given. Dust density profiles of several waves are shown in (b), (d), with time taken with respect to (a) and (c). The dust acoustic waves showed linear growth until the waves saturated at around t 0.16 s in (b) and t 0.08s in (d). Plots (a) and (b) corresponds to the experimental run shown in Fig. 4.4 and 4.5(a) Theoretical and observed frequencies and growth rates plotted vs wavelength for iron dust. The observed growth rate and frequency are shown with experimental uncertainty. Theoretical values are plotted for the detailed experimental parameters Theoretical and observed frequencies and growth rates plotted vs wavelength for silica dust. The observed growth rate and frequency are shown with experimental uncertainty. Theoretical values are plotted for the detailed experimental parameters Axial floating potential measured with an emissive probe in the absence of dust. Note the absence of potential structure. The resulting electric field was 180 V/m Aseriesofsingleframevideoimages,at2.5sintervals,ofthesecondary dust cloud evolution during the transitional phase. The first image (a) corresponds to the start of the transitional phase (t=0) with subsequent images taken (b) t=2.5 s, (c) t=5 s, (d) t=7.5 s, (e) t=10 s, and (f) t=12.5 s. In these images, the anode is seen on the right of the dust cloud. The bright portion is the primary (dense) cloud, and the dimmer, less dense secondary cloud is to the left. These images were obtained using spherical iron particles and show that as time progressed, the wavelength decreased xi

18 5.3 Evolution of the transient wavelengths, wave speeds, and amplitude. The time scale is set with respect to the onset of the transitional mode. The measurements were obtained by tracking peaks in the density profiles. The wavelengths were averaged when possible. Two experimental runs are shown, taken with iron and kaolin dust at 60 frames per second. The semi-log plot shows that the wavelength, wave speed, and amplitude decayed with a characteristic time / 12 sec 1. The wavelength, wave speed, and decay time were independent of the dust type Atemporaldustdensityplotalongwithaspace-timediagramofthe transient phase. The dust density plot (top) is taken from the red dashed line in the space-time diagram (bottom). Taken with iron dust and recorded at 60 frames per second Average density for transient and stationary mode. (a) Comparison of the average transient density profile with the stationary mode density profile. (b) Average transient dust suspension. (c) Average stationary dust suspension Robustness of the stationary cloud structure over 45 s. (a) Dust density profiles from images at (b) 0 s and (c) 45 s. In (b) and (c) the anode can be seen to the right of the images with the dust structure to the left. Taken with iron dust Tomographic reconstruction of the standing mode. The axis orientation is shown in (a). The cloud is illuminated by the laser sheet in the x-z plane. Three dimensional views can be seen (b), (c), and (d) as the point of view is rotated (in the direction shown by the blue arrow in (a). The tomographic reconstruction shows the nonplanar nested conical shell structure of the zero-frequency dust density striations. The anode was removed from the images to improve clarity Zero-frequency structure formations at di erent discharge currents. Images averaged over 200 frames and recorded at 60 frames per second. The electrode current is set at (a) 19 (b) 22 and (c) 25 mamp. Corresponding density profiles are shown to the right over the selected regions. The contrast and brightness of the image has been adjusted to improve clarity. Taken with hollow glass particles. As the wavelength increases fewer striations fit into the dust cloud due to finite size e ects. This imposed boundary condition can introduce rarefaction and compression into the density striations xii

19 5.9 Wavelength vs discharge current/plasma density. P 150 mtorr, B z 3 mt. Separate experimental runs are denoted by their trial number. Wavelengths were obtained by averaging over the density peaks. The dashed line is taken from D Angelo s model for a growth rate of! i =21.5 (rad sec 1 ). Taken with hollow glass particles Growth rate vs wavelength predicted by D Angelo s model for the ionization/ion-drag instability. Growth rates correspond to a zerofrequency mode with discharge currents from ma, corresponding to plasma densities of ( ) m 3. The dashed line indicates the growth rate taken for the dashed line in Fig Onset of the polarization instability. The shaded portion is the parameter region where t > 6.4 andstationarydustacousticwavesare predicted to exist [14] Experimental and theoretical (from D Angelo s ionization instability model) wavelengths of the zero-frequency mode taken at various pressures. Wavelengths were obtained by averaging over the density peaks. The dashed line is taken from the modified version of D Angelo s model for a growth rate of! i =21.5 rad sec 1.Takenwithhollowglassparticles. Error bars indicate the range of wavelengths observed in each cloud. The discharge current to the anode was 22 mamp and the dust used was glass microspheres Theoretical growth rates calculated from the modified version of D Angelo s model, modified to include neutral gas dependence in the ion creation and loss. Calculated for experimental conditions listed in Table 5.1. Dashed line indicates growth rate value for Fig xiii

20 LIST OF SYMBOLS Constants B Magnetic field e Electron charge E Electric field " 0 Permittivity of free space G Gravitational constant Boltzmann constant k B Plasma Parameters d Neutral gas drag coe cient T Ratio of the Coulomb radius of interaction between thermal ions and dust grain and the Debye radius Epstein drag force coe cient Coulomb coupling parameter I d Discharge current m Mass µ Ion-drag coe cient from collisions µ crit Critical ion-drag for ionization instability µ tot Ion-drag coe cient from collisions and scattering n Density N d n d /n do n e,f,v Density of fast ionizing electrons with velocity v? Normalized dust viscosity e Fast electron flux Q Charge Q Ion creation coe cient 0 Threshold ionization cross-section c Dust collisional cross-section T Temperature T i /T e d Dust kinetic shear viscosity Dust charge number Z d Potentials ' d grain plasma g Potential Potential di erence between dust particle and plasma Potential of dust particle Potential of plasma Normalized potential, e /T e Gravitational potential xiv

21 Frequencies and Times f Frequency Heat capacity ratio n Particle-neutral collision frequency! Angular frequency! i Imaginary part of angular frequency (growth rate)! jd Jeans frequency! p Particle plasma frequency! r Real part of angular frequency! rf Radio frequency of rf discharge devices c Ion creation time Ion loss time l Miscellaneous Denotes plasma species: dust (d), ions (i), electrons (e), or neutrals (n) C Dust grain capacitance F pol Polarization force (v i ) Coulomb logarithm I Pixel intensity I p Pixel intensity from a single dust particle I tot Total pixel intensity Particle susceptibility Z( ) Plasma dispersion function (! kv o + i n )/ p 2kv T Lengths k mfp D D MT 0 (v i ) r d r s Shock width Wavenumber Wavelength Particle mean free path Plasma Debye length Particle Debye length Wavelength corresponding to maximum growth for ionization instability [88] Coulomb radius of interaction between thermal ions and dust grain Dust radius Inter-particle distance (dust grains) xv

22 Velocities C da u u rd u ds v v e,f v T v receiver v source v s Dust acoustic phase speed Particle drift velocity Reverse dust drift speed Secondary dust cloud drift speed Normalized dust fluid velocity Particle velocity Velocity of fast electrons Particle thermal velocity Velocity of reflected waves Velocity of source waves Shock velocity xvi

23 1 CHAPTER 1 INTRODUCTION 1.1 Dusty Plasmas: In Space, the Ionosphere, and Fusion Reactors The breadth of plasma physics speaks to the variety of environments where plasmas are found. Plasmas, which are roughly defined as ionized gas, have a large impact on everyday life, from lighting and entertainment to technology and medicine. Often, dust grains, sub-micron to micron in size, are found in plasmas. Immersed in a plasma, dust can acquire charge in a variety of ways. When dust become charged a new type of plasma can form, termed a dusty, or complex, plasma. Dusty plasmas are found ubiquitously in astronomical settings. Dark bands in nebulae, caused by the extinction of light upon dust particles, show the vast quantity of dust in the universe [1]. Dust has a pronounced a ect on plasma dynamics and plays an integral part in the development of interstellar clouds and planetary formations [2]. AspectacularexampleofadustyplasmasuspensionaretheringsofSaturn,presented in Fig. 1.1, whose spokes in the B-ring are attributed to dust-plasma interactions [3, 4]. Additional interplanetary in situ observations include satellite detection of micron-sized dust particles [5] and high resolution images of dust in comet tails [6,7]. Charged dust particles are also an important topic for lunar settings; large clouds, likely composed of highly charged dust grains, have been observed floating o the moon s horizon [8]. More terrestrially, dusty plasmas form noctilucent clouds in the lower ionosphere [9]. Noctilucent clouds, composed of charged dust and ice particulates, are correlated to various phenomena, including polar mesospheric summer echoes [10]. In laboratory plasmas, charged dust grains were first noticed in 1989 [11]. Here, in plasmas used for microchip fabrication, small particulates agglomerated into larger particles that subsequently contaiminated the microchip devices. More recently, dust particles have become an issue in fusion reactors where wall ablation by high energy ions can lead to

24 2 Figure 1.1: Detailed evolution of the spoke structure in Saturn s B ring. The smaller images on the left were taken at 10 minute intervals by Voyager 2, The evolution of the spoke structure is attributed to dusty plasma interactions (Image courtesy of Calvin Hamilton.) particle injection [12]. 1.2 Charging of Dust Particles Depending on the plasma environment, dust particles charge in a variety of ways. There are three basic mechanisms that lead to dust charging: dust particle interactions with plasma particles, secondary electron emission due to interactions with energetic plasma particles, and dust particle interactions with photons. Briefly, dust particle interaction with plasma particles results in absorption of ions and electrons by the grain and, generally, leads to negatively charged dust, due to higher electron mobility. When plasma constituents are energetic, dust-plasma collisions cause secondary electron emission and lead to positive charging of dust particles. Lastly, photon interactions with dust grains (photon absorption by dust particles) can result in photoemission of electrons and positively charged dust grains. There are additional charging mechanism,

25 3 which include thermionic emission of electrons from dust particles and charging by radiative sources. As a dust particle acquires charge it will develop an electric potential and sheath, whose characteristic length is the plasma Debye length. For a typical laboratory dusty plasma, where T i T e and v Ti >u i,theplasmadebyelength, D can be approximated as the ion Debye length, Di = p " 0 k B T i /n i e 2. Here T i(e) is the ion (electron) temperature, v Ti is the ion thermal velocity, v Ti = p k B T i /m i, u i is the ion drift velocity, and m i is the ion mass. The sheath shields the surrounding plasma from the dust particle s potential. This shielding is nonlinear and often approximated in a variety of di erent ways. The degree of nonlinearity in the ion-grain interaction can expressed as the coe cient T,theratiooftheradiusofinteractionbetweenthermal ions and the dust grain and the Debye length, T = Z d e 2, (1.1) 4 " 0 D m i vti 2 where Z d is the charge number of the dust grain. For a typical laboratory dusty plasma, T ranges from 1 to 30. Larger values indicate increased nonlinearity [13, 14]. When dust charging is dominated by the collection of less-energetic plasma particles, typical in laboratory dusty plasmas, the final charge on the dust grain is determined by the net current of ions and electrons to the grain, which must be zero. Dust charging can be simplified by considering a single isolated dust grain. For a single grain, the model most often applied to calculate the rate of ion/electron absorption is the Orbital Motion Limited Approach (OML). OML theory was originally developed for probe physics [15], where the collisional, or collection, cross-section of the probe is calculated using conservation of energy and momentum. For a dust grain, the

26 4 collection cross-section derived using OML theory is: c = r 2 d(1 2Q d ), (1.2) 4 " 0 m v 2 where denotes the plasma species (here ions or electrons), r d is the dust grain radius, d = grain plasma is the potential di erence between the grain and plasma, Q is the particle charge, and v is the particles velocity distribution. By integrating the collisional cross section over the species velocity distributions, the collection currents can be calculated. Finally, setting the ion and electron currents equal, d/q d can be solved (Q d = d C,whereC is the grain capacitance). The assumptions made in the formulation of the OML approach are collisionless ions, r d D imfp (ion mean free path), and isolated dust grains. For a typical laboratory plasma, r d 1 µm, d 70 µm, and imfp 350 µm. When there are multiple charged grains, inter-particle interactions must be considered. One of the first interactions to consider is the simple Coulomb interaction, which results in coupling between dust particles. Ignoring nonlinear shielding e ects, the magnitude of inter-particle coupling can be expressed as the Coulomb coupling parameter, which is the ratio of the dust Coulomb potential energy to the dust thermal energy: = Q 2 d 4 " 0 r s k B T d, (1.3) where r s is the inter-particle spacing between dust grains. For large values ( 1), dusty plasmas are strongly coupled and exhibit solid-like, or crystal, states. For smaller values (. 1), dusty plasmas are moderately to weakly coupled and exhibit liquid-like to gaseous states.

27 5 1.3 Laboratory Dusty Plasmas Perhaps the most popular laboratory dusty plasmas are radio frequency (rf) coupled dusty plasma. Radio frequency coupled discharge plasmas rely on radio frequency varying electric potentials ( 13 MHz) between parallel conducting plates to break down neutral gas into a plasma. Since these temporal potentials vary on a time scale much faster than the ion plasma frequency,! rf! pi where! p = (n Q 2 /" 0 m ) 1/2 and n is the particle density, only the electrons take part in the bulk ionization process. In a terrestrial setting, a dc potential across the plates is used to counter the Earth s gravitational force on the particles. The confining potentials of rf devices under gravity generally result in single and double layer dusty plasma suspensions, which are useful for studying crystallization and a variety of other phenomena. Single and double layered rf coupled dusty plasmas at pressures greater than 500 mtorr are typically strongly coupled. Another type of apparatus commonly used to study dusty plasmas are dc, or anodic glow discharge machines. Like rf coupled plasma devices, dc discharge devices utilize an electric potential, although in this case steady state, to drive an electron current that ignites the plasma. Usually, the dc bias is created between an anode and the chamber wall. When a relatively strong positive potential applied to the anode, an outward ion-flow is established. The potential structure in anodic dusty plasmas allow for large 3-D dust clouds to be trapped, making anodic dusty plasmas a suitable environment to study bulk dusty plasma phenomena. A dc discharge apparatus was used for the work in this dissertation and is detailed in Fig. 1.2 Akeydiagnosticfordustyplasmaisopticalimaging. Dustsuspensionsare easily illuminated using a laser sheet and, once illuminated, are visible to the eye. Additionally, dust grains react on much longer time scales than their plasma counter parts due to their massive grain sizes,! pd! pi.thesetraitsareuniquetodusty

28 6 g Side View Laser Plasma Anode Electrically Floating Dust Tray B-Field Top View Laser Plasma Anode Electrically Floating Dust Tray CMOS Camera Figure 1.2: DC discharge experimental apparatus. The direction of gravity and the magnetic field are indicated. A postive bias is applied, as shown, between the anode and chamber wall.

29 7 plasmas and have become even more valuable with the recent advent of high-speed video recording devices, which have the capability to capture dust dynamics. For cameras that have a linear response to light intensity, resulting images can be used to determine dust densities in suspensions when dust inter-particle spacing is not measurable. The fact that dusty plasmas can be optically imaged allows for convenient investigation of various processes occurring in plasmas, such as nonlinear waves, structure formation, and wave-wave interactions. Important wave information (wave spectra, wavelength, wave speed, wave amplitude, and wave frequency) can be obtained from optical images and video recordings. 1.4 Waves in Dusty Plasmas There are a variety of waves that are sustained in dusty plasma suspensions. In strongly coupled plasmas, where 170, dust acoustic, transverse shear, and lattice waves can exist. In moderately to weakly coupled systems, apple 1, acoustic and cyclotron waves can exist [16, 17], whose low frequency limits are the dust acoustic and, in dusty plasmas having magnetized dust, electrostatic dust cyclotron modes. The work in this thesis will focus on DAWs. The dust acoustic wave was first analyzed by Rao et al. in 1990 [18], where both linear and nonlinear dust acoustic waves (DAWs) were predicted. The first observations followed in 1995 by Barkan et al. [19]. Since then, the onset and parameter space where DAWs are stable has been well investigated both theoretically and experimentally, summarized by Merlino in 2009 [20]. Dust acoustic waves are generally excited by anetion-flowwithrespecttothedust[20]andaremodeledwellbybothfluidand kinetic theories [20, 21]. A sample image of a dust acoustic wave observed using the apparatus in Fig. 1.2 is shown in Fig From a simple fluid description of dusty plasmas, it can be shown that DAWs obey, to some approximation, the same momentum and continuity equations as

30 8 anode 2 cm Figure 1.3: Image of dust acoustic waves. The dust acoustic waves were recorded at 500 frames per second in the dc discharge dusty plasma machine detailed in Fig Taken with kaolin dust. The anode is located on the right of the image with the dust cloud to the left. The dust acoustic waves are propagating from the right of the image to the left.

31 9 ordinary sound waves and can be thought of as visible sound waves [22, 23]. Taking continuity and momentum equations for negatively charged dust particles, Boltzmann distributions for electrons and ions, and charge @x (n dv d )=0, (1.4) n d m + n dm d v + k BT d ez d =0, (1.5) k B T =0, i k B T + =0, (1.7) r 2 = 1 " 0 (q i n i q e n e Q d n d )=0, (1.8) and following the usual linearized fluid derivation (taking first order quantities to vary as e i(kx!t) ), a simple expression for DAW sound speed can be obtained [24]. Here d is the ratio of the specific heat at constant pressure to the specific heat at constant volume of the dust. In the long wavelength limit, the DAW speed can be expressed as: C da =! k =[ dk B T d m d + n d0 n i0 Z 2 d k BT i /m d 1+( T i T e )(1 n d0 n i0 Z d ) ] 1 2. (1.9) The subscript 0 indicate initial unperturbed values. 1.5 Goals for this Dissertation While dusty plasmas are found in virtually all plasma environments, fundamental experimental research only began in the 1990 s. With the recent availability of high

32 10 speed video recording, fundamental DAW physics can now be investigated. To this end my dissertation focuses on experiments designed to investigate fundamental DAW phenomena. To develop experiments to investigate DAW phenomena, the e ect of floating objects in a plasma can be used. When introduced into a plasma, a floating object, or probe, will develop a sheath (non-neutral plasma regions that delineate plasma boundaries) and negative potential profile. In a dusty plasma, negative potential profiles result in dust voids. The geometry of the potential and resulting void depends on the objects geometry. By inserting various floating objects into dusty plasmas, such as rods, apertures, slits, and grids, I was able to investigate a variety of fundamental DAW phenomenon. Additional experiments can be constructed without the use of foreign potentials. For the most part, DAWs occur in moderately to weakly coupled dusty plasmas, that is a dusty plasma whose dust particles kinetic energy is equal to, or greater than, their potential energy. Although weakly/moderately coupled dusty plasmas lack the formation of lattice structures found in their strongly coupled counterparts [25], they can still exhibit structure formation. By adjusting the discharge parameters, these structure forming instabilities can be investigated. The work in my thesis focuses on four sets of experiments that I have divided into four chapters. In Chapter 2, I present DAW di raction followed by a series of wave scattering experiments designed to study topological DA wave defects. These experiments serve to give a qualitative view of topological defects and are the preliminary attempt to characterize a nonlinear DAW phenomenon. In Chapter 3, I focus on nonlinear DAW experiments. I begin with experiments that utilized the potential formed by a single slit to create a shock-tube like system that naturally excited DA shock waves. The resulting shock waves were compared with nonlinear DAW models. Next, I examine nonlinear wave phenomena that developed without

33 11 the use of an external potential. The work in Chapter 4 details DAW growth. Briefly, by using a more involved technique and directly controlling the electrical bias of the foreign object, secondary dust clouds and drifting dust clouds were created. By exploiting this technique, the third set of experiments were designed to measure temporal DAW growth in a quiescent drifting dusty plasma. The measured growth rates and frequencies were compared to kinetic and fluid models. Finally, Chapter 5 presents experimental observations of dusty plasma structure formations. Under appropriate parameters, the ion-drag force, ionization e ects, and polarization force can set up purely growing aperiodic dust density striations.

34 12 CHAPTER 2 SCATTERING AND TOPOLOGICAL DEFECTS IN DUST ACOUSTIC WAVES 2.1 Dusty Plasmas as a Tool to Study Basic Fluid Dynamics Strongly coupled dusty plasmas, & 170, are used to study crystal dynamics, such as the melting transitions between solid and liquid-like states. Moderately coupled dusty plasmas, which correspond to a more liquid-like state with 1, can similarly be used to study fluid dynamics. Quantitative motivation for modeling fluid systems with moderately coupled dusty plasmas arises in the comparison of long-wavelength compressional perturbations in a plasma with those in a neutral gas, which was first recognized by Montgomery in 1967 [26]. The similarities between acoustic perturbations in a fluid and a dusty plasma can be examined directly starting with the continuity and momentum equations for a three + r (n v )=0, (2.1) + n m (v r)v + k B T rn + Q n r Q n v B =0, (2.2) where is taken over the dust, ions, and electrons. By taking velocities parallel to B, closing with a Boltzmann distribution (k B T rn = Q n r ), and linearizing, the momentum equation can be reduced to: m d n 0 + X k B T rn 0 =0, (2.3)

35 13 where first order parameters are primed. The fluid equations for small amplitude perturbations in a neutral gas are identical and can be written in the + n 0r u + C2 s n 0 rn =0. (2.5) For DAWs, C s is the DAW speed, n 0 the unperturbed dust density, and u is the perturbed dust velocity. An expression for the DAW speed, C da, is given in Eq For fluids, C s is the acoustic speed, n 0 the unperturbed fluid density, and u is the perturbed fluid velocity. In a neutral gas, the acoustic speed is C s =( n k B T n /m n ) (1/2). Obeying identical fluid equations, dusty plasmas and neutral gas should exhibit similar wave phenomena, such as di raction. The di raction, or scattering, of sound waves o a cylinder was first considered by Lord Rayleigh [27]. Di raction is observed in many system and the e ect is most prominent when the wavelength is comparable to the obstacle length. In 1982, di raction experiments on plasma waves were performed by Okutsu et al. [28]. In Okutsu s experiment, a plate with a 5 cm hole was placed in front of ion-acoustic waves. Like sound and regular electromagnetic waves, ion-acoustic waves were observed to undergo di raction, expected sincere Huygens principle holds. From this experiment, DAWs can be expected to undergo di raction since ion-acoustic and DA waves are both compressional plasma waves The interaction of dusty plasma with an obstacle The e ect of probes on dusty plasmas has been studied by Thompson et al. and Thomas et al. [29, 30]. In 1999, Thompson et al. introduced an unbiased probe into a dusty plasma and investigated the dependence of the dust void s geometry on the probe s velocity [29]. Later, Thomas et al. examined the e ects of a biased probe, at

36 14 various potentials, on dust void size [30]. The important aspect of these works is the reaction of particulates to a probe and the sharp dust density gradient and void that can be created, which, for DAWs, is akin to a hard boundary in acoustics. The size of the void created depends on the electric potential of the object. As discussed by Thomas et al., dustvoiddynamicscanbeunderstoodintermsofanion-dragforce and electric force resulting from the probe/void s potential. While both Thompson et al. s and Thomas et al. s experiments were able to observe dust voids induced by obstacles, neither group investigated the e ects of probe induced potentials and voids on DAWs. 2.2 Experimental Design and Apparatus To observe DAW di raction, I began with the dc discharge apparatus shown in Fig With a background gas pressure less than 10 mtorr, I introduced argon gas until the chamber pressure was 150 mtorr. With a small axial magnetic field of 7 mt for electron confinement, I applied a positive bias of 275 V to a 3.2 cm diameter electrode disk, located on the axis of a horizontal grounded vacuum vessel 90 cm in length and 60 cm in diameter. The large positive bias drew an electron current and ignited a plasma. After burning the plasma for period of time at a low discharge current ( 10 ma), I increased the discharge current to 40 ma. The increased discharge current creates a denser plasma that facilitates the incorporation of dust grains from the electrically floating tray into the anode glow. The dust is trapped through a process described by Trottenberg et al. and further discussed in Ch. 5.3 [31]. For this experiment I used polydisperse kaolin dust that has a nominal radius of 0.5 micron. After collecting dust into the fire-rod, I lowered the discharge current to 10 ma, where the discharge is more stable and less turbulent. Due to an ion-dust streaming instability, DAWs were self-excited. The experimental parameters are summarized in Table 2.1.

37 15 Table 2.1: Experimental Parameters for Di raction, Scattering, and Topological Defects PARAMETER VALUE REMARKS n e m 3 Double probe measurement n d m 3 Image analysis T i ev Taken as neutral gas temperature T e 2.5 ev Emissive probe measurement T d ev Experimental fit [32, 33] Argon pressure 150 mtorr Measured Z d 2000 e Estimated using OML theory Nominal kaolin dust radius 0.5 µm Measured The 3-D dust suspension was illuminated with a 2 mm thick vertical laser sheet. The dynamics of the dust cloud were recorded using a high speed camera at 250, 500, and 1000 fps. To use captured high-speed video as a diagnostic for plasma density the linear response of the camera to scattered light intensity was verified, as shown in Fig Once images of the dust cloud are captured they can be used to obtain the wave spectra, wavelength, and dust density. The wave spectrum was calculated using the method described by Noskenko et al. [34]. First, a rectangular spatial slice of the dust cloud images is chosen. Ideally, the selection should contain multiple wavelengths, be parallel to k (the direction of the DAW propagation), and in a region where the DAW phase velocity is constant. The raw pixel intensity, I(x, y, t), is binned across the slice in each video frame to create an averaged spatial dust density profile, Ĩ(x, t). Finally, a double Fourier transform over time and space is performed on Ĩ(x, t) to obtain the wave spectrum, which maps the kinetic wave energy in wavenumber (k) and frequency space. Dust densities in complex plasmas, particularly in strongly coupled suspensions, are typically obtained directly from the inter-particle spacing. The current experiment does not allow for direct measurement of the inter-particle spacing. A technique that does not require resolving individual particles involves measuring the extinction

38 16 Figure 2.1: The linear response of the high-speed camera to scattered light intensity. The image brightness is in arbitrary units. To measure the light sensitivity of the highspeed camera the illuminating laser intensity was varied and the resulting scattered light was recorded. of light (the amount of light scattered and absorbed by the dust) through the dust cloud [35, 36]. The later method is also insu cient as it calculates the average density for the region the light passed through and cannot be easily applied in our experimental apparatus. To obtain a dust density in our apparatus I employed a combination of these methods. To calculate the dust density in some region of the dust cloud the total pixel intensity, I tot,overthatportionismeasuredandsummed.theselectedregion stotal pixel intensity is compared with the total pixel intensity of light scattered by a single particle. The pixel intensity from a single particle is measured when dust particles can be individually identified, which is often possible in less dense regions of dust. Taking the total pixel intensity of a single particle, I p,thetotalnumberofdustparticles illuminated can be approximated, n d volume=i tot /I p. Using the volume of the region

39 17 Top View Glass Rod Laser Sheet Plasma Anode Dust Tray B-Field CMOS Camera Figure 2.2: Experimental configuration with a glass rod. Top view. (the illuminated cloud width is the laser sheet width, 2 mm), the dust density is calculated. When individual dust grains cannot be isolated, pixel intensities can still be used to calculate the relative dust densities between di erent portions of the dust suspension as well as N d = n d /n d0.similartechniquestoobtaindustdensityhave been used before [37]. To observe di raction and scattering of DAWs, I introduced a series of obstacles into the dust suspension, beginning with a 3 mm diameter glass rod. When a su cient dust cloud had formed (n d ), the glass rod was inserted into the middle of the dust suspension so that DAWs would impinge on it, outlined in Fig When manipulated into the plasma, the glass rod became negatively charged. The potential profile of the charged rod created a dust void, increasing the e ective rod diameter to 4mm.

40 18 anode topological defect secondary waves rod y x 1 cm Figure 2.3: Di raction of dust acoustic waves about a circular rod. The anode is on the left of image, with the dust cloud to the right. The circular glass rod is located in the center of the image surrounded by a dark dust void. Dust acoustic waves are propagating from the left of the image to the right. Primary waves begin to the left of the rod. Secondary waves can be seen originating to the right (downstream) of the rod. Taken with kaolin dust. Recorded at 500 frames per second.

41 Dust Acoustic Wave Di raction about a Cylinder Once a dust cloud was established, about 6 cm in length, with DAWs propagating through the suspension I introduced the glass rod. Upon encountering the glass rod the DAW fronts split and propagated around the obstacle. As the waves traveled past the rod s void they bent and di racted. Downstream of the obstacle secondary DAWs formed. An experimental image of DAWs di raction about the glass rod is shown in Fig There are three prominent features of interest: (1) the general di racting pattern of the DAWs, (2) the secondary waves, and (3) the topological wave defects, which will be examined in Ch The di raction pattern of DAWs about a cylinder clearly shows a transition from semi-planar waves to strongly curved cylindrical waves to planar waves as the DAWs propagate from before the rod to around the rod to past the rod. The general di raction pattern of DAWs can compared to that of sound waves since both sets of waves obey identical fluid equations for small amplitude perturbations, demonstrated in Eq. 2.4 and 2.5. The analogous acoustic system is in the regime where the wavelength is comparable to the radius of the di racting rod, a/2, and neither the source nor the observation region are far from the object. Here, a is the radius of the rod. Additionally, the rod can be treated as acoustically hard, justified by the sharpness of the dust void. In this acoustic regime, an analytical solution of acoustic plane waves scattering o a cylinder has been derived by Morse [38]. Beginning with the initial pressure contours for the source waves (planer acoustic waves expanded as a sum of cylindrical waves) and the reflected waves (outward cylindrical waves from the rod), Morse matched the boundary conditions at the rod, v source = v receiver =0,andsolvedfortheresultingpressurecontour. Theresulting pressure contour was evaluated for the parameter Ka =2 a/,whichcorresponds

42 20 (a) Contours of acoustic pressure (b) 2 DAW intensity Dimensions in Wavelengths Figure 2.4: Di raction of pressure waves by a cylinder. (a) Two-dimensional acoustic pressure contours for di raction by a cylinder. (b) Scaled image of dust acoustic wave di raction by a cylinder. The di racting cylinder located downstream at (0,0). x and yareinunitsofwavelength. =1 and a=2. to the desired acoustic system = a/2. The resulting pressure field P (r,, t) is plotted in Fig. 2.4(a). A corresponding image of DAW di raction is shown alongside in Fig. 2.4(b). The similarities between the analytical solution for scattering acoustic waves and the observed experimental DAW di raction are immediate. While Morse begins with plane waves and the experimental DAWs are semi-planar, the scattered DAWs show the same bending and di raction of acoustic waves. The most notable di erence between the theoretical pressure contours and the DAW di raction observations are the secondary waves that formed behind the rod in the dusty plasma. From the recorded images I calculated the wave spectra for the primary and secondary waves for regions of the dust cloud at equal distances from the anode. The phase velocity for the primary waves and secondary waves were 17 mm/s and 10 mm/s, respectively. Since the dust density of the two

43 21 regions is approximately equal, the di erence in phase velocities suggests the secondary wave is either the result of a wave-wave interaction between the upper and lower primary waves or due to an additional dusty plasma phenomena. The geometry of the dust suspension was dependent on the distance between the anode and the rod. By increasing and decreasing the distance between the anode and rod (the geometry of the dust suspension) the angle of incidence between the upper and lower primary waves could be varied. The geometry dependence of the primary wave-wave interaction allowed me to investigate the relationship between the primary and secondary waves. From a series of experiments over a range of primary incidence angels, the phase velocity of the secondary waves was determined to be independent of the primary wave-wave interaction angle. An important di erence to note between DAWs and acoustic waves lies in the respective energy source of the two waves. For DAWs, there is an extended background energy source available in the ion drift, where ordinary sound waves are produced by a localized energy source. 2.4 Topological Defects Topological defects in solids and waves are characterized by a dislocation of the lattice structure and wavefronts and are a nonlinear phenomena. In my experiments, topological wave defects, or dislocations, were observed concurrently with di raction. Analogous lattice structure dislocations have been studied in strongly coupled dusty plasmas [39]. In these dusty plasma suspensions, depending on the shear stress applied to the crystal structure, edge dislocations have been observed to propagate supersonically [40, 41]. In moderately coupled dusty plasmas, DA wave defects, while ubiquitous, have only recently garnered attention [42]. By introducing external potentials in front of the anode, DA wave dislocations can be investigated in a reproducible and controllable setting. The observed dislocations took the form of bifurcated wavefronts (edge dislocations), an example is circled in Fig It has been

44 22 Figure 2.5: Topological wave defects due to scattering by a rod. The bifurcation of wavefronts are circled in red and the spatial defect line is denoted by the dashed yellow line. Images (a)-(f) are taken at 0.01 sec intervals. The bifurcated defects propagate up the dust acoustic wavefronts. demonstrated that topological DA wave defects can be a demarcation of frequency boundaries [42,43]. In addition to being frequency discontinuities, topological defects are characterized by an abrupt change in phase [43]. Correspondingly, when topological defects are present there is no conservation of wavenumber. The observed bifurcated wavefronts were typically unstable and reverted back to unbroken wavefronts. I will present three instances of DA wave defects obtained from a series of DAW scattering experiments along with the subsequent wave-defect evolutions and decays. The experiments serve as a preliminary qualitative investigation and progress from ordered to disordered DAW systems. The three processes that were observed to transform bifurcated waves into plane waves are a zipper-like process (where the defect climbs along the wavefront reducing the bifurcated wave into a single wave, Fig. 2.5), a recombination process (where one of the bifurcations splits from the wave to form an independent wavefront, Fig. 2.6), and a wave breaking process (where the bifurcated wavefront breaks up into smaller segments and decays, Fig. 2.7). The first and most ordered of the observed wave dislocations developed in DAWs

45 23 di racted by a rod. These topological defects were observed forming as a forced wave-defect, mediated by the obstacle s potential. As the DAWs bent around the rod, the portions of the wavefront closest to the rod merged. Once formed, the defect propagated along the wavefronts, joining the two wavefronts into one. By tracking the dislocation point through the video sequence I was able to establish a spatial line where wavefronts joined and became bifurcated, marked in Fig. 2.5(a). The spatial line is evidence of a frequency discontinuity, supported by the frequency on both sides of the topological defect. Upstream of the spatial defect line, DAWs had a frequency of 49.6 ± 0.5 Hz and after the defect boundary the frequency dropped to 26 ± 1 Hz. In addition to the frequency discontinuity, there was a discontinuity in k/ k across the spatial defect-line. Downstream of the wave defects, the direction of wave propagation shifted downward, toward the rod s wake. In the next examples, where the systems become more disordered, defects were not always simply frequency discontinuities. It is well known that topological defects can lead to disorder and mediate turbulence in asystem[44]. By replacing the rod with a double slit, additional wave-defects were investigated. The double slit was constructed out of 2 mm thick aluminum sheets coated in insulating black paint. The center was 6.5 mm wide with 5.5 mm silts above and below. The obstacle was left floating and inserted so that the plates making up the double slit were parallel to and 8 mm from the face of the anode. The experiment was conducted as described in Chapter 2.2. As the semi-planar DAWs collided with the double slit they are compressed through the gaps, exiting as a pair of o set cylindrical waves. As the cylindrical waves exited the slits, wavefronts from the two line sources coalesced with each other and formed an array of topological wave defects. Downstream from a planar obstacle, the wave defects resulted from wave-wave interactions and were not a forced wave defect in an axially asymmetric potential (seen in the previous experiment with the rod). As the wave defects evolved in time they decayed through wave-wave

46 24 Figure 2.6: Double slit di raction and wave-defect evolution. The wave-defects are circled in red. In (a) and (f) the waves that make up final wavefronts are colored for clarity. Images (a)-(f) are taken at sec intervals. The green wavefronts in (a) break o of the blue and yellow wavefronts and join together (f). interactions. When defects aligned with neighboring defects, the dislocation broke and the detached wavefronts joined, forming non-bifurcated waves. An example is shown in the montage in Fig When DAWs became more disordered, a third type of wave defect decay was observed. Disordered DAWs were mitigated by introducing a glass plate in front of the dust suspension. The glass plate was attached to a rod so that it could be rotated, changing the angle of incidence of the density waves. The angled glass, 35 degrees o of the horizontal in the presented case, created an artificial boundary and caused purely forced wave defects, Fig. 2.7(a). These wave defects were turbulent in nature and bifurcated. The forced wave dislocation was observed to decay through a perpendicular breakup of the bifurcated wavefront. Dust density fluctuations along the wavefront appeared to seed the wave decay. The perpendicular wavefront breaking

47 25 glass plate Figure 2.7: Wavefront breaking of bifurcated waves due to a wave scattering from a glass sheet. Images (a)-(f) are taken at sec intervals. The topological defects are circled. The glass plate is labeled in (a). Density striations perpendicular to k are visible in the wavefronts in (a). As the waves evolve with time the density depletions between the striations grow (b)-(e) until the bifurcated wave vanishes (f). is reminiscent of turbulent cascade where large scaled structures break into smaller scaled structures. This third set of wave defects completes the illustration of ordered to turbulent DAW defects. The cascade in length scale and the turbulent breaking of wavefronts can also be seen in nonlinear high amplitude DAWs, which will be presented Chapter Summary The e ect of various objects on dusty plasmas was used to study DAW di raction, scattering, and topological wave defects. Dust acoustic wave di raction by a rod was observed and compared to sound wave di raction in a neutral gas. As DAWs di racted around the rod, two sets of features absent in acoustic di raction were observed, a secondary set of waves downstream of the rod and topological wave

48 26 defects. The secondary waves were determined to be independent of the angle of incidence between the upper and lower primary waves. The topological defects were determined to be forced wave defects and marked a discontinuity in frequency and wavenumber. This set of topological defects decayed through a zipper-like process. Two additional topological defects were studied. The first was due to DAW di raction through a double slit. These topological defects formed and decayed through wavewave interactions downstream of the double slit. The final topological defect was more turbulent and formed by a glass plate in front of the DAWs. These forced wave defects were observed to decay through a breaking of wavefronts. The three cases of topological defects due to the introduction of a potential structure help develop a categorical understanding of how topological wave-defects can develop and evolve and show a general trend from an ordered to turbulent nonlinear system. While speculative in nature, the topological wave defect experiments serve as a preliminary investigation to classify and qualify a common nonlinear DAW phenomenon.

49 27 CHAPTER 3 NONLINEAR DUST ACOUSTIC WAVES 3.1 Dust Acoustic Shock Waves, Solitons, and Other Nonlinear Waves Dusty plasmas are opportune environments to study nonlinear plasma waves. The evolution of nonlinear waves in plasmas was first modeled for ion acoustic waves in 1967 with a two fluid theory [26]. Here Montgomery demonstrated the connection between linear sound waves in a neutral gas and ion acoustic waves in a plasma. This connection led Montgomery to apply nonlinear Riemann equations used for acoustic waves to plasma ion acoustic waves. For dusty plasmas, Rao, Shukla, and Yu predicted the existence of nonlinear DAWs, specifically DA solitons, in 1990 [18]. Nonlinear DAWs were further investigated theoretically by Eliasson and Shukla, who, using a simple fluid model, numerically solved for the evolution of dust density pulses into DA shock waves (DASWs) [45]. When dust charge variations are included, the additional dissipation of wave energy can result in stable shock fronts [46, 47]. Similarly, Mamun and Cairns showed that stable shock fronts can form in strongly coupled plasmas due to dissipation from particle correlations [48]. Dissipation, from neutral drag, particle correlation, variable charge, etc., can prevent DAWs from evolving into DA solitons. A variety of driven nonlinear DAWs and DASWs have been investigated experimentally [49 51]. These experiments used a range of methods to investigate artificially generated nonlinear DAWs and shocks. Nonlinear DAWs and DASWs can be, however, self-exciting. For the liquid phase of dusty plasmas, 1, nonlinear growth and wave-particle dynamics have been observed [52]. Here, Teng et al. used an rf discharge to obtain a many layered dusty plasma crystal. By increasing the ion flow, Teng et al. melted the crystal and transitioned the dusty plasma from a strongly coupled to a moderately coupled, spontaneously exciting DAWs in the process. These DAWs became nonlinear as the ion current, a free energy source for DAWs, was increased.

50 28 In a similar rf discharge, Flanagan et al. observed the onset of linear DAWs and the transition to nonlinear DAWs by reducing neutral gas drag damping [53]. In addition to developing into shocks, high amplitude DAWs may trap dust particles [52]. Teng et al. observed that as the DAW amplitude increased the oscillations of the background dust also increased. When these oscillations are large enough, dust grains propagated with the wave. The trapped dust particles traveled some distance with the density perturbations before they were released and drifted back to an equilibrium position. I will show that grain trapping and high amplitude oscillations may result in additional nonlinear wave phenomena. 3.2 Experimental Design and Apparatus I began with an unobstructed dust cloud in the initial experimental setup described in Ch. 2.2 and shown in Fig The plasma parameters are identical to those listed in Table 2.1. For these parameters, the DA speed is C da mm/s, depending on the dust temperature. The experimental diagnostics remained primarily optical; I relied on a 532 nm 2 mm thick laser sheet for dust illumination and a high-speed camera to capture DAW dynamics. Under a discharge current of 10 ma with an 8 mt axial magnetic field (n e m 3 ), DAWs were excited. With a stable dust cloud supporting self-excited DAWs (n d m 3 ), I introduced a single slit in front of the anode for the DAWs to pass through, similar in design to the double slit used to observe di raction and topological defects discussed in Chapter 2. The slit was constructed from two aluminum sheets coated with nonconducting black paint to minimize scattered laser light. The slit width was 1 cm with a length much longer than the diameter of the anode. Note: The e ective slit width was smaller than 1 cm due to the sheath and dust void that developed. The gross geometry and cloud size was modified by changing the distance between the slit and the anode, allowing us to study two systems of nonlinear waves, Fig. 3.1.

51 29 Figure 3.1: The e ect of a single slit on dust acoustic waves. (a) Unobstructed dust acoustic waves, (b) slit located 11.5 mm from the anode, (c) slit located 15.0 mm from the anode. A pair of high and low amplitude dust acoustic waves are marked by a bracket in (b), with the high amplitude wave to the left and the low amplitude wave to the right. A dust acoustic shock wave is indicated in (c). The electric potential of the single slit created a shock-tube like environment, a high density system traveling into a lower density system. The shock-tube environment is observed directly in the density profile of the dust cloud, Fig Incoming DAWs were compressed through the slit and, as the DAWs (dust cloud) passed through the slit, the cloud expanded and the dust density rarified. Systems that undergo similar pressure and density changes are typically unstable to nonlinear wave growth. To study the evolution and development of nonlinear wavefronts, I developed a wave tracking routine. The routine tracks the wave peaks and the leading and trailing wave troughs of a discreet set of waves through a series of video frames and allowed for quick direct measurement of wave speed, amplitude, and shock width. 3.3 Observation of Dust Acoustic Shock Waves Two systems of nonlinear DAWs were observed with the single slit, coalescing DAWs and DAWs that evolved into fully developed DASWs. In the first system, where the slit was closer to the anode, the slit s shock-tube like potential created both high and low amplitude DAWs, Fig. 3.1(b). The high and low amplitude DAWs traveled at 100 mm/s and 65 mm/s, respectively. When the high amplitude waves collided with their slower counterpart, the two waves coalesced and formed a single wavefront.

52 30 (a) Intensity Profile (arb. units) (b) Plates creating single slit Anode Distance from Anode 0 Figure 3.2: The average dust density with a single slit. The shock-tube like system is apparent in the intensity profile of the single-slit configuration, (a). The averaged dust cloud is pictured in (b). The region the dust density profile was taken over is marked in red. Large gradients in the intensity profile correspond to large gradients in the dust density. The coalesced wavefront propagated at the speed of the high amplitude wave and developed into a DASW. The coalescing waves peak positions and amplitudes, taken with respect to time, are plotted in Fig Wave confluence is a trait of nonlinear waves [54]. (Solitons, while a nonlinear wave, do not coalesce.) Additionally, the lack of increase in wave amplitude between the fast wave and the coalesced wave indicates the fast wave was already saturated and nonlinear. The systematic nature of these observations show the inherent nonlinear nature of DAWs. To better observe DASWs, the slit was moved farther from the anode. When the electrically insulated slit was placed farther from the anode, DAWs were compressed in passing through the shock-tube like throat potential, Fig. 3.1(c). Measuring the temporal wave amplitudes of multiple DAWs at a range of distances (beginning approximately 1.1 cm from the slit), nonlinear DAWs were observed to evolve from Gaussian density profiles into fully developed shocks, Fig The observed wave amplitudes and waveform evolutions were very regular. The initial non-sinusoidal wave profiles are an indication of nonlinearity. As the initial high

53 31 Figure 3.3: Confluence of high and low amplitude dust acoustic waves. The wave peaks locations are plotted with respect to time. The fast (high amplitude) wave and the slow (low amplitude) wave are indicated with the blue and red lines, respectively. The coalesced wave is marked in green. Also plotted are the amplitudes (in arbitrary units) of the waves. The plots clearly show the coalesced wave traveling at the fast wave s speed as well as no significant amplitude di erence between the fast and coalesced waves. amplitude DAWs propagated, the leading edges steepened and the trailing edges elongated. Once fully developed DASWs, the wavefronts spread and the DASWs dissipated as they propagated through decreasing dust density. The shock width, defined in Fig. 3.5, and amplitude are plotted with respect to their propagation distance in Fig The wave amplitude of a cylindrical wave in a non-dispersive medium falls o as distance 1/2. A much steeper fall o is observed. The steeper observed fall o, approximately 4 times greater than what would be expected in a non-dispersive medium, can be attributed to dissipation. 3.4 Comparison with Theoretical Work The evolution of the observed DASWs can be compared to a numerical solution for nonlinear non-stationary DAW evolution that was derived by Eliasson and Shukla [45]. In their calculation, Eliasson and Shukla showed that DASWs can develop with time

54 Figure 3.4: The observed temporal dust acoustic wave density profiles. Wave amplitudes were taken 2.68, 2.82, 2.95, and 3.10 cm from the anode over a 0.16 s period. The image intensity saturated for the wave peaks 2.68 cm from the anode, creating a false flattening of the wave crests. The dust acoustic waves develop into shocks as they propagate away from the anode. Two wavefronts are identified and connected between observation locations with a dashed line. Taken over multiple wave cycles to show regularity of phenomena. 32

55 33 Figure 3.5: Shock amplitude and width,,of a single dust acoustic shock wave plotted with respect to position. due to a self-steepening potential. Beginning with Boltzmann distributions for ions and electrons (n ( )=n o exp(q /T ), charge neutrality, and Poisson s equation (Eq. 1.8), the normalized dust density, N d = n d /n d0,canbereexpressed: N d =[ exp( ') exp(')]/( 1), (3.1) where = n i0 /n e0, = T e /T i,and' = e /T e. Rewriting the fluid equations for dust particles (Eq. 1.4 and 1.5) in terms of N d, Eliasson and Shukla derived a set of equations for the evolution of a DAW s potential =0, =0. (3.3)

56 34 Here +(') = + p [ exp( ') exp(')]/[ exp( ')+exp(')] and,thedust fluid velocity, is normalized by C da.fromeqs.3.2and3.3,aninitialdustdensity wave pulse at a specified width and amplitude can be evolved with time. Solving these equations, I evolved a nonlinear wave (closely resembling the initial observed wave profile) under the experimental parameters, plotted in Fig. 3.6(a). To compare with my experimental observations, the spatial wave profile evolution of a single observed DAW is shown in Fig. 3.6(b). The observed DASW profiles were converted from intensity profiles to normalized dust density profiles, N d,bytakingtheaveragedintensityas the unperturbed dust density, n d0.thenumericalwaveformevolutionisaclosefitto the observed waveform evolution. Key di erences between the model and experiment are due to simplifications and limitations of the numerical approximation. Eliasson and Shukla s model contains no damping mechanism which results in a continuous steepening of the shock front. The model evolves a single density pulse rather than awavepacket. Lastly,thenumericalsolutionrequiresinitialbackgrounddensities of N d =1 rather than the observed preceding and trailing wave trough densities of N d 0.5. For the larger evolution times, the numerical solutions began to break down. In these cases, to obtain more meaningful results, the solutions were fitted with a cubic spline. In spite of the limitations, the model captures the propagation speed and the overall wave profile evolution, both spatially and temporally. Shock waves are characterized by their width. As the shocks propagated and evolved they stabilized to a minimum shock width of mm, Fig. 3.5, which is roughly the inter-particle spacing, r s =0.3to0.6mm. Theobservedshockthickness can be compared to the thickness of gas-dynamic shocks, which is approximately the mean free path of atom-atom collisions. In DASWs, the relevant mean free path is determined by the dust-neutral collisions, which is dust temperature dependent. Taking the dust temperature from previous work (where it was estimated by fitting the DAW dispersion relation to the observed DAW phase velocity [32, 33]), the mean

57 Figure 3.6: Evolution of dust acoustic shock waves. (a) Numerical solution for the evolution of non-dispersive dust acoustic shock waves. Calculated following Eliasson and Shukla s derivation [45].A cubic spine fitting of the solution is plotted for 30, 40, and 60 ms. (b) Shock wave development of a single observed normalized dust acoustic wave. The di erence in trough dust densities is due to limitations of the numerical model. 35

58 36 free path for dust-neutral collisions is mm. A better fitting theoretical shock width was derived by Mamum and Cairns. Mamum and Cairns showed that inter-particle correlation in strongly coupled dusty plasmas can provide a source of dissipation for DASWs and predicted a shock thickness of = d /v s,[48],or the kinetic shear viscosity over the shock velocity. Using a normalized viscosity of? =1.04, given by Kaw and Sen for moderately coupled dusty plasma [55], results in a kinematic shear viscosity of d 23 mm 2 /s for the experimental parameters. This gives a predicted shock thickness of 0.3 mmforashockspeedof74mm/s, corresponding to the experimental observations. Finally, Asgari et al., afterincluding variable dust charge and finite dust temperature, expanded the DAW fluid equations into a nonlinear Burgers equation [56]. By including dissipation into the nonlinear DAW equations, Asgari et al. were able to accurately predict the shock width of the observed self-excited DASWs [32, 56]. 3.5 Backward Drifting Nonlinear Dust Acoustic Waves In addition to developing into DASWs, high amplitude DAWs were observed to instigate a fine structured backward propagating dust density wave in an unobstructed experimental configuration. By increasing the spatial resolution per pixel to mm/pixel and recording at 500 fps, I investigated this nonlinear wave process. The observed process can be summarized as follows: As high amplitude DAWs propagated (N d & 2, shown in Fig. 3.7(a)), some of the dust streamed backward from the trailing edge of the waves. While the dust drifted back towards the anode, small amplitude fine structured density waves formed. These backward drifting density waves propagated until colliding with incoming high amplitude DAWs. The fine scale restoration waves were broken perpendicular to the wavefronts Fig. 3.7(b). In addition to the perpendicular wave breaking, the fine structure mode contributed directly to the perpendicular wavefront breaking of the primary waves when the two waves collided,

59 37 similar to that seen in Fig The reverse drift mode had a lab frame phase velocity 0 mm/s. From the lab frame phase velocity, the dust drift velocity was estimated, u rd C da.theacousticnatureofthereversedriftwavescanbeseeninthewave spectrum, calculated as described in Ch. 2.2 and plotted in Fig A simple wave process, based on observations of dust particle dynamics in high amplitude DAW by Teng et al. [52], was developed to explain the observations. If the amplitude of DAWs is large enough (N d 2), the dust oscillation distance is on the order of the wavelength and the dust particles are no longer localized. As the dust particles oscillate backward to their initial positions and if the drift density is large enough, secondary DAWs may be excited. (For DAWs excited in a drifting dusty plasma see Chapter 4.) Additionally, the dust trapped in high amplitude DAWs can contribute to backward dust drift when it is released. Typically, the oscillating and trapped dust speed is on the order of the DA speed, u rd C da.fromthisdrift speed, the secondary DAW s lab frame velocity is expected to be 0mm/s,which was observed. 3.6 Summary By creating a shock-tube like potential structure with a single slit for DAWs to propagate through, I observed self-excited nonlinear DAWs. Two nonlinear wave phenomena were observed by changing the distance between the slit and the anode, coalescing waves and shock waves. The coalescing waves consisted of high and low amplitude wave pairs and formed when the anode was closer to the slit. The high amplitude waves propagated at much high wave speeds compared to the proceeding low amplitude waves. As the high amplitude waves passed the low amplitude waves, the waves coalesced and propagated as a single wavefront. The second nonlinear wave phenomena was the development of DASWs and were best observed when the anode was farther from the slit. The development of the DASWs was characterized and

60 38 (a) (b) 1 cm Figure 3.7: Fine structure in high amplitude dust acoustic waves. The density profile of the high amplitude waves is shown in (a) in units of Nd = nd /nd0. A sample image of high amplitude dust acoustic waves with the reverse drift mode is shown in (b). The non-propagating waves are circled in yellow. The wave spectrum in Fig. 3.8 was sampled across the white rectangle. The region corresponding to the density profile in (a) is outlined in red Acoustic mode Reverse drift mode 100 I2/ f k frequency f (Hz) 400 Acoustic mode wave number k (mm-1) Figure 3.8: The wave spectrum of fine structure in high amplitude dust acoustic waves over the highlighted region in Fig. 3.7(b). The acoustic and fine structure modes are labeled. The wave spectrum was sampled over 1024 frames recorded at 500 frames per second.

61 39 compared to Eliasson and Shukla s theoretical model for DASW evolution. Next, the shock width was compared to theoretical predictions. A third nonlinear wave process was observed in an unobstructed configuration. By examining high amplitude DAWs, a secondary backward dust density wave mode was observed. The wave mode was examined in light of Teng et al. s observations of dust dynamics in high amplitude DAWs. I proposed that the secondary mode may be the result of the restoring dust drift that occurs after large dust oscillations in high amplitude DAWs.

62 40 CHAPTER 4 TEMPORAL DUST ACOUSTIC WAVE GROWTH 4.1 Wave Growth Dust acoustic wave growth occurs when the free energy source, typically a net ionflow with respect to the dust, is large enough to overcome the inherent damping found in dusty plasma suspension, which can be due to inter-grain correlation, dust-neutral collisions, etc. [20]. The growth rate of a wave is computed from its dispersion relation, which details the dependence of the growth rate and frequency on the wavenumber. The DAW dispersion relation has been derived using both fluid theory [14, 46, 57] and kinetic theory [21, 58, 59] to various degrees of complexity. Experimental investigation of the real part of the DAW dispersion relation has been conducted by numerous groups [31,33,34,60 62]. In these experiments, DAWs were driven by a low frequency oscillating current modulated through a range of frequencies. The resulting wave numbers were measured, and the wave numbers and frequencies were constructed into a wave-spectrum that could be compared to theoretical dispersion relations. These frequency synchronization experiments dealt directly with the real part of the dispersion relation and did not investigate the imaginary part, the growth rate. The growth rate has been experimentally measured through various techniques [31, 37, 63]. In these experiments spatial DAW growth was measured in dust clouds already experiencing dust density fluctuations from previously excited DAWs. Recently, by transitioning a plasma from an over-damped to an underdamped dusty plasma suspension for DAWs, DAW growth was observed at the onset of the DAW instability by Flanagan and Goree [64]. Here, in an rf discharge device with dust confined into a three dimensional cloud, Flanagan and Goree excited and measured the growth of DAWs as the neutral pressure (dust-neutral collision rate) was reduced, verifying damping of DAWs by dust-neutral collisions in the process. In

63 41 their experiment the DAWs were strongly damped (with neutral argon gas pressures above 400 mtorr), the dust experienced strong coupling e ects, and the spatial growth rates were measured in a dust suspension that supported about 3 wavelengths. Previous experiments investigating fundamental DAW phenomena have used external potentials to manipulate dusty plasmas [22, 32]. When a probe (object) is introduced into a dusty plasma it develops a sheath and the probe s electric potential creates a dust void [29, 30]. In the present work, I developed a technique, using a similar induced potential created with a biased mesh, to trap and release a secondary dust cloud, far from the primary dust cloud. When released, the secondary dust cloud drifted toward the anode. When the cloud was a certain distance from the anode, DAWs became excited. A linear growth phase followed by wave amplitude saturation was observed. The measured growth rates of the excited waves were compared with fluid and kinetic models. The new experimental technique of forming a drifting dusty plasmas allowed me to follow the growth of DAWs from nearly thermal fluctuations. Also, this technique allowed me to perform temporal growth measurements of weakly damped DAWs in a moderately coupled quiescent dusty plasma of su cient size to support many wavelengths. 4.2 Experimental Design and Apparatus The experiment was conducted in the described anodic discharge apparatus, modified to include a 12 cm diameter circular mesh shown in Fig. 4.1(a). A plasma was established under a 5-7 ma (300 V) discharge current between the 3.2 cm diameter anode and the vacuum chamber wall with a 4 mt axial magnetic field (for electron confinement) and an argon gas pressure of 150 mtorr (20 Pa). An axial discharge electric field of 200 V/m, measured with an emissive probe in the absence of dust, produced a net ion-flow. Dust particles, located below the anode on an electrically floating tray, became charged, lifted, and incorporated into the anode glow. The

64 42 (a) top view side view sheet laser CMOS camera g mesh plasma B-field anode electrically floating dust tray (b) primary cloud biased mesh secondary cloud 1 cm z x anode Figure 4.1: Experimental apparatus designed to trap a secondary dust cloud. (a) Schematics of the experimental apparatus with a 12 cm diameter mesh. The inter-wire spacing of the mesh allowed laser light to pass through unobstructed. The mesh was designed with a variable bias with respect to the chamber, permitting for a secondary dust cloud to be trapped and later ejected when the bias was removed. The distance between the mesh and the anode was adjustable. (b) Image of the dusty plasma suspension with the biased mesh and a trapped secondary dust cloud. The primary and secondary clouds as well the anode, mesh, and coordinate system are labeled. Here the mesh is 15.5 cmfromtheanode.

65 43 experiment was repeated for two species of dust, spherical iron and monodisperse spherical silica powder. The iron dust had a size range of r d 0.5-5µm andthesilica dust had a radius of r d 0.5 µm. Once a su ciently dense dust cloud (n d m 3 )wascollected,acircularmesh(12cmindiameter)wasmovedintotheplasma, 14 to 15 cm from the anode, shown in Fig. 4.1(a). The mesh had an inter-wire space of 0.87 mm. Once the mesh was in place, a -50 V bias was applied between the mesh and the chamber wall, creating a non-monotonic electric field. The bias produced a plasma glow around the mesh and the resulting potential configuration trapped dust grains, creating a secondary dust cloud adjacent to the anode side of the mesh, shown in Fig. 4.1(b). The axial electric potential profiles of the experimental apparatus with a biased mesh and a floating mesh are shown in Fig The positive potential well created by the biased mesh is not as deep as the potential well created by the anode. Consequentially, dust trapped in the secondary cloud can be expected to be colder than dust trapped in the primary cloud. The topology and density of the secondary cloud was controlled by adjusting the anode discharge current, the potential applied to the mesh, and the distance between the mesh and the anode. When the bias was released from the mesh (i.e. the mesh was allowed to return to its floating potential), the electric field became monotonic again at 210 V/m (in the region where the dust drifted) and the secondary dust cloud drifted to the positively biased anode. Dust acoustic waves spontaneously appeared in the drifting dust cloud. By increasing the distance between the anode and the mesh the streaming dust channel could be transformed from a large/wide channel to a narrow channel or jet. For longer distances the dust had larger drift velocities, well above the DA phase velocity, C da, with Mach numbers (with respect to the primary dust cloud) above 2. The optical diagnostics were similar to Chapters 2.2 and 3.2. The dust particles were illuminated with a 2 mm wide 532 nm laser sheet at 300 mw and the dust dynamics were recorded using a lens filter, to eliminate background light, at 250

66 44 Figure 4.2: Floating potential taken with an emissive probe in the absence of dust. The mesh was located 14.2 cm from the anode. The potential well created by the biased mesh that traps the secondary dust cloud was located from 9 to 13 cm from the anode. frames per second with a Photron (FASTCAM 1024 PCI) CMOS camera, which has alinearresponsetolightintensity.theimagesizeswere1024by1024pixels,giving pixel resolution between 1.8 and 2.2 mm/pixel. The laser light scattered by the mesh was subtracted from each frame. Image slices parallel to k (the direction of wave propagation, in this case z) weretakenfromthecapturedframesandconvertedinto an intensity array, (z j,i zj ), where z j is the distance from the anode and I zj is the corresponding image intensity in frame number j. Theestablishedcoordinatesystem is shown in Fig. 4.1(b) (x is parallel to the laser sheet). The camera s linear response to light intensity allows for pixel intensities to be transformed into dust densities, I / n d,sothattheintensityarrayscouldbeconvertedintoarraysofnormalizeddust density N d 1, N d 1=(I I ave )/I ave =(n d n d0 )/n d0. Once the images were formatted, a tracking routine was used to obtain the bulk motion of the secondary

67 45 dust cloud and the wave speed and amplitude of the spontaneously excited DAWs. While individual particle tracking in the described experiment is impossible due to the pixel resolution constraints, an average dust velocity was obtained by tracking the mean streaming dust cloud position. The plasma density and electron temperature were measured in the absence of dust (probes disturb dusty plasmas [29]) with a double Langmuir probe axially from the anode. In the region where DAW growth was observed n i ranged from (2 4) m 3 with an electron temperature T e 2.5 ev. The ion temperature is estimated as the neutral gas temperature, T i T n ev. The experiment with iron particles was conducted with a 7 ma discharge and the experiment with silica dust was conducted with a 5 ma discharge. With both iron and silica particles, DAW growth was observed in regions with similar plasma density, the di erence in discharge current was o set by the di erence in distance from the anode where wave growth was observed. The axial plasma density profiles for the 5 ma and 7 ma discharge currents are shown in Fig Estimating the dust density is more challenging and is described in detail in Ch Briefly, the dust density is calculated using image analysis and relies on the camera having a linear response to light intensity. Using this technique the observed streaming dust had densities n d ( ) m 3. Dust particles may be heated by thermal electric field fluctuations in the background plasma and via DAWs driven by an ion-flow instability [65 67]. I expect dust trapped in the potential well of the mesh to have a much lower kinetic energy than dust trapped in the primary cloud near the anode due to the weaker confining potential and the absence of DAWs (i.e. an ion-flow). The absence of DAWs leaves only thermal electric field fluctuations as a potential heating source. From a thermodynamics calculation, Avinash et al. estimate a dust temperature in a quiescent dusty plasma of T d (1 + Z 2 dn d /2n i )T i, (4.1)

68 46 Figure 4.3: Axial plasma density in the absence of dust for 5 and 7 ma discharge currents, corresponding to the silica and iron dust experiments, respectively. The plasma density falls o exponentially from the anode. giving the streaming dust a temperature 5 10 ev [67]. For my theoretical comparison, I took T d = 5 ev. While wavelengths and frequency of DAWs can be taken directly from the video recordings, space-time plots allow for more e cient measurements and give a better description of wave dynamics. Space-time plots map spatial variations with respect to time. A slice, similar to the one selected for calculating the wave spectrum, is taken across the entire cloud. By plotting the spatial slices sequentially, temporal dust density dynamics across the spatial slice can be clearly identified, such as DAW propagation. 4.3 Observations and Discussion While the mesh was biased, the secondary dust cloud was stationary, stable, and free of DAWs. When the bias voltage was removed, and the mesh was floating, the secondary dust cloud began drifting toward the anode. Once the secondary dust

69 47 (a) anode (d) mesh (b) streaming cloud primary cloud (e) (c) (f) x z Distance from Anode (cm) Distance from Anode (cm) Figure 4.4: Images of the drifting dust cloud with spontaneously excited dust acoustic waves taken at 0.08 second intervals. Taking t = 0sfromthefirstobservabletraces of dust acoustic waves, the streaming dust cloud is shown in (a) at t = 0.05 s. Early dust acoustic wave growth is shown in (b) at t =0.03 s. Fully developed dust acoustic waves are shown in (c) at t =0.11 s. A curvature in the wavefronts of can be seen in (c) with a sample wavefront highlighted in yellow. Plots (d)-(f) show the corresponding dust density profiles taken across the dotted line in (a). The spatial slice in (a) is also the line that the space-time plot in Fig. 4.5(a) was taken over. Taken with iron dust.

70 48 cloud travelled a certain distance from the mesh, DAWs became excited. Images of the drifting cloud and the excitation of DAWs along with the corresponding dust density profiles are shown in Fig Since the dust cloud was drifting, the DAWs were Doppler shifted with respect to the lab-frame. Relative to the dust frame, a Doppler frequency shift is measured by the moving (lab) receiver. The ions drift away from the anode (DAWs naturally propagate along the ion flow direction, as they do in the primary cloud) while the secondary cloud drifts towards the anode. Evidence for a Doppler shift can been seen in the wavefront and dust density profiles; in the lab-frame there is a curvature of the wavefronts opposite to the direction of propagation, Fig. 4.4(c), as well as a steepening of the wave s trailing edge. Both of these traits are indicative of semi-planar DAWs propagating in the opposite direction, showing that the DAWs are strongly Doppler shifted. To better examine the drifting dust cloud, the one dimensional spatial path of the secondary cloud to the anode and the resulting DAWs was plotted in a space-time diagram. Sample space-time diagrams taken from typical experimental runs are shown in Fig. 4.5(a) and (b). The following description of a typical experimental run corresponds to the space-time diagram in Fig. 4.5(a). Initially, the dust cloud begins to expand and accelerate towards the anode from t 0to0.3s. Att 0.3 s the dust cloud has reached its terminal velocity. At t 0.5 s DAWs first appeared. As the drifting dust cloud approaches the anode it slows (along with the lab-frame phase speed of the drifting DAWs) until the drifting DAWs propagate with the primary DAWs, seen in Fig. 4.5(a) at t 1 s at a distance of z 5 cm. Once the secondary dust cloud stopped streaming, the primary dust cloud grew in size and density. Wave-wave interactions between the drifting and primary DAWs were observed. The nature of the wave-wave interactions depended on the drift speed of the secondary cloud. Lower drift speeds resulted in the smooth transition from drifting DAWs to non-drifting DAWs, seen in Fig. 4.5(a) from t 0.9 to2s. Larger

71 49 (a) mesh secondary cloud primary cloud ion-flow region of interest anode Time (secs) cm Mesh Anode 2.2 (b) Distance from Anode (cm) Time (secs) cm Mesh Anode Distance from Anode (cm) 2 0 Figure 4.5: Space-time diagram of the spontaneous excitation of dust acoustic waves in a streaming dust cloud. The primary dust cloud is towards the right and the secondary dust cloud is seen towards the top left of the image (note the absence of waves). The anode is the line on the right of the image and the circular mesh is on the left. The region of wave growth is marked and the direction of ion-flow is indicated. The time is taken from when the bias was removed from the mesh. The spatial slice that the space-time plot in (a) was taken over is indicated in Fig. 4.4(a). Average dust cloud drift speeds for (a) and (b) from 0 to 0.5 s are 6.1 and 9.8 cm/s, respectively. The smooth transition between drifting DAWs and non-drifting DAWs seen in (a) from 0.9 to 2 s is due to lower dust drift speed. The wave collisions seen in (b) from 0.5 to 1.4 s are due to a larger dust drift speed. Taken with iron dust.

72 50 drift speeds resulted in wave-wave collisions between the drifting and non-drifting DAWs, seen in Fig. 4.5(b) from t 0.5 to1.4s. As the drifting DAWs were excited their density perturbations were seen growing in time. Converting slices of the video frames into averaged density arrays (z j,n dzj ), plotted in Fig. 4.6(b), the amplitudes for the growing waves, N d,weremeasuredfrom trough to peak through time. From these amplitude evolutions the growth rate was measured directly by fitting an exponential of the form N d = A t exp(! i t) (where A t and! i are the fitting parameters) to the temporal amplitude growth. Parameter A t takes into account the initial amplitude as well as the time di erence between the initial wave growth and when measurements were taken. Examples of the observed amplitude growth along with the growth rate fittings are shown in Fig. 4.6(a) and (c). When silica dust was used, the observed growth rates fell between 20 to 30 s 1 with an average of 26 ± 4s 1. For iron dust the observed growth rates fell between 10 to 20 s 1 with an average of 17 ± 3s 1.Theexponentialfitsusedtocalculatethe growth rates typically had correlation coe cients, R, greater than The waves exhibited linear growth until the wave amplitudes began to saturate and nonlinear wave steepening was observed. The linear to nonlinear transition of the growing waves can be seen in the wave profiles of Fig. 4.6(b), between 0.10 and 0.16 s, and of Fig. 4.6(d), between 0.06 and 0.09 s, where the waves profiles steepen from sinusoidal to non-sinusoidal wave amplitudes. Nonlinear waves were observed after the growth rate saturated. The inertial streaming DAW frequency depends on the DAW drift velocity. The average secondary dust cloud velocity, u ds,wasusedforthedawdriftvelocity.this approximation was found to be self-consistent by comparing DAW speeds before and after the secondary dust cloud slowed and merged completely with the primary cloud (as u ds! 0, C da(lab) = u ds + C da! C da,wherec da is the DA speed in the dust frame). The method proved valuable since the lab-frame DA speed, for both the

73 51 (a) (b) Fe d d (c) SiO 2 (d) -1-1 Figure 4.6: Examples of the observed dust acoustic wave growth. In (a),(c) the amplitude measured between the peaks and troughs of single dust acoustic waves (boxed in (b) and (d)) are plotted vs. time for iron and silica dust, respectively. The portions of the observed amplitude growth used to calculate the growth rate are indicated and the exponential fits used to calculate the growth rates are given. Dust density profiles of several waves are shown in (b), (d), with time taken with respect to (a) and (c). The dust acoustic waves showed linear growth until the waves saturated at around t 0.16 s in (b) and t 0.08s in (d). Plots (a) and (b) corresponds to the experimental run shown in Fig. 4.4 and 4.5(a).

74 52 drifting and non-drifting waves, and the average secondary dust cloud velocity were easily observable. With the streaming drift velocity, the dust-frame frequency can be calculated, f lab = f dust (u ds + C da )/C da (note: lab = dust = ). The lab-frame frequency, wavelength, and wave speed were measured directly from the wave profiles and averaged over single experimental runs. The experimental values and uncertainties along with the inertial-frame frequencies are given in Tables 4.1 and 4.2 for the iron and silica dusts, respectively. The frequencies and velocities are taken with respect to the z-direction. Table 4.1: Experimental Results: Iron Dust Parameter Value* Method/Expression Measured f lab 11 ± 1.5 Hz Image analysis 2.7 ± 0.2 mm Imageanalysis u ds 6.1 ± 0.2 cm/s Image analysis C da(lab) 2.9 ± 0.3 cm/s Imageanalysis! i 17 ± 3s 1 Image analysis Computed f dust 12 ± 5 Hz f lab C da /(u ds + C da ) C da 3.2 ± 0.4 cm/s C da(lab) u ds *Taken in the z-direction Table 4.2: Experimental Results: Silica Dust Parameter Value* Method/Expression Measured f lab 24 ± 4 Hz Image analysis 3.0 ± 0.3 mm Imageanalysis u ds 10.8±0.3 cm/s Image analysis C da(lab) 7.1 ± 0.4 cm/s Imageanalysis! i 27 ± 4s 1 Image analysis Computed f dust 12 ± 4.3 Hz f lab C da /(u ds + C da ) C da 3.7 ± 0.3 cm/s C da(lab) u ds *Taken in the z-direction

75 Comparison with Fluid and Kinetic Models There are three proposed free energy sources for DAWs that are relevant to the observed wave growth. While DAWs are typically excited by an ion-drift with respect to the dust particles in laboratory dusty plasmas, D Angelo [68] and Shukla et al. [69] suggested two additional potential free energy sources, both related to the acceleration of dust particles by gravity. During the short period of my experiment when the dust acceleration and dust velocity is the greatest there were no visible DAWs present. D Angelo s model requires a threshold dust particle velocity for DAW excitation [68] and does not fit the experimental observations as DAWs are not observed when the streaming dust velocity is maximal (Fig. 4.5(a) at t =0.2 s,z =11cm). Asimilar argument applies to the Shukla et al. model [69] as the free energy source. There was no DAW growth until the dust cloud is within some minimum distance of the anode, where the ion-flow is stronger, as expected if the free energy source is from an ion-dust streaming instability. If the free energy source was due to accelerating dust grains the observed wave growth would be expected to occur sooner. Additionally since DAWs were completely absent in the trapped secondary cloud, I assume there was an insu cient ion-flow at large distances from the anode to excite DAWs. The experimentally measured growth rates were compared with theoretical values taken from both fluid and kinetic models. The models chosen include a static electric field, collisions with neutrals, and finite dust temperature. For the fluid model Merlino provided a dispersion relation [20] that has been used for comparison to experiments under similar conditions [20, 70]. The ions, electrons, and dust particles were treated as fluids with the continuity and momentum equations and the system of equations was closed with Poisson =0, (4.2)

76 n + k Q n E = n n m v, = e " 0 (n i n e Z d n d ). (4.4) Here is taken over the three components, the ions, electrons and dust. These equations include collisions with neutrals, electron and ion inertia, as well as zeroth order drifts, which account for the background electric field, v 0 = u 0 / E 0.Linearizing and solving Eqs , assuming all first-order quantities vary as e i(kx!t),yields the dispersion relation [20]: where 1 X! 2 p & =0, (4.5) & = ( + i n ) k 2 v 2 T. (4.6) Here =! ku 0,and in(en) = n n in(en) v Ti(Te) is the particle-neutral collision frequency. The dust-neutral collision frequency was taken from Liu et al. [71], dn = 8 p 2 m n n n r 2 d v Tn/3m d with =1.26. The ion-mobility and ion drift velocity were taken from Robertson et al. [72]. The electron drift velocity was taken as u e0 = ee 0 /m e en and the dust drift velocity was taken from the experimental data. The kinetic model used for comparison is a modified version of models derived by Rosenberg et al. [62] and Rosenberg [73]. Rosenberg et al. s model included finite dust temperature, collisions, and drifting Maxwellians for ions and electrons and has previously been used for comparison of DAWs in a similar experimental setup [62]. Briefly, starting with the linear dispersion relation: 1+ X =1+ X [1 + Z( )] (k 2 2 D ) p =0, (4.7) 1+(i n 2kvT )Z( )

77 55 where =(! ku 0 + i n )/ p 2kv T and Z( )istheplasmadispersionfunction, Rosenberg et al. [62] simplified Eq. (4.7) for the long wavelength regime where u e0 v Te, ku i0 > in!,!>kv Td, e 1, 2 i & 1, and dn!. Since Iexpect dn!, thelastapproximationisnotvalid. Instead,Itookthedust susceptibility, d, from early work by Rosenberg [73] for DAWs in collisional dusty plasmas where d 1, which is a reasonable approximation for my experiment for & 2mmwithT d < 10 ev. The resulting dispersion relation is: 1+ 1 k 2 2 De! 2 pi k 2 u 2 i0 + 2 in r 1 u i0 i exp k 2 2 Di 2 v Ti u 2 i0 2v 2 Ti! 2 pd A d =0, (4.8) where A d!(! + i dn ) i dn k 2 v 2 Td! + i dn. (4.9) The calculated fluid and kinetic dispersion relations are plotted with the experimental observations for iron and silica in Fig. 4.7 and 4.8, respectively, for both the real frequencies and growth rates along with experimental uncertainties. For the calculations the following parameters were used: n i = m 3, T d = 5 ev, r d =0.5 µm, n d =3 10 9,furtherdetailedinTable4.3.Forthecaseoftheirondust,therange of dust radii may significantly change the modeled dispersion relation. The models predict values for both the real frequencies and growth rates that agree well with the measured values. 4.5 Summary By introducing a mesh cathode that could be switched on and o, I were able to trap and release a secondary dust cloud. When released, the secondary cloud streamed towards the anode and primary cloud, exciting DAWs when within a certain distance of the anode where free energy in the ion-flow was su cient These streaming waves

78 56 Figure 4.7: Theoretical and observed frequencies and growth rates plotted vs wavelength for iron dust. The observed growth rate and frequency are shown with experimental uncertainty. Theoretical values are plotted for the detailed experimental parameters. Figure 4.8: Theoretical and observed frequencies and growth rates plotted vs wavelength for silica dust. The observed growth rate and frequency are shown with experimental uncertainty. Theoretical values are plotted for the detailed experimental parameters.

79 57 were Doppler shifted. Temporal DAW growth was observed and the growth rates were measured in a quiescent dust cloud large enough to support many wavelengths. The growth rates of DAWs in silica dust and iron dust were measured and compared to kinetic and fluid theories. The dispersion relations derived from both the fluid and kinetic models predicted real frequencies corresponding to maximum growth that agreed well with the observed frequencies. Growth rates obtained from the kinetic theory were in better agreement with the measured growth rates as compared to those obtained from the fluid theory.

80 58 Table 4.3: Experimental Parameters for Dust Acoustic Wave Growth Parameter Value Expression Remark Measured n i (2 4) m 3 Double probe (no dust), axial density shown Fig. 4.3 n d (3±2) 10 9 m 3 Image analysis r d iron µm Microscope* r d silica 0.5 ± 0.1 µm Microscope T e 2.5 ± 0.2 ev Double probe (no dust) E-field 210 ± 20 V/m Emissive probe (no dust) B-field 4 mt Magnetometer pressure 150 mtorr Bartron gauge, argon gas Known iron 7860 kg/m 3 Manufacture specifications silica 2000 kg/m 3 Manufacture specifications Assumed T n ev Room temperature T i ev Room temperature in m 2 en m 2 Computed n e m 3 n e = n i Z d n d Charge neutrality Z d r d Estimated with OML theory dn 8 p 2 rd 2NV nt m n /3m d Using =1.26 [71] T d 5eV T d (1 + Zd 2n d/2n i )T i *Taken as 0.5 µm.

81 59 CHAPTER 5 STRUCTURE FORMATION IN DUSTY PLAMAS 5.1 Structure Forming Processes In plasma physics, spatiotemporal pattern formations have been observed in ordinary gas discharges, such as those used for lighting sources or gas lasers [74, 75], and in dielectric barrier discharges [76 79]. Dusty plasmas exhibit similar complex behaviors, such as the formation of voids [80 82]. As noted, due to the nature of the electron and ion influx on the surface of dust particles, laboratory dusty plasmas are open systems that require a constant source of ionization to replenish ions and electrons lost to dust particles [15]. The constant flux of charged particles onto dust grains results in the grains having a variable charge [83]. These characteristics lead dusty plasmas to be inherently non-equilibrium and nonlinear systems. It is well known that such systems can exhibit self-organizing structures [84 86] and pattern formation [87]. A variety of structure forming instabilities in dusty plasmas, due to ion-drag, gravitational, polarization and magnetic forces, have been predicted or observed [2,88 90]. Such structure formation due to the self-organization of dust is an important process in interstellar dust molecular clouds, protostars, planetary rings [91] and planet formation [2]. Perhaps the most famous structure forming instability in plasmas is the Jeans instability. The Jeans instability for dusty plasmas results when the attractive longrange gravitational interactions between dust particles overcome the repulsive Coulomb interactions. For critical densities, the Jeans instability can lead to the collapse of dust clouds, resulting in stellar formations, planets, and stars. The long-range interparticle gravitational interactions can be accounted for, when using fluid theory, with an additional term in the dust momentum equation for the gravitational potential,

82 g /@x, sothatthedustmomentumequationbecomes: n d m d ez d n =0, (5.1) with r 2 g =4 Gm d n d. The attractive gravitational potential allows for a purely growing zero-frequency mode. For the long-wavelength approximation (k 2 2 D 1) the dispersion relation simplifies to:! 2 = k 2 C 2 da! 2 jd. (5.2) Here! jd is the Jeans frequency,! jd = p 4 Gm d n d.forlaboratorydustyplasmasthe Jeans frequency is on the order of 10 7 sec 1,whichwouldrequiredustcloudstobe2 to 3 orders of magnitude larger than experimentally grown for a zero-frequency mode to be supported (neglecting external forces on the dust suspension). Smaller scale interactions (the ion-drag force, polarization force, etc.) that modify the dust momentum equation in a similar fashion are also expected to support purely growing zero-frequency modes. I will present observations of a analogous structure forming instability. These observations detail a transitional mode, with slowly propagating dust density striations, and a stationary mode, with aperiodic dust density striations. Relevant to the following experimental observations are the ionization instability, which results from ionization and ion-drag e ects, and the polarization instability, which results from the polarization and deformation of the dust Debye sphere Ionization instability Formations due to ionization instabilities are well known in non-dusty plasmas and can result in propagating and non-propagating striations [92]. In dusty plasmas, a similar instability exists. The mechanism, described by D Angelo [93] and by

83 61 Morfill and Tsytovich [88, 94], can be briefly summarized as follows: If a fluctuation decreases the dust density in a region of a homogeneous dusty plasma, there will be a lower electron absorption in that region and, in turn, a higher electron density. The higher electron density leads to a higher ionization rate, further increasing the plasma density. This results in an enhancement of the outward ion-drag force on the dust, further reducing the dust density in that region. The voids created by the outward ion-flow from the ionization region have sharp dust walls, theoretically predicted and observed [81]. The large solitary voids found in rf discharge dusty plasmas under microgravity conditions are attributed to this instability [80]. The ionization instability can also lead to the formation of ordered structures in which regions of high dust density are separated by dust voids. At the heart of the ionization instability is the ion-drag force, the momentum imparted to dust particles by ions. A fundamental interaction in dusty plasmas, the ion-drag force can be separated into a collective, or collisional, force and a scattering, or orbital, force [13]. The collisional term of the ion-drag force can be modeled using the OML approach. With the collisional cross-section (Eq. 1.2 with a grain potential d = Z d e/4 " 0 r d ) and the scattering collisional cross-section the total ion-drag force can be expressed as: F id = m i Z vf i (v)( ci + scat )dv. (5.3) The scattering cross-section term, scat, ismorecomplicatedduetothepotentialand screening profile of the dust grain. Barnes et al. used the Debye length as the collection (scattering) impact cuto parameter for modeling the ion-drag force in 1992 [95], the model used by D Angelo. The resulting orbital contribution for the ion-drag force was orders of magnitude smaller than the collisional contribution and ignored by D Angelo in his modeling of the ionization instability. The range of scattering interactions has since been extended

84 62 well beyond the Debye length [13,96]. If the Coulomb radius, 0 (v i )=Z d e 2 /4 " 0 m i vi 2, is larger than the Debye length, the scattering cross-section can be much larger than the Debye cross-section, changing the magnitude of the orbital contribution dramatically. (The ratio of the Coulomb radius of interaction to the Debye length has been previously used as a measure of nonlinearity in Eq. 1.1.) Increasing the collection impact cuto parameter to more accurately reflect numerical simulations and experiments, the scattering cross-section becomes scat(v i )=4 2 o (v i ) [13]. Here, the Coulomb logarithm, (v i ), is defined as (v) = ln[ 2 o(v i )+ 2 max(v i ) 2 o(v i )+ 2 min (v i) ]1/2 (5.4) where 0min is the minimal distance before ions no longer scatter and collide with the dust grain and 0max is the maximal interaction length approximated by 0max = D(1 + 2 ) 1/2. Finally, to solve for Eq. 5.3, a Maxwellian distribution is used for f i (v i ). The improved modeling of the ion-drag force is important for correcting D Angelo s model [93]. To include ionization and ion-drag e ects into the dispersion relation for DAWs, D Angelo used continuity and momentum equations for the ions and dust particles, the Boltzmann relation for the electrons, and a charge neutrality condition. The ion continuity equation included a source term for the creation of ions through ionization of neutral atoms by a small (compared to the thermal electrons) component of fast electrons, and a loss term due to absorption of ions by the dust and walls of the plasma container. The ionization source term contained an electron energy dependent ionization cross-section. The ionization instability occurs since, for acousticlike perturbations, regions of elevated electron density also correspond to regions of elevated plasma potential. Since the ionization cross-section is a rapidly increasing function of electron energy just above the ionization potential, electrons in the wave

85 63 crests will be more energetic than those in the wave troughs and thus ionization proceeds more rapidly in the wave crests. The dust momentum equation included momentum loss due to dust-neutral collisions and a term due to the drag of the ions on the dust particles, originally estimated using the expression derived by Barnes et al. in 1992 [95]. I modified D Angelo s dispersion relation so that the ion-drag coe cient is consistent with current and more accurate models of the ion-drag force. From this updated dispersion relation I compared my forthcoming experimental observations to theoretical wavelengths, etc. In addition to the ion-drag force, it is natural to include neutral gas drag on the dust particle [97]. I also included finite dust temperature into D Angelo s model to better reflect the experimental conditions. The resulting modification of the original fluid description for dust plasmas, Eq , to include gas drag e ects (ionized and neutral gas), ion creation and loss, and finite dust temperature to the dust momentum equation and ion continuity and momentum equations are summarized as d n d m d dm d v + dn d m d v d +µ tot n d m d (v d v i )+k B T =0, i n i m + i im i v + i BT + + Qm iv i =0, (5.6) @x (n iv i )+ n i l Q =0. (5.7) Here µ tot is the updated total ion-drag coe cient, d is the neutral drag coe cient, Q accounts for ion creation, and n i / l accounts for ion extinction. By plotting! r and! i vs µ tot,acriticalion-dragcoe cientµ crit. is observed where the DAW becomes purely growing and non-propagating (! r =0,! i > 0). This result was shown by D Angelo (see Fig. 5 of ref. [88]). It should be noted that in D Angelo s model, the dust is taken

86 64 with a constant charge ez d and ion-neutral collisions and ion viscosity are ignored. Similar to D Angelo s derivation, Morfill and Tsytovich [88] used force balance as well as the continuity and momentum equations for the electrons, ions, and dust to obtain the ionization instability dispersion relation and noted that the maximum growth rate occurred at a wavenumber: k 2r d, (5.8) 2 Di Tsytovich et al. also investigated the e ect of sharp dust density profiles on ionflow [98]. The sharp void wall can result in a discontinuity of the ion-flow velocity, creating a shock-like dust density profile termed dust di usive shocks. Other examples stationary structure formations at velocity discontinuities are hydraulic jumps in fluids as well as bow shocks [99, 100] Polarization instability Recently, in 2009, Khrapak et al. demonstrated theoretically the aperiodic e ect that the polarization force has on DAWs [14]. Predicted by Hamaguchi and Farouki [101, 102], the polarization force results from the interaction between a dust grain and the deformation in the surrounding Debye sphere, which occurs when there is a nonuniform plasma. Deformation of the Debye shielding cloud surrounding the particle creates an additional local electric field and force on the dust particle in the direction of the decreasing Debye length. By taking a Boltzmann distribution for the electrons and ions, n = n 0 exp( Q (x)/k B T ), for the simple linear potential, (x) =xe, thereisagradientintheplasmadensity,rn = n 0 Q E/k B T.The charge distribution in the Debye sphere is directly a ected by the gradient in the plasma density and results in the polarization force: F pol = Q2 d r D 8 " 0 2 D. (5.9)

87 65 The expression for the polarization force can be approximated using Poisson s equation. Assuming a non-neutral plasma with a charge variation Q d on an length scale l d, the polarization charge of the Debye sphere becomes roughly q dpol Q d D /l d.since r D 1/l d, an approximation for the polarization force is obtained, F pol Q d E dpol Q 2 d r D/ 2 D. The dependence of the polarization force on Q2 d can result in a relatively large modification to the dust momentum equation, for large particles. 1 D 2 Di + 2 De 1 Di For dc discharge dusty plasmas, when T i T e, the Debye length can be q approximated with the ion Debye length, =. When the iondrift velocity is slower than the ion-thermal velocity the Debye screening is dominated by ions. The ion-drift velocity depends on the background potential and can be calculated using Robertson and Sternovsky s model of ion mobility [72]. By including the polarization force into the dust momentum equation and solving, assuming T i T e,khrapaket al. derived the dispersion relation [14].! 2 2 D k (1 <)! pd 2 2 (1 + k 2 2 D ) (5.10) The e ect of the polarization force is contained in the term (1 <), where < 1 ( Q 4 d e/4 " 0 D T i ) (1 T i T e ). For T i T e, < 1 4 T.Duetothee ectofnonlinear screening around the dust grain, Khrapak et al. [14] found that the onset of the unstable non-propagating mode occurs at T 6.4, where T is defined in Eq Experimental Design and Apparatus The experiment was conducted using a dc discharge device, detailed previously in Fig Under an argon gas pressure of 150 mtorr (20 Pa), a 275 V positive bias was applied to the 3.2 cm diameter electrode disk. The discharge current was ma. An axial magnetic field of 3 mtwasapplied,whichtransformstheanode glow discharge (fire-rod [103]) to a cylindrical region protruding several cm from the anode.

88 66 Dust particles are introduced into the anode glow plasma from an electrically floating tray located approximately 4 cm below the anode. When the discharge is initially formed dust particles are lifted o of the tray and become trapped within the plasma. The process of dust incorporation and trapping in the discharge occurs over atimescaleofseveralminutesafterwhichawell-confineddustsuspensionisformed and the dust tray can be removed. The processes involved in the dust confinement in an anodic plasma have been discussed by Trottenberg et al. [31]. The dust particles are confined by a combination of electric forces due to the self-consistently formed potential structure of the anode and ion-drag forces. The phenomena was observed using various dusts and sizes; however, the measurements presented were taken using either spherical iron particles with a narrow size distribution of 0.5 to 2 micron radius, hollow glass microspheres with a broad size >1 micron, and kaolin dust with a nominal radius of 0.5 micron. When non-monodisperse dust particles are used the distribution of dust sizes that are actually trapped in the suspension may be much narrower than the distribution in the dust reservoir [29]. The dust particles charge number was approximately Z d 4000 r d 10 6,estimatedusingOMLtheoryforelectronandion (argon) temperatures of 2.5 ev and ev, respectively. The electron temperature and plasma density in the absence of dust particles 1 was measured using a double Langmuir probe and were found to be in the range: T e ( ) ev, and n i (2-4) m 3 for discharge currents from ma. By measuring the plasma density through a range of discharge currents with a double probe the linear dependence of the plasma density on the discharge current was found to be 1.36 ± A 1 m 3.Thedoubleprobemeasurementsweretaken3cm from the anode. The ion temperature in these experiments is typically approximately 1 Electrical probes usually cannot be inserted into a dusty plasma due to the substantial disturbance they cause to the dust suspension, as documented for example in Thompson et al. [29]

89 67 equal to the neutral atom temperature, T i T n ev. Taking r d =1µm, the Havnes parameter, Z d n d /n i, is between 0.1 to 0.2. All following calculations preserve charge neutrality and use the value of n i measured in the absents of dust. Although the measurements were taken in the absence of dust, the ion density is not expected to be very significantly e ected at the observed dust density and the measurements give averygoodindicationoftheiondensityinthepresenceofdustaswellastheplasma density dependence on the anode discharge current. The experimental parameters are summarized in Table 5.1. Since there are no propagating DAWs in the dust trapped in the secondary cloud, the dust temperature is expected to be much lower compared to the primary cloud. Assuming dust heating was due to thermal electric field fluctuations, the dust temperature was estimated to be approximately 5 to 10 ev, Eq. (4.1). For theoretical comparison, the dust temperature was taken as 5 ev. Table 5.1: Experimental Parameters for Dusty Plasma Structure Formations PARAMETER VALUE REMARKS n i (1 5) m 3 approx. from I D,measured n d m 3 image analysis T i.025 ev neutral temperature T e 2.5 ev measured T d 5 10 ev Eq. 4.1 E-field 180 V/m emissive probe B-field 3 mt measured argon pressure 150 mtorr measured Z d (iron) e OML theory iron dust diam. 1-4 µm measured hollow glass diam. < 30 µm measured The plasma potential distribution along the axis from the anode (in the absence of dust) was measured with a floating emissive probe and is shown in Fig There are no indications of striations or localized potential structures in the initial plasma. The potential falls o monotonically from the anode resulting in an axial electric field

90 68 Figure 5.1: Axial floating potential measured with an emissive probe in the absence of dust. Note the absence of potential structure. The resulting electric field was 180 V/m. 180 V/m.Thiselectricfieldproducesanionflowawayfromtheanode. The dust cloud is was illuminated by a 250 mw 532 nm laser sheet and dust cloud dynamics were captured using a 532 nm lens filter at 30 fps and 60 fps, and analyzed using wavefront tracking, discussed in Ch Experimental Observations and Theoretical Comparison of Structure Formations Trapping of the dust in the anodic plasma was achieved by temporarily increasing the discharge current to 40 ma. During the trapping phase of the dust suspension, DAWs became spontaneously excited. Once a well-confined suspension was formed, the discharge current was reduced to 20 ma. Subsequently, the dust cloud underwent a transitional phase, in which the dust cloud split up into a primary cloud (of higher dust density) and a secondary cloud (of lower density and farther from the anode). The transitional, or transient, phase evolves in the secondary cloud before the stationary

91 69 Figure 5.2: A series of single frame video images, at 2.5 s intervals, of the secondary dust cloud evolution during the transitional phase. The first image (a) corresponds to the start of the transitional phase (t=0) with subsequent images taken (b) t=2.5 s, (c) t=5 s, (d) t=7.5 s, (e) t=10 s, and (f) t=12.5 s. In these images, the anode is seen on the right of the dust cloud. The bright portion is the primary (dense) cloud, and the dimmer, less dense secondary cloud is to the left. These images were obtained using spherical iron particles and show that as time progressed, the wavelength decreased. zero-frequency mode is readily apparent. The transient phase and the stationary phase have been determined to be independent phenomena and have both been observed independent of the other; the transient phase occurring for quick, and absent for gradual, transitions of the operating discharge current Transitional mode The structure formation observed in this experiment displays propagating and standing striations in the dust suspension. The transitional phase is composed of slowly propagating nonlinear dust density fluctuations. As the dust density waves in the transitional phase evolved, their wavelength decreased, as shown in Fig. 5.2, and, as the wavelength decreased to some minimal value, the waves slowed dramatically, to nearly a stationary state. The transient state was triggered whenever the cloud was disturbed, i.e. when the discharge current or magnetic field are adjusted or when the cloud is disturbed physically by inserting a probe. From the analysis of a series of

92 70 Figure 5.3: Evolution of the transient wavelengths, wave speeds, and amplitude. The time scale is set with respect to the onset of the transitional mode. The measurements were obtained by tracking peaks in the density profiles. The wavelengths were averaged when possible. Two experimental runs are shown, taken with iron and kaolin dust at 60 frames per second. The semi-log plot shows that the wavelength, wave speed, and amplitude decayed with a characteristic time / 12 sec 1.Thewavelength,wave speed, and decay time were independent of the dust type.

93 71 single frame video images, the evolutions of the transient wavelength, wave speed and wave amplitude were obtained, as shown in Fig The evolution of the waves in the transitional phase has the appearance of the bellows of an accordion being closed. As the waves proceed away from the anode, their wavelength and wave speed decreases as more and more wavefronts fill the dust cloud. At the end of the transitional phase, the wave motion is barely perceptible and the waves disappear into the background with the gradual emergence of stationary dust density structures. I note in Fig. 5.3 that the wavelength, wave speed, and wave amplitude of the transient decay with the same characteristic time scale, 12 s 1. This transition has been observed for all types of dust used. The measurement was also repeated for di erent dust under the same plasma conditions, resulting in the same characteristic time scale of approximately 12 s 1 with very similar wavelengths and wave speeds. Interestingly new waves formed at constant rate of 1 Hz, shown by the space-time diagram of the transient phase in Fig Since the final stationary dust structures that emerge after the transient phase have typical wavelengths of 5mm,whichareconsiderablylongerthanthewavelengths that are found at the end of the transient phase, the question naturally arose as to whether the stationary dust density structures are the final stage of the transient phase or represent an independent phenomenon. To investigate the relationship between the transitional and stationary dust density perturbations I performed an additional image analysis. The video frames covering the transient phase were averaged to obtain an average dust density profile. The result, shown in Fig. 5.5, reveal the presence of the stationary dust density structure, Fig. 5.5(b). For comparison, an average over the final structure is shown, Fig. 5.5(c), along with a plot of the density profiles, Fig. 5.5(a), for both cases. The presence of the stationary dust density structure in the transient phase suggests that the two phenomena are independent. Additionally, the damping of the finite frequency transient waves is much smaller than is expected

94 72 Dust Density (arb) 0 Frequency=1.1 Hz Distance from Anode (cm) Primary Cloud Secondary Cloud Time (s) Figure 5.4: A temporal dust density plot along with a space-time diagram of the transient phase. The dust density plot (top) is taken from the red dashed line in the space-time diagram (bottom). Taken with iron dust and recorded at 60 frames per second. for finite frequency waves in D Angelo s model, Fig. 5.10, and much smaller than would be expected from damping due to neutral gas drag, which suggests there is an additional driving mechanism for the transient mode that is not accounted. Recently, I have observed the formation of the stationary structure without the transient mode. The characteristic time and constant frequency of the wave formation in the transitional period may give clues to the nature of the transient phenomena. Since there are clear indications that the transient mode is independent of the zero-frequency mode (the stationary mode is present during the transient phase and the two modes can occur independently of each other) and the transient phase occurs after the cloud had been physically disturbed, the transient wave evolution is similar to what would be expected if the primary cloud was undergoing an underdamped compressional oscillation. If the primary cloud was undergoing a breathing mode (resulting from the original disturbance) with a frequency of around 1 Hz that decayed with a time scale of

95 73 Figure 5.5: Average density for transient and stationary mode. (a) Comparison of the average transient density profile with the stationary mode density profile. (b) Average transient dust suspension. (c) Average stationary dust suspension. 12 s 1, the decreasing amplitude of the oscillation would be expressed as waves forming in the secondary cloud at a rate of one wave per second with decreasing wavelengths (due to the damping of the primary clouds compressional oscillation). If the waves in the secondary cloud were the result of a decaying oscillation in the primary cloud they would also experience a decreasing wave amplitude, as observed, since less dust would be perturbed by the primary clouds decreasing oscillating amplitude. The decreasing perturbation by the primary cloud would also result in a decreasing wave velocity in the secondary cloud as the secondary cloud would be pushed a smaller distance through each oscillation, which was observed. Additionally, the dust suspension in the current experiment is an underdamped system for dust density waves, supported by the observation of the zero-frequency mode Stationary mode After the primary cloud split into a high density cloud and a lower density cloud the stationary mode appeared. The dust in the lower density cloud spontaneously

96 74 self-organized into a non-propagating pattern of alternating regions of high and low dust density. The high density regions were much broader than the low density regions. The stationary structures had a wavelength 5mm,withtypically3 4regions of enhanced dust density. To illustrate the longevity and robustness of the dust structures in the standing mode I show in Fig. 5.6 two single frame video images taken 45 seconds apart along with the corresponding density profiles. The structures shown in Fig. 5.6 were obtained in a dusty plasma using spherical Fe particles with diameters in the range of a few microns, although these structures were also observed using glass microspheres and kaolin dust. If the formation of the standing dust structures were related to a local increase in the ionization rate in the plasma, we might expect to observe an enhancement in the plasma light emission. There are, however, two factors that a ected the ability to observe this enhanced emission: first, the plasma actually extends well beyond the boundary of the dust cloud, and second, the stationary dust structures are not planar but, rather, nested conical shells. The 3-D structure of the dust cloud is shown in Fig. 5.7, which was created by a tomographic reconstruction of aseriesofcross-sectionstakenthroughtheentiredustcloud. The dependence of the spatial scale (wavelength) of the structures on discharge current was investigated. Figure 5.8 shows single frame video images and corresponding intensity profiles for structures formed in a dusty plasma using hollow glass microspheres for discharge currents of 19 ma, 22 ma, and 25 ma. There is a systematic decrease in structure wavelength with increasing discharge current. A summary plot of wavelength vs. discharge current is shown in Fig The origin of the dashed line (theory) will be discussed later. The data of Fig. 5.9 may also be seen as a decrease in wavelength with increasing plasma density, since the plasma density increases as the discharge current is increased. As the wavelength decreases (increases), more (fewer) standing waves are able to form in the cloud, which leads to additional compression (rarefaction) of the dust,

97 75 (a) (b) t=0 s (c) t=45 s Figure 5.6: Robustness of the stationary cloud structure over 45 s. (a) Dust density profiles from images at (b) 0 s and (c) 45 s. In (b) and (c) the anode can be seen to the right of the images with the dust structure to the left. Taken with iron dust. (a) (c) y z y z x dust tray x (b) z x (d) z y 1 cm Figure 5.7: Tomographic reconstruction of the standing mode. The axis orientation is shown in (a). The cloud is illuminated by the laser sheet in the x-z plane. Three dimensional views can be seen (b), (c), and (d) as the point of view is rotated (in the direction shown by the blue arrow in (a). The tomographic reconstruction shows the nonplanar nested conical shell structure of the zero-frequency dust density striations. The anode was removed from the images to improve clarity.

98 76 (a) 19 ma (b) 22 ma (c) 2 cm 25 ma Distance from Anode (cm) Figure 5.8: Zero-frequency structure formations at di erent discharge currents. Images averaged over 200 frames and recorded at 60 frames per second. The electrode current is set at (a) 19 (b) 22 and (c) 25 mamp. Corresponding density profiles are shown to the right over the selected regions. The contrast and brightness of the image has been adjusted to improve clarity. Taken with hollow glass particles. As the wavelength increases fewer striations fit into the dust cloud due to finite size e ects. This imposed boundary condition can introduce rarefaction and compression into the density striations.

99 77 Figure 5.9: Wavelength vs discharge current/plasma density. P 150 mtorr, B z 3 mt. Separate experimental runs are denoted by their trial number. Wavelengths were obtained by averaging over the density peaks. The dashed line is taken from D Angelo s model for a growth rate of! i =21.5 (rad sec 1 ). Taken with hollow glass particles. as seen in Fig This e ect could result in additional nonlinearity in the system. The dust charge in the compressional zones may be lower than the charge on the dust in the rarefaction zones [83]. To compare the experimental observations with theoretical models, I solved the modified version of D Angleo s dispersion relation. I used a dust of radius r d =1µm, a dust-neutral collision frequency of 10 s 1,andanion-dustcollision frequency in the range of ( ) s 1 (depending on the ion density/discharge current). The ion-dust collision frequency was calculated using the theory of Khrapak et al. [96]. For these parameters a growing, zero-frequency mode with wavelengths greater than 1.5 mm was found, shown in Fig where I plot the growth rate (! i ) vs wavelength ( )forvariousvaluesofthedischargecurrent. Forthevaluesofthe

100 78 Figure 5.10: Growth rate vs wavelength predicted by D Angelo s model for the ionization/ion-drag instability. Growth rates correspond to a zero-frequency mode with discharge currents from ma, corresponding to plasma densities of ( ) m 3.Thedashedlineindicatesthegrowthratetakenforthedashedlinein Fig parameters in my experiments, D Angelo s model predicts growing non-propagating modes. Furthermore, the results of Fig. 5.9 indicate that growth shifts to shorter wavelengths as the discharge current (plasma density) is increased, as shown by the dashed line, which is in reasonable agreement with the measured wavelengths. The timescales for the appearance of the dust structures is also inline with the calculated growth rates. The characteristic size of the structures formed (the wavelength corresponding to 2 maximum growth) based on Morfill and Tsytovich s model is given by MT : Di /2r d [88]. For an ion temperature and density, T i ev, n i m 3 and r d in the range of (1 4) µm, MT (1 5) mm, which corresponds roughly to the sizes of the structures which I have observed. The inverse scaling of MT with ion density is also in qualitative agreement with the observations presented in Fig. 5.9.

101 79 Figure 5.11: Onset of the polarization instability. The shaded portion is the parameter region where t > 6.4 and stationary dust acoustic waves are predicted to exist [14]. It has been shown theoretically that the ionization instability gives rise to sharp boundaries between dust clouds and voids due to the nonlinear nature of the instability [98]. Tsytovitch et al. [98] predicted the formation of shock-like dust density structures due to changes in the dust charge and ion di usion flux. The dust density profiles obtained in the present experiment show that the region between the dust density maxima are not complete voids, but rather regions of greatly reduced dust density. Certain assumptions of the Tsytovich et al. theory, however, are not met in the experiment. The theory assumes T d =0andmonodispersedust,neitherofwhich is the case in the actual experiments. These additional characteristics may lead to smoothing of the dust-void boundaries. From the discussed work of Khrapak et al. [14], I can consider if, under the conditions of my experiments, growing non-propagating modes might be expected to occur due to the polarization force. As shown, the onset of the polarization