Physics and Applications of Complex Plasmas

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Physics and Applications of

Complex Plasmas

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Physics and Applications of

Complex Plasmas

s v vladirnirov K Ostrikov A A Sarnarian University of Sydney, Australia

Imperial College Pres

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE

Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 U K ofice: 57 Shelton Street, Covent Garden, London W C W 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

PHYSICS AND APPLICATIONS OF COMPLEX PLASMAS Copyright 0 2005 by Imperial College Press All rights reserved. This book, or parts thereoj may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 1-86094-572-4

Printed in Singapore by World Scientific Printers (S) Pte Ltd

To our families

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Preface

The physics of complex plasma systems containing a colloid “macroscopic” particle component (“dust”) is a rapidly emerging area at the forefront of the physics and chemistry of plasmas and gas discharges, space physics and astrophysics, and materials science and engineering. Complex plasma systems with nano- and microscopic particle inclusions is now a hot topic for many research areas. Such plasma systems present an excellent example of complex systems because of the many and varied constituents, and the space and time scales involved, with extensive interaction between them. In our decision to write this book, we were mostly motivated to introduce, in a systematic and easy-to-follow manner, our understanding of the fundamental physics and industrial applications of complex plasma systems. The recent progress in the field has been so remarkable that several novel directions and paradigms in complex plasma research have emerged. Therefore, we have decided to focus on the most important (as we see them) current topics, and new paradigms in the research on and applications of complex plasma systems. We also review the role of “dust” in laboratory plasmas and discuss various challenging applications of the nanoand micrometer-sized particles in high-tech industries. We systematically present the current state of research and the physical insights, including the advanced theoretical models and results of extensive computer simulations, complemented with the laboratory experiments specifically designed to elucidate the fundamental physics of complex plasmas. This book provides a broad perspective and opens up future development of this rapidly expanding field to interested researchers normally working in various areas. Even though the main attention in this book is given to the conditions relevant to the laboratory gas discharges and industrial plasma reactors, most of the fundamental concepts discussed here are also

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Physics and Applzcations of Complex Plasmas

applicable to space and astrophysical plasmas. A specialized and comprehensive description of the most recent theoretical, experimental, and modeling efforts to understand the unique properties of complex plasma systems, including the stability, dynamics, and self-organization of colloid particles and their associations, is given. Special attention is paid to the physical concepts and most recent technological advances in various industrial applications of the micrometer- and nano-sized particles. The first chapter introduces complex plasmas as a new and unusual state of matter with fascinating physical properties. Chapters 2 and 3 present the fundamentals of the theory of interactions of the colloid particles with ionized gases and experimental methods of production and diagnostics of complex plasmas. Topics important to the physics of strongly and weakly coupled particle-plasma. systems are discussed in the following three chapters. In Chapter 4, key attention is paid to the particle dynamic phenomena, as well as particle arrangement and stability in complex ionized gas systems. Chapter 5 covers the self-organized gaseous, liquid and crystal-like structures, and phase transitions in dusty plasmas. Collective waves and oscillations in weakly-coupled colloidal plasmas are reviewed in Chapter 6. The concluding Chapter 7 focuses on the advanced industrial applications of micron- and nano-sized particles. The knowledge is presented in a concise and comprehensive manner, with a categorized overview of the underlying physics followed by up-todate details. Leading references are given to key original research results describing the unique features and high-tech applications of complex plasma systems. As such, this book is suitable for undergraduate and graduate students, as well as for researchers who work either inside or outside the field. We thus expect that our book will be interesting to researchers and students working in the areas of gas discharge and plasma physics, applied physics, space physics and astrophysics, soft condensed matter physics, and materials and electrical engineering. The authors are deeply thankful to many of their colleagues, in particular to N. F. Cramer, V. E. Fortov, J. Goree, 0. Ishihara, G. Morfill, M. Nambu, 0 . F. Petrov, L. Stenflo, H. Sugai, V. N. Tsytovich, 0. S. Vaulina, S. Xu, and M. Yu (in alphabetical order) for fruitful collaborations in some of the works discussed, the many other colleagues for fruitful collaborations and discussions, as well as all the authors of the original figures for their kind permission for reproduction. K. 0. also thanks C. H. Diong for the stimulating discussions and support at Nanyang Technological University, and R. G. Storer for hospitality at Flinders University. This work was

Preface

ix

essentially supported by the Australian Research Council, and partially supported by the University of Sydney and the Flinders Institute for Research in Science and Technology (Australia), the Agency for Science, Training, and Research, and the Lee Foundation (Singapore), the Max Planck Society and the Alexander von Humboldt Foundation (Germany).

S. V. Vladimirov K. Ostrikov A. A. Samarian

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Contents

vii

Preface

1

1. Introduction

1.1 Complex Plasmas in Nature, Laboratory and Technology . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Dust particles in plasmas . . . . . . . . . . . . . . . . 1.1.2 Weakly coupled and strongly coupled dusty plasinas 1.1.3 Experiments on complex plasmas . . . . . . . . . . . 1.2 New Paradigms and Directions of Complex Plasma Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Basics of Plasma-Dust Particle Interactions

17

,

2.

,

,

2.1 Fundamentals of Dust Charging, Main Forces on the Grains and Dust Levitation . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Negatively and positively charged grains . . . . . . . 2.1.2 Charging by plasma currents . . . . . . . . . . . . . 2.1.2.1 Orbit Motion Limited approximation . . . . 2.1.2.2 Charging of small and non-spherical particles 2.1.3 Particle charge a new kinetic variable . . . . . . . 2.1.3.1 Dust charging rate . . . . . . . . . . . . . . 2.1.3.2 Kinetic theory of dusty plasmas with variable charge . . . . . . . . . . . . . . . . . . . . . 2.1.4 Main forces acting on dust particles in a plasma . . . 2.1.5 Dust levitation in a collisionless sheath region . . . . 2.2 Developments of the Theory of Dust Charging . . . . . . . 2.2.1 Charging dynamics and charge fluctuations . . , . . ~

xi

1

1 7

8

18 18 23 23 28 30 30 31 35 38 42 42

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Physics and Applications of Complex Plasmas

2.2.2 Charging in the plasma stream . . . . . . . . . . . . 2.2.3 Effect of trapped ions on dust charging and shielding 2.2.4 Charge as a function of the grain size . . . . . . . . . 2.3 Interactions of Grain Particles with Ion Flows . . . . . . . . 2.3.1 Ion wake behind a charged point-like particle . . . . 2.3.2 Dust Cooper pairing: Hamiltonian description . . . . 2.3.3 Wakes behind dipoles and non-spherical particles . . 2.3.3.1 Wakes behind dipoles . . . . . . . . . . . . . 2.3.3.2 Wakes behind elongated particles (rods) . . 2.3.4 Subsonic plasma wakes . . . . . . . . . . . . . . . . . 2.3.5 Simulations on plasma wakes . . . . . . . . . . . . . 2.3.5.1 Wake of a single test particle . . . . . . . . . 2.3.5.2 Wake of two particles . . . . . . . . . . . . .

3. Production and Diagnostics of Complex Plasmas

3.1 Discharge Plasmas . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 RF-capacitive systems . . . . . . . . . . . . . . . . . 3.1.2 DC discharges . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 Sheath . . . . . . . . . . . . . . . . . . . . . 3.1.2.2 Striations . . . . . . . . . . . . . . . . . . . 3.1.2.3 Double layers . . . . . . . . . . . . . . . . . 3.1.3 Inductively coupled plasmas . . . . . . . . . . . . . . 3.2 Complex Plasmas with Positive Grains . . . . . . . . . . . . 3.2.1 UV-induced plasmas . . . . . . . . . . . . . . . . . . 3.2.2 Thermal plasmas . . . . . . . . . . . . . . . . . . . . 3.2.3 Nuclear-induced complex plasmas . . . . . . . . . . . 3.3 Traditional Diagnostic Techniques in Complex Plasmas . . 3.3.1 Probe measurements . . . . . . . . . . . . . . . . . . 3.3.2 Spectroscopic techniques . . . . . . . . . . . . . . . . 3.4 Detection and Diagnostics of Dust Particles . . . . . . . . . 3.4.1 Detection in laboratory and processing plasmas . . . 3.4.2 Dust structure diagnostics . . . . . . . . . . . . . . . 3.4.3 Methods of measuring charges on grain particles . . 3.5 Dust Grains as a Diagnostic Tool . . . . . . . . . . . . . . . 3.5.1 Plasma sheath diagnostics . . . . . . . . . . . . . . . 3.5.2 Spatial profiles of main plasma parameters . . . . . .

43 45 48 52 55 60 65

66 69 73 80 82

85 93

95 97 98 99 102 103 105 107 108 110 113 114 115

119 123 124 126 130 133 134 138

...

Contents

4.

5.

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Particle Dynamics in a Complex Plasma

141

4.1 Dynamic Phenomena . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Modeling dust particle dynamics in a plasma sheath 4.1.2 Dynamics of particles in chemically active plasmas . 4.1.2.1 Dynamics of fine particles in silane plasmas 4.1.2.2 Levitation in fluorocarbon plasmas . . . . . 4.1.3 Oscillations of a particle in a plasma sheath . . . . . 4.1.4 Diffusion of dust particles . . . . . , . . . . . . . . . 4.2 Arrangements and Instability of Confined Dust Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Modeling stability of dust particles confinement . . . 4.2.2 Experiments on particle arrangements and stability . 4.2.3 Self-excitation of vertical motions of dust particles .

141 142 150 151 152 158 162

Structures and Phase Transitions in Complex Plasmas

191

5.1 Self-organized Plasma-Particle States . . . . . . . . . . . . 5.1.1 Dust-plasma sheath as a self-organized structure . . 5.1.2 Dust voids: observations and main mechanisms . . . 5.1.3 Modeling of dust voids . . . . . . . . . . . . . . . . . 5.1.3.1 Collisionless dust voids . . , . . . . . . . . . 5.1.3.2 Voids in collision-dominated plasmas . . . . 5.2 Liquid and Crystal-Like Structures . . . . . . . . . . . . . . 5.2.1 Structures in capacitively coupled RF discharge . . . 5.2.2 ICP and DC discharge structures . . . . . . . . . . . 5.2.3 Ordered structures in thermal plasmas . . . . . . . . 5.2.4 Dust structures in nuclear and UV induced complex plasmas . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Phase States and Phase Transitions. . . . . . . . . . . . . . 5.3.1 Order controlling parameters in a complex plasma . 5.3.2 Criteria of phase transitions . . . . . . . . . . . . . . 5.3.3 Experimental observations of phase t,ransitions . . .

192 192 199 202 202 207 213 214 219 223

6. Waves and Instabilities in Complex Plasmas 6.1 Waves in Arrays of Colloid Particles . . . . . . 6.1.1 Dust-lattice waves . . . . . . . . . . . . 6.1.2 Experiments on dust-lattice waves . . . 6.1.3 Dust-lattice transverse waves with out-of motions . . . . . . . . . . . . . . . . . .

169 170 178 185

227 233 233 237 241 245

. . . . . . . 247 . . . . . . . 247 . . . . . . . 251 plane . . . . . . . 256

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Physics and Applications of Complex Plasmas

6.1.3.1 Oscillation modes in one-dimensional chains of particles . . . . . . . . . . . . . . . . . . . 6.1.3.2 Oscillation modes in two vertically arranged one-dimensional chains . . . . . . . . . . . . 6.1.4 Dust-lattice waves in the arrays of rod-like particles . 6.1.5 Waves in chains of magnetized particles . . . . . . . 6.1.5.1 Compressional and bending modes . . . . . 6.1.5.2 Angular magnetic lattice modes . . . . . . . 6.2 Waves in Weakly Coupled Unmagnetized Complex Plasmas 6.2.1 Langmuir waves . . . . . . . . . . . . . . . . . . . . . 6.2.2 Ion-acoustic waves . . . . . . . . . . . . . . . . . . . 6.2.3 Dust-acoustic waves . . . . . . . . . . . . . . . . . . 6.2.4 Surface waves . . . . . . . . . . . . . . . . . . . . . . 6.2.4.1 Ion-acoustic surface waves at the dielectriccomplex plasma interface . . . . . . . . . . . . . . . . 6.2.4.2 Electrostatic surface waves at. the dusty plasma-metal interface . . . . . . . . . . . . 6.2.4.3 Electromagnetic surface waves in a dustcontaminated large-area plasma source . . . 6.3 Waves in Weakly Coupled Magnetized Complex Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Alfvkn resonance . . . . : . . . . . . . . . . . . . 6.3.2 Waves in a weakly non-uniform plasma . . . . . . . . 6.3.3 Surface waves . . . . . . . . . . . . . . . . . . . . . . 6.3.3.1 Dispersion . . . . . . . . . . . . . . . . . . . 6.3.3.2 Damping . . . . . . . . . . . . . . . . . . . . 6.4 Instabilities in Weakly Coupled Plasmas . . . . . . . . . . . 6.4.1 Dust ion-acoustic instability . . . . . . . . . . . . . . 6.4.2 Dust-acoustic instability . . . . . . . . . . . . . . . . 6.4.3 Buneman “dust-ion streaming” instability . . . . . . 6.4.4 Parametric and modulational instabilities in magnetized complex plasmas . . . . . . . . . . . . . 6.4.4.1 Parametric pumping . . . . . . . . . . . . . 6.4.4.2 Modulational instabilities . . . . . . . . . .

6.4.5 Other instabilities . . . . . . . . . . . . . . . . . . . .

257 259 264 269 272 275 276 276 280 283 287

287 289 291 294 296 297 300 300 302 304 304 308 313 316 317 318 320

Contents

Fine Particles in Industrial Applications 7.1 Growth and Characterization of Nano- and Micron-Sized Particles . . . . . . . . . . . . . . . . . . . . . 7.1.1 Origin and mechanisms of growth of clusters and particulates in reactive plasmas. . . . . . . . . . 7.1.1.1 Silane plasmas . . . . . . . . . . . . . . . . . 7.1.1.2 Hydrocarbon plasmas . . . . . . . . . . . . . 7.1.1.3 Fluorocarbon plasmas . . . . . . . . . . . . 7.1.2 Effects of fine particles on discharge stability . . . . . 7.1.3 Particle growth in plasma-enhanced sputtering facilities . . . . . . . . . . . . . , . . . . . . . . . . . 7.1.4 Problems of particle detection in the nanometer range . . . . . . . . . . . . . . . . . . . . 7.2 Deleterious Aspects . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Particulate powders as process contaminants in microelectronics . . . . . . . . . . . . . . . . . . . 7.2.2 Removal and growth suppression of dust particles . . 7.2.3 Role of dust particles in fusion reactors . . . . . . . . 7.3 New Applications of Nano- and Micron-Sized Particles . . . 7.3.1 Nanoparticles in the plasma-assisted assembly of carbon-based nanostructures . . . . . . . . . . . . . . 7.3.2 Nanopowders in plasma-enhanced chemical vapor deposition of nanostructured silicon-based films . . . 7.3.3 High-rate cluster and particulate deposition on nanostructured surfaces: a new paradigm in thin film fabrication . . . . . . . . . . . . . . . . . . . . . 7.3.4 Particle size as a key factor in nano-scale technologies . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Other industrial applications of nano- and micronsized particles . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Concluding remarks . . . . . . . . . . . . . . . . . .

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375 38 1 386 394

Conclusions

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Bibliography

403

Index

435

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Chapter 1

Introduction

1.1 Complex Plasmas in Nature, Laboratory and Technology Complex plasma systems containing condensed particles (“dust grains”) have recently been actively investigated in the physics and chemistry of plasmas, ionized gases, space physics and astrophysics, and materials research. These particles can be macroscopic in the sense that typically they are significantly larger than other plasma particles (electrons, ions and neutrals). On the other hand, they appear as “microscopic” objects since their size is usually much less than typical sizes of the plasma system. However, the smallest “macroscopic” particles can have the sizes comparable to those of nanometer-sized clusters that merely contain a few tens to a few hundreds of atoms/molecules. Conversely, the “microscopic” particles in some space environments can be much larger than any typical plasma device in a terrestrial laboratory. In any case the particles represent an important additional charged component that strongly complicates the physics of fundamental plasma phenomena and makes these plasma systems extremely complex. Thence, such plasma systems have recently been referred to as the “complex plasmas”.

1.1.1 Dust particles in p l a s m a s The particles with different sizes, from a few nanometers to tens of micrometers, have been observed for many years in various laboratory experiments on the complex plasmas. Meanwhile, there are numerous reports on the occurrence and the role of charged dust particles in the interstellar space, planetary atmospheres, ring structures, cometary tails, etc. For example, 1

2

Physics and Applications of Complex Plasmas

charged grains play an important role in the synthesis of molecular species in diffuse nebulae and directly participate in various astrophysical processes [Gould and Salpeter (1963)l. Planetary atmospheres and planetary rings are yet another example of the complex plasma system in space environments [Goertz (1989); Whipple (1981); Northrop (1992)l. Under terrestrial conditions, fine particles are the important parts of various colloidal suspensions, charged aerosols, high-pressure ionized gas systems, nucleation catalysts, and many others. Particulate powders with sizes in the nano-/micrometer range have also been detected in a number of experiments with low-temperature plasmas for materials synthesis and processing applications. Dust is also very common in radio-frequency (RF) gas discharges, where fine powders appear in the mixtures of chemically active (reactive) gases that generate electronegative free radicals [Garscadden et al. (1994)l. In all the situations discussed above, the macroscopic dust particles not only range widely in size but can also represent quite different forms of chemical organization (e.g., micro-crystallite, atomic/molecular cluster, complex agglomerate etc.) and feature different elemental composition (e.g., be a pure material or a composite), shape and architecture (spherical, fractal, cylindrical, etc.), and physical properties (e.g., be dielectric, conducting or semiconducting, optically transparent, magnetized, etc.) . For this reason, there is a large number of terms attributed to these particles, each of them usually reflects the specific case. For example, the terms “microparticles” or “nanoparticles” apparently reflect the certain size range, whereas “particulates”, “agglomerates” and “grains” may emphasize the organization of the particles concerned. Furthermore, it is also common to refer to the particles as “colloid particles” since the latter are immersed in a plasma “fluid” similar to colloidal suspensions of solid particles in conventional fluids. In the situations when a large number of the particles of smaller sizes is concerned, the terms “powder particles” or “fine powder” might be appropriate. The latter case is common for chemically active plasmas (reactive plasmas), which have recently been of a remarkable interest in a number of hightech industries that use various plasma-based surface modification technologies. The processes involved include but are not limited to ultra-fine surface modification and microstructuring, patterning and selective etching, plasma enhanced chemical vapor deposition (PECVD) of protective coatings and interlayers, data storage, sensing, guiding, emitting, converting and other functionalities of the numerous optoelectronic and photonic devices, etc.

Introduction

3

It was also noted that the low-temperature plasma discharges are very efficient as electrostatic traps for the grains. For example, in conventional parallel-plate reactors, where both electrodes are charged negatively due to the high mobility of plasma electrons, the inter-electrode ambipolar field also confines the dust grains. One should expect intensive interactions between the reactive plasma environments and the fine powders being grown. Under certain conditions, dense clouds of fine powders (with the number densities of up to 10l1 cm-3 and even higher) have been detected. Depending on the prevailing experimental conditions, the grains feature various chemical organization and structure, surface morphology and bulk architecture, gas-kinetic temperature, equilibrium positions in the plasma reactor, etc. [Garscadden et al. (1994)l. Furthermore, the fine particles not only range widely in size but can also be quite different because of the underlying physics and chemistry of their origin, growth, surface properties, and self-organization. Accordingly, their global role in the entire complex plasma system can vary significantly. Here, our major focus is on the laboratory complex ionized gas systems, although many of the discussed basic phenomena and fundamental results are also applicable to space dusty plasmas. In most of the cases considered, the additional particle component is able to affect the global and microscopic properties of the complex plasma system including the stationary spatial profiles and the temporal evolution of the electron/ion number densities and temperatures, the plasma potentials, and the electromagnetic fields in the discharge chamber. The complex plasma systems are highly capable for self-organization and can dynamically adjust their microscopic and global parameters through self-consistent nonlinear responses to changes in the external control parameters (e.g., gas/plasma composition, pressure, or input power) or in the internal colloidal subsystem (e.g., the dust charge and/or size variation). In particular, the effect of the growing dust component on the discharge strongly depends on the specific growth stage, with remarkably different density, structure, size, shape, and equilibrium electric charge of the dust grains. Evolution of the dust component in chemically active plasmas usually begins with the growth of nanometer-scale protoparticles that appear as a result of the homogeneous (purely gas-phase) or heterogeneous (particle/catalyst-induced) nucleation processes. The homogeneous nucleation is usually triggered by specific negative or neutral precursor radicals and proceeds through numerous chemical polymerization chains. On the other hand, the heterogeneous process can be initiated by substrate/wall

4

Physics and Applications of Complex Plasmas

flakes released as a result of the reactive etching or the physical sputtering or, alternatively, the externally injected (e.g., metal) nanoparticles. At this stage, the fine particles are usually either charge neutral or carry a positive/negative charge of up to a few elementary charges. Their number density typically exceeds 10' cmP3 and in the strongly electronegative (with high densities of the negative ion precursor radicals) cases can reach 10l1 cmP3 and even higher.

Fig. 1.1 High-resolution SEM micrograph of a particulate agglomerate grown in chemically active He+SiH4 plasmas [Shiratani et al. (1994)].

Fine particles of the intermediate (a few ten to a few hundred nanometer) size often appear as an agglomerate of a complex shape with fractal surfaces, as can be seen in Fig. 1.1. They are a result of the relatively fast gas-phase agglomeration of smaller particles, followed by a significant re-distribution of the electric charge in the overall charge neutral plasma system. The number density of the particles in this size range is typically a few orders of magnitude lower (lo6-lo8 ~ m - than ~ ) that of the nanometersized particles. Furthermore, the roles of the particles with different sizes in the complex plasma system are physically different. Indeed, due to the much larger surface area (S oc p.', where a is the particle radius, in the simplest case of a low fractal surface), the intermediate-sized grains become

Introduction

5

capable to collect larger numbers of electrons and ions from the adjacent plasma and accumulate the net negative charge of tens to hundreds of electron charges due to the high electron mobility. Thus, the negative charge is re-distributed between the electron plasma and the dust components. This phenomenon can lower the electron density and, hence the rates of the prevailing electron-impact elementary processes in the plasma. Meanwhile, the complex plasma system tends to self-organize itself in order to “reinstate” the number of the plasma electrons lost to the dust and re-establish the pristine charge/particle distribution. Typical changes in the plasma system include variations of the ionization/dissociation rates, electron temperature, changes in the ambipolar (confining) electric field, electron/ion fluxes to the walls/dust, density/temperature profiles of the plasma species, dust-induced transitions of the discharge operation regimes, plasma instabilities etc. The above changes affect the number densities, energies, and fluxes of the neutral/anion precursor radicals, thus modifying the basic conditions for the dust nucleation and growth. Therefore, the fine particle-plasma system is a typical example of an open dissipative system with the dynamically coupled sub-components. Furthermore, the variations of either the external conditions or the internal parameters of the plasma and/or colloidal subsystems dynamically affect the global power and particle balance in the entire complex plasma system, which in turn influence the dust generation and growth processes. On the other hand, larger ( u > 1 pm) particles are also quite common for many thin film deposition processes and laboratory experiments with externally injected dust. In the latter case, specially designed experiments usually employ spherical monodisperse particles. In processing discharges, the micron-sized particulates can naturally evolve from the submicron-sized particles via a relatively slow process of collection of the neutrals and ions from the reactive plasma environment. This process can be regarded as a microscopic plasma-assisted chemical-vapor deposition (CVD) of thin films on a dust grain. In this range, the equilibrium particle charge is also controlled by the dynamic balance of microscopic electron/ion currents flowing onto the dust surface. A complex plasma with the micron-sized grains is quite stable, and self-organization processes usually develop over longer time scales (such as those associated with large and therefore heavy colloid particles). In the micrometer size range, a relatively large (a few thousand or ten thousand electron charges) .electric charge favors the charge-induced self-organization of the plasma-dust system into strongly coupled colloidal liquid-like and

6

Physics and Applications of Complex Plasmas

ordered crystal-like particle structures [Thomas and Morfill (1996)]. The latter type of structures is also called Coulomb crystal and is observed in numerous experiments in the gas discharge plasma, for example, in radio frequency capacitively coupled and inductively coupled discharges, direct current discharges, hollow cathode discharges, cryogenic discharges. Some typical examples of such structures in a radio frequency capacitively coupled discharge are displayed in Fig. 1.2.

Fig. 1.2 Optical images of a typical dust cloud (left) and a self-organized crystal-like structure (right) in a low-pressure capacitively coupled plasma at the University of Sydney.

Furthermore, under certain conditions the complex plasmas are subject to various phase transitions in the gas-, liquid-, and crystalline-like states. The ordering forces are mostly electrostatic, however, the dipolar effects, the (anisotropic in some cases) plasma shielding, the focusing of the ion flows, etc., can play a significant role. The complex plasmas are of fundamental interest since the individual dust particles can be easily visualized and the collective processes can be studied at the most fundamental kinetic level [Thomas and Morfill (1996)l. Thus, the dynamic self-organization of the entire system as a result of the evolution of the colloidal subsystem or variation of the external control parameters is one of the most important fundamental features of the complex plasmas. Furthermore, the dynamic self-organization and collective phenomena in the complex ionized gas systems with the nano- and micron-sized particles are becoming the most critical issues for the lowtemperature plasma-based synthesis and surface modification of advanced nano-materials.

Introduction

1.1.2

7

Weakly coupled and strongly coupled d u s t y p l a s m a s

When the particle size is more than (approximately) hundred nanometers, its (negative) charge, in the typical case of charging by plasma currents in a gas-discharge plasma, becomes large, of the order of hundreds electron charges. Moreover, in the micrometer size range the grain can be charged up to a few thousands or even tens of thousands of e. Such high charges accumulated on the macroscopic particles lead to strong electrostatic interactions between them as well as with the ambient plasma. For the conditions when the potential energy of the electrostatic interactions between the particles far exceeds the energy of their kinetic motion, the system of these particles reveals qualitatively new phenomena. In particular, the highly charged macroscopic particle component can acquire the liquidlike or crystal-like properties [Thomas and Morfill (1996)l. Such types of systems are referred to as strongly coupled complex plasma systems. The main parameter characterizing the coupling of the Coulombinteracting particles is the Coulomb coupling parameter

which is the ratio of the Coulomb potential energy of the particle interaction to the kinetic energy of their thermal motion (here, d ~ nd1/3 is the mean intergrain distance, n d is the particle number density, zd is the dimensionless dust charge (such that Q = e Z d ) and the dust kinetic temperature T d is in energy units so that the Boltzmann constant equals unity). If to assume that the particles are screened by the plasma, the Debye (or Yukawa) type of screening can be incorporated into rc as well. It was found from experiments and numerical simulations that the system of Yukawa-interacting particles crystallizes when rc is of the order of one hundred (for example, in a one-component electron plasma numerical calculations give rc M 170 for three-dimensional systems and rc M 106 for two-dimensional systems). This is an example of a strongly coupled complex plasma system. It is important to note that under certain conditions the multicomponent (with the macroscopic particles) plasma can also be non-ideal in a sense of a limited number of dust grains within the plasma Debye sphere [Tsytovich et al. (2002)]. For smaller size (and, hence, charge) particles and/or for rarefied dust component, the coupling parameter is typically much less than unity. This is the weakly coupled regime, which might correspond (although not necessarily always) to an ideal multi-component plasma case.

8

Physics and Applications of Complex Plasmas

Indeed, the number of particles within the Debye sphere is approximately given by

where AD is the plasma Debye length. Here, we stress that the plasma Debye length includes the contribution of all the plasma species. For example, if the number of dust particles in the Debye sphere is large, they participate in the plasma screening, and the plasma Debye length is mainly determined by the dust contribution, i e . , AD z ( T d / n d e 2 Z i ) 1 / 2 .In this case we have

One can see that the ideal complex plasma with N D >> 1 also satisfies the definition of the weakly coupled complex plasma system, where I? << 1 holds. However, the opposite is not necessarily true. For example, the weakly coupled complex plasma can be non-ideal at low dust number densities. Indeed, if N D < 1the above simple relation between the number of the dust particles in the plasma Debye sphere and the coupling parameter does not hold. The reason is that in this case only the electrons and ions participate in the plasma screening and the Debye length is essentially determined by the ion component. The case of a strongly coupled complex plasma system typically corresponds to a highly non-ideal dust component although the above simple relation also does not hold in this case. Some properties of the complex plasma in the weakly coupled case (e.g., collective plasma modes propagating in such a system) can be qualitatively understood on the basis of the results known for the multi-ion-species plasmas. However, these analogies are not always applicable, in particular because of the noted above specifics of a complex plasma as an open dissipative system. In the strongly coupled complex plasma, the collective excitations can be analogous to those in liquids and/or crystals (e.g., lattice modes, see Chapter 6).

1.1.3

Experiments on complex plasmas

The first experiments on a complex plasma were in fact performed by Langmuir in the twenties of the last century. A few decades later, complex plasmas were studied as a part of the magnetohydrodynamics (MHD) problem. At that time, the complex plasmas were termed as the plasma with the condensed dispersed phase (CDP). It was noted [Yakubov and Khrapak (1989)I

Introduction

9

that the plasma flow in the working channel of the coal MHD generator contains impurity particles that noticeably affected the conductivity. [Selwyn et al. (1990)l were among the first to report the occurrence of clouds of submicron- and micron-sized powder particplates in the plasma processing experiments. However, the real boost in the specially designed experimental investigations of complex plasmas with added micrometer-sized dust component began after the discovery in 1994 of dust plasma crystals in the near-sheath areas of capacitively coupled RF discharge plasmas [Thomas et al. (1994); Chu and I (1994); Hayashi and Tachibana (1994)l. Shortly thereafter, the formation of the ordered dust structures was revealed in the DC discharge plasma [Fortov et al. (1997a)], Q-machines [Merlino et al. (1998)], thermal plasmas operated at atmospheric pressures [Fortov et al. (1996a); Fortov et al. (1996b)], UV-induced [Samarian and Vaulina (2000)], and nuclear-induced plasmas [Fortov et al. (1999b)l. New opportunities for studying the amazing properties of complex plasmas appeared with the recent advances of the experimental research under the microgravity conditions [Fortov et al. (1998); Morfill et al. (1999); Morfill et al. (2002)l. Under typical laboratory conditions, the particle structures levitate under the Earth’s gravity balanced by the electric field of the discharge. The gravity limits the variety of possible experimental situations as the research of complex plasma can proceed within a narrow range of parameters. On the contrary, experiments under the microgravity conditions allow one to study a much wider set of phenomena and increase the size of dust particles up to hundreds of micrometers, which would be impossible in the laboratories on Earth. The experimental investigations of complex plasmas are conducting in various types of gas discharges. The discharges are typically sustained in the power range of 0.03 to 3 Torr with the input powers of up to a few hundreds of Watts. Such is essentially a non-equilibrium plasma, with the average electron temperature T, 1-7 eV and the highest ion temperature Ti to be approximately the same as the neutral gas temperature, Ti = T, 0.03 eV. The electron and ion number densities (n, and nil respectively) and T, are usually obtained from the Langmuir probe measurements in the absence of the dust and can be cross-referenced by the modeling of the discharge parameters. Typical experimental conditions correspond to n, = ni los10’ cmP3. The basics of the plasma diagnostics and the description of the specific experimental facilities for the complex plasma research are given in Chapter 3 . N

N

-

10

Physics and Applications of Complex Plasmas

The radii a of the particles usually used in the laboratories on Earth are within the range of 0.5 to 10 pm (in some cases larger particles with the radii of up to 30 pm are possible). As mentioned above, due to the high mobility of electrons, the particles in the plasma discharge acquire a significant negative charge of about i03-105 elementary charges. Discharges are widely used for the research of the properties of complex plasmas in the electric fields since the negatively charged particles can be confined in both the vertical (along the gravitational field Fg), and in the radial (perpendicular to F,) directions. The sheath electrostatic force is the major force that balances the force of gravity. In the direction perpendicular to the gravitational field, the cloud of particles is confined by the radial electric potential trap. In RF discharges, it can be established by, e.g., a groove in or a metal ring of a few millimeters high on the lower electrode [Melzer et al. (1994); Ivlev et al. (2000b)], a ring-shaped electrode [Samarian et al. (2001b)], or segmented electrode [Cheung et al. (2003)l. In DC discharges, the radial electric trap originates due to the space charge distribution in the polarization field, the latter appears as a result of the ambipolar diffusion of the plasma electrons and ions onto the walls of the discharge chamber. Other forces (e.g., thermophoretic, ion drag, etc.) also contribute to the formation of the trap for the negatively charged particles. However, for typical experimental conditions under terrestrial conditions, these forces are often small (although their contribution can be important in qualitatively new phenomena, e.g., dust voids often sustained by the force balance including, in particular, the ion drag force, see Chapter 5). Thus, the electric and the gravity forces appear as the major contributors to the balance of forces on the dust grains. Under certain conditions, the particles form highly-ordered crystal structures such as those in Fig. 1.2. On Earth, the Coulomb crystals are observed in the sheath of RF discharge plasmas, as well as in the striation region and the double layer of DC discharges. Usually, the plasma crystals can contain up to several thousands of particles with the interparticle distances from 30 pm for the particles of the radius a < 0.5 pm up to a few millimeters for larger particles with a 10 pm. In most cases, the Coulomb crystals formed in the sheath region have a quasi-two-dimensional structure, as opposed to the three-dimensional clouds formed in DC discharges. The hexagonal-type lattices were observed in most of the experiments. However it should be noted that the formation of the three-dimensional lattices of various types in RF discharges was also reported in a number of experiments, especially in microgravity.

-

Introduction

11

Along with experiments in gas discharge plasmas, irregular structures were found in thermal plasmas at atmospheric pressures, for example, in the flow of gas combustion products and in the condensed region of the propellant combustion [Samarian et al. (2000b)l. Such structures have also been found in nuclear-induced plasmas [Fortov et al. (1999b)I and a two-component plasma induced by the sun radiation [Fortov et al. (1998)l. The structures were formed without any radial electrostatic confinement and appear similar to some common structures in liquids and gases. The dust particles in these experiments were mostly positively charged with the charge dependent on the efficiency of the thermion and photoelectron emissions. It is worthwhile mentioning that the experiments with complex plasmas have inspired the studies of various fundamental processes in plasmas. Some examples include the plasma transport properties, phase transitions, plasma waves and instabilities. Furthermore, novel methods of particle diagnostics and generation of environmentally friendly plasmas can also be developed in the course of these studies.

1.2

New Paradigms and Directions of Complex Plasma Research

The main focus here is on the most important current fundamental topics and new paradigms of dynamic self-organization phenomena in the complex ionized gas systems. Such systems are perfect examples of complex systems because of the many and varied constituents, with the extensive and intensive interactions between them, as well as numerous space and time scales involved. The fundamental interest is related to the kinetics of processes in open dissipative s y s t e m s of interacting particles. The applications are related to the origin, growth, and self-organization of micro- and nano-scale particles of the desired size and shape, with the major focus on the physics of those processes that allow one to efficiently manipulate the structure and basic properties of the matter at the corresponding length scales. The current state and the most important physical advances of the fundamental research activities are systematically presented. The latter include the advanced theoretical models and extensive computer simulations, complemented by the laboratory experiments'specifically designed to elucidate the fundamental physics of the complex plasmas.

12

Physics and Applications of Complex Plasmas

The optoelectronic, photonic, nano- and biomaterials technologies are extremely important for progress. Thus, the fundamental physics and kinetics of multi-scale dynamic processes in complex ionized gas systems that contain nano- and micrometer-sized solid particles is becoming a matter of the utmost importance for various industrial applications. At present, there is a vital demand for conceptually new approaches to the analysis of hehavior and manipulation of such systems, based on the advanced physical models of the interaction of the cluster/particulate matter with their ionized plasma environments. One of the aims of such attempts is to tailor the size, composition, and architecture of the nano-sized particles as building units of various nanoscale assemblies. Chemically active environments of the complex ionized gas systems have proved to be very efficient in generating charged fine particles. It is expected that these systems will become one of the major means of producing nano-scaled particles for numerous industrial applications. However, the application of plasma-particle systems to nano-scale processes is still at an early stage, and a number of fundamental physical problems are yet to be solved before the scheme can become commercially feasible. First, the nano-assembly process cannot be reliable without a precise control of the composition, size, and architecture of the plasma-grown building units. The latter have to preserve their unique features and need to be deposited on the nanostructured substrates in a non-destructive fashion. The optimal parameters of the ionized gas phase allowing a predictable and controllable fabrication process have yet to be established and linked to the process control parameters. The above goals cannot be achieved without solving a number of fundamental physical problems at different space and time scales. The building units originate at nanometer scales as a result of the elementary gasphase reactions, and frequently agglomerate into larger particles. The gasphase grown nano-sized particles do affect the entire ionized gas system by distorting and sometimes destabilizing the balance of electrons and reactive species, which in turn dynamically affects the cluster/nanoparticle origin and growth conditions. Thus, the processes at multiple scales become dynamically coupled through a number of elementary processes and self-consistent plasma responses, which are yet to be properly understood. In this book, in a systematic manner, we present the theory, experiments, simulations and applications of micro- and nanoparticles in various complex ionized gas systems. We discuss a number of dynamic collective phenomena inseparably associated with self-organization of the particle-

Introduction

13

plasma systems. In particular, we discuss the physics of the plasma-dust interaction, levitation and dynamics of macroscopic particles in the plasma, new self-organized structures involving the particles and the plasma, as well as a number of collective plasma phenomena (e.g., the ion wakes, the plasma waves and oscillations, and the collective motions in arrays of charged particles in the plasma sheath/pre-sheath areas) important for the particle interactions and formation of many-particle structures. When considering the basic physics of complex plasmas, special attention is paid here to the open character of this system. This openness is related to the continuous flows of electrons, ions, and radicals onto the surfaces of dust particles. The latter appear as important sink of plasma, since the electrons and ions usually recombine on the grain surfaces [Vladimirov (1994); Tsytovich (1997); Tsytovich et al. (2002)l. We note that in some earlier (and sometimes even current) investigations on collective phenomena in dusty plasmas, the fact that complex plasma systems are practically always thermodynamically open was either overlooked or side-stepped by invoking unspecified sinks and further assuming that their total effect is negligible. However, the details of these sinks are nevertheless important for the proper and rigorous treatment of the problem. On the other hand, sinks of the plasma electrons and ions resulting from their capture by dust should also be properly accounted for by introducing the corresponding capture processes into the electron and ion conservation equations. However, merely accounting for the plasma particle capture processes is not sufficient, since without a balancing source for the plasma particles there cannot exist a self-consistent stationary state. Generally, the energy and plasma particles have to be supplied externally to maintain the system in the equilibrium state [Tsytovich et al. (2002)l. It is therefore necessary to take into account the creation of the plasma particles, for example, by an ionization source. For most plasmas it is then also necessary to include related plasma-particle loss and transport mechanisms such as ambipolar diffusion and volume recombination, which usually develop over the same time scale with the ionization process [Ostrikov et al. (2000a)l. Since the ionization, transport, and recombination are all density dependent, all electrostatic processes in the system can be strongly affected. This opens the possibility that the presence of dust can affect the entire discharge system through modification of the ionization-transport-recombination balance. Many unique self-organization phenomena in complex plasmas are affected by or originate due to the openness of the system. These phenomena include, for example, various collective excitations in the background

14

Physics and Applications of Complex Plasmas

plasma, charging a.nd collective motions of dust grains, inter-particle and dust-plasma interactions, formation of numerous self-organized structures, such as dust-plasma sheaths, dust voids, etc., considered in this book. Thus, the openness is a remarkable intrinsic feature of complex plasma systems, which definitely qualify as a fascinating and unusual state of matter [Tsytovich et al. (2002)l. The openness of complex plasma systems implies numerous interactions between the two major components, namely, the solid particle and the plasma components. One of the major factors in the dust-plasma interaction is its electromagnetic character. Indeed, the charged particles move under the action of the electric and magnetic forces in the plasma medium and can also be confined by the self-consistent electrostatic potentials. On thc other hand, the electric field of the dust does affect the motion of the plasma species, especially in the vicinity of the grain surface. Furthermore, this seemingly simple process involves numerous complex processes on the surfaces of macroscopic particles, such as neutralization of the ions with the subsequent return to the gas phase or adsorption on the surface, adsorption/attachment and desorption/detachment of saturated and radical neutral species, activation of the surface layers followed by the slow (as compared with the charging time scales) process of the film deposition on the grain surface as well as many other elementary processes in the plasmasurface interaction. The problem of the dust-plasma interaction is therefore of a constantly increasing importance to the development of the fundamental theories of the complex plasma systems. In this regard, one has to always keep in mind that the two (dust and plasma) subsystems are continuously interacting. We thus arrive to the fundamental concept of the two interacting subsystems (both represent two different phases of the matter) in the open complex plasma system. The complex plasmas appear as open systems with strongly interactive subsystems, which also allow us to qualify them as Ynhomogeneous systems”. We note that from the thermodynamic point of view the complex plasma systems are usually “small” in a sense that limited volumes and particle numbers are involved. Indeed, a typical number of macroscopic particles in complex plasmas is small compared to the number of atoms and molecules (the latter can be estimated by the Avogadro number mol-I at normal conditions). Since the characteristic range 6 x of (mostly electrostatic) interactions is often comparable with the macroscopic sizes of majority of the common dust-plasma structures, one can also

-

Introduction

15

informally term a complex plasma as a non-extensive system. Furthermore, the presence of natural and/or artificial boundaries as well as non-uniform spatial distributions of the species (including macroscopic particles) makes the complex plasma system a classical example of a non-uniform system. In this case, the conventionally used thermodynamic limit would be inappropriate since it implies infinitely large and infinite (boundless) extensive systems. Furthermore, the so-called canonical ensemble approach, employed for the modeling of various thermodynamic systems with shortrange interactions, would fall short in the complex plasma case because of the prevailing long-range interactions between the macroscopic particles. In this case the details of non-uniformities and statistical fluctuations become indispensable for the adequate description of the system. Moreover, the role of the boundary and non-uniformity effects further increases when the sizes of the structures and the number densities and/or the sizes of dust grains shrink below a certain limit when the surface effects become dominant. Indeed, the particles near the (usually sharp) boundaries would then affect the intrinsic properties of smaller non-extensive complex plasma systems, such as dust voids and clouds, various liquid- and crystal-like structures, etc. From the internal organization point of view, the electronic and other properties of the dust particle subsystem also critically depend on the size and organization of the particle. For example, the ability of the nanoparticle agglomerates to interact with the larger (“plasma”) subsystem strongly depends on the number of protoparticles/clusters forming the agglomerate. The smaller the number of constituent particles in the agglomerate, the more important is the role of the surface effects that are determined by the arrangement and sizes of the protoparticles/clusters. Hence, the conventional “canonical” thermodynamic models are not applicable for the bounded complex plasma systems with a limited number of dust particles. In this case, the alternative microcanonical thermodynamics approach [Gross (2001)l can be employed. This approach has been used in the investigation of numerous fundamental physical processes. For example, the Ising model was proposed as a basic model for the magnetism. Some extensions of this model such as Potts, Heisenberg, and “decorated” models were applied for the description of phase transition in compressible multicomponent mixtures, complex fluids, glasses, and colloids [Hansen et al. (1991)]. Amazingly, the optimal functioning of all components of the nervous system is also controlled by the basic laws of microcanonical thermodynamic of colloidal systems. Meanwhile, the microcanonical approach

16

Physics and Applications of Complex Plasmas

has been successfully used in the studies of the thermodynamic properties of solutions or suspensions of colloidal particles, multi-fragmentation in nuclear decay, globular protein solutions and several others. From this point of view, the complex plasmas can be regarded as an ideal model system for the in-depth studies of “small” non-extensive and strongly interactive systems. To this effect, the obvious advantage would be in the unique properties of the complex plasmas such as the relative ease of the generation and manipulation, short time scales of relaxation to the equilibrium state, rapid responses to t h e changes of external parameters, etc. Moreover, the particles can be directly and individually detected and monitored by relatively simple optical techniques, which is almost impossible to do in the case of common liquids and solids. Another obvious benefit of using the complex plasma is the possibility to control the size (and the number of macroscopic particles in) of the system, as well as the character and the range of the interparticle interactions involved. Finally, the investigation of a wide range of the complex plasma parameters would provide one the opportunity to elucidate the fundamentals of the physical behavior of open dissipative systems.

Chapter 2

Basics of Plasma-Dust Particle Interactions

The fundamental question of importance for the understanding of the processes involving formation and evolution of various dust structures in a complex plasma, is the interaction of dust grains with other grains and the surrounding plasma. In most of space and laboratory plasmas, dust grains are charged. The electrostatic charge on a grain is the key element of most of the dusty plasma problems ranging from basic phenomena like interparticle interactions and shielding to self-organization effects in grain dynamics in dust clouds and formation of complex strongly electrostatically coupled systems. Knowledge of the charge is in the foundation of the character of self-organized structures observed in a complex plasma. Therefore, the basics and the most important developments in the theories of dust charging are given the most attention in this chapter. The electrostatic charge appears to be one of the most essential parameters that controls the dynamics of dust particles both in laboratory (with intrinsic natural boundaries) and space (usually without externally imposed boundaries) plasmas under the action of various forces that include but are not limited to the electrostatic force, force of gravity, ion and neutral drag forces, and thermophoretic forces. However, the physics of the plasma-dust interactions is not merely limited by the basic dust charging and action of the forces on the grains (considered in Sec. 2.1). The dust particles generally do not have a constant charge, their interaction with the surrounding plasma is complex, and therefore the dust charge is a non-trivial function of plasma conditions (Sec. 2.2). Moreover, collective effects in the ambient plasma as well as in ensembles of dust grains play important role in the physics of complex plasmas. For example (Sec. 2.3), intcractions of dust particles with ion flows established in the near-electrode areas of laboratory gas discharges often results in the attraction of the like electrostatic

17

18

Physics and Applications of Complex Plasmas

point charges, a situation impossible from the point of view of the standard electrodynamics in vacuum.

2.1

Fundamentals of Dust Charging, Main Forces on the Grains and Dust Levitation

Here, the most established approaches and the most important features of the charging of colloidal particles as well as forces acting on them in a plasma are presented. First, we consider the main mechanisms of dust charging leading to the negative or positive net charge (Sec. 2.1.1). Then we focus on the basic mechanism of dust charging by plasma currents and discuss various phenomena influencing the dust charge (Sec. 2.1.2). Fundamental principles of the kinetic theory of complex plasmas with variable charge dust grains are considered in Sec. 2.1.3. The main forces acting on dust particles in the plasma are studied in Sec. 2.1.4. Since in a typical laboratory discharge the particles are negatively charged and usually levitate in the sheath or pre-sheath region under the balance of gravitational, electrostatic (due to the sheath electric field) and plasma (such as the ion drag) forces, the basic problem of dust particle levitation in the collisionless sheath region is presented in Sec. 2.1.5. 2.1.1

Negatively and positively charged grains

The charging process of the macroscopic particles in a plasma is governed by the contributions of all currents entering (or leaving) the grain surface, involving the plasma electron and ion currents, the photoemission and thermionic emission currents, etc.,

dQ dt

-

=cr.+ i

,

(2.1)

where Q is the grain charge, I, are the currents absorbed and I3 are the currents emitted by the charged particle (with the appropriate sign). The charging equation (2.1) enables one to calculate the dynamic variations of the electric charge on macroscopic particles. In particular, in the stationary (or quasistationary) state ( d Q / d t = 0), all the currents balance each other Ii = -Cj I,) and one can obtain the equilibrium value of the dust charge. We note that the knowledge of the dust charge is one of the most fundamental milestones in the theories of the plasma int,eractions with

(Ci

Basics of Plasma-Dust Particle Interactions

19

macroscopic particles, where the electrostatic interaction plays the most essential role. The specific expressions for the microscopic charging currents entering (1%) or leaving (Ij)the particle surface depend on many factors, including the type and origin of the current (e.g., the flow of the plasma-generated species to or from the grain or the flow of the species released from the grain surface as a result of the interactions between the plasma or external radiation with the macroparticle surface) and model assumptions about the solid particles, the plasma medium, and the interactions between them. For example, it is very important to identify the way the plasma particles move in the vicinity of the dust particles and how they interact with the particle surface. In this regards, the knowledge of the dust particle shielding and interaction with the plasma is important. The most common and basic theory of the dust charging by the microscopic plasma currents was introduced by [Whipple (1981)I. The model involves the individual solid grains and the plasma electrons and ions. The motions of the plasma species in the areas adjacent to the dust particles are limited by the (collisionless in the simplest case) orbital trajectories consistent with the energy, momentum, and angular momentum conservation laws. For this reason, the model of the plasma species motion in the process of the dust charging has been widely accepted as the Orbit Motion Limited approach [Whipple (1981); Havnes et al. (1990b); Goree (1994); Tsytovich (1997)I. This approach has been common for many years in the modeling of the collection of the plasma species by the electrostatic probes of various shapes [Chen (1965)l. Further developments in the field have revealed a number of complex phenomena that make the calculation of the dust charge rather complicated. Here, we will just mention the effects of trapped ions [Lampe et al. (2003)], complex dependence of the grain charge on its size [Samarian and Vladimirov (2003)], stochastic charge fluctuations [Matsoukas et al. (1996); Kortshagen and Bhandarkar (1999)], quantum effects on the grain capacitance and the ability of dust to collect the plasma species [Krcmar et al. (2000)], collective effects in the ensemble of a large number of macroscopic particles [Goree (1994); Verheest (2000)l and several others. Some of these effects are considered in more detail below. In a gas discharge plasma the charge on a dust particle is usually negative. This happens when a non-emissive initially neutral dust grain is charged by the plasma electron and ion fluxes. In this case, the macroscopic particle becomes negatively charged due to much higher mobility

20

Physics and Applications of Complex Plasmas

of the plasma electrons compared to that of the plasma ions. Under typical experimental conditions in a gas-discharge plasma, the micrometer-sized grains acquire the net negative charge of 103-105 elementary charges and the equilibrium surface (floating) potential of a few electron volts below the potential in the plasma bulk. However, in some cases dust particles can have a positive electric charge. Such a situation can develop, e.g., when electrons are emitted from the particle due to electron emission that originates because of the electron impact, UV exposure, thermionic emission, and field emission. The electron emission constitutes a positive current with respect to the particle and decreases its negative charge; if it is large enough, it can cause the particle to be positively charged, in contrast to to the above case when the charging is dominated by the plasma currents only. Note that even if the particle is not always positive, the charge on it might sometimes fluctuate to a positive level: Moreover, due to the electron emission, a two-component system where the electrons are emitted from the dust particles might exist. In this case, the equilibrium potential (charge) of a dust particle can be achieved as a result of the balance between the absorption of the electron current on the grain surface and the emission current from the surface. Such a system can be used as a simple model system for the investigation of various processes related to the emission-controlled charging of dust grains. The secondary emission yield 6 depends on both the impact energy and particle material. The yield is generally much larger for the electron impact than for the ion impact. However, the secondary electron emission due to the ion impact can be important for ion energies above a few keV, which can be the case in the plasma immersion ion implantation and ion extraction experiments. The secondary emission from small particles can be significantly enhanced above the value of bulk materials. This was shown by [Chow et al. (1994)], whose theory also included relevant geometric effects. Scattered electrons escape more easily from a small particle than from a semi-infinite slab of material, so 6 is also enhanced. The expressions for the secondary electron currents caused by impinging electron currents are normally complex functions of the impact energy, maximum-yield energy, temperatures of the plasma and emitted (secondary) electrons, surface potential of the grain, and are presented for the case of Maxwellian electron energy distribution functions by [Goree (1994)I. Sub-micron and nano-sized particles also feature a pronounced field emission due to strong electrostatic fields near strongly curved (or sharp) surfaces. The critical fields for the ion (for positively charged particles) N

Basics of Plasma-Dust Particle Interactions

21

and electron (for negatively charged dusts) emission vary depending on the material and are typically in the range 106-10s V/cm. This mechanism has recently been of a great importance due t o the fabrication of carbon-based electron field emitting structures in chemically active complex plasmas. The temperature of emitted electrons Tee strongly depends on the material of the particles. For example, in a rarefied two-component plasma that consists of the macroscopic particles and the emitted electrons, the recombination rate on the particles can exceed the rate of the thermal energy loss due t o collisions of the plasma electrons with the neutrals, Teecan be of the same order or even close to the temperature of the plasma electrons T,. For some value of the positive charge, the emitted electrons cannot overcome the potential barrier on the particle surface. Thus the equilibrium dust charge of the particle in this case is controlled by the balance of the emission and the external electron currents. On the other hand, if the particles are charged negatively, the electric field is directed in such a way that the electrons are accelerated from the surface of the particle. In such a way the resulting emission current can be enhanced. In thermal plasmas the electron current can be emitted thermionically, which results in the positive dust charge [Fortov et al. (1996a); Rosenberg et al. (1999)]. The thermionic emission is thus of particular importance for the charging processes in thermal complex plasmas. In the equilibrium state, the plasma can be characterized by the effective temperature T , which is usually equal to the temperatures of all the plasma species (T = Te = Ti = T,, where the subscripts e , i , and n stand for electrons, ions, and neutrals). Apparently, the thermionic current increases with the grain surface temperature. In this case the electron density in the thermal plasma with the presence of the dust particles is determined by two processes. The first is the thermionic emission and the electron absorption on the surface of the particles. The second is the ionization-recombination process in the gas phase. The ionization potential for most gases exceeds 10 eV. Therefore, a t the temperatures up to 3000 K, the density of the charged species is determined by the concentration of the easily ionized impurities. The presence of strongly emissive alkaline metals can cause an additional electron flux onto the particles. As a result, the sign of the grain charge can be reversed. The photoelectron emission can also play a significant role in the charging process. It is caused by intensive UV-fluxes that release photoelectrons and thus lead to positive grain charging currents. If the dust is a subject of an intense photon radiation with the energy hu exceeding the photoe-

22

Physics and Applications of Complex Plasmas

mission work function, the number of the photoelectrons emitted from the surface is proportional to the intensity of the incident radiation I % where , J = Pi/hv is the number of quanta incoming per unit surface area per unit time. The maximum kinetic energy of the photoelectrons Kmax = mev2/2 grows linearly with the frequency of the radiation and does not depend on the intensity: K,,,max= hv - W,, where W, = hvo is the work function and vo corresponds to the long-wave limit of the photoeffect. A typical value for the photoelectron energy W, for most materials does not exceed 6 eV. For example, under low and moderate radiation fluxes (Wi < 1 W/cm2) W, is usually 2-3 eV for metals and is slightly higher for most semiconductors and dielectrics. Thus, photons with the energy hv < 12 eV can charge the particles without ionizing the working gases, such as neon or argon. The resulting plasma will consist of the positively charged macroscopic particles and the emitted electrons. Note that the main problem for the experimental observation of the structures made of the positive macroscopic particles in a complex plasma is the absence of an effective trap for the positive charges. Strong UV fluxes can also originate in low-temperature plasmas of gas mixtures containing highly-radiant gases with high excitation and UV emitting capacity. In SiH4+Ar gas mixtures of many plasma processing tools, UV-radiation from excited argon atoms has shown to be a factor affecting the coagulation (via the equilibrium electrostatic charge) of nanometersized particles [Kortshagen and Bhandarkar (1999)]. If the flux of the emitted photoelectrons is strong enough, the grain can charge positively. The photoemission depends on the material properties of the grain particle (its photoemission efficiency). It also depends on the particle surface potential, because a positively charged particle can recapture a fraction of its photoelectrons. Taking this into a,ccount, the photoemission current is given by [Goertz (1989)]

where T, is the uv flux and yph is the photoemission efficiency (typically, yph M 1 for metals and yph M 0.1 for dielectrics). The above expressions assume an isotropic source of UV radiation and Maxwellian distribution of the photoelectrons with temperature T,. Note that in very low-density plasmas of various space environments, the

Basics of Plasma-Dust Particle Interactions

23

photoelectric charging mechanism can be a dominant one. For example, the photoelectric charging measurements were performed on isolated grains and grains near a photoemissive surface in vacuum [Sickafoose et al. (2000)l. Isolated grains reach a positive-equilibrium floating potential, dependent upon the work function of the particle, which causes the emitted electrons t o be returned. Dust particles near a photoemitting surface can reach a negative floating potential, which reaches the equilibrium when the sum of the emitted and collected currents is zero. The particles tested are 90106 pm in diameter and are composed of Zn, Cu, graphite, and glass. Examples given in this section do not cover the overwhelming variety of the charging mechanisms and scenarios of nano- and micron-sized particles in low-temperature plasmas. Some more discussions on the physics of dust charging are given by [Mendis (2003)l.

2.1.2

Charging by plasma currents

2.1.2.1

Orbit Motion Limited approximation

Most charging theories for dust grains are based on the theories of the electrostatic probes in a plasma [Chen (1965)l. The probe theories predict the electron and ion currents to the probe. The currents are termed “orbitlimited” when the condition a << X D ~<< ,fX, mfp applies, where a is the particle radius, X D ~is the ion Debye length, and,fX, mfpis a collisional mean free path between neutral gas atoms and either electrons or ions [Goree (1994)]. In this case, the plasma currents can be calculated assuming that the electrons and ions are collected if their collisionless orbits intersect the probe surface. The Orbit Motion Limited approach has two major assumptions, specifically [Tsytovich (1997)l: 0

0

independently of the structure of the electrostatic potential in the plasma near the dust particle, any plasma particle which initially is far from the dust particle, can reach the surface of the dust particle, and if it is allowed by the conservation laws, will attach to the dust particle; for spherical dust particles, the limiting impact parameter of a plasma particle (when the plasma particle is absorbed by the dust particle) corresponds t o a tangential trajectory to the dust particle.

An obvious advantage of this model is that the cross sections of the plasma particle attachment to the dust can be found using the energy and

24

Physics and Applacations of Complex Plasmas

momentum conservation laws only, independently of the complexity and nonlinearity of the plasma potential close t o the dust particle [Tsytovich (1997)l. For the collection of Maxwellian electrons and ions characterized by the temperatures Te and Ti, the orbit-limited charging currents for an isolated spherical particle are given by [Whipple (19Sl)l (2.2)

(2.3)

(2.4)

(2.5) where 4, is the surface potential of the particle relative to the local plasma potential and Zi is the electronic charge of the ions. The coefficients loe and Ioi represent the current collected at d s = 0, and are given by (2.6) where na is the number density of the plasma species cy (in the simplest case, electrons and ions). Here, fol(w,V~,) is a complicated function of the thermal velocity V T = ~ (Tol/ma)1/2 and the drift velocity between the plasma and the particles. Simple expressions can be obtained in the cases of small ( ~ / V << T 1) ~ and large ( ~ / V T ,>> 1)drift velocities. The first case corresponds t o the situation when the dust charging is due to the spherically symmetrical flows of the plasma species with the thermal velocity VT, and the expressions for the microscopic electron/aon flows are

I,, = 4nu 2 n,qa(T,/2nm,)1’2

,

(2.7)

where it was noted that the plasma particle collection cross-section is 4 n a 2 due to the spherical symmetry of the problem. The dust charge Q is related t o the particle surface potential ds by Q = C @ swhere , C is the capacitance of the particle in the plasma. For a spherical particle with a << X D ~ , the capacitance is C cc a. The case of large w / V T ~>> 1 drift velocities is considered in detail in Sec. 2.2.2.

Basics of Plasma-Dust Particle Interactions

25

The standard model of dust charging by plasma currents assumes that the grain charge varies due to collection of electron and ion charging currents from the adjacent plasma according to 2.1

dQ dt

- = XI,

(2.8)

In equilibrium, one can set XI, = 0 and obtain the equilibrium particle surface potential qbSo and the particle equilibrium charge Qo. Generally speaking, since the electrons have higher thermal velocities and higher mobility than the ions, the electric charge and floating surface potential of the particle are negative. Note that $s is independent on the particle size, whereas the charge in this model is directly proportional to the particle radius, Q c( a. The solution for the equilibrium dust charge significantly depends on whether the charging for an individual particle (or a rarefied dust cloud) or for a dense ensemble of particles is considered. Specifically, this is reflected in the quasineutrality condition (the negative ions are neglected for simplicity here)

where P, = ndlZdl/ne [Havnes et al. (1990b)I stands for the ratio of the charge density carried by the dust particles t o the electron number density. Here, zd, n d , and ne are the dimensionless dust charge (in units of the electron charge), and number densities of dust particles and electrons, respectively. If P, << 1, we deal with the individual grain charging case, otherwise the effects of dense particle clouds should be taken into account. Indeed, when Pe > 1, the charge and the floating potential can be noticeably diminished as compared to the values obtained in the case when P, << 1. The effect of the dust charge proportion Pe on z d and qbs can be analyzed using the steady state equation for the microscopic grain currents XI,, = 0, which can be rewritten into a convenient non-dimensional form for the electropositive plasma with the single ion species [Tsytovich (1997)] exp(-Z)

=

(1

+ Pe)(. + 2),

(2.9)

v 6

where Z = IZdle2/aTe,r = Ti/T,, and p = mi/m,. Here, me, and mi are the electron and ion masses. The simplest illustrative solution of Eq. (2.9) for a hydrogen plasma with T , = Ti and P, << 1 yields 2 = 2.5. In an RF discharge, the dust density is often high enough to yield Pe >> 1.

26

Physics and Applications of Complex Plasmas

-

For example, for the representative parameters of the experiments of [Boufendi and Bouchoule (1994)l described in Sec. 7.1 (ni 5 x lo9 ~ m nd 10’ ~ m - T~e , 2 eV), the parameter P, appears to be approximately 3.2. Hence, from Eq. (2.9) one can deduce that the particle charge can be 60% smaller than the charge of an individual dust particle under the same conditions [Goree (1994)]. In principle, grains in dense dust clouds could have larger, as compared to the case of individual particles, capacitance. Each of the grains can be regarded as a tiny capacitor with an inner conducting spherical core and a near-grain plasma sheath as an outer capacitor shell. The closer dust packing results in a smaller interparticle spacing, which can become comparable to X D ~ . In this case the near-grain sheaths shrink, which increases the effective capacitance of the grain-sheath structure. An increase of the capacitance can also be regarded as an effective increase in the dust size a and the effective area of collection of the plasma particles. Hence, each grain in a dust cloud would be capable of collecting more plasma electrons than any individual particle of the same size. However, the number of electrons in the plasma is controlled by the established creation-loss species balance in the discharge and the average number of electrons available per grain is smaller than it would be in the case of an individual grain, which does result in a substantial decline of the average dust charge as well as the grain surface potential. Furthermore, an increase of the relative charge density of dust particles P, enhances the charge imbalance between the electrons and ions in the plasma and elevates the overall electrostatic potential of the cloud [Verheest (2000)l. The experimental procedure of [Barkan et al. (1994)l (see also in Sec. 3.3.1) involved the measurement of the dust charge. Variation of the electron density n,o resulted in significant changes in the ratio between the -113 intergrain spacing d nd and the electron Debye length X D ~ ,which, in turn, affected the dust grain charge values due to the “close packing effect”. Specifically, for the parameters of the experiment, the measured value z d approaches to the charge of the isolated grain of M 3000 when the ratio d/Xo, 5 and decreases to z d M 30 when d/Xo, N 0.5. Thus, the reduction of the dust charge in a dense dust cloud compared to an isolated dust grain has been confirmed experimentally. The charge reduction phenomenon in clouds of dust particles affects the formation of ordered complex plasma structures. The efficiency of dust coupling is proportional to Z i / d , where d is the interparticle spacing. Hence, the reduction of the average grain electrostatic charge in the ensemble can

-

N

-

-

Basics of Plasma-Dust Particle Interactions

27

-

be interpreted as a natural reaction of the system to impede the formation of the coupled state. For this reason, very dense (up to n d lo1’ ~ m clouds of small particles (such as those generated in the plasma processing experiments discussed in Chapter 7) carrying only a few electron charges most often remain weakly coupled. However, in situations when the interparticle spacing becomes small enough, the coupling effects prevail provided that the dust charge exceeds a certain value which can be approximately estimated from rc 170 [Ikezi (1986)l. It has also been shown by [Avinash et al. (2003)I that the dust grains, when packed closely (intergrain spacing is less that the Debye length), shield each other via charge reduction. In other words, dust charge reduction is a consequence of the mutual screening of grain charges. Since in the laboratory dusty plasma experiments various (conducting as well as dielectric or dielectric-coated) particles are usually used, the basic charging equation (2.9) can be generalized to adopt the case of insulating surfaces and account for the adsorption/desorption, surface recombination and diffusion of incoming charge carriers. In the simplified model, the stationary case was determined by the balance between the number densities of the adsorbed charge carriers and those that desorb or recombine. In this case, the balance equations of the adsorbed particles can be written as [Kersten et al. (2000)]

-

(2.10)

(2.11) where f l ( e d , i d ) are the electron/ion number density on the particle, j(e,i) are the current densities towards the particle (given by the OML theory), S(,,i) are the sticking probabilities, T ( R ~ , R(Tp) ~ ) are the residence times of the adsorbed particles, O(,,i) are the fractions of surface coverage, CYRis the recombination coefficient, and Tp is the particle temperature. The net negative charge on the particle surface is gp

= ged

-

cid = Q/Ap ,

where Q and A, are the charge and surface area of the grain particle. The simplest dust charging model discussed above can be extended into more complicated cases of charging in multiple-species plasmas (by including negative ions), streaming effects, charge fluctuations above the equilib-

28

Physics and Applications of Complex Plasmas

rium value, dust polarization phenomena, geometrical/shape effects, and so on. Some of the above situations are considered below in this chapter. Despite a significant progress in the studies of the dust charging mechanisms, there are still many unsolved problems. For example, very little is known about the charging of dust grains in magnetized plasmas. In particular, this is due to the overwhelming complexity to self-consistent,ly incorporate the effects of finite Debye lengths, collisions, and magnetic fieldaffected particle trajectories near the grain surface [Laframboise and Sonmor (1993)l. However, the simplest expressions for the electron and ion currents (2.2)-(2.5) are fairly accurate in the limit of weak magnetic fields when p~ >> a, where p~ = V T ~ / W , , is the electron Larmor radius, and weeis the electron cyclotron frequency [Ostrikov and Yu (1999b)l. Another apparent challenge in the development of the adequate models of dust charging in non-stationary plasmas. An example is the so-called dust “decharging” (neutralization) phenomenon in the discharge afterglow (after switching the discharge power off) [Morfill et al. (2002)l. 2.1.2.2

Charging of small and non-spherical particles

For nanometer-sized particles (nanoparticles), which are often merely macromolecules or atomic/molecular clusters, one has to consider the specific cross-sections for the electron/ion capture. Therefore, one cannot a priori predict the charge of an arbitrary nanometer particle, and one can also expect noticeable charge fluctuations [Chunshi and Goree (1994)]. The latter continuous fluctuations arise from the stochastic nature of the charging process (e.g., stochastic nature of encounters between particles in the plasma) and the discrete nature of the electric charge [Matsoukas et al. (1996)] (details of the charging dynamics and charge fluctuations is given in Sec. 2.2.1). Another effect that complicates description of the charging of very small (nanometer-size) particles is an instantaneous charge separation on the nanoparticle surface under the action of external electrostatic fields such as the fields of the charged plasma species. Thus, the finite polarizability of the grains gives rise to induced (image) forces. The latter forces are always attractive and contribute to the enhancement of the nanoparticle collision cross-sections with the electrons and ions [Matsoukas et al. (1996)l. Moreover, highly polarizable conductive nanoclusters and nanoparticles are strongly affected by the long-range image forces as a result of their

29

Basics of Plasma-Dust Particle Interactions

interaction with the low-energy plasma electrons. The resulting electronnanoparticle interaction potential Vpol = -adiPe2/2r&, which is proportional t o the nanoparticle dipole polarizability Q d i p and inversely proportional to the distance between the electron and nanoparticle T,,en thus contributes to the larger electron capture cross sections and has recently been invoked in the explanation of the enhanced electron attachment t o C60 fullerenes [Kasperovich et al. (2000)]. The resulting size-dependent electron capture cross-section is oe(€)= m2 ( 2 7 r 2 a d i p e 2 / E ) 1 / 2 and represents a sum of the nanoparticle hard-sphere area and the Langevin cross-section. This result has been experimentally verified via precise measurements of the absolute cross-sections for the capture of low-energy electrons by the large Nag000 nanoclusters ( a 4 nm) [Kasperovich et al. (2000)l. Furthermore, the image forces modify the charge and variance of a 1nm particle by 30% [Matsoukas et al. (1996)]. However, this effect is reduced to 15% for 10 nm particles. Naturally, the enhanced electron capture by highly polarizable nanoparticles results in the collection of larger amounts of negative charge and hence to the shift of the charge distribution to more negative values. In some circumstances, a precise quantification of the electronic levels in individual nanoparticles/nanoclusters may be in order. For example, the results on the electronic properties of silicon nanocrystals (nc-Si) grown on Si(100) [Baron et al. (2001)], where the charging and confinement energies of single hemispherical quantum dots were deduced from I ( V ) curves recorded using scanning tunneling microscopy. Nevertheless, simpler semiclassical considerations have been recently used to estimate the capacitance and polarizability of nanospheres and nanocylinders in a Thomas-Fermi approximation [Krcmar et al. (2000); Krcmar et al. (2003a); Krcmar et al. (2003b)l. In this case the nanoparticle electronic level structure is dense enough to warrant a semiclassical treatment but not so dense that the contribution of the finite density of states is negligible. The corresponding Thomas-Fermi capacitance CTF is a complex function of the ratio a l l s , where 1, is the charge screening distance that depends on the density of states per unit volume at the Fermi level. It is notable that CTF < Co, where CO= a is the classical nanoparticle self-capacitance. The addition of surface localized (bound) charge states drives the actual nanoparticle capacitance towards the classical limit. These results are relevant to metallic and semiconductor nanoparticles and various carbon-based nanostructures. Interaction of nano-sized particles with the high-energy plasma electrons can be even more complicated due to the tunneling effects, when

+

-

-

-

30

P h y s i c s and Applicatzons of Complex Plasmas

electrons can pass through the grain. However, the tunneling currents are non-negligible for the electron energies & > 1 0 4 d m eV [Verheest (2000)l. The particle shape is another important factor affecting the charging process. This has been clearly demonstrated by modeling and experiments on the levitation of rod-like dust particles in a plasma sheath. For example, in the experiment by [Annaratone et al. (2001)] two kinds of nylon microrods of the diameters 7.5 and 15 pm and the typical lengths of 0.3 mm, with some as long as 0.8 mm, were used. The microrods were levitated in a collisional sheath of the RF plasma. The rods below a critical length settled vertically, parallel to the electric field, while the longer rods floated horizontally. In the case of the levitation of 7.5 pm thick microrods in a 52 P a discharge in krypton with the characteristic electron temperature of 2.2 eV and the electron number density of 3 x lo9 cmP3, the critical orientation length appears to be 0.37 mm. The typical intergrain distance was varied in the experiments but remained within the range of 0.3 to 1.0 mm. Charging of rod-like particles can be modeled assuming that the charge redistribution over the particle surface requires much shorter time scales compared with the charging time scales due to the plasma particle absorption [Ivlev et al. (2003)l. In this case the surface potential can be considered as a constant and the problem is solved through the calculation of the total ion flux onto the particle surface.

-

2.1.3

Particle charge - a new kinetic variable

The process of dust charging by plasma currents discussed above in Sec. 2.1.2 leads to new and unusual phenomena in complex plasma systems. In particular, it causes specific dissipation, associated with the dust charging rate, in a number of collective plasma excitations. 2.1.3.1

Dust charging rate

Consider the basic charging equation

where Q is the charge residing on a dust grain, and the current I on the dust particle is a sum of electron and ion plasma currents. In the state of

Basics of

Plasma-Dust Particle Interactions

31

equilibrium, we have

Ieq(Qeq)z IZq(Qeq)+ IF(Qeq)= 0, The charging dissipative process is described by the charging frequency

ud

(2.12) which is a characteristic of the process of charging of the dust particles when the deviations from the equilibrium are small. Naturally, Eq. (2.12) contains contributions from both the plasma electrons and ions. Comparison of the rate (2.12) with the plasma collision frequencies reveals that the charging process can be one of the most important dissipative process in a complex plasma. Another characteristic is the rate of capture of the plasma electrons by the dust grains: (2.13) which is particularly important when the ion contribution can be neglected [Vladimirov (1994)l. 2.1.3.2 K i n e t i c theory of d u s t y p l a s m a s with variable charge The kinetic theory of dusty plasmas with variable charges on fine particles can be developed [Tsytovich and Havnes (1993); Vladimirov (1994)l by involving the dust charge as an additional dynamic variable. Specifically, this is done by introducing the d u s t distribution f u n c t i o n

where the electrostatic charge q is included as an additional independent variable. The first moment of the distribution function nd =

1

fddq

= const

,

(2.14)

gives the density of dust particles, and (2.15)

32

Physics and Applications of Complex Plasmas

is the averaged charge on a dust grain. The corresponding kinetic equation can be written as

(2.16) where it was assumed that the grains have infinite masses and, hence their velocity is negligible (vd = 0). The next momcnt of the distribution function

(2.17) yields the averaged current flowing onto the particle surface. From Eqs. (2.15) and (2.17) it is easily seen that (2.18) which in the equilibrium state transforms to

J ( Q = -Zde)

=0,

(2.19)

where the equilibrium state is described by the equilibrium distribution function fdeq of charges on dust particles. The corresponding kinetic equations for the electrons or ions taking into account charging collisions is given by

where fa is the electron/ion distribution function. In the equilibrium state, one has the equilibrium electron (ion) distribution fzq. Now, introducing (small) perturbations of the above distribution functions

where ldfdl << I f d l , 16faI << Ifcu\, and assuming that the capture of electrons and ions by the dust particles is compensated (in the zero-order approximation) by external sources, one can linearize kinetic equations (2.16) and (2.17) with respect to these perturbations arid obtain the following equation (2.20)

33

Basics of Plasma-Dust Particle Interactions

for the dust particle distribution function, where

(2.21) For the electron (ion) distributions one obtains (2.22) where

l/,d

is the rate of electron (ion) capture, (2.23)

After Fourier-transform

J'

Akw = - A(r, t ) exp(-ik. r

+iwt)drdt ,

one finds from Eq. (2.22) that

(2.24) Now, Eq. (2.20) can be integrated over the charge domain to obtain J'afddq

=0

,

(2.25)

which is consistent with n d = const. Furthermore, one can multiply Eq. (2.20) by q and integrate 0ver.q to find

=

-U?nd6Qk,

f

.I

bI(q)kwfiqdq,

(2.26)

where v:" is the charging frequency (2.12). To obtain the last expression in the r.h.s. of Eq. (2.26), the particular dependence of the cross-section CT on u was used [Vladimirov (1994)]. Therefore, the last equality of Eq. (2.26) is correct only for those CT which are linear functions of u.

34

Physics and Applications of Complex Plasmas

To calculate the last term in the r.h.s. of Eq. (2.26)' the perturbed charging current SI(q) can be split into the two parts

where

and

(2.27) where (2.28) In deriving the last equality in the r.h.s. of Eq. (2.27), we again have used the linear dependence of on v. This is the reason for the factor (T' [given by Eq. (2.28)] to appear in Eq. (2.27). Thus, for the perturbation of the averaged charge on dust grains one obtains (2.29) where the factor Gkw in Eq. (2.29) is given by

(2.30) and we have used Eq. (2.23),

and

Qeq

= -Zde is

the equilibrium charge of a dust particle.

Basics of Plasma-Dust Particle Interactions

35

Finally, for the linear perturbation of the electron (ion) distribution function one finds

(2.31) Expressions (2.29) and (2.31) contain all the necessary information for further calculation of the linear dielectric permittivity and other parameters of dusty plasmas with variable charges on dust particles. The resulting dielectric functions feature the imaginary parts related to the grain charging process and do not persist in the constant dust charge approximation.

2.1.4

Main forces acting on dust particles i n a plasma

The first apparent force that significantly affects the dust particle dynamics is the gravity force. It has the form Fg = m d g , where md is the dust mass and g is the gravitational acceleration. The second one, electrostatic force FE = Q E , is exerted by the electric field E . It is important that the ion velocity that enters the expression for the microscopic ion current charging the grain is a local function of the electric field (and can be related to E via the ion mobility). Hence, the electric charge of the particle is also indirectly depending on the local value of E . The third one, friction force Ffrresults from collisions with neutral gas atoms/molecules (in a gas discharge chamber, it is therefore proportional to the working gas pressure). Interactions between the neutrals and the dust particles can be assumed to be hard sphere, elastic collisions. Using the relevant momentum cross-sections, the average momentum transferred per neutral is given by the average relative neutral velocity multiplied by the reduced mass (which is close to the neutral's mass m,, since m d >> mn) and the cross-sectional area. Thus, the neutral drag force is approximated by [Barnes et al. (1992)] (2.32) where n, is the density of neutrals, v~ is the relative particle-neutral velocity, and un-d is the cross-section for the dust particle-neutral interaction, which can approximately be taken as r a 2 . When the grain is drifting, this force is in the direction opposite to the motion. Alternatively, when there

36

Physics and Applications of Complex Plasmas

is a net flows of neutral gas molecules, there is a momentum transfer to the particles in the direction of these flows. The ion drug force F d r is caused by the momentum transfer from the positive ion current driven by the electric field. This force consists of two components: the collection force F& and the orbit force F& [Barnes et al. (1992)l. The collection force represents the momentum transfer from all ions that are collected by the particle. Each ion that impacts the grain transfers its original momentum m,vi. Hence, this component is given by (2.33)

+

where ni and mi are the ion density and mass, respectively, u, = (v: v$i)1/2 is the mean ion speed that includes the directional (in the electric field) zli and the thermal v ~ =i ( 8 T i / ~ m i ) components. ~/~ The collection impact parameter is given by

where pp - 'ps is the difference between the local plasma (pp)and the particle surface (ps)potentials. The above expression is based on the orbit motion limited probe theory. The orbit force is given by

F& = 4.irnivsmivib:12R,

(2.34)

where b,lz = e Q / m i v ; is the impact parameter with the asymptotic orbit angle of .ir/2 and

is the Coulomb logarithm integrated over the interval from b, to Xni. The contribution from the orbit force is negligible when the collection impact parameter is greater than or equal to the ion Debye length. Note that the standard calculation of the ion-dust elastic scattering (orbital component) cross-section could fail in a complex plasma due to larger than the Debye length range of the ion-grain interactions [Khrapak et ul. (2002)]. The .conventional expression of \Barnes et al. (1992)) uses the Debye length as a cut-off in the impact parameter. The above forces acting on the dust particles critically depend on the ion velocity, particle size and specific position within the discharge. In particular, each of the forces has a different power-law dependency on the

Basics of Plasma-Dust Particle Interactions

37

particle radius suggesting that the dominant force changes as the particle grows. The electrostatic force FE is proportional to the particle radius since the charge depends linearly on the radius. The momentum transfer forces Ff, and F d r , are proportional to the particle surface area and hence the radius squared. Finally, the gravity force Fg is proportional to the particle's mass, which is proportional to u 3 . The forces on the particles are very sensitive to the location within the sheath. [Barnes et al. (1992)] show that in argon discharge with neutral 10 gas pressure of 100 mTorr, plasma density of 10" ~ m - T, ~ ,= 2 eV, Ti = 0.05 eV for 100 nm particles with the densities twice the density of the water, the ion drag force moves the particle towards the wall/electrode until it senses the electric field of approximately 14 V/cm in the sheath. At this point the electrostatic and ion drag forces balance each other. For a 1 pm particle, this point occurs for an electric field of approximately 35 V/cm. Finally, the 10 ,urn particles are pulled towards the lower wall surface by the gravity force until the electric field of 82.6 V/cm is reached. In the low-density plasmas, the ion drag force is not a dominant force for the particles of this size. Many experimental observations suggest that some complex plasma structures can be supported by the thermophoretic force FT. The latter originates due to the temperature gradient of the neutral gas VT, from the electrode towards the bulk plasma. This force can lift the dust particles from the near-electrode area back to the central part of the discharge. The thermophoretic force opened up several new opportunities for the observation of the microparticles in the plasma bulk and facilitate the investigation of complex plasma structures such as dusty plasma voids. The force FT acting on a spherical particle in a monoatomic gas at a low pressure is given by

FT

=

-

3.33(a2/aat)VT, ,

where it was assumed that the mean free path is much larger than the particle radius. Here, cat is the gas kinetic cross-section for the atomic scattering. Note that temperature effects can result in additional forces acting on the dust particles. For instance, the supersonic ion flows yield inhomogeneous temperature distributions on the grain surface. In this case the interaction of the grain with the neutral gas results in a force similar to photophoresis, although the latter is provided by a plasma recombination on the grain surface [Ignatov and Amiranashvili (2001)].

38

Physics and Applications of Complex Plasmas

A series of elegant experiments on the qualitative determination of the ion drag, thermophoretic, and electrostatic forces was reported by [Zafiu et al. (2003)I. The principle of these investigations was to study the free falling particles where the force of gravity is balanced in the vertical direction by the neutral gas friction at a constant particle velocity, whereas all other forces act in the horizontal direction. In this way, it appeared possible to decouple the effects of the forces in two mutually perpendicular directions. This case is important to understand the frequently observed formation of dust voids in microgravity experiments. A dedicated experiment was performed to characterize the forces that act upon the dust particles in the void region of a complex plasma [Thomas e t al. (2002)] (more on dust voids is in Sec. 5.1). A dust void in an RF-generated complex plasma was produced in a laboratory experiment with an externally applied temperature gradient. Laser flashing techniques were used to identify the dust and to obtain two-dimensional maps of the particle velocities. From these measurements] two-dimensional maps of the acceleration and force acting on the grains in the void were obtained [Thomas et al. (2002)]. 2.1.5

Dust levitation in a collisionless sheath region

The physics of the sheath region features strong electric fields, plasma density gradients, and ion flows. According to the Bohm criterion, the average velocity of the plasma ion flow in the sheath region exceeds the ion sound speed. The standard model of the sheath [Chen (1984)I considers the Boltzmann distributed electrons and the ballistic (cold) ions. The discharge pressure is assumed to be low enough that the ion collisions with neutrals and other species can be neglected. If the number density of dust particles is low enough, the influence of the dust grains on the field distribution in the sheath region can be ignored. The ion continuity equation gives the ion density ni in terms of the density no and velocity wo of the ion flow in the plasma bulk, i e . , far away from the electrode]

where $ ( z ) is the sheath potential. The collisionless sheath structure depends on the ballistic ions within the sheath with the distribution function fi o ( 6(v1)6(vZ - vi(z)), where vi(z) is the ion streaming velocity at the

39

Basics of Plasma-Dust Particle Interactions

distance z from the electrode. The ion velocity determined from the energy balance equation is

The sheath potential is found from Poisson's equation (2.35)

where the total charge contributed by the dust grains is neglected. The case of a higher pressure, with a sheath structured by ions drifting through the background neutrals, is considered in Sec. 4.1. Equation (2.35) can be integrated once to give (applying the boundary conditions E ( z = 0;)) = cp(z = 0;)) = 0) the squared sheath electric field (2.36)

UP

where = T,/mi is the squared ion-sound speed, and LI = 1 (2ecp(.~)/T~)(v,/uo)~. Assuming the electrode has a potential of -4 V, typical of dust plasma experiments in the collisionless regime, Eq. (2.36) can be numerically integrated to give the dependence of the potential, and thence of the sheath electric field, on the distance from the electrode. Dependence of the Mach number M ( z ) = wi(z)/us of the ion flow on the distance from the electrode is presented in Fig. 2.1.

=

Fig. 2.1 Dependence of the Mach number of the ion flow M ( z ) vi(z)/vs on the distance h = z / A D from the electrode [Vladimirov et al. (1999a)I. Here we have MO = vo/vs = 1, AD = 3 x lo-' cm, Te = 1 eV, and m,/me = 40 x lo3.

40

Physics and Applications of Complex Plasmas

Linearizing Eq. (2.36) with respect to small potential and field variations, one can find

where the electric field EO is at z = 0. Now, if to assume that the sheath electric field near the position of the dust grains can be linearly approximated, one obtains

Thus the effective width of the potential well trapping dust particles can be written as (2.37) The charge Q of the dust particles (which depends on the sheath parameters, in particular, on the local electric sheath potential) is found from the standard condition of a zero total plasma current onto the grain surface I ( Q ) = l e ( Q )+ & ( Q ) = 0. For the collective processes investigated on the time scale of the characteristic frequencies of the order of a few tens of s-l, which are much lower than the charging frequency [Vladimirov (1994)l (that can be of the order of lo5 s-'), the (re)charging of dust grains is practically instantaneous, and their charging dynamics can be ignored. The electrons are Boltzmann distributed, and the possible changes of the electron temperature in the plasma sheath are neglected. The inter-grain distance is assumed not less than the Debye length, so that the ion trajectory is affected by only a single grain. Thus the charge of a dust particle in the sheath region is determined by

(2.38)

If the electrode potential is sufficiently negative, and a dust grain is very close to the electrode, its charge can become zero and, possibly, positive. Since the electrode is negatively biased (due to the self-bias and/or the externally applied bias), this means that the (positively charged) particle cannot levitate at this distance and falls onto the electrode. For a particle

Basics of Plasma-Dust Particle Interactions

41

Fig. 2.2 Dependence of the charge q = - ( Q / e ) x 10-3 of the dust grain, levitating in the sheath electric field, on its size [Vladimirov et al. (1999a)l. Here, Mo = 1, AD = 3 x lop2 cm, T, = 1 eV, milme = 40 x lo3, p = 1 g/cm3, and amax= 0.375 x lop3 cm.

I

Fig. 2.3 Dependence of the position h = z / X D ~ of the dust grain, levitating in the sheath electric field, on its size [Vladimirov et al. (1999a)l. The sheath plasma parameters are the same as in Fig. 2.2; the position, corresponding t o a,,, is zmin = 1 . 6 4 A ~ o .

levitating in the sheath field, the sheath electrostatic force on the grain is FE = Q ( z ) E ( z ) ,where the instantaneous transfer of charge onto and off the dust grain at any grain position in the sheath is assumed, such that Eq. (2.38) is always satisfied. The balance of forces in the vertical direction also includes the gravitational force Fg = m d g . Thus in the simplest case the levitation of the grain appears as a result of the balance between the electrostatic force and the gravitational force, namely,

42

Physics and Applications of Complex Plasmas

Solution of this equation together with the charging equation (2.38) gives the dependence of the charge of the grainas a function of its size, an example is shown in Fig. 2.2. For the levitating dust particle, there is a one-toone correspondence of its size to its equilibrium position of levitation in the sheath, as shown in Fig. 2.3. Note that the equilibrium solutions for positions closer than z,in = 1.64X0, where the grain has a critical size a = acr, are unstable with respect to vertical oscillations, see Sec. 6.1.

2.2 2.2.1

Developments of the Theory of Dust Charging

Charging dynamics and charge fluctuations

The charging equation (2.8) was obtained by assuming the continuity of the charging process and the equal electric charge on the same size grains. This is apparently not the case for small nano-sized particles that typically carry a positive or negative charge of a few to few tens of electric charges. The charge fluctuations can be quantified by the probability p , of the grain to carry the electric charge qe and accounting for stepwise changes of the grain charge in the positive and negative directions upon collisions with the plasma electrons and ions. In the model of monodisperse cloud of identical non-interacting cloud, the discrete population balance equation is [Matsoukas et al. (1996)l (2.39)

+

where J ( q 1 4 ) = Ie,q+lPq+l - Ii,qpq and J ( q - 1 4 ) = &,,-ipq-i Ie,qpq are the net flows of particles from the charging state q 1 to q and from q - 1 to q, respectively. The solutions for each of the charging states q are found recursively by succssive iterations by noting that in the equilibrium d ( q 1 + q ) = J ( q - 1 --+ q ) = 0 and enforcing the appropriate normalization conditions. Note that the charging states of larger particles with relatively slower changes of f q with charge can be treated as a quasicontinuous function f ( q ) . In this case Eq. (2.39) can be reduced to the Fokker-Planck equation: +

+

+

+

(2.40)

where the second term on the right hand side of Eq. (2.40) is the source of fluctuations around the eqyilibrium value qo obtained from the equality of

Basics of Plasma-Dust Particle Interactions

43

electron and ion currents. The term proportional to the current imbalance Ii - 1, is responsible for the restoring force, which tends to bring the charge to the equilibrium. Further assuming that in the vicinity of QO the electron and ion currents vary slowly with charge, one can arrive to the following Gaussian charge distribution (2.41)

where the physical details of the charging process are incorporated into the mean ( 4 0 ) and the variance (04) of the distribution function. The ratio c q / q 0 was scaled with the grain size as 1/& by [Matsoukas et al. (1996)]. Thus, the relative magnitude of charge fluctuations is smaller for larger particles. Furthermore, for the particles larger than 100 nm, the assumption of the invariable mean charge is usually valid. On the other hand, treatment of the charging of nanoparticles smaller than 10-20 nm will most likely require the consistent discrete population balance approach with the charging equation (2.39). In this case, when the variance oq is comparable with 40, the fluctuations can reverse the sign of the grain charge [Chunshi and Goree (1994); Matsoukas et al. (1996)l. An example of the use of this technique in the modeling of silane plasma discharges contaminated by the plasma-grown nanoparticles is given by [Denysenko et al. (2003a)].

2.2.2

Charging in the plasma stream

Consider a special case of the charging of macroscopic particles in the presence of directed flows of the’plasma species. For example, external DC electric fields can drive the charged plasma species along certain directions determined by the electric field topography. Most commonly, this happens in the vicinity of the chamber walls or internal electrodes in the laboratory gas discharges, where the plasma sheaths sustain a DC electric field that accelerates the positive ions and strongly impedes the motion of the electrons and negative ions towards the walls/electrodes. Other laboratory examples include but are not limited to the transport of intense electron and/or ion beams, ion extraction, electron guns/emitters, plasma thrusters, magnetrons, as well as various plasma jets and plasmatrons. Directed streams of the plasma species are also quite common in the field aligned ionospheric currents. On the other hand, the relative dust-to-plasma drift can originate due to the circumplanetary dust streams, regular orbiting of macroscopic

44

Physics and Applications of Complex Plasmas

particles in various space plasma environments such as planetary rings, ejection of solid grains by volcanic moons, or expansion of dust clouds through the circumplanetary space. As was mentioned in Sec. 2.1.2, the expressions for the microscopic currents of the plasma species (2.6) depend on the relation between the directed and thermal velocities of the species. Recall that Eq. (2.7) is valid in the limit when the thermal velocities Z I T , are much larger than the stream velocities w. In this case there is no preferential direction for the plasma species to enter the grain. Hence, the process of the plasma-grain interaction is spherically symmetrical and the effective plasma species collection area can be approximated as the surface area of a spherical particle of radius a, which is 47ra2. The presence of directed flows of charged plasma species results in significant changes in the physics of the charging process. Apparently, the plasma species collection area becomes different due to the preferential flow direction, which means that different surface areas of the grain interact with the plasma in a different way. In the simplest approximation of a unidirectional electron/ion current, one can fairly accurately use the particle cross-sectional area 7ra2 as the plasma species collection area. In this case the general expression (2.6) is reduced to

I,

= 7ra%,q,w(l

-

2qa(ps/rnaw2)

(2.42)

where ~ / V T>> , 1 [Whipple (1981); Goree (1994)]. Note that under the typical conditions of low-temperature plasma discharges of most interest here, the electron thermal velocity is at least one order of magnitude higher than the typical electron drift velocities in moderate electric fields sustained in the plasma sheaths. Therefore, it is usually quite accurate to consider the electron fluxes onto the grain surface spherically symmetrical and use the expression (2.7) of Sec. 2.1.2. However, the ions with the directed velocity uo in a typical plasma sheath with the potential drop not exceeding a few tens of Volts can be both subthermal (uo< w T i ) and superthermal (uo > wTi). Thus, in the strongly subthermal case (uo << u T i ) the ion flow on the dust surface is due to the spherically symmetrical thermal fluxes and can be described by Eq. (2.7). However, if the ion flux is strongly superthermal (wo >> w ~ i and ) unidirectional, Eq. (2.42) is to be used instead. This case was termed a “monoenergetic beam limit” by [Barnes et al. (1992)l. In the intermediate case when the ion directed velocity is of the order of v ~ ione , can fairly accurately use Eqs. (2.2)-(2.4) and (2.7) provided that

45

Basics of Plasma-Dust Particle Interactions

U T ~is

replaced with the ion mean speed the negatively charged grains we have

Vi =

d w .In this case for (2.43)

which is valid for singly charged positive ions and other notations are the same as in Sec. 2.1.2. However, this approach should be used cautiously in the problems requiring a great degree of accuracy in the dust charge value. Indeed, if one further formally sets ~ l T i= 0 in Eq. (2.43), the resulting ion current would be a clear overestimate as compared with the correct expression (2.42) derived from more general principles [Whipple (1981); Goree (1994)l. The reason behind this discrepancy is in quite different characters of the plasma-grain interactions in the sub- and superthermal cases. Generally, the plasma Debye length Yd is defined (see Sec. (1.1.2)) as a characteristic length of the plasma-dust interactions and includes the contributions from both the electron and ion components. In the “subthermal” case AD FZ AD^. However, in the case of the strongly superthermal ion flows (vg >> w ~ i ) the , characteristic formation time X D ~ / W T ~of the ion Debye shield around the dust grain becomes much longer than the corresponding ion traverse time X o i / v o . Physically, this means that the corresponding ion species are effectively excluded from the electrostatic shielding of the macroparticle. In this case the dust shielding is due t o the plasma electrons and the characteristic scale of the dust-plasma interaction X o will be approximately equal to the electron Debye length AD,. The details of the charging mechanisms in the plasma streams are especially important in the studies of more complex plasma-dust interactions such as plasma wakes and other collective phenomena considered in Secs. 2.3 and 6.1. 2.2.3

EfSect of trapped i o n s on dust charging and shielding

In the previous sections, the effects of electron and ion collisions on the electrostatic shielding and charging of macroparticles in a plasma have been neglected. Here, we show that under typical discharge conditions, the account of ion charge-exchange collisions can result in significant amounts of the trapped ions (ions with negative energy), which can dominate in the shielding cloud around a dust grain and also increase the positive ion current to the grain [Lampe et d. (2001)] (see also a review by [Lampe et d. (2003)I). Note the neglect of any collision effects on the electrostatic shield-

Physics and Applications of C o m p l e x P l a s m a s

46

ing and charging of small spherical collectors in a plasma can be justified if the ion/neutral mean free path is greater than the plasma Debye length, Le., Xmfp >> AD. In this case (in many low-temperature plasma discharges) all the incoming electrons and ions apparently have a positive energy and can either hit or bypass the collector surface. However, if positively charged ions lose their energy in the processes of collisions, there is a chance that some of them will be trapped in a potential well around a negatively charged grain and attain a net negative energy. The build-up of the negative-energy ion cloud can be eventually limited by a fallout of the trapped ions on the surface or other collisional processes. An analytic method can be introduced [Lampe et al. (2001); Lampe et al. (2003)] to calculate self-consistent distributions of trapped n t ( ~and ) untrapped n,(r) ions, as well as the electrostatic potential # ( T ) around a spherical dust grain. The results of this model convincingly suggest that under typical low-temperature discharge plasma conditions the inner part of the shielding cloud is mostly made up of the trapped ions, which significantly modifies the potential profiles around the dust particle and enhances the ion fluxes onto its surface. The model considers a single stationary grain of radius a in a steadystate nonflowing plasma with Maxwellian-distributed neutrals, positive ions, and electrons. It is assumed that the plasma Debye length is small compared to the ion/neutral mean free path and much larger than the grain size (a (< << ;)X,f,mfp the ion charge exchange collisions with an energy-independent collision frequency exch are dominant; and any significant centrifugal potential barriers are absent [Lampe et al. (200l)l. Thus, the trapped ions are created by the ion-neutral charge exchange collisions. As a result of such collisions the pristine ions are neutralized and the newly created ions can be described by the spherically-symmetric phase-space distribution function h ( ~v,,8;r’). Here, T , r’, and v are the current location, “birthplace”, and velocity of the trapped ions, respectively, and 8 is the angle between r and v. Keeping in mind that the collision frequency is energy independent and the orbit period of a trapped ion is much shorter than the time between collisions, and since the trapped ions are born Maxwellian, it is reasonable t o adopt the distribution function h(r,u,8; T’) = C(T’)exp(-E/T)

,

(2.44)

which is determined by the constants of the motion, namely, the total energy E = m v 2 / 2 e4(r) and the angular momentum L = mvr sing.

+

Basics of Plasma-Dust Particle Interactions

47

The distribution function (2.44) of trapped ions has the same form as the one of the positive-energy ions. However, the main difference with the collisionless case is in partial population of h(r,w,O;r’). In other words, only those values of Y and 0 that correspond to trapped ions, are to be retained. The relevant integration limits can be found from the following four conditions. First, the total energy of the trapped ions must be negative, i e . , & < 0. Second, since a trapped ion was created at r’ with a positive kinetic energy, the total energy & ( T ) must exceed e4(r’). Third, the ion must have enough angular momentum so that its trajectory does not intersect the grain surface ( r = a ) , which yields the minimum ion velocity min Finally, there must not be any potential barrier in the radial effective w,in. potential U ( r ) = e$(r) L 2 / m r 2 ,which prevents an ion born at r’, with angular momentum L and energy &, from reaching the radial position T . Factor C(T’)in Eq. (2.44) can be obtained by integrating h(r,w,0; T ’ ) over all allowed (by the above conditions) values of T , w, and 0 and further equating the number densities of the created and lost trapped ions at the radial position r’. The total number density of the trapped ions ~ ( is then obtained by integrating h(r,~ , O ; T ’ ) over (w,O;r’). The density of untrapped ions is obtained from the results of OML theory keeping in mind the positive values of their total energy. Further assuming that the electron number density can be approximated by a Boltzmann factor, one can obtain the electrostatic potential distribution around the grain from the Poisson’s equation

+

d 4 = 47re[n,(r) -1_d r 2 -

r2 dr

dr

+ nt(r)

-

(2.45)

supplemented by the two boundary conditions ~ ( o o = ) 0 and 4 ( a ) = @float, where is the floating potential on the grain’s surface. The above set of equations can be solved numerically for typical conditions of low-temperature gas discharges. For example, assuming TITe= 0.04 and a / X D = 0.015, one can obtain that in the vicinity of a grain the density of the trapped ions is much greater than the deviations of the untrapped ion An, and electron An, densities from the bulk plasma density, ie., nt >> An, >> An,. It is notable that out to radial distances of approximately 0 . 7 X ~the grain shielding is mostly due to the trapped ions. Furthermore, the trapped ions significantly affect the electrostatic potential distribution around the dust particle, as shown in Fig. 2.4. In the semilog plot of Fig. 2.4 an unshielded Coulomb potential would appear as a horizontal straight line, and the Debye-shielded potential would appear as

48

Physics and Applications of Complex Plasmas

Fig. 2.4 Plots of -(./u)e4(r)/Te for three different models: self-consistent potential including trapped ions (solid line), potential with trapped ions neglected (dashed line), and Debye potential (dotted line) [Lampe et al. (2OOl)l.

the oblique dotted line. When the trapped ions are not included (dashed curve), the potential is quite far from the Debye-shielded one (dotted line). The trapped ions introduce additional shielding to bring the resulting potential closer to the Debye-shielded form. The numerical results of [Lampe et al. (2001)l also suggest that the account of negative-energy ions results in an increase of the positive ion current onto the dust particle. This is a consequence of the loss of angular momentum of newly created ions (which can thus fall onto the grain) as a result of charge exchange collisions. Therefore, the account of the trapped ions can result in significant changes in the interaction of a macroparticle with other particles and external forces. This opens up yet another opportunity to reveal the possibility of the net attractive forces between negatively charged dusts in a plasma. 2.2.4

Charge as a function of the grain size

Since complex plasmas are open systems, parameters of the macroscopic particle component such as particle charge appear as a function of not only their “internal” characteristics (size, shape, material, etc.), but also as a function of “external”, with respect t o that component, plasma conditions (e.g., plasma density and temperature). From Sec. 2.1 we recall that the charge can be reasonably predicted by the OML theory, where the dust grain is considered as a spherical probe and the charging is due to the plasma currents onto the grain surface. The currents are calculated by assuming that the electrons and ions are collected when their (mostly

Basics of Plasma-Dust Particle Interactions

49

collisionless) orbits intersect the probe surface. Usually, the electrons are assumed to be Boltzmann distributed, and the ions are shifted Maxwellian, taking into account their possible drift velocity in the external field. We also recall that in the simplest approximation of small ( a << X o i ) spherical particles their charge is Q = up,. If ps = const, one can expect that the particle charge is directly proportional to its radius. It is a common experimental practice to levitate the dust particle structures in the sheath region of RF discharge plasmas. The experimental data demonstrate nonlinear dependency of the particle charge on its size. Among the possible contributions to the particle charging, the effects of can singled out [Samarian and Vladimirov (2003)l. Note that the contribution of superthermal electrons (STE) to the particle charging can cantribute to the nonlinear behavior of the charge on the size of a levitating particle. The experimental investigation of dust charge dependence on the particle size were carried out in a capacitively coupled RF discharge in argon. The experimental setup is described in detail by [Samarian and Vladimirov (2003); Samarian et al. (2001b); Samarian and James (2001)l. The typical plasma parameters of the experiments are: the plasma density R, 2-8 x 10’ cm3 and the electron temperature T, N 1-1.5 eV. The particles used in the experiments were spherical of different material: melamine formaldehyde ( p = 1.5 g/cm3, radius a = 1.45, 2.12, 2.83, 3.05, 3.52 pm), carbon ( p = 2.1 g/cm3, a = 1.05 pm), corundum (AI2O3, p = 4.05 g/cm3, a = 2.45 pm), and glass balloons ( p , ~= 0.8 g/cm3, a = 5 pm). The charges on the particles were measured by two techniques successfully used previously, namely, by the vertical equilibrium technique [Samarian and James (2001); Tomme et al. (2000b)l and by the vertical resonance technique [Tomme et al. (2000b); Melzer et al. (1994)l. The results of these methods are in a good agreement. Some other techniques for the dust charge measurement are discussed in Sec. 3.4. Circles in Fig. 2.5 represent the experimental dependence of the charge on the levitating melamine formaldehyde particle on the size for the input power of 60 W and the pressure of 18.3 Pa (Fig. 2.5a) and of 12.1 Pa (Fig. 2.5b). The obtained dependencies are strongly nonlinear and have the exponentials 1.85 (Fig. 2.5a) and 1.66 (Fig. 2.5b). This result is in agreement with the data (the exponentials are within the range from 1.7 to 2.5) obtained by [Tomme et al. (2000a); Zafiu et al. (200l)l. In general, the particle charge cam be written as Q = F(a)p,, where the function F ( a ) is not necessarily linear. On the other hand, the surface potential reflects the plasma parameters “probed” by the parti-

-

50

Physics and Applications of Complex Plasmas

Fig. 2.5 Dependence of the charge of the levitating particle on the size. Circles represent the experimental results, rectangles the results of theoretical modeling with STE, and triangles the theoretical modeling without STE [Samarian and Vladimirov (2003)l.

cle at the point of levitation. Indeed, from the current balance equaThe plasma pation, the potential appears as ps = fV(n,/ni,T,,wi). rameters at the point of levitation are the functions of the particle size, ie., ne/ni = f"(hlev), T, = fT(hley), and ui = f"(hlev), where, in turn, the levitation height hlev = f h ( a ) . Thus the surface potential is pS = f v [ f n ( f h ( u )f)T, ( f h ( u ) )f," ( f h ( u ) ) ]and the particle charge appears as a complex function of its size via the size dependence of the levitation height and the height dependence of the plasma parameters. To elucidate the contribution of different functions (f,", f ; , and f:) ) into the charge dependence on the radius of a levitating particle, the charge can be calculated on the basis of the self-consistent hydrodynamic model where a one-dimensional (ID) discharge configuration is considered for typical experimental parameters of the dust levitation and equilibrium in the collisional plasma sheath taking into account the ionization processes. All variables of interest are calculated self-consistently in the sheath as functions of the distance z from the electrode and given plasma parameters in the bulk, namely, the sheath potential p ( ~ )the , electric field E(z) = ZE(z) = -dp(z)/dz, the ion flow velocity v~(z)= ZV~(Z)and density n i ( z ) , and the main electron density n,(z), which is supposed to be Boltzmann distributed. The temperature of the main electron popu-

Basics of Plasma-Dust Particle Interactions

51

lation is assumed uniform in the entire region of interest. The fraction of STE, with the ratio of the STE density to the ion density at the electrode as a boundary condition, is determined by the secondary emission yield and is taken as neh(0)/ni(O)= 0.045 [Phelps and Petrovic (1999); Shaw et al. (1999)]. Furthermore, the model space distribution is taken in the form close to the step function in space, namely, neh = (no/a)(tanh[(zZ O ) / Z ~ ]+ l), where the shift of the step is zo z,h where z,h is the sheath and pre-sheath width, and the width of the step z1 AD,. The ion dynamics is governed by the continuity and momentum equations. The continuity equation for the ions takes into account plasma production; the main mechanism of ionization is assumed to be electron impact ionization with the additional contribution of STE so that the effective plasma ionization source is proportional to the neutral gas, and contains “standard” contribution of plasma Boltzmann electrons with exponential of the inverse of the electron temperature [Lieberman and Lichtenberg (1994); Vladimirov and Cramer (2000)], and the additional STE contribution depending on the STE density and the STE temperature. Solution of the equation for the balance of forces together with the charging equation gives the dependence of the charge of the grain, levitating in the sheath electric field, as a function of its size, see rectangles in Fig. 2.5. Note a strongly nonlinear dependence for the experimental and simulation curves, with the exponentials to be sufficiently close (1.85 and 1.66 for the experiment, and 1.74 and 1.57 for the simulation). As an example of the contribution of STE, the triangles on Fig. 2.5(a) show the simulated charges of the levitating particles in the absence of hot electrons. This effect demonstrates the nonlinear dependence of the levitating particles on the grain size when bigger and therefore heavier particles levitate deeper into the sheath (and closer to the electrode) where the fraction of energetic electrons is higher because of the secondary emission from the electrode. On the other hand, in the absence of STE, the closer the particle to the electrode is, the more pronounced is the deficit of thermal electrons because of the electrode electric field. Indeed, by fitting the data without STE we see that the actual power index is 0.92, ie., the slope of the dependence is decreasing. The analysis of this simulations demonstrates that out of various contributions to the dependence of Q on the particle size one can single out the effect off:.fT It is common to assume that the main electron temperature is not changing in the sheath region. Therefore the change of f: = f;, f & E is due to the increased number of STE closer to the electrode and the observed nonlinear dependencies are due to the different N

N

+

52

Physics and Applications of Complex Plasmas

levitation heights of the particles with different sizes (and masses). Even a small proportion of STE is able to significantly influence the properties of the sheath. In the experiment [Takamura et al. (2001)], the presence of STE was attributed to the features of the filament discharge. In the sheath of an RF discharge plasma, despite vast number of experiments, possible presence of STE was not discussed yet. One of the possible reasons for that is that the standard models for the capacitively-coupled RF discharge plasma usually do not take into account the role of the secondary emission electrons since most of the electron current through the sheath is the capacitive displacement current. However, the ion-induced secondary electron emission from DC biased plasma electrodes is a well known phenomenon required to sustain a DC discharge [Cobine (1958)l. Although for normal RF biases (-10-15 V) the yield coefficient is relatively small, less than 0.05, the influence of superthermal electrons on the sheath properties and especially on the charging of macroscopic particles is profound even in this case. The simulations discussed above indeed show that the sheath size and other characteristics such as plasma density distributions are strongly affected by STE. This, together with the effect of STE on the particle charge, leads to the significant change of the levitation heights. The strong dependence of the surface potential on the levitating particle size reflects the dependence of the surface potential on the levitation distance from the electrode. This opens an opportunity to employ the particles as microscopic probes for determination of the electron distribution function (see Sec. 3.5.1); this goal, however, needs more elaborated model for the sheath region. We stress here that the observed character of the charge (or the surface potential) on size dependence can provide the information on the presence of energetic electrons in the sheath. Indeed, in the absence of STE, the surface potential shows a distinctively. different behavior. Note also that the maximum possible levitation radius decreases in the absence of STE.

2.3

Interactions of Grain Particles with Ion Flows

In the sheath region, the ion flow to the electrode naturally provides a distinctive direction, with the plasma properties different in the direction parallel or perpendicular to the flow. The ion flow, dragging dust particles (and supporting, in particular, the dust voids, see Chapter 4.2.3), is also responsible for the generation of associated collective plasma processes.

Basics of Plasma-Dust Particle Interactions

53

The latter strongly influence the formation of the plasma-dust crystals. In a typical experiment, the dust levitates in the sheath region where the balance between the gravitational, electrostatic, and ion drag forces is established, with often observed aligning of the dust particles in the direction of the flow. A theory taking into account these features suggests that in the case of supersonic velocities of the flow the ion wake field is formed behind the levitating dust grain [Vladimirov and Nambu (1995); Vladimirov and Ishihara (1996); Ishihara and Vladimirov (1997)I'; it was later developed in a number of studies [Benkadda et al. (1999); Schweigert et al. (1996); Lemons et al. (2000); Lampe et al. (2000); Ignatov (2003)l. Thus here we first study of the problem of shielding of a point-like grain in the presence of a supersonic ion flow that appears in the sheath and demonstrate how the wake potential is formed (Sec. 2.3.1). The plasma wake strongly modifies interactions of dust grains between themselves and with the plasma; in particular, supporting a Cooper-pairing-like attraction of the grains of the same sign of the charge. The Hamiltonian description (Sec. 2.3.2) elucidates this analogy. According to the majority of the experiments, the most studied (theoretically, numerically, and experimentally) cases correspond to spherical dust grains. However, experiments [Annaratone et al. (2001); Mohideen et al. (1998); Molotkov et al. (2000)] studied rod-like grains and the theory [Ishihara et al. (2000); Vladimirov and Nambu (2001)] predicted the wake formation as generated by the ion flow in the cases of particles with dipole moments and elongated (rod-like) particles. Thus we consider in Sec. 2.3.3 supersonic ion wakes behind dipoles and non-spherical (rod-like) particles. The plasma ions, accelerated by the sheath electric field in the nearelectrode area and, according to the Bohm criterion [Riemann (1991)], achieving supersonic velocities, can excite longitudinal plasma oscillations in the sheath region, as known from numerical simulations and experimental observations [Lieberman and Lichtenberg (1994); Allen (1985)]. The influence of the longitudinal random fields leads to the broadening of the ion distribution in the direction of the ion flow. In this case, the width 'Note that similar attractive forces appear between moving dust particles [Nambu et al. (1995)l involving the collective interactions via the low-frequency electrostatic perturbations. The effect is analogous to the Cooper pairing [de Gennes (1966)l of electrons in superconductors, and has been also studied for the two-component electronion plasmas [Nambu and Akama (1985)l in which the possibility of electron attraction has been demonstrated.

54

P h y s i c s a n d Applications of C o m p l e x P l a s m a s

of the ion distribution in the direction parallel to the ion flow (“downstrea.m”) should be larger than that in the perpendicular direction. Thus the ion drifting distribution can be highly anisotropic. Moreover, the downstream parallel ion distribution should be wider than the ion distribution in the opposite direction (“upstream”). This is due to the fact that the oscillations excited upstream act mostly on the downstream particles. It is noted also that the electric field of the sheath produces self-consistently an ion distribution downstream/upstream asymmetry in the widths [Emmert et al. (1980)l. Both arguments (the excited plasma turbulent fields and the influence of the electric field on the ion distribution) suggest that the ion distribution in the sheath can be asymmetric. This feature should also be taken into account when considering the charging and shielding of dust particles. There have been not many experimental measurements of the ion distribution in the sheath (e.g., an investigation of the ion velocity distribution in the sheath by means of a laser induced fluorescence [Bachet et al. (1995)l). There are indications that the general observed features of the ion distribution asymmetry agree qualitatively with the theoretical predictions [Emmert et al. (1980)] in which the effect of the excited plasma fields in the sheath was neglected. However, the measured absolute value of the width appears to be much larger than the theoretical result [Emmert et al. (1980)l. A candidate to explain this anomalous width can the sheath turbulence. The turbulence also modifies the regular electric field in the sheath and therefore can affect the dust charging in this way. The important observation of the experiment [Bachet et al. (1995)] is that the observed width of the ion distribution perpendicular to the flow is at least one order of magnitude less than the upstream (or downstream) width. The ion flow can thus finally appear to be moderate as compared to the thermal ion velocity parallel to the flow. The ion thermal velocity in the direction perpendicular to the ion flow is almost the same as in the bulk plasma and is therefore small compared with the ion flow velocity. The collective field that can broaden the ion distribution, can be created mainly in the direction of the ion flow. The charging and shielding of dust particles in this case is considered in Sec. 2.3.4. The complete problem of the plasma dynamics around a macroscopic body in the presence of plasma flows is highly nonlinear and therefore its numerical analysis is of a major importance. Various numerical methods can be employed to solve the problem. Among them, direct integration of the equations of motions of the plasma particles represents a numerical

Basics of Plasma-Dust Particle Interactions

55

experiment with significance approaching experiments in the laboratory. The particle-in-cell (PIC) simulations [Winske et al. (2000)l clearly demonstrated formation of the wake potential with the potential wells attracting the grains underneath. The ab initio three-dimensional molecular dynamics (MD) studies [Vladimirov et al. (2001a); Vladimirov et al. (2003a); Vladimirov et al. (2003c)I also show the region of the strong ion focusing behind the grain for the case of a single grain as well as two grains aligned parallel and perpendicular to the ion flow. In Sec. 2.3.5, we present the most important cases of three-dimensional MD simulations of the ion wakes of one dust particle and two particles aligned parallel to the ion flow. The ion wake theory proved to be very successful in explaining the observed experimental phenomena (e.g., dust particle vertical alignments), and was confirmed in specifically designed dedicated experiments [Takahashi et al. (1998)], experiments on pairing of two dust particles [Melzer et al. (1999); Steinberg et al. (2001)], and experiments and simulations [Hebner et al. (2003a); Hebner et al. (2003b); Hebner et al. (2004)l. 2.3.1

I o n wake behind a charged point-like particle

Here, we present a possibility of the charged particle attraction in dust plasma systems with the finite ion flow [Vladimirov and Nambu (1995); Vladimirov and Ishihara (1996)l. We consider the situation close to laboratory experiments, namely, static dust particles which can interact through .the ion low-frequency oscillations in the ion flow, which velocity vo exceeds the ion-acoustic velocity u s. Physically, the generation of the ion wake fields in this case is the same as the generation of the electromagnetic fields for the charge at rest placed in a medium moving faster than light. The electrostatic potential of dust particle can be written as

(2.46) where Q and v d (lvdl s 'ud << VT,) are the charge and velocity of the test dust particle, respectively. The dielectric response function of the plasma in the presence of a finite ion flow with the speed VO ( V T ~ << vo << V T ~ where ' U T ~=, (Tz,e/rni,e)l/a ~ is the electron (ion) thermal velocity, T, << T,) is calculated under the condition

(2.47) where the z-axis is directed along the ion flow.

,

56

Physics and Applications

of Complex Plasmas

Note that two different cases appear here. The first one corresponds to the situation when the influence of the dust on dielectric properties of the plasma can be neglected; the polarization of plasma is determined by the contribution of electrons and ions. The second case corresponds t o the sufficiently high dust density. For the latter situation, the dust particle is considered at rest, i e . , I W = k . v d J<< k v ~ dwhere , V T d = ( T d / m d ) 1 / 2 is the dust thermal velocity. The plasma dielectric response is given by (2.48)

YDe = ( T , / 4 ~ n , , e ~ )is~ /the ~ plasma Debye length for the where YD = AD, first case (of an isolated test particle), Yd = Yde + for the second situation (when condition (2.47) takes place), YDd X D ~= ( T d / 4 ~ n d Q ~ ) i]s the dust Debye radius, Q is the charge of the other dust particles. The negative contribution due to the ion flow is the most important for the present consideration. From Eq. (2.48), we can also see that the relative contribution of the plasma electrons is of less importance. The inverse of the function (2.48) can be written as (2.49)

+

where w, = kv,/(l k2X&)1/2 is the frequency of the oscillations in the ion flow, and v, = X D W , ~ is the ion sound velocity. Substituting Eq. (2.49) into Eq. (2.46)) the total electrostatic potential can be represented as the sum of the two potentials

@(x,t) = @ D ( X ) t ) f

@W(X,t)r

(2.50)

where

Q

(2.51)

@ D = - exp(-r/Ao)

r

is the usual static screened Coulomb (Debye screening) potential. The additional wake potential involving, in particular, the collective effects (the ion wake) caused by the oscillations in the ion flow is given by

=

s

Q

k2X&wiexp(ik. x - i w t ) S(W - k,vo)2 - w,2]

2.ir21c2 (1+ k”xg)[(w

-

k . v d ) d k d w . (2.52)

From Eq. (2.52), it follows that the potential changes its sign due to the overscreening depending upon whether w - kzvo is larger or smaller than

57

Basics of Plasma-Dust Particle Interactions

- k , u ~is close to this frequency, strong resonant interaction between the oscillations in the ion flow and the test particulate appears. The important point is that even for the static test charge, the potential "behind" it oscillates as a wake field, if the speed of the ion flow exceeds the ion-acoustic velocity. Therefore the formation of the quasilattice structures can be possible since the periodic regions of attractive and repulsive forces between the particulates of the same charge appear. Furthermore, Eq. (2.52) is integrated over frequencies assuming k . v d = 0. Integration over angles can be proceeded using an expansion in spherical harmonics [Nambu and Akama (1985)l. Assuming that Ic:X& YD<< ( M 2- 1) as well as k l X ~<< 1, where M = uo/u, is the Mach number of the flow, one can obtain

ws.In the case when w

X

cI'

1- M - 2

(

sin k l z /

7

1

M 2 -1 dki,

(2.53)

where z and p are the cylindrical coordinates of the field point, and JO is the Bessel function. The main contribution to the integral (2.53) is for k i X ~ 1. Then at ~ approximate expression for distances p > AD and IzI > X D V ' the the wake potential is given by N

(2.54) where

and zh = IzI p d m > 0 (note that the oscillating potential exists only in the wake of the test particle, ie., for z < 0 and IzI > p d m ) . Obviously, this function oscillates for changing p or z . Because the oscillating potential (2.54) is proportional to the same test dust particle charge Q as the static Debye potential (2.51), and contains no screening exponential, there are regions in space corresponding to the change of the effective potential sign and, hence, to the most probable positions of the particulates (note that these regions are not only on the line p = 0). For small distances in the perpendicular direction, k l p << 1, and for IzI > X ~ d m the,

58

Physics and Applications of Complex Plasmas

Fig. 2.6 The wake potential and the potential minima in ( p , z ) plane for various Mach numbers [Ishihara and Vladimirov (1997)l. The potential minima are shaded.

main contribution to the stationary wake potential is given by [Vladimirov and Nambu (1995)l

(2.55) where Ls = X ~ d ism the effective length. Thus the wake potential is attractive for cos(IzI/L,) < 0. Note that this result cannot be found from

Basics of Plasma-Dust Particle Interactions

59

Eq. (2.54) because of the different asymptotics used. The effective periodic spacing in the plane perpendicular to the flow is of the order of the Debye length; note that this can be seen not only from the approximate expression (2.54), but also from a more exact formula. Numerical integration [Ishihara and Vladimirov (1997)l gives the wake potential distributions of Fig. 2.6. Thus the collective interaction of a test dust particle with the lowfrequency perturbations in the ion flow can provide the attractive wake potential. The mechanism is similar to the Cooper pairing of electrons. We note that in the Cooper-pairing effect the test electron polarizes the medium by attracting positive ions. The excess positive ions, in turn, attract the second electron. If this attraction is strong enough to overcome the repulsive screened Coulomb interaction, the superconductivity phenomenon occurs. For example, the model [Pines (1958)] leads to an expression similar to Eq. (2.49) where the first term is due t o the screened Coulomb repulsion, whereas the second term corresponds to the attraction forces. For the effective attraction, the speed of moving charged particles (in the frame where the background medium is at rest) should exceed the sound velocity. However, the mechanism is the most effective when the particle speed is quite close to the latter (in this case the resonant interaction with phonons is maximum). In the situation with the static charged dust particles, the moving ions of the flow create the polarization necessary for the resulting attraction. In order for the latter to be effective it is required that one has continuous ion flow with the velocity exceeding the ion sound speed. As we already noted above, this condition can be observed in the sheath region of low-temperature laboratory plasmas. The test particles can attract to each other forming the quasi-lattice structures with the characteristic period of the order of X o d m . In principle, any charged plasma species can create similar attractive potentials under certain conditions. However, the speed of plasma electrons is usually too high (this corresponds to the condition W T ~>> v,) for the effective attraction t o be realized. On the other hand, the ion velocity is too small in the absence of the flow. For the model considered here (with the ions moving as a flow), we have to compare vo with the characteristic velocity of collective perturbations in such kind of plasma. Note that for ion-acoustic waves in the flow the (relative) speed of the ions is too small (which corresponds t o the reference frame moving with 210). Therefore we can conclude that the interaction of the static dust grain with the ion oscillations in the flow provides the most effective contribution for the situation considered. For quantitative comparison with experimental

60

Physics and Applications of Complex Plasmas

results, other factors should be taken into account. In particular, the potential of ensemble (in contrast to the isolated test particle) of dust particles might be calculated. This can be done by either adding the contributions of the isolated particles (if their density is not high), or introducing their distribution function (when dust collective effects become important). Furthermore, contribution of other forces acting on the dust, as well as such factors as an inhomogeneity of the ion flow should be considered to obtain more detailed picture.

2.3.2 Dust Cooper pairing: Hamiltonian description The Hamiltonian formalism is one of the most favored in physics since it is very useful in statistical and interaction analyses as well as in numerical simulations [Totsuji et al. (1997); Otani and Bhattacharjee (1997)l. The knowledge of the Hamiltonian is essential for the description of propagation and interaction of modes and especially for calculation of the free energy to study phase transitions and critical phenomena. Although the dust-plasma systems (e.g., two- and three-dimensional structures as well as phase transitions observed are thermodynamically open, the Hamiltonian description can provide useful insights, especially for the cases when the energy exchange with the external sources/sinks is insignificant. Here, we derive the semi-classical Hamiltonian which describes the interaction with external fields, screened Coulomb potential, and the effective interaction of dust particles by exchanging virtual phonons [Ishihara and Vladimirov (1998)]. This is motivated by the presence of the oscillating stationary wake behind a dust particle in the presence of the ion flow. Interaction of the particles in the wake field appears to be similar t o the Cooper pairing of electrons in superconductors [de Gennes (1966)]. Furthermore, we demonstrate how the oscillations of the grains, namely, longitudinal and transverse lattice modes (considered in detail in Sec. 6.1), can be also obtained within the framework of the present formalism. Consider an ensemble of charged test particles, with the coordinate xj and the momentum pj, in a background plasma and interacting with longitudinal collective plasma fields. The kinetic energy of the test particles is given by the particle Hamiltonian x j p ; / 2 m j . The interaction of test particles with the longitudinal electric fields E(x,t ) can be conveniently expressed in terms of the longitudinal vector potential A by replacing pj by pj - (Zje/c)A(xj) in the Hamiltonian. The test particles interact not only with the longitudinal electric fields, but also through forces derivable from

Basics of Plasma-Dust Particle Interactions

61

the external potential Vext such as the sheath potential. The Hamiltonian for our system is [Bohm and Pines (1953)] (2.56)

2mj

where the summation is over the test particles with masses mj , momenta pj, and charges Z j e , A(x, t ) is the longitudinal vector potential (in the assumed gauge the scalar potential q5 is zero), and E(x,t ) is the longitudinal electric field. Introducing

the Hamiltonian (2.56) can then be written as

H

=Hp

+ H f + H j l ) + Hj 2) + Kxt,

(2.57)

where V is the volume of the system. The kinetic energy of the particles is

and the energy of the electric field is

The interaction terms can be written as

and

where & ( t ) = d q k ( t ) / d t = -63-k. The short-range Coulomb static interaction term can be explicitly obtained from Eq. (2.57) by applying a unitary transformation

62

Physics and Applications of Complex Plasmas

where the summation over k is restricted for J k ( > ,YD-1 A', and E(k,O) = YD-2 is the static plasma form factor. In this case the Hamiltonian 1 JkJ-2A,2 transforms to

+

H

---f

V'HU

= Hp

+ H f + H;') + H j 2 )+ Kxt

The terms including the factor (l/[E(k,O)]'/') for (k(>' ,A are neglected since the Debye screening makes the plasma wave propagation impossible in the wavenumber range JkJ >' ,A (the effects of dynamic screening will be included below). The Hamiltonian appropriate to a set of harmonic oscillators which represent the collective field of the plasma wave can be written as

where k = -k' for H j 2 ) . Here, p i = -P-k and qc = -Q-k because of the reality condition for the electric field, and wpj = ( 4 ~ 2 , 2 e ~ / V r n j ) 'is/ ~ the frequency of the collective particle motion. Taking into account the dynamic screening of the wave fields and setting

one can obtain

and

(2.58)

63

Basics of Plasma-Dust Particle Interactions

where E = E(k, w) is the linear plasma dielectric permittivity. The solution of the dispersion equation &(k,u ) = 0 gives the eigenfrequency of the plasma waves w = wk. Below, we assume that there is no plasma wave damping, and consider the positive wave eigenfrequencies, W k > 0. Next, the canonical transforms xj = exp(--iS/h)Xj exp(iS/fi), p j = exp(iS/h)Pj exp(iS/ti), . ' . , is applied, where

and Qjk =

z3e mj

47rh

[Vlk12wk(dw&/dw),,

]

k . (Pj - hk/2)

a;

,

to obtain the set of the new variables ( X j , P j , A k , A i , ' F I ) from ( x j , p j , a k , a i , f f ) . Here, R', = W k - ( k . P j / m j ) -t (ti(kI2/2mj). The new canonically transformed interaction Hamiltonian includes the static Debye contribution X D , the external potential Vext, the term H I ' ) corresponding to and the interaction parts of 'lip and 'liosc in the lowest order, i.e.,

(2.59)

where (2.60) The last three terms in Eq. (2.59) can be combined and expressed as veff=

e2 y2y4Vrnimj I27rzi kl 2Wk(aWE/dW)wk Z j

-

i#j

k

(2.61)

Physics and Applications of Complex Plasmas

64

Here, the effective interaction between particles through the exchange of virtual plasma waves or quasi-particles is described. Note that the dynamic collective effect of the background plasma is included through the dielectric function &(k,w ) . Consider a pair of dust particles interacting via the plasma ion-acoustic waves. Let a pair of dust grains have masses ml,m2 and charges Q1 = Z l e , Q 2 = Zze, and consider the quasiclassical limit fi + 0. When the particles move in the same direction z with the velocities 211 and u2, the effective potential energy derived from the interaction Hamiltonian is

where the integration is limited t o the range Ikl < X i ’ , R = X1 - X2,and k2 = k$ lk1I2. We see that V,.,“ = 0 for 211 = 0 or 212 = 0 and

+

(2.62)

+

+

for u1 = w2 = W O , where Ic:,l = f ( ( 1 - M-’)XE2 Ikl12)/2 [Ikl(2M-2X62((1- M-2)X;2 lk112)2/4]1/2.The vanishing effective potential was investigated in the context of the study of molecular-ion beams interacting with metals [Kitagawa and Ohtsuki (1977)l. The effective potential (2.62) is similar to the wake potential of a test dust particle in the ion flow [Ishihara and Vladimirov (1997)I. It is noteworthy that the effective potential vanishes when one dust grain is stationary while another one is moving with respect to the ambient plasma. The exchange of quasiparticles (virtual phonons) between a pair of dust grains is only possible when the grains are moving together (or alternatively they are stationary in the presence of plasma flow). Note that the addition of the first term in the square brackets in Eq. (2.62) and X D , given by Eq. (2.60), forms the complete Debye (static) interaction potential V D = C i p j V $ / 2 , where

+

+

65

Basics of Plasma-Dust Particle Interactions

and the summation is performed within the entire range of possible values of k. For M > 1, the contribution from the poles at f k l in the k z plane gives the oscillatory wake potential. Assuming a cylindrical symmetry k . R = IklIRl cos 4 k z Z and integrating over k z in Eq. (2.62), the approximate expression for the oscillatory interaction potential energy at the distance Ri > AD and 1 2 1 > AD is given by

+

Jm

(2.63)

*

where Z* = 121 R l d m > 0, and w+ are defined after Eq. (2.54). The oscillating potential exists only in the wake of the test particle, ie., for 2 < 0 and (21> R l d m [Vladimirov and Ishihara (1996); Ishihara and Vladimirov (1997)l. On the other hand, for R l < AD and 121 > A ~ d m we, recover [Vladimirov and Nambu (1995)] (2.64)

The Debye static interaction potential can be expressed explicitly as

VD = 23

QiQj

JXi-XjI

IXi-XjI

.

The derived Hamiltonian describes the effective potential produced by a pair of moving dust grains due to their interaction ,with the external fields, the Debye screening potential, and the exchange of virtual phonons in the ion flow. The wake potential vanishes if one of the pair dust particles is stationary with respect to the ambient plasma. The Hamiltonian can be also applied to oscillations in a coupled system of dust particles, and characteristic frequencies of both longitudinal and transverse modes can be therefore derived [Ishihara and Vladimirov (1998)]. 2.3.3

W a k e s behind dipoles and non-spherical particles

The theory presented above is based on a test dust particle which is characterized by a point charge. However, the plasma crystals can be composed of finite size non-spherical dust grains. These grains, in addition to the electric charge, can have a dipole moment. Here, we present the results of the theoretical predictions of the plasma wake calculated for a particle

66

Physics and Applications of Complex Plasmas

with a dipole moment [Ishihara et al. (2000)l and for a rod-like particle [Vladimirov and Nambu (200l)l. 2.3.3.1

Wakes behind dipoles

The appearance of a large dipole moment for a dielectric dust grain in a supersonic flow has been studied in the context of the charging process on the surface of the dust grain, but using an unscreened Coulomb potential neglecting the plasma (and therefore wake) effects [Ivlev et al. (1999a)I. Here, we point out the effect of a dipole moment on the formation of the wake potential behind a dust grain in the presence of a flowing plasma and consider the ion wake potential produced by a single dust grain which is characterized by a given charge Q and a dipole moment p. For simplicity, we do not consider the process of the dust charging, thereby assuming that the test dust grain with a constant charge and a constant dipole moment is placed in the ion flow in a collisionless plasma, which supports the plasma collective modes [Ishihara et al. (2000)]. The potential of the distributed test charge in a plasma without an ion flow can be written, retaining the monopole and dipole contributions, as

where p is the dipole moment of a test charge q, er is a unit vector in the direction r, T = lrl, and AD is the plasma Debye length. As has been shown .previously, the first term should be modified in a plasma with an ion flow in a way to produce a wake potential behind a point test charge. Oscillations of the potential occur due to the excitation of an ion-acoustic standing wave in the flow downstream of the dust grain. Here, we show how the dipole moment affects the wake potential behind a test dust grain in an ion flow. Let the dust grain be made up of distributed point charges q j , so the charge density is (2.65)

where t j = r-rj -vjt, V is the volume of the system, and rj and vj are the location and the velocity of the distributed dust charges, respectively. We assume a small displacement Arj about the point ro for each distributed charge, or rj = ro Arj and set vj = v. Expanding exp(-ik . Ar) M

+

67

Basics of Plasma-Dust Particle Interactions

1 - i k . Ar, one obtains from Eq. (2.65)

1

p(r,t) = - x Q e x p [ i k . [ ]

-

' k

1 -xik.pexp[ik.E] , ' k

where E = r - ro - vt, Q = such a grain is given by

C jq j

and p =

C jArjqj.

The potential of

(2.66)

Note that the dipole correction appears in such a way as to effectively modify the charge Q in Eq. (2.66). The poles in Eq. (2.66) are from w - k . v = 0 and E(k,w) = 0. The contribution to the dipole potential term, due to the electron dielectric function, can be written, in the limit V + m, as

'F&,

t)

= 47r

1fi +" (243 1

k2x;

(-ik. p) exp [ik . (1,

(2.67)

where the subscript I 0 indicates the contribution from the pole w - k . v = 0. After the integration, the total potential in the range zo vot < z < zo + wt and Iz - zo - wtl > p ( M 2 - 1)1/2,in the near-field approximation becomes

+

1

sin
,

(2.68)

which is accurate for k ~

AD. Here, Ct = ( z - zo v t ) / A D d m . If Iz - zo - vtl < A D , one obtains

-

Equation (2.68) indicates that the dipole moment plays a major role in forming the wake potential if the magnitude of the dipole moment p approaches the order of I Q l X , J m . Around z-zo-vt = -(p/2)Xo(M2l)'j2,the dipole moment modifies the potential structure in a way to distort the oscillatory character. An analysis of the dipole moment of a dust grain of the radius a placed in a supersonic plasma flow showed that the magnitude of the ion flow-induced dipole moment can be N ulQl/2, due to the ion flow producing a strongly nonuniform charge distribution on the

68

Physics and Applications of Complex Plasmas

-

surface [Ivlev et al. (1999a)l. Thus the condition for the dipole moment to play a dominant role in forming the wake potential is a AD. The dipole moment p~ induced by the sheath electric field (in the same direction as the ion flow, i e . , P E .e, < 0 or p~ < 0) can be only neglected in comparison with the dipole moment caused by the ion flow for a grain size < 40 pm. Note that the moment p~ is nonzero also for a conducting grain, in contrast to the case of the dipole moment induced by the plasma flow.

Fig. 2.7 Wake potential 4 , normalized by 2M21QI/(M2 - l ) 3 ' 2 X ~ , at p = 0, against distance downstream of the grain, normalized by AD. M = 1.1 and p = 0 (solid curve), p = IQ~XD(dashed curve) and p = - l & l X ~ (dotted curve) [Ishihara et al. (2000)].

Figure 2.7 shows the normalized wake potential for a negatively charged grain a t p = 0, plotted against the normalized distance downstream of the grain Iz - zo - wtl/Xo, for M = 1.1, and for p / l Q l X ~ = 0 (no dipole moment), as well as for (extreme) cases ~ F ~ / I Q I X D= 1 (ion flow induced dipole moment) and p ~I Q/I YD == - 1 (electric field induced dipole moment). The downstream potential maxima, where other dust grains can reside in the stable equilibrium, are stronger and more distant from the original grain if the ion flow induced dipole moment is included, as is expected for this orientation of the moment, whereas the maxima are stronger and closer to the grain for the electric field induced dipole moment. These characteristics of the oppositely directed dipole moments may provide a way for experimental observations to distinguish between the mechanisms of creation of the dipole moment in the grains. Thus a charged dust grain with a dipole moment creates an oscillatory wake potential behind the grain, analogously to the monopole case. Note that when the.size of the dust grain becomes comparable to the Debye

Basics of Plasma-Dust Particle Interactions

69

length, the dipole moment plays an essential role in the structure of the wake potential. This structure is found to depend on the directionality of the dipole moment. The potential of a dust grain is given by Eq. (2.68) behind the particle in the presence of a supersonic ion flow when the ion acoustic wave (standing in the dust grain reference frame) is generated within the Mach cone. If the wake potential contributes to the alignment of grains in a dust crystal, the separation of the grains in the ion flow direction will be larger due to the ion flow induced dipole moment. Finally, we note that the wake potential will in turn modify the process of dust charging (and inducing the dust dipole moment), making the fully selfconsistent problem of dust charging and screening as well as the dust-dust interaction in the presence of an ion flow highly nonlinear. 2.3.3.2

Wakes behind elongated particles (rods)

Consider a finite size (in one direction) rod-like particle of the length l d is immersed in an ion flow of the speed VO. The angle between the direction of the flow and the rod is 00. The reference frame is chosen with the x-axis along the rod, and the ion flow is in the xz-plane, i e . , vo = (wg cos Oo,O, -wo sin 0 0 ) . The y and z sizes of the particle are zero. The angle 00= 7r/2 corresponds t o the ion flow in the negative z-direction, perpendicular to the rod, and the angle 00= 0 corresponds to the ion flow in 2-direction, parallel to the rod; note that in the experiments [Annaratone et al. (2001)l both possibilities were observed. The three-dimensional charge density of such a particle is given by

where pl is the linear density of the charge distribution on the rod. Below, we specify the following linear charge distributions: (a) charges distributed uniformly along the rod, p~ = (Qo/ld)e(z)O(ld - x); (b) charges distributed non-uniformly along the rod, p1 = (2Qox/l,3O(x)O(ld - x); (c) charges located a t the ends of the rod, p~ = Q1S(z) (Qo - Ql)S(x - I d ) . The total charge on the rod is QO for all cases. Note that cases (b) and (c) correspond to the nonzero dipole moment of the rod: PO = Q&/6 for the case (b), and Po = (Qo - 2Ql)ld for the case (c) (we assume Q1 # Qo/2). For the Fourier-transforms of the (stationary) particle charge density we have p d ( k , w ) = 27rpd(k)S(w), where: (a) pd(k) = iQo[exp(-ik,d)d) -- l]/k,d; (b) pd(k) = 2Q0[(1+ik,d) exp(-ik,d) - l]/k$d2; (c) m(k) = Q 1 + (Qo - Q l )exp(-ik,d).

+

70

Physics and Applications of Complex Plasmas

For the static rod in the presence of the ion flow, the electrostatic potential is given by

Separating the Debye potential and the oscillating wake potential, @(r)= @D(r) @w(r)[Vladimirov and Nambu (1995); Vladimirov and Ishihara (1996); Ishihara and Vladimirov (1997)], the wake potential @w(r)can be written as

+

@w(r)= R e

J'

WE exp(ik. r)

dk pd(k)k2X& 27r2k2 1 k 2 X &

-

+

+

Re

1

(2.69)

+

where R e = (-k,zlo cos00 ktu0 sin00 XI)' - ui and W k = /klus/(l + k2X&)lI2is the characteristic frequency of the oscillations in the flow. The potential (2.69) describes the strong resonant interaction between the oscillations in the ion flow and the test rod when 1k,'u0 cos 00- kZuosin 00 is close to the eigenfrequency w k of the ion oscillations in the flow. Equation (2.69) can be integrated over kk in a standard way and one can obtain analytical solutions in the limiting cases lrll > AD (wake field far away from the particle) and / r l /<< AD (near-particle wake field), and assuming near-sonic ( M M 1) and strongly-supersonic ( M >> 1) velocities of the ion flow. For the charge distributions (a) and (b), approximate expressions for the potential can be presented for the most important special cases 0 0 = 7r and OO= -7r/2, i.e., when the rod is oriented along the flow and perpendicular to the ion flow, respectively. Note that these cases are of the most interest since current experiments demonstrate that 0 0 = 7r and 0 0 = -7r/22 correspond to the observed stable equilibrium positions of rods. For 00= 7r, the wake potential is (2.70)

where for the case (a) A d ( 7 ) = 1 for the case (b) A d ( 7 ) = 27-/d, and for the case (c) Ad(r)= d ( Q l / Q o ) b ( ~ ) d(1 - ( Q ~ / Q o ) ) b (-r d). For 0 = -7r/2 2 if l d << X D ~ ,a simple expression is obtained

+

Basics

0.f

Plasma-Dust Particle Interactions

71

I"

Fig. 2.8 The surface plot of the wake potentials for the case (a) and three different angles between the rod and the direction of the ion flow [Vladimirov and Nambu (2OOl)l. Other parameters are: AD, = 300 pm, Te = 1 eV, M = 1.1, QO= 5000e.

Note that this expression coincides in this limit with the wake field potential generated by a point charge. The obtained approximate analytical expressions cannot describe the character of the wake potential for the interesting cases when the rod length is of the order of (or exceeds) the plasma Debye length. Thus here we present results of numerical integration for the rod lengths l d l = X0,/6 6 = 50 pm, id2 = AD, = 300 pm, and ld3 = 2x0, = 600 pm [Vladimirov and Nambu (2001)]. The first case ld = l d l is interesting for the comparison since the wake character in this case is practically close to that of a point charge.

72

Physics and Applications of Complex Plasmas

The dependence of the wake field on the orientation of the rod with respect to the direction of the ion flow is the most pronounced for the flow velocities close to the speed of sound (A4 I), when the first maximum of the generated wake is close to the rod. We note that in general, the larger are the ion velocities, the stronger are the ballistic effects in the ion focusing; this is because fewer ions participate in the wave-particle resonance, and the Mach cone is narrower. For M = 1.1, we present on Fig. 2.8 the result for the case (a) (the surface plots for other cases look similar) and three orientations of the relatively long rod, 00= K , 57r/4,3~/ (other parameters are: AD, = 300 pm, T, = 1 eV, M = 1.1, QO = 5000e, and Q1 = Qo/3 = 5000e/3). From these plots, one can clearly see the wave character of the wake potential. Note that the more is the angle between the rod and the direction of the flow, the closer is the first maximum of the wake potential to the rod. This type of behavior is more pronounced for longer rods with the big charge separation. Physically, it is clear that the shorter is the rod length, the closer is the character of the wake potential to that of a point charge. The basic feature of the wake in the case of the supersonic velocities of the ion flow, namely, its oscillating character, first reported for spherical particles, takes also place for the rods. The longer are the rods, the more is the difference in the characteristics of the wake potential as functions of the rod lengths and/or charge distribution along the rod. Regarding the formation of the wake potentials of rod-like particles, the plasma electron Debye length is the most important characteristic scale: for rods much shorter than the Debye length, the wake potential is almost the same as for the point particles; on the other hand, for rod lengths comparable or exceeding the Debye length, the characteristics of the wake potential develop the dependence on the orientation of the rod with respect to the ion flow as well as on the character of the charge distribution along the rod. The actual difference, e.g., for the location of the potential first minimum in the case of long (of the order of or exceeding the plasma electron Debye length) rods with a large charge separation can be of the order of the electron Debye lengths, i e . , of the order of a half of a millimeter for the typical discharge parameters which is an easily detectable effect in the laboratory. These results are important for the present and future experiments involving cylindrical particles. Furthermore, oscillations of rods as well as specific lattice modes propagating in the “liquid plasma crystals”, will be affected by the wake. Indeed, for the rotationally oscillating (in the plane parallel to the ion flow, i e . , in the vertical plane in the experiment) rods N

Basics of Plasma-Dust Particle Interactions

73

the motion of the wake potential minimum will take place thus strongly influencing the location of the downstream particles as well as the vertical arrangements and the vibration modes in the downstream chains. Note that additional (rotational) degrees of freedom will lead to new effects in experiments with cylindrical particles. Indeed, in the crystal lattices composed of spherical particles, vibration modes associated with the longitudinal of the transverse (with respect to the propagation of the wave) motions of grains can be excited. The “liquid crystal” lattices composed of rods will have new rotational modes associated with their rotational motions, similar to those in liquid crystals [Vladimirov and Tsoy (2001); Hertzberg et al. (2003a)l. Excitation and interactions of all these modes introduce new phase transitions and affect those existing in lattices composed of spherical grains. As was pointed out recently [Ivlev and Morfill (2001)], the wake can affect the interaction of the horizontal (longitudinal) and vertical (transverse) modes for spherical grains. Thus similar (or more complex) effects will exist for cylindrical particles, affecting their interaction and involving also the rotational modes. 2.3.4

Subsonic plasma wakes

Here, we consider shielding and charging of dust in the plasma sheath using the ion distribution function with a marginal (subsonic) velocity which is of the order of the characteristic broadening of the ion distribution in the longitudinal direction [Benkadda et al. (1999)l. For the moderate longitudinal ion flow, the L a n d a u d a m p i n g of the fields of the dust particle moving through the ion plasma (in the frame of ions at rest) should be rather strong. In this framework, the physics is strongly modified as compared to the case of superthermal (and especially supersonic) ion flows. The main mechanism in this case is closely related to the Landau damping of collective plasma perturbations excited by flowing ions. The case of the subsonic ion flow velocities is physically different from the wake excitation by a supersonic ion flow, considered above, as the wake field cannot be excited in this case. In the subsonic case, the stopping power of ions by the Landau mechanism changes the ion distribution around the dust particle and produces the ion bunching as well as the local excess of positive charges around the dust particle. The latter can create potential wells not only in the direction parallel to the ion flow but also in the direction which is almost perpendicular to it.

74

Physics and Applications of Complex Plasmas

The ion distribution function fi is characterized here by three temperatures: Ti is the temperature perpendicular t o the direction of the ion flow, TI is the temperature which corresponds t o the direction parallel to the ion flow (which we call the downstream direction) and T2 is the temperature in the direction antiparallel t o the ion flow (which we call the upstream direction). Thus the ion distribution function can be written as

where V O is the ion flow velocity, u corresponds t o the ion velocity component along the flow direction and V _ Lcorresponds to the velocity component perpendicular to the flow, the subscript 1 corresponds to the downstream part of the ion distribution where u > V O , and the subscript 2 corresponds to the upstream part of the distribution function where v < ' 0 ; ' U T ~ is the thermal ion velocity characterizing the width of the ion distribution perpendicular to the flow, ' u ~ 1and 'uT2 (it is assumed Ti << T2 < T I )are the ion downstream and upstream thermal velocities, characterizing the width of the distribution function downstream and upstream in the direction of the flow, respectively. Finally, VT = ( 2 1 ~ 1 UT2)/2 is the averaged thermal velocity along the ion flow. The linear dielectric function for this distribution can be written in the form

=d m

+

fld3v,

(2.71)

'Ti

where k and u are the components of the wave vector and ion velocity along the direction of the flow respectively while k l and the v~ are the components perpendicular to it. The f,' is the ion distribution in the frame of ions at rest, and, finally, ADi = V T ~ / Wis~the ~ ion Debye length determined by the transverse ion thermal velocity. In the ion frame (of ions a t rest) the plasma is moving with the velocity -VO and thus only the upstream part of the ion distribution can produce the strong Landau damping. Introduce also the Debye length for the "averaged" longitudinal temperature (2.72)

Basics of Plasma-Dust Particle Interactions

75

and the relative flow velocity (2.73) where w = v g / f i u T 1 and tal = T2/T1. Other dimensionless parameters are r = Ti/Te,71 = Ti/Tl, and z = Zde2/aTe. The dielectric function for the screening at large angles to the ion flow is given by

1

+

(2.74)

where UO is positive, i.e., w1 > 0 , w2 > 0 , and the imaginary part of Eq. (2.74) can be expressed as usual in the analytical form. The dielectric permittivity can be rewritten as 1

where k

W ( w )= WR(W)- i-W1(W), Ikl

(2.75)

(2.76)

and (2.77)

The negative sign in the imaginary part of Eq. (2.75) shows that the plasma perturbations propagating along the direction of ion flow are amplified while the perturbations propagating against the direction of the flow are damped. Both damping and amplification are strong for the moderate flow (w of order of 1) and can not be described as the wave excitation since the flow velocity is subsonic. This effect is the manifestation of the strong Landau damping (since the amplification is just the inverse Landau damping).

76

Physics and Applications of Complex Plasmas

The electrostatic potential of a particle normalized to its Coulomb potential is given by (2.78) The presence of k/lkl in the dielectric permittivity changes the screened potential in the upstream and downstream directions. The component of the wave vector along the r-direction is k'. This component is equal to k for the downstream direction and -k in the upstream direction. This can be seen by introducing the spherical or cylindrical coordinates in the r-space where the flow is directed along the positive z-axis for the downstream direction and the negative z-axis for the upstream direction. The potential ++ represents the downstream potential while the potential in the upstream direction is +-. Introducing y = k l r , x = k'r (ICY = k l ) and r = r / X D , one finds

x [(y2

+ x2 + r2WR)cos(xcos 0) T W1r2sin(z cos Q)] . (2.79)

This expression is useful to find the distribution of the potential perpendicular to the direction of the flow

41 = ?7r l r n d x L

+ x2 + T 2 W R ) Jo(Y)YdY + x2 + r2WR)2+ ( T ~

(Y2 (y2

W ' ~ ) ~

(2.80)

Integrating over J: is easily performed by converting the integration interval to [-co+ca] and determining residues in the complex 2-plane. For the directions along the flow, it is more convenient to use the spherical coordinates and integrate over the angles of the wave vector

(2.81)

The first integral in Eq. (2.81) can be computed by the pole residue method in the complex plane while the second one should be calculated numerically. The first term of Eq. (2.81) reduces to unity in the case when one neglects the shielding and Landau damping and the second term is completely determined by the Landau damping. The condition of validity of Eqs. (2.79)

Basics of Plasma-Dust Particle Interactions

77

and (2.80) is T ~ / ( Z >> ( VT~/VT~ which , T ~ defines the corresponding solid angle. Thus the validity domain of Eq. (2.81) is outside the cone defined by this solid angle. The parameter P, = (ni - n,)/n, can be used to characterize the difference between the ion density ni and the electron density n,; in the sheath P, > 0.

Fig. 2.9 The potential well [Benkadda et al. (1999)l. (a) For t z l = 1/2 and w = 0 (dash-dotted curve); w = 1 / 2 (dotted curve); w = 1 (dashed curve); w = 3 / 2 (solid curve). (b) For w = 1 and t 2 1 = 1 (solid curve); t z l = 1/2 (dashed curve); t z l = 1/5 (dotted curve).

In order to investigate the dust shielding, the real and imaginary parts of the dielectric function Eq. (2.71) have been calculated. The negative values of the dielectric function appear mostly in the range of the strong Landau damping. The appearance of the damping implies the existence of the collective stopping power for the incoming ions which allows an additional accumulation of the ion charges around the dust particle. The usual process of screening in the absence of the flow ends when the charge of the particle is totally compensated by the ambient plasma. The Landau damping provides corrections which are of the order of w for w << 1. For w of the order of unity, the stopping power related to the Landau damping is appreciable and the incoming ions from the flow can create a net positive polarization charge

78

Physics and AppZications of Complex Plasmas

around the particle. This charge can be larger than the negative charge of the particle itself leading t o the attraction of other negatively charged dust particles. The potential (normalized to the non-screened Coulomb potential) of the particle in the transverse direction is shown in Fig. 2.9. The normalized potential is presented as a function of the ratio of the distance from the dust particle to the Debye length XD defined by Eq. (2.72). One can see that the potential well indeed exists and increases with the increasing w and/or decreasing t 2 1 . Expressions for the potentials created by the dust particle close to the upstream and downstream directions but outside the narrow cone where the Debye shielding is present were also analyzed [Benkadda et al. (1999)l. In principle, this cone narrows with decreasing ZIT$,and the expressions used are approximately valid for the almost upstream and downstream directions. The potential well increases with the flow velocity. The tendency for the upstream case is the opposite: with the increase of the ion flow velocity, the dust particle becomes more shielded and the potential well never appears. This has a simple explanation since the stopping power in the direction opposite to the ion flow is not sufficient to create significant ion space charges t o change the sign of the shielding cloud. For comparison, the Debye shielding potentials for these cases are also plotted. Figure 2.10 illustrates the fact that the smaller the angle between the direction of the flow and the direction of the screening, the deeper is the potential well and the larger is the distance between the dust particle and the position of the potential minimum.

I

I

Fig. 2.10 Dependence of the dust potential (normalized to the Coulomb potential) on the angle with respect to the ion flow (such that 0 = 7r/2 corresponds t o the downstream direction) [Benkadda et aZ. (1999)l. The dashed curve is for 0 = 37r/10, and the dotted curve is for 0 = 27r/5.

79

Basics of Plasma-Dust Particle Interactions

To conclude this section, we first comment on the relation between the wake field potential wells and the Landau potential well. In the case of the large (supersonic) ion flow velocity, the dust particle velocity in the ion frame exceeds substantially all the ion thermal velocities (along and perpendicular to the direction of the flow). Then the Cherenkov condition for the excitation of the plasma modes by the particle is fulfilled (in the case such modes exist in the ion plasma at rest) and the emission of these modes as a wake is possible. For the moderate subsonic longitudinal ion flows, the presence of the strong Landau damping is important since there are no excited wake fields. In the case of the strong Landau damping, the potential wells are mostly concentrated in the range of wave vectors belonging to the near zone where the electric field produced by the dust particle is of dissipative character (but not the wave field as in the case of the wake excitation). Creation of this field is independent of whether the wake field is created (since it is in the near zone) and is even independent on whether the ion plasma can have weakly damped modes. The role of nonlinearities produced by the dust particles field is an important issue. Without detailed calculations, we present here the dimensional analysis which can give the order of magnitude of the distances where the plasma polarization charge can change and screen the dust charge independently of the sign of this polarization charge. Within the narrow angular interval with respect to the flow, the characteristic distance is of order X o i (note that for the isotropic ion distribution the only characteristic length is X D ~ ) . Nonlinear effects are always strong close to the dust grain. Indeed, for the isotropic case the condition for the nonlinearity t o be weak on the dust surface is

Zde2 aTi

-=

T, << 1 , Ti

(2.82)

-2

where z = zde2/aTe is the dimensionless dust charge. Since in most of the sheath experiments the ratio TJTi is high (- lo2), the inequality (2.82) can hardly be satisfied. However, the linear approximation can correctly describe the shielding if the nonlinearities are at least weak within the shielding radius. The ratio of the nonlinear charge density & to the linear charge density pk = -.Zdeb(w kv0)/27r2 of a dust particle in the ion frame shows that the nonlinearity in the direction of the ion flow is small (with k l an/.) when

+

-

(2.83)

80

Physics and Applications of Complex Plasmas

where a M 5 for w = 1/2 and t z l = 1/2, and a M 1.5 for w = 1 and t 2 l = 112. For r M 6x0 (which is the approximate position of the minimum of the potential well according to the numerical results), ni M 2 x lo9 cm-3 and AD, = 5 x lop3 cm. Then the right hand side of Eq. (2.83) is of the order of lo3, which can correspond to the charges observed for not too large dust particles. The above estimate can thus be considered as the first approach to the problem since the nonlinearities in the dust particle shielding could be important, especially for larger particles. The linear approach can still have a range of applicability in the sheath where the ion flow is mainly regulated by the potential in the sheath. The change of the ion flow produced by a single dust particle is small for the dust particles of a small size as compared to the size of the wall producing the sheath. However, this statement is incorrect if the distance between the dust particle and the position of the minimum of the potential well is much less than the distance between the dust particle and the wall (electrode). For a more detailed description it is necessary to take into account the dependence of the ion flow velocity on the position of the dust particle with respect to the wall. This effect can be especially important in the case when the flow velocity at the position of the dust particle significantly differs from the velocity of the flow a t the wall.

2.3.5

Simulations o n plasma wakes

The problem of plasma kinetics in the presence of a macroscopic body is also related to the charging of the body. As we already discussed, several models of particle charging can be developed, the OML theory being the basic one. Also, a number of experiments have been performed to elucidate the character of the charge of an isolated particle; most of experimental techniques are complicated, requiring special measurement procedures, and, on the other hand, do not always give the precise results. Moreover, it is especially difficult to determine the particle charges for two particles, especially in the sheath region in the presence of the ion flow. In this case, the ab initio numerical simulation, being one of the most complete model description, can provide very important information on the character of the plasma kinetics and particle charging. We note here that the complete problem of plasma dynamics around a macroscopic body in the presence of plasma flows is highly nonlinear and therefore its numerical analysis is of a major importance. Among various numerical methods, direct integra-

Basics of Plasma-Dust Particle Interactions

81

tion of the equations of motions of plasma particles represents a numerical experiment whose significance approaches experiments in the laboratory. Various models of dust in the presence of plasma flows were numerically considered by, e.g., [Sommerer et al. (1991); Boeuf et al. (1994); Graves et al. (1994); Schweigert et al. (1996); Winske et al. (2000); Melandso and Goree (1995)]. Note that most of the models considered deal with the fluid description of a plasma [Melandso and Goree (1995)], the kinetic ( i e . , coupled Poisson-Vlasov equations) case [Graves et al. (1994)l without collisions in the vicinity of a dust grain, as well as with particle-incell simulations [Boeuf et al. (1994)l of a uniform, steady state DC discharge plasma where plasma particle losses are assumed to be exactly balanced by a constant ionization source, or a hybrid model [Sommerer et al. (1991)l combining Monte Carlo with fluid simulation, with the latter ignoring the equations of motion of the plasma particles. In a non-self-consistent threedimensional description [Schweigert et al. (1996)], the ion-ion interactions were neglected and the charge of the particle was fixed. In PIC simulation [Winske et al. (2000)], the structure of the wake potential behind a stationary point-like grain with a constant charge was studied on ion time scales using particle-in-cell simulation methods; the ion dynamics (on a background of Boltzmann distributed electrons) was studied in one and two dimensions. The self-consistent three-dimensional molecular dynamics simulation ( M D simulation)[Maiorov et al. (200l)l of plasma kinetics around a single stationary grain demonstrated a strong ion focusing, with the ion density in the focus exceeding the ion density in the flow by a factor of 5-6. Thus the charging of the second dust grain located behind the first one, will be strongly affected by this (highly nonlinear) effect. On the other hand, the plasma dynamics around and behind the second particle is a function of its charge, and therefore the full self-consistent simulation of such an arrangement should necessarily take into account the charging of the second dust particle in the wake of the first one. It is natural to expect that the charge of this second particle will therefore differ from the charge of the first particle. The anisotropy of the forces is important for the proper modeling of processes in these structures levitating in the flow [Vladimirov et al. (1997a); Vladimirov and Samarian (2002)l; also, the ion charge accumulated behind dust grains, associated with the plasma wake, can also drive some instabilities in the dust chains [Ivlev and Morfill (200l)]; on the other hand, the analytical modeling can hardly provide the relevant number because of the high nonlinearity of the processes involved.

82

Physics and Applications of Complex Plasmas

In this section, we first present the kinetics of plasma electrons and ions around a single stationary grain in the presence of an ion flow studied using a three-dimensional molecular dynamicssimulation method [Maiorov e t al. (2001)l. The model is self-consistent, involving the dynamics of plasma electrons and ions as well as charging of the dust grain. The effect of ion focusing is investigated as a function of the ion flow velocity, and distributions of electron and ion number densities, and electrostatic plasma potential, are presented. Furthermore, the characteristics of plasma particle kinetics in the presence of ions flowing around two stationary grains aligned in the direction of the flow are studied using a three-dimensional molecular dynamics simulation code [Vladimirov et al. (2003a)l. The dynamics of plasma electrons and ions as well as the charging process of the dust grain are simulated self-consistently. Distributions of electron and ion number densities, and the electrostatic plasma potential are obtained for various intergrain distances, including those much less, of the order of, and more than the electron Debye length.

2.3.5.1

W a k e of a single test particle

The technique of studying the properties of a classical Coulomb plasma involves numerical integration of the equations of multi-particle dynamics [Hockney and Eastwood (1981); Maiorov et al. (1995)l. The core of the method includes the consideration of the time evolution of the system consisting of Ni positively (“ions” with Zi = 1) and N , negatively (“electrons”) charged plasma particles confined in a region 0 < z < L,, -L,/2 2 < y < L,/2,2 -L,/2 2 < z < L,/2, together with a macroscopic absorbing grain of the radius a with the infinite mass and an initial (negative) charge Q = -Zde, where -e is the electron charge. The grain is placed at IC = 5h,, y = 0, and z = 0. For the characteristic lengths we have (for most calculations unless otherwise specified) L,/4 = L, = L, = lOh,, with the characteristic grid step in the presented results h, = 4h, = 4h, = 1.077 pm. Other initial values are summarized in Table 2.1. The walls bounding the simulation region are elastic for electrons; for ions, they are elastic in the y and z directions, i.e., at y = (-L,/2, Ly/2) and z = (-LZ/2,L,/2). For the given values, the characteristic lengths in the plasma are: the electron Debye length AD, = 5.256 pm, the ion Debye length AD^ = 0.831 pm, and the Landau length for scattering of the ions on the dust particle by the angle 7r/2 is r L = Zde2/miug; for the above parameters rL = 0.6/M2 pm. The ions are introduced in the system a t the plane x = 0 as a uniform flow

83

Basics of Plusma-Dust Particle Interactions

with the Mach number M = vg/v, (vg > 0) and the temperature Ti; at x = L, the ions are removed from the system. The paths of the ions and electrons are determined through numerical integration of the equations of f k l and the Coulomb motion: d 2 r k / d t 2 = F k / m k , where F k = force is given by frci = 4 k q L r k l / l r k i 1 3 .

x:+Ni+Ne

Table 2.1 The intial values for the dust grain and plasma particles. mp = 1842me is the proton mass, me is the electron mass, and e is the (absolute)electron charge [Maiorov et al. (200l)l. MacroparticleIons Ions Electeronsons

Charge

Mass Number Temperature

-10000e kdjf 1 n/a

e

-ee

4mp

100me

10000 0.025eV

9000 1eV

The direct particle-particle interaction is in the core of the most appropriate method for the numerical integration of the equations of motion in studies of basic plasma kinetics. The particle-particle integration method is flexible enough but involves a high computational cost. In an implementation of this technique, an algorithm taking advantage of the specific properties of classical Coulomb plasmas was elaborated to significantly decrease the number of arithmetic operations. Central to the method is the determina,tion of the particles nearest to every particle and inclusion of their interaction by using a computational scheme of high-order accuracy (Runge-Kutta method of the 4th order with an automatically chosen time step). Simulating the charging process, the real electron/ion mass ratio is effectively taken into account (note that in the computational model the electron mass was assumed to be 100m,, see Table 2.1) by renormalizing the absorbed charge in the process of the electron-dust charging collision, so that the charge appearing on the grain corresponds to its value for the real electron/ion mass ratio. In the simulation, the ion number density was calculated by averaging within the spherical layer around the macro-particle. The total simulation time of the computed physical processes is 3 . 3 6 lo-’ ~ 9s which is approximately equal to the half of the oscillation period of plasma ions oscillating with the ion plasma frequency rPi = 6.76 x lo-’ 9 s. The charge of the dust particle was found to fluctuate around Zd FZ 1000 1100 weakly depending on the ion flow velocity.

+

84

Physics and Applications of Complex Plasmas

Fig. 2.11 Contour plot of the ion density, showing ion focusing, for three different velocities of ion flow [Maiorov et al. (2001)]. The plot is presented in the grayscale topograph style; regions A correspond to the normalized ion densities below 1, and regions B correspond to the normalized ion densities above 1. The ions are focusing behind the grain thus forming the region with the highly enhanced ion density. The distances are given in the units of h, N 1.077 pm used in the calculation; the physical distance corresponding to the electron Debye length AD, ( T D ~in the Figure) is presented.

In Fig. 2.11, the contour plots of the ion density ni normalized to no = Ni/L,L,L,, z, are presented for three values of the velocity of the ion flow (one is subsonic with M 2 = 0.6, and two supersonic, with M 2 = 1.2 and M 2 = 2.4). For better visualization, parts of the simulation volume where n i / n ~< 1 and nilno > 1, respectively, are presented in the (grayscale) topograph style. A strong ion focus, with nilno N 6.5 a t the maximum, is formed a t the distance of a fraction of the electron Debye length behind the dust grain. The maximum value of the density a t the ion focus is almost independent of the flow velocity, whereas the characteristic distance of the ion focus from the dust grain increases with increasing flow velocity, being approximately equal to 0.5X0, for M 2 = 2.4. This characteristic spacing corresponds to an ion focus effect in the near zone of the dust grain, which is a purely kinetic effect [Benkadda et al. (1999)] not associated with the collective wake field formation. Note that the oscillating wake field potential

85

Basics of Plasma-Dust Particle Interactions

in the wave zone behind the grain cannot form here for the considered simulation time (half of the period of the ion oscillations). Another kinetic effect seen from Figs. 2.11 is the appearance of precursors in front of the dust grain, which can be attributed to those ions reflected backwards within the radius (around the z-axis) of the order of the Landau length. 2.3.5.2

W a k e of two particles

Here, we present the results of a self-consistent three-dimensional MD simulation of the kinetics of plasma electrons and ions around two aligned (in the direction of the flow) dust grains, taking into account the dust charging and the supersonic ion flow [Vladimirov et al. (2003a)l. The numerical method involves simulation of the time evolution of the fully ionized (with the singly charged ions) plasma consisting of Ni positively (ions) and N , negatively (electrons) charged plasma particles confined in a simulation box 0 < x < L,, 0 < y < L,, 0 < z < L,, together with two macroscopic absorbing grains (dust particles), each of the radius a = 0.5 pm, with infinite masses and the initial (negative) charges Q1,2 = -Zd1,2e, where -e e is the electron charge. The details are given in Table 2.2. Table 2.2 The intial values for the dust grain and plasma particles [Vladimirov et al. (2003a)l.

Charge MAss

Number Temperature

Macroparticale

Ions

Electrons

-1250Ee B 1

e 4mp

-ee 100me

5000

2500

0.205eV

1eV

n/a

As in the previous section, the electron mass is assumed t o be loom,, see Table 2.2, but the real electron/ion mass ratio is taken into account by renormalizing the electron current and therefore the absorbed charge in the process of the electron-dust charging collision, so that the charges appearing on the grains correspond t o their values for the real electron/ion mass ratio [Vladimirov et al. (2003a); Maiorov et al. (2001)J. The ions were introduced in the system at the plane x = 0 as a uniform flow in the xdirection with the Mach number M = v0/Vs (ZIO > 0) and the temperature Ti; at x = L, the ions are removed from the system. The walls bounding the simulation region are elastic for electrons; for ions, they are elastic in

86

Physics and Applications of Complex Plasmas

the y and z directions, ie., a t y = (0, L,) and z = (0, L z ) . This means that electrons are specularly reflected from the walls in all directions; on the other hand, since the ions are moving in the x direction, for them the specular reflection condition applies in the y and z directions. The total numbers of the electrons and ions in the system are fixed; it is chosen on the basis of a test simulation runs to satisfy the given number densities and t o make sure that the system is neutral as a time average. In the place of the ion absorbed by the dust grains (or a t the back wall of the simulation box), a new one is introduced at a random point on the front wall with the chosen velocity distribution function (the latter is assumed to be shifted Maxwellian). The dust grains are placed at x = xo = L,/4 and x = xo D , such that D is the distance between the grains, with the other coordinates being y = yo = L,/2 and z = zo = L z / 2 ; thus the grains are aligned in the direction parallel to the ion flow. The initial distributions of the coordinates of the plasma electrons and ions are chosen to be homogeneous within the simulation box; the initial velocity distributions correspond to Maxwellian for electrons and shifted Maxwellian for ions a t infinity. Depending on the distance to the colloidal particles, the distributions are distorted because of the interactions with the macroparticles. Thus the initial distributions do not include finite ion orbits which can strongly affect the kinetic characteristics under certain circumstances [Lampe et al. (200l)l. The trajectories of the plasma electrons and ions are determined through numerical integration of the equations of motion. For the Coulomb force at very small distances we used the corresponding expression for finite (small) size mutually penetrating spheres [Hockney and Eastwood (1981)l. The equations of motion are solved by the Runge-Kutta method of the fourth order with an automatically chosen time step. For the characteristic lengths we have (for most calculations unless otherwise specified) L,/22 = L, = L , = 20h, with the spacing h, = 2h, = 2h, = 0.5375 pm. For the given values, the characteristic lengths in the plasma are: the electron Debye length AD, = 5.256 pm and the ion Debye length X D ~= 0.831 LLm. The ion number density is ni = 2 x 10l2 ~ r n - ~and , hence the ion Debye length in term of the average ion-ion distance X ~ i n i ’= ~ 1.06; the number of ions in the ion Debye sphere is approximately 5. Note than since the ions are supersonic, their energy exceeds T, and they are really weakly coupled. For electrons, we have XD,n:l3 = 5.25, which corresponds to more than 500 electrons in the electron Debye sphere, and the electron-ion system can be considered as an ideal plasma. Finally, the electron and ion

+

Basics of Plasma- Dust Particle Interactions

87

Table 2.3 The charges on the dust grains depending on the distance between them [Vladimirov et al. (2003a)l.

Distance D Charge 1Q11 1390 0.06Lx = 0.25X~e 1420 O.lOLx = 0.41X0, 0.15Lx = 0.62X~, 1390 1430 0.20Lz = 0.82X~e 1470 0.25L, = 1.03X0, 1500 0.35Lx = 1.45X0, 1410 0.40Lz = 1 . 6 4 X ~ ~ 1480 0.50Lz = 2.05X~e 0.60Lx = 2.46X~e 1450 0.65Lx = 2.67X0, 1430 l.OOL, = 4.10Xoe 1460 1450 00

Charge IQzI 840 860 840 1010 1040 1080 1020 1130 1230 1180 1200

nla

number densities are chosen t o be higher than those in real experiments for numerical reasons (to decrease the plasma Debye length). The total simulated time of the physical processes is 9.2 x lo-' 9 s which should be compared with the inverse ion plasma frequency rpi = l / w p i = 3.4 x lo-' s. The time step of the numerical simulation is 4 x lo-'' 12 s which is not only much less than the inverse ion plasma frequency, but is also much less than the electron plasma frequency rpe= 1/upe= 8 x lo-'' 10 s. The speed of the ion flow corresponds to the Mach number M 2 = 2. Table 2.3 demonstrates the dependence of the charges accumulated on the dust grains as functions of the intergrain distance. When the particles are very close t o each other, their charges are influenced by the presence of the other particle. This influence is especially strong for the second ( i e . , downstream) particle; its charge is significantly (typically, 40%) less than the charge of an isolated particle (see the last line of the table). As soon as the distance between the particles is increased, the second charge exhibits a noticeable increase; it is interesting to note that the first charge is increased, too, although by a lesser value. We also see that when the interparticle separation exceeds the electron Debye length ( D = 1.03X0, 1.45Xge), the increase of the charge of the first particle stops; on the other hand, the increase of the charge accumulated on the second particle located downstream continues to grow until the distance exceeds two electron Debye lengths ( D = 2.46X0,). ). We can attribute this phenomenon to the fact

88

Physics and Applications of Complex Plasmas

Fig. 2.12 Surface plot of the normalized ion density, showing ion focusing, for three different separations D between two dust grains [Vladimirov et al. (2003a)l. The plot is presented in the grayscale topograph style; regions A correspond t o the normalized (to the unperturbed ion density nio) ion densities below 1, and regions B correspond t o the normalized ion densities above 1. The distances are given in the units of the total length of the simulation box in the direction of the ion flow L , % 4 . 1 X ~ , used in the calculation; the physical distance corresponding to the electron Debye length AD, is also presented.

that the ion wake of the first particle is spreading at distances significantly exceeding the electron Debye length; on the other hand, the influence of the second (ie.,downstream) particle on the charge of the first one in the simplest approximation is limited t o distances of the order of the electron Debye length. Another interesting phenomenon is that the charge of the

Basics of Plasma-Dust Particle Interactions

A

1 .

89

I

Fig. 2.13 Contour plot of the plasma potential for five different distances between two dust grains [Vladimirov et al. (2003a)l. The plot is presented in the greyscale topograph style. Note that the potential well (region B) is formed behind the dust grain and starts to form between the grains when the separation exceeds the electron Debye length.

second particle, at the considered distances (up to D = 4.1Xoe), is always less than the charge of the first particle, thus confirming the long-range influence of the ion wake. To compare the results for two particles with the case of an isolated particle, in the last line of Table 2.3 the result of the special simulation run for the charge of the first particle when the second particle is removed from the system ( D = cm) is presented. The particle charge in this case (Q = 1450e) coincides with other numerical MD [Maiorov et al. (200l)l as well as PIC [Vladimirov et al. (2001a)l simulations; moreover, this result also agrees to the model OML calculation for a particle levitating in the sheath region in the presence of the ion flow [Vladimirov et al. (1999a)I. In Fig. 2.12, the surface plots of the ion density ni normalized to nio = Ni/L,L,L,, z, for three different distances between the charged colloidal particles are presented: the first one (Fig. 2.lZ(a)) corresponds to the

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Physics and Applications of Complex Plasmas

short distance of D = O.25Xoe, the second one (Fig. 2.12(b)) is of the order of the electron Debye length: D = O.82Xoe, and the third one (Fig. 2.12(c)) corresponds to the relatively large ‘distance exceeding the electron Debye length: D = 2.7X0,. For better visualization, parts of the simulation volume where ni/nio < 1 and ni/nio > 1, respectively, are presented in the (grayscale) topograph style, i e . , part A is for ni/n,o < 1 and part B is for ni/nio > 1, so that the change from lower (with respect to nio) to higher densities is clearly seen. A strong ion focus is formed at the distance of a fraction of the electron Debye length behind the first dust grain; depending on the position of the second grain, the wake maxima are either combined, see Fig. 2.12(a), or clearly separated, see Fig. 2.12(c). Figure 2.13 gives the contour plot of the plasma potential (in V) for five different distances between the grains: (a) corresponds to a short distance which is much less than the electron Debye length, D = O.25Xoe, (b) is for the increased distance D = O.82Xoe, (c) is for D = 1.43Xoe, (d) is for D = 2.1Xoe, and (e) is for a distance relatively large with respect to the electron Debye length D = 2.7Xoe. For short distances (Figs. 2.13(a) and 2.13(b)), the wake is almost the same as that of one (combined) particle; on the other hand, for distances of the order of (Fig. 2.13(d)), or more than (Fig. 2.13(e)) the electron Debye length, the formation of quasi-wake features can be seen after the first grain, ie., before the second one. The characteristic distance for the region of the attractive potential to appear in the %-directionis of the order of the electron Debye length, thus coinciding with the linear theory (Vladimirov and Nambu (1995)]. We stress that the ion wake strongly influences the charge of the second grain located downstream with respect to the first particle. This was also proved by experimental measurements of the charges on two vertically aligned particles in ICP argon plasma [Prior et al. (2003)I. It was found that the charge on the bottom particle is 0.7 of the charge on the top particle. This influence is especially strong for the intergrain distances small compared to the electron Debye length. It is interesting that there is also an influence of the downstream particle on the charge of the particle located upstream; this influence, however, is limited to distances of the order of the electron Debye length, in agreement with the Debye approximation. The charge of the second particle, for the distances considered (up to four electron Debye lengths), is always less than the charge of the first particle, and this is attributed to the long-range influence of the plasma wake. The simulated plasma electron and ion densities as well as their cross-sections provide details of the corresponding distributions within and outside the

Basics of Plasma-Dust Particle Interactions

91

wake. Note that [Vladimirov et al. (2003c)l presented MD simulations of the wake behind two particles aligned perpendicular to the ion flow.

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Chapter 3

Production and Diagnostics of Complex Plasmas

A laboratory complex plasma can be categorized by the method it is produced. Of particular importance is the ionization source used to generate the background plasma as well as the mechanism of dust charging. We use these criteria to distinguish different types of complex plasmas. In Fig. 3.1, the corresponding classification sketch is presented. As mentioned above in Sec. 2.1.2, in most cases a dust particle in a plasma is charged negatively due to the higher mobility of electrons. This type of charging can be called an L L a b ~ ~ r p one t i ~ because e” it is determine by absorption of plasma electrons and ions on the grain surface. This occurs in discharge plasmas formed with various kinds of gas (or mixture of gases). There are numerous observations of negatively charged dust particles, mostly in capacitive RF discharge (such as [Thomas et al. (1994); Chu and I (1994); Melzer et al. (1994); Pieper et al. (1996)]), as well as in inductive RF discharges (such as [Fortov et al. (2000b); Hwang et al. (1998); Cheung et al. (2003)I) and DC glow discharges (e.g., [Fortov et al. (1997a); Nunomura et al. (1998); Thomas and Watson (1999)]), Sec. 3.1. In the case when emission mechanisms play the major role, positively charged dust grains appear. Ordered liquid-like structures of dust particles charged positively by the thermionic emission have been observed in the laminar jet of a thermal plasma [Fortov et al. (1996a)l and in the boundary region of a propellant combustion plasma [Samarian et al. (2OOOb)l. Positively charged particles have been also observed in the glass tube under the solar UV radiation [Fortov et al. (1998)] and in abnormal DC discharge where the charging was mainly due to photoemission [Samarian and Vaulina (2000)l. Positive grains are believed to have occurred in nuclear induced plasma due to secondary emission [Fortov et al. (1999b)I. In Sec. 3.2, we present the cases of complex plasmas with positive dust grains.

93

94

Physics and Applications of Complex Plasmas

Fig. 3.1 Classification of laboratory complex plasmas. The lower part corresponds to the “absoptive” charging regime when negatively charged dust appears; the upper part corresponds to the “emissive” regime when dust is charged positively.

Since one of the main objectives of the complex plasma research is the study of self-organized structures of grain particles, an efficient trapping of dust is necessary. For experiments done in terrestrial laboratories, one of the main forces acting on dust is the Earth’s gravity. Therefore a potential trap set in such kind of experiment should always presume a force (acting

Production and Diagnostics of Complex Plasmas

95

on the charged dust particles) t o compensate the influence of gravity. This leads to the levitation of dust. The electrostatic force is the perfect balancing force, especially in a gas discharge plasma. However, other forces, such as the thermophoretic force, can also balance the gravitational force. A typical potential trap for negatively charged particles is formed in discharge plasmas by the negatively charged chamber (tubular) walls. In the experiments, stable dust structures of various forms (see below in Secs. 5.2.1 and 5.2.2) have been observed in the region of strong electric field, at the edge of the plasma sheath, near the electrode, as well as in the striation region and in the artificially created electric double layers. An effective trap for positively charged particles does not yet exist (the trap formed in abnormal DC which is discussed below can hold up to tens of particles compared to the thousands for the negatively charged particles). The absence of an effective electrical trap capable of supporting positively charged particles and forming stable structures has so far hindered the corresponding experimental investigations. As a result, a common feature of complex plasma with positive dust is the formation of weakly correlated structures as the particles move in dynamic streams (see in Secs. 5.2.3 and 5.2.4). The lifetime of this complex plasma is determined by the dynamics of relaxation (propellant plasma and UV induced plasma in the glass tube) or plasma flow (combustion product plasma). The diagnostics of complex plasma can be divided into three major areas: (i) diagnostics of plasma (by classical probe, laser and spectroscopic techniques) taking into account the influence of dust particle on the measurement performed, see Sec. 3.3; (ii) diagnostics of dust and dust structures, Sec. 3.4; (iii) and the usage of dust as diagnostic tools for plasma and particles diagnostics, presented in Sec. 3.5.

3.1

Discharge Plasmas

In discharge plasmas, the background plasma is formed by the glow discharge in air or noble gases (He, Ar, Ne, Kr, Xe). Typically, the electron temperature T, ranges from 0.5 to 8 eV and the plasma density ranges from lo7 cm-3 33 to lo9 ~ m - ~The . dust particle size ranges from 10 nm to 40 pm, and the dust charge ranges from a few electron charges t o 106e. Appearance of dust in discharge plasmas can be proceeded in two ways. First, the preliminary produced dust of various shapes and sizes is distributed into the plasma from a dust shaker or from a tray. In the first

Physics and Applications of Complex Plasmas

96

method, dust stored inside the shaker is released with a shaking motion generated by an electric pulse or a mechanical jolt. Normally, the shaker (see Fig. 3.2) has a small (tens of micrometers in diameter) hole or multiple holes in its base. The hole ensures that only a small number of particles comes out. A gentle mechanical push causes the shaker to sprinkle the particles into the desired region. Likewise, a mesh can also be used for experiments with a larger number of particles. The second one, tray method, involves dust particles originally placed on a tray. They are released into the plasma by mechanical vibration of the tray or by applying the negative bias pulse to the (metallic) tray. The number density of dust particles is determined by the plasma and the potential trap parameters (typically, nd lo4 c 1 r 3 for a micrometer-size particles).

-

Single mesh

Multiple mesh

Disk with Disk with single hole multiple holes

Another way to obtain a complex plasma is to produce dust grains in the plasma. This can be done by sputtering of the electrodes or a special biased substrate. The particles produced in such a way are polydispersive, of irregular shape, with sizes up to 1 pm. The number density of dust particles is determined by the efficiency of the sputtering process ( n d is up to lo6 cmP3). In a chemically active plasma the particles usually grow

Production and Diagnostics of Complex Plasmas

97

in the plasma bulk. The details of the fine particle origin and growth in plasma-enhanced sputtering facilities and reactive plasmas can be found in Sec. 7.1.

3.1.1

RF-capacitive systems

Most laboratory experiments on complex plasmas are performed in a parallel plate RF capacitive discharge. Such a set-up in most cases is based on the GEC reference cell. The experimental set-up is shown in Fig. 3.3. A CCD camera

Lense

Upper electrode Particles Laser

Powered electrode Confining /electrode Fig. 3.3

Set-up for parallel plate RF discharge.

The discharge plasma is generated in the cylindrically symmetric system at a pressure in the range from 1 to 100 mTorr. A disk electrode is used

as the lower electrode with the 13.4 MHz RF signal applied to it. A ring or grid is used as the upper electrode which is grounded. The ring or grid are necessary to provide a better top view of dust structures. Each electrode has the diameter of the order of a few centimeters, and the outer diameter of the ring (or grid) is usually larger than the disk diameter. The electrode separation is about a few centimeters (2-8 cm) and can be

98

Physics and Applications of C o m p l e x P l a s m a s

adjusted to the experimental needs (the electrodes can be moved vertically). The ring and disk are both connected to electrical feedt,hrnughs. To produce the sinusoidal RF voltage, a waveform generator is connected to a power amplifier with a matching box to reduce the reflected power. The peak to peak voltage is up to 100 V, and the input power is up to tens of Watts. Typically, due to the asymmetry of the discharge] there is a DC self-bias of the powered electrodc up to -70 V.

Dust structui

Fig. 3.4 Set-up for RF discharge [Chu and I (1994)l

The parallel plate discharge is not the only one used for complex plasma investigations. One of the first complex plasma experiments was performed in a cylindrically symmetric plasma RF capacitive discharge (see Fig. 3.4). The set-up consists of a hollow outer electrode and the grounded center electrode with the ring shaped groove on the top for trapping particles [Chu and I (1994)]. Usage of a specially shaped electrode for improving trapping of particles is common in RF discharge experiments. In order to confine particles, an additional ring confining electrode is sometimes placed on top of the lower electrode. In other configurations, a groove or a rectangular additional electrode are used.

3.1.2

DC discharges

In a DC discharge, dust structures are observed in various regions. Most frequently] they appear in the striations] but can also be seen in the sheath as well as in an artificially created electric double layers (such as anode double layers in Q-machine, magnetic discharges, variable cross section discharge tubes).

Production and Diagnostics of Complex Plasmas

99

3.1.2.1 Sheath It is well-known, that the strongest electric field that potentially can be used to balance the gravity force and therefore to support the particle levitation, is in the cathode sheath. However, the cathode sheath can hardly be used for effective dust trapping because of the dramatic reduce of the dust charge (that could even change its sign) due to the sharp decrease of the electron number density towards the cathode. Thus the majority of experiments are performed in the auxiliary anode sheath. This sheath is created in the auxiliary discharge volume outside the main discharge between the anode and a specially added electrode to ensure the particle levitation (see Fig. 3 . 5 ) . DC glow discharge plasmas are generated in the metal chamber or glass tube using both a biased anode and a biased cathode or a biased cathode and a grounded anode. The anode and cathode are usually circular disks, each with the diameter of a few centimeters. Both the anode and cathode can be moved horizontally for convenience of trapping and visualizing the dust clouds. cathode anode PIV camera

Fig. 3.5 Sketch of the electrode set-up for dust levitation in the auxiliary anode layer of a DC discharge [Thomas and Watson (1999)l.

As was mentioned above, the dust powder is loaded onto a tray or filled into a shaker (container). The tray or shaker can be moved vertically in the chamber. Once the plasma is formed, the dust particles on the tray become charged, usually negatively, see Sec. 2.1.2. When the charge is sufficient, the electric force due to the sheath electric field can overcome the gravitational and adhesive forces that keep the grains onto the tray. Thus the particles leave the tray and enter the plasma volume. Then the particles are trapped within the auxiliary anode sheath that is situated

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Physics and Applications of Complex Plasmas

under the anode, see Fig. 3.5. This layer is typically a few centimeter in the diameter. Dust clouds trapped in this region have sharp boundaries. The shape of a dust cloud generally corresponds to the boundary of the anode sheath.

Fig. 3.6 Set-up of (a) the segmented triode; (b) the segmented triode with an auxiliary discharge.

The auxiliary anode sheath can be also formed in the so-called segmented triode. In this case, similar to the set-up described above, the cathode and anode disks are centered with the central apertures, Fig. 3.6(a). The lower segmented electrode is comprised of an annulus confining electrode and a circular levitation electrode. Typical voltage settings are: the anode is grounded, the cathode is at -300 V, the levitation electrode is at -20 V, and the confining electrode is at -100 V. Under these parameters, the DC glow plasma is very stable, reproducible, and able to trap the dust particles readily. A small amount of dust is placed on the levitation electrode. By inducing an arc between the levitation electrode and the anode, the dust

0 0

0

Production and Diagnostics of Complex Plasmas

101

particles are embedded into the plasma region. They are trapped into the plasma sheath directly above the levitation electrode. The dust clouds formed have an inverted conical shape with the size of several millimeters revealing the potential structure of the sheath. In order to enhance the possibility to manipulate the complex plasma parameters (e.g., the dust charge), more complicated set-up with an auxiliary anode can be used, Fig. 3.6(b). In this case, dust particles levitate in the anode layer which is created between the main and auxiliary discharges. More complicated device allowing to operate at very low gas pressures, less than 10 mTorr, which is important to reduce the damping in the particle structures is a filamentary DC discharge. The plasma is generated by the DC discharge between hot filaments and the anodes, as shown in Fig. 3.7. The diffused plasma in the central area is confined by multiple cusps of the magnetic field generated by a number of permanent magnets. Dust particles are trapped in the plasma-sheath boundary area above a negatively biased mesh electrode. These particles form a two dimensional Coulomb crystal inside the ring electrode.

Illuminated

camera

Fig. 3.7 DC filamentary discharge: (a) discharge set-up [Misawa et al. (ZOOl)]; (b) dust levitation set-up [Nunomura et al. (1999)].

102

3.1.2.2

P h y s i c s a n d Applications of C o m p l e x P l a s m a s

Striations

The striations are the waves of plasma ionization appearing in the positive column of a DC discharge under certain conditions [Raizer (1991)l. In particular, a natural standing striation can be produced. Striations are characterized by periodic changes of the electron density, the electric field and the potential along the axis of the discharge. An important feature of a striation is the strong electric field (up to 30 V/cm) on a level with a small deviation from electroneutrality (&/no < 0.1%) in its head (the most luminous part of the striation). This leads to large charges on dust and therefore to the effective electrostatic trapping.

C

Fig. 3.8

DC discharge tube [Fortov e t al. (1997a)l.

The DC discharge is formed in a cylindrical glass tube positioned vertically with cold electrodes and diameter much less than the distance between the electrodes (see Fig. 3.8). The typical discharge current is up to a few mA and the pressure range is from hundreds of mTorr to a few Torr. The electron temperature in such a discharge is around 1-4 eV, and the plasma density is of order 10’ cmP3. Dust particles are dispersed into the plasma by a dust shaker at the top of the glass tube. The particles are trapped and suspended in the head of the striation and are seen as a cloud levi-

Production and Diagnostics of Complex Plasmas

103

tating at the center of the striation head. The size of the luminous part and the separation between striations depend on the discharge conditions. For example, in the experiment [Fortov et al. (1997a)l the length of the luminous part increases from 10 mm to 25 mm as the pressure decreases from 1.2 Torr to 0.2 Torr, and it weakly depends on the distance between the striations, which are within the range 35-50 mm. Sometimes, several clouds of particles are observed in different striations, see Sec. 5.2.2. By varying the discharge parameters (such as current and/or pressure), it is possible to merge the structures of the neighboring striations into one long, extended formation - a filamentary structure, Sec. 5.2.2. 3.1.2.3 Double layers

A double layer is a region of a non-neutral plasma consisting of two adjacent space-charge layers of the opposite sign (see insert in Fig. 3.9). There is a relatively strong electric field between the layers and a negligible field outside the layers. This strong electric field can be utilized in complex plasma experiments as a trap for dust particles.

Cathode Fig. 3.9 A double layer in the tube with variable cross-section

Double layer can be produced in a variable cross-section tube. It i s well known that plasma parameters can be changed by varying the transverse cross section of the positive column. If we use a vertical discharge tube with the variabIe cross section, a double layer of the space charge separating the two plasma regions with different electron temperatures and electron den-

104

Physics and Applications of Complex Plasmas

sities appear at the mouth of the contraction (see Fig. 3.9). The narrow cathode part has higher values of T, and n,. The potential drop U and the electric field E in the double layer depend on the gas pressure, the radii R, and R, and their ratio R,/R,, and the discharge current [Allen (1985)I. For a typical DC discharge tube, the potential drop is about 10 V and the size of the double layer is about 10 mm. This means that the electric field of the double layer is large enough to support a dust particle. In experiment [Lipaev et al. (1998)], levitation of dust particles at the mouth of the narrow part have been reported. The contraction (R, = 3.5 mm and R,/R, = 4.3) was created by introducing an additional cylindrical glass tube with a variable inner diameter into the discharge tube. The wider part of the variable tube was placed over the cylindrical cathode. The obtained structures contain a few particles and are of a low order. In Q-machine, dust particles were trapped in a specially formed anode double layer [Barkan and Merlino (1995)l. The plasma is created by surface ionization of potassium atoms on the hot tantalum plate (7' 2500 K) which also provides neutralizing electrons. The plasma column is confined by a longitudinal magnetic field. The electrons and K+ ions have approximately equal temperatures, T, T, 0.2 eV, and the densities are within the range 107-109 cm-3 As we mentioned before, an appropriate electric field configuration must be established within the plasma column in order to suspend the negatively charged particles. In this case, the electrostatic potential trap is provided by the anode double layer near the end of the plasma column [Barkan and Merlino (1995)l. The double layer is formed in the Q-machine by establishing a glow discharge on a small anode disk located a t the end of the plasma column. When the anode disk is biased to a positive potential, electrons from the ambient plasma are accelerated to the anode. If the potential somewhere in the anode sheath exceeds the ionization potential of the neutral gas, the accelerated electrons can ionize the gas. When the ion density in the sheath is comparable t o the electron density, and the sheath detaches from the anode with the potential drop now occurring a t the boundary between the ambient plasma and the glow discharge plasma a t the anode. The plasma parameters within the double layers can be quite different from those in the ambient plasma. Typically, T, 2-4 eV, Ti 0.2 eV, and the plasma density is of order lo9 cmP3. The dust charged up to lo6 electron charges is effectively trapped in the double layer forming well-defined structures. One of the main features is formation of stable dynamic structures in the form of dust-acoustic waves [Merlino (1997); Merlino et al. (1997)l. N

N

N

N

N

Production and Diagnostics of Complex Plasmas

3.1.3

105

I n d u c t i v e l y coupled p l a s m a s

R F inductive discharge is used for background inductively coupled p l a s m a (ICP) production. As we mentioned before, a potential trap is necessary in order to confine dust particles in a plasma. In the electrode discharge of any type, electric fields are related to the current in the external circuit. They play a substantial role in the formation of the potential trap. Electric fields of such a kind are absent in the RF inductive discharge. Here, the potential trap for dust particles is formed due to the ambipolar diffusion that leads to violation of electroneutrality. In turn, this is caused by the difference in the rates of the diffusion and the mobilities of plasma electrons and ions. Thus, formation of the electrostatic trap is determined by both the bulk plasma processes as well as the processes of the charge recombination on the walls. Vacuum chamber

TOP

RF coil

Bottom

Around

Fig. 3.10 Possible positions of the inductor for ICP production.

The inductive antenna can be placed on the top or bottom of the chamber (tube) or can be curled around the chamber (tube), see Fig. 3.10. The schematics of a typical ICP discharge which was used for the complex plasma study [Fortov e t al. (2000b)l is presented in Fig. 3.11. In this case the plasma is generated in a vertically aligned glass tube. A circular inductor consisting of several rings around the discharge tube is used. The size of the plasma region in the radial direction is restricted by the tube wall. In the vertical direction, the size depends on the gas pressure and the input power, and can be varied from a few centimeters up to the size comparable to the vertical size of the tube. The discharge tube can be moved freely inside the inductor and therefore there is a possibility of changing the distance from the lower boundary of the plasma region to the bottom of the tube while keeping the discharge parameters fixed. Dust particles are injected from the dust shaker at the top of the discharge tube.

106

P h y s i c s a n d A p p l i c a t i o n s of C o m p l e x P l a s m a s

Circular inductor nductive RF plasma

P D u s t structure

Fig. 3.11 A typical set-up for ICP discharge [Fortov et al. (ZOOOb)].

Because of the absence of electrodes in the discharge, levitation of particles can be observed in a plasma periphery, see Fig. 3.11. This region is near the lower boundary of the plasma, namely, in the transition region between the bright luminous plasma and the neutral gas. Levitation of particles cannot be observed in the central part of the discharge because of the weak vertical electric field not sufficient to compensate the gravity. Furthermore, the falling particles avoid the central part of the discharge thus indicating the absence of any possible electrostatic trap. At low pressures, the particles exhibit a random motion in the trapping region. As the gas pressure is increased, ordering of the particles appears accompanied by formation of dust particle structures. Similar to other types of discharge complex plasmas, variations of the input power and gas pressure affect the structures. In particular, under certain conditions, dust density waves can be observed, usually in the lower part of the dust structure. The pressure growth damps the waves, and rather stable dust structures appear (see Sec. 5.2.1). The RF inductive discharge in a chamber with a planar spiral RF coil on the top (see Fig. 3.12) was used to study Coulomb clusters. The bottom of the chamber has a feed-through consisting of multiple tungsten pins through a ceramic disc, providing connections to the electrode. The dust is trapped above the confining electrode introduced to define the location and type of dust structures. The experiments involves creation of plasma crystals of various sizes (from a millimeter to tens of centimeters) and shapes

Production and Diagnostics of Complex Plasmas

107

Camera

RF coil

B

, ; r

Gas inlet

Fig. 3.12

IC discharge in a chamber with a planar spiral RF coil [Cheung et al. (2003)].

(ellipsoidal, circular, annular, as well as Coulomb clusters of various configurations). Different types of the electrode arrangements provide different potential wells and therefore can be used to produce a certain type of structure. The electrodes are made from a conventional PCB (printed circuit board): electrically isolated annular regions were created by etching or machining narrow circular grooves in the copper; electrical connections to various regions are implemented by pins through the insulating sheet. Using an auxiliary cylinder electrode placed on the top, one can obtain prolonged (ellipsoidal) structures. To produce a confining potential well, a positive voltage is applied to the central disc while the outer rings are grounded. For example, to produce annular dust crystals, a positive voltage is applied to the intermediate annular region while the central and outer annular regions are grounded. For Coulomb clusters consisting of a few dust particles, the electrode is a centered disc surrounded by three concentric annuli. To create a confining well, the central disc is powered, with all other regions grounded. The structures produced are discussed in Sec. 5.2.1.

3.2

Complex Plasmas with Positive Grains

When emission of electrons from dust particle surface is unimportant, the equilibrium charge is negative as discussed above in Secs. 2.1.1 and 2.1.2. The emission of electrons can ensure the positive charge. Such a difference

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Physics and Applications of Complex Plasmas

in the sign can dramatically modify properties of a complex plasma. In particular, in a plasma with positive dust particles the electron density is larger than that of the ions. The electron emission from the particles affects the electron Debye length and thus the condition for plasma crystal formation. Also a number of new instabilities and waves can appear. Since the positive dust occurs in a plasma in the presence of a strong ultraviolet (UV) radiation or fast electrons, the plasmas with positively charged dust are common in space as well as in the Earth’s mesosphere.

3.2.1

UV-induced plasmas

When particles are irradiated by UV, plasma with positively charged particles is formed as a result of photoemission if the energy of photons exceeds the work function of photoelectrons emitted from the particle surface. The characteristic value of the photoelectron work function for the majority of materials does not exceed 6 eV. Therefore, photons with an energy less than 12 eV can charge the particles without ionizing a buffer gas. UV-induced plasmas can be produced by external as well as internal UV sources. In the case of an external UV source, the dust particles are exposed to UV radiation from the source outside the plasma volume. That can be UV laser, UV lamp or sun. To provide sufficient UV flux, special windows (for example, quartz or uviol) are used. Note that the gas pressure in the system is limited by the UV light absorption. Another case is realized when the UV source is inside the plasma volume such as the self-radiation of abnormal hollow cathode discharge. The strong radiation from the cathode region is a distinctive feature of a hollow cathode gas discharge. The abnormal discharge with a hollow cathode has been used as an effective UV source to produce complex plasma [Samarian and Vaulina (2000)]. In this experiment, the air glow discharge is formed in a cylindrical stainless steel chamber with a specially shaped cold electrode (see Fig. 3.13). The cathode is consisted of two copper bracelets coupled by a molybdenum wire, and the anode is a metal foil disk. The discharge current is varied from 0.5 mA to 15 mA, while the air pressure is varied from 0.2 to 2 Torr. The trapping of the dust particles is determined by the formation of the potential well due to an increase of electron number density towards the current core and towards the cathode. The density gradient creates an electrostatic field which allows formation of a cloud of tens of particles (see Sec. 5.2.4) at the distance of a few millimeters above the center of the anode.

Production and Diagnostics of Complex Plasmas

109

Fig. 3.13 Experimental set-up for UV-induced plasma producing an abnormal hollow cathode discharge [Samarian and Vaulina (2000)].

UV-induced plasma under the external solar radiation can appear in space. In an experiment, such a plasma is produced by exposing a dust cloud to the sunlight onboard a space station. The bronze particles coated by a monolayer of cesium (cesium has a lesser work function than bronze) have been placed inside the glass tube by [Fortov et al. (1998)l. One of the end surfaces of the tube is a flat uviol glass window; it provides illumination of the particles with the solar UV radiation, see Fig. 3.14.

CCD camera

Fig. 3.14 The glass tube in the experiment (Fortov et al. (1998)l.

Neon was selected as the buffer gas because of its chemical inertness towards the bronze and the cesium coating of the particles, its spectral

110

Physics and Applications

of Compl ex Plas m as

transparency, and its high ionization potential. The experiment was performed under the two different gas pressures, namely, 0.01 and 40 mTorr. This gave the possibility t o observe the dynamics of the structure formation for different values of the particle charge and the neutral gas friction. Dynamics of dust structures induced by solar radiation in a n experimental chamber with finite dimensions is determined by the charging time, the damping time, the time of formation of dust particle structures, and the time that particles spend in the experimental chamber. The structures observed under such conditions are presented in Sec. 5.2.4. 3.2.2

Thermal plasmas

Laminar jet of combustion product plasma as well as the boundary region of a propellant flame can be used to produce structures of dust particles charged positively by thermionic emission. In this case, the dust structures are formed in the quasi-neutral thermal equilibrium plasma without any trapping.

~

‘outer flame

Fig. 3.15 Sketch of a thermal plasma source.

The main part of the plasma source is a two-flame burner with propane and air fed into its inner and outer flames (see Fig. 3.15). Dust particles are introduced into the inner flare of the burner. In this case the refractory and chemically inert particles have to be used. Another requirement is the

Production and Diagnostics of Complex Plasmas

111

low contamination of the sodium and potassium compounds. The presence of alkali metals in the dust particles leads to the appearance of atoms with low ionization potentials in a plasma. This leads to the increase of plasma density and therefore to the decrease of the particle’s charge and decrease of the plasma screening length. Thus the particle electrostatic interaction is weakened and strongly coupled structures are not possible. In experiments, thermal plasmas with two types of chemically inert particles, A1203 and Ce02 were produced. The main components of the plasma in one case are charged CeO2 particles, electrons, and singly charged Na+ ions, and, in the other case, charged A1203 particles, electrons, and Naf and Kf ions. The burner design (with two flames) makes it possible to create a laminar stream of plasma with uniform parameter distributions: temperature, electron and ion densities in the region in the inner flame. In the experiments [Fortov et al. (1996a); Fortov et al. (1996b)], the velocity of the plasma stream was varied over 2-3 m/s and the electron density was within the range 109-1011 ~ m - ~The . combustion product plasma is at atmospheric pressure. Under these conditions the temperatures of the electrons and ions are equal and varied within the range T, Ti T, -1700-2200 K. The plasma temperature is changed by varying the fuel/air ratio 0.95-1.47. The temperature Td of the particles is close to the gas temperature for the transparent particles (as Ala03). For absorbing particles (as CeOz), some difference of 30-100 K (Td < T,) due to the radiative cooling is observed. Being colder than the gas, a particle stimulates a heat flow onto its surface and therefore a temperature gradient occur. The thermophoretic force appearing in this case plays significant role in dust structure formation (see Sec. 5.2.4). The complex plasma in the propellant combustion product is created by burning samples of synthetic magnesium and aluminum propellant. The propellant samples are cylindrical with various lengths and diameters. Pellets with the size dependent on the burn rate of the fuel (chosen so as to provide an adequate time for the measurements, 20-40 s) are fabricated. Heaters are preferably used for controlled and safe ignition of the pellet (for example, it can be a nichrome wire connected to the power supply). An experimental procedure is as follows. Synchronized measurements of the main plasma parameters are started at the moment when the fuel is ignited. Different fuel types give different flame zones. The magnesium fuel produces a strongly nonuniform combustion zone and numerous sparks as well as the nonstationary character of the combustion. This excludes the possibility of any kind of a dust structure to be observed. For the aluminum

-

N

112

Physics and Applications of Complex Plasmas

fuel (producing A1203 dust particles), there are three characteristic regions of the flame (where the dust structures can be found): the high-temperature zone (with the plasma temperature Tp 3000 K and np 5 lo2 cmP3 in the core of the flame), the boundary region (T, 2000 K and np = lo3los cmP3), and the condensation zone (T, 600 K and np 2 lo4 ~ m - ~ ) , see Fig. 3.16.

-

- -

Condensation

Fig. 3.16 Plasma in a propellant combustion product [Samarian et al. (ZOOOb)].

In the experiments [Samarian et al. (2OOOb)], the aluminum fuel produced a 10-30 mm high brownish-colored flame. The distinct condensation zone can be seen above the flame, where the exotic dust flows have been found, while the ordered structure formation was observed in the boundary region (see Sec. 5.2.3). The charge composition of the solid fuel combustion product plasma as well as that of the laminar jet of thermal plasma depends strongly on the easily ionized alkali-metal impurities (Na and K ) , that are present in the synthetic fuel and end up in the combustion products. The density of the alkali-metal atoms, the gas temperature and the particle surface temperature, as well as the work function of the thermal electrons from the surface of the dust particle determine the electric properties of the thermal dusty plasma and have a profound effect on the magnitude and sign of the dust charge and therefore on the formation of the ordered structures.

Production and Diagnostics of Complez Plasmas

3.2.3

113

Nuclear-induced complex plasmas

Nuclear-induced complex plasma is formed as a product of nuclear reactions passing through gas exciting and ionizing its atoms and molecules. In most cases, the energy of nuclear particles is large enough to penetrate into a dust particle with a radius of several microns. As a result, the dust can acquire a positive charge due to the secondary electron emission. Besides, the dust particle itself can become radioactive and emit charged particles (electrons) in the process of nuclear reactions. Dust particles, placed into a nuclear-induced plasma, are also charged by plasma currents. Thus, a radioactive dust particle is charged by various mechanisms which affect the value as well as the sign of the charge. Therefore a thorough investigation of the charging process is necessary for any particular case of dust in a nuclear-induced complex plasma. There are two possibilities to produce a nuclear induced complex plasma. Firstly, an a-particle source placed at the top or bottom of an experimental chamber can be used. There is secondary electron emission where a-particles and fission fragments sputter electrons from near-surface regions of dust grains. Secondly, preliminary activation of dust particles (which radioactively produce the plasma) can be done in a reactor with a p-source. The number of electrons ejected by each original nuclear particle varies from several units up to hundreds. The scheme of the experimental set-up realizing the first possibility (with the a-particle source) is presented in Fig 3.17. The source of aparticles and fission fragments is a thin layer of 252Cfwith the intensity of lo5 fissions/s and 1.6 x lo6 a-decaysls have been used [Fortov et al. (1999b)l. The spherical monodisperse melamine formaldehyde (MF) particles and polydisperse cerium oxide (CeOa) particles have been used. The experiments were performed in neon and argon at sub-atmospheric pressures within the range (0.25-1) x lo5 Pa. The dust particles were injected into the chamber volume from a shaker placed above a hole in the upper electrode or thrown in from the bottom by an air jet. The experimental installation for producing the complex plasma by pactive dust particles is similar to the installation described above. The main difference is that instead of the 252Cf,CeOz particles were as the source of gas ionization. The particles have been activated in the nuclear reactor. The activation occurs according to the reaction

114

Physics and Applications of Complex Plasmas

Dust shaker

. -I__-_.-

-/

Electrodes

To pump

Fig. 3.17 Experimental set-up with the a-particle source [Fortov et al. (1999)]

During the experiments, 0-particles are emitted according to the reaction 1 4 1 ,141 ~ ~

Pr

+e + v/

where v’ is the electron antineutrino. The measured intensity of the 0-decay in the experiment was of the order of lo9 decays/s which corresponds to the output of fast electrons from one CeOZ particle with the rate of 0.1 decay/s. Experiments were performed in air a t the atmospheric pressure. The charge of the levitating dust ranges from 2001el to 4001el in the cy case and 3001e1-5001el in the 0 case. The experiment with the a-source demonstrates that it is possible to control the position of charged dust particles near the electrode. Conical cloud of dust is formed near the hole in the upper electrode when the electrode has a positive polarity. In the case of reactor-activated particles, broad regions of levitating dust were observed in the central part of the inter-electrode space where the electric field is less than 30 V/cm. When the electric field exceeds 30 V/cm, there is no stationary levitation of particles, and the vort,ex motion of the grains begins (see Sec. 5.2.4).

3.3

Traditional Diagnostic Techniques in Complex Plasmas

Traditionally, common diagnostics of a plasma include: various probe measurements, emissive/absoptive spectroscopic methods, as well as laser diag-

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115

nostic techniques (such as laser induced emissive fluorescence method). Tn a complex plasma, the most commonly used are probe measurements (in gas discharge and in thermal plasmas) and the emissive/absorptive spectroscopy (in thermal, UV-induced, and discharge plasmas). As we already mentioned, there are two aspects in the application of traditional diagnostic techniques in complex plasmas. Firstly, there is an influence of dust particles on the measurements of plasma parameters. Secondly, there is a possibility (though not always direct) of the measurement of dust parameters (such as charge, number density, size, and surface temperature). 3.3.1

Probe measurements

The probe technique is the most common and simplest diagnostic method in a plasma [Chen (1984); Raizer (1991)l. The plasma parameters are extracted from I-V probe characteristics using some model approach (such as Langmuir model for gas discharge plasmas, hydrodynamic model for high pressure plasma, etc.). The presence of dust particles might affect the probe characteristics. Therefore, the influence of dust has to be taken into account for correct plasma diagnostics. Moreover, the presence of the biased probe can disturb the dust distribution. The best way to determine the influence of dust on plasma parameters is to provide a preliminary diagnostics (for example, by mapping of the discharge or doing measurements under the same plasma condition just before introducing the dust). The probe characteristics mostly affected by the dust is the electron saturation current. There are two reasons for that. Firstly, the dust particles accumulate (or emit) electrons and therefore the number of free electrons in the plasma is reduced (or increased). Secondly, the dust particles (nonconductive in most cases) can cover the probe surface and decrease the collection area of the probe. To provide an observable effect on the probe, the dust number density should be sufficiently high. The first possibility realizes when the parameter P, > 1. In this case for negatively charged dust the electron saturation current is smaller than the current measured without dust. This is due to the fact that electrons attached to grains of extremely low mobility (as compared with mobility of plasma ions/electrons) are not collected by the probe. There are several ways to determine the dust charge from probe characteristics. The simplest one is to find the floating potential. Generally, the floating potential is the difference between the plasma potential and

116

Physics and Applications of Complex Plasmas

the zero current potential. Note, that the plasma potential is sometimes assumed to be zero; in this case, the floating potential is naturally equal to the zero current potential (which is not always the case, especially when data are taken from probe measurements in which all potentials are related to a reference electrode). Once the floating potential is obtained, the dust surface potential is also known, and the dust charge can be determined. If one cannot obtain the proper probe characteristics and, correspondingly, the floating potential, one can just measure the electron saturation currents in two cases, namely, with dust and without dust. From the data obtained, the value of the floating potential U f can be extracted. This approach was used by [Thomas and Watson (2000)]. For the zero current potential, the current balance equation can be written in the case T, >> Ti and ni = const as (1 +

g)

(z)1’2exp

(2) (2) =

1-

.

(3.1)

From the experimental measurement of the difference in the electron saturation currents, the ratio of the plasma (ion) to dust densities is a known quantity (3.2) where Ipst stands for the electron saturation current in a dusty plasma. Using the experimentally determined value of the ratio ( 3 . 2 ) , one can solve Eq. (3.1) and find the value of the floating potential. The results of the electron saturation current measurements with and without dust is shown in Fig. 3.18. It was observed that in the presence of dust particles, there is a decrease in the electron saturation current. Furthermore, the largest difference in the electron saturation currents is observed in the region where the dust density is the highest. Another way to find the dust charge from the probe measurements is to analyze the difference in the electron saturation currents by employing the charge neutrality condition (with the known dust number density) [Barkan et al. (1994)l. The experiment utilized, as the basic plasma source, a Qmachine in which a fully ionized, magnetized potassium plasma column of 4 cm diameter and 80 cm length is produced by surface ionization of potassium atoms from an atomic beam oven on a hot (N 2500 K ) tantalum plate. The basic constituents of the ambient plasma are K+ ions and electrons with approximately equal temperatures T, M T, M 0.2 eV N

N

Production and Diagnostics of Complex Plasmas

117

Fig. 3.18 Electron saturation currents in a plasma with (open circles) and without (solid squares) dust [Thomas and Watson (2000)].

and densities in the range of 105-1010 ~ m - ~To . dispense the hydrated aluminum silicate (kaolin) dust particles into the plasma, the plasma column was surrounded over its portion by a rotating dust dispenser. The grains had a size distribution in the range of 1-15 ,urn with an average grain size of 5 pm. The dust number density was measured via estimating the dust flux from the dispenser to the collector and via measuring the spatial decay of the ion density within the dust cloud, and was approximately 5-6 x lo4 ~ r n - ~ . The main diagnostic tool of the plasma was a Langmuir probe, which is a 5 mm in diameter tantalum disk. The Langmuir probe enabled one to determine how the negative charge in the plasma is divided between free electrons and negatively charged dust grains. Fig. 3.19 shows the Langmuir probe characteristics obtained under identical conditions except for the absense (upper curve) or presense (lower curve) of dust, with the electron portion of the characteristic shown as a positive current. When the dust is present, the electron saturation current I , to a positively biased probe is smaller than the current Ieo measured without dust. This is due to the fact that the plasma electrons, which attach to dust grains of extremely low mobility are not collected by the probe. The ratio L = ( l ~ s t / l , n O ) / I ~ S is then the measure of the fraction of the cumulative negative charge where the of free electrons in the dusty plasma, ie., L = ratio IpSt/Iznocontains the probe ion currents with and without dust, respectively. N

npst/nyst,

118

Physics and Applications of Complex Plasmas

Fig. 3.19 Langmuir probe characteristics obtained for the same conditions, except for the absence (upper plot) or presence (lower plot) of dust [Barkan et al. (1994)l. When the dust dispenser is turned off (dust off arrow on a diagram), the lower plot reproduces the upper one.

npst

The plasma density = ne0 can be determined by using the well known relation Ieo = Aeneovre,where A is the collecting area of the probe. From the measurements of L and the plasma density no we can obtain the quantity & n d / e = no(1 - L ) taking into account the overall charge neutrality. If we find L as a function of the plasma density no, for constant n d , a, and T d , then the dust charge can be determined. A reduction of the probe collecting area because of covering it by the dust particles is another reason for the electron saturation current to decrease. Careful checks should be made to ensure that the probe functions properly by the return of the electron saturation current to the “no-dust” level when the dust inlet is abruptly turned off. For example, in the thermal plasma experiments the dust charge is positive and therefore one can expect an increase of the electron saturation current due to additional electrons emitted form the particles. However, the decrease of the electron saturation current takes place. This decrease can be more pronounced on the initial stage of the measurement, and then it saturates. The probe surface exam-

Production and Diagnostics of Complex Plasmas

119

ination reveals numerous particles attached onto the probe surface. Under some condition, particles cover the probe surface in a discharge plasma also. Therefore, careful examination has to be done to ensure that the probe functions properly in the complex plasma environment. 3.3.2

Spectroscopic techniques

Spectroscopic diagnostics in a complex plasma is used to determine plasma parameters as well as grain parameters such as dust surface temperature, grain size and number density. The presence of dust particles can considerably influence optical properties of a plasma. Most of spectroscopic diagnostics are based on the measurements of the intensity of plasma emission and extinction of radiation produced by extcrnal sourccs. Thus one can determine plasma temperature and density. In a complex plasma, there are difficulties in interpretation of the experimental data associated with the influence of dust on the measured radiation intensity. Also, additional difficulties of the use of spectroscopic techniques for particle diagnostics in dusty plasmas are related to the inverse problems of the scattering theory, which are usually ill-posed. The temperature of gas and surface temperature of particles can be obtained by the emission-absorption method involving measurements on a gas emission line. In general, in order to determine the gas temperature, calculations of the radiation transport in the plasma and reliable preliminary data on the optical characteristics of a dust cloud are required. In real situations, especially in thermal plasmas, optical characteristics of dust grains are usually unknown and are to be determined experimentally. Here, a number of problems arise, in particular, the reliability of reconstructing the gas-phase temperature by the inversion method and the determination of the atom density by the conventional total-absorption method. A conventional method for determining the temperature of a thermal plasma is the generalized reversing method. The plasma density is usually deduced from the data measured by the emission-absorption method. The gas temperature is calculated by the formula (3.3) where SLP is the signal from a (light) source after transmitting through a dusty plasma, Sp is the emission from the plasma, SL is the signal from a reference source, TL is the brightness temperature of the source filament at

120

Physics and Applications of Complex Plasmas

the diagnostic wavelength A, ko is the extinction coefficient for the source radiation a t the wavelength A, and Cz = 1.44 cm . K. The charge composition of a thermal complex plasma strongly depends on easily ionized (alkali) atoms. Therefore, knowledge of their number density is necessary. Standard routine of determination of the density of alkali atoms includes calculation of so-called equivalent absorption width Ax given by

(3.4) where AX is the spectral interval in which the measurements are carried out. Then, from the growth curves, the absorption coefficient xo for the purely Doppler contour can be found. To apply this procedure to a mixed (Voigt) contour, it is necessary to know the Lorentz width of the spectral line. This width, within the 5% error limit over a rather wide range of thermal-plasma parameters (the temperature from 1500 to 2400 K, and the atmospheric pressure), is equal to 0.04 A. The density of alkaline atoms is determined by the formula (3.5) where ma is the mass of the atom, 6X is the Doppler width, and f is the oscillator strength. Rewriting formula (3.5) in the form convenient for experimental data processing, we obtain (3.6)

where 6X = 7.16 x 1 0 W 7 X m , p is the atomic weight, and 1 is the thickness of the absorbing layer. The temperature is measured in K , 6A, A, and x in cm-l. Analysis of the effect of dust grains on the measurements of the atom density by the emission-absorption method shows that, in the presence of dust grains, expression (3.4) is transformed to

A(X)

=

[SLexp(--rp)

+ SP

-

S L P ~ X / Sexp(--rp)]. L

(3.7)

It is easy to see that the only difference from the case of a pristine plasma is the factor exp(-rp) at the reference-lamp signal (here, -rp = -In [(SE, - SF)/SL] is the optical thickness of the dust cloud, and the quantities SEp and SF are measured outside the spectral-line contour),

Production and Diagnostics of Complee Plasmas

121

which describes the additional extinction of the signal by the dust cloud. To calculate the atom density in the complex plasma, one can employ the conventional calculating algorithms developed for a pure gas. The required value of the Doppler broadening SX in Eq. (3.6) can be calculated from T,. A detailed analysis of the influence of dust grains on the measurements of the gas-phase temperature shows that when the condition (3.8)

is satisfied (here, ag(X) is the absorption coefficient of the gas atoms), the effect of the dust grains can be ignored and the temperature can be calculated by formula (3.3). In this case, the error in T,, which is caused by the effect of the dust grains on the results, does not exceed 5%. For the case when condition (3.8) is not satisfied, an algorithm was proposed [Nefedov et al. (1995)] to derive the gas-phase temperature from the measurements of three signals at two wavelengths, with X lying either within the spectral line or outside the spectral line. A similar procedure for determining the temperature was also considered by [Bauman (1991)], where the gas temperature was calculated by (3.9)

+

where F = s p / ( S ~ Sp - S L ~ )the , superscript 0 corresponds to the quantities measured within the spectral-line contour. Thus, to carry out simultaneous measurements of the gas-phase temperature and the density of alkali atoms in a dusty plasma, one should measure three signals Shp, S p , and SL at several (at least, two) wavelengths. Since the parameters of plasma can vary, these simultaneous spectral measurements at several wavelengths should be conducted by using a proper number of photodetectors. In this case, it is necessary to compensate for the fluctuations of the optical characteristics of the complex plasma, which leads to rather sophisticated designs of spectroscopic devices. On the other hand, successive measurements in several spectral regions with the use of a simple measurement scheme (e.g., based on a monochromator) lead to significant errors. Random errors appear because of the finite time of the spectrum scanning, whereas the parameters of the plasma often vary rapidly. Systematic errors are caused by an unavoidable backlash of the monochromator mechanism. The mentioned difficulties can be avoided if a spectrometer is used as a spectral instrument and a charge-coupled-device

122

Physics and Applications of Complex Plasmas

(CCD) array is used as a detector in order to measure the emission intensity in different spectral regions simultaneously. Furthermore, the CCD array can provide more complete information about the emission-absorption line as compared to a monochromator, which makes it possible to investigate media with a nonuniform temperature. The restoration of several unknown parameters from measurements on a gas emission line is a complicated diagnostic problem, which requires measurements in narrow spectral band regions and considerable effort in order to avoid significant experimental errors. One of the possible solution of this problem is to measure particle surface temperature and their parameters in a wide wavelength range beyond the gas emission line. The measurements over a wide spectral band region provide the improved resolution and give higher accuracy in determining the temperature. The classical pyrometric two-color method can be employed for the temperature measurements in an optically gray medium, the emissivity of which is independent of the measurement wavelengths. The general difficulties for this method are concerned with the necessity of a knowledge of the spectral emissivity &(A) of a dusty cloud. Determination of &(A) requires preliminary information on the particle sizes and the refractive index. In many cases of practical interest, however, these quantities are not known. In this case, the temperature of the particles can be obtained by the spectroscopic method involving the spectral approximations of &(A) [Vaulina et al. (1998)l. This method is based upon solving a system of radiative transfer equations for the light intensity measurements of the signals 5'1,S p , and SL at several wavelengths X i (i = 1 , . . . , N ) . Assuming that multiple in-scattering is negligible, we have

&(A) = [l - w ( X ) ] (1- exp[-~(X)])

(3.10)

Here, w ( X ) is the single scattering albedo, .(A) is the optical depth. The error estimate 6' of the emissivity &(A) (by neglecting the in-scattered radiation) can be easily calculated using the the measured .(A). ). For a dust cloud with .(A) ) < 0.9 - 1.2, the 6 value is less then 2% for particles with the size parameters 2a/X 2 1. In this case the problem can be reduced to the choice of an appropriate model for the spectral dependence of the single albedo w ( X ) . We consider the spectral approximation W(X) for the function 1 - w ( X ) (3.10) as

W(X) = C/XA."(X).

(3.11)

Production and Diagnostics of Complex Plasmas

123

Here A , B , and C are parameters, which values are independent on the wavelengths. Relative measurements allow us to eliminate C. The obtained data lead to reduced problem of an empirical inversion for two unknown parameters (Tpand A). Therefore the particle surface temperature, T p ,as well as the appropriate approximation of &(A), ( i e . , the parameter A ) can be determined by minimizing the mean-square error between the experimental and the calculated data. Previous information for selecting the &(A) approximation can be obtained by two means: from an analysis of the spectral behavior of the optical depth .(A) or by using this method in combination with the techniques for determination of the particle mean size [Vaulina et al. (1998)l. The spectroscopic measurement can also be used tor determine the particle size. In this case, the spectral dependence of &(A) can be obtained from (3.12) where B(Tp,X)is the Planck blackbody function, Tl is the temperature of the reference source. For the case of (2a/X) > 5, &(A) is dependent on the grain size only, and is weakly dependent on the grain refractive index for absorbing particles with k > 0.4, where k stands here for the imaginary part of the refractive index. Then from measurements of &(A) the size and k of dust particles with 0.001 < k < 0.1 can be obtained. The spectral distribution of the albedo, w ( X ) is determined by the particle sizes provided that the refractive index is weakly dependent on A. The w ( X ) value at X = A0 is uniquely determined by k . Therefore, with the known particle sizes and real refractive index, the spectral absorption index, k ( X ) , can be easily measured. For determination of the particle size, number density, and the real refractive index, a technique based on measurements of the forward angle scattering transmittance (FAST) at different aperture angles of the detector can be used [Nefedov et al. (1997)].

3.4

Detection and Diagnostics of Dust Particles

In this section, we focus on conventional experimental techniques of detection and characterization of dust particles as well as structures they form. The current state of the diagnostics of nano-sized particles is discussed below in Sec. 7.1.4. Here, we present methods of dust particle detection (Sec. 3.4.1), diagnostics of dust structures (Sec. 3.4.2), and,

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Physics and Applications of Complex Plasmas

finally, in Sec. 3.4.3, techniques to determine the main dust parameter the equilibrium grain charge. Detection of fine particles and dust structures is most commonly based on the optical diagnostics. The optical methods of diagnostics are nonintrusive (not disturbing an investigated object), and allow real-time automated data acquisition and processing.

3.4.1

Detection in laboratory and processing plasmas

When the grain size is relatively large (about few micrometers, as in the majority of complex plasma laboratory experiments), dust particles can be directly detected by optical means. This significantly simplified application of diagnostics is based on identification of individual particles and tracking of their trajectories. Visualization of dust particles is usually carried out by their illumination (in the horizontal or vertical planes) by a laser beam which is formed as a flat laser sheet with the thickness of 100-300 pm by a cylindrical lens or rotating mirrors. The scattered (by the particles) light is observed with the help of CCD-camera and then the images obtained are stored in a computer or recorded on a tape. The subsequent analysis of the images obtained allows one to reveal all main parameters of dust structures and dust particles. One of the main features of the visualization of dust particles is that it opens new diagnostic possibility of investigation of complex plasma phenomena on the kinetic level. The ability to follow every individual particle and determine its trajectory allows us to find one of the main parameters of the complex plasma-dust system - the velocity distribution function (VDF), which contains information on the dust kinetic temperature and on the forces acting on a particle. For correct definition of VDF, the high temporal and spatial resolution of the diagnostic equipment is necessary. For example, the time interval between consecutive positions of particles should be less than the characteristic collision time l / v f r . Otherwise, the displacement of a dust particle is caused by diffusion that leads to underestimation of the particle’s speed. Experimental measurements in discharge plasma have demonstrated, that VDF is Maxwelian with an effective dust kinetic temperature T d . The latter is close to the temperature of the neutral component (T, 0.03 eV) for high and medium pressures (down to 0.3 mBar) and rapidly increases (up to tens of eV) for lower pressures. In the case of CC discharge plasma, due to the anisotropy of the plasma, VDF has different temperatures in the vertical and horizontal directions. N

N

Production and Diagnostics of Complex Plasmas

125

The method of visualization of dust particles makes possible a realization of essentially new methods of diagnostics of parameters of dust particles and the plasma. However, application of this method is not always possible even in laboratory conditions; for example, in the cases of intensive plasma self-radiation, small sizes, and/or high number densities of dust particles. In particular, in thermal and nuclear-induced plasmas, one's ability to visualize dust particles is limited. In these cases, for measurements of dust parameters, the absorptive-emissive spectroscopy (see Sec. 3.3.2), the laser scattering technique [Nefedov et al. (1997)], or methods of the correlation spectroscopy [Cummins (1974)] are used. For example, in a complex plasma VDE' can be determined by measurements of the autocorrelation function for photons scattered off dust particles. As was mentioned above in Sec. 3.3.2, the majority of spectroscopic methods developed for diagnostics of sizes and number densities of particles are based on measurements of their radiation, or measurements of extincted and scattered light from an external source. These methods are based on the Mie theory and BouguerLambert law. If the number density is high enough for effective dust selfradiation as well as for transmitting the radiation from the external source, the multiple-scattering corrections have to be applied [Zardecki and Tam (1982)l. The laser light scattering (LLS) diagnostic methods can be subdivided [Garscadden et al. (1994)] into spatially resolved detection of the dust from its Mie scattering using a broad area beam and CCD video cameras, angular variation of the scattered light, intensity from a small focal volume, and scattering depolarization using a polarized laser (or white light) beam. One should always keep in mind that the Mie scattering formalism allows one to infer the particle size and the number density unambiguously when the complex refractive index is known, the dust particles are spherical, and the size distribution is monodisperse. However, the ex situ examination of the dust by the SEM can be used to verify the latter two assumptions making the analysis unambiguous. We note that the concentration of the particles can also be obtained in situ by examining the fluctuations of the scattered optical signal. The ratio of the fluctuations to the total signal scales as -1/2 nd , where n d is the number of fine particles in the scattering volume. This method works fine for the number densities of the particles exceeding N lo7 cmP3. Unfortunately, the Rayleigh scattering (a6/X4, where X is the laser wavelength) law imposes strict limitations on the accuracy of the LLS technique in the small grain size limit. For example, using the blue (488 nm) line of the Ar+ laser, the scattered intensity becomes observable

126

Physics and Applications of Complex Plasmas

after - 2 s into the discharge run [Boufendi and Bouchoule (1994)l. At this time, the average particle size in silane discharges (see Sec. 7.1) can reach approximately 20 nm. On the other hand, the particle detection limit for the 647 nm Krf laser is about 60 nm. Thus, as will also be detailed in Sec. 7.1.4, this method falls short in detecting the nanometersized grains. The dynamics of dust particles can also be studied using the Doppler velocimetry [Schabel et al. (1999)]. To end this section, we note that in particle-generating plasmas the particle detection methods, at different stages of particle growth, include, e.g., the mass spectrometry, the photo-detachment, the infrared absorption, the microwave cavity measurements, the Mie laser scattering. After the particles have been collected on a solid surface (substrate), a number of conventional solid-state characterization methods [SEM, TEM, FTIR, Raman spectroscopy, X-ray photoelectron spectroscopy (XPS), X-ray diffractometry (XRD), etc.] can be used.

3.4.2 Dust stmcture diagnostics Visualization technique, laser counting technique, and measurement of the optical diffraction are used for the dust structure diagnostics in a complex plasma. The ranges of applicability of these techniques are shown on Fig. 3.20. The imaging technique is the most often used technique in

Fig. 3.20

Applicability ranges of different dust structure diagnostics.

127

Production and Diagnostics of Complex Plasmas

complex plasma studies. It is based on an analysis of dust structure images obtained by illuminating the structure by the specially collimated laser beam. This technique is common in gas discharge, UV- and nuclear-induced plasmas. Laser counters and diffraction methods are applied for thermal complex plasmas. It is well known [Drude (1959)] that illumination of an ordered structure leads to the appearance of a diffraction pattern. This phenomenon can be used for diagnostics of the ordering of dust structures in a complex plasma if obtaining of an image the structure is impossible. Analogous techniques are used in the X-ray spectroscopy to study the atomic structure of materials. If one illuminates an ordered structure, the kinematic approximation yields the following expression for the intensity Is(&) of the monochromatic radiation with the wavelength X scattered at an angle a to the direction of the incident radiation [Vaulina et al. (2000)]:

1

( g ( r ) - I)rsin(rq)dr/q ,

(3.13)

where q = 47rmsin(a/2)/X, m is the refractive index of the medium, and lo(a)is the intensity of the radiation scattered by a random particle space distribution. Equation (3.13) is valid, when the radiation scattered by the dust grains is small compared with the intensity of the incident radiation, and the (mean) particle size is less than the characteristic distance between the particles. The intensity ratio S ( a ) = Is(a)/I~(a) is referred to as the structure factor. Measurements of the angular dependence of the structure factor allow one to determine the correlation function g ( r ) by the inverse Fourier transform of the function S(q). To observe the scattering pattern, the spatial dispersion V = A / q , where r1 is a characteristic interparticle distance (the useful one is the most probable distance r = r1, which determines the position of the first maximum of the correlation function). The value of is different for various types of lattices,

where C = 1for the cubic (c), C = 3&/4 for the body-centered cubic (bcc) lattices, and C = fi for the face-centered cubic (fcc) lattice. The upper boundary (V < 1) is determined by the condition of existence of second and higher scattering maximums. The lower boundary V > sin(a0) is given by the finite size of the probing laser beam, which prohibits detection of the scattered radiation at angles below a minimum value ao. Taking into

128

Physics and Applications of Complex Plasmas

-

account that the spatial dispersion is limited by V 0.005-1.0 (a0 = 0.3"), we can estimate the maximum number density of dust grains permitting observation of the scattering pattern. Thus, for the visible range X = 0.40.8 pm, the number density of grains forming the cubic crystal must be within the range 1 x lo5 - 9 x 10l2 ~ m - for ~ ,fcc structures, this density is within the range 1.2 x l o 5 5.5 x 1012 cmP3, and for liquid-like structures, 3 x lo5 - 8 x O I 2 ~ m - ~ . -

Fig. 3.21 Experimental set-up for the diffraction technique [Vaulina et al. (2000)]: (a) overall view; (b) superposition of the CCD array and the diffraction pattern.

A typical experimental set-up for diffraction technique is presented in Fig. 3.21. The small angle scattering measurements are made by illuminating a complex plasma by a laser beam and recording the space intensity of the scattered light on CCD-array. The restricting diaphragm, filter, and wide-aperture lense are used for better signal/noise ratio. Image-processing software yields the spatial distribution of the scattered light. The angular dependence of I ( a ) can be obtained using the geometric parameters of the optical scheme directly from the scattering light time series which was smoothed to reduce the noise. In the case of a weakly coupled particle system, the angular dependence of the scattered radiation is determined by

Production and Diagnostics of Comples Plasmas

129

particle parameters (the size and the refractive index). If a complex plasma is strongly coupled, the scattering pattern is changed and the angular dependence of the scattered radiation is determined the particle characteristics as well as by the ordered structures parameters. The light scattering maximums can be easily interpreted if we take into the account that the interparticle distance 7-1 >> a. Because of that, the small angle scattering (if cr < 3') is determined only by the light diffraction on the ordered structures.

Fig. 3.22 (a) The angular distributions of the scattered light: (1) T = 1800 K , ( 2 ) T = 2200 K. (b) The structure factor S ( q ) [Vaulina et al. (ZOOO)].

The angular dependence for ordered and disordered structures observed [Vaulina et al. (2000)]is presented in Fig. 3.22(a). Using the light scattering measurements, the values of the structure factor S(q) can be found. Typical function S(q) is shown in 3.22(b). Then the obtained structure factor can be inverted into the pair correlation function.

130

3.4.3

Physics and Applications of Complex Plasmas

Methods of measuring charges on grain particles

Numerous experimental procedures are used to determine the grain charge Q. In most cases, the dust charge can be experimentally estimated without applying an external action on the dust-plasma system. The simplest approach is based on an analysis of the equilibrium position of dust grains (such as the particle levitation height). The main forces acting on a dust grain levitating in the sheath region were considered above in Secs. 2.1.4 and 2.1.5. They include the gravity F,, electrostatic F E , thermophoretic F T, and ion drag Fdr forces. In equilibrium, the total force acting on the particle is zero. Since FE and Fdr depend on the particle charge, the equilibrium equation can be solved giving the charge (if other parameters are known). Other possibilities associated with the passive diagnostic techniques include an analysis of natural modes in dust structures (for example, the particle diffusion or phonon spectra), as well as Brownian motion or transport properties of dust particles. For example, measurements of the pair correlation functiop and VDF and particle displacement allow one to determine the transport properties and therefore to estimate the dust charge (as well as the plasma screening length). The same parameters can be obtained from the investigation of plasma crystal phonon spectra or chaotic self-excited waves. As was mentioned above in Sec. 3.3.1, the charge on dust particles can be also determined from probe measurements (from the floating potential or variation of the electron saturation current). In some chemically active plasmas, a method based on laser-induced photodetachment of charges and subsequent detection by microwave interferometry methods or probes is used [Fukuzawa et al. (1996); Stoffels et al. (1996)l. The charge on the particles can also be measured by studying the afterglow diffusion [Childs and Gallagher (2000)]. When the particles are suspended in the diffused region of a low-pressure plasma discharge [Hayashi (2001)], the dust can be trapped due to the balance of the neutral drag and the electrostatic forces. The dust charge can then be determined from the equilibrium equation. The charge on dust particle can be also determined by collecting dust particles in the Faraday cup. Then the overall charge in the cup is divided by the number of particles collected, and the charge on an individual particle is obtained. Modification of this technique is to apply a pulsed voltage to the parallel electrode with the complex plasma inside. The dust charge can be determined from the current measurement, see Fig. 3.23. After the voltage has been applied, the current appears in the electric circuit. Due

Production and Diagnostics of Complex Plasmas

131

Fig. 3.23 T i m e dependence of t h e current.

to the presence in the plasma of charged particles with different mobilities, the current has the time profile. At first stage, the current is determined by the most mobile plasma species, ie., electrons, and then by heavier particles such as ions and dust grains. Changing the duration of the pulse, it is possible to elucidate the current peaks corresponding to the different charge carrier. With the knowledge of the dust flux on the electrode, the dust charge can be determined. This technique is useful when the sign of the dust charge is unknown, thus the latter can be easily determined by the sign of the electrode to which the particles are attached. More precise methods are based on measurements of the dust dynamic response to various external perturbations. They involve an analysis of consecutive images of perturbed dust particles. These techniques have been mostly implemented for diagnostics of complex discharge plasmas. However, they can be extended for other complex plasmas as well. The procedures include driven particle shifts and/or oscillations in the vertical and horizontal directions, excitation of various lattice waves involving compressional and shear perturbations, formation of Mach cones behind moving dust particles, and stimulated collisions of dust particles. The charge on dust particles is then obtained from the corresponding equations such as equations of motion of dust particles under the action of known external and interparticle forces and/or wave dispersion equation.

132

Physics and Applications of Complex Plasmas

The most convenient and applicable technique consists in excitation, observation, and analysis of vertical vibrations of dust particles levitating in the sheath region near the bottom electrode in an RF discharge plasma. As a source of periodic excitation, a low-frequency modulation of the bias on the electrode (in this case, all particles oscillate) or modulated laser radiation (which can be focused on an individual particle) are used. The charge on dust particles is determined by relation Q2 = wg/EI, where wo is. the resonance frequency of the vertical oscillations and El is the derivative of the sheath electric field with respect to the vertical coordinate. Since this method needs preliminary knowledge of (or assumption on) the electric field profile in the plasma sheath, its accuracy can be relatively poor (typically, 20-40%) but still providing a reliable particle charge. Analysis of oscillations excited in the horizontal plane in a particle chain (lattice), allows one to determine the charge Q as well as the plasma screening length AD. In this case, under the assumption that the particles interact via the screened Coulomb potential, the amplitude and phase of dust particle oscillations as a function of the excitation force (usually provided by the pin electrode) are determined. The oscillations can be excited by the sinusoidal bias on the pin electrode or by the modulated laser action on the edge particle in the chain. The real and imaginary parts of the complex wave number k of the excited oscillations are measured. Dependence of k on the excitation frequency is analyzed on the basis on the appropriate chain lattice wave model; this gives the dust charge and the screening length [Pieper and Goree (1996)]. This technique can be applied to twodimensional crystal structures as well, but in this case a more complicated analysis is required to extract the particle charge. The interparticle potential and, correspondingly, the dust charge and the plasma screening length AD, can be determined by a collision technique based on analysis of particle trajectories after their collision. In experiment [Konopka et al. (1997)], this technique was applied to find the interaction potential of dust particle in a RF capacitive discharge. It was found that in the horizontal plane at the distances up to 5 - 6 X ~ the interpaticle potential with a good accuracy is the screened Coulomb (Yukawa) potential. The Mach cones can be excited in a two-dimensional crystal behind a fast moving dust grain [Samsonov et al. (1999)I or by a laser beam [Melzer et al. (2000)], and the charge can be derived from the analysis of these excitations. In experiments [Merlino et al. (1997); Samarian et al. (2001c)], the charge has been determined from the measured characteristics of dust-acoustic waves using their dispersion relation.

Production and Diagnostics of Complex Plasmas

133

All the methods based on periodic excitation of particle oscillations are applicable for relatively low pressures (up to approximately 100 mTorr) . Fbr higher pressures, dissipation of the oscillation energy is increasing, and stronger external forces need to be applied. This can be either impossible or leading to destruction of dust structures. The analysis of aperiodic motions (shifts) should be used for these cases. The dust particles can be shifted from their equilibrium positions by the laser beam of the pin electrode. The charge on the particle is deduced from the equation of motion of the particles moving back to the equilibrium under the action of a restoration force (such as the electrostatic force in most cases) and the neutral drag force. More accuracy is achieved if the charge is determined from the balance of these forces at the point where the particle acceleration is zero. This method was used, for example, by [Fortov et al. (2001)], to determine the dust charges in the striation region of DC discharge.

3.5

Dust Grains as a Diagnostic Tool

Interactions between the plasma and the dust particles is used for diagnostics such as the characterization of electric fields in the plasma sheath (particles as micro-probes), energy fluxes in the plasma and towards surfaces (particles as micro-calorimeters), and plasma-wall interactions (particles as micro-substrates). The most common use of dust particles as a micro-probe is generally based on the fact that the particles are charged in a plasma (in particular, by collecting plasma electrons and ions). Thus the dust particle can be considered as an electrostatic probe with good spatial resolution. As was mentioned before in Sec. 2.1, the particle charge is the function of the bulk plasma parameters such as n,/ni, T,, zli. The motion and the equilibrium position of the particle is strongly dependent on the local plasma conditions. The diagnostic technique is relatively simple as the prime measurements are those of the position of the particle, and/or its motion following a perturbation. In Sec. 3.5.1 we discuss in detail how this approach can be used for the sheath profile measurements, and in Sec. 3.5.2 we consider a more general case of the determination of spatial potential distributions in a discharge plasma. The dust particle as a micro-calorimeter (thermal probe) for measurement of the thermal flux were successfully employed in discharge plasmas [Kersten eC al. (2000); Daugherty et al. (1993)l. The equilibrium

134

Physics and Applications of Complex Plasmas

temperature of the particles, which is controlled by the energy fluxes from the argon plasma, has been estimated to be of the order of 100-200°C by the ternperature-dependent fluorescence of laser dye mixed with tde dust powder [Kersten et al. (2000)]. In this case, the particle was heated predominantly by the charge carrier recombination, while the losses were mostly due to radiation and conduction. The thermal energy flux towards the particles was estimated to be approximately 0.1 J/(cm2 s). To utilize the particle behavior as a diagnostic tool, a good understanding of its interaction with the surrounding plasma is required. At any time, only a single particle could be used, causing the minimal plasma perturbation. A simple access to the plasma, for illumination of the particle and recording its motion, is merely required. The simplicity of the diagnostic sct-up and maintenance makes it possible to conduct measurements in a short period of time for a wide range of plasma conditions. Because of its simplicity, such a technique makes it promising for measurements in modern processing devices [W. W. Stoffels et al. (1999a); Kersten et al. (2000)].

3.5.1

Plasma sheath diagnostics

Understanding of the nature of the plasma sheath is important for many applications. For example, in CC discharges commonly used for plasma etching and plasma-assisted thin film deposition, the processes are critically dependent upon the ion fluxes to the surface, which are the result of the ion acceleration in the sheath electric field. Models have been developed to relate the product characteristics (e.g., thin film properties) to the plasma and sheath parameters [Sobolewski (2000); Godyak and Sternberg (1990); Bornig (1992)]. In real processing devices, however, application of such theories is limited by uncertainties in our knowledge of the plasma and sheath conditions due to the absence of, or unreliable a t best, in situ plasma sheath diagnostics. Various experimental methods have been used to investigate the plasma sheath region. The use of probes is severely limited by their ability to significantly perturb the potential profile in the sheath. Passive spectroscopy is of a limited ability to use since the sheath region is characterized by the paucity of emitted light. Direct measurements of electric fields can be made by using the optogalvanic spectroscopy and laser-induced fluorescence (LIF). LIF is also used for time-resolved investigations of RF sheaths. Such investigations, however, involve complicated diagnostic techniques, and are

135

Production and Diagnostics of Complex Plasmas

possible for only a small number of parameter sets, and also usually involve an apparatus specially constructed for the measurement rather than devices actually used for applications.

Plasma

Presheath

Sheath

Fig. 3.24 Sketch of the particle positions in the sheath

(TI

< TZ < r g )

By analyzing the position, motion, and oscillations of a dust particle in the sheath, one can reveal the sheath profiles. As was mentioned in Sec. 2.1.5, the equilibrium position of the test particle depends on the sheath electrostatic field and the particle charge. As shown in Fig. 3.24, dust particles of different sizes levitate in different positions within the sheath, corresponding to particular values of the non-uniform sheath electric field. This allows one to use particles of various sizes for measuring the vertical potential profile in the sheath. For this, one needs to measure the particle charge. That can be done by VTR technique (see Sec. 3.4.3) by using the laser excitation or the electrode potential modulation. Then, with the data for particles of different sizes (levitating at different heights in the sheath, see Fig. 3.25), the values of Q and Q d E / d z can be obtained as a function of the position within the sheath.

Fig. 3.25 Video images of levitating dust particles of the radius: (a) a = 3.1 pm; (b) a = 2.1 p m ; (c) a = 1.4 p m ; (d) a = 1.0 pm.

By observing the motion of a single dust particle as it falls through the plasma, and undergoing damped oscillatory motion as it approaches an equilibrium position in the RF sheath (where the electric force on the

136

Physics and Applications of Complex Plasmas

particle is sufficiently large to balance the gravity), we are able to get a quantitative information about the sheath potential profile [Tomme et al. (2000b)l. The results obtained confirm the common assumption on the parabolic nature of the sheath.

Particle radius [pm]

Fig. 3.26 Normalized difference between the grain equilibrium position and the sheath edge position [(hb- heq)/hb] as a function of the grain radius for different values of the ion plasma frequency: wpi = lo' s-' (solid curve); wpi = 5 x lo6 s-' (dotted curve); wpi = lo6 s-' (dashed curve) [Samarian and James (ZOOl)].

Figure 3.26 shows that for a particle with the radius less than 500 nm, the equilibrium position and the location of the sheath edge differ by less than 5 percent with respect to the width of the sheath, and this difference decreases with decrease of the grain radius. This gives the possibility to use test fine dust grains for measurement of the sheath edge location in an R F discharge. [Samarian and James (200l)] used this approach to diagnose the sheath in an argon plasma, which generated at pressures in the range 20-100 mTorr. The effect of the gas pressure and the RF power on the sheath width was investigated. The obtained results (Fig. 3.27a) show the expected trends and agree with the data obtained from other diagnostic techniques. In particular, the dependence on the RF power is in good agreement with that obtained by [Orlov et al. (2004)l from measurement of the ion energy distribution function (Fig. 3.2713). Analysis of large amplitude oscillations around the equilibrium position also gives us a possibility to determine the sheath profile. The method employs images of vertical oscillations of a single particle in the sheath of a low-pressure RF discharge. It was found [Ivlev et al. (2000a)l that the oscillations become strongly nonlinear as the amplitude increases. Using

Production and Diagnostics of Complex Plasmas

137

Fig. 3.27 Dependence of the sheath widths on the input RF power. (a) Obtained by the sheath visualization technique at the University of Sydney. (Solid curve: pressure 30 mTorr. Dashed line: 65 mTorr.) (b) Comparison with the dependence obtained from measurement of the ion energy distribution function [Orlov et al. (2004)l.

the anharmonic oscillations theory, the first two terms in an expansion of the sheath potential around the particle equilibrium were found. Clouds of nano-sized particles can be generated using sputtering or plasma chemistry to investigate the influence of the electrode shape on the sheath. In contrast to the single particle measurements providing local information, the nano-clouds give a global picture of the relation between the sheath shape and the electrode configuration. It was shown [James et al. (2002)] that nanoparticles (with the radius less than 300 nm) cannot find equilibrium within the sheath; instead, they fill the positive column of the discharge, leaving the sheath devoid of particles. This phenomenon provides a simple way to visualize the sheath in the case of complicated

138

Physics and Applications of Complex Plasmas

Fig. 3.28 Visualization of the potential distribution by test dust particles of diameter less than 300 nm, generated in the chamber by sputtering from the RF-powered cathode. The sheath regions are devoid of particles: (a) the upper (grounded) electrode is a disk with a central hole, a ring on the lower powered electrode is visible; (b) detail of the sheath at the edge of a simple disc-powered electrode.

electrode structures (see Fig. 3.28). This situation is of interest, e.g., for plasma immersion ion implantation (PIII) where the ions are implanted into the surface layers as a result of their acceleration by the sheath electric fields [Lieberman and Lichtenberg (1994)l. For an irregular target, information about the sheath shape, obtained using the target as an electrode for a steady discharge, is helpful for locating auxiliary structures to ensure the uniformity of implantation.

3.5.2

Spatial profiles of m a i n plasma parameters

Dust particles can be used to measure radial profiles of discharge plasma parameters. The corresponding techniques employ ideas similar to those in the sheath diagnostics, e.g., tracking the particle motion under an external perturbation, analyzing the particle equilibrium positions or a dust structure characteristics (such as interparticle distance). Knowledge of the dust charge is necessary for this type of diagnostics. For many situations, this charge can be taken as almost constant thus significantly simplifying the analysis. These situations include the particles in the plasma bulk where the dust charge can be estimated from the floating potential easily determined by, e.g., probe measurements. However, even in the sheath, due to more homogeneous condition in the horizontal plane, the dust charge is almost constant for horizontal motions. In the latter case, radial profiles of the plasma sheath parameters can be obtained. One of the simplest is the method based on an analysis of the inhomogeneity of the particle arrangement in dust structures (one-dimensional and two-dimensional) formed in the horizontal plane. The information about the plasma parameters is extracted from the difference of the equilibrium

Production and Diagnostics of Complex Plasmas

139

distances between the dust particles. The advantage of this static diagnostics is the convenience of measurements including obtaining images of the structure and their subsequent analysis to determine the particle interaction parameters and the characteristics of plasma. The separations between the nearest neighbors (the quasi-lattice constants) are, in general, functions of the distance and direction from the center of the crystal. Deformation of a dust structure in the fields of the traps depends on the particle characteristics (such as the charge) and on the plasma parameters. Therefore, the response of a dust structure to a static external disturbance can be used as a diagnostic tool for the surrounding plasma. It should be noted that the most general consideration of the equilibrium is based on the set of the coupled kinetic equations for plasma species and Poisson’s equation, involving the self-consistent electric field without the separation between an external field and the interparticle interaction fields. The equilibrium positions for the dust particles are the points where the interparticle electric field is balanced by the external trap field. In general, other forces (such as the ion drag and thermophoretic forces) can also contribute to the equilibrium. However, for practical purposes, simple approaches are used. For the horizontal plane, the typical simplification excludes the ion drag force, uses the nearest-neighbor approximation, and neglects a possible dependence of Q on the particle position [Podloubny et al. (1999); Hebner et al. (2002)l. The radial potential profile can be also measured using the transient motion technique (TMT) [Thomas and Watson (1999)l. According to the latest version of this technique [James et al. (2002)], a dust particle levitating in the sheath is shifted horizontally from the equilibrium position by applying the negative voltage to two pin electrodes. When the voltage switches off, the particle is transiently moving to its original position due to the action of the electrostatic force. Tracking the particle motion, the velocity and acceleration values are obtained. Then, using the equation of motion and the value of the dust charge measured separately by, for example, the vertical resonance method (see Sec. 3.4.3),the potential difference between the two points can be found. The obtained profile for a planar RF discharge is shown (see triangles) in Fig. 3.29. This is complemented by the profile obtained by the RF-compensated singe Langmuir probe. The TMT method can be further improved using a laser to push the particle instead of the pin electrode. In this case the plasma is not disturbed and the dust charge is less affected. These techniques can be extended to measure spatial distributions not

140

Physics and Applications of Complex Plasmas

Fig. 3.29 The radial potential profile of a planar RF discharge measured by the Langmuire probe technique (squares) and the transient motion technique (triangles) [James et al. (2002)].

only in the horizontal plane. In this case, the particles should move in the vertical direction. Measurements of the velocity of the particle in two vertically separated points provide the change in the kinetic energy. From the balance of energy, this change in the kinetic energy plus the change in the potential energy is then equal to the work done by all other nonpotential forces (having the projection Fnp onto the vertical axis) upon the dust particle between those two points. Taking into account the change of the potential energy determined by the electrostatic potential profile, the plasma electric field can be determined as (the vertical axis is directed upwards)

E=

AT

-

mdgAh

+ sf F,,dh dh

QLh

For example, [Thomas and Watson (2000)] applied this approach to obtain the plasma potential in the vicinity of a dust cloud, and [Thomas et al. (2002)l - to measure the electric profile inside the void under the microgravity condition.

Chapter 4

Particle Dynamics in a Complex Plasma

Electrostatic interactions between highly charged grains is very strong thus often making the complex plasma a highly non-ideal system. The combined effect of interactions of the grains between themselves as well as with the ambient plasma leads to formation of various structures which can be strongly correlated (like clusters and crystals composed of dust particles), with complex behavior exhibiting various oscillations, regular and chaotic motions, etc. To understand these phenomena, such fundamentals of dust particle dynamics as levitation and arrangement in a small model systems, and Brownian motion and diffusion should be understood. Thus in this chapter we first consider dynamic phenomena of dust grains in a plasma (Sec. 4.1). Particle arrangements and their stability are then investigated on example of a simple system of two particles (Sec. 4.2). We demonstrate that even in such a simplified system, many fundamental features of dustdust and dust-plasma interactions can be seen.

4.1

Dynamic Phenomena

It is well known that the charge of dust particles, being one of the most important characteristics for the trapping and interaction of dust grains, appears as a result of various processes in the surrounding plasma, mainly (under typical laboratory conditions) due to the electron and ion currents onto the grain surfaces (see Sec. 2.1). 'Thus the first step in any research on the properties of dust in a plasma is an adequate description of the surrounding plasma. In general, there are two situations of interest: (i) dust particles do not affect significantly the properties of the plasma they are embedded in (this usually corresponds to low number densities of the dust component, i e . , to a small number of dust particles), and (ii) the 141

142

Physics and Applications of Complex Plasmas

dust component is relatively dense, thus significantly changing the field and density distributions of the surrounding plasma. Note that the second case corresponds to such interesting self-organized dust-plasma structures as voids and clouds in the dust-plasma sheaths (see Sec. 5.1). In this section, we start with the simplest problem of modeling of dynamics and levitation of an isolated dust particle in a plasma sheath (Sec. 4.1.1). Here, we are mainly concerned with a plasma with a rarefied dust component, ie., assume that dust charges and electric fields do not change the plasma parameters significantly. We therefore consider the case of essentially isolated dust grains (the intergrain distance exceeds the plasma Debye length), with a low total number of dust particles. Then, dynamics of fine particles in near-electrode areas of discharge plasmas relevant to the attempts to increase the deposition rates for a number of semiconductor microfabrication processes is considered (Sec. 4.1.a). Here, dynamics of dust in silane plasmas as well as levitation of submicron-sized particles in fluorocarbon plasmas is presented. Oscillations of an isolated particle in the plasma sheath are studied in Sec. 4.1.3 where we consider vertical oscillations of a grain, taking into account the dependence of the particle charge on the local sheath potential. We show that the equilibrium of the dust grain close to the electrode can be disrupted by large amplitude vertical oscillations. Finally, diffusion of dust particle is considered in Sec. 4.1.4. 4.1.1

Modeling dust particle dynamics i n a plasma sheath

The first step in considering dynamics of dust particles is the modeling of those plasma regions where the dust particles are trapped. The most important are the sheath and pre-sheath regions of a discharge. Due to relatively high neutral gas pressures (often more than 50 mTorr for typical dust-plasma experiments), the laboratory plasma is strongly influenced by the effects of ion-neutral collisions. Thus the simplest mathematical approach relevant for collisionless plasmas [Chen (1984); Lieberman (1988); Godyak and Sternberg (1990)l (see Sec. 2.1.5) is not fully appropriate in this case. On the other hand, the correct description of collisional effects involves the speed of the ion flow and therefore naturally depends on the properties of the region (sheath or pre-sheath) we are interested in. While in the sheath region, where the speed of the ion flow is expected to exceed the ion sound velocity, a simple approximation [Nitter (1996)l describing ion-neutral collisions can be used, in the total pre-sheath/sheath region

Particle Dvnamics in a Complex Plasma

143

more sophisticated approaches are necessary [Sheridan and Goree (1991)I. Thus an advanced model of momentum transfer between the ion and neutral species, which describes the ion-neutral collisions on the basis of kinetic theory, without semi-empirical approximations [Horwitz and Banks (1973)I was employed [Vladimirov and Cramer (2000)l. Another important issue is the ionization rate. It was demonstrated experimentally (see, e.g., [Samsonovand Goree (1999b); Morfill et al. (1999)I) that an increase of the ionization rate leads to an increased size of the dustfree void region, moving the equilibrium position of the dust cloud closer to the electrodes. It is thus reasonable to expect that even in the case of dust in a plasma, with a negligible influence of the dust on the plasma and sheath parameters, the equilibrium positions of the grains are affected by the ionization process, which is included in the basic set of equations. Note that various numerical models of dust particle levitation in the low-temperature plasma discharge sheath region can be developed. Some of the models deal with the collisionless [Ma et al. (1997); Vladimirov et al. (1999a)I or collisional [Tsytovich et al. (1999)] fluid cases without ionization, the kinetic (2. e., coupled Poisson-Vlasov equations) case [Graves et al. (1994)]without collisions in the vicinity of a dust grain, as well as with particle-in-cell simulations [Boeuf et al. (1994)l of a uniform, steady state DC plasma where plasma particle losses are assumed to be exactly balanced by a constant ionization source, or a hybrid model [Sommerer et al. (1991)] combining Monte Carlo with fluid simulations, with the latter ignoring the equations of motion of the plasma particles. Possible vertical motions of the dust can lead to the disruption of the equlibrium position of the grains. Note that some analytical models considering vertical lattice vibrations as well as numerical models studying phase transitions [Totsuji et al. (1997)] in the dust-plasma system, dealt with dust grains of a constant charge. However, dependence of the dust particle charge on the sheath parameters has an important effect on the oscillations and equilibrium of dust grains in the vertical plane, leading to a disruption of the equilibrium position of the particle and a corresponding transition to a different vertical arrangement (such as in the model of a collisionless plasma sheath with supersonic velocities of the ion flow, was used [Vladimirov et al. (1999a)I). On the other hand, the whole range of possible velocities of the ion flow can be studied [Vladimirov and Cramer (2000)l. Furthermore, the sheath problem should be treated self-consistently, which allows one to study possible dust trapping in a collisional plasma with an

144

Physics and Applications of Complex Plasmas

ionization source, as well as the disruption of the equilibrium. The latter can occur at various positions corresponding to not only supersonic, but also subsonic ion flow velocities at the position of the dust grain. The charge Q of the particles (depending on the plasma parameters, in particular, on the local electric sheath potential and the velocity of the ion flow) can be found in a standard way from the condition of zero total plasma current onto the grain surface. Numerical solutions [Vladimirov and Cramer (2000)l for the dust charge, as a function of the particle position z , are presented in Fig. 4.1, for the example of a dust grain of radius a = 4 pm.

Fig. 4.1 Dependence of the charge q = - ( Q / e ) x l0W3 of the dust grain, of radius a = 4 pm, on the grain position h = z / X o i [Vladimirov and Cramer (ZOOO)]. Here, (a) uion/upi= 0.1, (b) ~ ion/wpi i ~ = 0.01, ~ / and ~ (c) ~ u i io n / ~ p= i 0.001. The characteristic values of the charge at various positions, as well as the position for the maximum possible charge, are summarized in Table 4.1. The vertical and horizontal lines indicate the position of maximum charge, and the charge where ui = us.

144 Physics and Applications of Complex Plasmas

Table 4.1 The characteristic numbers for the dust charge calculation (Fig. 4.1) [Vladimirov and Carmer (2000)].

Ionization '

frequency Vion/Wpz

0.1 0.01 0.001

Charge at electrode

Maximum charge

qo ( 1 0 ~ ~qmax )

(-1 1.64 3.46

Position

of qmax ( 1 0 ~ ~zqm ) (in A D % )

56.77 67.77 83.60

27.19 18.96 14.91

Charge at vi = v, qVs

(10~4

24.84 16.46 11.31

The characteristic values of the charge at various positions, as well as the position for the maximum possible charge, are summarized in Table 4.1. The vertical and horizontal lines indicate the position of the maximum charge, and the charge where the Mach number of the ion flow becomes unity, ie., II; = u s , Figs. 4.l(a), (b) and (c). An extra line in Fig. 4.l(a) indicate that in general there are two positions for a charge of less than the maximum charge. Note that the higher is the input power ( L e . , the higher T, and 4on the stronger are the ion fluxes, and, correspondingly, the lower is the size of the negative charge on a grain placed very close to the electrode. The dust charge can even become positive; in the case qon= O.lwp;, the dust charge becomes positive near the electrode, with the result that no (equilibrium) particle levitation is possible. The maximum possible size of the charge is larger for a higher level of ionization rate; the position of the maximum charge size becomes closer to the electrode as the ionization rate increases. We note also that the negative gradient of the equilibrium charge ( i e . , d Q ( z ) / d z < 0) can lead to an instability of dust particles with respect to their vertical oscillations due to delayed charging [Nunomura et al. (1999)l. For a particle levitating in the sheath field, the force acting on the grain includes the sheath electrostatic force, the ion drag force, and the gravity force: (4.1)

+

where the ion drag force Fdr(z)= F&(z) F&(z) includes two components discussed in Sec. 2.1.4, namely, the collection force F & ( z ) and the orbit force F&(z). Note that the force (4.1) includes the z-dependence of the grain charge Q , since we assume an instantaneous transfer of charge onto and off the dust grain at any grain position in the sheath. The balance of forces in the vertical direction thus relates the electric

146

Physics and Applications of Complex Plasmas

field force with the gravity force and the ion drag force. It is given by the equation

Q ( z ) E ( z )= m d g + Fdr(z).

(4.2)

Solution of this equation together with the charging equation gives the dependence of the charge of the grain, levitating in the sheath electric field, as a function of its size. For the levitating dust particle, there is therefore a one-to-one correspondence of its size to the equilibrium position of particle levitation in the sheath, as shown in Fig. 4.2. Table4.2 The characteristic numbers for the dust grain radius and the position of dust levitation (Fig. 4.2) [Vladimirov and Cramer (2000)].

Ionization frequency Vion/Wpi

0.1 0.01 0.001

Maximum grain radius amax(pm) 12.77 8.11 5.62

Position of

Positions

grain amax zam (in h i ) 24.93 26.55 28.87

of a 4 pm grain (in X D ~ ) 58.14(2.93) 64.11(0.78) 66.41 (0.43)

Unstable position avs (in X D ~ ) 12.97 15.89 24.93

Here, the lines corresponding to various sizes of dust grains are plotted: as an example, a = 4 pm, as well as the maximum possible sizes, and the sizes corresponding to a grain levitating at the position where the Mach number of the ion flow is unity ( i e . , at wi = u s ) , as summarized in Table 4.2. Note that there are no equilibrium solutions for a > amax,the latter being a function of the ionization rate (see Table 4.2). The absence of an equilibrium means that the particles with such sizes will fall down onto the electrode. From Fig. 4.2 and Table 4.2, one can conclude that the greater is the ionization rate, the closer is the equilibrium position of a levitating dust grain to the electrode. This fact agrees well with the experimental observations [Samsonov and Goree (199913); Morfill et al. (1999)I showing that the size of a dust void is directly proportional to the ionization rate. The void corresponds to a dust-free region where the electron impact ionization rate is relatively high, producing an outward electric field and ion flow, thus dragging the dust particles outwards. In the case discussed here, a higher ionization rate again gives a stronger ion flow, dragging the dust grain closer to the electrode. Note that if there are two positions for a grain of a given radius (e.g., 4 pm in Fig. 4.2), the one with a negative

Particle Dynamics in a Complex Plasma

147

Fig. 4.2 Dependence of the size of the dust grain (in pm), levitating in the sheath electric field, on its position h = z / X o i [Vladimirov and Cramer (2000)]. Here, (a) vion/upi = 0.1, (b) vion/wpi = 0.01, and (c) vion/upi = 0.001. The lines correspond to various sizes of dust grains: as an example, a = 4 pm, as well as the maximum possible sizes, and the sizes corresponding to a grain levitating at the position where the Mach number of the ion flow is unity (ie., at wi = u s ) , as summarized in Table 4.2.

derivative da(z)/dz is stable, while the one with a positive derivative is unstable. We also note that the maximum possible radius for the grain levitation increases with the increase of the ionization rate, and its position also shifts closer to the electrode. Finally, we see that the smaller is the ionization rate, the smaller is the maximum possible size amaxof a grain capable of levitating, and therefore the greater is the proportion of dust (if there is a dispersion of grain sizes) levitating in the region of subsonic ion flow velocities, i.e., in the pre-sheath region.

148

Physics and Applications of Complex Plasmas

It is instructive to find the total “potential energy”, relative t o the electrode position, of a single dust particle of given size at the position z in the sheath electric field: (4.3)

Note that the total energy in this case contains not only the electrostatic energy Q(z)cp(z),but also the terms associated with d Q / d p which represent, because of the openness of the system, the work of external forces which change the dust charge. The dependence of the total potential energy on the distance from the electrode is shown in Fig. 4.3.

-50

Fig. 4.3 The total interaction energy Ut0t as a function of the distance h = ‘ / A mDi from the electrode for the different sizes of a dust particle and the different ionization rates [Vladimirov and Cramer (2000)j: (a) uion/upi = 0.1, (b) vion/upi = 0.01, and ( c ) uion/upi = 0.001. The curves correspond to: 1) a = amax 1 pm; 2 ) a = amax; 3) a = avs and 4) a = 4 pm. See also Fig. 4.2 and Table 4.2.

+

Particle Dgnamics in a Complex Plasma

149

It can be seen that the potential has a maximum and a minimum, corresponding to the two equilibrium positions found above. The minimum (the stable equilibrium) disappears if a > amax(curve 1 in Fig. 4.3). Other effects, such as an electron temperature increasing towards the electrode, may serve to increase the negative charge on a grain, and so preserve an equilibrium. The critical (maximum possible for levitation) radius appears also in Fig. 4.2; for the decreasing ionization rate, amaxalso decreases (see Table 4.2). For a > amax,the minimum of the potential energy curve disappears thus indicating that there is no equilibrium position for such grain sizes. Thus, for a collisional plasmas with an ionization source, for a grain size a less than the critical radius amax,there is a stable equilibrium position close to (or in) the pre-sheath; for sufficiently high input powers (within a certain range of grain sizes, see Fig. 4.2), there can also be an unstable equilibrium position deeper inside the sheath. For a greater than the critical radius amax,there is no equilibrium position. Note that possible vertical oscillations about the stable equilibrium may develop high amplitudes, thus leading to a fall of the oscillating grain onto the electrode when the potential barrier (see Fig. 4.3) is overcome. Here, we note on how the change of the ionization power can affect size distributions of dust particle levitation in an experiment [Vladimirov and Cramer (2000)l. Suppose that we start with a low ionization power and then increase it in the process of the experiment. Since the maximum possible size of particles capable to levitate in this case is only increasing, no change of dust size distribution occurs, with dust levitating closer to the electrode as the input power increases. However, if in the next experiment the ionization power is decreased, the possible size of particles capable to levitate is also decreasing, with the heaviest grains (whose size and, correspondingly, mass does not satisfy the condition for levitation) falling down to the electrode. Thus the dust size (and mass) distribution can be changed in this way, leaving only smaller particles levitating. Note also that another experimental possibility to force bigger particles (whose sizes are close to the critical one) to fall down to the electrode is to apply a low frequency modulated voltage (with the frequency close to the resonant frequency of vertical vibrations of the dust grains around the equilibrium position [Vladimirov et al. (1999a); Vladimirov et al. (1999a); Vladimirov et al. (1998a)l) t o the lower electrode thus forcing particles to oscillate. When their amplitude (and therefore the energy of oscillations) becomes large enough to overcome the potential barrier, see Fig. 4.3, they fall down.

150

Physics and Applications of Complex Plasmas

Above, it was assumed that the dust number density is small enough not to influence significantly the plasma parameters (such as the electric field, etc.). The performed study has also demonstrated some qualitative features of dust trapping and equilibrium for those experiments (e.g., on void formation), where the dust density is higher. One can therefore qualitatively conclude that the effects of single particle trapping and equilibrium are important also for the levitation of dust distributions. For dust clouds, there are several physical effects important for the particle trapping and equilibrium, among which are: the change of the plasma electric field, density, etc., distributions due to the charged dust cloud [Goree et al. (1999); Tsytovich et al. (1999)], and the change (decrease) of the charge of a dust particle due to the presence of neighboring grains in the cloud [Whipple et al. (1985); Havnes et al. (1987)l. The quantitative answer to the question of the differences between the dense dust cloud and the isolated dust particle trapping can be done only after the corresponding theory for the dust distributions is developed. Another issue is the influence of the ion flow effects on the stability and equilibrium of dust grains. For an adequate description of the effects of the ion wake, see Sec. 2.3, a treatment taking into account the plasma ion kinetics is necessary. It is physically clear that for a monolayer dust distribution (in the plane of the electrode) the ion wake effects are less important. However, we note that, generally speaking, the wake influence exists not only strictly below the particles (downstream the ion flow), but also in the perpendicular direction (for example, within the Mach cone in the case of the linear ion wake). On the other hand, for a highly collisional plasma the wake effects are weaker if the ion-neutral mean free path is of the order of the plasma Debye length, when the ion focusing is destabilized by frequent ion-neutral collisions. For significantly dense dust distributions the plasma ion kinetics can still be effective for a layer near the dust cloudvoid boundary, of the order of the ion-dust (or the ion-neutral, whichever is smaller) mean free path.

4.1.2

Dynamics of particles in chemically active plasmas

In this section, we discuss forces acting on dust particles and supporting the particle levitation as well as dynamic and trapping phenomena involving submicron-sized particles appearing in silane and fluorocarbon plasmas. The particular mechanisms of origin and growth of such particles in chemically active plasma environments is discussed in detail in Chapter 7.

151

Particle Dynamics in a Complex Plasma

4.1.2.1 Dynamics of fine particles in silane plasmas Experiments [Seeboeck et al. (1994)l aimed to study experimentally how the gas flow conditions and temperature gradient govern the trapping and fallout of the particles in RF parallel-plate discharges between parallel plates and to understand this behavior by means of a simplified theoretical modeling. The experiments were conducted in a parallel-plate 13.56 MHz capacitively coupled plasma reactor with externally heatable electrodes. The plasma discharges were sustained in silane at 20 Pa with 15-80 W RF powers. To view the dust particles in the discharge, the scattered light of a He-Ne laser was collected by a video camera. With the help of a complicated mirror system, the fine particles were visualized in various regions of the reactor. It was found that the motion and trapping of the particles is strongly affected by the gas flow conditions and the temperature gradient in the discharge. To understand the dynamics of submicron-sized particles in the plasma, the following simplified set of equations was used (4.4)

(4.5)

mddu,/dt = -mdg

-

+

K T V ~T67rurl,~((21, ~ -u ~ f )FE(z),

(4.6)

(4.6)

where the four terms in the r.h.s. of Eq. (4.6) correspond to the gravity, thermophoretic, neutral friction, and electrostatic forces, respectively. f the velocity of the gas flow, veff and T,(z) are the dynamic Here, u ~ is effective viscosity and the temperature of silane gas, and KT is the thermophoretic coefficient. In the case of a constant flow velocity in the axial direction and the constant thermophoretic force, Eqs. (4.4)-(4.6) can be decoupled. Outside the sheath, where the effect of the electric force can be neglected, the analytical solution of Eq. (4.6) can be presented as

where r, and

=

md/67rave~,uot is the axial component of the initial velocity,

152

Physics and Applications of Complex Plasmas

is the assymptotic grain velocity at t 3 00. The corresponding solutions of E q ~ ( 4 . 4 and ) (4.5) can be written in the exponential form. The validity of the friction force is verified by monitoring the sinks of the particles after switching the discharge off when the electric force disappears. [Seeboeck et al. (1994)I have calculated the particle positions and velocities as a function of time for the grain radii of 150 and 100 nm. When the discharge was on, the light particles were trapped at the sheath boundary. When the temperature gradient was applied (heating the lower electrode), the lighter particles were driven upwards by the thermophoretic force and were trapped at the sheath edge. Finally, the downward-directed gas flow (feeding gas through t)he upper electrode) of 30 cm/s can strongly impede the trapping of particles at the lower sheath boundary, as was confirmed by the experimental observations. The electrode heating, together with the adjustable gas flow, can result in a complete de-trapping of the fine particles in the inter-electrode region. Therefore, the proper control of the thermophoretic and drag forces can contribute to the solution of the particle contamination problem in the design of industrial PECVD reactors.

4.1.2.2

Levitation in fluorocarbon plasmas

Inductively coupled (ICP) and surface wave sustained (SWS) plasmas generated in the C4Fa+Ar gas mixtures have an outstanding potential for ultra-fine and highly selective etching of large-area silicon wafers [Kokura et al. (1999); Sugai et al. (1996)l. Physically, it appears possible to achieve high dissociation rates of the feedstock gas and the high densiand higher) of neutral radicals at the low (-J 20 mTorr) ties ( w pressures. An interesting peculiarity of fluorocarbon plasmas is that fluorine atoms and other fluorocarbon radicals relatively easily become negatively charged, in particular, due to the electron attachment. Recent data on the laser photo-detachment measurements in high-density CdF8+Ar plasmas suggest that under certain conditions the negative ions can constitute a substantial proportion among other discharge species [Kono and Kato (2000)l. The negative ions modify the power and particle. transport, the potential distribution in the discharge, as well as directly participate in the etching process. Furthermore, the fluorocarbon plasmas are prone of plasma polymerized nanoparticles that can also carry a substantial proportion of negative electric charge. Thus, the problem of the particle dynamics and trapping is of a vital interest for a number of plasma etching technologies.

Particle Dynamics in a Complex Plasma

153

Using a simple model of [Ostrikov et al. (2001a)], we consider here a one-dimensional near-electrode region of the width z p s , with an electrode placed at z = 0, and plasma occupying the region z > 0. In the plasma bulk z > xPs,all plasma parameters are uniformly distributed. The plasma composition is taken from the etching experiments in the mixtures ~ ) argon and includes the electrons, of octafluorocyclobutane ( c - C ~ F and and the positive Ar+, CF+, CF;, and CF:, as well as negative F- ions. The charging, the forces acting on, and the trapping of an individual single particle are considered. In the plasma bulk, the overall charge neutrality is assumed. The specific plasma parameters and proportions of positive ions have been taken as typical values from the experiments on SiO2 wafer etching in 2.45 GHz SWS and 13.56 MHz IC plasmas at low pressures [Kokura et al. (1999)l. In particular, in 90% Ar and 10% C4F8 gas mixture at 20 mTorr, the proportion of argon ions varies from 55% in the surface-wave plasmas sustained with 400 W to about 90% in the ICP (- 1.5 kW) [Kokura et al. (1999)I. Both plasmas feature dominant CF+, CFZ, and CF: fluorocarbon ion radicals in a wide range of the R F input powers. The proportion of the negative ions in electronegative C4F8+Ar discharges varies within 15% and 45% from the total number density of negative charge in some of the etching experiments in fluorocarbon-based plasmas [Kono and Kato (2000)]. Because of the overwhelming complexity of the gas-phase reactions and the large number of elementary processes in the fluorocarbon plasmas with negative ions, a realistic choice of the near-wall potential 4(x) was made [Ostrikov et al. (2001a)l. The density profiles within the sheath/pre-sheath were computed according to the following Poisson’s equation

-

d24(x) d22

= 4 T e [ n e ( z ) f n F - (z)-nA~+(z)--Cp+(z)-ncFz+(z)-nc~3+(z)1 ,

(4.7) where it was assumed that the electrons and negative ions are Boltzmann distributed, (4.8) (4.9) and T F - the the temperature of the negative fluorine ions. Here,

154

Physics and Applications of Complex Plasmas

+ncF+( x p S )

+ ncF:

(%pS)

f ncF,'

(xps)

is the combined number density of negative or positive plasma species in the plasma bulk (x = x p s ) ,and ( Y F - = n ~(xps)/no. We now turn to the discussion of the ion motion in the RF-driven sheath. It is assumed that TRF& ~ , h ,where ~ , his the ion sheath traverse time, and TRFis the R F period. In this case, the ions respond to the time-averaged (over the time scales ~ , h t >> T R F )electric field. The positive ion fluid velocities v(,)f satisfy the following momentum equation

-

(4.10) where v(,), = n n ~ ( , ) n v (is , ) the rate of the ion-neutral collisions, c ~ ( , ) ~ is the cross-section for the resonance charge exchange in argon, n, is the density of neutrals, y,)= [u?,)~+8T(,)/~rn(,)]~/', and m(,) and T(,) are the ion mass and temperature, respectively. The profiles of the non-dimensional number densities of the charged particles, spatial coordinate-dependent charges and the major forces acting on them have been computed for the following main plasma/dust parameters: the electron temperature of 2 eV, the dust size of 300 nm and 2 pm, the particle mass density p = 1.5 g/cm3, the plasma bulk density no = 4 x 10l1 ~ m - the ~ , ion and neutral temperatures of 0.067 eV and 0.026 eV, respectively, and various positive/negative ion proportions. From the computed distributions of the charged species in the nearelectrode area one can conclude that the number density of negative ions dramatically decreases towards x 0.6xp,, and is negligible in the areas closer to the electrode. Furthermore, the computation suggests that all ion velocities w(,), being equal to the ion thermal velocity vT(j) = &i"(,)/~rn(,) in the plasma bulk, become equal to the ion-acoustic veTe/m(j) within the area 0.2 < x / x p s < 0.3. Thus, it locity ws(,) = ,/-Te/m(j) sounds reasonable to tentatively term the area 0.6 < x/xp, < 1 as the electronegative region, or the pre-sheath with the negative ions (Region I), x, < x / x p , < 0.6 as the electropositive region, or the pre-sheath without negative ions (Region 11), and 0 < z / x p s< x , as the sheath itself (Region 111). In this case, the uncertainty in the sheath edge location x, falls within the limits 0.2xp, < x, < 0 . 3 ~ The ~ ~ exact . location of the sheath edge can be obtained by applying the Bohm sheath criterion for a multi-component plasma. In Region I, the positive ions are accelerated towards the electrode, and the negative ions contribute t o the sheath structure. In Region 11, the positive ions are further accelerated, whereas the effect of the negative ions

-

Particle Dynamics in a Complex Plasma

155

on the sheath structure is already negligible. In the near-wall Region 111, the electron and ion densities diminish, the latter remains higher sustaining the electrostatic field in the sheath. This approximate model thus reflects basic features of the plasma sheaths in electronegative gases [Lichtenberg et al. (1994)l. The value of the average particulate charge is determined by the dynamic charging equation

(4.11) where Q = -1Zdle is a negative dust charge, and C [ j ( + , - ) ~ l [ j ( + , are the combined microscopic currents of the positive and negative plasma species [Ostrikov et al. (2001b)l. The microscopic currents onto the dust grain were computed using the OML theory discussed above in Sec. 2.1. The equilibrium state of the dust charge (Qo = const) was calculated by invoking the stationary balance of the positive and negative plasma currents on the grain. The results for the spatial profiles of the equilibrium charge are presented in Fig. 4.4 for 300 nm and 2 pm particles, respectively. Figure 4.4

a, c

0

'

Fig. 4.4 Profiles of the equilibrium charge on (a) 300 nm and (b) 2 pm particles in the near-electrode region for aAT+= 0.65 [Ostrikov et al. (Zoola)]. Curves 1-3 correspond to a F - = 0.15, 0.3, and 0.45, respectively.

156

Physics and Applications of Complex Plasmas

reveals that the particulate charge has a tendency to grow in the near-wall region, and starts t o decline after reaching maximum at x 0 . 1 2 ~ ~ ~ . Likewise, the dependence Zdo(x) features a distinctive minimum in the electronegative region, which reflects the dust charge depletion caused by the negative ions. If the proportion of the negative ions is low (curve l),the dust charge is almost independent on the local position within 0.5xps < x. In the region 0 . 1 2 < ~ ~x ~< 0 . 5 the ~ ~charge ~ rises, which is a consequence of the increased electron/ion currents onto the grain. Physically, the diminishing of the potential lowers the potential barrier for the plasma electrons, which can easier be collected by dust. Meanwhile, the strong electron/ion density depletion within the sheath results in the weakening plasma fluxes on the dust grains, and hence, of lZdl. It is notable that the particulate charge is lower when the number density of the negative fluorine atoms in the electronegative region is higher (Fig. 4.4). Above all, a variation of bp(essentially in Region I) affects the particulate charge in the sheath (Region Ill). Physically, the negative ions affect the formation of the ion flows originating in the electronegative region, and hence, the grain currents within the sheath. Knowledge of the ion velocity and the dust charge distributions allows one to compute the total force acting on the fine particles N

which includes the gravity force F,, the electrostatic force FE,and the drag forces due to positive FZ” and the negative F Z - ) ions. The numerical results [Ostrikov et al. (2OOla)l demonstrate that nearly in the whole electronegative and electropositive pre-sheaths the particles are pushed by the positive ion drag force towards the electrode without any serious counteraction by other forces. In the electronegative region F:+)(x) (X)increases, reaches maximum in Region I1 at x 0.5xp, and declines towards the electrode thereafter. The flex point in the ion drag force corresponds to the area of the significant depletion of the positive ion number density. One can further infer that the resulting force on the particulates Ftotis directed towards the wall in the entire region xtr < x and reverses a t x ztr,where xtr 0 . 2 5 for ~ ~300 ~ nm particles, and xtr 0 . 1 8 for ~ ~ 2~ pm particles. Furthermore, the gravity force makes a noticeable contribution only in the plasma bulk. The electrostatic force, negligible in the plasma bulk, becomes compa~ . the rable with the positive ion drag force at positions x < 0 . 3 5 ~ ~Inside N

N

N

N

157

Particle Dynamics in a Complex Plasma

sheath (Region 111) the force on a particle is essentially electrostatic and pushes it outwards. Indeed, the electric field becomes strong (in our example it is 200 V/cm in the electrode proximity), while the ion drag force diminishes further. At the equilibrium (particle levitation/trapping position) z = xeq the electrostatic and negative ion drag forces balance the positive ion drag force and the gravity force

-

As evidenced by the numerical results, the equilibrium position of the larger (2 pm) particle appears to be closer to the electrode (zeq 0.18zps),which is consistent with the experimental results on the dust void formation [Samsonov and Goree (1999b)I. One can thus presume that the effect of the positive ion drag force is stronger for larger particles. In the near-electrode areas of low-density (- lo9 crnp3) hot cathode discharges, the effect of the ion drag force on micron size dust grains is negligible [Arnas et al. (2000)l. We emphasize that due to the negative dust potential, the collection impact parameter, and hence the microscopic grain current of the negative fluorine atoms, appear to be small. Hence, the major effect of the negative ions on the equilibrium particulate charge is associated with the electron density depletion in the electronegative Region I. Indeed, in the plasma bulk/electronegative pre-sheath, 2, is 10-30% lower than in the absence of fluorine ions (Fig. 4.4). Likewise, the negative ions are not expected to affect the particulate trapping in the Region 111. However, as Fig. 4.4 suggests, the dust charge within the sheath depends on the fluorine ion number density in the plasma bulk. Physically, this can be regarded as an example of the action of the ‘pre-history’ effects associated with the formation and acceleration of ion flows in the electronegative pre-sheath region. In high-density fluoroca.rbon plasmas the negative ion drag force is weak, so is the gravity. Furthermore, the orbit component dominates in the positive ion drag force. The gravity effects for both 0.3 and 2 ,urn particles appear to be important in the plasma bulk only. Within the region x < 0.9, the competition between the positive ion drag and electrostatic forces controls the particulate dynamics. The ion drag force peaks at z O.45zp,, which reflects the dynamic balance between the counteracting effects of the positive ion acceleration towards the electrode and depletion of their number density in Regions I1 and 111. Finally, we note that the sheath/presheath structure, the ion fluxes, and hence, the particulate charging and trapping processes, are very sensitive t o the temperatures of the electrons,

-

N

158

Physics and Applications of Complex Plasmas

the positive/negative ions, as well to the electron energy distribution functions, which often appears to be non-Maxwellian in etching experiments.

4.1.3

Oscillations of a particle an a plasma sheath

The charge of dust particles, appearing as a result of electron and ion current onto the grain surfaces, strongly depends on the parameters of the surrounding plasma. Dependence of the dust particle charge on the sheath parameters has an important effect on the oscillations and equilibrium of dust grains in the vertical plane, leading to a disruption of the equilibrium position of the particle and the corresponding transition to a different vertical arrangement [Vladimirov et al. (1999a)I. Consider vertical vibrations of a particle of mass m d levitating in the sheath region. The charge Q of the dust particle is found, as usually, from the condition of a zero total plasma current onto the grain surface. Since we are interested in the collective processes on the time scale of the characteristic frequencies (of the order of a few times 10 s-'), which are much lower than the charging frequency (of the order of lo5 s-'), assume that (re)charging of dust grains is practically instantaneous, and therefore neglect the charging dynamics. The plasma electrons are Boltzmann distributed. The discharge pressure is assumed to be low enough that ion ' collisions with the neutrals and other species can be neglected (this corresponds to experiments in a low-pressure discharge where a spontaneous excitation of vertical vibrations was observed [Nunomura et al. (1999)l). Thus, in contrast to the electron distribution, we consider collisionless, ballistic ions in the sheath. The charge of a dust particle in the sheath region is then determined by Eq. (2.38). The sheath potential is found from the Poisson's equation and defined by Eq. (2.35) (where the total charge contributed by the dust grains is neglected) and the sheath electric field is given by Eq. (2.36). The balance of forces acting on the particles in the vertical direction (4.2) includes the gravity force and the sheath electrostatic force. Solution of this equation together with the charging equation (2.38) gives the dependence of the charge of the grain, levitating in the sheath electric field, as a function of its size, as shown in Fig. 2.3. There is no equilibrium solution for a > amax= 3.75 pm. Note also that the equilibrium solutions for positions closer than zmin = 1 . 6 4 X ~ 0where , the grain has a critical size a = acr < amax,are unstable in relation to vertical oscillations. Consider small oscillations of an isolated dust grain around the equilib-

Particle Dynamics in a Complex Plasma

159

rium position. The equation of motion is given by (4.12) where the coupling constant is dE yC= EQdcp

+ E ~dQ -. dcp

(4.13)

Note that all the derivatives (as functions of the sheath potential cp and the particle charge Q ) in Eq. (4.12) can be found analytically from Eqs. (2.38) and (2.36). It is also important to note that the second term on the right hand side of Eq. (4.13) is negative and becomes dominant for larger dust size. The function yc(u) is presented in Fig. 4.5.

Grain radius (microns)

Fig. 4.5 Dependence of the coupling constant yc of the dust vertical vibration on the dust size [Vladimirov et al. (1999a)I. The critical radius when yc = 0 is acr = 0.372 x l o r 3 cm.

We see that for u > ucr = 3.72 pm, the coupling constant is negative and therefore no oscillations are possible. This case corresponds to an instability of the equilibrium particle levitation in the sheath field because the heavy (large) particle is positioned too close to the electrode where the charging by plasma electrons is insufficient because of the electron density depletion. The equilibrium charge as well as the sheath potential and electric field at the position of the dust grain, and thence the derivatives in Eq. (4.13), can be found numerically by solving Eqs. (2.38) and (4.2) simultaneously. For example, with the parameters [Vladimirov et al. (1999a)I ADO = 3 x cm, T, = 1 eV, A40 = WO/W, = 1, milme = 40 x lo3, p = 1 g/cm3 and u = 0.35 x l o p 3 cm, the resulting characteristic long-wavelength

160

Physics and Applications of Complex Plasmas

frequency is (4.14) and the equilibrium charge is 4 = - ( Q / e ) x 10-3 G 5.4. In general, the characteristic frequency of the dust vertical vibrations is a function of the dust size, as shown in Fig. 4.6; the frequency becomes zero for a = ucr = 3.72 pm.

Fig. 4.6 Dependerice of the characteristic frequency f c h (in Hz) of the oscillation of dust on the size a (in microns) of the dust grain [Vladimirov et al. (1999a)I. The critical dust radius i s a,, = 0.372 x l o r 3 cm.

It is also instructive to find the total “potential energy”, relative to the electrode position, of a single dust particle of a given size at the position z in the sheath electric field: (4.15) The total energy in this case contains not only the electrostatic energy Q ( z ) ~ ( zbut ) , also terms associated with dQ/dp which represent, because of the openness of the system, the work of external forces which change the dust charge. Solving Eqs. (2.38) and (2.36), the dependence of the total potential energy on the distance from the electrode can be found, as shown in Fig. 4.7. For comparison, in Fig. 4.7 the energy in the case of a constant Q placed at the same equilibrium (or marginal equilibrium) position is also plotted. The potential always has a minimum for the case of Q = const, but in the case of a variable charge there can be a maximum and a minimum, corresponding to the two equilibrium positions. The minimum (the stable equilibrium) disappears if u > ucr (the curve c in Fig. 4.7). A

Particle Dynamics in a Complex Plasma

161

Fig. 4.7 The total interaction energy Utot as a function of the distance h = Z/XD from the electrode for the different sizes of a dust particle [Vladimirov et al. (1999a)l: (a) a = 0.35 x 10W3 cm, (b) a = acr = 0.372 x 10-3 cm, and (c) a = 0.4 x lop3 cm. The dashed lines correspond to the case of a constant charge at the equilibrium (or the marginal equilibrium) position: (a) Q = -5.42e x lo3 and (b) Q = -4.93e x lo3.

similar result has been found for the collisional sheath case [Nunomura e t al. (1998)l. Other effects which have been neglected here, such as an electron temperature increasing towards the electrode, may serve to increase the negative charge on a grain, and so preserve an equilibrium. The critical radius can be found numerically [Vladimirov e t al. (1999a)I by solving Eqs. (2.38) and (4.2) simultaneously with the condition w = 0. For the parameters considered here, for a = acr = 0.372 x lop3 em, the minimum disappears. This is close to the critical radius observed experimentally [Nunomura et al. (1999)l. Thus for the collisionless sheath, for a less than the critical radius, there is an unstable equilibrium position deep inside the sheath, and a stable equilibrium position closer to the pre-sheath. For a greater than the critical radius there is no equilibrium position. Vertical oscillations about the stable equilibrium may develop high amplitudes (e.g., because of an instability in the background plasma). This may lead to a fall of the oscillating grain onto the electrode when the potential barrier (see curve a in Fig. 4.7) is overcome. Such a disruption of the dust motion has been observed experimentally [Nunomura e t al. (1999); Samarian et al. (2001b)l. One can note that the charge, position, and vertical oscillations of a dust grain levitating in a collisionless sheath field of a horizontal negatively biased electrode strongly depend on the parameters of the sheath, in particular, the sheath potential. The dependence of the particle charge on the potential is also important for the characteristics of the oscillation mode as

162

Physics and Applications of Complex Plasmas

well as for the equilibrium of the dust particle. Large amplitude vertical oscillations of the dust grains may be responsible for experimentally observed disruptions of the equilibrium of the dust crystal as well as with numerically demonstrated phase transitions associated with vertical rearrangements of the grains.

4.1.4 Diffusion of dust particles Diffusion of dust grains is one of the most important processes for mass transfer in a complex plasma [Vaulina and Vladimirov (2002)l; it determines the energy losses (dissipation) as well as the system dynamic characteristics, formation of structures, and phase transitions. Note that for various complex systems such as colloidal suspensions [Lowen et al. (1993)], complex quasi-2D plasmas [Juan and I (1998)], sandpile models [Carreras et al. (1999)], turbulent flows [Solomon et al. (1993)], diffusion can exhibit anomalous character associated with nontrivial topology of the phase space of the system and spatio-temporal correlations [Zaslavsky et al. (1993)l. The main manifestation of the anomalous diffusion is in the nonlinear time dependence of the mean square displacement, in contrast to the linear character for the normal diffusion process [Zaslavsky et al. (1993)l. Complex plasmas provide a natural example of a system of strongly interacting particles with possible anomalous character of the grain diffusion. The hydrodynamic approach makes it possible to describe the macroparticle diffusion in a complex plasma only for the case when the short-range interactions dominate. On the other hand, when the particle interactions are stronger than in (ideal) gases, the relevant kinetic equation is difficult to derive. Essentially, the theory of diffusion in molecular liquids developed in two directions, one of which is based on general ideas of statistical physics while another one utilizes the analogies between the liquid and solid states and gives for the diffusion of molecules the relation [Frenkel (1976)l: D~

=

d2 -exp 6-rO

(-F) ,

(4.16)

where d is the average distance between the molecules, ro is the characteristic time, defining the frequency vo of flights (“jumps”) of the particles from one stable position to another, and W is the energy threshold for the particle flights. The exponential character of the temperature dependence for the self-diffusion of macroparticles was obtained for dissipative Yukawa systems [Vaulina and Khrapak (2001)]. Note that the experimen-

Particle Dynamics in a Complex Plasma

163

tal verification of this result in a complex plasma is difficult since in the laboratory change of any parameter of a plasma-dust systems inevitably leads to the self-consistent change of other parameters determining the grain dynamics. To solve the problem, a proper determination of basic functional dependencies under certain approximations for the self-diffusion rates of grains in Yukawa system is necessary; this has been the subject of extensive studies [Ohta and Hamaguchi (2000); Frenkel (1976); Vaulina and Khrapak (2001); Swinney (1974); Cummins (1974); Pusey (1974); Hofman et al. (20OO)l. The correct expression for the diffusion rate in dissipative systems of macroparticles interacting via Yukawa potential (4.16) is important not only from the point of view of determining the dynamic characteristics of a complex plasma, but also for the analysis of various kinetic processes in molecular biology, medicine, polymer chemistry, etc. [Frenkel (1976); Vaulina and Khrapak (2001); Swinney (1974); Cummins (1974); Pusey (1974); Hofman et al. (2000)l. Various approximations are mostly based on either virial expansions of the thermodynamic functions, e.g., the effective viscosity in the Einstein expression DO = qT [Cummins (1974); Pusey (1974); Hofman et al. (2000)], or on the analogies with the critical phenomena in gases [Ohta and Hamaguchi (2000); Swinney (1974)l. Thus, for some conditions the rate D can be expressed as a sum of the first terms in an expansion like Do (1 ( 0 ) or as a power function like (1+(2 c2 (T/Tc- I)$, where To is the temperature at the melting point, and the parameters C0, (1, (2, and $ can be determined for particular values of the complex plasma parameters from experimental or numerical simulation data. To model the motion of dust grains in a weakly ionized plasma, the Brownian dynamics method is invoked which is based on the solution of ordinary differential equations with the stochastic Langevin force Fbr , taking into account random collisions with the plasma neutrals [Vaulina and Vladimirov (2002)l. The equation of motion is given by

+

(4.17)

where the force Fint(l) = -eZ&$,/dl dl accounts for the pairwise particle interaction in the system. The characteristic friction frequency can be written as ufr[s] Cvpo[Torr]/(p[g/cm3] . a[micron]) [Yakubov and Khrapak (1989)], where a is the particle radius (in microns), p is the particle density (in g/cm3), po is the neutral gas pressure (in Torr), and C, is a dimensionless parameter, defined by the nature of the neutral gas, e.g.,

164

Physics and Applications of Complex Plasmas

for argon C, 2 840, and for neon C, g 600. The equilibrium charge of a dust particle (in the approximation of charging by plasma currents) is Z d FZ C,a[micron]T,[eV] [Goree (1994)], where C, E 2000 for the majority of experiments in noble gases. Three-dimensional equations of motion (4.17) are solved with the periodic boundary conditions. The full number of particles in the modeled system is N p = 125 x 27, and the number of independent particles is 125. The cutoff of the interaction potential is im-1/3 . posed at distances equal or more than 4d, where d = nd is the average interparticle distance. The time step is At = min(l/vf,, l/w*)/20, where w* = eZd(~~d/rnd)~/'(l + R + K ' / ~ ) ~ / ' exp(-~/2) is the normalized dust frequency, and R = d/Ao. The total simulation time is ZOOO/uf, for uf, < w * , and 20OO~f,/(w*)~ for u f r > w* being in the range from 2 x lo5& to 2 x 106At depending on the parameters of the system. The self-diffusion rate of the dust particles can be considered as the parameter determining the dynamic behavior of the complex plasma system. It is defined by (4.18) where l(t) is the displacement of an isolated particle, and (. . .) stands for the ensemble ( N ) and time ( t ) average, respectively. The self-diffusion coefficient (4.18) is calculated for various pressures of the neutral gas (uf,), the characteristic dust frequencies w*,the dust temperatures T d , and the Debye screening lengths AD ( K = 2.4 and K = 4.8). Note that the particular choice of the plasma screening length is limited by the condition L >> AD necessary for the correct modeling of the dynamics of Yukawa systems [Farouki and Hamaguchi (1992)l; in simulations [Vaulina and Vladimirov -1/3 (2002)] that was L 5 x nd > 12 - 2 4 A ~ . The ratio between the interparticle interaction and dissipation in the system is defined by the scaling parameter ( = w * / u f , , i e . ,

< = e~(nd/rnd)'/~(l+ + ~~/2)l/~exp(-~/2)/vS,, K

(4.19)

The particular numbers for this parameter can be chosen utilizing data of gas-discharge plasma experiments; for the typical experimental param700, /G eters ( p E 4 g/cm3, Te E 1.5 eV, C, a), one can obtain from Eq. (4.19) ( M 10-3(nd[cm-3]/r [mi~ron])~/~(p~[Torr])-~, and for the particles of the radius a = 2.5 micrometer when their number density n d is changed from lo3 cm-3 to l o 5 cmP3 and the neutral gas pressure p o is changed from 1 Torr to 0.01 Torr, one obtains the range of values

Particle Dynamics in a Complex Plasma

6 = 0.02-4.2.

Accordingly, it was chosen [Vaulina and Vladimirov (2002)l.

6 = 0.055, 0.166, 0.5,

165

1.5, 4.5

Fig. 4.8 Dependence of the first maxima of the structure factor S1 and pair correlation function 91, (a), as well as their positions d s l / q I and d g l / q , (b), on the parameter I?* = ryu [Vaulina and Vladimirov (ZOOZ)]. The ranges of the deviations of the computed functions are plotted for various runs when 6 = (0.166-1.5) and K = (2.4,4.8).

To analyze particle ordering in the modeled system, the pair correlation function g ( r ) and the structure factor S ( q ) can be used. Thc dcpendcncics on the factor

r*= ryu= Q ~ ( I +K. + ~ . ~ /exp(-K)/dTd 2) (see Sec. 5.3.1) of the first two maxima of these functions (91, 5'1) as well as their positions ( r = d g l , q = d s l ) are presented in Figs. 4.8(a) and 4.8(b), respectively. The ratio of the diffusion coefficient D of the charged dust particles to the diffusion coefficient DO = Td/uf,md of the non-interacting (Brownian) particles is presented in Fig. 4.9(a) for different values of the parameters K. and E . For the range from r: % 102 to ?:ITm =1545106 we have abrupt changes of g1 and S1,see Fig. 4.8(a), as well as of the ratio D / D o (cf. Fig. 4.9(a), D + 0 for I?* + 106). With increasing I?* + r:, the bodycenter cubic (bcc) crystal structure has been formed. The positions of the first maxima of the functions g and S for the crystal lattice also correspond to the bcc-type lattice (d,l q l = 2n(find)i/3, r l = ( 3 & / 4 r ~ d ) l / ~d, s l see Fig. 4.9(b) as well as [Vaulina et al. (2000)l. The diffusion coefficient D of the particles in strongly non-ideal Yukawa systems exhibits exponential dependence on the parameter I?*, namely, D exp(-clr*/r;). To illustrate that, Fig. 4.9(b) presents the functional dependence of the diffusion rate D on the factor I?* in the logarithmic scale. It is easy to see that the dependence D(I?*)for I?* within the range of 50 N

166

Physics and Applications of Complex Plasmas

Fig. 4.9 Dependence of the (a) ratio DIDO and (b) the diffusion coefficient D (b) on r’ for different K (filled symbols correspond to K = 2.4; others, to K = 4.8) and E [Vaulina and Vladimirov (ZOOZ)]. x’s correspond to E = 0.055; circles, to [ = 0.166; triangles, to [ = 0.5; diamonds, to E = 1.5; and boxes, to = 4.5.

Table 4.3 The best fit of the calculated parameters c1 with the precision f4% for the considered values of

c

I

c1

(IF.

CI

(IF.

= 2.4) = 4.8)

E

and

IF.

1 0.055 I

I

3.12 -

[Vaulina and Vladimirov (2002)l. 0.166

3.13 3.13

0.5 2.92 2.96

1.5 2.89 2.96

4.5

2.93 3.05

1

to 102 is almost linear. Moreover, different curves corresponding to the different values of E and K have almost similar inclination ( i e . , c1 const). The best fit of the calculated dependencies D(J?*) gives c1 2 3 with the precision f 4 % for all considered 5 and K , see Table 4.3. Figure 4.10 shows the character of particle diffusion in the modeled system and presents the difference between the ensemble average A, = ,/(l(t) - 1(0))&/d2and its time average A h = ((l(t)- l(O))&),/d2 near the crystallization line. The time-average diffusion rate D of the interacting particles can be determined by the proper choice of the characteristic time 70 in relation (4.16). It is assumed that the characteristic frequency vo of the transitions between the neighboring quasistable states is related t o random collisions between the colloidal macroparticles with the characteristic frequency w 2 czw*, where c2 = const, as well as t o collisions with the molecules of the ambient neutral gas with the characteristic frequency vfr. Also, 70 M 2/v0 can be written as 7 0 = 2(w + v f r ) / w 2 ,since the frequency vo tends to w for E << 1, and vo tends to w 2 / v f r in the case 5 >> 1 [Fortov et al. (1998)l. The

Particle Dynamics in a Complex Plasma

167

Fig. 4.10 Dependence of the normalized mean square displacement A N / d on the normalized time vfrt for the system of macro-particles with = 0.5 and r; = 0.4 [Vaulina and Vladimirov (2002)l. The bold line corresponds to the time averaged normalized displacement A h / d for I‘* = 92.

<

unknown constant c2 can be found by the best fit to the numerical results; the procedure of the minimization of the mean square deviation between the relation (4.16) and the numerical data within the range of I?* from 50 to 102 gave c2 1/&. Thus taking into account the values of the coefficients obtained for expression.(4.16) as well as equality ( ~ * d=) (Tdr*)/md, ~ the diffusion coefficient is (4.20) The error of the approximation of the numerical results by the expression (4.20) does not exceed 2.5% for r* within the range 50 to 102; with I?* decreasing down to 40, the error increases up to 7-13%, and for I?* = 30 the error is 25-30%. The dependence of the ratio D / D o , where D is determined by Eq. (4.20), on the value of I?* is given on Fig. 4.9(a) for various E . The characteristic collision frequency between the dust particles in a liquid state wz = w*/& can be obtained if to consider the interparticle force in the dust-plasma system F = (eZd)’ exp(-l/XD)(1+Z/XD)/l2, taking into account that the electric fields of all particles except the nearest ones are fully compensated [Vaulina and Khrapak (2000)l. Then the frequency wz determined by the first derivative d F / d l at the point 1 = d is given by wf = 4 r ~ d ( e Z dexp(-K)(l+ )~ K rc2/2)/4nrnd. It is interesting to note that oscillations of particles with the frequencies close to wz = w*/& can be

+

168

Physics and Applications of Complex Plasmas

observed for particle motions in the system even for relatively high viscosity when = 0.5. The detailed investigation of these oscillations is beyond the scope of this review, however, we note that this type of regular motion disappears with decreasing I'*.

Fig. 4.11 (a) Condensation of the colloidal clusters, (b) dependence on I?* of the diffusion coefficient D and (c) the ratio D I D O for various E : circles correspond to = 0.166, and boxes, to = 0.5; filled circles and boxes are for IF. = 2.4, others, for IE = 4.8 [Vaulina and Vladimirov (2002)].

<

<

Finally, note the effect of the particle condensation (Coulomb clusters of dust particles ), Fig. 4.11(a), which was observed in numerical simulations [Vaulina and Vladimirov (2002)], for the parameter 'I'* approximately equal to 23.5. The formation of groups of correlated particles is also reflected in the displacement of the position of the first maximum of the pair correlation function, see Fig. 4.8(b), as well as in the rapid change of the diffusion rate, Figs. 4.11(b), 4.11(c). It can be suggested that this phenomenon is the second-order phase transition of the system from the strongly correlated liquid state to the non-ideal gas state. However, the attraction forces were not explicitly introduced into the set of equations (4.17), and the diffusion coefficient has not exhibited the tendency to disappear at the critical point rg M 23.5 [Maekawa et al. (1998)]. The observed effect can be also related to the symmetry properties of the particle interaction potential; then the value of I'g M 23.5 should define such conditions when the system properties related to the particles' identity are developed. Indeed, the mean free path of a particle in the liquid phase I,-, - p 54 = (6Dto)lI2is assumed to be equal to the Wigner-Seitz radius aws = ( 4 ~ n d / 3 ) - ~ then / ~ , for the critical parameter I?* the value 26 can be obtained, which, within the considered accuracy of Eq. (4.20) in the range I?* < 30, coincides with r$.We remind N

Particle Dynamics in a Complex Plasma

169

that for I?* < 40, the considered system cannot be modeled by simple relations obtained under the assumptions of the theory of “jumps”, and the scaling parameter [ given by Eq. (4.19) cannot be used for the analysis of the particle dynamics. Note that the mean free path of the dust particle-particle collisions lP-, is independent on the dust particle-neutral collisions in this approach. We can also estimate it from other considerations. For example, if to assume the mean free path as 1,-, P = (8Td/.irw *2 md)1/2 equal to the Wagner-Seitz radius aws = ( 4 ~ n d / 3 ) - l / then ~ for the critical parmeter at the point of the gas-liquid transition, we obtain r’*g-l l M 21 for the Maxwellian particle velocity distribution. On the other hand, if to assume the particle velocity as the quadratic mean (3Td/md)1/2,we find l?*g-l l M 26, similar to the above estimate on the basis of the diffusion coefficient. This condition can be analogous to the condition of finding one particle in the sphere with the radius l p P p and can serve as a criterion of the transition of the system to the strongly non-ideal state. Thus at present it is hardly possible to say definitely whether the observed effect is the (second order) phase transition or not. Nevertheless, it is obvious that at the point I?* M 23.5 the simulated systkm qualitatively changes it dynamic characteristics. We note that the investigated transition is the transition from one non-ideal gas-like state to another non-ideal liquid-like state; it can also be related to the “flow and floe” state [Thomas and Morfill (1996)l.

4.2

Arrangements and Instability of Confined Dust Particles

Stability and arrangements of macroscopic particles as well as the properties of excitations propagating in strongly coupled particle structures (twodirxiensional and three-dimensional dust-plasma crystals, one-dimensional chains of grains) observed in a complex plasma is a subject of a significant recent interest [Thomas and Morfill (1996)l. Recall that in the laboratory experiments, the micrometer-size highly charged grains usually levitate in the sheath region of the horizontal negatively biased electrode where there is balance between the gravitational and electrostatic forces acting in the vertical direction as well as externally imposed confining potential applied in the horizontal plane. In this region, arrangements of dust particles are also influenced by the strong ion flow, and the effects of the plasma wakes

170

Physics and Applications of Complex Plasmas

were extensively discussed above in Sec. 2.3. It was shown theoretically [Ivlev and Morfill (2001)] that the ion focusing associated with the wake can induce instabilities in the horizontal chain of dust grain related to interaction of transverse and longitudinal modes via the plasma ions focused in the sheath below the levitating grains. However, an instability of particle equilibrium may appear even for two particles [Steinberg et al. (2001); Vladimirov et al. (2002)] when obviously we cannot relate it to any cooperative lattice mode. The vertical confinement involving the gravity force and the electrostatic force acting on the dust particles with variable charges is a complex process exhibiting oscillations, disruptions and instabilities [Vladimirov et al. (1999a); Nunoniura et al. (1999); Vladimirov and Samarian (2002); Melzer et al. (1999); Steinberg et al. (2001)l. Major factors affecting formation of plasma crystals include the particle interacting potentials, plasma fields, as well as external forces. Here, we can distinguish the “external” (such as the confining fields) and “internal” (such as the particle interacting potentials) factors. The latter, due to the openness of the dust-plasma system, include not only electrostatic interaction of charges, but also consequences of their interactions with the plasma environment. The nature of particle arrangements in systems containing large number of particles can be understood by considering a simpler test systems allowing to elucidate the character of these influences.

4.2.1

Modeling stability of dust particles confinement

The stability of the vertical and horizontal confinements of dust particle levitation in a complex plasma appears as a non-trivial interplay of the external confining forces as well as the particle interactions and plasma collective processes such as the wake formation. As as representative case allowing to see the characteristic features of dust particle arrangements, the stability of the combined vertical and horizontal confinement of two dust grains can be studied. Here, we demonstrate that the potentials confining particles in the directions perpendicular to the particle motions can disrupt the equilibrium and discuss qualitative consequences for the experiments [Vladimirov and Samarian (2002)l. Consider vibrations of two particles of mass m d l , d 2 and charges Q ~ J , separated by the distance x d horizontally ( i e . , aligned along the z-axis), see Fig. 4.12(a) or z d vertically (aligned along the z-axis), see Fig. 4.12(b). In the simplest approximation, the particles interact via the screened Coulomb

Particle Dynamics in a Complex Plasma

171

Fig. 4.12 Sketch of possible particle configurations for two confined particles [Vladimirov and Samarian (2002)].

(Yukawa or Debye screening) potential q 5 ~= Q1Q2 exp(-lrl/AD)/lrl. Note that for particles levitating in the plasma sheath, the interaction potential in the vertical direction is actually such that the forces between them are asymmetric because of the ion flow towards the negatively charged electrode. However, it is also instructive to consider the case with Debye only interaction even in the vertical direction; there are two reasons for that. Firstly, in the microgravity experiments, such as those onboard of the International Space Station, the dust particles can levitate in the plasma bulk where the effects associated with the ion flow can be negligible. Secondly, consideration of the effects associated with the symmetric Debye screening allows us to elucidate the role of more complex asymmetric potentials. Thus consider two cases of the interaction in the vertical direction: 1) when the interaction potential is symmetric of the screened Coulomb type; and 2) when the interaction potential is asymmetric. The latter can be of a different physical origin, here, it is sufficient to assume only that it can be parabolically approximated near the equilibrium position of a dust particle. As an example of the asymmetric potential, the wake potential can be considered; it has the following approximate expression along the line (the z-axis) connecting two vertically oriented particles (see Sec. 2 . 3 ) : @w = 2Qcos(Izl/L,)/lzl(-l - 1/M2), where L , = X ~ d mNote . that this expression is only applicable on the line behind the dust grain; generally, within the Mach cone the wake potential has more complex structure, while outside the Mach cone the particle potential can be approximated by the Debye formula. Therefore the potential acting on the upper particle due to the lower particle, see Fig. 4.12(b), is a simple Debye repulsive potential. The balance of forces in the horizontal direction involves the action of the external (horizontal) confining potential as well as the Debye repulsion. We note that in experiments, the symmetric horizontal potential can be ob-

172

Physics and Applications of Complex Plasmas

tained using the ring or disk electrodes. For example, the glass cylinder was used to create the "square well", that is, the parabolic confining potential by [Steinberg et al. (2001)l; in the experiment [Melzer et al. (1999)] the circle grid electrode was used for this purpose; and in the experiment [Samarian et al. (2005)], copper and glass rings were used (see Sec. 4.2.2 below). Thus for the external horizontal potential it is assumed that the external confining force acting in the x-direction can be written as Fext= -yx(x - ZO), where yx QdEzxt/dx is a constant and xo is the equilibrium position of a single dust particle or two vertically aligned particles, Fig. 4.12(b); for further convenience xo = 0. The equilibrium distance xd for the case of two horizontally aligned particles, Fig. 4.12(a), appears as a result of the action of the external confining and Debye repulsion forces (note that for the horizontal alignment of two levitating particles they should be identical, ie., Q1 = QZ and mdl = md2 = md, see also below)

-

(4.21) The balance of forces in the vertical direction, in addition to the electrostatic Debye and the wake potential forces, includes the gravity force Fg = mdg as well as the sheath electrostatic force FE = QEZxt(z) acting on the grains. In equilibrium, the interparticle vertical distance zd is assumed to be small compared with the distance between the lower particle and the electrode (as well as small compared with the width of the sheath), therefore the sheath electric field for distances near the position of the equilibrium can be linearly approximated so that we write FE - mdg = -yZ(z - ZO), where yz N QdEZxt/dz is assumed to be a constant and zo is the equilibrium position of a particle of mass md due to the forces mdg and FE only. We stress that the position 20 corresponds to the actual vertical positions of the horizontally aligned two identical particles (see Fig. 4.12(a), mdl = mdz = md and Q1 = QZ = Q). On the other hand, for the vertically aligned particles, such as those in Fig. 4.12(b), the lower and upper equilibrium positions are z01 and 202 = 201 zd, respectively. In this case, the equilibrium balance of the forces in the vertical direction acting on the lower particle and the upper particle is more complicated and potentially includes the force due to the ion wake. Therefore, it can be written as D W F',1(2) (zO1(z))-md1(2)g+F~(~) (z~z-zoi)= 0, where Ff;" are the forces of the interaction between the particles due to their interaction Debye and/or asymmetric (wake) potentials @ D and/or @ w ,respectively: F ? ( Z ~ Z - Z =

+

173

Particle Dynamics in a Complex Plasma

Q d @ ~ ( I ~ l ) l d1121=zd b I and F : ' W ( ~ ~ 2- zoi) = - QdQ.o,w/(lzl)dl.lll,i=,, including either Debye or wake effects. In the case when the two interacting particles are identical, and there is only Debye-type interaction between them, the equation similar to Eq. (4.21) is obtained (4.22) In the case of the asymmetric potential, the equilibrium condition for the levitation of two identical particles gives (4.23) where zw is the distance between the minimum of the asymmetric attracting potential characterized by y2 and the upper particle (for the wake potential, zw = T L , and in the parabolic approximation 7," = Qd2Qw / d z 2 12=2,). Now, consider the first case of two horizontally aligned particles located at the positions ( - 2 d / 2 , ~ 0 ) and ( 2 d / 2 , zo), see Fig. 4.12(a). As we already noted, to achieve the horizontal alignment, the particles are assumed to be identical. First, introduce small horizontal perturbations Szi, where i = I,2, assume that the vertical displacements are zero (note that in the linear approximation the vertical and horizontal modes are decoupled) and include the phenomenological damping p d due to the friction of particles with the neutral gas. Linearly expanding the interaction forces, two oscillation modes appear. Their frequencies are given by the frequency (4.24) for the two particle oscillating in phase with equal amplitudes A1 and by the frequency

=

A2,

(4.25)

for the two particles oscillating counter phase with equal amplitudes (A1 = - A 2 ) . The both modes are always stable. The counter phase mode provides (if excited) a good diagnostic tool to determine the plasma parameters (such as Debye length and the neutral friction): by knowing the experimental values of the in-phase and counter-phase frequencies, together with the

174

Physics and Applications of Complex Plasmas

equilibrium interparticle distance, we are able to determine the unknown plasrna parameters (or at least their ratios). The next case to consider involves vertical vibrations of two horizontally aligned particles, Fig. 4.12(a). In this case, the two oscillation modes have the frequency (4.26) for the two particle oscillating in phase (A1 = AZ), and the frequency (4.27) for the two particles oscillating counter phase (A1 = -A2). We see that while the first mode is .always stable, the counter phase mode can now be unstable, depending on the ratio yz/yz. We stress that this instability arises because of the action of the confining potential in the direction perpendicular to the direction of particle oscillations. This instability can lead to disruption of an initially stable horizontal arrangement in an experiment if the relative strength of the vertical to horizontal confining potentials is changed. By introducing small vertical perturbations 6zi of the vertically aligned particles at equilibrium positions (O,zoi), where i = 1 , 2 , and expanding the interaction forces, one obtains for the case of Debye only interactions the equations analogous to the first case of horizontal vibrations of horizontally aligned particles (for simplicity, the particles are again assumed to be identical; the corresponding generalization to the case of different charges/masses is trivial). There are two oscillations modes; the first one has the frequency (4.26) for the two particle oscillating in phase with equal amplitudes A1,2, and the second mode’s frequency is given by (4.258)

for the counter phase oscillations, A1 = -A2. Again, both modes are always stable and the counter phase mode provides (if excited) similar diagnostic tool to determine the plasma parameters (such as Debye length and the neutral friction). If the asymmetry of the interaction potentip1 (e.g., the plasma wake) is taken into account, the equation of the vertical motion of the upper

Particle Dynamics an a Complex Plasma

175

particle (number 2) is in the Debye potential; motion of the lower particle now involves the wake potential. There are two oscillation modes in this case; the first one, for the particles moving in phase with equal amplitudes A1 = A2, has the frequency (4.26); the second frequency is now given by

(4.29) for the counter phase oscillations; their amplitudes are not equal in magnitude and now related by

(4.30) Again, both modes are always stable and the counter phase mode provides (if excited) a diagnostic tool to determine the plasma and the wake parameters (such as Debye length and the position of the first potential minimum). A very useful information can also be obtained by measuring the amplitude ratio of this type of oscillations. Now, consider horizontal oscillations of two vertically aligned particles. In the first case, when the particle interaction is symmetric (and of Debye type) the equations of motion are similar to the case of the vertical vibrations of the horizontally arranged particles (with the obvious change of z to z).Thus, two modes of oscillations appear, the first one corresponds to Eq. (4.26), when the particles oscillate in phase (with equal amplitudes), and its frequency is equal to the frequency (4.24). The second one is similar to Eq. (4.27), with the frequency (4.31) and A1 = -A2. While the first mode is always stable, the counter phase mode can be unstable, depending on the ratio Tz/yz. The condition for this instability is opposite to the condition of the instability of the mode of vertical vibrations of two horizontally arranged particles, see Eq. (4.27). Finally, consider the case of horizontal oscillations of two vertically aligned particles taking into account the plasma wake. The equation of horizontal motion of the upper particle in this case is the same as for the symmetric Debye only interaction, while the lower particle is oscillating in

176

Physics and Applications of Complex Plasmas

the wake potential characterized by 7," which is its horizontal strength in the parabolic approximation. Here, it is sufficient to assume that 7," is a positive constant of the order of (or slightly more than) r,", see, e.g., numerical results in Sec. 2.3.5.1 [Maiorov et al. ( 2 0 0 l ) l . For the two oscillatory modes, the frequency of the first one coincides with the frequency (4.24) while the frequency of the second mode is given by (4.32)

Now, we see another important feature: the wake potential can stabilize the possible horizontal instability of two vertically algned particles (this can be easily seen for the case y y = y r ) ; note that for the supersonic wake potential this stabilization occurs only within the Mach cone. The amplitudes of the second mode of oscillations are related by

(4.33) Thus for the asymmetric interaction potential, the second mode of oscillations does not correspond to the counter phase motions: the vibrations of particles are in phase now, with unequal amplitudes. Here, we see another powerful experimental tool to determine the character of the interaction potential experimentally: for the pure symmetric interaction potential of repulsive Debye (or Coulomb) type, the oscillations of the second mode are counter phase, while for the asymmetric repulsive-attractive potential the oscillations are in phase (with unequal amplitudes). The proposed mechanism can be related to experimentally observed phenomena, for example, for the two-particle system in planar R F discharge [Melzer et al. (1999); Steinberg et al. (2001); Vladimirov et al. (2002)], involving horizontal oscillations of two particles aligned in the vertical string [Vladimirov and Samarian (2002)l and hysteretic phenomena in the disruptions of the3orizontal and vertical arrangements [Vladimirov and Samarian (2002); Steinberg et al. (200l)], see below Sec. 4.2.2. For simplicity, in the analysis only symmetric Debye interactions of particles are considered, and the stability diagram for the two-particle system is presented in Fig. 4.13 [Vladimirov and Samarian (2002)l. There are two extreme regions: one is the region I where yz > 7, + m&4, corresponding to the vertical string unstable with respect to the horizontal motions of the particles, another is the region I11 where rz > yz + md/32/4 corresponding to the horizontal string unstable with respect to the vertical motions of the particles, as well

Particle Dynamics in a Complex Plasma

177

/

Fig. 4.13 Stability diagram of the particle arrangements for two confined particles [Vladimirov and Samarian (2002)]. The m a s of dust particle m d is denoted as M in this figure.

as the central region I1 where both structures are stable. Realization of the particular arrangement depends on the initial conditions (for example on the particle inserting technique). Since for the sheath conditions of planar R F discharge yz >> yz [Vladimirov and Samarian (2002); Steinberg et al. (2001)], the vertically aligned two-particle system is in this case in the region I of Fig. 4.13, and the instability with respect to the horizontal motions and stability with respect to the excitation of the vertical oscillations should be expected. Indeed, the self-excited horizontal but no vertical oscillations were observed in this case. Also, it was observed that the decrease of the input power leads to the stabilization of the system with respect to the horizontal motions; the decreasing input power is usually accompanied by the decreasing strength of the vertical confinement yz,while the strength of the horizontal confinement yz does not change significantly; according to Fig. 4.13, this means that the system enters the stability region 11. The hysteretic phenomena in disruption of the vertical and/or horizontal alignment of two particles observed in experiments [Melzer et al. (1999); Steinberg et al. (2001); Samarian et al. (2005)I can be qualitatively explained by Fig. 4.13 [Vladimirov and Samarian (2002)l. Let us start with

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Physics and Applications of Complex Plasmas

the horizontally arranged particles under the conditions of the (stable) region 11. Then, if to decrease the input power and therefore decrease the rn&,2/4) the reratio yz/yz the system enters (at the point yz = yt gion 111, where only the vertical arrangement is stable, that is, the transition from the horizontal to the vertical arrangement takes place. When reversing the process, the transition from the vertical to the horizontal arrangement occurs only at the point yz = yz - m&/4 and the hysteresis is observed. The strength of the hysteretic behavior A y z , z can be written as (if the asymmetric wake potential is taken into account) Ay, Ay, rn&; 4[yT - z w / z d ) ] and can be used to estimate the plasma and confinement characteristics. An interesting observed phenomenon, a “particle jump” [Steinberg et al. (2001); Vladimirov et al. (2002)] can be attributed to the point where the particle changes the region from the repulsive symmetric Debye interaction potential to the region where asymmetry in the particle interaction exists - for example, crosses the boundary of the Mach cone of the wake potential.

+

+

4.2.2

-

+

yy(1

Experiments o n particle arrangements and stability

The simplest experiment, similar to the above model, involves two particles arranged vertically or horizontally and maintained by an electrostatic confining potential in the horizontal direction and plasma sheath/gravity potentials in the vertical direction [Melzer et al. (1999); Steinberg et al. (2001); Vladimirov and Samarian‘(2002); Ticos et al. (2003)]. Due to its simplicity, the main features of the particle arrangements in the sheath region, such as the influence of the ion wake, can be directly obtained. It was first found that depending on the relative strength of vertical and horizontal confinements, the arrangement of particles changed from the vertical to horizontal ones and vice versa reveals hysteretic phenomena, that can be attributed to the influence of the wake potential. In the experiments [Samarian et al. (2005)], mono-dispersed melamine formaldehyde dust particles of diameters 2.79 (~k0.06pm) and 1.79 (f0.04 pm) were used to introduce into an argon RF discharge. The plasma was generated at pressures in the range 18-60 mTorr and a 15 MHz signal was applied to the powered electrode. To produce the sinusoidal RF voltage, a waveform generator was connected via a power amplifier with a standard matching box to reduce the reflected power. The peak-to-peak voltage was 15-100 V. The electron density is (2-9) x lo8 cmP3, the electron temperature is about of 2 eV, and the plasma potential is 40-80 V. In order

Particle Dynamics i n a Complex Plasma

179

to apply a radial confinement, a 20.9 mm diameter copper ring was used; the ring was placed on the lower electrode t o provide a confining potential of a well-shaped form. The DC voltage was applied to the ring in order t o change the radial electrostatic confining. The relative strength of the vertical and horizontal confinements depends on the controlling discharge parameters (pressure and peak-to-peak voltage). It can be also affected by changing the radial electrostatic confinement (by DC bias) without changing the vertical one. Therefore two principally different sets of experiments can be conducted.

Fig. 4.14 Structural diagram of the particle arrangements versus the peak-to-peak voltage and the pressure [Samarian et al. (2005)l. Regions correspond: I to vertically aligned particles; I1 to horizontally aligned particles; I11 to the transition region between the vertical and horizontal alignments; IV to horizontal rotation of the upper particle in the vertical alignment; V to circular oscillations of horizontally aligned particles. Triangles and squares correspond to the parameters at the transition (solid symbols stand for VHT; open symbols, for HVT): triangles, due to the pressure changing; squares, due to the peak-to-peak voltage changing. Open circles stand for rotation of vertically aligned particles, solid circles stand for circular oscillations of horizontally aligned particles.

In Fig. 4.14, the structural diagram of the particle arrangements versus the system controlling parameters (the peak-to-peak voltage and the pressure) is shown. Two main regions (I and 11) with different arrangement of the two particles can be clearly distinguished, namely, the vertical and horizontal alignments. The transition region I11 exists between these two. Also, regions with different types of rotations and oscillations can be highlighted (see regions IV and V). The first region (I in Fig. 4.14), when a pair of dust particles is vertically

180

Physics and Applications of Complex Plasmas

aligned, is found throughout the whole range of the peak-to-peak voltage for low pressures (P 5 40 mTorr); at higher pressures it occurs only in the lower voltage region (V, 5 20 V). The particle random oscillations (not correlated with each other) in the horizontal plane are typical for this region. The amplitude of these oscillations is small (0.12-0.24d) as compared to the interparticle distance d; their frequency is of the order of 10 Hz. The particles arranged in the horizontal plane (region I1 in Fig. 4.14) have been found when P and V, achieved the upper end of the range (typically, for P > 40 mTorr and V, > 50 V). It was observed that two particles aligned in a horizontal string in most cases oscillate in phase as a rigid body. Depending on the control parameters, the amplitude of these oscillations is of the order of the interparticle distance. The frequencies of these oscillations are a few Hz. The transition (region I11 in Fig. 4.14) between the vertical and horizontal alignment was observed at intermediate gas pressures within the range 35-50 mTorr and peak-to-peak voltages of 20-50 V. At the first stage of the transition, the angle cx between the vertical direction and the main string axis changed continuously, and at the second stage it changed discontinuously (the lower particle “jumped” to the new position). The transition can be induced by changing either V, or P . The triangles in Fig. 4.14 represent the transition occurring due to the pressure changing, and the squares represent the transition occurring due to the peak-to-peak voltage changing. The reading were taking at the point where the “jump” occurred. The clearly manifested hysteretic phenomena were observed for this transition. The transition from the vertical to the horizontal arrangement always took place under the condition different from that of the reverse transition from the horizontal to the vertical arrangement. The solid triangles and squares in Fig. 4.14 represent the vertical-horizontal transition (VHT), and the open triangles and squares correspond to the horizontal-vertical transition (HVT). One can clearly see that HVT is “better defined” than VHT (such as a “boundary line” between the region I and virtual region I11 is created by threshold parameters). Moreover, the thresholds for KVT triggered by changing the pressure and peak-to-peak voltage are close to each other while the P- and V,-triggered thresholds for VHT are different. An interesting behavior was discovered for the vertically aligned particles when the pressure was about 30-35 mTorr and V, achieved the upper end of the range (V, > 40 V). The transition between the random oscillations (region I) and the rotary motion (region IV) was observed. During the transition process, the amplitude of the oscillations increased more and

Particle Dynamics in a Complex Plasma

181

more, then particle started t o oscillate in phase until the upper particle started to rotate around the lower one, which even seems t o be stabilized by the rotation of the upper particle. The radius of the rotation of the upper particle is of the order of the interparticle distance (d 200-600 pm). The frequencies of these rotations depend on the plasma parameters and are about 0.1-2 Hz. This motion, however, is not a uniform rotation but with an oscillatory angular velocity. By analyzing the z-position plotted vs the y-position of the particles, it was concluded that while the trajectories of the particles were on a rotational orbit, the centers of the orbits were oscillating back and forth. The particles were rotating on the opposite sides of the their orbits. The overall direction of these rotations was counterclockwise. To analyze this type of rotation, it is necessary to calculate the angle of both particles with respect to the center of mass. Taking into account that the particles are monosized (the same mass) and spherical, the center of mass of the particles was determined by averaging their spatial position. It was observed that the particles are always in phase with each other. This is an indication of the “rigid” structure maintained during the rotation of the system. There is no sharp transition between the vertical and horizontal arrangement in this case (between regions IV and V in Fig. 4.14). When the pressure increased, the particles placed in the horizontal plane. When the jump occurred, the rotation degenerates into the circular rotary oscillations. By changing the controlling parameters of the discharge, the conditions of the dust particle-plasma systems are changed. Thus practically all the values characterizing the dust particle interactions (such as the particle charge, the plasma screening length, the parameters of the ion wake) are changed. Nevertheless, the main features of the particle arrangements and transitions can be seen. The first one is that the particle arrangement depends on the ratio of the confining potential strengths. It should be noted that region I1 corresponding to the horizontal arrangement (and observed for high pressures and peak-to-peak voltages) takes place for high values of 7; = (-yZ/rz)’/2. The transition (HVT) to the vertical arrangement occurs with the decrease of 7; when 7,” 4. The reverse VHT happens at a different 7; ranging approximately between 3 and 7. Furthermore, the hysteresis in the cyclic HVT and VHT, i e . , in the way the transition between the vertical and horizontal alignments occurs for, e.g., increasing or decreasing V,, was observed (see Fig. 4.14). TOelucidate the influence of 77; on the particle arrangement, it would N

-

182

Physics and Applications of Complex Plasmas

be advantageous to change the confinement strength ratio without affecting other system parameters. The best way t o realize that is to change the horizontal confinement by applying an additional DC bias to the confining electrode (which does not affect the discharge [Raizer (1991)]. Thus 7; decreases with increasing v b . The particle arrangements can therefore be investigated for various qz without changing the main discharge parameters. The experiment was performed for P = 45 mTorr, V, in the range 150-300 mV, and the (negative) bias voltage V, in the range 0-20 V. The dust particles used by [Samarian et al. (2005)] were 2.79 pm and 1.79 p m (naturally, the lighter particle levitated in the higher position while the heavier particle levitated in the lower position) in diameter.

Fig. 4.15 Hysteresis angle a when changing V b . Open circles stand for increasing Vb; solid circles, for decreasing vb [Samarian et al. (2004)l.

Dependence of the angle a vs V b is presented in Fig. 4.15. Starting from the zero bias, a! slowly increases with increasing v b from 48" to 62" until V b = 8.8 V, and then jumps up to a! = 86" corresponding to the lower particle moving below the upper and to formation of the vertically aligned arrangement. When decreasing v b , the angle a practically does not change until v b = 5.9, and then jumps down to a = 64" (when disruption of the vertical alignment occurs). After that, the angle slowly decreases back to the original value a t v b = 0. Similar behavior was observed for other values of V, (with different values of the threshold V b ) . From the experimental data, we can clearly see that the arrangement of a pair of dust particles indeed depends on q:. Figure 4.16(a) shows the values of Oz,zat the transitions. The straight line corresponds t o 0, = 0,. All transitions correspond to q: > 1 ( i e . , 0, > aZ).We note that all VHT

Particle Dynamics in a Complex Plasma

183

Fig. 4.16 Frequencies O z , z corresponding to the transitions when: (a) the pressure and peak-to-peak voltage were changed; (b) the DC bias voltage was changed. Solid squares correspond to VHT; open squares, to HVT [Samarian et al. (2004)l.

correspond to values of 7 > .I," > 3.8 that are higher than those for HVT. All HVT occur for nearly similar values of qg 4; this is reflected by the fact that all these transition points are almost on a straight line. For VHT, the value of q: where the transition occurs, cannot be generally specified. If to assume, that the interaction of vertically arranged dust particles is symmetric (ie., the upper particle acts on the lower one in the same way as the lower one acts on the upper one), the transition frequencies differ only by the friction of dust particles with the neutral background: ie., 7: # 1when the transition from the horizontal to the vertical (or vice versa) arrangement takes place. However, the influence of the ion wake makes the N

184

Physics and Applications of Complex Plasmas

interaction of the vertically arranged particles highly asymmetric. The friction and the wake effects can contribute to the hysteretic behavior although for different discharge characteristics. Indeed, in the case of higher pressures when the ions are slowed down because of friction with the plasma neutrals, the wake effects are generally weaker. In this case, the friction (which is, on the opposite, higher) can define the hysteresis. In the opposite case of lower pressures when plasma ions are faster and the wake effects are more pronounced, the influence of friction can be negligible. In experiment, we see (Fig. 4.14) that the width of the hysteresis increases with decreasing pressure thus indicating on stronger influence of the wake phenomena. This is natural to expect because of the essentially nonlinear character of the wake parameters (while the dependence on the friction is linear, see Fig. 4.13). However, the wake also affects the stability of the arrangements further away from the transition regions. Indeed, for the vertical arrangement, the lower particles is stabilized by the excessive positive ion charge of the wake. On the other hand, in the case of the horizontal alignment, the wake contributes to disruption of the arrangement. The different role of the wake (stabilizing vs destabilizing) for different arrangements leads to the hysteretic behavior of the lower particle in the cyclic (HVT-VHT) transitions. Furthermore, HVT is more pronounced (or better defined for certain system parameters) than VHT, see Fig. 4.16(a). This can be also attributed to the different influence of the wake: when the system is first aligned horizontally, that transition occurs because of repelling between the particles combined with attracting t o the wake ions, while in the case of the vertical alignment the position and stability of the lower particle (located in the wake) is affected by the vertical and horizontal profiles of the wake potential. The wake characteristics strongly depend on the parameters of the discharge and can fluctuate significantly. Thus a n attempt to trigger VHT by changing the discharge pressure and/or power may succeed with a wider range of 17;. This conclusion can be proved by analyzing data of VHT and HVT in the case of the transition triggered by an additional bias Vb. Figure 4.16(b) shows the values of at these transitions. Again, the straight line corresponds to R, = 0,. We see that both VHT and HVT are well defined now and each occur for close values of the respective 172. We also observe the hysteresis in this case when Vb is cyclic changed.

Particle Dynamics in a Complex Plasma

4.2.3

185

Self-excitation of vertical motions of dust particles

Enhanced level of dust particle oscillation has been reported in many experiments [Molotkov et al. (1999); Nunomura et al. (1999); Samarian et al. (2001~);Thomas and Watson (2000)]. As was already mentioned in Sec. 2.1. the dust particles achieve electrostatic equilibrium with respect to the plasma by acquiring a negative charge. This charge is not fixed but is coupled self-consistently to the surrounding plasma parameters. For this reason; new instabilities can develop in a complex plasma, leading to different wave modes and oscillations. One of the most interesting type of oscillation is the vertical vibrations of dust particles. Vertical dust density fluctuations were observed in current driven DC discharge experiments [Molotkov et al. (1999); Samarian et al. (20Olc)]and the developed instability was attributed to Buneman ion-streaming instability (see Sec. 6.4.3). The self-excited oscillations of individual particles were reported in filamentary DC discharge experiment [Nunomura et al. (1999)]. In experiment [Samarian et al. (2001b)],the self-excited vertical oscillations of dust particles trapped in the sheath region of an RF discharge plasma were investigated. It was found by [Samarian et al. (200lb)l that when the pressure was decreased below a critical value the dust particles began to oscillate spontaneously in the vertical direction. Figure 4.17 shows typical

Fig. 4.17

Side view of a self-excited vertical oscillation of dust particles [Samarian et a1 (2001b)l. Input power P= 35 W; pressure (a) 2.9 Pa, (b) 4.0 Pa.

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Physics and Applications of Complex Plasmas

images of the vertical oscillations a t different pressures. The amplitude and frequency of the oscillations are several millimetres and greater than ten Hz, respectively. When the R F input power was decreased, the oscillation amplitude was found to increase. Figure 4.18 shows the dependence of the oscillation amplitude on gas pressure and R F power. For pressures below 4.5 P a the oscillation amplitude increased dramatically. This increase is greater for lower RF powers.

Fig. 4.18 The variation of the vertical oscillation amplitude (mm) as the function of the RF power and the pressure [Samarian et al (2001b)l.

For RF powers of more than 80 W the effect was not observed. In this case, when the pressure was decreased there were similarly increasing amplitudes of both horizontal and vertical oscillations. This is consistent with the well-known fact that when the power is increased or the pressure is reduced in a dust plasma system a melting transition from ordered to fluid states can be induced, see Sec. 5.3. The frequency and amplitude of oscillations were independent of the number of particles in a monolayer structure consisting of a few to a few tens of particles. When the number of particles in the structure reached hundreds, the dust particles formed several layers where the self-excited oscillations were also observed. The excitation of vertical vibrations and the transition from one to several layers (triggered by them) is related to the self-confinement instability considered above. To excite the oscillations the energy input must be sufficient t o over-

187

Particle Dynamics in a Complex Plasma

come the gradual damping by friction. One possible way for a particle to gain energy is the delayed charge effect, proposed by [Nunomura et al. (1999)l. The energy gain for the one oscillation cycle can be estimated as Wgain = AQEL, where E and L are respectively the electrostatic field at the equilibrium height and oscillation amplitude, and AQ is the difference in the charge for a particle moving up, compared to a particle moving down. This difference can be written in the form AQ = AQ"4 exp(-v,h/wo) where wo is the frequency of the dust oscillation, vch is the steady-state dust charging frequency, and AQeQis is the difference of the equilibrium charge in the extreme points of oscillations. The energy losses, which primarily occur due to friction with gas molecules, can be estimated as W~, = 7rr~udm,u,P,L/kT,, where r d and ud are respectively the radius and velocity of the dust particles, u, and m, are respectively the thermal velocity and mass of the gas molecules, and P, and T, are respectively the pressure and temperature of the neutral gas. We can obtain the velocity of the dust particles and the oscillation amplitude from analysis of the images. To estimate the values of E and Q we use the model by [Vladimirov and Cramer (2000)l described in Sec. 4.1.1. Based on data obtained from probe measurement (T, = 1.8 eV, n, = 1.4 x 10' cmP3) the model was used to calculate of the spatial dependencies of E and Q for a pressure of 4 Pa and an electrode potential of -10 V. The results indicate that at the levitation point E = 3.5 V/cm, Q = 4350e and AQ"q = 0.1Q for the oscillation amplitude L = 1 mm. For the experimental condition [Samarian et al. (2001b)l (Ar, P = 4 Pa, T, = 300 K, rp = 1 pm, L = 1 mm, E = 3 V/cm, and AQ = 0.lQ) we obtain that energy losses is W1,, lo-'' J. In our case the oscillation frequency is w0/27r 14-16 Hz, the charging frequency vch lo5 s-'. This means that the above rough estimate gives us that only negligible portion of energy (less than 1OpZo J) can be gained. For a more detailed analysis of this mechanism consider the equation of motion for a single particle in the vertical direction

-

-

-

(4.34) where = (4Pr~/3md)(27rm,/T,)1/2 is the damping rate due to neutral gas friction, and F, is the net force acting on the particles. [Ivlev et al. (2OOOb)l presented a model of dust particle vertical oscillation taking into account for the net force only electrostatic F, = QE and gravity Fg = mdg forces. They suggested that the threshold for the instability due to the

188

Physics and Applications of Complex Plasmas

charging delay in the case of the regular charge variation and the stochastic charge fluctuations can be evaluated as: for the regular charge variation (4.35)

where Im w is the damping rate, the prime denotes the derivative at the equilibrium point z = zeq. We note that the instability develops when Imw > 0, and, correspondingly, Wgain> Wloss. For the stochastic charge variation (4.36) where reis the energy damping rate, 0 is the dimensionless dispersion of the charge distribution. Taking into account the values of A Q / Q 0.1 and A E / E 0.2, we obtain that the second term in the right side of equation (4.35) is about 2 x lo-’ s-’. This means that we should expect the instability when the pressure is less than 0.01 Pa. A similar analysis of equation (4.36) for CT = 0.1 gives the pressure threshold for stochastic fluctuation as 5 mPa. We can consider the net force in Eq. (4.34) taking into account the ion drag force

-

-

(4.37)

where v, is the velocity of ions in the sheath, and XI and contributions given by

x2

are given by

(4.38)

The resulting equation can be analyzed numerically. However, some analytical results can be derived in several limiting cases for the ion drag force. If we consider only the orbit force contribution to Fi = 7re2ni(z)Q2(z, t ) / m i v z , the damping rate is given by (4.39)

For Fi including only the collection force, Fi = 27ru2rnini(z)v,2(z),the

Particle Dynamics in a Complex Plasma

189

damping rate is

(4.40) For both cases the pressure threshold is less than 0.4 Pa (for purposes of estimation we use the values An,/ni 0.2 and Avi/v, 0.35), in disagreement with the experimental results [Samarian et al. (2001b)l. A similar value for the pressure threshold can be obtained if to use the model considered by [Fortov et al. (2000a)l for instability due to charge variations in the presence of external charge-dependent forces together with the ion drift effect. Thus the observed self-excited oscillations cannot be explained only on the basis of the time dependence of the dust particle charge. Other effects, such as spatial charge dependence and external confinement should be taken into account for the corresponding theory. N

N

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Chapter 5

Structures and Phase Transitions in Complex Plasmas

Complex plasma is a system where self-organization occurs in many ways. Complex plasma systems represent a unique mutual arrangement of two ensembles of charged species, namely an ionized gas (plasma) and a solid disperse phase (also termed colloidal dust grains). Depending on the specific environment (for example, it can be space or laboratory plasmas) and prevailing conditions of the system, the two charged particle species can be arranged differently with respect to each other. For example, when fine dust particles nucleate in a laboratory discharge, a weakly interacting dust subsystem is fairly uniformly dispersed over the plasma bulk. However, under the action of various forces exerted by the plasma subsystem the originally uniform dust clouds can evolve into unusual structures of dust particles in a plasma such as the self-organized dust clouds and voids with sharp boundaries (Sec. 5.1). The clouds of dust particles change the sheath properties in a self-similar way leading to formation of the so-called dust-plasma sheath. One of the most striking examples of the self-organization of micrometer-sized particles was a discovery of dust-plasma crystals and liquids by a number of research groups around the world [Thomas e t al. (1994); Chu and I (1994); Melzer et al. (1994); Hayashi and Tachibana (1994); Barkam and Merlin0 (1995); Fortov e t al. (1996b); Fortov e t al. (1996c)l. Under certain conditions, the particles form regular arrays exhibiting various lattice-like structures (Sec. 5 . 2 ) . In most of the laboratory experiments, the gravity pulls the relatively heavy particles down in the sheath until the gravitational force (as well as other forces acting in the same direction, e.g., the ion drag force) is balanced by the sheath electric field force. Because of the clear distinction between the horizontal (parallel to the electrode) and vertical directions (where the plasma is non-uniform

191

192

Physics and Applications of Complex Plasmas

and the ions are flowing towards the electrode), the system structures in a quasi-two-dimensional manner, with often clear two-dimensional features, e.g., the hexagonal-type lattice cells [Thomas and Morfill (1996)l. Phase states of the structures observed can change in the processes of phase transitions (Sec. 5.3). The ability to trace the particles directly makes the complex plasma a good model system for visualizing the kinetics of phase transitions. The stability of complex plasma systems, in contrast to the “standard” solid and liquid matter, is influenced by a riiimber of intrinsic as well as extrinsic parameters that induce transitions between different phases. These parameters are different for different kinds of complex plasmas. Therefore a wealth of phenomena associated with different phase states and transitions between them that is triggered by changes of such parameters as the pressure. input power, UV flux, and impurities, is observed in various complex plasmas. The richness of the phase behavior also makes the ordered structures in a complex plasma interesting from the fundamental point of view.

5.1

Self-organized Plasma-Particle States

The feature that unifies such structures as dust-plasma-sheaths (see Sec. 5.1.1) and dust voids (Sec. 5.1.2) structures is that they are intimately related to the open character of a complex plasma, namely, they are created and supported not only by interactions of charged particles but also by processes in the surrounding plasma. One of the most important features of these structures is the sharp boundaries between the regions occupied by the particles and the dust voids. Obviously, such distinctive inhornogeneities can be created and supported by external sources of energy; in the case of voids this is the ionization source. Below, we discuss the physics of these structures in more details. We also present the theory of dust voids (Sec. 5.1.3) explaining the main void features.

5.1.1

Dust-plasma sheath as a self-organized structure

In the presence of the particles in the near-wall region, there are two physically different possibilities: 1) when the dust influence is relatively weak (typical for a rarefied dust component), the sheath electric field is mostly determined by the plasma-wall boundary conditions, and 2 ) when the dust strongly influences all the parameters in the sheath thus creating a specific

Structures and Phase Transitions in Complex Plasmas

193

structure in the near-wall region. Here we focus on the latter possibility [Tsytovich et al. (1999)l. The formulation of the problem is not trivial since the main parameter, viz. the electric field in the sheath, deiends not only on the boundary conditions, but also on the dist,ribution of tlie particles which therefore must be found simultaneously with the electric field distribution. Paradoxically, the new physics introduced by the dust can make the problem somewhat simpler to describe mathematically. One of the most important effects is that the grains absorb plasma particles (which recombine on the grain surfaces) thus creating a sink for the plasma species. This makes the dust-plasma sheath highly dissipative. If the plasma particle mean-free path is shorter than the width of the dust layer, the cloud constitutes a dispersed virtual wall, in addition to the real wall bounding the plasma. Thus the Bohm criterion [Chen (1984)l is changed in the presence of dust, and the dust-plasma sheath appears as an open dissipative system where the plasma fluxes are created consistently by the dust structure and the wall. Simple estimates show that the width of the dust-plasma sheath can significantly exceed the width of the plasma sheath, however, the self-organization restricts its size [Tsytovich (1997)l. In the absence of dust, there are two regions in the near-wall plasma [Chen (1984)l: the plasma sheath itself where the main drop of the electric field potential occurs, and the pre-sheath region where the drop of the potential is rather small and where the ionization and the ion acceIeration take place. In the presence of dust, the main drop of the electric field potential is in the dust layer. Here, the simplest model is discussed where the ionization is only in the pre-sheath, and there are three different layers: the plasma boundary layer (containing no dust, but the ionization is absent), the dust cloud, and the wall-plasma layer (containing no dust; this layer does not always exist) [Tsytovich et al. (1999)]. It is important physically that tlie dust layer can have sharp boundaries. The drop of the electric field potential is small both in the wall-plasma and plasma layers. The next problem is to determine the domain in the parameter space where the equilibrium state is possible. Indeed, the dust-plasma sheath can support only specific Mach numbers of the ions flowing toward the dust cloud and the wall, thus creating the ranges of allowed ion flow velocities. Moreover, the self-organized dust sheath can contain only a certain number of dust particles per unit surface of the sheath. These results are important for dust-plasma experiments, especially for those on dustcrystal formation. The electrons are assumed to be Boltzmann distributed

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Physics and Applications of Complex Plasmas

noexp(ecp/T,), where no is the unperturbed electron density in the region where quasineutrality holds (Le., in the pre-sheath), and cp is the electric field potential (which is almost zero in the pre-sheath). Then the electric field distribution should be found, with the total electrostatic field E including the fields created by the grains as well as the field due to the charged wall. The forces acting on the particles are the electric field force -ZdeE, where Zd is the dimensionless dust charge, and the ion drag force. For simplicity, other forces such as the gravity and thermophoretic forces, etc., are ignored, and the electron and ion temperatures are assumed to be constant in the sheath region. The equations to solve are those for the ion force balance, the stationary ion continuity equation, and the balance equation for the electron and ion charging currents onto the grain surface. These equations are complemented by Poisson’s equation to determine P, = ndZd/ni. In the case of a subthermal ion flow, the jump of P, at the boundary of the sheath is small, and the set of model equations, using the quasineutrality condition, is converted to a set of three equations for the changes of the ion density and flow as well as for the change of the dust charge. This case is in fact not much interesting for applications since according to the Bohm criterion the ion flow is expected to be at least superthermal. Below, we consider only the latter limit which covers both the subsonic and supersonic cases. The dust-plasma sheath can have sharp boundaries; solutions outside those boundaries must be found without dust. These solutions, which are matched at the boundaries with the solutions inside the dust cloud, also depend on the boundary conditions a t the walls. Thus the result inside the dust cloud is a function of the plasma-wall boundary conditions as well. There are several alternatives, in particular: 1) the boundary conditions on the wall can be satisfied inside the dust sheath; in this case the dust cloud exists everywhere up to the wall and the grains cannot levitate inside the discharge, 2) the boundary conditions can be satisfied outside the dust sheath, e.g., for the specific values of such parameters as the ion flow velocity uo far from the sheath; in this case the dust levitation is possible. In the presence of one wall, the dust-plasma boundary conditions are different for the plasma side of the dust cloud and for the side oriented to the wall. However, the presence of the wall influences the conditions at the opposite boundary by affecting the ion flow. The ion continuity, the balance of forces equations, and the adiabaticity assumption for the electrons in the plasma bulk are invoked to find the first integral of Poisson’s equation and t o obtain the electric field. ne =

195

Structures and Phase Transitions in Complex Plasmas

Table 5.1 Two first regions of the allowed Mach numbers for the ion flow (all Values between Mmin and Mmax are allowed). The results are given for argon Plasma [Tsytovich et al. (1999)].

Temperature ratio 7

Ti/Z 0.001 0.01 0.001 0.01

1

size

First zone

a/XDi

Mmin,I

0.5 0.5 0.1 0.1

0.708 0.710 0.709 0.712

Grain

First zone Mmax,~

1.094 1.096 1.104 1.106

Second zone Mmin,~

1.742 1.743 1.746 1.748

Second zone Mmax,2

2.550 2.297 3.363 3.342

The set of the boundary conditions is solved numerically to determine the parameter ranges allowed by the boundary conditions [Tsytovich et al. (1999)l. These ranges correspond to the equilibrium states of the dust sheath. The numerics are performed for 1 > (Ti/T,) > 0.001, 0.01 < M < 10, where M is the Mach number of the ion flow, and 0.001 < ( u / X o i ) < 0.5 for various gases (hydrogen, argon, and krypton). It was found that the ranges of the plasma parameters corresponding to the equilibrium states exist. Table 5.1 gives a summary of the first two allowed zones for the initial Mach number in argon plasmas with various values of T = Ti/T, and a. It can be seen that the allowed values of the Mach number change from 0.7 to 3, ie., the sheath can be either subsonic or supersonic, in contrast to the Bohm criterion in the absence of dust. This is not a surprise since the charged dust layer strongly affects the electric field distribution in the sheath. Important experimental consequences are that by changing the system control parameters, one can easily obtain or exclude the equilibrium dust-plasma sheath (the latter can be desirable in applications to fusion plasmas where high-power loading can lead to significant dust contamination from the walls. The obtained equilibrium ranges look like the quantization zones. The model relations describe the balance of the electrostatic pressure, the thermal pressure of electrons and ions, and the ram pressure of electrons. In fact, they describe the electrostatic confinement of electrons, ions, and dust in the sheath similar to the widely known magnetic confinement, where the magnetic pressure is balanced by the thermal pressure of the plasma. The set of equations for the quasi-neutral sheath is solved first, followed by the solution of more accurate equations without the quasineutrality assumption. The calculations are done for the broad ranges of the Mach

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numbers and temperature ratios. The calculations start at the dust-plasma boundary of the sheath followed by solution of the nonlinear model equations in the region occupied by dust. Then the solution is found in the region between the dust cloud and the wall (if the levitation criterion is satisfied). The solutions at the first boundary are used to obtain the drop of the potential in the region. The total thickness of the plasma-dust sheath can be calculated as a. sum of three regions: the plasma layer between the pre-sheath and the dust cloud, the dust cloud, and the region between the dust layer and the wall.

Fig. 5.1 The dust layer in argon plasma [Tsytovich et al. (1999)]: the normalized ion density n/no (solid curves; the upper line gives the ion density as a solution of the wallboundary condition, the lower solid line gives the actual distribution), the normalized electron density n,/no (dotted curve), the normalized dust density ndaTe/nOe2 (dashdotted curve), and the parameter Pi = n d Z d / n o (the dashed curve). The initial Mach number is Mo = 1.743 corresponding to the second allowed range (see Table 5.1). Other parameters are: Ti = O.OOlT,, a = 0 . 5 X ~ i .

Figure 5.1 shows the result (without the quasineutrality assumption) for the dust layer in argon where the boundary with the plasma is in the left and the boundary with the wall-sheath layer is in the right. The upper solid line gives the ion density as a solution of the wall-boundary conditions; since it does not intersect the lower solid line (representing the ion density in the sheath), the dust levitation is possible. The thickness of the dust layer is h d = 8 . 2 0 a 2 / X ~ i .Note the sharp peak of the dust density near the wall. The total number of dust particles per unit area is Nd = 1.96, the drop of the potential is Acpd = -5.878Te/e. The calculation also gives the values of the parameters a t the near-wall boundary of the dust layer: n d u , = 0.015~~0, ?zn,,dw = 0.003n0, and Mdu, = 2.720. The results for the

Structures and Phase Transitions in Complex Plasmas

197

thickness and the drop of the potential in the plasma-wall layer, as well as the final Mach number at the wall are Axw = 0.28U2/X~ilApw = -0.004Te/e, and Mw = 3.396, respectively. The thickness of the plasma layer between the pre-sheath and the dust layer is Axps = 0.1h2/XDi, and the potential drop is AppS= -0.104Te/e. The total width of the plasma, the total potential drop dust sheath in this ease is Ax,d = 8 . 5 9 a 2 / X ~ iwith a p p d = -5.986Te/e.

Fig. 5.2 The same as Fig. 5.1, but for Mo = 0.708 corresponding t o the subsonic ion flow in the first allowed range (see Table 5.1) [Tsytovich et al. (1999)].

Figure 5 . 2 shows the result for argon plasma when the initial Mach number is the lowest allowed in the first range. The solid line gives the ion density distribution, other lines are the same as in Fig. 5.1. The thickness of the dust layer is h d = 30.0a2/Xoi. Note the completely different (shocklike) structure of the dust density distribution as compared with Fig. 5.1. The total number of dust particles per unit area is Nd = 0.61, the drop of the dimensionless potential is a ( p d = -3.481Te/e. The parameters a t the near-wall boundary of the dust layer are: n d w = O . O l l n 0 , n e , d w = 0.009n0, and h f d w = 2.381. The thickness and drop of the potential in the plasma-wall layer, as well as the final Mach number a t the wall are Axw = 0.69a2/X~i,Apw = -0.007Te/e, and Mw = 3.593, respectively. Solution of the equations for the near-wall layer gives its thickness at the point where the ion density curve reaches the value corresponding t o the ion density a t the wall. The results for the total thickness, total drop of the potential, and the final Mach number a t the wall are Axw = 8.481a2/Xoi, A p w = -5.882Te/e, and Mw = 3.396, respectively. The final problem is to find the solution in the pre-sheath to find the thickness of the plasma layer

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between the pre-sheath and the dust layer, and the potential drop. The thickness of the plasma layer between the pre-sheath and the dust layer is Axps = 0.108a2/ADi, with the potential drop in it ApPs = -0.104Te/e. The total width of the plasma-dust sheath is Ax,d = 8 . 5 8 9 a 2 / A ~ i with , the potential drop in it A ( p p d = -5.986Te/e. Thus the most important general properties of the plasma-dust sheaths as appearing from the extensive numerics are: 0

0

0

0

0

the plasma-dust sheath is a self-organized structure determined by the limited set of external parameters such as the initial Mach number of the ion flow, the size of the dust grains, and the electron to ion temperature ratio; the plasma-dust sheath can exist only within the certain ranges of the Mach numbers; the plasma-dust boundaries are sharp in the limit of the cold grains (the zero dust, temperature); the size of the plasma-dust sheath is much larger than the usual sheath size when a 2 / X o i >> AD,; for the sufficiently high Mach numbers, the grains cannot levitate; the size of the particles influence the sheath structure in such a way that the smaller the size of the grain is, the less probable is the levitation. Also, when the mass of the plasma ion is smaller, the range of parameters corresponding to the absence of the dust levitation is wider.

The possibility to find the solution is related to the new (compared to the usual plasma-wall problem in a dust-free plasma) physics involved, which is particularly because of the high dissipation introduced in the system by the dust grains. The solutions describe the electrostatic equilibrium so that the the electrostatic confinement appears due to the balance of the electrostatic force with the pressure force including the ram pressure of the flowing ions. The balance of the electrostatic pressure, the thermal pressure of the electrons and ions, and the ram pressure of the flowing ions in the dustplasma sheath is similar to the well known magnetic confinement balance, where the magnetic pressure force is compensated by the plasma thermal pressure force. It is important, however, that this electrostatic confinement is not imposed externally and is created by the plasma-dust sheath itself in a self-similar way, which can be considered as a new phenomenon of self-confinement.

Structures and Phase Transitions in Complex Plasmas

5.1.2

199

Dust voids: observations and main mechanisms

Dusty plasmas in a gas discharge often feature stable dust voids, i.e., a region free of dust. This occurs under conditions relevant to both plasma processing discharges and plasma crystal experiments [Samsonov and Goree (1999b); Morfill et al. (1999); Praburam and Goree (1996); Melzer et al. (1998); Dahiya et al. (2002); Annaratone et al. (2002); Rothermel et al. (2002); Mikikian et al. (2003); Nefedov et al. (2003)l. In particular, that as particles in a sputtering plasma grow in diameter, an instability develops in two stages [Praburam and Goree (1996)l. At first, there is a sudden onset of a “filamentary mode” oscillating with the frequency 100 Hz, in which the ionization rate and dust number density are modulated. This mode develops into the second stage, a void. N

Fig. 5 . 3 Image of the dust void in a carbon sputtering plasma [Samsonov and Goree (1999b)l. A dusty plasma is between the two electrodes. Darker grays correspond to higher particle number densities. Note the sharp boundary between the void and the surrounding dust cloud.

The void is usually a stable centimeter-size region completely free of dust. The void has a sharp boundary with the surrounding dusty plasma, as shown in Fig. 5.3 [Samsonov and Goree (1999b)l. The electron density and ionization rate are enhanced in the void, compared to the surrounding dust cloud. Quite similar voids are also found in a silane deposition plasma [Melzer et al. (1998)l. Using much larger particles to form a stronglycoupled dusty plasma in a microgravity, one can produce a centimeter-size void (Fig. 5.4) [Morfill et al. (1999)l. For some operating conditions the voids exhibit a 1-1.5 Hz relaxation oscillation, termed the “heartbeat”, in which the void shrank drastically and then expanded to its original size. The experiments are usually carried out in low-temperature plasma discharges and feature voids with sharp boundaries.

200

Fig. 5.4

P h y s i c s a n d Applzcations of Complex P l a s m a s

Cross-sectional view of the dust void structure in the microgravity experiment

[Morfill et al. (2002)l.

Two mechanisms are required to explain the voids: a force balance on a dust particle and maintenance of a sharp boundary. The balance of the electrostatic and ion drag forces involves the electron depletion and the electron-impact ionization. The electron depletion. i. e., the reduction of the electron number density in the dust cloud, is due to absorption on the particles. This can reduce the electroii-impact ionization rate within the cloud. In a void, the coniparatively higher ionization rate leads t o an electric field that is directed outward from the void’s center. This yields an outward ion flow, which exerts an outward ion drag force on the dust particles, as sketched in Fig. 5 . 5 . In equilibrium, there is a balance of forces on a dust particle: an inward electrostatic force and an outward ion drag force. The second mechanism required to explain the voids is the maintenance of a sharp boundary. Sharp boundaries is a common feature of dusty plasmas. For example, in etching plasmas, dome- and ring-shaped dust clouds are formed above electrodes, and these clouds have sharp edges [Selwyn (1994)l. Planetary rings and noctilucent clouds in the lower ionosphere also have sharp boundaries [Havnes e t al. (1990a)I. Advanced theories of dust voids [Goree e t al. (1999); Tsytovich e t al. (200l)l include the plasma fluxes and ionization processes and are able to explain the sharp boundaries,

Structures and Phase Transitions in Complex Plasmas

with electron impact ionization

201

without electron impact ionization

Fig. 5 . 5 Sketch of a void (left) and its converse (right) [Goree et al. (1999)l. In the presence of the electron-impact ionization, a positive space potential develops, creating an outward ambipolar electric field that drives ions outward, applying an outward ion drag force, which can maintain a void. In the absence of the electron impact ionization, for example in a space plasma where plasma is generated far away, the dust cloud forms with its boundary sustained by an inward ion drag force driven by an inward electric field.

In experiments, the void arises from a uniform dust cloud as a result of an instability [Samsonov and Goree (1999b)l. This instability develops as follows. Suppose that in a uniform dust cloud in a gas discharge, there is a spontaneous fluctuation in the dust number density. In the region of reduced dust density, there is less electron depletion by the dust. This leads to a higher electron density there, and hence to a higher ionization rate. This high ionization spot develops a positive space charge with respect to the surrounding medium. The resulting force balance involves two forces acting on the negatively-charged particles: the inward electrostatic force and the outward ion drag force, Fig. 5.5. For a small particle size, the inward force will dominate and fill the spot again with the dust, and the fluctuation will disappear. This is the initial stable situation. However, if the particle size exceeds a critical size, the outward ion drag force exceeds the electrostatic force. The region of the reduced dust density will then expel more dust particles, and the fluctuation will grow. This is an instability that yields the ‘filamentary mode’. The threshold for the instability is determined by the particle size and the electric field strength. The particle size is an independent parameter, whereas the electric field is determined self-consistently by the electron and ion transport mechanisms and Poisson’s equation. This initial stage of the linear growth of the instability can be modeled [D’Angelo (1998)]. Then the mode becomes nonlinear and the instability saturates. The final nonlinear state can be a stable void. We note here that the linear theory cannot answer the question what is the

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Physics and Applications of Complex Plasmas

final nonlinear stage, is there a void or another nonlinear structure such as that observed at the filamentation stage. Only fully nonlinear treatment can answer this question.

5.1.3

Modeling of dust voids

5.1.3.1

Collisionless dust voids

Discuss how a stable equilibrium of the voids can be modeled in the collisionless case [Goree et al. (1999)]. The actual mechanism that led to the formation of the void, whether by an instability as described above or by some other process, is not considered here. Since the equilibrium develops through nonlinear effects, we use a nonlinear treatment of the relevant fluid equations. The only collisions taken into account are the dust-electron and dust-ion collisions inside the dust region. This model can be used to predict the conditions for the void’s existence, its size, and its sharp boundary with the surrounding plasma. The void existence requires a local source of ionization of a background gas. This means that the problem is mostly applicable to gas discharges, where the ionization is due to electron impact, although photo-ionization and other sources of ionization could have a similar effect. In the absence of the local ionization, the converse of a void appears, as sketched in Fig. 5 . 5 . The latter problem is applicable, for example, to astrophysical dust clouds [Tsytovich (1997)l. The electron depletion within the dust cloud causes the cloud to acquire a negative space charge, which attracts ions. The developed ion flow causes the void to develop. The one-dimensional model is based on the set of fluid equations, Poisson’s equation, and the charging equation for the dust. We assume that the problem is symmetric around x = 0, which is the system’s center. The electric potential and the ion flow velocity are zero at the center. If the void appears, its center is at x = 0. Thus, two regions are modeled: the void with 1x1 < x, and the surrounding dusty plasma with 1x1 > x,, where x, indicates the void boundary. Three main forces act on the dust: the electrostatic, the ion drag, and the neutral gas drag forces. The latter force is proportional to the dust particle velocity, which is zero in the stationary state, but we retain it here to include the possible slow motions of the void. For the ion drag force, it is assumed that (for the considered here collisionless case) the ion velocity in the dust region is superthermal. At the center, the electric field E = 0, so that the potential is parabolic for small 2 . This is the first boundary

Structures and Phase Transitions in Complex Plasmas

203

condition to be satisfied. Another boundary condition is applied at the void’s edge, x = xu. In the region occupied by the dust, the dust charge is taken into account in Poisson’s equation. The charge is found from the charging equation, which depends on the electron and ion parameters. The number density is obtained using the equations of motion and continuity for the dust. Under the assumption of the steady-state conditions the dust inertia can be neglected and the force balance on a dust particle is

FE

+ Fdr + Ffr = 0 ,

(5.1)

where FE = -ZdeE, Fdr = miuiv, and F f T = 3n,a2v~,m,vd dvare the Coulomb, ion drag, and neutral drag forces, respectively. Here, v is the collision frequency for the momentum transfer between the ions and dust. The electric field includes the field of the dust particles. The particle charge z is determined in the steady state. The dust continuity equation is trivial when the dust is stationary. On the other hand, for a void with moving dust, for example, when the void is expanding or contracting in size, the dust continuity equation must be included in the model. For the case of a void that is expanding or contracting with time, the boundary’s motion should be taken into account. Assume that x = x u ( t ) describes the boundary motion, where xv(t) is the boundary’s position a t time t. The dust density near the boundary can be written as n d ( x , t ) = nd(x - x u ( t ) ) .It is then obtained from the dust continuity equation that

where n d and vd are functions of x and t. The dust velocity a t the boundary equals the boundary velocity dx,(t)/dt, and (5.2)

Note that a slow motion is assumed, i e . , the electrostatic equilibrium is maintained. In this case, the boundary’s motion, but not the dust motion inside the dust cloud, can be investigated. Numerous experiments have shown that the dust-void boundary is a sharp, discontinuous interface [Samsonov and Goree (1999b); Morfill et al. (1999); Praburam and Goree (1996); Melzer et al. (1998); Dahiya et al. (2002); Annaratone et al. (2002); Rothermel et al. (2002);

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Physics and Applications of Complex Plasmas

Mikikian et al. (2003); Nefedov et al. (2003)l. In the steady state this is because the force balance acting on dust particles requires a jump in the dust number density at the boundary, provided that the dust kinetic temperature is zero. First, we must recognize that at the boundary between the dust cloud and the void, the ion and electron parameters, and the electric field, are all continuous. The electric field E is continuous because the particle cloud is a diffuse body that can sustain no surface charge on its boundary. The ion and electron densities and the ion velocity are therefore also continuous. Moreover, the dust charge is also continuous, since it is determined by the ion and electron parameters. Second, the dust force balance equation can be generalized to include a nonzero dust pressure, n d T d . The dust kinetic temperature T d describes a dust particle’s random motion (and should not be confused with the particle’s surface temperature, which can be different). In dimensionless units the dust pressure is the product of the dust number density and the dust dimensnionless temperature T d = T d e 2 / a T 2 = T d Z / T , z d . If T d is small, then the dust number density is discontinuos. This means that the parameter P, = n d Z d / n , is also discontinuous. Under these conditions, one can use a simplified force balance for a stationary void. The jump in P, is positive, LP, = P, > 0, since Z d > 0 (the negatively charged dust). This criterion is necessary to determine whether a void can exist. This explanation applies to a stationary void. It is also applicable to a void with a moving boundary, provided that the motion is sufficiently slow so that the force balance is maintained a t all times. Consider now what would happen if one started with a void that had a smooth profile, rather than a sharp edge. As the ions flow from the void into the dust cloud, they are gradually absorbed by the dust. The ion flux will therefore diminish with depth in the dust cloud. Thus, the ion drag force will act weaker on the second dust layer than on the first, and for subsequent layers it will decrease continuously. In other words, the ion drag pressure is most severe on the first dust particles that encounter the ion flow from the void. This pushes the first layer back toward the others, compressing the dust so that a sharp edge is developed. Note that there are several effects that can smoothen the sharp profile, in particular, the dust particle temperature, the particle size dispersion, and the dust particle inertia. Numerical results valid inside a void in a dusty argon plasma are presented in Fig. 5.6. These show the structure of the potential +(z), the ion

Structures and Phase Transitions in Complex Plasmas

205

Fig. 5.6 Solutions in the void region [Goree et al. (1999)l showing the spatial profiles of (a) normalized electrostatic potential (b) normalized electric field E , and (c) the Mach number of the ion flow. Note that the ion Mach number is almost exactly unity at the void edge. The ion thermal velocity indicated by a dashed line in (b) has a Mach number of 0.376. $J,

velocity M ( z ) and the electric field E ( z ) in the void. The void's edge is marked in these profiles; its location was found from the boundary conditions. The ions flow from the void center and accelerate toward the void boundary. The ions attain a speed almost exactly equal t o the ion acoustic speed when they reach the boundary. This condition is analogous to the Bohm criterion for a collisionless sheath. The ions have subsonic velocities in the void region, and they reach sonic velocities as they enter the dust cloud. The potential in Fig. 5.6(a) varies smoothly, with a nearly parabolic dependence on 2 in most of the void region. There are small oscillations

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Physics and Applications of Complex Plasmas

superimposed on the otherwise smooth monotonic shape of the potential profile. These oscillations are also apparent in the ion velocity in Fig. 5.6(b), but they are most prominent in the electric field, Fig. 5.6(c). These spatial oscillations reveal the presence of a stationary ionization striation; they are suppressed when.the ion-neutral collisions are included in the model. Near the void edge, the potential and the electric field vary rapidly with x. This occurs beyond the point where the oscillatory regime dies out.

Fig. 5.7 Solutions for an expanding or contracting void [Goree et al. (1999)], showing the spatial profiles of: (a) the normalized velocity of the void edge d x v / d t , (b) the normalized dust charge z at the void edge, and (c) the jump of the dust number density parameter P at the void edge. The phase diagram (a) shows a stationary point. Larger voids contract and smaller voids expand. There is a maximum size for a contracting void; it occurs where the dust density jump becomes zero.

Structures and Phase TTansitions in Complex Plasmas

207

By including a moving boundary between the void and dust cloud, but keeping the void center stationary at x = 0, the void that can expand or contract with time can be modeled. The motion should be slow enough so that the equilibrium force balance on dust particles is maintained. Under these conditions it is still possible to find the velocity of the void’s edge, because the force balance includes the dust-neutral drag force, which depends on the dust velocity. Results are shown in Fig. 5.7. The parameters assumed here are a relatively large dust size a = 0.1 and a low ionization rate l/xoi = 0.09. The phase diagram shown in Fig. 5.7(a) is a plot of velocity vs. position of the void edge x,. It reveals a single stable equilibrium void size, where the velocity of the void edge dx,/dt is zero. Smaller voids will grow, because their edge will have a positive velocity. Larger voids will shrink, with an edge moving with a negative velocity. This stationary point of the phase diagram was found for all the parameters tested. It indicates a single stable equilibrium for a given particle size and ionization rate. This is a noteworthy result that we compare to experimental results below. Contracting voids with a size larger than a maximum one cannot exist. Near the maximum size, the velocity of the void’s edge is always directed inwards, i.e., the void is contracting. The maximum possible void size in all investigated cases was larger than the size of a stationary void. The dust charge in Fig. 5.7(b) has a maximum for a void size slightly larger than the stationary void size. The charge decreases rapidly for the voids that are large and shrinking, i.e., near the maximum void size. Under some operating conditions voids exhibit a 1.5 Hz relaxation oscillation [Morfill et al. (1999)l. This mode was termed the “heartbeat” because the particles underwent a throbbing motion like a muscle in a heart. The void repeatedly collapsed to a no-void condition and then reversed t o the original size. In the simulations, when a void is expanding or contracting with time, it always approaches the equilibrium stationary size. If the ionization rate in the void varies in time in some self-consistent manner with the void size and other plasma parameters, it is possible for the void size to oscillate in a repetitive cycle as observed in the microgravity experiment [Morfill et al. (1999)l.

5.13 . 2

Voids in collision-dominated plasmas

Recall that dust voids usually appear from a uniform dust cloud as a result of an instability associated with increased local ionization in spontaneously appearing depletions of the dust number density. The size of the

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Physics and Applications of Complex Plasmas

void is often comparable or even larger than the ion mean free path in the ion-neutral collisions [Samsonov and Goree (199913); Morfill et al. (1999); Praburam and Goree (1996); Melzer et al. (1998)]. In this case, one should use the collision-dominated dust void approximation [Tsytovich

et al. (aool)]. Introduce here the dimensionless quantities relevant to the high ionneutral collision rates, The ion density ni and the electron density R, are normalized to the ion density in the void's center, i.e., n = ni/nio and n, = n,/nio. The electrostatic potential is 4 = ep/T,, and the dimensionless electric field E and the distance x are normalized as E = EeX&,/aT, and z = xa/A&, respectively. For the collision-dominated case, a new characteristic length related to the ion-neutral mean free path appears. Since all the lengths are normalized to the same distance X&/a, the dimensionless electric field is measured in units of the field in which an electron receives the energy T, on the distance equal t o the ion mean free path in their collisions with the dust particles. The velocity vi of the ion flow is normalized as u = vi/ JZvTi. The number density of the plasma neutrals is usually much higher than the ion number density ( L e . , the plasma has a low degree of ionization) and the change in the neutral distribution due t o the ion-neutral collisions can be neglected. The ion-neutral collisional friction decelerates the ion flow and for the low ion flux velocities the friction force is proportional to the ion velocity. Thus for the case when only the electric field and the friction with neutrals are important, the total dimensionless force acting on the plasma ions is given by 2u F=--+E, Xn

(5.3)

where z n / 2 is the ion mobility in the dimensionless units. The experimental data show that with the increase of the electric field E the mobility starts to depend on E and for large field u 0: [Frost (1957)l. Therefore the expression for the friction force (leading to the mentioned changed in the dependence of the mobility on the electric field in the presence of the electric and friction forces only) can be written as U

Fu -- --(2

+

auu). xn The balance of the electric and friction forces then leads to

(5.4)

(5.5)

Structures and Phase Transitions in Complex Plasmas

209

which provides both the limits (z,cyu.E << 1 a.nd z,a,E E >> 1) with the known dependence of the mobility on the electric field. If a void is in the intermediate regime where the ion-neutral collisions (although being important) do not dominate the ion ram pressure forces, the calculations should take into account the friction, the ram pressure, and the electric field forces. In this case Eq. (5.5) becomes invalid and the ion balance equation appears as an additional cquation. In the collision-dominated case, it is necessary to account for the dependence of forces acting on the dust particles (such as the ion drag force) on the ion flow velocity when the latter is close to the ion thermal velocity [Tsytovich et al. (2001)l. The elect,ron drift velocity in the void region cannot be larger than the ion flow velocity, i e . , it is much less than the electron tlrerrrial velocity, arid thus the electron friction with neutrals is much less than the electric and the electron pressure force. According to the known dependence of the ionization rate for the electron impact ionization, the ionization rate is proportional to the electron density. The length xi is the distance on which the ionization makes the dimensionless electron density to become unity. The size of the void can be found as a simultaneous solution of two boundary conditions: the continuity of the electric field and the dust charges a t the edge of the void. The electric field a t the void side is calculated numerically by solving the set of equations inside the void region. The electric field at the dust side can be calculated once the dust charges at the void surface are known. There are two dimensionless parameters of the system describing the void, namely, D , which is the ratio of the void size to the electron Debye length, and xi,which is the ratio of the ionization length to the ion-neutral collisions mean free path. Note that the parameter zi does not depend on the neutral gas pressure and depends only on the ionization power while the parameter D is proportional to the neutral gas pressure. In numerical solution in the void region one can use either equations for the quasi-neutral void which depend only on one parameter, xi, or for a void smaller or of the order of the electron Debye length, the set of equations depending on both D and xi. In addition to the solution of the equations in the void region it is necessary to satisfy the boundary conditions at the void edge which additionally depend on x,, i e . , the ratio of the ion-neutral mean free path to the iondust collision mean free path times the parameter Pi. The value of Pi a t the boundary of the void determines the discontinuity of the dust density. The boundary conditions give two values: the dust charge at the void boundary

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Physics and Applications of Complex Plasmas

and the size of the void. The electron and ion densities a t the void surface and the value of the ion flow velocity are found straightforwardly. The calculations start in the center of the void x = 0. The void is assumed to be planar and symmetric, and the one-dimensional nonlinear equations were solved a t IC > 0. At the center of the void, z = 0, the electric field is zero, E = 0, and the dimensionless ion density is n = 1. For the quasi-neutral voids n, = n and the dimensionless electron density is also equal to 1. For the non-quasi-neutral voids the proper asymptotics in the center of the void is used: Q, 4x / x i , E 4 2x/xi for x 4 0 which implies that n, 41 - 2D2/xi. Since n, > 0 in the center, the non quasi-neutral void can exist only for xi > 2D2. This condition is taken into account for numerical solution of the non-quasi-neutral dust voids. For fixed D and x, for non-quasi-neutral voids or for fixed xi for quasineutral voids, they exist if x, exceeds some critical value. In some cases this takes place within the certain range between xn,,in and x , , ~ while ~ ~ , for other conditions there are several ranges of the parameter 5 , . The boundary conditions also depend on r = Ti/T, and ad = a/Xoi. The calculations were performed for the parameter range most interesting for the existing experiments: 0.02 < T < 0.1 and a d = 0.1. In principle, the boundary conditions can restrict the range of these parameters as well (for a void to exist). Thus an increase of 7- can lower the threshold for z, as can be seen from the results given below. The results are presented for the parameter z, exceeding the threshold for the smallest value of T used. In this case the results can be presented in a compact form as surface plots. All calculations are performed for argon plasma. In the case of a quasi-neutral void the calculations were done for z i equal t o 1 / 5 , 1 and 5. The first case corresponds to the high ionization rate and the third case corresponds to the low ionization rate. Figure 5.8 shows the results for the allowed range of the void sizes for the first case. Figures 5.8(a)-(d) show the size of the void x, as a function of x, and T in the range z, > xn,,in and 0.02 < r < 0.1; the dust charges z, at the surface of the void in the same range; the boundary jump P, of the parameter P ; and the ion flow velocity u, at the void boundary. The calculations were done until z , = 10 (note that the latter value does not violate the boundary conditions). Note that the void size can significantly exceed the ion-neutral mean free path. The void size grows proportionally to the mean free path. Indeed, the size of the void is approximately equal to the ion-neutral collision mean free path divided by r. The voids larger than this size do not exist since the ion

Structures and Phase Transitions in Complex Plasmas

211

Fig. 5.8 Dependence of the collisional void parameters on xn and T in the range zn,,in < xn < 10 and 0.02 < T < 0.1 [Tsytovich et al. (200l)l. The results are ob, ~ the i void ~ size , tained for quai-neutral voids when xi = 1/10. The value of ~ ~ where is close to zero, is 1.6 for T = 0.02; for larger T the minimum xn decreases. The figure shows that the size of the void at lowest xn increases with T .

drag force does not increase with the size of the void (the ions accelerated in the void will be stopped by the ion-neutral collisions). This behavior is found for all calculations performed for the quasi-neutral voids. The critical value of x, for which the void starts to form is shown in Fig. 5.8 as the minimum value of the x axis. The minimum value x,,,in increases gradually with xi (with the decrease of the ionization power). This minimum value also decreases with the increase of the temperature ratio r . The jump of the parameter Pi at the void boundary, P,, is the highest when the void size is close to the x,,,in. in The ion flow velocity at the surface of the void is of the order of the ion thermal velocity, the minimum value being about 0.2 and the maximum value less than 2. With the increase of the ionization length xi the velocity of the ion flow increases. As soon as the size of the void becomes comparable to the electron Debye length the condition of the plasma quasi-neutrality is violated. The two series of numerical calculations were performed: one for D = 0.5, xi = 1 which corresponds to the initial electron density n, = 0.5 and another for

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Fig. 5.9 The parameters of non-quasi-neutral void for zi = 1,D = 0.5; z,,,in [Tsytovich et al. (ZOOl)].

in

= 2.8

D = 3, xi = 40, which corresponds to the initial electron density n, = 0.45. The case of D = 0.5 corresponds to the marginal violation of plasma quasineutrality, while the case D = 3 corresponds to the substantial violation of the quasi-neutrality. Some results are shown on Fig. 5.9. In the absence of the quasi-neutrality, the void can exist within the two zones of the possible values of 2,. The second zone not shown on Figs. 5.9 is for the relatively narrow range 1.1 < x, < 1.2. For the second set of numerical calculations for a large violation of the quasi-neutrality, with D = 3 and xi = 40, it was revealed that the two zones of possible values of x, determine the conditions for a void to exist. One of the zone is for the narrow range of x,, namely, 4.7 < x, < 5.06. Another zone appears a t rather low values of x,, namely 0.69 < x, < 3.4. For this zone, the value of the ion flow velocity at the surface of the void becomes very large when the parameter 2 , is approaching the largest possible values in the zone. The calculated size of the void is of the order of the mean free path (with respect to the ion-neutral collisions) divided by the ion/electron temperature ratio, i e . , x,/r, which corresponds to the void sizes observed in the experiments [Samsonov and Goree (1999b); Morfill et al. (1999);

Structures and Phase Transitions in Complex Plasmas

213

Melzer et al. (1998)l. Also, the void size increases with the ionization power, in accordance with the experimental results. We note that a “collisionless” void [Goree et al. (1999)] can be sustained by the ion drag force when the ions have a velocity greater than the ion thermal speed, u >> 1. In the collisional case, see Fig. 5.9, it appears that u 5 1 almost everywhere in the void. That means that the ion drag force is in the regime where the force increases with u,which is different from the collisionless case considered above [Goree et al. (1999)l. In particular, in the collisional case, the electric field and the ion drag force both increase with the ion speed. However, the faster increase of the electric field E can lead to the force balance, especially for u 1 when a further increase (with the distance off the center of the void) of the ion drag force actually stops. On the other hand, the charge of a dust particle also enters the force balance equation, and also is a function of the distance. The result is a complex interplay of a number of functions of the distance from the center of the void (the electric field, dust charge, speed of the ion flow, etc.), see Fig. 5.9. In the above, we have not discussed a possible effect of the thermophoretic force associated with temperature gradients [Rothermel et al. (2002)l. The gradient of the neutral temperature in the void region is produced by the flowing ions colliding with the neutrals. The flow velocity of the ions increases towards the boundary of the void. We finally note that the presented theory does not provide the initial transition stages of the void formation but rather demonstrates the existence of stationary solutions in the case of bordering void-dust regions. Instabilities leading to the void formation should be studied separately [Ivlev et al. (199913); Avinash (2001); Dahiya et al. (2002)l. N

5.2

Liquid and Crystal-Like Structures

In this section, we present mostly experimental results on formation of liquid and crystal-like structures in various plasmas. As we already stressed, self-organization of structures in a complex plasma is inseparably associated with the plasma conditions. Thus the variety of structures observed is directly related to variety of complex plasmas produced in experiments. First, we consider here the most common case of RF discharge capacitively coupled plasmas, Sec. 5.2.1, when the the highly ordered and strongly correlated structures are observed. They are typically two-dimensional or quasi-two-dimensional, because of the condition of dust trapping and levi-

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tation (in the sheath region which is fairly homogeneous in the horizontal plane). Next, we present results for direct-current and inductively-coupled discharges, Sec. 5.2.2. The majority of structures observed in this case are three-dimensional, they are less ordered, and numerous stable dynamic structures (such as vortices) appear. The most important factors influencing self-organization in such plasmas are various types of potential wells where the particles are trapped, with inhomogeneities in all directions. Sections 5.2.3 and 5.2.4 are devoted to structures in thermal, nuclear, and UV induced plasmas. A characteristic feature of dust in such plasmas is a positive charge on grains. As was mentioned above in Introduction and in Sec. 3.2, there is no effective trap for positively charged particles. Therefore the most structures involving positively charged dust are dynamic and have a limited life time. Due t o the lesser absolute charges appearing in the case of emission charging, the electrostatic coupling of the particles is weaker and liquid-like structures are representative for thermal, nuclear, and UV induced plasmas.

5.2.1

Structures in capacitively coupled RF discharge

Dust particle structures formed in an R F discharge capacitively coupled plasma (CCP) in most cases demonstrate strong ordering with twodimensional character,. as opposed to those observed in other complex plasmas. The reason, as was mentioned before in Sec. 3.2, is that these structures are usually observed in the sheath region of a lower horizontal electrode. The sheath region is strongly inhomogeneous in the vertical direction. In such a condition, monodispersed dust particles can find an equilibrium only at certain heights [Samarian and Vladimirov (2003)l and therefore form a monolayer structure. In most experiments in R F CCP discharges, a hexagonal lattice was observed [Chu and I (1994); Melzer et al. (1996a); Thomas et al. (1994)I. As an example, images of dust structures in R F discharge plasmas obtained by different researches are presented in Fig. 5.10. Formation of the hexagonal lattice is predicted by the W i p e r model for a classical one-component plasma. In complex plasmas, such structures appear as a compromise between the particle-particle interaction and the particle-well interaction. The potential well is formed by the external radial confining electric field which plays the role of a compensating background. The hexagonal structure is sometimes called also a triangle lattice because the triangle cells appear as structural units of such type of lattice. If a

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215

Fig. 5.10 Hexagonal lattice structures formed in an RF capacitively coupled discharges: (a) observed by [Thomas et al. (1994)]; (b) observed by [Chu and I (1994)l; (c) observed by [Melzer et al. (1996a)l; (d) observed at the Complex Plasma laboratory, the University of Sydney.

number of particles in such a structure is decreased, the Wigner lattice will then be replaced by the shell (ring) structure, see Fig. 5.11. The crystals with a small number of particles are called Coulomb or plasma clusters. Configurations of Coulomb clusters observed in a dusty plasma agrees well with theoretical predictions for a system of charged particles confined by an external field [Lozovik and Mandelshtam (1990); Lozovik and Mandelshtam (1992); Lozovik and Pomirchy (1990); Peeters and Scweigert (1995); Lai and I (1998)]. This is proved by a comparison of the theoretically calculated numbers of particles in the shells and the experimentally observed clusters, see Fig. 5.12. In the case when the number of dust particles exceeds a critical one that can be trapped in one layer, a multilayer structure forms as simulations [Totsuji et al. (1997)l demonstrate. Such a transition triggered by the interpaly of the particle-particle interaction vs the particle-well interaction, can be clearly seen even in the simplest case of just two particles, see Sec. 4.2. The structures formed are quasi-three- (or 2.5) dimensional because they are extended in the horizontal plane and typically have only a few layers in the vertical direction. The structure of these crystals is quite similar to that of graphite: the simple hexagonal cells are placed directly on top of each other and the particles are arranged on a triangular lattice in the horizontal plane see Fig. 5.13. The interlayer distance is typically 1.5-2

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Fig. 5.11 The Coulomb clusters [Cheung et al. (2003)].

times more than the lattice constant. This is in contradiction with simulation results for the three-dimensional Yukawa system, but such a structure can be explained by different interparticle forces acting in the horizontal and vertical directions. Thus the anisotropy appearing in the experiments (and having various physical reasons such as the action of gravity, the inhomogeneity of the sheath distributions, the ion flow) leads to formation of the stable simple-hexagonal structure which does not appear in the isotropic three-dimensional Yukawa case when bcc and fcc structures are formed. It was also found in dusty plasma experiments that different crystal lattice types can exist and even coexist in one structure. In addition to the most common hexagonal lattice, there are other lattices (like bcc- and, less often, fcc-type) observed in experiments under special discharge parameters. In general, these structures are formed when the RF power is decreased and the vertical size of the dust structure increases. In Fig. 5.14,

Structures and Phase Transitions in Complex Plasmas

Fig. 5.12 clusters.

217

Calculated numbers of particles in the shells and the experimentally observed

I _ _

Fig. 5.13 The structure of 2.5-dimensional plasma crystal.

we can see the images of different dust structures and sketches showing the different lattices. The images in Fig. 5.14 reveal structures that are bcc with (110) axis in the vertical direction, (b), and simple hexagonal in which particles are vertically aligned, (a). The coexistence of two different structures is possible because there are two crystalline configurations with the minimum potential

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Physics and Applications of Complex Plasmas

Fig. 5.14 Images and sketches of the different plasma crystal structures [Chu and I (1994)l: (a) hexagonal; (b) bcc with (110) axis in the vertical direction; ( c ) fcc with (111) axis in the vertical direction. The bars correspond to 200 pm.

energy with respect to a small displacement in the particle position. The simple-hexagonal structure can exist since it reduces the potential energy by bringing particles into the attractive potential well of the ion focus created by a particle located above. On the other hand, the square bcc lattice has particles not vertically aligned, but staggered instead thus minimizing the repulsive part of the electrostatic potential. Both structures coexist and stable, and the one favored energetically is more likely to be observed. This depends on the strength of the ion focus potential, which in turn is the function of discharge conditions. The order determined from the static structural analysis technique is characterized by the pair, Go, translational, gT(lr - r’l) = (exp[iGo(r r’)]),and bond-orientatzonal, gn(lr-r’l) = (exp in[O(r)- O(r’)]), correlation functions. In some cases, and especially in the case of thermal and/or UVinduced plasmas, the structure factor S(k) is also useful. Both correlation

Structures and Phase Transitions in Complex Plasmas

219

I

--

I

01

Fig. 5.15 The typical correlation functions: (a) pair [Quinn et al. (1996)], (b) translational and (c) bond-orientational [Pieper et al. (1996)l.

functions gT and gn decay exponentially with the distance thus yielding the corresponding correlation lengths. The typical pair, translational and bond-orientational correlation functions are presented in Fig. 5.15. 5.2.2

ICP and DC discharge structures

The structures of dust particles in ICP and DC discharges have a few typical features different from those observed in CC discharges. Firstly, they are predominantly three-dimensional. Secondly, they are less ordered. And lastly, stable dynamic structures such as vortices are observed. The difference in the dust structures observed reflects the difference in the plasma conditions. The shape of a dust cloud obtained generally reflects the shape of the potential well formed in the discharge. Thus for, e.g., the striation condition of DC discharge in a vertical tube, an ellipsoidal cloud with the diameter of 5-10 mm and the length of 2-3 cm can be observed for relatively large

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Physics and Applications of Complex Plasmas

particles of the size of about 50 pm [Lipaev et al. (1998)l. The diameter of the cloud reaches up to 20 mm for smaller particles (of order 5 pm). Figures 5.16 show typical images of the dust cloud cross-sections for these two cases ((a) and (b), respectively). In the vertical plane, the structural ordering appears as a formation of particle chains similar to those in an RF discharge. However, one distinctive structural feature observed in the gas discharge running in the vertical tube was the occurrence of an extended vertical (filamentary) structure which was formed from few one or more vertical lines (see Fig. 5.17). In the ellipsoidal cloud, the particles are organized in 10-20 layers for larger particles, and even more layers for smaller particles. Another distinctive feature of the structures in DC discharges is that the distance between the layers is less than the interparticle distances in the horizontal plane, namely, 250-400 pm and 350-600 pm, correspondently, for structures presented in Figs. 5.16. Similar structures where the particles are aligned in vertical chains have been observed in ICP plasma [Fortov et al. (2000b)]. The number of chains is reduced as the distance from the central part of the discharge is increased. Such filamentary structures have been generated by particle injection into the plasma at a fixed pressure. The number of particles held in such formations is limited. As the chain reaches a certain length, the structure is becoming unable to catch a new particle, ie., it becomes Horizontal cross-section

Vertical cross-section

Fig. 5.16 Cross-sections of a dust cloud in the striation region of a DC discharge [Fortov et al. (1997a)I.

Structures and Phase Transitions in Complex Plasmas

Fig. 5.17 (1997a)l.

22 1

Cross-sections of a filamentary structure in the DC discharge [Fortov et al.

“saturated”. Extra particles introduced are either not caught or replace the particles in the structure. The size of the structure in the vertical direction is reduced as the gas pressure is increased. Simultaneously, the average interparticle distance decreases moderately. As seen from Fig. 5.17, the interparticle distance in the filamentary structure at a fixed gas pressure decreases as we move from the head to the tail of the structure. A structural analysis verifies that the structures in a DC discharge plasma are crystalline with a short correlation length (3-4 lattice constants, which is less than typically in a CC discharge palsma). The two-dimensional pair correlation functions g h ( l ^ ) in the horizontal plane of the particle structures are presented in Fig. 5.19(a). The first sharp peak, along with the trailing descending peaks, manifests the crystalline structure. The correlation length is about four times of the interparticle distance. The pair correlation function in the vertical plane gv ( r ) displays typical fluid-like

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Physics and Applications of Complex Plasmas

Fig. 5.18

Cross-sections of a filamentary structure in ICP [Fortov et al. (200Ob)l

30

Fig. 5.19

Typical pair correlation functions for DC discharge dust structures.

decaying oscillations (see Fig. 5.19(b)). The interparticle distance strongly depends on the discharge current and therefore on n,; it differs qualitatively from the dependence of the distance on the input power and, accordingly, on the electron density n, observed in RE' discharges. A distinctive feature of structures typically observed in DC and ICP plasmas is the coexistence of structures in different phase states. Moreover, coexistence of a steady state and stable dynamic structures (such as vortices and acoustic waves) were also observed. A typical complex structure with a solid crystal on the center top of it, liquid in the middle, a vortex structure

Structures and Phase Transitions in Complex Plasmas

Fig. 5.20 (1999)l.

223

Typical complex dust structure in a DC discharge plasma [Molotkov et al.

on the top left and right corners, and dust acoustic waves at the bottom is shown on Fig. 5.20. Similar structures with a solid crystal in the central region and vortices at the edge can be observed in an ICP as shown in Fig. 5.21 .

Fig. 5.21 Typical complex dust structure in an ICP obtained at Flinders University [Prior and Samarian (200l)l. (RF discharge 17.5 MHz, pressure from 560 mTorr, the input voltage from 500 mV, with melamine formaldehyde particles 6.21 mm, in argon plasma T, 2 eV and n, 108 ~ m - ~ . ) N

5.2.3

N

Ordered structures in th.ermal plasmas

Only a limited number of experimental studies of ordered dust structures in thermal plasma was done (in contrast t o the large number of such experiments in discharge plasmas). In early works, spatial distributions of dust particles in the dispersed phase were investigated using the electron

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Physics and Applications of Complex Plasmas

microscopy of samples extracted by a probe introduced for a certain period of time into a plasma flow. Such photomicrographs have been obtained [Dragan et al. (1984)] in the flame of a synthetic aluminum-coated propellant. The arrangement of the particles indicates the presence of an ordered structure of a polydispersed particles on the surface of the sarnpling probe. Thus, the authors have concluded that struct,ures of particles are present in the plasma. Later, formation of ordered structures of particles was found in a weakly ionized, laminar, plasma flow at atmospheric pressure and temperature near 1700 K [Fortov et al. (1996a); Fortov et al. (1996b)l. In the experiments, a dusty plasma was produced by introducing particles into the inner flame of a two-flame Mekker-type burner. This gave a narrow range of variation of the plasma parameters and a relatively short plasma lifetime (of the order of 10 ms). Thus only liquid-like structures were found under such conditions. Figures 5 . 2 2 (a)-(c) show typical pair correlation functions g ( r ) for CeOz particles in an aerosol stream at the room temperature (Tg 300 K ) and in plasmas (Tg 2170 K and 1700 K).

-

-

Fig. 5.22 Typical pair correlation functions g ( r ) [Fortov et al. (1996a)l for CeOz particles in an aerosol stream at the room temperature, (a) T, ,-, 300 K , and in plasmas, (b) Tg 2170 K, and (c) Tg ,-, 1700 K).

-

Structures and Phase Transitions in Complex Plasmas

225

It is quite evident that the correlation functions g ( r ) for the plasma with the temperature Tg 2170 K and the particle density np = 2.0 x lo6 cm3 and for the aerosol stream are essentially the same. This means that the particles in the plasma interact weakly and formation of ordered structures is impossible. This is also confirmed by the plasma parameters. The average interparticle distance (. = 50 mm) under these conditions is approximately 3.5 times more than the plasma Debye length ( T D = 14 mm), while the interaction Coulomb coupling parameter rpis about 40. Then an estimate of the screened parameter rS,which accounts for Debye shielding, yields rs > 1. For the lower plasma temperature Tg = 1700 K and the particle density np = 5.0 x l o 7 cmp3, Fig. 5.22(c) shows that the binary correlation function g ( r ) exhibits the short range ordering. Under these conditions, the ion density (ni 10’ ~ m - ~ is )approximately an order of magnitude lower than the electron density (n, lo1’ omp3), while the particle charge Q is determined by the quasineutrality condition as Q n p = en, (ni << n e ) . The calculated values of rp and rs are 120 and 40, respectively, which correspond to a system of strongly coupled particles. The formation of ordered structures was observed only at sufficiently high ( l o 7 ~ m - ~particle ) densities. Lowering the concentration of the dust particles (leading to the increase of the average distance between the particles) leads to a reduction in the Coulomb interaction energy. Then an ordered structure does not develop, as shown in Fig. 5.22(b) ( n d = 2.0 x lo6 cmP3). An analysis of the distributions of the relative interparticle distances ./(.) r) obtained under the same conditions as the correlation functions showes that regions with the ordered structures (domains) as well as regions with random particle positions exist. Figures 5.23 (a) and (b) show histograms for the aerosol (Tg 300 K , the “gaseous” plasma) and the plasma (Tg= 1700 K). For a plasma temperature of 1700 K , the histogram becomes substantially narrower (by a factor of 4-5), its peak shifts toward smaller interparticle distances (TI(.)) 0.3, and the maximum value rises from 6-7% to 10-11%. In this case the observed structure of the particles differs dramatically from the “gaseous” case, which suggests strong correlation of the particle positions. At the same time, the presence of randomly placed particles leads to the appearance of the broad pedestal characteristic of the “gaseous” plasma in the histogram. The development of domains can be explained by the thermophoretic attractive forces which appear due to a temperature difference between the surrounded gas and the dust particle [Fortov et al. (1999a)I. It turns out that the thermophoretic attraction of dust particles leads N

-

-

-

N

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Physics and Applications of Complex Plasmas

to a nonperiodic Jeans-type instability. Figure 5.24(a) shows the numerically simulated final arrangement of dust particles in the simulation box. Thus the presence of the thermophoretic force and its effect on the formation of ordered structures can lead to the coexistence of regions with a chaotic arrangement of particles and regions of ordered structures (domains). Particles in the domains are positioned with the distances less than the average interparticle distance in the whole dust-plasma system. The measured and calculated pair correlation functions show a good agreement, see Fig. 5.24(b). Ordered structures of particles of a condensed dispersed phase (characterized by appearance of solid or liquid particles in a plasma) formed in a boundary region between the high-temperature and condensation zones (see Fig. 5.25) was observed for samples of aluminum-coated propellants with a low content of alkali-metal impurities. Dust formations with different shapes, observed in the condensation zone, are shown in Fig. 5.25(a,b). Analysis of the correlation functions for these structures does not show even a short-range order in the particle arrangement (see Fig. 5.26). One of the reasons is that gas-dynamic processes in the background flow determine the nature of the observed structures (as we already described above in Sec. 3.2.2, a distinct condensation zone is formed above the flame). Analysis of smaller (V < 10-4 cmV3)structures of particles of the condensed dispersed phase, which form in the boundary region where the temperature is still high enough (T 1500 K) and regions where a high particle number density exists. (see Fig. 5.25(c,d,e)),shows a different picture. The short-range order (see Fig. 5.26) is observed in these structures when the particle density is sufficiently high ( n d 107-108 ~ m - ~

-

-

b

Fig. 5.23 Histograms [Fortov et al. (1996a)l for: (a) the “gaseous” plasma with Tg 300 K, and (b) the plasma with Tg = 1700 K.

N

Structures and Phase Transitions in Complex Plasmas

227

Fig. 5.24 Arrangement of dust particles in the case of thermophoretic attraction [Fortov et al. (1999a)l: (a) numerically simulated final stage; (b) calculated (solid curve) pair correlation function for the final stage (for comparison, the experimentally measured correlation function (dashed curve) is also presented).

~~

Fig. - 5.25

Images combustion dusty plasma - of ordered structures in a propellant . [Samarian et al. (ZOOOb)].

5.2.4

Dust structures in nuclear and UV induced complex plasmas

Generally, structures in nuclear and UV induced complex plasmas reveal the same features as those in ICP and DC discharge plasmas. However, because of strong inhomogeneities, the variety of dynamic phenomena in complex structures formed in such plasmas is wider. Figure 5.27 show clouds with well-defined boundaries which formed in nuclear induced plasma [Rykov et al. (2002)l. The cloud has the shape of

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Physics and Applications of Complex Plasmas

Fig. 5.26 Pair correlation functions for structures in a propellant combustion dusty plasma [Samarian et al. (2000b)l.

a truncated cone with the base parallel to the plane of the upper electrode and the top near the radioactive source. The cloud formed has a complex structure: the entire volume of the experimental chamber can be divided into five regions in which the particles behave differently. In region I, the particles are almost stationary. In region 11, the particles move slowly, creating an ascending flow. Their velocities in the flow decrease as the

Fig. 5.27 Dust clouds formed in a nuclear induced plasma [Rykov et al. (2002)].The snapshots have been taken: (a) 2 min, (b) 4 min, (c) 4 min 30 s, and (d) 4 min 45 s after the injection of the dust.

Structures and Phase Transitions in Complex Plasmas

229

particles approach the boundaries of the structure and its upper part. In the middle part of the flow, the velocities are maximum, equal to 0.6 cm/s. In region 111, the particles move downwards along the generator of the cone at the velocity 1.2 cm/s; some of the particles are deflected to the axis of the structure in the vicinity of the lower electrode and then move upwards. Another fraction of particles get into region V. Region IV contains a very small number of particles whose velocities a t the boundaries of the structure is equal to the velocities of particles in region 111. When dust particles approach the lower electrode, they are deflected to the walls, where they ascend to form closed trajectories. In region V, all particles fall down a t a rate of approximately 1 cm/s.Thus, there are two types of vortices: one is characterized by the relatively fast motion of particles with a low number density a t the periphery, while the other by the slow motion within the structure. These vortices are separated by a sharp boundary near which the velocities of particles are identical on both sides. The momentum transfer from ions to neutral particles islarge enough for initiating the motion of the gas. This appears as one of the main reasons for the emergence of vortices [Fortov et al. (1999b); Rykov et al. (2002)l. Another reason can be an appreciable recombination of ions and electrons on dust particles, which gives rise to large density gradients and auxiliary flows of plasma particles. The ionization and recombination of the plasma (including that on dust particles) occur at the highest rate in the vicinity of the source of the ionizing radiation. The ion flow reducing the diameter of the structure near this source is directed precisely at this region. At the lowest part of the structure (region V), the charge of particles becomes positive, and they fall to the lower grounded electrode. This is another example of how inhoniogeneities of a complex plasma open an-

Fig. 5.28 Pair correlation functions of structures in a nuclear induced plasma [Vladimirov et al. (2001b)l.

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Physics and Applications of Complex Plasmas

other channel of self-organization, namely, causing the appearance of stable dynamic structures. Structural analysis of the structures observed gives the pair correlation function (see Fig. 5.28). There is a clearly seen first peak, typical for liquid structures. The coupling parameter €or the dust component varies from 45 to 340 for particles of different sizes, which also speaks in favor of the liquid type of the obtained structures. The size distribution of particles, the fluctuations of their charges and the motion of the medium considerably hinder the formation of crystalline dust structures. Under a constant pressure and €or constant potentials of the electrodes, the cloud forms in a few minutes with the streamlined hemisphere upper part, Fig. 5.27(b). Then it gradually changes the shape of its boundaries and smoothly falls on the lower electrode, Figs. 5.27(c) and (d). At the same time, the vortex motion of the periphery particles, as if grinding the structure, creates a constriction at its lower part. In Fig. 5.27(d), the vortex motion also considerably affects the behavior of the upper part, exerting an additional pressure on it. After the structure falls on the lower electrode, the motion which was formerly typical of the periphery particles, embraces the entire volume.

Fig. 5.29 (2OOO)l .

Cloud of dust particles in abnormal DC disharge [Samarian and Vaulina

Formation of a cloud of several tens of particles can be observed under condition of abnormal DC disharge (see Fig. 5.29). Under such condition, a perfect crystal structure cannot be obtained. Firstly, under all discharge parameters the particles are moving, with the temperature of more than 1 eV. Secondly, the particles are not equidistant within one layer, or between layers. The reason for the first phenomenon is due to the fact that random charge fluctuations for the photoemission charging mechanism are relatively large. This leads to the effective heating of dust particles. Also, the particle kinetic temperature can increase because of the dispersive

Structures and Phase Transitions in Complex Plasmas

231

instability occurring in inhomogeneous dusty plasma, which can produce spatial fluctuations of grains similar to Brownian motion [Vaulina et ul. (2003)l. These spatial fluctuations cause the heating of the dust particles system. Considerable spatial inhomogeneity explains also the nonequidistant arrangement of particles. Another example of dust structures in UV induced plasma is ordered structures of dust particles charged by photoemission due to solar radiation and formed without electrostatic traps to confine the particles. These structures exhibit features similar to those in a thermal plasma jet because in this case the life time of the system is also less than the time necessary to form a crystalline structure. In experiment, the dynamic ensemble of particles under the action of solar radiation can be observed. Initially, the particles are on the walls of the glass chamber. Therefore, the typical experiment is carried out in the following steps: first, a dynamic disturbance (jolt) of the system, and subsequent relaxation to the initial state, Le., drift to the walls. Figures 5.30(a)-(d) show the successive states of the system of particles [Fortov et ul. (1998)I. The velocity vectors of the particles are randomly directed in the initial stage and the particles drift to the walls without a preferential

Fig. 5.30 Successive states of the system of dust particles in UV induced plasma ( P = 40 Torr) [Fortov et al. (1998)l.

232

Fig, 5.31

Physics and Applications of C o m p l e x P l a s m a s

Agglomerates of dust particles in UV induced plasma [Fortov et al. (1998)l.

direction. Subsequently, a preferential direction appears,and the motion along certain trajectories is more pronounced for a higher pressure. One of the interesting features of such kind of complex plasma is the formation of dust agglomerates, in which the number of particles varies from three or four to several hundred, Figs. 5.31(a) and 5.31(b). These agglomerates leave the walls in response to a weak dynamic disturbance. The results of the observations of the behavior of the particles show that the majority of agglomerates forms in the volume within a few seconds following the dynamic disturbance. The agglomeration of particles in the volume occurs because the particles initially acquire the opposite charges when illuminated: the positive charges appear as a result of the emission of photoelectrons, and the negative charges are due to the fluxes of electrons emitted from neighboring particles. Under such conditions, relatively high charges appear on dust particles, see Sec. 3.2.3. However, despite the high charges and the large value of the coupling parameter I? 104, there is no strong correlation in the particle cloud. The measured correlation functions exhibit appreciable deviations from the theoretical expectations. The typical measured correlation functions for the dark and illuminated structures complemented by the numerically computed ones are shown in Fig. 5.32. The difference between the N

Structures and Phase Transitions in Complex Plasmas

233

Fig. 5.32 Correlation functions for the illuminated (1) and dark (2) structures in UV induced plasma [Fortov et al. (1998)l.

experimental and simulated results can be due to several factors, namely, the polydispersity of the particles investigated, the formation of agglomerates, or the limited life time of the system.

5.3

Phase States and Phase Transitions

The structural ordering of dust particles in complex plasmas is thought to be similar to that of atomic systems, such as crystalline solid, liquid and/or amorphous matter. Although the main interparticle forces are of different nature in a complex plasma as compared to the above media, the resulting arrangements are often the same (see Sec. 5.2). Studies of these structures has shown that they exhibit a number of distinguished properties, which makes it possible to use them as model systems for investigation fundamental properties of other states of matter. A strong dependence of the dust structures on parameters of the bulk plasma as well as short relaxation times of these structures to equilibrium and short-time responses to external perturbations, allow dynamics of phase transitions to be studied. In this section, we first discuss the criteria of phase transitions in a complex plasma, Sec. 5.3.1. Then in Sec. 5.3.2 we consider experimental results on phase states and phase transitions, mostly in discharge plasmas. 5.3.1

Order controlling parameters i n a complex plasma

Ordering in any system of interacting particles basically arises due to a competition between the interaction energy U and the energy T of the particle thermal motion. For instance, in atomic solid state systems, an increase of the temperature lowers the ratio I? = U / T below a critical value

234

Physics and Applications of Complex Plasmas

and the ordered crystalline lattice melts into a disordered liquid state (the latter can further evaporate to the gas state). In a complex plasma, the parameter T appears as the controlling parameter (depending in a complex manner on numerous parameters affecting phase states and transitions). The number of the order parameters, describing such a plasma state, can exceed one, depending on the complexity of the interparticle interaction. Since the dominant interaction is associated with the electric forces between the charged species, the phase behavior in the first approximation can be described using the Coulomb coupling parameter

rc = Q 2 / d T d . Traditionally, models of a one-component plasma or a plasma with a screened (Debye or Yukawa type) potential have been used to describe phase transitions in a complex plasma. In the one-component model, the plasma is treated as an idealized ion system against a uniform neutralizing background, so that the system as a whole is electrically neutral. In this case, the interaction between charged grains is described by the Coulomb potential and the associated Coulomb coupling parameter rc. This system crystallizes when rc = 171, (for the three-dimensional case) and when rc M 106 for the two-dimensional case [Van Horn (1969); Plolloc and Hansen (1973); Slattery et al. (1980); Hamaguchi et al. (1997)l. The gas-liquid transition occurs when rc M 4. In the Yukawa model, it is assumed that the grain charge is screened by a charged background, which leads to the screened Coulomb (Debye) interaction potential. The plasma thermodynamics and, accordingly, the phasetransition conditions in the Yukawa model are described by two parameters, namely, by fc and K = d / X o . For simplicity, these two parameters can be combined into the screened Coulomb coupling parameter r D =

exp(-K)rc.

Then the critical OCP values can be used to take into account the plasma screening. The short-range order in such a model is established for TD > 1 as extensive numerical studies [Van Horn (1969); Plolloc and Hansen (1973); Slattery et al. (1980); Hamaguchi et al. (1997); Vaulina and Khrapak (2000); Robbins et al. (1988); Meijer and Frenkel (1991); Stevens and Robbins (1993)I demonstrate. This corresponds to the simplest two-body approximation when the interaction energy of two particles is taken as U .

Structures and Phase Transitions in Complex Plasmas

235

However, strictly speaking, the parameter TD cannot adequately describe either the interaction between two charged particles nor the phase state (or phase transition) of, e.g., lattice, composed of such interacting particles. Indeed, the state of a system of particles is determined by the forces between them. Thus the corresponding parameter should involve the derivative of the interaction potential (such that the additional factor 1 K appears). When considering the phase transitions in an ensemble of particles such as the lattice, the standard theory relates them with collective excitations becoming large (and/or unstable) at the transition. The natural mode in the complex plasma crystal is the lattice dust-acoustic wave (DAW) [Melandso (1996)I. The DAW description involves oscillations of dust particles in the field of their neighbors. The characteristic oscillation frequency, in the simplest approximation, is determined by the potential well of the fields of the nearest neighbors. The curvature of the potential well is proportional to the second derivative of the particle interaction potential. For Yukawa-type, this leads to the factor 1 K ~ ' / 2 . Thus the phase state in Yukawa model, as was also suggested by [Vaulina and Khrapak (2000); Vaulina and Khrapak (200l)l on the basis of the best fit of simulated data, is determined by the value of the coupling (non-ideality) parameter

-+

-

ryu= (1 + K + ~

+ +

~ /exp(-K)rc. 2 )

For example, its value rFu 106 on the melting line can be used as the melting criterion for the body-center cubic (bcc) lattice. The use of the modzfied coupling p a r a m e t e r r Y u allows one to illustrate the behavior of the melting curves for the transitions fcc-lattice-liquid and bcc-lattice-liquid, Fig. 5.33. The functional dependence relating rc and K with the critical value I?" = f(rc,K ) is presently unknown for the transitions of face-center cubic (fcc) lattice into the liquid as well as for the transitions between the bcc and fcc structures. Therefore, dependencies determined on the basis of computer simulations employing various interaction models are nowadays the only possibility to obtain the phase diagrams. The actual interaction potential of particles in a complex plasma can deviate from the screened Coulomb (Yukawa) type, especially for distances exceeding 4 - 5 X ~ . With increasing distance, the effect of the screening weakens and the asymptotic character of the potential U can follow the another (such as power-law) dependence [Vaulina et al. (2004)l. In this case, the natural generalization of ryuwill involve a factor proportional to

236

P h y s i c s a n d Applications of Complex P l a s m a s

Fig. 5.33 Dependence of ryu on K for various phase transitions [Vaulina e t al. (ZOOZ)]. (Circles: data of [Vaulina and Khrapak (2001)]. Diamonds: [Plolloc and Hansen (1973)l. Squares: [Van Horn (1969)]. Triangles: [Hofman e t al. (ZOOO)]. Filled symbols correspond t o formation of fcc lattice. 1: ryU= 106 (bcc+liquid). 2: fcctliquid. 3: bee-fcc. 4: fcc+bcc.)

the second derivative of the interaction potential I'G =

U"

-. 2dndTd

(5.6)

Note that the Yukawa model may be also incorrect under conditions of a dense grain cloud and in the sheath region of laboratory gas discharges, as well as this model does not take into account the ionization/recombination processes, the collisions of plasma electrons and ions with neutrals and many other factors. We stressed many times in the course of this book that complex plasmas are open dissipative systems. Therefore it is natural to expect that the dynamics of phase transitions in a complex plasma should depend on the dissipative processes in it. Currently, there is no adequate analytical description of the influence of dissipation. However, computer simulations [Vaulina et al. (2002a); Vaulina et al. (2004)l allow one to introduce an the so-called scaling factor auxiliary controlling parameter ~

(5.7)

where y is the characteristic damping (dissipation) rate. For the Yukawa model, the scaling factor is given by

Structures and Phase Transitions in Complex Plasmas

237

Obviously, the factor is related to the ratio of the oscillation energy of particles to the energy dissipation because of the friction with neutrals. This means that the neutral plasma component appears as an additional subsystem and a sink for energy of dust particle motions. In terrestrial experiments, vertical arrangements of dust particles are significantly affected by forces acting in that direction, such as the gravity force, the sheath electrostatic force, the ion drag force, as well as the plasma collective effects such as the ion wake formation (see Sec. 2.3). On the other hand, the horizontal patterns appear mostly as a result of the particle interaction potentials. In the simplest approximation, these potentials are of the Debye-Huckel (or Yukawa) type and involve the Coulomb interaction screened by the surrounding plasma. However, the openness of the dust-plasma system (see Sec. l.2), leads to more complex features of the interaction potential [Tsytovich and Morfill (2002)]. For example, the so-called shadow forces of Lesage type can lead to the change of the simple screened Coulomb potential and even to particle attractions at some distances [Ignatov (1996)]. Thus, experiments and/or ab initio computer simulations can further elucidate these unusual and intriguing complex phenomena.

5.3.2

Criteria of phase transitions

Currently, there is no experimental measured full phased diagrams in a complex plasma (at best, merely a parts of them, often representing isolated points, are known). Thus computer simulations provide important information on the character of phase transitions and phase state behavior. These simulations are usually done similar to simulations of chemical systems of, e.g., Yukawa interacting particles [Stevens and Robbins (1993)]. The melting line(s) in such systems is determined on the basis of commonly accepted empirical criteria of phase transitions. There are two known empirical rules for the first-order fluid-solid phase transition in three dimensions. The first one is the Hansen criterion which states that melting occurs if the first maximum of the liquid structure factor is less than 2.85. This numbers can vary (from 2.5 to 3.2) for different simulations and strongly depend on the definition of the structure factor in the systems with a finite number of particles. The second rule is the Lindemann criterion, which determines that the ratio of the root-mean-square displacement of a particle from its equilibrium position A, to the interparticle distance d on the melting line of the

238

Physics and Applications of Complex Plasmas

solid should be 0.15. Thus the ratio 6, = &‘Ao/d d must be equal to 0.21 on the melting line. Note that the particle displacement measured in numerical simulations, is usually less than 0.2d approaching 0.2d with increasing number of particles [Stevens and Robbins (1993)l. Various numerical simulations give for this Lindemann parameter the range from 0.16-0.19 for fcc lattices to 0.18-0.2 for bcc structures. For complex plasmas, a condition, analogous to the Lindemann criterion, can be obtained [Vaulina et al. (2002a)l with the assumption that the average volume of thermal fluctuations &f ( ~ Y , A for ) ~ bcc lattice should not exceed (1 - 7rfi/8)& M 0.32Vd, where a , = (47r/3)ll3, V d = n-l d -= ( ~ ~ u w and s ) uws ~ , = ( 4 ~ n d / 3 ) - ~is/the ~ Wigner-Seitz radius. For a stable fcc structure to exist, we have V,f < (1 - 7rfi/6)Vd M 0.26Vd [Hofman et al. (2000)l. Accounting for the possibility of counter displacements of particles, V,f 2 (201,A)~ (the factor a), we find that the value of the ratio A / d must either exceed 0.211 (&/d d 2 0.15) to melt bcc structure or 0.198 ( A o / d 2 0.14) to melt fcc lattice. The criterion for the transition between the bcc and fcc structures can then be obtained with the assumption that for the change of the bcc symmetry of the lattice, the interparticle distances should exceed AD (the grain interaction is in this case similar to that of “hard spheres” when formation of fcc structures is possible [Hoover and Ree (1968)l). Thus, we have the following expression for the line of transition between the bcc and fcc structures:

-

(5.8)

where (uws - )A, determines the effective size of the region where a displacement of one particle does not significantly influence other particles of the crystal lattice. This assumption is supported by numerical simulations [Hamaguchi et al. (1997)l where the fcc structure was not formed when uws < X even for r 4 m. The values of 6, and & / d for various phase transitions are presented in Table 5.2 and Fig. 5.34; the range of K between 5.8 and 6.8 defines the region with the triple (bcc-fcc-liquid) phase transition. The coefficient C, for the approximation w,” = Cwnd(eZd)2exp(--r;) of the characteristic oscillation frequencies is 4(1 /c (yi2/2))/7r for transitions started in bcc phase and is 2a;(-r; - a,) for transitions started in fcc phase, where cy, = ( 4 ~ / 3 ) ’ / ~K ,= d / A o . On the basis of simulations [Vaulina et al. (2002a)], the empirical

+ +

239

Structures and Phase Transitions in Complex Plasmas

Table 5.2 The ratio 6, of the most probable displacement A to t h e mean interparticle distance d , the factors of the coupling parameter rc = ( e Z d ) 2 n i / 3 / T d= C,(K, exp(-K))-' on the lines of various phase transitions [Vaulina et al. (2002)].

1 1 Transition

bcciliquid

6, = A/d

-

(1- ~ &2% /8)

CP 3n/(26,2) 106

-

'I3

0.211

fccjbcc fcciliquid

-

(1

0.27(1 -

(1-7rv'5/6) 2a, N

1 1 3~/(2.0.27')

bCCjfcC ~~T

(1-W5/6l1/3

%)

~

7) 'I3

0.198

Kn

-

3 / (0.272cy:) - 69.8 4

-

3/(&,2) 18.5

.

(1+IC+$)

(1

+ + $)(1 IC

R(1-

K, -

3

- %)2

3

cYT

Fig. 5.34 Dependence of Ao/d on K for various phase transitions lVaulina et al. (2002a)l. (1: bcc-liquid. 2: fcc-liquid. 3: bcc-fcc. Filled circle: K = 5.8. Circle: K = 6.8.)

rules are formulated to determine the normalized coupling parameter rn = K , e x p ( - ~ ) r as a value close to a constant C, at the line of different phase transitions (including the melting of cubic lattices and the transition between the bcc and fcc structures). The normalized coefficient K , and constant C, can be obtained from the relationship for the harmonic oscillator

&j = 3 T d / m d W : ,

(5.9)

where md is the particle mass and w, is the characteristic frequency of the particle vibrations in the lattice. The approximation (5.9) takes place when

Physics and Applications of Complex Plasmas

240

Td > > O D (00 is the Debye temperature) and the displacement A, can be characterized by the frequency w, not depending on the temperature. To determine this frequency, the most frequently used are the quasi-harmonic [Robbins. et al. (1988)] and/or the Einstein approximations [Plolloc and Hansen (1973); Robbins et al. (1988)] based on the calculations of the oscillation frequency of a particle about its equilibrium position when all other particles are fixed. For both cases, there is no analytical form for w,, and the results are usually additionally adjusted by the linear, quadratic, and/or cubic fits of the numerical results for various (sufficiently short) parts of the phase diagrams [Plolloc and Hansen (1973); Robbins et al.

(1988)]. In complex dust fluids and bcc lattices, the characteristic frequencies of particle oscillations are proportional to the dust-lattice wave frequencies. Thus the frequency w , = Wbcc for the bcc lattice can be obtained from the expression F = (eZd)’ exp(-l/XD)(l l / X ~ ) / l ’ for the intergrain force assuming that the electric fields of all particles except the nearest ones are fully compensated [Vaulina and Khrapak (2000)l: it is determined by the probability 8/47r of the intergrain collisions and by derivative d F / d l at 1 = d , and is given by

+

Wbcc =

eZd(4nd/~rnd)(1/2)(1+

+ K2/2)l/’ exp(-K/2).

Substitutng this expression into Eq. (5.9) gives rn = ryu and C, 3 37r/(262) M 106 in accordance with [Vaulina and Khrapak (2000); Vaulina and Khrapak (200l)l (here, 6, = &A,/d d = (1 - 7rfi/8)’/’/2aT z 0.211 at the melting line of the bcc-structure, see Table 5.2). On the other hand, the assumption that for the fcc structure w, = wf,, cc d F / d l leads to l?yu =const on the crystallization line for lattices of both types thus contradicting the results of numerical simulations, see Fig. 5.33. Suitable approximation w?,, M 2a;nd(eZd)’ exp(-K)(K-a,)/rnd m d can be obtained for a homogeneous system with the gradient dFc/dl of the sum Fc of the electrical forces estimated as dFc/dl CK nd(eZd)’exp(-K)(Ka,). Thus, assuming that 6, = (1 - 7r~‘5/6)’/~/2a, on the melting line of the fcc lattice, we find from Eq. (5.9) I?n = I?(& - a , ) e x p ( - ~ ) , Cp Z 3/(cr26,2) M 18.5 (see Table 5.2), and

r;

M

18.5(1+ K

+ ~’/2)/(r;

-

a,).

(5.10)

The use of the modified parameter I?; defined by Eq. (5.10) allows one to illustrate the behavior of the melting curves for the transitions fcc-lattice-

Structures and Phase Transitions in Complex Plasmas

241

liquid and bcc-lattice-liquid, Fig. 5.33. The normalized coupling parameter rn (see Table 5.2) and the modified coupling parameter ryua t the line of transition from the bcc structure to the fcc structure can be obtained from Eqs. (5.8)-(5.9) with w, = wbcc. We have

ryu

M

M K ~ ( K - a,)-2.

(5.11)

Taking into account that the possibility of the reverse transition from the fcc to bcc structure is defined by the frequency wf,,, as a criterion for this transition one can use

ryu

M

-

a,)-3(1

+ + 2p).

(5.12)

Note that condition (5.12) depends on the approximation of the frequency K 4 a,. However, calculations on the basis of Eqs. (5.11)-(5.12) (curves 3 and 4 in Fig. 5.33) fully determine the region of the triple phase transition ( K = 5.8-6.8) and agree well with the results obtained for A g / d (see Fig. 5.34). Since the determination of the coupling parameters on the lines of phase transitions is based on Eq. (5.9), their values are independent on the viscosity of the background gas. The ratio between the particle interaction and dissipation in the system is defined by the parameter C (5.7) which in the case considered here is given by = wbcc/uf,. Then for typical experimental con3 ditions ( p &’ 4 g/cm , T, 2 1.5 eV, C, 2 700, K S 2), E M (4nd[cm-3]/m[pm])1/2(pg[Torr])-1. If the particle radius is a = 2 pm, the number density n d is from l o 3 to lo5 ~ m - and ~ , the neutral gas pressure pg is from 1 to 0.01 Torr, the range is = 0.04-6.9. It should be noted that the obtained results (including the criteria for the phase transitions) are independent on the viscosity of the surrounding gas. Therefore one can conclude that the phase state of a complex plasma system is mostly determined by the value of the coupling parameter while dissipation affects dynamic relaxation processes during phase transitions [Vaulina et al. (2004)l.

wf,, and therefore can be incorrect for small

c

c

5.3.3

Experimental observations of phase transitions

The most of experimental observations of phase transitions in a complex plasma are being done in discharge plasmas. The detailed description of

242

Physics and Applications of Complex Plasmas

these significant experiments can be found in [Thomas et al. (2003)l. In such plasmas, melting can be stimulated in three different ways. First, it can be induced by increasing the input power to the plasma (increasing the current in the case DC discharge). This leads to increasing plasma density and therefore decreasing plasma Debye length, i. e., decreasing strength of the interparticle interaction. Thus the coupling parameter I? decreases due to the decrease of U . Observations show that the dust particles come closer together, the diameter of the particle cloud shrinks and the mobility of the particles increases. The dust kinetic temperature T d also shows some increase leading to further decrease of I?. Second, it can be triggered by decreasing the neutral gas pressure. This leads to the change of the viscosity of the buffer gas affecting the energy dissipation rate and therefore to the change of the parameter (. As was mentioned before, the change of C itself does not affect the phase state. However, the decrease of the gas pressure leads also to an adjustment of the plasma parameters, and to the increase of the dust particle kinetic temperature (obtained from the dust velocity distribution function which remain Maxwellian within a wide range of gas pressures). Despite the effect of the pressure change on the overall discharge parameters the pressure variation provides an easier experimental way to control a dusty plasma through the melting phase transition as compared to the changing input power. Finally, melting can occur with increasing the number of dust particles in the plasma crystal. In this case, the associated phase transitions are also related to the external confining potentials. The increase of the dust number density leads to the development of the confining instability as discussed in Sec. 4.2. The melting occurs at smaller interparticle distances, in apparent contradiction to the increased interaction energy U . However, the dust kinetic temperature T d is also increased in this case as a result of the confining instability, thus keeping I' decreased. Contrary to this case in a thermal dusty plasma, where no confining potential is applied, an increase of the particle number leads t o increased order the system consistently with increased r (since the interactions energy U is increased and the kinetic temperature T d is not changed). In a typical experiment on phase transitions in RF discharge complex plasma, the melting starts from the well-established crystalline state. Phase transitions are followed by continuous lowering of the gas pressure or input power. The change of state is monitored by determination of such structure

Structures and Phase Transitions in Complex Plasmas

243

and dynamic properties of the plasma crystal as translational and bondorientational correlation functions, thermal and transient particle motion, self-diffusion, viscosity, and interaction cross sections. On the basis of the analyses, the two transitional states, the “flow and floe” and the “vibrational”, can be identified. The “flow and floe” state characterized by the coexistence of clusters of ordered crystalline structures (floes) and differently directed multiple transient flows of particles. The particle thermal velocities correspond to the room temperature, and the transient flow velocities are typically of a half of the thermal velocity. The translational and orientational orders are decreased significantly as compared to the crystalline state, and occasional vertical particle migration to other lattice planes occurs. This state appears practically always when the phase transition is induced by the pressure change. If to change the input power, this state can be absent when the pressure is high enough: The “flow and floe” state can attributed to the “clusterization” phenomenon seen in numerical simulations when the diffusion rate is suddenly dropped, see Sec. 4.1.4. Thus for higher pressures (large dissipation rates) the particle condensation (clusterization) is suppressed. The “vibrational” state characterized by a return to a more orientationally ordered structure and diminished flow regions. The vibrational amplitudes, thermal energy, and vertical migration of particles increase. The translational order continues to decrease. This state is present for any scenario of inducing (decreasing pressure or increasing input power) the phase transitions. Finally, we note that in experiments, the phase transitions in complex plasma crystals are also affected by such features of the crystalline structure as, e.g., defects and the number of layers. Defects appear as starting points for instabilities [Schweigert et al. (1998)l; in multi-layer crystals, the lower layers are less stable than the upper ones [Samarian (2002)l; the vertical instability (see Sec. 4.2) also contributes to the transitions. One of the interesting cases of the influence of so-called strong defects is an example of an experiment with an additional dust particle placed below or above the lattice layer.

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Chapter 6

Waves and Instabilities in Complex Plasmas

Different arrangements of the dust particle component in a plasma result in different equilibrium states of the entire complex plasma system, which can be destabilized by some internal (e.g., fluctuations or drift of the plasma parameters) or external (e.g., laser beams or external bias) means. In such cases a broad spectrum of numerous collective oscillations and waves appears that can be separated into two major subclasses. The first one includes waves and oscillations originating due to collective motions in the plasma subsystem and affected by the colloidal phase (dust grains). On the other hand, collective motions of the solid grains in various dust structures (e.g., dust clouds, strings, liquids or crystals) sustain other waves and oscillations in complex plasma systems. In this case the plasma strongly affects the development of the dust-related collective motions. From previous chapters we know that the massive and negatively charged dust grains substantially modify properties of the plasma. In particular, they lead to the appearance of new eigenmodes and instabilities, modification of the existing ones, as well as other dust-specific phenomena. Besides the effects associated with their large mass and charge, which affects the overall charge balance, the dust also significantly increase the dissipative loss in the system. Moreover, the charge of the dust grains can fluctuate and results in the relaxation of dust charges. These processes make the picture of particle and energy exchange in a dusty plasma system rather complicated. A description of the dust charging processes requires a rigorous account of electron and ion capture by the dust. Furthermore, the motion of the electrons and ions is perturbed because of the strong momentum loss caused by the Coulomb interactions, the charge capture collisions, as well as friction with the neutrals. The resulting effective collision frequency appearing in the momentum conservation equations can

245

246

Physics and Applications of Complex Plasmas

be much larger than that of the electron-neutral and ion-neutral collisions which often dominate in dust-free laboratory plasma devices. Therefore the inclusion of the variable dust charge and related dissipative effects can be crucial in describing the collective excitations and instabilities in most laboratory and space dusty plasmas. Realizing the futility of the attempt to discuss all possible kinds of waves and oscillations in the complex plasma systems, we limit here the scope by discussing a few representative examples. Section 6.1 is focused on the dustlattice waves that develop over equilibrium arrangements of the colloidal particle ensemble in the plasma. These will include oscillations excited in one-dimensional chains of spherical or rod-like particles suspended in the plasma as well as waves in arrays of magnetized dust particles. The scope of Sec. 6.2 are various plasma waves propagating in weakly coupled complex plasma systems. The examples include waves that originate due to collective plasma motions a t the electron (Langmuir waves) or ion (ion-acoustic waves) time scales and specifically address the issue of the effect of dust. On the other hand, dust-acoustic waves are supported by the collective motions of (weakly coupled) dust grains and are affected by the plasma. Depending on the strength of inter-dust interactions (e.g., weakly-coupled or strongly-coupled cases) the waves excited in dust particle ensembles can be regarded either as “string-controlled” excitations (dust-lattice waves) propagating in particle chains (as is the case in solid crystal lattices) or a sequence of compressions and rarefactions of the dust density (dust-acoustic wave) propagating in an ionized gas, which is similar to acoustic waves in gases. The key focus of Sec. 6.3 is the effect of the dust particle component on some waves in magnetized plasmas. In particular, we demonstrate that charged dust grains can have a large effect on the dispersion characteristics of hydromagnetic Alfvkn and magneto-acoustic waves propagating a t frequencies well below the ion-cyclotron frequency, even if the proportion of the total charge on the dust is quite small (as, e.g., can happen in space and astrophysical plasmas, such as cometary atmospheres and interstellar molecular clouds). Wave energy propagating at oblique angles to the magnetic field in an increasing density gradient can be very effectively damped by resonance absorption process in a dusty plasma. It is shown that the Alfvkn resonance is strongly affected by the dust. The study of wave phenomena in complex plasmas would not be complete without a discussion of their stability and the means of their excitation in most common real situations. To this effect, we discuss some of the com-

Waves and Instabilities in Complex Plasmas

247

mon instabilities in complex plasma systems in Sec. 6.4. For example, the ion-acoustic and dust-acoustic waves are often excited as a result of development of the current-driven ion-acoustic (Sec. 6.4.1) or dust-acoustic (Sec. 6.4.2) instabilities. Buneman-type ion-streaming instability is discussed in Sec. 6.4.3. Parametric and modulational instabilities in magnetized dusty plasmas are considered in Sec. 6.4.4. Finally, in Sec. 6.4.5 we briefly overview other instabilities in weakly coupled complex plasmas.

6.1

Waves in Arrays of Colloid Particles

In this section, we present the dispersion properties of waves in regular arrays of dust particles. We start with an overview of the most common modes of oscillations of spherical electrically charged particles (Sec. 6.1.1) and typical experiments on dust-lattice waves (Sec. 6.1.2). Then we consider in more details the dust-lattice transverse waves with out-plane motions of grains (Sec. 6.1.3). Waves in arrays of rod-like particles exhibit new modes associated with rotational degrees of freedom (Sec. 6.1.4). Similar types of modes, but associated with rotational degrees of magnetic moments, exist in lattices composed of magnetized dust particles (Sec. 6.1.5). 6.1.1

Dust-lattice waves

Solid matter can support two types of low frequency modes, longitudinal (compressional) and transverse (shear) waves. A crystalline lattice in a complex plasma also can support vibrations of two kinds of dust lattice waves (DLW), namely, compressional and shear, with longitudinal and transverse polarization, respectively, and with different propagation velocities. The standard consideration of the wave propagation is based on assumption that dust particles interact via screened Coulomb (Debye or Yukawa) potential, and therefore the dust charge and the plasma shielding length enter the wave characteristics. The longitudinal and transverse wave velocities depend on the particle charge and reveal different functional dependence on the shielding parameter. The dust-acoustic lattice wave (DALW), in the simplest approximation, can be considered in a one-dimensional chain of dust particles taking into account the nearest-neighbor interactions. The dispersion relation is (6.1)

248

Physics and Applications of Complex Plasmas

where rg is the distance between the neighboring dust particles [Melandso (1996)l. In the long-wavelength limit, this gives an acoustic-type linear dispersion thus explaining earlier attempts to employ the known dust-acoustic wave (DAW) dispersion to characterize the compressional waves in dust lattices. For shorter wavelengths, the actual dispersion of DALW in complex plasmas strongly depends on the coupling parameter rc [Winske et al. (1999)] and significantly differs from DAW as well as can differ from the predictions of the simplest model (6.1) (note that models of a strongly coupled plasma in the gas phase [Rosenberg and Kalman (1997)] give similar acoustic-type dispersion in the long-wavelength limit as well as strong dependence on I‘). In Fig. 6.1, several curves corresponding to different models of DAW and DALW are presented.

Fig. 6.1. Frequency w , normalized t o the dust plasma frequency W d , versus wave number k , normalized t o the inverse ion Debye length ,:A, as computed from fluid theory (DAW, solid curve), fluid theory with strongly coupled corrections (dotted and dotdashed curve), and lattice dynamics (dashed curve) [Winske et al. (1999)l.

There are two kinds of dust transverse lattice waves (DTLW), which can be classified according to whether the particle motion remains in the plane or is out of the plane. The out-of plane transverse waves D T L W l are considered in more detail below in Sec. 6.1.1. Their dispersion relation in the simplest case of a one-dimensional chain taking into account the nearest-neighbor interactions [Vladimirov et al. (1997a); Vladimirov and Crarner (1998)] is (6.2)

where ys characterizes the confining potential in the vertical direction. Note the ‘minus’ sing in front of the dispersive term as well as the different factor

Waves and Instabilities in Complex Plasmas

249

+

(1 ro/Xo) before the exponential as compared with Eq. (6.1). Because of the negative sign of the dispersive term, this transverse mode is of optical character (the frequency decreases with the wave number), in contrast to the acoustic-type character of the compressional lattice wave. On the other hand, the factor (1 r o / X D ) is characteristic of the shear-type motion of screened Coulomb interacting particles. For in-plane (horizontal) motions in one-dimensional chain of dust particles, the dispersion relation of the transverse mode is similar to Eq. (6.2), with the term corresponding to the confining potential now containing ”ir, i. e., the potential strength in the horizontal direction. In two-dimensional lattices, the shear-type DTLWll mode characteristics depend on the type of the crystalline lattice. For example, in a hexagonal lattice, the dispersion relation can be written as [Wang et al. (2001)]

+

w

klro

= -2Q2

mdri

I,m

F(l,m)sin22 ’

(6.3)

where

+

and R2 = 1’ m2. Here, we see the interplay of the “pure compressional” and “pure shear” factors as reflected in the function F(1,m): indeed, the factor (3 + 3yiR yi2R2),yi = r o / X o , reminds (with the correction because of the hexagonal-type geometry) the corresponding expression in Eq. (6.1) while the factor -(l+rcR) refers to the same one in Eq. (6.2). This is easy to understand since in the case of in-plane motions, the particle displacements that are sheared to one neighbors, are compressional to other neighbors. Thus we see that dispersion relations for all the above modes of lattice vibrations depend on the dust particle charges as well as on the plasma screening. This fact can be effectively used for diagnostics of the dust particle parameters as well as of the plasma parameters. For example, observations of vertical motions of the dust are important for diagnostics of processes in the plasma sheath [Melzer et al. (1996a); Peters et al. (1996)], especially in the case of several vertically arranged horizontal layers when vertical oscillations are affected by the parameters of the ion flow [Vladimirov et al. (1998a)l. Note that the spontaneous excitation of vertical vibrations of dust grains was also experimentally observed [Samarian et al. (2001b); Nunomura et al. (1999)], and driven vertical oscillations were studied in a separate series of experiments [Homann et al. (1999); Zafiu et al. (2001)].

+

250

Physics and Applications of Complex Plasmas

On the other hand, molecular dynamic simulations [Totsuji et al. (1997)] clearly demonstrate a sequence of phase transitions associated with vertical arrangements of horizontal chains when the strength of the confining (in the vertical dimension) parabolic potential is changed. We note that while melting-type phase transitions in three-dimensional crystals are typically associated with excitation of acoustic mode fluctuations, the “structural” transition of the vertically arranged chains of dust particles is associated with excitation of the optic-character transverse mode (6.2). The vertical rearrangements of the dust grains are also directly connected with the possible equilibria of the system as we have seen in Sec. 4.2. There are experimental observations of the formation of colloidal structures composed of elongated (cylindrical) particles [Annaratone et al. (2001); Mohideen et id. (1998); Molotkov et al. (2000)] levitating in the sheath region of a gas discharge plasma. The experiments demonstrate that there are various arrangements of such grains, levitating horizontally ( i e . , oriented parallel to the lower electrode and perpendicular to the gravity force) and vertically ( i e . , oriented perpendicular to the lower electrode and parallel to the gravity force). In the case of rod-like dust particles additional modes appear due to the new (rotational) degree of freedom [Vladimirov and Tsoy (2001); Hertzberg et al. (2003a)]. The “liquid crystal” dust lattices [Molotkov et al. (2000)l composed of rods should therefore exhibit the rotational oscillation modes similar to those in liquid crystals. Excitation and interactions of all these modes lead to new types of phase transitions and affect those phase transitions existing also in lattices composed of spherical grains. New features appear in structures composed of charged particles with a non-zero magnetic moment levitating in a gas-discharge plasma in the presence of an external magnetic field, including agglomeration due to magnetic attraction of grains [Samsonov et al. (2003); Yaroshenko et al. (2003)I. Lattices composed of magnetized particles exhibit rotational oscillation modes [Vladimirov et al. (2003d)l related to the magnetic interactions amongst themselves and with the external magnetic field, analogous to those existing in ferromagnetics. Excitation and interactions of these magnetic oscillation modes affect the lattice dynamics and phase transitions by modifying those transitions existing in lattices composed of unmagnetized grains and introducing new transitions associated with the dust magnetic properties.

Waves and lnstabzlztzes zn Complex Plasmas

251

-

Fig. 6.2 Experimental setup for DLW investigation. (a) One-dimensional chain [Homann et al. (1997)l; (b) two-dimensional lattice [Nunomura et al. (2002)].

6.1.2

Experiments o n dust-lattice waves

An experimental observation of DLW is typically performed in a parallel plate rf discharge as that shown in Fig. 6.2. A linear chain of particles as well as purposely (typically, rectangular) shaped two-dimensional plasma crystal can be formed by a suitable potential traps. For this, small bars are laid on (or a groove can be made in) a lower electrode to create the required potential well. For studying waves in a two-dimensional plasma crystal the potential trap is modified using a rectangular groove or wider bar. The dust particles are observed in scattered laser light by means of CCD cameras from the top and from the side. Preliminary estimation of the friction coefficient and the value of the charge on the dust grains provide data for the analysis of the wave characteristics. In the one-dimensional chain, a DLW is exited by pushing the first particle in the horizontal direction for the compressional DALW and in the

252

Physics and Applications of Complex Plasmas

vertical direction for the shear DTLWl mode. In the two-dimensional lattice, the waves are excited by pushing the side chain of dust particles; a differently arranged (see Fig. 6.3) horizontally directed push excites DALW and DTLWII. A laser sheet expanded in the y-direction is used for a longitudinal wave excitation. To launch a transverse wave, the laser sheet is spread in the x-direction. To excite a plane wave in the two-dimensional case, much higher laser power is required. The laser beam has to be expanded into a sheet by means of a cylindrical lens, or rotating mirror which can illuminate an area of few millimeters. In order to hit the first few rows of particles only, the laser sheet is adjusted.

&3

Fig. 6.3 Method of wave excitation for DALW and DTLWI/ [Nunomura et al. (2002)].

The particles can be also pushed by a sinusoidal biased pin electrode placed close to the end of the chain (or plasma crystal). However, excitation with the electric force distorts the shape of the plasma trap and affects all particles simultaneously so that it is difficult to estimate the forces acting on each dust particle. Moreover, the negatively biased electrode can induce the ion flux which affects the interparticle potential and further complicates the wave analysis. Thus, the most convenient way of the wave excitation is to push the particle(s) by the radiation pressure of a laser beam (see Fig. 6.2). Typically the beam is larger than the particle diameter, therefore it is necessary to adjust the focus carefully to ensure that the most radiation is

Waves and Instabilities in Complex Plasmas

253

fallen on the first particle. By fine adjustment of the laser beam the other particles stay practically unaffected by the beam thus ensuring that their further motions are in the field of the wave. To excite a DLW, the laser beam should be modulated. This can be achieved by an internal modulation of the laser (for example, by using a laser diode with an alternative current) or by using an unmodulated beam with an external rotating (oscillating) chopper.

Fig. 6.4 Sequence of 15 snapshots for a linear chain arrangement of 12 particles [Homann et al. (1997)l. The time step between each image is 100 ms.

The typical frequency range of DLW is 0.5-15 Hz. Figure 6.4 shows the wave motion in a chain of 12 particles. The compression and expansion of the linear chain propagates as a damped wave into the linear chain. The particles at the far end stay practically unaffected by the excitation. This is different from the excitation by the pin electrode method, where the distortion of the potential trap affects all the particles. Analysis of the particle motion shows that the response of the particles is sinusoidal and by fitting the sine waves to the trajectories of the individual particles one can extract the phase change along the chain as well as the wave damping, and finally obtain the real and imaginary parts of the wave number. In

254

Physics and Applications of Complex Plasmas

-

-

experiment [Homann et al. (1997)], the wave dispersion at a gas pressure of 22 Pa, a particle charge of Q 14000e, and the coupling constant rc lo4 were obtained.

Fig. 6.5 Measured dispersion relation denoted by symbols and the theoretical dispersion relation of a DLW (solid line) and a DAW (dotted and dashed lines) are shown [Homann et al. (1997)l. For the DAW two sets of parameters are plotted. Dotted line: same parameters as DLW. Dashed line: best fit for A. In (a) the real parts and in (b) the imaginary parts are shown.

The wave number qr and the damping factor qi in the linear chain as a function of the wave frequency in comparison with the DLW model for different shielding factors are presented in Fig. 6.5. The DLW model represents very well the experimental points, whereas the DAW model is not in agreement with the experimental data although, as was mentioned above, in the long-wavelength limit the experimental data agree with DAW model as well. Using the excitation method presented in Fig. 6.3(b), DTLWll were experimentally investigated by [Nunomura et al. (2000)l. The dispersion relations measured in this experiment, reveal an acoustic-type mode with

Waves and Instabilities in Complex Plasmas

255

non-dispersive character over the entire range of wave numbers. Figure 6.6 presents the experimentally obtained dispersion relations. Note that the measured DTLW((linear dispersion demonstrates the above mentioned mixed (compressional and sheared) character of this mode.

Fig. 6.6 Dispersion relation of DTLW(( [Nunomura et al. (2000)]. Experimentally measured wave numbers k , and Ici are shown as filled and open circles, respectively. A solid line indicates the sound velocity.

Exp.

Fig. 6.7 Theoretical dispersion curve with an experimentally observed data point for DTLW [Takamura et al. (2001)].

A self-excited DTLW propagating along a one-dimensional dust chain was reported by [Takamura et al. (2001)l. In the experimental setup, a rectangular-shaped electrode on the biased mesh confines the onedimensional chain of levitated dust particles similar to that shown in

256

Physics and Applications of Complex Plasmas

Fig. 6.2(a). The frequency of the self-excited vertical oscillation of dust particles is about 10-15 Hz. Figure 6.7 shows the dispersion relation calculated for given experimental conditions. The wavelength obtained from the time evolution of the levitated vertical positions of dust particles, and the frequency obtained by FFT analysis yield an experimental point on the w - k plot. 6.1.3

Dust-lattice transverse waves with out-of plane motions

As we have mentioned already, motion on the dust grains in the vertical direction can provide a useful tool for determining the grain charge. The vertical vibrations of dust particles lead to a low-frequency DT L W l mode. The mode is characterized by an optical-mode-like inverse dispersion (2. e., its frequency decreases with the growing wave number) if kro << 1 where k is the wave number, TO is the interparticle distance in the chain, and only nearest-neighbor interactions are taken into account. As we have seen in Sec. 2.3.1, the charge of dust particles, appearing as a result of the electron and ion current onto the grain surfaces, strongly depends on the parameters of the surrounding plasma and therefore on the levitation height. Thus we also show here that the dependence of the dust particle charges on the sheath parameters has an important effect on the oscillations and equilibrium of the grains in the one-dimensional chain, leading t o a disruptions of the equilibrium positions similar to the case of an isolated particle considered in Sec. 6.1.3.1, and a corresponding transition to a different vertical arrangement, see Sec. 6.1.3.2. In most of the experiments, the Coulomb dust lattices consist of a few layers of dust particles levitating above the electrode. In particular, there are two low-frequency modes associated with vertical oscillations in the Coulomb crystal of dust grains arranged in two horizontal chains in a sheath region of a low-temperature gas discharge plasma and the dispersion relations and characteristic frequencies of the modes were found [Vladimirov et al. (1998a)l. Here, we consider vertical oscillations of dust grains in two vertically ordered one-dimensional horizontal chains. In the following we demonstrate that two low-frequency modes (associated with the vertical vibrations of the dust) can propagate in such a system. Both of the modes are characterized by the optical-mode-like inverse dispersion (2. e., its frequency decreases with growing wave number) if kro << 1 and only the nearest-neighbor interactions are taken into account.

W a v e s and Instabilities in Complex Plasmas

6.1.3.1

257

Oscillation modes in one-dimensional chains of particles

Consider vibrations of a one-dimensional horizontal chain of particles of equal mass md separated by the distance ro, see Fig. 6.8.

Fig. 6.8 Oscillations of dust particles in a one-dimensional chain [Vladimirov et al. (1997a)l. The mass of dust particle m d is denoted as M in this figure.

We assume that the interaction potential between the particles can be approximated by the screened Coulomb law. In addition to the electrostatic Debye shielded force, the gravity force and the sheath electrostatic force FE = Q E ( z )act on the dust grains in the vertical direction z . The balance of forces in the linear approximation with respect to small perturbations 6z of the equilibrium at z = 0 gives the equation of motion for the vertical oscillations

(6.4)

where (6.5)

and ys is a constant assuming a linear variation of the sheath electric field. Here, 62, is the vertical deviation of the n-th particle from its equilibrium position. Note that although in general particles oscillate in the vertical as well as in horizontal direction, see Fig. 6.8, in the linear approximation their transverse and longitudinal vibrations are not coupled. Substituting 62, = Aexp(-iwt iknr,) into Eq. (6.4) gives the dispersion (6.2) of the

+

258

Physics and Applications of Complex Plasmas

vertical vibrations. For k = 0, the characteristic frequency is given by w2 = - y s / m d , decreasing with the growing wave number when lcro << 1.

To estimate the effective width of the potential well -ys, consider the standard model of the sheath with the Boltzmann distributed electrons and ballistic ions, see Sec. 2.3.1. When e4o << T,, the characteristic frequency is approximately given by [Vladimirov et al. (1997a)l (6.6)

-

where AD M (T,/41rnoe~)~/’N 2 x l op2 cm and ug/uz 1.5. Figure 6.9 presents the numerical result for the frequency f o as a function of the dust charge Q. Thus, vertical oscillations of a one-dimensional chain of dust grains with a constant charge levitating in the sheath field of a horizontal negatively biased electrode can give rise to a specific low-frequency mode, which is characterized by inverse optical-mode dispersion when the wave lengths far exceed the inter-grain distance.

Fig. 6.9 Frequency fo in Hz vs dust charge IQ/eI x 10W4 for oscillations of particles with a constant charge in a onedimensional chain [Vladimirov et al. (1997a)l.

Considering now the equilibrium and oscillations of a horizontal chain of dust grains with variable charges, we note that the parallel (to the electrode) component of the interaction force acting on the particle at the position n due to the particle a t the position n - 1 (see Fig. 6.8) can now be written as q,n,n-1 = RIIE2(1+

El)exP(-ll)

>

(6.7)

where XD(Z) = AD,(z) = ( T , / 4 1 r n ( z ) e ~ ) ~ / ~=, IRI/AD(Z,-~), and & = &,(~,)Q,-l(z~-l)/IR1~. Note that now the dust charge Q and as well as the Debye length AD are functions of the vertical position of the dust

Waves and Instabilities in Complex Plasmas

259

particles. The component of the interparticle force in the z-direction is given by Fz,n,n-1 =

(zn - zn-1)&(1+

< I ) ex~(-E1).

(6.8)

Assuming only the nearest neighbor interactions, a small perturbation Sz of the equilibrium z = zo gives the equation of motion for vertical oscillations of the grains in the linear approximation

(6.9)

+

Substituting Sz, N exp(-iwt iknro) into Eq. (6.9), the expression for the frequency of the mode associated with the vertical vibrations at the position zo is obtained

(6.10)

This mode has an optical-mode dispersion similar to that studied above in the case of a constant grain charge. With the charge variation taken into account, the characteristics of the mode are strongly affected by the sheath parameters, in particular, by the sheath potential. Note that this equilibrium is stable only when the last term on the right hand side of Eq. (6.10) dominates over the first one. The case when both terms are equal to each other, corresponds to the phase transition associated with the vertical rearrangement of the type N + N+1, where N is the number of the one-dimensional chains in the vertical dimension. This type of transition is similar to the change of particle arrangements of two particles, Sec. 4.2. Note that in the case of a horizontal chain of interacting dust particles, with a non-negligible negative first term on the right hand side of Eq. (6.10), the oscillation frequency can become zero (and hence the equilibrium can be disrupted) for even smaller dust sizes than acr, presented above. 6.1.3.2

Oscillation modes in two vertically arranged one-dimensional chains

Here, we consider vibrations of the two one-dimensional horizontal chains of dust particles of equal mass md separated by the distance ro in the hor-

260

Physics and Applications of Complex Plasmas

Fig. 6.10 Forces acting on dust particles in two one-dimensional horizontal chains: the gravitational and the sheath electric external fields, as well as the wake Fl and Debye Fz interaction fields [Vladimirov et al. (1998a)l. The mass of dust particle m d is denoted as 111 in this figure.

izontal direction and d in the vertical direction, as shown in Fig. 6.10. In the simplest approximation, the particles interact with their neighbors in horizontal as well as in the vertical direction. Interaction in the horizontal direction leads to the low-frequency modes associated with the horizontal and vertical vibrations of dust grains. An important feature in the consideration of the interaction potential between particles in the vertical direction is that the forces between them are asymmetric because of the wake potential. Thus the (wake) potential in the vertical direction acting on the lower particle due to the upper one is (see Sec. 2.3.1) (6.11) where 121 is the distance between the dust grains in the vertical direction. We remind that the potential (6.11) is only applicable within the Mach cone while outside the Mach cone the particle potential can be approximated by the Debye formula. Therefore the potential acting on the upper particle due to the lower particle (as well as the interaction potential between particles in the same chain) is the simple Debye repulsive potential.

261

W a v e s and Instabilities in Complex Plasmas

The balance of forces in the vertical direction, in addition to the electrostatic Debye and the wake potential forces, includes the gravity force as well as the sheath electrostatic force acting on the dust grains. In equilibrium, since the interchain distance d is assumed to be small compared with the distance 201 of the lower chain from the electrode (as well as small compared with the width of the sheath), we can assume that the sheath electric field in the range of distances 201 to 2 0 2 = 201 d can be linearly approximated as

+

(6.12)

where yo is a constant and 20 is the equilibrium position of a particle of mass m d due to the forces m d g and FE only. We stress that the actual equilibrium positions of particles in lower and upper chains are 201 and 2 0 2 , respectively. The equilibrium balance of the forces in the vertical direction acting on the lower chain and the upper chain can be written as (6.13)

and (6.14)

where Ff,2 are the forces of interaction between the chains due to the potentials and @ 2 . Here, since 202 = 201 d ,

+

The equilibrium distance d can be found from the balance equations (6.13) and (6.14) after substituting Eq. (6.12)

By introducing small perturbations dzi,+ of the equilibrium at z ~ iwhere , i = 1 , 2 for the lower and upper chains, respectively, and including interactions with nearest neighbors in each chain, one obtains

-rohL +

- SZI,,)

,

(6.15)

262

Physics and Applications of Complex Plasmas

where, in the linear (small amplitude) approximation the coefficient B is

and

+

Substitution of SZi,n = Aiexp(-iwt iknro) into Eq. (6.15) gives the dispersion and amplitude relations for the two modes kro m d

A1

= A2

,

(6.16)

mdro

and (6.17)?

Ai = Az (71/ 7 2 ) . For k

=

0 the characteristic frequencies are given by w:

+

= To/md

and

w i = (70 71 - y 2 ) / m d 1 and they decrease with the wave number when kro << 1.

To estimate the effective width of the electrode potential well 70, again the standard model of the sheath, which considers Boltzmann distributed electrons and ballistic cold ions, is employed. For simplicity, it is assumed that the sheath electric field near the position of the dust grains can be linearly approximated. The electric field is found taking into account the balance of the electric force and the gravity at z = 20. Solutions for the sheath potential in the equilibrium position and for the equilibrium distance d in the vertical direction are found numerically [Vladimirov et al. (1998a)l. Assuming AD M 2 x lop2 cm, w ~ / wM ~1.5, Q/e = 2 x lo4, md = 0 . 6 lo-' ~ 9 g, the equilibrium vertical distance is given by d = 1.75X0, and 71/72 % -25.3. The characteristic frequencies f i = wi/27r of the two modes are f i ( k = 0) M 21.3 Hz and f 2 ( k = 0) M 63.5 Hz. Results of numerical calculation of the frequencies f l and fi as functions of the dust charge Q are presented in Figs. 6.11 and 6.12. Since the equilibrium distance d between the chains is almost independent of Q, the frequency f 2 is approximately directly proportional to Q. Note that the amplitude of dust grain oscillations in the lower chain

Waves and Instabilities in Complex Plasmas

263

for the second mode (when the grains oscillate with opposite phases) is much smaller than the amplitude of the oscillations in the upper chain, IAl/Azl = Jy1/y21M 25.3. For the first mode, the amplitudes are the same for the upper and the lower particles.

Fig. 6.11 Dependence of frequency f l of the first mode (the amplitudes are the same and in phase for the vertically arranged grains in the first and in the second chains) vs the dimensionless grain charge q = IQ/el x 10W4 [Vladimirov et al. (1998a)l.

Fig. 6.12 Dependence of frequency f2 of the second mode (the amplitudes are different and in counter phase for the vertically arranged grains in the first and in the second chains) vs the dimensionless charge q = [&/el x [Vladimirov et al. (1998a)l.

10-4

Thus the vertical oscillations of the two one-dimensional chains of dust grains levitating in the sheath field of a horizontal negatively biased electrode can give rise to two low-frequency modes which are characterized by the inverse optical-mode dispersion when the wave lengths far exceed the intergrain distance. The second mode can be especially useful for diagnostic purposes because its dispersion is almost directly proportional to the dust grain charge Q which in turn is a function of plasma parameters. We stress that while the interparticle distance d in the vertical direction is defined

264

Physics and Applications of Complex Plasmas

by the equilibrium balance of forces, the horizontal interparticle distance TO is a free parameter in the presented theory; in real experiments, it is determined by many factors (including the horizontal confining potential) and usually is of the order of the Debye length AD. Finally, we note that the charge on a dust grain is strongly dependent on the grain size as well as on the ambient plasma parameters. Although grains in many experiments are usually geometrically identical, plasma parameters can differ in the position of the upper and the lower chains thus leading to different charges of particles Q1 and Q 2 in the respective row. Different charges will change the coefficients y0,1,2for the upper and lower chains thus strongly affecting character of the modes. In particular, the first mode, corresponding to oscillations of upper and lower particles in phase with the same frequency f l , will be split into two modes with close frequencies if the difference in charges &I - Qz is small compared with these charges.

6.1.4

Dust-lattice waves in the arrays of rod-like particles

For oscillations in the one-dimensional chain consisting of rod-like dust particles levitating in a plasma, new oscillation modes associated with the rotational degrees of freedom appear. The simplest case corresponds to the rods with given (and static) charge distribution [Vladimirov and Nambu (200l)l along the rod length. More complicated is the case when the interaction of the rod-like particles between themselves and with the plasma is studied dynamically together with their charging, thus demanding that the problem of the charging of rods by the surrounding plasma should be first solved [Ivlev et al. (2003)l Here, we model the rod-like particle as the rotator having two charges (and masses) concentrated at the ends of the rod [Vladimirov and Tsoy (200l)l. For further simplicity, the charges are assumed to be fixed and the masses are equal. The rod of the length l d , connecting these two charges, has zero radius and mass. Consider the following geometry: the one-dimensional rod chain is along the z-axis, with the distance T O between the centers of masses of the (unperturbed) rotators, R" is the radius-vector of the center of mass of the n-th rotator (of equal masses m d at the rod ends, the center of mass is located in the center of the rotator, at the distance ld/2 from its ends), the angle 0" is between the n-th rotator and the z-axis, and the angle 4" is between the x-axis and the projection of the n-th rotator on the xy-plane. At the upper end of the n-th rod, there is a spherical particle (coordinate a") with the charge Q a , and mass m d , and at the lower end of the same

265

Waves and Instabilities in Complex Plasmas

rod there is another spherical particle (coordinate b") with the charge Q b and mass m d . The radius-vectors of the n-th rod ends are a" = nrge, R" l d S n / 2 and b" = w o e , R" - 1dSn/2, where S" = (cos 4" sin On,sin 4" sin On, cosO"). The four distances between the ends of the n-th and (n 1)-th rotators are given by

+

+

r:&lb = b" - a(b)"+' = -roez

+ c1

-

c2

,

+ +

(6.18)

where q1 = R" - Rn+' and c2 = l d ( s " ,"+')/a. Here, the upper sign on the right hand side corresponds to r,, or r b b , and the lower sign on the right hand side corresponds to rb, or r,b, respectively. For the distances between the (n-1)-th rod and the n-th rod (e.g., for r:;), we have similar expressions with the simultaneous change (n+1) + ( n- 1) and TO + -TO. The general equation of motion of rod-like dust particles can be derived by using the Lagrangian approach [Landau and Lifshitz (1969)l (6.19) where s = (RZ, O", 8)and the Lagrangian of the system C accounts for the oscillatory and rotational degrees of freedom and the interactions between the the nearest particles. For simplicity, only motions in the ( x , z)-plane are considered below, such that S" = (S,",O,5':) = (sin O", 0, cos O n ) , 4" = 0, and R" = (P, 0,z"). General equation of motion (6.19) can be split into two equations, one for the motion of the center of the mass in the x direction and the other one for the rotation (described by the rotation angle 0 ) of the n-th rod. These equations can be used to study the interactions of different oscillation modes. However, for small deviations from the equilibrium state, the oscillations can be decoupled, and the linear dispersion relations can be derived. According to the experiments [Annaratone et al. (2001)], there are two preferred equilibrium positions of levitating rod-like particles: when the rotators are oriented vertically (ie., along the z-axis in our geometry) and when the rotators are oriented horizontally, see Fig. 6.13. Thus below we consider the dispersion relations for the modes associated with small deviations around these two equilibrium positions.

266

P h y s i c s a n d Applications of C o m p l e x P l a s m a s

Fig. 6.13 Rod-like micron-sized particles levitated in krypton plasmas [Annaratone et al. (ZOOl)]. The dots are vertically aligned micro-rods.

If there are only small oscillations 6z << min(r0, I d ) of the centers of mass of vertically oriented (0 = 0) rotators in z-direction (which is horizontal), the equation of motion (6.19) can be simplified 2 m d 6 X n = -(Qapab

+ &bpba)(26xn

-

6zn"

-

6 ~ " - ,~ )

(6.20)

where

+

L d = (r," = (Qa/lrl)exp(-lrl/x,) is the interaction potential, and the primes denote a differentiation over Irl. The expression for (Pba can be obtained from (Pab by interchanging the subscripts a and b. For perturbations propagating along the z-axis, Eq. (6.20) gives the following dispersion equation of the lattice-acoustic type

w2 =

2 - [Qa@:(ro)

m d

+ &b@b'(ro)

+ 2 Q a47 @ : ( L , j )+

(6.21)

Ld

where the Debye interaction potential was assumed to satisfy Q a @ b = Q b G a . In the limiting case l d < < T O , from Eq. (6.21) the standard dispersion relation for the dust-acoustic lattice wave (6.1) in the chain of particles with

267

W a v e s a n d I n s t a b i l i t i e s in C o m p l e x Plasmas

the charge Qa

+ Q b and mass 2md can be straightforwardly recovered

w2 =

2

+

-((&a m d

Qb)[@;(To)

+

+

+

@:(To)]

+

sin

2

kro 2 ,

(6.22)

+

where @:(TO) @:(TO) = (Qa Q b ) ( 2 2 r o / b T ~ / A $ exp(-ro/Ao)/ri ) for the interaction potential of the Debye type. For the horizontally oriented rod chain (0 = 7r/2) the dispersion relation for the acoustic mode is 2 W2

+ Qb@:(TO)

= -[[email protected]:(To) m d

+ Qa@:(To

- Id)

+

Qb@:(Tg

kro 2 Again, in the limit

+

2 I d ) ] Sin

-

(6.23)

and we have noted that in this case ro > l d . l d << T o , Eq. (6.22) can be recovered from Eq. (6.23). For small oscillations in the vertical direction of the vertically oriented rotators ( i e . , parallel to the z-axis), assuming the parabolic profile of the external potential (depending only on z ) Q a a e x t ( a n . ) QbQext(bn) = yv(zn - 1d/2)’/2 %(z” ld/2)’/2, where Y a , b > 0, one obtains

+

+

+

(6.24) which is the dispersion relation of the optical-type wave mode propagating along the chain. Here,

and = e a b ( b ++ u ) . In the limit ld < < T O , the dispersion relation of the optical-like mode (6.2) propagating in the chain of particles can be recovered from Eq. (6.24): (6.25)

+

+

+

Qb)(l r o / X ~ exp(-ro/XD)/ri ) for where @ ; ( d ) / d @L(d)/d = - ( Q a the interaction potential of Debye type. In the parabolic external potential and Debye interaction potential ( Q a @ b = Q b @ a ) , the dispersion relation for the optical mode of the vertical oscillations of horizontally oriented rotators (note that T O > I d ) is

m d

m d

268

Physics and Applications of Complex Plasmas

(6.26)

+

and we have noted that in this case QaBext(an) Q b Q e x t ( b n ) = y h ( ~ In the limit l d << T O equation similar t o Eq. (6.25) can be recovered from Eq. (6.26). Note that a slightly different character of the external potential was assumed to allow for the cases of the stable vertically or horizontally oriented rotators, respectively. The dispersion relation of the small rotating oscillations around 0 = 0 (z.e., for vertically oriented rotators) in a similar parabolic potential is

(6.27) where the standard relation Q a @ b = Q b a u was used. Note that for Debyetype interaction potential, the factor at the oscillating term on the right hand side part of Eq. (6.27) is always positive. This means that although in general there is the frequency gap, similar (but not equal) to that of the vertical oscillations of the center of mass of the rotators, the dispersion character is different as compared with the case of the vertical vibrations: because of the sign of the factor in front of the dispersion term in Eq. (6.27), there is no anomalous dispersion (i.e., the mode frequency increases with the increase of the wave number). Note also the second term on the righthand side of Eq. (6.27): if positive, it allows rotational oscillation modes (in the za-plane) in the chain of the vertically oriented rotators even in the absence of an external confining (in the a-direction) potential due to the interactions with the nearest neighbors. For the Debye screening potential, this term can be either positive or negative depending on the relations between l d , d , and the plasma Debye length AD. This indicates that the system can be unstable with respect to rotations of rods on the angle 0 ; this can have important consequences for excitation of the corresponding mode and the related phase transition associated with the rotational (in)stability in the chains of rotators. Indeed, by changing the plasma characteristics, the originally stable vertically oriented equilibrium state of rotators can change its character and become unstable (and vice versa). Moreover, for marginally unstable equilibrium, because of the normal character of the wave dispersion, for some wavelengths the oscilla-

W a v e s and Instabilities in Complex Plasmas

269

tions still can be stable (ie. when the positive dispersive term exceeds the negative non-dispersive term). Finally, consider rotational oscillations of horizontally oriented rotators (around 0 = 7r/2). For the external potential supporting the horizontal orientation of the rotators (and, as usual, taking into account that for the Debye interaction potential Q a @ b = Q b a u ) , the dispersion is given by

(6.28) where the second term on the right-hand side originates from the nearest neighbor interactions. For the Debye interaction potential, it is always negative and can prevent the stable equilibrium of the horizontally'oriented rotators even in the case of the confining (in the z-direction) potential. Since the vertical confinement is usually associated with the properties of the plasma sheath (where the particles are levitated), the change of plasma parameters can lead to the change of the stability characteristics of the horizontally oriented rods. Depending on the particular character of the interaction (and on the plasma parameters), this feature can affect the excitation of the rotational oscillation modes of the horizontally oriented rotators. In the limit ld + 0, dispersion (6.25) can be recovered from Eq. (6.28). New features related to the rotational modes include an interesting interplay of interactions of the rotators with a plasma (formalized above by the terms containing the external potential) and with themselves [Vladimirov and Tsoy (2001); Hertzberg et al. (2003a)l. Combination of these interactions strongly affects the equilibrium positions and orientations of the rotators and therefore will influence phase transitions associated with such rotating modes. 6.1.5

Waves in chains of magnetized particles

When considering the equilibrium and oscillations in a one-dimensional chain of magnetized particles, i.e., particles having non-zero magnetic moment, we see that especially pronounced effects are for the cases when only the magnetic interaction can be singled out; for electrically charged spheri-

2 70

Physics and Applications of Complex Plasmas

cally symmetric magnetized particles these effects correspond to rotational modes associated with the directions of the magnetic moments which are absent in the case of purely electrically charged symmetric particles. Theoretically, each spherically symmetric particle is considered as having a charge Q, mass m d , and magnetic moment mo. For simplicity, we assume that the charges and magnetic moments are constant and the masses are equal. Similar to the previous cases, we consider a one-dimensional infinite linear chain of particles, with their centers evenly separated by the distance T O in equilibrium, along the x-axis. As before, the external potential Qext is a combination of the potentials due to gravitation and the electric field of the external electrodes; we also impose an external magnetic field Bo = (O,O, &) in the z-direction. The interparticle forces now include, in addition to the Coulomb electrostatic force, the magnetic force due to interactions of magnetic moments between the particles and with the external magnetic field. Since the electrostatically charged dust grains are shielded-by the surrounding plasma, the interparticle interaction is modeled by the standard plasma screened Coulomb potential; on the other hand, no plasma shielding for the magnetic moments is expected and therefore we have

(6.29) The magnetic moment of a spherical particle can be written as mi = 47ra3Mi/3 where Mi is the particle’s magnetization. Thus, the Lagrangian is now given by

(6.30)

We remind the reader that dot stands for the time derivative, the sum is over all the particles (except self-interactions), and I is the moment of inertia of each particle ( I = 2 m d a 2 / 5 for a spherical particle of radius a). Furthermore, 0 is the angle the magnetic moment makes with the z (vertical) axis, and Cp is the angle the projection of the magnetic moment onto the x - y (horizontal) plane makes with the x-axis. The summation implied in the calculation of QiQj for the ith particle is, in principle, over all the other particles. For simplicity, only the two nearest neighbor electrical and magnetic interactions are considered. The external potential is

Waves and Instabilities in Complex Plasmas

271

approximated by a parabolic potential; this assumption is valid for small oscillations, whose minimum lies at the center of mass of a particle (y = 0, z = O), i e . , Qiaext(ri) = (k,y2 kzz2)/2. The motion of the center of mass (we add the phenomenological friction 2yR proportional to the particle's velocity) is governed by the equations

+

a(mo .B,m+) d(mn .BZ) +

+

dRi

dRa

+

d(mo. Bo) , (6,31) dRi

where Ri* = Ri+l - Ri, BY = Bmo(R*),and @.',(r) = dQa(r)/dlr1 = -(Q/Irl)(A;' irl-')exp(-/rI/Ao). The last term on the right hand side of Eq. (6.31) equals zero if the external magnetic field is homogeneous. The angular motion for the angle 0 is governed by (we also add the rotational friction 2p6; note that p and y are not necessarily of the same order)

+

(6.32)

and similarly for 4. Equations (6.31) and (6.32) can be used not only to obtain dispersion relations for small amplitude oscillations, but also to study mode interactions for larger amplitudes. For small deviations from the equilibrium, the oscillations decouple, and linear dispersion relations can be derived. These general expressions for motion and rotation in all three dimensions are cumbersome and for simplicity we assume that motions are only in the (x,z)-plane, such that mi = (m;,0, rnt) = rno(sin O i , 0, cos O i ) , & = 0, and Ri = (xi,0, zi). Thus, considering angular O motions, rotations around the z-axis with constant angular velocity $ will be excluded from further consideration. In the studied case, there are two possible equilibria: (1) when all magnetic moments are aligned in the z-direction, i.e., parallel to the external magnetic field, and (2) when all magnetic moments are aligned in the z-direction, i e . , along the chain and perpendicular to the external magnetic field.

272

6.1.5.1

Physics and Applications of Complex Plasmas

Compressional and bending modes

For small oscillations 6x << T O of the centers of masses in the horizontal x-direction of vertically oriented (0 = 0) magnetic moments, we find from Eq. (6.31) the equation of motion

(6.33)

+

+

where @ ’ ( T O ) = Q(2 2r0/Xo rz/A&)exp(-ro/Xo)/ri. Equation (6.33) gives the dispersion equation for the lattice-acoustic compressional modes (DALW)

m21

QW’(rO) + 1 2 2 sin -

(6.34)

TO

The (repulsive) interaction of vertically oriented magnetic moments leads to some increase of the factor in front of the [sin(kro/2)] dispersion term; thus the lattice acoustic velocity in the long-wavelength limit increases in this case. Experimentally, this opens the possibility to check the strength of the magnetic interactions by comparing the lattice acoustic velocity of magnetized and unmagnetized particles of the same size. The external homogeneous magnetic field does not contribute to the dispersion relation (6.34) (this is not the case if there is inhomogeneity of the magnetic field); for experiments this means that the same, e.g., paramagnetic, particles can be used to check the dispersion of the lattice acoustic wave without and with the magnetic field to elucidate the influence of the magnetic effects. For the chain of horizontally oriented magnetic moments (0 = 7r/2), allowing small oscillations in 2-direction, we obtain the modified dispersion relation for the lattice-acoustic compressional modes (DALW) kr0 Q@”(rg)- 2 4 0 sin2 , m d

6

m2]

2

(6.35)

We note here the possible qualitative difference introduced by the interactions of the magnetic moments: an instability of an initial configuration may arise related to the attractive magnetic interaction of magnetic moments when the attractive force can overcome the electrostatic repulsion. Although the electrostatic interaction is usually dominant, because of the plasma shielding it can be effectively screened out thus opening the possibility for the effects of magnetic moments not screened by the plasma,

Waves and Instabilities in Complex Plasmas

273

see Eq. (6.29). The magnetic effects in this case can decrease the interparticle distance until the electrostatic interaction becomes large enough to stop the process. If the electrostatic interaction can be 'switched off' (e.g., by switching off the plasma which leads to the disappearance of the particle charges mostly supported by plasma currents), the magnetic particles (e.g., made of ferromagnetic material) can even merge together thus destroying the chain. Again, the external homogeneous magnetic field does not explicitly enter the equation of motion (if there is inhomogeneity of the external magnetic field, an extra force associated with the inhomogeneity will contribute to Eq. (6.35)). Note that the equilibrium orientation of the magnetic moments in the x-direction can be realized by the external magnetic field in the x-direction or appear spontaneously in the absence of the external magnetic field. For small oscillations in the vertical direction of vertically oriented magnetic moments (ie., parallel to the z-axis) we obtain the equation of motion

x ( 2 6 2 - 6Zi+l - 6zi-1),

(6.36)

+

where @ ' ( T O ) = -Q(1 r o / X ~ exp(-ro/XD)/rg. ) Thus we obtain for the bending lattice wave (DTLWI the following modified (with respect to the case of particles without magnetic moments) dispersion relation, which is of an optical character (6.37)

Although the factor in front of the dispersion term is modified (increased) in this case, the character of the wave dispersion is the same as in the case of pure electrostatic interactions. Because of the larger factor in front of the negative dispersion term in the presence of the magnetic field, the Brillouin zones (if they exist) will be changed thus indicating slightly changed parameters for the transition associated with excitation of the bending mode (this mode can be responsible for the transitions associated with the change from a one-dimensional chain to two vertically arranged chains). However, larger amplitudes of the vertical vibrations may lead to a new type of instability associated with the possible attraction of vertically oriented magnetic moments when their vertical separation is large enough. Indeed, the interaction of magnetic moments in a general form ( i e . , not assuming small

274

Physics and Applications of Complex Plasmas

amplitudes) is given by (because of the

( 5 ,z

) plane motions Ry,i&= 0)

+

(9R&+ - 6Rz,i+) (+

-)

1

.

(6.38)

Here, (+ + -) stands for the same term, where Ri+ changed to Ri-. We note that if horizontal motions are absent, Rx,i- = Rx,i+ = T O . Furthermore, for small vertical oscillations Rz,i+ << Rx,i+ = TO and the sign of the interaction of the magnetic moments is determined by the sign in front of Rx,ik. However, when the amplitude of the vertical oscillations exceeds 3ro/2, the sign of the interactions is changed - an attraction of vertically and horizontally displaced magnetic moments appears, and if this attraction is capable of overcoming the electrostatic repulsion of the charges, a vertically arranged configuration of particles appears leading to the possible merging of particles [Yaroshenko et al. (2003)];the latter was observed in experiments [Samsonov et al. (2003)l. To conclude this part, consider vertical oscillations of horizontally oriented magnetic moments and obtain for the lattice wave (6.39)

md

md

Note that magnetic effects can now change the character of dispersion for this type of oscillations. Physically, this is related to the magnetic moment attraction in the horizontal direction; again if the attraction is strong enough to overcome the electrostatic repulsion, the magnetic effects can be dominant. However, if sufficiently large amplitude vertical oscillations are excited, the attraction of magnetic moments can switch to repulsion. Indeed, we have for the horizontally oriented magnetic moments

(6.40) Again, for small amplitude vertical oscillations Rz,i* << Rx,i* = T O and the sign of the interaction of the magnetic moments is determined by the sign in front of Rz,i&;when the amplitude of the vertical oscillations exceeds 12ro/5, the sign of the interactions is changed.

275

Waves and Instabilities in Complex Plasmas

6.1.5.2

Angular magnetic lattice modes

Consider small rotating oscillations around 0 = 0 ( i e . , for vertically oriented magnetic moments). These motions are described by the equation of motion

(6.41) The external magnetic field oriented in the z-direction plays the role of the external parabolic potential for this type of small amplitude oscillation. For the lattice wave, we obtain the following dispersion relation: w2

+ 2 i p w = moBo

-

I

6mi/ri

+ 8mo sin IT0

lcro 2

-

(6.42)

Qualitatively new features appear in this equation. First, note the normal character of the wave dispersion which increases with the increasing wave number; however, a cutoff frequency associated with the interplay of the external (stabilizing) field and the nearest neighbor interactions exists. Second, in the absence of an external magnetic field (oriented along the z-axis) this mode can be unstable in the long-wavelength limit. Physically, this instability is related to the attraction force between the magnetic moments acting in the horizontal direction; this force component appears even for small 0 and the corresponding torque causes further increase of the angle 0. This implies that a stationary configuration involving vertically oriented magnetic moments may not be possible without a stabilizing vertically oriented magnetic field. However, if we compare the dispersion term with the first term on the right hand side of Eq. (6.42), we see that for some wavelengths the wave can become stable. Thus, Brillouin zones appear on the dispersion chart. Third, even in the presence of an external stabilizing magnetic field, the nearest neighbor interactions of the magnetic moments can still overcome the stabilizing influence, if the field is not strong enough. The Brillouin zones, however, are wider in the presence of the magnetic field. And finally, dispersion characteristics of this mode do not explicitly reveal electrostatic interactions; implicitly, however, the interparticle distance ro is related to the electrostatic repulsion. Indeed, for the finite chain in an external confining potential the interparticle distance is related to the interparticle interactions as well as to the strength of the confining potential. Finally, consider rotational oscillations of horizontally oriented magnetic moments. We introduce a new angle variable 8 = 7 ~ 1 2 0 and assume that

276

Physics and Applications of Complex Plasmas

the magnetic field points in the x-direction. We have

(6.43) This leads to the following dispersion equation:

(6.44) (6.44) Oscillations in this case are always stable, even in the absence of the external magnetic field; the same result holds for small angles if there is a vertically oriented (in z-direction) magnetic field. However, for larger angles the vertically oriented magnetic field plays a destabilizing role trying to align the moments along its field lines (in the vertical direction). Indeed, the corresponding force on the right hand side of Eq. (6.43) is proportional to moBo(1 - cos29) in this case, ie., to (619)~ for small angles. Again, the electrostatic effects are not reflected in the equation for small angular deviations.

6.2

Waves in Weakly Coupled Unmagnetized Complex Plasmas

In this section, we present the dispersion and damping properties of some of the common plasma waves in weakly coupled unmagnetized dusty plasmas. In particular, the attention will be focused on Langmuir waves (Sec. 6.2.l), dust-ion-acoustic waves (Sec. 6.2.2), dust-acoustic waves (Sec. 6.2.3), and different types of surface waves (Sec. 6.2.4). 6.2.1

Langmuir waves

Consider propagation of Langmuir waves in a self-consistent close system that includes non-isothermal plasmas and dust particles with variable charge. Here, the dust charging process, ionization, recombination, as well as collisional dissipation will be explicitly incorporated in the basic set of the electron/ion number density and momentum conservation equations [Vladimirov et al. (1998b)l. The above dissipative processes lead to a net damping of the Langmuir waves in typical complex plasmas. The size of the dust grains is assumed to be much less than the intergrain distance, the electron Debye radius, as well as the wavelength, so that they can be treated as heavy point masses. The charge of a dust grain

Waves and Instabilities in Complex Plasmas

277

varies because of the microscopic electron and ion currents flowing into the grain according to the potential difference between the dust surface and the adjacent plasma. The dust particles are treated as an immobile background since the time scale of charge variation is much smaller than that of the dust motion. Equations describing propagation of the Langmuir waves are

dtn,

+ V . (n,v,)

=

(6.45)

S,

(6.46)

V 2 p = -4ne(Z,n,

-

n,

-

(Zdlnd),

(6.47)

where p is the electrostatic potential, m3,n3 (including the stationary value no3),and vIIare the mass, density, and fluid velocity of the species j = el i, d for electron, ion, and dust, respectively. Furthermore, 2,e and -1Zdle are the charges of the ions and dust particles, y is the adiabatic constant of the electrons, and 'uT, = (Te/m,)1/2 is the electron thermal velocity. Likewise, S = -u&n, v,ne - pnz +ps2nz v . (D,Vn,), where ved is the collection rate of plasma electrons by the dust grains, v, is the ionization rate, p is the volume recombination rate, pSzis the step-wise ionization rate, and D, is the ambipolar diffusion coefficient. We have also defined the effective electron collision frequency v,ff = v, v,"' v,Ch, where v, is the rate of electron collisions with the neutrals and plasma particles, v,"' is the rate of elastic (Coulomb) electron-dust collisions, and v,Ch is the effective rate of collection of plasma particles by the dust. Assume that the wave perturbations behave like exp[i(kz - w t ) ] and use the Orbit Motion Limited model for dust charging (see Sec. 2.1). In the absence of perturbations, the system is assumed quasi-neutral. For Langmuir waves, the contribution of the perturbed (by the wave process) ion current I%Iis negligible with respect to the perturbed electron one I e l , since it is on the slower ion time scale. For perturbed dust charge we have

+

+

+ +

N

dtgdl

+ vchqdl = -i~eOlnel/neO,

(6.48)

where v& = UW&A/&VT,is the charging frequency of the dust particle defined by the equilibrium electron and ion microscopic currents (see Sec. 2.1). Here, VT, is the ion thermal velocity, T = T,/T,, A = 1 IT 2, and 2 = Zde2/uTe. The effective charging rate is v,Ch = v,hP(4 Z ) ( T Z ) / A 2 , and the electron capture rate (2.13) a t

+ + +

+

278

Physics and Applications of Complex Plasmas

the grain surface is [Tsytovich and Havnes (1993); Vladimirov (1994); Vladimirov and Tsytovich (1998)l: ued = v , h P ( ~ + + ' Z ) / d 2and , P = Z d n d o / n , ~ . The frequency of elastic electron-dust collisions is [Vladimirov and Tsytovich (1998)l vel = 4 f i Z ~ n d e 4 A / 3 r n ~ v $ , ,where A = ln(Xo,/a) is the Coulomb logarithm. The expressions for the rate u, of electron collisions as well as for vi, ,B, psi and D , can be found elsewhere [Biberman et al. (1982)l. Equations (6.45)-(6.47) and (6.48) describe the coupling between the high-frequency electrostatic Langmuir waves and the dust charge fluctuations. To determine the stationary electron plasma density assume that the pressure is not too low, such that recombination losses prevail over diffusion losses. The last term in S can then be ignored. From Eq. (6.45) the lowest order (steady state) electron plasma density ne0 = (vi - u e d ) / , B e ~can then be obtained, where p e = ~ /3 - psi. We note that the ionization rate must be high enough sucli that vi > u e d , otherwise no self-consistent stationary state exists. Linearizing with respect to the wave perturbations, and combining the above equations, we obtain the following dispersion relation of the Langmuir waves

(6.49) where D(w,k)= (w+iv,~)~-w&, 7 =w-iu,~+i~~vi-yk2v~,/(w-ti and Ez = 2 - 51, wpe is the electron plasma frequency. Here, the two models, namely, that of a density-fluctuation independent ionization rate, with 51 = 0, and that of density-fluctuation dependent ionization rate, with 51 = 1, are considered. Equation (6.49) describes the linear coupling of the high-frequency Langmuir plasma waves with the dust charging process w = -ivLh. It can be solved numerically for any given set of parameters. w e also note that if u:', v , " ~V,,d and (&,d/azd)z,, are set to zero in Eq. (6.49), one recovers the coupling equation derived earlier [Ma and Yu (1994)l. It is instructive to estimate the effect of electron capture (by the dust grain) and dissipative collisions on the Langmuir waves. For this purpose it is convenient to make the Ansatz w >> (veff, V e d , u&). By assuming w = w1+ 6;+is:, where w; = wge+yk2v$,, from Eq. (6.49) one can derive

s:

=-

2 - 2 (wpeuwl - a) V / 2 W l ,

(6.50)

279

Waves and Instabilities in Complex Plasmas

where 23 = a 3 n i o w $ A / v , h ,

and

(6.51) so that the frequency of the Langmuir waves is down shifted, and the waves are damped by most of the collisional processes included here. In the absence of the latter processes, the first term in Eq. (6.51) remains, thus indicating the possibility of the Langmuir wave instability [Ma and Yu (1994)]. Although the rate Ued of electron capture by the dust also has a positive sign, it is always smaller than the term -&vi involving ionization because vi > l/ed (required by existence condition for the stationary state) i62. From and & > 1. For the charge relaxation, we let w = -iz& Eq. (6.49) one then obtains

+

(6.52) which shows that the dust charging frequency is slightly reduced by the coupling with Langmuir waves. One can now estimate the average dust charge and the dissipation parameters v e ~v e,d , v,h and V for typical dusty plasmas. The factor eAy,/Te, which defines the average charge on a dust grain and can strongly affect the density ratio n d o / n , o through the quasineutrality condition, can be found from the condition of zero total current flowing into the dust in the absence of the high-frequency perturbations. For a typical dusty argon plasma, we have Te 10 eV, Ti 1 eV, a 5 pm, ne0 5 x lolo cme3, and n i o / n e o = 10. One then obtains eAy,/Te = -1.71, ZdO = -6.12 x lo4, and ndO/n,O = 1.74 x lop4. One can also show that the inequality V << w , where V = (veff, v,d, v&,, 6)represents the dissipative effects invoked here, is satisfied. In fact, we find V 3 x lo7 - 5 x lo8 sec-' and w lolo sec-l, which validate our Ansatz. It is also necessary to verify the existence condition ui > ved for the stationary state with the equilibrium density neO = (vi - V e d ) / @ & , which imposes the existence of the minimal electron temperature threshold, allowing for the stationary state to exist. Indeed, a decrease of Te would lead to a decrease of the ionization rate, and hence violation of the existence condition for the stationary state. We note that Tehresdepends weakly on the ion density. Furthermore, the threshold is affected by the degree of ionization (an increase of nio/Nn would lead to a downshift of Tehres the dust size and charge (Xehres increases with a and Z d o ) , as well as by the amount of electrons in the system (Tethres increases when n , o / n i o de-

-

-

N

-

-

-

280

Physics and Applacations of Complex Plasmas

creases). Furthermore, the threshold temperature can be decreased in cases when multi-step ionization processes become important. If ionization, recombination, and other dissipative processes are included, the linear coupling of the fluctuations and Langmuir waves leads to a damping and a frequency downshift of the waves. This result differs considerably from that where a uniform source is invoked to replace the electrons and ions lost to dust charging [Ma and Yu (1994)l. Thus, the actual ionization and recombination processes that maintain the total charge balance of a dusty plasma system can be important in investigations of other waves and instabilities in dusty plasmas (also occurring at much slower, ionic and dust time scales), which will be further evidenced in the following sections. 6.2.2

Ion-acoustic waves

Discuss now the self-consistent theory of ion-acoustic waves in dusty gas discharge plasmas with variable-charge dust particles [Vladimirov et al. (1999b)I that enables one to determine the stationary state of the plasma and the dispersion and damping characteristics of the waves on the ionic temporal scales. Here, the major dissipative and plasma particle-dust interaction processes are accounted for in a manner similar t o Sec. 6.2.1. Note that the effect of dust on acoustic waves in space and laboratory plasmas was studied by many authors [D'Angelo (1990); Merlin0 et al. (1997); Ostrikov et al. (1998); Ostrikov and Yu (1998)]. However, most theoretical models consider dust-charge variation without properly taking into account the change of the plasma particle number density arising from their capture (and release) by the dust grains, as was discussed in Sec. 6.2.1. For the relatively high frequency Langmuir waves considered in Sec. 6.2.1, the ion motion is unimportant. However, for the lower-frequency ion-acoustic waves (IAW), a self-consistent theory should include a proper account of the ion dynamics. Most of the basic assumptions here are similar to the ones used above in Sec. 6.2.1 and the basic set of Eqs. (6.45)-(6.47) is to be complemented by the following equations describing the balance and dynamics of the plasma ions:

(6.53)

(6.54)

28 1

W a v e s and Instabilities in Complex Plasmas

where E = -Vp is the electric field of the ion-acoustic waves, cp is the electrostatic potential, mi and ni are the mass and number density of the ions, respectively. The effective frequency of ion collisions can be presented as ufff = vi + uf uth, where ui = ui, + vie is the rate of ion collisions with the neutrals and plasma particles. The relation between the electron/ion charging collisions is

+

(6.55) while the rate of electron and ion capture by the grain is given by (6.56)

+

where a = A - 1, y = ( Z d ~ n d o / n , ~ ) c p ~ ~ /and c p ~0~ ,= 4 cpzl/cp",". Here, u:h = aw;,A/&v~i is the dust charging frequency, and we have defined A = 1 (Ti/T,) (cp:/cp","), ' p i 1 = Zde/a, cpLh = T,/e; v ~ = i ( T z / v ~ i ) is l/~ the ion thermal velocity, wpi is the ion plasma frequency. The rates of elastic electron- and ion-dust Coulomb collisions are

+

+

(6.57)

where A is the Coulomb logarithm. The electron- and ion-neutral, electronion and ion-electron collision rates (also ionization and recombination) are standard [Chen (1984); Biberman et al. (1982)]. The equilibrium electron density neo = (uion- u,d)/Peff has the same form as in Sec. 6.2.1. For the equilibrium ion density, one obtains nio = ( u e d / u i d ) n e o . The equilibrium electron and ion densities are different and are inversely proportional to the rates of electron and ion capture by dust. The same result follows from Eq. (6.56) and the overall charge neutrality condition. The dispersion relation of the ion-acoustic waves in the plasma with variable-charge dust grains is

(6.58)

282

Physics and Applications of Complex Plasmas

In the absence of dissipation and dust-charge variation we have from Eq. (6.58) w = k w s ~where , U S D = J(Te/mi)(nio/neo) Ti/mi is the ionacoustic speed in a dissipationless plasma with the constant-charge dust particles. If k2v$, >> [ w 2 ,vZff(40n- u i d ) ] , which is usually valid for ionacoustic waves, from Eq. (6.58) follows

+

(w

+ i@)

[(w

+ i v , " f f ) ( w+ iu*)

-

k27J;,]

(6.59)

where u* = u i d + vi,, - b e d . To solve the dispersion relation (6.59), one can assume that a wave frequency is real and a 'complex wave number can be derived in the form k = k' + ik". The solution is valid when the frequency of the ion-acoustic waves is of the order of or even less than the rate of capture of the plasma electrons by the dust grains. For high frequencies ( w >> v , d Z d n d o / n i o ) , the relation (6.59) describes the coupling of the ion-acoustic waves and the dust-charging process. One can then obtain the two equations, namely D l ( w , k ) = ( w + i u F f f ) ( w + i v * )k 2 v i , = 0 for the ion-acoustic waves, and D z ( w , k ) = w iij:h = 0 for the dust charging process. The solution of the dispersion equation D1 ( w ,k ) = 0 is

+

(6.60)

which describes damped ion-acoustic waves in a low-temperature dusty plasma. From Eq. (6.60) it follows that propagating waves exist if kvsD > (u,eff v*)/2. The real and imaginary parts of the frequency are w i A = [k2v:, - (ufff f ~ * ) ~ / 4 ]and ~ / w;A ~ , = -(@ v * ) / 2 . However, if kvsD < (u,eff+v*)/2,the solution is purely damped. The coupling with the dust charging process leads t o the variation ~ W I of A the IAW eigenfrequency

+

+

(6.61)

+

+

i ( 2 V e d - v i o n ) ( w 1 ~ iu,eff).From Eq. (6.61) where & = k 2 ( w i D - &) we see that the coupling of IAWs to the dust-charge fluctuations leads t o a frequency downshift and an increase of the damping decrement. The corresponding solution for the dust charging process is given by W D =

283

Waves and Instabilities in Complex Plasmas

-i,Gzh

f?

+ S W D , where

= k 2 (wSD 2

-

v$%)f (2v,d

-

u ; , , ) ( u ~~ fizh), and 2, = k2wiD

+ (@

-

,G:h)(u*- fi:h). We note here that the dust charge relaxation rate is reduced due to coupling with the ion-acoustic waves. Equations (6.52)-(6.61) describe propagation of ion-acoustic waves in a plasma with variable-charge dust particles for a wide range of parameters of interest. The frequency of the IAWs should not be too low, otherwise inclusion of temperature fluctuations and use of the full Braginski’s equations for strongly collisional plasmas would be necessary [Vladimirov and Yu (1993)l. This is especially true if the frequency of IAW is less than the effective rate of ion collisions. Finally, note that the above theory of the ion-acoustic waves can also be extended to multicomponent plasmas with negative ions [Vladimirov et al. (2003b)l.

6.2.3

Dust-acoustic waves

In this section we present a self-consistent theory of low-frequency dustacoustic waves in low-temperature collisional plasmas containing variablecharge dust grains. The generalized dispersion relation describing the propagation and damping of the dust acoustic waves is derived and analyzed. The presence of the massive and highly charged particles introduces new, very long time scale phenomena associated with dust motion. In particular, a low-frequency mode, namely the electrostatic dust acoustic waves (DAW), has been demonstrated theoretically and experimentally [Pieper and Goree (1996); Merlin0 et al. (1997); Rao et al. (1990); D’Angelo (1995)l. In the earlier studies on DAW the constant-charge approximation was often used [Rao et al. (1990)I. In this approximation, DAW are similar to ion acoustic waves propagating in multi-component plasmas containing negative ions. However, the charge on a dust grain can vary according to the local plasma potential, and electrons and ions can be captured or released by the dust grain. There is thus a corresponding variation in the electron and ion densities. A realistic description of DAW necessarily requires the inclusion of the electron/ion capture/release by the dust particles, as well as the relevant dissipative processes [Ostrikov et al. (2OOOa)l. The basic assumptions are similar to Sec. 6.2.1 and 6.2.2, and the corresponding electron and ion balance/motion equations (6.45)-(6.46) and

284

Physics and Applications of Complex Plasmas

(6.53)-(6.54) (with time-dependent terms dropped due to much slower, dust motion time scales) are complemented by the following equations for the dust and neutrals: (6.62)

(6.63)

(6.64) where E = -Vp is the electric field of the dust-acoustic waves, V d n E 4 m n n n a 2 V ~ , / m d is the effective frequency of the dust-neutral collisions, and is the thermal velocity of neutrals. The frequency of the neutraldust collisions appearing in Eq. (6.64) is given by vnd % ( n d m d / n n m n ) V d n . Other electron/ion rates entering the basic set of equations for DAW, are similar to those used in Sec. 6.2.1 and 6.2.2. In the intermediate pressure range, when the plasma particle loss is predominantly controlled by the volume recombination, the equilibrium values of the electron/ion number densities appear to be the same as described above in Secs. 6.2.1 and 6.2.2. The dispersion equation of the dust-acoustic waves (in the limit T d << Ti,Te)is then written as

(6.65)

+

+

+

+

where t p d = 1 - W 2 (W i v n d ) / W 2 [ W i ( v d n V n d ) ] ,(1 = 1 n e O v e d / n i O f i $ h , PdetT t 2 = 1 v e d / f i i h - Vi tl(2ved - Yion)/vi. Here, rle = YZff (Uion - v , d ) k2vge, and qi = vidvFff k2vgi. The three terms on the left side are the effective dielectric functions of the dust particles, electrons and ions, respectively. From Eq. (6.65) one can obtain the DAW eigenfrequency w = w' + iw", where

+

+

+

(6.66) w" = -vdn/2, and it was assumed that w >> Vdn,Un/nd. The dispersion relation (6.65) also allows for simpler solutions in several limiting cases.

Waves and Instabilities in Complex Plasmas

For example, if k2v$,

285

>> vZff(vion - V e d ) and k2Xg, << 1, it follows that (6.67)

+

where 61 = 1 V : f f V i d / k 2 ' U $ i . If k2v& >> vtffvid,the expression (6.67) can be simplified by noting 6 1 x 1. In the latter case, neglecting the effects of dust charging as well as the ionization-recombination balance of the electrons and ions in dusty plasmas, i e . , letting <1,2 = 1, one can recover the dispersion relation of the DAW in the collisionless limit [Rao et al. (1990)l. In the strongly non-isothermal limit we obtain under the same conditions

which is valid if T, >> Ti. For long-wavelength DAW satisfying k2X'$, << vzff(qon- V e d ) / w i , and k2XLi << V:'uid/W$, one can obtain w' = w, for the real part of the frequency, where (6.68) and B = W g z < l / V & ! , e f f . It is of interest to note that w, is a (lower) cutoff frequency of the DAW in the presence of sources/sinks and dissipation of ions and electrons. We note that usually wge >> vEff(uion- V e d ) , and the unity in Eq. (6.68) can be neglected. Thus, the cutoff frequency is much less than the dust plasma frequency, or (6.69) which can be attributed to an imbalance of charge densities due to the creation and dissipation of the electrons and ions in the DAW motion. Another interesting case is for k2v$, >> v,"'(von - V e d ) , but k2v$i << This situation is relevant to strongly non-isothermal plasmas or plasmas with large ion mass (such as argon or potassium).. Accordingly,

z&L/tff.

(6.70)

286

Physics and Applications of Complex Plasmas

is the real part of the DAW frequency. Note that in Eqs. (6.66)-(6.70) the frequency of DAW satisfies w < Wpd and w << kvTz. Figure 6.14 shows the dependence of the normalized eigenfrequency of the dust-acoustic waves on the normalized wave number in hydrogen plasma. The numerical results [Ostrikov et al. (2OOOa)] suggest that the frequency of the DAW increases with the electron-to-ion temperatiire ratio and that the dimensionless lower cutoff frequency is lcss in the heavy-ion (argon) plasma. Furthermore, in argon plasmas the lower cutoff depends weakly on the electron-to-ion temperature ratio. This is expected since the cutoff originates from the dynamical wave-induced charge imbalance for which the thermal motion of the ions plays a role.

Fig. 6.14 Dependence of the non-dimensional frequency of the DAW R = w / w p d on the non-dimensional wave number K = k r D , in hydrogen plasmas for zdondo/n,o = 0.3, U,d/vsh = 0.1, Uion/u,d = 1.5, vZff/U,"ff = 25, U : f f /Ups = 2, and U , d / W p z = 0.5 [Ostrikov et al. (2OOOa)l. Curves 1-6 correspond to T,/T, = 1.05, 1.5, 2.5, 5.0, 7.5, and 12.5, respectively.

In this model, the cutoff frequency (6.68) appears because of a charge imbalance of the light particles in the process of particle creation and momentum dissipation. We recall here that for low-frequency A1fvi.n waves in dusty plasmas a similar imbalance leads to uncompensated electron and ion Hall currents, see Sec. 6.3. The latter give rise to the low-frequency cutoff of the A l f v h wave dispersion. Likewise, we note that the model in question implies that the ionization source is unaffected by the wave oscillations.

Waves and Instabilities in Complex Plasmas

6.2.4

287

Surface waves

Here, we explore the dispersion properties and damping characteristics of the three different types of surface waves (SW) within a simplified constant dust charge approximation. We also discuss the possible effects of the grain charge variations and different dissipative mechanisms on the properties of SW and surface-wave sustained low-temperature gas discharges. 6.2.4.1 Ion-acoustic surface waves at the dielectric-complex plasma interface We now consider propagation of the linear ion-acoustic surface waves (IASW) at the interface between a dielectric and dusty plasma [Ostrikov and Yu (1998)l. The isotropic plasma with thermal electrons is bounded at z = 0 by a dielectric with dielectric constant € 0 , the dust grains are assumed to have a constant average charge q d = Zde < 0, and to be much smaller than the electron Debye length T D e as well as the inter-particle distance. Thus the dust grains are heavy (compared to the plasma ions) point masses with constant negative charge in this model. The linearized equations describing the IASW are given by dtnj

+ V . (noj + n j ) v j = 0 ,

(6.71)

(6.72) and

v . E = 4.ir C Zje(noj + n j ) ,

(6.73)

j

where E is the electric field of IASW, m j , n o j , n j , v j , V T ~and , q = Zje are the mass, unperturbed and perturbed densities, fluid velocity, thermal speed, and charge of the species j = e, i , d , respectively. In the unperturbed state the usual charge neutrality condition for a three-component plasma Zinio = ne0 Z d n d o is fulfilled. For simplicity, the dynamic effects of sources and sinks of the electrons and ions are not accounted here, which is fairly accurate within the invariable charge approximation. The equilibrium values of electron/ion number densities are assumed to be known and can be obtained from stationary particle and power balance equations of the relevant plasma discharges.

+

Physics and Applications of Complex Plasmas

288

Purely electrostatic wave perturbations (E = -Vp, where p is the IASW electrostatic potential) propagating in the z direction are considered, so that the dependencies of all the perturbation quantities are in the form f ( x ,z , t ) f ( x )exp[i(kz - w t ) ] . For T, >> Ti,T d , one obtains N

n,

=

fa d -v2cp, h e

(6.74)

and (6.75)

+

where Ei,d = 1 - w;i/w(w + ivi) - w;d/w(w i u d ) , wpi and W p d are the ion and dust plasma frequencies, ui and l/d are the effective collision frequencies of the ions and dust particles, respectively. Substituting Eqs. (6.74) and (6.75) into Eq. (6.73), we obtain for the IASW electrostatic potential (6.76)

+

where E = tpi,pd - wge/w(w iv,). The differential equation (6.76) is of the fourth order. In the plasma region (x > 0), the solution has the form cp

where 1; = k: - E W ( W the form

N

A1 exp(-kyz)

+ iV , )/€i, dV $, . p

+ A2 exp(-lzz), In the dielectric region (z

A3 exp(k3z)

(6.77)

< 0) it

has

(6.78)

1

which is to be matched to the solution (6.77) a t the interface IC = 0. Three boundary conditions have to be satisfied. Two of them are obtained directly from Eq. (6.76) by integrating it across the narrow interface region 0- < x < O+, yielding the conditions cplz=o- = plz=o+

and

Ei,d&(Plz=O-

= to&plz=o+

(6.79)

for the potential and its derivative a t the plasma-dielectric interface. The third boundary condition depends on the physical situation considered. For example, it can be in the form of a vanishing normal component of the electron fluid velocity a t II: = 0. This macroscopic model boundary condition has been widely used for bounded fluids and is valid t o a good accuracy for well-polished dielectric or metal interfaces [Gradov and Stenflo (1983);

Waves and Instabilities in Complex Plasmas

289

Azarenkov and Ostrikov (1999)]. Accordingly, the boundary condition u,,(x = 0) = 0 written in terms of the IASW electrostatic potential is [dz (1 -

which leads to I A I I / I A ~ = ( Z Z / ~ ~ ) ( W / W ~ ~<< ) ~ 1. Thus one can set and obtain from Eq. (6.79)

(6.80) A1 = 0

(6.81) which is the dispersion relation for the acoustic-type surface waves in dusty plasmas. The corresponding damping coefficient 6 (= Imk3) is given by (6.82) which shows that the damping (6 is negative definite) of IASW in a dusty plasma is determined by the ions. Equation (6.81) is valid for all ion and dust surface waves ‘satisfying T, >> Ti,Te. In the usual ion-acoustic frequency range w2 << w $ ( l E;)-’ and Re Ei,d >> € 0 , Eq. (6.81) can easily be reduced to ,kz = w2/&, where v,d = v,g is the modified ion sound speed in the dusty plasma, and g E ( 1 z&ndo/neo)1/2. Terms proportional to miZi/md << 1 have been neglected. Thus, although the heavily charged dust particles can significantly affect the charge balance, their motion does not contribute to the wave dynamics. Since for most dusty plasmas g 2 1 , the dust grains can cause a significant increase of the ion sound speed and the IASW phase velocity, as is the case of bulk ion-acoustic waves in infinite plasmas [Popel and Yu (1995)l. Clearly, the presence of dust also leads to a g-fold increase of the skin depth Askin % 2~112,and a corresponding increase of the IASW field localization region. Physically, these increases are caused by a reduction of the eIectron density in the plasma because of the presence of the dust, which compensates the negative charge imbalance without being moved by the ion-acoustic wave field.

+

+

6.2.4.2

Electrostatic surface waves at the dusty plasma-metal interface

In this subsection, dispersion properties of electrostatic surface waves at the interface between the complex plasma and an ideally conductive metal surface are considered. In this case, two boundary conditions are necessary

290

Physics and Applications of Complex Plasmas

to derive the dispersion relation [Ostrikov et al. (1998)]. One of them is obtained directly from Eq. (6.76) by integrating it over a narrow interface layer z = [-a, a] with a + 0, or (6.83)

$(z = 0) = 0 ,

so that the tangential component of the electric field vanishes a t the plasma-

metal interface. Similar to the SW a t the dusty plasma-dielectric interface, the second boundary condition (6.80) can be the vanishing of the normal component of the electron fluid velocity in the SW field at z = 0, or v,,(z = 0) = 0. Using the boundary conditions (6.83) and (6.80), the solution of Eq. (6.76) leads to the dispersion relation kz

(6.84)

K(W/Wpe)2

where the collisional losses are neglected. The surface waves exist in the frequency range w& < w2 << w;,. Moreover, the skin depths X I and Y2 satisfy XI >> A2 X D ~ The ~ SW . eigenfrequency can then be expressed as (w/wPe)' = k,X~,fi. We also note that uph/uTe N w / w p e f i , where Vph is the wave phase velocity. In the limit n d = 0 the corresponding dustfree dispersion relations [Azarenkov and Ostrikov (1999)] are recovered. For plasmas containing negatively charged dust the electron density can be lowered compared with the dust-free case by a factor y = [l + ( n d ~ / n , o ) Z d o ] ~ / ~This . implies that SW wave number k , will be larger, corresponding t o a decrease of Vph. This decrease of the wave phase velocity in turn leads to the possibility of more effective interaction between the SW and the plasma particles in the gas discharge. It also results in more prominent nonlinear effects, with the efficiency proportional to VE/Vph, where V , is the electron oscillation frequency in the wave field. In the presence of dust particles the electric skin depth A1 decreases, and the thermal skin depth A 2 increases. We now estimate the effect of the dust grains on the SW phase velocity using typical parameters for dust-containing plasmas. For the case studied we have Vph c( K y1I2,where y1I2is the relative variation of the wave phase velocity with respect to the dust-free case. For the estimates we take the two sets of parameters, which are relevant to RF plasma-assisted deposition processes and gas-target divertors, respectively. For a typical plasma-assisted deposition process in argon plasma we have (see, e.g., Chapter 7) n d o 2 x lo5 crnp3, n,o lo9 ~ m - SiOa ~ ,

-

-

-

291

Waves and Instabilities in Complex Plasmas

-

-

-

particles with a 5 pm, T, 2 eV, Ti 0.3 eV, and RF frequency w/27r 13.56 MHz. To estimate the charge of the dust grains we use -2de = (q$g - q$o)a(l a/TDe), which is valid for a conducting sphere in the electrostatic approximation (see Sec. 2.1). Assuming q$g - 40 10 V and noting that a << r D e we have Zd 7 x lo4. Taking into account that the charge of the real (usually a dielectric and not perfectly conducting) dust grain is less than that given by the electrostatic approximation, for the numerical estimates we take the slightly lower value Z d 5 x lo4. Under these conditions we have y = [l ( n d o / n e o ) Z d ] 1 / 2 RZ 3.32. Therefore, the SW phase velocity is approximately 1.8 times lower than in dust-free case. It should be noted that a similar effect for the ion-acoustic SW at a dusty plasma-dielectric interface considered in Sec. 6.2.4.2 is y1I2 1.8 times higher. Moreover, if we assume a 1 pm, then Z d lo4 and y In this case the SW phase velocity would be 1.19 times lower than in dusty-free plasmas. That is, smaller dust grains have less effect on the SW properties. Similarly, for typical parameters for a gas-target divertor configuration T, 20 eV, T, 3 eV, n e O 2 x 10l2 cmP3, HO 0.15 T, a 5 pm, n d o lo9 cm-3 one can obtain for the SW phase velocity a factor 2.26 decrease. Therefore, the dust particles can also affect the physical processes near the divertors of fusion devices. N

+

-

-

-

+

- a.

-

N

-

-

-

- -

-

-

6.2.4.3 Electromagnetic surface waves an a dust-contaminated large-area plasma source Above, we have discussed the electrostatic surface waves a t the interfaces of complex plasmas with dielectric or metal. Consider now purely electromagnetic surface waves [Ostrikov et al. (1999a)I that have recently been used to sustain planar microwave slot-excited plasmas widely used by the microelectronic industry for large-area semiconductor processing [Sugai et al. (1998)].

The contaminant dust particles strongly affect dispersion properties of standing electromagnetic surface waves [Ostrikov et al. (1999a)I in a structure modeling the slot-excited planar plasma source [Sugai et al. (1998)]. A vertical cylindrical waveguide of circular cross-section with radius R is short-circuited at its top ( z = 0) by a metal plate. The bottom of the waveguide is open. A dielectric layer of thickness d and dielectric constant Ed occupies the entire short-circuited region near the cylinder top. The effect of the slots and the bottom of the discharge chamber on the eigenmodes of the structure shall be neglected. This

292

Physics and Applications of Complex Plasmas

model structure is known as the two-interface model [Sugai et al. (1998); Ostrikov et al. (1999a)I. In this structure the plasma is produced below the dielectric window. The plasma is assumed cold and homogeneous, and can contain massive and highly charged dust, uniformly distributed over the entire volume of the plasma. Possible resonant power absorption due to the inhomogeneous transient layers near the dielectric window and the side walls of the chamber is neglected. We also assume that the conductivity of the short-circuiting plate and the side walls is infinite and that the average size of the dust grains is much less than the interparticle distance, the wavelength, and the skin depth of the SW. Since electromagnetic SW in the microwave frequency range are considered, the ion and dust motions can be neglected. The charge on the dust particles can vary and can affect the balance of electrons in the discharge. Since the SW in the high-frequency range are considered, the basic set of equations includes the electron balance and motion equations and Maxwellian equations for the electromagnetic fields. Using the appropriate boundary conditions, the following dispersion relation can be derived [Ostrikov et al. (1999a)I:

D

(€p/yp) -k

(Ed/Yd)

coth(’Ydd) = 0,

(6.85)

where ep is the dielectric constant of the plasma, y& = (jmn/R)2(W/c)’6p,d, and j,, the n-th root of Bessel function of m-th order. The solution of Eq. (6.85) is given by

(6.86) where wpe(0) is the electron plasma frequency in the absence of dust, and x d = Zdnd/ni is the proportion of the dust charge in the plasma. If the thickness of the dielectric layer is large compared with the SW skin depth in the dielectric ( ~ d >> d l),the corresponding expression for the one-interface model [Sugai et al. (1998)l is recovered (up to the dust terms), and the solution of Eq. (6.85) becomes

(6.87)

293

Waves and Instabilities in Complex Plasmas

To study the effect of dust on the resonant properties of the discharge plasma it is reasonable to assume that prior to the release of dust the electron density corresponding to the TM,, mn mode is nzn.In particular, for the one-interface model and the set of parameters used in the experiments [Sugai et al. (1998)], namely R = 11 cm, w/27r = 2.45 GHz, and Ed = 4, we have n:; = 1.3 x lot2 cmP3 for TM33 mode and = 1.0 x lo1’ cm- 3 for TM62 mode. The corresponding values of the “dust-free” resonant electron densities for the two-interface model are n:: = 2.13 x 1OI2 ~cm-3 r n -and ~ nz,”= values are easily obtained from Eqs. (6.86) and 1.4 x lo1’ ~ m - ~These . (6.87) after setting Xd = 0, j 3 3 = 13.02, and j 6 2 = 13.59.

nzi

Table 6.1 Possible values of Xd for mode transitions TMmn TMmn [Ostrikob et al. (1999a)]. m 3 3 3 3 3 3 3 3 6 6 6 6 6 6

n

m’

3 3 3 3 3 3 3 3 2 2 2 2 2 2

1 6 4 2 0 5 3 1 4 2 0 5 3 1

n’ 4 2 3 4 5 3 4 5 3 4 5 3 4 5

One-interface model

Two-interface model

0.115 0.185 0.33 0.375 0.39 0.46 0.485 0.5 0.18 0.235 0.26 0.33 0.365 0.38

0.19 0.3 0.485 0.54 0.555 0.62 0.645 0.67 0.27 0.35 0.37 0.455 0.49 0.51

In the presence of dust (ie., for finite values of Xd) and fixed initial value of the electron density ngn (corresponding to release of dust in the discharge), the same values of j,, no longer satisfy the dispersion relations (6.86) and (6.87). For this reason we replace the discrete quantities j,, mn in Eqs. (6.86) and (6.87) by the continuous quantity U , and for each given proportion xd of dust particles in the discharge numerically computed the values of U with Eqs. (6.86) and (6.87) satisfied. One can then show that if Xd = 0.3, the field pattern in the discharge is adjusted so that the initially excited TM33 mode is converted into a TM62 mode under the same operating conditions (power and frequency of the microwave generator, operating

294

Physics and Applications of Complex Plasmas

gas pressure, etc.). Other values of x d leading to mode conversions in the discharge are presented in Table 6.1. Thus, in the two-interface model, mode transitions occur for higher values of xd. This is because the twointerface model is characterized by higher values of the electron density. The dispersion properties of the ion-acoustic surface waves [Ostrikov and Yu (1998)I can be affected by nonlinear effects. For example, the nonlinear effects can be associated with density modulations caused by surfacewave induced anomalous ionization (ionization nonlinearity). The effect of the ionization nonlinearity arising from the ion-acoustic surface waves can result in the formation of surface envelope solitons. The ionization nonlinearity in dusty plasmas is usually characterized by larger values of the relative nonlinear wave number shift as compared to dust-free plasmas. In certain electron temperature ranges the effect of the ionization nonlinearity is stronger than that of the heating nonlinearity. The studies of the ionization and other common nonlinearities in dust-contaminated plasmas can be applied for the stability analysis and design optimization of the low-temperature plasma glows produced and sustained by surface waves [Ostrikov et al. (1999d)I. The variable dust charge and self-consistent variations of the equilibrium background plasma density can also be a decisive factor in the dispersion and damping properties of high-frequency electrostatic surface waves (with the resonant frequency wsw = wpe/d=). +ed). For example, a coupling of the surface waves and the dust-charge relaxation mode can result in the anomalous damping and frequency downshift of the SW [Ostrikov et al. (1999c)I. Furthermore, near-sheath particulate clouds substantially modify the RF power deposition [Ostrikov and Yu (1999a)I in gas discharges sustained by the high-frequency axially-symmetric surface waves.

6.3

Waves in Weakly Coupled Magnetized Complex Plasmas

Magneto-acoustic waves that propagate obliquely to the magnetic field in a non-uniform plasma can encounter the Al$&n resonance [Cramer (2001)], where the wave number perpendicular to the magnetic field direction becomes infinite (in the limit of zero electron temperature and no resistivity) and wave energy can be absorbed there. This process has been postulated to be responsible for the heating of the solar corona [Ionson (1978)], and has been used as an auxiliary heating mechanism for laboratory plasmas

Waves and Instabilities in Complex Plasmas

295

[Hasegawa and Chen (1976)l. The Alfvitn resonance process is modified by ion-cyclotron effects, and by the presence of multiple charged ion species. One example is a two-ion components plasma, where a second resonance, the ion-ion hybrid resonance, appears. This resonance can play an important role in the heating of fusion plasmas with the presence of a minority ion plasma component [Cramer and Yung (1986)I. Another example is a complex plasma [Cramer and Vladimirov (1996a)l. The Alfvkn resonance process is strongly modified by the presence of dust [Cramer and Vladimirov (1996a)], even if the proportion of charge residing on the dust is quite small. The presence of dust grains strongly affects the dispersion relations of collective waves in a magnetized plasma, and low-frequency electromagnetic waves, that would be described as shear and compressional A l f v h waves in a dust-free plasma, are strongly influenced by the non-cancellation of the ion and electron Hall currents. The right-hand circularly polarized mode experiences a cutoff due to the presence of the dust. Furthermore, for a finite charge on the grains the waves are better described as circularly polarized whistler or helicon waves extending to low frequencies. The resonance absorption of waves in dusty interstellar molecular clouds [Cramer and Vladimirov (1997)l can play a role in their energy balance and in the magnetic breaking of protostellar clouds. The boundary layers of fusion plasmas can be highly structured, with large density and magnetic field gradients. Thus surface waves may be easily excited in the edge plasma and affect wave absorption in the plasma interior [Elfimov et al. (1994)l. Here, we consider how the Alfvkn resonance condition is modified by the presence of charged dust (Sec. 6.3.1). The waves in a weakly non-uniform magnetized dusty plasma are considered in Sec. 6.3.2. The effects of dust on the dispersion of surface waves in highly structured dusty plasmas are presented in Sec. 6.3.3. In addition to modifying the dispersion relation of the waves, the dust can also affect the absorption of wave energy because of a damping mechanism. Here, we investigate the role of these additional damping mechanisms in the resonance absorption process and show that a short wavelength dissipative Alfvitn mode exists due to these dissipative processes. The resonance absorption can proceed via a mode conversion from the hydromagnetic fast wave into this dissipative mode, which then dissipates the energy rapidly. A surface wave on a sharp surface, and therefore with no resonance damping, will have a two-mode structure, with dissipation occurring primarily via the short wavelength dissipative mode.

296

6.3.1

Physics and Applications of Complex Plasmas

The Alfve'n resonance

We employ a three-fluid model of the cold complex plasma, consisting of the fluid momentum equations for plasma (singly charged) ions, electrons, and dust particles all with the same mass and with constant (negative) charge. The uniform background magnetic field Bo is assumed to be in the z-direction, and the electron, ion and dust grain number densities in the in the uniform plasma are n,o, nio, and n d o , respectively. The parameter P, = Zdndo/nio measures the proportion of the negative charge in the plasma residing on the dust. The plasma density is assumed to vary in the x-direction. We ignore here collisions between the dust particles and background neutral particles in order to focus on the processes of interest; their effects have been discussed by [Cramer and Vladimirov (1997)I. The equations are linearized, so the wave fields can be taken to vary as f ( x ) exp(ik,y i k Z z - i w t ) , where k , and w are assumed positive. The species momentum equations may be used directly, or the usual dielectric tensor components for a three-component plasma may be used. We neglect collisional and grain charge damping in the momentum equations, and assume the charge on the dust grains is not affected by the wave (although we later briefly discuss the role of the charging process). We define the local A1fvi.n wave speed using the plasma ion mass density, VA(X) = pi0, where pi0 = minio(x), and the normalized frequency f = w/Ri, where Ri = eBo/mic is the plasma ion-cyclotron frequency. We finally obtain the following two differential equations in 2 for the wave field components E y and B,,

+

d w

(6.88)

dB, -dx

iA ~ - D ~ =EY w A

k,D -Bz A

'

If there is the same charge on each grain, A and D are given by W2

A = -u;(

1- f 2

+ 1- f 2 / g 2 (6.89)

Ri D = v;- ( 1L- f 2 W

+1

-

f2/g2

-

k;.

Here, the ratio of the charged dust mass density to the plasma ion mass density is b = m dndo/minio,and the ratio of (magnitude of) the dust-

+

Waves and Instabilities in Complex Plasmas

297

cyclotron frequency I i d = ZdeBo/md to the plasma ion-cyclotron frequency is g = Rd/R,. The factors A and D are functions of x through their dependence on U A and b. However, if there is a continuous spectrum of dust grain sizes, the functions A and D involve integrals over the dust size. For a single dust particle size, A and D are real functions (provided f # 1 and f / g # l ) , whereas for a size spectrum, the wave frequency may become equal to a dust-cyclotron frequency within the integrals, resulting in an imaginary part to A and D and dust-cyclotron damping of the wave. A singularity of Eqs. (6.88) occurs where A = 0. For a wave of fixed frequency, the singularity occurs where the following resonance condition is satisfied: (6.90) where Q = k , v ~ / R i . For a given Q, Eq. (6.90) has two solutions for f 2 , so there are two resonance frequencies. For a cold single ion species plasma ( b = 0), there is a single resonance frequency from Eq. (6.90):

w2= Ic,2?&/(l+

Q2),

which is the generalized Alfvkn resonance. 6.3.2

W a v e s in a weakly non-uniform plasma

Before we proceed to a further discussion of the resonance absorption and surface waves in a highly structured non-uniform plasma, discuss the solution of Eq. (6.88) in a weakly non-uniform plasma. In a uniform plasma, the solutions are of the form E,, B, c( exp(ik,z), where the wave number k , is given by 2

k,

-k,

+A

~ - D ~ A '

(6.91)

Cutoffs occur where k& = 0. If k , = 0 also, the cutoffs correspond to the case of propagation of the wave purely parallel to the magnetic field, and are given by A f D = 0, which, after simple algebra, can be reduced to (6.92)

298

Physics and Applications of Complex Plasmas

Fig. 6.15 The frequency of waves in a weakly non-uniform complex plasma, normalized to the dust-cyclotron frequency, plotted against the normalized wave number, along the magnetic field, for cutoff in k , (dotted curves) and resonance in k , (dashed curves) [Cramer and Vladimirov (2000)l. Pi = lop6 and g = C&/L?i = lop6.

There are three positive frequency solutions of Eq. (6.92) for the parallel propagating modes for given a , as shown by the dotted lines in Fig. 6.15. The lowest frequency mode is the dust-cyclotronmode, which is right-hand circularly polarized, has a resonance in k , at w = !&,and has the dispersion relation at low k,:

where VAT is the Alfvkn speed based on the total mass density of plasma ions and dust particles. The next highest frequency mode is the ion-cyclotron mode, left-hand circularly polarized, with a resonance in k , at w = Ri, and with the same dispersion relation at very low k,, i.e., for w << Q, as the right-hand mode; the two modes together can form the linearly polarized Alfvkn wave with the phase velocity V A T . For the frequency range f l d << w << Ri, the lefthand mode has a whistler-type dispersion [Vladimirov and Cramer (1996)]

The highest frequency mode is a right-hand circularly polarized mode, which is the fast Alfvkn wave at high frequencies, with a k , cutoff at R, (1 b ) R d / ( l - Pi). The linear and nonlinear properties of this

-- +

Waves and Instabilities in Complex Plasmas

299

mode for stationary dust have been discussed in detail by [Vladimirov and Cramer (1996)]. A resonance in k, occurs, from Eq. (6.91), when A = 0, i e . , when Eq. (6.90) is satisfied. For given a this defines two resonance frequencies. The resonance frequencies are plotted in Fig. 6.15 (the dashed lines) against kz7/A/& = a/g for the case s = lkz/kgl = 1, g = 10-6 , and b = 1. The highest cutoff frequency occurs at w = Rm (= 2Rd in this case) for k , = 0. It is useful to consider two ranges of a for a discussion of the character of the resonances. In the limit of long wavelength along the magnetic field, such that a < g << 1, i e . , ~ , V A << f i d (the left part of Fig. 6.15), the lower of the two resonance frequencies is given by W A = VAT, i e . , the Alfvkn resonance frequency for the plasma with mass density contributed by the plasma ions and the dust particles. The second (higher) resonance frequency is called the dust-ion hybrid resonance frequency here, and in this limit it is given by W H = ((1- Pi)!&f12,)1/2 = (1 + b)1/2Rd. The dust-ion hybrid resonance frequency is the analogue of the ion-ion hybrid resonance frequency considered previously [Cramer and Yung (1986)] in connection with resonance heating of fusion plasmas with minority ion species, usually positively charged. In the shorter wavelength range fld < ~ , V A << fli (i.e., g < Q! << l),to the right in Fig. 6.15, the highest resonant frequency is given by (6.93) and the lower by (6.94) Which of the two resonance frequencies is to be interpreted as the Alfvkn resonance frequency now clearly depends on the size of b, or Pi. If the dust mass density is small enough that b < a 2 / g 2 ,we have w1 = ~ , v A ,which is the Alfvkn resonance frequency based only on the plasma ions, while the other (dust-ion hybrid) resonance frequency is M o d . For any value of b there is always a value of Q high enough that the dust motion is frozen out of the A1fvi.n resonance. If Q becomes comparable to 1, ion-cyclotron effects modify the Alfvkn resonance frequency. Even though the resonant frequency is not dependent on Pi in this case, is has bee shown by [Cramer and Vladimirov (1996a)I that the two cutoff frequencies on either side of

300

Physics and Applications of Complex Plasmas

the resonant frequency are highly sensitive to the value of Pi, and that this can lead to enhanced Alfvkn resonance absorption in a dusty plasma. This is also the case considered earlier work on surface waves in the dusty plasma [Cramer and Vladimirov (1996b)], where it was shown that the presence of dust strongly modifies the dispersion relation and damping of the waves, even though the resonance condition is independent on Pi. Finally, in the case b > a 2 / g 2 ,w2 M a/bl/’, which is the A l f v h resonance frequency based on the dust mass density, and the hybrid frequency is w1 M bl/’Rd. Consider a wave of fixed frequency and k , propagating in the xdirection. Around the Alfvkn resonance, a wave initially at a frequency just below the local resonant frequency and propagating into an increasing plasma density, will encounter the resonance a t the point in the density gradient where Eqs. (6.88) have a singularity. Thus wave energy will be absorbed at the resonance position where the wave frequency satisfies Eq. (6.90). In the collisionless case the resonance absorption in such a nonuniform plasma can be considerably enhanced by the presence of the dust, because the wave may be cutoff downstream of the resonance [Cramer and Vladimirov (1996a)l. Around the hybrid resonance, the wave can propagate at a frequency just below the resonant frequency, and again encounter the resonance in an increasing density profile. If there is a spectrum of dust sizes and charges, the dust is frozen out of or into the plasma motion when the resonant frequency is, respectively, either much greater or much smaller than the typical dust cyclotron frequency. In either case the resonant frequency depends simply on the effective plasma mass density. If, however, the resonant frequency is close to the typical f&, the resonance position will depend on the mass and charge spectrum of the dust, and the resonance will be smeared out. In addition, the dust cyclotron damping will contribute to the total dissipation. ’

6.3.3

Surface waves

6.3.3.1 Dispersion Consider now the solutions of Eqs. (6.88) localized about a narrow surface of width r separating the dusty plasma and a vacuum. The dispersion relation is found by requiring that the tangential components of the electric and magnetic field, E, and B,, are both continuous across the boundaries x = 0 and 11: = -r, i.e., that the homogeneous dusty plasma and vacuum solutions be matched with the solution inside the non-zero width surface

301

Waves and Instabilities in Complex Plasmas

[Cramer and Vladimirov (1996b); Cramer et al. (1998)I. The surface wave solutions mast have k; < 0 in the uniform plasma region z < - T , i.e., the wave fields there vary as exp(k,z), where k, = lkzl. In the vacuum (x > 0) they vary as exp(-k,x) where, since the phase velocity is assumed to be much less than the speed of light c, we have a good approximation k: = k i k:. The solution of Eqs. (6.88) is oblained by a perturbation technique. The wavelength in the plane of the surface is assumed much larger than the width of the surface transition layer, so that the parameter E = k,r is small, and the dispersion equation may be written as

+

+

(6.95)

D(G) = D,(G) EDI(G) = 0 , with G = w - iy,the damping rate y being of order EW.Here,

D,(G)=

D k: ,+-, A, - k$ kV

k,A,-k

(6.96)

and the correction Dl(1;))is given by

Fl 2 -a

Dl(1;))=

1

[-2D-

k, kV

+ A'--'k22

+ ( A k2)3] k,2 dx.

(6.97)

-

Here, A, and D, are A and D evaluated in the plasma at z = - T . The dispersion relation for the surface wave on a sharp dusty plasmavacuum boundary, as E + 0, is obtained by setting Do(W) = 0 in Eq. (6.96). This leads to the following quatric equation for f :

2 4 1 - Pi) (1 - Pi)2f 4

-

( l + s2 11 / 2 a

2

f3-

2 + s2 (1 + [m

1

+ (Pz+ gI2

f2

(6.98) where o = sign(k,) and (remind) s = /k,/k,l. In the dust-free case (Pi = 0 and g = 0) there is one possible positive frequency solution for each sign of the wave number component in the Bo x n or y direction, a fast wave ( f + ) for positive Icy, and a slow wave (f-)for negative k,. For mobile dust grains there can be two surface wave solutions for each sign of k , [Cramer et al. (1998)l. Consider here each sign of k , separately. For positive k g , there are two surface modes, and the larger the proportion Pa of negative charge on the dust, the greater is the separation in frequency between the two modes. The lower frequency surface wave stops

302

Physics and Applications of Complex Plasmas

at an upper frequency close t o the dust-cyclotron frequency, where a resonance in k , occurs, and so can be referred t o as the dust-cyclotron surface wave. For small a / g , this wave has the dispersion relation

(6.99) In the limit k , 4 0, when the quadratic term in k , can be neglected, this mode is the usual non-dispersive Alfvin surface wave, but in the plasma of combined ion and dust mass density. The higher frequency (fast) surface wave has the following dispersion relation for small wave number

(6.100) This mode has a frequency cutoff at w = 0, as k , + 0. The fast surface mode has no upper cutoff for the range of wave numbers shown, but a t much higher frequency (close to the ion-cyclotron frequency) it stops where a cutoff in k + p is encountered [Cramer and Vladimirov (1996b)l. A second surface mode exists generally in any two-ion species plasmas [Cramer and Yung (1986)], and can strongly influence the Alfvkn wave heating of plasmas with minority positive species. For negative k,, again there are two distinct modes, however, there is only a small overlap range of wave number of the modes. The lower frequency mode has the same dispersion relation, for k , + 0, as the corresponding positive k , mode. However, the negative k , mode stops a t the second cutoff of k,, rather than at a resonance. The higher frequency surface wave stops a t a lower wave number k, where it encounters the higher resonance. The higher frequency surface wave, at much higher frequency (close to the ion-cyclotron frequency), stops where a resonance (the Alfvkn resonance) is again encountered. As s decreases, the two modes are practically continuous, except for a narrow stop-band delineated by the second cutoff and the higher resonance in k,. The width of the stop-band between the two modes increases as the proportion of negative charge on the dust increases. 6.3.3.2 Damping

A surface wave of fixed frequency on a dusty plasma-vacuum interface can experience damping in the transition region between the plasma and the

Waves and Instabilities in Complex Plasmas

303

vacuum at either the Alfvh resonance or the hybrid dust-ion resonance, but not both. In the transition region, the plasma ion density decreases from its value at x = -r to zero at x = 0, and the local AlfvBn speed correspondingly increases, so that the local value in the dusty plasma at x = --T to oc) in the vacuum. Thus a surface wave of given frequency experiences resonance damping in the surface only if a resonance curve lies to the right of the surface wave solution curve. Whether the resonance encountered is to be interpreted as the AlfvBn resonance or hybrid resonance, depends on the k , dependence of the local resonance frequency. Resonance cannot occur if the frequency lies in the range separating the lower and upper resonance curves: !& < w < (1 + b)1/2Rd. The positive k, surface waves always encounter a resonance in the surface layer. Similarly, the higher frequency surface mode for negative k , always encounters a resonance. However, the lower frequency mode for negative k , extends into the above frequency range and so does not encounter resonance in that range, and indeed beyond it. If the resonance is not encountered, no Alfvkn resonance damping occurs, and the only damping will be due to global resistive, viscous or other collisional processes, or collisional Landau damping, or due to charging of the dust by plasma currents. The damping rate of the surface waves due to the non-zero width of the surface is proportional to the imaginary contribution to Dl,which comes from the pole of the integrand in Eq. (6.97), i e . , at that point in the surface transition where A = 0 which is the condition for the Alfvkn or dust-hybrid resonance. The resonant damping rate is independent on the dissipative mechanism in the plasma. It is well known that the actual dissipative mechanism depends on the plasma parameters, and can be resistive or viscous for collisional plasmas, or it can be mode conversion into a shortwavelength mode, which is subsequently Landau damped, in a collisionless plasma. All of these processes can occur in a complex plasma as well, but, as we already discussed in the course of this book, an additional unique dissipative mechanism in the presence of dust grains is the dust charging process (see Sec. 2.1). It was shown by [Cramer and Vladimirov (1998)] that the dust charging induces the existence of an additional short wavelength highly damped mode, into which a magneto-acoustic wave (such as the surface waves considered here) will convert as it encounters a resonance. In a sense, this short wavelength mode is the shear AlfvBn wave, modified by the charge perturbations in the plasma during the dust charging process. Another effect of the dust charging is that, even in the case of a sharp surface and thus with no resonance damping, the short wavelength mode

304

P h y s i c s a n d Applications of Complex Plasmas

will couple to the usual surface mode because of the boundary conditions at the surface and the existence of the electric field component E,, with a resulting modification of the dispersion relation and a global damping due to the dust charging. Provided the dust 'charging frequency is small compared with the wave frequency, these effects will not change the above results on surface waves significantly.

6.4

Instabilities in Weakly Coupled Plasmas

In this section, we consider a number of instabilities in a complex plasma such as the current-driven dust-ion acoustic (Sec. 6.4.1), dust-acoustic instabilities (Sec. 6.4.2), and Buneman-type ion-streaming instabilities (Sec. 6.4.3), complemented by the study parametric instabilities in a magnetized dusty plasma (Sec. 6.4.4). Finally, we briefly discuss the basics of some other of the most common instabilities in weakly coupled dusty plasma systems (Sec. 6.4.5).

6.4.1

Dust ion-acoustic instability

In a pioneering attempt to explore the effect of dust grains on conventional ion-acoustic instabilities [Merlino (1997)], it was shown that such an instability can develop in typical gas-discharge dusty plasmas at low or moderate external DC fields and introduced a notion of the dust-ion-acoustic wave (DIAW instability). In this case, the relative drift between the electrons and ions arising from the large difference in their mobilities can result in the excitation of unstable IAW. The instability threshold becomes lower when the dust grains absorb more electrons and electron-neutral and ion-neutral collisions dominate [Merlino (1997)I. A possible damping mechanism that can affect the development of the instability is the loss in average momentum of the injected ions that replace those which are lost to the dust grains [D'Angelo (1994)]. Later, dust-charge fluctuations were included in the description of the instability [Annou (2001)]. However, the problem was not treated self-consistently nor the practical realization of the DIAW instability was verified. Thus the DIAW instability problem was reconsidered and the most important momentum and particle loss mechanisms were included self-consistently [Ostrikov et al. (1999b)I. To explore the DIAW instability in a typical laboratory complex plasma, that is in a collisional plasma system with variable-charge dust particles,

Waves and Instabilities in Complex Plasmas

305

consider a three-component plasma in an external DC electric field EOalong the II: axis. The massive dust grains, containing a significant proportion of the plasma's negative charge, are treated as immobile point masses with a variable charge. The external DC electric field causes plasma electrons and ions to drift in opposite directions with different velocities because of the large difference in their mobilities. The dynamics of the electrons and ions is described by the fluid continuity and momentum equations [Ostrikov et al. (1999b)l

dtn,

+ d,(n,v,) =

--v,dn,

+ s,

(6.101)

(6.102)

dtn,

+ d,(n,v,)= - u n 2 + S,

(6.103)

and

dtv,

+ v,d,w, + vzffv, + (Tz/n,m,)dzn,= (e/m,)E,

(6.104)

where E is the total electric field in the plasma including the steady-state (zero-order) electric field Eo, and u,d and u,d are the rates of capture of electrons and ions by the dust grains. We have defined S = v,n,-pnz, where v2 is the ionization rate, and p is the coefficient of volume recombination. Note that in Eqs. (6.101) and (6.103) the ionization and recombination terms are the same since the electrons and ions always appear and recombine in pairs. For the effective electron collision frequency, we write uzff = u,, u,"' vzh, where v,, is the electron-neutral collision frequency, v,"' is the frequency of the elastic electron-dust Coulomb collisions, and u,Ch is the effective frequency of charging collisions (due to collection of plasma particles by dust). For the ion component one has u , " ~ = u,, u,"' v$, where u,, is the frequency of ion-neutral collisions, u,"' is the frequency of the elastic ion-dust Coulomb collisions, and is the rate of collection of ions by the dust grains. Additional terms affecting the sources and sinks of the plasma particles, such as step-wise ionization, ambipolar or anomalous diffusion, can be added to the right-hand sides of Eqs. (6.101) and (6.103). The charge neutrality condition n, = n, Z d n d , where z d is the magnitude of the negative dust charge, and Poisson's equation for the electric field complete the basic set of equations. The rate coefficients involved can be found in Sec. 6.2. The fluctuations of the average dust charge are described by the charge balance equation similar to Eq. (2.8) of Sec. 2.1.

+ +

+ +

+

306

Physics and Applications of Complex Plasmas

The set of Eqs. (6.101)-(6.104) is linearized and in the zeroth-order approximation one obtains ne0 = (vi - v,d)/p, nio = ( v e d / v i d ) n e O , z1o, = -(e/m,v,"ff)Eo, and vi0 = (e/miv,eff)Eo.Assuming that the perturbed quantities depend on the coordinate z and time t as exp[i(kz-wt)], combining the linearized Eqs. (6.101)-(6.104), (2.8), and quasineutrality condition, one can arrive to the following dispersion equation

(6.105)

where R = Ri = w - kvi. Furthermore, uo = W,O - vi0 is the relative drift of electrons and ions, and p = Ileolndo/en,o. The DIAW instability threshold can be estimated by considering the case of marginal stability [Merlino (1997); Annou (2001)]. Setting R = Rr + i y one derives the growth rate for the marginal case (y << w)

(6.106)

2 mine0

where v' = V i d - 2v,d+vi. It is seen that the first three terms in Eq. (6.106) are always negative and are responsible for the damping of the DIAWs. The only term that can be positive is proportional to 1 - kuo/Rr, where Rr N f k c s d , and V S d = JTi/mi TeniO/mineOis the ion-acoustic velocity in the dusty plasma. Since the drift uo is always negative, for 1- kuO/Or to be positive, one must have k < 0 and luol > vsd. In this case, the threshold (y = 0) Mach number h f t h z IUOlth/Vsd of the DIAW instability is

+

(6.107)

In Eqs. (6.105)-(6.107) the variations in the electron and ion sources and sinks caused by the fluctuations of the grain charge have been neglected. One can show that this is valid if

307

Waves and Instabilities in Complex Plasmas

where X is the DIAW wavelength. This inequality is usually satisfied when the currents flowing onto the grains are weak, and the dust number density, size, as well as the electron-to-ion temperature ratio are small. For the typical dusty plasma parameters p / w = 0.1, Ti/T, = 0.1, a = 1 pm, n,dO = lo5 cmP3, and =10 cm (in the experiments on DIAWs in dusty plasmas wavelengths up to 20 cm have been observed [Merlino et al. (1997)]), we have A O(10P3). However, this inequality is very sensitive to an increase in the dust size and concentration, and can eventually become invalid. Therefore, the effect of the electron (ion) capture-rate variation is important when the dust size and density are high. It is of interest to compare the threshold drift (6.107) for variable dust charge with the expression N

obtained in the case of a constant dust charge approximation [Merlino (1997)l. It is clear that the threshold (6.107) is much higher. Moreover, in most cases when ne0/nto is not too small, the unity in Eq. (6.107) is negligible. This is clearly not possible in the case of a constant dust charge since ui, << u,,. Meanwhile, the quantities @, u i d , u z f f ,v e d , and vih are of the same order, and usually u : ~ , >> vi, [Ostrikov et al. (1999b)l. For an argon or hydrogen plasma with T, = 2 eV, Ti = 0.2 eV, n,o/nio = 0.5, po = 10 mTorr, nio = 3 x lolo cmP3, N , = 3 x 1014 cmP3, a 1 pm, and 2 2, where po is the gas feedstock pressure, and N , is the concentration of neutrals, we have h f t h 1.1 x l o4 in Ar and 2.8 x 10' in Hz. These ratios correspond to 1 ~ 0 1 t h 6.6 x lo6 m/s in HZ and lu0lth 4.7 x lo7 m/s in Ar. We note that the electron thermal speed under these conditions is U T ~= 5.5 x lo5 m/s. The corresponding threshold electric field for Ar plasmas is EO lo3 V/m at po = 10 mTorr, EO 5 x lo3 V/m at po = 50 mTorr, and Eo lo4 V/m at po = 100 mTorr. Under the same conditions the instability for the constant dust charge case develops for Eo lo2 V/m a t PO= 100 mTorr, and less for lower values of the working gas pressures [Merlino (1997)l. The value of h f t h strongly depends on the relative amount n e 0 / n i o of electrons in dusty plasma. This dependence is almost linear when n,o/nio << u,h/P. We note that a similar tendency is also the case for the constant dust charge case. The constant dust charge regime is valid if the rates of dust charge relaxation and plasma particle capture are small compared with the frequencies of the electron- and ion-neutral collisions, which is the case, e.g.,

-

-

N

N

N

-

N

N

N

-

308

Physics and Applications of Complex Plasmas

when the working gas pressure is not too low. Otherwise the charge variation effects must be included. However, accounting for the dust charge relaxation only [Annou (2001)], is not self-consistent since most of the electron/ion capture (by the dust) processes take place at the same time scale as that of the instability. Furthermore, by neglecting the electron and ion density variations due to the charging processes one implies the existence of unspecified uniformly distributed particle sources that replace the lost electrons or ions. Moreover, an instantaneous increase of the negative charge on the dust grains means that more electrons than ions are absorbed by the grains. This effect can only be described by accounting for the variation of the plasma particle capture rate in the presence of dust charge fluctuations. 6.4.2

Dust-acoustic instability

The open character of a dust-plasma system also affects the current-driven dust-acoustic wave instability ( D AW instability) in a collisional plasma with variable-charge dust particles [Ostrikov et al. (2OOOc)]. We derive the instability thresholds affected by a number of particle sources and sinks, as well as by numerous dissipative mechanisms. In particular, we show that the current-driven instability is relevant to dust cloud filamentation at the initial stages of void formation in dusty RF discharge experiments discussed in Sec. 5.1.2. In many space and laboratory environments complex plasmas are affected by external DC electric fields, such as that in the sheath/pre-sheath regions, or that additionally introduced for electrode biasing, ion extraction, etc. In such cases the occurrence of directed plasma flows is unavoidable. The latter can give rise to current-driven instabilities, which develop at the dust-acoustic [Merlino et al. (1997); D’Angelo and Merlino (1996); Rosenberg (1996)l or dust-ion-acoustic [Merlino et al. (1997); Merlino (1997)l time scales. As has already been mentioned in previous sections, the electron and ion capture/release by the dust grains, usually occurring at the same time scale as that of dust charge variation, can strongly affect various collective phenomena in dusty plasmas. On the dust-acoustic time scale, the above effects can affect the dispersion properties and stability of the dust-acoustic waves. It is also noteworthy that ion drag on the dust grains is important for the instability and can lead to the formation of regions void of dust in the plasma (see Sec. 5.1). Below, we discuss the effects of the plasma particle capture/release by the dust as well as the ion drag force on the development

Waves and Instabilities in Complex Plasmas

309

of the current-driven instability at the dust-acoustic time scales and relate the results to void formation, dust cloud filamentation, and instabilities in low-temperature RF complex plasmas. Consider a three-component plasma in an external DC electric field Eoi. The external DC electric field causes the electrons and dust particles t o drift with different velocities in a direction opposite to the ions. Here, we list only those of the fluid continuity and momentum equations that differ from the basic set of Sec. 6.2.3: (6.108)

(6.109)

-+at and

d(ndvd)

dX

=0 ,

(6.110)

where E is the total electric field in the plasma including the steady-state (zero-order) field Eo, and other equations and notations are similar to the ones used above in Sec. 6.2. In Eq. (6.109) the term P i r a g ( V d - v,) corresponds t o the ion drag force acting on the dust grains. In the standard OML theory expressions for the microscopic ion currents on the dust particles, the contribution of the steady-state ion flow v , ~= eEo/rn,Z/,e'ff has been accounted for. For the electrons, the thermal energy is much larger than the directed energy associated with the steady electron flow v , ~= -eEo/mevzff due to the external field, so that the effect of the latter in the electron charging equation has been ignored. In Eq. (6.109) there are two dissipative terms that affect the dust dynamics, namely the rate of dust-neutral collisions u d n u d of Sec. 6.2, and the ion drag term p i r a g ( v d - v,), where pirag 4 m , n 0 , b 2 v , / m d mdis the ion drag coefficient, b a*(l 1 - Acp,/T,) is the ion-collection impact parameter (see Sec. 2.1)' and v, is the ion-acoustic velocity. For the stationary dust drift velocity, one obtains

-

-

(6.111)

u?tek,

where A = ZdOmivtff/md&ag. Note that since > (u,,,uin) the magnitudes of the electron and ion drift velocities are lower than those under the constant charge approximation. The second term in Eq. (6.111)

Physics and Applications of Complex Plasmas

310

is responsible for the pushing of the dust grains in the direction opposite to the electric force acting on the negatively charged dust particles. Thus, if the ion drag is weak, the dust particles drift in the negative x direction, and if the ion drag is sufficiently strong, they drift in the opposite direction. In particular, if A > 1, the negatively charged dust particles drift in the same direction as the plasma electrons. Otherwise, the dust particles and electrons drift in the opposite directions. The dispersion relation of the unstable dust-acoustic mode is [Ostrikov et al. ( ~ O O O C ) ]

(6.112) where fle0 = f l & - k6,, Q,o = f l d o - kd,, 6,= 'Ueo - U d o , and 6,= V,o - v d o . Here, rle = Reo+i(v,,,-ved+IC2v~,/veeff) and 7%= fl,o+i(v,d+k2V~,/v,eff). In the limiting case of negligible dust-specific electron and ion dissipation (by formally setting uc"zfiFee, 4 v(z,e)n and v(,,,)d -+ 0), dust-charge variations (vih 0), as well as the ion drag force (pirag+ 0), the dispersion relation (6.112) is reduced to Eq. (7) of the earlier work [D'Angelo and Merlino (1996)]. The analytical solutions for the frequency of the unstable mode can be obtained in the case when the equilibrium density of the ions much exceeds that of the electrons ( n , > ~> n , ~ ) This . may happen at the initial stage of the filamentary mode development [Samsonov and Goree (1999b)], when the charge density Z d o n d o of the dust particles constitutes a significant proportion of total plasma negative charge density. By assuming f i d = fli ifl:, from Eq. (6.112), it is possible to derive -+

+

(6.113) for the imaginary part of the frequency in a dust frame, where B = Zd0miv,eff/mdv&. One can see from Eq. (6.113) that there is no instability if k6i < R&. If, however, the relative ion-to-dust drift in the external electric field is such that 6i > fl&/k, the dust-acoustic waves can become unstable. The instability criterion strongly depends on the direction of the stationary dust drift in the external electric field (6.111). As already mentioned,

Waves and Instabilities in Complex Plasmas

311

one can see that the dust grains drift in the direction of the electric force (opposite to the ion drift) when the inequality A > 1 is satisfied. This means that the effect of the steady electric force on the dust particles overcomes that of the ion drag force. If the opposite inequality holds, the dust grains are pushed by the ion drag force and move in the same direction with the ions. Note that the inequality varies continuously with growing dust grains (see Sec. 7.1). It is possible that A > 1 is satisfied for initially small grains, and when the grains becoming sufficiently large the condition eventually becomes violated. We now examine the conditions when the current-driven dust-acoustic instability can develop [Ostrikov et al. (2OOOc)]. One can see that the terms in Eq. (6.113) leading to the instability are proportional to the effective frequency of the ion collisions. This means that ion collisions have a destabilizing effect on the dust acoustic waves. It is worth mentioning that the electron-neutral collisions in a dust-free plasma lead to resistive ion-acoustic instability in a similar manner. We emphasize that the competition between the electrostatic and ion drag forces on dust particles strongly affects the conditions for the instability. The ratio of these forces is d,which can be greater or less than unity. In fact, this ratio can be in a rather broad range from to 2.5 x lo4 [Ostrikov et al. (2OOOc)]. The conditions for the instability strongly depend on the value of parameter B. If it is small compared with unity but still large compared with p i r a g / v d n r then the relative ion-to-dust drift should be large such that Gthres O L / k B >> OL1.k. In this case we face the situation that the destaa bilizing effect of ion collisions is small, but the effect of the ion drag is even smaller. If the opposite inequality, B >> 1, holds, 6, should simply exceed n L / k , as is the case for the constant dust charge case [D’Angelo and Merlino (1996)j. This conclusion is expected since ion drag then becomes unimportant. For conditions representative of the experiments [Merlino et al. (1997); D’Angelo and Merlino (1996)], namely mi/m, 51600, ma/md rv 4.7 x l o p i 4 , nio lo9 ~ m - van ~ , 8.8 x lo4 s-l, N , 3 x 1015 ~ m - T, ~ , 3 eV, Ta 0.1 eV, T n 0.025 eV, and a 5 pm, one can obtain A 10’. Therefore, the effect of ion drag can be neglected and the instability seems to follow the weak ion drag scenario. On the other hand, one also obtains B 10-l. Thus, 6, must exceed lOnl,/k for the dust-acoustic wave instability to be realized [Ostrikov et al. (2000c)l. We now discuss the relevance of these results to the dust-void experiments [Samsonov and Goree (1999b)l (see also Sec. 5.1.2). In particular,

-

N

-

--

N

N

N

N

N

-

N

312

Physics and Applications of Complex Plasmas

it was assumed that the electric field is externally applied and uniform. Clearly, from the Poisson's equation it follows that electric fields sufficiently strong for the development of the instability can be generated because of charge fluctuations in the pristine dusty plasmas. Namely, for ne0 101010l2 cmP3, 13,/n,o 10-3-10-4, and characteristic filament size L 1 cm, an electric field Eo 1.5-150 V/cm can be generated. Measurements near the instability threshold clearly indicated the presence of the DC electric field of EO 20 V/cm [Samsonov and Goree (1999b)I. An important feature of the experiment [Samsonov and Goree (1999b)l is a relatively high operating gas pressure (400 mTorr), providing a large neutral gas density N , 1.4x 10l6 cmP3. For such an operating regime the assumption of the volume recombination controlled regime would seem to be fairly accurate. For lower operating gas pressures the diffusion terms in particle balance equations (6.101) and (6.103) should be taken into account. The high density of neutrals yields a relatively large rate v,, lo7 s-l of ion-neutral collisions in argon at po = 400 mTorr. This rate appears larger than the rates of ion-dust Coulomb (v,"') and charging ( u , " ~collisions. ) For EO 20 V/cm, T, 3 eV, and T, 0.05 T,, the characteristic ion drift velocity is W,O 4.4 x lo4 cm/s. For this value of V,O and the spatial scale 1 cm of the filamentation, the nonlinear term w,av,/dz in Eq. (6.108) is ignorable. However, this is valid only at the initial stage of the instability. At later stages the nonlinear term can be crucial in determining self-organized nonlinear dissipative structures of the dust void (see Sec. 5.1). Furthermore, near the filamentation threshold, assuming a 0.13 pm one can estimate the dust mass m d 9.2 x 10-15 g. From the condition of the equality of the equilibrium electron and ion grains currents on the dust particle one obtains Z d o 250. For n d lo8 cmP3 and T, 0.1 T,, it follows that the dust-neutral collision rate vd, is approximately 1.31 x l o 3 s-l. We remark that the latter exceeds the characteristic frequency of the oscillations associated with the plasma striations at the initial stage of the instability. There is however a fairly large uncertainty in determining the ion drag force, which depends strongly on the impact parameter b for the ion collection in the ion drag coefficient. Assuming a local potential difference 4% 3 eV, and T, 0.15 eV, for the ion drag coefficient one obtains pdrag 1.65 x lo3 s-l so that p&,g/vdn 1.26. For argon gas and 130 nni dust particles the key parameters are: ZdOmz/md = 1.95 x 10-6, Zdomzvzff/ r n d v d , = 0.015, and Zdomzu,"ff / r n d p & rag = 0.01, respectively. This means that the effect of the ion drag on the current-driven dustacoustic instability can really be important for the development of the

--

--

-

-

N

- -

N

-

N

-

-

-

N

--

N

-

N

Waves and Instabilities in Complex Plasmas

313

filamentary mode in dust void experiments [Samsonov and Goree (1999b)I. The sudden onset of the instability can be understood if we note that the threshold value of the electric field, appears to scale like EihresN 1/a [Ostrikov et al. (2oooc)]. This means that for small dust grains the instability threshold cannot be reached. When the grains grow in size, the threshold is greatly decreased and the instability onset becomes possible. To conclude this section, we emphasize that the conditions for the instability are very different for weak and strong ion drag. Furthermore, the threshold of the external electric field is la,rger for variable-charge dust particles compared to that for constant dust charge because of the large dissipation rate induced by the dust particles. Finally, the self-consistent plasma particle sources and sinks, as well as numerous dissipative processes strongly affect the onset and development of other plasma instabilities, including the drift wave [Vladimirov and Tsytovich (1998)] and ion acoustic [Ostrikov et al. (1999b)I instabilities. 6.4.3

B u n e m a n “dust-ion streaming” instability

In the classical case of the plasma comprising only electrons and a single ion species, a commonly known B u n e m a n instability can be caused by the streaming of the electrons with respect to the ions. In a complex plasma (such as, e.g., in the sheath region), the situation becomes more complicated: in addition to the ion flow and Boltzmann distributed electrons there are also (almost) stationary dust grains. However, the nature of the instability remains the same, namely, the energy of the ion flow is converted to the low-frequency oscillations of acoustic type [Vladimirov and Ishihara (1999)l. An important feature of such a dust-ion streaming instability in a complex plasma is that it can also appear in the case when the speed of the ion flow is less than the ion-acoustic velocity, which is important for many experiments (especially those in DC discharges, in RF discharges under microgravity conditions, and experiments with smaller particles, where dust particles can be collected in the plasma bulk), as well as for the plasma processing technology. In the first approximation, the dispersion relation gives (6.114) This relation gives the following equation for the parallel and perpendicular

314

Physics and Applications of Complex Plasmas

(to the ion flow) wave vector components (6.115)

From Eq. (6.115), we can easily see that oscillations with lkll < Jkll)can be excited only for Mach number M < 1 ( i e . , subsonic character of the ion flow). For parallel propagations, we have (6.116) (6.116) while for almost parallel, but oblique propagations we have (6.117)

The supersonic ion flow excites oblique perturbations with Ikl I the Mach numbers are sufficiently high, M >> 1, we have

2 lkll I.

If

(6.118) The excited perturbations are unstable with the real (6.119) and imaginary (6.120)

d )( 1 / 2 ) ( ~ ~ d / w parts of the frequency, where E = ( 2 3 ~ r ~ i ) / ( 2 n i r n= The growth rate of the excited perturbations for parallel propagations is peaked around the wave length given by Eq. (6.116) with a half-width in the range (6.121)

for the subsonic ion flow ( M < 1), and the growth rate is peaked around the wave number given by Eq. (6.118) with a half-width in the range (6.122)

315

Waves and Instabilities in Complex Plasmas

for the supersonic ion flow ( M > 1). For a typical value of the RF discharge dusty argon plasma (ni lo9 ~ m - experiments ~ ) and z d 3 x lo4, 00-12 g, we have y 104 s-l which is nd lo5 cmP3, m d 8 x lo4 quite high comparing with the inverse characteristic time of dust motion. With T, = 1 eV, we obtain the characteristic distance of the instability L = 1.9 cm, which is quite close to the scales of experimental devices. N

N

0

N

N

N

0 -

Fig. 6.16 Growth rates in the k-space for various Mach numbers [Vladimirov and Ishihara (1999)]. kll = k l i X ~ ,and k l = IkllXD,. w p i / w p d = lo3.

The numerical solutions are shown in Fig. 6.16 for various values of the Mach numbers. We set W p i / W p d = lo3 (or E = 0.5 x loP6) and the growth rates are shown in the k domain. For the subsonic ion flow, the instability is developed in a wide range of k space, i e . , 0 5 K I I5 0.8 at K I = 0 and 0 5 K I I5 1.1 at K I = 1.2 for M = 0.8, where K I I= I c l l X ~ , and K l = IklIXo,. The peak of the growth rate occurs at the wave numbers described by Eq. (6.117). For nearly sonic flows as shown in Fig. 6.16, the peak of the growth rates occurs at the wave numbers given by K I = K i , as expected by Eq. (6.117). In the supersonic flow, the instability is dominated by modes K I >> K I I . The concept of the dust-ion streaming instability has been invoked to interpret the observation of the instability in the striation of a DC glow discharge [Molotkov et al. (1999)]. Later, it was shown that the instability can be excited due to the variation of the grain charge in the presence of an external electric field [Fortov et al. (2OOOa)]. The experimental and theoretical analysis of the dust-acoustic wave instabilities in gas discharge

316

Physics and Applications of Complex Plasmas

plasmas was also extended to the diffuse edge of RF low-pressure inductively coupled plasmas [Fortov et al. (2003)l. The instability was spontaneously excited in the area of a free diffuse edge of the 100 kHz inductive low-pressure (1-120 Pa) discharge in neon. The dynamics of the suspended dust particles at the diffuse edge of the plasma was recorded by a specially oriented laser sheet. The wave number, frequency, and growth rate were measured by using a high-speed CCD camera, whereas the main parameters of the discharge plasma were measured by a Langmuir probe. These measurements have revealed that the plasma parameters within the dust cloud are more uniform than in the striation of DC discharges or in the near-sheath areas of capacitively coupled plasmas. The dispersion equation for unstable dust-acoustic modes was generalized to incorporate the effects of DC electric field, ion-neutral collisions, dustneutral friction, dust charge variation, and forces acting on the grains in the non-uniform areas of the discharge [Fortov et al. (2003)l. It has been experimentally confirmed that the DC electric field in the cloud is a necessary condition for the development of the instability. Furthermore, the instability originates as random fluctuations of the dust density, the latter eventually developing into regular dust-acoustic modes. Further growth of the wave amplitude signals the transition to the nonlinear regime of the instability. An interesting conclusion drawn from the discussed experiments is that the grain charge does not vary much during the development of the instability, which can be described fairly accurately within the framework of the constant dust charge approximation.

6.4.4

Parametric and modulational instabilities in magnetized complex plasmas

As we already mentioned, one of the most important effects in the presence of dust is the collection of electrons and ions from the background plasma by the charged grains, influencing the propagation of plasma and electromagnetic waves. Here, we investigate the propagation of plane hydromagnetic waves parallel to the pumped magnetic field, modified due to the presence dust [Hertzberg et al. (2003b); Hertzberg et al. (2004a); Hertzberg et al. (2004b)l. We assume all the dust grains are of the same size and mass, and have the same charge and highlight the effect of dust on the parametrically excited waves. Large amplitude parallel propagating modes are also considered, and modulational instabilities [Vladimirov et al. (1995)l of these modes are analyzed.

317

Waves and Instabilities in Complex Plasmas

We invoke the standard multi-fluid plasma model. The background magnetic field Bo is in the z-direction. We employ the parameters (which measure the distribution of charge amongst the species) bi = ne/Zini, where Zi accounts for a possible multiply charged ions, and b d = n , / z d n d . w e then may write the total charge neutrality condition as 1/& I/& = 1. Ignoring collisions, but including the effects of pressure, the nonlinear equations for the velocities, electric and magnetic fields reduce to:

+

dB -at= v x

(if i: (-+-)

1

('"

x B ) - ~ V X

B, ene

")

(6.123)

(6.124)

(6.125) Here P i , d = m i , d n i , d are the densities of each massive component of the plasma and !&d are the corresponding (signed) cyclotron frequencies. U, are the individual sound speeds, and a: = ZiUzme/mi and a: = Z d U z m , / m d are thermal speeds associated with the electron pressure. 6.4.4.1 Parametric pumping To study the parametric instability, the background magnetic field is modulated periodically, with B(O) = Bo(1 Ecos(k~z)cos(w0t))z. The effect of this magnetic pumping is to modify the velocity, density and charge imbalance to order E, and we obtain the following dispersion equation for the pump wave:

+

w; (w," -u :)= w k; (w," - x - Yk,"),

(6.126)

where /w,I is a hybrid cutoff frequency as k~ + 0. In the presence of a secondary species (dust) the fast magneto-acoustic wave gains an additional mode and the dispersion relation is dispersive. We now test the stability of the pump wave solution. We search for transverse plane wave perturbations, propagating in the z-direction, with ion velocities v 1 and v 2 , with a spatial dependence exp(ikz). The dispersion

318

Physics and Applications of Complex Plasmas

equation governing linear waves (with E = 0) propagating along the z-axis is given by

(6.127) For a negatively charged secondary species there are three modes of excitation. There are two right hand modes WRf and W R ~(with a resonance at the heavy species cyclotron resonance), and a single left hand mode WL with a cutoff. A parametric interaction occurs between the excited linear fields due to the pump fields. There will be a resonant parametric interaction when w wo or w - wo is near a natural mode of the system. It is found that the only choice for this resonance is for a left-hand polarized mode to interact with a right-hand polarized mode, or vice versa. The quantity T = ($/R1)’ was calculated as a function of pump frequency p = w 0 / 0 1 for the upper “U” and lower “L” branches of the pump dispersion relation for each mode of interaction [Hertzberg eC al. (2003b); Hertzberg et al. (2004a); Hertzberg et al. (2004b)l. It was found that “Uf’ and “Lf’ correspond to the combination between W R and w L f , that is the fast interaction. YJs” and “Ls”, refer to the slow interaction between WR and W L ~ .The dust species has a substantial number density in the plasma, with 60, = 1.1 or 6oi = 1.02. The “UP’ interaction is unstable and monotonically decreases, approaching the single species result “N”for large W O . The “Ls” interaction is also unstable, however it experiences a minimum, about which it turns over and approaches zero. The “Us” interaction is small and positive, and hence it is weakly stable. The “Lf’ interaction, which is only present in a warm plasma, is strongly stable. The instability growth rates are generally reduced compared t o the single species case.

+

6.4.4.2 Modulational instabilities Investigate now the stability of the parallel propagating waves. By considering the low and high p regimes we test for modulational and decay instabilities [Vladimirov et al. (1995)l. We now refer to the parallel propagating waves as the ‘pump’ wave. The linear solution is an exact solution of the nonlinear equations. The velocity expansions may be written VZ(Z,

t)

= FVlZ(Z, t )

+ &’V;Z(Z,

t )+ E’V;,(Z, t )

(6.128)

+

t ) f E’VLd(Z, t ) ,

(6.129)

Vd(2,t ) = E V l d ( Z , t )

E’V;d(Z,

Waves and Instabilities in Complex Plasmas

319

where E refers to the parallel propagating pump, and E’ refers to the excited fields. Substituting these fields into the nonlinear equations, we obtain the equation of motion governing the transverse velocities for each ion species. The presence of dust causes the resulting differential equations to contain non-derivative terms that would otherwise be absent. The dispersion equation gives a tenth-order polynomial. In the case of electron-ion plasma, the dispersion equation was found t o be a sixth-order polynomial. In the domain K = k / k o < 1 a modulational instability exists, which is also seen in the one species description. In Figs. 6.17(a) and 6.17(b) we plot the growth rate for a range of values of 601. This modulational instability is found to decrease in growth rate and shift to low values of wave number

004

0 03

Y 0 01

0 00 0

1

2

3

4

K

OM

0 03

Y

Oo2

0 01

0 00 0

1

2

3

4

K Fig. 6.17 The normalized growth rate Im(w)/Q for the full dispersion relation plotted against the normalized wave number K = k / k o , with K. = k o v ~ i / R i= 0.3 and wo(k0) the upper (positive) left hand mode [Hertzberg et al. (2004a)l. (a) Rd/Ri = 0.04 and boi = 1.05. (b) Rd/Ri = -0.04 and 60i = 0.9, for a range of boi. Here pi = O.l,p, = 0, Bi = 0.4 and TJ’ = 0.04.

320

P h y s i c s and Applications of C o m p l e x P l a s m a s

as 6oi moves away from 1. For K 2 1 there exists a very narrow instability. It is found that this shifts to higher values of wave number as the number density of the second species is increased. In the region K > 1 we see a strong decay instability. This is absent when Soi = 1, and hence this reveals an extra instability. It is the result of the two slow acoustic modes associated with the additional heavy species interacting. For high values of p, all four acoustic modes increase in magnitude. We only find instabilities for high values of when the pump wave is the upper (negative) right hand mode. Again a modulational instability appears. In the single species case this is the only instability found. However, we find two further instabilities, both of which are quite narrow. Firstly, the slow right handed mode interacts with the fast acoustic mode in the region

K21. Secondly, the right handed transverse mode interacts with the slow acoustic mode in a region such that K >> 1. If the second species is positive then these narrow decay instabilities are no longer present. Also, if the pump wave is chosen to be the slow mode (right or left handed) no additional instabilities with large growth rates are found [Hertzberg et al. (2003b); Hertzberg et al. (2004;); Hertzberg et al. (2004b)l. 6.4.5

Other instabilities

Various instabilities in weakly coupled dusty plasmas have been a very popular research topic in the last decade. A possible reason is that various plasma media often become unstable under external or internal action. In fact, most of the instabilities common to dust-free plasmas, have been revisited by incorporating micro- and nanoparticles as an additional plasma species with a constant or variable charge (see e.g., [Shukla et al. (1996); Shukla et al. (1997); Nakamura et al. (2000); Verheest (2000)] and references therein). For this reason it is really difficult to overview this area of complex plasma research in a short section. Nevertheless, below we discuss some of the instabilities that involve dust-specific processes. Generally speaking, a unique feature of dusty plasmas is that under certain circumstances the random or correlated collective excitations in the complex plasma system can become spontaneously unstable. In most of the cases, the heavy dust introduces a “longer” timescales comparable with those of dust motions and oscillations, which opens up a separate temporal niche for the interplay between the effects associated with electrons, ions, and heavy dust grains.

Waves and Instabilities in Complex Plasmas

321

Some examples of spontaneously excited regular or random motions of the ensembles of dust grains include anomalous grain “heating” in capacitive RF discharges [Quinn and Goree (ZOOO)], spontaneous dust oscillations [Nunomura et al. (1999)], “heartbeating” and vortices in capacitively coupled plasmas [Morfill et al. (1999)], and several others. Some of these phenomena have already been discussed above. We emphasize again that the ionization-recombination balance plays a critical role in the development of many interesting phenomena in complex plasma systems. In particular, an interplay between the effects of the ionization and ion drag force has been explored as a possible mechanism of excitation of the dust-acoustic ionization instability [D’Angelo (1998)I. The underlying physics behind this instability is in the difference between the ionization efficiencies of weakly ionized plasmas in the areas of high plasma densities and potentials (wave crests) and plasma rarefaction (wave troughs). Specifically, the electron-impact ionization of the neutral gas usually proceeds in the wave crests at a rate higher than in the wave troughs. Thus, if the energy dissipation mechanisms are not too strong, the wave amplitude might grow [D’Angelo (1998)l. However, the question about the excitation of the dust acoustic waves through the development of the ionization instability still remains open since within the parameter ranges of the experiments [Samsonov and Goree (1999b)I the DAW excitation conditions are not met even if the ion drag force is taken into account [D’Angelo (1998)I. On the contrary, an increase of the ion drag force only enhances the damping of the DAW. However, a zero-frequency (non-propagating) perturbation can be excited and grows when the ion drag on the dust grains overcomes the effect of the perturbation electric field [D’Angelo (1998)]. Note that streaming and other instabilities play a significant role in space dusty plasmas. In particular, streaming phenomena can be important in the interactions of the solar wind and cometary dust grains, interstellar dust clouds with dust size distribution, in the wake of intense shock waves traveling through the interstellar gas, in the study of drifting beams and shear flows in non-uniform planetary rings, for the understanding of ion heating and anomalous drift in the inner magnetosphere of Saturn, etc.

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Chapter 7

Fine Particles in Industrial Applications

Powder-contaminated plasmas pose a number of challenges to the microelectronic industry, materials science, and gas discharge research and development areas. For example, particulate powders with the sizes comparable to feature sizes of the semiconductor integrated circuits have become a troublesome factor in the semiconductor micromanufacturing. Dust in the plasma reactors often causes irrecoverable defects and line shorts in some ultra large scale integrated (ULSI) circuits, which can totally compromise the entire microchip fabrication process. However, the accents in the fine particle research in technology are gradually shifting from the traditional view on them as unwelcome process “killer” contaminants to often desired elements that can dramatically affect and even improve the basic properties of plasma-made thin films. In this chapter, we focus on the role of fine particles in the lowtemperature plasmas and discuss some applications of nano/micron-sized particles in a number of high-tech industries. For example, reactive plasmas are increasingly used for the synthesis of nano- and biomaterials, as well as various micro- and optoelectronic functionalities and devices. The use of nano-sized particles can lead to a substantial improvement of the efficiency of solar and fuel cells, the development of the entirely new classes of functional thin films and novel materials with the desired properties, drug delivery systems, environmental remediation and homeland security technologies, advanced applications in polymer catalysis, and several others. One of the key modern issues in the industrial applications of the complex plasma systems is in the tailoring of various properties of the plasmagenerated micro- and nanoparticles and nanoclusters in the ionized gas phase. Such particles and clusters can be regarded as building units in nanofabrication involving doping, structural incorporation or self-assembly

323

324

Physics and Applications of Complex Plasmas

processes. In particular, in the low-energy nanocluster chemical vapor deposition [Jensen (1999)], the cluster charge is critical for the growth of various silicon- and carbon-based nanofilms [Hwang (2000)l. Until recently, it has undoubtedly been presumed that atoms and molecular complexes play a pivotal role in the assembly processes a t nano- and microscales. However, the building block paradigm has recently shifted towards accepting the crucial role of nanoclusters and nanoparticles. It is interesting that the nanocluster scenario invokes the notion of the short-range interactions between the atoms of neighboring building blocks. To this end, the charge and size of the clusters are the crucial parameters for the explanation of the unique architectures of many nano-sized objects. Amazingly, the nanocluster charge appears to be a key reason for the highly anisotropic growth of ordered nanostructures, such as silicon nanowires and carbon nanotubes [Hwang (2000)l. This chapter is subdivided into three major sections. Sec. 7.1 introduces the details of fine powder particle origin and growth in low-temperature plasmas most commonly used for the industrial plasma processing and other applications. In Sec. 7.2, deleterious effects caused by the presence of dust particles in fusion reactors and plasma processing facilities and the relevant remediation methods are discussed. The chapter concludes with Sec. 7.3 detailing various advanced industrial applications of the fine solid particles.

7.1

Growth and Characterization of Nano- and Micron-Sized Particles

Generation of fine powders, ranging in size from a few nanometers to several tens of microns is frequently reported for various plasma processing facilities. The particulate matter in the processing plasmas has numerous implications for the semiconductor micro-fabrication and materials processing. Here, the underlying physico-chemical processes involved in the origin and growth of fine particles in reactive silane-, hydrocarbon- , and fluorocarbonbased plasmas, are discussed (Sec. 7.1.1). Despite a remarkable difference in the process kinetics and the plasma chemistry involved, the growth scenario can be quite similar. Indeed, the dust growth in chemically active plasmas starts with the formation of sub-nanometer/nanometer-sizedprotoparticles nucleated as a result of homogeneous or heterogeneous processes. Thereafter, agglomeration/coagulation processes result in the pronounced generation of particulates with the sizes in the few-tens of nanometers range,

Fane Particles in Industrial Applications

325

which quickly acquire negative electric charge as a result of collection of the plasma electrons/ions [Boufendi and Bouchoule (1994)]. These rapidly developing processes result in a significant reorganization of the entire plasma system due to pronounced changes in the particle balance (such as a depletion of the electron number density) induced by the growing dust grains (Sec. 7.1.2). Meanwhile, the plasma system reorganizes to compensate the enhanced electron losses onto the dust grains. As a result of this reorganization, the effective electron temperature and hence the major ionization/dissociation rates, increase. The dust growth then usually proceeds to sub-micron and micron sizes via a relatively slow process of accretion of neutral/ionic monomers (e.g., deposition of SiH, radicals on the grain surface in silane-based reactive plasmas). Chemical nucleation in the ionized gas phase is not the only possible mechanism of the dust growth in the processing plasmas. The particulate growth can also be induced by physical and reactive sputtering of the wall/electrode material in the plasma-assisted DC/RF magnetron and other sputtering facilities. The basic physics of the relevant processes is discussed in Sec. 7.1.3. Furthermore, since most of the fine particles feature the sizes in the nanometer or sub-micrometer range, this poses a number of apparent challenges for the in situ detection and ex situ characterization methods. Some of the most recent relevant techniques are discussed in Sec. 7.1.4.

7.1.1

Origin and mechanisms of growth of clusters and particulates i n reactive plasmas

7.1.1.1 Silane plasmas Plasmas of pure silane (SiH4) and its mixtures are widely used for applications in the semiconductor industry (e.g., integrated circuitry and siliconbased microchips, flat panel displys, amorphous silicon solar cells). It is believed that understanding of the fine particle generation processes in silane-based plasmas is the most comprehensive as compared to other reactive plasma chemistries [Perrin and Hollenstein (1999); Hollenstein (2000)l. Here, we discuss the most recent advances and current problems of the origin and growth of fine powder particles in low-pressure silane-based discharges. The initial stages of the particle growth in pure silane discharges can adequately be described by the steady-state homogeneous nucleation model [Kortshagen and Bhandarkar (1999); Gallagher (2000)]. The basic assump-

326

P h y s i c s a n d Applications of C o m p l e x P l a s m a s

tion of the model is that the particle growth process is triggered by SiH, anions and/or SiH, neutral radicals, which polymerize into Si,H, radicals with larger numbers R of silicon atoms. With an increase of R , large clusters, and eventually subnano-/nano-sized particles of hydrogenated silicon are generated. The apparent puzzle is to identify the precursor species and dominant gas-phase/surface reactions for the growth of larger (with > lo4 silicon atoms) particulates and relate the dust growth to the discharge control parameters. At present, the above problem seems t o be quite far from being solved and in most cases there is yet no consensus on the dominant precursors for the fine particles in the plasma. However, there exist only three major classes of possible catalyst candidates in the silicon hydride clustering process [Bhandarkar et al. (2000)], namely, anions (negative ions), neutrals, and cations (positive ions). Apparently, the underlying physics and chemistry of the powder origin in chemically active plasmas critically depends on the prevailing experimental conditions. For example, short lifetime neutral radicals SiHz can play the role at several stages of particulate growth [Watanabe et al. (200l)l. Neutral complexes are capable to incorporate into larger saturated molecules and can thus be considered as viable nanoparticle growth precursors in reactive silane plasmas [Hollenstein (2000)l. Hence, in the short residence time situations one could expect that short-lifetime, highly reactive neutral radicals can efficiently support numerous homogeneous nucleation processes. In particular, neutral radicals SiH, (rn = 0-2) can be responsible for the nanopowder formation in dense helium or argon-diluted silane discharges [Watanabe (1997); Koga et al. (2000)]. Likewise, positive ions can also be regarded as potential powder precursors despite high activation barriers preventing the formation of highermass cations. In particular, cationic silicon clusters that contain up to ten silicon atoms, have been detected in argon/hydrogen thermal plasmas by means of time-resolved mass-spectrometry [Leroux et al. (2000)l. On the other hand, the anionic pathway is another viable route for the powder generation in silane-based plasmas. Invoking a simple argument that the formation of particulates does require critically large clusters, one can conclude that typical residence times of the neutrals are not sufficient t o trigger the efficient dust growth process [Choi and Kushner (1993)I. However, the clustering process can involve negative ions trapped by the ambipolar potential in the plasma. Furthermore, the negatively charged

327

Fine Particles in Industrial Applications

intermediaries can increase the average residence time of the clusters and enable their growth to the critical size [Choi and Kushner (1993)I. Likewise, hydrosilicon anions can be efficiently confined in the nearelectrode areas and participate in the plasma-assisted clustering process. Thus, a large number of negative ions can accumulate and grow towards higher masses according to the homogeneous model [Gallagher (2000)l. Relevant time-resolved mass-spectrometry data have revealed that the anionic pathway is the most likely route for the nanoparticle generation in lowpressure RF silane plasmas [Hollenstein (2000)l. For example, the dust evolves from the molecular to the particulate form in low-pressure silane RF capacitively coupled plasmas [Howling et al. (1996)l. In this case the negative ions play a crucial role in the powder nucleation and growth process, and the entire range of negatively charged species, ranging from monosilicon anions through to nanometer-sized clusters, can be observed [Howling et al. (1996)]. Furthermore, the anion confinement correlates with the pronounced particle formation. Conversely, de-trapping of the negative ions strongly inhibits the entire growth process. In the above examples, it is likely that the negative ion clustering reaction SinHz

+ SiH4 + (Sin+lH;)* + (H) ,

(7.1)

leads to the efficient polymerization of the higher-mass anions (Si,+l H;)* in the excited state, where (H) denotes the hydrogen-bearing products [Hollenstein (2000)l. We note that silicon hydride clustering (7.1) involves silyl (x = 2n 1) and siluene (x = an) anions. The results of recent numerical simulation of dust particle formation mechanisms in silane discharges confirm that the anion SiH, is the most dominant primary precursor of the particle formation. In fact, over 90% of the silicon hydride clustering (7.1) proceeds through the silyl anion (SinHTn+,) pathway, starting from SiH,, whereas only 10% through the siluene anion (SinHTn) pathway, starting from SiH, [De Bleecker et al. (2004)l. This conclusion is valid for negatively charged silicon hydride clusters SinH; containing up to 12 silicon and 25 hydrogen atoms. The second phase of the particle growth can proceed via a rapid agglomeration of small clusters into larger (usually 40-50 nm-sized) particles [Kortshagen and Bhandarkar (1999)I. This process is accompanied by self-organization of the plasma-dust system and is considered in more detail in Sec. 7.1.2. After the agglomeration phase is complete, the grain size increases with the relevant thin film growth rates.

+

N

N

328

Physics and Applications of Complex Plasmas

We emphasize that the key dust nucleation and growth processes discussed above are most relevant to the plasmas of pure silane discharges. However, many real thin film fabrication processes require a substantial dilution of silane by hydrogen and/or argon. The offset and dynamics of the particulate growth appear quite different as compared to the pure silane case. In particular, silane dilution complicates the discharge chemistry and elongates the time scales required for the powder detection. Thus, the particulate size, bonding states, architecture, and surface morphology of the particles grown in the pure and buffer gas diluted silane plasmas can be quite different and critically depend on the reactive gas feedstock. Physically, by varying the gas composition one can control the residence time t,,, of the precursor species in the discharge. Moreover, t,,, appears to be a critical factor in the nanoparticle generation and growth. Specifically, there is a direct correlation between the residence time of the precursor radicals and the size of fine particles detected [Bouchoule and Boufendi (1993)I. The selective trapping model [Fridman et al. (1996)l assumes that the neutrals should reside in the ractor volume long enough to acquire a negative charge through the electron non-dissociative attachment and/or heavy particle charge exchange collisions. In this is the case, the nanosized particlcs can be trapped in the near-electrode areas, building up the minimum number density for the coagulation onset. The critical size of the particles that can be trapped and are capable to agglomerate appears to be acrit N 2 nm for the following set of parameters: the flow rates of Ar and SiH4 being 30 and 1.2 sccm (total gas pressure 117 mTorr); the gas and electron temperatures of 300-400 K and 2 eV; the electron/positive ion number density of 3 x lo9 cm-3 and 4 x lo9 cmP3, respectively [Fridman et al. (1996)l. We note that under the above conditions the neutral gas residence time is approximately 150 ms. The formation of dense (- 101o-lO1l ~ m - powder ~ ) clouds of fairly monodisperse, a few nanometer-sized fine particles immediately before the coagulation onset is usually experimentally confirmed by using the high-resolution transmission electron microscopy (TEM) and/or laser light scattering (LLS) techniques [Boufendi et al. (1999)l. In mixtures of reactive gases currently used for fabrication of various functional thin films and nanomaterials, the processes of fine powder generation are usually more complicated than in pure silane plasmas. For example, high-density plasmas of highly reactive SiH4+02+Ar gas mixture (involving two electronegative gases - silane and oxygen) are used for the fabrication of silica nanoparticles. In this regard, the current challenge

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for the adequate understanding of the fine particle nucleation and growth processes is in incorporating the effects of the complexity of the numerous chemical reactions involved, high reactivity of high-density (especially inductively coupled plasmas currently used as benchmark plasma reactors for semiconductor manufacturing) plasmas, and finite reactor size effects (most of the currently existing models deal with spatially averaged discharge models). From the plasma chemistry point of view, the polymerization of critical clusters can involve a combination of several clustering pathways. Indeed, the clustering process in high-density SiH4+OZ+Ar plasmas can proceed through various ion-neutral and neutral-neutral clustering channels [Suh et al. (2003)l. Ion-neutral clustering reactions

can involve either positive or negative ionic precursors. On the other hand, the neutral-neutral clustering develops through the self-clustering reaction

or by addition of SiO or SiOz radicals to the growing polymeric chain. An interesting peculiarity of dust-generating high-density plasmas of SiHd+Oz+Ar mixtures is that the rates of positive-negative ion neutralization are very high due to comparabe number densities of positively and negatively charged species. In this case, the neutralization of anions (e.g., SiH, currently believed to be one of the most important precursors for the dust growth in pure silane plasmas) occurs much faster than the clustering of neutral species. This explains why neutral clusters exhibit higher concentrations relative to anionic clusters [Suh et al. (2003)l. Thus, it becomes clear that in each particular case the plasma chemistry behind the dust nucleation and growth is quite different even in the presence of the same silicon-bearing precursor gas, which is an essential component for the fine particle nucleation. To this end, one cannot a priori state which particular precursor radical triggers the plasma polymerization of the critical clusters that nucleate into larger particles. However, it is quite possible to identify a few dominant clustering pathways and relative role of the species in any particular charge state (positive, negative, or neutral). In this regard, one can perform the sensitivity analysis, which can give an answer as to which reactions dominate the production and steady-state number densities of higher silicon oxides (e.g., anions (SiO,), or neutrals (SiO,)ll) [Suh et al. (2003)l.

330

7.1.1.2

Physics and Applications of Complex Plasmas

Hydrocarbon plasmas

The existing understanding of the nanoparticle growth in hydrocarbon (C,H,, e.g., methane, CH4 or acetylene, C2H2) discharges is still at an early stage as compared with the similar processes in the silane-based plasmas. However, the study of the plasma chemistry and growth of nano-sized particles in relevant ionized gas mixtures is gradually gaining momentum. For example, a numerical model of the nanoparticle clustering kinetics in the low-pressure RF discharge in acetylene [Stoykov et al. (2001)l incorporates numerous gas-phase processes including the electron impact dissociation, electron attachment leading to the negative ion generation, ion-ion recombination, ion-neutral clustering, chemical reactions involving the hydrocarbon (chain and aromatic) neutrals, as well as diffusion losses of the plasma species to the discharge walls. Based on numerous data from the reactive plasma, aerosol and combustion literature, it is usually assumed that the carbon hydride clustering process is triggered by the electron-impact abstraction of hydrogen from the acetylene monomer CzHz+e-CzH+H+e, followed by the efficient generation of C,H, radicals (with higher numbers of carbon and hydrogen atoms) via a chain of polymerization reactions [Stoykov et al. (2001)l. The model allows one to predict the most probable clustering pathways as well as the temporal evolution of the number densities of the major charged and neutral species. The most likely clustering process CiHj

+ C,H,

-+

Ci+mHjl,-l

+H

proceeds through the addition of the anion species CiHJ to the neutrals C,H, accompanied by the elimination of hydrogen and generation of the higher-mass anions. Eventually, the rapid chemical nucleation stage evolves into the equilibrium state, which can usually be reached when the particle loss to the walls is compensated by the production of the new species. The equilibrium state is strongly affected by the neutral gas temperature, RF power input, and working gas pressure. Similarly, depending on the external parameters, the particle nucleation process can either be enhanced or inhibited. Even though clustering occurs mainly through the formation of linear molecules, the proportion of aromatic hydrocarbons increases and becomes significant at higher working gas temperatures.

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1Oi3 10l6

Time (s)

Time (s) Fig. 7.1 Temporal dynamics of the clustering process involving (a) neutral and (b) charged species at T, = 300 K and the ionization degree of lop6 [Stoykov et al. (200 l)].

The results of numerical modeling of the clustering processes in acetylene plasmas [Stoykov et al. (2001)] are presented in Fig. 7.1, which shows a temporal evolution of the number densities of the neutral and charged species. To present the results transparently, the following notations are used [Stoykov et al. (2001)l. The number densities of all the chain species with the same number of carbon atoms are added together and plotted as single curves. For instance, the label 2 refers to the sum of the concen-

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Physics and Applications of Complex Plasmas

trations of C2H, C2H2, C2H3, C2H4, etc. species. All species that have a ring structure (regardless on the number of carbon atoms) are combined together and denoted as “rings”. Species with more than 10 carbon atoms, are lumped together as “particles”. From Fig. 7.1 one can see that after the initial increase in the species concentrations, the production rates slow down and eventually a steady state is reached. This indicates that a balance between the species production and diffusion losses is achieved. We note that the rates of the diffusion losses are proportional to the species concentration and this loss channel plays only a minor role at the early stages of particulate development. This certainly favors a quick initial rise of the number densities of the reactive plasma species. Eventually, the diffusion losses are balanced by the gasphase reactions that lead to the particle production and a steady state of the discharge can be established. A comparison of the number densities of the structurally similar neutrals and anions reveals that the anion concentrations are much lower (Fig. 7.1). Since the density of positive ions is an upper limit for the combined anion and electron densities, the above difference can be attributed to high grow rates of the neutral particles in acetylene plasmas. However, this does not necessarily mean that nano-sized particles are mostly neutral. In fact, one can note that most of the negative-species are formed in the particle form, with the number densities of the same order of magnitude as the concentrations of the neutral particles. The ratio between the number densities of neutral and negative particles is important for the understanding of the details of the further growth processes, which are affected by the grain charge [Kortshagen and Bhandarkar (1999)l. To this end, the formation of neutral particles is favored at lower temperatures, higher degrees of ionization, and higher pressures [Stoykov et al. (2001)]. Further details on the dependence of the ratio of the negative-to-neutral particle densities on the plasma parameters (e.g., gas temperature, degree of ionization, pressure) can be found in the original work [Stoykov et al. (2001)l. We now discuss the chemical composition and aromatic hydrocarbon content of nanoparticles in acetylene-based plasmas sustained at room temperatures and 100 mTorr pressure with 10-30 W RF powers in a 13.56 MHz capacitively coupled plasma reactor [Stoykov et al. (2001)]. The in situ infrared (IR) absorption measurements can be used to detect the particle generation in the discharge [Boufendi et al. (1999)I. In this case the IR absorption spectra of the particle-generating discharge are quite different

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from the reference spectra of the neutral acetylene gas feedstock. On the other hand, the plasma-grown nanopowders can be collected and analyzed ex situ, e.g., by the Diffuse Reflectance Infrared Fourier-Transform Spectroscopy (DRIFTS) and a Nuclear Magnetic Resonance (NMR) techniques [Stoykov et al. (2001)l. It is interesting that Fourier Transform IR (FTIR) spectroscopy data collected in situ and from the powder samples confirm a predominant production of the acetylenic compounds in the ionized gas phase, although the presence of aromatic compounds appears to be non-negligible. The recent mass spectrometry studies of the RF plasma in acetylene show the presence of aromatic compounds such as benzene, substituted benzenes and toluenes [Descheneaux et al. (1999)l. The above numerical and experimental results are in an agreement with the recent experiments on the dust particle generation, size-controlled growth, diagnostics and deposition in 13.56 MHz RF plasmas of Ar/CH4 and Ar/C2H2 gas mixtures in a Gaseous Electronic Conference (GEC) Reference Cell plasma reactor [Hong et al. (2002)l. We emphasize that the most efficient dust particle generation is commonly observed for the elevated RF power levels, which indicates on the importance of the adequate amounts of the particle growth precursors [Hong et al. (2002)]. However, similar to the silane-based plasma chemistries, the dominant precursors still need to be identified. It is also worthwhile to mention that recent in situ FTIR spectroscopy and the plasma-ion mass spectrometry measurements evidence the highlymonodisperse size distributions of nanoparticles grown in RF plasmas of Ar+C2H2 gas mixtures [Kovacevic et al. (2003)l. This conclusion is also cross-referenced by the scanning electron microscopy of the powder samples collected during different growth phases. Measurements of the intensity of the Rayleigh-Mie scattering of the infrared signal reveal that the process of the fine particle generation, growth and disappearance is periodic, as shown in Fig. 7.2. It is seen that the oscillation period of the infrared signal is approximately 35 min under prevailing experimental conditions [Kovacevic et al. (2003)]. The time scales when the electron-impact ionization is enhanced and the plasma parameters in Ar+C2H2 RF discharges noticeably change due to the dust growth 7 C 2 H 2 appear to be consistently longer than the corresponding time scales ‘TsiH4 in silane-based plasmas (see Sec. 7.1.1.1). The observed periodicity of the Reyleigh-Mie scattering signal can be explained by noting that negatively charged particles are confined in the

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Physics and Applications of Complex Plasmas

plasma potential as long as the different forces acting on the single particle are balanced [Kovacevic et al. (2003)l. Since the major forces (see Sec. 2.1 for details) scale differently with the grain radius, the actual particle confinement critically depends on their size. As soon as the particles reach the critical size, they are either dragged out of the plasma bulk or fall down onto the lower electrode, which results in a decrease of the scattered signal. A quick drop in the intensity of the above signal evidences a highly monodispersive character of the powder growth process in Ar+CH4 plasmas.

Fig. 7.2 Time evolution of Rayleigh-Mie scattering for the FTIR spectral line of 4500 cm-l. The time resolution is 1 min [Kovacevic et al. (2003)l.

Furthermore, the ex situ scanning electron microscopy suggests that the particles collected 10 min after the ignition of the discharge have a spheroidal shape with the particle diameter of about 150 nm and a fractal surface texture (Fig. 7.3). It is thus likely that that the accretion (uniform deposition of the neutral species onto the particle's surface) is probably a dominant particle growth mechanism. A very interesting observation relevant to the dust growth process is a very high consumption of the acetylene monomer for the plasma polymerization as evidenced by the neutral mass spectrometry [Kovacevic et al. (2003)]. This is consistent with the recent findings that acetylene as a monomer plays an important role in the fine powder formation in hydrocarbon plasmas [Descheneaux et al. (1999); Stoykov et al. (2001)]. This fact is highly relevant to the PECVD of various carbon-based nanostructures discussed in detail in Sec. 7.3.1.

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Fig. 7.3 SEM micrograph of particles collected 10 min into the discharge run [Kovacevic et al. (2003)l.

It is thus imperative to investigate the growth precursors and dynamics of the dust grain formation in argon-methane and argon-acetylene RF plasma discharges. It turns out that acetylenic compounds play a vital role in the dust nucleation and growth processes. There are two relevant experimental observations [Hong et al. (2003)]. First, in Ar+C2H2 plasmas the fine particles usually nucleate spontaneously at low discharge powers. On the other hand, the particle growth in Ar+CH4 plasmas usually starts only after a transient elevation of the RF power or a quick inlet of the C2Hz monomer into the discharge volume. This can presumably be attributed to different nucleation scenarios in Ar+CH4 and Ar+C2H2 discharges. Apparently, the procedure of adding more C2Hz or RF power to the discharge is required to trigger the nucleation of primary clusters and protoparticles, which is a quite slow process in Ar+CH4 plasmas. Once the cluster precursors are formed, the further growth process can proceed under normal discharge operation conditions. It is very interesting to note that the elevated abundance of the C2Hz monomer species in the ArfCH4fH2 inductively coupled plasmas for the PECVD of various carbon-based nanostructures [Denysenko et al. (2003b); Tsakadze et al. (2004); Tsakadze et al. (2005)] can be achieved by operating the discharge at elevated RF powers. One can thus presume that the relevant nanostructure growth process can be strongly affected by the pronounced formation of fine powder particles in the ionized gas phase (for details, see Sec. 7.3.1). On the other hand, the'dynamics of the dust formation in Ar+C2H2

336

Physics and Applications of Complex Plasmas

plasmas is periodic and follows the scenario: nucleation + further growth -+ development of dust-free regions (mostly due to the action of the ion drag force that pushes the dust grains out of the plasma bulk) + new nucleation in the dust-free regions. A possible explanation for the differences in the dust growth dynamics in methane-based and acetylene-based reactive environments is t>hatthe nucleation process strongly depends on the concentration of C2H- negative ions, which are efficiently generated in the Ar+C2H2 plasmas [Hong et al. (2003)l. Another way to trigger the dust generation in low-density methanebased plasmas is to use a pulsed Nd:YAG laser [W. W. Stoffels et al. (1999b)l. If the photon energy fits the dissociation energy of the C-H bond, the absorption of UV photons results in the rapid dissociation of methane molecules and creation of active radicals, which is otherwise inefficient in the pristine methane plasma. In this way, it appears possible to synthesize submicron-sized dust particles that can subsequently arrange into larger agglomerates and structures (such as ordered Coulomb lattices, freestanding and networked particle strings, V-shaped structures) levitating in the vicinity of the powered RF electrode [W. W. Stoffels et al. (1999b)l. Some of the resulting particle arrangements can be further deposited and continue growing on the surface. There are numerous indications that powder formation can also be induced by the surface and reactor contamination effects. For example, in pure methane discharges in a clean reactor chamber, the powder formation process takes at least a few hundred seconds. However, in a contaminated reactor, the fine particle appearance can be detected much faster. Thus, the powder formation might be affected by surface effects as is the case for SiN dusty plasmas. However, no high mass neutrals, cations or anions have been detected by the mass spectrometry, in contrast to the silane plasmas [Hollenstein (2000)l. Hence, it is very likely that large particles are formed via heterogenoeus processes. The latter processes are most common for the situations when the plasma species are non-reactive and direct gas-phase reactions leading to the formation of critical clusters are not efficient. In this case the fine particle growth can proceed via the electron-induced surface desorption of nano-sized clusters. The initially neutral clusters can migrate into the near-electrode/plasma sheath area where the probability of their excitation/ionization via collisions with high-energy electrons is quite high. Ion-molecular reactions can further contribute the particulate growth. Finally, a pronounced coagulation process can lead to the formation of larger agglomerates [Hollenstein (2000)l.

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Carbon nanoparticles can be formed in thermal plasmas of arc discharges. For example, shell-structured carbon clusters can be formed via the liquid cluster crystallization [Lozovik and Popov (1996)]. In this case, both the fullerenes (one-shell structure) and the carbon nanoparticles (multi-shell structure) can be generated.

7.1.1.3

Fluorocarbon plasmas

Fluorocarbon (C,F,) based plasmas have recently been widely used for ultrafine and highly selective etching of polysilicon and a number of PECVD processes including many common applications in microelectronic industry [Kokura et al. (1999)l. Furthermore, many plasma etching processes of silicon and its components, as well as deposition of chemically resistant barriers, dry lubricants, etc. involve CF4, CzFs, CHF3, CzF4, aromatic fluorocarbons, etc. [Buss and Hareland (1994)l. A gas phase particulate formation is possible in capacitively-coupled RF fluorocarbon plasmas [Buss and Hareland (1994)]. As usual, the laser light scattering can be used to monitor the appearance and trapping of particles. Likewise, the powder particles can be collected and analyzed e z situ by the FTIR and TEM tools. A sequence of monochromatic images of particulate suspension and growth obtained from a 13.56 MHz capacitively coupled vinilydene fluoride plasma at 27 mTorr sustained with 30W RF powers reveals that the time of the initial particle detection usually varies in the 10 to 250 s range [Buss and Hareland (1994)l. The TEM results show that in this case the particles are usually non-agglomerated, have an almost spherical shape and can accumulate during the extended discharge operation. The grain diameter typically ranges from 110 to 270 nm. Similar to silane- and hydrocarbon-based plasmas, the particles develop in size and evolve into a certain spatial pattern (e.g., particulate cloud) usually suspended between the two electrodes [Buss and Hareland (1994)]. The time of the first appearance of particles is quite sensitive to the total gas pressure and the discharge chemistry. The addition of hydrogen or hydrogen-containing gas (e.g., CH4) to a fluorocarbon discharge can result in an increase of the particle growth rate and the corresponding shortening of their first detection time. This effect can be attributed to the enhanced production of free radicals by hydrogen atom abstraction of fluorine. An important issue is to obtain direct experimental quantitative indicators of the fine particle growth process in fluorocarbon plasmas. The appearance time for particles in a C Z Fplasma ~ at 140 mTorr turns out to

-

338

Physics and Applications of Complex Plasmas

be approximately 110 s [Buss and Hareland (1994)]. Assuming a constant radial growth rate (and estimating a minimal diameter for the detection by the laser light scattering methods), one can obtain 0.5-1.4 nm/s for the particle growth rate. It is worth noting that the fluorocarbon film growth on the substrate placed on the lower electrode has a comparable rate of 2 nm/s. Meanwhile, RF discharges of octafluorocyclobutane-based gas mixtures also generate large amounts of highly polymerized molecules, which correlates with the plasma polymerization processes in the gas phase [Takahashi and Tachibana (2001a)l. It is worth noting that the chemistry behind the gas-phase nucleation processes can be quite similar in the silane- and fluorocarbon-based plasmas. Indeed, higher fluorocarbons polymerized in the ionized gas phase can act as efficient precursors for the generation of nano-sized particles and also take part in the thin film deposition processes. Solid grains and agglomerates can also be abundant in fluorocarbon plasmas for ultra-fine selective etching of SiOz and PECVD of low-dielectric constant polymeric films. There is a remarkable correlation between the polymerization in the ionized gas phase and the relevant surface processes, which can shed some light on the prevailing powder formation mechanisms in fluorocarbon plasmas. For example, fluorinated carbon particles can be generated in a parallel plate 13.56 MHz plasma reactor, where a capacitively coupled plasma of c - C ~ Fis ~sustained with the RF power density of 0.15 W/cm2 within the pressure range from 23 to 250 mTorr, which is typical to the PECVD of fluorinated amorphous carbon (a-C:F) thin films [Takahashi and Tachibana (2001b)l. Under such conditions, numerous nano-/micron-sized particles and agglomerates dispersed over the wafer surface can be observed. The diameter of the gas-phase grown particles, measured by the SEM, typically ranges from 0.5 to 2.3 microns. In the intermediate pressure range (> 50 mTorr), a pronounced generation of the agglomerates with the size in the few tens of micrometer range and composed of the primary spherical particles takes place [Takahashi and Tachibana (2001b)l. Note that the number of primary particles building up the agglomerates increases with pressure. A typical size of the fluorocarbon-based agglomerate at 250 mTorr pressure is about 30 pm. We stress that the gas-phase particulate polymerization can be inferred through the dependence of the film deposition rate on the gas feedstock pressure. Specifically, the film deposition rate decreases when the gas pres-

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sure exceeds 50 mTorr [Takahashi and Tachibana (2001b)l. Presumably, this can be attributed to the enhanced loss of the gas-phase polymer precursors to the particle generation processes. It is also worthwhile to mention that no significant particulate formation in the gas phase is observed in CF4 or CaFs-based discharges. Thus, it seems reasonable to conclude that stable CF4 and C ~ F molecules G cannot be efficient as precursors for the particle growth. Even though the trigger catalyst plasma species are yet to be conclusively identified, one can presume the nanocluster route for the fine particle growth. The following chain associative electron detachment polymerization reaction

can be regarded as a basic clustering pathway in the fluorocarbon plasmas. This mechanism also appears to be consistent with the elevated particle production rates at higher gas pressures. Furthermore, there is a correlation between the clusterizing rates and the gas-phase concentrations of the source gas molecules and the main products of the first-order reactions [Takahashi and Tachibana (2001b)l. We emphasize that similarly to SiH4 (Sec. 7.1.1.1) and C,H, Hn (Sec. 7.1.1.2) plasmas, the negative ions also play a crucial role in the clustering reactions in fluorocarbon plasmas. Thus, elucidation of the dust generation pathways, including a detailed experimental investigation of the catalyst species and gas-phase reactions for polymerization is an apparent forthcoming challenge for the coming years. For example, solid C:F particles can be polymerized in C4F8 plasmas under conditions of pronounced generation of molecular species CF4, C2F6, and C2F4 [Takahashi and Tachibana (2001b)l. Among them, the C2F4 molecule playes a leading role in the gasphase synthesis of dust grains [Takahashi and Tachibana (2002)l. A possible reason is that this molecule can be activated in the plasma and subsequently transformed into highly reactive species -CF=CF- and -CF=CF2 involved in numerous branching/polymerization reactions. Thus, high molecular weight compounds appear and act as the particle nucleation precursors. It is interesting to mention that under similar experimental conditions the particle growth and film deposition is usually not observed in pure CF4 or C2Fs plasmas [Takahashi and Tachibana (2002)l. This observation still awaits explanation in the future. At the end of this section, we note that dust growth under the plasma conditions is not merely limited by the silane-, hydrocarbon-, and

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Physics and Applications of Complex Plasmas

fluorocarbon-based chemistries. For example, carbon nitride particles with diameters o f a few hundred nanometers, can grow at room temperatures in RF capacitive discharges of N2+CH4 gas mixtures [Boufendi and Bouchoule (2002)l. Generally speaking, solid particle growth can certainly take place in many other reactive gas mixtures supporting polymerization/clustering processes in the ionized gas phase.

7.1.2

Effects of fine particles on discharge stability

Here, we consider critical phenomena in silane-based reactive plasmas following the protoparticle nucleation stage. Once the density of the powder particles has exceeded a critical value, being typically within the range of 101o-lO1l cmP3, a fast coagulation process starts. At this stage, the generation of new protoparticles is usually inhibited and 40-50 nm (can be up to 100 nm) aggregates are detected. The particle size and the number density also change with time. At the beginning of the process (within typically a few seconds into the discharge run), the particle size increases and their number density decrease monotonously. However, during the rapid (coagulation) phase, the plasma and dust parameters usually change discontinuously. Thereafter, the process reverts to the slow growth phase. The fine particle charging processes are physically different before and after the agglomeration stage. When a particle size is in a few nanometer range, the electric (usually negative) charge is due to the chain negative ionsupported polymerization processes. On the other hand, larger (a few tens of nm sized) grains are usually charged by the microscopic electron/ion fluxes originated due to the potential difference between the particulate surface and adjacent plasma. In the latter case, the dust charge can be calculated using the Orbit Motion Limited (OML) approximation [Whipple (1981); Barnes et al. (1992); Goree (1994)l. For more details on the dust charging mechanisms, the reader can be referred to Sec. 2.1. Note that the negative particle charge appears to be a strong limiting factor in the agglomeration process [Kortshagen and Bhandarkar (1999); Gallagher (2000); Hollenstein (2000)]. However, the agglomeration can be facilitated by pronounced charge fluctuations that sometimes result in broad charge distribution of nanoparticles. From Sec. 2.2.1 we recall that the resulting charge of very small ( w 1-5 nm) clusters and nucleates can be positive, negative, or neutral. A simple estimate of the critical dust density for the agglomeration onset can be derived by comparing the rates a t which the fine particles collide and N

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are lost to the walls. When the characteristic time of the (usually classical Brownian) diffusion losses 70 becomes smaller than the characteristic time of dust collisions T,, i e . , rc < To, the agglomeration onset is likely. Here,

where T, is the gas temperature (in energy units), and other notations are standard, namely, md is the mass of the dust particle, a is its radius, and n d is the number density of the dust particles. For example, for the room temperature argon plasma at the 130 Pa pressure, the density of the 1 nmsized particles should exceed nd,,,it = 6 x 10l1 cmP3 for the agglomeration process to start [Roca i Cabarrocas et al. (2002)]. It is worth noting that the agglomeration has two basic forms such as coagulation and flocculation. The major difference between the two forms is in the extent of fusion of the primary particles. A process when primary particles clump together with no or very little reduction of the combined surface area, is termed flocculation and usually results in the growth of porous skeletal or cauliflower-shaped agglomerates. On the other hand, the reduction of the combined surface area indicates on the coagulation, which is usually accompanied with sintering and spheroidization of the constituent particles. If the time required for the efficient fusion of the primary nucleates into a larger particle is comparable with the duration of the coagulation phase, one should expect smoother, and sometimes almost spheroidal shapes. The shape of the agglomerates is further modified by continuous deposition and attachment of reactive neutral and charged radicals to the surface (during the next, accretion, dust development stage), which eventually results in fairly smooth surfaces. As the powder size increases, the dust number density and other discharge parameters change rapidly during the agglomeration stage. This critical phenomenon leads to dramatic modifications of the power and particle balance in the discharge. The major reason for this reorganization is a newly emerged electron/ion sink channel onto the combined surface of the fine particles. In some cases, the electron/ion capture by the dust can become a dominant channel of the plasma particle loss (as compared to, e.g., ambipolar diffusion losses to the electrodes/discharge walls). In this case the plasma-particle system can become unstable giving rise to the a - y’ transition [Bouchouleand Boufendi (1993); Boufendi and Bouchoule (1994); Fridman et al. (1996)I. Hence, the enhanced loss of the electrons/ions is to be compensated by an additional ionization. Consequently, the plasma-

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Physics and Applications of Complex Plasmas

particle system generates a self-consistent feedback by elevating the population of higher-energy electrons responsible for the ionization/dissociation processes. This process is accompanied by notable modifications of the electron energy distribution function (EEDF), higher rates of electron-impact ioniiation/dissociation, and often higher effective electron temperatures in the discharge. It is quite common that a. number of silane-based discharges feature elevated ( T e ~ 10 eV) electron temperatures, as compared to T e ~ 1-2 eV in pristine (dust-free) plasmas [Bouchoule and Boufendi (1993); Boufendi and Bouchoule (1994); Fridman et al. (1996); Hollenstein (2000)]. An interesting insight into the coagulation of small protoparticles into larger particulates in low-temperature plasmas can be obtained by taking into account the details of dust charging [Kortshagen and Bhandarkar (1999)l. Furthermore, in addition to microscopic electron and ion currents, the UV photoelectron emission (photodetachment) can.strongly affect the particle charging and agglomeration. We note that it is quite typical for discharges in rare gases that up to 50% of the R F power can be transferred into optical/UV emission [Ashida et al. (1995)l. In this case the UV radiation usually features energies between 10 and 20 eV. Due to size-dependent electronic structure, nano-sized clusters and particulates are very sensitive to the photoelectron emission. Moreover, the intensity of the photoelectron emission from nanoparticles is approximately two orders of magnitude higher than from the respective bulk materials. In addition, electrons can be detached from the nanoparticles due to their collisions with the excited atoms (quenching process) and the secondary electron emission. The key equation of the model is quite common for the aerosol science and describes the dynamics of the coagulation process

-

-

N

(7.2)

where the first term accounts for the gain of particles within the volume [v,v dv] due to coagulation of smaller particles [Kortshagen and Bhandarkar (1999)l. The second term describes the loss of particles from the same element of volume due t o coagulation with particles of any volume, and p(v,v’) is the coagulation rate between the two particles with a volume v and v’. The corresponding rates can be found in the available aerosol

+

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literature [Reist (1993)]. Likewise, the plasma properties can be modeled by using a standard set of the positive ion and the power balance equations of the global discharge model [Lieberman and Lichtenberg (1994)]. The ion balance equation accounts for the ion loss to the walls and to the fine particles, the ion production by the electron impact, as well as the energetic electrons entering the discharge through the sheaths. The power balance equation suggests that the power provided by the RF field is dissipated in the electron collision processes and the ion acceleration to the plasma sheaths. The set of the basic equations is complemented by the overall charge neutrality ni = n d Z d ( a ) + n, and by the equation for the two resonant radiative levels 3P1 and ' P I , where Zd(a) (here it is emphasized that the grain charge is a function of the size) is the grain charge. For simplicity, the negative ions that can in reality strongly affect the major processes of the particle's formation and growth, have been neglected. The positive ion density appears to be a critical factor for the coagulation process. It is interesting to note that the coagulation is more likely to occur in a low-density plasma [Kortshagen and Bhandarkar (1999)]. On the other hand, the particle coagulation in the low pressure plasma can be enhanced as compared to coagulation in neutral aerosols due to the charge effects. However, the dust coagulation in the plasma follows the same time dependence as the neutral coagulation ( n d oc t - 6 / 5 ) . In this sense the charged particle coagulation can be considered as the neutral coagulation of particles with an effective cross-section depending on the particle charge distribution [Kortshagen and Bhandarkar (1999)]. Most recent results show that the process of particle coagulation is even more complicated. Recent experiments performed onboard of the International Space Station (ISS) reveal that the coagulation of micron-sized particles develops much faster than expected [Morfill et al. (2002)]. Meanwhile, huge agglomerates are formed while the aggregation among smaller clusters is still on the way. The clusters are charged, positively or negatively (the charge is measured onboard the ISS by applying a sinusoidal voltage to the chamber electrodes). Note that in this case the conventional coagulation equation ( 7 . 2 ) cannot explain the observed features of the aggregation kinetics and needs to be generalized taking into account the enhancement of the coagulation rate due to the charge-induced attraction [Morfill et al. (2002)l. We note that the role of charge fluctuations in the agglomeration of nanometer-sized clusters and particulates, although qualitatively understood, still needs to be properly quantified.

344

P h y s i c s a n d Applications of Complex P l a s m a s

Under certain conditions dust agglomeration can be enhanced by the plasma particle and neutral particle bombardment [Bingham and Tsytovich (2001)l. Generally, fine particles can agglomerate when the attraction forces due to the particle bombardment exceed the Coulomb dust repulsion forces. We recall that the coagulation process is usually accompanied by reorganization of the plasma discharge into another state with lower electron number density and higher electron temperature. This transition is a classic example of a self-organization process in the plasma and is commonly referred to as a cv - y’ transition [Bouchoule and Boufendi (1993); Fridman et al. (1996)I. We now have a semi-quantitative look at this interesting phenomenon and estimate the critical particle radius a, and the corresponding change in the electron temperature T, - T,o. A simplified electron balance equation is [Fridman et al. (1996)I

dn,/dt

= Ki(T,)n,n,

-

n,Da/R2

-

Katt(a)ndn,,

(7.3)

where nn, nd, and n,, are the number densities of the neutral gas, particulates, and electrons, respectively, Ki(T,) is the ionization rate, Katt is the rate of electron attachment to nano-sized particles, D, is the ambipolar diffusion coefficient, a is the particle radius, and R is the discharge dimension (e.g., space between the two RF electrodes in a parallel plate geometry). In a steady state (dn,/dt = 0), from Eq. (7.3),one can obtain

Koi exp(-Ui)n,

+ ndKoau2exp(-e2/hva),

= Da/R2

where Ui is the ionization potential. Here, hu is a characteristic quantum that an electron can transfer t o molecular vibrations of a small nano-sized particle [Fridman et al. (1996)l. Before the coagulation onset, the electron temperature can be determined by the balance of the volume ionization and electron losses to the walls. Physically, the cy - y’.transition happens when the electron losses on the particles become more essential than those on the walls/electrodes. The critical radius can thus be calculated from Eq. (7.3)

which yields the estimate a, M 3 nm for the representative parameters of the experiment IFridman et al. (1996)I. By using Eq. (7.3),it is also possible to obtain the following simple relation for the electron temperature evolution

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Fine Particles in Industrial Applications

as a function of the aggregate radius in the process of a

I/T,

-

-

y’ transition:

I/T,~= (e2/hv)(l/a - 1/uc) ,

(7.4)

where T,o is the value of the electron temperature immediately before the a - y’ transition. Before the transition, when the electron attachment is much slower than the ambipolar diffusion to the walls, the electron temperature can be estimated from Eq. (7.3) as T, M 2 eV. From Eq. (7.4) one can also estimate that during the particle agglomeration process the electron temperature rises and levels off at e2/hvac,numerically being T, 5-7 eV. One can also infer that in the process of the a - y’ transition, the electron number density decreases in at least one order of magnitude due to enhanced electron collection by large agglomerates. To summarize, the particles trapped and surviving in the plasma play a major role in the self-organization of the plasma-powder system. The aggregate formation resembles a phase transition and starts only when the particle density exceeds a certain threshold value. The coagulation process triggers the a - y’ transition, when the high rate of electron attachment to relatively big aggregates results in an essential electron temperature increase and a dramatic fall of the electron density. In the above, we have focused on the critical phenomena, which are presently best understood in silane-based discharges. However, fine particle agglomeration is also frequently observed in C,H, plasmas. For example, this process can develop in Ar+C2H2 radio-frequency discharges and be monitored by the time-resolved measurements of the spectral broadening of the H, optical emission line of atomic hydrogen [Stefanovich et al. (2003)l. We also note that the main parameters like voltage/power thresholds necessary for the discharge maintenance, can be strongly affected by dust particles dispersed over the volume of the plasma [Nonaka et al. (2002)l. Furthermore, dust grains can trigger transitions between the two different discharge states in various RF plasmas. Thus, it certainly becomes clear that further studies of the coagulation/agglomeration kinetics in lowtemperature gas discharges are warranted. N

7.1.3

Particle growth in plasma-enhanced sputtering facilities

Nano/micron-sized particles of various materials (graphite, titanium, copper, silicon, aluminum, etc.) can also be successfully generated in plasmaenhanced sputtering facilities [Buss and Hareland (1994); Samsonov and

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Physics and Applications of Complex Plasmas

Goree (1999a)I. Contaminant particles appear in several kinds of plasma discharges with various sputtering targets. In particular, silicon/silica, aluminum, copper, carbon, etc. particles can be synthesized in DC and RF plasmas of various gas feedstocks [Selwyn et al. (1990); Jellum et al. (1991); Ganguly et al. (1993)l. For example, submicron to micron-sized particles can be formed in the gas phase of sputtering capacitively coupled discharges by using a variety of target materials [Samsonov and Goree (1999a)I. The particulate clouds usually appear after a few seconds or minutes of the discharge and can be detected by a sensitive video camera. This period will further be referred as the particle detection time t d e t . As the particles grow in size, their distribution can be imaged by the laser light scattering technique (see Sec. 3.4). Initially, the particle cloud fills the entire volume between the electrodes except for the plasma sheaths, with the highest density near the upper (powered) RF electrode. Once the particles reach a critical diameter (which is approximately 120 nm in the experiments of Samsonov and Goree), the discharge becomes unstable. At the end of the instability cycle one can observe an empty region (void) in the particle cloud. The dust voids in complex plasmas are discussed in detail in Sec. 5.1. The void expands as the particles grow in size until the void fills in nearly the whole inter-electrode region. This marks the end of the growth cycle tgrowth. Typically, tdet varies from 15 s for copper to 10 min for aluminum, whereas tgrowth varies from 3 min for Torr Seal epoxy to 3 hr for titanium sputtering targets [Samsonov and Goree (1999a)l. After the end of the growth stage, the sizes of the graphite, titanium, stainless steel, and tungsten particles are usually in the submicron range (typically 300-400 nm in diameter), whereas aluminum and copper particles grow to micrometer (typically 1-5 pm) sizes, as suggested by the ex situ Scanning Electron Microscopy (SEM) analysis. Note that particles grown from different materials feature different . Some particles, such as copper or aluminium, are filamentary fractals. In contrast, carbon particles usually have a bumpy spherical shape. Other materials, such as titanium and stainless steel, form compact coagulants of a few spheres. On the other hand, tungsten particles form compact agglomerates. As compared with reactive (e.g., silane) plasmas, particle growth rates are usually lower in the sputtering discharges mostly because of the lower number densities of clustering/agglomerating species. However, the sputtering discharges have an obvious advantage that they can produce particles

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from almost any solid material that can be sputtered without decomposition. Similar to chemically active plasmas, the particle growth process in sputtering discharges also develops in several stages. However, in this case the particles originate as clusters released from the sputtering target or the walls. Afterwards, the clusters nucleate into primary particles, which can further agglomerate and form particulates of various shapes and architectures (e.g., spongy and filamentary or compact and spheroidal). Note that the electric charge is a critical factor in determining the shape of the plasma-grown nanoparticles [Huang and Kushner (1997)]. Indeed, when particles have a small (typically negative) charge and a high velocity, they can easily overcome Coulomb repulsion and form compact or spheroidal agglomerates. On the other hand, when the charge is larger and the velocity is lower, the electrostatic repulsion is stronger and an incoming particle is more likely to strike the end of a particle chain than the middle (mostly because of the plasma shielding of the distant elements of the agglomerate), and this process tends to promote a filamentary or fractal shape [Huang and Kushner (1997)]. The picture of the particulate growth is certainly more complex in magnetron sputtering discharges [Selwyn et al. (1997)l. The mechanisms for particle generation, transport and trapping during the magnetron sputter deposition are different from the mechanisms reported in etching processes in reactive plasmas, due to the inherent spatial non-uniformity of magnetically enhanced plasmas. Since the magnetron sputtering facilities are usually operated at low pressures, the contribution from the homogeneous mechanism (which is a dominant one in silane plasmas, see Sec. 7.1) is likely to be small. Hence, most contamination problems in magnetron sputtering processes can be attributed to heterogeneous contamination sources, such as wall flaking. Furthermore, highly non-uniform plasmas typical of magnetron sputtering processes are subject to simultaneous material removal and redeposition in different target regions. Thus, the formation of filament structures can be favored. Meanwhile, the filaments can be resistively heated by intense current flows, which can cause violent mechanical failures and the removal of the filament into the plasma bulk. Combined with the repulsion between the negatively charged filament and the sheath region, this process can result in an acceleration of the filaments away from the sputter target, which can be a source of hot and fast particles capable of damaging the substrate being processed [Selwyn et al. (1997)]. We note that DC/RF sputtering belongs to a larger group of particle generation mechanisms from the surrounding solid surfaces, encompassing

348

Physics and Applications of Complex Plasmas

the reactive ion etching (RIE) [Anderson et al. (1994)], the filtered cathodic vacuum arc deposition [Beilis et al. (1999)], the hollow cathode discharges, and some other processes. For example, in the anisotropic RIE controlled by directed ion fluxes, small columnary etch residues are usually formed. AS a result of a slight under-etching, the columns become thinner at their base and thus unstable. Since the structures are negatively charged, the Coulomb repulsion from the surface causes them to break off and to be ejected into the discharge. The split etch residues are finally trapped in the glow by the plasma force balance. We emphasize that the above mechanism is quite similar to the magnetically-enhanced sputtering systems discussed previously. Further investigation of the nanoparticle origin and growth mechanisms in the complex plasma systems with solid particles released from the walls/electrodes, is eagerly anticipated in the near future.

7.1.4 Problems of particle detection in the nanometer range At present, the most advanced common particle detection techniques (e.g., laser light scattering) (see Sec. 3.4) allow one to successfully monitor the growth and dynamics of submicron-sized particles. However, detection and diagnostics of nano-sized particles as well as investigation of the complex physical/chemical mechanisms leading to nucleation of particle precursors, is still an unresolved problem so far. Some common methods for the detection of micron-sized particles in a low-temperature plasma are discussed in Sec. 3.4. Here, we remark that in the studies of the particulate growth in reactive plasmas (see Sec. 7.1) the most common methods are the Laser Light Scattering based on the Mie light scattering theory. This method can be used to detect the particles and get an insight into the dust number density. However, the accuracy of the methods based on the Mie scattering theory is restricted to very small (typically larger than 20 nm in diameter) particles, due to the a6/X4 dependence of the intensity of the scattered signal, where X is the laser wavelength. Here, we address some new in situ methods of the detection of nanosized clusters and particles grown in chemically active plasma environments. Since such environments are common for industrial manufacturing processes, it is imperative to be able to detect and control the particles in the ionized gas phase. The double-pulse-discharge (DPD) method enables a highly sensitive in situ detection of very small (> 0.5 nm in size) clusters and particles [Koga et al. (2000)l. Using this method, one can simultane-

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349

ously measure a temporal evolution of the average size and density of the clusters during the nucleation and subsequent growth phases in silane RF discharges (see See. 7.1). In the DPD method, each discharge cycle includes three discharge phases, namely, the periods of the main pulse discharge Ton, the cluster diffusion to^, and the sub-pulse discharge ton.The nanoclusters nucleate and grow during Ton,and diffuse after the main pulse discharge is turned off. Assuming that the loss of plasma electrons due to their attachment to the neutral clusters is comparable with the electron loss due to diffusion, one can relate the cluster density nd to the decay rate of the electron density n,: (7.5)

where oatt is the cross-section of the electron attahment to the clusters, Tdif is the characteristic time of the electron loss due to diffusion, u, is the electron velocity, and the angular brackets denote the averaging over the electron distribution [Koga et al. (2000)l. The dynamics of the electron density decay provides a valuable information on the cluster diffusion during t,R. From Eq. (7.5) one can obtain %@off)

1 / T ( t o f f )- 1/Tdif

,

where r ( t o ~is)the characteristic time of the electron density decay after the subpulse discharge is turned off, which can be used to estimate Tdif. The characteristic decay time of the cluster density rpcan be deduced from the dependence of T on t O E . Having rp,one can obtain the cluster diffusion coefficient, which is related to the cluster size. The absolute value of nd can then be calculated by using the measured cluster size and fitting Eq. (7.5) to the time evolution of n, after the main pulse discharge. In the nanocluster detection experiments in 13.56 MHz parallel-plate capacitively coupled silane plasmas, the dynamics of the electron density decay (after switching the RF pulse off) is measured by a 9 GHz microwave interferometry [Koga et al. (2000)l. Figure 7.4 shows the nanocluster size distribution for different main pulse durations To, = 3, 10, and 100 ms. One can note that only small clusters exist for Ton= 3 ms. For To, = 10 ms, the larger clusters coexist with the small ones and there is a “bottleneck” in the size distribution of SinHz (n 4). This bottleneck clearly indicates that the large clusters are structurally different from the small ones. Presumably, the larger clusters are well nucleated. After the main discharge pulse of 100 ms is over, the large clusters grow further in a monodisperse way. On the other

-

350

Physics and Applications of Complex Plasmas

hand, smaller clusters with n M 4-6 are consumed by the large clusters. It is imperative that the latter process almost completely suppresses the nucleation of new large clusters with the intermediate number of silicon atoms. Thus, it indeed appears possible t o control the size and number density of the nanoclusters using the pulsed discharge technique.

Fig. 7.4 Size distribution of clusters for Ton = 3, 10, and 100 ms in a 13.56 MHz RF discharge sustained (with RF power densities of 0.18 W/cm2) in pure SiH4 at 13.3 Pa [Koga et al. (ZOOO)].

Another method of in situ particle detection is based on the analysis of the radio-frequency discharge impedance [Boufendi et al. (200l)l. It enables one to detect the occurrence of fine powders with the size of about 2-3 nm. In capacitively coupled non-symmetrical discharges (in silaneargon gas mixtures) the relation between the RF voltage and RF current is usually quite nonlinear [Lieberman and Lichtenberg (1994)]. Specifically,

Fine Particles in Industrial Applications

351

this nonlinearity generates higher harmonics in the discharge current. From the analysis of the nonlinear higher-harmonic currents one can infer a valuable information about the electron and ion dynamics. The spectral analysis of the RF current reveals two distinctive harmonics, namely, with the frequencies v1 = 13.56 MHz (fundamental frequency or the first harmonic) and v3 = 40.68 MHz (third harmonic). In pure argon plasmas, the amplitudes of the two harmonic components remain constant. However, in a dust-forming plasma, their amplitudes start to decrease a few tens of milliseconds after the ignition of the discharge owing to the formation of nanoparticles in the gas phase. This effect can thus be used as a powerful tool to detect the occurrence of the nano-sized powders in reactive plasmas. Capacitive probe measurements of RF current and voltage show that in the pristine (pure argon) plasma, the intensity of the 40.68 MHz nonlinear signal is invariable, while it is a clearly declining function of time in Ar+SiH4 dust-generating plasmas [Leroux et al. (2000)l. The time intervals when the discharge parameters change and the typical particle growth times (see Sec. 7.1) in similar gas mixtures are in a remarkable correlation. In particular, in a room-temperature discharge in the 30 sccm Ar+1.2 sccm SiH4 mixture at the 12 Pa total pressure, the 70 ms interval appears to be sufficient to detect the 2-3 nm-sized crystallites in the plasma [Leroux et al. (2000)l. Another interesting feature is that the intensity of the 40.68 MHz nonlinear signal after 200 ms diminishes linearly, revealing the linear character of the dust growth at later stages. On the other hand, the size of the initially formed particles remains constant during the first phase of the particle growth, when their number density increases up to a critical value (about 1011-1012 ~ m - for ~ ) the coagulation phase to start. Furthermore, the observed drastic decrease in the amplitude of the nonlinear signals can be attributed not only to the higher resistance of the dust-loaded plasmas but also to the drop of the electron density as a result of the electron capture by the dust grains. We emphasize that the above method is non-intrusive, with a temporal resolution in the microsecond range, very easy to implement, and can thus be used in industrial plasma reactors. The voltage, current and phase measurements of fundamental and harmonic radio-frequencies can be cross-referenced by the temporal evolution of the optical emission and scattered laser light from the gas-phase grown particles. It is interesting that the variations of the fourth harmonic of the RF current correlate with the optical emission intensity of the Si-H and H, emission lines [Ghidini et al. (2004)l. The measurements of the electrical

352

Physics and Applications of Complex Plasmas

and optical discharge parameters can thus be used to monitor the fine particle growth at different stages, making them good candidates for process monitoring in industrial devices. Note that even though the laser light scattering methods can be successfully used to obtain the in s i t u information on the growth processes and the spatial distribution of fine particles, the information regarding the size and the number density of the dust particles requires certain assumptions about their size distribution and refractive index. In some cases, the sensitivity of the laser light scattering appears insufficient to accurately monitor relatively low-density dust clouds. This situation is relevant to nanoparticle generation in C,H,-based plasmas discussed in Sec. 7.1.1.2. The problem can be overcome by using multiple-wavelength Rayleigh-Mie scattering ellipsometry for the in s i t u analysis of nanoparticles [Gebauer and Winter (2003)]. This technique enables one to accurately measure the size distribution and the complex refractive index of various (e.g., melamine formaldehyde) nanoparticles suspended in the plasma. It is worth noting that the Rayleigh-Mie scattering ellipsometry can be used for the analysis of the particles consisting of different layers (e.g., having a shell-like structure) in broad pressure ranges. Some of the existing problems related t o quantitative in situ measurements of the particle size distributions can be solved by using a differential mobility analyzer (DMA) system operated a t low pressures. The DMA can be coupled to a plasma reactor in the process of PECVD of amorphous silica (a-SiOz) films in tetraetylortosilicate (TEOS) and oxygen gas feedstock [Seol et al. (2001)]. The differential mobility analysis results can be cross-referenced, e,g., by the ex s i t u transmission electron microscopy

(TEM). Generally, the dust mobility measurements can be carried out by applying a variable voltage to the collecting electrode of the DMA. Only the charged particles controlled by the applied voltage, can successfully pass through the slit in the collecting electrode. The mobility distribution of the charged particles is obtained by measuring their total charge using a Faraday cup electrometer. In this way, the particle size can be estimated using conventional relationships between the particle mobility and the voltage appied to the DMA. This method provides a good qualitative agreement (within 14-50%) between the particle sizes (of a few tens of nanometers) obtained by using the DMA and TEM techniques [Seol et al. (200l)l.

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353

Another apparent challenge in the characterization of nanoparticles in reactive plasmas is to perform a real-time compositional analysis of small particles [Reents et al. (1994)]. For instance, a CW laser can be used to sense a particle and fire a pulsed light. The laser ablation produces secondary ions with a wide mass spectrum, the latter can be measured by the time of flight mass spect,rometry. The apparent disadvantage of this method is that only particles with a size of > 200 nm scatter enough light to be detected. Therefore, smaller particles cannot be detected in this way. With some modifications and without a CW laser, this technique can be used for the in situ analysis of the chemical composition of the particles ranging in size from 20 nm to > 10 pm in air [Reents et al. (1994)I. The system can also be adjusted to the analysis of the sumbmicrometer and nano-sized particles in high pressure low-temperature plasmas. A least direct but also quite reliable method of particle size detection is based on the use of the particle-size dependent dispersion relation of dustacoustic wuwes (DAWs) during the nucleation phase [Kortshagen (1997)l. The dispersion properties of the dust-acoustic waves depend sensitively on the particle size a. In particular, a size-dependent momentum relaxation frequency of nanoparticles, entering the dispersion relation of the DAW is .D(U)

27 52

= -7ru2-

m, md(a)nnwTn '

where m, and md are the masses of the neutrals and the nanoparticles, and nn and WT, are the densities and thermal velocities of the neutral gas atoms/molecules, respectively. In the case of particle generation in silane plasmas, a Si,H, particle with the radius of 1 nm usually consists of approximately lo3 atoms. In this case the composition ratio of Si to H atoms is 3 : 4 [Hollenstein et al. (1996)l. The particle mass entering the expression for the momentum relaxation frequency is

-

N

where msi and m H are the masses of the silicon and hydrogen atoms, respectively [Hollenstein et al. (1996)]. Therefore, the measurements of the dust acoustic speed can in principle be used to characterize the size of monodisperse nanoparticles. It should be noted that the Quadruple Mass Spectrometry (QMS) can also be used for the in situ analysis of hydrogenated nanophase silicon powders through the mesurement of mass spectra of the abundant Si,H,

354

Physics and Applications of Complex Plasmas

neutral and positively charged radicals that play a crucial role in fine powder nucleation and growth [Bertran et al. (1996)]. For instance, typical QMS spectrometers with the range of 200 amu can accurately detect highersilicon radicals SiGH$+, where superscripts “0” and denote neutral and positively charged states [Bertran et al. (1996)], Finally, we stress that in the above we have focused on the fine particle growth and associated phenomena in low-temperature processing plasmas. For the peculiarities and fascinating details of dust grain growth in astrophysical plasmas the reader can be referred t o the recent review [Bingham and Tsytovich (2002)l.

“+”

7.2

Deleterious Aspects

In the following two sections, we discuss various technological implications of fine solid particles in chemically active plasmas, including deleterious (such as the process contaminants in the semiconductor micro-fabrication and safety hazard for the future fusion reactors) and useful (building units and functional embedded particles for various thin film deposition and nanoassembly processes, etc.) aspects. For a number of years, the plasmagrown powder particles have been deemed as unwelcome process contaminants in the semiconductor industry. Indeed, particles with the sizes comparable with the typical feature sizes of the integrated circuitry elements caused line and interconnect shorts thus irrecoverably compromising the entire semiconductor wafer manufacturing process. Therefore, a number of remediation methods has recently been developed to remove the powder particulates from the processing volumes or suppress the dust growth a t the initial stages (Sec. 7.2.1 and 7.2.2). Meanwhile, radioactive nuclear-induced dust has recently become a major concern for the safety of operation of the future fusion reactors. The physical mechanisms of the origin of and the major problems associated with the radioactive dust in the fusion reactors are discussed in Sec. 7.2.3.

7.2.1

Particulate powders as process contaminants in microelectronics

We recall that our aim here is to discuss the dynamic processes in chemically active complex plasma systems presently used by the microelectronic and other industries, e.g for manufacturing semiconductor microchips and

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355

more complex integrated devices. Such systems are prone of fine particles, ranging in size from few nanometers to tens of microns [Duffalo and Monkowski (1984); Perrin and Hollenstein (1999)]. As a result of intensive research efforts of the last decade, it was proven that the particulate matter appears in ultra-large scale integration (ULSI) fabrication processes not because of the dust remaining in the clean rooms, but emerges as a result of numerous chemical and physical processes in the ionized gas phase. The fundamental processes of particulate formation include but are not limited to gas-phase polymerization leading to creation of protoparticles (large molecules), release of atoms/radicals of the substrate/wall material as a result of reactive sputtering, self-organization of the atomic/molecular matter into larger clusters and nucleates, typically in the nanometer range, and coagulation of fine particles into larger particulates, with the size ranging from tens of nanometers to a few microns. Presence of the solid particles in processing volumes inevitably results in compromising the product yield, e.g., limitations to plasma production of semiconductor integrated circuits and micro-devices. Indeed, the particles with the sizes comparable with the ULSI feature size can cause line shorts and have been termed in the literature as “killer” particles. Moreover, dust contamination can result in pinholes, delamination and interconnection shorts or opens in ULSI circuits. Optical data storage disks can also be affected by read-write errors, damaged sectors, and total failure caused by excessive powder contamination. About ten years ago, when the average feature sizes of ULSI circuits were in the micrometer range, nanoparticles appearing in the discharge volume, were simply too small to be of any concern for the semiconductor manufacture. On the other hand, an efficient solution for contaminant micron-sized particles was found to confine them in the near-wafer areas, and subsequently remove them from the processing volume by direct pumping, gas flows or other simple means. Efficient particle confinement was possible due to large sheath (near-wafer) potentials in capacitively coupled plasma (CCP) reactors, which were used as benchmark plasma processing tools in microelectronic industry at that time. In the last few years, the situation has changed. First, the current typical ULSI feature sizes shrunk to 0.13 microns, and it is expected that by 2010 they will be as small as 40-50 nm [Semiconductor Technology Roadmap (2003)I. Hence, even nanometer-sized particles have to be considered now as “killer” particles in the semiconductor technology. Secondly, the capacitively coupled RF discharges have been widely replaced by inductively coupled plasma (ICP) reactors featuring higher plasma densities and

356

Physics and Applications of Complex Plasmas

process efficiency. Unfortunately, the particle trapping capacity of ICPs is relatively weak due to smaller sheath potentials, which can in principle result in uncontrollable fallout of even nano-sized particles on the wafer. Thus, the existing solution based on the discharge trapping capacity is unlikely to work in the near future, and the problem of removal of nano-sized particles from the discharge volume or controlling their growth is becoming critical for the semiconductor industry. If the problem is not solved within the next couple of years, the development of the next generation of the integrated circuits can be substantially compromised [Kroesen (2002)]. It is interesting to note that fluorine-containing gas mixtures usually produce fewer particles during silicon etching process than chlorine or bromine-containing gases do because of the generation of high-pressure volatile etch products in fluorine-based plasmas [Lii (1996)l. On the other hand, recent detection of fine powders in fluorocarbon gases (discussed in detail in Sec. 7.1.3) suggests that a certain dust fallout prevention strategy should also be implementer by the microelectronic industry during fluorinebased silicon etching processes. The plasma etching tool design, especially the electrode, significantly affectes the overall cleanliness of plasma tools [Lii (1996)l. It is not thus surprising that during a number of years the main goal of early dusty plasma investigations was to obtain a good control of contamination in plasma-processing reactors, either by eliminating dust particles from the gas phase or by preventing them from getting into contact with the wafer surface. As a result of numerous studies of dust dynamics in industrial plasma reactors [Selwyn (1994); Boehme et al. (1994); Lapenta and Brackbill (1997)], process contamination by relatively large (> 100 nm) particles at present is well under control.

7.2.2

Removal and growth suppression of dust particles

Generally speaking, dust contamination can be managed through the proper discharge operation (e.g., by a periodic sequence of the discharge “on” and “of€” cycles), maintaining the desired process chemistry (e.g., suppression of plasma polymerization and powder growth) and appropriate plasma tool design (e.g., electrode shape). Particles suspended near the plasma/sheath boundary can be removed before shutting the plasma process off by using the proper changes in RF power, gas flow, and magnetic field near the end of the plasma etching stage. Furthermore, the fine particles can be purged to the pump line during the process by a proper

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tool design, whereas the dust generation and growth can be impeded or disrupted by controlling the number densities and reactivities of the dust growth precursor species [Lii (1996)l. More specifically, in order to minimize the deleterious effect of dust particles on thin film deposition and etching, it is important to develop either new processes completely suppressing the dust generation or process cycles in dusty plasmas without contamination over the substrate regions, which are sensitive to dust fallout [Kersten et al. (2001)l. There are several options in this regard: intelligent arrangement of electrodes and substrate holders, construction of special electrode shapes (e.g., “grooved” electrodes) [Selwyn (1994); Lapenta and Brackbill(1997)],square waveform plasma modulation [Koga eC al. (2000)], fast transport regimes of the reactive species, external electrostatic forces (e.g., Langmuir probe induced) [Uchida et al. (2001)], additional forces based on neutral drag (gas flow) or thermophoresis (temperature gradient) by external heating [Perrin et al. (1994)l or photophoresis (laser irradiation) [Klindworth et al. (2000)l. The basic idea for the introduction of square wave plasma modulation with “on-off’ cycles is that the small and negatively charged dust precursors are not allowed to grow in size and concentration during the “on” sequence and leave the plasma volume during the “off’ sequence. The introduction of special electrode shapes and additional forces usually results in changes of the equilibrium particle trapping positions. Using the above methods, dust can be effectively pushed outwards the sensitive areas of the plasma reactor.

A dust cloud trapped and guided out of the active zone of the plasma reactor by using a groove in the electrode [Selwyn and Patterson (1992)]. Fig. 7.5

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For example, by using a few millimeter deep grooves encircling the semiconductor wafers, one can divert the dust cloud from the active processing area and confine it in electrostatic potential wells over the grooves. The grooves can be extended further away from the active processing area and be used to guide the dust cloud to the pumping line as shown in Fig. 7.5. The advanced numerical simulations show that the grooves with circular cross-sections can trap the fine particles better than the commonly used grooves with square cross-sections [Lapenta and Brackbill (1997)l. It is interesting that even after the fallout from the gas phase, the dust grains can still be removed from the surface [Goree and Sheridan (1992)l. As usual, this process can be monitored by an situ laser light scattering measurements. The particles can be shed from the walls, substrates, test surfaces, etc. by using the same principle. For example, when the discharge is turned on, particulates can be rapidly released from the test surface coated with a layer of micron-sized particles [Goree and Sheridan (1992)l. However, this process stops during the plasma ‘‘off’ cycle. Thus, the plasma exposure is likely to cause the particulate shedding. Furthermore, the dust shedding rate increases with the plasma density. Physically, the grains become negatively charged due to the electron and ion fluxes onto the surface and are then pulled off the surface by the electric field in the plasma sheath. In this case an individual dust grain is shed when its charge becomes sufficiently negative [Goree and Sheridan (1992)I. Recently, a quite similar principle has been adopted to remove the fallen particles from the semiconductor wafers by using a low-power discharge in the plasma processing tool and direct mechanical agitation of the dust grains on the wafer surface [Selwyn (1998)l. The actual fine particle shedding process involves two steps. The first step includes mechanical agitation of the particles by the piezoelectrical excitation of ultrasonic surface vibrations. Once the vibrational forces overcome the surface sticking forces, the powder particles can be removed from the surface, acquire a negative electric charge, and eventually be directed to the pumping line by manipulating the discharge power [Selwyn (1998); Boufendi and Bouchoule (2002)l. One of the key problems of the fine powder confinement is to develop the appropriate particulate potential traps. The latter can naturally emerge near various three-dimensional physical and electromagnetic structures in plasma processing discharges, such as internal antennas (or coils), gas injection nozzles, sub- or super-substrate topography elements, as well as single-sided vacuum pump ports. These structures contribute to azimuthal

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assymmetries in reactant fluxes and can also create efficient dust particle traps due t o significant distortions of the electromagnetic pattern at distances well exceeding the geometrical sizes of the internal structures [Hwang et al. (1998)l. There are several other simple physical methods for removing charged particles from RF processing plasmas. Generally, most of the methods adopt control of the particle dynamics by varying input power, working gas pressure, applying external electrostatic or magnetic fields [Beck et al. (1994); Kim and Ikegawa (1996)l. Variation of the R F power deposition affects the location, shape, and depth of the potential wells, where the particles can be trapped. Likewise, if the gas pressure increases, the sheath edge usually moves closer to the substrate being processed. Since the particulate traps are usually formed in the vicinity of the plasma sheaths, the dust particles are thus confined closer to the substrate at elevated pressures. On the other hand, a negatively biased Langmuir probe can repel the negatively charged dust grains, and attract them when the bias polarity is reversed. The probe can also distort the sheath edge when the tip resides within the sheath. Meanwhile, external magnetic fields can change the characteristics of the particle traps and also exert additional forces on the charged dust grains. Specifically, E x B drift in crossed electric and magnetic fields can be used to remove silicon particles grown in silane discharges for the PECVD of large-area uniform hydrogenated amorphous silicon thin films. The magnitude of the external magnetic field directly correlates with the efficiency of dust removal [Fujiyama et al. (1999)l. Particle confinement in particle traps in processing plasmas can also be controlled by ultraviolet (UV) radiation, which normally reduces the negative equilibrium charge on fine particles and modifies the prevailing force balance [Rosenberg and Mendis (1996)I. In this way, the removal of the contaminant particles from the traps can be facilitated by using the UV radiation. The thermophoretic force (experimentally implemented by additional heating of the grounded electrode in parallel plate plasma reactors) can also be used to drive nano-sized clusters and particles away from the processing area. This method works well even for ultra-small particles above a few nanometers in size. Alternatively, growth of large size clusters can be significantly suppressed by strong dilution of silane by hydrogen (e.g., [Hz]/[SiH4] > 20). For example, pulse discharge modulation combined with the electrode heating can be very efficient to suppress the nanocluster growth [Shiratani

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e t al. (2000)l. A combination of several nanocluster control strategies (thermophoretic force, directed gas flows, and evacuation without stagnation) is used in recently implemented plasma reactor for the manufacturing of the device-grade aniorphous silicon films [Watanabe e t aZ. (2002)l. 7.2.3

Role of dust particles in fusion reactors

In this subsection, we consider radioactive dust as a major concern for the safety of operation of fusion devices. It has been common for a number of years that small dust particles do exist in magnetic confinement fusion devices [Winter (2000); Winter e t al. (200l)l. Their origin is mostly due to the plasma-surface interactions. The radioactive dust contains large amounts of hydrogen isotopes, with up to 50% in tritium. The main consequence of the dust contamination is the safety hazard due to the high chemical reactivity and mobile tritium inventory. The dust can also affect the plasma performance, stability, and the operation of the fusion devices. Physically, tritium, incorporated into carbonaceous dust, undergoes radiaoactive decay, and this can lead to dust charging and the formation of the nuclear-induced plasmas. In the plasma, charged dust particles can be transported and levitated. There are thus two major sets of problems related to particulate generation in fusion devices. One of them is related to the safety of operation of the fusion reactor, the other being related to the plasma parameters and stability. Specifically, dust-bound tritium inventory is a major safety concern for the future fusion reactors. The main problem in this regard is that dust can not be reprocessed together with tritium, thus increasing the site inventory. Dust is also a potential carrier of tritium in the case of a severe reactor failure. Furthermore, if the reactor cooling systems are damaged, large amounts of hydrogen can form an explosive mixture with oxygen from the environment. The key point of another aspect is that large amounts of dust can accumulate at the bottom of the device (which is usually a divertor area in tokamak and stellarator devices). Dust accumulation can impede the heat transfer to the cooling surfaces and also compromise specially designed gaps for electrical insulation or thermal expansion purposes. Such layers can sublimate when exposed to huge heat loads. On the other hand, this can lead to a source of plasma impurities adversely affecting the plasma parameters and stability. It is thus reasonable to pose the following question: what are the sources of origin and formation mechanisms of dust particles in fusion devices? Note

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that significant parts of the plasma-exposed surfaces (e.g., limiters, divertors, antennas for RF heating) are often coated with carbon-based materials, such as graphite or carbon fiber composites. However, carbon suffers from high erosion rates from intense physical sputtering and chemical erosion. As a result of the exposure to chemically active hydrogen, several forms of hydrocarbon are released from the surfaces into the plasma edge where they interact with the plasma and could be ionized or dissociated. Edge-localized modes (ELM) and pressure-driven instabilities or disruptions (quick and uncontrollable discharge quenching leading t o deposition of the plasma-stored energy onto the suface) at the plasma edge can lead to excessive heat fluxes onto the divertor surfaces. In the International Thermonuclear Experimental Reactor (ITER) device, the thermal load from a disruption is estimated t o be 100 GW/m2 during 1 ms. For this reason, recent discharge disruption experiments include the dust as a critical component. Thus, a complete understanding of the mechanisms responsible for particulate production from plasma-surface interactions in fusion devices is required. Moreover, this area has been highlighted by the US Fusion Safety Program as one of the priority areas of research. Recently developed plasma/fluid and aerosol models of disruption simulation experiments in the SIRENS high heat flux facility integrate the necessary mechanisms of plasma-material interactions, plasma and fluid flow, and particulate generation and transport [Sharpe et al. (2001)l. The model successfully predicts the size distribution of primary particulates generated in SIRENS disruption-induced material immobilization experiments. Meanwhile, the estimated erosion rate of the carbon material can be quite high, up to 2 x loz1 mp2/s in TEXTOR (Tokamak Experiment for Technology-oriented Research) fusion device. The eroded material is usually redeposited in a form of carbon-based layers in the areas of lower heat fluxes, and contains a large amount of radioactive hydrogen isotopes. The dust thus becomes radioactive and can carry a large proportion of tritium inventory in the future devices. Above all, P-decay of tritium may lead to charging of dust and formation of nuclear-induced plasmas, which may affect the initiation phase of a thermonuclear plasma. One can estimate the charge that can be accumulated by a carbonbased particle due to radioactive decay of tritium (half-lifetime of tllZ = 12.3 years with a maximum electron energy of 18.6 keV) [Winter (2000)]. Specifically, the number of @-decaysin a 5 pm-sized carbon particle carrying 0.4 hydrogen isotopes (with 50% tritium) per carbon atom, can reach up

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to 5 x 102 per second. Assuming that all @-electronsleave the particle, the secondary electron emission yield of unity, and mean charge lifetime of 1 s, one can obtain that such a particle can accumulate a positive equilibrium charge of Q = 5 x 102e. The electric field of 38 V/cm would be sufficient to confine such a particle near the surface. The size of particulates in fusion devices varies in broad ranges, from a few tens of nanometers to several millimeters. The estimates of the total amount of redeposited radioactive dust show that large amounts up to a few tens of kilograms can be generated in the ITER device [Winter (2000)]. The dust composition is mostly carbon but may also include all other materials used inside the vessel or for wall conditioning purposes (e.g., B, Si). In the TEXTOR experiments, a large number of almost perfect metallic spheres with diameters from 10 pm to 1 mm has been identified. The most likely formation mechanism is the reactor wall flaking (heterogeneous process) with subsequent coagulation of metal atoms on hot and non-wetting graphite surfaces. It is also interesting to note that very small, sub-100 nm carbon particles can be formed in the fusion devices as a result of CVD processes in carbon vapor. Formation of small globular clusters, fullerene-like materials, etc. is also possible. In the TEXTOR device, agglomerates of individual particles of about 100 nm in diameter are frequently observed. Another possible mechanism is the dust growth in the scrape-off layers (detached plasmas in the proximity of divertors and limiters), where the conditions are quite similar to those in chemically active low-temperature hydrocarbon plasmas (see Sec. 7.1). Under such conditions, the growth will probably proceed via negative hydrocarbon ions and multiple ion-neutral reactions. Large particles introduced into the plasma can also induce a disruption. However, usually if a discharge is fully developed, their effect on the discharge performance is weak. However, if particles pre-exist in the vessel prior to the plasma start-up, a significant amount of impurities can be released into the plasma volume. Indeed, the intensive impurity radiation is often observed during the start-up phase and may be due to the levitated dust. It is worth noting that the electron number density of nuclear-induced plasmas is typically about 5 x lo9 ~ m - When ~ . the gas pressure in the vessel increases to about lop3 mbar, the plasma breakdown takes place and the fast /?-electrons from T-decay ionize the gas along their track (of the order of lo3 m at this pressure). In this way, about 500 electron-ion pairs per pelectron can be formed. The plasma induced by radioactive particles can be

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formed in a simple parallel plate model reactor configuration even without any magnetic field. In this case, dust levitation and formation of ordered dust structures is possible [Fortov et al. (2002)l. Therefore, the study of the possible methods to remove from or minimize the consequences of the radioactive dust in the fusion reactors is very important. Note that use of the thermophoretic force (in a manner similar to what was discussed in Sec. 7.2.1) can be a viable route for the removal of the radioactive dust [Yokomine et al. (2001)l.

7.3 New Applications of Nano- and Micron-Sized Particles Nano- and micron-sized clusters and particulates attract a significant interest for a number of applications. In particular, dust-loaded plasmas offer very specific and unusual conditions for the processing of various surfaces and particles due t,o the highly efficient chemistry that can be achieved as a result of the enhancement of the electron energy [Boufendi and Bouchoule (2002)l. Therefore, complex plasmas can be even more efficient as compared with electronegative gases in terms of chemically active radical production and be suitable for various thin film technologies. The examples of the existing and potential applications include but are not limited to the self-assembly of various carbon-based nanostructures, deposition of nanostructured amorphous silicon films for solar cell manufacture, embedded particles for functional coatings, and many others. Some of them are discussed in the following sections.

7.3.1

Nanoparticles in the plasma-assisted assembly of carbon- based nanostructures

In this section, we address the issues relevant to the role of the plasmagrown nano-sized particles in the synthesis of various carbon-based nanostructures. Carbon nanotubes (CNTs) represent one of the most common nanostructured organizations of carbon. The CNTs were discovered in the early 1990s [Iijima (1991)] and represent a new and extraordinary form of carbon. Depending on chirality and diameter, CNTs can either be metallic or semiconductor and be useful for fabrication of metal-semiconductor and semiconductor-semiconductor junctions. Furthermore, they exhibit extraordinary electrical and mechanical properties and offer good potential for applications in electronic devices, computing and data storage technol-

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ogy, sensors, composites, storage of hydrogen or lithium for battery development, nanoelectromechanical (NEMS) systems, and as tips in scanning probe microscopy for imaging and nanolithography. Many common recipes for the synthesis of various carbon-based nanostructures are based on the plasma-enhanced CVD using the surface catalysis [Meyyappan (2000)]. Fine particles containing single-walled carbon nanotubes were originally synthesized using a carbon-arc reactor with the carbon cathode containing iron and observed in soot-like deposits on the chamber walls [Iijima and Ichihashi (1993)I. The length of single-walled CNTs synthesized by arc discharge methods is usually less than several hundred nanometers. Nevertheless, the nanotubes can also be grown up to a micrometer length with special arrangements for longer process duration. Different transitions between various forms of organization of nanostructured carbon are due to the reorganization of carbon-based nano-sized clusters. Furthermore, if the clusters were actually grown by the PECVD, the processes in the ionized gas phase can also critically affect the selforganization of nanostructured carbon. For example, under external irradiation, carbon nanotubes can transit to the diamond state through the intermediate carbon nanoonion state [Wei et al. (1998)l. The transition from the nanotube to the nanoonion structure is accompanied by a structure collapse of the tube and reorganization of the carbon clusters. This further supports the idea that nanoclusters are indeed the building units for the fabrication of various nanostructures, including the carbon nanotubes. Here we refer the reader to further discussion of the nanocluster route of nanomaterials fabrication in Sec. 7.3.3. However, the dynamics of the crystallization behavior [Ci et al. (200l)l of CNTs in various (including plasma-based) CVD systems still remains a challenge. One of the common nucleation models invokes a rearrangement of the metal (e.g., Fe, Ni, or Co) catalyst from a pre-deposited nanolayer into round-shaped nanoparticles. The nanoparticle size is a decisive factor in the nanostructure nucleation [Yudasako et al. (1997)l. We thus remark that CNTs usually grow within a narrow range of substrate temperatures, promoting the required reorganization of the catalyst layer. One needs to keep in mind however that carbon-based nanostructures is a separate and quickly emerging area of the solid-state physics and materials engineering [Dresselhaus et ul. (1996); Thostenson et ul. (2001); Ren et al. (1998)l. Moreover, even a discussion of various plasma-based methods of the nanostructure fabrication is worth a dedicated review (see e.g., [Meyyappan et al. (2003)l and references therein) due to the large and

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continuously increasing number of publications, we will mention only a few relevant works. In particular, well-aligned CNTs can be grown on metal nanoparticle catalyst precursors exposed to low-temperature hot-filament, microwave, and inductively coupled plasmas of several PECVD systems [Murakami et al. (2000); Delzeit et al. (2002)l. Our specific aim here is to discuss the details of the plasma-assisted growth of carbon-based nanostructures in various complex plasma systems and elucidate the actual role of the plasma-grown nanoparticles. For example, by injecting metal nanoparticles into a microwave surfacewave sustained plasma reactor one can grow various carbon-based nanostructures directly in the reactor volume, which is quite different from many conventional methods involving solid substrates [Hayashi et al. (200l)l. This process requires the microwave slot-excited plasma sustained in a methane-hydrogen gas mixture with 300-800 W microwave powers (2.45 GHz) in the pressure range 1-20 mTorr. One of the side ports of the chamber can be used to inject fine metal particles. The particulate clouds can be monitored using light scattering of a laser beam directed from another side port. A biased electrode positioned in the lower part of the plasma glow suspends fine metal particles to enable the efficient gasphase growth of carbon nanostructures. The electrode is also used to collect the nanopowder. At lower powers, the injected nickel ultra-fine particles form a cloud in the plasma bulk between the chamber window and the electrode. As the power increases, some of the particles escape the main cloud and move downwards to the electrode, presumably due to stronger ion drag force (which is higher at higher RF powers and plasma densities, see Sec. 2.1), where they can be suspended (due to the balance of forces acting on fine particles, also considered in Sec. 2.1) during the nanostructure growth phase (with an approximate duration of 20 min [Hayashi et al. (200l)l). At higher microwave powers, tubular structures resembling carbon nanotubes can be grown in the gas phase and further dropped on the particle collector when the generator is turned off. The above experiments support the idea that fine nano-sized particles grown or externally dispensed in the plasma reactor volume can act as the gas-phase catalyst particles that can support the growth of the carbon nanotube-like structures in the ionized gas phase alongside with the most commonly adopted scenario of the CNT assembly on rough catalyzed surfaces. Kinetic theory of the carbon nanotube nucleation from graphitic nanofragments [Louchev and Hester (2003)l also supports this possibility. In particular, a carbon nanosheet wrapping around amorphous carbon,

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fullerene-like, and carbonized metal catalyst nanoparticles that are present in the gas, can be regarded as the first step in the CNT nucleation process. The proposed mechanism of CNT nucleation thus invokes the nanosheetnanoparticle collision pathway, which is an alternative to the conventional carbon segregation pathway. Certainly, there is a room for speculations whether this mechanism is applicable or not for the PECVD systems witth the plasma-grown nanoparticles, which is apparently an interesting but yet unresolved problem.

Fig. 7.6 FESEM image of ordered carbon nanotip arrays for electron field emitting applications synthesized in low-pressure inductively coupled plasmas of CH4+H2 +Ar gas mixture on Ni-based catalyzed silicon substrates. Courtesy of Z. L Tsakadze, K. Ostrikov, and S. Xu (unpublished).

Another issue in the PECVD of carbon-based nanostructures (CNSs) is the actual role of the gas-phase grown nanoparticles discussed in detail in Sec. 7.1.1. A relevant observation from the growth of the CNSs for electron field emitting applications (a representative ordered carbon nanostructure is shown in Fig. 7.6) in the CH4+HZ+Ar [Denysenko et al. (2003b)l and other CH4-based reactive chemistries [Meyyappan et al. (2003); Hash and Meyyappan (2003); Hash et al. (2003)l is that the CNSs growth is accompanied by a very strong methane conversion (the gas feedstock dissociation) of up to 95% and even higher. There is direct evidence that in the dust-growth regimes, the methane dissociation can be enhanced by a factor of 6-8 as compared with the pristine plasma a t the same operating conditions such as the pressure and the RF power [Boufendi and

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Bouchoule (2002)l. The reason behind this is the rise of the electron temperature due to the dust generation (discussed in detail in Sec. 7.1.2), which increases the rates of dissociation of methane molecules. We emphasize that this effect is a notable illustration of the enhanced chemical efficiency of dust-contaminated plasmas [Boufendi and Bouchoule (ZOOZ)]. It is also interesting to point out that in many conventional thermal CVD processes the feedstock gas does not dissociate in the gas phase even at the temperatures commonly used for the single- and multiwalled carbon nanotube growth ( w 800-10OO0C), and the nanotube production is entirely due to surface reactions of the CH4 molecules on the catalyst surfaces [Hash and Meyyappan (2003)]. In contrast, significant amounts of acetylene, ethylene, a variety of C,H, radical and cation species, all of which contributing to the carbon nanotube assembly, are generated in low-temperature plasma reactors [Meyyappan et al. (2003); Hash and Meyyappan (2003)l. We note that PECVD of the carbon nanostructures is a complex process that involves numerous transformations of neutral and charged radical species both in the gas phase and on the surfaces of catalyzed substrates. A very intersting observation that favors the possible role of the dust grains grown in the ionized gas phase in the nanoassembly is that large amounts of the acetylene monomer C2Hz are generated both during PECVD of CNSs [Denysenko et al. (2003b); Meyyappan et al. (2003); Hash and Meyyappan (2003); Hash et al. (2003); Tsakadze et al. (2005)l and the dust growth in Ar+CH4 plasmas [Hong et ul. (2003)l. Indeed, it appears that the concentration of the C2Hz species is an important factor for the initiation of the dust formation [Descheneaux et al. (1999); Stoykov et al. (2001)]. Furthermore, in saturated and unsaturated hydrocarbon plasmas the polymerization process is triggered and proceeds at a higher rate after detection of large enough acetylene concentrations [Hong et ul. (2003)I. For further details of the dust grain nucleation process in the argon-methane and argon-acetylene plasmas refer to Sec. 7.1.1.2 and the original works [Kovacevic et al. (2003); Hong et al. (2003)l. One of the critical problems in the kinetics of the plasma-based growth of various CNSs is whether the gas-phase-grown carbon nanoparticles participate in the nucleation process as catalyst particles, nano-sized building blocks of the CNSs, valuable morphology elements of the nanostructured carbon films, or are just deleterious contaminants that can compromise

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the entire CNS assembly process. The answer to this question still remains open and will certainly depend on specific experimental conditions, including the gas feedstock composition, the RF power and gas pressure ranges, the substrate heating and biasing conditions, etc. Nevertheless, one of the clues for the solution of the above problem is the fact that the CNSs grown by the PECVD methods are mostly aligned vertically due to the near-substrate electrostatic fields [Hofmann et al. (2003)], which is a preferential direction for their growth. Furthermore, the electrostatic fields near sharp tips of many CNSs (e.g., CNTs) certainly modify the deposition conditions of most of the charged particles in the plasma (including the radical and ion species, and fine powder particles) leading to the selective deposition of them onto the CNS tips rather than the lateral surfaces and the inter-structure “valleys” [Tanemura et al. (200l)l. For example, for a carbon nanofiber with the radius of rg = 20 nm protruding from a biased (K = -500 V) substrate surface, the field strength at the tip F,, can be estimated approximately 5 x lo7 V/cm from a simple calculation widely used for the CNT field emitter nanotips F,, = V s / a g ~ O where , the geometrical factor ag = 5 [Tanemura et al. (200l)l. The presence of strong electrostatic fields near the surface of CNTs does require the understanding of the capacitive issues of the carbon nanotube-based systems and their ability to charge-up, store electric charge and dynamically respond to external variable fields [Pomorski et al. (2003)l. This is very important for a number of industrial applications of CNTs as scanning probes, non-volatile memory cells, and nanoelectronic devices. In this sense the notion of the electrostatic charging of carbon nanotube-like structures is very similar to that of dust grains considered in Sec. 2.1. It is important to note that surface-based methods of CNT growth usually require substrate temperatures of the order of 600-800°C. Thus, accurate control of the gas (and hence, the catalyst particle surface) temperature by varying the RF/microwave power could be instrumental in the synthesis of nanostructured carbon powder material in the ionized gas phase by using the intrinsic ability of low-temperature plasmas t o levitate colloidal particles in the near-electrode areas. Present and future challenges in the direction of the plasma-based production of various carbon nanostructures are discussed in a recent review [Meyyappan et al. (2003)] (see also references therein). Here, we would pose one more important question: what is the role of the plasma-nucleated nanoparticles and plasma polymerization processes in the assembly of various CNSs? This apparent puzzle is a further argument for the need of intensive theoretical and experimental

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research in the area in the coming years. Finally, we believe that research in this direction should certainly continue in the near future to demonstrate the outstanding potential of the complex plasma systems in the fabrication of various nanostructured materials.

7.3.2

Nanopowders in plasma-enhanced chemical vapor deposition of nanostrmctured silicon-based films

Design of efficient PECVD reactors for the coating of large-area glass plates (- 1 m2) in the flat panel display or solar cell manufacturing technologies raises a number of issues in physics, chemistry, materials science, mechanical, thermal, and electrical engineering. In such reactive discharge plasma glows, excited at the RF frequency from 13.56 MHz to 100 MHz, the uniformity of the thin film deposition is controlled by the gas flow distribution, as well as the local plasma perturbations, and other factors. Moreover, the film properties critically depend on the plasma chemistry involving formation and trapping of fine particles triggered by the homogeneous nucleation, the neutral radicals contributing to the film growth, the effect of ion bombardment, and other processes [Perrin et al. (2000)]. In particular, plasma assisted CVD of the amorphous silicon (a-Si) films is one of the most advanced methods of fabrication of thin film transistors, flat panel displays, solar cells, and other opto-electronic devices [Bruno et al. (1995)l. However, the characteristics of the films, and hence the deposition techniques are to be continuously upgraded in line with the increasing demands of the emerging optoelectronic industry. It is interesting that several parameters (including the transport and stability) of the a-Si films prepared in the powder-generating regime can be significantly improved as compared to those achievable in other regimes [Nienhuis et al. (1997); Hadjadj et al. (2000); Martins et al. (Zoola)]. It is apparent that this improvement can be attributed to silicon-based nanoparticles nucleated and grown in the chemically active environment of SiH4 plasmas. Conventional thin film characterization routines confirm that the films grown in the powder-generating regime have a nano-scaled structure featuring ordered arrays of silicon nanocrystals [Martins et al. (Zoola)]. Therefore, the efficient fine powder control is crucial for the development of viable methods of the nanostructured film deposition. Nanostructured Si-based films can be prepared by the PECVD in highly hydrogen-diluted silane discharges sustained a t higher working pressures that promote the low ion bombardment conditions for the better surface

-

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activation and/or passivation [Martins et al. (2001b)l. In this way, highquality nanostructured silicon films with low density of states and high mobility-lifetime product can be fabricated. Presumably, the film growth ’proceeds via the gas-phase nanopowder nucleation and subsequent embedding of fine particles in the amorphous silicon matrix. It is worth mentioning that since in the low-pressure plasma glows the dust particles usually acquire the negative electric charge (see Sec. 2.1), one can predominantly deposit either positive ions/radicals or anions/dust particles by reversing polarity of the substrate bias. Specifically, the positive bias favors embedding of the gas phase-grown grains into the matrix on the substrate, while the negative one strongly impedes this process. The films fabricated under conditions of the positive bias feature a higher density, smoother surface morphology, more compact arrangement of the grains over the surface, as compared to the negative DC bias case. It is remarkable that when the bias is negative, the films have a pronounced porous structure. Moreover, the elements of the surface morphology appear to be distributed randomly over the surface. Above all, the positive bias promotes clearly higher deposition rates. From the device quality point of view, the incorporation of the siliconbased nanoclusters widens the optical bandgap as well as lowers the density of states of the film material. Therefore, the film fabrication process can be remarkably improved by the nano-sized particles grown in the plasma and capable of embedding into the films being grown. It is thus imperative to continue the investigation into the underlying mechanisms of the dust-substrate interactions and the growth kinetics of the nanoparticleincorporated films. We emphasize that the nanostructured hydrogenated silicon films (nsSi:H), also frequently termed in the literature as polymorphous. (pm-Si:H), feature a high degree of the crystallinity due to the incorporation of nanosized (typically in the few nanometers range) crystallites grown in the ionized gas phase [Viera et al. (2002)l. The term “polymorphous” applies to silicon-based nanomaterials that consist of a two-phase mixture of the ordered and amorphous silicon material [Viera et al. (2002)l. A nanostructured amorphous a-Si:H matrix with the embedded silicon-based nanocrystallites grown in the plasma, is a typical example of a polymorphous thin film. To control the deposition rate and ensure the device quality of the nanostructured Si films, it appears instrumental to pre-set the plasma conditions to the powder-generating regime, which enables a simultaneous

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deposition of the plasma radicals that contribute to the formation of the amorphous matrix and the ordered plasma-grown nanoparticles that act as the nanometer-sized highly-crystalline “dopant” particles. Furthermore, the resulting crystalline structure appears quite different from the diamondlike structure of the bulk silicon [Viera et al. (2002)l. However, the fact that the Si nanoparticles appear as crystallites with a specific (usually fcc) crystalline structure under high argon dilution of the silicon carrier gas SiH4 still warrants the adequate explanation. It should be noted that in the presence of powder particles in the discharge, freestanding nanoparticles can be grown alongside with nanostructured films. It is quite common that the efficiency of nanopowder growth decreases with stronger dilution of silane in hydrogen. Other factors such as the deposition temperature, gas pressure and RF power can be kept at either low [Bertran et al. (1998)] or high [Roca i Cabarrocas et al. (1998)I levels. Meanwhile, device-grade pm-Si:H films can be deposited from square wave modulated (SWM) RF plasmas in the presence of powder particles in the ionized gas phase [Viera et al. (200l)l. Plasma modulation and gas temperature can be changed to control the powder development pathway [Viera et al. (2001)l. The square wave modulation of the RF plasma can control the selective incorporation of nanoparticles into the film being grown. During the plasma-on time of the modulation cycle, an amorphous Si film is deposited onto the substrate and, at the same time, nanoparticles nucleate and grow in the gas phase. During the afterglow periods, the dust grains leave the plasma and are deposited onto the amorphous Si film. Therefore, after a number of cycles, the nanostructured film contains numerous silicon nanoparticles embedded in the a-Si:H matrix [Viera et al. (200l)l. On the other hand, freestanding nanocrystalline Si particles can also be generated [Roca i Cabarrocas et al. (1996); Viera et al. (1999)l. Typically, the nucleation of freestanding nanoparticles requires high silane dilution in argon, moderate pressures, and higher RF powers. A typical nanocrystal size is 2-10 nm, whereas larger particles appear to be either amorphous or polymorphous. The intermediate pressure (500-800 mTorr) inductively coupled plasmas can also be used to synthesize single-crystal silicon nanoparticles with diameters between 20 and 80 nm [Bapat et al. (2003)l. Such nanoparticles are suitable as building blocks for various single-nanoparticle electronic devices. We can infer the following common features of the silane-based nanoparticle growth. First, the particle concentration is roughly proportional to the

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silane partial pressure. Secondly, highly oriented single-crystal particles are favored by longer plasma-on times, higher power density, and higher total gas pressure. And finally, fractal agglomerates, or even dendric or cauliflower structure particles are found at low discharge pressures and low input powers [Bapat et al. (2003)l. Meanwhile, the deposition rates can be efficiently controlled by enhancing or inhibiting the dust growth in powder-generating (such as silane-based) discharges. A model for high-rate film deposition from dusty dichlorosilane (SiH2ClQ) and tetrafluorosilane (SiF4) RF discharges considers the particulate generation as a multi-step process: generation of negative ions by electron attachment; clustering of negative ions; and growth of clusters by parent molecule addition [Tews et al. (1997)l. In this way the new growth techniques of amorphous, microcrystalline and polycrystalline thin films, with remarkably higher deposition rates, can be developed. It is interesting that the inlet of SiH2Cl2 feedstock correlates with the higher generation rate of dust precursors and is accompanied by an increase of the electron temperature presumably due to the loss of the low-energy electrons via the electron attachment (leading to an increase of the negative ion density) to the high-affinity chlorinated species. Likewise, small additions of dichlorosilane can be used to improve the overall film quality. Physically, termination of active bonds on the surface becomes more efficient once chlorine atoms become able to participate in this process alongside with hydrogen atoms. Hence, diffusion of film precursors such as reactive SiH3 species over the surface can be enhanced, which further results in the improved surface morphology and the film quality. Discussion of the film precursors, binding and surface diffusion of various radicals, surface roughness evolution, and other aspects of hydrogenated amorphous silicon and other silicon-based films is given by [Perrin et al. (1998); Dewarrat et al. (2003); Smets et al. (2003)I. On the other hand, a-Si:H films deposited at high deposition rates (which is usually achieved when the number densities of higher silane species Si,H, actively taking part in dust polymerization are high) often suffer from the pronounced photo-induced degradation. It is interesting that in this case the contribution of the Si,H, species is intimately related to the electron temperature in the discharge [Takai et al. (2000)l. Likewise, the photostability of a-Si:H films can be drastically improved when the contribution of the Si,H, species t o the film growth is low [Takai et al. (2000)l. From this point of view, a reduction of the electron temperature in the plasma during the film growth appears to be a key issue in the improve-

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ment of the photoinduced degradation properties in the high-rate grown amorphous silicon films. Under relevant experimental conditions, the electron temperature is intimately related to the dust size and number density. The results of computation of the equilibrium electron temperature for the capacitively coupled plasmas of the RF hydrogen-diluted silane discharge [Hori et al. (2001)l are displayed in Fig. 7.7. The computation parameters are: the gas pressure po = 300 mTorr, the neutral gas/substrate temperature Tn = 150°C, the electrode radius and spacing of R = 13 cm and L = 1.7 cm (parallel plate geometry), respectively; the total number density of the positive ions Eni = 10" ~ m - the ~ ; dust number density n d = lo7 - 1.1x lo8 cm-3 and the dust radius a = 50 nm. The equilibrium nanoparticle charge has been obtained from the balance of the microscopic electron and total ion currents onto the dust particles and in this particular example ranges from 76 to 83 electron charges.

Fig. 7.7 Electron temperature vs dust charge proportion [Hori et al. (ZOOl)] for the representative parameters of the experiments [Takai et al. (2000)l.

From Fig. 7.7 one can see that an increase in the charge proportion on dust particles
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(2003a)l shows that the dust strongly affects the electric field distribution through its charge and the plasma density through the recombination of the positive ions and electrons at the grain surfaces. Furthermore, the presence of dust enhances the deposition rate of the amorphous silicon because of the rise of the average electron temperature that affects the rates of the species production and near-substrate sheath potentials. Another interesting aspect is that 30-50 nm-sized nanocrystals (identified by HR TEM) can be grown in a dual-frequency PECVD process of silicon oxides in silane-based plasmas [Lin et al. (2000)l. In this case siliconbased crystalline nanograins are usually formed inside the silicon-rich oxides. We recall here that the silicon oxide nanoparticle growth dynamics can be quite different as compared to pure and hydrogenated silicon. Relevant discussion can be found in Sec. 7.1.1.1.

Fig. 7.8 Dependence of R, on cluster amount. Experimental conditions: SiH4 (100%), 30 sccm, 9.3 Pa, 13.56-60 MHz, 0.018-0.054 W/cm2 and substrate temperature 523 K. Cps means counts per second; film thickness is 600 nm [Shiratani et al. (2003)l.

Apparently, the controversy on whether the fine particles are favorable or deleterious for the fabrication of the hydrogenated amorphous silicon films is definitely not yet resolved and a number of fascinating results is expected in the near future. For example, high-quality a-Si:H films can be grown under either cluster suppressing [Watanabe et al. (2002)] or powdergenerating [Viera et al. (2002)l regimes. Nevertheless, it is certainly clear that fine powder particles do play an important role in thin film fabrication processes in silane-based (and other chemically active) discharges. On one

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hand, near the threshold of the powder generating (7)regime, dust particles originate and directly contribute to the high-rate film growth. On the other hand, higher silanes SizH, can compromise the photostability properties of a-Si:H films. What is certainly clear is that nano-sized clusters and solid particles generated in the ionized gas phase of silane-based discharges can affect various properties of the PECVD-synthesized silicon films. For example, there is a direct correlation between the amount of nanoclusters in silane plasmas and the microstructure parameter R, of amorphous hydrogenated silicon films [Shiratani et al. (2OO3)I. In particular, Fig. 7.8 shows that larger amounts of nanoclusters in the discharge correspond to more rough film surfaces (higher R,). It is interesting that the films represented in Fig. 7.8 feature microstructure parameters lower than 0.1, which is a conventional device quality limit. One can also notice from Fig. 7.8 that the actually achieved R, can be made of the order of 0.01 by suppressing the nanocluster growth [Shiratani et al. (2003)l. It is noteworthy that particulate formation usually elevates the effective electron temperature, which in turn affects the rates of the negative ion formation, the latter (e.g., SHY) being precursors for dust polymerization in reactive plasma (see Sec. 7.1). Thus, the complex plasma system becomes dynamically and self-consistently coupled and further self-organizes into the states with higher electron temperatures, which affects t,he powder growth conditions. Therefore, chemically active plasmas do offer a great variety of options for the thin film deposition. Specifically, one can deposit the films by predominantly using the atomic/radical units, which is usually the case in the regimes inhibiting a significant dust growth. Alternatively, it appears possible t,o tailor the film deposition process in the dust-generating and high-deposition-rate regime.

7.3.3

High-rate cluster and particulate deposition o n nanostructured surfaces: a new paradigm in thin film fabrication

Here, we discuss a new paradigm in the assembly of various nanostructured films that invokes a concept of the high-rate nanocluster/particle deposition. In particular, the Charged Cluster Model (CCM) [Jang and Hwang (1998)l can explain various observations in the diamond chemical vapor deposition (CVD) process, where charged diamond clusters are usually suspended in the gas phase during the diamond-like film deposition.

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According to the CCM, a large number of the commercially available thin films are actually cluster-assemled ones even though they have been believed to be grown by the atomic unit. The idea emerges from the study of the thermodynamic driving forces for the charge-induced nucleation of the (negatively charged) carbon-atom clusters in the process of diamond synthesis. In this case the short-range ion-induced dipole interarkion a,nd the electrostatic effects are mainly responsible for the unusual nucleation phenomenon. The Charged Cluster Model is quite generic and can thus be applied to virtually any process (e.g., in the ionized gas phase or on the substrate surface) that involves the charge-induced nucleation (see Secs. 7.1.1 and 7.1.2). The nanometer-sized charged carbon clusters nucleate in the gas phase of the diamond chemical vapor hot-filament deposition process [Hwang (1999)I. Similar processes are also common in the field of combustion and flame, where the charged carbon clusters are produced in the gas phase and are regarded as major precursors of the graphitic soot. Confirmation of the existence of charged carbon clusters in the gas phase ceratinly favors the hypothesis that CVD diamond is grown by the nanocluster rather than atomic/molecular unit. Molecular dynamics (MD) simulation can be ,used to examine the possibility of thin film growth by nano-sized clusters as buildig units. In the process of deposition of various-sized (yet in the nanometer range) Au clusters on the Au(001) surface that the epitaxial recrystallization of the clusters on the substrate surface is the best for the small (in the nanometer range) clusters containing a few hundred atoms [Lee et al. (200l)l. The epitaxial recrystallization behavior of small and large clusters suggests the possibility of the thin film growth by a cluster unit as suggested by the CCM. Therefore, according to the new, cluster-unit deposition paradigm of the thin film growth, the success of the epitaxial growth of the actual production line in the microelectronics industry critically depends on the cluster size. A supporting evidence for the CCM in the diamond CVD process is that less stable diamond phase frequently grows with a simultaneous etching of more stable graphite, which would violate the second law of thermodynamics if the deposition unit were an atom [Lee et al. (2001)l. This seemingly paradoxial phenomenon could be explained successfully by assuming that the gas-phase nucleation of the diamond clusters is followed by their subsequent deposition to form diamond-like films [Lin (2000)l. Efficient nucleation processes of silicon nanowires in reactive silane/hydrogen/hydroclorine e atmospheres can also be explained on a thermodynamically sound basis by the

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Fig. 7.9 SEM photographs of diamond films deposited in situ during the measurement of mass distribution of charged clusters at 6 Torr with (a) 1%CH4+99% H2 and (b) 3% CH4+97% H2. (a) Clusters of narrow size distribution containing mainly a few hundred atoms produce high quality diamond crystals with well-defined facets while (b) clusters of broad size distribution containing up to more than a thousand atoms produce poor quality diamonds with a cauliflower-shape structure [Jeon et al. (2001)].

cluster-unit deposition and the atomic/radical-unit etching [Hwang (2000)l. The CCM theory has two basic assumptions [Hwang (2000); Jang and Hwang (1998)], related to the spontaneous formation of charged clusters in the gas phase and their role as building units of various thin films. The charge of nanoclusters has two major implications on the gas-cluster-film system. First, the charged clusters maintain the nanometer size due to remarkably suppressed Brownian coagulation, which frequently leads t o skeletal or fractal-shaped microparticle formation. Secondly, the charge enhances the diffusivity of the atoms in the clusters and induces a selective landing of charged clusters on the nanostructured surface, which can further result in the efficient self-assembly packing. The cluster size and charge would be a decisive factor for the grain size control of thin films,

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which is a completely new paradigm of thin film growth and microstructure evolution. For example, by using size-monodisperso (usually negatively charged) clusters containing a few hundered carbon atoms, one can obtain a perfect diamond micro-crystalline structure shown in Fig. 7.9(a). On the other hand, poor quality (cauliflower-shaped) diamonds nucleate from larger (> 1000 atoms) clusters as seen in Fig. 7.9(b). It is thus quite probable that the carbon nanotube-like structures (discussed in Sec. 7.3.1) might also evolve as a result of the growth by charged carbon cluster units. In the deposition of silicon nanowires in the SiH4+HCl+HZ gas mixtures [Hwang (2000)], a highly anisotropic nanostructure growth can be attributed to the electric charge of the gas-phase grown nanoclusters. Indeed, if the clusters were not electrically charged, they would be subject to fast Brownian coagulation in the gas phase, leading to the porous fractal and skeletal structures. Thus, the highly anisotropic growth of nanowires is not expected from neutral clusters. Furthermore, it is quite common that the clusters land relatively easily on conducting surfaces but have difficulty in landig on insulating surfaces, which leads to their selective deposition. The selective cluster deposition results in the growth of structurally and otherwise different films. For example, under conditions of simultaneous deposition in the same reactor, diamond crystals (similar t o those shown in Fig. 7.9(a)) grow on a silicon substrate whereas carbonaceous soot-like nanoparticles (with a similar shape as in Fig. 7.9(b)) appear on a Fe substrate. The key to the highly anisotropic growth puzzle is the electrostatic interaction between the charged clusters and the nanostructures being grown. Indeed, the two conducting spherical particles carrying charges of the same sign interact with the interaction force [Jackson (1967); Hwang (2000)l

(7.6) where r j and q j are the radii and the electric charges of the spheres, and d, is the distance between their centers. From the above equation, one can conclude that if the size difference between the two conducting charged particles is small, the interaction is repulsive. However, if the size difference is large, both particles can attract to each other. Elongated particles of the cylindrical shape or nanowires typically have a small dimension in the radial direction but larger sizes in the axial diection. Therefore, small clusters might be subject to the electrostatic attraction from the nanostructure

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in the axial direction and at the same time be repelled from the structure radially. Thus, if a negatively charged conducting particle approaches axially to the nanowire/nanotip, the electrons in the nanostructure will move along the axial direction away from the approaching (negatively charged) particle. However, if the same particle approaches in the radial direction, the electrons in the nanowire/nanotip cannot move a long distance away from the approaching particle because of the limited size in the radial direction. It is also likely that once an elongated shape is initiated, a highly anisotropic growth will further accelerate to promote a pronounced growth of nanowires. Higher gas pressures can result in an enhanced gas-phase precipitation, which normally increases the cluster size. According to Eq. (7.6) the selectivity of the charged cluster deposition is weaker for larger clusters. Size distributions of clusters/nanoparticles suspended in the gas phase of various CVD (including PECVD) reactors can be studied, for example, by the differential mobility analyzer (DMA) and a particle number counter techniques [Reist (1993); Adachi e t al. (1996)l. Remarkably, in the process of CVD of silica films using tetraethylorthosilicate (TEOS) as a precursor, the films do not grow under conditions where clusters are not detected in the gas phase [Adachi et al. (1996)l. Note that various nanostrutured materials can be prepared by the low-energy deposition of gas-phase preformed clusters. The composition of the clusters can be various, e.g., noble metal clusters, transition metal-based clusters, silicon, carbon-based clusters, etc. [Perez e t al. (200l)l. Complex compounds can also nucleate in the gas phase. For instance, A1N nanocrystals can be grown in the gas phase of the AlClS+NH3+Nz CVD system via a homogeneous nucleation similar to that in silane-based discharges (see Sec. 7.1.1). Furthermore, A1N thin films can also be grown through the intermediate stage of the gas-phase powder formation [Wu e t al. (2000)l. Meanwhile, silicon-based nanocrystals (powder precursors) can be tailored at room temperktures in the gas phase of silane-based plasmas of square-wave modulated discharges. This result further encourages the investigation of the plasma/film parameters near the onset of fine powder generation [Roca i Cabarrocas (2000); Roca i Cabarrocas (2001); Chaabane et al. (2003)l. The ultimate aim of research in this direction is to deposit nanostructured silicon-based films under discharge conditions favoring the fine powder generation. In this case the silicon clusters, nanoparticles, and crystallites are formed in the plasma and contribute to the actual film deposition process. From the previous sections we recall that the plasma-grown nanocrystallites and/or clusters play a pivotal role in the deposition of the pm-Si:H

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Physics and Applications of Complex Plasmas

(polymorphous silicon) films [Roca i Cabarrocas (2000); Roca i Cabarrocas (200l)l. Therefore, tailoring the size and the number density of the nano-sized Si-based particles in the plasma opens up a new horison of the dust-involving processes in the nanoelectronic technologies including the passivation, surface modification and coating of fine particles followed by their integration into devices as valuable functional elements [Roca i Cabarrocas (2001)]. It is also interesting that the plasma-grown nanoparticles can be intentionally moved to or from the substrate, e.g., by using the thermophoretic force (see Sec. 2.1) [Fontcuberta and Roca (2001); Chaabane et al. (2003)I. Thermal low-temperature plasmas can also be used to produce hypersonic flows of ultrafine gas-phase nucleated micro-/nano-sized particles, which can be supersonically sprayed onto a temperature-controlled substrate to produce nanostructured materials with the desired properties [Rao et al. (1997); Di Fonzo et al. (2000)]. In this way, one can also fabricate nanocrystalline microstructures. The nanoparticles can be synthesized in a thermal plasma expanded through a nozzle, and then focused to a collimated beam. This approach is instrumental in the fabrication of high-aspect-ratio structures of silicon carbide and titanium and allows size selection of the particles that are deposited [Di Fonzo et al. (2000)l. Various nanostructured films can be tailored by using the cluster-unit deposition schemes. For example, it appears possible to control the film morphology and the structure of the cluster-assembled carbon at nano- and mesoscales. To do this, the cluster beam deposition method of carbon clusters onto Ag(100) and Si(100) surfaces can be used [Magnano et al. (2003)l. The choice of Ag and Si as substrates is motivated by the expected higher mobility of carbon clusters on conducting substrates as compared to dielectric surfaces, in accordance with the selective deposition effect discussed above. Another interesting example is the deposition of size-sele'cted clusters, which represents a new route for the fabrication of various nanometer-scale surface architectures, e.g., nanopores. For example, size-selected A u ~Ag7, , and Si7 clusters can be implanted into a model graphite substrate [Magnano et al. (2003)l. Interaction of the nanoclusters with the surfaces being processed shows a very strong size-dependent character. In particular, the Co nanoclusters can undergo a full contact epitaxy on landing or burrow into the Cu surface [Frantz and Nordlund (2003)]. Nanoclusters larger than a few nanometers in radius show a different, as compared to subnanometer-

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sized clusters, behavior due to the higher structural integrity. This is a further illustration of the nanocluster-based film fabrication pathway. To conclude this section, we remark that the CVD growth of various nanostructures involves a large number of physical and chemical phenomena occuring at various length scales that sometimes differ by a few orders of magnitude. For example, the global particle/power balance, the distributions of particle number densities and temperatures, usually develop over the length scales comparable to the reactor dimensions. On the other hand, the elementary gas-phase reactions or the atomic/cluster interactions with the surface involve processes at the atomic/molecular length scales. Hence, the modeling of cluster unit-based synthesis of advanced materials would require a multiscale approach [Masi (2001)l.

7.3.4 Particle size as a key factor in nano-scale technologies In this section, we discuss several examples showing that a particle size can indeed be a critical factor in several important applications. This sizedependent feature is the most impressive when the particle size is in the nanometer range. In the first example we address nano-sized particles for various optical applications. It is commonly known nowadays that spectacular changes happen when nanocrystals of various materials shrink in size. When the crystal size becomes smaller than the wavelength of the visible light, the coherence of the light scattered by the surface changes, which results in changes of the color. For instance, metals lose their metallic luster and change color usually to a yellow-brown hue. On the other hand, white crystals of semiconductors ZnO and Ti02 become increasingly colorless as the crystals shrink in size to below 15 nm [Mulvaney (200l)l. Another important effect is that surface effects increasingly perturb the periodicity of the “infinite” lattice of a regular crystal. These changes are known collectively as quantum size effects and typically occur in the 1-10 nm range. In the case of metals, their thermal and optical properties are determined by the electron mean free path, which typically ranges from 5 to 50 nm for most metals. If the crystal becomes comparable in size, the electrons are then scattered off the surface, which apparently increases the resistivity of the nanoparticles. Furthermore, if the grains become small enough, the conduction and valence bands then break into discrete levels. For semiconductors, the bandgap widens when the crystal size diminishes.

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Hence, pronounced molecular properties can emerge and cause the changes in the crystal color and luminescence [Mulvaney (2001)]. It is common nowadays that nanocrystals of almost any material feature various unique physical and optical properties primarily controlled by the nanocrystal size and shape. For example, gold can dramatically change color with the nanoparticle size. We emphasize here that gold nanoparticles with different size can be regarded as building units for the assembly of bulk gold material. It is a common knowledge that bulk gold is normally yellow. Smaller gold particles, 30-500 nm in size, appear blue to purple and red in color. This is largely attributed to geometric light scattering effects that can be described by the Mie theory (which is used for the detection of fine particles, see Sec. 3.4 for details). The subsequent changes of color from the reddish-blue to orange and even colorless happen when the particle size shrinks from 30 nm to 1 nm and are definitely attributed to strong quantum size effects. This opens up a totally new principle for the fabrication of bulk films using building blocks of different size and architecture (see also Sec. 7.3.3). For example, one can control the size and spacing between the building blocks, thus changing the interaction between them. Hence, one could assemble an object with tailored optical (and other) properties anywhere in between those of a nanoparticle and those of a bulk material. We note that modification of the optical absorption properties are presumably due to changes in the surface absorption band with the grain size. The surface plasmon frequency of nanocrystals changes drastically with the particle shape (e.g., rod-like or ellipsoidal/spherical) due to the changes in the restoring force on conduction electrons that are extremely sensitive to the particle curvature [Mulvaney (2001)]. For example, the polarzzabzlity of a rod-like gold particle is 4abc(&~,- E m ) Qx,y,t = 3Em

+ ~LX,Y,Z(EAU

-Em)

’

where a , b, and c ( a > b = c ) refer t o the geometrical sizes of the rod, E A and E~ are the dielectric constants of the gold and optical medium, respectively, and Lx,y,zis the depolarization factor for the respective axis. Metal nanoparticles are interesting from the application point of view because of the high electrical capacitance and because the color of the nanoparticle colloid is affected by the stored electric charge on the particles. For the nanoparticles/clusters of an arbitrary shape one should use a general definition of the polarizability tensor = a<,/aEj, where <, and Eg are

~

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the Cartesian components of the dipole moment and the applied electric field, respectively. For example, the polarizability of two-dimensional gold clusters with an arbitrary size and shape in the normal direction can be estimated using the density functional method and the fitting formulas [Zhao e t al. (2003)I. Thus, special attention should be paid to charge confinement and quantum size effects in metal nanoparticles and nanoclusters. In particular, by changing the overall size of a metal nanocluster, one can control the electron storage and redistribution of the electron population between the inner and outer populations [Hoshino et al. (2001)]. For example, the storage or release of electrons is allowed within f 2 e for the cluster consisting of 169 A1 atoms. Furthermore, the contribution of the atoms located in outer shells of the cluster in the overall cluster charging is the most important. Note that using the first-principle semi-empirical molecular orbital calculations one can estimate the effective capacitance of a 3.8 nm metal cluster of the order of 0.15 aF [Hoshino et al. (200l)l. Other examples of calculations of the electron energy levels and the electronic properties of nano-sized objects can be found elsewhere [Bednarek e t al. (2001); Li e t al. (2001a); Li et al. (2001b); Niculescu (2001)]. In the nanometer range, the electronic levels, the bonding states, and other important characteristics of the solid state critically depend on the size of building blocks. The latter can be numerous, e.g., atomic, molecular, large polymerized molecules, atomic clusters, and nanoparticles (NPs). Since the plasma-grown solid grains are of our primary interest here, we will focus on the nanoparticle-assembled films. One of the viable examples is the layer-by-layer assembly of multilayered optical coatings made of nanoparticles such as doped and undoped CdSe, core-shell and naked CdSe, CdTe, PbS, TiO2, Z r 0 2 , Au, SiOz, MoO2, FesO4 and several others [Kotov (200l)l. For multilayers made of NPs, the nanoparticle size and layer sequence are particularly important because their optical, electronic, and magnetic properties strongly depend on the interparticle connections and therefore, their nanometer-scale organization. In particular, magnetic phenomena of the ordered layers of nanoparticles can be substantially modified by insulating particle coatings (coating of nanoparticles in low-temperature plasmas will be discussed in Sec. 7.3.5). It is worth highlighting that one can fabricate stratified or graded microstructures by using the nanoparticles of the same chemical composition and different size. Indeed, stratified assemblies of nanoparticles can be made from only one parent material by organizing NPs by size [Kotov

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(200l)l. Such organization of the layers is particularly interesting for semiconductor quantum dots displaying strong dependence of the bandgap on the nanoparticle size. Since the absolute energy of both conduction and valence bands of semiconductors is affected by the confinement of the excitonic wave function, a film with a gradual change in particle diameter should display a similarly gradual evolution of the valence- and conduction-band energies. Such assemblies are the examples of graded semiconductor films, known for their exceptional capabilities as photodetectors, bipolar transistors, waveguides, and other photonic and electronic devices. PECVD methods are commonly used to fabricate such layers. However, a relatively high process cost can be an obstacle for immediate industrial applications. In particular, using layer-by-layer assembly of CdSe nanoparticles of different size, one can fabricate a graded “nanorainbow”, in which the inner layers are made of smaller, 2-3 nm NPs with the luminescence maximum in the green (495-505 nm) and yellow (530-545 nm) wavelength ranges and larger (4-6 nm) NPs in the orange (570-585 nm) and red (605-620 nm) sides of the assembly [Kotov (2001)l. In that way, one can engineer the polarizability, the refractive index, and other parameters at the nanometer scale so that the overall interaction of an electromagnetic wave with the film can be made very much different from many conventional optically uniform materials. Nanocrystal quantum dots (QDs) are yet another interesting and irnportant size-sensitive object for nanoelectronic applications. QDs in the strong-confinement regime have the emission wavelength that is a pronounced function of the size, adding the advantage of continuous spectral tunability over a wide energy range simply by changing the size of dots [Klimov and Bawendi (2001)l. In particular, QDs are very promising for the development of a new type of lasing elements often commonly termed as quantum dot lasers. The obvious advantage of such a technology is that nanocrystal QDs can be prepared as close-packed films or incorporated with high densities into glasses or polymers. Thus, they are compatible with existing fiber-optic technologies and are useful as building blocks for bottom-up assembly of various optical devices, including optical amplifiers and lasers [Klimov and Bawendi (2001)l. Semiconductor quantum dots can also be fabricated by using plasmabased techniques. For example, by using 13.56 MHz RF magnetron cosputtering of A1 and In targets in low-pressure (-5 mTorr) Ar+N2+H2

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plasmas, one can fabricate Al,Inl-,N ordered quantum dot structures on silicon or glass substrates [Huang et al. (2004)]. The abundance 2 of A1 in the ternary compound can be efficiently controlled by varying RF power coupled to each target. A typical STEM image of the resulting quantum dot structure is shown in Fig. 7.10. The Al,Inl-,N quantum dots in Fig. 7.10 feature a highly-monodispersive size distribution in the sub-10 nm range. The energy bandgap measured through the optical absorption spectroscopy increases with the elemental presence of A1 in the compound. Furthermore, a strong blue shift of the photoluminescence can be observed as the size of quantum dots and II: decrease [Huang et al. (2004)l.

Fig. 7.10 STEM image of ordered Al,Inl-,N quantum dot structures grown by RF magnetron co-sputtering deposition in low-pressure Ar+N2+H2 plasmas, z Y 0.1 [Huang et al. (2004)].

Colloidal particles in the plasma are similar to colloidal nanoparticles chemically synthesized in a solution [Shim et al. (200l)l. Since the chemical synthesis of QDs from nanoparticle colloids offers greater control over their chemical composition, shape, and size (as compared with surface growth methods), one can expect a similar flexibility for the plasma-based methods using complex plasma colloids [Shim et al. (2001)l. In addition, colloidal NPs can be doped by another element, which can introduce extra carriers and provide an impurity center that can interact with the quantum-confined electron-hole pairs. However, the apparent modern challenge in this direction is to introduce the impurity in the core (ie., not at the surface or interface) of the particle without compromising the quality of nanocrystals (e.g., high crystallinity, well-controlled size and monodispersivity ) .

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Finally, it appears that the main challenge is the creation of new materials through the manipulation of size-dependent effects in nanocrystals and their assembly into macroscopic objects [Mulvaney (2001)]. In this regard, recent reports on the synthesis of Si-based nanoparticles with selective size, composition and structure using low-temperature RF plasmas (see relevant discussion in Sec. 7.3.2) sound optimistic. In particular, depending on the discharge conditions, various ultrafine nanoparticles based on the Si-C-N system can be synthesized in the plasma of SiH4+CH4+NH3+Ar gas mixtures [Viera et al. (1999)l. Moreover, under conditions of fast particle development, highly monodisperse SiCN powders can be generated. The above plasmas are also able to generate (before the onset of the particle coagulation) non-agglomerated amorphous clusters smaller than 10 nm in size [Viera et al. (1999)]. We thus believe that plasma-based methods are promising for the production of the nanometer-sized building blocks for the applications that require selective size and architecture of the building blocks for various nano-scale assemblies.

7.3.5

Other industrial applications of nano- and micronsized particles

Apart from the common deleterious aspect discussed in Sec. 7.2, nano- and micron-sized particles have a number of applications in the material engineering, optoelectronic, optical, petrochemical, automotive, mineral and several other industries. For example, ultra-fine particles can be efficiently incorporated into polymeric/ceramic materials to synthesize a number of advanced nanostructured materials for the applications as water repellent, protective, fire resistant, functional and other coatings. Furthermore, fine powders of 10 nm-sized particles have been widely used as catalysts for inorganic manufacturing, ultra-fine UV-absorbing additives for sunscreens and other outdoor applications. Other applications include textiles, wear-resistant ceramics, inks, pigments, toners, cosmetics, advanced nanostructured and bioactive materials, environmental remediation and pollution control, waste management, as well as various colloidal suspensions for mining, metallurgical, chemical, pharmaceutical industries, and food processing. Meanwhile, nanoparticles have recently emerged as valuable elements of several technologies aiming to tailor the materials properties at nanoscales and manufacture novel nanoparticle-assembled materials with unique optical, thermal, catalytic, mechanical, structural and other properties N

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and featuring nano-scale surface morphologies and architectures [Roco et al. (1999)]. The rapidly emerging applications of nanoparticles include nanopatterned and nanocomposite films, nanocrystalline powders and consolidated structures, sophisticated nanoparticle assemblies that represent new forms of supramolecular crystalline matter. Further potential applications include but are not limited to nano-scale inorganic synthesis, dispersions and suspensions with the controlled fluid dynamics, and nano-sized single/few-electron data storage units. It is noteworthy that PECVD methods have recently been instrumental in the fabrication of various nanostructures wherein the gas-phase grown nanoparticles play a pivotal role. A few years ago, it was discovered that generation, confinement and processing of solid grains in a plasma can be potentially beneficial for a number of appplications [Kroesen (2002)l. From Sec. 7.3.2 we recall that by embedding the nanometer-sized particles grown in silane discharges into an amorphous silicon matrix, one can substantially improve the performance of silicon solar cells. Similarly, one can develop substantially new classes of particle-seeded composite materials [Stoffels et al. (200l)l. In addition, fine particles, advertently injected into the plasma, can be trapped and subsequently coated to enhance their surface properties (e.g., for catalytic applications) [Kersten et al. (2003a)l. Plasmas can also be used to oxidize (and thus eliminate) contaminant particles in a bio-gas. In this way, it is also possible to electrically charge and control nanometer-sized soot particles in Diesel engine exhausts. This is an example of the complex plasma treatment of soot and aerosol particles for environmental remediation. We emphasize that an entirely new class of materials can be synthesized in particulate form by using the plasma-based technology. Hence, the recent interest of a number of major industries to the dusty plasma technologies is quite understandable. This interest is supported by constantly increasing research and development efforts. Thus, it becomes possible to tentatively categorize the applied dusty plasmas research into the directions of coating, surface activation, etching, modification, and separation of clustered grains in the plasma environment [Kersten et al. (2001)]. Powders produced using plasma technology possess a number of interesting and potentially useful properties, e.g., very small sizes, uniform size distribution, and chemical activity. Size, structure, and composition can be tailored for the specific application [Stoffels et al. (2001)l. Specifically, unique objects, like coated or layered grains with the desired surface struc-

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ture, color, and/or fluorescent properties can be created via the plasma processing. Normally, particle coating is a complex, multi-step process, involving various reactive chemistries and/or support by external devices. A sophisticated approach for the coating of externally injected particles in argon R F plasma involves particle charging and confinement [Kersten et al. (1998); Kersten et al. (2003a)l. The particles can be coated by means of a separate DC magnetron sputtering source. This approach capitalizes on the unique ability of the complex plasma systems for the electrostatic trapping, control of the particle residence time and position. A common arrangement of an RF plasma reactor for particle coating uses a built-in magnetron sputtering system. Particles can be dispersed externally, e.g., via a manipulator arm: The head of a magnetron electrode is usually positioned opposite the RF powered electrode and releases the sputtered metal particles. To obtain a stable electrostatic trapping in the plasma glow, a metal ring is sometimes placed on the RF electrode. This arrangement affects the plasma sheath and enables a reliable radial confinement of the particle cloud. Laser light scattering is frequently used to localize the particles in the potential trap and obtain the required spatial distributions in the cloud. To avoid any significant plasma disturbance by the magnetron sputter source, a metal grid can be placed between the magnetron and the RF plasma. In such a way, one can also trap in an almost motionless position and further etch even a single particle [W. W. Stoffels et al. (1999a)l. By using DC magnetron sputtering systems, one can deposit various (e.g., Al, Ti, Cu) metallic films on micron-sized silicon oxide particles confined in an R F discharge in argon. It is interesting to note that the surface morphology of the coating appears quite different for different metals (similar to Sec. 7.1.3). This can be attributed of different sputtering yields and wettability/adhesion of the metals to the silica particles. The deposition rates can be straightforwardly controlled by varying the power supplied to the magnetron targets. One can deposit thin amorphous carbon (a-C:H) films onto micronsized SiOz grains [Thieme et al. (2002)l. A noticeable increase in the laser scattering during the process of the particle coating by a-C:H films signals an additional generation of small dust particles. After the examination of the collected particles by the scanning electron microscopy (SEM), a rather small amount of large coated SiOz grains and a large amount of small and monodisperse carbon dust particles ( w 100 nm) can be observed. As an alternative to the sputter-coating process, one can certainly use several metal-organic gas precursors. For instance, a metal-organic pre-

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cursor AT1 has been used to deposit an alumina layer on fluorescent (e.g., barium magnesium aluminate particles often used as the blue component in tri-color fluorescent lamps) high-brightness, high-maintenance phosphors to improve their stability and adhesive properties [Kersten et al. (2003a)l. Coating of ZnO particles (the most important constituent of sunscreens) with SiO, films by injecting ZnO grains into TEOS/Ar/02 plasma is yet another viable example of the plasma processing of ultrafine powders. The ongoing fundamental research related to industrial treatment of dust grains aims to reveal the underlying surface physics and chemistry of particulate processing [Stoffelset al. (200l)l. More specifically, the nature of plasma-particle interactions in chemically active plasmas is yet to be uncovered. In this regard, particle temperature and surface heat exchange are of the major interest for the modern applied dusty plasma research. One of the issues is to quantify the coating process of nano- and micronsized grains. This is particularly suited for highly-uniform ultrathin film deposition on fine particle surfaces [Cao and Matsoukas (2002)]. In this process the size distribution and the uniformity of deposition appear to be a function of the deposition time and particle size. The results are interpreted qualitatively via the film deposition model, which assumes the linear film growth rate

+

where at = p[(2~rn,/T,)~/~ & , ~ T ~ ~ ~ / VR V Ris] ,the plasma volume in the reactor im m3, Atot is the total surface area available for deposition, kdiss is the dissociation rate constant (m3/s), ne and [Mo] are the electron density and the concentration of the hydrocarbon gas (isopropanol) at the inlet of the reactor, respectively [Cao and Matsoukas (2002)l. Here, T, is the temperature of neutrals (K), m, is the mass of the radical (kg), T,,, = V R / J is ~ the ~ ~residence ~ time of the reactive radical in the plasma reactor, Jfl,, is the flow rate of the gas inlet, and p is the material density of the film (kg/m3). We note that the linear growth model is valid as long as the variations in the total surface area available for the film deposition due to the particle growth, are negligible. However, when the combined surface area of the growing particles is comparable with the surface areas of the walls and electrodes, the varation of Atot with the particle size should be accounted for. We now discuss yet another interesting application is the plasma-based synthesis of carbon nanoparticles. Specifically, nano-sized plasma-grown

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Physics and Applications of Complex Plasmas

crystallites do play an important role in the synthesis of ultrananocrystalline diamond films, which possess many unique and fascinating properties, such as extreme hardness, high sound velocity and excellent thermal conductivity, which are quite different from any other forms of diamond [Gruen (200l)l. A common approach to synthesize nanocrystalline carbon films is the PECVD in hydrogen-diluted hydrocarbon gas precursors. l t appears that diamond nanofilms can be grown in a 2.45 MHz microwive PECVD system, using diamond nanoparticles seeded over a heated substrate (including silicon, silicon nitride, silicon carbide, silicon oxide, aluminum oxide, titanium, molybdenum, and tungsten at temperatures ranging from 400°C to 800°C) in contact with the plasma [Gruen (2001)l. Ultrananocrystalline state of diamond features a highly dense nanocrystalline structure with a crystallite size distribution that peaks at 3-5 nm. Amazingly, the nanocrystalline diamond can also be synthesized in the atmospheres of carbon-rich red giant stars by a mechanism similar to the laboratory microwave PECVD synthesis [Gruen (2001)l. Nanostructuring of carbon-based films, presumably via the plasmagrown nanoparticle incorporation, results in a substantial improvement of the field-emission properties due to electric field enhancement created by highly non-uniform electronic properties over nanometer scales [Merkulov et al. (1999)l. Diamond field emitter cathode tips with a radius of curvature of less than 5 nm show a significant improvement in the emission characteristics [Kang et al. (2001)]. Furthermore, the self-aligned gate diamond emitter diodes and triodes can be developed using the silicon-based microelectromechanical system (MEMS) processing technology to achieve totally monolitic diamond field emitter devices on silicon wafers [Kang et al. (200l)l. Plasma-modified nanoparticles also have various biomedical applications. In particular, calcium-phosphate-based [ e.g., hydrozyapatite (HA)] nanoparticulate material produced in plasma spray [Kumar et al. (2001)l or sputtering [Long et al. (2002a); Long et al. (2002b)l RF plasma deposition systems is useful for the development of functional biocompatible coatings for the enhanced intimate bone ingrowth and rapid fixation for orthopedic applications. The observed calcium phosphate particulates usually range from 10 nm to 4 pm [Holt et al. (1996)I. The resulting surface of the HA coating features well-resolved nano-sized morphology elements as can be seen in Fig. 7.11. The nano-scaled surface morphology appears to be ideal to promote the osteoblast cell growth and proliferation on the surface of the hydroxyapatite-coated thin film depicted in Fig. 7.12 [Long et al. (2004)I.

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Fig. 7.11 Representative AFM image of the nano-scale surface morphology of hydroxyapatite biocompatible coatings prepared by RF magnetron sputtering deposition [Long

et al. (2004)l.

Fig. 7.12 Osteoblast cell growth and proliferation on hydroxyapatite-coated A1-Ti-V orthopedic alloy [Long et al. (2004)].

From Fig. 7.12 one can note a very smooth surface of the coating, which is consistent with the AFM image shown in Fig. 7.11. A general trend of diminishing the particle size and increasing the particle density with R F power is consistent with the experiments on particulate growth in other plasma reactors (see Sec. 7.1) and further supports the role of

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plasma grown/modified ultra-fine particulate powders in the deposition of nanocomposite biocompatible materials. From another application point of view, calcium phosphate nanoclusters have recently been proposed as reagents to control pathological calcification and calcium flows in tissues and biological fluids exposed to or containing high concentrations of calcium [Holt et al. (1996)]. Furthermore, nanoparticles coated with polybutylcyanoacrylate can be instrumental for the development of drug delivery systems for the treatment of disseminated and aggressive brain tumors [Kreuter (200l)l. Generally speaking, plasma-based synthesis of nanoparticles is quite similar to many gas-phase processes, common from other physico-chemical fields of research. The nanoparticles, smaller than 100 nm, have many properties, which differ them from the corresponding bulk materials, and make them attractive for many new electronic, optical or magnetic applications. The applications include quantum dots, luminescent materials, gas sensors, resistors and varistors, conducting and capacitive films, hightemperature superconductors, and thermoelectrical, optical, and magnetic materials [Kruis et al. (1998)l. Other present and potential applications of the plasma-treated fine particles include, e.g., 0

0

0 0

0

0 0 0

0 0 0

treatment of soot and aerosols for environmental remediation [Kogelshats et al. (1999)l; powder particle synthesis in high-pressure and low-pressure plasmas; lighting technology; enhancement of adhesive, mechanical and protective properties of powder particles for sintering processes in the metallurgy fragmentation of powder mixtures in order to sort them [Uchida et al. (200111; improvement of thin film hardness by incorporation of nanocrystallites in hard coatings [Veprek (1999)I; coating of lubricant particles [E. Stoffels et al. (1999)I; application of tailored powder particles for chemical catalysis; functionalized fine particles for pharmaceutical and medical applications; production of color pigments for paints; improvement of anti-corrosion properties of fluorescent particles; surface modification of toner particles [Kersten et al. (1998)I;

and several others [Kersten et al. (200l)l. By using plasma-grown nanopar-

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ticles, it is also possible to produce composite coatings, where the properties of va,rious materials are combined. An example is the deposition of wearresistant self-lubricated coatings [Kersten et al. (2001)]. The idea of such coatings is based on the incorporation of small grains of lubricant (e.g., MoSa particles) into a hard matrix (e.g., TIN). When the surface is exposed to friction and wear, small amounts of lubricant are released to form a thin protective film over the surface. Such hybrid coatings are efficiently lubricated and at the same time are environmentally clean. Manufacturing of hydrophobic/hydrophillic coatings in acetylenenitrogen and fluoroalkylorthosilicate (FAS)-based reactive gas chemistries is yet another industrial example where an intensive plasma polymerization is accompanied by a pronounced nano- and micro-powder formation [Boufendi and Bouchoule (2002); Takai (2002)l. In particular, C-N hydrophilic surface coatings of aluminum are used by the refrigerating and airconditioning industries [Boufendi and Bouchoule (2002)]. Moreover, water repellency has recently been required for many types of substrates (e.g., glass, plastics, fibers, ceramics and metals) in various industrial fields such as biomedical, automotive, mining, various consumer goods and appliances, etc. For instance, ultra-high water-repellency can be achieved by using FASbased (and similar) PECVD in microwave discharges at total pressures over 40 Pa [Takai (2002)l. In this case the surface becomes rough due to the growth (presumably, plasma-based) of fine fluorinated silicon-based particles. Physically, the surface roughness is intrinsically linked to the wetting angle. The latter should be in the range of 140-180 degrees to warrant the industrial applicability as a water-repellent coating. The efficient control of the surface morphology of the deposited films, which appears to be the most important to obtain highly-transparent ultra water-repellent films, is eagerly anticipated in the near future. At the end of this section we mention that a number of high-pressure discharge systems has recently been used for the synthesis of various nanoparticles. For example, metallic titanium nanoparticles can be generated in an atmospheric pressure plasmas containing argon, hydrogen, and titanium tetrachloride (TiC14). This is an alternative to conventionally used multi-stage production of metallic titanium from titanium ores such as rutile [Murphy and Bing (1994)l. Another interesting recent advance is the highly-efficient synthesis of silicon nanoparticles in atmospheric pressure microhollow discharges in argon-silane mixtures [Sankaran et al. (2003)]. It is also interesting to note that metal nanoparticles can be synthesized by the PECVD in highly-unusual environments such as micron-sized

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channels of microchannel glass. This can be achieved, e.g., by using a metal-organic precursor ferrocene (CsH5)2-Fe and differential pumping accross the microchannels [McIlroy e t al. (2003)]. Non-equilibrium atmospheric pressure plasmas in the dielectric barrier discharge configurations can also be used for electrostatic rupture of Gramnegative E. coli and Gram-positive Bacillus subtilis bacteria [Laroussi et al. (2003)]. The electrostatic disruption mechanism for cell rupture requires that the local electrostatic tension of the cell wall overcomes its tensile strength. The electrostatic charge on the surface of bacteria originates due to elecron and ion collection currents in a manner similar to solid dust grain charging in the plasma considered in Sec. 2.1. To this end, airborne bacteria exposed to high-pressure non-equilibrium plasmas can also be regarded as microscopic “dust” particles. From the point of view of mass production, high-pressure technologies do have advantages over low-pressure ones, the latter requiring sophisticated vacuum equipment. Furthermore, the plasma route for nanoparticle production in bulk quantities would certainly pose a number of technological and cost efficiency problems. Nevertheless, low-pressure plasma-based techniques offer a better deal of process control and precision, without damage or thermal overload and are promising for the production of small amounts of nanoparticles with the specific properties tailored for the required industrial applications. 7.3.6

Concluding remarks

In the last two decades, fine solid particles in plasma processing facilities have been considered from different points of view. On one hand, they are indeed deleterious process contaminants in microelectronics. To this end, a significant progress has been achieved in the development of various methods of removal and suppression of dust in various industrial PECVD facilities. It has also been recently discovered that radioactive dust can appear in the active zone of the toroidal plasma devices and, moreover, be a serious hazard for the safety operation of the future nuclear fusion devices. Most recently, the plasma-grown solid grains have become increasingly attractive for a number of thin-film technologies including but not limited to low-temperature self-assembly of ordered nanoparticle arrays, nanocrystalline and polymorphous silicon-based thin films for optoelectronic functionalities and devices, biocompatible calcium phosphate-based

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bio-ceramics for dental surgery and orthopedic applications, hard wearresistant, self-lubricating, UV-protecting and many other functional coatings. Several other aspects of applications of ultra-fine powder particulates and larger solid grains have been discussed in this chapter. We note that despite a notable progress in industrial applications of the plasma-grown micro-/nanoparticles, some of them still remain at the research and development stage. In view of the most recent advances in the science and applications of fine powders, control and manipulation (and eventually the adequate management strategies) of the plasma-grown solid particles is becoming the matter of utmost importance. However, this aim cannot be achieved without proper understanding of the underlying physics of the basics of plasma-fine particle interactions. For this reason, the dynamics and selforganization of the particulate matter, as well as various collective processes in low-temperature complex plasma systems are crucial. Moreover, without proper understanding of the fundamentals of basic dust-plasma (and wherever applicable dust-solid substrate/wall) interactions, it is impossible to adequately and self-consistently describe the real processes in the complex plasma systems with variable-size particulate matter of complex shapes and internal organization, such as nanoclusters and complex nucleates and agglomerates. On the other hand, knowledge of the collective phenomena involving charged fine particles is important in the studies of self-organization and critical phenomena in gas discharges that affect the origin and growth of fine powder particles. One can thus conclude that “fundamental” and “applied” aspects of the problems discussed in this book are inseparably associated with each other. Specifically, the deposition of the plasma-grown building units onto micro-patterned substrates is a typical example of the problem that still requires a major conceptual advance in the near future. New approaches to the analysis of the behavior and manipulation of the complex plasma systems, based on advanced physical models of the interaction of nano-sized clusters/particulates with their ionized reactive plasma environments and nanopatterned substrate surfaces become a vital demand for the coming years. The ultimate aim of such research is to tailor the size, composition, and architecture of the gas-phase grown clusters/nanoparticles as building units for the assembly of ordered nanoparticle arrays and nanostructured materials. We expect that the physical foundations of novel ordered nano-scale assembly concepts will employ reactive plasma-grown charged nanoparticles as building units and nanostructured materials as nanopat-

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terned substrates. New control strategies for such systems are anticipated t o advance the existing approaches for the development of new classes of nanoelectronic and photonic functionalities and devices. Control of ordering and site density of nanostructures over industrially viable surface areas remains a challenging task for the coming years and we anticipate that ordered plasma-grown individual nanoparticles and complex structures can play a pivotal role in the solution of the above problem. Furthermore, the question of the fundamental physical mechanisms leading to the ordered self-assembly processes a t nano-scales still remains open. Some of the key problems in this regard are related to the formation of the fundamental building units in the ionized gas phase, and the self-organization of them into the nano-sized ordered structures either in the plasma bulk or on thc solid surface. Another challenging problem is related to technological and economical limitations in lithographic technologies using optical, electron, or X-ray beams for the next stage of sub-100 nm ultra-large scale integrated semiconductor devices. For this reason, it is expected that in the near future, the development of ULSI compatible self-organization methods for the assembly of nanostructures, which do not require artificial time and labor-consuming pattern delineation routines, will become a problem of the outmost importance. However, application of plasma-particle systems to nano-scale processes is still a t an early stage, and a number of fundamental physical problems at different dynamically coupled space and time scales are yet to be solved before any particular scheme can become industrially viable. First, the nanoassembly process cannot be reliable without precise control of composition, size, and architecture of the building units. Furthermore, the building units have to be deposited on the nanopatterned substrate in a non-destructive fashion, preserving their unique features. The optimal parameters of the ionized gas phase allowing a predictable and controllable fabrication process have yet to be established and linked to the process control parameters. However, despite a notable recent progress in the growth of nanostructures, the working parameters of many nanoassembly experiments are still often chosen by trial and error. This practice should certainly be replaced in the near future by a number of highly-predictable (via appropriate modeling and simulation) processes. The existing knowledge of cluster/particulate dynamics in reactive plasmas is mostly limited to experimental detection, size measurement of particles, and attempts to growth control. The key obstacle to self-consistent

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modeling of particle dynamics in near-substrate areas of chemically active environments lies in the appropriate treatment of plasma-particle and particle-surface interactions, involving flows of numerous reactive species onto the grain/substrate surface, secondary electrons, reactive chemical etching of the substrate surface, particle size and substrate surface morphology variation due to deposition/re-deposition of thin films, etc. The relevant existing theories do not account for the chemical interactions of reactive species and the particle/substrate surfaces, and consider essentially stationary cases. However, the stationary models for particle-plasma interaction have limited relevance to chemically active plasmas, as the balance between microscopic electron/ion/radical flows on the particle/substrate surface varies continuously. Above all, the existing models are not applicable to nano-sized objects comparable to large molecules, when the electron confinement effects come into play. Thus, the complexity of the problem requires major conceptual advances in the description of the charging, dynamics, and interaction of nano-sized particles with nano-sized objects. Therefore, novel methodologies that will eliminate the existing conceptual shortcomings outlined above and will advance the existing knowledge on the complex self-organized systems involving reactive plasmas with nano-sized colloidal particles and nanostructured substrates targeted for ordered nanoassembly processes are highly warranted in the near future. From the experimental point of view, real-time detection, number density, size, and composition measurements still require greater precision and reliability to be suitable for immediate industrial applications. Thus, diagnostics of complex nanoparticle-loaded plasma systems should evolve into real-time, in situ monitoring of dynamic variations of grain size and elemental composition, alongside with real-time diagnostics of the plasma species and other plasma parameters.

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Conclusions

Here, numerous physical phenomena in complex ionized gas systems have been considered. We have presented a variety of problems related to solid dust particles in laboratory and industrial low-temperature plasmas. The peculiarity of complex plasma systems is related to the unique capacity of dust particles to self-organize themselves into ordered structures. Individual dust grains, particulate clouds and more complex structures can be confined in a discharge chamber by internal electrodes or substrates, or in the vicinity of the walls. This opens up numerous opportunities for processing and modification of dust grains as well as controllable deposition of ultra-fine powders onto micro-/nanopatterned substrates. To elucidate the underlying physics of the particle-plasma interactions, most of the fundamental theoretical, experimental, and numerical complex plasma research is currently conducted using model systems. Experimentally, they usually involve an inert ambient gas as well as externally dispersed over the plasma volume dust particles. In such a way, it becomes possible to investigate the fundamental physics of the particle-plasma and particle-particle interactions, collective phenomena and complex nonlinear dust-plasma structures (such as dust clouds, dust voids, ordered liquid and crystal-like structures). The basics of charging processes of the colloidal particles in lowtemperature plasmas can serve as a sound background for description of a wide variety of self-organization and collective processes in the complex plasma systems. We stress that knowledge on the the dynamic charging processes is indispensable for the self-consistent description of dynamics (involving various forces acting on the dust), and ordering (into one-, two-, or three-dimensional structures) of solid particles in the complex plasma systems. The dust charging also opens up a new channel of dissipation in

399

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the plasma. In this case the dust charge appears as a new dynamic variable in the kinetic description of a complex plasma. The presence of the charged solid component does modify the existing and create a number of new collective phenomena in the background plasma. In particular, the established flows of positive ions in the nearelectrode areas of low-temperature laboratory dusty plasmas result in the formation of the plasma wakes, which, in turn, dynamically affect the charging, ordering, interactions, and dynamics of dust grains. In the above, we have presented the basics of the plasma wake formation behind the dust grains of different (spherical and non-spherical) morphologies. Ordering of dust particles in complex plasmas allows one to draw analogies with complex liquid and solid systems. As an example, analogously to the Cooper pairing of electrons in semicoductors, the negatively charged dust grains can be coupled in the presence of the plasma wake. Note that in the case of subsonic plasma wakes, the attraction of dust particles can be attributed to the inverse Landau damping, which is another important collective plasma process. Complex plasmas can be produced in many different ways. It is important that the way the complex plasma is produced significantly defines its properties. In laboratory discharges, dust particles usually carry negative charge because of the main charging process by plasma currents. Ordered dust structures observed in discharge plasmas are trapped in potential wells formed in either the sheath, double layer or striation regions. The structures are supported by the discharge and can exist as long as the discharge is maintained. Dust in thermal plasmas as well as in UV-induced plasmas is charged positively because of the prevailing emission charging. The dust structures in such plasmas have limited lifetime related to drifts of the plasma supporting the structure or to the characteristic time of the particle diffusion towards the walls. The charge on dust particles is not fixed depending on the surrounding plasma conditions. This opens up a number of possibilities for plasma diagnostics. The clouds of nano-sized dust particles can be used for visualization of the complicated sheath structures while individual micrometer-sized dust grains can be used as micro-probes. The character of dust dynamics can provide information on such plasma parameters as the screening length and electron temperature. The complex solid particle-plasma system possess an outstanding capacity to self-organize itself into a number of ordered structures. The emerging structures are thermodynamically open and involve numerous processes of

Conclusions

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generation and loss of the plasma electrons and ions, in particular on the surfaces of the dust grains. The examples include dust clouds, plasma regions free of dust (dust voids), ordered Coulomb crystal-like lattices, liquidlike and gas-like ordered structures. Some of the above plasma-dust structures can be regarded as model systems for the study of solid-state crystals, non-crystalline solids, and even complex astrophysical objects. Visualization of the ordering and oscillations of the dust particles in such systems is crucial for understanding various physical processes at the most fundametal kinetic (microscopic) level. Plasma collective phenomena (such as plasma waves and oscillations at the plasma, i e . , electron and ion, time scales) are strongly modified due to the presence of dust. The presence of a charged heavy dust component in the plasma introduces a completely new time scale (and novel normal plasma modes) associated with the collective motion of the dust grains. We have also considered collective oscillations in the ordered arrays (relevant to the strongly coupled Coulomb lattices) of dust particles in the near-electrode areas. We emphasize that parameters of the background plasmas as well as external forces strongly affect the arrangements, equilibrium states and collective oscillations of the ordered dust particle arrays. We also note that vertical and horizontal arrangements of dust grains of different morphologies in the plasma sheath areas have direct relevance to the controlled deposition of the ordered nanoparticle arrays. The “applied aspect” of problems considered deals with the origin, growth, characterization, and industrial applications of fine particulate matter. This book certainly do not completely cover the full variety of the complex physical and chemical processes involved in the growth and selforganization of fine particles in the reactive ionized gas phase. We focused here on the established models of the particulate growth, self-organization, and discharge restructuring caused by rapid dust agglomeration processes. We have also emphasized the role of the plasma-grown nano-sized clusters and particulates in the fabrication of numerous funct.iona1nanofilms. Note that research and application of fine powders in low-temperature plasma discharges is a rapidly growing area, with the key focus gradually shifting towards nanometer-sized grains (fine crystallites) as the major structural elements in many advanced thin film fabrication technologies. Finally, challenges in theoretical and experimental studies, and new industrial applications presented here suggest that complex ionized gas systems have clear perspectives from both the fundamental knowledge and applications points of view and warrant further extensive research efforts.

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Index

a

-

CzHz monomer, 335, 367 C4FafAr plasmas, 338 CiH; anions, 330 capacitively coupled RF discharge, 49 carbon nanotubes, 363, 365, 368 carbon-based nanostructures (CNSs), 335, 364, 365, 367, 368 cell growth and proliferation, 390 charge density, 69 charge fluctuations, 42, 188, 278, 304, 305, 312, 340, 343 charge variation, 188, 189, 259, 280, 283, 304, 308 charged cluster model (CCM), 375, 377 charging currents, 19, 24, 25 chemically active plasmas, 2, 3, 130, 324, 328, 348, 354, 389 cluster deposition, 380 clustering process, 326, 327, 329, 330 clusters, 28, 326, 336, 340, 342, 347, 349, 363, 364, 370, 383 coagulation, 340-343 collection force, 36, 145, 188 collision frequency, 167 collisional plasmas, 143, 149 collisionless dust voids, 202 collisionless sheath, 38 combustion product plasma, 110-1 12 confining electrode, 98, 106 confining potential, 179, 248, 275 contamination, 336, 346, 355

y’transition, 341, 344

accretion, 325, 334 acetylene, 330-334 agglomerate, 4, 15 agglomeration, 327, 345, 347 agglomeration onset, 328, 340 Alfvkn resonance, 294, 295, 297, 299, 302 Alfv6n speed, 296, 298, 303 Alfvkn surface waves, 302 Alfvkn wave cutoff frequency, 298 Alfvkn waves, 286, 295, 298 amorphous silicon, 369, 373, 374 ArfCzHz plasmas, 333, 336 Ar+CH4 plasmas, 334 Ar+CH*+H2 plasmas, 366 Ar+SiH4 plasmas, 351, 371 aromatic hydrocarbons, 330, 333 auxiliary anode sheath, 99 bacterial inactivation, 394 biocompatible coatings, 390 bond-orientational correlation function, 218 Brownian dynamics method, 163 Brownian motion, 130 building units, 12, 371, 376, 377, 382, 383, 386 Buneman instability, 185, 313 C2H- anions, 336 435

436

Physics and Applications of Complex Plasmas

Cooper pairing, 59 Coulomb clusters, 106, 107, 168, 215 Coulomb coupling parameter, 7, 234 Coulomb crystal, 6 Coulomb interaction potential, 234 Coulomb logarithm, 36, 278, 281 criteria of phase transitions, 237 critical clusters, 326, 329, 336 critical phenomena, 340, 341 critical radius for coagulation, 344 DAW instability, 308 DC discharge, 98, 101, 102 device grade nanofilms, 370, 371 DIAW instability, 304, 306 differential mobility analyzer (DMA), 352 diffraction, 128 diffraction technique, 126 diffusion rate, 163-166 dipole moment, 66, 68 discharge disruption, 361 discharge plasmas, 95 double layer, 103 double-pulse-discharge method, 348 dust charge, 18, 21, 25, 132, 144, 373 dust charging, 18, 340, 342 dust charging equation, 18, 42 dust charging frequency, 31, 187, 277, 279, 281, 305 dust diffusion, 162 dust distribution function, 31 dust growth suppression, 350, 359, 375 dust kinetic temperature, 7, 124 dust lattice waves (DLW), 247, 251, 253, 254 dust origin and growth, 324, 325, 327, 329, 332, 347, 355, 360 dust particles in applications, 387, 392 dust shaker, 95, 102, 105 dust surface temperature, 112, 122, 123 dust transverse lattice waves (DTLW), 248, 249, 252, 254, 256,

273, 274 dust voids, 199, 209, 308, 346 dust-acoustic lattice waves (DALW), 247, 248, 251, 266, 272 dust-acoustic waves (DAW), 248, 254, 283-285, 308, 353 dust-cyclotron frequency, 297 dust-cyclotron mode, 298 dust-dust collisions, 169 dust-ion hybrid resonance, 299 dust-neutral collisions, 169, 309 dust-plasma interaction, 14 dust-plasma sheath, 193 dynamic relaxation processes, 241 EEDF, 342 electron capture frequency, 279, 281, 282 electron collisions, 343 electron confinement, 383, 384 electron depletion, 200 electron field emitters, 366, 390 electron flow, 24 electron temperature, 342, 372 electron-dust collision frequency, 278, 305 electron-dust collisions, 281 electrostatic force, 35, 130, 151, 156, 158, 172, 187, 257, 261 electrostatic potential of dust particle, 55, 76 electrostatic probe, 133 emission-absorption method, 119, 120 epitaxial growth, 376 equation of motion, 159, 163, 187, 257, 265, 266, 272, 273, 275 equilibrium position, 136, 261 Faraday cup, 130, 352 field emission, 20 filamentary structures, 220 film deposition, 338, 369, 376 film deposition model, 389 fine powders, 3, 324, 333, 335, 350 floating potential, 115 flocculation, 341

Index

flow and floe state, 169 fluorescent Ba-Mg-A1 particles, 389 fluorocarbon plasmas, 152, 337-339 freestanding nanocrystalline particles, 371 friction force, 35, 151 FTIR, 333, 337 functionally graded films, 384 generalized reversing method, 119 glow discharge, 95, 108 gravity force, 35, 130, 145, 151, 156, 158, 172, 187, 257, 261

H, emission line, 345, 351 Hamiltonian, 60, 62 Hansen criterion, 237 heterogeneous nucleation, 336, 347, 362 hexagonal lattice, 214 homogeneous nucleation, 325-327 horizontal alignment, 173, 177, 179 horizontal confinement, 170, 171, 178, 182 hybrid dust-ion resonance, 303 hydrocarbon plasmas, 334, 345, 352, 365, 367 hydrogenated silicon, 326, 327 hydrophobicjhydrophillic coatings, 393 hydroxyapat ite, 390 hysteretic phenomena, 176-178, 180 impact parameter, 36 impurities in fusion plasmas, 360, 362 inductively coupled plasmas, 105, 152, 329, 335, 355 infrared absorption, 332 input power, 145 International Space Station (ISS), 343 ion capture frequency, 281 ion drag force, 36, 130, 145, 156, 188 ion flow, 24, 44, 55, 67, 73, 142, 146, 150, 171, 313 ion wake, 55, 56, 90, 150, 183

437

ion-acoustic surface waves (IASW), 287, 289, 294 ion-acoustic waves (IAW), 280-282, 304 ion-cyclotron frequency, 297 ion-cyclotron waves, 298 ion-dust collisions, 281 ionization rate, 143, 145, 146, 278 ionization source, 13, 144, 149 ITER, 361 Lagrangian, 265, 270 Landau damping, 73, 74, 76, 303 Langevin force, 163 Langmuir waves, 276 laser counting technique, 126 laser light scattering (LLS), 125, 328, 338, 348 laser-induced fluorescence (LIF), 134 laser-induced nucleation, 336 layer-by-layer assembly, 383 levitation, 41, 50 levitation electrode, 100 Lindemann criterion, 237 liquid crystal dust lattices, 250 Mach cone, 72, 171 Mach number, 57, 83, 145, 146, 306, 314, 315 magnetic moment, 250, 270-273, 275 magnetic oscillation modes, 250 magnetic pumping, 317 magnetized particles, 250, 269, 272 magnetron sputtering, 347, 388 metal-organic precursors, 389 microcanonical thermodynamics, 15 microgravity conditions, 9 microstructure parameter, 375 modified coupling parameter ry,,, 235, 241 modulational instability, 316, 318-320 molecular clouds, 295 molecular dynamics (MD) method, 81, 82, 85, 89, 376 multi-fluid plasma model, 317

438

Physics and Applications of Complex Plasmas

nanocrystals, 371 nanomaterials, 369 nanoparticle deposition and incorporation, 371, 390 nanoparticle polarizability, 382 nanoparticle synthesis, 393 nanoparticle/cluster detection, 348, 350, 351 nanoparticles, 28, 328, 330, 332, 333, 337, 341, 342, 348, 352, 353, 363, 364, 367, 371, 378, 384, 393 nanostructure growth, 366, 376 nanostructure growth catalysts, 365 nanostructured films, 367, 369 nanostructured silicon, 370 negative ions, 152, 283, 326, 330, 332, 339 neutral drag force, 35 neutral-dust collisions, 284 non-extensive system, 15 normalized coupling parameter r,, 241 nuclear-induced plasmas, 113, 360, 362 nucleation, 325, 329, 330, 335, 347 numerical integration, 82, 86 one-component plasma, 234 open dissipative systems, 5, 11, 16 optical coatings, 383 optical depth, 122 optical-mode dispersion, 258, 259, 263, 267 orbit force, 36, 145, 188 Orbit Motion Limited (OML) approach, 19, 23, 155, 277, 340 pair correlation function, 165 parametric instability, 317 particle agglomerates/structures, 232, 336, 338, 340, 341, 343, 378 particle and power balance, 343 particle coating, 388, 389 particle condensation, 168 particle detection, 337 particle detection time, 346, 351

particle interactions, 378 particle levitation, 130, 143, 145, 146, 149, 159, 170 particle ordering, 165 particle shape, 334, 338, 341, 346, 362, 372, 377 particle size, 362, 374, 382, 390 particle size measurement, 352 particulate shedding, 358 PECVD, 335, 338, 364, 366, 367, 369, 387, 390 phase transitions, 6, 233 photoelectron emission, 21, 342 photoluminescence, 384 plasma etching, 152 plasma polymerization, 327, 329, 330, 334, 338, 339, 356, 367 plasma screening length, 8, 74, 132 plasma tool design, 356 plasma-assisted quantum dot fabrication, 384 plasma-surface interaction, 14 polymorphous silicon, 370, 380 positively charged particles, 93 potential trap, 10, 104, 251, 358, 388 powder clouds, 328, 337, 346 powder contamination management, 357 powder precursors, 326, 328, 333, 338, 339, 375 powder-generating PECVD regime, 369, 375 pre-sheath, 142 probe characteristics, 115, 117 probe technique, 115 propellant, 110, 111 Q-machine, 104, 116 QMS, 353 quantum dots, 384 radioactive dust, 113, 360, 361 Rayleigh-Mie scattering, 333, 352, 382 reactive species, 332 residence time, 326, 328

439

Index

RF capacitive discharge, 97, 98, 132 RF discharge impedance method, 350 RF inductive discharge, 105, 106 R F parallel-plate discharges, 151 rod-like dust particles, 30, 69, 250, 264, 265 rotational oscillation modes, 250, 268, 269, 275 Runge-Kutta method, 83, 86 safety of fusion reactors, 360 scaling parameter, 164, 169 screened Coulomb coupling parameter, 234 screened Coulomb potential, 56, 132, 170 secondary emission, 20 segmented triode, 100 selective cluster deposition, 378 selective trapping, 328 self-assembly, 377 self-excited oscillations, 185, 189 self-lubricating coatings, 393 self-organization, 3, 5, 6, 11-13 SEM, 333, 334, 346, 388 sheath, 38, 99, 132, 134, 137, 142, 154, 158, 161, 178, 188, 257 sheath diagnostics, 134 sheath electric field, 39, 135, 172, 257, 261 sheath electrostatic force, 145 sheath field, 145 SiH, anions, 326, 327, 329 SiH4+H2 plasmas, 373 SiH4+Oz+Ar plasmas, 328 silane plasmas, 325, 328, 329, 333, 340, 342, 349, 369 silicon nanowires, 376 simple-hexagonal structure, 218 single albedo, 122 single-crystal nanoparticles, 371 size-dependent optical properties, 381, 382, 384 solar cells, 387 spectroscopic diagnostics, 119 static structural analysis, 218

striations, 102, 103 strongly coupled complex plasma, 7 structure factor, 165 superthermal electrons, 49 surface morphology, 370, 390, 393 surface wave sustained plasmas, 152 TEM, 328, 337, 352 TEXTOR, 361 thermal probe, 133 thermionic emission, 21, 110 thermophoretic force, 37, 111, 130, 151, 359, 363 TOF mass spectrometry, 353 transient motion technique (TMT), 139 translational correlation function, 218 trapped ions, 45, 47 tray method, 96 triangular lattice, 215

ULSI technology, 355 ultrananocrystalline diamond, 390 UV radiation, 93 UV-induced plasmas, 108, 109 velocity distribution function (VDF), 124 vertical alignment, 177, 179 vertical confinement, 170, 178 vertical vibrations, 132, 158, 160, 174, 185, 186, 256-258, 263, 268, 273, 274 visualization of dust particles, 124, 125 vizualization technique, 126 wake potential, 56, 57, 67, 70, 71, 84, 171, 175, 176, 178, 260, 261 weakly coupled complex plasma, 8 Wigner-Seitz radius, 169 Yukawa model, 234 Yukawa systems, 165 ZnO nanoparticles, 389