Frontiers in Dusty Plasmas
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Proceedings of the Second International Conference on the Physics of Dusty Plasmas ICPDP-99
Frontiers in Dusty Plasmas Hal
Editors: Y. Nakamura The Institute of Space and Astronautical Science, Yoshlnodai, Sagamlhara, Kanagawa, Japan
T. Yokota Department of Physics, Faculty of Science, Ehime University, Bunkyo-cho 3, Matsuyama, Japan
P.K. Shukia Ruhr-Universitat Bochum, Bochum, Germany
2000 ELSEVIER AMSTERDAM • LAUSANNE • NEW YORK • OXFORD • SHANNON • SINGAPORE • TOKYO
ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands
First edition 2000 ISBN: 0 444 50398 6 Library of Congress Cataloging-in-Publication data A catalog record from the Library of Congress has been applied for. © 2000 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Rights & Permissions Department, PO Box 800, Oxford 0X5 IDX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail:
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PREFACE
The Second International Conference on the Physics of Dusty Plasmas was held during May 24-28, 1999 in the beautiful Fuji-Hakone National Park in Japan. The number of participants was 110 from 22 countries. The first Conference was held in October 1996 in Goa, India and its Proceeding was published as a book entitled '* Advances in Dusty Plasmas". The study of dusty plasmas is now in a vigorous state of development. Dust and plasma coexist in a vast variety of cosmic environment and their research received a major boost in early eighties with the Voyager spacecraft observations of peculiar features in the Saturnian ring system(e.g. the radial spokes) which could not be explained purely with gravitational terms. In addition, dust streams were measured by the Galileo spacecraft in the Jovian magnetosphere and charged dust in the earth's mesosphere was detected by a direct rocket experiment. Since then the area has been in a rapid state of progress with dedicated laboratory experiments verifying aspects of basic physics of charged dust grains in plasmas. For example, the very low frequency dust-acoustic wave has been observed visually by light scattering. Recent laboratory experiments could show even nonlineai' phenomena such as chaos, self-organization and shock waves, which had been known to exist in ordinary plasmas. The study of nonlinear physics together with instabilities in dusty plasmas will be rapidly developed in the near future. Talking of laboratory experiments, it is noted that industrial and dusty plasma processing are also lively investigated mainly in industrial laboratories from a point of applications. The condensation and transport of fine dust particles in discharge plasmas, leading to the yield loss of product, are significant problems being faced by industries. The most recent and fascinating development has been the crystallization of dusty plasmas in laboratories. Plasma crystals consist of Coulomb lattice of negatively charged dust grains interspaced with positive ions and electrons. It is now recognized that this new material of which structure is observed visually by a laser light scattering could be a valuable tool for studying processes such as melting, annealing and lattice defects. Some modes of the lattice wave are found even experimentally. To understand the processes physically, numerical simulations are also performed. Taking this occasion, we would like to propose a new terminology ''fineparticle plasma" for a laboratory dusty plasma when the material and size of dust
vi
Preface
grains are well defined. The invited review, topical and poster papers presented at the Conference ai^e included in this book. It is a pity that dynamical, vivid movements of dust particles shown by movies to the audience are not reproduced in this book. Finally we would like to express our sincere gratitude to Inoue Foundation for Science, International Union of Pure and Applied Physics and The Institute of Space and Astronautical Science for providing partial support for the Hakone Conference. This proceeding is supported by Grant-in-Aid for Publication of Scientific Research Result, Japan Society for the Promotion of Science. We acknowledge the help of the International Advisory Board for their suggestions regarding the invited speakers.
Yoshiharu Nakamura(Chairman) Institute of Space and Astronautical Science, Yoshinodai, Sagamihara, Kanagawa 229-8510, JAPAN
Toshiaki Yokota(Scientific Secretary) Ehime University, Faculty of Science, Department of Physics Bunkyo-cho 3, Matsuyama, Ehime 790-8577 JAPAN
Vll
INTRODUCTION This monograph consists of invited as well as poster papers that were presented at the Second International Conference on the Physics of Dusty Plasmas held in Hakone, Japan, during the period 24-28 May 1999. The physics of dusty plasmas (plasmas containing extremely massive charged microspheres or paiiiculates) has been an intense new area of research over the latest ten years following the discoveries of dust acoustic waves and dusty plasma ciystals. Dusty plasmas occur in space and astrophysics settings as well as in low-temperature laboratory dusty plasma discharges and in the microelectronic fabrication industry. Thus, they are investigated by diverse scientific communities ranging from astrophysicists studying stellar evolution and planetary rings to applied physicists and engineers attempting to either control particulate contaminations of semiconductor chips in the processing industry or to study exotic collective behaviors of weakly and strongly coupled dust grains in a controlled fashion. In laboratory dusty plasma discharges, dust grains can become highly charged so that the Coulomb potential energy between the dust grains can become comparable to, or larger, than their kinetic energy. Such a strongly coupled dusty plasma offers a unique opportunity to investigate the properties of dust coagulation, dust crystallization, phase transition, dust lattice oscillations etc. at room temperature. Thus, the field of dusty plasmas is truly multi-disciplinary in that it provides a link between the gaseous discharge and the space and condensed matter physics communities. The Hakone Conference provided a forum for researchers from diverse scientific communities ranging from astrophysics and space science to plasma processing and colloidal plasma science. The conference was attended by scientists from all over the world. There was a great deal of interactions between the speakers and participants who lively discussed and debated theoretical and computer simulation models, experimental observations and interpretations. The results of the conference are presented in this book in the foim of review and topical papers, in addition to several selected posters. The material contains new aspects of collective interactions in dusty plasmas. For example, discoveries of dustacoustic Mach cones, dust ion-acoustic shocks, great dust voids, vortex formation, dust crystallization imder microgravity, coexistence of positive negative dust grains in the mesosphere, dust in tokamaks, powder formation in industrial reactive RF plasmas, and trapping and processing of dust particulates in a low-pressure discharge are some of the highlights on the experimental side. On the other hand, theoretical and simulation studies focused on dynamical and structural properties and kinetic theories of strongly coupled dusty plasmas as well as on self-organizations and structures, in addition to identifying forces (viz. wakefields, electrostatic and dipolar interactions, etc.) which are responsible for charged dust grain attraction and phase transitions. The scientific program of the conference also included a ke5niote address on perspectives of collective processes in dusty plasmas and a popular talk on dusty plasmas:
viii
Introduction
far and near. Clearly, the material constitutes a mix of theory, dedicated laboratory experiments (both in weakly and strongly coupled regimes), numerical simulations, and industrial applications. We hope that the present monograph, which contains results that are in the frontiers of dusty plasma science, will be useful to researchers and graduate students who want to learn the recent developments in the rapidly growing field of dusty plasmas. It is our pleasure to thank all the participants for their stimulating oral and poster presentations, and who took the pain of preparing their material for publication in this volume, which should be a valuable contribution to the dusty plasma physics literature. We also sincerely acknowledge the help from the members of the International Advisory Board in selecting the speakers. Fxu'thermore, we express our sincere thanks to the International Union for Pure and Applied Physics (lUPAP), the Institute of Space and Astronautical Science (ISAS, Japan), the Inoue Foundation for Science (Japan), and Mrs Eikon Nakamura, for providing partial support for the Hakone Conference. This book is supported by the Japan Society for the Promotion of Science (JSPS) through the "Grant-in-Aid for Publication of Scientific Research Results". Finally, a special word of thanks to Professor A. Nishida, Director General, ISAS, who helped us to organize a very successful conference at Hakone which will remain a wonderful memory in our hearts for the rest of our life. P. K. Shukla, Ruhr-Universitat Bochum, Germany T. Yokota, Ehime University, Japan Y. Nakamura, The Institute of Space and Astronautical Science, Japan
IX
CONTENTS
Preface Introduction
v vii
Part I. Basic Dusty Plasmas Perspectives of collective processes in dusty plasmas: a keynote address RK. Shukla
3
Non-ideal effects in dusty plasmas N.N. Rao
13
The kinetic approach to dusty plasmas VN. Tsytovich and U. de Angelis
21
Dynamical and structural properties of strongly coupled dusty plasmas M.S. Murillo
29
Directional ordering and dynamics in dusty plasmas J.E. Hammerberg, B.L. Holian, M.S. Murillo and D. Winske
37
Multiple-sheath and the time-dependent grain charge in a plasma with trapped ions Y.-N. Nejoh
43
Surface charge on a spherical dust H. Amemiya
49
Numerical simulation model for dust expansion into a plasma G.S. Chae, W.A. Scales, G. GanguH, RA. Bernhardt and M. Lampe
55
Interaction potential between two Debye spheres J.X. Ma and E Pan
61
Coupling of waves and energy conversion in a slowly varying nonuniform dusty plasma M. Khan, S. Sarkar, T.K. Chaudhuri, A.M. Basu and S. Ghosh
67
X
Contents
Part II. Plasma Crystals - New Material Classical atom-like dust Coulomb clusters in plasma traps Lin I, Y.-J. Lai, W.-T. Juan and M.-H. Chen
75
Crystallography and statics of Coulomb crystals Y. Hayashi and A. Sawai
83
Monolayer plasma crystals: experiments and simulations J. Goree, D. Samsonov, Z.W. Ma, A. Bhattacharjee, H.M. Thomas, U. Konopka and G.E. Morfill
91
Nonlinear dust equilibria in space and laboratory plasmas VN. Tsytovich
99
Dust particle structures in low-temperature plasmas A.P.Nefedov
107
Experimental evidence for attractive and repulsive forces in dust molecules A. Melzer, VA. Schweigert and A. Piel
115
Self-organization in dusty plasmas S. Benkadda, VN. Tsytovich, S.L Popel and S.V Vladimirov
123
Dynamical properties of strongly coupled dusty plasmas S. Hamaguchi and H. Ohta
135
Structures and structural transitions in strongly-coupled Yukawa dusty plasmas and mixtures H. Totsuji, C. Totsuji, K. Tsuruta, K. Kamon, T. Kishimoto and T. Sasabe
141
Vortex chains and tripolar vortices in dusty plasma iBow J. Vranjes, G. Marie and EK. Shukla
147
Dynamical structure factor of dusty plasmas including collisions A. Wierling, VJ. Bednarek and G. Ropke
153
Melting of the defect dust crystal in a rf discharge LV Schweigert, VA. Schweigert, A. Melzer and A. Piel
159
Part III. Industrial Applications On the powder formation in industrial reactive rf plasmas Ch. HoUenstein, Ch. Deschenaux, D. Magni, F. Grangeon, A. AfiFolter, A. A Howling and P Fayet
169
Contents
xi
Trapping and processing of dust particles in a low-pressure discharge E. StofFels, W.W. Stoffels, G.H.P.M. Swinkels and G.M.W. Kroesen
177
Effects of gravity, gas and plasma on arc-production of fuUerenes T. Mieno
185
Formation of dust and its role in fusion devices J. Winter
193
The formation and behavior of particles in silane discharges A. Gallagher
199
Dust particles influence on a sheath in a thermoionic discharge M. Mikikian, C. Arnas, K. Quotb and E Doveil
207
Plasma deposition of silicon clusters: a way to produce silicon thin films with medium-range order? P. Roca i Cabarrocas
213
Part IV. Atmospheric and Astrophysics Formation of a dust-plasma cloud A.M. Ignatov
229
Effects of dust on Alfven waves in space and astrophysical plasmas N.F Cramer and S.V Vladimirov
237
Investigation of plasma irregularity generation in expanding ionospheric dust clouds W.A. Scales, G.S. Chae, G. Ganguli, PA. Bernhardt and M. Lampe
245
Mass distributions and self-gravitation in dusty plasmas F Verheest
253
Statistical description and 3D computer modeling of relaxing dusty plasmas Y.I. Chutov, O.Yu. Kravchenko, PPJ.M. Schram and R.D. Smimov
261
Regular structures in dusty plasmas due to gravitational fields N.L. Tsintsadze, J.T. Mendonca, PK. Shukla, L. Stenflo and J. Mahmoodi
269
A rocket-borne detector for charged atmospheric aerosols S. Robertson, M. Horanyi, B. Smiley and B. Walch
275
Paleo-heliosphere: effects of the interstellar dusty wind based on a laboratory simulation S. Minami and S. Miono
281
xii
Contents
Current loop coalescence in dusty plasmas J.I. Sakai and N.F. Cramer
289
Jeans-Buneman instability in non-ideal dusty plasmas S.R. Pillay, R. Bharuthram, F. Verheest, N.N. Rao and M.A. Hellberg
295
Part V. Basic Experiments Waves and instabilities in dusty plasmas N. D'Angelo
303
Dust charging in the laboratory and in space M. Horanyi and S. Robertson
313
Charging measurements and planetary ring simulation by fine particle plasmas T Ybkota
321
Structure controls of fine-particle clouds in dc discharge plasmas N. Sato, G. Uchida, R. Ozaki, S. lizuka and T. Kamimura
329
Structural formation and stability of Coulomb clouds in medium through low gas pressure range S. Takamura, N. Ohno, S. Nunomura, T. Misawa and K. Asano
337
Poster Session Poster Session A: Basic Pliysics of Dusty Plasmas Kelvin-Helmholtz instability in weakly non-ideal magnetized dusty plasmas with grain charge fluctuations N.N. Rao
347
Self-similar expansion of a non-ideal unmagnetized dusty plasma R. Bharuthram and N.N. Rao
351
Computer modeling of non-linear sheaths with dust particles Y.I. Chutov, O.Yu. Kravchenko, RP.J.M. Schram and VS. Yakovetsky
355
The gravitational effect on negatively-charged dust-grains in a plasma H. Yamaguchi, Y.N. Nejoh and N. Mizuno
359
Stationary equilibria of dusty plasmas R.T. Faria Jr., P.H. Sakanaka, T. Farid and P.K. Shukla
363
Contents
xiii
Photoelectric charging of dust particles A. Sickafoose, J. Colwell, M. Horanyi, S. Robertson and B. Walch
367
Influence of dust grains rotation on waves dispersion in plasmas J. Mahmoodi, N.L. Tsintsadze and D.D. Tskhakaya
373
Plasma-maser instability in magnetized dusty plasma B.J. Saikia and M. Nambu
377
Ion bursts in a negative ion plasma S. Yoshimura, M. Yohen and Y. Kawai
381
Nonlocal effects in an ion beam driven ion acoustic waves in a magnetized dusty plasma S.C. Sharma and M. Sugawa
385
Experimental observation of attraction of massive bodies in a plasma A.E. Dubinov, VS. Zhdanov, A.M. Ignatov, S.Yu. Kornilov, S.A. Sadovoi and VD. Selemir
389
The role of random grain-charge fluctuations in dusty plasmas S.A. Khrapak, O.F. Petrov and O.S. Vaulina
393
Shock waves in plasmas containing dust particles S.I. Popel and VN. Tsytovich
397
Investigation of the ordered structure formation in a thermal dusty plasma YK. Khodataev, S.A. Khrapak and O.F. Petrov
401
Separation of diamagnetic fine particles in a non-uniform magnetic field I. Tsukabayashi, S. Sato and Y Nakamura
405
Minimal charge asymmetry for Coulomb lattices in colloidal plasmas: effects of nonlinear screening 0. Bystrenko and A. Zagorodny
409
Characteristics of asymmetric ion sheath in a negative ion plasma K. Koga, H. Naitou and Y Kawai
413
Coherent interaction model of attractive forces in the plasma crystals 1. Mori, T. Morimoto and K. Tominaga
417
Charge distribution function of negatively and positively charged plasma dust particles B.F. Gordiets and CM. Ferreira
423
Control of ultra-fine particles in plasma by an electromagnetic field K. Takeda, N. Kawashima, T. Etoh, Y Takehara, H. Kubo and S. Bessho
427
xiv
Contents
Poster Session B: Strongly Coupled Dusty Plasmas Damping of collective excitations in Coulomb crystals VJ. Bednarek, A. Wierling and G. Ropke
433
Behaviour of particles released from inner walls in an ECR plasma etch tool T. Kamata, K. Miwa and H. Arimoto
437
Anomalous diffusion and finite size effect in strongly coupled 2-D dust Coulomb clusters W.-T. Juan, Y.-J. Lai and Lin I
441
Plasma crystals and liquids in DC glow discharge VE. Fortov, VL Molotkov and VM. Torchinsky
445
Formation of ring-shaped fine-particle clouds in a DC plasma G. Uchida, R. Ozaki, S. lizuka andN. Sato
449
Vertical spread of fine-particle clouds in a magnetized DC plasma S. lizuka, R. Ozaki, G. Uchida andN. Sato
453
Vertical string structure of fine particles in a magnetized DC plasma R. Ozaki, G. Uchida, S. lizuka andN. Sato
457
Determination factor of Coulomb crystal structure in dusty plasmas K. Takahashi and K. Tachibana
461
Instabilities of dust particles levitated in an ion sheath with low gas pressure T. Misawa, S. Nunomura, K. Asano, N. Ohno and S. Takamura
465
Instabilities of the dust-plasma crystal N.F Cramer and S.V Vladimirov
469
Some remarks on dust lattice waves in plasma B. Farokhi, N.L. Tsintsadze and D.D. Tskhakaya
473
Artificial fireball as dust-plasma cloud A.M. Ignatov, L.V Furov, VN. Kunin, VS. Pleshivtsev and A.A. Rukhadze
477
Quenching of a high-temperature phase of Fe nanoparticles by a microwave plasma processing K. Tanaka, T. Fukaya, K. Kawase, J. Fujita and S. Iwama
481
Observation of dust particles trapped in a diffused plasma produced by low pressure RF discharge N. Hayashi, T. Kimura and H. Fujita
485
RF potential formation in a magnetised plasma containing negatively charged particles P. Cicman, A. Kaneda, N. Hayashi and H. Fujita
489
Contents
xv
Formation of particle conglomerates in a methane discharge WW. Stoffels, E. StofFels, G. Ceccone and F. Rossi
493
Change of the potential relaxation instability in a plasma containing heavy C^Q ions D. Strele, C. Winkler and R. Schrittwieser
497
Experimental verification of dust particle's transport by ambipolar ExB drift Y. Maemura, M. Ohtsu, T. Yamaguchi and H. Fujiyama
501
Nuclear induced dusty plasma structures L.V Deputatova, VE. Fortov, A.V Khudyakov, VI. Molotkov, A.R Nefedov, VA. Rykov and VI. Vladimirov
505
Experimental investigation and numerical simulation of inductively coupled dusty plasma VS. Filinov, A.R Nefedov, VA. Sinel'shchikov, O.A. Sinkevich, A.D. Usachev and A.VZobnin
509
Dust wake in a collisional plasma D. Winske, WS. Daughton, D.S. Lemons, M.S. Murillo and WR. Shanahan
513
Measurement of electric charge of dust particles in a plasma M. Itoh, T. Fujimori, Y Komatsu and Y Nakamura
517
Transport of a polydisperse ensemble of dust particles in plasma I.V Schweigert and VA. Schweigert
521
Influence of the lattice symmetry on melting of the bilayer Wigner crystal I.V Schweigert, VA. Schweigert and F.M. Peeters
525
Poster Session C: Collective Effects and Astrophysics Effects of dust grains on planar RF discharges S. Nonaka, K. Katoh, Y Nakamura, S. Ikezawa and S. Takamura
531
Experimental studies of UV-induced dusty plasmas under microgravity VE. Fortov, A.R Nefedov, VP. Nikitsky, A.I. Ivanov and A.M. Lipaev
535
Parameters of dusty particles in plasma flows A.A. Samarian, O.S. Vaulina, A.V Chernyshev, A.R Nefedov and O.F. Petrov
539
Hard X-rays from high power density plasma "dust" Yu.K. Kurilenkov, M. Skowronek, G. Maynard and J. Dufly
543
From dusty plasma target to hard X-ray lasant media M. Skowronek, Yu.K. Kurilenkov and G. Louvet
547
xvi
Contents
Coupling between Jeans and X-modes in self-gravitating magnetized dusty plasmas M.A. Hellberg, R. Bharuthram, R.L. Mace, P. Singh and F. Verheest
551
Fluctuation electrodynamics of dusty plasmas VR Kubaichuk and A.G. Zagorodny
555
Measurement and modeling of particle spacing in strongly coupled dusty plasmas M.S. Murillo and H.R. Snyder
559
Part I. Basic Dusty Plasmas
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
Perspectives of collective processes in dusty plasmas: a keynote address p . K. Shukla Fakultat fiir Physik und Astronomie, Ruhr-Universitat Bochum, D-44780 Bochum, Germany There has been a great deal of interest in investigating numerous collective processes in dusty plasmas following the discovery of the dust acoustic wave and the observation of dusty plasma crystals. Our objective here is to review the status of waves, instabilities, and coherent nonlinear structures in weakly and strongly coupled dusty plasmas. Specifically, we describe the physics of dusty plasma waves and associated nonlinear structures such as solitons, shocks, and vortices in dusty plasmas. Also discussed are the wake-potential and the associated attractive force that is responsible for the Coulomb crystallization of charged dust grains. Finally, we point out observations of many collective processes in low-temperature laboratory i dusty plasmas. PCAS:52.25.Vy, 52.35.Mw
1
Introduction
The dusty plasma physics is a rapidly growing area of science. The physics of collective processes has grown many folds following the great discovery of the dust acoustic wave[l] and the dusty plasma crystal[2, 3]. The theoretical idea for the latter was envisioned by Ikezi[4]. The author was somewhat fortunate to join the dusty plasma club at the right time in order to contribute to that field in a significant manner. It was author's participation in the Capri Conference on Dusty Plasmas in the summer of 1989, where he theoretically discussed the possibility of the dust acoustic wave (DAW) which involves the dynamics of the dust grains. In the DAW, the pressures of the electrons and ions provide the restoring force, while the inertia comes from the dust mass. Thus, the DAW is an extremely low-frequency (in comparison with the dust plasma frequency) wave, and is not subjected to Landau damping in a plasma with equal electron and ion temperatures. The classic paper[1] on the linear and nonlinear dust acoustic waves was written in collaboration with Nagesha Rao of PRL (India) and Ming Yu of RUB (Germany). Furthermore, while attending the Topical Conference on Plasma Physics at the Abdus Salam International Center for Theoretical Physics in the summer 1991, the author and Viktor Sihn wrote a paper deahng with the dust ion-acoustic wave[5] (DIAW), which is the modification of the standard ion-acoustic wave in an electron- ion plasma with stationary charged dust grains. Both the DAW and DIAW have been spectacularly verified in many laboratory experiments[6, 7, 8, 9, 10, 11, 12]. On the other hand, in 1994 the groups led by Gregor MorfiU (MPE Garching) and Lin I (NUT) independently observed the formation of dusty plasma crystals in low-temperature laboratory discharges.
4
RK. Shukla/Perspectives of collective processes in dusty plasmas
Thus, important discoveries of the dust acoustic wave and the dusty plasma crystal boosted the field of the dusty plasma research, which is infancy now. Dusty plasmas are fully or partially ionized low-temperature gases composed of electrons, ions, and micron-sized extremely massive charged dust grains. The latter, which are billion times heavier than ions, acquire several thousands of the electron charge. The dust grain charging occurs due to a variety of physical processes including the collection of the background plasma electrons and ions by dust grains, the photo electron emission, secondary emission and sputtering, etc. Dust grains can be charged both negatively and positively. The grains act like a source when they are charged positively due the irradiation of the UV radiation. There are indications that both the positive and negative dust grains can coexist in laboratory and space plasmas. Dusty plasmas are ubiquitous in astrophysics and space environments (e.g. solar nebula, planetary rings, molecular clouds, cometary tails. Earth's mesosphere, etc), as well as in radio-frequency discharges, in plasma aided manufacturing of microelectronic industry, and in fusion devices. There have been arguments that a dusty plasma is similar to a multi-ion plasma. However, this assertion has to be refuted as a dusty plasma is significantly different from a multi-ion plasma in that the presence of massive charged dust grains is responsible for new collective phenomena which appear on a completely new time and space scales. Also the dust charge fluctuation and dust-dust interactions give rise to new effects. The attractive forces between charged particulates, which can exist in dusty plasmas, are responsible for the formation of dusty crystals and dusty molecules. There is a dust lattice wave whose counterpart exists only in solids. Thus, the knowledge of basic plasma physics, probe theory, statistical mechanics, as well as condensed matter and solid state physics is very essential for understanding collective processes in dusty plasma physics. Charged dust grains can be either weakly or strongly correlated depending on the strength of the Coulomb coupling parameter T = {QydTct)exp{—d/XD), where Qd = ZdC is the dust charge, Zd the number of charges residing on the dust grain surface, e the magnitude of the electron charge, d = (3/47171^)^^^ the intergrain spacing, n^ the number density of the dust grain, Td the dust temperature, and the Debye radius (A^) of the dusty plasma is given by[13] A^^ = A^^ + A^^, where Xoei^Di) is the electron (ion) Debye radius. A dusty plasma can be considered as weakly coupled as long as F < 1. However, when F > > 1 and the inter-grain spacing is of the order of XD^ charged dust microspheres strongly interact with each other, and we have the possibility of forming Coulomb lattices in a strongly coupled dusty plasma. According to Ikezi[4], the critical value of F for the Coulomb crystallization of charged dust grains is about 170. Strongly coupled plasmas are also found in a highly evolved star, in a white dwarf, in planetary rings (narrow rings of Uranus, incomplete rings of Neptune, etc.), in the Jovian interior, in laser implosion experiments, as well as in colloidal systems. In this paper, we highlight the present status of collective processes in dusty plasmas. We focus our attention on waves, instabilities, and nonlinear structures in weakly and strongly coupled dusty plasmas. The consequence of the dusty plasma wave spectra with regard to the Coulomb crystallization is discussed. Furthermore,
RK. Shukla /Perspectives of collective processes in dusty plasmas
5
we enlighten the physics of soUtons, shocks, and vortices in dusty plasmas. It is remarkable that both the dust ion-acoustic shocks and vortices are observed in laboratory dusty plasma devices.
2
D u s t y p l a s m a waves
We consider a multi-component dusty plasma composed of electrons, singly charged positive ions, and extremely massive negatively charged dust grains, in a neutral background. The dust grain radius R is usually much smaller than the dusty plasma Debye radius A^. When the intergrain spacing d is much smaller than \D, the charged dust particulates can be treated as massive point particles similar to multiply charged negative (or positive) ions in a multi-species plasma. On the other hand, for d < \D the effect of neighboring particles can be significant, whereas for d > > A^) > > J? the grains are completely isolated from its neighbors. The dusty plasma quasineutrality condition for negatively charged dust grains is {rte/rii) + P - 1 = 0, where P = Zdrtd/rii. Generally, the spectra of dusty plasmas waves are obtained by Fourier analyzing the Vlasov, Poisson, and Maxwell equations, supplemented by the dust charging equation. In unmagnetized and weakly coupled dusty plasmas with Maxwellian particle distributions, the electrostatic wave spectra are obtained from the dispersion relation 1+
$Z Xa + Xqde + Xqdi = 0,
(1)
where the dielectric susceptibility is given by[14] X. = ^ [ l
+ ^.Z{^a)]
V2kvta
(2)
Here, k^a — ^pal'^ta^ ^pa the unperturbed plasma frequency of species a, Vta the thermal velocity, Z the standard plasma dispersion function of Pried and Conte, ^a = {^ — kUaQ + ii^an) IV^kvta^ and tx^o the unperturbed streaming velocity. The linear susceptibility Xqds{^ — ^^ i) arising from the dust charge fluctuation[15] produced by electrostatic perturbations is[16] Xqds =
—
/
VCTs [qdO^ V) dv,
(3)
where fso is the unperturbed distribution function of species 5, Vph = u/k the phase velocity, rj = e\Io\ UCTe)~^ + [CTi — eqdo)~ the charge relaxation rate originating from the variation in the effective collision cross-section due to dust charge fluctuations at the grain surface as experienced by the unperturbed particles, leo the equilibrium electron current reaching the grain surface, C the grain capacitance, qdo is the equilibrium grain charge, Te{Ti) the electron (ion) temperature, and (is^e.i the collision cross-section.
6
P.K. Shukla/Perspectives of collective processes in dusty plasmas
Let us first ignore the equilibrium drift, collisions, and the dust charge perturbation [viz. we set -u^o = 0, z/an = 0, and Xqde.qdi = 0 in Eq. (1)]. The dusty plasma wave spectra can then be deduced from 1 + Xe + Xi + Xd — 0. First, we consider the DIAW with Vtd.vu « uo/k « Vte so that Xe ~ V^^'^De? Xi ^ -^lil
(^^ - '^^'^^u) and Xd ~ -^Idl
(<^^ ~ '^^'^^Id)' When the dust plasma
frequency ujpd is much smaller than the ion plasma frequency cjp^, we obtain the DIAW frequency[5] (nio/rieo)^^^ fcc^/(1 + 6e)^'^^ where c^ = {T^/rriif''^ is the ionacoustic velocity, and bg = ^^^DC We see that the phase velocity of the usual lAW in an electron-ion plasma is increased when a dust component is added. Second, we focus on the DAW for which Vtd « oj/k « Vtej^^u- Here, we have Xe ~ l/A:^A|)g, Xi ^ ^l^^^loi^ and Xd ^ -^Idl^'^- Hence, the frequency of the DAW is ct; = kXoUJpd/ (1 + k'^X'jj) ; the latter in the long wavelength (in comparison with XD) limit reduces to a; ^ kco, where the dust acoustic velocity is denoted by CD = XuOJpd = Zd {Ti/rudf^
{ndo/riio)^^'^ / [1 -h r (1 - Zdndo/rtio)]^^'^. Here, rrid is the
mass of the dust grains and r = Ti/Te. We stress that the DAW is a new type of extremely low-frequency (in comparison with the DIAW) sound-like wave in which the restoring force comes from the ion and electron pressures and the inertia is provided by the mass of extremely heavy charged dust grains. In relatively low-temperature laboratory plasmas[6, 7, 8, 10, 11, 12], the frequency of the DAW is in the range 10-30 Hz and the visual images of the DAW can be obtained due to the scattering of light from the dust grains. On the other hand, the frequency of the DIAW in a laboratory plasma[12] is of the order of 3-5 kHz. Clearly, experimentally observed frequencies[6, 7, 10, 11, 12] in unmagnetized dusty plasmas are in good agreement with those of the DAW[1] and the DAW[5]. The dust charge perturbation produces a coUisionless non-Landau-type damping[15, 16] of the DAW and DIAW. In a coUisional plasma, both the DIAW and the DAW suffer damping. For example, the spatio-temporal damping rates of the DIAW in the presence of iondust collisions are obtained from[17]
e = -L-Jifei±i^,
(4)
where i^id is the ion-dust collision frequency. The spatial attenuation rate is obtained by letting k = kr + iki^ where kr{ki) is the real (imaginary) part of the wavenumber k. On the other hand, the DAW dispersion relation in a partially ionized coUisional dusty plasma with Zdo^do ^ ^i^io is[18] (cj + iu) = k^c\^{u + ivn),
(5)
where z/ = z/^ + ^n, ^c = ^dn + pi^in/pd, ^n = i^ni + ^nd, Pi = nioTUi, pd = Ud^rrid,
Cda = [{Td+ZdoTi/Zi)/rridY^'^, and the subscripts d, i and n stand for the dust grains, the ions, and the neutral atoms, respectively. In deriving (5), we have assumed that the wavelength of the DAW is much larger than the ion Debye radius. Several comments are in order. First, for u » z/c? ^n> Eq. (5) yields u ^ kcda{'^—i^c/'^kcda)^
PK. Shukla/Perspectives of collective processes in dusty plasmas
1
which is the frequency of the damped DAW. Second, for uJ^ « u'^, i/„z/, we obtain LO ^ {k^c^J^n/^) (1 — ikcdaJ^c/^^^n)- Finally, the real and imaginary parts of k are U^
I
2
2 (a;2 + i/2) !_
2
2 c
1/2^
(u^ + uun) +u;^K
(6)
and " ^ " 2 ^ , ( 0 . 2 + ^2)'
^^^
where the wave vector, the wave and coUisional frequencies are normaUzed by tOpd/cda and cjpd, respectively. The properties of dust acoustic waves including dust-dust interactions and correlations have been investigated by Rosenberg and Kalman[19], Murillo[20], de Angelis and Shukla[21], and Kaw and Sen[22]. It is found that dust-dust interactions cause a spatial damping of the DAW. The effect of the plasma boundary[23] on the DIA and DA waves in weakly coupled dusty plasma has also been investigated. The boundary effect modifies the shielding term. For example, for the DIAW we have to replace k'^X^^ ^Y ^De(^^ + JI/RI), whereas for the DAW we must replace k^Xj^ by A|)(fc^+7^/i?o)- Here, 7^ is a root of zero order Bessel function JQ and RQ the radius of the cyhndrical waveguide. For jn » kRo^ the frequencies of the DIA and DAWs are ou ^ XDe^pi7n/Ro{^ + ^l^De/Riy^^ and u ^ \DUJpdlnRol{'^ + ll>^])IRlf'^^ respectively.
3
Dust lattice waves
The wave propagation in crystals is well studied in solid state physics. Here, we discuss the properties of the dust lattice wave (DLW) in a plasma crystal. The DLW arises due to the oscillation of charged microspheres under the action of a screened Coulomb potential which describes the interaction between neighboring dust particulates in the plasma crystal lattice. Using a linear chain model and considering only nearest-neighbor forces, we can write the equation of motion for the dust grains in a dusty plasma crystal as[24, 25] d T dv '^d-Q^ + rudi^dn-^ = OL (rn-1 " ^T^ + r^+i),
(8)
where r^ = A^/a, A^ is the deviation of the dust particles from its equilibrium position, a the mean lattice spacing, and the coupling constant is defined as a = (5^exp(—(^)[1 + (1 + 0^]/^^5 where we have assumed that negatively charged dust grains carry the same charge Qd, and ^ = a/A/). The constant Vdn = 2y/TxpgR?'Cs represents the dust-neutral collision frequency, where pg is the neutral gas density and Cs the thermal speed of the neutral gas atoms. The expression for Vdn holds as long as the mean free path for the gas molecules is much larger than the particle size. Making use of the Bloch condition, (8) can be written as[24, 25]
PK. Shukla/Perspectives of collective processes in dusty plasmas
^
+ i^an^
+ Pism\ka/2)ro
= 0,
(9)
where /3/ = Aa/ruda and k the wavenumber. The subscript 0 denotes the origin. The frequency of the DLW, which is deduced from (9), is given by uj'^ + ivdn^ - (5ism^{ka/2) = 0.
(10)
We note that the hnear dispersion relation (10), which describes the propagation and damping of the dispersive DLW, has been experimentally verified[26, 27]. The nonlinear propagation [24] of finite amplitude DLWs in a coUisional dusty plasma is governed by the modified Boussinesq equation ^
+ ^ . n ^ - % ^ ^ - t ; o ^ - Y ^ a — +
—
^
= 0,
(11)
where rjd represents a kinamtic viscosity, VQ = y/Pia is the phase velocity, u = dro/dz^ and 7z = {GQl/a"^) e x p ( - 0 ( l + ^ + ^V^ + ^V^)- For nonlinear DLW waves with phase velocity close to t;o) (H) ii^ ci moving frame (Z = Z — VQT^ t = T) can be written as the modified Korteweg-de Vries-Burgers equation
Equation (12) admits both rarefactive soliton as well as shocks. The rarefactive dust lattice soliton appear in a coUisionless dusty plasma with Vnd — 0 and ry^ = 0. On the other hand, in a coUisional plasma, (12) predicts monotonic and oscillatory dust lattice shocks. The latter can cause the melting of dust crystals due to the shock heating.
4
Instabilities of dusty plasma waves
The dusty plasma waves, as discussed in section 2, can be excited provided that there exist free energy sources. The latter include streaming beams of charged particles[28, 29, 30, 31], radio-frequency and laser beams, etc. In the following, we present two examples of linear instabilities that could be responsible for the excitation of dust acoustic waves. We first consider the kinetic instability of the DAWs in a coUisionless dusty plasma. Here, for Vtd « ^/k « Vte^vu and \u — kzUio\/k « Vu^ Eq. (1) reduces to 1
.
1
/7r\ V2 ijj - kujo _ (4d
Equation (13) admits an oscillatory instability of the DAW when Uio > (jOr/k » uji, where uOr = kX]:,ujpd/{l + A:^A|,)^/^ and the growth rate is given by
PK. Shukla/Perspectives of collective processes in dusty plasmas
UJI
1
\kUiQ-UJr\
, ..
Second, we focus on the ion-dust two-stream regime, where ^e < < 1? ^ind ^i, Cd » 1- For Ud « uj « Ue and Ui « |a;'|, where uj' = uj — kuio, (1) gives
where A = l + {l/k'^Xj)^).
In the absence of ion-dust colhsions, Eq. (15) for kuio
LO gives ujr '^ uji ^ (uJpiUJp^] /VA,
with a maximum growth rate at kuio ~
On the other hand, in a coUisional dusty plasma with uj « has the approximate solution[19] (1 + i) fu;^\'/'
1
»
ujpi/vA.
kUio ~ ujpi/y/A, i/j, (15)
..„.
which yields a dissipative instabihty.
5
Wakefield
In this section, we point out the importance of collective effects with regard to the generation of a wakefield in dusty plasmas. The wakefield, which is responsible for the attraction of charged dust grains of like polarity, arises due to the resonance interaction of a test dust charge which moves with a velocity close to the dustacoustic velocity Q . The wake potential of a test dust charge in the presence of the DAW in an unmagnetized plasma is found to be[32, 33] (t>w{p = 0,^t.t) = fcos{^t/L),
(17)
where ^t = \z — Vtt\, qt is the charge of the test particle L — Xjj [{vt — VQ)^ — c^] /Q is the lattice spacing, Vt is the test charge velocity, VQ is the equilibrium ion streaming velocity, and p and z are the radial and axial coordinates in a cylindrical geometry. For \vt — Vol ~ 30 c m / s , A^^ ~ 300/xm, and Q ~ 6 cm/s, we find L ~ 1 mm, which is in agreement with the observation[6]. The physics of the charged dust attraction is similar to the electron attraction in superconductors in which Coopers pairs are formed due to collective interactions involving phonons. In dusty plasmas, the latter are replaced by the DAWs, and negatively charged dust grains feel an attractive force in the negative part of the oscillatory potential (17), where the positive ions are focused. The wake-potential concept for charged dust attraction seems to be verified both in laboratory[34] and computer simulation[35].
10
6
RK. Shukla/Perspectives of collective processes in dusty plasmas
Nonlinear waves
The nonlinear waves in include solitons[l], shocks[36, 37], and vortices[38]. To the best of our knowledge, there are no observations of solitons in a dusty plasma. However, a recent laboratory experiment[37] has observed the compressive dust ionacoustic shock in a weakly coupled dusty plasma. The dynamics of that DIA shock is governed by
where $ = e(/>/Te, A^ = 5 + 3T, 5 = rtio/rieo, ^i = i^id/^pu Vi ^ ^iKnl^])e^ ^^d S = (3(5 — 1 + 12a)/5, Furthermore, (/) the electrostatic potential, Uid the ion-dust collision frequency, and A^ represents the effective mean-free-path. The time and space variables in (18) are in units of the ion plasma period {l/(jOpi) and the electron Debye radius X^e — '^tel^pe- An equation similar to (18) can also be obtained for the nonlinear dust acoustic waves. While the ion-dust drag is responsible for the monotonic dust ion-acoustic shock, the dust-acoustic shock can be formed even in the presence of dust charge perturbations[36]. It is suggested that future experiments should be designed to observe the dust lattice and dust acoustic shocks in an unmagnetized dusty plasma.
7
Discussion
In this paper, we have discussed the present status of waves, instabilities, and nonlinear structures that are observed in unmagnetized dusty plasmas. Both the laboratory experiments and computer simulations have conclusively verified the spectra of DIA and DA waves. The latter are excited by streaming ion beams in a dusty plasma. On the other hand, the DLW is excited by the radiation pressure of a modulated laser beam in rf discharges. Furthermore, we have discussed the physical mechanism for ion focusing and charged dust attraction in an oscillatory wakefield that is created by the motion of a test dust charge in the presence of DAWs in a weakly coupled dusty plasma. It may well turn out that the wakefield concept should still prevail in a strongly coupled dusty plasma in which the dielectric response function of the DAW is rather complex. However, the wakefield calculation has yet to be performed in a highly coUisional strongly coupled dusty plasma, taking into account dust-neutral and dust-dust interactions. Finally, we have also discussed the nonlinear properties of dust acoustic and dust lattice waves in a coUisional dusty plasma. The dynamics of experimentally observed dust ion acoustic shocks is governed by the KdV-Burgers equation. On the other hand, analytical and numerical models for vortical structures, which are observed in strongly coupled laboratory dusty plasmas (even under micro-gravity), have to be worked out. Vortices are formed in the presence of sheared plasma flows, the dust angular rotation, and the temperature and density gradients.
P.K. Shukla/Perspectives of collective processes in dusty plasmas
11
Acknowledgments: The author is grateful to Gregor MorfiU for useful discussions and support. This work was partially supported by the Max-Planck Institut ftir Extraterrestrische Physik, Garching.
References [1 N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 (1990). [2 J. Chu and Lin I, Phys. Rev. Lett.72, 4009 (1994). [3 H. Thomas, G. E. MorfiU, V. Demmel, J. Goree, G. Feuerbacher, and D. Molmann, Phys. Rev. Lett. 73, 652 (1994).
[4 H. Ikezi, Phys. Fluids 29, 1764 (1986). [5 P. K. Shukla and V. P. SiUn, Physica Scripta 45, 508, 1992; see also P. K. Shukla, Physica Scripta 45, 504 (1992).
[6 J. H. Chu, J. B. Du, and Lin I, J. Phys. D: Appl. Phys. 27, 296 (1994). [7 A. Barkan, R. L. MerUno, and N. D'Angelo, Phys. Plasmas 2, 3563 (1995).
[8] C. Thompson, A. Barkan, N. D'Angelo and R. L. Merlino, Phys. Plasmas 4, 2331 (1997).
[9] R. L. Merlino, A. Barkan, C. Thompson, and N. DAngelo, Phys. Plasmas 5, 1607 (1998).
[10 J. B. Pieper, and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). [11 H. R. Prabhakara and V. L. Tanna, Phys. Plasmas 3, 3176 (1996). [12 A. Barkan, N. D'Angelo, and R. L. Merlino, Planet. Space Sci. 44, 239 (1996). [13 P. K. Shukla, Phys. Plasmas 1, 1362 (1994). [14 K. Miyamoto, in Plasma Physics for Nuclear Fusion (MIT Press, Cambridge, 1989).
[15 R. K. Varma, P. K. Shukla, and V. Krishan, Phys. Rev. E47, 3612 (1993). [16 P. K. Shukla, in Physics of Dusty Plasmas, Editors: P. K. Shukla, D. A. Mendis, and V. V. Chow (World Scientific, Singapore, 1996), pp. 107-121.
[17 P. K. Shukla, in Physics of Dusty Plasmas, Editors: M. Horanyi, S. Robertson, and B. Walch (AIP, New York, 1998), pp. 81-96.
[18 P. K. Shukla, G. T. Birk, and G. MorfiU, Physica Scripta 56, 299 (1997).
12
P.K. Shukla/Perspectives of collective processes in dusty plasmas
[19] M. Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997). [20] M. S. Murillo, Phys. Plasmas 5, 3116 (1998). [21] U. de Angelis and P. K. Shukla, Phys. Scripta 60, 69 (1999).
Lett.
A 244, 557 (1998); Physica
[22] P. K. Kaw and A. Sen, Phys. Plasmas 5, 3552 (1998). [23] P. K. Shukla and M. Rosenberg, Phys. Plasmas 6, 1038 (1999). [24] F. Melands0, Phys. Plasmas 3, 3890 (1996). [25] H. M. Thomas, R. J. Jokipii, G. E. Morfill, and M. Zuzic, in Strongly Coupled Coulomb Systems, Editors: G. Kalman et al. (Plenum Press, New York, 1998), pp. 187-192. [26] G. Morfill, G., H. Thomas, and M. Zuzic, in Advances in Dusty Plasmas, Editors: P. K. Shukla, D. A.Mendis, and T. Desai, 1997, World Scientific, Singapore, pp. 99-142. [27] A. Homann, A. Melzer, S. Peters, R. Madani and A. Piel, Phys. Lett. A 242, 173 (1998). [28] M. Rosenberg, Planet. Space Sci 41, 229 (1993). [29] M. Rosenberg, J. Vac. Sci. Technol. 14, 631 (1996). [30] D. Winske, S. P. Gary, M. E. Jones, M. Rosenberg, V. W. Chow, and D. A. Mendis, Geophys. Res. Lett. 22, 2069 (1995). [31] D. Winske and M. Rosenberg, IEEE
Trans. Plasma Sci. 26, 92 (1998).
[32] M. Nambu, S. V. Vladimirov, and P. K. Shukla, Phys. Lett. A 203, 40 (1995). [33] P. K. Shukla and N. N. Rao, Phys. Plasmas 3, 1770 (1996). [34] K. Takahashi, T. Oishi, K. Shimomai, Y. Hayashi, and S. Nishino, Phys. Rev. E58, 7805 (1998). [35] G. Lapenta, Phys. Plasmas 6, 1442 (1992). [36] F. Melands0 and P. K. Shukla, Planet. Space Sci. 43, 635 (1995). [37] Y. Nakamura, H. Bailung, and P. K. Shukla, Observation of ion-acoustic shocks in a dusty plasma, Phys. Rev. Lett. 83, No. 6, in press, (1999). [38] P. K. Shukla, M. Y. Yu, and R. Bharuthram, J. Geophys. Res. 96, 21343 (1991); M. SalimuUah and P. K. Shukla, Phys. Plasmas 5, 4502 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
13
Non-Ideal Effects in Dusty Plasmas Nagesha N. Rao^ Theoretical Physics Division Physical Research Laboratory Navrangpura, Ahmedabad-380009 INDIA
Abstract. Electrostatic waves and instabilities in weakly non-ideal magnetized dusty plasmas have been investigated by incorporating the van der Waals equation of state as well as the grain charging equation. For the homogeneous case, the propagation of linear dust-acoustic waves (DAWs) has been considered. It is found that the volume reduction coefficient enhances the DAW phase speed while the molecular attractive forces lead to a decrease in the speed. In the high temperature limit, there is a net increase in the DAW phase speed while near the critical point the phase speed is reduced. Inhomogeneous magnetized dusty plasmas support density as well as temperature gradient driven electrostatic drift waves in the low-frequency regime. On the other hand, when shear flows are present, dusty plasmas are susceptible to (parallel) Kelvin-Helmholtz (K-H) instabilities. In addition to the usual density gradient driven K-H (DGKH) instability, we point out the existence of a new type of temperature gradient driven K-H (TGKH) instability. At higher frequencies near the dust-gyro frequency, electrostatic dust cyclotron waves (EDCWs) can be driven unstable due to sheared flows. In aU cases, dust charge fluctuations lead to damping effects, thereby reducing the instability growth rates.
I.
INTRODUCTION
The presence of finite-sized, charged particulate matter in plasmas leads to a host of novel collective phenomena in plasmas. The so-called dusty plasmas contain grains in the s u b - or super-micron range and are highly charged, typically to about a fev^ thousand electrons. In the ultra-low frequency regime, dusty plasmas support new kinds of waves and instabilities arising due to the dust collective dynamics [1]. Examples are the Dust-Acoustic Waves (DAWs) [2], the Dust-Ion-Acoustic Waves (DIAWs) [3], and the Kelvin-Helmholtz (K-H) instabihties [4,5]. In laboratory dusty plasmas, such waves have been observed, and the frequency of DAWs is usually about a few hertz [6,7], while in space and astrophysical plasmas it could even be much smaller. ^) E-mail :
[email protected]
14
N.N. Rao/Non-ideal effects in dusty plasmas
When the grains are in the super-micron range, dusty plasmas can exhibit n o n ideal behavior. First, large grains are highly charged and hence particle correlations as well as intergrain molecular forces come into play. Second, volume reduction contribution can be significant at higher gas densities, particularly for grains in the few tens of micron range. For example, for grains with a radius of about 50 microns, the volume reduction contribution is about 10% for dust gas density of about 10^ particles/cc. Third, grain charge can fluctuate because of the equilibrium charging processes as well as due to the wave induced perturbations in the electron and the ion currents flowing onto the grain surface. Research studies beginning in this decade have mostly concentrated on the ideal response of dusty plasmas, while parameter regimes wherein non-ideal contributions become signiflcant are within the scope of present day laboratory experiments on dusty plasmas. Recent analysis [8,9] of DAW propagation in non-ideal dusty plasmas has shown that the volume reduction enhances DAW phase speed while the molecular cohesive forces lead to a reduction in the speed. On the other hand, charge fluctuations are known to result in damping of waves [10,11], which otherwise would propagate as normal modes. In this paper, we present a systematic analysis of electrostatic waves and instabilities in a weakly non-ideal low-^ magnetized dusty plasma by including shear in the dust flow velocity as well as the charge fluctuation effects. Non-ideal effects are modelled through the van der Waals equation of state for the dust fluid. First, we consider the low-frequency regime and discuss electrostatic modes such as the DAWs, density and temperature driven drift waves and the electrostatic dust cyclotron waves (EDCWs). Next, we discuss the onset of Kelvin-Helmholtz (K-H) instabilities driven by shear flows. The existence of a new kind of K - H instability driven by dust temperature gradient is pointed out. Damping effects on the instability growth rates due to charge fluctuations have been computed.
II.
BASIC EQUATIONS
For low-frequency phenomena, we follow the dusty plasma model first suggested by Rao et al [2] wherein the electrons and the ions are assumed to behave like ideal massless thermal fluids in equilibrium at their respective temperatures, while the wave dynamics is governed by the heavier dust component. Accordingly, the electron and the ion number densities are given by the respective Boltzmann distributions rie = rieo exp {e(j)/KBT^),
n,- = riio exp {-e({)/KBTi),
(1)
while the dust component is governed by the fluid equations
(2)
{dnJdt) + V'{n,v,) = 0, n,m, [{dvjdt)
+ {v, . V)v,] = -n,q,
Vcf> + {n,qjc)[v,
X B,) - V p „
(3)
15
N.N. Rao/Non-ideal effects in dusty plasmas where the notations are standard [12]. The grain charge variable q^ is determined by the charge current balance equation
(4)
{dqJdt) + {v,-V)q, = h + Ii, where the average electron (/g) and ion (7^) currents are given by [13] /e = -TreR'ried
exp V Trme
—— , \f^B^eJ
7, = ireR riiJ V ^^i
1 V
— , f^B-^iJ
(5)
and ij; = qd/R is the dust surface potential relative to the plasma potential where R is the grain radius. The dust fluid is described by the van der Waals equation of state [14] {p, + Anl)
{l-Bn,)
= n,K,T„
(6)
where the gas constants A and B are given by A = dKaTc/Sric and B = l/Sn^; here, the subscript 'c' denotes the respective values at the critical point. Note that for A, 5 -^ 0, Eq. (6) reduces to the ideal gas law. Equations (l)-(6) are closed by the charge neutrality condition, qdn^ + eni — erie = 0, and thus constitute a complete system of governing equations to describe lowfrequency phenomena in non-ideal dusty plasmas. We now discuss the steady state equilibrium relevant to the (parallel) K - H configuration. Accordingly, there exists in equilibrium a transverse shear in the dust flow velocity component parallel to the magnetic field, BQ = BQZ. The equilibrium dust flow velocity is thus represented by, v^o = VyV + K^? where the components Vy and Vz are to be suitably evaluated consistent with Eqs. (2) and (3). The latter are identically satisfied provided the equilibrium quantities n^o? ^o-i Pao ^^d T4 are functions of X only, while V^ is a constant which is calculated from the i-component of Eq. (3) as, Vy = {c/Bo){d(l)o/dx) + {c/ndoqdoBo){dpdo/dx), Note that the right-hand —•
—*
side contains E x B a.s well as diamagnetic drifts. In equilibrium, the pressure gradients give rise to the dust diamagnetic drift defined by, V^ = yV^ where V^, = {dp^oIdx)j771^71^,0^1^0- In particular, when p^^ given by Eq. (6), we obtain V^ = ^{)^iVdn + ^2VdT) where Ai = 9(3 - T/)"^ - 9a7//4 and A2 = 3/(3 — r/) defined in terms of the non-ideahty parameters, a = T^Td and 7/ = rido/ric. Note that for an ideal dust fluid, a, 7/ —> 0, and hence Ai, A2 ^^ 1. Here, Vdn and VdT are, respectively, the characteristic drifts in the presence of density and temperature gradients, and are given by
In Eqs. (7), CDA = <^pd^D is the DAW phase speed [2], Upd = {inndoql^/may^'^, ^do = q,oBo/m,c, A;2 ^ ^ - 2 + x-2^ ^ ^ V,yCl^, and V^ = {K,Td/m,y/\ The corresponding diamagnetic drift frequencies are then defined by u^n = kyV^n and <^dT = K^dT-
16
N.N. Rao/Non-ideal effects in dusty plasmas
III.
GENERAL DISPERSION RELATION
For weak gradients in the equilibrium quantities, a general dispersion relation can be obtained by linearizing Eqs. (l)-(6) together with the charge neutrality condition. Assuming the perturbed quantities to vary as ~ exp [i {kyy + k^z — cjt)], we obtain the dispersion relation [12] Cs [u' - kld^C,)
{u' - ill,) -
= firfo {u^K' + K^doS)
^'kld^C^Cs [uVn + kyCl^CsCs)
+ unlkyVo,
where UJ = u; — k - v^o 'is the Doppler shifted wave frequency, S = characterizes the shear flow, and
(3 - r]Y
4
(cji - luj)
(8) {dVzldx)IVtdQ
C3
where the charging frequencies cji and LJ2 are given in Ref. [13]. The dimensionless parameter / = 47rndo^A^ is a measure of the dust grain packing, and the dust component may be considered as tenuous or dense according as /
IV. (A)
ELECTROSTATIC MODES Low-frequency regime : uJ
In the low-frequency regime, the general dispersion relation (8) admits d u s t acoustic waves (DAWs) and drift waves, which we discuss in the following.
(1) Dust-Acoustic Wave (DAW) Consider first the case of homogeneous dusty plasmas without the shear flow as well as the charge fluctuation. For parallel propagation {ky = 0), dispersion relation (8) yields [8,9], LJ^ = klCl{l + e), where Cl = Cl^ + V^l is the DAW phase speed including the dust thermal contribution, while the non-ideal contributions are given by e = e^r + ^c/ with e^^ = ^7/(6 — 7/)/(3 — 7/)^, e^f = —9a^7//4, and ^ = /?/(l + ^ ) . The quantities e^^ and e^^ represent, respectively, the contributions due to the volume reduction coefficient and the attractive cohesive forces. When charge fluctuation as well as sheared flows are included, we obtain the generalized DAW dispersion relation.
A^.A^' Rao/Non-ideal effects in dusty plasmas
17
w\\ + klpl.C) = klCl.C, - kyKCl.SCs,
(10)
where p^A = CD^I^dQ- Charge fluctuation lead to a damping of DAWs, which for weak damping can be calculated as earlier [10,11].
(2) Density Gradient Drift Wave (DGDW) Consider the low-frequency hmit of Eq. (8) for a cold inhomogeneous dusty plasma when the wave propagation is almost perpendicular to the magnetic field but with a small kz so that the electrons and ions can flow along the field lines to maintain quasi-neutrality. Equation (8) reduces to (11)
U=-kyVUCz^klpl,)-\
When the charge fluctuation effects can be neglected, that is, Cs ^^ 1, Eq. (11) becomes
a; =-u,l(l + ^ > L ) - ' = ^ 1 -
(12)
Equation (12) represents a density gradient driven drift wave in a cold dusty plasma [15]. For weak damping, we write cJ = cJ^. + Z7 with I7I
while the damping rate is given by
7 « -Mu;^[(l +fc>L)(^?+ ^l)]-'-
(13)
It should be noted that the damping rate vanishes for a tenuous dusty plasma (/-O).
(3)
Temperature Gradient Drift Wave (TGDW)
We now show the existence of a novel kind of drift wave driven purely by dust temperature gradient even when the density gradient is absent. For small kz^ dispersion relation (8) becomes u = -/3\2kyVdT
[Cs + A;>L(1 + l3CiC3)]~'.
(14)
For the case when charge fluctuations can be neglected, above equation yields LJ - -^A2a;^^ [1 + klpl^il
+ e ) ] " ' = uj"-,^,
(15)
The dispersion relation (15) is similar to (12), and represents temperature gradient driven drift wave (TGDW) [12] in a warm dusty plasma. For a tenuous plasma when the damping is weak, Eq. (14) yields real frequency as uJr ~ ^ T whereas the damping coefficient (7) is given by 7 « -fu,oJl
[1 + kliPpl,
where p^ = Cn/^do[cf. Eq. (13)].
+ epD] [{ul + Ul) {1 + klplil
+ e)}]"',
(16)
This is similar to the damping rate obtained for the DGDW
18
N.N. Rao/Non-ideal effects in dusty plasmas
(B) Dust gyro-frequency regime : u; ~ Qdo Consider first the case of homogeneous plasmas. For almost perpendicular propagation {ky ^ kl) without shear flows, Eq. (8) becomes
^
[uji + fuj2 - ^(^)
+
^y^DA
9^
9a/37]
(17)
(3 - IJY
which is a generahzed dispersion relation for the electrostatic dust cyclotron wave (EDCW) modified by dust charge fluctuations (the third term) as well as non-ideal contributions (the fourth term). For an ideal plasma with constant dust charge, we recover the usual dispersion relation for EDCWs [16,17], cJ^ = f]^o + ^I^DA0^ the other hand, for the non-ideal case with weak damping, we obtain
K =fi^o+ KCii^ + ^)'
2 , —2\-l
7 = -..fKci.^M
+ K)
(18)
Shear flow effects manifest themselves for finite, non-zero A:^. Accordingly, for homogeneous plasmas, Eq. (8) reduces to ^2 \
—2 7.2/-f2
(u;^ - klci.Oiu' - nl,) - uj^ci^c, = kyKni,cl,c,s,
(19)
which shows a coupling between DAW and EDCW. For the case when / -^ 0 so that charge fluctuation damping can be neglected, Eq. (19) has the roots LJ± given by u, —
mdo
2 L
(1 + k'pl^C)
± {(1 + Ppl.Cf
- Akyk.pl^K
-
S)}
1/2
(20)
where K = kjky, P = k^ + kl and C = (1 + ^ C i ) . The mode io = a;+ is the usual E D C W modified by the dust shear flow, while cJ == a;_ is a low-frequency mode which becomes unstable for S > K. This can be seen for the case when \K^ZPIAC{I^ - S)\ < (1 + k'^pljfif, which leads to 2^2
\-l
wl« ni, + k'ci.c + kyk,ci,c{s - K)(I + k'pl,c)-\ \-l
ui = -kyKCi,c{s - /c)(i + fcX.c)
(21)
(22)
Thus, a;_ is the purely growing zero-frequency unstable mode for S > K. The effects of inhomogeneities on the modes cJ^ and a;_ have been discussed elsewhere [12].
19
KN. Rao/Non-ideal effects in dusty plasmas
V.
KELVIN-HELMHOLTZ INSTABILITIES
Dusty plasmas are susceptible to (parallel) Kelvin-Helmholtz (K-H) type of lowfrequency (aJ
+ U [uj*,,Cs + 0C;' (1 + KS)r] - kyKCl.C,
{K - S) = 0, (23)
where r = (^I^^7I+'^2<^^T)- Equation (23) includes non-ideal and charge fluctuation effects, and has been investigated elsewhere [18] for the onset of K-H instability. We consider below two simpler cases depending on the nature of plasma inhomogeneity.
(A)
Density Gradient K - H (DGKH) Instability
For a cold dusty plasma having only density gradient. Eq. (23) becomes u' {C, + klpl,)
+ u;luJ - kyk,Cl,{^
- 5) = 0.
(24)
Neglecting the charge fluctuation damping, the DGKH instability occurs when the shear flow satisfies the condition
^ ? «+ Vi[4«CL(l +
fcJ/^L)]"'
for K^O.
(25)
The critical shear for the onset of the DGKH instability is given by, Sc = VdnlCoA = CDA/^doLn where we have assumed kyP^,^
(1 + |e + 1^) (1 + f6)-'/\
(26)
where 6 = L02liOi. Thus, thermal effects increase the critical speed while the sign of the non-ideal contribution e depends on relative values of a and r/. Denser dusty plasmas have smaller critical flow speed for the onset of DGKH instability.
(B)
Temperature Gradient K - H ( T G K H ) Instability
We now show the existence of a novel type of K-H instabihty driven purely by dust temperature gradient. In the low-frequency limit, Eq. (23) yields u' (l + klpl, C,) + ^A^a^^^C"' (1 + /c5) cJ - kyk^Cl^C,
{K-S)^
0,
(27)
which is similar to (24). When charge fluctuation can be neglected, TGKH instability is excited when the shear flow parameter satisfies
20
N.N. Rao/Non-ideal effects in dusty plasmas
S^K+|3'\lVJr[^KCl,^^{l
+ klpl,f^)]-'
for
K^O,
(28)
where // = {I + e){l + jS), The critical shear for the onset of the T G K H instabihty is given by Sc ~ l^^2CDA/y/l^^doLT' If A l ^ now denotes the relative flow speed between adjacent layers over scale-length L^^ the T G K H instability is excited provided AT4 ^ /^Cn- Since for typical dusty plasmas ^9
(29)
Comparing this expression with (26), we note that the contributions due to the non-ideal as well as thermal effects to the DGKH and T G K H instabilities are quantitatively opposite in nature. In both cases, charge fluctuations lead to damping effects and thus result in a reduction in the growth rate of the instabilities [5]. For the general case, the damping rate is obtained as 7 = - ( M o F , ) [2u;,(l + ixklpl^)
+ / / < + /?(1 + KS) (Aia;^„ + \^u*,^)]"',
(30)
where Ur is the real part of the frequency.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Verheest, F., Space Sci. Rev. 77, 267 (1996). Rao, N.N., Shukla, P.K., and Yu, M.Y., Planet. Space Sci. 38, 543 (1990). Shukla, P.K. and Silin, V.P, Physica Scripta 45, 508 (1992). Rawat, S.P.S. and Rao, N.N., Planet. Space. Sci. 4 1 , 137 (1993). Singh, S.V., Rao, N.N. and Bharuthram, R., Phys. Plasmas 5, 2477 (1998). Chu, J.H., Du, J.B., and Lin, I., J. Phys. D27, 296 (1994). Barkan, A., Merlino, R.L., and D'Angelo, N., Phys. Plasmas 2, 3563 (1995). Rao, N.N., J. Plasma Phys. 59, 561 (1998). Rao, N.N., Physica Scripta T75, 179 (1998). Varma, R.K., Shukla, P.K. and Krishan, V., Phys. Rev. E47, 3612 (1992). Melandso, F.J., Aslaksen, T. and Havnes, 0., Planet. Space Sci. 4 1 , 321 (1993) Rao, N.N., Phys. Plasmas 6 (6), June 1999. Rao, N.N., and Shukla, P.K., Planet. Space Sci 42, 221 (1994). Joos, G., Theoretical Physics, New York : Dover, 1986, p. 497. Shukla, P.K., Yu, M.Y. and Bharuthram, R., J. Geophys. Res. 96, 21343 (1991). Shukla, P.K. and Rahman, H.U., Planet. Space Sci. 46, 541 (1998). D'Angelo, N., Planet. Space Sci. 46, 1671 (1998). Rao, N., in Proceedings of the Second International Conference on the Physics of Dusty Plasmas, Hakone (Japan), 24-28 May 1999, Contributed Papers.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
21
The Kinetic Approach to Dusty Plasmas V.N.Tsytovich General Physics Institute, Russian Academy of Sciences, Moscow U . d e Angelis Department of Physical Sciences,University of Naples "Federico IF',Italy ABSTRACT: The formulation of a generalized Bogolyubov-Klimontovich kinetic approach to dusty plasmas is presented,pointing out basic differences with the physics of multicomponent plasmas,taking into account the absorption of plasma particles on dust grains. The theory is valid for a parameter regime where the binary plasma particle collisions can be neglected with respect to their collisions with dust particles.
In a dusty plasma the solid particles (dust grains) can acquire and maintain an electric charge absorbing plasma particles from the plasma fluxes to their surfaces. A dusty plasma is therefore an open system which requires a plasma source and the constant presence of plasma fluxes to the dust particles.The discreteness of the grains and this continuous charging process can produce important differences from a multi-component plasma: the dust charges fluctuate in response to plasma current fluctuations and this can alter the interactions appreciably; plasma fluctuations are induced by the dust fluctuations (even when the plasma discreteness is neglected and the plasma is treated as a continuous fluid) and these generate induced correlations between plasma particles; the particle absorption and the inelasticity of plasma-dust collisions can create non-thermal distributions of plasma particles and modify the screening which,especially for the attracted species,can change drastically from the usual Debye screening (as well known to people working with plasma probes); new "friction" and "attraction" forces can be present in the system[3]. All these new effects can only be investigated in the framework of a kinetic theory which should be consistently formulated with explicit account for the absorption of plasma particles. In this case the charge g on a grain depends on the whole time history of that particular grain.A way to avoid this problem is to consider the charge g as a new phase variable in dust phase-space.In Ref.[l] a consistent kinetic theory has been formulated introducing the distributions of the plasma particles (a = {e.i}) fp{r.t)
(normalized to
22
VN. Tsytovich, U. de Angelis/The kinetic approach to dusty plasmas
the particle densities Ua) and the distribution of dust particles fryt{iL^q.t) (normalized to the dust density rid)., where p = mcV,p' = m^v' are the momenta (the velocity and momentum of dust particles are always denoted with a prime in the following). These satisfy the Klimontovich equations
I + V . ^ + gE • A ^ /p'(r,g,t) + |{/e.. + E4)/^,(r,q,t) = 0
(2)
73
Ia{q,r,t)
= j e^a^{q,v)vf^{Y,t)—-^
(3)
where 5^ describes any external source of plasma particles, ao,{q^v) are the cross-sections for the charging collisions Jo,{q^T^t) are the plasma currents to a grain of charge q at position r and I^^t includes any other possible current (photoelectrons,secondary emission...). For comparison the Klimontovich equations for the case of a three-component plasma (dust as a plasma component with fixed charge erf)is given by Eqs.(l) with the right-hand side equal to zero and a = e, i, d. It is important to point out that the Klimontovich equation (2) for the dust distribution function follows from a generalized Liouville equation which includes the new phase variable q for the dust particles (see ref.[l] for details). In this approach the electron/ion binary collisions are neglected as compared to the collisions with dust and only the discreteness of the dust component is taken into account while the electrons and ions are treated as continuous fluids in 6(f phase space {p,r} interacting with the individual dust particles. The discreteness of the dust component creates fluctuations in all the components of the dusty plasma including electrons and ions which have only the fluctuations induced by dust fluctuations. The theoretical procedure is based on the generalization of the Bogolyubov - Klimontovich scheme [4,5]. The single-particle dust distribution function fluctuates and can be separated into the regular and fluctuating parts. The plasma particle distributions will
23
V.N. Tsytouich, U. de Angelis/The kinetic approach to dusty plasmas also fluctuate, due to the interactions with dust particles, so that for all components and for the electrostatic field it is possible to separate the regular and fluctuating parts :
/p = ^p + <^/p,^p=, /p'(9) = $p'(9)+<J/^'(9), *p'(?)=,E=<E>+JE where the brackets denote the statistical averaging on the ensemble of the dust particles since the fluctuations are related only with dust discreteness. For the average distributions (we omit the arguments r, f and take < E > = 0 for simplicity)the equations are :
^ { [h.i + Y^iUq))) ^^'(?) + Ti^lMSf'^M))^
(5)
where we have defined the plasma-dust collision frequency
(6)
t^
and SIct{q) are the fluctuating parts of the currents from Eq.(3). For a three-component plasma only the first term on the right-hand side of Eqs.(4) survives : it is known as the ''collision integral" .The charging process introduces new collision integrals in the theory,all of which depend on the cross-sections for the charging collisions. The equilibrium charge on a grain,5'eg?depends on the average plasma distributions and is the solution of the equation for the net current : 73
lext + J2{Ia{qeq))
= hxt + ^
I ^a^^a{qeq,v)<^'^——
= 0
(7)
The Qeq is a slowly varying function of r and t in accordance with the changes of $ p in space and time.This equation can be used for the study of the charging process self-consistently
24
VN. Tsytovich, U. de Angelis/The kinetic approach to dusty plasmas
when the plasma distributions are taken as the solutions of the kinetic equations above where the same charging process is taken into account. For the fluctuating parts we give the equations for the Fourier components:
where we have defined 0 = a' — k • V and :
<^ V,k, J-) = / ^^4?'. ')%').,M') ^
(10)
For a three-component plasma the fluctuating parts are given by Eqs.(8) (a = e^i^d) with jy^.a = ^ ^^rfaku; ~ ^* ^'^^ third term in the right hand side of (4) describes the average recombination of plasma particles on dust particles and the last term describes the plasma-dust interactions in the charging process. The terms with a g-derivative in Eqs.(5,9) arise from the charging process and are also completely new with respect to the theory of multi-component plasmas. The term x/^^a in Eq.(8) also arises from the charging process and leads to an effective damping of plasma fluctuations. In the derivation of Eqs.(8,9) higher order terms in the fluctuations
have been neglected and the regular parts of the distributions have been
assumed constant on the fast time and space scales of the fluctuations. The system of equations above is valid for any values of the dust charge fluctuations.In the following the assumption of small charge fluctuations around the equilibrium value is used,expanding in Aq = q — q^^ the average current. The procedure is to solve Poisson's equation for the fluctuating field and the equations for the fluctuations separating the solution into ''free fluctuations" (the solution of the homogeneous equation) and induced fluctuations : (J p ,
= <5/^P "'"^•^nk V ^^^Pi'^ss the latter and the field in terms of the free fluctuations
and then perform the statistical averages appearing in the collision integrals using the well known statistical property of the free fluctuations[4,5]. In the present theory only
25
VN. Tsytovich, U. de Angelis/The kinetic approach to dusty plasmas the dust particles have free fluctuations related to their discreteness while the plasma particles (a continuous fluid)have only induced fluctuations [Eq.(8)].Therefore the solution of Eq.(9) is taken in the form SfL{q) = 5/p, {q)+Sfp,
(q) and from the solution of the
homogeneous Eq.(9) (right hand side equal zero) it is possible to find the generalization of the basic statistical property (see Ref.[l] for details) in the form : ^^fi'^ij^'l^^rk.'(^'))
= *P"^(P' - P")'^(^ - k • ^')^i^ + '-')<5(k + k')<5(g - q')5{q - q^,) (11)
The solution of Eq.(9) for the induced part of the dust fluctuations can be found using the Green function for the differential equation in q , G{q, g', a; — k • v') ,in the form:
^^'^tl^'i) = IGM:^ - k • v')i?p,k,j9')d5'
(12)
where the Green function is given explicitly in Ref.[l] and -Rp/^^ stands for the right hand side of Eq.(9). The electric field fluctuations from the Poisson equation :
can then be written in terms of the free fluctuations in the form :
where the second equation gives the result for a three-component plasma with grains of fixed charge g^^. The expression (14) contains the dielectric function €j^^ (which includes the effects of the charging process)and an ''effective charge" c^J^ both defining the interactions in dusty plasmas. These quantities are given in Ref.[l]. Without the charging process {GQ, = 0) the dielectric function reduces to the well known form for three-component plasmas [denoted as ef
in (14b)] and the effective charge of interaction
reduces to q^^. The effective charge determines the strength of the interactions, can be quite different from q^q (as shown later) and depends on distance. Using these results all dust and plasma induced fluctuations can be expressed in terms of the free fluctuations so that the property(ll) can be used to calculate the collision integrals and write the coupled system of equations for the average distributions. The equation for the dust distribution
26
VN. Tsytouich, U. de Angelis/The kinetic approach to dusty plasmas
takes into account the changes of both momentum and charge in the interactions and its full expression is given in Ref.[l]. By integration over the dust charge q or over momentum it is possible to find equations for the reduced dust distribution functions :
$^,(r,t) = j¥p,{r,t,q)dq;
^'{v,t,q) = J ^^^,{T,t,q)-0^
(15)
the latter giving an equation for the distribution in charges (it will not be given here). The two coupled kinetic equations are conveniently written in the forms :
(17) For a three-component plasma the well known result is given by Eqs.(16) (for all components)with only the third term present on the right-hand side and the diffusion coefficient Dfj
given by the usual Lenard-Balescu expression. The diffusion coefficients
in dusty plasmas include the modifications of the Coulomb collision integral by the dust charging processes and differ from the Landau-Balescu expression (see Ref.[l]). The new "friction forces" in Eqs.(16,17) are determined completely by the charging process and represent a drag of electrons and ions by the dust fluxes.Their full expressions can be found in Ref.[l]. The term u^^ in Eq.(16) represents a renormalization of plasma particles absorption. As the diffusion coefficient and the friction forces,it can be expressed through the eflfective dielectric permittivity.effective charge and the new charging responses introduced
in Ref.[l]. The system of equations (16,17) form the basis for a self-consistent description of the physics of plasma particles and dust particles (influenced by the charging process) in a dusty plasma. The general formulas are valid for distribution functions which are self-consistent solutions of these equations and the new responses depend on these distributions. For an order of magnitude estimate of these effects we have calculated[2] the responses using (not
VN. Tsytovich, U. de Angelis/The kinetic approach to dusty plasmas self-consistently) thermal distributions with electron,ion and dust temperatures T^^Ti and Td. In this case all the responses can be calculated numerically and they are all found to depend on the parameters (for q^q = eZd) : z = Zdc'^/aTe^ P = ndZd/ue^ r = T'l/Te^ Td = Td{l + P)/{PZdTi).
We give some results for the screening,the effective interaction charge
and the energy diffusion of plasma particles (heating and cooling).The approximations for the validity of the following results are discussed in Ref.[2]. The screening can be calculated from ej^^.This contains plasma and dust contributions.For the first we find that for na^rid
and r
ing) disappears,while the electron screening is little modified from Debye screening.For k
where di is the ion-Debye radius and the /-functions represent the modification of Debye screening (which corresponds to f{k)
= l).It has been found numerically that /^ can
reach values of order 10 (for P < 1) and 0.5 (for P > l).The screening by dust dominates for k ^ na^ndz/r
where / ^ <^ 1. The effective charge,in the same approximations,has
been also calculated numerically[2] : for small k the ratio q^^^/qeq can be of order 10^ (for P = 0.01) and of order 10 (for P = 0.1).Thus the effective dust charges in the interactions can be quite different from the "fixed'' equilibrium value and,most important,they are a function of distance between particles,which can also be interpreted as an additional form of screening. The diffusion of plasma particles in momentum due to Coulomb collisions with dust produces changes in the plasma particle distributions due to the inelasticity of the Coulomb collisions caused by the charging process. The energy exchange is determined by the longitudinal part of the tensor Dij. For Ud^a -> 0 this vanishes but in the presence of charging collisions the energy exchange in Coulomb scattering does not vanish. We find[2] that the diffusion coefficients for energy exchange of electrons and ions can be estimated, for r
27
28
KN. Tsytovich, U. de Angelis/The kinetic approach to dusty plasmas
These can be used to calculate (for thermal distributions) the changes in energy.This is found relatively small for electrons but not for ions. We find,for the changes related to the inelasticity of collisions and to the charging process [the term —Vd,i^^ in Eq.(15)] : 1 dTp^^
2^2
^ ^
P
a
ldT^^_
2v^
2 4^
and the heating due to inelasticity dominates the cooling for P and z of order 1 and afdi > reconditions which are often fulfilled in laboratory experiments. We conchide that the differences with three-component plasmas are in general too important to be neglected : the charging process introduces essential new properties in the physics of dusty plasmas.
REFERENCES 1. V.N.Tsytovich and U.de Angelis :"Kinetic Theory of Dusty Plasmas.I General approach" Phys.Plasmas 6(4),1 (1999). 2. V.N.Tsytovich and U.de Angelis I'^Kinetic Theory of Dusty Plasmas.II Dust- Plasma particles collision integrals" Phys.Plasmas submitted (1999). 3. V.N.Tsytovich PhysicS'Uspekhi40{l),53
(1997).
4. Yu.L.Klimontovich The statistical theory of non equilibrium, processes in a plasma (Pergamon Press,London 1967). 5. V.N.Tsytovich Lectures on nonlinear plasma kinetics Springer,Berlin 1995).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
29
Dynamical and Structural Properties of Strongly Coupled Dusty Plasmas ^ M. S. Murillo Plasma Physics Applications Group (X-PA), MS B259 Applied Theoretical Division Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract. Dusty plasmas offer a unique method for testing dynamical wave theories in the strong Coulomb coupling regime. Recently, there have been many theoretical models, based on either a generalized, hydrodynamic or on kinetic descriptions, developed to describe dust acoustic waves under conditions of strong coupling. These theories attempt to extend the usual acoustic wave dispersion relation to the strong coupling regime and, in some cases, to finite frequencies and wavevectors beyond the hydrodynamic limit. Here a kinetic approach is used to obtain dispersion relations in terms of an approximate dynamic local field correction, and comparison is made with the viscoelastic Navier-Stokes description.
I
INTRODUCTION
Following the dusty plasma wave experiment by Pieper and Goree [1] - that produced dust acoustic wave dispersion relations in the strongly coupled regime - there has been considerable activity in describing longitudinal waves in strongly coupled dusty plasmas. The dusty plasma is typically modeled by treating the dust grains as an additional heavy ion component. Wang and Bhattacharjee [2] applied a viscous hydrodynamic description and obtained both correlation functions and a wave dispersion relation that is similar in form to the known weakly coupled dust-acoustic relation, but with a viscous damping term. Rosenberg and Kalman [3] applied the quasilocalized charge (QLC) method to dusty plasmas, and they predict a reduced phase speed and a negative dispersion regime for short wavelengths. Similar conclusions were drawn from a kinetic approach based on local field corrections (LFC) by Murillo. [4] More recently, a viscoelastic theory was apphed to dusty plasmas by Kaw and Sen. [5] Interestingly, the QLC and LFC methods both predict mode frequencies below the heavy ion (dust) plasma frequency for all wavevectors and a ^) This research was supported by the Los Alamos National Laboratory Directed Research and Development (LDRD) program.
30
M.S. Murillo /Dynamical and structural properties of strongly coupled dusty plasmas
negative dispersion region. The hydrodynamic approach of Wang and Bhattacharjee does not yield a negative dispersion region, and the viscoelastic approach yields mode frequencies exceeding the heavy ion plasma frequency. Evidently, different theoretical predictions are obtained from different approaches, with discrepancies occurring at finite frequencies and wavevectors. Experimentally, [1] waves were driven from zero frequency to about three times the dust plasma frequency. Here the LFC formulation [4] is extended to finite frequencies and wavevectors by utilizing an approximate form for the dynamic local field correction. First, the hydrodynamic formulation is described in terms of a viscoelastic Navier-Stokes equation for comparing the two approaches.
II
GENERALIZED HYDRODYNAMICS
The linearized Navier-Stokes equation for dust qrains is +
Vdn-
V V'nM
ri + H VV- j(r,<) =
Vn(r,f) + ^ F ( r ) , (1)
3nM
where P is the fluid pressure, r] is the sheeir viscosity, ( is the bulk viscosity, the current is j(r, i) = nou{r,t), no is the average grain density, Udn is the dust-neutral damping frequency, and n(r,t) is the density fluctuation that satisifies the usual continuity equation d_ n{r,t) + V-}{r,t) dt
(2)
= 0.
For simplicity, heat flow is neglected. Together (1) and (2) describe density fluctuations to second order in the time derivative and, through the viscous terms, second order in the spatied derivatives. The longitudinal collective mode solution to (1) and (2) can be written in dimensionless form in terms of the ion-sphere radius rj = [Z/AnndY^^, the dust plasma frequency a;^, and the chaxacteristic longitudinal viscosity rj* = nMr^Wj as (qra)
2 3^ + ^
r
1
f^ ^d
(dP\
,
.2 ,
g'
(3)
Note that this equation is quadratic in both u and 9, with the exception of the force term. Evidently, neutral drag and viscosity serve to damp the mode, with the viscous damping being wavelength dependent. All other strong coupling effects enter through the pressure gradient term. We can expect neutral damping to dominate in the hydrodynamic limit and to compete with viscous damping when {qra)
23
V +C 1*
^d '
(4)
31
M.S. Murillo /Dynamical and structural properties of strongly coupled dusty plasmas Although this suggests that waves will be quite damped in the short wavelength Umit, it also suggests that the longitudinal viscosity can be measured in a system with low neutral damping in that Umit. At sufficiently high frequencies dense fluids do not have time to flow and tend to display elastic behavior similar to that of a solid [6]. This phenomenon can be included in the hydrodynamic description by generalizing the viscosity to finite frequencies [5-7] as
3
1 — lUT
where r is the viscoelastic relaxation time that characterizes the time scale separating viscous flow from elastic waves and is related to the high-frequency shear modulus Goo = v/'^-
III
RESPONSE FUNCTION
THEORY
Consider now a multicomponent strongly coupled plasma, consisting of AT species, that is perturbed by external forces. These external forces will produce density fluctuations in each of the species of the form (n,(q,u;))=xi°)(q,a;)t/t.,(q,u;).
(6)
Here X^°^(q,^) is the screened susceptibiUty of species s and Utoti^^^) potential energy
^^ *^^ total
Utotici,uj = U.ici,u;) + X : K r ( q ) K ( q , a ; ) ) [1 - G „ ( q , a ; ) ] .
(7)
r
From (7) we see that the total potential energy of species s has an external contribution Ug and contributions from fluctuations of all species. Interactions between the various species are given here by V^r(q) and dynamic local field corrections (DLFC) Gari^i^) are included to describe strong couphng effects between species s and species r. Clearly (6) is one equation in a set of coupled equations for the multicomponent system and will yield the full collective mode structure of the multicomponent plasma, including electron Langmuir waves, ion acoustic waves, and dust acoustic waves. Here we are primarily concerned with dust acoustic waves and it is therefore preferrable to obtain a description that exploits the experimental situation of very massive grains embedded in a weakly coupled background. For typical dusty plasmas, it is possible to obtain a reduced description in terms of a single component description by obtaining an effective dust-dust interaction potential. In essence we are attempting to replace the full multicomponent system with a quasiparticle model of the form (0)
(-^) =1 -—rsuJfrr-^wTTf^^. _ J^W^ff J-
Xd
^dd
^dd
(«)
32
M.S. Murillo /Dynamical and structural properties of strongly coupled dusty plasmas
which describes a single component system with effective interactions given by V^^ . We begin by considering a weak perturbation Ud that acts only on the dust grains and determine how dust density fluctuations arise in the presense of the electrons and ions. To simplify the presentation we use the fact that dust grains are screened predominantly by the electrons [8], and we consider a two component system composed of dust grains, electrons, and a uniform positive neutralizing background. In this case the fluctuation in the dust density can be shown to be
M
(0)
Xd
-U. 1 - XTVM [e-i - Gai\
(9)
where ege = eee(9,ci;) is the dielectric response function of the electron subsystem alone, including DLFC Gee(9)^)- In order to relate this to a simple effective interaction we have neglected the electron-dust DLFC Ged(9j^) that includes contributions beyond linear screening. This result indicates that, by ignoring strong dust-electron correlations, the dust grains will respond to an external perturbation as if they interacted via the dynamic effective interaction 1
V^\q,^)[l-Gt\q,u;)]=Vaa
-Xq,^)
Gdd{q,i^)
(10)
where the explicit wavevector and frequency dependencies have been written. Alternatively, this result can be presented in an equivalent form in terms of the full response function Xd of the homogeneous strongly coupled dusty plasma without screening as {nd) =
Xd
;Ud.
1 - VddXd [e;
(11)
These results, which are only vahd for vanishing Ged(9,ti;), are in agreement with the early work of Postogna and Tosi. [9] The result of (11) is easily generalized to include ion polarizability with the replacement eee(9,^) -> 65(5,0;), with 65 representing the total dielectric response of the background species, and illustrates that the effective interaction for describing dust dynamics is the bare dust interaction dynamically screened by the background particles, provided the background particles are weakly coupled to the dust. Screening by the dust, as described by Wang and Bhattacharjee [10], occurs here implicitly through the dynamic evolution of the dust density fluctuations {rid) but does not appear explicitly in the effective potential. We can now use (11) to obtain a wave dispersion relationship by noting that dust density fluctuations occur for vanishingly small Ud when 1 - VddXd
= 0,
where ej, is used for the general case, and yields
(12)
M.S. Murillo / Dynamical and structural properties of strongly coupled dusty plasmas
U)
2
a.3
F(g,a;)
2
\
1+ 1+ ^ 7 ^ 1 . \ F{q,w)q',
(13)
where a;^ is the dust plasma frequency, qt is the DH screening wavevector for the background particles, and F{q,u;)^-^-G,,{q,u;).
(14)
The form of (13) has been obtained by expanding the dust susceptibility to second order and neglecting Landau damping; the result is thus vaUd for moderate frequencies and wavevectors. Unfortunately, the DLFC Gdd{q,^) is not well known and must be computed in some approximation. Previous studies have adopted the static local field correction (SLFC) Gdd{q) = Oddiq^O)- [4] However, it is possible to construct an approximate DLFC, vaUd for all frequencies and wavevectors, that is dependent on only a few free parameters. In this approach we use the asymptotic relations Um Gdd{q,u;) = l
(15)
]imGM{q,u;)
(16)
= GM{q)
UmG,,(g,a;) = /(g),
(17)
which are the generalized Niklasson relation, [11] the static limit, and the thirdmoment sum rule, respectively. Although I{q) is known for the OCP model, [12] we treat it here as an unknown function that scales as q^ at long wavelengths. Essentially Gdd{q) describes low-frequency structure whereas I{q) describes highfrequency structure. In using (15) we are assuming that gdd{0) = 0 and that Gdd{q^^) tends to an OCP Umit for very short wavelengths. Em.ploying the functional form used by Ichimaru [12] we can write r (n.A Gdd[q,^) =
Gdd{q)-i^T{q)I{q) : 7-T , (18) 1 — iu;T[q) where T{q) is the mode-dependent viscoelastic relaxation time. In the Kaw-Sen treatment this mode dependency was neglected, which Umits the vaUdity of their results to the long wavelength regime. We take the forms T{q) = r ( 0 ) e - " ' ^ ' r{q)I{q)
=
(19) fiWr{0)e-'<^\
(20)
The full DLFC correction is thus parameterized in terms r ( 0 ) , a^, and jS^ and interpolates smoothly between small and large frequencies and wavevectors. The forms given in (19) ignore any detailed structure contained in I{q) and the large
33
34
M.S. Murillo /Dynamical and structural properties of strongly coupled dusty plasmas
frequency limit of (18) is likely smoother than reality. The Gaussian form for r ( g ) is arbitrary, and other rapidly decreasing functions of q could also be used. The collective mode structure of (13) should be equivalent to that of (3) in the hydrodynamic limit. In this limit we find Re[Gdd{q,u)] =
\ 9d L
(21)
a; = —
Im[Gdd{q,^)]
(22)
which establishes the relations
(0) = - ( 1 , + C ) L T U J ^ )
-1
(23)
and dP
dP...
( 9{ndT)^
M-^dT)
I
+ 3.
(24)
Here Tdd = {Zd^Y/i'f'dT) is the dust Coulomb coupling parameter. This expression for r(0) illustrates that viscoelastic phenomena arise through viscous and excess thermodynamic properties of the dust grains. The following expression establishes properties of the pressure P , which originated in the Navier-Stokes equation, in terms of the excess pressure Pea?? which describes the pressure beyond the effective Vlasov pressure in the kinetic description. We see that the dust Vlasov pressure for this problem is 37x^7, as expected for one-dimensional adiabatic compressions.
IV
OCP ESTIMATES
Longitudinal collective modes can be found with (12) and (18), given that we have some estimate for the free parameters /3, the viscosity §7/ + C? s-nd a. At the present time values for these parameters are not known for screened strongly coupled plasmas. As a first estimate we can, however, obtain these parameters from the OCP model. In particular, we use the OCP estimates of excess internal energy Uex « -^dd, I{q) ^ 4(rd9)^/45, and take a « r^. When the relaxation time is short compared with a wave period, the dispersion relation is a; u;:
3g
-^"-^--Gdd[q)\[l+u.y{qmq)-Gdd{q)^^ .q' + qi q'd
-1
(25)
which differs from the SLFC form previously used [4] by the term ^q^/q^ and the second factor. For the physically interesting case of g^r^ = 0.5, Fig. 1 shows the
M.S. Murillo /Dynamical and structural properties of strongly coupled dusty plasmas
0
4 k
35
8
F I G U R E 1. Approximate dispersion relation for a Vdd — 200 dusty plasma. The top (dot-dash) curve is the weakly coupled dust acoustic result. The lower (dashed) curve [4] is based on the SLFC. The middle (solid) curve is based on the results of this paper and indicates the correction that arises from extending the theory to finite frequencies and wave vectors. Note that the gaps of the SLFC model are eliminated in the present model. Here the dimensionless wavevector k — qv^ is used.
dispersion relation (25). In Fig. 2 the long wavelength regime is shown to indicate the compressibility modification to the phase speed. For longer relaxation times (25) is a; ^A ^d
3g^
r
+ Qd + 9d
-m^
(26)
At present I{q) is not available for screened strongly coupled plasmas.
V
RESULTS AND DISCUSSION
Dust acoustic waves have been studied in the strongly coupled regime at finite frequencies and wavevectors. Through the use of the DLFC and sum rules, the dust response function has been obtained in terms of a few parameters. These parameters can be approximated by known unscreened results or obtained from molecular dynamics simulations. Damping of the waves by neutral drag, viscosity, and Landau damping is included in the theory and it has been shown that experiments with minimal neutral density may be able to measure the longitudinal viscosity in the strongly coupled regime. The result reduces to a weakly coupled description, a hydrodynamic description, and a viscoelastic description in the appropriate limits.
36
M.S. Murillo /Dynamical and structural pmperties of strongly coupled dusty plasmas
0.6
1 ' ' '
' 1 ' • • ' ' ' '
'
"^~"'
"•^
^ . •
^ ^ ^' _^-
0.4
y
^-
^
^
V
y^r
" 0.2 -
' -
y^
^
^ AT
r^l
-
y 1
0.00
1
1
1
1
1
I
•
1
1
1
1
0.10
, 1
_._1_
0.20
0.30
j_^
0.40
0.50
k F I G U R E 2. Same three curves of the previous figure, but with an expanded k scale to better reveal the hydrodynamic limit and the compressibility modifications to the phase speed. Here the dimensionless wavevector k — qvi is used.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). X. Wang and A. Bhattacharjee, Phys. Plasmas 4(11), 3759 (1997). M. Rosenberg and G. Kalman, Phys. Rev. E 5, 1862 (1998). M. S. Murillo, Phys. Plasmas 5(9), 3116 (1998). P. K. Kaw and A. Sen, Phys. Plasmas 5(10), 3552 (1998). J.-P. Hansen and I. R. McDonald, "Theory of Simple Liquids" (Aceidemic Press, New York, 1986) 290. S. Tanaka and S. Ichimaru, Phys. Rev. A 35, 4743 (1987). U. Konopka, L. Ratke, and H. M. Thomas, Phys. Rev. Lett. 79, 1269 (1997). F. Postogna and M. P. Tosi, II Nuovo Cimento 55B(2), 399 (1980). X. Wang and A. Bhattacharjee, Phys. Plasmas 3, 1189 (1996). A. Nakano and S. Ichimaru, Phys. Letters A 136, 227 (1989). S. Ichimaru, "Statistical Plasma Physics, Volume II: Condensed Plasmas" (AddisonWesley, New York, 1994) 15.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
37
Directional Ordering and Dynamics in Dusty Plasmas J. E. Hammerberg, B. L. Holian, M. S. Murillo and D. Winske Los Alamos National Laboratory Los Alamos, NM 87545 USA
Abstract. We use molecular dynamics (MD) simulation methods to investigate dusty plasma crystal structure, with the grains subject to a spherically symmetric Debye-Htickel potential, a uni-directional external potential and an asymmetric wake potential. The structure is studied as a function of Mach number and magnitude of the wake as well as with the strength of the rf input power, using parameters from a self-consistent sheath model.
INTRODUCTION Molecular dynamics (MD) methods have been shown to be very useful to understand dust grain dynamics and crystal formation. Such calculations include not only the Coulomb repulsion of the grains screened by the background plasma (1), but also the important contributions of gravity and the sheath electric field (2,3). Other simulations have show^n the need to also include the effect of ion flow in the sheath. Such flow effects can be included through a modification of the fluid equations describing the background plasma response (4), through an inherent asymmetric potential around each grain (5) or through a dipole-dipole force in the interaction between grains (2,6). All three methods demonstrate that including even a small asymmetry in the interaction between grains can significantly change the type of crystalline structure that is formed. Experiments show that the dust grains tend to align into vertical chains as they form crystals (7,8). Recent experiments (9) use a laser to perturb the grains in a crystal and suggest that wake effects which occur on the downstream side of grains immersed in an ion flow contribute to give the proper grain spacing. A similar conclusion has been reached by Murillo and Snyder (10), who use experimental data for the spacing between two vertically aligned grains to develop a self-consistent model for the charge and forces on the grains. The wake structure has been studied in a number of theoretical papers (11,12) as well as with simulations (13). In this paper we include a spatially asymmetric wake potential in MD simulations of grain dynamics. We show that the wake potential more accurately models the experiments, and is more physically justifiable, than using a dipole interaction.
38
J.E. Hammerberg et al. /Directional ordering and dynamics in dusty plasmas
SIMULATION MODEL AND RESULTS We consider a system of 1000 dust grains of identical charge Q, mass m, interacting via a screened Coulomb interaction and an anisotropic wake potential in the presence of a vertical (z) external potential that models the effect of gravity and the sheath electric field. The screening length is taken to be the electron Debye length, A^oeTaking the unit of length to be A.De and the energy unit to be Q^/XDQ, the normalized potential of interaction between two particles situated at ri and FJ is
(1)
where , ,
,
<^oifii) =
exp(
-ry)
—^
(2) ^w (Pij ,Zij)=-aM sinizy IM) exp(-vz^. / M) exp(-/7^. / M). (3) The expression for the wake potential, (|)w, differs somewhat from other forms (11, 12) and is discussed in (13). Here, M is the Mach number, pij and zij are the cylindrical coordinates of the relative particle positions, a is a scale factor, and v is a measure of the ion-neutral collision frequency (13), which we take as 0.1. The p dependence has been added so that the potential falls off smoothly from the z-axis. The wake potential acts only on the "downstream" particle (i.e., the one with the smaller z value). We consider a range of Mach numbers about M = 1.2, a value consistent with the recent experiments (10). Other parameters are taken from the table of values given in that paper; in addition, we use their model for the extemal potential. In what follows, we take the origin in z to be the point of zero extemal force for grains with radius of 5.2 |im. The simulations are begun with the grains on a simple cubic lattice with unit nearest neighbor distance. This is equilibrated at a temperature of 0.1 for 10 to. (The time unit to = (mX.De*^/Q^)^^^-) During this time, periodic boundary conditions are imposed in all directions for the 10 x 10 x 10 cube and no extemal or wake potential is applied. Under these conditions, the lattice of Yukawa grains melts. At this point, the periodic boundary conditions along z are removed and the extemal potential and wake potential are introduced. The equations of motion are then integrated with a NoseHoover thermostat included so that the temperature can be quenched to a much lower value. This temperature is taken to be the "effective" electron temperature in (10), which is a fraction of an eV. We have used the simulations to study the effects of (a) rf power, (b) Mach number and (c) strength of the wake potential on the resulting crystal structure.
J.E. Hammerberg et al. /Directional ordering and dynamics in dusty plasmas
39
The input power dependence is investigated using the measured and derived plasma conditions of Murillo and Snyder (10), at M = 1.2. The system configuration at t = 100 to is shown in Fig. 1 for the lowest (75 W) and highest (150 W) power levels. The higher power results in a stronger external potential that confines the system to a smaller region in z [Fig. 1(a), 2(d)]. The upper edge of the crystal is irregular, while the bottom edge (closer to the electrode) is sharper. Combined with the smaller X^Q for these plasma conditions, this leads to a decreased spacing (~ 20%) along z, consistent with experimental measurements (10). The formation of vertically aligned chains is evident in Fig. 1(c) and 2(f), but the transverse order, as seen in Fig. 1(b) and 1(e), is more fluid-like, at least at this temperature.
(b)
5
2.5
-2.5 -5
-2.5
0
2.5
-5
-2.5
^
2.5
5
(«)
- 4 - 2
0
2
4
FIGURE 1. Dust configuration at two powers (left panels, 75W; right panels, 150W). Top panels give view along x-axis, middle panels along z-axis, and bottom panels show a y-z plane over a narrow slice around x=0.
40
IE. Hammerberg et al /Directional ordering and dynamics in dusty plasmas
The Mach number dependence of the effect of the wake is shown in Fig. 2, which assumes the parameters of the 125 W experiment. The figure shows results for M = 1.0, 1.2, and 1.4. The effect of the larger Mach numbers is to have a more repulsive near field force along z. Thus, the higher Mach numbers give rise to a more extended structure along z as well as a lower equilibrium location [Figs. 2(a), (c), (e)] for stronger ion flow. Again, the upper crystal boundary is diffuse, while the lower boundary is much sharper. The structure in the transverse direction [Figs. 2(b), (d), (f)] is rather disordered at these times.
0>)
5
(d)
(*) -
2.5
^
*
"•
.
'
'I.
fc
..
I
.
r
n
/ , " . .
1
•
'
-
•
.
-
•
I
' • •
I
•
•
.
J '
'
P '
'
*
-•
.
p
•
'
-2.5
•
I
, I
.-.J
• ^
+
\
•
-5
5 -2.£
^
2i
r
r
5
J
£ -2i
I
I.
•
.
J
H 2i
'
•
•
'
.
'
•
'
.
•
,
.
-
• • ' ' - - - • ' •
2.5 • ••
-5
,
V 1
,
'
-
.
k
-2.5
ii
^ , .
'
2.5
.
p
•
•
.
•
,
•
• \
••
•
»•
'
1 "
.
-
-
. -
p.
.
• .
' I I
• -
I
I P
-5
I I
1
-2iS
jl
-
2.5
• "^
S
FIGURE 2. Mach number dependence: Dependence on M: (left panels) M = 1.0, (center panels) M = 1.2, (right panels) M = 1.4. Upper panels are views along x; bottom panels are views along z.
J.E. Hammerberg et al /Directional ordering and dynamics in dusty plasmas
41
Finally, we consider the effect of the strength of the wake potential at fixed rf power (125W) and Mach number (1.2). Figure 3 shows results with a = 0., 0.01, and 0.1, at somewhat later times, t = 300 to. The lower two strengths have the same ground state as observed previously, with a slightly different external potential (2). The main ordering occurs first in planes, followed by a fully three-dimensional structure at later times. However, for strengths beyond about a = 0.1, the ordering is first along the zaxis, and the resulting structure is dominated by strings. For a= 0.1, the linear features are more apparent, as seen by the smaller number of distinct particles in the transverse directions [Fig. 3(f)]. With smaller effective wake, the upper crystal boundary is better defined [Fig. 3(e)]. Thus, the overall structure is more akin to nematic liquid crystals than to 3-D crystalline order, as is commonly seen in experiments (10). (a)
(c)
zO
^o; il''^>ik> ••)_••—i^»-_ •••.•;-• --V -n^. ^
y^.
T
^
. fi-»a . ^ - < > i - O W " ^ K i r * *
^
-5 -5 -2.5
^1
2.5
(b)
5
2.5
-2.5 -5 -5 -2.5 \
-2.5
2.5
0
y
2.5
(d)
2.5
2.5
yo
yo
2.5
-2.5
-5 -5
2.5 jl 2.5 5
ff)
5
-5
^
-
^
1
'
"
i
'
^
;
? _ -JL
-
•
-2.5
jl
* , «
^
••
•
^-
•
•
2.5 5
FIGURE 3. Dependence on a: (left panels) a = 0.0, (center panels) a = 0.01, (right panels) a = 0.1. Upper panels are views along x; bottom panels are views along z.
42
J.E. Hammerberg et al /Directional ordering and dynamics in dusty plasmas
SUMMARY We have investigated the dynamics of charged dust grains trapped in a plasma sheath and gravitational potential. We have extended our previous calculations (2) to include an aysmmetric wake potential to model the flow of ions in the sheath. The addition of this potential changes both the structure of the crystal that results and how it is formed. In the absence of a wake potential (or if the scaling factor a is set to a small value), the crystal first forms into planes and then into an entire crystal. With the wake, the grains first form vertical strings and then a more liquid-like crystal is produced. Increasing the flow speed of the ions (i.e., the Mach number) increases the vertical spacing. Using parameters based on experiments (10), we also find that the vertical spacing decreases with rf power, in agreement with the measurements.
ACKNOWLEDGMENTS This work was supported by the Los Alamos Laboratory Directed Research and Development Program.
REFERENCES 1. Hamaguchi, S., Farouki, R. T., and Dubin, D. H. E., Phys. Rev, E, 56, 4671-4682 (1997). 2. Hammerberg, J. E., HoUan, B., L., Murillo, M. S., and Winske, D., "Molecular dynamics simulations of dipolar dusty plasmas," Physics of Dusty Plasmas, AIP, CP 446,1998, pp. 257-264. 3. Totsuji, H., Kishimoto, T., and Totsuji, C , Phys. Rev, Lett., 78, 3113-3116 (1997). 4. Melandso, F., and Goree, J., J. Vac. Set Technol. A, 14, 511-517, (1996). 5. Melandso, F., Phys. Rev. E., 55, 7495-7506 (1997). 6. Belotserkovskii, O. M., Nefedov, A. P., Filinov, V. S., Fortov, V. E., and Sinkevich, O. A., JETP, 88, 449-459 (1999). 7. Pieper, J. B., Goree, J., and Quinn, R. A., Phys. Rev. E, 54, 5636-5640 (1996). 8. Nunomura, S., Ohno, N., and Takamura, S., Phys. Plasmas, 5, 3517-3523 (1998). 9. Takahashi, K., Oishi, T., Shimomai, K., Hayashi, Y., and Nishino, S., Phys. Rev. E, 58,7805-7811(1998). 10. Murillo, M. S., and Snyder, H. R., This conference, 1999. 11. Ishihara, O., and Vladimirov, S. V., Phys. Plasmas, 4, 69-74 (1997). 12. Xie, B., He, K., and Huang, Z., Phys. Lett. A, 253, 83-87 (1999). 13. Winske, D., Daughton, W. S., Murillo, M. S., and Shanahan, W. R., This conference, 1999.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukia (eds.) © 2000 Elsevier Science B.V AH rights reserved
43
Multiple-Sheath and the Time-Dependent Grain Charge in a Plasma with Trapped Ions Yasu-Nori Nejoh Graduate School of Engineering, Hachinohe Institute of Technology Myo-Ohiraki, Hachinohe 031-8501, Japan Multiple-sheath structures and the time-dependent dust grain-charge in a dusty plasma with the electron emission and trapped ions are investigated by numerical calculation. Trapped positive ions, electron emission beam and dust grains modify the shape of the sheath, enhance the space charge density and give rise to the triple-electric field structure. It is found that the eflFect of trapped ions plays an important role in the formation of the new electric field structure in the sheath and shorten the charging time.
1. Introduction The investigation of the sheath is of current interest from the viewpoints of the various fields of dusty plasmas and the control of the potential of a substrate immersed in a reactive plasma. The presence of particulate contaminants in etching, sputtering and deposition processors remains a major problem in engineering plasma. Dust grains do not have well-defined mass, charge and size, and thus it is often quite difficult to understand plasmas having such dust grains or fine (ultrafine) particles. The particulates occur in the range of sizes, 0.01-100/im being the most common[l-2]. They tend to congregate into particle traps due to their electrical nature, sometimes referred to as electrostatic traps [1-4]. A sheath with electron emission from the wall or the electrode has been considered in a plasma with negative ions [5-7]. However, not many theoretical works on trapped ions and the time-dependent grain-charge have been done in dusty plasmas. We investigate trapped ions and the dust grains to understand the behavior of the sheath near the wall and temporal-evolution of the grain-charge in plasmas. We demonstrate the new properties of the sheath potential, space charge density and the electric field, and show the temporal-evolution of the dust grain-charge, comparing this model with the preceding results[7-9].
2. Theory We assume a one-dimensional model of a dusty plasma near the wall or the electrode. We assume that, from the right-hand side of the sheath edge, thermal electrons and dust grains penetrate the sheath, following the Boltzmann distributions in the sheath. A secondary electron beam produced at the wall by the
44
Y.'K Nejoh/Multiple-sheath and the time-dependent grain charge
thermionic emission or secondary emission is ejected from that place towards the plasma. The electrode may be a cathode or a substrate. The electron, positive ion and negatively-charged dust grain densities at the sheath edge denote ngo, ^to and Udo ; the potential in the sheath is — $ , the plasma potential is $ = 0 and the electrode potential is —$ = —^w] the initial velocity of positive ions entering the sheath is i^o; the electron, positive ion and dust grain temperatures are Te, Ti and Td] and the magnitude of the electronic-charge and mass are e and m^. The density of electrons is rie = neoex'p(—e$/Te), where n^o denotes the electron density at the sheath edge. The secondary electron beam density n^ produced by the thermionic emission or secondary emission, which form an electron beam current j ^ , is given by ^6 — (i6/^)\/^e/2e($^^ — $ ) . The positive ions enter the sheath with initial energy mivl/2^ the potential drop being $o- The ion density and the ion velocity at the sheath edge are n^o ^ind VQ^ respectively. We adopt the Al'pert model[10] for positive ions as n,(T,-,$)
= niQ<
|e$| 7r(l + x,)T^
^1 +
Xv
1 + Xi
1 + -exp
+
1-^2-
{i + x,y Ie$l ^^2(l + :r,)^,
V2-
-erf^
Ie$|
TT •'\| 2(2 + :c,)r,
(1)
where Xi = r/ri — l^r and r^- are the distance from the ion and the average radius of the ion. It is assumed that A^/r,- » 1 and e $ / T , <^ A^/r,-. The dust grain density is assumed to be the Boltzmann distribution, Ud — ndoexp{—Qd^/Te), where Udo is the initial dust grain density at the sheath edge. Here Qd = e|Z^|, and \Zd\ is the magnitude of the charge number. This holds for (f)^^ ^ Td/e. The Poisson equation in the sheath is ^2$ ^0
dx^
-en^oexp{--^)
- j^.
Qd^ rUe -h eni{Ti, $ ) - endoexp{), (2) 2e($^ - $)
where to and p are the dielectric constant in vacuum and the space charge density in the sheath respectively. We make the following normalization: the potential (f) = e$/Te; the positive ion density at the sheath boundary Si = n^o/^eo; the secondary electron beam current density J5 = jb/jw^ where j ^ , = eneo(T'e/me)^/^(2^-t^)^/^, the dust grain density at the plasma boundary 5d = n^o/^eoJ the distance from the cathode ^ = X/XD^ where A^ = {eoTe/e^UeoY^^ is the Debye length. By the normalization, the Poisson equation reduces to
dx'^
= p = —exp{—(j)) — 2 ^
3h4>n
A.
+ Si {Ni + N2 + N3) - Sdcxp
(3)
Y.'N. Nejoh/Multiple-sheath and the time-dependent grain charge
45
where N,
=
N2 = Ns
=
l<^l 1
-exp
6
'1 +
7r(l + ^i)T^(f)Q
\ I'/'l
1
1
l + 6 ^ 2 + eJr,c^o"^2r
l - V 2 - \
ir
-erf, • \2{l
+ (,)n(i)o
(1 + e,)^
l-V2-J-erf,
\\
TT •'\2{2
+
^,)n4>or
where r^ = Ti/Tg, Td = Td/Tg. Quasineutrality condition at the sheath edge is (^j = 1 + 2(l)yjJi, + ZdSd- The boundary condition is set as follows. The derivative of the space charge density is assumed to be zero at <j!) = 0: dp/d(j) = 0. Using this relation, we obtain (4)
r^[l_j, + M.y
where we used the definition 5 = 1 + (1 + i,/{l + C,))^^'^ and C = (1/2)(1 - (1 + Ci)~^Y • This equation coincides with the results for Ji, = 0,Sd = 0 obtained in Ref[ll]. When J{, — 0,Sd y^ 0, it reduces to the result of one-dimensional case[6]. The electron beam increases (^0 while the presence of dust grains decreases (/)o, if Td
The electric field is assumed to be E = 0 at the plasma-sheath boundary. Integration of (3) gives rise to
_{l_exp{-cl>)}-Uk^l{l-Jl-
2
-S,{Ai + A2 + A3) - ZdSdTd I 1 - exp
4'v Zd£ -
(5)
where A1+A2
=
B
( <\>Y ^
2iia \Ti(j)Q)
l^expjC) T
b
2aT,iy^o
Aexp{C) 4v^ 7T{ab- 1) [ where f = I - ^/2, g = y/^
1 — exp """' y
T,(jjo
and i) = 6 - l / 2 a .
1
,
46
Y.'N. Nejoh/Multiple-sheath and the time-dependent grain charge
the orbital motion limited theory, which states that the grain potential should be independent of the plasma density. In this case, the particle potential is determined by balancing the current of electrons, secondary electrons and positive ions to dust grains. Each current in the bulk plasma is described as I^ — — enr^ J8Te/7Tmene{4>) exp{e'$/Te), h = -tSmh, U = enr'^^JsTJ7Tmin,{Si,Ti,(f)) {^ - e ^ / T , ) , where * = Qd/^^^of''d', Qd = ^\Zd\ and rd is the average radius of dust grains. Here e and Sm is characteristic parameters of the grain material, which are defined as e = lA.8Te/Em and 6^^ = h/hi where E^ depends on the energy of primary electrons. The normalized equation of the conservation law of current reduces to = - ( 1 - lA,8t5,^) n^{(j))exp{adZd) + J—ni{5i,Ti,(l)) where T = t/to, respectively.
to = ^ul^d^pe,
(l
^-^) ,
(6)
^pe = (e^Ueo/eome)^/^ and a^ = e^lATTtor^T^,
3. Numerical Calculation We investigate the sheath structure because the sheath is closely connected with the control of the plasma by the potential of the electrode. In order to study the formation of the sheath due to the grain-charge, we perform the calculation for |Z(i|=:10, 20 and 30, and, in this case, (^^=1.8, 2.8 and 3.8, respectively. Figure 1 shows the sheath potential structure near the electrode (and/or wall), where the fixed electrode potential (/)^ = —100, (/>o = —0.01, r^ = 0.2, ^i — 0.03 and 5^ — 0.1. It turns out that the shape of the sheath changes in the range of 0.025A^ < i < 0.033Ai:) by the presence of trapped ions and an electron beam. We illustrate the space charge density in the sheath for |Z^| = 10 and 5i = 1.8 in Fig.2. In the comparison of our case with that of Boltzmann ions, as is seen in Fig.2, the space charge density enhances due to the trapped ions and an electron beam. The structure of the electric field in the sheath is shown in Fig.3, where we used the same parameters as employed in Fig.2. We see that the electric field forms a triple-layer, which is newly found in this sheath. Positive ions are accelerated and decelerated frequently by this electric field, and are drastically accelerated near the electrode. Figure 4 shows temporalevolution of the grain-charge, which implies that dust grains take ~ 0.4to=~ 1.1/is to saturate its charge, whereas ^ 1.0to=~ 2.8/is in the case of Boltzmann ions, where Tg^lOeV and Ugo = lO^^m"^. It is because the space charge density greatly enhances due to trapped ions in the range of 0.025XD < ^ < 0.0d3XD when the grain-charge number \Zd\ — 10. This tendency is similarly true in the cases of the charge number |Z^|=20 and 30. It turns out that the time take to attain the equilibrium charge of the grain is shortened by the trapped ions and beam electrons.
Y'N. Nejoh /Multiple-sheath and the time-dependent grain charge
30
47
1.x 10'
1.x 10"
<\>
Trapped ions 0.0175
0.02 0.0225 0.025 0.0275
Boltzmann ions
0.03 0.0325
X/\D
0.02
0.022
0.024
0.026
0.028
0.03
X/XD
Figure 1 Spatial profile of the potential in the sheath with Zd = 10(solid line), 20(dotted line), and 30(dashed line) for (p,u = -100, n = 0.2, Td = 0.1 and (f)o = -0.01.
Figure 2 Spatial profile of the space charge density in the sheath for (f>^u = -100, Zd = 10, Si = 1,8, n = 0.2 and (/)o = - 0 . 0 1 .
"
'
"
"
•
'
"
'
120 100 80
Zd
/
60 40 20
// /
0 2
0.4
0.6
Trapped ions 0.02
0.0225
0.025
0.0275
0.03
0.0325
X/\D
Figure 3 Profile of the electric field as a function of the distance, for 0^ = -100, Zd = 10(solid line), 20(dotted line) and 30 (dashed line), Ti = 0.2, Td = 0.1 and 0o = -0.01.
Boltzmann ions Figure 4 Temporal-evolution of the dust grain-charge for (f)^, = —100, Zd = 10, Si = 1.8, Ti = 0.2, Td = 0.1 and 00 = -0.01.
48
Y.-N. Nejoh/Multiple-sheath and the time-dependent grain charge
4. Concluding discussion We understand that positive ions are trapped in the localized region by our calculation. If a secondary electron beam is only calculated numerically, we cannot find the modification of the sheath and the formation of the triple-electric field. The effect of trapped ions gives rise to the remarkable change for the profiles of the sheath potential, space charge density and electric field, i.e., the modification of the potential profile, an enhancement of the space charge density and the occurrence of the triple electric field, respectively. Therefore this fact imphes that our results may be applicable to the control of the sheath by modifying the potential of the electrode.
References l]G.M.Jellum, and D.B.Graves, Appl. Phys. Lett., 57, 2077, (1990). 2]L.Boufendi, A.Plain, J.Ph.Blondeau, A.Bouchoule, C.Laure and M.Toogood, Appl. Phys. Lett., 60, 169, (1992). 3]G.S.Selwyn, J.Heidenreich and K.L.Haller, J. Vac. Sci. Techn. A., 9, 2817, 1991). 4]R.N.Carlile and S.S.Geha, J. Appl. Phys. 6 1 , 4785, (1993). 5]P.D.Prewatt and J.E.Allen, Proc. Roy. S o c , A 3 4 8 , 435, (1976). 6]N.St.J.Breithwaite and J.E.Allen, J. Phys., D 2 1 , 1733, (1988). 7]H.Amemiya, B.M.Annaratone and J.E.Allen, J. Plasma Phys., 60, 81, (1998). 8]H.Yamaguti and Y.Nejoh, Phys. Plasmas, 6, 1047, (1999). 9]Y.Nejoh, Austr. J. Phys., 52, 37, (1999). lOjYa.L.Al'pert, Spact Plasma^ vol.2. Chapter 12. Cambridge University Press, Cambridge, (1990). [lljD.Bohm, Characteristics of Electrical Discharges in Magnetic Fields^ Chapter 3. McGraw-Hill, New York, 271, (1949).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
49
Surface Charge on a Spherical Dust H. Amemiya The Institute of Physical and Chemical Research Hirosawa, Wako, Saitama-Pref, 351-0198, Japan 1. Introduction The dust suspending in the plasma influences the charge balance due to the surface charge. The dust has been treated as a small probe to estimate the charge. 1"^) Recently, the electric charge on the dust has been estimated by applying the probe theory, where positive ions are assumed to follow the radial motion theory.4) When the dust density is not small, the absorption radius of each dust will overlap with each other and the radial motion theory will break down. When the dust radius is small, positive ions would follow the orbital motion theory. Although the pervious theories do not assume the existence of negative ions, this is necessary not only for a negative ion containing plasma but also for a plasma containing dust because the dust has usually a negative charge and behaves effectively as a negative ion. Therefore, the dust charge should be calculated inevitably by including negative ions. The objective of this paper is to evaluate the dust charge by applying the orbital motion theory of probe to a negative ion containing plasma. The result would be more consistent than the previous theories in that the dust charge is a function of the charge itself. 2. Model Assume that a plasma consists of positive ions with a density N+o, thermal electrons with the density of Neo and the electron temperature Te, negative ions, and dust particles. We define N^Q as the negative ion density and N^Q as the dust density, and Tn as the negative ion temperature and T^ as the dust temperature. The dust potential can be obtained from the current balance between positive and negative charged particles. If Te and Tn are known, the current I_ of negatively charged particles to the dust with a radius r^ at a potential -Vp against the space potential (V=0) is given by
I - = e N „ S ( ( l - a „ ) V ^ e . p < - ^ H a ^ exp(-^)|,
(,)
50
H. Amemiya/Surface charge on a spherical dust
where 8=4^^2, a^=^JZ^^^-^ QNdo/ZN+o=«+5Q, 5=Ndo/ZN+o' oc^N^^ZN+o' Q is the dust charge and e is the electronic charge. The first term denotes the electron current while the second term the negative ion current, but unless 1-a is very near to (Tj^m/TgMjj)^/^, I. is almost given by the electron current. The positive ion current 1+ to a spherical dust is coupled with Poisson's equation as^)
where Ti=eV/KTe, r|p=eVp/KTe, ^=rAD, ?^D= (eoKTe/Noe^)!/^, No=ZN+o^ P=T^/Te, and i=I+/lA.; lA.=eZN+Q47iA-D2(2KTg/M)l/2. Signs + and - stand for outer and inner sides of the absorption radius at which r|/p=4i/pl/2^2-l. ^ is given by 4>(Ti)=(l-a-6Q)exp(-Ti) + aexp(-Yn) + 5Qexp(-YdQTl), (3) where T=Tjj/Tg and Yd^Td/Tg. It should be noted that <E> is a function of Q and 5. The conditions for absorption radius and quasi-neutrality are given respectively as Ya=Xa-2 -1 , (4) X-2=4{(l+Y)l/2 - 4>} . (5) where Y ^ / p and X=^/(2il/^/pl/'^). The absorption point in the quasi-neutral region is given by the crossing points between eqs.(4) and (5) as (1 + Y)l/2 =2.
(6)
In the plasma where Y is very small, we have Y=l/[2+4P{l+a(Y-lH5Q(l-YdQ)}]X2.
(7)
The following four cases may be considered. Case I: rd, Q and r|f (floating potential) or the surface electric field. We will deal with two limiting cases below. In the case II (TSL), we have I+=|iIo, \^ being given by ^^
lV^(k)2(i+3i),
(8)
51
H. Amemiya /Surface charge on a spherical dust
(5i)2=4cI>(Tl3){(l+^)l/2-(Tl3)}/(l+'li),
(9)
and Io^ZN+o47irp2(KTe/M)l/2,'t,s=rjX]^,^a='"a/^D' ^s and T|a are normalized potentials at sheath edge (plasma boundary) and absorption radius determined respectively by (l+H ) + _ ^ ^
2(T1)(1+^ )l/2 =0.
*I>(r|a) = (l+—)/2.
(10) (11)
P
To obtain the floating potential r|f, |LI should be balanced by I. given in the normalized form as
L,(,.„.8Q,y^exp(-„.>.
(12)
By assuming a capacitance in vacuum as 47C8Qr^, Q is determined as Q=47r8ordVf; Vf=KTeTif/e.
(13)
Introducing Q£)=47C8Q?i^(KTg/e), we can express Q=^i1fQj). This value is a minimum value due to a finite sheath thickness. Q^ may be defined as the Debye sphere charge. In the case IV (OML), ^^ corresponds to ^^j. Then
^ = 4^(1-^).
(14)
lo 4 p The dust potential r|f is determined from (l-a-6Q)^/^exp(-Tif) =V2p (1 + ^ ) . V Tim
ft
(15)
Besides eq.(13), the dust charge can be also calculated from dfx\ld'^ at the dust surface. The field must be continuous at ri=n-|a» ^=^a- Then, the surface should coincide with the absorption radius, i.e. — = - - ^ ^ = - — ( 1 + - ) ; r|=qF, ^=^d.
^^
I'
^
^
(16)
The surface charge Qp is given by QF=47trd28o— = 2 ^ / — (47reo^;iD).
ar
Then, the ratio Q/Qp becomes
QF/Q=2(Tif+P)/rif.
a^
(17) (18)
52
H. Amemiya/Surface charge on a spherical dust
3. Calculation of dust charge Calculation has been made for thin sheath and orbital motion limited cases under the condition that the dust radius is r^j^O.liim and the plasma ion species are Ar"*" and 02". First, the ion and electron currents were calculated for a tentative charge Q. Then, from the current balance, the floating potential, the surface field and a new charge were calculated. The process was repeated until Q coincided. Figure 1 shows Q vs P in both models for some values of 5 (T^ =leV, a=0). The case when 5 is small corresponds to "single dust case". As 5 is increased, Q departs from the value by the single dust model, i.e. Q decreases with 6. In the thin sheath model, Q decreases with P slightly up to about p=l, however, above P=l, Q falls with (3. This is because an increase of I^. due to the higher positive ion temperature causes a reduction of Vp. An abrupt fall of Q at p-^O.S for 6=2.5x10"^ is associated with an abrupt drop of the sheath edge potential by the dust behaving like a negative ion. Above 5=2.5xl0'3, some times no solution was found consistently. This means that it becomes difficult to sustain a stable plasma as the limit 6Q=1 is approached. Near this limit, all the electrons are absorbed by dust and Q becomes independent of the models. Such a "dust plasma" where the plasma consists of dust and positive ions (or positively charged dust) has been recently observed.^) In the OML model, Q is smaller than that for the thin sheath model by a factor of about 2 for small p and the variation against P is different. However, the charge Q F calculated from the surface field is larger than Q by eq.(18) especially at larger p. Q F VS P in the OML model is shown in Fig. 2 for some values of 6 (T^ =leV, a=0). At larger p, the region of "no solution" appears as 6 is larger. In such a region, a some sort of unstable plasma is predicted. Figure 3 shows Q vs 5 in both models for some values of a (T^ =5eV, p=0.1). It is seen that Q is decreased by negative ions and especially at a=0.9, a strong reduction of Q is seen. As 5 is increased, Q approaches asymptotically to the limit (l-a)/5 as shown by dotted curves. For a=0 in the thin sheath model, no solution was found before this limit is reached: it becomes difficult to sustain the plasma stably for 5>2xlO"3. In the presence of negative ions, however, it seems that negative ions can avoid the plasma destruction by dust even if the limit (l-a)/5 is approached. Figure 4 shows the dust charge Q vs a in both models for different values of 5 and T^ (P=0.01). The values of 5 are as shown labeled. 5=0 corresponds to the case of "single dust case" where the dust density does not affect the charge equilibrium. Some humps of Q appearing at (X=0.4-0.8 are due to an abrupt drop of the sheath edge potential caused by the dust and negative ions when a4-6Q reaches near 0.7.^) A reduction of the sheath
H. Amemiya/Surface
500
1
1—I—I I I I 11
T—r
I I i 11
1
1—I—I
53
charge on a spherical dust
2000
I I I1
-!
1
1—[ 1 1 I I I
Te=leV, a =0 Thin sheath
400
1
1
OML
IxlQ-^
1—I I I I I I
\
1
1—I M
M
Te=leV,a=0
1500
—ft
o
1000
1x10' 500 - • — » » »» • • -J
:
10-
I I I I 11
I
I
I
I I I I 11
10-'
-i
I
I I I I I
_i
10^
10"
r - r - r i i i j-
T
-
10^
•^ T
^ ^—•^^^
1—1' T"
10-^
10'
T
"*~^
1—
a=0.5
\ %
10-=
Te=5eV, p=0.1 . .......1 10-^
1
5x10-' i__j
,
' I 1111
I
10"
-
o
1 1 1 1 1 1 1 ^ p-n
1 1
Thin sheath Te=10eV
[Jr
a
•
1.
10-^
4
E OML • • ^ txhin sheath O D A h 1
10^
T-j-rTTT
*%/^Xa=0
a=0.9
I
10-'
'
I
I ' l l
10'
F
-
^
a
rTTj
,
I I 1111
--' no solution
Fig. 2. Surface chage Q^ determined by the surface field.
Fig. 1. Surface charge Q vs temperature ratio (3.
• r—' J
I
10-
• — • • »» • •.•
D
n
D-Q-
-g—.
^Q.
p
^
OML Te=
r 10'
—1
L..,X_I_
MI
1
1
10-^
Fig. 3. Surface charge Q vs dust density ratio 5.
I 1 1J. i "K
10--
10
0
Fig. 4. Surface charge Q vs negative ion density ratio a .
54
H. Amemiya/Surface charge on a spherical dust
edge potential causes an increase in the floating potential difference. For larger a above these humps, the current of negatively charged particles, eq.(l), also decreases with a and Q makes a smooth change with a. Q increases in proportion to T^ up to a^, a critical a, from which the effect of Q on the charge equilibrium becomes remarkable. As 5 is made larger, Q begins to decrease at a^, and as a approaches unity, Q drops steeply towards the limit (l-a)/5. In the thin sheath model, the ratio rjx^ remains almost near unity at p>0.3 while it increases in proportion to 1/p^ at p<0.1: the absorption radius increases as the positive ion temperature becomes lower. In order for the model to apply, the dust should not overlap with each other. The condition is given by 5"^/3>J.^(2N^.Q)^/3 The value of Q has been compared with some experiments. The experimental value of Q, Qexp=106-2xl06 for rd=10|Lim, Te=4eV, p-'O.OOTS, 5-0 in ref.(8) is reduced proportionally for the present model with r(i=0.1|im and Te=leV as Q=250-500 and Qexp=106 for rci-60|Lim, Te=4.8eV, p-'O.l, 6-0 in ref.(9) is reduced as Q=350. The values are consistent with each other. For the plasma consisting of fine particles of plus and minus signs ^0), it should be that 5Q=1 or QNcioQ=ZN+o4. Conclusion The electric surface charge on spherical dust has been calculated for the thin sheath and orbital motion models as functions of the ratios of negative ion density a, the ratio of positive ion to electron temperatures p and the ratio of dust density 5. It is assumed that dust behaves as multicharged negative ions and contributes to the charge equilibrium. The calculation is performed consistently by satisfying both the current balance on the dust and the charge balance in plasma. It has been found that the variation of Q with a, p, 5 depends on two models but in any cases Q is not held constant as 6 is increased but decreases from the value of single dust model. Near the critical value of 6, Q drops steeply with 6 towards the charge equilibrium limit where the model dependence becomes less. References 1. U. de Angelis, Physica Scripta 45, 465 (1992). 2. T.G. Northrop, Physica Scripta 45, 475 (1992). 3. J. Goree, Plasma Source Sci.Tech. 3, 400 (1994). 4. C.M.C. Naim, B.M. Annaratone and J. E. Allen, ibid. 7, 478 (1997). 5. J.E. Allen, Physica Scripta 45, 497 (1992) and references therein. 6. H. Amemiya, B.M. Annaratone & J.E.Allen, Plasma Source Sci.Tech.8, 179 (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
55
NUMERICAL SIMULATION MODEL FOR DUST EXPANSION INTO A PLASMA G. S. Chae*, W. A. Scales*, G. Ganguli^^, P. A. Bernhardt^, and M. Lampe^ "" Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, U.S.A. ^Plasma Physics Division, Naval Research Laboratory, Washington D.C., U.S.A.
Abstract. The physics of dust expansion into a background plasma has many important applications in space plasmas. This study will describe a new two-dimensional numerical model that has been developed to study physical processes associated with the expansion of dust into a plasma across a magnetic field. The background plasma electrons and ions are treated as a fluid whose density is reduced by dust charging. Fourier spectral methods with a predictor-corrector time advance are used to temporally evolve the background plasma electron and ion equations. The dust is treated with a Particle-In-Cell (PIC) model in which the dust charge varies with time according to the standard dust charging model. A dust charge fluctuation mode is investigated using our numerical model as well as a lower hybrid instability due to the streaming of the dust relative to the background plasma.
INTRODUCTION Recently, there has been interest in new waves and irregularities that result from an expanding dust cloud released into a plasma [1]. Effluents from large space platform and exhausts from rockets and the space shuttle can create negatively charged dust clouds. In such dust clouds, the dust particles can be charged due to the collection of electron and ion currents from the background plasma. During these processes, dust charge fluctuations may be produced due to the unique interdependence of dust grain charging and plasma instability and are an important distinctive feature of a dusty plasma [2,3]. We have developed a new numerical model to study the generation of plasma irregularities produced by an expanding dust cloud. We also consider dust charge fluctuations associated with these plasma instabilities. Dust charge fluctuations associated with plasma instibilitiies have not
56
G.S. Chae et al. /Numerical simulation model for dust expansion into a plasma
been studied extensively with numerical simulations. The theoretical model and simulation results will be discussed in the next section.
MODEL In this section we describe the main features of our two dimensional numerical model which is implemented for studying expansion of dust into a plasma across a background magnetic field and also for studying dust charge fluctuations. The electrons are treated as a massless fluid whose density is reduced due to the dust charging. The electron continuity equation is given by
dt ^
• (ne.e)
=
(1)
^ 1 char J
where
1 [dpde
dUe
dt
charging
^® L
dt
The electron momentum equation reduces to ^
E
f fi =
X
B
(2)
52
since the electrons are taken to be strongly magnetized. The ions are also treated as a fluid. The ion continuity equation is (3) charging
where drii ~di
dt
I charging
The momentum equation for the unmagnetized ions is
dt
(4)
rrii
The dust is treated with the Particle-In-Cell (PIC) method. The dust positions fj are advanced as follows dt
m\
(5)
G.S. Cfiae et al. /Numerical simulation model for dust expansion into a plasma 57
dt
(6)
= ^j.
where v-^ is the dust velocity. Note that dust dynamics are included in this model. The standard dust charging model is used [2]. For the f^ dust grain this becomes
lej = 7ra^ -
VTT/
T
'y hi
= na
(7)
— lej + i i j ,
dt
QerieVte exp
2/ ° \
1-
TT/
0f =
e0f kTe e0f
(8)
(9)
Qj(i) 47reoa
Poisson's equation is used to calculate the electrostatic potential. It is given by 6oV^0 = -{qeTle + qiU'i + Qd^d)-
(10)
where Qd^d is the dust charge density and Qd^d = pd = Pde + pdi where pde is the charge density due to electron charging current collection, pdi is the charge density due to ion charging current collection.
SIMULATION AND RESULTS We used spectral methods [4] to solve the background plasma equations. Spectral methods have been the natural choice for this type of simulation involving nonlinear terms such as turbulence and transition studies, ocean dynamics, and ionospheric instabilities. A predictor-corrector method [5] is applied to solve the temporal dependence of our simulation model. Our first investigation was a fundamental study of the dust charge fluctuation mode. This also served as a test for our numerical model. The dust charge fluctuation dynamics equation is given by [2,3] dQd , dt
A
\T \ f^^ Vriio
^e rieo
(11)
where the dust charge damping rate c<;chg is given by ^chg —
el/,eO C
\kTe
+
kTi-e(f):fO,
(12)
58
G.S. Chae et al /Numerical simulation model for dust expansion into a plasma
S)^
0.10
'
\
'
1
T
• - • a ^ = 0.05 • - a ' ' ^ = 0.25 — w^'^ = 0.50
r\
1\
chg
k \
k \ 1
0.15
1
-
\
r ^ i
-0.15 0.15
\
r ^ '
1' '
- -
Qdo
0.05
=0.05
chg
r ^^
-i
\^ '• \\ \
-y
-0.15 0.15
\
\
10
0
^dO
V
\\
CO t = 3 0
20
CO t = 6 0 pi
30 CO t pi
40
50
60
-0.15
64
"
128
x/A (b) (a) F I G U R E 1. (a) Dust charge fluctuation temporal decay for a specific dust particle (b) Dust charge spatial peturbation decay for (Dchg = 0.05.
where /eo is the dust electron equilibrium current, C is the grain capacitance, Te,i are the temperature of the electrons and ions, and 0fo is the equilibrium floating potential. Equation (11) shows that dust charge fluctuations may be produced by electron and ion density perturbations (fie , ni) such as those produced by plasma instabilities. The dust charge fluctuations damp without electron and ion density perturbations. The low-frequency damped mode is described as [2,3] UJ ^
— ICUchg-
(13)
Figure 1. (a) shows the dust charge temporal decay for a single dust particle. Figure 1. (b) shows dust charge spatial peturbation decay for cuchg/^pi = 0.05. These results agree well with the theoretical predictions. We have also considered a lower hybrid instability which is developed due to the streaming dust grains with drift velocity Vd- The growth rate 7 is given by [6] 7
V3
2V3 WLH
t^pd
u,Pl
(14)
where C^LH = ^pi/\/(l + (^pe/^e)^ ^ ^'^d (for maximum growth of waves). Figure 2. shows the field energy and floating potential which present how the waves grow. The beam strength parameter u^ld/^li = 0.001. Figure 3. shows the density fluctuations of each species. Note that well defined perturbations in the dust charge as well as
G.S. Chae et al /Numerical simulation model for dust expansion into a plasma
59
2000
1500 W
E
1000
200
F I G U R E 2. Field energy and floating potential at a)chg = 0.5 for the lower hybrid streaming instability. CO .t = 5 0
CO .t = 7 0 Ei
0.5 ELECTRON DENSITY
-0.5 0.5
CO . t = 1 0 0 Ei
ION DENSITY
iO
-0.5 3.0
-1.0 0.5
-0.5
DUST DENSITY
DUST CHARGE
128 0
x/A
128 0
128
F I G U R E 3. Electron, ion, and dust densities and dust charge at cDchg = 0.5 for the lower hybrid streaming instability.
60
G.S. Chae et al /Numerical simulation model for dust expansion into a plasma
plasma and dust densities are produced by the instability. Linear growth occurs to time cjpit = 100. The growth agrees well with equation (14) for the parameters under investigation. At later times nonlinear processes occur. These are discussed in more detail in [7].
SUMMARY AND CONCLUSIONS We developed a new numerical simulation model for studying dust expansion with dust charging. The dust charge fluctuations which arise from the difference in relative density fluctuation of electrons and ions are investigated using our simulation model. We also presented the damping of the dust charge perturbation due to the phase difference between the perturbed dust charge density and the wave potential. An important role of this study is that we verifled some features of dust charge dynamics using our numerical simulation. Our model is general and may be used for future investigations of linear and nonlinear development of instabilities in dusty plasmas.
ACKNOWLEDGEMENTS This work was supported by the U.S. National Science Foudation NSF and Department of Energy DOE under grant DE-FG02-97ER54442 and the U.S. Office of Naval Research ONR.
REFERENCES 1. Scales, W. A., Ganguli, G., Bernhardt, R A., Lampe, M., Physica Scripta T75, 238 (1998) 2. Jana, M. R., Sen, A., and Kaw, R K., Phys. Rev E48, 3930 (1993) 3. Varma, R. K., Shukla, R K., and Krishan, V. K., Phys. RevE47, 3612 (1993) 4. Canute, C , Hussaini, M. Y., Quarteroni, A., and Zang, T. A., Spectral Methods in Fluid Dynamics (1988) 5. Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994) 6. Rosenberg, M., Salimullah, M., and Bharuthram, R., Planetary Space Science accepted (1999) 7. Scales, W. A., Chae. G. S., Ganguh, G., Bernhardt, P. A., Lampe, M., this issue (1999)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
61
Interaction Potential Between Two Debye Spheres J. X. Ma and Feng Pan Department of Modern Physics, University of Science and Technology of China Hefei, Anhui 230027, P.R. China
A B S T R A C T . An analytic model for the interaction potential of two adjacent Debye spheres is presented. It is shown that for an equilibrium non-drifting background plasma, the potential is repulsive for the same sign of charges.
I. I N T R O D U C T I O N Debye shielding is a fundamental property of plasmas. The Coulomb potential of a charged particle immersed in a plasma is shielded out by the plasma in a distance of the order of Debye length, v^ith the resulting screened potential known as Debye-Hiickel (Yukawa) potential (1). When there are streams of particles, wake fields (oscillatory potentials) develop in addition to the static screened potential (2), which were recently investigated by a number of authors (3-6) to explain the formation of plasma crystals (7-9). If two charged particles approach each other, their Debye spheres overlap and plasma particles redistribute. It was shown by computer simulation that the mutual shielding of closely spaced dust particles results in the reduction of the dust charge (10). In this paper, we present an analytical method for the mutual interaction of the two Debye spheres. The linearized Poisson equation is solved in cylindrical coordinate and the analytical expression for the interaction potential is obtained in terms of Bessel integration. Though the method can be apply to include effects such as plasma streaming, only static equilibrium background plasma is considered for simplicity. The potential distribution is obtained in section II and the interaction force is presented in section III. The conclusion is summarized in section IV.
62
J.X. Ma, E Pan /Interaction potential between two Debye spheres
II. POTENTIAL DISTRIBUTION Let us consider two dust particles with charges Zit and ^26 placed at 2: = ±c?/2 in a static equilibrium background plasma, where e is the electronic charge and d is the separation. Assume the dust size is much less than both d and plasma Debye length such that they can be treated as point charges. The total electrostatic (ES) potential (j) of the system is governed by Poisson equation V V = 47re[noexp(e(^/re)-noexp(-e
rf/2)-Z2%,z-^/2)],
(1)
where UQ is the unperturbed plasma density at infinity and ^e,^ are the electron and ion temperatures. Although the Coulomb coupling parameter of the dust particles can be much greater than unity, we restrict our analysis to the case where \e(f)lTQ^i\ < < 1. To see it is reasonable, let us estimate the ratio of the ES energy of electrons (ions) in the bare Coulomb potential of the dust to the kinetic energy, ^i,2eV/?re,, - (Zi,2/3A^D)(Ae,,//9), where A^,, = {T.^.jiire'n^fl'^ is the Debye length of electron (ion) species, A^^ = 47rAg^no/3 is the number of particles in the Debye sphere, and p is the distance from a dust particle. Because of the screening, |(/>| is less than the bare Coulomb potential. Thus, as long eiS Nd >> ^1,2, the Poisson equation can be linearized V V - A^V = -47re[Zi^(r, z + d/2) + Z2^(r, z - d/2)] ,
(2)
where A^^ = {K^ + K^)~^^^ is the effective Debye length. In order for mathematical brevity, we normalize spatial coordinate with respect to XD^ i.e., R = r/A/^, Z = Z/\D^ and D = d/Xj:)^ and introduce dimensionless ES potential $ = (^/(e/A/^). Thus, Eq. (2) can be written in dimensionless form ( V ' - 1)$ = -i7r[ZrS{R,
Z + D/2) + Z^SiR, Z - 0/2)]
.
(3)
Equation (3) can be solved in cylindrical coordinate. The solution can be expressed as ^ ^ Z. e x p ( - , , ) ^ Z, e x p ( - p - ) ^^
(4)
^^^^^^^^
where the first two terms represent the screened potentials of isolated particles, p± = [R^ + {Z ± D/2yY^'^, and $p is the modification due to the redistribution of background plasma
$p=
r Jo
-[AriK)exY>{-K\Z+\) 1^
+ A2{K)exp{-K\Z.\)]Jo{KR)dK
,
(5)
J.X. Ma, E Pan /Interaction potential between two Debye spheres
63
where AC = (1 + K'^Y^'^^ Z± = Z ± J5/2, and Jo{x) is the zeroth-order Bessel function. To determine the coefficients Ai^2{K)-, we employ the asymptotic behavior of the total $ . We note that, since the background plasma only modifies the potential in the neighborhood of the Debye spheres, the asymptotic solution at z -^ ±00 approaches that of the sum of isolated Debye spheres. Expanding $1 and $2 according to Bessel integration and comparing Eq. (4) with the asymptotic solution, we obtain Zi sinh[(/c ± l)D/2] + Z2 sinh[(Ac T 1 ) ^ / 2 ]
^
(6)
smn^/cl/j
It is noted that when D ^^ 0^ Eq. (4) approaches (Zi + Z2) exp(—/))//9, which is exactly the potential of one particle with total charge Zi + Z2. The interaction potential can be defined as the total potential at the position of one particle minus the self potential of that particle. Thus, we have the interaction potential at the position of particle 2
$12 = ($1 + ^P)R=O,Z=D/2
= Zi exp{-D)/D
i:
+ /
[A^ exp{-KD)
+ A2]dK .
(7)
Figure 1 shows a typical potential distribution for the case of Zi — Z2 and d — 2\D. It is shown that $ is strongly peaked at Z = ± 1 , which are the positions of the two particles (In fact, it is infinity at these points). It can also be seen that the potential has week oscillations in the radial direction due to the Bessel function dependence on R. The latter is the result of finite upper limit of K in numeric integration. When the upper limit is larger, the oscillation disappears.
Z2
F I G U R E 1. The distribution of the dimensionless potential ^/Z^ Z/XD for the case of Zi = Z^ and d — lXi).
with respect to r/Xj) and
64
J.X. Ma, E Pan/Interaction potential between two Debye spheres
D FIGURE 2. The variation of ^12/^2 with respect to the inter-particle distance d/Au for the case of Z1 — Z2. The variation of the interaction potential at particle 2 with respect to the intergrain distance D is numerically shown in Fig. 2. The potential is monotonically decreasing with the increase of Z), which implies that interaction is repulsive. When the distance is more than a few Debye lengths, $12 tends to saturates and the modification due to background plasma becomes week.
III. INTERACTION FORCE The electric field can be obtained by — V $ . Because of the cylindrical symmetry, only z-component of the electric field contributes to the interaction force. The dimensionless field Ez — —d^jdZ is 00
Jo{KR)dK
,
(8)
where A[ 2 = ^1,2 + ^1,2- It is obvious that Ez is discontinuous at exactly Z = Z±. To overcome this problem, we make use of an alternative method to calculate the force. Since the particles are surrounded by Debye spheres, the force on the particle is equivalent to the force on the Debye sphere. To evaluate the force, we further assume, as an approximation, that the net charge in a Debye sphere uniformly distributes on the surface of the sphere, with the dimensionless surface charge density cri,2 = ^i,2exp( —l)/47r. Thus, the dimensionless force on particle 2 is obtained by integration over the surface oi R = sm6 and Z — D/2 = cos 0,
J.X. Ma, E Pan/Interaction potential between two Debye spheres
65
1 0.8F
0.4 0.2
0
D
F I G U R E 3. The dimensionless force Fi2/27rcr2Z2 versus inter-particle distance djXD for the case of Z\ — Z^'
I
Fi2 = 2ira2 /
^^^(sin 9, cos 0 + D/2) sin OdO
(9)
where we have used the total electric field instead of the field excluding the self field of particle 2 since the latter is spherical symmetric and does not contribute to the net force when integrating over the Debye sphere. A plot of the variation of the force F12 devided by 27r(j2Z2 with respect to the particle separation d/Xjj is numerically shown in Fig. 3 for the case of Zi = Z2. It is shown that the force is monotonically decreasing with the increase of c?, and the force is repulsive (positive) for the case of two charges having the same sign. This is consistent with the result of Fig. 2 for the interaction potential. If the two charges have opposite sign, Eq. (9) yields the attractive force (not shown here) as expected. When d exceeds a few Xjj^ the force approaches zero and the interaction is negligible.
IV. CONCLUSION In this paper, an analytic method is presented for the interaction potential of two closely-spaced Debye spheres in an equilibrium non-drifting background plasma. The linearized Poisson equation is solved to yield the potential distribution, and the interaction force is obtained by the net ES force on the Debye sphere. It is shown that for two charges of same sign, the interaction force is repulsive and decreases rapidly with the increase of inter-particle distance. When the distance
66
J.X. Ma, E Pan /Interaction potential between two Debye spheres
exceeds a few Debye lengths, the interaction becomes negligibly week because of the screening by background plasma. It should be point out that our model is based on the linearized Poisson equation, which is limited to the case where the number of electrons (ions) in a Debye sphere is much larger than the charge number of a single dust particle. The latter condition usually holds for laboratory plasmas and micrometer-sized particles with inter-particle distance d of the order of Debye length. If d is much less than Debye length, however, nonlinear effects dominate in determining the interactions. It was recently demonstrated in reference (11) that dust grains inside a Debye sphere can attract each other to form various clusters. Another effect not considered in the present paper is the plasma streaming. Since grains are usually levitated in sheath region, ions drift supersonically relative to the grains, which is the origin of oscillatory wake potential (3). Further investigation will include the effects of nonlinearity and plasma drifting.
ACKNOWLEDGMENTS This work is supported by the International Atomic Energy Agency (Contract 8933/Regular Budget Fund), the National Natural Science Foundation of China (Grant 19875051), and the Grant LWTZ-1298 and the Basic Research Program (Youth Science Fund) of Chinese Academy of Science.
REFERENCES 1. Ichimaru, S., Statistical Plasma Physics, Vol. I: Basic Principles, Redwood City: AddisonWesley, 1992, ch. 1. 2. Yu, M. Y., Tegeback, R., and Stenflo, L., Z. Physik 264, 341-348 (1973). 3. Vladimirov, S. V., and Nambu, M., Phys. Rev. E 52, 2172-2174 (1995). 4. Vladimirov, S. V., and Ishihara, O., Phys. Plasmas 3, 444-446 (1996). 5. Shukla, P. K., and Rao, N. N., Phys. Plasmas 3, 1770-1772 (1996). 6. SalimuUah, M., and Amin, M. R., Phys. Plasmas 3, 1776-1778 (1996). 7. Ikezi, H., Phys. Fluids 29, 1764-1766 (1986). 8. Chu, J. H., and I, Lin, Phys. Rev. Lett. 72, 4009-4012 (1994). 9. Thomas, H., et. al., Phys. Rev. Lett. 73, 652-655 (1994). 10. Choi, S. J., and Kushner, M. J., /. Appl. Phys. 75, 3351-3357 (1994). 11. Chen, Y. P., Luo, H., Ye, M. F., and Yu, M. Y., private communications.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
67
COUPLING OF WAVES AND ENERGY CONVERSION IN A SLOWLY VARYING NONUNIFORM DUSTY PLASMA
Manoranjan Khatiy Susmita A.M.Basu
Sarkaf", Tushar k.
and Samiran
Chaudhuri,
Ghosh
CENTRE FOR PLASMA STUDIES , FACULTY OF SCIENCE JADAVPUR UNIVERSITY , CALCUTTA - 7 0 0 0 3 2 , INDIA
ABSTRACT Propagation of electromagnetic and acoustic waves through nonuniform plasma and their mutual coupling due to various factors has important application in space and astrophysical bodies, ionosphere, etc. In this paper, we have studied the propagation of low frequency ionacoustic waves coupled with dust acoustic waves and electromagnetic radiation through a slowly varying unmagnetized dusty plasma. When low frequency ion-acoustic wave is incident at the boundary of each layer of the varying plasma medium, a part will be transmitted consisting of three components of waves such as ion-acoustic, dustacoustic and electromagnetic wave and other part will be reflected back consisting of these components stated above. The generation of the excited dust-acoustic wave and electromagnetic wave were evaluated using W.K.B. approximation for slowly varying plasma parameters.
^Permanent Address: Dept. of Applied Mathematics, University of Calcutta, 92, APC Road, Calcutta-700009, INDIA
68
M Khan et al /Coupling of waves and energy conversion
INTRODUCTION In a non-uniform plasma, compressional waves and e. m. waves are coupled and energy is transferred from compressional waves to e. m. waves and vice versa. In a slowly varying plasma, Bremmer [1], Tidman [2], Chakraborty [3] studied the wave propagation and their mutual coupling using the technique of W.K.B approximation. Also Khan and Paul [4] have investigated the coupling of electron-acoustic and e.m. waves in a slowly varying two component plasma and obtained the power flux of transmitted and reflected e. m. waves using W.K.B. approximation generated by electron acoustic wave. However , the presence of charged dust grains has an important impact on the total scenario of coupling of waves in a non-uniform plasma. Dusty plasmas produce some new electrostatic modes such as 'dust acoustic wave' [5], 'dust-ion acoustic wave' [6] in the very low frequency limit. In this paper, we have investigated the propagation of low frequency ion- acoustic wave through unmagnetized slowly varying dusty plasma medium using the technique of W.K.B approximation. When a low frequency ion-acoustic wave is incident at the boundary of each layer of the varying plasma medium, a part will be transmitted consisting of three components of waves such as ion-acoustic, dust-acoustic and e.m. waves and other part will be reflected consisting of those components. BASIC EQUATIONS We consider a three component non-uniform, collisionless, warm dusty plasma having electrons, ions and dust particles in the absence of any static background magnetic field. The dust particles are negatively charged satisfying the charge neutrality condition ri^^ + z^n^^^ = n^^. The non-uniformities of the plasma medium is due to the density gradient of the plasma constituents which varies slowly along the Z-direction and the scale length of variation (L) is much smaller than the wave length [X) of the incident wave. The basic equations are the equations of motion, the continuity equations for the three fluids, the Maxwell's equations. We assume that the wave phase velocity rt>/A:« V^^ so that we can neglect the electron inertia term from the corresponding momentum conservation equation. We assume that the charge on the dust grains are constant, so that we can neglect damping arising due to charge fluctuations.
M Khan et al./Coupling of waves and energy conversion
69
To find the linearized plasma equations, we consider the following first order field quantities as Ar = //.. + Sri^, u. = 0 + u^, £ = 0 + ^ Now following [3], we obtain the following linearized coupled equations between transverse field variables and the pressure perturbation for all relevant quantities, Ami (V;.)xV( £ > £ - ^ ) CCOjil m, m
(r-+kf)H
a
•d^Pd V
(V^+4) rri^^
SP,
IC\ V ( G ; M . ( V X / / )
'", J
^d
l^co' v(-»').v-••*'
^Pd
^Pd
a
TT^P.^ m}
JUO)^
(^'pd+0}p,
// = 1
*
+-
V; = xT, /m,,
(2)
AKCILUO
IC\
I /
+-
(3)
Am^ejuo)
,_co--co;, VJ
CO'
5p, = m/jai^,
(1)
^t
I
^{wl,)-W
(^'^klA-^^.m. m/-
V//x(Vx//)
Where zJ^^n^ho
CO,
are
respectively the Boltzmann constant, kinetic temperature, mass and equilibrium density of the jth species. W.K.B. SOLUTION
We assume that the ion pressure perturbation is the only source of excitation of the longitudinal waves and coupling occurs only by density gradient. We also assume that the equilibrium densities change slowly but continuously along OZ in such a way that the wave length {X] of radiated waves is much less than the density gradient scale length i.e. X«L [2], for which we can make the W.K.B approximation. Under this condition the equation (1) and (3) reduces to the following form, {V'-+kf)H
47rei CJUO)
V//xV
^ 'd^Pd
Sp, ^
V
mI
^d
J,
(4)
70
M Khan et al./Coupling of waves and energy conversion
i.Sp,
iy'^K)
m,
0) pd
mV,
5p-0
(5)
The disturbances propagate in the (z , x) plane and the space dependence of the field quantities are of the form ~ exp(/A:oX). The differential equation for the dust-acoustic wave excited by the incident ion -acoustic wave is {D'+kl)p,{z)^'^pXz), kl-kf,-kl (6) my; For W.K.B. solutions of the dust acoustic wave, using the method of variation of parameters [7], considering only real part we get dp J (z) = -y=exp/(A^()X + \k^dz - cot)
(7)
A_ r Sp/(z) = -j=Qxpi(kQX- ik^dz-cot)
(8)
Where A^ and A_ are the pressure amplitude of the transmitted and reflected part of dust-acoustic wave. The differential equation of the excited transverse e. m. wave is of the form {D'-^kl)H{z).
A/rei CJUO)
V//xV ^^dPd V ^^d
A^
kj - k^
k^
m.IJ
(9)
In the low frequency limit, the e. m. wave is attenuated. ENERGY FLUX DENSITIES
The z-component of the energy flux density of the incident longitudinal pressure wave is the product of pressure perturbation and velocity perturbation i.e. the energy flux density is S[ = (Rtui Re^^^)^, j=i,d. The time average of incident energy flux over the time period 27i;/co is o)B; K'^^'^o^M^^l' +ftj;,)cOS(^, -0,) 1 (^
1-1 +
M Khan et al/Coupling of waves and energy conversion
(ol^z,eklAB,(^ol,+(ol,)^\n{e,-9,)
71
\
+ 4co'n,^m,{&- -o)J, )^k, (k^ + k, )^k^
L
colz,k:A_BXco;,+col,)cos(0,+e,)
i
(10)
The z-component of the energy flux density of the dust wave is given by The time averaged energy flux of the transmitted and reflected part of the excited dust acoustic wave are given as S',
2n,,m,(o)^-0)1,)
4n,,m,co'(o)' -co;,)Mk,k,{k, -kj
cal,k^((ol,+(o;,)Al 4njQmjO)'((o' -o)lj)^,k,{k,-k,y
//
L' (11)
{s^4 (12)
DISCUSSION There is an extreme anomalous situation in the Saturn's F-ring, where the number density of free electrons is much smaller than the number density of ions [8]. Since for this type of wave mode Vtd, vti <<(D/k<< vte, a very little amount of electrons is required in the back ground of collisionless dusty plasma [6], this phenomena is relevance to low frequency noise in the F-ring of Saturn. The optical depth of F-ring is ~ 0.1 and there are micron to submicron ranged dust grains. Thus the number density of the dust grain varies as the grain radius. Hence Saturn's F-ring maybe considered as the slowly vaiying medium and the W.K.B approximation is valid in this medium.
72
M Khan et al./Coupling of waves and energy conversion
Thus for ion-acoustic wave incident on the Saturn's F-ring may generate the transmitted and reflected dust acoustic waves and the corresponding energies can be calculated by using the W.K.B technique as mentioned above. Due to the presence of varying grain distribution in such a media, the conversion of energy may be more efficient than the ordinary electron-ion plasma.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Bremmer, H. Physica, 15, 593(1949). Tidman, D.A. Phys.Rev,, 111, 366(1960). Chakraborty, B. JMatKPhys., 12,529(1971). Khan, M.R. & Paul, S.N. ApplSci.Res,, 3 3 , 209(1977). Rao, N.N., Shukla, P.K. & Yu, Y.M. PlanetSpace Sci„ 38, 543(1990). Shukla, P.K. 8& Silin, V.P., Phys.Scr,, 4 5 , 508(1992). Mathew, F. & Walker, R.L. ''Mathematical Methods of Physics'\ W.A.Benzamin Inc.New York, 1964, Sec. 1.1 . 8. Northrop, T. Private Communication.
Part II. Plasma Crystals - New Material
This Page Intentionally Left Blank
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
75
Classical Atom-like Dust Coulomb Clusters in Plasma Traps Lin I, Yin-Ju Lai, Wen-Tau Juan, and Ming-Heng Chen Department of Physics. National Central University, Chungli, Taiwan 32054, Republic of China The structure and thermal excitations of strongly coupled quasi-2D dust Coulomb clusters from small to large particle number ^V in a cylindrical rf plasma trap are reviewed and compared with the numerical results from the 2D strongly coupled Coulomb clusters with several different interaction forms. The common symmetry under the interplay between the central confining force and mutual repulsion leads to many generic structures and motions such as the concentric shells with the classical periodic packing sequence and angular motion dominated excitations at small A^; and the triangular lattice core surrounded by a few outer circular shells and the recovering to the isotropic vortex type excitations at large N. Changing the interaction and confining potentials affect the detailed packing sequence, the radial variation of the packing density, the positions of the shell-triangular core interface, and the density distribution of defects..
In a dusty plasma, the suspended dust particles can be strongly coupled due to their large negative charges (about 10"^ electrons per //m sized particle) to form large volume dust crystals (with sub-mm lattice constant) and liquid states (1-6). Quasi-2D hexagonal crystals can also be induced under the vertical ion flow. With this convenient experimental system, the formation, behaviors on the structures and dynamics of the small A^ (particle number) limit, and the transition to large N are certainly interesting issues. In this paper, after our first observation of the quasi-2D dust Coulomb clusters in a rf plasma trap, we give a brief review of our recent experimental and numerical studies on these issues (7-10). In general, this type of classical systems consisting of a finite number of elements interacting through Coulomb type repulsive interactions can be classified as strongly coupled Coulomb clusters (SCCC). J.J. Thomson's Classical Atoms with electrons imbedded in a neutralizing ion background, electron dimples on the liquid He surface, electrons in Coulomb blockades, flux lines in super-fluids, super-conductor domains and magneto-plasma traps, etc. , are the few typical 2D SCCC systems (11-16). Under the common features of the mutual repulsive interaction and the circular symmetry of the central confining force, this nonlinear system exhibits rich but generic behaviors regardless of the detailed interaction forms. For example, concentric shell structures with classical periodic table for packing, and nonuniform excitations at small A^ have been predicted theoretically and numerically for 2D clusters with I r interacting potential (17,18). Unfortunately, in the above laboratory systems, they have never been studied experimentallyfi*omsmall to large ^V. For a 2D Wigner crystal at infinite large N, triangular lattice is the basic stable
76
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
form shell structures. The competition between them leads to the generic structures with inner triangular cores surrounded by the outer circular shells. This can also be understood starting from the limit of hard sphere interaction (11). In a weak confining potential, the hard spheres can only be packed into different shells of the triangular lattice with 1 to 4 particles at the center. The cluster with a close-shell structure has six defects each with -1 topological charge at the six comers of the outmost shell and a defect free core. If the repulsion can be relaxed to a softer form, the outmost shell can be readjusted to a more circular shell to minimize the potential energy. The inner shells are still in the hexagonal fomi due to the weaker confining. Depending on A^ and the detailed forms of the confining and mutual repulsion, the radial distribution of the particle packing density may be different for minimizing the total potential energy and reaching the local force balance for each particle. Defects may appear in different places to accommodate the lattice defonnation caused by bending, interface matching between the shell and triangular structure, and the nonuniform density distribution. As temperature increases, the competition between thermal agitation and mutual coupling between particles generates different type of collective excitations. The excitation could be quite nonuniform at small N due to the strong boundary effect and should recover to the vortex type excitations in the large volume dusty plasma observed previously (9). In our molecular dynamics (MD) simulations (10), the equation of motion for the particle / includes the forces from the mutual interaction potential K^ due to the y-th particle and the confining potential Vd, in addition to the inertia and linear viscous damping terms. Note that in 2D, the parabolic confining potential is equivalent to the potential generated by a uniform frozen neutralizing ion background. The effects of different fornis for F^ (e.g. labeled by type I to V interactions in Table 1) are checked. A spatially and temporally 5 related Gaussian noise is also included for simulating the temperature effect. The ground state configurations can be determined through many annealing cycles by changing the intensity noise rj o. The experiment is conducted in a cylindrical symmetric rf dusty plasma system described elsewhere (8). Instead of the large radius groove in our previous experiment (1), a hollow coaxial cylinder with 3-cm diameter is put on the bottom electrode to confine 5 // m polysterine particles in the weakly ionized glow discharge generated in Ar at a few hundred mTorr using a 14-MHz rf power system. Vertically, the dust cluster is dragged by the gravitational force to the bottom part of the glow. It fornis a virtual cathode and induces strong vertical ion wind which generates dipoles and causes the formation of vertical dust chains. The cluster can be treated as a quasi-2D system. The particle positions in the horizontal middle plane are optically monitored. The micrographs of the different states (i'V/,iVj ,N-^....) with different occupation numbers N^ from the inner to the outer shells for the small A^ clusters from our experiment are shown in Fig. 1 and 2. Basically, the outmost shell is more circular. As iV increases, particles are alternately packed into different shells. Ni , A'., and N^ periodically oscillate between 1 and 5 , 5 and 11, and 11 and 17 respectively. The
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
11
above filling process shows adding particles into certain shell generates new rooms in the neighboring shells for adding the following particle. Table I shows the experimental packing sequence and similar numerical results with different forms of V^j and Vci. A few typical triangulated configurations are shown in Fig. 2.
17(1,5,11)
19(1,6,12)
29(4,10,15)
609
71
FIGURE 1. A few snap shots of the experimental micro-images of the typical clusters at different A. For the displaying purpose, the scales are not the same for the pictures. The typical inter-particle spacing is between 0.3 and 0.7 mm.
-3
(a) 5
£) ^
6(1,5)
7(1,6)
( ^ (^ ^
11(3,8)
12(3,9)
13(4,9)
14(4,10)
18(1,6,11)
19(1,6,12)
20(1.7,12)
21(1,7,13)
25(3,8,14)
O O
8(2,6)
15(5,10)
_ 22(2,7,13)
9(2,7)
^
10(2,8)
(^
16(5,11)
17(1,5,11)
23(2,8,13)
24(28,14)
^ (e)
26(3.9,14)
27(3.9,15)
28(4,9,15)
29(4,10,15)
30(4,10,16)
10(2,8)
15(4,11)
20(1,7,12)
25(2,8,15)
30(4,10,16)
10(2,8)
15(5,10)
20(1,7,12)
25(3,9,13)
30(5,10,15)
(b)
(c)
FIGURE 2. Triangulated configurations up to .V = 30. (a) The experimental results, (b) and (c) The few typical simulation results for the t\pe IV and III interactions respectively.
Depending on the detailed interaction forms, AM ^ M - M.y might deviate by one or two. The cluster with a shorter range repulsion (e.g. decreasing A D to 0.1) tends to
78
Lin I et al. /Classical atom-like dust Coulomb clusters in plasma traps rf
V, Vf
N ~3 4 5 8 7 8 9 10 11 12; 13 14 15 18 17 18 19 20 21 22 23 24 25 26 27 28 29 30
exp('r„/Xo) h Xo = 0.1 Xo^l-O 3 4 5 1.5 1.6 1.7 2.7 2.8 3.8 3.9 4,9 4,10 5, 10 1,5, 10 1.6.10 1,6,11 1.6.12 1,7,12 2.7,12 2,8.12 3, 8, 12 3.8.13 3.9,13 4,9,13 4, 9, 14 4, 10. 14 4,10,15 5, 10, 15 I
—4
4.10 5.10 1. 5,10 1,6,10 1.6,11 1.6,12 1,7.12 2,7,12 2,8,12 3.8,12 3.8.13 3. 9.13 4.9,13 4,9.14 4,10.14 5.10,14 5.10.15 g
1/r^
Xo-°o
14
~3 4 5 1,5 1.6 1.7 2.7 2,8 3,8 3,9 4,9
5 1,5 1,6 1.7 2,7 2,8 3.8 3.9 4,9
exp
rhrflTj \n(i/r^)
4. 10 5, 10 1,5,10 1,6,10 1.6,11 1.6,12 1,7,12 1.7,13 2,8.12 2,8.13 3,8,13 3,9,13 3,9,14 4,9,14 4, 10,14 4, 10.15 5, 10.15 ffl
14
5 1.5 1,& 1,7 1.8 2,8 3.8 3,9 4.9
4,10 5,10 5,11 1,5.11 1.6,11 1.6.12 1.7,12 1,7,13 1.7.14 1.8.14 2. 8.14 3.8,14 3,9.14 3, 9,15 4. 9,15 4,10,15 4.10,16 IV
^
"5
4 5 1.5 1,6
5 6 1.6 1.7 1.8 2.8 2,9 3,9
3, 10 4. 10 4.11 5, 11 5,12 1.5. 12 1.6.12 1.6, 13 1.7,13 1.7, 14 1.8, 14 2,8, 14 2,8, 15 3, 9, 14 3,9, 15 3,9, 16 3, 10, 16 4, 10, 16 V
2.6 2,7 2,8
1 1 1
4.9
1
3,8 3.9
4.10 5,10 5.11 1.5,11 1.6,11 1.6,12 1,7.12 1,7.13 2. 7. 13 2, 8, 13 2.8.14 3, 8. 14 3, 9. 14 3, 9. 15 4. 9,15 4,10.15 4.10.16
i 1 i \
VI
Table I The Mendeleev table for the particle packing sequences from N = 1 to 30 under different combinations of the confining and the mutual repulsion potentials. I to V are also used in other figures to represent the corresponding type of interaction and confinement, and type VI is the experimental results. In the type V confining potential, rcN is the cluster radius obtained in the type III interaction at the corresponding N. N = 23 Vij = I n 1/rjj (1,8,14)
-449.473
(2,8,13)
125.530
13.3122
0.64274
FIGURE 3. The change of the ground and the nearest meta-stable states at N = 23 for the type IV repulsion and the Yukawa type repulsion in a parabolic confining well with decreasing interaction range. The energy of each state is listed below each cluster, A and n represent the -1 and +1 topological defect respectively.
have a higher packing density in the inner region. This can be reversed by turning on a steeper confining force around the boundary of the cluster (e.g. adding a fourth order term on Va as shown in the type V Column of Table I, which corresponds to the case with a higher density of neutralizing ions around the cluster boundary). This trend can be manifested by the series of simulated structures for the N = 23 state under different interaction forms. The first and second rows of Fig. 3 represent the ground and first excited states respectively. The energy difference between the ground and the first excited states is very small. Note that the trend of increasing packing density in the
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
79
inner region can be reversed at the very small A D liinit. The structure is akin to the hard sphere with triangular packing over the entire cluster even to the outmost shell. Comparing with the packing sequences from our simulation, our experimental results stay between the cases with Vy = J/ry and ln(7/r;y).
(b)
_
«•
(C)
^
,
,
EXP
^
^>« 0 - 20 s
MD
(d)
^
".=• 0.063
(e)
^ ^ ^ * ^ ^ , J'.'Oia 1
' / '^ ^ . I
. s
^'/»005
«
s
»
-.
^
-»
*
•
At= 5000
MP
At= 2000
MD
^t= 2000
FIGURE 4. (a) and (b) Triagulated snap shots and trajectories showing the evolution of ±e (2,8,13) state from our experiment, (c) to (e) The trajectories from our MD simulation with increasing noise intensity- 7] o- The damping coefficient is kept at 0.15.
This strongly coupled system supports many interesting thermally induced collective excitations. Under the shell structure induced by the circular system symmetry, the collective inter-shell angular excitations predominate over other excitations, such as the intra-shell angular and radial motion, the radial vibration of the whole cluster relative to the confining center, radial breathing, etc.. Figure 4(a) and (b) give an experimental example of the time evolution of the N = 23 cluster. The elongated trajectories indicate the higher excitation energy for the radial motions. The sequential snap shots with triangulations show the relative particle and defect positions. From our simulation, the (2,8,13) cluster has three stable (meta-stable) configurations: a single 5-fold defect, a connected 5-7-5 defect pair, and a 5-7 defect pair with another 5-fold defect separated by one dust particle around the second shell. The center two particles prefer to line up with the two particles in the second shell. These three states are almost degenerated and the energy barriers among them are much smaller than the transition to other nearly degenerated states with different sets of Ni ,such as (1,8,14) and (3,8,12), in which the transition should be associated with the particle radial hopping. The experimental snap shots in Fig. 4 manifests that these three nearly degenerated states can be easily thermally accessed through the slight adjustment of the particle relative positions which causes the propagation, generation and annihilation of defects. Note that the state with defects at the center is unstable and seldom occurs. The triangularly packed center core is quite robust. The relative position of the second to the third shells is not that strongly locked and can be more easily excited. For the structure with magic packing such as the (2,8,14) state, the thermal defect can equally access to the entire cluster (20). These phenomena are also evidenced by our simulation results. Depending on the temperature and viscosity, the relative fluctuation amplitudes of the angular motions for
80
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
different shells may change. The MD results in Fig. 4 further shows that at high temperature, the inter-shell hopping can also be excited.
FIGURE 5. Triangulated configurations (MD results) for the larger N clusters with tvpe IV and III interaction forms at zero temperature. The two states for N = 180 (type IV) have the same total energyup to the sixth digit. Figure. 5 shows a few simulated ground state configurations at larger A^ for the cases of Vjj = ln(l/ry ) and IMy obtained at zero temperature. For the former case, the defects mainly stays around the outer shells and leaves a uniform defect free core. However, for the latter case, the core part is relatively defect free only for A^ < 150. The former case has an almost uniform radial packing density distribution through the cluster but the latter has lower density as radius r increases (the mean lattice constant can increase about 30% from the center to the boundary) (11). Note that, Vy = In(l/r,y) is the interaction between two uniformly charged long wires, Vy = l/r^ is the interaction between two point charges on a plane. Va = rr is the field from a uniform neutralizing ion background. The packing density should be uniform for the first case in order to null the coarse grained space charge. For the second case, the leaking of the electric field along the axial direction weakens the radial field. Particles in the inner region are compressed to higher density to reach the force balance (10). The easier bending at larger A^ makes the defects move inwards from the outmost shell. In addition to accommodating the interface matching between the outer circular shells and the inner triangular cores, the defects have to move further inward for the latter case to accommodate the lattice bending due to the nonuniform packing. In our experiment, the higher rf power for supporting the larger A^ cluster also makes the background fluctuations larger than the cases at small A^. The fluctuations can cause lattice deformation and make the defect move inward. Our experimental results (Fig. 6) show that, unlike the large increase of the mean lattice constant Ur with r for the case of Vy = I fry, Qr is uniform in the inner region and only increases less than 10% toward the boundary. Namely, our larger A^ cluster is more similar to the case of long wire interaction. The contribution from the particles along the vertical chain and the slightly ion rich double layer boundary could be the possible causes (note that there could be about 15 particles along one vertical chain for the large cluster but less than 10 for the small clusters. Our MD simulation results show that in addition to the intrinsic defect around the cluster boundary for the case of Vy = ln(l/r;,), the thermal excitations cause
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
81
the Spread of defects into the center part of the cluster (Fig. 7). N = 88 / V \y-.
Os
10s
^ .
0 - 10s
10-20 s
N = 179 •
K ' ' I ' ^ ' ' . V'.
v**;-*- i • ; . ^ ' t - f l ^ ^ •
• *^' ,^*^;--Vrt^'
...*-." M /
10s
0-IOs
10-20 s
N = 287
•::rn-^
•/.-.••''rSyti:.-.
Os
FIGURE 6.
"
10s
0-IOs
' " '
10-20S
The typical experimental results of the defect and particle trajectories at large .V.
(a) . V0.05
.••//.",• • V 0 ' ' 2
• • • • • ' • i ' •' •* '*>'.* ^'; -.' •' ^ ' ' ' - V - '•»
0.0001
0.0010
0.0100
0.1000
1.0000
FIGURE 7. (a) The trajectories and the corresponding snap shots of defect configurations showing the transition to the more disordered liquid state as noise intensity 7] o increases for the ideal 2D Coulomb interaction at A' =300 from our MD simulation, (b) The averaged power spectra of particle coordinate evolution (averaged over 100 particles in the core of the cluster).
Unlike the small A^ cases, the excitations become more isotropic and uniform as N reaches the level of a few tens because the system also has a larger inner triangular domain which provides space for the radial hopping. Figure 6 shows the typical vortex type collective excitations observed in our experiment. Only the particles along the outmost shell prefer angular excitation due to the circular shell structure. Similar to the
82
Lin I et al /Classical atom-like dust Coulomb clusters in plasma traps
previous observation in the very large N crystal (5,6), the cluster exhibits continuous excitation and relaxation of vortices with different sizes and life times, associated with the generation, propagation, interaction and annihilation of defects. The spatial and temporal correlation lengths of the excitations first increase with noise strength 7] o and then decrease as 77 ^ further increases to the liquid state (Fig. 7). The two peaks in the power spectra correspond to the center of mass vibration and the shortest wavelength acoustic modes. Increasing j] o causes the increase of the background floor and the slopes of the low frequency part. It corresponds to varying the temporal correlation length of the vortex type excitations which generates persistent diffusion (6). We also found that applying an axial magnetic field with a few tens of Gauss can cause rotation of the cluster along the ExB direction, with about 10"^ Hz frequency and no averaged angular velocity shear(9), where E is the radial (outward) space charge field in the sheath. Our rough estimate of the momentum transfer rates shows that the cluster is floating and drifling with the background neutral whose rotating speed is determined by the balance between the incoming angular momentum from ions and the viscous loss toward the surrounding walls. This research is supported by the National Science Council of the Republic of China under contract number NSC-88-2112-M008-008.
References [I] J.H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). [2] Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. 33, 804 (1994). [3] G.E. ^ Thomas, et al, Phys. Rev. Lett. 73, 652 (1994). [4] A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 (1994). [5] Lin I, W.T. Juan, and C.H. Chiang, Science, 272, 1626 (1996). [6] C.H. Chiang and Lin I, Phys. Rev. Lett, 77, 647 (1996). W.T. Juan and Lin I, Phys. Rev. Lett, 80, 3073 (1998). [7] W.T. Juan, Z.H. Huang, J.W. Hsu, Y.J. Lai and Lin I, Phys. Rev. E, R6947 (1998) [8] J.M. Liu, W.T. Juan, J.W. Hsu, Z.H. Huang, and Lin 1, Plasma. Phys. Control Fusion, 41, A47 (1999) [9] W.T. Juan, J.W. Hsu, Z.H. Huang, Y.J. Lai, and Lin I, Chinese J. Phys. 37, 184 (1999). [10] Y.J. Lai and Lin I, to be published. [II] J.J. Thomson, Phil. Mag. S. 6. 7, 39, 236 (1904). [12] H. Ikezi, Phys. Rev. Lett. 42, 1688 (1979). P. Leiderer, W. Ebner and V.B. Shikin, Surface Science, 113,405(1982). [13] Nanostructure Physics and Fabrication, edited by M.A. Reed and W.P. Kirk (Academic, Boston, 1989). [14] D. Reefman, H.B. Brom, Physica 183C, 212 (1991). [15] W.I. Glaberson and K.W. Schwartz, Phys. Today, 40, 54 (1987). [16] D.Z. Jin and D.H.E. Dubin, Phys. Rev. Letts, 80, 4434 (1998) [17] V.M.Bedanov and P.M. Peeters, Phys. Rev. B. 49. 2667 (1994). [18] V.A. Schweigert and P.M. Peeters, Phys. Rev. B. 51, 7700 (1995). [19] A. A. Koulakov and B.I. Shklovskii. Phys. Rev. B. 55. 9223 (1997) [20] J.W. Hsu, Master Thesis. National Central Umversit\' (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
83
Crystallography and Statics of Coulomb Crystals Yasuaki Hayashi and Akito Sawai Department of Electronics and Information Science Kyoto Institute of Technology Matsugasaki, Kyoto, Japan 606-8585
A b s t r a c t . Three-dimensional and two-dimensional Coulomb crystals have been formed in plasmas by growing carbon fine particles, of which size has been controlled using the Mie-scattering ellipsometry. Lattice constants of the crystals were determined from CCD images of top view and side view taken at the same time and at the same position. Smaller particles of L4 M m in diameter formed the structure of a three-dimensional Coulomb crystal, which was face-centered orthorohmbic. It is suggested that the ratio of the lattice constants of the crystal is decided so that Coulomb energy takes a minimum value under the condition of constant particle density in horizontal layers. Larger particles of 5 M m in diameter formed a two-dimensional crystal structure, simple hexagonal one. The result of the dependence of lattice constants with particle density for the crystal indicates that the force acting on particles in the vertical direction is repulsive for smaller value of the vertical lattice constant while attractive for larger one. The vertical directional forces are mainly external ones for smaller particles, however a directional attractive force between particles plays an important role for the formation of Coulomb crystal for larger particles.
INTRODUCTION Since Coulomb crystals in fine particle plasmas were experimentally discovered several years ago [1-3], many types of the structure have been reported to be observed. They are simple hexagonal, body-centered cubic (bcc), face-centered cubic (fee) and so on. Because structures of Coulomb crystals reflect forces acting on particles, the quantitative or qualitative analysis of the forces should be possible from the analysis of the structures. In order to determine the three-dimensional structure and lattice constants of t h e crystals exactly, top and side views have been taken with two CCD video cameras at the same time and at the same position.
84
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
EXPERIMENTAL A schematic drawing of an experimental system for the formation of Coulomb crystals is illustrated in Fig.l. Coulomb crystals were formed by the growth of monodisperse spherical carbon fine particles in methane/argon R F (13.56 MHz) plasmas. As in the previous experiments [1,4-6], the diameter of growing carbon fine particles was monitored by the Mie-scattering elhpsometry [7,8]. For the elhpsometry measurement and the observation of particle arrangement, argon-ion laser hght of 488 n m in wavelength was irradiated upon particles, which were suspended above an R F electrode and trapped in a potential bucket formed by a ring of 3 cm in inner diameter on the electrode. Three-dimensional structures of particle arrangement were observed by two CCD video cameras from the top and the side at the same time and the same position.
C( 2 D (Tc
Ar+ Laser *"""
CCD (Side)
Mie-Scattering Ellipsometry
-*iHii ilPl
(*-^
*^ 1
FIGURE 1. Schematic drawing of the experimental system.
RESULTS AND DISCUSSION Three-dimensional Coulomb crystal First we controlled the diameter of fine particles to 1.4 /x m by decreasing R F power from 5 W to 1 W with the use of the Mie-scattering ellipsometry. The
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
85
arrangement of the fine particles is shown in Fig. 2.
SOD am F I G U R E 2. Top and side views of formed three-dimensional Coulomb crystal.
TOP VIEW
SIDE VIEW
F I G U R E 3. Explantion drawing of three-dimensional crystal structure.
Laser Hght beam was directed to illuminate fine particles in a few lowest layers
86
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
and in several vertical layers. Thus, in the top view, fine particles in the lowest layer are indicated by bright spots and those in the second lowest layer by rather dimmed spots. By the comparison with the result of the two-dimensional MonteCarlo simulation, the particle arrangement was supported to be in the solid phase [9]. Some particles in the lowest layer are seen to be piled up by fine particles in another layer. From the correspondence to the side view, it is found that they are in the third lowest layer and the structure of the arrangement is face-centered or body-centered as shown in Fig.3. The following relationships hold among the lattice constants a, 6, and c: a = 6 = c for fee and 2^/^ a = 6 = c for bcc. The average lattice constants and the standard deviations of the formed crystal in the experiment were evaluated for 11 images during 10 seconds to be a = 106 ± 2.6 At m, 6 = 159 ± 4.1 n m, and c = 162 ± 2.4 A6 m (see Fig.4). Since the relationship among these three lattice constants is 2^/^ a
-^
1
1
1
1
1
1
1
1
1
r
E c
•*-^:ft>.
(0 % 150
c o
^
(ab)
o
o
. „ . - - . - -
1/2
^>».^a^.
(0 - 1 100 -5
o^^
"^.
0
5
Time (sec) FIGURE 4. Time variations of lattice constants a, h and c.
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
87
minimum Coulomb energy as a function of c/(a • 6)^^^ ratio is given at the c/{a • hyl'^ ratio of 1 and 2^/^ ( = 1.19). Crystal structures satisfying these conditions are fee and bcc respectively, because 6/a = 1 and cl{a • 6)^^^ = 1 for fee, and 6/a = 2^/2 ( = 1.41) and c/(a • hf'^ = 2^/^ for bcc. The values of hi a and cl{a • hf'' for the experimental result axe 1.50 and 1.25 respectively and is on the curve of hj a c/{a • &)^/2 ij^ Pig 5 vs Forces act on fine particles near plasma-sheath boundary in the perpendicular direction t o the R F electrode. They are mainly gravitational, ion-drag and electrostatic forces. Former two forces direct downward while the last force upward. It was demonstrated by the molecular dynamics simulations [10-12], as shown also in this experiment, t h a t fine particles compressed in such one-dimensional external forces and the isotropic and repulsive shielded-Coulomb force take the form of plain layers parallel to the electrode, regardless of whether the particle arrangement is in the solid phase or in the liquid phase. This result implies that fine particles are easy to be trapped in horizontal layers and move only in the direction. Thus the reconstruction of particle arrangement in the layers occurs more easily than in the vertical direction, i. e., the variations of the lattice constants a and 6 are larger t h a n t h a t of c or (a • 6)^/^. Distance between neighboring two layers, c/2 is decided by the relationship between the compressive strength of one-dimensional external forces and the repulsive strength of the mutual shielded-Coulomb force. For a given value of c or (a • 6)^^^, the b/a ratio takes the value for the minimum Coulomb energy as has been presented in Fig.5. • — '
1
"
'
I
•
I
I
•
1
•
0.0165 ^
jl
o (O
1.6h bcc
1/
h
1.4 h CO
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o
L_
i
L
t
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k).016
$
t
1
0)
c
lU
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1
HK).0155
1
J
1
1
1.2
^
1
c / (a-bf^
.
1
.
E o o
o
1
1.4
FIGURE 5. hi a ratio for minimun Coulomb energy and the Coulomb energy per particle against cl{a»hyl^. Coulomb energy is indicated in units of Q2/47r6ora, where Q, eo, and Xa means particle charge, dielectric constant of vacuum and the Wigner-Seitz radius respectively.
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
Two-dimensional
Coulomb crystal
Next a Coulomb crystal of another structure was formed by larger fine particles. Spherical and monodisperse carbon fine particles of larger diameter were grown in another RF plasma system which has the same arrangement as that illustrated in Fig.l except for the size of the system and the direction of side-view CCD and ellipsometry measurement. The inner diameter of a ring put on an RF electrode was 1.6 cm. The directions of laser light, side view by CCD and ellipsometry were set at every 60 degrees. Particle size was controlled using the monitoring of the Mie-scattering ellipsometry. An arrangement formed by 5 AC m-sized particles is shown in Fig.6. From the side and top views it is easily confirmed that the structure is simple hexagonal. The particle arrangement was supported to be in the solid phase by the same method as in the former. Particles are aUgned in the vertical direction, and the structure is two-dimensional formed from the aligned particle strings. In order to obtain the dependence of particle distances with the density of fine particles, the density was decreased by the blow of gas after the Coulomb crystal was formed. Lattice constants a and c, which are indicated in Fig.6, were determined from the top and side views of CCD images respectively and then the density N was obtained from the equation, A'' = 2/(3^^^ ' a' c). The change of lattice constants a and c of the crystal is plotted as a function of the particle density in Fig.7. It
i}—
<|i
'h:^4 4-41 kj -"t yt. I i
mMMm
500/1
<SideView> F I G U R E 6. Top and side views of formed two-dimensional Coulomb crystal.
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
89
is seen in the figure that the increase of the lattice constant c with the decrease of the density is larger than the lattice constant aiov N^ 7 X 10^ cm~^ or c < 220 n m, while smaller for iV < 7 X 10^ cm~^ or c > 220 IM m. The repulsive shielded Coulomb force isotropically acts on paricles in the plain parralel to an electrode, although the resultant of the Coulomb force and directional forces does in the vertical direction. Therefore the difference in the tendency of the change of lattice constant between a and c should be caused by such directional forces. The experimental result indicates that the force acting on particles in the vertical direction is repulsive for c < 220 M m while attractive for c > 220 /i m. Gravity, ion drag force, or electrostatic force is directional in the vertical direction, however they does not produce an attractive force between particles. If negatively charged particles and positive ions surrounding them are polarized in the vertical direction, distant particles attract each other while close particles repulse it in the direction. Such polarization may arise from electrostatic field near plasma-sheath boundary or Wakefield generated by positive ion flow toward an electrode [13]. CONCLUSION Coulomb crystals were formed by growing carbon fine particles in plasmas and lattice constants were determined from CCD images. Smaller particles formed the structure of a three-dimensional Coulomb crystal, while larger particles did that of a two-dimensinal one. The three-dimensional crystal structure was strictly facecentered orthorohmbic. It is suggested that the ratio of the lattice constants of the crystal is decided so that Coulomb energy takes a minimum value under the 1
^600 =1. ^^^ ^-» C
£(0
1
0
J
0
a
J
8
§400 O o
1—1—r-i
Q
(0 (0
0)
1
0
-
-"200 -
•
•
• :
•\
,
C •• —
10*
J
1
1—•-•
•
1
10"
.-3»
Particle Density (cm ) F I G U R E 7. Change of lattice constants a and c of the crystal with fine particle density.
90
Y. Hayashi, A. Sawai /Crystallography and statics of Coulomb crystals
condition of constant particle density in horizontal layers. T h e structure of the twodimensional Coulomb crystal was simple hexagonal. The result of the dependence of lattice constants with particle density for the crystal indicates that the force acting on particles in the vertical direction is repulsive for smaller c while attractive for larger c. T h e resultant of the isotropic shielded-Coulomb force and directional forces act on fine particles in the vertical direction. The directional forces are mainly external ones for smaller particles, however a directional attractive force between particles plays an important role for the formation of a Coulomb crystal for larger particles.
REFERENCES 1. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 33, L804 (1994). 2. Lin, I, and Chu, J., H., Phys. Rev. Lett. 72, 4009 (1994). 3. Thomas, H., Morfill, G., E., Demmel, V., Gorre, J., Feuerbacher, B., and Mohlmann, D., Phys. Rev. Lett. 73, 652 (1994). 4. Hayashi, Y., and Tachibana, K., J. Vac. Sci. Technol. A 14, 506 (1996). 5. Hayashi, Y., Takahashi, K., and Tachibana, K., Advances in Dusty Plasmas , Singapore: World Scientific, 1997, pp.153-162. 6. Hayashi, Y., and Takahashi, K., Jpn. J. Appl. Phys. 36, 4976 (1997). 7. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 3 3 , L476 (1994). 8. Hayashi, Y., and Tachibana, K., Jpn. J. Appl. Phys. 33, 4208 (1994). 9. Hayashi, Y., submitted . 10. Totsuji, H., Kishimoto, Y., and Totsuji, C , Phys. Rev. Lett. 78, 3113 (1997). 11. Totsuji, H., Kishimoto, Y., and Totsuji, C., Jpn. J. Appl. Phys. 36, 4980 (1997). 12. Hammerberg, J., E., Holian, B., L., Lapenta, G., Murillo, M., S., Shajiahan, W., R., and Winske, D., Strongly Coupled Coulomb System,s , New York: Plenum Press, 1998, pp.237-240. 13. Melandso, F., and Goree, J., Phys. Rev. E 52, 5312 (1995).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
91
Monolayer Plasma Crystals: Experiments and Simulations J. Goree, D. Samsonov, Z. W. Ma, A. Bhattacharjee Dept. of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52246
H. M. Thomas, U. Konopka, G. E. Morfill Max-Planck Institute for Extraterrestrial Physics, 85740 Garching, Germany Abstract. Experiments and simulations are reported for two kinds of monolayer plasma crystals. In a monolayer with a small number of particles, ranging from 1-19, we measured the microscopic structure of very small crystals. The crystals have concentric rings. For certain numbers of particles, corresponding to 'closed-shells' in the outermost ring, the configurations are the same as for hard spheres. Yukawa molecular-dynamics simulations accurately reproduce our experimentals results. By adding far more particles (-10,000) and operating at higher power, we discovered Mach cones. These are V-shaped shocks created by supersonic objects. They were detected in a two-dimensional Coulomb crystal. Most particles were arranged in a monolayer, with a hexagonal lattice in a horizontal plane. Beneath the lattice plane, a sphere moved faster than the lattice sound speed. The resulting Mach cones were double, first compressional then rarefactive, due to the strongly-coupled crystalline state. Molecular dynamics simulations using a Yukawa potential also show multiple Mach cones.
INTRODUCTION Many experimenters have reported Coulomb crystals in dusty plasmas (1-13). In this paper, we report experiments and simulations with monolayer Coulomb crystals. These v/crc carried out by suspending microspheres in a horizontal electrode sheath. Two experiments were performed. In the first experiment, we prepared crystals with small numbers of particles, ranging from 1-19 particles. These configurations are arranged in concentric shells. These arrangements can be categorized, as a function of particle number, as a "periodic table" of small plasma crystals. In the second experiment, which was carried out with -10,000 microspheres in the monolayer, we observed Mach cones. These are V-shaped shock waves produced by supersonic particles. Mach cones are familiar in the field of gasdynamics (14) where they are created, for example, by supersonic aircraft. Less commonly, Mach cones also occur in solid-state matter. For example, surface waves along a borehole, in seismographic testing, propagate faster than the sound speed in solid rock, thereby creating Mach cones in the rock (15). The existence of Mach cones in dusty plasmas was predicted theoretically by Havnes et al (16). Moreover, they predicted that Mach cones are produced in the dust
92
J. Goree et al /Monolayer plasma crystals: experiments and simulations
of Saturn's rings by boulders moving in Keplerian orbits. Dust moves at a different speed, nearly co-rotational with the planet. In the case of Saturn's rings, the dust is probably weakly-coupled, and the relevant sound wave would be the Dust Acoustic Wave (DAW).
PERIODIC TABLE EXPERIMENT In the experiment, we used a modified GEC reference cell with a capacitivelycoupled lower electrode. By filling the chamber to a low pressure of 55 mTorr, and applying a 166 V peak-to-peak radio-frequency voltage at 13.56 MHz, we produced a Krypton glow discharge. The dc self bias was -55 V, and the Debye length measured by a Langmuir probe and using the ABR method, is estimated as 370 |im. We used microspheres of diameter 8.9 ± 0 . 1 |im and density 1.51 g/cm^. By dropping these through a 1-mm opening in an upper electrode, we were able to introduce only one sphere at a time. After it came to equilibrium, we imaged it, using a 512 X 512 resolution digital camera equipped with a Nikon micro lens. Then we added a second particle, and so on. The particle charge was measured as 2 = -12,300 e, using a variation of the resonance method of Melzer and Trottenberg (4,5). Our variation avoids Langmuirprobe measurements of ion density, which can have uncertainties of a factor of 3 or more due to various factors including placing the probe at a location other than the particle height. Instead, we use measurement of the dc self bias, dc plasma potential, and particle levitation height, which can be measured more accurately. Using the assumptions that the dc electric potential in the sheath varies quadratically with height and that the particle height is determined by a force balance with gravity but not ion drag, we are able to compute the particle charge with a smal random error. Particle x-y coordinates were identified from the camera images, and then plotted, as in Fig. 1. We applied Delaunay triangulation to show the bond configuration. For comparison, we also show results from a Yukawa molecular dynamics simulation, which is described later. The simulation parameters were Q = -15,300 e, Xj) = 370 |Lim, m = 5.57 x 10"^" g, and the curvature of the confining potential was parameterized by /: = 1.15 x 10'^ g/s^. We also show results for a hard-sphere experiment, which was carried out simply by dropping metal balls into a spherical bowl. We found that plasma crystals at the smallest size are predictable. Their positions are accurately modeled by a Yukawa simulation. We believe that the Yukawa potential is suitable because the particles lie in a two-dimensional plane that is perpendicular to the ion flow. In a three-dimensional crystal, it is known that ion focusing leads to strongly non-isotropic potentials that cannot be modeled accurately with a Yukawa potential (17). Certain numbers of particles have multiple multiple stable equilibria. One might call these "isotopes." These were most easily identified in the numerical simulations, which we repeated for 100 different random initial seedings of particle positions.
J. Goree et al /Monolayer plasma crystals: experiments and simulations
soft disk triangulation (experiment)
soft disk
•
experiment
O
simulation
93
tiard disk
©
©
(•K ^1 • )
©
closed shell
© ®©
4
® ©©
^
o
P®
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ers)
03^1 85%
o®© ©
^ 98% closed shell
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FIGURE 1. Particle configurations from the dusty plasma experiment are shown with dots in the left column, and in the Delaunay triangulation in the center column, (continued next page)
94
J, Goree et al /Monolayer plasma crystals: experiments and simulations
FIGURE 1. (continued) Simulation results are shown as open circles in the left column. In some panels, a percentage is indicated, showing the fraction of simulation runs that yielded the equilibrium (isotope) shown; other runs resulted in different isotopes.Hard disk experiment results are shown in the right column. These results are the beginning of a series for particles 1-19. The complete results will be presented elsewhere.
Comparing the dusty plasma results to the hard-sphere analog, we find that the plasma crystal tends to arrange with an azimuthally-symmetric outer ring, whereas the hard spheres arrange in a perfect triangular pattern even if the outer ring is incomplete. For certain numbers corresponding to complete shells, the two are the same. This occurred for 3, 7, 12, and 19 particles. Further details of the these experiments and simulations will be reported elsewhere.
SIMULATION We performed molecular-dynamics simulations of the experiments. Particles were constrained to move in a horizontal plane. The particle equation of motion m d r / d f = - (2 V (jf) - / d r I dt was integrated for N particles. The electric potential (p consisted of a parabolic potential to model the radial confinement from the plasma, plus a Yukawa inter-particle repulsion, (p- -kr^l 2 - 2 (2 / r^) exp (-n / Xjj). Here r is the distance from the central axis, r/ is the distance to particle /, and the sum is over all other particles. The parameter k determines the curvature of the bowl-shaped confining potential. The particles were loaded with random initial positions, and then their motion was followed by integrating their equations of motion simultaneously, using a simple leapfrog integrator. Their motion eventually ceased, as the excess kinetic energy was dissipated by drag, / d r / d f, leaving the particles in a stable equilibrium.
MACH CONE EXPERIMENT Our experiments were carried out in a strongly-coupled plasma, with a particle separation that was smaller than the Debye length. Under these conditions, the sound wave is the Dust Lattice Wave (DLW), which is different from the DAW that propagates under other conditions. The results we report here are peculiar to the crystalline strongly-coupled state. We expect that certain Mach cone features, which we will identify, will be different in a weakly-coupled dusty plasma. We used a larger 230 mm diameter electrode, without a glass insert in the upper ring electrode. Krypton gas was used at the low pressure of 0.05 mBar. Higher pressures result in a damping rate too high to observe mach cones. Approximately 10^ microspheres were shaken into the plasma above the electrode. The rf input power was much higher than in the periodic table experiment. We operated at 50 W, yielding a self-bias of -245 V on the lower electrode. This bias levitated the negatively-charged
J. Goree et al /Monolayer plasma crystals: experiments and simulations
95
particles 6.5 mm above the lower electrode. In the radial direction a gentle ambipolar electric field trapped the particles in a disk approximately 40 mm in diameter. This disk, which we term the "lattice layer," was a two-dimensional lattice, with a particle spacing A = 256 |im. The Debye length was smaller, Xj)- 124 |Lim, with an accuracy of a factor of two. There was very little particle motion. Mach cones in the lattice layer were produced by charged particles moving in a second incomplete layer, 200 \im beneath the lattice layer. This lower layer was populated by less than 10 particles. Unlike the particles in the lattice layer, they moved rapidly in the horizontal direction. Presumably they were accelerated by a horizontal electric field in the sheath, which we cannot explain. Sometimes they traversed the entire disk in a nearly straight line. They may have been single spheres or agglomerates of two or three. Imaging the cones with the digital camera and a horizontal laser sheet, we observed the Mach cones in the lattice layer. These cones are easily seen in the moving video, but more difficult to identify in a still image. The video rate was 50 frames/sec. To process the data to produce a still image of a Mach cone, we carried out the following computational process. Images were analyzed to identify the x-y coordinates of the particles. Particles were threaded from one image to the next, and their velocity was computed as the change in position, divided by the 0.02 sec frame interval. Then, by computing the magnitude of the velocity vector, we produced a map of the particle speed. To reduce noise, we averaged this over nine consecutive frames, which we displaced spatially so that the position of the fast particle coincided. This yielded the image shown in Fig. 2.
2 mm FIGURE 2. Map of particle speed in the lattice layer. Darker grays con-espond to faster particles. A supersonic particle moving at 4 cm/sec moved toward the lower right, producing the Mach cones shown here.
96
1 Goree et al /Monolayer plasma crystals: experiments and simulations
Some peculiar features to note are the multiple cones and the rounded vertices. Analyzing the particle motion, we determined that the first cone is compressional, with particles displaced forward, while the second cone is rarefactive, with particles moving in the opposite direction. We attribute the presence of multiple cones to the strong coupling in the crystalline state. In our experiment, the particles were arranged in a crystalline lattice, and they were deformed elastically as the fast particle passed by. Unlike a gas atom, an atom in a crystal has a memory of its original position. When the crystal is deformed elastically, the atoms are restored toward this position. In our experiment, the particles over-shoot this equilibrium position, and oscillate about it. The oscillation is damped by the gas drag. The rounded vertices are probably due to the finite size of the Debye sphere surrounding the fast particle. In gasdynamics, it is well known that a finite-size supersonic object, such as a sphere, creates a U-shaped Mach cone, in contrast to a needle-shaped object, which produces a V-shaped cone. We carried out molecular dynamics simulations of these experiments. This was done using 2,500 particles, which we allowed to settle into equilibrium positions. We then injected an additional particle, which we constrained to move on a horizontal plane 200 microns below the lattice layer. These simulations also revealed multiple Mach cones. The simplicity of the physics in the simulation demonstrates the simple physical nature of the Mach cones. As predicted by Havnes (16), Mach cones can be used as a diagnostic of the dusty plasma. In our case, we measured the particle charge, 2, from the Mach angle // = sin"l (1 / M), where M = v / c is the Mach number of an object moving at speed v through a medium with an acoustic speed c. Doing this requires a model for the acoustic speed. In our case, with a strongly-coupled two-dimensional suspension and particles separated by a distance large compared to the Debye length, it is valid to use the DLW dispersion relation of Roman et al (9). In the long-wavelength limit, X > 2nA, the DLW has only weak dispersion. Thus, the acoustic speed is a constant, as required for Mach cones. It happens to vary linearly with the particle charge Q. By measuring the Mach angle and the speed of the fast particle from the video, the particle separation from a correlation function analysis of a still image, and the Debye length from a Langmuir probe, we find a particle charge. In our case, Mach angle \i = 33.5° ± 3° for the first second cone yields find 13,000 <-Q/e< 24,000, where the uncertainty arises from the uncertainty in the Debye length. This result can be compared to our measurement using the resonance method, -Ql e- 23,000. Further details of the Mach cone experiments and simulations will be reported elsewhere (18).
ACKNOWLEDGMENTS The experiments reported here were carried out at MPE-Garching. Simulations and data analysis were carried out at Iowa. Funding for the Iowa group was from NSF and NASA,
J. Goree et al /Monolayer plasma crystals: experiments and simulations
97
REFERENCES 1. J. H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). 2. H. Thomas, G. Morfill, V. Demmel, J. Goree, B. Feuerbacher, and D. Mohlmann, Phys. Rev. Lett. 73. 652 (1994). 3. Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. 33, 804 (1994). 4. A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 (1994). 5. T. Trottenberg, A. Melzer, and A. Piel, Plasma Sources Sci. Technol.4, 450 (1995). 6. O. Havnes, T. Aslaksen, T. W. Hartquist, F. Li, F. Melands0, G. E. Morfill, and T. Nitter, J. Geophys. Res.. 100, 1731 (1995). 7. R. A. Quinn, C. Cui, J. Goree, J. B. Pieper, H. Thomas, and G. E. Morfill, Phys. Rev. E 53, R2049 (1996). 8. J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). 9. A. Homann, A. Melzer, R. Madanio, and A. Piel, Phys. Lett. A 242, 173 (1998). 10. U. Konopka, L. Ratke, and H. M. Thomas, Phys. Rev. Lett. 79, 1269 (1997); and U. Konopka, unpublished. 11. S. Nunomura, N. Ohno, and S. Takamura, Phys. Plasmas 5, 3517 (1998). 12. Y. K. Khodataev, S. A. Khrapak, A. P. Nefedov, and O. F. Petrov, Phys. Rev. E SI, 7086 (1998). 13. D. A. Law, W.H. Steel, B. M. Annaratone, and J.E. Allen, Phys. Rev. Lett. 80, 4189 (1998). 14. H. W. Liepman and A. Roshko, Elements of Gas Dynamics (Wiley, New York, 1957). 15. N. Cheng, Z. Zhu, C. H. Cheng and M. N. Toksoz, Geophysical Prospecting 42, 303 (1994). 16. O. Havnes, T. Aslaksen, T. W. Hartquist, F. Li, F. Melands0, G. E. Morfill, and T. Nitter, J. Geophys. Res. 100, 1731 (1995). 17. F. Melands0 and J. Goree, Polarized Supersonic Plasma Flow Simulation for Charged Bodies such as Dust Particles and Spacecraft, Phys. Rev. E 52, 5312-5326, (1995). 18. D. Samsonov, J. Goree, Z.W. Ma, A. Bhattachrjee, H.M. Thomas, and G.E. Morfill, submitted io Phys. Rev. Lett. (1999).
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
99
Nonlinear dust equilibria in space and laboratory plasmas V.N. Tsytovich General Physics Institute, Vavilorm 38, Moscow 117942^ Russia, Abstract. Theory of equilibrium nonlinear dissipative structures in dusty plasma is presented. Nonlinear electrostatic and pressure equilibria is analyzed for dust clouds embedded in dust free plasma, dust voids and dust convection pattern. Homogeneous dusty plasma is shown to be instabile to formation of dust clouds surrounded by dust-free plasma. A single dust structure creates in self-consistent way the plasma flux toward the structure which compresses and self-confines dust in the structure. Exact solutions of nonlinear equations describing ID planar and spherical structures are found numerically. The structures are determined by only one parameter - the number of dust particles in the structure. Dust cloud and dust voids have sharp dust boundaries (calculated and investigated). Phenonmenon of critical flux velocity is found. The theory of voids in dusty plasma is presented both in the limit where the ion-neutral collisions are negligible and in the case where the ion -neutral collisions are dominant. The void equilibrium position is stable for large deviation from equilibrium for perturbations perpendicular to the void surface and are unstable for disturbances along the void surface,creating convection patterns.
INTRODUCTION Dissipative seif-organazed structures in dusty plasma should naturally appear due to openness of dusty-plasma system and absorption of plasma fluxes by dust particles (^). Energy and particles sources are necessary to keep the structures in equilibrium. The plasma flux on single dust particle is modified by shadoW' effect of other dust particles and lead to dust attraction (^). This binary dust particle attraction is proportional to the solid angle at which one particle shadows the flux of the other particle a a^/r^ (a-is dust size, r is the distance between the dust particles), the pressure nT and the surface area a a^. Thus the shadow^ force is a a^jr^ while the Coulomb repulsion force is a ZUr'^ oc a^/?^'^.Eventually the dust growing in size wdll reach the size w^hen the attraction wall dominate and this is the natural way to explain the observed dust agglomeration (^).For a dust cloud containing many dust particles apart of individual dust particles attraction there appear a global coherent flux of plasma to the structure w4iich can lead to ram
V.N. Tsytovich /Nonlinear dust equilibria in space and laboratory plasmas
100
pressure confining the dust paxticles in the cloud.
INI k)
I
pbsmi fhiitts
FIGURE 1 Schematic picture of a)dust structures b)dust voids. This coherent flux is created coherently by all particles in the cloud and its effect on dust and plasma particle distribution in the cloud will be the main subject of present report. The eifect of global flux on the structure results in global dust confinement in the structure observed both numerically {^) and experimentally (;^). The forces acting on dust particles are the electrostatic forces and the drug forces related to the ion momentum transfer by the flux. The sketch of a planar and spherical structures produced in this way are shown on Fig.la. The plasma production needed to create the flux is outside the structures. Fig.lb shows an opposite case where the ionization is large un central region from which the plasma fluxes originate and create the dust void region. Both type of structures are shown have sharp boundaries separating the dust containing regions from the regions where the dust exists. The nonhnear equations used are given in Appendix. Used are the following notations and dimensionless units: r = ra/cf^;E = Eedf/aTe;n = ni/ni^o]n^ = ne/n,',o;u = Ui/\/2vTi]z = Zde^/aTc]T = TijT^P = Zdnd/no^i]Vd = ^d^n'.^n = r = Ini{Lo)/ujpiV2{a/di);rnd = {md/mi){3no,i/TZdnn)]a = a/di where u,- is the ion drift velocity, nu.n^d are ion, electron, neutral particle and dust densities respectively, n,,o = ne,o is the ion density outside the structure or at the center of the void, Te^i^n are the electron, ion and neutral temperatures, a is the dust radius, di is the ion Debye radius and E is the strength of the electrostatic field. The normalization of dust mass used in the way that the dust inertia is comparable with dust friction on neutrals for rrid > L In these dimensionless units the impact electron ionization is described by ionization length xi in the ion continuity equation and the ion mean free path in ion-neutral collisions is described by Xn (see Appendix).
VN. Tsytovich /Nonlinear dust equilibria in space and laboratory plasmas
101
The ratio of the ionization length Xi to x^ is denoted as X^. The dust drag force is determined by the drag coefficient a^^, the absorption of plasma particles on dust particles is described by the coefficient Oc (see Appendix). The coefficient a^ describes the nonlinearity in the ion-neutral collisions.
STRUCTURIZATION IN DUSTY PLASMAS The simplest equilibrium state in dusty plasma is a homogeneous state related with balance of plasma absorption on dust and plasma creation by ionization. The ionization source, dust and plasma particle density distributions assumed to be homogeneous. This state can be shown to be unstable for separation on dust clouds and dust voids (^).
/l'jTrnTT-T>..,., I f t ^ f 1 !• i ^ i i f I f i
wssm
,
\ \?>>
^mm^^u^^^^^ } ii m ^''''^--^-:^§§^
I klH
r
FIGURE 2 Growth rate T for a) k = 0 b)]arge dust inertia m^ = 50 The ionization rate is assumed to be proportional to electron density as it is in most experiments with the volume RF ionization. The initial equilibrium state is described by balance of ionization and absorption on dust acPo = ne,o/xi (ion density is normalized to be 1) which allows to find the dust charges in the equihbrium state as function of x^ and then for given dust density to find the value of electron density. In the equilibrium state the ion and dust velocities are zero and the electric field is zero. The perturbations of this state are the ion, electron density, dust charge perturbations, the electric field and the velocities of ions and dust. Nonhnearities in friction are neglected, the perturbations are assumed to be quasi-neutral, the electrons in perturbations are adiabatic, the OML approach for dust charging is used. The solutions of the obtained in this way dispersion relation has the following properties: 1) the perturbations with k = 0 are always unstable ( an analogy with gravitational instability and is caused by the global flux perturbations), 2)the perturbations with large k are stable F = ik 3) the maximum growth rate corresponds to k of the order of 1 (of order of mean free path of ion-dust
102
V.N. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas
collisions), 4)dust inertia enlarges substantially the growth rate. For growth rate To corresponding fc = 0 the analytic expression is found (see (AlO)) Fig.2a shows the dependence of the growth rate at fc = 0 on PQ and r. The Fig.2b shows the dependence of the growth rate on k enlarged by dust inertia. This investigation supports the statement that the structurization should be always present in dust plasma systems.
DUST PLANAR AND SPHERICAL STRUCTURES For nonlinear stage of instability the set of nonlinear equation (see Appendix) is solved exactly with stationarity conditions imposed. The stationary dust velocity in the structure is 0, the dust friction on neutrals and the dust inertia are unimportant. In nonlinear equations the exact expressions for the drag and capture coefficients (see Appendix) are used. Ion drift velocity is assumed to be comparable with ion thermal velocity and in accordance with experimental data the ion neutral friction force is modified by the factor (1 + ua„) with the nonlinear coefficient a„ being of the order of 1. The planar ID structures with plasma fluxes from it both sides are calculated with assumption that ionization is absent inside the structure and in the outside close region. The only parameter which regulates the all parameters of the structures is the flux velocity UQ far away from the structure. This parameter is determined by the number Nd = J{P/z)dx- the total number of dust particles per unit surface area in dimensionless units. The larger is Nd the largest is uo, the maximum UQ which allows the nonlinear equilibria is l / \ / 2 and the maximum Nd is a number of order of 1. 1
" l^_
>c
t.s
L...,^,.^
n
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1
1 1
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•
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_
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. ..
1
F I GURE 3 Distribution of parameter in planar dust structures , x^ = oc The value of dust charge jump, of the parameter P , ion drift velocity, ion and electron densities at the surface of the structure are found from boundary conditions and solution of Poisson equation inside the dust region. The distribution of all the parameters inside the dust region are shown on Fig.'s Sab for the case where
V.N. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas
103
the ion-neutral collisions do not play a role and for two cases where the quasineutrality holds and whee it is not( the latter correspond to large jump of P at the surface). The theory is applied to the problems of dust planetary ring thickness and appearance of dust layers with sharp boundaries in lower ionosphere. The nonlinear set of equation was analyzed for 3D case assuming the spherical symmetry. In this case the ion drift velocity and the electric field are vanishing in the center of the structure. The solution of the nonlinear equations was started from the center of the structure r = 0 using the asymptotical properties of the quantities at r ~> 0. The calculations where performed up to the edge of the structure and the conditions for structure to exist were found from boundary conditions at the surface of the structure. It was found that the structure is regulated by one parameter Nd = {47T/:i)J{P/z)r'^dr. By increasing it one finds its maximum possible value when the structure can exist. It is of order of 1. The structures were investigated in broad range of parameters for the case where the ion-neutral collisions do not play any role and for the case where they are important. Fig.'s 4 illustrate the case of large ion-neutral collision rate; the ion density has a minimum inside the center of the structure and, increases to its periphery and than decreases to n = 1. The drift velocity has maximum close to the surface of the structures. The theory as applied to problems of proplanetary condensations in dust-molecular space clouds. The spherical structures in space are new astrophysical objects- "dust star", their maximum mass can be of the order of planetary mass.
F I GURE 4 Parameter in spherical dust structures a)a:„ = oo h)xn = 1.
DUST VOIDS Dust voids observed in recent experiments have sharp boundaries (^'^). They explanation can be based on same set of nonlinear equations assuming the presence of ionization in the central region. The ionization creates the drag force which moves the dust from central region crating a dust void. The theory of dust voids was formulated both in the limit where the ion-neutral collisions do not play important
104
VN. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas
role (/^) and in the case where they are important ^^ The corresponding cases are named the collision less voids and collision dominated voids, in analogy with collision less and collision dominated plasma sheaths. But for both types of voids the ion-dust collisions are very important and create the drag force which is the force responsible for the void appearance. The void boundaries are sharp. The voids appear if the ionization rate exceeds critical value. For collision less voids it is illustrated on Fig.5a where the dependence of the void size on the ionization rate Ijxi is shown. The theory of moving voids was created which takes into account the neutral-dust friction force. The phase diagram of the moving void shows the presence of one stationary void position for given ionization rate (Fig.5b). The hart beating voids or breathing voids are explained by an change in equilibrium ionization rate after void expansion or contraction. The jump of the parameter P at the void surface is minimum at the equilibrium position corresponding to the stationary void.
—
'
r-
V
1
1
a1
1
0,05
"
"1
•
r
1
0.01 i
1
—
i , , .-
1
,
\
F I GURE 5 a) void size iis function of l/x^ bjpha^e diagram of a moving void. The collision dominated voids are described by two parameters Xi and Xn^ The quasi-neutral void corresponds to x^ > dijy/r = ri^, the mean free path in ion-neutral collisions much larger than the electron Debye radius. The size of the quasi-neutral void appears to be proportional to Xn with numerical coefficient increasing with AV The Fig.6 shows the parameters of the quasi-neutral void for A^- = 0.1. The quasi-neutral voids for fixed Xi exist for Xn > Xn^rnin- The non quasi-neutral voids exist for ^e < Xny/2Xi = y/2xiX^, The global stability of ID structure including the voids for perturbations along the direction of the parameter distribution in the structure was proved numerically in computations of moving boundaries between the dust side and the void side of the structure. The structures can be shown to be unstable for creation of perturbations in the perpendicular direction which is related to appearance of dust convection (^). There exist a critical parameter related both with dust neutral collisions and ion-neutral collisions which determines the threshold of convection instability.
VN. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas
X
105
CUM
1.&
»^2
i
o. O.OM
t
,gi»P'--' S^mliis*--
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F I G U R E 6 Parameters of quasi-neutral coUisional void
APPENDIX The system of nonlinear equation used is: dm^Vd
1
— = —Vd - £• + nzuadr] dt z
T-
+ 2u . —u + —X- ] =E\dt dr ndrj dn dnu ^ I dt dr xi
1 One ^ _ g ^ Tie or
{A\)
1 + auu)
zodruP
(.42)
x„ ^P ^^ P^r dt z or z
(\
(A3)
106
V.N. Tsytovich/Nonlinear dust equilibria in space and laboratory plasmas
OjTp
The dependence of the drag coefficient adr and the dust capture coefficient ac on the ion drift velocity and the ratio t = r/z is given by the expressions
erf{u) 8ti3
—^
t{-l + 4u^ + 4tt'') + 2t{-l + 2u^) + 4ln T-)] +
'- \t{til + 2u^) + 2 - 4/n ; a,: = -^^^{1 L \aJi Su
+ t + ZuH)
The growth rate F of structurization instability for k = 0: '°
2v^(zo(^o + T + 1)(1 - Po) + Po(^o + r))
^l—-l 40r (A5)
^''''^
REFERENCES {') Tsytovich,V.N., Physics Uspekhi 4.0, 53 (1997). (^) Tsytovich,V.N.,Ya.K.,Khodataev, Bingham,R., and Resendes, D.,Comments on Plasma Pkys. and Contr. Fusion 17, 287 (1996). (^) Tsytovich,V.N.,Khodataev, Ya.K., Bingham,R., Morfill, G., and Winter, J., Comments on Plasma Pkys. and Contr. Ftision 18, 345 (1998). C) Tsytovich,V.N.,Khodataev,Ya.K.,Bingham,R.,and Tarakanov,V.A.,Jounia/Plasma Physics, 5, 32 (1998). i^) Morfill G.,and Thomas, H., J. Vac, Sci. TechnoL, A14, 490 (1996) (®) Bouchoule,A.,Morfill,G., and Tsytovich, V.}^.,Comments on Plasma Phys. and Contr. Fusion 21, 52 (1999). C) Morfill, G., and Tsytovich,V.N.,Pftys«C5 of Plasmas (submitted) (1999). (**) Goree, J.,and Samsonov, S., Phys. Rev, £" (accepted), (1999). n Meltzer, M.,Piel, A. et all in press, (1998). 0°) Goree, J.,Morfill,G.,Tsytovich,V.N.,and Vladimirov,S.V., Phis. Rev. E (accepted) (1999). (11) Goree, J.,Morfill,G.,Tsytovich,V.N.,and Vladimirov,S.V., Phis. Rev. E (submitted) (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
107
Dust Particle Structures in Low-Temperature Plasmas Anatoly P.Nefedov High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia
Abstract. Dust particle ordering is discussed in various types of low-temperature plasma, such as thermal plasmas at atmospheric pressure, DC glow discharges, inductively-coupled plasma, UVand radioactivity induced plasmas. The investigations of UV-induced dusty plasma were made under microgravity conditions. Experimental data are presented. Properties of ordered structures are discussed and the conditions of formation considered.
INTRODUCTION A plasma containing macroscopic particles or grains (often referred to as a dusty plasma) has the feature that a particle introduced into such a plasma or produced in it by, say, condensation may be charged by an electron or ion flux or by photo-, thermoor secondary electron emission. Electron emission from a grain surface produces a positive charge, increases the electron density in the gas phase and hence enhances its electrical conductivity. Capturing electrons makes dust grains charged negatively, producing the reverse effect, a reduction in the electron density (1,2). The distinguishing feature of a dusty plasma is that, owing to the relatively large size of a grain (from hundredths of a micron to tens of microns), its charge Zp may assume extremely large values of up to 10-10 elementary charges. Therefore the (Z/-dependent) Coulomb interaction energy of the particles may, on average, be much greater than the thermal energy, which makes the plasma strongly non-ideal. Equilibrium calculations show that under certain conditions a strong intergrain correlation leads to a gas-liquid- solid phase transition and makes the grains arrange themselves into spatially ordered structures similar to those existing in liquids and solids. The electron and ion gases remain ideal analogous to those in a Debye plasma. Crystallike structures that form in plasmas from low-thermal-energy, strongly electrostatically coupled charged particles have been analyzed theoretically by Ikezi (3) and have come to be known as Coulomb or plasma crystals. In later studies, the Coulomb crystals of dust particles were found in a weakly ionized plasma in a radio-frequency (RF) discharge at low pressure (4, 5).
108
A.P. Nefedov /Dust particle structures in low-temperature plasmas
In parallel with the study of the properties of plasma crystals under RF-discharge conditions, in recent years attempts to obtain extended, essentially three-dimensional ordered structures in the bulk of a quasineutral plasma have been made, and structure formation processes for various charging mechanisms, particularly secondary emission and photoemission, have been investigated. An example is the observation of macroscopically ordered structures in a bulk thermal plasma under quasineutrality conditions at atmospheric pressure and temperature of about 1700 K (6, 7). The plasma studied was homogeneous, relatively large in size (its volume of 30 cm""^ corresponding to a particle number of the order o
7
'^
10 and particle density 10 cm"), and free of external electric and magnetic fields. Owing to the large plasma volume and the availability of reliable plasma diagnostics techniques, various types of gas and particle measurements have been performed, plasma state parameters obtained and also comparisons with numerical simulation results has been made. Here we present our studies of the formation of a macroscopic ordered structure in strongly coupled plasmas over a wide range of plasma pressures and temperatures. The plasma was investigated under conditions of low pressure DC gas discharge, inductively-coupled plasma and under thermal plasma conditions. The plasma was also formed from the positively charged dust grains in the presence of a flux of ultraviolet (UV) photons and radioactivity radiation. The investigations of UVinduced dusty plasma were made under microgravity conditions. PLASMA CRYSTALS AND LIQUIDS IN THE DC GLOW DISCHARGE A glow discharge is a non-isothermal room-temperature low-pressure plasma. Experiments were made in neon over a pressure range of 0.1 to 1.0 Torr and an electron and ion density range of 10^ to 10^^ cm""^. The electron temperature ranged between 20000 and 50000 K, and the ion and atom temperature between 300 and 400 K. Under these conditions, strong fields of the stratified discharge kept particles within the volume of the plasma. Observations showed that under certain conditions in such a plasma, quasicrytalline structure forms in the positive column far away from the electrodes, in the ionization instability region (stratum) with a relatively high degree of quasineutrality (-'0.01%) (8). These structures are essentially three-dimensional and their vertical dimension may be as great as several tens of centimeters, with an intergrain spacing ranging from 300 to 400 |im. The grain charge may reach unusually high values (10^ e), which makes it possible to keep the grains in the relatively weak stratum field of order 10 V/cm (to be compared with about 100 V/cm in the near-electrode field in an RF-discharge). By varying the discharge parameters (pressure and current), dust cloud shapes in the range from nearly spherical to cylindrical can be obtained.
A.P. Nefedou/Dust particle structures in low-temperature plasmas
109
Four types of dust spherical particles were dispersed into the column which allowed us to vary their diameters in the wide range from 1.9 |Lim to 63 |am, masses from 5 10'^^ g to 10'^ g and their charges in the range from 10"^ up to 5 10^ e. In the presence of standing or weakly vibrating strata in the column, the particles appeared (usually) as ellipsoidal clouds in the center of their glowing regions. Usually, there are several dust clouds located in neighboring strata, several tens of centimeters away from the tube electrodes. The cloud diameter was 5 to 10 mm for glass microspheres and increased to 20 mm for AI2O3 particles. Note the ordered structure and nearly equal spacing of the grains. In the vertical plane the grains were seen to arrange in chains. In the elliptic case the dust grains arranged themselves in 10-20 (glass spheres) and more (AI2O3) planar layers. The interlayer separation ranged between 250 and 400 i^m and the intergrain spacing in the horizontal plane between 350 and 600 |Lim, corresponding to the particle densities rip^-lO^AO^ cm'^. Clearly, the observed particle pattern is quasicrystalline and essentially three-dimensional in nature. Experimental data show that a stratified discharge has regions of strong and weak electric field alternating periodically along the tube axis. In the region of a strong (10 V/cm) longitudinal electric field in a stratum, a potential well arises in the vertical plane as a result of the balance of the electric and gravitational fields. A similar potential well in the horizontal plane is formed by the high (30 V) floating potential on the walls of the tube. The conclusion to be drawn from these facts is that the dust grains are kept - indeed trapped - by a strong electric field. By varying the discharge parameters (pressure and current) it is possible to change the size of the potential well and hence the shape of the dust cloud. For instance, reducing the discharge current and pressure causes two neighboring elliptic clouds to gradually transform into a cylindric shape with a vertical dimension of several tens of centimeters. Varying the current may violate particle ordering thus causing the quasicrystal to 'melt'. In the case of small melamine formaldehyde particles, an action of the ion drag force beginning to play here an important role may lead to an origin of a convective movement of dust particles. As a result, the dusty plasma structures with a simultaneous existence of crystal, liquid and gaseous phases are observed. GRAIN ORDERED STRUCTURES IN AN ALUMINIZED PROPELLANT FLAME The formation of the particle ordered structure in a solid propellant flame was studied. In our experiment we use aluminized solid propellant. The flame is formed
110
A.P Nefedov /Dust particle structures in low-temperature plasmas
rjf
b) FIGURE 1. Video image (a) and pair correlation function g(r) (b) of the dust particle cloud in the boundary sheath. The bar corresponds to 100 jim.
by ignition of a propellant tablet ('-10 mm diameter and -^30 mm height ) with the aid of an electrical heater. The spectral measurements revealed that a plasma spray of particles contains sodium and potassium atoms with a low ionization potential. As a result, the basic constituents of the plasma studied are charged alumina (AI2O3) particles, Na^ and K^ ions, and electrons. As a diagnostic base to support our experiments, we measure basic plasma parameters such as the alkali atom and ion number densities, plasma temperature, and the diameter and number density of the grains. The plasma diagnosis has been taken for different heights h above a propellant tablet surface. The measurements were performed at a temperature of 1900-2250 K. Microscopic structure observations were made by illuminating a horizontal or vertical plane with a sheet of Ar"^ laser light, with a thickness of 30 |Lim and a breadth of 10 mm. It is adjustable to various heights. Scattered light was viewed at 70^ to the horizontal plane through a transparent wall of glass tube. To observe particles in the vertical plane, receiving optics was positioned at an angle of 90° to the vertical. Individual particles were observed with a charge-couple device (CCD) video camera fitted with a macro objective. A 58-mm macro objective with extension tubes provided magnification from x30 to 140. Our investigations show that propellant flame includes three zones: combustion region (/z-lO mm, rg'-2000 K, ^2p<10^ cm""^), condensation region (/z-'40-50 mm, rg'-'600 K, np>\(f cm"^) and boundary sheath between them (/2'-2-3 mm, rg'-19002000K,np-10^-10^CM-^). A numerical analysis of grain images in a single plane (horizontal and vertical) for combustion and condensation regions does not reveal the particle ordering. This fact may be explained by a low particle density in the combustion zone and a low grain charge in the condensation region, due to low flame temperature in the region. In contrast, in the boimdary sheath the numerical analysis of AI2O3 particle cloud
111
A.P. Nefedov /Dust particle structures in low-temperature plasmas
TABLE L Results of Plasma Measurements in the Boundary Sheath of Aluminized Propellant Flame T n„ cm'' No «K, cm'' n,K «N« cm"^ 3-10" 210'" 1 0.13 2000 2-10'^ 2 0.7 1940 1.5-10'^ 1.510" 4-10'" 3 0.08 1950 2-10" 3-10'^ 410'" 4 0.05 2000 1.510'^ 4.5-10" 9-10"'
image in a plane (see Fig. la) verifies it as ordered structure, and directly measuring the distance between particles, we obtained two-dimensional pair correlation function g{r) which reveals liquidlike structure (Fig. lb). The plasma parameters measured in the region are shown in Table I. Based on these data and quasineutrality of the plasma, the AI2O3 particles have a positive electric charge. This fact can be explained by the thermionic emission of electrons from the hot particles. The obtained data let us estimate the value of parameter Fs = 10 that corresponds to the plasma liquid state.
ORDERING OF GRAINS IN A NUCLEAR INDUCED DUSTY PLASMA The nuclear induced plasma is formed under an interaction of ionizing particles and radiation, which accompany nuclear reactions, with a gas. The presence of dust grains in such plasma may lead to new effects. Two ways of generating a charge on dust grains were used. In one of them grains were being charged during the P-decay.
g(r) 2 . 0 —\
1.5
-J
1.0 - j 0.5 H 0.0
500
a)
1000
b)
FIGURE. 2. Digitized videoimage (a) and pair correlation function (b) for monodisperse particles with diameter dp=1.87 |am (atmosphere air, radiation source - ^^^Cf, voltage between electrodes 20 V, particle charge -10^ e, F'-SO).
112
A.P. Nefedov/Dust particle structures in low-temperature plasmas
For this an activation of Ce02 particles with a 1 |im diameter was performed in a nuclear reactor. In the other case dust particles were being charged during a passage of a-particles and fission products of ^^^Cf through them. As a result of electron and a-particles impacts, the grains take a positive charge due to the secondary electron emission (3). The experimental cell was a plane capacitor. The upper electrode had a hole for injecting dust particles from a container into an interelectrode volume. In the case of experiments with ^^^Cf a thin layer of fissioning ^^^Cf with intensity of 10^ fission/s was placed at the lower electrode. The interelectrode distance was varied from 10 up to 40 mm. Electrodes and the container were disposed in a glass tube with an inner diameter of 36 mm. The visualization of dust particles was performed with the aid of CCD-camera and diode laser. Melamine formaldehyde particles of diameters 1.87-4.82 |Lim were used. Experiments were performed at atmosphere air with a DC interelectrode voltage Uei varying from 0 up to 400 V. When Uei was less than 100 V regions with levitating dust particles were observed in the interelectrode volume. Pair correlation functions obtained for 1.87 |im particles revealed a short-distance order with a first maximum at 200 |im (Fig. 2). With increasing Uei a movement of dust particles started. From the balance gravitational, electrostatic and neutral drag forces on a dust grain, the charge of dust particles was obtained. The charge value is (0.6-1.3) 10 e. The estimated value of Yp is about 30. The characteristic peculiarity was an appearance of stable vortex movements of dust particles. The movement character did not change when the Uei polarity was varied. Varying an angle between the cell axis and the vertical it was revealed that dust particles moved initially from the ^^^Cf source along the tube axis. EXPERIMENTAL SPACE STUDY OF UV-INDUCED DUSTY PLASMAS UNDER MICROGRAVITY The purpose of performing the space experiment under microgravity was to study the possibility of the existence of plasma-dust structures in the upper layers of the earth's atmosphere when the particles are charged by solar radiation as a result of the photoemission of electrons from their surface. The experiment was carried out with glass ampuls containing particles of bronze with a cesium monolayer (two ampuls)) in a buffer gas (neon) at different pressures. Immediately before the performance of an experiment, the required ampul was placed in the clamp of the working-chamber holder with its flat end surface toward the illuminator. For diagnostics of the ensemble of particles, the ampul was illuminated by a laser sheet (the width of the laser sheet was no greater than 200 |Lim), and an image was obtained using the CCD camera, whose signal was recorded on magnetic
A.P. Nefedov /Dust particle structures in low-temperature plasmas
113
tape. The experiments were carried out with three values of the neon pressure: ?i=0.01 Torr and ^2=40 Torr. The first stage of the experiment was confined to observing the behavior of the ensemble of macroparticles placed in the ampul under the action of solar radiation. In the initial state the particles were on the walls of the ampul; therefore, the experiment was carried out according to the following scheme: a dynamic disturbance (jolt) of the system and relaxation to the initial state, i.e., drift to the wall. Observations of the motion of the particles showed that the velocity vectors of the particles are randomly directed in the initial stage and that the particles drift to the walls without a preferential direction. Subsequently, a preferential direction usually appears, but motion along definite trajectories is displayed more strongly in the vessel with the higher pressure. It was concluded that particles are charged on the basis of observations of the changes in the trajectories of the particles when they come close to one another (collide) or approach the wall. It should also be noted that the particles move very slowly under ^2^40 Torr when the solar radiation is blocked and that acceleration of the motion occurs when radiation acts on the ensemble of particles. The charge of the macroparticles can be estimated by analyzing their dynamic behavior. The analysis results reveal that bronze particles acquired a charge of the order of 10"^ e (the coupling parameter y was about 10^) (9). Despite the high particle charges and the large value of the coupling parameter, no strong correlation between the interparticle distances could be observed. An analysis of the results obtained confirm the conclusion that the existence of extended liquidlike ordered structures of macroparticles charged by solar radiation is possible under microgravitational conditions.
INDUCTIVELY-COUPLED DUSTY PLASMA The distinctive feature of inductively-coupled plasma (ICP) is the absence of electrodes. In such plasma the trap for dusty particles is formed at the expense of static electrical fields arisingfi*oma violation of an electroneutrality, which, in turn, is called by differences of diffusion speeds and mobilities both ions and electrons. Firstly the possibility of levitation of micro-sized particles and formation of ordered structures from them in such type of plasma was shown in (10). The plasma was generated in a glass tube by power supply at working fi^equency 100 MHz with the circular inductor. Levitation of particles was observed in a neon pressure range fi-om 20 Pa up to 500 Pa. The level of input power did not exceed several watts. The further researches have shown that the inductive RF discharge has wide possibilities for investigations of strongly coupled dusty plasma, as it allows to generate the structures of different types. Let's take two examples, in our opinion, verifying this conclusion.
114
A.P Nefedou/Dust particle structures in low-temperature plasmas
a)
b)
FIGURE. 3. Digitized videoimage (a) and pair correlation function (b) for monodisperse polymeric particles with the size of 1.90 |im.
In Fig. 3a the image of horizontal cross-section of a structure formed by monodisperse polymeric particles with the size of 1.90 |Lim is shown. The number of horizontal layers in such structures was up to ten. The above mentioned structure was located above a flat bottom, under which the inductor was placed. The frequency of the generator was equal to 27 MHz. The neon pressure was about 120 Pa. The pair correlation function shown in Fig. 3b allows one to conclude that an ordered structure like "plasma crystal" is obtained under the considered conditions. ACKNOWLEDGMENT This work was supported by the Russian Foundation for Basic Research, Grant No. 97-02-17565 and Grant No. 98-02-16828.
REFERENCES 1. Sodha, M.S., and Guha, S., Adv. Plasma Phys. 4, 219 (1971). 2. Chow, V.W., Mendis, D.A., and Rosenberg, M., IEEE Trans, on Plasma Science 22, 179 (1994). 3. Ikezi, H., Phys. Fluids 29, 1764 (1986). 4. Chu, J.H., and Lin, I., Phys. Rev. Lett. 72, 4009 (1994). 5. Thomas, H., Morfill, G.E., Demmel, V. et al, Phys. Rev. Lett. 73, 652 (1994). 6. Fortov, V.E., Nefedov, A.P., Petrov, O.F. et ai, Phys. Rev. E. 54, R2236 (1996). 7. Fortov, V.E., Nefedov, A.P., Petrov, O.F. et al. Physics Letters A 219, 89 (1996). 8. Fortov, V.E., Nefedov, A.P., Torchinsky, V.M., et al., Physics Letters A 229, 317 (1997). 9. Fortov, V.E., Nefedov, A.P., Vaulina, O.S. et al, Journal of Experimental and Theoretical Physics 87, 1087(1998). 10. Gerasimov, Yu.V., Nefedov, A.P., Sinel'-shchikov, V.A., and Fortov, V.E., Pisma v Zhurnal Tehnicheskoy Fiziki, 24, 62-68 (1998) (in Russian).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
115
Experimental evidence for attractive and repulsive forces in dust molecules A. Melzer*, V.A. Schweigertt, A. Piel* *Institut fiir Experimentelle und Angewandte Physik, Christian-Alhrechts-Universitdt Kiel, 24098 Kiel, Germany ^ Institute of Theoretical and Applied Mechanics, Russian Academy of Science, 630090 Novosibirsk, Russia
Abstract. The net interaction force in a system of two single dust particles trapped in an rf discharge is determined quantitatively by direct laser manipulation. It is proven that a strong asymmetric attractive force between the particles exists that leads to a vertical alignment and thus the formation of dust molecules. Moreover, with increased discharge pressure a reversible transition to net repulsive forces between the particles takes place. This transition is explained in terms of our previous plasma crystal models.
INTRODUCTION Plasma crystals consist of highly negatively charged micrometer-sized particles immersed in a plasma environment. They are usually produced in the non-neutral sheath of rf discharges, v^here strong electric fields prevail that levitate the dust particles against gravity [1-3]. There the particles are trapped forming strongly coupled ordered particle arrangements. In plasma crystals a vertical alignment of the particles is observed, where the particles are located directly belov^ each other [1,4]. Moreover, a phase transition from the ordered structure via a liquid to an almost gas-like state v^ith reducing the gas pressure takes place [5,6]. From theoretical studies, the alignment w^as explained by the ion streaming motion in the plasma sheath forming an attractive wake potential downstream the dust particles [7,8]. Under such attractive forces, the dust particles can behave like 'classical' molecules. From a more complex analysis of plasma crystals and their sheath environment [9-11], we had argued earlier that the interaction between the particles in plasma crystals is attractive and asymmetric in such a way that it acts only on the downstream particles and that there is no backaction on the upstream particles. With these asymmetric forces the experimentally observed alignment and phase transition in plasma crystals has been explained quantitatively [10-12]. Although experiments on these forces in plasma crystals have been
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A. Melzer et al /Attractive and repulsive forces in dust molecules
performed [13], a direct and quantitative experimental verification of the existence of such asymmetric attractive forces is still missing. We report here on experiments with a 2-particle system, from which the repulsive or attractive particle interactions can be directly derived. The 2-particle system is used as a force 'probe', where the reaction of one particle to the motion of the other is analyzed. A further advantage of the 2-particle system is that manybody effects which lead to instabilities in plasma crystals [10] are not present. A selective external force that acts only on one of the two particles can be exerted with a focused laser beam.
EXPERIMENTAL SETUP AND RESULTS The experiments are performed in a parallel plate rf discharge operated in helium in the pressure range between 10 and 250 Pa. The lower electrode is powered at 13.56 MHz at discharge powers of 6 W. The dust particles (monodisperse M F particles) are immersed into the plasma and are trapped in the sheath above the lower electrode. They are illuminated by a HeNe laser and the scattered light is recorded by CCD cameras from top and from the side. A second laser diode (690 nm, 35 mW) is used as a manipulation tool [14]: the radiation pressure of the focused laser beam pushes the stroke particle in the direction of the beam. The laser beam can be focused on each particle individually (see Fig. 1).
b)
grid electrode illumination aser
^ d+
dust particle
[m)
/illl ^*_iaseiL
manipulation 1 laser RF voltage generator
dust particle
F I G U R E 1. (a) Scheme of the experimental setup, (b) Force diagram when the upper particle is pushed by the laser beam.
In this experiment, only two particles of different mass are immersed into the plasma. The first (second) one has a diameter of 2ai = 3.47 ^ m (2a2 = 4.81 fim) and a mass of mi = 3.31 • 10"^^ kg (m2 — 8.81 • 10"^^ kg). From standard measurement techniques [3,4,15], the dust charges are determined to be Qi = 2200 e
A. Melzer et al /Attractive and repulsive forces in dust molecules
117
and Q2 ~ 5860 e. Since the particles are trapped in the sheath by the balance of gravitational force and electric field force, i.e. rriig — QiE{zi)^ [g is the gravitational acceleration and E[z) the electric field) the two dust particles are trapped at different heights, and their vertical position is fixed by this force balance. Horizontally, however, the particles are free to move, and from the reaction of one particle to the motion of the other the effective interaction can be directly demonstrated (see Fig. 1). Therefore, these two particles act as a force probe which can be manipulated in a controllable manner by the focused laser beam.
120 Pa
150 Pa
(d)|
200 Pa FIGURE 2. Behavior of the two particles with varying discharge pressure. First, the behavior of the two particles without laser manipulation shall be presented. At an intermediate pressure of 150 Pa, the two particles are found in a separated state (Fig. 2a), which can be expected for repulsing particles. When the discharge pressure is reduced to 120 Pa the two particles suddenly j u m p into the aligned state (Fig. 2b,c), which is usually observed in plasma crystal experiments. The video sequence of this transition is analyzed in Fig. 3. The two particles are found to be separated horizontally by about 1250 ^ m at the beginning of the sequence (the vertical distance is c/ = 913 /im). Then at about t = 7 s the lower particle starts to approach the upper one and the two particles j u m p into
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A. Melzer et al /Attractive and repulsive forces in dust molecules
the aligned position. After t = 8 s the two particles move as a rigidly connected dust molecule. This behavior already shows the attractive interaction between the particles. When again increasing the pressure to 150 Pa the particles are still found to be aligned (Fig. 2d). At a pressure of 200 Pa the alignment breaks and the particles j u m p into the separated state (Fig. 2e,f). The sequence of this separation is analyzed in Fig. 4. One can see the molecule-like behavior in the beginning of the sequence, although the lower is already slightly shifted horizontally with respect to the upper one. At t = 18 s the two particles separate within a second. This ^dissociation' of the dust molecule clearly demonstrates that the interaction between the dust particles is repulsive as one would expect for two negatively charged particles. This loop (a-f) can be repeated perpetually. Such a behavior can also be viewed as a structural phase transition with hysteresis. In order to unambiguously show that the dust molecule formation is due to asymmetric and attractive forces the two dust particles are pushed by the laser individually. Fig. 5a shows the behavior of the two particles when the upper particle is hit by the laser beam. Then both particles move in the same way, they stay vertically aligned. This proves that the lower particle feels an attractive force mediated by the upper particle. When, however, the lower particles is pushed by the laser beam, the two particles are shifted with respect to each other (Fig. 5b), and, finally, they can even be separated by the laser beam. The upper particle does not follow the motion of the lower one. Therefore, the upper particle does not feel an attractive force mediated by the lower particle. Summarizing, the attractive force is asymmetric acting on the downstream particles, only.
DISCUSSION The asymmetric attractive force arises from the ion streaming motion in the sheath. Due to Coulomb collisions with the dust particles the ions are deflected downstream of the particles forming a region of enhanced ion density there. In our previous plasma crystal models [9-11], this ion cloud is replaced by a single positive point charge (5+ which is rigidly connected at a distance c/_|. downstream of the upper particle. The horizontal motion of the two dust particles can then be written as . . .
n •
7712X2 + m2P2^2
X
/-.x
—{xi - X2)
(1)
Q1Q2
rriixi + mifJiXi =
— \ ^ - ^)z
/
l^V^i ~ ^2)
47reo^^'
'
'
"
Qi {d-d+f
where /? is the Epstein friction coefficient due to dust-neutral collisions. The term on the RHS of Eq. (1) is the Taylor expansion of the Coulomb interaction between upper and lower particle. For the upper particle it is purely repulsive. For the lower particle there is an additional attraction denoted by the parameter e which is
A. Melzer et al. /Attractive and repulsive forces in dust molecules
119
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F I G U R E 3. Behavior of the two-particle system during transition into the aligned state at 120 Pa. In (a) the particle trajectories from the side and in (b) the horizontal position as a function of time are shown. the relative strength of attraction in units of the repulsion. Typically, the friction force dominates over the inertial force. Then the aligned situation is stable for
t >
tr
1 + mi/3i
1+
2.92
(2)
120
4. Melzer et al. /Attractive and repulsive forces in dust molecules
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FIGURE 4. Behavior of the two-particle system during transition into the separated state at 200 Pa. In (a) the particle trajectories from the side and in (b) the horizontal position as a function of time are shown. That means that for alignment the attractive force on the lower particle must exceed the repulsion by a factor of almost 3 under the conditions of this experiment. This instability is not the same as that responsible for the phase transition in plasma crystals. The melting instability is due to a collective interaction of many dust particles. That instability sets in as a direct consequence of reduced pressure since the energy transferred into the plasma crystal cannot be dissipated by dust-neutral
A. Melzer et al. /Attractive and repulsive forces in dust molecules
121
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*^
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F I G U R E 5. Horizontal position of the two particles at 120 Pa as a function of time when (a) the upper particle and (b) the lower particle is pushed by the laser beam. The shaded areas indicate the time when the laser beam was switched on.
collisions. Here, the instability is only due to the strength of the attraction and is only indirectly connected to the discharge pressure, as shown below. In the experiment the strength of the attractive force can be determined from following situation (see Fig. l b ) : when the upper particle is pushed by the laser, both particles move in the same direction. The attractive force on the lower particle (e— \-)QiQ2{^i — X2)/{4:7Teod^) is then balanced by the neutral drag m2(^2^2- Taking
122
A. Melzer et al /Attractive and repulsive forces in dust molecules
the values of X2 = 1.27 m m / s and {xi — X2) = 300 //m from the laser push in Fig. 5a {t = 15.4 s to 18.8 s) the drag force is determined as Fdrag = 6.87 • 10"^^ N and the repulsive force as 1.16 • 10"^^ N yielding e = 5.9. This value is decisively larger than the critical value of ec, thus showing that the observed aligned situation is in the stable region predicted by the model. The breakup of alignment at higher pressure can be understood by recalling how the attractive positive ion cloud arises: The streaming ions in the sheath are deflected below the dust particles by ion-dust Coulomb collisions. Ion-neutral charge exchange collision play an important role as a scattering process that tends to destroy this ion cloud [10]. With increasing discharge pressure the ion mean free path for these collisions is reduced and the ions are deflected closer to the dust particle, thus reducing the attraction provided by the ion cloud. When the attraction parameter e drops below its critical value the aligned situation becomes unstable and the interaction forces become net repulsive thus establishing the separated state. In conclusion, the two-particle system is a very useful tool for the quantitative determination of the interaction forces between dust particles. By laser manipulation of a two-particle force probe we have shown the existence of net attractive and asymmetric forces leading to the formation of dust molecules. With increasing pressure a transition to a net repulsion is found. This behavior has been explained in terms of our previous plasma crystal models.
REFERENCES 1. Chu J.H. and I Lin Phys. Rev. Lett. 72 4009 (1994) 2. Thomas H., Morfill G.E., Demmel V., Goree J., Feuerbacher B. and Mohlmann D. Phys. Rev. Lett. 73 652 (1994) 3. Melzer A., Trottenberg T. and Piel A. Phys. Lett. A 191 301 (1994) 4. Trottenberg T., Melzer A. and Piel A. Plasma Sources Sci. Technol. 4 450 (1995) 5. Melzer A., Homann A. and Piel A. Phys. Rev. E 53 2757 (1996) 6. Thomas H. and Morfill G.E. Nature 379 806 (1996) 7. Vladimirov S. V. and Nambu M. Phys. Rev. E 52 2172 (1995) 8. Melands0 F. and Goree J. Phys. Rev. E 52 5312 (1995) 9. Melzer A., Schweigert V.A., Schweigert I.V., Homann A., Peters S. and Piel A. Phys. Rev. E 54 46 (1996) 10. Schweigert V.A., Schweigert I. V., Melzer A., Homann A. and Piel A. Phys. Rev. E 54 4155 (1996) 11. Schweigert V. A., Schweigert I. V., Melzer A., Homann A. and Piel A. Phys. Rev. Lett. 80 5345 (1998) 12. Schweigert I. V., Schweigert V. A., Bedanov V. M., Melzer A., Homann A. and Piel A. JETP 87 905 (1998) 13. Takahashi K., Oishi T., Shimomai K., Hayashi Y. and Nishino S. Phys. Rev. E 58 7805 (1998) 14. Homann A., Melzer A., Madani R. and Piel A. Phys. Lett. A 242 173 (1998) 15. Homann A., Melzer A. and Piel A. Phys. Rev. E 59 3835 (1999)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
123
Self-Organization in Dusty Plasmas S. Benkadda^ V.N. Tsytovich^ S.I. Popel^ and S.V. Vladimirov^ ^ Equipe Dynamique des Systemes Complexes, CNRS- Universite de Provence, Centre de St Jerome, Case 321, 13397, Marseille Cedex 20, France ^ General Physics Institute, Vavilova 38, Moscow 117942, Russia, ^ Institute for Dyn. of Geosph., Leninsky pr, 38, hid. 6, 117979 Moscow, Russia ^ School of Physics, The University of Sydney, NSW 2006, Australia
A b s t r a c t . Dusty plasma systems as open and highly dissipative media have an inherent tendancy to self-organization resulting in formation of dissipative structures. Two types of self-organized structures are analyzed, the dustplasma sheaths and dust nonlinear drift vortices. It is shown that all properties of the dust-plasma sheath depend on only one parameter - the Mach number of the ion flow, which have allowed zones and can be less than unity, in contrast to the features of the plasma sheath without influence of dust. It is also shown that high rate of dissipation in dusty plasma increases the growth of drift vortices.
INTRODUCTION Self-organization processes in dusty plasma are expected to be very important since the latter is an open sytem with a high rate of dissipation. The latter is produced by dust particles which absorb plasma particles. Dust plasma systems can not survive in absence of either external sources of electrons and ions or plasma particle fluxes from the regions where there is no dust. The plasma particle fluxes recombine on the dust particles as well as the energy fluxes are absorbed by the dust particles. Thus external sources of energy and particles are necessary to exist to compensate the absorption in the plasma particles and absorption of the energy on dust particles [1]. The presence of a high rate of disssipation provides rapid developement of self-organization processes and formation of long lived dissipative structures. These structures should be investigated flrst as single isolated structures to uderstand they properties which will be alterated in presence of other structures in the system. We consider here two types of such dissipative structures : the
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S. Benkadda et al /Self-organization in dusty plasmas
dust-plasma sheath and the dust-plasma drift vortices. This choice is related to different applications such as plasma etching, dust crystals and turbulence in edge tokamak plasma and ionospheric plasmas. Nonlinear effects are very important for all these structures. These nonlinearities are very different from those in absence of dust and are related to large dust charges, as well as to the interaction of plasma particles with dust particles. The presence of high dissipation rates make these structures unique. We show that the dissipation lowers the possible dependence of the structures on parameters and often the dissipative structures depend only on very few parameters. The dustplasma sheaths depend only on the Mach number of the incident ion flux. Drift vortices in dusty plasmas become much more unstable than in absence of dust, then they evolve to the marginal stable state depending on few parameters. This tendency is natural since the other possible nonlinear states dead out due to the high rate of dissipation and only the structures for which the balance between dissipation and excitation occurs finally survive.
DUST-PLASMA SHEATHrA NEW DISSIPATIVE SELF-ORGANIZED STRUCTURE The presence of dust near the wall of a low-temperature plasma leads to formation of a dissipative structure confining dust and plasma particles in a selfsimilar way and creating the specific dust-plasma layer. This equilibrium structure is formed self-consitently with the fields of the dust particles. The problem of plasma-wall boundary is known in physics since the pioneering works of Langmuir in the 20s. The specific feature of the plasma sheath [2], which is formed near the conducting wall, is the existence of flows of plasma particles towards the wall, the latter being negatively charged because of the difference of electron and ion masses and temperatures. In the presence of a relatively high energy flux, the impurity contaminants, or dust^ consisted of macro-scaled (comparing with sizes of electrons and ions) grains can be ejected from the wall. Dust strongly influences all parameters of the sheath, in particular, the electric field distributions and the ion flow velocities. The electric fields of dust particles contribute to the distribution of the electric potential in the sheath and influence the ion flow; at the same time, the grain fields themselves strongly depend on the flow. In the absence of dust, there are two regions in the near-wall plasma [2]: the plasma sheath itself, where the main drop of the electric field potential occurs, and the pre-sheath, where the potential drop is small and ionization as well as ion acceleration occur to result in the plasma recombination on the wall. In the presence of dust, there are three distinguished layers: the plasma boundary pre-sheath layer (containing no dust), the dust cloud (there is the main drop of the electric field potential), and the wall-plasma layer (containing no dust). In contrast to the plasma sheath in the absence of dust, the dust-plasma sheath can support only spe-
125
S. Benkadda et ah /Self-organization in dusty plasmas
cific Mach numbers (which can now be less than unity) of the ions flowing toward the dust cloud and the wall, thus creating regions of allowed ion stream velocities. These results are important for dust-plasma experiments, plasma processing and fusion (formation of dust layers in edge plasmas). Consider Boltzmann distributed electrons rie — noexp(e(/p/Te), UQ is the unperturbed electron density in the region where quasineutrality holds (i.e., in the presheath), (/? is the electric field potential (which is practically zero in the pre-sheath), e is the electron charge, and Tg is the electron temperature in the energy units. The dissipation on dusts for electrons is negligible as compared to pressure and electric forces. The forces acting on the dust particles are due to the ion drag and the electric field —ZdeE^ where Zd is the dust charge in units of the electron charge e. For simplicity, we do not account for other forces such as gravity, thermophoretic force, etc., and suppose the electron and ion temperatures being constant in the sheath region. The ion drag force due to collisions of ions with dust can be written as ^dr = f^i^iUid = aPzuadr/TeXj)-^ where m^-, t;^-, and Uid are the ion mass, speed, and the effective collision frequency with the dust grains. Here, we have introduced the dimesionless quantities: the parameter P = UdZd/no^ where Ud is the dust number density in the sheath, the normalized dust charge z — Zdc'^/aTe^ where a is the dust grain size, the normalized ion speed u — v^|\/2vTi'> where VTI = {Ti/miY^'^ is the ion thermal velocity (with Ti being the ion temperature), X^i = {Ti/Anrioe'^y^'^ is the ion Debye length, and defined the drag coeflBcient adr- The force balance equation for dust is then P{-E + zunadr) = 0, (1) where E = eXj^-E/aTe is the dimensionless electric field (this is the field in which ions get the energy Tg at the distance Xj^Ja)^ and n = rii/no is the normalized ion density. Thus E = zuna^r when P 7^ 0. Equation (1) describes the balance of the electric field and ion drag forces; the important point here is that the electric field E is the self-consistent field which includes fields created by charged dust grains. Note that in our approximation (zero dust temperature) there can exist a step change of P from zero to a finite value. Since Zd is determined by the parameters of plasma particles, its value is continuous, and therefore rid can have a jump. To clarify that, generalize the force balance equation (1) for the nonzero dust pressure d fP\
P f
r.
2
\
X = xa/Xj)- is the dimensionless space variable and Td = Tde'^/aT^ characterizes the dust temperature Td. We see that if this parameter is small, then the boundary of the dust cloud is sharp. Eq. (1) is derived for r^f -> 0 when there are two solutions, P = 0 in the dust-free region, and P > 0 in the region occupied by dust. The balance of the electric and pressure forces acting on plasma electrons is given in dimensionless form by £' = —dne/riedx. Thus we have from (1) — —— = -zunadr^
(3)
126
S. Benkadda et al /Self-organization in dusty plasmas
The drag coefficient adr is a function of u and r jz^ where r = Ti/Te^ and includes the charging collisions as well as the Coulomb scattering collisions. In the limit T/Z
,,, 4
The third equation comes from the stationary ion continuity equation
dx
= -Pnach,
(5)
where the charging coeflScient ach depends also only on two parameters u and r/z. In the limit of super-thermal ions is [4] ach ~ [1 + (M^/22:)]/4u. The balance equation for electron and ion charging currents hitting the dust grain surface is [4, 5] 72eexp(—z) = 2^/^Tznach|y/T^i where m — milm^ is the ratio of ion to electron masses. For further convenience, cf. Eqs. (3)-(5), we differentiate it to obtain the fourth equation of the model 1 dza^Yi dz dx za^h dx
1 drie
1 dn
rie dx
n dx
1 dach du a^h du
dx
The system of equation (3)-(6) is not closed since it contains the additional parameter P which can be found explicitly as a function of n, li, z, ng by using the Poisson equation d 1 dne\ dx The dx J
T - r ^a? (^e - n - h P ) .
(7)
where a = a/X^i is the dimensionless dust size. It is important that the expression for P exhibits critical behaviour related to singularities for certain relation between the parameters involved. In the presence of a wall, the dust-plasma boundary conditions are different for the side of the dust cloud bordering the quasi-neutral bulk plasma (pre-sheath) and for the side oriented to the wall. In the plasma bulk the first integral of the Poisson's equation can be used 2 £;^ = —
r 1 -exp
n^
sG-i)}'
+ r n — T — 1 -f 2i(
(«'
where UQ is the initial ion velocity in the pre-sheath. Due to continuity of the electric field, we have at the border E = zunadr- The charging equation can be
127
S. Benkadda et al /Self-organization in dusty plasmas write as (note that exp
MQ
=
\/2TUO)
^ Mo A
^ m e (zn_ ^ MoX ^,^,^
1
2mi \Mo
^^^
n
and the condition that the j u m p of P is positive at the boundary, give the remaining boundary conditions. This system of boundary conditions was solved numerically to determine the ranges of parameters which are allowed by the boundary conditions and correspond to possible equilibrium states of the dust sheath. The numerics were performed in the ranges 1 > r > 0.001, 0.01 < MQ < 10, and 0.001 < a < 0.5 for various gases (Hydrogen, Silicon, Argon, and Krypton). The numerical solutions show that the ranges of plasma parameters corresponding to equilibrium states do exist; Table 1 gives a summary of the first two allowed zones for the initial Mach number in Argon plasmas with various values of r and a. It can be seen that the sheath can be either subsonic or supersonic, in contrast to the Bohm criteria in the absence of dust, viz. Mo > 1. The allowed regions for the Mach number correspond to those sets of parameters when an equilibrium solution exists. For other values of the ion flow velocities, the absence of the equilibrium means that the system becomes unstable and a transient process, involving, e.g., charging of dust grains and therefore changing the electrostatic field and the speed of the fiow, ends up for parameters' values from an allowed zone. The condition at the boundary of the sheath adjacent to the wall, is written for the same assumptions (Boltzmann electrons and super-thermal ion drift) for consistency of the model. The electron current is thermal, jg = —enet'Te/47r, while the ion current is determined by the ion drift, ji = nvi. Giving to the values at the wall the subscript w^ we write the boundary condition for the dimensionless variables as
When this condition can be fulfilled inside the dust layer, dust levitation becomes impossible; this can be checked by putting into (10) the values found inside the dust sheath. The solutions of the nonlinear equations inside the dust sheath taking into account the dust-plasma boundary in an allowed Mach number domain gives the parameters at the dust boundary oriented to the wall; we denote them with the subscript dw. Using the condition u ^ I and the Boltzmann distribution for electrons, we write the equation for the change of the ion density in the form
The second equation is the Poisson equation written as —
X
0 1 ne,dw '^e.dw 1
Ta-^ I
\ n
exp
2
I n2
-n
.
(12)
128
S. Benkadda et al. /Self-organization in dusty plasmas
Temperature ratio r = Ti/Te 0.001 0.01 0.001 0.01
Grain size a/Xoi 0.5 0.5 0.1 0.1
First zone M„,i„,i 0.708 0.710 0.709 0.712
First zone
Second zone
Second zone
^"max,l
Mn,in,2
Mma.x,2
1.094 1.096 1.104 1.106
1.742 1.743 1.746 1.748
2.550 2.297 3.363 3.342
Table 1: Two first regions of the allowed Mach numbers for the ion stream (all values between Mmin ^nd Mmax are allowed). The system of equations (11) and (12) is sufficient to find distribution of n^j and E^ from Udw and Edw as well as distribution of the electrostatic potential from the boundary of the dust sheath to the wall. Fig. 1 gives the result for the dust layer in Argon where the boundary with the plasma is at the left corner, and boundary with the wall-sheath layer is at the right corner. The initial Mach number is chosen to be MQ = 1.796, thus being in the second allowed region of the Mach numbers starting from Mmin2 = 1.743. The upper grey solid line gives values for the ion density as a solution of the wall-boundary relations; because it does not intersect the lower black solid line (representing the ion density in the sheath), levitation is possible. The dashed line represents the electron density ng, the dotted line - the parameter P , and the dash-dotted line - the dust density (~ P/^)- The grey dashed line corresponds to the ion number density for which the denominator in expression for P equals zero. The calculations cannot proceed for the distances beyond this point. The Mach number at this boundary is equal to 2.72. Solution of Poisson's equation in the region between this boundary and the wall gives the Mach number at the wall equal to 3.396. The thickness of the dust layer is Axd = 8.116A|)-/a. Note the sharp peak of the dust density near the wall. Fig. 2 gives the result for Argon plasma when the initial Mach number is MQ — 0.710 - the lowest allowed in the first range. The lower black solid line gives values for the ion density distribution, other lines are the same as on Fig. 1. The thickness of the dust layer is Axd = 30.0XjjJa. Note the completely different (shock-like) structure of the dust density distribution (dash-dotted line) as compared with the case on Fig. 1. The two figures represent two typical examples of numerical results. The most important properties of the plasma-dust sheaths obtained in series of numerical computations are: 1) The plasma-dust sheath is a self-organized structure determined only by the Mach number M of the ion flow; 2) The plasma-dust sheath can exist only within certain ranges of M; 3) The value of M is determined by the total number of dust particles in the sheath per it unit surface times Ana^; 4)the plasma-dust boundaries are sharp in the limit of cold dust grains (zero dust
S. Benkadda et al /Self-organization in dusty plasmas
0
1
2
3
4
5
6
129
7
Distance (normalised)
Figure 1: The distributions in the dust layer in Argon plasma as functions of the distance x — xa/X'jj- of the: normalized ion density n/rio (solid curves; the upper grey solid line gives the ion density as a solution of the wall-boundary relations, the lower black sohd hue gives the actual distribution), normalized electron density ^e/^o (dashed curve), normahzed dust density n^aTg/noe^ (dash-dotted curve), and the parameter P = ridZd/no (the dotted curve). The initial Mach number is Mo = 1.796 corresponding to the second allowed range (see Table 1). Other parameters are: Ti = O.OlTg, a = O.^Xoi-
130
S. Benkadda et al. /Self-organization in dusty plasmas
0.6
r
Ti
\
\
r
2
4
6
8
10
-i
12
r
14
Distance (normalised)
Figure 2: The same as Fig. 1, but for MQ — 0.710 corresponding to the subsonic ion flow in the first allowed range (see Table 1).
S. Benkadda et al /Self-organization in dusty plasmas
131
temperature);5) For M in the upper range larger than some critical value the dust levitation is not possible (gray sohd and solid Hues intersect) which shows a possibihty of catastrophic destroy of the dust sheath ; 6) The maximum value of P in the sheath is of the order of 1 (it is 0.3 on Fig's 1,2); comparison with dissipation less dust distribution found in [6] shows that neglecting of dissipation in [6] is possible for much lower P
NONLINEAR D R I F T VORTICES Two-dimensional drift vortices constitute another example of self-organisation in dusty plasmas. Their appearance is related to the fact that the drift waves are negative energy waves and any dissipation leads to their excitation. On the other hand, the increase in the dissipativity of a plasma-dust system (in comparison with the usual plasma) makes the drift waves in dusty plasmas much more unstable than in the case of the absence of dust. Thus the dust serves as a catalyst for excitation of drift vortices in the plasma. We consider the case of a plasma in an external magnetic field when the dust can be considered as stationary but their spatial distribution is inhomogeneous. The density gradient is supposed to be along the axis Ox, while the direction of the magnetic field is parallel to that Oz. The dissipative drift vortices in dusty plasmas are described by the modified set of Hasegawa-Wakatani equations for nonlinear drift waves [7, 8]:
dn
~
^.
+ a{n-zQ
di/j
, ,
X
r
,-,
+ — - Ce(V' - n) = {n, V'},
^ + PC = a{l + P)n,
132
S. Benkadda et al /Self-organization in dusty plasmas
where the notation {..,} denotes the Poisson brackets SA B \ - — — - — — dx dy dy dx ' In the above equations the dimensionless variables i, a:, and y are used which are related to the usual corresponding variables by the relationships: ^ Ps t -> ili—t^ ^n
^ y X —>- —, y -^ —. Ps
Ps
Furthermore, the following variables and parameters are introduced: Srie Ln
.
rieoPs'
SZd Ln
Zd Ps'
,
Cip Ln
^
T,ps'
PL - 1 d
{l + P){l +
-i\' P-PL^')
where Qi is the ion gyrofrequency, ps is the ion Larmour radius for the electron temperature, L~^ = dlnriQe/dx^ L^^ = dlnndZd/dlnrti^Q. The coefficient Cg describes the electron non-adiabaticity along the magnetic field lines due to collisions of electrons with dust particles:
i>ei^i{ps/Ln)' where k^ is the wave vector component parallel to the magnetic field, z>e is the frequency describing the momentum transfer from plasma particles to dust particles [8]. The coefficients a and /? describe the infiuence of charging and Coulomb collisions: 0>i{ps/Ln)' where v^.i are the collision frequencies in the continuity equations for electrons and ions (see [1]). To describe the excitation of the drift vortices we have to find the unstable solutions of the linearized set of the modified Hasegawa-Wakatani equations. The linearized set written in Fourier-components is
{-ioj + a + Ce)n - azC, + {iky - Ce)'il^ = 0, {-iuj + /9)C = a{l + P)n. These equations allow us to obtain the dispersion relationship for ui = w / a :
' \
{1 + P)a
a{l + P)[-iL0 + l + {cJa)-{z{l
+ P))/{-iu
+ (3/a)]
S. Benkadda et al /Self-organization in dusty plasmas
133
The dispersion equation has been solved for different limiting cases. The unstable solutions have been found. For example, for Cg ^ a (the case when the effect of non-adiabaticity is much weaker than that of dust charging) and P
^-— + l-
1+ P
/?
The real part of the drift wave frequency is approximately the same as in the absence of dust, but the growth rate is determined by the dust particle charging processes and can be higher significantly than that in the absence of dust. For Ce ; » a, P ~ 1, and if
a V l
+ Py
1+ P
a\
there exist the unstable solution /?2 Az 2 \^\| a2 + ^ P +
^
In this case the real and imaginary parts of the frequency are of the same order of magnitude (the case of strong instability). For Cg
_sk^ '^~
fc2
+'
( 1 + P ) Z P ( T + Z)
Thus the growth rates of the drift instability in the presence of dust can be substantially larger than those in the absence of dust. The investigation shows also that the thresholds of the instability can be much lower than those for the drift instability in the absence of dust. This confirms the qualitative result that dust serves as a catalyst for excitation of drift vortices in the plasma.
CONCLUSIONS The main conclusions of the present investigation are: (1) Dusty plasma is an open system where the rate of dissipation is high and there is a tendency to self-organization. (2) New types of nonlinear interactions are possible in dusty plasmas. They are related to both new time scales and the process of charging of dust particles. (3) The increase in the dissipativity of a plasma-dust system (in comparison with the plasma without dust) makes the drift waves in dusty plasmas much more
134
S. Benkadda et al. /Self-organization in dusty plasmas
unstable than in the case of the absence of dust. The dust serves as a catalyst for excitation of drift vortices in the plasma.
ACKNOWLEDGMENTS VNT is grateful to the Laboratory of Physics of Molecular and Ionic Interactions CNRS-Univesite de Provence and to the Humboldt Foundation for support. SVV was partially supported by the Australian Academy of Science and the Australian Research Council. Hospitality of G. MorfiU is gratefully appreciated by VNT and SVV.
References [1] V.N. Tsytovich, Physics-Uspekhi 40, 53 (1997) [Uspekhi Fizicheskikh Nauk 167, 57 (1997)]. [2] F.F. Chen, Introduction to Plasma Physics (Plenum, New York, 1974). [3] F.F. Chen, in: R.H. Huddlestone and S.L. Leonard (eds.), "Plasma Diagnostic Techniques'^ Academic, New York (1965), Chap. 4. [4] V.N. Tsytovich, Sov. Phys. Uspekhi 40, 53 (1997). [5] S.V. Vladimirov, Phys. Plasmas 1, 2762 (1994); Phys. Rev. E 50, 1422 (1994). [6] J-y. Lui and J.X.Ma, Phys Plasmas 4, 2798 (1997). [7] S. Benkadda, V. N. Tsytovich and A. Verga; Comments in Plasma Phys.Contr. Fusion 16, 321 (1995). [8] S. Benkadda, P. Gabbai, V. N. Tsytovich and A. Verga; Phys. Rev. E 53, 2717 (1996).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
135
Dynamical Properties of Strongly Coupled Dusty Plasmas S. Hamaguchi and H. Ohta Department of Fundamental Energy Science, Kyoto University, Kyoto 611-0011, Japan
A B S T R A C T Dynamical properties of strongly coupled dusty plasmas have been studied using molecular dynamics (MD) simulation. Systems of dust particles immersed in a plasma are modeled by a collection of particles interacting via Yukawa (screened Coulomb) potentials. Thermodynamic properties of Yukawa systems can be characterized by two parameters, i.e., K, (the ratio of the average interparticle distance to the Debye length of the background plasma) and F (the ratio of the interparticle Coulomb potential energy to the kinetic energy per particle). By examining velocity autocorrelation functions and mean square displacement for single particle diffusion, we have obtained self-diffusion coefficients of Yukawa systems for various AC and F values.
!•
Introduction
Small solid particles (e.g., the diameters less than 1 /xm) immersed in a plasma typically obtain negative charges as the mobility of the background plasma electrons is much larger than that of ions. Such a charged dust particle forms a Debye sheath around it and interacts with other particles via Coulomb potentials with some screening effects. We model the pair potentials by Yukawa (screened Coulomb) potentials given by ^ W "^ 'A
exp{-kDr),
(1)
where — Q = — Ze {Z ^ 1), r, and k^ denote the charge, radial distance between two particles, and the inverse Debye length of the background plasma, respectively.^ If all the particles have the same mass m and charge — Q, thermodynamic properties of the Yukawa system may be characterized by the two dimensionless parameters tz = a/A/) and F = Q^/47reoakT with a — (3/4?™)^/^ and n being
136
S. Hamaguchi, H. Ohta/Dynamical properties of strongly coupled dusty plasmas
2
3
K = p / ^D 4 5
6
• fluid-bcc • fluid-fcc ^ bcc-fcc 100 0.0
1.0
2.0
3.0 K = a / ^D
4.0
5.0
Figure 1: Phase diagram of Yukawa systems in the K-T plane. The circles are fluid-bcc phase boundary points {K < 4.3), the squares are fluid-fcc phase boundary points (AC > 4.3), and the triangles are bcc-fcc phase boundary points The solid lines represent curves fitted to these data points. From Ref. 2 with permission the Wigner-Seitz radius and the particle number density. The parameter K is the inverse of the normalized screening length and the parameter F is roughly the ratio of the (unscreened) Coulomb potential energy to the kinetic energy per particle. Several laboratory experiments and numerical simulations demonstrated such charged dust particles form crystalline structures (Coulomb crystals) when they are confined in a quiescent plasma (usually by external electric field potential wells) and their kinetic energy is sufiiciently decreased.^""^ Figure 1 shows the phase diagram of Yukawa systems obtained from Molecular Dynamics (MD) simulation.^ In the present paper, we shall examine self-diffusion processes of Yukawa particles in the fluid state. Motion of dust particles immersed in a plasma is strongly influenced by collisions with background particles (i.e., ions, electrons, and neutral species) in addition to Yukawa interactions with other dust particles. Here, for the sake of simplicity, we consider dust particle motion due to the latter, i.e., diffusion due to Yukawa interactions. To determine the self-diffusion coefficients
137
S. Hamaguchi, H. Ohta / Dynamical properties of strongly coupled dusty plasmas of Yukawa systems, MD simulation is employed.
II.
M D simulation The effective pair potential^ used in our MD simulations is given by * ( r ) =
(2)
with (f){r) being the Yukawa potential given by Eq. (1). The potential above represents the interaction energy of particle i with particle j (at separation r = Tj—Ti) and with all periodic images of the latter. The infinite sum of (f) over integer vectors n = {l^m^n) represents the periodic images. In this way, we can emulate correct particle interactions even in the weak-screening regime, where the range of the interparticle forces is comparable to or greater than the side L of the cubical simulation volume. The infinite sum of Eq. (2) is approximated numerically by a tensor-product spline function.^ To have the system attain the desired temperature T (or F), we periodically rescale the velocity of each particle during the simulation until the system reaches the thermodynamical equilibrium.^""^
III.
Self—diffusion
coefficients
The self-diffusion coefficient is given by the well-known Einstein relation, i.e.,
where r(t) is the position of a particle at time t and < > denotes the statistical average. In MD simulation, the time average over a sufficiently long time period is used to evaluate < > after the system reaches a thermodynamical equilibrium and periodic rescaling of particle velocities is no longer employed. It is also know that the self-diffusion coefficient D may be evaluated from the velocity autocorrelation function Z{t) through
D = ll^Z{t)dt.
(4)
The velocity autocorrelation function Z{t) is defined by
Z(0 = (v(t).v(0)), where v(i) is the velocity of a particle at time t. We have evaluated self-diffusion coefficient D from MD simulations using both Eqns. (3) and (4).
138
S. Hamaguchi, H. Ohta/Dynamical properties of strongly coupled dusty plasmas
r
r
r
,—
r
0.14
r-
1
1—
A B
o + + G
,
0.12
~
0.1
-
0.08
Q
6 H
• \
G
o
-
s* * ®
0.06
6
9
n
•j
#*
0.02
-\ H
9
0.04
+
L
L
i_
5
T*
6
10
Figure 2: The normalized self-diffusion coefficients D* as a function of the normalized temperatures T* for various values oi K {OA < K < 5). A(o) and B(+) indicate D* values evaluated from Eqns. (3) and (4). Figure 2 plots the dimensionless self-diffusion coefficient D* = ^/SD/CJE^^^ for various K, values (0.1 < AC < 5), where u^ is the Einstein frequency for the fee lattice of Yukawa systems defined by CJ-
Here (f) is the Yukawa potential given by Eq. (1), and particles are assumed to be at the fee structure sites. The Einstein frequency UJE denotes the harmonic oscillation frequency of a particle around its equilibrium site (an fee site in our case) when all other particles are located at their equilibrium sites. The Einstein frequency UJE depends on K, and OUE —^ ^p (plasma frequency) as K —> 0, i.e., in the limit of the classical one-component plasma (OCP), which is a system of mobile charges immersed in a strictly uniform neutralizing background..^"^^ The abscissa of Fig. 2 is the Yukawa system temperature T normalized by the melting temperature (i.e., fluid-solid phase transition temperature) T^, i.e., T* — TjTm ~ r ^ / r . As shown here, when the system is in a fluid state close to
139
S. Hamaguchi, H. Ohta /Dynamical properties of strongly coupled dusty plasmas the melting point (T* < 10), the normahzed self-diffusion coeJB&cient D* almost hnearly depends on the normalized system temperature T* and hardly depends on n. Therefore we may scale
Z ) * - a ( r - l ) ^ + 7,
(5)
with a c^ 0.01, /? — 1, and 7 c^ 0.003 for the range 1 < T* < 10, all coefficients being independent of K. The scaling of Eq. (5) is consistent with earlier simulation results for OCP by Hansen et alJ and those for Yukawa systems with limited range of parameters (5 < ^c < 16, 0.5 < T* < 2) by Robbins et al}^ However, our simulation also shows t h a t the dependence of those coefficients on K (that we ignored above), especially that of /?, becomes pronounced for large T* ( > 50).
IV,
Summary and discussion
We have obtained the self-diffusion coefficients for Yukawa systems in a wide range of the thermodynamic parameters. When the system temperature is not too high above the melting point (T* < 10), the normalized self-diffusion coefficient D* is almost independent of K and its dependence on the normalized temperature T* is given by Eq. (5). The independence of D* on K, may be accounted in the following manner. When the system is in a fluid state close to the melting point, motion of particles may be seen as oscillation (at lea^t for a short duration of time) about their equilibrium sites and particle diffusion results from hopping motion of such an oscillating particle from one equilibrium site to another. This self-diffusion process may be characterized by the diffusion coefficient given by D = C A r ^ / A t , where C is a constant, A r is the oscillation amplitude, and A i = u^^ is a typical time scale. The Lindemann criterion^^ states that fluidsolid phase transition occurs when the ratio Ar/a is nearly constant regardless of the interparticle potentials of the system, we may consider systems of the same T* have approximately the same ratio JR = Ar/a^ regardless of /^ as long as T* is relatively small. Therefore D* oaD/uji^o^ — CB? is independent on n for a given T*. However, for larger T* ( > 50), our simulations show that the the K dependence of the parameters for Eq. (5) is more pronounced and cannot be ignored.
Acknowledgements The authors thank R. T. Farouki for useful discussion on MD numerical methods.
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S. Hamaguchi, H. Ohta/Dynamical properties of strongly coupled dusty plasmas
References ^S. Hamaguchi and R. T. Farouki, J. Chem. Phys. 101, 9876 (1994). ^S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phys. Rev. E 56, 4671 (1997). ^S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, J. Chem. Phys. 105, 7641 (1996). ^R. T. Farouki and S. Hamaguchi, J. Chem. Phys. 101, 9885 (1994). 5R. T . Farouki and S. Hamaguchi, J. Comp. Phys. 115, 276 (1994). ^S. G. Brush, H. L. Sahlin, and E. Teller, J. Chem. Phys. 45, 2102 (1966). ^J. -P. Hansen, I. R. McDonald, and E. L. Pollock, Phys. Rev. A 11,1025 (1975). *M. Baus and J.-P. Hansen, Phys. Rep. 59, 1 (1980). ^Strongly Coupled Plasma Physics, (F. J. Rogers and H. E. DeWitt, eds.). Plenum Press, New York (1986). i"R. T. Farouki and S. Hamaguchi, Phys. Rev. E 47, 4330 (1993). " M . O. Robbins, K. Kremer, and G. S. Grest, J. Chem. Phys. 88, 3286 (1988). ^^F. A. Lindemann, Z. Phys. 11, 609 (1910).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
141
Structures and Structural Transitions in Strongly-Coupled Yukiiwa Dusty Plasmas and Mixtures Hiroo Totsuji,* Chieko Totsuji, Kenji Tsuruta, Kenichi Kamon, Tokunari Kishimoto, and Takashi Sasabe (*[email protected]) Faculty of Engineering^ Okayama University, Tsushimanaka 3-1-1, Okayama 700-8530, Japan
Abstract. Regarded as Yukav^a systems in external confining potentials, structures and transitions in dusty plasmas have been analyzed theoretically and by numerical simulations. The one-dimensional confinement in three dimensions and the radial confinement in two dimensions are considered. The results of simulations have been reproduced to a good accuracy in both cases. It is shov^n that the inclusion of the correlation energy or the effect of strong coupling is of essential importance in the theory. A crossover from the surface freezing of Coulomb system to surface melting of systems of short-ranged interactions is observed. INTRODUCTION Physics of dusty plasmas, assemblies of macroscopic particles immersed in plasmas, is closely related to important practical problems in plasma processes of semiconductor manufacturing and also to subjects of basic statistical physics. Direct observations of structures such as crystals and their transitions have inspired us to ask what determines these structures. The purpose this paper is to answer this question through numerical simulations and theoretical analyses. We simplify the system as far as possible and try to find essential factors in the formation of structures and their transitions. The main message is that the contribution of the correlation (cohesive) energy in dusty plasma is of essential importance in these phenomena and we can reproduce them by taking the correlation energy into account properly. SIMPLE MODEL OF DUSTY PLASMA Let us consider the case where our dusty plasma is formed above a wide horizontal plane electrode as shown in Fig.l [1]. Dust particles are levitated by the electric field in the vertical gravitational field in the direction of ~z. We adopt the ion matrix sheath model and, for simplicity, assume that the density of charges in the sheath (except for those of dust particles) is given by erish, e being the elementary charge, and is nearly constant in the domain of interest.
142
H. Totsuji et ah /Structures and structural transitions in strongly-coupled Yukawa plasmas
The potentials for a dust particle of mass m and charge —q are written as mgz + 271 qerishZ^' In the domain z < 0, dust particles are in the potential well: (l>ext{z < 0) = mgz + 27rqenshz'^ = (l>ext{zo) + 2nqensh{z ~ z^f,
(1)
where ZQ = —{g/^T^^'^shYj^/Q) < 0. In the domain z > 0, we have (j)ext{z > 0) == mgz. In our model, the external potential (t>ext cannot keep dust particles afloat in the domain of neutral plasma. We regard dust particles as interacting via the isotropic repulsive Yukawa potential (5^/r)exp(—r/A), where ~q is the (negative) charge on a dust particle. It has been pointed out that there exists an anisotropic interaction coming from the ion flow in the sheath We may, however, expect to have the cases where the isotropic part of interaction potential plays the central role to determine the overall structure in z-direction, even if the configurations in the a:y-plane relative to adjacent layers are aff'ected by the anisotropic part. ROLE OF CORRELATION ENERGY IN STRUCTURES OF DUSTY PLASMA UNDER ONE-DIMENSIONAL CONFINEMENT In the case of one-dimensional geometry discussed above, we have analyzed the structure of the Yukawa system confined by the potential v^xti^) — {}./2)kz'^ by molecular dynamics simulations and theoretical approaches [2-6]. In this case, k — ATrqerish' At low temperatures, dust particles form layers conformal with the confining potential and the number of layers are determined by the parameters ^ = a/A, and T] = {TI^I'^/2)[{l/2)ka^/[q^/a)], a = {TTNS)-^^^ being the mean distance determined by the surface density Ns. The main results of simulations are expressed as the phase diagram shown in Fig.2. This phase diagram has been reproduced by our theoretical calculation and it is shown that the correlation (cohesive) energy of Yukawa particles makes an essential contribution in constructing the phase diagram [5]. TWO-DIMENSIONAL YUKAWA SYSTEM We now consider the case where dust particles are confined in a plane near the boundary between the plasma bulk and the sheath and they are also confined laterally by an electrode surrounding dust particles. We denote the coordinates as r = (R-^z)^ R being the xy components. The electrostatic potential of the surrounding electrode may be approximately expressed in the form {1/2)KR\
(2)
We thus have a two-dimensional Yukawa system confined by a parabolic potential. At low temperatures, this system is characterized by a single parameter
a = q^lK\\
(3)
H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas
143
Structures at Low temperatures (Simulation) We have performed molecular dynamics simulations at constant temperatures on this system. Some results for the structures at local minimum of the total energy at low temperatures are shown in Fig. 3. For smaller systems, the star-like structure becomes global minimum for N==6 and larger. With the increase of the number of particles, these structures gradually change into triangular lattice in the central part and surrounding circular structures. The surface number density p{R) is shown as a function of radius in Fig.4. In the case with relatively large number of particles, the distribution is characterized by this function. In the case of unscreened Coulomb interaction, two-dimensional clusters have been simulated [8,9]. Structures of dust clusters have recenly been observed experimetally [10]. Structures at Low temperatures (Theory): Role of Correlation Energy In the case of relatively large number of particles on a plane, we may describe the distribution of particles by the isotropic surface density p(R) — p{R) and p{R) = 0 for Rm < R. When the Yukawa particles are distributed uniformly on a plane z = 0 with the surface density po, the interaction energy per unit area is thus given by irq^Xp^. Based on this, we estimate the interaction energy Uint neglecting the edge effect. Together with the external potential, we have
Uint - / dRWMRf.
Ue,t = I dIi^KR''p{R).
(4)
We find p{R) and Km which minimize the value of Uint + Uext- The results are AV(^) = ^
{{Rm/Xy - (R/X?) ,
{ ^ y
- SaN.
(5)
When compared with results of simulations, this result underestimates the surface density, as shown in Fig.4 or particle distributions are more compact in reality. In this calculation, the correlation (cohesive) energy between particles has been neglected. Since the correlation energy is negative, particles can be distributed more closely when the correlation energy is taken into account. The correlation energy (per unit area) of the two-dimensional Yukawa lattice of the surface density po is expressed by a function ecoh{^/^pl ) as q'^pl Ccohi'^/^Po ) [6]. This expression provides us with approximate values of the cohesive energy of two-dimensional Yukawa system at low temperatures. When we take the cohesive energy between particles into account within the local approximation, we have finally the results are plotted in Fig.4. We observe that theoretical results for the density and the maximum radius are greatly improved and the results of simulations are almost reproduced. We here emphasize again that the contribution of the correlation energy is of essential importance for this improvement.
144
H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas
Surface Melting vs. Surface Freezing We have analyzed the melting of Yukawa system confined in the onedimensional geometry and have shown that with the increase of the temperature, the intra-layer melting and inter-layer melting occur in this order [7]. We here observe the melting of the laterally confined two-dimensional Yukawa system. The results of numerical simulations are shown in Fig.5 as orbits of particles in a certain duration of time. We note the crossover from the surface melting to core melting with the increase of the parameter a. When a
(6)
to represent the separation in z-direction. According to the values of r] and 5, we have four cases. When r] ^ \ and 5 1 and 5 :§> 1, separate two-dimensional Yukawa systems, each being composed of one species, when ry 1, two separate one-component Yukawa systems with finite thicknesses. We have performed molecular dynamics simulations on dust mixtures. As an example of dust mixtures, we take the one composed of species 1 and 2 where 91/92 ~ 1/2' mi/m2 — 1/8, and N1IN2 = 1. Here —g^, m^, and Ni are the charge, the mass, and the surface number density of the dust of species i. These conditions for charges and masses correspond to the case where both kinds of dust particles are of the same material, the ratio of radii is 2, and the electrostatic potentials are the same. As for the parameters ^, we assume ^ = 1 evaluating the mean distance a by the total surface density Ns = Ni + N2. We define the parameters for species z by 77^ = {e/qi){nsh/N/ ) and Si = ~Zi/ai — {g/4iTTenshO'i){mi/qi) where a^ = (TTA^I)"^/^ and Zi is the equilibrium
H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas
I
9
z=o
FIG. 1. Dusty plasma confined near the boundary of sheath and plasma bulk. FIG. 4. Radial distribution in two-dimensional dust crystals, simulation (thick lines), theory (thin lines), and theory with correlation energy (medium lines).
0.0001
FIG. 2. Phase diagram of confined Yukawa system: Number of crystal layers, simulation (symbols) and theory (thin lines).
a=10^
a=10-
FIG. 5. Surface melting (a » 1) vs. surface freezing (a <^ 1). ^1 rri
'"I"""
I
X 61=1.5/ 2 "2 •
82=6.0/ 2"2
S 'c X l: ^
0
^
XX-
-2
• ••
-4
• ••
*^^
XX
J^XXXJOk
X
> i
•
•
-6 kwJ
1
I
I M » "I
01 - 8 - 6 - 4 - 2
0
2
4
6
8
.8
-6
^
-2
0
2
4
6
8
FIG. 3. Structure of two-dimensional dust crystals. for
^2
piG. 6. Example of positions of layers Si = 1.5/2^/2 and 82 = 6.0/2^/2.
145
146
H. Totsuji et al /Structures and structural transitions in strongly-coupled Yukawa plasmas
position of species z, Zi = -{gI^T^^Tish){'^ilqi)In the above case, 7?i/r;o — {q2/qi){N2/Nrf/' = 2, and 81/82 = 1/4. Results are summarized in Fig.6 where the positions of layers normalized by ai are plotted as functions of r^^: Since the values of 77^ depend on species, the abscissa has two scales. When r^^'s are sufficiently large, both species are in the one-layer state. With the decrease of the parameters r/^'s, formations of multiple layers are also observed. The critical values of transitions have been compared with those for the onecomponent case and satisfactory consistency has been shown when appropriate interpretations are made. It is also noted that the low-lying heavier species can provide a support for the lighter species afloat in the domain of bulk plasma. This indicates that we may have dusty plasmas of light particles in the plasma bulk by intentionally adding heavy dust particles. In the bulk plasma, the ion flow is small and we may have a nearly ideal Yukawa system. CONCLUDING R E M A R K S We have analyzed structures and structural transitions in the dusty plasmas based on the model as the confined Yukawa system. The structures and transitions are easy to observe and critical parameters for transitions may be useful in determining the plasma parameters surrounding dust particles. The control of these structures may also applicable to structure formations related to plasma processing. This work has been partly supported by the Grant-in-Aid for Scientific Researches of the Ministry of Education, Science, Sports, and Culture of Japan, No.08458109.
REFERENCES [1] H. Totsuji, T. Kishimoto, C. Totsuji, and T. Sasabe, Phys. Rev. E 58, 7831(1998). [2] H. Totsuji, T. Kishimoto, Y. Inoue, C. Totsuji, and S. Nara, Phys. Lett. A 221 215(1996). [3] T. Kishimoto, C. Totsuji, and H. Totsuji, Proc. 1996 Int. Conf. on Plasma Physicsj Nagoya, 1996 (Japan Society of Plasma Science and Nuclear Fusion Research, Nagoya, 1997) Vol. 2, p. 1974. [4] H. Totsuji, T. Kishimoto, and C. Totsuji, Advances in Dusty Plasmas^ eds. P. K. Shukla, D. A. Mendis, and T. Desai (World Scientific, Singapore, 1997), p. 377. [5] H. Totsuji, T. Kishimoto, and C. Totsuji, Phys. Rev. Lett. 78, 3113(1997). [6] H. Totsuji, T. Kishimoto, and C. Totsuji, Jpn. J. Appl. Phys. 36, 4980(1997). [7] H. Totsuji, T. Kishimoto, and C. Totsuji, Strongly Coupled Coulomb Systems^ eds. G. J. Kalman, K. Blagoev, and J. M. Rommel (Plenum, New York, 1998), p.l93. [8] V. M. Bedanov and F. M. Peeters, Phys. Rev. B 49, 2667(1994). [9] V. A. Schweigert and F. M. Peeters, Phys. Rev. B 51, 7700(1995). [10] W-T. Juan, Z-H. Huang, J-W. Hsu, Y-J. Lai, and Lin I, Phys. Rev. E 58, R6947(1998). and references cited therein.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
147
VORTEX CHAINS AND TRIPOLAR VORTICES IN DUSTY PLASMA FLOW J. Vranjes, G. Marie and P. K. Shukla^ Institute of Physics, P.O. Box 57, Yu-UOOl Belgrade, Yugoslavia E-mail: [email protected] ^Institut fiir Theoretische Physik, Fakultat fiir Physik und Astronomie, Ruhr Universitat Bochum, D-44780 Bochum, Germany
In this paper we investigate strongly nonlinear perturbations in an inhomogeneous, no = no(x), magnetized dusty plasma virith a macroscopic flow perpendicular to the magnetic field lines VQ = Vo{x)ey. In such systems instabilities of two-stream, or Kelvin-Helmholtz type can develop, and, in the strongly nonlinear limit in principle they can saturate into several types of vortex chain structures or tripolar vortices that are found in this paper. We study linear and strongly nonlinear, lowfrequency, d/dt
= 0,
+ ViV]vi = —(E
j = i, d, +
ViXBo),
(1) (2)
The quasi-neutrality condition, in the equilibrium and in perturbed state ni{x) = rieix) + ZdUdix), and the massless electron limit, yielding ne/ueQ = e^/T^ will also be used.
J. Vranjes et al/Vortex chains and tripolar vortices in dusty plasma flow
148
The instability of two-stream type can be easily demonstrated. We may assume a homogeneous, and quasineutral plasma nio = rieo + Zdndo, with spatially homogeneous, but different flows of ions and dust grains in the background of hot electrons. According to the assumption of a homogeneous magnetic field we have no currents here which means that VioCy = {Zdndo/niQ)vdQey. For the perturbations of the form ^ exp{—iujt + iky), we readily find the following linear dispersion equation: 1+ P
.21,2
.
1 - cik {P.kvdo-cjy-^f
/i/?
+ {Vdok-ujy-^nl
= 0,
(4)
where we use notation /? = Zdndo/rieo, A = Zdrido/nio, /i = Zdmi/rridj ^i = eBo/rrii, Qd = ZdeBo/rrid, c^ = Te/rrii, The singular, fourth order for a;, Eq. (4) is typical for the problems of this type and the instability conditions can be easily found. In a more appropriate nonlocal treatment, and in the case of a streaming plasma, from Eqs. (2), (3) for the perpendicular perturbation of the moving plasma components we find: ^0
±where the signs —, + stand for the ion and dust components, respectively. Using Eq. (5), and substracting two continuity equations (1) for ions and dust we obtain: d . I ^
ai + W^' ^ ^ ^ * • Vx ($ - ap2vi$) - ap2 ,dt Bo
..(x)g-Vi4-..(xr-
a$ + VQ{X)^a$ = 0. + — V • t/i,|| + P—V • (i?i,|| - tTd,!,) + v^e{x)—
(6)
The primes denote the the x-derivative of the corresponding variables and « = — (7r+ -?)—]' rieo Viii \ld /
^=Pr' \k
•"•e =
UeO
"eO-
For the parallel motion of ions and dust particles we have, respectively:
dvm
(
d
_ „ \
e„ ,
(7) (8)
149
J. Vranjes et al./Vortex chains and tripolar vortices in dusty plasma flow
For small magnetic shear effects V|| • v^^ ~ {dldz)vg ~ f{x){d/dy)vz, perturbations of the form $(x,y) = ^{x)exp{-iut
and for
+ iky), linearizing Eqs. (5) -
(8) one can obtain an eigenvalue problem equation describing the Kelvin-Helmholtz instability:
V
rridJ VfivQ - Vf)
VQ - Vf
$(x) = 0,
(9)
where Vf = uj/k^ and the spatial variables are in units of a^/^p. A necessary condition for the instability of the Kelvin-Helmholtz type can be found in the following manner. Assuming a complex frequency uj = Ur + iooi^ we multiply Eq. (9) by a complex-conjugate $* and integrate across the flow. Separating real and imaginary parts of the integral obtained in that way, we find that the latter one vanishes only if the following condition is satisfied at any position across the flow: v'l{x) + v.,{x) + ( l + /? + / 3 ^ ) ^
^
(10)
= 0
In the nonlinear limit we look for travehng solutions, moving with a contant velocity u along the y-axis. We write d/dt = —ud/dy and integrate Eqs. (7), (8): Vi^z = nif{x) + F{^ + BQ(p-uBox),
Vd,z = -nj{x)
+ G{^ + Bo(p-uBox),
(11)
where F{^), G(^) are arbitrary functions of the given arguments, and we introduced VQ{X) = {d/dx)(p. On condition of vanishing perturbations for x ~> cx), and taking the above functions as linear F{^) = Fi • ^, G{^) = Gi • ^ we obtain Vi^^ = ^i * ^j Vd,z = Gi • $, and Qif{x) + Fi • {Bo(p - UBQX) = 0, -^df{x)
+ Gi • {BQ(P - UBQX)
which yields Gi = —FiCld/^i- Using the same notation Eq. (6) can be written as:
di ^ B~^' ^ ^-^^^ ^ ^^"^^ * ^^ -Bo(¥^ + ^ ) ^ 7 / ( x ) ] = 0,
7 =
^1
(12)
v.. = dx'
Looking for traveling solutions as above, Eq. (12) can be integrated yielding ( V i - 1)$ +
(13)
150
J. Vranjes et al./Vortex chains and tripolar vortices in dusty plasma flew
where we introduced the following normahzation: VVj.
-^ V j . ,
$,
$n
"• ^-*' t^^' y/oLfmB^
$n
= $0-
For Zd-^ 0 Eq. (13) becomes identical to our earlier derived equation in Ref. [2]. Tripolar vortex solution can be constructed by adopting that the functions (p{x), *(x), and f{x) satisfy (p{x) - x = aix^, (p"{x) + *(x) + f{x) = a2X^. Further, Eq. (13) is solved separately inside and outside of a circle with an arbitrary radius TQ, choosing the function J{^) in the linear form, i.e., J{^+aiX^) = Jo+Ji'{^+aiX^), and allowing for different values of Jb? Ji inside and outside the circle with the radius TQ, and on condition of localized solutions for r —^ oo. We use the following physically justified continuity conditions at r = TQ: continuity of the potential $, and the derivative (9/9r)$, and the continuity of the function J{^) together with the assumption that its argument is constant at r = TQ. Analytical tripolar solution of Eq. (13) in the cylindrical coordinates r = (x^ + x/^)^/^ and 9 = arctan(y/x), can be written in the form $(r,^) = boKo{f^ir) + b2K2{Kir)cos2e,
(14)
r > ro,
and $(r,0) = aoJo{f^2r) -a—
-b + 0^2^2(^20 "" ^
cos 20,
r < ro-
(15)
Here, JQ^^ = 0, J^^ = 02/01, and Ko^2, Jofi axe modified Bessel and Bessel functions of the given order, respectively, and we introduced the notations 1^2 _ 1 , ^out _ 1 , ^2
Ki = i+Ji
= H —Oi,
2 _
1
^in
/C2 = — 1—Ji
,
^ _ ^
o = OiH
01 + 02 K>2n—,'
0 =
-I(f-)-
The other unknown constants in Eqs (14), (15), and nonlinear dispersion equation, can be found from the boundary conditions listed above, yielding: ai-arl
ao
,
1 oirg
(16)
( J„K; + 1 j;/r.) = „ [ ( I - ^ ) K;+r„ I jifo bo\JoK;, +
«
-^j;,Koj=a
^ ^ 'i'^J'^^-'"^'
- ^ Jo,
(18)
J. Vranjes et al/Vortex chains and tripolar vortices in dusty plasma flow j,2
151
jf
j^i
b = aoJo - 60^0 - « if J «i^o T;^ = («i - a) ^0 ^ + 2a.
(19)
Choosing ai, a2, and ro, from Eqs. (16)-(19) we may find the other unknown constants. Vortex chain solutions of Eq. (13), for a dusty plasma described in the basic state in the following way ^(x) + f{x) = 0 can be found numerically by choosing the function J{^) in the form: (20)
J{^ + (p-x) = CA^exp - 7 ^ ( * + ¥?~x) LO Now we may rewrite Eq. (13) in the form:
(21)
( V i - 1)$ - T(x) ( e - ^ - 1) = 0, where
SA^
(22) JF(x) = g2Ax + g-2Ax + 2 ' Numerical, well localized (in the x-direction) solutions of Eq. (21) can be written in the form of a nonlinearly generated potential $(x) and a wave like perturbation, i. e., $(x,z/) = ^i{x) + 6^x)cos{ky),
\S^x)\ «
|$i(x)|.
(23)
More detailes can be found in Ref. [3]. An additional, analytical, vortex chain solution can be found by choosing ^(x) + /(x) = -(^(x) + x.
(24)
and
J{0 = -^ +
^expL2^^
In this case the solution of Eq. (13) can be readily written in the form: 1 $(x,j/) = X - (p{x) + Clog2 cosh(/ux) + y 1 ~" T^ cos{ky)
(25)
On condition
^hm^
(26)
Eq. (25) represents a row of vortices localized in the x direction and periodic along the z/-axis.
152
J. Vranjes et al/Vortex chains and tripolar vortices in dusty plasma flow
To conclude, we have derived nonlinear equations describing modes in a spatially nonuniform magnetized dusty plasma consisting of electrons, ions and dust component and in the frequency domain of drift waves. The macroscopic plasma flow in the basic state can serve as a source of free energy for the development of linear instability of streaming type. In the strongly nonlinear regime we find a new type of coherent solution in the form of a tripolar vortex, structure settled in the plasma flow and carried in the direction of it. By its form it resembles well known structures obtained recently in experiments with rotating fluids and observed also in the seas of our planet. The vortex chain solutions discovered here are of the same form as the ones found in our recent paper [3]. The structures of this type are known to exist in some laboratory and space plasma configurations. In tokamaks they can form a transport barier in the edge plasma region during transition from L to H regime.
References [1] S. V. Lakshmi, R. Bharuthram, M. Y. Yu, Astrophys. Space Sci 209, 71 (1993); Y. Chen, M. Y. Yu, Phys. Lett. A 185, 475 (1994). [2] J. Vranjes, D. Jovanovic, P. K. Shukla, Phys. Plasmas 5, 4300 (1998). [3] J. Vranjes, Phys. Scripta 59, 230 (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
153
Dynamical structure factor of dusty plasmas including collisions A. Wierling, V.J. Bednarek, G. Ropke Universitdt Rostock, FB Physik, Universitdtsplatz 3, D-18051 Rostock, Germany
A b s t r a c t . Based on a generalized linear response theory, an expression for the dielectric function of dusty plasmas is derived. It is shown, that the approach allows for a systematic inclusion of collisions. Results for the dynamic structure factor at moderate values of the nonideality parameter and for three-dimensional systems are presented. We focus on the impact of collisions on the dispersion relation of dust acoustic waves.
I
INTRODUCTION
Recently, the study of lov^ frequency modes in dusty plasmas has attracted much attention. Because of the high charge of the macro-particles, interaction effects play a dominant role, leading to a solid-like state with long range order. Under these conditions, dust lattice v^aves (DLW) are expected to occur as excitations of the ordered crystal-like structure [1]. Indications for the existence of DLW has been reported by [2]. Hov^ever, some of the experimental results have been attributed to the propagation of dust acoustic waves [3-5], which have been predicted as modes of a dusty plasma by Rao et al. [6]. This is a striking phenomenon due to the fact, that the plasma was found in a solid-like state, the dispersion relation of the collective excitation on the other hand was fitted by an approximation suitable for a weakly coupled plasma. There are several theoretical approaches improving the dispersion relation presented by Pieper at al. [3]. These are based on generalized hydrodynamics [7,8], the quasi-localized charge approximation [9] or the concept of static local field corrections [10]. While a great deal of attention is paid to static correlations, collisions are introduced by very simple arguments such as the relaxation time approximation neglecting any further frequency dependence [11]. In this article we want to focus on the dust dynamic structure factor. We will present a systematic microscopic approach based on a generalized linear response theory. Within this approach, the structure factor is given as a ratio of frequencyand momentum-dependent correlation functions allowing to describe collisions beyond the relaxation time approximation.
154
A. Wierling et al /Dynamical structure factor of dusty plasmas including collisions
II
GENERALIZED LINEAR RESPONSE THEORY
To describe a dusty plasma, we consider a system consisting of the four different species c, dust grains , neutral gas atoms, ions, and electrons, denoted d, n, z, e, respectively. The corresponding properties are the densities ric, masses rric, charges ec, and the temperature T = (A;B/?)~^. Electrical neutrality gives Y^c^c^c = 0We are interested in the dynamical properties of the system, in particular the time dependent correlation function of density fluctuations Sud = n^— < Ud > of the dust particles, what is measured in laboratory experiments. Also other combinations of species are possible. In thermal equilibrium, characterized by po? it is given by L^\T, t) = Tr[po (Jn,(f, i)(Jn,(0,0)]
(1)
and depends only on the differences in time and space. This quantity contains information about collective excitations and their damping. The Fourier transform is the dynamical structure factor -*
x""ik,u)=
rcx)
dte^'{ni{t)ni{0))
,
(2)
where nf{t) = fd^re^^'^Srhdif^t) denotes the Fourier transformed density fluctuation. In order to present an approach applicable to a wide range of systems we will use a quantum description, but most of the calculations will be taken in the classical limit. We introduce correlation functions as
{A; B) = (5+; A+) = ^ J^r Tr [Ai-ihT)B+po] , roo
{A;B),=
Jo
dte^^'{A{t);B),
(3)
with A{t) = exp{iHt/h)Aexp{-iHt/n) and A = i[H,A] . The averages are performed with the equilibrium statistical operator po = exip{—f3H + PEcl^cNc) /Trexipi^PH + pEcf^cNc)^ Making use of the conservation of the current Jjg = ^ ^ ^ JlpPz'i^p^k ? we can relate the dynamical structure factor to the current-current correlation function [12] X^'iKcj) = {nthi)^^i,
= -(3no-{4;4).^i,
(4)
with k is directed in e^ and QQ denotes the system volume. Furthermore, the Wigner transformed single particle distribution is given by rip^ = ^t,p-k/2^c,p-{-k/2, where ac,p, a'^^p are the annihilation and creation operators of specie c, respectively. Let us consider a set of observables P = { P i , . . . ,Piv}. According to [13] we can cast the current-current correlation in a form more suitable for a perturbation approach
A. Wierling et al /Dynamical structure factor of dusty plasmas including collisions
X^mt
0 Nln
^n/z
Nmk Mik
155
(5)
/\Mlk\
with Nmn = (Pm; Pn) and Mmn = -iz{Pm;
0
Pn) " {Pm; Pn) + {Pm; Pn)z +
{Pm;Pj).
l\{Pi;Po)z\- (6)
Here, we take as observables Pm the density fluctuations n% or the current densities J^. The correlation functions (3) can be related to Green functions and evaluated within perturbation theory, using diagram techniques. Due to the long range of Coulomb interactions, screening is of great importance. It can be described by summing ring diagrams. As well known from Coulomb systems [14], the sum over ring diagrams can be described by introducing the polarization function Y[{q^uS) and replacing the Coulomb interaction V{q) = l/{cjQeoq'^) by V'{q,u) =
V{q) l-V{q)U{q,u;)
(7)
As a consequence, only irreducible contributions to the correlation functions arising in (4), (5), and (6) have to be taken into account which do not disintegrate cutting only one Coulomb interaction line. We find X''{k,u)
=
U'^{k,u}) l-V{q)n{q,co)
(8)
•
The polarization function is given by the sum of the different combinations H{k,u)
= U^{k,u) + Y^eaeiW{k,u)
.
(9)
ah
Here, we neglect collisions between the electrons and the other components of the dusty plasma. We have (J|; Jj^) = 0 and {J^\Jk) = nc/{mcPO,Q). Neglecting collisions at all, the RPA result is reproduced. If collisions are included in Born approximation, we have to evaluate the force-force correlation function (J^; Jk)u-^iri' Considering the interaction between the dust particles and the neutrals, the result for the collision term in the low-frequency, long-wavelength limit reads
{jt; Jtr^s, = 7r/i '^Evin{Q)fifrs{E'p+,+Et, - E'^ - Er)^ ^^0 Ipq
^
2 '
(10)
where fp denotes the Boltzmann distribution function an Vqn an auxiliary dustneutral interaction potential. We do not calculate this force-force correlation function but rather use the result of Baines et al. [15], assigning the notation z/^n for this expression. Furthermore, collisions between ions and dust grains occur. These
156
A. Wierling et al. /Dynamical structure factor of dusty plasmas including collisions
collisions can be treated within a T-matrix approximation with respect to a static screened potential, leading to the Spitzer formula for the conductivity of plasmas [17]. In the long-wavelength limit ( fc -> 0), the current-current correlation functions differ in the mass factors and read I fdi, fduirred _
^
^d^i
''''id
^
^Hur
T
(^^\
(^o,^o)a,+i.-j^^;^(4^^(^^ypV27r-LsP
,
(11)
1 e5e? "Hd mj/' 1 ^ 4 ^d^i ^ ^/O^ ^ T {Jo , Jo Uir, - ^ ^ ^ , (4^^^)2 ^ 2 Jj^^jyT^ ^ 2 ^ 3 ^ ' ^
'
^^^^
/ jid,
Tzdxirred
I fdi, Tdzxirred __
^
{Jo , Jo Uin - ^^^^
^d^i
'^id
./o^
^
(4^,^)2 ^ ^ ^ ^ J^^ffF^
t^0\
T
V27r- LSP
(A'W
,
(13)
where Lsp = ln(3/2 F"^), F symbolizes the coupling parameter and m^d the reduced mass of the ion-dust system. As a short hand notation, we introduce corresponding collision frequencies Vahcd- The expression without the mass factor will be denoted v^i. This result can be compared with the collision frequency given in [16]. However, within our approach, the determination of these correlation functions at finite values of the wavenumber is also possible. In lowest order with respect to the interaction the current-current correlation functions can be calculated analytically [12] and the RPA expression for the dynamical structure factor is reproduced. Taking into consideration also collisions between ions and neutral atoms by a collision frequency z/^n, we obtain the following expression M,,(fc,a;) = ( 4 ; 4 ) : ^ , , [ ( 4 ; Jlf Mid{k,u)
= iyiddi , Mdi{k,u)
MddiK^) = {Jt Ji)Zli,
+ ( 4 ; 4 ) . + i , {v,,^ + ^in)]
,
= UdUd ,
[{Jk'. 4f
+ {Jt J'kUi,
iydidi + vdn^
•
Using Eq. (5), we get an expression for the polarization function 11^^ in terms of the matrix elements M^'^'.
Ill
DISPERSION RELATION OF DUST ACOUSTIC WAVES
Prom the different polarization functions we can determine the dispersion 0 = 1 — V{q) 11(^,0;). The large difference in mass of the ions and the macro-particles leads to a different behavior of the dimension-less parameter Zg. We study the following approximation. While we treat the ions and electrons in the static limit Ze.Zi <^1 the dust particle are treated in the long-wavelength limit Zd^l. In this limit, the zeroth order contribution to the current-current correlation simplifies {Ji;Ji)^+ir, = -iconi{enoy'
,
(14)
{JtJth+ir,
.
(15)
= ini{uQopmd)-'
A. Wierling et al. /Dynamical structure factor of dusty plasmas including collisions
157
Vdi=Vi„=0.2 C0p„
co/co,'pl.D
F I G U R E 1. Dispersion relation of dust acoustic waves. The experimental data are taken from [3] as well as the calculated dispersion according to Eq. (17). A dispersion relation accounting for dust-ion i/di and ion-neutral collisions ui^^ given by Eq. (16), is shown as well.
Finally, we arrive at the following dispersion relation 1 = zn^ -
^ 2 / - -
- -
i {ydi + ^dn) - %
k^
- - \
(16)
[^di + ^dn + q^^di + q^^in + 29Z/di] + % q^ {Udi + ^in) , A:2
given in dimensionless quantities k = k/K,u) = u)/u)p\^d and u = v/uo^i^d-, q = ^d'^dli^i'^i)- We first consider a plasma dominated by dust-neutral collisions, Vid = 9in = 0. Then, Eq. (16) reduces to the one used by Pieper and Goree, 1 + A; '^ = (u'^ + iVdn^^^
(17)
For finite values of the collision frequencies the dispersion can only be obtained by numerics. Results for Vid = Vm = 0.2 are shown in Fig. 1 using the value for Udn of Pieper. It is found that the inclusion of additional collisions leads to an increase in Rek as well as Imk. For high values of the frequency (a) > 1.5), the assumption of the long-wavelength limit is not correct any more. Instead, a more detailed calculation taking into account the ^-dependence of the collision integrals has to be performed, which is also feasible.
158
A. Wierling et al /Dynamical structure factor of dusty plasmas including collisions
IV
CONCLUSIONS
Based on a many particle approach the dynamical structure factor for a dusty plasma has been calculated taking into account collective effects as well as collisions. This was achieved using a consistent perturbation expansion based on a generalized linear response theory. The heart of our treatment is the systematic determination of the force-force correlation function which in turn leads to expressions beyond the relaxation time approximation. Making use of the structure factor the dispersion relation for collective modes in dusty plasmas can be derived. In particular, we study the dispersion for dust acoustic waves and obtain Eq. (16) as a result, taking into account dust-ion, ionneutral and dust-neutral collisions. In particular, for dust-neutral collisions the result of Pieper and Goree was re-derived. Since the proposed treatment is valid in the entire {k^uS) plane, improved dispersion relations can be obtained be higher order expansion beyond the static or long-wavelength limit. In this way, diffusion and/or the dispersion of the plasmon can be included.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
F. Melaiids0, Phys. Plasmas 3, 3890 (1996) A. Homann, A. Melzer, S. Peters, A. Piel, Phys. Rev. E 56, 7138 (1997) J.B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1997) A. Baxkan, R.L. Merlino, and N. D'Angelo, Phys. Plasmas 2, 3563 (1995) R.L. Merlino, A. Barkan, C. Thompson, and N. D'Angelo, Phys. Plasmas 5, 1607 (1998) N.N. Rao, P.K. Shukla, and M.Y. Yu, Planet. Space Sci, 38, 543 (1990) X. Wang and A. Bhattacharjee, Phys. Plasmas 4, 3759 (1997) P.K. Kaw and A. Sen, Phys. Plasmas 5, 3552 (1998) M. Rosenberg and G. Kalman, Phys. Rev. E 56, 7166 (1997) M.S. Murillo, Phys. Plasmas 5, 3116 (1998) A.V. Ivlev, D. Samsonov, J. Goree, G. MorfiU, V.E. Fortov, Phys. Plasmas 6, 741 (1999) G. Ropke, Phys. Rev. E 57, 4673 (1998), G. Ropke and A.Wierling, Phys. Rev. E 57, 7075 (1998) D.N. Zubarev, V. Morozov, and G. Ropke, Statistical Mechanics of Nonequilibrium Processes (Wiley-VCH, Berlin, 1997) L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics (Addison-Wesley, Redwood, 1989) M.J. Baines, LP. Williams, and A.S. Asebiomo, Mon. Not. R. Astron. Soc. 130, 63 (1965) T.K. Aslaksen, O. Havnes, J. Plasma Physics 51, 271 (1994) R. Redmer, Phys. Reports 282, 35 (1997)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
159
Melting of the defect dust crystal in a rf discharge I.V Schweigert\ V.A Schweigert^, A Melzer\ and A. PieP ^Institute of Semiconductor Physics, Novosibirsk, Russia, [email protected] ^Institute Theoretical and Applied Mechanics, Novosibirsk, Russia ^ Institut fur Experimentelle und Angewandt Physik, Christian-Albrechts-Universitat, 24098 Kiel, Germany Introduction Recently a lot of experimental and theoretical studies are devoted to phenomena of the Wigner crystallization of dust particles in the sheath of a rf discharge. The experimental observations of the dust crystal indicate that a Debye-Huckel model of isotropic interparticle interaction breaks down in moving plasma. The dust crystal in the sheath exhibits a specific crystalline structure in which the negatively charged particles are arranged in vertical chains, whereas in the horizontal plane the particles form the usual hexagonal lattices. This is not the most energetically favorite structure for the Coulomb (screened) isotropic potentials. The ion flux flowing through the sheath is focused by negatively charged particles that leads to formation of areas of enhanced ion density behind upperstream particles These areas provide the attractive force for lower particles and explain the vertical alignment of particles. The phenomena of particle attraction in ion flux was considered in [1] within a coUisionless approach and in [2], using a fluid approximation The trajectories of ions flowing through the sheath containing the bilayer crystal were calculated in [3] by the Monte-Carlo technique. It was established that the inter-particle interaction is anysotropic due to asymmetric screening of particles into ion flux. Model of the dust crystal in the sheath In order to study the dynamics of the dust crystal we developed the semi-empirical model with asymmetric inter-particle interaction [3]. In this model the inhomogeneous ion density around a particle is replaced by the isotropic ion distribution characterized by the screening length and an effective positive charge placed below a particle. The value of effective positive charge Zc and its coordinates for experimental conditions of Ref [4] were found in [3] in non-self consistent Monte-Carlo calculations. Late selfconsistent calculations of the dust bilayer crystal in the sheath were carried out with using 3D PIC MCC method (Three Dimensional Particle In Cell Monte Carlo Collisions) [5] It should be noted that 3D PIC MCC algorithm allows to obtain all the main parameters of the plasma-crystal system: a particle charge, the ion distribution, the inter-layer and the inter-particle distances. Besides, these calculations are very time consuming and it is not possible to apply this technique to dynamics problems such as melting or wave propagation. The important point of our semi-empirical model is that positive charges Zc are rigidly connected to their parent upperstream particles. In Fig. 1 the ion distribution is shown for the case when particles are in equilibrium position and
/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge
160
when the lower particle is shifted, ( see Ref. [5]). The ion distribution fast relaxes 2 0 0
2 0 0
H 150 H 100
H 50
3000
3200
3400
3600
3000
3200
3400
3600
z, miorons o
-3
Fig. 1 Ion density distribution (in units of 10 cm") for shifts 5x of the lower layer relative to upper one along the x axis: (a) 5x=0, (b) 5x=130 \im\ z=0 - cathode location. around a parent particle while its moving. It is apparent from Fig. 1 that the ion distribution around the upperstream particle does not depend on lowerstream particle position. We calculated forces acting on the particles into ion flux as function of their shifting and found coordinates and a value of positive charges, which fits well the attractive force acting on a particle in the horizontal plane. Fig. 2. 100
1o J 1
0,1
-0,08
0,1
0,2
0,3
X / e
0,4
Fig. 2 Restoring force acting on the particle in ion flux as fiinction of particle shifting for different ion free paths: 50|im(l), 100|im (2) 200|im(3).
0,10
0,15
0,20
0,01
Fig. 3 Particle kinetic energy in the upper layer (triangles), in the lower layer (squares) and in experiment (circles). The vertical dashed lines shows the instability (from linear analysis) and melting threshold.
Thus, the model dust crystal consists in the hexagonal lattices of negatively charged
/. V Schweigert et al. /Melting of the defect dust crystal in a rf discharge
161
particles and lattices of effective positive charges between them. The effective charges are rigidly connected to upperstream particles. The particles and effective charges move in the horizontal plane ( the experiment [4]). Melting transition. Linear analysis. The experimental observations [4,6] showed that the melting transition of dust crystals in the sheath is a non-ordinary process. The decrease of gas pressure causes rise of the particle kinetic energy Ek, even though the surrounding gas remains of the room temperature. What initiates the particle heating? The mechanism of particle kinetic energy growth was explained in [3] The linear analysis of particle motion equations has demonstrated that the system is unstable relative to short wave oscillations at gas friction less than some critical value. We have obtained the eigen vectors and eigen values of the dynamics matrix which are complex. It was established that with decreasing gas friction the unstable modes arise in the crystal that leads to substantial increase of Ek The sheath of a rf discharge containing the dust crystal is an open system and additional input of energy is provided by ion flux. The simulation input parameters were taken from the experiment [4]. The dust particle radius is R= 4.7|j-m corresponding to a dust mass M = 6.73-10"^"^ kg. The dust charge Z=16000 elementary charges, the interparticle distances a=450|j,m and the interlayer distances d=0.8a. The corresponding dust plasma frequency for the experimental conditions is (Op = (Z e /so M a ) =110 s" . The transition from the solid crystal structure to the gas-like state in the experiment is observed by reducing the pressure from 120Pa down to 40Pa corresponding to an Epstein dust-neutral friction coefficient n = 32s"^ down to lOs'V The critical friction of instability was found to be «=0.1575a)p. Applying the fluid approach, in [7] it was found also that the particle temperature rises at some critical gas viscosity. Melting transition. Molecular dynamics calculations. To understand the crystal melting in details we have performed the molecular dynamics simulations of particle behavior with decreasing pressure [8]. The asymmetry of interparticle interaction was described on the base of our model above. The simulation parameters also correspond to the experimental data [4]. We found that the particle heating is a two step process. In the crystalline state at higher gas friction the particles perform harmonic oscillations about equilibrium sites. Reducing gas pressure, we reach the first critical point of friction n=nins and the particle energy Ek increases quickly by orders of magnitude. Fig. 3. The crystal evolves to the state with developed oscillations, but the crystalline structure survives. Further with decreasing pressure at second critical point of friction n= «^^/the crystal transits to the isotropic liquid. Fig. 3. The melting is accompanied with a jump of the energy Ek- It was surprising that the parameter r = Z^/akfiT, (where Z is the particle charge and a is the inter-particle distance) for the 'hot' crystalline state was essentially smaller than the critical melting F of a single layer crystal with the same screened inter-particle interaction. The vertically aligned crystal in ion flux turned out to be more stable relative to melting than a single layer crystal. The particle velocity distribution function (PVDF) displays features of particle motion in the different regimes In crystalline regime PVDF is Maxwellian, Fig. 4(a).
162
/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge
0 , 0 0
0,0-1
eo
0 , 0 ^
0 , 0 0
n/^' = 0 . 1 e s
0 , 0 ^
0 , 0 s
0,1:2
i^/w„=0. 1 e i
2 3
# Ul
-40
1 O
2 0 (si)
Cb)
\
.^Wp=0. 1 Si
^/w
= 0 . 1
^ S
i -^
Cci>
Co)
0.0
O, 1
0 , 0
0,2
0 , 1
Fig. 4 Particle velocity distribution function at various frictions. The dashed curves are a Maxwellian distribution (a,d) and the distribution of harmonic oscillator (b). 2 0 0
i/w.. = 0 . 1
STe 1 SO
«/wp=o.:^i
Ni o 2 0
CQ>
i:k/w_=0. 1 : 2 s
Ck^) It/W_=0. 1
_..^
fthhmAtA
1
Cci>
Co) •v^V.,v^
1 .0
0 . 0
O.S
1
.0
>3VX'VX.''
Fig. 5 Spectrum of particle velocity autocorrelation function at various friction constants. The dashed curve (a) refers to a single layer crystal.
/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge
163
At n=nins which correlates with arising instability PVDF corresponds to the harmonic particle oscillations, Fig. 4(b). At lower gas friction the distribution becomes broader. Fig. 4(c), and eventually after melting PVDF looks like Maxwellian, Fig. 4(d). We calculated the particle velocity autocorrelation ftinction and, applying the Fourier transformation, derived the spectra of excited phonons at different gas pressures. At the crystalline state the dust crystal has usual spectrum for 2D crystals. Fig. 5(a). At «=««,* the spectrum has sharp peak shape. Fig. 5(b). A few modes with frequencies about the dust crystal frequency (Dp are excited. This is the reason of enhanced stability of the dust crystal, since the high frequency modes increase only particle energy ER. whereas the long wave oscillations cause the crystal to melt. In the liquid state the structure exhibits the reach phonon spectrum including zero mode. Fig. 5(d). We have explained heating of the dust crystal in the sheath by developing self-excited oscillations arising with decreasing pressure. Note, that the calculated Ek agrees with the measured one within an order of magnitude. However the experiment displays a more complex picture of crystal melting With decreasing pressure first some 'hot' spots are created and then streamline motion develops around heated parts of the crystal. Melting of the dust cluster with defects. In the case of simulation of the perfect crystal the transition to the 'hot' crystalline regime is accompanied with quick rise of Ek within a narrow interval of the pressure. In contrast to calculated data, in the experiment the particle temperature enlarges smoothly. We supposed that defects which are always present in the crystal even far from critical temperature are responsible for continuous melting. Two kinds of defects are considered. The first type of defects is point defects (vacancies and interstitials) and dislocations which are typical for the 2D systems and destroy the translational and the angular order. The second type of defects is additional particles surrounding the dust crystal. We study the melting transition of the defect bilayer crystal with asymmetric inter-particle interaction, using the Langevin molecular dynamics simulations. The crystal consists of 996 particles and in the crystal phase the particles and the positive charges are arranged into two parallel almost hexagonal lattices. In our earlier works [3,8] we have considered an infinite in the horizontal plane bilayer crystal. We took the fragment of the crystal and used the periodic boundary conditions. Here we take a finite cluster in order to create point defects and dislocations. The main reason of defect formation in the crystal phase is the presence of the external boundary of the system. The particle motion is described by the following equation —-i-= — F -n—- + —Fj dtM ' dt M ^ M
WU(r).
where ri==(xi^H-yi^)^^, Xi, yi are the transverse coordinates of the ith particle, M the mass of a particle, Fi is the electrical force, n the friction constant and F\ is the Langevin force which refers to the room temperature of gas. The electrical force includes interparticle repulsion and attraction between particles and positive charges. U is the external confining potential in the horizontal plane. Note, that each particle does not interact
/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge
164
with its downstream positive charge which mimics the asymmetric ion distribution around a particle, but interacts with all other particles and positive charges. The electrical force acting on the particle can be written as
F,=Z'Y^ ^
'^\e^'^
r. - r.. I"'
(1 + k
:, I)
ZZ^
^!-~'"\e-' r-^^A^l^k\r-r„\) 13
r - r^
where \/k is the screening length, j,n denote the summation over particle layers and layers of charges, respectively, j = 2, n = 2 for the crystal with point and extended defects, j = 4, n = 4 in the case of additional particle defects. The coordinates of charges ^-^/: where dc the distance between positive charge and the lower layer in the vertical direction and k = 2/a [9]. In this simulation, the positive charge is taken to be Zc= 0.5Z with a vertical distance dc = 0.6a [3]. Vacancies, interstiteals and dislocations are formed during slow numerical cooling of the system from the high temperature liquid state. Initially the particles are placed randomly within area r
L I 1 T 1T IJT
10h 5h
1 1 1 11 1 1 1 1 • 1 • ' ' ' 1 j
E (eV)
hcb •
H T 10 A
•,V'-.'vi •'•-•' \i. vi.' \.*x * J. * J. vi. *X* 1 * iVj »Tt"rvr i;«7. \^\ > YVT < I O" »J«I *'!- * "L * V»""!"«"J •^ Cri/jxi.*I*iM'«'i"»i'TV'r_ • T i l * ' i V f ' i ' r » ' i » T » " i *'' * i''
•^•:/.v;-:^';-/
t
L It
^I».]>rrr..v.ivi.r
\ #1
-10 -10
-5
0
10
K>\ o
4i
1
J
•
\m
o
•
L *^ V '
*
'
'
*
•
^1 A
•
•
J
•
•
\
o
o^
^0,1
5
o oi
•
1
•
•
'
'
l i l t . . I . 1 . _L l_ 1
0,10 0,15 0,20 0,25 n/w Fig. 6 Particle trajectories in the crystal with point defects and dislocations (I-type defects) at «=0.0721a)p. (Top view)
]]
0,30
Fig.7 Kinetic energy of particles as ftinction of friction for the crystals with I-type defects (solid circles), with extra particles (open circles), and the experiment data (squares).
/. V Schweigert et al /Melting of the defect dust crystal in a rf discharge
165
It contains uncorrelated dislocations with Burgers vector equal to 2. The crystal has also a lot of defects near the external boundary, since the hexagonal structure can be fitted in a circle only with the certain number of defects. The second layer is supposed to be of the same structure. First at given gas friction we allows the system to achieve quasi-equilibrium state during 2-10"^ ^6-10*^ MD steps in which the particle temperature stays constant while calculation. The mehing scenario of the bilayer crystal with first type defects turns out similar to a non-defect crystal melting described above. The change of ER with decreasing pressure is shown in Fig. 7 (solid circles). At gas friction n < «ins=0.105(Dp the system exhibits the crystalline structure, Fig. 7. Note, that even at friction n ^ Wins, the particle trajectories do not display some marks of occurrence of dislocations. Fig. 6. At fi = Wins an instability begins to develop. Within friction interval nins< n
(a).
' *• ' .
•
[
>
'
:
-
'
*
•
'
•
•
•
-
•
•
-
'
•
•
•
•
•
•
'
* % '^ *
*^ . ^ -•. ' . ^ . ' . •. ' . •* - - . - '
Fig. 8 Particle trajectories in the crystal with extra particles for (a) n=0.15 ©p, Ek=10eV, (b) «=0.14 Qp, Ek=12eV, (c) «=0,13 ©p, Ek-14 5eV, (d) «=0.12 ©p, Ek=17eV.
166
I.V Schweigert et al /Melting of the defect dust crystal in a rf discharge
transition takes place. As followed from our results the occurrence of points defects and dislocations does not make the substantial contribution to the particle heating. Let us consider melting of the bilayer crystal with extra particles defects. The additional particles are placed randomly above and below with distance a from the bilayer crystal (added to the crystal shown in Fig. 6). The number of these particles equals 45 or 5% of the total number of particles. Our calculation showed that the heating evolution of the bilayer crystal with extra particles has interesting features. At sufficiently high friction the particles perform harmonic oscillations near their equilibrium sites with small amplitude, and, however, the positions of additional particles are clear seen. In Fig. 8(a) the particle trajectories of the upperstream layer of the bilayer crystal is plotted. The enhanced particle motion areas point out the locations of the additional particle placed above the bilayer crystal. At lower friction the bilayer crystal exhibits fragments of streamline motion and also crystalline regions. The crystal begins to melt locally and liquid fragments appear first under additional upper particles. Fig. 8(b-d).The occurrence of additional particles modifies the heating dynamics and leads to a visible increase of ER, Fig. 7 (open circles). The instability relative to short wave oscillations appeares also at higher gas friction. Now two step melting of the bilayer crystal with the asymmetric inter-particle interaction is smeared out by the local heating and it is difficult to identify the melting point. The excitation spectra of the bilayer crystal with extra particles have principal features. In the crystalline state the spectrum has a second sharp peak about © = 0.6(Dp. which corresponds to high-frequency phonons excited by the additional particles placed above the crystal. In conclusion we have developed the new model of asymmetric inter-particle interaction in the sheath of a rf discharge. Using this model linear analytic analysis and MD simulations of the dust crystal melting were carried out. It was shown that an increase of the dust particle kinetic energy with decreasing gas pressure is explained by instability related to excitation of short wave oscillations. The coexistence of liquid and crystalline fragments in the bilayer crystal observed in the experiments is explained by influence of additional particles situated above the crystal, whereas point defects and dislocations do not practically affect the particle temperature. [1] S.V. VladimirovandM. Nambu, Phys. Rev.E 52,2172(1995). [2] F. Melandso and J. Goree, Phys. Rev.E 52, 5312 (1995). [3] V.A. Schweigert, I.V. Schweigert, A.Melzer, A. Homann, and A.Piel, Phys. Rev.E 54, 4155 (1996); A. Melzer, V.A. Schweigert, I.V. Schweigert, A.Homann, and A.Piel, Phys. Rev.E 54, 46(1996). [4] A. Melzer, A. Homann, and A. Piel, Phys. Rev.E 53, 2757 (1996). [5] V.A. Schweigert, V. Bedanov, I.V. Schweigert, A.Melzer, and A.Piel, JETP 88, 905 (1999). [6] H. Thomas and G.E. MorfiU, Nature 379, 806(1996). [7] F. Melandso and J. Goree, J.Vac. Sci. Technol.A 14, 511 (1996). [8] V.A. Schweigert, I.V. Schweigert, A.Melzer, A. Homann, and A.Piel, Phys. Rev. Lett. 80, 5345 (1998); I.V. Schweigert, V.A. Schweigert, A.Melzer, and A.Piel, JETP 87,905(1998). [9] A. Homann et al., Phys. Rev.E 56, 7138 (1997); A. Homann, A. Melzer, R. Madani, and A. Piel, Phys. Lett.A 242, 173 (1998).
Part III. Industrial Applications
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
l69
On the powder formation in industrial reactive RF plasmas Ch. HoUenstein, Ch. Deschenaux, D. Magni, F. Grangeon, A. Affolter, A.A Howling P. Fayet Centre de Recherches en Physique des Plasmas Ecole Polytechnique Federale de Lausanne CH-1015 Lausanne Switzerland Tetra Pak (Suisse) SA 1680 Romont Switzerland Abstract. Particle formation in reactive plasmas plays an important role in industrial plasma processes. In the present study, inia-ared absorption spectroscopy, emission spectroscopy and mass spectrometry have been simultaneously applied to dusty organosilicon and hydrocarbon plasmas. Powder formation and its dependence on important process parameters in methane, acetylene, ethylene and m helium/oxygen diluted hexamethyldisiloxane (HMDSO) gas mixtures have been investigated. The influence of the chemical structure of the monomer gas on the neutral and charged plasma species and on possible polymerization reactions leading to powder precursors is discussed. In addition, cavity ring down has been applied to these different dusty plasmas in order to investigate in detail the powder precursors, their origin and then- consequences.
INTRODUCTION Powder or particles are found in deposition and in etching plasmas. However, the origin of these particles might be of quite different nature. Powder formation in plasmas used for film deposition, in particular (diluted) silane RF plasmas, have been intensively investigated during the last few years (1). The different stages of the powder formation such as the nature of the powder precursors, the agglomeration and accretion phases have been identified and investigated experimentally and theoretically. Besides this very prominent reactive plasma, many other plasma processes employed in industry show powder formation. In most cases, the powder formation leads to problems either in the quality of the film or to process interruptions due to prolonged maintenance and cleaning of the reactor. Hexamethyldisiloxane (HMDSO)/helium/oxygen plasmas, for example, are used for the deposition of silicon dioxide as a permeation barrier in the packing industry. Nano- to micrometer-sized Si02 particles are also formed in these plasmas (2). Another important category of plasma where strong powder formation can be observed are hydrocarbon plasmas. These plasmas find wide applications in plasma polymerization and in the production of hard carbon coatings. Various diagnostics have been applied to investigate the powder formation. These include in-situ particle diagnostics and diagnostics covering the plasma composition and parameters (3). In the present study, mass spectrometry, infrared absorption
170
Ch. Hollenstein et al/On the powder formation in industrial reactive rf plasmas
spectroscopy and emission spectroscopy have been applied to these plasmas in order to elucidate the origin and the different phases of the powder development and powder composition as a function of various external parameters and monomer types. From the point of view of the production of nanoparticle-seeded coatings (4), diagnostic methods measuring nanometer-sized particle and their density are needed. Besides exotic methods such as the laser explosion technique, no in-situ methods are available to measure the size and density of the (sub-) nanometer sized proto-particles. Such a diagnostic would also be of great interest to understand the transition from the large clusters to the proto-particles. In this paper we report on the cavity ring down method applied to dusty plasmas (5) in order to obtain new information on the early phases of the powder development.
EXPERIMENTS The experiments were performed in two different reactors with different substrate sizes. The Balzers KAI 1 reactor allows deposition on 35 x 45 cm substrates (6), whereas the substrate size was limited to 10 x 10 cm in the second reactor. For these investigations, both reactors were operated at an excitation frequency of 13.56 MHz and at approximately the same RF power density. For monomers silane, hexamethylsiloxane (HMDSO), methane, ethylene, aceteylene and oxygen were used. The influence of different diluting gases such as argon and helium on the powder formation was also investigated. Infrared absorption spectroscopy of the dusty plasma was performed by means of a commercial FTIR instrument and is described in more detail elsewhere (2). In addition to the infrared absorption measurements, optical emission spectroscopy and quadrupole mass spectrometry were applied at the same time. A Balzers PPM 422 plasma monitor with a mass range up to 512 amu was used with the probe head positioned adjacent to the electrode gap at 1 cm from the plasma boundary. Rayleigh-Mie scattering, using a He-Ne laser or an Argon ion laser as light source, was utilized to qualitatively visualize the powder within the discharge volume. The ring down cavity consists of two high reflectivity dielectric mirrors (>99.99%) in vacuum. To avoid instabilities of the confocal arrangement, the distance between the mirror was 120 cm. A pulsed dye laser tuned at 566 nm, operated at 10 Hz (4 ns pulse length) is introduced into the back face of the entrance mirror.
RESULTS AND DISCUSSION Silane discharges High mass anions up to 1760 amu containing at least 60 Si-atoms have been detected in silane plasmas (7). From various investigations it was concluded that these negative ions are the most probable polymerization pathway for powder precursors in pure RF silane plasma at low or moderate power densities. The regular nature of the mass distribution in the mass spectra suggests that a simple statistical approach might be sufficient to explain the form of the mass spectra. A simulation of the mass spectrum
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by means of random bond theory shows good agreement with the measured mass spectrum (8). As in the case of the pure silane discharge, the oxygen diluted silane plasma also shows considerable powder formation. Mass spectrometry has also been applied to the SiH4/02 plasmas (9) and also in this case very high mass anions were detected, whereas neutrals and positive ions remain limited to moderately high masses. For high masses the rich variety of Si-H-0 radical chemistry completely changes the regular structured spectrum found for the pure silane plasma case. Higher abundances of some mass values in the anion spectra may indicate clusters preferentially formed within the plasma. The masses of the preferential clusters coincide with hydrosilasesquioxanes(Si2nH2n03n n=3-6) which consist of a polyhedral cage made up of Si-O-Si linkages. A simple iterative model (9) has been adopted to explain the cation spectra in the SiH4/02 plasmas. Similarly to the anions for the pure silane plasma case, a statistical model including isotope effects reproduces the overall features of the observed cation spectra. However the complexity of the anion spectra in this condition implies that additional reaction mechanisms must also be included in the model.
HMDSO discharges In-situ infrared absorption spectroscopy was used to investigate the chemistry in diluted HMDSO plasmas. The admixing gases investigated were oxygen, helium and argon respectively. For the dilution with oxygen the infrared spectra reveal the presence of the oxidation steps of the methyl-groups: formaldehyde (COH2), formic acid (CO2H), CO and CO2 and with water as combustion by-product. The presence of methane (CH4) and acetylene (C2H2) found in the dilutions with argon, helium and oxygen indicate also the presence of hydrocarbon chemistry in these plasmas. As in the case of silane/oxygen plasma, silicon oxide (SiOx), particles were also formed in argon diluted HMDSO plasmas. This is in contrast to helium and oxygen diluted plasmas where no visible particles have been observed under similar conditions. Metastables produced in helium are supposed to induce a Penning dissociation breaking the polymerization precursors and thereby inhibiting the formation of particles. From the infrared absorption spectra, the HMDSO depletion was calculated based on three characteristic absorption bands for the Si-O-Si, Si-CHs and CH3 bonding within the HMDSO molecule. It was found that the depletion calculated from the different absorption bands gives new insight into the fragmentation of the HMDSO molecule and its dependence on the type of the diluting gas. In the case of helium and oxygen diluted HMDSO plasmas, the three absorption bands lead to similar depletion values as a fiinction of the dilution. However considerable differences are found in the case of the argon diluted HMDSO plasma. The depletion calculated from the Si-O-Si absorption band is 100% at high dilutions whereas the other two absorption band give about 30-40% less. We suppose that for the case of the argon diluted HMDSO plasma the molecule is fragmented into small radicals still containing Si-CHs bondings and
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CH3. This partial fragmentation might also be responsible for further polymerization which in turn can lead to the powder formation as observed in these plasmas. In the case of the oxygen and helium dilution a high degree of fragmentation of the HMD SO molecule is reponsible for reducing the powder formation.
Hydrocarbon discharges The combination of neutral mass spectrometry and FTIR absorption spectroscopy clearly shows that the production of acetylene is one of the most important processes in all three hydrocarbon plasmas investigated. In all experiments, we observe formation of acetylenic compounds, which appear to be very well-favoured products for the polymerisation reactions in the plasma. In the cases of methane and ethylene, the production of acetylene under the present discharge conditions is most probably not sufficient to influence strongly the chemistry of the anion and cation formation. In the acetylene plasmas, the triple C=C bonding remains mainly intact, imposing the characteristic even carbon number spectra of the ionic components, whereas double bonds from the ethylene are cracked within the plasma and play an insignificant role in the ongoing plasma chemistry. The main bonding type of the solid particles within the different plasmas is the sp bonding. Only limited evidence for the presence of some sp bonding in the powder could be found for the case of the acetylene plasma. This fact might be also of interest for astrophysical topics. Powder formation in the sp bond-containing acetylene is found to be strong whereas single and double bonds in accordance with previous reports in the literature show less powder formation. This might be partially explained by the production of hydrogen, observed in each case, which may prevent or delay the formation of powder. The limited production rate of acetylene might be a second reason for restricted powder formation. In the anion spectra of all three gases the absence of any CHx' groups has to be mentioned. This absence might indicate that C2Hx' rather than CHx" anions are produced by electron attachment. The observed formation of larger anion clusters might proceed by recombining C2Hx' anions with CH4 or most probably with C2HX. The different tendency to powder formation of the gases investigated might therefore be related to the different reaction rates for the production of the neutral C2HX (in particular C2H2) and their corresponding attachment rates. It might therefore be concluded that the formation of acetylene and subsequent electron attachment to this molecule leads to larger anions, which by analogy with the silane plasma could end in powder formation. In-situ powder diagnostics by cavity ring down technique The cavity ring down technique has been applied to investigate the powder formation in argon diluted silane plasmas, diluted HMDSO plasmas, and in pure methane plasmas. The change in the ring down time of the cavity is due to absorptions of
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various origins. Besides photodetachment of negative ions, and line absorption due to electronic excitation, the main losses of the laser beam are due to absorption within the nanoparticles and scattering. The extinction due to small particles is w^ell described by Rayleigh scattering theory and depends on R (R particle radii) whereas absorption depends on R"^ (volume fraction) and on the refractive index. In general absorption is the dominating effect for very small particles such as in the case of silicon- and carboncontaining particles. However the Si02 particles must be treated as non absorbing due to their very small imaginary part of the index of refraction in the visible. The time development of the powder formation in (diluted) silane, HMDSO and methane plasmas has been measured. In each case the scattered intensity at 135° and the single pass extinction of the Ar-ion laser beam was simultaneously monitored. The appearance of any scattered intensity indicates the presence of powder particles in the range of about 40-50 nm.
1000
E
c
time (s)
FIGURE 1. Powder formation in pure methane plasmas. Particle size and density for cleaned reactor (circles), after two (squares) and four (triangle) time developments of the powder formation.
Fig. 1 shows the formation of particles in a pure methane plasma. In a cleaned plasma reactor the powder formation takes a few hundred seconds. In a contaminated reactor, powder appearance is much faster, since powder formation might be triggered by particles emitted from the particle-contaminated electrodes. It can be concluded that contamination may strongly influence the powder formation. Therefore care must be taken in order to avoid artefacts due to this effect. The particle size and particle number density in this case were determined by assuming that absorption is dominant for very small particles and that the volume fraction is constant during the development. The particle size can be estimated from the ratio of the extinction due to absorption measured early in the particle development, to the measured absorbance (proportional to R ) and from this finally the number density can be obtained. For the case of Si02 particles this estimation cannot be applied due to the non-absorbing character of the particles. In this case only the behaviour of NR^ can be given. For
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larger particles, scattering is the dominating process. This leads to an increased extinction and therefore large particles even at low density might strongly influence the absorbance and dominate over the contributions from small particulates. In Fig.l the presence of a large particles is observed late in the time development. These large particles arrive near the plasma sheath due to rearrangement of the powder as was visually observed.
time
(s)
FIGURE 2. Addition of oxygen to an argon diluted HMDSO plasma (15sccm Ar, 2 seem HMDSO) a) Extinetion eoeffieient from eavity ring down method, b) extinetion eoefficient determined from Ar ion laser beam (single pass).
Fig 2 shows the absorbance in an argon-diluted HMDSO plasma. In these plasmas, first indications of powder formation appear after about 50 seconds. However, on adding 0.7 seem respectively 2 seem, of oxygen, very fast powder formation is observed. Recently the cavity ring down technique has been applied to determine the negative ion density in oxygen plasmas using photodetachment (5). These investigations show that the electronegativity of the oxygen results in a large fraction of negative charge carriers being negative ions. Therefore also in a dusty electronegative plasma many negative charges must be negative ions. This leads to a reduction of the particle charging and its consequences on the coagulation (larger particles) and powder dynamics (larger neutral forces and smaller electrostatic forces). It was found that Si02 particles in these plasmas are in the range of 400-500 nm, whereas powder particles in other reactive plasmas such as silane plasmas are typically only around 100 to 200 nm
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(10); this is probably a consequence of reduced charging of the particles in these plasmas. In the case of powder formation in very low power argon diluted silane plasmas, periodically varying absorbances have been observed (5). These oscillations are characteristic of powder creation/growth/elimination cycles in the reactor as already shown by other diagnostics. However, at low silane dilutions the ring-down signal is rapidly lost. The absorbance presented in fig. 3 follows an exponential law.
1 0- 4
5 0
F I G U R E 3 . Extinction coefficient in a argon diluted silane plasma (500 seem argon, 15.4 silane, lOW and 0.2 T).
Contrary to Si02, amorphous silicon shows considerable absorption at the dye laser wavelength and the beam extinction is rather given by the absorption than by scattering losses. The absorbance signal can therefore be interpreted for particle sizes up to about 40nm (about Mie scattering onset) as the total volume fraction of the powder in the plasma. An exponential increase of the volume fraction can be explained by a simple theory including coagulation and the growth of particulates and where the volume of the particulates remains unaffected by the coagulation (11). Up till now most of the theories applied to the coagulation in the plasma (1,12,13) assume constant volume fraction. In order to interpret the measurements, more sophisticated theories including simultaneous nucleation, condensation and coagulation in the plasma should be applied. A model including the simplest particle forming system by gas to particulate conversion (14) allows to describe the dynamic behaviour of such a system. In addition the influence of the negative ions on the charging of the particles needs clarification since charging infiuences not only the coagulation and the expected size, but also the governing forces on the particles and therefore the powder dynamics. For the powder formation and also for powder processing important information on changes in the polydispersity due to the coagulation might be obtained. Further investigation is under way.
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CONCLUSION Infrared absorption spectroscopy, mass spectrometry and emission spectroscopy have been applied to investigate the powder formation in different hydrocarbon and organosiUcon RF plasmas. The plasma composition and the powder formation depends on the chemistry of the monomer. Noble gas dilution can strongly influence the fragmentation of the monomer and with this the tendency to powder formation. The presence of negative ions such as negative oxygen ions will strongly affect the particle charging and in consequence the powder dynamics. The cavity ring down technique is a powerful method to investigate the early phases of the powder genesis.
REFERENCES 1. A. Bouchoule ed., "Dusty Plasmas between Science and Technology," (Wiley, 1999) 2. C. Courteille, D. Magni, A.A. Howling, V. Nosenko and Ch. Hollenstein, "Infra-red absorption spectroscopy of Si02 deposition plasmas", 40th Annual Technical Conference Proceedings of the Society of Vacuum Coaters , 304-3308 (1997). 3. Ch. Hollenstein, Plasma and Polymers, to be published 4. J. Dutta, H. Hofinann, C. Hollenstein and H. Hofineister, "Plasma-produced silicon nanoparticle growth and crystalization processes," in Nanoparticles and Nanostructural films, edited by Fendler (Wiley-VCH, Weinheim, 1998). 5. F. Grangeon, C. Monard, J.-L. Dorier, A.A Howling, Ch. Hollenstein, D. Romanini and N. Sadeghi, Plasma Sources Sci. TechnoL, to be published 6. L. Sansonnens, A.A. Howlmg and Ch. Hollenstein, Mat.Res.Soc.Symp.Proc. 507, 541-546 (1998). 7. A.A. Howling, C. Courteille, J.-L. Dorier, L. Sansonnens and Ch. Hollenstein, Pure & Appl. Chem. 68 (5), 1017 (1996). 8. Ch. Hollenstein, W.Schwarzenbach, A.A. Howling, C.Courteille, J.-L. Dorier and L. Sansonnens, J. Vac. Sci. Technol. A 14, 535-539 (1996). 9. Ch. Hollenstein, A.A. Howling, C. Courteille, J.-L. Dorier, L. Sansonnens, D. Magni and H. Muller, Mat.Res.Soc.Symp. Proc. 507, 547-557 (1998). 10. C. Courteille, D. Magni, C. Deschenaux, A.A. Howling, Ch. HoUenstem and P. Fayet, "Gas phase and particle diagnostics of HMDSO plasma by infrared absorption spectroscopy," 41st Annual Technical Conference Proceedings 1998 Society of the Vacuum Coaters, 327-332 (1998). 11. M.M.R Williams and S.K. Loyalka, Aerosol Science Theory and Practice (Pergamon Press, 1991). 12. C. Courteille, Ch. Hollenstein, J.-L. Dorier, P. Gay, W. Schwarzenbach, A. A. Howling, E. Bertran, G. Viera, R. Martins and A. Macarico, J. Appl. Phys. 80 (4), 2069-2078 (1996). 13. U. Kortshagen and U. Bhandarkar Phys. Rev. E submitted (1999). 14. S. E. Pratsinis, J. Colloid Interface Sci. 124 (2), 416 - 427 (1988).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
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Trapping and Processing of Dust Particles in a Low-Pressure Discharge E. Stoffels, W.W. Stoffels, G.H.P.M. Swinkels, G.M.W. Kroesen, Department of Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]. tue. nl
Abstract. Formation, behaviour and modification of dust particles in a low-pressure plasma are discussed. Coulomb interactions of negatively charged particles together with other forces in the plasma result in efficient trapping. A single particle can be kept motionless in the plasma, and fully controlled by the experimentalist. We are able to accurately determine the particle properties by angle-resolved Mie scattering, and modify them by etching or deposition in the plasma. This research is related to the industrial demand for particles with specially tailored properties. We show that a low-pressure discharge is well suitable for production or surface modification of special dust particles.
INTRODUCTION Dusty plasmas are nowadays a vast research field. Numerous aspects of the formation, interactions and consequences of dust particles in plasmas have been investigated, both from the fundamental and industrial side. The fundamental research covers the problems of charging and dynamics of dust particles in plasmas. In particular, formation of ordered structures called Coulomb crystals has been investigated, and a variety of wave phenomena in dusty plasmas have been described (1). Moreover, dust particles in the interstellar space attract much attention in relation to the rings of Saturn. The applied side emerged in the late eighties, and triggered dynamic development of the dusty plasma research (2). Its origin is the semiconductor processing industry, where sub-micrometer particles are formed in situ in the processing reactor. This is a common phenomenon in plasma enhanced chemical vapour deposition (PE-CVD) reactors, e.g. during deposition of amorphous silicon in the fabrication of solar cells. In deposition plasmas particle formation and co-deposition on the substrate can lead to layer defects. In the reactive ion etching (RIE) of semiconductor elements the purity of the surface is the major issue; there the consequences of particle formation are much more severe. Dust particles, when deposited on the processed surface, are the major cause for device failure. The major aim has been to minimise particle
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contamination, either by avoiding particle formation or by preventing their deposition. Many advanced diagnostic techniques for in situ contamination control have been developed to meet the increasing demands from the industry. Progressive miniaturisation of semiconductor elements implies that even nanometer size particles can be dangerous contaminants, and their presence must be well diagnosed. At the moment, detection of nanometer size particles is possible by means of inelastic laserparticle interactions (2). An immense research effort has resulted in the elucidation of in situ particle formation mechanisms in deposition plasmas, as well as the mechanisms of surface sputtering and flaking in etching plasmas. The powerful experimental diagnostics, the understanding of particle origin and the knowledge of various forces and interactions of particles in the discharge allow now a good control of reactor contamination. A dust particle has become a predictable object, whose behaviour and properties can be manipulated by the researcher. In recent years new trends have emerged, in relation to potentially interesting properties of plasma-produced particles. The knowledge acquired during the combat with dust contamination is now being used to explore the novel applications of dusty discharges. Two trends can be observed. First, particles formed in situ or injected externally, are processed in the discharge in order to adjust their properties (3). Applications include coating of particles with active layers for various purposes and separating large particle conglomerates in a discharge in order to obtain homogeneously coated grains, etc. Fabrication of catalysts and pigments, or improvement of toners for copying machines are just a few examples. For such applications it is essential to gain a good control of the particle behaviour in the discharge, and to understand plasma-particle interactions, like charging, etching and deposition processes on the particle surface. Previous studies were mainly concerned with systems containing many particles, e.g. with characterisation of particle clouds formed in the plasma. However, it is difficult to study surface modification processes when large amounts of not well-defined particles are involved. Alternatively, some investigators injected well-defined particles into the discharge in order to study plasma-particle interactions. Particle dynamics, force balance and formation of Coulomb crystals were intensively studied (4). Here we shall briefly describe particle charging and trapping phenomena, before introducing the particle processing. In order to be able to study the mechanisms of plasma processing on a clear model system, we propose a single particle experiment. A well-defined particle is introduced into the plasma and pinpointed in a specially constructed potential well. We can accurately determine the particle properties and influence them in a controlled way by means of plasma processing. In particular, angle-resolved Mie scattering is employed to monitor the particle radius and the particle size is adjusted by means of etching in a low-power radio-frequency oxygen discharge. The second major trend in dust particle applications is the production of hybrid materials by particle sintering or particle enclosure in a solid material. Small particles formed in the discharge can be embedded in a layer on the surface, in order to improve the layer quality. For example, the quality of amorphous silicon solar cells
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deposited in a silane plasma can be significantly improved by co-deposition of nanometer size silicon particles formed in the same discharge (5). One of the recent applications of hybrid coatings is the production of hard layers for the protection of mechanical tools. In most cases, lubricating the contact surface is needed to reduce friction in order to improve the mechanical yield of the system and reduce the wear of the parts in contact. It has been shown that coatings with an addition of lubricants like M0S2, combine wear resistance with a significantly lowered friction (6). Addition of M0S2 poses a technical problem, because of its poor adhesion and a bad oxidation resistance. To circumvent this problem, a coating where M0S2 in the form of nanometer particles is immersed has been proposed. However, production of nanometer particles and their co-deposition within a layer is not a straightforward task. In the second part of this paper we will describe the recently developed method of plasma-enhanced production of M0S2 particles and their handling in a PE-CVD reactor.
PARTICLE TRAPPING The characteristic feature of an object immersed in the plasma is the negative electrical charge it acquires. In the steady state, the fluxes of positive ions and electrons from the plasma towards the solid surface must be equal. As electrons have a higher mobility than positive ions, an appropriate negative surface charge and electric field will be established to accelerate the positive ions and repel the electrons. Such effects occur for bounding surfaces and floating objects in the plasma. The potential difference between the particle and the plasma (Vp) is a function of the electron temperature (Tg). In low-pressure discharges Vp is usually two to three times kTe/e, i.e. about 10 eV for typical electron temperatures of about 3 eV. The potential difference determines the energy of positive ions, reaching the particle surface as well as the number of elementary charges on the particle (Z). The latter can be estimated from the classical expression Ze = 47isoa Vp, where 47rsoa denotes the capacitance of a sphere with a radius a. This simple expression for the particle charge yields about 3 elementary charges per nanometer radius of a particle: Z ^^ 3-10^ a. A particle in the plasma is subject to various forces, the most important being the electrostatic force, gravitation, neutral and ion drag and thermophoretic forces. Coulomb repulsion between the negative charged particle and the electrodes provides trapping of the particle in the plasma glow. This in combination with the other forces allows sustaining the particle in a fixed position. Previous experiments have shown that one can easily control the particle position by influencing the force balance. Coulomb and ion drag forces are tuned by varying parameters like plasma power and pressure or changing the electrode geometry, thermophoretic force is influenced by introducing temperature gradients, and gas dynamical forces are determined by the gas flow pattern (4,7). The ability of controlling the particle position is essential in a study of particle processing in the discharge.
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ETCHING OF A SINGLE PARTICLE Below we describe trapping and etching of a single particle in a low-pressure oxygen discharge (8). A schematic overview of the experiment is shown in Figure 1. We use commercial melamine-formaldehyde particles with a diameter of about 12 |Lim and a refractive index of 1.68. For the particle injection, we use a particle-filled sieve, which is mounted on a manipulator arm. The measurements are performed in a GEC reference cell: a parallel-plate capacitively coupled 13.56 MHz radio-frequency reactor. An aluminium ring (diameter: 2 cm, thickness: 1.5 mm) is placed on the lower powered electrode in order to create a potential trap, so the particles shed from the sieve are immediately "caught" at the glow-sheath edge above the metal ring. In this way it is possible to prepare a Coulomb crystal, but also to "freeze" a single particle in a fixed position. Angle-resolved Mie scattering is applied to monitor the particle size and refractive index as a function of time during plasma processing. The light source is an argon ion laser, linearly polarised in the direction perpendicular to the detection plane (see Fig. 1). The scattered light is collected by an optical fibre, passed through an interference filter and fed to a photo-multiplier. The optical fibre is mounted on a moveable stage. This allows continuous scanning of the detection angle in the range of 1 to 15 degrees for forward scattering, with an angular resolution of 0.06 degree. The angle-resolved scattering intensities are fitted to a numerical model for Mie scattering using the particle radius and refractive index as tuning parameters. A typical angle-resolved measurement and its fit to the data are shown in Figure 2. As can be seen, there is a good agreement between the measured data and the fit. The fitted particle radius (a = 5.90 |im) and refractive index (n = 1.68) agree with the size determination using SEM pictures and the data provided by the particle manufacturer. The particle size can be determined with accuracy better than 1% by the angleresolved scattering technique. In order to demonstrate a good control of the particle radius and the sensitivity of the diagnostics, and to study surface modification of particles, we use a low power oxygen discharge to etch the organic polymer, of which the particles are made. There is a large difference between the standard etching of a substrate placed on the electrode (RIE) and the etching of a free floating object in a plasma. In the sheath of the powered electrode a high potential in the order of 1 kV accelerates positive ions towards the surface. Sputtering and etching of the material is performed by highenergy ions, reaching the electrode surface. In contrast, the potential difference between the plasma and a floating particle is much lower, in the order of 10 V. Therefore, plasma chemical effects due to low-energy ions and radicals are expected to be most important for microscopic particle etching. In order to check the influence of physical ion sputtering, we monitored time changes in the angle-resolved scattering signals of single particles trapped in argon. In argon, where no chemical effects can occur, no remarkable changes in particle properties were observed even after hours of plasma operation. In an oxygen plasma substantial variations of the scattering signal were recorded already after a few minutes of processing.
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detection plane
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argon ion laser
moving detector
FIGURE 1. A scheme of the experimental setup. A single particle is trapped in the plasma above the rf electrode and irradiated by an Ar ion laser. The angle-resolved scattered light intensity is collected by an optical fiber mounted on a moving stage and fed to a photomultiplier.
angle (d^)
100
^
FIGURE 2. Angle-resolved Mie scattering data of a single melamine-formaldehyde particle (squares), trapped in a radio-frequency oxygen plasma. The data are fitted by a theoretical scattering curve for a particle of 5.90 [am radius and 1.68 refractive index. FIGURE 3. Time-dependent angle-resolved scattering intensity of a single melamine formaldehyde particle treated in a 0.2 mbar oxygen plasma. The particle is processed for 30 minutes in a 1.5 W plasma, then for 30 minutes in a 5 W plasma and finally for 60 minutes in a 7 W plasma. The etch rate increases with increasing plasma power from 0.06 to 0.13 nm/s.
Oxygen is a well-known etching gas suitable for etching of organic polymer materials. Ion and radical chemistry in radio-frequency oxygen discharges was extensively studied in the past (9). Free oxygen radicals and oxygen ions provide an ideal composition of active species to etch organic polymer particles trapped in the discharge. The etching process is determined by various plasma parameters. Tuning of these parameters provides a means to obtain a good control of the etch rate and consequently of the particle radius. In Figure 3 the time evolution of the angleresolved Mie scattering signal is shown for a particle treated in a 0.2 mbar oxygen discharge with a varying plasma power. The characteristic Mie fringes are clearly visible, and it is evident that the fringes shift towards higher angles as the etching
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proceeds. In terms of the Mie theory, this implies that the particle size decreases in time. Using the numerical model the time dependent particle size as a function of time is deduced. At a given power level, the particle size variation is fairly linear with time, and the corresponding power-dependent etch rate can be determined. The densities of reactive species in the discharge increase nearly linearly with increasing power, and the same trend is expected for the etch rate. The particle shown in Figure 3 is injected into a 1.5 W oxygen discharge, and processed for about 30 minutes. At this power level only slight changes in the angle-resolved scattering intensity can be observed, which implies that the particle radius hardly changes (etch rate is 0.06 nm/s). After about 30 minutes the plasma power is increased to 5 W. At this power level etching of the particle proceeds faster (0.10 nm/s), which is reflected by time variations of the angle-resolved scattering intensity. Next, the plasma power is increased to 7 W. The plasma activity is further enhanced, resulting in an increased etch rate of 0.13 nm/s. Similarly, varying the pressure of the processing gas allows to etch the particles at a desired rate. By tuning of the plasma parameters, the etch rate can be varied from 0 up to 1 nm/s with an accuracy and reproducibility of about 10%. Similar etch rates were found for particle clouds (10). Thus, the proposed experiment of single particle processing is a simple and accurate model system for studying large scale particle processing. In fixture it will be extensively used to control particle coating in deposition plasmas.
PRODUCTION OF PARTICLES IN THE PLASMA Hybrid layers, containing a particle suspension, find many applications in the modem coating technology. Here we describe a part of the work aiming on fabrication of hybrid titanium nitride coatings with embedded molybdenum sulfide particles (11). The lubricating properties of M0S2 in combination with wear-resistant properties of a TiN film results in a unique self-lubricating hard coating. Production and codeposition of nanometer particles in CVD processing as well as in a plasma environment is a challenge, as such small grains are not commercially available and very difficult to handle ex situ. Moreover, when externally produced particles are used, there are always problems with contamination with water and oxygen. Therefore, an integrated process is required, combining particle formation, layer growth and seeding in one closed system. Particle production is generally performed in a different chemical and physical environment than the deposition process. Typically, particles are formed at high pressures, either using thermal plasmas or ovens, while for PE-CVD processes the pressure must be kept low. Thus, new recipes should be developed for particle formation, which can be incorporated in a PE-CVD environment. Particle formation at low pressures encounters many difficulties. First of all, the reaction rates are lower as a consequence of lowered densities of reagents. Especially three-body association reactions, which are essential for the formation of large molecules and their condensation into particles, are inefficient at low pressures. Moreover, in the commonly used flowing systems the residence times of the species
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FIGURE 4. SEM micrograph of M0S2 particles produced in the plasma at 0.5 mbar, using H2S, M0CI5, H2 and Ar. The micrograph shows particles, which are conglomerated into larger structures. The diameter of individual particles is smaller than 100 nm.
in the reactor are limited, typically under one second. This implies that many-step processes, leading to particle growth, are hindered. A possible solution to these problems is the use of a low-pressure plasma. Rapid formation of macroscopic particles in low-pressure discharges has been observed in many chemical systems (2,4). The non-equilibrium character of such plasmas and the participation of charged species in the formation processes facilitate particle production. The non-equilibrium character is reflected by a large difference between the neutral gas temperature (ambient) and electron temperature (a few eV). All chemical processes are initiated by electrons, which have sufficient energy to dissociate molecules and a high mobility to undergo frequent collisions. Other problems of particle formation at low pressures, namely the residence times which are too short to allow for particle growth, can be also overcome. In the plasma the particles acquire a negative charge at a very early stage of their growth. By means of Coulomb interactions they remain trapped in the plasma sufficiently long to reach even sub-millimeter sizes. The size can be adjusted to one's wishes by controlling the plasma duration time, plasma power and gas flows. In the considered example, M0S2 particles are produced from M0CI5 and H2S in the following reaction: M0CI5 + 2H2S + I/2H2 ^ M0S2 + 5HC1. In the neutral gas, no particles can be produced at pressures below 10 mbar. In contrast, abundant particle formation has been observed already at 0.5 mbar in the plasma containing evaporated M0CI5, H2S and excess of hydrogen. This is a typical pressure used for PE-CVD of TiN layers, which will facilitate codeposition of these species. The quality of plasma produced particles meets the requirements for the production of hybrid coatings: they have a monodysperse size distribution and the desired size of < 100 nm. During plasma operation the particle suspension levitates above the substrate, and particles can be easily collected on the surface after plasma termination. A typical SEM micrograph of plasma-produced particles is shown in Figure 4. The procedure of embedding particles in the coating will thus consist of M0S2 production and collection, and switching the chemistry for the TiN deposition.
CONCLUSIONS In this paper we discuss handling and processing of particles in the discharge, relevant for industrial particle tailoring. First, fundamental aspects, like particle trapping and etching have been treated, and an experimental model system for the
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E. Stoffels et al /Trapping and processing of dust particles in a low-pressure discharge
Study of particle processing has been described. We have shown that a single particle can be efficiently trapped in a discharge, and its size can be accurately regulated. We discuss etching as an example of such controlled particle treatment, but the presented technique opens many other possibilities. Particles can be coated in deposition plasmas, e.g. using silane or methane. Fine tuning of the particle size, at a welldefined rate, has undeniable benefits. For example, particle modification process can be studied on a relative simple, single particle system, and extended to the situation of particle clouds. Moreover, the forces acting on a particle are size-dependent, so an accurate determination and modification of particle size can prove extremely helpful in a study of the force balance and particle motion in the plasma. Furthermore, we discuss the advantages of production of nanometer particles using a low-pressure discharge. Advanced coating technology involves co-deposition of small grains within a layer, under low-pressure conditions. An integral process of in situ particle production and seeding in a PE-CVD reactor has been proposed and efficient production of M0S2 particles has been achieved.
ACKNOWLEDGMENTS We appreciate the contribution of Drs. F. Rossi and G. Ceccone. This work is supported by the European Commission under Brite- Euram contract BRPR CT97 0438 (HALU), by the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) and by the Dutch Technology Foundation (STW). The research of Dr. W. W. Stoffels has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW).
REFERENCES 1. D.A. Mendis, C.W. Chow, P.K. Shukla, Proc. 6* Workshop The Physics of Dusty Plasmas, La JoUa, Califomia, 1995, publ. World Scientific, Singapore, 1996, ISBN 981-02-2644-6. 2. Dusty Plasmas, ed. by A. Bouchoule, to be published by Wiley&Sons, 1999. 3. H. Kersten, P. Schmetz, G.M.W. Kroesen, Surface and Coatings Technology 108-109, 507 (1998). 4. Proc. Dusty Plasmas '95 Workshop on Generation, Transport and Removal of Particles in Plasmas, Wickenburg, Arizona, 1995, publ. in J. Vac. Sci. Technol A14(2), 1996. 5. P. Roca i Cabarrocas, P. Gay, A. Hadjadj, J. Vac. Sci. Technol. A14, 655 (1996). 6. H. Suhr, R. Schmid, W. Sturmer, Plasma Chem and Plasma Proc. 12, 147 (1992). 7. E. Stoffels, W.W. Stoffels, PhD Thesis, Eindhoven University of Technology, The Netherlands (1994). 8. W.W. Stoffels, E. Stoffels, G.H.P.M. Swinkels, M. Boufhichel, G.M.W. Kroesen, Phys. Rev. E59, 2303 (1999). 9. E. Stoffels, W.W. Stoffels, D. Vender, M. Kando, G.M.W. Kroesen, F.J. De Hoog, Phys Rev. E51, 2425 (1995). 10. G.H.P.M. Swmkels, E. Stoffels, W.W. Stoffels, N. Simons, G.M.W. Kroesen, F.J. de Hoog, Pure and Applied Chemistry 10(6), 1151-1156 (\99Sy 11. E. Stoffels, W.W. Stoffels, G. Ceccone, F. Rossi, R. Hasnaoui, H. Keune, G. Wahl, accepted for publ. in J. Appl. Phys. (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukia (eds.) © 2000 Elsevier Science B.V All rights reserved
135
Effects of Gravity, Gas and Plasma on Arc-Production of FuUerenes Tetsu Mieno Department of Physics, Shizuoka University, Shizuoka-shi 422-8529, Japan Abstract. In orda^ to improve production efficiencies of many kinds of ftillerene families, production characteristics of fullerenes in an arc discharge are investigated. Effects of gravity, impurity and magnetic field in the arc production are investigated and, key conditions and more efficient methods to produce fullerenes in the arc discharge are found.
INTRODUCTION Mtoy kinds of fullerenes and nanotubes are produced in large quantity by an arc discharge in helium gas atmosphere. While, their molecular process of self-organization in arc region is not made clear because many compUcated reactions are mixed and many parameters influence on the reaction. Usually, the reaction process is decided by arc condition and control of reaction parameters such as reaction time, reaction volume, reaction temperature is not carried out well. Here, microscopic and macroscopic properties of fuUerene synthesis are investigated and important basic parameters are pointed out. !» ^ In order to improve tiie synthesis, effects of gravity, gas pressure and impurities, and magnetic field are investigated and effective results to improve fuUerene production have been obtained. Efficient production of these new carbon materials is demanded for various kinds of applications.
BASIC CONDITION OF FULLERENE SYNTHESIS Basic configuration of an arc reactor to produce fullerenes is shown in Fig. 1. 3 Empirically production rates of fullerenes strongly depend on many parameters of arc discharge. For example, production characteristics change with discharge current, discharge voltage, gap distance, diameters of electrodes, gas pressure, gas species, impurity density in the gas, etc. 1* ^ By arc heating, sublimated carbon atoms from an anode react in hot gas atmosphere and they flow upward by heat convection and deposit on a upper wall of the chamber. Elucidation of basic growing process of fullerenes in the arc discharge is hard because their reaction occurs in high pressure and high temperature condition, and synthesized molecules are in high density. Instead of the arc discharge, molecular process in laser-ablated carbon gas, "^ which is expected to have similar growing process, is precisely investigated by use of the mass-spectrometric technique and the ion drift methods and important result is obtained by Bower's
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T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
VACUUM CHAMBER (STAINLESS STEEL)
FIGURE 1. Example of an arc reactor to produce fuUerenes.
group. 5, 6 j h e evaporated carbon ions are mass separated by a sector type magnet and selected mass species is injected into a drift cell, in which molecules of different shapes are separated like a chromatography method and difference of transit time signal is recorded. From this result and theory of the molecular stabiUty, shapes of synthesized ion species for each mass are determined. As a result, it is found that carbon molecules grow via chain, ring and multiple ring structures into fuUerene type cage structure. For cluster size of more than 50, more than 90 % of produced clusters have fuUerene structure. This growing process is simulated by the molecular dynamics method by Maruyama' s group and similar result is obtained theoretically. ^ These clusters are partly stable and alive in the air condition, because by the pentagon rule of molecular stability, ^60' ^70 and C2n (n> 35) are only stable in the air ^ and collected as fuUerene samples. In order to clarify the region where the growing reaction takes place in the arc discharge, a simple experiment is carried out A large carbon block is located on an arc flame and its distance from the arc center ZCB is changed. ^ Cgo content in produced soot versus Z^g is measured and shown in Fig. 2, where/?= 300 Torr and 7^= 60 A. At ZcB< 2.5 cm and temperature of higher than 900 ''C, Cgo content seriously decreases, which means that fuUerenes are effectively synthesized only in the arc region at higher than 900 ^C. From the investigation of the molecular process and result of macroscopic experiment, important conditions of high-efficiency production of fuUerenes in the arc discharge are obtained as hollows, 1) High sublimation rate of carbon atoms from anode material by controlling a discharge current and anode size. 2) Long reaction time of carbon molecules in high temperature He gas, which is
T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
15
187
1500
' I • ' • • I ' ' ' ' I
^X(C6o) CARBON BLOCK
5
1000
10
o
t
o
(+)
(-)
^v-=r
4^500
><
0
1
2
3
4
5
6
7
8
ZcB(cm)
FIGURE 2. C5Q content versus carbon block position and vertical profile of gas temperature. p= 300 Torr, I^ 60 A and 4od- ^-^ ^^"^^
controlled by He pressure and He heat convection. 3) High coUision frequency of carbon-to-helium coUision to anneal the carbon clusters. 4) Low impurity density in He gas to ehminate another carbon reaction. Usually this reaction is automatically decided by the arc discharge condition. In order to obtain more efficient production of fuUerenes, active control of the arc reaction is needed.
EFFECT OF GRAVITY In the arc reaction with He gas, strong heat convection takes place and reaction time in the high temperature gas is limited by the convection speed. ^ Therefore, if the arc reaction is carried out in gravity-free condition, high-temperature reaction time is Arc Reaction with Heat Convection
Arc Reaction without Heat Convection T< 1,000 0 Carbon clusters are stabilized T= 1,000-5,000 C Carbon clusters are formed and anealed
T= 5,000-10,000 c C, C2 Molecules are dominant * Carbon molecules are quikly folwn up by the heat convection.
*Bigger and heavier molecules stay longer time depending on the thermal diffudion
FIGURE 3. Models of flow of sublimated carbon vapor for two gravity conditions.
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T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
G=0
SIDE BOTTOM TOP
G=1gn
i'.
SIDE
,
BOTTOM I
I
I
6
8
10
12
14
P,(g/m2)
FIGURE 4. Schematic of a 12 m vertical
FIGURE 5. Mass density of deposited soots wing tower (VST), at three parts for two gravity conditions. p= 300 Torr, discharge time T^ 30 min., 4od~ ^ °^°^-
prolonged and higher fuUerenes and endohedral metallo-fuUerenes would be more efficiently produced. Model figures of two discharge conditions are shown in Fig. 3. The convection speed of He gas at 0.5 atm is calculated to be about 1 m/s 1^ and the reaction time for each carbon clusters is about 10 ms. For larger fullerenes, its thermal diffusion speed becomes much smaller than the convection speed. In order to examine this hypothesis, a 12 m high vertical swing tower (VST) was constructed in the universit>^ campus and arc production in gravity-free discharge is examined!! Figure 4 shows the schematic of experimental set-up. A vacuum vessel with 1.6 L in volume and 6 kg in weight is suspended by a rubber rope and a steel wire. By applying periodic tension, the reactor moves as a periodic oscillation. By applying pulse-modulated discharge current (7^= 20-40 A), carbon is sublimated only in the gravity-free condition. In this experiment, the swing amplitude is about 4 m peak-to-peak and swing period is 2.3 s, in which gravity-free time is 1.1 s. For discharge time of 30 min, integrated gravity-free time is 13 min. ^^ In this experiment, production rate of C50 is about 0.4 g/h (using 5 mmcj) anode) and C50 production rate is about 17 mg/h.
T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
189
G= 0
G=ign 0
0.2
0.4
0.6
0.8
1
1.2
W^(g/h)
FIGURE 6. Production rates of total carbon soot for two gravitational conditions. p= 300 Torr and T^ 30 min and 4od~ ^ °^^0.15
C78
>'
0.1
t og„
00
i 0.05
^
19„
5^
-m—=ffl200
^B^
ffl- —
0.05
-ffl
:ffl_ 400
600
800
200
P (Torr)
400
600
800
P (Torr)
FIGURE 7. Relative intensities of higher fullerenes versus pressure measured by the LD-TOF-MS for two gravity conditions. 4od- ^ ^ ^
For two conditions; gravities of G= 0 and G = I^Q (normal gravity), fullerenes are produced for 30 min and deposited soot densities on the inner wall is measured as shown in Fig. 5, where p- 300 Torr and rfrod= 6 mm. For gravity-free condition, almost isotropic diffusion of carbon soot is obtained, while for normal gravity condition, strong heat convection flows carbon clusters to upper direction. Total production rates of soots are measured and shown in Fig. 6. For gravity-free condition, the production rate becomes twice. Higher fuUerene contents in the soot are measured by a laser desorption time-of-flight mass analyzer (LD-TOF-MS). The soot sample bound with xylene is measured and relative intensities of positive ion signal (100 shots averaged) as a function of He gas pressure are shown in Fig. 7, where dxo^- 5 mm and the swing amphtude is 2 nip-p- For higher fullerenes such as C7g and Cg4, the relative intensities considerably increase compared with those of the normal gravity condition. ^^ Next, collection rate of endohedral La metallo-fuUerenes is compared. Fullerenes in the soot are soxhlet-extracted by xylene and La@Cg2 molecules are separated by a high pressure liquid chromatography (HPLC) method and its content is measured by absorbance at A=
190
T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
G=0
G=1go
G=2go 0
2 4 6 8 10 Collection Rate of LaCgj (arb. units)
FIGURE 8. Collection rate of La@C82 for three gravity conditions. /?= 300 Torr, T^ 30 min.
300 nm, where Nacarai-tesque Co., 5PBB column wifli xylene is used. From the La@C82 content and the production rate of the soot, collection rate of La@C82 is obtained as shown in Fig. 8, which shows that the collection rate increases about 14 times at gravity-free condition, whereas for high gravity condition (G= 2gQ), the collection rate is seriously reduced. Therefore, production efficiency of La@C82 strongly depends on the gravity. Production rate of carbon nano-tube also considerably increases at the gravity free condition.
EFFECT OF GAS PRESSURE AND IMPURITY When the He gas pressure is reduced to less than 50 Torr, fuUerene production is seriously reduced, and it suddenly increases by increasing gas pressure than p= 100 Torr. 1 At/7= 300 Torr, the coUision mean-free-path is about 0.5 fxm and coUision frequency is about 3 GHz, therefore collision number in the arc reaction is about 2x10^ where reaction zone is estimated to be spherical and its radius is 1 cm. Impurity effect of hydrogen is examined. Production rate of CgQ as a function of H2partial pressure is measured and shown in Fig. 9, where total pressure is 300 Torr, 0.3 [T 0.25 O)
0.2
o O
0.15
G
0-1 0.05
• * '•• l \
] j
1\
!
^
;
"v^
;
tvJ
>
0
10 P(H2) (Torr)
FIGURE 9. Production rate of C^Q versus H2 partial pressure. Total pressure ;?= 300 Torr, /j= 70A, and4od= 6.5 mm.
T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
191
/(j= 70 A, and anode diameter (irod= 6.5 mm. When more than 5 % of H2 gas is included, fuUerene production is strongly obstructed. Impurities of N2 and Si also reduce the production rate, while impurities of O2, Al, Fe, W, Au do not reduce it. Near the normal carbon stars, hydrogen content in their atmosphere is large, which would obstruct fuUerene synthesis in space, though much abundance of Cgg"*" ions are reported to be observed in space. 13
EFFECT OF MAGNETIC FIELD In order to increase the production efficiency of fuUerenes and reduce the deposition of carbon material on a catiiode, a steady magnetic field {^ 30 Gauss) is apphed perpendicular to the arc current By the JxB force, arc plasma and carbon gas are jetted out to the JxB direction. 14, 15 By this method, production rate of C50 considerably increases and deposition rate to the cathode decreases. By using the JxB arc method, chip-carbon material injection type fuUerene automatic producer has been produced as shown in Fig. 10.1^ An anode is made of a carbon crucible, in which carbon chip or carbon grain materials are dropped in and they are submitted by the/xfi arc discharge. By applying the magnetic field, the production rate is about 6 times increased. Using this machine, fuUerene production from plant materials and used carbon-based materials are examined, and from charcoal, activated carbon, used synthetic rubber, and used toner, CQQ, C70 and higher fullerenes are successfully produced.
VACUUM CHAMBER
CARBON ROD
FIGURE 10. Schematic of a carbon-chip-injection type /xfi arc reactor.
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T. Mieno/Effects of gravity, gas and plasma on arc-production offullerenes
SUMMARY 1) Fullerenes are synthesized from carbon atoms via chain and ring structures in arc region. Reaction temperature is between 1,200 - 5,000 K. 2) High subhmation rate of carbon atoms from anode, long reaction time of carbon molecules in high temperature gas, high collision frequency with He to be annealed, low impurity density in space, are important for efficient production. 3) By means of a 12 m VST, 13 min of the integrated gravity-free discharge time is obtained. Collection rate of La@Cg2 at the gravity-free discharge is about 14 times higher than that of the normal discharge. 4) Hydrogen impurity strongly obstructs fuUerene synthesis. 5) Magnetic field improves production efficiency of fullerenes.
ACKNOWLEDGMENTS I would like to thank Professor Eiji Oosawa of Toyohashi University of Science and Technology for useful comments.
REFERENCES 1. T. Mieno and D. Yamane, J. Plasma Fusion Res. 74, 1444 (1998). 2. T. Mieno, H. Takatsuka, E. Kumekawa A. Sakurai and T. Asano, J. Plasma Fusion Res. 69, 793 (1993) [in Japanese]. 3. T. Mieno, J. Phys. Soc. Jpn. 62, 4146 (1993). 4. H. W. Kroto, J. R. Heath, S. C. O'Brein, R. F. Curl and R. E. Smalley, Nature 318, 162(1985). 5. G. vonHelden, M. Hsu, N. Gotts andM. T. Bowers, J. Phys. Chem. 9 7 8 , 8182(1993). 6. N. G. Gotts, G. von Helden and M. T. Bowers, Int. J. Mass Spectrum. Ion Proc. 149/150,217(1995). 7. Y. Yamaguchi and S. Maruyama, Chem. Phys. Lett. 286, 336 (1998). 8. R. E. Smalley, Ace. Chem. Res., 25, 98 (1992). 9. S. Aoyama and T. Mieno, Jpn. J. Appl. Phys. 38, L267 (1999). 10. S. Usuba etal, Proc. 14th FuUerene Sympo., Okazaki, 1998, p.24 (in Japanese). 11. T. Mieno, Jpn. J. Appl. Phys. 37, L761 (1998). 12. T. Mieno, Jpn. J. Appl. Phys. 35, L591 (1996). 13. B. H. Foing and P. Ehrenfreund, Nature 369, 296 (1994). 14. T. Mieno, FuUerene Sci. Technol. 3, 429 (1996). 15. T. Mieno, A. Sakurai and H. Inoue, FuUerene Sci. Technol. 4, 913 (1996). 16. T. Mieno, T. Asano and A. Sakurai, Advanced Materials '93,1/B, 1201 (1994).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
193
Formation of Dust and its Role in Fusion Devices J. Winter Institut fur Experimentalphysik II, Ruhr-Universitdt, D 44780 Bochum, Germany
Dust particles are produced in fusion devices by various plasma-surface-interaction mechanisms. This paper discusses properties of dust collected from fusion devices, some dust generation mechanism and the interaction of dust with the cold edge of a fusion plasma. The potential problems for operation and performance associated with the dust-plasma-interaction are outlined. First attempts aiming at the in situ observation of dust in fusion plasmas are presented.
1. Introduction Although it is known since long that dust particles are formed by plasma-surfaceinteractions in fusion devices (1), the problems associated with their presence have not been addressed in detail until recently (2-5). In the framework of the ITER project (Intemational Thermonuclear Experimental Reactor) it became evident that for conditions of large plasma fluences (steady state operation) the fomiation of dust raises serious safety problems (6,7). ITER, as most of the present large devices, will have carbon based wall components (graphite, Carbon Fiber Composites). The incorporation of tritium in carbon dust can be as high as 2 T atoms/C atom (6) giving rise to T inventories of up to several kg. Reactions of the highly activated radiation damaged dust with steam from a leaking cooling line may liberate H in large amounts creating an explosion hazard. In addition, the existence of dust may also influence the plasma operation and performance of the device (4). It is not only the dust particles formed during the specific discharge which may cause problems (1,4) but also the repeated interaction with dust particles accumulating on the bottom of the device. Experiments (8) and theoretical considerations (9) on the problem of dust shedding and levitation make it very likely that some of the existing particles may be levitated and sucked into a discharge. Depending on the confinement in the plasma edge, dust may even accumulate in certain areas of the cold edge plasma during the plasma discharge (2,4).
2. Characterization of dust and of its sources The particles collected from existing fusion devices with carbon wall components TFTR, D-III-D, JET, TEXTOR (10, 11,2) span a wide size range from several 10 nm up to the mm range showing essentially a log-normal distribution (10). Dust in the context of this paper is defined as loose material with particle diameters up to about 0.1mm. In the case of the D-III D tokamak, the mass concentration was found to be in
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J. Winter / Formation of dust and its role infusion devices
the range from 0 . 1 - 1 ^lg cm"^ on vertical surfaces, 10-100 |xg on the floor and lower horizontal surfaces. The total amount is estimated to be between 30 - 120 grams (10) for an integrated plasma exposure time of less than a few hours. The dust composition is dominated by carbon but may also include all other materials used inside the vessel or those for wall conditioning (B, Si (12)). Scanning Electron Microscopy (SEM) on the dust from TEXTOR reveals that the majority of particles are flakes of failed films produced by redeposition. The average H (D) flux at the leading edge of the graphite limiter of TEXTOR is 10^^ m"^ s\ The averaged erosion yield (including both sputtering and chemical erosion) is 2x10'^ resulting in a primary loss rate of 2x10^^ m"^s C atoms. This corresponds to a gross erosion of Im per year plasma exposure at the limiter tip. Almost all of this material is redeposited. Since the location of redeposition is not identical to that of erosion redeposited layers will accumulate. C incorporates hydrogen upon redeposition. The material is brittle with a layered or columnar structure and many stress induced cracks. Finally flakes fall off the surfaces. Figure 1 shows a SEM picture of a large flake from the deposition dominated area of the limiter. The majority of flakes found on the bottom of TEXTOR is similar to that in figure 1. This source of dust increases proportional to the plasma fluence. In the case of TEXTOR about 15% of the dust was
0.1 mm FIGURE 1. Large flake of redeposited carbonaceous materialfroma TEXTOR limiter
found to be ferromagnetic. Metal atoms are enriched in the flakes to preferential reerosion of the pure carbonaceous matrix (4). Flakes of typically a few 100 nm thickness may be liberated from spalling of thin films deposited for the purpose of wall conditioning (12). The films usually have a very good adhesion but may flake when the vessel is exposed to moist air during an opening. Following plasma operation may then liberate flakes by plasma-surfaceinteractions. In TEXTOR a significant number of almost perfect metallic spheres with diameters from 0.01 to 1 mm were identified. The metal spheres are formed most probably by coagulation of metal atoms on hot graphite surfaces (13). They are first evaporated
J. Winter / Formation of dust and its role infusion devices
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from metal wall elements by unipolar arcing, transported via the plasma to the limiter surface and, as their surface mobility increases when the limiter gets hot from to the thermal load of the plasma, they coagulate. They are then released by plasma-surfaceinteraction from the limiter. Arcing occurs mostly during the start-up phase of the plasma or at disruptions (uncontrolled rapid loss of the plasma). Electric fields are strongest during these phases leading to an easy local breakdown of the plasma sheath. Coagulation of particles from the vapor phase during arcing may be another possible mechanism for formation of the metal spheres (see below). Thermal fatiguing and thermal overloading of wall components is another source of dust. There exist power transients in fiision devices. Edge LocaUzed Modes (ELM's), plasma pressure gradient driven instabilities occurring at the plasma edge of divertor tokamaks can periodically deposit 2-5 % of the total stored plasma energy [14]. In the case of JET this was found to be up to a few GWm"'^ during 100 |LIS occurring at a rate of up to 100 Hz (15). Another transient is disruptions during which the total stored plasma energy is deposited on part of the wall. The deposited power in ITER is estimated to be lOOMJm''^ during 1 ms., i.e. power densities of 100 GWm"'^ on the divertor plates are expected. This leads to local evaporation of large quantities (several kg) of material. Even in present large devices the material loss during disruptions is significant. The failure mode for repeated exposure to low powers is a loosening of the material structure by propagating cracks as consequence of the large compressive stress. For graphite finally the ejection of grains may occur. In the TEXTOR dust graphite grains were easily distinguished from flakes by their facetted appearance. They have an average diameter of typically 5-20 |im. For higher power fluxes the evaporation or sublimation of material is dominant. The vapor has a high density close to the surface and the particles have consequently a short mean free path. Coagulation processes from supersaturated vapor leads to the formation of small particles (16,17) with average diameters of several 10 to 100 nm. This mechanism was verified in a laboratory experiment in which different carbon materials were repeatedly exposed to power loads of 0.5 - 1.5 GW m"^ using a high power electron gun (18). Transmission electron microscopy of the released material shows small globular clusters, see figure 2 left side, and evidence for the formation of fiiUerene like materials. Quantitative image analysis shows that most of the particles have sizes below 500 nm. Interestingly, a similar type of particles is identified in the TEXTOR dust (figure 2 right side) consisting of agglomerates of individual single globular particles of about 100 nm diameter. They too may have been formed by coagulation from C vapor during transients. Another possible mechanism is the growth of dust in the scrape off layer (2,5) involving negative hydrocarbon ions and multiple ion-neutral reactions. Hydrocarbons are released from the wall as a consequence of chemical erosion and their concentration may be as high as 10 % of the plasma density. The electron temperatures can be below 5 eV and electron densities about 10^^ m'^ close to the wall. Under these conditions the formation of negative ions by dissociative electron attachment is likely and conditions are close to that of a methane process plasma in which the formation of nanoparticle precursors was studied recently (19). Negatively
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J. Winter / Formation of dust and its role infusion devices
30 nm
30 nm
200 nm
FIGURE 2 . Left side: Transmission Electron Micrograph of a globular carbon particle likely to be formed by coagulation from dense C vapor (from ref. 18). Right side: Coagulated dust particle collected from TEXTOR (4).
charged ions and agglomerates are confined in the plasma edge: the sheath potential repels them from the surface, confinement in radial direction is provided by the magnetic field. A fiiction force from the background plasma is acting on these particles, driving them away from the stagnation point. Thus probable locations of dust particles are close to limiter-like protrusions of the wall, where an effective trap from the superposition of these forces exists (4). The high flux of UV photons in the fiision plasma tends to photo-detach the excess electrons, however. The balance of attachment and detachment rates will critically depend on geometrical factors and is difficult to assess a priori. No experimental proof of this mechanism exists to date.
3. Possible impact of dust on the performance of fusion plasmas Large particles falling into a fiision plasma can induce a disruption. Narihara et al.(3) used the Thomson scattering set up in JIIPT-2U to study the influence of small carbon particles with diameters < 2|Lim on the performance of tokamak discharges. Dust was dropped from the top of the machine and the authors concluded that an amount of about 10 particles of 2 ^im do not affect a fiiUy developed discharges, but that such particles existing in the main volume before start up of the plasma lead to increased initial impurity concentrations. Most particles fall to the bottom of the device at the end of a fiision plasma discharge. After some period a significant reservoir may have accumulated which increases with plasma fluence. Light particles may be re-injected into thefiisionplasma by magnetic and also by electric forces when dust flakes are charged by plasma contact. They may then be levitated close to the wall (8,9,4). Magnetic particles experience a VB-force and may be sucked into the main vessel volume upon rising the toroidal magnetic field. Plasma
J. Winter / Formation of dust and its role infusion devices
197
breakdown and bumthrough may be scuuu&iy impeded under these conditions. The intense impurity radiation often observed during the start-up phase of tokamak plasmas may be due to levitated dust. Erosion rates are often determined involving measurements of the line radiation of neutral atoms or ions. In building up clusters by agglomeration, neutral impurity atoms or molecules will be consumed before they can radiate, thus escaping emission spectroscopy. Divertor armor madefi-omberyllium was deliberately overheated in JET and Be atoms evaporated at a large rate. The data show still unexplained large differences between gross erosion and spectroscopically measured fluxes (20). This discrepancy may very well be due to agglomeration and Be dust formation. Fine particles can be transported far away from their point of origin and are subject e.g. to thermal forces, unlike e.g. massive splashes of molten metal, which are deposited close by. Dust particles are a sink for electrons and, for large concentrations, will change the balance between electrons and protons in the edge plasmas. This will result in a different sheath potential and heat transmission factor and in a different dynamics of the edge plasma.
4. Laser light scattering from particles in TEXTOR The experimental set up for the light scattering experiment on TEXTOR is described in detail in ref (5). The observation concentrated on the plasma edge region. Not knowing the location, the spatial distribution and density of the particles, a volume as large as possible was investigated with adequate spatial resolution compromising however the laser power density. The beam of an Ar-ion laser (70 mW cw) was guided through a fiber to a tilting mirror on top of TEXTOR sweeping it across a poloidal cross section (diameter 1.04 cm) with a width of about 2 cm at the bottom of the vessel. The observation of the scattered light fi-om almost the fiiU poloidal cross section was made through a tangential port, toroidally about 1.5 m away by a CCD camera with Hght amplification and a narrow band interference filter for the laser wavelength. The geometry was dictated by the available existing ports. When the vessel was vented to nitrogen moving particles levitated from the bottom by gas convection, were observed in the lower 1/3 of the vessel when the pressure reached values > 400 mbar. Light scattering by particles was also observed in the initial pump down phase from atmospheric pressure. Attempts to measure scattering signals during the presence of tokamak discharges failed. The background intensity of plasma light is orders of magnitude larger than the intensity from the previously identified scattering events. However few scattering signals from particles in the lower part of the vessel could be identified uniquely before discharge initiation and also right after the discharge. Though the data are scarce, the results strongly suggest that existing dust is levitated and may interact with the fiision plasma.
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J. Winter / Formation of dust and its role infusion devices
5. Conclusions Dust particles found in existing tokamaks can be correlated with various production mechanisms involving plasma-surface-interactions and gas phase agglomeration. Plasma induced growth is a potential mechanism, too. Failed redeposited layers are a main source of dust. The dust quantity increases with plasma fluence. In addition to safety hazards, dust is likely to impact operation and performance of fusion devices. Dust is thus an important issue for future long pulse devices.
Acknowledgement The author greatfully acknowledges the help of G. Gebauer in performing the laser light scattering experiment.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Ohkawa, T., Kako Yugo Kenkyo 37, 117(1977) Winter, J., Proc. 24* EPS Conf. Controlled Fusion and Plasma Physics, EPS Conf. Abstr. 21 A, 4,1777(1997) Narihara, K., Toi, K., Yamada, Y., et al., Nucl. Fusion 37, 1177 (1997) Winter, J., Plasma Phys. Contr. Fusion 40, 1201, (1998) Winter, J., Gebauer, G., Proc. 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Federici, G., et al., 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Piet, S.J., Costley, A., Federici, G., et al., Proc. 17* lEEE/INPSS Symp. Fusion Engineering, Oct. 6-10, 1997, San Diego, USA, IEEE 97HC35131,Voll, 167 (1998) Sheridan, I.E., Goree, J., Chiu, Y.T., Rairden, R.L., Kiessling, J.A., J. Geophys.Res. 97, A3, 2935 (1992) Nitter, T., Aslaksen, T.K., Melandso, F., Havnes, O., IEEE Trans. Plasma Science 22, 159 (1994) McCarthy, K.A., et al., Fusion Technology 34, 728 (1998) Peacock, A.T., et al., 13* Int. Conf. Plasma Surface Interactions in Controlled Fusion Devices, May 18-22, 1998, San Diego, USA, J. Nucl. Materials, in press Winter, J., Plasma Phys. Contr. Fusion 38, 1503 (1996) Behrisch, R., Borgesen, P., Ehrenberg, J., et al, J. Nucl. Materials 128+129, 470 (1984) Leonhard, A.W., Suttrop, W., Osborne, T.H., et al., J. Nucl. Materials 241-243, 628 (1997) Lingertat, J., Tabasso, A., Ali-Arshad, S., et al., J. Nucl. Materials 241-243, 402 (1997) Schweigert,V.A., Alexandrov, A.L., Morokov, Y.N., Bedanov, V.N., Chem.Phys.Lett. 235, 221 (1995) Schweigert,V.A., Alexandrov, A.L., Morokov, Y.N., Bedanov, V.N., Chem.Phys.Lett. 238, 110 (1995) Bolt, H., Linke, J., Penkalla, H.J., Tarret, E., Proc. 8* Int. Workshop Carbon Materials, Sept. 34, 1998, Jiilich, FRG, Physica Scripta, in press, and private communication Winter, J., Leukens, A., Proc 14* Int. Conf. Plasma Chemistry, August 2-6, 1999, Praha, Czech Republic Campbell D.J. and the JET Team, J.Nucl. Materials 241 -243, 379 (1997)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
199
The Formation and Behavior of Particles in Silane Discharges Alan Gallagher JILAy University of Colorado and National Institute for Standards and Technology Boulder, CO 80309-0440 ABSTRACT: Particle growth in silane RF discharges and the incorporation of particles into hydrogenated-amorphous-silicon (a-Si:H) devices is described. Measurements of particle density and growth in a silane RF plasma, for particle diameters of 8-50 nm, are described. A decrease in particle density during the growth indicates a major flux of these size particles to the substrate. Particle densities are a very strong function of pressure, film growth rate and electrode gap, increasing orders of magnitude for small increases in each parameter. A full plasma-chemistry model for particle growth from SiHm radicals and ions has been developed, and is outlined. It yields particle densities and growth rates, as a function of plasma parameters, which are in qualitative agreement with the data. It also indicates that, in addition to the diameter >2 nm particles that have been observed in films, a very large flux of SixH^ molecular radicals with x >lalso incorporate into the film. It appears that these large radicals yield more than 1% of thefilmfor typical device-deposition conditions, so this may have a serious effect on device properties.
INTRODUCTION In silane (SiH4) and SiH4-H2 discharges, a-Si:H film deposition is initiated by electron collisions that dissociate SiH4. This yields primarily neutral-radical fragments, but is accompanied by a small fraction of positive ions (cations) and a very small fraction of negative ions (anions). The neutral radicals Si, SiH, SiH2, and H rapidly react with SiH4 to form stable molecules or SiH3, while SiH3 reacts only with the a-Si:H film. Thus, film growth is primarily due to the reactions of SiHs radicals with the surface. Cations induce only a few percent of the growth, but as they typically strike the surface with energies in the 1-30 eV range this can be important. The anions can not reach the growing film due to sheath fields, so for many years these were neglected as a contributor to film electronic properties. However, it has now become clear that anions can profoundly influence the film, for they cause Si particles to grow in the plasma. Since anions (SixHn~) are trapped in the plasma, there is plenty of time for SiH3anion reactions to induce their growth, and this can lead to enormous values of x, the number of Si atoms. Once x exceeds perhaps 30, these anions are beUeved to be relatively spherical with a structure similar to that of the a-Si:H film: clustered Si at near-crystalline density with occasional H inside and an H terminated surface. Thus, these larger clusters are generally described as "particles." From standard "particle-in-plasma" theories, particles with x>200 (radius Rp>l nm) are expected to be negatively charged and consequently trapped in the plasma by the sheath fields.^ (The number of negative charges increase Unearly with Rp.) With gas flow, these charged particles are dragged away to the pump once they grow to 0.1-1 |im size (x=10^-10^^) and the film does not suffer. However, we discovered several years ago that particles of 2-15
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A. Gallagher/The formation and behavior ofparticles in silane discharges
nm diameter escape the plasma and incorporate into the films.^ These particles constituted 10"'-10' of the film volume, and their surface bonds to the surrounding film exceed the electronic defect density of a-Si:H by many orders of magnitude. Here I report that when smaller particles not visible in that experiment are included, particles probably constitute 110% of the film volume and their bonds to the surrounding film are a significant fraction of all Si-Si bonds within the film. These particles are a major concern, as it appears likely that voids and strained bonds occur at particle-film interfaces.
MEASUREMENTS OF PARTICLES IN SILANE PLASMAS Since the original detection of particles in silane plasmas by Roth et al.,^ the most common method of measuring particle suspended in plasmas is still Hght scattering. However, important insights have also been gained from measurements of negative ions escaping the plasma afterglow,"^ and from scanning tunneling microscope^ and transmission electron microscope (TEM) measurements of particles collected on the electrodes.'^ Unfortunately, most of these measurements were not carried out in pure SiH4 or SiH4-H2, or for conditions that yield device-quality a-Si:H films. Thus, to understand particle growth under device production conditions, I have utilized primarily the insights, not the particle data, from these experiments. In my laboratory we have concentrated on the conditions and gases used to produce devices. Initially, we studied the accumulation of larger particles (Rp >50 nm) at the downstream end of the RF discharge and their escape to the pumps.^ These particles can influence the discharge operating conditions, but they do not incorporate into devices. Thus, in the last few years we have studied^ the very small particles that form directly below the substrate and are indicative of those that incorporate into films. It is difficult to study the very small particles (Rp <20 nm) that incorporate into films, because their light scattering cross section is very small and the scattering signal only establishes UpRp^ where Upis the particle density. To establish Rp we measure particle diffusion, since neutral particles diffuse out of the discharge afterglow in a time proportional to Rp'^. An example of the scattering signal versus time is shown in Fig. 1. The exponential decay well after discharge termination results from fundamental spatial-mode diffusion of 20 nm particles. With Rp thus determined, the size of the scattering signal yields Up. This method works for 4nm
A. Gallagher / The formation and behavior of particles in silane discharges
201
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linear growth result is thus quite reasonable, and in contrast to most previous data, demonstrating the importance of studying the low power conditions and gases appropriate for device production. A second result is that the measured particle density decreases with discharge operating time, as the particles grow from Rp = 4 to 25 nm. Particle agglomeration rates are far too small to explain this, so particles must be escaping the plasma to the electrodes. Indeed, the escape rate implied by the data is consistent with measurements in Ref. 2. This rate of particle escape requires that a major fraction of particles are neutral, so that they can escape the plasma. To understand how and why this happens, as well as particle formation and growth, we have developed a full plasma-chemistry model that is outlined in the next section.
A MODEL FOR PARTICLE GROWTH AND ESCAPE TO THE ELECTRODES In this steady-state model, SiHs, SiHn"^ and SiH3~ are formed by electron collisions, and these radicals grow and alter their charge due to collisions between themselves and with electrons and SiH4. Much of this growth results from consecutive additions of one Si atom. The negatively charged complexes, SixHm", are trapped in the plasma, but neutral and positively charged SixHm radicals can diffuse to the electrodes. Thus, there is a continual attrition of complexes as they grow in size. The model only uses x and charge z to describe all SixHrn"" radicals or particles. With increasing x, particle-diffusion slows, the cross section for adding a Si atom increases, and the particles can accommodate increasing numbers of negative charge. Thus, diffusive loss is most severe for small x values, and the density of
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A. Gallagher/The formation and behavior ofparticles in silane discharges '^4^
visible (x>10 ) particles depends strongly on the fraction of small-x particles that are lost to diffusion before they can grow to this "large" size. The competition between growth and diffusive loss depends strongly on discharge conditions. At higher powers and pressures, the radical density is high and particle growth is rapid, while diffusion slows with increasing pressure and discharge gap (L). Thus, at high P,L and G many particles remain in the plasma and visible particle densities are high. One might reduce particle incorporation into films by simply lowering P and L, at low P and L ion bombardment of the film increases in energy, and this can also cause loss of film quahty. I will now describe the model in more detail, using the notation n(x,z) for the density of particles with charge z and containing x Si atoms. The first requirement is to establish the densities n(l,z) of SiHm radicals and ions with charge z=0,l, and -1. Since SiHs diffusing to the surface produces most of the film growth rate (G), this estabUshes the SiH3 production rate (Fo), and the density n(l,0) then follows from the SiH3 diffusion coefficient and the electrode gap. Next, based on electron collision coefficients I assume that SiH3, SiHm^ and SiHs" are produced in the ratio Fo/Fi/F.i = 125/5/1. This yields n(l,l) = Fi/R(l,l) and n(l,-l) = F.i/R(l,-1), where R(x,z) is the loss rate for SixHm^, due to transforming into other x,z species and diffusion to the electrodes. Next, I calculate the particle densities n(2,z) from these n(l,z), based on the rate of Si atom addition from collisions with SiHm^, SiH3 and SiH4. The diffusive loss rate for these x=l molecules is included in this x=l to 2 step; it competes with the growth rate to x=2, causing some loss of x=l particles. This step-by-step (x-x+1) particle growth is repeated to x=10'* or 10^, a visible-particle size where comparison to experiment can be made. At appropriate xi values where Si particles carrying i>l negative charges become stable, these additional charge states are included. Iteratively calculating only x-x-i-1 steps is an approximation, as some collisions add more than one Si atom. However, it greatly simplifies the calculation and is reasonable because particle growth is dominated by collisions with SiH4 for x<100, and by collisions with SiH3 for x>100. An example result is shown in Fig. 2 for discharge conditions that are typical of those in Ref. 7. In Fig. 2, first consider the neutral particles that are labeled n(z=0) and flux(O). These neutral particles are molecular radicals, with x=l representing SiH3. The x=2 radicals result from collisions between a pair of SiH3, and the x=3 primarily result from collisions between x=l and x=2 radicals. As x increases from 1 to 2, n(x,z=0) and F(x,0) drop a factor of -2.5 and -3.5 respectively. As x further increases, a smaller decrease per x-x+1 step occurs, due to an increase in the rates of growth/diffusive loss. Even before considering the larger particles, the heavy-radical incorporation indicated in Fig. 3 could have significant effects on film properties. (This has been pointed out and modeled previously.^) The rate of F(x,0) falloff versus x is very sensitive to plasma parameters; it slows if G, L or P is increased and visa versa. For x>20, n(x,0) begins to follow the negative ion density, n(x,-l), as these populations become strongly coupled by electron and cation collisions. Turning now to the anions, labeled n(-l) in Fig. 2, their density is almost independent of X for x<50 due to very rapid growth and only minor loss by cation coUisions. Reactions with silane cause this rapid growth, in a process that has been observed"^ to occur for x<~30. For x>100 this anion growth rate slows and diffusive loss of neutral particles causes the coupled densities of z=0 and -1 particles to decrease.
A. Gallagher/The -
formation and behavior of particles in silane discharges
203
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X (dimensionless) Figure 2. Densities of particles containing x Si atoms and z charges, and flux ot z=U ana +i paiticles to the electrodes, in units of lOVs and cm'' of plasma volume. The RF plasma conditions are 0.3 Torr pure SiH4, T=300K, 0.3 nm/s film growth rate, and 1.3 cm electrode gap, as in Ref. 7. To simplify the figure only the x=l to 730 range is shown here; for x>730, pardcles with z = -3 also occur. The charged pardcle densifies are Ug = 3xl0^cm"^n. n+ = 3.6x10^" c m l 10^3 03 CO
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204
A. Gallagher / The formation and behavior ufparticles in silane discharges
Considering next the cations, labeled n(+l) and flux(+l) in Fig. 2, the density of x=l-3 is severely lowered by very rapid growth reactions with SiH4. However, these reactions slow at x=3 and essentially terminate by x=6, and only slower reactions with radicals cause further cation growth. The result is a peak in n(x,l) at x=5. For x>30, the cations also become closely connected to the z=-l and 0 particles by charge-changing collisions, and all three densities fall together. The cation-silane reaction rates for x<8 are measured values.^ However, this x=5 peak contradicts mass-spectrometer measurements through the substrate electrode of silane RF discharges."^'^^ The most Hkely explanation for this discrepancy is that the larger-x ions exist in the neutral plasma region, but breakup while traversing the sheath due to energetic collisions with SiH4. This is indirectly supported by comparing mass spectrometer observations from our laboratory, taken at the side of the neutral plasma region and through the cathode of a silane discharge. ^^ Particles carrying two negative charges become possible at x=60, and, as can be seen, these play a minor role for these conditions. This, as well as the fact that n(x,0)/n(x,l)>l for x>100 in Fig. 3, is due to n+ > Ue, where n+ is the total cation density and Ug the electron density. This is a neutral plasma, but a charge n. > Ue is carried by the negatively-charged particles, mostly by the anions with x<50 in Fig. 3. The ratio n(x,0)/n(x,-l) is set by a balance between electron attachment to n(x,0) and cation attachment to n(x,-l), so it is a strong function of n+ZUe. The unusually large neutral fraction results from n+Zug^ 100 for this plasma. Figure 3 again shows particle densities and (total) flux to the electrodes for the plasma conditions of Fig. 2. The x range is extended to 10"^ (Rp = 3.6 nm), higher charge states are included, and the total density is also plotted as a density of particles per nm of radius. (All other densities are per unit x.) Charge states -5 and -6 also occur in this x range, but are unimportant for these conditions and are not plotted. Note that the particle density per nm of radius is approaching a constant at large x. This is the steady-state result of a constant dRp/dt growth rate, when diffusive loss is minor. Very large particles thereby accumulate in this homogeneous plasma model, whereas in the flowing-gas plasmas used to produce devices these would normally be carried to the downstream end. However, particles may trap and grow to large size above irregularities in the electrode surfaces, since these can produce a local maxima of the plasma potential that traps negative particles. To estimate the particle sizes that occur below substrates without such irregularities, the typical measured Rp growth rates from the experiment described above should be multiplied by gas dwell times in the reactor. This might typically yield lOnm/s times 0.5s = 5 nm for a device reactor. The particle density at x>10'^ is a very sensitive function of P,G, and L, as is shown in Fig. 4, where n4 is the total particle density per nm of radius at x=10'^. Since the total particle negative charge cannot exceed the cation density, most particles become neutral once the particle density exceeds n+. Particles then agglomerate and escape the plasma with a high probabiUty, so this sets a Umit on total particle density. The data is plotted versus P«L^^ because this reduces results at different P and L values to a relatively common curve. The extreme sensitivity to the plasma parameters is consistent with the results of the experiment described above, as is the range of P,L and G that yield the measured particle densities. However, the exact values of P,L and G for which the model yields these densities is not meaningful, as many collisional rates must be assumed.
A. Gallagher/The formation and behavior of particles in silane discharges
205
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Figure 4 LogioCnVcm''^) as a function of RF discharge parameters. G=film growth rate in A/s. L=electrode gap in cm, P=silane pressure in Torr at 300 K, and n4 is the density of 3.6 nm radius (x=10'^) particles per nm of radius.
CONCLUSIONS In order to mitigate particle incorporation into a-Si:H devices, we first set out to understand why this incorporation occurs. Our model has shown that, in addition to the 2-15 nm radius particles that can incorporate at lO'"^ to 10"^ of the film density, large SixHm complexes with x>10 incorporate into films as more than 1% of the volume. It seems likely that some of the interfaces between these particles or large complexes and the remaining film are not well structured or passivated. Particle growth in the plasma is an inevitable consequence of the strong electron-attachment energies of Si-containing molecules and particles. SiHs" production and growth into SiiooHm" (Rp=0.8 nm) occurs within 1 ms of discharge initiation, and cannot realistically be avoided. Particle growth from x=100 to 10"^ (Rp = 3.6 nm) or larger can be mitigated by using on-off modulation or low P,L and G, but this is achieved by allowing the growing particles to escape to the electrodes before they reach "large" size. As there is not much one can do to prevent the growth of particles, it is important to ask why they incorporate into the film, and if this might be avoided. Alternatively, if one wishes to incorporate crystalline siHcon particles into the film, can we control this? Only neutral or positively charged particles diffuse to the electrodes, so it is important to understand why a large fraction of the particles are neutral or positive. The reason is that n+ and n. are much larger than Ue; this occurs because n+ and n+ build up to a value where cationanion collisions balance the rate of producing anions. There doesn't seem to be much one can do about this in SiH4 or SiH4-H2 gases; a large fraction of particles will be neutral and will escape to the surrounding surfaces before being dragged to the downstream end of the reactor.
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A. Gallagher/The formation and behavior ofparticles in silane discharges
There may be something one can do about where the particles escape to, which need not be equally to all surfaces or regions of the electrodes. One possibility is to drive particles to the RF electrode with the thermophoretic force, which causes all size particles to drift toward colder regions with a velocity proportional to the temperature gradient. (The very small ones partially back diffuse against this drift.) Since the substrate is normally heated to --2400, this temperature gradient would naturally occur if the RF electrode were cooled. A second method of controlling where particles escape is to design advantageous variations in plasma potential. If the discharge runs harder in some regions between the electrodes, the plasma potential is larger in these regions, yielding a potential well for negatively charged particles in the plane between the electrodes. It may be possible to guide particles out to the plasma edges with certain electrode structures.
ACKNOWLEDGEMENTS I wish to thank M. A. Childs and A. V. Phelps for many valuable ideas and suggestions. This work has been supported in part by the National Renewable Energy Laboratory under contract DAD-8-18653-0L
REFERENCES 1. Daugherty, J. E., Perteous, R. K., Kilgore, M. D. and Graves, D. B., J. AppL Phys. 11, 3934 (1992). 2. Tanenbaum, D. M., Laracuente, A. L., and Gallagher, A., Appl Phys. Lett. 68, 1705 (1996). 3. Roth, R. M., Spears, K. G., Stein, G. D., and Wong, G., Appl. Phys. Lett. 46, 253 (1985). 4. Howling, A., Sansonnens, A. L., Dorier, J.-L., and Hollenstein, Ch., J, Appl. Phys. 75, 1340(1994). 5. Boufendi, L. Plain, A., Blondeau, J. Ph., Bouchoule, A., Laure, C., and Toogood, T., Appl. Phys. Lett. 60, 169 (1992). 6. Jelenkovic, B. M., and Gallagher, A., J. Appl. Phys. 82, 1546 (1997). 7. Gallagher, A., Barzen, S., Childs, M., and Laraquente, A., "Atomic Scale Characterization of Hydrogenated Amorphous SiUcon Films and Devices" NREL/SR-520-24760, Golden, Colorado (June 1998); Childs, M. A., and Gallagher, A., in preparation. 8. Gallagher, A., J. AppL Phys. 63, 2406 (1988). 9. Perrin, J., Leroy, O., and Bordage, M. C , Plasma Phys. 36, 1 (1996). 10. Haller, I., J. Vac. ScL Technol. Al, 1376 (1983). 11. Gallagher, A., and Scott, J., "Diagnostics of a Glow Discharge Used To Produce Hydrogenated Amorphous SiUcon Films," Annual Report to Solar Energy Research Institute (now NREL) for Contract XJ-0-9053-1, Golden, Colorado (April 1981). * Member, Quantum Physics Division, NIST. E-mail address: [email protected]
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
207
Dust particles influence on a sheath in a thermoionic discharge M. Mikikian, C. Amas, K. Quotb* and F. Doveil Equipe Turbulence Plasma - Lab. de Physique des Interactions loniques et Moleculaires, UMR 6633 - CNRS / Univ. de Provence, Centre de Saint Jerome, case 321, Avenue Escadrille Normandie Niemen, 13397 Marseille cedex 20, France *Laboratoire d'Astronomic Spatiale, Traverse du Siphon -12eme Arr./BP 8, F14476 Marseille cedex 12, France
Abstract. The basic characteristics of dust particles trapped in the sheath of a plate embedded in a plasma produced by emissive hot filaments, are reported. For different plate biases, taking into account the measured plate sheath potential profiles, the high negative dust charge is determined. This charge value is compared to the one predicted by the Orbital Motion Limited model when the contribution of the primary electrons emitted by the filaments is considered. When several dust particles levitate, a sheath potential steepening is observed. The analysis of a collision between two dust particles yields the shielding length at the levitation height.
INTRODUCTION We report experiments on dust particle levitation in the sheath of a conducting horizontal disc plate embedded in a continuous discharge plasma. The ionization sources are hot filaments emitting energetic primary electrons. In these conditions, using the Orbital Motion Limited (OML) model (1), we show that the main contribution of the high negative dust charge is due to the primary electrons. Using a differential emissive probe diagnostic, we measure the sheath-presheath profile of the disc plate, for different negative plate biases. Taking into account the obtained potential distributions, the charge of an isolated dust particle is established by measuring the levitation height h. When the negative plate voltage is increased, h increases too. For any plate bias, the equilibrium height corresponds to the same sheath potential (same electric field) where the balance between the electric and gravitation forces is achieved. The charge found in this way is compared to the value predicted by the OML model. Using the non-intrusive Laser Induced Fluorescence (LIF) diagnostic, we observe that: i) the dust particles levitate below the sheath edge, in a layer where the ions are supersonic and ii) their presence produces a steepening of the sheath potential profile. Using the classical Coulombian scattering relations for a collision analysis between two dust particles and taking into account the measured parameters like scattering angles.
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M. Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge
impact parameter, minimum approach radius, we can find the shielding length at the levitation height.
EXPERIMENTAL ARRANGEMENT The experiments are performed in a multipolar device in series with a continuous discharge circuit, operating at low argon pressure: Par =10" mbar (coUisionless plasma). The primary electrons emitted by hot tungsten filaments are accelerated toward the grounded wall by a negative voltage VD = - 4 0 V (the plasma appears at VDO =-35 V). In the plasma center, we have set a conducting disc plate which can be left to the negative floating potential or can be more negatively biased by an extemal power supply. The dust particles are hollow glass micro-spheres of radius: rd = (32 ± 2) |im, with a mass density: pa ~ 110 kg/m^. In our standard conditions, the dust particles levitate in an horizontal plane (2), parallel to the plate (liquid phase). Their trajectories are rectilinear, with sudden direction changes in the horizontal plane due to their collisions. In order to establish the potential profiles perpendicularly to the plate, we use two diagnostics: i) the differential emissive probe (DEP) system (3), displaced perpendicularly to the plate, for five different plate biases and ii) for a given plate bias, the non intrusive LIF diagnostic (4), in presence of dust particles. In this case, the laser beam (3 mm diameter) propagates perpendicularly to the plate and the collection optics system is displaced vertically. A CCD camera is used to measure the levitation heights.
DUST PARTICLE CHARGE A dust particle acquires a negative charge when the balance of the ion and electron currents on its surface is reached (5,6). These currents are given in a reasonable approximation by the OML model where it is assumed that the dust grain behaves like a spherical probe in a coUisionless plasma and verifies: r^ « X^D, A.D being the Debye length. So, the dust charge must fulfill the following electrostatic equilibrium equation: I i ( V ) + I e ( V ) + Ipe(V) = 0
(1)
where Ii is the ion current (monoenergetic ions flowing from the plasma to the plate), Ig is the current of the background plasma electrons (Maxwellian population) and Ipe is the current of the primary electrons well represented in our experimental conditions, by an isotropic drifting MaxweUian population (2). V is the difference between the dust surface potential Vd and the local potential. The charge of the grain simply is: (Qd)oML= CVd where C is the capacitance of a sphere with radius r^j. The standard conditions of our experiments give: V(i= -17 V. This high negative value is mainly due to the contribution of the primary electrons and for r^= (32 ± 2) [im, the resulting negative charge is: (Qd)oML= (3.7 ±0.3) 10'e-
(2)
M Mikikian et al/Dust particles influence on a sheath in a thermoionic discharge
209
EXPERIMENTAL RESULTS Experimental dust particle charge Figure 1 gives different potential profiles in function of the distance to the plate in mm, obtained with the DEP diagnostic, for five plate biases: the floating potential -26.5 V and for-32,-38,-44 and-50 V.
presheath
fitting functions
• Vbias = -50 V o -44 V ° -32 V V -26.5 V -60
^ -38 V
H — I — I — \ — I — I — \ — \ — I — I — I — I — I — I — \ — I — I — I — I — h
0
1
2
3
4
5
6
7
8
Distance from the plate (mm)
9
10
FIGURE 1. Plate sheath-presheath vertical profiles, for five plate biases. The symbols are the measurements and the curves, the best fits. The plate biases are: floating potential -26.5 V and applied voltages- -32 -38 -44 and-50 V. & . , ,
The effect of increasing the plate voltage is to shift the potential distribution toward the plasma reached, at about 20 mm. Each set of measurements is well fitted by a function: V(z) = -a.exp(-b.z) + c
(3)
where z is the vertical direction. The parameters a and b depend on the plate voltage and c has a common value equal to the plasma potential: Vp = -1.5 V.
FIGURE 2. Superimposition of five CCD camera frames showing the levitation height increase of the same dust particle when the plate voltage is increased from -26.5 V to -32, -38, -44 and -50 V.
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M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge
TABLE 1. For each plate bias, dust height, sheath potential and electricfieldand dust charge values Vbias (V)
-26.5
-32
-38
-44
-50
hexp (mm)
1.90
2.33
2.62
2.90
3.34
Veq (V)
-3.88
-3.60
-3.89
-4.25
-4.40
Eeq (kV/m)
2.11
1.62
2.11
2.17
2.40
Qd (10' e-)
4.3
5.6
4.3
4.2
3.8
The charge can now be estimated by measuring the levitation height. Using the empirical function (3), the distance from the plate gives directly the sheath potential where the electric force Fg due to the negative plate balances the gravitation force Fg. Because we can inject a unique dust particle in the plasma, we can study its position change over the plate. In Figure 2, five CCD camerafi-amesare superimposed, each one showing the position of the same levitating isolated dust grain for a given plate voltage. We observe that when the plate bias is increased, the levitation height rises. For each Vbias? we have reported in Table 1 the measured levitation height hexp. Each one corresponds to almost the same equilibrium sheath potential Vgq and local electric field Egq. For each case, the dust charge found by writing Fg = Fg is calculated and provides approximately the same value. So, we conclude that an isolated dust particle of radius 32 |im is in equilibrium in the plate sheath where the mean local potential is Veq = (-4.01 ± 0.12) V. Its mean charge is Qd = (4.4 ± 1.2) 10^ e". This value is slightly higher than the one predicted by the OML model given by solution (2). We assume that this discrepancy can originatefiromtwo facts: i) the use of a DEP diagnostic could induce an artificial flattening of potential profiles, producing a decrease in the local electric field and (or) ii) a local sheath potential steepening in presence of dust particles could appear. So, with our measurements, in order to balance the gravitation effect, the dust particle charge must be higher. We have checked the assumption ii) through measurements with the LIF diagnostic when several dust particles levitate.
Sheath potential steepening in presence of dust particles When the optics collection system of the LIF diagnostic is displaced in the sheath plate, perpendicularly to the plate, biased at Vbias = -38 V, the ion velocity distribution function f(vi) is shifted toward higher ion velocities. We measure the ion drift velocity v^ax corresponding to the maximum of f(vi), versus the distance h to the plate (Figure 3-a), firom 2.3 mm to 3.2 mm, the sheath layer where dust particles are trapped. The full circles are the data with dust particles and the open triangles without dust particles. For a given h, the drift velocity is higher when dust grains levitate (ion acceleration). The error bar, due to the estimated error of v^ax position is the same for all the data and it is given at h = 2.6 mm. The dust particles behavior in this experiment was the following: i) at each time, they were six to eight dust grains crossing the laser beam and ii) they were oscillating in a vertical plane (vertical oscillation observed for high primary electron emission).The mean levitation height was: ho=(2.7±0.05) mm and the oscillation amplitude: Ah = 0.5 mm.
M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge
I - ^ With dust particles 3.0E+03
A NO dust particle
J
211
f-^ With dust particles A NO dust particle
Vbias = -38V T
^ ^A
1 I 2.5E+03
•
A
T
A " " «
j 2.0E+03 -1 2.2
A - ^ .
a) 1 - H — —t 2.4
h-
2.6
1
1
1
2.8
h (mm)
1
11
3.2
2.4
2.6
2.8
h (mm)
FIGURE 3 a). Ion drift velocity vmax, versus the distance to the plate (full circles: with dust particles, open triangles: without dust particle), b) corresponding sheath potential profiles (full circles: with dust particles, open triangles: without dust particle).
This oscillation amplitude corresponds approximately to the width of the sheath layer (oscillation influence) where the curves presented in Figure 3, are separated. Moreover, for Te = 1.8 eV, the ion sound speed is Cg ~ 2.1 10^ m/s. The ions reach this velocity at a plate distance of the order of 3.1 mm. So, the dust particles levitate in a sheath layer where the ions are supersonic. The corresponding potential profiles are found writing that the energy of an ion of velocity Vmax? in the potential V is: E = (m^ v^^x) / 2 + V . The ion energy conservation yields: E = eVp. So, V can be deduced and it is given in Figure 3-b) versus h. For a given h, the potential is higher in absolute value with dust particles than without. In other words, the slope of the curve with dust particles is higher than the slope of the one without dust particles (except at each end where both curves begin to superpose). This result is the signature of an electric field increase in this region.
Local shielding length determined through collisions The analysis of a collision (7) between two dust particles can provide the local shielding length. In the presented example, before the collision, one of the dust grain is moving (dust 1) while the second one is motionless (dust 2). Making the following assumptions: a) the collision is elastic (no charge lost) and b) the interaction force between both particles is a central force, we have checked the momentum and energy conservation during the collision. In the laboratory frame, the classical scattering of a charged particle on another, provides the equation: tg(Xo) =
sin(2yo) mi Ivci2 -COS(2V|/Q)
(4)
linking the projectile deviation angle Xo and the target trajectory angle VJ/Q, after the collision, mi,2 are respectively, the mass of the dust grain 1 and 2. Measuring (Xo ? H^o)' we
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M Mikikian et al /Dust particles influence on a sheath in a thermoionic discharge
have checked that (4) is fulfilled. According to assumptions a) and b) given previously, the following equation also must be fulfilled:
1-^-4 1
2
r^
=0
(5)
where p is the impact parameter of the dust grain 1, |i is the reduced mass of the dust particle system, r^ is the minimum approach radius, vi the dust grainl velocity before the collision and (|)(r) the interaction potential between both particles. Equation (5) must be fulfilled whatever the variation law of ^(r). Taking a screened Coulombian potential interaction, ^(r) is given by: (^(r) = ^ ^ e x p ( - - L )
(6)
where Zi^2^ are respectively the charge of dust grain 1 and 2 andX^g is the local shielding length. Measuring p, r^, vi and calculating the dust charges with the OML model, the only unknown parameter in equation (6) is X^.ln this experiment, where the plate bias is: Vbias ^ -50 V and the levitation height: h ~ 3.3 mm (no vertical oscillations), we find: X^ = (0.46 ±0.12) mm. The measured plate sheath potential profile allows us to estabUsh the electron and ion densities at any position from the plate and then the corresponding electron and ion Debye lengths. At h - 3.3 mm, X^ is close to the electron Debye length.
ACKNOWLEDGEMENTS The authors would like to thank V. N. Tsytovich and S. V. Vladimirov for helpful discussions during their visits in Equipe Turbulence Plasma. This work was supported by a grant from the Direction des Recherches et Etudes Techniques.
REFERENCES 1.Bernstein, I. B . , and Rabinowitz, I. N., Phys. Fluids 2, 112-121 (1959). 2. Amas, C , Mikikian, M., Quotb, K., and Doveil, F., "Dustparticle levitation experiment in a hot cathode discharge at low argon pressure", presented at the ICPP&25^ EPS Conf on Contr. Fusion and Plasma Physics, Praha, 29 june-3 July, ECA 22C, 2493-2496 (1998) 3. Yao, W. E., Intrator, T., and Hershkowitz, N., Rev. Sci. Instrum. 56, 519-524 (1985) 4. Goeckner, J., Goree, J., and Scheridan, T. E., Phys. Fluids B 4, 1663- 1670 (1992) 5. Whipple, E. C , Rep. Prog Phys. 44, 1197-1250 (1981) 6. Havnes, O., Aanesen, T. K., and Melands0, J. Geophys Res. 95, 6581-6585 (1990) 7. Konopka, U., and Ratke, L., and Thomas, H. M., Phys. Rev. Lett. 79, 1269-1272 (1997)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) 2000 Elsevier Science B.V
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PLASMA DEPOSITION OF SILICON CLUSTERS: A WAY TO PRODUCE SILICON THIN FILMS WITH MEDIUM-RANGE ORDER ? P. ROC A i CABARROCAS, Laboratoire de Physique des Interfaces et des Couches Minces, (UMR 7647 CNRS), Ecole Polytechnique, 91128 Palaiseau, Cedex, France ABSTRACT The growth of hydrogenated amorphous silicon films is often explained by the arrival of SiHx radicals on the substrate and the subsequent cross-linking reactions leading to an homogeneous material which can be described by a continuous random network. Here we summarize our recent work on a new class of silicon thin films produced under plasma conditions where silicon clusters and radicals contribute to the deposition. The main aspects are: i) silicon clusters with sizes of the order of 1-5 nm are easily formed in silane plasmas; ii) these silicon clusters can contribute to the deposition and lead to the formation of films with mediumrange order ("polymorphous silicon"); iii) despite their heterogeneity, the films have improved transport properties and stability with respect to a-Si:H. The excellent transport properties are confirmed by the achievement of single junction p-i-n solar cells with efficiencies close to 10 %. INTRODUCTION Among thin film semiconductors, hydrogenated amorphous silicon (a-Si:H) has experienced a rapid development in the last three decades. Many of today's ubiquitous consumer products (e.g. solar cells, flat panel displays, and photodetectors) were made possible by the development of this material. Even though much progress has been achieved in the understanding of film growth, two fundamental questions still remain open: 1. What is the structure of a-Si:H ? Two main descriptions have been proposed: on the one hand, a-Si:H is considered an homogeneous material which can be described by a continuous random network (CRN) [^]; on the other hand, a-Si:H is often viewed as an heterogeneous material in which silicon clusters are embedded in an hydrogenated amorphous matrix [\\'*]. 2. What are the growth mechanisms of a-Si:H films ? They have been largely discussed on the basis of the physics and chemistry of silane plasmas and the current models consider SiHs to be the main film precursor [W even though there is some controversy on this subject [\ In fact, the two questions are related. The understanding of the growth mechanisms, sketched in Figure 1, can indeed throw some light on the film structure. Focusing on the gas phase reactions, one usually considers the primary processes, i.e. the dissociation of the silane molecules after inelastic collisions with electrons [\ Because of the complex plasma chemistry most of the models used to describe the a-Si:H deposition are limited to these primary processes. However, even in this simple case the increase of atomic hydrogen flux towards the substrate achieved through: i) a high dissociation of silane [\ ii) a high dilution of silane in hydrogen ["^], iii) the use of a layer-by-layer technique [ \ results in the growth of an heterogeneous material with long-range order: microcrystalline silicon (|ic-Si:H). Many models have been proposed to explain the growth of this latter material, but most of them fail to explain the formation of crystallites and the long term evolution of the film properties [^% Now, given these two materials (a-Si:H and |ic-Si:H) which can be deposited in the same reactor, one can ask whether there is a sharp transition between the amorphous and the microcrystalline silicon materials. In other words, is it possible to grow silicon-thin-film materials with different degrees of medium-range order ? One might think that just by choosing plasma conditions at the border between a-Si:H and |ic-Si deposition, silicon films with medium-range order should be obtained. However, thermodynamic considerations suggest a
214
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
discontinuous disorder/order transition, taking place for crystallite sizes of about 3 nm \^\ Nevertheless, some experimental observations support the existence of ordered domains with sizes below 3 nm {'\'W In order to progress in the understanding of the deposition and structure of silicon thin films, we like to stress the importance of secondary reactions. Indeed, the SiHs hypothesis which may be reasonable under "soft" plasma conditions, becomes less plausible under conditions of high silane dissociation (needed to achieve high deposition rates) where secondary reactions and powder formation take place ]^\ This phenomenon, which is a real drawback in the microelectronics industry because it can lead to a marked decrease in production yield, has attracted much attention in the last five years [^\ Detailed studies of powder formation in silane plasmas have allowed to separate the two main stages of the process, namely the initial formation of a high density of nanosized silicon particles followed by their coalescence to produce powder into five steps \^\ Moreover, studies based on TEM analysis of the particles produced in silane-argon discharges at room temperature before the coalescence stage have shown that the 2-nm silicon particles are crystalline [^"]. Therefore, the incorporation of these crystallites in an a-Si:H matrix should allow to synthesise a new class of silicon thin films with a degree of order intermediate between that of a-Si:H (short-range order) and crystalline silicon (long-range order). Here we give evidence of the contribution of clusters formed in the plasma to the deposition of silicon films consisting of an amorphous matrix in which these nanoparticles are embedded. Such films will be referred to in the following as polymorphous silicon (pm-Si) [^'].
Electrical Power Electron density Energy distribution
SiH^
[-•(
Primary reactions
e-+SiH4
)^[Pumps
Secondary reactions
Increasing RF power. Pressure, Geometry, Flow
.•
SiHx+SiH.
SinHm
Clusters, Crystallites polymers. Powder
|a-Si:H ^ic-Si pm-Si
Surface mobility Chemical equilibrium
Substrate Temperature Figure 1. Schematic representation of the plasma processes (gas phase) and solid phase reactions involved in the growth of silicon thin films.
215
p. Roca i Cabarrocas /Plasma deposition of silicon clusters EXPERIMENT
The films were produced by the decomposition of silane in a multiplasma-monochamber reactor [^^]. Different process parameters (substrate temperature, RF power, total pressure, and dilution of silane in H2, Ar, or He) have been studied in order to achieve plasma conditions where silicon clusters but not powder are formed. The plasma conditions necessary to be at the onset of the formation of powder were determined from Mie scattering measurements. The optical properties of the films were characterized by optical transmission, spectroscopic ellipsometry in the UV-visible range, and photothermal deflection spectroscopy (PDS) measurements. The hydrogen content and hydrogen bonding were deduced from infrared transmission measurements [^^]. High Resolution Transmission Electron Microscopy (HRTEM) measurements were performed in selected samples to confirm the presence of ordered domains, which was also deduced from the kinetics of crystallization of the films. The defect density of the films was studied by PDS measurements and their transport properties by dark and photoconductivity measurements, from which the r^t product was derived. Further information concerning the transport properties and stability of the polymorphous silicon films are given in references ^^ and ^^ respectively. In order to confirm the device quality of the polymorphous silicon films, single junction p-i-n solar cells have been produced and their stability has been studied. The accelerated light-soaking tests on films and solar cells were performed at 80 °C under a 350 mW/cm^ light from a 1-kW Xe arc lamp filtered by- a 150-nm a-Si:H film to ensure uniform creation of defects through the film thickness. RESULTS Low temperature growth: From powders to nanoparticles Figure 2 shows a schematic diagram of the genesis of powder formation. The evolution of the size of the particles, related to the time elapsed from the beginning of the discharge, is very fast. Detailed studies have shown that the time scale is in the range of seconds to minutes and that the increase in size is accompanied by a drastic decrease of the particle density, from lO^^ cm-^ at the beginning down to 10^ cm"^ after agglomeration [^^ ^\ A general review on particle formation, from monomers to macroscopic particles, has been given by J. Perrin and Ch. Hollenstein in the chapter II of the book « dusty plasmas » (Wiley & Sons, 1999). Radicals Molecules
Macromolecule
X^
Clusters, Crystallites
Agglomeration
A
^Sf 1 nm
Hr
M\y—I
10 nm
V ^
Powder
0.1 ^m
h*
1 |iim
Figure 2. Schematic representation of the genesis of the formation of powder. Because of its simple implementation, we have used Mie scattering to determine the plasma conditions (pressure, power, temperature...) which are close to the limit of powder formation
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
216
) ^ \ Moreover, because powder formation is favoured at low temperatures, we have performed studies at 30 °C in silane-argon mixtures, a largely investigated system for which the kinetics are known in great detail [20]. Figure 3 gives an example of the evolution of the scattered light intensity as a function of the plasma duration in a 7 % silane-in-argon discharge at room temperature with an RF power of 20 W. One can see that the increase of pressure results in a sharp transition between a pristine and a powdery plasma. Moreover, the formation of powder is an extremely fast process, as indicated by the strong increase of the signal after a few seconds at 67 Pa. Now, the fact that a scattered light is not detected does not necessarily imply the absence of particles in the plasma. However if there are any, they are present in a very low density or, rather, they are too small to produce a detectable signal. To prove the presence of particles at 20 Pa, we used modulated discharges with a microscope grid located on the substrate holder. As shown on figure 4, nanometer-size crystalline particles are evidenced on HRTEM micrographs in these conditions (silane-argon discharge at room temperature). Similar nanometric particles have been evidenced by other groups using different plasma conditions ) ^ \ ]^\ Although individual silicon particles with sizes in the nm range are not easily detected, this is precisely the range of sizes we are interested in, because their incorporation into the growing film leads to polymorphous films (with a medium-range order), intermediate between a-Si:H and |jc-Si.
0
10
20
30
40
50
60
Time (s) Figure 3, Mie scattering measurements performed in a silane-argon discharge at 30 °C with an RF power of 20 W.
Figure 4. HRTEM of a microscope grid
An important step towards the deposition of polymorphous silicon was the demonstration that films with similar optical properties can be deposited under continuous and modulated plasma conditions [28]. As a matter of fact, this result is implicit in the data shown in Figure 3. Because the nanoparticles are formed before the transition to macroscopic powder and do not grow to form powder, they are not trapped in the discharge and can therefore contribute to the deposition. For this to be possible, one has to admit that the silicon clusters are not always negatively charged, or else agglomeration could not happen. The contribution of nanoparticles to the deposition of a-Si:H has also been observed by scanning tunnelling microscopy ^ \
217
p. Roca i Cabarwcas /Plasma deposition of silicon clusters
At this stage we have shown that the deposition of siUcon thin films can result from the contribution of radicals and positive ions (left side of Fig. 1) as well as from that of silicon clusters and crystallites (the products of secondary reactions in Fig. 1). Therefore, under the latter conditions the growth of heterogeneous materials can take place. Now, what about the optoelectronic properties of such films ? The optimization of a-Si:H deposition conditions has been often driven by the desire to obtain dense and homogeneous films where all kinds of microstructure are avoided ^\ Moreover, there is a general consensus about deposition at room temperature leading to poor quality a-Si:H, even though low-defect density a-Si:H films have been obtained at 50 °C ['']. Figure 5 shows the infrared transmission spectra of films deposited at 30 °C by the dissociation under an RF power of 20 W of a 7 % silane-in-argon mixture to which an increasing amount of hydrogen was added. The film deposited without addition of hydrogen shows two strong absorption bands at 840-890 cm~^ a signature of the presence of (SiH2)n groups, while the Si-H stretching band is shifted towards 2100 cm"^ The addition of hydrogen to the silane-argon mixture produces a dramatic decrease of the (SiH2)n groups as can be seen in the spectra of the films deposited with the addition of 50 - 100 seem of hydrogen. Their stretching bands are shifted to 2010 cm*^ while the bands at 840 - 890 cm"^ are absent. As a matter of fact, these spectra are comparable to those of polymorphous silicon films deposited at 250 °C [^]. On the contrary the spectra of the films produced from argon or with only 10 seem of hydrogen added to the argon-silane mixture correspond to those of a-Si:H films which are generally considered as porous [^^]. However, very similar spectra have been reported for microcrystalline silicon films produced by reactive sputtering at 100 K ^% Despite the large changes in the hydrogen bonding configurations, the hydrogen content of the films, deduced from the integrated absorption of the wagging band at 630 cm-^ remains in the range of 21 % to 24%. Ar + 100 seem H.
c^ '-^w 2 1 % /
A
Ar + 50 seem H AT+10 seem H
B
'^24
%//
e - \l\ y i c
^^^^Ar
y^
o
S h
2
H
•
:3
\22%/
500 600 700
800 900 1000 1100 1200
1900
2000
2100
2200
Wavenumber (cmO Wavenumber (cm"^) Figure 5. IR transmission spectra of a-Si:H films deposited at 30 ^Cfrom a7 % silane-inargon gas mixture to which an increasing flow ofH2 was added. The above results show that even at 50 °C it is possible to produce dense a-Si:H films. Note that no oxygen band (• 11(X) cm"^) related to the ambient contamination [35] is detected in these IR spectra.
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
218
The changes in the hydrogen bonding are also reflected in the absorption of the films deduced from PDS measurements (Fig. 6). One can see that the addition of hydrogen produces a shift of the high energy part of the spectra towards lower energy. 100 seem H, annealed 100 seem H^ as-deposited Argon
S
0.8
1.2
1.4 1.6 1.8 Energy (eV)
2.2
Figure 6. Absorption coefficient measured by PDS in samples deposited at 30 ^C by the decomposition ofa7% silane-in-argon gas mixture.
The absorption coefficient of the film deposited without hydrogen addition has a larger optical gap (Eo4 = 2.15 eV) than the film produced with the addition of 100 seem of hydrogen. As for the high absorption coefficient values in the 1.3 - 1.7 eV range, they could be attributed to a high defect density. However, they can also be due to the presence of crystallites in the film. The fast decay for photon energies below 1.2 eV, along with the fact that annealing the film for 1 hour at 200 °C did not change the absorption coefficient (not plotted for clarity), are in favor of that hypothesis. A quite different behaviour is observed in the film produced under hydrogen dilution. This film has a smaller optical gap (E04 = 2 eV) and we observe a strong effect of annealing during 1 hour at 200 °C. Indeed, the absorption coefficient values strongly decrease over almost the whole spectral range. While the decrease in absorption in the 0.8 - 1.6 eV range can be related to the annealing of deposition-induced metastable defects [17], the decrease in the exponential region indicates that large structural rearrangements have occurred. The standard analysis [37] of the absorption spectra in the annealed state gives a defect density of 3.10^^ cm'^ and a disorder parameter of 60 meV. The different shapes of the absorption edge as well as the different evolution of both types of films under annealing could be explained by the different size of the ordered domains. In the argon case stable silicon crystallites are formed in the plasma (Fig. 4), while smaller (unstable) paracrystallites may be formed with the addition of hydrogen, known to reduce powder formation. Therefore, and not surprisingly, the addition of hydrogen results in denser a-Si:H films with lower defect density. The question is whether nanoparticles are still present in the film, or whether the addition of hydrogen has completely suppressed their formation. Figure 7 shows the HRTEM micrograph of a film deposited at 50 °C from a hydrogen-silane mixture in its asdeposited state and after annealing in the microscope at 425 °C. One can see that nanometersize ordered domains with circular and fringe-like features are indeed present in the film. These nanoparticles can act as seeds for the crystallization of the film at a temperature as low as 425 °C. Interestingly enough, HRTEM studies performed on silicon powders produced from pure
p. Roca i Cabarrocas /Plasma deposition of silicon clusters
219
silane and annealed between 300 °C and 600 °C for 1 hour [15] have shown the formation of ring-like and fringe-like contrast features similar to those reported in Figure 7. Moreover, we have reported similar results for films deposited from silane-hydrogen mixtures at 100 °C [38, 39].
Figure 7. HRTEM of a silicon film produced by the decomposition of a 2 % silane4nhydrogen mixture at 50 °C in its as-deposited state (left) and after annealing in the microscope chamber at 425 ^Cfor one hour. The above results show that the formation of nanometer-size ordered particles in silane plasmas is a quite general feature. As a matter of fact, care must be taken to choose plasma conditions where no secondary reactions take place, though this tends to decrease the deposition rate. Therefore, we tend to think that many studies dealing with a-Si:H films deposited at thigh rates are actually related to nanostructured or polymorphous films as, in fact, claimed by some authors [2-4]. Towards high temperature growth of polvmorphous silicon films Even though low-defect density films can be deposited at 50 °C, their high hydrogen content makes them unstable and fragile. So far we have focused on the formation of polymorphous films at low temperatures. According to Figure 1, the change from a pristine plasma to a plasma containing silicon clusters or powder can be obtained in different ways. As a matter of fact, lowdefect density a-Si:H films have been produced at 250 °C under conditions of powder formation [17]. By adjusting the plasma parameters just below the onset of the formation of powder, it is possible to produce pm-Si films in a wide range of deposition conditions. Figure 8 shows the effect of pressure on the imaginary part of the pseudo-dielectric function (epsilon 2) of silicon films deposited at 250 °C from the decomposition of a 2 % silane-inhydrogen gas mixture under an RF power of 20 W. The increase of the pressure from 66 Pa up to 160 Pa results in strong changes of epsilon 2 spectrum. In particular, the shoulder at 4.2 eV, characteristic of pc-Si films, disappears for the films deposited at high pressures. The analysis of the data by within the framework of the Bruggeman effective medium theory gives a crystalline
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
220
volume fraction of 67 % for the film deposited at 66 Pa. For the films deposited at high pressures, even though their spectra are similar to those of dense a-Si:H films, we cannot model the whole spectrum by a mixture of amorphous silicon and voids, in particular the low energy part (interferences). We have recently shown [34] that nanometer-size ordered domains are also present in these films, with features similar to those presented in figure 7. Moreover, as in the case of the deposition at 50 °C, the existence of ordered domains is revealed by the crystallization of these films at lower temperatures, or equivalently, by their faster kinetics of crystallization. In Figure 9 we compare the evolution of the conductivity of silicon thin films produced at 250 °C (under the same conditions as for the films in figure 8) to that of a standard a-Si:H film; the conductivity was measured at 560 °C in order to determine the kinetics of crystallization. We can see that in the case of standard a-Si:H films there is a long incubation time (to) during which the conductivity does not change. On the contrary the conductivity of the |ic-Si film produced at 120 Pa starts to increase from the beginning of the experiment, i.e. there is no incubation phase. As for the polymorphous films, they display an intermediate behaviour which appears as a signature of the presence of domains with medium-range order. The solid-phase crystallization kinetics shown in Figure 9 can be simulated by two characteristic times: the incubation time to and the crystallization time tc which represents the time above which 63 % of the material has been crystallized [^^]. The inset in Figure 9 shows that the incubation time is reduced in the pm-Si films, but that there are large fluctuations in the crystallization times. While in the case of the film deposited at 133 Pa both to and tc are shorter than for standard a-Si:H, tc is higher in the pm-Si films deposited at 160 Pa and 213 Pa. This suggests that, as the pressure increases, there are changes in either the density or the nature of the domains (ring-like versus fringe-like?). Then, even though the presence of domains with medium-range order (clusters or paracrystallites) reduces the incubation time, their interaction with the growing crystallites in the solid phase crystallization process may result in a slowingdown of the kinetics. 10
—I—I—I—I—I—I—I—I—I—r-—t—I—\—I—I—I—I
1—
P(Pa) 5 120 133 160 213 t^^(s) 9485 — 2103 3891 2130 t"(s) 6852 2926 4952 1060 9919 a-Si:H (5 Pa)
2.5
3
3.5
4
4.5
Photon energy (eV)
Figure 8. Effect of pressure on the <£2> of films produced from the dissociation of 2% films silane-in-hydrogen at 250 °C and 20 W,
10000 15000 Time (s)
20000
Figure 9. Evolution of the conductivity during the crystallization of silicon thin at 560 °C under vacuum.
Because most of the results presented above were obtained with silane-hydrogen discharges, one could suspect hydrogen of being responsible for the formation of domains with medium-
p. Roca i Cabarrocas /Plasma deposition of silicon clusters
221
range order; i.e. consider that the ordered domains have nothing to do with gas phase reactions. Indeed, |ic-Si films are formed through hydrogen-mediated solid-phase reactions leading to the complete rearrangement of the a-Si:H matrix. As a matter of fact, we have shown that there are two routes for the production of nanostructured silicon films ^^\. i) The use of a high hydrogen flux with respect to the flux of SiHx radicals (solid phase reactions in Fig. 1). ii) The direct deposition of SiHx radicals and silicon crystallites resulting respectively from the dissociation of silane and from the reactions of the radicals in the gas phase (gas phase reactions in Figure 1). These processes are schematically described in Figure 10. Hydrogen dilution has been reported as a way to improve the quality (decrease the disorder) in a-Si:H films ^\ Moreover, the use of high dilution of silane in hydrogen allows to produce |ic-Si:H films. The control of the flux of atomic hydrogen with respect to that of silicon radicals is easily achieved by the layer-by-layer technique [11]. As shown in Figure 10, the increase of the hydrogen exposure time (dilution) results in the formation of a highly porous and disordered film, before the transition to |Lic-Si growth occurs; as a matter of fact, the nucleation of crystallites takes place in the highly disordered porous phase ^% Moreover, a further increase of the hydrogen exposure time (dilution) results in no film deposition on a glass substrate [11], but in the formation of a more disordered material (a film with medium-range order) when the process is performed on an a-Si:H substrate. This is explained by the long-range effects of hydrogen which induces the rearrangement of the whole a-Si:H substrate ]^]. PLASMA Gas phase reactions (^•TT Crystallites Clusters .
I2 o
— Hz -^^Ar -o-He
-
W) 1 o
C/3
|iC-Si
pm-Si
Flux of atomic Hydrogen Solid state reactions (Layer-by-layer growth) Fig. 10. Schematic diagram of the disorder in silicon thin films as a function of the flux of polyatomic hydrogen impinging on the growing film.
1 . . .
0.8
1 . . .
1 . . .
1 . . .
1 . . .
1.2 1.4 1.6 Energy (eV)
1 . .
.
1.8
Fig. 11. Absorption coefficient measured by photothermal deflection spectroscopy on morphous films produced by the dissociation of silane diluted in either H2, Ar ,or He.
Now, to further support the assumption of the contribution of clusters formed in the plasma to the deposition, as opposed to solid phase reactions, we substituted hydrogen by either argon or helium because in these cases the smaller concentration of hydrogen makes solid state reactions driven by hydrogen improbable. Then we adjusted the plasma conditions in order to be close to the onset of powder formation. It is striking to see that the three films deposited at 250
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
222
°C from different gas mixtures have almost the same Urbach energy (52 - 55 meV), subgap absorption (defect density • LIO^^ cm-^), as well as transport properties and stability (see Fig. 12). We suggest that since in the case of Ar or He dilution hydrogen can hardly be accounted for in solid phase reactions, the growth of films with similar properties is indeed related to the gas phase reactions, i.e. the contribution of clusters produced in the gas phase to deposition. However, further studies are still necessary to analyze in detail the structure of the 1-2 nm clusters (paracrystallites) as a function of the gas used. Stability and Devices Further support for the existence of medium-range order in polymorphous silicon films is given by the study of their metastability [25]. Figure 12 shows the evolution of the r||ax product as a function of the light-soaking time for different silicon thin films. The initial r||iT value of the pm-Si films lies between that of standard a-Si:H and that of |ic-Si films with crystalline fractions above 60 % [25]; i.e. irrespective of the gas added to silane (H2, Ar or He), the pm-Si films have higher r||XT values than standard a-Si:H films. If we now consider the evolution with time of the r||iT product, we see that the |Lic-Si film with a crystalline fraction of 96 % is stable, while the standard a-Si:H film experiences a decrease of the r[\xx product, following a stretchedexponential law. The kinetics of the pm-Si films are quite different from those of a-Si:H. We observe a fast decrease of the r||iT product in the initial stages of light-soaking. However, despite this strong light-induced decrease of the T^T product, by about a factor of 100, the steady-state y\\ix values of the polymorphous films produced with a He or a H2 dilution remain higher than the annealed-state value of a-Si:H. Another interesting aspect is the evolution of the r||iT product in the microcrystalline film with a crystalline fraction of 30 %. Its x\\ix product shows no degradation during the first 100 s of light-soaking, then decreases very fast and stabilizes at a value close to that of the pm-Si film produced with hydrogen dilution. In our opinion this is a further indication of the presence of medium range-order in the pm-Si films. 12
10^ic-Si (Fc = 96 %) -oo - - - O- - -O - - O- - O- - -O , - 0- - O- - - • - pm-Si (Ar)
10-i-40
pm-Si (0.45 nm)
10
-
- - • - pm-Si (He) - - 0 - pm-Si ( H )
^-^
^ 10-^
10-^
HC-Si (Fc = 30 ^ 'O.
a-Si:H
^ ^ j
10-
w
-ft:.
0
10^ itf 10^ 10^ 10^ Time (s) Fig. 12. Evolution with time of the ruit product of silicon films put under an accelerated lightsoaking test at 80 °C and 350 mW/cm^ of light filtered by a ISO-nm thick a-Si:Hfilm . \{f
a-Si:H (0.5 nm) 4^ • . ^ _
10^
10^
4J
I
10^
. MMMI
aiul
X I Miml
10^ 10^ Time(s)
10^
10^
Figure 13. Evolution of the efficiency of P'i-n solar cells submitted to the same conditions as the films in figure 12.
p. Roca i Cabarrocas /Plasma deposition of silicon clusters
223
We have produced single-junction p-i-n solar cells in which the plasma conditions of the intrinsic layer have been switched to those used to obtain pm-Si films. As in our previous studies on solar cells with intrinsic layers deposited under different plasma conditions \^% we had to adapt the interface between the p-layer and the pm-Si layer in order to avoid damage on the p-i interface due to the strong plasma conditions used to obtain the pm-Si films. We have thus been able to produce solar cells with initial efficiencies of above 9 %. The high efficiency as well as the high values of the FF achieved (0.72 - 0.74) demonstrate that indeed pm-Si films are an interesting alternative to a-Si:H films. Moreover, a comparative study of the metastability of p-i-n solar cells put under accelerated light-soaking conditions, similar to film ageing conditions, has shown that pm-Si based solar cells have an unusual stability. In Figure 13 we compare the evolution of the efficiency of a p-i-n solar cell based on our standard a-Si:H material to that of two devices based on pm-Si films with two different thicknesses. It can be seen that the thinner pm-Si solar cell (0.45 |im) has a stable efficiency close to 10 %, while the efficiency of the thicker pm-Si solar cell (0.75 |im) decreases but remains higher than that of the standard a-Si:H solar cell (0.5 |im). Further studies are necessary to understand the stability of these devices. CONCLUSIONS Thirty years of research on a-Si:H and |Lic-Si films have produced a tremendous progress in the understanding of the structure and optoelectronic properties of these materials. The leading principle for the optimization of a-Si:H has often been the deposition under conditions far from those leading to the formation of powder and resulting in films with a degree of microstructure as low as possible. On the other hand, |ic-Si films are produced under high power conditions and have a high degree of microstructure. We have shown that by running the plasma in a regime close to the formation of powder, it is possible to produce a new type of silicon thin films (polymorphous silicon) which results from the simultaneous deposition of radicals and nanoparticles (silicon clusters and/or paracrystallites). Despite being heterogeneous, polymorphous silicon films exhibit improved transport properties and stability with respect to aSi:H. Moreover, their implementation in single junction p-i-n solar cells has resulted in stable efficiencies close to 10 %. Further work is necessary to better understand the structure of the ordered domains with sizes in the range of 1-2 nm. ACKNOWLEDGEMENTS This paper is the result of many collaborations, in particular with the French laboratories involved in the coordinated research action on polymorphous silicon and the European teams involved in a Brite-Euram contract on micropowder processing (BE 7328). Special thanks to J. Costa and H. Hofmeister for the HRTEM pictures in Figures 4 and 7 respectively, to P. St'ahel for the light-soaking experiments, and S. Hamma for ellipsometry analysis of the films. The research has been funded by Brite-Euram BRE2-CT94-0944 and ECODEV/ADEME contracts. REFERENCES 1. L. Guttman, in in Semiconductors and Semimetals Vol. 21 A edited by J. I Pankove, Academic Press Inc. (1984) pp. 225-246. 2. J.A. Reimer and M.A. Petrich in Amorphous Silicon and Related Materials, Vol. 1, edited by H. Fritzche, 1989 World Scientific Publishing Company, p. 3-27. 3. Bellisent, same as reference 2, pp. 93-121. 4. J.C. Philips, J. Appl. Phys. 59, 383 (1986). 5. J.R. Doyle, D.A. Doughty, and A. Gallagher, J. Appl. Phys. 71, 4771 (1992).
224
6.
P. Roca i Cabarrocas /Plasma deposition of silicon clusters
A. Matsuda, K. Nomoto, Y. Takeuchi, A. Suzuki, A. Yuuki, and J. Perrin, Surface Science 227,50(1990). 7. S. Veprek, Thin Solid Films; 175, 129 (1989). 8. J.P.M. Schmitt, J. Non Cryst. Solids 59&60, 649 (1983). 9. M. Scheib, B. Schroder and H. Oechsner, J. Non Cryst. Solids 198-200, 895 (1996). 10. A. Matsuda, S. Yamasaki, K. Nakagawa, H. Okushi, K. Tanaka, S. Izima, M. Matsumura, and H. Yamamoto, Jpn. J. Appl. Phys. 19, L305 (1980). 11. N. Layadi, P. Roca i Cabarrocas, B. Drevillon, and I. Solomon, Phys. Rev. B 52, 5136 (1995). 12. S. Hamma and P. Roca i Cabarrocas, J. Appl. Phys. 81, 7282 (1997). 13. S. Veprek, Z. Iqbal, and F.A. Sarott, Phil. Mag. B 45, 137 (1982). 14. A. Chenevas Paule in Semiconductors and Semimetals Vol. 21 Part A edited by J.I Pankove, Academic Press Inc. (1984) pp. 247-271. 15. H. Hofmeister, J. Dutta, and H. Hofmann, Phys. Rev. B 54, 2856 (1996). 16. Y. He, Ch. Yin, G. Cheng, L. Wang, and X. Liu, J. Appl. Phys. 75, 797 (1994). 17. P. Roca i Cabarrocas, J. Non Cryst. Solids 164-166, 37 (1993). 18. For a recent review see the Proceedings of the Dusty Plasma Workshop published in J. Vac. Sci. Technol. A 14 pp. 489 - 666 (1996). 19. J. Perrin, Ch. Bohm, R. Etemadi and A. Lloret, Plasma Sources Sci. Technol. 3, 252 (1994). 20. L. Boufendi and A. Bouchoule, Plasma Sources Sci. Technol. 3, 262 (1994). 21. This denomination refers to a material which has different crystalline forms. It has already been applied to silicon films by R. Alben et al. in Phys. Rev. B 11, 2271 (1975). 22. P. Roca i Cabarrocas, J.B. Chevrier, J. Hue, A. Lloret, J.Y. Parey, and J.P.M. Schmitt. J. of Vac. Sci. and Technol. A9, 2331 (1991). 23. A.A. Langford, M.L. Fleet, B.P. Nelson, W.A. Lanford, N. Maley, Phys. Rev. B 45, 13367 (1992). 24. C. Longeaud, J.P. Kleider, P. Roca i Cabarrocas, S. Hamma, R. Meaudre and M. Meaudre, J. Non Cryst. Solids (to be published). 25. P. St'ahel, S. Hamma, P. Sladek, and P. Roca i Cabarrocas, J. Non Cryst. Solids (to be published). 26. Ch. Hollenstein, J-L. Dorier, J. Dutta, L. Sansonnens, and A. A. Howling, Plasma Sources Sci. Technol. 3, 278 (1994). 27. Y. Watanabe, M. Shiratani, H. Kawasaki, S. Singh, T. Fukusawa, Y. Ueda, and H. Ohkura, J. Vac. Sci. Technol. A 14, 540 (1996). 28. P. Roca i Cabarrocas, P. Gay, and A. Hadjadj, J. Vac. Sci. and Technol. A14, 655 (1996). 29. C. Courteille, J.L. Dorier, J. Dutta, Ch. Hollenstein, A.A. Howling, T. Stoto, J. Appl. Phys. 78, 61, 1995 30. T. Ifuku, M. Otobe, A. Itoh, and S. Oda, Jpn. J. Appl. Phys. 36,4031 (1997). 31. D.M. Tanenbaum, A. L. Laracuente, and A. Gallagher, Appl. Phys. Lett. 68, 1705 (1996). 32. D.L. Williamson in Mat. Res. Soc. Symp. Proc. vol. 377, 251 (1995). 33. P. Roca i Cabarrocas, Appl. Phys. Lett. 65, 1674 (1994). 34. P. Roca i Cabarrocas, S. Hamma, S.N. Sharma, J. Costa, and E. Bertran. J. Non Cryst. Solids (to be published). 35. E. Srinivasan, D. A. Lloyd, and G. N. Parsons, J. Vac. Sci. Technol. A 15, 77 (1997). 36. S. Furukawa and T. Miyasato, Phys. Rev. B 38, 5726 (1988). 37. W. B. Jackson and N.B. Amer, Phys. Rev. B 52, 555 (1982). 38. H. Hofmeister, J. Dutta, and H. Hofmann, Phys. Rev. B 54, 2856 (1996). 39. G. Viera, E. Bertran, P. Roca i Cabarrocas, J. Costa, S. Martinez. MRS Symp. Proc. Spring vol. 467, 313 (1998). 40. G. Viera, P. Roca i Cabarrocas, J. Costa, S. Martinez and E. Bertran, 1'^ ICPDP conf., 1997
p. Roca i Cabarrocas/Plasma deposition of silicon clusters
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41. R. Bisaro, J. Margarino, K. Zellama, S. Squelard, P. Germain, and J.F. Morhange, Phys. Rev. B 31, 3568 (1985). 42. P. Roca i Cabarrocas and S. Hamma, in Proceedings CIP'97, Le Mans, 25-29 Mai 1997, edited by Societe Frangaise du Vide, p. 172 (1997). 43. S. Sugiyama, J. Yang, and S. Guha, Appl. Phys. Lett. 70, 378 (1997). 44. P. Roca i Cabarrocas, N. Layadi, B. Drevillon, and I. Solomon, J. Non Cryst. Solids, 198200, 871 (1996). 45. S. Hamma and P. Roca i Cabarrocas, Thin Solid Films 296, 11 (1997). 46. R. Meaudre, M. Meaudre, S. Vignoli, P. Roca i Cabarrocas, Y. Bouizem, and M. L. Theye, Philos. Mag. B 67, 497 (1993). 47. J. C. Knights and R. A. Lujan, Appl. Phys. Lett. 35, 244 (1979). 48. P. Roca i Cabarrocas, M. Favre, P. Morin, P. Sladek, and P. St'ahel, Proc. of the 13th European Photovoltaic Solar Energy Conference, Nice, 23-27 October 1995, p. 605.
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Part IV. Atmospheric and Astrophysics
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
229
Formation of a dust-plasma cloud A.M. Ignatov General Physics Institutej 117942J 38, Vavilovstr., Moscow, Russia Abstract. We discuss the conditions under which a compact dust cloud in a steady- state plasma may exist and, particularly, how the attraction of likely charged grains may arise. It is shown that interaction of two like charges in a quasineutral plasma is always repulsive. The most suitable for a low-pressure dusty plasma mechanism of attraction is found to be provided by plasma recombination at the grain surface. A kinetic model of a compact dust cloud is developed.
Introduction A single dust grain immersed in a plasma carries some charge. The value of the charge is determined both by the state of surrounding plasma and by the properties of the grain material. In a weekly ionized low-pressure plasma, the grain charging is provided mostly by the balance between electron and ion currents entering the grain surface. Due to the higher electron thermal velocity the current balance is maintained for a negatively charged grain. Additional factors Uke, e.g., the thermoelectric emission in a dense thermal plasma or radioactivity of a grain etc. often influence the current balance that may result in altering the sign of the grain charge. It is commonly beUeved that the charges in a steady-state plasma interact via the screened Coulomb potential. Recent theoreticcil and computational studies demonstrated that the situation is much more complicated (for the review see, e.g., [1]). The main question we focus on throughout the paper is whether there are any attractive interactions yielding the formation of some complexes, clusters or clouds in dusty plasmas. It should be noticed that similar problems arise also in other areas of physics, for example, in the physics of colloidal suspensions [4].
Interaction of Grains The most familiar to plasma physicists reason for attraction of like charges is altering the sign of the static dielectric permittivity in a plasma with relative motion of species. Seemingly, in appHcation to dust-laden plasmas, this process is responsible for the formation of the dust crystal in the sheath area [2]. Other possibilities discussed below are the nonUnear Debye screening and specific interaction arising due to plasma recombination at the grain surface.
230
A.M. Ignatov /Formation of a dust-plasma cloud
F I G U R E 1.Layout and surface of intergation. Qualitatively, the idea of the nonlinear screening may be outlined as follows. Two suificiently large similar chairges can collect a Icirge amount of plasma species of the opposite sign. Evidently, it is more favorable for the species to gather between the external charges forming a cloud of the opposite sign which may attract the charges. To clarify whether it is really possible let's pose a problem in a more systematic way. Suppose there are two spheres of the same radius, a, each carrying equal charge, Q, immersed in an equilibrium plasma. Let the coordinates of the spheres be z = ±R/2 (see. Fig. 1). For simplicity, we assume that the spheres are perfectly conducting and the plasma is collisionless. Under these assumptions, the plasma is described by the Vlasov equation 5/a(<,r,p) dt
p g/a(t,r,p) H rria ;:; dv
, ^,^,dfa{t,r,p) r ea^jyr) = U, 5—p
(1)
where the subscript, cc, distinguishes the plasma species, combined with the Poisson equation A<^ = -47r f Y^ eana{(j>) - ph\ •
(2)
Here pi is the charge density of the neutrciUzing background and na(<^) = y < i p / a ( r , p ) .
(3)
The boundary condition for the Poisson equation (2) corresponding to the conducting spheres is given by
231
A.M. Ignatov /Formation of a dust-plasma cloud *
dsV^ <^|
-^
-47rQ,
(4)
0,
(5)
where the integral in Eq. (4) is carried over the surfaces of the spheres, Si,2To start with, we assume that the boundary conditions for the Vlasov equation (1) correspond to the mirror reflections from the spheres: ISi,2,Pn>0
II;i,2,Pn<0
where pn is the normal component of the particle momentum, and distribution functions at infinity are Maxwellian with the unperturbed densities, rioa, and the temperatures, To,: fa. -^ foa = noa{2iTmaTa)~^^^ exp{-p^/2maToc). ir—^oo
Eqs. (1,2) guarantee the momentum conservation that in the steady-state plasma is written as Vjllij = 0, where the stress tensor is
m/
=
^—-
Siip, + 5ii-^
n!f = X:/dp^/„(r,p).
(7)
(8)
A force acting upon a grain is, by definition, the momentum flux through its surface. In evaluating the momentum flux at the upper grsdn, it is convenient to choose the integration surface as a combination of the upper hemisphere of sufficiently large radius, iZo, and the circle of the same radius at the z = 0 plane (Fig. 1). Thus the net force, which is evidently parallel to the z axis, is written as
F,=
[
J y/x^
dxdyU^A
\z=0
-Rl f dnUrA ^ . J ""^^
\r=RO
(9)
It is worth mentioning that the force (9) is not proportional to the derivative of the net energy with respect to R. The solution to the Vlasov equation with boundary conditions (6) is / a ( r , p) = /oaexp(—ea/ra), that is, the stress tensor (8) is diagoucd:
nSj^ = <J.iE«o«r.exp(-^j
(10)
Taking into account that due to the evident symmetry the potential, (/9, z), is an even function of z, and making use of Eq. (5), in the limit RQ ^ oo the force (9) is reduced to
232
A.M. Ignatov/Formation of a dust-plasma cloud
2-Kj pdp
W(P,0)) + ^
(11)
where
G(<^) = M + E
(12)
For the boundary condition (5) to be consistent with Eq. (2), plasma at infinity should be neutral, i.e., ph = Z)a ^a^Oa- Making use of the latter constraint one can easily check that the function (12) is nonnegative. Therefore, in the framework of this model, the force between grains is always repulsive. There are a few factors that may alter this conclusion. First, the dispersion in grain size may give rise to attractive interaction - Ukely charged spheres of different size may attract even in vacuum. Then, if a number of grains is sufficiently large, the electron and ion densities in the ambient plasma are not equal. Thus, in a more reaHstic model, to describe interaction of two grains we have to use Eqs. (1,2) with pb = 0 keeping in mind that overall neutraUty is provided by other grains. In this case we can no longer use boundary condition (5) and pass to the Hmit RQ -^ oo in Eq. (9). Although we are unable to evaluate the force (9) analytically, evidently, it may become negative. The similar conclusion has been drawn in studying the effective interaction in colloidal plasmas [3]. It was shown that while the dependence of the effective macroion interaction potential (i.e., the free energy) on the distance is always monotonic, the influence of surrounding macroions may give rise to the attractive branch of the potential. However, the value of the macroion charge required for attraction is hardly achievable in dusty plasmas. It should be also noted that the connection between the effective potential and the force on a grain is not evident. Seemingly, this effective attraction may be regarded as the influence of surrounding grains, which pushes the two neighboring particulates together. The third cause that may result in attraction is the interaction of plasma particles with the grain surface. Being reasonable in simpHfied models, the boundary condition (6) seems unreaHstic in low-pressure dusty plasmas. Since the grain charging is provided by the plasma recombination at its surface, we should rather use the condition of the complete absorption:
The Vlasov equation (1) combined with this boundary condition describes an open dissipative system with permanent plasma flow. To maintain the steady state we assume that far from the grains the plasma is sustained by some external means, and its distribution function at infinity is Maxwellian. Consider once more two neighboring grains. Evidently, the boundary condition (13) results in
233
A.M. Ignatov /Formation of a dust-plasma cloud reduction of plasma density compared to Boltzmann distribution. For example, the cones in the momentum space shadowed in Fig.l are empty. Consequently, the plasma pressure (8) also decreases, moreover, it becomes anisotropic. The corresponding attracting force may be evaluated exactly in neglecting the electric field i n E q . (1) [6]: (14)
F. = -\-^,n,T,
It was also shown that with the charge of the grains taken into account, the assymptotic expression for the force is the same times a factor of the order of unity. Physics of this interaction is quite transparent. Since each grain creates spherically convergent plasma flow, it should attract any other object in its vicinity [5]. In other words, a number of particles arriving the grain from the direction to the neighboring grain is less than from the opposite direction, and this give rise to the nonzero net momentum flux through its surface. The electric field of a single grain decreases at large distance as 1/r^, therefore the attraction is dominant. The same conclusion follows from the computer experiments [8]. Of interest is that nearly the same process was proposed more than two hundred years ago by G.L. Lesage for explaining the universal gravitation. For this reason in what foUows this interaction is referred to as the Lesage gravity. It should be noted that generally this interaction is non-pairwise and can't be treated as a potential
one [6].
Dust-Plasma Cloud In this section we discuss the influence of the Lesage gravity at the ensemble consisting of large but finite number of dust grains [7]. We start with the following set of kinetic equations for plasma species
^
^
^
+ ^
}
^
^ + e . E ( r ) 5 ^ ^ = /W(r,p) + /W(r.p).
(15)
Since the charge of each grain is determined by the state of the surrounding plasma, in using the kinetic approach it is treated as an additional intrinsic variable of the dust component, i.e., the dust distribution function, / j , besides the coordinate and the momentum depends on the grain charge, Q:
dt
+^
Fr
^^
5i
-Mr,P,Q),
(16)
Electron and ion variables here are distinguished with the lower index a = e,i, the charge of the dust is —Q < 0, other notation is standard. The electric field is governed by the Poisson equation
234
A.M. Ignatov /Formation of a dust-plasma cloud
V E = 47ry d^
fee„/«(r,p)
- /dQQU{v,p,Q)\ .
(17)
For simplicity only two of coUisional processes are taken into account: (i) inelastic collisions of plasma particles with dust grains resulting in absorption of electrons and ions, and (ii) elastic scattering of plasma particles in the electric field of a grain. The collision terms in the right-hand sides of Eqs. (15,16) are obtained considering the probabilities of corresponding processes [7]. Assuming that all dust grains are of the same size, a, small compared to the Debye length, A/p, and m« < < Tn^, the collision terms in Eqs. (15,16) are given by:
/^^)(r, p)^^ldQ
Va(T{va, Q)nd{r, g ) / « ( r , p);
/ , ( r , P, Q) - - ^
( A ( r ) / 4 r , p , Q)) - ^
a = e,i
( J ( r , Q ) / , ( r , p , Q)),
(18) (19)
where Va = p/fria,
M^^Q) =
Jdpfd{r,p,Q)
J{r,Q) = J^ejdp a
A(r,g)
=
Va(^a{Va, Q)fa{^,
P)
(20)
^
E/'^PP/a(r,P)(^a<^aK,g) + ^ ^ ^ ^ ^ ^ ^ ) ,
(21)
and L = ln(A£)/a). The physical meaning of these quantities is evident: J ( r , Q) is the net electric current at a dust grain with the charge Q situated at the point r, while A ( r , Q) is the force acting upon the grain. This force is the momentum flux at the grain consisting of two parts: the Lesage force provided by plasma absorption [the first term in Eq. (21)] and the drag force due to Coulomb collisions. The absorption cross-section, a^, is usually obtained assuming that a grain traps all particles coming at the distance smaller than its radius from the point charge —Q with the Coulomb potential: cr«(T;,Q) = Tra^^ (1 -f- J ^ ^ j - Coulombian collision term for the plasma component in Eq. (15) is given by the Landau integral. Kinetic equations (15, 16) combined with collision terms (18, 19 ) and the Poisson equation (17) form a complete set. This set guarantees the conservation of the total charge and momentum of the plasma-dust system, but the energy and the number of plasma particles are no longer conserved. As it was cilready mentioned, to describe a steady state of a dusty plasma within this model one should take into account external plasma sources of some kind. In what follows we consider a compact dust pattern consisting of finite number of grains and assume that these sources 2tre removed to infinity where the plasma distribution function is MaxweUian.
A.M. Ignatov /Formation of a dust-plasma cloud
235
Let us consider an ensemble of AT (AT > 1) immobile dust grains randomlydistributed in space. Assuming that the plasma is steady, i.e.^ all time derivatives are zero, the first two momenta of Eq. (16) are written as
J ( r , Q ) n , ( r , g ) = 0,
(22)
( g E ( r ) - A ( r ) ) M r , Q ) = 0,
(23)
Eq. (22) guarantees the absence of the net electric current in the equilibrium state. Its only solution is ^^(r, Q) — nd{r)S{Q — Qo(r)), where Qo(r) is the self-consistent charge of the dust component obeying the relation J{r.,Qo{v)) = 0. The value of the grain charge is estimated as Q ~ aT^e. It is determined mainly by the electron temperature and its dependence on other plasma parameters is fairly weak. In what follows we ignore the spatial variation of the grain charge. Assuming the spherical symmetry of the cloud, the solution to Eq. (15) was obtained in powers of the dust density [7]: / a ( p , r ) = foM
( l - ^a(^a)A(r, v„) - g(r)) ,
(24)
where
and / ( r , v) = |r x v\/v. The deviation of the distribution function (24) from its equiUbrium value is provided, first, by the electric potential, (r), and second, by the plasma absorption. The latter contribution is introduced by the non-local dependence on the dust density because the probabiUty of absorbing a plasma particle depends on a path it reaches the observation point. Straightforward integration of Eq. (24) results in expressions for the plasma density and the Lesage force acting upon the dust component. The latter is independent of the electric field and determined by the dust density only: 1 } Ar = -^-jr'UT'nd{r'),
(26)
0
Provided that the net number of grains is not too large, N
236
A.M. Ignatov /Formation of a dust-plasma cloud
This means that the force (26) behaves like Ar ~ (3N/47rr^. As it follows from Eqs. (17,24), the electric field Er oc 1/r^. Therefore, the Lesage force attracting grains at large distance always exceeds the repulsing electric force. In this situation, the only way to satisfy the momentum bcdance equation (23) is to put nd{r) = 0. However, the dust density is not identically zero, that is, there should exist some Umiting radius, RQ^ such that nd{r) = 0, QEr < Ar for r > RQ^ and nd{r) ^ 0, QEr = Ar for r < RQ, In other words, the plasma-dust cloud is not a diffuse object with smooth density profile, but it is a compact formation with a sharp boundary. To determine the radius of the dust cloud we consider the force balance inside it. Assuming that Ro ^ ^D^ we implement Eq. (26) and the plasma neutraUty condition Yla ^cx'^a — Q'^d = 0 to rewrite Eq. (23) as 1 d ^dndjr) r^dr dr
^
ndjr) b^ '
^
'
where
!>' = ^
(28)
fixes the spaticil scale of the cloud. According to the estimations written above, b^ « A^/a. Eq. (27) is easily solved resulting in , ,
N
Mr) = 4^.^3
sm(r/b)
V^
„
- < ^-
,^^,
(29)
Since the dust density is nonnegative, the cloud radius should be RQ = irb. The range of vedidity of the expansion in powers of the dust density may now be written as N <^ Xo/a. With larger number of dust greiins, the absorption of slow ions inside the cloud becomes significant and Eq. (27) fails.
References [1] Tsytovich V.N., Physics - Uspekhi,167, 57 (1997). [2] Vladimirov S.V. Nambu M., Phys.Rev. E, 52, R2172 (1995) [3] Schram P.P.J.M., Trigger S.A., Contrib. Plasma Phys., 37, 251, (1997) [4] Lowen H., Phys. Rep., 237, 249 (1994) [5] Ignatov A.M., P.N. BuU. Lebedev Physics Inst., no. 1/2 (1995) [6] Ignatov A.M., Plasma Phys. Rep., 22, 648 (1996) [7] Ignatov A.M., Plasma Phys. Rep., 24, 677 (1998) [8] Khodataev Ya., Bingham R., Tarakanov V., Tsytovich V., Plasma Phys. Rep. 22, 1028 (1996).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
237
Effects of Dust on Alfven Waves in Space and Astrophysical Plasmas N. F. Cramer and S. V. Vladimirov Department of Theoretical Physics and Special Research Centre for Theoretical Astrophysics, School of Physics, The University of Sydney, N.S.W. 2006, Australia
A b s t r a c t . Even if the proportion of the total charge on dust grains (usually negative) is quite small in space and astrophysical plasmas, such as cometary atmospheres and interstellar molecular clouds, it can have a large effect on the dispersion characteristics of hydromagnetic Alfven and magnetoacoustic waves propagating at frequencies well below the ion-cyclotron frequency. The reason for this is the existence of the dust grain cyclotron frequency. A number of effects of the dust are considered here. Wave energy propagating at oblique angles to the magnetic field in an increasing density gradient can be very efficiently damped by resonance absorption processes in a dusty plasma. It is shown that, as well as the usual Alfven resonance, a low frequency dust-ion hybrid resonance also occurs. The effects of dust on the dispersion of surface waves in highly structured plasmas, damping due to dust charging, and nonlinear and shock waves may also play roles in such plasmas.
INTRODUCTION Dust can be a major component of space and astrophysical plasmas, such as cometary atmospheres and interstellar molecular clouds, and it may be charged due to electron attachment from the surrounding plasma or due to photoionization. For interstellar molecular clouds v^ith electron attachment the grain charge is negative. For HII regions there can be several hundred electrons per grain, v^hile for HI regions there are only a fev^ per grain [1]. The effects of dust on small amplitude magnetohydrodynamic v^aves [2,3] and on shock waves [4] in the clouds have been investigated previously. The properties of hydromagnetic waves in interstellar clouds, such as their speeds and damping rates, are important because the waves transport angular momentum if the cloud collapses to form protostars. There is some evidence of these hydromagnetic waves in such clouds from the observed widths of CO emission lines from molecular clouds [5]. Alfven wave propagation parallel to the magnetic field in a dusty interstellar cloud was first investigated in [2], and was applied to the problem of wave propagation in the dustless and dusty regions of stellar outflows [6]. Oblique
238
N.E Cramer, S. V. Vladimirov / Effects of dust on Alfven waves
propagation in interstellar clouds was considered in [3]. The basic properties of circularly polarized electromagnetic waves propagating parallel to the magnetic field in a plasma with static dust grains, in particular the case of frequency much less than the ion-cyclotron frequency were first studied in [7,8]. The effects of dust on the propagation of shock waves in interstellar clouds, including the rotation of the magnetic field in an obliquely propagating shock, have been considered in [4,9]. Alfven waves are excited by ion ring-beam instabilities in the plasmas surrounding comets as they approach the sun, and have been observed in the form of turbulence in spacecraft observations. The ring-beam distribution of ions is produced by pickup ionization of cometary molecules as they they stream from the comet in the solar wind. Resonant and non-resonant (firehose) instabilities of Alfven waves are produced. The eflfects of charged dust on these instabilities have recently been investigated [10,11], and it has been found that the dust has its greatest eflfect on the firehose instability. Even if the proportion of negative charge on the dust grains compared to that carried by free electrons is quite small (typically ^ 10"^ in interstellar clouds and in cometary plasmas), it can have a large eflFect on hydromagnetic Alfven waves propagating at frequencies well below the ion-cyclotron frequency [2,12]. In particular, the right hand circularly polarized mode experiences a cutoflF due to the presence of the dust. With a negligibly small charge on the dust grains, the waves have the usual shear and compressional Alfven wave properties, while for a non-zero charge on the grains the waves are better described as circularly polarized whistler or helicon waves extending to low frequencies [7]. Waves that propagate obliquely to the magnetic field in a nonuniform plasma can encounter the Alfven resonance, where the wavenumber perpendicular to the magnetic field direction becomes infinite (in the limit of zero electron temperature) and wave energy can be absorbed. This process has been postulated to be responsible for the heating of the solar corona [13], and has been used as an auxiliary heating mechanism for laboratory plasmas [14]. Alfven resonance absorption of waves in dusty interstellar molecular clouds could play a role in their energy balance and in the magnetic braking of protostellar clouds. The low frequency MHD waves observed upstream of comets may undergo resonant absorption in the structured plasma near the cometary bow-shock, and so give rise to the observed rapid heating of the solar wind protons [15]. The Alfven resonance process has been shown to be strongly modified by the presence of dust [12,3]. We show here that there is another resonance, introduced by the heavy dust grains, the dust-ion hybrid resonance, which may play a role in resonance absorption, but at much lower frequencies than Alfven resonance.
THE FLUID EQUATIONS We first consider small-amplitude waves in a plasma consisting of neutral atomic and molecular species, the ionized atomic and molecular species, the electrons.
N.E Cramer, S. V Vladimirov /Effects of dust on Alfven waves
239
neutral dust grains, and negatively charged dust grains. The plasma is allowed to be nonuniform in a direction perpendicular to the magnetic field. In interstellar clouds, collisions between the different species are important, so such terms are included. The plasma is assumed cold, so that the gas pressures of all the species may be neglected. For simplicity, all dust grains are assumed to have the same mass and charge. To describe resonant absorption, a 4-fluid model of the plasma is used here, as in [2,3], which employs the linearized fluid momentum equations for plasma ions (singly charged), electrons, neutral molecules and charged dust grains: - ^
= ^
- ^
= - t ^ m ( v „ - Vj) - f „ d ( v „ - Vd),
- ^
= 0 =
—6
( E + Vj X B o ) - Vin{Vi - \n)
" i^id(Vi - Vrf),
(1) (2)
— ( E + Vd X B o ) - Udn{yd - Vn) " 2^dz(Vrf - V^),
(3)
( E + Ve X B o ) - Veniye
(4)
" V„) - Ved{^e " Vrf),
where E is the wave electric field, m^ is the species mass and v^ is the species velocity in the wave. Vst is the collision frequency of a particle of species s with the particles of species t. We have neglected electron inertia and momentum exchange between ions and electrons, but have included ion and neutral molecule inertia terms because we are mainly interested in the frequency regime above the dust cyclotron frequency, where the ion and neutral molecule dynamics are important. To complete the system of equations. Maxwell's equations ignoring the displacement current (assuming the phase speed to be much less than the speed of light) are used. The background magnetic field BQ is assumed to be in the z-direction, and the direction of nonuniformity is the x-axis. The steady electron, ion and dust densities are rigo, n^o and n^o- The parameter 5 = neo/riio measures the charge imbalance in the plasma, with the remainder of the charge residing on the dust particles, so that the total system is charge neutral, with —erieo + enio — Zderido = 0, and —ZdC is the charge on the dust grain. Linearizing the equations, the wave fields are assumed to vary as exp{ikzZ — icut). We assume initially that the charge on the dust particles is not affected by the wave, i.e. we neglect the dust charging effects discussed later. A dense molecular cloud is considered, in which the density of molecular hydrogen Uno = 10^ cm~^, nio/uno = 10"*^, the magnetic field is 10~^G and the dominant ion species is HCO^ [2,3]. It is assumed that 1% of the mass is contained in spherical dust grains of radius 10~^cm composed of material with a mass density of 1 g cm~^. We then have 1 — 5 == 0.8 x 10~^. The temperature of the cloud is assumed to be 20 K, so that the thermal pressure is approximately 10% of the magnetic pressure, and we are justified in neglecting the pressure gradient terms. We define the Alfven speed VA = JBI/ii^pi^ where pi = miUio^ the ion-cyclotron frequency Qj = Boe/nii and the dust-cyclotron frequency fi^ = B^e/md^
240
N.E Cramer, S.V Vladimirov/Effects of dust on Alfven waves
RESONANT ABSORPTION The above equations can be reduced to the following 2nd-order differential equation for hz, the wave magnetic field component parallel to the equilibrium magnetic field:
dPK ^
{kl-D^){kl-D.)
The coeflScients D± are complicated functions of the collision frequencies, and are given in [3]. For the case of no collisions between species, they reduce to v\
u ± ltd
where, Vtrn = ^z(l — 5)/5. With a local approximation, where the fields are assumed to vary as exp(zA:a;x), the left hand side of Eq. (6) is replaced by —klbz^ so defining the local xwavenumber kx satisfying the dispersion relation of the fast Alfven wave branch. Cutoffs of kx (where kx = 0) can then be defined, given by
kl-D± = 0.
(7)
This is the parallel propagation in a uniform plasma case treated in [2], and for the case of no collisions [8,7] the parallel propagation dispersion relation is, from Eq. (7),
k2=^-
v\
\
---f^-rny
cj ± J7d
(g)
For a given value offc^^,there are three modes of parallel propagation derived from Eq. (8). If kz is fixed and there are no collisions, there are resonance frequencies at which kx -^ oo, given from Eq. (5) by the condition kl = (D+ + D_)/2.
(9)
As discussed in [3], Eq. (9) yields the Alfven resonant frequency LVA- This frequency is given to first order in Qd by VAkz , SQ^d^m ^A ^ — ^ + ^ , • S 2vAkz
/.^x 10)
This is the generalization to the dusty plasma of the well-known Alfven resonance in dust-free plasmas (eg. [14]). There are two cutoffs in kx determined by the vanishing of the two factors in the numerator of Eq. (5), bracketing the Alfven resonance frequency. In the collision-free case, and for small fid ^nd Qrm the
N.E Cramer, S.V Vladimirou /Effects of dust on Alfven waves
241
F I G U R E 1. The three cutoffs in kx (dotted curves) and the two resonances in kx (dashed curves).
cutoffs occur at a; ^ a;^ ± flrn/'^, i-^- there is a cutoff-resonance-cutoff triplet of frequencies [12]. Eq. (9) yields a quadratic equation in a;^, so there is a second resonant frequency, given by CJH ^ ^d for t;^^:^ > ^d- A cutoff in kx occurs just below LJH- The second resonance is analogous to the ion-ion hybrid resonance introduced by minority species, which is of importance in controlled fusion plasmas, and is called the dustion hybrid resonance here. Figure 1 shows the two resonance frequencies defined by Eq. (9) (the dashed curves), as well as the three cutoff (parallel propagation) frequencies defined by Eq. (8) (the dotted curves), as functions of A:^. The lowest frequency parallel propagating wave, corresponding to a right hand circularly polarized, dust-cyclotron mode, occurs just below the lowest resonant frequency, at frequencies less than Q^- The next highest frequency parallel propagating wave is left hand polarized, and becomes the ion-cyclotron wave at frequencies approaching Qi. For the highest frequency parallel propagating wave, which is a right hand polarized mode, a cutoflF frequency where A;^ = 0 occurs at a; = ^rn + ^d- For a given value oi kz, a plot of kl against frequency shows the two resonances with their associated cutoflFs: Figure 2(a) for the Alfven resonance (for VAkz/^d = 10) and Figure 2(c) for the dust-ion hybrid resonance (for VAkz/^d = !)• For parameters applicable to interstellar clouds, cuA/^d = 0.83 x 10^. At very long wavelength for the two lowest frequency modes, i.e. LJ
242
N.E Cramer, S. V. Vladimirov /Effects of dust on Alfaen waves
2U0U
(c)
(a) :
r' \
1000
0
1000 2000 0.4
0.6
0.8 1.0 1.2 Frequency xlO*
1.4
1.6
-6l__ 0.90
1.00 Frequency
0.6 1.0 1.2 Frequency xlO*
F I G U R E 2. Square of the wavenumber perpendicular to the magnetic field against normalized frequency (Real part solid curves, imaginary part dotted curves), (a) Alfven resonance, no collisions (b) Alfven resonance, collisions included, (c) Hybrid resonance, no colHsions (d) Hybrid resonance, collisions included.
the presence of collisions: the cutoff-resonance-cutofF triplet is still discernible and strong absorption occurs over a wide range of frequencies about the Alfven resonant frequency. For the interstellar cloud parameters considered here, with a low degree of ionization (n^o ^ lO'^nno), the ion-neutral collision frequency is ^ 10^ times Vtd, so that around the hybrid resonance, the wave is completely collisionally damped, as is shown in Figure 2(d). However, for clouds with higher degree of ionization (n^o ^ lO'^rino), the wave can propagate at frequencies just below the resonant frequency. The Alfven resonance frequency is ^ 10^yr~\ for a wavelength along the magnetic field of ^ 10~^pc(?^ 20 AU). At that frequency, the damping length is ^ 10 times the wavelength for parallel propagation (and the damping time is '^ 10"Vr for 2L real wavenumber), while for oblique propagation at the resonant frequency the wavelength and damping length are both ^ 20 AU. Another view of the resonance absorption mechanism is gained by considering a wave of fixed frequency and kz propagating into a plasma of varying ion density p in the x-direction. The wave equation Eq. (5) can then be written
dPh, dx^
{kl-Fp){kl-Gp) bz, kl - Hp
(11)
where F , G, and H are independent of x. Around the Alfven resonance, a wave initially at a frequency just below the local resonant frequency and propagating into an increasing plasma density, will encounter the resonance at the point in the
N.E Cramer, S. V Vladimirov /Effects of dust on Alfven waves
243
density gradient where Eq. (11) has a singularity, i.e. p = kl/H, Thus wave energy will be absorbed at the resonance position where the wave frequency satisfies Eq. (10). 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 [12]. The same will occur in the coUisional case, the only difference being that the resonance absorption occurs via the coUisional damping processes. 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.
DISCUSSION The damping of the waves shown in Figure 1(b) is due to effects of collisions, between the charged particles and the dominant neutral molecules, on the fast Alfven wave. Other effects can introduce a second, short wavelength, mode into which the fast Alfven wave mode converts at the resonance. Such effects are electron-ion collisions, which introduce the resistive Alfven mode, and thermal effects, which introduce the kinetic Alfven wave. Another effect, which is unique to dust grains, is the grain charging process, which leads to the appearance of an imaginary part in the dispersion equation [16], implying a damping of the wave in addition to the collisional damping. However such a damping depends on the electric field component parallel to the magnetic field, which is small for Alfven and magnetoacoustic waves in a low gas pressure plasma as considered here, except close to the resonance. This effect has been shown [17] to also introduce a second, short wavelength, dissipative mode. This mode may be pictured as an electrostatic modification to the shear Alfven wave due to the charge perturbations in the plasma during the grain charging process. The effect of this mode on Alfven resonance absorption has been shown in [17], but it should also arise at the hybrid resonance. Another effect which should be included is the range of dust grain masses or charges. If the resonant frequency is either much greater or much smaller than the typical dust cyclotron frequency, the dust respectively is frozen out of or into the plasma motion. In either case the resonant frequency depends simply on the effective plasma mass density. If however the resonant frequency is close to the typical ^2^, the resonance position will depend on the mass and charge spectrum of the dust, and the resonance will be smeared out. In addition, dust cyclotron damping will contribute to the total dissipation. Finally, we note that structured plasmas such as highly inhomogeneous interstellar clouds may also support surface waves, localized on regions of rapidly varying density or magnetic field. Surface waves may be damped by resonant absorption processes if the surface has non-zero width. The effects of dust, including the effects of the existence of the Alfven and the hybrid resonances, on surface waves and their damping have been considered in [18]. If dust charging is included, the short wavelength dissipative mode will couple to the surface wave and provide a damp-
244
N.E Cramer, S.V Vladimirov / Effects of dust on Alfven waves
ing mechanism, even when the surface is very sharp and no resonance damping can occur. A c k n o w l e d g e m e n t s Support for this work has been provided by the AustraUan Research Council.
REFERENCES 1. Spitzer, L., Jr., "Physical Processes in the Interstellar Medium", p. 168. (John Wiley, New York) (1978) 2. Pilipp, W., Hartquist, T. W., Havnes, O. & MorfiU, G. E., Astrophys. J. 314, 341 (1987). 3. Cramer, N. F. & Vladimirov, S. V., Publ. Astron. Soc. Australia 14, 170 (1997). 4. Pilipp, W., & Hartquist, T. W., Mon. Not. R. Astron. Soc. 267, 801 (1994). 5. Arons, J. & Max, C. E.,Astrophys. J. 196, L77 (1975). 6. Havnes, O., Hartquist, T. W. & Pilipp, W., Astron. & Astrophys. 217, L13 (1989). 7. Mendis, D. A. & Rosenberg, M., IEEE Trans. Plasma Set. 20, 929 (1992). 8. Shukla, P. K., Phys. Scripta 45, 504 (1992). 9. Wardle, M., Mon. Not. R. Astron. Soc. 298, 507 (1998). 10. Verheest, F. & Meuris, P., Phys. Scripta T75, 84 (1998). 11. Cramer, N. F., Verheest, F. & Vladimirov, S. V., Phys. Plasmas 6, 36 (1999). 12. Cramer, N. F. & Vladimirov, S. V., Phys. Scripta 53, 586 (1996). 13. lonson, J. A., Astrophys. J. 226, 650 (1978). 14. Hasegawa, A. & Chen, L., Phys. Fluids 19, 1924 (1976). 15. Sharma, A. S., Cargill, P. J. k Papadopoulos, K., Geophys. Res. Lett. 15, 740 (1988). 16. Tsytovich, V. N. &: Havnes, 0., Comments Plasma Phys. Contr. Fusion 15, 267 (1993). 17. Cramer, N.F. & Vladimirov, S.V., Physica Scripta T75, (1998). 18. Cramer, N. F., Yeung, L. K. k. Vladimirov, S. V., Phys. Plasmas 5, 3126 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
245
INVESTIGATION OF PLASMA IRREGULARITY GENERATION IN EXPANDING IONOSPHERIC DUST CLOUDS W. A. Scales*, G. S. Chae*, G. Ganguli^, P. A. Bernhardt^, and M. Lampe^ 'Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, U.S.A. ^Plasma Physics Division, Naval Research Laboratory, Washington D.C, U.S.A.
Abstract. Plasma instability processes associated with small-scale irregularity generation produced by the expansion of localized dust clouds into the earth's ionosphere are considered. These processes may produce irregularities on the meter scale size and in the lower hybrid frequency range which have been observed during some past space experiments. Aspects of the linear and nonlinear development of these processes are studied by using theoretical and nimierical simulation models.
INTRODUCTION Natural and artificial expanding dust clouds occur in the ionosphere in many situations. In recent years, several sounding rocket experiments have involved the release of electron attaching agents that produce expanding heavy negative ion clouds [1]. These experiments mimicked the release of dust into a background plasma since electrons are captured to produce electron depletions and heavy negative ion clouds that expand into the backgroimd plasma. Consistently, the production of small-scale plasma irregularities was associated with these experiments. An example of these irregularities is shown in Figure 1. This data shows the development of meter-scale electron density irregularities in the relatively thin (100 meters or less) boundary between the heavy negative ion cloud produced and the background plasma. Also, recently, radar retiurns have been observed from the space shuttle exhaust [3]. It has been proposed that the shuttle exhaust produces a dust cloud of ice particles that charges and expands into the background plasma. Associated with the expansion of the dust into the backgroimd plasma is the production of
246
WA. Scales et al /Plasma irregularity generation in expanding ionospheric dust clouds
30 Moy \%%1 170,2 t 15»e ny^t 2a3,80 km
<02
,0i
.Ofl
.OS
.10
.12
30 yoy 1992 tSO.O % 1306 n^t 29i.2l km
.$ .S .7 .8 Oisfoiict (km)
KO 1.1
1.2
F I G U R E 1. Electron density irregularities observed by an ionospheric sounding rocket experiment where an expanding heavy negative ion cloud was produced in the F-region. From [2].
plasma irregularities that scatter the radar signal. In this case, the irregularities are also on the meter scale. In general, irregularities produced by expanding ionospheric dust clouds have relevance to remote sensing of ionospheric processes as well as the creation of natural and artificial dusty plasmas in space. The objective of this work is to consider several models for the production of these irregularities.
PLASMA INSTABILITIES One likely source for the free energy required to drive plasma irregularities during the expansion of dust across a magnetic field into a background plasma are the inhomogeneities in the boundary between the the backgroimd plasma and dusty plasma produced by the dust charging. Steep boundary layer electron density gradients may be produced by electron capturing agents such as dust. Such a configuration may lead to highly sheared electron E x B fiows in the boimdary [4]. If these electron fiows have shear scale lengths of the order of the ion gyroradius and shear frequencies of the order of the lower hybrid frequency, the Electon Ion Hybrid EIH instability may result [5]. The local dispersion relation for this instability [6] including the effects of dust charge fiuctuations is given by rA
S'lS
^ +\ +
a-i'.
-Tk
6i
+ ia)chg ]u;'+[-l
(f-
0i\7
+ iuchg -^^
Si
+ (1 + 7)
6iS' ^ Oii
r,2
u
+ I —k - lUchs ( 1 + 7 ) ) ' ^ + ia)chg(l + 7)^A; = 0
(1)
WA. Scales et al /Plasma irregularity generation in expanding ionospheric dust clouds
EIH Instability
3.5
247
LHS Instability
Dispersion Relation Growth Rate
3
0
0 k V^ /CO .
(a)
d
pi
F I G U R E 2. Growth rate calculations for the Electron Ion Hybrid EIH and Lower Hybrid Streaming LHS instabilities for cjchg = 0.
where u = (j/cjpi and k = kL^, The ion plasma frequency and electron velocity shear scalelength are denoted by uj^i and LE- It should be noted that a;ih « a;pi under the assumption of strongly magnetized electrons used here. The other parameters are given in [6]. The dust charging decay constant [7,8,10] is denoted by ujchg and 7 = |/eo|^do/(e^eo^chg) where /eo? ^do, ^eo are the equilibrium dust grain electron current and equilibriiun dust and electron densities. Equation (1) is a generalization of past work on the EIH instability to include dust charge fluctuations. Figure 2a shows mmaerical calculations of Equation 1 for parameters typical of past experiments. The frequency, growth rate and wavelength at maximum growth scale as (Jr ~ cjih, uJi ~ cjih, A ~ LE. The effect of increasing uj^hg = ^chg/^pi is to reduce the growth rate and frequency slightly and narrow the bandwidth of the growing waves. Streaming of the dust relative to the background plasma at velocities larger than the ion thermal velocity may provide free energy to the production of irregularities in the lower hybrid range as well [9]. The dispersion relation under the assumption of strongly magnetized electrons can be shown to be ;,-,3 rA - 2fccD^ d;^ + {k'^-l-ul^)Cj^
+ 2fcd; - P = 0
(2)
where a) = uj/u^i, k = kvd/ujpu and (Dpd = a;pd/ct;pi. The dust plasma frequency and streaming velocity are denoted by a;pd and v^i respectively. Figure 2b shows a numerical calculation of (2) for typical experimental parameters (a;pd/a;pi < 0.01). The frequency, growth rate, and wavelength scale as a;r ~ c^m, uJi ^ u;ih(a;pd/a;pi)^/^, A ^ Vd/f\h' It is noted that the growth rate of the LHS instability may be substantially smaller than the EIH instabiUty due to the smallness of a;pd/a;pi. It is
248
WA. Scales et al /Plasma irregularity generation in expanding ionospheric dust clouds
found that dust charge fluctuations have neghgible effects on the Unear growth of the LHS instabiUty in the parameter regimes considered here.
NUMERICAL SIMULATIONS A new numerical simulation model has been developed to study the evolution of plasma instabilities in expanding dust clouds. The model and the numerical methods utilized are described in more detail in [10]. The model and numerical methods are an extension of previous work used to study the expansion of heavy negative ion clouds into a background plasma [6]. The model is two-dimensional and it treats the background plasma as a fluid and the dust particles with the Particle-In-Cell PIC method. Presently, the model allows the study of dust expansion in the plane perpendicular to a background magnetic field. The electrons are treated as being strongly magnetized and the ions and dust as unmagnetized. The dust dynamics as well as self-consistent dust charge fluctuations are included in this model. The model allows the study of the nonlinear development of plasma instabilities relevant to expanding dust clouds. It is also possible to perform numerical investigations of the nonlinear evolution and effects of dust charge fluctuations produced by these instabilities. The model and numerical methods are quite general and the study of a wide range of plasma instabilities in dusty plasmas is planned in future investigations. The objectives here are to discuss the qualitative behavior of dust expansion processes rather than to consider speciflc experimental parameters. The EIH instability has been investigated with our simulation model over a wide range of parameters. In the results to be presented, the dust is allowed to expand across the magnetic fleld into the background plasma at a speed of approximately 0.3?;ti where t^ti is the ion thermal velocity. Many applications may involve the case where the expansion velocity exceeds va, however, simulation results in this regime show that the instability is still excited and only minor quantitative differences are observed. A thorough study of this regime will be left for a future investigation. The dust is taken to have an equilibrium charge of 300 electrons. It is observed as the dust expands across the magnetic fleld, the charging of the dust and reduction of electron density produces a well-deflned boundary layer between the dusty plasma and the background plama. An ambipolar electric fleld develops across the boundary which produces a highly sheared electron E x B flow velocity along the boundary. These are the conditions necessary for the development of the EIH instability. This velocity shear-driven instability produces vortex-like structures of wavelength of the order of the velocity shear scale size. In the case of a dusty plasma, the effects of charging and dust charge fluctuations should be taken into account on the EIH instability as was discussed in the previous section. Figure 3 shows the results of two numerical simulations of the EIH instability produced by the expansion of dust across the background magnetic fleld. Mode 4 is choosen to be excited in the simulations. The flrst of these simulations has a relatively low charging rate with cDchg = 0.03 and the second has a relatively
WA. Scales et al. /Plasma irregularity generation in expanding ionospheric dust clouds
high charging rate case cDchg = 1.00. Note that the low charging rate case produces negUgible fluctuations in the dust grain charge Qa while in the high charging rate case well defined vortex structures are observed to develop in the dust charge as well as the electron density. This is due to the fact that the charge fluctuations axe able to follow the electron and ion density fluctuations more faithfully since the charging period is comparible with the instability growth period. The results indicate the importance of irregularity production in the dust charge as well as the electron and ion densities. The growth rate and frequency observed in the simulations generally agrees well with the theoretical predictions discussed in the previous section. The lower hybrid streaming instability [9] may lead to growth of irregularities in expanding dust clouds due to the streaming of the dust relative to the background plasma at speeds larger than va. Here we present some preliminary results of numerical simulations of this instability. We consider the case of a weak beam with the classical beam strength parameter ^1^/(j^1i = 0.01. In this case, the dust charge is taken to be 1% of the background charge density and the equilibrium charge is 1000 electrons. Future investigations will consider a wider spectrum of relative dust charge densities. The dust streaming velocity is taken to be IQva, The charging parameter (Dchg = 0.5. The LHS instability produces irregularities in the background plasma that have been proposed to produce radar signal returns from the space shuttle exhaust [3]. The development of irregularities by this instability is discussed in [10]. Structuring in the electron, ion and dust density as well as the dust charge have been observed in the numerical simulations and agree well with the predictions of the linear theory. We wish to point out some interesting effects here that involve the nonlinear evolution of the instability and in particular the effect of the instability on the dust grain charge. Typically associated with streaming instabilities is the modification of the velocity distribution function. This may be observed in the background plasma and/or beam plasma. The modification of the velocity distribution is one means of saturating the instability [11] and an important consequence of the nonlinear development of the instabilities in general. Modification in the velocity distribution of dust grains is observed in the simulations. We also observe an interesting new eff'ect due to the incorporation of dust charge fluctuations. Considering a phase space of dust charge, we also observe nonlinear evolution of charges on the dust grains. Figure 4 shows the evolution of the phase space simultaneously for dust velocity and charge. Mode 1 is choosen to be excited in this simulation. These are shown at three different times during the simulation. At early times {uj^it = 40), the LHS instability can be seen to grow in both the dust velocity and charge by producing a small amplitude sinusoidal perturbation. Later {uj^it = 50) some of the dust grains start to become trapped in the growing potential of the instability. Finally at late times (a;pit = 100), the charge phase space shows considerable scattering as well as the velocity phase space. Therefore nonlinear effects are seen to have fundamental effects on the dust charge.
249
250
WA. Scales et al /Plasma irregularity generation in expanding ionospheric dust clouds 0 ) , =1.0
( 0 , =0.03 chg
Electron Density
chg
H250
Htfew^^^^.
Ion Density
^
k
190
210 x/Ax
FIGURE 3. a Numerical simulation results for the Electron Ion Hybrid EIH instability showing electron and ion density and dust charge with cDchg « 0.03. The dust cloud expands in the positive x direction and produces a highly sheared electron E x B flow in the —y direction. Note weak dust charge fluctuations in the slow charging rate case.
FIGURE 3. b Numerical simulation results for the Electron Ion Hybrid EIH instability showing electron and ion density and dust charge with cDchg ^ 1-0. Again the dust cloud expands in the positive x direction and produces a highly sheared electron E x B flow in the —y direction. Note strong vortex dust charge fluctuations in the fast charging rate case.
WA. Scales et al/Plasma irregularity generation in expanding ionospheric dust clouds
C0p.t= 40
-0.5 0.05 Dust Charge
V = 50
251
C0p.t= 100
^m^
-0.05
64 128 0 64 128 0 64 x/A x/A x/A F I G U R E 4. Numerical simulation results showing the development of the lower hybrid streaming instability. Note charge phase space as well as velocity phase space modification produced by the nonlinear evolution of the instability.
In these preliminary results, typical dust charge perturbations of ±5% are observed as well as charging and discharging in the instability produced potential wells. Another important effect observed in the simulations is the increase in the saturation amplitude of the instability due to the dust charge fluctuations. This increase is due to the decrease in negative dust charge which allows the wave potentials to grow to larger amplitudes before trapping dust grains. The decrease in negative dust charge is due to the growth of the instability ion density irregularities. It is therefore important to note that even though the effects of dust charge fluctuations may be small in the linear development of the irregularities, the effects may be important in the nonlinear development. A much more thorough study of this process is under investigation.
SUMMARY AND CONCLUSIONS Two instabilities relevant to the production of irregularities produced by expanding dust clouds in the ionosphere have been considered. These two processes will produce waves in the lower hybrid frequency range as well as meter size irregularities in the ionosphere. Our results also point out some important aspects of the nonlinear evolution of these instabilities as well as the nonlinear evolution of
252
WA. Scales et al. /Plasma irregularity generation in expanding ionospheric dust clouds
dust charge fluctuations associated with these instabiUties. It is believed that the general results here may have applications to situations other than the ionosphere in association with dust expanding into a background plasma. Also the nonlinear evolution of dust charge fluctuations is a topic of a fundamental nature in dustyplasmas that will be pursued in more detail in future investigations with the models and methods presented here.
ACKNOWLEDGEMENTS This work was supported by the U.S. National Science Foundation NSF and Department of Energy DOE under grant DE-FG02-97ER54442 and the U.S. Ofiice of Naval Research ONR.
REFERENCES 1. Bernhardt, R A., et al., J. Geophys. Res. 100, 17331 (1995) 2. Scales W. A., Bernhardt, R A., Ganguli, G., Siefring, C. L., and Rodriguez, R, Geophys. Res. Lett. 21, 605 (1994) 3. Bernhardt, R A., Ganguh, G., Kelley, M. C., Schwartz, W. E., J. Geophys. Res. 100, 23811 (1995) 4. Ganguli, G., Bernhardt, R A., Scales, W. A., Rodriguez, R, Siefring C. L., and Romero, H., Physics of Space Plasmas (1992) SPI Conf. Proc. Reprint Ser.^ 12, edited by T. Chang and J. Jasperse, 161 (Scientific, Cambridge, 1992). 5. Ganguli, G., Lee, Y. C , and Palmadesso, R, Phys. Fluids 31, 2753 (1988) 6. Scales, W. A., Bernhart, R A., and Ganguli, G., J. Geophys. Res. 100, 269 (1995) 7. Varma, R. K., Shukla, R K., and Krishan, V., Phys. Rev E47, 3612 (1993) 8. Jana, M. R., Sen, A., and Kaw, R K., Phys. Rev E48, 3930 (1993) 9. Rosenberg, M., SalimuUah, M., and Bharuthram, R., Planet. Space Sci^ accepted (1999). 10. Chae, G. S., Scales, W. A., Ganguli, G., Bernhardt R A., Lampe, M., this issue^ (1999) 11. Winske, D., Gary, S. R, Jones, M. E., Rosenberg, M., Mendis, D. A., and Chow, V., Geophys. Res. Lett, 22, 2069 (1995)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
253
Mass distributions and self-gravitation in dusty plasmas Frank Verheest Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, B-9000 Gent, Belgium
Abstract. In the present review dust grain distributions are discussed, by adopting either models with a discrete number of different grains species, or by using a continuous size range. The changes in dispersion laws for some of the better-known dusty plasma modes are given. In addition, we review how plasma thermal effects, dust mass distributions and magnetic field pressure alter the Jeans instability lengths in a mixture of plasma electrons and ions together with different charged and neutral dust species. Very few self-consistent kinetic descriptions are available.
INTRODUCTION Plasma and dust are key ingredients of our universe. Combinations of both are encountered in circumsolar dust rings, in noctilucent clouds, in the asteroid belt and cometary comae and tails, in the rings of the Jovian planets and in interstellar dust clouds. These have been reviev^ed recently [7,9,17]. Dust grains immersed in plasma and radiative environments become electrically charged. Mixtures of electrons, ions and charged dust distinguish themselves from ordinary plasmas, and a consistent modelling of charged dust grains runs into interesting difficulties compared to standard plasmas. Besides dealing with particles v^hich can have fluctuating charges, dust sizes range almost continuously from macromolecules to asteroids. For the heavier dust grains self-gravitation could become important, maybe even comparable to electromagnetic forces. Whereas usually charged dust grains are treated as mono-sized, point-like heavy (negative) ions, more is needed. The daunting task to describe a variety of sizes and masses by any form of tractable distribution has so far yielded only preliminary and incomplete results. Needless to say, fev^ attempts at a proper kinetic theory for such complex systems have been made. Hence the study of dusty plasma modes still needs a lot of fundamental effort. Here v^e will review how mass distributions have been dealt with, by adopting either discrete models with a number of different grain species, or by using a continuous differential density in a bounded size range, and indicate the changes in dispersion laws for some of the better-known dusty plasma modes.
254
E Verheest/Mass distributions and self-gravitation in dusty plasmas
PARALLEL MODES We start with parallel modes in dusty plasmas with various grain mass distributions. In the absence of fully self-consistent kinetic theories, we will adopt a poor man's perspective and represent charged dust by a limited number of discrete species, or else as continuously distributed in a hmited size range.
Dust-acoustic Jeans modes Many authors have devoted quite a lot of attention to streaming effects on dusty plasma waves, including self-gravitational influences, and hence we can omit Buneman-type instabilities [11,12] from the present overview. For parallel waves possible self-gravitation only affects the longitudinal modes with dispersion law [3]
The high-frequency limit gives Langmuir modes, while the lowest-frequency branch leads to the ordinary Jeans instability. For a combination of electrostatic and gravitational effects we simplify (1) by using obvious inequalities like UJ^UJ']^ <^ ujpeUJjyiLjjeUjji
^]i Y. ^Id < ^ J IZ ^Jrf' d
d
^Pi^Ji Y. ^pd^Jd < ^ J I ] ^M^ d
d
(2)
because in all known dusty plasmas the mass-per-charge ratio for the dust species is much larger than for ions. Analogous relations hold even more strongly for electrons [10,11]. Now we take intermediate phase velocities so that Csd <^ Re cj/k <^ Cgi^ CgeWithout self-gravitation, this is precisely the regime for the dust-acoustic mode [13]. Introducing A = /c^Ay(1 + k'^Xl), with 1/A^ = l/Xj^^ + 1/A|)- referring to a global Debye length, we simphfy (1) to
^ + {^Mg - ^ ^Idg) ^^ - ^ E Yli^Pd^Jd' " ^pd'^Jd)^ = 0.
(3)
Global dust plasma and Jeans frequencies are defined for all dust species together, by putting u^^g = Ed^ld and uj^g = Ed^h- The product of the roots u'^ of (3) is negative, so that there is always an unstable solution, with a;^ < 0. The roots are given by
\(^A-B±^{A-By ^pdg
with
+ 2ACy
(4)
E Verheest /Mass distributions and self-gravitation in dusty plasmas
5 =^ > 0 , ^pdg
C ^ = 4 - E Yli^vd^Jd' - ^vd'^Jd? > 0. ^pdg
d
255
(5)
d'
Both coefficients are very small for micron-sized dust grains, and it is only for highly charged millimeter-sized dust that B c:i\. C can only vanish if for all dust species ^pd/^jd = ^pd' /^Jd'^ requiring all species to have the same charge-to-mass ratio, a very special case indeed. Moreover, A = 5 < 1 or /c^A|) = ^'jdg/i^ldg ~ ^Jdg) determines a critical wavenumber. With the dust-acoustic frequency given through (jj^^ = AuUp^g, the description of the modes is as follows [11]: • When A > B or uuda > ^Jdg, the plasma effects dominate. The plus sign in (4) gives a generalized dust-acoustic wave, at a higher frequency due to the dust distribution. The minus sign leads to an unstable mode, the frequency of which vanishes when C and the dust mass distribution go to zero, and hence called a dust distribution instability. • In the special case that A — B, the dust distribution through C leads to uj^ — ±ijJpdg^JdgyC/2, tending to zero-frequency modes for mono-sized dust. • For A < B the self-gravitational effects dominate, and the plus sign gives a new stable mode that does not exist without dust mass distribution. The minus sign leads to a modified Jeans mode, with an increased growth rate. Since B and C are small for micron-sized dust grains, there is only a very narrow fc-range (with correspondingly large structures) in which both the new stable mode as well as the Jeans instability itself can exist. Similar results are obtained when one considers charged and neutral dust together, for then C = 2uPj^JuJ^^^^, with subscripts c and n for charged and neutral dust. \i A > B, the plus sign in (4) gives the dust-acoustic mode, whereas the minus sign is essentially the Jeans instabihty in the neutral gas. The latter cannot be counterbalanced, as we have omitted here all dust thermal effects. For A < B the plus sign is a stable mode that does not exist without neutral dust, and the minus sign gives a Jeans mode, with an enhanced growth rate due to the neutral gas.
Continuous mass distribution Another possibility is that the charged dust radii a come in a continuous range [^min, ^max], with au associated differential density distribution Udici) [5,6,10]. When we suppose for all grain sizes that a
256
E Verheest/Mass distributions and self-gravitation in dusty plasmas
^—
nd{a)ada
I
(7)
nd{a)da
which conserves the total charge on all the dust grains together, by dint of the linear relation between charge and size. We now look at Langmuir oscillations described by the electrostatic part of (1),
in which dust thermal effects have been kept but self-gravitation not. For a size range we replace the sum over several dust species by an integral [4], so that 1
<
<
cc;2 _ /c^c^^
a;^ - k'^cl^
r -
^^ ^ ^^
ri,ia)qj{a)
Jamin eoUj'^md{a) - k'^SoKTd
assuming a common (low) temperature for the different dust grains. Expressing mass and charge as functions of size, the integral obviously is of the form
. - . npa^^^ _ Jamin
a^
- 0
.-a. fi^ Jz^,^
^ ^ .... f(^
Z - b
Jzmin
^ "
_ »
We have introduced z = a^ and notice the immediate analogy with the Landau integral in velocity space. Hence a new kind of damping occurs if the pole in z = b can be accessed, needing ^mm ^ b < Zmax- There are thus three damping mechanisms for Langmuir waves, the (velocity space) Landau damping, the Troms0 damping due to dust charge fluctuations [8], not discussed here, and a dust distribution damping [4]. Analogous results have been obtained for streaming instabihties [21], and for electromagnetic modes through the gyrofrequencies [15].
Nonlinear dust-acoustic modes When we turn our attention to small but finite-amplitude dust-acoustic modes, we use general Sagdeev pseudo-potential results [18],
1{^J + l^i^)^" + l^i^)^' + \c(V)^' = 0'
(11)
up to fourth order in the electrostatic potential
My) =
E^-^^
3^2 '
E Verheest/Mass distributions and self-gravitation in dusty plasmas
257
For A we have used the common knowledge that only slightly supersonic solitons are possible, defined here relative to the generalized dust-acoustic velocity Qa = ^D^pdg^ SO that V = Qa(l + /^^) ^^d /x ~ 0{ip). In principle, both rarefactive and compressive solitons are possible [13], but only the rarefactive ones are compatible with weak nonlinearities [16], and given by 6/i^sech^«) a+ — a_ tanh (/^^) where K = /x/A/^v^ and a± = SA^^ ± XD^B^XI+
9fi^C.
Circularly polarized electromagnetic modes To study low-frequency electromagnetic modes [14], we write the dispersion law in the presence of one or more charged dust species as
uj^ = c^e + - — ^ + — - ^ + J2 -TFT'
14
The modes are supposed of low frequency in the sense that cc;
^'^l
^ ^ ' T (1 - Snoo - eVl
= 0,
(15)
where VAP is the plasma Alfven velocity, computed without charged dust. The RHCP modes are given by ^~
\ {(1 - m^ ± v/(l-(^W+4Wi^ } .
2
(16)
For the further discussion we assume all the different dust species together to be negatively charged {5 <\) and distinguish two extreme wavelength regimes. At small wavenumbers (4A:^V^^
L2T/2
Our approximations in deriving (15) and (16) require for the forward propagating mode that \Q.d\
258
E Verheest/Mass distributions and self-gravitation in dusty plasmas
[6 < I), For the backward propagating mode equally stringent conditions apply, since the ordering of small terms requires |f^d| ^ ^^^ip/(l — <^)^2 ^ (1 — ^)^i. On the other hand, the large wavenumber regime is easier to deal with, implying (1 - J)2Q2 ^ 4A:2y2p, SQ ^hat t h e n
ujc^\{l-6)^i±kVAp.
(18)
Compared to Alfven waves in uncontaminated plasmas, the modes remain dispersive due to the presence of the charged dust species.
MAGNETOSONIC MODES In the most generic case when both charged and neutral dust grains are present, the dispersion law for perpendicular propagation gives high-frequency modes, which we will not address, and also several low-frequency waves, part of the X mode [19,20]. The dispersion law for different charged and neutral dust species is then
[^' - k'Vl) (l + E j l ^ ) + ^'y^^ + -3.. = 0.
(19)
The electrons and plasma ions are supposed light enough so that we can neglect their self-gravitational effects. B represents the modifications due to dust mass distributions, discussed in (21) and below. The magnetosonic velocity Vms is defined here through y2
_ T/2
, / 2\ _
^^0
, E o^^a'^a^^s AU^
, ^a
(20)
Both the Alfven velocity VA and the mass-weighted average for the thermal velocities squared are defined for all the charged species together. When the mass density is mostly in the charged dust grains, VA essentially is the dust Alfven velocity V^ci, much smaller than the plasma Alfven velocity. Also, (c^) tends to a dust-acoustic velocity squared, if the plasma species provide most of the thermal pressure. The influence of dust mass distributions manifests itself in the term
B = j;yT,E-Wu
(l - ^)\
(21)
which is clearly positive, and vanishes when the dust species all have the same charge-to-mass ratios. In the absence of neutral gas (19) becomes u' = k-'Vl,{\-B)-u],^,
(22)
and hence the system is unstable at smaller lengths than before. The critical Alfven-Jeans wavenumber isfc^j— ^jdgl^ms^J^ — ^- Opposition to gravitational
259
E Verheest /Mass distributions and self-gravitation in dusty plasmas
collapse comes not only from the thermal pressure effects, as in the standard Jeans treatment, but also from the magnetic pressure, because the external magnetic field resists gravitational squeezing. Hence we have a restoring force against gravitation even in the complete absence of thermal effects [20]. The neutral gas terms, on the other hand, introduce new modes for each gas species. If we restrict ourselves for simplicity to one neutral gas species, we obtain from (19) that [^' - k'VLi}
- S) + a;^,) (^^ - k^c^ + c.}J = (a;},, + feV^fi) u%,
(23)
This clearly shows how the Alfven-Jeans mode in the charged dust fluids and the pure Jeans mode in the neutral dust fluid are coupled together. As the discriminant of this bi-quadratic equation is always positive. A = ;-^L. - ^% + k'c% - fcVKl -8)]\
Aul (a;L, + ^¥^8)
> 0,
(24)
o;^ is real and unstable modes require o;^ < 0. A negative constant term occurs in (23) when L2
^ 7,2
'^ ^ '^AJdg
_
2 ^Jdg
y2 (\ — tZ]
,
2 ^Jg
r} (\ — B\
^
i2
, p
^AJ,charged dust ' '^J,neutral gas*
(cy^\
v^"^/
The direct influence of the mass distribution is to increase kAjdg and hence to lower the instability lengths. Usually Csg
KINETIC EQUATIONS FOR M A S S D I S T R I B U T I O N S Only a few, partial self-consistent kinetic descriptions are available in the literature, pioneered by the Troms0 group [1,2]. In these papers, the dust grains are supposed to have the same mass, but different charges are described as ionization levels, and hence the charge is an additional phase space variable. Generalizations of the standard kinetic equations have been obtained, by using a Krook-like collision integral to describe the interactions between the various charge levels, but without real charge fluctuations.
CONCLUSIONS We have reviewed the consequences of dust mass and size distributions for some of the well-known modes propagating in dusty plasmas. For lack of better alternatives
260
E Verheest/Mass distributions and self-gravitation in dusty plasmas
in these extremely complicated problems, we have used very simple models with either a discrete number of different dust species, or with a continuous, but bounded range of sizes. Little could be said about self-consistent kinetic descriptions, for which only partial results are available, and where clearly much more needs to be done, although the complexities are intimidating. A final remark when discussing Jeans-type instabilities concerns the common use of the Jeans swindle. It is assumed that a homogeneous equilibrium exists or can be used, even though we know that masses cannot be shielded and nonuniform initial states are unavoidable. Here also much needs to be done, in order to circumscribe the ranges of validity of our models.
Acknowledgments Thanks are due to the Bijzonder Onderzoeksfonds (Universiteit Gent) for a research grant, and to the Fund for Scientific Research (Flanders) for a travel grant to attend the 2nd International Conference on the Physics of Dusty Plasmas. Stimulating discussions with M.A. Hellberg and P.K. Shukla are gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Aslaksen, T.K., J. Plasma Phys. 54, 373-391 (1995). Aslaksen, T.K. and Havnes, O., J. Plasma Phys. 51, 271-290 (1994). Bliokh, P.V. and Yaroshenko, V.V., Sov. Astron. 29, 330-336 (1985). Bliokh, P.V. and Yaroshenko, V.V., Plasma Phys. Rep. 22, 411-416 (1996). Brattli, A., Havnes, O. and Melands0, F., J. Plasma Phys. 58, 691-704 (1997). Havnes, O., Aanesen, T.K. and Melands0, F., J. Geophys. Res. 95, 6581-6585 (1990). Horanyi, M., Annu. Rev. Astron. Astrophys. 34, 383-418 (1996). Melands0, F., Aslaksen, T. and Havnes, O., Planet. Space Sci. 4 1 , 321-325 (1993). Mendis, D.A. and Rosenberg, M., Annu. Rev. Astron. Astrophys. 32, 419-463 (1994). Meuris, P., Planet. Space Sci. 45, 1171-1174 (1997). Meuris, P., Verheest, F. and Lakhina, G.S., Planet. Space Sci. 45, 449-454 (1997). Pandey, B.P. and Lakhina, G.S., Pramana 50, 191-204 (1998). Rao, N.N., Shukla, P.K. and Yu, M.Y., Planet. Space Sci. 38, 543-546 (1990). Shukla, P.K., Physica Scripta45, 504-507 (1992). Tripathi, K.D. and Sharma, S.K., Phys. Rev. E 53, 1035-1041 (1996). Verheest, F., Planet. Space Sci. 40, 1-6 (1992). Verheest, F., Space Sci. Rev. 77, 267-302 (1996). Verheest, F. and Hellberg. M.A., Physica Scripta, in press (1999). Verheest, F., Hellberg, M.A. and Mace, R.L., Phys. Plasmas 6, 279-284 (1999). Verheest, F., Meuris, P., Mace, R.L. and Hellberg, M.A., Astrophys. Space Sci. 254, 253-267 (1998). 21. Yaroshenko, V.V., Plasma Phys. Rep. 23, 433-439 (1997).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
261
Statistical Description and 3D Computer Modeling of Relaxing Dusty Plasmas Yuriy I. Chutov*, Olexandr Yu. Kravchenko*, Pieter P J . M . Schram^ and Roman D. Smimov* ^Faculty of Radio Physics, Taras Shevchenko Kiev University, Volodymyrs'kaStr. 64, 252017Kiev, Ukraine Department of Physics, Eindhoven University of Technology, The Netherlands
Abstract. The Liouville equation is considered for relaxing dusty plasmas consisting initially of equilibrium electrons and ions as well as neutral dust particles. Plasmas relax due to a collection of electrons and ions by dust particles. The free and collected electrons and ions are considered as a common sub-system described by the common whole distribution function. It is shown that this sub-system is described by the complex integral-differential equation, which can be expected to lead to Non-Markovian kinetics. Computer modeling show that the selective collection of electrons and ions by dust particles causes a deviation from the initial equilibrium of free electrons and ions in plasma crystals.
INTRODUCTION Dusty plasmas are complex statistic systems due to the intensive charge exchange between dust particles and background electrons and ions. Usually these plasmas are considered as statistical systems consisting of at least three sub-systems (components), namely: electrons, ions, and dust particles. According to this consideration, the bounded electrons and ions creating the dust particle charge are not include to the subsystems of free electrons and ions after their collection by dust particles. However they were free before their collection and act on other free particles in exactly the same way as before collection. This circumstance causes some non-logic of this consideration, especially in the case of non-stationary dusty plasmas. Non-stationary dusty plasmas has to be considered from this point of view as statistical systems with variable numbers of particles (1) due to a collection of background electrons and ions by dust particles. Dust particles can be considered here as some specific boundary distributed over the entire dusty plasma. In the general case, such non-stationary systems have to be non-equilibrium because the probabilities of all processes providing a change of particle numbers depend on particle energies. In particular, a selective collection of electrons and ions by dust can cause a nonequilibrium state of dusty plasmas (2-4). However, modem theoretical investigations of dusty plasmas including their non-stationary behavior (5-7) are usually carried out
262 Y.L Chutov et al /Statistical description and 3D computer modeling of relaxing dusty plasmas
on basis of the assumption of equilibrium background electrons and ions, because of the large complications of their non-equilibrium description. The aim of this work is a consideration of relaxing dusty plasmas as integral statistical systems and a computer modeling of the relaxation phenomena in a plasma crystal investigated intensively during recent years (8,9). Some results of these investigations were published earlier in (10,11).
STATISTICAL DESCRIPTION Let us consider a system of Ne electrons and Nt ions (or N =Ne +A^/ particles) which is written by some whole distribution function^A^ (X, t) where X is the whole set of coordinates r^j, ^^2 '• • -^aN ^^^ momentum p^^ ,p^^ ^--PN components of electrons and ions. Index a = ej denotes here electrons and ions. This system is relaxing due to a collection of electrons and ions by some M immobile identical grains (dust particles) in a volume consisting of these electrons and ions. In this case, electrons and ions can be divided into two groups changing in time, namely: K^ (t) free electrons and ions as well as L^(t) electrons and ions coupled to grains. As a result, the following relations hold: (1)
K,(t) + L,(t) = N„ According ^a,j3-~
to
^(^^^aj ^
this
division,
the
summary
potential
energy
""^jskO can be divided into several kinds, namely: the
\
potential energy of an interaction between free particles {E^j p^), between free and bounded particles (Eaf,/3b)^ ^^^ between bounded particles (£'«^^^). Indexes/and b denote free and bounded particles, respectively. As a result, the Hamiltonian can be written for this system in the following form:
«»w= S [^+^.ft/)J+ I Paj
^-^-^U (r ) +
\
\
J
(2)
where these energies can be written as
^ \
^
Eab,pb=-
l
Y.^(l>l7aj^'^pkf) ^
\
(3)
Y.I. Chutov et al /Statistical description and 3D computer modeling of relaxing dusty plasmas 263
Triple indexes are necessary here for the mark of free (index f) and bounded (index b) electrons and ions at the condition of the main double index {aj.fik) conservation for the equations of free particle motion. Let us introduce the number iSamC^am'O electrons and ions collected to some grain to the instant t so that the effective charge Qm(Rm, 0 of a grain can be written as: a
Obviously, the following relation takes place:
4 ( 0 = YQan,(K,0
(5)
\<m <M
Let us also introduce some effective inner energy Em(RmJ) of a grain including the kinetic energy of bounded particles as well as the potential energy of an interaction between particles bounded by this grain, and the potential energy of these bounded particles in external fields. Therefore, the following expression can be written
TE^{K^t)= X l^-^UotM^-
\<m<M
\
^2m
^
^
TA/r^j-r,j)
(6)
\
Here indexes ajbm and fijbm denote the electrons and ions bounded by mgrain. In this case, the equation (2) can be rewritten in the following form: (p' \
^
1
2m
\
+2^ /Z1e-^-^m . ( ^^ . ,m^0
\<m<M
rn\
I^Q.(K,t)QSl,tm^Rn,-KO
"^ \<m,n<M m^n
It is very important that this Hamiltonian is constant in the time although free and bounded particles have the variable potential and kinetic energies. This constancy is provided by the exchange between free and bounded particles. It means that the effective charge and inner energy of dust particles depend non-directly (through the time) on the set of coordinates ry,r2,...r^ and momentum PJ,P2,"'PN components of free particles. Taking into account the equation (7), it is possible to get from the Liouvile equation the following formal equation for the whole distribution function fN(Xt)
264 Y.L Chutov et al /Statistical description and 3D computer modeling of relaxing dusty plasmas
f
SQ„ ^ \<m<M
+
^Paj
\
\<m<M
m„Q„)(t>
Sfn dr.aj
J)
+ dt \
(8) aj
^\
S^aj
^ \<m<M
df. ^P aj
f
•I \
^^a
aj
" \
«7
It is necessary to note that three last terms in the equation (7) depend on the whole distribution function fy (X t) and therefore on the whole set of coordinates ^ei^Ki^-^N,^ ^ P 4 V . . 4 , and momentum Pei^Pei^-'PN,^ Pii^Pa^'-PN, components as well as on the position Rm of a grain. These integral dependencies are very complex, as can be seen, for example, from the following expression for the number particles Qm(Rm> t) collected by some grain during the time t:
(9) where w{X^j,R^)
is a collection probability of some free ay-particle by a grain
located in the point Rm . Therefore the equation (19) is the complex integraldifferential equation which together with the equation (20) can be expected to lead to Non-Markovian kinetics. The integration of the equation (8) gives a principal possibility to get equations for other kinds of distribution function including the equation for the one-particle distribution function used in the Boltzman kinetic equation with a collision integral. However this integration is very complex in the general case and therefore it is very useful for the investigation of relaxing plasmas to use the computer modeling allowing to trace an evolution of every computer particles during a relaxation of dusty plasmas.
COMPUTER MODEL The choice of a model is caused by dusty plasma parameters, which can change within very large boundaries (12). In typical cases of laboratory plasma crystals, the density of neutral atoms (-10 cm') exceeds essentially densities of dust particles (-10"^ -10^ cm"^) and electrons (-10^^ cm""^). In the last case, the electron
Y.I. Chutov et al. /Statistical description and 3D computer modeling of relaxing dusty plasmas 265
temperature, the average grain size, and the average intergrain distance is equal about 2eV,l jum, and 2-4 LD {LD is the electron Debye length), respectively. Of course, the ion temperature is equal about the atom temperature that is equal about to 0.025 eV, Simple evaluations show that the equilibrium charging time (13) of a dust particle is essentially less than collision times of electron and ions with neutral atoms even in typical cases of laboratory plasma crystals. The normalized ion-atom cOp^r^^ and electron-atom cOp^r^a collision characteristic times are equal about 13.5 and 9, respectively, where co^^ is the ion plasma frequency. Therefore these collisions can be not taken into account and relaxation phenomena due to charging processes, can be investigated in plasma crystals without taking into account neutral atoms. In this model, some 3D cube crystal initially consisting of motionless neutral circular dust particles of radius R^ and background equilibrium electrons and ions with initial densities n^ and temperatures T^^ and T^^ as well as masses m and M, are considered. Dust particles of a density n^ are separated by some distance d between the centers of these particles. Relaxation phenomena start after the start of an interaction of electrons and ions with dust particles. Collisions between electrons and ions are not taken into account, because the relaxation time is less than the electronion collision time, owing to the choice of plasma parameters. Of course, there is a cube crystal cell around each dust particle. The periodic structure of the crystal gives on the boundaries of this cell some periodic boundary conditions providing an equality of all parameters in corresponding points of these boundaries. The 3D PIC method is used for the modeling of relaxation phenomena in a crystal cell around a dust particle. This cell is divided into cube simulation cells where electrons and ions are presented by large macroelectrons and macroions of a corresponding square cross-section. These macroparticles move according to a selfconsistent electric field and are collected by a dust particle if trajectories of their centers cross a surface of this dust particle. Macroparticles have to cross opposite cell boundaries simultaneously (enter - exit) as a result of the influence of neighboring crystal cells. Poisson's equation is solved using the Fourier transform method with periodic boundary conditions.
COMPUTER RESULTS Some typical obtained results are shown in Fig. 1 - 3 for 3D cube plasma crystals with d= 1, R^= 0.2, {T^o I ^w) ""40 where spatial coordinates and all line sizes are normalized by the initial Debye length >^ ^ = ikT^^ 147rn^e^] , and time t is multiplied by the initial ion plasma frequency co ^^ = {47rn^ I M )
.
266 Y.L Chutov et al /Statistical description and 3D computer modeling of relaxing dusty plasmas
N/N 1,0
0,5 h
0,0 FIGURE 1. A temporal evolution of the total number of electrons N^ and ions Nj (a) as well as a dust particle charge Q^ (b) in a plasma crystal. Here A^^ is the initial number of electrons and ions, Q^ = euo/rid' where rio and rid is the initial densities of ions and dust particles, respectively.
As can be seen in Fig. 1 {a), the total number of electrons A^^ and ions N^ in a crystal cell decrease monotonously due to their collection by the corresponding dust particle, i.e. a dusty plasma relaxation takes place. This decrease is stronger for electrons at first due to a more intensive electron flux that charges the dust particle negatively so that electron and ion fluxes are about equal. The temporal evolution of a dust particle charge Q^ is shovm in Fig. 1 {b). This charge increases at first and therefore the electron flux decreases, so that the ions flux a fQ\N exceeds it, and the charge Q^ decreases after having reached some maximum value. However the charge Q^ can not reach its equilibrium value due to the decrease of the total number of electrons A^^ and ions A^, in a crystal cell. The dusty plasma relaxation is accompanied by an essential change of electron and ion velocity distribution fiinctions that is confirmed by Fig. 2 where the mean v^^ and v.^ dependent fiinctions fex and fix are plotted for some times co^^t after a relaxation start. The initial density rio and the characteristic velocity v,o = (kTeo/M) are used here for the normalization. Corresponding Vy dependent fiinctions fey and fy are identical. As can be seen from Fig. 2, the electron velocity distribution fiinction f^ is impoverished by fast electrons during a relaxation contrary to the ion velocity distribution fiinction f^ impoverished by slow ions. These effects are caused by the energy dependences of the electron and ion fluxes to dust particles. Indeed, slow electrons can not reach dust particles charged negatively, and slow ions are collected by these dust particles more effectively than fast ions due to the strong inverse dependence of the effective cross-section on the ion energy. Besides these distribution fiinctions are averaged through whole non-equipotential crystal cell and therefore they can change due to a motion of electrons and ions in the self-consistent electric field depending also on the dust particle charge.
Y.I. Chutov et al. /Statistical description and 3D computer modeling of relaxing dusty plasmas 267
fix/CV^io)
fex/(VVio)
a
2 h
0,01 h --
0,00
-100
V
O
/v
0.0 0.5 1.5 5.0 15.0
0
100
-0,5
0,0
0,5
o of the mean electrony^;^ (^) and ion fix (b) distribution functions in a FIGURE 2. A temporal ex: evolution plasma crystal. Here v^ = J^'^eo / ^
is the characteristic velocity where Teo and M are the initial
electron temperature and the ion mass, respectively.
Of course, the temporal evolution of distribution functions causes a temporal change of mean electron (Eke) and ion (Ekt) kinetic energies shown in Fig. 3. An effective electron cooling as well as an effective ion heating takes place because these functions are impoverished by fast electrons and slow ions. Both these processes are especially effective at first when the change of distribution functions is very strong and a corresponding strong change of the dust particle charge Q^ takes place (Fig. 1).
^k/^keo
^ke^^keo
1,0 L
0,04
0,00 O
0,5 L
a
0,02
<^iot
10
15
0,0
<^iot
o
J
10
I
I
15
FIGURE 3. A temporal evolution of the mean electron Eke (a) and ion Eki (b) kinetic energy m a plasma crystal. Here Ekeo is the initial electron kinetic energy.
268 Y.I. Chutov et al /Statistical description and 3D computer modeling of relaxing dusty plasmas
The plasma relaxation is accompanied also by a change of mean potential energies of electrons (E^^) and ions (E^^) because of the temporal evolution of the dust particle electric charge Q^. In several cases, the ratio of the potential and kinetic energies E^^ I Ej^- of ions can be larger than unity even in the case of a large number of ions in a Debye cube. It means that ions are non-ideal components of relaxing dusty plasmas during this time although their initial numbers in the Debye cube can be much larger than unity and these ions have to be an ideal gas without dust particles. The indicated non-ideality of ions is caused by a change of dust particle charging accompanying any change of plasma parameters for example in some oscillations or waves. It means that the usual condition of the plasma non-ideality taking into account only the potential energy of interaction between electrons and ions cannot be used in plasmas with non-neutral dust particles with strong mutual interaction ACKNOWLEDGMENTS This work was partially supported by INTAS (Contract No 96-0617) and by a grant from the Ukrainian Committee of Science and Technology.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
9. 10.
11. 12.
13.
Bowley, R. and Sancher, M., Introductory statistical mechanics, Oxford: Clarendy press, 1996, ch. 9, pp. 179-200. Chutov, Yu. I., Kravchenko, A. Yu., and Schram, P., "Expansion of a bounded plasma with dust particles" J. Plasma Physics. 55 (part 1), 87-94 (1996). Chutov, Yu. I., Kravchenko, A. Yu., and Schram, P., "Evolution of an expanding plasma with dust particles" PhysicaBUS, 11-20(1996). Chutov, Yu. I., Kravchenko, O. Yu., Schram, P., and Smimov R. D., "Expansion of plasma layers with dust particles" Ukrainian Journal of Physics 42 (8), 996-1002 (1997) (in Ukrainian) Lee, H.C., Chen, D.Y., and Rosenstein, B., "Phase diagram of crystals of dusty plasma" Phys. Rev. E56 (4), 4596-4607 (1997). Tsytovich, V.N., "Plasma dust crystals, drops and clouds" Uspekhi fisicheskikh nauk 167(1), 57-99 (1997) (in Russian) Wang, X., and Bhattacharjee, A. "Hydrodynamic waves and correlation functions in dusty plasma" Phys. Plasmas 4 (11), 3759-3764 (1997) Morfill, G. E., Thomas, H. M., and Zuzic, M., "Plasma crystals - a review", in: Advances in Dusty Plasmas, (ed. P.K.Shukla, D.A.Mendis and T.Desai), Singapure, New Jersy, London, Hong Kong: World Scientific, 1997, pp. 99-141. Roman, A., Melzer, A., Peters, S. and Piel, A., "Determination of the dust screening length by laser-exited lattice waves" Phys. Rev. E56, 7138-7143 (1997). Chutov Yu., Kravchenko O., Smimov R., and Schram P.P.J.M., "Relaxation of dusty plasmas," in: Strongly Coupled Coulomb Systems (eds.G. J. Kahnan, K. Blagoev, J. M. Rommel) New York, London, Moscow: Plenum Press, 1998, pp. 209-212. Chutov Yu., Schram P., "Relaxation of background electrons and ions in plasma crystals" Contrib. Plasma Phys. 39 (1-2), 127-130 (1999). Mendis D.A. "Physics of dusty plasmas: an historical overview," in: Advances in Dusty Plasmas. (ed. P.K.Shukla, D.A.Mendis and T.Desai), Singapure, New Jersy, London, Hong Kong: World Scientific, 1997, pp. 3-19. Goree, J. "Charging of particles in a plasma" Plasma Sources Sci. Technol. 3, 400-406 (1994).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V Allrightsreserved
269
Regular Structures in Dusty Plasmas due to Gravitational Fields N.L.Tsintsadze^, J.T.Mendonca^ RK.Shukla^ L.Stenflo^ and J.Mahmoodi^ ^ Tbilisi State University, Chavchavadze 3, Georgia ^Centro de Fisica de Plasma, Instituto Superior Tecnico, 1096 Lisboa Codex, Portugal ""Institute fiirTheoretische Physik IV, Fdkultat fur Physik und Astronomie, Ruhr-Universitdt, Bochum, D-44780 Bochum, Germany "^Department of Plasma Physics, Umea University, S-90187 Umea, Sweden ^IPM, P.O.Box 19395-5531, Tehran, Iran We cosider dusty plasmas where the gravitational field plays an important role. We have considered the formation of nonlinear structures due to gravitational field and their stabihty in the dusty plasma. We have also generalized the problem of Jeans instability.
1. INTRODUCTION In recent years the subject of dusty plasmas has received enormous attention [1 — 5]. Such interest arise since dusty plasmas are rather ubiquitous in space, from interplanetary to intergalactic media. The presence of dust grains in plasma processing experiment causes major contamination of material surfaces; also of importance are formation of the dusty crystal structure.. Here v^e present a model making the assumption that the dust-plasma system is hot and charge neutral in the equilibrium state. All particles are under the influence of both gravitational and electrostatic forces. The organization of this paper v^ill be as follows: In section 2, we shall treat the nonhnear equilibrium state. In section 3, we shall derive a plasma wave dispersion relation and then discuss the various instabihties which arise. Finally, a brief summary of our results will be given in the last section.
2. BASIC EQUATIONS To investigate the equilibrium and stability of the dusty plasma, we shall make use of hydrodynamic equations, assuming the temperature of particles to be constant. The conservation equations of momentum and density of particles are
d
m,^ \dt ^
+ V«.V
\ J
Va = ZaCaE
T
riot-Vria
" TTlaV^
(l)
270
N.L Tsintsadze et al /Regular structures in dusty plasmas due to gravitational fields
^
+ V.(n.v„) = 0
(2)
where Z^ and T^ are the charge and temperature of the particle of species a\ Zo, is in unit of electron charge. E — — V(^ where 0 is electrostatic potential and ijj is the self-gravitational field. Poisson's equations for the self-gravitational and electrostatic potentials are: A'^ = ATrGrriDnD
(3)
A(f> = A7re{ZDnD + ne - ZiUi).
(4)
In this section we treat the nonUnear structure of the dusty plasma in equilibrium state, when the system as a whole does not undergo rotation. Since the characteristic gravitational scale length is much larger than the Debye length of dust grains LQ^ LD^ we can simpUfy Eq.(4) and we obtain a nonlinear equation for the self-gravitational potential ip : AVb + A2exp(KG) = 0
(5)
where
^ =_ _ ^ ! ^
and X' = j ^ = t^i^^f;^
(6)
Here we have new length scale which involves both gravitational as well as electrostatic parameters, LcE = La^
^^^^
.
(7)
Let our plasma have shape of a disk and assume that the function VG depends on the z coordinate only. Then Eq.(5) has an exact solution, giving the density of dust grains as follows: ^D{Z)
= „
rioD
. 2/ A ^^
cosh'(^z)
(8)
For the cylindrical plasma we have:
where Vo is the effective radius of the plasma given by: 2 2 ZjjTi + ZiTi) r^ = - ^ ^ ' ^ ^ ""
TT
ZiGm^rioD
(10)
N.L. Tsintsadze et al/Regular structures in dusty plasmas due to gravitational
fields
271
3. NONSTATIONARY WAVES AND THEIR STABILITY Here we analyze the stability of an isothermal plasma in which matter rotates in circular orbits about a center, with constant angular velocity fi. Consider the propagation of small disturbances in a homogeneous plasma; to this end, we linearize the set of Eqs.(l-4), and we seek solutions which have the following Fourier decomposition: (11)
F= F{r)exp {iW + ikz - iui)
where u; is a mode frequency, and / is an integer. In that case the set of Eqs.(l-4) will yield the following set of equations: i (a; - m) 5vr + 2n5v0 = V^^v or
- ^ (^^ - S^jj] or \ rriD J
{iv - in) 6v0 + 2inSvr = vdiy
- - (^ ^
r
- Si/A
r \ rriD
(a; - in) Sv, = kV^u - k (^^
(12) (13)
J (14)
- S^A
/ i d (to — in) u — kSvz = -Svo 7^ (rSvr) r ror - ^ r — 5 V ^ - ffc2+ — J (^^ = AT^GmDUoD^
(15) (16)
where u — ^^^ . Due to the complexity of the Eqs.( 12-17), we treat several special simpUfied cases relevant to rotating dusty plasma problem. There are axially symmetrical waves propagating only along the radial direction r; i.e. / = fc = 0. In this case the equation for 6vr can be written in the following form: (PSVr
1 dSVr
/ r2
dr^
r dr
\
1 \ r
^
/ X
r^
where ^
=
2
Ct
'
^^^
C« =
mD\
\^D
+
rr
I 72
UoeTi +
^
ZfUoiT,
The solution of Eq.(18) that vanishes for r = 0 is therefore: Svr = const. Ji {6r), where Ji is the Bessel function of order 1 .
(20)
272
N.L. Tsintsadze et al /Regular structures in dusty plasmas due to gravitational fields
This motion comprises of regions between coaxial cylinders (see Fig.l) with radii between Svp and dr^^i such that: 1,2,3,
(21)
u;' = 2n^ + Ky,
(22)
Svp = Xp,
p=
where the pth root of Ji. Prom Eq.(33) it follows that:
where Kn= ^ ^ Here we consider the spiral waves in a thin flat plasma disk. In this treatment, we assumed that the equatorial radius R to be much larger than its characteristic thickness sR ; £
[(a; - mf - 4n^ (a; - iny - ev^ + f (u; - mf
2fi2
k^'V?
2fi2
wpD
1 + '^
1 + k'^T - ^ + W^1 UJ, pD
+
(23)
/47rZ^eVD\2
is the Langmuir frequency of where rj, = 4.e(noe7HZ^o.7^) ^^^ ^^^ =" [ rriD the dust species. Let us examine the dispersion (22) for case / = 0 . In this case we obtain: 2
where i?^ =
2
1,2 2
1 + k^rl + '^ ^^
- 2fi2
(24)
\ ^ 4 ^Jn^IT^^
Eq.(23) is the modification in the Jeans equation. Let us examine this equation for two cases: k'^r%
^' = cl{k' - k'j)
(25)
where kj is the Jeans wave number. In the second case, i.e. when k'^r^ > 1 , Eq.(23) will reduce to: u;2 = u;l^ (l + k'Rl)
- 2^' c. cv^j, (l +
k'Rl)
(26)
Next we shall examine the propagation of the spiral electrostatic-gravitational waves and their stability. For this, we assume that I ^ 0, and k^r^ » 1 in Eq.(22). Then this equation reduces to:
r_fcV|_ (w - inf = Wpo [l + k^rl + k'Ri
2n'
(27)
N.L. Tsintsadze et ai/Regular structures in dusty plasmas due to gravitational fields 273
The Eq.(26) is similar to the Eq.(23) with the following these differences: when right hand side is negative, Eq.(23) yields Jeans type instabiUty, while the Eq.(26) yields an oscillation instabiUty.
4. SUMMARY In this work we treated the nonUnear equilibrium structures of the dusty plasma. We found that for formation of some structures a self-gravity is very important. Then we examined the time dependent perturbations and new types of the radial waves were found. We then the studied the propagation of the spiral charge density waves and have showed that there exists the Jeans oscillatory instabiUty. We take opportunity to thank F. Kazeminezhade for valuable discussions.
REFERENCES 1. Mendis, D. A. et al., Electrodynamic processes in the ring system of Saturn, edited by Gehrel, T. and Matthews, M. S., P.546, University of Arizona Press. Tucson (1984) 2. Tsytovich V.N., Dust plasma crystals, drops, and clouds, Phys. Uspekhi 40(1), 53 (1997) 3. Hunter C. and Toomre, A., Astrophys. J. 155, 747 (1969) 4. Bolmforth, N. J.,Howard, L. N. and Spiegel, Mon. Not. R. Astron. Soc. 260, 253(paper I), (1993); SIAM, J. Appl. Math. 55, 298 (1995)
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
275
A Rocket-borne Detector for Charged Atmospheric Aerosols S. Robertson, M. Horanyi and B. Smiley Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309-0392
B. Walch Department of Physics University of Northern Colorado, Greeley, CO 80639
Abstract. A rocket-borne charge-collecting probe has been developed to detect charged atmospheric aerosols. A magnetic field is used to prevent detection of electrons and low-mass ions. In a test flight above White Sands, New Mexico, a narrow layer of positive charge was detected at 86.5 km altitude, 495 m in vertical extent, and with a density of approximately 2000 charges/cm^.
INTRODUCTION Evidence for aerosols in the polar mesosphere includes noctilucent clouds (NLC) [1] and strong radar backscatter called polar mesospheric summer echoes (PMSE) [2,3,4]. Electrostatic probes collecting electrons find narrow layers ('bite-outs') v^ith reduced electron densities [4] which may arise from the attachment of electrons to macroscopic particles. NLC, PMSE and bite-outs occur at altitudes of 80-90 km and have been observed to occur simultaneously. These phenomena are thought to be caused by the presence of ice crystals with diameters below ^70 nm [5,6]. The increased frequency of NLC sightings may be an indication of anthropogenic changes in atmospheric composition [7,8]. Several experimental campaigns have been aimed at understanding the polar summer mesopause [9,10]. Observational techniques include ground based visual observers, radar backscatter, lidar, satellite observations and rocket-borne probes. Rocket flights of mass spectrographs have found water cluster ions with masses up to about 500 amu [11]. Cluster ions of higher mass may be present but would have a mass beyond the range of the spectrograph. The identification of mass peaks corresponding to H"^[H20]n is taken as confirmation that NLC particles are indeed water clusters. Electric field probes [12] find signals at NLC layers which appear to be due to aerosol impacts and aerosol detectors have been constructed in which the created charge is used to
276
S. Robertson et al /A rocket-borne detector for charged atmospheric aerosols
detect impacts [13]. Charged aerosols have also been detected by charge-collecting probes designed to separately collect the current of charged aerosols [14]. The measured aerosol charge density is several 10^ cm~^ and both positive layers and negative layers have been detected. The distribution in masses remains an open question and the motivation of our research is the development of a mass-selective instrument.
TEST FLIGHT OF THE P R O B E The charged aerosol probe is a modification of an instrument used to detect charged dust particles which have fallen through a laboratory plasma [15,16,17]. In this instrument, the charged dust detector is a Faraday cup connected to a sensitive amplifier and placed in a recessed volume below the plasma. The field of a small permanent magnet prevents detection of electrons and ions from the ambient plasma. This field defiects these lighter particles more strongly and thus acts as a mass-selective filter passing only the heavier dust particles. In the rocket version of the instrument, the Faraday cup is replaced by a rectangular patch on the skin of the payload that is electrically isolated and connected to an amplifier. The field of an adjacent permanent magnet prevents collection of ionospheric electrons and low-mass ions. The rocket velocity, collector area and collected current are used to calculate the charge density. Figure la is a schematic of the charged aerosol detector. The current collecting plate is a rectangle with dimensions of 2 cm x 3 cm. The collecting plate is made from graphite because of its low secondary electron yield [18] and low photoelectron yield [19]. The plate is mounted at the surface of an aluminum enclosure 6 x 6 x 3.5 cm and the mass of the finished detector is approximately 250 grams. Measurements of the magnetic field above the surface indicate that the best-fit dipolar field has an origin 28 mm below the collecting surface and a strength of 2730 Gauss-cm^. In Figure la the detector is shown mounted at the tip of a payload for measurement on ascent and in Figure lb the detector is shown mounted on the side of a payload for measurement on descent. An opportunity for a test fiight was made available on a rocket used for solar imaging in the extreme ultraviolet. The fiight was a NASA Terrier - Black Brant sounding rocket launched from White Sands on November 2, 1998. The ultraviolet images required an apogee of 350 km and a velocity of 2 km/s at the mesopause. The ascent was spin stabilized and the descent had attitude control for pointing at the sun. The detector was mounted so that it pointed downward on the descent as in Figure lb. The long axis of the payload was pointed 42 degrees from horizontal thus aerosols were incident upon the surface of the detector at an angle of 42 degrees from the normal with a velocity of 2.2 km/s. This detector for the test fiight is designed with a low magnetic field in order to maximize the probability of a signal so that proper operation of the detector and amplifier could be verified. The magnetic field is oriented so that positive
S. Robertson et al./A rocket-borne detector for charged atmospheric aerosols
Direction of travel
4V Y
J
,
111
Light particle trajectory
^_._ Heavy ^~"^ particle •-^trajectory Magnetic field V V
Direction of travel Heavy particle trajectoty
A)
B)
F I G U R E 1. Schematic diagram of the charged aerosol detector. At left the detector is shown mounted at the payload tip to measure on ascent and at right the detector is shown mounted on the side of the payload to measure on descent.
ions and aerosols are deflected toward the collector. The collection efficiency as a function of mass is determined by computer simulations. Particles uniformly spaced on a square grid upstream of the collector are followed using the Lorentz equations of motion and their footprints are located on the surface of the nosecone. Those particles falling within the collecting patch are those that are collected. The collection efficiency, Figure 2, is defined as the fraction of particles in the volume upstream of the collector which strike the surface. This fraction approaches 100% at high masses (>100 amu) because the magnetic force is insufficient to cause deflection. At very low mass (< 2 amu) the ions are deflected sufficiently to prevent detection. At intermediate masses, the collection efficiency exceeds 100% because ions are deflected into the collector that would have fallen outside the collector in the absence of the Lorentz force. The mass distribution of aerosols would be found by flying several such detectors with differing mass thresholds. The flow of air around the rocket carries with it low-mass aerosols but heavier aerosols are not entrained in the flow and may strike the surface. This effect has been studied with a gas kinetic computer code [19] but not yet for the conditions of our test flight. Preliminary work indicates that the temperature behind the shock is suffcient to cause evaporation of ice clusters below about 10 nm in radius. The collected current as a function of altitude is plotted for our test flight in Figure 3a and b. A narrow layer appears in the data at an altitude of 86.5 km, approximately the location of the mesopause. From the rocket velocity we flnd that
278
S. Robertson et al. /A rocket-borne detector for charged atmospheric aerosols
100
10
10000
1000
proton masses FIGURE 2. Collection efficiency as a function of mass for the conditions of the test ffight. Efficiency is defined as the number of particles collected divided by the number in the volume directly upstream. Positively charged particles are deflected toward the collecting surface thus the efficiency can exceed 100%.
90 85 4 km 86.5 4
km 80 4 75 4 I ' 70 -0.4 -0.2
' ' I
0
0.2
0.4
detector signal (nA)
0.6
0
0.1
0.2
0.3
0.4
0.5
detector signal (nA)
FIGURE 3. Current collected on descent as a function of altitude. The plot at right shows the detailed structure of the layer. Data points are spaced 14 m apart.
S. Robertson et al. /A rocket-borne detector for charged atmospheric aerosols
279
the charge density is 2,150 cm~^ if we assume that the flow field about the rocket does not alter the aerosol density and that the collection eflSciency is 100%. The duration of the signal, 0.23 seconds, indicates a layer thickness of 495 m. Formation of ice clusters is extremely unlikely at the latitude of New Mexico thus this layer may either be ions or meteoric dust. The standard mechanisms for ion production in the ionosphere have scale heights of many tens of kilometers, thus if the detected layer is ions some mechanism must be invoked for vertical convergence.
FUTURE WORK Work on this topic is continuing in several directions. Computations are being made of the flow fleld around the rocket for the conditions of our test flight. The goals are to find the effect of shock heating on the evaporation rate of ice clusters, and the effect of both flow and evaporation upon the detection efficiency. In the laboratory, we have constructed a facility for creating beams of ice clusters for testing and calibration of rocket instruments and for fundamental studies of clusters such as the determination of rates of electron attachment and photoionization. This facility consists of a vacuum chamber having at one end a supersonic nozzle that creates a spray of argon and water vapor that forms ice clusters during expansion into vacuum. The velocity of the clusters (~550 m/s) is comparable to the velocity of sounding rockets. A magnetic mass spectrograph has been constructed for determining the distribution in masses of the clusters. Initial results indicate that the creation of clusters with masses of four to eight water molecules.
ACKNOWLEDGMENTS The authors acknowledge support from the National Aeronautics and Space Administration and the Department of Energy. We thank Don Hassler, Southwest Research Institute, and Tom Woods, Laboratory for Atmospheric and Space Physics, for the test flight.
280
S. Robertson et ah /A rocket-borne detector for charged atmospheric aerosols
REFERENCES 1. Thomas, G. E., Rev. Geophys. 29, 553, (1991). 2. Hoppe, U.-R, T. A. Blix, E. V. Thrane, F.-J. Lubken, J. Y. N. Cho and W. E. Swartz, Adv. Space Res. 14, 139 (1994). 3. Cho, J.Y.N, and J. Rottger, J. Geophys. Res. 102, 2001 (1997). 4. Ulwick, J. C., K. D. Baker, M. C. Kelley, B. B. Balsley and W. L. Ecklund, J. Geophys. Res. 93, 6989 (1988). 5. Witt, G., Space Res. 9, 157 (1969). 6. Thomas, G. E. and C. R McKay, Planet. Space Sci. 33, 1209 (1985). 7. Thomas, G. E., J. Atm. Terr. Phys. 58, 1629 (1996). 8. Thomas, G. E., Adv. Space Research 18, 149 (1996) 9. Goldberg, R. A., E. Kopp and G. Witt, Adv. Space Res. 14, 113 (1994). 10. Liibken, F.-J., K.-H. Pricke and M. Langer, J. Geophys. Res. 101, 9489 (1996) 11. Bjorn, L. G., E. Kopp, U. Herrmann, P. Eberhardt, P. H. G. Dickinson, D. J. Mackinnon, F. Arnold, G. Witt, A. Lundin and D. B. Jenkins, J. Geophys. Res. 90, 7985 (1985). 12. Zadorozhny, A. M. A. A. Vostrikov, G. Witt, 0 . A. Bragin, D. Yu. Dubov, V. G. Kazakov, V. N. Kikhtenko and A. A. Trutin, Geophys. Res. Lett. 24, 841 (1997). 13. WachU, U., J. Stegman, G. Witt, J.Y.N. Cho, C. A. Miller, M. C. Kelley and W. E. Swartz, Geophys. Res. Lett. 20, 2845 (1993). 14. Havnes, O., J., Tr0im T. Blix, W. Mortensen, L. L Naeshim, E. Thrane and T. T0nneson, J. Geophys. Res. 101, 10839 (1996). 15. Walch, B., M. Horanyi and S. Robertson, Phys. Rev. Lett. 75, 838 (1995). 16. Robertson, S., Phys. Plasmas 2, 2200 (1995). 17. Horanyi, M., J. Gumbel, G. Witt and S. Robertson, "Simulation of rocket-borne particle measurements in the mesosphere," to appear in Geophysical Research Letters, 1999. 18. Gibbons, D. J., in Handbook of Vacuum Physics, Vol. 2., A. H. Beck, editor (Pergamon Press, Oxford, 1966), p. 301. 19. Feuerbacher, B. and B. Fitton, J. Appl. Phys. 43, 1563 (1972).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
281
Paleo-heliosphere: Effects of the Interstellar Dusty Wind Based on a Laboratory Simulation Shigeyuki Minami* and Shigeo Miono ^ *Dept. Electrical Engineering, Osaka City University, ^ Dept. Physics, Osaka City University Sumiyoshi Osaka 558-8585 JAPAN Abstract. The heliosphere is produced as an interaction between the super sonic (solar wind) plasma flow and interstellar supersonic magnetized plasma flow (local interstellar medium:LISM) in relative to the solar system. Our solar system is also moving with respect to the galactic spiral structure. The density changes in time having the period of 0.6 billion years by a differential rotation. The highest 7
-3..
neutral density of the LISM is the order of 10 cm Thus it is possible to see the temporal change of LISM. A few weight percent of dust component exists with the neutral. In this paper, the effects of such high density neutral/dust in the LISM are considered to form the heliosphere based on a laboratory
simulation
considering
the
ancient
electromagnetic
environment,
called
the
paleo-heliosphere. The neutrals of the simulated LISM can be controlled by changing the applied voltage of plasma gun. The laboratory interaction between the spherically expanding solarwind plasma flow and another uniform supersonic and super/sub Alfvenic plasma flow have qualitatively revealed the important role of the LISM magnetic field and the neutral component of the LISM on the structure of the heliosphere. The experimental result of the partially ionized LISM plasma flow suggested a lack of the pressure balance at the nose of the heliosphere and the diffusion of the magnetic field. The structure of the heliopause might be changed due to the long term variation of rich neutral and dusts of LISM because the solar system is moving through the various conditions of our galaxy
INTRODUCTION The solar wind spreads radially outwards at supersonic velocities throughout the solar system. The wind collides with the ambient interstellar medium (ISM) through which the solar system is moving with respect the galactic frame. The region is called the heliosphere. The outer boundary called the heliopause is still a postulated separatorix. The heliocentric distance is predicted to be 100 to 500 AU.
282 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation
The local ISM parameters are also not yet known directly, especially the magnetic field intensity is uncertain to be 0.01 nT to 1 nT (1). The density of neutral and plasma are known to be 0.1 and 0.01 [cm-3] respectively. One of the most important thing about the ISM is its variability in space. The recent millimeter radio waves from the interstellar CO gas has revealed a clear evidence of irregularity forming spiral arms as shown in Figure 1(1).
"P ••••to
FIGURE 1 ••«•
»?••«
The distribution of the CO cloud in
the Galactic plane (after Dame et al.)
It is important to examine the role of neutral gas to the structure of the heliopause. It is known that our solar system is rotating as a period of 0.2 billion years, while the gas cloud on the solar system diameter takes 0.3 billion years for one rotation. So the solar system encounter the same structure of the cloud having the period of 0.6 billion years. The enhanced neutral density is regarded to be about 1 million times higher than the present local ISM. Although the electromagnetic environment of the heliosphere can not be recorded historically on the earth, it could be estimated, because the interstellar parameters especially the neutral density is recorded in space. So we named the structure as the paleo-hehosphere.
LABORATORY EXPERIMENTS Our experiment is performed using a plasma emitter powered by an intense capacitance bank power supply, which can produce luminous supersonic Ba plasma flow spherically expanding from a certain point; this flow simulates the solar wind plasma flow. The density at 5 cm from the center of the emitter inside of the terminal shock is about lO^'* cm^ which is measured without the LISM. The temperature is about 2 eV, and the obtained nose distance is about 10 cm. The velocity of the simulated solar wind (50 km/s) is measured by the time evolution of the spherically expanding plasma images taken by a time-resolved camera. The LISM (argon plasma; supersonic, but sub- or super-Alfv^enic; density of 10^^ cm'^ temperature of 2
S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 283
BxCOIL
BxCOIL B2COIL
FUGURE 2 Experimental et up. The LISM flow comes from the left. LISM magnetic field is controlled by the two pairs of coils.
SUN PLASMA L ^
eV) is produced by a plasma LISM gun. The LISM magnetic field FLOW of a maximum 450 G can be applied in any direction to the BzCOIL ^ LISM plasma flow, permitting the experimenter to change the AlfVen Mach number. The experimental design for similar laboratory simulations of terrestrial shock formations has already been described by Minami (2). The experimental set up is illustrated in Figure 2. The time evolution of the interaction is recorded by a time-resolved camera. The variable magnetic field of 0 - 450 G can be applied to magnetize and to change the Alfvenic Mach number, MA, of the LISM. The LISM plasma flow can be magnetized because the magnetic field is applied before the particle ionization at the plasma gun. The LISM plasma flow is regarded to be a frozen-in condition. According to the measurements, the LISM can flow toward the simulated sun even when MA < 1 according to measurements. GUN
EXPERIMENT 1 (With fully ionized LISM flow) Our experiment is performed using a plasma emitter powered by an intense capacitance bank power supply, which can produce luminous supersonic Ba plasma flow spherically expanding (FIGURE 3) from a certain point; this flow simulates the solar wind flow. FIGURE 5 shows the structure of the simulated heliopause by illumination in images taken by a 1 u s time resolved camera at r = 50 /x s; (a) with the magnetic field parallel to the LISM flow and MA =0.3, (b) With the LISM magnetic field tilted (about 30 degrees) toward the LISM plasma flow and MA = 0.1, and (c) without the magnetic field, then MA = 10. The fast-mode Mach number, MA = vo / V" ( (VA^VS^) , is used for case (c) because this Mach nimiber controls the shock structures, where the quantities vo, VA, VS are the solar wind, Alfv^en, and the sonic velocities, respectively. The LISM comes from the left. The emitted plasma that simulates the solar wind interacts with the magnetized LISM plasma flow (3).
284 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation
The illumination created by the Ba plasma line simulating the solar wind plasma so that the boundary of the outflowing solar wind plasma, the heliopause, can be seen optically. The heliopause structure obtained by this illumination shows a clear axiasymmetric structure to the LISM flow when MA < 1 and the LISM magnetic field is tilted. The LISM plasma density is almost constant for the different LISM magnetic fields.
FIGURE 3 Simulated solar wind plasma gun
FIGURE 4 photographs Expanding solar wind for 5 jU s, 15/is, 2 5 / i s after the ignition. FIGURES (a),(b),(c) Simulated heliosphere without neutral LISM.
EXPERIMENT 2 (With partially ionized LISM flow) Figure 4 shows the effects of the neutral component of the LISM showing a collapsing heliosphere. A neutral gas plume of several cm"^ at 1 bar is injected into the plasma gun. When the applied gun voltage is low, the simulated LISM flow is
S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 285
partially ionized. The experiment is performed by applying insufficient voltage to the plasma gun. The result shows significant difference to the result of fully ionized LISM flow. At the beginning of the ionization of the neutral LISM flow, the nose of the contact surface, heliopause, is extended due to the ionization of the neutral flow by the solar wind. The initial ionization along the LISM magnetic field at the nose can be explained by the intrusion of the LISM magnetic field into the heliosphere.
FIGURE
5
Laboratory
heliosphere
structure for different ionization rate in %. LISM flow comes from the left.
0.5 H
10%
50%
100% lonizatjoi^ Rate
DISCUSSION The preliminary laboratory interaction between the spherically expanding solar wind plasma flow and another uniform supersonic and super/sub AlfVenic plasma
286 S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation
flow have qualitatively revealed the important role of the LISM magnetic field and the neutral component of the LISM on the structure of the heliosphere. It is also known that the magnetospheric protector, called the Solar magnetopause is also collapsed when the wind —> neutral is rich as shown in FIGURE 6. The center is the the simulated earth. The soalr wind comes from the left, (a) without (a) Without fast neutral. neutral, (b) with netural.
Solar wind
FIGURE 6. The simulated magnetopshere (a) without neutral flow (b) with neutral flow allowing the intrusion (b) With fast neutral.
of the solar wind partilces into the magnetopause.
FIGURE 7
Illustrated paleo-magnetosphere when the rich interstellar neutral/dust are
encoutered.
The rich neutral brings dusts together (a few perent mass weigh ratio), so the rich
S. Minami, S. Miono/Effects of the interstellar dusty wind based on a laboratory simulation 287
dust particles might precipitate to the solar system and also to the earth. At that time, every day meteor shower could be seen in the sky. FIGURE 7 shows a possible electromagnetic environment of the ancient earth showing a meteor shower. The study of the paleo-heliosphere and the paleo-magnetosphere has just started. The research was supported by the Grant-in-Aid for Scientific Research (03238214 and 04222215) of the Ministry of Education and Culture in Japan.
REFERENCES 1. Dame et al., Astrophys. J., 305, 892 (1986). 2. Minami, S., Geophys. Res. Let, 21, 81-84, (1994). 3. Minami S., et al., Geophys. Res. Lett., 13, 884 (1986).
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
289
Current Loop Coalescence in Dusty Plasmas J. I. Sakai and N. F. Cramer* Laboratory for Plasma Astrophysics, Faculty of Engineering Toyama University, 3190, Gofuku, Toyama 930-8555, Japan * Theoretical Physics Department and Research Centre for Theoretical Astrophysics, School of Physics, The University of Sydney, N.S.W. 2006, Australia
Abstract. The weakly ionized plasmas that occur in protostellar disks and in the cores of molecular clouds generally have a dust component. We consider the effects of the presence of charged dust on magnetic reconnection process. We investigate the dynamics of two current loops coalescence, for both cases; partial and complete magnetic reconnection. It is shown that the field-free gas produced during the two loops coalescence with complete magnetic reconnection would be able to initiate star formation.
I
INTRODUCTION
The w^eakly ionized plasmas that occur in protostellar disks and in the cores of molecular clouds generally have a dust component. The ionization fraction of molecular clouds is typically only ^ 10"'', and the dust may contribute ^ 1% of the mass of the cloud. It has been shown that the ion-neutral drift leading to ambipolar diffusion can lead to the steepening of the magnetic field profile and to the formation of singularities in the current density of hydromagnetic fluctuations [1,2] and Alfven waves [3,4]. Recently Bulanov and Sakai (1998) [5] investigated in datails magnetic reconnection process in weakly ionized plasmas. In the present paper we consider the effects of the presence of charged dust on magnetic reconnection process. We investigate the dynamics of two current loops coalescence, for both cases; partial and complete magnetic reconnection. It is shown that the field-free gas produced during the two loops coalescence with complete magnetic reconnection would be able to initiate star formation.
290
J.I. SakaU N.E Cramer/Current loop coalescence in dusty plasmas
II
BASIC EQUATIONS
We consider molecular clouds consisting of neutral atomic and molecular species, the ionized atomic and molecular species, the electrons, and negatively charged dust grains. We start a 4-fluid model of the plasma, which employs the fluid momentum equations for plasma ions (singly charged), neutral molecules, charged dust grains and electrons: A f - ^ + Vz • V v J = -Vpi + UiC (E + Vi X B) -Pi^ini^i - Vn) - pmdi^i - Vd),
(1)
P„(^+v„.Vv„)=-Vp„-p,.„.(v„-v.) -Pn^nd(Vn-Vrf), 'dVd
Pdi-gf+^d'
(2)
V v J = ZdUde (E + Vd X B) -Pd^dni^d - Vn) - Pd^di{^d - VO,
0 = - n e e ( E + Ve X B ) - pe^eni^e
" V ^ ) " pe^ed{^e
" ^d) " Pe^ei{^e
(3) " V^),
(4)
where E is the wave electric field, rris is the species mass, Ps is the species mass density and V5 is the species velocity, pi and Pn are the ion thermal and neutral thermal pressures, and u^t is the coUision frequency of a particle of species s with the particles of species t. We have neglected electron inertia in (4), momentum transfer to ions from electrons in (1) and to dust grains from electrons in (3), and the dust thermal pressure gradient in (3). The parameter S = Ue/rii < 1 measures the charge imbalance of the electrons and ions in the plasma, with the remainder of the negative charge residing on the dust particles, so that the total system is charge neutral. -eue + erii - ZdCUd = 0.
(5)
A typical value of 5 for molecular clouds is (^ = 1 — 10~^. The charge on each dust grain is assumed constant, and for simplicity we also assume that 5 is constant, even though Tin and thus n^ are variable. The neutral mass density obeys the continuity equation
^
+ V • (p„v„) = 0.
(6)
To complete the system of equations. Maxwell's equations ignoring the displacement current are used, with the conduction current density given by j = e{niVi - UeVe - ridZdVd).
(7)
291
11. Sakai, N.E Cramer/Current loop coalescence in dusty plasmas where equilibrium charge neutrahty is expressed by (5). Equations (4) and (7) lead to the following generahzed Ohm's law: X. J X B X B = ^^ e
e
e
We now use the strong coupling approximation (Suzuki and Sakai 1996) [3], whereby the ion inertia term (the left hand side) and the ion thermal pressure term are neglected in (1), leaving a balance between the remaining terms. At this point it is useful to normalize the magnetic field by a reference field 5o, and define the Alfven speed based on the field BQ and the ion density: VA = Bo/{iioPi)' Eq. (1) may then be written, using (8) and Faraday's law neglecting the displacement current, - ^ ( V X B ) X B = Qrni^i
- Vrf) X B + Uin{Vi - V^) + Uid{Vi - V ^ ) ,
(9)
where fi^ = Cli{l — S)/6 and Qi is the ion cyclotron frequency, Qi = Boe/rrii. The presence of dust introduces the first and third terms on the rhs of (9). For strongly coupled dust grains and neutral gas, such that v^ = v^, the ion equation of motion gives the following expression for the ion-neutral relative drift velocity: VD - V, - V, = F^
[(V X B) X B - (i?/p)((V x B) x B) x B],
(10)
where Ui = Uin + Uid, R = ^mpn/^ip^. B^ = B^ + B^ + B^^ and F = l/{l + R^B^/p'^), For a small dust number density, Uid « i^m, so we approximate ui = Vi^. Summing the momentum conservation equations (1,2,3) of the ions, neutrals and dust, the equation of motion for the neutral velocity is obtained. We normalize the density by PQ, defining p = {pn + Pd)/po, and the pressure by po- The velocity is normalized by the Alfven velocity based on po, VA — Bo/(/ioPo)^^^7 and space and time are normalized by LQ and TA = LQ/VA- The result is p [ ^ + v - V v j =-p\/p+{V
xB) x B .
(11)
The magnetic induction equation gives, using (8) with the coUisional electron momentum transfer terms neglected, and neglecting the Hall term, dB ^ , — = V X (v X B) + ADS/
X - (((V X B) X B) X B + {R/p)B\V
where J5 = |B| and Ajj = Pn/j^i^APi-
x B) x B )
(12)
292
J.I. Sakai, N.F. Cramer/Current loop coalescence in dusty plasmas
(a)
(b)
0 -1^^^^^^P^^^H*^®*''
50 H
50 -
>^ 100 H
00 -
150 H
50 -
^-^^^H ' ^'^"^^H
-
\
^
X
50
100 Y
-0.5
0.0 Bz
150
^—
2.5 5.0 7.5 10.0 12.5 D e n s i t y and V x - V y v e c t o r p l o t s
-1.5
-1.0
0.5
1.0
1.5
(d)
(C) 0 -
50 -
!><1
100
100 -
150
150
^
-
1
'
1 50
'
1 100 Y
'
1 150
'
^ ^ ^ ^ ^ -1 ' \ ' 2.5 5.0 7.5 10.0 12.5 15.0 -0 . 02 -0,01 0.00 0.01 0 . 02 0.03 Temperature Jz F I G U R E 1. (a) Density distribution overlapped with velocity field [vx -Vy), (b) magnetic filed Bz, (c) current J^, and (d) temperature distribution at t = S.ITA for complete reconnection.
11. Sakai, N.E Cramer/Current
loop coalescence
in dusty
plasmas
293
(a)
N ^ ^ ^ \ >\/ V y y ^ ^ ^ ^ ^ '^ ^ ^ ^ \l \f S W N / y w l ^ ]l\/^ i^ 1^
50
N
100 H
X ^ >j \ \/ \f sL J/\L
NZ- i^
^>i'ii^i^>\f^i4'
V \l^ ]/' ^ ^
-»^ -?
-:»
50 -A
14 ^ ^ ^ ^
X
«
y ;^ f ^ ^ '^k ^
^
^ -^^^^^^ / 7\ f^ M\f
150 H
/ /
100 H
7? ; f ;?1 ^
15 0 H
/ . >l / l ' ^ ^ f f. f yjK } ^
/v- V ^ 1^ 1^ A f f^ f f^ f n / /\ /\ /\ f ^ ^ ^ 1^ \ ^ Kj^ ^ ^ y ^ ^ f f f f f^ -t—I f ^ ^ |A- t-f^
50
100 Y
150
-t-
2.5 5.0 7.5 10.0 12.5 •1.0 Density and Vx-Vy vector plots
T
-0.5 0.0 0.5 Bz and Density Contour
50 -\
X
X
100
150 H
100 H
150
0.01
0.02
0.03
0
T
5 10 Jz Temperature F I G U R E 2. (a) Density distribution overlapped with velocity field {v^ — Vy)^ (b) magnetic filed Bz with density contour plots, (c) current J^, and (d) temperature distribution at t — 8.1r^ for complete reconnection, with self-gravity eflPect. 0.00
1.0
294
J.I. Sakai, N.E Cramer/Current loop coalescence in dusty plasmas
III
SIMULATION RESULTS
We use a 3-D simulation code of the above equations in which the numerical scheme is the modified 2-step Lax-Wendroff method. The system sizes are 0 < X = y < GirL, and 0 < 2: < O.birL in the x, y and z directions, respectively. The mesh points are A^^; = 200, Ny = 200 and A^^ == 10 in the x, y and z directions, respectively. We used free boundary conditions (first derivatives of all physical quantities are continuous) for the x and y directions, while periodic boundary condition is used in the z direction. We take an initial each current loop, which is placed along the z direction to satisfy a force-free condition. Other parameters used here are P = 0.01 and AD = 0.5. Fig. 1 shows four snapshots ait = S.ITA- (a) density distribution overlapped with velocity field {v^ — Vy), (b) magnetic filed Bz, (c) current J^, and (d) temperature distribution. As seen in Figs. 1(a) and (b), during two loops coalescence the density increases about 13 times of the initial state. This is contrast to the case of the partial reconnection, where only poloidal magnetic field produced by two loop currents can dissipate. As seen in Fig. 1(a), the density accumulation appears near the region where the magnetic reconnetion occurs and the magnetic field is almost free. Next we investigate the effect of self-gravity. Fig. 2 shows the simulation results for complete magnetic reconnection case with the self-gravity at t = 8.1r^. As seen in Fig.2 (a) there occurs strong density accumulation with about 13 times larger than the initial value near the region where the magnetic fields are almost free, due to complete magnetic reconnection. The field-free gas produced during the two loops coalescence with complete magnetic reconnection could be able to initiate star formation.
IV
CONCLUSION
We investigated the effects of the presence of charged dust on magnetic reconnection process for both complete and partial magnetic reconnection during two current loops coalescence. We showed that the field-free gas produced during the two loops coalescence with complete magnetic reconnection could be able to initiate star formation. We thank the Cosel and Densoku company for the support.
REFERENCES 1. Brandenburg, A. and Zweibel, E., Ap. J. 427, L91 (1994). Brandenburg, A. and Zweibel, E., Ap. J. 448, 734 (1995). 2. Mac Low, M. M. et al., Ap. J. 442, 726 (1995). 3. Suzuki, M. and Sakai, J.I., Ap. J. 465, 393 (1996), 4. Suzuki, M. and Sakai, J.I., Ap. J. 487, 921 (1997). 5. Bulanov, S. V. and Sakai, J.I., Ap. J. Suppl. 117, 599 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
295
Jeans-Buneman instability in non-ideal dusty plasmas S.R. Pillay\ R. Bharuthraml'^ F. Verheest^ N.N. Rao^ and M.A. Hellberg^ ^Department of Physics, University of Durban-Westville, S.A., "^M.L. Sultan Technikon, Durban, S.A., ^Sterrenkundig Observatorium, Universiteit Gent, Belgium, "^Physical Research Laboratory, Ahmedabad, India, ^Faculty of Science, University of Natal, Durban, S.A.
Abstract. The behaviour of Jeans-Buneman instabilities in a non-ideal dusty plasma is examined, by deriving the dispersion law and solving it numerically for a range of realistic parameters. The thresholds and growth rates for the Jeans and Buneman instabilities are discussed in terms of the relative drift between ions and charged dust, of the dust-to-ion temperature ratio and of the deviations from the ideal gas law for the dust. We find that for weak dust self-gravitation non-ideal effects have a significant influence on the growth rate of the pure Jeans instability, whereas for stronger dust gravitation the departure from the ideal gas behaviour is less prominent. On the other hand, non-ideal effects have little or no influence on the Buneman instability.
INTRODUCTION Dusty plasmas are encountered in a wide range of environments, from astrophysical situations to industrial and laboratory devices. Their increased observation has resulted in a growing interest in the behaviour and properties of dusty plasmas. Various reviews of the field have covered dusty plasmas form the solar system [4] to waves and instabilities in dusty plasmas [7,13]. On the other hand, the Jeans instability (JI) of a self-gravitating system has been well known for a long time [6]. In a dusty plasma, on the other hand, the JI has come into focus more recently [1,2,9,14], and it has also been found that in the presence of particle streaming, both the JI and the usual Buneman instability (BI) can overlap [8,10]. Further investigations have examined the influence of particle size distributions on the Jeans-Buneman instability, concluding that the growth rate was enhanced both for discrete particle size distributions [8] as well as for continuous size distributions [11]. One of the outstanding features of dusty plasmas is that the dust is of finite size, making these plasmas non-ideal [3,12]. The commonly used ideal gas law is primarily valid for particle sizes in the submicrometre range and for dilute dusty plasmas. However, this approximation breaks down for grain sizes in the supermicrometre
296
S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas
range, as the mean particle separation is smaller and interactions between neighbouring particles are enhanced [12]. In this paper we examine the Jeans-Buneman instability in a non-ideal dusty plasma, by incorporating the Van der Waals equation of state for the dust [12], whilst retaining the ideal gas law for the ions.
MODEL EQUATIONS AND LINEAR DISPERSION LAW Our dusty plasma consists of electrons, ions and charged dust. The electrons are light enough so that inertial effects can be neglected and their density is Boltzmann distributed. Full ion dynamics is retained, on the other hand, with a possible equilibrium streaming compared to the background dust species. Finally, for the dust both non-ideal and self-gravitational effects are incorporated. Ion self-gravitational effects are omitted, as these have been shown to be negligible [11,14]. The description of parallel electrostatic modes starts from the continuity equations for the ion and dust species, ^
+ ^ M i ) = 0
{j = hd),
whereas the electrons are assumed to be Boltzmann, i.e., rig = In addition, we need the ion and dust momentum equations, dvj dt dVd dt
dvi^ _ _ jiksTj drij _ j e^ ^dcj) ^ ' dx miTii dx rrii -^ ^-^^ dx' d^p Qd dcj)_ ^^_r_ ^ _ _ 1 9Pd _ ^iu_^^ rrid dx ndTUd dx rriddxdx dxdx
(1) Neexip(e(l)/kBTe). .^. ,^x
Here cj) and T/? are the electrostatic and gravitational potentials respectively, the labels j = e^i^d refer to electrons, ions and charged dust grains, and the other notations are standard. Finally, the description is closed by the electrostatic and gravitational Poisson's equations, ^ 0 ^ + H ^3^3 = 0.
-Q^ = AnGndTrid,
(4)
3
together with the Van der Waals equation of state for the dust {pd + Anl){l
- Brid) = Udi^BTd^
(5)
The constants A and B in (5) are calculated in terms of the critical parameters [5], by requiring that dpd/drid — 0 and d'^pd/d'^Ud — 0, yielding A = 9kdTc/Snc and B = l/3nc, where the subscript c denotes the respective values at the critical point. Using uppercase letters for the equilibrium values, charge neutrality in equilibrium requires that A^^- = Ne + ZdNd^ assuming that the dust is negatively charged and carries Zd electron charges.
S.R. Pillay et al. /Jeans-Buneman instability in non-ideal dusty plasmas
297
We linearize and Fourier transform the relevant equations, invoke 'Jeans swindle' in ignoring the zeroth order of the gravitational potential in looking at local perturbations, and obtain the general dispersion law as 1 1 + —1 ^ k^\l^
=
o;^uj'^. H! . 2i (a; - kVoY - k'Cf ^ a;2 + u;j, - k^Cj - k^Cl,'
(6) ^^
We have introduced the different plasma frequencies through LJ^J = NjOj/sorrij^ the dust Jeans frequency through CJJ^ = AnGNdrnd^ the thermal velocities through Cj = ^yjKBTj/mj and VQ is the ion drift speed. In addition, there is a non-ideal part to the dust thermal effects, given by Cj^^ = V — Q^ with V = BNdkdTd{2 — BNd)/md{l - BNdf and Q = 2ANd/md. We briefly discuss some limiting forms of this dispersion law. In the limit when (a; - kVoY ^ k'^Cf, the ions also are Boltzmann-like, i.e. ujli/{to - kViY " ^^^i — — 1/k'^Xjy-. With the help of the ion Debye length X^i we introduce a global Debye length XD through l/Xj^ = 1/A^g -|- l/Xj^-^ so that (6) then reduces to
P
l + k^Xl
+ CI + CI,,
(7)
Without dust gravitational effects this is precisely the dispersion law given by Rao [12]. On the other hand, for an ideal gas A = 5 = 0, and we have a generalization of the results of Avinash and Shukla [1] to warm dust. Expanding (6) we have the following normalised form of the dispersion relation, 0 = u^ai -f u^{-2KVoai)
+ ^'^[K^a^a^ - (1 + a2) + ax{l3'^a2 - K'^a^)]
+ u{2KVo[a2 - ai{^^a2 - K^a^)]} + K^a^[ai{f3^a2 - K^a^) - a^] -{f3^a2-^K'a3),
(8)
where ai = I + Ne/K'^Ni, a2 = ZdNdrrii/Nimd, a^ = miTd{l + e^r + ^cf)/mdTe, a4 •= V^ — Ti/Te^ and e^,^ = 7y(6 — ry)/(3 — r/)^, Q / = —9ar]/A^ represent respectively the contributions to the volume reduction coefficient and the molecular cohesive forces. Also we have that rj = Nd/ric^ a = Tc/Tdj K = kX^ and uj is the normalised frequency. The following normalisations have been introduced: distance by A^ = (soTe/iVie^)^/^, time by uj~^ and speed by c^ = {Te/miY^'^. The factor /? is the ratio of the dust Jeans frequency to the dust plasma frequency, i.e. /3 = ^jdl^pd-
RESULTS AND DISCUSSION The dispersion relation (8) is solved numerically for the following fixed parameters: milrrie = 1836, Zd = 700, Ne/Ni = 0.5 and Td/Te = 0.1. To aid our understanding of the low frequency nature of the fiuctuations, the frequencies are plotted in terms of the dust plasma frequency ujpd- Note that in all figures the continuous line represents the ideal case. Figure 1 examines the growth rate as
298
S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas
a function of the non-ideality parameter rj at kXi) = 0.2, Ti/Te = 0.1, /3 = 0.1 and a = 0.1, 1.0 and 2.0, respectively, for stationary ions (i.e. K = 0). Clearly this mode is a pure JI. We note that at a = 0.1, the growth rate decreases with increasing rj. For both a = 1.0 and 2.0 there is an initial increase in the growth rate with r/, before decreasing below that observed for an ideal gas. The growth rate remains greater than that of an ideal gas over a larger range of rj for higher a. This can be interpreted as follows. With Td/Te fixed, increasing a corresponds to an increase in the critical temperature, Tc. Hence one notices that for a greater critical temperature, the combined non-ideal effects of volume reduction and cohesive forces initially enhance the growth rate of the pure JI, before exerting a strong damping influence. Furthermore, for large a, there is a large range of 77-values over which the factor ^ = (1 -h e^^r + ^c/) < 1- On inspecting figure 1 we notice that the enhancement in the growth rate in the non-ideal case corresponds to the state when (^ < 1, i.e. the cohesive forces dominate over the volume reduction effect. Figure 2 repeats the investigation illustrated in figure 1, but now at /3 = 0.5, corresponding to stronger dust gravitation. The behaviour of the curves for increasing a is very much the same as for /? = 0.1. However, the enhancement in growth rate due to non-ideal effects is significantly lower here than in figure 1. Hence it appears that in the presence of stronger dust gravitation the growth rate of the pure JI is less sensitive to the non-ideal nature of the plasma. We also note from figures 1 and 2 that the corresponding drop in the growth rate at high r/ is less significant. Figure 3 now examines the growth rate as a function of rj for an ion drift speed, Vo = 0.5. At this drift speed both the JI and the BI occur [11]. Here kXD = 0.2, /3 = 0.1 and T^/Tg = 0.1. The observed behaviour in the growth rate for increasing a is similar to that observed in figure 1. Comparing figures 1 and 3 one notes that the departure from ideal behaviour is more prominent at K = 0, i.e. in the absence of the BI. It has been shown that the transition from the JI to the BI occurs when K > Vti [11], which in normalized form implies Vo > {Ti/TeY^'^. In figure 3 we have Vo = 0.5 > \ / 0 T . Hence we conclude that the instability seen in figure 3 is the BI, whilst that seen in figure 1 (for Vo = 0) is the pure JI. As it has been shown that the BI is a much stronger instability than the JI [11], the growth rate in figure 3 (for the BI) is much larger than that in figure 1 (for the JI). Figure 4 examines the growth rate as a function of the ion drift speed K at /? = 0.1, Ti/Te = 0.1, kXD = 0.2, T] = 2.0 and a = 0.1, 1.0 and 2.0, respectively. Only marginal differences are observed in the growth rates at the respective a values. Figure 5 examines the behaviour of the growth rate under the same conditions, but over a range of low ion drift speeds (i.e. corresponding to the state when the JI is still present). One now notes a more significant dependence of the growth rate on the non-ideal parameters. Thus, when the BI dominates, as is the case at high ion drift speeds, the influence of the non-ideal parameters is negligible. Figure 6 examines the growth rate as a function of kXD for 1^ = 0, /? = 0.1, Ti/Te = 0.1, 7/ = 2.0 and a = 0, .1, 1.0 and 2.0, respectively. Clearly, this is once again the pure JI. Here, the departure from ideal gas behaviour is more prominent for low a, whereas at high a one approaches the ideal gas limit. There is a maximum
S.R. Pillay et al. /Jeans-Buneman instability in non-ideal dusty plasmas 0.080
299
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Figure 4:Normalized growth rate y/o)^ vs. V^ /C^
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0.05
Figure 6:Normalized growth rate y/cOp^ vs. k?i^. a values are indicated on the curves.
300
S.R. Pillay et al /Jeans-Buneman instability in non-ideal dusty plasmas
kXn value, beyond which the JI does not exist due to gravitational collapse. This limit decreases only marginally with decreasing a.
CONCLUSION We have investigated the influence of non-ideal effects on the Jeans-Buneman instability. We note that for weak self-gravitation of the dust, non-ideal effects have a significant influence on the growth rate of the pure JI. There is a marked damping of the growth rate of the JI at high rj and low a {= Tc/Td). For stronger dust gravitation, the departure from the ideal gas behaviour is less prominent. With the onset of the BI, the departure from ideal gas behaviour becomes negligible. Hence non-ideal effects in the dust component have little or no influence on the BI. As in the ideal gas case, we have a critical Jeans length for the JI, which decreases marginally for low a.
Acknowledgements This work was supported by the Flemish Government (Department of Science and Technology) and the (South African) Foundation for Research Development in the framework of the Flemish-South African Bilateral Scientific and Technological Cooperation on the Physics of Waves in Dusty, Solar and Space Plasmas.
REFERENCES 1. Avinash, K. and Shukla, P.K., Phys. Lett A 189, 470-472 (1994). 2. Bliokh, P.V. and Yaroshenko, V.V., Sov. Astron. 29, 330-336 (1985). 3. Fortov, V.E. and lakubov, I.T., Physics of Nonideal Plasmas^ New York : Hemisphere, 1990. 4. Goertz, C.K., Rev. Geophys. 27, 271-292 (1989). 5. Joos, G., Theoretical Physics^ New York: Dover, 1986, p. 497. 6. Kobb, E.W. and Turner M.S., The Early Universe^ Reading: Addison-Wesley, Reading, 1990, p. 342. 7. Mendis, D.A. and Rosenberg, M., Annu. Rev. Astron. Astrophys. 32, 419-463 (1994). 8. Meuris, P., Verheest, F. and Lakhina, G.S., Planet. Space Sci. 45, 449-454 (1997). 9. Pandey, B.P., Avinash, K. and Dwivedi, C.B., Phys. Rev. E 4 9 , 5599-5606 (1994). 10. Pandey, B.P. and Lakhina, G.S., Pramana 50, 191-204 (1998). 11. Pillay, S.R., Bharuthram, R. and Verheest, F., Proc. 1998 Int. Conf. Plasma Physics^ Geneva: EPS, 1998, 22C, pp. 2497-2500. 12. Rao, N.N., J. Plasma Phys. 59, 561-574 (1998). 13. Verheest, F., Space Sci. Rev. 77, 267-302 (1996). 14. Verheest, F., Shukla, P.K., Rao, N.N. and Meuris, P., J. Plasma Phys. 58, 163-170 (1997).
Part V. Basic Experiments
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
303
Waves and Instabilities in Dusty Plasmas N. D'Angelo Department of Physics and Astronomy The University of Iowa, Iowa City, Iowa 52242
Abstract. The first section of the paper presents a summary of the work performed by our group at the University of Iowa on waves and instabilities in dusty plasmas, emphasizing the very close connection between wave phenomena in these plasmas and those in plasmas with large percentages of negative ions. The second section examines the effects of negatively charged dust grains on an ionization instability in plasmas of low degree of ionization, in which electrons are present which have an energy slightly above the ionization energy of the neutral gas.
I. This section of the paper presents a summary of the experimental and theoretical work on v^aves and instabilities in dusty plasmas, performed by our group at the University of Iowa during the last decade. Our work has generally proceeded along two parallel lines, with the same wave modes and/or instabilities being studied both in dusty plasmas and in plasmas with large percentages of negative ions. Since in typical laboratory dusty plasmas the dust grains are negatively charged, the close connection between wave phenomena in these plasmas and those in negative ion plasmas should not come as a surprise. Dusty plasmas, on the other hand, in addition to much smaller charge-to-mass ratios have certain properties which are not shared by negative ion plasmas. In the first place, in a dusty plasma many charge-to-mass ratios for the dust grains occur at the same time, unless one is dealing with the so-called "monodisperse" grains. Secondly, in the presence of a wave the dust grain charge may be variable, as pointed out, e.g., by (1). And finally, in a steady state plasma, the ions lost to the grains must be replaced by ions produced by ionization of a neutral gas. This introduces the wave-damping mechanism referred to as "creation" damping (2). Table 1 lists papers by our group, divided according to the particular wave mode investigated, for both negative ion plasmas and dusty plasmas. If we confine our attention to "low-frequency" electrostatic modes in magnetized dusty plasmas, we find (3) that four such modes exist, two being of an ion-acoustic
N. D'Angela/Waves and instabilities in dusty plasmas
304
TABLE 1. Negative ion plasma Dusty plasma Wave mode/ (refs.) (refs.) Instability 4,11 2, 9,10 Ion-acoustic (fast) 4 10, 12, 13 Ion-acoustic (slow) 10,18 15, 17 EIC (fast) 19 15, 17 EIC (slow) 23,24 22 PVSI 26 Rayleigh-Taylor 25 28 29 PRI 31 Shocks
type and the other two of an electrostatic ion cyclotron (EIC) type. The first two modes correspond to the so-called "fast" and "slow" modes in a negative ion plasma analyzed by D'Angelo et al (4) and experimentally investigated by Wong et al (5), Sato et al (6), and Nakamura et al (7). The "fast" mode in a dusty plasma is simply an ion-acoustic mode modified by the presence of a nearly stationary dust. It is also called a DIA mode (dust ion-acoustic mode). The presence of negatively charged dust increases the phase velocity of the wave, thereby decreasing the wave coUisionless (Landau) damping (see, e.g., Rosenberg (8). Properties of grid-launched DIA waves are illustrated in Figure l(a,b) which show (a) the wave (normalized) phase velocity and (b) the ratio of spatial damping rate to wave number, Ki/Kr^ as functions of eZ or (eZ^), where e = rid/n^ is the ratio between dust density and positive ion density, while Z = Q/e (or Zd) is the ratio between a dust grain charge and the electron charge (9,10). The variation of the phase velocity and of the damping rate with 6, for the case of grid-launched waves in a negative ion plasma is shown in Figure 2(a,b) (11). 1.5
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FIGURE 2. Grid-launched ion-acoustic waves in negative ion plasmas (a) phase velocity vs. e (b) the damping of the fast mode. The quantity /^,Vi,th/^ vs. e (ref. 11). The "slow" ion-acoustic mode was observed by Sato et aL (6), and Nakamura et al. (7) in negative ion plasmas and by Barkan et aL (12) and Thompson et al. (13) in dusty plasmas. This dust mode has been named "dust-acoustic" (DA) by Rao et ah (14). It has typical frequency in the tens of Hz range and phase velocities of ~ 10 cm/s. Figure 3 shows single frame video images of dust-acoustic wave crests at frequencies of 16, 22 and 30 Hz (13). Figure 4 shows the experimentally determined dispersion relation for the DA wave, i.e., the wavenumber A^(cm~^) vs. angular frequency u;(s"'^) (13). To come next to the ETC modes (slow and fast), they were considered theoretically by D'Angelo and Merlino (15) and, more recently, by Chow and Rosenberg (16). Experimental work in plasmas with negative ions (K'^,SF^,e) was reported by Song et al, (17), who found that the frequencies of both the slow and the fast mode increase with increasing e, the percentage of negative ions, while the critical drift velocities for excitation of either mode decrease with increasing e. Figure 5, from Song et ah (17), shows the measured frequencies of the SF^ and K"*" modes as functions of e. Figure 6, also from Song et al. (17), shows, as functions of 6, the critical drift velocities of the electrons for excitation of the K"^ and SF^ modes. In dusty plasmas, the fast EIC mode was investigated experimentally by Barkan et al. (18) who found that, in a manner similar to negative ion plasma, the fast EIC mode is more easily excited when the concentration of the negatively charged dust is increased. Figure 7 shows the wave amplitude with dust divided by the amplitude without dust as a function of eZ. The slow EIC mode in dusty plasmas (referred to also as the electrostatic dust-cyclotron or EDC mode) has never been observed in the laboratory because of the difficulty in obtaining magnetized dust grains, i.e., grains with a gyroradius considerably smaller than the transverse size of the plasma column. The mode has been analyzed theoretically by D'Angelo (19), also for possible relevance to phenomena in comet tails.
306
N. D'Angela/Waves and instabilities in dusty plasmas
16 Hz
22 Hz
0 30 Hz
50
100
150
200
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CO [s-^ ] F I G U R E 4. The dispersion relation of the dust-acoustic wave, K vs. uj (ref. 13).
F I G U R E 3. The "slow mode" dust-acoustic wave. Single frame video images of dust-acoustic wave crests at frequencies of 16, 22 and 30 Hz (ref. 13).
The parallel velocity shear instability (PVSI) can be excited when ions flow parallel to the magnetic field but with a velocity that changes from one magnetic field line to another (20). In early experiments D'Angelo and von Goeler (21) demonstrated the existence of the instability. Much more recently D'Angelo and Song (22,23) analyzed the conditions for excitation of the instability both in dusty plasmas and in negative ion plasmas. In the case of dusty plasmas with negatively charged and immobile dust grains, an increase of eZ produces an increase of the critical velocity shear for generating the instability. This is understandable, since for instability the velocity difference between adjacent layers should be on the order of a characteristic wave speed in the medium and, as we have seen above, the ion-acoustic speed increases with increasing eZ. In a negative ion plasma the same conclusion is obtained if the ratio m-/m^ between negative and positive ion masses is very large (> 10^). On the other hand, for much smaller m_/m^^ negative ions have a destabilizing effect on the PVSI. For example, for fixed values of the plasma
A^. D'Angelo/Waves and instabilities in dusty plasmas
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density gradient transverse to the magnetic field, of the parallel and perpendicular wavenumber and of the velocity shear parameter S = {l/ujc+)dv+o^/dx (where Uc+ is the angular ion gyrofrequency and dv+ojdx is the variation of the ion flow velocity, V+Q^, parallel to the magnetic field B = Bz in the transverse, x, direction) the instability growth rate was shown to increase with increasing e. This and other theoretical predictions were verified in a Q machine experiment with K+ positive ions and SFg ions, by An et al. (24), using essentially the same experimental setup employed many years before by D'Angelo and von Goeler (21) (see Fig. 8). An analogous experiment has been planned for a Q machine dusty plasma. The Rayleigh-Taylor instability may occur when a magnetic field acts as a light fluid supporting a heavy fluid (the plasma). If either negative ions or very massive negatively charged dust grains are added to the plasma, the range of unstable wavelengths is reduced, that addition thus having a stabiUzing effect (25,26). The potential relaxation instabihty (PRI) is excited when the cold endplate of a single-ended Q machine is biased positively (27). For suflaciently large positive biases, coherent oscillations are excited at a frequency on the order of a few kilohertz. Time-resolved measurements of the plasma potential oscillations show that they are associated with a moving double layer that appears when the endplate is biased sufficiently positive. When negative ions (SFg) are added to the plasma in sufficient concentration, the PRI is quenched, but a current-driven ion-acoustic (lA) instability is observed. As the negative ion concentration is increased, the frequency of the lA waves increases and the critical electron drift needed for their
308
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FIGURE 7. The fast EIC wave in a dusty plasma. The wave amplitude with dust divided by the amplitude without dust (ref. 10). excitation decreases. Figure 9, from Song et al. (28), shows the frequency of the ion-acoustic waves vs. the SFe gas pressure. Something entirely similar happens if, instead of negative ions, negatively charged dust grains are added to the plasma. Figure 10, from Barkan et ah (29), shows the lA wave frequency vs. eZ, The conditions under which large density pulses might develop into sharp fronts or "shocks" were investigated in a Q machine by Andersen et al, (30). Under normal operating conditions, with equal electron and ion temperatures, Tg = T^, Landau damping prevented the formation of a shock, and only a "spreading" of the pulse was observed. However, when the ratio T^jTi was made as large as about 3 or 4,
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FIGURE 9. The PRI in negative ion plasmas. The frequency of the ion-acoustic wave vs. the SFe gas pressure (ref. 28).
FIGURE 10. The PRI in dusty plasmas. The ion-acoustic wave frequency vs. eZ (ref. 29).
309
N. D'Angela/Waves and instabilities in dusty plasmas
by cooling the ions through ion-neutral collisions, shock formation was observed. A reduction of the coUisionless (Landau) damping can also be realized by the introduction of negative ions into the plasma. As the percentage of negative ions is increased, the phase velocity of the fast ion-acoustic mode increases, and the damping is reduced (4). The effect of negative ions on shock formation has been studied in a Q machine by Luo et al. (31) using a Cs+ plasma and SFg negative ions. About 30 cm from one of the hot plates a grid was inserted into the plasma column with its plane perpendicular to the magnetic field (Fig. 11(a)). The grid is normally biased at ~ - 6 V with respect to the grounded hot plate and absorbs most of the ions from the hot plate, resulting in a plasma density distribution of the type shown in Figure 11(b). By suddenly changing the grid bias to ~ - 2 V, the grid "opens up," launching a plasma density pulse toward the other hot plate. Figure 12 shows the curves of ion density, n, vs. time (from oscilloscope traces) at various axial distances from the grid for e = 0 (no negative ions present) and e = 0.95. One observes the pulse steepening effect by the negative ions. In, addition, measurements of this type provide the pulse propagation velocity vs. e,
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310
A'^ D'Angela/Waves and instabilities in dusty plasmas
shown in Figure 13. A similar experiment is under way, using negatively charged dust grains instead of negative ions. This may have relevance to the possible existence of an electrostatic shock in the ionopause of comet Halley, as pointed out by Mendis and Rosenberg (32).
II. In the second part of this paper we are no longer dealing with a comparison of wave phenomena in negative ion plasmas and in dusty plasmas. We examine, instead, the effects of negatively charged dust grains on an ionization instability which may be relevant to certain wave phenomena in dusty plasmas. In the course of an investigation of double layers produced by ion injection into the target chamber of a double-plasma (DP) device, Johnson et at. (33) observed that low-frequency (^ 0.5 to ~ 2 kHz) waves were spontaneously excited. The experiments were performed in a device consisting of an aluminum tube 200 cm long and 33 cm in diameter, provided, by means of a set of ten coils, with a longitudinal magnetic field of up to ^ 100 G. Plasma was produced by discharges in noble gases in two end chambers, each 63 cm long and 75 cm in diameter. The device was divided into "driver" and "target" chambers by a large grid located near one end of the central chamber, the grid being biased at —75 V so that electrons from either chamber could not enter into the other. The potential of the driver chamber could be raised to ^ 100 V above that of the target chamber, thus allowing the injection of an ion beam from the driver into the target. This would produce a stable double layer near the middle of the device. The gases which were studied were He, Ne, Ar, Kr and Xe, in the pressure range 10""^—10""^ Torr. At these pressures the ion beam injected from the driver into the target does not survive for long after crossing the grid separating the chambers, since the charge exchange mean-free path is only of the order of centimeters. Thus, several centimeters away from the grid into the target, the only ions present are "thermal" ions. On the other hand, energetic electrons (15 — 25 eV) would be present, produced by the double layer and with mean-free paths comparable to or longer than the length of the device. The ^ 0.5 kHz to ^ 2 kHz waves were observed in Xe, Kr and Ar, but never in Ne and He. These waves had the general features of ordinary ion-acoustic waves, and their excitation could be attributed to an ionization instability whose main features can be explained as follows. Suppose that an ion-acoustic perturbation of very small amplitude is present in a plasma of low degree of ionization. Regions corresponding to the crests of the wave (higher plasma density) are also regions of higher potential. If electrons are present which have an energy just above the ionization threshold of the neutral gas, with the ionization cross section being a rapidly increasing function of the electron energy over an energy interval of some 20 — 30 eV, ionization of the neutral gas will proceed in the wave crests at a rate larger than in the wave troughs. Thus, the wave might grow if the damping mechanisms which are present are not too strong. A theory of the instability was
N. D'Angela/Waves and instabilities in dusty plasmas
311
developed and, in particular, a relation was found between the growth rate and the quantity da/dE just above the ionization threshold of the neutral gas, a being the ionization cross section and E the energy of the ionizing electrons. Rather large values of da/dE accounted for the fact that the instability was observed in Xe, Kr and Ar gases, while no oscillations were seen in Ne and He gases which have a much smaller da/dE. In examining this instability in dusty plasmas (34), two questions were asked: (a) what is the influence of negatively charged dust grains on the characteristics of lA waves excited through the ionization instability, and (b) can the low-frequency dust-acoustic mode also be excited by the instability? As for (a), the presence of negatively charged dust grains was found to increase, at any given wavenumber K, both the frequency and the growth rate of the lA wave excited through the ionization instability. This was expected, since the removal of free electrons by the dust grains would increase the "spring constant" of the oscillations. The remaining free electrons are less able to neutralize the wave space charge and to reduce the wave electric field. As for (b), it was found that the dust-acoustic mode is not excited by the ionization instability, under the plasma conditions envisaged in ref. 34. A further step in this investigation (35) considered the effect of ion drag on the negatively charged dust grains as a possible mechanism of excitation of the dustacoustic ionization instability. It was found that the DA waves are, in fact, more and more damped as the coefficient of ion drag increases from zero to some critical value, beyond which a zero-frequency (nonpropagating) perturbation grows, when the drag of the ions on the dust grains overcomes the effect of the perturbation electric field. This is illustrated in Figure 14, where the real and the imaginary part of the frequency are shown as functions of the ion drag coeflacient, /i. At a /icrit ^ 160 one observes a transition from a damped DA wave to a growing, nonpropagating perturbation. It seems possible that this growing perturbation may represent the initial (linear) stage of the perturbations producing dust "voids" in some experimental situations (36). 250
~1
1
'
1
'
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I50h 3
3
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-50'[
120
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140
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1
160
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1 180
FIGURE 14. The ionization instability in a dusty plasma. The Ur and a;,- vs. the ion drag coefficient (ref. 35).
312
N. D 'Angelo/Waves and instabilities in dusty plasmas
ACKNOWLEDGMENTS Work supported by ONR and NSF.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.
Melands0, F., T. Aslaksen, and 0 . Havnes, J. Geophys. Res. 98, 13,315 (1993). D'Angelo, N., Planet. Space Sci. 42, 507 (1994). D'Angelo, N., Planet. Space Sci. 38, 1143 (1990). D'Angelo, N., S. von Goeler, and T. Ohe, Phys. Fluids 9, 271 (1966). Wong, A. Y., D. L. Mamas, and D. Arnush, Phys. Fluids 18, 1489 (1975). Sato, N., et al., Phys. Plasmas 1, 3480 (1994). Nakamura, Y., T. Odegiri, and I. Tsukabayashi, Plasma Phys. Control. Fusion 39, 105 (1997). Rosenberg, M., Planet. Space Sci. 41, 229 (1993). Barkan, A., D'Angelo, N., and Merlino, R. L., Planet. Space Sci. 44, 239 (1996). Merlino, R. L., et al., Phys. Plasmas 5, 1607 (1998). Song, B., D'Angelo, N., and Merlino, R. L., Phys. Fluids B 3 , 284 (1991). Barkan, A., Merlino, R. L., and D'Angelo, N., Phys. Plasmas 2, 3563 (1995). Thompson, C , et al., Phys. Plasmas 4, 2331 (1997). Rao, N. N., Shukla, P. K., and Yu, M. Y., Planet. Space Sci 38, 543 (1990). D'Angelo, N., and Merlino, R. L., IEEE Trans. Plasma Sci. 14, 285 (1986). Chow, v . , and Rosenberg, M., Planet. Space Sci. 43, 905 (1995). Song, B., et al., Phys. Fluids B l , 2316 (1989). Barkan, A., D'Angelo, N., and Merlino, R. L., Planet. Space Sci. 43, 905 (1995). D'Angelo, N., Planet. Space Sci. 46, 1671 (1998). D'Angelo, N., Phys. Fluids 8, 1748 (1965). D'Angelo, N., and von Goeler, S., Phys. Fluids 9, 309 (1966). D'Angelo, N., and Song, B., Planet. Space Sci. 38, 1577 (1990). D'Angelo, N., and Song, B., IEEE Trans. Plasma Sci. 19, 42 (1991). An, T., Merlino, R. L., and D'Angelo, N., Phys. Lett. A214, 47 (1996). D'Angelo, N., Ann. Geophys. 11, 494 (1993). D'Angelo, N., Planet. Space Sci. 41, 469 (1993). lizuka, S., et al., Phys. Rev. Lett. 48, 145 (1982). Song, B., Merlino, R. L., and D'Angelo, N., Phys. Lett. A153, 233 (1991). Barkan, A., D'Angelo, N., and Merlino, R. L., Phys. Lett. A222, 329 (1996). Andersen, H. K., et al., Phys. Rev. Lett. 19, 149 (1967). Luo, Q.-Z., D'Angelo, N., and Merlino, R. L., Phys. Plasmas 5, 2868 (1998). Mendis, A., and Rosenberg, M., Ann. Rev. Astron. Astrophys. 32, 419 (1994). Johnson, J. C , D'Angelo, N., and Merlino, R. L., J. Phys. D: Appl. Phys. 23, 682 (1990). D'Angelo, N., Phys. Plasmas 4, 3422 (1997). D'Angelo, N., Phys. Plasmas 5, 3155 (1998). Samsonov, D., and Goree, J., Phys. Rev. E 59, 1047 (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and P.K. Shukla (eds.) 0 2000 Elsevier Science B.Y All rights reserved
313
Dust Charging in the Laboratory and in Space MihAly HorAnyi and Scott Robertson Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309-0392
Abstract. The study of dusty plasmas bridges traditionally separate subjects: celestial mechanics and plasma physics. Dust particles immersed in plasmas and UV radiation collect electrostatic charges and respond to electromagnetic forces in addition to all the other forces acting on uncharged grains. Simultaneously they can alter their plasma environment. Dust particles in plasmas are unusual charge carriers. They are many orders of magnitude heavier then any other plasma particles and they can have many orders of magnitude larger (negative or positive) time dependent charges. In this paper we summarize our ongoing laboratory experiments that are aimed t o verify dust charging theories for plasma and UV bombardment. These measurements are useful in the data analysis of recent Galileo observations (imaging, plasma science and dust) where the laboratory experiments helped t o show that small dust grains acting as active plasma probes - allow for a unique consistency test of our models of planetary magnetosphere.
INTRODUCTION Dust particles immersed in plasmas and UV radiation collect electrostatic charges and couple to the electric and magnetic fields of their environment. The traditional probe theory to calculate the charge on a dust grain have a number of limitations. For example, the shape of the dust particles, the presence of electric and magnetic fields, and collisions are all thought to influence the charging processes. The charge on the grains is of fundamental importance to calculate dust-dust coupling or the electromagnetic forces acting on the grains. To date, electrostatic charges on dust grains in space were never measured. In this paper we first summarize or laboratory charging measurements and show that probe theory can be used, at least in these simple experiments. We also show, using probe theory, the expected charging of dust particles in the magnetospheres of Jupiter and Saturn. In these cases, the electromagnetic forces acting on small charged dust particles do lead to unusual dynamical effects, like rapid transport, capture and ejection of dust.
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M. Hordnyi, S. Robertson/Dust charging in the laboratory and in space
LABORATORY SETUP We have constructed an experiment where individual dust grains can be exposed to a thermal plasma background and a flux of fast electrons. We have conducted experiments using glass, copper, graphite, and silicon particles and also grains from MLS-1 and JSC-1 lunar substitute materials [1-4]. The apparatus is shown in Figure 1, it is the double plasma machine that we used in our earlier experiments. Fast electrons are created at one end of a cylindrical vacuum chamber (30 cm diameter by 30 cm long) by an emissive tungsten filament biased to a negative potential (—100< —U< —10 V). These electrons impinge on falling dust particles and also ionize the background neutral argon gas. Dust grains are dropped into the chamber at the top and are collected in a Faraday cup mounted in a diagnostic arm below the chamber. We placed a small permanent magnet outside the exit hole to prevent electrons and ions from entering the Faraday cup. The Faraday cup is connected to a sensitive electrometer for measuring the charge on the grains. The dropping mechanism, a vibrating plate with a small hole, is adjusted to drop particles infrequently so that the majority of Dust dropper
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rt <
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FIGURE 1. d){top) Schematic diagram of the experiment used for dust charge measurements with fast electron fluxes. The dust particles enter the plasma from the dropper at the top and fall through the plasma into the Faraday cup below the chamber, b) {bottom) The modified plasma chamber for photoelectric charging experiments {see Sickafoose et a/., in this volume).
M Hordnyi, S. Robertson /Dust charging in the laboratory and in space
315
events are due to individual grains. Signals from multiple grains are easily identified by their waveform and are not used. The filament current is set to 2 mA, which is suflScient to make the charging time for the dust (a 30 /im radius grain takes 50 ms to charge to 30 V) much shorter than the dust transit time through the apparatus (250 ms). The filament is surrounded by a grounded wire mesh so that the potential within the chamber is not distorted by the filament potential. The chamber has a base pressure 4 x 10~^ Torr and the experiments were performed with 2 x 10"^ torr of argon. The fast electrons also ionize the background argon gas creating a thermal plasma background. The characteristics of the plasma are measured by a Langmuir probe [5], placed close to the exit hole of the grains. Due to the energy dependence of the electron-neutral ionization cross section [6] and the changes in the fast electron fluxes entering the apparatus, the plasma conditions change with the bias potential (i.e., the energy of the fast electrons). In our data analysis we used the measured flux of fast electrons (Jj), and the temperature (Tg) and the flux of thermal electrons (J^) as a function of the bias potential. We repeated these Langmuir probe measurements before and after the data were taken for each bias potential setting. There were no noticeable drifts in the plasma parameters during the data taking period of a sample material.
EXPERIMENTAL RESULTS At each beam energy eC/, 50 charge measurements were made. These are shown in Figure 2a. In regions where the distribution of the 50 measured charges indicated a bimodal distribution due to the presence of two stable charge equilibria, we have split the data points into two groups and treated them independently. To analyze these measurements, we used the secondary electron yield function 5{E) [7] 5{E)
= 7A5M{EIEM)
exp [-2{EIEMY''')
(1)
where the maximum yield 5M and the optimum energy EM (for which 5 — 6M) are material parameters. For a given fast electron energy (et/), we used the measured charge Q, to minimize J/(Q, 5M. EM) + Je{Q) + Ji[Q)
(2)
for all the fast electron settings simultaneously, by varying the material parameters 5M and EM- In theory, for a grain in charge equilibrium equation (2) should yield 0. Our fits are shown in Figure 2b. These fits indicate, that for a given plasma environment the expected charges on small dust particles can be estimated with confidence using probe theory to calculate the charging currents. Figure lb shows our modified experimental setup where the the emitting filament was replaced with a quartz window to allow for UV illumination. Our most recent UV charging experiments are discussed by Sickafoose et al.j in this volume.
316
M. Horanyi, S. Robertson/Dust charging in the laboratory and in space
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F I G U R E 2. a) (left) Measured charges (small dots) as a function of the beam electron energy. The large dots represent the average of the measured charges, and the horizontal line separates the regions where the distribution of the measurements indicates multiple charge states, b) (right) Average of the measured charges (dots) with error bars to show one standard deviation and our fits setting EM = 400 and J ^ ^ = 3.1 for MLS-1 (top), 5 ^ ^ = 3.4 (middle), and (J^*^ = 3.2 for the Apollo 17 (bottom) soil sample.
DUST CHARGING AT JUPITER AND SATURN The most prominent feature of the Jovian plasma environment (Figure 3) is the plasma torus maintained by the strong production of oxygen and sulfur atoms from the volcanos on the moon lo (located at r ^ 6 i?j, i?j = 7.1 x 10^ km, is the radius of Jupiter). The plasma density peaks close to the orbit of lo v^ith n « 3000 cm~^ and the plasma temperature shows a strong minima kTe = kTi ?^ 1 eV at r = 5Rj sharply rising to kTe ^ 50 and kTi ?^ 80 eV at r = SRj [8]. The characteristic surface potential in this region varies in the range of —30 < <> / < +3 V. The change in the sign of 0 is mainly due to changes in Tg, which controls the relative contribution of the secondary electron current (Figure 3). In the vicinity of the moon Enceladus ( at r ?^ 4i?5, Rs = 6 x 10^ km, is the radius of Saturn) Saturn's plasma environment is characterized by a bi-Maxwellian electron distribution ('cold' and 'hot' electron population) with 100 cm "^, j^rpcoid ^ 5 gY^ ^hot ^ 0.2 cm-^ kT^°^ = 100 eV. The ion population consists of H+ and 0+ with UH « 10 cm-^ kTn « 10 eV, n,, ^ 60 cm-^ and kTo ^ 50 eV [9]. Here the expected surface potential of a dust particle is in the range of —8 < (/» < —4 V [10,11]) (Figure 3).
M Hordnyi, S. Robertson /Dust charging in the laboratory and in space
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F I G U R E 3. a) (iop 0 leji) The electron density and temperature as function of distance in Jupiter's magnetosphere. b) (to'p & right). The plasma parameters for Saturn. The indexes c, h correspond to the cold (thick lines) and hot (thin lines) plasma population, respectively. In a) and b) the parameters for ions are not shown, c) (bottom & right) The equilibrium surface potential of dielectric dust particles moving on an unperturbed circular Kepler orbits in the equatorial plane of Jupiter as function of distance from the planet. We assumed a secondary electron production parameter of EM — 500; 5^ — ^ (dot-dashed line), 1 (continuous line), and 3 (dashed lines). The radial dependence of the equilibrium potential primarily reflects the variations in the plasma parameters, d) (bottom & right) Same as in c), but for the case of Saturn. In c) and d the vertical dashed lines mark the location of moons lo (6i?j), Dione and Helene {Q.ZRs) and Rhea {8.7Rs). These moons are likely sources of dust grains due to meteoroid bombardment.
318
M Hordnyi, S. Robertson/Dust charging in the laboratory and in space
Generally, the distribution of astrophysical plasmas is not uniform. The plasma composition, density and temperature might all exhibit spatial and temporal variations. Consequently, grains can have complicated charging histories, their charge will not only depend on the instantaneous plasma environment, but also on the previous charge states. In all planetary magnetospheres, for example, dust grains will collect only a modest charge and the resulting electrodynamic forces acting on cm or bigger grains are negligible compared to gravity. However, toward the lower end of the mass distribution (a < few microns) the Lorentz force might become the most important perturbation resulting in significant deviation from Kepler orbits. This size range of dust particles is frequently represented in planetary rings: Saturn's E-ring, the Jovian ring, Neptune's arcs are examples where a significant portion of the optical depth is attributed to micron sized grains. Electrodynamic perturbations often couple with other perturbations (oblateness, radiation pressure, plasma and neutral drag, etc.) and can lead to unusual dynamics: transport, capture and ejection. In situ measurements of the electrostatic charge on a dust particle is very diflficult and as of yet, it was never accomplished. The laboratory experiments are important to build confidence in using simple probe theory to estimate dust charging using measured plasma parameters.
ACKNOWLEDGMENTS This work was supported by the Magnetospheric Program of NASA and the Department of Energy.
REFERENCES 1. Walch, B., M. Horanyi, and S. Robertson, Charging of dust grains in plasma with energetic electrons, Phys. Rev. Lett., 75^ 838, 1995. 2. Horanyi, M., S. Robertson, and B. Walch, Electrostatic charging properties of simulated lunar dust, Geophys. Res. Lett, 22, 2079, 1995. 3. Horanyi, M., B. Walch, S. Robertson and D. Alexander, Electrostatic charging properties of Apollo-17 lunar dust, J. Geophys. Res. 103, 8575, 1998 4. Walch, B., M. Horanyi and S. Robertson, Electrostatic charging of Lunar Dust, imPhysics of Dusty Plasmas, eds: M. Horanyi, S. Robertson and B. Walch, American Institute of Physics 446, 271, 1998 5. Hershkowitz, N., How Langmuir probes work, in Plasma Diagnostics, edited by 0. Auciello and D. Flamm, pp. 113, Academic, San Diego, Calif., 1989. 6. McDaniel, E.W., Collision Phenomena in Ionized Gases, John Wiley, New York, 1964. 7. Sternglass, E.J., Backscattering of kilovolt electrons from solids, Phys. Rev., 95, 345-58, 1954.
M Hordnyi, S. Robertson/Dust charging in the laboratory and in space 8. Bagenal, F., Empirical model of the lo plasma torus: Voyager observations, J. Geophys. Res., 99, 11,043, 1994. 9. Richardson, J.D., and E.G. Sittler, Jr., A plasma density model for Saturn based on Voyager observations. J. Geophys. Res. 95, 12019, 1990. 10. Horanyi, M., J. A. Burns, D. Hamilton, The Dynamics and Origin of Saturn's E ring, Icarus 97, 248, 1992. 11. Jurac S, A. Baragiola , R.E. Johnson, E.G. Sittler Jr., J. Geophys. Res. 100 14,821, 1995.
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
321
Chai^g Measurements and Planetary Ring Simulation by Fine Particle Plasmas Toshiaki YOKOTA Faculty of Science, Department of Physics, Ehime University Bunkyo-cho 3, Matsuyama, Ehime 790-8577 JAPAN E-Mail: [email protected]'U.ac.jp
ABSTRACT. Fine particle plasmas were generated by the ultraviolet light irradiation foe aluminum particles gwierated by a boat method. This plasma was composed of two components; positively charged particles and negatively charged particles. Fine particle plasma density was measured by the attenuated and the scattered light intensity of He-Ne laser light, and was in order of 10^ cm~^. The charge of particle was measured by the distance of beam split after pass through the electric fields or the magnetic fields, and estimated to be ~±25 electrons.^'^^ It was estimated that these plasmas were in a strong coupling state. Simulation experiments for planetary ring formation was performed by the rotation of sphere with radius of 4 cm in which permanent magnet was set along the rotating axis. When the sphere rotates in the fine particle plasmas, the electric fields are introduced by the unipolar induction and this field interacts with the charged particles around the rotating sphere. It is our scenario of simulation that the fine particle ring will be created by above interactions between charged particles andfields.The magnetic field strength of sphere was about 300 Gauss at both poles. The rotatingfi-equencywas controlled by control of motor current and was measured by a choppingfi-equencyof laser diode light. The simulation result and some of elementary process that particles take charges are described in this article.
INTRODUCTION It is much interest to know the mechanisms that particles get charges. The methods for generating the fine particle plasmas (dusty plasmas) are several ways: The first is to immerse the particles into gaseous plasmas. Particles take negative charges, usually, this plasma is in the mixed state of negatively charged particles, electrons, and ions (three or four component plasmas). The second is the ultraviolet light irradiation for particles. Particles emits photo-electrons and emitted electrons collide with another particle, so, this plasma is composed of only positively charged particles and negatively charged particles (two component plasmas). The third is the irradiation of electron beam. The second interest is to know the mechanisms which the particles capture the electric charges in the space. The particles (dust) are in various circumstances. It is
322
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas
well known that the dust is omnipresent in the space. The presences of dust are interstellar clouds, circum-stellar clouds, super nova, interplanetary dust, earth's magnetosphere, comets, and planetary rings. In the interstellar clouds, solar system, comets and planetary rings, especially, the ultraviolet radiation and gaseous plasmas will take important roles, because particles are irradiated by the radiations from the central star such as the sun and are immersed in the solar wind plasmas. The third interest is to know the mechanisms why the planetary rings are created. Planets such as Jupiter and Saturn have magnetic fields and are rotating around its axis. We can guess that the electric fields are generated according as the rotation of planet by the unipolar induction. The simulation experiment was performed to check that the rings are created or not by this formation. The resuh is reported in this paper.
CHARGING MECHANISMS OF PARTICLES The elementary processes that particles capture charges are complex, and mainly depend on the circumstances around particles. The elementary processes are as follows; (1) interaction with photons, (2) interaction with electrons, (3) interaction with gaseous plasma particles, (4) interaction with surrounding particles, and (5) the others (size dependence etc.). 1. INTERACTION WITH PHOTON: The yield of photo-emission depends on the particle size. The yield is in order of 10"* for ordinary surface but its value increases to 1 according as the particle size is decrease to 10 nm size.^^ The particles emitted photoelectrons become positive. The emitted electrons colhde with another particles and are captured by particles. These particles become negative. As a results, fine particle plasmas become to two component plasmas. 2. INTERACTION WITH ELECTRON: The slow electron was captured by particle, but the fast electron can emit another electrons (secondary electrons) when colhde with particles. 6{Ey=JJJ^ indicates secondary electron yield parameter, where J^ and J^ are incident electron current and secondary emitted electron current, respectively. 6{E) is given by following equation."^) ll
,(£).Zd«M(£)e,p ^m
J-^''
E and E^ indicate incident electron energy and its maximum, 6^ is the value of 6 at E=E^. 6(E) depends on the material of particle. 3. INTERACTION WITH GASEOUS PLASMA: When the fine particles are mixed in gaseous plasmas, particle charge Q is shown by following continuity equation.
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas
323
where subscript b means ion and electron. If particle is sphere with radius a, then Jh^'^ita^n^Q^Vj^, ^j^{T^lm^^^^, The total currents flowing into particle become to zero, JQ'^J^"^ 0, then particle reaches some potential ^. If gaseous plasma is in Maxwell distribution, this potential is given by
In case of hydrogen plasma, ^—2.5ir-/e, and in case of oxygen plasma, V ^ 3,6TJe.^^ That is, particles immersed in gaseous plasma have negative charge. As a result,fineparticle plasmas (dusty plasmas) are composed of three components. Many of dusty plasmas (fine particle plasmas) in the universe and the laboratory are this type of plasma. The charging time r that particle potential becomes rp is shown by *^
X =
^pe
^
where A^ is Debye length and a is particle radius. Moreover, charge of particle decreases according as particle density in gaseous plasma increases.^^ This form is shown by following equations, ip = M(i + x) a R 1 + JC
\
^D
4. INTERACTION WITH SURROUNDING PARTICLES: If particles collide with each other, or collide with vessel wall, it seems that particles catch charges. In case of mutual collision of particles. The charge of particle will be given by
AG«a5|l-exp(~—U where. At is contact time, S is contact area, r is time constant of electrification. When particles collide with wall, particles catch charges by the effect of surrounding particles and its charge will be given by
where N is coUision number, Q^ and Q^ are initial and final charge. N^ is constant depending on vessel size and particle density.^^ 5. SIZE DEPENDENCE: When particle size is larger than the order of 1 jim, above
324
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas
charging mechanisms are almost correct. If particle size becomes to smaller than the order of 1 ^im, charge of particle shows some offluctuation.^^Moreover, particle size becomes the order of 10 nm, such particle cannot get charge, because of high coulomb energy.^^
PLASMA GENERATION Aluminum fine particles were generated by the boat method, and fine particle plasmas were created by the irradiation of ultraviolet light by using halogen lamp in our laboratory. Particle size was determined as the fimction of gas pressure and boat current by using a sampling method. ^^ The charge of particle was estimated from the separation offineparticle beam which pass through a static electricfieldor static n H a l o g e n Lamp
0
CCD Camera [
g
'
:Pump
iHe.Ne Laser
I I JAC H>VJ " - ^ Electric Field or Magnetic Field
Ar Gas
FIGURE 1 Experimental setup for measuring the particle charge and photograph of splitted particle beam by the electric field. magnetic field. Experimental setup to measure the plasma parameter is shown in Fig.l. Charged particle beam was clearly split into two beams after pass through the static electric and magnetic field. This indicates that fine particles irradiated by ultraviolet light were composed of positively charged particles and negatively charged particles (two component plasmas). Observed charge of particle are in range of ±20 + 50electrons.^^ Spatial density distribution of plasma was estimated from the scattering intensity distribution of He-Ne laser light and average plasma density was measured by observation of transmittance of He-Ne laser light with good accuracy. The observed density of particle is in order of 5 X10^ cm""^ (Fig.2). ^'^^^
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas FPP Density
10'
2m 3 0 s
325
1996.11.22
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10"
1 Q 3
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•
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FIGURE 2 Density distribution of fine particle plasmas measured by scattering intensity. Horizontal axis is distance from center in cm and vertical axis is density in cm"^.
RING SIMULATION EXPERIMENT Aluminum fine particle plasmas are used for a planetary ring simulation experiment. This plasma is two component plasma with charge of ±20 -- ±50 electrons, density of--^ 5 X10^ cm"^, and radius of 0.5 jim. The small magnetized sphere was set in this plasmas and rotated by a small motor as shown in Fig.3. The small permanent magnet was set in the plastic sphere with radius of 4 cm along the rotating axis for the planetary ring simulation experiment. The magnetic field distribution around miniature satellite is shown in Fig.4. The sphere is connected to a small DC motor in the chamber by a bamboo and brass rod. The rotating fi^quency of sphere was measured by a choppingfrequencyof laser diode light passing through a hole of brass rod. The motion of fine particle plasmas was observed by using a TV camera and was recorded on a video recorder. Some of typical pictures were took photographs by a steal camera and VTR camera as shown in Fig,5.
SIMULATION RESULT AND DISCUSSION The experiment was performed under the condition of particle radius 2a=0.5 jim, plasma density A^ --5 X10^ cm"" ^, and particle charge ±20 - ±50 electrons using
326
T. Yokota/Charging measurements and planetary ring simulation byfineparticle plasmas
f\ Halogen Lamp Motor
FIGURE 3 Miniature satellite put into the fine particle plasma for simulation experiment.
FIGURE 4 The magneticfielddistribution around miniature satellite. Field strength is in Gauss unit.
FIGURE 5 Simulation image of planetary ring in laboratory experiment.
halogen lamp power of 18 W. When the rotation of sphere was started, fine particle cloud occurred disturbance at first. After these disturbance, it was observed that particles maderingshaped beh around the rotating sphere, and this form is resemble to planetary ring. Typical photograph around the sphere is shown in Fig.5. Magnetic field B at the equator is given by following equation with magnetic dipole
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas
321
Br=PJ47tr^ ,
moment P^m'
Electric field by unipolar induction is E^^Q rB according to rotation of magnetic dipole. By using i>=2;r IT, T is period of rotation, electricfieldis rewritten as follows;^ ^^ £ = -%2rT The Saturn is taken as a simulation ofringsimulation. The ratio E and Ey^ are ^ 5
E^ % 2rM^7)M
The suflBx S and M mean the Satum and miniature sphere for simulation. The parameters of the SatumareP =4.4XIO^^ Gauss-cm^ 7=38361.6 s, and r =1.12 X10^" cm. Conditions for creation of ring is given by this equation and calculated result is shown in Fig.6. Magnetic dipole moment P^^^ of sphere was 1.265 x 10^ Gauss • cm^, so, we can .1
• • •'
' • ' »
30000
J
1
25000
1
20000 PlSOOO 10000 5000 0
1 i i II
0
20
iiii
itii
ii
III
»
I
II 1
40
60
80
•' 100
1/Tm
FIGURE 6 Estimation of unipolar induction. estimate how many rotating frequency is adequate for this simulation experiment. It is estimated that the revolution of sphere in the fine particle plasmas was adequate around 10 Hz in this experimental setup. This is almost compatible to experimented value. The fine particle plasmas generated by ultraviolet light were in the strong coupling state. Because, F-value seems to be order of 10 when plasma density is 5 X10^ cm" ^, particle charge is around 30 electrons, and temperature is a little higher than room temperature.^)
328
T. Yokota/Charging measurements and planetary ring simulation by fine particle plasmas
CONCLUSION We can confirm the optical methods for measuring the fundamental parameters of the fine particle plasmas; the density and charge. Thefineparticle plasmas generated by the irradiation of ultraviolet light were two component plasmas which were composed of positively charged particles and negatively charged particles, and were in strongly coupled state. We can generate the planetary ring form around the rotating sphere by the unipolar induction by using aluminum fine particle plasmas when plasmas were in strongly coupled state. The real time information of particle size is inevitable in the laboratory experiments. Measurement was performed by forward scattering intensity of He-Ne laser light. Details of this experiment will be reported in another report.
ACKNOWLEDGEMENT This work was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan (No. 10680461).
REFERENCES 1. T. Yokota and K. Honda; J. Quant. Spectrosc. Radiat. Transfer, 56,761,1996 2. T. Yokota and S. Manabe; J. Quant. Spectrosc. Radiat. Transfer, 61,219,1999 3. P. Aanestand andE. M. Purall; Ann. Rev. Astron. Astrophys., 11, 309,1973 D. C. Morton; Astrophys. J., 193, L35,1974, and 197, 85,1975 4. N. Meyer-Vemet; Astron. Astrophys., 105,98,1982 U. de Angelis; Physica Scripta, 45,465,1992 5. T. G. Northrop; Physica Scripta, 45,475,1992 6. C. K. Goertz and W. -H. Ip; Geophys. Res. Letts., 11,349,1984 7. W. John, G. Reishl and W. Devor; J. Aerosol Sci., 11,115,1980 8. T. MatsuokaandM. Russell; J. Appl. Phys., 77,4285,1995 9. R. Kubo; J. Phys. Soc. Japan, 17,975,1962 10. T. Yokota; CP446 Physics of Dusty Plasmas, Edited by M. Horanyi et al. The American Institute of Physics, pp.49-52, 1998 11. P. Bliokh, V. Sinitsin and V. Yaroshenko; Dusty and Self-Gravitational Plasmas in Space, Kluwer Academic Publishers, Dordrecht, 1995 M. Horanyi; Geophys. Res. Letts., 24,2175,1997
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
329
Structure Controls of Fine-Particle Clouds in DC Discharge Plasmas Noriyoshi Sato, Giichiro Uchida, Ryoichi Ozaki, Satoru lizuka, and Tetsuo Kamimura* Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan "^National Institute of Fusion Science, Toki 509-5292, Japan Abstract As a series of our experiments on fine-particle clouds in plasmas, we have carried out various kinds of experiments on structure controls of the fme-particle clouds in low-pressure dc discharge plasmas. Here a short review is presented of essential points of the results on vertical and radial profiles, phase transition, azimuthal votices and rotations, vertical spread of particle clouds, and vertical strings of periodic alignment of particles.
INTRODUCTION Fine particles in plasmas are of current interest in plasma researches and applications. In plasmas, fine particles are negatively charged and form so-called "Coulomb lattice" under the strong Coulomb interaction among the particles [l]. The first demonstrations of the Coulomb lattice and almost all subsequent experiments, which have shown many interesting features of the phenomena, have been performed in plasmas produced by low-pressure rf discharge [2-7]. Generally speaking, however, it is difficult to control plasma conditions in the rf plasmas. Our experiments [8] have been carried out in lowpressure dc discharge plasmas to form and control fine-particle clouds which are in liquid or in solid (Coulomb lattice) state, depending on the conditions. Our work is different from other works performed under the dc situations [9,10] in the sense that our emphasis has been always on active controls of structures and dynamics of the particle clouds. We are interested in global structures of fine-particle clouds and their macroscopic dynamics in the radial, azimuthal, and vertical directions, in addition to their fine structures and phase transition. Here we present essential points of our experiments on static and dynamic features of the fine-particle clouds in plasmas. Some of them were presented at the Workshop, Boulder, 1998 [8].
EXPERIMENTS Standard Situation: Vertical Profiles A standard situation is schematically described in Fig. 1. A plasma is produced at Ar gas pressure around 200 mTorr by applying a negative dc potential of about 300 V to a cathode (stainless steel 6 mesh/inch grid) of 85 nmi in diameter with respect to a ring anode
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N. Sato et al /Structure controls offine-particleclouds in dc discharge plasmas
(outer and inner diameters are 80 and 95 mm, respectively) which is separated 10 mm downward from the cathode, being electrically grounded. The hole of the anode is often covered with a grid. The discharge current is in the range 0.2 -- 2 mA, yielding the electron density of 10'^ -- 10^ cm'^ and temperature of 1 ~ 3 eV, respectively. Mono-dispersive methyl methacrylate-polymer particles (specific gravity: 1.17 -- 1.20 g/cm^) of 10 (± 1.0) |Lim in diameter are injected into a glow discharge region through the mesh cathode. There are stainless steel plates for vertical levitation and radial confinement of the particles in the experimental region below the anode. The 20-nmi-diam plate (LE) for the levitation is set at a position of 15 mm downward from the anode. The plate (CE) for the confinement with a 20-mm-diam hole at its center is set at a position of 1 mm upward from the LE. The particles are observed by a CCD camera detecting Miescattering of He-Ne laser Ught spread with horizontal width of 5 mm. When the CE is biased more negatively than the LE, there appears a hill-shaped radial potential profile above the LE. Then, the particles levitate, being confined radially above the LE, as shown schematically in Fig. 1. The electric charge on each particle is estimated to be (10^ -- 10^) e (e: electronic charge), yielding the coupling coefficient T around 10^. The particle clouds observed has an almost hexagonal lattice structure with interparticle distance of a few hundred [im, although the particle positions fluctuate. The fluctuations are often so large as to yield liquid behaviors especially when the background neutral gas pressure is decreased at a fixed discharge current. As we increase the negative potentials of the LE and CE with keeping the radial potential profile constant, the vertical position of the particle clouds increases, for example, from a few to about 10 mm above the LE. The particle clouds consist usually of several layers as observed in the rf plasmas [2-7]. With an increase in the negative potential of the CE at a fixed potential of the LE, however, the layers increases, for example, from 3 to 15, being followed by a gradual change of the shape to V-shaped cone (see Fig.l). This is ascribed to an increase in the height of the hill-shaped radial potential profile. The Vshaped cone was reported in Ref. [9], but there was neither physical explanation for its formation nor control of the shape. The shape change is due to the radial potential profile becoming broad in the upward direction. In Fig. 1, we also find that the interparticle distance decreases gradually in the upward direction. This is because the plasma density increases and thus the Debye shielding length decreases in this direction. PARTICLES C A T H O D E T ANODE "PARTICLES"
CONFINEMENT ELECTRODE
LEVITATION ELECTRODE FIGURE 1. Left: standard experimental setup. Right: V-shaped fine-particle cloud observed.
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For a further increase in the negative bias of the CE, the particles pushed up fall down in the upper layers of the clouds because the electric field is too small for the particles to levitate there. Repetition of "up and down" particle motion induces collective oscillations of the particles, generating fine-particle acoustic waves propagating downward.
Segmented Plates: Radial Profiles In this case, the vertical levitation and radial confinement of the fine-particle clouds are accomplished by a plate segmented radially, as shown in Fig. 2. Since the particle positions are sensitive to the radial potential profile, a radial distribution of the particle clouds is controlled by using this segmented electrode (SE) biased negatively. A hill-shaped radial potential profile above the SE is produced by applying negative potentials to the segmented parts of the SE, which increase gradually in the radial direction. We can change the height and width of the potential profile by varying the negative potentials applied to the segments [11]. The particle clouds are confined in the region of the potential hill and has the radial spread depending on the potential profile. When the potential profile has a well at the radial center and a hill at some radial position, we can observe a ring-shaped fine-particle cloud. A typical example of the particle rings is shown in Fig. 2. Multi-ring structures are observed to be generated by potential profiles with multi-peaks in the radial direction. We can change the radii and widths of the rings. A rosary-shaped particle ring can be formed in an extreme case. In the presence of a weak magnetic field in the vertical direction, the rings of the particle clouds are observed to rotate in the azimuthal direction. Details of this rotation is described later. A temporal change of the radial potential profile is provided by applying a sinusoidal potential variation to the central part of the SE. When hill- and well-shaped potential profile are alternately formed, we can observe a clear periodic radial motion of the particle clouds as far as the frequency of the sinusoidal variation is much smaller than the plasma frequency of the particles. With an approach of the frequency to the plasma frequency, however, there appears a complicated feature of the particle motion which also includes vertical oscillations of the particles. PARTICLES C A T H O D E T ANODE
ELECTRODE SEGMENTED FOR LEVITATION & CONFINEMENT FIGURE 2. Left: setup with segmented electrode for particle levitation and confinement. Right: ringshaped fine-particle cloud generated.
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N. Sato et al/Structure controls offine-particleclouds in dc discharge plasmas
Electron Shower: Phase Transition It is important to control negative electric charges on the particles for a phase transition of the particle clouds. A schematic of our setup for this control is shown in Fig. 3. Here, the levitation electrode (LE) is a fine grid with 300 mesh/inch, below which an auxiliary dc discharge plasma (AP) is produced in addition to the main plasma. This plasma is used for supplying electron shower (low-energy electron beam) on the particles levitating above the LE, which is provided by biasing the AP negatively for electrons in the AP to pass through the LE. A typical example of the top view for an effect of the electron shower on the particle clouds is presented in Fig. 3, where we can also find a radial fine structure of the particle clouds at the auxiliary discharge currents /a=1.2 mA. Here, the experimental condition is set to yield the particle clouds in liquid-like state in the absence of the electron shower (/a=0). As found in Fig. 2, the particles are arranged to form the Coulomb lattice with hexagonal structure in the presence of the electron shower supplied at this discharge current. The pair correlation function g{r) of the particles at a distance r is measured with la as a parameter. At /a=1.2 mA, there appear four peaks in g{r), corresponding to the regular particle arrangement of hexagonal structure, within a distance of 1 mm, in Fig. 3. With a decrease in la, however, the number of the peaks decreases and the peak widths become broad. There is only one peak with quite broad width at /a=0.4 mA. Since the charge on the particles is almost proportional to la, the results here mean that, with a decrease in the negative charge on the particles, there occurs a gradual phase transition of the particle clouds from solid (Coulomb lattice) to liquid state. This double-plasma method used here for supplying electron shower on fine particles is quite useful for levitation of particles which are much larger and/or heavier than those in this experiment. We are able to levitate glass balloons of 50 |im in diameter and are now ready for levitation of glass balloons of 100 |im in diameter, which, off course, fall down in the absence of the electron shower. We are interested in levitation of much bigger particles by using this method. PARTICLES C A T H O D E T ANODE CONFINEMENT .ELECTRODE ELECTRONS..
ANODE CATHODE
FIGURE 3. Left: setup supplying electron shower on particles. Right: a top view of fine structure of particle clouds at la (auxiliary discharge current) = 1.2 mA.
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333
Vortices and Rotations We have employed two methods for generating particle flows in the particle clouds. One of them is an electric method and the other is a magnetic method. In case of the electric method, a small electrode (SE) with sharp edges is situated just on the LE, as shown in Fig. 7. When the SE is biased, there appear large potential gradients around the sharp edges, which give rise to forces acting on the particles and generate particle flows in the particle clouds. An example of the particle flows is described in Fig. 4, where the SE with triangle shape is biased negatively. There appear vortices with velocity shear in the horizontal plane above the LE, just as in case of usual liquids. In general, one edge drives two vortices, generating six vortices in this case. The speeds of the vortices are in the range up to a few mm/s, depending on the position. The measurements for different shapes of the SE with n (integer) edges show that there appear 2n vortices in the particle clouds. The vortex size depends on the size and/or position of the SE. We can generate similar vortices in computer simulations for the particle clouds. In case of the magnetic method, a vertical magnetic field B up to 400 G is appUed to the standard setup in Fig. 1. In the presence of B, there appears an azimuthal rotation of the particles in the particle clouds. The rotation is in the diamagnetic direction. The angular frequency, which is of the order of 0.1 rad/s (much larger than the fine-particle cyclotron fi-equency), is found to be almost independent of the radial position and increase with an increase in the radial potential drop for the particle confinement. During this rigid-body rotation, the interparticle distance is kept to be almost constant. The rotation depends on the particle density. When the density is extremely low, there appears no particle rotation. The rotation starts when the particle density becomes high enough to provide the strong Coulomb interaction among the particles. The rotation frequency increases with an increase in the particle density. This means that this rotation is generated in the presence of the interaction among the particles under the magnetic field.
CONFINEMENT ELECTRODE
ELECTRODE FOR VORTEX FORMATION
LEVITATION ELECTRODE MAGNETIC FIELD FOR VORTEX FORMATION
20 mm
FIGURE 4. Left: setup for vortex formation. Right: particle flows in vortices generated by biasing a small triangle-shaped plate negatively in the horizontal plane.
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A'^ Sato et al /Structure controls of fine-particle clouds in dc discharge plasmas
Our results cannot be explained by the ExB drift of electrons, being different from the results in the experiment of Fujiyama et al. [12], where an alternative magnetic field was applied in the direction perpendicular to a dc electric field E. Anyway, the magnetic field is so weak that there is no direct effect of B on the fine-particle orbits, i.e., we can neglect the cyclotron motion of the particles. But, we cannot neglect magnetic effects on electron and ion orbits. A rotation mechanism could be understood by taking account of a potential profile modified by the effects of B on ions and/or electrons, which drive the particle cloud in the diamagnetic direction. We must be also careful for self-rotation of each particle to understand phenomena observed in a magnetic field, although this effect is negligibly small under our weak magnetic field.
Vertical Spread of Particle Clouds In the usual experiments [2-7] on fine particles, the particles levitate with several layers in the narrow vertical region above the LE. In our experiments with the setups in Figs. 1 and 2, a vertical spread of the particle clouds increases with an increase in the height of the hill-shaped radial potential profile above the LE, as mentioned before. But, this radial potential structure cannot penetrate upward into the plasma away from the sheath above the LE. In this direction, the radial profile becomes gradually broad, yielding a radial spread of the particle clouds, and finally cannot confine the particles radially, Umiting the vertical spread of the clouds. A further spread of the particle clouds in the vertical direction would be provided if we could keep the hill-shaped radial potential profile far above the LE. In our experiment for increasing this spread of the particle clouds, the auxiliary dc discharge plasma (AP) is produced in addition to the main dc discharge plasma (MP), as in case of the situation for supplying the electron shower. Biasing the AP negatively with respect to the MP, we have two diffusing plasmas with different potentials in the experimental region between the two discharge regions, which are separated radially as shown schematically in Fig. 5. PARTICLES CATHODEf
FLOATING ELECTRODE ANODE
^^1
LEVITATION
•^^^P^ELECTRODE
CATHODE\ MAGNETIC FIELD FIGURE 5. Left: setup for vertical spread of particle clouds. Right: column-shaped particle cloud.
A^. Sato et al /Structure controls of fine-particle clouds in dc discharge plasmas
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The radial potential profile can be controlled by changing the potential difference applied between the plasmas. In the vertical direction, there is no appreciable change of this profile, especially in the presence of a vertical magnetic field [13]. Under such a situation, the particles confined radially spread into an almost all vertical region up to a position just below the anode of the MP, as demonstrated in Fig. 5. Now the vertical spread is much larger than the radial spread given by the diameter of the LE, forming a shape of long column in the vertical direction. Since there is a vertical magnetic field, the column rotates in the azimuthal direction as in case of the thin particle clouds. Depending on the conditions, this three-dimensional particle cloud becomes unstable, showing a big dynamic motion. Then, there appear fine-particle acoustic waves propagating toward the LE and/or vortices associated with "up and down" motions of the particles in the particle clouds, which were quite similar to the vortices induced by a small probe inserted in the particle clouds [14].
Vertical Alignment of Particles When the width of the radial potential profile is decreased under the double-plasma configuration mentioned above, the column diameter decreases. A further decrease in the profile width, which is provided by decreasing the diameter of the LE, yields a vertical alignment of the particles in rows. They line up with almost equal distance between the neighboring particles in the vertical direction. This structure of the fine particles was reported by Mitchell and Prior at the Workshop, Boulder, 1998 [14]. In our case, there appears a azimuthal rotation of the rows in the diamagnetic direction in the presence of a weak vertical magnetic field. This rotation is accompanied by a small radial oscillation. The number of the rows is observed to increase, for example, from one to ten or so with an increase in the particle number supplied. In Fig. 6, the observations from the top and side are presented in case of four rows of the particles in the vertical direction. Computer simulations demonstrate almost the same alignment of particles in the region of the potential hill in the radial direction, being consistent with the experimental results.
PARTICLES CATHODE
\
ANODE FLOATING ELECTRODE ANODE
iij/^LEVlTATION -^ ELECTRODE
CATHODE FIGURE 6. Left: setup for vertical alignments of particles. Right: vertical alignments of particles in four rows (inset: view from top).
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N. Sato et al /Structure controls of fine-particle clouds in dc discharge plasmas
It is quite interesting to observe a temporal process of the vertical and horizontal alignments of the particles which are injected one by one in the experimental region. This process is useful to understand the Coulomb-lattice formation and would be also important for physical clarification of the particle rotation observed in such a weak vertical magnetic field as there is no direct magnetic effect on the fine particles
CONCLUSIONS Active structure controls of the fine-particle clouds in plasmas, which are in liquid or in solid (Coulomb lattice) state, have been presented under several different dc situations. We have described essential points of the phenomena observed up to now. Our experiments have clarified new features of fine-particle clouds in plasmas. On the basis of the particle motions presented here, we can now establish a simple method for removal of fine particles in dusty plasmas. This short review of our experiments is a new version of our paper in Physics of Dusty Plasmas [8], which includes recent topical results.
ACKNOWLEDGMENTS The authors would like to thank T. Hatori for his useful discussions and comments. We also thank R. Hatakeyama and M. Inutake for their interests in our work.
REFERENCES 1. Ikezi, H., Phys. Fluids 29, 1764-1766 (1986). 2. Chu, J. H. and I, L., Phys. Rev. Lett. 72, 4009-4012 (1994). 3. Thomas, H., Morfill, G. E., Demmel, V., Goree, J., Feuerbacher, B., and Mohlmann, D., Phys. Rev. Lett. 73, 652-655 (1994). 4. Hayashi, Y. and Tachibana, K., Jpn. J. Appl. Phys. 33, L804-806 (1994). 5. Pieper, J. B., Goree, J., and Quinn, R. A., /. Vac. Sci. Technol. A14, 519-524 (1996). 6. Melzer, A., Homann, A., and Piel, A., Phys. Rev E 53,2757-2766 (1996). 7. Tsuji, K., Yokoyama, A., Sakawa, Y, and Shoji, T., Double Layers (edited by Sendai "Plasma Forum"), Singapore: World Scientific, 1997, pp. 100-104. 8. Sato, N., Uchida, G., Ozaki, R., and lizuka, S., Physics of Dusty Plasmas (edited by Horanyi, M., Robertson, S., and Walch, B.), New York: American Institute of Physics, 1998, pp. 239-246 and the references therein. 9. Nunomura, S., Ohno, N., and Takamura, S., Jpn. J. Appl. Phys. 36, L949-951 (1997). 10. Fortov, V. E., Nefedov, A. P., Torchinsky, V. M., Molotkov, V. I., Petrov, O. F., Samarian, A. A., Lipaev, A. M., and Khrapak, A. G., Advances in Dusty Plasmas (edited by Shukla, P. K., Mendis, D.A., and Desai, T.), Singapore: World Scientific, 1997, pp. 195-203. 11. Tsushima, A., Mieno, T., Oertl, M., Hatakeyama, R., and Sato, N., Phys. Rev. Lett. 56, 1815-1818 (1986). 12. Fujiyama, H., Kawasaki, H., Yang, S-U., and Matsuda, Y, Jpn. J. Appl. Phys. 33, Part 1, 4216-4220 (1994). 13. Sato, N., Nakamura, M., and Hatakeyama, R., Phys. Rev. Lett. 57, 1227-1230 (1986). 14. Law, D. A., Steel, W. H., Annaratone. B. M., and Allen, J. E., Phys. Rev. Lett. 80, 4169-4192 (1998). 15. Mitcell, L. W. and Prior, N. J., "Vertical Structures in a Dusty Plasma," presented at the Seventh Workshop on the Physics of Dusty plasmas, Boulder, 1998.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
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Structural Formation and Stability of Coulomb Clouds in Medium through Low Gas Pressure Range S. Takamura, N. Ohno, S. Nunomura, T. Misawa and K. Asano Department of Energy Engineering and Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan Abstract. In a hundred mTorr gas pressure, two kinds of electrostatically coupled dust cloud with a heterogeneous and a discrete layered structures formed in a plasma-sheath boundary in a dc glow discharge are clearly observed, depending on the dispersion in size distribution of dust particles. A single particle analysis gives the equilibrium position, but is not suflFicient to explain detailed internal structure. A more sophisticated 3-D particle simulation including repulsive Coulomb interaction taking the radial electric field into account reveals a physical mechanism for formation of curious funnel-shaped dust cloud. In a low gas pressure range less than ten mTorr, mono-layered Coulomb crystal formed horizontally on a negatively biased mesh has an outstanding local growth of vertical oscillation of dust particles, sometimes they drop onto the electrode. A physical mechanism of such an interesting spontaneous positional instability is discussed by suggesting an energy gain of particles traversing over the range of electrostatic potential variation with delayed charging. It is pointed out that it may become an energy source of self-excited transverse dust lattice waves which have never been identified in the experiment. The particle simulation mentioned above reproduces the dispersion relation theoretically obtained.
I. INTRODUCTION Coulomb crystals in dusty plasmas have been observed in many capacitively-coupled rf devices (1-3) and several dc glow discharges (4,5) at a high gas pressure more than 100 mTorr. Structural formation was analyzed using a molecular dynamics particle simulation (6,7) in which the system parameters make transitions in its structure. Such a high pressure condition inhibits the observation of collective dynamic behaviors of the Coulomb crystal because the motion of dust particles is restricted by a strong friction with gas molecules. Therefore, the gas pressure in Coulomb crystal is crucial for study of dynamic behaviors, including the lattice oscillation and the wave phenomena. As regards the instability, however, few experimental investigations are reported so far (8). Coulomb crystals in dusty plasmas are quite attractive since it may give a new field of physics in which the parameters, like charge on the particle, change depending on the interaction between the crystals and the plasma. A charge on a dust particle changes in time, referred to as a charge fluctuation. One of the origins is thought to be caused by a movement of dust particles in an inhomogeneous plasma-sheath. Self-excited instability may destroy the crystal structure, associated by a coupled dust lattice wave.
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S. Takamura et al. /Structural formation and stability of Coulomb clouds
In this paper, we focus on the physics of stmctural formations and dynamic behaviors of electrostatically coupled Coulomb cloud under the external electricfielddetermined by the generation of plasma and the configuration of biased electrodesfi*oma medium through a very low gas pressure environment. n. FORMATION OF DUST CLOUDS WITH INVERSE CONE STRUCTURE Two different types of dust clouds have been observed in the experimental device shown in Fig. 1(a) with a high gas pressure of about 0.1 Torr. Both have a shape like a 3Dfiinnelwith its vertex at the bottom, and their vertical thickness is 1.5-2.5 mm. The cloud in Fig. 1(b) is composed of dust particles with a large dispersion in size distribution: r^ = 0.1-1.0 fim. The heterogeneous vertical structure seems to be made of many Dust feeder
, Illuminated dust cloud
FIGURE 1. (a) Schematic view of the experimental setup. Ar plasma is produced around the negatively biased cathode with dc glow discharge. The dust cloud is trapped at the center of the biasing ring in the plasma-sheath boundary, (b), (c) Two images of typical 3-D funnel-shape dust clouds with two different: (b) large, (c) small, dispersions of particle size distribution.
23 24 z[mm] FIGURE 2. (a) Space potential V^ (broken line) in the cathode sheath with respect to the plasma potential (V^ = 0), and the dust charge Q^ (solid line). Q^ becomes positive at a little bit inside of sheath, (b) The profile of potential energy W^^^ for a single dust particle coming from the gravitational and the upward electrostatic forces in the plasmasheath boundary, (c) The equilibrium position of dust particles as a function of dust radius where the unstable position is shown by a broken line corresponding to the maximum for the potential curve in (b).
S. Takamura et al /Structural formation and stability of Coulomb clouds
339
horizontal layers, each of which has almost the same sized particles. The size seems to be large deep into the cone. In the case of small-disperse particles (r^= 1.5 db 0.25 ^im), the cloud is made of a few separated horizontal layers periodically located in the vertical direction. Figure 2 gives a single particle analysis for an external field given by a modified ChildLangmuir expression for coUisional cathode sheath. The following electron and ion currents onto the particle are employed 4^j
exp
(e(V^Vl] hTe
I. = Tzr^en
N
eKf ks^e 1 - m. m,v//2+e|FjJ
where the floating voltage F^is defined as the potential difference between the surface of dust particle and the space around the dust, and is related to the charge(2^by e^ = 47r/^F.. A stable equilibrium height for a heavy particle is deep in the sheath, which basically explains the heterogeneity of Fig. 1(b). For mono-disperse particles, we have to consider the Coulomb coupling energy among particles in addition to the potential energy coming from the external field. In the numerical analysis the internal potential energy is divided into the screened Coulomb potential energy between different layers and that among particles on the same layer. The potential minimum ^ 2 6 . 2 "a" 26.0 is obtained in the 2-D parameter J , 25.8 space of layer number and cloud "25.6 thickness. The experimental 0.0 condition gives the layer number 0.0 x[inm] x[min] of 6 and the thickness of 2mm, which agrees well with FIGURE 3. Vertical and horizontal structures obtained with 3-D particle simulations, (a) mono-disperse particles experimental observation. -2 (r^ = 2.5^m) in a weak radial electric field, E^ = -3.3 x 10 3-D particle simulation for {r[mm]} •^'^[V/mm]. (b) mono-disperse particle (r^= 2.5 mono-disperse particles with a ^un) in a relatively strong radial electricfieldE^ = -0.53 little bit large radius gives the {r[mm]} •^'^[V/mm]. (c) dust particles with dispersive (r^ similar structure shown in Fig. 3 in = 2.5±0.3^m) distribution in the same E^ as (b). /? = 0.1 which screened Coulomb Torr, A^^ = 250 (a number of the simulated particles), i^= interactions among particles are all 0.2mm. The sheath edge is located at z = 26.0mm, where considered under the given g . = 8.6x10 C.
S. Takamura et al /Structural formation and stability of Coulomb clouds
340
externalfielddetermined by the plasma conditions shown in Fig.2. Three different E^. are assumed: for (a) E^ is an order of magnitude smaller than E^ which can sustain the particle and is defined by E^ =^^g/Q^= 15 VImm, and for (b) and (c) they are comparable. The simulation reproduces the discretely layered structure for dust particles with non-dispersive size distribution, and the diffusively layered one for those with size distribution. We can see a weak inverse cone structure since the particles at the edge of deep layers are unstable due to repulsive Coulomb forces. It is also noted that the selfgravity effect makes the multi-layer penetrate deep into the sheath (7). m. FORMATION OF SINGLE-LAYERED COULOMB LATTICE IN LOW GAS PRESSURE A. Structural Formation Single-layered Coulomb lattice in a low gas pressure is formed in a dc discharge with multiple cusp of magneticfieldas shown in Fig.4. Dust particles are trapped near the sheath edge above the negatively biased {V^ = -5 V) mesh. A radial electricfieldfor a horizontal confinement is made by a negatively biased (-10V) ring. These dust particles are arranged with a regular interval on the horizontal plane, forming a 2-D triangular lattice. The mean interparticle distance is about 430 |im. A conventional theory on coUisionless sheath gives the vertical distribution of the potential energy and the equilibrium charge as shown in Fig. 5. The position of minimum potential energy gives the levitation height which depends on the biasing voltage, the plasma density and FIGURE 4. (a) Schematic view of experimental setup. The dc discharge between hot filament cathode and the anode covering magnets produces Ar plasma at a gas pressure less than 10 mTorr. T'-lcYandn --10^ V ^ Dust e
e
particles are trapped in the central area of the ring above the negatively biased mesh electrode. (bMd) Images of 2-D Coulomb crystal composed of dust particles with 2.5 ± 0.5^im in radius, (b) top view and (c) side view. Image (d) shows traces for vertical oscillation of dust particles.
S. Takamura et al /Structural formation and stability of Coulomb clouds
341
the radius of particle. The comparison of equilibrium position between the numerical analysis and the experimental observation is successfully obtained. B. Self-Excited Vertical Oscillation When decreasing the gas pressure less than a certain critical value, dust particles start to oscillate spontaneously in the vertical direction. Figure 4(d) shows a typical image of such a vertical oscillation, in which 5 images are superposed for 0.17s. The oscillation was also observed by decrease in plasma density as shown in Fig.6. An example of temporal evolution for a growing oscillation is shown in Fig. 7(a). A typical time scale of growth is
0)
4
^
0
a -8
Plasma J 2
^ 1 I?
0 r. ^ds.EVQd>oi |-r(j=2.5nm ne=10i4m-3
-3
-2
-1 z [mm]
EVQd < 0 ^ -1
0
FIGURE 5. Vertical profiles for the equilibrium charge on a dust particle (solid line) and its potential energy (broken line) in the plasma-sheath boundary. The sheath edge is at r = 0.
3.0
p = 2.9 mtorr rd = 2 . 5 ± 0 . 5 urn
5 4
rUnstable
?3
! Stable
T
r-
1
I
4 6 p [mtorr] Unstable
Ii;
¥3 E " 2
I
8
5 4
! Stable
•
2
4
n J I O ^ * m-3]
1
!'• I ' I
5 10 Time [s]
10
15
(b)
L p ~ 4.6 mtorr 0
y
J
rj ~ 2.5nm ne~4X10<"m-3
J
Dropped
(a)
m—
E " 2
S
6
FIGURE 6. Levitation height from the mesh surface and the amplitude of vertical oscillation as a fimction of (a) gas pressme and (b) plasma density.
0
5 10 Frequency [Hz]
FIGURE 7. (a) Typical temporal evolution for the vertical instability, (b) FFT spectra for four different growing stages: 1) 0-2s, 2) 4-6s, 3) 810sand4)11.5-13.5s.
342
S. Takamura et al. / Structural formation and stability of Coulomb clouds
around 10s. The dust falls down from the trapping area when the amplitude exceeds around 2 mm. The FFT analysis gives the frequency of 10-14Hz. The potential curve shown in Fig. 5 is approximated as a parabolic around the well, giving an eigen frequency of about 14 Hz, and a maximum amplitude of about 2.1mm. They are very close to those experimentally observed. We should note that Q^ changes the sign deep inside the sheath, causing a release of trapping. Figure 8(a) shows contour plots for the oscillation amplitude in the 2-D parameter space of gas pressure/? and bulk plasma density n^. It was found that the amplitude tends to increase as decreasing/? and n^. Especially we should note a critical boundary for the spontaneous excitation. In other words, the stable crystal formation needs highp and n^. We should consider the physical mechanism of energy supply against the dissipation due to friction with gas molecules. Let us look again at Fig. 5 where the equilibrium charge changes over the range where dust 5 -{ particles may oscillate around the potential well. The charging time dQ^ /dlis very short 4 H 35 |Lis compared with the oscillation period of CO 67 ms. However, it is not zero but finite. The E instantaneous charge on the dust is given by . I
0
We can follow it during the movement of dust particle in the inhomogeneous sheath. The Q/t) may differ from the equilibrium value Q^,^^, so that the dust may be accelerated or decelerated on the way of periodic motion. The acceleration is obtained on the path where E'WQ^_^^<0, while the particle coming into the deep inside of sheath is slowed down by the electrostatic field because of E'WQ^_^^>0. Figure 8(b) shows numerically calculated contour plots for the saturated amplitude of self-excited oscillation taking such a delayed charging into account. In the saturation phase, the energy gain from the electrostatic acceleration balances the energy loss induced by friction with gas molecules and by electrostatic deceleration in deep in side the sheath. The qualitative behavior on the instability condition is similar to that experimentally obtained (8). By looking carefiiUy at Fig.4(d) again, we note that the amplitudes of oscillation differ among particles, and that several particles stay
" 2 c 1 H 3.0 3.5 p [mtorr]
1 2 p [mtorr]
3
FIGURE 8. (a) Contour plots for the amplitude of self-excited dust oscillations in 2-D parameter space of the plasma density and the gas pressure, (b) The same as (a) but obtained by numerical analysis taking the effect of delayed charging into account.
S. Takamum et al /Structural formation and stability of Coulomb clouds
343
quietly at the same time. In addition, dust particles sometimes fall down from the trapping area. It is not reproduced by the above single-particle analysis. Since the particles in the dust crystal are coupled with strong Coulomb interactions, the particles exchange their energy through such a collective coupling. Therefore, the kinetic energy of a specified particle can be increased by an energy transfer through Coulomb interaction from neighboring particles. This is thought to be a reason why a particle drops down. The spatial variation of oscillating amplitudes in the vertical direction implies the collective interaction among particles: excitation of transverse dust lattice waves. Detailed correlation analysis between two oscillations at different locations has been done and is presented in this Conference (9). C. Transverse Dust Lattice Wave The transverse dust lattice wave is not identified in the experiments so far although the longitudinal dust lattice wave was observed using the radiation pressure of laser light for an excitation tool (10). The dispersion relation of transverse wave is described (11) by / , \ / \ ( \ Q', ^0 ' 2 kr^ 2 Y sm exp 4
2mfy
[ 2j
,
, }^ is defined by -y6z=F{ZQ
L (a-2) 11 Hz
+ 6z) - m^g, r^ is the
mean interparticle distance and /?is Epstein friction coefficient (12). The relation is plotted in Fig. 9(b) under the following parameter condition: V^ = -lOV, 7; = leV, T. = 0 . 1 e V , w ^ = 1 0 ' V , / 7 = 50 mTorr {13= 5 Hz), r^= 1.5^m, and 14 /w^=1.7xl0"\g. 2-D particle simulation has been carried out to investigate the propagation of transverse dust lattice wave under the above experimental condition, in which the external field, like that in Fig.5, is assumed for 21 particles being able to move on the horizontal x and the vertical z directions. The particle at an edge is artificially oscillated with the amplitude of 0.5mm to excite the collective mode. The examples of wave propagation is showm in Fig.9(a).
^D)
h» • • • • • » » f » 4 .
.6.49s 6.51s -6 -4 -2
x[mm] (b-1) real part T—r k=7r/a
2 3 4 5 kj.[min"^]
(b-2) imaginary part
0.1 0.2 0.3 0.4 0.5
k.[mm"^]
FIGURE 9. Results of 2-D particle simulation. A horizontal electricfieldis assumed to confine 21 particles at a limited horizontal range, (a) Some propagating pattem of transverse lattice wave when the right-most particle has a forced oscillation ± 0.5 mm. (b) The dispersion relations of transverse lattice wave. The broken line shows the theoretical curve taking the coUisional dissipation into account.
344
S. Takamura et al. /Structural formation and stability of Coulomb clouds
At the frequency below the eigen frequency of single particle, the wave propagation can be detected. A high gas pressure inhibits a reflection of the wave at another edge to see clearly damped propagating waves. A fairly good agreement between the theory and the simulation is obtained as shown in Fig.9(b). Movie can be seen on the Web site, in which the program is written in Java (13). IV. DISCUSSIONS AND CONCLUSIONS Static and dynamic structural formations of Coulomb dust cloud have been investigated fi*om medium through low gas pressure. The configuration of external electricfieldis so important for determination of the structure that the equilibrium may be roughly determined by a single particle analysis. But the detailed internal structure is very much influenced by the Coulomb coupling energy among particles. And also we should consider the reaction of the strongly coupled Coulomb cloud to the weakly coupled background plasma. Structures of two kind of Coulomb clouds with the shape of inverse cone are discussd in terms of size distribution of particles and the energy principle. 3-D particle simulation suggests the origin of such a funny shape. A positional instability of dust particles is obsereved in low gas pressure. Its origin is discussed in terms of the delay of dust charging induced by the particle motion over the inhomogeneous sheath. However, it is not suflHcient to explain the experiment. We invoke an excitation of transverse dust lattice wave which may enhances the vertical oscillation due to, for example, a standing wave. 2-D particle simulation identifies the propagation of transverse dust lattice wave, in which the dispersion relation obtained in the simulation agrees well with the theoretical prediction. REFERENCES 1. H. Thomas, G.E. MorfiU, V. Demmel, J. Goree, B. Feuerbacher and D. Mohlmann, Phys. Rev. Lett. 73, 652 (1994). 2. J.H. Chu and I. Lin, Phys. Rev. Lett. 72, 4009 (1994). 3. Y. Hayashi and K. Tachibana, Jpn. J. Asppl. Phys. 33, L804 (1994). 4. V.E. Fortov, A.P. Nefedova, V.M. Tochinskii et al., JETP Lett. 64, 92 (1996). 5. S. Nunomura, N. Ohno and S. Takamura, Jpn. J. Appl. Phys. 36, 92 (1997); Phys. Plasmas 5, 3517 (1998). 6. H. Totsuji, K. Kishimoto and C. Totsuji, Phys. Rev. Lett. 78, 3113(1997). 7. K. Asano, S. Nmiomura, T. Nisawa, N. Ohno and S. Takamura, Czech. J. Phys. 48/S2, 239 (1998). 8. S. Nunomura, T. Misawa, N. Ohno and S. Takamura, submitted to Phys. Rev. Lett. 9. T. Misawa, S. Nunomura, K. Asano, N. Ohno and S. Takamura, in this Conference. 10. A. Homann, A. Melzo, S. Pets and A. Piel, Phys. Rev. E56, 7138 (1997). 11. S.V. Vladimirov, P.V. Shevchenko and N.F. Crane, Phys. Rev. E56, R74 (1997). 12. P.S. Epstein, Phys. Rev. 23, 710 (1924). 13. http://www.ees.nagoya-u.ac.jp/^asano
Poster Session Poster Session A: Basic Pliysics of Dusty Piasmas
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
347
Kelvin-Helmholtz Instability in Weakly Non—Ideal Magnetized Dusty Plasmas with Grain Charge Fluctuations Nagesha N. Rao^ Theoretical Physics Division Physical Research Laboratory Navrangpura, Ahmedabad-380009 INDIA
Abstract. Parallel Kelvin-Helmholtz (K-H) instability in weakly non-ideal dusty plasmas having transverse shear in the dust flow velocity parallel to the magnetic field has been investigated by using the van der Waals equation of state and the grain charging equation. At higher temperatures, the critical shear for the onset of the density gradient driven K-H instability increases due to non-ideal and thermal contributions. Dust temperature gradient gives rise to a new type of K-H instability when the relative flow speed between the adjacent layers exceeds a threshold value which is much smaller than the dust-acoustic wave (DAW) phase speed. Typically, grain charge fluctuations lead to damping eff'ects, thereby reducing the instability growth rates.
I.
INTRODUCTION
Waves and instabilities in dusty plasmas have been extensively studied beginning in this decade [1]. In the ultra-low frequency regime, dusty plasmas admit new types of wave modes not found in the usual electron-ion plasmas. The first such normal mode arising solely due to the dust dynamics was theoretically predicted by Rao et al [2] who named it as the "Dust-Acoustic Wave" (DAW), and subsequently confirmed by laboratory experiments [3]. On the other hand, dusty plasmas with sheared flows are susceptible to Kelvin-Helmholtz instabilities when the relative flow speed between adjacent layers exceeds a critical value which is typically of the order the DAW phase speed [4,5]. Dusty plasmas exhibit non-ideal behavior for grain sizes in the super-micron range. For example, for grains with a radius of about 50 microns, the volume reduction contribution is about 10% when the dust gas density is about 10^ particles/cc. Large grains are highly charged, in which case particle correlations as well as cohesive forces between the grains may be significant. Recent analysis [6,7] of ^^ E-mail : [email protected]
348
N.N. Rao /Kelvin-Helmholtz instability in weakly non-ideal magnetized dusty plasmas
DAW propagation has shown that the volume reduction enhances the DAW phase speed while the molecular cohesive forces lead to a reduction in the speed. In the present work, we analyze K - H instability in a dusty plasma by incorporating the non-ideal contributions as well as the charge fluctuation effects.
II.
FORMULATION
Consider a dusty plasma with electrons, ions and dust grains embedded in a magnetic field BQ = BQZ, Following the dusty plasma model first suggested by Rao et al [2], we assume the electrons and the ions to behave like ideal massless thermal fluids, while the wave dynamics is governed by the heavier dust component. The basic governing equations for the system are rie = rieo exp {e(j)/KBTe), n,m, [{dvjdt)
n,- = n^o exp {-e(j)/KBTi),
+ {v, • V)v,] = -n,q,
{drid/dt) + V • {ridVd) = 0, (1)
V(^ + {n,qjc){v,
X A ) - Vp„
(2)
where the notations are standard [8]. The grain charge variable q^ is determined by the charge current balance equation
(3)
{dqJdt) + {v,-V)q, = h + Ii, where the average electron (/g) and ion (7^) currents are given by [9] /e = -T^eR n^\
exp
——
,
7, = ireR UiJ
1
— ,
(4)
and -0 = qdlR is the dust surface potential relative to the plasma potential. The dust fluid is described by the van der Waals equation of state [10] (p, + An^) {l-Bn,)
= n,K,T,,
(5)
where the gas constants A and B are given hy A = d^dTc/Sric and B = l/Sric] here, the subscript 'c' denotes the respective values at the critical point. Equations (l)-(5) together with the charge neutrality condition, namely, qaUd + erii — erie = 0 form a complete set of governing equations. For the (parallel) Kelvin-Helmholtz instability, we assume in equilibrium a transverse shear in the dust flow velocity component parallel to the magnetic field and represent, Vdo = Vyy + VzZ^ where Vy and Vz are suitably calculated consistent with Eqs. (1) and (2). The latter are identically satisfied provided the equilibrium quantities Udo^ (f>o^ Pdo and Vz are functions of x only, while V^ is a constant given by Vy = {c/Bo){d(f)o/dx) + {c/ndoqdoBo){dpdo/dx)^ where the right-hand side contains —*
-•
E X B a,s well as diamagnetic drifts.
N.N. Rao /Kelvin-Helmholtz instability in weakly non-ideal magnetized dusty plasmas
349
III. DISPERSION RELATION Equations (l)-(5) together with the charge neutrality condition are Hnearized and Fourier analyzed by assuming perturbed quantities to vary as ~ exp [i {kyy + k^z — Lot)]. This yields a general dispersion relation [8]. For frequencies much larger than the grain charging frequency (ct;i), the dispersion relation approximates to D^ + iDi = 0 where Di = fuJ2^ and Dr = a;2 ( l + klpl^fi)
+ u [/ic^l + ^ ( 1 + KS)V] - kyhCl.fi
{K - S),
(6)
where uJ = LJ — k ^ v^o^ K, = k^/ky^ P z=z ky + fc^, S = {dVz/dx)/i}do characterizes the shear flow, Qdo = qdoBo/m^c, p^A = CoA/^do, /^ = (1 + ^){^ + ^), ^i = 9(3 - 7/)-2 - 9a7//4, A2 = 3/(3 - r/), Cl = C ^ + %% C^A = U;,AD is the DAW phase speed [2], u;^, - {iTrndoql/m.y/^ Vtd = {t^,Tdlm,fl\ /3 = (3/{l + /3), and (3 = V^^jC^^^, The non-ideal contributions are given by e = e^^ + e^^ with ^vr = /S^(6 - 7/)/(3 - TJY, e^f = -9a/?7//4 where a = T^Ta and 7/ = Udoln^. Note that a,7/ —> 0 for the ideal case. The quantity / = A:T:ndQR\\ is a measure of the dust grain packing, where A~^ = A~g + A~?. The diamagnetic drift frequencies are defined by u;^^ = kyVdn and u;^^ = kyVdj where
^'--LJho^ ^''--L^o'
^'^-"^"i^j
' ^^-^\-d^)
• ^^^
and p = (Aicj^y^ + A2CJ^^); expressions for the charging frequencies a;i and LJ2 are given in Ref. [8].
IV. KELVIN-HELMHOLTZ INSTABILITIES In the presence of the density and temperature gradients, the condition for the -*
excitation of the K - H instability is obtained from Drip-, fc) = 0 as
Kn/^ + KnAi + ui,\^) ^(1 + ^S)f + ^k,K^iCl,(l + klpl^^i){K - 5) < o. (8) The corresponding critical shear for the onset of the instability is given by Sc = {Cnjn,o)
[(// + ^Xi)L-'
+ I3\2L-']
[//(I + klpl^^^)] ~'^'.
(9)
The imaginary part Di gives rise to damping effects, which for weak damping yields 7 = - ( M u ; , ) [2cJ,(l + fiklpl^)
+ /iu;^„ + ^(1 + KS) ( A i o ; ! + X^LO*,^)] " ' .
(10)
To estimate the critical flow speeds for instability onset, we consider next the case when the wave frequency is much smaller than the charging frequency so that charge fluctuation damping can be neglected.
350
N.N. Rao/Kelvin-Helmholtz instability in weakly non-ideal magnetized dusty plasmas
(a)
Density Gradient K-H (DGKH) instability
For the onset of DGKH instability, the critical shear is given by
Sc = iCnjn,oLn) {6 + ^Ai) [^(1 + f8)]-^'^,
(11)
where 0 = fi + (^fi ^ 1)/^- K AT4 denotes the relative flow speed between adjacent layers over a scale length L^, that is, dVz/dx ^ AVz/L^^ DGKH instability is excited provided AT4 > (AV^)c where, in the lowest order, (Ay.)c « Co. (1 + |e + 1^) (1 + f6)-'/\
(12)
Thus, the critical flow speed for the onset of the DGKH instabihty increases due to dust thermal contribution, while denser plasmas have lesser critical speed. The contribution (e) due to the non-ideal effects can either be positive or negative depending on the relative values of a and 7/ [6].
(b)
Temperature Gradient K-H (TGKH) instability
On the other hand, for the TGKH instabihty, the critical shear is given by Sc = {X2^Cojn,oLr)
m
+ fS)]-'^'.
(13)
If AVz now denotes the relative flow speed between adjacent layers over a scale length LT, then the critical speed ( A K ) , « \,^Co^ (1 - le -1^) (1 + f8)-'f\
(14)
Comparing this expression with (12), we note that the contributions due to the non-ideal as well as thermal effects to the DGKH and TGKH instabilities are quantitatively opposite in nature.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Verheest, F., Space Sci. Rev. 77, 267 (1996). Rao, N.N., Shukla, P.K., and Yu, M.Y., Planet. Space Sci. 38, 543 (1990). Barkan, A., Merlino, R.L., and D'Angelo, N., Phys. Plasmas 2, 3563 (1995). Rawat, S.P.S. and Rao, N.N., Planet. Space. Sci. 41, 137 (1993). Singh, S.V., Rao, N.N. and Bharuthram, R., Phys. Plasmas 5, 2477 (1998). Rao, N.N., J. Plasma Phys. 59, 561 (1998). Rao, N.N., Physica Scripta T75, 179 (1998). Rao, N.N., Phys. Plasmas 6 (6), xxx (June 1999). Rao, N.N., and Shukla, P.K., Planet. Space Sci 42, 221 (1994). Joos, G., Theoretical Physics, New York : Dover, 1986, p. 497.
FRONTffiRS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and P.K. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
351
Self-similar Expansion of a Non-Ideal Unmagnetized Dusty P l a s m a R Bharuthram^ and NN Rao^ ^ML Sultan Technikon, Durban, South Africa, "^Physical Research Laboratory, Ahmedabad, India
INTRODUCTION Most of the recent studies on thermal dusty plasmas have assumed an ideal equation of state for the dust component. Such models are valid for sub-micron and micron size dust grains and dilute plasmas where the characteristic Coulomb potential energy is much less than the mean thermal energy. However, for larger grains in the super-micron range and higher grain densities, the reduced mean inter-particle distance enhances the interaction between neighbouring dust grains. Hence, such dusty plasmas are essentially strongly coupled and the effects due to the non-ideal nature of the plasma cannot be neglected [1,2]. In this paper we consider the self-similar expansion into vacuum of a non-ideal dusty plasma filling semi-infinite half-space. The plasma is unmagnetized. To simplify the complex set of equations governing the dynamics of the system, we restrict our analysis to a one dimensional expansion of a collisionless plasma in which the dust grains have constant charge. The effect of plasma parameters, such as temperatures and densities of the charged species, on the expansion profile is examined by solving the derived set of self-similar equations. Our results are compared with those obtained previously for an ideal dusty plasma [3].
THEORY Our model consists of a three-component plasma consisting of electrons, ions and non-ideal dust grains. The electrons and ions are treated as point particles, as such they are assumed to be in electrostatic equilibrium, with their densities given by the Boltzmann distribution, viz.. n j = njoexp{-ej(f)/KBTj),
(1)
where j = e{i) for the electrons(ions), ej = —e(-he) is the electron (ion) charge, Ujo the equilibrium density, Tj the temperature and (j) the electrostatic potential. The dynamics of the dust component is governed by the equation of continuity
352 R. Bharuthram, N.N. Rao /Self-similar expansion of a non-ideal unmagnetized dusty plasma 9n
d r
.
r.
(2)
the momentum conservation equation .d
d .
_
d(j>
dp
(3)
dx ' and the van der Waal's equation of state for a non-ideal dust component [2] {p + An'^){l - Bn) = nK^T,
(4)
where A = a/L'^ = 9KdTc/{Snc) and B = b/L = l/(3nc). Here a, 6 and Kd are the usual gas constants and n = L/vm-, with L is Avagadro's number and Vm is the molar volume. The system of equations is closed with the quasi-neutrality condition Tie = Hi + Zn. In the above equations, n is the dust number density, v the dust fluid velocity, p the dust pressure, T the dust temperature and Z the charge number of the dust particles. Note that Z > 0 ( < 0) for the positively (negatively) charged dust particles. For a dusty plasma occupying at initial time i = 0 the semi-infinite half-space X < 0, we consider its expansion into vacuum in the x-direction at later times. The equations governing the self-similar expansion of the plasma are obtained by letting n = n(
(5) H = ^^[ ~ "'!!!,-' + e[(l - ^n/3)-2 - (9/4)r^n] ,
(6)
(a(J + nr+')
drii
nivH
(7)
(8) and F = ZS{a + ifui'^iaS where 8 = neo/nio,a and A = JuBTilm.
+ n,-^+^)-2 + tO^^ ^ ^'"^' Ziii"^'^
= Ti/Te,T
= Tc/Td,d
'^ [3 (1 - ^n/3)3
= nio/nc,e
9
4r
= K^Td/KBTi^fi
(9) = neouf^
In addition, the normalized parameters are defined by n^ =
R. Bharuthram, N.N. Rao /Self-similar expansion of a non-ideal unmagnetized dusty plasma 353 ^t/^t'o7^e = rie/nio.h = njuio^v = vjX and p = p/ricKdTc^ with rig = 5/ni^^h = (^ -> n,"+^)/(Zn," ) and p = ( ^ n / r ) ( l - ^ n / 3 ) ' ^ - (9/8)^2n2. For real solutions, h > 0, requires n,- < (>)(J^/('^"^^) for Z > ( < ) 0 . In addition, we require H > 0.
NUMERICAL RESULTS The set of equations (5) — (9) are numerically solved for chosen values of a, ^, e, 6 and r. In each case, the equations are solved using a numerical code which employs the Runge-Kutta-Fehlberg algorithm [4], The fixed parameters are 5 = 0.1 and a = 1.0. Then for Z = - 1 and A = Po/riioTi = 0.5 of ref. 3, we find e = 0.555. Further setting r = 0.1, figures 1-3 show the behaviour of the expanding plasma. The parameter labelling the curves is 0, The curve corresponding to ^ = r = 0 in figure 1-3 represents the ideal plasma case and is in total agreement with figures 2 and 3 of ref. 3 for an isothermal plasma. Figure 4 shows the variation of the dust density with ^ for dust charge Z = —100 (corresponding to e = 55.56). The effect of the r - variation is studied for Z = — 1 in figures 5-6 by fixing ^ = 1.0. From the above figures, it is seen that a non-ideal plasma expands over a much larger distance than the isothermal ideal plasma. It is seen that the variation of 9 (figures 1-4) has a much more significant effect on the plasma expansion than that of r (figures 5-6). As 9 is increased from 0 to 3 the expansion distance (when h drops to zero) trebles for Z = — 1. A similar behaviour is seen in figure 4 for 9 increasing from zero to 300. A comparison of figures 2 and 4 shows that a plasma with highly charged dust grains expands over a significantly larger distance before the dust density drops to zero (corresponding to (^ = 15 for Z = —1,9 = 3 and ^ = 150 for Z = —100,^ = 300). It is found that the range of paramieters for which real solutions are possible is primarily restricted by the requirement H > 0, For example, for Z = — 1, when ^ = 1.0, the upper limit of r for jff > 0 is found to be r = 1.0. When we have ^ = 0.1, the threshold is r = 9.0.
REFERENCES 1. Fortov, VE and lakubov, IT, Physics of Nonideal Plasmas (Hemisphere, New York, 1990). 2. Rao NN, Linear and non-linear dust acoustic waves in non-ideal dusty plasmas, J. Plasma Physics, 59, 561-574 (1998). 3. Bharuthram R and Rao NN, Self-similar expansion of a warm dusty plasma - I. Unmagnetized case. Planet Space Sci,, 43, 1079-1085 (1995). 4. Burden, RL and Faires, JD, Numerical Analysis (3rd Edn, Chap.5, p.230, Prindle, Weber and Schmidt, Boston, 1985).
354 R. Bharuthram, N.N. Rao /Self-similar expansion of a non-ideal unmagnetized dusty plasma
1.2 r 1.0
,,,,,.,-
P^\ \\
- u \
0.4
-
0.2
~
0.0
•
°e -
0.8 0.6
1
0«0/0-l
3.0
-
^
_
\ \ ^'^
-
NOST—2.0
:^^
on 1
-
-^Tn
-
1
10
15 Fig. 2: Dust density n versus ^ for different 0 values (Z = -1,T = 0.1)
Fig. 1: n^ and iij as a function of % for different 6 values (Z =-1,1 = 0.1) 0.010
0.002
150 Fig. 4: Dust density n versus ^ for different 0 values (Z =-100,1 = 0.1)
Fig. 3: Dust fluid velocity v versus % for different 0values(Z = -l,T = O.l)
0.004
0.000
Fig. 5: Dg, Oj and n as a function of % for different T values (Z = -1,6=1.0)
20
40
Fig. 6: Dust density n versus ^ for different i values (Z =-100,0=1.0)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
355
Computer Modeling of Non-linear Sheaths with Dust Particles Yuriy I. Chutov*, Olexandr Yu. Kravchenko*, Pieter P.J.M. Schram^, and Volodymyr S. Yakovetsky* ^Faculty of Radio Physics, Taras Shevchenko Kiev University, Volodymyrs 'ka Str. 64, 252017 Kiev, Ukraine "Department of Physics, Eindhoven University of Technology, The Netherlands
Abstract. Computer modeling of sheaths with dust particles performed by means of the modified PIC method with self-consistent dust particle charge as well as electron and ion velocity distribution functions, show that dust particles can strongly influence the properties of these sheaths. This influence is caused by a selective collection and a scattering of electrons and ions by dust particles with self-consistent electric charge. That causes a change of both the well known Bohm's criterion on the sheath boundary and the other sheath parameter distributions.
INTRODUCTION The non-linear electrostatic sheaths are always existing on cold walls with which the plasma is in contact (1). These sheaths determine an interaction of plasmas with walls including heat and charged particle flows from the plasmas to the walls. Intensive investigations of electrostatic sheaths have been carried out during recent years (2-5), especially with respect to sheaths in magnetic fields. The well known Bohm criterion (1) is valid on the coUisionless sheath boundary providing a continuous change of plasma parameters in positively charged sheaths. However, this criterion should be replaced in case of collisional sheaths by the Person's criterion (3), which allows the inverse relation between the velocities mentioned. In many important practical cases, including plasma processing (6), dust particles can be created close to a surface of condensed matter and therefore can be located in sheaths. These particles can strongly influence various plasma properties (7,8). Of course, these dust particles have to influence on the properties of the electrostatic sheath due to both a collection and a scattering of electrons and ions, which penetrate into the sheath from the plasma. However, theoretical investigations of sheaths with dust particles (5) do not take into account an influence of electron and ion collection by dust particles on properties of the electrostatic sheaths. The aim of this work is to study non-linear sheaths in plasmas with dust particles using the computer modeling with self-consistent velocity distribution functions of electrons and ions in order to investigate the influence of these particles on sheaths.
356
YI. Chutov et ah /Computer modeling of non-linear sheaths with dust particles
COMPUTER MODEL One-dimensional slab plasma consisting of equilibrium electrons and ions with densities n^^ = ^to - ^o ^^^ temperatures Teo and Tto creates an equilibrium sheath in front of an electrode to which a large negative potential (f) is applied. According to Bohm's sheath criterion (1), the ion drift velocity u^ has to satisfy the boundary condition u^ ^{kT^ I Mf'^ close to the sheath boundary where M is the ion mass. Dust particles with a density N^ = A^^^ exp(-x^ / x]) and a radius R^ appear in this sheath at some initial time and both collection and scattering of electrons and ions by these dust particles starts here. These processes cause an evolution of the sheath to a new steady state. The PIC method is used for computer modeling of sheaths, taking into account the dynamics of the dust particle charge in plasmas with self-consistent energy distribution fixnctions of electrons and ions (9). The Coulomb scattering of electrons and ions is taken into account in the framework of the Monte-Carlo method with 3D velocity distribution fimctions for electrons and ions.
RESULTS Results obtained at various parameters of dust particles, show that the influence of dust particles on sheaths strongly depends on relations between some characteristic times, namely: the time r^ of an ion penetration through a sheath, the ion collection time T^, and the ion scattering time r^. Of course, this influence is very small in the case Tp « T^+T^. In the opposite case, r^ « T^+T^, dust particles create a barrier between the plasma and the electrode due to a strong collection and scattering of electrons and ions by dust particles, so that electrons and ions cannot reach the electrode at all. In this last case, there is no space electric charge close to the electrode and the sheath shifts from this electrode inside the plasma space. Of course, the most interesting phenomena are realized in the case T^ - r^ ^ r^ when electrons and ions can reach the electrode but their collection and scattering are still effective. Some typical results of computer simulations are shown for this case in Fig. 1 - 3 for N^^ =1,R^ = 0.1, x^=16A^, = -10 where A^^ is the number of dust particles in a Debye cube, i?^ is the radius of a dust particle divided by the initial Debye length A ^= (kT^ 147myy^. In these figures, spatial distributions of various plasma parameters are plotted for various times t after the start of the evolution from initial equilibrium distributions due to the appearance of dust particles in the sheath at Q)p^t = 5. Here, the spatial coordinate x is divided by the initial Debye length /I ^, the time t is multiplied by the initial ion plasma frequency CD^^ =(47m^e^ I M), and the potential ^ is divided by the characteristic value (p^=kT^/e, As can be seen from these figures, all plasma parameters change essentially due to the influence of dust particles in a sheath. This influence is especially clear from the evolution of the ion drift flux /. shown in Fig. la. Initially, this flux is practically
Y.I. Chutov et al./Computer modeling of non-linear sheaths with dust particles 357
J/J
J/Jo 1,0
uofr
a ^pit
0,5
-
\...//'y' ••••.••••
/
/
/
/
20
f' /
O
10
0,5
130
^"^'"'^X.
0,0
10
20
30
0,0
100
200
FIGURE 1. a) Spatial distributions of the ion flux / • in a sheath with dust particles for various times t after the appearance of dust particles in a sheath where CO p^ is the initial ion plasmafrequency,A ^ is the initial Debye length, j ^ = nJJiT^^ I M) is the Bohm's flux, b) A temporal evolution of the electric current J in an external cu-cuit after the appearance of dust particles m a sheath. Here J^ is the initial current.
uniform because it corresponds to the case of a usual steady-state electrostatic sheath without dust particles (1). However, dust particles decrease this flux and cause its essential heterogeneity due to different cross-sections of collection and scattering of ions in different points of the sheath due to a dependence of these cross-sections on ion energy. Corresponding changes of the electron drift flux take place also. An evolution of electron and ion drift fluxes in a sheath causes corresponding changes of the electric current in an external circuit shown in Fig. lb, where d is the sheath size. It is possible to note in Fig. la and Fig. lb that a new steady state sheath is formed during about 25 ion plasma periods. The main properties of this new sheath are both the non-uniform distribution of the ion drift flux in the sheath and a boundary condition for this flux which is different from the well known Bohm's sheath criterion (1) due to the influence of dust particles. The sheath evolution is accompanied by a change of distributions of ion (n^) and electron ( « J densities in the sheath. Some increase of these densities takes place close to the boundary of sheath and plasma which is caused by a scattering of the ion flux by dust particles close to this boundary. Therefore the ion flux is changed here according to condition of a continuous change of plasma parameters and does not correspond to an initial ion flux through the boundary of a sheath without dust particles from a non-disturbed plasma (1). Of course, distributions of the electric potential , shown in Fig. 2a, evolve according to the evolution of electron and ion densities. These distributions can be non-monotonous during their evolution times, but the final distribution is monotonous always. Note the Child-Langmuir law (1) is not valid for sheaths with dust particles because the ion flux is not conserved in this case, as can be seen in Fig. la.
358
Y.I. Chutov et al. /Computer modeling of non-linear sheaths with dust particles
¥<^a
qy^o
o h
-5
0,5
0,0
h
-0,5 h 10
10
20
10
X/XH
20
FIGURE 2. Spatial distributions of the self-consistent electric potential ^ (a) and the dust particle charge Qd (b) in a sheath with dust particles for various times t after the appearance of dust particles in a sheath. Here n^ and Qo are the plasma density and the plasma ion charge in a Debye cube, respectively.
As can be seen in Fig. 2b, spatial distributions of the dust particle charge q^ are strongly non-uniform so that some dust particles even change the charge sign. Positively charged dust particles have to move to the electrode according to this selfconsistent electric field. Therefore it can be thought that dust particles produced by an electrode have to return due to an electrostatic sheath influence w^hich is always present at an electric current to this electrode. ACKNOWLEDGMENTS This work was partially supported by INTAS (Contract No 96-0617) and by a grant from the Ukrainian Committee of Science and Technology.
REFERENCES 1. Chen, F. F., Introduction to plasma physics, New York and London: Plenum Press, 1985. Beilis, I.I., and Keider M., "Sheath and presheath structure in the plasma-wall transition layer in an oblique magneticfield"Phys. Plasmas 5(5) 1545-1553 (1998). Chen, X. P., "Sheath criterion and boundary conditions for an electrostatic sheath" Phys.Plasmas 5(3)804-807(1998). Cohen R. H., and Ryutov D. D., "Particle trajectories in a sheath in a strongly tilted magnetic field" Phys. Plasmas 5(3) 808-817 (1998). 5. Ma J. X., and Yu M. Y., "Electrostatic sheath of a dusty plasma" Phys. Plasmas 2(4) 1343- (1995). 6. G. M. W. Kroesen, "Dusty plasmas: industrial application" m Advances in Dusty Plasmas (Eds.: P.K.Shukla, D.A.Mendis and T.Desai), Singapure, New Jersy, London, Hong Kong: World Scientific, 1997, pp. 365-376. 7. Chutov, Yu. L, Kravchenko, A. Yu., and Schram, P., "Expansion of a bounded plasma with dust particles" J. Plasma Physics. 55 (part 1), 87-94 (1996). 8. Chutov, Yu.L, Kravchenko, A.Yu., and Schram, P., "Evolution of an expanding plasma with dust particles" Physica B128, 11-20 (1996). 2.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
359
The gravitational effect on negatively-charged dust-grains in a plasma H. Yamaguchi*, Y.N. Nejoh^ and N. Mizuno* * Department of Physics, College of Humanities and Sciences, Nihon University, Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan ^ Graduate School of Engineering and Departm.ent of Electronic Engineering, Hachinohe Institute of Tex:hnohgy, Myo-Obiraki, Hachinohe, Aomori 031-8501, Japan
A b s t r a c t . The effect of the gravitational force on dust grains in a dusty plasma is studied. The result of simulation shows that the terrestrial gravitation affects not only the density and fluid velocity of dust grains but the dust-charge. This is interpreted by the comparison between the Debye length and the average intergrain distance.
INTRODUCTION It has been found that the instability due to the external gravitation might be observed in interplanetary space and laboratory experiments. In fact, the particle motion is governed by the gravitational and electromagnetic forces. In actual situations, the gravitational effect is considered to be significant wrhen a fall of dust grains causes the damage of the device in engineering-aided plasmas. Since not many studies on the gravitational effect have been considered in dusty plasmas, we have investigated its effect on negatively-chargai dust grains in a dusty plasma. In our previous work [1], we demonstrated temporal and spatial evolution of the dust-charge and clarified the charging process of micron-sized dust grains in plasmas. Since we did not treat the motion of dust grains in the previous study, we consider it and the effects of external forces in the present paper. This work is useful for understanding the effect of the terrestrial gravitation on dust grains, and also will be applied to study on the effects of other external forces and the interaction between electrostatic waves and dust grains in dusty plasmas.
MODEL We consider a three component plasma consisting of electrons, ions, and dust grains, and assume that electrons and ions have Maxwell distributions with constant
360
H. Yamaguchi et al /Gravitational effect on negatively-charged dust-grains in a plasma
temperatures. In order to describe the behavior of dust grains, we use the continuity equation and the equation of motion; diiA
at
d ,
_
+ _(„,„,) = 0,
ox
m^n^ ox
(1)
m^
where rid, Vd^ rrid, Td, and Qd represent the density, fluid velocity, mass, temperature, and charge of dust grains, respectively, t is time, and x is the space coordinate parallel to the gravitational acceleration g. We also assume that the charging of dust particle arises from plasma current due to electrons and ions reaching its surface, and that other charging effects are negligible. In this case, the dust-charge Qd is determined by the charge current balance equation [2];
where I^ and I\ denote the electron and ion currents [3,4]. In order to consider the effect of charged dust grains on the electron density, we use the condition of charge neutrality; ne = n i - l^dlrid,
(4)
where n^ and n^ represent electrons and ions density, respectively. Z^ (= Qd/^) is the charge number of dust particles measured in units of electron charge e. According to the experimental result in the dusty plasma device [5,6], we employ the following assumptions: (a) dust grains begin to fall from the vacuum region (x < 0), (b) at the vacuum-plasma boundary (x = 0), dust grains are supplied continually with rid = ^^do ^nd Vd — ^do ^nd the dust-charge is zero (Zd = 0), (c) the ion density is constant (rii = no) and the temperatures of ion and electron are also constant in the plasma region {0 <x < Xmax). The set of equations is solved by the finite difference method over the domains 0 < X < Xmax- At the bottom of the plasma column {x = Xmax), the natural boundary condition is employed in the present model.
RESULT AND DISCUSSION We assume the input parameters as shown in Table 1, which are typical results of experiments in the dusty plasma device [5]. Figure 1 illustrates spatial evolution of (a) the dust to plasma density ratio n^/riQ, (b) the ratio of dust fluid velocity to its initial value t'd/'^do? (c) the dust-charge number |Zd|, and (d) the electron
H. Yamaguchi et al /Gravitational effect on negatively-charged dust-grains in a plasma
361
TABLE 1. A summary of the input parameters. (Potassium plasma) Plasma temperature Plasma density Ion mass (Dust particle) Density Mass radius
T(eV) no (cm ^) "M (g) nd (cin~^) "Id (g)
r (cm)
0.2 10^ ^ 10^° 6.5 X 10-23
5 x 10^ 10-^ 5 X 10-^
density normalized by the plasma density ne/rio, when no = 10^ cm"*^, t^do = 40 cm/s, Td = 0, and E — 0. In this figure, the results in the absence and presence of the gravitational force in the plasma region {0 < x < Xmax) are plotted by solid and broken lines, respectively. This figure indicates as follows: while the dust density n^ and the dust fluid velocity v^^ do not change in the absence of the gravitational force (g = 0), rid decreases with increasing v^ when g ^ 0. If g' = 0 in the plasma region, the dust charge number |Zd| builds up rapidly within a very small region near the top of the plasma column, and |Zd| reaches its equilibrium value {\Z^\ = 1701). On the other hand, in the presence of the gravitational force {g ^ 0), \Z^\ increases through the plasma region and |^d| = 2380 at Xmax ( = 4cm). The spatial evolution of electron density is determined by |Zd| and rid as described in Eq-(4) Firstly, we discuss the behavior of dust grains. In the absence of the gravitational force, since external forces expressed in the right hand side of Eq.(2) do not act on dust grains, the dust fluid velocity keeps its initial value. On the other hand, since dust grains are accelerated by the gravitational force when p 7^ 0, the fluid velocity of dust grains increases. From the continuity equation, since we derive that ridt'd is constant at the steady state {d/dt = 0), the dust density remains constant if 5^ = 0 and decreases with increasing v^ when g ^ 0^ Secondly, we interpret the spatial evolution of the dust-charge by the comparison between the average intergrain distance, d = n ^ , and the Debye length. AD = (r/47rnoe2)V2 ^ 3.3 x lO'^ cm. If 5 = 0, d (= 2.7 x 10"^ cm) becomes constant, because the dust density is constant in the plasma region. Thus, the dust-charge does not change in the plasma region after it reaches the equilibrium value. On the other hand, when g' 7^ 0, the average intergrain distance increases with increasing X, because the density of dust grains decreases as shown in Fig. 1 (a). In this case, we estimate that d ( = 2.7 x 10"^ cm) < AD at the top of the plasma column (x = 0), and d {= 3.6 x 10"^ cm) > AD at the bottom of the plasma column {x — Xmax)Since the average intergrain distance changes near the Debye length, \Z^\ changes over the plasma region. Even though the dust density decreases due to the effect of gravitational force, we expect that the dust-charge does not change in the plasma region if we can treat dust grains as isolated particles {d ^ AD)- Although it is not presented here, we confirm this prediction by changing the plasma density, and that |Zd| at equilibrium is coincident with the isolated grain value.
362
H. Yamaguchi et al. /Gravitational effect on negatively-charged dust-grains in a plasma
5.0x10*
(a)
(c) 20001-
2.5x10
1000 ^
0.0
n /m (d)
(b)
e C 1.0 \
0.5
2
0.0
2
X (cm) FIGURE 1. Spatial evolution of (a) the dust to plasma density ratio na/no, (b) the ratio of dust fluid velocity to its initial value Vd/^do, (c) the dust-charge number |Zd|, and (d) the electron density normalized by the plasma density ne/no, when no = 10^ cm~^, V^Q — 40 cm/s, T^ — 0, and £• = 0. Solid and broken lines represent the results in the absence and presence of the gravitational force in the plasma region, respec:tively. X (cm)
Since we also study the effects of other external forces such as the external electric field, the^e results will be presented in the conference. In future investigations, we will study the interaction between electrostatic waves and dust grains in dusty plasmas by applying the present work. In conclusion, we have demonstrated the spatial evolution of the density, fluid velocity, charge of dust grains in a dusty plasma. From this result, it is found that not only the density and fluid velocity of dust grains but also the dust-charge are affected by the gravitational force. We explained the result of simulation by the comparison between the Debye length and the average intergrain distance. This work was s u p p o r t a l by the Joint Research Program of the National Institute for Fusion Science.
REFERENCES 1. 2. 3. 4. 5. 6.
H.Yamaguchi and Y.-N.Nejoh, Phys. Plasmas 6, 1048 (1998). F.Melandso, F.T.Askalsen, and O.Havnes, Planet Space ScL 4 1 , 312 (1993). J.R.HiU, and D.A.Mendis, Moon and Planets 2 1 , 3 (1979). E.C.Whipple, Rep. Prog. Phys. 44, 1197 (1981). A.Barkan, N.D'Angelo, and R.L.MerUno, Phys. Rev. Lett. 73, 3093 (1994). W.Xu, B.Song, R.L.MerUno, and N.D'Angelo, Rev. Sci. Instrum. 63, 5266 (1992).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
363
Stationary equilibria of dusty plasmas R. T. Faria Jr., P. H. Sakanaka Institute de Fisica "Gleb Wataghin", Universidade Estadual de Campinas, 13083-970 Campinas, Sao Paulo, Brazil, T. Farid, and P. K. Shukla Institut fiir Theoretische Physik IV, Ruhr-Universitat Bochum D-44780 Bochum, Germany Abstract Our objective is to present an investigation of self-consistent stationary equilibria of multi-component dusty plasmas. For this purpose, we employ Hamiltonian and guiding center models to obtain the profiles for the plasma number density, the plasma flows and currents as well as magnetic and electric fields, by choosing appropriate distribuction functions for the dusty plasma particles. We assume that during the charging process, most of the background plasma electrons are attached onto the dust grain surface. The coupled nonlinear equations are solved numerically to obtain self-consitent solutions which may include a vortex in which the centrifugal force of the dust grains (or ions) is balanced by the pressure gradient force. The relevance of our investigation to stationary nonhnear equilibria of low-temperature laboratory dusty plasmas has been pointed out. PACS: 52.25.Vy, 52.35.Nx, 52.55.Dy
1
Introduction
New types of dusty plasma waves and vortex structures have been observed[l, 2] in low-temperature space and laboratory dusty plasmas. Specifically, recent laboratory observations have conclusively demonstrated the formation of dusty plasma vortices without and with magnetic fields. It is shown that when the equilibrium ion pressure gradient and the gradient of the dust charge density ZdTid are nonparallel to each other, we have the possibility of spontaneous generation of magnetic fields in dusty plasmas. We discuss a class of self-consistent dusty plasma equilibria. In order to understand these equilibria in electromagnetic fields, we have adopted the Hamiltonian and guiding center approaches for the plasma particles and have constructed appropriate distribution functions.
2
Equations
We use a multi-component dusty plasma: ions, electrons, and negatively charged dust particulates. In the unperturbed state, we have the charge neutrality riio — ^eo + ^d'^dO' The dust grains attaches most of the electrons creating a global quasineutrahty condition n^o ~ 2'^^do- Then it is possible to consider the dusty plasma
R.T. Faria Jr. et al. /Stationary equilibria of dusty plasmas
364
as a two-component system. The evolution of the magnetic field is predicted by Faraday's law, as in ([3]) ^ (l - A^^V^) B = - T ^ V f t X V(Z,n,) + V x (v, x B) eZjnl dt + — V X Vi X V X Vd ^V^B e \ rui
+^ ^ V rud
X [(V X B) X B] ,
(1)
where v^ is obtained from Ampere's law n^Vi = ZdUdyd + (l/^/^o)V x B, v^ is the dust fluid velocity, A^ = c/ujpd is the dust skin depth, ujpd — {'Zj\e^ndjvfid^^^^'^ the dust plasma frequency, and A^ the ion skin depth. In order to justify equation (1), we follow Ref. [4] and present a class of stationary dusty plasma equilibria in which the plasma density, the electric potential, the plasma current densities, and the magnetic fields are related in a nonlinear way. The dust particulate dynamics in the electromagnetic fields is governed by the Hamiltonian ^•^ = ^ f ( ^ + ^
7^ exp
fd =
where Td is the dust temperature and a^ is a constant. We have for the number density of the dust, nd{R)^ and the 6 component of the current density Jd{R) rid
I fd
,
T
Zdo^ido
[VR^A^ - Z^ecj))
exp
(4)
where T = aaZj e^/[2md (1 + adR'^)] and R0fd IMiJd = fJ'oe I —^dpndpe
_
=
adR^A -±,[VR'A'-Zdecf) c" (l + adi22)3/2 exp
to: '^pd
5)
We use the guiding-center Hamiltonian for the ions because we understand that the Larmor radius of singly charged positive ions is much smaller than the scale size, and therefore, Hi — fiujci + ecf), where /i (== miv\/2uci) is the magnetic moment. The canonical variables are di.s ii^ 9 =^ J Uddt^ R, and 6. The ion distribution function is
f,{^^,e,R,e)^fdH,,R)
=
'^exp
{fiuci + e(f))+ g (R) •'•I
Here, g (R) acts in the function at R larger.
(6)
R.T. Faria Jr. et al./Stationary equilibria of dusty plasmas
365
Following the reference ([3]) we have for the Ampere's law
de
ao
1 + aoC
,2
{u — Ad) exp
1 + dQi
(7)
where r ^ Rupd/c, AQ = sj{l + r]d)2mdTd/r]d/Zde, ao = c^ctdl^ld ^^d introduce the symbols A* = A/AQ, ^ = r^ A^ = Updy]ridTdl2md{l + r]d)/{uJcdcZd), and u = rA\ We have integrated equation (7) radially inwards on a logarithm scale from log ^ = 10.6, by choosing typical parameters that are relevant to the ionospheric dusty plasma. Accordingly, we have taken rid ^ 5 x 10^ cm"^; negatively charged grains have Zd ~ 1000. Moreover, we have taken rrid ~ 10"^"^ g; Td ~ 300 K] Ti - 0.2 eV, Te ~ 2 eV, and a^ = 10"^ m-\ Figure 1 shows the stationary nonhnear solution of the equation (7) depicting the z-component of the magnetic field generated by the dusty plasma; the central magnetic field strength is 7.5 mG and the profile radius is about 7 km. Figure 2 shows the normalized profiles of the ion number density, the electric potential, the plasma current density, and the magnetic potential times radius in a dusty plasma; their respective maxima are: ion number density = 10^^ m~^, electric potential = 2.82 eV, current density = 2.18 x 10"^^ A/m^ and RAmin = ~17.8 x lO""^ T. If the ions were singly ionized nitrogen then the ion Larmour radius would be 322 m.
Figure 1: The stationary nonlinear solution of the equation (7) depicking the zcomponent of the magnetic field generated by the dusty plasma; the central magnetic field strength is 7.5 mG and the profile radius is about 7 km.
3
Conclusions
To summarize, we have presented a class of stationary nonlinear dusty plasma equihbria in which the plasma number density, the electric potential, the plasma current density, and the magnetic field are connected in a specific fashion. This problem has been addressed by employing the Hamiltonian and guiding center models and by choosing appropriate dust and ion distribution functions that yield the desired form for a stationary dust vortex which is required for the maintenance of the magnetic
366
R.T. Faria Jr. et al / Stationary equilibria of dusty plasmas
fields that could be created by the barocUnic vector in a dusty plasma. The results of our investigation should be useful in understanding not only the dusty laboratory device equilibrium, but also the equilibrium of dusty stars and other astrophysical objects.
1.0
A
/
J/
0.8 0.6 0.4 0.2
0
\
\
/
\
\
/ / 1/1
\ n 1 ^ r^ 1 ^#* "LI
A v^
*••y yJ /
\
y
V •'•
-••*
A
\\ \
!
\\ 10
VX ^
15
R[km]'
Figure 2: The normahzed profiles of the ion number density, the electric potential, the plasma current density, and the magnetic potential times radius in a dusty plasma. The scale for the u-profile is to start with — 1 at JR = 3 and ends with 0 at i? = 20. Acknowledgments: This work is partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 191, the Deutscher Akademischer Austauschdienst (DAAD) and the Brazihan Agencies, Fundagao Coordenagao de Aperfeigoamento de Pessoal de Nivel Superior (CAPES) and Fundagao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP). P. K. Shukla acknowledges the support of International Space Science Institute (ISSI) at Bern through the project "Dust Plasma Interaction in Space". He also thanks Professor Bengt Hultqvist for the warm hospitahty at ISSI, where a part of this was carried out.
References [1] A. Barkan, R. L. Merlino, and N. DAngelo, Phys. Plasmas 2, 3563 (1995). [2] H. Fujiyama, S. C. Yang, Y. Maemura ei a/., in Douhlt Layers: Potential Formation and Related Nonlinear Phenomena in Plasmas, edited by Sendai (World Scientific, Singapore, 1997). [3] R. T. Faria Jr., Tahir Farid, P. K. Shukla, and P. H. Sakanaka, Phys. Plasmas 6, 2950 (1999). [4] A. Hasegawa, M. Y. Yu, P. K. Shukla, and K. H. Spatschek, Phys. Rev. Lett. 41, 1656 (1978)] ibid. 42, 412 (1979).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
367
Photoelectric Charging of Dust Particles A. Sickafoose, J. Colwell, M. Horanyi, S. Robertson Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80309-0392
B. Walch Department of Physics University of Northern Colorado, Greeley, CO 80639
A b s t r a c t . Laboratory experiments have been performed on the photoelectric charging of dust particles which are either isolated or adjacent to a surface that is also a photoemitter. We find that zinc dust charges to a positive potential of a few volts when isolated in vacuum and that it charges to a negative potential of a few volts when passed by a photoemitting surface. The illumination is an arc lamp emitting wavelengths longer than 200 nm and the emitting surface is a zirconium foil.
INTRODUCTION Natural and man-made objects in space charge to a floating potential determined by the local plasma environment [1]. The dominant charging currents are the flux of electrons and ions from ambient plasma, electrons created by secondary emission and photoelectrons. An object floats at the potential at which the sum of the currents is zero. The charging v^ithin planetary magnetospheres is usually determined by the flux of magnetically trapped charged particles and secondaries. In interplanetary space and at geosynchronous orbit, on the other hand, the charging is usually dominated by photoelectric emission. In this case there is a positive floating potential at which nearly all of the photoelectrons are returned to the surface. This situation is altered only slightly by the small flux of solar wind particles. For objects of centimeter scale and larger, the local plasma environment is dominated by photoelectrons. Dust on larger objects such as the Moon [2] or asteroids [3,4] may be charged, levitated and transported by the sheath of the parent object. We have constructed an experiment to investigate photoelectric charging of isolated grains and those near surfaces. Experimental work on the photoelectric effect has been focused upon determination of the photoelectron energy distribution function through the use of a retarding potential and upon measurement of photoelectric yields [5]. For example, photoelectron yields as a function of wavelength have been determined for spacecraft materials [6]. The existence of a photoelectron sheath
368
A. Sickafoose et al./Photoelectric charging of dust particles
has been inferred from measurements made on spacecraft. Electron energy analyzers have observed low energy photoelectrons which originate from the spacecraft and are returned to the surface by the sheath potential [7]. These analyzers also see ambient plasma particles accelerated toward the spacecraft by the spacecraft charging potential [8]. The electron sheath was first analyzed theoretically for a thermionically emitting surface such as the cathode of a vacuum tube [9]. The emitted electrons in this case have a Maxwell-Boltzmann distribution with an energy of order 0.1 eV determined by the cathode temperature. These may create a potential well near the surface of a few tenths of an eV which would be difficult to detect experimentally. For the photoelectron sheath, however, the energy distribution has a width of several eV and the electrons have a well-defined high energy cutoff determined by the diflFerence between the work function of the material and the short-wavelength cutoflF of the spectrum. The sheath potential profile is found by solving simultaneously Poisson's equation and the Vlasov equation. For thermionic emission, the potential approaches the potential at infinity asymptotically, however, for the photoelectron sheath this potential is reached in a finite distance as a consequence of the finite spread in the distribution function. Solutions for the photoelectron sheath have been given for several model distribution functions [10, 11, 12].
EXPERIMENTAL SETUP AND RESULTS The experiments are performed in a device (Fig. 1) used previously for measurements of dust charging in plasma [13,14,15]. This device consists of two aluminum cylinders 30 cm in diameter and 30 cm long placed end to end. Experiments are carried out in one of the two sections. The chamber is pumped by a diffusion pump to a base pressure of 4x10"^ Torr.
Dust dropper
XE
Photoemitter Lens
1^'<='''"P
V
Faraday cup
n
Anode grid
FIGURE 1. Schematic diagram of the experiment. The dust (dotted arrow) falls from a dropper at the top of the vacuum chamber then falls past a photoemitting surface. The charge on the dust is measured by a Faraday cup below the chamber. The photoemitter and the anode grid may be removed to determine the photoelectric charging of isolated grains.
369
A. Sickafoose et al./Photoelectric charging of dust particles
Photoelectric emission is induced by a 1 kW Hg-Xe arc lamp. The light is collimated by a lens and directed through a window into the vacuum system. The collection efficiency of the optics is such that about 10 % of the lamp emission falls upon the photoemitter. All optical components are of quartz so that wavelengths down to 200 nm are passed. Approximately 1.4 % of the lamp spectrum is in the wavelength band 200-250 nm. The photocathode is an electrically isolated 12.5 cm diameter zirconium foil disc. Zirconium and hafnium have the lowest work functions ('^4 eV) of elements that are neither radioactive nor reactive in air [16]. Zirconium was found to give a slightly higher photoelectric yield with up to 20 /xA being obtained. For some experiments, an anode grid of nickel wires 15x15 cm^ is placed parallel to the photoemitter. The energy distribution of the photoelectrons is determined by a retarding potential analysis. All potential measurements are made relative to a copper grounding strap. The potential within the grounded aluminum chamber differs from this ground due to contact potentials. The potential in the vacuum adjacent to the grounded photoemissive surface, for example, is about one volt positive relative to the potentials adjacent to other surfaces because the photocathode has a lower work function. For the retarding potential analysis, the photocathode is swept in voltage and the emitted current is measured. The anode mesh is spaced 2.5 cm from the photocathode and is held at -4.5 volts so that any low energy electrons from beyond the mesh are accelerated toward the wall. The foil is illuminated in a central region 8 cm in diameter to minimize electron losses from the edge. Figure 2a shows the photocathode current as a function of bias potential. For this measurement, the photoemission is reduced to 2 fxA to reduce space-charge effects which would alter the measurement. At a photocathode potential 1 volt more negative than the anode nearly all of the emitted electrons pass to the anode. At a potential 1 volt more positive, nearly all of the electrons are returned. This point would occur at a potential two volts more positive if there were no contact
-6
-5
-4
Bias potential (V)
-6
-5
-4
-3
-2
Voltage [arb. zero]
F I G U R E 2. a) (left) Current emitted by the zirconium photoemitter (in microamps) as a function of its bias potential. The anode is at -4.5 volts, b) (right) The electron energy distribution obtained from the derivative of the curve in a). The low energy end of the distribution is on the left side.
370
A. Sickafoose et al /Photoelectric charging of dust particles
potential. Figure 2b shows the derivative of the retarding potential curve which gives the distribution of electron energies perpendicular to the surface. The full width of the curve is approximately 2 eV which indicates a mean electron energy of approximately 1 eV. A maximum electron energy of 2.15 eV is expected from the difference between the short wavelength cutoff in the spectrum of illumination (6.2 eV) and the work function of zirconium (4.05 eV). The density of photoelectrons above the surface can be estimated using the emitted electron current density and the mean electron velocity. If we assume an effective emission area of 10 cm in diameter the emission is 2.5x10"^ A/m^. If we further assume a mean electron velocity of 6x10^ m/s corresponding to an energy of 1 eV, the density of photoelectrons above the photoemitter is 2.6x10^ cm~^. This value is doubled if the surface is biased relative to the surroundings so that electrons are returned. The dust particles are of powdered zinc that is sieved to obtain particles with diameters of 53-63 microns. In the absence of the photoemitting surface, these particles become charged positively by their own photoemission. Experiments with a zinc surface in the place of the zirconium surface indicate that the emission from zinc is 10% of that from zirconium or 2.5x10"^ A/m^. The photoelectric work function of zinc (4.3 eV) is near that of zirconium so we expect that the zinc will charge positively by two volts. The capacitance of the grains (3.2xl0~^^ F) is such that we expect a charge of 2.0x10^ electrons per volt or approximately 4 xlO^ electrons. The time for the zinc grains to charge can be calculated from the expected emission current and the capacitance and is approximately 0.01 seconds, which is short in comparison with the time the dust is within the beam (0.1 sec). The particles are dropped through the illumination beam and captured in the Faraday cup. The signal from the Faraday cup has been calibrated to yield charge
u s. o
Q/10A6|e|
FIGURE 3. a) (right) Histogram of charge (in number of elementary charges) for 150 zinc dust particles dropped through the illumination beam with the photocathode removed, b) (left) Histogram of charge with the photocathode present and illuminated. Positive charge is a positive number for these graphs and the dust diameters are 53-63 microns.
A. Sickafoose et al /Photoelectric charging of dust particles
371
as a function of pulse height. The detected charge is sorted into bins with a width of 2 xlO^ electrons (Fig. 3a). The two histogram bins adjacent to zero cannot be used because of false triggers from the circuit noise. In the absence of illumination, dropped dust results in no triggers. With illumination and no photoemitting plate, each of 150 dropped particles has a positive charge with the bin from 4 x 10^ to 6 X 10^ having the most particles. The larger charge on some particles is probably a result of the circuit noise adding to the true signal. A careful statistical analysis is required to remove this effect or a better detection system. Data for grains dropped past the photoemitting plate are shown in Figure 3b. In this case each of 150 particles charges negatively with the greatest number of charges being between 4x10^ and 6x10^. The current density above the zirconium surface is about ten times larger than the current density emitted from the zinc surface, thus the charging is dominated by electron collection rather than by photoemission from the grains. Again the data are affected by circuit noise. We are encouraged by the approximate match between our experimental results and the theory.
FUTURE EXPERIMENTS We plan to continue these experiments using materials of space research interest. We will use simulated as well as real lunar Apollo 17 samples to understand the charging and dynamics of levitated dust clouds on the Moon. We also plan to conduct experiments using simulated Mars regolith. We are currently building a new experimental chamber where the photoemitting surface will be horizontal so that the sheath electric field can balance gravity on the charged dust particles. The properties of the sheath will be studied as function of the dust density and size distribution. This setup is the laboratory analog for future space station experiments to study dusty plasma sheaths in microgravity environments.
ACKNOWLEDGMENTS The authors acknowledge support from the National Aeronautics and Space Administration and the Department of Energy.
372
A. Sickafoose et al /Photoelectric charging of dust particles
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
E. C. Whipple, Rep. Prog. Phys. 44, 1197 (1981). S. F. Singer and E. H. Walker, Icarus 1, 112 (1962). P. Lee, Icarus 124, 181 (1996). T. Nitter and O. Havnes, Earth, Moon and Planets 56, 7 (1992). A. L. Hughes and L. A. DuBridge, Photoelectric Phenomena, McGraw Hill, 1943. B. Feuerbacher and B. Fitton, J. Appl. Phys. 43, 1563 (1972). E. C. Whipple, Jr., J. Geophys. Res. 81, 815 (1976). S. E. DeForest, J. Geophys. Res. 77, 651 (1972). The Collected Works of Irving Langmuir, C. G. Suits, ed., Pergamon, New York, 1961. Vols. 3 and 4. S. F. Singer and E. H. Walker, Icarus 1, 7 (1962). E. Walbridge, J. Geophys. Res. 78, 3668 (1973). R. J. L. Grard and J. K. E. Tunaley, J. Geophys. Res. 16, 2498 (1971). B. Walch, M. Horanyi, and S. Robertson, IEEE Trans. Plasma Sci. 22, 97 (1994). B. Walch, M. Horanyi and S. Robertson, Phys. Rev. Lett. 75, 838 (1995). M. Horanyi, B. Walch, S. Robertson and D. Alexander, J. Geophys. Res. 103, 8575 (1998). H. B. Michaelson, J. Appl. Phys. 48, 4729 (1977).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
373
Influence of dust grains rotation on waves dispersion in plasmas J.Mahmoodi'''*, N. L. Tsintsadze^ D. D. Tskhakaya^ ^'Institute for studies in Theoretical Physics and Mathematics,
P.O.Box
19395-5531,
Tehran, Iran. ^Institute of physics, Georgian Academy of Sciences, P.O.Box 380077, Tbilisi, Georgia.
The dispersion relation for plasma containing elongated and rotating dust particles is obtained. Dipole moments of dust grains are assumed to be not zero. It is shown that the longitudinal waves with frequency closed to the angular frequency of dust grains rotation are unstable. The dusty plasma with nonspherical dust grains is not investigated till now [1]. The plasma with elongated and rotating dust grains acquires a new characteristic freciuency, equal to the angular frequency of dust grains. We show that the dispersion properties of such plasma can be drastically modified - the waves with frequencies in the range closed to the rotational frequency of dust grains become unstable. We construct the kinetic equation for charged, rotating dust particles. We assumed the dust particles to have non-zero dipole moments. The dispersion relation for plasma, where elongated dust grains rotate with some preferred angular frequency, is found. In conclusion it is suggested an indirect way for the detecting of dust grains rotation by the scattering of the radiation on the exited
fluctuations.
To simplify the description let us suppose that dust grains are ciuite elongated, like a stick. The angular moments of rotating dust grains we assume to be directed along the some direction - along the z-axis. The dust grains system can be described by the distribution function /^(R, v, f^, ^, t), which enumerates the particles according to the coordinate R and velocity v of center of mass, angular frequency Ct. The azimuthal angle 0 determines the direction of the grain's elongation axis. The charge density produced by grains is defined by the grains distribution function as follow p(r,t) = / d R d r p ( r - R , ( / ) ) / r f ( R , v , f ^ , 0 , i ) ,
dT = dv dQ dcj).
(1)
374
J. Mahmoodi et al /Influence of dust grains rotation on waves dispersion in plasmas
Here p describes the charge distribution in a single grain. Outside of the grain's volume p = 0. The dust grains are identical. Every given direction of the grains elogation axis, determined by angle 0, can be considered as a final position of the turning of the axis from the direction with 0 = 0. We have p(r - R,(/.) = p{~A ((/>)(r - R),0) = p{A (r - R)),
(2)
where A () is the tensor of turning by the angle 0. Further we will assume the grain's size to be smaller than the scale of plasma inhomogeneity, a « I. Subtituting (2) in (1) and expanding the distribution function f\i about the point r we obtain p(r, t)=ql
d r / , ( r , v, Q, (/), t) - | dr(d. V)/,(r, v, ^, (/>, t)
(3)
Here q = J drp{r), and d = ^ ((/>) / d r rp(r) are the total charge and the dipole moment of the grain respectively. The motion of the charged dust grain in the E, B-electromagnetic field can be described by the Hamiltonian H = - ^ { P - ^A + i [ d X B]r + ^ ZrUd
c
c
21
+ q- (d.E),
(4)
where rrid is the mass of the grain, P is the generalized momentum, p^ = 10. is the angular momentum, / is the moment of inertia of the grain and A, (j) are potentials of the field. By means of this Hamiltonian the kinetic equation for the grains distribution function can be written as
+ Q | ^ + [d X ( E + -[v X B ] ) ] , ^ = 0.
(5)
The Eq. (5) and kinetic equations for electrons and ions, jointly with Maxwell's equations allow us to describe dispersion properties of the dusty plasma with rotating dust grains. We write the perturbed distribution functions as fd — fdo + ^fd^ fao = fao + ^fa^ a = e,i and suppose fdo » Sfd, fao » Sfao- Sfd, 6fa ^ exp{-iut + zk.r). In the unperturbed state we choose /^o as Maxwellian distribution functions. For the unperturbed gas of rotating dust grains we can use [2] /.o = n , o ( 2 7 r m . r , ) - 3 / 2 ( 2 ^ / T , ) - V 2 e ^ p { - ^ _ i ^ ^ - ^ } , ZrUdld
21 Id
(6)
J. Mahmoodi et al /Influence of dust grains rotation on waves dispersion in plasmas We assumed the grains to rotate with the preferred angular frequency Vt^, p^po = /r^o- Expressing the induced current density through perturbations of distribution functions we can find the dielectric tensor for the dusty plasma by the standard method [3]: e,A^,k)
= ^ e ' ( a ^ , k ) + {% - ^ l ^ ' C ^ . k ) + {S., - ^ } f ' ' ( ' ^ . k ) ,
(7)
e'(.,k) = l + i ; ^ { l - / . ( j | ^ ) K
(8)
.Vk) = l - i : $ W j | ; ) } .
(0)
/3 = e, i, d, ujp(j and Vrp are the plasma frequency and the thermal velocity of the particle, Ijri^) is well-known function [3], /? = e,i,d. The influence of the grain's rotation is described by e^{uj,k). It equals r2
(10) where Qr = [47rd^nrfo/2/]^/^, /^^ = rrid/I, k = [/C^ + K^]^/'^ ^ K. In deriving of (10) for the components of the dipole moment we use d^ — d cos[(j)), dy = d svn{(f)). Further we shall consider the frequency range, /cV^d < < oo « kVTe, (^ — ^o >> kVrd' According to (7)-(10) we find: a) for the transversal waves, polarized along 7/-axis, E = (0, E, 0), the dispersion relation
For the waves with frequencies close to QQ (<^pd ^ ^r) we obtain
So the ordinary transversal waves become unstable. Let us note that in the plasma without the dust gas the low frequency transverse oscillations decay aperiodically due to their collisionless obsorption by the electrons [3]. b) for the longitodinal waves, k^(? » uP', the dispersion relation has the form 1
^ k^rl
Cj2
k\
u^
k^ {uj -
^l
fio)'
^ ^
375
376
1 Mahmoodi et al /Influence of dust grains rotation on waves dispersion in plasmas
where a
From (13) it follows that the grains rotation gives the contribution only for waves with fcx 7^ 0. If Qo = 0 this contribution is expressed in the change of the dust characteristic frequency
. = ^^,+|o^r/Mi+^r/'^
(14)
The equation (13) has the complex solution for any rotating frequency QQ, satisfying the condition
^0 < a;,.[l + ^V"[l
+ i^^Y^r'-
(15)
The equality of f^o to the right hand side of (15) defines the boundary of stability of longitudinal waves. If cUp^/Ql ^ 1 + \/k^r\,
for the increment of instability we
find cj ~ S7o + ^7, 7 < < OQ 7
.A;2 Q? ^31/2 2-4/3 [ ! ^ q i ] l / 3 ^ ^ . • F 00,
(16)
Thus the energy of the dust rotation can flow into plasma oscillations that leads to the instability of the latters. It is well-known that the cross section of the scattering of transverse electromagnetic waves in a plasma has sharp maxima near the natural plasma low frequencies [3]. For the case considered above (see (12)) the cross section will have the sharp maximum at a; ^ r^o- When the grain's rotating frequency QQ approaches a critical value, defined by the right hand side of (15), the fluctuations
of longitudinal waves sharply increase and the scattering cross section
must also sharply grow. Thus the existence of a preferred frequency of the grains rotation can be found by the scattering of transverse electromagnetic waves in a dusty plasma.
1. Advances in Dusty Plasmas, edited by P. Shukla, D. A. Mendis and T. Desai (World Scientific, Singapore, 1997). 2. L. D. Landau and E. M. Lifshitz, Statistical Physics(Pergamon Press, Oxford,1989). 3. A. F. Alexandrov, L. S. Bogdankevich and A. A. Rukhadze, Principles of Plasma Electrodynamics (Springer-Verlag, Berlin, 1984).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
377
Plasma-Maser Instability in Magnetized Dusty Plasma Bipul J. Saikia* and Mitsuhiro Nambu"^ * Centre of Plasma Physics, Saptaswahid Path, Dispur, Guwahati 781 006, India. "^Tokyo Metropohtan Institute of Technology, Hino, Tokyo 191-0065, Japan.
A b s t r a c t . The plasma-maser theory of Langmuir waves in presence of low frequency dustion acoustic oscillations in a magnetized dusty plasma is presented. The up-conversion of wave energy in a dusty plasma is much enhanced owing to the increased acceleration of electrons by the dust-ion acoustic mode.
1
INTRODUCTION
The nonlinear interaction of a resonant mode (v^hich satisfies the Cherenkov resonance condition a; — k • v = 0) v^ith a nonresonant mode (which neither satisfies the Cherenkov resonance condition fi — K • v 7^ 0 nor the scattering resonance condition Q — a; — (K — k) • v / 0) in a turbulent plasma is known as plasma-maser instability in the literature(1,2). We mention on the recent numerical simulations on plasma maser. The first simulation study of the generation of high frequency Langmuir waves in the presence of the enhanced low-frequency whistler mode fluctuations by the plasma maser is done recently(3). It is confirmed that the plasma maser mechanism does not require an electron beam component for the growth of high-frequency nonresonant Langmuir waves from the enhanced low-frequency resonant whistler mode fluctuations driven by temperature anisotropy. In the second simulation study(4), the generation of R-mode electromagnetic waves from whistler turbulence by plasma maser effect is studied. Simulations using a two dimensional electromagnetic and relativistic particle code show that high frequency electromagnetic waves with right-handed polarization (R-mode) can be generated by a plasma maser mechanism from low frequency whistler waves excited by electron temperature anisotropy. The third one(5) reports that high frequency electromagnetic waves with right-handed polarization (R-mode) can be generated by a plasma-maser mechanism from electrostatic Langmuir waves excited by electron beam. Recently, a novel low frequency dust-ion acoustic wave has been reported in a coUisionless dusty plasma(6). This wave appears in the presence of static charged grains such that the free electron number density is much smaller than the ion number density, which could happen because the charged grains collect electrons from the background medium. In the long wavelength limit (viz. kXoe ^ 1, where Xoe = jTe/Airriee'^ is the electron Debye length), the frequency of the wave becomes a; = /cc*, where c* = JniTe/riemi is the effective sound speed of the wave.
378
BJ. Saikia, M. Nambu/Plasma-maser instability in magnetized dusty plasma
and TIQ i arc the number density of free electrons and ions. The anomalous situation Tie <^ Ui occurs in the planetary rings, in particular, in the F-ring of Saturn. Accordingly, the phase velocity uj/k of the dust-ion acoustic mode is increased in comparison with that of the conventional ion sound waves in fully ionized plasmas, where n^ — rii. This means that the effective electron acceleration due to the dust acoustic field is also much enhanced. Here we consider the plasma-maser instability of Langmuir waves in the presence of dust acoustic turbulence in a magnetized dusty plasma. The effective dielectric function of the Langmuir waves is obtained in Sec. 2. The plasma-maser effect of Langmuir waves by electrons scattered by dust-ion acoustic fields is investigated. Conclusions are summarized in Sec. 3.
2
FORMULATION and GROWTH RATE
We consider a three component homogeneous magnetized dusty plasma consisting of electrons, ions and charged dust grains. We assume that the dust ionacoustic wave propagates along the external magnetic field. The interaction of a test Langmuir wave (or Bernstein mode) with the dust acoustic waves is governed by the Vlasov-Poisson equations. It is straightforward to show that the nonlinear dispersion function of the high frequency electrostatic wave with frequency Q and wave vector K {— i(^_L,0, Ky) in the presence of the low frequency dust acoustic fluctuation with frequency u and wave vector k (= 0,0, k\\) can be written as, cr . , - p W
(p)
j^M)
/.x
where s^ j ^ is the linear part and ^ L K ^^ ^^'^ direct mode-coupling term.
Q — LU — {K\\ — k\\)v\\ — sCle \ y± dv± 1 d xik\\v\\ —^—.^fi''dv^ u + iOdv\
dv
(2)
and e[^j^ is the polarization mode-coupling term, ,2 \ 2 /
,
,.,2
X 2
4"K = ( ^ ) ( ; ^ ) E l ^ . . P ^ , ^ r i ^ ^ | r ^ S K , K - . . x 5 K . . K . - . . where '5K,K-k,k = a,s,n-' J2 TTT -
Jl{x) Kl\V\\-ane
d
jKx)
dv\\ Cl — (JU — {K\\ — k\\)v\\ — sQ.e
(3)
BJ. Saikia, M. Nambu /Plasma-maser instability in magnetized dusty plasma
X
X
d
afie d + vx_ dvj_
' sQp d
v±_ dv
379
{K\\-k{) " " dv\
1 d u) — k\\V\\ + i^dv
(4)
and ^K-k,K,-k = %,K-k,k(f^ ^ fi - a;, K ^ K - k, a; ^ - o ; , k ^ - k ) .
(5)
Here, Ja^Js^ Jn are the Bessel functions, Upe and fig are the electron plasma frequency and gyrofrequency, and x = K±vj_/Q,e' The linear dispersion relation for Langmuir wave for magnetized plasma reduces tofi^a;pe§[exp(-/?)/o(/3)]'/'. Next, we calculate the growth rate of the Langmuir wave trough the plasmamaser interaction. The growth rate originates from two different processes: the direct nonlinear coupling [Eq.(2)] and the polarization coupling [Eq.(3)] of the waves. Case (A) Growth rate from the direct couphng term. We keep n = s = a = 0 in Eq.(2). Partial integration of Eq.(2) leads to 2 / e N2 oo, pe
Y
rUe
a}_^\Eu,M\ Hj
d
-^i
a;,k
/f(^ll)d^ll,
uj — k\\V\\ — iOdv\
(6)
with a = f^ J^{x)f^^\v^)27rv^dv'_LHere /W(^||) - yJme/27rTeexp(-me^f/2Te) and f^^\v±) = yJme/27rTe X exp(—met'5./2Te), respectively. In deriving Eq.(6) we assumed Q. > K\ Thus we obtain the growth rate due to the direct coupling term as,
^
c pe
^ c, , , ^x . ,^x-,-.'^/2^5r^... r^\^\\\ = o^VJaS [exp(-/J)/„(/?)l-'^5:W,""Mexp a;,k
kl
(7)
here, Wds — (^5k/'^^^e7'e)(fce/^||)^ is the normalized dust acoustic-wave energy. Case {B): Growth rate from the polarization term. The dominant contribution for the Langmuir wave occurs from the n = s = a = 0 terms in Eq.(3) which is justified by the assumption fig > ^pe- The neglected terms are smaller by a factor {ajpe/Qe)^' Under this assumption, Eqs.(4) and (5) reduce to
5'K,K-k,k
=
/
d
a + fl — K\\v\\ dv\\ [Cl — (jj — {K\\ — k\\)v\\
xA/f)(„,|)
UJ — k\\v\\ + iO (8)
380
B.J. Saikia, M. Nambu/Plasma-maser instability in magnetized dusty plasma
M % , K - k , k X 5K-k,K,-k] = h{b-a)^fi
"^ "
^'^^
"
^— e
X —exp
(9)
Thus we obtain the growth rate due to the polarization couphng term as „(P)
. r. . 2
o
(10) From Eqs.(7) and (10), it emerges that the growth rate of the Langmuir wave is much enhanced in dusty plasma in comparison to usual electron-ion plasma, because the effective sound speed(c* = JniTe/riemi) is much enhanced in dusty plasma in comparison with that of the conventional electron-ion plasma.
3
CONCLUSIONS
We have studied the effect of the low-frequency electrostatic dust-ion acoustic wave on the plasma-maser instability in a magnetized plasma. It is found that due to the increased electron acceleration by the dust-ion acoustic fields, the growth rate of the Langmuir wave due to the direct nonlinear coupling as well as polarization coupling is much enhanced. Our results should be useful in understanding the enhanced emission of anomalous high frequency radiation in multi-component plasmas containing charged dust particulates such as those in planetary rings, interstellar clouds and earth's mesopause.
REFERENCES 1. Isakov,S.B., Krivitskii,V.S. and Tsytovich,V.N., Zh. Eksp. Teor. Fiz. 90, 933 (1986) [Sov. Phys. JETP 63, 545 (1986)]. 2. Nambu, M., Bujarbarua,S. and Sarma,S.N., Phys. Rev. A 35, 798 (1987). 3. Eda,M., Sakai,J.I., Zhao,J., Neubert,T. and Nambu,M., J. Phys. Soc. Japan 66, 525(1997). 4. Bujarbarua,S., Nambu,M, Saikia,B.J., Eda,M. and Sakai,J.I., Phys. Plasmas 5, 2244 (1998). 5. Nambu, M., Saikia,B.J., Gyobu, D. and Sakai,J.I., Phys. Plasmas 6, 994 (1999). 6. Shukla,P.K. and Silin,V.P., Phys. Scr. 45, 508 (1992).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
381
Ion Bursts in a Negative Ion Plasma S. Yoshimura*, M. Yohen ^ and Y. Kawai National Institute for Fusion Science, Oroshi, Toki 509-5292, Japan Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
Abstract. Ion species in a negative ion plasma is measured by a simple method utilizing ion bursts excited by a pulsed mesh grid. The burst velocities show a good agreement with the theoretical values for SF6'. Furthermore, the transient sheath width is estimated from the time-of-flight data of the ion burst. The result is qualitatively reasonable that the sheath width increases due to the effect of negative ions.
INTRODUCTION In the experiments on ion acoustic wave propagation, many researchers have observed a faster propagating signal in addition to ion acoustic waves. The velocity of the faster signal is dependent on the excitation voltages of the grid, FQ, according to the relationship u = (leVJm^^'^, where m^ is the ion mass(l). Thus the faster signal is identified as an ion burst accelerated by the excitation voltages. In other words, the ion species is determined from the velocity of the ion burst. In the previous study, we experimentally investigated the positive ion species in an Ar/SF^ mixture plasma from the burst velocities and found them to be Ar^, SF3^ and SF5^(2). In this paper, we propose a simple method for a negative ion species measurement utilizing the ion burst. We also discuss the possibility of estimation of the transient sheath (3) width from the time-of-flight data of the ion burst.
EXPERIMENTAL A schematic of the experimental arrangement is shown in Fig. 1. The experiments have been carried out using double plasma (D.P.) device, 100cm in length and 50cm in
382
S. Yoshimura et al /Ion bursts in a negative ion plasma
diameter. Argon and sulphur heaxafluoride (SF^) are introduced separately into the chamber by two mass flow controllers. The Ar pressure is 3X10"^ Torr and the flow rate of SF^ is 0-0.2sccm. The discharge voltage and current are 50V and 40mA, respectively. An 8cm-diam mesh grid (100 lines/inch) is used to excite ion bursts. The negative ion bursts are excited by a positive voltage pulse of 2.4|isec duration time. The perturbed part of the ion saturation current is received by a movable mesh grid, which is biased at -24V. Though the bias voltage is negative enough to collect almost only positive ions, several negative ions accelerated by sufficient excitation potential can reach the receiver grid. Therefore, the negative ion bursts, which are observed as the rarefaction of the positive ion saturation currents, can be measured by the time-offlight technique using an oscilloscope. Typical plasma parameters in the experimental region are measured by the Langmuir probe. The plasma density and the electron temperature are 3 x lO^cm"' and 0.4eV, respectively.
RESULTS AND DISCUSSIONS When the positive voltage pulse is applied to the exciter grid, the rarefactive or negative signal, which is excited by the fall part of the pulse, is observed. The ion
Oscilloscope
FIGURE 1. Schematic of the experimental an-angement. The D.P. plasma is produced by dc discharge between tungsten filaments and the magnetic cage.
S. Yoshimura et al /Ion bursts in a negative ion plasma
383
acoustic wave signals are also detected in these experiment. However, the velocity of the ion acoustic waves are always much smaller than that of the negative signals, so that we can neglect them. The velocity of the negative signal increases with increasing the excitation voltage. The dependence of the negative signal velocities on the excitation voltages is shown in Fig. 2. It is evident that the experimental results show a good agreement with the theoretical burst velocities for SF^', not for SFs". Since any other negative signals cannot be observed, it is concluded that the principal negative ion species in these experiments is SF^". This result should be examined using a quadrupole mass spectrometer (QMS). It is now under progress. Several authors have reported the existence of F" ions in a similar device(4). It would be explained by the difference of the operation parameters, since F" ions also have been detected in our device by changing the SF^ flow rate(5). The relation between the negative ion species and gas flow rate is considered to be a new subject in this field. Next, let us discuss the estimation of the transient sheath width by using the time-offlight data of the ion burst. When the interval between the grids decreases, the velocity of the ion burst shows a slight deceleration. It is considered that the total sheath width, which is the sum of the Langmuir sheath of the receiver grid and the transient sheath of the exciter grid, exceeds the interval. Thus the transient sheath width can be estimated from the inflection point of the time-of-flight data, since the Langmuir sheath width is constant for given applied voltage. The results are shown in Fig. 3. Since the theory of transient sheath in a negative ion plasma is not established, the theoretical values for Ar plasma is adopted to examine the validity of the experimental results. It is clear that
o
0) (0
E o
o
SFgburst (theory)
i
J
Negative ion concentration = 0.37 10
20
30
40
50
60
70
80
90
100
Voltage(V) FIGURE 2. Comparison of the experimental results with the theoretical curves for SF5" and SF^" burst.
S. Yoshimura et al. /Ion bursts in a negative ion plasma
384 5 4
Negative ion concentration = 0.14
31-
E o
2[
CO ^
jz — S O
^ I-
1F 0.8 I0.7 0.6 10.5 [
Langmuir sheath(0.89cm) + Transient Sheath for Ar plasma
0.4 h 0.3
8
20
9 10
30
40
50
60
Voltage (V) FIGURE 3. Comparison of the total sheath width estimated from the ion burst with the theoretical curve. Increasing the excitation voltage, increase in the sheath width is observed.
the total sheath width increases as the excitation voltage increases. In addition, the experimental value deviates from the theoretical curve for Ar plasma. It is well known that the sheath width increases under the existence of negative ions. Thus the result obtained here is qualitatively reasonable. To improve this analysis method, a theory which quantitatively includes the effect of negative ions on the transient sheath is required. It will form our future work. Finally, we suggest a possible application of this method to dusty plasmas. So far, there is no investigation of the ballistic effects in a dusty plasma. However, if a dust grain burst can be excited by external mesh grid, the specific charge of the dust grain can be determined. We consider that it may work for dust charge measurements.
REFERENCES 1. Lonngren K., Montgomery D., Alexeflf I., and Jones W. D., Phys. Lett. 25A, 629-630 (1967). 2. Yoshimura S. and Kawai Y, Jpn. J. Appl Phys. 37, L254-256 (1998). 3. Widner M., Alexeff I., and Jones W. D., Phys. Fluids 13,2532-2540 (1970). 4. Nakamura Y, Ferreira J. L., and Ludwig G. O., J. Plasma Phys. 33, 237-248 (1985). 5. Kawai Y.,private communication (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
385
Nonlocal Effects in an Ion Beam Driven Ion Acoustic Waves in a Magnetized Dusty Plasma Suresh C. Sharma and M. Sugawa Department of Physics, Faculty of Science, Ehime University, Matsuyama 790-8577, Japan Abstract An ion beam propagating through a magnetized dusty plasma cylinder destabilizes electrostatic dust ion acoustic waves. A nonlocal theory of this process has been developed. 1. Introduction The fluctuations of the dust grain charge are found to be a source of wave damping or growth^"^. Barkan et al? reported experimental results on ion acoustic waves in dusty plasma. In this case the phase velocity of the ion acoustic fast mode increased with an increasing e (the concentration of the negatively charged dust grains ) while the wave damping decreased with increasing e. 2. Instability Analysis Consider a cylindrical dusty plasma column of radius a^ with equilibrium electron, ion and dust particle densities as n^^, n^^ and n^^ immersed in a static magnetic field BjZ . The charge, mass and temperature of the three species are (-e,m,TJ, (e,mi,Ti) and (-qji,m^,Td), respectively. The plasma is collisionless. An ion beam with velocity VQIJZ ,
mass m^,, density n^b and radius r^ (=ai) propagates through the dusty plasma
along the magnetic field. Following Sharma et al}^ in the limit of short parallel wavelength {(x>«]^^
and a)«(i)^, we obtain perturb electron density [cf. Eq.(7) of
Sharma etal]. In the limit (i)«a)^, k^Vt.<(i)^, i.e., kj^vy((i)-a)J«l or k ^ p i « l , Eq. (8) of Sharma et al? is modified as
386
S.C Sharma, M. Sugawa/Nonlocal effects in an ion beam in a magnetized dusty plasma
Cl
where a)^=e BJmf and (j^^^-Aitn^f^lm'^. The dust particle density fluctuation can be neglected, however, their charge fluctuation could be significant. Following Jana et aO and Sharma et at? we obtain dust charge fluctuation ^
^
l^ocl
,^li
^les
^u-^-Z—r^(—~—) 1(G)+1T1) n^.
n^
.^.
•
(2)
where the natural decay rate TI is same as Eq.(16) of Sharma et al? . Substituting the values of Uj^ [from Eq.(7) of Sharma et al] and n^^ [from Eq. (1)], Eq. (2) can be rewritten as
i(c^^ni)
(o^m. co^m. mv^
The response of cold ion beam can be obtained by solving the fluid equations of motion and continuity which on solving and linearization yields the beam density perturbation [cf. Eq. (22) of Sharma et al\ Using the values of n^^ from Eq.(7) of Sharma et al, n^j [from Eq.(l)], (\^^ from Eq.(3) and n^, from Eq.(22) of Sharma et al in the Poisson's equation, we obtain
Ar2
r 5r
2
2
(4)
where 2
p =
2
i
2
1
'
(^>
S.C. Sharma, M. Sugawa/Nonlocal
effects in an ion beam in a magnetized dusty plasma
387
coupling parameter ^ is same as Sharma et al? and ix^^^-ATtn^^t^lm^. Now we attempt to solve Eq.(4) using perturbation theory and following the procedure of Sharma et al?, we obtain
where 12 2
,
,
k-c,n 7n„
iB
0)
pi Ei 21 .
k_c.
a(l+—i^)
{U—±-2L)a
a=l..
, 2 2
Pn rn 1:L.+ 1 2 0 2 2
iP ip i^^
2 2
^"pifloeP p i "oe *^n ^ ^ ^ 2 * n% i _1-2 2 2
.
(8)
^ ^
3. Results and discussions In the calculations we have used plasma parameters for the experiment of Barkan et al?. We solve numerically the set of equation [viz. Eq.(6) by writing a)=(i)r+iY] for the following parameters : potassium ion plasma density noj-lO^-lO^^ cm"^, electron plasma density Uo^.-lO^ cm"^, relative density of negatively charged dust (5(=noj/noJ=2.0, temperatures T^— Tj~0.2 eV, plasma radius a^-^Z cm, guide magnetic field 85-4x10^ Gauss, dust grain density n^^Sxl^f cm"^ average dust grain size a-SxlO"^ cm, potassium ion beam energy Ei,=4 eV, beam density nob=lxlO^ cm"^ and mode number n=l. We find out that the instability has the largest growth rate y when ^-12
cm'^ Using k2=1.2 cm"^ we have plotted in Fig.l, the phase velocity Vp,^
[^(i^jkj^m cm/sec)] of ion acoustic mode as a function of relative density of negatively charged dust d. From Fig. 1 it can be seen that the phase velocity of the unstable wave increases with the relative density of negatively charged dust 6 in compliance
388
S.C. Sharma, M. Sugawa/Nonlocal
effects in an ion beam in a magnetized dusty plasma
with the experimental observations of Barkan et al^. Using 1^=1.2 cm"^ we have plotted in Fig. 2, the normalized imaginary part of the frequency (Y/<*>d) as a function of relative density of negatively charged dust d. From Fig. 2 it can be seen that the growth rate increases by a factor -1.48 when 6 changes from 1 to 2.0 m compliance with the theoretical results of Sharma et al. References 1. M. R. Jana, A. Sen and P. K. Kaw, Phys. Rev. E 48, 3930 (1993). 2. Suresh C. Sharma and M. Sugawa, Phys. Plasmas 6, 444 (1999). 3. A. Barkan, N. D'Angelo and R. L. Merlino, Planet. Space Sci. 44, 239 (1996).
0.08
?-
0.04h
(5 (=noi/noc)
4 d (=no/noc)
8
Fig.l Phase velocity v^^ [sto^k, (in cm/sec)J of unstable
FigJ Normalized imaginary part of frequency (Y/WJ as a
mode as a function of relative density of negatively charged
function of relative density of negatively charged dust 6 for
dust 6. The parameters are given in the text.
the same parameters as Fig.l.
389
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
Experimental observation of attraction of massive bodies in a plasma Dubinov A.E. \ Zhdanov V.S. ^ Ignatov A . M A Kornilov S.Yu. \ Sadovoi S.A. \ Selemir V.D. ^ ^Russian Federal Nuclear Centre, 607190 ^ 37, Mira pr.,Sarov, Russia ^General Physics Institute, 117942, 38, Vavilov str., Moscow, Russia Abstract We discuss the results of experiments demonstrating that massive bodies in a glow discharge attract each other. The value of measured force exceeds theoretical predictions.
Experimental setup It is known that dust grains immersed in a plasma often arrange into ordered structures like crystals or clouds. Under normal conditions of the glow discharge, all grains are negatively charged, so they have to repulse each other at the distances smaller than the Debye length. However, to provide the stability of any structure with a free surface, there should exist some attractive forces compensating the Coulomb repulsion. Although there was a number of theoretical and computational works concerning different mechanisms providing the attraction of like-charged grains (see, e.g. [1]), very little was done experimentally. It seems extremely difficult to measure directly the real force acting upon a single micrometer size grain. Recently we made an attempt to increase the spatial scale and to measure the force between much larger bodies in a glow discharge [2]. In the present paper we report the results of the more detailed force measurement.
14
wH
W-
VK-
F I G U R E 1. Experimental setup
g
390
A.E. Dubinov et al /Observation of attraction of massive bodies in a plasma
FIGURE 2. Sheet mounting. All dimensions are in mm.
The experimental setup is depicted schematically in Fig. 1. Experiments are carried out in the glass chamber 180 mm in diameter, the distance between electrodes is 700 mm. The air pressure inside the chamber can be varied from 0.01 to 0.5 Torr; the net discharge current is up to 0.5 A. The plasma parameters are controlled with the electrostatic probe. To measure the force between bodies we use a device resembling the electroscope. Two square sheets (25x25 mm) are hanged across the discharge at thin (0.1 mm) copper wires 1 cm apart as depicted in Fig. 2. The sheets are cut either of paper with surface mass density 80 g/m^ or of aluminum foil (110 g/m^). The design of the suspension provides reasonable response towards horizontal forces and prevents the sheetsfromrotating about the vertical axis. The electroscope is well insulated from the chamber wall. The electrical circuit between the leaves provides measurement of the voltage bias.
Experimental results In the absence of the discharge, the sheets hang vertically. With the growing discharge current the leaves draw together, so that at the sufficiently high current magnitude, the lower edges of the leaves join. The effect is fairly stable: when the current is fixed the leaves stay in the same position for hours. Finally, when the current is off, the leaves quickly move apart, then, in a few minutes, return to the vertical position.
A.E. Dubinov et al / Observation of attraction of massive bodies in a plasma
10
F,„ dynes
Fp dynes
*
391
8
10
10 t 20
L,iimi
4 20 +30 2 i
^
0 0
8 1 A^
10
12
14
-3
n^, 10 cm FIGURE 3. The dependence of the distance (left) and the force (right) for aluminum (•) and paper (•) leaves.
The experimental dependence of the distance between lower edges of the leaves on the plasma density is depicted in Fig.3, where the calculated horizontal component of a force on a leave is also shown. There is no visible dependence of the force on whether the leaves are short-circuited or not. The force is also independent on the orientation with the respect to the direction of the discharge current. However, when the leaves are turned perpendicular to the current, there appears a small, of the order of 1 V, voltage drop between them. Discussion Seemingly, none of the theories of attraction agrees with our experimental data. The experiments clearly demonstrate that the observed attraction cannot be concerned with electrostatic interaction. The effective pressure pushing the leaves together is of the order of Po^l dyne/cm^. The mechanism of attraction due to plasma recombination at the solid surface [3,4] predicts the pressure which, under our experimental conditions, is about 0.01 Po. Actually, the plasma density was measured far from the electroscope, while there is some visible increase in the glow intensity near its leaves. Nevertheless, one can hardly believe that the plasma density near the surface increases to such an extent.
392
A.E. Dubinov et al /Observation of attraction of massive bodies in a plasma
On the other hand, the pressure of the neutral gas is at least 100 Po, i.e., to explain the observed effect one should assume that the pressure between leaves drops by 1%. This may really happen if, by some means, the temperature of the surface is less than the temperature of neutrals [4,5]. It was proposed in [5] that the cooling of the surface of the solid body might be provided by radiation. However, the cooling process discussed in [5] is independent of plasma, while our experiments show that the attraction is triggered by the discharge and it depends on the plasma density. We don't know how plasma can cool down the surface of a solid body. To clarify the situation further experiments are required.
1. 2. 3. 4. 5.
References
Tsytovich V.N., Physics - Uspekhi, 167, 57 (1997). Dubinov A.E., et al. Bull. Lebedev Physics Inst. No. 7-8 ( 1997) Ignatov A.M., Bull. Lebedev Physics Inst., No. 1/2 (1995) Ignatov A.M., Plasma Phys. Rep., 22, 648 (1996) Tsytovich V.N., Khodataev Ya.K., Morfill G.E., Bingham R.,Winter D.J., Comm. Plasma Phys., 18,281(1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
393
The Role of Random Grain-Charge Fluctuations in Dusty Plasmas Sergei A. Khrapak, Oleg F. Petrov, and Olga S. Vaulina High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia
Abstract. Random charge fluctuations are always present in dusty plasmas due to the discrete nature of currents charging the dust particle. These fluctuations can be a source of heating of dust particles system. Such unexpected heating leading to the melting of the dust crystals was observed recently in several experiments. In this paper we examine the role of random charge fluctuations in dust particles heating.
A dust particle in a plasma acquires electric charge by collecting electrons and ions from the plasma (v^hen emission processes are unimportant). Electron and ion currents consist of individual electrons and ions which arrive at the particle surface at random times and by random sequences. For this reason the charge on a particle will fluctuate about the steady state value Z^. Some studies appeared in recent years that addressed various aspects of random charge fluctuations (1, 4). We have developed a model (4) to study quantitatively these fluctuations. This model is based on the equation describing the random-walk process in the one-dimensional space of particle charge. The useful characteristic of charge fluctuations - their temporal autocorrelation function (TAF) can be obtained within the framework of our model. In Ref (4) it was shown that independently of the charging mechanism, the TAF of random charge fluctuations has the form {sZ,{t)dZ,{t')) where 6Z^ = Z^{t)-Z^,
= (6Z/)exp(-pk - ^1)
(1)
P"^ is the characteristic time of charge fluctuations, and
(dZ^ I = OLJ\Z^ I is the amplitude of fluctuations. For example, using the standard orbital - motion limited (OML) theory to compute the charging currents yields e'^ 1 + T + ZT al^
1 + zT
1 + ZT Z(1 + T + ZT).
1/2
(2)
394
S.A. Khrapak et al/The role of random grain-charge fluctuations in dusty plasmas
where /^Q = ^JSna'^n^Vj^^ exp(z^e^ / ^ ^ ) is the steady state electron current, a is the dust particle radius, T^, m^, and n^ are electron temperature, mass, and number density respectively, Vj^^ =[T^ /mj
is the thermal velocity of electrons, and the
dimensionless parameters z and x are introduced by: z = ———, z = T^ / T^, The other charging mechanisms, including charging by thermionic and photoelectric emission are considered in Ref. (4). Equations (1) and (2) can be used to examine the effect of random charge fluctuations on the dust particle dynamics in a plasma. The point is that the charged particles are under influence of mutual Coulomb interaction. As it was mentioned in (2, 5), charge fluctuations lead to a fluctuating intergrain potential, providing an effect similar to the random motion additional to the thermal one for a strongly coupled system. Thus, random charge fluctuations can, in principle, heat the particles above neutral gas temperature (in spite of the effective cooling by neutral gas friction in lowionized plasmas). The additional kinetic energy associated with charge fluctuations can be evaluated as (6)
r,=^.(vj)/2J
l^^^^p
(3)
where m^ and n^ are the dust particle mass and number density, r| is the friction frequency of the neutral gas. In the free molecular regime r| = —
, where
r^, «„, and v^n are neutral gas temperature, number density, and thermal velocity respectively. We have also neglected screening in obtaining Eq. (3). The real dust temperature is 7^ = 7^ + 7^. The neutral gas temperature T^ arises in the last equation due to effective coupling between the particles and gas, so that in the absence of charge fluctuations and other heating mechanisms the particles are in the equilibrium with the neutral gas T^ =T^. The charge fluctuations induced heating becomes important when y = 7y / 7^ > 1. It is easy to find with the help of Eqs. (2) and (3) that in a case when T^ -T^, and Vj^^ - Vj^ 3
Z^T^(1 + ZT) (nrTy(rt^'(n^ d_
^ ^ 6 4 7 C ( 1 + T + ZT)^
e'
"l
ynj
(4)
independently of particle size. For usual for laboratory plasma parameters z - 3, T ~ 40, T, -I eV, «i -^10^ cm-\ n^ ^10'^ cm"^ we obtain from Eq. (4) that y > 1 for w^ > 5x10^ cm'\ Such dust concentration is rather large for experiments on plasma crystal formation. Thus, random charge fluctuations can not heat dust particles by this way.
S.A, Khrapak et al /The role of random grain-charge fluctuations in dusty plasmas
395
However, in addition to mutual interactions, dust particles are under the influence of various external forces which depend on dust charge. The random charge fluctuations thus lead to the appearance of random forces which can heat the particles. To analyze the role of this effect we consider the conditions common for experiments on dust crystallization in rf plasma. Namely, we assume that the dust particles are trapped in the sheath edge region, where there is balance between the gravitational m^g and electric force Z^eE (thus neglecting ion drag, which is important for sufficiently small particles). In a steady state m^g-hZ^eE = 0, However, due to random charge fluctuations a random force / = eE?>Z^{t) acts on the particle. This random force causes the particle to vibrate in the direction of electric field acquiring an energy from the field and loosing energy through neutral gas fiiction. The additional kinetic energy of an isolated particle associated with this process can be evaluated as (6)
(5)
r , ~ ^ ^
For an isolated dust particle this additional energy is concentrated in the direction of the extemal electric field (charge fluctuations do not change the particle energy in the direction perpendicular to the electric field). To study the effect of strong interaction between dust particles we have performed molecular dynamic (MD) simulation of a system of particles with fluctuating charges (6), based on Eq. (1). We have found that the particle velocity distribution fiinction is the anisotropic Maxwellian fimction, characterizing by two temperatures (corresponding to different directions) T^^ (x- is the direction of electric field), and T^y {y -is the direction perpendicular to the field). It should be noted that T^y is always less than T^^^. This is because the energy is supplied in our system basically in the direction of the extemal field. However, due to intergrain interactions (particles collisions) the particle kinetic energy in the ydirection can also be sufficiently high. For example, MD simulation shows that in the limit of strong interaction, the total kinetic energy is distributed almost equally between x- and y - directions. However, the main result of MD simulation is that the total particle temperature T^ = T^^ + T^^ obtained in simulation is close to the analytical result for an isolated particle Eq. (5). To evaluate numerically the role of the effect considered we have to made some assumptions about the sheath structure. We assume that the dust particles are trapped near the sheath edge, where ions are accelerated by a presheath potential drop to the velocity VQ » v^i. Redefining T as x = J^ / (AW^VQ / 2 | we arrive to P = 7ia^w,.Vo—r[l + T + Tz], a =
1 + TZ
ll/2
Lz(l + T + Tz)J
(6)
396
S.A. Khrapak et al /The role of random grain-chargefluctuationsin dusty plasmas
Further, we assume that VQ = v^ = ^T^ I m^ (Bohm criterion), so that x = 2. We also assume Ar plasma with parameters typical to rf plasma experiments 7^ = leV and n^ =10^ cm^ Quasineutrality is assumed to hold up to the sheath edge, so that n^ « n^ +\zj\n^. The latter equation takes into account the possible effect of electron depletion in the dust cloud if dust particle concentration is sufficiently high. To describe quantitatively the effect of electron depletion one can introduce the so - called rrt
Havnes parameter P = —j--^
which is roughly the ratio of the charge density of the
dust particles to that of electrons. When P>\ (this can be expected for some experiments on dust ordering) the charge is significantly diminished with respect to its value for an isolated particle (P « 1). Assuming also the dust material mass density p =5 g/cm , and the friction jfrequency r[-25/ ar \ [s" ] (corresponding to a pressure P - 0.2 Torr of a background gas Ar at room temperature) we obtain that Td is varied from approximately 0.1 eV to 0.3 eV for a = 5 |im, and from 1.7 eV to 8.3 eV for a = 25 |im with the increase ofP from 0 to 5 (6). These energies are significantly higher than the thermal energy (- 0.03 eV at room temperature). Thus, random charge fluctuations can be important when considering the reason of dust particle heating, observed experimentally (7, 9). This heating is important because it can lead to the melting of the dust crystals by reducing the coupling parameter T = Z^e n^ / T^. The role of the effect considered increases with the increase of dust particles size and concentration and with the decrease of the neutral gas pressure.
ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, Grant No. 98-02-16825 and Grant No. 97-02-17565.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
Morfill, G., Grun, E., and Johnson, T., Planet. Space Sci. 28, 1087 (1980). Cui, C, and Goree, J., IEEE Trans. Plasma Sci. 22, 151 (1994). Matsoukas, T., and Russel, M., J. Appl. Phys. 11,4285 (1995). Khrapak, S., Nefedov, A., Petrov, O., and Vaulina, O., Phys. Rev. E 59, 6017 (1999) Morfill, G.E., and Thomas, H., J. Vac. Sci. Technol A 14, 490 (1996). Vaulina, O.S., Khrapak, S.A., Nefedov, A.P., and Petrov, O.F., submitted to Phys. Rev. E. Fortov, v., Nefedov, A., Torchinsky, V. et al. Phys. Lett. A 229, 317 (1997). Thomas, H., and Morfill, G.E., Nature 379, 806 (1996). Melzer, A., Homann, A., and Piel, A., Phys. Rev. E 53, 2757 (1996).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and P.K. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
397
Shock Waves in Plasmas Containing Dust Particles S.I. Popel(«) and V.N. Tsytovich^^) ^^^ Institute for Dynamics of Geospheres, Leninsky pr. 38, Building 6, 117979 Moscow, Russia ^^^ General Physics Institute, Vavilova 38, Moscow 117942, Russia A b s t r a c t . An anomalous dissipation originating from the dust particle charging processes results in a possiblity of existence of a new kind of shock waves in a dusty plasma. The possibility of the experimental study of the shock waves related to the dust charging process in active space experiments in the Earth's magnetosphere is discussed. In dusty plasmas dust particles acquire big charges during very short periods of time. The charging process can be due to field emission, ultraviolet irradiation, microscopic plasma currents flowing into the dust grains, etc. If the charging is due to the microscopic currents then the dust particles which appear in a plasma soon acquire very large charge corresponding to the floating potential. The charges of dust grains are variable because the magnitude of the charge depends on plasma parameters and their perturbations. Shock waves often arise in nature because of a balance between wave breaking nonlinear and wave damping dissipative forces [1]. CoUisional and coUisionless shock waves can appear because of friction between the particles and wave-particle interaction [2], respectively. The presence in the plasma of the macro particles with variable charges results in the new mechanism of dissipation. An anomalous dissipation appears which originates from the charging processes. This dissipation can, in particular, give rise to shock waves. They are coUisionless in the sense that they do not involve electron-ion collisions. However, in contrast to the classical coUisionless shock waves, the dissipation due to dust charging involves interaction of the electrons and ions with the dust grains in the form of microscopic grain currents. The shocks related to this dissipation can be very important in space dusty plasmas. This is associated with different astrophysical applications [3]. For example, according to modern concepts [4], the formation of stars occurs mainly in interstellar dust-molecular clouds after compression shock waves propagate through them, creating the initial density condensations for further gravitational contraction. The presence of the dust in the interstellar clouds can influence significantly even the magnitude of the sound velocity, not to mention the shock wave propagation. The
398
S.I. Popel, VN. Tsytovich/Shock waves in plasmas containing dust particles
investigation of shock waves related to the dissipation originating from the dust particle charging processes can also be important [5] for the description of shocks in supernova explosions, particle acceleration in shocks, formation of dusty crystals, etc. The basic results, which demonstrate the new physics in formation of the shocks in dusty plasma, are obtained in [ 5 - 7 ] . In [6] the case of the nonlinear d u s t acoustic waves appearing as a stationary shock wave structure has been considered. These waves are not intensive. They assume that the ion speed perturbation in the wave is smaller than the ion thermal velocity. In space one can expect the significant role of more intensive shock waves. For the situation when in the plasma there are only electrons, ions, and dust grains the exact solutions in the form of shock waves have been found [7] for time scales corresponding to ion acoustic wave propagation. However, in astrophysical situations one has to take into account the presence of neutrals in the dusty plasma in addition to dust particles, electrons, and ions. The way how one can develop the hydrodynamics for description of stationary shock waves in a space dusty plasma which contains electrons, ions, neutrals, and dust grains is presented in [5]. The shocks in stationary media can be described by conservation laws for particle momentum and energy fluxes through the surface of the shock. In a dusty plasma the situation is more complicated because the dusty plasma is an open system. Therefore, the consideration [5] is restricted only to the case when at the front of the shock wave the dusty plasma is stationary and not moving. This means that there should be some external ionization which supports the density of the plasma particles (in the absence of the source of electrons and ions the plasma is recombined very rapidly on dust [8]). The ionization can be produced by electromagnetic radiation of burning stars, by cosmic rays, by electron beams, etc. We assume that behind the shock front there is a source of the plasma particles with the same rate of particle generation as at the front. The hydrodynamics equations [5] allow us to find the generalized Hugoniot relations for the shock waves in dusty plasmas as well as the width of the shock wave front. These generalized Hugoniot relations are much more complicated than the Hugoniot relations appeared in the conventional gas dynamics. The new relationships appear. For example, in order to find the relationship between the dust charge behind the shock front and the Mach number of the shock wave it is necessary to solve a set of equations (including the transcendential ones). This problem can be solved only numerically. Here we mention only the width of the shock front for the case when the Mach number of the shock wave M ^ 1. In accordance with [3] the width is determined by the mean free path of electrons and ions in collisions with the dust: ^i
^ A^O^eo/tt^dO^dO,
(1)
where A^^ = A^^ -f A^^, \De(i) is the electron (ion) Debye length, a is the dust particle size, n^i^d) is the electron (dust particle) density, Zd is the charge of the dust particles in the units of electron charge, the subscript "0" denotes the unperturbed
S.L Popel, V.N. Tsytovich /Shock waves in plasmas containing dust particles quantities. The magnitude of A(^ is much less than the particle mean free path for binary (electron-ion, electron-electron, ion-ion) collisions. We emphasize that the process of the collisions of electrons and ions with the dust particles results in the charging of the latter. Thus the dissipation which gives rise to shock waves is related to the dust charging process. Let us discuss the possibility of observation of shock waves related to the dust charging process. The width of the front of the shock waves related to dust charging can be very large (see, Eq. (1)). For example, for \DQ ^ 1 cm, a ~ 1 //m, Zdo^do ~ ^e07 the estimate for the shock front width gives A<^ ~ 10"^ cm. It is difficult to carry out the investigation of such shock waves in laboratory plasmas. Thus the problem of interest is to investigate the possibility of an appearance of charged dust particles in active rocket experiments which involve the release of some gaseous substance in near-Earth space and the formation of shock waves related to dust charging in such experiments. We pay the main attention to the experiment assuming that its altitude is 600 km and the scheme is analogous to the experiments which have been carried out by the Institute for Dynamics of Geospheres of the Russian Academy of Sciences at the altitudes of 150 km at night time [9]. The source for the charged particle release in the magnetosphere in these experiments is the generator of high-speed plasma jets. Macro (dust) particles in the experiment at the altitude of 600 km appear as a result of condensation [10]. The period of the formation of the centers of condensation is very short, and all drops have approximately the same size a. The charge acquired by the macro particles can be estimated as the unperturbed charge of the dust grain from the balance condition of the electron and ion currents into the grain. For the parameters of the magnetosphere at the altitude of 600 km {T^ ^ 0.3 eV, Te ^ 0.3 eV, ng ~ 10^ cm"^ (at night time), and the prevailing ions 0"^ in the ion composition) the estimate gives Z^ ^ 3 • lO^a, where a is expressed in units of //m. The estimates of the size of the macro particles formed in the process of condensation have been carried out for two situations. The calculations in the first one have been fulfilled by N. Artem'eva. This situation corresponds to the air jet. The characteristic expansion speed of molecular nitrogen is [/ = 0.3 — 0.5 k m / s . The condensation starts when the jet passes the distance of the order of 10 cm. The degree of condensation is equal approximately to 0.72, while the macro particle size is a = 1.5 /im. The charge acquired by the macro particle is 5 • 10^ times larger than the electron charge. The second situation corresponds to the case when the gaseous substance is iron. The estimates are carried out with the use of the data [10]. The degree of condensation is x ?^ 0.44. The important result here is the dependence of the macro particle size on the expansion speed U of the gas injected in the magnetosphere. The size a decreases significantly with the increase of the speed (for U — 9.2 km/s a = 60 /im; for U — 15.5 k m / s a = 0.33 /im; for U — 21.4 k m / s a — 0.006 /im; for U — 27.2 km/s a — \Q~^ /im). If the speed U is approximately equal to 10 km/s
399
400
S.L Popel, VN. Tsytovich /Shock waves in plasmas containing dust particles
then a > 10 /im and correspondingly Zd > 10^, i.e. the dust particle charging effect is significant. The increase of the speed assumes the weakening of this effect. For the speeds U ^ 30 k m / s the magnitude of Zd becomes less than unity, i.e. the macro particle does not acquire the charge. Let us find the characteristic distance which is necessary for the macro particles to acquire significant charges. For the above parameters of the magnitosphere the estimate of the dust particle charging frequency (see, e.g., [8]) gives Uq ~ 5a, where a is expressed in units of /im while Vq in units of s~^. The distance from the region of injection to the place where the charging effect becomes significant is L ~ U/i/q. For the first situation (air expanding with the speed U = 0.3 — 0.5 km/s) the distance L is of the order of 100 m. For the second situation (iron) and U = 9.2 / we find L < 100 m. However for the velocity U = 15.5 k m / s we obtain the distance L exceeding several kilometers. Thus the optimum velocities to manifest the charging effect of the macro particles are U < 10 k m / s . The obtained distances L are reasonable from the viewpoint of the active experiments. The estimates show that the active experiments which involve the release of some gaseous substance in the Earth's magnetosphere can be very useful for investigation of the shock waves related to dust charging in such experiments.
ACKNOWLEDGEMENTS This work is supported by INTAS (grant no. 97-2149).
REFERENCES 1. Zel'dovich, Ya. B., and Raizer, Yu. P., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, New York: Academic, 1967. 2. Sagdeev, R. Z., in Reviews of Plasma Physics, edited by M. A. Leontovich, Vol. 4, New York: Consultants Bureau, 1966, p. 23. 3. Spitzer, L. J., Physical Processes in the Interstellar Medium, New York: John Wiley, 1978. 4. Kaplan, S. A., and Pikel'ner, S. B., Interstellar Medium, Cambridge Mass.: Harvard Univ. Press, 1982. 5. Popel, S. I., Tsytovich, V. N., and Yu M. Y., Astrophys. Space Sci. 256, 107 (1998). 6. Melands0, F., and Shukla, P. K., Planet. Space Sci. 43, 635 (1995). 7. Popel, S. I., Yu, M. Y., and Tsytovich, V. N., Phys. Plasmas 3, 4313 (1996). 8. Tsytovich, V. N., Physics-Uspekhi 40, 53 (1997) [Uspekhi Fizicheskikh Nauk 167, 57 (1997)]. 9. Adushkin, V. V., Zetzer, Yu. I., Kiselev, Yu. N., et al., Doklady Akad. Nauk 331, 486 (1993). 10. Raizer, Yu. P., ZhETFSl, 1741 (1959).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
401
Investigation of the Ordered Structure Formation in a Thermal Dusty Plasma Yaroslav K. Khodataev, Sergei A. Khrapak, and Oleg F. Petrov High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya J 3/19, 127412, Moscow, Russia
Abstract. The dynamics of the ensemble of interacting dust grains under conditions of thermal plasma is investigated by means of Brownian dynamics simulation. In particular, the time evolution of the pair correlation function is studied. By comparing simulation results with experimental ones it is found that the structure obtained in the experiment is far from the stationary state because the plasma flight time is less than the time of structure formation. It conforms to the emergence of the experimental pair correlation function characterized by a sharp main peak with no high-order ones.
Shov^n recently was the formation of dust ordered structures in a thermal dusty plasma under atmospheric pressure and temperatures of about 2000 K (1, 2). The experimental facility incorporates the plasma device and the diagnostic instrumentation for determination of plasma and gas parameters. The dusty plasma device produces the laminar spray of thermal dusty plasma under atmospheric pressure. It is possible to make a number of measurements of dust/plasma parameters such as the electron n^ and ion n^ number densities, plasma temperature Tg, and the diameter a and number density n^ of particles. The time-of-flight laser system v^as employed to analyze the particle structure in the thermal spray and then to compute the radial pair correlation function. This experiment differs essentially from other experiments on dust ordering in (3-5), carried out in a gas discharge plasma. First, dust particles (Ce02) with a ^ 0.4 |am are charged positively to about 500 elementary charges due to thermionic emission. The latter process is absent in gas-discharges where particles charge is negative. The viscosity of surrounding gas plays an important role in conditions of the thermal plasma (as the pressure is atmospheric), as compared to the conditions of low-pressure gas discharges. But the main distinctive features is that the system is not in a stationary state. The point is that the structure was formed in a laminar flow of thermal plasma, and diagnostic measurements (including pair correlation function) were made at a height of approximately 35 mm over bumer surface (where flow is laminar). Taking into account the spray velocity
402
Y.K. Khodataev et al. /Ordered structure formation in a thermal dusty plasma
(about 5 m/s) the estimate for the dusty plasma flight time is ^^ ^ 7 ms. Thus, the time for structure formation is severely limited. This circumstance may explain why the structure obtained in the experiment is characterized by a sharp main peak with no high-order ones (weak correlation), in spite of the fact that the corresponding coupling parameter is sufficiently high F = Z^e n^ / T^ - 100 (where Z^'-- 500, n^ 5x10^ cm'\ and T^ = T^- 2000K are the dust particles charge, concentration, and temperature respectively). Latter we will show that the possible reason of this fact is that the plasma flight time is less than the time of structure formation. To investigate the dynamics of ordered structure formation in a thermal dusty plasma the numerical simulation have been carried out using Brownian dynamics method in 2D geometry. It includes solution of the equation of motion for each dust grain taking into account the interaction between dust grains, neutral drag force, and random force arising from asymmetric molecular bombardment (Brownian force):
^d^-
I^Wl '•=|r/t-ry|
'
m.^^
+ F,,
(1)
where m^ is the grain mass, rj is the neutral friction decrement, F^^ is the random force providing the Brownian motion. 0(r) is taken in the following form: 0(r) = - Z ^ e ^^d ^^ -
r
^d^^
2
r"
•
exp(- r IXA .Here ^^ is the Debye potential
^ - '
and X^ is the Debye length. The computation area is of square form with the side length LQ. In order to emulate an infinite system we use periodic boundary conditions. Initially charged dust grains are situated in random positions inside the computation area after which the process of self-organization starts. Such consideration corresponds to the real process in experiment where initially neutral and disordered dust grains come into the plasma region, acquire electric charge very quickly (6) and start interacting. Following parameters were obtained in experiment n^ (n^)= 7x10^° (4x10^^) cm'\ n^ = 5x10^ cm^ so that in numerical simulation we have used A^^j - 10 |Lim, and interparticle distance was / ~ 15 ^im. For the neutral gas fiiction the expression appropriate for the free molecular regime was used T] =
[inm^ I TA, where P is the neutral gas pressure and m^ is the mass of
neutral gas atoms or molecules. This choice is supported by photon correlation spectroscopy measurement of the dust particles diffusion coefficient (6). The dust mass is m^ « 1.5x10"^^ g, leading to r| « 9.6x10"^ s'\ The time step of simulation T^ should be less than r|"^ to simulate Brownian motion accurately, in particular r^ = 0.03/Vf, was taken with 250000 time steps being executed during the simulation. This is the reason why the 2D approach was utilized with relatively small number of grains
403
Y.K. Khodataev et al/Ordered structure formation in a thermal dusty plasma
(A^ = 200). A larger number would require too much computing time. Nevertheless this seems to be enough to take into account that the media with moderate value of F was investigated where the correlation radius does not exceed several /. Simulation has shown that after some relaxation process the system approaches the final stationary stage corresponding to the liquid-like state. This agrees with the values of the coupling parameter P. The time evolution of the pair correlation function R{r) calculated from the grains positions at the moment taken is presented on Figure 1 (dashed line corresponds to the pair correlation function measured in the experiment). The last picture in Fig. 1 was obtained through time averaging of R which is possible because on the final stage of the simulation (40 ms < ^ <70 ms) the system approach to the equilibrium state and the pair correlation fiinction does not evolve in time. The process of dust ordering can be examined and compared with the experimental results using Fig. 1. One can see that first of all, the particles at small distances disappear with the area of zero correlation function at small r being formed {t = 0.7 ms). This process takes place very quickly because the electric repulsive force t =5 ms
0.7 ms
t =7 ms
od
2
3
4
0
1
2
3
4
r/1 r/1 FIGURE 1. The time evolution of the conelation function in the numerical simulation. Dashed curve correspond to the experimental correlation function.
404
Y.K. Khodataev et al /Ordered structure formation in a thermal dusty plasma
at small distance is very strong. Then the sharp nearest-neighbor peak develops (^=5 ms). Further, this peak grows and simultaneously the high-order peaks develop {t = l ms). The final correlation function is characterized by many sharp oscillations (/ > 40 ms). Thus, the time of dust ordered structure for our conditions can be estimated as t^ - 40 ms. As it was mentioned earlier the flight time of dust grains in the experiment is ^ft = 7 ms < /f. This means that the pair correlation function measured in the experiment corresponds to the forming structure. However, the plasma flight time is large enough for the short order to appear in the system. From Fig. 1 we see that at ^ = 7 ms the first peak is already close to its final form and the process of high - order peaks formation is on the initial stage. Thus, the absence of high-order peaks of the experimental pair correlation function can be caused by the non-stationary character of the structure concerned. The second specific peculiarity of the experimental correlation function is that the peak is extremely wide. However, the mechanism leading to this effect remains unclear now and requires special analysis. In conclusion, we will make some remarks about the role of enhanced neutral gas friction in the experiment considered. When the friction is unimportant there is only one time scale for the ordered structure formation - the inverse dust plasma frequency -1
/
2 2
\-l/2
_.
^ pd - I^TiZ^e n^ I m^\ . For our dust parameters co^^ is '-- 0.2 ms that is rather small as compared to the simulation results. However, we can evaluate t^ as the time needed for a dust particle to travel the average interparticle distance n^ in the force field of the neighboring particle. This gives rise to another time scale which reads as (neglecting screening) /y ~ r|co^j ~ 5 ms for our conditions. This evaluation is in reasonable agreement with the simulation. The latter time scale for ordered structure formation is more appropriate when dealing with "high - pressure" plasmas.
ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Researches under Grant No.97-02-17565 and Grant No.98-02-16825.
REFERENCES 1. 2. 3. 4. 5. 6.
Fortov, v., Nefedov, A., Petrov, O. et al, Phys. Lett. A 219, 89 (1996). Fortov, v., Nefedov, A., Petrov, O., et al, Phys. Rev. E 54, R2236 (1996). Chu, J.H., and I, Lin, Phys. Rev. Lett. 72, 4009 (1994). Thomas, H., MorfiU, G.E., Demmel, V. et al, Phys.Rev.Lett. 73, 652 (1994). Fortov, v., Nefedov, A., Torchinsky, V. et al, Phys. Lett A 229, 317 (1997). Khodataev, Ya., Khrapak, S., Nefedov, A., and Petrov, O., Phys. Rev. E 57, 7086 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
405
Separation of Diamagnetic Fine Particles in a Non-Uniform Magnetic Field Isao Tsukabayashi*, Sugiya Sato* and Yoshiharu Nakamura* Nippon Institute of Technology, 4-1 Gakuendai, Miyashiro-machi, Saitama 345-0826, Japan The Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan
Abstract. The possibility of the control of the diamagnetic fine particles by the high gradient magnetic field is examined. The possibility of separation and control of diamagnetic substances is shown by means of a fine mesh in accordance with the high gradient magnetic separation method.
I . INTRODUCTION In the dust plasma experiment the position control such as the hold and transport of the fine particle is a fundamental problem. And also the same kind of controls and separation of the impurity particles from the mixture dust atmosphere are important in the application of industrial process. In order to introduce the dust particles of nonelectrification state, the sublimation method, the spout by a vibrator and the free fall under gravity are used. The transport by atmosphere gas and the pressure of soimd are also used in the low vacuous situation. susceptibility 1 chemical The control by the electromagnetic substance symbol [xlO-*cm'/g] forces is considered when the dust Copper Cu -0.09 1 particles are charged. In that case, such as dust plasma, the electric field is the Mercury -0.18 1 Hg object of an interest as a main control Carbon C -0.49 1 parameter in formation of the Coulomb crystal by self-organization and so on. Benzene -0.71 1 CfRe Control by the direct electric field is 1 Water H2O -0.72 1 difficult because of the sheath formation P Phosphorus -0.86 1 in plasma condition. The research of the active control of the ultra-fine particle Bismuth Bi -1.34 1 that tumed into the plasma by the Hydrogen H -2.00 electromagnetic field is carried out by Kawashima et al. recently [1]. Black lead C? -3.66 1 It is generally thought that the direct Cholesterol C27H46O -8.18? 1 control by the magnetic field is difficult, and the object has been limited to the J A B L E h Magnetic susceptibility of main ferro- or para- magnetic substances. The diamagnetic substances.
406
/. Tsukabayashi et al /Separation of diamagneticfineparticles
substances whose magnetic susceptibility is small are being used mainly as the dust, and the direct control by the magnetic field have not been considered. By the way, all of the substances of the nature have the diamagnetism that is originate in motion of the electrons. Because the magnetic moment of the paramagnetism by the spins is generally bigger than that of the diamagnetism, the effect of the diamagnetism is not admitted conspicuously. Therefore much attention have not been paid about the application of the diamagnetism. However, the research for such weak magnetism is advanced due to the progress of the recent high magnetic field technology. For instance, the water is a famous diamagnetic substance and the magnetic levitation of it is demonstrated by means of superconducting magnets [2]. And also the research in the relation with the living body is advanced. The bismuth is representative diamagnetic matter in metal. Also most of the organic compounds show diamagnetism. The magnetic susceptibility of main diamagnetic matter was shown in Table 1. On the other hand, there is the high gradient magnetic separation method as the separation method that used magnetism. This method is used to the purification of the water and the separation of the fine metal particles as application. And recently, even the possibility of the more advanced magnetism chromatography is studied. The difference of magnetic and the grain size of the fine particles, flowing in a high gradient magnetic field, are used in that method. The carbon compounds are conceivable as the dust fine particles and many of them have comparatively strong diamagnetism. Therefore it is useful if the control and selection of the particles are able to do by utilizing this nature. Moreover, there is the possibility that brings the new reaction similar to the experiment in space, because that the magnetic levitation of the diamagnetic substance is equivalent with a gravity free state essentially.
II. REPULSIVE FORCE TO DIAMAGNETIC SUBSTANCES Magnetization M of the matter of the magnetic susceptibility z^ in a magnetic field H is as follows. The magnetic energy of such a diaiiiagnetic substance in the magnetic field is given by
In this case the force that the magnetic substance receives becomes; F = -grad U
Where only the direction of the magnetic field is considered . In the case of diarnagnetic substance y,.^ < 0, so that this force is repulsive for the diam^.gnetic substances. The formula above shows that this repulsion force depends on both the strength and the gradient of the field. That is, no force appears in the homogeneous field even if the strength of the field is high. Conversely speaking even comparatively
407
/. Tsukabayashi et al /Separation of diamagnetic fine particles
weak magnetic field strength can produce strong force if there is a high magnetic gradient. In the magnetic levitation of water, the magnetic gradient is optimized together with strong magneticfieldthat generated by superconducting magnets. From this point of view, the idea of the high gradient magnetic separation method is to put a fine mesh in a magnetic field. Then it creates magnetic curvature and high gradient of thefield.The condition that magnetic repulsion force balances to be gravity, in short the condition of levitation, for several diamagnetic substances lo^r I __^ 1 1 -^ , _ _ , '*'- \. ^ 'X. ^ are shown in Figure 1. It is lO^h ^ Bismuth H \AA^ understood from this figure that is D Graphite \V^' able to produce the force to Water J V^ ° \ \ r-JO^j ti. \ 0 s compete with gravity even in the X magnetic field strength of several 6m .J X \ uX^ kGauss, if 10^-10^ Gauss/cm of the 03 '\ X N \ \\ magnetic gradient is achieved. lO^h X X^ H This magnetic field strength is the 10'h range where is able to form a \ X \ usual strong permanent magnet. _J 1 1 i\ X 1 10»i So that it is expressed below we W 10^ 10^ 106 10^ 10« \CP W W did the verification experiment of grad B [Gauss/cm] the repulsion effect of the F I G U R E 1. Levitation condition of diamagnetic magnetic field to a diamagnetic substances in the magnetic field, gradient fine particle. _ j
* ••-
j
N . "^
X ^
W
I
*•••
X '^ X N
V ^
mm-
X. N
«
1
1
M. VERIFICATION EXPERIMENT OF DIAMAGNETIC EFFECT In this experiment the bismuth fine particle is used as the diamagnetic substance. A cylindrical neodymium magnet (the surface magnetic field strength is 3 kGauss) is used as the source of the magnetic field. The mesh disk made of the stainless steel ( SUS430F) was put on the magnet to produce a high magnetic field gradient locally. At this time the magnetic field gradient is estimated about 10^ [Gauss/cm] , showed in Figure 1 as X mark, and was thought with sufficiently to see the effect of repulsion force although the condition of levitation is not filled. The bismuth fine particle was sprinkled to the thin paper that put it on the mesh. The pattern that appeared when it is oscillated is shown in Figure 2. The figure shows that it is flocculating to the aperture part (the holes) of the mesh by repulsion force, although it is expected that it gather on the line of the mesh if it is paramagnetic FIGURE 2. Condensation pattern of substance powder.
diamagnetic fine particle in the magnetic field. gradient
408
/. Tsukabayashi et alJ Separation of diamagneticfineparticles
IV. DIAMAGNETIC DUST PLASMA EXPERIMENT (A PLAN) The diamagnetic dust plasma experiment will be carried out with the device that is shown in Figure 3. The dust chamber main frame is the flat cylinder made of nonmagnetic stainless steeel (SUS304) of the outside diameter 0 165 mm and of the height 60 mm. In the upper part of chamber the plasma source and the dust storage are attached. The dust sprinkler that causes dust fallen with the oscillation method is incorporated to dust storage A. In the dust sprinkler, the bismuth fine particle of grain size 10 // is loaded on the stainless steel mesh of 300 mesh with the line diameter of 0.04 mm. And it is oscillated with a bolt bundled Langevin type ultrasonic oscillator (45kHz) to cause the dust fallen. Argon plasma is formed by the hot cathode filament supported dc discharge with the argon gas pressure of the order of 10'^ Torr in plasma source B. C is the neodymium magnet of the outside diameter 0 39 mm, inside diameter 0 19 mm, the height of 7 mm and be adopting the ring style in order to introduce bismuth dust to high gradient magnetic field space. The magnet E in the lower part is a disk magnet of the diameter 0 30 mm, the height of 15 mm and the surface maximum magnetic field strength is 3.5 kGauss. Ferromagnetic material filter D for high gradient magnetic field formation was arranged in the intermediate department of C and E magnet. The high gradient magnetic filter is the disk made of SUS430F with the outside diameter 0 30 mm, thickness of 1 mm and has about 900 holes of the diameter 0 0.60 mm with the pitch of 1X1 mm. Although it is not shown in Figure 3, the steel yoke is attached between C, E magnet for the purpose of the magnetic flux leakage inhibition and magnetic flux density reinforcement. The working ports (ICF 0 70X3 and ICF 0 34 X 3) are attached to the ,, NJ 1 circumference of the dust chamber main frame. These ports are used as for the evacuation, the instrumentation and the control of FIGURE 3. Setup of a diamagnetic dust experiment. the dust chamber.
REFERENCES 1. N. Kawashima, K. Takeda, T. Etoh, S. Besshou and H. Kubo, "Control of Ultra-Fine Particles in Plasma by an Electromagnetic Force", Proc. of The Second International Conference on the Physics of Dusty Plasmas (1999). 2. E. Beaugnon and R. Toumier, "Levitation of Water and Organic Substances in High Static Magnetic Field", J. Phys. Ill France 1, 1423 (1991)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
409
Minimal Charge Asymmetry for Coulomb Lattices in Colloidal Plasmas: Effects of Nonlinear Screening O.Bystrenko and A.Zagorodny Bogolyubov Institute for Theoretical Physics, 252143, Kiev, Ukraine
A b s t r a c t . The effects of nonlinear screening of grains on liquid-solid phase transtions in colloidal plasmas are studied, based upon the Yukawa model. It is shown, that there is a minimal charge asymmetry needed for crystallization of a Coulomb lattice and its value depends on the packing fraction of the colloidal component. The smaller packing fraction, the higher charge asymmetry, and, respectively, the stronger coupling in colloidal component are required to reach the solid state.
In the present work we are going to consider the properties of colloidal plasmas (CP), i.e. the plasmas containing large number of colloidal (macro) particles. By virtue of various physical processes, the colloidal particles (grains) can attain the charge which exceeds the electron charge by orders. As a consequence, the potential Coulombic energy in colloidal component may be much greater than its kinetic energy, which results in a possibility of a condensed state in colloidal component [1-3]. In particular, a liquid-crystal freezing transition with formation of Coulomb lattices in the colloidal component may be observed. The basic reference systems of CP based on the notion of effective interaction [4-6] is the so-called Yukawa-system (YS) with the interparticle effective potential V{x) = \exp{-^)
(1)
with two dimensionless parameters: the coupling F and the screening length A; x is the dimensionless distance. Thermodynamical properties of the YS, in particular, its phase diagrams, are investigated rather well by extensive computer simulations [7,8]. The connection between the dimensionless parameters F and A of a YS and the microscopic parameters of CP can be established in the following way.
410
O. Bystrenko, A. Zagorodny /Minimal charge asymmetry for Coulomb lattices
u&o g^l.OO o O oO-SO
0.00 0.00
0.20
GrQ\n radius, o/ro
FIGURE 1. Relative effective charge Z* vs. grain radius, the nonlinear PoissonBoltzmann theory [12]. Z = 25; the coupling in the plasma component Tp = 0.1(A); 0.05(5). The line (C): the linear (DLVO) approximation. Let us consider a two-component asymmetric strongly coupled plasma, which is a simplest example of CP. A good microscopic model for it is a system of charged hard spheres interacting by Coulomb forces. In the case that the size of a plasma particle is negligiblly small, such a system can be described by three dimensionless parameters: - packing fractions for colloidal component: v = n7ra^/6; - charge asymmetry Z; - coupling constant of one component to the other x = 2Ze^/{kBTa). Here T is temperature, n is concentration of colloidal component; a is the diameter of a grain, /c^ is the Boltzmann's constant, e is the charge of a plasma particle. Under the assumption that the screening can be described in terms of the linear Debye-Hiickel theory for point charges, one can easily, get the effective interaction in the form (1), with the parameters of YS specified as F = Z'^e^/{ksTd)^ and A = TD/d with X = r/d. Here d = (47rn)~^/^ is the average distance between colloidal plasma particles; TD = {A:T[nige^IksT)'^!'^ is the Debye screening length with n^g being the background plasma concentration. The potential energy in Eq.(l) is measured in units of ksT, A more accurate approach with account of a finite size of grains leads to the well known DLVO theory [9,10] resulting in a similar effective interaction, except that the charge Z in Eq.(l) must be replaced by effective charge Z* = Zexp{a/2TD)/{1 + (J/^TD)However, recent studies performed within the nonlinear Poisson-Boltzmann the-
O. Bystrenko, A. Zagorodny /Minimal charge asymmetry for Coulomb lattices
411
ory and via microscopic Monte-Carlo simulations [11-12] have shown that the effects of nonlinear screening essentially affect the behavior of the effective charge Z* at strong plasma-grain coupling (for x > 4). For illustration, we give the relevant dependencies in Fig. 1. In the case of small grain sizes, there is a sharp decrease in effective charge as compared to the linear DLVO theory. The physical reason for it is the effect of 'plasma condensation' near grain surfaces. According to experimental estimates of parameters of CP {v :^ 10~^ — 10~^ and x — 1 — 10), the above nonlinear effects in screening may significantly influence the properties of CP. Below we discuss the influence of nonlinear effects on melting curves within the Yukawa model. Our consideration is based upon the results of the works [11,12] and the idea, that the properties of CP can be described by effective pair interaction in the form (1) even in the case that the nonlinear screening is significant [11]. In these works it was shown that the effective screened potential in this case retains the Debye-like form at distances. However, in this case the effective charge should be found from exact solution of the relevant PB equation; the background density n^^, which determines the Debye length, is assumed to be equal to the average plasma background concentration. Within such an approach, the account of nonlinear effects reduces to re-scaling the well known melting curve for YS [7] with the use of the relevant effective charge Z* instead of the hare charge Z. Namely, in order to determine the state (liquid or crystal) associated with a given point in the r — A plane, we solved numerically the nonlinear PB equation for a single grain in one-component plasma background in a spherical cell with relevant parameters. After that, the effective charge Z* and the effective coupling F* were evaluated from comparing the known linear Debye-Hiickel solution and that obtaind by numerically solving the nonlinear PB problem. Finally, the point (F*, A) was checked to belong the area of crystal or liquid state in the known YS phase diagram. The results are given in Fig. 2a. The grain radius is the function of the packing fraction, thus, we obtain a set of meling curves for different values of v. Due to the connection between the microscopic parameters of two-component asymmetric plasmas and the parameters of the Yukawa model one can obtain important qualitative conclusions about a minimal charge asymmetry Zmin needed for formation of Coulomb lattices in CP. Namely, under the above assumptions, the following relation can be obtained
z = ^A^ which makes it possible to transfer the melting curves onto the Z — A plane (Fig. 2b). As can be seen from the figure, there exist a minimal charge asymmetry Zmin = 350 needed to obtain a crystal state. In this connection Ref. [5] should be cited, where the same conclusion and a close value of Zj^m ~ 360 was obtained based on the Lindemann melting criterion for the case of specific effective graingrain forces. However, the account of nonlinear screening results in shifting melting curves to higher values of charge asymmetry Z at small packing fractions of colloidal component.
O. Bystrenko, A. Zagorodny /Minimal charge asymmetry for Coulomb lattices
412
v=0.05 v=0.005 v-0.0005
2500
a)
b)
_ v=0.05 _ v-0.005 __ v=0.0005
M2000 4>
c
I 1500
8000H
CRYSTAL 0) 1000 O
4.000-
FLUID FLUID 0.00
JJ^T^
1.20
I I I I M I I l| I
0.00
O.SO .
Debye length. A
111111111111111111111111111111111
1.00
1.90
2.00
Debye length, A
FIGURE 2. Melting curves for colloidal plasmas: a) T - A plane; b)Z - A plane.
ACKNOWLEDGMENTS The authors thank Prof. P.Schram for valuable discussions. The financial support for the work was provided in part by State Fund for Fundamental Research of Ukraine (Project No. 2.4/319) and by International Association INTAS (Projects No. 96-0617 ajid No. 95-0133).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Pieransky, P. Contemp.Phys. 24, 25 (1983). Thomas, H., et al. Phys. Rev. Lett. 73, 652 (1994). Chu, J.H., and I, Lin. Physica A205, 183 (1994). Lowen, H., Madden, P.A., Hansen, J.P. Phys.Rev.Lett. 68, 1081 (1992). Schram, P.P.J.M, Trigger, S.A. Contrih.Plasma Phys. 37, 251 (1997). AUayarov, E., Lowen, H., Trigger, S. Phys.Rev. E57, 5818 (1998). Robbins, M.O., Kremer, K., Grest., G.S. J.Chem.Phys. 88, 3286 (1988). Dupont, G., et al. Mol.Phys. 79, 453 (1993). Derjaguin, B.V., Landau, L. Acta Physicochimica (USSR) 14, 633 (1941). 10. Verwey, E.J., Overbeek, J.Th.J. Theory of the Stability of Lyophobic Colloids, Amsterdam: Elsevier, 1948. 11. Alexander, S., et al. J.Chem.Phys. 80, 5776 (1984). 12. Bystrenko, O., Zagorodny, A. Phys.Lett. A255, 325 (1999); Cond.Matt.Phys. 1, 169 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
413
Characteristics of Asymmetric Ion Sheath in a Negative Ion Plasma Kazunori Koga ^ , Hiroshi Naitou^ and Yoshinobu Kawai Dept. of Advanced Energy Engneenng Sciences, IGSES, Kyushu Umv., 6-1 Kasuga-kouen, Kasuga, Fukuoka 816-8580, Japan ^Faculty of Engeneenng, Yamagucht Umv. Tokiwadat 2557, Ube, Yamaguchi 755-8611, Japan
Abstract. The suppression of the ion sheath instabihty is observed by introducing negative ions. It is confirmed that the current at the presheath plays an important role in the excitation (also suppression) of the instabihty. A new oscillation mode of the ion sheath is observed at high negative ion concentrations. Moreover, it is found that the species of dominant negative ions changes from F~ to SF^ with increasing SFe gas flow rates.
INTRODUCTION It has been pointed out [1] that there are numerous negative ions in reactive gas plasmas, such as silane and fluorocarbon plasmas, and that negative ions play an important role in plasma processing. How^ever, the behavior of negative ions in such reactive gas plasmas has not been clarified. Thus, the study of negative ions becomes one of the important subjects in plasma processing. Interesting phenomena [2] about an ion sheath were experimentally investigated using a double plasma (D.P.) machine [3]. It was pointed out that the current at the presheath region plays an important role in the excitation of the instability [4]. Recently, the ion sheath instability was observed in a negative ion plasma [5]. It was found that the instability was suppressed by introducing negative ions. In this paper, we firstly discuss the suppression mechanism of the ion sheath instability by measuring the profiles of the potential and ion saturation current. Secondly, we investigate the instability at high negative ion concentrations. Finally, we report the experimental results of the compositions of negative ions by the mass spectra analysis. ^^ present address: Graduate School of Information Science and Electrical Engineering, Kyushu University, 6-1-10 Hakozaki, Higasiku, Fukuoka, Fukuoka 812-8581, Japan
414
K. Koga et al /Characteristics of asymmetric ion sheath in a negative ion plasma
EXPERIMENTAL APPARATUS The experiments were carried out with a D.P. machine. In this experiment, Ar and SF« gases were used and precisely introduced into the chamber with a mass flow controller (MFC). The negative ion concentration a was estimated from the ratio of the electron saturation current with negative ions to that without negative ions. The separation grid was located at the center of the D.P. machine in order to divide into two plasma regions (the driver and target region). Fluctuating parts of the separation grid current were obtained from the voltage drop across the resistor and were analyzed with a spectrum analyzer. The plasma parameters were measured with a 6 mm-diam.-plain Langmuir probe. In the pure argon plasma (SFg: 0 seem), the electron temperature and plasma density was 1.0 eV and 5xl0^cm~^, respectively in the target region. The mass spectrometry was achieved using HIDEN EQP [6] where the separation grid was removed from the D.P.
RESULTS AND DISCUSSION When the separation grid was negatively biased and the plasma density in the target region was much denser than that in the driver region, an oscillation in the grid current with high cohearency was observed at the frequency close to half the ion plasma frequency, that is, the ion sheath instabihty was excited in a negative ion plasma. The frequency and amplitude of the oscillation depended on both the grid bias V^ and a [5]. In order to investigate the behavior of the instabihty in detail, we measured the axial profiles of the sheath structure. When a was increased, the space potential tended to decrease and the instability was suppressed [5]. This tendency was stronger in the driver region rather than in the target region. The potential swell was also observed at the sheath edge in the driver region. Local structures in the ion saturation current (/is) profiles were formed near the sheath edge in both the target and driver regions [5]. The peak value of the I\^ in the driver region was independent of a. This result suggests that positive current is constant in the presheath region of the driver region. In our previous report, it was found that the space potential was oscillated by the ion flow into the presheath in the driver region and negative ions exist in the presheath region. The total current in the presheath region decreased as a increased and the potential oscillations in the driver region were suppressed, so that the time-averaged space potential decreased. Since the oscillation amplitude was larger in the driver region than that in the target region, the decreasing tendency of the space potential was stronger in the driver region. As a result, the ion sheath instability was suppressed. This result is consistent with the excitation mechanism of the instabihty ,which has been proposed in Ref. [4]. In previous experiment [5], the instability was easily suppressed for any a. However, the instabihty tended to be excited even for higher a when the density difference between the two regions is large. Here, we investigated the behavior of the
K. Koga et al /Characteristics of asymmetric ion sheath in a negative ion plasma
300 Q
300
I I I I I I I I I I I I I I
(a)
290 CP
0.8
N
X :^280 o c
0.6
P
-I 0.4 •§
g.270 260 250
c 3
3
*
0}
0
415
n
•Q.
E
i
Q
•
_L 0.01
0.02
_L 0.03
0.2 03
• -L-LL.
0.04
• I • •
0.05
J] 0.06
SF Flow Rate (seem)
FIGURE 1. (a)The dependence of the frequency (o) and the amplitude (•) of the instability on the SFe gas flow rate. (b)The dependence of the frequency on the electron density in the target region A^eT- The grid bias \Vg\ is kept constant at 80 V.
instability at high SFe flow rates (high a). Up to 0.001 seem [a — 0.053 in the target region), the instabihty was rapidly suppressed. The amplitude normalized to one at the 0 seem deereased to about 1/10 and was kept nearly eonstant up to the 0.06 seem {a — 0.55 in the target region). On the other hand, the frequeney inereased from 256 to 273 kHz for the flow rate between 0.01 and 0.06 seem. In the previous experiment [2], it was observed that the frequeney is proportional to the eleetron density in the target region. As seen in Fig. 1(b), the frequeney is proportional to the eleetron density whieh deerease as the flow rate inerease, over 4 x lO^em"^ (SFe flow rate: below 0.01 seem). The instabihty observed in the previous experiment was exeited for the flow rate between 0 seem and 0.01 seem. However, at 4 x lO^em"^ (over 0.02 seem), the frequeney deviates from the straight hne as shown in Fig. 1(b). It seems that the new oseillation mode is exeited within the ion sheath. In Ref. [5], the dominant negative ion was determined by the propagation of the ion aeoustie wave. It was found from the dispersion relation of the ion aeoustie wave that the dominant negative ion is the SF^ or SF^ ion. In order to study quantitatively the instabihty, it is neeessary to determine the eomposition of the ion speeies. We attempted to determine them by mass speetrometrie analysis. Figure 2 shows the negative ion speetra for different flow rates. Firstly, F~ ions appear and beeome dominant at the low flow rate(0.005, 0.01 seem). The speetrum of F~ ion peaks at eertain flow rate and deereases, whereas the speetrum of SFg and SF^ inereases with inereasing the flow rate. Espeeially, SFg speetrum takes a large eount rate when the flow rate is inereased. The fraetion of SF^ ions is nearly equal to that of F " at 0.005 seem. Above 0.005 seem SF^ ions beeome dominant. It was found that the eomposition of negative ions depends on the SFe flow rate. Sinee the phase veloeity of the ion aeoustie wave beeomes more sensitive to a at higher a, dominant ions are determined in the high a region (a = 0.4 - 0.8), that
416
K. Koga et al /Characteristics of asymmetric ion sheath in a negative ion plasma
SP- (146 amu)
F-(19) h i
• L
SF;(127)
SF flow rate: 0.3 seem
2.5 1Cf r # ^ 2 10®&1.5 1Cf & -
1 lO^g^
5 lO^ffiy 0 1 0 ° ^
50
100
mass (amu)
150
200
^0.01 seem __ 0.005 seem — 0 seem
FIGURE 2. Negative ions mass spectra for different flow rates. is, high SFe flow rate. Therefore, the dominant negative ions were regarded as SF^ or SF^ in the previous experiment [5]. This result well consists with that of the mass spectra analysis.
CONCLUSION We provided a model of the suppression mechanism of the ion sheath instability by measuring the profile of asymmetric ion sheath. The decrease of the total current at the presheath in the driver region caused the suppression of the instabihty. This model consists with the excitation mechanism of the ion sheath instabihty in pure argon plasmas. It was found that a new oscillation mode is excited at high SFe flow rates. Furthermore we attempted to determine the composition of the negative ions by the mass spectra analysis. It was found that the dominant negative ions are F~ at low flow rates and SFg ions take the place at high flow rates, that is, the composition depends on the SFg gas flow rate.
REFERENCES 1. 2. 3. 4. 5. 6.
H. Amemiya, J. Phys. D 23, 999 (1990). N. Ohno, A. Komori, M. Tanaka and Y. Kawai, Phys. Fluids B 3, (1991) 228. R. J. Taylor, K. R. MacKenzie and H. Ikezi, Rev. Sci. Instrum. 42, 1675 (1972). K. Koga, H. Naitou and Y. Kawai, J. Phys. Soc. Jpn, (to be published in vol.68). K. Koga and Y. Kawai, Jpn. J. Appl. Phys. 38, 1553 (1999). P. Charbert, R. W. Boswell and C. Davis, J. Vac. Sci. Technol. A 1 6 ( l ) , (1998) 78.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
417
Coherent Iiiteractioii Model of Attractive Forces in the Plasma Crvstals Ichiro Mori, Toshifumi Morimoto* and Kikuo Tominaga Fa c u Ity of Engin eering Th e Un iversity of Tokush im. a Mmami-josanjima 770 Tokushima Ja/pan *Takinna Radio Technical College Kohda, Takuma^ 761-11, Kagawa, Japan E-mail :mori@ee. tokushirna-u. ac .j p
Abstract A new nonlinear tlieoi\y for creation of an ion wave by an electron flow, which will be expected to bring about an attractive force through the action of the ion wave in a plasma crystal, is proposed. The difference of the new theory compared with the common theories, is an intensity dependent theory with coherent interaction. It is shown that an electron flow emits the elementarj/' excitations which are orderly lined up with time, when the flow collides to the negative front of the ion wave. The experiments of ion wave excitation by an electron flow are performed with an electron beam-plasma interaction system. The ion waves appear as triangular wave or as an ion solitary wave. The strong coupling between the ion wave and the elementary excitation is confirmed. As the results, existence of ordered arrangement of the elementary excitation is realized from the theory, while the presence of the strong coupling is verified experimentally. From the results, the crystallization seems to be possible. Keywords : nonlinear theory, ion wave, attractive force
1. Introduction Recently, several authors have experimentally shown the co-existence of the Coulomb plasma crystals and the enhanced low-frequency fluctuations^^), while there have been theories where an ion flow plays significant role for generation of ion waves which cause attractive forces between dust particles^'^^ It seems easier to explain, however, that an electron flow generates the ion waves when its speed exceeds the sound velocity, because of its light mass. This mechanism is quite similar to the explanation*'-^) of the auroral kilometric radiation (AKR) which occurs due to hremsstrahlung interaction between aurora beam electrona and electrostatic ioncyclotron turbulence.
2. The nonlinear theory To explain the experimental phenomena, one must consider 3'"^'' order nonlinear process since a soliton (elementary excitation) appears, therefore the Green's function, which means a 'kerneF connecting the initial stage to the final stage of interaction, must at least second order quantity for calculation of soliton's electric field.
418
/. Mori et al./Coherent interaction model of attractive forces in the plasma crystals
Thus the mechanism describes the interaction between two waves and one particle. We examine to describe the interaction by using the renorm,alization technique which was developed by several authors^^^ For a soliton one must put emphasis on coherent interaction, i.e., fei =fe,while the theory of Weinstock dealed with the case I fell > > |fe|, and if we use the relation ki — k in the theory of Kono, Ichikawa, their diffusion coefficient degrades to that of the quasi-linear theory. Then we selected a Gausian-type Green's function proposed initiahy by Horton^''' and define a diffusion tensor D,%^, then the Green function becomes : Ga{k,v,i\k',v',i')
= —i I —7^kikjDij{k,v)t\
• exp
ki{v - v')ikj{v - v')j AkikjDij{k,v)t
kikjDij{k,v) 12
o
' .^f^'(v^v)^ ^k,k'
i
Fourier transformation of the above Geen^s function is described as : ^-^^ m!
k^
li-uj + c)
• ^ 3 m - l { 2 \ / ^ « ( - ^ + c ) } . b^^'K.,^.' _ l^ijv - v')ikj{v
Q —
— -
—
—
- v')j -
4kikj-Dij{k,i'))
,
_ kikj-Dij{k,v)
0 —
—-
12
,
_
C— K
V —
(1) tl^ai
where the function K^^-,_L{X) is the modified Bessel function and m = 0 term is very important and K_L{X) - KL{X) = y^7r/{2x)'exp{-x). The terms niT^O are proportional to fe^"\ For the quantity kikjDij/k\ we use D^j simply since the kikj/k'^ is 0 ( 1 ) . We get the following one set of self-consistent equations : D,j{k,v.Lj)=
(—Xy
'-^^^^\Eikr^LV,)\'G.Ak
(2)
- k,^v
dk' -^(^^y^JdvJdv'J-^G,{k,v,co;k',v',co'yfi°\k',v\u>')
E{k,c.) = Uk,v,iu)
f
=
S„(fc,t.,u;)= ( ^ ) G^-=°Hk,v,.;k',v\.')
Jdv'G.Ak,v,w;k',v',w'yfi"\k',v',iv')
(4)
(5)
T[^E{k,,u;,)~{G^{k-k^,v,i,-LO,)-Ei-k,,-u;,)-^} \v — v' = -^-^.^---^.ex-p (^-i-^.y^P^^TOJ
(3)
-^fc.fc'*...'
(1')
where the E{k,uj), fi^\k\v',UJ'), C, E,^(fe,^^u/') are the electric field of elementary excitation, the initial beam distribution, c - k-v-iT^^^ and the self energy term respectively. The ReEa is collision frequency and the hnEa corresponds to frequency broadening. Since the eq.(5) is operator, we multiply the eq.(5) by eq.(l') and after integration with respect to cJi, we divide the result by the eq.(l') and get the value of the EQ,.
/. Mori et al /Coherent interaction model of attractive forces in the plasma crystals
419
We t a k e afterwards |jE;(i^i,u."i)|' as ion wave intensity a n d a s s u m e d t o b e c o n s t a n t or slowly varying q u a n t i t y c o m p a r e d with carrier frequency a n d also t a k e t h e v' or 1^0 as b e a m velocity, t h e n t h e t r a n s f o r m a t i o n of Ea(fe,v,u;) b e c o m e s as follows : kiihi
ki
ZTT"
\Eik,,iv,)\'
-exp
-exp{-iiJit
- i{k -ki)-
vt - Ej}
(6)
To solve t h e above e q u a t i o n s w i t h respect t o t h e electric field, W e used only m = 0 - t h t e r m , which is t h e m o s t i m p o r t a n t one, t h e n we get E{k,t) in initial value p r o b l e m s by inverse t r a n s f o r m a t i o n . E(kJ,)
Y.
1
i'^-ni.vo
f(0
/r'(fc
•exp(—ik-vot)-sinn 1 ai = iCi\E\'t DiAk,v,t)~
w h e r e /?.&, f[^\k),
\Ef = \E{ki,-^i)\\
Ci —
C\ =
\E{ki,wi)
1 2^
2
kijkxj
/TT^a
(7)
g=1.3,
k\
nic \V — T)
/A,
Iml •2k
l\2
•exp •exp{-iLOii - i{k - ki) • vi
:j}
\v - i/|, T{q,x) a r e t h e b e a m density, no-dimensional form factor of
t h e b e a m , correlational length in velocity space, i n c o m p l e t e T-function, respectively. T h e A;j, ^a in t h e right h a n d side of eqs.(6), (8) are zero-th order q u a n t i t i e s come from e q s . ( l ) , (1') a n d we a s s u m e d firstly to be c o n s t a n t s , t h e n we get a relation : ReEaik^vJ)
ReD,j{k,v,t) = 5--
(9)
1
'
1
1
'
200 -
T h e eq.(9) is t h e effective coUision fre-
1
-\
q u e n c y of particles a n d waves o b t a i n e d by Tsytovich^^^ if we ignore t h e factor 5 a n d exchange our correlational length in velocity space, \v-v'\,
w i t h v. At t h e
factor a p p e a r e d in t h e eqs.(6),(8) : exp{-iujit
- i{k - ki) • vt - ErJ}
lij Q
0
QL
< 200 -
-
If in t h e case (a) : (ki^uJi) belongs t o u p p e r h y b r i d wave b r a n c h a n d UJI - ki • v, t h e n t h e factor oscillates quickly with frequency k • v a n d soliton will n o t ap-
1
3 TIME (sec)
1
4
.
5 . [x10-^]
Fig.l. Calculated results : Amplitude of the elementary excitation (soliton) is shown. If the (ion wave frequency) a n d t h e direction quantity T^a is a fiuiction of the ion plasma freof (k - k^) is p e r p e n d i c u l a r t o v ( b e a m quency, then the electron ermts the elementary ^ / r -n • 1 excitation. |i? - i/1 = 9.8xlO'^rn/s) is used. velocity), t h e n t h e factor oscillates with
p e a r , while in t h e case (b) : if u;i = uJi
420
/. Mori et al /Coherent
interaction model of attractive forces in the plasma
ion wave frequency then the soHton will appear. At resonance of nonlinear Landau-damping (c): {LU
- uJi) - {k - ki) • 1? + iT^a = 0,
The factor fluctuate with the width of broadenning, /mEa, where / m E a - 0(u.',:) and the amplitude of envelope soliton depends on its width times t, as shown in the hyperbohc-sine term in the eq.(7). Figure 1 shows periodical behavior of the soHton which is similar to tha,t of experiment, and figure 2 represents resonant interaction between the ion wave (wave number ^-i) and two upper hybrid waves k and {k - ki). If the beam velocity is very large compared with ion sound velocity, then the ion waves which propagate to almost all direction can couple to the electron beam.
(a)
(b)
Fig.2(a):Triangular representation of the Stokes frequency generation, and (b): t h a t of the aiiti-Stokes are shown, when the ion wave interact with two upper hybrid wa.ves.
3. Experimental results The experiment is performed with argon plasma. The argon is fed from 10"^ to 7x10"^ Torr. A beam of 8mm in diameter, 1.9keV-18mA, is injected into the argon gas. The soliton appears as a burst wave, the carrier of which is :^400 MHz. When the gas pressure is c^5.6xl0"^ Torr, i.e., at high plasma density, coupled ion wave with the soliton appears. Figure 3 shows the strong coupling state between the sohton and the ion wave. ^ Fig.3. The couphng between the soliton and the ion wave ' is shown. The narrow peaks are sohtons and the wide waves are 1 MHz-cornponent (with 300KHz-BW) of the ion waves.
;50mV/div.
crystals
/. Mori et al /Coherent interaction model of attractive forces in the plasma crystals The ion wave has a spectrum of the width 0 ~10 MHz, and has the maximum intensity at 200kHz. Power spectrum is 0.2V/MHz in 50-O system at IMHz, calibrated as impulse noise. As the 300KHz band width is used in fig.3, then amphfication of the spectrum analyzer becomes 4-times (in Volt), while the sohton is not amphfied and directly calibrated. The ion waves are surrounded by a pedestal consisting of dispersive wave whose width are almost same as that of the solitons. Figure 4 represents also the coupling of sohton and ion wave. Sohton has pointed peak and the ion wave appears as triangular .x.ow^ r... fl.^ 1 ^ T?r 1 wave on the beam current. Figure 5 shows ion wave generation on the pulsed beam current. The beam electron is scattered out by the ion wave and gradually diminishes in
50 Time [lis) Fig.4 Coupling between the ion waves and the elementary excitation (soliton) is shown.
"^^^^ rectangular signal represents the ion ^^^^ (ac-component of beam current), and measured by right side scale. The narrow P^^l^^ ^^^ envelope of the solitons detected by a mixer and measured by the left side
scale. In the measurements^ no active devices are used therefore the both scales are equal to the voltages getting at two t h e i n t e r a c t i o n . A s for t h e c o u p l i n g b e t w e e n antennae positions themselves. T h e beam .1 1-. ] ^1 • ^ . 1 electron emits the soliton when it colhdes t h e s o l i t o n a n d t h e i o n w a v e s , w e t r e a t a l s o to the steep negative front of the ion wave by Bremssirahlung. Such the well ordered p r e v i o u s ly'^'. structure by the solitons and the ion waves will compose the possible mechanism to References construct the plasma crystals. ( l ) H . T h o m a s , G.E.Morfill, V.Deminel, J.Goree,B. >, Feuerbacher and D.Mohlmann: Pl1ys.Rev.Lett.73 .t^ t (1994)652; F.Verheest: Advances in dusty ^ I 'isiy c plasmas, P.K.Shukla, D.A.Mendis and T.Desai, Q World Scientific, ibid. (1996) 61 (2)M.Nambu, S.V.Vladmhrov and P.K.Shukla: Phys. Lett. 203A (1995) 40; S.V.Vladiimrov and M.Nambu: Phys.Rev. 5 2 E (1955) R2172 (3)P.B.Dusenbery and L.R.Lyons: Geophys.Res. 87(1982)7467; S.Bujarbarua and M.Nambu: J. Phys.Soc.Jpn.52(1983)2285
^1 0^ I
\0h^>^*^*^**f^
0 Time (500xis/div.) (4)LM.Artshul and V.LKarpman:Sov.Phys. J E T P . 22(1966)361; T.H.Dupree: Phys.Fluids.9(1966) Fig.5 Figure shows ion wave generation, 1773; J.Wemstock: Phys.Fluids. 1 2 I 1 9 6 6 ) 1045 when the beam-pulse is injected into the ; M.Kono and Y.H.Ichikawa: Prog. Theor. Phvs dilute plasma which is made bv Ar-gas. 49(1973)754 The upper trace is light intesity of plasma and assumed to be proportional to plasma (5)C.VV.Horton; 'Long time Prediction in Dina\ density. T h e lower trace manifestes the mics John Wiley (1983)311 pulsed b e a m current increases downwards (6]V.N.Tsytovitch: Nonhnear Effects in Plasmas. started at zero of horizontal scale. The (Trans.by M.Ha,mberger),p.l71 New York London: beam is scatterd out by the ion wave and Plenum Press (1970) diminishes gradually, on which phases of the ion waves are initially uncertain but (7)I.Mori and K.Ohva: Phvs.Rev.Lett. r.Q (1989) it becomes accurate with time by the pull 1823 ^ 'oy V - ; into action with nonlinear character.
421
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) 2000 Elsevier Science B.V
423
CHARGE DISTRIBUTION FUNCTION OF NEGATIVELY A N D POSITIVELY CHARGED PLASMA DUST PARTICLES
B. F. Gordiets^ and C. M. Ferreira Centra de Fisica de Plasmas, Instituto Superior Tecnico, 1096 Lishoa, Portugal t -On leave from Lebedev Physical Ins. of the Russian Acad, of Sci., Moscow This article is an extension of our previous work ^ v^here the charge distribution function (CDF) of dust particles has been obtained as a function of a discrete negative charge Z = 0; —1; —2;... and various plasma and grain parameters. Here, the CDF is derived not only as a function of the discrete negative charge (the usual typical case) but also as a function of the positive charge Z = +1; +2; ... Positively charged grains can be formed v^hen secondary electron emission, that is, ionization of the grains by electron impact, is sufficiently important to change their charge. As in Ref.l an expression for the CDF is derived from the master equations for the densities N!^z\^, ^^z^ ^^ negatively or positively charged monodispersed dust particles w^ith Z = =[=1, ^ 2 , ... discrete elementary negative and positive charges and radius R. The change of charge |AZ| < 2 in the one collision of primary electron v^ith dust particle is taken into account in master equations. These equations are following:
-5f>|",7 + 5gj^,iV-%
(1)
424
B.E Gordiets, CM. Ferreira/Charge distribution function
for negative Z and \Z\>
^
1.
= [{Sf_, + Rz.,) Nfl, - AzNT] - [(4'^ + Rz) NT - Az+iiVf^] +
(2)
St2m"-2 - S^z^Nr
for positive Z and Z > 2. Note that N^z-2 = ^o for Z=2, where A^o is the density of neutral dust particles with radius R. For positive Z = 1 the equation will be similar (2) but instead part 5 j l 2 ^ z - 2 mast be Sf\^N\'^l^ The probabihties (in sec~^) of electron attachment A\z\^ Az, single and double ionization (secondary electron emission) SLL Sz
^ind S'L, 5^^ as well as reas-
sociation of one charged positive ions with the dust particles R\z\ are taken into account in these equations. The approximate analytical solution of quasi stationary Eqs.(l),(2) ( dN^z\l^^ ^ °; dNf'Idt
?^ 0 ) are following:
A^izT ^ ^0 n ^Tz'v ^ r ^ A^o n ©r, |ZM=1
(3)
Z' = l
where ^„e,
iVPz? ^ -^1^1 - R\z\ + sl{S\z\ + R\z\f
^r.r-i
+ 45gj^,"^
25fij,,
npos _ ^ ^ r ^ g^-i + Rz-i + ^/(gz-l + Rz-if
+ 45g!2Az
,,,
Here S^z\ = S\l] + 5 ^ , Sz = 4^^ + 5^^". The neutral particle density A^o is obtained from the condition N^ =No + E | = i N^"' + Eiz|=i N\z\^ where iV^ is the total density of grains with radius R. The average charge of dust particles is =
1 ^,
Z=l
|Z|=0
^ E ^ / r - E i^i/izTZ=l
|Z|=0
(6)
425
B.E Gordiets, CM. Ferreira /Charge distribution function
where f^
= iVf VAT^, f^^^^ = N^z\l^^
is the normalized CDF.
The "one quantum" charge probabiHties A\z\^ R\z\ and Az^ Rz for, respectively, negatively and positively charged dust particles have been calculated in Ref. 1,2. Here, we will, for simplicity, neglect polarization interaction, use Maxwellian EEDF and give the expression only for S\^^, 5 | | | , Sz\
$fi,c.rf^^,exp{-^^};
S^-
r i . r f W e ( l
We have obtained:
+
e'Z
AnReokT^
(8)
where v^ = {SkT^/TTrrieY^'^ is average electron velocity; N^, T^ , rn^ and e are the electron density, temperature, mass and charge, respectively; k is the Boltzmann constant; 6o is the vacuum permittivity. For positively charged dust particles
of
gf
(1 - ttz) Iz [ oiz {iz - 1) + (1 - «z) (2 -iz)
0 [ (1 - az) (TZ - 1)
if 7z < 1 if 1 < 7z < 2
if 7z < 1
(9)
(10)
if 1 < 7z < 2
The similar expressions will also for case of negatively charged grains. Herein, 7|z| j 7z and a\z\^ az are, respectively, the averaged on Maxwellian EEDF the secondary electron yields from dust particles which before collision with primary electron had the negative and positive charges \Z\^ Z and the fractions of the electron fluxes to the negatively and positively charged grains that are actually absorbed. We have used the simple model for calculation secondary electron emission coeflScients assuming, as in [3]. The presented in [3] method for calculation of 7^ is improved by taking into account the energy threshold Ei^j^ for ionization and absorption of not only secondary but primary electrons too. To calculate electron absorption coefficients ot\z\') otz we have also accepted that the primary electrons in the region where them
B.F. Gordiets, CM. Ferreira/Charge distribution function
426
energy E is small {E ^ Ei^) are not absorbed completely (as it was adopted in [3]) but can move in any direction as a secondary electrons. I I I I I I I I I I I I I I 1 I I »» 1 I I I I I I I I I I I I" f
-50-40-30-20-10
0
10
.«
Z
^ ^e ' ^V
Fig.l
The CDF for different Tg. Long dash line - calculation without the secondary electron emission.
Fig.2
Average dust particle charge < Z > as a function of the electron temperature Te for grain with different
radii. Solid lines - negative charges, long dash - positive charges, dotted Hues - negative charges for calculations without secondary electron emission.
The examples of a calculations are given in Figs.1,2. In these calculations, the following parameters are used: Eion = 15 eV, the "temperature" of secondary electrons kTsec = 5 eV, and the plasma electron and ion densities were assumed identical. Ion temperature and mass were taken 1000 K and 1 a.u.. We also have used for calculations the realistic values for ionization cross-section and assumed that the absorption lenth for electron L{E) ^ LQ x EI Eion, LQ = 10"^ cm. As seen from Figs.1,2, secondary electron emission can strong influence CDF and < Z >. The similar sharp change of grain potential was found in [3,4] where the effect was analyzed using the classical electron and ion currents to the grain, that is, without investigation of CDF. References ' B.F.Gordiets and C.M.Ferreira, J.Appl.Phys., 84,1231, (1998). '' B.T. Draine and B. Sutin, Astrophys. J. 320, 803, (1987). ^ V.W. Chow, D.A. Mendis and M. Rosenberg, J.Geophys.Res. 98, 19085, (1993). '^ C.K.Goertz, Review of Geophysics 27, 271, (1989).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
427
Control of Ultra-Fine Particles in Plasma by an Electromagnetic Field K. Takeda, N. Kawashima, T. Etoh, Y. Takehara (Kinki Univ.) H. Kubo (Nagasaki Inst of Appl. Science) and S. Bessho (Kyoto Univ.) ABSTRACT In order to utilize the weightlessness environment in the Space Station for the production of new material using ultra-fine particles (UFPs), a method is proposed to control ultra-fine particles (UFPs) in a plasma by a pulse magnetic field utilizing a charge separation electric field generated. The experimental setup has been completed. The particle size distribution has been obtained experimentally by measuring the velocit>^ distribution of falling particles in a gas pressure around 3,000 Pa and concluded that the UFP flov/ is composed of single particles. 1. Background Ultra-fine particles (UPFs) are chemically very active because of its large surface area compared with its volume. Now, in plasma physics, high research acitivity is going on as a dust (y) plasma. There the interest is on a strongly coupled plasma, that is, the electric potential energy between particles. However, the practical application of UPF is not much developed, only toner, magnetic tapes etc. The final objective of this experiment is to utilize the weightless environment of the Space Station to create a new material fi'om UFPs. On the ground, the UPFs are affected by the earth's gravity and it is difficult to make UPFs floating in space, particularly in vacuum. Then they drop on the bottom of the container and stick with each other. In the weightless environment, it is easy to make UPFs floating in space, however, it is not easy to isolate ultra-fine particles from the wall and control them as wanted such as the contraction or the translation of them. 2. Control of Ultra-Fine Particles (UFPs) UFPs are often charged up and it is rather easy to control their motion by the electric field. Hov/ever, when UFPs are charged up, they soon are scattered out by their mutual repulsion force. To avoid it, it is one way to put UTPs in a plasma to neutralize them. Then, it is difficult to control UPFs by the electric field because the electric field is shielded by the plasma and only penetrates into the plamsa within the sheath. Most of dusty plasma experiments have been done in the region of the sheath. For a physics experiment, it would be sufficient, but for the purpose of a new material production, it is essential that the bulk of UFPs in plasma can be controlled. In order to achieve it, it is proposed here to utilize the magnetic field as shown in
428
K. Takeda et al. /Control of ultra-fine particles in plasma by an electromagnetic field
Fig. 1. In general, the effect of the magenetic field on UFPs is so weak unless either the magnetic field is so strong or the velocity of UFPs is ver>^ high. In order to make the magnetic field effective, 5cm^Ultra-fme particle a pulse magnetic field shall reservoir be applied and the resultant Meshes vibrated by PZT charge separation electric field is to be used. When a Plasma Production ^ ^ Microwave plasma column with UFPs RF JiOJ^gBft^ in ii is contracted by the DC glow discharge magnetic field, the plasma ^ ^ is contracted but UFPs Ultra-fme remains. Then between the particle reservoir plasma column and UFPs, a charge separation electric Fig. 1 Control of UFPs using a charge separation field is produced and LTPs are attracted towards the contracted plasma column. As a result, both the plasma and UFPs can be contracted by the magnetic field. 3. Experiment Setup The experiment setup is shown in Fig. 2. A glass piping system of 5 cm in diameter is used as a vacuum chamber. UFPs (3 /i m in diameter) are dropped to the experimental region through a couple of fine filters by the ultra-sonic vibration excited by a pzt. A plasma is produced by either DC discharge (2 kV 300 mA), 13.65 MHz RF discharge or Ultra-fine particle Q_ 0 S - band microwave Plasma discharge. Ultra-fine particles A pulse magentic field in plasma (rise time ^ about 5 msec) is applied to the plasm^a Plasma is contracted column by a pair of coils by a pulse Electic fded magnetic field generated by excited by a 20 kJ capacitor the charge separation discharge (lkV,50000 /JL F ) . Untra-fine particles UFPs drop as a free fall and are attracted towards O pass through the plasma the contracted plasma with a velocity of about 1.8 m/sec. The time constant of Fig, 2 Schematic layout of the experiment the magnetic field is determined by a condition that the motion shall be observ^able in the experimental region.
K. Takeda et al /Control of ultra-fine particles in plasma by an electromagnetic
field
429
ultra-fine particles The measurement of the Green YAG laser motioPx of UFPs is done excited by Glass vacuum tube using a YAG green laser semi-conductor laser A 532 nm and a high speed TV camera as shown in Fig. 3. The high speed TV camera has been developed by one of the authors (T. E.) and its characteristics are shown in Table L This experiment is to demonstrate that the idea Fig. 3 Diagonstics of UFPs motion proposed above should work. Since the experiment is done on the ground, it is more difficult than in space. In other word, if it is verified here, it can be used in space rather easily.
4. Measurement of the Particle Size Distribution In the experiment, it is important that UFP partices drop separately as a group of single particles. It should be avoided that a part of UFPs are composed of 2 or 3 particles sticked with each other. The particles are monitored by iluminating with a green laser light and it is observ^ed by a high speed TV camera as stated. The resolution of the camera, however, is not enough at all to judge whether UFPs are composed of single particles, since one pixel corresponds to 40 /i m. In vacuum^, all particles drop with the same velocity as a free fall. In an atmosphere higher than certain pressure, the partice velocity is determined by the balance of 4 TT r p g and friction F s = 6 TT 77 V. gravity : Fg
(Stokes law)
From there, the velocity is obtained as _
2
2r p g 9 7)
This m e a n s that the 20 dropping velocity of UFPs ^^ is proportional to their size. ^^ Consequently, the particle ^ 12 size distribution can be g 10 obtained when their ^ 6 dropping velocity 4 distribution is measured. 2 0
II 1 1 1 1 1 1 1.1 1 1 1 llJii i l l l l l l l l - l . i . X I t i l l
l-L-l.l-IJ-l 1 1 1 i M J 1 1 t J 1 1 J 1 i 1 1 1 r-L.l,i_L-LJ i..l
Ococsju^oqT-x}^i^cococqa>csjiqoqT--"^r«.
particle velocity[m/s] The result is shown in Fig. Fig.4 (a) The particle velocity distribution (at IPa) 4. Fig.4 (a) is in vacuum
430
K. Takeda et al /Control of ultra-fine particles in plasma by an electromagnetic field
and Fig.4(b) is at the 30 pressure of 3000 Pa. In v a c u u m . The particle velocity distribution is con20 centrated at a single velocity 15 as expected. Even when the gas pressure is at 3000 Pa, 10 the p a r t i c l e v e l o c i t y d i s t r i b u t i o n is a l s o IMJMI concentrated at a single v e l o c i t y , t h o u g h the particle velocity[m/s] dispersion is a litlle bigger Fig.4 (b) The particle velocity distribution than in vacuum. It is (at 3000Pa) concluded that a biggger dispersion of the distribution peak shoves the dispersion of the size of each particle itself No particles, however, are observed at either multiple nor a fraction of that velocity and it shows that the UFP flow is composed of single particles.
J
5. Discussion A method to control ultra-fine particles (UFPs) in a plasma by a pulse magnetic field is proposed utilizing a charge separation electric field generated. The experimental setup has been completed : free-falling UFP flow, plasma production, pulse coil and capacitor bank to excite the magnetic field etc. The particle size distribution is measured by measuring the dropping velocity of particles in a gas pressure higher than 1000 Pa. This method is a very simple way to know in a free-falling UFP flow system that the flow is composed of single particles. Here we should note that the falling of UFPs through a fine mesh is the same in vacuum as in a gas. The experiment to control UFPs by the magentic field is now under way and we will soon get a result to verify our proposal. 6. Acknowledgement This work is supported by a grant for laboratory experiments toward the utilization of the space station.
Table 1 Characteristics of High Speed TV Camera : 4,500 frame/sec Frame Rate 256 X 256 No. of Pixels MCP type Image intensifier Imager : 100 mix Brightness 100 ns (fastest) Shutter speed :
Poster Session B: Strongly Coupled Dusty Plasmas
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
433
Damping of Collective Excitations in Coulomb Crystals V.J. Bednarek, A. Wierling, G. Ropke Universitdt Rostock, FB Physik, Universitdtsplatz 3, D-18051 Rostock, Germany
A b s t r a c t . Within a generahzed linear response theory, an expression for the response function of a Coulomb crystal to an external electric field is given. It can be used as a starting point for a perturbative treatment of collisions. Anharmonicity effects are included via phonon-phonon collisions. Implications for the dispersion of collective modes are discussed.
I
INTRODUCTION
Since macro-particles in a dusty plasma are highly charged, such a plasma represents a strongly coupled plasma, i.e., the physical properties of the system are dominated by interaction effects. The relative importance of these interaction effects is expressed with the nonideality parameter F, v^hich is defined as the ratio of Coulomb energy to kinetic energy. Coulomb crystallization is expected to occur at high values of F. For example for a three-dimensional one-component plasma, simulation results predict crystallization if F > 170 [1]. H. Ikezi suggested certain conditions, under which dusty plasmas should crystallize [2]. In 1994 a so called dusty plasma crystal was observed for the first time [3-5]. The range of the lattice constants is 100 — 200 /xm and the particle sizes varies from 2 — 10 //m. Important properties of this macroscopic lattice are described by the dynamical structure factor. The possible existence of dust lattice waves (DLW) and their dispersion relation has drawn much attention lately [6-8]. We propose an expression for the dynamical structure factor which also accounts for collisions and is based on generalized linear response theory. The dispersion relation of the modes can be obtained from the location of the maxima in the dynamical structure factor. Here, we are focusing on small derivations from periodicity, thus performing a perturbation expansion by including anharmonic contributions of the lattice Hamiltonian. Within a quantum statistical description, this corresponds to the inclusion of phonon-phonon interactions.
434
II
VJ. Bednarek et al/Damping of collective excitations in Coulomb crystals
RESPONSE FUNCTION OF A DUSTY PLASMA CRYSTAL
We investigate the response function of a dusty plasma crystal under the influence of an external electric field E{t), The total Hamiltonian H^oiit) = i f + i?ext(i) contains the system Hamiltonian H and the interaction with the electric field ifg^(i) = E{t)M{t), which acts on the dipole moment of the crystal M{t) as its corresponding dynamical variable. Under the influence of the external potential, a deviation of the position of the dust grain from its equilibrium position will occur, leading to a change of the dipole moment. We introduce a Fourier representation of the displacement vector Ui{r} of a dust grain with components i at position r
where z/ represents the polarization direction and the momentum integration is carried out over the first Brillouin zone. Since we focus on a linear chain, we suppress the index z/ in what follows. Characterizing the nonequilibrium state of the system by the mean values {Pn{r)y of a set of relevant observables {Pn{r)}^ the response function (dipole susceptibility) can be written as [9,10] x(/c,a;) =
0
MoniKco)
Mmo{k,Uj)
Mmn{k,U))
l\Mmn{k,Uj)\
Here, the matrix elements are formed by correlation functions of the relevant observables: Mon{k,Uj) Mmn{k,u)
= {Pn]Mk) = (Pm\ Pf)^^.^
,
,
Mmo{k,Uj)=P{Mk]Pm)
~ i^ {Pm] Pn)^^irj
These equilibrium correlation functions are defined via the Kubo scalar product {A- B) = (B+; A+) = ^iyTTr
[A{-ihT)B+po\
,
and its Laplace transform {A; 3)^- We focus on a one moment approach, using Fourier components of the dipole moment (displacements) as a relevant observable p^ = Mk. We now use the exact relations [9] {Mk] Mk)^ = {Mk, Mk) - iu {Mk] Mk)u; , and the complex conjugated relation to convert the response function to the following form
435
VJ. Bednarek et al /Damping of collective excitations in Coulomb crystals
,, X{k,u;)
,
iLj{Mk,Mk){Mk;Mj;)^ = —7
;
X
^
1
iuj ((Mfc; M , ) , + (M,, M,)) + (M,; M , ) ,
.
(1)
As shown in [10], this expression can be used as a starting point for a perturbative treatment, calculating {Mk, Mk) and {Mk] Mk)u in zeroth order with respect to the interaction, whereas collisions are considered in {Mk]Mk)Lj-
III
EXPLORATORY EXAMPLE: LINEAR CHAIN
We illustrate our approach by calculating the response of a one-dimensional, infinite chain of negative charged dust grains immersed in a positive background. Following indications from experiment, the potential between the dust grains is assumed to be of Debye form Vdd{x) = ey(47reo)exp(—x/rd)/x, where r^ denotes the Debye screening length, and e^ the charge of the dust grain. Furthermore, d denotes the inter-particle distance. We expand up to fourth order with respect to the deviation from the equilibrium position, including anharmonic contributions beyond the harmonic approximation of an ideal crystal. The Hamiltonian ^ = Y.^k [a^dk + - | + $]V(^n^i. ^2,^:3) {ak,ak,atk, + + Y.V^^\kuk2,ks,k4){ak,ak2ak^atk^
+ ak,atkA3^U4
ak.atkA,} + ^kia^^^^
.
is given in terms of phonon creation and annihilation operators a'^^ak, respectively. Momentum conservation is guaranteed within V^^^ and V^^). The crystal dipole moments can be determined in the following way M{k) = Y.k^\ A/A^/OT Jti/{2u{k)) {ak + ttk) and the phonon frequency obeys the dispersion relation Lj{k) = (JOQ\/1 — kd, with a;o being the oscillation frequency of the harmonic approximation. The coupling strength is given by the derivative of the dust potential Vdd with respect to the position. These anharmonic parts are considered as perturbations of the ideal Hamiltonian and we calculate the correlation functions {Mk] Mk), < Mk]Mk >, and < Mk,Mk > with respect to the ideal part, while the correlation function < Mk\Mk > is calculated in first Born approximation. The following correlation function are obtained in zeroth order with respect to the interaction ^^k
(M,;Mfc)S,,--^pifcBr,
^k
[(^ + ^r|)
-^k
J, {M,,M,)%^ = -u^^,kBT ^ ^^+/^^ ^ {u^iT}) -uj^ [uj + iri) -uf. Taking into account only this zeroth order contribution, the phonon dispersion relation is reproduced. (Mk^Mk) contains the following contribution due to the three phonon interaction
436
VJ. Bednarek et al /Damping of collective excitations in Coulomb crystals
{[Hs,M,]; [Hs,Mk]Ui,
= i e ^ - ^ksT ^
Y. V\kuh^ I
-k) V'{^ku
^h^k)
n{(^ki) - ri{uk2)
+ interchanged terms . -1
Here, n{u) = [C^I^BT _ A jg ^.j^^ Bose distribution function, N denotes the number density of the dust grains, m their mass and T the temperature. Inserting this expression into eq. (1) allows the calculation the dynamical structure factor.
IV
CONCLUSIONS
For high values of the coupling parameter F, dust grains in a dusty plasma exhibit a solid-state like structure with long range order. Whereas for small values of F, a perturbative treatment with respect to momentum eigenstates is well-suited, here, the ideal crystal represent an appropriate basis. A systematic approach to the dynamical structure factor of a Coulomb crystal was formulated, which is based on a generalized linear response theory and links the response function to equilibrium correlation functions. While formally equivalent to the Kubo formula, it is more suited for a perturbative expansion. A perturbative expansion is proposed starting from the ideal crystal as the unperturbated system and expanding the Hamiltonian up to fourth order in the displacement. In zeroth order with respect to the interaction the RPA description was reproduced. The dispersion relation is given by the phonon dispersion. Taking into account collisions in first Born approximation the corresponding correlation function can be determined. An evaluation for a linear chain is feasible. The impact of self-energy effects as well as damping on the dispersion relation can be studied.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
S. Hamaguchi, R. T. Farouki, D. H. E. Dubin: Phys. Rev. E 56, 4671 (1997) H. Ikezi: Phys. Fluids 29, 1764 (1986) J. H. Chu, Lin I: Phys. Rev. Lett.72, 4009 (1994) H. Thomas, G. E. MorfiU, V. Demmel, J. Goree, B. Feuerbacher, D. M"ohlmann: Phys. Rev. Lett.73, 652 (1994) A. Melzer, T. IVottenberg, A. Piel: Phys. Lett. A 192, 301 (1994) F. Melands0: Phys. Plasmas 3, 3890 (1996) A. Homann, A. Melzer, S. Peters, A. Piel: Phys. Rev. E 56, 7138 (1997) N.F. Otani, A. Bhattachaxjee, and X. Wang: Phys. Plasmas 6, 409 (1999) D. Zubarev, V. Morozov, G. Ropke: Statistical Mechanics of Nonequilibrium Processes . Volume 1,2. VCH-Wiley, Berlin 1996,1997. G. Ropke: Phys. Rev. E 57, 4673 (1998) G. Ropke, A. Wierling: Phys. Rev. E 57, 7075 (1998)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
437
Behaviour of particles released from inner walls in an E C R plasma etch tool Takeshi Kamata*, Kazuhiro Miwa^ and Hiroshi Arimoto* * Fujitsu Laboratories Ltd., 10-1, Morinosato-wakamiya, Atsugi 243-0133, Japan ^ VLSI Laboratory, Fujitsu VLSI Ltd, 1500, Mizono Tado-cho, Kuwana-gun, Mie 511-01, Japan
A b s t r a c t . We observed the behaviour of particles (> 3/im) released from a reactor chamber innner walls during polysilicon etch processing with CI2/O2 by using in situ laser light scattering in a commercial etch tool with ECR plasma source and simulated their trajectories. We found that few particles were observed during the etch processing, while they were frequently observed during a gas-only, no plasma test. We think this is because a particle can fall on a wafer following a process gas flow without plasma, while the particle falls along the inner wall with plasma due to ions drag force, which is greater than neutral drag force, so that the particle can not fall onto the wafer.
INTRODUCTION In ULSI fabrication process, particles generated in a plasma itself cause to decrease production yields. To maximize the yields, the process-induced particles must be suppressed. One of the process-induced particles are those released from a residual film deposited on an inner wall of a reactor chamber during plasma processing. Recently, there are studies using in situ\a.seT light scattering [1],wafer-level particle count [2] and simulation [3]. Hence,the behaviour of the particles was observed during plasma etching process by using in situ laser light scattering in a commercial etch tool with an E C R plasma source, which is one of representative high-density, low-pressure plasma sources. In this work, the investigation was made under a specific conditions in CI2/O2 discharges used polysilicon gate etch processing. Inner walls of the reactor chamber were heavily contaminated by long etching bare silicon wafers prior to the investigation so that a large amount of particles release from the inner walls during etch processing.
438
T. Kamata et al. /Particles released from inner walls in an ECR plasma etch tool A<-wave
Opt i caI fiIter
Ar ion laser
Wafer
i^or
''*^*'' *^^*^*
Video recorder
Figure 1
(b)
(a) Plasma! off I
o"
| off
PiasmaL
Time ( sec. )
Time ( sec. )
Figure 2
Figure 3
(b)
^ E
550p
^
50oL
^
400^
8
r-
1
1
«
h
g 200h
L looh
r-i
1-
-
r
3 •" 300^
S2
1
rr' 1
\
II
8 X I IT
«4-
-200
-100
0
Distance from wafer center ( rrni )
-200
-100
0
Distance from wafer center ( mm )
Figure 4
O
-200
-100
0
1
o oh « -50h|_ 4J
Distance from wafer center *^ ( mm )
Figure 5
1
-200
1
I
-100
I
1
0
Dittanc* from mf«r e«nt«r ( «• )
T. Kamata et al /Particles released from inner walls in an ECR plasma etch tool
439
EXPERIMENTAL SETUP AND PROCEDURE Figure 1 shows the schematic diagram of the experimental setup. We used the commercial ECR plasma etch tool for polysilicon gate etch using CI2 and O2. The quartz internal bell jar surface is frosted (Ra = l~10/xm). The hight of reactor chamber is 500mm and its diameter is 440mm. The process gases are introduced through the gas distribution ring around the waferstage and evacuated by a turbomolecular pump through the square cross section exaust port of 440mmx 120mm and to the side of the wafer stage. 200mm diameter wafers are introduced into the chamber through a separately pumped load lock and set onto the wafer stage. So, particles from an environment of cleanroom could not affect the measurement. Radiofrequency power (13.56MHz) was applied to the wafer. We made the particle generation ready prior to the measurement using following procedure. We made the chamber clean by cleaning all internal chamber parts and then the internal bell jar was heavily contaminated by etching bare silicon wafers for 6 hours using an etching process parameters 50sccm CI2, 8sccm O2, 2.5mTorr, IkW incident microwave power, and 20W RF power. As a result, a residual film was observed to deposit in a ring on the internal bell jar above the wafer stage as shown in Fig.l, 4000 particles greater than 1 /xm were measured on a wafer after etch processing of 60s by using a laser surface particle detector (Surfscan 6420). A scattering light from the particle was measured by using both the Ar ion laser(A=514nm, CW,2W, ^ = 2 m m ) and the video camera with high sensitive camera tube ( sensitivity 8/xA//xW, sensitivity limit I p W ) during etch processing. The laser light was introduced in parallel to the wafer,through its center, at 20mm above it. A plasma background light except at 488nm was suppressed by an optical filter so that the particles was made observable under plasma on conditions.
RESULTS AND DISCCUSSION The particle measurement was performed running gas-only (no plasma) test and etch processing (plasma on) alternately to investigate a plasma effect for particle behaviours. Figure 2 shows a photograph of a particle, which is falling onto a wafer. Figure 3 shows particle counts during one cycle of an etch processing and gas-only test. We found that few particles were observed during etch processing, while they were frequently observed during gas-only test. We speculated that this is because the particles experience different kinds of force between plasma on conditions and no plasma conditions. The neutral drag force and the gravitational force act on the particles without plasma, while the ion drag force and electrostatic force act on it with plasma in addition to them , which is greater than them in magnitude. Since there is difference in direction and magnitude among positions in reactor chamber, it is very complex to figure out the particles behaviours. Hence, the particle trajectory was simulated
440
T. Kamata et al /Particles released from inner walls in an ECR plasma etch tool
In this simulation, we assumed spherical quartz particle(n=1.46,cr=2.6g/cm^), a particle is released from the rididual film as shown in Fig.l and its initial velosity is zero. We calculated that the particle detection limit is 3/xm by Mie theory [4] taking into account a free falling velosity (2m/s) and the camera sensitivity limit. Hence, we focused a 3/xm diameter particle. First, we simulated trajectories of the particle under gas-only conditions. Figure 4(a) shows the simulation model and simulation result. The following assumptions were made. (1) the gas is flowing in viscous range in parallel to a wafer at flow velosity of 6 m / s , which is estimated in the exaust port. (2) the gravitational force and the neutral drag [5] force act on a particle. Figure 4(b) shows the simulation result. We found that the particle can fall on the wafer following the gas flow without plasma. Next, we simulated trajectories of the particle under plasma-on conditions. Figure 5(a) shows the simulation model. The following assumptions were made. (1) There are a uniform argon plasmas (Te=4eV, Ni,Ne=10"^"^cm~^) in the reactor chamber. (2) There are presheath and sheath between the plasmas and the inner walls. (3) MEO dc sheath models [6] was used. (4) Ions are accelarated by an uniform electric fields so that their kinetic energy is Te/2 at sheath edge in the presheath, of which thickness is equal to ion-neutral mean free path. (5) The ion drag force [7] and the electrostatic force [5] act on it with plasma in addition to the neutral drag force and the gravitational force. Figure 5(b) shows the simuration result. We found that the particle falls along the inner walls with plasma due to the ion drag force. In conclusion, we found that few particles were observed during the etch processing (plasma on), while they were frequently observed during the gas-only test (no plasma). We think the reason is that the particle can fall on the wafer following the process gas flow without plasma, while the particle falls along inner wall with plasma due to the ion drag force. Therefore, we expect that we could suppress particles falling onto a wafer with increase in plasma density.
REFERENCES F.Uesugi et al, J. Vac. Set. Tech. A16, 1189 (1998). M G Blain et al, Plasma Sources Set. Tech. 3, 325 (1994). M D Kilgore et al, J. Vac. Sci. Tech. B12, 486 (1994). C F Bohren and D R Huffman, Absorption and Scattering of Light by Small Particles, New York: John Wiley & Sons Inc., (1983). 5. M D Kilgore et al, J. Appl. Phys. 72,3934 (1992). 6. A Metze, D W Ernie, and H J Oskam, J. Appl. Phys. 60, 3081 (1986). 7. M S Barns, Phys. Rev. Lett. 68, 313 (1992). 1. 2. 3. 4.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
441
Anomalous Diffusion and Finite Size Effect in Strongly Coupled 2-D Dust Coulomb Clusters Wen-Tau Juan, Yin-Ju Lai, and Lin I Department of Physics and Center for Complex Systems, National Central University, Chungli 320, Taiwan, Republic of China
The particle diffiision in a strongly coupled 2-D dust Coulomb cluster is studied.
In the center
region, the motion shows behaviors similar to the infinite large system, such as the isotropic motion with an initial caged antipersistent diffusion and the transition to the persistent diffusion driven by the vortex type excitation at larger time. In the outer region, the motion is no longer isotropic. The central confining field suppress the radial diffusion and make the motion anisotropic. This effect propagates radially inwards and becomes more significant as time increases.
The two dimensional strongly coupled Coulomb cluster (2D SCCC) with finite particle number A^ is an interesting nonlinear model system connecting the Wigner crystal with iV= oo to the few body limit.. In addition to the early theoretical study by J.J. Thomson on the structure of his Classical Atom v^th electrons imbedded in a neutralizing background. Coulomb clusters have also been found in a few laboratory systems such as electrons in the Coulomb blocked, ions in the electromagnetic traps, flux lines in superfluids or magneto-plasmas, etc.(1-6). Recently, the formation of the quasi-2D SCCC with dust particles in a plasma trap was demonstrated. The generic packing rules, structures and topological properties have been systematically investigated (7-10). The proper system size also offers an advantage of directly monitoring the particle motion over other SCCC systems. It is found that the cluster has a center core with triangular structure and surrounded by the circular shells. Namely, the system preserves the features of the infinite 2D Wigner crystal in the center region and the boundary region is affected by the lattice bending due to the central confining force. In our previous experimental studies we found that, under the thermal agitations, the large volume dusty plasma exhibits vortex type excitations through the strong mutual coupling, which leads to the anomalous diffusion with transition from the initial caged regime to the persistent diffusion regime (11,12). The exponents H of the
442
W'T Juan et al /Anomalous diffusion and finite size effect
mean square displacement (MSD = < A R^( r )> = < (R(t + r ) - R(t))^ > ^ t" ) both deviate from one. Since the particle motions are strongly sensitive to the local environment, the finite size and boundary effect of 2DSCCC certainly alter the uniform and isotropic motions observed at the large N limit. In this paper, their impact on the diffusion properties is addressed. The experiment is conducted in a cylindrical rf discharge system as described elsewhere (7). A hollow cylinder with 3-cm diameter and 1.5-cm height is put on the bottom electrode to confine 5-/z m polystyrene dust particles in the weakly ionized glow discharge generated in Ar at 200 mTorr using a 14 MHz rf power system. At very low rf power, the particles are confined in the small center glow region surrounded by a thick double layer adjacent to the wall, which has strong outward radial space charge field. The vertical ion flow generates dipole field and turns the system into a quasi-2D system through forming vertical particle chains. N = 288
0-28
FIGURE 1.
0-103
The typical snap shot of the triangulated configuration and the trajectories with different
exposure times under a fix operating condition. The motion anisotropy changes with different exposure time.
At small A^, the cluster shows concentric shell structures. The motion is basically anisotropic. Angular inter- and intra- shell vibrations and hopping are the dominant excitation processes (7-9). We are not going to address their behavior here. As N increases to a few hundreds, the cluster has the inner triangular lattice surrounded by a few circular shells due to the easier bending in the outer region (7-10). We focus our study on the anomalous diffusions for a cluster with A^ = 288. In Fig. 1, the piecewise circular outer shells can be easily observed through the triangulated plots. The defects (represented by the triangles and squares for the -1 and +1 topological defects respectively) induced by thermal motions are plotted. The typical trajectories with different exposure times show interesting behaviors. For the very short exposure
W-T. Juan et al. /Anomalous diffusion and finite size effect
443
(2 sec), most particles show quite short and isotropic trajectories. For the medium exposure time with a few sec, the vortex type excitation can be observed. The particles in the outmost shell start to show anisotropic motions with larger azimuthal amplitude. For the exposure time longer than 10 sec, even the particles in the second and the third outmost shells show elongated trajectories mainly along the azimuthal direction. Fig. 2 shows the mean square displacements MSD along the radial and azimuthal directions for the particles in the six radially equally spaced bands. TTTTin—I
T1mnj—r^1 IIIIIH
I 11 n i l
-d
0.1
T 1 iiiiii| T n 6
-
0.1 10
r
"d 5
S 0.01
S 0.01
J
0.001
1
0.001 lud
10 Time (S)
100
J
' • '•"••'
10 Time (S)
100
•
_^_-^——-,—- luLUui
L.LI mill
.. 4 3 ^—r- .J 2
\ -^
1 111 mil
10 Time (S)
100
FIGURE 2. The time evolution of the mean square displacements (normalized by the mean lattice constant a ) along the azimuthal and radial directions for the particles at different radius r. Radially, the cluster is divided into six bands with equal spacing, numbered by 1 to 6fromthe center in (a) to (c ) . (c) The ratio of the above two MSDs.
The deviation from 1 reflects the deterioration of the motion
anisotropy affected by the boundary.
Basically, except curve 6, most of the curves show quite common features of regimes with different slopes, i.e. the anti-persistent regime at the small time and the transition to the persistent region at the larger time. This observation is quite similar to our previous observation for the very large dusty plasma system (12). Antipersitent regime corresponds to the caged motion confined by the surrounding particles and the persitent regime is due to the vortex type excitations. The presence of the finite boundary does affect the uniformity of the collective excitations. The azimuthal MSDs are quite similar. It only slightly decreases for the two to three outmost shells where the radial MSD drastically decreases with increasing radius (Fig. 2). For a infinite large system vydthout a preferred orientation, the ratio of the azimuthal MSD to the radial MSD should equal to 1. Their ratio versus time plot indicates that the motions in the center three bands are isotropic. The isotropy in the
444
W'T. Juan et al /Anomalous diffusion and finite size effect
outer three bands deteriorates with increasing time and increasing radius. For example, for the fourth bands, it deteriorates to 1.5 at intermediate time, and to about 2 at longer time. It increases to 20 for the particles in the outmost band at large time. This change of anisotropy can be manifested in the trajectory plots shown in Fig. 1. In addition to the more preferred angular motion for the outer shells, the strong space charge field can generate quite steep radial confinement which could suppress the radial excitation. This effect is not too obvious in the short time regime where the motion amplitude is small and the motion is mainly caged by the surrounding particles . As time increases, the outmost shell particles can not radially penetrate the steep confining well when the excursion amplitude becomes larger. Therefore, unlike other curves, the diffusion is anitpersistent over the entire time for curve 6 in Fig. 2, though it slope does increase in the longer time regime. The finite boundary prohibits the radial motion starting from the particles in the outmost shell. On the other hand, the periodic azimuthal boundary condition does not suppress the azimuthal motion. It can support many soft excitations with long wavelengths. This effect gradually propagates radially inwards, generates longer range correlation, and causes the deterioration of the motion isotropy for the particles in the second and the third outmost bands at longer time scale. This work is supported by the National science Council of the Republic of China under the contract number NSC 88-2112-MOO8-06.
REFERENCES [I] J.J. Thomson, Phil. Mag. S. 6. 7, 39, 236 (1904). [2] H. Ikezi, Phys. Rev. Lett. 42, 1688 (1979). P. Leiderer, W. Ebner and V.B. Shikin, Surface Science, 113, 405 (1982). [3] Nanostructure Physics and Fabrication, edited by M.A. Reed and W.P. Kirk (Academic, Boston, 1989). [4] D. Reefinan, H.B. Brom, Physica 183C, 212 (1991). [5] W.I. Glaberson and K.W. Schwartz, Phys. Today, 40, 54 (1987). [6] D.Z. Jin and D.H.E. Dubin, Phys. Rev. Letts, 80, 4434 (1998) [7] W.T. Juan, Z.H. Huang, J.W. Hsu, Y.J. Lai and Lin I, Phys. Rev. E, R6947 (1998) [8] J.M. Liu, W.T. Juan, J.W. Hsu, Z.H. Huang, and Lin I, Plasma. Phys. Control Fusion, 41,A47(I999) [9]. Juan, J.W. Hsu, Z.H. Huang, Y.J. Lai, and Lin I, Chinese J. Phys. 37, 184 (1999). [10] Y.J. Lai and Lin I, to be published. [II] J.H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994). [12] C.H. Chiang and Lin I, Phys. Rev. Lett, 77, 647 (1996). W.T. Juan and Lin I, Phys. Rev. Lett, 80, 3073 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
445
Plasma Crystals and Liquids in DC Glow Discharge V.E. Fortov, V.I. Molotkov, V.M.Torchinsky High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia
Abstract. Results of the experimental investigation of dusty plasma structures being formed in striations of a positive column of the dc low-pressure glow discharge in the wide range sizes of dust grains from 5 lO'^^g to lO'^g are presented. Depending upon a relation between forces acting on dust particles (gravity, electrical, ion drag) different type of dusty plasma formations were observed: plasma crystals, structures with a short-distance order and with a convective movement of dust grains. The molecular dynamics simulation performed showed a good agreement with the experimental data.
The discovery of plasma crystal was made in the case of radio-frequency lowpressure discharges [1,2,3]. Recently we have demonstrated a possibility to study dusty plasma structures in striations of the dc glow discharge [4,5,6]. The main difficulty of forming the structures in the dc discharge is that in the near-cathode region of a dc discharge the electric field increases simultaneously with a decrease of the electron density. This leads to a sharp decrease of the particle charge and a reduction of the electrostatic force balancing gravitation. In the stratified discharge the situation is qualitatively different. Here, a sharp increase of the field in the striation head is accompanied by only an insignificant decrease of the electron density and it became possible to form an ordered structure of dust grains in the striation. In Fig. la there is a fragment (8.8 x 9.6 mm^) of the horizontal section of the cloud of monodisperse (diameter 1.87 ± 0.05 |Lim) melamine formaldehyde particles (/?= 1.5 g/cm^) levitating in the standing striation of the low pressure dc discharge in a neon-hydrogen mixture at pressure/?=0.8 Torr and discharge current 7 = 1 . 1 mA. The structure was obtained at the installation described elsewhere [5,6]. The cloud has a stable ellipsoidal form and there is a long range order in positions of dust charged particles that is testified by the distribution Sanction n(r) shown in Fig. lb. The fiinction n(r) is the particle number density at a distance r from some particle. In this work we performed the molecular-dynamic simulation of the system of dust particles present in the glow discharge plasma. The parameters of the electric fields have been chosen to be closed to the experimentally measured values [7]. The radial component of the potential was given by the expression: (JK,= (f>t (r/RtT, where Rt= 1.5
446
KE. Fortov et al /Plasma crystals and liquids in DC glow discharge
n(r)10^^m"^
1000
2000 r, |im
3000
FIGURE 1. Example of plasma crystal in the striation
cm is the discharge tube radius, ^r = ^o + (f)i/(l+((z - zi)/dif) is the wall potential at the height z, a, zi and d\ are parameters. The particle charge Zp was in a rigid connection with the floating potential (t)t and its radius Rp, Zp = Rp(^x- The interaction of the dust particles was taken into account with the help of the screened Coulomb potential. A dust particle was subjected along the vertical axis z to gravity and a random Langevin force ^(0, which satisfied to the condition <^> = l^Tlbd^ where p is the friction force of the buffer gas, T is the gas temperature and A^ -is the step of integration on time. Besides we took into account the ion drag force [8], whose role increased with decreasing the particle size. For the integration of the equation of movement the Shtermer method of 8-th order [9] was used. The number of dust particles was varied in the range from 1 to 800. In the experiments we observed a levitation of separate particles as well as chains consisting of several particles arranged on the axis of the discharge. In this case the particles were located not equidistantly and the maximum distance was in the lower part of the chain. The addition of separate particles into the chain was possible only when the number of particles n < 10. When a number of particles is greater than the number mentioned above, the subsequent particles found a place at the side of the central part of the chain. It is possible to achieve the quantitative agreement during the simulation only at the particular choice of parameters when along the striation in the direction of the anode firstly a maximum of the electron density takes place, then a maximum of the floating potential (the particle charge) and at last a maximum of the electric field. This is in a good agreement with measurements of a distribution of parameters in striations in the neon discharge [7] and testifies to the adequacy of the model chosen to the real experimental conditions. The analysis of a distribution of charged particles in the chain can be used as an indirect means for diagnosing properties of the glow discharge positive column. In the experiments we used four types of particles: particles of borosiHcate glass (p = 2.3 g/cm^), in the form of thin-walled, hollow spheres of diameter 50-63 |am with wall thickness 1-5 |im, AI2O3 particles (p = 4 g/cm^) with diameter 3-5 |im, monodisperse
VE. Fortov et al /Plasma crystals and liquids in DC glow discharge
FIGURE 2. a) Example of a vertical section of a structure with a convective movement of dust grains with 1.87 jj.m diameter, maximum vertical size of a structure - 3.5 mm; b) Vertical section of the structure calculated by MD simulation
melamine-formaldehyde particles of two diameters 10.24 ± 0.12 |Lim and 1.87 ± 0.04 |am. Masses and charges of particles were in the range from 5-10"^^ g to 10'^ g and from 10"^ e to 5-10^ e, respectively. In the case of heavy particles when their levitation is mainly due to an equality of the force of gravity and the electrostatic force and the ion drag force is low we observed stationary dusty plasma clouds with a well ordered structure [4,5,6], with increasing a number of particles in the cloud sometimes the local region with developed oscillations of dust particles along the vertical axis appeared. It seems to us that this phenomenon is due to the anomalous heating of the dusty plasma the explanation of which was proposed in [10]. With decreasing the particle size the gravity force decreases proportionally to the particle volume and the ion drag force, which is proportional to a cross section of the particle, appears in the forefront. In contrast to the gravity force directed strictly along the discharge axis the ion drag force has also the radial component which is connected with a flux of ions to the walls of the discharge tube. This can lead to an origination of convective flows of an ensemble of dust particles. An example of such a flow is given in Fig.2a where arrows indicate a direction of a movement of particles in the ellipsoidal cloud which is averaged in time. The numerical simulation allows us to obtain the qualitatively similar structure reproducing both a form of the cloud and dynamics of a behaviour of dust particles. This is presented in Fig.2b where traces of the movement of separate particles are seen. In the case of small particles an increase of their number leads at the specific discharge parameters to formation of structures where different regions coexist: the regions of the strong ordering (plasma crystal), the regions with convective and oscillatory movement of particles (dusty plasma liquid). These oscillatory movements in complex dusty plasma structures which we observed are waves of a density of particles.
447
448
VE. Fortov et al. /Plasma crystals and liquids in DC glow discharge
The frequency of these oscillations is about 25-30 Hz and the wave length is about 1000 |im with an average interparticle distance of about 200 \xm. The such complicated picture is associated with the pecuUar distribution of forces acting on dust particles: the ion drag force, the electrical force and a distribution of plasma parameters along the striation.
ACKNOWLEDGMENTS This work was made in part by Grant No.99-02-18126 from the Russian Foundation for Basic Research.
REFERENCES 1. 2. 3. 4.
Chu, J.H., I, L., Phys. Rev. Lett. 11, 4009 (1994). Thomas, H., Morfill, G.E., Demmel V. et al, Phys. Rev. Lett. 73, 652 (1994). Hayashi, Y. and Tachibana, K., Jpn. J. Appl. Phys. 33, L 804 (1994). Fortov, V.E., Nefedov, A.P., Torchinsky, V.M., Molotkov, V.I., Petrov, O.F., Khrapak, A.G. and Volykhin, K.F. JETP Lett. 64, 92 (1996). 5. Fortov, V.E., Nefedov, A.P., Torchinsky, V.M., Molotkov, V.I., Petrov, O.F., Samarian, A.A., Lipaev, A.M., Khrapak, A.G., Phys. Lett. A 229, 317 (1997). 6. Lipaev, A.M., Molotkov, V.I., Nefedov, A.P., Petrov, O.F., Torchinsky, V.M. Fortov, V.E., Khrapak, A.G., Khrapak, S.A JETP, Vol.85, No.6, P. 1110 (1997). 7. Golubowskii, Yu.B., Nisimov, S.U., JETP 66, 20 (1996). 8. Barnes, M.S., Keller, J.H., Forster, J.C, O'Neil, LA, Coultas D.K., Phys. Rev. Lett 68,313(1992) 9. Zhakhovskii, V.V., Anisimov, S.I., JETP 84, 734 (1997). 10. Zhakhovskii, V.V., Molotkov, V.I., Nefedov, A.P., Torchinsky, V.M., Fortov, V.E., Khrapak, A.G. JETP Lett., 66, 419 (1997).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
449
Formation of Ring-Shaped Fine-Particle Clouds in a DC Plasma Giichiro Uchida, Ryoichi Ozaki, Satoru lizuka, and Noriyoshi Sato Department of Electrical Engineering, Graduate School of Engineering Tohoku University, Sendai 980-8579, Japan
A b s t r a c t . We have observed various structures of fine-particle clouds by controlling the potential profile in a dc ion sheath. Fine particles in a strongly-coupled state are found to form a ring-shaped and a cone-shaped cloud. We have also observed wave propagations in the cloud. By compressing the fine particle clouds in radial direction with increasing the external electric field, the Coulomb crystal changes into an unstable state, and the particles begin to oscillate in vertical direction.
INTRODUCTION Dusty plasmas attract a great deal of attention not only in plasma physics but also in plasma material processing. Fine-particles in the plasma, negatively charge up due to fast electrons, are knov^n to form Coulomb crystal under strong repulsive force (1,2). Fine particles interact v^ith the plasma, and various collective behaviors, for example, dust vortex driven by the electric field or magnetic field, have been observed (3—5). The collective behaviors of fine particles, especially in stronglycoupled state, is very complicated, because the various forces, such as electrostatic force, electromagnetic force, ion drag force, and friction force v^ith gas, act on these particles. It is important to study the connection betv^een the fine particles and the electric field around the particles. In our experiment, fine particles of micronsize are injected into a completely dc plasma system(3—5). In order to control the potential profile in the particle levitation region, a segmented particle levitation electrode is set up. Because the external electric field acting on particle cloud is completely controlled, the system is quite simple to analyze the collective behaviors of the particles. In this paper, v^e have controlled various potential profiles, and investigated the structure and dynamics of particle clouds. Fine-particle cloud changes its shape to a ring and to a cone according to the potential profile. By increasing the external electric field confining fine particles, v^e have also observed collective oscillation of the particles accompanied v^ith v^ave propagation in a dc ion sheath.
450
G. Uchida et al. / Formation of ring-shaped fine-particle clouds in a DC plasma
EXPERIMENTAL APPARATUS A schematic diagram of the experimental apparatus is shown in Fig. 1. A dc argon discharge plasma is produced at 220 mTorr by applying negative dc potential of about —270 ^ —320 V to an upper cathode electrode with respect to a middle grounded mesh anode. The plasma diffuses downward through the mesh anode. Typical electron density and electron temperature of the diffused plasma are 1 x 10^ /cm^ and 1 eV, respectively. At 2.0 cm below the mesh anode, a segmented particle levitation electrode (SPLE) is set up, consisting of three electrodes. The electrode at center of the SPLE is a disc of 0.5 cm in diameter, and the two ring electrodes are set up around it. The one is the ring electrode with inner and outer diameters of 0.5 cm and 1.5 cm, respectively, and the other is with 1.5 cm and 19 cm, respectively. Different dc potentials K, Ki? and K2 can be applied to the three electrodes independently in order to control potential profile in the particle levitation region. The particles used are mono-disperse methyl methacrylate-polymer spheres of 1.17 - 1.20 g/cm^ and diameter of 10 (±1.0) /im. They are injected from a sieve into the glow region of the plasma through the mesh cathode and anode. Particles are observed by the Mie-scattering of He-Ne laser sheet. To investigate the particle behaviors, the CCD camera is used as a detector.
EXPERIMENTAL RESULTS In case of Vc = Ki = —20 V, a coUisional ion sheath of 10 mm in width is formed typically, in which the charged particles can levitate at 6-8 mm above the SPLE. Fine-particles have about (1—5) x lO^e charges, that is calculated from a balance between the gravitational force and the electrostatic force in the ion sheath. CCD CAMERA
s
DUST DROPPER
CATHODE
CCD CAMERA
FIGURE 1. The schematic diagram of experimental apparatus.
G. Uchida et al. /Formation of ring-shaped fine-particle clouds in a DC plasma
—1
1
I
1
'
-7.5 > •
,''
o
-8
'
^— 1
-5
' 1
0 r (mm)
1 •
""^-J '^N 1
.
•
1
451
5 mn
'^^^i"^^^m^^m.,^.;;sm^
5
F I G U R E 2. Radial profiles of floating potential above the particle levitation electrode (left). Dot line is obtained at Vc = K-i = - 2 0 V, and Vr2 = - 5 0 V, and solid line is obtained at Vc — Vr2 — - 5 0 V, and Vri = - 2 0 V. The image of ring-shaped fine-particle cloud (right).
Figure 2(left) shows radial profiles of floating potential at 7 mm above the SPLE. The potentials of the center and two ring electrodes of the SPLE are fixed at Vc = Vr-i = — 20 V > Vr2 = —50 V, and a potential hill with single peak for confining the negatively charged particles is formed as shown by dot line in Fig. 2(left). When the center-electrode bias Vc is decreased from —20 V to —50 V, the potential profile changes from single peak to double peaks as shown by solid line in Fig. 2(left). Fine particles levitating in the center region move outward in radial direction due to the repulsive electric field, and a ring-shaped fine-particle cloud is formed at the radial position of about 5 mm as shown in Fig. 2(right). Figure 3 shows a photograph of the ring-shaped cloud from the top. Two or three fine particles are confined in radial direction with a regularity in order. By decreasing the number of particles, we can also produce a ring-shaped particle string. The mean interparticle distance is about 400 /im. Coulomb coupling parameter F is 18-112, indicating that the ring-shaped fine-particle cloud is in the strongly-coupled state. Decreasing the bias Vr2 under keeping Vc = Ki = - 2 0 V, the dust cloud is pressed radially toward the center, because the ion sheath region formed by the ring electrode moves toward the center. By this compression, the interparticle distance decreases, and the particles are pushed up in vertical direction, resulting
F I G U R E 3. Top -view image of a ring-shaped fine-particle cloud. A few particles are confined in radial direction.
452
G. Uchida et al /Formation of ring-shaped fine-particle clouds in a DC plasma
10 O CD C/)
—I
1
1^
1
r
,
Stable region : unstable region \
o
s>
> 1
one wave T length J !
2
I
100
50
N
X
two wave lengthL
3
Zd (mnfi)
FIGURE 4. Phase velocity and frequency of the wave propagating in fine-particle cloud as a function of the width Zd of particle layers (left). The image of wave propagation (right). The wavelength is about 2 mm . in an increase of total layer width Zd and dust density n^. When Zd > 2 mm, and rid > 10^ /cm^, which corresponds to the interparticle distance of about 100 /im, the Coulomb crystal changes into an unstable state and the particles start to oscillate in vertical direction as shown in Fig. 4. Further, we have observed a wave propagation toward the levitation electrode, in the direction of ion flow in the dc ion sheath. The wavelength is about 2 mm, which is almost equal to the total width of layer Zd. The phase velocity and frequency are about 7 cm/sec and 35 Hz, respectively. Increasing the total width up to 2^^ — 3 mm, the wavelength changes to Zd/2 2:^ 1.5 mm. This instability is closely related to the potential structure in axial and radial directions in the ion sheath.
CONCLUSIONS We have investigated a relation between the structure of fine-particle clouds and the external electric field. By controlling the potential profile by the separation electrodes, various structures of ring-shaped fine-particle clouds are formed. Increasing the external electric field in radial direction, the clouds change to an unstable state, and wave propagation has been observed in the dc ion sheath.
REFERENCES 1. Hayashi, Y. et a/., Jpn. J. Appl Phys. 33, L804-L806 (1994) ; Thomas, H. et a/., Phys. Rev. Lett. 73, 652-655 (1994); Chu, J. H. et a/., Phys. Rev. Lett. 72, 4009-4012 (1994). 2. Tsuji, K. et a/.. Double Layers (edited by Sendai"Plasma Forum"), Singapore: World Scientific, 1997, ch. 1, pp. 100-104. 3. Sato, N. et a/., Pysics of Dusty Plasma Seventh Workshop^ edited by M. Horanyi et al.., New York : American Institute of Physics, 1998, pp. 239—246. 4. lizuka, S. et al.., Pysics of Dusty Plasma Seventh Seventh Workshop, edited by M. Horanyi et al, New York : American Institute of Physics, 1998, pp. 175-178. 5. Uchida, G. et a/., " Dust Vortex in DC Discharge Plasma under a Weak Magnetic Field" in Proceedings of the 1998 International Congress on Plasma Physics, 1998, pp. 2557-2560.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
453
Vertical Spread of Fine-Particle Clouds in a Magnetized DC Plasma Satoru lizuka, Ryoichi Ozaki, Giichiro Uchida, and Noriyoshi Sato Department of Electrical Engineering, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
Abstract. Vertical spread of fine-particle clouds is observed in a weakly ionized argon dc plasma by applying axial magnetic field in vertical direction, using a double-plasma method. The fine particles are trapped radially to form a large-volume fine-particle cloud along the magnetic field. The fine-particle cloud rotates around its vertical axis, as a whole, under the presence of the vertical magnetic field. We observe a small-scale convection motion of fine particles near the surface of the cloud. Changing the discharge conditions also excites unstable fluctuations propagating along the magnetic field.
INTRODUCTION Dusty plasmas have been investigated theoretically and experimentally in relation v^ith the space environment and modem industries since the observation of Coulomb crystal [1]. The experiments are usually carried out using thin layers of dust particles spreading mainly in horizontal direction. However, in order to investigate 3dimensional behaviors of strongly-coupled dust particles, it is necessary to build up a long dusty plasma-column filled with the particles. Therefore, the formation of a large-volume dusty plasma extending both in vertical and radial directions is of crucial importance for the investigations of the 3-dimensional, for example, dust vortices, convection, instabilities, and particle transportation. Up to now, we have observed collective behaviors of the fine particles [2], forming vortex flows on a levitation electrode, driven by the electric field provided by a biased small plate placed on the levitation electrode in a dc-discharge plasma [3, 4]. We have also found a formation of dust vortex flow driven by the magnetic field applied in vertical direction [5]. In this paper, we report a formation of large-volume dusty-plasma clouds spreading in vertical direction in a dc glow discharge plasma. Here, we use a doubleplasma method to make a hill-type radial potential profile for the particle confinement. We also investigate the stability of the fine-particle clouds by changing the discharge conditions. Our method proposed here provides a new method for production of large volume dusty plasmas.
454
S. lizuka et al./Vertical spread offine-particleclouds in a magnetized DC plasma
EXPERIMENTAL APPARATUS The schematic of the experimental apparatus is shown in Fig. 1. We produced double plasma (DP) which consists of main plasma (MP) and auxiliary plasma (AP) produced between anode 1 and cathode 1, and between anode 2 and cathode 2, respectively. The anode 1 has a 2-cm-diameter hole at the center, covered with a mesh through which the MP diffuses downwards and is terminated by a 2-cm-diameter levitation electrode which has a rim of 2 mm both in width and height at the edge. The levitation electrode can be biased at the different potential from the anode 2. Usually, the levitation electrode is biased at -70 V. The AP also diffuses upwards through the mesh anode 2 and is terminated by a rind electrode 1 with a 2-cm-diameter hole at the center. The potential of the ring electrode 1 is as same as the anode 2 which is usually biased at -100 V. The potential of anode 1 is fixed at grounded. Owing to the axial uniform magnetic field of 180 G applied in vertical direction, both plasmas can contact each other at a sharp boundary, which is situated around radial position between the hole edge of the ring electrode and the outer radial edge of the levitation electrode. The particles used are mono-disperse methyl methacrylate polymer spheres of 1.17-1.20 g/cm^ and diameter of 10( ±1.0) fx m. They are injected from a sieve into the MP region through a center hole of cathode 1. Particles levitating above the levitation electrode are observed by Mie-scattering of the He-Ne laser light sheet with a breath of 5 mm, incident from horizontal direction. To investigate the particle behaviors, a CCD camera is used as a detector of particle positions, which are recorded on videotape and are processed by a personal computer. A wire probe measures the plasma parameters, which is movable in radial and axial directions. Cathodel
I
I
I
MP t AP
ZJ
I
Anodel
=1
Dust^HB^^^^ Electnodel Levitadon Electiode Cahode2
FIGURE 1. Schematic of the experimental apparatus.
EXPERIMENTAL RESULTS Figure 2 shows the radial profiles of floating potential of the double-plasma at different axial positions above the levitation electrode. We observe a hill-type
S, lizuka et al. / Vertical spread offine-particleclouds in a magnetized DC plasma 455
potential profile extended in axial direction over more than 10 mm above the levitation electrode, by the effect of the vertical magnetic field. Therefore, we have produced a quite homogeneous potential profile in vertical direction. A typical photograph of a fine-particle cloud in a stable state, formed in the potential structure plotted in Fig. 2 is shown by the left figure in Fig. 3. We find an axis-symmetrically-cylindrical structure of a strongly coupled fine-particle cloud spreading in vertical direction. Although the diameter is 3-4 mm, the axial length is more than 10 mm. Since the average distance of Coulomb lattice spacing is 200-300;/ m, the axial length corresponds to 300-500 lattice layers, which is extremely large compared with the conventional width of ^ 1 0 layers. Here, it should be emphasized that this fine-particle cloud is rotating, as a whole, around its vertical axis in the direction of electron gyration. That is, a right-handed circular rotation against the vertical magnetic field is generated in the cross-sectional horizontal plane. The rotational angular frequency is a; ^ 0.8 rad/s, which is varied by the magnetic field and the discharge current. The properties of this diamagnetic rotation are quite similar to those observed in the thin particle layers [3,5]. Another notable phenomenon observed in this fine-particle cloud is a formation of several tiny convection vortices generated just underneath the surface of the cloud. We find several small-scale convection flow patterns, in which inner downward particle flows turn to upward flows near the surface, forming a tiny vortex in the vertical cross section. The size of the vortex is -^2 mm which is roughly 8 times as much as the lattice size. By changing the discharge current for the MP production, for example, the fineparticle clouds becomes unstable as shown by the right figure in Fig. 3, where density fluctuations of the fine particles are excited. The fluctuations propagate along the magnetic field lines from the top to the bottom of the clouds, with the velocity of 5.1 cm/s. In our experimental conditions, however, the collisions between neutral gas and
-30 - 2 0 - 1 0 0 10 20 30 r (mm) FIGURE 2. Radial profiles offloatingpotential with axial positions as a parameter.
456
S. lizuka et al/Vertical spread of fine-particle clouds in a magnetized DC plasma
FIGURE 3. Side-view photographs of the fine-particle clouds in stable (left) and unstable (right) states.
fine-particles are so dominant that it would be difficult to excite so-called dust acoustic waves. The observed frequency of the instability is 30-40 Hz that is the order of the dust plasma frequency. Therefore, the fluctuation observed might be related to the vertical particle oscillation coupled with the dust plasma frequency.
CONCLUSIONS Vertical spread of strongly coupled fine-particle clouds is formed in weakly ionized argon dc plasma by using a double-plasma method. Owing to the vertical magnetic field, a well-type plasma potential is produced for the particle confinement. The fine-particles are, therefore, trapped within a radial potential well that is extended along the vertical magnetic field. We find that the fine-particle clouds rotate around its vertical axis, as a whole, under the presence of the vertical magnetic field, in which several tiny convection motions of fine particles are observed near the surface of the clouds. Changing the discharge conditions also excites unstable fluctuations propagating along the magnetic field. The method proposed here is quite usefiil for investigating the 3-dimensional dynamics of strongly coupled fine particles.
REFERENCES 1. Chu, J. H. et al, Phys. Rev. Lett. 72, 4009-4012 (1994); Thomas, H. et a/., Phys. Rev. Lett. 12>, 652655 (1944) ; Hayashi, Y. et a/., Jpn. J. Appl Phys. 33, L804-L806 (1994). 2. Sato, N. et al. Physics of Dusty Plasmas, edited by M. Horanyi et al. New York : American Institute of Physics, 1998, pp. 239-246. 3. Uchida, G. et al, "Generation and Control of Vortex Flow of Fine Particle with Coulomb Lattice," in Proceedings of the 15th Symposium on Plasma Processing, 1998, pp. 152-155. 4. lizuka, S. et al. Physics of Dusty Plasmas, edited by M. Horanyi et al. New York : American Institute of Physics , 1998, pp. 175-178. 5. Uchida, G. et al, "Dust Vortex in DC Discharge Plasma under a Weak Magnetic Field" in Proceedings of the 1998 International Congress on Plasma Physics, 1998, pp. 2557-2560.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
457
Vertical String Structure of Fine Particles in a Magnetized DC Plasma Ryoichi Ozaki, Giichiro Uchida, Satoru liziika, and Noriyoshi Sato Department of Electrical Engineering, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan
Abstract. Vertical string structures of fine particles are observed in a weakly ionized argon dc plasma by applying axial magnetic field in vertical direction. The particles are confined in an extremely narrow radial potential well produced by a double plasma method, which enables a formation of a chain-like structure of particles in vertical direction. Several vertical strings are coupled with each other to constitute 3 dimensionally characteristic and fundamental lattice structures. It is noted that these coupled vertical strings are rotating, as a whole, around the axis that is directed along the vertical magnetic field.
INTRODUCTION Fine particles in plasmas, which are negatively charged up by electrons, have been demonstrated to form a Coulomb crystal in plasmas under strong interaction among the particles. Usually the fine particles constitute a very thin layer extended in horizontal direction and the properties of the crystal formation are actively investigated theoretically and experimentally Recently it is reported that crystal structures consisting of single chain-like string of fine particles, which is extended in either vertical [1] or horizontal [2] direction, are formed in an rf discharge plasma in the absence of the magnetic field. On the other hand, the collective motions, such as the potential driven vortex flow of fine particles [3,4], are observed in fine particle clouds in a dc plasma as well as a vortex formation under the weak magnetic field [5,6]. Li this paper, we have succeeded in a formation of a chain-like structure of fine particles, i.e., a vertical string, which is extended in vertical direction in a dc glow discharge plasma along vertical magnetic field. Here, we use a smaller levitation electrode in order that a shaper hill-type potential profile is formed in radial direction. Several fine particles are trapped in this narrow potential well and are arranged to form a chain-like vertical string, because such a radial potential profile can be kept to extend in vertical direction along the magnetic field by using a double plasma method. By increasing the number of fine particles, we can also control the number of the vertical strings.
458
R. Ozaki et al./Vertical string structure of fine particles in a magnetized DC plasma
EXPERIMENTAL APPARATUS The schematic diagram of the experimental apparatus is shown in Fig.l. We produced a double-plasma (DP) which consists of an inner main plasma (MP) of higher plasma potential and an outer auxiliary plasma (AP) of lower plasma potential, surrounding the MP radially. Both the MP and AP plasmas are produced by a dc glow discharges between upper mesh cathode 1 and ring anode 1, and between disk cathode 2 and mesh anode 2, respectively, at the argon pressure of 270 mTorr. The outer diameters of these electrodes are 10 cm all, but the cathode 1, anode 1, and ring electrode 1 have a hole of 1 cm in diameter at the center. Only the anode 2 has a hole of 8cm in diameter at the center, which is covered with a mesh. Anode 1 is biased at 0 V, and anode 2 and ring electrode 1 are biased at -100 V, in order that the plasma potential of AP is lower than that of MP. Using this DP plasma method, we have two diffusing plasmas with different plasma potentials in the experimental region where the levitation electrode (LE) biased at -100 V is situated. The hill-type plasma potential profile in the radial direction is produced in this way and the profile can be kept along axial direction because the vertical magnetic field of 180 G is applied. Here, we used a small LE of 10 mm in diameter, therefore a sharp hill-type potential is sustained in the experimental region over 15 mm above the LE. The particles used are mono-disperse methyl methacrylate-polymer spheres of 1.17-L20 g/cm^ and diameter of 10 (±1.0) ju m. They are injected from a sieve into the MP region through the center hole of mesh cathode 1. Particles are observed by Mie-scattering of He-Ne laser light sheet, with a breath of 5mm incident from horizontal direction. To investigate the particle behaviors, a CCD camera is used as a
Calhodel MP
Anodel 3
r c
Dust Particles _
E
Ring Electrodel
I-
^^^X)de2 Levitaticm Electrode
CMxxte^ 100mm
Figure 1. Schematic diagram of experimental apparatus
R. Ozaki et al/Vertical string structure offineparticles in a magnetized DC plasma
459
detector of the signals, which are recorded on videotape and processed by a personal computer. The plasma parameters are measured by a wire probe, which is movable both in radial and axial directions.
EXPERIMENTAL RESULTS Figure 2 shows a typical side-view photograph of single vertical string in which we can see eleven fine particles are lining up in vertical direction along the magnetic field. Fine particles making a string are arranged with an almost equal distance of about 300// m. Top-right figure in Fig. 2 shows a top-view photograph of the string. The string is situated around the radial center of the LE at about 15 mm above the LE. By injecting more fine particles, we can produce plural vertical strings. In case of four vertical strings, they are located on the horizontal plane at the apex of a square with side length of about 400// m. It is noted that the vertical positions of particles in the diagonal strings are in phase, i.e., the corresponding particles in diagonal strings are situated almost at the same vertical positions. The particles of adjacent strings, however, are situated out of phase, i.e., the particles are arranged at the vertical position in the middle of the closest particles in adjacent strings. Therefore, all the particles are trapped at the stable positions generated by the electrostatic repulsive forces of the particles. In case of more than two strings, we find that these strings start rotating in azimuthal direction. The rotation direction is as same as the electron gyration motion against the applied vertical magnetic field, i.e., in the diamagnetic direction of electrons. The azimuthal velocity v^ in case of four vertical strings is about l-3mm/sec, which corresponds to angular rotation frequency of about 1-2 Hz. We also find that the particle rotation in azimuthal direction is accompanied with a radial fluctuation of about 20Hz.
Figure 2. Typical side-view photograph of single verical string. Top-right figure shows a cross sectional view of the string.
460
R. Ozaki et al./Vertical string structure of fine particles in a magnetized DC plasma
-80
-100
-120
v., (V)
-140
-160
Figure 3. Dependency of rotation velocity on the anode 2 potential in case of four vertical strings.
The bias voltage of anode 2 varies the azimuthal rotation frequency as shown in Fig.3. The rotation velocity v^ is increased with decreasing the anode 2 potential. Here, the potential of anode 1 is fixed at 0 V It is noted that the radial potential gradient between the MP and AP is increased by this procedure, because the plasma potential of AP is decreased with the anode 2 potential Therefore, the increase of the azimuthal rotation velocity seems to be related with the increase of the potential gradient.
CONCLUSIONS Vertical strings of fine particle are observed in weakly ionized argon dc glow discharge plasma by using a double plasma method under vertical magnetic field. In case of plural vertical strings, the strings rotate in the electron gyration direction against the vertical magnetic field. The rotation velocity depends on the potential gradient between the MP and AP, which is controlled by the anode 2 potential.
REFERENCES 1. Mitchell L. W. et al. In Abstract of Seventh Workshop on the Physics of Dusty Plasmas, 1998, p. 30. 2. Homann, A. et al, Phys. Rev. E, 56, pp. 7138-7141 (1997). 3. Sato, N. et al. Physics of Dusty Plasmas Seventh Workshop, edited by M. Horanyi et al. New York : American Institute of Physics, 1998, pp. 239-246. 4.1izuka, S. et al. Physics of Dusty Plasmas Seventh Workshop, edited by M. Horanyi et al. New York : American Institute of Physics, 1998, pp. 175-178. 5. Uchida, G. et al, "Generation and Control of Vortex Flow of Fine Particle with Coulomb Lattice," in Proceedings of the 15th Symposium on Plasma Processing, 1998, pp. 152-155. 6. Uchida, G. et al, "Dust Vortex in DC Discharge Plasma under a Weak Magnetic Field" m Proceedings of the 1998 International Congress on Plasma Physics, 1998, pp. 2557-2560.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
461
Determination Factor of Coulomb Crystal Structure in Dusty Plasmas Kazuo Takahashi and Kunihide Tachibana Department of Electronic Science and Engineering, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
Abstract. In dusty plasmas, solid particles fonn Coulomb crystals, which have face-centered cubic, body-centered cubic, so called three-dimensional (3D), and simple hexagonal, two-dimensional (2D), structure. These structures depend on particle diameter and density of particle cloud. The simple hexagonal crystal is formed due to a wake potential, in addition to a Coulomb potential between particles. In this study, the condition is discussed, where the wake potential is effective and the 2D structures are formed.
INTRODUCTION Many researchers have been studying crystallization phenomena in strongly coupled Coulomb systems of dusty plasmas (1, 2, 3, 4). The formation of Coulomb crystals attracts interests of plasma, condensed matter and solid state physics. Negatively charged solid particles can form 3D and 2D crystal structures in non-equilibrium cold plasmas. The 3D structures should be formed by the Coulomb repulsive force in particle cloud confined by some forces from external field. In the formation of the 2D, simple hexagonal, structures, where particles line up in straight lines, hov^ever, an attractive force plays an important role in addition to the repulsive force. The attractive force is proposed to be caused by the wake potential, which is generated by ion flows around particles (5). Furthermore, experimental results of analyses of the attractive force by optical manipulations implied that the attractive force was the wake force (6). On the other hand, 3D-2D transition was observed in particle growth process of a carbon plasma (7). The transition seemed to be induced by dominant particle interaction change from the repulsive force to the attractive one. In this paper, the condition and the reason are discussed, where the simple hexagonal structures are formed, why the wake potential becomes effective in particle growth process.
STRUCTURAL TRANSITION FROM 3D TO 2D Coulomb crystals formed by growing particles changed their structures from 3D to 2D
462
K. Takahashi, K. Tachibana/Determination factor of Coulomb crystal structure
Structures. As the particles grow, their diameter increases and density of particle cloud decreases. At the beginning of the process, where the diameter and the density were small (-- 1.0 |Lim) and high (-- l.OxlO^cm'^), respectively, the 3D structures were formed. After that, low density (< 1.0x10^ cm-^) particle cloud consisting of big particles (> 2.0 |Lim) formed the 2D structure. In the low density cloud, an attractive force from a wake potential was effective. The wake potential can be caused by ion flow in pre-sheath or sheath region, where particles are levitated by external forces. Then, ion drift velocity must exceed phase velocity of ion acoustic wave, i.e., ratio of the drift velocity v. to the phase velocity v^ vJv^=M>L
(1)
If the potential is effective, a particle draws other particles, which are located in the lower reaches of ion flows, under itself (8). The wake force, which can be stronger than the Coulomb repulsive force, makes particle alignments in the direction parallel to ion flows. We note the reason why the wake potential becomes effective in low density cloud consisting of big particles and dominant force of particle interaction changes from the Coulomb repulsive to the wake force.
NUMERICAL ANALYSIS Here, Coulomb crystal structures are investigated, which depend on particle conditions, i.e., particle diameter and cloud density. Particles in a plasma are levitated by some forces from external field, electrostatic, ion drag and gravitational force. Although neutral gas drag and thermoforetic force can actually act on particles, there are no gas flow and no thermal gradient in this analysis. Then, particle charge Q^ is determined by two conditions. One is that the total current flowing into a particle is zero, i.e., the sum of electron current J and ion one J e
I
Je + Ji=0.
(2)
where energy distribution function of electron and ion obey Maxwell-Boltzmann distribution. The other is charge neutralization condition of plasmas, -eN^+eN^ + Q,N,=0,
(3)
where e is elementary electric charge and, A^^, A^. and A^^ are density of electron, ion and solid particle, respectively. Forces acting on a particle of the charge Q^, electrostatic and ion drag force are estimated in finite ion flows. Their forces are expressed as fiinctions of ion drift speed (9). Balance of those forces and gravity gives an equilibrium position of the particle. Ion velocity at the position is calculated in the condition, where A^., ion
K. Takahashi, K. Tachibana/Determination factor of Coulomb crystal structure 463
1
1
\
r
3 4 5 Particle Diameter (|Lim)
6
\
2
1
\
Figure 1. Mach number M at an equilibrium position of particles
temperature T. and electron temperature T^ are 1.0x10^cm ^ 3.0x10"^ eV and 3.0 eV, respectively. Ion velocity at an equilibrium position of a particle with its diameter from 0.5 to 6.0 |Lim was calculated. Then, particle number density from 1.0x10"^to l.OxlO^cm'^ was used for charge calculation. Figure 1 shows Mach number M at the equilibrium position. When particle diameter increases and density of the cloud decreases, i.e, particle growth process progresses, M increases. At the beginning of the process, M at the position of the particles which form 3D structures is less than 1. When 2D structure is formed, ion speed at the position, however, exceeds ion acoustic wave velocity. In a region of low density cloud consisting big particles, i.e., M>1, the wake potential should be effective in particle interaction, therefore, simple hexagonal structure must be stable after growing of the particles. At the boundary, M=l, the crystal structures may drastically change from 3D to 2D. The wake potential from a test particle at z=0, in the direction parallel to ion flows, is given by
Q] 2Ke^(\-M-^)
COS(|Z|/A^,VM^-I)
(4)
where z and A^^ are distance from the test particle and electron Debye length, respectively (8). On the other hand, the Yukawa-type Coulomb potential is given by
464
K. Takahashi, K. Tachibana /Determination factor of Coulomb crystal structure
where r and A^ are the distance and effective Debye length in dusty plasmas, respectively. In the ion flow direction, a particle, separated from the test one for an inter-particle distance, is affected more strongly by the wake potential than by the Coulomb potential. Because the correlation length of the wake potential is longer than that of the Coulomb one. The wake force on a particle can be stronger than the Coulomb repulsive force at intervals of an inter-particle distance. Therefore, it seems that vertical alignments in the direction parallel to ion flows apt to be formed at first.
CONCLUSION The mechanism of the 3D-2D transition in particle growing process was noted. With particle growth and decreasing of cloud density, particles move to high ion speed region. When ion speed at an equilibrium position of the particles exceeds ion acoustic wave velocity, simple hexagonal Coulomb crystal is formed due to wake potential. In this study, a simple calculation of forces on particle presented the phase boundary between 3D and 2D structure in Coulomb crystal formed by growing particles. Further analyses with the wake potential is necessary to reveal the structural transition and formation of 2D Coulomb crystal. Especially, it will be interesting that the simple hexagonal Coulomb crystal are analyzed as one of the complicated systems composed by two particle interactions.
REFERENCES 1. Hayashi, Y., and Tachibana, K., Jpn, J, Appl. Phys. 33, L804-L806 (1994). 2. Thomas, H., Morfill, G. E., and Demmel, V., Phys. Rev. Lett. 73, 652-655 (1994). 3. Chu, J. H., and I, Lin, Phys. Rev. Lett. 72, 4009-4012 (1994) 4. Melzer, A., Trotenberg, T, and Piel, A., Phys. Lett. A191, 301-308 (1994) 5. Vladimirov, S. V., and Nambu, M., Phys. Rev. E 52, R2172-R2174 (1995) 6. Takahashi, K., Oishi, T, Shimomai, K., Hayashi, Y., and Nishino, S., Phys. Rev. E 58, 7805-7811 (1998) 7. Hayashi, Y, and Takahashi, K., Jpn. J. Appl. Phys. 36, 4976-4979 (1997) 8. Ishihara, O., Phys. Plasmas 5, 357-364 (1998) 9. Barnes, M. S., Keller, J. H., Forster, J. C , O'Neill, A., and Coultas, D. K., Phys. Rev. Lett. 68, 313-316 (1992)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
465
Instabilities of Dust Particles Levitated in an Ion Sheath with Low Gas Pressure T. Misawa, S. Nunomura, K. Asano, N. Ohno, and S. Takamura Department of Energy Engineering and Science, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan abstract. Correlation between vertical oscillations of dust particles, which are spontaneously excited at a very low gas pressure, is investigated using a high speed ICCD camera. It is found that the correlation among particles in neighborhood is strong. The frequency and wavelength obtained from experimental observations are in agreement with the dispersion relation theoretically predicted for transverse dust lattice wave.
I. Introduction In recent years, the investigation of wave propagation phenomena in dusty plasmas has received considerable attention. Especially, the dust acoustic wave (DAW) and dust lattice wave (DLW) have been widely investigated in theoretical and experimental works. The DAW, theoretically predicted in (1), has been observed in several plasma experiments. The DLW, propagated in crystal-like structure of dust particles, has been studied theoretically. Two low-frequency modes associated with horizontal (longitudinal waves) (2) and vertical oscillations (transverse waves) (3) were predicted in a onedimensional horizontal dust chain. In laboratory experiments, the longitudinal waves in DLW mode were found to be excited by a laser pulse in a one-dimensional dust chain formed in RF sheath region (4). On the other hand, the transverse waves DLW, however, has not been observed yet. In our recent experiments, dust plasma crystals can be formed in an ion sheath region in front of the negatively biased horizontal electrode with a low gas pressure of several mtorr (5,6), where the friction force between dust particle and neutral gas particle is not dominating compared with gravity force and electric forces on dust particles. This experimental condition differs completely from those in conventional RF discharge and DC glow discharges with a high neutral gas pressure. Furthermore, it is clearly observed that the dust particles start to oscillate vertically, in a parameter range depending on plasma density HQ and neutral gas pressure P. In this paper, we report detailed analysis on self-excited vertical oscillation of
466
T. Misawa et al /Instabilities of dust particles levitated in an ion sheath
particles on a one-dimensional dust chain taken by a high speed intensified chargecoupled device (ICCD) camera. These oscillations seem to have a strong relation to the transverse waves in DLW mode.
II. Experimental setup The experiments are performed in a Filament large, unmagnetized plasma devices (Cathode) "KAGEROU" with 42 cm in diameter and 110 cm in height, equipped with multidipole magnets for surface plasma He-Ne Laser confinement, as shown in Fig. 1. Argon plasma is generated by the DC discharge between the hot-filaments and the tungsten anode electrodes enclosing each with permanent magnet. Typical plasma parameter are TQ- 0.5 eV and A2O~ 5X10^ 5x10^ cm'"^. The plasma density can be FIGURE 1. Experimental setup controlled by changing discharge current. Dust particles can be levitated above the negatively biased meshed electrode and confined on a horizontal plane by the negatively biased ring electrode. The rectangularshaped electrode also put on the meshed electrode can make one-dimensional chain of levitated dust particles. The diameter of the employed dust particle is about 5 |i,m. The observation of dust particles was carried out by using the high-speed ICCD camera with a macroscopic lens, imaging the scattered light from the dust particles by the HeNe laser expanded vertically with a cylindrical lens. This high-speed ICCD camera system can record of 125 image frames every second Therefore, the sampling rate of this camera is 8 msec, which is necessary to analyze the dynamic behavior of the dust crystals, because typical frequency of the self-excited vertical oscillation of the dust particle is about 10 Hz.
III. Experimental results and discussion Figure 2 (a) shows the typical image taken by the high-speed ICCD camera. It is found that nine dust particles trapped in plasma-sheath boundary almost aligned in a single row. The distance between dust particles d is ranged from 0.5 mm to 1.3 mm. The time evolution of the levitated position of dust particles is shown in Fig. 2(b). The dust particles in the right hand side are found to oscillate vertically with large amplitude, compared to those in the left hand side. More careful observation gives us
467
T. Misawa et al. /Instabilities of dust particles levitated in an ion sheath
that two dust particles in neighborhood almost oscillate in out-of-phase ( see No.l ~ No. 4 ). This indicates that dust particles in neighborhood are strongly coupled by repulsive Coulomb interaction. It may be possible to regard the wavelength as twice of an interparticle distance (X~1.6mm) if this time evolution is assumed to be related to the propagation of transverse DLW. Now, consider the No. 1 ~ 3 dusi ( a ) P=1.8mtorr n^-0.5xl#cm-^ particles in detail. Figure 3(a) shows that the time evolution of vertical positions of No. 1 ~ 3 dust particles for several period. In Fig. 1mm / t ^ , 3(b), frequency spectra unformed from the No.4 No.3 No.2 No.l data shown in Fig. 3(a) by using fast Fourier transform (FFT) show that the eigenfrequency of the oscillation is about 12 Hz, but there is a slight difference of eigenfrequency between these three dust particles because the particle size distribution is not mono-disperse. This deviation of the 6.5 eigenfrequency is likely to influence the t = 16 ms phase relation of the dust particle oscillation • I I I 1—L 6.0 in a long period, which may leads that the .^^•' $:5 6.5 32 ms
c^
O
13 o
v.
>
6.0
-i-J
I
L
•N./'
6.5
\ .
t = 48 ms
6.0 6.5
t =^ 64 ms
6.0 /^—-
6.5 6.0 20
30
40
Frequency [Hz]
50
60
80 ms 0
2.0
4.0
6.0
Horizontal Position [mm]
FIGURE 3. (a) Time evolution of oscillating
FIGURE 2. (a) Typical images of one-
dust particles, and (b) FFT spectra obtained by
dimensional dust chain, (b) time evolution of
analyzing the wave patterns of (a)
the dust chain.
T. Misawa et al /Instabilities of dust particles levitated in an ion sheath
468
phenomena of transverse DLW propagation become more complicated. The dispersion relation of transverse DLW is theoretically derived by the following equation (1); (O
kd. -4a(J)sin^(—),
(1)
where d is the interparticle distance, y ^s the coefficient under the linear vibration, given by the potential well structure for the vertical dust confinement, a{x) is quantity related the screened interparticle potential. Figure 4 shows the dispersion relation calculated from Eq. (1) by taking the experimental condition into account. The wavelength obtained from Fig. 2 and the frequency from Fig. 3(b) gives an experimental point on the (xyk space shown in Fig. 4. This experimental data is found to be in reasonable agreement with the theoretical model of the DLW wave excitation.
2
3
4
5
6
A: [mm"'] FIGURE 4. Dispersion relation of transverse DLW. Calculated result is shown by solid {d=\3mm) and broken {d=0.5mxx\) lines. The closed circle is experimental result.
IV. Summary Detailed observation on the self-excited vertical oscillation in a one-dimensional dust chain has been carried out in the experiments, which suggests that the transverse DLW may be excited. For much clearer verification of the transverse DLW, experiments using monodispersion fine particles should be required.
REFERENCES 1. 2. 3. 4. 5.
N. N. Rao, P. K. Shukla and M. Y. Yu, Planet. Space Sci. 38, (1990) 543. F. Merandsct), Phys. Plasmas 3, (1996) 3890. S. V. Vladimilov, P. V. Shevchenco and N. F. Cramer, Phys. Rev. E56, (1997) R74. A. Homann, A. Melzer, S. Peter and A. Piel, Phys. Rev. E56, (1997) 7138 S. Nunomura, T. Misawa, K. Asano, N. Ohno, S. Takamura, Proc. of 1998 ICPP & 25th EPS Conf on Contr. Fusion and Plasma Phys., Praha, EGA Vol.22C, (1998) 2509 6. S. Takamura, N.Ohno, S. Nunomura, T. Misawa and K. Asano, Invited talk in this Conference
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
469
Instabilities of the dust-plasma crystal N. F. Cramer and S. V. Vladimirov Department of Theoretical Physics and Special Research Centre for Theoretical Astrophysics, School of Physics, The University of Sydney, New South Wales 2006, Australia
A b s t r a c t . The dispersion characteristics of the low frequency mode associated with vertical oscillations of dust grains in a quasi-two-dimensional dust-plasma crystal are calculated, taking into account the dependence of the dust charge on the local sheath potential. It is shown that the equilibrium of the dust grains close to the electrode may be disrupted by large amplitude vertical oscillations.
The negatively charged dust grains in a dust-plasma crystal usually levitate in the sheath region near the horizontal negatively biased electrode v^here there is a balance of gravitational and electrostatic forces on the grains [1]. It has been demonstrated that vertical vibrations of dust particles may cause the propagation of lattice modes with optical-like dispersion [2]. Observations of vertical motions of the dust are important for diagnostics of processes in the plasma sheath [3], especially in the case of several vertically arranged horizontal layers v^hen vertical oscillations are affected by the parameters of the ion flow [4]. The spontaneous excitation of vertical vibrations of dust grains was recently experimentally observed
[5].
Previous analytical models considering lattice vibrations [2,4,6] dealt with dust grains of a constant charge. Here we demonstrate that the dependence of the dust particle charge on the sheath parameters has an important effect on the oscillations and equlibrium of dust grains in the vertical plane, leading to a disruption of the equilibrium position of the particle. Consider vertical vibrations of a one-dimensional horizontal chain of particulates of equal mass m^ separated by the distance ro in the horizontal direction (see [2] for details). The equilibrium charge Q of the dust particles (which is dependent on the local electric sheath potential) can be found from the condition of a zero total plasma current onto the grain surface. We assume that (re)charging of dust grains is practically instantaneous, and we therefore neglect the charging dynamics. The electron and ion currents onto the dust grain are given by ^ a ( Q ) = I ] / ^afaCTaiv,
Q)vdv.
(l)
Here, the subscript a = e,i stands for electrons or ions, e^ and fa are the charge
470
N.E Cramer, S.V Vladimirov /Instabilities of the dust-plasma crystal
and distribution function of the plasma particles, with te = —ei = - e , v = |v| is the absolute value of the particle speed v, and aa is the charging cross-section given by [7]. The electrons are assumed to be Boltzmann distributed, with a temperature Tg and density no in the bulk plasma. We consider coUisionless, ballistic ions within the sheath with the distribution function fi a S{\^)5{vz - Vi{z)), where Vi{z) is the ion streaming velocity at the distance z from the electrode. The charge of a dust particle in the sheath region is then determined by the equation \
arriivl (1 - 2eip{z)lmivl) ^ exp
LTe
[^[z)-vQ{z)la)
(2)
where v^ = Te/rrii is the squared ion sound speed, and (p{z) is the external potential in the sheath relative to the potential in the bulk plasma. The charge can become zero for a strong enough sheath potential, such that the ion current dominates, so the dust particle cannot levitate and must fall onto the electrode. The sheath potential can be found from Poisson's equation, neglecting the total charge contributed by the dust grains. Assuming the electrode has a potential of -4 V, typical of dust plasma experiments [5,8], the dependence of the potential, and thence of the dust grain charge using Eq. (2), on the distance from the electrode can be found. A plot of the charge against distance from the electrode is presented in Figure 1.
2 4 6 8 Normalized Distance
10
F I G U R E 1. The dependence of the charge q = -{Q/e) x 10~^ of the dust grain on the distance h = Z/XD from the electrode. AD = 2 x 10~^cm, Tg = leV, M^ = VQ/VI = 1.5, mi/rrie = 40 x 10^, p = Ig/cm^, and a = 0.25 x 10~^cm.
Considering now the equilibrium and oscillations of a horizontal line of dust grains, we note that the interaction energy between particles at the n and n — 1 positions can be approximated by the Debye law
N.E Cramer, S.V Vladimirov / Instabilities of the dust-plasma crystal
exp
Un,n-l =
,
[—^)
471
(3)
where Xo is the Debye screening length of the dust charge Q by plasma particles (in our case XD = Xoe = (Te/47rnoe^)^/^). The electrostatic energy of a dust grain due to the sheath potential is Uei = Q{z)^{z), The force on a grain is then found from the total potential energy. Assuming only nearest neighbour interactions, the balance of forces in the vertical direction gives the equation of motion for linear vertical oscillations of the grains dv
r5
V D
XD
E'^iSZn-l+SZn+l),
(4)
where F depends on the sheath potential. Substituting 5zn ~ exp(—za;i + ikur^) into (4), we obtain the expression for the frequency of the mode associated with the vertical vibrations:
rrid
(5)
where 7 is a function of the sheath potential. This mode has an optical-mode-like dispersion similar to that studied earlier [2] in the case of a constant grain charge. The total potential energy of a dust particle (including the mutual interaction energy of the grains), for A: = 0, is t/tot = Q{z)ip{z) -F magz + 2 ^ ^ e - ^ « / ^ ^ .
(6)
The dependence of the total potential energy on the distance from the electrode is shown in Fig. 2. For comparison, we also plot the energy in the case of a constant Q. We see that the potential always has a minimum for the case of Q =const, but in the case of a variable charge the minimum can disappear (the upper curve). For a = 0.338 X 10~^cm, the minimum disappears. This is close to the critical radius observed experimentally [5]. 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 presheath. Vertical oscillations about the stable equilibrium, with frequencies given by (5), may develop high amplitudes (because of an instability in the background plasma or a driving force). This may lead to a fall of the oscillating grain onto the electrode when the potential barrier is overcome. Such a disruption of the dust motion has been observed experimentally [5,8].
472
N.E Cramer, S.V Vladimirov /Instabilities of the dust-plasma crystal
2 4 6 8 Normalized Distance
10
F I G U R E 2. The total interaction energy U — C/tot/^e as a function of the distance h = Z/XD from the electrode for the different sizes of a dust particle: lower curves a — 0.25 x 10~^cm; upper curves a ~ 0.34 x 10~^cm. The dashed lines correspond to the case of a charge constant at the large z value: q = 0.46 (lower) and q = 0.67 (upper).
ACKNOW^LEDGMENTS This work was supported by the Australian Research Council and a University of Sydney Research Grant.
REFERENCES 1. J.H. Chu and Lin I, Phys. Rev. Lett. 72, 4009 (1994); H. Thomas, G.E. MorfiU, V. Demmel, J. Goree, B. Feuerbacher, and D. Mohlmann, Phys. Rev. Lett. 73, 652 (1994); Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. 33, L804 (1994); A. Melzer, T. Trottenberg, and A. Piel, Phys. Lett. A 191, 301 (1994); H.M. Thomas and G.E. MorfiU, Nature 379, 806 (1996). 2. S.V. Vladimirov, P.V. Shevchenko, and N.F. Cramer, Phys. Rev. E 56, R74 (1997). 3. A. Melzer, A. Homann, and A. Piel, Phys. Rev. E 53, 2757 (1996); S. Peters, A. Homann, A. Melzer, and A. Piel, Phys. Lett. A 223, 389 (1996). 4. S.V. Vladimirov, P.V. Shevchenko, and N.F. Cramer, Phys. Plasmas 5,4 (1998); S.V. Vladimirov and N.F. Cramer, Phys. Scripta 58, 80 (1998). 5. S. Nunomura, T. Misawa, K. Asano, N. Ohno, and S. Takamura, Proceedings of the Int. Congress on Plasma Physics, (Prague, 1998), Europhysics Conference Abstracts, 22C, 2509 (1998). 6. F. Melands0, Phys. Plasmas 3, 3890 (1996). 7. L. Spitzer, Physical Processes in the Interstellar Medium (Wiley, New York, 1978). 8. L. Mitchell and N. Prior, private communication (1998).
FRONTffiRS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and P.K. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
473
Some Remarks on Dust Lattice Waves in Plasma B. Farokhi^ N. L. Tsintsadze^ and D. D. Tskhakaya^ "Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5531, Tehran, Iran. ^Institute of Physics, Georgian Academy of Sciences, P.O. Box 380077, Tbilisi, Georgia The dependence of the dust grain charge on the grain potential is taken into account. The Poisson equation for small potentials takes then the form of the Schrodinger equation. The spatial distribution of the potential in the lattice includes the effect of whole system of dust particles. Such a self consistent description gives the dispersion relation for the dust lattice waves, different the expression found earlier. It is shown that for the existence of ideal lattice the dusty plasma parameters must satisfy the definite relation. I. I N T R O D U C T I O N Recently several investigations [1-4] have dealt with the lattice oscillations in the dust plasma crystal.
In [4] most consistent construction of the dust-lattice
wave (DLW) theory has been presented. In the mentioned paper the interaction only between nearest dust-grains is taken into account. Furthermore, the charge of dust-grains is assumed being fixed. Under this assumption the main Eq. (8) of [4] has obtained the form of an inhomogeneous equation, that restricts the application of methods, known from the solid state physics, for the description of oscillatory phenomenon in dust lattice. In this paper we investigate DLW on the basis of Kronig-Penney model [5]. The charge of dust grain is connected with its surface potential, which depends on the potential in plasma [6]. We assume grains being spherical conductors with the radius a^, thus the grain charge Q^ is proportional to its potential. Such a dependence of grain charge on its potential changes the type of main equation, obtained from the Poisson's equation. For small values of potentials it becomes the form of the Schrodinger equation for potential. The calculations are carried out for the one-dimensional lattice putting the cyclic boundary condition on the chain of dust grains. We found: 1. the spatial distribution of potential along the lattice chain. 2. the relation between dust plasma parameters, when the ideal dust crystal (dust grains are spaced with equal distances) can exist. 3 . the dispersion
474
B. Farokhi et al /Some remarks on dust lattice waves in plasma
relation for DLW in the case of small potentials. This dispersion relation differs from the relation found in [4]. II. MODEL OF D U S T Y PLASMA The massive, negatively charged dust grains are considered as discrete particles, while the electrons and ions assume to be distributed by Boltzmannean with the same temperatures, T^ = Ti — T. Assuming the grains to be conductors, we have Qn = dn^n, whcrc $^ is the potential on grain's surface. When the grain's size is smaller than the grains separation distance, a^ < < d, the grains position Zn can be described by 5-function. Below we consider the one-dimensional case only. This approximation looks rather artificial, however its results can give some notion for the more general, three-dimensional case. In one-dimensional case the Poisson equation gives ^
^
- klSinhmz))
+ X: hn6{z - ZnMz) = 0,
(1)
where RD — (T/STre^no)"^/^, $ = e(/)/r and hn — N^an/S is the size of dust grain, averaged over the area S, perpendicular to the linear lattice, A^o is the number of grains embraced by the area S. The Eq. (1) is obtained by the integration of the three dimensional Poisson equation over the area S. The solution of Eq. (1) and its derivative must satisfy the following boundary conditions: (z)|,,+o = <E>L-o,
(2)
Apparently the symmetry of the dust crystal is determined by the shape of plasma boundaries and the minimum of the potential energy of grains interaction in our case corresponds to the equal space between the grains. III. S T R U C T U R E OF POTENTIAL FIELD IN LATTICE Let us consider at first the small potential case, $ < < 1. Eq. (1) can then be written as ^
^
- 4 $ ( z ) + Y. bnS{z - z^mz)
= 0.
(4)
B. Farokhi et al /Some remarks on dust lattice waves in plasma
475
In the region, z^ < z < 2:^4-1, Eq. (4) has the solution, $(z) = {^nSinh[kD{zn^i-
z)] + ^n+iSinh[kD{z-
(5)
Zn)]]Sin^^
According to (3) the grains potentials are connected with following relation: ^n{tanh[^{zn
- Zn-i)] + tanh[^{zn+i
- z^)] - bn/ko} =
= ($^_i - ^ri)Sinh-^[kD{zn - Zn-i)] + {^n+1 " $n)5zn/i~^[A;^^(z,^^+i - Zn)], (6) The solution in the region Zn-i < z < z^ has the analogical form. It can be found from (5) by the shift n -^ n — 1. For the lattice with identical grains, (they have the same surface potentials, $^ — $Q = const and bn = b^ = const) from (6) it follows that distances between the grains are equal and the following relation is satisfied (7)
tan{kDd/2) = bo/2kD,
where d — \zn^\ — Zn\ is the separation between two consecutive grains. In the general case, when the grain's potentials are not equal, the relation (6) connects the potentials of diflFerent grains with each other. Therefore the potential (5) can be seen to arise from the effect of grains' system. IV. DISPERSION RELATION FOR DLW Let us assume the dust particles to execute small oscillations around their equilibrium position. The grains are assumed to have the same potential $0 (and charge Qo) and an uniform separation distance d. We consider the case, when the dust particles maintain their equilibrium potential (and charge) during oscillations. It means we assume the oscillation frequency cj to be larger than the characteristic frequency of dust particles charging. The potential (5) we can then present in form (z) = $oCos/i[^(z„+i + z„ - 2z)]Co5/i-i[^(z„+, - z,,)l Zn < Z <
(8)
Zji^i.
This expression gives the total electrostatic energy of interaction for the dust grains' system, which yields '^ == EnQn^{Zn)
^
kn = Qo^oY^Cosh[-—{Zn+i
k + Zn-l - 2Zn)]Cosh~^[^{Zr,,^i
- Zn-l)]-
(9)
476
B. Farokhi et al /Some remarks on dust lattice waves in plasma
The grains execute small oscillations, Zn = Zno + <^n, around their equilibrium position Zno- As usually, we seek the wave-train solution ^^ — Anexp[i{knd — out)]. Under the condition, ko^ < < 1, we obtain the dispersion relation op r \(i. J"Sin[kndl2), (10) mQaoCosh{kDa) where mo and ao are the mass and the radius of dust grain respectively. The maximum frequency equals ^ = ±^o[
COm = Qo[
^ % ,
J^'.
(11)
The expressions (10), (11) differs from expressions, found in [8]. The results of [8] are valid for small potential in cases, when the dust separation distance is of the order of or larger than plasma Debye length {k^d > 1). These conditions allows to use the approximation, considering the interaction only between neighbor particles. Our expression (10) is valid for the case, k^d < 1, also. In both cases, kod » 1 and kjjd ^ 1, the frequency (10) is {d/aoY^'^ » 1 times larger than the frequency found in [8]. In conclusion we want to notice that for the ideal lattice, where the spacing and charges of grains are identical, the dusty plasma parameters must satisfy the definite relation (7).
REFERENCES
1. A. Homann, A. Metzer, S. Peters, and A. Piel. Phys. Rev. E 56, 7138 (1997). 2. G. MorfiU, H. Thomas and M. Zuzic. in Advances in Dusty Plasmas^ edited by P. K. Shukla, D. A. Mendis and T. Desai (World Scientific, Singapore, 1997), PP. 99-142. 3. M. R. Amin, G. E. MorfiU and P. K. Shukla. Phys. Plasmas 5, 2578 (1998). 4. F. Melandso, Phys. Plasmas 3, 3890 (1996). 5. C. Kittel, Introduction to solid state physics (John Wiley and Sons, Inc, New York, Chapman and Hall, Ltd, London, 1956), chapter 11. 6. E. C. Whipple, T. G. Northrop and D. A. Mendis. J. Geophys. Res. 90, 7405 (1985).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
477
Artificial Fireball as Dust-Plasma Cloud Ignatov A.M.^ Furov L.V. *, Kiinin V.N. *, Pleshivtsev V.S. * ,Rukhadze A.A. "^ ^General Physics Institute, 117942, 38, Vavilov str., Moscow, Russia "^Vladimir State University, 600027, 87, Gorkogo str., Vladimir, Russia
Abstract. We discuss the experiments aimed to creating the toroidal dusty plasma vortex. The mass of the dust trapped by the vortex and the grain charge are estimated.
Introduction A fireball is one of the most intriguing and mysterious atmospheric phenomena. Although numerous attempts were made to reveal its nature as long as to reproduce it in the laboratory (for the review see, e.g., [1]), there is no exhaustive conception of the subject. Li the present paper we discuss the experiments aimed at creation and study of stable plasma toroidal vortex, behavior of which reproduces many features of a natural fireball [2]. The experiments were carried out at Vladimir State University using large-scale facilities that provide discharge parameters close to atmospheric ones. Experimental setup The artificial fireball (AF) is created by electric explosion of a thin metal diaphragm. An inductive accumulator, from which energy was brought to a plasma gun, was used. The latter gun is an electrode system made up of a ring current conductor, a metal diaphragm, and a central pin conductor (Fig. 1). On a textolite table (1), a conducting diaphragm (2) is fastened with the help of a mandrel (4) and the lower ring conductor (3). On the mandrel (4) is placed a dielectric (7), on which the upper ring conductor (6) is arranged. An electric pulse is delivered from current-carrying buses (9, 11), clamped together by fluoroplastic insulators (10). The diameter of the current-directing buses is 60 cm. The current-carrying buses 9 and 11 are joined by wires 8 and 12 to mandrel 4 and the upper ring conductor, respectively. The diaphragm is coimected in the electric circuit of the plasma gun via the lower ring conductor pressed against the diaphragm and via a system of copper wires, which are connected at one end to the upper ring conductor and at the other end are joined together into a bunch to form a central electrode pressed against the diaphragm at its center. Such a design of conductors to the diaphragm ("spider") forms a toroidal coil, and when the current is flowing, there arises a toroidal magnetic field (magnetic torus) in this coil.
478
A.M. Ignatov et al. /Artificialfireballas dust-plasma cloud
9
B
5
6
7
10
FIGURE 1. General view and design of a plasma gun.
FIGURE 2. Artificialfireballimages.
The diaphragm is metal (steel, copper, aluminum) in the form of a thin round disk. In most of the experiments, diaphragms made of seven layers of aluminum foil were used. The thickness of each layer was 8 mem. In different experiments, the central electrode consisted of four, six, or eight copper wires. The lower ring conductor (3) was made of nonmagnetic material (stainless steel) with an inner diameter of 100 mm.
Experimental results The experiment is perceived visually in the following way. At the first moment, a bright flash arises. This flash appears as a blinding sphere about a meter in diameter. The sphere quickly goes out; since the adaptation time of a human eye is rather long, it is impossible to see anything further. Since the stable glowing formation forms inside this blinding sphere, we did not succeed in observing it. However, if the area above the plasma gun is blocked with an opaque screen up to a height of about a meter, we can see the evolution of the glowing formation. From behind the screen, a blinding sphere 30-35 cm in diameter appears (Fig. 2). This sphere is smooth and completely nontransparent and seems to be liquid instead of gaseous. It is golden yellow, similar to the color of the sun or egg yolk. On the surface of the sphere, darker stains with a violet tint and irregular, quickly changing forms run at random.
479
A.M. Ignatov et al /Artificial fireball as dust-plasma cloud
n \/l V V
I,kA \ 15 (a) 10 / \ 5 \ L 0 LA 0 80 P, rel. Units
160
1-
240
/ /
0.4
LL
P, rel. 0 Units
80
160
lA10
t.ms
0.2
D 0
1320
r
0
2
1 0
-4
0.6
2
(b) °
«AMW« ' • ^ M ^ '
U,V t,s 75 50 1-2 25 1,0 0 t,ms (^) 0.8
1400
1480
1560
10
12
IJ:A
t,ms
FIGURE 3. Time dependence of current and voltage drop (a), radiation flux (b). Dependence of afterglow duration on maximum current strength (c)
The sphere is surrounded by a comparatively transparent gray-violet shell, which has a smooth extemal surface. The shell is pierced by dark points chaotically moving within. The sphere moves upward with a velocity on the order of one meter per second. At some height, the sphere disappears, and in its place arises a smoky white torus, which reaches the ceiling of the laboratory, stops and slowly increasing in size, disintegrates over a period of several seconds, forming under the ceiling a smoky layer 30-40 cm thick with a smooth, dense lower surface. The characteristic dependencies of current, voltage drop and radiation flux are shown in Fig.3. In approximately 10% of experiments the disintegration of the AF is accompanied by the increasing brightness (the lower chart in Fig.Sb). The duration of brightness as a function of current strength of the electric impulse that explodes the diaphragm has the threshold character. For a diaphragm diameter of 100 mm and a current of less than 10.5 kA, the brightness duration is short and weakly depends on the current strength (^r/;/^/=6.88xl0"^ s/kA). For high currents, the AF lifetime very quickly grows with currents (dtth/dI=0A6 s/kA).(Fig. 3c).
Radiative cooling and grain charging To estimate the dust density inside the AF we consider the radiation losses. The measured spectrum corresponds to the black-body radiation with the temperature of the order of 4500 K. Cooling time for a single grain with this surface is about 0.01 s. To explain the observed lifetime we have to assume that the AF contains sufficiently large amount of grains, that is, it is opaque to the respect of the black-body radiation. The necessary condition for itis 4 d R a^ nD>l, where R is the radius of the AF, a is a radius of a single grain, and no is the dust density. The latter may be expressed in terms of the net dust mass, M, and the mass density of the grain material, n. This yields the critical value of the net mass
480
A.M. Ignatou et al /Artificial fireball as dust-plasma cloud
M,,=^mR'p,
(1)
that is, if M<Mcr, then the dust cloud is transparent for radiation. Under the typical experimental conditions, the critical mass is about 0.5 g, i.e., for the cloud to be opaque at least 50% of the sprayed diaphragm material should be trapped by the vortex. A dust density at that is about no --^ 10^ cm"^. The characteristic time of radiative cooling is estimated as > 0.1 s. Evidently, the grain charging is provided mostly by the thermoelectric emission. To estimate the grain charge we assume that the charge balance is provided by the electron thermoemission and the current collected by a grain. The thermoemission current is given by
h where A is the work function (for estimations we put A=4 eV) and O is the grain potential. Since the grain in thermal plasma is positively charged, it attracts the negative charges. To estimate the current collected by the grain we use the OML theory 4=4m^W-\—^ 1+—,
(3)
where n^~^ is the density and m^'Hs the mass of the negatively charged plasma species. Implementing the neutrality condition Zn^^ =n^~\ where Z = aO/e, we solve the current balance equation with respect to Z. Suppose first that negative plasma particles are electrons, then the grain charge is estimated as Z'-'1500. However, a part of electrons emitted by a grain may be trapped by atmospheric oxigen and this may influence the current balance, hi ultimate case, all electrons are trapped, so the plasma consists of grains and negative ions. The grain charge may grow up to Z-^SOOO. This results in increasing flow of oxigen ions at the grain surface and, in its tum, in enhanced oxidation. Although there is no selfconsistent model of this process, perhaps, it anticipates the desintegration of the AF. References 1.
Sinkevich O.A., Teplofizika Vysokich Temperatur (EngUsh translation: Hight Temperature), 35, 651 (1997); 35, 698 (1997) 2. Kunin V.N., Pleshivtsev V.S., Furov L.V., Teplofizika Vysokich Temperatur (English translation: Hight Temperature), 35, 866 (1997);
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukia (eds.) © 2000 Elsevier Science B.V All rights reserved
481
Quenching of a High-Temperature Phase of Fe Nanoparticles by a Microwave Plasma Processing K.Tanaka, T.Fukaya, K.Kawase, J.Fujita* and S.Iwama Department of Applied Electronics and "^Department of Electrical Engineering, Daido Institute of Technology, 2-21 Daido-cho, Minami-ku, 457-8531 Nagoya, Japan
Abstract A heating model of nanoparticles in a plasma is proposed in order to explain the quenching of the high-temperature phase of iron (y-Fe). The temperature rise of Fenanoparticles with the size ranging from 10 to lOOnm is calculated by using the measured electron temperature and ion density, the maximum values of them being 6.6eV and l.TxIO^^m"^, respectively. It is found that in one heat cycle of pulsed plasma with repetition of 120Hz the temperature of Fe-nanoparticles varies drastically from gas temperature to 2460K with the increase of the heat input from the plasma.
INTRODUCTION We developed the flowing gas evaporation technique(FGE technique) to prepare metal nanoparticles(l). By applying a microwave plasma processing to the FGE technique, a high temperature phase of iron can be quenched to room temperature in the form of y-Fe (Austenite) nanoparticles(2,). However, the heating mechanism of nanoparticles in a low temperature plasma is not clear yet. In the present paper a heating model of nanoparticles in a plasma is proposed. It is very important to make clear the heating mechanism of nanoparticles in a plasma not only for the mechanism of phase transition but also for a precise application such as the surface decoration of nanoparticles.
HEATING MODEL OF Fe-NANOPARTICLES FLOATING IN A PLASMA Iron nanoparticles transported in a low temperature plasma together with the carrier gas of Ar are heated by the plasma. Temperature increase d77d/ of the nanoparticle is represented by the following equation,
TrrVpef = 47rr^|a-c7(r^-r;)-^(r-rJ^}
(1)
482
K. Tanaka et al / Quenching of a high-temperature phase ofFe nanoparticles
where the nanoparticle is regarded to be a sphere of radius r(m), c(J/kgK) and /}(kg/m^) are the specific heat and the density of the material, respectively, gp(W/m^) the heat gain of nanoparticle from the plasma, o(W/m^K^) the Stefan-Boltzmann constant, 7i?(K) gas temperaturein in the plasma, A:(J/K) the Boltzmann constant, njja^) and v^rCm/s) density and mean velocity of Ar gas, respectively. The second and the third terms in the right-hand side correspond to the heat loss from the nanoparticle due to the radiation and the collision with neutral gases. The |2p consists of three heat fluxes supplied from electron bombardment, ion bombardment and the consequent recombination of electron with ion on the nanoparticle, and can be given by the following formula
a=
i+-
—+^f+^^ A
(2)
where x(m) is the sheath thickness, TJ^tW) the electron temperature, F^V) the sheath potential, E^^ the ionization energy of Ar andyis(A/m^) the saturation ion current. The term x/r is introduced from a simple assumption that all ions reached to the spherical sheath surface can bombard the nanoparticle, together with a basic assumption of Maxwell-Boltzmann distribution for electron energy. From the floating condition of nanoparticles in the plasma, the same numbers of electrons and ions reach to the nanoparticle in a time. Then, the relation between x and Ff is written by the following formula 4 7rr'^exp|
kTe
= 4;r(r + x ) ' 7 ,
(3)
y
where 7es(A/m^) is the random electron current. The temperature of nanoparticles floating in a plasma can be calculated from above equations by using the measured parameters; T,J^,J,„ r and 7;.
EXPERIMENTAL Iron nanoparticles prepared by the flowing gas evaporation method(l) were transported into the microwave plasma region with the carrier gas of Ar including a small amount of CH4. The probe characteristics were measured by the double probe TABLE 1. Plasma processing conditions Gasflowrate Ar 3 1 /min
CH4 2-10 ml /min
Microwave Total gas pressure 530 Pa
Power
Frequency
200~600W 2.45GHz
K. Tanaka et al/Quenching of a high-temperature phase ofFe nanopartides483 method, the plasma processing conditions being Usted in Table 1. The thus processed nanopartides were analyzed by X-ray diffraction and the electron microscopy.
RESULTS AND DISCUSSION Figure 1(a) shows an electron micrograph of Fe-nanoparticles formed at a typical plasma processing condition of 6ml/min in CH4 flow rate and 500W in microwave power, the particle size distribution being a broad one from 5 to 80nm. The X-ray diffraction pattern of the same lot of nanopartides collected on a glass substrate is shown in Figure 1(b), where the both peaks of Y-Fe(lll) and a-Fe(llO) are identified. Mean particle size of the two phases is estimated to be 37nm for a-Fe and 13nm for yFe from the broadening of diffraction profiles. The relative amount of y-Fe to a-Fe depends on CH4 content in Ar which is the major constituent gas in the plasma and also on the incident microwave power(2). In every case of those, mean particle size of yFe was found to be always smaller than those of a-Fe. Figure 2(a) shows the variation of the electron temperature T^ and the ion density n^ in one cycle of the pulsed discharge for 3.4msec. These were derived from the probe characteristics in the plasma processing condition of 2ml/min in CH4 flow rate and 600W in microwave power. Substituting the time-resolved T^ in equations (2) and (3), the time-resolved Q^ in equation (2) can be calculated. Among three factors contributing to gp, the recombination energy between electron and ion was found to play a dominant role in the present experiment. Figure 2(b) shows the calculated temperature of Fe-nanoparticles in the plasma region for three different particle sizes of 15, 30 and lOOnm. In the present calculation, the gas temperature 7^ was fixed at 600K throughout the plasma region from results in the previous paper(l). The hatched zone in the graph represents A. transition temperature zone, which distinguishes the low and high temperature phases in the temperature-composition diagram for metastable iron- cementite system(3). It should be noted that the temperature of nanopartides varies drastically in accordance
"100nm
41
43 45 2 6 (deg.)
47
FIGURE 1. (a) An electron micrograph and (b) X-ray diffraction profile of plasma processed Fe-nanoparticles.
484
K. Tanaka et al /Quenching of a high-temperature phase ofFe nanopartides 40
^ 3000
35
; 2500
(b)
<0 15nm naonm AlOOnm
30 " E 25 "2 X
^ 20 ^Ion Density
2.0 2.5 3.0 Time t (msec)
i 1500
r-Fe
: 1000 H 10 5
1.5
i 2000
4.5
c
AiTrarvsition Z(Ae
> I
i. 500
0.0
0.5
1.0
1.5
2.0 2.5 3.0 Time t (msec)
3.5
4.0
4.5
FIGURE 2. (a) Plasma parameters of T^ and n^ and (b) calculated temperature T of Fe-nanoparticles in one cycle of the pulsed plasma.
with the ion density profile, and all the Fe-nanoparticles are heated above the A3 transition temperature for a period of about 1.8msec. This means that the present heating model explains the formation of y-Fe nanopartides by the plasma processing. However, the fact that a-Fe phase is detected in the present experiment may reflect the shell structure(2) composed of the surface layer with carbon inclusion and a core without carbon inclusion. It is known that the quenching of y-Fe to room temperature is difFicuh without any carbon inclusion. Out of the plasma region, the particles were cooled to room temperature within about 1msec even in larger particles. The cooling rate between the maximum of 2460K and room temperature is roughly estimeted to be 2x10^- 6xlO^K/sec, which being larger by two order of magnitude than those necessary for martensitic transformation in the bulk Fe-C system. Nevertheless, no martensite phase has been recognized in the present Fe-nanoparticles. It is strongly suggested that the martensitie start point Ms is lower than room temperature in the case of nanopartides.
ACKNOWLEDGEMENTS The authors would like to express their thanks to Mr. Nakashima, M. and Mr. Nakane, H. for their support in the double probe measurement and also to Mr. Suzuki, T. and Mr. Kojima, S. for their helpful assistance.
REFERENCES 1. Hayakawa, K., andlwama, S.,/. Cryst. Growth, 99, 188-191 (1990). 2. Iwama, S., Fukaya, T., Tanaka, K., Ohshita, K., and Sakai, Y, in printing in NanostructuredMater., 12 (1-8), (1999). 3. Bentz, M.G., and Eliot, J.F., Trans. A.I.M.E., 111, 323- (1961)
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
485
Observation of Dust Particles Trapped in a Diffused Plasma Produced by Low Pressure R F Discharge N. Hayashi, T. Kimura, and H. Pujita Faculty of Science and Engineering^
Saga University,
Saga 840-8502,
Japan
Abstract. Submicron dust particles (Cu, 0 0.05 /um) were observed to be trapped in a radio frequency plasma at relatively low pressure, 20 mTorr. Dust particles were localized in the diffused region of the chamber for a period longer than three hours after injection of particles. The suspension mechanism of the particles may be the electrostatic force by the potential structure. The long-term time evolution of dust particles was found in plasma parameters, laser light scattered by particles and dispersion relation of dust acoustic waves.
INTRODUCTION The suspension mechanism of dust particles has been studied for several years in connection with various forces acting on dust particles. Dust plasma experiments have been performed utilizing an electrode to generate the electric field for particle sustaining against the gravity force [1,2]. The trapping of dust particles around a sheath region of powered electrode without the external field was also studied [3,4]. In order to clarify the temporal behavior of dust particles, particles are required to be trapped for a period longer than the time scale of dust plasma frequency. Also, particles must be suspended v^thout the external field for suspension to avoid disturbance. In this work, we report the experimental results on the submicron dust particles trapped for a long period in a relatively low pressure radio frequency plasma without the external field. The suspension mechanism was investigated concerning the potential formation. Temporal evolution of dust particles was investigated by saturation current, intensity of scattered laser light and dispersion relation of dust acoustic wave.
EXPERIMENTAL APPARATUS A schematic diagram of the experimental apparatus is illustrated in Fig. 1. The chamber with 16 cm in diameter and 50 cm in length was separated into a source
486
N. Hayashi et al / Observation of dust particles trapped in a diffused plasma
region and a diffused region by the magnetic filter at Z = 0 cm. An aluminum rf electrode of 6 cm diameter was set at the end of the chamber Z = —20 cm in the source region. The rf plasma (13.56 MHz) with the power of 1.1 W/cm^ was produced by introducing the Ar gas for the pressure of 10 ^ 60 mTorr. Multipole magnets forming multicusp mirror configuration were set in the source region. The magnetic filter that produces a diffused region was assembled of three parallel rodtype permanent magnets, and set perpendicular to Z axis. The submicron particles (Cu, (j) 0.05 /im) were supplied by the speaker system set on the diffused region. The electron density Ue and electron temperature Te were measured with a small cyUndrical Langmuir probe (L = 1 mm, ^ Q.l mm) : Ue ^ 10^ cm~^, Te ^ 3 eV in the source region and Ue = 10^ cm~^, Te ^ 1.5 eV in the diffused region. The plasma potential was obtained by the inflection point method using an emissive probe. The intensity of the scattering light of He-Ne laser (A = 633 nm) was measured by a spectroscope and a CCD camera that are set on the window in the diffused region and perpendicular to the chamber axis.
RESULTS A N D DISCUSSION Suspension of submicron dust particles The injected dust particles were trapped without an electric field for suspension at relatively low pressure, 20 mTorr. The scattering light of laser by dust particles was observed for a period longer than three hours after switching off the speaker vibration. Particles were localized at Z = —6 cm ^ 3 cm at 5 minute after injection, and the trapped region decreases to Z = —3 cm ~ 1.5 cm at 120 minute, as shown in Fig .2. -20 I
-10
10
-^ i-i-iiiStHWiYiiYTifflViWH^
source regicm
||
20 '
Z (cm)
•
diffused region
magnetic filter stainless steal frame permanent magnets
7777
H 160min
Permanent multi-pole magnet rod
FIG. 1. Experimental apparatus.
j ^ ^^^^^
FIG. 2. Trapped region of dust particles.
A^. Hayashi et al. / Observation of dust particles trapped in a diffused plasma
487
In order to clarify the suspension mechanism, we measured axial profiles of plasma potential at r = —3 cm, where r is the vertical distance from the chamber axis. The plasma potential in the diffused region was higher than that in the source region to result in the potential gap formation with the corresponding electric field of 6 V / c m in front of the magnetic filter. The dust particles would be attracted to this potential gap region. We estimated the electric field for suspension of dust particles under the competition of the electrostatic force and the gravity force. The gravity force acting on the Cu particle with the diameter of 50 nm is 4.6 x 10~^^ N. If an electron attaches to the particle, the electric field for the force balance is estimated to be E = 2.9 V/cm. Therefore, the electric field of E > 2.9 V / c m is required for suspension of the Cu particle. The vertical profile of the potential around the magnetic filter indicated that the potential decreased in front of the chamber wall forming ion sheath with an electric field of 40 V/cm. Therefore, the electrostatic force would act vertically on the negatively charged particles against the gravity force.
Temporal behavior of dust particles Time evolution of the scattering light intensity of He-Ne laser by dust particles was measured by a spectroscope. The speaker for particle injection worked at t == 0 '^ 5 minute. The intensity after injection of particles measured at r = — 5 cm (lower part of trapped region) increased gradually for two hours, as shown in Fig 3. While, the intensity at r = —2 cm (upper part of trapped region) was kept constant. We also measured the temporal variation of electron density after pulsed 1.8
n
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100
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^ 60
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30 minutes
20
60 minutes Z/md= 1.0 X 10-2 C-kg^i
k i — I — ^ — \ — I — I — I — \ — I — L
0
20 ^
60
80 100 120 140 160 180
Tune (minute) FIG. 3. Time evolution of intensity of scattered laser.
2
4 6 8 Wave number k (cm )
FIG. 4. Dispersion relation of dust acoustic wave.
10
488
A^' Hayashi et al / Observation of dust particles trapped in a diffused plasma
injection of particles. Probe measurements indicated that the electron density at Z = 1 cm and r = —5 cm decreased gradually from 3 xlO^ cm~^ to 1.9x10^ cm~^ and became constant after t = 100 minute. These results indicate that particles of lower part would aggregate and electron attachment to particles was enhanced for about two hours. Therefore, particles observed here may be supported from below by the electrostatic force. The density fluctuations of dust particles were found to be excited spontaneously in the diffused region. The fluctuation was measured by CCD camera as the spatiotemporal variation of the scattering laser intensity at lower part of the trapping region. The phase of the fluctuation was varied spatially and propagated to the downstream. The frequency and wavelength obtained from the intensity fluctuations in video images at after injection of particles at t = 5 minute was 10 Hz and 0.9 cm, respectively. Those at t = 60 minute became 4 Hz and 1.9 cm. The reduction of frequency suggests that the mass of a particle increased for two hours due to the particle coagulation. The dispersion relation of a dust acoustic wave (DAW) deducted using a long wave length approximation and assumption of cold particles (T^ = 0) is given by [5],
i-i—T' k
\ m^ J
«
where, Z and m^ are a charge and mass of charged dust particles, and ion temperature Ti is assumed to be 0.4 eV. The measured dispersion relations agree with eq. (1) with a specific charge Z/rrifi — 1.5 x 10"^ C-kg"^ for t = 5 minute and 1.0 xlO~^ C-kg~^ for t = 60 minute, as indicated in Fig. 4. Therefore, the specific charge of particles increased and the dispersion relation changes temporally.
CONCLUSION The Cu submicron dust particles were trapped for a long period without the electric field for suspension. The balance between electrostatic force due to potential structures and gravity force would be the best candidate for the suspension of the particles. The results of saturation current, scattering light intensity and dispersion relation of DAW suggest the long-term time evolution of submicron dust particles.
REFERENCES 1. Y. Hayashi and K. Tachibana, Jpn. J. Appl. Phys. 33, L804 (1994). 2. J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1996). 3. K. Tachibana, Y. Hayashi, T. Okuno and T. Tatsuta, Plasma Source Sci. Technol. 3, 314 (1994). 4. Jin-yuan, Ma, Phys. Plasmas 4, 2798 (1997). 5. N. D'Angelo, J. Phys. D Appl. Phys. 28 1009 (1995).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
489
RF potential formation in a magnetised plasma containing negatively charged particles P. Cicman , A. Kaneda, N. Hayashi and H. Fujita Faculty of science and Engineering, Saga University Honjo-machi 7, Saga, 840-8502, Japan
Abstract: RF potential formation has been investigated in magnetically confined electropositive and electronegative diffused plasmas at pressure Imtorr. Oscillation amplitude change and big potential jump in radial direction have been observed using uniform magnetic field configuration, while a uniform type potential structure was detected under divergent field for both electropositive and electronegative plasmas.
1. Introduction Potential formation is one of the most important phenomena to understand plasma physics and also processing plasma. The importance of studying negative ion containing plasmas appears in plasma research because negative ion plasmas are used widely in industry [1][2]. Diffuse plasmas confined by magneticfieldare often used as processing plasmas such as for plasma etching and deposition. Most processing plasmas are produced by capacitively and inductively coupled rf discharges but experimental studies of the negative ion plasmas confined by magneticfieldhas been developed with relation to negative ion sources without theoretical understanding of the magnetically confined negative ion plasma [3] [4]. 2. Experimental set-up Experiments were done in a stainless chamber with 16 cm in diameter and 116 cm in length as shown in Fig. 1. Plasma was produced by asymmetrical RF discharge in the region near the powered electrode and passed through the mesh (diameter 7 cm) in the diffusion region. Here, the chamber was grounded and themesh was biased at-25V. In the diffusion region the plasma was confined by the magneticfield,generated by the coils placed outside of chamber. Experiments were carried out using pure argon and argon-SFe mixture. The ratio of SF6 to Ar+SF6 was chosen at 80%. Working pressure was 1 mtorr. Using the uniform and divergent magnetic field configuration in the diffusion region, rf potential oscillation profiles in both, axial and radial directions have been obtained using an inflection point method of the emissive probe which allows the reduction of the probe heating current [5]. Using this method we detected VH and VL plasma potentials at the crest and through phases in the rf oscillating field, respectively. Radial profiles of rf potential oscillations were measured at distance z =5 and 37 cmfi-omthe mesh (z=0) where the magnetic field geometry should be marked and appear, respectively. Time averaged plasma potential Vs, electron density ne, electron temperature Te were roughly estimated using movable cylindrical langmuir probe with diameter 0.1 mm and length 1 mm. Typical argon plasma parameters in the source region were Vs = 20V, Te = 6 eV, ne = 6x10^ cm"^, while in the diffusion region Vs = 5 V, Te = 3V and ne = 3.5x10^ cm'^.
490
P. Cicman et al /RF potential formation in a magnetised plasma Id Ml
mm
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Fig. 1. Experimental apparatus (upper part) and magnetic field strength for uniform and divergent case (lower part) In order to evaluate negative ion behaviours, the ratio R of negative charge saturation current I - to positive one I + collected by the probe was measured. Using mesh, the electron temperature in the diffusion region was well reduced (6eV -^ 3eV) [6] [7] enhancing the negative ion production by dissociative electron attachment to SFe. 3. Results and discussion Figure 2 shows axial potential profiles for argon and argon-SF6 plasmas for both, uniform and divergent magneticfieldconfiguration. Uniform magneticfieldconfiguration provides an uniform axial profile of the time averaged potential (VH + VL)/2 and a constant rf amplitude VH-VL. This oscillation amplitude is bigger in a SF6 containing plasma. Increase of the rf amplitude in electronegative plasma is due to the reduction of electrons consumed by dissociative attachment to SFe molecules. It is known, that electrons can prevent any potential oscillation in the body of plasma due to shielding effect, but while the electron density decreases in the electronegative plasma, the oscillating amplitude should increase. Fromfigure2 it can be seen that value of VH potential for Ar plasma and Ar+SFe plasma is roughly the same. This however is not so for all ratios of SF6 / Ar+SF6, as already observed and mentioned in [8] where the VH value decreases to its minimum value at about 40% SF6 concentration and for higher SF6 concentration rising up to roughly the value that at 0% SF6 concentration. In addition, the potentials in a negative ion plasma are more negative as those in an electropositive plasma reaching negative absolute values. Usually without magnetic field, the plasma potential is determined by the wall potential. However in the present experiment, due to plasma confinement by magneticfield,effective walls are no more chamber wall but mesh and end plate. Thus changing the mesh and end plate bias, it is possible to control the value of plasma potential. Figure 3 shows the VH, VL profiles as afimctionof mesh bias for z = 5 cm. Electron temperature under uniform magnetic field is roughly constant in axial direction, having values of 3 eV for argon and 4.5 ev for argon SF6 plasmas. The reason is that low energy electrons were consumed to the negative ion production and only the e;ectron with higher energy remain in plasma, because at this energy the negative ion formation efficiency is very low.
p. Cicman et al. / RF potential formation in a magnetised plasma •
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Fig. 4. Radial potential profiles under uniform (on the right) and divergent (on the left) magneticfieldconfiguration for Argon and Argon-SF6 plasmas. Dotted line correspond to the mesh radius. For divergent magneticfield,axial profiles show the decrease of plasma potential as going far from mesh. While the amplitude in argon plasma does not change for the whole distance, in Ar+SF6 plasma oscillation amplitude decreases. Current ratio R for Ar+SF6 plasma decreases with increasing distance z, leading to conclusion, that negative ion density increases in axial direction. Decrease in rf potential oscillation amplitude with increasing negative ion density has been reported in [8] and might be caused due to a change in the negatively charged particles. Fig. 4 shows radial potential profiles for uniform and divergent magneticfieldcase at two different distances z = 5 cm (a, c) and z = 37 cm (b, d). It is seen that the radial potential profile
492
P. Cicman et al / RF potential formation in a magnetised plasma
forms a well-type shape except the profile at z = 37 cm (d in Fig. 4) in the divergent field case. For ArH-SF6 plasma the rf potential oscillation amplitude VH - VL is bigger than for Ar plasma. The well-type profile would be formed by a fact that non magnetized positive ions which diffuse across the magnetic field lines are lost to the chamber wall, resulting in the ambipolar diffusion of electrons. Here, the potential gap was created around the mesh edge (r = 3.5 cm). As a result outside the plasma column (beyond r = 3.5 cm) a plasma exist with lower plasma density than in central part. In fact, this potential gap was measured as a function of the magnetic field strength to show, that the gap was proportional to the field strength and was found to increase linearly with increasing field strength. In case of the argon-SFe plasma, the potential difference between central and outer part bigger than in argon case. Current ratio measurement shows, that negative ion density increases as going to chamber wall. One possible explanation is as follows. Although the negative and positive ions can diflSase across the magnetic field some electrons are still needed to preserve charge neutrality in outer part. However in central part we have now only electrons with higher electron energy and bigger potential jump is needed to draw these electrons to outer region. For the divergent magnetic field Fig. 4 d) the well type radial oscillation no longer exists and we can observe almost uniform-type potential oscillation. This is caused by the fact that plasma is expanded as a magnetic field is weaker. Current ratio for both plasmas is radially constant corresponding to plasma expansion and existence of radially homogenous plasma with lower density. 4. Conclusions RF potential profiles in electronegative and electropositive plasmas have been investigated under different magnetic field configurations. Radial potential formation with a potential jump has been observed for electronegative and electropositive plasmas, providing the higher density plasma in the central part and low density one for outer part for uniform magnetic field configuration. Potential jump increases with adding the electronegative gas. Decreasing the magnetic field strength potential jump disappears and radially uniform plasma can be detected. However negative shift time averaged plasma potential is at present unclear and will be subject of further investigation. References [1] K. Ayoyagi, I. Ishikawa, H. Kusunoki and Y. Saito and S. Suganomata, Jap. J Appl. Phys 36 (1997)1268 [2] N.L. Basset and D.J. Economou, J. Appl. Phys, 75 (1994) 1931 [3] O. Fukumasa, J. Appl. Phys, 71 (1992) 3193 [4] K.N. Leung, K.W. Ehlers and M. Bacal, Rev. Sci. Instrum. 54 (1983) 56 [5] E.Y.Wang, N. Hershkowitz, T. Intrator and C. Forest: Rev. Sci. Instrum 57 (10) 2425 [6] K. Kato, S. lizuka and N. Sato, Appl. Phys. Lett. 65 (1994) 816 [7] M. Nasser and H. Fujita, Jpn. J. Appl. Phys, in press [8] M. Nasser, Y. Ohtsu, G. Tochitani and H. Fujita, Jpn. J. Appl. Phys, 36 (1997) 4722
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
493
Formation of Particle Conglomerates in a Methane Discharge W.W. Stoffels, E. Stoffels, G. Ceccone*, F. Rossi*, Department of Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]. tue. nl '^Institute of Advanced Materials, Joint Research Centre, Via E. Fermi, 21020 Ispra (VA), Italy.
Abstract. Particle nucleation and growth in a low-pressure radio-frequency discharge in CH4 are studied. Nanometer size particle precursors are formed during irradiation of the plasma by a high-power laser. Laser-produced particles acquire negative charge and remain trapped in the plasma, where particle growth and coalescence takes place. As a result of coalescence, long particle strings are formed. A large structure of few centimetres in diameter, consisting of connected particle strings is observed. This negatively charged particle network is floating in the plasma and eventually it deposits on the electrode surface.
INTRODUCTION Formation of macroscopic grains under low-pressure conditions attracts much attention in plasma physics, astrophysics and surface processing technology. Plasmaproduced particles can be either dangerous pollutants of the surface in plasma processing or useful components in catalysis, ceramic industry and coating technology. On the fundamental side, particle charging and interactions in ionised gases have been extensively studied (1). In many laboratory discharges levitating particle suspensions have been observed. Coulomb interactions between negatively charged particles and confinement in the positive plasma glow lead to formation of ordered structures, called Coulomb crystals (2). Gas phase particle nucleation as well as dust production by sputtering and flaking of the surface have been described (3). A separate category is laser-assisted particle formation in the gas phase. In this paper we describe formation, grovv1;h and coalescence of particles in a methane plasma, used for diamond deposition (4). We show that the initial step of nucleation of particle precursors is a laser-assisted process, and the further development of particles occurs in the plasma.
494
WW Stoffels et al. /Formation of particle conglomerates in a methane discharge
EXPERIMENT Particle formation is studied a low-pressure capacitively coupled 13.56 MHz plasma, in an open configuration. The radio-frequency electrode is 20 cm in diameter, the reactor walls act as a grounded electrode. The output of the generator is 1 kV, 0.25 A. A standard methane flow of 100 seem is introduced at the top of the reactor. Particle suspension above the powered electrode is visualised by light scattering using a He-Ne laser or tungsten band lamp. Particle formation is initialised by a Nd:YAG laser with 5-6 ns pulse duration and wavelength 355 nm; the power density during the pulse is 2-10 W/m at the maximum laser output. Apart from initialising particle synthesis, the NdiYAG laser is simultaneously used as a diagnostic tool to detect nanometer size particles by means of laser-induced particle explosive evaporation (LIPEE) (5). After plasma termination, particles are deposited on the surface. They are analysed by a SEM microscope with EDX facility for elemental analysis. B: particle cloud with Coulomb lattice
A:Particle nucleation Nd:YAG
glow
• • •
glow sheath
sheath • rf electrode
- rf electrode
C: Formation of strings D: parachutes
• •
•••••••
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glow
llillilllllllli
M Vy V Vy Vi I
sheath - - - ^ - — - " - — - - — ^ ^ ^ - ^ ^ — — rf electrode
E: floating network glow sheath • rf electrode
sheath • rf electrode
F:growth on electrode
•V
glow
V •
^Ar^t^
FIGURE 1. Six phases in particle formation and coalescence (see text).
sheath • rf electrode
WW Stoffels et al /Formation of particle conglomerates in a methane discharge
495
RESULTS AND DISCUSSION In Figure 1 six phases in particle evolution are schematically depicted: Fig. 1(a). Synthesis of nanometer size particle precursors by laser interactions with the plasma species. In absence of laser irradiation, no particles are formed. The laser photon energy (3.5 eV) is resonant with the dissociation energy of a C-H band. During irradiation a high radical density, needed for particle nucleation, is created locally. We can observe the appearance of particles smaller than 50 nm by means of laser-induced continuum emission (LIPEE). Particle formation has a threshold of about 8-10^^ W/m^ of laser power, and the time scale of nucleation is a few seconds. Fig 1(b). Particle conglomeration and the formation of ordered structures (Coulomb crystals). After the nanoparticles are formed in a laser-assisted process, they grow in the plasma. They acquire negative charge and accumulate in the plasma-sheath region
-^—
10 |Lim
FIGURE 2. Left: network of particle strings after deposition on the surface. The plasma is on and the strings are still negatively charged. The diameter of the structure is about 5 cm. Individual strings are about 1 cm long. Some loose particles are floating in the discharge (right above the deposited structure). FIGURE 3. Right: a SEM micrograph of the fine structure of the particle stringsfi*omFigure 2. The bar corresponds to 10 jim.
above the powered electrode. When the particle density is sufficiently high, ordered structures are formed due to interactions between the particles. Fig. 1(c). Coalescence of grains and formation of strings. At a particle size of 100 nm, coalescence occurs due to instabilities within the Coulomb crystal. Particles form charged strings, which are aligned vertically above the plasma sheath by Coulomb interactions and by the ion drag force. Fig. 1(d). Condensation of the strings. When the strings become too long and heavy, their alignment becomes unstable and they collide with each other to form characteristic V-structures. Fig. 1(e). Condensation of V-structures into a single network. The large network, of several centimetres in diameter is suspended in the plasma. It can remain floating for several hours until it collapses due to plasma instabilities and its own gravity.
496
WW Stoffels et ah /Formation of particle conglomerates in a methane discharge
Fig 1(f). Deposition of the network on the electrode. After the collapse, individual particle strings of about 1 cm stretch through the plasma sheath, perpendicularly to the surface. A photograph of the deposited structure is shown in Figure 2. Deposited particles have been analysed ex situ. In Figure 3 a typical SEM micrograph of particle strings is shown. The particles are transparent in visible and IR, they consist of fairly pure carbon.
CONCLUSIONS Particle formation in a low-pressure methane discharge is initiated by a highpower pulsed laser, most likely due to photo-dissociation of the parent gas. Particle precursors remain trapped in the discharge. When the particle density is sufficiently high, Coulomb crystals and subsequently string shaped particle conglomerates are formed. At a later stage they join into a single network. This structure, as large as several centimetres in diameter, remains floating in the plasma until it collapses due to gravitation and plasma instabilities.
ACKNOWLEDGMENTS This work is supported by the European Commission under Brite- Euram contract BRPR CT97 0438 (HALU), by the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) and by the Dutch Technology Foundation (STW). The research of Dr. W. W. Stoffels has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences (KNAW).
REFERENCES 1. D.A. Mendis, C.W. Chow, P.K. Shukla, Proc. 6*^ Workshop The Physics of Dusty Plasmas, La Jolla, Califomia, 1995, publ. World Scientific, Singapore, 1996, ISBN 981-02-2644-6. 2. Y. Hayashi, K. Tachibana, Jpn. J. Appl Phys. 33, L804 (1994). 3. Proc. Dusty Plasmas '95 Workshop on Generation, Transport and Removal of Particles in Plasmas, Wickenburg, Arizona, 1995, publ. in J. Vac. Sci. Technol A14(2), 1996. 4. W.W. Stoffels, E. Stoffels, G. Ceccone, F. Rossi, submitted to J. Vac. Sci. Technol. (1999). 5. L. Boufendi, J. Hermann, A. Bouchoule, B. Dubreuil, E. Stoffels, W.W. Stoffels, M.L. de Giorgi, J. Appl. Phys. 76, 148 (1994).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
497
Change of the Potential Relaxation Instability in a plasma containing heavy C^Q ions Daniela Strele, Carsten Winkler and Roman Schrittwieser Institute for Ion Physics, University of Innsbruck, A-'6020 Innsbruck, Austria 1. mXRODUCTION The fullerene Ceo has a very large cross section for electron attachment over a wide range of electron energies, from thermal energies up to 10 eV [1]. This feature can be utilized to produce a plasma containing C^Q ions. In a Q-machine the electron temperature is typically 0.2 eV. By introducing Ceo molecules into the plasma column of a Q-machine single ionized negative ions are formed due to the reaction C^Q + ^ -> C^Q. It is possible to replace up to 90% of the electrons by C^Q ions [2]. The charge to mass ratio of a single ionized C^Q ion is -^//w = -1.3-10^ C / k g . Since the charge to mass ratio of a typical dust particle and an electron is -0.15C/kg and -1.76-10^^ C / k g , respectively, a plasma containing C^Q ions offers the chance to investigate the transition between a conventional electron plasma and a dust plasma. We present the influence of the C^o ions on the Potential Relaxation Instability (PRI) [3]. This instability is a large amplitude relaxation type oscillation of the plasma potential as well as of the plasma density. PRI-like oscillations can be observed in any low density current carrying bounded plasma system. In a Q-machine the PRI can be excited by biasing the cold end plate (CP) with a positive voltage. Since the driving mechanism of the PRI is the loss of the negative charge carriers by the positive voltage of the CP and the inertia of the positive charge carriers, the PRI will be strongly influenced by the C^Q ions, since their mobility is much lower than the mobility of the electrons. 2. EXPERIMENTAL SETUP The experimental investigations have been carried out at the single ended Q-machine at the University of Innsbruck. A schematic of the experimental setup is shown in Fig. 1. A potassium plasma is produced by surface ionization of K-atoms on the 6 cm diameter hot tungsten plate (HP). The plasma is terminated by the electrically floating cold endplate (CP). For most of the experiments presented here the distance between the HP and the CP, representing the system length, was 33 cm, but it could be varied. The radial confinement of the plasma is achieved by an axial magnetic field with a field strength in between 0.1 - 0.22 T. Usual plasma densities are about 10 cm . C6o
498
D. Strele et al /Potential relaxation instability in a plasma containing heavy C^Q ions
powder is evaporated in a source mounted at a distance of 17 cm from the HP. In order to minimize the disturbance of the plasma the source is kept on floating potential. At temperatures of about 450^C €50 evaporates and heavy negative ions are formed. To excite the PRI, the CP was biased with a positive voltage in the range between 3 - 10 V. ]
B-field
33 cm CP 6 cm
Probe
K-Oven
C«n Source
k.
5
Fig. I: Experimental setup.
To determine the plasma parameters an indirectly beatable Langmuir probe was used [4], which could be moved axially as well as radially. Since the mass number of the C50 ions is very high (720 amu), their contribution to the total negative current is negligible and from the decrease of the electron saturation current the negative ion density can be estimated. 3. RESULTS By biasing the CP with +3 V the PRI is excited with a frequency of 5.5 kHz. Under these conditions €50 molecules are added to the plasma by heating the €50 source. In Fig. 2 the frequency spectrum of the AC component of the CP current is shown for two different cases, (i) in the potassium plasma without C^Q ions and (ii) in the plasma where 60% of the electrons are replaced by C50 ions. In the potassium plasma without C^Q ions the PRI together with its 2**"* and 3'"^ harmonic is excited. With an increasing density of C^o ions, equivalent to a decreasing electron density, the PRI is quenched and simultaneously an oscillation at a frequency of about 12 kHz rises. Its amplitude increases with an increasing density of C^Q ions. Since this additional oscillation is clearly related to the C^Q ions it will henceforth be called C60 oscillation. Although the total number of negative charge carriers does not change in the plasma containing C^Q ions, the effective loss of negative charge carriers due to the CP bias is reduced, since the mobility of the C^Q ions is lower than that of the electrons by orders of magnitude. Thus, the quenching of the PRI becomes understandable.
D. Strele et al /Potential relaxation instability in a plasma containing heavy C^Q ions
0.3
•J "-2
Ceo Oszillation
. PRI
. 1 1 11
<
PRI
j j
n c" = 60%
1 2nd y Harmonic
(D Z3
1
n p- = 0%
00 '" 0
'
1 3rd
1 Harmonic
»
i
1 Harmonic
-1
10
20
30
0
10
20
J
30
Frequency [kHz] Fig. 2: Frequency
spectra in the potassium plasma and in the Ceo plasma.
\
For a further characterization of the €50 oscillation the system length was varied. Similariy as in Fig. 2 the 1st harmonic of the PRI, the Ceo oscillation and the 3rd harmonic of the PRI could be seen in the spectrum of the CP. It is well known that the frequency of the PRI is inversely proportional to the system length. Fig. 3 shows the inverse frequency of the various oscillations as a function of the system length. Multiplying the freguency of the 1st harmonic of the PRI with 2 and with 3 yields the frequency of the 2" and the 3""^ harmonic of the PRI, respectively. Fig. 3 shows a good agreement in the case of the 3'*^ harmonic, however in the case of the 2""^ harmonic for greater system length, a clear difference is observed between the Ceo oscillation and the PRI ground frequency times two. Therefore, the Ceo oscillation is not an amplification of the 2""^ harmonic of the PRI. But also this oscillation shows a linear dependence on the inverse system length. 0.14
Ceo Oscillation
N
PRI (Ground Frequency x 2)
X
0.10 cr
P R I 3rd Harmonic PRI (Ground Frequency x 3)
0.06 30
34
38 42 S y s t e m length [cm]
46
Fig. 3: Inverse frequency of the various oscillations versus the system length.
499
500
D. Strele et al /Potential relaxation instability in a plasma containing heavy C^Q ions
A variation of the magnetic field strength shows only a weak influence on the fi-equency of the Ceo oscillation. The frequency decreases for higher magnetic field strength. By increasing the magnetic field strengthfi*om0.1 T to 0.22 T the fi-equency decreases of about 1 kHz. The same behavior is observed for the PRI, and explained by in increasing effective system length due to sheath effects in front of the CP [5]. More interesting is the behavior of the amplitude of both the PRI and the Ceo oscillation. Since the gyroradius of the C^Q ions is about 4 mm for B = 0.22T and 8 mm for B = 0.1 T a strong radial loss for weak magnetic fields and therefore a quenching of the Ceo oscillation is expected. This is true for large system lengths (L = 39 cm). But for smaller system lengths (L = 33 cm) the behavior is inverse. For low magnetic fields the Ceo oscillation is amplified and the PRI is strongly quenched. Using the Langmuir probe it is possible to obtain spatially resolved information of the amplitude of the oscillations. Since the PRI is a current driven oscillation it is present across the entire plasma column, reaching a maximum in the center where the electron density has its maximum as well. In contrast, spatially resolved measurements indicate that the Ceo oscillation occurs at the edge of the plasma column. For decreasing magnetic field strength the maximum of the oscillation is shifled to larger radii, since the magnetic confinement of the C^Q ions decreases. 4. CONCLUSION By introducing C^Q ions into the plasma column of the Q-machine the PRI is strongly quenched and in addition an oscillation at a frequency about twice the PRI frequency occurs. The frequency of this Ceo oscillation shows a linear dependence of the inverse system length. However, the slope is steeper than that expected for the 2nd harmonic of the PRI. By increasing the magnetic field strength the frequency of the Ceo oscillation is slightly shifled to lower frequencies and the amplitude decreases. Spatially resolved measurements indicate a maximum of the oscillation at the plasma edge, that is shifled to larger radii as the magnetic field strength is decreasing. It will be subject of further investigations to develop a model together with a detailed explanation of the nature of the additional oscillation that is observed in the Ceo plasma. ACKNOWLEDGMENT: This work has been supported by the Fonds zur Forderung der wissenschafllichen Forschung (Austria) under grant No. P-12145, and has been part of the Association EURATOM-OAW under contract No. ERB 5004 CT 96 0020. REFERENCES [1] Smith D., Spanel P., Mark T.D., Chem. Phys. Lett., 213 (1993) 202. [2] Strele D., Winkler C, Krumm P., Schrittwieser R., Plasma Sources Sci. TechnoL, 5 (1996) 162. [3] Schrittwieser R., Int. J. Mass. Spectrom. Ion Processes, 219 (1993) 205; Phys. Lett A, 95 (1983) 162. [4] Strele D., Koepke M., Schrittwieser R., Winkler P., Rev. Sci. Instrum., 68 (1997) 3751. [5] Popa G., Schrittwieser R., Plasma Phys. Control. Fusion, 38 (1996) 2155.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
501
Experimental Verification of Dust Particle's Transport by Ambipolar ExB Drift Yoko Maemura, Mikio Ohtsu, Torn Yamaguchi and Hiroshi Fujiyama Faculty of Engineering^ Nagasaki University, UH Bunkyo, Nagasaki 852-8521, JAPAN
Absti'act. In order to verify the transport of dust particles in the direction opposite to E X B drift of plasmas, Cu dust particles were injected into dc argon plasmas and the drift velocities of dust particles were estimated from the results of time resolved laser light scattering intensities of dust particles. The drift velocity of dust particles depended on crossed magnetic field strength and the driving force acting on dust particles toward in the ~ E x B direction was discussed by taking into account ambipolar E x B drift of dusty plasmas.
INTRODUCTION Recently, the growth mechanisms and behavior of dust particles have been investigated in processing plasmas used in thin fdm deposition and etching [1,2]. In addition to the importance of particle contaminations in industrial applications, the problems are great interesting by their physical properties such as the relationship between the presence of dust particles and the discharge structures and plasma parameters [3,4]. In our previous experimental researches, it was observed the dust particles generated in silane discharges were transported in the direction opposite to E X B drift of plasmas [5]. In the present paper, in order to study the transport mechanisms of dust particles in more simplified system without the mention of the growth processes seen in silane plasmas, we tried to inject the small Cu particles into a parallel-plates dc argon discharges and the drift velocities of dust particles in the — E x B direction were measured experimentally. Furthermore, the space charge electric field enhanced by the E x B drift of electrons was calculated as a driving force acting on the dust particles and we finally presented the comparison of the dust particles' drift velocities between the results from the calculations and experiments.
502
Y. Maemura et al./Verification of dust particle's transport by ambipolar ExB drift
EXPERIMENTAL APPARATUS AND METHODS A schematic diagram of the experimental apparatus is shown in Fig.l. In cyHndrical vacuum chamber, two parallel electrodes of 90mm in diameter were placed at a separation of 30nmi. DC power supply was used for a generation of glow discharges and the typical experimental condition is as follows: Ar gas pressure, gas flow rate a.nd discharge voltage were P=0.12 Tbrr, F.R=10sccm a.nd V=500V, respectively. A homogeneous magnetic field of 0-50 Gauss in the axial direction of the cylindrical chamber was generated by two solenoid coils mounted outside the vacuum chamber. Cu particles (dust particles) were injected from out of the discharge space at a distance of 6cm from electrode edges by using the vibration of the speaker to give kinetic energy to dust particles, as shown in Fig. 1(a). The size and mass of dust particles is 50nm in diameter and 5.8 x 10~^^kg, respectively. In order to measure the drift velocity of dust particles, the spatio-temporal evolution of dust particles in the discharge space were observed by laser light scattering (LLS) method using an He-Ne laser beam (75mW at 632.8nm). The system for laser light scattering measurements is shown in Fig. 1(b). Cyiindricai Lens
lOOcm
pf] • ^
He-Ne Laser (632.8nm)
(a)
(b)
F I G U R E 1- Experimental set-up: (a) side view of chamber and (b) system for laser light scattering measurements.
RESULTS AND DISCUSSION Figure 2 shows the image data of injected Cu dust particles. Fig.2(a) and (b) show the cases that non magnetic field and magnetic field of 40Gauss are applied
Y. Maemura et al / Verification of dust particle s transport by ambipolar ExB drift:
503
after powder injection, respectively. The dust particles are balanced at the edge of the lower cathode sheath to get a upward electrostatic force against the force of gravity. The number densities of dust particles residing between electrodes changed from lO^^cm"^ to lO^^cm"^ for 5sec after powder injection. When the magnetic field applied, the density of dust particles rather decreases by 1 order of magnitude (not shown here). From Fig.2(b), dust particles drifting towards —E x B direction were also confirmed in this more simplified system than silane plasmas. Figure 3 shows the time resolved 1-D distributions of LLS intensity on the discharge radial axis (y-direction) after the powder injection. The LLS intensity in Fig.3 is one dimensionally integrated value on the discharge column axis (xdirection) in Fig.2(b). The magnetic field was applied at the same time of powder injection and the applied magnetic field strength was set at 40 Gauss. When the magnetic field is not applied, the distribution of LLS intensity is axially symmetric as seen in Fig.2(a). On the other hand, it is able to discriminate that the dust particles are transported in the direction of —E X B with a applied crossed magnetic field and furthermore quickly removed with an increase in magnetic field strength. The drift velocity of dust particles was obtained from the results of time resolved 1-D profiles of LLS intensity.
0 Gauss
40 Gauss
xB Direc
FIGURE 2. Image data of injected Cu dust parti- FIGURE 3. Time resolved distributions cles. (a) non magnetic field and (b) magnetic field of of dust particles on the discharge radial 40Gauss are applied, respectively. axis (y-direction). The magnetic field of 40 Gauss is externally applied.
These transport behaviors are explained in terms of an increase in electrostatic force acting on dust particles due to a space charge electric field enhanced by the E x B drift of magnetized electrons. Under the present experimental conditions, the plasmas is weakly magnetized and the electron Hall parameters are around 1. The space charge electric field therefore can be enhanced between the preceding electrons and the positive ions and negatively charged dust particles. Then, we calculate the space charge electric field from the ambipolar conditions of electrons, ions and dust particles. In this calculation, the charges of dust particles are estimated by the spherical capacitor model and the charge neutrahty of plasmas and it is assumed that the magnetized species is only electron. From the equations of motion, the drift velocities perpendicular to the magnetic field on y-direction can be derived.
504
Y. Maemura et al / Verification of dust particle s transport by ambipolar ExB drift
Figure 4 shows the drift velocities of dust particles as a function of the magnetic field strength in the direction of — E x B. The solid lines and the dots were obtained from the calculations and the experiments, respectively. The drift velocities increase with magnetic field strength. The calculation results agree quahtatively with the results from experiments. nd=108cnT3_
0)
"11^=5x10^_cmi. nc!=10'cnT3 Q
CO
Q
Calc.
10•7 h
J
'Expt. I
20 40 B [Gauss]
L
60
F I G U R E 4. Drift velocity of dust particles in the direction of — E x B as a function of magnetic field strength from experiments and calculations.
CONCLUSIONS When the magnetic field from 0 Gauss to 50 Gauss was applied, the injected Cu particles were removed from discharge space in the direction opposite to E x B drift of plasmas with a drift velocity from 2 mm/sec to 15 mm/sec. The drift velocities from the experiments agree qualitatively with the results obtained from the calculation model based on the ambipolar condition of electrons, ions and negatively charged dust particles, w^here the space charge electric field is enhanced by the magnetized electrons and it acts on the particle drift towards the —ExB direction in weakly magnetized plasmas.
REFERENCES 1. G. S. Selwyn, J. Singh and R. S. Benett, J. Vac. Sci. Technol. A7, 2758 (1989). 2. Y. Watanabc, M.Shiratani, Y.Kubo, I.Ogawa and S.Ogi, Appl. Phys. Lett. 53, 1263 (1988). 3. H.Hayashi and K.Tachibana, Jpn. J. Appl. Phys. 33, 804 (1994). 4. H. Ikezi, Phys. Fluids 29, 1765 (1986). 5. H.Fujiyama et.al, Jpn. J. Appl. Phys. 33, 4216 (1994).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
505
Nuclear Induced Dusty Plasma Structures Lidia V. Deputatova, Vladimir E. Fortov, Alexandr V. Khudyakov, Vladimir I. Molotkov, Anatoly P. Nefedov, Vladimir A. Rykov, Vladimir I. Vladimirov. High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, MOSCQW, Russia Abstract. We have performed for the first time experiments in the nuclear-induced plasma with micron-sized dust macroparticles injected into it. The plasma was formed in the atmospheric air by alpha-particles and fission fragments of ^^^Cf as well as by beta-active particles. In both cases there were observed ordered liquid-type structures of dust particles levitating in the interelectrode space.
Introduction The nuclear induced plasma is formed by products of nuclear reactions travelling through a gas and producing in their tracks electrons, ions as w^ell as exited atoms and molecules. The injection of micron-sized dust particles into a such plasma may change its properties and lead to a number of new^ effects. Li most cases an energy of nuclear particles is quite enough to shoot through a macroparticle with a radius of several microns. As a result the macroparticle will acquire a positive charge due to secondary electron emission. Besides, the microparticle itself may become radioactive and emit charged particles after nuclear conversions. The dust particles, placed into the nuclear induced plasma, experience flows of electrons and ions. Since there is a great difference in masses and temperatures of electrons and ions the dust particle begins to be charged negatively acquiring an equilibrium charge. Therefore, a dust particle charge may be obtained as a result of number of processes during which not only a charge value may change but its sign as well.
Experimental Results The experiments were performed with two kinds of the nuclear induced dusty plasma. In one case for plasma producing there was used a phenomenon of the electron betadecay. For this solid particles were activated in a nuclear reactor chaimel. After activating dust particles became beta-active. In the other case charges on finely divided solid particles appear after passing alpha-particles and fragments of fission of califomium-252 nuclei through the substance. In both cases there is an effect of secondary electron
506
L. V. Deputatova et al. /Nuclear induced dusty plasma structures
L B L E I . Maine!laracteristics o:f ionizers. Ionizer Mean initial Full path in solid energy, substance MeV (CeOj),
Full path in air (p=l atm), cm
Number of electrons for one original particle
|im
Beta-particle
0.138
58
5,6
~5
Mean fission fragment Alpha-particle
90
5,5
2,3
--250
6
20
4,7
-10
I
emission when beta-particles, alpha-particles and fission fragments eject electrons from a near surface layer of dust grains. The number of electrons ejected for each original nuclear particle varies from several units up to several tens and even hundreds (Table The installation for experiments with the radioactive Cf source (2) consisted of the ionizing chamber placed under a glass cover, a source of ionizing particles, means of pumping out, an electrometer and a dc voltage supply. As a source of alpha-particles and fission fragments there was used a flat layer of ^^ Cf with an intensity of 10^ fissions/s. The layer diameter was 7 mm. The solid angle in which ionizing particles escaped was near to 271. The visualization of dust particles was performed with the help of a diode laser sheet with a stretching thickness in the center of the chamber equal to 150 |Lim. The light scattered by dust grains was imaged by CCD - camera the output signal from which was registered by a video tape-recorder. We used spherical monodisperse melamine formaldehyde particles of 1.87 |Lim diameter. The experiments were performed in air at atmospheric pressure. Using the measured current between electrodes and known drift velocities of positive ions n^ and negative ions n' it was revealed that the ion density averaged on a volume n^~n'~10 cm' . Dust particles were injected into the chamber volume from the dispenser placed above a hole in the upper electrode or thro wed in from the bottom by an air jet. Experiments were performed both in glass tubes with diameters in the range from 10 up to 36 mm and with a height equal to a distance between electrodes varying from 10 up to 40 mm and without tubes with walls limiting the experimental volume. The experimental installation for researching a behavior of beta-active dust particles was similar to the installation described above. The main difference is that instead of the Cf layer utilized to create the nuclear-induced plasma we used Ce02 particles with a mean diameter of 1 |Lim. The particles were activated in the nuclear reactor. The activation occurred according to the reaction During experiments beta-particles were emitted according to the reaction a half-value period of which was equal to 32.5 days: '^'Ce^'^'Pr+e- + v', where v' is the electron antineutrino.
L V Deputatova et al /Nuclear induced dusty plasma structures
507
FIGURE 1. Video image of the vortex formation of monodisperse particles with a diameter of 1.87 |am. The area shown is 9.5x11 mm^.
The measured intensity of the beta-decay in the experimental chamber was about 10^ c'^ that corresponded to an output of fast electrons from one Ce02 particle 0.1 c'\ Experiments were performed in air at atmospheric pressure. Li the case of the radioactive ^^^Cf source when an electric field existed in the interelectrode space vortex movements of dust particles were observed in a plane of the laser sheet. We observed the similar vortex movement for single dust grains as well. Fig.l presents a general view of the vortex formation of particles with a diameter of 1.87 |Lim. The particles rotated clockwise in the interelectrode space at a voltage between electrodes equal to 200 V. The mean diameter of the vortex was about 5 mm. Note that a character of a movement of dust particles did not change under changes of the electric field direction. The angular velocity of particles' movement increased as the voltage increased. We performed also the experiments when an angle between an axis of the tube and a vertical was equal to 45°, 90° and 180°. In all cases the initial direction of a rotation of dust particles was from the radioactive source along the tube axis. When the voltage was switched off or when the radioactive source was closed dust grains fell on the lower electrode under an action of gravitation. Thus, the movements of dust particles described above could not be due to a gas movement caused by convection. It should be noted that in parallel with regions with a rotational movement there also were observed regions with dust particles levitating for several seconds. The processing of video images of a movement of particles revealed that a velocity of particles changed from 0.5 mm/s up to 10 mm/s with varying a voltage from 100 V up to 300 V. In the case of monodisperse spherical particles we evaluate a dust particle charge from a balance of a gravity force, an electrical force and a friction of neutral gas assuming a constancy of the particle charge. For particles of 1.87 |Lim diameter the charge obtained was about 10^ e. Fig. 2 presents a 2-dimensional distribution function n(r) (3) of a levitation region for particles of 1.87 |Lim diameter. Fig.2 testifies that there is a short-range order in the structure with a radius of the first correlation sphere equal to about 200 |Lim. In the case
508
L V Deputatova et al /Nuclear induced dusty plasma structures n(r),
jjM
1
' r, foM
FIGURE 2. Distribution function of monodisperse levitating particles of 1.87 jam diameter
of beta-active dust particles in the central part of the interelectrode space there were observed broad regions with dust particles levitating for several minutes. The estimate of the particle charge gave a value o f ( 0 . 3 - 0 . 5 ) 1 0 e. The distribution function for the levitation region of Ce02 beta-active particles at a voltage between electrodes equal to 20 V reveals a short-range order of the structure of polydisperse particles. The radius of the first correlation sphere is about 150 |Lim. When an electric field exceeds 30 V/cm there is no levitation of particles and the vortex movement of dust grains begins.
Conclusion For the first time there have been performed experiments in the nuclear induced dusty plasma formed in the atmospheric air by alpha particles and fission fragments of Cf as well as by beta active particles. When the electric field intensity was less than 20 V/cm regions of levitation of macroparticles forming ordered liquid-type dust structures have been observed in the interelectrode space. With increasing the electric field the vortex movement of macroparticles occurred. The initial direction of a rotation of dust particles was from the radioactive source.
REFERENCES 1. Rykov, V.A., and Dyachenko, P.P., ^tom/c Energy 83, Part 4, 266 (1997). 2. Fortov, V.E., Vladimirov, V.I., Deputatova, L.V., Molotkov, V.I., Nefedov, A.P., Rykov, V.A., Torchinskii, V.M., Khudayakov, A.V., Doklady Akad. Nauk (in Russion) 366, 1-4 (1999). 3. F.M.Kuni, Statistical Physics and Thermodynamics (Nauka, Moscow, 198) p.362
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
509
Experimental Investigation and Numerical Simulation of Inductively Coupled Dusty Plasma Vladimir S. Filinov, Anatoli P. Nefedov, Vladimir A. Sinel'shchikov, Oleg A. Sinkevich, Alexandre D. Usachev, Andrei V. Zobnin High Energy Density Research Center, Russian Academy of Sciences Izhorskaya, 13/19, Moscow, 127412, Russia Abstract. The results of an experimental research and numerical simulation of a behaviour of microparticles in plasma of inductive RF discharge are presented. The experiments were fulfilled in neon with monodisperse polymer particles. The discharge was initiated by radiofrequency power supply at 100 MHz with circular inductor. The calculations carried out by Monte-Carlo method have shown a possibility of formation of the chain structures from microparticles, which are similar to the structures observed in the experiment.
INTRODUCTION In most cases the experimental research of properties of dusty plasma and, first of all, the study of conditions of formation of ordered structures, so-called "plasma crystals", w^as conducted in RF capacitive discharge [1-3] or in strata of dc glow^ discharge [4]. hi [5] information on levitation of particles and formation of ordered dusty structures in plasma of inductive RF discharge was reported for the first time. Notice that the electrodeless discharge is rather attractive both for a number of the technological applications and for performance of fundamental research. For confining microparticles in plasma the formation of a potential trap is necessary. Li the electrode discharge, where always there are a fixed extemal electrical field and an extemal electrical current, just availability of an electrical field, which acts on charged dusty particle, allows to compensate influence of gravity. Li electrodeless RF discharge a constant extemal electrical current and constant extemal electrical field are absent. Li this case the essential role is played only by static electrical fields formed owing to violation of an electroneutrality, which, in tum, is caused by difference in speeds of diffusion and mobilities of electrons and ions.
EXPERIMENT In this work, as well as in [5], the plasma was generated in the vertically located glass tube with diameter of 3 cm and length of about 50 cm by power supply at frequency 100 MHz with the circular inductor. The level of input power didn't exceed several watts. The plasma filled all transversal cross-section of the tube. In a vertical
510 VS. Filinov et al /Investigation and numerical simulation of inductively coupled dusty plasma
direction the size of plasma formation depended on gas pressure and input power and could be changed from several centimeters up to size equal to the size of the tube in a vertical direction. The monodisperse polymer particles with diameter of 1.90 ± 0.05 microns were used. The system for particle observation was similar to that used in [4,5] and consisted of a diode laser and CCD-camera. Li the range of neon pressure from 30 up to 80 Pa the stable ordered structures of particles were observed. Their images are presented in Fig.l. The levitation of monodisperse particles was observed only in peripheral region of discharge, namely, near a lower boundary of plasma in transient region between a brightly luminous plasma formation and neutral gas. An increase of input power caused a growth of plasma volume in a vertical direction. In so doing the ordered structure moved together with the boundary of the discharge without any changes so long as it came closer to a bottom of the discharge tube. The structures shown in Fig.l were generated by the particle injection in plasma at fixed pressure. The particles were grouped together in chains extended in a vertical direction. The number of chains was reduced as the distance from a central part of discharge was increased. Note that on the boundary of luminous region of the discharge the sharp modification of an amount of chains in an ordered structure was observed. The size of a structure in a vertical direction was reduced as the neon pressure was increased (see Fig. 1). In this case the distance between particles decreased also and the structure expanded in a transverse direction (see Fig. 2). The quantity of particles held in such formations was limited. At attainment of a certain length, which depended on pressure, the structure ceased to catch new particles, i.e. became "saturated". The new particles either were not caught or displaced other particles from a structure and occupied their place.
FIGURE 1. Images of the vertical cross-section of the structures formed at different pressures of neon: a) p = 30 Pa; b) 50 Pa; c) 80 Pa. The area shown is 7.5x17 mm^.
vs. Filinov et al /Investigation and numerical simulation of inductively coupled dusty plasma 511
FIGURE 2. Images of a vertical cross-section of the structures formed at different pressures of neon: a) p = 30 Pa; b) 50 Pa; c) 80 Pa. The area shown is 3.4x5.2 mm^.
NUMERICAL SIMULATION From an analysis of the processes of a charge transfer as well as from experimental investigations the electrostatic trap was supposed to be created by the positively charged plasma formation having the shape of an elongated ellipsoid of revolution and negatively charged walls of the discharge tube. The interparticle potential took into account a spatial dependence of a particle charge, a plasma screening of dusty particles, a focussing of the drift ion current by negatively charged dusty particles and a possibility of destruction of ion clouds at small interparticle distances. In our model the negative particle charge was supposed to be proportional to the floating potential of the walls. So charge of the particle depended only on the vertical coordinate of the particle. According to the [6,7] the effective charge of ion clouds was assumed to be proportional to the particle charge. The distance between a center of the ion cloud and the dusty particle was taken equal to the Debye radius while the maximum value of the effective charge of ion cloud could be equal to the one third of the particle charge. The numerical simulation was performed using the standard Monte-Carlo method. The number of particles in Monte-Carlo cell varied within the limits of 300 -1000. The cell size was chosen to be 80 Debye radius. On Fig. 3 the vertical cross-section of the typical dusty particle structure is presented. The scale of one tip is equal to 8 Debye radius. The conditions of levitation of particles and formation of ordered structures superimposed rather rigid restrictions on a choice of parameters of the numerical model. The interparticle distance and the form of ordered structures, simulated and observed in the experiment, were in qualitatively agreement at the following values of parameters of the numerical model: a particle charge of the order lO^e, charge of a plasma elipsoid 3x1 O^e, focal length of a plasma elipsoid of the order 8 Debye radii.
512 VS. Filinov et al /Investigation and numerical simulation of inductively coupled dusty plasma
• • • • •
, • • •
1 '
«
•
>
• < • • • • • ••
•
• 4• • •• • • • • • • i
•
FIGURE 3. Vertical cross-section of a structure calculated by Monte-Carlo method. The scale of one tip is equal to 8 Debye radius.
CONCLUSION The experimental research and numerical simulation of formation of ordered structures in plasma of inductive RF discharge have been performed. The qualitative agreement of computational and experimental resuhs proves that the theoretical model suggested describes satisfactorily the dusty particle structures experimentally obtained in the inductively coupled plasma.
ACNOWLEDGMENTS The authors thank Yurii V.Gerasimov for preparation of the experimental setup.
REFERENCES 1. 2. 3. 4. 5.
J.h. Chu and Lin I., Phys. Rev. Lett. 72,4009-4012 (1994). Y.Thomas, G.E.MorfiU, V.Demmel et. al., Phys. Rev. Lett. 73, 652-655 (1994). Y.Hayashi and K. Tachibana, Jpn. J. Appl. Phys., Part 2 33, L476 (1994). Lipaev A.M., Molotkov V.I., Nefedov A.P. et al, JETF 85,1110-1118 (1997). Gerasimov Yu. V., Nefedov A.P., Sinel'shchikov V.A., and Fortov V.E., Pisma v Zhumal tehnicheskoy fiziki (in Russian), 24, No. 19, 62-68 (1998). 6. Belotserkovski CM., Zaharov I.E., Nefedov A.P.et al, JETF, 188,449, (1999). 7. Schweigert V.A., Schweigert I.V., Melzer A. et al, Phys. Rev. E, 54,4155 (1996).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V. All rights reserved
513
Dust Wake in a CoUisional Plasma D. Winske, W. S. Daughton, D. S. Lemons, M. S. Murillo and W. R. Shanahan Applied Theoretical and Computational Physics Division Los Alamos National Laboratory Los Alamos, NM 87545 USA
Abstract. We study the interaction between a stationary dust grain in a flowing, collisional plasma. An expression for the wake potential is derived and compared with results from particle-in-cell simulations that treat the flowing plasma ions as discrete particles which are collisionally coupled to the background neutral gas.
INTRODUCTION The flow^ of plasma around dust grains in many experiments is typically on the order of the ion sound velocity. This flow gives rise to a quasi-periodic wavetrain, or wakefield, downstream of the grain (1,2). The wake is thought to contribute to the vertical alignment and grain spacing in plasma crystals. More generally, the problem is very similar to that of a test particle moving through a plasma, which has been the subject of many studies (3,4). In most dusty plasma experiments, the plasma is weakly ionized. The ion flow through the region where grains are trapped is often moderately collisional. In this paper we include the effects of ion-neutral collisions on the grain-plasma interaction. As our principal interest is in the oscillatory character of the downstream wake, we employ a simplified analysis to derive its properties. We compare results of the theory with particle-in-cell simulations of the wake evolution.
THEORY AND SIMULATIONS To derive the basic oscillatory structure of the wake, we consider a steady-state, one-dimensional system with cold flowing ions, ion-neutral collisions, and Boltzmann electrons. Linearized fluid equations combined with Poisson's equation give the well known dispersion relation for ion acoustic waves:
514
D. Winske et al /Dust wake in a collisional plasma
1 + ^
^
=0
(1)
where co is the frequency, k is the wavenumber, Vo is the ion flow speed, Vi is the ionneutral collision frequency, X.De is the electron Debye length and (Oi is the ion plasma frequency. If wave propagation is allowed in two dimensions, k = (ki^+k||^)^'''^ and kVo -^ k|| Vo. For Mach number M, which is the ratio of the flow speed to the sound speed, less than one, phase standing (co = 0) waves downstream of the grain are possible. For M > 1, phase standing waves occur only if oblique propagation is allowed. Including collisions, the potential downstream of the grain (with surface charge a) has the form: (p
; — ^ ^ sm
x ( M ~ ' - l ) Ml
exp(-x(;./FJ
(2) We use simulations to verify the overall scahngs implied by (2). Generally, we confirm that the magnitude of the wake varies as the grain charge, and as shown below, the spatial damping varies as v\. We also find that the wake exists only for M < 1 in 1-D simulations, with wavelength scaling ~ 27iMA.De /(1-M^)^^^. In 2-D, for M > 1, the wavelength scales ~ 27rMA.De- These results, and comparison with analytic theory, will be presented elsewhere. Here, we describe the basic simulation methodology and a few representative results. The simulation system consists of a plasma region of length Lx and width Ly, in which is embedded a dust grain at a fixed position. The grain is a point particle with a fixed charge. Thus, effects associated with ions focussed behind a finite-size grain or charging are ignored. Plasma flows in the left x-boundary, and out the right xboundary. The plasma ions are treated as discrete particles, with charge Zi, mass mi, and temperature Ti=miVi^/2 and are injected to maintain a uniform inflow. The electrons are treated as a linearized Boltzmann fluid. The sound speed is defined as Cs 1 /9
= (Te/mi) and thus the inflow speed is given in terms of the Mach number M=Vo/Cs. The ions are treated using standard particle-in-cell (PIC) methods (5). The ion dynamics are represented by several hundred macroparticles per computational cell; the particles are collected to determine the charge density. Poisson's equation is then solved on this grid, and the local electric field that each ion experiences is also determined from linear interpolation of the grid-based field quantities. Collisions of the plasma ions and the background neutrals are treated using a Langevin approach, which essentially tries to keep the ion flow at a particular drift speed and also maintains the ion temperature at a fixed value. The calculations use parameters characteristic of rf discharges. We assume argon ions with number density ni = 10^ cm"^ and an ion temperature of 0.03 eV. The electron temperature Tg is fixed, with the ratio TJ1L\ = 128, giving a sound speed of Cs = 8 Vi. The electron Debye length is thus 460 |Lim, and the ion Debye length is 40 |Lim. For a 1 |Lim radius dust grain, the charge is roughly 8000 electronic charges. The 2-D simulations use 400 x 128 cells, corresponding to Lx = 50A,De,Ly = 16A.De.
D. Winske et al /Dust wake in a collisional plasma
515
Plotted in the top panel of Figure 1 is a contour plot of the normalized electrostatic potential, O = Q^I2T\, at cojt = 40, from a 2-D simulation in which the grain is placed at x = 25?LDe in a flow with M = 0.8 and Vj/coi = 0.03. The potential shows winged structures at the potential extrema behind the grain due to a combination of parallel and obliquely propagating, phase standing waves. In the middle panel, we plot the potential versus x, at fixed y =8A.De5 along the grain axis. We also plot, as dashed lines, the results of a 1-D calculation that uses both the same physical parameters as well as numerical parameters. The structure of the potentials and the wavelength of the oscillations of the wavetrain are similar in the two cases. The bottom panel superimposes the spatial profile of O at cojt = 40 for 1 -D runs at M = 0.8 with several values of the collision frequency. The wake spatial damping rate is (to within 10%) vi/Vo, consistent with Eq. (2).
FIGURE 1. [Top panel] Contour plot of O from a 2-D simulation. [Middle panel] O versus x at the grain position in y, compared with corresponding 1-D result. [Bottom panel] O versus x for runs with different V;.
516
D. Winske et al./Dust wake in a collisional plasma
SUMMARY The theory and simulations we have presented show that the principal features of the trailing wake are retained with the inclusion of ion-neutral collisions. The collisions add spatial damping to the wake. For the M < 1 flows considered in this paper, wakes are seen in both 1-D and 2-D simulations. In experiments, such wake effects are suggested by the presence of strings of dust grains (6) and the vertical alignment seen in plasma crystals (7,8). The wake potential structure indicates that the effect of a flowing plasma is to create an ion focus region some distance downstream of the grain (9). For M < 1, this focusing distance can be comparable to the observed particle spacing (1,2). The simulations here show that the distance is not modified by collisional effects, but the overall interaction length is limited to a few wavelengths due to spatial damping. Molecular dynamics simulations (10) that include the wake potential behind each dust grain give aligned strings of grains and crystals with reasonable particle spacings. The calculations also indicate that the wake effect is a more physically justifiable correction to the grain interaction potential than a dipole.
ACKNOWLEDGMENTS This work was supported by the Laboratory Directed Research and Development Program.
REFERENCES 1. Ishihara, O. and Vladimirov, S. V., Phys. Plasmas 4, 69-74 (1997). 2. Xie, B., He, K., and Huang, Z., Phys. Lett. A 253, 83-87 (1999). 3. Sanmartin, J. R., and Lam, S. H., Phys. Fluids 14, 62-71 (1971). 4. Chenevier, P., Dolique, J. M., and Peres, H., J. Plasma Phys. 10, 185-195 (1973). 5. Winske, D., Murillo, M. S., and Rosenberg, M., Phys. Rev. E 59, 2263-2272 (1999). 6. Murillo, M. S. and Snyder, H. R., This proceedings (1999). 7. Pieper, J. B., Goree, J., and Quinn, R. A.., Phys. Rev. E 54, 5636-5640 (1996). 8. Takahashi, K., Oishi, T., Shimomai, K., Hayashi, Y., and Nishino, S., Phys. Rev. £58,7805-7811(1998). 9. Schweigert, V. A.. Schweigert, I. V., Melzer, A., Homann, A., and Piel, A., Phys. i?ev. £54,4155-4166(1996). 10. Hammerberg, J. E., Hohan, B. L., Murillo, M. S., and Winske, D., This proceedings (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) 2000 Elsevier Science B.V.
517
Measurement of Electric Charge of Dust Particles in a Plasma M Itdi ^, T. Fujimori ^, Y. Komatsu ^, and Y. Nakamura t
Tokyo Gakugei University, Nukuikitamati, Kognanei, Tokyo 184-0015, JAPAN * Institute of Space and Astronautical Science, Sagamihara, Kanagawa 229-8510, JAPAN Abstract The electric charge of dust particles (grains) immersed in a plasma, whose diameters are lager than " /x m" were measured. It is detected by a charge sensitivity amplifier, and was analyzed with a spectroscopy system. The particles are found to be negatively charged and their electric charge is about 10 electrons. The measured values are close to a theoretical prediction.
INTRODUCTION In the laboratory and in space, it's well known that small dust particles are ordinarily negatively charged. The pioneering theoretical works of the currents that contribute to charging include those due to incident electrons and ions, secondary electrons induced by high-energy electron and ion impact and photoemission. Measurement of electric charge of dust particles in a plasma is one of important experiments for dusty plasmas. But it's difficult because dusts charge is influenced many factors, for example, the electric temperature, floating potential, particles radius and density of dust.
EXPERIMENTAL PROCEDURE The schematic diagram of the experimental setup is shown in Fig.l [1]. The device was made of Pyrex glass tubes of 15 cm in diameter and was composed of three parts, a dropping system of dust particles, plasma generation part, and measuring part of electric charge. An ultra sonic vibrator drops dust particles from the top and then they pass through the plasma [2]. The charged particles were detected with a charge sensitive pre-amp. Then charge sensitive pre-amp output was applied (Fig.2) to a low noise amp. A low pass filter was used for an improvement of the input signals to the pulse height analyzer The pulse height analyzer measures the height of pulses. We investigated two plasma conditions. The plasma was generated by discharge between tungsten filaments and the grounded cage of muhi-dipole magnets. The working gas was Argon and its pressure was 2.5 x IC^orr. The discharge voltage was lOOV. The plasma parameters were measured by a Langmuir probe. The single plane probe of 3mm in diameter was used. One plasma density n^ was lO^^lO^cm and the electron temperature Te is about 7 eV and the other was ne = 1 0 ^ ^ and Te is 0.5--1 eV. Under the magnetic cage of 50cm in length, a stainless steel disc is set up, and the dust particles pass through the center hole of this disc. The dust particles whose diameter are 5 /x m are shown in Fig. 3. The shape is sphere, and they are gold-coated polymer.
M. Itoh et al /Measurement of electric charge of dust particles in a plasma
518
ppectro-scopy System
Ultra Sonic Vibrator
Fig.2. Typical signals of dust particles.
Charge Sensitive Pre-Amp
Fig. 1. Experimental setup.
Fig.3. A picture of gold-coated polymer spheres whose diameter is 5 JJLWL.
EXPERIMENTAL RESULT AND DISCUSSIONS We calculate the electric charge of dust particles. When dust particles are isolated each other, there is no interaction between dust particles. Then electric charge Q of dust particle is estimated as follows (spherical capacitor model) [3]. Dust particles are considered to be spheres of radius r, so that their capacitance C is given by C = 4;r£,r. 0) When the surface potential of the particles is (f), Q is estimated by the relation Q=C=4^£or(^. (2) Since the particles arefloating,thisfloatingpotential is determined from the condition that the sum of the electron current le and the ion current li must be zero, which are given by /, = ATO-'en, ^TJ2mn,{\ -e^lT,), (3) I^=-4m-'enjTJ2mi, exp(e^/TJ,
(4)
519
M Itoh et alJ Measurement of electric charge of dust particles in a plasma
where me and mi are mass of electrons and ions, respectively m and Ti (ne and Te) are the density and temperature of ion (electron). The dust particles are assumed to be isolated each other, because we sufficiently reduced the event numbers of dust fall, and we also made the measurement repeatedly. Fig.4 and Fig.5 show experimental results of the electric charge for different discharge currents. Since the charge sensitive amplifier was very sensitive to the plasma noise, we recorded only signals above the average noise level. When the discharge current is less than 9mA(Fig4), peak values of Q/e are about 60,000 (Fig.6). The measured floating potential was about -24V(from the Langmuir probe, Te = about 7eV), so the estimated value is about 42,000. When the discharge current, however, was increased (11mA in this case), the peak value of electron number of Q/e was decreased to about 25,000. When the discharge current was increased farther (14mA in this case), the height of the signal was not measured since it was lower than the threshold (15,000). In such a higher discharge current, sometimes positively charged particles were observed. The experimental results are understood as follows. When the discharge current is increased, the density of primary electrons increases. Then secondary electrons from particles by the collision of primary electrons are increased so that the negative charge of the particles is reduced[4~6]. Improving the magnetic cage, we could obtain a low noise plasma condition and the low electron plasma temperature of 0.5-^1.OeV. When the discharge current is 10mA, Q/e is about 6,000--10,000. ( Te = 0.8eV, then estimated value Q/e is about 3,000.) Accordingly the electric charge of dust particles are influenced by the electron temperature. Experimental results are summarized in the following. 1. The distribution of electric charge of dust particles is Gaussian. As we see in Fig. 3, the dust radius is almost uniform. Therefore this distribution depends only on the plasma state. 2. The electric charge of dust particles are influenced by electric temperature (or, floating potential). But when the filament emission current is large, the charge is reduced due to the secondary emission. 1
1
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Fig.4 and 5. Charge distributions of dust particles for different discharge currents obtained by the pulse height analyzer. Data under 14,000(Q/e) are eliminated, because of plasma noise.
520
M. Itoh et al./Measurement of electric charge of dust particles in a plasma
60000 r
•
•
I
40000 o
20000
0
5 10 15 Discharge current(mA)
Fig.6. Measured electric charge of dust particles as a function of discharge current. When the discharge current is increased, the density of primary electrons increases. Then secondary electrons from particles by the collision of primary electrons are increased so that the negative charge of the particles is reduced.
REFERENCES [1] Y.Nakamura and T.lto, " Charging of Dust Grains in Laboratory Device", Advances in Dusty Plasmas, World Scientific, 71-76(1997). [2] D.P.Sheehan, Carillo, and W. Heidbrink, "Device for dispersal of micrometer and submicrometer-sized particles in vacuum" Rev.Sci.Instrum, 61,3871(1990). [3] T.Yokota, "Charging process in the particle plasmas" Journal of Plasma and Fusion Research (in Japanese), 73,1222(1997). [4] B.Walch, M.Horanyi, and S.Robertson, "Measurement of the Charging of Individual Dust Grains in a Plasma" IEEE TRANSACTIONS ON PLASMA SCIENCE, 22,97(1994). {5] B.Walch, M.Horanyi, and S.Robertson, "Charging of Dust Grains in a Plasma with Energetic Electrons" Phys. Rev Lett., 75, 838(1995). [6] T.Sato and K.Watanabe, "Simulation Study on the Growth of Grains in Dusty Plasmas" Journal of plasma and Fusion Research (in Japanese), 73,1257(1997).
E-mail: [email protected]
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
521
Transport of a polydisperse ensemble of dust particles in plasma. IV. Schweigert\ V.A. Schweigert^, ^Institute of Semiconductor Physics, Novosibirsk, Russia, ischweig@isp nsc.ru ^Institute Theoretical and Applied Mechanics, Novosibirsk, Russia The collective transport of different size dust particles in a gas discharge is studied for the various coupling parameters F The influence of Coulomb screened interaction of particles on their mobility in a small electric field is revealed with using the multi-liquid (MF) approach and the Langevin molecular dynamics (MD) simulations. The comparison of results of these approaches shows that the multiliquid model is valid for the description of particle transport at F<10 Using MD simulations we found that at some critical Y the transition from the liquid state to the solid state takes place. The influence of particle dispersity over size on solidification of the system is considered. Model system. We study the strongly coupled dust particle system, which is either a binary mixture or ensemble with uniform distribution over size. The particle move in a small electrical field in viscous gas We assume that particles interact through the isotropic Yukawa potential. The coupling parameter of polydisperse particle system is
-"~± where n^ is the density of ith group of particles with the radius Ri and the charge Zi , n-the total particle density, N the number of different groups of particles, and ai =(3/47i:ni)^ ^ the inter-particle distance. In our calculations we consider the mofion of silicon particles with radius R= 130^200 nm in argon at the pressure P=20 Pa. The particle potential (p is varied from 3eV to 5eV. The particles of different radii are supposed to have the same potential and the particle charge is proportional to the particle radius Zi ~ Ri. The electrical field E is appHed in the z-direction and action of electrical force on a particle is balanced by gas friction force. In the case of a weak coupling the particle velocity Vi ~ 1/Ri, and groups of different size particles move practically independently. In the case of strong coupling, particle velocity is determined with a particle radius and interaction with the other species. The Debye radius is supposed to be 17 \\m and the ion density equals 5.0-10 cm". The total particle density varies from 10" to 10' cm' and sets the coupling parameter F, which changes from 1 to 180. Multi-fluid approach In the MF model the Coulomb interaction between particles is replaced by momentum motion exchange. The system of particle motion equations can be written as
522
/. V Schweigert, V.A. Schweigert/Transport of a polydisperse ensemble of dust particles
ViMiVi+Zvij ^'""P nj(vrvj) = eZjE
(1)
where Mi is the particle mass of ith group, and rij is the particle density of jth group. The gas friction frequency is Vi =167iRi^pg (MgeTg/(27r)y^ VMi\ where pg, Mg, Tg are the gas density, the atom mass and the gas temperature, respectively. The momentum motion exchange between ith and jth groups of particles is set with the frequency Vy'T which is Vi/'^^ = 16My(eTg/(27iMii)f'Si/S, where My is the effective mass My = MiMj/(Mi+Mj), and Sij is the momentum transfer rate coefficient. We assume that particles have a Maxwellian distribution over velocities f(v) and the momentum transfer rate coefficient can be calculated as Sy = J aij(v)f(v)dv, where aij(v) is the momentum transfer cross section. In order to define aij(v) for two particles with different negative charges we have applied the classical orbit method. In Fig. 1 the calculated momentum transfer cross section 0{\ (closed circles) is shown as function of parameter L=ZiZ2e^/47i8oEp, where Ep is the particle energy. We have constructed the analytical form for the definition of the momentum transfer cross section aa=Ci?L^(ln(l+C2LA?L)f,
(1)
where Ci, C2 are the fitting parameters. In Fig.l we give also the momentum transfer cross section Gat calculated in [1] (open circles) for ion - negative particle interaction.
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I I I 1111
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Fig. 1. Momentum transfer cross section of two negatively charged particles. In our calculations L/A. =1-1000 and the analytical form (1) (Fig. 1, solid line) approximates well aij calculated by the method of classical orbits. Different groups
/. V Schweigert, V.A. Schweigert / Transport of a polydisperse ensemble of dust particles
523
of particles are assumed to have of equal densities. Solving the Eq.(l) with the momentum transfer cross section (2), we found the particle velocity of different species in the electrical field E=0.1V/ciii. The calculated mobility of particles with radii 180 nm and 130 nm is shown in Fig. 2 (dotted lines) as function off. Molecular dynamics simulations. Here the influence of Coulomb interaction on transport of particles in binary mixtures and in polidysperse ensembles is studied with using Langevin molecular dynamics simulations. We took the 3D box with N particles (from 250 to 432) and used periodic boundary conditions. The initial coordinates of particles are set randomly. First we allow the system to achieve the equilibrium state executing 10"^ ^10^ MD steps and then obtain the particle motion characteristics averaged over time at a fixed parameter F. Fig. 2 shows the particle mobility obtained from MD simulations (circles) of the binary mixture. It is seen from Fig. 2 that Coulomb
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Fig. 2 Mobility of particles in the binary mixture (180nm and 130nm) as function of coupling parameter. MD simulations results are denoted with solid (180nm) and open (130nm) circles. The MF results are given by dotted lines. Insert picture in more details shows solidification of the system at the critical parameter F. interaction affects the mobility of particles even for small F and the collective transport takes place. We found that the MF approach with the momentum transfer cross section (2) well describes the particle motion up to r<10. Within next interval 10 69 the system
524
/. V Schweigert, V.A. Schweigert/Transport of a polydisperse ensemble of dust particles
transits to the solid state. In calculations we take the equal densities of different species, therefore in the crystalline state at F > 69 the particle system exhibits the bcc structure. Let us consider the ensemble of particles with sizes 120
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Fig. 3 Fhe critical coupling parameter as function of polidispersity. uniformly distributed between Rmin and Rmax. We study the solidification of the system at different values of dispersity. Fhe critical parameter F of solidification is found as function of parameter A, which is A =(Rmax-Rmin)/(Rmax+Rmin) In calculated variants we fixed the mean particle radius R, which is R=150 nm and Rmax, Rmin are varied. For an example, R max=190 nm, Rmin^HO nm, A =0.27. Note, that parameter F remains practically the same for an electrical fields E=0 and E=0.1 V/cm. In all cases at F>F the solidification of the system takes place within narrow interval of the coupling parameter and particles move with some effective velocity. We found the rapid transition between-two regimes of particle transport. Fhe critical F as function of dispersity A is shown in Fig. 3 Fhe critical coupling parameter increases monotonically within a range of A =0.07 - 0.27 and steeply rises for A ^0.3. In conclusion we have studied the transport of strongly coupled polydispersive dust particle ensemble using the multi-fluid model and the molecular dynamics calculations. For the MF description we have calculated the momentum transfer cross section of two negatively charged particles of different radii. Fhe comparison of particle mobility obtained in the multi-fluid calculations and in MD simulations demonstrates that for F<10 the multi-fluid approach precisely describes the particle motion. It is shown that solidification of the system takes place at critical F which depends on dispersity of particle ensemble. [1]. M.D. Kilgore, J.E. Daugherty, R.K. Porteous, and D.B. Graves. J. Appl. Phys. 73,7195-7203 (1993).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
525
Influence of the lattice symmetry on melting of the bilayer Wigner crystal. LV Schweigert^ V.A. Schweigert^, and F. M. Peeters^ ^Institute of Semiconductor Physics, Novosibirsk, Russia, [email protected] ^Institute Theoretical and Applied Mechanics, Novosibirsk, Russia Universiteit Antwerpen (UIA), Antwerpen, Belgium The melting transition of the five different lattices of a bilayer crystal is studied using the Monte-Carlo (MC) technique. We found the surprising result that the square lattice has a substantial larger melting temperature as compared to the other lattice structures, which is a consequence of the specific topology of the temperature induced defects. A new melting criterion is formulated which we show to be universal for bilayers as well as for single layer crystals Model system. In the crystal phase the particles are arranged into two parallel layers with close packed symmetry The particles interact through an isotropic Coulomb (screened) repulsive potential V(r,.rj) =(q-/s„|rrrj|)e''""'^ where q is the particle charge, r^.rj the particle coordinates, l/k= ao/l the screening length. The type of lattice symmetry at temperature T=0 depends on the dimensionless parameter n =d/dn) where d the interlayer distance and ao =l/(7i:N/2)^^ ( N is the particle density), is the mean interparticle distance In [1] it was found that the bilayer Coulomb crystal exhibits five different types of lattices as function of the interlayer distance at T=0: n <0.006- hexagonal, 0.006< n <0.262-rectangular, 0.262< n <0.621-square, 0.62K n <0.732-rhombic, and n >0.732—hexagonal. Melting phase diagram and a new melting criterion. In order to identify the point of melting we calculated the potential energy of the system as function of temperature for all types of lattices. In the crystalline state the potential energy of the system first increases linearly and then exhibits a jump at some critical temperature. Fig. 1. This denotes the beginning of mehing and is related to the unbinding of dislocation pairs. It is turned out that the square lattice bilayer crystals have a substantial higher melting temperature, and consequently is more stable against thermal fluctuations than the other crystals. The maximal temperature of melting belongs to the square bilayer crystal with n =0.4 and its change of the energy is about a factor 2 larger than for a hexagonal lattice. Fig. 1. To characterize the order in the system we calculate also as function of T the mean square particle displacement, the first peak of the pair correlation function and the translational Gn and the bondangular order factors Gang [2]. The behaviour of these values point out the melting temperature and allow the construct the melting phase diagram of the bilayer crystal. Moreover, our numerical resuhs show that for all five types of lattices the bondangular order factor: 1) decreases linearly with increasing temperature (except very
526
/. V Schweigert et al. /Influence of lattice symmetry on melting ofbilayer Wigner crystal
close to the melting temperature), and 2) it drops to zero just after it reaches the value 0.45. We checked this implication for the bilayer crystal with screened and pure Coulomb interaction and for a single layer crystal with a Lennard-Jones V= 1/r^^-l/r^ and a repulsive V = 1/r^^ interaction potential. The dependence of factor Gang on the reduced temperature T/Tmei, where Tmei is the melting temperature is given in Fig. 2. It is clear seen that all types of inter-particle interaction behaviour of the factor Gang is similar. From the present numerical results for Gang we formulate a new criterion for melting which we believe is universal: melting occurs when the bond-angle correlation factor becomes F- 0.45.
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Fig. 1 The potential energy as function of temperature
Fig. 2 Bond -angular order factor as function of temperature
Our results for the mehing temperature are summarised in the phase diagram of Fig. 3. For n =0 and / =0 we obtained the critical F =132. As seen in Fig. 3 the hexagonal (I and V), rectangular (II) and rhombic (IV) lattices melt at almost the same temperatures. For the square bilayer crystal (phase III) has the largest Tmei
/. V Schweigert et al /Influence of lattice symmetry on melting of bilayer Wigner crystal
521
Given the above mentioned criterion for melting we calculated Tmei using the harmonic approximation. Therefore, we numerically diagonalized the Hessian matrix [3] to obtain the eigenvalues. Tmei is then derived by linear extrapolating Gang to the value 0.45 In this way we obtained analytically Tmei for different types of lattices which agrees with our MC calculations within 10%. Topology of defects To understand why the square lattice bilayer crystal has a considerable larger melting temperature, we investigated various temperature induced isomers of a single layer crystal and compared them with those of the square lattice bilayer crystal with n =0.4 which has the largest melting temperature. At given temperature we found that during the MC simulation the system transits from one metastable state to another. They differ by the appearance of isomers in the crystal structure which appear with different T„e,/ToXlO^
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E/keT,
Fig. 4 The bond-angular (solid squares) and the translational order factors (circles) of the different defects in (a) a single layer crystal (#i =0), and (b) in the square lattice bilayer system ( n =0.4).
probabilities We found these isomers by freezing instant particle configurations during our MC steps. The topology of the defects, their energy and the bond-angular and the translational order factors of these configurations are determined. Each point in Fig. 4(a,b) represents one configuration containing an isomer in a single layer and
528
/. V Schweigert et al /Influence of lattice symmetry on melting ofbilayer Wigner crystal
the square lattice bilayer crystals, respectively The qualitative behaviour of both crystals during melting is similar although the energy of the defects in both lattice structures is substantially different For the single layer (w =0, Fig. 4(a)), all isomers at Ti=0.00756T() (the location of temperature is given in Fig. 1) just before melting, and at T2=0.0076T() just after melting, were obtained. Note, that for the square lattice (« =0.4, Fig. 4(b)), we took Ti-0.01076T() and T2=0.01078To. First, at T=Ti the quartet of bound disclinations, point defects and correlated dislocations are formed. The point defects appear in pairs in our MC calculations, which are a consequence of the periodic boundary conditions. Note that in a single layer crystal the total energy of a non bounded pair of a centred vacancy' and a centred interstitial' is E=0.29kBTo- In the square bilayer crystal the energy of unbounded pair of vacancy' and the interstitial' is E=0.315 keTo. The disclinations bound into a quartet and point defects produce only a negligible effect on the periodic lattice structure and Gang =0.8-^0.9 and Gtr=0.85^ 0.95 (group A in Fig. 4(a,b)).It should be noted that in spite of prolonged annealing of the system during 5-10'^ MC steps at T=Ti, which is just below melting, we did not find more complex isomers than point defects and quartets of disclinations. At T==T2 uncorrelated extended dislocations with non-zero Burgers vector and unbounded disclination pairs are formed which causes a substantial decrease of the translational order (group B in Fig. 4(a,b). At this temperature single disclinations appear and the system looses order, both order factors become small and the system transits to the isotropic fluid (group C in Fig. 4(a,b)). Fig. 4(a,b) clearly illustrates that for a square lattice the defects which are able to destroy the translational and orientational order have a substantial larger energy than those of a single layer crystal with hexagonal symmetry. As a whole the localised and extended dislocations as well as disclinations in the square bilayer crystal are defects with a higher energy as compared to the ones in the hexagonal bilayer crystal. Thus, the square type bilayer crystal requires larger energies in order to create defects which are responsible for the loss of Gang, Gtr and thus for melting of the crystal. In conclusion, we studied the melting temperature of the five lattice structures in a bilayer crystal and found evidence that the melting temperature depends on the crystal symmetry. A square lattice has a substantial larger mehing temperature than e.g. a hexagonal lattice. In order to understand this resuh we investigated the defect structures responsible for melting and found that the defects in a square lattice have a larger energy as compared to those in a hexagonal structure and consequently larger thermal energy is required to create them We also formulated a new melting criterion, in two dimensional layers and bilayers melting occurs when the bond-angular order factor is Gang =0.45, which is independent of the functional form of the interparticle interaction. [1] G. Goldoni and F.M. Peeters, Phys. Rev. B 53, 4591 (1996). [2] B.I. Halperin and D.R. Nelson, Phys. Rev. Lett. 41, 121 (1978); DR. Nelson and B.I. Halperin Phys. Rev. B 19, 2547 (1979). [3] V.A. Schweigert and F.M. Peeters, Rev. B 51, 7700 (1995).
Poster Session C: Collective Effects and Astrophysics
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FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V Allrightsreserved
531
Effects of Dust Grains on P l a n a r RF Discharges Shigehiko NONAKA'\ Kouji KATOH'\ Yoshiharu NAKAMURA^', Shunjiro IKEZAWA^' and Shuichi TAKAMURA""' l):Toyota Technological I n s t i t u t e , Hisakata 2-12, Tempaku, Nagoya 468-8511, Japan. 2):Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan. 3):Electronic Engineering, Chubu University, 1200, Matsumoto, Kasugai, Aichi 487, Japan. 4): Department of Energy Engineering and Science, Nagoya University, Nagoya 464-8603, Japan. Abstract Effects of dust grains on capacitive RF discharges are disclosed analytically. In this study, dielectric dust grains with a constant radius are assumed to be set in the plasma. The results obtained are: (1) A new dipole resonance of the plasma surrounding the dust is discovered in low gas-pressure. (2) Along with a geometrical resonance, the new dipole resonance can also produce the plasma. (3) The produced plasma density and the RF voltage threshold for discharges are obtained graphically. 1. Introduction Capacitive RF discharges between parallel-electrodes are widely used in 13.56 MHz operation for material processing in the micro-electronics industry and research[l]. It has been reported recently that, when using mixture gases in low gas-pressure, growing dust particles[2] often give rise to defects in the products of large-scale integrated (LSI) circuits [1]. This paper deals with dust effects on the capacitive RF discharges. In this study, we discovered a new dipole resonance of the plasma surrounding the dielectric dust particle. From numerical calculations, splitting of a pure normal resonance by the dusts and the plasma production by the new dipole resonance. The results obtained will shed some light on dusty RF plasmas in material processing. 2. Modeling for Planar dusty RF Discharges An idealized model for dusty RF discharges is shown in Fig. 1. Each electrode (x=±b) is covered by a lossless dielectric material with a relative permittivity e g. Subscripts, g and p, Metai—. denote the barriers and the plasma (-a^ Dieiectric-» ^^^ x^a).
The plasma is assumed to be
Dusty
PlAsna
K^J^(2)
A^ Y,y ^^p(iwt) *^ ^ ^ _ «I—^^
_
1^
•-^r n
d Dust
a uniformly produced, and dust grains are • • • also assumed uniformly dispersed. For %etai——J^-^-!^ simplicity, we tentatively ignored effects of dust-surface charging and secondary electrons from the dust surface. Fig. 1: Model of RF discharge
532
S. Nonaka et al /Effects of dust grains on planar RF discharges
Neglecting motions of d u s t s and ions and using a Wagner's t h e o r y [3], an e f f e c t i v e , macroscopic p e r m i t t i v i t y Keti- of d u s t y plasmas is o b t a i n e d as: 5 o( /c o - X"^ - J 77 X * ) ( /c N - X * - j 77 X * ) KeTT
=
(1) /c 1 - X* - j 77 X*
where, 5 = ND(4;r/3)(D/R)^ do = 2(l-(^ )/(2+5 ), j =/^TT, /c o = 1, /c 1 = 1 + {(l-5)/(2+(5)}£ d ( ^ 1 ) , /CM = 1 + {(l+25)/2(l-(5)}£
A = KG/So + KeTi-/5o, KG = ( a / d ) £ o,
(2) U = e o(Vo/2d), KGO = (a/d) e 3/do
On t h e o t h e r hand, an RF c u r r e n t Jx in t h e plasma is obtained, u s i n g Eps, as; Jx = e oKe-r-r(9Ep2/3t) = j(o a OKSTI-EP2. 3. Characteristics o f Pure Modified Resonance In t h e l o s s l e s s case (77 =0), t h e condition A = 0 in eq. (2) gives r i s e t o a p u r e r e s o n a n c e ; t h a t is, 5 oA( 77 =0) = KGO + ( A : O - X ) ( / C N - X ) / ( A: 1 - X) = 0. Then, we h a v e two s o l u t i o n s X(±) for X; X<±> = ( 1 / 2 ) { A : o + /c N + KGO ±
y
(/c
o + /c N + K G O ) ^ - 4 ( A ; N A : O +
/ci KGG)
}
( > 0)
(3)
The s i g n s ( ± ) c o r r e s p o n d t o t h e ones of t h e s q u a r e r o o t . When 5 app r o a c h e s z e r o ( i . e . , 5 o - * l ) , t h e y have d i f f e r e n t limittlng v a l u e s as follws; Xc+) -* 1 + KG X<-) -> 1 + (1/2) e d
(= XGO, for G-resonance) (= XDO, f o r D-resonance)
(4) (5)
The r e s o n a n c e XGO c o r r e s p o n d s t o a s o - c a l l e d "Geometrical r e s o n a n c e " in t h e d u s t - f r e e system, whereas t h e one XDO c o r r e s p o n d s t o a "Dipole r e s o nance" of t h e plasma around a s i n g l e dust p a r t i c l e . The D-resonance has
533
S. Nonaka et al /Ejfects of dust grains on planar RF discharges
n o t been p o i n t e d o u t in t h e l i t e r a t u r e , d e s p i t e of a simple resonance of a s p h e r i c a l s u r f a c e wave a t t h e d u s t plasma i n t e r f a c e . In Fig. 2, Xc±), which a r e a l s o stamped as XG or XD, a r e shown as f u n c t i o n s of KG u n d e r a fixed e rj. The r e s o n a n c e p o i n t s Xc±) c r o s s each o t h e r a t t h e d e g e n e r a t i n g p o i n t where XGO=XDO. AS 6 i n c r e a s e s , t h e c u r v e s g r a d u a l l y d e v i a t e from t h e c u r v e and l i n e of XGO and XDO, and moreover move r e p u l s i v e l y each o t h e r .
10^
1 ,
J
,1
3 L
*
Q.
3
L_^-U
—
1
1
1 — 1
0 ( X o o . Xoo ) : < 5 - 0.01\(X '16 " 0.1 i X •
10'
: 5 -
1
1
1.
> ^
.'''"/^ L
1-4
*
1
Hi
10°
l^lCo).
/—.XDO
[
K.
10^
1
10<^
10'
Fig. 2; Pure modified r e s o n a n c e s
4. Solution of Dusty RF Plasmas The RF power Pab absorbed by e l e c t r o n s p e r u n i t e l e c t r o d e - a r e a is obtained, by i n t e g r a t i n g p r o d u c t of Epsand Jx, as follows; Pab = (l/2)Re{ / -a^Re[Jx • Ep2]9x}. While, an e l e c t r o n - e n e r g y loss Qioss p e r u n i t time and e l e c t r o d e - a r e a s h a l l be p r o p o r t i o n a l t o t h e e l e c t r o n d e n s i t y Ue in low g a s p r e s s u r e as follows; Qioss = 2aneQ = o) e oaQoX, where Qo = 2a) (m/e^)Q [3]. The power b a l a n c e is t h e n obtained, equating Pab and Qioss, as follows; f)
(/c 1 - X*)^ + (/c M-/C i)(/c
1-A:
o) + 77 ^X^
Qo (6)
5 o ( l + 77^)
Y ^.^ + 77 ^X^^( 2X^ - /c Q -
A: N
-
KGQ)'
U"
where, Y * = / C N A : O + A : IKGO - (/C O + A: N + KGO)X'' + (1 - 77 ^)X*^ . The l e f t - s i d e of (6) d e n o t e s t h e normalized RF power Pab/(a; a e oU^X). This is s h o r t l y w r i t t e n by Pab/X h e r e a f t e r . Equation (6) gives t h e normalized p l a s ma d e n s i t y X f o r maintaining t h e RF plasma. 5. Discharge Characteristics 5.1 Peak s p l i t t i n g due t o 5 : From experimental p o i n t s of views, c o n v e n i e n t v a r i a b l e s a r e t h e RF v o l t a g e Vo (or, U), gas p r e s s u r e (or, 77) and d u s t d e n s i t y ND (or, d ). Fig. 3 shows s e v e r a l resonance curves, which is t h e l e f t - s i d e of eq. (6) as a function of X under fixed 6 (curves A: 6 =0, B: 6 =0.2, and C: 6 =0.6) n e a r l y a t a d e g e n e r a t i n g p o i n t KG =£ d/2. In t h e d u s t - f r e e case ((5=0), t h e a b s o r b e d power Pab/X e x h i b i t s a normal s i n g l e G-resonance c u r v e a t X = XGO. When 8 i n c r e a s e s , t h e G-peak s p l i t s i n t o two peaks. These two peaks c o r r e s p o n d t o two modified resonances X<;±) a t t h e d e g e n e r a t i n g p o i n t in Fig. 2. S e v e r a l p r o p e r t i e s of t h e d i s c h a r g e can be found q u a l i t a t i v e l y from b o t h t h e r e s o n a n c e c u r v e (Pab/X vs. X) and loss l e v e l Qo/U" in Fig. 3; for examples, an RF v o l t a g e t h r e s h o l d f o r d i s c h a r g e from t h e maximum peak l e v e l , and t h e p r o d u c e d plasma d e n s i t y X as a function of \S/f^ that appears at the region a(Pab/X)/ax^O.
S. Nonaka et al /Effects of dust grains on planar RF discharges
534 X N Xao«2 ^ 10. h XDO»2 O
5 1 KA sB
1 J7-0.01
O 3
/I.y\/\
0.1
i y\ V \
1
A: 5 = 0 J D: 5 » 0 . 2 1 C: 5 = 0 . 6 1
;1_Q,/V_.
'»
i
0.00001
U 0.1
0.5
1—'• ; t 111 T -; 5-0 -: d - 0 . ] 1 Xoe» - 1 . 5 -^i 7 - 0 . 0 1 XDI
D o
\
O 3
—I
K'
0.01
A' A : X O O - 3 ^
0.0001
Xoo Xoo
^^^_i^..
1
O
O LllV----—*'''* 0.001 h
L
X (V
^s.\
**.,
^i._rTT.i.., 1 ^ - x , 5
X ( H
10.
50. 100.
10
- T - j r-T "1
-T
r-
A'ft
t'i"
» ' 1
\
/ '' k>'''
1 Xo
*\\
Ij
11
1
B' B : X o o - 8 _ C* C:Xoo-30
.. i
11 0.5
0.1
[ 5
1
10.
cjp^/o;'^)
Fig. 3; Resonance curvesCXGo^Xpo) 5-2 XGo-dependence of resonances : Similarly, Fig. 4 shows Xoo-dependence of r e s o n a n c e c u r v e s (A, B and C) u n d e r fixed v a r i a b l e s Xoo, d and 77 ( < 1). In t h e c a s e A', B' and C where 5 =0, o n l y t h e G-resonance a p p e a r s , b u t n o t t h e G~resonance. A l l t h e p o s i t i o n of t h e G-peaks s h i f t t o t h e l a r g e r dens i t y side with i n c r e a s e of 5 . Only when 5 =?^0, t h e D-resonance a p p e a r s n e a r l y a t t h e p o i n t X=XD without s h i f t ing of t h e peak. These p r o p e r t i e s a r e i d e n t i c a l t o t h o s e in Fig. 2.
Fig. 4; Variation of r e s o n a n c e 50.
.
,—^
,—p
D:»-0.01
(Xoo-2.Xoo»5) E:V-0.1 (XOO-S.XDO-IO) F:I»-1.5
10.
(X00-2.X00-IO) « -0.01
nt
>< 2 [-
D...W
Xoo
1 r
0.1
0.5 I
Ail 5 10
5 0 100
u ir^
Fig. 5; X a s a function of U/y"Qo^ 5.3 Detection o f h y s t e r e s i s ; Fig. 5 r e p r e s e n t s t h e produced plasma d e n s i t y X a s a function of U//n55^ under t h e fixed parameters K G O < X D O , 8 =0.01 and 77 =0.01-^-1.5. The dip p o i n t s of U/V Qo^ c o r r e s p o n d t o peaks of t h e modified Gand D-resonances in Pab/X c u r v e . The s t a b l e s o l u t i o n X a p p e a r s a t t h e u p p e r p a r t s of t h e dips; c[X/2l((U//'"5o")^0. This p r o v e s t h a t t h e D-resonance is p o s s i b l e t o p r o d u c e t h e plasma when XGO<XDO and 77 < 1 . Fig. 5 a l s o shows 77 - d e p e n d e n c e of t h e h y s t e r e s i s phenomenon which a r i s e s with an RF v o l t a g e going up and down under t h e conditions XGO<XDQ and 77 < 1 . However, t h e h y s t e r e s i s f i n a l l y d i s a p p e a r s in t h e range 1< 77 as seen in c u r v e (F). In summary, t h e d i p o l e r e s o n a n c e s of t h e plasma around t h e d u s t g r a i n s a r e p o s s i b l e t o p r o d u c e t h e plasma. References 1) ''Dusty plasmas";J. Plasma and Fusion Res., vol.73, no.ll(1997)pp. 1220-1274. 2) "Phenomena on d i e l e c t r i c s " , ed. Inst. E l e c t r i . Engin. Jpn.(1973)pp. 143-146. 3) S. Nonaka; J o u r n . Phys. Soc. Jpn, vol.59,no.9(1990)pp.3217-3226.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
535
Experimental Studies of UV-induced Dusty Plasmas under Microgravity Vladimir E. Fortov, Anatoli P. Nefedov, Vladimir P. Nikitsky, Alexander I. Ivanov, and Audrey M. Lipaev
High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia
Abstract. The dynamics of the formation of ordered structures of macroparticles charged by photoemission under the action of solar radiation under microgravitational conditions without the use of electrostatic traps to confine the particles is studied experimentally and theoretically. An analysis and comparison of tile results of the experimental and theoretical investigations permit drawing a conclusion regarding the possibility of the existence of extended ordered formations of macroparticles charged by photoemission under microgravitational conditions.
The purpose of performing the space experiment under microgravity was to study the possibility of the existence of plasma-dust structures in the upper layers of the earth's atmosphere when the particles are charged by solar radiation as a result of the photoemission of electrons from their surface. The working conditions needed for the formation of structures of charged macroparticles in the experimental chamber (the type of buffer gas and its pressure, as well as the concentration, material, and size of the particles) were selected as a result of a preliminary numerical analysis of the problem posed; the particle charges induced by solar radiation and the interparticle interaction parameter were determined, and the times specifying the dynamics of the formation of an ordered system of macroparticles (the charging time, braking time, and dispersion times of particles in the working chamber and the times for establishing ordered dust structures) under microgravitational conditions without the use of electrostatic traps to confine the particles were calculated. The calculations were performed for particles of different materials and sizes (1-100 |Lim) with variation of their concentration and the pressure of the buffer gas. Experimental investigations of the behavior of an ensemble of macroparticles charged by solar radiation were performed under microgravitational conditions on board the Mir space station (1) (see also the Internet address: http://www.redline.ru/-ipdustpl/).
536
VE. Fortov et al. /Experimental studies of UV-induced dusty plasmas under microgravity
^
=
^
Glass tube
Videocamera
Particles
Solar radiation
FIGURE 1. Schematic of the experimental setup for UV-induced plasma under microgravity conditions.
The experiment was carried out with glass ampuls containing particles of bronze with a cesium monolayer (two ampuls)) in a buffer gas (neon) at different pressures (Fig. 1). The glass ampuls had the form of glass cylinders, one of whose end surfaces was a flat uviol glass window and was intended for illuminating the particles with solar radiation. Immediately before the performance of an experiment, the required ampul was placed in the clamp of the working-chamber holder with its flat end surface toward the illuminator. For diagnostics of the ensemble of particles, the ampul was illuminated by a laser sheet (the width of the laser sheet was no greater than 200 |j,m), and an image was obtained using the CCD camera, whose signal was recorded on magnetic tape. The field of vision of the video camera had the form of a rectangle measuring roughly 8x10 mm.
a) t=0.2 sec
b) t=2.4 sec
FIGURE 2. Successive states of the system of bronze particles in the ampul with Pi =0.01 Tonfollowing dynamic disturbance of the system.
VE. Fortov et al. /Experimental studies of UV-induced dusty plasmas under micrograuity
a) t=2 sec,
537
b) t=20 sec
FIGURE 3. Successive states of the system of bronze particles in the ampul with ^2=40 Torr following dynamic disturbance of the system.
The experiments were carried out with three values of the neon pressure: Pi=0.01 Torr and ^2=40 Torr. The first stage of the experiment was confined to observing the behavior of the ensemble of macroparticles placed in the ampul under the action of solar radiation. In the initial state the particles were on the walls of the ampul; therefore, the experiment was carried out according to the following scheme: a dynamic disturbance (jolt) of the system and relaxation to the initial state, i.e., drift to the wall. Figures 2a,b show the successive states of the system of particles in the ampul with Pi=0.01 Torr following dynamic disturbance of the system, and Figs. 3a,b show the state of the system in the ampul with ^2=40 Torr. Observations of the motion of the particles showed that the velocity vectors of the particles are randomly directed in the initial stage and that the particles drift to the walls without a preferential direction. Subsequently, a preferential direction usually appears, but motion along definite trajectories is displayed more strongly in the vessel with the higher pressure. Vibration of the particles on a background of the overall translational motion was observed in several experiments, and the treatment of the particle trajectories revealed periodic variations of the magnitude of the particle velocity in all the experiments. These variations of the particle velocity can be associated with fluctuations of the particle charge or with the dynamic action of microscopic accelerating forces arising on board the space station. Variation of the visibility of the particles was observed (one possible cause is rotation). It was concluded that particles are charged on the basis of observations of the changes in the trajectories of the particles when they come close to one another (collide) or approach the wall. It should also be noted that the particles move very slowly under ^2=40 Torr when the solar radiation is blocked and that acceleration of the motion occurs when radiation acts on the ensemble of particles. The charge of the macroparticles can be estimated by analyzing their dynamic behavior. The analysis results reveal that bronze particles acquired a charge of the order of 10"^ e (the coupling parameter y was about 10"*).
538
VE. Fortov et al. /Experimental studies of UV-induced dusty plasmas under micrograuity
1
1000
1
1
2000
3000
r
4000
FIGURE 4. Experimental correlation functions g(r) obtained as a result of the treatment of images without illumination (1) and with solar irradiation (2).
Despite the high particle charges and the large value of the coupling parameter, no strong correlation between the interparticle distances could be observed. The typical correlation functions obtained as a result of the treatment of experimental images without illumination and with solar irradiation are shown in Fig. 4. In summary, the results of observations of the behavior of an ensemble of macroparticles charged by photoemission under the action of solar radiation under microgravitational conditions have been presented. An analysis of the results obtained confirm the conclusion that the existence of extended liquidlike ordered structures of macroparticles charged by solar radiation is possible under microgravitational conditions.
ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research, Grant No. 98-02-16828.
REFERENCES 1. Fortov, V.E., Nefedov, A.P., Vaulina, O.S. et al. Journal of Experimental and Theoretical Physics 87, 1087 (1998).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
539
Parameters of Dusty Particles in Plasma Flows Alex A. Samarian, Olga S.Vaulina, Alex V. Chemyshev, Anatoli P.Nefedov, and Oleg F.Petrov High Energy Density Research Center, Russian Academy of Sciences, Izhorskaya 13/19, 127412, Moscow, Russia
Abstract. Here we present a combination of optical methods for determining the particle temperature, Tp, mean size, D^i, concentration, A^^, and refractive index, m(X)=n(X)'ik(A), in high-temperature flows. The unknown particle parameters are obtained by minimizing a meansquare error between the measured and calculated data. Calculations were performed by using the Mie theory.
THEORY 1. Determination of Particle Temperature The temperature of particles can be obtained from emission-absorption measurements in an emission line of the gas. A restoration of several unknown parameters from measurements on a gas emission line is a complicated diagnostic problem w^hich requires measurements in narrow band spectral region and considerable efforts in order to avoid significant experimental errors (1-3). Therefore, it is preferable to measure the particle temperature, Tp, and their parameters in a wide wavelength range beyond the gas emission line. The classic pyrometric two-color method can be employed for the determination of Tp in optically grey medium. We consider the temperature measurements for particles which are optically nongrey at the experimental wavelengths. The general difficulties for this task are related to the necessity of a knowledge of the spectral emissivity, £(X), Determination of ^A) requires a priori information of the particle parameters. In many cases of practical interest, however, these quantities are not known. The temperature of particles can be obtained by the spectroscopic method involving the approximate relationships for s{X) (4). This method is based on solving a system of radiative transfer equations for light intensity measurements of the signals of reference lamp plus flame, 5^/, flame only, Sp, and lamp. Si in the absence of the flame at several wavelengths At (i=\'N). Assuming that multiple in-scattering is negligible yields:
540
A. A. Samarian et al /Parameters of dusty particles in plasma flows
s{X) = (l-cy(A))(l-exp{-r(A)})
(1)
Here o)(X) is the single scattering albedo, T(/1) is the optical depth. The case of turbid media was detailed in (3). The error estimate 6^^~^^ of the emissivity e(X) (Eq. 1) from neglecting the in-scattered radiation (A>0) can be easily calculated using the value of the measured optical depth, r{/l) (4). For confined scattering medium with T(A) < 0.91.2, the 5^^"^^ value is less then 2% for particles with the size parameters p>l (3). In this case a problem can be reduced to choosing an appropriate model for the spectral dependence of single albedo co(A.). We consider the spectral approximation W(X) for the function {1'0}(X)} (Eq. 1) as W(A) = c / { r r(;i)'}
(2)
Here a, b, c are some coefficients, the values of which are independent of the wavelengths. The relative measurements eliminate the c value from consideration. Then the particle temperature, Tp, as well as the appropriate approximation of £{X), (i.e. coefficients a, b) are obtained by minimizing a mean-square error between the experimental and calculated data.
2. Determination of Particle Mean Sizes and Refractive Index The spectral dependence of s{X) can be obtained from the following equation: Sp{X)/SiX) = €{X) B{Tp^ X)/B{TuX)
(3)
where B{Tp, X) is the Planck blackbody fimction, Ti is the temperature of the reference lamp. For the case of > 5, €(X) is dependent on 0^2 only, and is weakly dependent on m for absorbing particles with k > 0.4 (4). Then the e{Z) measurements can be inverted for D^2 ^^d k of particles with 0.001< k > 1. Assuming single scattering, the problem can be reduced to empirical inversion of co{X) (4). Dependencies of the albedo cy(/lo) on the particle Sauter mean diameter, D^2^ and absorption index k are shown in Figs. la,b for the Gaussian distribution with the standard deviation SQ = 25%. The spectral distribution of the albedo, co{X) is determined by the particle sizes provided that the refi'active index is weakly dependent on X. The co(yi) value at Ar^A^ is uniquely determined by the k value.
A. A. Samarian et al. / Parameters of dusty particles in plasma flows
CO
541
1.00
1.00
0.90 + 0.80 + 0.70 + 0.60 0.50
0.60 2.50 4.50 6.50 D32, Mm
0.00 0.01 0.02
0.03
k b)
a)
FIGURE 1. Dependencies of the albedo co on the particle mean diameter D^2 (^) ^^^ the image refractive index k (b) for the Gaussian distribution with the standard deviation SQ=25%: a) w=1.7-0.01z; X\ 1- 0.5 |Lim, 2- 1.0 ^m; b) n=\.l, D^2. 1- 0.5 jim, 2- 1.0 jam
Therefore, with the known particle sizes and real refractive index, the spectral absorption index, k{X), can be easy measured. For determination of the particle mean size, concentration, and real refractive index, we use a technique based on measurements of forward angle scattering transmittance (FAST) at different aperture angles of the detector (5, 7). Because of the detector's finite field of view, the contribution fi-om forward-scattered light to the detector irradiance can not be completely eliminated, even when multiple scattering can be neglected. In this case, true optical depth, T, is distinguished fi-om the apparent (measured) optical depth, T*. Taking into account that some light scattered at angels 0<0^ reaches the detector, the relative value of measured extinction cross section can be expressed as follows q(d^,m,D^2)==T*/T
(4)
By varying the detector aperture {0^, one can measure an angular distribution of q{0^,m,D^^. In order for the least mean-square fit to provide the correct determining of the particle diameters D^j^ it is necessary that
e^
(5)
542
A. A. Samarian et al /Parameters of dusty particles in plasma flows
TABLE 1. Results of Measurements of Mean Size, 7)32, Real Refractive Index «, Temperature, Tp, and Retrieved Coefficients a, b (Eq. 2) for Ce02, and Ash Particles No / ) „ |iin Z)j, ^m n Particles a b 7>,K r^K CeOj
ash
1
1.23
1.17
2.05
0.20
si
2020
2025
2
1.17
1.20
1.98
0.22
=1
2230
2240
3
0.33
-
-
0.85
0.85
1810
1820
5
1.65
1.73
1.68
0.32
=1
1922
1930
6
1.57
1.64
1.68
0.32
si
2090
2095
In many cases of practical interest the determination of Dy2 can be made from FAST measurements at angles G^ < ff^ without knowledge of the PSD and the refractive index, m.
EXPERIMENT The polydispersions of particles ofSi02, Ce02, and ash in a flow of a propane-air flame combustion products were studied in the spectral range from 0.5 to 1.1 |im. The particles were introduced in the iimer flame of the Mekker combustor. The flame temperature was varied from 1800 to 2250 K. The measured optical depth T{X) < 0.35, because the multiple scattering was negligible (4). The particle temperature, (Tp), number density, (Np), complex refractive index, (m=n-ik), and mean diameter {D22), were obtained. The results of measurements are given in Table 1.
ACKNOWLEDGEMENTS This work was supported by the Russian Foundation for Basic Research, Grant No. 98-02-16825.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Bauman, L.E., Combust, and Flame 98, 46 (1994). Philip, P.H., and Self, S.A., Appl. Opt. 28, 2150 (1989). Vaulina, O.S., Nefedov, A.P., and Petrov, O.F., High Temp. 32, 521 (1994). Nefedov, A.P., Petrov, O.F., and Vaulina, O.S., JQSRT 54, 435-470 (1995). Vaulina, O.S., Nefedov, A.P., and Petrov, O.F., High Temp. 30, 817 (1992). Hottel, H., and Sarofim, A.F., Radiative Transfer, McGraw-Hill, New-York, 1967. Nefedov, A.P., Petrov, O.F., and Vaulina, O.S., Appl. Opt. 36, 1357-1366 (1997).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
543
Hard X-rays from High Power Density Plasma ^^Dust^^ Yu.K.Kiirilenkov\ M.Skowronek^, G.Maynard^ J.Dufty"^ ^Insliiule for High Temperatures (IVTAN), Russian Academy ofSciences, Moscow 127 412, ^Laboratoire des Plasmas Denses, Universite P.&M. Curie, F-75252 Paris Cedex 05, France ^Laboratore de Physique des Gaz et des Plasma, Universite Paris-Sud, Orsay ^University of Florida, Gainesville FL 32611, USA Abstract - This work concerns some new aspects of dust}^ plasmas physics - high powder density microplasma "dust". Our prime goal is to determine tlie optimal ways for simple experimental production and stud>^ of ensembles of hot microplasmas -cold grains created by an intense energy deposition into the cold solid density dust "target" in vacumn discliarge. Absorbing (stopping) power of correlated media of dust grains (clusters) with vety small volmnes and near solid densities is essentially liiglier tlian botli tlie plasma (gas) and tlie sohd planer targets (estimated energy densities deposited anomalously may be up to ~ 10^ J/cm^ at nanosecond time scale). Supratliermal x-ray emission, generation of energetic ions, x-ray trapping due to multiple scattering and the x-my bursts related ("random" laser) are discussed.
1. BASIC REMARKS The controlled generation of high power density matter (HPDM) with extreme parameters in vacuum is a complex interdisciplinary problem. Rather often occasional dots or spots have been observed [1,2]), which could be attributed to dense plasmas in vacuum discharges, especially under high energetics (- kJ or higher). Our particular aim is to generate HPDM under just - 1 J stored, and try to operate with x-ray dots in a well determined or predictable manner. The expansion process of the anode flare and supersaturating of anode vapors concentrated on the axis is accompanied by the compact collecting of nucleated clusters, microdrops, accelerated microparticles, grains (real "dusty" matter) at particular parts of interelectrode space. The resulting degree of condensation in anode flare (cluster, microparticles) is varied and may be estimated as high as 0.2 - 0.5 [1]. Thus, a unique combination of both solid density and gas-plasma as "target" medium may be "prepared" behind the anode flare front before the arc phase of discharge. Immediately after the end of breakdown, the main anomalous deposition of external energy into this condensed phase obtained in vacuum starts up (ns-scale electron beam, but mainly Joule overheating of small volumes of solid density grains by post breakdown high density current). The percolation of current may create a definite number of microplasmas with extreme temperatures and densities from the part of grains ( Te,i « 1-5 keV from TOF (time of flight) data, and electron density is n^ -10^^ -10^^ cm"^), meanwhile, the rest of the clusters and grains will stay as "cold" ones. Correspondingly, the processes of the explosive destruction of dusty grains (hydrodynamic expansion) is accompanied by x-ray radiation during expansion, cooling and recombination of the
544
Yu,K, Kurilenkov et al /Hard X-rays from high power density plasma "dust''
dense hot microplasmas created as well as by generation of energetic ions (with energies about ~ 0.1 -1 MeVfromTOF measurements). The x-ray hot dense granular matter (before overlapping due to expansion and during further plasma density oscillations in space) will be surrounded by highly scattering, reflecting and refractive media of cold dusty microparticles of different sizes. Generally speaking, these "cold" grains which are mixed with plasma "dust" may provide weak or strong multiple scattering and reflecting (both specular and diffusive one) for x-ray photons emitted by hot microplasmas. The mixture of "cold" grains (reflectors and scatterers ) and expanding "hot" grains (HPDM as active media) have to represent, in general, an ensemble of specific random media with x-ray amplifying prop^ies. These properties will depend on the relation between characteristic amplifying and absorbing lengths lamp / W , for a particular ensemble [3,4]. Thus, potentially the arc phase of discharge may transform the anode "dusty" plasma into particular amplifying media, if gain could overcome absorption and loses. Looking forward, to describe or analyze these mixtures we may introduce the parameters a, p (characterizing the mean intergrain distances to their mean diameters) and r[ (relation between numbers of cold and hot grains): a = Nc"^^^ / 2ac, p = Nh"^^^ / 2ah , Ti = Nc / Nh, where Nc, Nh are the densities of cold and hot "grains", ac,ah averaged values of their radii, respectively. In principle, any experimental variation of parameters like a , p, and r\ may allow for different levels of reflection, scattering and pumping in ensembles of cold and hot grains. As a result, it would be possible to get even various types of x-ray lasing media or systems on the basis of these ensembles.
F I G U R E la) hard x-ray image of thermal "ball" b) Suprathermal x-ray from merged microplasmas.
Subsequent manipulation with amplifying media of hot microplasmas and cold grains under background of very fast thermal processes suggests new opportunities. In particular, the motion of plasmas dust in interelectrode volume may be accompanied by scattering (Fig.la) or collection of cold grains and hot microplasmas in space. For example, transformation of HPDM of hot microplasmas into almost homogeneous macroplasma "ball" may increase the x-ray output up to superradiance (Fig. lb, ASE lasing regime [5]). A second case is the controlled partial focusing of grains and
Yu.K. Kurilenkov et al / Hard X-rays from high power density plasma "dust" 545 microplasmas, emitting fast ions under hydrodynamic expansion during overheating of particular grains. This combination of partially "trapped" hard x-rays (fig.2a ) and fast ions at interiors of the cold - hot grain ensembles allows perhaps to operate with some particular nuclear processes related to any hot dense matter originating from composite anode (like gamma - lasers pumping, novel energy conversion schemes, etc.).
•A- ' .4>'',^' ^
FIGURE 2. Partially trapped hard x-ray, losses are still liigher than gain (a) and diffuse regime of photons due to multiple scattering with x-my burst ("random" laser)
2. TRAPPING OF X-RAYS. HARD X-RAY "RANDOM'' LASERS Generally speaking, such contrasting cases as shown on figs.la) lb) represent the basis and the top level of an imaginary "pyramid" of x-ray ensembles with different sets of parameters a, p, and r|. Intermediate cases between merged and distributed microplasmas may have a special interest too, and might be reaHzed if needed for some particular purposes. In this Section, as an example of particular interest, we will demonstrate experimentally the realization of a laser scheme when the x-ray photon is diffused by multiple scattering on grains immersed into active HPDM media ("random" laser [3,4]). This scheme was studied in detail theoretically about 30 years ago [3]. Letokhov [3] considered the case when the mean free path of photons due to scattering Isc =l/NoQs, ensemble dimension R and the wavelength X are related as X « Isc « R , and No ~^^^ » X (Qs -scattering cross section). Being trapped in disordered system, light makes a long random walk before it may leave the medium from near surface area. The laser generation threshold(accompained by x-ray burst) may be achieved when the volume gain becomes larger than surface losses, at some ensemble volume which is above critical one, or at radius R > Rcr » 7C (Isc / 3 y ) ^^^ in the case of sphere ( y is the gain, cm'^). We can change effectively the values of parameters a, p, and r] in our x-ray ensembles, i.e. regulate the levels of scattering and pumping inside the system. Our x-ray mixtures have the space scale of R -- 0.1-0.5 cm, which is a reasonable order of magnitude of the critical radius in our real experiments. Let us
546
Yu.K. Kurilenkov et al /Hard X-rays from high power density plasma "dust"
suggest that number of scatterers in the expression for mean free path Isc =l/NoQs is equal No = Nc + Nh « Nc, and assume Qs ^ TVC^ 2ii ka» I. For example, in this case, for the values No -- 10 ^"^^ cm "^ we estimate kc ~ 0,01 - 0.1 cm. For this value of photon mean free path and assuming a gain of y « 1-5 cm~^ the critical radius may be about Rcr - 0.2 -0.3 cm, which is accessible to experiment. Collection of plasma dust on fig.2a shows just the beginning of x-ray trapping. Further, an example of hard x-ray burst which could be identified with random diffuse laser is given on fig.2b. The merged part of the intense x-ray image corresponds to an effective radius which is about - 0.2-0.25 cm. The "halo" of this core may be due to "sublightening" of the shell of grains. In spite of the bright, large and oversaturated character of the image, oscillograms related show very modest x-ray intensities (comparing with fig la). It may suggest that radiation is partially trapped, and just the surface layer (with gradient of radiative flux (r)) provides the x-ray intensities - kc V O (r), represented by oscillograms.
3. CONCLUDING REMARKS AND DISCUSSION We have used several different stages to create high power density matter with 10 " W/cm (by depositing - 10 J/cm at ns scale) in a vacuum discharge with hollow cathode Hard x-ray generation beyond the electrodes, formation of hard x-ray "balls" originated from hot microplasmas and cold grains ( with power -10^-10^ W and total photons number about 10^^'^ per shot), suprathermal level of hard x-ray emissionn were observed The effects like intergrain correlations, self-organization, x-ray multiple scattering, x-ray trapping (or even localization) and guiding are helping to complete the picture of a variety of intriguing phenomena which may be investigated on the basis of ensembles of cold grain - hot microplasmas. Many questions remain, however, "soft" physics and small-scale experimental design for efficient hard x-ray source based on vacuum discharge illustrate also promising new opportunities for nanotechnologies, for studies of energy conversion, superchemistry, self-organisation effects or modeling of dense astrophysical plasmas and their evolutions: hot dust around stars, clusters of stars (like on fig.2a), anomalous gamma-ray bursts [6] (for example, cluster of stars as "random" laser, fig.2b), x-ray from comet tails, dusts self-organisation, etc.
REFERENCES 1. G. A. Mesyats and D. I. Proskurovsky, "Pulsed Electrical Discharge in Vacuum'\ Berlin: SpringerVerlag, 1989; G. A. Mesyats, ''Ectons\ "Nauka", 1993 2* R. L. Boxman, S. Goldsmaii, and A. Greenwood, Twenty-five years of progress in vacuum arc research and utilization, IEEE Trans. Plasma Sci., vol. 25, no. 6, pp. 1174-1186, 1997. 3. V. S. Letokhov, Generation of light by a scattering medium with negative resonance absorption, Sov, Phys. JETP, 1968, vol. 26, no. 4, pp. 835-840 4. D. S. Wiersma and A. Lagendijk Liglit diffusion with gain and random lasers. Phys. Rev.E, 1996 vol. 54, no. 4, pp. 4256-4265. 5. R. C. Elton, "X-rqv Lasers'\ New York, Academic Press, 1990. 6 Physics World, 1999, v. 12, No 15, p.5.
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
547
From Dusty Plasma Target to Hard X-ray Lasant Media M. Skowronek, Yu. K. Kurilenkov*, and G. Louvet Laboratoire desPlasmasDenses,
Univ. P. etM. Curie F'7S252 Paris cedex 05 (France)
* Institutefor High Temperatures, Russian Academy of Sciences, Moscow 127 412(Rusda) Abstract We have studied experimentally a high power density medium of hot microplasmas collected in the interelectrode space in «vacuum» discharges produced with an hollow cathode geometry. Our aim is to reaUse some effects of high local power density and to manipulate the hot microplasmas originating from clusters and dust grains in chosen parts of the space. Using a IJ , 50 Q 70kV Marx generator , we have detected the hard x-ray emission in 4 perpendicular directions during 10 to 30 ns, having an energy above 50 keV. In certain conditions, the lasing effects are observed: anisotropy, and an enhancement of emission about 3 to 10 times. Time of flight measurements show a production of energetic ions, up to 1 MeV.
1. INTRODUCTION The description and understanding of the phenomenon arising in vacuum and in low pressure discharges has progressed a lot during the last three decades [1]. It is interesting to note that these discharges allow the study of very dense plasmas like cathode micro drops and hot spots, that is a matter under extreme conditions . The occurrence and the physics of anode spots or anode explosive centers which are still less investigated, are the main object of the work described in the following. This particular field requires the use of results accumulated in different domains: dense strongly coupled plasmas [2]; dusty plasmas of cold fine grains immersed into a low temperature plasma produce a dusty plasma (whose properties have been studied and discussed intensively), target physics (the anomalous absorbing power phenomenon provide to use this medium for high power sources of radiation including hard x-ray), driven granular medium dynamics; physics of clusters and novel cluster sources of radiation, laser cluster interaction [3], gamma- and x-ray lasers [4], physics of disordered media where multiple light scattering allows lasant effects, pseudo-spark (due to the geometry).
2. EXPERIMENTAL SET-UP A specially designed (Europulse) Marx generator has an impedance of 50 Q. It is charged under 20 kV. It stores only IJ and the output voltage is about 70 kV. It is applied, through a 50 Q cable to the experimental cell.(see figure 1) [5]
548
M Skowronek et al /From dusty plasma target to hard X-ray lasant media
PM(2) FAST IONS PIN(1)|g^ HARD X-RAY MICROPLASMATXJSr
FIGURE 1. Experimental Set-up The discharge cell has dimensions very similar of that of the cable. The hollow cathode(C) and the anode(A) have few millimeters diameter and their distance can be adjusted within 0.1 mm precision by means of a screw. Three mylar windows allow the x-ray intensity measurement in three perpendicular directions (side-on: right, left, upper) in the plane corresponding to the anode edge. Another mylar window allows the end-on measurement through the hollow cathode. Calibrated PIN diodes having a 1 to 2 ns rise-time are used to measure the x-ray output. A photo-mukiplier(PM) associated to a NE 102A scintillator is also used. Different metal attenuators prevent the detector saturation. The vohage and the current are recorded using a Rogowski coil and a voltage divider. A 4-ways 500 Mhz oscilloscope(osc) (Tektronix) records the different signals in each case (see figure 2). The emission has a very short duration. Depending of the conditions, distance and pressure its duration varies from 10 to 30 ns. Due to the metal coverage of the different detectors we are mainly interested in highly energetic radiation : 50 keV and upper.
t, ns
FIGURE 2 . Oscillogram of the x-ray emission. Ch 1: left PIN diode +0.1 mm Cu ; Ch2 : end-on. Photo-muhiplierH- 0.5 mm Cu -I- 0.6 mm Fe; Ch3 : upper PIN diode + 0.1 mm Cu; Ch4: instant of camera opening The image of the x-ray sources is obtained by means of a pinhole (PH) (0= 0.1mm) bored in a lead screen. It is registered using a scintillator sheet (NE102A) by a
M Skowronek et al /From dusty plasma target to hard X-ray lasant media
549
low noise CCD camera designed by Lhesa. As the camera is protected against spurious light by a black scotch sheet and the pinhole is covered by an Al foil, the lower limit of the x-ray energy recorded is about 5 keV. The camera is able to give one view taken during 5 ns using the dynamic regime. In the static regime, the camera is open during 40 ms. The observed image (see figure 3) shows a great number of hard x-ray dots : a few hundreds by shot. Their diameter is Umited usually by the pinhole diameter. Using a precise reference of the electrode position, it has been shown that the main x-ray emission originates from outside the anode. As the emission is very short in time, no image blurring has been observed, even in the static regime.
FIGURE 3 . Typical hard x-ray image from ensemble of cold grain -hot microplasmas. The image definition has been studied vesus the variation of the scintillator thickness between 0.2 to 4mm. No big difference is noticeable. Evidently the best defined image is obtained with the thinner scintillator. The figure 3 shows a typical example of the emission taken at the initial stage of arc phase the discharge in the dynamic regime during 5 ns.
3, LASANT MEDIUM. TIME OF FLIGHT MEASUREMENTS A 60 cm long tube is installed either side-on or end-on,.at a window place. A copper target transforms the electrons or the ions emitted into X-rays which are recorded trough a lateral mylar window. An immediate saturated pulse is oberved due to the electrons. A fainter pulse follows having a greater with related to the ion velocity distribution. The ion velocity is of the order of Vion= 5*10 ^ -10^ m/s. The temperature is estimated Tei'^1-3 keV. The hot microplasmas emitting the hard x-rays may be distributed in interelectrode space (fig 4a) or collected. A strong x-ray emission anisotropy is rather often observed in the different intensities recorded by the detectors. The next step was concerned to tranform this medium in an homogeneous one in certain conditions. We then obtain x-ray superradiance in any direction from unifom x-ray balls obtained by the merging of the multiple hot-spots. This process of merging (fig.2b) may increase in a
550
M. Skowronek et al. /From dusty plasma target to hard X-ray lasant media
result the intensity of the x-ray recorded by the camera with a factor 2 to 10. This superradiance effect is similar to another phenomena: the lasing occuring in strongly scattering and absorbing media (see figure 4b, see next paper in this volume)
:M. ^' i «-^-^
»- *_ ^ - -/
V
.- •-* **-.
4a
4b
FIGURE 4 a)Spraying of hard x-ray ensemble and b) Collecting of microparticles to create the uniform "ball" for hard x-ray superradiance.
5. CONCLUSION The discharge of a Marx generator in a vacuum hollow cathode produces an accumulation of microplasmas wich can give rise to a scattering hard x-ray laser. The pressure in the discharge chamber has been varied from 10'^ to lO'^ mbar without great changes. Different gases (air, Ar, H) have been used and also different metals: (W, Cu , Fe Sn, Pb) whithout great changes. As the charge exchange is of the order of |j.C the electrodes can support thousands of shots. This real table top experiment alow to reach plasma.temperatures in the keV range.
6. REFERENCES 1 R.L. Boxman, S. Goldsman and A. Greenwood, «Twenty-five years of progress in vacuum arc research and utihzation», IEEE Trans.Plasma Sci., vol 25 (1997), ni6, pp. 1174-1186 . 2. Yu. K. Kurilenkov, MA. Berkowski in «Transport and Optical Properties of Non-ideal Plasmas « ed. G.A Kobzev et al. Plenum Press, NY 1995. 3. T. Ditmire, T. Donelly, R.W. Falcone and MD. Peny, «Strong X-ray Emission from High Temperature Plasmas Produced by Intense Inadiation of Clusters»,Phys. Rev. Lett.( 1995) 75, p3122. 4. B.R. Benware, CD. Macchietto, C.H. Moreno, ans J. J.. Rocca ((Demonstration of a H i ^ Average Power Tabletop Soft X-ray Lasen> Phys. Rev. Lett. (1998) 81p. 5804 5. M. Skowronek and P. Romeas,»Properties of a miniature x-ray source», IEEE Trans.Plasma Sci., vol PS15(1987) pp. 589-598;
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
551
Coupling between Jeans and X-modes in self-gravitating magnetized dusty plasmas M.A. Hellberg^ R. Bhamthraml R.L. Mace^'^*, P. Singh^ and F. Verheest'* ^ University of Natal, Durban, South Africa, "^M.L. Sultan Technikon, Durban, South Africa, ^University of Durban-Westville, Durban, South Africa, '^ Universiteit Gent, Gent, Belgium, *Also at Bartol Research Institute, University of Delaware, USA
INTRODUCTION Large self-gravitating systems can collapse under their own gravitation if typical lengths exceeds a critical (Jeans) length, i.e. they are subject to the Jeans instability, if the gravitational forces exceed the pressure. For charged systems self-gravitational forces can compete with and even exceed inter-particle electromagnetic forces. The coupling and competition between electromagnetic and gravitational fields in a self-gravitating dusty plasma is one of its more interesting properties. For MHD modes propagating in a self-gravitating plasma in any direction other than strictly perpendicular to the magnetic field, the Jeans length is the same as for a neutral gas cloud — that is, the collapse is unaffected by the presence of a magnetic field [1]. On the other hand, for electromagnetic wave propagation perpendicular to an ambient magnetic field, gravitational collapse involving a cloud of charged dust and plasma is strongly inhibited by the magnetic field and in particular by the extraordinary-mode (X-mode) wave, leading to a longer effective Jeans length [2,3]. Together with the general result given by Chandrasekhar [1], this implies the formation of pancake-shaped structures, fiattened in the plane perpendicular to the magnetic field [4]. Using a quasi-inertialess model for the lighter species a single dispersion law was derived for Alfven-Jeans waves with frequencies below the ion gyrofrequency [4],
^^ = ^^(v^L + 4 + c L ) - E < ,
(1)
where VAd is the dust-Alfven speed, Cda a generalized dust-acoustic speed and Csd the dust thermal speed. The sum is over all the species and represents the gravitational effects, dominated by the dust. Like density perturbations in a neutral gas, which travel typically at the local sound speed, electromagnetic waves in the X-mode may
552
M.A. Hellberg et al /Coupling between Jeans and X-modes
either produce catastrophic collapse of a density perturbation, or act to "erase" that perturbation, depending on their wavelength. Whereas the typical speed in the conventional Jeans instability is the speed of sound, in the case of the X-mode, it is essentially the magnetosonic speed Vms given through V^^ = V^^ + c^^. For a system including both charged and neutral cold dust of the same kind, the critical wavenumber follows from (1) approximately as
f'ljn.^(^-^ \
+ ^)^i O
(2)
If bo /
Here a — Nd/{Nd + Nn), with A^^ and Nn the charged and neutral dust densities, respectively, c^ a common thermal velocity for both dust components, and the total Jeans frequency is given by uj — AixG[N(i + A^nj'^d- Because of the smaller denominator associated with the first term for typical parameter values, the neutral dust tends to dominate unless there is little neutral dust present.
MODEL AND RESULTS We consider the case of a dusty plasma comprised of four stationary warm fluids, viz. both neutral and negatively charged massive dust particles, as well as finitemass positive ions and electrons. We investigate magnetosonic waves propagating perpendicularly to the static magnetic field BQ, and assume that the wave frequency is much larger than the charging frequency of the dust grains. We consider the equations of continuity and motion for each of the four species, together with Maxwell's equations and the gravitational Poisson's equation. Linearization and Fourier transforming leads to a complex dispersion law, similar to that presented in [3]. Differences in the tensor elements from that work arise because (a) we have included an additional (neutral) dust species, (b) we have ignored beams, and (c) we have allowed for finite dust temperature. We first consider a three-fiuid system, ignoring neutral dust. An important parameter is the normahzed dust lower hybrid frequency L = cJdih/^d — (^Jd/^Ji, which also represents the relative contributions to self-gravitation of the dust and the plasma. If most of the mass is concentrated in the dust, the more common situation cosmologically, then L 3> 1. In terms of the normalized frequency Q = cj/H^, the dispersion law for higher frequency waves with u ^ Cld^^dih is approximately n^ ^ (k'^V^s/^'d) + ^^ P]- The stable modified dust magnetosonic wave is decoupled from the Jeans mode at these frequencies. Numerical evaluation of the full tensor dispersion law has confirmed this form, for instance for L = 100. If, with other parameters constant, one reduces the dust density ratio, Nd/Ni, L decreases. Examination of the curves reveals empirically that fi -^ L^ + 1 as A: ^ 0, and the dispersion curve is well-represented by (fi — 1)^ = k'^V^^/Q,'^ + L"^. This effect is clear for L < 10, and an example is shown in Figure 1, for L = 2. With lighter dust grains (m^ = 1.67 x I0~'^^kg) Figure 2 shows a typical low frequency stable X-mode with Cld < ^ < ^dih) which tends to the dust lower hybrid frequency (L = 10).
M.A. Hellberg et al. /Coupling between Jeans and X-modes
553
a a
kpd(10-5)
kpd (10-^) Fig. 1: Standard parameter values : 5^ =3 x 10'
Fig. 2: Plot of Q vs. kpj for L = 10, /w^/wy =10^
T,
A^^/A^/=10"'^.
Ti = 10^ K, 7-^=10^ A:, r^= 30 K, Nd= 10"^ w"^, Z^ =10'*e, m^ = 4 X 10
kg{\ \im water ice). Plot
-12 of Q = coj. /Q^j vs. kp^ for L = 2, N^/N-= 1.7 x 10
o
o
—1
1.842
1.844
1.846
1.848
1.850
0.95
0.96
0.97
1—
0.98
0.99
1.00
kpd(io-9) Fig. 3: Plot of n vs. kp^ for L = 5000, N^/N^ = lO"
Fig. 4 : Plot of k^j.j^P(i vs. a for L = 5000.
554
M.A. Hellberg et al /Coupling between Jeans and X-modes
For waves with uo < Vt^^ the coupHng of the Jeans mode with the magnetosonic wave in the Alfven-Jeans mode may be pronounced. One finds a cutoff in k, below which the roots of the dispersion law are imaginary. This reflects the effect of selfgravitation leading to the Jeans instability [2-4]. Depending on parameter values, kcrit niay be governed mainly by the dust or the plasma component. An example (L = 5000) is shown in Figure 3. We see that at higher values of k, and hence for n > 1, the curve goes over into the usual Alfven mode. It is interesting to note that for L = 0.01, similar behaviour is observed, even although the typical speed is then dominated by the plasma component. Next we include a component of neutral dust particles with the same mass as the charged dust. This increases the self-gravitational instability while not changing the electromagnetic inhibition of the instabihty. We vary the ratio a of the charged fraction of the dust particles, and from dispersion curves of the type shown in Figure 3, explore the variation of kcHt- Figure 4 confirms the analytical prediction [4], underlining the importance of the neutral dust component: kcritPd falls by several orders close to a = 1, but does not —> 0. Results for L = 10 are indistinguishable.
CONCLUSION We have generalized our model [3] to include a neutral dust component and finite temperature effects, and evaluated the full dispersion law numerically. In addition to confirming the analytical results reported earlier, it has been possible to extend studies to the region cj ~ Q^ which is not accessible analytically. Dispersion curves reveal the transition from the Jeans unstable regime to the Alfven-magnetosonic waves. We have confirmed numerically the significant role of a neutral dust component in diminishing the stabilizing effect of charged dust.
Acknowledgements This work was supported by the Flemish Government (Department of Science and Technology) and the (South African) Foundation for Research Development in the framework of the Flemish-South African Bilateral Scientific and Technological Cooperation on the Physics of Waves in Dusty, Solar and Space Plasmas.
REFERENCES 1. Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford: Clarendon Press, 1961, pp. 588-595. 2. Mace, R..L., Verheest, F. and Hellberg, M.A., Phys. Lett. A 237, 146-151 (1998). 3. Verheest, F., Meuris, P., Mace, R.L. and Hellberg, M.A., Astrophys. Space Sci. 254, 253-267 (1997). 4. Verheest, F., Hellberg, M.A. and Mace, R.L., Phys. Plasmas 6, 279-284 (1999).
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
555
Fluctuation Electrodynamics of Dusty Plasmas V.P.Kubaichuk, A.G.Zagorodny Bogolyubov Institute for Theoretical Physics^ 252143 Kiev 143, Ukraine
Abstract. Basic principles of fluctuation electrodynamics in dusty plasmas are studied for the case of grain charging by plasma currents taking into account the dynamics of grain charging associated with electron and ion absorption by grains. Both dielectric response functions and correlation functions of the Langevin sources are calculated in terms of the transition probability matrix of a system without self-consistent electric fields. General relations for the correlation functions of electron density fluctuations are derived and detailed numerical analysis of their spectral distributions is performed. The fluctuation spectra are shown to be considerably influenced by the self-consistent grain charging dynamics.
BASIC SET OF EQUATIONS We start from the equations for the fluctuation of the plasma particle distribution function, Sfa{T, v, i), and the grain density fluctuation, 6fg{r^ v, t), which have been obtained in a manner, similar to the analysis in Refs. [1-3]. We have, respectively,
{a«+^a;}*^'(^'"-^^**(''') -u^,^UX,t)
dv
- {5ul + M ) / . o ( v ) ,
(1)
v^here z/^^ = rigvaa{qeq,v), 5vl = rigV^^^^^Sq, 5u^ = Sngvaa{qeq,v), Ga is the charging cross-section for dust prticles, and q^q is the eqilibrium grain charge [1]. When deriving the right-hand part of Eq. (1), v^e did not take into account the plasma regeneration. This is v^hy Eq. (1) and the relevant equation used in Refs. [4,5] do not coincide. Obviously, Eq. (1) is valid for fluctuations v^ith correlation times smaller than the plasma regeneration time. The fluctuation potential 5$(r,t) is governed by the Poisson equation
556
VP Kubaichuk, A.G. Zagorodny / Fluctuation electrodynamics of dusty plasmas
A<J$(r,i) = -47r Y. ^PA^^t),
(3)
a=e,i,q
where
5p^(r,i) = e^n^ / d\5f(j{X,t),
a = e,z, (4)
6pg{r,t) = egng I dv5fg{X,t)+ng6q{r,t). and Sq{r,t) is the fluctuation of the grain charge (see, for example Ref. [2]). ELECTRIC FIELD AND PARTICLE DENSITY FLUCTUATIONS
In the koj-representation, the self-consistent solution of Eq. (3) is given by
where
e{k,u) = l+ ^ XcT{k,oj)= -
52
c',a"=e,i,g
X
a=e,i,g
^ "^2" ^ ^^
'^<^''^
^
^
W;T'
{ 1 - Sag) Saa' + h' ( v ,
/.'(v,a;) = V + (1 - V) —
d\'R„^{\\uj) X
a = e,i,g;
(6)
Uj)6ag,
^JTi^'
^^^
' w — kv + iua*^ a; — kv + iua^ {g{k,u){ui + iv^ r eo.-n^/g,r'(^0 Y^ ^^ f dvv'^fQan{v)a„n{v)5atg 1 agSgig 1 Lw - kv + ii/^? J^e.i " -^ ('^ ~ kv + ii^^»)(a; - kv + iO)\ u —"kv+ iuy
Similarly,
(5p<,k(^ = J ]
I ^c<^'
"n ' \ I ^^PSL >
(9)
VP. Kubaichuk, A.G. Zagorodny / Fluctuation electrodynamics of dusty plasmas
557
where ^p^k^; is the spontaneous ("source") part of the charge density fluctuations. The next step is to calculate the correlation functions of the Langevin sources ^Pcrkcj- "The final result is given by 0-10-2
J
J
X [n,. Joa2(v')W"^ic72ka;(v, v ' ) + U^Joa, MW*^arkui'^'^)]
•
(10)
It should be pointed out that, in the case of a dusty plasma, there occur crosscorrelations between the Langevin sources associated with various particle species. Eqs. (5) - (9) make it possible to calculate the correlation functions of any fluctuation quantities. For example,
iSp.S,,.)^ = E (^.. - m )
(*.. - ^
)
(Vff*«>.„.
(11)
NUMERICAL ANALYSIS Eqs. (5) - (11) describe the fluctuation spectra for arbitrary values of plasma parameters. In order to illustrate the eff'ect of charging dynamics on fluctuations, in what follows we put rrig = oo, i.e., we neglect the fluctuations produced by the random grain motion. Below we present the calculated results for the electron density fluctuation spectrum in terms of arbitrary units S'(k,a;) = (^?^e)ka;(^pe/^€) as a function of the dimensionless frequency u/upi^ the wavenumber k/ke, {kl = ATrelue/Te), and the grain density A^^ = {TTUga^)/{keaY (a is the grain radius). Both cases of isothermal and nonisothermal plasmas were studied in the low-frequency range (cj ^ Upi) where the contribution of charging processes is likely to be the most important. The value of the grain charge Cg was calculated from the quasineutrality and zero-charging-current conditions, i.e., within the equation \ltme
t+Z ^
*
^^
'
ZiTe'
Figs. 1-4 show the a;, A^^-dependence of fluctuation spectra for given values of wavenumbers and ion to electron temperature ratio. Figs. 1-3 are related to the case k/ke = 0.1 for i = 1, t = 0.5, and t = 0.2, respectively. Fig. 4 illustrates the case A;/A:e = 0.01,^ = 0.2. As follows from the Figures, the fluctuation spectra are very sensitive to the grain concentration. The most important features of such influence are the existence of additional ion-sound resonance (Figs. 2-4) and the critical dependence of spectra on the quantyty Ng at zero frequency (Figs. 1,2). The critical value of A^^ decreases with the decrease of the wavenumber k and the temperature ratio Ti/Te.
558
KP. Kubaichuk, A.G. Zagorodny / Fluctuation electrodynamics of dusty plasmas
S(k, CO)
0.05 0.2
'^--Cr^:u.-'^.0.02
F I G U R E 1.
F I G U R E 2.
F I G U R E 3.
F I G U R E 4.
A_
ACKNOWLEDGMENTS This work was funded in part by the INTAS grant 96-0617 and the projects of the State Fund of Fundamental Research of Ukraine 24/319 and 24/688.
REFERENCES 1. Tsytovich V.N., Havnes 0.,Comm.Plasma Phys.Contr. FusionlA, 267 (1993) 2. Sitenko A.G., Zagorodny A.G., Tsytovich V.N., In: Proc.Int.Conf. on Plasma Physics, ICPP 1994, AIP Conf.Proc, 345, 311 (1995) 3. Sitenko A.G., Zagorodny A.G., Chutov Yu.N., Sdiram P., and Tsytovich V.N., Plasma Phys. Contr. Fusion 38, A105 (1996) 4. Li F., Tsytovich V.N., Phys.Rev. E , 53(1), 1028 (1996) 5. Li F., In: Advances in Dusty Plasmas. Proc.of Intern.Conf.Phys.Dusty Plasmas, Singapore-NewJersey-London-Hong Kong, World Scientific, 1997, p.539
FRONTIERS IN DUSTY PLASMAS Y. Nakamura, T. Yokota and RK. Shukla (eds.) © 2000 Elsevier Science B.V All rights reserved
559
Measurement and Modeling of Particle Spacing in Strongly Coupled Dusty Plasmas^ M. S. Murillo* and H. R. Snyder t * Plasma Physics Applications Group (X-PA), MS B259 ^Plasma Physics Group (P-24), MS E526 Los Alamos National Laboratory, Los Alamos, NM 87545
A b s t r a c t . Vertical strings of dust grains were studied using a two-dimensional array of grains in a rf Ar plasma in a two-phase experiment. First, the sheath was characterized by fitting a sheath model to single grain behavior and Langmuir probe measurements. Then, the intergrain spacing was measured over a range of powers for spherical glass grains of diameter 10.4/x77i. A one-dimensional energy model for the arrays which includes the sheath model, gravity, and all intergrain interactions is compared with the data. Various asymmetric potentials were employed to account for possible dipolar effects. It is shown that the bulk plasma temperature greatly exceeds the effective temperature (defined in terms of the charging equation applicable in the bulk plasma) in the sheath and that existing interaction potential models do not agree with the data.
I
INTRODUCTION
Dusty plasmas are plasmas that contain impurities of very massive solid grains. In a typical laboratory rf discharge these particles axe micron-size and hence about ~10^^ times more massive than the ions in the discharge. The grains acquire a very large negative charge (~10^ times the electronic charge), allowing them to be suspended against gravity in the plasma sheath above the lower electrode. Moreover, the temperature of the dust grains is also relatively low ( < 1 eV), so that large values of the Coulomb coupling constant F = Q^/vgT are achieved. As a consequence, the grains tend to form into crystaUine structures and exhibit phase transitions, characteristic of strongly coupled Coulomb systems. [1] Experiments indicate that strongly coupled grains do form crystal structures, but these structures are inconsistent with structures predicted by standard strong coupling theories. [2-4] Rather than forming a bcc or fee lattice, the grains are observed to be vertically aUgned above the electrode and, as a consequence, form ^) This research was supported by the Los Alamos National Laboratory Directed Research and Development (LDRD) program.
560
M.S. Murillo, H.R. Snyder/Particle spacing in strongly coupled dusty plasmas
a hexagonal lattice. [5-9] The alignment process can be modeled in molecular dynamics simulations [10] as a dipole force added to the usual screened Coulomb interaction between the charged dust grains. The simulations [10] show that various types of crystalline structure are possible, depending on the strength of the dipole, consistent with a phase diagram analysis by Lee et al.. [11] Moreover, the dipole model appears to be consistent with recent experimental observations [12]. The dipole may result from either the wake formed by the flow of ions through the sheath toward the electrode, [13,14] asymmetrically grain charging from the flow, [15] or the charge on the grain can also be redistributed by the electric field in the sheath, [12] generating an intrinsic dipole moment.
II
EXPERIMENT AND MODEL
We have performed experiments with one-dimensional Coulomb crystals similar to those seen previously, [12] but with intrinsic dipole moments three orders of magnitude smaller. To eliminate experimental uncertainties we use preformed spherical grains of known diameter and mass density and perform the experiment in two phases. First, we measure positions of individual leviated dust grains to characterize the sheath parameters and determine the grain charge. Subsequently, we measure the spacings between grains in vertical strings and fit to a simple theoretical model in order to determine the effective magnitude of the dipole. The dusty plasma was produced by applying a 13.55 MHz rf voltage to an Al electrode in Ar gas. Vertical strings were produced by introducing an Al plate with a narrow groove to trap grains. Although the trap formed a two-dimensional axray of dust grains, the grains were seen to form independent, vertical one-dimensional crystals. A laser was used to illuminate the particles and a video detection system imaged the particles. Before introducing plasma into the chamber a ruler was first placed in the field of view at the intended location of the particles resulting in an accurate distance per pixel calibration of the video system. In the first phase of the experiment, very dilute grains were used to probe the sheath environment. The sheath electron density Tig is assumed to be the same as in the plasma and was measured using a Langmuir probe. This assumption is partly justified since the grains leviate near the sheath edge where the electron density is near the bulk density, although in general the electron density is lower in the sheath. However, we do not assume the sheath temperature Tg is equal to the plasma temperature Tp^ although temperature measurements were made in the plasma. Rather, we define an effective sheath temperature Teff by assuming the force balance on a single grain is given by Mg = -2000rdTeffeE{z).
(1)
The sheath is assumed to produce a Unear field of the form E{z) = —Eo{l — Z/ZQ)^ where EQ and ZQ are parameters fit to the data and E{z) = 0 for z > ZQ. TO obtain EQ we integrate E{z) from the electrode (taken to be at
M.S. Murillo, H.R. Snyder/Particle spacing in strongly coupled dusty plasmas
561
TABLE 1. Experimental data for characterizing the sheath parameters. Eo{V/cm)
P{W)
267 350 371 425
75 100 125 150
ZQ{rn'rn) TeffieV) 3.79 0.70 3.69 0.63 3.69 0.58 3.56 0.60
Zeff 7313 6548 6001 6289
ne{cm ^) T,{eV) 2.09-lO^ 2.6 2.19-10^ 3.43 3.52 2.28-10^ 3.75 2.36-10^
z = 0) into the plasma to obtain EQ = 2{Vp — Vo)/zo. Here VQ is the potential at the electrode and Vp is the plasma potential. To determine ZQ we separately suspend two grains of differing radii ri and 7*2 and measure their equilibrium heights Zi and Z2 above the electrode. A trap was not used during this part of the experiment. The force balance equations, ignoring ion drag forces, for the two grains are Mig = 2000riTseEo ( l " - ) .
(2)
which yields, by taking the ratio of Eqn. (2) for two grains i = 1 and i = 2, zo
T]ZI
r\Z2
(3)
rr - n
for grains of equal composition and shape. The sheath model is now chaxacterized and the effective sheath temperature can now be found with Eqn. (1). This is equivalent to defining the magnitude of the effective grain charge as ^e//
Mg eEo{l -
(4)
z/zo)'
The results for a lOAfim grain are shown in Table 1. Note the difference in the effective sheath temperature and the plasma temperature. Clearly the effective temperature required to levitate a single grain is much less than the actual plasma temperature. This indicates a failure of the charging equation used in Eqn. (1) and may result from nonuniform charging throughout the rf cycle and/or significant violation of the quasineutraUty condition assumed in the usual charging equation. We model the dust structures as one-dimensional lattices with constant lattice spacing a and height b above the electrode. The total energy of this system can be written as N U{a, b)^Y:
N MgZn
+ E
N UnmiZnm)
n<.m
71=1
+ E n=l
^ ^ ^
' Zo " + ' '"
Zo
2z„
(5)
where Zn = b+ {n- l)a and Znm = Zn- Zm- The first term in Eqn. (5) is the total gravitational energy of the system, the second term is the sum over all pair-wise interactions between grains, which we separate as ^nmyZri')
^m)
Q'
e x p ( - | z „ „ | / A e ) + C/i^^)(^„,z„).
(6)
562
M.S. Murillo, H.R. Snyder/Particle spacing in strongly coupled dusty plasmas 0.5
f M=1.2-^ A M=1.3 --h data -^
0.4
?
V
^--^... ^^-----
£
J
-*-,
"^0.3
0 z
o <0.2 CO
o.u 60
80
100
120
140
160
180
POWER (W) F I G U R E 1. Grain spacing versus delivered power. The data is shown (top curve) with theoretical predictions. The mach numbers used were 1.2 (cross) and 1.3 (plus).
Here U!^"^ contains "nonideal" plasma interactions such as induced dipoles, nonuniform charging, ion wakes, etc. The final term is the energy associated with the sheath field, with the boundary condition of zero potential energy at the sheath edge ZQ. The number of grains N is chosen to agree with that seen in the experiment and the energy ?7(a, 6) is minimized with respect to a and 6 for various forms of U!^^^\zr,,Zm). The results are shown in the figure for various values of the mach number. All theoretical curves include an induced dipole, which is of no consequence. In conclusion, we have performed an experiment that isolates the interaction of two grains in the vertical direction. To minimize experimental uncertainty we have used spherical grains, of a known composition and size, and employed a sheath model. A two-step method to estimate the charge without resorting to a charging equation, that may not be valid in the sheath, was used. However, we did assume the electron density in the sheath is approximately that of the bulk and that the charge on the grain scales Unearly with the radius. A minimum-energy model with various effective interactions then gave predictions of the particle spacings. For the small grains we used, ~10.4//?n, the induced dipole interaction was found to be negligible. Only with a wake interaction did the predictions approximate the data, although the agreement was less than satisfactory. In the future we will employ improved wake potentials and include the mach number self-consistently.
M.S. Murillo, H.R. Snyder/Particle spacing in strongly coupled dusty plasmas
563
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
H. Ikezi, Phys. Fluids 29{6), 1764 (1986). W. L. Slattery, G. D. Doolen, and H. E. DeWitt, Phys. Rev. A 21(6), 2087 (1980). M. O. Robbins, K. Kremer, and G. S. Grest, J. Chem. Phys. 88(5), 3286 (1988). S. Hamaguchi, R. T. Farouki, and D. H. E. Dubin, Phys. Rev. J? 56(4), 4671 (1997). J. H. Chu and Lin I, Phys. Rev. Lett. 72(25), 4009 (1994). H. Thomas, G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher, and D. Mohlmann, Phys. Rev. Lett. 73(5), 652 (1994). A. Melzer, A. Homann, and A. Piel, Phys. Rev. £^53(3), 2757 (1996). J. B. Pieper, J. Goree, and R. A. Quinn, J. Vac. Sci. Technol. 14(2), 519 (1996). S. Nunomura, N. Ohno, and S. Takamura, Phys. Plasmas 5(10), 3517 (1998). J. E. Hammerberg, B. L. Holian, M. S. Murillo, and D. Winske, Physics of Dusty Plasmas, AIP CP446, 257 (1998). H. C. Lee, D. Y. Chen, and B. Rosenstein, Phys. Rev. £^56(4), 4596 (1997). U. Mohideen, H. U. Rahman, M. A. Smith, M. Rosenberg, and D. A. Mendis, Phys. Rev. Lett. 81(2) 349 (1998). F. Melandso and J. Goree, J. Vac. Sci. Technol. 14(2), 511 (1996). O. Ishihara and S. V. Vladimirov, Phys. Plasmas 4(1) 69 (1997). G. Lapenta, Phys. Rev. Lett. 75(24), 4409 (1995).
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