Complex plasmas

Physics Reports 421 (2005) 1–103 www.elsevier.com/locate/physrep
Complex (dusty) plasmas: Current status, open issues, perspectives
V.E. Fortova, A.V. Ivlevb,∗, S.A. Khrapakb, A.G. Khrapaka, G.E. Morfillb
aInstitute for High Energy Densities, Russian Academy of Sciences, 125412 Moscow, Russia
bMax-Planck-Institut für Extraterrestrische Physik, D-85741 Garching, Germany
Accepted 30 August 2005
Available online 7 October 2005
editor: S. Cowley
Abstract
The field of complex (dusty) plasmas—low-temperature plasmas containing charged microparticles—is reviewed: The major
types of experimental complex plasmas are briefly discussed. Various elementary processes, including grain charging in different
regimes,interactionbetweenchargedparticles,andmomentum exchangebetweendifferentspeciesareinvestigated.Themajorforces
on microparticles and features of the particle dynamics in complex plasmas are highlighted. An overview of the wave properties in
different phase states, as well as recent results on the phase transitions between different crystalline and liquid states are presented.
Fluid behaviour of complex plasmas and the onset of cooperative phenomena are discussed. Properties of the magnetized complex
plasmas and plasmas with nonspherical particles are briefly mentioned. In conclusion, possible applications of complex plasmas,
interdisciplinary aspects, and perspectives are discussed.
© 2005 Elsevier B.V. All rights reserved.
PACS: 52.27.Lw
Keywords: Dusty plasmas; Complex plasmas
Contents
1. Introduction........................................................................................................... 3
2. Types of experimental complex plasmas ................................................................................... 4
2.1. Complex plasmas in rf discharges................................................................................... 4
2.1.1. Ground-based experiments .................................................................................. 4
2.1.2. Microgravity experiments ................................................................................... 4
2.2. Complex plasmas in dc discharges .................................................................................. 6
2.2.1. Ground-based experiments .................................................................................. 6
2.2.2. Microgravity experiments ................................................................................... 7
2.3. Other types of complex plasmas .................................................................................... 7
2.3.1. Complex plasmas at cryogenic temperatures.................................................................... 7
2.3.2. Thermal complex plasmas ................................................................................... 8
2.3.3. Nuclear-induced and track complex plasmas ................................................................... 9
3. Charging of particles in complex plasmas .................................................................................. 9
3.1. Charging in isotropic (bulk) plasmas ................................................................................ 9
∗Corresponding author.
E-mail address: [email protected] (A.V. Ivlev).
0370-1573/$-see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physrep.2005.08.007

2V.E. Fortov et al. / Physics Reports 421 (2005) 1–103
3.1.1. Motion of a particle in the central field ........................................................................ 9
3.1.2. Orbit motion limited (OML) approximation .................................................................... 12
3.1.3. Accuracy of the OML approximation ......................................................................... 14
3.1.4. Charging in a weakly collisional regime ....................................................................... 15
3.1.5. Charging in a strongly collisional regime ...................................................................... 18
3.1.6. Transitions between the charging regimes ...................................................................... 19
3.2. Charging in anisotropic plasmas .................................................................................... 19
3.3. Other effects important for particle charging .......................................................................... 23
3.4. Role of microparticles in plasma charge balance ...................................................................... 24
3.5. Fluctuations of the particle charge .................................................................................. 25
4. Electrostatic potential around a particle .................................................................................... 26
4.1. Isotropic plasmas ................................................................................................ 26
4.2. Anisotropic plasmas .............................................................................................. 28
5. Interaction between particles in complex plasmas ........................................................................... 30
5.1. Isotropic plasmas ................................................................................................ 30
5.2. Anisotropic plasmas .............................................................................................. 31
5.3. Experiments .................................................................................................... 31
6. Momentum exchange in complex plasmas ................................................................................. 32
6.1. Momentum transfer cross section ................................................................................... 33
6.2. Momentum exchange rates ........................................................................................ 36
6.2.1. Grain–electron collisions .................................................................................... 37
6.2.2. Grain–ion collisions ........................................................................................ 37
6.2.3. Grain–grain collisions ...................................................................................... 38
6.3. Momentum exchange diagram ..................................................................................... 38
7. Forces on particles in complex plasmas .................................................................................... 40
7.1. Ion drag force ................................................................................................... 41
7.1.1. Binary collision approach ................................................................................... 41
7.1.2. Kinetic approach........................................................................................... 42
7.1.3. Complementarity of the two approaches ....................................................................... 45
7.2. Other forces ..................................................................................................... 45
8. Dynamics of single particles and particle ensembles ......................................................................... 47
8.1. Single particle dynamics .......................................................................................... 47
8.2. Role of charge fluctuations in the particle dynamics.................................................................... 47
8.3. Dynamics of ensembles with spatially varying charges ................................................................. 49
8.4. Complex plasmas as non-Hamiltonian systems........................................................................ 49
9. Waves and instabilities in complex plasmas ................................................................................ 50
9.1. Wave excitation technique ......................................................................................... 50
9.2. Waves in ideal (gaseous) complex plasmas ........................................................................... 51
9.2.1. Major wave modes ......................................................................................... 52
9.2.2. Damping and instabilities of the DIA and DA mode ............................................................. 55
9.3. Waves in strongly coupled (liquid) complex plasmas ................................................................... 58
9.3.1. Longitudinal waves ........................................................................................ 58
9.3.2. Transverse waves .......................................................................................... 60
9.4. Waves in plasma crystals .......................................................................................... 60
9.4.1. One-dimensional string ..................................................................................... 60
9.4.2. Two-dimensional triangle lattice .............................................................................. 61
9.4.3. Three-dimensional plasma crystals............................................................................ 63
9.4.4. Instabilities in plasma crystals ............................................................................... 65
9.5. Nonlinear waves ................................................................................................. 66
9.5.1. Ion solitons and shocks ..................................................................................... 66
9.5.2. Dust solitons and shocks .................................................................................... 66
9.5.3. Mach cones ............................................................................................... 68
10. Phase transitions in strongly coupled complex plasmas ....................................................................... 69
10.1. Strong coupling of dust species..................................................................................... 69
10.2. Phase diagram of Debye–Hückel (Yukawa) systems ................................................................... 71
10.3. Experimental investigation of phase transitions in complex plasmas ...................................................... 72
10.3.1. Melting .................................................................................................. 73
10.3.2. Crystallization............................................................................................. 74
11. Fluid behaviour of complex plasmas ...................................................................................... 76
11.1. Transport properties .............................................................................................. 76
11.2. Hydrodynamic instabilities ........................................................................................ 77
12. Onset of cooperative phenomena in complex plasmas ........................................................................ 79

V.E. Fortov et al. / Physics Reports 421 (2005) 1–103 3
12.1. Coulomb clusters ................................................................................................ 79
12.2. Nanofluidics .................................................................................................... 83
13. Magnetized complex plasmas ............................................................................................ 83
14. Complex plasmas with nonspherical particles............................................................................... 86
15. Possible applications ................................................................................................... 88
16. Interdisciplinarity and perspectives ....................................................................................... 89
17. Conclusion ........................................................................................................... 91
Acknowledgements ........................................................................................................ 91
References ............................................................................................................... 91
1. Introduction
“Dusty”, or “complex” plasmas are plasma containing solid or liquid particles (dust) which are charged. The charges
can be negative or positive, depending on the charging mechanisms operating in the plasmas. Dust and dusty plasmas
are quite natural in space. They are present in planetary rings, comet tails, interplanetary and interstellar clouds [1–3],
found in the vicinity of artificial satellites and space stations [4,5], etc. Also, dusty plasmas are actively investigated
in laboratories. Currently, the term “complex plasmas” is widely used in the literature to distinguish dusty plasmas
specially “designed” for such investigations.
The presence of massive charged particles in complex plasmas is essential for the collective processes. Ensembles of
microparticles give rise to new very-low-frequency wave modes which represent the oscillations of particles against the
quasiequilibrium background of electrons and ions. Overall dynamical time scales associated with the dust component
are in the range 10–100Hz. The particles themselves are large enough to be visualized individually and, hence, their
motion can be easily tracked. This makes it possible to investigate phenomena occurring in complex plasmas at the
most fundamental kinetic level.
Micron size particles embedded in a plasma do not only change the charge composition, they also introduce new
physical processes into the system, e.g., effects associated with dissipation and plasma recombination on the particle
surface, variation of the particle charges, etc. These processes imply new mechanisms of the energy influx into the
system. Therefore, complex plasmas are a new type of non-Hamiltonian systems with the properties which can be
completely different from those of usual multicomponent plasmas.
Dust plays an exceptionally important role in technological plasma applications, associated with the utilization of
plasma deposition and etching technologies in microelectronics, as well as with production of thin films and nanopar-
ticles [6–8]. To control these processes, it is necessary to understand the basic mechanisms determining, e.g., transport
of dust particles, influence of dust on plasma parameters, etc.
Due to large charges carried by the grains (typically, of the order of thousand elementary charges for a micron-size
particle),theelectrostaticenergyofthemutualinteractionisremarkably high.Hence,the strongelectrostaticcoupling in
the dust subsystem can be achieved much more easily than in the electron–ion subsystem. In complex plasmas, one can
observe transitions from a disordered gaseous-like phase to a liquid-like phase and the formation of ordered structures
of dust particles—plasma crystals. The first experimental observation of the ordered (quasicrystalline) structures of
charged microparticles obtained in a modified Paul’s trap was reported in 1959 by Wuerker et al. [9]. The possibility of
dust subsystem crystallization in a nonequilibrium gas discharge plasma was predicted by Ikezi in 1986 [10]. The first
experimentalobservationsoforderedparticlestructures werereported in1994in rfdischarges[11–14].Lateron, plasma
crystals were found in dc discharges [15], thermal plasmas at atmospheric pressure [16], and even in nuclear-induced
dusty plasmas [17].
The enormous increase of interest in complex plasmas was triggered in the mid 1990’s, by the laboratory discovery
of plasma crystals. Today, the physics of complex plasmas is a rapidly growing field of research, which covers various
fundamentalaspects oftheplasma physics,hydrodynamics, kineticsofphase transitions,nonlinearphysics,solidstates,
as well as the industrial applications, engineering, and astrophysics. More and more research groups throughout the
world have become involved in the field, and the number of scientific publications is growing exponentially. In this
review, we have made an attempt to provide a balanced and consistent picture of the current status of the field, by
covering the latest development in the most important directions of the experimental and theoretical complex plasmas,
and outlined the perspective issues to pursue in future. We decided to omit the discussion of the space and atmospheric

4V.E. Fortov et al. / Physics Reports 421 (2005) 1–103
dusty plasmas, chemical plasmas with growing particles, etc. —their properties are so diverse and the whole field is so
huge that one cannot span it in the framework of a single review.
2. Types of experimental complex plasmas
Since the discovery of plasma crystals in 1994, the experimental investigations of complex plasmas have been going
on in dozens of laboratories throughout the world. In this section we briefly discuss the major types of the experimental
setups employed to study various phenomena occurring in complex plasmas.
2.1. Complex plasmas in rf discharges
2.1.1. Ground-based experiments
Plasma crystals were discovered in a capacitively coupled low-pressure rf discharge in inert gases almost simulta-
neously in several laboratories [11–14]. The schematic of a typical experimental setup [18] is shown in Fig. 1. The
experimental apparatus includes a lower electrode 5–10cm in diameter capacitively coupled to an rf generator (at
frequency 13.56MHz) and an upper grounded electrode. The electrodes are placed in a vacuum chamber. Micron-size
dust particles fill a container and can be introduced into the discharge through a metallic grid. The electrode separation
is typically 3–10cm. The particle visualization is performed with the use of laser illumination. The laser beam is
transformed into a sheet approximately 100m thick using a lens and illuminates the particles in the horizontal or
vertical plane. The particles introduced into the discharge become (highly) charged negatively and levitate in the pre-
sheath or sheath region above the lower electrode, where the electric force balances the gravity. To confine the particles
horizontally, a metallic ring with an inner diameter of 3–6cm and a height of 1–3mm is placed on the lower electrode.
Sometimes, instead of a ring, a special curved electrode is used. The light scattered by the particles is recorded by a
video camera.
Under certain conditions, the particles form ordered structures consisting of several fairly extended horizontal layers.
The number of particles in such crystalline structures can be as high as ∼104–105, and the number of layers can vary
from one to a few dozen [19]. Usually the particles assemble themselves into hexagonal structures in each layer.
Vertically, they often settle strictly below each other, forming a cubic lattice between neighbouring layers—the so-
called “vertically aligned” hexagonal lattices. Such an arrangement is believed to be a consequence of ion focusing
downstream from the particles—the wake effect (see Sections 5.2 and 10.3).
For reference, we summarize typical parameters of rf discharges and complex plasmas in laboratory experiments:
Particlesizesareintherange∼1–30m.Theneutral(typicallyinert)gaspressure∼1–100 Pa.Theplasmaconcentration
in the bulk of a discharge is ne∼ni∼107–1010 cm−3, the electron temperature is Te∼1–5eV, and ions are typically
assumed to be in equilibrium with neutrals (usually at room temperature), so that Ti∼Tn∼0.03eV. The sheath is a
region of a positive space charge where the ion concentration exceeds that of electrons even in the absence of dust. The
ion velocity at the edge of the collisionless sheath satisfies the Bohm criterion uiCIA ≡√Te/mi. For the collisional
sheath, the ion directed velocity can be lower than the velocity of sound, but the ions are still typically superthermal:
uivTi. The characteristic spatial scale of the charge screening in the sheath is given by the electron Debye radius De,
because fast ions do not contribute substantially to the screening. The interparticle interaction is essentially anisotropic
in this case.
2.1.2. Microgravity experiments
Complex plasmas formed under microgravity conditions have been intensively investigated in rf discharges during
the last few years. A typical sketch of the experimental setup is shown in Fig. 2. The results of the first sounding rocket
(TEXUS) experiments (with about 6 min of microgravity) and their qualitative analysis were reported by Morfill et al.
[20]. Currently, the “Plasma Crystal” (PKE-Nefedov) laboratory is in operation onboard the International Space Station
(ISS), created within the framework of the Russian–German scientific cooperation program [21]. The major tasks of
the PKE-Nefedov include investigations of dusty plasma crystals, phase transitions, wave phenomena, properties of
boundaries between different plasma regions, etc. in a three-dimensional isotropic dusty plasma at the kinetic level. The
first “basic” experiments designed to study the behaviour of the dust component over a broad range of dusty plasma

V.E. Fortov et al. / Physics Reports 421 (2005) 1–103 5
CCD video camera
Glass window
Grounded electrode
Sheet of laser light
RF electrode
Copper ring
Probe for
excitation
Up
Fig. 1. Sketch of typical experimental setup used for laboratory investigations of complex plasmas in a rf discharge [18].
Ground
Ground Dust dispenser
RF electrode
Insulator
RF
electrode
Plasma and particles
Field
of view
Glass
Glass
3.00 cm
4.20 cm
10.00 cm
Fig. 2. Sketch of the rf parallel plate discharge used in microgravity experiments [20]. Monodisperse particles of different sizes can be injected into
the plasma chamber from two dispensers mounted in the upper and lower electrodes. The microparticles are illuminated by a thin (about 150 m)
sheet of laser light perpendicular to the electrode system. Slow-speed scanning of the laser into the depth of the plasma chamber is used to measure
the 3D positions of the microparticles.
parameters were performed at the beginning of March 2001. Currently, most of the investigations are in the active phase
or in the stage of data analysis.
The typical static and dynamic behaviour of complex plasmas under microgravity conditions is illustrated in Fig. 3.
The figure shows the trajectories of microparticles. The dominant features are: (i) a microparticle free “void” in the
centre of the system for most experimental parameters, (ii) a sharp boundary between the void and the complex plasma,
(iii) demixing of complex plasma clouds formed by microparticles of different sizes, (iv) crystalline structures along
the central axis, and (v) torus-shaped vortices in different areas away from the central axis.
The microparticle-free centre between the electrodes—the void—can be explained by the balance of forces acting on
the particles in the discharge (see, e.g., [22–24]). Since the dominant force on earth— gravity—is reduced by orders of
magnitude the weaker forces start playing major role. These forces are the electrostatic force arising from the electric
field in the discharge and pointed toward the centre (plasma potential has a maximum in the centre), and the ion drag
force which pushes particles to the periphery. It is well established that even a relatively small number of particles
is repelled from the centre, suggesting that the void formation is not a collective particle effect. At present, the ion
drag is commonly believed to be the most probable mechanism responsible for the void formation (see, e.g., Ref. [25]
and references therein). The ion drag force and the electrostatic force have different dependencies on the grain size.
This causes demixing of different particle sizes: an equilibrium position of smaller particles is closer to the centre than
that of larger ones. The void can be closed under special experimental conditions. These conditions are neutral gas