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Basics of dusty plasma

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DOI: 10.1134/1.1856707

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Plasma Physics Reports, Vol. 31, No. 1, 2005, pp. 46–56. Translated from Fizika Plazmy, Vol. 31, No. 1, 2005, pp. 52–63.
Original Russian Text Copyright © 2005 by Ignatov.

DUSTY
PLASMA

Basics of Dusty Plasma

A. M. Ignatov
Prokhorov Institute of General Physics, Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991 Russia
Received June 30, 2004

Abstract—The paper presents an introductory review of the basic physical processes in dusty plasmas. The
topics to be addressed are dust charging, forces acting on dust grains, interaction between dust grains, and dust–
plasma structures. © 2005 Pleiades Publishing, Inc.

1. INTRODUCTION containing aerosols. A medium composed of electrons,

ions, and neutral atoms is called, for simplicity, a pure
Dusty plasma (also called complex, colloidal, or
aerosol plasma) is an ordinary plasma contaminated plasma. The more formal term “complex plasma” is
with a certain amount of condensed (solid or liquid) regarded as a synonym for dusty plasma. Aerosol par-
particulates (grains). To start with, such a definition is ticulates are commonly called dust grains. When speak-
quite sufficient because there is currently no exact def- ing about colloidal plasma, we imply just water colloi-
inition of plasma in general. The main objective of the dal solutions, which are not addressed here.
present review is to provide a clearer insight to some
aspects of the problem. Tremendous advances in dusty plasma physics were
made in the late 1980s and were triggered mainly by the
Aerosols in gas discharges were observed under lab- needs of plasma surface treatment, where the formation
oratory conditions as early as the beginning of the of dust plays a major and, as a rule, harmful role. On the
twentieth century. In 1912, A. F. Joffe measured the ele-
other hand, synthetic dust grains produced in plasma
mentary charge with the help of metallic grains that
were synthesized in a gas discharge and acquired an allow one to create new materials with numerous bene-
electric charge. The grains then fell down into a capac- ficial properties. From the physical standpoint, the most
itor with a dc electric field counterbalancing gravity so stimulating discovery was the theoretical prediction of
that each grain steadily held its position over a long dust condensation and, in particular, the formation of
period. This experimental setup anticipated many mod- Coulomb crystals composed of dust grains. Experimen-
ern installations. tal observations of dust–plasma crystals have given
Astronomical observations of various objects such impetus to thorough investigations on this subject.
as nebulae or Saturn’s rings had began much earlier. We Since the early 1990s, the number of papers on
now realize that these are also examples of dusty dusty plasma has grown exponentially without any ten-
plasma. However, dusty plasma physics is a very young dency to saturation. A few thousand papers on this sub-
science. When the first issue of Plasma Physics Reports
was published, the term “dusty plasma” had not yet ject are now published annually. It seems that it is time
come into use. Although astrophysicists always under- to strike a balance. This is why many review papers
stood the importance of dust grains in space, only a few have been written during the last few years. Plasma
papers on this subject were published annually. Physics Reports has recently published an encompass-
ing series of articles on dusty plasmas [1–4]. The basic
About thirty years ago, the first experiments on
plasma with a condensed disperse phase (CDP) were physical processes are discussed in the reviews [5, 6]
performed. Such a plasma is actually a hot flame with and in the monograph [7]. Processes in chemically
an admixture of aerosol particles emitting electrons. In active dusty plasmas are addressed in [8–11]. Astro-
addition, water solutions of charged colloids have been physical aspects are discussed in [12], and the state of
investigated for a long time. Many phenomena in CDP the art in colloidal plasma physics is described in
plasmas and in colloidal solutions have analogues in [13, 14].
gas-discharge dusty plasma, so it is sometimes pro-
posed that a single term to unify all these physical sys- In the present paper, I try to briefly describe a few
tems be used. For example, the terms “colloidal” or physical processes occurring in dusty plasmas that are
“complex” are applied. However, there are important most important in my opinion. Considerations of space
distinctions between all these systems; therefore, in preclude a comprehensive account of the problem. The
what follows, the term “dusty plasma” will be used in list of references is minimal; a much more complete
its narrow sense to designate a low-temperature plasma bibliography may be found in the above reviews.

1063-780X/05/3101-0046$26.00 © 2005 Pleiades Publishing, Inc.

 BASICS OF DUSTY PLASMA 47

2. TYPICAL EXPERIMENT
Most of the dusty plasma experiments have been
performed with a discharge chamber like that sketched
in Fig. 1. The plasma is produced by applying a voltage
to electrodes 1 and 2. With an rf discharge, there may
be only one (lower) electrode. Various gases with pres-
sures varying over a very wide range are used. In esti- QE
mating the characteristic plasma parameters, we will
imply that the buffer gas is argon at a relatively low (<1
torr) pressure. There are also a lot of other means to mg
produce plasma; the details may be found in the
reviews cited above.
Dust grains may form spontaneously from the gas-
eous or plasma phase or appear due to sputtering of the
electrodes. The grains thus produced are polydisperse
and have very different dimensions and properties. Fig. 1. Sketch of the discharge chamber in which most of
ground-based dusty plasma experiments have been per-
Artificial grains with well-controlled dimensions are formed.
also often injected into plasma. The total number of
grains, in this case, may vary from one to a few tens of
thousands. Obviously, the physical conditions in the plasma
Grains with a size larger than a few microns can eas- bulk and in the sheath region are quite different and,
ily be observed by optical methods. The relatively slow accordingly, different processes may dominate in them.
grain motion may be recorded with a video camera. By As a result, dust structures formed in the plasma bulk
processing the video record, one can determine the differ from those formed in the sheath. This issue will
velocity and position of each grain. This yields unique discussed below in more detail.
information about the dust component as a whole. For
example, one may observe various structures formed by 3. DUST CHARGING
dust grains, study phase transitions, etc. However, mea-
suring the plasma parameters in the region occupied by The main difference between dusty plasmas and
dust is a much more difficult task; therefore, many aerosols in a neutral gas lies in the huge charges of the
experimental works deal only with estimated values of dust grains. A neutral grain placed in plasma acquires a
these parameters. negative charge since the electron flux onto its surface
exceeds the ion flux due to the higher electron mobility.
A distinctive feature of dusty plasma is that various The negative charge reduces the electron flux and
nonelectric forces can play a very important role in it. increases the flux of positive ions. The steady state is
In ground-based experiments, gravity dominates for achieved when the net electric current at the grain sur-
grains with a size larger than a few microns, so these face becomes zero,
grains fall. Near the lower electrode, the weight of rel-
atively light grains is counterbalanced by the electric I = Ie(Q) + Ii (Q) + … = 0, (1)
field and they gather in the electrode sheath (see Fig. 1). where Ie, i are the electron and ion currents, which
There are various ways to confine grains in the bulk depend on the grain charge Q, and the ellipsis stands for
of the plasma. First, Brownian motion is capable of sus- other currents (e.g., those of negative ions), which
pending submicron-size grains; however, it is rather sometimes are also of importance. This process is sim-
difficult to observe the motion of such grains by optical ilar to the charging of a floating probe; however, the
means. The second way is to use thermophoresis in a charge of the probe is of little interest for plasma diag-
neutral gas. Cooling the upper electrode (or/and heating nostics, while in dusty plasmas, the grain charge plays
the lower one) creates a heat flux in a neutral gas, which the major role.
supports grains in the bulk of the plasma. In a stratified In order to evaluate the equilibrium grain charge, we
dc discharge, potential wells in the central part of the have to solve Eq. (1); i.e., it is necessary to know the
plasma column are formed; this also results in the trap- explicit dependence of the currents on the grain charge.
ping of grains. Finally, the most radical way to get rid As applied to probes, this problem has been discussed
of the gravity force is to place the experimental cham- for a number of decades; nevertheless, comprehensive
ber on a ballistic missile or on an aircraft flying over a analytical theory of charging is still lacking. To esti-
parabolic trajectory or to perform the experiments mate the equilibrium grain charge in an isotropic
onboard the orbital space station. All these ways of plasma, the so-called orbital motion limited (OML)
embedding grains in the main plasma have been suc- model is often used. This approximation is based on the
cessfully used (in particular, in experiments performed following assumptions: First, collisions between elec-
onboard the International Space Station). trons, ions, and neutral atoms are ignored. Second, it is

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

48 IGNATOV

supposed that a charged plasma particle hitting the collisionless model, the potential well is empty. How-
grain is either absorbed by the grain or recombines on ever, even at an arbitrarily low collision frequency, the
its surface, thus producing a neutral atom. Finally, it is well is gradually filled with ions and the resulting
assumed that, if the laws of conservation of energy and potential distribution may substantially differ from that
angular momentum allow a particle to reach the grain given by the OML model. In particular, numerical sim-
surface, then it does reach this surface. This is enough ulations show that the equilibrium charge may be
to evaluate the currents at the grain surface: halved as compared to the OML model.
Plasma absorption by the grain surface leads to a
2 T e eϕ0 /T e specific distribution of the electric field around the
I e = – 4πea n e ------------
-e , (2)
2πm e grain. Since electrons and ions are absorbed, there
exists a converging plasma flow in the vicinity of the
Ti  grain and the plasma density perturbation δne, i at large
- 1 – --------0 .
2 eϕ
I i = 4πea n i ----------- (3) distances from the grain behaves as δne, i ~ 1/r2.
2πm i  Ti 
Accordingly, the electric potential also decreases in
Here, it is assumed that the plasma consists of electrons inverse proportion to the distance squared, ϕ(r) ~ 1/r2.
and one ion species; me, i , Te, i, and ne, i are the electron Therefore, due to plasma absorption, the grain electric
and ion masses, temperatures, and densities, respec- field penetrates into the ambient plasma to a depth
tively; and ϕ0 is the surface electric potential of a spher- much exceeding the Debye length.
ical grain of radius a. Substituting these expressions In order to evaluate the currents at the grain surface,
into Eq. (1), one may calculate the equilibrium surface the effective charging (or absorption) cross section is
potential ϕ0. However, knowing currents (2) and (3) is often introduced. In the OML model, this cross section
yet insufficient to evaluate the grain charge or, equiva- can be easily calculated for an arbitrary spherically
lently, the electric field on its surface; it is also neces- symmetric distribution of the electric potential, assum-
sary to know the self-consistent distribution of the elec- ing that all the plasma particles approaching the grain
tric potential around the grain. It was found that the center a distance smaller then a are absorbed:
capacitance of a sufficiently small grain (a
λDe ≡
σ α ( q, v ) = πa  1 – ----------------
2Qe α 
θ ( am α v – 2Qe α ). (4)
2 2
 2
2
T e /4πe n e ) in plasma is close to its vacuum value; am v α
i.e., Q = aϕ0. Moreover, a rigorous analysis shows that
the basic assumptions of the OML model are satisfied Here, α = e, i; v is the particle velocity at infinity; and
in this limit only. In evaluating currents (see expres- eα = ±e is the charge of a plasma particle. The Heavi-
sions (2), (3)), it was supposed that the grain charge is side step function θ applies here to the repulsive poten-
negative; i.e., it repels electrons and attracts ions. The tial only, i.e., to the plasma species some particles of
difference between expressions (2) and (3) is due to the which cannot reach the grain surface. The current den-
additional assumption that there are no trapped ions sity of the particles absorbed by the grain surface is
(i.e., ions with a negative total energy), which follow given by the integral
finite orbits around the grain. If, for some reason, the
grain charge becomes positive, then expressions (2) and ∫
j α = e α dvv σ α ( q, v ) f α ( v, r ), (5)
(3) should be interchanged. It should also be stressed where fα(v, r) is the distribution function of the parti-
that for the OML model of charging to be applicable,
the grain size should not be too small. We may regard a cles of the α species at the grain location. Evaluating
grain as a solid body and plasma as a continuous integral (5) with a Maxwellian distribution, one obtains
– 1/3
expressions (2) and (3). Absorption cross section (4) is
medium only if a  n e, i . Otherwise, the dust should also often used to evaluate the grain charge in more
be treated as an additional microparticle species. complicated anisotropic situations, e.g., when the
It follows from balance equation (1) and expressions plasma drifts relative to the grain. It should be stressed
(2) and (3) for the currents that the equilibrium charge that such an approach is groundless. In anisotropic
may be represented in the form Q = –zaTe /e, where the plasmas, we cannot assume that the potential distribu-
tion around the grain is spherically symmetric, and the
dimensionless coefficient z depends weakly on the analysis of the conservation laws underlying the OML
plasma parameters and usually ranges from 2 to 5. For model fails. The influence of the deviations from spher-
typical plasma parameters and micron-size grains, the ical symmetry on the dust charge has not yet been esti-
normalized grain charge Zd = |Q |/e may be as large as mated even in the case of weakly anisotropic plasmas.
Zd ≈ 104–105. It could be even larger; however, at Zd ≈ Thus, we may accept as a reasonable estimate that the
105–106, the negative electric field pressure at the grain grain charge is equal to Zd = zaTe /e2, where the factor z
surface becomes comparable to the ultimate strength of is on the order of unity and depends on the properties of
the grain material and the grain is destroyed. the ambient plasma; however, the accuracy with which
As was noted above, a negatively charged grain this factor is evaluated in various theoretical models
forms a potential well for positive ions. In the idealized should not be exaggerated.

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

 BASICS OF DUSTY PLASMA 49

It follows from the above estimates that a single Therefore, the grain charge must be considered as an
grain absorbs nearly all the electrons from a plasma additional inner degree of freedom.
region with a characteristic size of L0, defined by Besides the charging due to plasma absorption,
1/3
L 0 n e ~ Zd, i.e, L0 ~ ( aλ De ) . If a
λDe, then the size
3 2 there are also other mechanisms for dust charging. This
is, e.g., photoemission, which is often dominant under
of this region is much larger than the grain radius. space conditions and results in a positive grain charge.
Strictly speaking, it is only to this case that the dusty If the temperature of the grain surface is sufficiently
plasma concept is applicable. Otherwise, we are deal- high (as is the case with grains in flame), the thermal
ing with an absorbing body that changes the density of electron emission becomes efficient. The surface heat-
the surrounding plasma only near its surface. ing may also be caused by plasma recombination; in
For a sufficiently high dust concentration, when the this case, the equilibrium charge is determined by both
average distance between grains is about L0, a consid- the current and heat balance [16].
erable part of plasma electrons is absorbed by the dust. For comparison, we also mention the charging in
Since, on average, plasma is quasineutral, the condition colloidal suspensions. Colloidal particles in water
of the zero net charge, Zd nd + ne = ni , where nd is the acquire an electric charge due to electrochemical reac-
dust density, is satisfied. The relative dust concentration tions at their surface. Although the charge of colloidal
is conveniently characterized by the dimensionless particles (Zd ~ 100) is smaller than the grain charge in a
parameter P = Zd nd /ni , which indicates what fraction of gas-discharge plasma, colloidal plasma is a more non-
electrons is absorbed by the dust. Under experimental linear medium. The ratio eϕ0 /Te in a dusty plasma is on
conditions, P may be close to unity. The average elec- the order of unity, while in a colloidal plasma at room
tron density and the grain charge are then considerably temperature, the characteristic surface potential of a
reduced. This happens at a relatively small dust concen- particulate is about one volt, i.e., eϕ0 /T 1. Another
tration, nd /ni ~ 10–6–10–5. The difference between the important distinction that does not allow one to con-
electron and ion densities influences the dispersion of sider dusty and colloidal plasmas from a common
some plasma oscillations (e.g., Alfvén waves) whose viewpoint is that colloidal systems are in thermody-
frequency is much higher than the characteristic fre- namic equilibrium, while dusty plasmas are always far
quency of dust motion. from equilibrium.
The grain charge also varies over time. The rate of
charge relaxation toward its equilibrium value is deter-
mined by the derivative of current (1) with respect to 4. FORCES ACTING ON A GRAIN
the charge. If the charge is close to its equilibrium IN PLASMA
value, Q = Q0 + δQ, where I(Q0) = 0, then we have Unlike pure plasmas, there are many different forces
dδQ/dt = –νchδQ, where the charging frequency is νch = acting on a grain in dusty plasma. A characteristic effect
–I'(Q0). In the OML model, from Eqs. (2) and (3) we is the so-called ion drag. In low-pressure gas-discharge
obtain plasmas, surface recombination occurs predominantly
at the chamber wall; as a result, there are always
a z+1+τ
ν ch = ω pi ------- -------------------- , (6) directed plasma flows that drag dust grains. Evidently,
λ Di 2π interaction with ions plays the governing role. In the
central part of the discharge, the flow velocity may be
where τ = Ti /Te (as a rule, τ
1) and ωpi is the ion much smaller than the thermal velocity, while near the
plasma frequency. Under typical laboratory conditions, wall or the electrodes, it can exceed the ion sound
the charging frequency substantially exceeds the char- velocity.
acteristic frequency of dust motion, which is usually
about a few hertz. For this reason, the charge of a dust Ion drag arises due to both the ion absorption by a
grain moving through an inhomogeneous plasma is per- grain and the ion scattering in its electric field. Accord-
manently changing. This results, e.g., in the specific ingly, the ion wind force is a sum of two parts, the first
damping of dust waves. The character of the electric of which is related to the momentum flux of the ions
interaction between dust grains may also change [15]. absorbed by the grain and the second is related to the
momentum flux of the scattered ions. The net force act-
Fluctuations of the grain charge are of great impor- ing on the grain may be written as
tance. Since the charging is a discrete Markovian pro-
cess, the charge fluctuations are proportional to the
square root of the grain charge, i.e., 〈δZd〉2 ~ Zd. For suf- ∫
F d = m i dvvv f i ( v ) [ σ i ( v ) + σ s ( v ) ], (7)

ficiently small grains with Zd ~ 10, this leads to a ran- where σi (v) is the ion absorption (or collection) cross
dom change in the sign of the grain charge; this effect section given, e.g., by Eq. (4) and σs(v) is the scattering
plays an important role in the coagulation and synthesis cross section. The latter may be evaluated analytically
of dust grains from the plasma phase. At larger values only for a Coulomb potential by cutting off the diver-
of Zd, the charge fluctuations correlate with the fluctua- gent integrals at large and small scales. Since the cut-
tions of the ambient plasma and the grain velocity. off parameters are somewhat arbitrary, the resulting

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

50 IGNATOV

drag force may vary by many times [17]. When the flow phoretic force is Fth = –8a2nnλ—Tn, where nn and Tn are
velocity u is much smaller than the thermal velocity, the the density and temperature of the neutral gas, respec-
drag force is written as tively, and λ is the mean free path. This force pushes the
grains toward a colder gas regions. Producing the
F d = a m i n i v Ti u ( K a + K s ),
2
(8) proper temperature gradient in the neutral gas, one may
where Ka and Ks are dimensionless coefficients corre- easily counterbalance the grain weight and hold the
sponding to absorbed and scattered ions, respectively. grains in the plasma bulk.
Under the conditions typical of a gas-discharge plasma, Experimentalists often use laser radiation to control
both coefficients in Eq. (8) are large, Ka, s 1. By using the motion of individual grains [21]. Using a laser
different approximations, different authors obtain dis- beam, one can push a grain in an ordered structure (e.g.,
similar results. For example, when the ion scattering in a crystal), thus exciting a sound wave. The light may
with impact parameters exceeding the Debye length is influence the grain motion via two mechanisms. First,
taken into account, it turns out that the scattering dom- the grain may drift toward the maximum of the electro-
inates, Ks Ka [17], while ignoring such scattering
magnetic field due to the ponderomotive force that acts
yields Ks ~ Ka. Although there are experimental indica-
across the laser beam. Second, when the grain is illumi-
tions in favor of the second alternative [18], the ques- nated by a laser, a force acting along the laser beam
tion about the drag force still remains open even for low
drift velocities. The situation with drift velocities appears. Although this force is usually attributed to the
exceeding the ion thermal velocity is much more inde- light pressure, in most cases the photophoresis pro-
terminate. Thus, the possibility of a negative friction vided by the radiometric force dominates. The latter
force directed opposite to the ion flow was discussed in arises because the grain surface is headed nonuniformly
[19] and was recently confirmed by numerical simula- by the laser and the neutral gas pressure at the hotter
tions [20]. side is larger than at the colder one.
In this context, it is worth returning to the discussion Since both thermophoresis and photophoresis are
of the grain charge. As was already pointed out, dust provided by the heat exchange between the grain sur-
grains in a low-temperature plasma acquire a negative face and the ambient medium, the presence of plasma
charge. It is sometimes reasonable to regard the grain may drastically influence these processes. In an aniso-
and the accompanying perturbation of the ambient tropic medium, the heat flux Φ at the grain surface can
plasma as a whole, i.e., as a quasi-atom. Due to be represented as Φ = 〈Φ〉 + δΦ, where 〈Φ〉 is the heat
quasineutrality, the net charge of a quasi-atom is nearly flux averaged over the grain surface. In a steady state,
zero. When an external electric field is applied to the
the condition of the zero net flux, 〈Φ〉 = 0, determines
plasma, two forces act on the grain: the electric field
force, directed against the field, and the ion drag force, the average equilibrium temperature of the grain, while
directed along the field. It follows from the above esti- the anisotropic part δΦ is responsible for nonuniform
mates that the ion drag force is much larger than the heating. A dust grain exchanges its energy with both the
electric field force. In other words, the net force acting plasma and the neutral gas. With reasonable accuracy,
on a quasi-atom, i.e., the electrophoretic force, is we may suppose that, when a neutral atom hits the grain
directed along the external electric field. In physics, the surface, its energy is completely accommodated, i.e.,
charge is usually defined as a force-to-field ratio; if we the atom wastes its energy on the grain heating and
accept this definition, then we should consider dust leaves the grain with the energy corresponding to the
grains as positively charged objects. local surface temperature. An ion hitting the grain sur-
In low-temperature plasmas, the degree of ioniza- face recombines, and a considerable amount of energy
tion is small and the interaction of dust grains with a that is on the order of the ionization potential (10–
neutral gas influences their motion. Two processes are 20 eV) is transferred to heat. Moreover, due to the large
most important here. First, the friction on the neutral grain charge, the ions falling on the grain surface gain
buffer gas affects the grain motion. The friction force additional kinetic energy. All this increases the energy
may be approximately found from Eq. (8) by substitut- flux from the ion component by two or even four orders
ing Ka = Ks = 1 and replacing the ion density and the ion of magnitude. As a result, the average grain temperature
thermal velocity with the corresponding parameters of may achieve a few hundred degrees. The anisotropic
the neutral gas. Under typical laboratory conditions, the plasma heat flux may also result in the nonuniformity of
friction force is comparable to the grain weight; as a the surface temperature. For example, when the plasma
result, after several centimeters of free fall, the grain anisotropy is caused by the ion heat flux, the plasma
moves uniformly. thermophoretic force is directed along the ion tempera-
The thermophoretic force produced by the neutral ture gradient and a grain drifts to the hotter plasma
gas also can play an important role. In the free-molecu- region [22]. Seemingly, the presence of plasma may
lar regime, thermophoresis is caused by the heat flux also influence the photophoresis but, so far, this prob-
that distorts the atom velocity distribution. The thermo- lem has not been investigated.

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

 BASICS OF DUSTY PLASMA 51

5. INTERACTION BETWEEN GRAINS (a)

z
Above, we discussed various macroscopic forces
acting on a single grain. Analogous forces also result in
0
the interaction between grains. Two ultimate cases
should be distinguished. At a sufficiently low dust con-
centration, the dust has little effect on the parameters of
the ambient plasma. In this case, one can talk about –2
pairwise interactions between grains. In contrast, at a
high dust concentration, the dust determines the dis-
charge structure and the plasma parameters, thereby
resulting in collective interactions. –4
We start with pairwise interactions. Evidently, the
electrostatic interaction between grains is of great
importance. In an isotropic plasma, the electric field of
an external charge is screened on the spatial scale on the –6
order of the Debye length. This leads to the repulsion
interaction between grains at intermediate distances
between them. As was already mentioned above,
plasma absorption at the grain surfaces results in a –8
power-law radial profile of the grain potential. How- –4 –2 0 2 4
x/λDe
ever, this effect yields a rather weak interaction force,
so other (nonelectric) forces dominate at large inter- Φ, arb. units
grain distances. (b) 80
At intergrain distances smaller than the Debye
length and comparable to the grain size, the interaction 60
is more complicated. In this case, the grains can no
longer be considered as point charges and their electric
40
field is described by essentially nonlinear equations
that cannot be solved analytically. Moreover, the grain
charge depends on the distance to the neighboring 20
grain. It should be noted that there have been many
attempts to derive the attraction between like charges
due to the nonlinear electrostatic interaction. However, –15.0 –12.5 –10.0 –7.5 –5.0 –2.5
a rigorous analysis of the momentum balance in colli- –20 z/λDe
sionless plasmas shows that this is impossible [23].
Attraction may arise due to some other nonelectric
forces, e.g., those caused by plasma absorption by Fig. 2. (a) Two-dimensional distribution of the electric
grains. potential around a unit charge in the presence of an ion flow
In an anisotropic plasma, the situation is quite dif- with u/vs = 2 and (b) the z profile of the potential at x = 0.
ferent. Supersonic ion flows are always present near the
electrodes and the chamber wall. The dielectric permit-
tivity of a homogeneous plasma under these conditions which is often called ion focusing, results in interesting
has the form (ω, k) = 1 – ω Li /(ω + kz u – i0)2 +
2
features of the grain interaction. A bounded pair forms
when the second grain is placed in the wake excited by
1/ k λ De , where it assumed that the ions move along the
2 2
the first grain [24]. This interaction is asymmetric: if
u axis and their directed velocity u points downward. one push the upper grain with the help of a laser pulse,
Since the static dielectric function (0, k) may change the lower grain will follows the upper one. However,
its sign, a point charge at rest produces an ion sound the lower grain acts on the upper one through a
wake with a spatially alternating electric potential. Fig- screened Coulomb potential; therefore, when the lower
ure 2 shows a typical potential distribution around a grain is shifted, the position of the upper grain remains
unit charge located at z = 0 and x = 0; here, the darker unchanged. This elegant experiment demonstrates that
regions correspond to the lower electric potential. The the pairwise interaction provided by the ion focusing is
potential oscillates in the region located below the essentially nonpotential and, moreover, Newton’s third
charge and confined by the Mach cone. Outside the law is violated in this case.
Mach cone the electric potential decays exponentially In a homogeneous plasmas, grains situated at the
with increasing distance from the charge. same altitude repel one another. However, if they are
Downstream from a negatively charged grain a local placed near a conducting wall, the electrostatic images
maximum of the ion density is formed. This effect, of their Mach cones may provide attractive interaction

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

52 IGNATOV

Fig. 3. Shadow interaction between two dust grains. The ions with velocities lying within the shadow cones do not reach the grains.

1 2 3

Fig. 4. Shadow interaction among three dust grains.

[25, 26]. In this case, the interaction force between the temperature determines the grain charge and, accord-
grains is an oscillating function of distance. ingly, the net current of the absorbed ions. As was men-
tioned above, the electric potential of the grain behaves
In addition to the electric forces, the forces caused
by the ion wind also affects the grain interaction. As asymptotically as 1/r2 (i.e., the electric field decreases
was mentioned above, the plasma is absorbed by a grain with radius as E ~ 1/r3); hence, at large distances, the
and there is a converging plasma flow around it. Using electric repulsion changes with the shadow attraction.
Eq. (3), one can readily estimate the flow velocity far To the best of my knowledge, the attraction between
from the grain (r  λDe): vr ~ vs za2/r2, where vs = two isolated grains has not yet been observed experi-
mentally; however, experiments demonstrated the
T e /m i is the ion sound velocity. If another grain is attraction of grains toward more massive bodies. Thus,
placed a certain distance r from the first one, then it is in [28], a negatively biased wire was placed in a dusty
dragged by the ion wind force (8). This drag force is plasma. The grains placed near the wire were repelled,
estimated as F ~ zTe ni Ka4/r2, where K = Ka + Ks is the while the grains situated at larger distances were
net dimensionless coefficient in Eq. (8). Such an inter- attracted. This effect was interpreted as an attraction
action results in attraction between grains. The origin of caused by the ion wind, i.e., as the LeSage force.
this attraction is sketched schematically in Fig. 3. Since The shadow force exemplifies nonpairwise interac-
grains absorb the plasma, part of the ions moving tion. Let us suppose that there are three neighboring
toward, e.g., the left grain are trapped by the right one. grains (Fig. 4). The force acting on grain 1 is then inde-
As a result, the ion velocity distribution function at the pendent of the position of grain 3. The change in the
grain surface is zero inside a certain cone. This, in turn, plasma momentum flux that results in the shadow force
reduces the plasma pressure in the gap between grains. is provided by the second grain only, while the third
Attraction provided by plasma absorption is called grain is invisible from the surface of the first grain. This
the shadow force or the LeSage gravity, after the French example shows that the LeSage force may be screened
scientist who proposed a similar explanation of univer- by other grains.
sal gravitation in the 18th century [27]. Although the At a sufficiently high dust concentration, the
shadow force is caused by the redistribution of the ion shadow forces become collective. In the absence of
momentum flux, its magnitude is proportional to the dust, the transport processes in a discharge plasma are
electron temperature. The reason is that the electron mainly governed by the collisions of charged particles

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

 BASICS OF DUSTY PLASMA 53

with neutral atoms. The cross section for ion–dust col-

lisions is very large. Under typical experimental condi-
tions, cross section (4) for thermal ions is σi ~ 10–5–
10−4 cm2, which exceeds characteristic collisional cross
sections in pure plasmas by approximately ten orders of
magnitude. Therefore, even a moderate dust concentra-
tion is sufficient for the gas-discharge structure to be u
governed by the ion–dust collisions, rather than ion–
neutral or ion–ion collisions; in this case, the plasma
mainly recombines in the bulk of the discharge rather
than on the discharge chamber wall. The theory of grain
interaction [4] under these conditions takes into QE
account the plasma production by an external source,
the bulk recombination, and the electric forces. Accord-
ing to the theory, the characteristic spatial scale (analo-
gous to the Debye length in pure plasmas) above which
collective processes become important is Lcr = λ Di /aP.
2

A number of other mechanisms of grain interaction Fig. 5. Formation of a dust void.

have been discussed in the literature: the thermophore-
sis provided by the heat exchange between dust grains
and the neutral gas, the polarization forces analogous to standing striations of a dc discharge in [5]. As the dis-
van der Waals forces acting between neutral atoms, and charge current grows, the crystal melts and a phase
others. Under typical laboratory conditions, all these transition to a short-range order fluid state occurs. As
forces are much smaller than the electric or shadow the current grows further, a transition to a gaseous state
forces, but one cannot exclude that they may manifest is observed. An interesting feature of the gaseous state
themselves somewhere. is the anomalous heating of dust. The dust temperature
defined as the average kinetic energy of grains may
6. STRUCTURES IN DUSTY PLASMAS exceed the electron plasma temperature. The reasons
for this are not quite understood, and a number of theo-
As is well known, the degree to which a pure plasma ries has been proposed to explain this phenomenon.
is nonideal is characterized by the so-called coupling Anyway, the temperature difference between the dust
parameter Γ = U(n–1/3)/T, which is the ratio of the aver- and the ambient plasma shows that the traditional con-
age interparticle potential energy to the average kinetic cepts of Brownian motion are inapplicable to dusty
energy. For charge particles (with the charge eZ) inter- plasmas.
acting via the Coulomb potential, the coupling parame-
ter is equal to Γ = e2Z 2n1/3/T. A pure plasma is regarded Another interesting feature is the structural instabil-
as nonideal if Γ > 1; as a rule, the coupling parameter ity of dust in a gaseous state, as is depicted in Fig. 5. Let
for pure plasmas does not substantially exceed unity. us suppose that the gravity force is insignificant, the
The main difference between dusty and pure plas- plasma is produced by an external ionizer, and the dust
mas is the large variety of the interaction processes in homogeneously fills a spherical chamber. At a suffi-
the former. As was already pointed out, the interaction ciently high dust concentration, the plasma density is
between dust grains is often nonpotential and nonpair- determined by the balance between ionization and bulk
wise and, strictly speaking, there is no exact analogue recombination on dust grains. Let there accidentally
of the coupling parameter for dust. Nevertheless, the appear a region with a reduced dust concentration. The
coupling parameter is often used to characterize dusty plasma density in this region then increases, and the ion
plasma. It is usually introduced by assuming that the flows extruding the dust arise. This leads to a further
interaction between grains is described by the Debye– reduction in the dust concentration. However, the dust
2 2 cannot settle down at the chamber wall since the latter
Hückel or Yukawa potential, U(r) = e Z d exp(–r/λD)/r. is negatively charged. As a result, a peculiar plasma
For the dust grains in the plasma bulk, the predictions bubble (void) forms (see Fig. 5). The void is character-
of this model are in qualitative agreement with experi- ized by a sharp interface between the pure plasma and
mental data. Dusty plasmas are characterized by very the region occupied by the dust. When a grain is at the
large values of Γ (up to tens of thousands). For this rea- void boundary, the electric field force pushing it toward
son, the dust is often strongly correlated and the grains the center is counterbalanced by the ion drag force and
form various ordered structures. by the excess plasma pressure. The thickness of the
The grain ordering depends the plasma parameters transition layer is determined by the dust temperature,
and the method for the gain trapping. Three-dimen- which should be small, otherwise no void blows up.
sional crystalline dust structures were observed in The void formation is related to the universal instability

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

54 IGNATOV

z0 z1 z arrange in an (N – 1)-sided polygon. As the number of

E grains increases, more complex clusters consisting of
a set of concentric polygons are formed. This resem-
bles the filling of the electron shells in an atom and
2 there exist an analogue of the periodic table for dust
E0 clusters [29]. Investigating oscillations of dust clus-
u = vs
ters, one can get more precise information on the grain
interaction.
1 A large number of grains (N  1) arrange in a layer
with a hexagonal ordering; i.e., a two-dimensional
plasma crystal appears. Such a crystal has all the
attributes of a solid. For example, waves of different
polarization (and, accordingly, different dispersion)
Fig. 6. Electric field profile near a floating electrode (1) in
may propagate through the crystal. When the number of
the absence of dust and (2) in the presence of a dust layer grains reaches a certain critical value and the average
with a sufficiently high dust concentration. Here, E0 = distance between grains becomes small enough, the
−md g/Qd and the vertical dashed line shows the Bohm crystal melts. The second layer situated above the first
boundary of the electrode sheath, where the directed ion one then starts to grow. Curve 2 in Fig. 6 shows the field
velocity is u = vs. distribution near the electrode at a sufficiently large sur-
face charge density of the dust layer. It can be seen that,
near the dust layer, there are two additional equilibrium
of dusty plasma [4] and is eventually provided by the positions for a single grain. Only the upper position
above collective shadow force. (z = z1) is stable; it is this position where the second
Under the action of the Earth’s gravity, heavy grains layer starts to grow. By adding grains to the discharge
settle down near the lower electrode. Curve 1 in Fig. 6 chamber, it is possible to grow a multilayer dust crystal
schematically shows the spatial distribution of the elec- consisting of more than ten layers. Due to the above
tric field at a distance of a few Debye lengths from the discussed anisotropy of the electric forces in the sheath,
floating electrode. Qualitatively, electrons in the elec- the grains arrange in vertical chains, while in the hori-
trode sheath obey a Boltzmann distribution; i.e., the zontal plane, hexagonal symmetry is conserved.
electron density is inhomogeneous. Hence, the grain
In most models implemented so far to describe two-
charge also depends on the distance to the electrode. If
dimensional dust crystals, grains situated at the same
the mass md of a grain is not too large, there is an equi-
height were assumed to interact via the Debye–Hückel
librium height at which the electric field compensates potential. This assumption usually agrees with experi-
for the grain weight, md g + Qd(z0)E(z0) = 0. mental data. However, there have been experiments
As the number of grains of equal masses increases, indicating that some other forces may play an important
they gather at the same height. Ions drift toward the role in the sheath. In [30], a void in a two-dimensional
electrode at a velocity that is larger or comparable to dust layer was observed. The origin of this phenome-
the ion sound velocity. Under these conditions, the non is still unclear: it may be attributed to some external
electrostatic forces repel the grains from one another in influence or to an alternating interaction potential.
the horizontal direction (Fig. 2). Apparently, in most
experiments, there are no intergrain forces that might Until now, we have discussed various structures in
confine grains in a horizontal plane. In order to prevent dusty plasmas under the Earth’s gravity conditions.
grains from pushing out from the electrode, the latter Under microgravity conditions, a much more fascinat-
is made a little concave or, as is shown in Fig. 1, a con- ing picture is observed that has not yet been studied in
ducting ring is placed along the electrode perimeter. detail. The PKE-Nefedov facility (named in honor of
This forms a potential well that prevents grains from A.P. Nefedov, one of the pioneers and developers of the
flying away. project) has been operating onboard the International
Space Station since 2001. Although most of the results
By investigating the horizontal motion of a single obtained are still being processed, the first papers have
grain, one can measure the parameters of the potential already appeared (see, e.g., [5, 31, 32]). As in ground-
well and the friction force acting on the grain. With one based experiments, the central part of the plasma turned
grain placed at the bottom of the well, one can throw out to be free of dust. The dust boundary sometimes
another grain and follow the process of scattering. This performs periodic anharmonic oscillations resembling
allows one to find in the dependence of the intergrain the heartbeats of the central void. Closer to the elec-
force on the distance. Evidently, such experiments trodes, the dust may be in various phase states. Unlike
require extremely proficient technique. conditions on Earth, three different crystalline phases
A small number of grains (N ≤ 5) in equilibrium with pronounced interfaces between them have been
form a regular polygon in a horizontal plane. When N observed. Moreover, there are regions filled with dust in
= 6, 7, or 8, one grain settles in the center and others a liquid state. Finally, dust vortices are formed in spite

PLASMA PHYSICS REPORTS Vol. 31 No. 1 2005

 BASICS OF DUSTY PLASMA 55

of the absence of considerable plasma flows. Although ACKNOWLEDGMENTS

several theories have already been proposed to explain
the latter phenomenon, there is still no complete under- This work was supported in part by the Russian
standing of the vortex generation. Foundation for Basic Research (project no. 02-02-
16439), the Netherlands Organization for Scientific
Research (NWO) (grant no. 047.016.020), and the RF
7. CONCLUSIONS Presidential Program for State Support of Leading Sci-
entific Schools (project no. 1962.2003.2).
So, what is dusty plasma? The answer depends on
the level of description and on the kind of phenomena
with which we are dealing. In the simplest definition, it REFERENCES
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