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First Observation of Electrorheological Plasmas
A. V. Ivlev,
1
G. E. Morfill,
1
H. M. Thomas,
1
C. Ra
¨
th,
1
G. Joyce,
2
P. Huber,
1
R. Kompaneets,
1
V. E. Fortov,
3
A. M. Lipaev,
3
V. I. Molotkov,
3
T. Reiter,
4
M. Turin,
5
and P. Vinogradov
5
1
Max-Planck-Institut fu
¨
r extraterrestrische Physik, 85741 Garching, Germany
2
University of Maryland, College Park, Maryland 20742, USA
3
Institute for High Energy Densities, Russian Academy of Sciences, 125412 Moscow, Russia
4
European Astronaut Centre, 51147 Cologne, Germany
5
RSC Energia, 141070 Korolev, Russia
(Received 24 August 2007; published 6 March 2008)
We report the experimental discovery of ‘‘electrorheological (ER) complex plasmas,’’ where the control
of the interparticle interaction by an externally applied electric field is due to distortion of the Debye
spheres that surround microparticles (dust) in a plasma. We show that interactions in ER plasmas under
weak ac fields are mathematically equivalent to those in conventional ER fluids. Microgravity experi-
ments, as well as molecular dynamics simulations, show a phase transition from an isotropic to an
anisotropic (string) plasma state as the electric field is increased.
DOI: 10.1103/PhysRevLett.100.095003 PACS numbers: 52.27.Lw, 83.80.Gv
‘‘Conventional’’ electrorheological (ER) fluids consist
of suspensions of microparticles in (usually) nonconduct-
ing fluids with a different dielectric constant [1,2]. The
interparticle interaction, and hence the rheology of ER
fluids, is determined by an external electric field, which
polarizes grains and thus induces additional dipole-dipole
coupling. The electric field plays the role of a new degree
of freedom that allows us to ‘‘tune’’ the interaction be-
tween particles. This makes the phase diagram of ER fluids
remarkably diversified [3,4].
So far, colloidal suspensions have been the major focus
for ER studies, providing a wealth of information [1–6].
The discovery that complex plasmas also have electro-
rheological properties adds a new dimension to such re-
search—in terms of time or space scales and for studying
new phenomena: Ensembles of microparticles in complex
plasmas can act as an essentially single-species system
with very weak damping [7]. This is very different from
colloids [8] (it is a consequence of the fact that the neutral
gas density in complex plasmas is 10
6
–10
8
times smaller
than the fluid density in colloids). Therefore, complex ER
plasmas cover new physics and enable us to investigate
previously inaccessible rapid elementary processes that
govern the dynamical behavior of ER fluids—at the level
of individual particles. In particular, such investigations
may allow us to study critical phenomena accompanying
second-order phase transitions [9].
Laboratory complex plasmas are low-pressure gas-
discharge plasmas containing monodisperse microparticles
that are highly charged due to absorption of ambient elec-
trons and ions [10,11]. Complex plasmas are charge-
neutral and optically thin. In contrast to conventional ER
fluids (e.g., colloids) where the induced dipoles are due to
polarization of microparticles themselves, in complex plas-
mas the primary role is played by clouds of compensating
plasma charges (mostly, excessive ions) surrounding nega-
tively charged grains. A schematic illustration of the par-
ticle potential is shown in Fig. 1.
Quantitatively, the (field-induced) interparticle interac-
tion in ER plasmas can be determined from the linearized
dielectric response formalism based on the solution of the
kinetic equations for the plasma species (the formalism is
applicable as long as the perturbations of the ion density
induced by a charged grain are weak at the relevant dis-
tances, see [11,12] for details). Such an approach allows us
to calculate self-consistent wake potential as a function of
the ion drift velocity. Without an electric field, the inter-
action is via the Debye-Hu
¨
ckel potential characterized by
the particle charge Q and ion screening length . The
external ac field E causes (mobility-limited) ion oscilla-
tions with the velocity u
i
i
E, where
i
’ 1:6
10
5
=p cm
2
=Vs for Ar ions in an Ar parent gas. Far-field
asymptotics for the potential can be expanded into a series
over (small) u
i
(with the angular dependence of the first
three coefficients of expansion being proportional to the
corresponding multipoles, i.e., charge, dipole, quadrupole).
Furthermore, all ‘‘odd’’ terms (/u
j
i
with odd j) are pro-
portional to linear combinations of the odd-order Legendre
polynomials, whereas ‘‘even’’ terms are combinations of
the even-order polynomials. Thus, for an ac field Et with
hEi
t
0, all odd-order terms disappear in the time-
averaged potential hi
t
, which becomes an even function
of coordinates. In spherical coordinates, the effective en-
ergy Qhi
t
of the time-averaged pair interaction is
Wr; ’Q
2
e
r=
r
0:43
M
2
T
2
r
3
3cos
2
1
; (1)
where is the angle between E and r and M
2
T
hu
2
i
i
t
=v
2
T
is the (squared) ‘‘thermal’’ Mach number normalized by the
thermal velocity of ions (equal to that of neutrals), v
2
T
T
n
=m
i
[12]. Thus, the interaction consists of two principal
contributions: The first ‘‘core’’ term represents the spheri-
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cally symmetric Debye-Hu
¨
ckel (Yukawa) part, whereas the
second term is due to the interaction between the charge of
one grain and the quadrupole part of the wake produced by
another grain. The charge-quadrupole interaction is iden-
tical to the interaction between two equal and parallel di-
polesofmagnitude ’ 0:65M
T
Q—thereforewerefer to the
second term in Eq. (1) as the ‘‘dipole’’ term. This implies
that for small M
T
the interactions in ER plasmas are
equivalent to dipolar interactions in conventional ER
fluids.
The phase variables commonly used to describe ER
colloids are the particle volume fraction and the electric
field strength [3,4]. For ER plasmas, where the screening
length is much larger than the particle size, the so-called
‘‘screening parameter’’ = is a more convenient
phase variable (here and below n
1=3
is the mean
interparticle distance), and the electric field can be mea-
sured in units of M
T
. The core coupling is measured with
the ‘‘coupling parameter’’ Q
2
=T [11].
Investigations of the phase states in ER colloids have
largely concentrated on solids, because of the rich variety
of possible crystalline states [1,3,4], whereas relatively
little research of the fluid phase exists. In particular, the
dynamics and details of the phase transition between iso-
tropic and ‘‘string’’ fluids is still unexplored [13,14].
In order to quantify the expected ‘‘isotropic-to-string’’
phase transition, a suitable order parameter has to be
employed that is sensitive to the changing particle struc-
tures. Conventional approaches, e.g., binary correlation or
bond orientation functions, Legendre polynomials, etc.,
were too insensitive in our case. A much more satisfactory
order parameter is the anisotropic scaling index — a local
nonlinear measure for structure characterization (for de-
tails, see, e.g., [15]), with which any symmetry changes
can be quantified by using ‘‘principal axes’’ distributions:
P
x
, P
y
, and P
z
, where P
q
d Pf 2
; d; qg. The emergence of anisotropic structures
is reflected by decreasing for particular direction q.In
our case the problem has uniaxial anisotropy (field is in the
z direction), so that P
x
and P
y
are statistically the
same. Therefore, longitudinal and transverse distributions
defined as P
k
P
z
and P
?
1
2
P
x
P
y
are employed below.
The MD simulations [16] of a weakly coupled ER
plasma (using 45 000 particles) were performed for differ-
ent combinations of and that ensure sufficiently small
magnitude of the pair interaction energy, viz., e
& 1.
We found that at sufficiently small M
T
the distributions P
k
and P
?
practically coincide. This suggests an isotropic and
homogeneous distribution of particles, unaffected by the
electric field. At larger M
T
the transverse distribution
remains practically unchanged, whereas the longitudinal
distribution is noticeably shifted towards smaller . This is
an indicator for the emergence of stringlike patterns along
the field—for aligned 1D structures the scaling index
should decrease towards unity [15].
A quite natural (and simple) scalar order parameter to
characterize the onset of the isotropic-to-string transition is
the difference between the transverse and longitudinal
scaling indices averaged over the ensemble,
R
P
?
d
R
P
k
d. Figure 2 summarizes the simula-
tion results for e
’ 0:24 (orange or light gray bullets
FIG. 1 (color). Particle potential in a complex plasma. The relevant (ion) screening length and, hence the effective radius of the
polarizable cloud ‘‘attached’’ to a microparticle is typically 1-to-2 orders of magnitude larger than the particle size. Without an external
field (a) the cloud is spherical (the so-called ‘‘Debye sphere’’), when a field is applied (b) the cloud becomes asymmetric and acquires a
fairly complicated shape. The ‘‘center’’ of the cloud—which is then called ‘‘ion wake’’—is shifted downstream from the grain, along
the field-induced ion drift. In this case the pair interaction between charged grains is generally nonreciprocal (i.e., non-Hamiltonian),
because the wakes ‘‘belong’’ to the surrounding plasma and therefore play the role of a tenuous ‘‘third body’’ [19]. The nonreciprocity
of the interaction could only be eliminated if the wake potential were an even function of coordinates, i.e., rr. A simple
‘‘recipe’’ to create such a reciprocal wake potential is as follows: One has to apply an ac field of a frequency that is (i) much lower than
the inverse time scale of the ion response (ion plasma frequency, typically 10
7
s
1
) and, at the same time, (ii) much higher than the
inverse dust response time (dust plasma frequency, typically 10
2
s
1
). Then the ions react instantaneously to the field whereas the
microparticles do not react at all. The effective interparticle interaction in this case is determined by the time-averaged wake potential
(c). In the framework of the linear response formalism, the resulting interaction is rigorously reciprocal (Hamiltonian), so that one can
directly apply the formalisms of statistical physics to describe ER plasmas.
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and line) and e
’ 0:06 (blue or gray bullets and line).
The calculated values of the order parameter versus the
control parameter M
T
are fitted by power-law functions.
The fit yields an offset M
cr
T
—a measure of the critical
field—of 0.22 and 0.33, respectively. Note that if we
were to enforce M
cr
T
0, we would obtain unreasonably
high values of the chi-square parameter (10 larger than
those for the curves shown in Fig. 2), which strongly
suggests that we have a second-order or a weak first-order
phase transition between isotropic and string fluids.
We can make a simple analytical estimate for the ther-
modynamic stability of weakly coupled ER plasmas and
derive a rough criterion for the isotropic-to-string phase
transition: Weak coupling implies a rarefied (gaseous)
ensemble of particles, where triple interactions play a
minor role. Thermodynamic properties of such systems
can be understood in terms of the second virial coefficient
[17]: B
R
1
0
R
1
1
1 e
~
W
r
2
drdx, where x cos
and
~
Wr; xW=T. The core interaction radius in
Eq. (1) (radius of excluded volume, in units of )is
approximated by
~
R
ex
ln1 (for typical experi-
ments 10
3
–10
5
and 1–10). For M
2
T
&
~
R
3
ex
=
the integration yields the following estimate:
3
1
B ’
2
3
~
R
3
ex
0:04M
2
T
2
=
~
R
3
ex
. At larger Mach numbers the
attractive interaction provides an exponentially large con-
tribution to B and, hence, the virial coefficient rapidly
becomes negative. Thermodynamic stability requires
2nB * 1 [17], and the violation of this condition sug-
gests the occurrence of a phase transition to a string fluid.
This happens when M
2
T
* 3
1 N
1
ex
p
~
R
3
ex
=, where
N
ex
4
3
R
3
ex
n & 1 is the number of particles per excluded
volume. For the MD simulations shown in Fig. 2 this
simple estimate gives critical values M
cr
T
’ 0:4 (blue line)
and ’ 0:6 (red line)—a good agreement for such an
‘‘order-of-magnitude’’ evaluation.
Experiments with ER plasmas were performed with the
‘‘PK-3 Plus’’ laboratory [18] under microgravity condi-
tions, on board of the International Space Station (ISS). We
usedmicroparticles of different sizes (diameters ’1:55 m,
6:8 m, and 14:9 m), and Ar gas at pressures between 8
and15 Pa. To ensure different number densities n, the num-
ber of injected particles varied as well. Sinusoidal out-of-
phase signals were applied to the rf electrodes at frequency
100 Hz, with the peak-to-peak voltage between 26.6 and
65.6 V varied in steps of 2.2 V. In each experiment an ac
field was first ramped up, and then ramped down. At weak
fields charged particles form a strongly coupled (e
30–100) isotropic fluid phase with typical short-range or-
der. As the field is increased above a certain threshold,
particles start to rearrange themselves and become more
and more ordered, until eventually well-defined particle
strings are formed. The transition between isotropic and
string fluid states is fully reversible—decreasing the field
brings the particles back into their initial isotropic state.
The trend to form strings increases with particle size,
which is in line with our above theoretical estimates.
However, the quantitative comparisons with the estimates
are meaningless due to the rather strong coupling and,
hence, multiple particle correlations. On the other hand,
the MD simulations performed with similar parameters
demonstrate remarkable agreement with the experiment.
Figure 3 shows the experimental results with 6:8 m
particles at p 10 Pa and the comparison with the MD
simulations. (Parameters of the experiment are estimated
as follows: ’ 0:05 mm, n ’ 3 10
4
cm
3
, Q ’10
4
e,
and T ’ 3 10
2
eV.) The structural order of the well-
developed strings is quite evident in both experimental and
simulation data shown in the first two rows. The lower two
rows show the corresponding distributions P
k
and
P
?
. As the electric field is increased, P
?
remains
practically unchanged whereas P
k
is shifted towards ’
1, suggesting strong 1D ordering (strings). Note that the
local amplitude of the electric field in the experiment
cannot be measured or directly calculated, due to unknown
plasma screening. However, from the comparison with the
MD simulations one can deduce the corresponding M
T
FIG. 2 (color). Onset of the isotropic-to-string plasma transi-
tion. Order parameter (average difference of transverse and
longitudinal scaling indices) versus control parameter M
T
(ther-
mal Mach number of the ion oscillations) is obtained from MD
simulations for two cases of weakly coupled ER plasmas:
coupling parameter 530 (orange bullets) and 133
(blue bullets), screening parameter 7:7 for both cases. A
two-parametric least-squares fit /M
T
M
cr
T
for M
T
>
M
cr
T
and 0 for M
T
M
cr
T
yields M
cr
T
0:22 and 2:3
(orange line), and M
cr
T
0:33 and 1:7 (blue line). The inset
shows an example of histograms for longitudinal (red triangles)
and transverse (green squares) distributions of the scaling in-
dices, P
k
and P
?
, calculated for 530 at M
T
1:3.
Note that a uniform and isotropic set of points makes P
peaked at ’ 1:9.
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and, hence, the effective magnitude of the field, which
turns out to be a factor 3–10 smaller than the ‘‘vacuum’’
value U=H (here U is the ac peak-to-peak voltage at the
electrodes and H is the distance between them).
Complex plasmas enable us to study elementary dy-
namical processes by observing the fully resolved motion
of individual particles, and we expect that future ER
plasma investigations will make ample use of these unique
properties. Such studies should provide us with essential
knowledge about generic phenomena that govern the be-
havior of ER fluids in general.
This work was supported by DLR/BMWi Grant
No. 50WP0203, and by RFBR Grant No. 06-02-08100.
We would like to thank the firm Kayser-Threde, RKK-
Energia, the Mission Control Centre in Korolev, and, fi-
nally, the Yuri Gagarin Cosmonaut Training Centre and the
cosmonauts for their perfect work.
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FIG. 3 (color). Formation of developed
strings in ER plasmas. First row:
Microgravity experiments (6:8 m par-
ticles, raw data), microparticles are illu-
minated by a thin (less than mean
interparticle distance) laser sheet parallel
to the applied ac electric field. Examples
of ‘‘low’’ (first column), ‘‘intermediate’’
(second column), and ‘‘high’’ (third col-
umn) fields are shown, the peak-to-peak
voltage of the ac signal (applied to two
parallel horizontal electrodes) is indi-
cated. Second row: MD simulations,
the same configuration, field is measured
in units of the thermal Mach number M
T
(scale bars 2 mm). Third and forth rows:
Histograms for longitudinal (red) and
transverse (green) distributions of the
scaling indices, P
k
and P
?
, calcu-
lated for the experiment and simulation,
respectively. (Note that at higher den-
sities the particle positions in the neigh-
boring strings became highly correlated.)
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