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Shock waves in a complex (dusty) plasma

D. Samsonov, S. Zhdanov† and G. Morfill†
Dept. of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill, Liverpool, L69
† Max-Planck-Institute for Extraterrestrial Physics, Garching, 85741, GERMANY
Abstract. Shocks with a Mach number 3.4 were studied experimentally in a two dimensional complex (dusty) plasma.
Monodisperse microspheres formed a hexagonal monolayer lattice in a plasma sheath of an rf discharge. An electrostatic
pulse excited shocks in the lattice, which was laser-illuminated and video-imaged at 1000 fps. The lattice was melted by the
shock. The structure of the shock was resolved at the kinetic level, with motion of every particle traced. The macroscopic
parameters, such as number density, kinetic temperature and particle flow velocity were determined.
Keywords: complex plasma, shock wave, phase transition
PACS: PACS:52.27.Lw,52.35.Tc,62.50.+p,47.40.Nm
A complex plasma is a mixture of macroscopic (micron-sized) grains with an ion-electron plasma. The grains charge
up and interact with each other collectively, they can be also levitated and confined in a gas discharge. Complex
plasmas can exist in solid, liquid, and gaseous states (similar to colloids) [1] and exhibit phase transitions. They can
be easily observed in real time at the kinetic level. Due to a low damping of the grain motion, a range of dynamic
phenomena can be observed in complex plasmas (unlike colloids), such as waves [2, 3], solitons [4], Mach cones [5],
shocks [6, 7]. Complex plasmas can be used as an easy to study dynamic model system of solids, liquids, and gases.
Basic properties of transport phenomena can be modelled using complex plasmas. Examples include heat transfer[8],
diffusion[9], and shear flows [10].
Grains in ground based experiments either form two-dimensional clouds (affected by gravity) or have to be very
small (. 1mm). Small grains form only weakly coupled (gaseous) complex plasmas due to their small electric charge
and they are very difficult to observe individually. PKE-Nefedov [11, 12], an experiment on board the International
Space Station (ISS) allows the investigation of complex plasmas under microgravity conditions. It is based on a
radio-frequency discharge cell. Microgravity experiments were performed on particle decharging [13], structure and
dynamics of complex plasmas [14], fluid flows and crystallization [15], void properties [16]. Shocks in a 3D complex
plasma under microgravity were studied in Ref. [6].
We performed the experiments in a 13.56 MHz capacitively coupled rf discharge in a setup (Fig. 1) identical to that of
Ref. [4]. A powered lower electrode and a ring upper electrode were placed in a vacuum chamber. The upper electrode
and the chamber were grounded. A rf power of 10 W (measured as forward minus reverse) was applied to the lower
electrode. The working pressure of 1.1 Pa was maintained by a flow of argon at a rate of 2 sccm. Monodisperse
plastic micro-spheres 8.90:1 mm in diameter were levitated in the sheath ‘ 9 mm above the lower electrode. They
were confined radially in a bowl shaped potential formed by a rim on the outer edge of the electrode and formed a
monolayer hexagonal lattice. The particle cloud was about 6 cm in diameter with the particle separation in the lattice
of 65050 mm. A horizontal thin (0.2-0.3 mm) sheet of light from a doubled Nd:YAG diode-pumped laser (532 nm)
illuminated the particles, which were imaged by a top view 1 megapixel digital video camera at 1000 frames/s. The
field of view was 27:327:3mm and it contained about 2000 particles.
In order to excite shock waves, a horizontal tungsten wire 0.1 mm in diameter was stretched above the lower
electrode, about 2 mm below the particle layer and roughly half way between the center and the edge of the electrode
(Fig. 1). The wire was normally grounded and it did not affect the particles. A short negative pulse applied to the
wire repelled the negatively charged micro-spheres breaking the lattice above the wire and creating a pulsed one.

FIGURE 1. Sketch of apparatus. (a) Oblique view. Spherical monodisperse particles charged negatively and formed a monolayer
levitating in the plasma sheath above the lower electrode. (b) Side view. The grounded wire was placed below the particles. Short
negative pulses were applied to the wire to excite shocks. Reprinted from Ref. [4]
dimensional compressional disturbance, propagating horizontally perpendicular to the wire. The particle cloud also
oscillated in the vertical direction with a small amplitude which caused a periodic change of brightness of the particles.
The amplitude of the excitation pulse was -19 V and duration 100 ms. The purturbation caused by the pulse was
stronger than in Ref. [4] because the wire was positioned closer to the lattice layer. We allowed about 1 min cooling
time between the experimental runs.
The recorded video sequences were analyzed by a program which identified particle positions. Tracing the particles
from one frame to the next yielded their velocity. It was then averaged in 50 narrow bins parallel to the wire. The
kinetic temperature was calculated from the standard deviation of the particle velocity in the bins which depends only
on the particle random motion. We used triangulation to determine the nearest neighbors for each particle. The local
number density was determined as the inverse area of Voronoi cells. The compression factor was computed as a ratio
of the number density to the unperturbed number density. All measured quantities were binned in order to reduce the
influence of random fluctuations.
We performed a molecular dynamics simulation in order to support the experimental results. A monolayer lattice of
721 particles was formed in a three-dimensional confining potential. The lattice was strongly confined in the vertical
direction and weakly in the horizontal. The particles interacted via a screened Coulomb (Yukawa) potential and their
motion was damped by friction. Initially placed at random positions, the particles were equilibrated by running the
code, until a round hexagonal lattice was formed and cooled down. Then a pulsed excitation force was applied to
produce shocks.
The unperturbed lattice was in a crystalline state with hexagonal symmetry. It had about 10% defects, mostly pairs of
five- and seven-fold cells. This indicated a relatively good crystalline structure.
After the excitation pulse was applied, the grains were swept away from the wire, and the shock was formed (fig. 2
at 100 ms). As the shock propagated into the lattice, its amplitude decreased due to the frictional drag of the neutral
gas. The Mach number of the shock decreased from 3.4 to 1.9 as the shock passed across the field of view.
The lattice was melted behind the shock front, with the defect fraction about 50 %, which indicated a liquid state.
The detailed shock structure is revealed in fig. 3. The number density had a jump at the shock front (with a small
overshoot at the front), then it slowly decreased due to the particle backflow toward the wire. The kinetic temperature
jumped from a few eV to about 100 eV, which is above the melting temperature of a similar lattice found in the
experiments of Refs. [3, 8]. There was also a jump in the flow speed and defect density.
The flow rate across the shock front remained constant indicating that no particles were lost at the front. Particle
loss from the field of view was a major problem in earlier experiments. It happened due to a slow recording speed of

FIGURE 3. Number density (the unperturbed value is marked by the dashed line) (a), kinetic temperature (b) particle count (c),
flow velocity in the direction of the shock (d), and particle flow rate (e) in the shock wave versus distance to the excitation source
at 500 ms. The shock front (positioned at 21 mm) was manifested by a jump in the number density, kinetic temperature, particle
count, and flow speed. The particle flow rate remained constant.
the video camera (see [7]), when fast moving particles at the front overlapped and coudl not be traced correctly. The
backflow of the complex plasma was also reduced, making the number density jump more pronounced.
Stable and unstable shock regimes were also observed. A stable shock (like in fig. 2) with a well pronounced linear
front was formed when a long (100 ms) excitation pulse was applied. A short excitation pulse (10 ms) with higher
amplitude produced an unstable front which had not a linear but a zig-zag shape.
The molecular dynamics code we used did not include any explicit plasma. The plasma was in the code only in the
form of confining potentials and grain to grain interaction. This simplified treatment produced a very good agreement
with the experiment [2, 3, 7].
Molecular dynamics simulation reproduced both stable and unstable shocks and helped explain how they are formed.
The excitation force rapidly decreases with the distance to the wire. Particles do not move significantly during a short
excitation pulse. Those far from the wire are weakly accelerated and only those few close to the excitation region get
strongly accelerated and acquire large speeds. The fast particles penetrate the lattice, get scattered and break the lattice
in an irregular pattern. Long excitation pulses allow the particles to move significantly during the excitation. They end
up far from the wire where the force is weak. The overall effect is that more particles have nearly the same speed as
they form a linear shock front.
A shock wave with Mach number of 3.4 was obtained in a two-dimensional complex plasma. It exhibited a jump in
number density, kinetic temperature, and flow speed. The particle flow rate across the shock front remained constant.
Shocks with a stable and an unstable fronts were observed. The experimental results were reproduced by a molecular
dynamics simulation which took into account the particle interaction and confinement potentials.
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