Role of the ion wake effect on stability of plasma crystals
V. V. Yaroshenko, H. M. Thomas and G. E. Morfill
Max-Planck-Institut für Extraterrestrische Physik,
D-85740, Garching, Germany
Keywords: ion wake, dust-lattice modes, plasma crystal
PACS: 52.27.Lw, 52.27.Gr, 52.35.Lv, 52.35.-g
A set of solid particles introduced into a low-temperature discharge plasma can spontaneously exhibit ordered crystalline
structures – so-called “plasma crystals”. Such systems have attracted much interest in fundamental physics,
plasma science as well as technological applications (see e.g. ). In ground-based experiments, the strongly-coupled
formations are usually observed in the sheath of the lower electrode, where the upward directed electrostatic force
balances the gravitational force on microparticles. The particles mostly arrange themselves in a flat crystal with a diameter
of a few hundred interparticle distances and a thickness of up to a few tens of layers. The flatter systems exhibit
the thermodynamically expected two-dimensional hexagonal order. In the vertical direction the microparticles are then
found to be aligned, forming vertical chains [2, 3, 4]. Note that if only repulsive interparticle forces dominate, externally
confined grains should organize themselves in a close-packed structure (the minimum energy state for repulsive
forces), not in aligned chains. As an explanation, the attraction force due to the formation of a positive space-charge
region underneath the suspended microparticles, the so called ”ion wake”, has been discussed [4, 5, 7, 6].
Observations of the plasma crystals, which reach equilibrium very rapidly and can be easily turned between their
ordering and disordering states, stimulate theoretical investigations of the processes underlying the solid-to-liquid
phase transition. In this context, the instability of the horizontal oscillations in adjacent horizontal layers due to
the asymmetry in the particle-wake interactions has been given great attention in the past, but which so far have
only been investigated quantitatively for double layer crystals, not multi-layers [4, 5, 10]. Observations of multi-layer
crystals from the side (see ) suggest that pre-melting particles displacements and oscillations are isotropic (within
the statistical limits). This implies that compressive instabilities are important. We therefore consider the stability of
the vertical particle chain (it highlights the major aspect of the multi-layer crystals) in the presence of perturbations
along and transverse to the direction of the ion flow. We introduce a ”hybrid” type of particle interactions, containing
the screened Coulomb repulsion of the charges and attraction due to the positive charge of the ion wake. It turns
out that the attractive forces can overcome the Coulomb repulsion even for small effective wake charges. They can
also be responsible for the development of a dipole instability below a certain threshold value of gas pressure (i.e. gas
damping). Such an instability could be a direct precursor of the melting transition in plasma crystals.
BINARY PARTICLE INTERACTIONS
To model the particle-wake interactions we introduce an electric dipole formed by a point-like charge (the effective
wake charge). This means that each “particle” in a flowing complex plasma can be considered as a negative charge
Qn with the electric dipole moment Pn = qnl, where qn denotes the effective wake charge, l =lz0, l a characteristic
length scale of the ion focusing and z0 the unit vector in the direction of the ion flow. Accordingly, the force describing
dust-dust interactions (between any pair j and n grains ) can be represented as a combination of the electrostatic force
due to the repulsion of the like particle charges F(Q)
jn = ¡Qj(Ñjn)r=r j and a dipole force F(p)
jn = (PjÑ)En due to the
streaming ions, so that
Fjn = ¡Qj(Ñjn)r=r j +(PjÑ)En: (1)
The electrostatic potential of each particle is taken as a screened Debye potentialjn (r)=Qn exp(¡jr¡rnj=lD)=jr¡rnj,
where lD refers to the screening length. The electric dipole field is En (r) = 3((r¡rn)¢Pj) (r¡rn)=jr¡rnj5 ¡
Pj=jr¡rnj3. The use of model (1) implies that the interparticle distances are large enough, viz.
justify the dipole approximation.
Introducing the hybrid type interactions through (1) immediately modifies the character of binary particle interactions,
making them strongly asymmetric. Indeed, the mutual interactions between two identical particles are now
determined by a force, which can be attractive or repulsive, depending on the plasma parameters and relative positions
of the two dipoles. In particular, the force component along the radius-vector between two particles is given by
where k = D=lD is the “lattice parameter”, D is the interparticle distance, c is the angle between the horizontal
direction and radius-vector connecting two particles. Finally, the dimensionless coefficient V = q2l2=(QlD)2 specifies
the real value of the electric dipole moment. When (1¡3sin2c) > 0; so that jcj < 0:62; the particle are repelled since
both forces act in the same direction, while for jcj > 0:62, the electrostatic and dipole forces compete with each other
and the resulting force corresponds to either attraction or repulsion.
As an extreme case, we consider two vertically aligned particles (c = p=2). It can be easily verified that the
particle attraction due to the short-ranged dipole force µ ¡1=D4 (F < 0) changes to repulsion (F > 0) only if
exp(¡k)=24. The latter gives Vcr · 0:3 for typical values of k » 1 ¡ 4. For such V ,
the dipole term prevails at either small interparticle distances (k · 1¡2:5) or large distances (e.g. k ¸ 4¡6) (the
latter is due to screening of Coulomb interactions of the particle charges). Therefore, in this model, the interactions
are repulsive only for a rather narrow region of k; which is determined by specific values of V (e.g. k » 1:5¡4
for V » 0:3;0:2). The obtained values of k are comparable to those observed in the experiments on the particle
“pairing”instability [8, 9].
possible because of the available free energy. The source of the free energy is provided by the streaming ions, which
are focused in the wake downstream and thus create the effective positive charge region. These charges attract a given
particle asymmetrically: the wake below the upper grain attracts more effectively than the wake of the lower particle.
Thus, whilst a dipole representation is only an approximation that allows a quantitative treatment analytically, it catches
the essential topology and physics quite well.
In laboratory experiments the typical values of the characteristic frequencies are W0 » 50¡100 s¡1; and Wv »
100¡200 s¡1. The ratio 2W21
is expected to be of the order of » 1¡0:5 , leading to gcr · W2q
=(kWv). This gives a range of the critical damping coefficient » 50 ¡ 300 s¡1, and correspondingly the
vertically aligned structures formed by micron-sized particles should remain stable above a neutral gas pressure
10¡40 Pa in Krypton, 15¡50 Pa in Argon, 20¡90 Pa in Neon, and 30¡160 Pa in Helium. These estimates
correspond well to typical experimental parameters. To obtain more precise estimates of the threshold value we use
parameters for particular plasma-crystal structures observed in a Krypton discharge : Q»1:7£104e;M »3£10¡10
g D ‘ 2£102m m , k » (3¡3:5). These yield gcr » (60¡80) s¡1 or equivalently a neutral gas pressure 25¡35 Pa.
Note that this threshold is in good agreement with the experimental limit pcr » 32¡29 Pa, where crystal melting was
To summarize, we have shown that ion streaming around dust grains can modify binary dust-dust interaction creating
regions of attractions and repulsion for grains oriented along the ion flows. Moreover, a simple 1D model of a particle
chain oriented along the ion flow predicts an instability (termed “dipole instability”) of longitudinal perturbations due
to the asymmetry in the particle wake interactions downstream. Neutral gas damping may quench this instability.
Using realistic experimental conditions the damping threshold for the dipole instability is found to be in the range of
gas pressures 10¡150 Pa. The instability mainly develops at wavenumber k0D ‘ p=2. It is quite possible that the
dipole instability could trigger (and thus be responsible for) phase transitions in plasma crystals from solid to liquid
when the gas pressure is reduced. Available experimental data are compatible with this suggestion.
Finally, we briefly discuss the dipole instability in relation to the ion-induced instability trigger for plasma crystal
melting [4, 5, 10]. The estimates of the dipole instability threshold are of the same order of magnitude (or even higher)
than those obtained for the instability leading to the onset of the horizontal (transverse) oscillations in horizontal
layers of plasma crystals also due to particle-wake interactions. This means that the dipole instability, which produces
vertical perturbations in aligned structures, could be as important as a precursor (or initiator) of the melting the crystal
structures as the onset of the horizontal oscillations, which have been given great attention in this context in the
past, but which so far have only been investigated quantitatively for double layer crystals, not multi-layers [4, 5, 10].
The dipole instability could be the critical process for multi-layer crystals which so far had not been identified. If
the scenario developed here is correct, then it would become a “kinetic blueprint” for other anisotropic systems (e.g.
magnetized matter) and thus assume a relevance of generic proportions.
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