VOLUME 88, NUMBER 24 PHYSICAL REVIEW LETTERS 17 JUNE 2002
Criteria of Phase Transitions in a Complex Plasma
O.S. Vaulina,
1
S. V. Vladimirov,
2,
* O. F. Petrov,
1
and V.E. Fortov
1
1
Institute of High Temperatures, Russian Academy of Sciences, 13/19 Izhorskaya Street, Moscow 127412, Russia
2
School of Physics, The University of Sydney, New South Wales 2006, Australia
(Received 23 October 2001; published 3 June 2002)
New empirical rules for different phase transitions (including the melting of cubic lattices and the tran-
sitions between body-centered-cubic and face-centered-cubic structures) are proposed. The arrangements
of charged macroparticles in a complex “dusty” plasma are numerically investigated for the conditions
of laboratory experiments on weakly ionized gas discharges.
DOI: 10.1103/PhysRevLett.88.245002 PACS numbers: 52.27.Lw, 82.70.Dd
Complex plasmas containing large (compared with
the sizes of electrons and ions), highly charged colloidal
particles (“dust”) recently attracted wide attention [1–4].
In the majority of laboratory experiments the grains were
immersed in a weakly ionized plasma; the combined
effect of interactions of grains between themselves as
well as with the ambient plasma led to the formation of
structures exhibiting liquidlike behavior [1], as well as
crystals [1,2], clouds and voids [3], and clusters [4].
The simplest model for the three-dimensional (3D)
particle interaction takes into account plasma screening
and therefore the electrostatic interaction potential is of
the Yukawa type
f
D
eZ
l
exp
μ
2
l
l
∂
, (1)
where eZ is the particle charge, l is the plasma screen-
ing length, and l is the interparticle distance. It should be
noted that the Yukawa-type approximation might be un-
suitable for the vertical direction in those laboratory ex-
periments where grains levitate in the sheath region with
strong plasma flows forming the wake potential [5]. Nev-
ertheless, the potential of the Yukawa type (1) can be used
with very good accuracy in the horizontal direction [6], and
it is especially important for the analysis of dust dynamics
in a complex plasma under microgravity conditions where
grains can levitate in the plasma bulk.
The nonideality is usually characterized by the cou-
pling parameter G eZ
2
dT which is the ratio of the
Coulomb potential energy of the particle interaction to the
kinetic energy of their thermal motion (here, d n
213
is the mean intergrain distance, n is the particle number
density, and the temperature T is in energy units). It is
also well known that phase transitions in Yukawa systems
are determined by two dimensionless parameters: G
and k dl. The extensive numerical studies [7–14]
demonstrate that in a Coulomb system of particles the
short-range order appears for G ¿ 1, with the critical
value G
m
106 on the melting line [10 –13] [for complex
plasmas, the assumption of plasma screening (1) leads to
larger G
m
]. The studies [14,15] suggest that the condi-
tion of the constant (normalized) nonideality parameter
G
1 1k1k
2
2 exp2kG (namely, G
m
106)
can be used as the melting criterion for the body-centered-
cubic (bcc) lattice. However, the functional dependence
relating G and k with the critical value G
m
f G, k is
presently unknown for the transitions of face-centered-
cubic (fcc) lattice into the liquid as well as for the tran-
sitions between the bcc and fcc structures. Some authors
suggest various linear approximations of numerical data
for different parts of the phase diagram [7,9]; these ap-
proximations usually appear as a result of the best mathe-
matical fit, though sometimes being not fully justified
physically. In this Letter, we propose the criteria of phase
transitions in the Yukawa system by employing simulation
data [7,9,13] obtained for systems without dissipation,
as well as on the basis of new original simulations of
a 3D Yukawa dissipative system, with parameters close
to those in experiments on laboratory weakly ionized
gas-discharge plasmas.
There are various phenomenological criteria for phase
transitions in the Yukawa system used for a complex
plasma. The most popular is the Lindemann criterion
stating that the melting occurs when the ratio of the
root-mean-square displacement D
0
of a particle from its
equilibrium position to the average interparticle distance
d achieves 0.15. Since in numerical simulations the
displacement D
p
2 D
0
of a particle from the center
of mass is usually computed, the ratio d
c
Dd on the
melting line should be expected about 0.21 (for the
majority of real solids Dd 0.2 0.25 at the melting
point). However, various numerical simulations give for
the Lindemann parameter the range from 0.16 0.19 for
fcc lattices to 0.18 0.2 for bcc structures. These numbers,
less than 0.21, may be related to the insufficient number
of particles N
p
in the modeled systems; we note that
Dd ! 0.2 with the increase of N
p
for the melting of
the both types of lattices [9]. Another popular criterion,
proposed by Hansen and Verlet [16], defines the value of
the first maximum S
1
of the structure factor in the liquid
state to be less than 2.85. These numbers can also vary
(from 2.5 to 3.2) for different simulations and strongly
depend on the definition of the structure factor in the
systems with a finite number of particles.
We obtain the condition, analogous to the Lindemann
criterion, with the assumption that the average volume
245002-1 0031-9007 02 88(24) 245002(4)$20.00 © 2002 The American Physical Society 245002-1