Nonlinear Theory and Simulation of Void
Formation in Dusty Plasmas
A. Bhattacharjee, C. S. Ng, K. Avinash, S. Hu, and Z. W. Ma
Space Science Center, University of New Hampshire, Durham, NH 03824
Abstract. We present new developments in the theory and simulation of a recent nonlinear
time-dependent model for void formation [1]. For experimentally representative initial
parameters, the model describes the nonlinear evolution of a zero-frequency linear instability
that grows exponentially in the linear regime, near-explosively in the nonlinear regime, and
eventually saturates to form a stable void. The model has been extended to two dimensions
using a more complete set of dynamical equations, and qualitative features of void formation are
shown to be robust with respect to different functional forms of the nonlinear ion drag operator.
Several colloidal (or dusty) plasma experiments, in laboratory as well as under
microgravity conditions, have shown the spontaneous development of voids[2-6]. A
void is typically a small and stable centimeter-size region (within the plasma) that is
completely free of dust particles and characterized by sharp boundaries. In the
laboratory[3], the void is seen to develop from a uniform dust cloud as a consequence
of an instability when the dust particle has grown to a sufficient size. It was suggested
in [3] that the ion drag force plays a crucial role in causing the initial instability.
Recently, we have proposed a basic, time-dependent, self-consistent nonlinear
model for void formation[1]. This basic model contains three elements: (a) an initial
instability caused by the ion drag force Fd , (b) a nonlinear saturation mechanism for
the instability, and (c) the void as one of the possible nonlinearly saturated states,
dynamically accessible from the initially unstable equilibrium. For the initial
instability, we choose a simple variant of the zero-frequency mode described by
D'Angelo[7]. The saturation mechanism for the instability relies on a crucial nonmonotonic property of the ion drag force that appears to be a qualitatively robust
feature of the force, independent of the regime of collisionality. In the collisional
regime, Fd initially increases with the ion velocity vi , attains a maximum for vi = vthi ,
where vthi is the ion thermal velocity, and decreases for vi > vthi [8,9]. As the linear
instability grows, the ions are initially accelerated in the growing electric field, and Fd
initially increases. Eventually, as the ions are accelerated to speeds larger than the ion
thermal speed, Fd decreases to balance the electric field Fe and thus saturate the
instability. The functional form of the ion drag force and where it peaks in velocity
space changes in the collisionless regime, but it retains the qualitative feature of nonmonotonicity seen in the collisional regime[10].
CP799, New Vistas in Dusty Plasmas: Fourth International Conference on the Physics of Dusty Plasmas
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We have extended the results presented in [1] in two ways: first, by using a more
complete system of equations than used in [1] and extending the one-dimensional
treatment presented in [1] to two dimensions, and second, by showing that the
qualitative features of the dynamical relaxation to the void solution is robust with
respect to other functional forms of the nonlinear drag operator[10]. Due to space
restrictions, we are unable to describe in detail these extensions here, and provide a
brief summary instead. The system of equations used in the first part of the study is:
&
)
"v d
a
+E % , 0 v d % - d $n d
+ v d # $v d = (
%1
3
"t
nd
(' b + vi
+*
!
!
!
,
(2)
"ni
= #$ % ( ni vi ) + Ani # Cne
"t
,
(3)
nE
"ne = # e
$i
vi = µE
!
!
(1)
"n d
= #$ % ( n d v d ) + D0$ 2 n d
"t
" # E = ni $ n d $ ne
!
,
,
(4)
,
(5)
.
(6)
Here all symbols have their usual notations, and the equations are written in
normalized dimensionless form following the conventions of [1]. To be specific, all
distances are in units of "DiTe /Ti , t is in the unit of " #1
pd , u is in the unit of ion thermal
1/ 2
!
!
speed v thi = [Ti /mi ] , the dust velocity v d is in the unit of "DiTe# pd /Ti , the electric
field E is in the unit of Te /e"Di , and Fd is in the unit of "Di md Te# 2pd /Ti . Note also that
!
! frequency and " d = Td Ti /Te2 Z d . The first
dust-neutral collision
" 0 is the normalized
for the dust fluid, represents
! the momentum equation
! term on the right of (1), which is
!
a model nonlinear
form
of
discussed
in
[9].
We
also
include the convective
F
!
! d
!
nonlinearity, omitted in [1]. When D0 = 0 , it can be shown that in steady state the
!
dust pressure gradient is balanced by the dust convective nonlinearity. Equation (3) is
a continuity equation for ions and has two constant coefficients A and C in its source
!
terms. It replaces the assumption ni = constant made in [1].
!
We integrate (1)-(5) numerically in two-dimensional cylindrical geometry,
beginning from a homogeneous field-free, stationary equilibrium.
We use
experimentally representative parameters which yield a = 100 , b = 1.6 ! 0 = 2 ,
!
" d = 0.01, " i = 10 , µ = 10 , D0 ! 10 "2 , A = 0.8, C = 1. The system evolves through a
linear exponential phase and a nonlinear near-explosive phase before relaxing to a
stable void solution. Despite the fact that the numerical
simulation
allows for radial as
!
!
!
!
!
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well as angular dependencies, the final relaxed state turns out to be nearly
axisymmetric. This is shown in Fig. 1, where we show an image plot of the dust
plasma density in the saturated state, exhibiting a void with a sharp boundary. We
FIGURE 1. Image plot of the dust density in the relaxed state of a two-dimensional solution,
exhibiting a void with a sharp boundary.
have checked that a steady-state solution, assuming axisymmetry, shows good
agreement with the numerical solution. In Fig. 2, we compare the two-dimensional
axisymmetric solutions (solid lines) with the one-dimensional solutions (dotted lines)
for the same set of parameters, demonstrating that qualitative features of the void
solutions are similar in both geometries.
Finally, we have also computed time-dependent void solutions using the ion drag
operator developed by Khrapak et al.[10] which, expressed in our dimensionless
coordinates, cam be written as
0* 1
a0 E
1 # $ u '"u 2 / 2
Fd =
e
"
erf& )/ 3
1
,
u 2 2 % 2 (.
(1+ c1 E )1/ 2 u 2+ u
2 6
# $ u '
erf & ) 2 + u 2 2(1+ z4 ) + u 2 + ue"u / 2 7
2 % 2(
8
1/ 2
where u = c 0 E /(1+ c1 E) , and a0,z,c 0, and c1 are parameters specified by the
theory[10]. The nonlinear theory does appear to yield qualitatively similar dynamics
and void features (seen in Fig. 2) when we use the Khrapak operator. While this is
reassuring, it should be mentioned that there are quantitative differences between the
!
! predictions of various operators.
These discrepancies do matter in assessing the
accuracy of various operators for a specific experiment, and also in determining
whether ion drag force is the dominant mechanism controlling void formation in the
experiment.
This research is supported by the Department of Energy.
[
]
1+ 2z4 " 4z 24 2 ln 5 +
!
[
][
]
147
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FIGURE 2. Axisymmetric solutions (solid lines) compared with one-dimensional solutions (dotted
lines) for the same set of parameters as in [1].
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