Effect of dust non‐linear charge and size‐distributio n on dust-acous tic double‐layers in dusty plasmas M. Ishak-Boushaki , R. Annou, and R. BharuthramCitation: Physics of Plasmas 19, 033707 (2012); doi: 10.1063/1.3684230 View online: http://dx.doi.org/10.1063/1.3684230 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/19/3?ver=pdfcov Published by theAIP PublishingArticles you may be interested inDust-acoustic Gardner solitons and double layers in dusty plasmas with nonthermally distributed ions of two distinct temperaturesChaos 23, 013147 (2013); 10.1063/1.4794796 Large amplitude double-layers in a dusty plasma with a q-nonextensive electron velocity distribution and two- temperature isothermal ionsPhys. Plasmas 19, 042113 (2012); 10.1063/1.4707669 Effects of flat-topped ion distribution and dust temperature on small amplitude dust-acoustic solitary waves and double layers in dusty plasmaPhys. Plasmas 17, 123706 (2010); 10.1063/1.3524562 Dust-acoustic solitary waves and double layers in a magnetized dusty plasma with nonthermal ions and dust charge variationPhys. Plasmas 12, 082302 (2005); 10.1063/1.1985987 Dust-acoustic solitary waves and double layers in dusty plasma with variable dust charge and two-temperature ionsPhys. Plasmas 6, 3808 (1999); 10.1063/1.873645 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 69.161.253.236 On: Thu, 17 Mar 2016 17:17:22
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Dust-acoustic solitary waves and double layers in dusty plasma with variable dust charge and two-temperatureions
Phys. Plasmas 6, 3808 (1999); 10.1063/1.873645
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Effect of dust non-linear charge and size-distribution on dust-acousticdouble-layers in dusty plasmas
M. Ishak-Boushaki,1 R. Annou,1 and R. Bharuthram2
1 Faculty of Physics, USTHB. B.P. 32 El Alia, Bab-ezzouar, Algiers, Algeria2University of the Western Cape, Modderdam Road, Bellville 7530, South Africa
(Received 23 October 2011; accepted 7 January 2012; published online 14 March 2012)
The investigation of the existence of arbitrarily large amplitude electrostatic dust-acoustic double
layers is conducted in a four-component plasma consisting of electrons, two distinct positive ion
species of different temperatures, and massive negatively-charged dust particles that are assumed
spheres of different radii distributed according to a power-law. The dependence of the dust grain
charge on its size is considered to be nonlinear. The number densities of electrons and ions are
assumed to follow a Boltzmann distribution, whereas the dynamics of charged dust grains is
described by fluid equations. Comparison is conducted between plasmas containing size-distributed
dust grains and those containing monosize dust grains, while examining the criteria for the
existence of dust-acoustic double layers along with the dependence of their amplitudes and Mach
numbers on plasma parameters. VC 2012 American Institute of Physics. [doi:10.1063/1.3684230]
I. INTRODUCTION
A dust-acoustic double-layer (DL) is a structure consist-
ing of two space-charge layers of opposite charges. Conse-
quently, the potential experiences a drop which is necessarily
greater than the thermal energy per unit of charge of the cold-
est plasmas bordering the layer. Hence, the electric field is
stronger within the double layer, whereas quasi-neutrality is
violated in the space-charge layers.1 Double layers may be
considered resulting from solitons having an asymmetry that
is caused by motion. As a matter of fact, the potential having
a drop would be due to the reflection of the low energy com-
ponent by the potential barrier of the soliton and the transmis-
sion of the high energy one.2
These electrostatic structures
(DLs) have a tremendous role to play in space plasmas as well
as laboratory plasmas. Indeed, double layers are considered
the appropriate candidate to interpret charged particles accel-
eration to high energies in plasmas, e.g., the auroral region of
the ionosphere.3 Double-layers may be formed by way of
spacecr aft-ejected electr on beams,5 shocks waves in a
plasma,6 laser radiation,7 in jection of non-neutral electr ons
current into a cold plasma,8 or by electrical discharges.9 In
dusty plasmas, the characteristics along with the existence cri-
teria of DLs may be affected by the presence of dust particu-lates having high charge and mass.10
This type of plasmas is
believed to be the rule, as they are encountered almost every-
where in situations spanning from astrophysical to industrial
ones. So far, the dust particulates have been taken monosized,
whereas in real situations, they present a size distribution due
to grain-grain collisions that lead to fragmentation and coales-
cence11,12 which tend to produce a power law size distribution
(PLD), for which the differential density distribution is of the
form13 f (r d )dr d ¼ Cr d p dr d , where r d that is the dust grains ra-
dius is in a given range [r d min, r d max ]. Actually, as noted by
Liu et al.,14 dust size distribution is strongly connected to the
natural environments, e.g., space plasmas such as F and G
rings of Saturn, cometary environments, interstellar galactic
clouds,12,14
where the existence of size-distributed dust grains
according to a PLD has been indeed observed, the values of
the parameter p, being p ¼ 4.5 for the F-ring of Saturn, p ¼ 7
or 6 for the G-ring and a value of p ¼ 3.4 for cometary
environments,15 as well as experimental conditions in the lab-
oratory where the study is conducted. Hence, grain size-distri-
bution is an additional element to be taken into account while
modeling a plasma. Indeed, Ishak-Boushaki et al.16,17 have
investigated dust-acoustic solitons when ions are adiabatically
heated and dust grains are size-distributed, and found that sol-
utions experienced a translation from solitary waves to Cnoi-
dal waves. Moreover, they found that the grain size-
distribution affects the modes supported by the plasma along
with the growth rate of some parametric instabilities.18
Besides solitons and parametric instabilities, Ishak-boushaki19
in a study devoted to coherent structures sheds some light on
the effect of grain size distribution on dust acoustic double
layers (DADL) in a plasma consisting of Boltzmannian elec-
trons, size-distributed dust grains, and two types of Boltzman-
nian positive ions having different temperatures. Plasmas with
two ion species may occur in industrial processing of materi-
als, low temperature plasma devices, ionospher ic modification
experiments, and astrophysical situations.
20,21
In these plas-mas, the particle distribution function has a fast component
that excites a beam plasma instability (Buneman instability)
that is at the root of current carrying double layers.22 – 25 As a
matter of fact, double layers are common in current-carrying
plasmas. The effect of the non-linear dependence of the grain
charge on the grain radius on dust-acoustic double layers is
also investigated. In this paper, the work is augmented and
many aspects are revisited.
The paper is organized as follows. In Sec. II, the model
is presented, whereas in Sec. III, the results are discussed.
The last section is devoted to some concluding remarks.
1070-664X/2012/19(3)/033707/9/$30.00 VC 2012 American Institute of Physics19, 033707-1
PHYSICS OF PLASMAS 19, 033707 (2012)
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where r ¼ r d =r o and dn ¼ ntot f d ðr d Þdr d ¼ ntot f ðr Þdr are the
number of grains having radii between r and r þ dr :The mass and charge of a dust grain that is assumed
spherical may be connected to its radius through the relations
mdj ¼ ð4=3Þpqdj r 3dj and Qdj ¼ Z dj =e ¼ Cdj V 0, where qdj is
the grain mass density, V o is the grain electric surface poten-
tial at equilibrium, and Cdj is the grain capacitance that is
given in cgs units by, Cdj ¼ r dj . For a hydrogen plasma for
instance, one has V o ¼ 2:5 for T i ¼ T e ¼ 1 eV. However,
taking into account the parameters of the surrounding
plasma, some authors found that the grain charge does
depend non-linearly upon the grain radius rather, that isQdj / r
bdj , where 1 < b < 2 (c.f. Refs. 28 – 32).
To implement the model, we consider a power-law size-
distribution that is the case in space plasmas, viz., f ðr Þ¼ C pr p, where
C p ¼ p 1
1 r pþ1m
for ð p 6¼ 1Þ;
¼ ½lnðr mÞ1for ð p ¼ 1Þ:
Since for such a distribution, dust number density is maxi-
mum at minimum grain size, and we have r 1
¼(r d 1 /
r o) ¼ (r dmin / r o) ¼ 1 and r 2¼ (r d 2 / r o) ¼ (r dmax / r o) ¼ r m.
033707-2 Ishak-Boushaki, Annou, and Bharuthram Phys. Plasmas 19, 033707 (2012)
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where r m ¼ 10 and C4¼ 3003.The condition (a) in Eq. (15) is clearly satisfied by the
Sagdeev potential V ð/; M Þ, as the quasi-neutrality is retrieved,
namely,
@ /wð/; M Þ/¼0
¼ N e0 þ N c0 þ N h0 g1ð p; r mÞ ¼ 0; (17)
where gbð p; r mÞ ! g1ð p; r mÞ ¼ C41r 2
m
2
.
In addition, applying the condition (b) in Eq. (15), we
obtain
A
ð/m
Þ þC4 M 2
2/m
H ð/mÞ ln
r m þ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffir 2m þ H ð/mÞp 1 þ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi
1 þ H ð/mÞp " #
ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffir 2m þ H ð/mÞ
e of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 69.161.253.236 On: Thu, 17
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffir ð3bÞm þ2/= M 2
p 1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ2/= M 2p
#and RðbÞ¼ a þ b expðb=cÞ, with
a ¼ 0.664, b ¼ 0.156, and c ¼ 0.737.
The condition (a) in Eq. (15) is clearly satisfied again,
@ /wð/; M Þ/¼0
¼ N e0 þ N c0 þ N h0 gbð p; r mÞ ¼ 0: (23)
FIG. 1. (Color online) (a) Sagdeev potential V(/, M ) versus / for N e0 ¼ 0
and ( N c0 / N h0) ¼ 0,11. Dust grains are described by power-law distribution.
The parameter labeling the curves is the ratio of cool to hot ion tempera-
tures (Tc /Th) for b ¼ 1. (b) The double layer potential profile /ðnÞ versus n
associated with the Sagdeev potential in Fig. 1(a) and (Tc /Th) is the ratio of
cool to hot ion temperatures, for b ¼ 1.
FIG. 2. Variation of the DLs amplitude /m and the corresponding Mach
number M versus the ratio of cool to hot ion temperatures (T c=T h), for
power-law size-distribution and b
¼1, by opposition to the monosized one,
where N e0¼ 0 and ( N c0 / N h0) ¼ 0,11.
033707-4 Ishak-Boushaki, Annou, and Bharuthram Phys. Plasmas 19, 033707 (2012)
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ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffir ð3bÞm þ H bð/mÞ
q Þ
8><>: þ 1
ðr ð3bÞm þ H bð/mÞÞðr
ð3bÞ=2m þ
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffir ð3bÞm þ H bð/mÞ
q Þ2
1
r ð3bÞ=2m ðr
ð3bÞm þ H bð/mÞÞ3=2
1
ð1 þ H bð/mÞÞð1 þ ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ H bð/mÞp Þ2
1
ð1 þ H bð/mÞÞ3=2
ð1 þ ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ H bð/mÞp Þ
þ 1
ð1 þ H bð/mÞÞ3=2); (27a)
FIG. 5. (Color online) (a) Sagdeev potential V(/, M ) versus / for N e0 ¼ 0
and ( N c0 / N h0) ¼ 0,11. The parameter labeling the curves is the ratio of cool
to hot ion temperatures (T c / T h) for b ¼ 1; 83. (b) The DLs potential profile
/ðnÞ versus n associated with the Sagdeev potential in Fig. 5(a).
FIG. 6. (Color online) (a) Sagdeev potential V(/, M ) versus / for N e0¼
0
and ( N c0 / N h0) ¼ 0,11. The parameter labeling the curves is b for (T c /
T h) ¼ 0,03 and power law (PL) distribution, by opposition to the monosized
one. (b) The DLs potential profile /ðnÞ versus n associated with the Sagdeev
potential in Fig. 6(a).
033707-6 Ishak-Boushaki, Annou, and Bharuthram Phys. Plasmas 19, 033707 (2012)
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e of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 69.161.253.236 On: Thu, 17
M. Ishak-Boushaki, S. Bahamida, and R. Annou, Phys. Plasmas 10, 3418
(2003).17
M. Ishak-Boushaki, T. Daimellah, and R. Annou, in 34th EPS Conference
on Plasma Physics (Warsaw, Poland, 2007), p. 5.055.18
M. Ishak-Boushaki, R. Annou, and B. Ferhat, Phys. Plasmas 8, 5040 (2001).
19M. Ishak-Boushaki, Doctorate thesis, University of Science and Technol-ogy Houari Boumedienne, USTHB, Algiers-Algeria, 2009.
20R. Bharuthram and P. K. Shukla, Phys. Fluids 29, 3214 (1986).
21R. Bharuthram and P. K. Shukla, Planet. Space Sci. 40, 465 (1992).
22P. Leung, A. Y. Wong, and B. H. Quon, Phys. Fluids 23(5), 992 (1980).
23A. V. Gurevich, B. I. Meerson, and I. V. Rrogachevskii, Sov. J. Plasma
Phys. 11(10), 693 (1985).24
W. L. Theisen, R. T. Carpenter, and R. L. Merlino, Phys. Plasmas 1(5),
1345 (1994).25
R. L. Stenzel, W. Gekelman, and N. Wild, Geophys. Res. Lett. 9(6), 680,
doi: 10.1029/GL009i006p00680 (1982).26
M. Ishak-Boushaki and R. Annou, in 37th EPS Conference on Plasma
Physics (Dublin, Ireland, 2010), p. 2.329.27
P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics
(Institute of Physics, Bristol, 2002).28A. A. Samarian, A. V. Cheryshev, A. P. Nefedov, O. P. Petrov, Y. M. Mikhai-
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FIG. 10. The corresponding Mach numbers to the DLs structures versus b,
for N e0 ¼ 0 and ( N c0 / N h0) ¼ 0,11. The parameter labeling the curves is the ra-
tio of cool to hot ion temperatures (T c / T h).
033707-8 Ishak-Boushaki, Annou, and Bharuthram Phys. Plasmas 19, 033707 (2012)
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