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Effect of dust non‐linear charge and size‐distribution on dust-acoustic double‐layers in

dusty plasmas

M. Ishak-Boushaki, R. Annou, and R. Bharuthram 

Citation: 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 the AIP Publishing 

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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 

numerous mechanisms, e.g., currents driven instabilities,4

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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II. BASIC EQUATIONS

We consider a collisionless, unmagnetized four-compo-

nent plasma consisting of electrons having a temperature T e,

two distinct groups of ions having temperatures  T c (cold spe-

cies) and   T h   (hot species), and negatively charged dust

particulates assumed to be spheres of various radii.26

The de-

pendence of dust grain charge on its size is taken non-linear.

The number densities of electrons and ions are given by theBoltzmann distribution,

ne ¼ neo exp   e/=T eð Þ;   (1)

nc ¼ nco exp e/=T cð Þ;   (2)

nh ¼ nho exp e/=T hð Þ:   (3)

The quasi-neutrality condition is given by27

nco þ nho ¼ neo þX N 

 j ¼1

 Z djondjo;   (4)

where nco,  nho, ndjo, and  neo  are the unperturbed cold ion, hot

ion,   j th dust grain, and electron number densities, respec-tively, and   Z djo   is the unperturbed charge number of the   j th

dust grain.

Let us adopt the following normalization, viz., the space

coordinate   x   is normalized by the effective Debye length

k Dd  ¼   T eff =4p Z ontot e2

1=2, the dust fluid velocity is normal-

ized by the effective dust acoustic speed defined by

Cd  ¼   Z oT eff =mo

1=2, time   t   is normalized by the effective

dusty plasma period defined by  x1 pd  ¼   mo=   4p Z 2o ntot e

2 1=2

,

the dust density is normalized by  ntot  ¼P N 

 j ¼1 ndjo   (total num-

ber density of all dust grains), the ion and electron densities

 N so ¼ nso=ntot  Z o  are normalized by ntot  Z o, and the electrostaticpotential   U   is normalized by (T eff = Z oe), where

ð Z 2o ntot =T eff Þ ¼ ½neo=T e þ nco=T c þ nho=T h   and   as ¼ T eff = Z oT s   (for each species). Moreover, the dust charge   Z dj   and

mass mdj  are normalized by the charge and mass corresponding

to the grain of the most probable radius   r o, viz.,  Z o ¼  Z ðr oÞand   mo ¼ mðr oÞ. The quasi-neutrality reads then,

ae N eo þ ac N co þ ah N ho ¼ 1, where  N s   is the normalized den-

sity for the species (s).

The above mentioned normalization taken into account,

the dynamics of grains is governed by the continuity and mo-

mentum equations, namely,

@  N dj 

@ t   þ   @ 

@  x  udj  N dj 

¼ 0;   (5)

@ udj 

@ t  þ udj 

@ udj 

@  x ¼   Z dj 

mdj 

@ /

@  x :   (6)

The closure relationship is nothing but Poisson’s equation,

@ 2/

@  x2 ¼  N e  N c  N h þ

X j 

 Z dj  N dj :   (7)

Assuming the physical quantities to depend on  n ¼  x  Mt ,

where   M   is the Mach number, the stationary solutions of 

Eqs. (5) and (6) are given by

 N dj  ¼   M  ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi M 2 þ 2/ð Z dj =mdj Þ

p    ;   (8)

where we have used the boundary conditions,  /; ðd /=d nÞ;u j  ! 0, and N dj  ! 1 corresponding to unperturbed plasmas at

n ! 1.

Substituting for the particle number densities from Eqs.

(1) – (3) and Eq. (8) into Eq. (7), then integrating the resultingequation, we obtain

1

2

d /

d n

2

þwð/; M Þ ¼ 0;   (9)

where

wð/; M Þ ¼  N eo

ae

1 expðae/½ þ N co

ac

1 expðac/½

þ N ho

ah

1 expðah/½ W ð/Þ   (10)

is the Sagdeev potential, where

W ð/Þ ¼X

 j 

 M 2mdj 

 ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi1 þ 2

  Z dj 

mdj 

  /

 M 2

s   1

" #:   (11)

When the size distribution is continuous, discrete summation

is replaced by an integration, and Eq. (11) reads as

W ð/Þ ¼  M 2ntot 

ð r d 2

r d 1

r 3d 

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi1 þ 2r 

b3d 

/

 M 2

r   1

" # f d ðr d Þdr d ;   (12)

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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When dust size-distribution and nonlinear dependence

of dust charge on its size are taken into account simultane-

ously, the charge quasi-neutrality reads

 N co þ N ho ¼  N eo þ gbð p; r mÞ;where

gbð p; r mÞ ¼ C p

1

r  pþðbþ1Þ

m

 p ðbþ 1Þ ;   for   ð p 6¼ 1;bþ 1Þ:   (13)

For instance, in meteor plasmas, one has  p ¼ 4, yielding

gbð p ¼ 4; r mÞ ¼   33b

  1r 

ðb3Þm

1r 3m

. Besides, when all the dust

grains are of the same size and have a charge linearly de-

pendent on its size, i.e.,  r m ! 1 and  b ! 1, we retrieve the

mono-sized case,33 gbð p; r mÞ ¼ 1.

Furthermore, the Sagdeev Potential in Eq.   (10)   reads

then as follows:

wð/; M Þ ¼  N eoae

  1 expðae/Þ½ þ N coac

  1 expðac/Þ½ þ N hoah

  1 expðah/Þ½

C p M 2ð rm

1

r ðb5Þ=2

 ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffir 3b þ 2/= M 2

q   dr 

ð rm

1

dr 

r 

9>>=>>;:   (14)

The formation of a double layer demands (c.f. Refs.  20, 21,

and 34)

aÞwð0; M Þ¼@ /V ð0; M Þ¼0 for all   M bÞwð/m; M Þ¼@ /V ð/m; M Þ¼0 for some  /m; M 

cÞwð/; M Þ<0 for    M in ðbÞ and 0<  /j j<  /mj j

9>=>;:   (15)

A. The size linearly dependent grain capacitance case(b ¼ 1)

For size-distributed grains according to a power-law,where   p ¼ 4 (meteor plasma), integration of Eq.  (14)   leads

to the following Sagdeev potential:

wð/; M Þ ¼  N eo

ae

1 expðae/Þ½ þ N co

ac

1 expðac/Þ½ þ N ho

ah

1 expðah/Þ½

C4 M 2 lnr m þ  ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi

r 2m þ 2/= M 2p 

1 þ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi

1 þ 2/= M 2p 

" #

 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffir 2m þ 2/= M 2

p   r m

þ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

1 þ 2/= M 2p 

  lnðr mÞ !

;   (16)

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

p   r m

þ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

1 þ H ð/mÞp 

  lnðr mÞ!

¼ 0;   (18a)

 M 2 ¼   2/m

 H ð/mÞ ;   (18b)

where

 Að/mÞ ¼   N eo

ae

1 expðae/mÞ½ þ N co

ac

1 expðac/mÞ½

þ N ho

ah

1

exp

ðah/m

Þ½ ;   (19a)

 H ð/mÞ ¼   C4

r 2m B2ð/mÞ   C4 þ C4r 2m þ 2r m ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi

C24 þ r 2m B2ð/mÞ

q  ;

(19b)

and

 Bð/mÞ ¼  N eo expðae/mÞ þ N co expðac/mÞ½þ N ho expðah/mÞ:   (19c)

For a given set of density and temperature, the resolution of 

Eqs.  (18a)  and (18b)   yields the value of  /m   along with the

associated Mach number M.

Furthermore, the limiting condition   @ 2V ð/; M Þ@ /2   < 0 at

/ ¼ 0 and   / ¼ /m   imposes a range of acceptable Mach

numbers, given by the following inequalities:

 M 2 >  C4

4

  1

ae N e0 þ ac N c0 þ ah N h0

C4

4  ;   (20a)

 M 2 > C4

 Fð H ð/mÞÞ Dð/mÞ   ;   (20b)

where

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 Fð H ð/mÞÞ ¼   1

ðr 2m þ H ð/mÞÞ3=2ðr m þ  ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffir 2m þ H ð/mÞp    Þ

þ   1

ðr 2m þ H ð/mÞÞðr m þ  ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffir 2m þ H ð/mÞp    Þ2

(

  1

r mðr 2m þ H ð/mÞÞ3=2   1

ð1 þ H ð/mÞÞð1 þ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffi

1 þ H ð/mÞp 

  Þ2

  1

ð1 þ H ð/mÞÞ3=2ð1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffi

1 þ H ð/mÞp 

  Þ þ  1

ð1 þ H ð/mÞÞ3=2)

  (21a)

and

 Dð/mÞ ¼   N eoae expðae/mÞ þ N coac expðac/mÞ½þ N hoah expðah/mÞ:   (21b)

B. The size non-linearly dependent grain capacitancecase (b 6¼ 1)

Let us now investigate the effect of the non-linear de-

pendence of the grain charge on the grain size, namely, we

consider the following cases,   b ¼ 1:3,   b ¼ 1:5, and

b ¼ 1:83. The integration of Eq. (14) leads to the following

expression of the Sagdeev potential:

wbð/; M Þ ¼  N eo

ae1 expðae/Þ½

þ N co

ac

1 expðac/Þ½ þ N ho

ah

1 expðah/Þ½ C p M 2  I bð/; M Þ lnðr mÞ

;   (22)

where   I bð/; M Þ¼ RðbÞ"   ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 2/= M 2p 

   ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffi

r ð3bÞm   þ2/= M 2

r ð3bÞm

r 

þ ln  r 

ð3bÞ=2m   þ

 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.

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Applying the condition (b) in Eq. (15), we obtain

 Að/mÞþC4 M 2  2/m

 H bð/mÞ   RðbÞ   lnr ð3bÞ=2m   þ

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir ð3bÞm   þ H bð/mÞ

q 1þ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ H bð/mÞp 

2

4

3

5

0

@

8<:

 ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffir ð3bÞm   þ H bð/mÞ

q r ð3bÞ=2m

þ ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi

1 þ H bð/mÞq  1

A lnðr mÞ9=; ¼ 0

(24a)

and

 M 2 ¼   2/m

 H bð/mÞ ;   (24b)

where   /m   is the double layer amplitude associated to the

Mach number  M , and

 Að/mÞ ¼   N eo

ae

1 expðae/mÞ½

þ N co

ac

1 expðac/mÞ½ þ  N ho

ah

1 expðah/mÞ½ ;

(25a)

 H bð/mÞ ¼   C4 RðbÞr ð3bÞm   B2ð/mÞ

C4 RðbÞ þ C4 RðbÞr ð3bÞm

þ2r ð3bÞ=2

m

 ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiC2

4 R2ðbÞ þ r 

ð3bÞm   B2ð/mÞ

q  ;   (25b)

and

 Bð/mÞ ¼  N eo expðae/mÞ þ N co expðac/mÞ½þ N ho expðah/mÞ;   (25c)

whereas the concavity condition of the Sagdeev potential

curve, viz.,  d 2wð/; M Þ

d /2   < 0 at  /

¼0 and  /

¼/m, leads to a

modified range of acceptable Mach numbers, namely,

FIG. 4. (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 the ratio of cool

to hot ion temperatures (T c / T h) for  b ¼ 1; 5. (b) The DLs potential profile

/ðnÞ versus n  associated with the Sagdeev potential in Fig. 4(a).

FIG. 3. (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; 3. (b) The DLs potential profile

/ðnÞ versus n  associated with the Sagdeev potential in Fig.  3(a).

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 M 2 >  C4 Rðb; r mÞ

ae N e0 þ ac N c0 þ ah N h0

C4 Rðb; r mÞ   (26a)

and

 M 2 > C4 RðbÞ Gð H bð/mÞÞ D

ð/m

Þ  ;   (26b)

where

 Rðb; r mÞ ¼  RðbÞ   13

18þ   2

9r ð3bÞm

  5

9r 2ð3bÞm

( );

Gð H bð/mÞÞ ¼   1

ðr ð3bÞm   þ H bð/mÞÞ3=2ðr 

ð3bÞ=2m   þ

 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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and

 Dð/mÞ ¼   N eoae expðae/mÞ þ N coac expðac/mÞ½þ N hoah expðah/mÞ:   (27b)

III. RESULTS AND DISCUSSIONS

We consider a dusty plasma model in which most of back-

ground electrons are collected by the negatively-charged dust

grains, a situation quite realistic in environments such as Sat-urn’s F-ring.35

Thus, without loss of generality, we set  N e0¼ 0

and calculate typical forms of the Sagdeev potential as well as

the associated double layer structures. Setting   N eo   strictly to

zero allows us to closely examine the role played by the tem-

perature ratio (T c / T h) of the two ion populations along with the

nonlinear dependence of the dust grain charge on its size,

when the power law is considered to describe the distribution

in size of the charged dust grains. Figures 1(a) and 1(b) depict

the Sagdeev quasi-potential  wð/Þ   and the associated double

layer structure /ðnÞ, respectively, for (b¼ 1) and different val-

ues of cold ion temperature to hot ion temperature ratio, viz.,

ðT c=T h

Þ¼0.03, 0.04, and 0.05, where   N eo

¼0, ( N co / 

 N ho) ¼ 0.11, ac ¼ ½ N h0 þ N c0ðT h=T cÞ1

, and ah ¼ acðT h=T cÞ.

The variation of the DL amplitude   /m   and the Mach

number M with respect to the ratio ðT c=T hÞ   is depicted in

Figure 2, where a comparison is made between the monosize

FIG. 7. (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   b   for (T c / 

T h)

¼0,04 and PL distribution, by opposition to the monosized one. (b) The

DLs potential profile  /ðnÞ versus n  associated with the Sagdeev potential inFig. 7(a).

FIG. 8. (Color online) (a) Sagdeev potential V(/, M ) versus /  for  N e0¼ 0 and

( N c0 /  N h0) ¼ 0,11. The parameter labelling the curves is   b   for (T c / T h) ¼ 0,05

and PL distribution, by opposition to the monosized one. (b) The DLs potential

profile /ðnÞ versus n  associated with the Sagdeev potential in Fig. 8(a).

FIG. 9. The DLs amplitude /m  versus  b, for  N e0¼ 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).

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case and the power law distribution case (b¼ 1). It is shown

that for a given value of the ratio ðT c=T hÞ, the DL is stronger 

(/m  is higher) for size-distributed grains in opposition to the

uniform grain size case, whereas the requirements on the

Mach number for DLs existence are lowered. Moreover, it is

revealed that a new feature of a size distributed population

of grains, that is, subsonic solutions are allowed in this

model, whereas in the monosized case, only supersonic solu-

tions are predicted.

When the dust size-distribution and the nonlinear de-

pendence of dust charge on its size (1 <b<2) are taken

into account simultaneously, typical forms of the Sagdeev

quasi-potential wð/Þ  and the associated double layer struc-

tures  /ðnÞ  are calculated and plotted in Figures  3(a),   3(b),

4(a), 4(b), 5(a), and 5(b) for different values of  b  and differ-

ent ratios ðT c=T hÞ. Whereas in Figures 6(a), 6(b), 7(a), 7(b),

8(a), and   8(b), we illustrate the Sagdeev quasi-potential

wð/Þ and the associated double layer structure /ðnÞ, respec-

tively, for different ratios ðT c=T hÞ and different values of  b,

by opposition to the monosized case. As the Sagdeev poten-

tial  wð/Þ  along with associated DL structures depend on  b

and  a, we kept  b  constant and varied a, then kept  a  constant

and varied   b   in an attempt to illustrate the effect of ener-

getic particles as well as the charge collected by the grainson  wð/Þ  and  /ðnÞ. It turns out that in a dusty plasma con-

taining size distributed grains, DLs amplitude increases

with respect to the uniform grain radius case, viz.,  /m;b¼1

/m¼

1.14 (1.18) for   ðT c=T hÞ¼ 0.05 (0.03) and  /m;b¼1:83

/m¼2.04

(2.19) for   ðT c=T hÞ¼ 0.05 (0.03). In the latter case, the

charge collected by a grain of radius   r d   is higher than the

one corresponding to a linear grain capacitance.

Moreover, in Figures (9) and   (10), the DL amplitude

/m   along with the Mach number   M  are plotted versus the

coefficient  b. It is shown that for lower values of the ratio

ðT c=T hÞ   that is for highly energetic ions, double layers are

stronger as well as the required Mach numbers for their 

formation.

IV. CONCLUSION

In this work, we present a study of the existence of arbi-

trary amplitude dust acoustic double layers in an unmagne-

tized dusty plasma with a couple of Boltzmann distributed

species of ions having different temperatures, i.e.,  T c  and T h,

and a cold fluid of dust grains of different sizes described by a

continuous power law distribution with a nonlinearly size-de-

pendent charge. Most of the background electrons are col-lected by the negatively-charged dust grains. Such plasmas

may exist in both laboratory and space environments. The

results of this paper confirm that only compressive DLs are

possible, not only in the particular case of monosized grains

(c.f. Ref. 20) but also in the general case when the grain size

distribution and the non-linear dependence of charge on the

grain size are taken into account. Besides, it is shown that the

size distribution of dust grains enhances the double layer struc-

ture as the amplitude  /m  increases. Moreover, as the non-lin-

ear dependence of the grain charge upon the size (capacitance)

increases, that amplitude   /m   increases further. The double

layer amplitude increase may well be an efficient tool for par-

ticles acceleration. Finally, it is worthwhile noting that the

introduction of the dust grain size distribution allows the exis-

tence of subsonic double layers, while in the monosized grains

case, only supersonic solutions are to propagate in the plasma.

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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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