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Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013, Article ID 389365, 6 pageshttp://dx.doi.org/10.1155/2013/389365
Research ArticleEffect of Size Distribution on the Dust Acoustic Solitary Wavesin Dusty Plasma with Two Kinds of Nonthermal Ions
Samira Sharif Moghadam1 and Davoud Dorranian2
1 Physics Department, Science Faculty, North Tehran Branch, Islamic Azad University, Tehran, Iran2 Laser Lab., Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Tehran, Iran
Correspondence should be addressed to Samira Sharif Moghadam; [email protected]
Received 24 May 2013; Accepted 14 July 2013
Academic Editor: Mahmood Ghoranneviss
Copyright Β© 2013 S. Sharif Moghadam and D. Dorranian. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.
Effect of dust size,mass, and charge distributions on the nonlinear dust acoustic solitarywaves (DASWs) in a dusty plasma includingnegatively charged dust particles, electrons, and nonthermal ions has been studied analytically. Dust particles masses and electricalcharges are assumed to be proportional with dust size. Using reductive perturbation methods the Kadomtsev-Petviashvili (KP)equation is derived and its solitary answers are extracted.The coefficients of nonlinear term of KP equation are affected strongly bythe size of dust particles when the relative size (the ratio of the largest dust radius to smallest dust radius) is smaller than 2. Thesecoefficients are very sensitive to πΌ, the nonthermal coefficient. According to the results, only rarefactive DASWs will generate insuch dusty plasma. Width of DASW increases with increasing the relative size and nonthermal coefficient, while their amplitudedecreases. The dust cyclotron frequency changes with relative size of dust particles.
1. Introduction
The study of dusty plasmas represents one of themost rapidlygrowing branches of plasma physics. A dusty plasma is anionized gas containing small particles of solid matter, whichacquire a large electric charge by collecting electrons andions from the plasma. Dust grains or particles are usuallyhighly charged. Charged dust components appear naturallyin space environments such as planetary rings, cometarysurroundings, interstellar clouds, and lower parts of earthβsionospheres [1β4]. Fusion reactors, film deposition reactors,and ions courses are other examples of dusty plasma systemsin laboratory. Since dust particles are relatively massive, thefrequency of DASW is typically few Hz and propagates ata speed of a few cm/s. This wave is spontaneously excitedin the plasma due to ion drift relative to the dust. In mostinvestigations, reductive perturbation method has been usedfor deriving the Korteweg de Vries (KdV) or modified KdVequations in one-dimensional studies [5β8] or Kadomstev-Petviashvili (KP) equation for two-dimensional cases [9, 10].
There aremany reports on the formation and propagationof DASW in dusty plasma. But far from ideal condition, thereare several phenomena which affect the DASW. Presence ofnonthermal particles and size distribution are two of them.A nonthermal plasma is in general any plasma which isnot in thermodynamic equilibrium, either because the iontemperature is different from the electron temperature, orbecause the velocity distribution of one of the species doesnot follow a Maxwell-Boltzmann distribution. The effects ofdust temperature and dust charge variation in a dusty plasmawith nonthermal ions have been investigated by Duan [11].Pakzad and Javidan have worked on the problem of variabledust charge in a dusty plasmawith two-temperature ions withnonthermal electrons [12, 13]. Dorranian and Sabetkar haveinvestigated the characteristic features of dust particles in adusty plasma with two ions at two different temperaturesand have observed the effect of nonthermal ions on theamplitude and width of DASW [14]. All of these theorieshave focused on dusty plasma containing only monosizeddust grains. In fact, the dust particles have many different
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2 Advances in Materials Science and Engineering
species of size [15β17]. The dust size distribution can bedescribed by a power law distribution in space plasma andgiven by a Gaussian distribution in laboratory plasma [18].The effects of dust size distribution on the dust acousticwaves,solitons, or instabilities in a dusty plasma have been studiedby several researchers previously [15, 17]. The effect of dustsize distribution for two ion temperature dusty plasma inone dimension (KdV equation) was studied by Duan and Shi[19]. He et al. observed the effect of dust size distribution onthe nature of DASW in two dimensions (KP equation) [20].In this work the collective behavior of dust particles is notconsidered, and the effect of relative size is not taken intoaccount.
Continuing the former works, we have added the effectof dust size distribution to the model which contains twononthermal ions at two temperatures, including Maxwellianelectrons. In this paper, characteristic features of dust par-ticles with variable size, mass, and electric charge in a non-thermal two-temperature ions dusty plasma are investigated.The mass and charge of dust particles are assumed to beproportional with their size.The paper is organized as follow:following the introduction in Section 1, in Section 2, wepresent the model description and the theoretical aspectsof the model. The KP equation and its solitary answer arederived and described in this section. Section 3 is devoted toresults and discussion, and Section 4 includes the conclusion.
2. Model Description
2.1. Basic Equations. We consider the propagation of DASWin collisionless, unmagnetized dusty plasma consisting ofhighly negatively charged dust grains, electrons, and two-temperature nonthermal ions. The dust particles are muchlarger and extremely more massive than ions or electrons,but the sizes of dust grains are different. We shall assumethat there are π number of different dust grains sizes whosemasses are πππ(π = 1, 2, . . . π). In this case we have differentspecies of dusts with different size, which is counted withπ. Plasma is assumed to be quasineutral. The state of chargeneutrality can be written as
ππ0 +
π
β
π=1
πππ0πππ0 β πππ0 β ππβ0 = 0 (1)
in which ππ0, πππ0, and ππβ0 are the unpertubrbed number den-sity of electrons, low-temperature ions, and high-temperatureions, respectively. πππ0 is the unperturbed number of chargesresiding on the πth dust grainmeasured in the unit of electroncharge. The dust grains are assumed to be much smallerthan electron Debye length πππ as well as the intergraindistance. Thus, we can treat the dust grains as heavy pointmasses with constant negative charge [21]. The governing
equations including the normalized Poissonβs equation, con-tinuity equation, and the equation of motion describing thedynamics of the dust particles in the plasma are
π2π
ππ₯2+
π2π
ππ¦2= ππ +
π
β
π=1
ππππππ β πππ β ππβ, (2a)
ππππ
ππ‘
+
π (ππππ’ππ)
ππ₯
+
π (πππVππ)
ππ¦
= 0, (2b)
ππ’ππ
ππ‘
+ π’ππ
ππ’ππ
ππ₯
+ Vππππ’ππ
ππ¦
=
π§ππ
πππ
ππ
ππ₯
, (2c)
πVππππ‘
+ π’ππ
πVππππ₯
+ VπππVππππ¦
=
π§ππ
πππ
ππ
ππ¦
. (2d)
In these equations, π is the electrostatic potential; ππ, πππ, andππβ are the number density of electrons, low-temperature ionsand high-temperature ions respectively. πππ and πππ are thecharge number and the density number of the πth species ofdust particles, respectively. π is the basic charge magnitude.π’ππ and Vππ are the components of the velocity of πth speciesof dust particles in the π₯- and π¦-directions, respectively. Forthe case of dusty plasma with two kinds of ions at differenttemperatures, πeff, the effective temperature is
πeff = [1
πtotπ§π0(
ππ0
ππ
+
πππ0
πππ
+
ππβ0
ππβ
)]
β1
, (3)
in which ππ, πππ, and ππβ are the plasma electron temperatureand the temperatures of plasma ions at lower and highertemperatures, respectively. πtot is the total number densityof dust grains which can be written as πtot = β
π
π=1ππ0π; π§π0
is determined by the equation π§π0πtot = βπ
π=1ππ0ππ§π0π. π is
the average radii of dust size given by π = βπ
π=1ππ0πππ/πtot.
The dust density is normalized by πtot; π§πππ is normalizedby π§π0. The space coordinates π₯ and π¦, time π‘, the compo-nents of dust velocity π’ππ, Vππ, and the electrostatic potentialπ are normalized by the effective Debye lenght ππ·π =
(πeff/4ππtotππ0π2)1/2, the inverse of effective dust plasma
frequency πβ1ππ
= (ππ/4ππtotππ02
π2)1/2, the effective dust
acoustic speed πΆπ = (π§π0πeff/ππ)1/2, and πeff/π, respectively,
where ππ is the average mass of dust grains determined bythe equation of ππ = β
π
π=1ππ0ππππ/πtot; πππ is normalized
byππ.Ions in the plasma are assumed to be nonthermal, so their
density can be described by the known Maxwell Boltzmanndistribution function. For the dimensionless density number
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Advances in Materials Science and Engineering 3
of electrons, ππ, lower temperature ions, πππ, and highertemperature ions, ππβ, we have
ππ =
1
πΏ1 + πΏ2 β 1
exp (π½1π π) , (4a)
πππ =
πΏ1
πΏ1 + πΏ2 β 1
(1 +
4πΌ
1 + 3πΌ
π π +
4πΌ
1 + 3πΌ
(π π)2) exp (βπ π) ,
(4b)
ππβ =
πΏ2
πΏ1 + πΏ2 β 1
Γ (1 +
4πΌ
1 + 3πΌ
π½3π π +
4πΌ
1 + 3πΌ
(π½3π π)2) exp (βπ½3π π) ,
(4c)
in which π½1 = πππ/ππ, π½2 = ππβ/ππ, π½3 = π½1/π½2, πΏ1 = πππ0/ππ0,πΏ2 = ππβ0/ππ0, and π = πeff/πππ = πΏ1 + πΏ2 β 1/πΏ1 + πΏ2π½3 + π½1. πΌis the nonthermal parameter that determines the number offast (nonthermal) ions [22, 23].
2.2. The KP Equation. In this part by using the basicequations, the so called KP equation in the plasma withvariable dust particle electric charge is derived. KP equationis a fundamental equation, and its solution describes thecharacteristic of DASW in detail. In order to simplify theequations based on the reductive perturbation theory, it isuseful to introduce three independent variables [24]:
π = π (π₯ β ππ‘) , (5a)
π = π3π‘, (5b)
π = π2π¦, (5c)
in which π is the dimensionless expansion parameter andcharacterizes the strength of the nonlinearity of the system.π is the phase velocity of the wave along the π₯ direction.Different parameters of the dust particles can be expandedin terms of π as
πππ = πππ0 + π2πππ1 + π
4πππ2 + β β β , (6a)
π’ππ = π2π’ππ1 + π
4π’ππ2 + β β β , (6b)
Vππ = π3Vππ1 + π
5Vππ2 + β β β , (6c)
π = π2π1 + π
4π2 + β β β .
(6d)
By substituting (5a), (5b), (5c), (6a), (6b), (6c), and (6d) in(2a), (2b), (2c), and (2d) and collecting the coefficient of thelowest order of π, we have
πππ1 = β
πππ0π§ππ
π2πππ
π1, π’ππ1 = β
π§ππ
ππππ
π1,
πVππ1ππ
= β
π§ππ
ππππ
ππ1
ππ
,
(7)
π =[
[
π
β
π=1
πππ0π§2
ππ
πππ
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
]
]
1/2
. (8)
Tending πΌ β 0 results are the same as the resultspresented by He et al. [20], and neglecting the effect ofsize distribution, equations are the same with the resultspublished by Dorranian and Sabetkar [14]. For the higherpower of π, we have
ππππ1
ππ
β π
ππππ2
ππ
+
π (πππ0π’ππ2)
ππ
+
π (πππ1π’ππ1)
ππ
+
π (πππ0Vππ1)
ππ
= 0,
(9a)
ππ’ππ1
ππ
β π
ππ’ππ2
ππ
+ π’ππ1
ππ’ππ1
ππ
=
π§ππ
πππ
ππ2
ππ
(9b)
πVππ1ππ
β π
πVππ2ππ
+ π’ππ1
πVππ1ππ
=
π§ππ
πππ
ππ2
ππ
, (9c)
π2π1
ππ2=
π
β
π=1
πππ2π§ππ +
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
π2
β
π 2(πΏ1 + πΏ2π½
2
3β π½2
1)
2 (πΏ1 + πΏ2 β 1)
π2
1.
(9d)
Using (9a), (9b), (9c), and (9d) the KP equation for aunmagnetized dusty plasma with two ions can be derived as
π
ππ
[
ππ1
ππ
+ π΄π1
ππ1
ππ
+ π΅
π3π1
ππ3] + πΆ
π2π1
ππ2= 0, (10)
in which
π΄ =
π
2
(πΏ1 + πΏ2π½2
3β π½2
1) (πΏ1 + πΏ2 β 1)
(πΏ1 + πΏ2π½3 + π½1)2
(πΏ1 + πΏ2π½2
3β π½2
1)
(πΏ1 + πΏ2 β 1)
Γ
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
β
3
2π3
π
β
π=1
πππ0π§3
ππ
π2
ππ
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
,
(11)
π΅ =
π
2
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
, (12)
πΆ =
π
2
. (13)
For dust grains with radii π in a given range [πmin, πmax], thedifferential polynomial expressed distribution function is ofthe form [19]
π (π) ππ = πΎπβπ½ππ. (14)
It is found that values of π½ = 4, 5 for the F-ring ofSaturn, while for the G-ring values of π½ = 7 or 6. For
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4 Advances in Materials Science and Engineering
cometary environments, we recall a value of π½ = 3, 4. πΎ isa normalization constant, and π is the radii of dust particle,which should satisfy the following equation [19]:
πtot = β«πmax
πmin
π (π) ππ. (15)
Outside the limits πmin < π < πmax, we have ππ = 0. Inthe equations π§ππ = ππ§ππ and πππ = πππ
3
π, ππ§ = 4ππ0V0/π
is the electrical potential of the surface of dust particles atequilibrium. ππ = 4/3πππ, in which ππ is the mass density ofdust particles, ππ is taken to be constant for all dust particles,V0 is the electric surface potential at equilibrium, and π0 isthe vacuum permittivity [25]. Integrating (8), (11), using theprevious definitions, we have
π = [
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
Γ
πΎπ2
π§
πππ½
(πβπ½
min β πβπ½
max)]
1/2
,
π΄ =
π
2
(πΏ1 + πΏ2π½2
3β π½2
1) (πΏ1 + πΏ2 β 1)
(πΏ1 + πΏ2π½3 + π½1)2
(πΏ1 + πΏ2π½2
3β π½2
1)
(πΏ1 + πΏ2 β 1)
Γ
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
+
3πΎπ3
π§
2π3π2π(π½ + 2)
(πΏ1 + πΏ2π½3 + π½1) (1 + 3πΌ)
(1 + 3πΌ) π½1 + (1 β πΌ) (πΏ1 + πΏ2π½3)
Γ (πβπ½β2
max β πβπ½β2
min ) .
(16)
The general steady state solution of (10) for π1 has the form[14]
π1 = ππsech2(
π
π€
) , (17)
in which the new variable π = π + π β π’π is the transformedcoordinate relative to the frame which moves with thevelocity π’ [14]; ππ = 3(π’βπΆ)/π΄ andπ€ = 2βπ΅/(π’ β πΆ) are theamplitude and width of soliton, respectively. The coefficientsof KP equation would change to the results presented in[14, 20] in case neglecting of nonthermal effects and sizedistribution effects, respectively.
3. Results and Discussion
Effects of ions temperature and density on the coefficientsof KP equation are very small. In other words, variations ofions temperature and density lead to negligible effects on thecoefficients of KP equation in this model.
To plot our data, πmin is taken to be unity. In order toavoid the effects of ions density and temperature, π½1 and π½2are taken to be equal to 0.25, and πΏ1 and πΏ2 are both taken tobe 2.
Γ106
A
0
β2
β4
β6
β86
42
0 00.5
1
πΌc
Figure 1: Variation of π΄ versus π and πΌ.
1500
1000
500
06
42
0 00.5
1
πΌ
B
42 0 5
c
Figure 2: Variation of π΅ versus π and πΌ.
For a typical dusty plasma [19], ππ0 βΌ 105β1010 cmβ3,
ππ βΌ ππ βΌ 0.1 eV, ππ0 βΌ 105 cmβ3, ππ βΌ (β10
4 e)β(β105 e),ππ βΌ 10
β15β10β12 π βΌ 109β1012m, ππ βΌ 1 g/cm3, and
π βΌ 0.1β1 πm, so it can be estimated that ππ = 4, ππ§ = 108,
andπΎ = 10β12. Here π½ is taken to be 6.
The variation of the nonlinear coefficient, π΄, with themaximum dust size ratio π = πmax/πmin and the nonthermalcoefficient πΌ is presented in Figure 1. π΄ increases withincreasing π. For all the ranges of 1 < π < 5 and 0 < πΌ < 1, thenonlinear coefficient π΄ is negative, mean that in this range ofvariation only rarefactive solitons will be generated in dustyplasma, and it is not useful to continue our work for modifiedKP equation. This point is similar with the presented resultsin [20] in which the effect of size distribution is taken intoaccount. But in Dorranian and Sabetkarβs work [14], in thepresence of thermal effect without size distribution effect, justin the case of small πΌ, rarefactive soliton will appear in thedusty plasma medium. As the function of π, π΄ varies fasterwhen 1 < π < 2. When the magnitude of the maximum ratioof dust size is larger than 2,π΄ is approximately constant.Withincreasing πΌ also π΄ increases. Variation of π΄ with πΌ is largerwhen 1 < π < 2. With increasing π, the effect of πΌ decreases.Effect of πΌ is noticeable when π is small.
Effects of π and πΌ on π΅ are presented in Figure 2. Incontrast with π΄, effect of πΌ on π΅ is noticeable when π is large.
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Advances in Materials Science and Engineering 5
80
60
40
20
064
20 0
0.51
πΌ
C
c
Figure 3: Variation of πΆ versus π and πΌ.
For π β 1, the coefficient π΅ is not changed with πΌ. The sameresult can be seen for π. When πΌ is small, π΅ is not changedwith π.When nonthermality is high, increasing themaximumratio of dust size leads to increasing the coefficient π΅ of KPequation.
The same effect are occurred for the coefficient πΆ. Effectof π on πΆ is noticeable when πΌ tends to 1, and πΆ increases fastwith πΌ when π tends to 5 as is shown in Figure 3.
Width of generated soliton in this model stronglydepends on π and πΌ.This dependence is shown in Figure 4. Inany case, solitons are broadened with increasing π and πΌ. Thebroadest soliton generates when πΌ and π are maximum at thesame time.The same results have been obtained inDorranianand Sabetkarβs paper when the effect of size distribution wasnot taken into account [14]. The rate of increasing solitonwidth with πΌ is smaller in the presence of size distributioneffect.
Profiles of generated solitons in dusty plasma regardingthe requirements of our model are presented in Figures 5and 6. In Figure 5, the effect of πΌ is shown, when π = 5.With increasing πΌ, the width of generated solitons increaseswhile their amplitude decreases. This result is in goodagreement with the results of works in which the effect ofsize distribution is neglected [14]. With increasing π also theamplitude of generated solitons decreases but their width isincreased.
πΌ is the nonthermal coefficient which changes the distri-bution function of ions from Boltzman distribution. Increas-ing πΌ leads to increasing the energy of ions in the dustyplasmamedium or decreasing the dust particles energy.Withdecreasing the energy of dust particles, the amplitude ofsolitons decreases. Less energetic particles aremore localized,so the width of generated solitons increases with increasingπΌ.
π is the ratio of the maximum dust particle size tothe minimum dust particle size. With increasing π thewidth of the size distribution function of dust particles isincreased, while their amplitude decreases. Increasing π isactually increasing the dust species. In this case, we havelarger variety of dust particles accompanied in solitonicoscillation. Increasing the ratio of dust particles size leadsto increasing the perturbation in their uniform motion. Theresult will be reducing the amplitude of solitons. Different size
201510505
43
2 1 0 0.2 0.4 0.6 0.8 1
πΌ
w
c
Figure 4: Variation of width versus π and πΌ.
β5 β4 β3 β2 β1 0 1 2 3 4 50
0.5
1
1.5
2Γ10β3
πΌ = 0.4
πΌ = 0.6
πΌ = 1
π
π
Figure 5: The profile of soliton at different πΌs, when π = 5.
particles moving with different velocities lead to generatingless localized oscillations.
4. Conclusion
In the present paper, nonlinear dust acoustic solitary waves(DASWs) in a cold unmagnetized dusty plasma containingnegatively charged dust particles, electrons, and nonthermalions are investigated. The size of dust particles is takento be variable, and the charge and mass of dust particlesare supposed to be proportional with their size. For thispurpose, a reasonable normalization of the hydrodynamicand Poissonβs equations is used to derive the Kadomstev-Petviashvili (KP) equation for our dusty plasma system.
In this model DASWs are found to be rarefactive forall magnitudes of the maximum ratio of dust size π andπΌ. In this case, soliton nature is under the influence ofsize distribution rather than nonthermal ions, since in thepresence of nonthermal ions without taking into account theeffect of size distribution in both rarefactive and compressivesolitons have been observed [14]. But, the results of themodelin which the effect of size distribution is taken into account isonly generation of rarefactive solitons is reported [26].
For 1 < π < 5, amplitude of DASW varies noticeablywith π rather than temperature and density of two species ofdusts particles. With increasing π, the amplitude of DASWdecreases sharply, and for π > 2 it is almost independent ofπ. With decreasing the nonthermal coefficient, the amplitude
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6 Advances in Materials Science and Engineering
β10 β8 β6 β4 β2 0 2 4 6 8 10β2
β1.8
β1.6
β1.4
β1.2
β1
β0.8
β0.6
β0.4
β0.2Γ10β5
c = 2
c = 3
c = 4
π
π
Figure 6: The profile of soliton at different πs, when πΌ = 0.2.
of DASW increases. The same results have been reportedby Dorranian and Sabetkar [14]. Decreasing the ratio ofmaximum size of dust to its minimum magnitude leads togeneration of more localized solitons, that is, decreasing thewidth of DASW. The most energetic, localized solitons arewhen π = 1 that is, there is not any size, mass, or chargedistribution for dusts. Since in this case the dusty plasmamedium is uniform, it leads to formation of less perturbationin the medium.
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