Nonlinear dust-ion-acoustic waves in a multi-ion plasma with trapped electrons

— journal of August 2011

physics pp. 357–368

Nonlinear dust-ion-acoustic waves in a multi-ion plasma with trapped electrons

S S DUHA^1,∗, B SHIKHA²and A A MAMUN¹

1Department of Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh

2Department of Mathematics, Shaikh Burhanuddin Postgraduate College, Dhaka-1100, Bangladesh

∗Corresponding author. E-mail: shadab22@yahoo.com; sduha@yahoo.com

MS received 13 September 2010; revised 22 December 2010; accepted 12 January 2011

Abstract. A dusty multi-ion plasma system consisting of non-isothermal (trapped) electrons, Maxwellian (isothermal) light positive ions, warm heavy negative ions and extremely massive charge fluctuating stationary dust have been considered. The dust-ion-acoustic solitary and shock waves associated with negative ion dynamics, Maxwellian (isothermal) positive ions, trapped electrons and charge fluctuating stationary dust have been investigated by employing the reductive pertur- bation method. The basic features of such dust-ion-acoustic solitary and shock waves have been identified. The implications of our findings in space and laboratory dusty multi-ion plasmas are discussed.

Keywords. Dust charge fluctuation; trapped distribution; solitary waves; shock waves.

PACS Nos 52.35.Tc; 94.20.wf; 94.05.Bf

1. Introduction

Significant attention has been devoted to the study of the wave propagation in dusty multi- ion plasmas because of its vital role in understanding different types of collective processes in space environments [1–4] as well as in laboratory devices [5–10]. The presence or dynamics of charged dust grains in a plasma not only modifies the existing plasma wave spectra [11–14], but also introduces a number of novel eigenmodes, such as the dust-ion- acoustic (DIA) waves [14,15], the dust-acoustic (DA) waves [16], the dust lattice waves [17], etc. About 18 years ago, Shukla and Silin have first theoretically [15] shown that due to the conservation of equilibrium charge density ne0e+nd0Zde = ni 0e, and the strong inequality ne0 ni 0 (where ni 0, ne0 and nd0, respectively are the ion, electron and dust number density, Zd is the number of electrons residing onto the dust grain surface and e is the magnitude of the charge of an electron) a dusty plasma (with neg- atively charged static dust) supports low-frequency DIA waves with phase speed much smaller (larger) than electron (ion) thermal speed. The dispersion relation (the relation

between the wave frequencyωand the wave number k) of the linear DIA waves is [15]

ω² = (ni 0/ne0)k²C_i²/(1 + k²λ²_De), where Ci = (kBTe/mi)^1/2 is the ion-acoustic speed (with Tebeing the electron temperature, mi being the ion mass, k_Bbeing the Maxwellian constant) andλDe =(kBTe/4πne0e²)^1/2is the electron Debye radius. When a long wave- length limit (viz. kλDe 1) is considered, the dispersion relation for the DIA waves becomesω = (ni 0/ne0)^1/2kCi. This spectrum is similar to the usual ion-acoustic wave spectrum for a plasma with ni 0=ne0and Ti Te(where Tiis the ion temperature). How- ever, in dusty plasmas we usually have ni 0 ne0and Ti Te. Therefore, a dusty plasma cannot support the usual ion-acoustic waves, but can support the DIA waves of Shukla and Silin [15]. The DIA waves have also been experimentally observed [6]. The linear proper- ties of the DIA waves in dusty plasmas are now well understood from both theoretical and experimental points of view [3,6,15,18]. The nonlinear features of the DIA waves have also received a great deal of interest in understanding the basic properties of localized electro- static perturbations in space and laboratory dusty plasmas [3,6,19–22]. The DIA solitary waves (SWs) have also been investigated by several authors [20,23–26], but the works [20,23–26] are valid only for a dusty plasma with cold ions and isothermal (Maxwellian) electrons. Recently, Mamun et al [27] have studied the dust-negative ion-acoustic (DNIA) shock waves by considering Maxwellian electrons, light positive ions, heavy negative ions and extremely massive (few micron sized) charge fluctuating stationary dust. Duha [28]

has then extended the work of Mamun et al [27] to include the effect of charge fluctuating stationary positive dust on the DNIA shock structures. The works of Mamun et al [27] and Duha [28], where the shock structures are only focussed, are valid for isothermal electrons but not for non-isothermal (e.g. trapped) electron distribution which is very important for many space and laboratory plasma situations [29–33]. It has been shown that the expres- sion for the electron (dust charging) current is significantly modified by the trapped electron distribution [34,35]. This modified electron current, in turn, may cause a drastic change of the nonlinear features of the DNIA waves. Therefore, in our present work, we consider trapped and free electrons, light positive ions, warm heavy negative ions and extremely massive (few micron sized) charge fluctuating stationary dust, and study the properties of the DNIA solitary and shock waves in such a dusty multi-ion plasma.

The paper is organized as follows. The basic equations governing the unmagnetized dusty multi-ion plasmas are given in §2. The modified KdV equation and the modified Burger’s equation are derived by employing a reductive perturbation method in §§3 and 4, respectively. A brief discussion is finally presented in §5.

2. Governing equations

We consider a one-dimensional, collisionless, unmagnetized dusty multi-ion plasma system consisting of non-isothermal (trapped and free) electrons, isothermal (Maxwellian) single- charged light positive ions, warm heavy negative ions (in comparison with positive ion mass) and extremely massive (few micron sized) charge fluctuating stationary dust. We are interested in the propagation of a purely electrostatic perturbation mode on the time scale of the ion-acoustic waves. Thus, the frequency (ω) of this perturbation mode is much higher than the dust-plasma-frequency (ωpd), i.e. ωpd ω. Therefore, dust are assumed to be

stationary. The macroscopic state of such a dusty multi-ion plasma system in the presence of purely electrostatic perturbation mode is described by

∂nh

∂t + ∂

∂x(nhuh)=0, (1)

∂uh

∂t +uh

∂uh

∂x = zhe mh

∂φ

∂x − V_{T h}² nh

∂nh

∂x , (2)

∂²φ

∂x² =4πe

ne−ni+zhnh− Qd

e nd0

, (3)

where nh is the heavy negative ion number density, uh is the heavy negative ion fluid speed, zhis the number of excess electrons in the negative ions,φis the electrostatic wave potential, V_{T h}² = Th/mh, Th is the heavy negative ion thermal energy, mh is the heavy negative ion mass, ne(ni) is the electron (light ion) number density and Qd is the charge of the stationary dust.

The phase speed of a purely electrostatic perturbation mode is very low compared the electron and positive ion thermal speeds. Thus, the electron number density (ne) of the electrons following trapped (vortex-like) distribution [29–33], and the number density (ni) of ions following Maxwellian distribution [27], are given by

ne=ne0

1+ eφ

Te f

−4(1−βe) 3√

π eφ

Te f

3/2

+1 2

eφ Te f

2

, (4)

ni =n_{i 0}exp

−eφ Ti

, (5)

where Ti is the light ion temperature (in energy unit),βe is a parameter determining the number of trapped electrons and its magnitude is defined as the ratio of the free hot electron temperature Te f to the hot trapped electron temperature Tet in energy unit, i.e.

βe = Te f/Tet. We note that we have directly used the expression for newhich has been derived in details elsewhere [36,37].

The dust charge Qd (appeared in (3)) is not constant, but varies with time according to the dust charging equation [3]

∂Qd

∂t =Ie+Ii+I_n0, (6)

where Ie(Ii)are the trapped electron (Maxwellian ion) currents and I_n0is the negative ion current at equilibrium. We have assumed here that the negative ion current fluctuation is much less than the electron or ion current fluctuation. This is a good approximation for any typical dusty multi-ion plasma [38–40], where Th/Ti 0.125 and mh/mi =μh146/39 (light ion is K⁺and heavy negative ion is SF6). The electron and ion currents (Ieand Ii) are given by

Ie= −4πrd²ne0e Te f

2πme

_1/2

exp(βeχ)+

1− 1 βe

χ+1+ 1 βe

,(7)

Ii =4πr_d²nie Ti

2πmi

_1/2

1−e Qd

rdTi

, (8)

whereχ =(e/Te f) (Qd/rd+φ)for Qd <0. It may be noted here that we have directly used the expressions for Ie and Ii which have been derived in details elsewhere [36,37].

It is important to note that (1)–(8) form a complete set of equations to describe the purely electrostatic perturbation mode under consideration, and that in the continuity equation for heavy negative ion fluid (1), which is coupled with (6) via (2) and (3), we have neglected the source and sink terms (which do not lose any physics insight) as neglected by many authors [3,27,28,41–48]. The equilibrium state of the dusty multi-ion plasma system under consideration is defined as

ni 0−ne0−zhnh0+qd0nd0/e=0, (9)

Ie0+Ii 0+In0=0, (10)

where qd0 is the dust grain charge at equilibrium, and Ie0 = Ie(φ = 0, Qd =qd0)and Ii 0 = Ii (φ = 0, Qd = qd0), are electron and ion currents at equilibrium. It is noted here that in order to have a non-zero negative ion current, i.e. In0 > 0, we must have

|Ie0|/Ii 0<1 which reduces to ne0/ni 0=ηe< η0, where η0=√

μσifiat, (11)

in whichαe,i = q_d0e/rdTe f,i, μ = me/mi, σi = Ti/Te f, at = 1/[exp(βeαe)+(1− 1/βe)(1 +αe +1/βe)], fi = 1+αi and −ezd = qd0. We have graphically shown how η0 varies with zd and rd for typical laboratory dusty multi-ion plasma parameters [38–40] Ti = 1.5 eV, mi = 39mp (light ion is K⁺) (figure 1). Now, using Qd = qd0 +qd and Ie = Ie0 + ˜Ie, Ii = Ii 0 + ˜Ii (where I˜e (I˜i) is the perturbed part of the trapped electron (Maxwellian ion) currents) (4), (5), (7), (8) and (10), we can express (6) as

∂qd

∂t = ˜Ie+ ˜Ii, (12)

4600 4700

4800 4900

5000 zd

3.5 3.75

4 4.25

4.5

rd

0 0.05 0.1 0.15 η0

0 4700

4800 zd 4900

Figure 1. Showing the variation ofη0with zdand rd(inμm) forβe= −0.3, Ti=1 eV and Te f =5Ti.

where

I˜e

Ie0 =at

exp

eβe

Te f

qd

rd +φ

+

1− 1 βe

eqd

Te f + eφ

Te f +1+ 1 βe

− 1

at , (13)

I˜i

Ii 0 = − eqd

firdTi +

1+ eqd

firdTi −eφ Ti + 1

2!

e²φ² T_i² − · · ·

. (14)

3. Modified KdV equation

We now derive a dynamical equation for the nonlinear propagation of the electrostatic waves from (1)–(5) and (12)–(14) using the reductive perturbation technique [49]. For that, first we have to introduce the stretched coordinates [50]

ξ =^1/4(X−V0t), (15)

τ =^3/4t, (16)

whereis a smallness parameter (0 < <1) measuring the weakness of the dispersion and V₀is the phase speed of the perturbation mode, and expand nh, uh,φand qdabout their equilibrium values in power series of, viz.

nh =nh0+n⁽h¹⁾+^3/2n⁽_h²⁾+ · · ·, (17) uh=u⁽_h¹⁾+^3/2u⁽_h²⁾+ · · ·, (18) φ=φ⁽¹⁾+^3/2φ⁽²⁾+ · · ·, (19) qd =q_d⁽¹⁾+^3/2q_d⁽²⁾+ · · ·. (20) Now substituting (13), (14) and (15)–(20) into (1)–(3) and (12), we develop equations in various powers of. Denoting n⁽¹⁾h by N , u⁽¹⁾_h by U , q_d⁽¹⁾by Q andφ⁽¹⁾bywe have for the lowest order of:

U = − V0zhe

mh(V₀²−V_{T h}² ), (21)

N = − nh0zhe

mh(V₀²−V_{T h}² ), (22)

Q= −β1rd, (23)

V₀²=V_{T h}² + ηnC_n² 1+ fq

, (24)

whereβ1 = fi(1+β0)/(1+ fiβ0),β0 =σiηe(1+βe−1/βe)/η0,μ =√

me/mi, C_n² = zhTi/mhσ1,σ1=1+ηeσi,v²q=e²zh/rdmh,ηn =zhn_h0/ni 0=1−ηe−ηd,ηd =n_d0/ni 0

and fq = ηdβ1(C_n²/vq²). Equation (24) represents the linear dispersion relation for the ion-acoustic waves associated with negative ions, which are significantly modified by the presence of the charge fluctuating stationary dust. It implies that for no dust (ηd =0), cold heavy negative ion (Th =0) limits, the phase speed of this wave is equal to Cn

√n_h0/ni 0. It is clear that for fixed light ion number density, i.e. ni 0=constant, the phase speed of these waves increases (decreases) as the heavy negative ion number density increases (decreases).

To the next higher order of, one obtains

∂N

∂τ −V0

∂n⁽²⁾_h

∂ξ +nh0

∂u⁽²⁾_h

∂ξ =0, (25)

∂U

∂τ −V₀∂u⁽h²⁾

∂ξ = zhe mh

∂φ⁽²⁾

∂ξ −V_{T h}² nh0

∂n⁽h²⁾

∂ξ , (26)

∂²

∂ξ² =4πe

ni 0C1φ⁽²⁾− 4 3√

π(1−βe) e

Te f

3/2

+zhn⁽²⁾_h −1

end0q_d⁽²⁾ , (27)

q_d⁽²⁾= −β1rdφ⁽²⁾, (28)

where C1 =σ1e/Ti. Now, using (21)–(28), one can eliminate n⁽²⁾_h , u⁽²⁾_h , q_d⁽²⁾andφ⁽²⁾and can finally obtain the modified Korteweg-de Vries (KdV) equation describing the nonlinear propagation of the DNIA solitary waves in a dusty multi-ion plasma.

∂

∂τ +A√

∂

∂ξ +B∂³

∂ξ³ =0, (29)

where the nonlinear coefficients A and B are given by A = (1−βe)

(Te f)^3/2 e

π

1/2mh(V₀²−V_{T h}² )²

V0nh0z²_h , (30)

B = 1

8πe²

mh(V₀²−V_{T h}² )²

V₀n_h0z²_h . (31)

It is important to note that the modified KdV equation (29) is only valid for a trapped excavated electron distribution (forβe < 1). This equation is not valid for a Maxwellian hot electron distribution (βe=1) because the nonlinear effect caused by the term(eφ/Te f)² (which is usually small compared to that caused by the term(eφ/Te f)^3/2 forβe <0) has been dropped out due to the ordering we have used.

The stationary solitary wave solution of the modified KdV equation (29) can be obtained by introducingζ =ξ −U0τ, where U0is the wave speed (in the reference frame) and by imposing the boundary conditions for localized perturbations, viz. → 0, d/dζ →0, d²/dζ²→0 atζ → ±∞. This leads us to write the steady-state solution of (29), as

=msech⁴(ζ/), (32)

wherem=(15U₀/8 A)²and=√

16B/U₀are, respectively, the height and thickness of the DNIA solitary waves. Equation (32) reveals the existence of small but finite amplitude of DNIA solitary waves with a negative potential. It is obvious that formation of such DNIA solitary waves is due to the charge fluctuation of the negatively charged stationary dust. To interpret the basic features (polarity, height and thickness) of these DNIA solitary waves we consider some approximations, viz. zh =1 andηeη0, which are completely valid for both space and laboratory dusty negative ion plasmas [38–40], where U0 Ch

is considered. It is obvious thatis always negative for any possible values ofη0and fq. This means that the DNIA solitary structures exist only with<0 (i.e. with Q<0, since Q= −β1rdandβ1 >0). To have some numerical appreciations of our results, we have also numerically analysed the amplitude, the width and the solitary profiles for the potential () and for the corresponding perturbed dust charge (Q = −β1rd) by using the general expressions for the coefficients A and B (i.e. by using (30) and (31)). We have chosen the parameters corresponding to the recent laboratory dusty multi-ion plasma experiments [38–

40], viz. ηe =10⁻²– 0.1,ηd =10⁻²– 0.3, rd =10 – 500μm, Te f Ti =0.25 – 1.5 eV, Th =5Ti, zh =1, mi =39mp(K⁺) and mh =146mp(SF₆), where mpis the proton mass.

The results are displayed in figures 2–4.

-0.5 -0.4

-0.3 -0.2

e 4

4.5 5

5.5 6

rd

-0.03 -0.02 -0.01 e_Φm 0

Ti

5 -0.4

β^e -0.3

Figure 2. Showing the variation ofm = φ⁽¹⁾with U₀ = 0.1,ηd =0.2,η0 = 0.1, ηe=0.01 andσi=0.1.

1 1.5

2 2.5

3 rd

-0.4 -0.3

-0.2 -0.1

β^e 0

2 Δ 4 rd

1 1.5

2 rd 2.5

Figure 3. Showing the variation ofwith rdandβefor U₀=1,η0=0.2, zd(in 10⁴), ηe=0.01 andσi=0.1.

0 2

4 6

8 rd

-0.45 -0.4

-0.35 -0.3

-0.25

βe

-0.04 -0.02 Q 0 qdo

0 2

4 rd 6

Figure 4. Showing the variation of q_d⁽¹⁾= Q with rd(inμm)andβe, forηe=0.01, η0=0.2 andσi=0.1.

It is obvious from figures 2–4 that the DNIA solitary structures exist only with negative potential (φ <0), i.e. only with negative dust charge fluctuation (qd <0). Figure 2 shows that the amplitude of the solitary waves increases with increasingβe(trapped parameter) and decreasing rd (dust size). The width of the solitary waves increases (decreases) with decreasing (increasing) value of rd(βe) has been shown in figure 3. Figure 4 shows that the charge (q_d⁽¹⁾=Q) increases by decreasing the dust size (rd).

4. Modified Burgers’ equation

Now, we consider another different scaling of the stretched coordinates [27,50]

ξ =^1/2(X−V0T), (33)

τ =T, (34)

and follow exactly the same mathematical procedure that we have applied above. The linear response for U , N , Q and V0 have been found to be exactly the same as before, and are given by (21)–(24).

To the next higher order of, one obtains same as (25)–(27), only different

−V0κ∂

∂ξ =q_d⁽²⁾+β1rdφ⁽²⁾, (35)

whereκ=β1Tifi/Ii 0(1+β0fi). However, we have obtained a different form of nonlinear dynamical equation, using (25)–(27) and (35), which is given by

∂

∂τ +A√

∂

∂ξ =C∂²

∂ξ², (36)

where the nonlinear coefficient A is given by (30), and C is given by C =κηdmh(V₀²−V_{T h}² )²

2z²_he² . (37)

Equation (36) is the modified Burgers’ equation describing the nonlinear propagation of the DNIA shock waves in a dusty multi-ion plasma, which includes the effects of trapped

parameter (βe). It is obvious that the dissipative term is due to the effect of dust charge fluctuation and the strong nonlinearity is due to the effect of trapped distribution (through

√). We are now interested in looking for the stationary shock wave solution of (36), by introducingζ =ξ−U0τandτ=τ, where U0is the shock wave speed (in the reference frame). This leads us to write the solution of (36), under the steady-state condition∂/∂τ= 0, as

=m

1+tanh(ζ/δ)2

, (38)

wherem =(3U0/4 A)²andδ =4C/U0. Equation (38) represents a shock-like localized solution of eq. (36), and this new type of shock solution arises due to the combined effects of the nonlinear term (containing√

) which arises due to the trapped electron distribution, and the dissipative term (containing C) which arises due to the dust charge fluctuation.

The solution (38) shows that as U0 increases, the amplitude|m|increases whereas the widthδ decreases. To have some numerical appreciations of our results, we have also numerically analysed the amplitude, the width and the shock structures for the potential () and for the corresponding perturbed dust charge (Q = −β1rd) using the general expressions for the coefficients A and C (i.e. by using (30) and (37)). We have chosen the parameters corresponding to the recent laboratory dusty multi-ion plasma experiments [38–

40], viz. ηe = 10⁻²– 0.1,ηd =10⁻²– 0.3, rd =100 – 500μm, Te Ti =0.15 – 1.5 eV,

-2 0

2 3

3.5 4

4.5 5

rd

-0.15 -0.1 -0.05 e m 0

Ti

-2 0 ξ 2

Figure 5. Showing the variation of shock wave profilemwithζ and rd for U₀=1, ηd =0.2,η0=0.1,ηe=0.05,αe=1.4,σi=0.1 andβe= −0.4.

-4 -2 0

2 4

-0.5 -0.4

-0.3 -0.2

β^e -0.15

-0.1 -0.05 e m 0

Ti

-4 -2 0

2 ξ 4

Figure 6. Showing the variation of shock wave profilemwithζ andβefor U₀=1, rd=300μm,ηd =0.2,η0=0.1,ηe=0.05,αe=1.4 andσi =0.1.

-0.6

-0.4

-0.2

e

1 2

3 4

rd

0 2 4 δ 6 rd

-0.6 βe -0.4

Figure 7. Showing the variation of widthδ withβe and rd (in μm)forηd = 0.2, η0=0.1,αe=1.4,σi =0.1 andηe=0.01.

Th = 5Ti, zh = 1, mi = 39mp (K⁺) and mh = 146mp (SF₆), where mp is the proton mass.

The results are displayed in figures 5 and 6. It is obvious from figures 5 and 6 that the DNIA shock structures exist only with negative potential (φ < 0), i.e. only with nega- tive dust charge fluctuation (qd < 0). Figure 5 shows that the amplitude of these shock structures are increasing with increase in the size of the dust(rd). Figure 6 shows that the amplitude of the shock profile significantly is modified with the change ofβe(trapped parameter). Figure 7 shows that the width of the shock profile increases with increase in bothβe(trapped parameter) and rd(dust size).

5. Discussion

To summarize, the nonlinear propagation of the low phase speed of the DNIA waves in a dusty multi-ion plasma containing trapped electrons and light positive ions, which follow the Maxwellian distribution, mobile negative ions and extremely massive and large (few micron size) charge fluctuating stationary dust has been theoretically investigated by the reductive perturbation method. At equilibrium, dust are charged and qd =q_d⁽¹⁾is found to be a fraction of|q_d0|because the presence of heavy negative ions causes a significant electron density depletion, and consequently makes the electron collection current insignif- icant compared to the ion collection current. This means that the presence of negative ions significantly reduces the magnitude of the charge of negatively charged dust, and under a certain condition [38], it can cause the charge neutralization of dust particles. This predic- tion agrees with the recent experimental observations of Merlino and Kim [39] and Kim and Merlino [38]. We note that our analysis, which is not valid for qd comparable to or greater than|qd0|, is not able to show a complete neutralization of dust particles or a transition from negative to positive dust. However, it can be expected that an analysis of arbitrary ampli- tude DNIA waves, which is beyond the scope of our present work, will be able to show a complete charge neutralization of dust particles as observed by the laboratory experiment of Merlino and Kim [39].

It has also been shown that the basic features (polarity, height and thickness) of such DNIA solitary and shock structures are different from those of the DIA solitary and shock

structures observed by the previous experiments. We, therefore, expect that the result of our present investigation should help us in understanding the localized electrostatic disturbances in space and laboratory dusty plasmas with electrons following the trapped distribution.

Acknowledgements

One of the authors (S S Duha) acknowledges the financial support of the University Grants Commission, Bangladesh. The author also acknowledges Ministry of Education, Bangladesh for granting his deputation during the course of this work.

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