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Effects of two temperature non-extensive electrons on the sheath of dusty plasma
Materials Today: Proceedings xxx (xxxx) xxx
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Effects of two temperature non-extensive electrons on the sheath of dusty plasma O. El Ghani a,⇑, I. Driouch a,b, H. Chatei a a b
Laboratory of Physics of Matter and Radiations, Department of Physics, Faculty of Science, University Mohammed I, B.P. 717, 60000 Oujda, Morocco National School of Applied Sciences, B.P. 03, Ajdir Al-Hoceima, Morocco
a r t i c l e
i n f o
Article history: Received 1 June 2019 Received in revised form 21 July 2019 Accepted 22 July 2019 Available online xxxx Keywords: Plasma sheath Dust grains Nonextensive electrons Two temperatures Hot electrons
a b s t r a c t In this paper, the problem of sheath is investigated using the fluid model in a magnetized fourcomponent dusty plasma system comprising positive ions, negatively charged statistic dust grains and two species of electron populations. These electrons are assumed to be a sum of two q-nonextensive electrons distribution (i.e. the electrons evolve far away from their Maxwellian distribution (q ¼ 1) with two different temperatures (cold and hot). The effects of population ratio and the temperature ratio of hot to cold non-extensive electrons on the plasma sheath parameters are studied numerically. A significant change is observed in the quantities characterizing the sheath as sheath thickness, sheath potential, electron and ion densities and dust velocity in the presence of nonextensively (case superextensively) distributed two-temperature electrons. Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
1. Introduction It is well known that interest in the sheaths with dust particles has been of great importance to the plasma physics community for a long time. It can be found in various technological devices, such as the device for plasma assisted material and chemical processing, plasma discharges, as well as controlled fusion devices [1,2]. Many studies have focused on the effect of parameters that modify the plasma sheath characteristics, for example the action of an external magnetic field, collisions force, multi-species ions, positive or negative two electrons temperature and ion temperature [3–11]. In all of these studies the particles distribution has been considered to be Maxwellian (i.e, in Boltzmann-Gibbs (BG) statistics). But it is well known that the Maxwell distribution is believed to be valid universally for the macroscopic ergodic equilibrium state [12,13]. Moreover, Maxwell distribution is inadequate to describe the systems in non-equilibrium state with long range interactions. In this case space plasmas observations and discharge plasmas [14,15] clearly indicate the presence of the particle populations distribution functions may be different from the Maxwellian distributions. For that, Tsallis statistics or non-extensive ⇑ Corresponding author. E-mail address: [email protected] (O. El Ghani).
statistics [16], as a generalization of the Boltzmann–Gibbs–Shannon (BGS) entropy, has attracted much attention. This statistic is very important to describe the systems of particles with long range interaction such as plasma and dusty plasmas. In non-extensive statistics of Tsallis, the plasma particles are described by a power-law q-distribution function or called q-nonextensive distribution. q is a number real characterizes the degree of nonextensivity of the systems. It is important to note that the distribution function, with q < 1 indicates the system with more superthermal particles (superextensive) i.e more fast particles compared with Maxwellian distribution (q = 1), for q > 1 the system containing a large number of low-speed particles (subextensive). This q-nonextensive distribution has been successfully employed in plasma physics, such as collisionless thermal plasma [17], the dispersion relation for electrostatic plasma oscillations [18,19], dust charging processes [20,21], dust ion acoustic waves or dust acoustic waves in a dusty plasma by taking non extensive ions [22,23], or nonextensive electrons [24–27], or both to be extensive [28]. Recently, a great deal of attention has been paid to the Bohm criterion and the sheath structures in plasma which nonextensive distributed electrons [29,30]. For example, Hatami [31] studied the sheath formation criterion in a collisional plasma consisting non extensively distributed electrons and thermal ions, and
https://doi.org/10.1016/j.matpr.2019.07.599 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
2
O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
discussed the effect of nonextensive electrons on the behavior of the density distribution of charged particles. Liu et al. [32] investigated the bohm criterion in a dusty plasma in the presence of electrons nonextensively distributed. Recently, Shalini et al. [33] studied the amplitude modulation of ion-acoustic wave (IAW) packets in an unmagnetized electron-ion plasma with twotemperature (cold and hot) electrons obeying Tsallis nonextensive q distribution. However, the sheath characteristics for magnetized dusty plasma with two temperature electrons in the context of the nonextensive (superextensive) q distribution have not been studied yet. Therefore, in the present work, we have presented a more generalized investigation on the sheath characteristics considering the q-nonextensive distribution for two electrons population i.e. cold and hot superextensive electrons by using a multi-fluid model. So, our model is extended to include the effects of population ratio and the temperature ratio of hot to cold nonextensive electrons on the sheath structure as well as the characteristics of dust particles in the sheath. To do this, we consider the near wall region of a magnetized multi-component plasma which consists of two superextensive electrons with different temperature cold and hot, cold fluid negatively charged statistic dust grains and cold fluid ions. The paper is organized as follows. In the next section (Section 2), we present our assumptions and the governing equations. The results and numerical simulations are presented and discussed in Section 3. Finally, conclusions are given in Section 4. 2. Basic equations and assumption In this work, we consider a magnetized stationary (dt = 0) dusty plasma sheath in contact with a planar wall (Fig. 1). The sheath is studied in a one-dimensional coordinate space system and a threedimensional speed space system. The external magnetic field, which is spatially uniform and constant in time, lies in the x z plane and makes an angle h with the x-axis. The region of interest lies between x ¼ 0 (the sheath edge), and the wall which can be located anywhere in the region x > 0. We assume that the physical parameters change only along the x-direction (normal to the wall, see Fig. 1) in the sheath region. At the sheath edge x ¼ 0, the potential is assumed to be zero, and the ion, the dust, the cold and hot electron species densities ni0 , nd0 , nec0 ; neh0 satisfy the quasi-neutrality condition:
e ni0 neh0 nec0 þ qd0 nd0 ¼ 0
ð1Þ
where e is the electron charge and qd0 is the dust charge at the sheath edge. We assume that the two electron species (hot and cold) obey the q-nonextensive distribution function defined as [34].
1 me v 2e e/ ðq1Þ f e ðv e Þ ¼ Aq ½1 ðq 1Þ Te 2T e
ð2Þ
Where the constant of normalization is
8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Cð1q Þ me ð1qÞ > > for 1 < q < 1 < ne0 Cð 1 1Þ 2pT e 1q 2 Aq ¼ ffiffiffiffiffiffiffiffiffiffiffiffi q 1 1 Cð1qþ2Þ me ð1qÞffi > > : ne0 1þq for q > 1 1 2p T e 2 Cðq1 Þ
ð3Þ
e is the unit electric charge, T e is the electron temperature in eV, / is the electrostatic potential and C denotes the standard gamma function. The density of the two electron species nej (where j ¼ c;case of the cold electrons and j ¼ h; case of the hot electrons), takes the form
"
nej ¼ nej 0 1 þ qj 1
# qj þ1 e/ 2ðqj 1Þ T ej
ð4Þ
qj stands the non-extensivity of electrons, In the extensive limiting case (qj ¼ 1), the density distribution reduces to the density Maxwell Boltzmann distribution, qj –1 describes nonextensivity of the electrons under consideration, and, we note here that the condition (qj < 1) refers to the superextensive case, whereas the opposite condition (qj > 1) refers to the subextensive case. The ions are treated as a fluid; the basic equations of ions are the continuity
@ðni v ix Þ ¼0 @x
ð5Þ
and the momentum conservation equations
v ix
@ v ix e @/ ¼ mi @x @x
ð6Þ
where v i , ni and mi are the velocity in the depth direction, density and mass of the positive ion species in the sheath, respectively. The dust grains are treated as cold fluid obeying the sources free. The electric, magnetic and gravitational forces are considered. The dust fluid equations are written as follows:
@ðnd v dx Þ ¼0 @x md v dx
! @vd @/ ! ! ! ! x þ qd v d ^ B þ md g x ¼ qd @x @x
ð7Þ ð8Þ
where qd ¼ zd e is the charge of dust grain, zd is the charge number of ! the dust, md , nd and v d are the mass, the density and velocity of the dust grains respectively. The system of equations is completed by the Poisson’s equation which relates the electric potential to the density of dust grains, two electrons and ions as follows:
@2/ ¼ 4pðeðni neh nec Þ þ qd nd Þ @x2
ð10Þ
Fig. 1. The geometry of considered sheath model.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
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O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
In order to solve the system of Eqs. (1)–(10), we introduce new dimensionless variables. We normalize each charged particle density to its value at the sheath edge: N i ¼ ni =ni0 ; N d ¼ nd =nd0 ; N ec ¼ nec =ne0 ; N eh ¼ neh =ne0 ðwherene0 ¼ neh0 þ nec0 Þ. Also we normalize the positive ion velocity to the positive ion pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sound speed ci ¼ T ec =mi , and the dust velocity to the dustpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi acoustic speed cd ¼ T ec =md respectively i.e. uix ¼ v ix =ci and ud ¼ v d =cd . Furthermore, we scale the x-coordinate and the sheath potential as follows: n ¼ x=kD and g ¼ e/=T ec , where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kD ¼ T ec =4pne0 e2 is the electron Debye length and T ec the cold electrons temperature. ! ! ! The magnetic field is of the form B ¼ B0 cosh x þ sinh z and by substituting the above dimensionless variables into Eqs. (1)–(10), we obtain:
Neh
qh þ1 P g 2ðqh 1Þ ¼ 1 ðqh 1Þ 1þP T hc qc þ1 2ðqc 1Þ
ð11Þ
Nec ¼
1 ½1 ðqc 1Þg 1þP
Nd ¼
Md udx
ð13Þ
Ni ¼
Mi uix
ð14Þ
@uix 1 @g ¼ uix @n @n
ð15Þ
udx
@udx @g ¼ zd þ czd sinðhÞudy þ f g @n @n
ð12Þ
ð16Þ
udx
@udy ¼ czd ½cosðhÞudz sinðhÞudx @n
ð17Þ
udx
@udz ¼ czd cosðhÞudy @n
ð18Þ
@2g @n2
qc þ1
2ðqc 1Þ 1 ½1 þ ðqc 1Þg 1þP
qh þ1 P g 2ðqh 1Þ 1 þ ðqh 1Þ þ ð1 dÞzd Nd 1þP T hc
¼ dNi
ð19Þ
1
c ¼ eB0 kD =ðmd T e Þ2 ; T hc ¼ T eh =T ec ; f g ¼ gkD =c2d ; P ¼ neh0 =nec0 Here the boundary conditions are as follows: at the wall ðn ¼ dÞ,
gðdÞ ¼ gx ; at the sheath edge ðn ¼ 0Þ, the boundary conditions ui ðn ¼ 0Þ ¼ M i , udx ðn ¼ 0Þ ¼ Md , and aregðn ¼ 0Þ ¼ 0, udy ðn ¼ 0Þ ¼ udz ðn ¼ 0Þ ¼ 0. To better understand the effect of the presence of the hot electron on the characteristics of the sheath, we definite the density ratio of hot electrons neh0 to cold electrons nec0 , where neh0 and nec0 are the electrons species densities in the plasma region. Also, we definite the temperature ratio of hot and cold electrons T hc ¼ T eh =T ec . From the definition of P it is clear that P ¼ 0 is the case where magnetized collisional plasma sheath consists of only one species of electrons (cold electron), so the increase of P leads to increase of the concentration of hot electron populations. Now, from the Poisson equation (Eq. (19)) if (P ¼ 0 and T hc ¼ 1) the sheath consists of only one electron populations (cold electron), therefore if (P–0 and T hc –1) is the case when the sheath consists of two electron populations. Eqs. (11)–(19) with appropriate boundary conditions are solved using a Runge–Kutta method of the fourth order [30].
Fig. 2. (a) The normalized spatial distributions of electrostatic potential; (b) the normalized electron density; (c) the normalized ion density; (d) the normalized spatial distributions of dust velocity in the depth direction versus x under various values of P and for the parameters: qc ¼ 0:7 and qh ¼ 0:65.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
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O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 3. (a) The normalized spatial distributions of electrostatic potential; (b) the normalized spatial distributions of dust velocity in the depth direction versus x under two different T hc values, the other parameters are the same as Fig. 2.
3. Numerical results and discussion To solve numerically the normalized multi-fluid Eqs. (11)–(19) we use ne0 ¼ 109 cm3 ,T e ¼ 2 eV, M i ¼ 3, Md ¼ 5, d ¼ 1:01 and @ g=@nðn ¼ 0Þ ¼ 0:01 as boundary conditions at the plasma-sheath interface ðn ¼ 0Þ. The normalized magnitude of magnetic field
c ¼ 0:3with angle h ¼ 30 . Also, in our calculations, we consider the dust as a spherical particle of radius rd ¼ 4 lm with uniform mass density qd ¼ 2 g:cm3 and zd ¼ 1000. The numerical solutions of Eqs. (11)–(19) are displayed in Figs. 2 and 3 for various P and T hc parameters. Fig. 2(a)–(d) depict the distribution of the normalized sheath potential, the electron and the ion densities and the normalized dust normal velocity across the sheath consisting of superextensively distributed electrons. This figure illustrates the structure of the sheath on the presence of the hot electron populations. Firstly, Fig. 2(a) shows that by increasing P, the normalized potential of the sheath increases which is due to the increasing of energetic electrons (hot electron populations). Another notable point observed from the same figure is that the sheath thickness decreases when P increases. In addition, the effect of hot electrons is noticeable when density number of cold electrons is high (low values of P). Fig. 2(b) and (c) demonstrate that the density distribution of the both electron species (cold and hot) and that of the ions respectively, increase when the hot electron populations are weaker, we can explain that for low values of hot electron populations, the electric force becomes less effective, which leads to the increase of the electrons and ions densities. Fig. 2(d) shows the behaviour of normalized dust velocity under various values of P, where it is shown that the dust normal velocity evolves the same way near the sheath edge because of the dominance of the gravitational force on grains of dust, while when one moves away from the edge, the effect of hot electrons becomes more obvious (the electric force becomes more effective) and the dust velocity decreases in higher values of hot electron densities. Now, we study effects of the ratio temperature T hc ¼ T eh =T ec of hot and cold electrons on the structure of the sheath. Fig. 3(a) and (b) show the profiles of the normalized sheath potential and that of dust velocity. We can see that with a fixed value for P ¼ 0:3 and P ¼ 1:7, the sheath potential increases for large values of the temperature ratio T hc , while the normalized x-component
velocity of dust grains decreases in increasing of T hc . A simple observation shows that the effect of the temperature ratio is obvious in higher values of hot electron populations (case of P ¼ 1:7). 4. Conclusion In this paper, we have investigated a magnetized sheath of dusty plasma consisting of cold ions and two-temperature electrons featuring nonextensive distributions (case of superextensive distributions). The study interests to the effect of hot superextensive electron populations on some quantities characterizing the magnetized plasma sheath such as the electrostatic potential, the sheath thickness and dust dynamic. Based on multi-fluid model, the sheath potential and the x-component dust velocity have been investigated for different values of the density ratio of the hot electron to the cold populations and temperature ratios of hot to cold electrons. Our results show that the behaviour of the sheath depends in particular on the ratios of densities and temperatures of hot and cold electrons. Specially, it is observed that the electrostatic potential increases with increasing the hot electron populations, also, by increasing the temperature ratio of hot to cold electrons. Consequently, the hot electron populations decrease the sheath thickness. Furthermore, our numerical results reveal that the higher values of the density ratio of the hot electron to cold electron densities and the temperature ratio of hot to cold electrons, decrease the dust velocity. Therefore, the effect of hot electron populations on the sheath structure is more obvious when the cold electron populations dominate. References [1] A. Bouchoule, Dusty Plasma: Physics, Chemistry and Technical Impact in Plasma Processing, Wiley, New York, 1999. [2] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol, 2002. [3] M.Y. Yu, H. Saleem, H. Luo, Dusty plasma near a conducting boundary, Phys. Fluids B 4 (1992) 3427–3431. [4] C. Arnas, M. Mikikian, G. Bachet, F. Doveil, J.X. Ma, C.X. Yu, Sheath modification in the presence of dust particles, Phys. Plasmas 7 (2000) 4418–4422. [5] T. Nitter, Levitation of dust in rf and dc glow discharge, Plasma Source Sci. Technol. 5 (1996) 93–111. [6] Z. Xiu, Characteristics of dust plasma sheath in an oblique magnetic field, Chin. Phys. Lett 23 (2006) 396–698. [7] J.Y. Lieu, D. Wang, T.C. Ma, the charged dust in processing plasma sheath, Vacuum 59 (2000) 126–134.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx [8] Z. Wang, Y. Liu, L. Ren, J. Liu, X. Wang, Bohm criterion for electronegative dusty plasmas, Thin Solid Films 506–507 (2006) 637–641. [9] Zheng-Xiong Wang, Jin-Yuan Liu, Yue Liu, Xiaogang Wang, The electrostatic sheath in an electronegative dusty plasma, Phys. Plasmas 12 (2005) 012104– 012109. [10] G. Foroutan, Fluid simulation of an electrostatic plasma sheath with two species of positive ions and charged nanoparticles, Phys. Plasmas 17 (2010) 123711–123722. [11] I. Driouch, H. Chatei, M. El Bojaddaini, J. Plasma Phys. 81 (2015) 1–9, 905810104. [12] W.M. Moslem, W.F. El-Taibany, E.K. El-Shewy, E.F. El-Shamy, Phys. Plasmas 12 (2005) 052318. [13] M. Asaduzzaman, A.A. Mamun, Phys. Rev. E 86 (2012) 016409. [14] M.P. Leubner, J. Geophys. Res. 87 (1982) 6335. [15] S.K. Mishra, S. Misra, M.S. Sodha, Europhys. J. D 67 (2013) 210. [16] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [17] J.A.S. Lima, R. Silva, J. Santos, Phys. Rev. E 61 (2000) 3260. [18] R. Silva, J.S. Alcaniz, J.A.S. Lima, Physica A 356 (2005) 509.
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Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Effects of two temperature non-extensive electrons on the sheath of dusty plasma O. El Ghani a,⇑, I. Driouch a,b, H. Chatei a a b
Laboratory of Physics of Matter and Radiations, Department of Physics, Faculty of Science, University Mohammed I, B.P. 717, 60000 Oujda, Morocco National School of Applied Sciences, B.P. 03, Ajdir Al-Hoceima, Morocco
a r t i c l e
i n f o
Article history: Received 1 June 2019 Received in revised form 21 July 2019 Accepted 22 July 2019 Available online xxxx Keywords: Plasma sheath Dust grains Nonextensive electrons Two temperatures Hot electrons
a b s t r a c t In this paper, the problem of sheath is investigated using the fluid model in a magnetized fourcomponent dusty plasma system comprising positive ions, negatively charged statistic dust grains and two species of electron populations. These electrons are assumed to be a sum of two q-nonextensive electrons distribution (i.e. the electrons evolve far away from their Maxwellian distribution (q ¼ 1) with two different temperatures (cold and hot). The effects of population ratio and the temperature ratio of hot to cold non-extensive electrons on the plasma sheath parameters are studied numerically. A significant change is observed in the quantities characterizing the sheath as sheath thickness, sheath potential, electron and ion densities and dust velocity in the presence of nonextensively (case superextensively) distributed two-temperature electrons. Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
1. Introduction It is well known that interest in the sheaths with dust particles has been of great importance to the plasma physics community for a long time. It can be found in various technological devices, such as the device for plasma assisted material and chemical processing, plasma discharges, as well as controlled fusion devices [1,2]. Many studies have focused on the effect of parameters that modify the plasma sheath characteristics, for example the action of an external magnetic field, collisions force, multi-species ions, positive or negative two electrons temperature and ion temperature [3–11]. In all of these studies the particles distribution has been considered to be Maxwellian (i.e, in Boltzmann-Gibbs (BG) statistics). But it is well known that the Maxwell distribution is believed to be valid universally for the macroscopic ergodic equilibrium state [12,13]. Moreover, Maxwell distribution is inadequate to describe the systems in non-equilibrium state with long range interactions. In this case space plasmas observations and discharge plasmas [14,15] clearly indicate the presence of the particle populations distribution functions may be different from the Maxwellian distributions. For that, Tsallis statistics or non-extensive ⇑ Corresponding author. E-mail address: [email protected] (O. El Ghani).
statistics [16], as a generalization of the Boltzmann–Gibbs–Shannon (BGS) entropy, has attracted much attention. This statistic is very important to describe the systems of particles with long range interaction such as plasma and dusty plasmas. In non-extensive statistics of Tsallis, the plasma particles are described by a power-law q-distribution function or called q-nonextensive distribution. q is a number real characterizes the degree of nonextensivity of the systems. It is important to note that the distribution function, with q < 1 indicates the system with more superthermal particles (superextensive) i.e more fast particles compared with Maxwellian distribution (q = 1), for q > 1 the system containing a large number of low-speed particles (subextensive). This q-nonextensive distribution has been successfully employed in plasma physics, such as collisionless thermal plasma [17], the dispersion relation for electrostatic plasma oscillations [18,19], dust charging processes [20,21], dust ion acoustic waves or dust acoustic waves in a dusty plasma by taking non extensive ions [22,23], or nonextensive electrons [24–27], or both to be extensive [28]. Recently, a great deal of attention has been paid to the Bohm criterion and the sheath structures in plasma which nonextensive distributed electrons [29,30]. For example, Hatami [31] studied the sheath formation criterion in a collisional plasma consisting non extensively distributed electrons and thermal ions, and
https://doi.org/10.1016/j.matpr.2019.07.599 2214-7853/Ó 2019 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
2
O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
discussed the effect of nonextensive electrons on the behavior of the density distribution of charged particles. Liu et al. [32] investigated the bohm criterion in a dusty plasma in the presence of electrons nonextensively distributed. Recently, Shalini et al. [33] studied the amplitude modulation of ion-acoustic wave (IAW) packets in an unmagnetized electron-ion plasma with twotemperature (cold and hot) electrons obeying Tsallis nonextensive q distribution. However, the sheath characteristics for magnetized dusty plasma with two temperature electrons in the context of the nonextensive (superextensive) q distribution have not been studied yet. Therefore, in the present work, we have presented a more generalized investigation on the sheath characteristics considering the q-nonextensive distribution for two electrons population i.e. cold and hot superextensive electrons by using a multi-fluid model. So, our model is extended to include the effects of population ratio and the temperature ratio of hot to cold nonextensive electrons on the sheath structure as well as the characteristics of dust particles in the sheath. To do this, we consider the near wall region of a magnetized multi-component plasma which consists of two superextensive electrons with different temperature cold and hot, cold fluid negatively charged statistic dust grains and cold fluid ions. The paper is organized as follows. In the next section (Section 2), we present our assumptions and the governing equations. The results and numerical simulations are presented and discussed in Section 3. Finally, conclusions are given in Section 4. 2. Basic equations and assumption In this work, we consider a magnetized stationary (dt = 0) dusty plasma sheath in contact with a planar wall (Fig. 1). The sheath is studied in a one-dimensional coordinate space system and a threedimensional speed space system. The external magnetic field, which is spatially uniform and constant in time, lies in the x z plane and makes an angle h with the x-axis. The region of interest lies between x ¼ 0 (the sheath edge), and the wall which can be located anywhere in the region x > 0. We assume that the physical parameters change only along the x-direction (normal to the wall, see Fig. 1) in the sheath region. At the sheath edge x ¼ 0, the potential is assumed to be zero, and the ion, the dust, the cold and hot electron species densities ni0 , nd0 , nec0 ; neh0 satisfy the quasi-neutrality condition:
e ni0 neh0 nec0 þ qd0 nd0 ¼ 0
ð1Þ
where e is the electron charge and qd0 is the dust charge at the sheath edge. We assume that the two electron species (hot and cold) obey the q-nonextensive distribution function defined as [34].
1 me v 2e e/ ðq1Þ f e ðv e Þ ¼ Aq ½1 ðq 1Þ Te 2T e
ð2Þ
Where the constant of normalization is
8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Cð1q Þ me ð1qÞ > > for 1 < q < 1 < ne0 Cð 1 1Þ 2pT e 1q 2 Aq ¼ ffiffiffiffiffiffiffiffiffiffiffiffi q 1 1 Cð1qþ2Þ me ð1qÞffi > > : ne0 1þq for q > 1 1 2p T e 2 Cðq1 Þ
ð3Þ
e is the unit electric charge, T e is the electron temperature in eV, / is the electrostatic potential and C denotes the standard gamma function. The density of the two electron species nej (where j ¼ c;case of the cold electrons and j ¼ h; case of the hot electrons), takes the form
"
nej ¼ nej 0 1 þ qj 1
# qj þ1 e/ 2ðqj 1Þ T ej
ð4Þ
qj stands the non-extensivity of electrons, In the extensive limiting case (qj ¼ 1), the density distribution reduces to the density Maxwell Boltzmann distribution, qj –1 describes nonextensivity of the electrons under consideration, and, we note here that the condition (qj < 1) refers to the superextensive case, whereas the opposite condition (qj > 1) refers to the subextensive case. The ions are treated as a fluid; the basic equations of ions are the continuity
@ðni v ix Þ ¼0 @x
ð5Þ
and the momentum conservation equations
v ix
@ v ix e @/ ¼ mi @x @x
ð6Þ
where v i , ni and mi are the velocity in the depth direction, density and mass of the positive ion species in the sheath, respectively. The dust grains are treated as cold fluid obeying the sources free. The electric, magnetic and gravitational forces are considered. The dust fluid equations are written as follows:
@ðnd v dx Þ ¼0 @x md v dx
! @vd @/ ! ! ! ! x þ qd v d ^ B þ md g x ¼ qd @x @x
ð7Þ ð8Þ
where qd ¼ zd e is the charge of dust grain, zd is the charge number of ! the dust, md , nd and v d are the mass, the density and velocity of the dust grains respectively. The system of equations is completed by the Poisson’s equation which relates the electric potential to the density of dust grains, two electrons and ions as follows:
@2/ ¼ 4pðeðni neh nec Þ þ qd nd Þ @x2
ð10Þ
Fig. 1. The geometry of considered sheath model.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
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O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
In order to solve the system of Eqs. (1)–(10), we introduce new dimensionless variables. We normalize each charged particle density to its value at the sheath edge: N i ¼ ni =ni0 ; N d ¼ nd =nd0 ; N ec ¼ nec =ne0 ; N eh ¼ neh =ne0 ðwherene0 ¼ neh0 þ nec0 Þ. Also we normalize the positive ion velocity to the positive ion pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sound speed ci ¼ T ec =mi , and the dust velocity to the dustpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi acoustic speed cd ¼ T ec =md respectively i.e. uix ¼ v ix =ci and ud ¼ v d =cd . Furthermore, we scale the x-coordinate and the sheath potential as follows: n ¼ x=kD and g ¼ e/=T ec , where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kD ¼ T ec =4pne0 e2 is the electron Debye length and T ec the cold electrons temperature. ! ! ! The magnetic field is of the form B ¼ B0 cosh x þ sinh z and by substituting the above dimensionless variables into Eqs. (1)–(10), we obtain:
Neh
qh þ1 P g 2ðqh 1Þ ¼ 1 ðqh 1Þ 1þP T hc qc þ1 2ðqc 1Þ
ð11Þ
Nec ¼
1 ½1 ðqc 1Þg 1þP
Nd ¼
Md udx
ð13Þ
Ni ¼
Mi uix
ð14Þ
@uix 1 @g ¼ uix @n @n
ð15Þ
udx
@udx @g ¼ zd þ czd sinðhÞudy þ f g @n @n
ð12Þ
ð16Þ
udx
@udy ¼ czd ½cosðhÞudz sinðhÞudx @n
ð17Þ
udx
@udz ¼ czd cosðhÞudy @n
ð18Þ
@2g @n2
qc þ1
2ðqc 1Þ 1 ½1 þ ðqc 1Þg 1þP
qh þ1 P g 2ðqh 1Þ 1 þ ðqh 1Þ þ ð1 dÞzd Nd 1þP T hc
¼ dNi
ð19Þ
1
c ¼ eB0 kD =ðmd T e Þ2 ; T hc ¼ T eh =T ec ; f g ¼ gkD =c2d ; P ¼ neh0 =nec0 Here the boundary conditions are as follows: at the wall ðn ¼ dÞ,
gðdÞ ¼ gx ; at the sheath edge ðn ¼ 0Þ, the boundary conditions ui ðn ¼ 0Þ ¼ M i , udx ðn ¼ 0Þ ¼ Md , and aregðn ¼ 0Þ ¼ 0, udy ðn ¼ 0Þ ¼ udz ðn ¼ 0Þ ¼ 0. To better understand the effect of the presence of the hot electron on the characteristics of the sheath, we definite the density ratio of hot electrons neh0 to cold electrons nec0 , where neh0 and nec0 are the electrons species densities in the plasma region. Also, we definite the temperature ratio of hot and cold electrons T hc ¼ T eh =T ec . From the definition of P it is clear that P ¼ 0 is the case where magnetized collisional plasma sheath consists of only one species of electrons (cold electron), so the increase of P leads to increase of the concentration of hot electron populations. Now, from the Poisson equation (Eq. (19)) if (P ¼ 0 and T hc ¼ 1) the sheath consists of only one electron populations (cold electron), therefore if (P–0 and T hc –1) is the case when the sheath consists of two electron populations. Eqs. (11)–(19) with appropriate boundary conditions are solved using a Runge–Kutta method of the fourth order [30].
Fig. 2. (a) The normalized spatial distributions of electrostatic potential; (b) the normalized electron density; (c) the normalized ion density; (d) the normalized spatial distributions of dust velocity in the depth direction versus x under various values of P and for the parameters: qc ¼ 0:7 and qh ¼ 0:65.
Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599
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O. El Ghani et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 3. (a) The normalized spatial distributions of electrostatic potential; (b) the normalized spatial distributions of dust velocity in the depth direction versus x under two different T hc values, the other parameters are the same as Fig. 2.
3. Numerical results and discussion To solve numerically the normalized multi-fluid Eqs. (11)–(19) we use ne0 ¼ 109 cm3 ,T e ¼ 2 eV, M i ¼ 3, Md ¼ 5, d ¼ 1:01 and @ g=@nðn ¼ 0Þ ¼ 0:01 as boundary conditions at the plasma-sheath interface ðn ¼ 0Þ. The normalized magnitude of magnetic field
c ¼ 0:3with angle h ¼ 30 . Also, in our calculations, we consider the dust as a spherical particle of radius rd ¼ 4 lm with uniform mass density qd ¼ 2 g:cm3 and zd ¼ 1000. The numerical solutions of Eqs. (11)–(19) are displayed in Figs. 2 and 3 for various P and T hc parameters. Fig. 2(a)–(d) depict the distribution of the normalized sheath potential, the electron and the ion densities and the normalized dust normal velocity across the sheath consisting of superextensively distributed electrons. This figure illustrates the structure of the sheath on the presence of the hot electron populations. Firstly, Fig. 2(a) shows that by increasing P, the normalized potential of the sheath increases which is due to the increasing of energetic electrons (hot electron populations). Another notable point observed from the same figure is that the sheath thickness decreases when P increases. In addition, the effect of hot electrons is noticeable when density number of cold electrons is high (low values of P). Fig. 2(b) and (c) demonstrate that the density distribution of the both electron species (cold and hot) and that of the ions respectively, increase when the hot electron populations are weaker, we can explain that for low values of hot electron populations, the electric force becomes less effective, which leads to the increase of the electrons and ions densities. Fig. 2(d) shows the behaviour of normalized dust velocity under various values of P, where it is shown that the dust normal velocity evolves the same way near the sheath edge because of the dominance of the gravitational force on grains of dust, while when one moves away from the edge, the effect of hot electrons becomes more obvious (the electric force becomes more effective) and the dust velocity decreases in higher values of hot electron densities. Now, we study effects of the ratio temperature T hc ¼ T eh =T ec of hot and cold electrons on the structure of the sheath. Fig. 3(a) and (b) show the profiles of the normalized sheath potential and that of dust velocity. We can see that with a fixed value for P ¼ 0:3 and P ¼ 1:7, the sheath potential increases for large values of the temperature ratio T hc , while the normalized x-component
velocity of dust grains decreases in increasing of T hc . A simple observation shows that the effect of the temperature ratio is obvious in higher values of hot electron populations (case of P ¼ 1:7). 4. Conclusion In this paper, we have investigated a magnetized sheath of dusty plasma consisting of cold ions and two-temperature electrons featuring nonextensive distributions (case of superextensive distributions). The study interests to the effect of hot superextensive electron populations on some quantities characterizing the magnetized plasma sheath such as the electrostatic potential, the sheath thickness and dust dynamic. Based on multi-fluid model, the sheath potential and the x-component dust velocity have been investigated for different values of the density ratio of the hot electron to the cold populations and temperature ratios of hot to cold electrons. Our results show that the behaviour of the sheath depends in particular on the ratios of densities and temperatures of hot and cold electrons. Specially, it is observed that the electrostatic potential increases with increasing the hot electron populations, also, by increasing the temperature ratio of hot to cold electrons. Consequently, the hot electron populations decrease the sheath thickness. Furthermore, our numerical results reveal that the higher values of the density ratio of the hot electron to cold electron densities and the temperature ratio of hot to cold electrons, decrease the dust velocity. Therefore, the effect of hot electron populations on the sheath structure is more obvious when the cold electron populations dominate. References [1] A. Bouchoule, Dusty Plasma: Physics, Chemistry and Technical Impact in Plasma Processing, Wiley, New York, 1999. [2] P.K. Shukla, A.A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics, Bristol, 2002. [3] M.Y. Yu, H. Saleem, H. Luo, Dusty plasma near a conducting boundary, Phys. Fluids B 4 (1992) 3427–3431. [4] C. Arnas, M. Mikikian, G. Bachet, F. Doveil, J.X. Ma, C.X. Yu, Sheath modification in the presence of dust particles, Phys. Plasmas 7 (2000) 4418–4422. [5] T. Nitter, Levitation of dust in rf and dc glow discharge, Plasma Source Sci. Technol. 5 (1996) 93–111. [6] Z. Xiu, Characteristics of dust plasma sheath in an oblique magnetic field, Chin. Phys. Lett 23 (2006) 396–698. [7] J.Y. Lieu, D. Wang, T.C. Ma, the charged dust in processing plasma sheath, Vacuum 59 (2000) 126–134.
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Please cite this article as: O. El Ghani, I. Driouch and H. Chatei, Effects of two temperature non-extensive electrons on the sheath of dusty plasma, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.599