- Email: [email protected]
Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions
Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions Aziza Asserghine a,⇑, Morad El Kaouini a,b, Hassan Chatei a a b
LPMR, Department of Physics, Faculty of Science, Mohammed I University, Oujda, Morocco Department of Physics, Polydisciplinary Faculty of Nador, Mohammed I University, Nador, Morocco
a r t i c l e
i n f o
Article history: Received 1 June 2019 Received in revised form 18 June 2019 Accepted 17 July 2019 Available online xxxx Keywords: Plasma Bohm criterion Magnetized sheath Negative ions Nonextensive electrons
a b s t r a c t Sheath region of an electronegative magnetized plasma consisting of q-nonextensive electrons, Boltzmann distributed negative ions and positive ions with finite temperature is investigated by using a steady state fluid model. Considering Sagdeev’s pseudo potential method, a modified Bohm criterion is derived. Taking into account the new formation criterion, the fluid model is then solved numerically and the density distribution of charged particles in the sheath region is studied for different values of the initial positive ion velocity at the sheath edge. 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 When a plasma is in contact with a wall, a localized non-neutral space charge layer, called the plasma sheath, is formed near the wall boundary. Plasma sheath with electronegative ions has lots of importance in many applications such as material surface treatment, plasma chemistry and nuclear fusion energy [1–3]. Many studies have been developed to describe the formation criterion and properties of a plasma sheath considering the effect of the presence of negative ions [4–7]. Ming Li et al. [4] studied the sheath structure of an electronegative plasma, they found that the sheath thickness is strongly affected by the plasma electronegativity and the temperature ratio of electrons to negative ions, and it varies nonmonotonically with the plasma electronegativity. In a collisionless magnetized electronegative plasma sheath with two species of positive ions, Hatami et al. [5] concluded numerically that by increasing the density of the negative ions in the plasma, the net density distribution of charged particles in the sheath region is nonmonotonic in the presence of negative ions while in the absence of negative ions it is monotonic. In references [6,7], the sheath formation criterion in a unmagnetized and magnetized electronegative plasma has investigated with considering the effect of collisions. The modified sheath formation criterion was derived, and it was shown that there are upper and lower ⇑ Corresponding author. E-mail address: [email protected] (A. Asserghine).
limits for positive ions velocity to enter to the sheath region, and the parameters of the negative ions only affect the lower limit. In these numerical calculations related to the Bohm criterion in the magnetized electronegative plasma sheath, the initial ion velocity components parallel to the wall are ignored or considered as fixed. However, in other work developed for collisional electronegative plasma sheath formation criterion, Li et al. [8] have examined the effect of velocity components parallel to the wall at the edge on the formation criterion and properties of the sheath. It has been shown that the sheath formation criterion in an external magnetic field when the ions enter the sheath with an oblique incident angle is different from that with ions entering the sheath edge perpendicular to the wall. Considering the effects of the mentioned components, it was found that the exclusion of the component parallel to the wall, evaluated by considering the drift velocity at the sheath edge, from the boundary conditions, lead to a decrease in the critical value of Bohm criterion. By including the two velocity components parallel to the wall in the boundary conditions, the numerical results show that, the sheath characteristics profiles change from oscillatory (or peak) structure to smooth structure. Most of these studies are carried out for plasma sheath with Maxwellien Boltzmann electrons distribution which is inadequate to describe the systems in non equilibrium state with long range interactions including plasma [9,16]. Recently Borgohain and Saharia [16] investigated a collisionless unmagnetized electronegative plasma sheath consisting of
https://doi.org/10.1016/j.matpr.2019.07.439 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: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
2
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
q-nonextensive electrons, it is found that the Bohm velocity depends on the nonextensivity parameter, negative ion temperature to nonextensive electron temperature ratio, and negative ion density. However, as far as we know, there is no report on the formation criterion and the characteristics of electronegative plasma sheath consisting of electrons following nonextensive q-distribution in the presence of an external magnetic field. In the present study, following the previous works [8,17] for Maxwellien Boltzmann electrons distribution case, we investigate the formation criterion of an electronegative nonextensive plasma sheath in presence of an oblique magnetic field. By using the fluid model, we examine simultaneously the effects of the parameters of the negative ions, the nonextensivity parameter q, as well as the external magnetic field and the positive ion temperature, on the Bohm sheath criterion and examine the charged particles density distributions in the plasma sheath region when the sheath criterion is fulfilled or not. The outline of this paper is as follows: Introduction in Section 1, the theoretical model and basic equations are formulated and the modified Bohm criterion is done analytically with its examination in some interesting physical conditions in Section 2. The numerical results and the corresponding discussions are presented in Section 3 and a brief summary is presented in Section 4.
! where q, Te , /, me , and v e are, respectively, the nonextensivity parameter, the electron temperature, the electrostatic potential, the mass and velocity of electrons. The constant of normalization Cq is given by:
Cq ¼
8 > 3q1 > > < ne0 2ð1qÞ
1=q1 me v e 2 e/ðxÞ ! f x; v e ¼ C q 1 ðq 1Þ kB T e 2kB T e
v e max ¼
3=2
ðq1Þme 2pkB T e
;
1 3
3=2
;
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2kB T e 1 e/ q 1 kB T e me
ð2Þ
Integrating the three-dimensional velocity distribution function (Eq. (1)) over the velocity space and after more detailed calculations the nonextensive electron density is obtained as follows:
ne ¼ ne0
ð1Þ
Þ
ð1qÞme 2pkB T e
where ne0 is the equilibrium density of electrons and C denotes the standard gamma function. It is useful to note that in the case of q < 1=3, the q-nonextensive distribution function is unnormalizable. In the extensive limiting case (q ¼ 1), the distribution function (Eq. (1)) reduces to the well-known Maxwell-Boltzmann distribution function. Furthermore, for q > 1, the distribution function exhibits a thermal cutoff on the maximum value allowed for the electron velocity, given by:
The geometry of the simulated region is presented in Fig. 1. We assume that the wall is electrically isolated and the sheath region to lie between x ¼ 0 (the sheath edge) and the wall. We assume also that the wall is infinitely long in they and z-directions, which means that the physical properties change only in the x-direction (normal to the wall), and hence allow us to reduce the nabla vector ! ! operator r to ð@=@xÞ x . This assumption is well known in sheath study. It is adopted by many authors to make the model simpler [5,9]. The external magnetic field is exerted in the x y plane, making an angle h with the x-axis, it can be written as: ! ! ! B ¼ Bcos h x þ Bsin h y , where B is the magnitude of magnetic field. The plasma sheath contains electrons, singly charged negative ions, singly charged positive ions with finite temperature. The electrons are supposed to follow the Tsallis statistical mechanics, so their three-dimensional velocity distribution function is given by [11,18]
Cð
1 1 1q 2
1 þ1 > Cðq1 > 2Þ > : ne0 ð3q1Þðqþ1Þ 1 4ðq1Þ Cðq1 Þ
2. The theoretical model and sheath criterion 2.1. The model and basic equations
1 Cð1q Þ
e/ 1 þ ð q 1Þ kB Te
3q1 2q2
ð3Þ
Negative ions are assumed as the Boltzmann distributed particles, their density is expressed by the Maxwell-Boltzmann distribution distribution [19–22]
e/ n ¼ n0 exp kB T
ð4Þ
where n0 and T are, respectively, the equilibrium density and temperature of negative ions. The positive ions, treated as fluid, are governed by the continuity and momentum balance equations.
! ! rðnþ v þ Þ ¼ 0
ð5Þ
! ! !! ! ! kB T þ ! mþ v þ r v þ ¼ e r / þ e v þ ^ B r nþ nþ
ð6Þ
! where m+, n+ and v þ are the mass, density and velocity of positive ions, respectively. The Poisson’s equation that relates the electron, negative and positive ions densities to the electric potential is:
@2/ e ¼ ðnþ ne n Þ @x2 e0
ð7Þ
To normalize the above equations, we adopt the dimensionless variables as follows: ! ! W ¼ ke/ ; u þ ¼ VCsþ ; Ne ¼ nne0e ; Nþ ¼ nne0þ ; N ¼ nne0 ; n ¼ B Te
; q ¼ kRDg ; T ¼ TTþe ; c ¼ nn0 ; and b ¼ TTe where Rg ¼ e0 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðmþ kB T e Þ=ðB e Þ is the positive ion gyro-radius and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kD ¼ ðe0 kB T e Þ=ðne0 e2 Þ is the electron Debye length for q ¼ 1. Cs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the ion acoustic speed given by Cs ¼ kB T e =mþ . Considering the quasineutrality at the plasma sheath boundary ðn ¼ 0Þ,ðnþ0 ¼ ne0 þ n0 Þ, we get the normalized equations as follows : x kD
Fig. 1. Magnetized plasma sheath configuration.
Ne ¼ ½1 ðq 1ÞW3q1=2q2
ð8Þ
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
3
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
N ¼ c expðbWÞ
ð9Þ
@ðNþ uxþ Þ ¼0 @n
ð10Þ
@uxþ 1 ¼ @n 1 uT2
uxþ
xþ
uxþ
uxþ
@uzþ ¼ q uxþ sin h uyþ cos h @n
ð13Þ
0
ðN e þ N N þ ÞdW is the Sagdeev potential
5q3 2 1 ½1 ðq 1ÞW2q2 5q 3 Z W c ux0þ þ ð1 expðbWÞÞ ð1 þ cÞ dW b uxþ 0
potential at the sheath edge, i.e.,
ð16Þ
ð14Þ
ð15Þ
W0 sin h
q
ð18Þ
We substitute Eq. (18) into (17), the Bohm criterion takes the following form:
the sheath edge into the sheath, one can obtain:
W þ 2V ðW; ux0þ Þ ¼ W0
ð17Þ
0
W0 ¼ Wðn ¼ 0Þ ¼ 0, W0 ¼ W ðn ¼ 0Þ–0, and Nþ0 ¼ Nþ ðn ¼ 0Þ ¼ 1 þ c, 0 and multiplying both sides of Eq. (14) by W and integrating from 2
< 0, we can obtain the
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u 2 ð 1 þ c Þ q u sin h z0þ 1 tT þ 0 2cb þ 3q 1 W0
uz0þ ¼
0
0
@ W2
where uz0þ is the drift velocity in the zdirection and it given at the sheath edge by [24]:
By using the following boundary conditions at the sheath edge, 0
@ 2 V ð0;ux0þ Þ
Bohm formation criterion for magnetized electronegative plasma sheath with q-nonextensive electrons and thermal positive ions as
ux0þ ¼ Nþ Ne N
2.2. Sheath formation criterion
02
RW
By considering the condition for maximizing the Sagdeev
ð12Þ
2
VðW; ux0þ Þ ¼
ð11Þ
@uyþ ¼ quzþ cos h @n
@2W @n
@W quzþ sinh @n
where V ðW; ux0þ Þ ¼ [23] given by:
ux0þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ cÞcos h2 Tþ 2cb þ 3q 1
ð19Þ
Fig. 2. Variation of the modified Bohm velocity as function of b for different values of c at q ¼ 0:7, h ¼ 30 , and T ¼ 0:2.
Fig. 4. Variation of the modified Bohm velocity as function of b for different values of q at c ¼ 5, h ¼ 30 , and T ¼ 0:2.
Fig. 3. Variation of the modified Bohm velocity as function of b for different values of T at q ¼ 0:7, h ¼ 30 , and c ¼ 5.
Fig. 5. Variation of the modified Bohm velocity as function of b for different values of h at q ¼ 0:7, c ¼ 5, and T ¼ 0:2.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
4
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 6. The Normalized densities of negative charged particles N e þ N , and positive ions N þ , as a function of the normalized distance from the sheath edge for q ¼ 0:7, c ¼ 5, b ¼ 15, T ¼ 0:2 for (a) h ¼ 25 , and (b) h ¼ 60 .
In the limit of an unmagnetized extensive case with one species of ions (cold positive ions), i.e., h ¼ 0, c ¼ 0 , and T ¼ 0, our derived Bohm criterion reduces to the Maxwellian counterpart [25].
The normalized velocity of positive ions at the sheath edge is fixed to ux0þ ¼ 0:49 that satisfy or do not satisfy the inequality (19). In Fig. 6(a) and (b), the angle of magnetic h values are chosen
3. Results and discussion In the following numerical calculations, A low-pressure magnetized plasma containing q-nonextensive electrons, thermal positive ions (Ar + ) and one species of negative ions is considered. The other parameters are ne0 ¼ 108 cm3 ,T e ¼ 1eV, B ¼ 0:35Tesla, and we assume the boundary values of the electric potential and the electric field at the sheath edge ðn ¼ 0Þ as W0 ¼ 0 and 0
W0 ¼ 0:01, respectively. To investigate the effect of the main parameters characterizing the proposed plasma on the modified Bohm criterion, we have plotted the initial modified Bohm ion velocity at the sheath edge as a function of b for different values of c; T; q; and h. Such parameters can modify the structure and properties of plasma sheath near the wall used in many applications such as fusion device and material surface treatment. Furthermore, to verify the validity of our derived Bohm criterion, we have studied the behavior of charged particle distributions in the sheath region, for two different situations of plasma by varying the orientation of magnetic field. Fig. 2 depicts the lower limit of the Bohm criterion versus the nonextensive electron temperature to negative ion temperature ratio b for different values of negative ion densityc. It is clear, from Fig. 2, that as b and c increase, the positive ions entrance velocity at the sheath edge ux0þ decreases. This result is in agreement with the results obtained by Borgohain and Saharia [16] in a collisionless electronegative plasma sheath without the presence of an external magnetic field. Moreover, it is observed, from Fig. 3, that by increasing values of positive ion temperature, the normalized velocity of positive ions at the sheath edge increases. The effect of the ratio of positive ion to electron temperature T becomes high when the negative ion temperature decreases. However, as can be shown in Fig. 4, the influence of nonextensivity qparameter on the Bohm criterion can be ignored. It has also been found, from Fig. 5, that as the angle of the magnetic field h increases, the positive ions entrance velocity at the sheath edge decreases especially for the lower values of b. In order to discuss the effects of magnetic field orientation with the modified Bohm criterion in electronegative magnetized collisionless plasma sheath containing q-nonextensive electrons, we illustrate in Fig. 6 the normalized density of positive ions and total negatively charged particles (electrons and negative ions) distributions, as a function of normalized distance from sheath edge, for two different values of the angle of magnetic field.
as h ¼ 25 and h ¼ 60 , respectively. Corresponding to inequality (19), their relevant minimum Bohm velocity would be ux0þ ¼ 0:5150 and ux0þ ¼ 0:4689. So, in Fig. 6(a), the velocity of positive ions at the sheath edge is lower than minimum Bohm velocity and the sheath criterion is not satisfied although situation in Fig. 6(b), satisfy inequality (19). Therefore, since the initial ion velocity in depth direction is less than the Bohm criterion in the Fig. 6(a), the number density of ions falls below that of electrons near the sheath edge. However, the ion density distribution is always greater than that of electrons in Fig. 6(b) and the sheath is formed. 4. Conclusion In this paper, we investigated the sheath formation criterion of q-nonextensive plasma, in the presence of an external oblique magnetic field and negative ions. Using a steady state fluid model, the equations of the fluid model for the thermal positive ions inside the magnetized electronegative plasma sheath are solved numerically and a modified Bohm criterion was obtained and examined for different parameters characterising the plasma sheath. The results have shown that the sheath formation criterion strongly depends on the parameter of negative ions, the magnetic field and the ratio of positive ion to electron temperature. References [1] E. Stoffels, W.W. Stoffels, G.M.W. Kroesen, Plasma Sources Sci. Technol. 10 (2001) 311–317. [2] Jinyuan Liu, Feng Wang, Jizhong Sun, Phys. Plasmas 18 (2011) 013506. [3] K. Annou, N. Saoula, R. Tadjine, Int. J. Eng. Res. Technol. 2 (2013) 882. [4] M. Li, M.A. Vyvoda, S.K. Dew, M.J. Brett, I.E.E.E. Trans, Plasma Sci. 28 (2000) 248. [5] M.M. Hatami, B. Shokri, A.R. Niknam, Phys. Plasmas 15 (2008) 123501. [6] H. Ghomi, M. Khoramabadi, P.K. Shukla, M. Ghorannevis, J. Appl. Phys. 108 (2010) 063302. [7] M.M. Hatami, B. Shokri, Phys. Plasmas 20 (2013) 033506. [8] J.J. Li, J.X. Ma, Zi-an Wei, Phys. Plasmas 20 (2013) 063503. [9] I. Driouch, H. Chatei, Eur. Phys. J. D 71 (2017) 9. [10] L.A. Gougam, M. Tribeche, Phys. Plasmas 18 (2011) 062102. [11] N.N. Safa, H. Ghomi, A.R. Niknam, Phys. Plasmas 21 (2014) 082111. [12] M.M. Hatami, Phys. Plasmas 22 (2015) 013508. [13] Y. Liu, S.Q. Liu, L. Zhou, Phys. Plasmas 20 (2013) 043702. [14] D.R. Borgohain, K. Saharia, K.S. Goswami, Phys. Plasmas 23 (2016) 122113. [15] N. Navab Safa, H. Ghomi, A.R. Niknam, J. Plasma Phys. 81 (2015) 905810303. [16] D.R. Borgohain, K. Saharia, Plasma Phys. Rep. 44 (2018) 137–144. [17] J. Ou, N. Xiang, C. Gan, J. Yang, Phys. Plasmas 20 (2013) 063502. [18] R. Silva Jr., A. Plastino, J. Lima, Phys. Lett. A 249 (1998) 401–408. [19] R.F. Fernsler, S.P. Slinker, Phys. Rev. E 71 (2005) 026401. [20] Y. Ghim Kim, N. Hershkowitz, Appl. Phys. Lett. 94 (2009) 151503.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx [21] S.V. Berezhnoj, C.B. Shin, U. Buddemeier, I. Kaganovich, Appl. Phys. Lett. 77 (2000) 800. [22] R.N. Franklin, J. Snell, J. Plasma Phys. 64 (2000) 131.
5
[23] B. Alterkop, J. Appl. Phys. 104 (2008) 123301. [24] M. Elkaouini, H. Chatei, J. Fusion Energ. 31 (2012) 317–324. [25] F.F. Chen, Introduction to Plasma Physics, Plenum Press, New York, 1974.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions Aziza Asserghine a,⇑, Morad El Kaouini a,b, Hassan Chatei a a b
LPMR, Department of Physics, Faculty of Science, Mohammed I University, Oujda, Morocco Department of Physics, Polydisciplinary Faculty of Nador, Mohammed I University, Nador, Morocco
a r t i c l e
i n f o
Article history: Received 1 June 2019 Received in revised form 18 June 2019 Accepted 17 July 2019 Available online xxxx Keywords: Plasma Bohm criterion Magnetized sheath Negative ions Nonextensive electrons
a b s t r a c t Sheath region of an electronegative magnetized plasma consisting of q-nonextensive electrons, Boltzmann distributed negative ions and positive ions with finite temperature is investigated by using a steady state fluid model. Considering Sagdeev’s pseudo potential method, a modified Bohm criterion is derived. Taking into account the new formation criterion, the fluid model is then solved numerically and the density distribution of charged particles in the sheath region is studied for different values of the initial positive ion velocity at the sheath edge. 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 When a plasma is in contact with a wall, a localized non-neutral space charge layer, called the plasma sheath, is formed near the wall boundary. Plasma sheath with electronegative ions has lots of importance in many applications such as material surface treatment, plasma chemistry and nuclear fusion energy [1–3]. Many studies have been developed to describe the formation criterion and properties of a plasma sheath considering the effect of the presence of negative ions [4–7]. Ming Li et al. [4] studied the sheath structure of an electronegative plasma, they found that the sheath thickness is strongly affected by the plasma electronegativity and the temperature ratio of electrons to negative ions, and it varies nonmonotonically with the plasma electronegativity. In a collisionless magnetized electronegative plasma sheath with two species of positive ions, Hatami et al. [5] concluded numerically that by increasing the density of the negative ions in the plasma, the net density distribution of charged particles in the sheath region is nonmonotonic in the presence of negative ions while in the absence of negative ions it is monotonic. In references [6,7], the sheath formation criterion in a unmagnetized and magnetized electronegative plasma has investigated with considering the effect of collisions. The modified sheath formation criterion was derived, and it was shown that there are upper and lower ⇑ Corresponding author. E-mail address: [email protected] (A. Asserghine).
limits for positive ions velocity to enter to the sheath region, and the parameters of the negative ions only affect the lower limit. In these numerical calculations related to the Bohm criterion in the magnetized electronegative plasma sheath, the initial ion velocity components parallel to the wall are ignored or considered as fixed. However, in other work developed for collisional electronegative plasma sheath formation criterion, Li et al. [8] have examined the effect of velocity components parallel to the wall at the edge on the formation criterion and properties of the sheath. It has been shown that the sheath formation criterion in an external magnetic field when the ions enter the sheath with an oblique incident angle is different from that with ions entering the sheath edge perpendicular to the wall. Considering the effects of the mentioned components, it was found that the exclusion of the component parallel to the wall, evaluated by considering the drift velocity at the sheath edge, from the boundary conditions, lead to a decrease in the critical value of Bohm criterion. By including the two velocity components parallel to the wall in the boundary conditions, the numerical results show that, the sheath characteristics profiles change from oscillatory (or peak) structure to smooth structure. Most of these studies are carried out for plasma sheath with Maxwellien Boltzmann electrons distribution which is inadequate to describe the systems in non equilibrium state with long range interactions including plasma [9,16]. Recently Borgohain and Saharia [16] investigated a collisionless unmagnetized electronegative plasma sheath consisting of
https://doi.org/10.1016/j.matpr.2019.07.439 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: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
2
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
q-nonextensive electrons, it is found that the Bohm velocity depends on the nonextensivity parameter, negative ion temperature to nonextensive electron temperature ratio, and negative ion density. However, as far as we know, there is no report on the formation criterion and the characteristics of electronegative plasma sheath consisting of electrons following nonextensive q-distribution in the presence of an external magnetic field. In the present study, following the previous works [8,17] for Maxwellien Boltzmann electrons distribution case, we investigate the formation criterion of an electronegative nonextensive plasma sheath in presence of an oblique magnetic field. By using the fluid model, we examine simultaneously the effects of the parameters of the negative ions, the nonextensivity parameter q, as well as the external magnetic field and the positive ion temperature, on the Bohm sheath criterion and examine the charged particles density distributions in the plasma sheath region when the sheath criterion is fulfilled or not. The outline of this paper is as follows: Introduction in Section 1, the theoretical model and basic equations are formulated and the modified Bohm criterion is done analytically with its examination in some interesting physical conditions in Section 2. The numerical results and the corresponding discussions are presented in Section 3 and a brief summary is presented in Section 4.
! where q, Te , /, me , and v e are, respectively, the nonextensivity parameter, the electron temperature, the electrostatic potential, the mass and velocity of electrons. The constant of normalization Cq is given by:
Cq ¼
8 > 3q1 > > < ne0 2ð1qÞ
1=q1 me v e 2 e/ðxÞ ! f x; v e ¼ C q 1 ðq 1Þ kB T e 2kB T e
v e max ¼
3=2
ðq1Þme 2pkB T e
;
1 3
3=2
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2kB T e 1 e/ q 1 kB T e me
ð2Þ
Integrating the three-dimensional velocity distribution function (Eq. (1)) over the velocity space and after more detailed calculations the nonextensive electron density is obtained as follows:
ne ¼ ne0
ð1Þ
Þ
ð1qÞme 2pkB T e
where ne0 is the equilibrium density of electrons and C denotes the standard gamma function. It is useful to note that in the case of q < 1=3, the q-nonextensive distribution function is unnormalizable. In the extensive limiting case (q ¼ 1), the distribution function (Eq. (1)) reduces to the well-known Maxwell-Boltzmann distribution function. Furthermore, for q > 1, the distribution function exhibits a thermal cutoff on the maximum value allowed for the electron velocity, given by:
The geometry of the simulated region is presented in Fig. 1. We assume that the wall is electrically isolated and the sheath region to lie between x ¼ 0 (the sheath edge) and the wall. We assume also that the wall is infinitely long in they and z-directions, which means that the physical properties change only in the x-direction (normal to the wall), and hence allow us to reduce the nabla vector ! ! operator r to ð@=@xÞ x . This assumption is well known in sheath study. It is adopted by many authors to make the model simpler [5,9]. The external magnetic field is exerted in the x y plane, making an angle h with the x-axis, it can be written as: ! ! ! B ¼ Bcos h x þ Bsin h y , where B is the magnitude of magnetic field. The plasma sheath contains electrons, singly charged negative ions, singly charged positive ions with finite temperature. The electrons are supposed to follow the Tsallis statistical mechanics, so their three-dimensional velocity distribution function is given by [11,18]
Cð
1 1 1q 2
1 þ1 > Cðq1 > 2Þ > : ne0 ð3q1Þðqþ1Þ 1 4ðq1Þ Cðq1 Þ
2. The theoretical model and sheath criterion 2.1. The model and basic equations
1 Cð1q Þ
e/ 1 þ ð q 1Þ kB Te
3q1 2q2
ð3Þ
Negative ions are assumed as the Boltzmann distributed particles, their density is expressed by the Maxwell-Boltzmann distribution distribution [19–22]
e/ n ¼ n0 exp kB T
ð4Þ
where n0 and T are, respectively, the equilibrium density and temperature of negative ions. The positive ions, treated as fluid, are governed by the continuity and momentum balance equations.
! ! rðnþ v þ Þ ¼ 0
ð5Þ
! ! !! ! ! kB T þ ! mþ v þ r v þ ¼ e r / þ e v þ ^ B r nþ nþ
ð6Þ
! where m+, n+ and v þ are the mass, density and velocity of positive ions, respectively. The Poisson’s equation that relates the electron, negative and positive ions densities to the electric potential is:
@2/ e ¼ ðnþ ne n Þ @x2 e0
ð7Þ
To normalize the above equations, we adopt the dimensionless variables as follows: ! ! W ¼ ke/ ; u þ ¼ VCsþ ; Ne ¼ nne0e ; Nþ ¼ nne0þ ; N ¼ nne0 ; n ¼ B Te
; q ¼ kRDg ; T ¼ TTþe ; c ¼ nn0 ; and b ¼ TTe where Rg ¼ e0 ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðmþ kB T e Þ=ðB e Þ is the positive ion gyro-radius and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kD ¼ ðe0 kB T e Þ=ðne0 e2 Þ is the electron Debye length for q ¼ 1. Cs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the ion acoustic speed given by Cs ¼ kB T e =mþ . Considering the quasineutrality at the plasma sheath boundary ðn ¼ 0Þ,ðnþ0 ¼ ne0 þ n0 Þ, we get the normalized equations as follows : x kD
Fig. 1. Magnetized plasma sheath configuration.
Ne ¼ ½1 ðq 1ÞW3q1=2q2
ð8Þ
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
3
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
N ¼ c expðbWÞ
ð9Þ
@ðNþ uxþ Þ ¼0 @n
ð10Þ
@uxþ 1 ¼ @n 1 uT2
uxþ
xþ
uxþ
uxþ
@uzþ ¼ q uxþ sin h uyþ cos h @n
ð13Þ
0
ðN e þ N N þ ÞdW is the Sagdeev potential
5q3 2 1 ½1 ðq 1ÞW2q2 5q 3 Z W c ux0þ þ ð1 expðbWÞÞ ð1 þ cÞ dW b uxþ 0
potential at the sheath edge, i.e.,
ð16Þ
ð14Þ
ð15Þ
W0 sin h
q
ð18Þ
We substitute Eq. (18) into (17), the Bohm criterion takes the following form:
the sheath edge into the sheath, one can obtain:
W þ 2V ðW; ux0þ Þ ¼ W0
ð17Þ
0
W0 ¼ Wðn ¼ 0Þ ¼ 0, W0 ¼ W ðn ¼ 0Þ–0, and Nþ0 ¼ Nþ ðn ¼ 0Þ ¼ 1 þ c, 0 and multiplying both sides of Eq. (14) by W and integrating from 2
< 0, we can obtain the
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u 2 ð 1 þ c Þ q u sin h z0þ 1 tT þ 0 2cb þ 3q 1 W0
uz0þ ¼
0
0
@ W2
where uz0þ is the drift velocity in the zdirection and it given at the sheath edge by [24]:
By using the following boundary conditions at the sheath edge, 0
@ 2 V ð0;ux0þ Þ
Bohm formation criterion for magnetized electronegative plasma sheath with q-nonextensive electrons and thermal positive ions as
ux0þ ¼ Nþ Ne N
2.2. Sheath formation criterion
02
RW
By considering the condition for maximizing the Sagdeev
ð12Þ
2
VðW; ux0þ Þ ¼
ð11Þ
@uyþ ¼ quzþ cos h @n
@2W @n
@W quzþ sinh @n
where V ðW; ux0þ Þ ¼ [23] given by:
ux0þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 þ cÞcos h2 Tþ 2cb þ 3q 1
ð19Þ
Fig. 2. Variation of the modified Bohm velocity as function of b for different values of c at q ¼ 0:7, h ¼ 30 , and T ¼ 0:2.
Fig. 4. Variation of the modified Bohm velocity as function of b for different values of q at c ¼ 5, h ¼ 30 , and T ¼ 0:2.
Fig. 3. Variation of the modified Bohm velocity as function of b for different values of T at q ¼ 0:7, h ¼ 30 , and c ¼ 5.
Fig. 5. Variation of the modified Bohm velocity as function of b for different values of h at q ¼ 0:7, c ¼ 5, and T ¼ 0:2.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
4
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 6. The Normalized densities of negative charged particles N e þ N , and positive ions N þ , as a function of the normalized distance from the sheath edge for q ¼ 0:7, c ¼ 5, b ¼ 15, T ¼ 0:2 for (a) h ¼ 25 , and (b) h ¼ 60 .
In the limit of an unmagnetized extensive case with one species of ions (cold positive ions), i.e., h ¼ 0, c ¼ 0 , and T ¼ 0, our derived Bohm criterion reduces to the Maxwellian counterpart [25].
The normalized velocity of positive ions at the sheath edge is fixed to ux0þ ¼ 0:49 that satisfy or do not satisfy the inequality (19). In Fig. 6(a) and (b), the angle of magnetic h values are chosen
3. Results and discussion In the following numerical calculations, A low-pressure magnetized plasma containing q-nonextensive electrons, thermal positive ions (Ar + ) and one species of negative ions is considered. The other parameters are ne0 ¼ 108 cm3 ,T e ¼ 1eV, B ¼ 0:35Tesla, and we assume the boundary values of the electric potential and the electric field at the sheath edge ðn ¼ 0Þ as W0 ¼ 0 and 0
W0 ¼ 0:01, respectively. To investigate the effect of the main parameters characterizing the proposed plasma on the modified Bohm criterion, we have plotted the initial modified Bohm ion velocity at the sheath edge as a function of b for different values of c; T; q; and h. Such parameters can modify the structure and properties of plasma sheath near the wall used in many applications such as fusion device and material surface treatment. Furthermore, to verify the validity of our derived Bohm criterion, we have studied the behavior of charged particle distributions in the sheath region, for two different situations of plasma by varying the orientation of magnetic field. Fig. 2 depicts the lower limit of the Bohm criterion versus the nonextensive electron temperature to negative ion temperature ratio b for different values of negative ion densityc. It is clear, from Fig. 2, that as b and c increase, the positive ions entrance velocity at the sheath edge ux0þ decreases. This result is in agreement with the results obtained by Borgohain and Saharia [16] in a collisionless electronegative plasma sheath without the presence of an external magnetic field. Moreover, it is observed, from Fig. 3, that by increasing values of positive ion temperature, the normalized velocity of positive ions at the sheath edge increases. The effect of the ratio of positive ion to electron temperature T becomes high when the negative ion temperature decreases. However, as can be shown in Fig. 4, the influence of nonextensivity qparameter on the Bohm criterion can be ignored. It has also been found, from Fig. 5, that as the angle of the magnetic field h increases, the positive ions entrance velocity at the sheath edge decreases especially for the lower values of b. In order to discuss the effects of magnetic field orientation with the modified Bohm criterion in electronegative magnetized collisionless plasma sheath containing q-nonextensive electrons, we illustrate in Fig. 6 the normalized density of positive ions and total negatively charged particles (electrons and negative ions) distributions, as a function of normalized distance from sheath edge, for two different values of the angle of magnetic field.
as h ¼ 25 and h ¼ 60 , respectively. Corresponding to inequality (19), their relevant minimum Bohm velocity would be ux0þ ¼ 0:5150 and ux0þ ¼ 0:4689. So, in Fig. 6(a), the velocity of positive ions at the sheath edge is lower than minimum Bohm velocity and the sheath criterion is not satisfied although situation in Fig. 6(b), satisfy inequality (19). Therefore, since the initial ion velocity in depth direction is less than the Bohm criterion in the Fig. 6(a), the number density of ions falls below that of electrons near the sheath edge. However, the ion density distribution is always greater than that of electrons in Fig. 6(b) and the sheath is formed. 4. Conclusion In this paper, we investigated the sheath formation criterion of q-nonextensive plasma, in the presence of an external oblique magnetic field and negative ions. Using a steady state fluid model, the equations of the fluid model for the thermal positive ions inside the magnetized electronegative plasma sheath are solved numerically and a modified Bohm criterion was obtained and examined for different parameters characterising the plasma sheath. The results have shown that the sheath formation criterion strongly depends on the parameter of negative ions, the magnetic field and the ratio of positive ion to electron temperature. References [1] E. Stoffels, W.W. Stoffels, G.M.W. Kroesen, Plasma Sources Sci. Technol. 10 (2001) 311–317. [2] Jinyuan Liu, Feng Wang, Jizhong Sun, Phys. Plasmas 18 (2011) 013506. [3] K. Annou, N. Saoula, R. Tadjine, Int. J. Eng. Res. Technol. 2 (2013) 882. [4] M. Li, M.A. Vyvoda, S.K. Dew, M.J. Brett, I.E.E.E. Trans, Plasma Sci. 28 (2000) 248. [5] M.M. Hatami, B. Shokri, A.R. Niknam, Phys. Plasmas 15 (2008) 123501. [6] H. Ghomi, M. Khoramabadi, P.K. Shukla, M. Ghorannevis, J. Appl. Phys. 108 (2010) 063302. [7] M.M. Hatami, B. Shokri, Phys. Plasmas 20 (2013) 033506. [8] J.J. Li, J.X. Ma, Zi-an Wei, Phys. Plasmas 20 (2013) 063503. [9] I. Driouch, H. Chatei, Eur. Phys. J. D 71 (2017) 9. [10] L.A. Gougam, M. Tribeche, Phys. Plasmas 18 (2011) 062102. [11] N.N. Safa, H. Ghomi, A.R. Niknam, Phys. Plasmas 21 (2014) 082111. [12] M.M. Hatami, Phys. Plasmas 22 (2015) 013508. [13] Y. Liu, S.Q. Liu, L. Zhou, Phys. Plasmas 20 (2013) 043702. [14] D.R. Borgohain, K. Saharia, K.S. Goswami, Phys. Plasmas 23 (2016) 122113. [15] N. Navab Safa, H. Ghomi, A.R. Niknam, J. Plasma Phys. 81 (2015) 905810303. [16] D.R. Borgohain, K. Saharia, Plasma Phys. Rep. 44 (2018) 137–144. [17] J. Ou, N. Xiang, C. Gan, J. Yang, Phys. Plasmas 20 (2013) 063502. [18] R. Silva Jr., A. Plastino, J. Lima, Phys. Lett. A 249 (1998) 401–408. [19] R.F. Fernsler, S.P. Slinker, Phys. Rev. E 71 (2005) 026401. [20] Y. Ghim Kim, N. Hershkowitz, Appl. Phys. Lett. 94 (2009) 151503.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439
A. Asserghine et al. / Materials Today: Proceedings xxx (xxxx) xxx [21] S.V. Berezhnoj, C.B. Shin, U. Buddemeier, I. Kaganovich, Appl. Phys. Lett. 77 (2000) 800. [22] R.N. Franklin, J. Snell, J. Plasma Phys. 64 (2000) 131.
5
[23] B. Alterkop, J. Appl. Phys. 104 (2008) 123301. [24] M. Elkaouini, H. Chatei, J. Fusion Energ. 31 (2012) 317–324. [25] F.F. Chen, Introduction to Plasma Physics, Plenum Press, New York, 1974.
Please cite this article as: A. Asserghine, M. El Kaouini and H. Chatei, Investigation of magnetized plasma sheath in the presence of q-nonextensive electrons and negative ions, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.07.439