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Effect of dust–neutral collisions on the dust characteristics in a magnetized plasma sheath
Vacuum 83 (2009) 1031–1035
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Effect of dust–neutral collisions on the dust characteristics in a magnetized plasma sheath S. Farhad Masoudi a, *, G. Reza Jafari b, Hossein Akbarian Shorakaee a a b
Department of Physics, K.N. Toosi University of Technology, 41, Shahid Kavian St., P.O. Box 15875, 4416 Tehran, Iran Department of Physics, Shahid Beheshti University, G. C. Evin, Tehran 19839, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 May 2008 Received in revised form 31 January 2009 Accepted 2 February 2009
The characteristics of dust in a plasma sheath are investigated in the presence of an external magnetic field and taking into account the dust–neutral collision force. The continuity and momentum equations of ions and dust particles are solved numerically with various magnitudes of collision force by using the fluid model. The numerical results have revealed that the collision force reduces the dust gyro radius, changes the positions of the extrema of the dust density and the velocity in the depth direction. It is shown that the collision force reduces the dust kinetic energy which has no fluctuation even in a strong external magnetic field. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Plasma sheath Dust particles Collision force Magnetic field
1. Introduction The problem of plasma flowing into a wall has previously been studied [1]. Investigation into the characteristics of the space charged region in front of the wall is one of the oldest problems in plasma physics. This region which shields the bulk plasma from the wall is named the ‘‘sheath’’. Studying the sheath region has continued to remain of the current interest because of its practical importance in plasma dynamics [2–6]. Recently, the study of the plasma sheath in the presence of dust has become an important research area due to its common observance in laboratory and space plasmas [7–11]. Compared with other charged grains, the dust has considerable charge and mass. The typical dust particles have wmicron sized, w1000e negative charges and w1000 kg/m3 mass density. Moreover, as these characteristics are dynamic variables, the behavior of the dust particles modifies the plasma dynamics and their effects may have properties different from those caused by negative ions. During the past decades, several studies have been developed to investigate the structure of dust plasma sheath [12–15]. They include for example, the fluid model [14,16], isolated particle model [17], electrostatic probe model [18] and a new one called the dust acoustic wave model [19]. These studies have considered the effect of parameters which modify the dust plasma characteristics, such
* Corresponding author. E-mail address: [email protected] (S.F. Masoudi). 0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.02.003
as the action of electrostatic, gravitational force and the Lorentz force. Two important factors which affect the structure of the dust plasma sheath are the collision force and external magnetic field. Only a few of the studies have considered the external magnetic field [13–15,20,21]. It has been shown that the structure of the dust plasma sheath without magnetic field is considerably different from the dust plasma sheath in an external magnetic field. In the present study, we simultaneously consider the effects of the electrostatic, gravitational, external magnetic field and the collision force on dust characteristics. Based on some earlier studies [14,15,22,23], the fluid model is used for a dust plasma sheath which has one-dimensional coordinate space and three-dimensional velocity space. Solving the basic equations of ion and dust movement, we investigate the effect of the magnitude of the collision force on some characteristics of dust grains like gyration movement, velocity, kinetic energy and the density of dust. We assume that the collisions between ions (which are treated as a cold fluid) and neutrals can be neglected. The sheath is generally investigated in two limiting cases; collisionless (li > >le) and collisional (li <
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S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
2. Basic equations of the model In our model, we have considered a dust plasma sheath consisting of isothermal electrons, fluid ions and cold charged dust grains. The x-direction is the depth from the sheath edge to the wall. The constant magnetic field is embedded in the x–z plane and makes an angel q with the x-direction. The geometry of the model is illustrated in Fig. 1. Before considering the basic equation of the model, we introduce some parameters as follows. The normalized forms of these parameters are used in numerical simulation. The variable physical parameters (which we assume change only along the depth direction):
f; the electrostatic potential which is zero at the edge of the sheath x ¼ 0 Ne; the electron density with ne0 value at sheath edge Ni (nd); the ion (dust) density with ni0 (nd0) value at sheath edge vi (vd); the ion (dust) velocity Constant parameters: qd; the charge of dust mi (md); the mass of ion (dust) Te; electron temperature cis ¼ (Te/mi)0.5; ion sound speed cds ¼ (Te/md)0.5; dust sound speed sd ¼ md/eB0; dust cyclotron period rd ¼ cdssd ¼ ðTe md =e2 B20 Þ0:5 ; dust grain gyro radius le ¼ (30Te/ne0e2)0.5; electron Debye length and the dimensionless quantities: Mi ¼ vix0/cis; ion Mach number Md ¼ vdx0/cds; dust Mach number g ¼ le/rd di ¼ ni0/ne0 Zd ¼ qd/e; dust grain charge number di ¼ ni0/ne0 dd ¼ nd0/ne0 h ¼ ef/Te z ¼ x/le ud ¼ vd/cds Ne ¼ ne/ne0 Ni ¼ ni/ni0 m ¼ glemd/Te
Fig. 2. The velocity of dust in depth direction under various a values (g ¼ 0.1 and q ¼ 50).
Using these parameters, the basic equations of the model can be expressed as follows:
1: The Boltzmann relation for isothermal electrons:
ne ¼ n0 expðef=Te Þ
(1)
2: Equation of continuity and momentum for ions:
Plasma
x=0 y⊗ x
z B
Sheath
θ
Wall Fig. 1. The geometry of the model.
Ni ¼
vix0 vix
mi v2ix v2ix0 ¼ 2hTe
(2)
(3)
In equations (2) and (3), we have taken into consideration the fact that the mean effect of the magnetization of ions and electrons on their movement is zero. This is due to the fact that the movement of dust particles is far slower than that of the ions and electrons, and so the ions and electrons are excessively magnetized.
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
1033
Fig. 4. The density of dust for collisionless (a ¼ 0) and collisional plasma sheath (a ¼ 5) (g ¼ 0.1, q ¼ 50 and Zd ¼ 1000).
udx
vudy ¼ gZd ðcos qudz sin qudx Þ audy vz
udz
vudz ¼ gZd cos qudy audz vz
(9)
(10)
where a is a dimensionless parameter which characterizes the magnitude of the collision force and can be expressed as a ¼ lessnn (where nn is the neutral gas density and ss is the collision crosssection measured at dust sound speed). The model is completed with the Poisson equation which relates the electrostatic potential to the density of dust, ions and electrons as follow:
V2 f ¼ ðqd nd þ eni ene Þ=30
Fig. 3. The velocity of dust in depth direction for collisionless (a ¼ 0) and collisional plasma sheath (a ¼ 5) (g ¼ 0.1 and q ¼ 50).
By using the dimensionless parameters, the Poisson equation can be expressed as follow;
d2 2
dz
Combining equations (2) and (3), we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ni ¼ Mi = Mi2 þ 2h
(4)
(11)
h ¼ Zd dd Nd þ di Ni Ne
(12)
Now we solve the basic equations of the model (equations (1),(4),(7)–(10),(12)) numerically to find the characteristics of dust in the plasma sheath in the presence of the external magnetic field and various magnitudes of collision force.
3: Equation of continuity and momentum for dust:
! V $ðnd ! v dÞ ¼ 0 ! ! ! md ð! v d $ V Þ! v d ¼ qd V f þ qd ! v d B 0 þ md g b x md n! v
3. Numerical calculation and discussion
(5) (6)
The last term of equation (6) corresponds to the collision force in which n is the dust collision frequency. In constant collisional mobility (where the cross-section for collision between dust and neutral has an inverse ratio to velocity) n is constant. Using the dimensionless parameters, equations (5) and (6) can be expressed as follows:
Nd udx ¼ Md
udx
vudx vh ¼ Zd þ gZd sin qudy þ m audx vz vz
(7)
(8)
We adopt the following parameters for numerical calculation; Mi ¼ 1.1, Md ¼ 5.0, di ¼ 1.01, dd ¼ 105 and the original electric field is allowed to take on a finite value vh/vzjz¼0 ¼ 0.01. These parameters are considered the same as used by Liu et al. [14] enable us to compare the results for the collisionless case with their calculations. However, Liu et al. have not defined the form of their dust mass. In our numerical calculation, we consider the dusts as spherical particles with uniform mass which have 4 mm radius and 2 g/cm3 density. Using the density and radius of dust and adopting Te ¼ 2 eV and ne0 ¼ 109 cm3, the value of dimensionless parameter corresponding to mass is m ¼ 5.46 103. In Fig. 2, we calculate the dust velocity in the depth direction versus the variation of a value for Zd ¼ 1000 and Zd ¼ 500. Fig. 3 shows the same calculation for two specific cases a ¼ 0 and a ¼ 10.
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S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
Fig. 5. The relive difference of velocity between collisionless and collisional case.
Fig. 7. The relative difference of kinetic energy of dust for different directions and magnitudes of magnetic field.
As the figures show, in the collisionless case (a ¼ 0), the dust velocity fluctuates as has been reported in Ref. [14]. There are more fluctuations corresponds to bigger jZd j value. The bigger jZd j value becomes, the more fluctuations occur. As the fluctuations are due to the external magnetic field, the results show that the effect of magnetic field is greater for bigger jZd j value. The effect of collision force on the dust velocity can be neglected near the sheath edge. By increasing the distance from the sheath edge, the collision force decreases the dust velocity. The larger the distance from the sheath edge becomes, we observe lesser the dust velocity in depth direction. The same result can be found for vdy and vdz by numerical calculation. As a result of collision force, the extrema of dust velocity occur nearer to the sheath edge. The degree of shifting of extrema is in direct ratio to the distance from the sheath edge. This means that at a certain distance (for instance from sheath edge to z ¼ 20 in Fig. 3) there are more fluctuations in collision regime compared with collisionless regime. Because of the inverse relation between the
velocity in depth direction and the dust density (equation (6)), the results can be related to the density of dust (see Fig. 4). However, as the dust velocity decreases in the collisional plasma sheath model, the density of dust increases. In Fig. 5, the relative difference in velocity between a collisional and collisionless plasma sheath (Dvdx ¼ (vdx(a ¼ 0) vdx(a ¼ 5))/ vdx(a ¼ 0)) is shown for different g and q values. For a higher magnetic field, the relative difference has more fluctuations since the bigger magnetic field corresponds to more fluctuations of dust velocity [14]. The fluctuations of the relative difference are due to the fact that the collisions change the positions of the extrema. For example, in Fig. 3 (Zd ¼ 1000), between z ¼ 12 and 15, the velocity increases in collisionless case whereas the velocity decreases in the collisional case. So in the interval z ¼ 12–15, the relative difference increases. Fig. 5 shows that the shifts in the positions of the extrema of the velocity and density (and changes in their amplitudes) depend strongly on the magnitude of magnetic field in z-direction.
Fig. 6. The dust kinetic energy versus depth direction in different a values.
Fig. 8. The dust helical movement in collisional (solid line) and collisionless (circles) plasma sheath.
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
1035
In Fig. 8, the dust helical movement is investigated in a collision mode. The figure shows that the collision force decreases the dust gyro radius and makes more helical movements in specific distance from the sheath edge. The larger the distance from sheath edge becomes, the smaller dust gyro radius results. The electrostatic potential versus the depth direction is calculated numerically in Fig. 9. As the figure shows the collisions between dust and neutrals do not affect the electrostatic potential. This implies that the effect of collision on electron density can be neglected. 4. Conclusion The dust characteristics were investigated numerically in a magnetized plasma sheath. In this model, we consider the electrostatic, gravitational, external magnetic field and the different magnitudes of collision force simultaneously. The numerical results reveal that the collision force decreases the dust velocity, changes the positions of the extrema of the velocity in depth direction and the density of dust. The results show that the effect of collisions increases when the magnitude of magnetic field in depth direction is decreased. Fig. 9. The electrostatic potential in depth direction under different magnitudes of magnetic field.
In Fig. 6, the dust kinetic energy per mass ðK ¼ v2d =2Þ is calculated numerically as a function of depth for various magnitudes of collision force. As the figure shows, there are no fluctuations in kinetic energy. We consider g ¼ 0.5 which corresponds to a larger magnetic field. In this case, the fluctuations of dust velocity in the depth direction between z ¼ 0 and 20 are approximately 4 times more than that of the case in which the quantity of g is 0.1. However, the kinetic energy (or the magnitude of the velocity; v2dx þ v2dy þ v2dz ) does not fluctuate. This result is reasonable since the forces arising from the magnetic field do not contribute to any energy change. The relative reduction of dust kinetic energy per mass (between the collisional and collisionless case; (DK ¼ (K(a ¼ 0) K(a ¼ 10))/ K(a ¼ 0))) is plotted in Fig. 7. This figure shows that the effect of collision on the kinetic energy, for a specific q value, is approximately the same for different magnitudes of magnetic field. However, the effect of collisions increases when the magnitude of the magnetic field in the depth direction decreases.
References [1] Chen FF. Introduction to plasma physics and controlled fusion. New York: Plenum; 1984. [2] Riemann KU. Phys Fluids 1981;24:2163. [3] Masoudi SF. J Phys D Appl Phys 2007;40:6641. [4] Franklin RN. J Phys D Appl Phys 2003;36:309. [5] Tskhakaya DD, Shukla PK, Eliasson B, Kuhn S. Phys Plasmas 2005;12:103503. [6] Thornhill LD, Desai PV. J Plasma Phys 1998;59:505. [7] Selwyn GS, Heidenrich JE, Haller KL. Appl Phys Lett 1990;57:1876. [8] Liu JY, Wang DZ, Ma TC. Vacuum 2000;59:126. [9] Mendis DA, Rosenberg M. Ann Rev Astron Astrophys 1993;32:419. [10] Nitter T. Plasma Source Sci Technol 1996;5:93. [11] Wang ZX, Liu JY, Liu Y, Wang X. J Plasma Phys 2004;70:577. [12] Duan P, Liu J, Gong Y, Liu Y, Wang X. Plasma Sci Technol 2007;9:394. [13] Kalita P, Das GC. J Plasma Phys 2003;69:551. [14] Liu JY, Zhang Q, Zou X, Wang ZX, Liu Y, Wang XG, Gong Y. Vaccum 2004;73:687. [15] Das GC, Kalita P. J Phys D Appl Phys 2004;37:702. [16] Liu TY, Ma JX. Phys. Plasmas 1997;4:2798. [17] Wang ZX, Wang W, Liu Y, Liu J, Wang X. Chin Phys Lett 2004;21:697. [18] Ma JX, Yu MY. Phys Plasmas 1995;2:1343. [19] Barkan A, Merlino RL, Angelo D. Phys Plasmas 1995;2:3563. [20] Baishya SK. Phys Plasmas 1999;6:3678. [21] Salimullah M, Nitta H, Islam MK, Shukla PK. J Plasma Phys 2005;71:157. [22] Kang SW, Min KW, Lee Es, Seon. J Surf Coat Technol 2003;171. [23] Riemann KU. Phys Plasmas 1994;1:552. [24] Zou Xiu. Chin Phys Lett 2006;23:396.
Contents lists available at ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Effect of dust–neutral collisions on the dust characteristics in a magnetized plasma sheath S. Farhad Masoudi a, *, G. Reza Jafari b, Hossein Akbarian Shorakaee a a b
Department of Physics, K.N. Toosi University of Technology, 41, Shahid Kavian St., P.O. Box 15875, 4416 Tehran, Iran Department of Physics, Shahid Beheshti University, G. C. Evin, Tehran 19839, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 May 2008 Received in revised form 31 January 2009 Accepted 2 February 2009
The characteristics of dust in a plasma sheath are investigated in the presence of an external magnetic field and taking into account the dust–neutral collision force. The continuity and momentum equations of ions and dust particles are solved numerically with various magnitudes of collision force by using the fluid model. The numerical results have revealed that the collision force reduces the dust gyro radius, changes the positions of the extrema of the dust density and the velocity in the depth direction. It is shown that the collision force reduces the dust kinetic energy which has no fluctuation even in a strong external magnetic field. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Plasma sheath Dust particles Collision force Magnetic field
1. Introduction The problem of plasma flowing into a wall has previously been studied [1]. Investigation into the characteristics of the space charged region in front of the wall is one of the oldest problems in plasma physics. This region which shields the bulk plasma from the wall is named the ‘‘sheath’’. Studying the sheath region has continued to remain of the current interest because of its practical importance in plasma dynamics [2–6]. Recently, the study of the plasma sheath in the presence of dust has become an important research area due to its common observance in laboratory and space plasmas [7–11]. Compared with other charged grains, the dust has considerable charge and mass. The typical dust particles have wmicron sized, w1000e negative charges and w1000 kg/m3 mass density. Moreover, as these characteristics are dynamic variables, the behavior of the dust particles modifies the plasma dynamics and their effects may have properties different from those caused by negative ions. During the past decades, several studies have been developed to investigate the structure of dust plasma sheath [12–15]. They include for example, the fluid model [14,16], isolated particle model [17], electrostatic probe model [18] and a new one called the dust acoustic wave model [19]. These studies have considered the effect of parameters which modify the dust plasma characteristics, such
* Corresponding author. E-mail address: [email protected] (S.F. Masoudi). 0042-207X/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2009.02.003
as the action of electrostatic, gravitational force and the Lorentz force. Two important factors which affect the structure of the dust plasma sheath are the collision force and external magnetic field. Only a few of the studies have considered the external magnetic field [13–15,20,21]. It has been shown that the structure of the dust plasma sheath without magnetic field is considerably different from the dust plasma sheath in an external magnetic field. In the present study, we simultaneously consider the effects of the electrostatic, gravitational, external magnetic field and the collision force on dust characteristics. Based on some earlier studies [14,15,22,23], the fluid model is used for a dust plasma sheath which has one-dimensional coordinate space and three-dimensional velocity space. Solving the basic equations of ion and dust movement, we investigate the effect of the magnitude of the collision force on some characteristics of dust grains like gyration movement, velocity, kinetic energy and the density of dust. We assume that the collisions between ions (which are treated as a cold fluid) and neutrals can be neglected. The sheath is generally investigated in two limiting cases; collisionless (li > >le) and collisional (li <
1032
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
2. Basic equations of the model In our model, we have considered a dust plasma sheath consisting of isothermal electrons, fluid ions and cold charged dust grains. The x-direction is the depth from the sheath edge to the wall. The constant magnetic field is embedded in the x–z plane and makes an angel q with the x-direction. The geometry of the model is illustrated in Fig. 1. Before considering the basic equation of the model, we introduce some parameters as follows. The normalized forms of these parameters are used in numerical simulation. The variable physical parameters (which we assume change only along the depth direction):
f; the electrostatic potential which is zero at the edge of the sheath x ¼ 0 Ne; the electron density with ne0 value at sheath edge Ni (nd); the ion (dust) density with ni0 (nd0) value at sheath edge vi (vd); the ion (dust) velocity Constant parameters: qd; the charge of dust mi (md); the mass of ion (dust) Te; electron temperature cis ¼ (Te/mi)0.5; ion sound speed cds ¼ (Te/md)0.5; dust sound speed sd ¼ md/eB0; dust cyclotron period rd ¼ cdssd ¼ ðTe md =e2 B20 Þ0:5 ; dust grain gyro radius le ¼ (30Te/ne0e2)0.5; electron Debye length and the dimensionless quantities: Mi ¼ vix0/cis; ion Mach number Md ¼ vdx0/cds; dust Mach number g ¼ le/rd di ¼ ni0/ne0 Zd ¼ qd/e; dust grain charge number di ¼ ni0/ne0 dd ¼ nd0/ne0 h ¼ ef/Te z ¼ x/le ud ¼ vd/cds Ne ¼ ne/ne0 Ni ¼ ni/ni0 m ¼ glemd/Te
Fig. 2. The velocity of dust in depth direction under various a values (g ¼ 0.1 and q ¼ 50).
Using these parameters, the basic equations of the model can be expressed as follows:
1: The Boltzmann relation for isothermal electrons:
ne ¼ n0 expðef=Te Þ
(1)
2: Equation of continuity and momentum for ions:
Plasma
x=0 y⊗ x
z B
Sheath
θ
Wall Fig. 1. The geometry of the model.
Ni ¼
vix0 vix
mi v2ix v2ix0 ¼ 2hTe
(2)
(3)
In equations (2) and (3), we have taken into consideration the fact that the mean effect of the magnetization of ions and electrons on their movement is zero. This is due to the fact that the movement of dust particles is far slower than that of the ions and electrons, and so the ions and electrons are excessively magnetized.
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
1033
Fig. 4. The density of dust for collisionless (a ¼ 0) and collisional plasma sheath (a ¼ 5) (g ¼ 0.1, q ¼ 50 and Zd ¼ 1000).
udx
vudy ¼ gZd ðcos qudz sin qudx Þ audy vz
udz
vudz ¼ gZd cos qudy audz vz
(9)
(10)
where a is a dimensionless parameter which characterizes the magnitude of the collision force and can be expressed as a ¼ lessnn (where nn is the neutral gas density and ss is the collision crosssection measured at dust sound speed). The model is completed with the Poisson equation which relates the electrostatic potential to the density of dust, ions and electrons as follow:
V2 f ¼ ðqd nd þ eni ene Þ=30
Fig. 3. The velocity of dust in depth direction for collisionless (a ¼ 0) and collisional plasma sheath (a ¼ 5) (g ¼ 0.1 and q ¼ 50).
By using the dimensionless parameters, the Poisson equation can be expressed as follow;
d2 2
dz
Combining equations (2) and (3), we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ni ¼ Mi = Mi2 þ 2h
(4)
(11)
h ¼ Zd dd Nd þ di Ni Ne
(12)
Now we solve the basic equations of the model (equations (1),(4),(7)–(10),(12)) numerically to find the characteristics of dust in the plasma sheath in the presence of the external magnetic field and various magnitudes of collision force.
3: Equation of continuity and momentum for dust:
! V $ðnd ! v dÞ ¼ 0 ! ! ! md ð! v d $ V Þ! v d ¼ qd V f þ qd ! v d B 0 þ md g b x md n! v
3. Numerical calculation and discussion
(5) (6)
The last term of equation (6) corresponds to the collision force in which n is the dust collision frequency. In constant collisional mobility (where the cross-section for collision between dust and neutral has an inverse ratio to velocity) n is constant. Using the dimensionless parameters, equations (5) and (6) can be expressed as follows:
Nd udx ¼ Md
udx
vudx vh ¼ Zd þ gZd sin qudy þ m audx vz vz
(7)
(8)
We adopt the following parameters for numerical calculation; Mi ¼ 1.1, Md ¼ 5.0, di ¼ 1.01, dd ¼ 105 and the original electric field is allowed to take on a finite value vh/vzjz¼0 ¼ 0.01. These parameters are considered the same as used by Liu et al. [14] enable us to compare the results for the collisionless case with their calculations. However, Liu et al. have not defined the form of their dust mass. In our numerical calculation, we consider the dusts as spherical particles with uniform mass which have 4 mm radius and 2 g/cm3 density. Using the density and radius of dust and adopting Te ¼ 2 eV and ne0 ¼ 109 cm3, the value of dimensionless parameter corresponding to mass is m ¼ 5.46 103. In Fig. 2, we calculate the dust velocity in the depth direction versus the variation of a value for Zd ¼ 1000 and Zd ¼ 500. Fig. 3 shows the same calculation for two specific cases a ¼ 0 and a ¼ 10.
1034
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
Fig. 5. The relive difference of velocity between collisionless and collisional case.
Fig. 7. The relative difference of kinetic energy of dust for different directions and magnitudes of magnetic field.
As the figures show, in the collisionless case (a ¼ 0), the dust velocity fluctuates as has been reported in Ref. [14]. There are more fluctuations corresponds to bigger jZd j value. The bigger jZd j value becomes, the more fluctuations occur. As the fluctuations are due to the external magnetic field, the results show that the effect of magnetic field is greater for bigger jZd j value. The effect of collision force on the dust velocity can be neglected near the sheath edge. By increasing the distance from the sheath edge, the collision force decreases the dust velocity. The larger the distance from the sheath edge becomes, we observe lesser the dust velocity in depth direction. The same result can be found for vdy and vdz by numerical calculation. As a result of collision force, the extrema of dust velocity occur nearer to the sheath edge. The degree of shifting of extrema is in direct ratio to the distance from the sheath edge. This means that at a certain distance (for instance from sheath edge to z ¼ 20 in Fig. 3) there are more fluctuations in collision regime compared with collisionless regime. Because of the inverse relation between the
velocity in depth direction and the dust density (equation (6)), the results can be related to the density of dust (see Fig. 4). However, as the dust velocity decreases in the collisional plasma sheath model, the density of dust increases. In Fig. 5, the relative difference in velocity between a collisional and collisionless plasma sheath (Dvdx ¼ (vdx(a ¼ 0) vdx(a ¼ 5))/ vdx(a ¼ 0)) is shown for different g and q values. For a higher magnetic field, the relative difference has more fluctuations since the bigger magnetic field corresponds to more fluctuations of dust velocity [14]. The fluctuations of the relative difference are due to the fact that the collisions change the positions of the extrema. For example, in Fig. 3 (Zd ¼ 1000), between z ¼ 12 and 15, the velocity increases in collisionless case whereas the velocity decreases in the collisional case. So in the interval z ¼ 12–15, the relative difference increases. Fig. 5 shows that the shifts in the positions of the extrema of the velocity and density (and changes in their amplitudes) depend strongly on the magnitude of magnetic field in z-direction.
Fig. 6. The dust kinetic energy versus depth direction in different a values.
Fig. 8. The dust helical movement in collisional (solid line) and collisionless (circles) plasma sheath.
S.F. Masoudi et al. / Vacuum 83 (2009) 1031–1035
1035
In Fig. 8, the dust helical movement is investigated in a collision mode. The figure shows that the collision force decreases the dust gyro radius and makes more helical movements in specific distance from the sheath edge. The larger the distance from sheath edge becomes, the smaller dust gyro radius results. The electrostatic potential versus the depth direction is calculated numerically in Fig. 9. As the figure shows the collisions between dust and neutrals do not affect the electrostatic potential. This implies that the effect of collision on electron density can be neglected. 4. Conclusion The dust characteristics were investigated numerically in a magnetized plasma sheath. In this model, we consider the electrostatic, gravitational, external magnetic field and the different magnitudes of collision force simultaneously. The numerical results reveal that the collision force decreases the dust velocity, changes the positions of the extrema of the velocity in depth direction and the density of dust. The results show that the effect of collisions increases when the magnitude of magnetic field in depth direction is decreased. Fig. 9. The electrostatic potential in depth direction under different magnitudes of magnetic field.
In Fig. 6, the dust kinetic energy per mass ðK ¼ v2d =2Þ is calculated numerically as a function of depth for various magnitudes of collision force. As the figure shows, there are no fluctuations in kinetic energy. We consider g ¼ 0.5 which corresponds to a larger magnetic field. In this case, the fluctuations of dust velocity in the depth direction between z ¼ 0 and 20 are approximately 4 times more than that of the case in which the quantity of g is 0.1. However, the kinetic energy (or the magnitude of the velocity; v2dx þ v2dy þ v2dz ) does not fluctuate. This result is reasonable since the forces arising from the magnetic field do not contribute to any energy change. The relative reduction of dust kinetic energy per mass (between the collisional and collisionless case; (DK ¼ (K(a ¼ 0) K(a ¼ 10))/ K(a ¼ 0))) is plotted in Fig. 7. This figure shows that the effect of collision on the kinetic energy, for a specific q value, is approximately the same for different magnitudes of magnetic field. However, the effect of collisions increases when the magnitude of the magnetic field in the depth direction decreases.
References [1] Chen FF. Introduction to plasma physics and controlled fusion. New York: Plenum; 1984. [2] Riemann KU. Phys Fluids 1981;24:2163. [3] Masoudi SF. J Phys D Appl Phys 2007;40:6641. [4] Franklin RN. J Phys D Appl Phys 2003;36:309. [5] Tskhakaya DD, Shukla PK, Eliasson B, Kuhn S. Phys Plasmas 2005;12:103503. [6] Thornhill LD, Desai PV. J Plasma Phys 1998;59:505. [7] Selwyn GS, Heidenrich JE, Haller KL. Appl Phys Lett 1990;57:1876. [8] Liu JY, Wang DZ, Ma TC. Vacuum 2000;59:126. [9] Mendis DA, Rosenberg M. Ann Rev Astron Astrophys 1993;32:419. [10] Nitter T. Plasma Source Sci Technol 1996;5:93. [11] Wang ZX, Liu JY, Liu Y, Wang X. J Plasma Phys 2004;70:577. [12] Duan P, Liu J, Gong Y, Liu Y, Wang X. Plasma Sci Technol 2007;9:394. [13] Kalita P, Das GC. J Plasma Phys 2003;69:551. [14] Liu JY, Zhang Q, Zou X, Wang ZX, Liu Y, Wang XG, Gong Y. Vaccum 2004;73:687. [15] Das GC, Kalita P. J Phys D Appl Phys 2004;37:702. [16] Liu TY, Ma JX. Phys. Plasmas 1997;4:2798. [17] Wang ZX, Wang W, Liu Y, Liu J, Wang X. Chin Phys Lett 2004;21:697. [18] Ma JX, Yu MY. Phys Plasmas 1995;2:1343. [19] Barkan A, Merlino RL, Angelo D. Phys Plasmas 1995;2:3563. [20] Baishya SK. Phys Plasmas 1999;6:3678. [21] Salimullah M, Nitta H, Islam MK, Shukla PK. J Plasma Phys 2005;71:157. [22] Kang SW, Min KW, Lee Es, Seon. J Surf Coat Technol 2003;171. [23] Riemann KU. Phys Plasmas 1994;1:552. [24] Zou Xiu. Chin Phys Lett 2006;23:396.