http://dx.doi.org/10.12988/ams.2015.411970
The Impact of the Dust Characteristic on the Plasma Sheath
S. Lahouaichri, A. Dezairi, J. Louafi, D. Saifaoui, R. Moultif and S. Mizani
Laboratory of condensed matter, Faculty of sciences ben M`sick (URAC.10) University Hassan II- Mohammedia Casablanca, Morocco
Copyright © 2014 S. Lahouaichri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The study of the plasma sheath in the presence of dust has continued to remain of the current interest because of its practical importance in plasma dynamics. The aim of this paper is to study the behaviour of the sheath structure in a plasma with collisions, to determine analytically the expression of the sheath thickness and the dust impact energy in the three directions (radial, poloidal and toroidal).
Keywords: Dust, Sheath thickness, Impact energy
1. Introduction
During the past decades, several studies have been developed to investigate the structure of dust plasma sheath [1-6].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 physics [7–10].
Recently, the study of the plasma sheath in the presence of the dust has become an important research area due to its common observance in laboratory and space plasmas [11–16].
The presence of dust near the wall leads to formation of a dissipative
structure confining dust and plasma particles and creating the specific dust-
plasma layer. The presence of dust particles in the sheath changes the plasma
parameters and affects the collective processes in such plasma systems. In
particular, the charged dust grains can effectively collect electrons and ions
324 S. Lahouaichri et al.
from the background plasma. Thus in the state of equilibrium, the electron and ion densities are determined by the neutrality condition which is given by
−en e + eZ i n i − e ∑ Z d n d = 0 (1) Where n e , n i and n d are respectively the density of the plasma electrons (with the charge –e), ions (for simplicity, we consider singly charged ions), and dust particles. The charge of the dust particle −eZ d can vary significantly depending on plasma parameters. For many dusty plasmas, this charge is negative (i.e.,Z d > 0) and large (Z d ~10 2 − 10 3 ), thus because of the neutrality condition (1) it is possible to have in such plasmas n e ≪ n i [17].
Several authors have recently considered the effects of collisionality in the dusty plasma sheath [18-20]. Vladimirov and Cramer studied the dynamics of motion and the charging of a macroscopic dust particle in plasma with an ion flow [21]. M. Samir, M. Eddahby and A. Dezairi developed the model for the ion-neutral and dust-neutral collisions in plasma sheath on only the radial coordinate [22-24]. However, we studied the particles collisionalty on the sheath in the three directions (radial, poloidal and toroidal).
In this paper, we consider magnetized dusty plasma, which is in contact with physical boundary like a wall. It is well known that a sheath is created to separate the plasma from the surface. The second section provides the mathematical formulation, the basic equations that are used to describe the sheath in dusty plasma. The third section provides the dust characteristics, which were investigated analytically in a magnetized plasma sheath.
Graphical representations are presented in the fourth section but the Numerical results of simulations are presented in the fifty section. In the last section we comment the result obtained.
2. Mathematical formulation
The specific feature of the plasma sheath, which is formed near the conducting wall, is the existence of flows of plasma particles towards the wall, the latter being negatively charged because of the difference of electron and ion masses and temperatures. In the presence of a relatively high energy flux, the impurity contaminants, consisting of macro-scaled grains can be ejected from the wall. The dust grain has considerable charges and mass, comparing with other charged grains.
In our survey, the standard models of magnetic field in tokamak take the following form:
𝑩 = 𝐵 𝜃 𝒆 𝜽 + 𝐵 𝜑 𝒆 𝝋 (2)
𝑩 = 𝐵 0 ( 𝑞(𝑟)𝑅 𝑟 𝒆 𝜽 − (1 − 𝑅 𝑟 𝑐𝑜𝑠𝜃) 𝒆 𝝋 ) (3)
Consists of the ploidal magnetic field component and the toroidal magnetic field component of which r, R are the minor and major radius of plasma, θ and φ are respectively the ploidal and toroidal angle, B 0 is the magnetic field on the principle toroidal axe, and q(r) is the safety factor.
The sheath is consisting of isothermal electrons and fluid ions, in the presence of a number of cold charged dust grains. The electrons are assumed to be in thermal equilibrium with the density given by
n e = n 0 exp ( K eϕ
B T e ) (4) Here, e is the elementary charge, K B the Boltzmann constant and T e the electron temperature.
The ions fluid equations are
∇. (n i 𝐯 𝐢 ) = 0 (5)
m i (𝐯 𝐢 . ∇)𝐯 𝐢 = −e∇𝛟 -𝐅 𝐜𝐢 + e𝐯 𝐢 ∧ 𝐁 (6) The cold dust fluid obeys the source free, steady state continuity equation:
∇. (n d 𝐯 𝐝 ) = 0 (7) Dust particles are described by the standard hydrodynamic equations of momentum, which in a purely electrostatic and magnetic case take the form:
m d (𝐯 𝐝 . ∇)𝐯 𝐝 = Z d e𝛁ϕ-𝐅 𝐜𝐝 − Z d e𝐯 𝐝 ∧ 𝐁 (8) Where m d is the mass of dust particle.
The collisional effects between the particles and the neutrals are introduced.
We use the collisional force term F cd which is given by the equation [24]:
𝐅 𝐜𝐝 = m d n n v d 2 σ s (𝐯 𝐝 ⁄ ) C d γ (9) Here, n n is the neutral gas density and σ is the momentum transfer cross
section for the collisions between the charged grain and neutrals. C d =
√k B T e ⁄ mv d is the dust acoustic speed and γ is a dimensionless parameter ranging from 0 to -1.
The system is completed with the Poisson equation:
∇ 2 ϕ = −4πe(n i − n e −Z d n d ) (10) Combining eqs (2) to (10), we find four coupled differentials equations describing the sheath structure as:
v dr ∂v ∂r dr = −Z d m e
d (1 − B B 0 (1 − R r cosθ)) ∂ϕ ∂r − n n σ s v dr 2+γ C
s γ (11) v dθ ∂v ∂θ dθ = Z d m e
d (1 + B B 0 (1 − R r cosθ)) ∂ϕ ∂θ − rn n σ s v dr 2+γ C
s γ (12)
326 S. Lahouaichri et al.
v dφ ∂v ∂φ dφ = Z d m e
i
∂ϕ
∂φ − Rn n σ s v C dr 2+γ
s
γ (13) ∂
2 ϕ
∂r 2 = −4πe(n i − n e − Z d n d ) (14)
The governing equations can be transformed to the dimensionless form by an appropriate choice of variables. The electric potential ϕ is scaled by the electron temperature, η = − k eϕ
B T e and the dust velocity v d is transformed to dimensionless parameter by the dust acoustic speed, u d = C v d
d .also the parameters Z d n n d0
e0 = δ − 1, δ = n n i
e0 and θ = T T e
i . The degree of collisionality in the sheath is parameterized by α = λ λ D
mfp = λ D n n σ s , where λ mfp = n 1
n σ s is the mean free path of dust and α is proportional to the neutral gas density n n .
Based on these non-dimensional parameters, the basic equations reduce to:
u dr u dr ′ = −Z d (1 − B B 0 (1 − R r cosθ)) η ′ − αu dr 2+γ (15)
u dθ u dθ ′ = Z d (1 + B B 0 (1 − R r cos θ)) η ′ − rαu dθ 2+γ (16) u dφ u dφ ′ = Z d η ′ − Rαu dφ 2+γ (17) η" = δexp(θη) − exp(−η) + (1 − δ) u u d0
d (18)
3. Analytical resolution
In the limit of strong collisions, the collision parameter is large; the equations of motion are simplified by neglecting convective term on the left hand side. The equation (15) becomes:
Z d K η ′ = αu dr 2+γ (19) Where K = (1 − B B 0 (1 − R r cosθ))
We obtained that
η r ′ = − ( Z α
d K ) u dr 2+γ (20) Similarly to equations (16), (17) we find:
η θ ′ = ( Z rα
d K′ ) u dθ 2+γ (21)
η φ ′ = − ( Rα Z
d ) u dφ 2+γ (22) Where 𝐾 ′ = (1 + 𝐵 𝐵 0 (1 − 𝑅 𝑟 𝑐𝑜𝑠𝜃))
By neglecting the electron term exp (−η) into the Poisson’s equation (18) we find:
𝜂" = (1 − 𝛿) 𝑢 𝑑0 𝑢 𝑑 𝜂" = (− ( Z α
d K ) u dr 2+γ )′
− ( Z α
d K ) (2 + γ)u dr ′ u dr 2+γ = (1 − δ)𝑢 𝑑0
𝑢 𝑑𝑟 2+𝛾 = [(𝛿 − 1)𝑍 𝑑 𝐾 𝑢 𝛼 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (23) Similarly, for the poloidal and toroidal directions we obtained that:
𝑢 𝑑𝜃 2+𝛾 = [(1 − 𝛿)𝑍 𝑑 𝐾′ 𝑢 𝑟𝛼 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (24)
𝑢 𝑑𝜑 2+𝛾 = [(1 − 𝛿)𝑍 𝑑 𝑢 𝑅𝛼 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (25) By combining equations (20)-(25):
𝜂 𝑟 ′ = ( 𝑍 𝛼
𝑑 𝐾 )
1
3+𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (26)
𝜂 𝜃 ′ = ( 𝑍 𝛼𝑟
𝑑 𝐾′ )
1
3+𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (27)
𝜂 𝜑 ′ = ( 𝑍 𝛼𝑅
𝑑 𝐾 )
1
3+𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ) 𝜉]
2+𝛾
3+𝛾 (28) Equations (26)-(28) leads to
𝜂 𝑟 = 5+2𝛾 3+𝛾 ( 𝑍 𝛼
𝑑 𝐾 )
1
3+𝛾 [(𝛿 − 1)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )]
2+𝛾
3+𝛾 𝜉 5+2𝛾 3+𝛾 (29)
𝜂 𝜃 = 5+2𝛾 3+𝛾 ( 𝑍 𝛼𝑟
𝑑 𝐾 ′ )
1
3+𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )]
2+𝛾
3+𝛾 𝜉 5+2𝛾 3+𝛾 (30)
𝜂 𝜑 = 5+2𝛾 3+𝛾 ( 𝑍 𝛼𝑅
𝑑 𝐾 ′ )
1
3+𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )]
2+𝛾
3+𝛾 𝜉 5+2𝛾 3+𝛾 (31)
The sheath thickness, founding by invoking the boundary condition 𝜂 = 𝜂 𝜔
, 𝜉 = 𝑑 , is
328 S. Lahouaichri et al.
𝑑 𝑟 = ( 5+2𝛾 3+𝛾 )
3+𝛾
5+2𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )] −
2+𝛾 5+2𝛾 ( 𝑍 𝛼
𝑑 𝐾 ) −
1
5+2𝛾 𝜂 𝑤 5+2𝛾 3+𝛾 (32)
𝑑 𝜃 = ( 5+2𝛾 3+𝛾 )
3+𝛾
5+2𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )]
2+𝛾 5+2𝛾 ( 𝑍 𝛼𝑟
𝑑 𝐾 ′ ) −
1
5+2𝛾 𝜂 𝑤 5+2𝛾 3+𝛾 (33)
𝑑 𝜑 = ( 5+2𝛾 3+𝛾 )
3+𝛾
5+2𝛾 [(1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 )] −
2+𝛾 5+2𝛾 ( 𝑍 𝛼𝑅
𝑑 𝐾 ′ ) −
1
5+2𝛾 𝜂 𝑤 5+2𝛾 3+𝛾 (34) We next wish to find the dust impact energy which can be written using eq. (28) as
𝜀 𝑤 = 1 2 𝑢 𝑑𝑟 2 = 1 2 ( −𝑍 𝛼 𝑑 𝐾 𝜂 𝑤 ′ )
2
2+𝛾 (35)
Evaluating η ω ′ using eq. (30) we find:
𝜀 𝑤 𝑟 = 1 2 [ 5+2𝛾 3+𝛾 ( 𝑍 𝑑 𝛼 𝐾 ) 2 ((1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ))
3+𝛾 2 𝜂 𝑤 ]
2 5+2𝛾
(36) Similarly to poloidal and toroidal directions we obtained that:
𝜀 𝑤 𝜃 = 1 2 [ 5+2𝛾 3+𝛾 ( 𝑍 𝛼𝑟 𝑑 𝐾 ′ ) 2 ((1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ))
3+𝛾 2 𝜂 𝑤 ]
2 5+2𝛾
(37)
𝜀 𝑤 𝜑 = 1 2 [ 5+2𝛾 3+𝛾 ( 𝑍 𝛼𝑅 𝑑 𝐾 ′ ) 2 ((1 − 𝛿)𝑢 𝑑0 ( 3+𝛾 2+𝛾 ))
3+𝛾 2 𝜂 𝑤 ]
2 5+2𝛾
(38)
4. Graphical representation and Discussion
In order to study the effect of collisionality in dusty plasma, we have set the numerical values which coincided with ITER’s characteristics [24].
Fig.1. Graphical representation of the sheath thickness width collisional parameter α for the three direction for plasma in the presence of dust.
0 10 20 30 40 50 60 70 80 90 100
2 3 4 5 6 7 8
radial
poloidal
toroidal
alpha
d
Fig.2. Graphical representation of the sheath thickness width wall potential for the three direction for plasma in the presence of dust.
Fig.3 . Graphical representation of the sheath thickness with number of charges 𝑧 𝑑 for the three direction for plasma in the presence of dust.
In Figs. 1, 2, 3 we plot the sheath thickness 𝑑 as functions of the collision parameter α, the wall potential 𝜂 and the number of charges 𝑧 𝑑 for the three direction. We show that by increasing the collisions the sheath thickness decreases, we note also that the sheath width increases when the wall potential increases.
The increase of the number of charges 𝑧 𝑑 brings the increase of the sheath thickness (see Fig.3).
Fig.4. Graphical representation of the impact energy width collisional parameter α for the three direction for plasma in the presence of dust.
0 10 20 30 40 50 60 70 80 90 100
0 1 2 3 4 5 6 7 8 9 10
radial
poloidal toroidal
potential
d
0 10 20 30 40 50 60 70 80 90 100
0 1 2 3 4 5 6 7 8
radial
poloidal toroidal
Zd
d
0 10 20 30 40 50 60 70 80 90 100
0 100 200 300 400 500 600 700
radial
poloidal toroidal
w330 S. Lahouaichri et al.
In Figs. 4, 5 we plot the dust velocity 𝑢 𝑑 and the wall potential 𝜂, as functions of the collision parameter α for the three direction.
We show that by increasing the collisions the dust velocity decreases, although 𝜂 decreases and approach a limiting asymptote.
Fig.5. Graphical representation of the dust velocity 𝑢 𝑑 as function of width collisional parameter α for the three direction for plasma in the presence of dust.
Fig.6. The approximate solution of the of the average dust impact energy at the wall as function of the collisional parameter and the sheath width.
Fig.7. The approximate solution of the wall potential as function of the collisional parameter and the sheath width.
0 10 20 30 40 50 60 70 80 90 100
1 2 3 4 5 6 7 8 9 10
radial poloidal
toroidal
Ud0 20
40 60 80 100
0 5
10 0 10 20 30 40
d
Ud0 2 4 6 8 10
0 50
100 0 100 200 300
d
dIn Figs. 6, 7 we plot the dust velocity 𝑢 𝑑 and the wall potential 𝜂, as functions of the collision parameter α and the sheath thickness d.
We show that by increasing the collisions the dust velocity decreases, and then this would bring increase of the sheath width.
We show also that the collisions between dust and neutrals do not affect the electrostatic potential, although 𝜂 decreases near the sheath.
Fig.8. Graphical representation of the velocity of dust with number of charges 𝑧 𝑑 and sheath width.
Fig.9 . Graphical representation of the density of dust with number of charges 𝑧 𝑑 and sheath width.
In Figs. 8, 9 we plot dust velocity 𝑢 𝑑 and the density of impurities 𝑁 𝑑 as functions of the number of charges 𝑍 𝑑 and the sheath thickness d.
We show that the charge number of the dust particles and the sheath width increases with increasing of the velocity of dust.
We notice that for large value of number of charges 𝑧 𝑑 the density of dust 𝑁 𝑑 is unaffected, but for the small values of 𝑧 𝑑 the density of dust 𝑁 𝑑 increases. Also, we see that the density of dust 𝑁 𝑑 increases when the sheath width increases (see Fig.9).
0 2 4 6 8 10
0 50
100 0 5 10 15
Z d d U
d0 5
10
20 0 60 40
100 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
d Zd
Nd
332 S. Lahouaichri et al.
Fig.10. the approximate solution of the dimensionless energy of dust as function of the number of charges 𝑧 𝑑 and the wall potential 𝜂 𝑑 .
In Fig. 10 we plot the energy of dust and as function of the number of charges Z d and the the wall potential η d .
We show that the energy of dust increases with increasing of the potential, we note also that the number of charge increases when the energy of dust increases.
5. Numerical Simulations and Discussion
In order to study the collision’s effect on the sheath’s thickness and impact energy and the effect of potential, number of charges and density of impurities on the sheath thickness, we solved the governing equations with impurities [Eqs.(15-18)] by using the numerical Runge-Kutta routine.
We used and simulated the ITER tokamak parameters
0,01 0,1 1 10
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
1,6
radial
poloidal toroidal
she ath w ithd
collisional parameter
Fig.11. The sheath thickness width collisional parameter α for the three direction for plasma in the presence of dust.
In Fig. 11 we plot the sheath width and as function of the collisional parameter for the three direction, this plot shows that sheath width decrease with increasing collisionally.
0 20 40 60 80 100
0 5
10 0 5 10 15 20
Z
dd
wThree regimes are evident for the radial and poloidal directions, but in the toroidal direction the sheath width decreases quickly for the collisional parameter.
0,01 0,1 1 10
0,0 0,2 0,4 0,6 0,8 1,0
radial poloidal toroidal
im pa ct e ne rg y
collisional parameter
Fig.12. The impact energy with colisional parameter for the three directions, for plasma in the presence of dust.
This plot shows that dust impact energy decreases when the collision parameter increases, for the three direction.
We notice that the dust impact energy varies in the same way for radial and poloidal directions, but in the toroidal direction, the impact energy decreases quickly for the collisional parameter.
Fig.13.The sheath width with wall potential for dusty plasma.
In Fig. 13 that the sheath width for the three direction radial, poloidal and
toroidal with a wall potential are shown. From the figure we can see that the
sheath width increases when the intensity of magnetic field. We notice that
the dimensionless sheath width increases quickly for the potential 𝜂 = 1 and
takes an approximately constant value.
334 S. Lahouaichri et al.
1E-4 1E-3 0,01
0 20 40 60 80 100 120 140
radial poloidale toroidale
dens ity of char ges
sheath width
Fig.14.The sheath width with density of charges.
In Fig.14 we note that the sheath width increases when the density of charge the dust particles increases for the three direction, also we note that the sheath width increases when the charge number of the dust particles increases (see Fig.15).
10 100
0,000 0,002 0,004 0,006 0,008 0,010
