A sheath model of a collisional dusty plasma

A Sheath Model of a Collisional Dusty Plasma

M. Samir, ¹ M. Eddahby, ¹ A. Dezairi, ¹ D. Saifaoui, ² O. El Haitami, ¹ and M. El Mouden ²

1

Laboratoire de physique de la mati` ere condens´ ee, Facult´ e des sciences Ben M’sik, B.P. 7955, Casablanca, Morocco

2

Laboratoire de physique th´ eorique, groupe de physique des plasmas, Facult´ e des sciences Ain chock, Casablanca, Morocco

(Received June 27, 2006)

The aim of this paper is to study the behaviour of the sheath structure in a plasma with collisions and to simulate the effects of collisionality on the plasma sheath using the Runge- Kutta routine. We have considered here the near wall region of an unmagnetized dusty plasma which consists of electrons, ions, micron size dust particles, and neutral particles.

Since the dust particles are much heavier than both the electrons and ions, the latter can be assumed to be in non-thermal equilibrium with the dust as a cold fluid. The neutrals are taken as immobile. Exact numerical solutions of the model are used to determine the collisional dependence of the sheath width and the impact energy at the wall.

PACS numbers: 52.40.Kh, 52.27.Lw, 52.27.Cm

I. INTRODUCTION

The behavior of a plasma in contact with a negatively biased surface has been quali- tatively studied for many decades. It is known that a strong localized electric field appears between the plasma and the surface. This ion rich boundary layer, called the sheath, confines electrons within and expels ions from the plasma [1].

The sheath is composed of ions, atoms, electrons, and dust particles. Sheath forma- tion at the plasma-boundary interface separating the quasi-neutral plasma is ubiquitous in bounded plasmas. Accurate sheath modeling is of considerable interest to the effective design of ionized flow in wide ranging applications in plasma processing: in ion cyclotron heating, in electric propulsion devices, and in controlled thermonuclear fusion. Recently, the study of dusty plasmas has represented one of the most rapidly growing branches of plasma physics research. The relevance of dusty plasma to both the space plasma and industrial plasma communities has led to the rapid growth of a variety of laboratory experiments.

Dust grains in a plasma are not neutral, but are charged due to their interaction with electrons, ions, and background radiation [2]. The mass of dust grains can have a very high value, up to 10 ⁶ – 10 ⁸ times the proton mass. The presence of the charged dust grains alters many of the physical properties of the plasma, including its charge distribution and potential. In addition, dusty plasmas facilitate the investigation of many unique phenomena including the formation of dust crystals and new collective modes.

The basic mechanism by which a dust grain interacts with a plasma is electrical charge accumulation on the surface of an insulating particle. In the space environment, this

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charge accumulates via photo ionization, secondary electron emission due to impacts with energetic particles, and collisions with the background thermal plasma. For the laboratory experiments on dusty plasmas, the principle charging mechanism is the flux of charged particles (electrons, ions) from the plasma to dust particles residing on a plasma exposed surface. Due to the mobility of the electrons, the grain becomes negatively charged.

Once the surface charge of the dust grain is large enough, the electric force, due to the sheath electric field, can exceed the combination of the gravitational and the adhesive forces that bind the grain to the surface. Under these conditions, the dust particle is accelerated through the sheath and passes into the main plasma.

The presence of a dust grain in a plasma reactor, under a radio-frequency for engraving in micro-electronics, or for the deposit of a thin layer is a critical phenomenon. It is thus of practical interest to investigate the interaction between a dusty plasma and a solid boundary.

Several authors have recently considered the effects of collisionality on the sheath.

Sheridan and Goree derived a simple criteria for the amount of ion collisionality. Vladimirov and Cramer studied the dynamics of motion and the charging of a macroscopic dust particle in a plasma with an ion flow [3]. Tsytovich, Vladimirov, Morfil, and Goree developed the theory of dust voids in a collisional dominated plasma for the one-dimensional case [4].

However, we develop the model for the ion-neutral and dust-neutral collisions in order to determine the sheath thickness and the grain impact energy at the wall

In this paper, we consider an unmagnetized dusty plasma, which is in contact with a physical boundary like a wall, probe, or substrate. 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 a dusty plasma when both ions and dust grains are out of thermal equilibrium. The exact numerical solutions for the sheath thickness, and the average dust and ion energies at wall impact as a function of the collisional parameter are presented.

In the last section we comment on the different results obtained from the coupling equations.

II. SHEATH MODEL OF A COLLISIONAL DUSTY PLASMA

II-1. Governing equations

The goal of this section is to study the effect of collisionality in a dusty plasma.

We consider an unmagnetized, charge-neutral plasma in contact with a planar wall. The

potential in the sheath is φ, and the wall is held at a negative potential φ ω . Consequently,

a sheath forms to separate the plasma from the wall. Ions enter the sheath as a cold beam

with a velocity v 0 and strike the wall with a velocity v _ω . In order to estimate the sheath

characteristics, we adopt a one-dimensional model of the sheath. The sheath surface is

represented by the oyz plane, and the ox axis is perpendicular to the surface. We assume

that all the physical variables (density, velocity, potential, etc.) depend only on the x

coordinate, and we suppose a steady process (i.e., the variables do not depend on time).

The boundary between the plasma and the sheath is at x = 0, and the sheath thickness is D, that is, the wall is at x = D. The sheath is assumed to be source free.

It is well known that dust grains are common in both space and laboratory plasmas [5–

7]. Dusty plasma consists of neutral gas, ions, electrons, and micron size particles that have a negative charge [8]. Since the dust particles are much heavier than both the electrons and ions, the dust acts as a cold fluid. The neutrals are taken as immobile. Here, since we have used the fluid theory for the dust, we can ignore the variations in shape, size, and charge separations among the individual dust particles [9]. This is so because these variations are sufficiently small that they can not be distinguished on the fluid element.

The collisions of charged particles in a plasma are of two types: collisions with other charged particles and collisions with neutral atoms and molecules. To plasma physicists the collisions with neutral particles are of practical interest, because they are dominant in low-temperature plasmas, where the degree of ionization is only a few per cent [10]. The opposite case—where the degree of ionization is high—is called a fully ionized plasma. In that case the collisions with other charged particles tend to dominate over collisions with neutral particles. We shall assume a partially ionized gas. The plasma-neutral collisions usually determines the kinetics of the motion [11].

Several forces act on the dust particles. In our paper, the only significance forces are neutral gas drag and the electric force F _el . All other forces are assumed to be negligible.

The electric and ion drag forces are driven by the electric field, but in opposite directions. When the particles are negatively charged, as in most gas discharges, these two forces will point in opposite directions. Due to the negative charge, F _el will point opposite to the electric field, whereas the ion flow direction, and therefore F c , will be in the same direction as the electric field.

The forces scale differently with particle radius r, F _el ∝ r [12] and F _c ∝ r ² [13].

The electric force will thus dominate for small particles, while ion drag will dominate for larger particles. Indeed, the resulting electric field will impose two forces on the negatively charged dust particles: an inward electrostatic force and an outward ion drag force. For a small particle size, the inward force will dominate [14]. Samsonov and Goree found that the curves for F _el and F c crossed at a critical diameter of 160 nm [14]. In this work, we shall assume that the dust diameter is less than 120 nm, so the ion drag force is assumed to be negligible.

The dynamics of the electrons are treated as a neutral background in the plasma and are assumed to obey the following Boltzmann relation [15, 16]:

n _e = n _o exp eφ

k _B T _E

, (1)

where e is the elementary charge, k _B is Boltzmann’s constant, and T _e is the electron tem- perature.

The cold dust fluid obeys the source free, steady state continuity equation:

∇ · (n d v d ) = 0 , (2)

and the momentum transfer equation:

m d (v d ∇) v d = Z d e∇φ − F cd , (3)

where m d , n d , v d , and Z d are respectively the dust particle mass, density, velocity, and charge number and n _d 0 , v _d 0 are respectively the dust density and velocity at the sheath edge.

The collisional effects between the dust and the neutrals are introduced. We use the collisional force term F _cd , which is given by [17]

F cd = m d n n v ² _d σ = m d n n v ² _d σ s

v d

C _d γ

. (4)

Here n _n is the neutral gas density, and σ is the momentum transfer cross section for the collisions between the charged dust grains and neutrals. C _d = q

k

T

m

is the dust acoustic speed. γ is a dimensionless parameter ranging from 0 to −1.

The rate of ion production in a dusty plasma is determined by the ionization fre- quency. The rate of ionization is given as S i = k i n n n e . Here k i is the ionization coefficient.

The probability of recombination is S r = k r n i n e , where k r is the recombination coefficient.

The ions can be described by a cold fluid responding to the plasma potential according to the continuity equation

∇ (n i v i ) = S i − S r (5)

and the momentum transfer equation

m _i (v _i ∇) v _i = −e∇φ − F _ci . (6)

Here m i , n i , and v i are respectively the ion particle mass, density, and velocity and n i0 , v i0

are the ion density and velocity at the sheath edge. As the ion fluid travels through the sheath it experiences a drag force F _ci , which is given by

F _ci = m _i n _n v _i ² σ = m _i n _n v _i ² σ _s v _i

C s

γ

. (7)

The electron and ion densities are then included in the Poisson equation [18]:

∇ ² φ = −4πe (n i − n e − Z _d n _d ) . (8) For strong dust-neutral and ion-neutral collisions the movement of dust is mobility limited.

Therefore, we are here interested only in the constant dust mobility and ion mobility case (i.e., γ = −1), hence we do not consider the case of constant dust mean free path (i.e., γ = 0).

Combining Eqs. (1) to (8) we find four coupled differential equations describing the sheath structure:

v _i d

dx v _i = − e m i

d

dx φ − n _n σ _s v ²⁺ _i ^γ C s ^γ

, (9)

∇ (n _i v _i ) = k _i n _n n _e − k _r n _i n _e , (10) v d

d

dx v d = Z d e m d

d

dx φ − n n σ s

v _d ^{2+ γ}

C _d ^γ , (11)

∇ ² φ = −4πe (n i − n e − Z d n d ) . (12) The governing equations can be made dimensionless by an appropriate choice of vari- ables [19]. The electric potential φ is scaled by the electron temperature,

η = − eφ k B T e

.

The distance x is scaled by the Debye length, ξ ≡ x

λ D

, where λ _D =

q ε

k

T

n

e

. The ion velocity v _i is scaled by the ion acoustic speed, u _i ≡ v i

C _s .

In addition, the dust velocity v d is made dimensionless by the dust acoustic speed u d ≡ v d

C _d .

Along with the parameters Z _d ⁿ _n

eo

= δ − 1, and d = _λ ^D

D

.

The degree of collisionality in the sheath is parameterized by α = _λ ^λ

mf p

= λ D n n σ s , where λ mf p = _n ¹

n

σ

is the mean free path of dust, and α is proportional to the neutral gas density n n . The collisionless case (α = 0) is the limit of zero gas density. If the gas density is high enough, or the Debye length short enough so that the ion mean free path is one Debye length, then α = 1. The average number of collisions in the sheath, which will prove to be a useful quantity, is given by _λ ^D

mf p

= αd.

Based on these non-dimensional parameters, the basic equations reduce to

u i u ⁰ _i = η ⁰ − αu ²⁺ _i ^γ , (13)

dn i

dε = − η ⁰ n i

u ² _i + αn i u ^γ _i + λ D

c S u i

k i n n n 0 exp (−η) − λ D

u i c S

k r n i n ² 0 exp (−2η) , (14)

u d u ⁰ _d = −Z d η ⁰ − αu ²⁺ _d ^γ , (15)

η ⁰⁰ = n i

n _e 0

− exp (−η) + (1 − δ) u do

u _d . (16)

The prime represents the derivative with respect to the spatial coordinate ξ.

II-2. Numerical solutions

The governing equations were solved exactly for the electric potential η ω , the ion velocity u i (ξ), and the dust velocity u _d (ξ), by integrating them numerically with a Runge- Kutta routine [20].

In Figures 1, 2, and 3 we plot the sheath thickness d, the ion impact energy, and the dust impact energy as functions of the collision parameter α and the wall potential η ω ; these plots show three regimes of sheath collisionality. For small α, collisions are negligible, and d, ε _wd , and ε _wi are nearly independent of α. For large α the ion and the dust motion is collisionally dominated: d, ε ωd , and ε ωi decrease and approach a limiting asymptote.

Between the collisionless and collisional regimes there is a transition regime. Approximate analytic expressions for d and ε _ωi can be derived. The transition regime is much more difficult to treat analytically. Consequently, the numerical result in Figs. 1, 2, and 3 are most valuable for their accuracy in the transition regime.

FIG. 1: The exact numerical solutions of the governing equations for the dimensionless sheath thickness as a function of the collisional parameter α for various wall potentials. Three regimes are evident. We have assumed γ = −0.5, AKi = 1.5e − 15, and Akr = 1.e − 14.

In the special case of thermal equilibrium without both ionization and recombination, the ions are assumed to follow the Boltzmann relation:

n _i = n _i 0 exp (−eφ/k _B T _i ) . (17)

We find two coupled, differential equations describing the sheath structure:

v _d d

dx v _d = Z d e m _d

d

dx φ − n n σ s

v _d ^2+γ

C _d ^γ , (18)

and

∇ ² φ = −4πe (n i − n e − Z _d n _d ) . (19) Based on the non-dimensional parameters, the basic equations reduce to

u d u ⁰ _d = −Z d η ⁰ − αu ²⁺ _d ^γ , (20)

FIG. 2: The exact numerical solution for the average ion energy at wall impact as a function of the collisional parameter. We show the results for γ = −0.5, three regimes are evident: a collisionless regime where ε

is nearly independent of the collisional parameter, a collisionally dominated regime where ε

approaches a limiting asymptote, and a transition regime. We have assumed u

= 100, γ = −0.5, AKi = 1.5e − 15, and Akr = 1.e − 14.

FIG. 3: The exact numerical solution for the average dust energy at wall impact as a function of the collisional parameter for various wall potentials. In this figure, three regimes are evident:

a collisionless regime where ε

is nearly independent of the collisional parameter, a collisionally dominated regime where ε

approaches a limiting asymptote, and a transition regime. We show also that the dust impact energy increases when the electric potential increases.

η ⁰⁰ = δ exp (ηθ) − exp (−η) + (1 − δ) u do

u _d . (21)

The governing equations were solved exactly by integrating them numerically with a Runge- Kutta routine.

In Figure 4 and 5 we plot the sheath thickness d and the dust impact energy as

functions of the collision parameter α for various wall potentials η ω . These plots show

three regimes of sheath collisionality: collisionless, collisional, and a transition regime. We

show also that both d and ε ωd decrease and approach a limiting asymptote with increasing

collisionality.

FIG. 4: The exact numerical solutions of the governing equations for the dimensionless sheath thickness as a function of the collisional parameter α for various wall potentials. Three regimes are evident. We have assumed γ = −0.5.

FIG. 5: The exact numerical solution for the average dust energy at wall impact as a function of the collisional parameter for various wall potentials. In this figure, three regimes are evident: a collisionless regime, a collisionally dominated regime where ε

approaches a limiting asymptote, and a transition regime. We show also that the dust impact energy increases when the electric potential increases.

II-3. Comparison

The dust impact energy profile for two cases of ions is shown in Fig. 6. We note

that the evolution of the dust impact energy is correlated with the collisional parameter

evolution. This plot shows that a presence of dust charged grains in a plasma influenced

the characteristic behavior of the plasma sheath. The presence of three regimes of sheath

collisionality is also shown in this figure. In the limit of strong collisions the decrease in the

dust impact energy approaches a limiting asymptote. We find also that the dust impact

energy at the wall is more in the case of ions out of thermal equilibrium than in case of

ions in thermal equilibrium.

FIG. 6: A comparison between the exact numerical solutions for the dust energy at wall impact in thermal equilibrium and out of thermal equilibrium. This plot shows that the dust energy at wall impact is more in the case of ions out of thermal equilibrium.

III. SUMMARY

The exact numerical solution of our model provides information about the behavior of dust energy at wall impact. We have demonstrated that the sheath thickness d, the dust impact energy, and the ion energy at wall impact decrease with increasing collisionally. Our plots shows three different regimes: a collisionless regime, a collisionally dominated regime where ε ωd approaches a limiting asymptote, and a transition regime. We showed that the dust impact energy increases when the electric potential increases. It is observed also that the dust energy at wall impact is more in the case of ions out of thermal equilibrium than in case of ions in thermal equilibrium.

We note that the evolution of sheath width is correlated with the collisional parameter evolution. In the limit of strong collisions the decreases in the sheath thickness approach a limiting asymptote. We conclude that a sheath always develops if there are collisions at some point, and also that the presence of dust charged grains in plasma influences the characteristic behavior of the plasma sheath.

Our model can be applied to study the sheath in various material plasma processing techniques where negatively charged dust particles are usually found to be present.

References

[1] T. E. Sheridan and J. Goree, Phys. Fluids B 3 (10), (October 1991).

[2] A. A. Mamun and P. K. Shukla, Phys. Plasmas. 10, No. 5 (May 2003).

[3] S. Vladimirov, S. A. Mariov, and N. F. Cramer, Phys. Rev. E 63, 045401.

[4] V. N. Tsytovich, S. V. Vladimirov, G. E. Morfil and J. Goree, Phys. Rev. E 63 , 056609.

[5] C. K. Goertz, Rev. Geophys. 27 , 271 (1989).

[6] D. A. Mendis and M. Rosenberg, Annu. Rev. Astron. Astrophys. 32, 419 (1994).

[7] P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics, Institute of Physics Publishing Ltd. (Bristol, 2002).

[8] O. S. Vaulina and S. V. Vladimirov, Phys. Plasmas 9, 835 (2002)

[9] M. K. Mahanta and K. S.Goswami, Pramana J. Phys. 56, No.4 (April 2001).

[10] R. J. Goldstone and P. H. Rutherford, Introduction to Plasma Physics, (IOP Publishing Ltd., 1995) pp 146–7).

[11] S. Roy and B. P. Pandey, Phys. Plasmas 10. No. 6, (June 2003).

[12] J. Goree, Plasma Sources Sci. Technol. 3, 400 (1994).

[13] M. Barnes, J. H. Keller, J. C. Forster, J. A. O’Neil, and D. K.Coultas, Phys. Rev. Lett. 68 , 313 (1992).

[14] D. Samsonov and J. Goree, Phys. Rev. E 59 , 1047 (1999) [15] F. Melandsφ and J. Goree, J. Vac. Sci. Technol. A14(2) [16] J. B. Pieper and J. Goree, Phys. Rev. Lett. 77, 3137 (1990).

[17] S. K. Baishya, G. C. Das, and J. Chutia, Pramana J. Phys. 55, No. 5 & 6 (Nov. & Dec. 2000).

[18] N. N. Rao, P. K. Shukla, and M. Y. Yu, Plant. Space Sci. 38, 543 (1990).

[19] F. F. Chen, Introduction to plasma physics and controlled fusion, 2

ed, Vol. 1, pp. 290–297.

[20] C. E. Roberts, Jr., Ordinary Differential Equation: A computational Approach, pp. 111–116.