Simulation of interacting particles in a cylindrical

plasma reactor

M. Davoudabadi∗, B. Rovagnati∗ and F. Mashayek†

842 West Taylor Street, Chicago, IL 60607

In this paper, the structure of a RF plasma glow discharge in a cylindrical geometry isscrutinized through numerical solution of the well-known local field approximation modelequations. Implementation of the extension of our previous 1D formulation to the 3Daxisymmetric case is validated by choosing the corresponding boundary conditions in a 2Dplane case. In the cylindrical case, boundary conditions which closely reflect the physicsof the plasma particles transport are applied both on the electrodes and the surroundingdielectric wall surface. Temporal behavior of the various currents on the electrodes andtemporally-averaged electric potential on the surrounding wall of the reactor are post-processed. After the quasi-steady-state plasma variables distribution in space are achieved,trajectories of a number of micron-sized dust particles injected into the plasma from thetop electrode are calculated in a Lagrangian framework by solving the particles equationsfor their motion and charge, accounting for the various forces acting on each particle. Effectof particle-particle Coulomb interactions on the particles dynamics is investigated.

Nomenclature

~E Electric fielde Electron chargeer,z,θ Unit vectors corresponding to r−, z−, θ− directions in cylindrical coordinate systemf RF frequency~Fm

e Electric force acting on the mth particle~Fm

g mmp ~g, Gravity force acting on the mth particle

~Fmid Ion drag force acting on the mth particle

~FCmn Particle-particle Coulomb interaction force exerted on the mth particle by the nth particle

~Fmnd Neutral drag force acting on the mth particle

~Fmt Total force acting on the mth particle

g Gravitational constantGe,i Electron, ion generation source termH Interelectrode gapIme,i Electron, ion current towards the mth particle~Je,i Electron, ion current densitykB Boltzmann constantLe,i Electron, ion loss termmi Ion massmm

p 4π(rmp )3ρm

p /3, The mth particle massn Unit vector perpendicular to the boundary surface and directed towards the electrodes or wall

of the reactorne Electron number densityni Ion number densitynn P/kBTi, Neutral number density

∗PhD Student, Department of Mechanical and Industrial Engineering, AIAA Student Member.†Professor, Department of Mechanical and Industrial Engineering, AIAA Associate Fellow.

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AIAA 2007-790

Copyright © 2007 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

P Reactor pressureQm

p Charge of the mth particleR Radius of the electrodes~r Particle distance (from the origin (r,z)=(0,0)) vectorrmp Radius of the mth particle

Te Electron temperatureTi Ion temperature~vi Ion velocity~vm

i Ion velocity at the location of the mth particle~vm

ip (~vmi − ~vm

p ), Ion velocity relative to the mth particle~vm

p Velocity of the mth particle

vms

√v2

thi+ (vm

ip)2, Ion mean speed

vthe

√8kBTe/πme, Mean electron thermal velocity

vthi

√8kBTi/πmi, Mean ion thermal velocity

vthn

√kBTn/mn, Neutral thermal velocity

~xmp Position of the mth particle

α Townsend coefficient for ionizationβ Townsend coefficient for recombinationγ Secondary electron emission coefficient~Γe,i Electron, ion fluxε0 Permittivity of free spaceλm

De

√ε0kBTe/e2ne, Electron Debye length at the location of the mth particle

λmD 1/λm

D =√

e2/ε0[nme /kBTe + nm

i /(kBTi + mi(vmip)2)], Effective (linearized) Debye length at the

location of the mth particleρm

p Mass density of the mth particleσ Charge density on the wall of the reactorφ Plasma electric potentialφn Debye-Huckel screened potential around the nth particleφp Dust particle surface potentialφdc Applied DC voltage to the powered electrodeφrf Applied RF voltage to the powered electrode

I. Introduction

Numerical simulation of Plasma Enhanced Chemical Vapor Deposition (PECVD) process for coating ofsubmicron-sized particles can help us understand the process and, through parametric study, examine thepossibilities of eliminating its drawbacks such as nonuniformity of the coating layer on the particles. Thisstudy provides insights also into particle contamination of wafers during plasma processing of microelectronicdevices. In the course of our study,1–4 we simulated the low pressure RF plasma glow discharge in a 1Dparallel plate reactor using the local field approximation model based on which a description of the electronsand ions kinetics coupled with Poisson’s equation for electric field were obtained. Furthermore, dynamicsof a single dust particle injected into the plasma was elaborated in a Lagrangian framework by temporalintegration of the particles equations of motion due to the sum of appropriate forces and charge due to thesum of electrons and ions currents.

As a continuation of our study, we now extend our simulations to the three-dimensional axisymmetriccylindrical geometry. In the particle phase, we look into the dynamics of a number of interacting particlesinjected into the plasma from the upper electrode, after the glow discharge reaches its quasi-steady-state.Each of these particles is tracked through the bulk and sheath in a Lagrangian framework. We are interestedin the entire process of the particles motion from the moment they are injected into the reactor until they areeventually trapped and become stationary. In the course of the particles transport and trapping, particle-particle interaction becomes significant in the regions where their spacing approaches the local plasma Debyelength. This interaction dictates the structure of the assemblies that particles form at the end of their motions

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(i.e. equilibrium).In the following section we present the plasma model used in this study. Dust particles equations of

motion together with the various forces acting on them are explained in section III. The results of thesimulations are presented and discussed in section IV. Finally, some concluding remarks are provided insection V.

II. Plasma model

We consider a cylindrical RF glow discharge plasma reactor as shown in figure 1. We set the parametersas close as possible to the experiment by Cao and Matsoukas5 in which argon is used as the backgroundcarrier gas. Several points must be mentioned regarding the differences between our simulations and thisspecific experimental set-up. There is an asymmetry between the electrodes areas, with a larger hollow diskat the top and a smaller bottom electrode having a depression, which conduces to a large negative DC biasacross the sheath of the lower electrode. Moreover, in the experiment, there is a hole through the upperelectrode of the cylindrical reactor which delivers the particles in the plasma and disrupts the horizontalsheath at the top. Consequently, at this point, we do not seek any quantitative comparison of our resultswith those of this more involved experimental configuration; the results of our study on the dynamics of theparticles are expected to show qualitative agreement.

The drift-diffusion model equations with local field approximation consists of the following set of equa-tions:6

∂ne,i

∂t+ ~∇.~Γe,i = Ge,i − Le,i, (1)

∇2φ =e

ε0(ne − ni), (2)

where,~Γe = −neµe

~E −De~∇ne, (3)

~Γi = +niµi~E −Di

~∇ni, (4)

~E = −~∇φ, (5)

Ge,i = α | ~Γe |, (6)

Le,i = βneni. (7)

Equations (1), (3), and (4) form a set of two convection-diffusion transport equations, while equation (5)couples them to the Poisson equation (2) for electric potential. Equations (6) and (7) represent, respectively,forms of the source and loss terms in the first equation. In the cylindrical coordinate system,

~∇ ≡ ∂

∂rer +

∂

∂θeθ +

∂

∂zez,

~∇ · ~Γ ≡ 1r

∂(rΓr)∂r

+1r

∂Γθ

∂θ+

∂Γz

∂z,

∇2φ ≡ 1r

∂

∂r

(r∂φ

∂r

)+

1r2

∂2φ

∂θ2+

∂2φ

∂z2.

As the geometry under consideration suggests, there is symmetry around the axis (corresponding tor = 0) and thus, variables do not vary with θ and all the terms containing ∂

∂θ can be dropped and theproblem is reduced to a 3D axisymmetic one. We make use of this feature and restrict our computationaldomain into only the plane surrounded by the electrodes on top and bottom (0 ≤ z ≤ H), the wall on theright and the axis of symmetry on the left (0 ≤ r ≤ R).

On the reactor axis the boundary condition for any variable ξ = ne, ni, and φ satisfies the symmetrycondition:

∂ξ

∂r

∣∣∣∣r=0

= 0. (8)

Inside the sheath, close to the electrodes or reactor wall where the largest electric field is expected, thediffusion term (−D~∇n) in the particles flux expression is negligible comparing to the drift term (±nµ~E)

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Figure 1. Schematic of the problem.

caused by the electric field. Hence, the gradients of plasma particles number densities are set to zero at theboundaries. Furthermore, since there is no ion detachment from the electrodes or wall, total ions’ outgoingnormal flux from the electrodes towards the bulk plasma is set to zero,

~Γi · n =

{+niµi

~E · n, ~E · n ≥ 00. ~E · n < 0

(9)

The electrodes are assumed to be perfectly absorbing. Thus, similarly,

~Γe · n =

{0, ~E · n ≥ 0−neµe

~E · n. ~E · n < 0(10)

On the surface of the reactor wall the only incoming (with respect to the electrode) electrons are thethermalized ones. Meanwhile, as a source of the outgoing electrons, there is the secondary electron emissionphenomenon which arises from ions impact on the electrode. Therefore, electrons’ outgoing normal drift fluxis proportional to the ions’ incoming normal drift flux, with γ being the proportionality factor:

−neµe~E · n =

{γniµi

~E · n, ~E · n ≥ 014nevthe .

~E · n < 0(11)

As boundary conditions for the electric potential, the upper electrode is grounded, and the lower electrodeis powered by a periodic voltage with frequency f . The external circuit is not considered and no DC biasvoltage is applied to the electrodes. Therefore,

φ(z = 0) = φdc + φrf cos(2πft), (12)

φ(z = H) = 0. (13)

The charge density distribution along the dielectric wall is found by temporally integrating the summationof electrons and ions normal current densities at each node on the wall,7,8

~Je = −e~Γe, (14)

~Ji = +e~Γi, (15)

∂σ

∂t= ~Je · n + ~Ji · n. (16)

Then, the normal electric field is imposed on the surrounding wall,7,8

~E · n =σ

ε0. (17)

Note that the normal electric field will be nonuniform along the dielectric wall. Moreover, we are assumingthe insulator wall of the cylinder to be a surface (with no thickness). For the case where the radial wall has

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a finite thickness, the above relation should be modified to incorporate the effect of the normal electric field(due to the polarization of dielectric material) at the interface from the wall side. Hence, Poisson equationneeds to be solved within the wall too. This case is beyond the scope of the present study.

There exist singular points on the circle where the electrodes are attached to the chamber wall. Tocircumvent this singularity problem we linearize the potential along the electrodes from its value on the wallto its magnitude on the electrodes within a distance of 5 percent of the radius from the singular points. It isworth mentioning that this is a more natural way than linearization of the potential on the wall where thevalue of the normal electric field (and not the potential itself) is imposed.

The Numerical scheme together with the nonuniform staggered-grid mesh stencil employed to solvethe above set of equations and boundary conditions is elaborated for the one-dimensional case elsewhere.3

The discretization of the above equations leads to a system of coupled algebraic equations for the discretevariables. At each time step, these coupled equations are solved iteratively until a reasonable convergence ofthe variables is reached. Inside each internal iteration, a Gauss-Seidel method with relaxation is employedto solve each of the equations. The convergence of the full system to the periodic quasi-steady-state solutionis assumed to be reached once the change in the average conduction current on the electrodes over one RFcycle becomes less than 0.001 percent.

III. Dust particles motion

Tracking a swarm of spherical dust particles (which could be of different mass and radii) is possible inLagrangian framework. In addition to the particles equations of motion, we account for the charging processof the particles by solving the transient charging equation. The Lagrangian equations for the mth particleposition, velocity, and charge are described, respectively, as

d~xmp

dt= ~vm

p , (18)

mmp

d~vmp

dt= ~Fm

t , (19)

dQmp

dt= Im

e + Imi . (20)

In the present analysis where we simulate the discharge in a cylindrical geometry, the above equations areaccounted for in r− and z− directions. These equations are integrated using second order Adams-Bashforthmethod with specified initial conditions.

Since the dust particles are much heavier than the plasma species, they do not respond to the fast varyingplasma properties in one RF period. Furthermore, considering a limited number of particles (dilute particlephase) to avoid locally dense collection of particles, we neglect any effect of the particles on the plasmavariables. Thus, the coupling between the plasma and particle phases is one way, i.e., only the plasma affectsthe particles transport. Therefore, after the temporal and spatial variations of plasma variables are resolvedvia the plasma model simulations, they are averaged over the RF period. In fact, the necessary time stepsize to resolve the particles motion represents the time interval during which the particles are assumed tohave reached equilibrium with respect to all the plasma parameters. Such a time step is larger than the RFperiod, but could be smaller than the particle charging time. The former statement confirms the validityof employing the temporally averaged values for the plasma quantities involved in the particles equations.Using these local values, depending on the location of the spherical particles in the plasma, different forcesacting on them can be calculated. Ions velocity which includes both advection and diffusion of the ions areextracted from the plasma variables using

~vi =~Γi

ni.

The main forces on the dust particles in a discharge comprise of the electric, gravity, ion and neutral dragforces, and particle-particle interaction force. In the present work, we assume that no temperature gradientis present in the discharge, hence, the thermophoretic force is not taken into account. (For a detailed analysisof the effect of this force, refer to Ref.4) Therefore, the total force acting on the mth particle is obtainedthrough ~Fm

t = ~Fme + ~Fm

g + ~Fmid + ~Fm

nd + ~Fmmn. A description of each of these terms follows.

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In the case under consideration, in order to exclude the effect of ‘closely packed’ particles on the particlescharge and their experienced ion drag, for each particle m and at each time step we make sure the inter-particle distances | ~rm− ~rn | to be greater than the local effective Debye length λn

D. This implies that duringits motion, each particle is always fully screened by the plasma particles, and use of the following OMLexpressions9 (with a range of applicability rm

p ¿ λD) to determine the currents in equation (20) remainsas robust as in the case of an isolated particle. It is emphasized that effect of degree of collisionality of theplasma (enhanced by ion-neutral charge-exchange collisions) on the ion current is not included in this work.

Ime = −eneπ(rm

p )2vthe exp[e(φm

p − φ)kBTe

], (21)

Imi = eniπ(rm

p )2vs

[1− 2e(φm

p − φ)mi(vm

s )2

], (22)

Qmp = 4πε0r

mp (φm

p − φ). (23)

From equations (20), and (21)–(23), the potential difference between the mth particle and the surroundingplasma, (φm

p −φ), and the mth particle charge, Qmp , can be evaluated at each time step. With the knowledge

of the plasma electric field at the particle location and the amount of charge accumulated on the particle,the electric force on the mth particle is simply obtained from10,11

~Fme = Qm

p~E

1 +

(rm

p

λmD

)2

3(1 + rm

p

λmD

)

. (24)

In our case, rmp ¿ λm

D , and the effect of the last term which represents the enhancement in the surfacefield of macroscopic bodies (as opposed to point charges) vanishes. Nevertheless, we keep this term in ourformulation for the sake of completeness.

Despite its significant role in dusty plasmas, a self-consistent model for the ion drag force (especially incollisional regime) has not yet been presented. In our recent work,3 we compared and contrasted two of thewell-known models by Barnes et al.9 and Khrapak et al.,12 and concluded that for the parameters understudy adopting the standard approach of Barnes et al. with electron Debye length would more closely agreewith the results observed in the experiments,13,14

~Fmid = πniv

ms mi~v

mip(bm

c )2 + 4πnivms mi~v

mip(bm

π2)2ln Γm, (25)

where the collection impact parameter is defined as

bmc = rp

[1− 2e(φm

p − φ)mi(vm

s )2

] 12

. (26)

Moreover, we found in our simulations that the inclusion of ions thermal energy (refer to the definition ofvs in nomenclature) in calculation of the collection impact parameter is vital in capturing the physics ofparticle dynamics. Also,

bmπ2

=e(φm

p − φ)mi(vm

s )2(27)

is the orbital impact parameter corresponding to a 90o deflection, and

ln Γm =12

ln

[(λm

De)2 + (bm

π2)2

(bmc )2 + (bm

π2)2

](28)

is the Coulomb logarithm.We assume a specular reflection of the neutral particles after their collision with the particle. Thus, the

following Epstein expression is utilized since physically, the typical ratio vmp /vthn ¿ 1:15,16

~Fmnd =

83

√2π(rm

p )2mnnnvthn(~vn − ~vmp ). (29)

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Under the circumstances where the spherical symmetry of the particle sheath remains essentially intactthe Debye-Huckel (Yukawa) potential adequately describes the sheath around the particle.11 This condition issatisfied in the regime of subthermal ion flow (e.g. central part of the discharge) if the inter-particle distancesare always greater than the effective Debye length (| ~r − ~rn |≥ λn

D). If so, the plasma around the particlesremain isotropic. Violating the above conditions, two difficulties arise in modeling the process under study;first, the particles should remain far enough from each other (no ‘closely packed’ particles should be allowed),and second, approaching the sheath the drift velocity of the ions increasingly exceed their thermal velocitywhich results in the anisotropy of the plasma through the ion wake phenomenon downstream of the particle.An easy solution for the former is to sequentially release the particles after the inter-particle distances becomelarger than the Debye length and to ensure that the particles remain sufficiently far apart during the wholeprocess. And to circumvent the latter, extending the recent works by Yaroshenko et al.,17,18 we theoreticallysuppose the problem to be a superposition of four interactions: symmetrical monopole-monopole interactionwhere the Debye-Huckel potential holds, dipole-dipole interaction, and two monopole-dipole interactions.(Yaroshenko et al. account only for the first two.) In the present work, however, we only include thefirst interaction and postpone the study of the rest to the future publications. In fact, it is experimentally19

investigated that the interaction between the particles trapped in the same horizontal plane is only by meansof Coulomb force. Dust particles, however, at different vertical positions interact by net attractive forces.This observation leads to the conclusion that the other three mentioned forces become important in caseswhere multi-layer structures exist and this is not the focus of our present study.

Taking the electrostatic screened potential around the nth particle as the Debye-Huckel form,

φn =Qn

p

4πε0 | ~r − ~rn | exp(−| ~r − ~rn | −rn

p

λnD

), (30)

one can readily calculate the repulsive Coulomb force on the mth particle exerted by the nth particle in thefollowing manner:

~FCmn = −Qm

p

∂φn

∂~r

∣∣∣∣~r= ~rm

=Qn

pQmp

4πε0| ~rm − ~rn |2exp

(−| ~rm − ~rn | −rn

p

λnD

)[ | ~rm − ~rn |λn

D

+ 1]

~rm − ~rn

| ~rm − ~rn | . (31)

In order to track each particle in the computational domain and find its host computational cell, weemploy an efficient generalized iterative algorithm for searching and locating particles in arbitrary meshes.20

The algorithm uses the Newton’s method to search for the particle within a reference element mapped fromthe quadrilateral computational mesh elements, together with a criterion to efficiently move from element toelement in the mesh. To calculate the forces, at each time step, the corresponding plasma variables presentin the above expressions are interpolated from the host element grid points to the particle location using atwo-point Lagrange interpolation in r− and z− directions.

IV. Results and discussion

A. Validation of the code

Imposing a gradient free condition on all the variables along the left and right boundaries, we conducted avalidation study by solving the planar 2D equations for both plasma phase and one single particle within arectangle of length L = 1 cm and height H = 2 cm. In such a case, it is trivial to analytically show thatequations (1)–(7) will give the same results as the 1D (in y−direction) case. Figure 2(a) depicts averaged(over one RF period) profiles of concentrations of plasma particles and electric field along y-direction inthe reactor. Figure 2(b) shows the trajectory of the single particle initially released from the top electrode(y = H). As expected the results of the present 2D planar case coincide (which vary only along y− direction)with the ones from previous 1D case. Noteworthy is that in the 2D code, we tested the cases with nonuniforminitial conditions for plasma variables, and also cases where the particle was released at a position away fromx = 0. The former ended with the same results as starting with a uniform spatial distribution of plasmavariables. The latter yielded identical trajectory compared to the one where the particle is released at x = 0.

B. Plasma phase

In this study, we assume that all the temperatures, and electrons and ions transport coefficients are constantwith time and space. The values of transport coefficients and various parameters of the discharge used in

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y [cm]

Ave

rage

dn i,

n ex1

0-10

[cm

-3]

Ave

rage

dE y

[KV

/cm

]

0 0.5 1 1.5 20.02

0.04

0.06

0.08

0.1

0.12

-0.1

-0.05

0

0.05

0.1

(a)

time [s]

y p[c

m]

0 0.025 0.05 0.075 0.1 0.125 0.150

0.5

1

1.5

2 (b)

Figure 2. Validation of the code. (a) Time-averaged (over one RF period) electrons (green) and ions (blue)densities, and electric field (red) along y-direction in the discharge, (b) temporal variation of particle verticalposition for one single particle of radius rp = 1.0 µm released from the top electrode. Lines show the resultsfrom the present 2D planar case. Symbols show the results from the previous 1D case.3

the simulation are tabulated in Table 1. After a resolution study, we decided to choose a nonuniform grid inboth r− and z− directions with 60×60 grid points and finer grid-spacing close to the electrodes and wall.Figure 3(a)-(d) depicts, respectively, averaged (over time) profiles of concentrations of ions, electrons, electricfield in r− and z− directions in the reactor. The structure of the contours of plasma particles densities isalmost circular. From the figure it is evident that plasma sheaths form not only in the vicinity of electrodes,but also in the region adjacent to the surrounding wall. Naturally, within the sheaths of the electrodes themagnitude of the electric field in z− direction is large and the electric field in r− direction is not noticeable.This trend is reversed inside the sheath of the radial wall. Moreover, the averaged ion r− and z− velocitiesare in the direction of the averaged electric fields in the corresponding directions. One can conclude that,in the average sense, the ions flow towards the electrodes and the radial wall. Another interesting feature(repeatedly observed in the discharge experiments) captured by the physical boundary conditions used inthis simulation is the negative averaged electric potential along the radial wall with a value of ∼ −3.7 V inthe central regions of the wall (see figure 4).

C. Particles

We choose 10 particles with ρp = 500 Kg/m3 and rp = 1.0 µm. As initial conditions for equations (18)–(20),we release the particles from the top electrode but at different radial positions ranging from r = 0 to r = 0.5cm, with zero charge and velocity. As was explained earlier, particles are consecutively injected into theplasma in the following manner: After the injected particle reaches the center of the interelectrode gap, thenext particle sitting on the top electrode at some distance apart from the reactor axis is released.

From figure 5(a) and (b) it is seen that the particles exhibit damping oscillations in both directions.These oscillations are more pronounced in the radial direction in the first phase of their motion from themoment they are injected until they approach the lower electrode sheath. Then, the particles sway betweenthe sheath and bulk plasma and finally become trapped in the sheath, and reach their equilibrium position(which corresponds to the location of the potential energy minimum). In r− direction, the ion drag forcepushes the particle away from the central parts of the reactor towards its radial wall. The farther the particleis released from the electrode the closer it becomes to the reactor wall. However, due to the strong repulsiveradial electric force in the wall sheath region, the particle never reaches the wall. In the case of interactingparticles, throughout their motion, the Coulomb force applied by one particle on another becomes significantwhen their interparticle distance becomes less than a few tens of Debye length (mostly in the last part oftheir motion while they are being trapped in the sheath).

Without including the Coulomb interaction, all of the particles are trapped literally at the same location(figure 5(c)). Their equilibrium position, in fact, would be a function of neither r nor z. This means thatthe potential well is located right on the reactor axis and in the electrode sheath. Whereas with Coulombinteraction included in the model, particles form a parabola-like structure at the end of their motion (see the

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Table 1. Parameters used in the simulations for plasma phase

parameter valueµeP 3×105 ( cm2

Vtorr)

DeµekBTe

e (cm2

s )µiP 1.4×103 ( cm2

Vtorr)

DiP 3×105 ( cm2

storr )αP 29.22 × exp

[−26.64× (P

E )12

]( 1cm)

β 0 (cm3

s )f 13.56 (MHz)H 2 (cm)R 1 (cm)P 200 (mTorr)Te 4 (eV)

Ti = Tn 300 (K)γ 0.05

φdc 0 (V)φrf 40 (V)

inset of figure 5(d)). From one particle to another one, there is not noticeable variations in their equilibriumz position, and in fact, the structure is seen flat from a larger view (figure 5(d)). Further, it is interestingthat although the particles are released in one side of the axis they are redistributed symmetrically aroundthe axis. As particle number n approaches particle number m from right, it pushes the particle m towardsleft and so the particles redistribute in the above-mentioned manner.

V. Conclusions

Dynamics of a limited number of dust particles released into a cylindrical plasma reactor is examinedthrough implementation of a comprehensive model taking various forces acting on the particles into consid-eration. Spatial distribution of the particles at their equilibrium state is proved to change from a localizedtrapping point in the case of non-interacting particles to an extended curve in the case of interacting onesthrough the Debye shielded Coulomb force.

Incorporating the electron energy equation into the plasma model, parametric study of the presentmodel (e.g. injection of particles with different radii and mass densities), taking into consideration the fullinteraction of particles in the framework of multi-layer structures, and study of the effect of the dust particleson the plasma particles transport (two-way coupling) where the particle phase will be denser and larger countof particles will be used comprise subjects for our future studies. Last, but not least, temperature gradientin the discharge field can be incorporated and its effect on the motion of the particles be taken into accountvia the thermophoretic force.

Acknowledgments

This work was supported by grant CTS-0422900 from the U.S. National Science Foundation.

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r [cm]

z[c

m]

-1 -0.5 0 0.5 10

0.5

1

1.5

28.50x10+08

7.37x10+08

6.24x10+08

5.10x10+08

3.97x10+08

2.84x10+08

1.70x10+08

5.68x10+07

(a)Avg. ne [cm-3]

r [cm]

z[c

m]

-1 -0.5 0 0.5 10

0.5

1

1.5

28.57x10+08

7.43x10+08

6.29x10+08

5.15x10+08

4.02x10+08

2.88x10+08

1.74x10+08

6.02x10+07

(b)Avg. ni [cm-3]

r [cm]

z[c

m]

-1 -0.5 0 0.5 10

0.5

1

1.5

2175.6

125.4

75.3

25.1

-25.1

-75.2

-125.4

-175.6

(d)Avg. Ez [V/cm]

r [cm]

z[c

m]

-1 -0.5 0 0.5 10

0.5

1

1.5

2232.5

166.1

99.6

33.2

-33.2

-99.6

-166.1

-232.5

(c)Avg. Er [V/cm]

Figure 3. Spatial distribution of time-averaged (a) electrons, and (b) ions number densities, and electric fieldin (c) r−, and (d) z− directions in the reactor.

r [cm]

z[c

m]

-1 -0.5 0 0.5 10

0.5

1

1.5

221.40

16.85

12.29

7.74

3.19

-1.37

-5.92

-10.48

(a)Avg. φ [V]

zp [cm]

Avg

.φ[V

]

0 0.5 1 1.5 2-14

-12

-10

-8

-6

-4(b)

Figure 4. Spatial distribution of time-averaged electric potential (a) within and (b) along the wall of thereactor.

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rp [cm]

z p[c

m]

-1 -0.5 0 0.5 10

0.4

0.8

1.2

1.6

2 (d)

-0.2 -0.1 0 0.1 0.2

0.343

0.344

0.345

rp [cm]

z p[c

m]

-1 -0.5 0 0.5 10

0.4

0.8

1.2

1.6

2 (c)

rp [cm]

z p[c

m]

-0.2 0 0.2 0.4 0.6

0.4

0.8

1.2

1.6

2 (a)

rp [cm]

z p[c

m]

-0.2 0 0.2 0.4 0.6

0.4

0.8

1.2

1.6

2 (b)

Figure 5. Trajectories of the particles (a) without interaction, and (b) with interaction. Particles initiallocations and final positions of (c) non-interacting, and (d) interacting particles.

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References

1Davoudabadi, M. and Mashayek, F., “Dust Particle Dynamics in Magnetized Plasma Sheath,” Phys. Plasmas, Vol. 12,No. 7, 2005, pp. 073505–1–073505–10.

2Davoudabadi, M., Rovagnati, B., and Mashayek, F., “Lateral Motion of a Dust Particle in Magnetized Plasma Sheath,”IEEE Transactions on Plasma Science, Vol. 34, No. 2, 2006, pp. 142–148.

3Davoudabadi, M. and Mashayek, F., “Dust Particle Dynamics in Low-Pressure Plasma Reactor,” J. Appl. Phys., Vol. 100,No. 8, 2006, pp. 083302–1–083302–10.

4Davoudabadi, M. and Mashayek, F., “Particle Mobilization in PECVD Reactor Using Thermophoretic Force,” AbstractBook of the 33rd IEEE International Conference on Plasma Science, Traverse City, MI, June 4-8 , 2006.

5Cao, J. and Matsoukas, T., “Deposition Kinetics on Particles in a Dusty Plasma Reactor,” J. Appl. Phys., Vol. 92, No. 5,2002, pp. 2916–2922.

6Boeuf, J. P., “Numerical Model of RF Glow Discharges,” Phys. Review E , Vol. 36, No. 6, 1987, pp. 2782–2792.7Boeuf, J. P. and Pitchford, L. C., “Two-Dimensional Model of a Capacitively Coupled RF Discharge and Comparisons

with Experiments in the Gaseous Electronics Conference Reference Reactor,” Physical Review E , Vol. 51, No. 2, 1995, pp. 1376–1390.

8Suetomi, E., Tanaka, M., Kamiya, S., Hayashi, S., Ikeda, S., Sugawara, H., and Sakai, Y., “Two-Dimensional FluidSimulation of Plasma Reactors for the Immobilization of Krypton,” Computer Physics Communications, Vol. 125, 2000, pp. 60–74.

9Barnes, M. S., Keller, J. H., Foster, J. C., O’Neil, J. A., and Coultas, D. K., “Transport of Dust Particles in Glow-Discharge Plasmas,” Phys. Rev. Lett., Vol. 68, 1992, pp. 313–316.

10Daugherty, J. E. and Graves, D. B., “Derivation and Experimental Verification of a Particulate Transport Model for aGlow Discharge,” J. Appl. Phys., Vol. 78, No. 4, 1995, pp. 2279–2287.

11Daugherty, J. E., Porteous, R. K., and Graves, D. B., “Electrostatic Forces on Small Particles in Low-Pressure Discharges,”J. Appl. Phys., Vol. 73, No. 4, 1993, pp. 1617–1620.

12Khrapak, S. A., Ivlev, A. V., Morfill, G. E., and Thomas, H. M., “Ion Drag in Complex Plasmas,” Phys. Review E ,Vol. 66, No. 4, 2002, pp. 046414–1–046414–4.

13Zafiu, C., Melzer, A., and Piel, A., “Measurement of the Ion Drag Force on Falling Dust Particles and its Relation to theVoid Formation in Complex (Dusty) Plasmas,” Phys. Plasmas, Vol. 10, No. 5, 2003, pp. 1278–1282.

14Zafiu, C., Melzer, A., and Piel, A., “Response to “Comment on ‘Measurement of the Ion Drag Force on Falling DustParticles and its Relation to the Void Formation in Complex (Dusty) Plasmas’ ” [Phys. Plasmas 10, 4579 (2003)],” Phys.Plasmas, Vol. 10, No. 11, 2003, pp. 4582–4583.

15Nitter, T., “Levitation of Dust in RF and DC Glow Discharges,” Plasma Sources Sci. Technol., Vol. 5, No. 1, 1996,pp. 93–111.

16Shukla, P. K. and Mamnun, A. A., Introduction to Dusty Plasma Physics, IOP Publishing, London, 2002.17Yaroshenko, V. V., Annaratone, B. M., Antonova, T., Thomas, H. M., and Morfill, G. E., “The Dipole ‘Instability’ in

Complex Plasmas and its Role in Plasma Crystal Melting,” New Journal of Physics, Vol. 8, No. 4, 2006.18Yaroshenko, V. V., Annaratone, B. M., Antonova, T., Thomas, H. M., and Morfill, G. E., “Dynamics of Cluster Particles

in a Dense Plasma,” New Journal of Physics, Vol. 8, No. 9, 2006.19Melzer, A. and Piel, A., “Experiments on Particle-Particle Interactions in Dusty Plasma Crystals,” AIP Conference

Proceedings, Non-neutral plasma physics III , Vol. 498, 1999, pp. 383–388.20Allievi, A. and R., B., “A Generalized Particle SearchLocate Algorithm for Arbitrary Grids,” J. Comp. Phys., Vol. 132,

No. 2, 1997, pp. 157–166.

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