Particle-in-Cell Modeling of Low-Temperature Plasma

Dmytro SydorenkoUniversity of Alberta, Edmonton, Canada

2

Motivation

Particle-in-cell simulations are important numerical tools in studying plasma properties.

They are invaluable in understanding kinetic effects, especially when experimental measurements are difficult to obtain or interpret.

3

Outline

Possible methods of kinetic numerical description of a plasma.

PIC basics: Charge and force weighting Leap-frog explicit algorithm Instability of the explicit algorithm

Direct implicit method Particle emission from walls Monte-Carlo model of electron collisions with

neutrals Coulomb collisions

4

Recommended literature

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, Adam Hilger, Bristol and New York, IOP Publishing, 1988.

C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Bristol and Philadelphia, IOP Publishing, 1991.

5

Kinetic description required for plasmas with low collision frequency Plasma density 1011-1012 cm-3 . Neutral gas pressure 1-100 mTorr . Electron temperature 1-50 eV,

corresponding thermal velocities 6x105-4.2x106 m/s .

Frequency of electron-neutral collisions 106-109 s-1.

Size of a device 10 cm. Electron mean free path 10-3-1 m.

3 kW Hall thruster in PPPL(from htx.pppl.gov)

ICP for material processing (University of Saskatchewan Plasma Physics Laboratory / PLASMIONIQUE Inc )

Fluid description

Kinetic description

6

The distribution function represents the number density of particles in 6-D coordinate-velocity space.

One can use EVDF to calculate density flux pressure

EVDF is described by the Vlasov-Boltzmann equation

Electron velocity distribution function),,( tvxf

),( vx

vdtvxftxn 3),,(),(

vdtvxfvtxj 3),,(),(

vdtvxfvmtxp 32 ),,(3

),(

COLLt

fvfBvE

mq

rfv

tf

v

dvvfdN )(

dv

f(v)

7

Direct solution of the Vlasov equation

Define EVDF on a grid in the coordinate-velocity space.

Advance EVDF using e.g. splitting scheme [Cheng and

Knorr, J.Comput.Phys., 1976] . flux-corrected transport

algorithm [Boris and Book, J.Comput.Phys., 1973].

semi-Lagrangian algorithm [Sonnendrucker et al., J. Comput.Phys., 1999].

The ideal method of advancing should preserve monotonicity, be conserving, do not disperse sharp gradients, adapt resolution to reproduce small scales.

Example of 1-D f(L,v)

Result of advancing with systematic interpolation error.

8

Lagrange (water-bag) scheme

A “water-bag” is a contour in the plane “coordinate –velocity” along which the EVDF is constant.

Such a contour is represented as a set of markers. Each marker is advanced according to the equations of motion.

From [Berk and Roberts, Phys. Fluids, 1967].

2/)()()( 2,

2, fvvLJ icici

9

The water-bag method: advantages and shortcomings

Good: No interpolation errors. Integration of EVDF is very

easy. With many contours, one can

approximate a smooth EVDF.

Bad: Contour processing

algorithms are complex. This method is very difficult to

extend to higher dimensions.

10

a water bag

11

Solving Vlasov equation for multiple dimensions is numerically very costly Consider a plasma with Te=1eV, ne=1011 cm-3, size L=1cm.

thermal velocity is 5.9x105 m/s, plasma frequency is 1.78x1010 s-1, Debye length is 3.3x10-3 cm, ratio L / Debye length is 300.

Assume that the desired resolution is 300 points (along each coordinate direction), 200 velocity points (along each velocity direction).

The number of variables is 1d1v :: 300 x 200 = 6x104, 1d3v :: 300 x 2003 = 2.4x109, 2d3v :: 3002 x 2003 = 7.2x1011, 3d3v :: 3003 x 2003 = 2.16x1014.

Compare: 3d3v PIC code with the same spatial resolution and 1000 particles per cell requires 6 x 3003 x 1000 = 1.62x1011

particle variables.

12

It’s easier to use particles…

Plasma is represented as a set of macroparticles.

Each macroparticle is a charged “cloud” representing many real charged particles (electrons or ions) “glued” together.

A macroparticle has the same charge-to-mass ratio (q/m) as the real charged particle.

Equations of motion are solved for each macroparticle.

The electric and magnetic fields are calculated self-consistently using charge densities and currents produced by the macroparticles.

ne=1012cm-3

L=10cm

Ne=1013

Q/e=106Nm=107

13

Particle-mesh method: Charges and currents are calculated

in a set of predefined points (nodes of computational grid).

Poisson’s and/or Maxwell equations are solved and give electromagnetic fields in the nodes.

The fields are then interpolated into the positions of the particles.

Particle-particle method: Electric and magnetic fields of a system of charged particles are (in a

static limit):

The numerical cost of calculating the individual fields grows with the particles number N as N2.

How to calculate the fields?

jjj BE

,,

)( BvEmq

dtd

i i

ii xx

xxqxE 30 ||4

1)(

i i

iii xx

xxvqxB 30

0

||4)(

jj J

,

14

Particle and force weighting

Distribute particle charge to neighbor nodes.

Get force, which is defined in the grid nodes, in the location of the particle.

15

Particle and force weighting Linear weighting to the grid:

This corresponds to a triangular shape function S.

Grid values are linearly interpolated into the particle’s location:

xxx

QQx

xxQQ ji

ijij

ij

11 ,

)(),()( 11

jijjijj xxSx

QxxSx

Qx

)()(

)(

11

11

ijjjij

jij

ijjii

xxSExxSExxx

Ex

xxEExE

0101

||0)(

hifxhhifxh

xhifhS

16

Bi-linear weighting on a 2D grid

17

Bi-linear weighting on a 2D grid

18

)()()()(

)()()()(

1,1

111,111,

kiyjixkjkiyjixjk

kiyjixkjkiyjixkj

yySxxSyx

QyySxxSyx

Q

yySxxSyx

QyySxxSyx

Q

Bi-linear weighting on a 2D grid

0101

||0)(

hifxhhifxh

xhifhSx

0101

||0)(

hifyhhifyh

yhifhSy

)()()()(

)()()()(

111,111,

1,1

kiyjixkjkiyjixkj

kiyjixkjkiyjixjki

yySxxSEyySxxSE

yySxxSEyySxxSEE

19

General sequence of a Particle-in-Cell (PIC) algorithm

Initial state

For all particles:accelerate+move

For all particles:collect grid values

j, Jj

For all grid nodes:calculate fields

Ej, Bj

t = t + t

20

Explicit leap-frog scheme

21

Explicit leap-frog scheme

Coordinates, charge densities, and fields are calculated at “integer” times .

Velocities are calculated at “half-integer” times

Equations of motion

Poisson’s equation

vdtxd

Fdtvdm

EqF

0

E

tvxx ni

ni

ni 2/11

txFvv ni

ni

ni )(2/12/1

2

011 2 x

njn

jnj

nj

xE jjn

j

211

tntn

tntn )2/1(2/1

22

Motion equation with magnetic field

To advance the velocity obtain v- from vn-1/2, rotate v- to obtain v+, obtain vn+1/2 from v+.

)( BvEqdtvdm

BvvEmq

tvv nn

nnn

2

2/12/12/12/1

22/1 t

mEqvv n

22/1 t

mEqvv n

Bvvmq

tvv

)(2

Centered-difference form

ttmqB

c

2

arctan2

Rotation by angle

This substitution cancels E completely [Boris, 1970].

22tan t

mqB

yxyyxx cvsvvsvcvv ,

212sin

s 2

2

11cos

c

For B directed along the z-axis, the rotation iswhere

23

Stability of the explicit leap-frog methodA harmonic oscillator can be described with the leap-frog method as follows

vdtdx

xdtdv 2

0

tvxx nnn 2/11

txvv nnn 20

2/12/1

220

11 2 txxxx nnnn

Assume that the solution is of the form then)exp( tnix 22

sin 0 tt

If then is complex and the solution grows exponentially.0

2

t

For the leap-frog explicit algorithm, the timestep must be small enough to resolve the electron plasma frequency: t<2/pe. Another important condition is the Courant criterion: t<x/v. Here x is the mesh size and v is the fastest speed of propagation of either a wave or a particle in the system. This criterion prevents a numerical instability and is applicable in many numerical schemes.

24

How to increase the time step?

Numerical schemes stable for large time steps, tpe>>1, are usually implicit.

In implicit schemes, calculation of the updated positions xn+1 requires knowledge of the fields En+1 at the same time.

Below we will consider a direct implicit algorithm for electrostatic simulations described in [Gibbons and Hewett, J.Comput.Phys., 1995].

25

Electrostatic direct implicit algorithm (1)

The finite difference equations are

Bvvmtqatvv nnnnn

2/12/12/12/1

2

2/11 nnn vtxx

11

21 nnn E

mqaa

Note that we must know En+1 to get xn+1.

The velocity equation can be transformed as follows:

1112/12/1

22nnnn E

mtqatAvKv

Here matrices K and A-1 contain coefficients depending on the magnetic field.

vdtxd

Fdtvdm

Note, the unknown updated electric field is separated.

26

Electrostatic direct implicit algorithm (2)

Acceleration and displacement split in two steps. Step 1 (pre-push, all values in the RHS are known):

Step 2 (final push):

The advanced electric field must be found between the two steps.

1112/12/1

22nnnn E

mtqatAvKv

112/1

2~

nn atAvKv

vtxx n ~~

vtxxn ~1

11

2

nEAmtqv

vvv n ~2/1

27

Electrostatic direct implicit algorithm (3)

To define En+1 it is necessary to predict the charge density n+1.

.)()(~2

)(~~~

~~

~)~()(

112

,,,,

,,

,,,

,,,

1

sj

nsjs

s

sj

isisj

jisisj

s

isisj

isisisj

s

isisisj

sj

n

xEAxmqtxxxS

xvtxxS

xq

xxSx

xxxSx

qxxxSx

qx

Here S is the shape function which defines charge distribution into the grid nodes, subscript s denotes particle species, subscript i denotes particle number.

Introduce implicit susceptibility: .

The field equation becomes .

ssjs

s

sj Ax

mqtX 1

2

)(~2

~10 X

28

29

Advantages and limitations of implicit algorithms Implicit simulation are usually less prone to numerical

instability than the explicit ones. Implicit algorithms are more complex and more costly

numerically. Time step may be increased but:

Debye shielding will not be reproduced correctly if kvtht>1. If pet>>1, waves must be longer than the Debye length, kD<<1

[Langdon, Cohen, and Friedman, J.Comput.Phys., 1983]. There is always the Courant criterion t<x/vmax.

With improper resolution in space and time, an implicit simulation may look stable but the results will be unphysical.

30

Main code features: Active material surfaces with secondary

electron emission induced by incident electrons and ions.

Monte-Carlo collisions between electrons and neutrals.

Electron-electron and electron-ion collisions. 107 particles, 1000s spatial cells, 1000s

particles per cell, abundant diagnostics. MPI parallelization.

The code was used to study EVDF anisotropy and plasma-wall interaction in Hall thrusters [Sydorenko, Kaganovich, Raitses, and Smolyakov, Phys.Rev.Lett, 2009].

Recently, the code was used to study two-stream instability and multi-peak EVDF formation in dc-rf discharges.

Schematic of PIC simulations,plane geometry approximation

Hall thruster, cylindrical geometry

EDIPIC – electrostatic 1d3v code based on the direct implicit algorithm

31

Processing of probabilistic events

Let an event has probability P<1. To define whether the event occurs or not: take a random number 0<R<1, the event occurs if R<P.

Let some variable is described by a probability density function f(v). To select the value of this variable: take a random number 0<R<1,

solve for v.

1)(

dvvf

v

dvvf ')'(

v

dvvfR ')'(

To reduce numerical cost, one can precalculate sets of vi(Ri) and interpolate: for random number R, Ri<R<Ri+1, ii

ii

ii

ii RR

RRvRRRRvv

11

1

1

32

Secondary electron emission

Emission coefficient , where 1 is the primary electron flux

and 2 is the secondary electron flux.1

2

The primary electron can be absorbed, be reflected elastically, be reflected inelastically, cause the emission of “true”

secondary electrons.

The secondary electron emission is an important process which modifies the structure of near-wall sheath and affects intensity of plasma cooling due to wall losses.

1

,,;2,,

iet

iet

iet

The partial emission coefficients are

The total emission coefficient is

iet ,2,2,22

33

Partial emission coefficients For primary electron energies above 10s of eV,

the total emission coefficient is [Vaughan, IEEE Trans.Electron. Dev., 1989]:

Total emission coefficient

Incident electron energy

Total (1) and partial emission coefficients: elastic (1), inelastic (2), true (3). Markers are BN experimental data [Dunaevsky, Raitses, Fisch, Phys.Plasmas, 2003]

20max,max

0max

0

2max

1)(,)(

),(

),(1exp),(2

1),(

s

ksV

kwwww

wwwv

wvwvkw

max

max

25.062.0

wwifwwif

kw and are the primary electron’s energy and angle of incidence.

At high energies, elastically / inelastically reflected electrons comprise abour 3% / 7% of the emitted current [Gopinath, Veboncoeur, Birdsall, Phys.Plasmas, 1998].

VtViLEeVe w 9.0,07.0),(03.0 ,

Additional term which increases elastic reflection at low energies.

34

Angular and energy distributions of secondary electrons Elastically reflected electrons:

energy w2=w1 specular reflection random reflection

emission angle 2= arcsin(R) azimuthal angle 2= R2

Inelastically backscattered electrons: energy w2=Rw1 emission angle 2= arcsin(R) azimuthal angle 2= R2

True secondary electrons: energy correspond to a half-Maxwellian

distribution of temperature Tt, emission angle 2= arcsin(R) azimuthal angle 2= R2

From [Seiler, J.Appl.Phys, 1983]

Energy spectrum of secondary electrons in simulation

Everywhere, R is a random number, 0<R<1.

35

Probabilistic model of secondary electron emission Assume that no true secondary electrons are emitted if the primary

electron reflects either elastically or inelastically. A particle collide with a wall. Then

calculate particle energy w and angle of incidence calculate the total and the partial e, i, t emission coefficients, return if =0 take a random number R, 0<R<1 if R<e inject elastically reflected electron, return if R< e+i inject inelastically backscattered electron, return if <1 then

if R< e+i +e inject true secondary electron return

calculate *=t / (1-e-i), must be *>1 inject INT(*) true secondary electrons take random number R, 0<R<1 if R< *-INT(*) inject a true secondary electron return

36

Electron-neutral collisions

37

Electron-neutral collisions

Collision processes: elastic scattering, excitation, ionization.

Assumptions: only two-particle collisions occur, only a small fraction of particles collides at each

timestep, a particle cannot collide more than once during

the timestep.

38

Probability of an electron-neutral collision is where is the collision frequency, t is the time interval, u is the particle velocity, T is the total scattering cross-section, nT is the density of neutrals (targets).

A straightforward and numerically ineffective way is to check whether each particle at each timestep makes a collision: calculate probability P, take a random number R, collision occurs if R<P.

Alternatively, if we know how many particles collide at each timestep, we can randomly choose these particles and perform collision procedures only for them.

Monte-Carlo model of electron-neutral collisions

)exp(1 tP TT nmuu )2/( 2

39

Calculate maximal frequency of collisions max and corresponding probability Pmax.

Calculate the number of colliding particles

Prepare the “accumulated probabilities” Pk.

At each timestep: select randomly Ncoll particles take random number R collision of type k occurs if

if the collision to occur is the null collision, do nothing.

The null collision methodTotal cross sections of elastic (1) excitation (2) and ionization (3) collisions in Xenon.

Accumulated “probabilities” of collisions [normalized]. The zero corresponds to the null collisions.

1;

;;;0

4max

3213

max

212

max

110

PP

PPP

[Vahedi and Surendra, Comput. Phys. Comm., 1995]

totcoll NPN max

kk PRP 1

40

Differential cross-section and the scattering angle are: [Surendra, Graves, and Jellum, Phys.Rev.A,

1990], w is the energy in eV

[Okhrimovskyy, Bogaerts, and Gijbels, Phys.Rev.E, 2002], is w/27.21

Scattering angle is .

Selection of scattering angles

vinc is the electron velocity before scattering, vsc is the velocity of scattered electron, and are the scattering angles.

)1ln()]2/(sin1[4)(),(

2 www

ww

www R)1(22cos

2cos441481

)(),(

)1(8121cos

RR

2R

Everywhere, R is a random number, 0<R<1.

41

Rotation and energy transfer

The vector velocity scattered by angles and is

To account for the energy transfer, the scattered velocity must be multiplied by factor vinc is the electron velocity

before scattering, vsc is the velocity of scattered electron, and are the scattering angles.

cossinsin||

sinsinsin

cosinc

incincincincsc v

kvvkvvv

cos121 Mm

42

Processing an inelastic (excitation) e-n collision The energy of the scattering electron decreases by the

excitation threshold. For example, for xenon it is wexc=8.32 eV.

The processing sequence is as follows: calculate the modified energy wsc = winc – wexc , calculate scattering angles and with wsc , rotate the initial velocity vector (use same formula as

in an elastic collision), multiply the rotated vector by a factor .

inc

exc

ww

1

43

Energies of primary and secondary electrons in ionization collisions Only electrons with energies above the ionization

threshold w>wion can do the ionization. For xenon wion=15eV.

The spectrum of secondary electrons is described as

Constant B is known from measurements (for example, B=8.7eV for xenon).

The energy of a secondary electron can be found as follows [from a mini-course “Particle-in-Cell Technique by J.P.Verboncoeur, 2002”]:

The energy of the scattered primary electron is then

Differential cross-section as a function of the secondary (ejected) electron energy. From [Opal, Peterson, and Beauty, J.Chem.Phys, 1971].

22

21

121

2arctan

)(),(wB

Bww

Bwwwion

i

BwwRBw ion

2arctantan 1

2

R is the random number, 0<R<1.

21 wwww ionscat

44

Processing an ionization collision Once the energies of scattered and

secondary electrons are found, the processing sequence is as follows: get the secondary electron velocity v2 :

calculate 2 and 2 , rotate v1 by angles 2 and 2 , multiply by factor (w2/w1)1/2;

get the scattered electron velocity vsc : calculate 1 and 1 , rotate v1 by angles 1 and 1 , multiply by factor (wsc/w1)1/2;

take ion velocity from the velocity distribution of the neutral gas.

Angular distribution of secondary electrons as function of their energy. From Opal, Peterson, and Beauty, J.Chem.Phys, 1971.

45

Collisions with neutrals are treated largely like scattering of spheres

46

Coulomb collisions need different approach The differential cross section for Coulomb scattering is [see e.g.,

Lieberman and Lichtenberg, Principles of plasma Discharges and Materials Processing]

Straightforward integration gives an infinite total cross section.

Compared to electron-neutral collisions, electron-electron [Coulomb] collisions are much more frequent and are characterized by small scattering angles.

Application of the traditional Monte-Carlo approach is both ineffective numerically and physically inappropriate.

)2/(sin)8(),( 44

022

0

42

0

vmeZv

R

0

0 sin),(2 dvtot

47

Electron-electron Coulomb collisions The effect of Coulomb collisions can be represented as a result of

dynamical friction and stochastic diffusion – scattering on many particles can be substituted by scattering off the grid [Jones, Lemons, Mason, Thomas, and Winske, J.Comput.Phys., 1996; Manheimer, Lampe, and Joyce, J.Comput.Phys.,1997].

)()(21)()(

2

vfvDvv

vfvFvt

fd

ee

)(4

)( 220

4

vHvm

nevFd

)(4

)(2

220

4

vGvvm

nevD

|~|)~(~2)( 3

vvvfvdvH

|~|)~(~)( 3 vvvfvdvG

Fokker-Planck equation for e-e scattering:

Dynamic friction

Velocity diffusion coefficient:

describes change of electron mean directed velocity.

describes electron spreading in the velocity space.

H and G are the first and the second Rosenbluth potentials [Rosenbluth, MacDonald, and Judd, Phys.Rev., 1957]

48

Drag force and diffusion coefficients

Calculations simplify if the EVDF is isotropic in the electron flow frame ue.

||

0

2222

0

4

)~(~~||

12||

)(euv

ee

ed wfwwd

uvmen

uvuvvF

euvw

~~

w

w

wfwwdwfwwwwdwm

enwnDwDwD0

222322

0

4

12211 )~(~~2)~()~3(~~13

)()()(

w

w

wfwwdwfwwdwm

enwnDwD0

4322

0

4

333 )~(~~)~(~~13

2)()(

The drag force

Velocity of a scatterer electron in the flow frame The diffusion coefficient tensor becomes diagonal in the [primed] frame where the 3rd axis is directed along : euv

49

Velocity correction due to diffusion

The Fokker-Planck equation is equivalent, to the first order of accuracy, to the Langevin equation:

QtFv d

tDQQ

tDQ

DDtQ

11

22

21

33

23

2/13311

2/3 2''

2'exp

)2(1)'(

)2/'()( 3332

'

3

3

tnDQgydydydyRQ

x

dxg )exp()( 22/1

)(2

2

tdtxd

dtxdm

The original Langevin equation describes Brownian motion:

drag force noise term

In the “primed” frame coordinates Q’1,2,3 of a vector Q’ correspond to the distribution

To find, for example, component Q’3, one has to solve equation

where R is a random number, 0<R<1, and .

To reduce numerical load, solution of R=g(x) can be tabulated.

50

Transformation to the laboratory frame

Vector Q’ must be transformed to the laboratory frame as follows:

z

y

x

z

y

x

QQQ

QQQ

'''

cossin0cossincoscossin

sinsinsincoscos

where

2222

22

cos,sin,cos,sinyx

y

yx

xzyx

ww

w

www

ww

www

and wx,y,z are the components of the scattering electron velocity in the flow frame.

euvw

51

Electron-electron Coulomb collisions, algorithm summary Calculate the electron flow velocity ue. Calculate the EVDF f(w) in the flow frame, w is the absolute

value of the velocity. Tabulate drag force coefficient Fd and the velocity diffusion

coefficients D1 and D3 as functions of w. For each electron

calculate velocity correction due to the drag force Fdt calculate velocity correction due to the velocity diffusion Q, apply the corrections, accumulate the kinetic energy of all electrons before (Wbefore) and

after (Wafter) the corrections are applied. For each electron

multiply the velocity by the factor (Wbefore/Wafter)1/2 to ensure energy conservation.

52

Electron-electron Coulomb collisions, examples

In this test, the Coulomb collisions transform the initially rectangular anisotropic EVDF into a Maxwellian isotropic EVDF.

Average energy of electron motion in x (1), y (2), and z (3) directions vs time.

Initial 1-D EVDFs.

Final1-D EVDFs.

Initial (left) and final (right) electron velocity phase space.

53

Electron-ion coulomb collisions The procedure is similar to the one for the electron-electron

collisions. The integrals in coefficients H and G are simplified because the ion velocity is much slower than the electron one.

Then

To avoid very large corrections for particles with small velocities, electrons with speed v<vthr are scattered off ions with constant coefficient Fd,thr=Fd(vthr), where the threshold is obtained from condition :

||,/1)( vvvvH vvG )(

222

0

4 14

)()(vm

envvvnF

vvvF dd

vmenwnDwDwD 1

4)()()( 22

0

4

12211 0)()( 333 wnDwD

)( thrdthr vtFnv 3/1

220

4

4

metnvthr

54

55

ReferencesC. Z. Cheng and G. Knorr, J. Comput. Phys., 22, 330-351 (1976).J. R. Boris and D. L. Book, J. Comput. Phys., 11, 38-69 (1973).E. Sonnendrucker, J. Roche, P. Bertrand, and A. Ghizzo, J. Comput. Phys., 149, 201-220 (1999).H. L. Berk and K. V. Roberts, Phys. Fluids, 10, 1595-1597 (1967).J. P. Boris, Princeton University, PPL Report MATT-769, March 1970.M. R. Gibbons and D. W. Hewett, J. Comput. Phys., 120, 231-247 (1995).A. B. Langdon, B. I. Cohen, and A. Friedman, J. Comput. Phys., 51, 107-138 (1983).D. Sydorenko, I. Kaganovich, Y. Raitses, and A. Smolyakov, Phys. Rev. Lett, 103, 145004 (2009).V. P. Gopinath, J. P. Veboncoeur, C. K. Birdsall, Phys. Plasmas, 5, 1535-1540 (1998).A. Dunaevsky, Y. Raitses, and N. J. Fisch, Phys. Plasmas, 10, 2574-2577 (2003).H. Seiler, J. Appl. Phys., 54, R1-R18 (1983).V. Vahedi and M. Surendra, Comput. Phys. Comm., 87, 179-198 (1995).M. Surendra, D. B. Graves, and G. M. Jellum, Phys.Rev.A, 41, 1112-1125 (1990).A. Okhrimovskyy, A. Bogaerts, and R. Gijbels, Phys. Rev. E, 65, 037402 (2002).J. P. Verboncoeur, Particle-in-Cell Technique, 2002.C. B. Opal, W. K. Peterson, and E. C. Beauty, J. Chem. Phys., 55, 4100-4106 (1971).M. A. Lieberman and A. J. Lichtenberg, Principles of plasma Discharges and Materials Processing,

published by John Wiley & Sons, Inc., Hoboken, NJ, 2005.M. E. Jones, D. S. Lemons, R. J. Mason, V. A. Thomas, and D. Winske, J. Comput. Phys., 123, 169-

181 (1996).W. M. Manheimer, M. Lampe, and G. Joyce, J. Comput. Phys., 138, 563-584 (1997).M. N. Rosenbluth, W. M. MacDonald, and D. L. Judd, Phys. Rev., 107, 1-6 (1957).