Computational Plasma Physics

Kinetic modelling: Part 2

W.J. Goedheer

FOM-Instituut voor PlasmafysicaNieuwegein, www.rijnh.nl

Monte Carlo methods

Principle: Follow particles by - solving Newton’s equation of motion - including the effect of collisions - collision: an event that instantaneously changes the velocity

Note: The details of a collision are not modeled Only the differential cross section + effect on energy is used

Example: Electrons in a homogeneous electric field Follow sufficient electrons for a sufficient time Obtain distribution over velocities etc. f0,f1

Monte Carlo methods: Equation of motion

Leap-frog scheme

Monte Carlo methods: B-field

Problem with Lorentz force: contains velocity, needed at time tSolution: take average

The new velocity at the right hand side can be eliminated by taking the

cross product of the equation with the vector

Monte Carlo methods: Boris for B-field

Equivalent scheme (J.P.Boris), (proof: substitution):

Monte Carlo methods: Collisions

Number of collisions: NMtot = 1/ per meter.

(x) = (0)*exp(- NMx) = (0)*exp(-x/)

dP(x)=fraction colliding in (x,x+dx)=exp(-x/)(1-exp(-dx/))=(dx/)exp(-x/)

P(x)=(1-exp (-x/))

Distance to next collision: Lcoll=-*ln(1-Rn) (Rn is random number,0<Rn<1)

Number of collisions: NMtot v= 1/ per second.

Time to next collision: Tcoll=-* ln(1-Rn)

Monte Carlo methods: CollisionsAnother approach is to work with the chanceto have a collision on vt: Pc=vt/

Ensure that vt<< to have no more than one collision per timestep Effect of collision just after advancing position or velocity introduces only small error

When there is a collision:

Determine which one: new random number

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

fracti

on

L/

no collision colliding once colliding twice colliding> twice

Monte Carlo methods: Null Collision

Problem: Mean free path is function of velocity Velocity changes over one mean free path

Solution: Add so-called null-collision to make v*tot independent of v Null-collision does nothing with velocity

Mean free path thus based on Max (v*tot)

Is rather time-consuming when v*tot peaks strongly

Monte Carlo methods: Null Collision

v

v*

v*2

v*1

v*3

v*tot

Max

v*0

1

1+2

1+2+3

1+2+3+..N

1+2+3+..N+ 0

’s normalized to maximum:Draw random number

Monte Carlo methods: Effect of collision

Determine effect on velocity vector

Retain velocity of centre of gravity

Select by random numbers two angles of rotation for relative velocity

Subtract energy loss from relative energy

Redistribute relative velocity over collision partners

Add velocity centre of gravity

Monte Carlo methods: Effect of collision

v1,v2 velocities in lab-frame prior to collision, w1,w2 in center of mass system

Monte Carlo methods: Effect of collision

A collision changes the size of the relative velocity if it is inelastic

A collision rotates the relative velocity

Two angles of rotation: and

usually has an isotropic distribution: =Rn*

has a non-isotropic distribution

Hard spheres:

Monte Carlo methods: Rotating the relative velocity

Step 1: construct a base of three unit vectors:

Step 2: draw the two angles

Step 3: construct new relative velocity

Step 4: construct new velocities in center of mass frame

Step 5: add center of mass velocity

Monte Carlo methods: Applicability

Examples where MC models can be used are:

- motion of electrons in a given electric field in a gas (mixture)- motion of positive ions through a RF sheath (given E(r,t))

Main deficiency: not selfconsistent

- electric field depends on generated net electric charge distribution- current density depends on average velocities- following all electrons/ions is impossible

Way out: Particle-In-Cell plus Monte Carlo approach

Particle-In-Cell plus Monte Carlo: the basics

-Interactions between particle and background gas are dealt with only in collisions

-this means that PIC/MC is not! Molecular Dynamics

-each particle followed in MC represents many others: superparticle

-Note: each “superparticle” behaves as a single electron/ion

-Electric fields/currents are computed from the superparticle densities/velocities

-But: charge density is interpolated to a grid, so no “delta functions”

Particle-In-Cell plus Monte Carlo: Bi-linear interpolation

xi=ix xi+1=(i+1)x

xs, qs=eNs

i:=i+(xi+1-xs)qs/xi+1:=i+1+(xs-xi)qs/x

zi=iz

zi+1=(i+1)z

xj=jx xj+1=(j+1)x

ij:=ij+(zi+1-zs) (xj+1-xs) qs/(x z)

zs

xs

Particle-In-Cell plus Monte Carlo:Solution of Poisson equation

Boundary conditions on electrodes, symmetry, etc.

Electric field needed for acceleration of particle:(bi)linear interpolation, field known in between grid points

Particle-In-Cell plus Monte Carlo:Full cycle, one time step

Collisionnew v

Interpolatecharge to grid

Solve Poissonequation

Interpolate fieldto particle

Check lossat the walls

Move particlesF v x

Particle-In-Cell plus Monte Carlo:Problems

Main source of problems: Statistical fluctuations

Fluctuations in charge distribution: fluctuations in Eaverage is zero but average E2 is not numerical heating

Sheath regions contains only few electrons

Tail of energy distribution contains only few electronslarge fluctuations in ionization rate can occur

Particle-In-Cell plus Monte Carlo:Problems

Solutions:

-Take more particles (NB error as N-1/2 ) , parallel processing!

-Average over a long time

-Split superparticles in smaller particles when neededrequires a lot of bookkeeping, different weights!

Particle-In-Cell plus Monte Carlo:Stability

Plasmas have a natural frequency for charge fluctuations:

The (angular) Plasma Frequency:

And a natural length for shielding of charges:

The Debye Length:

Stability of PIC/MC requires:

Power modulated discharges

Modulate RF voltage (50MHz)with square wave (1 - 400 kHz)

Observation in experiments UU)optimum in deposition rate

Modulated discharges

Results from a PIC/MC calculation: Cooling and high energy tail

1-D Particle-In-Cell plus Monte Carlo Simulationof a dusty argon plasma

Dust particles with a homogeneous density distribution are present in two layers

This resembles certain experiments done under micro-gravity

Dust particles do not move, they only collect and scatter plasma ions and electrons

The charge of the dust results from the collection process

The charge of the dust is defined on the grid needed for the Poisson equation

RF

Void

Crystal (21010 m-3)7.5 m radius

1-D Particle-In-Cell plus Monte Carlo Simulationof a dusty argon plasma

Capture cross section

Scattering:Coulomb, truncated at d

L/4L/8

w is energy electron/ion

Charging of the dust upon capture of ion/electron

The total charge is monitored on the gridpointsCharge of captured superparticle is added to nearest gridpointsDivision according to linear interpolationSuperparticle is removedLocal dustparticle charge is total charge divided by nr. of dust particlesThis number is: density*dz*a2, with a the electrode radiusFor Monte Carlo the maximum v is computed for all available dust particle chargesNull-collision is used

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00

2.50x1015

5.00x1015

7.50x1015

1.00x1016

1.25x1016

1.50x1016

Ne

N+

Den

sity

(m

-3)

x/L

Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1x1015

2x1015

3x1015

4x1015

5x1015

6x1015

NdQ

d/e

Ne

N+

Den

sity

(m

-3)

x/L

0 5 10 15 20 25 30 35 40100

101

102

103

104

105

106

107

108

L/2 L/4 3L/16 L/8 L/16

EE

DF

(ar

b.u

n.)

Energy (eV)

0 5 10 15 20 25 30 35 40100

101

102

103

104

105

106

107

108

L/2 L/4 L/8 L/16 L/32

EE

DF

(ar

b.u

n.)

Energy (eV)

dustfree with dust

Vd6V

Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cm

dustfree with dust

0 5 10 15 20 25 30 35 40100

101

102

103

104

105

106

107

108

109

L/2 L/8 L/16 L/32 0

IED

F (

arb

.un

.)

Energy (eV)

0 5 10 15 20 25 30 35 40100

101

102

103

104

105

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108

109

0 L/16 L/8 3L/16 L/4 L/2

IED

F (

arb

.un

.)

Energy (eV)

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.0000.0

0.5

1.0

1.5

2.0

2.5

3.0

Av. El. Energy

Ion.Rate

3kT

e/2 (

eV),

Ion

.Rat

e (a

rb.u

n.)

x/L

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.0000.0

0.5

1.0

1.5

2.0

2.5

3.0

Av. El. Energy Ion.Rate

3kT

e/2 (

eV),

Ion

.Rat

e (a

rb.u

n.)

x/L

Simulation for Argon, 50MHz, 100mTorr, 70V, L=3cmGeneration of internal space charge layers

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-1x1014

0

1x1014

2x1014

3x1014

Net

ch

arg

e / e

x/L

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000-20000

-15000

-10000

-5000

0

5000

10000

15000

20000

Av.

Ele

ctri

c F

ield

(V

/m)

x/L

An internal sheath is formedinside the crystal

Ions are accelerated beforethey enter the crystal

This has consequences forthe charging + shielding

Particle-In-Cell plus Monte Carlo:What if superparticles collide?

Example: recombination between positive and negative ions

Procedure: number of recombinations in t: N+N-Krec t

corresponds to removal of corresponding superparticles randomly remove negative ion and nearest positive ion but: be careful if distribution is not homogeneous

A more sophisticated approach: Direct Simulation Monte Carlo

DSMC: Basics

Divide the geometry in cells

Each cell should contain enough testparticles (typically 25)

Newton’s equation: as before, but keep track of cell number

Collisions: choose pairs (in same cell!) and make them collide

Essential: the velocity distribution function is sum of -functions Only small fraction of pairs collides in one time step

DSMC: Choosing the pairs

Add null collision

Chance of collision of particle i with j is Pc=(Npp/Vcell)*Max(v)t

Number of colliding pairs: n(n-1)* Pc/2

Select randomly particle pairs (make sure no double selection)

See if there is no null collision (again with random number)

Perform the collision

DSMC: An example

0 16 32 48 640

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3200

# p

art

icle

s

Energy (arb.un.)

25 50 75 100

Relaxation of a mono-energetic distribution to equilibrium20000 particles, hard sphere collisions. All particles are inthe same cell. Distribution at various time steps