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
106
107
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
400
800
1200
1600
2000
2400
2800
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
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