Charged particle kinetics by the Particle in Cell / Monte

Carlo method

Savino Longo

Dipartimento di Chimica dell’Università di Bari and IMIP/CNR

The system under examination

A gas can be ionized under non equilibrium conditions (too low temperature for equilibrium ionization) with constant energy

dissipation, like in electric discharges, photoionized media, preshock regions, and so on.

The result is a complex system where the nonlinear plasma dynamics coexists with chemical kinetics, fluid dynamics,

thermophysics and chemical kinetics issues

Basic phenomenology

The gas is only weakly ionized

Molecules are only partially dissociated and exhibit their chemical properties

The electron temperature is considerably higher (about 1eV) than the neutral one (< 1000K)

Velocity and population distributions deviate from the equibrium laws i.e. Maxwell and Boltzmann respectively

Items to be included in a comprehensive model

Neutral particles and plasma interaction

Chemical kinetics of excited states

Plasma dynamics

Plasma dynamics

II

The charged particle motion is affected by the electric field, but the electric field is influenced by the space distribution of

charged particles (space charge)

The problem of plasma dynamics

F qE / m

2 4

t

solve Poisson Equationfor the electric potential

Integration of equationsof motions, moving particles

Particle to gridInterpolation

Grid to particleInterpolation

The method is based on the simulation of an ensemble of mathematical “particles” with adjustable charge which move like real particles and a

simultaneous grid solution of the field equation

Charge density

E field

Particle in Cell (PiC) method

Vlasov equation

3 3

11

sim D sim D

Vg

n N

2 30

( , , ) 0

ion

ef t

t m

e fd v

r v

Ev r v

E

Ideal plasma

Particles propagate the initial condition moving along characteristic

lines of the Vlasov equation

Particle/grid interpolation: linear

Particle move: “leapfrog”

v vqEm

t

x x v t

Dx

1/ plt

( ) ( )p pp

q i x q S x i x

Plasma oscillation

Plasma dynamics+

Neutral particles and plasma interaction

IIII

Vlasov-Boltzmann equation

2 30

3max

max

3 3

3

( , , )

11 / ( )

( , ; , ) ( , )

( , )

ion

v v

v v

v v

ef t Cf

t m

e fd v

Cf f d vp f

p d wd w v w v w gF

d v p

r v

Ev r v

E

v

r w

r v

Vlasov equation

Vlasov/Boltzmann equation

medium

event

“free flight”

Initial moves alongcharacteristic lines -->deterministic method

(PIC)

Dispersion of the initial -->“choice” -->

stochastic method(MC)

Lagrangian Particles as propagators

tcoll ?

v’ ?

(1) Sampling of a collision partner velocity w from the

distribution F(r,w)/n

(2) rejection of null collisions with probability 1-ng(g)/ max

Statistical sampling of the linear collision operator

(3) kinematic treatment of the collision event for the

charged+neutral particle system

Af 1 r,v max

f r ,v 1

maxd3 v p v v f v

p v v d3wd3 w ( v ,w ;v, w )| v w| F(r ,w)

A ‘virtual’ gas particle is generated as a candidate collision partner based on the

local gas density and temperature.

The collision is effective with a probability

Test particle Monte Carlo

ngasg

max(ngasg)

For an effective collision the new velocity of the charged particle is calculated

according to the conservation laws and the differential cross section

A random time to the next candidate collision is generated

Preliminary test: H3+ in H2

reduced mobilities of H3+ ions as

a function of E/n compared with experimental results of Ellis2 (dots)

11

12

13

14

15

16

17

10 1 10 2

K0 (c

m2 V

-1s

-1)

E/n(Td)

T=600 K

T=300 K

10 -2

10 -1

10 0

10 1

10 100

mea

n en

ergy

(eV

)E/n(Td)

T=600 K

T=300 K

mean energy of H3+ ions as a

function of E/n

2H. W. Ellis, R. Y. Pai, E. W. McDaniel, E. A. Mason and L. A. Vieland, Atomic Data Nucl. Data Tables 17, 177 (1976)

Example: H3+/H2 transport* in

a thermal gradient = 500 K/cm, costant p = 0.31 torr

f(x,y,0)=(x) (y) (x) E/N=100 Td

* only elastic collisions below about 10eV

7s no field 7s withE field

0

0,2

0,4

0,6

0,8

1

1,2

-2 -1 0 1 2

h(y)

(cm

-1)

y (cm)

7s no field

E field

f(x,y)

f(y)

t

solve Poisson Equationfor the electric potential

Integration of equationsof motions, moving particles

Particle to gridInterpolation

Grid to particleInterpolation

space charge

E field

Monte Carlo Collisions

Particle in Cell with Monte Carlo Collisions (PiC/MCC) method

After R.W.Hockney, J.W.Eastwood, Computer Simulation using Particles, IOP 1988

Making the exact MC collision times compatiblewith the PIC timestep

Plasma turbulence due to charge exchange in Ar+/Ar(collaboration with H.Pecseli , S. Børve and J.Trulsen, Oslo)

2 component (e,Ar+) 1.5D PIC/MC106 superparticles

Initial beam: = 4 1013 m-3

< > = 1eV T = 100 KL = 0.05 m

Ar background:T = 100K, p= 0.3torr

vx

x

t = 0

The electron density is calculated as a Boltzmann distribution, this produces a

nonlinear Poisson equation solved iteratively

14e

2

nion ne0 exp(e / kTe)

vx

x

inertia

collisions

electrostatic repulsion

The collisional production of the second (rest) ion beam can lead to a two stream instability

Two stream instability

The propagation of two charged particle beams in opposite directions is unstable under density/velocity perturbations and can

lead to plasma turbulence

vv

rr

2 1(v v ) pl

0.050 0.100 0.150 0.200 0.250

-2000

0

2000

4000

vx (m/s)

x (m)

1 10 (log)

Simulation time. 2 10-5 s

VRFsin(2RFt)

Capacitive coupled, parallel plate radio frequency (RF) discharge

strong oscillating fieldregions (sheaths)

ambipolar potential energy well = -e

electron density

electrons

negativecharge

negativecharge

Simplified code implementation for nitrogen

2 particle species in the plasma phase: e, N2+

more than one charged speciesmore than one charged species

Process probability = relative contribution to the collision frequency

Selection of the collision process based on the cross section database

Particle position/energy plot

1013

1014

1015

1016

0 0.01 0.02 0.03 0.04

Vrf = 500 V, p = 0.1 torr, f =13.5 MHzele

ctro

n a

nd ion d

ensi

ty (

m

-3)

position (m)

ions

electrons

0.1

1

10

0 0.01 0.02 0.03 0.04

Vrf = 500 V, p = 0.1 torr, f =13.5 MHzele

ctro

n/ion m

ean e

nerg

y (

eV

)

position (m)

ions

electrons

-1 104

-5000

0

5000

1 104

0 0.01 0.02 0.03 0.04

Vrf = 500 V, p = 0.1 torr, f =13.5 MHz

ele

ctro

n/ion d

rift

velo

city

(m

/s)

position (m)

ions

electrons

Plasma dynamics+

Neutral particles and plasma interaction+

Chemical kinetics of excited states

IIIIII

Kinetics of excited states

e A e A*

A* A hA*B A(h)B

e A* e(h) A

e A* 2e A

A2* A2(v) h

A*B A B

e A2(v) A B

rcXcc1

N c rcXcc1

Nc

1rNr

Numerical treatment of state-to-state chemical kinetics of neutral particles (steady state)

(1) gas phase reactions:

are included by solving:

Dc2nc x x2

rc rc kr fe t r n c

rc

c

E.g.:

HH2(X,v) HH2(X, v )

(2) gas/surface reactions:

As r1A1r2A2 ... E.g.:

H(wall) 1

2H2

s1

4

8KTms

s

Dss

rss

r r s

r s rs s

are included by setting appropriate boundary conditions

BoundaryConditions

Poisson Equation

Reaction/DiffusionEquations

Charged Particle Kinetics

space charge

eedfelectr./iondensity

electricfield

gas composition

surfacereactions

(wall)

absorption,sec.emission

solve Poisson Equationfor the electric potential

Integration of equationsof motions, moving particles

Particle to gridInterpolation

Grid to particleInterpolation

Spacecharge

E field

Monte Carlo Collisions

Chemicalkinetics

equations

code implementation for hydrogen

5 particle species in the plasma phase: e, H3+, H2

+, H+, H-

16 neutral components: H2(v=0 to 14) and H atoms

Charged/neutral particle collision processes

electron/molecule and electron/atom elastic, vibrational and electronic inelastic collisions, ionization, molecule dissociation, attachment, positive ion/molecule elastic and charge exchange collisions, positive elementary ion conversion reactions, negative ion elastic scattering, detachment, ion neutralization

Schematics of the state-to-state chemistry for neutrals

e + H2(v=0) e +H2(v=1,…,5) e + H2 H + H+ + 2e

e + H2(v=1,…,5) e +H2(v=0) e + H2 H2+ + 2e

H2(v) + H2(w) H2(v-1) + H2(w+1) H2+ + H2 H3

+ + H (fast)

H2(v) + H2 H2(v-1) + H2 H2(v>0) – wall H2(v=0)

H2(v) + H2 H2(v+1) + H2 H – wall 1/2 H2(v)

H2(v) + H H2(w) + H e + H 2e + H+

e + H2(v=0,…,14) H + H- e + H2 e + H + H(n=2-3)

e + H2(v) e + H2(v’) (via b1u+, c1u) e + H- 2e + H

e + H2 e +2H(via b3u+, c3u, a

3g+, e3u

+)

Simulation parameters:

Tg = 300 K

Vrf = 200 V

p = 13.29 Pa (0.1 torr)

rf = 13.56 MHz

L = 0.06 m, Vbias = 0 V

v = 0.65, H = 0.02

1012

1013

1014

1015

0 0,01 0,02 0,03 0,04 0,05 0,06

num

ber

dens

ity

(m-3

)

position (m)

H2

+

e

H3

+

H+

H-

-

primary positive ions

H2 H2 H3

Hsecondary ions from:

charged particle density

1012

1014

1016

1018

1020

1022

0 2 4 6 8 10 12 14

0.6 cm1.2 cm1.8 cm2.4 cm

num

ber

dens

ity (

m-3

)

vibrational quantum number

plateau due toradiative EV processes

relatively lowT01 (~1000K)

eedf

10-6

10-5

10-4

10-3

10-2

10-1

100

0 5 10 15 20 25 30 35 40

0.6 cm1.2 cm1.8 cm2.4 cm3.0 cm

eed

f (e

V-3/2)

energy (eV)

Double layer

O. Leroy, P. Stratil, J. Perrin, J. Jolly and P. Belenguer, “Spatiotemporal analysis of the double layer formation in hydrogen radio frequencies discharges”, J. Phys. D: Appl. Phys. 28 (1995) 500-507

Bias voltage

p = 0.3 torr L = 0.03 m H = 0.0033 V = 0.02

A. Salabas, L. Marques, J. Jolly, G. Gousset, L.L.Alves, “Systematic characterization of low-pressure capacitively coupled hydrogen discharges”, J. Appl. Phys. 95 4605-4620 (2004)

Conclusion

A very detailed view of the charged particle kinetics in weakly ionized gases can be obtained by Particle in Cell simulations including Monte Carlo collision of

charged particle and neutral particles.

Items to study in the next future (students)

Charge particle kinetics in complex flowfields

Collective plasma dynamics in shock waves

Development of new MC methods for electrons matching the time scale for electron heating

….