Savino Longo

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Charged particle kinetics by the Particle in Cell
/ Monte Carlo method

• Savino Longo
• Dipartimento di Chimica dellUniversità di Bari
  and IMIP/CNR

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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
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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 (lt 1000K)
• Velocity and population distributions deviate
  from the equibrium laws i.e. Maxwell and
  Boltzmann respectively

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Items to be included in a comprehensive model
Plasma dynamics
Neutral particles and plasma interaction
Chemical kinetics of excited states
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I
Plasma dynamics
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The problem of plasma dynamics

• 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)

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Particle in Cell (PiC) method
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
Integration of equations of motions, moving
particles
E field
Particle to grid Interpolation
Grid to particle Interpolation
t
D
Charge density
solve Poisson Equation for the electric potential
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Ideal plasma
Vlasov equation
Particles propagate the initial condition moving
along characteristic lines of the Vlasov equation
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Particle/grid interpolation linear
Particle move leapfrog
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Plasma oscillation
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II
Plasma dynamics Neutral particles and plasma
interaction
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Vlasov-Boltzmann equation
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Lagrangian Particles as propagators
Vlasov equation
Initial d moves along characteristic lines
--gt deterministic method (PIC)
Vlasov/Boltzmann equation
medium
Dispersion of the initial d --gt choice
--gt stochastic method (MC)
tcoll ?
v ?
event
free flight
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Statistical sampling of the linear collision
operator
(3) kinematic treatment of the collision event
for the chargedneutral particle system
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Test particle Monte Carlo
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
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
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Preliminary test H3 in H2
reduced mobilities of H3 ions as a function of
E/n compared with experimental results of Ellis2
(dots)
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)
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Example H3/H2 transport in a thermal gradient
500 K/cm, costant p 0.31 torr
E/N100 Td
f(x,y,0)d(x) d(y) d(x)
only elastic collisions below about 10eV
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f(x,y)
f(y)
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Particle in Cell with Monte Carlo Collisions
(PiC/MCC) method
Integration of equations of motions, moving
particles
E field
Particle to grid Interpolation
Grid to particle Interpolation
t
D
space charge
solve Poisson Equation for the electric potential
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Making the exact MC collision times
compatible with the PIC timestep
After R.W.Hockney, J.W.Eastwood, Computer
Simulation using Particles, IOP 1988
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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/MC 106 superparticles
vx
t 0
Initial beam r 4 1013 m-3 lt e gt 1eV T
100 K L 0.05 m Ar background T 100K, p
0.3torr
x
The electron density is calculated as a Boltzmann
distribution, this produces a nonlinear Poisson
equation solved iteratively
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vx
x
The collisional production of the second (rest)
ion beam can lead to a two stream instability
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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

v
r
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Capacitive coupled, parallel plate radio
frequency (RF) discharge
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Simplified code implementation for nitrogen 2
particle species in the plasma phase e, N2
more than one charged species
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Selection of the collision process based on the
cross section database
Process probability relative contribution to
the collision frequency
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Particle position/energy plot
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ions
electrons
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electrons
ions
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ions
electrons
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III
Plasma dynamics Neutral particles and plasma
interaction Chemical kinetics of excited states
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Kinetics of excited states
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Numerical treatment of state-to-state chemical
kinetics of neutral particles (steady state)
(1) gas phase reactions
E.g.
are included by solving
(2) gas/surface reactions
E.g.
are included by setting appropriate boundary
conditions
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Boundary Conditions
Poisson Equation
space charge
electric field
Charged Particle Kinetics
Reaction/Diffusion Equations
eedf
electr./ion density
gas composition
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Chemical kinetics equations
Integration of equations of motions, moving
particles
E field
Particle to grid Interpolation
Grid to particle Interpolation
Space charge
solve Poisson Equation for the electric potential
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code implementation for hydrogen 5 particle
species in the plasma phase e, H3, H2, H,
H- 16 neutral components H2(v0 to 14) and H
atoms
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• 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(v0) ? e H2(v1,,5) e H2 ?
H H 2e e H2(v1,,5) ? e H2(v0) e
H2 ? H2 2e H2(v) H2(w) ? H2(v-1) H2(w1)
H2 H2 ? H3 H (fast) H2(v) H2 ? H2(v-1)
H2 H2(vgt0) wall ? H2(v0) H2(v) H2 ?
H2(v1) H2 H wall ? 1/2 H2(v) H2(v) H
? H2(w) H e H ? 2e H e H2(v0,,14)
? H H- e H2 ? e H H(n2-3) e H2(v) ?
e H2(v) (via b1?u, c1?u) e H- ? 2e
H e H2 ? e 2H(via b3?u, c3?u, a3?g, e3?u)

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charged particle density
Simulation parameters Tg 300 K Vrf 200 V p
13.29 Pa (0.1 torr) nrf 13.56 MHz L 0.06 m,
Vbias 0 V gv 0.65, gH 0.02
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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
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Bias voltage
p 0.3 torr L 0.03 m gH 0.0033 gV 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)
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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.

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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
• .