Charged particle kinetics by the Particle in Cell

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 1 e. V) than the neutral one (< 1000 K) Velocity and population distributions deviate from the equibrium laws i. e. Maxwell and Boltzmann respectively

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

I Plasma dynamics

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)

Particle in Cell (Pi. C) 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 Grid to particle Interpolation Dt Particle to grid Interpolation solve Poisson Equation for the electric potential Charge density

Ideal plasma Vlasov equation Particles propagate the initial condition moving along characteristic lines of the Vlasov equation

Particle/grid interpolation: linear Particle move: “leapfrog”

Plasma oscillation

II Plasma dynamics + Neutral particles and plasma interaction

Vlasov-Boltzmann equation

Lagrangian Particles as propagators Vlasov equation Initial d moves along characteristic lines --> deterministic method (PIC) Vlasov/Boltzmann equation medium tcoll ? v’ ? event “free flight” Dispersion of the initial d --> “choice” --> stochastic method (MC)

Statistical sampling of the linear collision operator (1) Sampling of a collision partner velocity w from the distribution F(r, w)/n (2) rejection of null collisions with probability 1 -ngs(g)/a max (3) kinematic treatment of the collision event for the charged+neutral particle system

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

Preliminary test: H 3+ in H 2 reduced mobilities of H 3+ ions as a function of E/n compared with experimental results of Ellis 2 (dots) 2 H. mean energy of H 3+ ions as a function of E/n W. Ellis, R. Y. Pai, E. W. Mc. Daniel, E. A. Mason and L. A. Vieland, Atomic Data Nucl. Data Tables 17, 177 (1976)

Example: H 3+/H 2 transport* in a thermal gradient = 500 K/cm, costant p = 0. 31 torr f(x, y, 0)=d(x) d(y) d(x) E/N=100 Td * only elastic collisions below about 10 e. V

7 ms no field 7 ms with E field 7 ms f(x, y) E field f(y) no field

Particle in Cell with Monte Carlo Collisions (Pi. C/MCC) method Monte Carlo Collisions Integration of equations of motions, moving particles E field Grid to particle Interpolation Dt Particle to grid Interpolation space charge solve Poisson Equation for the electric potential

Making the exact MC collision times compatible with the PIC timestep After R. W. Hockney, J. W. Eastwood, Computer Simulation using Particles, IOP 1988

Plasma turbulence due to charge exchange in Ar+/Ar (collaboration with H. Pecseli , S. Børve and J. Trulsen, Oslo) 2 component (e, Ar+) 1. 5 D PIC/MC 106 superparticles vx t=0 Initial beam: r = 4 1013 m-3 < e > = 1 e. V T = 100 K L = 0. 05 m Ar background: T = 100 K, p= 0. 3 torr The electron density is calculated as a Boltzmann distribution, this produces a nonlinear Poisson equation solved iteratively x

vx electrostatic repulsion inertia collisions x 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 v r

Simulation time. 2 10 -5 s

Capacitive coupled, parallel plate radio frequency (RF) discharge

negative charge strong oscillating field regions (sheaths) electrons electron density ambipolar potential energy well = -e negative charge

Simplified code implementation for nitrogen 2 particle species in the plasma phase: e, N 2+ more than one charged species

Selection of the collision process based on the cross section database Process probability = relative contribution to the collision frequency

Particle position/energy plot

ions electrons

electrons ions

ions electrons

III Plasma dynamics + Neutral particles and plasma interaction + Chemical kinetics of excited states

Kinetics of excited states

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

Boundary Conditions surface reactions j(wall) Poisson Equation absorption, sec. emission Reaction/Diffusion Equations electric field eedf electr. /ion density gas composition space charge Charged Particle Kinetics

Monte Carlo Collisions Chemical kinetics equations Integration of equations of motions, moving particles E field Grid to particle Interpolation Particle to grid Interpolation solve Poisson Equation for the electric potential Space charge

code implementation for hydrogen 5 particle species in the plasma phase: e, H 3+, H 2+, H 16 neutral components: H 2(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 + H 2(v=0) e +H 2(v=1, …, 5) e + H 2(v=1, …, 5) e +H 2(v=0) H 2(v) + H 2(w) H 2(v-1) + H 2(w+1) H 2(v) + H 2(v-1) + H 2(v) + H 2(v+1) + H 2(v) + H H 2(w) + H e + H 2(v=0, …, 14) H + He + H 2(v) e + H 2(v’) (via b 1 u+, c 1 u) e + H 2 e +2 H(via b 3 u+, c 3 u, a 3 g+, e 3 u+) e + H 2 H + H+ + 2 e e + H 2+ + 2 e H 2+ + H 2 H 3+ + H (fast) H 2(v>0) – wall H 2(v=0) H – wall 1/2 H 2(v) e + H 2 e + H+ e + H 2 e + H(n=2 -3) e + H- 2 e + H

secondary ions from: 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, g. H = 0. 02 primary positive ions

relatively low T 01 (~1000 K) plateau due to radiative EV processes

eedf

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 g. H = 0. 0033 g. 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 ….