Monte Carlo Collisions in Particle in Cell simulations Konstantin Matyash, Ralf Schneider HGF-Junior research group COMAS : Study of effects on materials in contact with plasma, either with fusion or low-temperature plasmas; Development of computational multi-scale tools
Particle in Cell model dx = v dt dv q = E + v B dt m ( ) Δt E = ϕ 1 Δ ϕ = ε E ρ ( x ) E qi ρ = S x x i i ΔxΔy x y 1 1 S( x) Δx Δy = x Δx and y Δy ( ) E E
Two-particle interaction (1D) David Tskhakaya, Heraeus-Sommer School "Computational Many Particle Physics, Greifswald 18.9.6
Coulomb collisions Motivation: 1. Classical PIC simulates only 5 4 PIC 1D D 3D macro fields and neglects particle collisions. 3 ~1/r ~1/r. Inside grid cells the interaction between particles deviates from the Coulomb law 1 ~r ~const..4.6.8 1 r/ Δr Interaction force between two particles inside the grid cell
Fokker-Planck Coulomb collision model Test particle hitting the field particles with distribution function f ( v ) friction Δ v = + ( 1 m ) L φ v and diffusion Δv Δ v = L ϕ v v φ ( v ) 1 = 4π f v ( v) 3 d v v v 1 8 v v π f v d v, ( ) = ( ) 3 ψ are Rosenbluth potentials ( ε ) L = e e / m lnλ ln Λ Coulomb logarithm B. A. Trubnikov, Particle interactions in a fully ionized plasma, Reviews of Plasma Physics 1 (1965) 15-4. M. N. Rosenbluth, W. M. MacDonald and D. L. Judd, Fokker-Planck Equation for an Inverse-Square Force, Phys. Rev. 17 (1957) 1-6
Fokker-Planck Coulomb collision model If the distribution of the field particles is Maxwellian : the friction and diffusion coefficients can be calculated analytically: mv 3 f (v ) = ( π kt m ) exp kt friction: ( 1 m ) ( ) Δ v = A + n G l v A Δ v = n l G( l v) Δ v = n H ( l v) diffusion:, v = v v A v G( x) erf ( x ) x erf ( x ) = H ( x) = erf ( x) G( x) x ln Λ,,, 4 A = 8π e m l = m T R. Chodura, Particle simulation of the plasma-wall transition. Proceedings of the 8th Europhysics Conference on Computational Physics, Computing in Plasma Physics, Garmisch, 1986
Fokker-Planck Coulomb collision model Treating diffusion as a Brownian motion we can calculate the change of particle velocity : Δ v = Δv Δ t + R Δv Δt Δ v = R 3 5. Δv Δt x 1 y,z, Δt is the time step Random numbers R 13,, are taken from Gaussian distribution with R i = R = 1 i If distribution is not Maxwellian the energy and momentum are not conserved! for particles of the same kind = after collision velocity of Center of Mass and the temperature are not the same u u T 1 T 1 CM CM Solution force the momentum and the energy conservation by hand: after collisions for every particle we subtract the difference in the center of mass velocities v ( 1,i = v,i ucm ucm ) and scale it to ensure energy conservation v,i = ucm + ( v,i ucm ) T T 1 Introducing large error if distribution is far from Maxwellian!
Takizuka-Abe binary Coulomb collision model Takizuka-Abe approach nonlinear collisional operator! T. Takizuka, H. Abe, A binary collision model plasma simulation with a particle code, J. Comput. Phys. 5 (1977) 5. Binary collision model 1. grouping the particle in cells. randomly changing the particles order inside the cells 3. colliding the particles (3a. the same type; 3b. - different types)
Takizuka-Abe binary Coulomb collision model v 1 v Sampling Fokker-Planck collisional operator: θ θ P( θ ) = exp θ θ Δt u = v v 1 Δt relative velocity θ μ = qqn lnλ = Δt 3 8πε m u R. Shanny, J. M. Dawson, and J. M. Greene, One-dimensional model of a Lorentz plasma, Phys. Fluids, 1 (1967) 181 m mm + m reduced mass u ϕ u t+δt θ = θ ln R Δt 1 ϕ=π R scattering angle azimuth angle Random numbers R 1, are taken from uniform distribution with m = + Δ m v 1 = v Δu m + m v1 v1 u m1 + m 1 1 R i θ Scheme explicitly conserves energy and momentum! qqn lnλ Limitation: ν νc = cδt 1 θ 1 3 πεmvt
Takizuka-Abe binary Coulomb collision model Relaxation of non-maxwellian distribution Relaxation of Maxwellian distribution with different parallel and perpendicular temperatures m(v + v ) 3 mv f (v ) = ( π k m e ) T T exp kt kt e x y e z EVDF, a.u. 6 5 4 ν t = ν t =. ν t = 1 ν t = 3 Gauss ΔT/ΔT 1..8 PIC MCC result analytical result 3.6.4 1. -4-3 - -1 1 3 4 v ex, v te. 1 3 4 5 ν i t
Nanbu binary Coulomb collision model Nanbu approach optimization of Takizuka-Abe scheme ( accumulation of many small angle collisions ) K. Nanbu, Theory of cumulative small-angle collisions in plasmas Phys. Rev. E 55 (1997). u = v 1 v u,u,u...u 1 N uu cos( θ N ) = u coth N ( A) A 1 = exp ( s) relative velocity after N collisions accumulated scattering angle after N collisions Accumulated scattering angle is sampled from the distribution allows larger Δt initial relative velocity qqn lnλ s = Δt 3 4πε m u in practice, in order to resolve relaxation dynamics νcδt 1 f ( N ) ( θ ) = exp A cos ( θ ) N A 4π sinh( A) Pro: lower computational cost Contra: complicated implementation
Collision happens with probability σ ( E ) Sampling the collision event: Max-Planck Institut für Plasmaphysik, EURATOM Association, 17491 Greifswald, Germany - cross-section of the process R Collisions with neutrals 1 < ( ( ) ) P = exp un σ E Δt P 1 col Random number is taken from uniform distribution 1 R 1 R i u ϕ u t+δt u = v v 1 Elastic collisions relative velocity energy in the Center of Mass system: E CM = m, u μ = as no energy is lost in the elastic collision we just need to rotate the relative velocity m mm + m - reduced mass no scattering angle distribution data is available for most of the species we use isotropic scattering θ cos ( ) θ = 1 R ϕ=π R 3 scattering angle for isotropic scattering azimuth angle Random numbers R 3 are taken from uniform distribution with, 1 R i m = + Δ m v 1 = v Δu m + m v1 v1 u m1 + m 1
Collisions with neutrals Ionization collision e - + N => e - + N + Splitting 3-body process in two -body processes: u ϕ θ u t+δt 1. Inelastic uniform isotropic electron-neutral collision with loss of energy E i mm mm u = u + E e n e n t+δt i e n e n ( m + M ) ( m + M ) cos ( ) θ = 1 R ϕ=π R 3 electron velocity after 1st step: scattering angle for isotropic scattering azimuth angle energy conservation in the Center of Mass system Random numbers R 3 are taken from uniform distribution with, dividing neutral into the ion and the secondary electron: 1 * M i t+δt m e ve = ve + Δu neutral velocity: vn = vn u me + M Δ n me + Mn t t t t v +Δ +Δ = v v = v t+δt i n e" n R i final velocity of primary electron:. Elastic uniform isotropic electron-electron collision * u = v e ve" u is rotated according to isotropic scattering t t * v +Δ e = v 1 t t e + Δu v +Δ e" = v 1 final velocity of secondary electron: e" Δu Scheme explicitly conserves particles, energy and momentum!
Reactions included in methane model ee - - + CH CH 4 4 CH CH 4+ + 4+ e e - - ee - - + CH CH 3 3 CH CH 3+ + 3+ e e - - ee - - + CH CH CH CH + + + e e - - ee - - + CH CH CH CH + + +e +e - - ee - - + CH CH 4 4 CH CH 3+ + 3+ H + e e - - ee - - + CH CH 3 3 CH CH + + + H + e e - - ee - - + CH CH CH CH + + H + e e - - ee - - + CH CH C + + H + e e - - ee - - + CH CH C + H + + e e - - ee - - + CH CH 4 4 CH CH 3 + 3 H + ee - - ee - - + CH CH 3 3 CH CH + H + ee - - ee - - + CH CH CH CH + H + ee - - ee - - + CH CH C + H + ee - - ee - - + CH CH 4+ 4+ CH CH 3 + 3 H + + ee - - ee - - + CH CH 4+ 4+ CH CH 3+ + 3+ H + ee - - ee - - + CH CH 3+ 3+ CH CH + H + + ee - - ee - - + CH CH 3+ 3+ CH CH + + + H + ee - - ee - - + CH CH + + CH CH + H + + ee - - ee - - + CH CH + + CH CH + + H + ee - - ee - - + CH CH + + C + H + + ee - - ee - - + CH CH + + C + + H + ee - - ee - - + CH CH 4+ 4+ CH CH 3 + 3 H ee - - + CH CH 4+ 4+ CH CH + H H ee - - + CH CH 3+ 3+ CH CH + H ee - - + CH CH + + CH CH + H ee - - + CH CH + + C + H ee - - + C C + + e e - - H + + CH CH 4 4 CH CH 4+ + 4+ H H + + CH CH 3 3 CH CH 3+ + 3+ H H + + CH CH CH CH + + + H H + + CH CH CH CH + + H H + + C C + + H ee - - + H H + + + e e - - ee - - + H H + H + ee - - ee - - + H H + + e e - -
PIC simulation: capacitive RF discharge Parallel plate RF discharge (Univesity of Bochum) f RF = 13.56 MHz, RF peak-to-peak voltage ~ -16 V Gas : CH 4 H mixture, pressure p = 1-1Pa, electron density n e ~ 1 9-1 1 cm -3 potential
PIC simulation: capacitive RF discharge electron and CH 4+ ion density CH 4+ ion energy distribution electrons reach electrode only during sheaths collapse energetic ions at the wall due to acceleration in the sheath
PIC simulation: capacitive RF discharge n H = 9. 1 14 cm -3, n CH4 = 7 1 14 cm -3, p = 11 Pa Stochastic (Fermi) heating n H = 9. 1 15 cm -3, n CH4 = 7 1 15 cm -3 p = 11 Pa Ohmic heating
PIC simulation: capacitive RF discharge simulation electron energy probability function experiment eepf (ev -3/ cm -3 ) 1 11 1 1 1 9 1 8 1 7 1 6 T 1 =.39 ev, n 1 = 1 1 cm -3 T = 3 ev, n = 1 9 cm -3 T T 1 5 1 15 electron energy (ev) V.A. Godyak, et al., Phys. Rev. Lett., 65 (199) 996. bi-maxwellian distribution due to stochastic heating
Capacitive RF discharge high pressure n e = 1 1 cm -3, n H = 9. 1 15 cm -3,n CH4 = 7 1 15 cm -3, p = 11 Pa potential n' ei, cm -3 c -1 1.x1 16 electron-impact ionization rate 1.x1 16 8.x1 15 6.x1 15 4.x1 15.x1 15. 3 64 96 18 Y, λ D field reversal after sheath collapse ionization localized in the sheaths
Capacitive RF discharge high pressure Y, mm 1 8 6 4 electron-impact ionization rate simulation n' ei, cm -3 c -1 7E16 6E16 4.8E16 3.6E16.4E16 1.E16 653.3 nm excitation rate experiment 4 6 8 1 1 14 time, ns C.M.O. Mahony et al., Appl. Phys. Lett. 71 (1997) 68. double peak structure due to sheath reversal