Multi-Scale Methods for Plasma Simulation

Transcription

1 Multi-Scale Methods for Plasma Simulation Juan Pablo Trelles Department of Mechanical Engineering University of Massachusetts Lowell Center for Atmospheric Research (UMLCAR) September 28,

2 Outline 1. Introduction Modeling, Plasmas, Non-Equilibrium 2. Plasma Modeling Models, Equations 3. Multi-Scale Methods Variational Multi-Scale 4. Applications & New Directions Thermal Plasma Processing Semiconductor Manufacturing Hybrid Field Particle Methods 2

3 Preface knowledge Scientific & Technological Advancement through Modeling & Simulation: knowledge k Physical Mathematical Numerical Solution k+1 Phenomena Models Models Algorithms 3 q B m d v dt = q( E + v B) E = d B dt v k+1 v k+1 m Δt E i =...? Ω e Ax = b bottleneck for Multi-Scale problems e.g., plasmas

4 Plasmas Gas mixture with charged species, e.g. Ar + M Ar + + e - + M +99% of observable mass in the universe 4 th state of matter: solid liquid gas plasma solid liquid gas plasma cold temperature hot Span a wide range of scales 4

5 Types: UMLCAR This talk natural * Plasma Science: From Fundamental Research to Technological Applications, The National Academies Press, 1995 technological 5

6 2. Plasma Modeling - Plasma Models - Non-Equilibrium - Fluid & Electromagnetic Modeling - Plasma Flow Model 6

7 Plasma Models m more accurate Particle (discrete) s d 2 xs 2 dt = Fs Quantum, ab initio Molecular Dynamics Particle-in-Cell Monte Carlo j j rate of change in phase-space D [ x, v] = + Dt t t [ x, v] Boltzmann Multi-fluid Chemical & thermal non-equilibrium Chemical & thermal equilibrium Fluid (continuum) Df Dt s c = f s change due to collisions this talk less expensive 7

8 Non-Equilibrium E in E out What energy do we use? E in E out S in few high energy photons S out many low (thermal) energy photons S in << S out Useful Energy due to 2 nd law Non-Equilibrium 8

9 Thermodynamics & Equilibrium Df Dt s c f s <<< Thermodynamics temperature, pressure, concentration, Df Dt s c << f s Local Thermodynamic Equilibrium 2 evolution of moments fsdv, fsvdv, fsv dv fluid flow models (incompress., compressible, reactive, ) Df Dt s c f s Non-Equilibrium Discrete: Molecular Dynamics, Kinetic Monte Carlo, Continuous: Boltzmann, Fokker-Planck, Hybrids: Particle-In-Cell, Sectional, Nodal, Needed for complex non-equilibrium & multi-scale problems 9

10 Fluid Modeling 1. Boltzmann Eqns. 2. Moments à Conserv. Eqns. 3. Composition - Mass action law 4. Thermodynamic Prop. - Kinetic theory 5. Transport Prop. (closure) - Chapman-Enskog Df Dt s c = f s f s dv t ρ s +... f sv dv t ρ su +... f s v 2 dv t ρ s e s +... n s + n r n t +... e s = e s (n s, n r, n t,...) p s = p s (n s, n r, n t,...)... µ = µ(n s, n r,...,t h,t e,σ sj,σ sk,...) At the end, if time permits 10

11 Thermodynamic & Transport Properties Example: Ar, T h T e, (θ = T e /T h ) ~ 10 2 ~ 10 1 ~ 10 3 > 10 7 Thermodynamic Transport 11

12 Electromagnetic Modeling Maxwell s equations: 1) Ampere s law: (Macroscopic) 2) Faraday s law: E p effective field for generalized Ohm s law 3) Ohm s law: 4) Gauss law: 5) No magnetic monopoles: B = µ 0J q E p = B t J q = σ ( E p + u B) J q = 0 B = 0 3) in 1) in 2): Magnetic Induction eqn. B ( u B) + ( η B) = 0 t B = 0 or, using potentials: B = A ( B = 0 a priori) E p = ϕ p A t (suggested from J q = 0) A t + ϕ p u A +η 2 A = 0 (σ ϕ p )+ σ ( A t u A) = 0 12

13 Plasma Flow Model 1. Mass cons. 2. Species cons. 3. Momentum 4. Energy Heavies 5. Energy Electrons 6. Current cons. Transient + Advective - Diffusive - Reactive = 0 ρ u ρ + ρ u 0 0 t y s chemical nonequilibrium c ρ ρu ys J s ρs t u ρ ρu u p τ J q B t hh ' Dph ρ ρu h q Q h h + eh t thermodynamic Dt he ' Dpe ρ non-equilibrium ρu h q Q Q e e eh + t Dt 0 0 J q 0 A t 2 7. Ampere s law φ p u A η A 0 Vector form: A0 Y + A Y ( K Y) ( S1Y + S0) = R ( Y) = 0 t i i i ij j transient advective diffusive reactive J Q 13 r

14 3. Multi-Scale Methods - Multi-Scale Problems - Variational Multi-Scale Methods 14

15 Multi-Scale Phenomena Paradigm: chemical reactions, nucleation y turbulence, wave scattering L 1 L 2 boundary layers, sheaths, interfaces x or t * Ed. Wiley, 2000 L 1 << L 2 (membrane, electrodes, ) << (reactor, collector, ) shocks, chemical fronts, phase change 15

16 Canonical Multi-Scale Problem Supersonic flow over a sphere different length scales mean flow l O(L) boundary layer l O(L/Re) ½ flow direc@on turbulent wake l O(L to L/Re) L shocks l O(λ) * M. Van Dyke, An Album of Fluid Mo@on, 1982 Tremendously difficult to model with today s methods! Recall: Re = ρul µ 16

17 Multi-Scale & Transport Problems Transport Problem (e.g. Plasma Flow): A0 Y + A Y ( K Y) ( S1Y + S0) = R ( Y) = 0 t i i i ij j transient advective diffusive reactive Multi-Scale Problem: (any term) >> (other terms) Advective >> Diffusive (Re >> 1) e.g. Turbulence Advective >> Transient (Ma > 1) e.g. Shocks Reactive >> Advective (Da >> 1) e.g. Chemical fronts more complicated for multi-physics systems 17

18 1. Mass cons. 2. Species cons. 3. Momentum 4. Energy Heavies 5. Energy Electrons 6. Current cons. ρ t ys ρ t u ρ t hh ρ t he ρ t 0 A t Plasma Flow Model Transient + Advective - Diffusive - Reactive = 0 B: B: A: u ρ + ρ u ρu y ρu u p ρu h ρu h 0 J Ampere s law φ p u A η A 0 s h e J A: B: B: B: A: A: B: A: A: A: A: B: A: B: τ q q ' h ' e s q Dp h Dt Dp Dt e J + Q Q ρ q eh eh c s B 0 + Q B: J Q r A/B: Reynolds A/B: Peclet A/B: Mag Reynolds A/B: Prandtl A/B: Schmidt A/B: Lewis A/B: Enthalpy num A/B: Damköhler A/B: Mach Highly Multi-Scale! 18

19 Numerical Methods Finite Differences Finite Volumes Finite Elements R ( Y ) = 0 R ( Y) dω = 0 Ω W R ( Y) dω = 0 Ω (i,j+1) (i-1,j) (i,j) in v 1 out e 3 e 4 (i+1,j) e 1 e 2 (i,j-1) Challenge for all methods: Multi-Scale Problems y exact solution discrete approximation x or t 19

20 Variational Multi-Scale Methods 1, 2 Variational form: W R ( Y) dω = ( W, R ( Y) ) = 0 Ω Scale decomposition: Y = Y + Y' and W = W + W' Standard Finite Element Methods total = large + small Large scales: solved ( W, R ( Y)) + ( L W, Y' ) = 0 Small scales: modeled Y' = τr ( Y ) large = f(small) small = f(large) 1. Hughes et al, Comp. Meth. Appl. Mech. Eng.166, 3-24, Hughes et al, Multiscale and Stabilized Methods, Encyclopedia of Computational Mechanics, Vol. 3, John Wiley & Sons,

21 Small Scales Modeling Model: τ 1 L = ( A0 t + Ai i ikij j S1 ) 1 transport operator = ( τt + τa + τd + τr 1 1 Proposed: τ ) time scales τ ~ ~ 1 ~ ~ 1 ~ ~ 1 ~ ~ (( A G A ) + ( A G A ) + ( K G : G K ) + ( S ) ) = 0 t 0 i ij j ij ij ij ij 1 S1 1 Final Model: (W,R (Y)) (L W, τr (Y)) + ( W, K DC i ij j Y) = 0 large scales small scales discontinuity capturing ~ 0 if solution is smooth 1. J. P. Trelles, Variational Multi-Scale Modeling of Thermal Plasmas, 2012, in preparation 21

22 Developed Software T-PORT: TransPORT solver Solves 1, 2, 3D systems of transport equations Scalar transport, incompressible, compressible flows, equilibrium and non-equilibrium thermal plasmas C++, parallel with OMP, Linux, OS X, Windows XNL: X-Numerical Library Generic numerical library; support for Scalars, Vectors, Matrices, linear algebra (BLAS 1, 2, 3), Linear, Non-Linear, and Time-Stepper solvers C++, serial, Linux, OS X, Windows HTPLFLOW: High Temperature & Plasma Flow Solver for equilibrium & non-equilibrium thermal plasmas Developed in Matlab, serial, Linux and Windows At the end, if time permits 22

23 4. Applications & New Directions - Thermal Plasma Processing - Semiconductor Manufacturing - Hybrid Fluid Particle Methods 23

24 Thermal Plasma Processing Spraying, cutting, welding, metallurgy, toxics remediation, gasification, fuel reforming, etc. jet coating E.g., spraying of coatings: wear, corrosion, thermal resistance torch powder * 1, 2 1. J. P. Trelles et al, Journal of Thermal Spray Technology, (2009), online 16 June J. P. Trelles, Ph.D. Thesis, University of Minnesota (2007) 24

25 Simulation of Plasma Torches Temperature & arc attachment location ~ experimental 1 spot 2 3 Experimental large scale structures captured 1. J. P. Trelles et al, Journal of Physics D: Applied Physics (2007) Vol 40, H.-P. Li and E. Pfender, J. Phys. D: Appl. Phys. 36 (2003) J. P. Trelles et al, IEEE Transactions on Plasma Science (2008) Vol. 36, No. 4, August 2008, pp

26 Vision: Untamed Local Energy Source Renewable resources: Non-local Transmission, storage à significant cost to renewables solar Need compensate fluctuations wind Local Energy Source: transmission Consumption: Local Dumpsters (urban waste) Agriculture (biomass) + Plasma gasification = Hydrogen city 26

27 inputs Plasma Gasification cathode outputs Electricity Electric arc anode syngas Syngas (H 2, CO) Fuel Energy Waste / Biomass organic inorganic material Pros: Compact, local, CO 2 negative Cons: Efficiency 27

28 Plasma Gasification (cont.) Complex arc dynamics, turbulence, chemistry, non-equilibrium 28

29 Non-Thermal Plasma Gasification (cont.) Goal: Minimize Energy Consumption & Maximize H2 production Approach: Non-thermal plasma Low Th + High Te + High E field + High Vorticity Optimum utilization of energy for breaking bonds 29

30 Semiconductor Manufacturing Highly Multi-Scale: ~10 9 transistors/die, 10 2 dies/wafer, 10 2 wafers/reactor Recipe & Equipment Film reactor wafer die layout feature Compressible, reactive fluid flow filtering feature extraction film evolution 10 mm 10 nm 10 cm 1 m 1 um 30

31 Physical Vapor Deposition target High Non-Equilibrium: ballistic transport atoms + ion - finite rate surface chemistry + substrate Barrier & Seed over Dual Damascene: ~10 nm ~2 nm 1 1. R. Arakoni, J. P. Trelles, D. Kim, M. Khabibullin, and S. Nikonov, 58th Int. Symp. & Exh., Nashville, Tennessee, Oct.30 Nov 4,

32 Hybrid Field-Particle Models Need: Complex non-equilibrium macroscopic phenomena Solution: Field-Particle - selectively describe degree of non-equilibrium Field problem (Eulerian) (A 0 t + A i i i (K ij j ) S 1 )Y f S = 0 0 R f (Y f,y p ) Variational multi-scale FEM Particle problem (Lagrangian) (Md t N)Y p Q = 0 R p (Y f,y p ) Macro-particle Shape function Coupling: Projectors: field à particle, particle à field Vision: Generalization of Particle-In-Cell (PIC) methods Unstructured Meshes + Time Implicit P p f P f f = (P f p ) 1 = P p p = I 32

33 Field-Particle Projectors Consistency: Y f t = P f p Y p t Y p t = P p f Y f t X Y f X P f p Yf Yp f P p p P f p P f Y p X Y p P X f p Y f Y f p P f Y p Total field: X Total particle: X Y t f = Yf p P f Y p X X X X Y t p = f P p Y Yp f X X X X X X To be continued X X 33

34 Summary & Conclusion Plasma Modeling & Simulation à complex challenges BUT great promise Key: understanding of Non-equilibrium & Multi-Scale phenomena Variational Multi-Scale Methods - comprehensive methodology for Plasma Modeling Applications in Mat. Processing, Energy, Semicond. Hybrid Field Particle Methods for Non-Equilibrium & Multi-Scale Thank You 34