Modelling of low-temperature plasmas: kinetic and transport mechanisms L.L. Alves llalves@tecnico.ulisboa.pt Instituto de Plasmas e Fusão Nuclear Instituto Superior Técnico, Universidade de Lisboa Lisboa, Portugal http://www.ipfn.ist.utl.pt Atelier Fonctionnalisation de surfaces par plasmas: Techniques et Applications 6-8 juin 2017, GREMI Orléans
Modelling of low-temperature plasmas Goal: understand and predict 2
Modelling of low-temperature plasmas Interaction between species (in volume and at the wall) Interaction with surface electrons Rotational excitation Vibrational excitation Electronic excitation Fragmentation Chemistry Ionization heavy-species 3
Modelling of low-temperature plasmas Modelling approaches Statistical models Kinetic models Fluid models Hybrid models (combination of the above) 4
Modelling of low-temperature plasmas Outline Kinetic modelling The electron Boltzmann equation Fluid modelling The hydrodynamic transport equations Hybrid modelling Joining the pieces Example results Final remarks and questions 5
Modelling of low-temperature plasmas Key references Kinetics and Spectroscopy of Low Temperature Plasmas J. Loureiro and J. Amorim, 2016, Springer International Publishing Principles of Plasma Discharges and Materials Processing M. A. Lieberman and A. J. Lichtenberg, 1994, John Wiley Lecture Notes on Principles of Plasma Processing F. F. Chen and J.P. Chang, 2003, Kluwer Academic / Plenum Publishers Plasma Physics, Volumes 1 and 2 Jean-Loup Delcroix, 1965 / 1968, J. Wiley Motions of Ions and Electrons W. P. Allis, Handbuch der Physik, vol. 21, 1956, S. Flugge, Springer-Verlag Berlin Electron kinetics in atomic and molecular plasmas C. M. Ferreira and J. Loureiro, Plasma Sources Sci. Technol. 9 (2000) 528 540 Fluid modelling of the positive column of direct-current glow discharges L. L. Alves, Plasma Sources Sci. Technol. 16 (2007) 557 569 6
Kinetic modelling The electron Boltzmann equation
Kinetic modelling The master kinetic equation Inclusion of an energy description Definition of boundary conditions Complete problem : 6D long run times F(r,v,t) is the distribution function, representing the number of particles per unit volume in phase space (r,v), at time t. 8
Kinetic modelling The electron Boltzmann equation Rate of F in configuration space due to collisions in time in velocity space Force acting upon particles The total electric field acting on electrons dc space-charge field hf field at frequency ω 9
Charge separation at the boundaries The space-charge sheath 10
Charge separation at the boundaries The space-charge sheath 11 Space-charge field, E s
Charge separation at the boundaries The space-charge sheath 12 Space-charge field, E s
Charge separation at the boundaries The space-charge sheath 1.0 n e / n e (0) ; n p / n e (0) 0.8 0.6 0.4 0.2 0.4 0.3 0.2 0.1 0.0 0.98 0.99 1.00 0.0 0.0 0.2 0.4 0.6 0.8 1.0 r/r Source: LL Alves, PSST 16 557 (2007) 13
The electron Boltzmann equation The two-term approximation Disregard the space-charge electric field acting on electrons dc space-charge field hf field at frequency ω No external magnetic field The electron distribution function F is expanded in spherical harmonics in velocity space in Fourier series in time Isotropic component (energy relaxation) Anisotropic components (transport) 14
The homogeneous electron Boltzmann equation Input data: working parameters Independent parameters 15
The homogeneous electron Boltzmann equation Input data: collisional data ν = c Nσ c ( 2eu m) 1/ 2 ( 2eu ) 1/ 2 q, ν = Niσ m ij N = i= 0 N Gas density N i 0 Chemistry model (heavy-species kinetics) 16
Input data Excitation / ionization mechanisms 17
Input data Cross sections Data from Bibliography Databases (e.g. LXCat: www.lxcat.net) 18
Input data The LXCat database 19
The homogeneous electron Boltzmann equation The electron energy distribution function (EEDF) Courtesy of Bolsig (https://www.bolsig.laplace.univ-tlse.fr/) 20
Fluid modelling The hydrodynamic transport equations
Fluid modelling The hydrodynamic transport equations No energy description Use of the hydrodynamic approximation moments of a kinetic equation rate balance equations (particle / energy) flux equations (charged particle / neutral species) Definition of boundary conditions Short run times 22
Fluid modelling The moments of the Boltzmann equation Particle balance equation Energy balance equation Particle flux equation Energy flux equation 23
Fluid modelling Summary of the hydrodynamic equations The particle balance equation (continuity equation) The flux equation (drift-diffusion approximation) The energy balance equation
Fluid modelling Poisson s equation Poisson s equation for the space-charge field r r E = e ( ) n i n e ε 0 10 3 10 2 50 ma 200 ma Space-charge field, E s E r /N (10-16 V cm 2 ) 10 1 10 0 10-1 p = 0.3 torr p = 5 torr Helium dc discharge 10-2 0.0 0.2 0.4 0.6 0.8 1.0 r/r Source: LL Alves, PSST 16 557 (2007)
Fluid modelling Boundary conditions: plasma-surface interaction Wall-loss probability β = 1 ζ : total loss probability γ: recombination probability s : sticking probability film 1-β β s γ = β-s
Fluid modelling Boundary conditions: plasma-surface interaction Recombination probability Dynamical MC (BKL algorithm) X Dedicated exp Model Source: V Guerra and D Marinov, PSST 25 045001 (2016)
Fluid / Chemistry modelling Collisional Radiative Models (CRM) n r r Γ = t r r Γ = D n n ν I System of coupled equations for the various plasma species n t = D 2 n n ν I transport kinetics
Fluid / Chemistry modelling Collisional Radiative Models (CRM) Transport rates D D 2 n Λ 2 n For electrons, the transport parameters can be related with the EEDF Electron free diffusion coefficient D e = 0 2 v 3ν c F n 0 0 e 4πv 2 dv = function ( E N ) Electron mobility µ e = 0 ev 3mν c 1 n e 0 F 0 2 4πv dv v = function ( E N )
Fluid / Chemistry modelling Collisional Radiative Models (CRM) Reaction rates (for collisional-radiative kinetic mechanisms) n ν I = n ν j j j n k ν k For electrons, the collision frequencies can be related with the EEDF Electron rate coefficients C j ν j n e N = 0 σ v j 0 F 0 2 4πv dv n = e function ( E N )
Fluid / Chemistry modelling Collisional Radiative Models (CRM) Kinetic data: example for nitrogen Source: LL Alves et al, PSST 21 045008 (2012)
Fluid / Chemistry modelling Collisional Radiative Models (CRM) Kinetic data: example for nitrogen Source: LL Alves et al, PSST 21 045008 (2012)
Hybrid modelling joining the pieces
Closing the calculation Eigenvalue calculation of the excitation field n t = D n n N DN 2 2 ν I 2 1 n nν I = 0 = N 2 Λ PDE solution (with boundary condition for the charged-particle density) 2 n n = 1 Λ 2 = const Electron gain/loss balance (Schottky condition) ν I N DN = ( N Λ) 2 E N EIGENVALUE
Hybrid modelling Joining the pieces E/N (initial value) Electron cross sections Electron energy distribution function Schottky condition satisfied? E/N (update) Kinetic module Boltzmann solver for electrons Fluid module Chemistry solver for neutral / charged heavy-species Electron rate coefficients transport parameters Other rate coefficients Densities and fluxes of charged particles
Hybrid modelling The LisbOn KInetics code (LoKI) Modular tools with state-of-the-art kinetic schemes and transport description, including Boltzmann solver Chemistry solver
Example results
Kinetic calculations EEDF for argon Influence of - excitation frequency - e-e collisions A B A: ω/n = 0 E/N = 3x10-16 V cm 2 B: n e / N 0, full 10-4, dashed ω/n = 2x10-6 cm 3 s -1 E/N = 3x10-15 V cm 2 Source: CM Ferreira and J Loureiro, PSST 9 528 (2000) 38
Kinetic calculations EEDF and VDF for nitrogen E/N = 10-15 V cm 2 ω/n = 0 6000 K T V = 2000 K 6000 K 6000 K 4000 K 3000 K 2000 K T V = 2000 K E/N = 10-15 V cm 2 ω/n = 0 Source: CM Ferreira and J Loureiro, PSST 9 528 (2000) 39
Kinetic calculations Swarm studies for oxygen (influence of rotational mechanisms) DCO CC-CAR discrete collisional operator continuous approximation rotations (with Chapman-Cowling correction) Source: MA Ridenti et al, PSST 24 035002 (2016) 40
Hybrid calculations dc discharge in helium Space-charge potential and electron mean energy pr (torr cm) 1 10 0 p = 0.4 Torr 150 ma simulations (x 0.5), 50 ma 200 ma 10 V (V) -10 ε(0) (ev) -20 simulation measurement [Kimura et al (1995)] -30 0.0 0.2 0.4 0.6 0.8 1.0 r/r 1 measurements Kaneda (1979),,, Dote and Shimada (1982), Sato (1993) 1 10 NR (10 16 cm -2 ) Source: LL Alves, PSST 16 557 (2007) 41
Hybrid calculations capacitive discharge in N 2 and N 2 -H 2 Self-bias potential and radiative transition intensities - V dc (V) 175 150 125 100 75 50 25 N 2 1mbar 0.5 mbar 0.2 mbar 0 0 5 10 15 20 25 W eff (W) Intensity SPS(0,0) (a.u.) Intensity FNS(0,0) (a.u) 1.0 0.8 0.6 0.4 0.2 N 2 1mbar 0.2 mbar 0.0 0 1 2 3 4 z (cm) 2.5 2.0 1.5 1.0 N 2 -H 2 1.2 mbar 0.6 mbar Source: LL Alves et al, PSST 21 045008 (2012) 0.5 0 1 2 3 4 5 H 2 /(H 2 N 2 ) (%) 42
Hybrid calculations microwave discharge in air Absolute densities of N 2 species N 2 (B) --- n e increased 20% T g 80K N 2 (C) N 2 (B) 129 W 129 W Source: GD Stancu et al, JPD 49 435202 (2016) 43
Hybrid calculations microwave discharge in air Absolute densities of atomic species N (3p 4 S) 129 W O (3p 5 P) 129 W --- n e increased 20% T g 80K Source: GD Stancu et al, JPD 49 435202 (2016) 44
Hybrid calculations microwave discharge in air Vibrational temperature of the N 2 (C) state Influence of the wall deactivation coefficient γ N2 = 1.1x10-3 γ N2 = 0.02 129 W Source: GD Stancu et al, JPD 49 435202 (2016) 45
Verification of codes Round-robin call A round robin test is an interlaboratory test (measurement, analysis, or experiment) performed independently several times. This can involve multiple independent scientists performing the test with the use of the same method in different equipment, or a variety of methods and equipment. In reality it is often a combination of the two, for example if a sample is analyzed, or one (or more) of its properties is measured by different laboratories using different methods, or even just by different units of equipment of identical construction. (Source: Wikipedia, April 2017) Following the GEC16 workshop on LXCat The need for a community wide activity on validation of plasma chemical kinetics in commonly used gases has been clearly identified. ( ) we would like to propose a round-robin to acess the consistency in results of calculations from different participants in a simplified system. (coordination: Sergey Pancheshnyi) 46
Verification of codes Round-robin exercise General conditions Gas temperature (constant) 300 K Species Gas pressure (constant) 0.1 bar e electrons Initial density of electrons 1 cm -3 Ar neutrals Initial density of ions 1 cm -3 Ar positive ions Time interval 0 0.01 s Ar* excited states Processes and rates e Ar e Ar scattering cross section provided e Ar e Ar* scattering cross section provided e Ar e e Ar scattering cross section provided Ar e e Ar e C (cm 6 s -1 ) = 8.75x10-27 Te (4.5) Input parameter E N ( Td) = 43 t(ms) exp t [ (ms)] 47
Final remarks
Modelling of low-temperature plasmas Final remarks and questions Modelling tools are formidable aides for understanding and predicting the behaviour of low-temperature plasmas (LTPs) The main difficulties in deploying LTP models are - describing the plasma excitation and transport (particularly if spatial effects are relevant) - defining a kinetic scheme for the plasma species (and finding the corresponding elementary data) Modelling tools need - verification, e.g. based on crossed-benchmarking, round-robin exercises (the community is currently starting a collective to meet this goal) - validation of results, by comparing simulations with experiment (collaboration with experimental teams is most welcome) Questions? 49