modelling of low-temperature...
TRANSCRIPT
Atelier Fonctionnalisation de surfaces par plasmas:
Techniques et Applications
6-8 juin 2017, GREMI – Orléans
L.L. [email protected]
Instituto de Plasmas e Fusão Nuclear
Instituto Superior Técnico, Universidade de Lisboa
Lisboa, Portugalhttp://www.ipfn.ist.utl.pt
Modelling of low-temperature plasmas:kinetic and transport mechanisms
Modelling of low-temperature plasmasGoal: understand and predict
2
Modelling of low-temperature plasmasInteraction between species (in volume and at the wall)
3
Rotational excitationRotational excitation
Vibrational excitationVibrational excitation
Electronic excitationElectronic excitation IonizationIonization
Interaction with surfaceInteraction
with surface
FragmentationChemistry
FragmentationChemistry
electronselectrons
heavy-speciesheavy-species
Modelling of low-temperature plasmasModelling approaches
• Statistical models
• Kinetic models
• Fluid models
• Hybrid models (combination of the above)
4
Modelling of low-temperature plasmasOutline
5
• Kinetic modellingThe electron Boltzmann equation
• Fluid modellingThe hydrodynamic transport equations
• Hybrid modellingJoining the pieces…
• Example results
• Final remarks and questions
Modelling of low-temperature plasmasKey references
6
• Kinetics and Spectroscopy of Low Temperature Plasmas
J. Loureiro and J. Amorim, 2016, Springer International Publishing
• Principles of Plasma Discharges and Materials ProcessingM. A. Lieberman and A. J. Lichtenberg, 1994, John Wiley
• Lecture Notes on Principles of Plasma ProcessingF. 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 ElectronsW. P. Allis, Handbuch der Physik, vol. 21, 1956, S. Flugge, Springer-Verlag – Berlin
• Electron kinetics in atomic and molecular plasmasC. M. Ferreira and J. Loureiro, Plasma Sources Sci. Technol. 9 (2000) 528–540
• Fluid modelling of the positive column of direct-current glow dischargesL. L. Alves, Plasma Sources Sci. Technol. 16 (2007) 557–569
Kinetic modellingThe electron Boltzmann equation
• Inclusion of an energy description
• Definition of boundary conditions
• Complete problem : 6D ⇒ long run times
Kinetic modellingThe “master” kinetic equation
8
F(r,v,t) is the distribution function, representing the number of particles
per unit volume in phase space (r,v), at time t.
Rate of F
in time
in configuration space
in velocity space
due to collisions
Force acting upon particles
Kinetic modelling The electron Boltzmann equation
9
� The total electric field acting on electrons
dc space-charge field hf field at frequency ω
Charge separation at the boundariesThe space-charge sheath
10
+
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+
++
+
+
+
+
+
+
+
+
+ ++
++
+
+
+
+
Charge separation at the boundariesThe space-charge sheath
11
+
+
+
+
+
+
++
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+
++
+
+
+
+
+
+
+
+
+ ++
++
+
+
+
+
Space-charge field, Es
Charge separation at the boundariesThe space-charge sheath
12
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+
+
+
+
+
+
+
+ ++
+
+
+
+
+
+
++
+
Space-charge field, Es
Charge separation at the boundariesThe space-charge sheath
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
r/R
ne / n
e(0
) ; n
p / n
e(0
)
0.98 0.99 1.000.0
0.1
0.2
0.3
0.4
13Source: LL Alves, PSST 16 557 (2007)
� Disregard the space-charge electric field acting on electrons
dc space-charge field hf field at frequency ω� No external magnetic field
The electron Boltzmann equationThe two-term approximation
14
� 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)
⇒ Independent parameters
The homogeneous electron Boltzmann equationInput data: working parameters
15
The homogeneous electron Boltzmann equationInput data: collisional data
( ) 2/12 meuN cc σν =
0
0
≠
= =
i
i
N
NN ⇒ Gas density
⇒ Chemistry model (heavy-species kinetics)
16
( ) 2/12, meuNq ijiσν =
Input dataExcitation / 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 equationThe electron energy distribution function (EEDF)
20Courtesy of Bolsig+ (https://www.bolsig.laplace.univ-tlse.fr/)
Fluid modellingThe 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
Fluid modelling The hydrodynamic transport equations
22
Particle balance equation
Energy balance equation
Particle flux equation
Energy flux equation
Fluid modelling The moments of the Boltzmann 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
Poisson’s equation for the space-charge field
Fluid modelling Poisson’s equation
( )0ε
ei nneE
−=⋅∇rr
0.0 0.2 0.4 0.6 0.8 1.010
-2
10-1
100
101
102
103
p = 5 torr
p = 0.3 torr
50 mA
200 mA
r/R
Er /
N (
10
-16 V
cm
2)
Helium dc discharge
Source: LL Alves, PSST 16 557 (2007)
++
+
+
+
+
+++
+
++
++
+
++
+
+
+
+
+ ++
+ ++++
+++
+
+
+ ++
++
+
+++
Space-charge field, Es
Fluid modelling Boundary conditions: plasma-surface interaction
1-βγ = β-s
sβ
film
Wall-loss probability
β = 1 – ζ : total loss probability
γ: recombination probability
s : sticking probability
Dynamical MC (BKL algorithm)
X
Dedicated exp
Model
Fluid modelling Boundary conditions: plasma-surface interaction
Recombination probability
Source: V Guerra and D Marinov, PSST 25 045001 (2016)
Fluid / Chemistry modelling Collisional – Radiative Models (CRM)
transport kinetics
∇−=Γ
=Γ⋅∇+∂∂
nD
nt
nI
rr
rrν
InnDt
n ν+∇−=∂∂
⇒ 2
System of coupled equations for the various plasma species
Fluid / Chemistry modelling Collisional – Radiative Models (CRM)
nD
nD2
2
Λ−≈∇
Transport rates
For electrons, the transport parameters can be related with the EEDF
( )NEdvvn
FvD
ec
e function43
0
20
0
2
== ∫∞
πν
( )NEdvvv
F
nm
ev
ec
e function41
30
20
0 =∂
∂−= ∫∞
πν
µ
Electron free diffusion coefficient
Electron mobility
Fluid / Chemistry modelling Collisional – Radiative Models (CRM)
∑∑ −=k
k
j jjI nnn νννReaction rates (for collisional-radiative kinetic mechanisms)
For electrons, the collision frequencies can be related with the EEDF
Electron rate coefficients
( )NEdvvn
Fv
n
NC
e
j
e
j
j function40
20
0 ==≡ ∫∞
πσν
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 modellingjoining the pieces…
Closing the calculationEigenvalue calculation of the excitation field
2
22
2 10
Λ≡=∇
⇒=+∇−=∂∂
NDN
N
n
nnnD
t
n II
νν
� PDE solution (with boundary condition for the charged-particle density)
const1
2
2
=Λ
=∇n
n
� Electron gain/loss balance (Schottky condition)
( )2Λ=
N
DN
N
Iν
EIGENVALUE
N
E
Hybrid modelling Joining the pieces
Kinetic moduleBoltzmann solver
for electrons
Fluid moduleChemistry solver
for neutral / charged
heavy-species
Other
rate coefficients
Electron
rate coefficients
transport parameters
E/N (initial value)Electron cross sections
Schottky condition
satisfied?
E/N (update)
Electron energy
distribution function
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
A:
ω/N = 0
E/N = 3x10-16 V cm2
B:
ω/N = 2x10-6 cm3 s-1
E/N = 3x10-15 V cm2
Kinetic calculations EEDF for argon
ne / N
0, full
10-4, dashed
A
B
38
Influence of - excitation frequency- e-e collisions
Source: CM Ferreira and J Loureiro, PSST 9 528 (2000)
TV = 2000 K
3000 K
4000 K
6000 K
E/N = 10-15 V cm2
ω/N = 0
E/N = 10-15 V cm2
ω/N = 0
2000 K
6000 K
6000 K
TV = 2000 K
Kinetic calculationsEEDF and VDF for nitrogen
39Source: CM Ferreira and J Loureiro, PSST 9 528 (2000)
Kinetic calculationsSwarm studies for oxygen (influence of rotational mechanisms)
40
DCO … discrete collisional operatorCC-CAR … continuous approximation rotations (with Chapman-Cowling correction)
Source: MA Ridenti et al, PSST 24 035002 (2016)
Hybrid calculations – dc discharge in heliumSpace-charge potential and electron mean energy
41
0.0 0.2 0.4 0.6 0.8 1.0-30
-20
-10
0
p = 0.4 Torr
150 mA
simulation
measurement [Kimura et al (1995)]
r/R
V (
V)
1 101
10
50 mA
200 mA
simulations (x 0.5)
,
measurements
Kaneda (1979)
, , ,
Dote and Shimada (1982), Sato (1993)
pR (torr cm)
NR (1016
cm-2)
ε(0)
(eV
)
1 10
Source: LL Alves, PSST 16 557 (2007)
Hybrid calculations – capacitive discharge in N2 and N2-H2Self-bias potential and radiative transition intensities
42
0 1 2 3 40.0
0.2
0.4
0.6
0.8
1.0
Inte
nsi
ty S
PS
(0,0
) (a
.u.)
z (cm)
N2 1 mbar0.2 mbar
0 1 2 3 4 50.5
1.0
1.5
2.0
2.5
H2/(H
2+N
2) (%)
Inte
nsi
ty F
NS
(0,0
) (a
.u)
N2-H2
1.2 mbar0.6 mbar
0 5 10 15 20 250
25
50
75
100
125
150
175
- V
dc (
V)
Weff
(W)
N2
1 mbar0.5 mbar0.2 mbar
Source: LL Alves et al, PSST 21 045008 (2012)
Hybrid calculations – microwave discharge in airAbsolute densities of N2 species
43Source: GD Stancu et al, JPD 49 435202 (2016)
N2(B)
N2(C)
N2+(B)
--- ne increased 20%… Tg–80K
129 W 129 W
Hybrid calculations – microwave discharge in airAbsolute densities of atomic species
44Source: GD Stancu et al, JPD 49 435202 (2016)
O (3p 5P)
N (3p 4S)
--- ne increased 20%… Tg–80K129 W
129 W
Hybrid calculations – microwave discharge in airVibrational temperature of the N2(C) state
45Source: GD Stancu et al, JPD 49 435202 (2016)
γN2 = 1.1x10-3
γN2 = 0.02
Influence of the wall deactivation coefficient
129 W
Verification of codes Round-robin call
46
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)
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)
Verification of codes Round-robin exercise
47
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 (cm6 s-1) = 8.75x10-27 Te –(4.5)
Input parameter
[ ])ms(exp)ms(43)Td( ttN
E −=
Final remarks
49
Modelling of low-temperature plasmasFinal 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 ?