icops_minicourse_yuki.pdf

University of California, Berkeley

37th International Conference on Plasma Science

Yuki Sakiyama, Ph.D. ([email protected])Research AssociateDepartment of Chemical Engineering,University of California, Berkeley

MINICOURSELow Temperature Plasma Modeling &

Simulation and Applications~ June 25 (Fri) 14:00-17:00 ~

2

Outline

6. Overview of available codes for simulating low-temperature non-equilibrium plasmas

2. Fluid modeling of atmospheric pressure plasmas

3. Simulation of atmospheric pressure plasmas using COMSOL and MATLAB

5. Neutral gas dynamics in atmospheric pressure plasmas

4. Plasma chemistry in atmospheric pressure plasmas

1. Problem setting and goals

3

1. Problem setting and goals-1

• helium RF plasma needle discharge

visible emission CCD image (cross sectional view)

(Images courtesy of Prof. John Goree)

Predicted emission distributiondark bright• Understanding the governing

equations and boundary conditions

• COMSOL and MATLAB

• Plasma chemistry

• Gas flow and plasma interaction

4

glow-mode (100 mW)corona-mode (1 mW)

Y. Sakiyama and D.B.Graves, J.Phys.D 39 3451 (2006) and J.Phys.D 39 3644 (2006)

1. Problem setting and goals-2

5

2.1 Introduction

2.2 Governing equations

2.3 Necessary parameters

2.4 Boundary conditions

2.5 Local field approximation (LFA)

2. Fluid modeling of atmospheric pressure plasmas

6

2.1 Introduction

e in

• species: electrons (e), positive heavy ions (i),neutrals (n)

• geometry: 1-D parallel plate, gap 2mm• external voltage: RF(= 13.56 MHz)• gas pressure: 1 atm (= 760 torr), static• gas temperature: room temperature

problem setting

• Which equations to be solved?

• What is the physical meaning of the governing equations?

• What is appropriate boundary conditions?

• How and where are the necessary parameters obtained?

7

2.2 Governing equation-1

( )

,

e-N

0

( , , ) (eq-1)

( , , ) (eq-2)

5 5 (eq-3)3 3

( , , ) (eq-4)

jj j l

l

j j j j j

ee e e e

j jj

nR j e i n

tn D n j e i n

nn D Q

tq n j e i n

μ

εε ε

ε

∂+ ∇ ⋅ = =

∂= ± − ∇ =

∂ ⎛ ⎞+ ∇ ⋅ − ∇ = − ⋅ −⎜ ⎟∂ ⎝ ⎠∇ ⋅ = =

∑

∑

Γ

Γ E

Γ Γ E

E

species continuity equation:

drift-diffusion approximation:

electron energy equation:

Poisson’s equation:

from (eq-1)… ,

0

0 (eq-1 )

(eq-4')

jj j j j l

j j j l

jj

j

nq q R

t

nq

t tε

∂′+ ∇ ⋅ = =

∂

∂∂∇ ⋅ =

∂ ∂

∑ ∑ ∑∑

∑

Γ

Efrom (eq-4)…

(eq-1’) + (eq-4’)…0 0 0 (eq-5)j j j j

j jq q

t tε ε

⎛ ⎞∂ ∂∇ ⋅ + ∇ ⋅ = ∇ ⋅ + =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

∑ ∑E EΓ Γ

total current continuity equation

total mass needs to be conserved!

8


, ( , , ) (eq-1)jj j l

l

nR j e i n

t∂

+ ∇ ⋅ = =∂ ∑Γspecies continuity equation:

change in time

due to motion (convection/diffusion)across the control volume

local creation/loss

xx+Δxx

n Δn

( )( )x t

n x t

Δ

= Δ

Γ

u

( )( )x x t

n x x t

+ Δ Δ

= + Δ Δ

Γ

u( )( )( )

1

3 1

6 1

[ ]

[ ]

[ ]

r rl l

r rl l

r rl l l

R k n k s

k n n k m s

k n n n k m s

−

−′

−′ ′′

=

=

=

Reaction term:

9


drift-diffusion approximation:

‘drift’ term(motion induced by electric field) ‘diffusion’ term

(motion induced by density gradient)

( , , ) (eq-2)j j j j jn D n j e i nμ= ± − ∇ =Γ E

mnt

∂∂

u ( )mn+ ∇ ⋅ uu (eq-6)mqn p mnυ= − ∇ −E u

(eq-7)m m m m

qn p q kTn n nm m m m

n D nυ υ υ υ

μ

∇= = − = − ∇

= ± − ∇

EΓ u E

E

Note: from momentum conservation equation to drift-diffusion approximation…

10


( )e-N

5 5 (eq-3)3 3

ee e e e

nn D Q

tε

ε ε∂ ⎛ ⎞+ ∇ ⋅ − ∇ = − ⋅ −⎜ ⎟∂ ⎝ ⎠

Γ Γ E

electron energy equation (EEE):

change in time

electron energy flux

electron heating

collisional energy loss

e-N 3 ( ) (eq-8)th r ele bl l i i i e g e g

l g

m kQ E k n n n k n n T Tm e′ ′′= + −∑

elastic loss with background gas

inelastic loss (reaction, vibrational excitation)

G.J. M. Hagelaar et al., Plasma Sources Sci. Technol. 14, 722 (2005)R.E. Robson, et al., Rev. Mod. Phys. 77, 1303 (2005)

32 b ee k Tε = : electron temperature

collisional energy loss with background neutral:

11

2.3 Necessary parameters-1

f (ε) or f (T)f (ε) or f (T)f (ε)inelastic collision rate coefficient (kr) [s−1, m3s−1, m6s−1]

00f (ε)elastic collision rate coefficient (kel) [m−3s−1]

f (T)f (μ, T)f (ε)diffusion (D) [m2s−1]

mobility (μ) [m2V−1s−1] 0const.f (ε)

neutrals (n)ions (i)electron (e)

12

10-6

10-4

10-2

100

EED

F

6050403020100energy [eV]

2.3 Necessary parameters-2: electrons

( ) (eq-9)ve

f ef f R ft m

∂+ ⋅∇ − ⋅∇ =

∂u E

Boltzmann equation for electrons:

collision cross section(with helium)

10-2310-2210-2110-2010-19

σ [

m2 ]

10-2 10-1 100 101 102 103 104

electron energy [eV]

Elastic 23S 21S 23P 21P 3SPD 4SPD 5SPD Ionization

EEDF (electron energydistribution function)

If EEDF is Maxwellian…

Actual EEDF is non-Maxwellian !

13

2.3 Necessary parameters-3: electrons

fitting functions or numerical lookup tables

( ) ( )( ) ( )

,

,e e e eel el r r

D D

k k k k

μ μ ε ε

ε ε

= =

= =from electron energy equation

parameters for electron (in He, 1 atm, and 300 K)

10-20

10-18

10-16

10-14

10-12

kr [m

3 /s]

0.12 4 6 8

12 4 6 8

102 4

mean electron energy [eV]

elestic collision

stepwiseionization

ionization

excitation

0.5

0.4

0.3

0.2

0.1

0.0

μ e

[m2 /s

V]

0.12 4 6 8

12 4 6 8

102 4

mean electron energy [eV]

1.0

0.8

0.6

0.4

0.2

0.0

De [m

2/s]

mobility

diffusion

14

2.3 Necessary parameters-4: ion mobility

1. Experimental data from literature:H.W. Ellis, et al, At. Data Nucl. Data Tables 17, 177 (1976 ) H.W. Ellis, et al., At. Data Nucl. Data Tables 22, 179 (1978)H.W. Ellis, et al., At. Data Nucl. Data Tables 31, 113 (1984)L.A. Viehland, et al., At. Data Nucl. Data Tables 60, 37 (1995)

2. Estimation from theory3

230

3.6 10 1 /[m /Vs] (eq-10)

( / )

g ii

g

m m

p m aμ

α

−× +=

[atm] [g-mol] polarizability

Bohr radius

Yu.P. Raizer, Gas Discharge Physics, (Springer, Berlin,1997) page 25

2.28H2O+1.83He2+

2.69OH+2.18O2+

2.69H2+2.25O+

3.39H+1.60N4+

1.78NO2+2.28N2

+

1.84N2O+2.19N+

2.15NO+1.16He+

ion mobility in helium [10−3 m2/Vs]

15

2.3 Necessary parameters-5: diffusion of ions

GER (generalized Einstein relation)

2( ), (eq-11)

5 3i g g ib i b i

i i i gm i i g b

m m mk T k TD T Tm q m m k

μμ

υ+

= = = ++

E

effective heating from electric fieldfrom (eq-7)

H.W. Ellis, et al, At. Data Nucl. Data Tables 17, 177 (1976 )

1

10

100

T i / T

g

1042 4 6 8

1052 4 6 8

1062 4 6 8

107

electric field [V/m]

N2+

He+

effective ion temperature in He gas

16

2.3 Necessary parameters-6: diffusion of neutrals

classical gas kinetic theory

collision integral [-]

1/ 23 3/ 2

211.8583 10 (eq-12)n g

n gn g LJ LJ

m mD T

m m pσ− ⎛ ⎞+

= × ⎜ ⎟⎜ ⎟ Ω⎝ ⎠[atm]

Lennard-Jones radius [Å]

R.B. Bird, et al., Transport Phenomena (Wiley, New York, 2002), page 526R.J. Kee, et al., Sandia Report SAND86-8246 (1986)

ΩLJ [-]σLJ [Å]ΩLJ [-]σLJ [Å]

0.78983.338O30.74873.0985N2

0.79813.038NO20.8882.591H2O

0.80923.202N2O0.73672.663O

0.7553.017H2O20.72982.937N

0.75463.017O20.62572.576He

0.74873.099NO0.77132.313H

17

2.4 Boundary condition-1

electrons: ( )81 (eq-13)4

b ee e s e e s i i

ie

k Tn nm

α μ α γπ

′⋅ = + − ⋅∑Γ n E Γ n

1 ( 0)0 ( 0)

1 ( 0)(eq-16)

0 ( 0)

s

s

α

α

⋅ ≥⎧= ⎨ ⋅ <⎩

⋅ <⎧′ = ⎨ ⋅ ≥⎩

E nE n

E nE n

81 (eq-14)4

b ii i s i i

i

k Tn nm

α μπ

⋅ = +Γ n E

switching function

81 (eq-15)4

b nn n

n

k Tnmπ

⋅ =Γ n

ions:

neutrals:

Boundary conditions for species continuity equations…

G.J.M. Hagelaara, et al., Phys. Rev. E 62, 14521454 (2000)Y.B. Golubovskii, et al, J. Phys. D 35, 751 (2002)

E

i

e

E

i

ee

n n

18


secondary electron emission coefficientfrom ions and metastables

( )th~ 0.016 2 (eq-17)Eγ ϕ−

Yu.P. Raizer, Gas Discharge Physics, (Springer, Berlin,1997) pages 68-71

E

ie

E

ie

5.32Pt

4.54W

4.5Ni

4.31Fe

4.25Alϕ [eV]

work function

γγ

0.03O4+0.15N+

0.07N4+0.20He2

+

0.03O2+0.13He2

*

0.09N2+0.23He+

0.1O+0.16He*

secondary electron emission coefficient (ϕ = 5.0 eV)

(semi-empirical formula)

19


option 1: ε = const. (0.5 or 1.0 eV)

option 2: ( )w5 (eq-18)3 e s i i

iε ε ε α γ

⎡ ⎤⎧ ⎫⋅ = ⋅ − ⋅⎨ ⎬⎢ ⎥

⎩ ⎭⎣ ⎦∑Γ n Γ n Γ n

inward flux from secondary electrons (εw~5 eV)

1.2

0.8

0.4

0.0

dens

ity

[10

17 m

-3]

2.01.51.00.50.0x [mm]

ne (BC-1) ne (BC-2)5

4

3

2

1

0

elec

tron

ener

gy [e

V]

2.01.51.00.50.0x [mm]

ε (BC-2)

ε (BC-1)

1D RF discharge in helium (Φ = 320 V)

Boundary conditions for electron energy equation (EEE)…

20

option 1: Φ = Φext at a powered conducting electrode= 0 at a grounded conducting electrode

option 2: on a dielectric surface

0 0 (eq-19)

(eq-20)

r d s

sj j

j

ed qdt

ε ε ε σσ

= +

= Γ∑

E E

Ed E


from Gauss’s lawdielectric

interface (Φs, σs)

gas

x0 ld

for perfect dielectric:

0sd

dE

LΦ −

= −

Boundary conditions for Poisson’s equation…

21

2.5 Local field approximation (LFA)-1

( )e-N

5 5 (eq-3)3 3

ee e e e

nn D Q

tε

ε ε∂ ⎛ ⎞+ ∇ ⋅ − ∇ = − ⋅ −⎜ ⎟∂ ⎝ ⎠

Γ Γ E

3 200

F dε ε ε∞ ′ ′= ∫EEDF (F0)

Two options to calculate mean electron energy…

EEE:

LFA: ε = f (E)

a fitting function or lookup table

0.01

0.1

1

10

ener

gy

[eV

]

102 103 104 105 106 107

electric field [V/m]

Table Fitting

Electron energy dependence of field

He, 1 atm, 300 K1015

1016

1017

1018

dens

ity

[m

-3]

2.01.51.00.50.0x [mm]

ne ni nnEEE LFA

comparison between EEE and LFA

22

2.5 Local field approximation (LFA)-2

ne

x

Y. Sakiyama et al., J. Appl. Phys. 101, 073306 (2007)V.R. Soloviev et al., J. Phys. D 42, 15208 (2009)

e e e en D nμ ∇EE

e

e( ) 0e e e e en D nμ− ⋅ = − − − ∇ ⋅ <Γ E E E

heating rate

electrons are cooled by field…!?

( )entε∂

∂5 53 3e e en Dε ε⎛ ⎞+ ∇ ⋅ − ∇⎜ ⎟

⎝ ⎠Γ e-N ~ 0e Q= − ⋅ −Γ EEEE:

Not negligible

A problem of the LFA in atmospheric pressure plasmas…

23

3.1 Introduction

3.2 How to set up and run a model in COMSOL

3.3 COMSOL with a MATLAB script

3.4 Example: plasma needle simulation (updated!)

3. Simulation using COMSOL and MATLAB

24

3.1 Introduction

e in

Problem setting

• How to set up and run a model in COMSOL?

• How to evaluate the simulation results?

• How to control the current/power, instead of voltage?

( )

,

e-N

0

( , , ) (eq-1)

( , , ) (eq-2)

5 5 (eq-3)3 3

( , , ) (eq-4)

jj j l

l

j j j j j

ee e e e

j jj

nR j e i n

tn D n j e i n

nn D Q

tq n j e i n

μ

εε ε

ε

∂+ ∇ ⋅ = =

∂= ± − ∇ =

∂ ⎛ ⎞+ ∇ ⋅ − ∇ = − ⋅ −⎜ ⎟∂ ⎝ ⎠∇ ⋅ = =

∑

∑

Γ

Γ E

Γ Γ E

E

25

• A convenient and powerful platform to solve reactive plasma equations: usually sets of coupled, nonlinear PDEs (partial differential equations) with associated initial and boundary conditions

• Treat charged and neutral species as continuum fluids (fluid model)

• Use either predefined modules (e.g. convection and diffusion, Helmholtz equation, etc.) or general PDE form

• Matlab scripts offer more flexible control of COMSOL ( e.g. solving equations sequentially and iteratively)

3.2 Setting up and running a model-1

General idea about COMSOL Multiphysics

26

solve Boltzmann equation

solve plasma equations

cross section

reaction rate μe, De

1. drift-diffusion approximation

νm >> τRF−1

νm: ~1011-1012 [s−1] for electrons~109-1010 [s−1] for ions

2. electron energy equation

Before starting to build a model…

νε (~109) >> τRF−1


27

Step 1: Input constant parameters and variables

Step 2: Draw simulation domain and generate meshes• use of symmetry (3D → 2D → 1D)

Step 3: Add the governing equations and the boundary conditions• general PDE mode rather than predefined modules• Lagrange quadratic element mostly works• finer meshes at electrodes and coarser meshes at the center

Step 4: Select time dependent solver• UMFPACK (default linear solver), or PARDISO (memory efficient)• Absolute tolerance: 0.0001, Relative tolerance : 0.001


28

Initial conditions:•continuity equation: low and uniform density (e.g. ~1012 m−3)•electron energy equation: low and uniform (e.g. ~1 eV)•Poisson’s equation: linear potential profile between electrodes

A few more tips before running the simulation…

Periodic steady state:•Running for 100-1000 RF cycles•Recording transient data of all variables to see the convergence at a fixed point (e.g. at the center)

Run the simulation!


29


2.0

1.5

1.0

0.5

0.0

norm

aliz

ed v

aria

bles

5004003002001000number of RF cycles

ne

ni

nnε

Φ

Transient behavior of variablesat the center of gap (= 1 mm)

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

dens

ity

[10

17 m

-3]

2.01.51.00.50.0x [mm]

5

4

3

2

1

0

electron energy [eV]

ne

ni

nn

ε

Phase-averaged propertiesafter 500 RF cycles

30


total current continuity:

computational environment• CPU: dual AMD Opteron 250• memory: 12 GB• OS: Linux (OpenSuse)

load of the model• number of meshes: ~250• number of DOF: ~4000 60 RF cycles per hour

(~5 hours until a steady state)memory usage: ~ 600 MB

After running the simulation…

total 0 const. (eq-5 )j jj

j qt

ε ∂ ′= + =∂ ∑E Γ

mesh size dependency: doubling (or halving) the mesh size

grid Peclet number (for ions): grd ~ 2 (eq-21)i

i

hP

Dμ

= <E

31

Example: a fixed current density

solve for a single RF cycle

3.3 COMSOL with Matlab-1

A sample Matlab script to control the COMSOL solver

calculate phase-averaged current density

total 00

1 (eq-22)RF

j jjRF

J q dtt

τ

ετ

⎛ ⎞∂= +⎜ ⎟⎜ ⎟∂⎝ ⎠

∑∫E Γ

adjust magnitude of the voltage

( )0 goal total (eq-23)c J J′Φ = Φ + −

32

V-I curve predicted by a fluid model

voltage control

2.0

1.5

1.0

0.5

0.0

curr

ent d

ensi

ty [

mA

/mm

2 ]

5004003002001000

voltage amplitude [V]

current control

3.3 COMSOL with Matlab-2

lost convergence

33

•power: <1W• voltage: ~300Vpkpk

• frequency: 13.56MHz• gas: helium

Images of plasma needle discharge

E. Stoffels et al., Plasma Phys. Control. Fusion 46, B167 (2004)

3.4 Plasma needle-1

without surface with surface

http://medicalphysicsweb.org/

Applications in biomedicine

34

1 mm

needle

Mesh size: 3 ∼ 140 μmNumber of mesh: 3,000 ∼ 5,000Shape function: Lagrange-quadraticNumber of DOF: 70,000 ∼ 90,000

RF (13.56 MHz)

φ = 30 μm

axis of symmetry

1 mm

3.4 Plasma needle-2

35

[1017 m-3]

0

1

2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

1015

1016

1017

3.4 Plasma needle-3: discharge at 1 mW

36

-0.2

-0.1

0.0

0.1

0.2

curr

ent

[mA

]

1.00.80.60.40.20.0f t

-1.0

-0.5

0.0

0.5

1.0

normalized voltage

Vext

Idsp

Ie

Iion

Itotal

Phase-averaged particle density Current properties

1015

1016

1017

1018

1019

1020

dens

ity

[m-3

]

1.00.80.60.40.20.0

z [mm]

electron He* He+ He2* He2+ N2+


37

[1025 m-3s-1]

0 2 4 6 8 10

ionization rate on the symmetry axis

Sheath

16

12

8

4

0

elec

tron

ener

gy

[eV

]

0.300.200.100.00z [mm]

-60

-40

-20

0

electric field [105 V

m-1]

4

3

2

1

0

-1heat

ing

rate

[1

09 Wm

-3 ]

0.300.200.100.00z [mm]

joule heating inelastic loss elastic loss


38

[1019 m-3]

0

1

2

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

1017

1018

1018

1019

1019


39

Phase-averaged particle density Current properties

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

curr

ent

[mA

]

1.00.80.60.40.20.0f t

-1.0

-0.5

0.0

0.5

1.0

normalized voltage

Vext

IdspIe

Iion

Itotal

1016

1017

1018

1019

1020

1021

dens

ity

[m-3

]

1.00.80.60.40.20.0z [mm]



40

[1027 m-3s-1]

0 3 6 9 12 15

ionization rate on the symmetry axis

Sheath-150

-100

-50

0

50

100 electric field [105 V

m-1]

0.200.150.100.050.00z [mm]

20

15

10

5

0

elec

tron

ener

gy

[eV

]

4

3

2

1

0

-1heat

ing

rate

[1

011 W

m-3

]

0.200.150.100.050.00z [mm]


-150

-100

-50

0

50

100 electric field [105 V

m-1]

1.00.90.80.7z [mm]

20

15

10

5

0

elec

tron

ener

gy

[eV

]

4

3

2

1

0

-1heat

ing

rate

[1

011 W

m-3

]

1.00.90.80.7z [mm]



41

1relative computational cost4,000Number of finite element3 μmMinimum mesh size

2D axisymmetric

1.5 mm

1 m

m

Needle Trea

ted

surfa

ce

1 mm

Needle tip

Treated surface

1D spherical3 μm2001/50

3.4 Plasma needle-9: from 2D to 1D

42

< low power (1 mW) > < high power (1000 mW) >

1015

1016

1017

1018

1019

1020

dens

ity

[m-3

]

1.00.80.60.40.20.0distance from inner electrode [mm]

metastables

electrons

positive ions

1017

1018

1019

1020

1021

1022

dens

ity

[m-3

]


metastables

positive ions

electrons

3.4 Plasma needle-10: 1D spherical model

43

< phase-averaged total ionization rate >

1022

1023

1024

1025

1026

1027

1028

ioni

zatio

n ra

te [

m-3

s-1]

1.00.80.60.40.20.0distance from the inner electrode [mm]

100 mW

300 mW

1000 mW

1 mm

(He flow: 2 m/s)

He

He

Photo: collaborative work with Dr. E.Stofflesin Eindhoven University of Technology

3.4 Plasma needle-11: 1D spherical model

44

< high power condition >< low power condition >

1016

1017

1018

1019

1020

dens

ity

[m-3

]


metastables

positive ions

electrons

1017

1018

1019

1020

1021

dens

ity

[m-3

]


metastables

positive ionselectrons

3.4 Plasma needle-12: 2D and 1D

45

20

15

10

5

0

pow

er

[mW

]

20016012080

voltage amplitude [V]

2D axisymetric 1D spherical

Power-voltage curve

1017

1018

1019

1020

1021

dens

ity

[m-3

]

1.00.80.60.40.20.0z [mm]

1017

1018

1019

1020

1021

dens

ity

[m-3

]


Time-averaged density

1D spherical

2D axisymmetric(on the axis)

3.4 Plasma needle-13: 2D and 1D

46

4.1 Introduction

4.2 Chemical reactions in fluid model

4.3 Example-1: simplified chemistry model for helium with impurity

4.4 Example-2: plasma chemistry in air

4.5 Tips for simulation with detailed chemistry model (advanced)

4. Plasma chemistry at atmospheric pressure

47

4.1 Introduction

Mass spectrometry in helium plasma needle discharge

E. Stoffels et al., IEEE Trans. Plasma Sci., 36, 1441 (2008)

E. Stoffels, et al., Plasma Sources Sci. Technol. 15, 501 (2006)

48

4.2 Chemical reactions in fluid models-1

solve Boltzmann equation

solve plasma equations

μe, De

cross sections for various species and paths

various electron impact reaction rate coefficients

reaction rate coefficientsfor neutrals/ions

10 exp( / ) (eq-24)cr th

bk c T E k T= −

(c1= 0: Arrhenius equation)

• classical gas kinetic theory• transition state theory• empirically…

Maxwellian?

Yes!

No… (= electrons)

49

f(Tg)f(Tg)f(Tg)f(Tg)f(ε)f(ε)f(Tg)f(Tg)f(ε)f(ε)f(ε)f(ε)

ion recombinationA+ + B− + M → A + B + M

electron impact dissociationA2 + e → A + A + 2e

A + B + M → C + D + M

A+ + B → A + B+

A− + B → A + B + eA + e → A−

A+ + e + M → A + MA* + B → B+ + A + eA* + A* → A+ + A + e

A* + e → A+ + 2eA + e → A+ + 2eA + e → A* + e

neutral-neutral reaction

charge transferelectron detachmentelectron attachmentelectron recombinationPenning ionizationassociative ionization

stepwise ionizationelectron impact ionization electron impact excitation

f(ε): rate constant obtained by solving Boltzmann equation


50

• Cross section data set by A.V. Phelps in JILA (http://jila.colorado.edu/~avp/)• Cross sections available in BOLSIG+• GAPHYOR online database (http://gaphyor.lpgp.u-psud.fr/)

• M. Capitelli, et.al, Plasma kinetics in atmospheric gases (Springer, Berlin, 2000).

• L.M. Chanin, et al., Phys. Rev. 128, 219 (1962).• T. D. Mark, et al., Phys. Rev. A 4, 1445 (1971).• H.W. Ellis, et al., At. Data Nucl. Data Tables 22, 179 (1978).• C.B. Collins, et al., J. Chem. Phys. 68, 1391 (1978).• J.W. Parker, et.al, J. Chem. Phys. 75, 1804 (1981).• J.M. Pouvesle, et.al, J. Chem. Phys. 77, 817 (1982).• H. Bohringer, et al., Int. J. Mass Spectrom. Ion Phys. 52, 25 (1983).• J.M. Pouvesle, J. Chem. Phys. 83, 2836 (1985).• F. Emmert, et al., J. Phys. D 21, 667 (1988).

Resources for reaction rate coefficient and cross sections data set


51

• H. Matzing, Adv. Chem. Phys. 80, 315 (1991).• I.A. Kossyi, et.al, Plasma Sources Sci. Technol. 1, 207 (1992).• M.J. Kushner, J. Appl. Phys. 74, 6538 (1993).• P.C. Hill, et al., Phys. Rev. A 47, 4837 (1993).• T.L. Williams, et al., Mon. Not. R. Astron. Soc. 282, 413 (1996).• R. Atkinson, et al., J. Phys. Chem. Ref. Data 26, 1329 (1997).• G.S. Voronov, At. Data Nucl. Data Tables 65, 1 (1997).• O. Eichwald, et al., J. Appl. Phys. 82, 4781 (1997).• H. Tawara, et al., NIFS DATA-51 (1999).• V.G. Anicich, J. Phys. Chem. Ref. Data 22, 1469 (1999).• S. Rauf, et al., J. Appl. Phys. 88, 3460 (1999).• L.W. Sieck, et al., Plasma Chem. Plasma Process. 20, 235 (2000).• J.T. Herron, et al., Plasma Chem. Plasma Process. 21, 459 (2001).• I Stefanovic et al., Plasma Sources Sci. Technol. 10, 406416 (2001).• F. Tochikubo, et al., Jpn. J. Appl. Phys. 41, 844852 (2002).• R. Dorai, et al., J. Phys. D 36, 666 (2003).• C.D. Pintassilgo et al., J. Phys. D 38, 417430 (2005).• K.R. Stalder, et.al, J. Appl. Phys. 99, 093301 (2006). etc. etc. etc…


52

A. Ricard et al., Surf. Coat. Tech. 112, 1 (1999)

OES in helium glow DBD

4.3 Helium plasmas with impurity-1

He DBD for material processing

measured waveform

53

Y B. Golubovskii, et al.,J. Phys. D 36, 39 (2003)Y. Sakiyama et al., J. Appl. Phys. 101, 073306 (2007)


N2+ + e → N2R13

He2+ + N2 → N2

+ + He2*R12

He2* + N2 → N2

+ + 2He + eR11

He* + N2 → N2+ + He + eR10

He2+ + e → He* + HeR9

2He2* → He2

+ + 2He + eR8

2He* → He2+ + eR7

He2* + M → 2He + MR6

He+ + 2He → He2+ + HeR5

He* + 2He → He2* + HeR4

He* + e → He+ + 2eR3

He + e → He+ + 2eR2

He + e → He* + eR1

ReactionIndex

e

He*

He He+ +R2

R1R3

He2*

R4

eHe2+ +

R7

R8

N2+ e+

R11R10

54


T. Martens et al., Appl. Phys. Lett. 92, 041504 (2008)

Volume-averaged particle densityfor different impurity level (helium DBD)

X. Yuan, et al., IEEE Trans. Plasma Sci., 31, 495 (2003)

Particle density distributionswith 0.5ppm N2 (helium RF)

55

4.4 Plasma chemistry in air-1

48 species• 11 negative particles: e, O−, O2

−, O3−, O4

−, H−, OH−, NO−, N2O−, NO2

−, NO3−

• 16 positive particles: N+, N2+, N3

+, N4+, O+, O2

+, O4+, NO+, N2O+,

NO2+, H+, H2

+, H3+, OH+, H2O+, H3O+

• 21 neutrals/radicals: N, N*, N2, N2*, N2**, O, O*, O2, O2*, O3,NO, N2O, NO2, NO3, N2O5, H, H2, OH,H2O, HO2, H2O2

630 reactions

• 21 electron impact excitation/ionization/dissociation• 76 electron recombination/attachment• 159 charge transfer• 245 ion recombination• 129 neutral-neutral reactions

56

• pulse-like plasmas• 0D simulation (spatially uniform plasmas)• pressure: 1 atm• gas temperature: 300 K• gas concentration: air with 30% humidity• computational time: ~10 hours

(Dual Opteron 250, 12GB Mem, comsol3.5a)


,j

j ll

nR

t∂

=∂ ∑

on (100 ns)

1 cycle (100 μs)

E = 3×106 V/mne = 1017 m−3

Various air DBD devices for biomedicine

Drexel, US Max-Planck, DEBerkeley, US

Model description

57

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

dens

ity fr

actio

n

10-9 10-7 10-5 time [s]

neutralsOx

NxOy

HxOy

Nx

10-9 10-7 10-5 time [s]

positive ions

Ox+HxOy

+

10-9 10-7 10-5 time [s]

negative ions

NxOy-Ox

-

electronHxOy

-


time development of species density (in periodic steady state)

58

4.4 plasma chemistry in air-4

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

dens

ity fr

actio

n

O3

N2O5

N2OH2O2O2

*

ONO2HO2

OH NOH2

NNO3

(charged particle density < 10ppb)

phase-averaged species density (in periodic steady state)

59

74%

O + O2 + M → O3 + M

6%: O3 + OH → HO2 + O2

6%: e + O3 → O + O2 + e

4%: O2* + O3 → O + 2O2

3%: NO + O3 → NO2 + O2

Ozone reaction paths


60

30%: O + NO2 → NO + O2

41%: NO + O3 → NO2 + O2

8%: N* + O2 → NO + O*

5%: NO + HO2 → NO2 + OH

4%: N + OH → NO + H


NO reaction paths

61

40%: NO + O3 → NO2 + O2

29%: O + NO2 → NO + O2

8%: NO2 + NO3 + M →

N2O5 + M

7%: O + NO2 + M →

NO3 + M

4%: NO + HO2 → NO2 + OH


NO2 reaction paths

62

NO2 NO

O3

O

NO3

N2O5

HO2

N

OH

N*

Oe, O2

*

O2


63

r

r

s rs s s s

N1 211 1 1 1

N1 222 2 2 2

N N1 2N N N N

n R R Rtn R R Rt

nR R R

t

∂+ ∇ ⋅ = + + +

∂∂

+ ∇ ⋅ = + + +∂

∂+ ∇ ⋅ = + + +

∂

Γ

Γ

Γ

K

K

M

K

Before starting simulation with hundreds of reactions…

Ns

Nr

For example…• 40 species/700 reactions (air)• one-dimensional, RF excitation• 12 GB, Dual Opteron 250

4.5 Tips for complex chemistry model-1

highly nonlinear problems(stiff problem)

heavy computation

wait for a few months until reaching steady state?

64

1. Sensitivity analysis


H. Rabitz et al, Ann. Rev. Phys. Chem. 34, 419 (1983)

To reduce the computational time…

jj

dndt

+ ∇ ⋅Γ ( , ) (eq-1 )

(eq-25)

0 (eq-26)

r rl l l l

jr r r r r r

jl l l l l l

jljlr

jl

R k t k n n

nd n R R R t R Rdt t nk k k k k k

dS R R Sdt nk

′ ′′ ′′= =

⎛ ⎞⎛ ⎞ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂= = + = +⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂

= + =∂∂ sensitivity coefficient

jl rl

nSk∂

=∂

(steady state assumption)

1

(eq-27)jl rj l

R RSn k

−⎛ ⎞∂ ∂

= −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠(j×l) matrix (l×1) matrix

• low computational load• perturbation around equilibrium state• sensitivity ≠ dominant reaction path

65


To reduce the computational time…2. Reference calculation

jj

dndt

+ ∇ ⋅Γ ( , ) (eq-1 )r rl l l lR k t k n n′ ′′ ′′= =

solve ODEs in several typical conditions (e.g. given ne, ε)

calculate contribution matrix (normalized reaction rates for each species)

s

r s r

11 1N

N 1 N N

(eq-28)lj

R R

R R

⎡ ⎤⎢ ⎥

Λ = ⎢ ⎥⎢ ⎥⎣ ⎦

L

M O M

L

eliminate unimportant reaction paths (e.g. 1% threshold)

• medium computational load• directly evaluating reaction rate• need of finding typical conditions

species

reaction

normalized

66

5.1 Introduction

5.2 Interaction between neutral gas flow and plasmas

5.3 Example-1: Plasma jet and flow (one-way coupling)

5.4 Example-2: RF plasma needle and flow (two-way coupling)

5. Neutral gas dynamics

67

5.1 Introduction

Old Dominion, US

Loughborough, UK

Eindhoven, NL

• gas: rare gas, air, hydrocarbon

• power: DC, RF, Microwave

• application: etching, thin film deposition, biomedicine

Greifswald, GE

feed gas

plasmas

air

68

5.2 Plasma-flow interaction-1

( )( )( )( )

0

p

air air

0 (eq-29)

( ) (eq-30)

(eq-31)

0 (eq-32)(eq-33)

iu p g

T c T Q

Dp RT

η

ρ

ρ ρ ρ

λ

ρω ρ ω

ρ

∇ ⋅ =

∇ ⋅ = −∇ − ∇⋅ − −

∇ ⋅ − ∇ + =

∇ ⋅ − ∇ =

=

u

u τ

u

u

Governing equations for neutral gas flow:total mass conservation:

total momentum conservation:

total energy conservation:

mass conservation of air:

perfect gas law:

(need compressible N-S equation !)

R.B. Bird, et al., Transport Phenomena (Wiley, New York, 2002)

Assumption: laminar flow, no thermal radiation

Re Re (~ 2000: inside a tube)cudυ

= <

69

plasma fluid model


neutral gas heat and mass transfer model

• gas temperature• gas velocity• gas component

• momentum transfer• energy transfer

• background gas density• mobility and diffusion coefficients for electrons/ions• diffusion coefficients for neutrals• reaction rate coefficients (including elastic energy loss)• flux of species

70


Momentum transfer from plasmas to neutral gas flow

J.P. Boeuf et al., J. Appl. Phys. 97, 103307 (2005)C.C. Leiby, Phys. Fluids 10, 1992 (1967)J-S Chang, IEEE Trans. Diel. Electr. Insul. 1, 871 (1994)

body force (ion wind): pls ~ (eq-34)i if q n E

NN

N

Ni

E

pls ,,

,

( ) (eq-6)

( )

m

j m j jj i e

j j jj j j j j j j

j i e

mn mn qn p mnt

f mn

m nq n p m n

t

υ

υ=

=

∂+ ∇ ⋅ = − ∇ −

∂=

∂⎧ ⎫= − ∇ − − ∇ ⋅⎨ ⎬

∂⎩ ⎭

∑

∑

u uu E u

u

uE u u

Under typical conditions: , ( ) othersi i e e b e eq n q n k n T> ∇ >E E

from ions

71


Energy transfer from plasmas to neutral

external electrical power

from plasmas to neutral

(=Joule heating)

pls e-N N-N~ Γ (eq-35)i iQ Q q Q+ ⋅ +Eheating source:electrons (eq-8) ions

Γ Γe e i iP q q= ⋅ + ⋅E E

source terms for EEE

e-N e-iΓe e eS q Q Q= ⋅ − −E

chemical reactions

source terms for IEE

e-i i-N

i-N e-i

Γ (~ 0)~ Γ

i i i

i i

S q Q QQ q Q

= ⋅ + −

⋅ +

EE

pls e-N i-N N-N

e-N e-i N-NΓi i

Q Q Q Q

Q q Q Q

= + +

= + ⋅ + +E

elastic collision

Energy transfer between electrons, ions, and neutrals

72

Assumption: heat/mass transfer from neutral ( < ~10−3 s)>> τRF (external electric field oscillation)


solve time dependent plasma equations for one RF cycle

solve steady state neutral gas flow equations

pls pls0 0

1 1,RF RF

i i i iRF RF

f q n dt Q q n dtτ τ

τ τ= = ⋅∫ ∫E E

73

X.Liu and M.Larousi, J. Appl. Phys. 100, 063302 (2006).

N. Merciam-Bourdet et al, J. Phys. D 42, 055207 (2009)

M. Teschke et al, IEEE Trans. Plasma Sci. 33, 310 (2005)

Observed ring-shaped emission pattern

5.3 Plasma jet-1: introduction

74

He

N22D steady state

neutral gas flow

r

N2 densityc

• He: 7 slpm• 7 kV pulse excitation• 8 kHz repetition

1D plasma dynamics in cylindrical coordinates(cross sectional view)

r

N2 density distribution

5.3 Plasma jet-2: one way coupling

75

( )( )( )

2 2N N

0

0

i

D

u p

ρ

ρω ρ ω

ρ

∇ ⋅ =

∇ ⋅ − ∇ =

∇ ⋅ = −∇ − ∇⋅

u

u

u τ

Compressible N-S equation

: total mass continuity

: continuity for N2

: total momentum continuity

0 10 20 30 50

[mm]

40

z

r

-10-20

10

20

0

Mole fraction of airN2 rich

He rich

He channel

0.4

0.3

0.2

0.1

0.0

mol

e fr

actio

n of

N2

2.01.51.00.50.0r [mm]

5.3 Plasma jet-3: neutral gas flow

76

r

0

nN2

(mass continuity)

(drift-diffusion)

(Poisson’s eq. in r-direction)0

( )1

sgn( )

( )1

i ii

ii i i i r i

ri i

i

n r St r r

nq n E D

rrE q n

r r

μ

ε

∂ ∂ Γ+ =

∂ ∂∂

Γ = −∂

∂=

∂ ∑

Fluid model with local field approximation

N2

Hechannel

• species: e, He*, He2*, He+, He2

+, N2*, N2

+

• rate coefficients: from local Boltzmann equation (local field and N2 concentration)

• pressure: 1 atm• temperature: 300 K• solver: COMSOL and Matlab

5.3 Plasma jet-4: plasma dynamics

77

r

0

nN2

Ez

125 μs (8 kHz)

Ez = 3×105 V/m

0

400 ns

2 2r zE E= +E

Given electric field (not self-consistent!)

givenPoisson’s eq.

( )k f= Ereaction rate:

r

5.3 Plasma jet-5: given electric field

78

given electric field

particle density distributionin radial direction

5.3 Plasma jet-6: time evolution of bullet

79

5.3 Plasma jet-6: time evolution of bullet

80

1023

1024

1025

1026

1027

1028

1029

reac

tion

rate

[m

-3s-1

]

2.01.51.00.50.0r [mm]

excitation direct ionization Penning ionization

1023

1024

1025

1026

1027

1028

1029

reac

tion

rate

[m

-3s-1

]

2.01.51.00.50.0r [mm]

excitation direct ionization Penning stepwise associative

early stage (200 ns) late stage (400 ns)

43210E z

[105 V

/m]

10008006004002000time [ns]

eHe*

He He2* N2

+

N2

e

5.3 Plasma jet-7: Penning ionization is a key

81

amplitude : 7 kVpulse width : 2 μsrepetition rate : 8 kHz

inner diameter: 3 mm

Time resolution: 1 nsSpatial resolution: ~50 μm

Helium flow: 7 slpm

5.3 Plasma jet-8: for comparison with simulation

82

6000

5000

4000

3000

2000

1000

0

inte

nsity

800700600500400300

wavelength [nm]

OHN2

N2+

N2+

N2+

He

He He

He

He

O

2 2u gB Σ X Σ+ +→ 3 , , 2 ,S P D S P→

N2+(B)

He*, He2* (40%)

N2+(X)

3SPD

2S

2Pe

kem

kex

Emission rate = kem~ kex (kem >> kex)~ kiz (kiz ~ kex)

He2+ (75%)

kPen

kchg

Emission rate = kem~ kPen+ kchg (kem >> kPen, kchg)

kem

Effective emission rate = kiz + kPen+ kchg

5.3 Plasma jet-9: ionization = emission?

83

1.0

0.8

0.6

0.4

0.2

0.0

norm

aliz

ed in

tens

ity

-2 -1 0 1 2r [mm]

OES Model

0 20 mm

zr

40 mm

integration time for OES:100 ms (800 bullets)

Y. Sakyiama et al, Appl. Phys. Lett. 96 (2010) 041501

5.3 Plasma jet-10: ring shaped pattern

84

J.Goree, et al, J. Phys. D. 39, 3479 (2006) and IEEE Trans.Plasma Sci. 34, 1317 (2006)(Images courtesy of Prof. John Goree)

• RF(13.56 MHz)-excited• gas: He • gap distance: 2.5 ~ 4 mm

Killing pattern

light intensity

1 2 3 4 5 6 7 8

radial position

0.3 m/s

5 mm

1 2 3 4 5 6 7

radial position

1.0 m/s

5.4 Plasma needle-1: introduction

85

( )

0

sgn( )

5 53 3

ii i

i i i i i i i

ee e e e

i i

n St

q n D n nn

n D Qt

q n

μ

εε ε

ε

∂+ ∇ ⋅ =

∂= − ∇ +

∂ ⎛ ⎞+ ∇ ⋅ − ∇ = − ⋅ −⎜ ⎟∂ ⎝ ⎠∇ ⋅ = ∑

Γ

Γ E u

Γ Γ E

E

Neutral Gas flow

( ) ( )( )( )

air air

p

0, 0

i i i

i i el

D

u p q n

T c T q Q

ρ ρω ρ ω

ρ

λ

∇ ⋅ = ∇ ⋅ − ∇ =

∇ ⋅ = −∇ − ∇⋅ +

∇ ⋅ − ∇ + = Φ + +

∑∑

u u

u τ E

u Γ E

Plasma dynamics

(mass conservation)

(momentum conservation)

(energy conservation)

(mass conservation)

(drift-diffusion)

(electron energy)

(Poisson’s equation)

He flow

air (diffusion)

Y. Sakyiama et al, Plasma Sources Sci. Technol. 18, 025022 (2009).

5.4 Plasma needle-2: two-way coupling

86

insulator

5 mm

5 m

m

1 mm 2.5 mm

r

zneedle

glass plate

N2 (1atm)

N2 (1atm)

He (1.5 m/s)

Unknown variablesHe/N2 concentrationgas pressuregas flow velocity (r and z)gas/needle temperature

5.4 Plasma needle-3: neutral gas flow

87

needle

insulator

2 mm

1 m

m

1.5 mm

0.85 mm

glass plate

Unknown variablesdensity: electron, He*, He+,

He2*, He2

+, N2+

electron energyelectrical potential

5.4 Plasma needle-4: plasma dynamics

88

Gas temperatureMole fraction of air (log scale)

5.4 Plasma needle-5: gas flow field

89

-0.5

-1

0

0.5

z [m

m]

1018

10171016

electrons

10181017

1016

He*

1018

1019

0

-0.5

-10.5 1 1.5 2

0

0.5

z [m

m]

r [mm]

1018

1017

1016

He2+

r [mm]0 0.5 1 1.5 2

10171016

N2+

5.4 Plasma needle-6: particle density

90

0

-0.5

-10.5 1 1.5 2

0

0.5

z [m

m]

r [mm]

0 0.2 0.4 0.6 0.8 1

[1023 m-3s-1]

gap:

3 m

m4 mm

needleinsulator

He He

Experimental resultsby J. Goree et al

Predicted emission intensity

5.4 Plasma needle-7: ring-shaped emission!

eHe*

He He2* N2

+

N2

e

91

Humid air concentration

needle Insulatortube

humid air concentration:2D neutral flow simulation

(N-S equation, convection-diffusion)

1D fluid model with detailed chemistry in spherical coordinates

on-axis(1mm gap)

off-axis(2mm gap)

Which is the major species hitting the surface?

S. mutans S. mutans

on-axis off-axis

nair/nHe<10-5 nair/nHe>10-3

5.4 Plasma needle-8: one-way coupling (again)

92

1016

1017

1018

1019

1020

1021

dens

ity

[m-3

]

2.01.51.00.50.0r [mm]

1016

1017

1018

1019

1020

1021

dens

ity

[m-3

]

1.00.80.60.40.20.0r [mm]

on-axis (1mm gap) off-axis (2mm gap)

He2+

N+,N2+

He2+

N+,N2+

O+,O2+

NO+

O+

H2O+,H+,OH+

NO+

5.4 Plasma needle-9: charged particle density

93

1016

1017

1018

1019

1020

1021

dens

ity

[m-3

]

2.01.51.00.50.0r [mm]

1016

1017

1018

1019

1020

1021

dens

ity

[m-3

]

1.00.80.60.40.20.0r [mm]

on-axis (1mm gap) off-axis (2mm gap)

He*,He2*

N,N*,N2*

He*,He2*

N,N*,N2*

O,O*,O2*

O,O*,O2*

H,OH,H2

H,OH,H2

NO

5.4 Plasma needle-10: neutral density

94

On-axis (1mm gap) Off-axis (2mm gap)

e

He*

He2*

N2+ O2

+He2+

He*

N2+ O2

+

H

H2O+

O N

NOOH

O2*O* N*

N2**

N2*

e

5.4 Plasma needle-11: reaction kinetics

95

12

10

8

6

4

2

0

flux

[10

19 m

-2s-1

]

e

NO2

+

H2O+

N2+

O

O2*

NOH OH

12

10

8

6

4

2

0

flux

[10

19 m

-2s-1

]

eN2

+

O2+

He2+

On-axis (1mm gap)

Off-axis (2mm gap)

S. mutans

S. mutans

5.4 Plasma needle-12: flux to a surface

96

6. Available codes for plasma simulation-1

1. Bolsig+:• Boltzmann solver for electrons• freeware• developed and managed by Dr. Hagelaar in LAPLACE

( G.J. M. Hagelaar et al., Plasma Sources Sci. Technol. 14, 722 (2005))

2. ELENDIF: • Boltzmann solver• Kinema Research & Software (http://www.kinema.com/)

3. HPEM:• solver for low pressure plasma processing reactors (ICP, RIE, ECR, etc)• developed and managed by CPSEG in U. Michigan

(http://uigelz.eecs.umich.edu/)

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4. CFD-ACE+:• general PDE solver• plasma physics module available• ESI Group (http://www.esi-group.com/)

5. ANSYS Fluent:• general fluid dynamics solver • applicable to low pressure CVD simulation• ANSYS Inc. (http://www.ansys.com/)

6. COMSOL Multiphysics:• FE (finite element) solver• ~20 pre-defined application modules from fluid dynamics to mechanics• plasma module included in the latest version 4.1• Comsol, Inc. (http://www.comsol.com/)


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7. SIGLO and SIPDP series:• plasma fluid solver in 1-D and 2-D from AC to RF• Kinema Research & Software (http://www.kinema.com/)

8. XPDP1, XPDP2, XPDS1:• particle-in-cell (PIC) solver• freeware• developed and managed by PTSG group in UC Berkeley

(http://ptsg.eecs.berkeley.edu/)

9. VORPAL, OOPIC Pro:• PIC solver• Tech-X corp. (http://www.txcorp.com/)

10. LSP Suite:• 2-D and 3-D PIC solver• Alliant Techsystems Inc. (http://www.mrcwdc.com/LSP/)


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cp: specific heat [J kg−1 K−1]D : diffusion coefficient [m2s−1]e: electron charge (=1.60×10−19) [C]Eth: ionization/excitation energy [eV]fpls: body force from plasmas [kg m s−2] g: gravity (= 9.81) [m s−2]J: current density [A m−2]h: grid size [m]kinel: reaction rate coefficient [s−1, m3s−1, m6s−1 ] kel: elastic collision rate coefficient [m3 s−1]kb: Boltzmann constant (=1.38×10−23) [J K−1]m: mass [kg]n : density [m−3]p: pressure [Pa]Qpls: heating from plasmas [W m−3]Qe-N: collisional energy loss [W m−3]Qη: viscous heat dissipation [W m−3]

Notation

R : reaction rate [m−3s−1]S: sensitivity coefficientt : time [s]T: temperature [K]

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ε : mean electron energy [eV]ε0 : vacuum permittivity (=8.85×10−12)

[CV−1m−1]εr : dielectric constantΦ: electrical potential [V]γ: secondary electron emission

coefficient [-]ϕ: work function [eV]λ: thermal conductivity [W m−1 K−1]ρ: neutral gas density [m−3]σs: surface charge [Cm−2]τRF: RF period [s]μ : mobility [m2 V −1 s−1]νm: momentum transfer collision

frequency [s−1]νε: electron energy relaxation frequency [s−1]ωair:mass fraction of air

E: electric field [V m−1]Γ: flux [m−2s−1]n: unit surface vector [-]τ: stress tensor [Pa]u: neutral gas velocity [m s−1]

Notation-2

Subscriptse : electroni : heavy positive ionsn: neutralsg: background gas

icops_minicourse_yuki.pdf

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