modélisation des plasmas : photoionisation et effet...

Anne BOURDON1, Pierre SEGUR2

1Laboratoire EM2C, Ecole Centrale Paris, Châtenay-Malabry 2Université de Toulouse, LAPLACE, Toulouse

Atelier A3: « Transfert radiatif dans la modélisation des plasmas »

Modélisation des plasmas :photoionisation et effet photoélectrique

6èmes journées du réseau plasmas froids, 2-5 octobre 2007

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In collaboration with…

E. Marode (LPGP, Supélec)

S. Célestin (EM2C, Ecole Centrale Paris)J. Capeillère (LAPLACE, Toulouse)

N. Liu (CSSL, Penn State University, USA)V. Pasko (CSSL, Penn State University, USA)

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Radiative transfer in plasma discharges

Plasma discharges usually radiate in a wide spectral range

The light emitted is often used to characterize the plasma⇒ Measurement of concentrations of active species and excitation temperaturesin the medium

For some discharges, the radiation emitted by the discharge has a significantinfluence on the structure of the discharge⇒In this talk we will discuss mainly about two radiative processes that have asignificant influence on the structure of non-thermal plasma discharges atatmospheric pressure : photoionization (volume) and photoemission (surface)

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Outline

• Non-thermal plasma discharges at atmospheric pressure

• Role and modeling of photoionization

• Role and modeling of photoemission

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Applications of non-thermal discharges atatmospheric pressure

Applications in industry• Electrostatic precipitation• Ozone generation

Recent applications in industry• Surface treatments at atmospheric Pressure• Exhaust gas treatment and odor treament at atmospheric pressure : combination of anon thermal plasma and heterogeneous catalysis

Research studies on non thermal plasmas at Patm• Plasma coatings at atmospheric pressure• Plasma assisted combustion• Flow control• Sterilization, medical applications

Interest for industry:• Easier and cheaper to implement than low-pressure plasmas• Unique properties

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Non-thermal discharges at atmospheric pressure

Using electromagnetic waves [R.F. (10

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Discharge between two metallic electrodes in DC, AC or pulsed modes(with no transition to arc)

Example : « corona discharge » at atmospheric pressureOne electrode has a very small radius of curvature in comparison to thesecond one => the electric field is very high close to the thin wire or point

RISK : transition to arc => thermal plasma

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Dielectric Barrier Discharge (DBD) AC or pulsed modes

plane-plane reactor LPGP Orsay (France)

H.Russ et al (1999)

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Filamentary discharge : high concentration of electrons in a filament (radiusof the order of 100µm) ⇒ high concentrations of active species. However,local heating may be significant

Diffuse discharge : small concentration of electrons, large volume of thedischarge and heating is weak

At Patm, discharges may be filamentary or diffuse

(Photo courtesy of V. Schulz-von der Gathen

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+

+

+

++

+++

+

+++++

+ +

- --

- -

-

--

--

-

+

• Radius of the discharge=20-100µm,• Velocity 107-108cm/s•Nonuniform discharge : filament andstreamer head

Propagation of a positive streamer

Propagation of the discharge in thedirection opposite to the drift ofelectrons ⇒ Need for a source ofelectrons ahead of the streamer head :

+

conductive filament

positively chargedstreamer head

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- preionization

+

++

+++++

+

++++++ +

- --- -

---

---

+

-

- -

1

+

In the streamer head, the electricfield is the highest :

⇒gas ionization : production ofsecondary electrons

⇒avalanche ⇒propagation

+

+

+

++

+++

+

+++++

+ +

- --

- -

-

--

--

-

++

++

+++++

+

+++ +

- --- -

---

---

+

--

+

++++

- ------

-- --++ +++

++

+

2 4

+++

+ +

+

++

+++++

+

+++ +

- --- -

---

---

+

--

+

++++

- ------

-- --++ +++

++

+

3

+++

+

-

--

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- preionization

- photoionization

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++

+++++

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++++++ +

- --- -

---

---

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1

+

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- preionization

- photoionization

-In the streamer head, the electric

field is the highest :

⇒ production of excited molecules

⇒ radiative deexcitation in alldirections

⇒gas ionization : production ofsecondary electrons

⇒avalanche ⇒propagation

+

+

+

++

+++

+

+++++

+ +

- --

- -

-

--

--

-

++

++

+++++

+

+++ +

- --- -

---

---

++

++

+++++

+

++++++ +

- --- -

---

---

+

--

+

++++

- ------

-- --++ +++

++

+

hν

-

-

1 2 4

+

+++

+ +

+

++

+++++

+

+++ +

- --- -

---

---

+

--

+

++++

- ------

-- --++ +++

++

+

3

+++

+

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Preionization in air

Repetitive discharges (Pancheshnyi (05)) :

⇒Uniform preionization caused by propagation of preceding streamers andaccumulation of O2- ions

⇒ the level of preionization increases with growth of the repetition rate

⇒ Agreement between calculated and measured streamer parameters for a0.5-10Hz repetitively pulsed discharge in a 2cm gap with a preionization levelof 105 to 107 cm-3 between 350 and 760 Torr.

Naidis (06) has mentioned that in this case, the change in the air composition

has to be considered

In the literature different uniform preionization levels have been used tosimulate streamer propagation : e.g. Dhali and Williams (87) =>108 cm-3

Preionization level depends on experimental conditions

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Experiments on photoionization in air

Low pressure (few Torr) experiments (Teich (67), Penney and Hummert (70)) :

Measurements of current produced due to gas ionization in N2/O2 mixtures andair by radiation- from corona discharges (Penney and Hummert (70))- from electron avalanches in a uniform field (Teich (67))as a function of gas pressure, discharge current and distance between thedischarge and the collector of photoions

Atmospheric pressure experiments (Aints et al. (02)) :

Emission from positive and negative corona discharges in dry and moist air wasrecorded on photographic film as a function of the distance between thedischarge and the film

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Model of photoionization in air

Widely used model in air : Zheleznyak et al. (82)

Radiation emitted in the 98 -102,5 nm range by N2* => photoionization of O2

For λ

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Simulation of a filamentary discharge withphotoionization or preionization

Photoionization : a source term is added to the transport equations

Preionization ; Sph =0

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Modeling of Photoionization in air

Photoionization source term in dV2 : ⇒Radiation emission from volumes dV1⇒part of the radiation is absorbed on the distance r⇒ part of the radiation which arrives at the volume dV2 will lead to ionization

Non local term

Photoionization at a given point is due to emission in the whole volume

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Photon distribution function of frequency at position indirection and time t

Isotropic part of thedistribution function

=> Quadrature over the volume and the frequency range

Photoionizationcoefficient

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Integration to derive :

Stationary radiative transfer equation

Source term absorption

where

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Integral modeling for photoionization in airThe photoionization source term becomes :

For convenience, the photoionization coefficient is assumed to be proportional tothe absorption coefficient :

Assuming that coefficients are independent of the position in the gas:

=> To calculate the Sph term, it is necessary to know the spectral dependence ofthe different coefficients

Example : Main steps of the widely used model derived by Zheleznyak et al. forphotoionization in air : Radiation emitted in the 98 -102,5 nm range by N2*⇒ Photoionization of O2

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Zheleznyak’s model for photoionization in air

As the frequency range is narrow, the integral is written as the product of twointegrals:

In the interval 98-102.5nm, the absorption coefficient of O2 is a sharp function offrequency of the form:

where

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The emission term is assumed to be proportional to the ionization term

Finally :

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-A 3D integral has to be calculated at each point and ateach iteration : (NrxNz)x(NrxNz)

-Pre-calculation and tabulation of the part of the integralwhich depends on geometry (Kulikovsky (00))

=> still very expensive to calculate

Numerical difficulty :

- In the literature : limitations of the integration volume

(Kulikovsky (00), Hallac et al. (03), Liu and Pasko (2004)….) :complex to implement and accuracy of these approximatemodels have not been rigorously evaluated

Finally :

where

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Differential models of photoionization in air

Direct solution of approximated radiative transfer equations:

Basic idea : avoid the calculation of the global quadrature over the simulationdomain

Two recent approaches

Helmholtz differential equations (Luque et al. (2007)):The absorption function of the gas is approximated in order to transform theintegral expression of the photoionization term into a set of Helmholtz differentialequationsLuque et al. (2007) : two-exponential fitBourdon et al. (2007) : three-exponential fit

- Djermoune et al. (1995) : monochromatic approach, 1st order (Eddington)approximation- Ségur et al. (2006) monochromatic approach, 1st (Eddington) and 3rd order(SP3) approximation- Bourdon et al. (2007) 3-group approach, SP3 approximation- Direct solution of the radiative transfer equation

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Basic idea : use the differential radiative transfer equationHow to take into account the wavelength dependence?

Let us consider j=1,Ng effective monochromatic radiative transfer equationswith the same source term but different absorption coefficients :

The isotropic part of the distribution function is :

We assume that the isotropic part of the total distribution is :

This approach is similar to the Gaussian-type quadrature generally used inthe correlated-k method

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Then

Both equations are identical if

To use this approach in air, this photoionization source term has to becompared with the Zheleznyak integral expression:

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1-Group fit, Ng=1, j=1,Ng

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3-Group fit, Ng=3, j=1,Ng

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The problem is now to solve the monochromatic radiative tansfer equation

The function depends on the location but also on directions.

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Hopefully, in many problems related to radiative transfer the photondistribution function does not manifest a strong dependence with the direction⇒ approximate approach can be used

First (Eddington) approach : The distribution function is represented by the two first terms in a sphericalexpansion

where the isotropic part is given by

where the first order anisotropy correction is given by

Finally, the function is the solution of :

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Larsen et al. (02) tested the accuracy of higher order approximations ⇒third order approximations appeared to be a good compromise betweenaccuracy and complexity

Third (SP3) order approximation :

Larsen et al. (02) have shown that it is possible to write the third orderapproximation as a set of two equations:

Two uncoupled diffusion equations => can be solved with a Poisson solver

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Boundary conditions

Third (SP3) order approximation (Larsen, 2002):

First (Eddington) approximation :

⇒ Consistent set of equations and boundary conditions

Classical boundary conditions derived by Marshak for various conditions. Forexample, for the case of a boundary surface with no reflection or emission (i.e.the boundary surface is transparent to the radiative flux emitted in the medium)

=> Weak coupling between both equations

Equations are solved independently and then a few iterations are carried out withβ coefficients => very rapid convergence

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Direct solution of the radiative transfer equation :

When absorption coefficients are very small, asymptotic methods can nomore be used. It is then necessary to solve the complete radiative transferequation.

This situation occurs for example at the edges of line profiles (in a multigroupapproach) and in the case of photoemission calculations.

Conditions of validity of approximate approaches :

When absorption coefficients are high and/or gradients of the source terrm issmall

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- Although many different techniques are available in the literature, the directnumerical solution of the radiative transfer equation for a general geometry, isessentially reached with the help of the discrete ordinate method (SNmethod).The basic idea of the SN method is to build up a discrete approximation thatmaintains the main major conservative properties of this equation as far aspossible.This method was first initiated by Lathrop and Carlson (1965,1970) in thefield of neutron transport theory and later extended to radiative transferproblems (Modest,2003)=> Problem of oscillations in cylindrical geometry (« ray effect »)

- In this work, we have used a finite volume method (FVM) (Chui, Raitbyand Hughes (92))

Differents methods :

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In a two dimensional cylindrical geometry, the distribution function ψ dependson five variables, namely two positions variables (z,r), two angular variables(θ,φ) and the frequency.

In the following, we will only consider the case of a monochromatic situation.

Monochromatic radiative transfer equation :

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rr

r

z

Cte=0

!

rer

zer

0!er

!r

!

!

x

y

Cathode

Anode

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Comparison of the FVM and the Eddingtonmethod for a Gaussian source

axe de symétrieGaussienne

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0,6 0,7 0,810

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1019

1020

1021

z (cm)

Ph

oto

ion

isati

on

so

urc

e t

erm

( c

m-3

s-1)

1012

1013

1014

Gau

ssia

n s

ou

rce te

rm

(cm

-3)

300 cm-1

130 cm-1

20 cm-1

FVM methodEddington

Comparison of the FVM and the Eddingtonmethod for a Gaussian source

monochromatic tests => best results obtained with the FVM method

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Comparison of the Zheleznyak integral model and theEddington method for a Gaussian source

Comparison with the Zheleznyak model : in the regions where the ionization sourceterm is small : small differences between the integral and the approximate model=> Very interesting for streamer propagation

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Validation and streamer simulations

Streamer simulations : two test-cases1) Streamer propagation in strong external field (>Ek, the conventionalbreakdown threshold field) : double headed streamer

2) Streamer propagation in weak external field ( best results obtained with the direct integralmethod or the finite volume method- comparison of approximate models (Eddington, SP3, Helmholtz) withthe integral Zheleznyak model=> The best agreement with the reference integral Zheleznyak model isobtained with the 3-group SP3 model

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Propagation of a double headed-streamer

Liu and Pasko (04)

Two remote electrodes establish a uniformLaplacian field Eo=4.8x106V/m

Neutral density is No=2.688x1025 m-3

⇒Eo/No=178.6Td

Initial neutral plasma cloud in the middle of thesimulation domain:

Computational domain : Number of cells :

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Propagation of a double-headed streamer

axe de symétrieGaussienne neutre

•Uniform Laplacian field E0 = 4.8 × 106Vm−1=>E0/N0 = 178.6Td (1Td = 10−17 Vcm2)• Initial neutral plasma cloud : Gaussian distribution in spacewith σr = σz = 0.02 cm and n0 = 1E20 m−3. • The size of the computational domain is 1.4×0.125 cm2.

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t=0 to t=3.5ns, with a timestep of 0.5ns.

Bourdon et al. (2007)

=> Very good agreement between all approximate differential models

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t=0 to t=3.5ns, with a timestep of 0.5ns.Bourdon et al. (2007)

=> Very good agreement between with the reference integral model

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Outline

• Non-thermal plasma discharges at atmospheric pressure

• Role and modeling of photoionization

• Role and modeling of photoemission

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Models for photoemission

Generally, in photoemission the absorption coefficient is very small=> asymptotic methods cannot be used.

Integral methods or direct control volume methods can be used.

The photon flux at the cathode can be obtained with the finite volume method=>To calculate the flux more points are required than in the photoionizationcase (photoemission nθ,nϕ =80 for photoionization nθ,nϕ =15)

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With the integral method (without absorption), the flux at the cathode canbe written:

! ! ! "++=

L R

zrrrrz

dzrQdrrdzzr

0 0 0

2/3222 )'cos'2''(

')','(''''2)(

#

$

$%

Integral model for photoemission

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Conclusions

In cylindrical geometry-Integral model : Nz x Nr x Nr’=Nz x Nr x Nr-Finite volume method : Nz x Nr x Nθ x Nϕ=Nz x Nr x 80 x 80=> To calculate the fluxes Nθ, Nϕ =80 whereas for photoionization Nθ, Nϕ =15are enough

Integral method is more efficient than the finite volume methodPhotoemission

Extension to 3D : finite volume method becomes more efficient

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ConclusionsPhotoionization:

•Integral method : (NzxNr)2•For high values of µ : Eddington αx(NzxNr) or SP3 ( 2αx(NzxNr))•For low values of µ : finite volume method : NzxNrxNθxNϕ où Nθ,Nϕ=15 Very difficult for the integral model

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Conclusions

Open questions:-The detailed process of photoionization in air is still not well understood-Need for further experiments on photoionization in air -Photoionization in other gases?-Preionization?-Photoemission coefficients

modélisation des plasmas : photoionisation et effet...

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