Capacitively coupled radio-frequency discharges in nitrogen at low pressures

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nitrogen at low pressures

L. L. Alves, L. Marques, C. D. Pintassilgo, Gaëtan Wattieaux, Et-Touhami

Es-Sebbar, J. Berndt, E. Kovacević, Nathalie Carrasco, L. Boufendi, Guy

Cernogora

To cite this version:

L. L. Alves, L. Marques, C. D. Pintassilgo, Gaëtan Wattieaux, Et-Touhami Es-Sebbar, et al..

Ca-pacitively coupled radio-frequency discharges in nitrogen at low pressures. Plasma Sources Science

and Technology, IOP Publishing, 2012, 21 (4), pp.045008. �10.1088/0963-0252/21/4/045008�.

�hal-00716113�

nitrogen at low pressure L L Alves 1 , L Marques 1;2 , C D Pintassilgo 1;3 , G Wattieaux 4 , Et Es-sebbar 5;6 , J Berndt 4 , E Kova evi 4 , N Carras o 5 , L Boufendi 4 and G Cernogora 5 1

InstitutodePlasmaseFus~aoNu lear,InstitutoSuperior Te ni o,Universidade

Te ni adeLisboa,1049-001Lisboa,Portugal

2

CentrodeFsi adaUniversidade doMinho,UniversidadedoMinho,4710-057,

Braga,Portugal

3

DepartamentoEngenhariaFsi a,Fa uldadeEngenharia,UniversidadedoPorto,R.

Dr. RobertoFrias,4200-465Porto,Portugal

4

GREMIUniversited'OrleansBP6744-45067OrleansFran e

5

LATMOS-UVSQ-CNRS11Bdd'Alembert78280Guyan ourt,Fran e

6

Clean CombustionResear hCenter,KingAbdullahUniversityofS ien eand

Te hnology(KAUST), Thuwal23955,SaudiArabia

E-mail: llalvesist.utl.pt

Abstra t. Thispaperusesexperimentsandmodellingtostudy apa itively oupled

radio-frequen y dis harges in pure nitrogen, at 13:56 MHz frequen y, 0:1 1 mbar

pressuresand 2 30W oupledpowers. Experiments performedon twosimilar (not

twin)setups,existingintheLATMOSandtheGREMIlaboratories,in ludeele tri al

and opti al emission spe tros opy (OES) measurements. Ele tri al measurements

give the rf-applied and the d -self-bias voltages, the ee tive power oupled to the

plasmaand theaverage ele trondensity. OES diagnosti smeasure the intensities of

radiativetransitionswiththenitrogense ond-positiveandrst-negativesystems,and

withthe811:5nmatomi lineofargon(presentasana tinometer). Simulationsusea

hybrid ode that ouplesa two-dimensional time-dependent uid module, des ribing

the dynami s of the harged parti les (ele trons and positive ions N

+ 2 and N + 4 ),

andazero-dimensional kineti module, des ribingthe produ tionand destru tion of

nitrogen(atomi andmole ular)neutralspe ies. The ouplingbetweenthesemodules

adoptsthelo al meanenergyapproximationto dene spa e-timedependent ele tron

parameters for the uid module and to work-out spa e-time average rates for the

kineti module. The model gives general good predi tions for the self-bias voltage

and for the intensities of radiative transitions (bothaverage and spatially-resolved),

underestimatingtheele trondensitybyafa torof3 4.

1. Introdu tion

Radio-frequen y (rf) dis harge plasmas produ ed in nitrogen and nitrogen mixtures

are be oming in reasingly popular, be ause they exhibit a high hemi al rea tivity

leadingto the produ tionof a tiveradi als and ions, even at lowpressure and ambient

temperature. These plasmas an be produ ed using various rf-ex itation te hniques

(diele tri barrier,indu tive ouplingand apa itive oupling), and they are frequently

usedforthepro essing,modi ationandfun tionalizationofdierentkindsofmaterials,

su h as fullerenes [1, 2℄, nanotubes and nanobres [3, 4, 5℄, polymers [6, 7, 8, 9℄

and textiles [10, 11℄. Nitrogen- ontaining rf-plasmas are also used to grow GaN thin

lms [12, 13℄ and in the produ tion of hydrogenated arbon nitride materials, from

thin lms [14, 15, 16, 17, 18, 19, 20, 21℄ to nanoparti les [22, 23℄. Nitrogen-doped

arbona eous materials an exhibit numerous interesting properties, su h as extreme

hardness[24℄orin reasedbio ompatibility[25,26,27,28℄. Theuseofnitrogen/methane

mixtures to produ e apa itively oupled radio-frequen y ( rf) dusty plasmas (with

CH

4

on entrations below 10%) is another very a tive resear h eld. The purpose

of these investigations is the simulation in laboratory of the hemistry of Titan's

atmosphere,in ludingthesynthesisofanaloguesofitsorgani solidaerosols[29℄. These

analogues (namedtholins)are probablyprodu edby photo hemistrypro esses indu ed

by the solar radiation upon harged parti lesfrom Saturn's magnetosphere. To ensure

thatthedustparti lesprodu edinalaboratoryplasmaenvironmentarerepresentatives

ofTitan'saerosols,oneneedsto hara terizetheplasmasour e ontrollingtherea tivity

of the N

2 -CH

4

mixture, omparing it with the photon sour e that provides the energy

required to grow the solid parti les of Titan. Several experimental works have been

arried out with this intention, involving a wide number of dierent N

2 -CH

4

plasmas

produ edbydire t- urrentdis harges(d )[30,31,32,33, 34℄,mi rowavedis harges[35℄

and their afterglows [36℄, indu tively oupled radio-frequen y dis harges [37℄, spark

dis harges [32℄, orona dis harges [38, 39℄, diele tri barrier dis harges [40℄ and rf

dis harges[41℄. Theinfraredpropertiesofdustparti lessynthesizedwithin rfnitrogen

/ methane plasmas,at dierent mixingratios, are studied forexample in [42, 43℄.

Regardless ofthe spe i appli ationintended, a ontroluponthe densitiesand/or

energies of the main plasma spe ies (via a modi ation in the operating onditions

-pressure and applied voltage or power) is essential to indu e hanges in the plasma

rea tivity, inview ofeither the rea tor's optimization (interms ofpro essing rates and

uniformity) or the tailoring of dust produ tion. With this respe t, the study of pure

nitrogendis hargesisamandatoryrst steptounderstandthe strong ouplingbetween

the dis harge operating onditions and the plasma parameters, before moving to the

(even more) omplexkineti s hemes asso iated with N

2

mixtures.

Although nitrogen plasmas have been studied for many years, and despite their

growinginterestforappli ations,there isonlyapartialknowledgeabout theirbehavior

under rf dis harge onditions. In fa t, the majority of papers on N

2

52, 53, 54, 55, 56℄ and indu tively oupled dis harges [57℄, and fo us essentially on

their kineti des ription. Usually, these are zero-dimensional self- onsistent models,

oupling the two-term ele tron Boltzmann equation (or just assuming a Maxwellian

ele tron distribution fun tion) to the rate balan e equations of dierent vibrationally

and ele troni ally ex ited (mole ular and atomi ) states, yielding the ele tron energy

distribution fun tion (eedf), the vibrational distribution fun tion (vdf) of the

ground-state nitrogen mole ules, and the populations of the ex ited spe ies onsidered. These

kineti models play animportantrole indetailingthe mainme hanismsof (vibrational

and ele troni ) ex itation, disso iation, and ionprodu tion with N

2

plasmas, but they

oer only partial des ription of the harged and neutral spe ies' dynami s, whose full

analysis requires a multi-dimensional spa e-dependent approa h. Other authors have

adoptedone-dimensionalMonte-Carlosimulationstostudytheele tronkineti s[58℄and

the oupling between the ele tron and the vibrational kineti s [59℄ of rf dis harges

in nitrogen. The approa h was extended to a hybrid Monte-Carlo- uid model [60℄,

des ribing the ele tron kineti s and the hemistry of these dis harge plasmas, and to

Parti le-in-Cell Monte-Carlo models al ulating the eedf, to analyze its oupling with

the vdf [61℄ orthe ele tron heatingme hanismwithin the spa e- harge sheaths [62℄.

Intheparti ular aseofnon-equilibrium rfdis hargesinnitrogen,aself- onsistent

modelling strategy must a ount for the interplay between the transport of parti les,

inthe presen e of density gradients and the rf eld, and their produ tion /destru tion

due to kineti me hanisms involving both ele trons and heavy spe ies. This modelling

requires a two-dimensional(2D) des ription of the harged parti le dynami s, to allow

the al ulationof the self-biasvoltage(inthe ase of asymmetri dis harges,with more

surfa es grounded than driven) [63, 64℄, and to provide more reliable predi tions for

the prolesof the ele tron density and energy(hen e, forthe oupled ele tri alpower).

Also, in the ase of mole ular gases like nitrogen, the use of a omprehensive kineti

s heme is preferable: (i) to adequately des ribe ele tron-neutral ollisions (espe ially

with vibrational ex ited states), playing an essential role in oupling the plasma-gas

system and establishing the nal gas-phase hemi al omposition; and (ii) to dedu e

self- onsistent ele tron parameters (transport and rate oeÆ ients), whose al ulation

depends onthe gas hemi al omposition via the eedf.

This paper studies rf dis harges (at f =13:56 MHz frequen y) in pure nitrogen,

produ ed within ylindri alparallel-plate rea tors (with radii R =65 and 69mm, and

interele trodedistan esd=40and33mm,fortheexperimentalsetupswiththeGREMI

and the LATMOS laboratories, respe tively, the latter orresponding to the PAMPRE

experiment [29℄),atV

rf

'100 300 Vappliedvoltagesand p'0:1 1mbar pressures.

Underthese onditions,theneutralgasisfound losetoroomtemperature(T

g

'350K),

as dedu ed from the rotational spe trum of the nitrogen se ond-positive system [41℄.

Figure1representsas hemati diagramofthe plasma hamberandtheexternal ir uit

with su h rea tors. In both rea tors, the rf voltage is applied to the upper ele trode

throughanL-typemat hingnetworkandablo king apa itorC

B

onlyofthe blo king apa itor but alsoof thestray apa itan eand ables);agrounded

ounter ele trodeshieldstheba kofthepoweredele trodeandxes a ylindri allateral

grid that onnes the plasma. Mass ow ontrollers adjust the inje tion, through the

driven ele trode, of nitrogenand ofargon tra esaddedat onstantAr/N

2

owratio(of

about5%),forthepurposeofmonitoringtheevolutionofele tron- ollisionme hanisms.

Thenitrogen owiskept equalto25s mintheGREMIrea tor,whileintheLATMOS

rea tor nitrogen is inje ted at ow rates between 4:6 98 s m to produ e a pressure

variation of 0:2 1:14 mbar.

This work was driven by the need to provide answers to several hallenging questions. (i) Is it possible to join a sophisti ated 2D spa e-time dependent dis harge ode to a omplex kineti s heme to model rf dis harges in pure nitrogen? (ii) Is it ne essarytoadoptsu ha ompletedes riptiontoprovideanadequatemodellingofthese dis harges? (iii) Is it reasonable to use su h a highly- omplex s heme, onsidering the un ertainties asso iated with some of the ollisional data required? (iv) What are the model predi tions for the disso iationdegree of nitrogenand the evolution of the main radiativetransitions withthe ex ited N

2

spe ies? The strategy adoptedto larifythese issues involves model validationby omparisonbetween simulationsand measurements performed on two similar (not twin) experimental setups and a sensitivity study of model results tovariationsof some key parameters.

Experimentaldiagnosti s allowmeasuring the following quantities: (i)the applied

rf voltage,using a high-voltage probe; (ii)the self-bias voltage; (iii)the ee tive power

oupled to the plasma, taking into a ount ir uit losses; (iv) the average ele tron

density, using a resonant- avity te hnique; (v) the intensities of radiative transitions

with the nitrogen se ond-positive system (SPS) and rst-negative system (FNS), and

with the 811:5 nm atomi line of argon, using opti al emission spe tros opy (OES)

diagnosti s. Modelling uses a hybrid ode that ouples a 2D time-dependent uid

module,des ribingthedynami sof hargedparti les(ele tronsandpositiveionsN

+ 2 and N + 4

),andahomogeneous(0D)kineti module,des ribingtheprodu tionanddestru tion

ofnitrogen(atomi andmole ular)neutralspe ies. The ouplingbetweenthesemodules

adopts the lo al mean energy approximation [65, 66℄ to dene spa e-time dependent

ele tron parameters for the uid module and to work-out spa e-time average rates for

the kineti module. Model resultsyield the self- onsistent d -bias voltage, the ee tive

power oupled to the plasma, and the 2D spatial distributions for (i) the rf plasma

potential,(ii)the densitiesand uxes with the hargedparti lesand the ele tron mean

energy, and (iii)the densitiesof the most relevant nitrogenspe ies.

2. Experimental diagnosti s

The experimentalanalysisof the rfdis harges inpurenitrogenresorttothe following

diagnosti s: (i) ele tri al and plasma measurements of the applied rf voltage, the

SPS and FNS band intensities, and of the 811:5 nm atomi -argonlineintensity.

2.1. Ele tri al and plasma measurements

The applied rf voltage V

rf

, one of the main parameters dening the experimental

work onditions, is measured onto the driven ele trode by using a high-voltage probe

onne ted to a digital os illos ope. On e the plasma has rea hed its steady-state

situation,the self-bias voltage V

d

is measured with a 10M voltmeter.

Thepower oupledtotheplasmaW

e

isobtainedadoptingtheso- alledsubtra tive

method [63, 67, 68, 69℄, whi h a ounts for the power losses in the external ir uitry

by measuring the power at the output of the rf generator, for the same V

rf

, with and

withoutplasma. NotethatW

e

isthequantityto onsiderindeningequivalentworking

onditions, for the omparison between simulationsand measurements.

The diagnosti of the average ele tron density n

e

uses the fa t that the rea tor

hamber, with its lateral metal grid, a ts as a resonant avity in the mi rowave range.

Wepreferthisspa e-averagediagnosti tolo alprobemeasurementswith ompensation, sin e the latter are mu h more intrusive under rf onditions. By pla ing two

axi-symmetri ele tri ally-insulatedloopantennaatthe bottomof the system[41℄,one an

measure the shiftinthe resonan efrequen y of the avity,with and withoutplasma,to

dedu e the on-axis ele tron density n

e0

a ording to the relation[70, 71, 72, 73℄

n e 0 =A8 2 f 2 res
f res f res0 m e " 0 e 2 : (1) In equation (1), f res and f res0

are the resonant frequen y with and without plasma,

respe tively,
f jf

res f

res0

j is the frequen y shift,e and m

e

are the ele tron harge

andmass, respe tively,"

0

isthe va uumpermittivity,andAisageometri alfa torthat

depends on the ele tron density prole and on the ele tromagneti mode adopted in

measurements whi h, inour ase, is found inthe range 1 1:8. Forsimpli ity,we have

taken A=1(whi h orresponds toassumea onstantele tron densityprole)sin ethis

approximation ae ts the value of n

e 'n e 0 by less than 10%. The value of f res0

has proven to be a riti al parameter in providing a good

estimationoftheele trondensity. Forexample,theresonan efrequen y an hangedue

tothe smalldilationof the metalli hamberae tedbygas heating,and thereforeitis

measuredherejustaftertheplasmaextin tion. The hoi eofthe ele tromagneti mode

isalsoof onsequen e. Inapreviouswork [41℄wehaveadoptedthe TM

210

mode,whose

resonan e frequen y isfound nearthat of othertransverse magneti (TM) modes; here,

we preferred the TM

010

mode be ause it appears isolated and be ause its ele tri eld

distribution presents anon-axis maximum, oin iding with the position of the ele tron

densitymaximum,hen eallowingformeasurementswithhighersensitivity(inthis ase,

rea hing n e values aslowas 610 13 m 3 ).

2.2. Spe tros opi measurements

OESdiagnosti s useanUV-VIS-NIRmono hromatorwitha60 mfo allengthinboth

the LATMOS and the GREMI laboratories. These systems measure the intensities

of the following radiative transitions: (i) the 370 438 nm bands with the SPS

N 2 (C 3  u ,v) ! N 2 (B 3  g ,v 0 ) and the FNS N + 2 (B 2  u ,v 0 ) ! N + 2 (X 2  g ,w 0 ) [74℄. Of these, some spe i (v-v 0

) transitions are monitored in LATMOS: for the SPS, transitions at

vv

0

v=2;3,for v=0;1;2 and at
v=4,for v =0;1;2;3;4,of whi h the most

intense one isthe 380:5 nm SPS(0-2); forthe FNS,transitions (0-0), (0-1)and (1-2), of

whi h the most intense one is the 391:4 nm FNS(0-0); (ii) the 811:5 nm Ar(4p[5=2℄

3 )

! Ar(4s[3=2℄

2

) atomi line of argon, hosen be ause it presents no overlapping with

nitrogenbands. Asmentioned,argonisadmittedat5%Ar/N

2

owratioinLATMOS

or atper entages that ensurea onstant 25s m nitrogen ow inGREMI. Argon a ts

asana tinomer: be auseitslevelsare populated by ele tron- ollisions,theevolutionof

their lineintensities isassumed torepresent that of ele tron me hanisms.

3. Model des ription

To model rf dis harges in pure nitrogen, we have developed a hybrid ode oupling

a 2D (r, z) time-dependent uid module (assuming azimuthal symmetry due to the

ongurationofthe rfrea tor) toa very omplete 0Dkineti module. The uidmodule

solves the ontinuity and momentum-transfer equations for ele trons and positive ions

N + 2 (X), N + 2 (B) and N + 4

, the ele tron mean energy transport equations (the ions are

assumedtobethermalizedatgastemperature),andPoisson'sequationfortherfele tri

potential. Thekineti modulesolvesthetwo-termhomogeneousandstationaryele tron

Boltzmannequation(a ountingforinelasti ollisionsfromground-statemole ulesand

atoms, and inelasti and superelasti ollisions involving vibrationally ex ited states)

and the rate balan e equations of 45 vibrational ex ited states with the ground-state

N 2 (X 1  + g ,v=0 45), 7 ele troni states N 2 (A 3  + u , B 3  g , C 3  u , a 01  u , a 1  g , w 1
u , a 001  + g

) with the nitrogenmole ule, and 3 ele troni states N(

4 S, 2 D, 2 P) with nitrogen atoms.

3.1. The uid module

The uidmoduleissimilartothatusedtomodel rfdis hargesinhydrogen[75,76℄. In

parti ular, the uxequations (for both parti leand energy) adopt the stationary

drift-diusion approximation, with the introdu tion of an ee tive eld [77℄ in the spe i

ase of the ions. Details about the validity of this approa h are dis ussed in[65℄.

The ontinuityand the momentum-transfer equationsfor harged-parti les (=

e;i for ele trons and ions, respe tively) are given by

n  = 1(r r )  z +S  (2a)

 q = (  N)n  E  q N 1 N [(D  N)n  ℄ q (fore;i) : (2b) Here, n 

(r;z;t) is the parti le density;

q

(r;z;t) are the q = r;z omponents of the

parti le ux; S

(r;z;t)isthenetprodu tionrateofparti lesduetokineti me hanisms;

N = p=k

B T

g

is the gas density (k

B

is the Boltzmann's onstant); (

N)(r;z;t) and

(D

N)(r;z;t) are the redu ed mobility and diusion oeÆ ient, respe tively; and

(E e

q

=N)(r;z;t) is the q omponent of the redu ed ee tive ele tri eld. For the

ele trons E e e q = E q

, orresponding to the rf ele tri eld, whereas for the ions E

e i q is al ulated using [65℄ (E e i q =N) t = 1  i N  v i r  r +v i z  z  v i q  i  i N v i q + i E q N E e i q N ! ; (3) where v iq  iq =n i

is the q omponent of the i-ion drift velo ity, 

i  S i =n i is the

i-ion net produ tion frequen y, and 

i

is the ion-neutral momentum-transfer ollision

frequen y. Thelatterrelatestotheionmobility,a ordingto

i N =e=[m i ( i =N)℄with m i

the i-ion mass.

The ele tron meanenergy transport equations are given by

(n e ") t = 1 r (r " r ) r  " z z ~ e
~ E S " (4a) " q = ( " N)"n e E q N 1 N [ (D " N)"n e ℄ q ; (4b)

where"(r;z;t)isthe ele tronmeanenergy,

"

q

(r;z;t)isthe q omponentofthe ele tron

energy ux, (

"

N)(r;z;t) and (D

"

N)(r;z;t) are the redu ed mobility and diusion

oeÆ ient for energy transport, respe tively, and S

"

(r;z;t) is the net power density

lost by the ele trons due to (elasti ands inelasti ) ollisions. Equation (4a) des ribes

the rate of hange of the ele tron energy density n

e

" as the result of (by order of the

termson theright-and sideof the equation) onve tion [ orrespondingtothe transport

of energy due to the drift-diusion ele tron motion, see equation (4b)℄ ondu tion

( orresponding to the transfer of energy from the rf ele tri eld to the ele trons) and

fri tion ( orresponding tothe ele tron energy dissipation in ollisions).

The uid module is losed by Poisson's equation for the rf potentialV(r;z;t)

1 r  r  r V r  +  2 V z 2 = e " 0 X i n i n e ! : (5)

Equation(5)relatesthespa e-timeseparationof hargedparti lestotherfele tri eld

~ E(r;z;t)= ~ r r;z V(r;z;t).

Equations (2a)-(5) are solved within the plasma rea tor ( orresponding to a 2D

workspa e delimitedbythe dis harge axisatr =0, the groundedlateral gridat r=R ,

thedrivenele trodeatz =0,andthegroundedele trodeatz =d,seeFig.1),subje tto

the following boundary onditions. At the rea toraxis symmetry boundary onditions

are given by

n

(n e ") r =0 (6b) V r =0 : (6 )

Flux onditionsfor totallyabsorbing boundaries are imposedatthe dierentele trodes

and walls using [65, 66℄

( e ) ? = 1 2 n e hvi  e X i ( i ) ? (7a) ( " ) ? = 1 2 n e hvui  e " X i ( i ) ? (7b) ( i ) ? = 1 4 n i v th i + i n i ( i N) E e i ? N : (7 )

In theseequations,? relatestothe ux omponent perpendi ulartothe boundary;hvi

and hvui represent the average values of v and vu over the eedf, respe tively; 

e

is the

se ondary ele tronemission oeÆ ient(here,weassumethatse ondary emissiono urs

at the driven ele trode only, with 

e = 0:1 1); v th i = (8k B T g =m i ) 1=2 is the thermal

velo ity for the i-ion spe ies, and

 i = ( 1 if E e i ?

is dire ted towards the wall

0 if E

e

i

?

is in the opposite dire tion tothe wall .

Note that onditions (7a)-(7b), for the ele tron parti leand energy, are obtained using

the expansion that writes the ele tron distribution fun tion as a ombination of two

omponents(isotropi andanisotropi )[66℄,whereas ondition(7 )fortheionsseparates

between their thermal(isotropi ) and drift (anisotropi )motion. Finally,the potential

atea h physi alboundarysatises

V = ( V d +V rf os (!t) at driven ele trode

0 at grounded ele trode and walls

(8)

where ! = 2f, V

rf

is the peak voltage applied to the driven ele trode, and V

d

is an

oset potential termed self-bias voltage, that develops in the ase of an asymmetri

rea tor(withmore surfa es groundedthan driven). The valueof the self-biasvoltageis

updated, afterea h RF periodT

rf , using V d =V d (t)+ Z t+T rf t 1 C B I (t 0 )dt 0 ; (9) whereI (t)=C B (dV C

(t)=dt)isthe urrenta rosstheblo king apa itorC

B

(seeFig.1).

This urrent an be evaluated by integrating the axial omponent of the ondu tion

urrent density over the surfa e of the driven ele trode

I (t)= Z R 0 J z (r;0;t)2rdr ; (10) introdu ing ~ J =e X i e ! ' ~ E+ ~ J di ; (11)

where   e( e n e + i  i n i

) is the total plasma ondu tivity (dened by assuming

~ E e i ' ~ E)and ~ J di

isthe net diusion urrent dened a ording to [ f. equation (2b)℄

~ J di  e X i (D i N) ~ r N n i +e ~ r N [(D e N)n e ℄ : (12)

The total urrent I

t

is obtained by adding the ontributions of the displa ement

urrent I

D

and the ondu tion urrent I

I t (t)=I (t)+I D (t)= Z R 0 J z (r;0;t)2rdr+ Z R 0 " 0  E z t  (r;0;t)2rdr ; (13)

and it isused to evaluate the ee tive ele tri alpower oupled to the plasma

W e  1 T rf Z T rf 0 [V d +V rf os(!t)℄I t (t)dt : (14)

Theele tron parameters(transport andrate oeÆ ients)inequations(2a)-(4b)are

al ulated asintegralsof ross se tionsover the eedf(see referen e [66℄for moredetails

on these expressions). As mentioned, the spa e-time dependen e of these parameters

is dened by adopting the so- alled lo al mean energy approximation [65, 66℄, whi h

assumesthatthespa e-timevariationofthe eedf, f(u;r;z;t),isintrodu edviathelo al

ionization degree and the ele tron mean energy prole, i.e. (u is the ele tron kineti

energy) f(u;r;z;t)=f(u) j"(r;z;t);n e (r;z;t) : (15)

The pro edure is the following: rst, the homogeneous and stationary eedf, f(u), is

al ulated from the kineti module (see se tion 3.2) for various values of n

e =N (as input) and of " eedf = R 1 0 u 3=2

f(u)du (as output); se ond, the eedfs obtained are used

to al ulate thedierentele tron parameters,whi hare thentabulated asafun tionof

n

e

=N and"

eedf

;third,thelo alvaluesoftheele trondensityandmeanenergy,n

e

(r;z;t)

and "(r;z;t), obtained from the solution to the ele tron parti le and energy balan e

equations(2a) and (4a),are used inthe table onstru tedtodene thespa e-time map

of the ele tron transport and rate oeÆ ients.

The ion transport parameters in equations (2a)-(2b) are as follows. For the ion

mobilities,adependen eontheredu edee tiveele tri eldwasintrodu ed, a ording

to[78, 79℄(valuesinV 1 m 2 s 1

,referredtoastandardgas densityof2:6910

19 m 3 ; 1 Td =10 17 V m 2 )  N + 2 (X ;B) = 8 > > > > > > > > < > > > > > > > > : 1:90 if E e N + 2 7Td 0:29+0:71exp " E e N + 2 2531:01 # +0:93exp " E e N + 2 # if E e N + 2 7Td

 N + 4 = > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > : 2:31 if E e N + 4 7 Td 2:25+0:55exp " E e N + 4 30:93 # 0:10exp " E e N + 4 3:44 # 0:49exp " E e N + 4 30:06 # if 7Td E e N + 4 100 Td 0:47+1:15exp " E e N + 4 2531:01 # +1:50exp " E e N + 4 257:37 # if E e N + 4 100 Td .

Forthe iondiusion oeÆ ientswe have used thevaluesproposed in[80℄ DN

N + 2 (X ;B) = 1:710 18 (T g =273) 1=2 and DN N + 4 =1:510 18 (T g =273) 1=2 m 1 s 1 .

3.2. The kineti module

Thekineti modulesolvesthespa e-timeaverageratebalan eequations,forthedierent

neutral(mole ular and atomi ) spe ies onsidered.

Generally,theratebalan eequationforspe ieskisgivenby(fortheaxis-symmetri

geometry onsidered here)

n k t + 1 r (r k r ) r +  k z z =S k ; (16) where n k and k q

are the spe ies density and ux q- omponent,respe tively, and S

k is

a sour eterm a ounting forthe net produ tionof k-spe ies

S k = X ;l C l ;k n  n l n k X  C tot k n  + X l ;m K l m;k n l n m n k X l K tot kl n l + X l A l ;k n l n k A tot k : (17) In equation (17), C l ;k and K ml ;k

are the rate oeÆ ients for the produ tionof spe ies

k from ollisions of neutral spe ies l with harged spe ies  and neutral spe ies m,

respe tively; C tot k and K tot kl

are the rate oeÆ ients for the total destru tion of spe ies

k from ollisions with harged spe ies  and neutral spe ies l, respe tively; A

l ;k

is the

produ tion frequen y of k-spe ies from the radiativede ay of l-spe ies; and A

tot

k

is the

total destru tion frequen y of k-spe ies by radiative de ay.

In order to limitthe al ulation runtime (due to the high number of spe ies and

kineti pro esses onsidered), the kineti module solves the stationary spa e-average

version of equations(16) 1 (R 2 =2)d Z d 0 R r k (R ;z)dz+ Z R 0 [ z k (r;d) z k (r;0)℄rdr  = 1 (R 2 =2)d Z R Z d S k dzrdr : (18)

The use of a stationary approa h to des ribe the neutral (heavy) spe ies dynami s is

justied, onsidering the long relaxation times asso iated with both their transport

[n k =(D k r 2 n k )  Æ 2 =D k

 0:4 ms, for a diusion oeÆ ient D

k  10 3 m 2 s 1

and a thi kness of the spa e- harge sheath Æ  1 m℄ and their kineti me hanisms

[1=(n

e C

ek

)  0:1 1 s, for an ele tron density n

e

 10

9

m 3

and ex itation rate

oeÆ ients C ek  10 8 10 10 m 3 s 1

℄, when ompared with the rf period of

1=f 74ns.

Theboundary uxesinequation(18)aresettosatisfytheMilne's ondition[81,82℄

q k j wall = D k n k q     wall = 0 k v th k 4 n k j wall (q=r;z) ; (19) where 0 k  k =(1 k =2) with k

the k-spe ies' wall lossprobability.

By using equations(17) and (19) into(18) one obtains

1 (R 2 =2)d k 1 k =2 v th k 4 Z d 0 Rn k (R ;z)dz+ Z R 0 [n k (r;d) n k (r;0)℄rdr  ' X  " X l C l ;k n  n l C tot k n  n k # + X l " X m K l m;k n l n m K tot kl n k n l # + X l A l ;k n l A tot k n k ; (20)

where the average quantities X (X =n

k ;n k n l ;C l ;k n  ,...) are dened as X  Z d 0 Z R 0 Xrdrdz (R 2 =2)d : (21)

The spa e-averages in equation (20) are obtained as follows. The averages of

the harged-parti le ollision frequen ies (C

l ;k n  and C tot k n 

) are al ulated using

the spa e-dependent distributions obtained from the solution to the uid module; the

averagesof the heavy spe ies densities[n

k

, n

k n

l

,and the integralson theleft-hand side

of equation (18)℄ are al ulated imposing the following density proles: (i) a onstant

prole for ground-state spe ies N

2 (X 1  + g ,v) and N( 4

S); (ii) a two-region prole (see

Fig. 2) that distinguishes between the plasma bulk (with a spatially homogeneous

density) and the plasma sheaths (where the density de reases linearly), for spe ies

N 2 (A 3  + u , a 01  u , a 1  g , w 1
u ) and N( 2 D, 2

P), that are transported by diusion and

are totally re ombined at the walls (i.e., for whi h

0

k

= 1). The 2D version of this

prole isgiven by n k (r;z)=n b k h k (r)g k (z) (22a) h k (r)= ( 1 ; 0r<R Æ R 1 1 f r k Æ R (r R+Æ R ) ; R Æ R rR (22b) g k (z)= 8 > > < > > : f 0 k + 1 f 0 k Æ 0 z ; 0z Æ 0 1 ; Æ 0 <z <d Æ d 1 1 f d k Æ (z d+Æ d ) ; d Æ d z d ; (22 )

where n

b

k

is the bulkdensity; Æ

R

, Æ

0

, and Æ

d

are the sheaththi kness nearthe grid, the

driven ele trode, and the grounded ele trode, respe tively; and the fun tions f

r k , f 0 k , and f d k

orrespond to the ratio of the wall density to the bulk density, al ulated at

r=R , z =0,and z=d using boundary ondition (19), i.e.

f x k  n k j wall n b k = 4 0 k v th k 4 0 k v th k + Æ x D k (x=r;0;d) : (23)

As we have asso iated the boundary layers for neutral parti les with the dis harge

spa e- harge sheaths, the quantities Æ

R

, Æ

0

, and Æ

d

are estimated in the uid module

asthe time-averagethi knesses of the regionswhere the rf eld exhibits strong relative

gradients. Inparti ular,thesheathedgesaretakenatpositionswherethe orresponding

(axial or radial) redu ed ele tri elds are equal to 10% of the wall eld (E

z =N)(0;0), (E z =N)(0;d) or (E r

=N)(R ;d=2). Note that equations (22a)-(23) allow al ulating the

spa e-averageintegralsinequation(20)alsofortheheavy spe ieswith onstantdensity

proles, by settingÆ

x =0.

Asmentioned,thekineti modulesolvestheratebalan eequationsof45vibrational

states (with the ground-state mole ule) and 10 ele troni states (7 for mole ules and 3

for atoms), oupled to the two-term homogeneous and stationary ele tron Boltzmann

equation,a ountingforinelasti ollisionsfromground-statemole ulesandatoms,and

inelasti and superelasti ollisionsinvolving vibrationallyex ited states (onlythe rst

10statesare onsidered, seeTable1). This moduleadoptsthe kineti s hemeproposed

in referen e [80℄,where details about the hoi es of the dierentme hanisms and rates

an be found. Tables 1-2summarizethe kineti rea tions for the mole ularspe ies and

the atomi spe ies, respe tively, used in writing the sour e terms (17) and in solving

the ele tron Boltzmann equation. In these tables, the double arrow ! indi ates

that se ond-kind ollisions are also onsidered for the rea tions where it appears. The

ross se tions for the various ele tron-neutral ollisional pro esses (appearing in the

ele tron Boltzmann equation) are taken from [83, 46, 47℄ and referen es therein. The

ross se tions for superelasti ollisions are obtained from those for the orresponding

inelasti pro esses by using Klein-Rosseland's formula[84℄.

The vibrationalstates of ground-stateN

2 (X 1  + g

,v)mole ules play a entralrole in

nitrogen dis harges. The kineti s of vibrational states in ludes ex itation/deex itation

me hanisms due to ele tron impa t ollisions (e-V) transitions and heavy spe ies

ollisions, involving an energy transfer via vibrational-translational (V-T, with both

mole ulesand atoms)andvibrational-vibrational(V-V)intera tions. Moreover, several

vibrational states are involved in heavy-parti le rea tions, some of them leading to

disso iation (see Tables 1-2). This work adopts the SSH (S hwartz, Slawsky and

Herzfeld) theory [84, 85, 86, 87, 88℄ to write the rate oeÆ ients for (i) the V-T

me hanismswith mole ules

P v ;v 1 =v  1  N2 e 1  N2 v  F(Y v ;v 1 ) F(Y 1;0 ) P 1;0 (24a)

P v 1;v =P v ;v 1 exp
E v ;v 1 k B T g (24b) Y v ;v 1 = L ~  M 4k B T g  1=2
E v ;v 1 (24 )

with the normalization[89℄

P 1;0 ( m 3 s 1 )= 1:0710 12 T 3=2 g 0:2772T g 80:32+35:5 v 1 39
0:8 F(Y 1;0 ) ; (24d)

and (ii)the V-V me hanisms

Q w 1;w v ;v 1 =vw  1  N 2 e 1  N 2 e v   1  N 2 e 1  N 2 e w  F(Y w 1;w v ;v 1 ) F(Y 0;1 1;0 ) Q 0;1 1;0 (25a) Q w ;w 1 v 1;v =Q w 1;w v ;v 1 exp "
E w 1;w v ;v 1 k B T g # (25b) Y w 1;w v ;v 1 = L ~  M 4k B T g  1=2
E w 1;w v ;v 1 (25 )

with the normalization[89℄

Q 0;1 1;0 ( m 3 s 1 )= 6:3510 17 T 3=2 g  (25d)  = ( 39:065 1:5625max[v ;w℄ ; max[v ;w℄<10 25:2+24:1  max[v ;w℄ 10 30  3 ; max[v ;w℄10 : (25e)

The rate oeÆ ients for the V-T me hanisms with atoms onsider the al ulations of

referen es [90, 91, 92, 93℄ todedu e the approximated formulae

P N 2 N v ;w <v = ( A 0 exp A1 v + A2 v 2
; v w5andvv ? 0 ; otherwise (26a) P N 2 N w <v ;v =P N 2 N v ;w <v exp 
E v ;w <v k B T g  (26b)

where the quantities A

0 , A 1 , A 2 and v ?

are given inTable 3, forboth rea tive (with an

atomi ex hangebetweenthetwo ollisionpartners)andnon-rea tive(dire t) ollisions.

In equations (24a)-(26b) F(Y)= ( 1 2  3 exp 2Y 3
 exp 2Y 3
; 0Y 20 8  3
1=2 Y 7=3 exp 3Y 2=3
; Y >20 ; (27)
E v ;v 1 = ~! N 2 (1 2 N 2 e

v) is the energy dieren e between two onse utive levels

v and v-1;
E w 1;w v ;v 1 = ~! N 2 2 N 2 e

jw v j is the energy variation with the transition

v to v-1 and w-1 to w;
E

v ;w <v

is the energy dieren e between levels v and w<v;

! N2 = 4:443  10 14 s 1 and  N2 e = 6:073 10 3

are parameters hara terizing the

anharmoni Morse'sos illator;L=2  10

11

mistherangeofintermole ularfor es;M is

themassoftheN

2

and V-V rea tions were obtained from (24a), (25a) and (26a) by detailed balan ing.

Note also that only single-quantum transitions (the most likely ones) are onsidered

for V-V and V-T rea tions with mole ules, whereas multi-quantum transitions are

a ounted for in V-T ex hanges with atoms. Vibrational disso iation by V-V and

V-T pro esses is also in luded, as a transition from v = 45 to a pseudo-level in the

ontinuum [94℄.

Thekineti s hemepresented inTables 1-2dependsonquantitieswhoseknowledge

is somewhat limited, su h as the se ondary ele tron emission oeÆ ient 

e , the wall loss probability 0 k of ground-state spe ies N 2 (X 1  + g ,v) and N( 4

S) (whi h may depend

on the working onditions and / or the spe i surfa e properties), and the bran hing

ratiosb

ion

fortheasso iativeionizationrea tions(23)-(24)andb

atom

fortheatomi

wall-deex itations (47)-(48). There is a dieren e in the nature of the walls, between the

GREMI and the LATMOS rea tors: both are essentially made of stainless-steel, but

the ounter-ele trode with the LATMOS rea torisin aluminum alloy. Withnospe i

data for

0

k

, a ounting for the intera tion of spe ies with dierent rea tor walls, the

quality of omparison between model results and measurements is probably ae ted.

Forthe simulationsinse tion4wehavetaken 

e =0, N 2 =4:510 4 , N =10 3 ,and b ion = b atom

= 0:5. The sensitivity of the model results to these input parameters will

be dis ussed in se tion4.

3.3. Model solution

The solutionto the model iteratesbetween the kineti and the uid modules. Figure3

presents the ow hart of the numeri al work ow used in the simulations.

Typi ally the kineti module runs every ten rf periods, knowing the

spa e-time-average values of the harged parti le densities and of the ele tron rate- oeÆ ients /

rates for the produ tion /destru tion of ea h neutralspe ies. Redu ing, by a fa tor of

ve, the frequen y adopted for updating the kineti module speeds up al ulations by a fa tor of two, with negligible hanges (below 2%) in the main quantities al ulated. Resultsarealsonotae tedifthekineti moduleisupdatedmorefrequentlyinstead,but the al ulation timeis heavily degradedin this ase. The stationarysolutiontothe set

of (non-linear) rate balan e equations (20) is obtained ombining asemi-impli it

time-relaxation te hnique with an iterative matrix-inversion method. Convergen e ensures

relativevariationsofless than10

6

forthedensitiesofallneutralspe iesandisa hieved

after several hundred iterations. The new hemi al omposition of the gas phase is

then used as input data to the homogeneous ele tron Boltzmann equation, allowing

updating the set of ele tron parameters (transport and rate oeÆ ients). The

spa e-timedependen eofthe ele tronparametersisdenedby thelo alele tronmeanenergy

approximation(seese tion3.1),andthespatialvariationoftheneutralspe iesdensities

is tailored using the proles given by equations (22a)-(22 ); then, these quantities are

5%,8%and2:5%forthe hargedparti ledensities,theself-biasvoltageandthe oupled power, respe tively, and a 2% variationinthe density of neutral spe ies.

The harged parti le transport equations (2a)-(4b) and Poisson's equation (5) are dis retized using a se ondorder nite dieren e representation that in ludes boundary onditions (6a)-(8). In order to orre tly des ribe the spa e- harge sheaths, near the integrationboundaries,the uxequations(2b),(4b)aredis retizedusingthe S harfetter-Gummelexponentials heme[95℄. Inthiswork,Poisson'sequationisnumeri allysolved by dire t matrix methods, and the harged parti le transport equations are integrated by adopting a semi-impli itsplitting method, that uses a Crank-Ni holson algorithm withanintegrationtime step ontrolledby the Courant-Friedri hs-Lewy ondition [96℄. The ode is usually solved in a 3316(r;z) point grid, for typi al 1000 4000 time steps within ea h rf period. The hoi e of ner grids leads to a severe degradation in the al ulation time, with minor hanges in the results. In general, 500 1000 rf y les are needed tomeet the global onvergen e riterion: relative hanges between two onse utiveperiods,below(i)510

4

forthe hargedparti ledensities,theele tron mean energy, the plasma potential and the self-bias voltage; (ii) 10

3

for the neutral parti les, orresponding to al ulation times of 1 3 hrs using a Fortan ompiler ona Intel-Xeon E5520 (2:26 GHz) CPU.

As a result, the model yields the 2D proles of the harged parti le densities n

and uxes

~

, of the ele tron mean energy " and ux

~

"

, and of the rf eld

~

E and

potential V; the values of the self-bias voltage V

d

and of the ee tive power oupled

to the plasma W

e

; and the average densities n

k of N 2 (X 1  + g ,v=0 45) (yielding the vdf), N 2 (A 3  + u , B 3  g , C 3  u , a 01  u , a 1  g , w 1
u , a 001  + g ) and N( 4 S, 2 D, 2 P), satisfying

the losure ondition

P k n k + P i n i =N.

4. Results and dis ussion

This se tion presents numeri al and experimental results hara terizing the operation

of rf dis harges in pure nitrogen, at 0:1 1 mbar pressures and 2 30 W oupled

powers ( orresponding toapplied rf voltages in the range 100 300 V).

An overall good agreement (qualitative and often also quantitative) is found

between simulations and measurements, whi h exhibit the same order of magnitude

and the same kind of evolution trends as a fun tion of the power oupled to the

plasma, for various pressures. Moreover, the LATMOS and the GREMI rea tors have

similarbehaviours (hen e justifyingapresentation here of sele ted resultsonly,namely

onsidering the kind of measurements made in ea h rea tor), although their dierent

dimensions (radii and spe iallyinter-ele trode distan es) an explain some distin tions

in the operationfeatures.

Figure4(a)-(b)presentsthe time-averageaxialproles (atdis harge axis)ofE

z =N and of n e , n N + 2 , n N + 4

, respe tively, al ulated for the LATMOS rea tor at various

pressures. For the same rea tor, at 1 mbar, gure 4( )-(d) shows 2D (r,z) plots of the

An observation of gure 4(a) onrms that the thi kness of the spa e- harge sheaths

orresponds approximatelyto 2=3 of the interele trode distan e (see the regions where

E

z

=N 6= 0), and that this thi kness de reases with an in rease in the pressure due to

a more limited ele tron transport. Although there is no net harge-separation in the

entral part of the dis harge (hen e justifying E

z

=N = 0 in this region), gure 4(b)

reveals that the harge parti le density proles are quite diverse. In parti ular, the

relative importan e in the N

+

2

to N

+

4

iondensities is highlydependent on the pressure

(at1mbar,forexample,bothionshavesimilarpopulations),but then

N +

2

axialproleis

theonewhoseshape hangesthemostwithvariationsinthepressure. Theseprolesare

a onsequen eoftherelationshipbetweenthepopulationsofele trons,ionsandneutrals,

interlinkedthrough themain harged-parti le produ tion/destru tionme hanisms: (i)

dire t and stepwise ele tron-impa t ionization [rea tions (6) and (7), see table 1℄; (ii)

asso iativeionizationinvolvingthe ex ited states N

2

(A) and N

2

(a') [rea tions (23) and

(24)℄; (iii) the ion onversion from N

+

4

to N

+

2

and the three-body ion onversion from

N + 2 to N + 4

[rea tions (25) and (26), respe tively℄; and (iv) the ele tron re ombination

with N + 2 and N + 4

ions [rea tions (10) and (11), respe tively℄. Me hanisms (i) and (ii)

are dominant and their relative importan e depends on the pressure: the ontribution

of the asso iative ionizationtothe harged-parti le produ tionis 35% at0:2 mbar and

10% at 1 mbar. The results show alsothat transport ee ts are less important for N

+

2

than for N

+

4

, espe ially at 1 mbar (noti e that 

N +

4

is about 20% higher than 

N +

2 ,

see se tion 3.1); onsequently, the N

+

2

ion is lost at the same positions where it is

reated, and its density prole re e ts that of the ionization rate by ele tron impa t

[its main produ tion me hanism at 1 mbar, see gure 4(d)℄. Figure 4( ) shows that

the ele tron mean energy exhibits maximum values in the sheath regions, with the

onsequent enhan ement of rea tivity (i.e. produ tion / destru tion rates) in these

regions [this is onrmed in gure 4(d) for the ele tron ollisionsme hanisms℄. Noti e

that the maxima of the ele tron mean energy (within the spa e- harge sheaths) are

displa ed fromthoseof the hargedparti ledensities, whi h o urhalfwaybetween the

ele trodes due to transport [see gure 4(b)℄. Noti e further the ex eptional magnitude

ofh"i atthe orner(r=R , z =0)(12eV forthese onditions),due totheverysmall

gap between the driven ele trode and the grounded grid.

Figure 5(a)-(b)presents the axial proleof radiative intensities with the SPS(0,0)

band and the 811:5nm atomi argon line, measured (alongthe dis harge rossse tion)

and al ulated (applying a time- and radial- average) for the GREMI rea tor at

various pressures. This gure shows that the 2D model yields an axial distribution

for both the nitrogenof the argonintensitiesingoodagreementwith spatially resolved

spe tros opi measurements. The latter are diÆ ult to ontrol, not only due to the

extreme dependen e of the emitted light intensity with position but also be ause of

the limitations in dening a pre ise referen e frame between the ele trodes. These

diÆ ulties an justify the dieren es observed in the positions of the al ulated and

The self-bias voltage, measured and al ulated for both rea tors at various

pressures, is depi ted in gure 6(a)-(b) as a fun tion of the power oupled to the

plasma. Notethat ain rease in the oupled powerand ade rease inthe pressure, both

lead to a more asymmetri dis harge operation (hen e to a higher self-bias voltage),

due to an enhan ement in the harged-parti le uxes towards the ele trodes. A good

agreement is found (both quantitative and qualitative) between V

d

simulations and

measurements for the LATMOS rea tor. For the GREMI rea tor, the model predi ts

the orre t evolution trend for V

d

with variations in both W

e

and p, yet yielding

values that seem systemati ally deviated with respe t to the experiment, whi h might

beduetoun ertaintiesinthewalllossprobabilitiesadopted. In any ase,the qualityof

the agreement for a global parameter su h as the self-bias voltage, obtained using two

dierentexperimentalsetups, onstitutes another su essful test for model validation.

Figure 7 presents the spa e-time average ele tron density, as a fun tion of the

power oupled to the plasma, measured and al ulated for the GREMI rea tor at

various pressures. As for V

d , one observes hn e i to in rease with W e (at onstant

pressure) simply due to the enhan ed energy involved in plasma maintenan e. Noti e

that a de rease inp (at onstant power oupled) leads to anin rease in hn

e

i (for most

onditions), whi h is asso iated with an in rease also in ", probably to ompensate

for higher (parti le and energy) losses. Ultimately, this variation is responsible for a

signi antin reaseintheionizationdegree, whenpde reases at onstantW

e

. Figure7

shows thatthe al ulatedele tron density isunderestimatedbyafa torof about4with

respe t to experiment, whi h is a typi al limitation of uid simulations when applied

to the modelling of rf dis harges. This is probably asso iated with the fa t that the

ele tron uxequation(2b)negle ts the orrespondinginertiaterm,whi hoverestimates

the energy budget required to maintain the rf sheaths leaving less energy available for

the ele tron produ tion. However, even with this limitation, the model is apable to

provide orre t qualitativepredi tions for the evolutionof hn

e

i with W

e

and p.

Figure 8(a)-( ) presents the spa e-time average radiative intensities, as a fun tion

ofthepower oupledtotheplasma,ofthe SPS(0,2)andthe FNS(0,0)bands,and ofthe

811:5nmatomi argonline,measuredand al ulatedfortheLATMOSrea toratvarious

pressures. Measurements were arried out inthe vi inity of thedriven ele trode, where

the emitted light intensity is maximum, and thus they are ompared with

spatially-averaged simulations results, al ulated over the entire dis harge ross se tion (in the

radial dire tion) and over 1=3 of the interele trode distan e (in the axial dire tion,

below the driven ele trode). The experimental observations reveal that the transition

intensities are highly dependent on position (within mm, see gure 5), whi h makes

extremely diÆ ult to ontrol the measurement of spatially-averaged data to ompare

with the simulations. Within ea hgure 8(a)-( )the model resultsare normalizedtoa

singleexperimentalpoint,be ausenoabsolute alibrationoftheopti aldevi ehasbeen

done and thus the OES data are in arbitrary units. A fair agreement is found between

the pressure orthe oupled power, issimilartothat ofthe ele tron density [see gure7

and the inset in gure 8(b)℄,thus onrming the relevant role of the ele tron ollisions

in the produ tion of N

+

2

and the oheren y of the present study (the n

e

measurements

were made in the GREMI rea tor, whereas the spatially-averaged OES measurements

were performed in the LATMOS rea tor). Note that the SPS(0,2) and the Ar(811nm)

intensities exhibit dierent behaviours with hanges in p and W

e

. The argon line

followsthe same variationtrends asn

e

(de reasing with anin rease inpand in reasing

with W

e

), whereas the SPS(0,2) band in reases with p and tends to saturate as W

e

in reases. Thesedieren es anbeexplainedby a ombinationof fa tors: (i)the argon

ex itation is assumed to pro eed via dire t ele tron ollisions only, whereas the N

2 (C)

ex ited state is reate not only by the dire t ele tron ex itation (4a) but also by the

poolingrea tion(15b)(seeTable1),whose ontributiontothetotalN

2

(C)produ tionis

5% at1mbar and 26%at0:2 mbar; (ii)thedieren es, bothinthreshold and inshape,

betweentheele tron-impa tex itation rossse tionsofN

2

(C)andofArleadtodierent

spatial distributions of the orresponding time-average ele tron rate oeÆ ients, hen e

to dierent lo al produ tion rates with these me hanisms whose values, in any ase,

de rease with the pressure.

Figure 9(a)-(b) shows the time-average ele tron energy probability fun tion

f(u)u

1=2

as a fun tion of u, al ulated for the LATMOS rea tor at p = 1 mbar and

W

e

=7:4 W and at p=0:5 mbar and W

e

=5 W ( orrespondingto V

rf

'200 V).By

using the lo almean energy approximation(see se tion 3.1) the eedfbe omesspatially

resolved via its dependen e with " and n

e

[see equation (15)℄, and thus results are

obtainedinthedis harge entre(r=0andz =16:5mm)andinthespa e- hargesheath

with the driven ele trode (r=0 and z =4:1 mm), where the OES measurements were

done(seegure8). Figure9revealsthata hangeinthepressureindu esa onsiderable

modi ationof the eedfinthe spa e- hargesheath, butthe same isnot observed inthe

plasmabulkwheretheeedf'sshapeisnotalteredbypressurevariations. Inbothregions

a de rease in p yields an in rease in the tail of f(u)u

1=2

and a simultaneous de rease

in its body, but these modi ations are parti ularly signi ant in the sheath where "

hanges from 5:9 to 9:7 eV when p goes from 1 to 0:5 mbar. Noti e that the in rease

in the eedf's tailwith the de rease in the pressure, although implying anenhan ement

in the ele tron ollision rate oeÆ ients, does not augmentthe relative ontribution of

the ele tron impa tme hanismsfor harged parti leprodu tion,atlowpressures. This

(apparent) ontradi tion is asso iated with the extraordinary in rease observed in the

N

2

(A) and N

2

(a') populations when the pressure de reases [see gure 10(a)-(b)℄.

Figure 10(a)-(b) presents the relative densities of the N

2

(A) and the N

2 (a')

metastable states, as a fun tion of the power oupled to the plasma, al ulated for

the LATMOS rea torat various pressures. These spe ies play animportantrole in the

ele tron produ tion [via the asso iative ionization rea tions (23) and (24), see table 1℄

and in the population of the N

2

(C) state responsible for the SPS transition [via the

poolingrea tion(15b),seetable1℄. Thedensitiesn

N 2 (A) =N andn N 2 (a 0 ) =N be omelarger

thein reaseinbothn

e

and". In identally,thesamekindofvariationwasreportedabout

measurements of the N

2

(A) density inindu tively oupled plasmas (at lower pressures

andhigher oupled powers)[97℄. Noti e thatthemain produ tionme hanismsofstates

N

2

(A) and N

2

(a') pro eed, respe tively, via the states N

2

(B) [pooling rea tion (16a)

and radiative transition(28)℄ and N

2

(a) [pooling rea tion (18) and ele tron ex itation

(4a)℄, whi h are reated via ollisions with N

2

(A) itself [vibrational de-ex itation (14)℄

and with ele trons [rea tion (4a)℄. The vibrationalex ited states N

2

(X;v=2 10) are

alsopopulated essentiallybydire tele tronex itation. Ultimately,the ele tronkineti s

ontrols the populationof the metastables, justifying their variationswith the working

onditions.

Figure 11 plots the disso iation degree of nitrogen n

N =(n N2 +n N ), as a fun tion

of the power oupled to the plasma, al ulated for the LATMOS rea tor at various

pressures. One observes weak disso iationdegrees, varying between 10

3

and 410

2

for in reasing W

e

values (at standard

0 N 2 = 4:5 10 4 and 0 N = 10 3 onditions)

and exhibiting little hanges with p. The results displayed show also that the

disso iation degree de reases with an in rease in the wall loss probabilities for atoms

(as expe ted, a signi ant ee t is observed) and vibrationally ex ited mole ules,

0

N

and

0

N2

, respe tively. These results an be explained by noti ing that the main

disso iation me hanisms of nitrogen involve ollisions with highly ex ited vibrational

states [above v> 10, see rea tions (22) and (41) in tables 1 and 2℄, whose kineti s

is essentially governed by N 2 -N 2 V-V and N 2

-N V-T me hanisms [rea tions (38) and

(13), respe tively℄ that are favoured by an in rease in both the oupled power and the

pressure. Consequently, the N(S,D,P) populations are roughly proportional to p and

the disso iation degree presented in gure 11 displays negligible variations with the

pressure. This is in ontrast with the results of gure 7, whi h suggest an important

variation of the ionizationdegreewith the pressure (in oheren y with the evolution of

n N 2 (A) =N andn N 2 (a 0 )

=N vs. pplottedingure10),regardlessthefa tthattheionization

energy is higher than the disso iation energy in nitrogen. This apparent ontradi tion

puts forward the existen e of dierent me hanisms, asso iated with the ele tron and

the vibrational kineti s, in ontrol of the ionization and the disso iation of nitrogen,

respe tively.

Asmentionedinse tion3.2,the kineti s heme presentedinTables 1-2depends on

several parameters (b ion , b atom , N , N2 , and  e

), whose values an in uen e the model

results. Simulations tests show that: (i) the bran hing ratios b

ion

and b

atom

an vary

between 0 1with littlein uen e uponthe results;(ii) the wall atomi re ombination

probability has an(obvious) dire t ee t uponthe disso iationdegree, whi h de reases

by a fa tor of  10 2 when 0 N in reases from 10 3 to 1. This variation in 0 N yields

also a maximum in rease of  12% in n

e

, due to a redu tion in the destru tion of

the metastableN

2

(A) by atomi impa t(re allthatthe asso iativeionizationinvolving

N

2

(A) is one of the main ele tron produ tion hannels); (iii) an in rease in the wall

impa t upon the disso iation degree ( ontrolled by the vibrational kineti s) and the

ele tron produ tion ( ontrolled by dire t / stepwise ele tron impa t ionization, whi h

is favoured by the quen hing of vibrationally ex ited states down to the

ground-state). The disso iation degree and the ele tron density de rease  93% and in rease

 45%, respe tively, when 0 N 2 varies from 4:5 10 4 to 1 at 0:5 mbar and 10 W;

(iv)variationsinthe se ondary ele tron emission oeÆ ientprodu e negligible hanges

in the ele tron density, although ae ting the self-bias voltage through hanges in the

boundary ondition(7a). Byin reasing

e from0to0:5,jV d jredu es10%at0:5mbar and 10 W. 5. Final remarks

This paperhas studied rfplasmadis hargesinpure nitrogen, usingboth experiments

and simulations. Experiments obtained relevant parameters (the rf-applied and the

d -self-bias voltages, the ee tive power oupled to the plasma, the average ele tron

density, and the intensities of radiative transitions with the nitrogen se ond-positive

and rst-negativesystems andwith the 811:5nm atomi lineof argon),re orded inthe

LATMOS and the GREMI laboratories. Measurements were made independently and

with dierent equipment upon two similar (not twin) experimental setups ( ylindri al

parallel-plate rea tors surrounded by a lateral grounded grid), at p = 0:1 1 mbar

pressures and W

e

= 2 30 W ee tive powers oupled to the plasma ( orresponding

toappliedrf voltagesV

rf

=100 300 V).The oupled powers were measured using the

subtra tive method(with and withoutplasma),thusa ounting for the power lossesin

the external ir uitry (the mat hbox, the oaxial ables and the dierent onne tors).

Simulations used a hybrid ode that ouples a 2D (r, z) time-dependent uid module

to a very omplete 0D kineti module. The uid module solvesthe ontinuity and the

momentum-transfer equations for ele trons and positive ions N

+ 2 (X), N + 2 (B) and N + 4 ,

the ele tron meanenergy transport equations, andPoisson'sequationfor the rf ele tri

potential. Thekineti modulesolvesthetwo-termhomogeneousandstationaryele tron

Boltzmannequation(a ountingforinelasti ollisionsfromground-statemole ulesand

atoms, and inelasti and superelasti ollisions involving vibrationally ex ited states)

and the rate balan e equations of 45 vibrational ex ited states with the ground-state

N 2 (X 1  + g ,v=0 45), 7 ele troni states N 2 (A 3  + u , B 3  g , C 3  u , a 01  u , a 1  g , w 1
u , a 001  + g

) with the nitrogenmole ule, and 3 ele troni states N(

4 S, 2 D, 2 P) with nitrogen atoms.

The ode was validated by omparison between simulations and measurements,

yielding a good agreement (within the experimental un ertainties) for the self-bias

voltage and for the intensities of radiative transitions (both average and

spatially-resolved), at dierent pressures and oupled powers. This validation showed that it

ispossibletojoinasophisti ated2Dspa e-time dependent dis harge ode toa omplex kineti s heme to model rf dis harges in pure nitrogen. Moreover, the validation

ex eptional ben hmarking onditions not only for this work but also for future works.

Finally,be ausemodelvalidationhasbeenbasedalsoontheanalysisofopti alemission

spe tros opy diagnosti s, it further allowed to larify the me hanisms ontrolling the

kineti s of the main ex ited spe ies with nitrogen. In parti ular, experimental SPS

results are only mat hed if the N 2

(A) density, n e

and the ele tri al parameters are orre t, whereas the FNS emissionis onlyproperly modelledif both the al ulated N

+ 2 density and the vdf are a urate. Noti e that we have hosen here to des ribe the

evolution of the relevant physi al quantities as a fun tion of W

e

instead of V

rf

. This

hoi e is be ause, in the present model, neither the inertial terms in the ele tron ux

equation (2b) were a ounted for, nor the external power- ir uit was onsidered in a

self- onsistent way, whi h an alter the phase between V(t) and I

t

(t), and thus the

relationship between V

rf

and W

e .

Results exhibit a strong spatial non-homogeneity that depends parti ularly on

the operating pressure. These features are well predi ted by the 2D transport ee ts

onsidered in the model, whi h adopts the lo al mean energy approximation to dene

spa e-time dependent ele tron parameters for the uid module and to work-out

spa e-timeaverageratesforthekineti module. Thesu essofthisspatialdes riptiondepends

mostly on the al ulated eedf, whose tailis more populated in the spa e- harge sheath

region than in the plasma bulk region, espe ially at low pressure. Transport ee ts

justify also that the maxima of the harged parti le densities o urs at the rea tor entre, displa ed fromthe maximaof their produ tionrates (lo ated within the spa e- harge sheaths). These results show the need to adopt the omplete 2D spa e-time des ription onsidered here toprovide anadequate modellingof these dis harges.

Simulations tests show that the model predi tions are not signi antly ae ted (variationsbelow10%)by hangesin(i)thebran hingratiosfortheprodu tionofatoms and of ions, (ii)the se ondary ele tron emission oeÆ ient, (iii) the frequen y adopted for updating the nitrogen kineti s in al ulations and (iv) the proles imposed in the model for the mole ularand the atomi ex ited spe ies. As expe ted, the probabilities forthe atomi re ombinationand thevibrationaldeeex itationatthe wallhaveadire t ee t upon both the disso iation degree and the ele tron density. However, even by settingthese probabilitiestounitythe disso iationdegreeremainslowand n

e

in reases notmorethan45%. Thisshowsthatthehighly- omplexkineti s hemeproposedhereis ableto apture themainplasmafeatures(withinexperimentalerrors),even onsidering the un ertainties asso iated with some of the ollisionaldata.

As usually observed in uid models applied to rf dis harges, simulations

underestimate the ele tron density by a fa tor of 3 4, yet yielding orre t qualitative

predi tionsfor the evolutionof hn

e

iwith W

e

and p. Ele trons are produ edmainlyby

dire tandstepwiseele tron-impa tionizationandbyasso iativeionizationinvolvingthe

metastables states N

2

(A) and N

2

(a'); these me hanisms yield a relatively low ele tron

density (hn e i'510 8 510 9 m 3

)due tothelimited oupled-power hara terizing

thesedis harges. At10Wtheionizationdegreerangesfrom6  10

8

at1mbarto4  10

7

and N

2

(a'). Even if the ionization of nitrogen is essentially ontrolled by the ele tron

kineti s,itsdisso iationisgovernedbythe vibrationalkineti sand by theatomi losses

at the walls. The disso iation degree is very weak, varying between 10

3

and 410

2

for in reasing oupled powers (these are maximum values, obtained for a vanishingly

small atomi wall-loss probability), being asso iated with atomi populations that are

roughly proportional to the pressure. Sin e the populations of atoms are not easily

measurable,this is animportantmodel result. The reliabilityof thepredi tions for the disso iationdegreeisindire tlyensuredbythegoodagreementbetween simulationsand measurements for the intensities of radiative transitions, sin e both results depend on the a ura y of the kineti des ription for the ex ited N

2

and N spe ies.

The present work is part of a more extended study aiming hara terizing rf

dis harge plasmas produ ed in nitrogen-methane mixtures (for CH

4

on entrations up

to10%),forthelaboratorysynthesis ofanaloguestoTitan'ssolidaerosols. The

step-by-step methodologyadopted wasinitiatedwiththe study ofpure N

2

dis hargespresented

here,whi hallowed toset well-groundedphysi albasis forunderstandingtheseplasmas

(in terms of the dis harge produ tion features, the transport des ription adopted, and

the kineti me hanisms onsidered), before moving to more omplex situations. This

strategywill ontinuewith thestudy of N

2 -H 2 and ofN 2 -CH 4

rf dis harges,the latter

in ludingdust formation.

A knowledgments

Work supported by a PICS Cooperation Program, nan ed by the Portuguese

Foundation for S ien e and Te hnology (FCT) and by the Centre National de la

Re her he S ientique(CNRS). The al ulationswere performedonSeARCH(Servi es

&Advan edComputing with HTC/HPC) funded by FEDER through the COMPETE

programand by the Portuguese FCT under ontra t CONC-REEQ/443/EEI/2005. Et

Es-sebbar thanks the ANR programme (ANR-09-JCJC-0038 ontra t) for his

Figure aptions

Figure 1. S hemati diagram of the plasma hamberand the external ir uitwith

0 1 1-d /d 0 /d f d 1 f 0 g ( z ) z/d

Figure 2. Axial densityproleg(z)[spatiallyhomogeneousin theplasma bulkand

linearlyde reasingintheplasmasheaths,seeequation(22b)℄,adoptedinthisworkfor

N 2 (A 3  + u ,a 01  u ,a 1  g ,w 1
u )andN( 2 D, 2 P).Inthisgure,Æ 0 andÆ d

arethesheath

thi kness near thedriven ele trodeand the groundedele trode, respe tively; f

0 and

f

d

0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 <n e I > (10 16 cm -3 s -1 ) 0 0.006 0.012 0.018 0.024 (d) z (cm) r ( c m ) 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 < > (eV) 1 3 5 7 9 11 13 (c) z (cm) r ( c m ) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.00 0.02 0.04 0.06 (b) n e , n N 2 + , n N 4 + ( 1 0 1 0 c m -3 ) z (cm) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -600 -400 -200 0 200 400 (a) < E z / N > ( 1 0 -1 6 V c m 2 ) z (cm)

Figure4. Cal ulatedtime-averageproles(fortheLATMOSrea tor)ofthefollowing

quantities: (a)theaxialredu edele tri eldand(b)thedensitiesofele trons(),N + 2 ()andN

+ 4

(N)ions,asafun tionofzatdis hargeaxis(r=0),forp=1mbar(solid line)andp=0:2 mbar(dashed);( ) theele tronmeanenergyand(d)theionization ratebyele tronimpa t,as afun tion ofrandz forp=1mbarpressure. Thepower oupledto the plasmais 7:4 W at 1mbar and2:3 W at 0:2 mbar, orresponding to V '200V.

0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 (a) I n t e n s i t y S P S ( 0 , 0 ) ( a . u . ) z (cm) 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 (b) I n t e n s i t y A r ( 8 1 1 n m ) ( a . u . ) z (cm)

Figure 5. Time- and radial- average radiativeintensities of the SPS(0,0)band (a)

and the 811:5 nm atomi argon line (b), measured (points) and al ulated (lines)

betweenthe ele trodes of the GREMI rea tor, for the following work onditions (at

V

rf

' 145 V): p = 1 mbar and W

e

= 4 W (solid line and ); p = 0:2 mbar and

W

e

0 5 10 15 20 25 0 25 50 75 100 125 150 175 (b) -V d c ( V ) W eff (W) 0 5 10 15 20 25 0 25 50 75 100 125 150 175 (a) -V d c ( V ) W eff (W)

Figure6. Self-biasvoltage,asafun tionofthepower oupledtotheplasma,measured

(points) and al ulated(lines)in theLATMOS (a)and theGREMI(b) rea tors,for

0 5 10 15 20 25 10 -2 10 -1 10 0 < n e > ( 1 0 1 0 c m -3 ) W eff (W)

Figure7. Spa e-timeaverageele trondensity,as afun tionof thepower oupledto

theplasma, measured (points) and al ulated (lines)in theGREMI rea tor, for the

followingpressures(inmbar): 1(solidlineand);0:72(dashedand);0:5(dotted

0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 (c) I n t e n s i t y A r ( 8 1 1 n m ) ( a . u . ) W eff (W) 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 (a) I n t e n s i t y S P S ( 0 , 2 ) ( a . u . ) W eff (W) 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 (b) I n t e n s i t y F N S ( 0 , 0 ) ( a . u . ) W eff (W)

Figure8. Spa e-timeaverageradiativeintensities,asafun tionofthepower oupled

to the plasma, of the SPS(0,2) band (a), the FNS(0,0) band (b), and the 811:5 nm

atomi argonline( ),measured(points)and al ulated(lines)intheLATMOSrea tor,

forthe following pressures(inmbar): 1 (solidline and ); 0:6(dashed and ); 0:2

(dottedandN). Theinsetingure8(b)isjustitssemilogrepresentation. Withinea h

subgure (a),(b) and ( ), all simulation values are normalized to one experimental

point obtained at:  7 W and 0:6 mbar for the SPS(0,2);  7 W and 0:2 mbar

for the FNS(0,0);  7 W and 0:2 mbar for the Ar(811 nm). Simulations were

averagedradially,overthe entire dis harge rossse tion, and axially,over1=3ofthe

0 2 4 6 8 10 12 14 0.00 0.05 0.10 0.15 0.20 0.25 (a) f u 1 / 2 ( e V -1 ) u (eV) 0 2 4 6 8 10 12 14 0.00 0.04 0.08 0.12 0.16 (b) f u 1 / 2 ( e V -1 ) u (eV)

Figure 9. Time-average ele tronenergy probability fun tion f(u)

jh"i;hn e i u 1=2 , as a

fun tionofthekineti energyu, al ulatedfortheLATMOSrea torandthefollowing

work onditions (at V

rf ' 200 V): p = 1 mbar and W e = 7:4 W (solid lines); p=0:5mbarandW e

=5W (dashed). Theresults orrespondto: (a)thedis harge

entre (atr =0and z=16:5 mm),where the time-average ele tronmean energyis

h"i'2:4and2:7eVforp=1and0:5mbar,respe tively;(b)thespa e- hargesheath

0 5 10 15 20 25 10 -5 10 -4 (a) n N 2 ( A ) / N W eff (W) 0 5 10 15 20 25 10 -6 10 -5 10 -4 (b) n N 2 ( a ' ) / N W eff (W)

Figure10. Redu edpopulations(relativetothegasdensity)oftheN

2

(A)(a)andthe

N

2

(a')(b)spe ies,asafun tionofthepower oupledtotheplasma, al ulatedforthe

0 5 10 15 20 25 0.00 0.01 0.02 0.03 0.04 x10 C B D i s s o c i a t i o n d e g r e e W eff (W) A

Figure11. Disso iationdegreeofnitrogenn

N =(n N 2 +n N

) ,asafun tionofthepower

oupledtotheplasma, al ulatedfortheLATMOSrea torandthefollowingpressures

(inmbar): 1(solidline);0:5(dashed);0:2(dotted). Thedashed urveswereobtained

usingdierent walllossprobabilities:

0 N 2 =4:510 4 and 0 N =10 3 (A, standard onditions); 0 N2 =1and 0 N asinA(B); 0 N =1and 0 N2 asinA(C,resultsmultiplied