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Coupled Electron and Molecular Vibrational Kinetics in a 1D Particle-in- Cell Model of a Low Pressure, High
Frequency Electric Discharge in Nitrogen
S. Longo, K. Hassouni, D. Iasillo, M. Capitelli
To cite this version:
S. Longo, K. Hassouni, D. Iasillo, M. Capitelli. Coupled Electron and Molecular Vibrational Kinetics in a 1D Particle-in- Cell Model of a Low Pressure, High Frequency Electric Discharge in Nitrogen.
Journal de Physique III, EDP Sciences, 1997, 7 (3), pp.707-718. �10.1051/jp3:1997133�. �jpa-00249608�
J Phys III FYance 7 (1997) 707-718 MARCH 1997, PAGE 707
Coupled Electron and Molecular Vibrational Kinetics in a ID Particle-in- Cell Model of a Low Pressure, High Frequency Electric
Discharge in Nitrogen (*)
S. Lange (~), K. Hassouni (~), D. Iasillo (~) and M. Capitelli (~,**)
(~) Centro di Studio per la Chimica dei Plasmi del Consiglio Nazionale delle Ricerche and Dipartimento di Chimica dell'Universith, Via Orabona 4, 70126 Bari, Italy
(~) LIMHP, Universitd de Paris Nord, CNRS, Villetaneuse, France
(Received 2 July 1996, revised 7 November 1996, accepted 13 December 1996)
ARCS.52 20 -j Elementary processes in plasma PACS.52.80.Pi High-frequency discharges
PACS.82.40 Ra Plasma reactions (including flowing afterglow and electric discharges)
Abstract. The interaction between state-to-state vibrational and charged particle kinetics
m a low-pressure, high frequency electric discharge m N2 is investigated by using a PIC-MCC
(Particle-in-Cell with Monte-Carlo collisions) model assuming a one-dimensional bounded sys- tem, while the vibrational kinetics is introduced by solving a set of reaction and diffusion equa- tions. The catalytic activity of the electrode surfaces is considered in the boundary conditions.
The coupling of the two kinetics has an effect on the Electron Energy Distribution Function
(EEDF) and consequently on the rate of vibrational and electronic excitation of N2 molecules.
the charged particle distribution and the electric field.
1. Introduction
The importance of a better theoretical understanding of electric discharges in terms of state- to-state kinetics of atomic and molecular species and kinetic description of charged particles is
now generally recognised [1-3]. This interest is due both to the large use of electric discharge
reactors for industrial purposes and to intrinsic interest of these far-from-equilibrium systems.
The discharge models fall into two categories:
(I) detailed state-to-state kinetics ill with a spatially independent transport equation or
macroscopic balance equation description of charged species kinetics, and;
(2) configuration space or particle description of charged species, in particular PIC-MCC
[4-6] (Particle-in-Cell with Monte-Carlo collisions) methods.
Due to technical difficulty and high demand of computational time, only few attempts exist in the literature to couple the approaches (I) and (2). In particular, detailed discussion of the effects of interaction between chemical kinetics and charged species kinetics has been performed only in the framework (I). In fact, molecules are excited and de-excited due to electron impact
and also to molecule-molecule, molecule-atom and molecule-surface collisions. At the same
(*) Complementary paper of the PISE special issue published m September 1996 (**) Author for correspondence
@ Les #ditions de Physique 1997
time, the presence of excited species in the mixture has an effect on electron dynamics, through super-elastic (or second kind) electron-molecule collisions and multi-step ionization processes.
All these phenomena have been studied extensively by using numerical simulation, but the studies presented in the literature refer mainly to uniform systems, because in this approxi- mation one can use the well-known method by Rockwood [7] to solve the Boltzmann equation for electron transport in the two-term approximation [8]. Very little work has been done to show the importance of these effects outside a uniform, two-term solution of the Boltzmann equation. Part of the work available can be applied to non uniform systems when the so-called local approximation is applicable: in this approximation the local solution of the Boltzmann equation is obtained by considering a uniform plasma with the local values for the electric
field, the ionization degree and, more generally, the state-to-state gas composition. Unfortu-
nately, the local approximation does not hold anymore when the gas pressure is low enough to allow a "mixing" of local electron properties by diffusion, or, at even lower pressures, simply by free-flight motion. These is the case of many electric discharges that are interesting for applications, in particular low-pressure glows produced by high-frequency oscillating voltages.
It is probably useful to stress again that the statement "self-consistent solution" of chemical kinetics and electron kinetics means that the electron-neutral collision frequencies are coherent at any time with the composition of the gas mixture; and not only for t
= 0. Most of the works devoted to extended plasma media in the literature do not satisfy this last requirement, unless the local approximation is used.
In recent works [9,10] it was studied the effect of a simplified, but state-to-state vibrational kinetics on a Monte-Carlo description of the electrons for a one-dimensional bounded system.
A closed expression for the electric field as a function of space and time was assumed. On the other hand the simplified vibrational kinetics contained e-V (electron-vibration) energy exchange processes, while a phenomenological bulk term was used to describe the deactivation of vibrationally excited molecules either m gas phase or in the interaction with the container
walls. No diffusion was considered.
The model presented here differs from the those above because the electron kinetics is taken
into account by using a PIC-MCC (Particle in Cell with Monte-Carlo Collision) model and
also because the ID vibrational kinetics includes diffusion and surface deactivation.
The application of PIC/MCC approach to an high frequency nitrogen discharge has been realised by Turner and Hopkins ill] which were able to reproduce the experimental results
for the EEDF m a wide range of gas pressures. What follows can also be considered as a
generalisation of the theoretical part of their work, in which nitrogen molecules were considered
"cold" Ii.e. Tvjb
= Ttransl " 300 K).
The outline of the present work is as follows. In Section 2 we discuss the electron kinetics
through the PIC-MCC model, while the coupling of the electron kinetics with the vibrational kinetics including diffusion is reported in Section 3. Section 4 presents the results having
as parameter the deactivation probability of vibrationally excited molecules on the electrode surface. Finally Section 5 reports some conclusions and possible future improvements.
2. Modelling of Electron Kinetics
To study the electron kinetics we have built a PIC-MCC model program. The PIC /MCC model is based on simultaneous solution of the particle dynamics equations for charged particles and Poisson equation for the self-consistent field due to space charge, while taking into account the electron (ion) collisions with neutral particles by the time-of-flight Monte-Carlo method.
Different methods can be used to integrate the equations above and to relate (interpolate) the
N°3 lD PIC/MCC AND VIBRATIONAL KINETICS IN A RF DISCHARGE IN N2 709
particle position and applied force to the mathematical grid used to solve the Poisson equation.
Details can be found in references [4-6j.
This approach is able to give a full account of the space and time dependent evolution of the Electron Energy Distribution Function (EEDF) f(s,x,t), defined in such a way that s~/~f(s, x, t)ds is the fraction of electrons with kinetic energy in the range (s,s + de) in the position x at the time t. The knowledge of this last function is essential to apply the methods of chemical kinetics to the discharge plasma.
In our model any particle is represented by the set of numbers (i, r(I), v(i), tc(I), s(I)). Here identifies the i-th simulated particle, r and v are the position and velocity vectors assigned
to the particle, tc is the time left to the next Monte-Carlo collision event, and s is a flag representing the nature of the particle: for example if s(i)
= I the i-th particle is an electron,
if s(I)
= 2 it is a positive ion, and so on: different particle species are therefore mixed in the
particle list. Any particle can be removed from the simulation (for example if it is absorbed
by the reactor walls) by putting s(I)
= 0.
The statistical weight w per particle (the same for all the particles) is introduced: if the number of simulated particles exceeds 2Njn, where Njn is the initial number of simulated
particles, one half of the particle ensemble is randomly removed and the statistical weight of any particle is multiplied by 2. If, on the contrary, the number of simulated particles is less than 0 5Njn, the particle ensemble is doubled and the weight reduced by a factor of 2.
The model is one-dimensional in space, the equations of motion for the particles are inte-
grated by using the leap-frog method [4-6j, and the electric field is calculated by solving the Poisson equation in the 1-2-1 scheme [4]
,
with a space charge obtained by using a first-order
particle-grid interpolation, or DC (Cloud-in-Cell) [4-6].
To implement MCC, we use the standard time-of-flight Monte-Carlo method with null col- lisions [5, 6]: the collision time tc is calculated for any particle after a collision event by using the formula
tc = ln r, (I)
"tot
where r is a real random number extracted from an ensemble with uniform distribution in [0,Ii, and utot is a constant collision frequency given by
utot = max l~~ ~j Np(x)ap(s)
,
(2)
0<x<1, o<c<cj,m me
~
P
where the sum runs over collision processes, ap(s) is the cross section, Np the number density
of the collision partner, is the discharge gap length, and slim is the limit of the scale of kinetic energy used to interpolate cross sections.
After any collision event, the kind of collision (elastic, inelastic vibrational, etc.) is also selected by a Monte-Carlo method according to the local collision frequency set
lynx), u~ix),..., utot ~j upix)1, 13)
p
where the last process is the "null" one, which has no effect on the electron motion.
The electron-neutral collisions are assumed isotropic and the momentum transfer cross sec- tion is used instead of the elastic scattering one: the validity of this approximation in the
present conditions has been directly tested by using the treatment of reference [12].
3. Coupling of the PIC-MCC with Vibrational and Diffusion Kinetics
The main point of this work is the possibility- to take into account in a self-consistent way the vibrational kinetics of the gas molecule inside the reactor.
In this paper we apply our approach to the same discharge studied experimentally and
numerically by Turner and Hopkins ill]. It is a discharge in pure nitrogen, produced into
a parallel-plate high frequency reactor1v.ith the two plates separated by 4 cm. The pressure assumed in our study is p
= 0.I torr (at T
= 300 K)
One of the plates (the so-called "grounded electrode") is constantly kept at zero voltage,
while the other one (the "powered electrode" is driven by an external generator, not explicitly simulated, to an oscillating voltage V(t)
= ~jf sin 27rft. The values 200 V and 13.56 x 10~ s~~
are selected for ~[f and f, respectively. Only electrons and N) ions are moved as particles, Nj
ions are neglected according to the results of reference [25], where it is shown that the con-
centration of Xl is only 10% of that of N) in typical RF conditions. As regards N) and N+
ions, we have neglected them consistently with both references [25j and IIIi this point is open to discussion (specially for N+ ions in the near-electrode regions) and the related treatment
should be improved in future works, but we believe that it is reasonable at the moment to
keep the particle kinetics near to reference IIIi, in order to compare results obtained with and
~vithout a self-consistent vibrational kinetics. The collision cross sections used are consistent with references [9,10j
The following collision processes are considered.
la) for the electrons. elastic collisions with N2 molecules, inelastic processes leading to rota-
tional, vibrational, electronic excitation of N2 molecules, and electron impact ionization of N2.
The set of cross section used is selected from [13j. The electronic excitation cross sections con-
sidered correspond to the following states and threshold energies: A~E(6.17 eV), B~II(7.35 eV), W~1h(7.36 eV), B'~E(8.16 eV), a'~E(8.~0 eV), a~II(8.55 eV), w~lh(8.89 eV), C~II(11.03 elf), E~E(11.88 eV), a"~E(12.25 eV), and a lumped set of singlet states (13 0 eV). Superelastic
vibrational cross sections are calculated from the corresponding inelastic ones in order to fulfil detailed balance in the equilibrium case [7j;
(b) for N) ions, elastic and charge-exchange collisions with N~ molecules, assuming the cross sections of reference [14j.
During the calculations, the composition variables (Np(x)) must be updated by solving the
equations for their reaction and diffusion kinetics, taking into account that the rate coefficients in these equations will depend on the local electron energy distribution Function (EEDF)
f(s, x, t), this last resulting in turn from the PIC-MCC simulation.
In reacting plasmas usually the relaxation times of the chemical kinetics are by far higher
that the corresponding ones for the electron kinetics, therefore an adiabatic approach to the solution for the steady state can be applied
Our method consists in solving the vibration and diffusion kinetics up to the steady state at different times (tn ) during the PIC-MCC simulation (which is by far more computationally expensive). At each tn the space-dependent values of rate coefficients and electron density are
obtained from the PIC-MCC, time-averaged over the period tn tn-i The solution of the
vibration and diffusion kinetics gives a new state-to-state composition and related electron- molecule collision frequencies for the PIC-MCC.
As regards the vibrational kinetics, we assume a very- simplified one, consisting of vibra- tional excitation and deactivation by electron impact between a ground state N2 molecule
N°3 lD PIC/MCC AND VIBRATIONAL KINETICS IN A RF DISCHARGE IN N2 711
and 8 vibrationally excited levels:
e + N2(0) ++ e + N2(v), v
= 1-8.
To these processes we add a pseudo-first order deactivation process leading to the ground state:
N2(v) ~ N2(0),
which takes place only on the electrode surface. It is characterized by the deactivation proba- bility +~, i.e. the probability of deactivation for a molecule hitting the surface.
In our model the vibrational distribution function (n~) is obtained by finding a stationary
solution for the following set of 9 one-dimensional reaction and diffusion equations
~'~~
= n~ ((kj~nj kuni + D~~~' ~~~
0t
~_~
assuming partial vibrational deactivation of excited molecules hitting the limiting surfaces at
x = 0 and z =1
where n~ is the particle density of molecules in the i-th vibrational level, ne the local electron
density, D is the diffusion coefficient, lI the mass of a N2 molecule, T the gas temperature,
+~is the deactivation probability, and k~j are the rate coefficients for the electron impact transition
between vibrational levels and j, given by
k~(x)
=
~ f(s.x)a~j(s)sds, (6)
Rle ~~
where f(s, x) is the local electron energy distribution function, a~ (s) is energy-dependent cross
section for the vibrational inelastic process I ~ j, s)~ is the related threshold energy. In our
case only coefficients of the form ko~ and k~o are used.
The system (4) is linear because we have neglected any VV relaxation process, i-e- colli- sional exchange of vibrational quanta between molecules (which would introduce terms of the form ni x nj). these processes become very important for a pressure range (p > I torr) not considered in the present paper. In any case V-V energy exchange processes affect the high lying vibrational levels, those belonging to the so-called plateau of vibrational distribution,
having a small role in affecting the low lying vibrational levels Iv < 10) those considered in the present paper (see for example [lj).
&loreover the model assumes that the deactivation of vibrationally excited N2Iv) molecules in gas phase is very small. This is true in our case: the characteristic relaxation times
r = (Nki,o)~~ for the reactions N2Iv
= 1) + N2 ~ N2Iv = 0) + N2
N2(v=1)+N~N2(v=0)+N
assume for the conditions studied in the present paper (T
= 300 K and p
= 0,1 torr) respec- tively the values of 2.6 x 10~ s and 3.2 x 10~ s. These values were obtained by using the rate
j~-1
~-s d
$ ~
~
~
~o8 a
o 5 lo 15
electron energy (eV)
Fig. I. Time averaged EEDF calculated in the discharge centre at steady state for different values of the deactivation probability. (a)
+~ = I, 16) 1= 10~~, (c)
+~ = 5 x 10~~, (d)
+~
=10~~.
coefficients ki,o reported in closed form in reference [lsj and assuming for the number density of nitrogen atoms a value 3.3 x 10~~ cm~~ equal to I% of the molecules. These relaxation times
are much larger compared to the deactivation with electrons and on the walls. Of course the relaxation times for higher vibrational levels decrease with increasing the vibrational quantum number but are still higher than the corresponding relaxation times for e-V processes and gas wall interaction for v < 10.
We have solved the system (4-6) for the steady state by a relaxation method: first the equa- tions are decoupled and solved by using the usual 1-2-1 technique, then the particle densities
are normalised in order to get a uniform and constant total density, and so on. Usually we
obtain convergence after few iterations
4. Discussion of the Results
We show the results obtained by using our model: typically we simulated about 10~ particles (electrons and N) ions) for a time up to t GS 3000 f~~ and the system (4, 6) was solved
10-20 times during the simulation in order to reach the steady state.
The fluctuation level was tolerable: better results could be obtained by using different inte-
gration schemes allowing to work with more particles [4j. The PIC time step used is 10~~° s, which ensures the stability of the leap-frog method for the plasma oscillations in these condi- tions [4-6]. The mesh used to solve the Poisson equation contains 200 grid points, while the
one for reaction and diffusion equation contains only 30 grid points, in order to reduce the statistical errors on the rate coefficients.
In Figure I it is shown the stationary and time averaged EEDF calculated m the centre
,
of the discharge for different values of the deactivation probability. In all cases we observe a
~strongly non-Maxwellian behavior of the EEDF.
N°3 ID PIC/MCC AND VIBRATIONAL KINETICS IN A RF DISCHARGE IN N2 713
v=0
V=0
nf/s q£ v=1
~/ 8
b vZl ~
~ .t
~ ~
Q~
~
l
£ O
~ 'n
-
( ~~
O
~ ~
v=8
v=8
0 2 3 4 0 2 3 4
X (Cm) X (Cm)
aj bj
Fig. 2. Space-dependent populations of vibrational states of N2, calculated assuming (a) i
= I,
16 +~ =
10~~. The
upper line refers to N2Iv
= 0), the lower one to N2Iv = 8). Intermediate lines from the top refer to
u = 2,
,
7 respectively.
Let us concentrate our attention on the EEDF calculated assuming the higher deactivation rate Ii
= I). This extreme case was considered as a reference: in this case, the N2 molecules
are excited due to electron impact and deactivated on the electrode surfaces: the EEDF is
practically the same that the one calculated disregarding vibrational kinetics, because the
population of vibrationally excited states is negligible. In Figure 2 it is shown the steady state
density of molecules in different vibrational states as a function of position: one can observe that the density profile of excited states for
+~ = I is controlled by the boundary conditions.
These results must be compared now with the other curves in Figure I.
The effect of the EEDF-vibration coupling at low values of the deactivation rate is first to
"fill" the dip due to vibrational excitation collisions, and then (for lower
+~) to raise the EEDF in the energy range between the vibrational inelastic and electronic inelastic thresholds, at the expense of the EEDF value in the low-energy region. Also it can be observed that for low values of i the concentration of vibrationally excited molecules becomes independent on position, because the diffusion process is very fast in this case Different results should be
obtained for higher values of the p/I ratio, at the same
+~.
The effect on the EEDF reported above is interesting for a comparison with the work of Turner and Hopkins ill], because it can explain the lower depth of the dip due to vibrational inelastic collisions in the experimental EEDF (see also [16]) with respect to the calculated one.
It is interesting to compare with our results the values of reference II?] and [18j for vibrational temperature of N2, estimated from experimental results. The values suggested are 2000 K and 2900 K respectively, differing on the conditions and the estimation technique. From our results
