22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Reduction of reaction mechanisms in plasmas containing oxygen
T. Ibehej, V. Hrachova and R. Hrach
Charles University, Faculty of Mathematics and Physics, Department of Surface and Plasma Science, Prague, Czech
Republic
Abstract: Processes in chemically active plasmas were studied by computer simulation,
the main object of study being a mixture of Ar and O 2 . The simulation technique was the
macroscopic kinetic approach. As a resulting set of balance equations consisted of
hundreds of equations, the technique based on differential weight factors and leading to the
reduction of both chemical reactions and active species was applied.
Keywords: chemical kinetics, reduction, plasma chemistry, oxygen
1. Introduction
Chemically active plasmas are subject of interest in
many application areas as in plasma processing of
materials, air pollution modelling, plasma chemistry, etc.
The presence of reactive species in plasma significantly
affects its properties. These effects were investigated
both experimentally and using computer models, e.g.,
[1, 2]. However, accurate models that correctly predict
studied chemical processes are typically extremely
complex and involve a large number of reacting species.
Therefore, it is desirable to reduce model complexity
without significant loss of its preciseness. The kinetic
equations describing the studied chemical process
represent the set of ordinary differential equations for
time dependencies of specie concentrations. A large
number of elementary reactions can occur among the
species – some of these reactions are fast and some are
slow and these multiple time scales cause severe stiffness
of the resulting set of equations.
2. Methods of reduction
The application of proper reduction method decreases
the complexity of the system and thus improves the
performance of model. The aim of simplified kinetics
modelling is to derive the simplest reaction system which
retains the essential features of full system. In the
literature many types of reduction methods can be found.
The classical chemical kinetics reduction approaches such
as the Quasi-Steady-State Approximation QSSA [3] or
the Partial Equilibrium Approximation PEA [4], have
generally relied on intuition. Further classical technique
is a sensitivity analysis, e.g., [5, 6], which studies the
change in the species concentration for small
perturbations of rate constants of chemical reactions in the
model. If a reaction is slow and unimportant, it can be
identified in this way. Another group of reduction
techniques is based on time scale analysis and separation
[7, 8]. These methods range from the computational
singular perturbation CSP method [9], intrinsic lowdimensional manifolds ILDM [10] to ideas of integer
optimization [11]. Apart of these methods, a powerful
geometrical approach based on the decomposition of the
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phase space of system’s dynamics into a fast and slow
subspace has been suggested [12]. And in the last years a
special attention is devoted to modern approaches based
on low invariant manifolds [13].
In our laboratories in Paris (Université Pierre-et-Marie
Curie) and in Prague (Charles University) the discharges
in hydrocarbons have been studied for a long time both
experimentally and by computer modelling. The main
task was to analyse the conversion of methane in more
suitable products, which is one of challenging problems
of plasma chemistry. The decomposition of methane into
simpler hydrocarbons was performed in a N 2 /CH 4
mixture in flowing afterglow conditions. The kinetic
mechanism of methane decomposition was analysed by
the computer experiment based on a macroscopic kinetic
approach. The first version of the model consisted from
166 chemical reactions between 46 active species – this
model is described in detail in [14]. The final version of
the model included 274 reactions between 50 species.
The resulting set of differential equations was solved by a
semi-implicit extrapolation method, see Fig. 1.
Fig. 1. Time dependence of concentrations n of various
species produced by the dissociation of methane [15].
1
The complete set of balance equations describes the
conservation of individual species and consists of creation
A i and disappearance B i processes due to chemical
reactions and of drift and diffusion terms. When both
drift and diffusion can be neglected for given
experimental conditions, the balance equations reduce
from partial differential equations to ordinary differential
equations
dn i /dt = A i (n 1 ,…n m ) – B i (n 1 ,…n m ), i = 1,…m,
where n i is the concentration of i-th specie and m the total
number of species.
As the complete reaction scheme consisted of several
hundreds of chemical reactions, the simplification of the
system of kinetic equations must be performed first. We
suggested the reduction technique suitable for afterglow
conditions [14]. This technique was based on the
comparison of reduction rates, which enabled us to
reorder the list of chemical reactions according to their
importance in the studied plasma chemical process. We
introduced the weight factors of i-th reaction w i (t) as a
sum of absolute values of all terms k i x n a x n b with the
same rate constant k i , where the terms k j x n a x n b (as
well as k j x n a and k j x n a x n b x n c ) represent the
contribution of given chemical reaction to the creation
and disappearance processes A i and B i . With the help of
these weight factors we simplified the kinetic scheme of
methane decomposition from 166/274 reactions to less
than 50 reactions, which can be thereafter analysed in
detail – see Fig. 2.
The main studied problem was the dependency of
O 2 /Ar plasma composition on the ratio of initial densities
of Ar and O 2 . In order to analyse the processes in plasma,
we applied the technique of computer simulation. The
first version of models of active oxygen and oxygenargon discharges consisted of 152 chemical reactions
between 16 species (electrons, 8 neutral species: O 2 ,
O 2 (a), O 2 (b), O, O(D), O 3 , Ar and Ar*, and 7 charged
species: O 2 +, O 2 -, O+, O-, O 3 -, Ar+ and Ar 2 +).
Some rate constants in balance equations can be found
in the literature – either in the form of explicit functions
of electron and gas temperatures or as constant numbers.
The second part of necessary rate constants was obtained
by the integrations of corresponding cross-sections from
the literature and electron energy distribution function.
Therefore in the model two types of calculations were
used:
• Determination of EEDF’s for given experimental
conditions with the help of solver of Boltzmann kinetic
equation BOLSIG+ [17].
• Solution of the system of balance equations based on
the macroscopic kinetic approach.
4. Results
An example of the output from first type of calculations
is shown in Fig. 3.
Fig. 3. Mean electron energy for various initial amounts
of argon in the mixture and corresponding EEDF’s
denoted in small graph. External field 60 Td.
Fig. 2. Kinetic scheme of methane decomposition in
flowing afterglow conditions, presented in ISPC 19 [15].
3. Study of active discharge in O 2 /Ar gas mixture
In our laboratory we studied the processes in oxygen
plasma. The experimental conditions correspond to the
positive column of DC glow discharge. The plasma
consisted of oxygen both pure and in the mixture with
rare gases. For the study of plasma properties optical and
probe diagnostics were used. Some of obtained results
are summarised in [16].
2
Again, as in the case of the analysis of methane
decomposition, the obtained set of balance equations was
too complex and the interpretation of obtained data was
too complicated, therefore the simplification of kinetic
scheme by the reduction method was applied. For this
purpose we used the technique based on the weight
factors of individual reactions w i (t) suggested for
afterglow plasma [14], modified for the study of active
discharges.
The analysis of the weight factors enabled us to derive
the importance of particular processes in particular time
of discharge ignition. In the Fig. 4 an example of
obtained results for the balance of atomic oxygen is
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shown.
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Fig. 4. Balance diagrams of atomic oxygen after 0.01 s
(top) and 1 s (bottom) of the discharge. The numbers
under the times denote actual densities of O atoms and the
numbers at the bottom denote the amount of O created
and disappeared in cm3 in one second. Calculated for
external field 60 Td.
5. Conclusion
Similarly, as in the case of the analysis of methane
decomposition in the flowing afterglow conditions [15],
we proved that the reduction technique based on weight
factors of individual chemical reactions brings a very
convenient tool which allows reducing model complexity
without significant loss of its preciseness.
6. References
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[5] R.W. Carr Jr., D.G. Peterson and K.K. Smith.
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